Author: Распопов В.Я.
Tags: общая механика механика твердых и жидких тел приборостроение измерительные приборы учебное пособие
ISBN: 5-217-03360-6
Year: 2007
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15
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nYCKaKJT H npO,l\alOT 50 l1pM. 3a ro,l\ HX
peaJIl13yeeTcH 100 MJIH, a npM6bIJ1b OT
npo)J,a)l(H COCTaBJIHeT -10 MJIp,ll ,[IOJIJI. CIlIA.
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'lMCJIe MHKpOMCXaHI1<J:ecKliX ITpH60pOB, Tec-
HO CBH3aHO C 06naCTHMH llX HCnOJIb30BaHIDI.
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CH M0611J1bHbIe P060TbI rproK,lJ,aHCKOro M BOCH-
HOro npl1MeHeHIDI. O)l(H,l\aJJOCb, qTO B 2005 r.
cYMMa ITpo,[la:>K M06HJlbHbIX P06oTOB ,[IOJDKH8
6billa COCTaBMTb He MeHee 20 MJIpJl. ,llOJIJI.
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,l\aT'IHKaM 11 KOHTp0J1bHO-H3MepHTeJJbHblM R
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flepe<JeHb QJI1311<JecKI1X BeJJH<JHH, KOTOpbIe
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npI1Mep, B 3aBHCHMOCTI1 OT THITOp8.3MepOB
CaMOJIeTOB o61lJ,ee <JHCJ10 TOJIbKO llaT'iI1KOB
,l\aBJ1CHIDI Ha O,llHOM CaMOJ1eTe COCTaBJU!eT
50...90 UlTYK.
B EBpone 3a 1996--2005 rr. 1l0JIIKHO
6bIJ10 6bITb npOI13Be,[leHO 8000 rpa)l{,[(aHCKHX
CaMOJIeTOB (MToroBhIe .n:aHHble ITOKa OTcyr-
CTBYIOT), T.e. O,llHHX mlUlb llaT<JHKOB llaBJIe-
HIDI He06xo,lll1MO 6bIJ10 60J1ee 500 TbIC lliT.
Tpe60BaHIDI K Maccora6apUTHbIM xapaKTe-
pl1CTllKaM npH60pHoro 060pY,llOBaHMR B
aBlIar.tH11 BCCbMa :>KeCTKHe. TaK KaK Y
M3MC-KOMITOHeHTOB B HanpaBJJeHHH MM-
HHaTlOp1I3al\HH B 0603pHMOM 6Y.QYIl\eM He
6Y)J,eT KOHK)'peHTOB, a lIX TO'lHOCTb M 3KC-
nJIyaTal\HOHHble xapaIcrepl1CTHKH HMelOT
YCTOH<JMBYKJ TeH)J,eHl\HIO K yrry1JrneHHIO no-
Ka3areJIeH, MO:>KHO O:>KI1,[1aTb, H 31'0 Y:>Ke ceH-
'lac Ha6JIIO,[IaeTCH B <JaCTM MA H M)I.l];, npo-
rpecCHpytOllJ,Cro BHe,l\peHH.SI MHKpoMeXaHM-
qecKMX npM60pOB B aBROHMI<)'.
B 3aKJIIOQeHHe OTMemM cJIeJJYlOlI{ee.
113-3<1 BbICOKOH HHTCrpal\HH H Me)l{,[(HCQI1IT-
J1HH8pHOH npHpo)J,bl M3MC TpY)J,HO OT,[Ie-
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M3rOT0BJ1eHIDI. CJIe;:r.OBaTeJIbHO, "BbICOKHH
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,lI)1S1 npoCKTHpOBaHHH. M3MC-YCTpoHCTBa.
KpoMe Toro, 3Ha'lHTeJIbHbIe BpeMH M cpC;:r.CT-
Ba 3aTpa1JllB81OTCSI Ha C03.n:aHU OlIbITHoro
o6pa3l\a. ITOCKOJIbKY ycnernHIDI pa3pa6oTKa
YCTpOHCTBa Tpe6YeT (j>R3W1ecKoro H HMJI'I'a.qH-
OHHoro MOD.eJ1M.POB'a.Hwr, BroKHO, 'lT06L1
M3MC-npoeKTHpOBmHKH RMeJ1H ,l\ocryn K
COOTBeTCTBytOIl\HM aHaJIHTIltJecKlW RHC1pY
MeHTaM. .IT03TOM)' 6oJIce'MoIJWble HHCTpyMCB-
Tbl HMHTaIiHOHHQro " "(j>tJ3HqCCKOm - Mo.nena-
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.
16
BBEAEHHE
B.2. HCTOpliH pa3BHTliH M3MC
rO!l C06bITH)j
1958 OOSIBIiJH1Cb KpeMHneBble, KOMMep'leeKH )J,OcTynHb!e CpenCTBa I13MepeHI1H
liB OCHOBamm MHoro MeeTS", - 3aH.BI1Jl PIi'lap.n c1>eHHMaH B KamIcP0pHliii-
1959 CKOM HHCTliT)'Te TeXHOJIOmU Ii npellJIO:>K!iJl 1000 .nonn. CiliA TOMY, KTO
C03)J,aCT 3JIeKrpnlleCk"HH ,llBHraTeJIb, MeHbIllHH 1164"
1961 )J.eMOHCTpHpOB8J1CSI nepBbIll KpeMHUeBb!H )J,aT'lHK ,llaBlleHHR
Ih06peTeHa nOBepXHOCTHaJI MIiKpoMeX8HH'leeKaH. 06pa6oTKa. c1>HpMa
1967 Westinghouse (CiliA) C03)J,aJI8 pe30HaHcHbIH ITOJIeBOH BeHTHJIbHbIH TpaH-
3Hcrop. Om1caHo HCITOJlb30B8HHe l:>KepTBeHHOrO" MaTepMaJIa, '<T06b! oc-
B060JUITb MHKPOMCX8Hll'leeKHC YCTpOHCTB8 OT KpeMHHeBoH nO)J)IO:>KKH
1970 )J.eMoHcTpHpOBaJICR nepBblH KpeMHHeBbIH aKCeJlCpoMeTp
1979 Ope1lJlO>KCHO ITepBOe MHKpOMeXaHMqeCKOe cTpYHHoe conJIO
C03.!IaH 0)J,HOpa30BbIH .napmK KpoBSIHoro ,l\aBJICHIDI. IToSlBI1JIaCb ITepBaJI
rry61lHKalJ,l1}1 0 KpeMHHH KaK 0 KOHCTp 1l0HHOM MaTCpHaJIe [26].
1982 Pa3pa60TaHa LIGA- TeXH0J10rHSI (peHTfeHOnHTorpa IDI, raJIbBaHHK8 11
cPopMoBKa) ,IIIDI M3rOTOBJIeHIDI llCnOJIHMTeJIbHbIX KOMnOHeHTOB MHKpO-
npu60poB
1983 BnepBble B CCCP ony6nHKoBaH MaTepHaJI 0 pa3pa6OTKe HHTerpaJIbHbIX
TeH3onpco6pa30BaTCJIeH [5]
1988 CocrO$UIaCb nepBWI KOHl}>epeHl.lIDI ITO np06JICMaTllKe M3MC
ITpe,l\CTaBJIeH MHOrOnOJ1b30B8TeJIbCKHH npoQeec M3MC (MUMPS),
1992 cnoHcHpyeMLIH YnpaBJIeHHeM nepCITeKTMBHblX Hccne.nOBaTeJlbCKHX ITpo-
rpaMM (DARPA) (CIlIA)
ITpellJIO)KeH nepBbIH MHKpoMeXaHJ1qCCKMH IllapHHp
IIpo.naH nepBbIH aKcenepoMeTp, BbIDOJIHeHHbIH C HCnOJIb30B8HHeM no-
1993 BepXHOCTHOH MHKpoMeXaHMl.JeCKOH TeXHOJIOfHH (<<pHpMa CIlIA Analog
I>evices,AD>OL50)
3anaTeHTOBaHO rn:y6oKoe peaKTHBHOe HOHHOC TpaBneMlle; ITepBWI Me:>K-
1994 .uyHapoJtHaJi KOH«pepeHI.lIDI no HHTerpl1pOBaHHbIM HaBMraQHOHHblM CHC-
TeMaM (CaHKT-ITerep6ypr, I..(HI1H 13JIelClJ>onp 60p")
HaqanOCL KOMMepllecKoe HCnOJIb30BaHHe MHKporHpocoKonoB. AKTHBHO
1995 pa3BHBaIOTCH J>HoM3MC; aKTHBU311p}'JOTCJl p83pa60TKH B 06JIaCTH MHK-
poreXHOJlOmH B POCCKH. YKpaHHe H EenopyccHH
Ony6JlliKOBaHa CTaTMI, B KOTOpOH HaMeqeHbI nepcneKTllBbI HCITOJ1b30Ba-
1996 "WI MUKpOopH60poB JJ}JjI peweHHJl 3a)J.aq cOBpeMeHHoH aBToHoMHOH Ha-
B B POCCKH [19]
2000 OJfI"WJecme M3MC CTaHOBJITCJl KOMMepqeCKOH npo.nyKl.lHeH
2005 PoccHHClCHe pa3p 3aJIBUJIH 06 onLlTHLlX naPTIDIx MMKpOaKCeJle-
poMeTpOB D MHJq)ompocKonoB
BBE)::(EHI1E
17
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4. "CeHcopHaH" nep'laTKa (nJfTb MA Ha KOH'lliKax rraJIblleB nep'laTIUI H O,lUm
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1. CHCTeMbl KOHTpOJU! ,llaBJ1eHlIJI, pacxo.na (M ro H TeMnepaT)'pbI TennOHD-
CHTeJ]eH 11 XJ1a.n;areHToB.
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pOBaHa, Ha MeM6paHy nCHCTBYeT C Ula )J,aB-
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H B pe3YJIbTaTC ee nCpCMell.\CHWI C nOMOllJ,blO
3J1eKTpo.na 4 Bblpa6aTbIBaeTcH ynpaBJISllO-
ll.\HH Cl1fHaJI Ha aKTlOaTOp, BKJIlO'ialOIlJ,HH
CHJ10BbIC 3JIeKTpo.ll.bl 2, 3, KOTopble HaXO,llHT-
CH ITO)J, HaITpH)!(eHl1eM U on , .Il.OCTaTOI.JHblM )J,JIH.:
C03)J,aHI1H KOMITeHcHpYlOllJ,eH 3JIeKTpoCTaTH-
I.JCCKOH CI1JIbI F3 = FA, cnpaBC.Il.JJHBO paBeH-
CTBO
?
EO 8:) U;;n S
2h 2 - Po M' (1.7)
o
rne EO = 8 85 . 1O 12 c1:I . 8. - nJIOllIaJ1.b CH-
, M
JIOBOrO 3JICKTpO.ll.a; ho - 3a30p Me:>K,l1y
3J]eKTpO.Il.aMH aKTlOaTopa.
.n:J]H.: cxeMbl, npIlBe.ll.CHHOM Ha pIlC. 1.23,
a, 8 M = 8 3 , 11 113 (popMYJ1bI (1.7) CJIc.n.yeT Be-
JIM'iIlHa HaITpH)i(eHHSI, npH KOTOpOH BbInOJI-
HJ1eTCH.: paBeHcTBo (1.7):
U on = ho .J 2po I EO . (1.8)
ITpH Po = 10 5 TIa 11 ho = 10-6 M M3 COOT-
HOUIeHHH.: (1.8) ITOJ]y'laeM U on = 150 B, '11'0
COBcpilleHHO HC,llO!IYCTI1MO )J,JlH.: Iill3KO-
BOJIbTHOH 3J1eK'IpOHI1KH.
ITpocTOH H 3 CKTHBHbIH CIIoco6 C03-
)J,aHHJI Hc06XO,llI1MOH KOMneHcIIpYIDm eH
CUJIbI IIpIl CHH)!(CHI1H onopHoro HanpIDKCHIDI
,llO ,llorrycTIIMOrO ypOBHJ1 3aKJIIOlffiCTCSI B
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31'OT cIIoco6 IIJIJIlOCTpllPYeTCH.: cxeMoH
M)UJ. Ha pl1c. 1.23, 6, B KOTOpOM MCM6paHa
BbI!IOJ1HHCT TOJIbKO YHKI.:(l1lO Q3, BOC!IpIl-
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C l!CHTpOM MeM6paHbI 1 CBH.:3aH IIO,l\-
BI1:>KHbIH 3J1CKTpO.ll. 2 CHJIOBOH pa3rpY3KH, a
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Koprrycc 5. I1nOlI.{a.ll.H CHJIOBbIX 3JICI\'Tpo,!{OB
MOry1' 6bITb 3Ha t ilneJJbHO 60JIbUIC nJIOIlla.nH
MCM6paHbI. IIpl1 3TOM H)')KHO HMeTb B BH.1JY,
'11'0 IIO,l\BH:>KHbIH 3JIeKTpO.ll. 2 co Bcex CTOpOH
IICnbITbIBaCT ,l\aBJICHHC Po. T.e. O.ll.HOHanpaa-
40
r:JaBa L KOHCTPYKUIIH H rIPIIHIUlflbl PAEOTbl IlPI!60POB
1
2 3-1
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1--
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LlaJl.rreHIIe
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:2 - pe30H3Top: 3 - BaJ>.-YY IHCU! Ka lepa;
4 - 1eTa.l...1H3I1pOBaHHble KOlrraKTbl; 5 - 1I30:ulUlUI
li3 OKcn a Kpe IHIUI
.:IeHHWI CI1:ta OTCVTCTBVeT. B :noii cxe!>{e
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eMKOCTb 3_1el\-rpo.:lOB 'I)"BCTBIITe:IbHOCTI! 4
TaKiKe MOiKeT ObITb YBe:IUqeHa. LlmI 3TOli
cxe!>{bI n.10IllaJ.H 5" "" 53' Ii U3 paBeHCTBa
( 1.7) c.le.JyeT BblpaiKeHIIe
2p o S",
Con = he. (1.9)
E05,
B COOTBCTCTBIIH c OpM}'-10H (1.9)
onopHoe HanpSliKeHHe ;I..'15I cxe!>!bI MM no
p HC. 1.2 3, 6 MOiKeT 6bITb )'!>!eHblIIeHo B
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eM. He06xO;IHMbIM ;I..'UI M)U]" BbInO!lHeHHo-
ro no cxeMe pllC. 1.23, G. HaITpnMep, LL'UI
S.. = 20 MK.\i.20 MK\i = 4.10.10 M 8 3 =
= 200 MKM.2oo MKM = 4-10- 8 no BblpaiKe-
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poil U3:\tepHe:\toro .D:aB lcmjj{.
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41
Pue. 1.25. KOHCTP)'KUlIH rll6pIIlIHoro
n.3T<{UKa il3B.aeHIIH:
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"leCKH BbIr01(HOH. B ueHTpe MOCTOBOH TCH-
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C060H KBa.rrparnylO paMKY, pacnOJIO:>KeH
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CSI MSI HarpeBa H nOMCp:>KaHHSI TCMITepary-
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42
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npe06pa30BaTeJIHMl1 Ha OCHOBC nJIOCKI1X
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,!I,alOT J1 ,l\aTlfHKl1 ,l\aBJJeHUSI Ha 6a3e q3 B
BH)J,e MeM6paH C :>KeCTKHM lteHTpOM.
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puc. 1.31, HCnOJIb30BaH B KOHcrpYKUUU,
pa3pa60TaHHoH B I1eH3eHcKoM rocY,l\apCT-
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MeM6paHa 1 (puc. 1.32, a) COCTOllT 113
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pbI om,e,l\HHeHbI B MOCT YHTcToHa. Ha KOH-
ltax .u.opO:>KeK CI1JOPMHpOBaHbl KOHTaKTHbJe
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46
fnaBa l. KOHCTPYKl{HU B nPIIHl{HI1bl PAIJOTbI IlPHl>OPOB
aJ I 2
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pea.'lH3)'JOm:WI npHHlrnn KOMnCHcaUHOHHoro
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1 - MeM6paHa; 2 - ::melCTpO.n H3MepmellbHoii
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4 - nO.I1BIDKHblH cHlIoBoii 311elCTpO.n.;
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,
(200x200 MK!\c) TOnllU1Hoii 1,2 MKM, npu-
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,
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nJ1aCTHHa ITO,ll)J,cp:>KUBaeTCSI no KpWlM paB-
HOMCpHO paCnOJ1O:iKeHHbIMU CTblpbMSI 6aJI-
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47
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ocek CnCJ1.oBaTeJIbHO, HX MOIKHO KJIaCCH-
IPHLJ,HpOBaTb no 1.JMCJIY H3MepHreJThHblX
OCCH: 001'10- HJIH oeyxKo.l"l11oHeHmHblu.
2. 1fHCJlO IIHepl.lnOHHblx Macc. Ba:iK-
HeHwHM 3JICMeHTOM Mf' JlBJISleTC 11M, no-
cryITaTeJIbHOe )J,Bl1IKeHHe KOTOpOH ITpHBO)J,HT
K nOSlBJIeHl11O KOJIH'leCTBa D:BHIKeHHSI, a
BpallJ,aTeJIbHOe - K MOMeHTy KOJIH1.JeCTBa
ABmKeHI1.11. ITo 3TOMY ITp113Hary MO)KHO p83-
JII11.JaTb OOHO- H .MH020AlaCC06ble ()J,Be 11 60-
nee) Mf. ITpM3HaK OAHHaKOBO cnpaBe,nmm
L\ml BCCX TImOB Mf.
3. THn nO,I\Beca. H3BecTHbI )J,aa mna
nOABeCOB: JHeXmmI/eCKUe (KOHl11GKmHble) 11
HeKOHmaKnmble. MexaHH1.JecKl1e nO)J,BeCbI
peaJIH3)'1OTCSI B BMlI.e YITpymx MHKPOCTPYK-
ryp p83JIH'lHOH KOH nrypaLJ,1111 (cTep:>KHe-
Bble, neTJIeBble, CnHpaJIbHble H lI.p.). 3JIe-
Melffbl ITOlI.BeCa MorYT paCITOJIaraTbCSI ITO
ITepl1MeTpy HM, H TaKoH nOJ1.BeC MO:>KHO
H83bIBaTb HapY)[(HbIM, HnH BHeIIIHHM. ITOlI.-
Bec MO)[(eT p83Me aTbcSI B npoCTpaHCTBe
car.-IOH HM, JIM60 MelKlI.Y HHepLJ,HOHHbIMI1
MaCCa!\1l1 B MHorOMaCCOBbIX Mr. TaKHe ITOL\-
BeCbl MO:>KHO H83bIBaTb BHYTpeHHHMI1.
[lpI1HLJ,Hnl1aJIbHO B03MO:>KeH JII060H BMlI.
HeKOHTaKTHOrO nO)J,Beca, peaJIH30BaHHblH Ha
IP1l31l1.JeCKOM npHHUHne, o6eCne1.JI1BalO eM
JIeBHTau,HIO ("napeHHe") HM. HeKOmaKT-
HbIH ITOJ1.BeC MO)[(eT 6bITb H83BaH no 1131l-
'lCCKOMY npHHunny ero pa6OTbI: 3neKTpO-
CTaTI1'lCCKI1H, MarHHTHblH 11 D:p.
4. HaJIlf'lne KHHeMaTn'lecKnx CBH-
3ell. H3BCCTHO, 'lTO AJISI pa60TbI Mf' He06-
XO,nI1MO nO)J,)J,epIKI1BaTb nOCTOSlHHble lJaCTO-
TY II aMnJIMTYJtY Kone6aHMH HM (HJII1 Macc).
B OJ1.HOMaCCOBbIX cxeMax ,nJI.II pellleHH.II: 3TOH
3a)J,a1.J11 I1CITOJIb3yIOTCj{ ,I\OnOJIHI1TCJIbHble
3JIeKTpOHHbie u,enH H )J,aT'lMKll. B MHorOMac-
COBbIX cxeMax HapSI,1.I,y co cpe,l\CTBaMH 3JIeK-
TpOHllKH 06eCITC'lHTh paBeHCTBO 1.JaCTOT 11
aMITJII1TY.n MO)[(HO npMMeHeHHeM KHHeMaTH-
1.JeCKOH CBH3H Me:>K)J,y HM. KuHeMaTJ1qeCKasi
CB.II3b JlBJIHeTCJI aHaJIOrOM cnapHHKa, npl1Me-
mer.-fOrO B KJIaCCH'leCKl1X ,l\ByxrMpocKonHblx
CXeMax. KHHeMaTH1.JeCKasi CBJl3b MO:>KCT 6bITh
48
rJI8Ba l. KOHCTPYKIJ;nl1 II IlPIlHIJ;HrIbl PAEOThl IlJ>11EOPOB
pCaJIH30BaHa a P}1. B PLJ II:III 6 OOOItX pe-
.J/Cll.lfa\" OOH06pe.ueHHO. CYlllCCTBYlOT II ..QPY-
rue Tlmbl KJlHeMaTII'leCKIiX CBSl3eu Me)K):\Y
11M, HaITpHMep ,!VISI YBe..1utJeHw/ _\lacul11w6-
H020 "03(jJf/JlIllueHma.
5. Blffl nepel\lelll,eHIIJI II:M. B031110iK-
Hbl .IlBa BH.Lla B3aJlMHOrO nepeMellJ,eHHH. 11M
B p.n. If PlJ. B nepBo 1 CJlYllaC B 060nx pe-
JKHMax ,!:lBlliKeHiUl 11M (IIJIII Macca) nepe:'\le-
IillllOTCJI B O.:lHOH nJIOCKOCTH. BO BTOpOM
c.'1)"Iae 11M B pq BbIXO.Lln 113 nJIOCKOCTII JlX
nepCMellJ,eHIUi B P.ll
6. Tnn nplfBOll,a (06paTHble npeo6pa-
30BaTeJIH). B Mr MOIyr 6bITb UCITOJIb30Ba-
Hbl :IlO6ble TltnbI o6paTHblx npe06pa30BaTe-
.:Ie" (aKrlOaTopbI). 06eCne'lIiBaIOIl{IiX npllBo..Q
11M B pCi!atMC .1BffiKeHIlSi C 33.LlaHHbIMIJ na-
paMeTpaMIi: !\'iarnHT03.1el\'TpIf'leCKIIe. 3JleK-
TpOMarHIITHble. nbe303JIel\'Tpl1'ieCKIfe. B
paccMaTpliBaeMbIx Tl1nax Mf HaH60JIee
pacnpocrpaHeH 3J1eK"rpOCTaTH'leCKHH npn-
BO.1. BbInOJIHeHHbJH B BHlIe rpe6eHQaTblX
crp)'KIYp.
7. Tun .:IaT'IUKa C"bCMa ClfrHa.rJa (06-
paTHblc npeo6pa30BaTe.1Ju). B Mr MOI)'T
6blTb HCnOJIb30BaHbI JIlO6bIe TI1ITbI npSlMblx
ITpe06p83oBaTeneH (llaTlJHKlf C'beMa CHfHa-
J1a), KOTopble Bblpa6aTbIBalOT HH opMaLlHlO
06 H3MepSieMbIX yrnoBbIx CKOpOCTSIX: 3JIeK-
TpOCTaTU'lCCKHe. MarHI1T03J1eKTpHqeCKHe,
nbe30pe3HCTIiBHhIC, OnTll'leCKHe Ii .Ilp.
Mf npC;J.CTaB:llUOT C060H 06'bCMHbJe
MHOrOCJIOHHble MHKpOcrpYKlJ'PbI, li3rOTOB-
JIeHHbIe H3 KpliCTaJlJIliqeCKoro MaTepHaJIa,
pa60qHe npoLlcccbl B KOTOpbIX nOJUl,ep:nm-
BalOTCH 3JIeKTpOHliKoH. BbInOJIHeHHOH no
J1JI8HapHOH TeXHOJIOrliH Ha O)J,HOM (HJIH He-
CKOJIbKI1X) 113 CJ1DeB MHKpOCTpYKTYPbl. B03-
MO O IOrOTOBJIeHlie Mf H no rH6pu,lJ,HOH
TeXHOJIOrHH - pe3oHaTop H3rOTaBJIlIBaeTC5I
no TeXHOJIOmli M3MC. a 3JIeKTpoHHKa -
Tpall,lfWfOHHO H3 OT,UeJIbHbJX IU1aTax.
AMl1JUfry,na BTOPHqHblX KOJle6aHMH
11M OqeHb Mana. n03TOMY Tpe6yercSi pe30-
HaHCHWI HaCTpOHI<a, npH KOTOpoH 'laCTOThJ
nepBJ1qHblX H BTOPHqHhIX KOJIe6aHHll H coG-
CTBeHHWl qacrOTa pe3oHaTopa 6JlH3KJ.I Me)K-
J!Y COGOH.
Mf MoryT paooT3Tb B pelKmmx [[PH-
Moro II KOMneHcaUiloHlioro npcoopa1oRaHilSi
(H3!11epeHIIII ).
HapSl)J,Y C paccMoTpeHHhIMII lI11,aaMII
pa1pa6aTbIBaIDTCJJ M f, KOTopbIe MO:>KIIO 011-
peJJ:eJIHTb KaK KaMepTOHHble II BOJIIIORJ.lC.
OTJIl1<JIlTenbHhIM nplBHaKOM KaMeprol1HblX
Mr (KMf) SlBJISleTCSI HaJIlI<Jlle CTCpiKlICRhlX
CTPYl\.l)'P ("HmKeK"). CYlllecTBeHHo IJ 10,
<JTO HM HOiKCK. KaK npaBliJIO. paBHoMt:pHO
pacnpeJJ:eJIeHa B,aOJIb liX )J,JII1Hbl Ii HMceTCSI
cB060,nHbIH, He3aKpenJIeHHblH KOHeu. I [pH
nOSlBJIemlH ITepeHOCHOH yrnoBou CKOpOCTI1.
BeKTOp KOTOpOH nepneHJII1KYJI»peli K BeKTO-
paM KOJ1J1qeCTBa )J,BliiKeHII» neMeHTaplihIX
Macc. paCITpenCJIeHHbIX B).lOJIb HOlKeK. B03-
HIIKaIOT CHJIbI IIHepLlll11 KOpJ1OJII1Ca, rCIICpl1-
pYlOIl{lie BTOpl1'lHhle Kone6aHliSi HO:>KeK
KMr.
OTJ]HQI1TeJIbHhIM npl13HaKoM BOJIHOBbIX
Mr JJBJISleTC5I HaJIlt'llle pe30HaTopOB, liMelO-
1l\IIX OPMY KOJIbua, KOTopoe C nOMOIl{blO
YITpyrl1x 3JIeMeHTOB nO)J,Beca CKpeITJIeHO C
KopnycoM, JIH60 4JOPMY CTep:>KIDI, 'laKpenne-
Hl1e KOToporo B Kopnyce He npenSlTcTByer
r
cro npOJJ:OJIbHbIM 11 IIonepe<JHbIM KOJIeOaJlI1-
SlM.
KOJIbLlO nYJIbCl1pyeT B ABYX B3aJ1MHO-
nepneH)J,MKYJISlpHbIX HanpaBJIeHl1H.X (none-
pC'lHaJI ynpyraSl BOJIHa), T.e. ITCplioJJ:l1qeCKIi
npllHI1MaeT OPMY OBaJIa_ I1pH BpallJ,eHI1I1
KOJIbLla OTHOCIITeJIbHO OCH, ITepneHJ1.I1KYmIp-
HOH K ero ImOCKOCTI1, CKOpOCTb apaIlleHI1S1
OCI1, BJ1.0JIb KOTOpOH nyJ1bClipyeT KOJIbLlO
(60JIhlllWi OCb OBaJIa), 6y,Uer MeHbllie CKOpO-
CTH BpallJ,eHH5I Kopnyca, a 3Ha<JIIT 11 KOJIbLla.
YrOJl OTCTaBaHMSI HeceT 11H4JopMaLlulO 06
yrJIe nOBopoTa OCHOBaHJ.IlI OTHOCliTCJIbHO
I1HepLll1aJIbHoro npOCTpaHcTBa.
B ,naHHOM cJI}"iac I1MeeT MeCTO reHe-
pMpoBaHl1e CT05l'leH BOJlHbI, KOTOpaJI 3aTeM
npeLleccl1pyer ITpl1 n05lBJICHI1H nepeHocHoH
YJ'JIOBOH CKOpOCTH OCHOBaHHSI. npoLlecc
rCHepl1pOBaHIUl CTOSl'leH BOJIHhI Ha6JIlO,naeT-
C5I II B TaK Ha3hIBaCMhlX CTcp:>KHeBbJX rllpo-
cKonax.
Tl1nOBhIMH xapaKTeplICTHKaMI1 Mf 5IB-
JlJUOTCH: ,UJ.IaIT830H 113MepeHIIH. qyBCTBU-
TCJIbHOCTb, nOJIoca ITponycK3HHJI qaCTOT,
rHPOCKOIlhl
49
\JaClllTa6I1bll! K03 (lmUHCIIT I1 ero cTa6l1JIb-
HOCTb, nepeKpeCTIIaH. <JYBCTBHTCJlbHOCTb,
r
wy . TeMncpa rypH3S1 CTaOI1:II,HOCTb xapaK-
repHCTlIK H llpyme 'JKCrlJIyaTaL(110HHbIC na-
paMCT})bI 11 xapaKTcpHCTHKI1.
1.3.2. MnK'pornpocK'onLi LL-TMna
KOHCTPYKTUBIIhlM Y3JlOM, OITpelleJISlIO-
1lJ,1IM YHKUI10Ha.lbHble B03MO:>KHOCTJ1 Mr,
SlBJISlCTCH 43. 3 Mr MOiKHO Ha%IBaTb 11M
(I1:IH taCCbl) B nO,/I,BCCe C npl1, BO,il,OM KOTO-
phIl! 06eCne'lHBaeT HM P.Q, Ha KOTOpbIH npl1
HaJUI'llHl nepCHOCHOH yrJIOBOH CKopocn1
BCJleJ1.CTBHe B03HI1KaIOIl{ero YCKopeHHJI Ko-
pl10JUIca 11 COOTBCTCTBYIOIlJ,HX eMY CHJI
11HepUHI1, reHepl1pylOTCSI BTOPH'lHhlC KOJIC-
6aHItSI (P ). Ha :nOM OCHOBaHW1 Mr HHorlla
Ha3bIBaJOT npl160paMI1 )J,JlSl H3MCpeHl1S1 YCKO-
peHHSI KOpHOJIHca.
Y CKOpCHl1e KOpl10JIl1Ca onpC,/I,eJIJ/cTCSI
BeKTopHbIM npoH3Be)J,eHI1eM a K = 2(nxv)
(n - BeKTop MfHoBeHHoH yrnoBoH CKOpOCTH
BpaIl{CHHSI ITOllBH:>KHOii CI1CTeMbI KOOp,l\I1HaT;
V - BeKTop MrHoBeHHoH JII1HeHHOH CKOpoCru
Tena B nOllBl1:>KHOH CHCTeMe KoopllHHaT).
Y cKopeHHe KOpHOJIHCa SlBJISlCTCSI BCKTOpOM,
,l\JIHHa KOToporo paBHa
a K = 2nv sin cp
( l.l 0)
3
y
2
QKy
maKx
z _____
m
1
maKy
n:
(<p - yroJI Me)J(llY BeKTOpaMH n 11 v) H KOTO-
pbIH HanpaBJICH ITCpnCHlll1KY.lUlpHO K BeKTO-
paM n H v B TaKYIO CTOPOHY, IJT06bI KpaT-
'laHIIIMH ITOBOpOT OT n K v Ka33J1CSI Ha6nlO-
.aaTeJIIO, CMOTpSlI1.!CMY C KOHua BeKTOpa aK'
HllYIlJ,I1M npOTHB lJaCOBOH cTpemrn.
A. O)J,HOl\laCCOBbIC I\IllK'pOmpOCK'Onbl
Ha pHC. 1.34 npHBe.D,eHa npHHUlmH-
aJIbHaH. cxe 1a q3 Mr, KOTOphlH COCTOHT 113
I1M I (BenH'lI1HOH m) H nOJ1.Bcca H3 ynpymx
3:ICMCHTOB 3, 4, CKpCnJICHHbIX C OCHOBaHHCM
5 (nOKa3aHO YCJIOBHO). 3neMeHTbI 206ecne-
'lHBaJOT ueJIOCTHOCTb H )J(CCTKOCTb B MeCTax
COnpSl:iKeHHH. ynpymx 3JIeMeHTOB. IIpHBo.D,
KaI<OH-JIH60 H3H'leCKOH npHpollbl (3JJeK1pO-
CTaTI1'leCKHH, 3JICK1pOMarHHTHbIH H T.)J,. - Ha
pl1CYHKe He nOKa3aH) YHKUHOHaJIbHO BXO-
AHT B COCTaB q3.
B 06I1.!eM cJIY'lae npHBO)J, pa3BMBaeT
CHJIY Fosin pt (Fo, p - COOTBeTCTBeHHO aM-
nJIHTY,l\a ClillbI I1 'laCTOTa ee reHepaQHH),
KOTOpaJI HaITpaBJIeHa nOA HeKOTOpbIM Ma-
JIbIM yrnoM E (sinE:::: E, COSE::::: E, t - BpeMSI) K
OCH x H co06maeT EM Kone6aHHH. x =
= xosinpt (xo - aMIIJIHTY,l\a KOJIe6am1ii).
MrHoBeHHblH BeKTOp JIMHeiiHOH CKOpOCTH
11M B P I1MeeT npocKUlUi Ha OCH X 11 Y:
V... = V cosE::::V; vy=-vStnE:::::-VE.
5
4
QlU;
x
v
E
1 ...
P
PRe. 1.34, npnHIUInJtaJlbHaH eXCMa O.ll:nOMaeeOBoro 113 Mr:
I -11M; 2 - )KeCTKHH 3neMem; 3, 4 - ynpyrMe 3neMembI nO.D;BeCa;
5 - OCHOBaHHe (nOKa3RHO YCJIOBHO)
50
rnasa 1. KOHCTPYKUHlI 11 IlPHHUl1rIbI PA60Tbi rIPlIbOPOB
DpH nOSlBJICHUlI ITepCHOCHOH yrnoBoll
CKOpOCTII .0.:, MrHOBeHHhIll BeKTOp KOTOpOM
HanpaBJICH B nOJIO:>KIlTeJIhHOM HaITpaBJIeHlU1
OCH Z, B03Hl1KaCT YCKOpeHl1e KOpJ1OJIl1Ca
BJ1.0JIb OCCll X H Y:
QKx = 2.0.:\'£.
aK.'" = 2n z v,
BCJIe,QCTBHe 'lero 11M OKa3hIBaeTCSI noD, )1eH-
CTBHe!\1 CILl I1HepwUl KopIlmIJ1ca: mQ}(,
B,QOJIh OCJ1 X. Ii maK;,' B,QOJIb OCJI Y.
C yqeTo 1 CHJI IlHepLl.JlII, )J,eMm]mpOBa-
HWI J1 ynpyrnx CIEI. .1eHCTBYlOlllJlX Ha 11M,
ypaBHeHIUJ J.BIDKeHIUJ B ITpocTeifllJeM
cJIyqac J1MeIOT BH.1
mT + bxx + G xX = Fo sin pt - 2m\'0=£:
m) + b y .\' + G 1'y = -(Fo sin pi}: - 2mvO:,
(l.I1)
r)J,e b x , b). - K03cp£l!HLl.l1eHTbl ,QeMn npOBaHJUf
11M B HanpaB eHlm cooTBeTcTByIOIlU1x
oceii; G x , G., - :>KeCTKOCTJI YITpyroro no.nseca
B HanpaB.;"1CHHJI cOOTBeTCTBytOIlll1X oceH.
ITepBoe ypaBHeHl1e CI1CTeMbI (1.11)
Onl1ChIBaeT P,[I" a BTopoe - pq, 113 KOToporo
cne;IYeT, liTO ITepeMellJeHl1C 11M B,QOJIb OCII Y
non. n.eMCTBHcM c w l1HepLl.HH KOpHOJIJ1Ca
llCKaA\aeTC npoeKLl.Heii CMJIhl npl1Bo)J,a Ha
:ny jKe OCb, liTO ITpHBOlJ,HT K 0lIIH6Ke B Mr
l13MepeHHl1.
OueHHM BJIIDIHHe BeJIJllil1HbI £ Ha TOIi-
HOCTb H3MepeHIDI no!. Tal< KaK
v = X = xoP cos pI, YCKopeHHe KOpHOJIHCa
aK}' = 20zxop cas pt . DpOeKLl.HH YCKopeHHH
11M B pejKJlMe .!lBHjKeHIDI Ha OCb Y paBHa
a y = X£ = -£xop2 sin pt = F.Xop2 cos (pt+ 90°)
H H8 31)' COCTaBJISlIOIllYIO YCKopeHIDI HM
pearHpYeT TaK :iKC, KaK M Ha aK}', liTO npHBO-
,!UIT K 01llH6Ke M3MepeHIDI, KOTOpyJO npIlH -
TO Ha3bIBaTb KBa,l\pa-rypHOH, nOCJ<OJIbKY Me-
)Kl(Y YCKopeHMHMH a y l1 aK}' cymeCTBYeT
CJtBMr no a3C Ha 90°.
3anHllIeM OTHOllleHMe aMnJIH1)',n YCKO-
peHMH
a y IOK}' = £P 120.: ,
J13 KOToporo CJIe,QyeT, 1.JTO npu Oll.I1HaKOBOM
nopSl)J,Ke YCKopeHIISI TO'UlOCTb c06mOJ1,CIII1J1
ncpneH,QI1KYJISIpHOCTII MelK,!ly HanpaBJICHIISI-
Mil P,[I, J1 P 1I.0JI:iKHa 6bITh O<ICHb BhICO-
KO". HanpJ1Mep, npll 0: = 0.05 pan/c. p =
= 10 5 pan.!c, a,. = llK,'= I )J,onycTlIMoe 3Ha<le-
Hlle yrJIa £ = 10 6 pan, 'ITO npaKTlit'cCKII
06eCneQliTb lie y)J,acTcSI.
KBa.rrpaTypHhlif Cl1rHaJI IIMeeT T)' lKe
'laCTOTY, liTO II npUBO)],. 3TO J1,CJIaCT 3aTpy,Q-
HIITCJIbHblM nJIhTpaUHIO nOMeXI!. O,llHaKO.
113-3a C)J,Bnra Ha 90° KBall.paTypHblH CJ1I'liaJI
MO:iKeT 6bITb 1.JaCTIfI-IHO IICKmO'leH C nOMO-
lllhIO a30'lYBCTBI1TeJIbHoro ,QeTeKTopa. '3 -
eKn1BHOCTh TaKO" UJIbTpaUUII 3aBI1CJlT OT
Toro, HaCKOJIhKO T01.JHO a30BOC COOTHOlIIe-
HJ1e MOIKCT noJt)J,ep:iKI1BaTbCSI 3JIeKlpOHJ1KOH.
CJIenyeT 06paTIITh BHl1MaHl1e lia TO,
'lTO B Mr C O,!lHOMaCCOBbIM 43 Tpy)],HO OT-
)J,eJIMT1. nOJIC3Hb1H CHI'HaJI, 06YCJIOBJIeHHblM
YCKopcHueM Kopl1oJIl1ca, OT CJ1rHaJIa, B1.I-
3BaHHoro JIJ1HeHHbIM YCKopeHJ1eM. BeKTOp
KOTOpoI'O I1MCeT COCTaBJISIIOIlJ,YIO BDOJIb OCM
BTOPllljHblX KOJIe6aHJ1i1( BbIXO,QHaSI OCb).
Ha pJ1C. 1.35 B 1I.0ITOJ]HeHJ1e K pJ1c. 1.34
npJ1Be)J,eHbl npUHLl.nnmL'IbHble cxeMbl Q3.
KOTopbIe npHMeHeHbl B 60JlblIIJ1HCTBe 113-
BeCTHblX KOHCTpYKUlIH OD,HOMaCCOBbIX Mr.
B CXCMe Q3 no pJ1C. 1.35, a nO)J,Bec 11M 1
OTHOCJITeJIbHO OCHOBaHJ1S1 5 BbJITOJIHeH B
BJ1)J,e ynpymx 3JIeMemOB 3, 4, pacnOJIO:>KCH-
HbIX BHe KOHrypa 11M. B OTJ1n1.JUe OT CXCMbl
q3 no pHC. 1.34, r)],e BCC ynpyme 3JIeMeHTbI
ITO,!lBeCa CBSl3aHbl MC:iK,lly C060H 1.Jepe3 11M,
43, BbmOJIHeHHbJM no CXCMe pJ1c. 1.35, a,
HMeeT ConpSl:iKeHHhIe C nOMOlll1.1O :>KeCTKJ1X
3neMeHTOB 2 napbl ynpymx 3JIeMCHTOB noJ1.-
Beca.
B CXCMe no pnc. 1.35, 6 3 COJ1.ep:>KnT
11M 1, KOTOpWI YIIpyrnMn 3JIeMeHTaMJI 3
COeI\HHeHa C 3JIeMeHTOM 2 nO)],Beca B opMe
paMKn, a OH ynpyrnMl1 3JIeMemaMH 4 CBSl3aH
c OCHOBaHHCM 5.
Bo Bcex cxeMax q3 H3MepHTeJIbHOH
HBJISleTCSI OCb z, BOl<pyr KOTOpOH I\eHCTByeT
n3MepSieMaJI ymoBWI CKOpOCTb n z . B cxeMax
no pHC. 1.35, a, 6 P,[{ MO:iKeT 6hITb OpraHH30-
rl1POCKOnbI
51
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B03MO:>KHOCTb 1136c:>KaTb BJIliXHMX nOCTOSlH-
Horo I1 BH6poYCKOpCHJ.!1I Ha BbIXOJJ.HOH Cl1r-
HaJI rnpOCKOITa.
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PIIC. 1.39. nplllluHlIHa.JLHaH cxenta Mr cJlHPMLI Samsung Electro. Mechanics:
1,2 - HM; 3, 5, 8 - aHKcpHhIC 3J1eMeHTbI; 4 - rpe6eH'laTaJi c-rpYKTYpa CMKOCTHoro H3MepHTem;
6, 7 - ynpyrlle JneMCHThI nO Beca; 9 - rpe6CH'IaThIe CTpYKIypbl 3ncKTpocraTI1'lCCKoro npllBo a
BbICOKO'laCTOTHbIH CHrHan, nonyqeH-
HbIH C eMKOCTHbIX M3Mepl1TeJIeH (,/:\aT'lMKoB)
ncpeMC eHI1H, nocTYITaCT Ha KaCK8).1.bl YCM-
JIeHIDI M ,/:\eMo.nyJIHIJ;HH, ITpe06pa3ylO He era
B BblXO).l.Hoe Hanp'DKCHI1C, nponopUMOHaJIb-
Hoe H3MepSieMoi1 ymoBOH CKOpOCTM. B co-
CTaB MI1KpocxeMbl BXO,/:\l1T )J,aT'lMK TeMnepa-
TypbI ,/:\ml KOMneHCaIJ;I1M TeMnepaTypHblX
norpelliHOCTeH H KaJIH6pOBKH, a TaK:>Ke npc-
lI.M3MOHHbIM MCTO<IHHK onopHoro HanpSl-
)!(eHMSI.
rupOCKonbI ADXRS I1MelOT cTa6l1nb-
HbIH BbIXO)J,HOH CMmaJI npH YCKopeHHHx ,/:\0
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pupoBaHHbIx C GPS CI1CTeMaX HaBl1ral{MI1, B
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cnY'la x.
npMHIJ;MnMaJIbHWI cxeMa Mr l1pMbI
Samsung Electro Mechanics (CIlIA), aH8JIO-
rM'lHWI cxeMe no pl1c. 1.35, 6, npMBC,l\eHa Ha
pHC. 1.39.
Mr BblnOnHCH B BI1)J,e nnaHapHoH KOH-
CTPYKl{MM Ha KpCMHMCBOH ITOAJ!O:>KKC M
BKJHO'laeT B cc6S1 HapY'A<HyIO 1 II BHYTPCH-
HIOIO 2 I1M, MC:>K).I.y KOTOpblMH HaXO,WITCSI
'JeTblpe YITpymx 3JICMCHTa 7 nO,/:\Beca, o6ec-
ne'lMBalOllJ,MX B03MO:>KHOCTb ncpeMcllleHHH
11M 2 B HanpaBneHMM OCM Y. lIeTbIpe ynpy-
mx 3JIeMeHTa 6 nO,l\BeCa, pa3Me eHHbIx
MC)K)J,y aHKepHbIMM 3JIeMeHTaMH 3 H 11M I,
rapaHTHpYIDT COBMeCTHoe nepeMcllleHMe
11M B HanpaBJIeHMH OCM X 3JIcKTpoCTaTll-
'leCKHe npMBo)J,bI pacITOJIO)!(eHbI B,l\OJIb OCM Y
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,/:\ep:>K8T rpC6eH'laTbIe CTPYKTypbI 9, 06'be).l.M-
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3neMeHTOB 8.
3neKTpOCTaUNccKHH npMBO,l\ 06ccnc-
'JMBaeT B P,L( nepeMe eHHSI 06eI1X 11M co
CKOpOCTbIO V B HaITpaBJIeHMI1 OCM X. ITO.SlB-
neHMe YfJIOBOH CKOpOCTH Q;: ITpHBO,l\HT K
BM6pauMoHHbIM ITepeMC eHlUIM 11M 2 no,/:\
,l\CHCTBHeM CHJI MHepQMM KOpMOJ1llCa B Ha-
ITpaBnCHHM OCM Y (pq). B pq ncpCMeIllCHMJ1
11M 2 MKCMPYIDTCSl 'JeTblpbM.Sl eMKOCTHbIMH
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CTPYKTYPbl 4 3JIeKTpo.noB KOTOpbIX .SIBJUIlOT-
CSI qaCTblO 11M 2 H qaCTbIO aHKepHbIX 3JIe-
MeHTOB 5.
ITo cxeMC pHC. 1.35, 6 BbInOJIHCHa O,nHa
H3 pa3pa60TOK HHCTH'1)"I'8 HSG-IMIT (<1>Pr)-
rnpocKon MARS-LL, CXCMa KOToporo nOKa-
3aRa Ha pHC. 1.40. Mr H3rOTOBJIeH no nna-
HapHoi1 TeXHonorHH, H Bee ero 3J1CMeHTIiI
paCnOJ1aralOTCH HM nOAJ!O)KJ(OH B OAHOH
nJIOCKOCTU. 11M I pa3MellJ,aeTCH BHYl1m pa-
56
fnaBa L KOHCTPYKU,H.H 11 IlPI1Hu,nrIb] PAEOTb] rIPHEOPOR
5
7
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PIIC. 1.40. CxeM3 !\1r MARS-LL:
1,2 -11M; 3 -/KeCTlutii (He.llecpopMHpyeMbIH) :mCMelIT nOllBeca; 4 - aHKepbl;
5 - rpe6eHQaTble cTpYI\l)'PbJ 3JJeKTpolloB 3nel\1pocTaTlI'leCKlJX npHBolloB; 6 - eMKOCTHble IBMepHTeJ1l1
nepeMemeHlJM; 7 - ynpyrne 3neMeHTbl nO.llBeCa
MO'ffiOH 11M 2. Ynpyrnc 3jlCMeHTbI 7 nO}.l.BC-
ea 06ecnC'lHB81OT B03MOiKHOCTb nepeMell.le-
HHH IiM 1 " 2 B HarrpaB.iJCHMH OCH x.
Ynpynle 3JleMCHTbI nO)J,Beca 3aMbIKa-
IOTCSI Ha aHKepbI 4 'lepe3 :>KeCTKlie (HeJ1.e-
$opMMpyeMbIe) 3JIeMCHTbl 3. 3neKTpOCTaTH-
'leCKHe npMBO..:lb! C rpe6eH'laTblMH CTpy"--ry-
paM" 3j]CKTpD;I:OB 5 06eCneQHBalOT P)J, 06e-
I1X 11M. IIOHB!leHHC ncpeHocHOH ymoBoH
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CR!I HHepUHH KOpHOJIHCa, BbI3brnalOllUiX
;J.BIDKeHHe 11M 1 B pq, KOTopoe I}>HKCHPYeT-
CJI eMKOCTHbIMH H3MepHTemlMM nepcMell.le-
mdi6.
I1pllHllHIIll3JlbHaJI cxeMa Mf HpMbl
Bosch (fepM8HIDI), COOTBeTCTBytOll.la51 CXCMe
no pHC. ) .35, a. npHBe.neHa Ha pMC. 1.41. Mf
BbUlOJ1HeH no l1JIaHapHOH, KpeMHHeBoH Tex-
HOJIOrHli, H BCSI MHKpOCTp)'KTYpa pacnono-
jJ(eHa H3 HeKOTopoM pacCTOJJHHH Ha.D, no.n-
ncrACKoH. 11M 1 HMeeT 'leTblpe ynpyrnx 3JIe-
MeHTa 3 no KOTopbIe 06eCne'lHBaIOT
B03MOiKHOCTb ee nepeMew.eHWI B HanpaBJJe-
HHH OCM X OTHOCmeJIloHO onopHblx paMOK 6,
9. nOCJle He. B CBOIO OqepeJtb. MOryI' ne-
peMemaTbCJJ B HanpaBJJeHHH OCH Y 6naro.n.a-
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II, CB C nO.MmtCI<OH 'lepe3 aHKepbI 10.
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2, 7 MO:ryr Mcnonb30BaTbCSI B O)J,HOM 113 }.I.BYX
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CTPYKTYP M nO)J,BH:>KHbIe 3JIel<Tp0}.l.bI, SlB-
mnOWHCCH 'laCTbIO 11M J, npMTSlrHBaJOTCSI
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C nO)J,JIO:iKKOM aHKepaMM 8.
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3JJeKTpo.noB 2, 7 MO:>KeT BbInOnH5ITb YHK-
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a .npyra51 - YHKUMIO CMKOCTHOro M3MepM-
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MelllCHHSI B HanpaBJJCHMH OCM Y (PlI). Ena-
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3JICMCHTOB 3 B HanpaBJJCHHH OCll Y BMeCTe C
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eT 11M 1. H3MepeHMe nepCMell.leHHH B Pl£
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PHC. 1.41. fiplIHQllmlaJlbHaH CXCMa Mr f1JHPl\lbI Bosch:
I -11M; 2, 7 - rpe6eH'lffi'bIe CTpYKTypbl 3neKTpOnOB; 3, II - ynpyrne 3neMCHTbI
nOJJ:Beca; 4 - e [KOCTHhJH H3MepMTenh ncpeMelI!eHlIH; 5, 8,10 - aHKCpbl; 6, 9 - onopHble paMKI1
JISI:\IH 4, KOTopble :\IOrYl 6blTb BKJIIO'leHbI B
)J,M epeHLlMaJIbHYIO cxeMY. TIO,Il,BM)!(Hble
JIeKTp0,llbI M3MepMTeJIeH nepeMCll.{eHHH }lB-
JISiIOTCSI 'laCTbIO CTPYKTYP onopHbIx paMoK, a
HenO)J,8I1:>KHble 3JIeKTpO,Il,bI, 06'be,l\HHeHHble B
e)J,I1HbIC CTPYKTYpbI, CBSl3aHbl C ITO,Il,JIO:>KKOH
aHKepaMM 5.
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TIpHHLlHITMaJIbHWI cxeMa Mr cpHPMbI
Microsensors (CIlIA), aHaJIOrH'lHWI pMC.
1.35, B, npMBe)J,eHa Ha pHC. 1.42.
Mr CO,Il,ep:>KHT .nae 11M (J I1 2), COell,H-
HeHHble Me:>K,II,Y C060H 'leTblpbWl napaMM
yrrpymx 3JIeMeHTOB 6, 7 no.nBeca. KmK bIH
113 3JIeMeHTOB 7 CB}l3aH C KpeMHl1eBOH l1JI11
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1, 2 -11M; 3 - 3JleKTpOCT8m'leCKHH npHBo.n C rpe6eH'laTOH C1'PYKTYPoii; 4, 5, 9, J 1, 13 - 8HKepbl;
6 H 7 - ynpyrHe 3neMeHTbI nO,llBeca; 8 - npHBo.n. KOHTypa KOMneHcaLlIDI; 10 - CMKOCTHblC
1i3MepHTeJlll nepeMell.\eHIiH c rpe6CH'laTOH C1'PYKTYPOH; 12 - npllBo.n ypaBHoBClUHBaHHH Ii
TeCTHpoBaHHH c rpe6euQaTOH C1'PYKTYPOH
58
[mlBa 1. KOHCTPYKlI.Hn 11 IlPI1HU,UIlbl P AbOTbl rIPHGOPOB
113rOTOBJICHHOH M3 pyroro Marepna.rI8 nO}1-
JIOIKKOH aHKepoM 5. HM I HMeeT OPMY
paMKH, II ee Macca 60J1bme, <reM 11M 2. B pn
)J,Ba 3JIeKlpOCTanI'leCKIIX npUBO}1a 3, Hcno)J,-
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KOTOpbIX 38l\.-penJlCHbl Ha ITOnJIOiKKe aHKcpa-
MM 4, oC)'ll1eCTBmnoT Bu6poncpeMcmeHiUl co
CKOPOCTbl() v B,llOflb OCU X TOJIbKO 11M 1.
IIpM 3TOM 1l3rn6HbiM }1C OpMallluIM ITon-
Bep:>KCHbI lIumb 3!leMeHTbl 6.
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1, no,il. HX ;:r,CUCTBUCM H8'lHHaKJTCSI BH6pone-
pCMeIlleHIUl nOC:Ie.1Heu B.10Jlb ocn Y (pq).
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O<rCBJL1HO, qTO OTHomeHlfe JIHHCHHbIX
ncpeMellleHMH Ii.M 2 K 11M 1 pasHo OTHO-
llleH1UO HH 3 eMeHTa B KOTOpOM ero
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TO<IKH l\.-peIl:"lCHI1H C 11M 2 no aHKepa 5. 3TO
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HaCTpoihcy pe30HaHCHoH QaCTOTbi 11M 2,
CCJIH OHa He 06ecneQeHa KOHCT)JYKUHeH.
Mr MMeeT TalOfCe KOH1}'pbI ypaBHOBe-
lliHBaHHJI H TeCTMpOBaHHSI C rpe6eHQ8TbiMU
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npl1Mep, Ka'leCTBa pa601'bl I13MeplneJ1eiI 10.
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60TKM MpMbI The Charles Stark Draper
Laboratory (CIlIA) npI1Be)J,eHa H8 pMC. 1.43.
Mr MMeeT )J,BC HM ], ITO)J,BelllCHHble Ha
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8. B pe>KMMe ,l\Bll:>KeHHSI 06e HM nepeMe-
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pbl B.nOJIb OCM X. fIpM nOSlBJ1eHMH yrJIOBOH
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npOTUBO a3HbIC nepeMel11eHHSI HM B.QOJIb
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60
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Rockwell International (CUJA). ):{mI aBTO-
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Analog Devices (CiliA) npMBe)J,cHa Ha
pMC. 1.45.
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MeHTOB KOHCTpYKI{HH C ITOMOllJ,blO npl1Bo)J,oB
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JICHHH OCCM X H Y. TIpH nOHBJIeHliH YfJIOBOH
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pa30M peanmYeTCJI P4 BOKpyr oeH Z.
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a - nepeMemeHID! 11M B P 11. H PLJ B OJlHOH nJ10CKocm; 6 - TO :iKe, B pa3HhIX nJ10CKOCUIX;
6, 2 - HanpaB!JeHJHI P l1. H pq: 1 -11M; 2 - ynpyme 3JIeMeHTbJ nO,!J.Beca P 11.; 3 - :iKeCTKHe JJIeMeHTLI
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B npellIloJIOA<eHMM, 'ITO BCC I1M O)Ui-
HaKoBLl, KOHcrpyKUIDI q3 CI1MMeTpI1'IHa,
ynpynte CBOHCTBa Bcex 3JICMCHTOB 2 11 3ne-
MeHTOB 4 Me C060H l1)J,CHTJ1IlHbl, npo-
CTeHmWl CHCTCMa ypaBHcHHH. onHCbIBafO-
IWiJJ mmeHHLIe nepeMclllcHIDI OMOH 11M B
P)::(, HallpHMep, no KoopJUmaTe X M yrJIOBbIe
KOJIe6aHIDI q3 Boxpyr OCM Z no IWOp)J,l1HaTe
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mi + b;xX + G;xx = Fo sin pI;
(1.20)
Jq>ifJ+ b lp ip+G",<fJ=-8mvQ.zRo ,
r,l\e bJ<' OJ< - Kro4>l}>HWfeHT AeMnl}>HpoBaHIDI
M :iKeCTJCOCTh ynpyrHX 3JICMeHTOB 2 AIDJ 0)1.-
HOM 11M; Fo, P - napaMeTpbl CHnbI. rCHepI1-
pYlOllleH P,lI; J - MOMCHT HHepttHH BceH
MI1KpOCTPYKTypbI q3 OTHOCHTcnbHO OCH Z;
b'f!, G - K03c1> I1ttI1eHT ,llCMn HpoBaHHSI 11
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YpaBHeHI1e ,llBIDKeHWI. 11M B P.n; aHa-
nOrI1QHO ,llJIjj BCCX paCCMOTpCHHblX paHce
CXCM Mr. 3)1.CCb Ha)J,O HaITOMHI1Tb elllC pro,
'lTO :>KeJJaTeJIbHO I13MCpCHHe CKOpOCTH v = x
11M B P.n;, TaK KaK 3Ta BeJUIQHHa Heo6xo.AH-
Ma ,llJIJI BbIQMCJIeHIDI YCKopeHIDI KOpHOnI1Ca.
113 ypaBHcHIDI ,l\BH:>KCHHSI JUlSi KOOp,l\l1-
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BbIpaiKeHHe
rHPOCKOIlbl
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\tHHI1Ma.'1bHOe "lI1CJIO KOHCTPYKTllBHbIX Y3-
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6apl1 rHblX pa3MepoB.
Pa60Ta q3 no pHC. 1.46, 6 ITpOHCXO,D.HT
CJ1C.nyIOIllHM 06pa30M. OpHBO,l\ 06ecne'mBa-
CT nlllxpOHHOC, npOTI1BO a3HOe )J,BH:>KeHHe
I1M 1 B HaITpaBJIeH1111 OCM X. KHHCMaTH'lC-
CKml CBSl3b 6, KOTopaSi no npHHQHny pa60Tbi
alla.iwrl1qHa aHTHnapanJle.iIOrpaMMY, cno-
C06CTBYCT CI1HXpOHH3aQMH npOTMBO a3HbIX
,llBH:>KCHHH I1M B P,[(. KIIHeMarn'leCKa
CRSl3b 6 He06S13aTenbHa.
TIpH nOSlBJIeHHH yrnoBoH CKOpOCTH f2,;
B03HHKalOT CHJIbI HHepHHH KOpHOJIMca. ,[(JISI
npHHSlToro MrHOBeHHoro HanpaBJ1CHMSI BeK-
TOpOB v 11 Q= )J,JISI npaBoM 11M CHJIa F K
HanpaBJICHa nepncH)J,I-mynHpHo K nJIOCKOClli
XZ B OTpl1UaTCJIbHOM HanpaBJIeHHI1 OCH Y, a
,llJISI nCBOH - Ha060poT. B pe3YJIbTaTe BOKpyr
OCH Z B03HHKaCT MOMCHT c m HHCp HH Ko-
pHOJIHCa Mz = -4mvD.;,Ro, BeKTOp KOTOporo
HanpaB:IeH B OTpHUaTeJIbHYIO CTOPOHY OCH Z.
TIpH H3MeHeHHI1 HanpaBJIeHIDI BeKTO-
pOB v H3MeHHTCSI H HanpaBJICHHe BeKTOpa
A1r., '1TO BbI30BeT KOJIe6aTeJIbHbIe ):IBH)I(eHHH.
q3 B P BOKpyr OCH Z. O'leBH):IHO, '1m B P4
ynpyrHe 3JICMCHTbI 4 pa60TalOT KaK TOpCHD-
Hbl, T.C. Ha KpY'leHHc BOKpyr OCH Z
(pHC. 1.46, z).
YpaBHeHIDI ,D.BH:>KCHI1S1 11M B P,[( aHa-
JIOrH'lHbI ncpBoMY ypaBHeHHIO CHCTeMbI
(1.20), a ypaBHcHHc q3 B pq - BTOPOMY
ypaBHeHHIO 3TOH CHCTeMbI:
J cpCP + bcp<p + Gcp<P = -4m v o.zRo,
r):lC G - )l(CCTKOCTb YITPYrnx 3neMCHTOB 4
Ha KpyqCHHe BOKpyr OCH Z; OCTaJIbHblC na-
paMCTpbI Te :>KC, '1TO H B CllCTCMC (1.20).
"3aKpYTKa" 3 BOKpyr OCH Z oITpe;J.e-
nSleTCJI opMynoH (1.21) C 3aMcHoH K034J-
4JHUHeHTa 8 Ha 4.
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'1TO CXCMa Ha pHC. 1.46, 6 JlBJIHeTCSI 06rueH H
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TeJIbHO, ecnH BbIITOnHHTb 3JIeMeHTbI 4 C Ma-
JIOH mrn6HoH :>KeCTKOCTblO BOKpyr OCH Y, TO
npH H3MepCHHH yrnoBoH CKOpOCTH BOKpyr
:nOH OCH ):IBM)I(eHHe q3 6y)J,CT B nJIOCKOClli
XZ. QHcno 11M B 3TOM CJIY'lae MO:>KHO YBe-
JIH'IHBaTb, KaK 06 3TOM YITOMHHaJIOCb BbIwe.
PaCCMOTpHM KOHcTpYKLUm Mr, Q3 KO-
TOpbIX COOTBeTCTBYIOT CXCMaM Ha pHC. 1.46.
061lJ,HH BH,ll KOHCTpYKUHH Mr, IDlHe-
MaTHKa 43 KOToporo OTBeqaeT cxeMC Ha
pHC. 1.46, a, ITOKa3aH Ha pHC. 1.47.
q3 COCTOMT H3 BOCbMH 11M 1, KOTopbIe
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pOBaHbI C KpyroBOH paMKoH 3, 3aKpenJ1eH-
HOM, B CBOIO O'lepc)J,b, nOCpe,D.CTBoM 'lCTbIpCX
pa)U1aJIbHbIX ynpyrHX 3JIeMCHTOB 4 Ha aHKe-
pax 5, CBSl3aHHbiX C nO)J,JIo)l(Koii, B pC:>KHMe
)JBH:>KeHHH BCC 11M, 06pa3YI01l\HC q3, C no-
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,llBHfaTeJIeH C OpMHpoBaHbl BMeCTe C 11M, a
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IIpM yrnoBoii CKOpOCTl! 0. BOKpyr OCH, ITep-
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TOpbIC C03lJ;aIOT MOMCHT CHJI, BbI3bIBalO1l\HH
pa3BOpOT paMKH BMeCTC C q3 OTHOCHTCJJbHO
no.nnO:>KKH. npM 3a,D.aHHb1X Ha pl1C. 1.47
MrnOBeHHblX HarrpaBJIeHHblX CKopocTeH v H
0. Bpall\aJ01l\HH MOMCHT HanpaBJJCH no qaco-
80H CTpCJIKC. YrnoBble nepCMCllJ,eHHH. paMKH
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01{1l pOBKH Ha pMC. 1.46 H 1.47 COOT-
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J - 11M; 2 - ynpyme 3J1eMeHTbI nO,ll.Beca 11M; 3 - )KeCTKllll paMO'lHbIH 3J1eMeHT
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npHHUHnHaJIbHWI cxeMa Mr HpMbI
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tJ3, aua..;10rMtJHIiIM CXCMe no pilC. 1.46, O.
npMBc,neHa Ha pilC. 1.48.
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BJIO.m, OCII X OTHOCHTeJIbHO :>KeCTKHX, HC,llC-
q.OpMHPYCMhIX 3JICMeHTOB 3 nO)J,Beca, oO'be-
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XPOIlH33IUUl npoTHBo 83Hh1X ABmKcHHH
MM. B TO'IJre nepece'IeHIDI 3JIeMeHTOB 13 J1
14 npoHCXOJIHT pa3BOpoT 3neMeHTOB J 4 KaK
e,I:{I1Horo CTepJICIDI BOKpyr Boo6paiKaeMOH
OCH. Bc" MHKpOCTpytcrypa C nOMOlI.1h1O TOp-
CHOHOB 4, T.e. ynpyrHX 3neMCHTOB, 06ccne-
'IHB8J01QJ1X B03MOJICHOCTL ee BpalI{aTeJIbHb1X
)J,BAJKeHHH BOKpyr OCM Z, YCTaHOBneHa Ha
amcepax 5 Ha HeKOTopoM B03BhimeHHH Ha,l\
KpeMHHeBoH nOJUIOJKKOH.
TIpOTHBO a3HOC ,D.BH)j(eHHC 11M co
CKOpOCTblO v (P L(), CHHXpOHH31fpyeMoe KH-
HeMaTH'leCKOH CBSl3blO 14, o6eCnC'lHBaeTC1I
rpc6eH'laTbIMH CTpYKTypaMH :mCKTpOCTaTH-
IJCCKHX npHBO,llOB 8 (JICBOrO H ITpaBoro).
I1H OpMaI..lHIO 06 3THX ABMJKeHIDIX BbI)J,alOT
CMKOCTHbJe H3MepMTem1 ITepeMeIl.{eHHH 10,
KOTOpbIC MOryr 6bITb BKJIlOIJCHbI B KOHTYP
ynpaBJJCHHH. P,[(.
TIpH B03HHKHOBCHHH yrnoBoH CKOpO-
cm D. z BOKpyr OCH Z Ha KaJK)J,YIO 11M AeHcT-
BYJOT CHJlbI HHCpl.lHH KOpHOJIHCa, KOTOpbIC
C03.nalOT BH6pauHoHHblH BpaIl.{alOIllHH MO-
MCHT, Bb13bIBalOIlJ,HH yrnoBb!C )J,BHiKeHHSI
3JIeMCHTOB 3 BMCCTe C M}J BOKpyr ocH Z
(pq). I10)J, 3JIeKTpO,l\aMH, C OpMHpoBaHHbI-
MH Ha ITJlaCTHHaX 11M, pacnOnm)(CHbl 3nCK-
TpO.nbJ 11 CMKOCTHbIX H3MCpHTCJ1CH yrnoBblx
nepcMCIl.{CHHH 11M B pq. Ha KpeMHHeBoH
IIOJUIO:>KKC (ocHoBaHHc Mf') BbIpameH CJIoi1
,l\HOKCH,lla KpeMHHH, Ha KOTOpOM HaXO)J)lTCSI
KOHTaKTHbIe IIJIOIl.{a)J,Kll 6, 7, 9, 12 H H.llYlI{HC
OT HMX TOKOnpOBOJUIlI{He AOpO:>KKH K COOT-
BeTCTBYIOlI{HM 3JIeMeHTaM KOHCTPYKWm.
IIpHH.QllnMaJIbHWI CXCMa Mf Pa3-
pa60TKH I}mpMbI The Charlcs Stark Drapcr
fI1POCKOrIhI
65
5 6
/2
7
/ 8
// J y
9 .......... 9 ,\
.:;..:.:.:.
/4 -;:::::? :. JO P4 I;q
II
Z x
6
4
L
PIIC. I.4S. npHHlUID1I8:lbH8H CXCM8 MI fjJllpMbl Somsung Electronics:
J MM; 2 ynpyme ')neMeHTbl nO,!:!Beca; 3. J 3 - 3,leMeHTbI rro,!:!Beca; 4 -TOpCHOHbl; 5 - aHKepbl;
6, 7, 9, 12 KOHTaKTHbre I1nOl1\aL\KH; 8 - rpe6eH'laTaJ/ c-rpYKTypa 3J1eKTpoCTaHr'leCKOrO npHBo,!:!a;
10- Ii3MepUTenb rrepeMemeHI1M P ,It 11 - 3neKTpo,Ubl eMKOCTHbIX H3MepmeneH rrepeMel1\eHHM P1.f;
/4 - JnCMeHTbI KI1HeMaTH'leCKOfi CB!l3M
Laboratory, 3 KOTOpOro COOTBeTCT-
nYeT cxeMe Ha pl1C. ] .46, 0 (6e3 KMHeManI-
qeCKoi1 CBSl3H Me)K)J,y EM) npMBe,QeHa Ha
pHC. 1.49.
43 COCTOHT J13 )J,BYX 11M I, KOTopble c
nOMOWblO ynpyrHx 3JIeMeHTOB 2, 4 nO,QBeca
I1 }KeCTKOrO 3JIeMCHTa 3 CBSl3aHbI C aHKepaMH
5, CKpCnJIeHHbIMH C ITOJtJ]O:>KKOM. Ha ITO-
CJ1e,QHei1 paCITOJIQ}KeHbI TalOKe cTaTopbl
4 5
109
0;
rpe6eH'laTbIX CTPYKTYP 3JIeKTpOCTaTH'le-
CI<I1X npHBOllOB 6, 7, 9 (npaBoro, ueHTpaJIb-
Horo Ii nCBoro) H HenO,QBIDKHbIC 3JICKTpO,l:(bI
8 eMKOCTHbJX ,QanliKOB (H3MepliTeneH) ITC-
pCMern:eHHH, nO,l\BIDKHble 3neKTpollbl KOTO-
pblX pa3Mern:eHbI Ha 11M. TIonBon 3neKTpo-
ITHTaHlliI H C:beM HH OpMattHOHHbIX CHrHa-
JIOB oCYIIIeCTBJIReTCH 'lepe3 TOKonO,l:(BO,QbI
10 H KOHTaKTHbIC IlJIOrn.a,l\KH 11.
}'
X pq
Z X
z
Puc. 1.49. npHHl1.nnUOJIbUBH cxellfB Mr pa3pB6oTKH Draper Laboratory:
1 - 11M; 1, 4 - ynpyme 3JIeMeHThl nO,!:!Beca; 3 - >KeCTKHH 3JICMeHT nO.llBeca; 5 - aHKep;
6, 7, 9 - 3JIeKTpoCTam'leCKHe npHBo,!:!bl; 8 - HcnO.L\BIi>KHbIH 3neKTpo,lI; eMKOCTHOrO I1"3MeplITeJIH.
ITepeMel1\eHI1H; 10 - TOKorrO,lI;BO,!:!; J 1 - KOHTaKTHaJI nnOIlla,IJ;Ka
66
fnana 1. KOHCTPYKU.1l11 II rIPIIHU.llflbl PAEOThl IlPIII,OPOR
Pe>KI\M n.BI\>KCHIiSi 11I'v1 B npOTltBO a3e
OCYl..U.eCTBJ1S1eTCSI B lianpaBJ1CIiIIll OCII X co
Cl\OpOCTblO v. npll nOI!BJICHlllt yrnoBofi CKO-
pOCTII !1= BOKpyr OCII Z (OCb '!)'BCTBIITeJIbHO-
CTII) 11M no.!!. ,!..I.eikTBllcM CliJI llHeplJ,11It Ko-
puonuca BbIXO.nSlT 113 nJIOCKOCTIl .\7 B npo-
TUBo a3e. Bb13blBaSi )'moBble K0J1e6aHIHI
paMK\!. IIMeSi B BII.1V, '!TO 06ecne'!eHa Ma.rIaSi
>KeCTKOCTb Ha Kpy'leHUe BOKpyr OCII Z )'ITpy-
rnx 3JIeMCHTOB 4. Ko.IIe6a1i1lSi 11M UKCU-
pylOTCSI e IKOCTHbIMU 113MepllTcnSlMIL
nepBbIe 06pa3lJ,bl ITpu60pa I1Mem1
.:lpeit 0.5__. I °/c. KOTOpblH B .!J.a.rIbHei1IlIeM
6bIn )'MeHbIlIcH. <DupMa Boeing (CiliA) oc-
BOII.la BbmycK rlipocKonoB II )'cnelliHo npu-
MeHIt.la ItX B aBTOM061tJIbHoit ITpOMhlIlIneH-
HOCTIt. n0.106HbJe fIIpocKonbl 6bInli ItCITOnb-
30BaHbl B pa3pa6OTaHlloit ltpMO" The
Charles Stark Draper Laboratory CI1CTeMe
.1..'UI )'npromeHIUI Ii HaBC,!..I.eHUSI peaKTltI3HblX
cHapSl)J,oB BM<D CiliA. .D:pei1cp mpocKonoB,
BXO;uII..U.ItX B CHCTeMV. COCTaBJJSlJI 1000 °/4.
-
1.3.4. MIIKpOntpOCKonbl RR-Tlma
B Mf RR-mna ,l\BlfiKeHItC 11M B PLt It
P4 HOCHT BpalllaTCJlbHbli1 xapanep. CO'le-
TaHMe OTHOCHTenbHoro, BpalllaTenbHoro
.nBHiKeHIIR 11M B P.D: It ee nepeHocHoro,
BpallJ,aTeJIbHoro )J.BHiKeHItSi B P4 npHBo)J,I1T K
BOJHHKHOBeHlllO MOMeHT8 CHJI HHeplJ,lili Ko-
pHO!1l1Ca, KOTOpbIH HOCHT Ha3BaHHe "rupo-
"
CKonHileCKHH MOMeHT .
A. rMpocKOmt..ecJwii MOMeHT II
t.J3 MIIKpOfHpocKonoB
B TeopMM rHpocKonoB BBe)J,CHO nOHSI-
THe "PCry.'UlPH8SI npelleCCMSI" Bp8l.L{a ro l..U. ero -
cst TI ep.noro Ten8, 3aKpenneHHoro B O,l\HOH
TO'lKe. PeryIlJlpHOH npelJ,eccHeH Bparn.aIOl..U.e-
rocS! C nOCTORHHOH CKOpDCTbro !1 BOKpyr CBO-
eH OCII Z TBep.noro Tena (pOTopa) (pHC. 1.50,
a) H83LJBaeTCH TalCoe ,l\BHiKeHHe, npH KOTO-
pOM OCb Z BpawaeTcSI C nOCTOJlHHOH CKOpO-
CThIO ro BOKpyr HCITO.nBHjf{HOH OCII x, coxpa-
HIDI C HeH O):\HH II TOT :>Ke YTOJl. OCb Z onll-
CblBaeT B npOCTpaHCTBC KOH}'C. OCb x "}Toro
KOH)'ca Ha3bIBaeTCJI OCblO npeueCCI1M, a ro -
yrnoBo CKOpOCTbIO npeu eCCHH .
Hafi)],c t Clt:I)' 1 IIICplll II I KnpllOJlllca.
.!!.eliCTBYIOU\YIO \la npOIl3Do.%II)'IO 10'11\)'
pOTopa. 13paU\.UOIllCrOI;Jl 1:0 I:I\0pOC'I1>I0 .0
BOKpyr OCB ;: II 0)J,1I0BpCl\te1l1l0 npall :uolI e-
rOC5l BOKpyr OCII x CO CKOpOCThlO (I) (pIlC.
1.50. 6). OCII x II .: 13'!aIlMHO IH plleIlJUII\Y-
JIRpHbl, a CKOpOCTl, !1 60llbUJaSl 110 cpaullc-
HIIIO c m.
TaK KaK OCh': SlB:meTCSI OCbIU CII II\It:I-
1'1111 TCJ1a, TO LlCIUpo6e)f(Hble ClIllhl 0111lKII-
TenbHoro ,llBII>KeHIIS! InallMHO ypaBlloBCIIIII-
BaIOTC5l. UCIITp06e>K1I hIM II CIIJIaMIl IIcpClloe-
Hora Bpml\eI-IIISJ mmn.y Ma.rIOelll Yl'JIOBoii
CKOpOCTIl W ITpcllc6peraeM. YCKOpCHIIC 1<0-
pllonuca el" ,lI.I1Sl 011.HOfi TO<II<II, II MCIOIucii
OTHOCItTeJIbHYIO CKOpOCTb V, .or (r pac-
CTOSlHIIC TO'IKII OT OCII I3pmueHIUI .:), II<lXO-
,llIlTC5l ITO OpI\lYlIC
aK = 2w V, sin (jJ = 2wf2 r sin (p.
')TO YCKopCHlie HanpaBJIeHO lIapaJl-
JIeJIbHO OCII Z. 3JIel\teHTapHaSi clIJra IIHeplUIU
KOpllOJII1Ca )J,JISI .noil TO'iKI1 HanpaBJIClla 13
CTOPOHY, npOTItBOnOJIO>KHY]{) YCKopelllllO
UK, 11 paBHa
dF K ::= (2!1wrsin Ip)dm = 2!1wxdm, (1.22)
r)J,e x = rsin <po
fIpe,l\CTaBHM pOTOp B 13H)J,e )J,HCKa co
cpenHeii (3KBUTOpltaJl bHOH) nJIOCKOcTbIO.
COBMel..U.eHHOii C nJIOCKOCTblO xY (pIlC. ).51),
r,ll.e paCnOJIOiKeHa npOli3BonhHaSi TO'IKa, HU
KOTOPYro nei1CT13yeT CIIJIa, Onl1CaHHaR (IJOp-
MynoH (1.22).
npoeKllHii 3JIeMeHTapHoH CHJIbl [CM.
opl\tyny (I.22)] Ha OCIi Koop,ll.lmaT
dFz ::= -2!1wxdm;
dFy = 0; dFx = 0,
a JIeMeHTapHbJe MOMeHTbl
dM... ;; 2!1w),ydm;
dMJ' ;; 2!1(J)x 2 dm; dM z = O.
n03TOMY ,WJSI Bcero pOTopa
Fz ;; -20w JXdm; Fy;; F,;;; M z ;; 0;
M x ;; 20w Jxydm; My = 2!1(J) Jx 2 dm,
rHPUCKOrIbl
61
x
x
, -y
,
,
a) y 6)
-
-
I'llc. 1.50. PcrynllpJlJtIl npCItCCCll1I pOTopa:
1/ KOHYC lIpCl\CCCIIII; 6 - onpc,Z:!eneHlIc cllnLIIIHepL!"" Kopl1oJlllca
.[{J1S1 IIIITCrpaJlOB HMCIOT MeCTO paBeH-
crnn:
JXdm = O. I aK KaJ( LlM ne}KIIT IIn
OCI-! Z;
f\)'dl1l = 0 - Bcne,QCTBlle ClIMMcrplHI;
)J,Ci1CT13l1TCJlbJ-lO, Ka)I\)..(OH TO<IKC B ncpnOM 'leT-
IJcpnl C KOOp.!l.HllaTaMII (+x, v) COOTBeTCTBY-
lOT B JJ.pynlx 'IeTBCpTSIX CIIMMeTpl1'lHhle TO'lKH
C lCOOp,QI1IJaTaMH (+X. -/y; +y, -x; -x, -y).
O<leBl1,QHO. 'ITO )..(n :lTHX 'leTblpex 1'0-
'leK J. vdm = 0, a TnK KaK BeCb pOTOp MO)l(-
110 paCCMaTpliBaTlJ COCTOSlUJ.MM 111 COBOKYIT-
1-IOCTeH no 'IeTLlpC TaKHe 'fO'IKI1 B Ka>I\)..(oil,
ro H ,QJUI Hero cnpaBC)J,JllInO pnBeHCTBO
J-'Ydm = 0 .
BCJlCl CTBlle CI-IMMeTpHII SlCIIO, ITO
JX:!dm = JY:!dlll, II no:noMY
J;: = C = J r 2 <1m = J{\,,2 + y2 )dm =
= JX:!dm+ Jy 2 dm = 2 Jx 2 dlll'
OTKY1 a J x:! dm = C, r,Qe C - OCCBOH MO-
MCIIT I1I1Cpl(l1H pOTopa.
C y<ICTOM H6H,D,CHHbIX BenM'IUH HHTC-
"paJIOR nOJly<IHM OKOH'JaTenbJ-lO
F. =Fy =Jvl =M z =0
II
M =C.ow=H','
\ .u ,
r,Qe H = cn - KMHeTM'leCKI1M MOMCHT po-
TOpa.
Cnc,QonaTeJIbHO, npel\eCClUI CIlMMeT-
pH'lHoro Tena BpallJ,eHWl Bbl3hlBaCT CHJ1bI
lIHepuml KOpHOJIUCa, KOTOpblC ,QaIOT ITOCTO-
SlHHblH no BCJ1H'IHHe II HanpaBJIeHHIO MO-
MeHT.
MOMCHT, c03,QaBaeMbIH CI1JIaMH HHCp-
l\Hl1 KOpMOJIHCa, Ha3blBaeTCSI rHpOCKOnH'le-
CKI1M MOMeHTOM, 3rp CKTOM HnH peaKl\HCH M
npoSlBnSleTcSI B ,QaBJICHMH OCM BpalllalOlllcro-
CSI Tena Ha onopbl. fnpOCKOITM'leCKI1H MO-
MeHT ,lI)UI cnY'IM, Kor,l\a OCb pOTopa z COC-
TaBnSleT C OCblO npcueCCHH x ITOCTOSlHHbIH
X
m!!rsin<P
z
Puc. 1,51. OnpC CJICIIHC MOMenTa CUJI
HIICPUHH KOpHOJIHCa
68
rnaBa I. KOHCTPYKUI1H 11 rIPHHUlIrIbl PAEOTbI rIPI1EOPOB
a) Mr=CQm sin a 6)
x
ffi
Z
ro (90°
90° Q
0 Mr=CQ(J)
x
(!)
Mr=CQffi sin a
PHC. 1.52. rHpOC OnH'IeC Hii IOMeHT
CH!\UteTpH'IHOrO pOTopa B c.'1Y'Iae ero
npel1eCCHH no OH 'CY
)TO.'! ex (PIiC. 1.52). H8.XOilSlT. ITpoennpySl
ymoB)'JO CKOpOCTb npeueccu:n <D Ha HanpaB-
.1eHHe OCH pOTopa = Ii HanpaBJIeHHe. nep-
ne Hl\.J pHoe K HeM. rHpOcKonH<JecKHH
MOMeHT Bbl3blBaeTCSI TonbKO nepneH!l.I1I<y-
illIpHOH cocTaBlliIJOllleij msina, H BeJIH<JHHa
ero BbI<£l:!CJ"UleTCH ITO opMy.rre
M r = CQ<Dsina.
( 1.23 )
HanpaB.leHlife MOMeHTa Jt.1 r onpene -
ercSi no O.i1HoM)' H3 cne.nyrolllHx npaBIiJI
(PHC. 1.53):
1. Ec.;m MCnc.rry OCblO c06CTBeHHoro
BpaIlI,eHIDI pOTOpa H OCblO npeueccHH MMeCTCSI
yroJ1 (pHC. 1.53. a), B03HHKaeT MOMCHT. CTpe-
MJnI{mIcH COBMecmTb no KpaT'IaHllJeM)' nyrn
OCh TeJIa C OCblO ITpeUCCCI1II TaK. 'lT06hI ooa
Bpaw.eHliSi COBCplllaJIHCb B O,[{HY CTOPOHY.
2. ECJ\II OCIi C06CTBCliHOro npalllcHHSI
pOTopa II ero npelleCCIIII BJaHMHO ITeplleH-
)J,HKYJ1S1pHbI, TO HanpaBJIeHl1e rHpOCKomltle-
CKoro MOMeHTa nOJ1Y<Jae-rcSI, ecml BeKTOp
yrnoBoll CKOpOCTlI npcueCCHIi (!) ITOBepllYTb
BOKpyr OCIi pOTopa B CTOpOH)' ero BpalllCIIIISI
Ha 90° (pnc. 1.53, 6). 3TO :>KC ITpammo MOJl(-
HO npHMeHHTb 11 K C.1)"IaJO HCITepneH,[{lil<)'-
.lUlpHJ,lX oceH, .L1.Jl5I qero npeiwapHTC.lbHO npo-
eKTHpyeTcSI yrnoBIDI CKOpOCTb ITpeneCCI1H na
HanpaBJIeHlie OCIi pOTopa Ii Ha HanpaBJJeHUe,
nepneH)J,ill.')'JIHpHOC K HcH. HanpaBJIClII1e.
Be1l.-ropa rHpOCKOnU'leCKOrO MOMeHTa ITony-
<JaeTCSI ITpH ITOBOpOTe nepnCHtJJII<)'JISlpHOH
COCTaBJUIJOllleH BOKpyr OCIi pOTopa B CTOPOHY
ero BpalllCHHSI Ha 90° (CM. pliC. 1.52).
TaKHM 06proOM, npcLlcccHSI BbI3bIBaeT
rlipOCKOnH'leCKUH MOMCHT Mr, H. Ha OCHO-
BaHHH 3aKOHa 0 paBeHcTBe )J,eHCTBHSI H npo-
TMBo)J,eHcTBHSI n cornaCHO Ha<JaJIY 'AnaM-
6epa, cne.!l:YCT, <JTO MOMCHT Ms BHCWHHX CHJI
)J,on:>KCH ypaBHoBecHTb rHpocKonJ1'leCKliH
MOMeHT. T.e. Ms + Mt = 0, HJlH Ms = -Mr.
l1TaK, CCIlH MbI Ha6JIIO,llaeM ITPCQCCCHIO
<D BpaJllalOlllerocSi C yrJlOBOH CKOpOCTblO .Q
pOTopa, TO MOMeHT BHelllHHX CHJI Ms, SlB-
10111HHCSI ITpH<JHHOH npcQcccHH, paBeH
rHpOCKonH<JCCKOMY MOMeHTY I1 HaITpaBneH B
npOTMBOnOJIO:>KH)'1O CMY CTOpOH)'.
PHC. 1.53. OnpCJlCJ1eHHe HanpaBJ1eHHH rHpOCKOnH"IeCKOrO MOMeHTa:
a - Me-A(.!(}' OCHMH cOOCTBellHoro BpameHIDI H npeueCCJUf pOTopa JfMeeTCR npoH3BonbHbIH yroJ1 a;
6 - OCH co6crBeHHoro BpameHI1J1 H npeIteCCI1H pOTOpa B3aHMHO nepneHnHKyllilpHbl
rHPOCKOTJbI
69
a)
J'
Al r2
Wt
x
6)
y
H
x
PIIC. 1.54. POTOp Ha BpmU310lllelllcn OCHOBaHHH
1-la,ao 06paUITb BHI1MaHHe Ha TO, 'lTO
eCHI1 Flpm.uaIOI.l!HMC pOTOp HaxO HTCSI Ha
BpalllcUOllleMCSI OCHOBaHI1H. TO 3TO ITepeHOC-
IiOc BpallleHHe OCHOBaHMSI ):VlSl pOTOpa aHa-
;:)01 H4HO npeueCCI1H 11 mpOCKonH4eCKHM
MOMell1 OITpe eJISleTCSI no ITpHBeneHHbJM
Bhlwe npaBHJIaM.
I-la pHC. 1.54 pOl'Op C KI1HeTH'leCKHM
MOMeHTOM H 3aKJIIO'lCIi B paMKY. KOTOpllil
pa3JIHQHbJM 06proOM MOlKeT npl1BO HTbCSI BO
BpalllCHl1e OTHOCHTeJIbliO ;no60R OCH, HO He
COBnaL\aJOLl.\eR C OCblO c06CTBeHHoro Bpa-
llleHHSI pOTopa. Ha pHC. 1.54, a (OCb C06CT-
BeHHoro BpaLQCHl1J! ropmOHTaJJbHa) H 6 (oHa
nep'fI1KaJJbHa) nOKa3aHbI BeKTOpbI yrnoBbIx
cKopoCTeH ITepCHOCHoro BpaLQCHWI WI H 0)2
BMeCTe C paMKOH H COOTBeTCTBYlOllJ,l1e HM
BeKTopbI rl1pOCKonH4eCKHX MOMeHTOB, KO-
TopbIe B COOTBCTCTBl1H C opMynoH (1.23)
paBHbI
M ri = HO)\; M r2 = H0)2'
MOMCHT M rl Ha pHC. 1.54, a, 6 cTpe-
MHTCSI pa3BcpHYTb paMKY C pOTOpOM BOKpyr
OCH X, HO B pa3Hblx HanpaBJIemrnx. MOMeHT
Md Ha pHC. 1.54, a CTpCMHTCH pa3BepHyTb
paMKY C pOTOpOM BOKpyr OCH y, a Ha
pHC. 1.54, 6 - BOKpyr OCH Z.
B Mr RR-THITa pCaJJH3YCTCH KOJIc6a-
TeJIbHOC )J,BH:>KCHHC 11M (pOTopa) B pC:>KHMax
nepBH'lHbIX H BTOpH'lHbIX Konc6aHHit Ha
pHC. 1.55 pOTOp J YKpCITJICH OTHOCH'ICJJbHO
onopbI-ocHoBaHHSI 3 Ha ynpyrHx 3JICMeHTaX
2 nO Beca, KOTOpbIC HMelOT MaJIble iKCCTlCO-
cnI Ha KPY'leHHe BOKpyr OCH YMHa H3m6
BOKpyr OCI1 Z. )l{CCTlCOCTb ynpyrHx 3JICMeH-
TOB BOKpyr OCH x 3Ha<II1TeJIbHO 60JIbllJe. Ta-
KHM 06pa30M, B03MO:>KHbI KOJIc6aHIDI pOTopa
BOKpyr oceH Z H y.
TIpHBO)J, 06eCITe'lHBaCT K0J1c6aTeJIbHbIC
)J,BHIKCHHSI pOTopa co CKOpOCTblO l' = n 80-
Kpyr OCH Z TaKHM 06Pa30M, 'iT0 ITCPBYIO no-
JIOBHHY nepl1o a KMHeTMqeCKHH MOMCHT HI
HarrpaaJIeH B nOJIO>KHTeJIbHYJO CTOPOHY OCH
Z, a BTOPYlO nOJIOBHHY H 2 - B OTjJl1uaTeJIb-
HYJO. ,lI,ocTaTO'lHO TO<IHO c06mO,llaCTCSI
paaeHCTBo HI = H 2 = H = JA (.ly - oceBOH
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HI1H Me){()J,y o)J,I1HaKoBbIMH TOIJKaMH B PSl)J,y
Hll3bIBaeTCR 3.::JeMCHTapHOlI (KpaT'laHIlIeH)
TpaHCJIHl.{HeH, HJIII nepHO,l\OM l1)J,eHTH<JHO-
CTl1; I1Hor.QR ynOYpe6JIHIDT Ha3BaHHSI "nepH-
O)J, TpaHCJISIUlII1", I1JII1 "napaMeTp pSl.Qa".
ECJIH C.QBMHYTb TO'lKI1 6eCKOHeIJHoro
pH.lJ.a Ha O)J,HH nepHO;:J, H)J,eHTI1'lHOCTH B;:J,OJ1b
HaITpaBJleHHSI TpaHClliIUHI1, TO BCC O)J,HHaKO-
BbIe TOIJKH nepe;:J,BHHYfCji HR O,lJ.HHRKOBble
PRCCTOjiHHH, pj],lJ. COBMCCTI1TCji caM C C060H,
TaK IJTO BH)J, cro He HapYWllTCji. CMMMeT-
pHIJHOe npe06pll30BaHl1e, C nOMOll.J.blO KOTO-
poro TO'lKa ITOBTOpjieTCji B npOCTpaHCTBe,
HMeHyeTCji npe06pll30BaHMeM C ITOMO blO
TpaHCJIH.QHH, HJIH YpaHCJIHl.{Hei1.
rlOBTOpSiSI KaKYIO-JIH60 TOIJl\.')' nocpe.Q-
CTBOM TpaHCJIRQMH, nonY'1aeM 6CCKOHe<JHbJH
nepI10.QHIJCCKHH pji,lJ. I.lI,CHTI1IJHbIX TOIJeK Ha
paCCTOSlHHSlX a, 2a, 3a,..., na. XapaKTepH-
CTHKOH ::JToro pMa SlBmleTCSI KpRT'lRHlliaSl
TpaHCJIHIUIR a. O.QI1HaKOBble TO'lKH, CBH3aH-
Hble MC:H():(y c060i1 TpaHCJIm..tlUlMM a B 6cc-
KOHC'lHOM pM)', Hll3bIBalOTCSI Y3naMH pR.lI,R.
Y3J/bZ He 06mame.nbHO OOJ/:JICHbZ C06naoamb
C MamepUaJlbHbl.iWlI lJaC111UllaMli 6eulecm6a,
3mo Mozym 6bzmb OOllHaK06bze mO'lKU .Ue:JIC-
oy 'laC1111111Cl.iI,I11.
84
r..mB8 2. TEXH0J10rW-lECKHE OCHOBbI llrOH3UO,llCTBA
z
a
b r
a
);:
PIIC. 2.1. nsps:I.'1e.'Iemme.a. nocTpoeHHblu
HS ).'IeMeIITspHblX Tpsnc.'ISIlUUIX
TIOBTOpSiSI OlU1HaKOBbIe TO'lKll c nOMO-
lllbfO ;J:pyroif TpaHcnHUUl1, He napanneJlbHO
nepBOH. nOJIY<JItM llByMepHYIO nJIOcK)'1O
cen:y, KOTOpWi nom1OCThfO onpe.ne.ileua
llBYMH 3neMeHTapHbIMIt TpaHCJISlUIU!MH: a 11
b. D.06aBJISlH TpeTblO. nepneH.!I.I1KynHpHYIO K
ITepBblM ..!lBYM OCb C TpaHCJISlUl1eU c. nOJ1Y-
<JMM npOCTpaHCTBeHHYfO TpaHC.ilSlUI1IO (pC-
llleTh'Y ).
I1apa."Lle.:Ienl1ne.1l, nOCTpoeHHblH Ha
-rpex 3!1eMeHTapHbIX TpaHClIHUl1SIX: a, b, c, -
Ha3bIBaeTCH JJIeMeHTapHbIM napaJ1lIenenl1-
ne.!10M. H::m 3.ileMeHTapHOH Sl'leHKOU
(pl1c. 2.1).
Bbl60p OCHOBHbIX TpaHCJISlLU1H B
crpyKType ICpHCTaJI.JIa OIJeHb BIDKeH, nOToMY
'ITO JiMH onpe.neillllOTcSI Kpl1cTannorpa I1'le-
CKHe OCH l\OOp.nHHaT. B 06111eM cJIYIJae 3TO
KocoyrOJIbHbIe CHCTeMbJ Koop,IDiHaT C He-
O.nJfHaKOBbJMM M8CUJTa6HbIMl1 oTpe3KaMI1 no
OCRM: a* b *c. a * P*r *90°.
HanpaBneHHe KpI1CTaJIJIorpaqm<JecKHx
acC" KOOP.nHH8T COOTBeTCTBYeT HanpaBJIe-
HHRM pe6ep 3JIeMeHTapl10H illieHKJ1 Kpl1-
CTaJ1JI8, a MacUJTa6Hble 0-rpe3KJ1 no OCHM
KOOp.nHHaT - }tJ1HHaM 3THX pe6ep. TaK KaK
ICpMCTaIIJl 6ecKoHetJeH, TO MO:>KHO BbI6paTb
6ecKOHe<JHoe tJHCJIO KpHCTanJJorpa HtJeCKHX
CHCTeM KoopromaT.
I1TaK. npocmpaHCmeeHHGfl peWEmKa -
3mo zeoMempUileCKOe nocmpoeHue, c nOMO-
"lblO Komopozo 6 Kpucma/UlOzpacjJU'leCKOM
npocmpaHcm6e 6b1J18J1JllOmCJl oOUHaK06ble
mO'lKU.
lJTo6hl OXap<1KTCpIl30B<1Tb nOJIO>KCHlle
IIJ],CaJlbllO !1JlOCKOii rpmm KpIlCTLUUJa. pac-
nOnO>KeHHOH IT8paJIJlenhHO nJIOCKOCTJlM
KpnCTaJJ1l11'ieCKOii pc ill CTKII, 06hl ' IHO HC-
nOJlb3)'IOT OTpC3KH. OTceK8CMhlC rpaHJ,IO Ita
TpCX KOOp!lHH8TllbIX OCJlX. 51pl<O BbJpmKCII-
IIblM (110 Hco6S13aTeJlbHbIM) npH31laKOM MIIO-
rnx Kp1lCTalIJ10B SlBJISICTCH IIX CBoilcTBO 06-
pa30BbIBaTh MHOrOl"paHHlIKI1 (nOJIlnJJ.pbJ),
KOTOpblC OrpaHII'ieHbl 60JlcC 11.1111 :\<ICHec
rJIaJ],KI1I1H1 nnOCKOCTSlMI1. B o6111eM CJ1Y4ac
'l)aHb Kpl1CTanJla nepeceKaeT Bce TPIl OCII
ITPliHSlTOiI CIICTeMbl KoopllmmT.
OTpe3KII Ha KplicTaJlJ10rp8 I1IJeCKIiX
OCSIX no TpeM HanpaBJICHI1SlM I13MepHIOTCJI He
O,lJ,I1HaKOBbIM MacwTa60M, a COOTBeTCTBYIO-
lllllMI1 nepl10.naMII TpaHCJISlUl111 a, b, C H!1I1
KpaTHblMIf 11M OTpe3KaMII ilia, lib II pc, r,lJ,C
m. II, p - ueJlble 4I1CJIa.
CHMBOJI aTOMHOH nnOCKOCTI1 IIII eK-
CLI MIIJIJIepa (11. k, /) - onpe,lJ,eJISIIOTCSI 113
COOTHOUJel1l1S1 BeJIHIJl1H, 06paTHbiX Benli41111e
OTpe3KOB Ha OCHX:
h : k: / = 11m: 1/11 : lip.
CHMBOJI 3aBl1CI1T TOJlbKO OT COOTHOUJe-
HUSI OTpC31<OB Ha OCSlX, TaK KaK B KIDK,lJ,OM
KpI1CTaJlJIe cO;J:ep>KI1TCSI 6eCKOHCIJHOe MHO-
:>KCCTBO HlleHTH4HbiX nnOCKMX ceTOK, KOTO-
pble npOXO,llJlT ITapanJIeJIbHO APyr llpyry.
TaKI1M 06pa30M, rpaHb, ITOKa3aHH8SI Ha pMC.
2.2, a, HMeeT CI1MBOJl (II I), nOCKOJ1bKY OT-
pe3KI1 Ha OCSIX COCTaBJISIIOT 1 a, 1 b, 1 C.
TpH I1H,lleKCa MUJ]nepa - 3TO Ha60p
HaHMeHbWHX ueJIblX IJHCCJI. OTpe3KI1, KOTO-
pbIC rpaHb OTceKaCT. ,llOJl>KHbl 6b1Tb KpaTHbl
uenoMY IJl1cny JtJIHH.
Cne.noBaTeJIbHO, rpaHb, ITpellCTaBJleH-
HaJI Ha puc. 2.2, 6, l1MeeT CI1MBOn (323), TaK
KaK OTpe3KI1 Ha OCilX COCTaBJlHfOT 2a, 3b, 2c
(I1HlleKCbl (323) H nOJlY'leHbl OHI1 nYTeM yM-
HO:>KeHI1i1 KaiK,lloro CHMBOJ]a Ha Hal1MeHbllil1H
06111Hi1 .uenHTeJIb: 6/2, 6/3, 6/2; 3:2:3). ECJII1
OTpC3KH Ha OCHX KaKoii-HI16Y,l\b rpaHI1 COCTa-
BSlT, HanpHMep, m = 2, n = 3, p = 4, TO CHM-
Bon rp8HM 6Y,l\eT cne.nYfOlllI1M: h : k : I =
= 1/2 : 1/3 : 1/4 = 6 : 4 : 3 (643). TIpOM3HO-
CI1TCH: "UJecTb", "'ieTbIpe", "Tpl1".
MA TI PI1AJlbl
fj5
(/J
"/
1('
(III)
Ih Y
(J)
L
2c
(323)
3h Y
IJllc. 2.2. HlIllUUllponallllc rpullcii I\pJlCTaJIJIU:
CI IIIIi1CI\Chl :Y111JIJICpa (III): 6 - IfII1\CI(CbI MII1JJICpa (323)
HH;J,CKCbl (643) 61,1:111 nOJlY<lcllhI nyrc:\1
y IHO)!(CHIHI Ka)!(; Oro CII\mOJ1a lIa H<UlMCHb-
umii 06111llfi t eJIIITCJlb. nocJ(OJlbKY IIc06xo-
)J,HMO nO.1Y'IIITb llCIIhlC 'lIIcsm: 12/2 : 12/3 :
:12/4=6:4:3.
ECIHI rpallh lIapamlCJlbl1a oell, TO OTce-
KaCMhlM OTpC30K lIa OCII CT<mOBIITCJI paBHblM
00, a COOTI3CTCTBYIOlllllii IIII).J.CKC parUlblM O.
Ka)[()J,aJl 113 IIJCCTH q1allcM Ky6a nepeccKaeT
TOJlbKO O}J;HY KOOp,l\IIH<lTHYIO OCb 11 ITpOXO-
)J,HT napaJI11CJlhllO )J,BYM )J,pyrIlM.
KpOMC TOro, BCC IUCCTb rpaHeii nonap-
HO ITap8J1J1CJlbllbl )J,pyr ).J.pyry. CJle)J,OB<lTCJIb-
110, TPH rpalll1 KY6a IIMCIOT CHMBOJlbl (100),
(0 I 0), (001), a npOTlIBOITOJIOIKHble 11M rpaHIl
ITpIl06pCT8IOT OTpl111aTcnbHbl i 31faK: (100).
- -
(010), (001 ) (pIlC. 2.3, ll, 6, (j). .D:JUI cnY'laH..
Kor}J;a rulOCKoCTb nap8J1JIeJlbHa OCII Z II OT-
ceKacr OT occii X II Y, Hanpl1MCp, no e).J.HHH-
ue, HMeeM 111 : 11 : p = I : I : 00, OTKY.l1a CJIe-
).J.yeT (Ilk!) = (110) (pHC. 2.3, ?).
.D:a.lce, HanpllMCp, CIIMBOJI nJIOCKOCTlI,
napa.1JIcJlhllOH ocm,l X II Y 11 oTccKalOlllcM
Tp 1 C.l1I1J1HUbl Ha OCI1 Y, onpe;:teJIJleTCSI TaK:
In : 11 : p = 00 : 3 : 00. oTK}'JJ.a 1/m : 1/11 : IIp =
= 0: 1/3 : 0 = 0: I : O. 3Ha ' IHT, (hkl) = (010)
(CM. pHC. 2.3, 6).
B OTJII1'lHC OT CIIMBOJIOB rpaHei1, .l1JlJl
KOTOpbIX IICnOJlb3YIOTCSI Kpyrnble cK06KII,
JtJISi CHMBOJ10B H:iITpal3nCUI1H ITpllMeHSIlOTcSI
KBa)J,paTHbIc cK06KII. '3TII CIlMBOJIbl cnymaT
xapm<TCpHCTHKOH m060ro KpIlCT8J1JlOrpa H-
'lecKOro H:inpal3J1eHI1S1. B 3TOM cJIY'lac B oc-
HOBY n0J10mCHa npSlMa$1 nponOpUHOH8J1b-
HOCTb oTpe3KoB Ha OCSlX, a He 06paTHaSl, KaK
y CI1MBOJIOB rpaHcii. HanpaBJ1eHHe onpc).J.e-
J1S1eTCSI 0;:tH03Ha<mo, ecJUI 113BeCTHbl KOOp-
;:tlmaTbl KaKOii-HI16y.nb T01.)1(11 Ha npSlMOH.
npll JJ.BH:>KeHHH TO'IKII B)J,OJIh npSlMOH OTHO-
welUH! KOOp,llHHaT OCTalOTCR Bcer).J.a ITOCTO-
H.HHbIMH.
a) L 6) Z 2) z
(100) (110)
r r b Y
PIIC. 2.3. HII.!tCKCLI MltJlJlCpa.IIJIA pa1J111'lIIbIX rpllilcii KY6n:
u - CIiMIJOn nnOCKOCTIl (100), (100 ); 6 - CHMBOJI nnOCKOCTH (010), ( 0 I 0 );
(J - CMMBOJI nJlOCKOCTlI (00 I). ( 001 ); Z - CHMBOJI nJ10CKOCTU (110)
86
rnaBa 2. TEXHOJIOrH ECKIIE OCHOBbl rIPOl I1nOLl,CTBA
1010)
1010
r
POOl
x
PIIC. 2.4. CII!tIBO.'1bl H8npaBJICH llii B !1.BYMCPIlOii peIllCTJi:C 8 IInOCIi:OCTIl (001)
Bo MHOfllX CJI)"-I8SlX pa3JlJI'lalOlllUCCSI
UlU:eKCaMll MHJJ.!lepa !1.!]OCKOCTl1 3KBIIBa-
JIeHTHbl KaK B KpIfCTa!mOrpaqHl'IeCKOM, TaK
U B 113I1qeCKOM CMbIC;Je. HanpuMep, KPH-
CTa.JL'lorpa K'-IeCKaJI 3KBlmaJICHTHOCTb nJ1OC-
KOCTe", CJlYA\8.IlJ,IiX rpaHSIMu Ky6a: (100),
- - -
(010), (001), (100), (010). (Dol). npoSlB.'lSl-
eTCM B TOM, 'ITO OHU COBMellJ,alOTCSI .II.pyr C
:1pyroM npl1 nOBopOTe BOKpyr COOrneTCT-
BYlOIlUIX oceM Ha yrOJI, KpaTHblH 90°.
<1>H3UIJeCK<Ul 3KBJ1Ba!IeHTHOCTb COCTOIn
B TOM, <iTO Bce OHIt xapaKTepH3yIOTCSI 0}J,11-
HaKOBOH crpYh.1YPOH B pacnonO:>Kemm Y3-
JIOB pe1IleTKH 11 n03TOMY 06na.!lalOT O.II.I1Ha-
KOBbIMH M3MqeCKJ1MIf CBOHCTB3MI1. CeMei1-
CTBO 3KBMBaJIeHTHhlX nJ]OCKOCTCH 0603Ha'1a-
eTCH MrypHbIMH cKo6KaMI1, Hanpl1Mep
{ 100 } .
J(jiB nOMCHeHI1M CI1MBOnOB HanpaBJIe-
HMH paCCMOTpI1M ,llBYMepHYlO perneTR:y B
rIJIOCKOCTH (001) (pMC. 2.4). B 3TOM cJIyqae
1peTHH CI1MBOn paseH HynlO. KaiK.!l.ail ITPR-
MaJI Mo:a<eT ITepeMemaTbCiI napaJIJIeJIbHO
CaMOH ce6e. TaK <iTO B KOHl.{e KOHUOB npoi1-
.net qepe3 Ha'laJIO KOOpJlMHaT. HanpaBJIeHI1M,
npoXO.l])U.Lt He napanJIeJIbHO OCH X, nOJIyqalOT
CMMBOJI [100]. mmpaBJIeHHH, napanJIeJIbHbJe
ocn Y, - CHMBOJI [010]. napaJIJIeJIbHbIe ocn
Z,- CnMBOJI [001]. I1pOTHBononmKHoe Ha-
npaBJIeHHe 0603HaqaeTCH oTpHllaTenbHbJM
3HaKOM HaJJ. CMMBonOM.
CJIe.nyeT 3aMeTMTL. qTO eCJIH CHMBOnbJ
KpaTHbI, TO HX MO:>KHO COKpaTHTb Ha uenoe
nOJlOJKJ1TeJIhHOe 'lI1CJlO. HanpliMCp [330]
= [110].
CeMei1nBo 3KBI!BaJIeUTHbiX HanpaBJ1e-
Hili! npmulTo 060'3Ha'laTb JIOMUlibIMH Clco6-
KaMH, Hanpl1MCp <100>.
I1plll\lep bI
Onpeoe.nmlb CII.1I601 1f{//Jpa(J elll/JI. npoxo-
O}/ILWZO 'IepeJ lIa'iG.w I\oopr>ullam II 11/0 1 11\.1' c K
opOll1wmal/ll (aI8, 3b18. 5c18).
Hal1lIeM uenOtJlICJIeHHblc OTlmwemUl KOOp-
.!UiHaT: 1/8: 3/8 : 5/8 = I : 3 : 5.
3To cOOTneTCTByel nepeHocy 3alt3HHOrO
HanpanneHMH B 6.IJIln<ailwlll1 K Ha'Jany KOOp,!\MH81
Y3en KIn1CTaJIJUI'JecKoil peweTKIi C KOop.LllIlU1TaMIi
(135). 3Ha'1J1T, CJIMBOn 3a1I8I-IHoro H3npaunemUl
r 135].
Onpei>e!/llnlb C:1U160Q Ifanpa(melllf}/, npo:w-
o}//ljezo '1epel mO'i1\1I A (O, b12, el2) 1/ B (a/2,
0, el2).
Bbl'1J1T3.JI cooTBeTcTBYJOutHe KOOp.lIJlllaTbl
O,!\HOM TO'iKM 1I3 Koop,!\JlHaT lIpyroH. 4TO coomeT-
CTByeT nap3JIJ1enblloMY nepeHocy nenopa AB B
Ha'1aJIO KOOp.lIJIH8T, nonY'JaeM HOBhle Koop,!\lIua-
T1>J KOHLla neKTopa: -al2, b12, O. TaIClIM 06pa30M,
peuJemle :HOH 3a)].3411 cne,!\eHO K peWeHl1l0 npe-
llbllIymeiL 3aMeHHM nOJ1y'ieHHOe OTHowellHe
Lleno'mcneHlIhlM: -1/2 : 1/2 : 0 = -I : I : O. Tor,!\a
CMMBOnllanpaBnemul 6Y,!\eT [ 110].
2.1.2. Kpel\fHHH
MOHOKpI1CT3.JlJlli'ICCKI1l1 KpeMHHH HaH-
60nee tjaCTO I1cnonb3YeTCR B M'3MC-
YCTpoHcTBax. MOHoKpeMHI1l1 I1MeeT CTpYK-
MATEPJ1AJlbl
7
I YPY pCIIICTKIl nma aJIMa3a (puc. 2.5), X<l-
pat(Tepw YIoIllYI{)CH HaJl H'lHCM TeTpa')JlpHtJC-
CKIIX clHlleli: Y l(aiK;J,oro aTO\ta eCTb <lcTbIpe
GJII1iKaOIlIllX coccna. B1aHMo)J,eHcTnYloIllIIX
CH.1HJ\HI I(() 13aJIC II I UblX CnH.3eM. '1TO BblI .1H}1H I
1<11(, K<lK eCJ1II 6101 B )JleMeHT<lpHYIO rpaHc-
IlCIITpHponaHuYlo Ky6ntJecl(Yl0 1 J"leHKY
n:lBIIIIYJIII i!pyrylO, IIOBCPHY1YID OTHOCH-
IC.]bHO ncpBoii ua 'lcrBep'lb nocToSlHUoii
,
pCIIICTKIC. TaKYlo iKC CTPYKIYPY IIMeCT II
[ epMallIIii.
TaKaSi ,lJ.OnOnbHO cnO)JUlml cTPYKTypa
KPC\!HHH CBH;J.CTC.lbCTBYe1 0 60J1ce rlJlOHrOIl
yn<lKOBKe HTOMOU B nJlOCKOCTII ( III) 110 cpaB-
IICIIIIID, IIHITpHMCp, C IIJIOCKOI."TI,1O (100) (Ha-
npaBncHI1CM [100]). [lJIOCKOCTIi (III) II (100)
CTPYKTYpbI KpCMIIHSI nOK83aHbi Ha pHC. 2.6.
npo6JIcr.m 60.1bwoli Ba:>KIIOCTH IJ KpeM-
111m - :mcprCTH<ICCKHJj 30Ha CT)JYKT}'PbI. B
ITepBoif nonoBime XX CTOJleTHH Y'leHbIC,
IIccne)J,YIOlllilC KBaHToBYID McxaHHKY, 06Ha-
PY)J{HJII1. 'ITO :JJIeKTpOHbl B TCJIaX MorYT
IIMeTb TonhKO }1HCKpeTllbIc ypOBHI1 1HeprHH.
3TH HcpnfH pa1;J,eJieHhl Ha pa3nH'lHble '30-
Hhl. npl1 caMOM HU3KOM nOTeHUHaJ1e Bce
3neKTpOIib] B TCJle 3aHilMaIDT nQ.TlOCY Ba-
neHTHOCTII, KOTopaSi COOTBcTCTByeT Op6IITaM
BanCHTHOCTH B aTOMax.
H)J,eaJlbHbIM, CB060,lJ.HbIM OT npllMeceH
KpeMHMM ITpl1 HH3KOM TeMnepaTYpc HMeeT
UCJ1HKOM 3anOJIlieHHYI0 3Heprenl'leCK}'1D
30HY (BaIIeHTHYIO), OTnCJIeHHYIO OT 6nH-
:>K<lHrneH CB060N10i1 (npOBO)J,SlllleH) 30Hbl
30HOH 3tmpellleHHbIX COCTOSlHHll (3aITpe-
IllCHHOM 30HOii). BCJIe,lJ.CTBHe 3TOro ITOJIY-
npOBO,lJ.HIIKOBbli:j MaTepHaII HBnH.eTCH. Ha ca-
MOM )J.ene H30nSlTOpOM, TaK KaK Bce ypOBHI1 B
naIICHTI-IOH 30HC 3aITOnHCHbl, a B 30He npo-
lJO)J,IIMOCTH HeT CB060)J,HbIX 3JIeKTpOHOB, T.e.
3nCKTpOHbI HC MorYT L\BllraTbCH. ITOn )J,eMCT-
BHeM BHcllJHero 3neKTpl1'lecKoro ITOnH. Ok
HaKo, CCJIl1 3J1eKTpOH nepc6paCbIBaCTeSi 113
BaneHTHoii 30llbI B 30HY ITpOBO,l\HMOCTH, 06a
cB060nHhlX HOCHTeJISi (3nCKTpOH B 30He ITpO-
BO.nHMOCTH H OCTaBWaSiCH. B BaneHTHOH 30He
z
PIIC. 2.5. CTPYII.-rypa KpIICT3JIJI3 TII1I3 aJI1\I3Ja
r:lTO 1bI B nnOCKOCTH (100) 'ICpHOHJ meTa, a
B IWOCI<OCll1 ( II )) - 'JaTeMHeHbI]
i!blpKa) )J,BlnKYTCH B 3JICKTpMtJCCKOM none II
y<laCTBYIOT B 06pmOBaHHI1 TOKa. B £1I-ICTbIX
nOJ1YITpOBO,lJ.HIIKax KOHUellTpaQHH HOCJlTCnCH
3apH.L\a - CB060L\HblX 3neKTVOHOB H nb1pOK -
COCTaBJUICT (10]6. " IOI8)/CM- Bell\ecTBa.
L(JISI CHH:>KCIU1S1 YL\eJIbHoro COITpOHIB-
neHHH ITOnYITpOBO)J,HHKa 11 npU)J,aHHSI eMY
onpC)J,eJleHlIoro nma 3nCICTponpOBO)J,HMO-
CTH: 3JIeKTpOHHOH B cnY'lae ITpe06nanaHHSI
CB060)J,HbIX 3JIeKYpoHoB I1JI 1 )J.bIpOtJHOii,
CCJ1I1 ITpe06JIaJJ:aIDT L\bIPKI1, - B '-II1CTble ITO-
JIynpOBOL\HHKM BHOCSIT He06xo)J.IIMble npH-
MeCH. TaKOH npOI.(CCC Ha3bIBaeTCSI nernpo-
BaHHeM, a COOTBCTCTBYIOrn.He nOJ1ynpoBo -
HliKOBble MaTcpHanbl - JIenIpOBaHHbIMI1.
B 3aBHCHMOCTl1 OT Toro, KaKI1M nmOM
OCHOBHblX HOCJ1TCJleH 3apSl,lJ.a onpe.nCJISleTCSI
npOBO,l\HMOCTb nOJIynpoBO)J.ffiuca, HX Hll3bl-
BaIDT n-Tl1na (OCHOBHble HOCHTeJIH 3apR a -
3JIeKTpOHbI) UJIH p-nma (OCHOBHble HOCHTe-
nH 3apRna - ,l\blpKI1). nonYITpOBO,lllUIK C
KOHl\CHTpallHeii npHMeCCH > IOIS/ CM 3 060-
+ + B
3Ha<meTCSI KaK 11 I1JIU p. Ka<leCTBC nerH-
PYIOll\HX npHMeceil npHMelUllOT 3JIeMeHTbI
III 11 V rpynIT nepl1o)1)1'lecKo i CI1CTeMbl
3nCMCHTOB .n;. 11. MeH,neneeBa. flerl1PYIOrn.He
3neMeHTbi III rpynnbl C03 alOT )J,bIpO'lHYlO
npOBOl1l1MOCTb 11 Ha3bIBaIOTCH aKuenTop-
HbIMH npMMecRMI1. 3neMeHTbI V rpynnbI-
I ATOMM rqJeMIII1H pacnOJIarmOTCH IJ Y3nax 1\:y6a l l.(eHTpaX ero weCTH rp8HeH. nplI'ICM a = b = c.
a=J3= y=90°.
2 PaCCTomUISI MC)1(,11Y llnYMH COCCllIUlMl1 aTOM8MU BAOJlb O,l:(HOro 113 Hanpannel-lIlH pemeTKH COCTaa-
IIHIOT 110CTOHHHYIO peWeTKH.
88
nJaBa 2. TEXHOJIOrH"4ECKI1E OCHOBbl IlPOI13BO,lJ,CTBA
oj
[001] Z
::i 1 : :
. : , -...s ..,.. (';
. . -: ...".-'v,;.........-.. .-
" ' '-1i OO - - :.'.
' ,,,,,.. ';I':',:.
.01.. .......v-......:.
"'.... ..;.. . -...J.. . .... -.
y
[010]
6.1
z
10011
PIIC. 2.6. n.1]OCKOCTlI n c-rpy""ype KpeJ\IIIHA (c uToJ\taJ\m):
a - nnOCKOCTb (l00); 6 - nnOCKOCTb (Ill)
3.1eK11'0HH)tO npOBO;uiMOCTb If Ha3bIBaIOTCSI
,llOHOPHblMI1 npHMecmm [15, 16].
BCJle;:I.CTBHe TOro qTO B cTpY"-"1)'Pe "--PH-
CTaJlJIa B pa3HbIX HanpaBj]eHliJI.X pa3JIH1.JHbI
paCCTOmIMSI H CH.1Ibl CBSl31f MeiKlIY 1.JaCTHua-
MH, OOJIbUlliHCTBO CBOHCTB h-pMCTaJlJIa aHl1-
30TpOnHO, T.e. pa3.m1'lHO B pa3HbIX HanpaB-
.1eHIDIX, HO O..<I:llliaKOBO B HaITpaBj]eHlUlx,
CHMMeTpH1.JHbIX ,llpyr .:IPyry.
PacCMOTpHM aHH30TpOnMIO MeXaHH'le-
CI<HX cBoikrB KpeMHMSI.
KaK H3BeCTHO, ynpyrlie CBOHCTBa 1130-
TponHbIX TBCp1IbIX Tej] npH TpexocHoM H8-
npJUKeHHOM COCTmunm OITpe.neJISllOTCH. Tpe-
M.SI napaMeTpaMM: Mo.uyJIeM IOHra E, K03qJ-
4IHlrneHTOM IIyaccoHa V H MOll)'JleM C,llBHra
G. a H3 HI1X He3aBHCHMbI, 8 -rpeTl1H BbI-
paAmeTCSI Qepe3 HMX 0)lH03HaqHO. }1,mI aHH-
30TpOITHbIX MaTepMaJIOB, K KOTOpblM npH-
Ha)l)lC"ACaT nOj]ynpoBO,llHHKJ1, ynpyrl1e CBOH-
CTBa onpe.neJlSllOTC Ha60poM ropa3,110
60JIbllIerO 'IHCJIa ynpyrux K03 4IHl.{HeHTOB.
CBSI3b MeJK.ZIY HanpX)KeHHeM H .nel}>opMa-
nIDIMM B aHH30TponHOM TeJIe YCTaHaBJIHBa-
eTCjJ 0606IlleHHblM 38KOHOM fYKa, KOTOpbIH
MOjf{HO 3anHCaTb B TeH30pHOM BM.ne B OJIRoi1
H3 JtBYX I}>OpM [5, 15, 16]:
0".. =C" /cm E km
!I IJ
(2.1)
HJ1I1
Ekm = skmlj-Clj,
(2.2)
rJ{e <J iJ' Ekm - TeH30pbl BToporo paHfa Ha-
npSl:>KeHHH Ii .nel}>opM81UfH.
KOMTIOHeHThI TeH30pa 1.JeTBepToro paH-
ra cijkm Ha.3bIBaKJTCR M0.DYJISlMH ynpyrOCTll, a
KOMnOHeHTbl sl..?ny - K03iPiPHUUeHTaMH no-
,!l8TJIHBOCTl1. TaK KaK Bce HH,lleKCbl i, j, k, m
nOCJIe)J,OBaTeJIbHO npHHl1M81OT 3Ha1.JeHHSI 1, 2
If 3, TO JtJISi onpe)J,CJIemDI Bcero HanpSl:>KeHHH
(11.i1lf ,lle OpMauHH) aHH30TponMoro MaTe-
pHaJIa B 06meM cJIY<18e Tpe6YeTcSI 81 K03 -
IfUHeHT cyl..?n I1nIl sl..?ny' KOTopble 06pa3Y-
lOT TeH30p 1.JeTBepToro paHra.
I1pH onepauHSlx C TeH30paMH 1.JaCTO
HCnOJIb3}'lOT COKp8meHHYlO 4IOPMY 38nHCH
HH,!:\eKCOB. K8:iK)J,ytO ITapy I1H,!:\eKCOB MO)KHO
3aMeHllTb O)J,HHM HH.neKCOM, KOTOpblH npo-
6eraeT 3Ha1.JeHHH OT 1 JJ.O 6, nOCKOJIbKY
BHyTpH ITapbl HH,lleKCbl pa3pernaeTCR ITCpe-
CTaBJISlTb Me:H{):(y c060i1. 38McHa HH,!:\eKCOB
0CYlIl.eCTBWl.eTCSI no cxeMe: 11 1, 22 2,
33 3, 23 4, 13 5, 12 6. B pC3yJIbT8TC
KOMnOHCHT)' TeH30pa K03 I1UHCHTOB ITO-
,118TJII1BOCTH sl..7ni] = 51323 COKpameHHO MO)K-
HO 3aITHC8Tb B 4IOpMC S pq = S54 = 545' a
KOMnOHeul)' TeH30p8 HanpIDKCHHH cry = crl2
TaK: cr q = 0"6.
qHCJIO HC3aBl1CHMblX K03 I1Ul1eHTOB
5 H C ,llJ)jI. 1l11I1CTaJJJIOB C onpeJJ;CJICHHOH
pq pq nj""
CHMMeTpHei1 yMeHbllIaeTCjI.. H3BeCTHO, ITO
ynpyrl1e CBoi1CTBa KpHCTaJJJIOB C Ky6HQC-
CKOi1 CHMMeTpl1ei1 (I< KOTOpblM OTHOCHTCSI M
KpeMHHi1) B CMCTeMe KOOp.nuH8T, onpe.n c -
JISleMoi1 KpHCTaJJJIOrpaclJwleCKHMH OCjl.MI1,
onHCbIBaJOTCjJ Bcero JIMllib TpeMSI HC3aBHCH-
MbIMH MO.n:YmlMI1 ynpyrocTM I1JIM TpeMjI. KO-
31}>I}>M MeHTaMH no.naTJIHBOCTH. MaTpHua
MA TEPI1AJIbl
89
KOJqJqJHUIfCHTOB nOJlaTJU1BOCTH WISI :noro
CJIYIJaH. fihlmSl,lll1T CJIe)J,YIOWI1M 06pa30M:
,\ I! Sl2 '12 0 0 1
.\h SII '\12 0 0
Sl2 S12 S11 0 0 0
'= . (2.3)
pq 0 0 0 0 0
844
0 0 0 0 S44 0
0 0 0 0 0 s44
AmUlOrl11.jHO MO>l<HO 3aITHCaTh MaTpHHY
MOJJ;y;lei1 ynpyroCT11. He3aBI1CI1:'vlbIe 3Ha'le-
mlSl K03QJqunwclIToB nO)JaTJIHBOCTI1 1\ MOJJ;y-
JICH ynpyroCTI1 I1MCIOT c'1c.IJy1OWl1e 3HalJe-
HHSI: SII = 0,768.10- 11 , S!2 = -0,214.10- 11 ,
844 = 1,256.10- 11 Oa-I; CII = 1,657.10 11 , CI2 =
= 0,639. I 0 1 I H C44 = 0,796.10 1 J TIa.
B IJpOH3BOJIbHOH Cl1CTeMe KOOpmmaT,
OCH KOTOpOH 06Pa3YIOT HeKoTopble yrnbl C
KpHCTaJJJIOrpaQJH'lCCKI1:\1H OCSlMH, TaKHX OT-
JII1'IHbIX OT HYJISI K03 qJl1l(I1CHTOB MO:>KCT
6bITb 21. O)J,HaKo Bce 3Tl1 K03QJqJHLlHeHTbi
MO:>KHO HaMTII 'lepe3 TpM ynOWIHYTblX KO-
3qJqJHLlHeHTa c nOMOllJ,blO cJIeJJ;yIOWcro npe-
06pa30BaHHSI:
x; = I,x] + mix2 + n,x3, j = 1,2,3, (2.4)
r)J,c x; - OCH npOH3BOJIbHOH CHCTeMbI KOOp-
,lJ.HHaT; I" m" n, - HanpaBJISlIOWl1e KOCI1HYCbl
ymoB Me)K)J,y j-H OCblO npOM3BOJIbH0i1 CHC-
TeMW Koop)].I1HaT 11 KpHCTaJIJIOrpa H'lecKM-
MI1 OCSlMH XI, X2, X3'
qaCTo npH paC'leTaX nOJIb3YIOTCSI Teo-
pHeH YITpyroCTH 1130TpOIIHbIX TCJI, n03TOMY
:>KenaTCJlbHO Bblpa3HTb ynpyme cBoHCTBa
KpeMHHSI B npHBbl'lHbIX )J...l.H H30TponHoH
cpe).(bl onpC}leJIeHHSlx: 'lepC3 MO)J,YJIb IOHfa,
Ko.3cjJq JH U H CHT llyaccoHa H MOJlYJlb CJlBHfa.
OOCKOjlbh")' KpC:\1HI1H SlBJISleTCSI aHH30Tp0n-
HblM MaTepHaJIOM, TO M 3Ha'leHWI 3TI1X MO-
,..
,llYlleH He oY)J,}'T nOCTOSlHHbIMI1. a 6Y)J,YT 3a-
BHCCTb 0'1' BbI6paHHoro HanpaBJICHW! B
nnOCKOCTH KpI1CTaJIJIa.
B Ta6JI. 2.1 npeJlCTaBJIeHbI aHaJIl1TI1'le-
CKI1C Bblpa:>KeHWI, xapaKTCpH3yiOIlJ,He aHH30-
TpOlIl1IO Mo,[(yml IOHra, K03 cjJHl(HeHTa Oy-
aCCOHa H MOJJ;yJ1S1 C,lJ.BHra )J,JIJi KpeMHHH B
llJIOCKOCTSIX (100), (I 10), (III) [5].
rpaQJ I'lCCKaH 11JI1IIOCTpaLlHSl H3MCHCHHSI
MO)J,YJIeH IOHra, C,lJ.BHra H K03 HLlHcHTa
TIyaccoHa B IIJIOCKOCTSIX (100), (110) H (111)
npeJlCTaBJICHa B TaM. 2.2.
PaccMoTpHM aHI130TpOIDIIO 3JIeJCTpHIJe-
CKoro conpOTIlBJIeHI1S1 KpeMHHSI.
TeopHSl nbe30pC3HCTHBHOrO 3 qJCKTa
YCTaHaBJJHBaCT CBSl3b Me)K)J,y l13MeHeHHeM
3JIeKTpl1IJeCKoro cOnpOTHBJIeHIDI nonynpo-
BO,lJ.HHKa 11 MexaHHlJecKI1M HanpIDKeHHeM B
HeM, KOTOpaJI B TeH30pHOM BH,!:\e MO>l<eT
6blTb npC,lJ.CTaBJIeHa CJIeJJ;ylOllJ,llM 06pa30M
[5, 13, 15]:
Elro = i, + 1[ljk1nOknJ" (2.5)
r,l\e i, i, k, In - KOMIIOHeHTbI BeKTOpa HanpSl-
:>KeHHOCTH 3J1eKTpH'leCKOrO nom: ro - Y.iJ.CJIb-
2.1. AHM30TpOUlIH cBoncTB KpeMHHH
IlnOCKOCTb
IlapaMeTp (100) (110) (III)
MO)J,YJIbIOHra E,I.10 1l H/M 2 11,36 15,15 1,69
7,73 +cos 46 9,3 + 1,33 CDS 26 + CDS 46
MO)J,YJIb c)J,Bl1ra G, I-lOll H/M 2 2,84 3,79 0,67
4,57 - CDS 46 5,77 -cos46
K03 I1I..{HeHT TIyaccoHa v 1,43 + cos46 2,24+cos46 0,358
7,73+cos46 9,3 + 1,33 CDS 26 + CDS 46
IT P 11 M e'l a H He. 6 - yrOJ1, xapaKTepH3YIOI11HH nOJ1O)KeHue np0113B0J1bHOH CHCTeMhl Koop.n IlHaT
OTHOCHTenbl!O CHCTeMb! KOOpAUHaT, 38,QaHHOH Kpl1CT8JIJlorpaclJH'IeCKIiMll OCJlMH.
90 fJlaBa 2. TEXHOJIOfI1LJECKHE OCHOBbI rIPOI13RO.lICTBA
2.2. rpalJ!n'leCKaD UJlJlIOCTpaLUID aHIBoTpolllIllMeXaHlI'leCKHX CBOUCTB Kpel\IIUlD
IlJIoc- (100)
KOCTb (110) (Ill)
t Z ,z Z
"'" ..,. 11'1'1'
1"11 . I I I ....
III, ,, .
(!) .." t"1:.J . ....\;'
:s: , . "
:s: ,
- I
.... 111.1
(!) b
:E 0
::I :.: Il1 h l
e., 0 .UI.1I
0 -'" )'
0 - 1-'. 1<>'-'
1""'1
:"1 "
:s: "'.
X/ Ilhl / ",
..
\"/
..... ,'OIC:
1"'1 . (1101
",' -'
u" c' ,.. ..,'
N - ",'
::E
---
-
. .
E ,.' ..
I
- pI 181. 0 ]
0 :1.,"
.J:: ""'
r::; ,
;;.., r. \ '")...;.
- './
"' .. ..
0 '" . ",
I
, ... ,, ",'
=" .
""[1 (lioJ It.!)
,,'
'""'
::E
--
-
.
<;;
t...
=
C 1001 (001] If10] IUDJ
c:::
U
.J::
-
;;..,
c::: ",.
0 ",,'
"'" ",,' In J
[lIO] UiO]
,a:" (001) T1" W' IHOI 1f
i ""
f-
-
U
... -
3- -
C pI JOOI
- 0 (lID] lilVl
.a- U
-9-:>-.
n-
0"-
I
.". "". »f ....r
",. ",'
pIJ I'IOJ 111 )
HOC COnpOTHBDeHHe MaTepl1aJIa npM MeXa-
HHqecKOM HanpJlACeHMH, paBHOM H}'nlO; jj -
KOMIJOHeHTbl BekTOpa ruIOTHOCTIl TOKa; akin -
TeH30pbl mmpJlACeHJ1SI; 1tijkln - TeH30pbI 'leT-
Bep1'oro paRra m.e30pe3HCTHBHbIX K03cpcpM-
eHTOB. npe)lCTaBIDIlOIItl1e C060H OTHOllle-
HHR M3MeHeHIDI COnpOTI1BneHI1S1, }J;eJICHHOrO
Ha HaqaJIhHOC COITpOTl-IBJIeHHC, K HanpsnKe-
HHIQ. BbI3bIBalQmeMY paCTSI:>KeHI1C. I1cnonb-
3YSl COKpamelmylO OPMY 3anl1CI1 I1H)leKCOB
aHaJIOrMqHO npe)),bI,llyureMY, MaTpl1I.{y nhe-
30pe3HCTI1BHbIX J(03 4>I1I.{l1eHTOB MO:>KHO
MA TEPJ1AJlbI
91
npCJIC'! aHI1Tb B BUile r 5. ] 3. 15]
11: 11 11: 11 11:(3 11: 14 11: 15 11: 16
11: 21 11: 22 11:13 11: 24 11: 25 11: 26
11:31 11:32 11: 1 -, 11:,4 11:'5 11:36
11:fJlf = 11: 4 1 11:4 11:43 11:44 11:45 11:46
11:- 1 11:5 iT,] 11:54 11:55 11:56
:-
71: 61 71: 62 1[61 11:64 11:65 11:66
1-13 YCJIOBHSI B3allMHOCTH nbe30pC311-
CTI1B1!hIX K03 HllIiCHTOB cne)J,YeT, 'lTO ne-
peC13HOBI(a HHJ],eKCOB HC MeHSIeT 3Ha'ieH11SI
KO'1QJQJHllHeHTa. TaKIiM 06pmOM, B 06meM
ny<me amnOTpOnHoro 18repMa.'a 06mee
Imc:JO lIe3amICH lbIX nLC'mpC3UCUIBHbIX 1<0-
cj)cjH1lllieHTOB iL1Si OmICaHI1S1 nbe30pe3l1CTI1I!-
1I0ro :JQJ eKTa B no.'ynpOBO!1HHI(e B ITpOm-
BOJlbHOH Cl1l.,eMe KOOp)J,HllaT paBHO 21.
):(JUl ,llOCTH:>KeHml MaI(CI1MaJIbHOH IiYB-
CTHHTCJlbHOCTl1 TeH30npc06pa30BaTeJIl1 'la-
me Bcero opHeHTHpYlD I B onpen:eJIeHHblX
KpI1CTaJLlOrpa H'leCKI1X HanpaBJ1emISlx.
KaI( npaBHJIo. opHcHTaUHSI npOl13BOJIb-
- / ' ", 3 '
HOI1 CI1CTe:\lbl KoopilliHaT ..!., . OT I<OTO-
pOH 3aBHCHT nOJIO>KeHHe pe3l1CTOpOB OTHO-
Cll1'eJlbHO KpHCTaJInOlpmjHl'leCl<OM CHCTeMbl
/,2.3, onpe,lleJIHeTcS! TpeMSI yrnaMH 3Hnepa:
<p, X, \jf, KaI( :no )J,eJ1aCTC$l )J,JUl BbIHBJIeHI1S1
nOJIO:>KeHMSI TBepJ],oro TeJIa (puc. 2.7).
Ha pHC. 2.7 BeKTOpbI, HaHeceHHbIe
B)J,OJIb oceM 3, N, 3, 0603Ha'laIOT HanpaBJle-
3
R
2'
\jJ
N
R
PUC. 2.7. OpIiCIIT311IlSi peJlICTOpOB
OTHOCIITCJ1bHO '-PJlCT3.1.-IOrpa4)11'lCCKOil
CIICTCJ\lbI KooplUlHaT
H11S1 ITOCnen:OBaTeJIbHbIX ITOBOpOTOB Tpex-
rpaHHHKoB oceH lia YlJIbI qJ, X, \jf, ECJII1
nJIOCKOCTb, onpe,lleJI5!eMaH OCRMH 1, 2, co-
BMemeHa C I(aI<OH-JlH60 KpHCTaJIJ10rpm))I1<Je-
CKOH ITJ10CI<OCTblO Kpe:'1Hl1eBOH nnacnlHbI,
TO ymbl qJ, X, \jf onpen:eJIHIOT npOCTpaHCT-
SCHHoe nOnO>KeliIiC pe311CTopOB R B nJ1a-
CTI1HKe. O'leBI1)J,HO, npH X = \jJ = 0 pe3I1CTO-
pbI HaXO}UlTC5! B l<pHCTaJIJlOrpa H'leCKOH
nJIOCKOCTI1 I1 HX nOJIO>KeHHe OITpCJ],eJ1eHO
ymoM qJ.
KOCHHYCbI ymoB MC:>K)),y JII06bIMII na-
paMH oceM 1,2,3 H 1', 2',3' 06pmYIOT MaT-
PHUY HanpaB11S1IOmliX KOCHHYCOB:
J I 111, 111
/2 1112 11'2.
J) 1113 11)
cos qJ cos X cos \jf - sin qJ sin \jJ sin qJ cos X cos \jf + cos qJ sin \jJ
- - COS qJ COS X sin \jf - sin qJ cos \jJ - sin IjJ cos X sin \jf + cos qJ COS \jJ
. ,
qJ X 1jJ X
. .
SIll X sm \jf
cosx
,
r)J,e /" m" 11, (i = 1, 2, 3) - COOTBeTCTBCHHO
HanpaBJISIIOIlJ,HC KOCHHYCbI yrJIOB OCCH 1, 2,
3 C Ka>K)J,OH 113 OCCH 1', 2', 3'.
,l],J1S1 KJIaCCa CI1MMeTpHM 1<)'6I1'lCCKllX
KpI1CTaJIJIOB, K KOTOpbIM npIIHa,wIe:>KMT
KpeMHHH, nOJIyqcHbl [5,16] BbIpa)l(CHHSI )lJlSI
nbe30pc3HCTHBHbIX 1<03 Hl.{I1CHTOB 1t mv (m,
V = I,..., 6) 'lepC3 TPH rJIaBHbIX nbc30pe3H-
- SIll X COS \jf
(2.6)
CTI1BHbIX K03 HuHcHTa 1t11, 1t12, 1t44 II Ha-
rrpaBJISllOIlJ,He KOCI'UYCbI Ii, lilt, 11; (T3611. 2.3).
B Ta6JJ. 2.3 BeJIl1'lMHa 1t n = 1t1 I - 1t12 -
- 1Lt4 SlBIDIeTCSI MCpOM aHH30TpOnHH 3 eK-
Ta. K03 l1llHCHTbl1t1 h 1C12, 141 HmbIBalOTCR
COOTBeTCTBeHHO npO,l\OJIbHbIM, nOnCpC'iHbIM
H C,QBHrOBbIM nbC30pe31lCTIlBHbIMH K031}J1}J11-
1.{I1CHTaMI1.
92
rml£la 2. TEXI-IOllonP.IECKIIE onlonbl IlPOII1BOJKTB/\
2.3. n.,CJOpC311CnumMc tW") .IIIIICUTM
K03!\J!\JI111IlCliT Bblpail\ClIlIC AllaJlC1II1'Ulhle 1\0 H!)(IJI II II ICII 1 101
, "II., ...... ...,., , ,
1t 11 1t 1l - 2n: , (/1- /11 1 + 111i IIi + Ill-In n . n33
, , ... ., ...... .,., , , , ,
1t I ::! = 1t::!1 1t ll + 1t a UI-/! + 1111111 + "1/2) 1t D ::: 1t.H' 1t::!J ::: 1t J :,
, "), 21t a (li'l"!. + 111illl"!. + I1j'II::! ) , ") , , '"},
1[16 = -1[61 1t:!., ::: _1t.!:!, 1[35 .:..1["
, - ") , I ?'
TI:!(1 - -1r:h 1t.n ::: -1t. 13
, ")'
1t" ::: _IT"
, ")' ") II (' .., .., , ? I I ?'
1t)(o ::: -1[1>3 _1[:\ ( I "!. ; + 1111 111:!t11i +11111:,113) IT I .\ ::: -1[-11' IT::!5 ::: _IT,:!
, , .., ..,
ir.\b == irb-l 2ITa (/1/5 / ) + t11111l!111J + 11 1 /2/13) , I , ,
IT'-I ::: 1[-15' IT(,5 ;::: IT S (.
, "" ... ....., .,.,. , ,
1[66 IT44 + 21[a (II-Ii + 111 1 - 1115 + 11)115) IT-I.\. 1[55
npe;J,nOJIOJKItM. 'l1'OOCII 1'. l' HaXO.!J.SlT-
CSI B m10CKOCTIi Kper.llmeBoH I1JIaCTHHhl. T.C.
Z = <P = O. It MaTpliUa (2.6) npHHIlMaeT BI1;1
I, In l "I cos <P Sllllp 0
'2 m:, II::! - - cos <P cos <P 0 (2.7)
/3111) 11} 0 0 1
YCJIOUIII\ICSI "HUnKC, 'ITO IIJIOCKOCTJ.
KpeMHIICBoii nJIaCTllllhl COBMClHellU C ollllOii
1t3 KpIiCTanJIOrpa(lm'ICCKIIX IlJIOCKOCTCIi
(100). (110) (pIlC. 2.8) IIJIII ( 111), II 1I0JlY'(J-IM
BblpaJKCHIiSi )J,JISI Ko:)(I) JllIlIICIiTOB IT' J" IT' '2.
IT'It> 'JCpC3 rllaBUhlC Ko)(I)(!JIIl\uCII'I bl It,,, IT,::!,
ItoM. Bocnonh30BamUilch n16JI. 2.3 II IIHrrpll-
l\eil (2.7), onpc)J,CJIlIM IlallllblC, IIpC,lJ,CTan-
neHHhle B Ta6n. 2.4.
2.4. nbe30peJHCTHBHbie KO')cJI HItHe.lTbl JJ,JUI KpHCTaJIJ]OI'pa II"lecKHx UJlocKoc..-eii
KpHCTa.'L'Jo- Km ) JlI1UiCIIT
rpa!\JullecKaH. , , ,
nnOCKOCTb 1t1l 1t)2 1t16
(J 00) 1[11 -(1t1l/4)(I-cos4lp) 1t 12 T (1[0 /4)(1 - cos 4lp) -(ITu /2)sin4lp
(110) 1t1J -(1t o /16)x 1[12 + (31t n /16)x (- 1[0 /8) x
x(7 -4cos2<p-3cos4<p) x (I -cos4<p) x (2sin 2lp + 3sin 4lp)
(Ill) 1t 1l - It o /2 1t12 - 1[u /6 0
3HaqeHHJi rnaBHbIX nbe'30pe3IiCTIiBliblX
Kool}>4>HlUteHTOB npHBe.nelibl B pMJJ.e IiCTOLf-
HHKOB. B IJaCTHOCTH. J1JUI KpeMHHJI H rcpMa-
HID! n - If P-THTIOB B COOTBeTCTBIiJ1 C pa60-
ToA [5] 3HaIJeHHR 3TJIX KOJ4>4>HItHeHTOB .na-
UN B Ta6n. 2.5, H3 KOTOpoli, B LfaCTHOCTIf,
cne.nyeT. <ITO ,WI1I KpeMHHJI P -THna npOBO-
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1I- Ita IIJIOClClJcnl (100) II 6 IIU IUIOCKOCTII (J 10)
2.5. nUlBIII.IC m,c30pc:JI1CTlllUlbIC KO:)( ( I1I\IICH'r1.l
MUTI:pmIJI p. (,10- 2 , OI\l'M KO:>I!J(!nU.\IICIIT, I.i/l-l
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11: 11 ,
1,5 -2,3 -3,2 -138.1
repMHIIHH 11- Tuna 5,7 -2,7 -3,9 -136,8
9,9 -4,7 -50 -137,9
,
16,6 -52 -55 -138,3
, ,
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,
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,
KpCMHl111 I1-Tlma 11,7 - \02,2 53,4 -13 6
,
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11:' 12, 11:' 16 MorYT 6hITb BblpmI<CHbl 1'OJIhKO
<ICpC3 rJIaBlJblH lCQ3<l)(!ml{l1ell1' 11:44 ,D,JUI
p-nma pC3lfCTOpOB H tlepC3 11:11 .nJUI pe3HCTO-
pOB n-nma (1'a6n. 2.6).
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IlHell1'OB 7t'III,,«p), HOpMHpOBaHHblX no maB-
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1'a6n.2.7.
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,
1t"JI'
(100)
(110)
(111)
p-mlm
, 0,257t4il- cos4q» 0,1 91t44 (2,33 - 1,33cos 2q> - cos4<p) 0, 51t 44
1t11
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7t12
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7t16
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7tIl
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7t12
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7t16
MA TEPHAJIbI
95
CPU31f<IeCKUe CBOUCTBa Kpel\lMMSI
KpHCraJIJJl1'leCKaM CTpYKlypa ....................__ ..... ................
o .
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KOJIII'lCC"1 BO aTO:\10B/ CM J
.. . .. . .. . .. . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. . . . .. .. . .. .. .. .. .. .. .. .. ...
I1JIOTHOCTb, r/CM 1 ....................................................
TO'IKa nJIaBJIeHl1H, °C
YnCJIbHaH TCnJ10CMKOCTb, .[J)KI(Kr'°C) ...... ....... .......... .....
MOJ\YJIb IOHra, rI1a .................................................
KOIiCTaHTLI )KCCTKOCTH, rTIa:
Cl I .................................. --. .. . ..... .. ...... .........
CI
..............................................................
C4d .......................................................................
KO)(l)(lmuHcHT I1yaccoHa . ....... ... ...... ... ..... ........ ..........
I1pcLleJI npO'IHOCII1 npH paCTSI:>KCHMH, MI1a .............. .......
T cnJIOnpORO)J,HOCTh, BTI (CM .oC) .................................
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TIhC30pC3HCTHBHhlC K03(1)
Hl.\HCHThI, TIa- l :
ll-THn:
11: II ...................... --. ..... ... ... .... ........
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p-THn:
11:44 ................................................
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11:12 .. .. . . ..... ... ................................ ..
7t44 . ..................................... .....................
Y.tJ..eJ1bHOC conpOTHBnCHHC, OM'CM .............. -.................
):{H3nCKTpH<ICCKaJI npOHMI.:(aCMOCTb ..............................n
KaK KOHCTPYKUl10HHblH f\ta'fepH8J1
KpeMHHH nOCTaBJJSlCTCR B BI1.nC ITJIaCTJm,
KOTopble )J,JIR pa3mIQHLIX )J,HaMeTpOB HMCIOT
pa3HYlO TOJILQI-my. TaK, npl1 ,l\HaMCTpc 60 MM
TOJIllJ,l1Ha nJIaCTHHbI paBHa 0,35 MM; npH
,llHaMcTpc 100 MM OHa HaxO.tJ..I1TC.fI B npc,llC-
JIax 0,46...0,6 MM; npH .tJ..I1aMeTpC 76 MM
COCTaBJISlCT 0,38 MM. I1JIaCrHHLI Tpc6YlOT
TLQaTeJ1bHOH 06pa6OTKI1 (PHC. 2.9).
,[(mI BH3y8J1bHOrO onpc)J,eJICHMH. OpHCH-
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Ha HHX BblllOJIHSlIOT 6a30BbIH II .tJ..onOJ1HH-
TeJIbHblH CpC3bI.
MapKHpoBKa nJIaCTHH HcceT HH
opMa-
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5,43
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2,32
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0,7
130
165,6
63,98
79.51
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3790
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,
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,
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102 . 10- 11
53,4 . 10- 11
13,6 . 10- 11
2,3. 10 5
11,7
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a HMCIOT nOKaJIbHbIC ,l\C
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neJILHOCTH (BLmYKJIOCTH, BorHyroCTI1), oc-
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reOMeTpH'leCKHe .:1e e"'TbI nJlaCTI1H 3a-
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noc.'le.Jy1OweM ynpaB..'ISieMOM TpaBneHIII1.
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nOnepe'IHHKe. XHMH'ICCKHe 3arpSl3HCHIiSi no-
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cop61um. TIpli 3TOM Ha nOBepxHocTIi TIJIa-
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npliMecci1. He npenSlTCTBY10Wax nOJIY'leHHlo
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n.TlaCTliH HCn0J1h3YIOT )Kl1.11.KOCTHble 11 cyxlle
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BaHI1C I1 TpaBJICHHC nJIaCHIH C 06S13aTeJIhHOH
npOMI>IBKOii nOCJIe KaiK)J,OH onepa1.(J1H. 06C3-
iKl1pHBaHI1C BbInOJIHXIOT B opraHI1 ' leCKI1X
paCTBOpax (lleTblpeXXJIOpl1CTblH ymcpo)J"
6eH30JI, TOJ1YOJI, H30npOnI1JIOBbIH cnHpT) I1 B
aKTIiBHblX cpelIa.x (nepeKI1CHO-aMMI1a<lHbIH
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6)
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PHC. 2.10. .Ae4JeKl'LI KpeMHHesLix onSCTHII:
0- HennOCKOnapaJIJlenbHOCTL; 6 - BorHyroCTL; 6 - BLlOYKJIOCTL; Z - Kopo6JJeliHe
MA fEPI1AJlbf
97
XIIMII'[CCKOe, OCHOBallliOe lIa XIIMH<JeCKHX
IIpeDpaLI CHmIX. npmICXO)]JILQHX npll 'JJIeK-
I pO.lll C.
CyxaSi O<mCTKa - 3TO OT))(llr, HOHl1Oe,
ra30BOC II n.lalMOXIIMI1<1eCKOe TpaB:IeHlle.
TIoCJ1e O'IIICTKI1 KOlnpOJIllpYIOT ee Ka-
'ICCTBO. o'lIJIIIM 113 wm60Jlce pacnpOCTpa-
I1eHHbIX JlKU!CTCSI Mero I<OHTpOJISI. OCHO-
BaHilblii Ha CMa'lIlBaeMOCTlI nJ1aCTlIH: nJIa-
CHIHa. CB060lUl8H. OT rpJl311. I3CJIe.D.C"1 BHe a!l-
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IIJIelIKY 130)].1,1. 3arpSl3HCHHSI HapYWalOT L\e-
JlOCTHOC rb n:ICHKI1 II ;lerKo o6uapY)l(u-
mOTCS'-
Bblwe Orl\!C<laJIOCb, <ITO HCnO;]b3YCMble
nJIaCTl1l1bl H:>ICIOT TOJIllU1HY 200 ... 600 MKM,
npl1<JeM p r36poc no TOJIUlI1He OT nJIaCTIIHbl K
U.laCTlIIIC \1Q)I(CT 6bITb 3Ha<n1TeJ1bHbIM. He-
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cnoc06a MO)J{JiO y)J,aJ1l1Tb XHMI1'ICCKHM Tpal3-
neHl1eM. Op,IWKO nOl3epXHOCTb, no.1YQaCMaH.
npll nOM, )J,OJI)I(H8 6bUb nOJIHpOBaHHOM,
<IT06bl, Bo-ncpsbIx, :VIOIKHO 6blJI0 npOBO.D.llTb
118 HC" (lJoToJIIITorpa(lmlO, a BO-BTOPbIX, HC
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CTJ1Hbl, ITOnY<lcHHYIO Hcnocpep,cTBcHHO no-
CJIe pe3KI1 CJlIITKa, C MHKpoHCpOBHOCTSlMH
ITOpR.D.Ka 13 HeCKO.:IbKO )].CCRTKOB MHKPOMCT-
pOB nO,llBCprHyrb nOJIllpYIOLQCMY TpaBJ1C-
HHIO, TO nOCJIC y,llaJ1CHlfR OKono 50...70 MKM
CJIOR KpeMHIISI y)].aeTCR nOJIY<IHTb nOJIHpO-
BaHHYlo CJICf!\8 BOJIl-IHCTylO nOBCpXHOGfb C
MHKpOHCpOBHOCTRMH nopSl)J,Ka B HCCKOJIbKO
MHKpOMCTpOB. XOTR Ka'lCCTBO nOBcpxHOCTII
npH 3TOM nOJIY<laeTcR :ma<IHTeJIbHO xY)J(C,
'ICM ITOCJIC cTaH,!J:apTHoH MCXaHI1'lCCKOH 06-
pa60TKH, TCM HC MCHCC, TIY nOBcpxHOCTb C
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nOJUtKpUCTaJIJIH'.CCKIIU KpCI\tHllU
B OTnl1'lHC OT MOHOKpHCTaJ1JIa nDJIH-
KpHCTaJIJI npe)J,cTaBJ1S1eT C060H COBOKyn-
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CKH OpllCHTllpoBaliHbJX MaJ1elibKIIX KPHCTRJI-
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TpOITHble cBoikTBa. T.e. O.D.I1HaKOBblc BO BCCX
HanpaBneHHJlx. OT)J,eJIbHbIe MOHOKpIlCTaJI-
JIII<JCCKHe 3epHa aHH30TpOnHbJ, a BeCb nonll-
KpI1CTaJ1JI. OC06CHHO McnK03cpHHCTbIiI, Be-
JJ.CT cc6S1 KaK 1130TpOITHhIH MaTepHaJ1. B TCX-
HOnOrHJlX 113rOTOBnCIiIlSi MHKpOnpI160poB.
IICnOJIb3YJOLQHX nOBcpxHOCTHYIO MHKpOMC-
xaHII<JCCKYIO 06pa60TKY. TOHKIIC nJICHKI1
KpeMHHR Hc06xo)].IIMbI KaK CTPYKTYPHblH
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3TH MaTepllaJIbl HaxO)J,H.T llillpOKOC
npl-fMCHCHHC B M3MC-npoMbilliJICHHOCTl1.
MCX8Im<leCKI1C CBOHCTBa nOJIHKpeMHI1CBblX
ITnCHOK 3aBIICSIT OT npoL\CCca ocaIK)J,cHI1R,
HCnOJIb3YlOLQCrOCSI npll IIX H3rOTOBJICHHI1.
TIncHKII )J,CMOHCTpHpyIOT H36I1paTeJIb-
HOCTb OpIlCHTaL\Hl1 3CpCH, KOTOpbJC H3MCIUI-
10TCSI C TCMncpaT)'pOH. TaK KaK H)J,CaJIbHWI
ITJICHKa HC 06HaPY:>KHBaeT 3aBHCHMOCTl1 OPH-
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OT nonHKpeMHI1CBOrO K aMoP<l)HoMY KpeM-
HHlO, - 3(1) cKTHBHbIH MeTO)J, npoll313Q1.J.CTBa
H30TponHblx nOJIHKpHCTanJJ CCKIIX nnCHOK.
npll 3"fOM TCMncpaType aMOp(lJHbIM KpCMHllM
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'ITO n03BOIDIeT nOJIY'lHTb nJIcHKH C no'lTU
paBHOMcpHblM MO,llyneM IOHra (165 rna.)
B nom1KpHcTaJIJIUlJeCKHX MaTepHanax
Pa3PYUlalOw.cC HanpSl))(CHI1C 3aBHC 1T OT )].BYX
(lJaKTopoB: pa3MCpa 3cpHa H nOBcpxHOCTHOH
3HCpfHH nCpeJIOMa.
B HCCKonbKllX IiCCJIC)].OBaHHSlX CPC)].HAA
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(nOpJl..D;Ka 2...3 rna), T.C. OHa MeHhlliC, '1CM
y MOHOKpHCTaJIJII1QCCKOrO KpCMHllJ1.
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MOlt)'.'Ib IOm'a (oobeMHoe 3HaQeHHe), rITa ................... .............. 75
K034>4muHeHT IlyaccoHa .......................... ........... .................. 0,42
KpHTHQecKUH npe.!lCJl npOQHOCTH npu pacnliKeHH", MIla. . .. . .... .. .. ... 125
Ten.;'IOIJpOBOllHOCTb, BT/(CM' cC) ............................................. 3,15
K03q,q,HUHem Ten.'IOBoro pacIIIHpeHlUl,cC- 1 ......... .......... ............. 14,2 . 1O-{;
Y;:J.e.'1bHoe conpoTHBJleHHe, OM . CM .......................................... 2,44 . 10-{;
30JIOTO nO'lTH BcerJl,a HaHOCHTCSI Ha
nOBepxHoCTb 60JIce TBep.lloro MaTepHaJJa
HJIH HCnOJ1L3yerclI B H3.11.eJImlx, He CBH3aH-
HbIX C Mexa.HM':lecKHM .nBIDKeHHeM. 30nOTO
HMeeT np06JIeMLJ aJU"C3HH C Si0 2 , HO eCTb
HeKOTOpLJe yCTaHOBJIeHHbIe MerOJJ.1>I JlIDI HX
06xo.na. OJUfH H3 Mero.nOB COCTOHT B TOM,
QT06h1 HCnOJII>30B8Th nocpeJUiH'lecKHH CJIOH
xpoMa K8K CB1I3}'lOUIHH M8TepHaJI, nOCKOJIb-
KY OH c1>opMHpyer CB1I3H Cr20] C Si0 2 , a
TaIOKC OTJlH'laeTClI CHJIbHI>IM Cl.{CnneHMeM C
30JIOTOM.
fn8BHbJH cmMyJI JlIDI HcnOJII>'30BaHHlI
30JlOTa B M3MC-Jl3J1,eJIIDIX 38KJIlOqaeTClI B
TOM, 'ITO OHO lIBID1eTCR nytflliHM 3J1eKTpHlIe-
CKHM npOBOJtHHKOM, IIeM 8JIIOMHHHH. KpoMC
Toro, 30J10TO He CP83Y OKI1CJlHeTCJI B aTMO-
c epe, qTO ,l\eJIaeT ero npHBJIeKaTCJIbHbIM
)lJUI npHMeHeHHH B 113.neJII1HX, CBJl3aHHblX C
Henocpe)J,CTBeHHbIM B03.lleHCTBI1eM aTMO-
C.pepbI.
Me.zu.
OCHOBHaJI npHqHHa HCnOJIb30BaHI1H
Me)l,li B M3MC COCTOHT B TOM, qTO OHa OT-
J1MqaeTC.II 6oJ1ee BbICOKOH ITpOBO,llHMOCTbID,
qCM aJIJOMHHMH H 30nOTO. Me)J,b - npe-
BOCXO.nHbIH npOBo.nHHK TenJIOTbI. O.nHaKO
H3-3a nJlOXOH aJl,re3HM c KpeMHMeM ol1a B
H8CTOllmeC BpeI\UI CJJY:>KHT TOJIbKO B KaqeCT-
Be npOBO)J,HHKa.
MA TEPI1AJIbl
103
<Dll3llQeCKHe CBOHCTBa MellH
rIJ10THOCTb, rlCM J . --......."........."..u
TOIJKa nJ18BJ1eHIIJI, °C ................"..
Y.ll.eJ1bH8J1 TenJIoeMKOCTb, ):\)K/(r .oC)
MO.ll.YJ1b IOHra (o6'beMHOe
3HatJeHlIe), rrIa .........__.............
KmqJljJuUHeHT IlyaccoHa ......... __'.
Ilpe.ll.eJ1 npOtJHOCTIi, MrIa ......."...
TerrJ10npOBO.ll.HOCTb, BTI( CM' 0c) "..' ...
K031jJljJuUHeHT TeITJIOBOrO pacwn-
° C - I
pel-lI1Sl. .o . e......................." e..
Y.ll.eJ1bHOe COnpOTl1BJ1eHne, OM'CM ..
8,89
1083
0,39
115
0,36
220
3,98
16,6.1O-{)
1.72- 1 O-{)
Cepe6po TaIOKe 06JI8).J.aeT xopOllleH
3neKTporrpOBOlIHMOCTbID H TCnJIOrrpOBO).J.HO-
CTblO H TaIOKe I-IMCeT ITJIOXYIO a).J.rC311ID C
KpCMHl1eM. CJIe).J.OBaTeJ1bHO, 11 3).J.eCb He06-
XOlIHM nO)J,CJIOH. qTO YCJIG:>KID1CT TeXHOnO-
mID rrOJIy<reHHjI UJIeHKll. KpOMC TOro, TaK
KaK aTOMbI ccpc6pa crroc06HbI Ml1rpl1pOBaTb
no nO)J,JlO)!(KC, B03MO:>KHO 06p830BaHl1e HH-
TeH H3 ccpe6pa Me)J()];y COCCJI,HHMH npoBo-
,l],SlIlJ,I1MI1 JII1HIUIMI1.
TIITaH H XpOM SlBJ1S1IDTCJI MaTepl1aJIaMH
)J,JlS! ITo)J,CJIOeB. TI1TaH OTJIUqaCTC5I BbICOKOH
MCXaHHQeCKOH rrpOQHOCTbID Ii KOpp03HOH-
HOH CTOHKOCTbID. O,l\HaKO, H3-3a BbICOKoro
y,l\enbHoro COITpOTMBJICHH}J 11 B03MO)!(HOCTH
06p830BaHI1S1 OKCI1)J,HOH nJIeHKM OH HCnOJIb-
3YeTCSI B Bl1,lle CI1CTeM: Si - Ti - Au; Si - Ni -
Mo - Au I1 )J,p. XpOM xapaKTepl13yeTcSI xo-
poweH 8).J.fe3l1eii C KpeMHl1eM, HO npl1 TeM-
ncpaType > 200°C E3aUMO,lleHcTByeT C lIH-
OKCH).J.OM KpeMHI1S1.
fiJianma, THTaH H I\IOJlHo.n.eH CJIY:>KaT
,l\JIS! C03)J,aHI1SI p83J1,eJIHTeJlbHbIX (6apbepHblx)
cnOCB. HmmYQWI1MH p83J1,CnllTeJIbHbIMH
CBoHCTBaMI1 06JIa)J.aeT ITJIaTI1Ha. Ee nJICHKa
TOJ1IlJ,HHOH 0,05 MKM HCKJIID'laeT B3aHMO,lleH-
CTBHe Me:>K)J.y 60J1bWI1HCTBOM MeTaJIJIOB,
Hcnonb3yeMblx B Ka'leCTBe KOHTaKTHbIX H
ITpOBO)J,SlllI.llX cnoeB. Co CBOHCTBaMl1 MaTe-
pl1aJIOB, npl1MeHSlCMbIX B npoH3Bo,l\CTBe KOH-
TaKTHblX CI1CTeM, MO:>KHO 03HaKOMHTbCS! ITO
nHTepaType [4, 17,20].
TIonYQeHl1e TonKHX MeTaJlJlHQeCKHX
nJleHOK B03MO:>KH0 O,l\HHM H3 cJIe.z:\YIOllI.Hx
crroc060B:
113I1QecKHM ocajJ{)J.eHI1CM HJIH KOH-
)J,eHcaQl1eH H3 ra30BOH CPC).J.bI (TepMoBaKY-
YMHoe MJII1 KaTO)J,Hoe pacnbmcHMe);
- XHMM'lCCKllM OCaJKl(eHHeM H3 ra30BoH
83bI (UHpOJIM3, peaKTHBHOe paCnWICHl1c);
- 3JICKTpOJIHTM'lCCKllM MJII1 raJIbBaHH-
qeCKHM OCa/K,l\CHHeM H3 paCTBopoB cOJIeH
MCTaJIJIOB (HaHeCeHI1C raJIbBaHI1QeCKHx UOK-
bITHH, XllMHqeCKOe Me,l\HeHHe);
- aHO)J,HbIM HJIH TepMHqeCKHM OKllCJIe-
HHe nOBepXHOCTI1;
- 110HHblM paCITbIJICHMeM.
llpoMbIllIneHHOCll, BbmYCKaeT YCTa-
HOBKH ,l],JISI UOJIY'lCHHjI TOHKI1X ITJIeHOK TCp-
MOBaKYYMHbIM HenapeHMeM, ITpl1HI.{Hll KOTO-
poro COCTOHT B TOM, qTO OCa:>K,l\aeMbIll MaTe-
pHaJI HarpCBOM nepeBO,llHTCSI B ITapOra30BYID
83Y" 06pa3YIOIlJ,l1HCSI napOra30BbIH nOTOK
paenpOCTpaID1eTCS! B BaKy)'MHOH KaMepe
IJpHMomIHeHHo I1 nona)J,aeT Ha nOlVlO:>KKY,
reMnepaTypa KOTOpOH Hll/Ke, 'leM napa.
ITPOllCXO,lJ,j1.T KOH,l\CHcaQ}lli H 06p830BaHl1e
ITJIeHKll.
MeTO,l], TOHKoro paCUbIJICHIDI n03BOJISI-
eT nOJIY'laTb nJICHKll M3 IJpOBO HX, lIl1-
31IeKTpHQeCKHX H nOJIyrrpoBO).J.HHKOBbIX Ma-
TepHaJIOB, a TaK)!(e M3 'I)'rOIL'IaBKUX H MHO-
rOKOMnOHCHTHbIX. llpl1HQl1n 3Toro MeTO).J.a
OCHOBaH Ha 60M6apJl,HpOBKe MHllleHH 113
OCa:>K)J,aCMorO M3TCpHaJIa 6blcrpbIMH qaCTH-
Q3Mll, HanpHMCp, nOnO:>KUTCnbHbIMH 110HaMH
aproHa. BbI6uTbIe 113 MHllIeHM QaCTHQbI 06-
P83YIOT nOTQK MaTepl1aJIa, KOTOpblH ocaJKl(a-
eTCJI Ha n0.ll.Jl0)KKaX B Bl1,llC TOHKI1X nJIeHOK.
llO,lJ,JIO)!(Kll pacrronaralOTCSI Ha nyTll nOTOKa
Ha HCKOTOpOM paCCTOJlHHH OT MHllleHll.
TOJIIl{HHa, 3J1e.....pHQeCKOe conpo-
THBlIeHHe II aL{l'e3HH - oeHOBHblC KOHTpO-
mIpycMbIC napaMeTpbI nJlCHOK. ,[{JIH. H3Mepe-
HliH TOlIllJ,HHbI MCnOJIb3YlOT MeTO,l],bI MMKpO-
B3BellII1BaHMJI, MHOrOJIY'leBOH MHTep CpO-
MeTplill 11 p83HOCTH qaCTOT KBap eBoro pC-
30HaTopa. KaQeCTBO a.II:re3HH nJICHKH C nOk
JIO:>KKOH 06bIQHO ITpoBepHeTcH no CliJ1e Ha
OTp B IJneHKH OT ITO,lJ,JIO:>KKll C nOMO lO
HanaJlHHOrO Ha nneHKY IlJIHH,llpa.
P83MepbI KOHTaKTHbIX nJIOlll.a,l\OK H
nJIeHOqHbIX npOBO,l],HllKOB H83HaqaIOTCSI C
yqeTOM OrpaHMqeHHH, 06YCJIOBJICHHhIX B03-
MO:>KHOCTHMH TeXHOJIOffill H HaKOnJIeHHbIM
OnbITOM. YKa:>KeM ,lJ,JIJI npHMepa, qTO MHHH-
104
rmma 2. TEXH0J10rI1QECKHE OCHOBbl IlPOH3BO.n.CTBA
MaJIbHOe paccToRHHe Me)K):\Y IIJICHO'iHbIMH
3JIeMeHTaMM M KOHTaKTHbIMIf ITJIOma)J,KaMH
300 MKM npH I1CnOJIb30BaHI111 MaCOK, Ii
50 MKM npH OTOJIMTOrpa l1l1, MMHliMaJJbHO
,lI.OnYCTUMble pa3MCpbI KOHTaKTHblX ITJIOIna-
,lI.OK COCT3.BJUUOT: .nml npHBapKH rn6KI1x BbI-
BO.nOB 200x150 MKM, a JI.,J"U3 npl1naHKn -
400x400 MKM.
,Z:lmI npMMeHeHIDI B M3MC-YCTpoHCT-
Bax paccMaTpHBalOTcR H Apyrne MaTepl1aJJbl.
Hanpl1Mep. n01maMIUJ.bI - KJIacc OpraHIi'le-
CKliX nneHOK, KOTopble MOI)'T KOHI,:ypliPO-
BaTh C .:mOKCHllOM KpeMHI1R, KaK H30nRTO-
pOM, T.e. 3TO HOBoe nOKOJleHHe .'lH3JIeJ..1pI1-
KOB C H1l3KOH 3JIeKTpI1'leCKOH npOHHnae-
MOCTbIO.
2.2. BbIP AIIJ,IlBAHHE If
nO OBAHHETOHKHX
llJIEHOK
Pa3BHTHe MIU..-pOTeXHOJIOfHl1 06ccne-
qHIIO T8K Ha3bmaeMbIH II.!JaHapHblH npOI!CCC,
HJIl1 nnaHapH)'lO TeXHO.iIOfHlO, CyTb KOTOpOH
3aK:-noQaeTCR B nocne):{OBaTeJIbHOM H3rOTOB-
JIeHMH CJIoeB C 3a):{aHHblM pHC)'HKOM, pacno-
nO:iKeHHLIX .npyr H3!( .npyroM M COCTOIDI\llX
H3 MaTepH8J10B C Pa3JIHQHbIMH 3J1eKTpI1'1e-
CKHMH CBOHCTBaMH.
C.'lOH C Pa3.1IHQHbIMH 3JIeKTpHIJeCKHMH
CBoikrBaMH MO:>KHO nOJI)"-IaTb, H3MeHAA
CBoHCTBa nO.!VIOj!(}lli, HanpMMep, ITyTeM ee
nernpOBaHIDI HJIl1 OKHcneHIDI, HJIH :>Ke OCroK-
.neHIDI Ha ee nOBepxHoCTh cnOR C nOMOlllblO
BHenmero HCTOQHHKa nocpe.nCTBoM Hcnape-
HIDI Ji.,"}M pacnbL:1eHIDI B BaKY)'Me. 3a){aHHblH
pHC)'HOK non)"-la.ercji' B npouccce OTOJ1HTO-
rpa4JHH.
I1pH 3TOM pHcyHOK C $OTorpaq,H'IeCKO-
ro -rpa4Japera npoeUHpYeTCJ:I Ha nOBepx-
HOCTb no.n.nOiKKH, npeJtBapMTeJIbHO nOKpbI-
TbIH. cnoeM OTope3MCTa. <1>oTOpe3HCTHBHble
MaTepHaJJbI 06na.n,aJOT .nBYMR cBoHCTBaMH.
O,ZtHO H3 HHX 3aKJIlO'laeTC1I B TOM, '1TO no.lI.
)J,ellCTBHeM CBeTa cnOC06HOCTb $OTOpe3HCT3.
pacTBopHThC5I B onpe,l\eneHHOM 1CJ1acce pac-
TBOplITeJIeii H3MeHJleTCJI. I10cne npo5lBJIeHIDI
B TaKOM pRCTBOpHTene cnpoeUHpoBaHHblH
pHcynOK OCTaerCH Ha nOBepXHOCTH nO.lJ)l()jj(-
KH. 'zwYToe CBOHCTBO COCTOHT B TOM, qTO
HepaCTBOpeHHbIC 06JlaCTH OTOpC3HcTa co-
BcpIIIeHHO He B3al1MoneiiCTBYlOT (pC3I1CTHB-
Hbl) C )J,pyrnM KnaCCOM paCTBOpIHCJ1CH, KO-
TOpblC enoc06Hbi TpaBIITb IIm1 l13MCmlTb
KaKJ1M-n1160 06pa30M HI1>KCJIC>KalllHM CJIOH
MaTepl1aJJa.
ECJII1 oLum II TOT >KC Marepl1aJJ He 06-
JIa.n.aeT o)J,HoBpeMcHHo 3TI1Ml1 .n.BYMSI CBOH-
CTBar-m, TO J1.JIR nOJIy'iCHHS! 3a.n.aHHoro PH-
CYHKa He06xo,llJ1MO )J,06aBl1Tb npOMe:>KyTOlJ-
H10IH cnoi1, I1!\1ClOmlfH pC3HCTI1BHblC CBOHCT-
Ba, HY:lKHble )J,JIS! TpaBJICHMji' nO)J,JIO:>KKI1 I1JII1
npyroro 113MCHeHI1S1 ee CBOHCTB.
2.2.1. 3mlTaKcHH
3nl1TaKCIDI - npouecc HapamHBaHMji' Ha
KpHCTaJlJ1H'ieCKOH nonnO:>KKC aTOMOB, yno-
pH.Il.O'leHHblX B MOHOKpI1CTaJJJII1'iCCKYIO
CTPYKTYPY, C TCM '1106bl cTpYKTYPa Hapa-
ll{IffiaeMOH ITJIeHKI1 nOJIHOCThlO nOBTOpS!J1a
IqmCTaJIJIl1'ieCKYIO Opl1CHTaUl11O nOnnO>KKl1.
ECJIn nOnnO:>KKa 11 HapaUU1BaeMWI nJICHKa
COCTORT 113 O,!XHoro Belll.CCTBa, TO npou ccc
Ha3bIBaIOT aBT03nMTaKCI18J1bHbIM, ccnl1 113
pa3JII1'1HhIX BemeCTB, TO reTep03nMTaKCI1-
aJIbHbIM. OCHOBHOC .n.OCTOI1HCTBO TCXHI1KI1
3rIHTaKCMI1 nOJIY'leHI1C QpC3Bbl'laMHO QI1C-
TbIX nneHOK npl1 COXpaHeHl111 B03MO>KHOCTI1
peryJIl1pOBaHIDI ypOBlUI JIernpOBaHI1R. JIerH-
PYlOmWl npl1MeCb MO>KeT 6bITb KaK n-, TaK 11
p-THna He3aBI1Cl1MO OT Tl1na nO)J,JIO>KKI1.
CxeMa YCTaHOBKH )J,JI}{ ra30BOH 3n11TaK-
CI1I1 ITOKa3aHa Ha pl1c. 2.12.
fa3006pa3HbIH BO,lI.OpO.n C npl1MeCblO
SiC1 4 KOHTpOJIHpyeMoH KOHueHTpau,n1 ITpO-
nYCKaeTCR qepe3 peal\."Top, B KOTOpOM Ha
rpa uToBoM OCHOBal1HH pacnOJ1o*eHbI
KpeMHl1eBble nnaCTI1HbI. HH.nYKUHOI1HbIM
HarpeBoM C ITOMOlll.bJO BbICOKOQaCTOTHbIX
KaryUJeK rpa uT ITporpeBaeTC}{ ,lI.0 BbICOKOi1
TeMneparypbI (> 1 000 0c). 3Ta TeMneparypa
He06xo,l\HMa 06ecne'leHu}{ npaBMJIbHOH
opHcHTaU Ull OCa:>K,l\aeMbIX aTOMOB B peIIIeT-
KC U nonY'IeHHR MOHOKpHCTaJIJIH'IeCKOH
nJIeHKU.
B OCHOBC npol.{ecca J1C)!illT peaKU HR
SiC1 4 + 2H2 = Si (TBep.n.bIH) + 4HCl.
BbIPAll{HBAHI1E H L\ErIOHHPOBAHME TOHKHX nJIEHOK
105
BLICOKO<IaCTOTHble KITI)'IllKH
j/ 1{:ur uarpCBa
.e'......
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SICI 4 +11 2 1:-
KpCMIIHCBbJe mtaCTI1HbI
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llo.L\JlmKKo.L\ep)KareJIb
PUC. 2.12. Cxcl\la YCTaUOBKU J1.J1H raJoBoii 3nUTaKCUU
JI.m! ITOJIY'lCHlU! 3ITI1TaKCI1aJIbHOro cnoSi
n-Tl1lIa I1CnOJIb3YIOTCX )KH.D:Kl1e (PCl, PBr3)
HJIll ra3006pa3Hble (PH 3 ) JIempY!Ollll1e Be-
ll{eCTBa, cO,l\Cp)KaIllI1C oc op HJIM )J,pyrnc
3J1eMeHTbl 3TOH rpYITITbI. CnoH p-nma nOJIY-
'la!OT B pe3YJlbTaTe JIernpOBaHIDI KpCMHI1S1
60pOM MJII1 cra coe)J,I1HeHI1S1MI1.
K ra30BOH 3ITI1TaKCHI1 OTHOCI1TCS! TIUOKC
KOH1{eHCal1I1H Ha n0)J,J10)KKe pa3pe)KeHHbIX
napoB BClllCCTBa. 3TOT CIToc06 Ha.3bIBaeTCR
ell{e BaKYYMHoH, MJII1 MOJIeKYJIHpHD-JIY'iC-
BOH, 3nmaRcl1eH. KpeMHllH I1cnapRCTcSI M3
)K11LJ,KOH I1JII1 TBep,l\oii a3bl H KOH,l\CHCl1py-
eTCSI Ha HarpeTY!O ,l\0 3a)J,aHHoH TCMnepary-
pbl MOHOKpI1CTaJIJII1'iCCKY!O TIOLJ,JIO)KKy. ATO-
Mbl KpeMHI1H BCJIe)J,CTBl1e BblCOKOH TeMnepa-
rypbI ,ll11 YHLJ,I1PYWT B ee nOBepXHOCTb B Te
MCCTa pellieTKI1, r,l\e MI1HI1MYM CB060)J,HOH
3HCprHI1. TaKI1M 06pa.30M 06pa.3YCTCR 3nl1-
TaKCl1aJIbHbli1 CJIoi1.
B npOMbIlliJICHHOCTI1 npMMCHRWT 0)),-
HOKaMcpHble 11 )J,BYXKaMepHble YCTaHOBKJ1
MOJICKYJH!PHO-JIYtJ:eBoi1 3nHTaKCI1I1. ITOCJ1C)),-
HIDI npOBO,[(HTCS! B CBepXBblCOKOM BaKYYMC
(10- 10 ...10- 11 TIa) 11 OCHOBaHa Ha B3aHMO-
)J,CHCTBMM HCCKOJIbKMX MOJIeKYJIRpHbIX nyq-
KOB C HarpeTOH MOHOI..-pI1CTaJlJIWIeCKOH nOA-
JIO)KKOH. 3TOT npOl1eCC I1J1JI!OCTpl1pyeTcSI Ha
pl1c. 2.13.
KIDK,llbIH HarpCBaTeJ1b C01{ep)KI1T TI1-
reJIb, SlBJISIWll{I1HCS! HCTO'iHHKOM o,ll,Horo H3
COCTaBHblX 3JIeMeHTOB nJIeHl<M.
TervUIcpaTypa Ka)KAOrO HarpCBaTeJIR
Bbl611paeTCH TaKHM 06Pa30M, 'iT06bl AaBJ1C-
Hl1e napoB I1CnapSieMbIx MaTepl1aJIOB 6bJJlO
LJ,OCTaTO'iHbIM ,D,IDI OpMHpoBaHIDI COOTBeT-
CTBY!Oll{HX MOJICKYIDIPHbIX ny'lKOB. Harpe-
BaTCJII1 pacnOJIaraWTCR TaK, lff06bI MaKCM-
MYMbI pacnpe,l\CJICHI1H MHTeHCI1BHOCTH. OT-
1{CJIbHbIX ny'-IKOB nepeCeKaJIllCb Ha nO,[(J10:>K-
loHoKpHcTannl ccKan
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PUC. 2.13. CHCTeMa HCTOlfUHlCoB-H8rpCB8TCJIeii J];JUI MOJIexYJIRpUO-JIY'leooii 3nnT91\:CHH
106
r.mma 2. TEXH0J10BiQECKHE OCHOBbl rIPOH3BO.llCTBA
Ke. flo,ll60pOM Ter.mepaT)'pbl HarpeBaTeneH J1
nOl!JIO:>KKll nOJIyqaHH nneHKIi co CJIOJKHb1M
XHMH"IeCKIiM COCTaBOM. JI,onOJIHl1TeJIbHOe
ynpaB.IleHJ1e npoueccoM BblpamJ1BaHuS! oCY-
lUeCTBmleTCR C nOMolUbro cneunanbHblX 3a-
CJIOHOK, p83Mell.{eHHbIX Me>K.IJ:Y HarpeBaTe-
neM H nOJ1.J1oJKlwii. lkIToJIb30BaHHe 31'1i
3aCJIOHOK n03BOJISIeT pe3Ko npepbIBaTb UJlH
B0306HOBJISTb ITOnan.aHl1e JIro60ro 113 MOJIC-
I\.")'JUlpHbIX n)"iKOB Ha nOllJlo "'-y.
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TcpH3yeTCH npOTeKaHl1eM Ha nOBepXHOCTIl
KpeMHlUI .!I.B)'X pcaKUltH: aHO.QHoro OKI1CJIC-
HHH. H KaTo oro BOCCTaHOBJIeHI1H OKHcnH-
TeJI}!. HaM60JIee 1llltpOKO HCITOJIb3YlOTCR
TpaBHTeJIH Ha oCHoBe CUCTeMbJ a30THOH
HN0 3 H tPTOpHCTO-BO,!J,OpO.!I.HOM HF KHCJIOT.
KUc.!IOTa HN0 3 BbmOJIHHeT tPYHKUillO OKHC-
"1HTe.jUl KpeMHIDI. a KHCJIOTa HF SlBJIHeTCR
KOMfi.'leKCoo6pa30BaTe.neM - paCTBopHTeneM
OKCH..!I.a.
npH XHMMQeCKQM MexaHH3Me TpaBJIe-
HIDI Ha nOBepXHOCTI1 "--peMHHH. ITpOTeKalOT
OlrnC.1J.HTeJIbHO-BOCCTaHOBHTeJIbHbJC peaK-
l!HH, CBH3aHHbJe C <JI1CTO XI1MUQeCKHM B3aH-
Mo.n.eiiCTBllCM MOJIeK)'JI TpaBHTCJISI C no-
BepXHOCTHbIMH aTOMaMH. CocraB TpaBHTe-
!leH, leaK npaBli.J10, ITo,!J,6I1paCTCSI onbITHbIM
nyreM. TpaBMTeJIM - 3TO CMeCll, COCTOSlI.UHe
Ji3 OKHCJ1MTeJ.l}!, KOMfi. 1 ICKCOo5pa30BaTeJISI
(06pa30BaHI1e paCTBopHMbIX coe}:\llHeHJiH C
OKCH1laMH). paCTBOpHTeJISl, YCKopHTeJISI HJ1H
3aMe)J.J'IJITeml peaKII:HH OKHCneHHH. H paCTBO-
peHIDJ OKCH.1a H Cnel.1liaJIbHbIX }:\o6aBOK,
OOecneQHBaJ01UHX, HanpMMep, H36HpaTCJIb-
HDe TpaBJIeHMe.
XnMJ1qecKoe TpaBJIeHHe nOJJ.jJa3,!1.CJISlIOT
ua H3OTpOnHOC, aHH30rp0nHoe M CeJJeKTHBHoe.
OJ
MacKa Si
6)
2.4.1. 1130rponHoe rpaB.lJeHlte
H30TpOITHOe TpaBJIeHHe RBnHeTCR Ok
HJiM 113 CaMbIX H3BeCTHbIX H pacnpocTpa-
HeHHbIX cnoc060B JIOKaJIbHOrO MI1KpOnpO-
IIJIHpOBaHIISI. XapaKTepHa51 oc06eHHOCTb
lI30TponHoro TpaBJ1CHHSI COCTOHT B OAl1Ha-
KOBOM B03.QeikTBHl1 BO Bcex KpllCTaJIJIOrpa-
l1'leCKltx HarrpaBJIeHJlSlX. .3TOT ITpOueCC
Ii3BeCTeH KaK nO)J,pC3aHlle (pl1c. 2.22 a, 6), a
TaK:>Ke KaK XI1MH<JeCKoe nOJIl1pOBaHl1e.
B Ka<JCCTBe 1130TpOnHbix TpaBI1TeJICM
HcnOJIb3YIOTCSI TpaBI-tTeJIH Ha OCHOBe ITJIaBI1-
KOBOH I1 a30THOH KI1CJIOT. MC.il.JICHHbIH Tpa-
BItTeJIb co}:\ep>KllT HNO] : HF : CH]COOH
(B COOTHOlilCHI1U 7 : 1 : 3) 11 npl1MCHSlCTCR
)LI1R nonyqeHHR MeJIKI1X penbe oB ITPH CKO-
pOCTH TpaBJICHIiR OKOJIO 0,1 MKM/MHH. EbI-
CTpbIH TpaBI1TeJIb I1MeeT TaKOH :>Ke COCTaB,
HO B COOTHOlliCHIfH 3 : 1 : 1 H C;1y:>KI1T ,l\JISI
ITOJIYQeHI1S1 my60Kltx peJIbe OB npH CKOpO-
cm - 4...8 MKM/MHH.
CKOpOCTb TpaBJICHI1R nOJIynpoBo.nHHKa
3aBHCHT OT 60JIblilOrO QHCJIa aKTOpoB:
- nma TpaBHTCJISI H cro TeMneparypbI;
- CKOpOCTH OTBo}:\a npo.QYKTOB peaKUI1H
H ITO,!J,BO,IJ.a peareHTOB K ITOBepXHoCTH nony-
npOBO}:\HHKa (T.e. CKOpOCTb nepeMelllHBaHI1R
TpaBHTeJISl);
- HaJIHQHH. I111H OTCYTCTBIUI )J,e eKTOB
KaK B caMOM nOJIynpoBo}:\HHKe, TaK H B 3a-
I.UHTHOH MaCKC, 06CCnCQI1BalOli\CH JIOK1UIb-
HOCTb TpaBJ1CHHSI;
- HcnapcHHSI TpaBI1TeJIH, npHBO,IJ.H.IlJ,Cro
K H3MeHeHHIO ero KOHl.1eHTpaUHH, H T.)J,.
EonbIlIHHCTBO 3THX 4JaKTOpOB nnoxo
nO.llJ(aeTCH KOHTpOJIIO, H B pe3YJIbTaTe MO:>KCT
D
2a
6j
Si0 2
(J>POJJT TpaBJ1eHIDI
Poco 2.22. Iborpo"uoe TpaBJleHHe:
a - c nepeMelUHBaHHeM TpaBHTeml; 6 - 6e3 era nepeMewHBaHHJ'I; (1 CTpYKTypa
c xapaKTepHblMI1 pa3MepaMH
TPABJIEHHE
])5
3a te,il:1IfTbCH CKOpOCTb TpaBnCHI1H, a B HeKO-
TOpbIX cnY'faHX (my60KHe 11 Y3KI1e KaHaJIbt)
0110 MO)KCT OClaHOBI1TbCSI. OlI.HaKO :nOT 3 -
llJeKT MO:>KHO J1l1KBI1)J,l1poBaTb nCpeMeWI1Ba-
Hl1e:\>! TpaBI1TCJ1H, lI.CJIaH. B CTPYKTYPax nO'fTH
T04HbiC H cKpymetUlble ITOBepXIIOCHI, )J,Ha-
MCTp KOTOpblX OITpe}.:\eJIHCTCSI no cpopMYJIe
(pl1c. 2.22, 6)
D = 2(a + VI),
r,Uc a - palJ.11Yc oTBepcTI-m B MacKe; v - CKO-
pOCTb TpaBJIelll1S1; t - BpCMH TpaBJTeHI1S1.
CITCLlH I1'feCKaJI cpopMa MllKpOITpO H-
.1H, a TalOKe CJTO>KHOCTb 06CCne'leHI1H JIO-
KaJIbHOH 3aJL(I1TbI OT )J,nHTCJIbHOrO B03)J,eikT-
BI1H. TpaBHTejlH He n03BO;IHIOT paCCMaTpH-
BaTb 1130TpOnHoe TpaBJIeHMe KaK nepcneK-
Tl1BHbIH cnoc06 MI1KpOnpO HJ1HpOBaHI1S1 ,lI.JlH
mrOTOBJIemUI npco6pa30BareJICH.
2.4.2. AHM30TponHoe Tp8BJIeHHe
Oc06eHHOCTblO aHH30TpOITHOrO TPaB-
JICHHH JlBJI5ICTCSI TO, 'iTO B pa3HblX KpI1CTaJI-
JIorpacpl1'feCKHX HaITpaBJIeHHSlX CKOpOCTb
YLIaJICHI1S1 aTor...!HbIX CJ10eB C nOBepXHOCTI1
TpaB:1eHHH, T.e. CKOpoCTb TpaBIJeHillI, H:\>tCeT
HeO)J,I1HaKOBbIe 3Ha<JeHHH. 3TO 06'bHCHHeTCSI
pa3JIl1'lHOH nJIOTHOCTblO ynaKOBKll aTOMOB B
pa3HblX ITnOCKOCTHX, a TaK>Ke pa3JIll'iHbIM
xapaKTepOM CBSl3H nOBepXHOCTHbIX aTOMOB
Me>KlI.Y C060H 11 C aTOMaMH, pacnOJIO>KeH-
HbIMH B 06'bCMe KpHCTaJJJIl-NeCKOH CTPYKTY-
pbI nonynpoBO)J,HHKa. CKOpoCTb TpaBJIeHIDI B
3aBI1CHMOCTH OT KpHCTaJInorpacpl1'ieCKOrO
HanpaBJIeHHJI COOTBeTCTBYCT CJIe}.:\YIOIL\CMY
pJl.ny:
Vt100] > V[I 10] > VPIO] > V[211] > V[221] > V[lll]'
)J.JI5I ITOJI}"IeHHH "43 MMKpo.naT'lHKOB
KpCMHHCBbIC 3aroToBKH B BH}.:\e nJIaCTMH 'ia-
lL\e Bcero opHeHTHpYJOT B llJIOCKOCTSIX (100),
(110), (111).
AHH30TponHbIc TpaBHTeJIH npe,l\CIaB-
JIJlIOT C060H MHorOKOMnOHeHTHbIe paCTBo-
pbl, COCTOSlllJ,l1e H3 OKHCJII1TCnJl KpCMHHSI }.:\O
rll}.:\paTHpOBaHHorO}.:\1l0KCH}.:\a,paCTBOpHTCJISI
ITOCJIe,l\HerO H 3aMe)J,JII1TeJISI HJ)H YCKopHTeJIJI
npOLlccca TpaB.leHillI. B COCTaBe TpaBI1TCneH
I1CnOJIb3YlOTCSI CI1CTeMbI: m.npa3l1H - BO)J,a;
rHLIpa3HH - 1130ITpOnHnOBbIH cnHpT - BOlI.a;
3TI1JICH.nHaMHH - nllpOKaTCXI1H - BO)J,a; C}.:\KOC
KaJIH - npOnHJIOBbIH cnMpT - BO)J,a; e)J,KI1H
HaTp BO)J,a; C}.:\KOC KaJIH - H30npOnHJIOBbli1
Cnl1pT - Bo}.:\a II )J,p. B :ITllX CHCTeMax 3TII-
JIeH)J,HaMHH, fl1)J,pa3I1H J1 C}.:\KOC KaJJH BbHIOJI-
HSiIOT po.1Ib OKI1CJlHTCJIJI: DHpOKaTCXHH, npo-
nUJTOBbIH H M30nponHJIOBbIH CITl1pT - KOM-
I1J]CKCHbIC arCHTbI. Bo}.:\a CJIY)!(}IT KaTaJIl13a-
TOpO:\>!.
ITPOLlCCC aHH30TponHoro TpaBJICHHH.
3aKJIIO'laCTCSI B n03TanHOM Y}.:\aJIeHl1J1 aTOM-
HblX CJIOeB (CJIOH 3a C.'IoeM) C ITOBepXHOCTI1
Kpm."TfLlna, T.e. B npOLlccce TpaBJICHJ1J1 Ha
nOBepXIIOCTH KpHCTaJIJIa 06pa3YiOTCH MHK-
pOCKOnl1'feCKlle cTyneHbKH. f103TOMY aHH-
3OTponHoe TpaBJICHHC He }.:\aCT 3CpKaJIbHbIX
nOBepXHOCTeH H ITOJIY'"leHHbIH 3neMCHT 06-
pa6aTbIBaiOT B Te'ieHHe 30 C B nomIpYlOlL\eM
pacTBopc H30TponHoro TpaBI1TeJIH, coc-
TOHIllero H3 CMeCH nnaBHKoBoH, YKCYCHOH 11
a30THOH KHC.'IOT, B3RTblX B nponoPQl1I1
1 : 1,2 : 6,2. 06pa6OTKB B nOJIl-IpYJOIllCM
pacTBope CfJIIDKHBaeT MHKpoHepoBHOCTll,
OCTalOlL\HeCH nOCJIe aHI13OTponHoro TpaBJIe-
HHH. B pe3YJIbIaTC nOBbIIllaeTCR TalOKe npe-
}.:\CJI npO'iHOCTH }.:\O pa3pyrneHHR q3 B 3-4
pa3a.
Ha CKOpOCTb TpaBJIeHHSI 3Ha'lMTeJIbHOe
BJIHJlHHC OI<a3bIBaIOT KOHl.(eHTpal.(HSI npl1Me-
ceii H J1X nm B KpeMHHI1. IT03TOMY npH
TpaBJIeHHH p- H n-KpeMHHJI B COCTaBbl aHM-
3OTpOITHbIX TpaBI1TCJIeH BBO}.:\H.T pa3JIH'iHbIC
np lca,l\KH. Pa3pa60TaHbl TaK:>KC pa3Ho06pa3-
HbIe caMOTopM03J1lL\l1eCR BH}.:\LI TpaBJICHIDI.
KOM6HHHpYR npc}.:\BapMTeJlbHblC }.:\Hcpct>y3HH
C aHH3OTpOnHbIMH H H3OTpOnHbIMJ1 TP8BH-
TCJIHMH, MO>KHO nOJIY'iaTb C 3a.n:aHHoM Pa3-
MepHoH TO'iHOCTbIO BeCbMa CJ10:>KHble 06'b-
eMHb1C MHKpo OpMLI q3.
CKOpOCTb TpaBJICHMSI lI.110Kc a KpeM-
HHSI BO Bcex npHMCHHCMblX TpaBHTCIDlx OKa-
3bIBaeTCSI 3Ha'iIITeJIbHO HH>Ke CKOpOCTH
TpaBJICHI1H KpeMHHH. II03ToMY npM JlOKaJIb-
HOM TpaBJIeHHH 3aIlll-ITa nOBcpxHOCTH
KpeMHHeBoH llJIaCTIIHLI OT TpaBJIeHMH MO:>KeT
116
f.'UlBa 2. TEXH0J10nt4ECKHE OCHOBbI rIPOH3BO)J,CTBA
a)
6)
46"3 l'
90 0
36°5'
\.
(100)
").......f :
- <.
....
3-f'
46<] l'
54()4'
72°27'
<
: ... ':.:;!:J,:;':<: . :(.
POCo 2.23. AHo30TponHoe TpaB.lJeHHe B n.ilOCKOCTIIX (100) II (110):
a - (I 00); 6 - (I 10)
OhTTb OCYllJ,CCTB!leHa C nOMOllJ,blO OKCll.QHOI1
nnemm. OKCHJIHbIC MaCKU nOKaJIJf3YIOT
npollecc TpaB.!leHJUI npoqnuII1poBaHHblx yr-
.ny6.:"leHHH. CKB03HblX OTBepCTliH II nB30B. Ha
pIiC. 2.23 nOKa3aHbI 1}>0pMbi yrny6JIeHHI1
[17] Ii COOT8eTCTBYlOll1Jfe IiM MaCKlf ITPI1
aHH30'IpODHOM TpalL'ieHHI1 Ha J1J10CKOCTSIX
(100) Ii (1l0).
TIpH B03,1eHCTBHH TpaBI1TeJISI Ha nJIOC-
ICOCTH (100) M opMeHT8llHI1 KpOMOK OKCH,ll-
HOH MacKH B HanpaBJIeHUIi [110) CKOpOCTH
Tpa8.J1eHIDI B HanpaBJIeHHH.X [100] II [111]
OTJ1H'laJOTC1I Ha ,lIBa nop.lUlKa M 60JIbllle. B
pe3ym.T8Te npoIIcxo.!U1T CMblKaHHe rpaHCH
(J 11) npH HC3Ha r une..'ibHOM no.nTpaBJIHBa-
HHM no.n MacK)' (pIiC. 2.24).
I1pI1 3TOM p83Mep .nHa LJ 06P83YlOllIei1-
CJI KaHaBKH onpe,1J.CJISleTcJI ypaBHCHHCM
LI = Lw - 2vlOo t( cosa - l/A)/sinB,
Lid
.
-; - . f . .;---- ,2.
." .' i
'.. . . ... i
. c... ... , ..
..'
. .... .'1;,
...
. . ...'. ."',,,,, '....
-. '.-i', . . ....
r)J,e L.. - pB3Mep llIHpHHbl KaHaBKM B MaCKC;
t - Bpe UI TpaDJIeHI1R; A = V lOoIv I II - CTenCHb
aHH30TponHH TpaBJICHIDI; V 100, VIII - CKO-
pOCTM TpaBJIeHMSI B HanpaBJIeHHSlX [1001 H
[111]; B - yroJI MeiKJIY nnOCKOCTHMI1 (100) H
(110).
CMbIKaHHe 60KOBbIX rpaHeH B03MO>KHO,
CCJIH A > lIcosB. BpCI\Ul, Hco6xo)J,HMoe ,l\JISI
CMbIKaHIDI, paBHO
tYp = L.,sinB/[2vlOo(cosB - lIA)].
rny6HHa TpaBJICHHSI B MOMCHT CMbIKa-
HHH onpC,llCJISIeTCR OPMYJIOH:
h 1 I' = LMsinB/[2(cosB lIA)].
TIpM BLICOKOH CTenCHM aHH30TpomIH
(VIOO » VIII) npJ.1BC,l\eHHaJI opMyna ynpo-
llIaeTCSI :
hYp = L.,tgB/2.
: .
I/, : .......
,.-" 1 .
: -: .- - :;?".:
.. . '. ." .
PJfc.2.24. ClJOpMLl4IHryp aHH:wrpODHoro TpllBJJellHH KpeMHHH HR nJIOCKOCTH (100)
TPAUJ1EHHE
117
He06xoAlIMbrii pa3Mep OKHa B .\laCKC
)J,JUI 'Ja..Qmmofi fJIy611HbI h,p onpe)J,eJ1S1eTCH.
3a1H1CHMOCThIO /,., = 1,41 h Tp ' 110.l.ITpaBJU1Ba-
HlIC nO,l MacKY onpC,leJUIeTCSI (!mpMYJTOH t.. =
= vwuf/(Asin&).
npl1BeIIeHHhle COO1'HOUleHI1S1 .\loryT
6h1Tb I1CnOJIb10BflHbi 11 npH paC'iere napa-
MeTpOB TpaBJIellll lIa ITJIOCKOCTH (110) ITpl1
nOAc I aHOBKC com Be rCTBYlOlllHX :ma'leHllii v
Ii &.
2.4.3. )]:pyrMe BIf,lJ,LI TpUBJIeHIIH
f1pu nlO6bIx Bl1,llax xm..m'ieCKJ1X TpaBI1-
TCJIcii 3 iPeKn1R1IOCTb ux B03 eik1'BlUI B03-
paCTacT npH nepc lelllHBaHuH pacTBopOB. B
XII UlKO-,I:\IfHUI\III'ICCKO:\t TpaBJTeHHH I1C-
nOJIb3YIOT MeruaJIKI1 JIOITaCTHOro nma H
YJIbTpa3BYKoBbIe Bl16paTopbl. J116I1paTeJIb-
1I0e y a.leHl1e cnoS! KpeMHHS! n )J,Pynrx Ma-
TepUaJIQB npl1 Ha.1H'iI1l1 3alllHnroH MaCKU
B03MO)l(HO C nOMOlllblO BaKYYMHO-nJl8J-
MeHHoro TpaBJIeHI1S1, KOTopoe 06eCne'iHBaeT
1136I1paTeJIbHOC'Tb, CKOpOCTb 11 aHU3OTpon-
HOC1'b TpaBJIeHII}l. CnOH KpCMHl1S1 Y)J,aJISle1'C}l
nYTCM XIIMI1'ieCKOrO B3alIMO.llciIcTBH}l HOHOB
11 pa)J,I1KaJIOB aKTUBHoro raJa C aTOMaMH
KpeMHI1S1 C 06pa30BaHHeM JICTY'lI1X Coe,l:\l1-
IrcIIHi1.
nJl33!\IOXII!\UI'IeCKOe TpaBJICHHC no,l:\-
pa3)J,enSleTC}l Ha nJl83MeHHoe H p8JUiKUJlb-
Hoe, B nepBOM cnY l f8e ITJIaCTI1Hbl KpeMHI1}l
HRXO,l\HTC}l HenOCpe,l:\CTBeHHO B nJIaJMC XH-
MI1<ICCKM aKTllBHbIX raJOB ['ieTbIpCXXJIOpli-
CTblM ymepo)J, CF 4 UJIH .llH TOP.llI1XJ]opMeraH
CF 4 Ch ( peoH-12)], a BO BTOpOM ITJIaCTIIHbI
HaXO,l:\H.TC}l B BaryYMHOH KaMepe, o1')J,eJIeH-
HOH OT ra30pa3pJl,l:\Hoi1 XHMH4eCKH aKTHBHOH
nJIa3Mbl ncpiPopupoBaHHbIMI1 MeTaJInH'ie-
CKI1MH 3KpaHaMH HJIH MarHHTHbIMIi H 3JIeK-
'Ip11'leCKIIMH ITOJI}lMH.
PropSl,l:\ B036 aeTCSI BbiCOKO'laCTOT-
HbIM reHcpaTopOM MOlllHOCTblO ,1:\0 300 BT H
"ropHT" BHe KaMepbl, B KOTOpOH HRXO,llHTCJ\
nJIaCTllHbI, )J,aBJIeHHe B KaMepe -0,01 lla.
CKOpOCTb TpaBJIeHI1}l -400 HM/MI1H. TIOCJIC
TpaBJIeHI1J1 npoMblBKH ITpaKTH'ieCKI1 He Tpe-
6yeTCJI.
3JleKTpo.1HTH1JeCKOe TpaBJTeHI1C pe-
KOMCH,I:\yCTC}l npHMeIDl1'b npl1 06pa60TKe
KpeMHHSI B nJIOCKOCTI1 (III). B Ka'ieCTBe
TpaBll1'eJICH HCnOJIb3YlOT pacTBopbI KMCJTOT II
llleJIOLfci1, HanpHMep ITJIaBlIKOBYIO KllCJIOry.
llleJIO% KOH.
IlpH 3JIeKTpOJIHHI'ieCKOM JIOKaJIbHOM
TpaBneHI1H BOCnpOI13Bo)J,HMOCTb OpMbl JlY-
HOK TpaBJICHI1}l onpe.lleJI}lCTCSI B OCHOBHOM
BOCITpOH3BO)J,I1MOC1'blO CKOpOCTII TpaBJIeHH}l,
KOTopaS! 3aBl1CHT KpoMC Bcex OTMe'laBUlI1XC}l
paHee aKTOpOB OT 3Ha4CHHSI 11 cnoc06a
nO,llBe)J,CHHSI HanpH)KCHlI}l MC)I(.L\y nOBcpxHO-
CTblO TpaBJ1emlSl 11 3JTeKTpOmITOM. Hanpll-
Iep. YBeJIH'ieHlIe fl.!lOTHOCTlI TOKa 06ccnc-
'iHBaCT, B 06m:CM, pOCT CKOpOCTM TpaBJIeHI1}l
H YJIyqrnaer Ka'ieCTBO nOBepXHOCTH. O,l:\HaKO
ITpl1 31'OM, KaK npaBHJIO, YXY,l:\lllaeTC}l paB-
HOMepHOCTb iPpoHTa TpaBJIeHlliI.
H3MeHeHHe aHO,l:\HOrO TOKa BJII1}lCT
TaIOKe Ha OTHOIlleHHe pa3MepOB 60KOBoro
nO)J,TpaBJII1BaHII}l ITO)J, KpaH 3am:I1THoi1 MaCKII
K r.rry6 IHC TpaBJICHM}l. llpu MaJIOH nJIOTHO-
CTH TOKa XapaKTep TpaBJTeHlliI npaKTl1'ieCKM
COBna,L\aer C H3OTponHbIM. B CJlY'iaC YBeJlH-
'ieHI1S1 TOKa TpaBJIeHl1e B my6HHY Ha'lUHaer
npeBaJIl1pOBaTb H8.L\ TpaBJIeHHeM no,l:\ t'.Jacry,
'lTO npOHCXOJtHT, no-BH,I:\HMOMY, H3-3a nepc-
paCITpe)J,eJIeHlUl aHO,l:\HOro TOKa MC:>K,L\Y Pa3-
JIH'iHbIMH Y'iaCTKaMH nOBepXHOCTI1 .rryHKll
TpaBJIeHIUL
ITpHMeHeHHe 3JIeKTpOllCKpOBoro
MIIKponpmJmJllfpOBaHIiH KpCMHlliI orpaHH-
'illBaeTCSI proMepoM HHCYpYMeHTa (::meK-
TpO)J,) H HaJIl1'iUeM Ha nOBepXHOCTH nnaCTl-IH
MHKpOTpelllJ1H. HaH60JIcc npHCMJICM KOM-
6HHHpOBaHHbIH cnoc06 06pa60TKI1: BHa'laJIe
3JICKTpOHCKpoBOH 06pa6OTKOH nonY'lalOT
npe)J,BapHTeJIbHbIe pa3Mepbl, a 3aTeM c no-
MOllJ,blO 3JIeKTp0J1HTH'iCCKOrO TpaBJ1CHHSI
HJIH nOJlHpylOm:HMH TpaBHTeJJ}lMH ,!J,OBO,l:\HT
p83MepbI ,1:\0 38,llaHHbIX 3Ha1JeHI1H.
TpaBJIeHUe nOJlUKpHCT8JlJlU'IeCKOrO
KpeMHHH. llpaKTH'iCCKH Bee TpaBHTeJIli,
B03.n.eHCTBYIOIlJ,Me Ha MOHOKpHCTaJIJ1M'ieCKMH
KpeMHHH, TpaBSIT H nOJIHKpHCTaJIJ1. O.lJ.HaKO
60JIee ITepCneKTI1BHbI Yp8BMTCJIlf, B KOTOpbIX
nJleHKa ,l:\110KCll,l\a KpeMHlU1 OK83bIB8eTCR
Ha,L\e:>KHOH 381lJ,HTOH IIPH JlOKaJIbHOM TpaBJJe-
118
rJIaBfi 2. TEXH0J10rll l IECKIlE OCHOBhl IlPOIl3BOJllTBA
mm nJ1aCTltHbl Ha 60JIbllIYlO rny6UHY. 113-
npHMCp nO'lTlt Ha BCIO TOJIlIUtHy nJIaCTIIHbl.
TIo cpaBHCHlt10 C 1i30TpOITIlblM TpaBJIC-
HUCM MOHOKpnCTaJIJIltlJCCKOrO KpeMHIUI
TpaB!leHHe nOJIltKpltCTaJJnlt'leCKOrO KpCMIUlSI
npHBOll.UT K HeCKOJIbKO xy,nwcii BOCnpOlI3-
BO.nHMOCTH 1}J0pMbl ynpyrnx 3JICMCHTOB
npe06pa.30BaTeJIeii 1t3-3a C)'UlCCTBCHIlO
60JIbWcii HCOJlHOpO,nHOCTIt nOJ1lIKpnCTaJ1JI3.
Ho B OTJIlitIHe OT U30TponHoro TpaBJICHItR
paCCt.1aTpMBaeMbIH npol\CCC xapalm plnYCT-
CH onpCJlCJIeHHOrr HanpaBJIeHHOcTblO. 80
Bee» CJIy"<UlX 601i:OBOe nOJlTpaBnlmaHne noli.
"paM MaCKI1 ITpOltCXOJ.l.ltT MeJlJICHHCC, 'leM
TpaBJIeHl1e B L1y6uHY.
3JIC"-JpoJIUTltl{eCKOe TpaBJIeHliC HapH)J,y
C H30TponHblM HB:1HCTCSI O.zUIl1M 1-13 C3MblX
pacnpoCTpaHeHHblx 11 xopowo OTpa60TaH-
HblX MCT0..'10B TpaB!1CHUH ITOJ1YITpOBO)J,HHIWB.
Mmcponpo4JHJ1HpoB8Hlle c8mlmpa
OCHOBaHO Ha TpaB::JeHuu B nOTOKe BO)J,opOJla
n MeTaHa npn TCMncparypc 1900 °C 11 JlaB-
.'1CHI1I1 1,3 na. J10KaJJbHOCTL TpaBJICHHSI
o6ecne..I1BaCTCSI MacKupoBaHlieM nOBepXHO-
CTH canI}JllpOBOH nnaCTltHbl C nOMOUlLlO
BO.lbcppaMOBOH n:1eHKU, BblpaUleHHoii In
napOI-a30BOH CMeCM. YKa3alfHoe TpaBJIeHMC
HOCHT aHlf3011'0nHblH xapatITep. OTHOllJeltl1C
CKOpocTH TpaBJICHI1H B rJly6l1HY K CKOpOCTM
6oKoBOro nompaIi.'1I1BaHmr paEHO 4. npH
TpaB!JCHJUi cancpHpa Ha my6MHY 100 MKM
60KOBOH pacrpaB < 25 MKM.
Tp8BJJeHHe 3aw.UTHbIX nOKpbITHit
J.\1uI 3aU1HTLI nOBcpxHOC111 nonynpoBo,nHM-
KOBOro KpeMHHJI HapSlJlY C OKCll.llHOH nnclt-
!COH SiO,2 HCnOJIb3YeTCH HMTpH)J, KpCMHMSI
Si 4, 8 laJOKC IUleHKH M3 aJUOMI1HJUl, 30no-
Ta, XpOM8 H Jlp.
)lnJl y CHHH OKCHnHOH nnCHKH npM-
MeWieTCH TaX H83blBaeMblH 6ycpepHblH Tpa-
BHTenb B COCTaBe: lJJJaBHKOB3J1 JCHCnOTa HF -
400/o-HWI. 4rrOpHCThlH aMMOHHH NH4F, BO,l\a
B COOTIiOWCHHH 2 : 7 : 1.
3aMeTHM. IJTO 3alUHTHLIe MeTaJIJJH..e-
CKHe nneHJOt pa3J1M"iHOIi TonlUHHbI MOtyl'
6bJTi> nonytfeHbl 3a HCCKonbKO .necHTKOB CC-
KYHJJ, MeT0J10M TCpMH'iCCKOro SaK)'YMHOro
HanbIJJeHHJI.
).lJUI p:nM plloru rpanJlClllIS1 MC [a;IJIII-
'ICCKIIX IIJI IIOK XOpOiIlO :Ji1p Kor\ICII)IOUaJ"1
cc6S1 u.eJIO'lJIbIC II KIICJlOTllhlC' TpaBIITCJIII,
2.4.4. KOlIl"pOJlb Im311tCplll,IX IIUpUIltC-a l)(In
"p" TIUlnJICllIUI
I-Ia p:136poc pa6n'II1X XapUI\fCpllCll1l<
npII60pa KpOM OGhl'lIIhlX TCXIIOJIOIII'ICCKIIX
)3KTOPOB nJllISl T P I)1 )10110:11111'1 C;lhllhIX,
nanpuMcp p:n6poc rCOMcTplltlCCKIIX p:nMC-
paB l.13 npco6p3'IOU3TCJISI. "IOT p:n6poc
onpc)J,eJI»CTCSI TO'lJIOCTblO lie npOllccca (110-
TOJllITOrpaqJllII, a IIpOIlCCCOB :IOK.1JJI.lloro
TpaBnclilul npu qJOpMllponalllHi In. P:npa-
60TIlHbi pa1JlII t lllblC CJlOC06bl KOIITpOJU! II
oocCnC'ICIJIHI BOCllpOIBBO}..\III\-1OCTII ){JlltllhI,
WllpJIIlbI 111111 JlI1aMCTpa l13. norpCIIIIIOCTb
BocnpOlllBOJUiMOCTII TOJIU{llIIbl MmJ(C"I L{OC-
Tllran, UeCIWJJhKIIX )ICCSI ("KOB IIPOl{CIlTOU.
OTCIO)J,a CTaliOBIITCJI Ot1C1U1)J,llOn l aJ!(IIOCTh
I<OHTpOJIR II BOenpOlI'UlO)J,llMOCTIt rOJIIHIIIII,I
,l.J,JISi YMCHbWCHIUI. P:I'IGpocn XapaKI."CpUCTlIK
npe06pa30BUTcnerr.
CJICJlYCT OTMCTHTh, o)J,HaKo, 'ITO n 0"1'-
nWUiC OT 06bl'1HblX I1I1TCrpaJIhllhlX CXCM n
ITpe06pa30BaTemiX nOBblWCliHC TO'IJIOCTI1
KOliTpOJIH TOJIUlIiIlh! II ec BocnpOU'UiO)J,I1MO-
CHi MCllbllJe BJIHSlCT lIa BbIXO){ ro,{lIbIX
CTpYK1)'p no J.(BYM npUtlllllaM: Bo-nepnblX,
113-13 Bccr,na CYIl\eCTBYJOUlCM ue06XO){lH,IO-
CTll llH)J,llBI1JlyaJlbUOH lIaCTpOHlm H q)UlIYI1-
pOBKI1 npe06pa30BilTCJIeH 11, BO-BTOphIX, 1-13-
3a Toro, 'lTO pa36paKOBKiI no TOJI1I{HIIC npo-
MCXO)J,HT lie no npllH1U1I1Y "ro)J,cu - Jlero-
JlCH", a nO npMHI{HITy "B KaKOH IUmcc ro-
)J,eH'?".
K IIUCTORllJ,eMY MOMeUTY pa' paGO'faIlO
HCCKOJlbKO cnoc060B KOIfTpOJIH oueCIIctlCIHlJ!
BocnpOH3BO.llMMOCTll pa3McpOB, Ha1l6oJIce
BaiKJtbIC 113 HHX:
- KOltTpOJ1b no BpeMCHI1 'I'paIUlCIII1JI;
- OnTH"ICCKllH clIoc06;
- KOifTpOJ1bHOe nOl TpaBnI1BaJII1C;
- HCIT0J1b30BaJIHC caMOTOpMO'JJ!IlJ,HXCJl
UM,l\OB 11'auneUHSI;
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MeH"ra.
TPABJ1E1JI1E
119
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np'IMhlM CIIl1C060M ICOIlI pom! TOnIHIIUhl LJ3
IIpco6pmOBaTcml HnmICTC)I KOIITpOJ1b IIC-
XU,Huoi! TOJlI1\11II1oI I\PCMIII[CIIOii nmlCTIIHh[ II
r:[y611I1LI rpUII:[CIIIUI 110 npCMClI1I Tpall:[CIIIHI.
<!>alCTOPUMII, onpCJ cmIlOIl IIMIl ro'lIIOCTb
'n-oro clloco6a, SIII1UIIOTCSI CJlC)JyIOIl HC: no-
rpCIIIHOCTb onpC;WJICIII 1)1 J\JIIITCJ1hIlOeTII
npmu.:cca 'I p3l111CIIII)I /).1, r.c, 1111'1 cpnaJ[ BpC-
MellU, n TC'ICIIIIC ICO'l'oporo [UlaCTIllIa nOMC-
UUICH::)I B TpUBIITCJIh II lI'UJJlCKaCTCJI 113 IIcro;
IIcnOCTomlcTHO C[COPOCfU TpaIlJ[CII 1II t!.v;
IICp1IBHOMCpllOCTb IICXO}\lIoii TOJlli\lIlIhl lIJ1a-
CTlUlhl ,'J.J-/. ECJIII TOJIUUllla IICXO}\1I0ii fma-
cnlllLI palma J J, )KCJJaCMa)l TOJILl\I111H ll3 11,
TO norpClIlHOCTL nOJIY'lCIIIHI TOJIII.UHIbi l/3
Ah npll PUCCM(lTp[maCMOM cnoc06c KOIlTpO-
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H -h
Nl = , v -I- vl:-.I -I- MJ .
v
OnTlI'lCCKllii Clloco6 KOIlTpOJUI. vl3-
ACCTIIO, 'ITO Y KpCMIII UI Iw')tIJCIIIIl\IICIIT no-
rJIOIqCHlUI onTII'ICCKOr() IIJ:lY'IC!H[H. cpaBHII-
TCJIhllO nJlallilO YBCJJlI'UmaCTC)I npll B03pac-
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- 8
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6
5
4
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1>lIc. 2.25. CXCI\III YCTIIIIOBKH KOIl'rpoJUI
1I1100.CCCII 'f1)IIIIJICIIIUI OIfTII'ICCKIII\I clloco6ol\l:
J nC'J'O'IIIIII( Cllcn); 2 - lIurpcnU'I'CJ\b;
3... 3Cpl<UJIO; 4 COCY) C nOl orpC'roi\ IJ01\OH;
5 - CM[(OCTIo C 'lpUUI-ITCJICM; 6 - np03pI1<lI[(\JI
IlJIaC'nIlIll; 7 - l(pCMHI1CnUJI nmlCl1l11n;
II - CTCI(JULHIIU!! I(PbIllJlffi
npOCBCT 6YJIC'I W1MClUJThCJI IIpll lI'lMCIICIII1II
TOJJlI\Hllhl IImlCTllllhl. OKpacKa I\1CIIJJCTCJI C
TCMIIO-l\paCHOli IIpl1 TOJIIl\llllC ["!opJJ}.(Ka
22 MKM !HI Opalli!(CBO-KpacIlYlo npl1 TOJIIWI-
IIC -10 MK:'v1 II MCHhllJC, HpaKTH'lcCKIl OKa-
3bIBaCTC)I B03Mo)KIII>IM C J.lOcTaTO'HIOH BOC-
ITpOll3nOllHMOCThlO KOIITpo:mpomrrL TOJIIUII-
IIY KpcMIIIlcnoii nJiaCTIIIII>I C 1I0rpeUJlloCTbi0
2,5 MKM. Ha ')TOM clJuikruc II OcHOlJaH OIT-
TH'ICCKUii MCTO,/.( KOIITpOJISJ TOJIli\IIHbl I\pCM-
IIHCOhlX ITJlaCTl1II 11 IIPOI\CCCC IIX TpaBJIeHI1S1.
CXCfo,W YCTHIIOBKII )J,JISI KOilTpom1 Ton-
U\ IIIh1 nJJaCTHIILI rrp IBC)J,Clla lIa p Jc. 2.25.
I-Ia6mo)J,aTcJI" '(Cpe:l CTCKJI'IHHYIO
I<pblLUK)' MOII(CT CJ!e)J,IITI, 3a H'3MeIlCIiHeM
oKpacKIl OTpa)KellllOro nY<II\a CBCTa B npo-
l\eCce TpaBJ1CHIIJI.
JJ,ocTOImcTBoM 3TOro cIToc06a JlBm1CT-
CSI cpaUlIIITem,Ho BbICOKaSI 'fO'IIIOCTh He3aBII-
CI1:'v!O 01' rny6mlbl TpaBJIeIlW1, lIe,/.(OCTaTI<OM -
ero ofpallwlclIHOCTh: TOJ!UUlHa KpCMH ICBbIX
l13 ::;; 25 MKM,
I(m....pOJlbliOe nO,!J,TpaBJIIIBlUlile.
3TOT cnoc06 3aKJIIO'I3L'TCfl B TOM, <no no
nepHMeTpy KpllCTaJlJIa C JIIIl cr!OH CTOpoHbl,
Ha KOTOpOJ:1 I13roTOBJ1J110TCSI aKTHBHble (npc-
06pa30B1neJIb11LIC) 3JleMeliTbl, npO 13BO.nH.T
KOHTpoJIbllOe nO.lI.TpaBJIIIBflIlHe B BI1,llC KOJ!b-
IlCBOH l\allal3l\l1 lIa rJly6wlY, paBIIYIO )KeJIac-
Moil TOJlli\I1He 3nCMCIrra npc06pfl30BnTenSi
(pHC. 2.26, 0). B ITp0I.l(..'Cce BbInOJIHelHl'I rny-
60KOro 'fpaBJ1CHWI B 30ne ITpe,l\BapHTCJ1bHO-
fO TpaBJIClUlSI 06pa3ycTc'I CKB03Hoe OKHO
(pHC. 2.26. 6) H npol1cXOjJJIT OTJ],eJ1eHlle 'IaC-
nU\bl l(pIlCTlliIna. '[TO JJBJ1HeTC'I CHrHaJ10M 0
npeKpaU CI1HI1 ,QaJ1bIlCHWCro TpaBJIeHI1S1.
K IIC,QOCTIlTKflM paCCMOTpclIHOro cno-
c06a O1'HOCSlTCJl Cne.lI.YIOllIHC. nOCKOJJbK)'
!jlopMl1pOBallHC 3JICMeHTa H OT,QCJ1CHliC KPH-
CTaJ1JIa 01' nJIflCTI1Hbl - 3aKJUO'lHTeJ1bHRJI
TCXHOJIOfH'ICCKflSl oncpflQliJ1, Heo6xo,QHMO
IiCnonb30BaTb cneQllaJ1bHbIC BH,l\bl MCTaJIJ1H-
3al.lHH AJJSI TOKOBe,QYIlJ,HX ,QOpO)KCK II KOH
TflI<THbIX nnOIlJ,8jJ,OK, ne CTpaBJIHBaeMbIX B
ITpOl\ecce TpaBJIeHHJI KpeMHHSI. ,L!PyroH He-
,QOCTaTOK - Tpe6yeTcSI CneL\HaJIbHOe 060-
PYAOBaHlle ,l\J1S1 aBTOMaTH'leCI(Qro y.n.aJ1eHHJI
120
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a, 6 - nOCJJelIOBaTe.'lbHOCTb 1paBJ1eHHH; J - npe.1l.BapllTeJJblWe 1paBneHlle;
2 - BCTpeQHOe 1paBJ1eHlle
OT.:{eJIeHHbIX OT lli18CTliHbl k-pI1CTaJIJIOB M3
TpaBMTe.;UI.
CaMoTopMo3Ru.meCH Blmb! TpaBJIe-
DU. B HaCTOHlltee BpeMH M3BeCTeH pH):{ Tpa-
BMTCJICii, CKOpOCTb TpaB.J1eHIDI KpeMHHH. B
KOTOpbIX onpe.n.eillleTcH ITpl1 npoql1x paBHbIx
YC.J10BHKX THTIOM npOBO.!lHMOCTII nonynpo-
BO;unuca 1L1H KOHLlcHTpaUHeii npllcYTCT-
B)'1OIIUfX B HeM ITpI1MCceH. O.LlHI1M M3 Hal1-
6o.:Iee umpOKO H3BeCTHblX 3.TJeKrpOJIHTJ1qe-
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paCTBOp IL1.aBHKOBOH ICHCJIOTbl. TIpH 113Y'le-
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no cym.ecrsy, onpeJ1eilllThCR p836poCOM
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eT, O)UlaKO. OTMeTHTb. ':ITO "PH Ji3rOTOBJJ(}-
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pOMCXaHHqCCKHC ycrpOHCTBa H3 o6"LeMHoH
qaCTH OCHOBaHllSi. B nOCJIe.n.Hlle rOl.{bI P83-
pa60TaHO HCCKOJlbKO BapHaHTOB 3Toro npo-
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PYeTCH TOHKOruJeHOQHbIH MaTepHaJI, KOTO-
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a - OKCIIJJ,lIhIH cnoH, BblpawellllblH H3 CKOnl!pOBaIlHoH Ha nOBepXHOCTb
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f!ly6ulle: (3 - BTOpoii BblpaweHllblll H CKOITllpOBaHHblH OKCI1.!!lIbIH cnOH: 2 - aHJJ30TpOnilOe
TpBBJ1elllle C Hcnonb30BallJIeM KOH
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TpaBHTcSI. TpaBJIeHl1C ero MO:>KCT 6blTb BbI-
nOJIHCHO :>KlInKl!M rum CYXIIM XI1Ml1'lecKHM
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TpaBJIeHI1S1 nOKa3aH Ha pl'lc. 2.27, a ):\BYCTO-
pOHHero - Ha pl1c. 2.28. npo ccc aH1130-
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nO,Qpa3,Q. 2.4.2.
rnaBHaH. oc06eHHocTb aHi130TponHoro
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npSlMoyroJIbHYIO OPMY C 60KOBbIMI1 CTeHa-
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113 nonHKpHCTaJIJln'leCKOrO KpCMHMSI Poly-Si,
HO B03MO:>K110 HCnOJlb30BaHMC OC opKBap-
[{CBOro CTCKJ1a, HHKeml, HI-!TpH,l\a KpeMHHSI
ShN 4 .
B Ta6n. 2.10 npl1Be)J,eHbI OCHOBHLIC
:nanbl H'3rOTOBJ1eHHSI 3J1eMCHTa THna MeM-
6paHbI H3 nOJIHKpItCTaJIJ]H'lCCKoro KpeMHH}I.
H31 OTOBJlEHI1E MHKPOCTPYKTYP
125
2.9. Tcxno.!JoHIH Il3fOTOB.!JeIIllH 1.13 MA
T eXHOJlOnl<JeCKliH npouec!.: I1JIJIlocrpaLlHSI npoLlecca
Xlnm'leCKaSl 06pa6oTKa MOHOKpCMIIHH (iU18- I t- Si
MeTp nnaCTl1l1loI 76 MM, TOJIIl\HHa 380 MKM,
K)J,E-O,O I < 100»
,lJ,BYCTopmlHec lIanlolJIelillC Cr ,'1"lH lIJOpMHpO- I t> e,
BamU! MaCKH LIllS! TpaBJICHl1S1
,lJ,BYCTOPOHHSlSI oTomnorpaqJMSI 11 npe)J,Ba- 6 ' <1>OTO-
p !TCJIbHOe TpaBJIeHHe lJ,l1H YTOHeHIISI "H3bl<J-
Ka" pC311CT
CHHTHe QJOTope311CTa 11 Cr C : I
,lJ,BYCTOpOHHee HanblJleHHC Cr .lIJI OpM lpO- I : : o
BaHI1S1 MaCKI1 )J,JISI TpaBRCHHSI Sl3bI'lKa CCHeopa
,lJ,BYCTOPOHHSISI OTOJIHTOrpa l1S1 H TpaBJIe- ;1> <J>OTO-
HHe Sl3bI<JKa pe3HcT
CHSlTl1e QJOTOpC311eTa 11 Cr D< ;:( I
HanblJlCHI1C Ti-Ni 'lCPC3 MaCK)' )J.JISI QJOPMH- D< ;:< =t>Ti-Ni
pOBaHI1S1 ITpOBO Sllllcro CJIOS! 11 KOHTaKTHWX
nJIow.alJ,OK
JIY)I(eHI1C nnow.a OK Ti-Ni C < ;< J>n-
XI1MI1<JCCKaH. o6pa6oTKa nO.l\llO:>KCK Si (CTeK- I I
JIa)
,lJ,BYCTopOHHee HanbIJICHl1e Ti-Ni ,llJI op- I: .1> Ti-Ni
Ml1pOBaHI1S1 Hl1)1(HeH 11 BepXHCH 06KJI8.n:OK
KOH)J,eHCaTOpOB 11 KOHTaKTHblX nnow.a.ll0K
I , : ,., I
)!BYCTOPOHIDISI OTOJII1TOrpa l1S1 no MCTaJI-
JlH3al.{1111 '--' ,
JIY:>KCHHC KOHTaKTHblX nJIOll.{a.llOI< C : T n p l1II01\
,
Pe3Ka nnaCTHH Ha KpHCTaJlJIbI -
C60pKa ceHcopa MeTO.llOM naHKI1 TpCX :me- IlanKa I :
Mcmos
126
rJlaaa 2. TEXHOJIOrI14ECKl1E OCHOBbl rIPOH3BO,lKTBA
2.10. TeXOOJIOTlIH HJrOTOB.T:leUUH 1:13 I\IIIKpOD:3T<JUK3 llaBJIeHlIH
TeXHO.'10fHQecKHii npouecc
OC8JKD.eHHe Ha rr..-:taCTlfHe KpeMHHSI :>KepT-
BeHHoro CJIO (SiO:!) Ii HaHeCCHlfe OTO-
pe3llCTa
<1>OTOJllfTOI-pa HH (n03I1TllBHbIH OTope-
3HCT); .pOpMlfpOBaHlle Komypa BHyrpeH-
HeH rrO;lOCTH MeM6paHhI
OCa)K!(CHlle ITOJIlfl\-plfCTaJIJIlf'leCKOrO
I\peMHIDI H HaHeceHlie OTOpe3IfCTa
<1>oTOJIHTOrpa4JIDI H OpMllpOBaHHe
BHeIllHero KOHT)'Pa Me 6paHbi
T paBJIeHHe :HCepTBeHHoro CJIOR H "OCBO-
6ojf\,D,eHHe" MeMOpaHbI
B KaqecTBe :iKepTBeHHoro CJIOH 3,!IeCb
HCnO..'IIo3YeTCJI SiO:!. OCHOBHblC TCXHOJIOfH-
qecKHe npoueccbI - OTOJIMToI-pa lfH 11
Tp qeHHe nOJIMl\.-pHCTaJIJIH'ieCKOrO KpeM-
HIDI. llO':Uuq)lfCTa.jL1M'ieCKHH KpeMHHH HlIH
l}>occpOpKB8pIICBOC CTeKJIO HaHOCMTCH nJIa3-
MO.ilHTM'iCCKHM H.rlll XHMJ1qeCKllM OCIDK.1(e-
HHeM H3 napoBou a3hI npH nOH CHHOM
,!laBJJeHHH. CTpYKTYPbI, H3rOTOBJJCHHbie C
nOMoIIIbJO nOBepXHOCTHOH MMKpo06pa6oT-
KH. HMelOT pa3MephI OT HecKOJIbKHX .n.eC5/T-
ICOB HJJ1{ COTeH MHKpOH B MllHY I1 OT ,llOneH
.no HCCKOJIbKHX MHKpOH B TOnlI.umy.
ooJIee nO.npOOHa5I TeXHOJIOrHSl nOBepx-
HOCTHOH MMKpOO6pa6oTKH Ha npHMepe M3ro-
TOBJIeHIDI q3 MHKpOrMpocKorra no pHC. 2.31,
KOHcrpyK11IDJ KOTOporo onHCaHa B n. 1.3.2
(CM. pHC. 1.36), HJIJIJOCT'PHpyeTCH pHC. 2.32.
BHa'Ia1le (PHC. 2.32, a) Ha nOMOiKKe 1
H3 MOHOKpeMHIDI .popMHPYeTCH IIJIeHKa 2 1f3
HHTpH)J,a ICpeMHIDJ TOJJII.UfHOH 1 MKM nyrcM
IlJIa3MeHHOrO HJIH TeMnepa-rypHoro XHMH'ie-
CKoro oca?JO(eHIDI H3 napoBoH (ra.30BOH)
«)a3Jd. llo.n IIJIeHKOH C nOMOlllI>1O HOHHOU
HJ1JIlQcrpaUIUJ npouecca
Si0 2
":,
POly-SI
. . ..,.- Poly-Si
IY-SI
3
2
J
A
Puc. 2.31. MHKpocrPYKTYpa 1J3 Mr:
J - MeCTa KpenneHHJI aHKepoB ynpyrnx
3neMeHTOB H HenOlIBHn<HblX 3neKTpo,noB
rpe6eH'laTb1X CT)JYKTYP; 2, 4 - 3J1eMeHTbl,
oCBo6<»KJ],aeMble OT nO,lJ,JIOJKKH; 3 - 3J1eMeHTbl
H8 nOAJIOJKJ<e (A-A - nHHHfi ce'!eHHJI)
HMnnaHTall.Hll HJ1H TeMncpaTypHoM .nu Y-
3MM BblIIOJIIDIlOTCJI MeTaJJJIH'ieCKHC JJ.l1 Y-
3HOHHhIe nnOlll.a)J,KM 4 Jl)JSI nOCJIcJJ.YlOllicro
KpenneHIDI Ha HUX aHKcpoB H HenO)J,BH:>KHbiX
IHr<nOBJIEHI1E MMKPOCTPYKTYP
127
(1)
4
3
2
1
5
3
eJ)
e)
4
6
3
" '."-."" ....... ccc: :E'
- -:-
-".' --.. 2
: tJ]fl fl ;I I j;]1 1
i')
6
........__... 0"" ".0'" p" _.... .........
-" .!III -':-_
. ' ,.. .
'4_;
6
,............"...... ........-....--...."'.......J'.
> .
- ' '. ' 2
_ 1 1 t:.;i .: :.!.;; ! I:i ;!I! i;f: 1
PIIC. 2.32. IhrOTOBJ1eHlIe IlfKpoC"rpYKTYPbI q3 MrMeTO OM
nOBepulOcTHOH MIIKpoo6pa6oTl,":
J - no.aJIOiKKa (:\IOHOKpeMIIHi1); 2 - rmeHKa HHTpH.lla KpeMHH.!I; 3 - nJIeHKa .llHOKCJI.Ua KpeMHIUl;
4 - MeTaJIJIH'IeCKaH, .llHCPCPy.3HOliliaH rIJIO a.IIKa; 5 - OKHa; 6 - nOnJlKpeMHHeBaH MHKpocTpyrrrypa
(ce<leIlHe no JIHHHH A-A 118 pHC. 2.31)
:meKTpOJ1.0B rpe6eH'-IaTbIX CTpYKryp npllBO,ila
H H3MCpHTCJICH ncpcMemCHHH. 3TOT :nan
3aKaH'-IHBaeTCSI HaHeCCHHeM nJICHKH 3 )J,BY-
OKCHl1a KpeMHHH TOJUUHHOH -1 MKM ITocpe,il-
CTBOM TeMncpaTypHoro XMMH'lCCKOrO oca:>K-
)J,eHHSI H3 ITapOBOH (ra30BoH) ctJa3bl.
CJIC)J,YlQlllHH :nan MHKpo06pa60TKH
(pHC. 2.32, 6) 3aKJIIQ'laeTCSI B 06p830BaHllH
OKOH 5 B nneHKax HHTpH)J,a I1 )J,110KCI1,L\a
KpCMHHSI HanpOTHB MCCT KPClllleHHSI aHKC-
pOB 11 HCITOJ1.BI1JKHbIX 3J1CKrpO)J,OB rpe6eHlIa-
TbIX CTp}'KTYP. BCKpblmc OKOH npOHCXO)J,l1T C
nOMOllJ,blQ pcaKTHBHoro HOHHoro TpaBJICHHH.
(cyXOC TpaBJIeHHC) 11 pe311CTHBHOH MaCKH.
)J,anee (pl1c. 2.32, 6) Ha nrrcHKY 311HOK-
Cl1)J,a KpeMHMSI H B OKHa 5 OC8JK,lJ.aeTC}! ITOJIH-
KpCMHl1eBaJI nJIeHKa 6 TOJIIlJ,I1HOH -2 MKM
nYTeM TeMnepaTypHoro XI1Mll'-1eCKoro oca)f(-
,lleHI1S1 113 napoBOH (ra30BOH) QJa3bI. TIpH He-
nOJIb30BaHI1H BaKYYMa BMCCTO nOJIHKpeMHlUI
MO:>KeT 6bITb npHMeHeH aMOPQJHbIH KpeM-
HHH.
CJIe,l\YIOIlJ,HH 3Tan (PHC. 2.32,2) 38KJ11O-
<JaercSi B OpMHpOBaHI1H ynpymx 3neMeH-
TOB nOJ1.Bcca, 11M C nO)J,BH:>KHbIMl1 3ncKTpo-
)J,aMM M HCITO,L\BH:>KHbIX 3JICKTpO)J,OB rpc6eH-
'-IaTbIX crpYKTYP npHBo)J,oB I13MepMTeJ1eH
nCpCMCIlJ,CHHH C nOMOllJ,b1Q cyxoro TpaBJJe-
HHH nOJIHKpCMHl1eBOH lllleHKU 6 npM HC-
ITOJIb30BaHMl1 pC3HcTMBH0i1 MaCKH, 3a)J,alQ-
IltCH KOH HrypaJ..{HIQ (eM. pHC. 2.31) Beex
3JICMCHTOB MI1KPOCTpYKTYpbI.
Ha nOCJIe)J,HeM nane (pl1c. 2.32, 0) ITpl1
ITOrpYJKCHHH BO QJTOPHCTO-BO)J,OpO,l\HbIH OK-
CH)], HJIH J],pyrOH TpaBI1TeJ1b BbITpaBJIHBaCTCSI
Beg nnCHKa 3 (:>KCpTBCHHbIl1 cnoH) H MI1KPO-
CTpYKTYpa OKa3bIBaeTCSl oCB060JK):\eHHOH Ha
paccToHHHH 1...2 MKM Ha)J, nOBcpXHOCTblQ
IlJICHKH 2.
)J,OCTOllHCTBaMH TeXHOJIOrnH nOBepx-
HOCTHOH MHKp006pa60TKH HBJ1 TCn OnHO-
CTOpOHHH5I 06pa6OTxa H XOpOlllWi COBMec-
THMOCTb C TCXHOJIorneH I1HTcrpanbHblX
MHKpocxeM. HC)J,oCTaToK - .zrOCTaTO<JHO
CJIO)KHbIH npo ecc 06p83oBaHHH He06xo)J,1l-
MbIX MHI<pOCTpYKTYp.
LIGA-TexHoJlorHH p83pa6OTaHa B
rCpMaHUH H npe,l\H83Ha<JeHa JtJIH I13rOTOBJIe-
128
rnaBa 2. TEXHOJIOn1 ECKI1E OCHOBbl rIPOH3BO.LI.lTBA
2.11. LIGA-TexHononIH nO,lly..eHHH nnaCTJIKOBOU ]\IIIKPOCTP)'KTJPbl
T eXHOJIOrMqeCKHU npouecc
PeHTreHOJIHTorpa<l>uSl B nOJIJ!-
MerHnMeTaKp JaTe
TIpOHB.!"1eHMe
HnmOCTpaUHH npouecca
j j j ! j j
MC:\16paH3 'IaCKIf
OI!
3.IC'ICHTh'
ar'Cop&.'I'"
<1>olOpcmc'l
DO;J.Hon;:Ka ([I)
[]]]
CTpYlnypa In
!)JoTope3lfCTa
raJIbBaHO OpMOBKa (3Jle)..-rpo-
OCIDK.:1eHHe)
MCTa.1JJ1 (NI, CU, Au)
IIo.JY'ieHHe maonOHa
LUa6noH-BCTanKa
3amap !)JapMbl
3a.JIHBKa ma6JJOHa (<l>opMOBKa)
Y.Ll8J1eHMe <l>OpMOBKH (KOHeI.J-
HblH npoJIYlIT)
HID! HCnOllHHTeJ1bHbIX KOMnOHeHTOB MHKpO-
npH60poB. TeXHOJIOJ"IDI BKJI)O<JaeT B ce6H
nponecchl peH'ITeHOJIHTorpa4JMH H ra.1IbBa-
HBKH. B pe3YJIbTaTe I<OTOpb1X MOryr 6b!Th
nOJJ}"leHLJ UJlacTIlI<OBb1e HJIH MeTaJIJ]HI.Je-
CIOfe crpylC1ypLl.
OCHOBHJ.Ie 3TaIIbJ LIGA-TeXHonorMH
Ha npHMepe nonyqeHIDI f11laCDIKOBbIX MMK-
pocrpyICI}'p DpHBe)!eHLI B Ta6n. 2.11.
XaparrepHLle pa3Mepbl crpYKTYP, H3-
roTOWIeHJll,le no LlGA-TeXHOJIorHH, - COT-
HH MKKpOH B ,rvUIlfY H .u.ecUKH MHKpOH B
TOJUIlHHy. .l{OCToHHCTBoM TeXHOJlOnfH 1IBmI-
erejJ B03MOiJCHOCTb nOJJ}"leHIDI MeTIL'lJIH'ie-
CKHX CIpyKTyp a He.u.OCTaTKaMH - .D;OBOJJbHO
6Qm.mHe pa3Mepl>l If DJIOxa1I COBMecTMMOCTb
C1JopMoIIKa
3aTBop !jJOTowa6J10Ha
nw
I
CrpYlnypa
1f3 nJJaCTHKa
c npOLtCCCaMI1 MHTCrpaJIb110H TeXHOJIOmH.
HaXOJUlT npHMeHcHHe H )J,pymc TeXHO-
JIOrHH opMo06pa30BaHl1JL
SIGA-TexHonomH (a66peBHaTypa 03-
HaI.JaeT - YJlbTpa<l>HOJleTOBlliI. JlHTorpa HSI,
raJIbBaHHKa H opMOBKa) n03BOJIJleT ynpaB-
mITb WHpHHOH npO HJlSl CTPYKTypbl 11 co-
BMCCTHMa c reXHOJIOrHeH TOHKHX ITJlCHOK.
TeXHOJIOrHH BKJI)o'iaeT B ce6S1 cne.nYIOIl\He
OCHOBHb1C npOLteCCbI: YJlbTpa HOJIeTOBYIO
JlHTorpa H)o, oca ,llCHl1C cnOCB H nJIa3MeH-
HOC TpaBneHHC, raJIbBaHHKy, nOJIyqeHHe
KonHH H3 nOJlHMcpOB HJlH MeT8J1JIa.
HART (high aspect ratio technologics)-
TeXHOJIOfMH C BblCOKl1M aCnCKTHbIM COOTHO-
weHHeM JJ.alOT B03MOn<HOCTb OpMHpOBaTb
H3iOTOBJ1EIIY!Ec MI1KPOCTPYKTYP
129
a)
I
2 6)
2 6)
3
PIIC. 2.33. JhroTOBJ1eHlle nonnecHoH CTp 'h"TYpbl B GaAs:
a 110lifHlH HMn.laHTaUI1S!; 6 - KOlIl!pOBaHI1e wa6.10l1a U3 Si J N 4 lIa nOBepXIIOCTb n.'1aCTl1IibI:
(j ceJJeK'1 mmoe Y,IJ,aJJeHHe HOIiHoro CJJOH; I - HMnnaHTnpoHaJlllaS! MaCKa; 2 - SiJN-I; 3 - GaAs
CTpYKTypbI C 60.:IbIlIH:\1 OTHOllIeHHeM BbICO-
TbI K 1l1HpHHe (Y3Kl1C 11 my60KHc KaHaBKI1).
B OCHORe Texllo."lOrI1ll ne:>KaT npoueccbI rny-
60KOro pCaKTI1BHO-HOHHoro TpaBneHI15I Ha
Cnel{l1aJIbHbiX YCTaHoBKax, KOTopbIe no3BOJUI-
lOT npORO,il.HTb TpaBJJCIII1C KpCMHI15I lIa rJIy6H-
HY AO HCCKOnbKUX COTeH MI1KpOH npI1 XO-
pOllieM npH6,1l1:>KeHHI1 CTCHOK K RepnlKa,lH.
MUMPS-npou.ccc TpeXCJIOHHOH nonu-
KPHCTaJJJIHYCCKOH nOBepXHOCTHOH MHKpO-
o6pa6oTKI1.
Ml1KpOCTepeOJIllTorpa(lmH - TeXHOno-
rI1S1 OpMI1pOBaHIUI H306pa)KeHIlli B cnoe O-
TOpC3HCTa, peanH3yeMWI B cneJ],YIOIl.{CH O'IC-
pCJ],HOCTH: Ha llJIaCTHHY nOcne)J,OBaTeJ1bHO
HaHOCSlTCSI cnOI1 cpOTope311CTa TOnllJ,HHOH
1. ..5 MKM H B Ka)K,!:\OM CJIoe cpOpMI1PYCTCH. CBoe
I1306pa)KcHMe. nocnc HaHCCCHHH. BCex CJIOCB
(MHTeJIbHOCTb 3aCBeTKI1 KIDK,lJ.OrO CJIOR 1 c),
KOTOpbIX MO:>KCT 6blTb 60nee 1000, cpOTOpC-
3I1CT npOHBJISICTCH.
nonyyeHHWI CTpYKTypa 113 ITOnI1Mepa
MO:>KCT J.1CnOJIb30BaTbCSI caMOCTOSlTeJ1bHO,
ecnI1 OHa Y)J,OBJICTBOpSieT no I13I1'lCCKUM
a)
6)
CBOHCTBaM, JIH60 CJIY)KI1Tb OpMOH B LIGA-
TCXHOJIOrI1l1.
I"Kpoo6pa6oTKa apCeHltDa rannBH
CTOJIb pa'mo06p83Ha, KaK 11 06pa60TKa KpCM-
11I1S1. YHl1KaJ1bHbIC CBOHCTBa GaAs 11 cro
Tp0l1'lHOro cnnaBa AIGaAs J1.onycKalOT
MCTO,llbI 06pa6crflOI, KOTOpbIC SlBJIJlIOTCSI Pa3-
RI1Tl1eM 06'bC:\1HOM MHKpo06pa60TIOI.
Ha pHC. 2.33 ITOKa3aHbi YKpynHcHHble
3Tanbl o6pa6oTKI1 llJIaCTllHbI GaAs. Ha
nepBoM :Hanc (PI1c. 2.33, a) ocylllccTllilJleT-
CH I10HHaJl I1MIIJIaHTau.l1S1 no MacKUpylOIl.{C-
MY CJ101O. rny6uHa npoHHKHOBeHIDI 110HOB
3aBI1CI1T OT YCKopHIOIl.{ero HanpIDKcHHSI.
06p83CU OT:>KHraCTCR npH TCMncpaTYpc
750°C B TC'leHHC -30 l\IllH H Ha BTOpOM
3TanC 06pa60TKH (pHC. 2.33, 6) KonHpycTcH
wa6noH CTpYKTYPbI 113 ShN.t. 3TOT npoLtCCC
peKpHCTaJIJIH3YCT HOHHbIH CJIOH, CKpbI-
TbIH B GaAs. 3aTeM Ha llJIaCTHHY I1cna-
pSiIOT SiOl, BbmonWlIOT aH1130TpOnHOe
TpaBJICHHC H y)J,aJISiIOT CKpbITbIM cnOH C
nOMOll.{blO CeneKTHBHorO TpaBJIeHllSi
(pHC. 2.33, 6).
3
6)
3
...
1
Puc. 2.34. IhnYfOBJICHUC MCM6paHbi 113 AIGaAs:
a - 3mITaKCHaJIbHOe Hapall!lIBaIiHe CJIOSI AIGaAs; 6 - HaHeceHHe MaCKH H3 Si J N 4 ;
6 - BbITpaBJIHBaHl1e GaAs; 1 - 3ITHTaKcHaJJbHb1H CJIOH AIGaAs; 2 - GaAs; 3 - Si 3 N 4
130
[nasa 2. TD..'HOJIOrH4ECKI1E OCHOBbl rIPOH3BOlJ,CTBA
,Z:{PYTOH MeTO)J, C03J1.aHllSi UlKPOCTpYK-
TYP I1CnOJIb3yeT pa3.!lI1'lne B XIf m'leCKliX
CBoHCTBax GaAs Ii AIGaAs. OCAOBHble 'na-
nbI MeTOJIa Ha npUMepe 1f3rOTOBJJeHlUJ MeM-
6paHbI IEl.!llOCTpnpyroTCE puc. 2.34. Ha ITep-
BOM 3Tane (PIiC. 2.34, a) BbInOJIHneTCSI :mn-
TaKCUaJIbHOe Hapalll..lfBaHlfe C!IOSI AIGaAs Ha
ITOBepXHocTH JUIaCTHHbI 1t3 GaAs.
Ha BTOpO f 3Tane Ha 06paTHYIO CTOpO-
HY ll..'1aCTlfHLI HaHOCSIT MaCI\")' 1t3 Si 3 N 4 , OIT-
peJIe!UlIOIU)'1O KOH l1rypauHlo MeM6paHbI.
Ha nOC:Ie;melll 3Tane. ItCnO.!lb3YH pacTBop
H;!O:!fNlLOH, YJ.a.1.SlIOT GaAs. OCTaB;I}tSl
MeM6paH)' H3 C.1.0JI AIGaAs.
Cne.nyeT HMeTb B BliJIY. 'ITO 3mnaKCH-
3.'lbHoe Hapau..UiBaHne - BbICOKOTeMnepaTYP-
HLIH npouecc. Bc.1.eJ.CTBlfe 3Toro JI<UKe npH
xopomeM COOTBeTCTBlfl1 I\-pHCTaJI.rH1'JeCKI1X
pemeTOK GaAs II AIGaAs. HO 113-33 P83JI11-
IfHH HX Te neparypHbIX K03 lfUHeHTOB
paclllHpeHHH, emit AIGa.A.s 6YJ1.eT HCIThITbl-
BaTh QCTaTOIJHOe TeMnepaTypHoe HanpSlA(e-
HHe.
2.5.2. MHKpo3.'le "TpOHHLIe 3.'leMeHTLI
IJpH cJ>OpMHpOBaHHH 3.'1eKTpOHHLlX
3.!1eMeIITOB C'beM8 HHcJ>OPMauHM (OMHIJCCKHe
KOHTCnCIbI. fmeHOqHble npOBOJJ.HHKH), a TaK-
e p3.3JlH'lHbIX BTOpHIJHb1X npe06prooBaTe-
.!Ieii. HHTerpHpoBaHHb1X C MHKpoMexaHllIJe-
CKHM tl3, He06xo.lI.HMO npHMcHeHHe TeXHO-
.nom", HCIlOJlb3yeMb1X B MMKpo:meKTpoHHKe.
P83BHrne MHlCpoTeXHOJJOnUi o6ccnc-
IfHJIO TaK H83b1BaeMb1H ImaHapHb1H npouecc,
MH llJIaHapHYIO TeXHOJI0fHIO, cyTL KOTOpOH
3aKJIJOqae-rc}t B nOCJJCAOBaTcnbHOM li3rOTOB-
neHHH C.iloeB C 3a;IaHMb1M pHCyHKOM, pacno-
JJOACeHHblX .npyr HaJI .npyrOM H COCTOJIlUHX
H3 MarepH8J10B C pa'3J1H'iHldMH 3J1eKTpHIJe-
ClCHMH CBOHCTBaMH. I1cno.Jlb'3YSI TaKYIO
CJIOHC1)'IO crpYK1YPY, li'3fOTaBJIHBaJOT P83-
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(aKTIOaTOp) H HCnbl1)'eMbIH 06pa3eu. C03)J,a-
HHC nO,ll06HOH e)J,l1HOH CHCTeMbl He SlBIDIeTCjI
ITpOCTOH 3ana'leii H3-3a He06xo)J.HMOCTM )'"le-
TS CJ1e1lYIOIlJ,HX 4>aJ<TOpOB: Bo-nepBbIX, TOJIbKO
3J!CKTpOCTaTI1'leCKl1e aKTIOaTOpbI MOryl' 6bITb
C03.llaHbI no O.nHOH TeXHOJlOrHH C 06p83uOM
)J,JISI HCnblTaHI1H, a, Me:il\JlY TeM, CHJIbI, C03,lla-
BBeMble nO)J,06Hb1Ml1 aKTlOaropaMM, BeCbMa
MaJIbI; BO-BTOPbIX, aJ<TIOaTOpbJ, .n.OJI:>KHbI P83-
BHBaTh KaJJH6pOBaHHYIO CHJIY.
Ha pHC. 2.45 nOKa3aHa npMHL(l1nMaJIb-
HaH. cxeMa H3MCpHTeJJbHOH CHCTeMbJ, B KO-
TOpylO BXO)J,l1T aKTIOaTOp, BKJIIO'laIOll(HH B
ce6S1 HenO.llBH)I(Hble J H nO.llBI1>KHble 2 rpe-
6eH'laTble )JIeKTpOCTaTH<1CCKHe crpYKTYPbJ,
06...e.nHHeHHb1e paMI\OH 3, nepe.naIOllleH pa3-
BI1BBeMYlO aKTIOaTOpOM CHJIy ':ICpC3 6aJIKY 4
Ha 06pa3eu 5 C MCTKOH-. PaMKa 3 OTHOCM-
TeJIbHO OCHOB8HHH no.n.BerneHa C nOMOlllblO
YITpyrHx )J!eMeHTax 7 Ha aliKepax 6. J1cnhI-
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pe He.naJIeKO OT MeTKH.
HCnhrt 1\111151 MEXAHWIECKI1X CBOl1CTB MA TEPI1AJIOB
139
/
5
6
PIIC. 2.45. npJIIIUHmla.-lbHaJi CXCl\I8 11]!\<ICPHTeJIbHOU CIICTCr.lbl
C IlHKJlH"lCCKII:\1 lIal PY)I\CHIIC:\I 06pa3ua:
I, 2 - Hen0l18HiKHbIe H nO,lBH)[(Hble rpe6eH'laTbIe CTPYf..'Typbl 3neKTpOCTaTII'leCKoro aKTlOaTOpa;
3 - paMKa; 4 - nepeL\alOL1laJI CIUY 6a.nKa; 5 - lIcnblTyeMblH o6pa3eu; 6 - aHKepbJ;
7 ynpyrlle 3I1eMeHTbI nOJ{Beca
HarpY)I(eHMe McnbrryeMoro 06pa3l(a
npOl1cxo,nl1T ,no era pa3JIOMa B paHoHe MeT-
KI1. .D:ml aHaJIl13a pe3yJIbTaTOB Ba>KHa lI3Me-
peHHaH ,ne opMal(HX CTPYKTypbl B paHoHc
pa3JIOMa. .lJ.nx 06pa6OTKlf ,naHHbIx, ITOJIYl.JeH-
HblX B xo,ne IICITbITaHI1H, nOJIe3HO HCITOJIb30-
BaHMe KOHel.JHOMepHoro MO)J,eJIHpOBaHIiX.
Pa3JUfI.JHble 6anO'lHbIe (CTep)l(HeBble)
MIIKPOCTPYKTYPbl HCITOJIb3YlOTCSI B MIIKpO-
MeXaHM'leCKHX npll60pax, oc06eHHO 3TO
OTHOCI1TCSI K Mf, KaK BM6paJ..tMOHHbIM
CTPYKTYPaM.
Pe30HaHCHaH. 'IaCTOTa MeXaHM'lCCKI1X
CIiCTeM H8XO)J,I1TCX B 3aBl1CMMOCTM OT
CBOHCTB, pa3MepOB II MaTepl1aJIOB CTpYKTy-
PbI. MO)J,YJIM IOHra MOryT 6blTb paCC'll1Ta-
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I13BCCTHbI ITJIOTHOCTb MaTepl1ana M TOI.JHaH.
reOMCTpHH. CTpyKTypbI. AHanOml.JHblM 06pa-
30M p83MepbI McnblT)'eMoi1 CTpYKTYPbI MO)l(-
HO ITOJIY'IMTb Ha OCHOBaHMI1 M3MepCHHOH
c06CTBeHHOH l.JaCTOTbl, Korna 113BeCTHbI Me-
XaHl1l.JeCKMe CBOHCTBa crpYKTYPbl. ECJm OHa
nOBpe)K)J,eHa B xO)J,e npOl(eCca McnblTaHMSI
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BeTCTBYlOllIl1M 06pa30M. KaK pC3YJIbTaT,
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OITpe)J,eJIeHO no 1I3MeHeHHIO pe30HaHCHOH
'laCTOTbl MexaHHl.JeCKOH CIiCTeMbI.
l£aCTb l13MepHTeJlbHoi1 CMCTeMbl nrur
MCCJle.n.OBaHMSI npOl(eCca ITOSIBJIeHlUI TpelllllH
nOKa3aHa Ha pHC. 2.46, nOJIyqeHHOM C no-
MOlllblO CKaHl1pYlOlllcro MI1KpoCKona.
B COCTaB H3 lepI1TeJIbHO" CI1CTeMbl
BXO)J,MT MHKpOCTpYKTypa, M3fOTOBJleHHaH. 113
nOJII1KpIiCTaJIJIlI'leCKOrO KpeMIiMSl, KOTOpaH.
BKJIlO'laCT B ce6x rpe6eHl.JaT)'1O CTpYKTYPY
3JIeJ\.'TpoCTanI'IeCKOrO 8KTlOaTOpa 1, rpe6eH-
'larylO cTpYK1YPY eMKOCTHoro 113MepllTeJIR
nepeMellJ,eHIIH 2, KOHTpOJIllpYlOlllero KOJIe-
6aHl1X nO)J.BM:>KHOH crpYKTYPbI, cKpenJIeHHoi1
C nO)J,JIO:>KKOM (OCHoB8HMeM) I1CnblT)'eMbIM
06Pa3l(OM 3 C MCTKOM, B paHoHe KOTOpOM II
npOHCXO,l\IIT 06pa30BaHHC TpellJ,l1H.
140
fJIaBa 2. TEXHOnOfUQECKHE OCHOBbI rIPOH3BO CTBA
-- . .. . : . =-:- *':; :i:-, c : ,:-
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I, 2 - rpeOeH'laTble C1p)'],.1)'pbl 3neJ..-rpoCTaTIl'leCKOrO aKTlOaTOpa 1I eMKOCTHoro 1I3MepllTeJU!
nepeMeIlleHHH COOTBeTCTBeHHO; 3 - HCnbIT)'eMbIU o6pa3eu C MeTKOi1.
B npouecce JlCnLoITaHHH nO.!UlepnrnBa-
eTCX pe30HaHCHblH penrnM Kone6aHHH C no-
MOlllblO KOH1)rpa o6paTHoH CBR3H. B03HHK-
HOBeHMe TpCIIUtHbI B paHOHe MeTKH ITpHBe-
,].eT K 113MeHeHHlO :iKeCTKOcn1 oOpa3ua, a
CJIe..nOBaTCJ1bHO, Ii pe30HaHcHoH '1aCTOTbI,
KOTOpaJI KOHTpOlIIiPYeTCH. M3MeHcHHe :iKe-
CTKOC"rn. H.."lH pe30HaHCHOH qaCTOTbI, MO:>KeT
6LITb Bhlpa:JKeHO B BI1.!{e amUlHTH'ICCKOH 3a-
BHCHMOCTH OT !LJIMH1>J TpeuUiHbI.
B 3aK..'IIOQeHHe OTMeTMM. '-ITO HcnbITa-
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.nyneH IOHra M npc.ncJIa npo'lHOcrn Ha P83-
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.nOBaHHJI npoueccoB 06pa30BaHHSI TpCIIUiH.
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pa36poc pe3yJILTaTOB, nonyqeHHl>lX pa'3HbT-
MIl BCCJJe,nOBaTeIDIMH, 'ITO K!lJIIOCIpHpye-rCH
Ta6n. 2.12.
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p83HhlMH HCCJJe.nOBaTemJMll npM npHMonH-
HeHHOM pacTIDKeHHH 06Pa3UOB, HO npH HC-
n0J1b30BaHHH p83JI1I'IHblX MeTO,l\OB 113Mepe-
HHSI HarpY)KalOllleH Cl1JIbI H )J,e opMau.I1M
(cMelIleHHH), a TaK:>Ke p83JIH'IHbIX cnoc060B
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npl1Be.neliHble pC3yJIbTaTbl neMoHcrpH-
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ITepI1MeHTOB. Cne.nyeT TaK)Ke U5paTHTb BHH-
MaHHC Ha 3aBHCHMOCTb MexaHWleCKHX
CBOHClB MOHoKpeMHHSI OT KpHCTaJInOrpacIJl-1-
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TOnOJIOrHH MHKpOCTpYKTypbl Ha KpHCTaJIJIe
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3YJIbTaT B OTHOllIeHHH npO'lHOCTl1 MMKpO-
npH60pa.
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nO.nBcprllIHXCSI MHKpo06pa60TKc, 3aBHCSlT OT
XHMl-1'1eCKHX ITpOUeCCoB, HCnOJIb'30BaHHbIX
npH I1X H3fOTOBJleHHIi. CJIc.nOBaTeJIbHO, M
TeXHOnOrH'ieCKHe npoueccbT H3fOTOBJJCHHSI
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IIC Ilbl'l AIIlUI MEXAHH ECKHX CBofIcTB MA TEPHAJIOB
141
2.12. PCJY.-II, TaTbl IICnbITaHIIU nOJ1I1Kpel\l HIIH
I1ccne.noBaTe.lb
Y C.'lOBIIH II
pe3ynbTaTM Umverslty Californm Fmlure Johns Hopkins
of California. InstItute of Analysis
University
Berkeley Teclmology Association
Tl1n HarpymeHI1 (,13m6 PaCT5I:>KeHHe yl.3fH6 Pacu:meHH e
TOJllWlHa n01ll1- 1.9 20 ],5 3,5
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qHCJIO 11CnbITaH- 90 3 12 19 24
HbIX 06Pa3UOB
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rIla
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rna H3JIOMa, Ce'leHlU!, pa3l1a 06p a3 l1 a
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142
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CTBCHH3J1 pClUeTJi:a.
2. CHMBOJlbl rpaHeii H HanpaUJICIU1iL
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nnOTHOCTH "ynal\oBKu" aTOMOB no KpI1CTtUl-
JIl1'IeCKHM IJJlOCKOCHIM.
4. OCHOBHblC cBoikTBa Si 11 SiO,.
5. OMHtJCCKHC I\aHTal,Tbl If nJICHO'I-
HLIC npoBO)J,HIU\Ii. MaTCpH3.Jlb1 II cxeMbi CQ-
C.llI1HCHIfH.
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.poTorna6.'10HOB H )'IX COBMemeHUc.
7. OcHOBHblC 3TanLI <)JOToJIlfTorpaqJII-
tJCCKOl'O npou,ccca npu n03HTUBHOM 11 Hera-
THBHOM qmTope3HcTc.
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npHHulin Ii OC06CUHOCTIf npouccca.
9. PCHTrcHOBCKaJl JIlfTorpa<)JI1J1. npUH-
Ulln H oc06eHHocTii npouecca.
10.lhoTponHoc TpaBJIClme. Oc06eH-
HOCTH qmpMHpOBaHHJI Mlil\pOnpO IIJIeH.
I 1. AHH30TpOnHoe TpaBJIeHI1e. OCO-
6eHHOCTI1 qmpMHpoBaHltSi MliKponpoqmJIeif.
12. Cnoco6hI 1\00rrpOJIH p:nMcpllhlX IHI-
paMcTpoB npll TIXlUJlClIIlI1.
13. OCUOUlII,IC 'rram,1 t<pcl\lIlIIclIoii 06-),-
CMHOH M11 Kpoo6pa6oTl\ll,
14. OCIlOImblc ')1'a II 1>1 KpCl\1l11lClmii 110-
BCPXIIOCTIIOii MIII'I)oo6pa6m KII.
15. OcnoBllhlc 'nallhl UGA-TCXIIOJIOIIIII.
16.0cHoBllhic n:lI1hl II'JI'OTOIUlCIIIIH
nn:UlapHoro )1110,1]:1 C P-Il-IICpeXO)J,OM.
17.3111ITaI\CIIH. CXC!I1hl YCT3110IJOK II
OC06CHIIOCTJI npo)\ccca.
I So ,lJ,1H!J{IJY3I1H. ('xcl\m YCl :11101.11\11 II
OC06CIHlOCTI1 npollecca.
19. l10HHaJl I1MI1JI<lIlTaHIlH. Oc06clIJlO-
CTII npouecca.
20. GCHoBlfL!e ,)1'al1bl C60PKII MHl\pO-
MexaHUtJeCKoro npll60pa.
21. Hcnl>1TaJHfJl 1\1CXallll'lCCKIiX cnoiiCTII
M3TCpUaJlOn MCTOLlOM 111ru6a.
22. I1cnblTaJtnH I\ICXamltJCCl\lIX cnoiic-rn
MaTeplfaJIOB MCTOTlOM paCTJOI\CHlfSi.
23. 11cfIhiTaHlfSi MCX3HlflICCKIIX cnoiicTn
MaTCpllaJIOn I\ICTO}1.01\1 IlIiKJIH t leCIWro
lfal-PY:iKCnuSi.
24. 3annCI1MOCTb PC3YJIJ.T3T0I3 01' YCJIO-
BJfIf licm,rrmmii.
r JUtBa 3
3J1EMEHTHAn IJA3A MHKPOMEXAHI14ECKI1X
nPHIJOPOB
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uplI6op, arCYI'CTUYIOT. JhoGoii MHI(pOMCXa-
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KOII JlalmCIIIU! (MIlLO - MCM6pallhi nocupH-
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11 pco6pa1013aTeJII I - :11'0 YCTpoikTl3a,
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3.1.1. Y npynlc IIO}J.BCCLI
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A. iKcCTI\OC"I"b 1I0}J.nCC08
rlOilUCC HM MA npell.CT:lBmlCT co6o 1
pa3JIH<lHhlC l<oM6I11-1all 1U ynpyrllX ,}JlCMeH-
TOIl Hilla 6<1J10I( (CTep>KHeii). npe06JI<I)J,alO-
LI.lee 60JIbWHlICTUO 1\01'OphiX nO}J.llep)J(eHO
tlC(jJOpM<lllIHIM H3n16a (::meMeIlTI,1 "pa60T1\-
10-1''' lIa Inrn6).
B 1'a6JI. 3.1 npllUeJ.lClibl Tll1IOBble cxe-
MbI lIarpy)[(elll1S1 ynpYnlx :>JIeMeHTOB C )I(e-
CTI<O 3aUlel\1Jlel-lllhlMli 1\00UlaMH. a TaI()I(C
npl1l1cJ].euhI 1II0PMYJIbI [24] ,llJISI BbJ'mCJIeHl1H
pe8KI-tI1H A", B" II MOMeHTon peal(Lllii1 A{".,
M/ \" 06YCJIOUJlCI-IIIbiX elllmH IHoii npoca.n.KoH
1aw.e!lo1Jlenuoro KOHua 6aJ1KH B 1'O IKe A, a
T8K)J(e (l>OpMYJIbl .nJJSI Bbl l IHCJ1eHlHl peaKUH}i
Am B« 11 MOMeHTOB pe8KLll1H M"o.. ft1 8ao 06y-
CJIOBJIemlblX C)J,I1IUI'll-IbIM nOBopoTOM 3n-
w.eMJlemmro 1\00Ula 6aJ1KH B 1'O<IKe A. Dpll-
ue)J,eHbI T81\II(e qlOPMYJIbI ,llJIJ1 BbI'lIicnemUI
nepepe3bIBalOllJ,HX CHn Q.
144 nJaB3 3. 3JIEMEHTHA5I EA3A MIlKPOMEXAHIf1.J.ECKHX rIPHBOP013
3.1. Ynpynle 3JIeMCHTbi C i1\ecTlW 38lUe1\l.'leIlHb)l\IlI IWHU81\111
CxeMb) HarpY)I\eHIUI; 3mopbl ;\f II Q
-\ 4J"
r;;}\fB...
-
B
x
A
I
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I
o
0M80.
B :x
' a
M lk
Q
OnopHblC peaKLum II MOMeHTI,I;
nepepe3bIB3JOllIIIe C\lnbl
12£1 12£1
A\. =: "\ ; B\ = - 1 ,
/- [-'
6£1
_"'1 -I = M 8 = ----,-- .
. . /-
0=- 12£1
- ,3
A = _ 6£1 . B = 6£1
0. 2' a 2'
, /
M = 4£1 . M _ 2£1
An , Bu ['
Q = _ 6£1
[2
Y c.n.OBHble 0603Ha<leHH"; E - Mo.'{)'m, ynpyrOCTH; J - MOMem HHepIJ.I1H nOnepe'lHoro ce-
'IeHHJI 6a.'1KH.
}L"JjI BCeX CXCM LI3 oceBbIX MA DO
pHC. 1.) npH YCJlOBHH, 'lTO CJLla npHJlO:>KCHa
s e 11M no OCM y, Harp}"KeHHe ynpy-
mx 3neMeHTOB COOTBeTCTByer e.nHHM'lHoMY
napawre.nLHOMY CMeIUeHHIO OlIHoro H3 KOH-
nos ynpyrOTO 3JIeMeHTa. ::hOM 1Ke cxeMe
Harp}"lreHIDI COOTBe'fCTBye-r H CJIyqaH npl1-
JImKeHHJl CHJlLI B)l:OJIL OCM x Ha pHC. 1.1, e.
EcnH TO'lKa npHJJO".&eHIDI CMJIbI K HM B q3
no cxeMaM puc. 1 J, a, 6 CMelUeHa OTHOCH-
TenbH() reoMe1pHQecKoro ueH1'pa, HanpH-
Mep. B)l:OJI" OCH X. TO HaI"p)'iKeHHe ynpyrHX
3neMeHTOB. pacnOJIO HHhlX BJJ;O.m. OCH x,
COOTBCTCTByer CJJyqalO eJ(MHHilHoro nono-
pOTa O)J,Horo H3 KOHuon, 8 ynpyrHe ::meMeH-
Tbl B)),OJIb OCB Z HcnblTblBaJOT )J,e opMaUHJO
KPY'lCHHSI.
0603Ha'lHM CYMM8pHble CMJIbl, nplllO-
:>KeHHbIe IC ynpyrOMY :meMelny B TO'lKe A,
'lCpe3 P, a cYMMapHbIc MOMeHTbl - qepe3 M.
PeaKUHH H MOMeHTb) npH e)J,BHHqHbIX Ha-
rpYiKCHHRX onpe,l\CJI"JOT K03 q)J[UHCHTbI
*CCTKOCTH 6anKH. Mcnonb3yeM PC3YJIbT8TbI
Ta6n. 3. I I1 nnC,l\eM 0603Ha'lCHH"
k ll = Ay, k l2 = k 2I = Ac.u k n = M A , (3.1)
o
H 3aIDIIJJCM MaTpHUY CHll H MOMeHTon, ,l\ei1-
CTBYJOlll1lX co CTapOHb! O,l\HOrO ynpyroro
YIlPyrHE rIOL\BECbI It! MEM6PAHhI
145
3JIC\1eHTa, paCnOllmKCHHOrO I!.QOIlL OCI1 x, Ha
11M npH ce ncpeMell eHHSlx ITO JlHHeilHoH
KOOpJlIIHaTC y u yrnoBoH u;
P
M
k l ! k 12 )'
k2.1 k n u
(3.2)
AHaJlOrHl.lllaSi mTpmla MO>KCT 6blTb
3alll1CaHa H ,llJl ynpyroro 3JIeMeHTa, pacno-
JImKeHHoro B;10Ilb OCI-! Z.
TaI\I1M 06Pa'30M, 06ycJlOBJ1eUHbIe )I(e-
CTKOCTbIO OIlHOH 6anKH CHIIbI H MO:"JeUTI,I,
)J,eHcTBYlOlUHe Ha nllaCnlHY l'IM, pallHbI
P = klty + k 12 U; Al= k 2l y + k 22 u. (3.3)
EC.'11 lleHTp Macc H remJeTpIi'leCKI-IH
uellTp !lJlaCTl1Hbl HM COI:IUanalOT H CJlJIa
npl1I1mKeUa no OCH y 1:1 UM, TO )KeCTKOCTb
nO)J,lIeca., COCTOSlLllero H3 'leTbIpex 6anOK,
IIJISI Bcex cxeM 3 no pHC. 1.1, a TaKlKe ,l1JUI
cxeMbI q3 no ptlc. 1.1, e npl1 lIeHcTBHH CH-
JIb! II,llOJIb OCH x onpe)J,eJISlCTCSI tPOPMYJIOH
GJ' := 4k ll = (48£1 )113 . (3.4)
I
ECJIH l{M II reOMeTpH'leCKHH IlCHTp
nJIaCTlIHbI 11M He COBna)J,aIOT, TO )J,J1J1 110..1-
ncca 3 no cxeMa 1 Ha pHC. 1.1, a, 6 MO:iKHO
npe;InOJIO>KI1Tb, 'lTO H3 'leTbIpeX 6anoK nO!l-
Beca )J,Be pa60TalOT Ha H3nI6, a )J,l:Ie Ha KPy-
l.leHHe, )!(eCTKOCTb nOJweca HRX01IHTCSI no
BbIpIDKeHI1IO
Ga. = G = 2k 22 + 2GJ<p = (8£I)/l + 2GJ<P,
(3.5)
r)J,e G KP - >KeCTKOCTb O)J,HOH 6anKH nO,llBeCa
Ha KpY':LeHHe.
JJ:JISI 6anKH npJlMoyrOJIbHOrO cetJeHHSI
b n x C n (lUHpHHa x BblcoTa) :>KeCTKOCTb Ha
KpY<leHHe BOKpyr OCH x HJIl1 Z (pMC. 1.1, a, 6)
BbJlmCJIHeTCJI no opMYJIe [24]:
G Kp = (Gk'bnc )ll , (3.6)
r)J,e G - MOIIYJIb C)J,BHra B cOOTBeTCTBytOLlleH
KpHCTaJ]JIorpa H'leCKOH nJIOCKOCn1, a KO-
3tPqmUHeliT k' 3aBHCHT OT OTHOllIeHHSI
b n 1 c n (Ta6JI. 3.2).
3.2. 3aBIICII!\10CTb k' = f(bn/c n )
bicn J 1,5 1,75 2 2,5 3 4 6 8 10,0 C()
k' 0,141 0,196 0,2]4 0,229 0,249 0,263 0,28] 0,299 0,307 0,321 0,333
TaKlIM 06Pa'30M, CI1JIbl I1 MOMeHTbI,
)J,eHcTBYlOllIlle co CTOpOHbI 01l.HOrO ynpyroro
3neMeHTa nOJtBeca Ha nJIaCTlmy 11M, OITpe-
)J,eJISlIOTCH tPopMynaMH (3.3), B KOTOpI>IX :>Ke-
CTKOCTb Bbl':LHCJISleTCSI C nOMoLllbIO Ta6JI. 3.1
B 3aBHCHMOCTI1 OT BHn.a Ae opMaUHH ynpy-
rHX 3neMeHTOB.
JJ:JlSl napaJIJIeJIbHO H nOCJIe)J,OBaTeJIbHO
coe)J,HHeHHblX ynpYfHX 3neMeHTOB CYMMap-
HaH. )1(eCTICOCTb HaXOAHTCJI H3 paBeHCTB
n
G L = L G j ; 11 G-r. = II q + 11 G 2 +... + 11 G" '
;=1
(3.7)
rAe G j - :>KeCTKOCTb i-ro ynpyroro 3nCMeHTa;
n - ':LHCJIO ynpyrnx 3JIeMeHTOB.
llpz/'\/ep 3.1
BbIl.nJCJUfM )!(eCTKOCTb ynpyroro nO)J.Beca
l.J3 aKCeJIepOMeTpa (PHC. 3.1). l.J3 aKCeJIepoMeTpa
mfOTOBJIeH H3 KpeMHl1eBOH fiJIaCUiHbl TOJIIUI1HOH
C K , nJIOCKOCTb xz KOTOpoif COBMell\eHa C KpHCTan-
norpa$H'leCKOH nnOCKOCTblO (100), a OCM xz co-
pHeHTl1pOBaHbl no KpHCTaJIJIorpa$lI'leCKHM Ha-
npaBJIeliHJlM [100].
C nOMOll\b1O aml30Tp0IIHoro TpaBJIeHllil
£PoPMHPYeTCH CMMMCTplf'fHaJI nynKa C HaKJIOHOM
rpaHeH 54°44', OpHeHTl1pOBaHHblX B ceMeik'rne
'leTbIpex WIoCKoCTeH (] J 1). 11M l.J3 B1>InOJlHeHa B
146 rJIaaa 3. 3JIEMEHTHA51 EA3A MHKPOMEXAHII ECKI1X rIPHEOPOB
I'
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4x>PMe lUIaCTHHbI C pa3MepaMI1 a..., b... c.... TOJI-
lIIHHiI ynpyroro 3.iJeMeHTa C m JlBillIlOlUerOCJI nOl1-
BeCON HM OJKe-r 6bITb 33..'13Ha no BpeMem1
TpaB..1:eHIDI. Ynpynlli nO;:J.Bec apOpMHpoBaH B
BI1lle 'Ie-n.Ipex CHMMeTplf'lHO pacnO.iJO)KeHHbIX
nepeMbl'leK (6a.1:0K).
Hcxo;uIbie !laHHble !lJIJI MaTepHaJIa: Eo 10) =
_ 11 ') _ 10 "t
-1,68.10 HIM'". G rl1o ,-6.17.1O HiM'".
Pa3Mepbl IL'laCTIDibl, M: a" = 9 . 10. 3 ;
b", = IO ; e" = 35 . IO- .
nonepe'lHoe Ce'leHHe ynpyroii 6aJIKlI. M:
b n = 4. 10-4; C n = 25 . 10-6; .!IJIHHa 1=2. 10- 3 .
Macca fI.:1:acTKHbI m n = pa",b"e" = = 7,5 . 10- 5 Kr,
r.ne p - D.'IOlHOCTb, r/CM 3 .
I. BbI'IHCJIJleM Ko3$$HlweHTbl jKeCTKOCTH
Ma-IpHUbI (3.2) (;t.'V! O.l1HOH 63;"lKH).
HMeeM
E(IICI)l =
E(IIO}hDC
12
=8,7S.IO- 8 H .",,2.
1,68.10 11 .4 .10- 4 (25.10- 6 r
12
B COOTBe'TCTBHH C cJI0pMy.naMH (3.1) H
Ta6n.3.1uonyqaeM
k _ 12EI 12.8,75.10- 8 - 13125 HI .
II - P .1O-3r ' M,
6.8.75-\O- - 656.10- 2 1-1'
( \2' ,
2.\O-3) .
k = 4EI _ 4.8,75.\0-8 =17.5.10-5
-- I 2.10- 3 .
k _ 6E1
12- P
I-I-M.
2. BbI'JHCJIJleM iKeCTKOCTh nOl1BeCa no
$opMyne (3.4) npH nepeMelllemHi nlJ3Cn1Hbi no
KOopJJ.liH3re .1':
G y = 4kJ I = 525 HIM.
3. OnpeJJ.eJIJ!eM iKeCTKOCTb nO.llBeCa no
$opMyne (3.5). J],nJl br/e" = 16 no n6n. 3.] J[ :>:
'" 0,33. no $opMYJIe (3.6) H3XO.lU1M
G(I 10) k'bnc
G KP - =
I
_ 6,17'10 10 .0,33.4.10- 4 (25.10- 6 r _
2.10- 3 -
=6,36,10- 5 H'M.
C}'MMapHlli! iKecncoCTb no rPoPMYJIe (3.5):
G u =2(k Z2 + G KP )=2{17,5+6,36}\o-s =
=47,72,10- 5 H'M.
YIlPYrl1E nOL\BECbI 11 MEMEPAHbf
147
3.3. YnpyrHe ::mel\1CHTbl C O.nHHM cBo6o)J,Hbll\1 KOHQOI\1
CxeMbl Harp)'IKeHH)j; 'JnIOpbl M)I Q
P A
B
<+-.,
l}
I
M H
c..,
- ""-..
Q
A_
B x
l}
M H
U
B MaSlTHIIKOBbIX MA q3 npenCTaBJISICT
c06oM: 11M C nO)J,BeCOM, BbillOJIHeHHbIM B
BI1,lJ,e KOHCOJIbHblX 6anoK (CM. pI1c. 1.11, a,
6), nO)J,BepraeMblx )J,e opMaul1.H.M J13fH6a,
JIH60 KaK ynpyrJ1e 3JIeMeHTbI THna TOpCHO-
HOB (pMC. 1.11, e), nO)J,BepraeMblx )J,e4JoPMa-
UHH KpY'leHJ1S1.
.D-eM:CTBJ1e HM (nJI8CTl1HbI) Ha ynpYrHM
3JIeMeHT nOMeca (6anKY) B TO'lKe A
(rn6JI. 3.3) MO>KHO paccM8TpHBaTb K8K 0,lJ,-
HOBpeMeHHo npHJlO:>KeHHbIe B TO'lKe A HeKO-
TOPYJO CI1JJY P J1 KPYTSlIlJ,Hfi MOMeHT M k l" B
060J1x cJIy [aSlJ( IIMeJOT MeCTO JIHHeHHWcfH
yrnoBbIe So nepeMCllJ,eHJ1H 6anKl1 B TO'lKe A,
BeJIH<IJ1HbI KOTOpbIX npI1Be.n.eHbl B rn6JI. 3.3
[24].
CYMMHPYH BeJIH'lHHbI JlHHeHHbIx H YT-
nOBbIX nepeMellJ,eHI1H B rn6JI. 3.3, nOJIyqaeM
f= Yr = Pall + M k l'0Zh
So = PO l2 + M0 22
HJIH B M8TpH'lHOH cPopMe:
IlepeMeUleIiH)j, MOMeHTbl H
nepepe3b1BalOUIHe CHJIbI
x
PI]
f = 3EI ;
PI]
So = -- (B TO'lKe A);
2EI
Q=-P;
Mil = -PI (B TO'lKe B)
MP
f- "P
. - 2EI '
Mki
So = - (B TO'lKe A);
EI
Q=-P;
M If = - M kl' (OT TO'lKH A ,lJ,0 TO'lKU B)
Yr
So
011
021
M '
"P
(3.8)
0)2
022
P
r,lJ,e Og (i = 1,2; j = 1,2) - K03 HQlleHTbI
BJIHHHI1.H., onpe,!J.eJIHCMble 3aBI1CHMOCTj{MH
(m6n. 3.3)
all = /3 1(3E1); 012::::: 021 = z2 1(2E1);
022 =//(E/). (3.9)
M8Tpl1l{e K03 I1l{I1eHTOB BJIllHHllH
011 0 1 2
V ::::: COOTBeTCTByeT 06paTH8.11
'021 022
M8TpHI.{a
k ll
K:::::
k 21
K03 Hl{HeHTOB
:>KeCTKOCTH
k l2
, T.e. HMeeT MeCTO M8TpJ1q-
k 22
HOC p8BCHCTBO I V II K I = I u I, r.ne U - e,l\H-
HJ1'lH8.11 M8TpHl{8.
148 rJ1aBa 3. 3JIEMEHTHA.5! EA3A MI1KPOMEXAHW.JECKI1X rIPHEOPOB
B COOTBeTCTBHl1 C onpe eneHHeM 06-
paTHoH MaTpmlbI nOJI)"-jaeM
011 012 1 k 22 - k r2
021 022 k Il k 12 - kil - k 21 kU
CJIenOBaTeJIbHO,
k n
.,
k Il k 22 - ki2
p
- 3EI .
OIl
_ k Il
U.,.1 -
k ll ;.,:::.: -k I 2 2
1
-
EI'
k"l
Up = 0.,. = -
- - k ll k 22 - k?2
[3
-
2EI'
k Il k 22 -k?2 = 1/(0 11 0 22 -oiJ-
)J.a.ilee 6y eM HMeTb
0.,., OIl
k I I - -., ; k 22 = 2 >
011022 - 0 12 011°22 - ° 12
k I2 = k 21 ;;.
°12
° 11 0 22 - 0 2
C )"ieTOM OPMYJlbl (3.9) nOJI)"iaeM
q.opMYJIbI ;I."VI K03cp MUHeHTon :>KeCTKOCTI1:
k _ 12El
11- P >
6EI
k 12 =k 21 =-7;
k 2 ? = 4£1 .
- [
(3.10)
y
---
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P ( '
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AHaJIOrll'mO MaTpl1ue (3.2) 3anUllJeM
06061UeHHble CHJIbl, 06YCJIOBJIeHHble )KeCT-
KOCTblO no.nneca. 11 MaTpH'JHoH cpopMe:
P k ll
-
M...-p k 21
k 12 y,.
k 22 S
(3.11)
IlplL\lep 3,2
PaCC"lUTaeM K03q,q,MQUeHTbl nO)l.aTnHBOCTH
II JKeCTKOCTIl .ilJJ 43, cxeMa KOToporo nOKa3aHa
Ha pnc. 3.2.
43 npellCTaBJUleT C060H nJ1aCTlmy C BHellJ-
HI1MH pa3Mepa!>UI OM' b,,, C." KOTopa» BMeCTe c
ynpyrHMH ::meMeHTaMlf BblTpaBJ1eHa 113 MOHOKpl1-
CTanna, 'laCTb KOToporo YCJ10BHO Ha3BaHa Kopny-
COM. B nJ1aCTllHe BbIDOJIHeHbI nepq,opaUlfOHHble
OTBepCTIDI rIO, H B lJ;M (TO'lKa C), COBna)J,aIOIlleM
C reOMeTplf'leCKUM QeHTpOM, pacnOJJO>KeHa )1.0-
nOnHlfTeJ1bHIDI Macca (rp)'3), KOTOpIDI MO>KeT II
OTC)'TCTBOBaTb. ,lJ,JISI aKcenepOMeTpOB KOMneHca-
QltOHHorO THna pOJIb rpy3a MO>KeT BbInOJ1H5ITb
KaTYlliKa MarHHT03J1eKTpJrqeCKOH 06paTHOH
CBg311.
Ynpyme 3neMeHn.1 (6anKH) MOryr liMen.
CJ10JKH}'lO KOHq,urypaUll1O KaK no BbICOTe (C n ), TalC
H no llJHpl1He (b n ). 311eCb 3TI1 pa3MepbI nplfIDITbI
nOCTOgHHbIMH. ,lI,nHHa I H TO'iKH A H B COOTBeT-
CTBYIOT Ta6]I. 3.3.
MOIiOKpMCTaJIJI!PleCKHH MIDITHHK H3rOTOB-
JIeH 113 KpeMHIDI, .ilJJ KOToporo: £(100) = 1,4 x
x 1011 H/ , P = 2,33 r/cM 3 . feOMeTpH"leCKHe
pa3Mepb] nJ1aCTHHb!, M: aM = 8562,5 . 10-6; a =
=4281,25.10-6; b.. = 9100.10-6; C.. = 350.10-6.
Pa3MepbI ynpyroi1 6aJ1KH, M: J = 805 . 10-6;
b n = 400 . 10-6; C n = 26,67 . 10-6. 06IllIDI 11M
MIDITHHKa [n!IaCTHlia C llBYMSI rpY3aMH (KaTyrn-
KaMH)] m = 0,29 . 10- 3 Kr.
r PY3 (KaTYWKa)
"
TInacl1IHa (MaJlTHHK)
" "
. ',-.. .... /'
':flr.-;
If t;
'-
"<'"
x
PRC. 3.2. CseMa 113 M3ftTHHKOBoro MA
YrIPYfHE rIOL\BECbl 11 MEMEPAHhI
149
MOMCHl liHCplllUi MaJlTHBKa OrHOOne,ILHO
OCH. npoxo wcli 'IepeJ M (To<IKa C) 11 naparr-
JICnbllOil oeli /.. paBCH
m (. 2 2 ) -'I 1
J c '" - a" I- C" = 1. 775 - 10 Kr'M .
]2
A npcmlo;!On(eHlIlf. 'II oReS! Maeca MaHl HI1-
Ka paeno.'10)l(eHa B QCHTpe nnaCHlllbl, MOMCHT
JlllepLlll11 MaHIIIIKa OlIiOCllTem.HO OClI, npoxo-
.r\JlLUeil 4CpeJ T04KY A II napan:JcnbHoH OCIi Z.
pascH.l 4 = J( + mal = 7,093 . 10- 9 KNi.
MOMelfr HIICPLllUl I10nepelJHOrO Ce'1CHIUI
6arrKII BOKpyr OCII Z, npoxoMlw:eii 4epCJ UCHTp
TSI)I(e<''T1l CClJCIlIUI, paBCIi
I =...!... l 3- 63 ? ]O 19 4
7nCn - . _ t .
]2
Bocnom;syeMcSI !)JopMynaMll (3.9) II (3.10)
II Bbl'JJ1CmlM K031jJ4mWleHTbl BmlJlHIUl B Kmq,qm-
LlueHTbl )I(CCTKOCTI1:
8 11 1,964.10-3 !\11H; 8 12 =8 21 =3,660 IIH;
0 22 = 9,093 .10 3 IIH'M;
k l1 =2.036-103 HIM; k l2 =k 21 =-0,819 H;
k 22 =4,398.10-4 H'M.
E. rJlaBHble OPMbl" qaCTOTbI MaJIblX
KOJle6aHldi 113
B COOTBeTCTBHH co cxeMaMH JJ.e opMa-
UHH ynPYrJ1X 3JIeMeHTOn (eM. Ta6n. 3.1 H
3.3) Ha pHC. 3.3, a ITOKa3aHO B03MO:>KHOe
IIOJIO:>KeHHe nJIaCTHHbJ l1M oceBoro MA B
IIJIOCKOCTH xy, I.{M C KOTOpOH He COBrra)J,aeT
C ee reOMCTpH'leCKHM uempOM 0, a Ha pHC.
3.3, 6 - 110J1O>KeHHe nJIaCTHHbI HM MaJlTHH-
Konoro MA, I.{M KOTOpOH H8XOnHTCR n ee
reoMeTpH'leCKOM ueHTpe.
aJ
HY)f(HO 06paTHTb BHlIMaHHe Ha TO, '-ITO
ITJIaCTIiHa MM MaHTHHKOBoro MA MO)f(eT
'3aHH!\1aTh ITOJIO>I<eHHe, COOTneTCTBYJOIllee
pHC. 3.3, a, eCJIH MbICJIeHHO nOO'lepe)J,Ho
y6HpaTb JIellbIH H npaBbIM yrrpyme 3JIeMeH-
TbI. TOlJKa A Ha pHC. 3.3 COOTBeTCTlIyeT TO'1-
KaM A B m6JI. 3.1 11 3.3, B KOTOpbIX npHJIO-
>KeHbI CHJIbI H MOMeHTbL OlJeBMAHO TaIOKe,
'1TO, ec:m UM n;mCUIHbI HM ocenoro MA
COBna)J,aeT C ee reoMCTpH leclClIM UeHTpOM 11
ynpyrHc 3JICMeHTbI HMelOT O)J,HHaKOllble >Ke-
CTKOCTH n HarrpaBJIeHI1H oeH y, B)J,OJ'lb ICOTD-
pOH npl1JIO:>KeHa BHClllHID! CHJIa, IIJIaCTHHa
HM 6Y)J,eT coneplllaTb nJIOCKorrapaJJJIeJIbHOe
JJ.BH>KeHHe.
CJIellYCT OTMeTHTb H TO, lJTO npH HeKO-
TOpbIX YCJIOBHSlX I1JIaCTHHa [-1M MaJlTHHKO-
1I0ro MA TaIOKe MO>KeT COBeplllaTb )J,IIH)Ke-
Hlie, 6mIJKOe K nJIOCKonapaJJJIeJIbHoMY.
IloJIo)f(eHHe nJIaCrnH 1aBHCHT OT JJ.nyx
KoopAJ-IHaT, KOTOpbIM COOTneTCTBYIOT )J,Ba
ypaBHeHH )J,BH)f(eHWI. ,l],JISI oceBoro MA -
3TO JIHHeHHaJI Koop)J,HHaTa y, onpe,llemIlOlll.aJI
nOJIO)f(eHlle TO<JKH C, 11 yrJIOBaH Koop)J,HHarn
a, onpe)J,emllOmaH TeKymHH pa3BOpOT nna-
CTl1Hb!. IlOJIO>I<eHlle nJIaCTHHbI HM MaJlTHH-
KOBoro MA 3aBHCI-IT OT mIHeHHOH KOOp -
HaTbI Yr paccToSlHHR TOlJKH A OT HeHTpaJJh-
HOro (BJJ.OJIb OCH x) nOJIO>KeHHH Ii yrJIOBOH
KoopJJ.HHaTbI 8- pa3nOpOTa nJIaCTIlHbI.
Y paBlieHHSI HCL\eMm}JHpoBaHHbLX ICOJIC-
6aHHH nJIaCTHH l1M HMelOT o HaKonyro
C-rpYKTYPY:
- .wm oceBoro MA
all
a21
ji P
a +n M
Kp
=0
,
(3.12)
an
a22
a
x
x
PUC. 3.3. nOJJOOI\CUUH nJIaCT"U HM MA:
a - oceaoro; 6 - MWlTHHKOBoro
150 rnasa 3. 3JIEMEHTHAJI EA3A Ml1KPOMEXAHHllECKHX IIPH150POB
r.ne all = m, al:! = a11 = mJ.., a:!2 = J a = J= +
+ mJ; (3.!tecb m - HM: J= - MOMCHT I1Hep-
l\HH nJIaCTHHbl OTHOCHTeJIhHO OCIi ::, npoxo-
,U)UlI.eH 'lepe3 TO'lKY 0; 1l - 'lHC.!J:O ynpyrnx
3JIeMCHTon): MaTpHua CJfJI H MOMeHTon Ha-
XOJ1.MTCX Ji3 nblproKcHIUI (3.2);
- MX MMTHHKOBOro MA
I all
a21
a22
\; p
_ r
.. + n = 0,
& M,,"p
(3.13)
aJ:!
f)J,C all = m, a12 = a2\ = ma, a2:! = .1.4 =
= J c + ma- (3.!lCCb Jc - !\1DMCHT IIHepUlili
llilaCTlfHbl OTHOCMTeJIbHO OCII. npoxoID!IlI.eii
'lCpC3 TO'll..)' C); M8TpHIla ClUJ II MOMeHTOn
onpe;I.e.:urCTC5I nbIproKeHHcM (3.11).
Bbl6epeM O,llHY 113 CliCTCM ypallHeHlIll,
Hanpl1Mep (3.13). Ii 3anHllleM ee II BliiI.C
anYr + a12& + n(klly, + k I2 &) = 0;
(3.14)
a2IJ.... + an& + l1(k 21 J'.. + knB) = 0 .
PCIllCHHC ClfCTe!\lbl (3.14) 6YiI.CM IiCKaTb
II 1lJ0PMC
}', = A Isin(pt-'-E); B -= A 2 sin(pt -L E), (3. I 5)
r)J,e A I, A 2, p, I:: - HCI13BeCTllbIC nOCT05l1iHhle.
BbIpIDKeHllx (3.15) 1II1ljJIlJepeHllHpycM
)J,BIDKllbl 110 BpeMeHII t II nOJIYtJeHHhie npo-
113nO!\Hble nO!\CTaBJlM B cl1CTeMy (3.14):
- a II A I P 2 sin(pt +E)- a21A2p2 sin(pt +E)+ n[kllAI sin(pt +E)+ k 12 A 2 sin(pt + t:)] = 0;
-a21AJp2 sin(pt + E)- a22A2p2 sin(pt +E)+ n[k 2I A l sin(pt +E)+ k 22 A 2 sin(pt + 1::)]= O.
QTo6bl nOJIY'lCHHbIe paBeHCTBa Y)J,OB-
JJeTBOpJL"IHCb ITPH mo6bIX t, He06xo)J,HMO
npHpaBlUITb uyJ110 1<03q,IlJHuHeHTbI npH
sin(pt + E):
") }
(nk ll - anP-) AJ + (nk I2 - a I 2p-) A 2 = 0
") ").
(nk 2 ] -a2IP-)A j + (nk 22 -a22P-)A 2 = 0
(3.16)
JIHHeiiHbIC anre6paH'lCCKI1e ypannemul
(3.16) OTHOCHTeJlbHO AI H A 2 .nOJliKHbI HMeTb
pemeHUe, OTJIH'lHOe OT HYJUI. ECJIH 6bl no
6LL'10 He TaK. TO COfJIaCHO nblpajKeHH5IM
(3.15) y, = 0, 9 := 0, <no COOTBeTCTnyeT no-
1<010, a He JIBHjKeHHIO.
CJIe,llOBaTCJIbHO, onpe)1.eJIHTCJIb CHCTe-
Mbj (3.16) ,l\OJIjKeH paBlUITbcSI HYJJlO:
nku - aup2 nk l2 - al2p2
nk 2J - a21P2 nk 22 - a22p2
=0.
(3.17)
PacKpbJB onpe.neJIHTeJIb, nOJIY'lHM
(nk l I - al )(nk22 - amlX-nk l2 - al")Jii = 0
(3.18)
IUJI1
(alla22 a ?) p4 -
.,
-n(aIlk22 +ankll-2aI2kI2)P- + (3.19)
2 k k 2
+n ( II 22 -k 12 ) = O.
YpanHeHl1S1 (3.17). ..(3.19) 3KBI1Ba-
JIeHTHbl, H JGl"0loe H3 HHX Ha3bmaeTCSI ypaB-
HeHHeM 'laCTOT, I1JIH neKonbIM ypanHeHHeM.
06a KOpH5I ypanHeHHSI 'lnCTOT OTHOCHTeJIbHO
p2 BeIl\eCTneHHbl H nOnO>KHTeJIbHbI.
PemeHHe 6HKBa.II.paTHOro ypaBHe-
HHSI (3.19) 3anHmeM n BlI,lJ.e
pt2 =n(b +.J b2 -4ac )/<2a), (3.20)
r.ne a = a] [a22 - af2; b = al1k n + a22 k ll -
- 2a12k12; c;:;: k) Ik22 - kf2 .
Ka)l(,llOMY nOnO:>KI1TeJIbHoMY 3HatJeHHlO
KOpHSI PI H P2, T.C. l<ajK,l\OMY H3 )J,nyx 3Ha'le-
HHH qaCTOT, 6Y,lJ.eT COOTBeTCTnOnaTb O,lJ.HO
'laCTlIoe pellleHHe (3.15) co cnOHMH 3HalJe-
HIDIMH nOCTOSlHHbIX A I. A 2 , E.
qaCTHble pemeHHX JIHHeHl-IO He3anHCH-
Mbl H D03TOMY 06Il\ec pemeHHe CHCTeMbI
YrIPyrl1E nO)1BECbl 11 MEMEP AHbI
151
ypaBHeHI1H (3.14) 6Y,lICT JIl-meHHOH KOM611-
HauHei1 pewcHHH (3.15):
Y r = All S.in(Pl f +1;1) + A I2 s.m(pZl + £2); }
B = A 21 sm(p(t + LI) + An sm(p2 t + £2)'
(3.21)
MC)!()J:Y 'lI1CJ1aMI1 A II 11 A 2 l. AI::! 11 A 22
HMceTCSI CBSl3b, KOTopaH onpe)J,eJlH.eTCSI
ypaBHCHI1S1MI1 (3.16). ECJlI1 B nOCJle.rume
ITO,l\CTaBHTb PI I1Jm P2, TO onpC,l\eJIHTeJIb
CI1CTeMbl (3.16) 06panITCH. B HYJIb.
CnC;I:OBaTe.lbHo, 113 )J,BYX ypaBHcHHH
He3aBI1CI1:v!0 TOJlbKO O,lIHO, npl1'lCM Jl[{)60e.
BbI6epeM, HaITpHMep, nepBOC ypaBHCHI1e 11
ITOJIY'lI1M 113 Hero OTHOlllCHI1e
A 21 _ nk ll _aIlP'l
All nkl2 - a 12 p 2
11, (i= 1,2).
Ka)!()J:OMY 3Ha'leHH[{) 'laCTOT PI 11 P2 co-
oTBeTcTBY[{)T 3Ha'lCHHSI 111 11 112:
_ nk lI -aIIP _ nk ll -allPi
III - 2' 112 - 2 .
nk 12 - a l2 PI nk l2 - aI2P2
(3.22)
TaKHM 06pa30M, A 21 = iliA", A 22 = 1l2A12.
06ll.\ee pellleHI1e CI1CTeMbI (3.21) npI1HHMaeT
BI1)J,
Y r = All sin(PI I + £1) + A)2 sin(p'lt -I- £2); }
B = I1A I sin (Pit + 1;1)+ 112 A I2 sin(P2 1 + £2)'
(3.23)
B peWeHI1H (3.23) 3Ha'leHlHf Ph P2, 1110
112 onpC,l\emr[{)TC 1JCpC3 H3BeCTHble napa-
MeTpbI q3, a 3Ha1JCHI1S1 All, A 12 , £1, £2 - npo-
H3B0J1bHble nOCTOSlHHbIe, onpe):(emleMbIc H3
Ha'laJJbHbIX YCJIOBHH ,l\BH:>KCHHH.
113 06ll.\ero peweHHH. (3.23) CJIe.nyeT,
1JTO ,llBH:>KeHI1C 1.13 OKOJIO nOJlO:>KeHHSI YCTOH-
'll1BOrO paBHoBccHSI COCTOHT 113 .u.BYx He3a-
BHCHMblX KOJIc6aHI1H:
Y r = y I) + y 2); B = 8(1) + 8(2) , (3.24)
r.u.e
(I) A . (p ) (2) A ' (p )
Yr = 11 sm 1 1 +£1 ;Y r :;: I2 sm 2 t +"2;
B(I) = III All sin(plt + "I); B(2) :;: 1l2A12 sinx
x (P21 +£2)'
nepBoe KOJIe6aHI1e ITO KOOp;lJmaTaM .l'r
M B rrpOHCXO,lIHT C 'laCTOTOH Ph a BTopoe - C
'laCTOTOH P2' 3TH Konc6aHI1R Ha3bIBalOTCSI
rnaBHbIMH. K03 I1UI1eHTbl 11] 11 112 mm.Sl1OT-
CR OCHOBHbIMI1 xapaKTCpHCTI1KaMH MaJIbIX
KOJle6aHI1i1 MaH.THHKa. Bcc KOOp,llHHaTbl B
KaJK,lIOM rJIaBHOM Kone6aHHI1 H3MernUOTCR
no rapMOHI14eCKOMY 3aKOHY Ii HMelOT 0)J,11-
HaKOBblC 'laCTOTbI II a3bI. 3TO 03Ha1JaeT,
'lTO KOOp,l\l1HaTbl O,llHOBpeMeHHO 06paIIIalOT-
CR B HYJlb Ii OJJ:HoBpcMeHHo JJ:oCTHralOT aM-
nJ1I1T)',l\HbIX 3Ha'leHI1H. AMnJlHT)'JJ:bl B Ka)!{-
)J,OM fJlaBHOM KOJle6aHHH HaXo)J,SlTCSI B no-
CTOSlHHOM OTHOWCHHH 111, KOTopoe HC 3aBH-
CI1T OT Ha1JliJlbHblX YCJIOBI1H.
flpu-\/ep 3.3
OnpeLIeIIHM C06CTBcHHble <zaCTOTbl H l}Jop-
MLI KOJIe6aHHH MMTHHKa c napaMeTpaMH no
npHMepy 3.2:
an = In = 0,29 . 10- 3 Kr;
- aM _ 0 29 10 3 8562,5.10-6
al2 - m - - .. -
2' 2
= I 24.lO-lI Vro'M
, ''',
a22=J A = 7,093.10- 9 KN1 2 ;
k ll = 2,036. 10 3 HIM; k]2 =k 2 ] =-0,819 H,
k 22 = 4,398 . 10-4 H'M.
IlpllMeM n = 3 H no l}JopMYIIe (3.20) nony-
t..IHM
2 _ 3(16,591:16,576)10-6 1
PI? -
,- 1,02.10- 12 c 2 '
OTKYAa cIIe,n)1OT 3Hat..leHJUI t..IaCTOT:
1
PI = 202,92 - = 32,31 rn;
c
1
P2 = 9876,59 - = 1572,71 fn.
c
B COOTBeTCTBUU C l}JopM)'naMH (3.22) 13bI-
1J1ICJIUM
A 21 a
III =-=-=
All Y r
_ 3.2,036.10 3 - 0,29 .1O 3 .0,057.10 6
-3.0,819-1,24.10-6.0,057.10 6
== -2,41.10 3 pa.n/M.
152 rJIaBa 3. 3nEMEHTHA5l .6A3A MHKPOMEXAHI1 ECKHX IlPI1.60POR
a)
AI
6)
---r 4 2
PHe:. 3.4. (l)OpJlll.l KO.TJe6sHRii MaHTHRKa:
a - nepB:Ul; 6 - BTOpax
.4."
/.1., =--=-=-=
- AI:! J'r
3.2,036.10 3 -0,29.10- 3 -83.193.10 6
-3.0819-124.10- 6 .83193.10 6
" ,
= 0,17 .1 0 3 p3.!l/M-
B COOTBeTCTBIUI C nOJI)"!:eHHbIMH pe3ynbTa-
TaMH ll!U1 nepBoii (Hli3mci1) 'laCTOTbI a.>.inmrry.n.a
Kone6a.HHH MaxnrnKa no yrny 9 B 2410 pa3
60Jlbrne ro.m.JIHT)'.!lhI Kone6wmii )'r TO'lKH A Kpen-
.9.eHIDI MWITHHKa C 6aJIlmll. Ha BTOpOi1 (Bblcweil)
'laCTOTe 3TO pa3.'lH'llie 3Ha<rnTe.'lbHO MeHbwe, T.e.
6o!Jee ODI)'11!MbI KOJIe6aHID1 no napa.>.ierpy Yr'
KpoMe roro, H3MeHWlaCb !)Ja3a yrJIa nOBOpOTa
M3J1THHKa.
cI>OpMLJ KoneOaHHH MIDITIiHKa nOKaJaHbI Ha
pRC. 3.4.
TaKHM o6pa30M. MOHOKpHCTaJIJIH'leCKHii
MaKfHHK C JIBYMJI CTeneIDIMH CBo6oJ)bI HMeeT .n.Be
r.9.aBHble cl>oPMhI KOJJe6aHHH OTHOCHTeJIbHO no-
:}()jKeHIDI YCTOH'lHBOrO paBHOBCCIUI, KOTOpble
xap3lCrepK3yiOTCjl pa3HhIMH OTHoweHWlMH yrJIo-
BOH aMIIJIHl)'JlbI JCOJle6a.HHH MIDITHHKa K a.>.inJ1H-
"I)'..!le JIHHeHHhIX KOJle6aHHH TO'lKH ero KpenJ1eHWI
C ynp)'lliMH '3J1eMeHTaMH (63JIKaMH) nOlI,Beca.
Z
O'JeBl1):\HO, HMeSl: B BII)J,)' I1Jl.CHTl1'JHOCTb
ypaBHeHuH (3.12) Ii (3.13), MO:>KHO 3aKJI}{)-
'JUTb, 'ITO nCpBaJl opMa Kone6allliH VIM
oceBoro aKCCJICpoMcTpa 6Y)J,CT COOTBCTcrBO-
BaTb nOCTynaTeJ1bHoMY nepeMCllJ,eHII}{) 11M
(nJIaCTIIHbI) no KoopnmlaTc )', a BTOpaJl
yrnoBoMY ,!lBIi:>KCHI1\{) no Koop,!lIlHaTC a. 3Ta
opMa KOJIc6aHUii nOKa3aHa Ha pUC. 3.3, a Ii
OTBC'laCT nOJImKeHlilO O)J,HOH 6amm, IIOKa-
3aHHoMY Ha pnc. 3.4, 6.
He06xo,!lIIMO 06paTiiTL BHI1MaHHe Ha
TO, 'lTO KaJK.LI.Oll nOITOnmlTeJIbHOH o606u{eH-
HOM Koop.n.IfHaTe, onpe)J,eJ1S1:KJllJ,eil nO':IO)KC-
Hlle 11M, COOTBCTCTBYIOT CBOSI opMa KOJIe-
6aHl1M 11 co6CTBeHHaH. 'laCTOTa. HanpHII'ICp, R
)J,ononHCHl1e K ,l\BYM nepBblM £popMaM KOJIe-
6aHuH 11M (MaJlTHHKa) (pHC. 3.4) MO:>KHO
YKa3aTb Ha )J,Be )J,pyrl1c, npenCTaBJIenHbIC Ha
pHc.3.5.
Ha pl1c. 3.5, a nOJIO)KeHHe MaSlTHl1Ka B
CI1CTCMe Koop,l\llHaT X",VkZk onpc):\eJICHO yr-
J10M k cra pa3BOpOT8 BOKpyr OCH Xk (A ,...
A3 - TO'IIill KpenJIeHI1S1 6aJIOK K nJIaCHIHe;
B 1 u.B 3 - T01fKH KpenJIeHHSI 6a!lOK K KOp-
nycy).
TIpH MaJIOM ymc k" II-!O:>KHO npc.nnoJIo-
iKHTb, 'ITO 6anIill AIB, I1 A3B3 nonBepralOTC}\
.n.c opMauHH 113rH6a. a CpC)J,HSISI - ,!lC£pOpMa-
UllH Kpy'IeHHSI. Ha pHC. 3.5, 6 nOJIO)KeHI1C
MaJlTHHKa onpe)J,encHo ymoM k). cro pa3BO-
pOTa BOKpyr OCM Yk. rHnOTeTI1'leCKI1 TaKoe
)J,BIDKeHHe B03MO:>KHO, eCJJH npMHSITb, 'lTO 6aJI-
Kl1 HMelOT, H8npI1Mep, KB8)J,paTHOC Ce'leHl1e. B
3TOM CJIY'l8e MO)KHO npe]JJIOnmKMTb' 'lTO Bce
68JIlG1 pa60Ta1OT Ha 113fH6, HO MaKCHMaJIbHblH
npom6 y Ka:>K,l\Oi1 H3 HMX 6Y)J,eT Pa3HbIH.
6)
Zk
Z
kJ'
Xk
PHCo 3.5. GOpMbI KOJJe6SHHM MaHTHHK8:
a - pa3BOpOT BOKpyr OCH x,t; 6 - pa3BOpOT BOKpyr OCH Yk
VnpYH1E nO,QBECbl YJ MEMEPAHbl
IS3
oj
,
'n .. . . ... ............... '. ".-.' .....;
4 n, 0 I;:
................." .... .m
.:..-:-:.;-:-:. "':.: :.....:,....... .
.: ::::::::; :: :. .:.:-:-: '.
;;1i r t ; 1 '.
,',"--,';'y :;.t:: r X
...... ..,. .'
..."4.......... ............"..
=-..:.....:-:.... [M'
'1 ; :::: :?-::::
.....:-: n n
: I >
:
......... .. ..- . ""-'-" -....1'0 ....<1.
I -
6)
Y
I
x d.
x
A
Yo
p
PHC. 3.6. K onpcJICJ1CHHIO BJIHHIIHH MaCCLI ynpyroro 3J1CMCHTa:
a CXCMa G3; 6 - cxeMa .n.eljJopMauHH yrrpyroro 3J1eMeH'f3
B. RuulHJle :\tacCbI ynpynlX :ueMeHTOB
Ita 'JaCTOry cOOCTBeHltblX KOJleOaltHH q3
)J:nSl HeKoTopblX cxeM '43 (CM. pHC. I. I ,
6, 2, 0) MA pa3MCpbI ynpymx 3JIeMCHTOB
nO;I:BCCa ,LJ,alOT OCHOBaHHe )J,YMaTb 0 BJIlUIHHH
I1X MaCCbI Ha qaCTOTY C06CTBCHHbIX Konc6a-
HHH '43. I1cxoJ1.H. H3 ::JHepreTH'lCCI<HX co06-
pa)l(CHHH [12], OITpClICJIHM 3TO BJIHHHHe Ha
ITpHMCpe q3 (PHC. 3.6, a), BbInOnHeHHoro no
cxeMC pHC. 1.1, 2. KmK.AbIH ynpymH 3JIC-
MCHT )J,JIl1HOH I Harpy)l(CH B TO'lKe A CMJIOH
P = mg/4 (m - Macca ITnaCTHHbI). CxcMa
Harp)'iKCHmI COOTBeTCTBYcr Ta6JI. 3.3, a )l(CCT-
KOCTI, klJ ynpyroro 3JICMCHTa B HanpaBJICHHll
OCI1)' onpe.QeJISleTCH. H3 ypaBHcIU1H (3.10).
KI1HeTl1'leCKWI 3HCprl11I q3 onpe)J,eJIH-
eTCH KHHCTlll1eCI<OH 3HeprHcH 11M nJIaCTl1HbI
11 3Heprl1eH yrrpyrHx 3JIeMCHTOB:
I
T =l.-my2 + 4 J Jldx(yxY , (3.25)
2 2
o
r,LJ,e Jl - Macca ynpyroro 3JIeMeHTa, npl1xo-
J1.H.IlIW1CH Ha e,l\HHH Y ero )J,JIHHbI.
MO)l(HO npl1HSlTb, liTO )J,JIH OTpe3Ka yn-
pyroro 3J1eMeHTa )J,JIHHOH dx (pl1c. 3.6, 6)
CIWpOCTb nepeMeIllcHHSI onpe)J,eJ1JIeTCH 113
paBeHCTBa
Yx I y = YXo I Yo.
(3.26)
113BeCTHbI [24] BbIpIDKeHI1H )J)lJI npom-
60B 6aJIKH (ynpyroro 3neMeHTa)
P 2 ( I J PI3
Y xo = 2EI x / -"'3 x ; Yo = 3EI '
nocJ1C nOlICTaHOBKI1 I<OTOpbIX B paBcHcTBO
(3.26) 11 3areM BeJm'll1HbI Y x B OPMYJ1Y
(3.25) nonYQI1M
1 .? ( 33 J
T=-y- m+-m ) . ,
2 35'
(3.27)
r,LJ,C my, -II-!acca O)lHOro ynpyroro 3JICMCHTa.
113 BbIpIDKCHI1S1 (3.27) cJIe.nYeT, liTO )J,JISI
BbIlIHCJICHHH KliHeTH'lCCKOH 3HeprHH 3 K
11M llJIaCTHHbI HY)I(HO )J,06aBMTb npl1McpHO
Maccy yrrpymro 3neMCHTa.
TIepHolI I<OJIe6aHHi1 q3 B 3TOM cJJY'lae
paBeH
T=2rr.
33
m+-m.. ,
35 J]
a 'laCTOTa C06CTBCHHblX K0J1c6aHHH onpe)J,e-
JIJIeTCH no £}JopMync
ro )'0 =
4k ll
33
m+-m y
35 >
)J:JISI 60nbwHHcTBa H3BeCTHbIX pa3Me-
pOB M M ynpyrllX 3neMeHTOB BnH HI1eM
MaCCbI YITpymx 3JIeMeHTOB H3 lIaCTOTy co6-
CTBeHHbIX KOJIe6aHHH 3 MO:>KHO npeHe6-
pe'ib. AHaJIOrWlHO MO:>KeT 6blTb BbInOJIHeH
aHaJIH3 BJ1ID1HHH MaCC ynpYTJ1X 3JIeMeHTOB
Ha lIaCTOTy co6cTBeHHb1X KOJIe6aJurn 'lJ:3 C
lIPYfHMH CXeMaMH no.nseca, B TOM lIHCJ1e H
)J,JISI Mr.
154 fJIaBa 3. 3JIEMEHTHNI EA3A MMKPOMEXAHI1QECKHX IlPH60POB
3.1.2. Ynpyrue nO)J;BecbI MHKporHpocKonOB
)l(eCTKoCTb nO)J.Beca M co6cTBeHHble
qaCTOTbI rllpocI<ona 3aBHCm OT KOH"--peTHoH
cxeMbI Mr H BapllaHTa llH OpManmHoro
.l1BH:>KeHIDlI1M.
A. MHKporHpocKoDbI LL-THna
Ha pHC. 3.7 npHBC.l1eHa CXCMa Mr, co-
OTBeTCTB)'IOIItWl pHC. 1.35, a, C O.!J.HOH Mac-
COM m, KOTOpWl nO)J,BellieHa B Kopnyce Ha
qeTbIpCx TOqKa.X (aHKepax) KpenneHHH. 3 C
nOMOIIUoIO o.nHHapHblx ynpYTIIX 3JIeMCHTOB
(ropcMOHOB) I H 2-
ECJIM H3MepgeTCSI yrnoBIDI CI<OpOCTb
0:, TO 11M no)J, .r!,eHCTBl1eM CHJlbI HHep1.\HH
KOpHOJIllCa 2m!l.zv nepeMelliaeTCSI B nJIOCKO-
C11i XY B HanpaBJIeHHH OCH Y. )l(ecTKocTb
no.il:Beca B HanpaBJIeHHH OCM X (PLO paBHa
JKecTKOCTll l.JeTbIpeX ropcMOHOB 2: G x = 4G 2x
(G 2x - )J(CCTKOCTb TOpCl10Ha 2 B HaITpaBJIe-
HHH OCM X), a c06CTBeHHIDI, HelIeMn I1pO-
f: ?
BaHHIDI qaCTOTa (0" = .
C m
)l(ecncOCTb l.JeTbIpeX TOpCHOHOB 1 B
H8IIpaBJIeHHH OCll Y (pe)f(J1M HH opManm-
HbIX Kone6amili. HJIll pC:>KHM <JysCTBMTeJIb-
Hoem): G)' = 4G t ;v> (G t }' - :>KeCTKOCTb TOp-
CHOHa I B HanpaBJIeHMH OCH Y), a C06CTBCH-
[G;;
HaSI'I3.CTOTa (OYo = --;;;- .
3
J
y
Pac. 3.7. CxeHa DOneca 113 C DOMOIUbIO
qeyypex Dap O.[UIB8pHWX yopyrHx :JJJeMeHTOB
(TopCHOBOB):
1,1- TopcHOHLJ; 3 - "TCJtIKa" KperuIeHIDI (aHKep)
CXCMa mpocKona )J,onycKaeT H3MCpC-
Hl1e yrnoBoH CKOpOCTM .0, BOKpyr OCH Y.
npl1 3TOM HH opMaTHBHoc )J,BH)KeHHC 11M
npOl1Cxo)J,HT no)J. ,l\eHCTBlieM CHJ1bI HHCp1.\HH
KOpMOJIMCa 2mD.vv B HanpaBJ1CHIiH OCH Z (113
nJ10CKOCTIi XY). B ,/J.aHHOM cJIY'lac napaJI-
JIeJIbHO BKJIIOI.JeHHbIe TOpCHOHbl 1 H 2 co-
ClIMHeHbl nOCJ1C,/J.OBaTeJIbHO ITO OTHOlIIeHHIO
.npyr I< )J,pyry. 3HaI.JI1T, cYMMapHa}i :>KeCT-
KOCTb nO)J.Beca B HanpaBJIeHliH OCl1 Z H co6-
CTBeHHaJI l.JaCTOTa onpe)J,eJUllOTcSI £}JopMYJIa-
MH (3.7)
G_ = 4G lz G 2z . (0 = 2
. G G ' Zo
lz + 2z
G lz G 2z
m(G lz + G 2z ) ,
r,/J.e G b G2:; - COOTBeTCTBCHHO )KCCTKOCTH
TOpCHOHOB 1, 2 B HanpaBJ1emlH OCH Z.
OCHOBHOH He)J.OCTaTOK CXCMbl ynpyro-
ro nOlIBeca HM no pHC. 3.7 3aKJJIO'laeTCSI B
HenOCpe,llCTBeHHOM l\.-pCnnCHIiH CHCTCMbl
TOpCHOHOB 2 K CHCTeMe TOpCHOHOB 1, 'lTO
npHBOJtl1T K B3aMMHOM)' BJ]]1JiHl1lO HX :>KeCT-
KOCTeH. 3TOT He,/J.OCTaTOI< YCTpaHcH B cxeMe
.nsoHHoro nO)J.Bcca HM, nOKa3aHHOH Ha
pHC. 3.8 (CM. pllc. 1.35,6).
11M m C nOMOIItblO 'leTblpeX nap TOp-
CHOHOB 2 nO)J.BeIliCHa K paMKe MaccoH mp,
KOTOpwr, B CBOIO OI.JepC)J,b, nO,llBellleHa B
Kopnyce Ha l.JeTblpeX TO'lKaX (aHKepax) l\.-pe-
nneHHSI 3 nocpe,l\CTBOM l.JeTb eX nap TOp-
CHOHOB I.
)l(eCTKocTH Ka:>K,l\OH napbl 1 HJ1H 2 ITO
OTHOlIIeHHIO )J.pyr I< ,llpyry " BKJIIO 'leHbl" na-
paJIJIeJ1bHO, a :>KeCTKOCTH TOpCl10HOB B
nape - nOCJIe.r!,OBaTeJIbHO.
3HaQHT, >I<eCTI<OCTb nO,l\BeCa paMKH B
HanpaBJIeHllM ocdi y, Z onpe,/J.eJJSeTCSI 3aBH-
CHMOCTblO
( Gly.lz ) G
G y . z =4 2 = 2 Iy,lz'
r)J.e G ly . G tz - COOTBeTCTBeHHO )KCCTKOCTb
TopcHoHa 1 B HanpaBJIeHHH ocei1 y H z.
AH8J10rnI.J110 BblI.JHCJ1SeTCSI: :>KeCTKOCTb
nO,l\BeCa q3 B HanpaBJJeHHH oceH x, z:
G x . z = 2G 2x ,2z,
r,l\e G 2x , G 2r - COOTBeTCTBeHHO :>KCCTKOCTb
TopcHoHa 2 B HanpaBJIeHI1H oeeH x Hz.
YrIPYrHE rrOJ\BECbI I1 MEMSPAHbI
155
PIIC. 3.8. Cxcl\la ,J80lrHOro fIO 8eC3 JIM (0603H3'lCHlUt CI\I. H3 pHC. 3.7)
TIpl1 1I3 lepCmm yrnoBoH CKOpOCTH n=
11M BMCCTC C paMKoH 1l0,LJ, .n;CUCTBHCM CHJIbI
liHepUl1H KopHonHca Ha CHCTCMe TOpCHOHOB
I COBeplllaCT HH opMaTHBHbIe KOJIe6aHHSI B
HanpaBJ1eHlH1 OCH y. C06CTBeHHaH. 4aCTOTa
3THX KOJIe6aHHH o npe,LJ,eJIH.eTC}I BbI pa:>KCHI1CM
ffivo '" J 2G 1y /(m +mp) .
MO:>KHO C4HTaTh, 'ITO B paccMoTpcHHbIX
cxeMax ynpyrne 3JIeMCHTbI nO,l\BCCOB no)J,-
BcpraJOTCSI TonbKO )J,e opMaLlHH 113rn6a.
TIplI 3TOM ynpyrl1e 3J1eMCHTbI no KOHLlall'!
iKeCTKO 3all\eMJleHbI. B 3TOM cJIY'lae JtJ]SI
BbI'IHCJICHI1S1 :>KCCTKOCTI1 Ha 113rH6 KaiK oro
ynpyroro 3J1eMeHTa MOiKHO BocnOJIb30BaTh-
CSI Ta6n. 3.1.
AHaJJorl1'lHO MorYT 6bITb BbI'IHCJICHbI
:>KeCTKOCTI1 11 C06CTBeHHblC 'laCTOTbI JtJ]SI
.n;pyrnx BaplIaHTOB CHCTeM ynpyrl1x nOJI-
BeCOB.
1>. MHKporHpocKonLl RR-nma
PaCCMOTpHM BbI'II1CJICHHe :>KeCTKOCTeH
H C06CTBCHHbIX 'laCTOT pOTOpHoro Mf
RR-THna, cxeMa KOToporo nplIBe.n.eHa Ha
pHC. 3.9 [23].
KOHllmrypaLlwI lJ:3 rnpocKona, BKJJJO-
'lalOll\cro B COOSi pOTOp 1, HCnetPopMl1pye-
MbIC 2 H )J,CtPopMHpyeMbIe 3, 4 3neMCHTbI
fIO,l\BeCa C aHKepoM 5, BbITpaBJIeHa B KpeM-
HlleBOH ITnaCTHHe. C nOMOll\blO aHKepa q3
COC,LJ,HHeH C no.n.lIOiKKoi1 (KopnycoM), Ha KO-
TopoH pacnOJIO:iKeHbI 3JICKTpOJ],bI CMKOCTHoro
ITpe06pa30BaTeJUI ITCpeMCll\eHHH, ,llaT'IJlKa CH-
JIbI KOHrypa KOMTIeHCaIUfH MOMeHTOB CHJI
HHCpLlHI1 KOpHOJIJ1Ca, a TrooKC CTaTopHble
3JICMeHTbI rpe6eH1.IaTbIX CTp)'KTYp 3J1eKTpo-
CTaTWICCKoro npI1Bo)J,a. TIoCJIe.n.HHe COBMeCT-
z
r
Poco 3.9. CXCM3 pOTOpHOro Mr:
I - pomp; 2 - He.D;etlJopMl1pyeMhle 3JICMeHThI
nO.D;BeCa; 3, 4 - ynPYrHe 3neMCHTbl noJIBCca;
5 -3neMeHT KpenJIeHlHI q3 K KOpnyCY (aHKep)
156 rJ1aBa 3. 3JIEMEHTHA5I EA3A Ml1KPOMEXAHH4ECKIIX nPIIEOPOB
HO C pOTOpHblMH cTpYKTYPaMII. pacnoJIo-
:>KeHHbIMH Ha pOTope, C031J.a1OT Bl16paQliOH-
HbIM Bparn.aIOlltllH t.1OMCHT BOKpyr OCII Z.
BC napbl ynpyrllx 3JIeMeHTOB 3 06ec-
ne'lHBalOT COe!l,ltHeHHC pOTopa C HeJIe Op-
MHPYCMblMH 3JIeMeHTaMl1 , KOTOpblC npe)J,-
nOJ1araIOTCR a6C0J1IOTHO >KCCTKHMIt. a ,llBe
napLl ynpymx 3JIeMCHTOB 4 oCYllleCTB.HSlIOT
Coe}1.HHeHHe 3JIeMeHTOB nOABeca 1 c aHKe-
pOM 5. DpH nOBopoTe pOTopa Bor..-pyr OCII X
Bce ynpynle 3JIeMeHTbI. pacnOJIO:>KeHHbIC
B.!IO.'Ib Hee. pa60TalOT Ha l\.1'Y'leHUC, a P83-
MemCHHLle BllOJIb OCH Y - Ha 1t3m6. Dpn
nOBopoTe pOTopa B,J.0.1b OCH r ynpyruc 3JIe-
MCHTbJ YHKUJfOHHPYIOT aHaJIOfl11JHO. DpH
p3.3BOpOTa.X pOTopa BOKpyr ocn Z BCC YITPy-
me 3JIeMeHTbJ nO!l,BCpralOTcSI ;!,e opMaQHn
H3rHOa.
npH nOSlIL1eHHH ncpeHOCHbJX yrJ10BblX
cKopoCTeH OCHOBamlR (Kopnyca) .ox I1 Q...,
KoropbJe J1.JUI rHpocKona SlBJIHIOTCSI npcQec-
CHeH, B03HHKaIOT ncpeMeHHbIC MOMeHTbJ CHJI
MHepUHH KOpHm1HCa, BbJ3bIBalOIJ.UoIe KOJIe-
6aHID1 poTopa no nepcr..-peCTHbIM OCRM (CM.
nO.!1pa:3.1. 1.3.4). Ha pl1c. 3.10 npHBe)J,eHa
cxeMa $parMeHTa nO.!1BeCa pOTopa.
MOlKHo C'liITaTh, 1JTO npH pa60Tc mpo-
CKona ll OpMllPYIOTCR TOJIbKO ynpyrHe
3JIeMeHTbJ .a,.,1HHoii I, KOTOpblX B nO,iJ,BeCe 8
wryK. B p poTOp cOBepmaeT KOJIc6a.HIDI no
Koop,!lHHaTe Y H Bce ynpyrHe 3J1eMeHThJ no,ll-
Beca pa60TaIOT Ha H3rH6. )l(eCTKocTb J<aj!\.u.o-
I'o 3JIeMeHTa Bbl'iHC.'VIeTCJI H3 OTHoweHHR
G"1 =PIIB,
r.ne P - CHJ1a, npHJ10EeHHWI Ha KOHlle ynpy-
J"oro 3JIeMeHTa; 9 - yron H3rH6a 3Toro 3JIe-
MeJrra. onpe.neIDIeMblH 3aBHCHMOCTbIO (CM.
Ta6n. 3.3)
p/2
9= ,
2E(np
r,nc E(n) - M0.z:\YJIb ynpyrocm ,WVI COOTBeT-
CTBylO eH KpHCTaJIJJorpa4mQecKoH nnOCKO-
CTH; J - MOMeHT MHepUHH nOnepe<:lHoro ce-
qeHIDI ynpyroJ"o 3JIeMeHT8.
M.
P
,
,
,
\...---....; ,
, f"
. ..
. .
AHI;:ep
r
.
.
.
.
I
z
Puc. 3.10. Cxellla nOJIBeCa l\1r RR-Tlma
1
c b'
HMeSi B BHJI Y , '1TO I = 11 n ITOJI Y -
12 '
'laeM:
3
G = E(ITJCnbn
y 61
}J:JISI nOCJ1C,l\OBaTeJIbHoro coe)J,HHeHHR
)J,BYX ynpyrnx 3JIeMeHTOB HX :>KCCTKOCTb
paBHa G y 12, a TaK KaK nap, pa60TalOll\HX
napaJIJIeJIbHO, B nO)J,Bece 4 WT., TO CYMMap-
HaJI :>KeCTKOCTb nO)J,Beca B p.n: G'E:r = 2G y , a
'laCTOTa c06cTBeHHblX K0J1c6aHHH
rc;;-
CO rO = vc;+c; .
B pe:>KHMe 1JYBCTBlneJIbHOCTH (pq) B
061l.lCM CJIY'lae npH Kone6aHI1S1X pOTOpa no
Koop)J,llHare ex. Hnl1 f3 ynpyrne 3JIeMeHTbl
B;!,OJIb O)J,HOH OCH nO,l\BCpralOTCR )J,e opMa-
UJ.IH KpY'leHMR, a no ,llpyroH - )J,c opMaQHH
H3rH6a. Dpe,unOJIO)!(HM lUlR npOCTOTbJ, 1JTO
ynpyrHe 3neMeHTbI nO)J,Beca nOJIY'laIOT TOJIb-
KO )J,e opMaUHlO KpY1JeHHR, a 3JIeMCHTbi
no.nBeca no nepeKpecTHOH OCH BeJ1.YT ce6R
KaK a6coJIloTHO )!(eCTKlle TeJIa.
)l(CCTKOCTb O)J,Horo ynpyroro 3J1eMeHTa
Ha J<PY<:leHHe G KP onpe)J,eJIReTCSI OPMYJIOH
(3.6). O<:leBH)J,Ho, <:ITO cyMMapHWI :>KecTKOCTb
no BbIXOJ].HbIM lcoop,l\HHaTaM G'f.a = G'f.f! =
== 4G KP ' a co6cTBeHHLJe 1JaCTOTbJ Kone6aHHH
paBl1bl
YrIPyrl1E rrOL\BEChI 11 MEMEPAHbI
157
La
WaO = O>fJ() = ,
Az+AI
me A 2 , A I - MG:\-JeH fhl HHepllHiI J:IeMCHTOB
ITOJIBCCa H pOTopa OTHOCI1TeJIhHO BhlxonHbIX
oCcii.
PaCCMOTpl1M BapnaHT Bbl'lHCJI':Hl1S1 >KC-
CTKOCTCI1 II C06CTBCHHbIX 'laCTOT rvrr RR-
nina, npeJJ.JIO:tKCIiHOrO UHI1M "3.1eKTpO-
npl160p" (r. CaHKT-ncTep6ypr). CxeMa ITO.lI-
Bcca :noro Mr npHBCnCHa Ha pHC. 3.11.
POTOp P rHpOCKOITa CKpCIlJICH C non-
JIO:>KKOM nocpe)J,CTBOI\.J aHKepa A. OTHOCH-
TeJIbllO KOToporo nO.l1.BeC pOTopa BbIITO.'IHeH
C ITOMOll!blO )J,BYX nap ynpyrl1x :JJICMeHTOB
(TOpCHOHOB). BH6paul10HHOC Bpall!eHl1e po-
Topa no Koop)J,I1HaTe y BOKpyr OCH Z, nep-
nCH)J,I1KYJISIpHOH K nJIOCKOCTH rHpOCKOITa,
OCYll!eCTBJISIeTCH pOTOpHbI:\UI H CTaTopHbIMH
cTpYKTYpaMH rpC6eH'laTblX )J,BHraTeJICH )1,.
)1,JIIiHa H wwpHHa TopcnOHOB pa3HbIe (noKa-
3al[0 Ha PHCYHKC). a TOJIll!l1Ha OllHHaKOBa C
TOJIIl!HHOH pOTopa.
)1,JISI P)1, no KOOpnHHaTc Y MO>KHO ITPH-
HSITh CXC:\IY 3a)J,CJIKH KOHl OB ynpyrux 3J1C-
MeHTOB H HarpY>KeHHH. TaKoH :>KC, KaK Ha
pHC. 3.10. TaKHM 06pa30M, nOJIaraH, liTO
BHYTPCHHHC KOHQhI TOpCHOHOB HMelOT :>KC-
CTK}'IO CBH3b C aHKcpOM, a CIUIbl npUJIO:>KeHbI
K HapY:>KHbIM KOHllaM, KOTOpbIC "cB060AHbI"
3a C'leT Bpall!aTeJIbHOrO )J,BH>KeHI1H. pOTopa,
aHaJIOfl1l1HO npCnbI,llYIlJ,CMY nOJIy<mM
[ b 3 b] )
G = E ( ) c --1!L+ n2 . <0 = --Y..... ( 3.28 )
y IT n I I ' Yo C
TI T2 1
)1,J1H. nBH:>KeHWr B pq no JII060H H3 KO-
op)J,HHaT IX HJIH 13, HMeSi B BH,llY, liTO npH
3TOM )J,Ba TopCHOHa pa60TaIOT Ha Kpy'leHHC,
a )J,Ba - Ha H3rH6, 3anHweM:
E c b]
G = 2G + (IT) n n2
a KPX I
T2
. '" = JG. .
'aO A'
I
; <OpO = .
(3.29)
_ E{nfnb l
G p - 2G KPY +
I TI
y
/3
f!j
....:...;-..
.. :h: .,,,. 'A:.::: . :;;: :.
. :''1.:;.....:. .f-:' '.. .'
.. .
." ../
-
a
x
IT I
b u 2
PIIC. 3.] 1. CXC/\I3 nO;Iscca Mr RR-nma
r,lle GKpx, GI<PY - :>KeCTKOCTH Ha KPY'lCHl1C
TOpCHOHOB, pacnOJIO>KCHHbIX COOTBeTCTBCH-
HO B,llOJIb oceH X Ii Y, OnpeneJISIeMble op-
MYJIOH (3.6).
TIpM B03MO:>KHOM nepeMCIlJ,CHMH poTopa
OTHOCMTeJIbHO aHKCpa no KOop,l\HHaTe Z Bee
TOpCHOHbI nO,llBCpraJOTCSI ,l\ecpopMalll1H H3-
ruGa I1 CYMMapHaH :>KCCTKOCTb no.nBeca (CM.
Ta6JI. 3.1) H C06cTBCHHaH. lIaCTOTa onpe)J,e-
JIHIOTCJi cpopMYJIaMH
G' 2E{+ { r +b,, (d }
<Ozo = . (3.30)
I1pH B03MO:>KHbIX nepeMemeHIUIX pOTO-
pa OTHOClITeJIbHO aHKepa BnOJIb OCH X HJIl1 Y
,l\Ba TOpC1l0Ha nOnBepraIOTCJi ne opMallJ.:UI
Ha H3ru6, a OnHH H3 TOpCHOHOB, pacn0J10-
:>KeHHbIX BnOJIb OCH nepeMellJ,eHll51 pOTOpa, -
n OpMaQl1ll pacT51:>KeHIDI, a npyroH - C:>Ka-
TJ1S1.
I1oJ1ara>1, '1TO :>KCCTKOCTH CTep>KHCBbIX
3JIeMeHTOB (TOpc1l0HOB) Ha C:>Karne H pac-
TJiiKeHHe O)J,llHaKOBbl, a 113rH6HbIe )KCCTKO-
CTH onpe.ncJ]JiIOTCSI aHaJIoru'lHO npe.nbIJIY
llJ,eMY, nOJIY'll1M PMYJIbI
158 rJIasa 3. 3JIEMEHTHNI EJA3A MHKPOMEXAHl14ECKl1X rIPHEOPOB
G =2E ( ) c [ ( bn2 ) 3 + b nI ] . w = fi x .
x n n I 1 '...0 ,
T2 Tl 111
a . =2E(n)c n [ ( b nl ) 3 + b n2 ] ; W-,O = Gy .
lYI lT2 m
(3.31 )
OqeBH,Z:I,HO, 'ITO npu HeKoropbIX 3Hal/e-
HIDIX Harp)'3KH Ha TOpCHOHbl B1\OJIb OCli X
HJIH Y, OHM TepSiKH ycrOH'lHBOCTb. Harpy:3-
Ka, nplf KOTOpOH npXMOJIHHeHHaJI opMa
crep)KIDI (ropcl1OHa) nepeCTaeT 6bITb YCTOH-
liHBOH, Ha3bIBaeTC}I KpHTHl/eCKOH. B 06ll.\eM
c.rryqae C:>KaTIDI CTep)KIDI k-pHTlll/eCKOC 3Ha-
'lemte Harp)'3KH onpe.ll.eJUleTCSI OPMYJ10H
[24]
p. = TIEl
Kp 2'
I
r.ne I - HaHMeHbillHH 1f3 rJ1aBHbIX l\eHTpaJ1b-
HbIX MOMeHTOB HHeproiU Cel/eHIDI CTep)KIDI;
1- .wnma crep)KIDI (TopcHoHa); 1') - K03 -
<pHI{HeHT KpI1TI1'leCKOrO 3HaqeHIDI Harpy:3KM,
ltiIll K03<P<PHlUieHT ycrOH'lHBOCTH.
3HaQCmte BeJIH'lliHbI 1') JJ.JU! KpeMHIDI B
.nOC'I)'IlHOH mrrepa1)'pe He llpHBO)J,HTCX. ):(JISI
CTaJIH, M0.z:\YJIb ynpyrOCTH KOTOpOH (E =
= 2 . toll ITa) COH3MepHM C Mo.nyJ1eM ynpy-
f'OCTH KpeMHIDI [E(lII) = 1,86 . 1011], a cno-
co6 33)J,eJIKH K0Hl10B HarpYA<aeMoro CTep:>K-
WI COBUa.naeT B OCHOBHOM C 3a)J,eJIKOH
KOHUOB ropCHOHOB, npHBO.!(HTCfi 3Hal/CHHe
11 = 4r.
[Jpu.o.tep 3.4
Pacc'IHTaeM 1KecTKOCTH nOJJ;Beca H co6CT-
BeHHLIe qacTOI1ol KOJle6aHHi! poTopHoro rnpo-
CiCona DO pHC. 3.11. MHKpOCTp}'lC1YPa BhlTpaBne-
sa H3 xpeMHHeSOH I1Jl3CTHHhI B I1JlOCICOCTH (100).
I1apaMe1JILI KpeMHHJI:
E(IDO} = 1.295 . 1011 H/"c; G(IOOJ = 0,79 x
x 10 11 H1"c. Porop HMee-r napaMe1JIbI: m =
= 2 . 10- 7 JCr, oceBOD MOMem HHeplUfH C I =
= 2,5 . ur 13 Kr'"c; 3kBaTOpHaJII>HLle MOMeHThI
HBepJ1KH: J... a; J y = AI = 10- 13 Kr'"c. Pomp H yn-
pyrHe 3JIeMeH1Dl (TopcHOHI>l) no.naeca HMe10T
TOmItHHY Cn = 20 . 10 M, a WHpHHa H D,JIHHa
TopcHOHOB CJ1e,ayJOJI(Jte, M: b ol = 20'1 ; 4, -
= 0.8. 10-3; bra = 15 . 10-6; la = 0,5 . 10- 3 .
1. Bhl'll1CmfM >KeCTKOCTh nOJ1seea no KOOp-
,nHHaTe Y If cooTBeTCTBYlOllI)'IQ eo6cTseHHYlO
'l8CTOry no !)JopMYJIe (3.28):
G y = 1,295 .10'1 .20 .1O-{j x
X [ (20'1O- 6f + (15.10- 6Y j =
0,8.10- 3 0,5.10- 3
"043.38.10- 6 HIM;
WyO =
43,38.1O-{j = 132.J 0 4 l/e::>< 2100 fll.
2 5.10 13 .
,
2. OnpelleJIliM >KeCTKOCTb nOllBeca no 1<0-
opnHHaTaM a H p H eOOTBeTCTBYIOI1!He C06eTBeH-
Hhle tJ:aCTOTbI.
Ilo !)JopMYJIe (3.6) HaMJ1eM ( k' = 0,2)
G _ 0,79.10 11 .0,2.20'1O- 6 (20.1O- 6 Y _
KP'" - 0,8,10- 3 -
= 3,16 .10- 6 HIM;
G == 0,79.10 1I .O,2.20.1O- 6 (15.Jo- 6 Y =
KPJ' 0,5.10- 3
=2,14.10- 6 H 'M.
)J,anee paCC'lHTaeM
G a =2.3,16.10-6 +
+ 1.295.10 11 .20.10- 6 (15.10- 6 r
05.10- 3
,
=(2.3,16+17,48)10- 6 =23,8.10- 6 H'M;
WaO =
23,8.10-6 =1,54.10 4 l!c::><2457
10- 13
fl\;
-6
G p =2.2,14.10 +
+ 1,295.10 11 .20.10- 6 (20.10- 6 r
08.10- 3
,
=(2'2,14+25,5)10- 6 =29,76.10- 6 H'M;
(0130 =
29,76.1O-{j =I,73.I0 4 1/c::::2747 fl\.
10- 13
YrIPYrHE nOJJ,BECbl 11 MEM6P AHbI
159
3. 13b1<IHCJU1M }{{CCTKOCTb nO,[(Hcca no KOOp-
ltlHlaTC Z IJ COOTHCICTBYIOlilYIO C06CTBCHHYI0
<mcTory 110 cjJopMync l3.30).
G z 2'1'2l)5'1011 [ 20'LO-6 ( 20'LO-6 1J +
0,8,10- 3 )
( 1
20. 10- 6
+ 15 .\0-6 _] J = 329,57 H/ l,
0,5 .10
329,57 4
O}=o = 1 =4,06.10 lIc"'='6458 fu.
2 10-
4. HaJ:\.neM }{{CCTKOCTH no.nscca no KOOp.nH-
HaTaM .r 11 Y If coumcrcTBYlOillHe c06cTBeHHbic
<mCTOTbJ no 4lOpMym: (3.31):
G,. =2.1.295.10 11 .20.1O-(,x
X [( ]5.10-(, ) "3 + 20.10- 6 ] = ]29 639,86 HIM;
0,5 -10-] 0,8,10- 3
129 639,86 = 80,5.10 4 II C "") 28200 IU;
2 .10- 7
G v = 2.1.295.]0 11 .20.10 6 x
X [( 20'LO- 6 ) 3 + 15.LO- 6 ] =]55480,93H/M;
08.10- 3 0 5.10- 3
, ,
(J)xo =
(J) y O= 155480' 1 93 =88,17-1041/c",=,140300ru.
2.10-
CJlC,llYeT 06paTI1Tb BHI1MaHI1C Ha CI1JIb-
HYlO 3aBI1CHMOCTb xapaKTepl1CTI1K Mr, B
lIaCTHOCTI1 lIaCTOT co6cTBeHHblx Kone6aHl1H,
OT rcoMeTpl1l1CCKI1X napaMeTpoB 1l0,llBCca.
B. BnUJlHue pacTHrHBalOwllx cun
Ha ",eCTlWCTb ynpyrux 3neMeHTOB
TIpH OTHOCI1TeJIbHOM CMCllIeHl111 >KeCT-
KO 3allJ,CMnCHHbIX KOHUOB ynpyroro 3neMCH-
T8 npol1C xo):\I1T era y.nnI1HeHHe
6/ ",=, 6y2 +/ 2 -I (t>.y - CMellleHHe :>KeCTKO
3aKpenneHHoro J(QHL a), BCJIC,llCTBl1e KOTOpO-
ro B ynpyroM 3neMCHTe B03HI1KaeT paCTJlfI1-
BaIOI.l.laH. CI1JIa
T=EbncntJll,
(3.32)
\'
- i
1'('()
F
1
A t
lV/A
Poco 3.12. CXCl\ta na'-PY)KCHU5I ynpyrom
3J1CI\ICOT3
r)J,e b n . Cn. I - Wl1pl1Ha, TOJIllJ,l1Ha 11 IV1I1Ha
HenecpopllmpoBaHHoro YITPYroro 3JIeMCHTa
(pl1c.3.12),
,[(J1S1 npoH3BonhHoro Ce'lCHIiSi yrrpyroro
3JIeMeHTa no ero }1.JII1HC I1MCeT MCCTO paBcH-
CTBO MOMeHTOB
M(x) = Fx Ty(x) -1v/.4,
(3.33)
r.ne
M(x)=-Ely" [24j; y'(x) = vl d"\"; (3.34)
1 - MOMeHT I1HCpUIiH nonepC'IHoro ce'leHIUI
ynpyroro 3JICMeHTa
06'bC):\I1HJlSI opMynbl (3.33) Ii (3.34),
nony<mcM ypaBHeHl1c
y" \")- y(x)=_.£.-x+ M A
El EI EI'
KOTopoe ,l\nH. HallaJIbHbIX YCJ10BllH y(O) -
.v' (0) = y' (l) = 0 I1MCeT PCWCHl1C
Yl-,; ) =
I
Ch--l ( )
F x FAX F
=--Ash-+-A ch--I +-x
TAT sh i AT'
A
(3.35)
r.n.c A = .J El IT.
)],JISI X = 1113 cpOPMYJIbI (3.35) cne)J,)'eT
2A( chi - I)
y(1)= F 1- A
T chi
A
(3.36)
BBe,lleM 6e3pa3MepHYIO OTHOCHTeJIb-
HYlO BenH'iHHY
E=TI Ebnc n .
(3.3 7)
160 fnaBa 3. 3JIEMEHTHNI 6A3A MHKPOMEXAHHI.J:ECKHX rIPl160POB
C Y'leTOM KOTOpOH napaMcTp A = C n I k
(k = JI2t; ) 11 Bblpa:>KeHlIe (3.36) npuHIIMaeT
BH)I.
( kl )
ch - - 1
1' ( / ) = FI 1- 2c n C n
. T kl kl
sh-
C n
(3.38)
11.3 Bblpa:>KeH1UI (3.3 7) Bhlpa3uM T, no,ll-
CTaBHM 3yY BenH'lliH)' B ypaBHeHHc (3.38) H
nOJI) eHHOe paBcHCTBo pa3pelllHM OTHOCH-
TeJIbHO onopHoH peaKliHH F, KOTOpaJI " on-
pe.n.emleT :iKecTKOCTb ynpyroro 3J1eMeHTa:
-I
( Ch kl -1 )
1- 2c n C n
kl sh kl
C n
..,
F-k' _ Ebncnk-
- II - 121
(3.39)
TIp" k -+ 0, T.C. nplI OTCyrCTBl1U pacTH-
rI1BaIDLUHX CUJI, >KeCTI(OCTb ynpyroro 3JlC-
MeHTa npH OTHOCHTeJIbHOM e.l1.IiHJ1lJHOM
CMemCHlln 3ameMJICHHbIX KOHQOB OITp e .l1. e -
mleTCSI Ii3 cOOTHOrneHHH (3.1):
k lI = 12£11/ 3 = £b l1 ( c; J
(3.40 )
BeJIU'lHHa (3.40) SlBJISleTCSI npe)J,CJIbHOll
,llmI paBeHcTBa (3.39) npH k -+ O.
llpu-\lep 3.5
BbI'UlCJ1HTh 3aBHCliMOCTb )KeCTKOCTH ynpy-
roro 3JIeMeHTa C napaMe1]JaMH E = 1,4. 10 t I HJr..f2.
/ = 5.10-4 M, h n = C n = 5.10-6 M B cpYHKl(1U1 OT
nepeMellleHIDI !>.y ero 3allleMJIeHHOrO KOHl(3.
Ilo cpopMYJIe (3.40) (npH !>.y = 0) Haxo.llHM
k Il = ki I = 0,7 HIM. Bbl'UlCJIeHHe no cpopMYJIe
(3.39) J],lliI 3a.n.aHHb1X 3HalJeHHH !>.y npHBe,neHo
HH:iRe:
f.l! 1.10-61\1 61,I.1O- 7 M T,1.10-4H !O:, ].10-4 k 1.10- 2 k//c n k ll , HIM
. , ,
5 0,25 1,75 0,5 2,45 2,45 1,117
10 1,0 7,0 2,0 4,9 4,9 2,342
15 2,25 15,75 4,5 7,35 7,35 4,325
20 4,0 28,0 8,0 9,8 9,8 7,034
}lID! Ha:rJUlll.l:lOCTH pe3YJIbTaTbI npe,nCTaBl1e-
H1d T3I0Ke Ha pilC. 3.13
Kax BH,!I.HO 83 npHMepa, npH nmlBJIeHHH
paculJ"l{BalOI.UHX CM B ynpyroM 3J1ef>1eHTe BCJlelI-
cmHe OTHOCmeID>HOrO CMeweHID! ero 3alIIeM-
kj
7
S
4
3
2
1
OSlO IS WAy.HO !of
PRe. 3.J3. 3aBHeHMOCTl> kit = /(Ay)
JIeHH X KOHUOB :iKeCTKOCTb ynpyroro 3JIeMeHT3
CT3HOBHTCj! HeJIHHeiiHOii .pYHKl(Heii.
Cne,nyer OTMeTHTb, lJTO npH'mHOH paCTSlm-
Ba10WHX CM 8 ynpyrOM 3JIeMeHTC MOryr 6h1Tb
TeXH0l10mlJeCKHC npollecc H3rOTOBJIeHH Ml1K-
pocrpYKTYp.
3.1.3. MeM6palibI MMKpo.naTIJHKOB
JIaBJIeHIIH
B MHKpO.l1.aTtIHKaX ,l\aBJIeHHJI (M)IJ1.)
MOryT npHMeHSlTbCJI KaK KPYfJIblC, TaK H
npJlMoyronbHble B nJ1aHe MeM6paHbl. TeopHSI
TOHKI1X lVIacTl1H ,l\aBHO H rny60Ko pa3pa60-
TaHa [25], TaR :>Ke, KaK II BapHaHTbl HX HC-
nOJIb30BaHHJI B Ka'lecTBC MeM6paH [5, 6, 10].
Ha npJ.!Mcpe KBa.llpaTHblX MeM6paH pac-
ynpYrl1E rIO.L\BECbl f.1 MEM6P AHbl
161
CMOTpl1:\1 XapaKTepHCTHKH, HaH6oJ1ee Ba)l(-
Hble no npl1MeHeHHIO B Ka4eCTBe 3 MM.
A. MCMOp3H3 O.nlfH3KOBOii TOJIll{lflfLI
3ameM;IeHHaSi no KOHTYPY KBM.paTHaJ1
MeM6paHa pa3MepOM AxA H TOJImHHOH Cn,
HarpY)I(CHHaJ1 paBHOMepHO paenpe)J,eJIeHHbIM
pa3HOCl HbIM aB:ICHHeM D.p, nOKa3aHa Ha
pHc.3.14.
ECJlH 4aCTOTa 113MeHeHHSI HHTeHCHBHO-
CTH D.p 3Ha<IHTeJIbHO MeHbllle 'laCTOTbl KOJIe-
6aHHH OCHOBHoro (HH3Illero) TOHa, YHKUHSI
nepeMemeHHSI TO'leK MeM6paHbI I1MeeT BH)J,
[22]
W{X,z) = A(l-COS 2; )(1 cos 2;). (3.41)
A nA 4 Ec 3
r,lle A = LJjJ [ D = n - WiJIHH)J,-
3271: 4 D 12(1-v 2 )
pH'lCCKaSi )l(eCTKOCTb nJIaCTHHbl (v - K03 -
HUHeHT nyaccOHa)].
O'leB HO, MaKCMMaJJbHOC ITepeMeIl1e-
HHC COOTBCTCTByeT UCHTpY MeM6paHbI C
KOOp)J,HHaTaMH x = A12; z = A12:
IJpA 4
W rnox = 4 .
871: D
(3.42)
Onpe)J.CJIHM )l(eCTKOCTb MeM6paHbl B
HanpaBJIeHHH OCH y
G = t-,pS _ I1pA 1
)'
W rnax W max
8rr 4 D
A 2
(3)J,eCb S = A-)
(3.43)
H 'laCTOTY OCHOBHOro TOHa MCM6paHbl:
! - fi V _rr1 £ D
- -;;; - A 1 PC n '
(3.44)
r)J,c m = SCnP - Macca MeM6paHbi (p - llJIOT-
HOCTL MaTepHana MeM6paHbl).
3aITHlUCM OTHOCHTeJIbHbIe y)J,JII1HeHlliI
MeM6paHbI B HanpaBJIeHHH oceH x H Z [22]:
x D.pS 2ID' ( 2nz )
e =-=-y cos- l-cos- .
x RJ 871: 2 D A A '
z 6pS 2m ( 21L\' )
e z = R 2 = -y 81t 2 D cos A l-cos A .
(3.45)
BbIpa)((CHHSI (3.45) n03BOJI.H.IOT oITpe)J,e-
mlTb MCCTa pacnOJIO:>KeHIDI pe311CTOpOB Rl,
R2 (TonOJIOrMIo) Ha MeM6paHe. ECJIl1, Ha-
npl1Mep, nJIOCKOC'Th HX pacnOJIO:>KeHIDI OIT-
pe)J,CJ1S1eTOI KoopJJ.HHaToH y = -c n /2, TO B
TO'iKaX C KOOp)J,HHaTaMH x = 0, Z = AI2 H
x = A, z = A12, a TaK:>KC Z = 0, x = AI2 11
z = A, x -A 12 HMelOT MeCTO MaKCHMaJJl>HbIC
)J,e opMauHH, onpe)J,eJIJ1eMble (j>OPMYJIOH
f¥JCn S
e rnax = J .
871:- D
(3.46)
04CBH)J,H0, B TOtIKe e KoopJJ.llliaTaMH x =
= A12, Z = AI2 JJ.e opMaul1S1 TaIOKe onpe,l\c-
JISlCTCR OPMYJIOH (3.46), HO co 3HaKOM "-"
y
A .. -4
Cn tR C n
.. R '.
'-'_ I ." .
D.P '-
........,.....
. ,"t:1K '.
.' ,
, I '" X
,
,
PHC. 3.]4. .lI.e opMsQHH KB8.QpSTHOH MeM6psHLI, HsrpYlKeHHoH ASBJleHHeM
162 rJ1!lBa 3. 3JIEMEfffl-lMI EA3A MI1KPOMEXAHl1l.J:ECKI1X rIPI1EOPOB
A
!t 1111 tt
y
3AI4
A
1-1Il} I 111
x
1
11 It 2 1 till
Pac:. 3.15. ,!(e4jJopMaQHH Cpe,rmHHOii D.'IOCKOCTH 3ameM.1JeHHOH no KOHTYP}' MeJlftipaHbI:
I - reoMeIpH'lecKoe MecTO TO'leK C HyneBoll OTHOCHTeJIbHOll JlellJopMaQHell;
2 - Koop.!UIHaTH!UI TO'lKa, COOTBeTCTB)'IOIU!UI HYJIeBOll OTHOCIITeJIhHOll JlellJopMaQlm;
3 - TO'lh.1i C MaKCHM8.TIbHOll OTHOCllTeJIbHOll JlellJopMaUHell
B TO'IKaX C MaKCHMaJJbHOH OTHOCH-
TeJIl,HOH .!lecpopMauHeH MCM6paHbI Heo6xo-
.;!!;HMO pacnoJlaraTb (HMI1J1aHTHpOBaTb) pe31f-
cropLl, BKJIIOqaeMhle, KaK npaBHJ10, B MOCTO-
Bble CXeMbl. B TO'llCaX C KOOp,lUfHaTaMH
x = A!4. z = Al2 II X = Al2. z = AI4, a T8IOKe
z = A/2, x = 3AI4 II Z = 3A/4, x = AI2 ):\e$op-
MaI1HH B)'JIeBLle, II B 3THX Mecrax CJIcJ:I)'eT
pacno.rraran. pe3HCTOpLI JtJISI TeMnepa'I)'p-
HLIX KOMDeHcarurn.
T O'lKH C H)'JIeBOH omOCHTeJlbHOH l1.e-
4x>pMaUHeH CO()'ffieTCTBYJOT TO'll<aM nepem-
6a JIJmHif. nOJJ)"laeMLlX B pe3YJILTaTC ce1JC-
HH.H JJ:ec)opMHpOBaHHOH cpellHHHoH llJIOCKO-
CTH MeM6pa.Hb! llJIOCKOCTIIMM, npoxOAALD:HMfI
1Jepe3 TO'lK)' C KOOp)J,HBaTaMH x = A/2,
z = Al2. II nepneHJuncYJIHpHLIX l( JlJIOCKOCTH
xz(pHC.3.15).
nO)J,CTaBHM mo6YJO M3 KOOpJ1.liHaT TO-
1JeK C HyneBoH OTHOCflTeJJbHOH ,l\eI}JopMaQI1-
CM B BLlpa:>KCHliC (3.41) If nOJ1ytfHM ncpCMC-
lltCHllC T01JCK ncpcrH6a, M:
b.jJA 4
(j)n - 161t 4 D .
(3.47)
nplL ep 3.5
J(mI Ipex KpeMHHeBblX nJIaCTHH (MeM6paH),
BbIpe3aHHbIX B IIJIOCKOCTH (100), co CTopOHaMH
KB8 1I. "a"'" M' A = 10- 2 . 0 5 . 10- 2 . 0 2 . 10- 2
l' ..a. . , ,., ,
HMelOIUHX C n = 2 . 10-4 H 10-4 M, pacCIJHTaTb :lKe-
CTKOCTH, MaKCHMaJIbHble OTHOCHTeJIbHbIe Jle<pop-
Ma H JlJlJl !:J.p = 1 KITa H I MITa H 'laCTOTbI co6-
CTBeHHblX KoneGaRHH. rIpH paC'le-rax HCnOJIb3Y-
JOTC!I CJIeJIYlOlllHe )laHHbIe; £(100) = 1,69 . lOll H/M 2 ,
v = 0,358, p = 2,33 . 10 3 Kr/M 3 . Pe3YJIbT8TbI pac-
'lerOB no <popMynaM (3.43), (3.44), (3.46) nOMe-
eHbJ B Ta6JIHUY.
I1 '1p"M eTpY A, I. 1O- 2 M G,.. 1 . 10' HIM !:J.p = I KI1a !:J.p = 1 MITa, In, I . 10 4 l/c
e"""" I . 10- 7 e max , 1 . 10-4
c.=2-1trM, 1.0 0.1 19,74 19,74 18,72
D=0.129 H-.. 0,5 0.4 4,94 4,94 75,35
0.2 2,52 0,79 0,79 464,87
c.= ItrM, 1.0 0,012 77,7 77,7 9,16
0,5 0,05 19,55 19,55 36,94
D=0,G16H'M 0.2 0.32 3,15 3.15 47,77
,
,
YrIPYrHE rIO.L\BECbI H MEMEP AHbI
163
113 IIpllBCHCHHOI 0 IIpl1 ICpa CJlC.ayeT. 'iTO
npn D.fJ = I MrIa MeM6paHLI CO CTOpOlIaMH A = I
n 0.5 CM TeOpen!'lecml '>10rYT H:\ICTb Ta!(JfC
60JlLWlle .L\cqJOpMaUIIII, 1\0 ropLle HB npaKTlII<e
npllBCJIII 6bl K IIX paJPYIIJCHHlO. C06CTBeHHbiC
'!aCTO"! hI I<OJlc6aHHH naCTOJlbKO lJeJUIKH, 'iTO npll-
IUIToe .aonyweHlle 0 B03MO}J(HOCII! paCCMOTpCHlH1
CTaTll'JeCl\O/"O pe}J(IIMa paGoT!.1 MCM6paHh! JjJlJl
PCWClIlIlI :m.=ra'Jll 0 ee .acljJopMalUlII BnO.THe on-
paB.L\alIHO.
E. MC 16paHa C "'CCTKIfM lIeHTpOM
KBaJIpaTHaSi MeM6paHa (n:mcnma) C
pa3MepaMI1 AxA no KOHTYPY HMeeT :>KeCTKI1H
Hej:\e<!)OpMIiPYC:\lbIH ueHlp C pa3MepaMI1 axa.
B :nOM cjJY'lae .pynKUHH (3.41) npHHHMaeT
Bn)J,
( 21L\ 0\ ( 2nz J
w(x,z)=A I-cos ) I-cos , (3.48)
A-a A-a
f)J,C
A _ D.j7(A-ar
32n 4 D
MaKeHM8JIbHOe nepeMemeHHc
.popMHpyeMoro )[(eCTKoro UCHTpa (x
A-a
) paBHo
2
He.1e-
=z=
{A-a)4
w max = D.p 4 '
8rc D
(3.49)
a :>KeCTKOCTb B HanpaBJIeHl-Il1 OCl1 y onpene-
J1HeTCSI OPMYJ10H
8n 4 A 2 D
G y = (A-a)4 .
(3.50)
flpll.\tep 3.6
,lIJIJI KpeMHlleBblX KBaJ1paTHbIX nJIaCTliH
(MeM6paH) C pil3Mepa:o.m CTopOH, M, A = 10- 2 Ii
0,5 . 10- 2 Ii HMelOlllllX TOJlunIHbl c n = 2 - 10-4 Ii
10-1 M pacc'nnaeM 1KeCTKOCTb 1I MaKCIiManbHbIe
nepeMe ellIDl ,!J;JIJI t.p = 1 KITa II ,!J;JIJI pa'3J1I1'IHbIX
3Ha'ieHHM OTHoweHIDI alA (alA = 0 03Ha'laeT, 'ITO
1KCCTKUi1 ueHTp OTCYTCTBye-r). lJ,HJIHHJlPH'JecKHe
>KeCTI\OCTIi B3RTbl H3 npeD;blD;ywero npliMepa.
Pe3YJIbTaTbl paC'ieTOB no qJOpMYJIaM (3.42),
(3.43), (3.49), (3.50) npeJ1CTaBJIeHbI B cJIe.!{ylOLQei1
Ta6nllue:
C =2-IO-4 M D = 0,129 H'M -4 D=00l6H'M
-' C n = 10 M
A, 1 . 10 - M alA n ,
G. 1 . 10 7 HIM (1)max, M G., 1 - 10 7 HIM Wrnax,M
0 0,1 )07 0,012 83,3 . 10- 7
1,0 0,2 0,24 4,1 - 10-8 0,029 32,92 . 10 8
0,5 1,6 6 2 . 10- 9 0,192 5,02 . 10- 8
,
0,8 62,5 1 6 . IO- w 7,5 0,13 . 10-8
,
° 0,4 06.10- 8 0,05 0,5. 10- 7
,
0,2 0,97 ° 25 . 10- 8 0,12 2,05 . 10- 8
0,5 ,
0,5 6,4 3 89 . 10- 10 0,6 0,31 . 10-8
,
0,8 250 9 96 . 10- 12 31,25 8.10 11
,
113 npHMepa Cne.l1yeT, 'iTa rrpH qJ1iKCHpO-
BaHHOM 3Ha'leHHH Co. 1i3MeHIDI OTHoweHHe Pa3-
MepoB a/A, MO>KHO nOJIY'IHTb q3 M.lUJ, .l1JIJ1 Wli-
pOKoro D;Hana30Ha Ii3MepJleMbIX .n;aBJIeHHH, npH
KOTOpbIX 3Ha'JeHlte W max OCTaeTCJI B JlOnYCTIiMbIX
npeJlenax.
O<leBH,l\HO, <ITO MeCTa MaxCllMaJIbHOrO
HanpH)!(eHHH MeM6paHbI pacnOJIaralOTCSI ITO
BHelliHeMY KOHTYPY ee 3allJ,eMJICHI1S1 H ITO
KOHTYPY ce rpaHH4bl C )!(eCTKHM ueHTp0M.
npH H8JIH'lHH )!(CCTKOrO ueHTpa npC,l\nOQTll-
TenbHec H3MepHTb He ,l\e!)JopMa4HM MeM-
6paHbI TeH30pe3HCTopaMM, a nepeMellJ,eHIUI
:>KeCTKoro UCHTpa eMKOCTHbIMH npeo6pa30-
BaTeJIHMH.
qaCToTa C06CTBCHHbIX KOJIe6aHHH MeM-
6paHbl C :>KeCTKHM ueHTpOM onpeJJ.enRCTCH
OPMYJIOH
rc;;-
W=V '
(3.51)
rJJ.e G y - :>KeCTKOCTb MeM6paHbI, BbI'IUCJUle-
MaJI no !)JoPMYJJe (3.50); mlj, m n COOTBeTCT-
BCHHO Macca :>KeCTKOro UCHTpa H ynpyrOH
nepeMbI'IKH.
164 rJIaBa 3. 3JIEMEHTHNI I>A3A MHKPOMEXAHH4ECKIIX rIPHEOPOB
OqeBHlUlO, npJf YCTaHOBKC JIOITOJIHIf-
TeJlbHbIX Mace Ha :>KCCTKOM ueHTpe 11X 3Ha-
':IeHHe JI06aB.mleTCSI K 3HaQeHlUO MaCCbI :>Ke-
CTKoro ueHTpa.
nplLuep 3.7
BbI'IHCJIHM 'lacroT)' e06CTBcHHhIx Konc6a-
HIDi n:laCTHHhl C )KecTKHM ueHTpOM 11 OUCHIfM
norpCIlIHOCTI, B Bhl'llleJJCHIUIX, eeJJll He Y411Thl-
B3Th Macey nCpeMhl'lKII. IlcxO..'1HhIC .aaIllILle: c n =
= 0.1 . 10- 3 M. C II = 35 . 10- 3 M (TOJJIllIIH3 iKCCTKO-
ro Uemp3). p = 2,3) . 10' Kr.lM 3 . 3H34CHlUI A. alA.
D. c.,. B3HTbi 113 npc.abm)'lllero npllMepa. Pe3y.TJL-
TIlTb! Bhl'me,1CHIlIt no (!JOpMYJle (3.51) npllBCJIeHbl
B TIl6mmc:
G,> f£
A. a!A D = 0.016 H'M. mU' 111m , . m . s.%
).10- 1 . M. c.,. ).10 HiM 1.IO . Kr 1.1O . Kr mU + I1In
.1 1 . 10.1 lie
1 . 10 lie
02 0,029 0,033 0,22 10,7 29,64 63,9
1,0 0,5 0,192 0,21 0,17 22.47 30.24 25.7
0,8 7,5 0,52 0.08 111.8 120,1 6.9
0,2 0,12 0,008 0,055 43,64 122,47 64,3
0,5 0,5 0,6 0,051 0,043 79,89 128,46 26,3
0,8 31,25 0,13 0.021 454,92 490,29 7,2
fu npHBell,CHHOrO npHMepa Cne.nyeT,
'ITO Macca ynpyrOH nepeMhI'IKH cymecTBeH-
HO B:"IIDleT Ha 'laCT01)' I<O!le6aHHH MeM6paHhI
C ]KCC'1"I<IfM nempoM.
3.2.llP.HMLIE llPEOEP A30BA TEJIH
3.2.1. EHKOC11ILle npeo6pa30BaTeIln
nepeMem:eBHH
Pa60Ta npeo6pa30sareJIeH 3TOro THna
OCHOBaBa Ha H3MeHCHHH eMKOCTeH Me)J(.!Iy
no.t1B 3JIelCTpOlI,OM, paenOJIaraeMhIM
06bl'1BO Ba DO.iIBIDKHOM 3JIeMeHTC npJf6opa,
H HeDO):(BHjiCHl.lMH 3JIelCTpO.!I,aMH, pa3Me-
II18eMIolMH Ha 3JIeMeHTaX I<opnyca
(pRe. 3.16).
EMKOCTH Me-Ar.!lY COOTBeTCTB)1Ol.\U1MH
n 3JJelCTpOJJ;OB ODpeJIeJIJUOTCj{ 3a.BHCJf-
UOCDfM:H:
c - £0 0 .
I ho+Ah'
c _ E.oES
2- ho -f)h'
(3.52)
r.ne E - ,rot3.JIeJ(1pJPIeCJCaR npoHHLtaeMOCTb cpe-
JU>I UeAGt.Y 3JIelCfjX)JIaMH; Eo = 8,85 . 10- 12 <1>JM;
S - 1VIO JtJ. B'3aIDfHOro nepeKpLl1'Wl 3JIeK-
1pONJIr. ho - HaqaJU,m.m 3a3Op Me7ICJtY 3JIeK-
1pO.naMH; Ah - H3MeHeHHe paccT()gJ{Hjl Me)I('-
JI.Y 3neICIpO,WiMH.
Be.JUI'lI1Ha !:JJl jIml flJmei-iHoro ITepeMe-
IlJ,CHlUI 03Ha'iaeT nocT)'naTe.1IbHOe ITepeMe-
llJ,eHl1e nO.lJ;BlfiKHOro 3JIeKTpOjIa, a ,1I,.rUI ymo-
Boro nepeMellleHJ.UJ OITpelleJI.IIeTC}J B COOT-
BeTCTBMH C pHC. 3.16,6:
!J.h = +(y r + aB).
oj
,-
-;,,-::! .?,,:,> '¥'; "'I:'.I 13
5 1
z I
.
.-
.;[
.'
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." (/
6)
-
--
13
o
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0""
..;:
PHC. 3.16. EMKoCTHLIe npeo6pa30B8TCJJH
nepeMemeHHii:
0- JUfHeHHbll:f; 6 - ymoBon; I H 2- nO.llBlIncHblH
H HeOoJtBHncHLIe 3neKTpoJ.tbl
rIP5IMbIE IlPEOEPA30BA TEJIYI
165
a)
CJIOH j),ml){lW3JU1 (fl')
1z;J. 10
Si
t
Si
" ."' ;Lt >.t_ ft .. {;. ! . :
6)
SiD 2
CJIoii MeTaJIJIH3aLU1Jl
t'::J flu
Puc. 3.17. no,aBHiI\Hblll H HenOLlBHiI\HblH )jleKTpO.l!bl, IlJrOTOB.leHHble:
a - .l!HC1Jl!>ymeii; 6 - MeTaiIJIH3al\lIeii
113 3aBlfCHMOCTeH (3.52) cJIe)),yeT, qTO
H3MCHeHHe napaMeTpoB E, S, I>.h npHBo}:\HT K
H3MeHCHHIO eMKocTCH npC06pa30BaTcnSl.
)lml HCKJIIOlIeHJHI BJIl1HHlUI H3MeHeHHH nJIO-
rna.LlH B3aHMHOrO ITepeKpblTHJI 06bIIIHO ITJlO-
rna)J,b O,ZJ;HOro 113 3J1eKTpO)J,OB, HaIIpl1Mep
ITO)J,BI1>KHOrO, AOJI>KHa 6bITb MeHbrne nJIOllJ,a-
11H 11pyrux 3neKTpO)J,0B. HCKJIIO'lCHI1C BJIIDI-
HI1J1 B03MO:>KHOrO H3MeHeHHH napaMerpa E
ocyllJ,eCTBJIHCTCH B 3JIeKTpOHHOM CXeMe npe-
06pa30BaHHH..
Ha,llo I1MCTb B BH)J,Y TaK:>Ke, <!TO MeJI<1lY
Ka)!()J.OH napOH 3JICKTpO.LlOB CYllICCTBYeT CI1-
JIOBoe B3allMOJ1.eHCTBHe (TJI>KeHl1e), onpe)J,e-
JIJleMOe OPMYJIOH
FT:=ql/(&o£s); i= 1,2,...,
(3.53 )
me q, - 3JIeKTpHIIeCKI1e 3apMbI Ha 3JIeKTpo-
11ax.
KpoMe Cl1JIbI TSDKeHHH. B }:\aT'lMKax C
nO)J,BH>KHbIMI1 Y3JIaMl1 oTpHQaTeJIbHYIO pOJIb
HrpaeT TIlK:>Ke 3$$eKT 3JIeKTpoCTIlTH'leCKOrO
"3aJIHnaHIDI" .
"llpOCTOH 11 3$ eKTI1BHbIH MCTO,l\ I1C-
KJIIO'ICHIDI "3aJIHnaHHH" 3aKJIIO<laeTCSI B BbI-
nOJIHCHl111 oxpaHHblX 3a3eMJIeHHbIX KOHT'Y-
pOB npOBO.D:HMOCTH, BbInOJIHSleMbIX Ha He-
IIO)J,BH:>KHbIX 3JIeKTpO)J,ax. PacnOJIO>KCHHe
oxpaHHbIX KOHTypOB )].OJI:>KHO 6bITb npOTHB
OcrpbIX pe6ep 11 KpOMOK ITO,l\BH>KHbIX 3JICK-
TpO}:\OB, T.e. B MecTax KOHUCHTpawrn Cl1JID-
BbIX JII1HI1H 3JIeKTpH'IeCKOrO noJISl, C ueJIbIO
OTBO,ZJ;a 6JIYJI<1laIOIlJ,I1X 3apSl)J,OB Ha "3eM-
JIIO" [6].
B CMKOCTHbIX npe06pa30BaTeJIJiX nc-
peMellJ,eHHH, KaK B JII06LIX KOHcrpYKI.{lUIX H3
pa3HOpO.QBhlX MaTepHaJIOB, Orn.yrHMhlMH
MOm 6bITb TeMneparypHble 01lIM6KH, 06y-
CJIOBJIeHHbJe JIHHCHHbIM paCllIHpCHHeM 3JIe-
MeHTOB npe06pa30BaHHH. TIOABMiKHble 3!1CK-
T"p011bJ 06b[lmo nonytJaIOT JIH60 .D:H $Y3HeH
npOBO,llSllQHX 06JIaCTeH, OT11ejUieMblX OT
KpCMHHeBOH IIJIaCHIHbI V -06pa3HbIMH Ka-
HaBKaMH, JII160 MeTa;lJI113aUl1eM Ha 60pOCH-
JIHKaTHbIX CTeKJIax, 'lTO lfJIJIIOcrpl1pYeTCSI
pl1C. 3.17, Ha KOTOpOM H306pIDKeHbI clJpar-
MeHTbI q3 aKCeJIepoMeTpa.
TOJIlQ 1Ha CJIO}! MCTaJIJIH3aQHl1 Bbl6Hpa-
eTCJI M3 YCJIOBHJI nOCTOJIHCTBa 3a30pa ho npl1
113MCHCHI111 TeMnepaTYPLI, qTO nplfHUl1nl1-
aJIbHO B03MO:>KHO npl1 paBeHCTBc npHpame-
HI1M 3a30pa H TOJIIUlfHbl CJIOJl MeTaJIJIH3aUHI1:
a T .... p
11" := 110
aT ""1' + aT "
(3.54)
,
r)J,e h" - TOJIII.{HHa CJIO}! MeT3mU13aQlU1;
kp' M - TeMnepaTYPHblc K03 HUI1CHTbI
JII1HCHHOro pacllIHpeHHH KpCMHIDI 11 CJIOJl
MeTaJIJII13aUI1H.
MeTa;1JI113aQIDI 06bI'lHO BbillomUleTCX
no nO,ll,CJIOIO, HanpHMep aJIIOMMHIDi no xpo-
MY, AJIJI CHIDKCHIDI TCMncpaTYPHoro HanpSl-
:>KeHIDI, onpe.LleJIJleMoro ITO $opMYJIe
E-
crT = I /j.a/!,.T,
I-v.
I
r.ne E;. v, - MO.nYJIb ynpyrOCTH H K03$$1l-
QHeHT TIyaccoHa i-ro KpHCTaJIJIorpaclJJ1qe-
CKoro HarrpaBJIeHIDI cOOTBeTcTBeHHO; 1>.0. -
p83HHUa MC)K)J,y TeMncpaTYPHbIMH KOO$$H-
l1HeHTaMH JIHHeHHOro paClliHpeHIDI conpx-
raeMLIX MaTepl18JIOB; I:1T - TeMnepaTYPHh1H
,l1J1arr83oH.
166 rnasa 3. 3JIEMEHTHAJI 6A3A MI1KPOMEXAHI1G.ECKI1X rIPI1EOPOB
a) -U on
+U ou
P1 I Cl R2 R4
Rl
41
C2 R3
q q 1 1 3 4 T q
I
q
K03 HI.lHeHT JIlmeHHOro pacmHpeHm!
nO,lJ.CJIOR orrpe,l\eJISleTCR KaK CpC,!lHeC 3Haqe-
HHe H3 K03 HUHeHToB J1HHeHHOro pacnm-
pCHIDI corrpSiraeMbIx CJIOeB.
llpwuep 3.8
PaCCqHTaeM TOJIIIIIIHY CJ10Jl MeTIlJlJlH3aL\1I1I
II.. amOMHHIUI II TOJIWUH)' C!)OJl .lI1I1}J1}Jy:31111 II:!
nerHpoBaHHoro Io..-peMHIIJI, BbInOJUIJleML)X H8
II.-peMHHII. K03<}JI}JIiUHeHThI JIHHeUHoro paclIIHpe-
HIDI: J)..'VI Io..-peMHIUI <X.r = 2,6 . 10-t. oc- I , ;WJI nerH-
poB8HHoro KpeMHIDI <X.r = 1.6 . 10-6 DC-I, .IJ.JIH. aJl}Q-
MHHID1 a,. = 23,8 . 10--6 °C-I. )J.ml 3Ha'leHHJI ho '"
'" 10 . IO-t. M no cpopM)'.i1e (3.54) nOJ)}"IJlM:
h,. =10.10- 6 2,6.10-{; - 0,98.10- 6 M
(2,6 + 23,8)10- 6
(-I MKM).
TOilUIHH3 C.'-IOJI ;J,Ucpl}Jy:3HH BLI'iHCmICTCJI
TalOKe no .popMyne (3.54):
-{;
II =10.10-6 2.6.10 - 6.l9.1O- 6 M
.3. (2,6 + 1,6 )10- 6 .
(-6 MKM).
EMKOC1Hhle rrpeo6pa30BaTeJIH nepe-
Mememm xopomo CO'ieTa.lOTCR C KOHcrpYK-
UHeH H TeXHOJIOrlUlMH M3MC H HaxOIDIT
mHpoKoc rrpHMeHeHHe.
r'
0)
/!x
MHTerpaTOp
UelIb o6parnoH
CBJl3H
Ha pHC. 3.18 npHBe)J,eHa CXC!.1a BlOIIO-
qeHHR CMKOCTHoro npc06pa3oBaTcnJl nepe-
MellJ,eHIIH [6], B KOTOpOH 06ecne<IeHo OTCYT-
CTBHC BJIHSlHHJI .n.H3JICKTpHqCCKOH ITpOHH-
llaeMOCTH CPCJ].bl Me>K.A)' 113MepHTeJIbHbl!\1H
3JICKTpo.n.aMH I1 HCKJIIO'lCHO TJDKCHHC Me)[()J,Y
113MepHTenbHblMlt 3neKTpO,!laMII.
no YHKU.J10HaJIbHbIM npH3HaKaM CXC-
MY MO)!{HO pa.3.n.eJIHTb Ha cnenYlOlllHe CO-
CTaBHble qacrn: 1 - reHcpaTop MCaHJ1.pa,
nUTalOll.(HH nocpc.n.CTBOM KJIIO'leBoro YCT-
pOHCTBa eMKOCTHbIH MOCT; 2 aKTI1BHblH
cYMMaTop CHfHaJIOB; 3 - CHHXpOHHblH )J,e-
TeKrop; 4 - liHTerpaTop Ii UeITb oTpliuaTeJIb-
HOH o6paTHoH CBJl3H.
,[{sa nne<18 eMKOCTHOrO MOCTa ITpe.l!-
CTaBJI.IIlOT J1.l1 tl>epeHUHaJIbHble H3MepHTeJIb-
HbJe KOH)J,eHcaTopbI C 1 Ii C 2 , a )J,Ba ,l\pymx
nJIeqa - I1CTO'lHHKH onopHoro HanpJl)!{CHHSI
+V on Ii -Van.
H3MepHTenbHbIC KOH)(eHCaTOpbI C] Ii
C 2 BKJIlOqCHbl nOCJlC,l\OBaTeJIbHO, Ii ITOcpe,l\-
CTBOM KJIlOqeBOH exeMbl Knl...Kn4 '3Ta ucnb
3a ITepBbIH nOJIYITepHO)J, MeaH)J,pa 3apR)!{aeT-
CSI OT l1CTO'lHHKa onopHOro HanpSl)!{eHHJI. a
38 BTOpOH, nCpeKJIIO'lHBWl1Cb, ITCpe3apSl)!{a-
eTe}] 3THM )!{e HCTOqHHKOM.
U"bIX
PIU:. 3.18. CxellfLi BICJIIo..eSHR ellflCOCTHoro DpeofipB30BBTeJlR nepeMellleuuu:
a- npHHWfllliaJIbIWI; 6 -crp}'xrypmur; OYI, OY2 - onepaL\HOHHLle YCHRHTeJm;
T - Tp3H3HCTop
IlPJIMblE rlPEOEPA30BATEJII1
167
I1HTerpaTop BbmOJ1HSler )J,BC cPYHKIJ,HH:
BO-nepBbIX, npl1 aeT aCTaTl13M ITpe06pa3o-
BaTemO, B pC3ynbTaTe 'lero TeMnepaTypHblH
peH{l) 3JIeKTpOHHblX Y3JIOB npSlMOH UenH,
OXBaljeHHbIX MeCTHoii 06paTHOH CBR3bIO, He
BJJHJIer Ha TO'lHOCTb ITpe06pa30BaHlUI; BO-
BTOpbIX, B nO)J,rOTOBHTcnbHOM TaKTe COBMe-
CI'HO C KJlIO'lOM CHHxpOHHoro .lIeTeKTOpa
OCYlIJ,eCTBJIHer {l)YHKl\HI1 aHaJIOrOBOH 3aITO-
MHHaIOllJ,eM SlqeiiKH pe3ynbTaTa ITpe06pa30-
BaHI1S1 Ha npe)J,bILlYlIJ,eM TaKTe.
BbIXO)J,Hoe conpOTHBJIeHHe H3Mepl1-
TeJ1bHOH J1.HaroHaJIH eMKOCTHo-pe3HCTOp-
Horo MocTa SlBJIHeTC.H BbICOKOOMHbIM, a ypo-
BeHb oone3Horo CHrHaJIa nOCJIe MOCTa - HC-
)J,OCTaTOqHbIM )J,JIJI )J,aJIbHeHIIIei1 06pa6OTKU.
B CB.H311 C 3TI1M MOCT J1.0nOJIHeH ITOBTOpHTC-
JICM 3apSl)J,a Ha OITepallHOHHOM YCHJIHTeJIe,
KOTOpblH o)J,HoBpeMeHHo CJIY:>KHT LlJIJI
'IpaHcc)JopMl1pOBaHH.H BbICOKOro I1Mne aHca
B HH3KOOMHbIH I1 epaBHeHJUI CHrHaJIOB MOCTa
H IJ,enH 06paTHOH CBJi3H. PC3HCTOpbI cYMMa-
Topa BbIOOJIHeHbI o HHaKoBbIMH (R I = R 2 ).
TJi)KeHHe MC}()J,y ITOLlBM)!(HbIM H Heno)J,-
BH)KHbIM 3JIeKTpo)J,aMH npe06pa30BaTemI
oTcyrCTByeT, TaK KaK npl1 nOCJIe,llOBaTeJIb-
HOM BKJIIO'ICHHI1 H3MepHTeJIbHbIX KOH)J,eHca-
TOpOB 3apJt)..\bl Ha HHX ql H q2 paBHbI.
qYBcTBHTeJIbHOCTb K 3HaKY ITepeMClIJ,C-
HHSI B cxeMe ,ll,OCTlIfaerCJi npHMcHeHHcM
CHHxpoHHoro eTeKTopa, BbInOJIHCHHOro Ha
KnlOqe Kn5. 3HaK YCMJ1IUeJIH. B CHHXpOHHOM
)J,eTeKTOpe onpeLlemleTCSI He nOJIH.pHOCTblO
HanpSl)KeHHJi Ha ero BXO)J,e, a BHellIHllM
YITpaBJIHIOIlJ,HM Cl1rHaJIOM. B Ka'leCTBe ITO-
CJIC HCrO CHfHaJIa Hcnonb3yerCSl TOT :>Ke
MeaH,L\p, KOTOpbIH YITpaBJIHeT KJIID'ICBOii
cxeMOH MOCTa. BbIXOLlHOH CHrHaJI CHHXpOH-
Horo LleTeKTOpa JiBJIHeTOI MeaH,L\pOM, qaCTO-
Ta KOToporo paBHa qaCTOTe 3aLlaIOlllero re-
HepaTopa C aMITJIHTY)J,OH, MO)J,YJIHpOBaHHOH
YCHJIeHHbIM HanpJi)KeHHeM paccomaCOBaHHSI
MOCTOBOH cxeMbI.
IlYJIbCaUHIO BbIXO Horo HanpH)KeHIDI
cxeMbl OT HCCYIlJ,CM ljaCTOTbl 38)J,aIOllJ,erO
reHepaTopa MO)KHO onpe)J,eJIHTb no aMnJIH-
Ty,ll,HOH qaCTOTHOH xapaKTepl1CTJ.IKe 3JIeK-
TpOHHOH 'IaCTH npe06pa30BaTCJISI, JiBJIHIOIlJ,C-
roeSl B )J,HHaMHqeCKOM OTHOllIeHHI1 aKTllB-
HblM HJIbTpOM ncpBoro nopSl)J,Ka.
Ilocpc CTBOM COOTBeTCTBYIDlIJ,ero BbI-
60pa napaMeT}JOB reHcpaTopa H nOCTOJtHHOH
BpeMCHH HHTCrpaTopa nYJIbCaUHH B BbIXOJ1.-
HOM CHrHaJIe MO)KHO CH!f3I1Tb )J,O CKOJIb
yro)J,HO MaJIOrO 3Ha'ICHHH..
B COOTBeTCTBHH C OITHCaHHeM pa60Tbl
H CTPYKTYPHOH CXCMOH (pHC. 3.18, 6) nepe-
)J,aTOlJHM YHKllHH. H3MepHTeJIbHOM UCITU
l1MeeT BH.a
w ( s ) - U BblX = Wl)3
If h Ts+l'
(3.55)
r e W q3 - nCpC,ll,aTOlJHWI YHKl\!fSl q3;
T = R4 . C 3 - nOCTOSlHHM BpeMeHI1 MHTe-
rpaTOpa; s - onepaTOp JIaITnaca.
TIcpC,D,aTO'IHM YHKIJ,HH 3 (CMKOCT-
HblH MOCT) onpe)J,eJIHeTcJt CJIe,llYIDIl(MM 06pa-
30M:
I3 = f1U 1M,
r,ll,e HanpSl)KCHHC B M3MCpMTeJIbHOH )J,Haro-
HaJIll MOCTa npH paBeHCTBe Me)K)J,Y C060H
a6COJIIOTHblX 3HaqCHMH Hanpn)KCHHH nOJIO-
IKHTeJIbHOrO H OTpHllaTeJIbHOrO HCTO'IHMKOB
onopHoro HanpJI)KeHlUI, paBHO
U = UOT! C 2 -C l . (3.56)
2 C 2 +C I
C yqeTOM OPMYJ1bI (3.52) BbIpa)KCHI1C
(3.56) npMHMMaer BH.l1
f1U = (Uon h )/2110 . (3.57)
113 4>OPMYJIbl (3.57) CJIe)J,yeT, 'ITO Ha-
npSl:>KCHMe B H3MepMTCJIbHOH ,D,MarOHaJIH
MOCT JIHHCHHO 3aBHCMT 01' M3MeHeHHH. 3a30pa
Me:>K)J,y 3JIeKTpo)J,aMH. 3HaqeHHe ,D,H3JIeKTpl1-
qCCKOH npoHHuaeMocnJ B 3aBl1CllMOCTH OT
Ii3M'1eCKI1X YCJIOBl1H ollIM6KU B ITpe06pa30-
BaHMC HC BHOCHT.
TaKlfM 06Pa30M, ITOJIarM B ypaBHCHHll
(3.55) s = 0, nOJIY'laeM CTaTU'IecKYlO xapaK-
TepMCTl1Ky npe06pa30BaTeJIJI:
h
U BLJX =U cn -' (3.58)
2110
113 c)JOpM}'JlbI (3.58) BH)J,HO, 'ITO OCHOB-
HbIMM npJ1qHHaMM TeMnepaTYPHoH norpellI-
168 rnaBa 3. 3nEMEHTHMI EA3A MHKPOMEXAHI1 ECKI1X rIPHEOPOI3
HOCTH H3MepMTCJIbHOH ueITH XBJISleTCX HC-
cra6HJIbHOCTb HCTO'iHHKa onopHoro Hanpx-
)KeHWI H 3830pa MeJK,!{j' 3J1eKtpO.ll.aMH.
TeMnepaTYPHytO norpewHOCTb MO)htto
pacc'UITaTb no $opMYjJe
=(<Xc- )I!J.T,
(3.59)
me <Xc - TeMTIepaTYPHblH K03 HuHeHT Ha-
npJDKeHIDI cra6HmrrpoHa; - TeMneparyp-
Hh1H K034>IfJHIU1eHT MaTepHaJIa. BJIHHlOlilCrO
Ha H3MeHeHHe 3a30pa.
TeMTIepaT)'pHblH K03 HlilleHT Ha-
npJDKeHWI cra6H.ilH1pOHa npHMepHo Ha no-
PJUl,OK npeBblWaeT Be..1H'iMHY .lI.pymx OlllH-
6oK. O..:mHM H3 cnoco6oB nOBblIlICHIDI TOq-
HOCTH eMKOCTHoro npe06pa30BaTeJIX XBml-
eTDI BWOOP cra6HJlIrrpoHa, 3HaK TeMITepa-
1)'pHOro Ko $H1.tMeHTa HanpSl:>KCHHH. CTa-
6iL'1H33IUill KOTOporo He COBnarraeT co 3Ha-
KOM re,.mepaTYPHoro K03$$HlilleHTa H3Me-
HelDUl Haqa.'lbHOrO 3830pa.
3.2.2. fipe06pa.30B8TeJJb nepeMeuteHHU
H3 M)J.II-Tp8U3HCTOpe
npe06pa30BaTejTh Ha ITOJleBOM 3 eK-
Te. KOTOpLlH KOHCTp)'XTHBHO XBJIXeTCSI
M.D;rI-TpaH3HCTOpoM C OTOpBaHHblM 3aTBO-
poM (pHC. 3.19), MOlICeT HCn0J1b30BaTbCX B
Kaqecme ..t{a'PIHJCa nepeMemeHHH q3 MA,
M)JJJ; M Mr (6, 7].
J1cTOK H CTOK rpaH311cTopa pacrroJIara-
IOTeX Ha Heno)J,BIDKHOH <mCTIi KpHCTanna, a
3aTBop - Ha nO.ll.BInKHOH 'laCTH lI3, Hanpll-
MCp Ha MeM6paHe .lI.aT'lHKa )J,aBJIeHlliL
KaHaJI npOBO)J,IIMOCTI1 Meil<J],y HCTOKOM
H CTOKOM 3aWlfweH 01' BHCIlIHIiX B0311eikT-
BMH TOHKOH nneHKOU Si0 2 . flpll ITOCTO H-
HOM HanpR:>KCHHI1 Ha 3aTBope U, ero npoBo-
iIliMOCTb U3MCHReTCS! B 3aBIICHMOCTI1 OT H3-
MeHeHHSI 3a30pa 11o, T.e. OT ITCpeMellJ,eHHSI
3aTBopa Ha BeJIH'lIiHY MI.
CxeMa BKJIIO'IeHl1R noneBblX TpaH311-
CTOpOB Tl, 1'2 AiISI Ii3MepeHHSI nepeMeweHIfSi
06mcro 3aTBOpa, pa3MellleHHoro Ha n011-
BH:iKHOM 3JICMeHTe, npHBe11CHa Ha pHC. 3.20.
TpaH3HCTOpbI Tl, 1'2 nocne)J,OBaTeJIbHO
BKJIIO'leHbI B uenb. nHTaeMYIO OT 11Byxno-
mlpHoro HCTO<JHHKa onopHoro HanpR)!(eHl1S1,
YITpaBJISICMOrO YCl1JUnCnCM OY no uenH
OTpHuaTCnbHoH o6paTHoll CBR3H.
,l:{mI .lI.M epCHlillanbHoH cxeMbl BKJIIO-
'leHIDI HMceM CJIe.ll.ytOlllee COOTHOllIeHHe:
v - -V /1R ( 3 )
BLIX - on R;;""[I - Ro 1(2R]) J ' .60
r.ll.C Ro - conpOTHBJIeHHe KaHaJIa ITPOBO)J,H-
MOCTH npH HeihpaJIbHOM nOnOiKeHl1H 3aTBO-
pa; /1R - H3MeHeHHC conpOTHBneHHR Ro ITPH
nepCMCIlJ,CHJ.iH 3aTBopa Ha aenl1<IHHY 1111.
U'3
JarBOp !
Si02 1
U BCl .:::::
<J
Vcr
KaHaJI
npoBOJ(HMOCTH 06e,WfeHHLJH
eJJOH
PIIC. 3.19.CUMa ope06paJOBaTeJJH oepeMe1l\eHHH HS M,lUI-Tp8H3HCTOpe
rIP .HMbJE rIPEOEP A30BA TEJII1
]69
TI 72
110 110
+V UII -V DbIX
R3
+ OY
U BblX
R2 -Von
-V on-VB!'/)(
Puc. 3.20. Cxell.Ja BKJUOqeUHH TpaH3HCTOpOB nH H3MepeHHH nepeMemeHHH
I1MeR B BH,l\Y, 'lTO Ro « R" BTOpbIM
cnaraCMbIM B 3HaMeHarene BblpalKCHI1R
(3.60) MO:>KHO ITpeHc6pe%. )J,anee MO:>KHO
ITpC)J,CTaBI1Tb
LV? = Jll-.h ; Ro = Jlho ,
f)J,C Jl - K03c»c»HL{l1eHT, 3aBHCR HH OT reo-
MeTpW!eCKHX pa3MepOB KaHaJIa npOBOJ],HMO-
CTH, cro )J,H3neKTpH'leCKOH nOCTOSlHHOH H
nOJ],BHIKHOCTH HOCI1TeJIeH npoBo)J,HMOCTH.
TaKHM 06Pa30M, CTaTlf'JeCKWI xapaKTe-
pHcTMKa npe06pa30BaTeJISI nepeMe cHHH Ha
noneBOM 3 eKTe I1MeeT BH)J,
M
V BblX =-V on -' (3.61)
ho
)J,Hana30H /),17 JIHHeHHOCTI1 xapaKTcpl1-
CTHKH (3.61) H ee 3aBI1Cl1MOCfb OT TeMnepa-
TYPbl TpC6YlOT CnClIHaJIbHOrO HCCJIeJ],OBa-
HHSI.
CpaBHeHHe xapaKTepHCTllK (3.58) 11
(3.61) nOIm3bIBaeT, <{f0 npH O,IDmaKOBbIX
napaMeTpax Van, ho KpyTI13Ha xapaKTepH-
CTHKH npe06pa30BareJISI nepeMemeHHH Ha
noneBbIX TpaH3HCTopax B 2 p83a 60JIbllIe,
qeM y eMKOCTHoro npeo6p830BareJIH.. CxeMa
BKJUOqeHI1H TaK:>KC npollJ,C, nOCKOJIbKY npH
3TOM He TpC6YeTCR 3allalOllJ,ero reHepaTopa H
cPl1J1bTpa ,llJISI crnaIKHBaHI1S1 nyJ1bCaUI1H Bbl-
XO)J,HOro HanpR:>KeHHSI.
CpC.lur )J,pymx THITOB npe06pa30BaTC-
JICH ITepCMCWCHHH CJ1CJ],YeT OTMeTHTb Mar-
HHTope3HCTMBHbIH 11 OnT03J1eKTpOHHbIH
npc06pa3oBaTeJ1H.
B pa60Te [1] paccMoTpcH MarHI1TOpe-
311CTl1BHbIH )]aT'lHK nCpeMellJ,eHl1H. Conpo-
THBJIeHl1e nOJIyrrpOBO.llHHKOBOro MarHHTope-
3HCTopa, BbInOJIHeHHOro Ha OCHOBC InSb C
J],o6aBJIeHHeM TCJIJIypa H pacnOJIO:>KCHHOrO
Ha no.nBI1:>KHOH QaCTI1 q3 npl160pa, Hanpl1-
MCp MCM6paHc )J,aT'!HKa J],aBJIeHWI. 113MeHJ1-
eTCH B 3aBHCUMOCTM OT H3MCHeHHH. HH)J,YK-
UHll MarHHTHOfO ITOJ1J1, c!f>OpMllpOBaHHoro
CITellHaJIbHbIM MarHJ1THhIM CJ10eM.
B pa60Te [5] paccMOTpcH 0IIT03JICK-
TpOHHbrn npC06pa3oBaTeJJb ncpcMclUCHHH
Ha 6a3e CBeTOJ],110J],a Ha KpeMHlleBOH no)J,-
J10:>KKe H J],BYX !f>OTO.nHO.nOB, pa3MellJ,eHHbIX
Ha O,l\HOH nJIaCTHHe. TIpH OTp8:>KeHl1l1 CBCTO-
Boro ITOTOKa OT no.nBH:>KHOrO q3 ITpH60pa.
HarrpHMep MaJlTHHl<a aKCeJIepoMeTpa, H3Me-
WleTCJ1 OCBCmCHHOCTb c»OTO.nHOJ],OB, T.C.
ITpC06p830BaTeJIb pa60TacT no CMCiUCHHIO
CBeTOBOfO nOTOKa. B03MO:>KHa cxeMa npeo6-
p830B8TeIDI II C ncpeKphITHcM CBCTOBoro
nOTOKa.
170 rJIaBa 3. 31lEMEHTHA5I EA3A MnKPOMEXAHH ECKI1X I1PI11l0P0I3
3.2.3. TeH30pe3HCTOpHble
npeo6pa30BaTe.JJH .l1e OpMaI\HH
Pa60Ta npe06pa30B8TeJ1eH :noro Tuna
OCHOBaHa Ha H3MCHeHHW xapaKrepl1eTHK
nonyrrpOBo.nHHKOBLIX MaTepnaJIOB B 3aBl1-
CHMOCTJi OT .n OpMau:lll1 q3. IhBeCTHo, liTO
BJIlliIHHe 1I, OpMaIUIH Ha conpOTl1BJ1eHlle
npOBO.tIHHKa CBSI3aHO KaK C 1l3MCHeHUCM
y.neJILHOH npOBOJ],HMOCTH ero MaTepnaJIa,
TaX H C H3MCHeHHCM .nnnHbl H nJIOIUa.!lH no-
nepe<rnoro Ce'leHIDI 06pa3Qa. B npOBOJ1.HI1-
J.."aX BJIIDI.Hlle 3THX H3MeHeHHH npHMcpHO
O.l1HHaKOBO. JI.;"UI nOJIyrrpoBo.nHI1KOB, TaKHX
lC8K repMaHHH Ge H J..-peMHHH Si, H3MeHeHHe
Y.l1e.JIbHOH npOBO)J,UMOCTl1 60JIce cyrn,eCTBeH-
HO, qeM H3MeHeHHe reoMeTpl1'1eCKHX pa3Me-
paB 06p83ua.
B MHKpOMeXaHH'IeCI<HX npl160pax (aKce-
nepoMeIpaX, HaKJlOHOMepax, .naT'lHKaX .naBJIC-
mm Ii .up.) H3.IllJIH npHMeHeHHe TeH3orrpe06pa-
30BaTeJIH .1J.Byx nmOB: lUiq,q,}'3HOHHbIC (HM-
lIillUITHpOBaHHl>Ie) H 3IIHTIlKClfaJIbHbIe.
.lb14MPY3lfOHHbJe TeH30pe:3HCTOpbl rrpc.n-
CTa.BJVIlOT co60H npHMccH n- HJIll p- THna
npoBOJJ,HMOCTH, KOTopLle B BHlI,C }'3KHX no-
JIOCOK BHe.npXlOT (HMlUJaHTIipYlOT) B npl1no-
sepXHOCTHLlH cnOH KpHCTaJI.JIa tjepe3 BCKpbI-
Thl'e OKHa B OKCH)le.
3IIHTaKCHaJIbHbIe TeH30pe3HCTopbI 113-
roTaBJJHBaIOT CJIe.uyroIIJ.IiM 06pa30M. BHa'la-
1
A
a)
JIC Ha nOBcpxHOCTII Kpl1CTaJIna BblpallIlilJalOT
3nllTaKCUaJIbHYlO nnCHKY Ii C ITOMOl.l\blO (1)0-
TOJIl1TOrpa nll HaHOCRT Ha Hee PIlCYHOK )J,O-
pO:>KeK TeH30pe3I1CTOpOB. 3a KOHTypOM KOTO-
poro OCTaJIbHylO <taCTb 3mITaKCMaJIbliOrO
cnoSi y.naJI JlOT.
Ha pliC. 3.2 I, a nOKa3aH Bapl1aHT pa3-
MellJ,emUI ljeTblpex "TOljeljHbIX" TeH30pc311-
CTOpOB 2 )J,lIctJ Y3110HHoro TUoa, I1l\1nJ1aHTU-
pOBaHHblx B MCM6paHY 3 J1.aT<JHKa .naBJICHI1H.
BbITpaBJIeHHYlo B Kpl1CTaJIJIe 1. Ha pnc. 3.21.
6 ITpl1Be)J,eH BapuaHT pa3MClliCHHSI 'leTbIpeX
")J,OpO:>KCK" 3nHTaKCHaJIbHblX TeH30pe3HcTo-
pOB 4, HaHeceHHblX Ha ynpymc 3JIeMenTbI
(6aJIKIl) 3, BbITpaBJICHHbIe BMeCTe C 11M 2
nHHellHoro aKCeJIepoMc-rpa B KpHCTaJIne 1
(ToKonOllBO.'I.SlllinC )J,OpO:>KKII K TeH30pe3H-
CTopaM He nOKa3aHbI).
)J.1l Y31l0HHbIH TeH30pe311CTOp KaK 6b!
co Bcex CTOpOH yrraKoBaH B KpHCTaJIn, ITO-
3TOMY Ba:iKHO COBna1l,eHlie UX KpliCTaJInl1<Je-
CKHX peDlCTOK. B npOTI1BHOM CJIYljae npH
TeMncpaT)'pHOM B03)J,eiicTBI1H yrrPYrHH 3ne-
Mem, HaITpHMep MCM6paHa, Kop06HTCR H B
TeH30pc3HCTOpax B03HHKalOT HanpSl:>KCHUR,
KOTOpbTC npHBO)J,HT K nORBJICHI1IO nO:>KHOrO
CMrHaJIa. 3nl1TaKCHaJIbHbIe TeH30pe3HCTopbI
C Tpex CTOpOH CB060)J,HbI, 11 OHH HMeIOT aHa-
JIorHIO C 06bltjHbIMH TeH30pe3HCTopaMH.
TeH30pe3HCTOpbI RBJISIIOTCR HeJIHHeH-
HblMH npe06pa30BaTenJlMM .l1C OpMaUHI1 B
AI
f :'-' ..- ........of; / ) if ... & & .;. .d. .-.: .
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3
4
6)
PHC. 3.21. TeH pe3HCTOpHble npeoOp830BaTeJlH .nelj>0pM8I(HH:
Q- ORHyJf [/- kpHCTaJIJJ (113 .naT'fHJCa JlaBJIeI1HH); 2 - TeH30pe'3l1CTOphI; 3 - MeM6paHa];
6 -3DHT3KCHaJIbHI.di [J - KJ>HCTaJIJJ (113 nHHeHHoro 8KCeJIepoMeTpa); 2 -11M;
3 - ynpymH 3JIeMeHT; 4 -7eH'30pe3HCTOpbI]
np}]MhIE nPEOEP A30BA TEJII1
171
npHparueHHSI COITpOTHKIeHHH. B pa60Te [5]
ITpHBeJl.CHa cJle)J,YJOllJ,aH. 3aBHCHMOCTb )J,JISI
cna60JIempOBaHHoro KpeMHH5I, KOTOpaH.
cnpaBe)J,JlHBa H ,lljUl CHJlbHOnCrHpOBaHHoro
KpeMHHH:
n
b.R - L K i
- .E
RE I'
,=1
(3.62)
r.lI.e R - COITpOTHBJIeHHe TeH30pc3HcTopa;
b.R - npHparn.c1IHC COnpOTHBJICHHSI TeH30pe-
3HcTopa ITO)J, .lI.e1icTBHCM )J,ectJ0pMauHH; L -
OTHOCHTeJIbHaH. .lI.ectJopMauwI; K, - KmctJctJH-
UHeHTbl, onpe.lI.emlCMble TeopCTH'-JecKH Ha
OCHOBe .lI.OnyruCHHSI CTanfCTHKH 1)0JIbUMaHa
WISI pacnpC.lI.eJIeHHH 3nCKTpOHOB Ha 3Hcpre-
TH'IeCK IX ypOBHSlX .lI.ectJOpMHpoBaHHoj;j KPH-
CTaJIJIH'-JCCKOH pCli1CTKH.
JLJlH YCTpaHeHHH HeJIHHeHHOcTeH TeH-
30pC3HCTOpOB ITPH C03.D.amm ITOJIHbIX MOC-
TOBbIX cxeM 06bl'IHO ITpHMeHSlJOT cne)J,YJO-
ll\HC cnoc06bI:
- BKJIIO'ICHHe 0.ll.HHaKOBbIX TeH30pe3H-
CTOpOB 113 CMe>I<HbIX nJIe'I Ha .lI.ectJ0PMaI.\Hll,
paBHble no MO.lI.YJIJO, HO HMeJOli.{HC pa3Hblc
3HaKH, T.e. O)J,Horo TeH30pe3HCTopa Ha pac-
TIDKCHHe, .lI.pyroro - Ha C:>Karne;
- BKJIIO'IeHl1e TeH30pe3HcropoB Ha
O.ll.I1H BH.D. .lI.cctJoPMauHH, npl1 KOTOpOM TeH-
30pC3HCTOpbI 11- H p-Tl1nOB C O.lI.HHaKOBOH
KOHueHTpaUl1ej;j ITpHMeCeH BKJIJOqeHbI B
npOTHBOnOJIO>I<Hble IUlC'IH Mocra.
B nHTeparypc OTMe'laJIOCb, 'ITO B03-
MO)l(HO YJIyqll1eHHC xapaKTCpl1CTHK TeH30pe-
3HCTOpOB TeXHOJIOfU'IeCKllMM cITOC06aMH.
JLml pac'IeTa TCMncparypHoro HanpSl-
)l(eHHH B MCCTax JIOKaJ1H3aUI1M HCO.ll.HOpO.ll.-
HOCTeH MaTepl1aJ10B no TeMncparypHbIM
K03 I1UHeHTaM JII1HeMHOro pacllmpeHHH.
cYllJ,eCTByer CJJc)J,YIOll\aH. 3aBHCI1MOCTb:
E-
aT = J b.a.b.T,
I-Vi
(3.63)
rAC b.a. - p83Hl1ua Me>K.!IY TCMneparypHbIMll
K03 ctJI1Ul1eHTaMH nHHcMHoro paCli1 peHHH
conpHraeMblX MaTepHaJIOB; b.T - TeMnepa-
rypHbIH Al1an830H; E i . Vi - COOTBCTCTBCHHO
MO.lI.YJIb ynpyrocTH H KmctJctJHUHCHT ITyac-
COHa ro KpHCTaJIJIorpactJH'-JCCKoro HanpaB-
JIeHI151.
HaH.LtCM MHHHMaJlbHyJO BeJIH'-JHHY lle-
ctJOpMal.\HH TCH30pC3HCTOpHOro npe06pa30-
BaTeJIH Emi , KOTOPYJO MO)l(HO H3MepHTb Ha
ctJOHe "TellJIOBbIX J1I)'MOB", T.C. Toro HanpSl-
)l(eHH5I BbIXO)J.l1OrO CHrHaJIa, KOTopoe HMeCT
MeCTO npH HarpCBe TCH30pe3HCTopa, HO 6e3
BuellIHeH HarpY3KH. O'IeBH.lI.HO, HanpSl)l(eID1e
nOJIC3HOrO CHrHaJla TeH30pC3HCTopa V n
)J,OJI>I<HO y)J,OBJIeTBOpSlTb COOTHOli1CHHJO
V n VT,IJ/O,£,
(3.64)
r)J,e V T W - HanpmKeHHe. COOTBeTCTBytOmee
TenJIOBOMY WYMY; 0'£ - OTHOCHTeJIbHaJ! eYM-
MapHaJ! 0llm6Ka TeH30npeo6pa30BaTeJIH.
HaITpSl:>KeHHe CHrHaJIa TeIUlOBblX li1Y-
MOB )J,JISI Beex nonynpoBO,llHHKOB
onpe.neJI CTCJ! opMynoH
V m = 2(kTRb.m)112,
(3.65)
r)J,e k = 1,3807 . 10- 23 ,[QIdK - nOCTOHHHaH.
1)0JlbUMaHa; T - TeMneparypa, K; 11m - llm-
pHHa ITOJIOebl npOITycKaHHH 'IaCTOT (l/c); R-
COITpOTI1BnCHHe TeH30pe3HCTopa, OM.
,[(lliI MOCTOBOH exeMbI C JtBYMSI TeH30-
,lI,aT'IHKaMI1, BKJIJO'IeHHbIMH ,l\J.1 ctJepCHl{H-
aJIbHO. HanpH)I(CHHC nOJIC3Horo eHrHaJla on-
pC,lleJIHCTCSI B BH.n;e
1 b.R 1
V =-U o -=-V o KE ( 3.66 )
n 2 R 2 '
f.lI.C K - K03 HUHCHT TeH30QysCTBHTCJ1bHO-
CTH [K = (MUR)/M,Ii T )]; E - OTHOCHTeJIbHaH
.lI.C OpMaI.\IDI TeH30pe3Mcropa (E = M.,fIT);
V o - HanpH>I<CHHC nHTaHml TeH30Mocm;
iT> to..l T - ,l\JJHHa H npHpall.\CHl1e AJIHHbI TCH30-
pe3I1CTopa.
,L(JISI nOJIynpOBO.lI.HHKOBbIX TCH30naT'lH-
KOB K = 50.. .200.
,[(onycTKMoe HanpSl)l(CHHe nHTaHHH
TeH30MOCTa MO)l(HO onpcnenHTb H3 YCJIOBHH.
HCKJIIOlJ:eHI1J! cro caMop83orpcBa:
uJ I R < 41 T b T [po].
(3.67)
172 rJI8Ba 3. 3HEMEHTHAJI EA3A MHKPOMEXAHl1l1ECKI1X rIPI1EOPOB
r.ne Lr H b-r - )J,JlHHa H llmpHHa TeH30pe3HcTo-
pa; [Po] - .nonyCTHMaH. y}:(enbHaH. MOLl(HOCTb
pacceSlHHSI [Po] < 5. 10 6 BT/M 2 .
06'be.nHHI1B OPMYJIbI (3.64)-(3.67).
llOJl)"iHM
2
MIR=KE=-
&
k T /).Ci)
iTbT[PoJ '
(3.68)
oTKy.!:la
2
E. =-
mm KO
E
k T tJ.Ci)
iT bT[Po] .
(3.69)
IlpU!uep 3.9
PaCCqHTaCM OrnOCHTeJIbHOe H3MCHCHue co-
lIpOTHB.'leHIDI "roqe'lHoro" TeH30pC3HCTopa ,!lID!
HCXO.znn,IX JXaHHblX:
4 = 0,1 . 10- 3 M; by = 0,01 . 10- 3 M; T =
= 300 K; !lro = 6,28 . 10 3 I1c; [PO] = 5 . 10 6 (BT/r.?);
=0.0l.
BJ.l'iHc-'IeHIDI no iPopM)'JIe (3.68) .;J.alOT pe-
3ym.TaT M/R = 1,44 . 10- 5 . l1.Me" B BH.!!)', '!TO
K = 50...200, no.'J)"lHM, '!TO lI..!Ul HCXOJ1HLIX .naH-
HLIX DpHMepa OTHOCHreJlbH ;J.e4J Maulli! 6Yl1eT
HaXO.!lHTLC" BlIHana30He Hr'...IO .
tI}'BCTBHTeJIbHOCTb TeH3ope3HcTOpOB
3aIIHCHT OT I1X KpHCTaJIJIorpa4PH'IeCKOrO Ha-
npaBJIeHIDI. M orrpe.1J.emIeTcSI TeH30pe3HCTHB-
HhIMH 1C034P4PHUHeHTa.. .m. AHH30Tp0nIDI Ma-
TepHaJIa, B KOropLrH HM!LllaHTHPYeTCSI TeH-
3OpC3HCTOp, TaIOKe BJIIDleT Ha ero IJ)'BCTBll-
TeJIbHOCTb (CM. n. 2.1.2).
TeH30pe3HCTOpHhlH npe06pa3oBaTeJIb
oepeMerneHHH MeM6paHbI npeJlCTaBJISleT
co6oif TeH30pe3HCTOpbI, BKJUOqeHHbIe B
MOCTOB)'IO cxeN)'. O'IeBHJ1.HO, HanpIDKeHHc
Ha BbIXOJJ.e MOCTOBOH cxeMLI, HMejJ B BH)lY
aHJD01]X>nHlO JILC3Ope3HCTH.BHbIX K03I}>I}>H-
lIHeHTOB H Marepuana MeM6paHbI, 3aBHCHT
aT TODOJIOnm TeH30pe:3HCTopoB Ha IDIOCKO-
em .MeM6pauu. B 6om..mHHCTBe CnyqaCB
OCHOBHbiM KpHTepHeM BLJ60pa BapHaHTa
TOnOJIOnm pe3HCTOpoB KBJIJleTCjJ MaKCJi-
MaJILHaJI 'l)'BCTBHTeJILHOCTb npeo6pa30BaTe-
JUl. KpynDua xapatrrepHCTHKH ('l)'BCTBH-
Tem..HOCTh) MeM6paHJ>t C TeH30pe3HCTopaMH,
B/(HI.?), onpe.u;eJIJleTCJI OTHoweHHeM:
S=U_lp,
(3.70)
rJle U "''' - BbIXO)J,Hoe HanpSlJKeHl1e TCH30pe-
3HcropHoro MOCTa.
06bJ'lHO TeH30pe3MCTopbl pacnonaraIOT
TaK, liTO y onHoi1 ITapbI TeH30pe3HcTopOB
OTHOCIlTeJIbHOC U3MeHCHue conpOTHBJICHHSI
nOJIOJKIITeJIbHOe (8 ), a Y J\PyroH OTpHI..la-
TeJ1bHOC (8-). OTHOCHTeJIbHOe H3MeHCHHe
COITpOTHBJICHWI HaXO)J,HTCSI H3 paseHCTBa
8=(R'-R)/ R = MI R,
r}:(e R,R' - COOTBeTCTBCHHO 3Ha'lCHHSI co-
npOTllBneHHSI He}:(c.popMHpoBaHHoro I )J,C-
.pOpMupoBaHHoro pC3HCTOpoB.
BbIxonHoe Hanp}lJKCHHC TeH30pe3H-
CTopHoro MOCTa onpCnemIeTC}I paBeHCTBOM
U BblX = O (1 0 +1+1 8 -1).
(3.71)
C CTOM Bblpa:>KCHHSI (3.71) nepenl1-
rueM OTHOUICHHC (3.70) B BI1)J,e
s = ; (10+1+10-1) .
(3.72)
113BCCTHO, 'ITO npH eTC Hanp}lJKCHHH
TOJIbKO B llJIOCI<OCTH ynpyroro 3J1eMeHTa
OTHOCllTeJIbHOe H3MCHeHllC COITpOTHBJIeHHSI
TCH30pe:mCTopa onpc,l\cJ1.!1eTC}I BblpaJKCHHCM
s:: " " , I
U = 11:1 JaI + 1I:J2 a 2 + 1I:16a6,
(3.73)
rne a;, a , a - 1<0MnOHCHTbI TeH30pa Me-
XaHHqCCKMX Hanp.!l:>KeHHH B CI1CTeMC KOOp-
.nHHaT, CBSl38HHOH C TCH30pe3HCTopOM
(pHC. 3.22).
Pa,l\HanbHOe crT H TaHreHQHanbHoe crT
Hanpj{)KeHHe B JII060H TOljKC KpyrJIOH MCM-
6paHbI P8.nHYCOM R MO:>KHO paCCqmaTb no
1}>0PMYJIaM [25]
crT = 3 [R 2 (l + v) - ,-2(3 + v)];
8e..
(3.74)
O"y= 3 [R 2 (I +v)-,-2(1 +3v)].
8e..
npH pacljeTax no opMynaM (3.74)
H}?KHO HMeTb B BHllY 3aBllCHMOCTb v(9) (CM.
np MblE nPEOEP A30BA TEJ1l1
173
D 1+1 .;! !_j EJ:= .
2R
--
PHC. 3.22. Pacnpe.neneHlle MeX3HIIIJ:eCKlIX
HanpSlmeHIIH H3 Kpyrnoii MCMopaHe
B TCU3ope3l1cTope
Ta6JI. 2.1). Ha pHC. 3.22 yroJI 0 xapaKTCpH-
3YCT aHH30TpOnl1l0 ynpymx CBOHCTB MaTe-
pH8J1a MCM6paHbI, a yroJ1 q> - aHI130TpOnHlo
ITbe30pe3HcTHBHbiX K03«P«PHQl1eHToB. OllC-
BH)J,HO paBeHcTBo
q> = 0 + '1', (3.75)
r)J,c \jf - yrOJI nOBOpOTa npo)J,OJIbHOH OCH TCH-
30pe3HcTopa OTHOCHTeJIbHO pa,lU1yca-BeKTopa
B TOllKC pacITOJIO>I<eHHH TeH30pe3l1CTopa.
HMCIDT MCCTO CJIeJlYIOllUIc paseHCTBa [5]:
I 2 . 2 .
0'\ = 0",. COS \jf + 0"1: sm \jf,
, , .,. ?
0'2 = O"r sm- \jf + 0", COS- '1';
O' = .!..(O", - O"r )sin 2\jf.
2
(3.76)
Ha npaKTllKC npllMeHSIlOT pMHaJIbHOe
H TaHrCHQH8J1bHOC pacno.,o cHHC TCH30pC-
3HCTOpOB. B COOTBeTCTBHH C pHC. 3.22 H
paBCHCTBaMH (3.76) nOJIY'lHM .LVI.II TCH30pC-
3HCTOpOB:
. pa).(ll8J1bHbIX ('I' = 0):
, , , 0
0'\ = 0' r; 0"2 = 0'1:; 0"6 = ;
. TaHreHl..!Hfu,bHbIX ('I' = 90°):
(3.77)
.....' .'...... ' 0
U1 = 0"" 0"2 = u,., v6 = .
(3.78)
flpu.:wep 3.10
PaCC'IHTaeM HanpJl)/(eHlle B TeH30pe3HcTope
c paAHanLHoH H T3HreHQHanLHOH opHeHTa HeH
.!I1IJI liCXOI(HLIX .n3HHbIX: nJIOCKOCTb MeM6p3HbI
(OOl);p = 10 6 IIa, R = 5. 10- 3 M; eM = 0,1 . 10- 3 M;
e = 0;45° H 90°.
Hanp.ll)/(eHHe pacc'iHmeM B eHTpe (r = 0) Ii
H3 KpalO (r = R) MeM6paHbJ.
B COOTBeTCTBliH C Ta6JI. 2.1 HaxO,!J)IM: )],lliI
e = 0 H 90° K03<p<plIl.\I1CIIT V = 0,278; )],lliI yrlIa
e = 45° K03<P<PH HeHT v = 0,064.
PC3YJIbTaTbI Bbl'UlCJIemlii no <popMYJIaM
(3.76) C Y<JeTOM paBeHCTB (3.77) H (3.78)
npeI(CTaBJ1eHbI B cJ1eA)'lO eli Ta6nl1 e.
KOOp.nHffaTbI '1'=0 'I' = 90°
TeH30pC3HCTOp3 0"' . 10 9 , H/M 2 , 9 2 0"; . 10 9 HlM 2 9 2
1 0'2 . 10 , HIM , 2. lD ,HlM
r=O 1,198
e = 0 11 90°
r=R -Q,521 -1,875
r=O 0,997
e = 45°
r=R -Q,12 -1,875
HCnOJIb3YH «POPMYJIbI (3.74), paBcHcTBa
(3.77), (3.78), rn6JI. 2.1 H Bblpa eHMc (3.73),
MO)!(HO Bbl1JI1CJIHTb OTHOCHTeJIbHbIC H3MeHe-
Hl1l1 conpDTHBJICHHlI TeH30pc3HCTOpoB 0,
HOpMHpoBaHHblc )J,Jl1l KpeMHIDI p-mna npo-
H3Be.neHHeM napaMcTpoB 1[44'Y, a JJ.JI1I n-n1-
na - npoH3Bc.z:r.eHHcM napaMeTpOB 1[1I'Y, npH-
1.JeM 'Y = (pR 2 ) I c .
BbITIOnHllM )'Ka3aHHbIe BLI1.JHCJ1eHHSI Ha
npMMcpe nJIOCKOCTH (001). BBe,l\eM BHa1J8JIe
CJIe.z:\YIDll{He 0603HaQCHHlI, nonarWl, 'iT0
174 rJ1aBa3.3nEMEHTHAJ! EA3A MHKPOMEXAHI14ECKI1X rIPHEOPOB
TCH30pe3Mcrop HMeeT MaJJblC pa3MepbI (TO-
lJe1.JHLIH pe3Hcrop):
OrO:::: (M I R )Ir=o - OTHOCUTeJIbHOe
H3MeHeHHe COnpOmBJIeHIUI pa)J,HaJJbHOrO
TCH30pe3HCTOpa B uempe MeM6paHbI;
O,R ::::(MI R)I,=R - TO )Ke, Ha Kp8KJ
MeM6paHLI;
01:0:::: (MI R)lr=o - OTHOCllTeJIbHOe
H3MeHeHHe COnpOTHBJIeHlUI TaHreHUHaJJbHO-
TO TeH30pe3HCTOpa B uempc MeM6paHbI;
0TR ::. (M I R) Ir=R - TO :>Ke, Ha KpaKJ
MeM6paHhl.
B COOTBCTCTBHH C «pOpM)'JIaMM (3.74) B
ueIITpe MeM6paHbl O"r = 0"1:, a Ha KpalO MeM-
3p ., 3p.,
6p aHlol 0" :::: -- R-' 0"_::. --vR- . 113
r "), L 4"
4c c;;.
, ,
-ra6JI. 2.6 CJ1C){)'eT, 'ITO 7t1 I :::: -7t1:2 ,Il.iUI
KpCMHIDI o6oHX TJmOB rrpOBO)J,MMOCTH. 3Ha-
<nIT, B DJIOCKOCTH (001) ,Il.iUI KpeMHIDI p- H n-
THIlOB npoBO.ilHMOCm B UeH1l'C MeM6paHbI
HMeeM
O"rO _ 0"1:0 :::: O"rO :::: 0"1:0 :::: O.
1[441 7t 44 1 1[11 1 7t Il )'
TaK K8K J1.JUI p8.!UiaJJbHoro TeH30pe3H-
cropa 'If = O. TO B COOTBeTCTBHH C ypaBHeHH-
eM (3.75) <p = 8 H)J,Jlj{ KpeMHHSI p-THna Ha
KpaJO MeM6paHLT [CM. Ta6JI. 2.6 H BbIpaiKe-
HHe (3.70)] nOJlY'iMM
cr rR 1 ( " " )
7t 1l ()1 + 1[120"2 ::::
"441 1[441
- I (1tjI O r +1tj2 0 1:):::: 0,25(1- cos48)x
1[441
) 3.0,25 ( )( )
x(cr, -crT = 4 l-cos4e 1- v .
Ha OCHOBaHHH -ra6JI. 2.7 MaKCHM8J1bHOe
3JIa1IeHHe OTHomeHIDI npHHHM810T npH
8 = 45°, 'lT0 COOTBeTCTBYeT KpHCTaJIJ]orpa-
HqecKOMY uanpaSJleHHIO [110]. 1-13 Ta6JI.
2.1 HMeeM v = 0,064 H H3 npeJU>I.n}'ll.(ero
BLlpIDKeHIDI Haif.D.eM
'R = -O,17(] - cos 48).
1[441
O'leBH,f.\HO, 'ITO
erR _
TCM1'
e YR
7t-l41'
BbmOJIHIIM aHaJIOrl1'lHbIC npe06pa30-
BaHUSI ,llJISI KpCMHIISI Il-TIma 11 nOJIy'ml\!
°rR :::: 3.0,37 :::: (1.67 + cos 40)(v _ I).
7t 4 -11' 4
B COOTBCTCTBIIH C T86JI. 2.7 M8KCH-
MaJIbHble 3Ha'lCHIfSi oTHollleHIlH 7t; I 17t) I Ii
7t;2/7t11 COOTBeTCTByKJT yfJIaM 8 :::: 0 (Ha-
ITpaBJIeHHC [100]) Ii 90° (H8ITpaBJIeHIie
[0 I 0]) J1 1t3 Ta6JI. 2.1 )J,JIj{ :nux 3Ha'lCHl1i1
YfJIOB nOJIY'lUM v :::: 0,278. CJIe)J,oBaTeJIbHo.
OrR = -o,2{l,67 + cos48). AHaJIOrH'lHO
7t 1l )'
o e
HalmeM :::: . BbInOJIHHM nO,f.\06HbIC
J1I11' J1 I1 1'
ITpe06pa30BaHHH .nnSi nJIOCKOCTCH (011).
(111) H pe3yJIbTaTbI npe.l.l.CTaBHM B Ta6.'I. 3.4.
Ta6JI. 3.4 onpC,f.\eJIHeT TOnOJIOrHKJ TeH-
30pe3HcropHbJX cxeM Ha ITJ10CKOCTH MCM-
6paHbI. PaCCMOTpHM HeKOTopbIC CXCMbL Ha
pHC. 3.23 nOKa3aHbJ BapIiaHTbl TOnOJIOfHH
TeH30pC3HCTOpHbIX cxeM Ha nJIOCKOCTH
(001) J1.JUI KpCMHMH p-nma.
1)0JIbWl1HCTBO )J,aT'lHKOB .l.l.8BJIeHHSI C
MeM6paHaMH 113 KpeMHHSI p-THna Ha ITJ10CKO-
CTM (00 I) HMeKJT TOn0J10rmo TeH30pC3H-
CTOpHbIX cxeM, npHHLI;lmHaJIbHO He OTJIH-
qaIOllUiXCSI OT npHBC,f.\eHHbIX.
Ha pMC. 3.24 nOK83aHbI B03MO:>KHble
BapllaHTM TOnOJIOfHH TeH30pc3HCTOpHMX
CXCM Ha ITJ10CKOCm (001) ,llJISI KpCMHHH 12-
Tl1na.
BapHaHTbI TOnOJIOrHH TeH30pe3HcTop-
HbIX CXCM Ha nJIOCKOCTH (111) npC,l\CTaBJIe-
Hbl H8 pHC. 3.25.
Ha pHC. 3.25, a nOK838H OCHOBHOH Ba-
pHaHT TOnOJIOrl1H Ha ITJ10CJ<OCTH (111) MCM-
6paHbI. PC3HCTOpbI C oTpI1 aTeJIbHbJM npH-
pallJ,eHHeM cOnpOTHBJ1CHll$1 (CM. Ta6J1. 3.4)
npH ,l\eHCTBHH )J,aBJ1CHMH Ha npo«pHJIHPO-
BaHHYIO CTOpOHY MeM6paHbl pacnOJIaraKJTCSI
paJJ,HaJJbHo, HanpHMep no HaITpaBJICHHKJ
IlPJlMblE DPEOEPA30BA TEJII1
115
3.4. OTHOCMTeJIbHble conpOTHBJIeHHH TeH30pe:mcTopoB
OTHlJCI1Tenhllhie
1I0pMlipOllallllhle
InMeHeHH
conpamBnCIIIUI
TcmopC3HCTOpOB
Kplfcnurnorpa4JIf'leCKaH nnOCKOCTb
(001 )
(011)
(I II)
p-mltn KpeMllw/
(\0 0 0,127(1 - cos28) 0.1 68
1t44Y
erR -O,I7( I - cos48) 0,164(2,15 - 1,157cos28 - cos48) -0,329
1t44Y
0-.: 0 0 0,12(1 + cos28) 0,168
-
1t44 Y
e TR -0,17(1 - cos48) 0,122(0,767 - 0,232cos28 - cos48) -0,007
1t44Y
n-mun Kpe.11HllR
01'0 0 0,178(0,33 + cos28) 0
1tIlY
erR -O,2( 1,67 + cos48) -0.15(1,38 + 1,84cos28 + cos48) {),12
1tlIY
0-.: 0 0 0,189(0,33 - cos28) 0
1tlIY
O-.:R 0,2( 1 ,67 + cos48) 0,152(0,609 + 0,511cos28 + cos48) 0,12
1t\lY
..--,
o
.-
Ot
'--'.
6)
..--,
o
8+
'--' .
a)
[100]
._._.
Puc. 3.23. BaplfaHTbl TOnOJlOmlf nOJlHblX MOCTOBbiX
TeH30pe:JlfCTOpUbIX cxeM ua nJ1OCICOCTU (001) p-Tuna:
a - O.!lHa pacnpe.!leneHHa.!! no KpalO MOCTOBaH. cxeMa; 6 - OIDia IfJIH HeCKonbKO
cOCpe,llOTo'JCl-/HbIX no KpalO nonHblX MOCTOBblX CXCM
IlP5IMbiE rIPE06PA30BA 1 EJIJ1
111
u)
5'
01
I
[100]
-
5'
C;+
tWO]
- .
6)
[100]
-
0 1
,
c)
1 I.90]
PIIC. 3.26. BapH3uTLI Tonononm nOJIIlblX :\IOCTOBbIX TeH30pe3ucTopHhlX cxeM
118 nJlOCKOCTII (011) p-THna:
G, 6. (I - TeH30pe3Hcropbl B LleHTpe 11 HB Kp!uo MeM6pBHbI;
i' - P8.11HaJJbHble H TIlHrCHIlI!a.'IbHble TemOpe3HCTOpbI Ha Kparo MeM6paHbI
Ha pMC. 3.26 ITpHBCAeHbI 'leTbJpe H3 MHO-
fO'IHCJICHHbIX BapHaHTOB TOnOJIOrl111 TeH30pe-
3HCTOpOB p-nma Ha nJIOCKOCTH (011).
Ha nJIOCKOCTH (00 I) B COOTBeTCTBHH C
m6JI. 3.4 MO:iKeT 6bITb npelIJIO)f(eHO HaM-
0)
o
-1
0'
,
6)
o
_1
o!
60J1bllIee 'lHCJIO BapHaHTOB pa3MellJ,eHm!
TeH30pe3HcropoB Kal< p-, TaK H II-Tllna. Ha
pHC. 3.27 D.aHbI '-ICTblpC 113 B03MO:>KHbIX Ba-
pHaHTOB p83MellJ,eHlffi TeH30pe3HCTopOB
II-TUna Ha nJIOCKOCTH (011).
6)
o
o.
.
[100]
-
o
-t
o.
c)
[ 100]
.-.
PUC. 3.27. BapU81fTbi TOnOJlOrIm nOJIIlblX MOCTOBb!X Teu30pe3uCTOpHb!X cxeM
. ua nJlOCKOCTH (011) n-THna:
a, 6 - TeH30pe3HcTopbI HB KpWO MeM6paHbl; 6, Z - TeH30pe3HCfopbl B LleHTpe II HB Kp!UO MeM6paHbl
178 rJlaBa 3. 3JIEMEHTHAJI 6A3A MI1KPOMEXAHH4ECKI1X rIPHEOPOB
3.5. K BbI'IUc.'leHIlIO IJYBCTBIITCJ1bHOCTH I\IOCTOBblX cxeM
Cxer.m no pl1c.
3.23 3.24 3.25, a 3.25,6 3.26, Q 3.26,6
., 0,3471:44 0,53471:11 0,24871:44
S c- 0,1271:\1 0,31671:44 0, 1551t 44
'"
U O R2
Cxe!\ta ITO pHC.
3.26. 6 3.26.z 3.27. a 3.27,6 3.27,6 3.27 ?
0,385141 0,36571:11 0,47871:11 0.43571:11 0,471:!1 0,2271:11
BOCnOill>'3)'eMOI OpM)'JIoi1: (3.72),
Ta6!l. 3.4 H onpe.:lemiM IJ)'BCTBHreJlbHOCTb MOC-
TOBbIX cxer.{. npHBeJl.eHHbIX Ha pHC. 3.23-3.27.
B COOTBeTCTBUH c pacnOJIO:>KeHHCM
TeH30pe3HCTopoB Ha pllc. 3.23, a MMeeM JI..iU!
pallHaJlbHblX TeH30pe3HCTopoB 8 = 45 0 H J1.JlSI
TaHfCHUHa.:lbHbIX 8 = 135°. ITo TaG)]. 3.4 .!I..iUl
IL"lOCKOCTH (OOI)p-rnna HaxO,!JJiM 18 rR I = 18tRI =
., ')
= 0,341441. 3HW!, 'ITO 1 = (pk')f c ' no opMY-
') .,
:Ie (3.72) no.:"I}"<JHM S = Uo(0,34:n: -)1 c: .
Ha cxeMax 3.24, Q-6 Ha l\.-p81Q MeM6pa-
RbI paJIHa.ilbHhle TCH30pe3HCTOpb! Opl1eHTH-
poBaHbI no HaIIpaBJ1eHHlO [010] (8 = 90°), a
.!lIDI cxeMbl 3.24, Z OpHeHTalIIDI TaHrCHllli-
aJIbHblX pe3HCTopoB onpe.n.eJISleTCH.: yrJIOM
e = 0, 8 pa.,rma..'lbHbIX - yrnoM 8 = 90°. 11:3
Ta6JI. 3.4 nony<raeM I orRI = ! OtRI = 0,53471:111,
Ii. CJIellOBaTCThHO, 'i)'BCTBHTeJIbHOCTb Bcex
MOCTOBblX cxeM paRHa
S = U o (0,53471: I1 R 1 )1 c .
BbInOJIHHM aHaJIon14HblC Bbl'IHCJICHl1S1
J1.JlH.: cxeM, npe)J,CTaBJIeHHblX Ha pMC. 3.25-
3.27, Ii Bce PC3YJIbTaTbI nOMeCTHM B
Ta6JI. 3.5.
CJIe.llyeT HMeTb B BJI.IlY, lITO paC'lCTHble
3aBI1CHMOCTl1 B Ta6JI. 3.4 HOCSIT npH6J1mKeH-
RbI" xapaKTep, n03TOMY 11 paC'lCT 4YBCTBH-
TeJIbHOCTCH TCH30pe3HCTOpHbJX MOCTOBblX
cxeM no npI1Be.n.eHHOH MeTo)J,HKe HOCHT
OueHO'lHbIH xapaKfep.
flplL1tep 3. J J
Pacc<nrraeM 'l)'BCTBIITeJJbHOCTh TeH30pe:m-
cropHbIX cxeM Ha nnOCKOCTH (011) JllliI HCXOJUjbIX
llaHHbIX: V o = 10 B. R = 5 . 10- 3 M. e" = 0,1 . 1O. 3 M.
., ') ,,3
Bbflll1CJ1HM napaMe1p (VoR")/ e = 25 . ] u B. 1-13
-ra6J1. 2.5 BblIUillleM 3HalJeHH.SI nbe30pe:3HCIOpHbIX
Kro$PllilHeHTOB .!lJI.SI KpeMHllii p-THna 1t.;4 = 138 x
x 10- 11 1H,,Wl.SI KpeMHH}\ lI-THna Inlll = -102,2 x
x 10- 11 M 2 /H. Pe3YJIbTaTbi BbllJHCneHliH nOMeCTI1.M
B Ta6J11111Y, B KOTOpyJO BnHcaHbl TaKJKe Heo6xo-
!lHMbIe !laHHbIe H3 Ta6J1. 3.5.
napaMeTp p-THI1; cxeMa no pHC.
3.26,6 3.26, i! 3.26, a 3.26,6
2
S C N , rllH 0.385"'44 O,365 1t 44 0,316144 0,] 551t44
V o R2
S, I . 10- s B/(Wrl) 3,452 3,272 2,833 ],289
£1apaMeTp n-nm; cxeMa no pl1.c.
3.27, a 3.21,6 3.27,6 3.21,z
2
S C M . rllH 0.4781t1l 0.43511:11 O, 4n u 0,22 1t 1l
V o R2
S, 1 . lO- s B/(Wrl) l,22J 1,1 JJ 1,021 0,562
IlP JIMblE IlPEOEP A30BA TEJIl1
119
TIepe aTO'lHa51 q)YHKL.(HH q3 MM, co-
CTOSlll\erO 113 MeM6paHbI C MOCTOBblMH TeH-
30pe3HCTOpHblMH CXeMaMM, B CTaTH<leCKOM
pe)KI1Me OITpC eJI5ICTCSI ero 'lYBCTBHTeJIbHO-
CTblO.
3.2.4. IIpeo6p330B3TeJIH e4JopM3I{HH
U3 nOBepxHocTHO-3KYCTJI"leCKHX
BOJIH3X H Ha cTpyue
Pa3pa6OTKa npeo6pa30BaTeJICH Ha no-
BepxHocTHo-aKYCTl1'leCKHX BOJIHaX (TIAB) H
CTpYHHbIX npe06pa'30BaTeJIeM 5IBmleTC5I nep-
CITCKTHBHblM HanpaBJIeHHeM pa'3BHTWI MHK-
pOMeXaHMKH.
TIpc06pa30BaTeJIb Ha I1AB (pHC. 3.28)
npe CTaBJI5IeT C060H CJIOH HJIH nJIeHKY C
nbe303J1cKTpH'leCKHMH CBoHcTBaMH (nbe30-
KBapL.(, HHTp a.mOMMHH5I, OKC L.(HHKa H
T. .), HaHeCCHHbIe Ha Tll\aTeJIbHO no rOTOB-
JIeHHYIO nOBepxHocTb nO JIO:>KKH, M3rOTOB-
JIeHHOH H3 MaTeplla.Jla C MaJIbIMH MCXaHH'le-
CKHMH nOTCp5lMH (KpCMHMH, CTeKJIO, Kcpa-
MHKa H T. .). 110 TCMneparypHoMY K03.p H-
L.(HeHTY J1MHeHHOro pacllIHpeHIDI HaHJIyq-
lIIHM 06pa30M comacylOTCJI KpeMHHcBaH
nO)J,JIo)[(J(a H nJICHKa 113 OKCM a L.(HHKa.
Aj
3J1CKrpOJ1.bI
-.... .. - .-- ..
::::: .:::i: ;: ::::"::;' ":. :;:::;...: .:,:= ::::;.;:;:::: :: ;:::: :::, :::: ::; .:;
...' ........: & t.-:- ....::-..-:........r_.........T._....'I...........
:.":;,.:::::::,.,:,,: .1{ n ' ...::;i1I: .-:« ::.f::::-;.':':::::' ;':-;;::;'..:"<: :": '::;:::-:",
.......... -'JI.!'. ....(....;.................-:..-:-.........%.;:.>.. .-...
:-:.: t ;::f{. . .:::: :: : : : :$:; :: ; : ;; t ::r:* t :;:.
[loA
I
L!J
d
B036)')1< aeMaJl
BOJIBa
PIIC, 3.28. CxeMbI npeo6ps30Bs'JeJIft Ha nAB
TIO)J,JlO)[(J(a MO)J(eT 6bITb 'laCTblO 43
npH6opa, HanpllMCp aT'lMKa aBJICHH5I MJIH
aKCeJ1CpOMeTpa. C ITOMOllJ,blO BcrpC'lHO-
llITblpeBblx 3JICKTpO OB Ha nOBepxHocTM
IIbC303JIeKTpH'leCKOH nJIeHKH B036Y)J()IaIOTCSI
3JICKTpH'lCCKHe KOJIe6aHIDI. Ha KOHl{bI nJIa-
CTHHbI HaHOCHT MCTaJIJIM'lCCKHe CJIOll .nml
ycrpaHeHIDI BJIHHHIDI OTpa:>KeHHblX OT KpaeB
nJIaCTMHbI BOJIH. llIar pellIeTKH BbI6HpalOT
paBHblM J1HHC B036Y)J( acMoH BOJIHbI d = A,
a ,l\JIMHY pa60Qero ITOJIJI - KpaTHoH llIary
pellIeTKH, T.C. I = nd.
I{mI B036Y)K,lleHIDI TIAB HCIIOJIb3YlOT
npSlMoH H 06paTHbIH llbe303 eKTbI. B03-
6Y)K,lleHllC I1AB- crpYKTYP OCYIIIecTBJI5lCTcH
no MeTO)J,y aBTOrCHCpaTOpa [13].
AMIlJIHry a CMellJ,eHllH: B HarrpaBJ1eHI1I1
HOpMaJIH K nOBepxHocTM nJIeHKH nOpJI)J,Ka
10- 5 A.
OCHOBHOH 3aBl1CHMOCTblO npH paCqeTC
ITpC06pa30BaTeJI5I SlBJ1JIlOTCJI paBCHCTBa
v = .JElp =w/ k,
(3.79)
r)J,e CD - 'laCTOTa KOJIe6aHHH; k - BOJIHOBOC
qHCJIO, 3aBHCHIIIce OT npO OJIbHOH CKOpOCTH
BOJIHbI; v - CKOpOCTb pacnpoCTpaHCHHH.
I1AB; E, P - MO)J,yJlb yrrpyroCTll H ITJIOT-
HOCTb IIbC30CJ10Sl COOTBeTCTBCHHO.
OnTHMH3aL.(HH xapaKTCpHCmKI1 npeo6-
pa30BaTeJISI )J,OCTHfaCTC5I Ha 'laCTOTe aKycm-
'lCCKOro CHHxpoHM3Ma: Ie ::::: vel d , r e V c -
CKOpoCTb CllHXpOHH3Ma.
qacTOTa CHHXpOHH3Ma C yqeTOM Bl>Ipa-
:>KCHM5I (3.79) onpC eJI5IeTCJI 3aBMCHMOCTb1O
Ie = .jElp(lI d).
(3.80)
O'lCBM HO, npM e<popMal{HM nO)J,JIO:>K-
KH C ITbC30CJ10CM H3MCH5ICTC5I pa3Mep d, a
CJIe OBaTeJIbHO, H 'laCTOTa!c. Ha OCHOBaHHH
.pOPMYJIbI (3.80) MO:>KHO 3aKJIlOqHTh, qTO
l13MCHCHHe qaCTOThI!c JIHHCHHO 38BHCllT OT
c4>opMal{llll q3 npH6opa.
CTPYHHbIH npe06pa30BaTCJlL (PHC.
3.29) npc)J,cTaBJIHCT C060H yrrpyrylO 6aJIKy.
H3rOTOBJICHHYlO Ha OCHOBC KpeMHIDI p-TMna
H 3nHTaKCMaJIbHbIMH CJ101IMll Ha 6a3e KpeM-
180 rJIaBa 3. 31IEMEHTHMI 6A3A MHKPOMEXAHI14ECKHX IlPH60POB
oj
.4!
. T
Si-p
;:o
.--
.,
- ._.' .', -'. -....'
I T .
.- . .
I
Si-II
lloA
I..
.-----"'-' -_._ ...
.,.
... - '-
6J I
. __0
C4
€ )
PHe. 3.29. CXeFtl8 CTp '""oro
npeo6pa30BaTCoi'UI (a) " pS3MeIQeHHe CTp 'Hbl B
3."letcrpOCTIlTHlJecKHX npeo6pa30BaTeJUlx (6)
HIDI n-rnna. CTpyHbl "MCIOT BH.!I. nOJIOCOK,
3aIl1eMJIeHHbIX no KpaHM. YnpYf8S1 6aJIKa
MO:iKeT 6LIT1> qaCThIO q3. HanpHMep M8SlTHM-
Ka aKce.;IepoMeTpa. B He,l\e OpMHpoBaHHoM
COCTO.!UIHH CHJ1a HarIpSDKeHIDI crpyHbI T = To
(const), a np" H3m6HblX KOJIc6aHIDIX 6aJIKH
CHJ1a HaTIDKeHIDI H3MeIDIeTCH.
CrpyHa. Bbmo;IHeHHaH H3 npoBo,Wuue-
ro M8TCpHaJIa, J:lBIDIeTCH nOJij3IDKHbIM 31JeK-
"I'pO.lI:OM, pa3MeIQeHHbIM MC:>KJI)' HenOJij3IDK-
HWMH 31JelCTpO.naMH KOH)J.eHCaTOpoB Cl, C2.
ABTOK03e6aHIDI crpYH B036}'JK.!taIOTClI 3JIeK-
Tp<>CTaTHtJecKHM reHepaTOpOM, KOTOpbIH
no.naeT onopHoe HanpSDKeHHe nooqepe)J,Ho
Ha 3J1elCTpO,ltld KOHJJ:eHC8TOpoB Cl H C2.
YnpaBIDleTCx reHeparop DO CHfHaJJaM C npe-
06pa30BaTemi nepeMemeHHii, BKJIIOqalOIQero
B CBOH COCTaB :melCTpO)J,bI KOHJJ:eHcaTOpOB
C3, C4.
Pe30HaHCHbIe 'laCTOTbI OCHOBHOH (l>op-
MLI KOJ1e6aH.Hli onpe,l\eIDIlOTCX 3aBHCHMO-
CThIO
F= 1Cn TO+T
1 pS'
(3.81 )
r.ne n - HOMep rapMoHHKH; p - ruJOTHOCTb
MaTepHaJIa C1'J'YHhI; S - DJIO IlIfI 1'(1) nonepeq-
HOro ceqeHIDl CTpYHLI.
B COOTBeTCTBHH C BlolpIDKeHHeM (3.81)
np" .ne4JopMHposaHHH ynpyroro 3JIeMeHTa "
1i3MCHCHJUi HanIiKCHlU1 CTPYHbl IBMeHHeTCSI
pe30HaHcHaH 'laCTOTa ee KOJIc6aHnii. 06pa-
60Tl\a 'laCTOTHbIX CltfHw"IOB BbmOJIHSleTCSI
H3SeCTHbIMH aITnapaTHblMH CpC.lI.CTBaMH [7].
3.3. 06P A THbIE IIPE06P A30BA TEJIII
(AKTIOA TOPhI)
3.3.1. 3JIeKTpOCT3TII'IeCKlle
npe06pa30B3TeJIII
110TeHUHa.nbHajJ ::JHcpmx 3apH.na Q, Ha-
KannnBaIOllJ,aHCH Ha KOH)J,cHcaTopc eMKO-
cnJO C(x, y, z), K I<OTOPOMY npHJ10iKeHO Ha-
ITpSliKeHne U, onpe,l\eJIHeTCSI paBCHCTBaMH
1 Q2 1
W =- ( ) = -U-C(x,y,z), (3.82)
2 C x,y,z 2
r)J,e x, y, Z - Koop,l\HHaTbI, onpe,l\eJIHIOIlI,HC
B3aHMHoe IJOJIOA<CHHe nJ1aCTHH KOH,lleHCa-
Topa.
BbI'lHCJIIDI 'laCTHble npoH3Bo.nHble OT
BbIpa:>KCHH» (3.82) ,llml nOTeHqHaJIbHOH
3HcpmH, MO:>KHO nOJI)"lliTb KOHcepBaTHBHble
31JcKTpocTaTI1'lCCKHC CMJIbl:
oW
F =--'
3.1: Ox '
oW oW
F;y =--; F3Z =--.
oy oz
(3.83)
ITpc.nnOJIO:>KHM, 'ITO B COOTBCTCTBHH C
pl1C. 3.30 nepeMeHHbIMI1, onpe)J,eJIHIOll\I1MI1
nOJIOiKCHHC nJIaCTI1HbI, »BJISlJOTClI KOOp.ll.I1Ha-
TbI x, y. B 3TOM cJ1Y'Iae eMKOCTb MC:>K,n:y nJIa-
CTI1HaMM KOHJ1.CHCaTOpa pasHa
C = EEoc(a + t.x) ,
Yo + y
(3.84)
r.ne c, a - pa3MepbI, OT I<OTOpbIX 3aBl1CI1T
Ha'laJJLHaH nJIOIQa,n:b ncpeKpWTMA nnaCTHH;
t.x, y - MaJILlC nepCMell\eHHSI B HanpaBne-
HMM COOTBeTCTBytOIQI1X I<Oop,l\HHaT; Yo
UaqaJJbHblH 3a30p MeiK,l{)' nnaCTI1HaMH.
113 cJ>OPMYJ1 (3.82) 11 (3.83) nOJI HM
EE cU 2
F.u = 2( :o + y) ; FJZ = 0;
EEoc(a + t.x)U 2
F3}' = ( 2
2 Yo + y)
(3.85)
OE)PA THbIE f1PEOE)PA30BA TEJIH
nn
z
i'1J
/r.
l'
N " :$ -=';'-"" ..' ...V".........:-::.i........Z-:.:;,:o,:.;J,; .:...01'....:.:..
:r.
. ,
.;
,.
. -', ..... "..
. .
: :.:.....:= ".:":":. . ...::':: " '". :::'.;.
.. .'.: ...... .......". ........
. . ..... ..
c.
x
..... ...
....-..- .
,
PIIC. 3.30. K BbIIJUCJJelllUO e!llKOCTII Kou)J,eHcaTopa
H3 paBeHcTB (3.85) C.'Ie)J,yeT. qTO 11JU!
paCCMaTpMBaer.1OI"O BapMaHTa HMeCTCSI co-
CTaBJUIlOlUaJI ,}JIeKTpOCTaTll t IecKoj:[ CHJIb!
BllOJIb OCM X, KOTOpaJI He 3aBI1CHT OT 3TOH
KOOp,llHHaTbl. COCTaBJlSl101l.{WJ B,UOJIb OCH Z
OTCYTCTByeT, a COCTaBJISIIOIl.{WJ B)J,OJIb OCH Y
rrpe)J,CTaBJUleT co60j:[ HenHHeHHYIO Cl1.ilY
3JIeKTpOCTaTMtIeCKoro ITpHTSl:iKeHMSI, KOTOpa.fl
KpMTJ1lIHa K ncpeMell\eHHHM no Koop)J,HHa-
TaM x, y.
I1pOCJIe)J,llM BHaqaJIe opr.mpOBaHMe
pa3HOCTHOH :meKTpocraTH<teCKOH CHnhI B
HanpaBJIeHHH KoopllHHaTbI y, <ITO HMeeT MC-
CTO, HanpllMep, B aKCeJIepOMeTpax HJ1H )J,aT-
'lHKaX I\aBneHH.fI C 3JIeKTpOCTaTHlJCCKoj:[ 06-
paTHoj:[ CBSl3blO. B COOTBeTCTBllH C pHC. 3.16
0603HaIJHM t>.Yu = ho, t>.y = t>.h (pMC. 3.31).
U oo
CI C2
, ::
;.
- U oo
+ U on
I F2
.
-++
I
I Fp
I
I
I
I
r;
1:111
ho
/to
PHC. 3.31. K eJIopMHpOB8HHIO p83HOCTHOii
:JJJetnpocT8THlJeCKOii CUJlbI
K HenOllBH:>KHbIM TIJIaCTHHaM (3JIeKTpO-
llaM) nO,llBC)J,eHO onopHoe HanpSl:>KeHHe
(+U ono -U OI ')' a K nO)J,BH:iKHoi1 nJIaCTHHe (q3
aKCeJICpOMeTpa HJIM MCM6paHa )J,aTIIMKa ,llaB-
neHll.f1) BbIXOI\Hoe HanpSl.iKeHHe 113MepM-
TeJIbHOH lIenH C 06paTHbIM 3HaKoM, T.e. Ha-
npSl.iKeHlle U oc OTpHUaTenbHoH o6paTHoH
CBSl3H. TaKHM 06p830M <popMHpyeTcSI 3JIeK-
TpocTaTHtIeCKaH. PC3YJIbTMPYIOIlIM CHJIa,
KOTOpa.fl cTpeMHTCSI. BepHYTb nOJtBMiKHYIO
ITJIaCTHHY B HCHTpaJIbHOe nOJIOiKeHMe.
TIYCTb eMKOCTb CI YBeJIMtIMBaeTCSI., M
npM 3TOM KaHaJI 06paTHoH CBSl.3H OPMHPYCT
Ha HeM HanpSl.iKeHlle UOl] - Uoc' EMKOCTb
KOH)J,eHCaTOpOB C2 npl1 3TOM YMeHblIIaeTCSI.,
a HanpSl)f(eHlle paBHO -( UOl] - U oc ). B COOT-
BeTCTBHM C <POPMYJIa.\111 (3.83), (3.84)
( A.. = 0; S = ca), nonytIMM )J,JISI KOH,!:leHCa-
TOpa Cl:
(u oo -U o J2 8 EOES_
2 8(t>.h) ho-M
_ EOES (U oo -U oc )2
2(11 0 -MY
F;=
IJ:m KOH,n,eHcaTopa C2 aHaJIOfHtJHO
F. (U oo +U m J2 8 EOES_
2:::: 2 15(M)ho+M
_ EOES (U on +UocY
- 2(h o +M)2
I1MeSi. B BH)J,y, qTO ho > M, Haii)J,eM
pa3HOCTHYIO CHny, .neHCTBYIOIlI,)'lO HR no,n-
BM:>KH)'IO nJIaCTHH)':
182 rnaBa 3. 3J1EMEHTHA5l 6A3A MHKPOMEXAHH ECKHX rIPH60POR
,
Fp = F 2 - f1 == -2E£ o SU on U oc I ho. (3.86)
TaR KaR U"YY. = -U oc , TO nepeLl8TO'rHaJI
4JYHKJ..tIDt K8Han8 o6pRmoi1 CBR31l llMeCT
BHJ1., H/B,
Jt = Fp I U Bl;).X = 2&&oSU on I hl, . (3.87)
<1>opM)'JIa (3.87) Onpe!lellileT TalOKe
l\-pyrn3H)' xapaKrepllCTl1J\Ii 3nel\."TpocTaTll'le-
CKOro rrpeo6pa30BaTeJU! (.1I.aT'mKa) CIiJIbI B
uenll 06parnOH CBR3H. TeMnepaT)rpHWI HC-
CTa6HJIbHOCTb pa60Tbl .aaT'IMKa CIiJIbl. TaR
)Ke, 1\.-aK II Y 3JIeKrpOCTaTl1'leCKOrO npe06pa-
30BaTeJUI nepeMellJ,eHllM, 3aBllCHT OT HeCTa-
61L'lbHOCTIf onopHoro HanpSDKeHHSl II 3830pa
Me)l(!t)' n.'1aCTliHaMH KOH!leHCaTopOB.
P8CCMOTpllM Tenepb OpMMpOBaHHe
3JleKTpOCTaTH'leCKOH CliJIbl B HanpaBJICHl1ll
DCM X (CM. pMC. 3.30), '1ro Ha6JIlO!laCTCR B
MHKpO!1BRnne!UlX, npHMelUleMblx B Mr .llIDI
C03!laHIDI BbICOKO'IaCTOTHbIX Kone6aHHH
11..\1.
Ha pHC. 3.32, a nOK83aH8 cxeMa MHK-
po.!IBHraTe:IH (CM. nOJ1.pa3J1. 1.3.2), KOTOpbli-i
COCTOHT H3 ABYX CTaTOpHbIX 1 H lI,BYX po-
TOpHbIX 3 rpe6eH'JaTbIx crpYKTYP, KOTopble
SIBIDllOTCX CTaTOpoM H poTOpOM MHKpO.!IB ll -
ICrre.IDl. 3JIeMeHTbI KpenneHIDI 2, 5 coe.nH-
lUIKIT CTaTOp H poTOp C onopHoM n.'IaCTHHOH
MHKpOCTpYJCTYPLl. POTOP Ha 3neMeHTax 5
CMOHTHpoBaH Ha nO.1J,Bcce, COCTORlUeM M3
'JeTb)pex nap ynPyrnx :meMeHTOB (6anoK) 4.
K K8JK.!(OH nOJlOBMHe MHKpO,U,BHraTeJJR no):\-
Be,l\e1l0 H8IIpjVf(eHHe
U I ==U on +U 1C ; U 2 =U on -U x , (3.88)
r.ne U on - onopHoe HanpJlJi(eHlle, V x - nepe-
MeHHoe HanpH)KeHHe, 06ecne'JMBalOllJ,ee pa-
6my neBOH M npaBOH nOJIOBHH MHKpO,l\Bllra-
TeJJJI.
Ha pllC 3.32, 6 nOK83aH 3J1eMeHT
C1J'ytcr)'pLl (O.nHH nanell poTOpa II ,lJ.Ba nanb-
na. crRTOpa) C pa3MepaMII, COOTBCTCTByKI-
IItHMH pHC. 3.30.
HMe1I B BJU1Y cJ>oPMYJIbl (3.82}-{3.85) II
pHC. 330, :mnHweM pa6oT)' 3JlelCTpOCTaTH-
"IecKI1X CHJI Ha B03MO:iKHbIX nepcMew. eHIDIx
pOTopa !!.X B HanpaBJIeHHH ocn x 11 !!.Y B Ha-
npaBJIemlll OCII y nJISI o.il:Hofi napbI na:JbIIcn
pOTopa II cTaTopa:
A= EEOC ( 1 + 1 ) (U l :! -U;)x
2 !!.Yo + 4" 4"0 - 4" -
X AT - E& ac [ (4"0 : 4")' (4" 0 4')' ]x
' ' A E & o CL\.V o ' ' )
x U - +u- , - - U - u- A._ +
1 :! J , 2 1 :2 LlA
4.1' ii -!!.Y
2Eoad9'o!!.Y (2 1. )
+ (, , ) 2 \UI + U:2 !!.y. (3.89)
\!,>Yo - !!.y-
(' Y'leToM BblpIDKeHI1H (3.88) paacHcTBa
(3.89) npMHIIMalOT BM)J,
A - 4&&oc!!.yo U U A. 4EEoac!!.yo!!.Y
2 ' on xLl.\" + ( )2 x
!!.Yo - L\.y- !!.Y1. - !!.y2 J
X (UJn + U;). (3.90)
ECJlH Bem1'mHa !!.y COCTaBJIHeT e,llI1HH-
I..{bl npOl..{eHTOB OT !!.yo, TO BbIpaJKeHl1e (3.90)
YITpOmaCTCSI:
E£OC
A = 4-U on U x fu-,
!!.Yo
OTKYJ1.a CJIe,l\YCT BblpIDKeHHC 3JIeKTpOCTaTI1-
'leCKofl CllJIbI B HanpaBJleHl1ll OCH x:
E£OC
F"jx = 4-U on U x '
!!.Yo
(3.91)
nOCKOJIbKY Ka>KJ.I.blH nanel..{ pOTOpa Ha-
XOJtl1TCSI Me>KJ.I.Y J1.BYMSI nanbl..{aMll CTaTOpa,
cyMMapHWI CHJla JlI1HeHHoro MHKpOJtBMraTe-
IDI, HanpaBJJeHHWI B O)J,HY CTOPOHY (nOJlOBH-
Ha I..{HKJJa pa6OTbl), onpCJI,eJIRCTCR 1}>0PMYJIOfl
F'f."JX :::; nF"jX '
(3.92)
r.ne n - CYMMapHOe (B o6ellX rpe6eHllaTbIX
crpYKTYpax) IIIICnO nanbqeB pOTOpa.
184 rnaBa 3. 3JIEMEHTHNI EA3A MMKPOMEXAHI11ffiCKHX npI1I)OPOB
3HeprIDI BbIXO,l\a onpe.!lemleTcSI CJIe-
.n.yromHM 06pa30M:
W BLIX = f(nFn -G1:ll.,)d\-=
o (3.95)
= ( F'£)X -
G
ll.'() M: .
T,l\e F'£)x paCClIHThlBaeTCSI no
opMYJIe
(3.92); G
- c)'MMapHaJI *eCTKOCTb 3ne-
MeHTOB nO.!lBCca.
llpu.wep 3.12
BhI'ilICJIHM napaMe1phI nHHeHHoro MIilI:pO-
1lBJU1iTeIDl no pRC. 3.32. MHIo:pocrpYKrypa Bbl-
'IpaB..'1eHa 83 nomucpeMHIDI, RMeIOmero
E = 1,4' lOll Ila, p = 2300 Kr/MJ. reoMeIpWlecKHe
pa3MephI no pRC. 3.32: C n = 3,5 MKM (rom1.\I1Ha
BceH MHl\.-poctpY!\.'T)'PhI). b = 4.0 MKM (wHpllHa
na..,Thua poropa. TIUaKe H Y CTaropa). a = 20 MKM
(B3aHMHoe nepeKpbITRe nJlaCT"H KOHlleHcaTO-
poB),
Yo = 2.0 MKM (38301' Me>K..!ty naJIbllaMH
rpe6eH'JaThIX CTpYl\.'T)'P), g = 30 MKM (.!lJlHHa
DaJIbua paropa, TIUaKe 8 CTaTopa), L = 112 MKM
(..!lJ"IHHa poropa), / = 120 MKM (nJIIIHa ynpyroro
3J1eMeaTa), b n = 2,0 MKM (wHpHHa ynpyroro 3ne-
MeHTa), d = 30 MKM, 1= 24 MKM, t = 12 MKM (Pa3-
Yephl coe.![HHJfre1IbHbIX 3neMeHTOB).
llpHMeM MO!l)'_'lH HanpxJKeHHH: U vn = U z =
= 30 B. tblc;lO na.'IblleB pompa n = 20 IllT.
BLl'Wc.'JeHIDI BbmOJIHHM B CJle.II)'JOllteH
OQepe.1IHOCTH:
I. Macca n01lBIDKHbIX 3J1eMeHTOB
m = pc(2Lt ... 2et + dt + 20bg) =
= 2300 . 3,5 . 10-6(2 . 112 . 12 + 2 . 24 . 12 +
+ 30. 12 + 20.4. 30)10- 12 = 4,85 . 10- 11 Kr.
2. lloJIarajl, 'ITO OlIHH KOHeu ynpyroro 3J1e-
MeHTa 3alUeMJleH, a ,!{pyrOH CB06o)J,HO ODepT H,
HMe$! B Bwry. 'ITO B n01lBece BOceMb :meMeHTOB,
ICOTopLle pa60raKrr napaJIJJeJIbHO. cyMMapHaH.
3I.'ecncocn. no.aseca pasHa
3
G =8 Ec n b n
I 41 3
8.1,4.10 11 .3,5.10- 6 (2.10- 6 r - HI
- .l r 4,54 M.
20.1O-6
3. IJacTara co6crBeHHbIX, He.ner.m4>HpOB8H-
OLIX ltone6aHJdi
CD= ,J GIlm = J 4,54/4,85.1O- JJ =
=305954 pa.D; =48,72 KI"u.
c
4. CYIIIMapHID! :melITpOCTaTII'"IeCKaJJ cnna
(3R nOnOBI1HY lluKna):
_? EEOC _
F1: 3_,--_0.4-U on U,--
.....yo
].885.10-]2.35.10-6.30 30
=20.4 ' , -
2.10- 6
=1.l2.1O- 6 H.
5. MaKCI1MaJIbHOe nepeMemeHlIe pm'opa
_ 17 IG _1,12.10-6 _ 0 ? 5 . IO -6 =
......" - rI:) x 1: - 4 54 ,- M
,
= 0,25 MKM.
6. 3HepniJI BXO,lla
Jl'BX = nU onU]CC =
? 8 -]2 -6 ?O 0 -6
_0.30.30.8, 5-10 .3,5.10 '_ .1
20.10- 6
= 5,57 .10- 12 LJ:JKlTaKT.
7. 3HeprIDI BbIXOAa
W BLIX = (F1: 3 ]C -
G1:......'(}it =
=(1,12.10- 6 -
4,54.0,25-1O- 6 )°,25.10- 6 =
= 0,138.] 0- 12 H. MlTaKT.
8. KIl)J, 11 - 0,138 100 =2,5 %.
5,57
llpellnOJIO)!G1M, 'HO nepeMbl'lKl1 coe)J,I1HRIOT
HaPYJKHhIe KOHllbJ ynpyrnx 3nCMeHTOB nonapHo,
'ITO JCIDlC):(aK napa ynpyrux 3neMeHTOB COe.lll1HeHa
nOCJlC1I.OBaTenbHO H '"ITO TaKHX nap, pa6oTa!OllII1x
nap3.Jl.lIenbHo,411IT.
3Ha<jHT, c)'MMapHID! )l(eCTKOCTb ynpyroro
nO,!{Beca Gr. = 2G] = 1,135 HIM (G. - JKeCTKOCTb
O)J,Horo ynpyroro 3neMCHTa); (0 = 24,36 lOll;
,,__ _ 1,26.10-6 _ 099 W. o 55 ?
LM , MKM; BLIX '" , _ x
11,35
}( 10- 12 H'M/TaKT H KTIJ]; 11 = 10 %.
0606ml1M pe3YJILTaTbl, nOJ1Y'IeHHLle
M'I JIHHeHHoro MHKpO,l\BMraTeJlJl, Ha cnY'IaM
MHKporomraTeJlJl ,nml c03,l\aHHR KOJIe6aTeJJL-
Horo spallJ,aTCJJLHOrO ,l\Bll:>KeHIDI.
06P A THblE rIPEOEiP A30BA TEJIB
185
3
2
t Y
'3
.-
x
PHC. 3.33. npHHltHml8.fILH8H cxeM8 IHKpO}],BH["8Te.fIH BpaU(eHHH:
1 - BM; 2 - 3neMeHTbl KpenneHIDI K II0}],J10>KKe 11M 1I CTaTOpHblx cTpYK'I)'p;
3 - ynpyme :meMeHTbl nO}],Beca; 4, 5 - pOTOpHble 1I cTaTopHble rpe6eWiaTbIc MlIKpOCTpYKTYPbl
Ha pHC. 3.33 npHBe.neHa npHHI1!inM-
aJlbHaJI cxeMa OliHoro H3 B03MO:>KHbIX Bapll-
aHTOB Mr, KOTOpbIH npe.n;CTaBJISIeT c060H
11M (poTOp) 1, HMelOrn.yIO C 3JIeMeHTOM
KpenJIeHIDI (aHKep) 2 K nOAf.lO:>KKe ynpyrym
CBR3b B BH.n;e ynPymx 3neMeHTOB 3. 11M
CO,l\ep:>KHT JtBa CCKTopa c rpe6eH'laTbIMll
crpYKTypaMH, B KOTOpLIX no aHaJIorml c
pHC. 3.32 HMelOTCJI poTopHble 4 H cTaTopHbIe
5 CTPYKTypbl. 3TH crpYKTypLl pa60TaJOT
nonapHO H C03 alOT KOJIe6aTCJIbHbIH 3JIeK-
TpOCTaTH'JeCKHH MOMeHT no KOop.nHIIaTe y.
BpallJ,aTeJ1LHblH MOMeHT (B o.n;uy eTO-
POHY) MHKpo,l\BHraTeJIJI npll YCJIOBHH, '1TO
l1JJOllJ,a,lJ,H B3aHMHoro nepeKpbITHj{ naJIbl\eB
rpe6eH'IaTbIX CTPYKTYP OJl,HHaKOBLI (KaK H B
JIHHeHHOM MHKpo.nBHraTCJIe), MO:>KeT BbI'IHC-
JISlTbCSI no opMYJIe
n;
ML3'Y = 2F 3 N L'i'
, l
(3.96)
r.ne F3 onpe.n;eJIJleTCR no tlJoPMyne (3.91);
nj - 'mCno naJIbl1eB pOTOpa B O)J,HOH rpe6eH-
'laToH crpYKType; N - 'IllCno CCKTOpOB C
rpe6cH'laThIMH cTpYKTYPaMH; rj - p8.!myc
i-ro naJIbl1R poTopa.
)J;nJl MHKpoJJ;BHraTenJl Ha pllc. 3.33
llMeeM nl = 3, N = 2 II no opMYJIe (3.96)
Mr.3'Y =4F 3 (rl H2 +rJ).
llpu.Mep 3.13
PaCCQHTaCM nap8MeTpLl MHKpOJIBHraTeJU!
Bpa eHIDI q3 MI' (PMC. 3.34). Mcxo}],HLIe .naH-
Hble: MaTCpHaJI - KpeMHHR B MOCKOCTH (110),
HMeIOImiHE(1I0)= 1,68 .10 11 H1 , p=2300KI"hf!;
186 rmuia 3. 31IEMEHTHNI EiA3A MI1KPOMEXAHWIECKHX IlPI1EOPOB
a)
.N
".:'.
. .
:'Jr- .
A" .
('.
q-
jj .:
!'-....
.... ::00.
"'I' ::", ...
6)
)0
x
-[j
Pile. 3.34. K npH!IIepy pac<jera:
a cxeMa 1..J:3 Mr; 6 - tlJparMeHT rpe6eH'iaTOH crpYK'rypbl;
.u. - Mecra .!L1UI rpe6eH'iaTblX cTpYJI."1)'p MlIKpO!lBllraTenJi
TOJIIlUIHa b.-peMHHeBoH fl.!Ia.crllHbl lBcex BbITpaB-
JJeHH.bIX MHKpOCTp)'KTyp) C n = 20 . IO-{i M; do =
= 0.55 . 10- 3 M (.:rna.>.IeTp OCHOBaHIIJI - aHKepa,
aHO.!lHO npHcoe.!I.HIDIeMOrO K creIGlllHHOH nO]1-
JIO:JKJre); d = 3 . W- 3 M (.llHaMCTp p<rropa); II = 0,7 x
x 10- 3 M, 1 2 = 0,35 . 10- 3 M (.iL'lHHbl ynpymx 3JIe-
MeHTOB nO,;:J,BeCa); b = 8 . ] O-{i M (lllHpHHa 3)'61.10B
rpe6eH'IaTblX crpyKl)1J); )'O = 2 . IO-{i M (3a:30p
MeJi\.!I.Y 3y6UaMH).
1laCToTa co6crneHHblx KOJIe6aHHH pOTOpa
no Koop.wmare y palma ro.,o = 3000 rl.l = 18,84 x
x 10> panic; napaMeTpbl nHTaHID! (no cxeMe Ha
pHC. 3.33) U Oil = U z = 5B.
BwrnCJIeHID! BblnonHHM B CJJe]1YJOllIeH
Oqepe.nHOCTH:
L OceBOH MOMeHT HHepllHH pOTOpa
J, 2 ,85. 10- 13 £T'';',
2. Ha OCHOBaHHH 4>OPMYJJbl ro')'o =
= J G'LY' J y BbJ'IHCJJlfM cyMMapHYIO n<ecrKOCTb
3neMeJlTOB ynPYTOro no.n.eeca:
G1:.y =Jyro =2,85.10- 3 68.84.10 3 f = 10-4 H'M.
3. Onpe.lleJIHM npH 3a,!laHHblX 3HaqeHHAA
110 1 2 , c" (AIIHHbJ H TOJJIJ.lI{Ha) nIHpHHY b n ynpyrHX
3neweHTOB DO.aseca, 06ecne'iHBaIOUlHX -rpe6ye-
M}'IO :IICeCTJroC'I1,. )l{ecncocn, o,W{oro ynpyroro
3neMeHTa (631lKH) Ha H3rH6 onpe.llemrerc$I tlJop-
3
_ E(IJO)Cnbn
MynoH Gr.')' = B npennonon<eHHH. 'ITO
61
BH}'I'peHHHH KOHeu 6aJIlCJ1 )KeCTKO 3allleMJ1eH Ha
aHKepe. a Hapy".ll<HblH cB060.nHO nepeMelllaeTCJ:I.
HMeJ:l B BH.U)', 'iTO Bce 6aJ1KlI pa6OTalOT napaJI-
neJJbHO II II = 2/ 2 , nonyqaeM:
3 b 3
E(IIO)Cnb n E() JO)C n n
Gr.'f =2 +2
6/) 6/ 2
3
E(110)c n b n
I)
OTKy.na Tpe6yeMaJlIImpllHa ynpyroro 3J1eMeHTa
10-4 -0 7 _10- 3
=3 '
1,68.10 4 .2.10- 5
Gr.-/I
b n =3
E() IO)C n
= 27,5 .10- 6 M.
4. BbI'iIlCJJHM 'iHCJIO 3y6uOB pOTOpa. KOTO-
d
poe MOn<HO pa3MeCTIl.Tb Ha .a.nHHe /1r =--rj.
2
d o 3
npllMeM rj =-+/ 2 = (0,275 + 0,35)10- -
2
= 0,625 . 10- 3 M II nonY'iHM /1r = (1,5 - 0,625)x
x 10- 3 = 0.875 . 10- 3 M. HaH)J.eM = 2(b + Yo) =
= 2(8 + 2)W..(, = 2 . 10- 5 M II onpe.neJIIIM
n=/1r' =0,875.1O-3'{2.1O-5)=43,75. jj
pacqeya ycnoBlIMC$I. 'iTO Ka)f{Jl.M rpe6eH'iaTaH
CTpYKTYPa HMeeT n= 40.
OEPATHbIE IlPEOEPA30BATEJIH
187
5. PaCC'lMTaeM MOMeHT MHKpOL1BHraTeml.
CIlJ1a, pa3BHBaeMaJI 0L1HI1M 3y6QOM pOTopa,
onpeJl,eJ1}[erc}[ no iPopMyne (3.91):
F = 4 EEOC U U =
:J A on x
'-'Yo
1.885.10- 12 .2.10- 5 .5.5 9
=4 ' 8.85.IO H
2.10- 6
HeTpYJl,HO OOKa3aTb, 'ITa C)'MMa Bcex pa-
JJ;liYCOB rpe6eH<IaTOi:i CTPYKTypbI O}J.HOro cerMeH-
Ta MHKpOL1BHrareml HaXOJJ;IITCJI CneJl,YlOIIU1M 06-
proOM:
d /I
L1j =?(1+11)- L>=1,5'1O 3(I+40)-
- 1=1
-2.10- 5 (1 +2+...+40)=45,1.10- 3 M.
Ilo iPopMyne (3.96) paCC'fHTaeM MOMeHT
MHKpOJl,BMraTeJUI
M Lgy =4F3L1i =4.8,85.10- 9 .45,1.10- 3 =
=1,6,10- 9 H'M.
6. OnpeJl,eJ1MM yron nOBOpOTIl pOTOpa B CTa-
TM'ieCKOM pe)KI1Me:
M L3y
Yo=
G Ly
1,6- 10- 9 16 10 -5 -
=,' pal1-
10- 4
= 5,5. 10- 2 ,.
.lJ,nJl Mr 06hl'lHO Tpe6ye-rcSi 8MUJ1HTYJJ;a KO-
ne6aHMH Yo "" 1 0 Ha 'laCTOTe COyo = 3000 rQ. Tpe-
6yeMoe 3Ha1.JeHlIe aMIU1HTY,I:\bI Kone6aHMM pOTopa
MOJKHO o6eCne1.JHTh TonhKO B pe30HaHcHoM pe-
JKHMe pa60TbI, npH KompoM 'laCTOTa reHepaTopa,
06eCne'fMBalOllIero Kone6aTeJIbHblM peIKHM, OT-
nM'laeTCSI OT 'faCTOTbI coyo Ha 1 0 %. TIpM 3'fOM
aMnmnYJl.a pC30HaHcliblx Kone6aHMH YOp = YoQy
(Qr - Jl,06porHOCTb no Jl,eMniPMpoBaHlIlO). Ilpll
Qr = 10 3 MOJKHO nonYLJHTb Tpe6yeM)'IO aMnJIlIT)'-
JJ.Y Kone6aHMH pOTopa YOp = 5,5 . 10- 2 .10 3 = 55'.
)J;JI5I 3JIeKTpOCTaTH<reCKHX JUIHeflHbIX H
MHKp011BMraTeJ1efl BpallJ,eHIDI KpHTH'JHL!
nOBepXHOCTHble .ne eKTbI Ii 'laCTHl\.bl nblJIll,
a TaKlKe MaJIhIe BepTHKa)IbHbIe pa3Mepbl, TaK
KaK TJJyMO 06ecne'iHTb Tpe6yeMoe 113MeHe-
IlHe eMKOCTll.
3.3.2. MarHHT03J1eKTpM'JecKlfe
npeo6pmOBaTeJ11f
Ha pHC. 3.35, a npMBe.neHa cxeMa Mar-
HHT03J1eKTpH'leCKoro !Ipe06pa30BaTeIDI ()J,aT-
'IRKa) CMJIbl Ha npHMepe MaJlTHHKOBOro
KOMneHC8L\HOHHoro aKCeJIepOMeTpa cepHH
AT [6, 22]. MarHMTHaJI CHCTeMa 06pa30BaHa
OlIHOpOl\HbIM MarHHTHbIM CJIOeM 5 C HaKO-
He'iHHKOM 4 H MafHHTonpOBOJ].OM 2. B 3a30-
pe 6 3aMKH}'Toro MarHHTOnpoBo)J,a pa3Me-
IIleHa B03Bparn.alO aJI (cMnOBaJI) 06MoTKa 3,
BbInOJIHeHHM 3JIeKTpOocroK,!.\eHHeM Ha
KpeMHHeBoM MaJITHHKe I. MarHMTHaJI CHC-
TeMa H3rOTOBneHa !IOCpen.CTBoM BbICOKO-
TeMneparypHoro HallbIJIeHH Ha nO,lI,JIO)!(KY
MarHHTHoro MaTepHaJIa B BH)J,e HeCKOJIbKI1X
eJIOeB H !IpoMe:>KYTO'iHhIX pa3MepHblx TpaB-
JIeHHM )J,JUI npM.D;aHH5I CHCTCMe 06"beMHoro
MHKpOpeJIbe a.
OlIHHM M3 nYQuII1x MaTCpHaJIOB ,llJI
HarrbIJIeHM MafHMTHbIX CJIOeB HBJIjjeTCjj co-
CTaB Al-Ni-Co 800, a )..\JISI BCTpaMBaeMbIX
)J,HcKpeTHbIX MarHMTOB - K06aJIbT-CaMapH-
eBhle CITJ1aBbl C ,ll06aBKaMH M3 pe.nKo3eMeJIb-
HbIX MaTepHaJIOB.
B03BpaIIlaIOLUa51 06MoTKa MO)KeT ObIT!'
H3rOTOBJIeHa cnep'YIOIlJ,MMH TeXH0J10rH'le-
CKHMM cnoc06aMH: BaKY)'MHbIM HarrbIJ1eHH-
eM HeCKOJIbKHX CJIoeB C H30JUIIllleH Me:>KJI.Y
HMMH. 3nHTaKCHaJIbHblM BblpmIlMBaHI1CM
'iepe,llYIOIlJ,I1XCH. p- H n-cnoeB llnH MeTon.OM
MHOrOKpaTHblX )J,1l Y31lH C pa3JlM'iHOH
npHMeCblO pa3J1H'lHOH npOBO!1HMOCTH. Cno-
c06 BbIIT0J1HeHll51 06MOTKH nOKa3aH Ha
pMC. 3.35, 6, B.
CnoM n Cny)KJ1T npOBO,l\HIIJ;CH OOMOT-
KOH, a cnoH p 5IBn5leTCjj M30J1HPYIDIIlMM.
Me:>KBHTKoBoe coelIHHeHHe ocym:eCTBJ1eHO
nocpe.l.I.CTBoM !1BYx lIl1!}J Y3HM pa3JIH'lHofl
npOBO,!U1MOCm (n03HIlIDI9 Ha pHC. 3.35,6).
TIHTaHHe 06MOTKa nonyqaeT 'Jepe3 KOHTaKT-
Hble llJIOllJ,an.Kll 7 II .nopOJKKY 8.
qHcno BMTKOB npH JII060M cnoco6e H3-
rOTOBJIeHIDI 06MOTKH orpaHll'leHO TeXHOllO-
fU'lecKHMH B03MO:>KHOCTHMH «< 10). O.nHa-
KO )J,JUI 60JIblllMHCTBa CJIY'laeB npHMeHeHIDI
3Toro OKa3l>IBaeTCK BnOllHe .nOCTaTO'IHO.
188 rnaaa 3. 3TIEMEHTHA>1 EA3A MI1KPOMEXAHUl{ECKI1X IlPI1EOPOB
aJ
6)
]
2
3
4
5
6
9
I
I
I
I
.
I
.
I
I
I
.
I
.
I
I
I
I
I
L._____._._________.
6)
11
p
Poco 3.35. CXCM8 M8rHHT03J"ICKTpH'iCCKOrO )laT'IHKa CHJ1bl (0);
B03BpamalOmaH o6MoTKa (6); YBeJIHQeHHbJii $panueHT COe)1nHeHHH CJlOeB (6):
1 - KpeMHHeBblH nOllBWJKHblH 3JIeMeHT; 2 - MarHlITOnpOBOJl; 3 - B03BpaIIIaWmIDI. 06MOTKB;
4 - nomocHblH CJlOH B MarHHTOnpOBOJle (HBKOHeQHliK); 5 - O,zrHOpOJlHhIH MarHlITHbIM eJIOH
(HilH nHCKpeTHblH MarHlIT); 6 - KOJIbl.leBoH 3a30p; 7 - KOHTaKTHIDI. IIJIOmanKa; 8 -lIHI}JI}JY3HOHHIDI.
,!10pO&:Ka B03BpaIllaWlUeH 06MOTKH; 9 - IIJ10IIIB.1um Me>KBHTKOBbIX coelIHHeHliH
BejJ}{lU{Ha pa3BHBaeMOH. MarHMT03JIeI<-
TpHqeCKOH CMJlbl oDpe,;!eJISleTCR H3BeCTHOH
3aBHCHMOCTbIO
FICJ = B 3 nlUI(R L + R H ),
(3.97)
r.ne B3 - MarHHTHWI B 3a30pe; n -
'llICJ10 BHTKOB; I - .nJIHHa O.ll.HOrO BHTKa; U -
HanpIDKeHHe, npHJIO:>KeHHOe K 06MoTKe (SlB-
ID1eTCB: O)1HOBpeMeHHo HHl}>opMa HoHHoM
BbIXOJUfOH BeJIM'IHHOH); R., - cODpDTMBJIeHHe
HafPY3KH; R L - COnpOTHBJIeHlle 06MOTKIf.
OCHOBHaH npo6neMa B pa3pa6OTKe
ManrnT03JIeK1pHQeCKOrO .n,aruuca - YCTa-
HOBJIeHIfe 3aBHCHMOCTH MeiK,ll,Y pa'3MepaMH
OCHOBHOro H llI)'HTHPYIOD.lerO cnOCB Mar-
HH11iOrO M8TepH8J1a H TeMDeparypHbIMH
J( HI{HeHTaMJ{ MarHHTHoH HH.lI.yKUHH
3nfX enoeB. B npeJUlOnO:iKeHHM, 'ITO npH
Docne.nOBaTeJlbHO.M COeJUiHeHHH MarHHTHbIX
CJIoeB B 33MKHyTOH CHCTeMe HaIlpj{:iKeHHOCTb
Mal"HHTHoro nOM B cnoJIX O.llHHaKOBa, MO:>K-
HO 3aIIHcaTh paBCHCTBO
B;V 3 = fJ.3 CJH ",B.i ' II' (B 3 V 3 ;: Bw V ru ),
(3.98)
r,lle B 3 . V 3 ; EM' VM; BUJ' VUJ - COOTBeTCTBeHHO
MarHIiTHWI HHn Hj! H o6"beM )J,JIH. 3a30pa,
MarHIfTHOrO cnoSl H lllYHTa; Jl3 - OTHOCH-
TeJIbHWI MafHHTHM npoHH aeMoCTb 3a30pa;
H.... - Hanpj!:iKeHHOCTb noJISI B MafHHTHOM
CJ10e; () - K03 m..tHeHT pacceSlHIDI (npH
OnTHMH3al{llll KOHcrpYKl{MM () 1).
3Ha'leHMe V 3 3aBHCHT OT Cpe,l.(HeH nJIO-
D.la.Lll1 H ,llJIHHLl pa60'lero 3a30pa. Cpe,l\HIDI
llJIOlI{a,lUo pa6o'lero 3a30pa onpe)J.eJISleTcj!
"CXO,WI Jf3 Tpe60BaHHi.i cBo60)J,Horo nepe-
Mel.I1;eHHSI o?MOTKll, a ,llJIJfHa 3a30pa - B03-
MO:>.KHOCTbIO C03.naHH nOCTOSlHCTBa llHJ1.YK-
l{HH. B LteJ1SlX o6ecne'lCHIDI JIllHeMHOCTH
MafHMT03JIeKTpH eCKoro npeo6p83oBaTeJIH.
,llJIHHa Y'IaCTK8 C nocToj!HHOH HH,l.(YI<l{llet'r
BhI6MpaeTCSl 06bltIHO Ha nOpSIJJ.OK 60JIbllle
MaKCMMaJIbHO B03MO:>.KHOrO nepeMellJ,eHHSl
06MOTKH.
MarHHTHblH MaTepMaJI xapaKTepH3yeT-
Cj! npOH3Be,lleHHeM BMH",. B CBOIO O'lCpe.z:\b,
'lHCneHHIDI BeJIM'IHHa 3TOro npoH3Be,lleHHj!, a
TaK:>Ke TeMneparypHbIH K03<lJelmuHeHT Mar-
H"THOH HHJtYKl\HH 3aBllCHT OT COOTHOIlleHHSI
HHrpe.llHeHToB. II0'3ToMY JVI" onpe.neJIeHHR
Ol)PATHhIE rIPEOl)PA30BATEJII1
189
BeJIHLHlHbI MarmUIIOH HHnYKHI1H B 3a30pC
rrpl1 pa3pa6OTKe naT'mKOB He06xo.l1HMO
'3HaTb HCTI1HHOe 3Ha'leHHe OCTaT04HOH I1H-
.n.YKUHI1. K03PUHTI1BHYIO CHJIY, a TaIOKe TeM-
rreparypHblH KOJiPiPHI..\HeHT MaTepHaJIa,
npHMeUSIeMoro )J,JJSI HaITbIJIeHI1S1 MamllTHOrO
CJIOH. HY:>KHO 3KcnepHMeHTaJIbHO H3MCpl1Tb
11 11 YKa3aTb B Ka'lCCTBe Tpe60BaHHi.t K Mar-
HHTHbIM MaTepHaJIaM paC'IeTHbIe 3Ha'ICHHH
::nHX BeJIH'mH, no KOTOpblM npH cepl1f!HOM
npOH3B0.l1CTBe 6y.l1YT npHrDTOBJIeHbI MaTe-
pHa.:"Ibl C COOTBeTCTBYlOllJ,llMH xapaKTe-
pHCTHKaMI1.
)J,JISI BCTpal1BaCMblX B MarHHTOrrpOBO,l\
,!IHCKpeTHbIX MamHTOB npe.l1Bapl1TeJIbHble
paC'ICTbI Y.D.06HO OCYll.{CCTBJISlTb C I1CnOJIb30-
BaHHeM Kpl1BOH pa3MarHH<mBaHI1S1, xapaK-
TepH3YIOllJ,eH KOHKpeTHbIH MaTCpHaJI
(Pl1c. 3.36).
KpHBaJI pa3MarHI1'mBaHlUl npC.l1CTaBJHI-
eT C060H BeTBb neTJII1 rHCTepe3I1Ca, pacna-
J10JKeHHYIO BO BTOpOM KBa.LI.paHTe OTHOCH-
TeJIbHO B H H. )J,JUI JIlO60rD MarnHTHOrD Ma-
Tepl1aJIa Ha KpHBOM pa3MarnH4HBaHIDI HMe-
eTCR xapaKTepHM TO<IKa C MaKCHMaJIbHblM
3Ha'leHHeM npOH3Be)J,CHHSI BH (TO'lKa a).
)J,JISI CllJIaBa AI-Ni-Co-800 MaKCHMaJIbHOe
3Ha'lCHHC (BH)mo., = 8,7 . 10 4 (TJ1-A)/M. Pa-
60'laJI TO'IKa onpe,neJISleTCR Ha nepece'leHHH
KpHBOfl pa3MarnM'mBaHIDI C rrpRMOH npoBO-
,l\HMOCTH. YrOJI HaKJIOHa npRMofl npOBO.l1H-
MOCTH BbI'mCJIReTCR no 1JI0pMYJIe, (Tn'M)/A,
I
t g a= LG.
S "
M
r,l\e 1 M , SM - JtJ]HHa H nJIO II a)l.b nOnepe'lHOrO
Ce'leHHR MarHHTa (O].l.HOPO].l.HOro MarHHTHO-
ro CJIDSI); G j - npOBO)J,HMOCTb ;-ro y'laCTKa
MarnHTHofl CHCTeMbl.
HeTpyJtHO nOKa3aTb, 'ITO OCHOBHaJI ].1.0-
ml conpOTHBJIeHHSI npOXO)f{l(eHHIO MarHHT-
Horo nOTOK8 npllXo.nI1TCSI Ha B03.l1YlllHbIe
3a30pbI, B KOTOpbIX paCnOJ1aralOTCSI B03Bpa-
llJ,alOllJ,He 06MOTKll.
TIpll BbI60pe pa60'leH TO'lKl1 B6JI1l3H
3KcrpeMaJIbHOrO 3Ha'leHHSI Ma'rHIITHWI CMC-
TeMa 06JIa.u.aer MllHHMaJIbHbIM o6'beMoM, <no
IOH.lIK-25
AI - NI- Co-8000 B T
, :1
1,4
1,2
1,0
0,8
0,6
0,4
Q
0,2
H.KN"1
500 400 300 200 100
o
PHC. 3.36. KpHBble p83Ma.-HII'lIlB8HIISI
SlBJUleTCSI Ba)l(HOM xapaKTepHCTHKoH ITPH
MI1KpOI1CnOJIHeHHI1. )J,J1S1 cpasHcHHSI Ha
pl1C. 3.36 npHBe,l\CHa KPHBM p83MarHH'lHBa-
HHH ,llJIH CnJIaBa JOH)J,K-25, OTKY)J,a BH)J,HO,
qTO ,lIJISI MaTepHaJIa C npHCa,lIKaMH H3 pe)J,Ko-
3eMeJIbHbIX MeTaJIJIOB K03PUHTHBHM CHJ1a
-
npeBocxo)J,HT OObl'lHbIl1 MarHMTHbIl1 MaTepH-
aJI, qeM 060CHoBblBaeTcSI ero BbI60p. O,lIHa-
KO onpe)J,eJIJlIOllJ,HM npH rrpoeKTHpOB8HHH
RBJ1.H.eTCR He MHHHMaJIbHbIfl pa3Mep MarHMT-
HOH CHCTeMbI, a MHHHMaJIbHM rrorpelliHOCTb
npe06p83oBaTellil.
MarHHTHYIO HHJ.{)'KQHIO B 3830pe B03-
MO:>KHO CneJIaTb He3aBllCI1MOfl OT 113MeHeHHH
TeMnepaTypbI nocpC,l\CTBOM no,n60pa COOT-
BCTCTByrollJ,HX TeMneparypHblx K03 l}>llQH-
eHTOB OCHOBHoro H rnYHTl1pYIOIIlero CJIOeB, a
TaIOKe HX Om,CMOB.
TIpHMepOM I1Cn0J1b30BaHHH. MafHHTO-
3JIeKTpM'lecKoro rrpe06p83oBaTeJUI CMJIbl B
Mr RBJUleTCR rnpOCKon C KOJIbueBLlM pe30-
HaTopoM [22], npllHI.{HllHaJIbHaJI cxeMa q3
KOToporo nOKa3aHa Ha pHC. 3.37, a.
q3 BbIITOJIHeH Ha KpeMHHeBofl rrJJa-
CTHHe 3 (pa3MepoM IOxlO MM), Ha KOTOpOH
pa3MeIIleH KOJIbueBOfl pe30HaTOp (KOllbUO)
,l\HaMeTpOM 6 MM H CC'leHHeM 120xlOO MKM.
Pe30HaTOp c rrOMOlItblO BOCbMH ynpynix
3J1eMeHTOB 2 nO,ltBeCa ce'IeHHeM 60x 1 00 MKM
coeJ.{HHeH C nJ1aCTHHoii. IlMn H3rOT0BJ1eH
MCTO,l\OM rrJI83MeHHOro H30TponHoro TpaB-
neHIDI. TOKOnpOBO)1,JlllJ,He npoBo.nHHKH HR-
HeceHbI TOJIbKO Ha BepXHlO1O nosepxHoCTb, a
KOH'faKTHble nJIOIlJ,a,zI:KH 4 JJ)IJI coe.mmeHIDI
190 fJIaBa 3. 3JIEMEHTHNI 6A3A MIIKPOMEXAHI1 IECKlIX nPH60POB
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Pac- 3.37. MaruHT03JICKTpRIJ:CClmU npco6pa1oBaTeJJb CRJIL! KOJIbQCBoro pe10JlaTopa:
a - npHHwmHaJ1LHaJI cxeMa 1J3 Mf; 6 - <}JparMeHT KOHCTpYKUHlI KOJ1LueBOro pe30HaTopa;
6 - npHBInlflHaJILHIUI cxeMa MarmiTo:melITpHLjeCKOrO npeo6pa30BaTeJIJI CHJIbI; J - KOJIbUeBOH
pe3OHaTOp; 2 - ynpyrnH 3JIeMeHT nO Beca; 3 - KpeMHl1eOaJl nJIacTl1Ha; 4 - KOHTaKTHaJi nJIOl1\8)IKa;
5 - nposoJU{HlC; 6 - BepXHHH H HlI)!(HHH nomocHble HBKOl1eLjHI1Klf; 7 - CTeKJUlHHaJi
nOJ1lJO ; 8 - fiJIan C 3JIeKTpOHI1KOH; 9 - Kopnyc; 10 - Kpbl111KB
c npoBOJ1HHKaM.H pa3MemeHbI Ha fUJacTHHe.
l-IMeercjJ BoceMb H,l\eHTH'iHbIX npoBo-
MUJ.HX ICOmypoB. lCailC.lXbdi H3 KOTOpLIX 06-
P83Yer neno<fKY: 1C0HTaKTHWI 1VI0UlaJ{Ka -
ynpyndi 3JJeMCHT nO.D,BeCa - 1/8 JV]HHLJ
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1C01ITaJC'l1UUJ W10 1l{a I\lCa . TaKHM o6pa30M,
H ynpyrnH 3JJeMeHT CO,l\epJfCHT ,l\Ba
npoBO no ollHoM}' OT JCajlCJ{oro CMe:>K-
Horo KOHTypa, B ,l\onOJIHeHMe K TpCTbeMY
npoBO,l\HHKY 5, KOTOpbIH Haxo.nHTCR Me>KJ1.Y
HHMH, 'IT06LJ Ml1HHMH31lpoBaTb eMKOCTHYJO
CBSl3b. KpeMHHeBWI nJIaCTHHa HMeeT CBR3b.
o6eCnC'IHBaIOUlYIO "3eMJISIHOH" CJIOH.
c1>pameHT KOHCTpYKllHH KOJIbQa c yn-
PymM 3neMeHTOM nO,l\BeCa II 'laCTblO KpeM-
MHeooH nnaCTHHbI nOKa3aH Ha pHC. 3.37, 6,
rJ{e xopowo npocMaTpHBaJOTCH. COOTHOWe-
OEPJ\THbIE rIPEOEPA30BATEJII1
191
HII}\ MC)I(,l\Y pR3MCpOM KOJIbL\R H '1JleMCIITRMH
ero nO;I,BeCR.
KpeMHHCB8J1 ITJIaCTHHa 3 aHO.l(HO co-
CjUlHClla co CTCKJISlln!Oii nOnJIml(JwH 7 (pHC.
3.37, (I). OHH TCMncpaTypHo COrJI8COB8HbI
MC)!()J,Y C060H. H8 nOnJIO>!(Ke pacITo.T]o)!(CH8
T8K)!(C MarHHTHaSi CI1CTCMa, B!UII04a10ll\aSl B
CC6}\ K06a."IbT-car-.mpHesbIH MarHHT (N-S) 11
MarHHTOnpOBo.l( 11'3 .n;BYX nOJIIOCHblX H8I(OHe4-
HHKOB 6, B 1a30pc MC)l(JJ.Y KOTOpbIMli H8XOJ],IfT-
C}\ KOJIbLlCBOH pe101laTOp J. CcpBHCHaH. 3JICK-
Tpm-noma IIaXOP:HTCJI Ha Cl-fraJlJ10BOH nJI8TC 8.
Bc}\ KOHCTpYKLlIISI c06p8l-Ia B rCpMeTHtlHOM
KOBapoBoM Kopnyce 9 C KpblWKOH J O.
Man!l1TO::JJICKTpH<lecKHH npeo6pa10Ba-
TCJIb COB:\-ICCTHO C :UCKTpOHHKOM 06eCITC4H-
BaCT 3JIJIHnnl4CCKYI0 MO)J,Y KOJIc6aHHH
KOJIbLla lIa 4aCTOTC ero :\-leXaHI14CCKOro pe-
30H8Hca [9].
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KOJIe68HHH C03.l(8IDTC}\ CJIC,llYIDll\HM 06pa-
30M. B ncpBbIH no."IyncpHOJJ. H8np8BJICHIIC
TOKOB B cCKTopax 111 V (noK83aHbI cTpCJIKa-
MH) BO B3aHMoJJ.cHCTBlIH C MarHIlUlbIM nOTO-
KOM C03.l(aIOT (ITO npaBHJ1Y JIeBOH PYKH) pac-
T}\fliBalOll\He CHJIbl, 8 B CCKTOp8X III H VII-
C)I(HMalOll\He CliJIbI. Bo BTOpOM ITOJIyncpHo-
JJ.C xapaKTcp CHJI :\-ICII}\eTC}\ Ha npOTl-lBono-
JIO:>KHbIH. Ka>[(.l(8}\ napa npOBO,llJlIlIMX CCKTO-
pOB KOJIbLla C03JJ.aCT CHJIY:
1trBU np
F. ""
M 4R
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(3.99)
mC r - pa)J,HYC KOJIbLl8; B - ManIHTHaH HH-
.n;yKLlH}\ B 3830pC Me:>K.l\y nOJIlOCHbIMH HaKO-
He4HI1KaMH ManlHTHOH CHCTeMbl; U np H8-
npSl:>KCHHC, npHJIO:>KCHHOe K npOBO.l(JlIlJ,CMY
CCKTOpy KOJIbLla; R" - 3Ha'lCHHe HarpY304-
Horo pe3HCTopa, BKJIIO'leHHOrO nOeJIe)J,OBa-
TCJIbHO c npOBO.D,JlIlJ,I1M ceKTopOM KOJIbLla.
3aMCTHM, 41'0 npOBO)J,HlltHC ceKTopa VI
H VIII KOJIbLl8, BH6pHPYIDIlJ,erO Ha (IaCTOTe c.o
no,l\ ,l\CHCTBHCM npe06p830BaTCJISI CHJIbI, Hr-
paIDl' pOJIb SlKOpSI HH,llYKLlHoHHoro npc06pa-
30B8TeJISI nepCMCllJ,eHHH. DpH 3TOM BbI.D,CJISI-
CTCSI HanpSl:>KeHHC
U ::::. 1trBc.of! ( 3.100 )
DL1X 4'
r.l(C f! - BH6ponCpCMellJ,CHHC KOJIbLla.
CeKTopa II H IV I1CnOJIb3yIOTC}\ B I<OH-
Type CHJIOBOH 06paTHoif CB}\3H )J,JISI YCTpaHC-
IIHSI JJ.c opMaUHH CMCllIel!l1}\. H3MepI1TCJIb-
HblC 11 KOMneHcaUHOHHblC ceKTopa B Ka:;f(llOIl!
nOJIyncpHO)J,e MeIISiIOTC}\ POJI}\MH [14].
qacToTbl c06cTBeHHbix KOJIc6aImil B
nJIOCKOCTH KOJIbL\a H MaKCnMaJIbHaSi JJ.c op-
MaLlH}\ KO."Ibua noJJ. Cj:jCTBI1CM CHJI, qJOpMn-
pyeMbIx 'leTbIPbM}\ npOBo.l(HIlIHMI1 cCKTopa-
MH KOJIbLla, onpeJJ.enmOTC}\ opMYJIaMH [24]
. = U 1 . o=0149 FMr3
(01 I 4 " ,
vI + ;2 pFr EI
(3.101)
rJJ.c i - LlCJIOC 4HCJIO, xapaKTepH3YIOlltee
cpOPMY KOJIe68HI1H (OCHOBHOM 'laCTOTe COOT-
BCTCTByeT i = 2); E - MOJJ.YJIb ynpyrocTH
M8TCpHaJI8 B COOTBeTCTBYIOlltCM KpHCTaJIJIO-
rp8 H'lCCKOM l!anp8BJIeHHH; 1, F - MOMeHT
HHepLlHH H nJIOllJ,a,lI.b nonepc4Horo CC4CHH}\
KOJIbLla.
flpWtlep 3.14
BbI'IHCJIllM napaMeTpbI Mr. I1CXOll-
HbIC ,l:\aHHble: MaTepUaJI (KpeMHuli) HMeeT p =
= 2328 Kr/M 3 , E[1ooJ = 1,295 . 1011 H/M 2 ; pa3Mepbl
KOJIbl,a: r = 3 . 10- 3 M (pan.nyc); b K = 12 . 10- 5 M
(Wl1pI1Ha), C K = 10--<1 M (TOJIIIIHfla); Urrp = 5 B.
,
R" = 100 OM, B = 0,3 B6/M-.
rIo opMYJIe (3.99) onpe,ll,eJ]HM CIfJIY, C03-
aBaeMYIO B HanpaBJIeHl1}JX paCTI[>KeHml 1I C>Ka-
Tlfll:
F. - 3,14.3.10- 3 .0,3.5 - 353-10- 5 H.
M 4.100 '
HaHj\eM
J =hb )/12= 10-4 'I _10- 15
::::144.10- 19 M 4 .
nOJIO>KHM i = 2 H nOJ]yqlfM
E b 3
=268 =?68 1100]b; -
(02' 4 -, 12pr 4-
12pcI(bl(r
= 2 68 1,295 . lOll (12.10- 5 r
, 12'2328 '1O-3 r
=7,69.10 4 paWc= 1,225. 10 4 fI.t.
192 rnaBa 3. 3nEMEHTHA5I BA3A MI1KPQMEXAHI11..J.ECKI1X rIPI1EOPOB
3aMeTHM. liTO npll TaKOll BblCOKon C06CT-
BeHHoM 'lacrOTe BHeUIHlle BH6paUHOHHble 803-
MymeHHH., KaK npaBlmo Ha MeHblllltX 'iRCTOTax,
MOJKHO He yqlITbIBan.
BLl'tHCJIHM MRKCIlMaJIbHYlO .ilecpopMaW-IlO
KOJIbUa:
0= 0.1 49 3,53 _10- 5 (3.10- 3 Y -
. 1,29S.IO ll .144.1O- 19
=7,6.10- 8 !II=7.6.1O- 2 MKM.
MaKCHMaJIbHOii .ae opMaI.Um KOJIbI.4a
Ha C06CTBeHHOH qaCTOTC COOTBeTCTBYCT
BMXo oeHanpIDKeHHe
7 _ 3,14-3.10- 3 .0.3.7,69.10 4 .7,6.10- 8
U Bwx'
4
=41,29.10 7 B.
JI.OCTOHHCTBOM paCCMOTpeHHOH KOHCT-
pytalHH )}B.iTI1eTCSI pa60Ta (BH6paLUUl) KOJIb-
lteBoro pe30HaTopa B OJ1.HOH llJIOCKocru, TaK
qro BC)} 06pa60TKa l\.-peMHIDI IUl3H3pHaJI.
KpoMe TOro, B .!1aHHOH KOHCTpYJ<I..lHM OTCyr-
CTBYeT npo6neMa 3aJIHnaHIDI, TaK KaK MeiK):{y
KO.JIhD.OM H OCTaJIbHbIMH 3JICMeHTaMH llOCTa-
TO'lHO 60JIblllHe pacCTOSHHSI.
3.3.3. 3J1eKTpOMarHBTHLIe
npeo6pa30BaTeJIB
lliBeCTHO, lITO .!I.Ba napaJIJIeJIbHbIX H
.lIOCTaTO'lHO MHHHbIX npoBo.nHHKa (B MHK-
poMexaHHKe - no cpaBHCHll10 C paCCTO)}HM-
eM Me'"&.ay npoBO.D.HHKaMH) C TOKOM B3aHMO-
.neHCTBYlOT .npyr C llpyrOM TaR, 'iTO eCJIH
TOICH HMelOT OJJ.I{HaKOBOe HanpaBJJeHHe, TO
OBB npIfDU1{BaJOTC)}, a ecJJM npOTliBOnO-
J10:iKHoe, TO o-rra..1KHBaJOTCjI. MaTeMaTM'ie-
CKoe BIoIpa4reHHe :noro 3aKOHa (AMnepa)
HMeer BH.n
F = J!J!tlI I 2 /
2n8 '
(3.102)
r)le J! - OTHOCHTeJIbHID1 MaI1fJITH.aj{ npOHH-
1!3eMOCTI> MaTepHana, Haxo.wnu,crOC)} Me)K.ny
npoBO.nnmwm; fJo = 1,257 . lO- s fHlM -
Ma.I'HJfI"HIDI nOCTO)}JIIUUI; II, h - CHJIbI TOKa B
DpOBO,IlIDOOIX; I - .n.mma npooolIHJfK8; 1) - 3a-
30p Me1IUIY npoBOwrnICaMH..
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1 - KopnycH8JI nnaCTl1Ha; 2, 3 - Hen01lBI1>KH8JI H
nO.!1BHJKH8JI 3neKTponpoBoIDIIUI.Je 1l0pO>KKlI;
4-HM
Ha pHC. 3.38 npl1Be.lleHa npl1HlumH-
8JIbHaJI cxeMa npe06p83oBaTeml [6], B KOTO-
pOM MO:>KeT 6blTb peaJIH30BaH 3aKOH (3.102)
l}>opMHpoBaHHSI 3neKTpOMarHHTHoH CHJIbl. B
KopnycHoH l\.-pCMHHCBOH nJmCTlmc 1 c op-
MMpoBaHa 11M 4 Ha ynpymx nO.llBCCaX C
ITpOBO,l\HlIU1MI1 1l0pO>KKaMI1 3. Heno,lJ,BHm-
Hble npoBO,llHlIU1e .nOpO:>KKH 2 HaHcceHbI Ha
llJIaCTHHe 1. B .nH CPCHQl1aJIbHOH cxeMe
BKJIIO'iCHIDI onopHoc HanpSI:>KeHHe V on MO-
:>KeT 6bITb ,l\OCTaTO'iHO 60nblllHM, TaK KaK
,l\OpO:>KKM 2 MOryr HMCTb 3Ha'lMTenbHOC no-
nepe'iHOe Ce'ieHHe. YnpaBJIjllOlQCe HanpSI-
iKeHHe U y orpaHH'ieHO TOKOM, KOTOpblH MO-
ryr npoITYCTHTb no.nBH:>KHbIe ,lJ,OpO:>KKH (npo-
BO,l\HHKH).
npH ncpHO,l\H'ieCKOM H3MeHCHHH 3HaKa
ynpaBJJSlJOmero Hanp)}:>KCHIDI MO:>KHO nOJIY-
qUTb KOJIe6aTeJIbHOe .nBH:>KeHHe 11M B.nonb
OCMX, KaK 3T0 npoHCXO.nHT B Mf.
Ha pMC. 3.39 nOK83aHa CXCMa 4>OpMH-
pOBaHHSI 3neKTpoMafHMTHoM CMJIbI B Ha-
npaBJJeHHH OCM Y, XapaKTepHoH )J,JJ)} aKCeJIe-
poMCTpOB. nO.nBM:>KHbIM npoBO,l\HJ.fK 1
pa3MemeH Ha KpeMHHeBOH DJlaCTHHe 11M, a
OEPATHhIE IlPE06PA30BATEJIYJ
193
y
t F
Puc. 3.39. IIPJlHlUlnURJIbUaH cxeMa 3JIeKTpOlllarnUTIIOro
npco6pa3OJmTCJ1H CIIJILI
HenO}J,BI1:>KHbIH npOBO,lJ,HI1K 2 Ha CTeKJUIH-
HOM OCHOBaHI1U. BCTBI1 npOBOnHI1KOB C TO-
KOM !10JI:>KHbI pacnoJIaraThc5J napaJIJIenbHO
.npyr !1pyry. ECJII1 npOBO!1HI1KI1 06pmYlOT
KB8J1.paT B ITJIaHe co CTOpOHOH /» 8, TO B
COOTBeTCTBl111 C BblpaJKCHlICM (3.102) nOJIY-
'lllM
F == 2JlJloI/2 / .
n8
(3.103)
B COOTBeTCTBHI1 C pl1C. 3.38 I1MeeM
IJ "" U v 1('- + R 2 ), r.l\C r - BHYTpeHHCC co-
ITpOTI1BJICHl1e ITpOBO.l\HI1Ka. ECJII1 R I = R 2 =
= R3 = R H , opMYJIa (3.103) npe06p83yeTcSI K
Bl1ny
F
4JloJlU v U on/
2 .
nB(r + R H )
(3.104)
YBenl1tJeHHe CHJIbI F MO:>KCT 6bITb ,lJ,OC-
TllrHYTO nYTeM C03naHI15J cnl1paJIC06pa3HOH
06MOTKI1, KOTOpWl B CJIY<1ae aKcenepOMeTpa
MO:>KeT pacnonaraTbCSI Ha HenpOBO.ll.5J CM
MWlTHI1Ke C )J,BYX CTOpOH. Cnl1paJIl1 COc.nl1-
H5J1OTCSI napaJIneJIbHO. Ha HCITO.nBI1JKHblX
KpbllliKaX C nBYX CTOpOH MaJlTHl1Ka pa3Me-
llJ,aIOTCSI Heno.nB(1)1(HbIe CITllpaJIeBI1.l\HbIe 06-
MOTKI1 co B3al1MHO napanJIeJIbHblMI1 CTOpo-
HaMI1.
KaK OTMe'laJIOCb, B HeKOTopbIX TeXHO-
JIOrl1S1X 113roTOBJlCHIDI MI1KPOYCTPOHCTB
(oc06eHHo xapaKTepHo ,l\JIJI UGA-
TeXHOJlOrl1l1) I1CITOJIb3YlOTCSI npl1eMbI raJIb-
BaHOTeXHI1KI1, T.e. B03MO)l(Ha pa60Ta C Me-
TaJIJlaMI1, TaKI1MI1, HanpM1\Iep, KaK HHKeJIb,
KOTOpbIH SlBJlSleTCSI cppOMafliI1THbIM MaTe-
pl1aJIOM. Y'IaCTOK C eppOMafHI1Tm,IM ITO-
KpblTI1eM, nOMellJ,eHHbIH B MarHI1THOe nOJIe,
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198 rnaBa 3. 3J1EMEHTHMI EA3A MI1KPOMEXAHl1 ECKHX ITPHEOPOB
TIonaraJI, 'lTO R3 = , It] OPMYJIbI
(3.106) ITOJIY'lHM
U =_ VoM K
I Ro M'
(3.107)
r)l.e Kt04 = RJRs.
Ha BTOpoH BXO}l ItHTerpaTOpa nocTyna-
eT Hanp >KeHHe -V o . HanpH)J(cHue Ha onop-
HOM BXO)J,e KOMnapaTopa UK = -V o . KOH,lI,eH
caTOp IIHTerpaTopa 3apH:>KaeTCH OT + V O )J,O
-V o (pHC. 3.43, 6), a Kor.na HaITpSl>KeHlle Ha
BbIXO,lle HHTerpaTopa cpaBHHeTCH C onopHbIM
HanpIDKeHHeM KOMnapaTopa, nOCJle.nHHH
nepelU1JOqaeTCH 113 H)'JIeBOrO COCTO HI1S1 B
e.u.HHH'lHOe 11 nCpeKJ1JO'laeT 3HaK HanpH)j{C-
HHj{ nHTaHHH TeH30MOCTa. TIPOI1CXO)J,HT
npou.ecc p83pj!J],a KOHJ1eHCaTOpa, a 3aTeM
UHlUl " 3apR )I,-P83pJlJ!." KOHJ1eHCaTOpa nOBTO-
pSleTCSI.
IIpollecc 3apMa KOH,l\eHcaTopa B pac-
CM01peHHOM llHKJle OnHCbIBaCTCSI ypaBHeHHcM
dV.,
RC --=- = +V I - U o
df ' ,
(3.108)
r)l.e V 2 - Hanpj["'.A<CHHe Ha BhIxo.ne HHTerpa-
TOpa.
C yqeTOM BhlpaJKeHMX (3. I 07) nOJIY'lHM
M
-2U o RC=-{}o-K M f] -Uo/] =
Ro
::;-{}ot{l+ KM J.
OTlC)'.lI.a c..'Je,r{)'eT BpeW! 3apR,l\a:
t] =2Rcj(I+ KI.IJ (3.109)
nOCKOJIhKy BpeMj! 38pSl..!I.a KOH.neHcaTO-
pa paBHo BpeMeHH ero P3.3pS1)J,8., nepMO)l. KO-
JIe6aHMH T = 2/3 H, '3Ha'lHT, 'IacrOTa BbIXO.n-
Horo CHfH8J1a pasHa
F =(1 + KM ) !4RC).
(3.110)
TaKHM 06P3.30M, J1IDI npHHJITOrO xapaKp
Tepa H3MeHeHH.sI 3HaqeHHH RI If Rz tjacTOTa
BLJXO.nHOrO CHnIan3 YBeJJlftjHBaeTCX no
cpasHeHHIO C ooopHOIt' qaCTOTOH 1/4RC.
fIpe,l\nOJIO)j{IIM Tenepb, 'lTO 3 I-IaLJCHI UI
TeH30pC311CTOpOB 113MeHHIOTCSI TaK: RI =
= Ro- M; R 2 = Ro + /)'R.
,l],JHI :noro BapllaHTa )J,eqmpMal(HIl TeH-
30pe3HCTopOB lIMeeM
VI =U o M K M .
Ro
B COOTBeTCTBI1H C c}mpMYJIOM (3.109)
nonY4HM
I] = 2RC/( 1- KM) .
CJIe.nOBaTeJIbHO, 'laCTOTa BblXO)J,Horo
cllrnana onpe}leml.eTCH Bblpa:>KCHlleM
F =(1- KM )/4RC, (3.111)
OTKY,l\a BHAHO, 'lTO H3MeHeHHC xap8l<Tepa
)J,c opMalllll1 Teli30pe3HCTopOB ITpHBeJIO K
YMCHbllleHHIO qaCTOTbl BbIXO,llHOrO CHrHana.
OTHOCI1TeJIbHaR oliJl16Ka no 'laCTOTC B
3aBHCMMOCTH OT H3MeHeHI1S1 TeMnepaTypbI
onpe.nenReTCSI Bblpa:>KeHHeM 8 F = M/ F =
= (UR - o.c)/)'T, r}lC UR, Uc - COOTBeTCTBeHHO
TeMnepaTypHble K03q)qmllHeHTbl conpoum-
JICHHSI. H eMKOCTH HIiTcrpaTopa.
B. CxeMbl C IllHM lOnynHTopoM
H3 Bblxo.ne
B HeKOTopblX KOHCTPYKllHSlX MHKpOMe-
XaHH'leCKI1X aKCeJIepOMeTpoB, HanpHMCp
ADXL202/ADXL210 ( HpMa Analog
Devices), HCITOJIb3ytOTCSI. BblXO,llHblC YCTpOH-
CTBa, ITOCTpOCHHhIe no npHHwmy ill 11M.
fIpMHI..lI1n pa60nI npOCTeHIllCrO 1lII1M no-
s.lCHReTCH pHC. 3.44.
llpocTeHlllHH illl1M COCTOI1T H3 KOM-
napaTopa H reHcpaTopa JIHHeHHO H3MCHHIO-
llJ,erocs.l Hanps.l:>KeHHs.I (rJJI1H). Ha HHq)OpMa-
llJ10HHhIH BXO.n KOMnapaTopa nocrynaeT
HanpSI:>KeHJie Vox H C BbIXO,l\a aKCeJJepoMeTpa,
a Ha BXO.n OITOpHOrO CHfHaJIa no.naeTCSI ITH-
JI006pa'3HOe HanpSl:>KeHHC U r C BblXO)l.a
fJII1H. B Ha aJIe llJ1KJIa npe06pa30BaHHSI
31'0 HaITpJDKCHl1e nCpeKJIIOl./aeT KOMnapaTOp
8 COCTOs.lHHe JIOrHLJeCKOH eroulHllbl, B KOTO-
"UEKTPOHHblr. CPE,lJ,CTBA OEPAEOTKI1 CMrHAJ10B
199
Un:>.
I
Bxo]J,
U r
IJIM"
I
T 2
U lJblX
71
I
T 2
KOMITa-
paTOp
-
BbIXO.n
PIIC. 3.44. K nOHcnClllllO pa60TbI IlIUM
pOM OH HaXOJ.\HTCJI )J,O Toro MOMCIITa, KOr,l\a
HanpSl:>KeHHe Ha BbIXO.ll,e rJIHH CTaHeT paB-
II)'[M HaITp5DKem1l0 Ha UHcPopMaUHOHHOM
BXO,lle.
DpH nOM KOMnapaTOp nepeKJIIOlJaeTCj{
B COCTOJlHHe nOflIlJeCKOrO "0", B KOTOpOM OH
HaXO,llHTCJI )J,O Tex nop, ITOKa HanpJl:>KeHHe Ha
BblXO,lle rJIHH )J,OCTHrHeT MaKcHMyMa. Do-
CJIe 3Toro Ha'IHHaeTCSI cnenYlOlllHi1 LlHKJI
ITpe06pa30BaHHJI.
TaKHM 06Pa30M, Ha BbIXO.ll,e KOMnapa-
Topa cpopMlIpyeTcJI HMnYJIbCHaJl nOCnC,llOBa-
TeJIbHOCTb, nepHo.n T 2 KOTOpOi1 paBCH ne-
pHO)J,y ITHJIoo6pa3HOrO HanpJl)j{eHHSI Ha Bbl-
XO.ll,e rJIHH, a )J,J)MTeJIbHOCTb HMnYJIbCOB TI
ITpOnOp4HOHaJIbHa HanpSI:>KeHmO Ha Bb[XO.ll,C
aKCeJIepOMeTpa. DpH HYJIeBOM HanpJl:>KeHHH
Ha BbIXO)J,e aKCCJICpOMCTpa )J,J)MTenbHOCTb
HMnynbca TI paBHa nOJIOBHHe ncpl1o)J,a HM-
nynbcHoH nOCJIC,IJ,OBaTeJIbHOCTl1 T 2 . ,l],nSl co-
xpaHcHHSI mmcHHoi1 3aBHCHMOCTH Me>K,l\Y
Bb[XO.nHbIM HanpJl:>KeHHeM 8KcenepOMe-rpa 11
,l\JIHTenbHocTbiO HMnynbca B KalJCCTBC pa60-
'lero Y'laCTKa HCnOJIb3yeTCJI TonbKO nOJIOBI1-
Ha ITepHO.ll,a T 2 .
Ha pHC. 3.45 nOKa3aHbJ npe,l\eJIbI H3MC-
HeHHJI )J,JlHTeJIbIIOCTH HMnYJIbCa TI lLIIj{ aK-
CeJIepoMeTpa ADXL202, HMClOllJ,ero ,lJ,Hana-
30H 113MepeHHJI:I: 2g.
ITepHo.n HMnYJIbCHOH nOCJIC,IJ,OBaTeJIb-
HOCTH MO:>KeT H3MeHJlTbCJI B npc,IJ,cJIax
0,5...10 MC nyreM H3MCHeHHJI BenHlJHHbI
pe3HcTopa, ITO,l\KJIlOlJaeMOrO K aKCeJ1e-
poMeTpy. YCKopeHHe, .nOJIH g, MO:>KeT 6blTb
BbIpa)KeHO lJCpe3 OTHOIlIeHl1e ,lI,JJHTenbHOCTU
llMnYJIbCa WHM K ITCpHO,llY HMnYJIbCHOH
nOCJIenOBaTeJIbHOCTH ITO opMync
a ( ;: - 0,5)/ K LUUM ,
r)J,e K WHM - KOO H41lCHT nepe,lJ,alJH WI1M
(B J(,aHHOM cJIy4ae K WHM = 0,125).
,l],JISI BBO)J,a CHrHaJIa llIHM B KOMnblO-
TCp MO:)J(HO Hcnonb30BaTb cxeMY, B KOTOpOH
H3McpeHHe )J,JIHTCJIbHOCTI1 HMnYJIbCa T[ npo-
BOJtl1TCH nYTeM 3aITOJIHeHHJI TaKTOBb[MM HM-
nYJIbCaMH BblCOKOH 4aCTOTb[. lIHCJIO ITO-
CJIe)J,HHX ITO)J,C'iHTbIBaCTCJI C nOMOlllblO ,lJ,BO-
HIlHoro C'leT<IHKa. aJIee nOJIY4eHHoc )J.BO-
H'lHOe lJHCJIO CIlHTblBaeTCH lJCpC3 lllHHY .naH-
Hb[X KOMnblOTepa H nporpaMMHbIM nYTeM
npe06pa3YeTcJI B .neCJlTUlJHOe IlHCJIO. ITpHH-
4HIT pa6OTb[ CXeMb[ TaKoro YCTpOHCTBa no-
IICHHeTCJI pHC. 3.46.
L 11
-2g -g g 2g
t
12
PHC. 3.45. 38BHCHMOCTb AIlHTeJ1bHOCTH
HJ\mYJlbCO CHfHOJJO mUM OT :mOqellHR
YCKopeHHH
200 ['JIaBa 3. 3nEMEHTHMI EA3A MHKPOMEXAHH'lECKHX rIPIlEOPOB
U ax 11
t
BxolI. pa3pelllemUI
12
UT.l1
t
TaIrfOBbIll BXO.J:
&
)J,BOIt'-IHbIH B hl):-,lnbl()TCP
C'leT'lUK
PHC. 3.46. K nOHCHeHll1O npHuuuna pa60TbI cxeMbl IIJ:\lepcHIiH
iJ,."'1HTe.lbHOCTH H1\In 'JIbCa UIHM
CxeMa COCTOMT 113 llBOH4HOrO C'leT'lH-
Ka H CXeMbI p83pellleHIDI Ha JlOrH'iCCKOM
3.!IeMeHTe 11. UMelOIl1CM ,nBa BXO)J,a. Ha C'leT-
HbIH BXO.n nO.naIOTCj{ TaKfOBble HMnynbcbI
UT.H BblCOKOH qaCTOTbL Ha BXO.a p83pelllCHHR
nocrynaeT CHrHaJI VB>' lllllM. Cl1rHaJI JIOrl1-
qecKOH "1" Ha BblXOt{e 111I1.M pa3pclllaeT
npOxOJl\,!1cHHe TaKroBblX HMnynbcoB Ha Bxon
.!1BOH'lHOrO C'ieT'lHKa.. I1pl1 nOSlBJ1eHHH Ha
BXOllC pa3pemeHHSl norWieCKoro "0" TaKro-
Bble UMIIY.ilbCbI nepeCTaIOT nocrynaTb Ha
BXO,l\ C'ieT'IHKa H J1.BOI1'lHoe 'lI1CJ10, nponop-
UHOHa..lbHoe YCKopeHHlO 11 3anI1CaHHOC B
C'leT'IHK, MO)l(eT 6bITb C'iHTaHO Ha IllI1HY
1I.aHHbIX. CqeTqJ1K TaKTOBblX I1MnyJIbCOB
MO)I(HO OpraHH30BaTb nporpaMMHbIM nYTcM
B KOMnbIOTepe HJIM MliKpoKoHTpOJIJ1epc.
MaxCHMaJIbHoe 'iHCJ10 HMnYJIbCOB, KO-
Topoe COOTBeTCTBYeT J1Han83oHY 113MepeHI1j{
aKce.!JepoMeTpa., BbT'lHC1U1eTCj{ no <POPMYJIC
n = 1j T3lcr , (3.112)
r,ae Inn - TaKTOBIDI 'lacrOTa.
lIo 3Ha'ieHJ1K) n MO)J(HO onpe)l.eJIHTb
"IHCJJO pa3pAAOB. ueOOxo.lJ,HMbIX }lIISI XpaHCHl1SI
.nBOH'IHoro -lHCJJa., a TaJOKe pa3pewaJOJUYlO
cnoco6HOC'J"L npeOOpa30BaTe.1JSI, T.e. 3Ha'ietillc
YCKopemur HJJH yrna OTKJIOHeHIDI OT BepTl1-
K8JII1, npI1XO.lUllUeecR Ha OJUfH pa3pJIJI..
lJpuMep 3./5
P8CC'IHT8eM pa3pewaJOwyJO cnoco6HOCTL
npeo6pa30BaTeJUI J1JlJI HCXOJI.HLIX .QaHHblX: nepuo-
.na HMnyJU.CHoJi DOCJIe,nOB3TCJIbHoct1f T 2 =
= 0,5-' 10- 1 H 10 - 10- 3 c; T3JCJOBOH <JaCTOTbIftuT =
= 10.10 6 ru; Omu = 2g.
rIpn OTCyrCTBIIH YCKopeHIUI HMeeM Tt =
= T /2 = 0,5 .. 10- 3 /2 = 0.25 . 10- 3 C H .IIml .HOro
CJ1Y'laJj no <popMYJ1e (3.112) onpeAem1M 'IIICJIO
HMnyJIbCOB, COOTBeTC'rBYlOwee am.":
11 = (0,25 . 10- 3 . 10 . 10 6 )12 = 1250.
Ha o,rum pa3pHD. C<ieT'IUKa npuxomnclI YC-
KopeHHe
1 paJp. = 2g/1250 = 1.6 . lO-J g ,
KOTopoe B03I-1I1K8eT npH HaKJIOHe OCHOBaHWJ C
aKCeJIepOMeTpot.f Ha yroJ1
sina '" a = [(1,6. 1O- J g)/g]57,3 . 60 = 5,5'.
Illar D.HCKpeTH3aW1l1 paBeH nepHOD.Y HM-
nYJ1bCHOH nOCJ1eD.OBaTeJ1hl1QCTI1 IllI1M, T.e.
0,5 MC.
TaK KaK B D.eCJlTIlpa3pHD.HhIH C<ieTqHK MO)!(-
HO '3anHCaTb 2 10 = 1024 lIr.mYJIhCa, a 11 = 1250
HMnynbcoB, TO paCC'lI1TaHHhle napaMeTpbl npeo6-
pa30BaTeJIH 3KBHBaJ1eIITHhl D.eCSITIlpaJpHD.HOMY
Al..lO. AHaJlOm'lHO nOJ]Y'laCM .JlJ]H Tz = I 0 MC
if = 10 M[,I.{) 'II1CJIO IIMnYJIbCOB II = 25 000. Ha
OD.HH pa3plUl. C'leT'IHKa npllXOD.I1TC5I YCKopeHue
1 pa3p. = 2g/25 000 = 0,08 . IO- J g,
lITO COOTBCTCTBYeT yrJIY H8KJ10Ha OCHOBaHHSI C
aKCeJ1epOMeTpOM Ha yroJ] a = 16,5". War D.HC-
KpCTlIJattl1H C'ICT'IHK8 10 MC, a TaK KaK B
14-paJpHD.HbIH C<ieT'iHK MO)!(eT 6blTh '3am1CaHO
16 384 HMnyJIbCa (n = 25 000), TO napaMeTpb!
npw6pa30BaTe1lll npeBblwalOT B03MO)!(HOCTH 14-
pa'3plUl.HOro Al.{n.
TaKHM 06pa30M, OCHOBHble napaMeTpbl
lliJ1M l1aXOJ1)lTCj{ l1a ypoBHe nYlillIl1X 06pa3-
UOB Al.{II, a no npOCToTc CXeMOTeXHI1KI1
"3Ha'iIiTeJIbHO npeBOcXOJ1)lT I1X.
r:)flEKTPOIIHbIE CPEL\CT13A OEPAEOTKYI CHfHAJIOB
201
z
5
u= <-- "'0
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X 2
J
7
PIIC. 3.47. MIII':POcTpYKTYpa rlipocKona LL-Tlma:
J )KeCTKHii :meMeHT nO.l\BeCa; 2 - 11M; 3, 4 - ynpymc 3JIeMeHTbI nO.ll.Beca; 5 - aHKcpbl;
6, 7 - 3JICKTpO.l\bl H3MepHTeJIH nepeMemcHIIH B P4; 8 - HenOllBII)Kf!ble 3JIeKI'j)ollbl
3;leKTpoCTaTlI L IecKoro npIlBo,na; 9, 10- JJIeKTp0,nbl H3MepHTeJIR nepeMCll\eHHH B P L\;
J J - nO.l\HH>KHbIH 3JIeKTp0,n npHBo,na; J 2 - ynop - orpaHIILJHTeJIb nepeMcmcHlIH;
J 3 - nOW!O)KKa OCHOBaHHH (OQI1QJpoBKa COOTBeTCTBye-r pi1c. 1.37)
3.4.2. MItli:pOrllpOcKonbI
3JIeKTpOHHbIe cpe)J,cTBa 06pa6OTKH
CHrHaJIOB B Mf paccMoTpHM Ha npHMepe
Mf LL-THna, oc06eHHocTH KOTOpbIX xapaK-
TepHbl H)J,JISI MT LR- H RR-nInOB.
A. 3aJIa<m, pernaCl\lbIe 3JIeKTpOHHbIlUli
cpe)J,cTBaJ\lII, H lUeTO.nbl IIX peaJIll3aQIIII
MHKpocTpYKTypa fHpocKona LL-THna,
KllHeManI'leCKaSi cxeMa KOToporo COOTBeT-
cTByeT cxeMe Mf no pMC. 1.37, npHBe)J,eHa
Ha pHC. 3.47 ( l1pMa CIlIA The Regents of
the University of California). OCHOBHaJI
'laCTb 11M 3aKJIIO'leHa B crpYKType 2,
HMef()llJ,efI paMO'lHYIO KOH l1rYP8l.\HIO H
BKJIIO'laf()llJ,eH B Ce6S1 rpe6eH'laTble CTpYK-
TypbI 3JIeKTpo.nOB 7, 9, 11. nO,ll,BeC 11M 2
OTHOCHTeJIbHO OCHOBaHMSI 13 BbInOJIHeH Ha
ynpyrl1X 3JIeMeHTaX 3, 4, Me>K)J.Y KOTOpblMM
BKJIIO'leHbl :>KeCTKMe (He)J,e OpMl1pyeMble)
3JICMCHTbI nO)J,Beca J.
I10,l\BM)f(HaH. CTpYKTypa Mr, a TalOKe
IfCITO.llBJi:>KHblC rpe6eH'laTbIe CTpyKTypbI
3JIeKTpO,l\OB 6, 8, 10 C nOMOllJ,blO aHKepOB 5
pacnOJIO)f(CHbl Ha paCCTO HllH -2 MKM Ha,ll,
OCHOBaHHeM 13. IIpH nepeMemeHllSiX no.n-
BM:>KHOii crpYKTYPbI B pe:>KMMe 'lYBCTBHTCJIb-
HOCTH (Fy.) 11M 1.I3 YBeJII1'JMBaeTCSI: 3a C'leT
MaCCbl 3JIeMeHTOB 1. npCBbIlllalOlllHe HOpMY'
nepeMellJ,eHIDI 11M B HanpaBJIeHMH OCH X
(pe)f(HM )J,BH)f(eHl1SI - Pro M OCM Y (P1.I) npM
nOSlBJIeHMM YfJIOBOH CKOpOCTH f!: BOKpyr
OCIf Z OrpaHH'lHBaIOTCSI ynopaMH 12.
BH6pMpYf()llJ,aH. MllKpocTpYKTYPa MO-
:>KeT 6bITb H3rOTOBJIeHa 113 MOHOKpeMHllSi C
TOJIlllHHOII Bcex 3JIeMeHTOB -2 MKM MeTO)J,OM
nOBepxHoCTHoH MMKpo06pa60Tlm. B03MO)f(-
HO ee M3rOTOBJIeHHe H3 nOJIHKpCMHl1SI MeTO-
)J,OM rny60Koro 1l0HHOrO pe8KTI1BHOrO TpaB-
JIeHHSI C TOJIIlJ,l1HOH BCex 3JIeMeHTOB -25 MKM.
nO)J,BM:>KHaH. (BH6pHpYlOllJ,aJI) crpYKTYPa Mr
C )'K83aHMeM p83MepOB 3JIeMeHTOB ,naHa Ha
pl1c. 3.48, a Ha pMC. 3.49 nOK83aHa crpYKTY-
pa HenO)J,BH:>KHbIX 3JIeKTpO,l\OB. Ynpyme
3JIeMeHTbl 4 (CM. pHC. 3.47) MOryr MMeTb
OPMY, OTJIM'lHym OT npHMOJIHHeHHOH (CM.
nyHKmp Ha pllc. 3.48).
PaCCTOSlHl1e MC Ka:>K)J,OH napoH
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no.nBM)f(HbIe 3JIeKTpO,ll,bl (naJIbQbl 3JIeKTpo-
)J,OB) nOJI:>KHbI 6bITb M3roTOBJJeHbl 1130.nHOro
CJIOSI I<pCMHIDI .no y,lI,aJ1CHIDI )f(CpTBCHHOro
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3JlEKTPOHHhIE CPE,[(CT13A OEPAI>OTKI1 Cl1rHAJIOB
203
0)
3
2
I
..-:-.;;:::;: Fi.f$.
'::}:- ;;.::: i::;1
I .'. . 1 -
:1; :: II
.:- .-.'
x
6)
U
0:-
I .-
n-t.U ,- Un + t.
-
:} : " F;.
:
tl tM fg
':-:«<,:t : -:-.,.=«,
ft;:;%=:::= "c' ."'.;.:..,:
. ::::::x : =-
* ; :: E ii{?:
:.:{:.",.--:,
it < pnt
.. '-
....
. - ...
:;=.......:-.:::.;:
:
= '::::
:.- :::
: .-:..-.:
:::. ; -,.
.-
'oj.
-r t.u '> =::: Un-t.
«.
If:
U
y
F;.
Un
U
x
PHC. 3.50. K nOSiCHeliHIO npHHU;lIna KOMneucaU;HII KB3.!\pa-rypuoii oWH6KH:
a - ,nBmKeHl1e 11M npll OTCyrCTBlIH KOMneHcHpYIOIIteM Cl1J]bl; 6 - TO )f(e, npH cm,n3Hlm
KOMneHcHpylOIIteM CIL'1bI; I -11M; 2, 3 - 3JleKTpO.D.hl
PaCCMOTpl1M npllH1.\Hn KOMneHcal.\I1H
Kl3a)J,paTYPHOM OUl1l6KH. Ha pl1c. 3.50, a no-
KmaHa 11M 1 C 3JleKTpO.D,aMH 2, pa3Me eH-
HblMll MC:>K,lJ.Y :'L"1eKTpO,llaMH 3 H3MepMTeJISI
nepeMellleHHH. 8 C IJIY :vtexaHl1'ieCKOM HC-
CI1MMe-rpllJI )KeCTKOCTM nO)J,l3eca HJIM 3JIeK-
Tpli'leCKOH HeCI1MMe-rpMH npMBop:a pe)KMM
,llBl1:>KeHHH. MO)KeT npoHcxo)J,HTb no)J, HeKD-
TOpbIM yr.'10M K OCll X, B pe3YJIbTaTe qero
nOSlBJISIlOTCSI COCTaBJISllO lle )J,BI1)KeHHH. 11M
13 HanpaBJIeHlIH OCH x - P}],y H OCll Y - P)J,}..
COCTaBJISllOIIJ.aH P)J,y H eCTb ITpl1'lllHa B03-
HI1KHOBeHMSI KBa)J,paTYPHOH 01ll116Kll (CM.
TaK:>Ke nO)J,pa3)J,. \.3.2, A).
ECJIH Me)K,lly 3JIeKTpO)J,aMH 2 II 3 C03-
p:aTb pa3HOCTb nOTeHl.\llaJIOB: (U on - t!U) -
- (U on + t.U) = -2t!U; (U on + t.U) - (V on -
-t!U) = 2t.U (V on - onopHoe HanpSl)KeHI1e),
TO 3TO npHBe.neT K B03HHKH013eHMlO KOMneH-
CHpylOllJ,HX 3JIeKTpOCTaTH'leCKI1X CHJI 0"
KOTopbIe B OTCYTCTBHe ymoBoH CKOpOCTll !1=
06ccrre'laT nepeMeIIJ.eHHe 11M TOJIbKO B,l\OJIb
OCI1 x.
OnpeneJIl-IM BeJ1ll'lI-iHY t!U, KOTOpaH.
C03.l(aeT CMJIY, KOMneHCl1pylOrn.ylO KBap:pa-
'I)'pHYlO 011lH6K)'.
TIOJIaraeM, 'lTO npllBo)J, 06eCne'll-IBaeT
)J,Bll)KeHl1e 11M no 3aKoHY
x = Xo sinrot,
(3.113)
r.D,e Xo, W - aMnJIHTyna I1 'laCTOTa COOTBeT-
CTBeHHO.
rIpll nOSlBJ1eHllll ymoBoH CKOpOCTI-1 !1 z
B03HHKaCT YCKopeHMe KOpHoj]llCa:
af( = 2Q z X o w cosrot.
(3.114)
)J,JISI Tl1nOBbIX 3HaqeHI1H napaMeTpOB:
Xo = 1 MKM, ro = 20 KrQ, Q= = 1 °/c YCKope-
HMe KOpMOJIHCa aK = 2 10- 6 .20.10 3 x
57,3
X 2n = 0,438.10- 2 M1lc:o: 0,44 Mg.
OepeMellJ,eHMC 11M B.l(OJIb BbIXO)J,HOH
OCll Y MO)KHO HaHTll KaK .l(OJIlO Jl 01' nepeMe-
IIJ.eHlUI (3.113):
Y = -IL"',
(3.115)
a KBaD.paTypHoe YCKopeHl1e onpencmleTCSI
Bblpa)KeHl1eM
aK = ji = J.LXoro 2 sinrot. (3.116)
OTMeTI1M, QTO YCKopeHHe Kopl1oJ1l1ca
CMellJ,eHO no a3e Ha 90° no OTHOlllCHHlO K
KOJIe6aHI1S1M (3.113), a KBa)J,parypHoe YCKO-
peHI-IC (olllH6Ka) ITponopUI-IOHaJILHO aMnJII-I-
TY.ne Xv H HaXO)J,I1TCSI B O,l\HOH a3C KOJIe6a-
HllH C npI1BO.l(OM.
3anl1lIICM OTHow'eHI-IC aMnJIl-ITY.n YCKo-
peHHSI
204 rnaBa 3. 3JIEMEHTHA.Sl EA3A MHKPOMEXAHI14ECKHX rIPHEOPOB
aK _ 20=X o (J) 2fL
a..-s ll-Y o(J)- 11(J)
(3.117)
OTh")'.na cne.ayeT, 'lTO ,D.aiKe nplf ada,," = 1
BeJIJ1'iMHa 11 = 20j(J) ,D.OJIiKHa 6bITb 'lpC3Bbl-
qaKHO MaJIO". LleHCTBIITeJIbHO. .i1JUI nplfHjJ-
TbIX Bblwe HCXOJ1.HbL'{ .naHHbIX. BeJII'JIUma
" = 2 1 ( 20.10 3 . 21t ) = 2 77.10- 1 T C
... 57.3 ". .
nepeMellleHHe 11M B.D;OJIb OCH Y ,D.O.:DKHO co-
CTaB.!llITb nopMKa .neCJlTHMHJI.JlHOHHOH qac-
rn OT a.\fllJIHT)'.D;bI Xo. O'leBM..!(HO, 'ITO TeXHO-
.rrornqeCKH BblfiOJIHMTb TaKoe COOTHOllJeHlle
HCB03MO:>KH0.
3.rreh'TpoCTaTH1JeCKaJI CHJ1a npHTIDKe-
HIDI MC){\,I.l)' 3.rrck"Tpo.na m npH HaJIM'lMH Ha
HHX pa3HOCTH nOTeHlU1aJIOB onpc)J,emreTcSI
3aBHCHMOCTblO (3.85):
_ Co \2 ( ) 2 ]
F,. -- U on +l1U J -\U OTJ -I1U =
o I1v
- 0 (3.118)
Co
=2-U on 6.U,
l1yo
fJ1.e Co - MaKCHMa.1}bHaJ:I eMKOCTb rpe6CH'la-
TbIX CTpYKryp 3JIeIcrponOB qyBCTBHTeJIbHO-
CTH; l1yo - HOMHHaJIbHOe paCCTOJlHHC Me)!(.D;y
3lIeKTpOllaMH.
lli paBCHCTBa man = F.,. (m - MaCCa
JIM) c yqeTOM q,opMyn (3.116), (3.118) cne-
.z:IYeT MaKCMMaJIbHWI BeJIH'IHHa pa3HOCTH
nOTeHl1HaJIOB, KOTOpaJ:I 06eCne'lHBaeT KOM-
neHcaIUfiO CHJILJ, 06ycnoBJJeHHOH KBa.npa-
"l)'pHblM YCKopcHHeM:
.,
l1U = mJlAYo(J) Xo . (3.119)
2C o U on
3aMe11IM, qJ'O BeJlM'lHHa AU nporrop-
I{HOHaJIbHa nepeM.ellleHH1O Xo, TalC >KC. KaK H
BeJIJIIIHHa aD' O eHKY 3Ha'IeHIDI BeJIHtJHHbJ
AU BI>UlOJIHHM wm DapaMeTpOB; m = 0,2 x
X 10- 3 xr, Ayo = 1 MKM. XO = 1 MKM, (J) =
=20 xr J.1 = 10- 1 , U on = JO B (06LJ'lHO
U OR = 1...JO B); 3Ha'ieHHe CO 3aJ3HCHT OT
reoMeTpHH H o611lero tIHcna nam. eB '!}'BCT-
BIn'eJJbHOCI'H H MO)Jre1' 6LIT& 06ecneqeHO
C o = 1 nt1J.
TaKIIM o6pa30M.
I1U =
= 0,2.10- 3 .10- 1 .10- 6 (20 0 10 3 .2" TIO- (,
2.10- 12 .10
= 15,77 .10- 3 B.
B )J,onOJIHeHHe K ycrpaHeHl110 KBa)J,pa-
TypHOll 01llu6KH CMCTCMa 3JIeKTpO)J,OB 1J.YB-
CTBHTeJIbHOCTU (113MepHTeJIb nepeMellJ,elU111
B pq) MOiKeT 06ecne'IIITb YCTpaHCHl1e BJ1lUI-
HHSI l..\eHTp06e:>KHblX CH]], .n:eikTBYIOIlJ,HX Ha
HM, eCJIll ee ueHTp Macc He COBna)J,aeT C
OCblO BpallleHIUI z Mr. lleHTp06c:>KHble CIUbI
YCTpaHSlIOTCSI npHnOiKCHlleM K 7JIeKTpo aM
'1YBCTBHTeJIbHOCTM. paCnOJ1Q)KeHHbIM no
p83HbIe CTOpOHbI OT HM, COOTBeTcTBylOlUl1X
CHJ1allf HanpIDKeHMSI. CJIe)J,YCT OTMeTMTb. '1TO
3JIeIITpI1'lecKI1H CllrHaJI, npOnOpl..\HOHaJIbHbIH
l..\eHTp06c}KHOH CHJIe, SlBJ151CTCSI HM3KO'laC-
TOTHbIM. il03TOMY )J,aHHbIH Cl1rHaJI MO)f(CT
6bITb OT HJ1bTpOBaH BbICOI(O'laCTOTHbIM
I1JIbTpOM, KOTOpblM nporryCTMT BbICOI(O'lac-
TOTHbJri. nOJIe3HbIH CHrHaJI, npOnOpl(HO-
HaJIbHblH YCKopeHlllO KOpMOJ1Mca.
HaKOHel..\, C nOMOllJ,b1O rpe6CH'IaTbIX
CTpYKTYP P'l Y)J,aeTCSI HaCTpal1BaTb 'laCTOTbl
KOJIe6aHHM MM. ,AeHcTBHTeJIbHO, 3J1eKTpO-
CTaTH'IecKYJO CHJ1Y B3allMO,D.ericTBllSi 3]]eK-
TpO,llOB MO}KHO 3anHcaTb, HMeSi B BH)J,Y
(3.90):
. = [( ., 2] X
l1yo l1y)- (11 11 )
o:)' (3.120)
l1y o C o U n _ 211y C U 2
x 2 - l1y 0 on'
r,ll,C l1y - rrepeMellleHHe 11M B HanpaBJIeHllH
OCHY.
3JIeKTpOCTaTH'leCKa51 }KCCTKOCTb onpe-
.neJUleTCR no opMync
ky =Fyl(l1y)=2COU f)/(I1Y5), (3.121)
a 'laCTOTa KOJIe6aHHH 11M B P'l paBHa
_ Gy ky
(J)y - ,
m
r.ne G y - CYMMapHaJ:I :>KeCTKOCTb ynpyroro
nO)J,Beca B HanpaOJIeHHH OCH y.
(3.122)
3JlEKTPOHHbIE CPE)J;CTBA OEPAEOTKI1 Cl1fHAJIOB
205
KaK cJIe)J,YeT H3 Bblpa:>Kemdi (3.120)...
(3.122), YBeJIH'leHHe UOI! ITpHBOnHT K
YMeHbllleHI1IO w Y ' HacTpoHKY <.IaCTaT Ha,l\O
Ha'lHHaTb ITpn m l , > W x H YBeJIH'leHMeM UOI!
yMeHbIlIaTb W V AO Tex nop, nOKa He 6Y,lleT
J).OCTHrHYTO :>KeJIaeMOe 3Ha'leHHe OTHOIlIeHiUl
W/W,. ,D;JISI H3BeCTHbIX OnIfcaHHli Mr Bcm1-
'lima UOI! HaXO)J,HTCSI B npe,llCJIax 1...10 B H
onpeJ).emleTCSI 3KcnepHMeHTaJIbHO )J,JUI 3a-
naHHOH paCCTpoHKI1 'laCTOT t!m = W y - WX'
E. PeiKlIM )J,BHiKeHHH
Pe)KMM )J,Bll)KeHHSI 11M 06eCne'lHBaeTCSI
3JIeKTpOCTaTI1'leCKllMH npHBOJ],aMM I1 H3Me-
pHTeJIeM nepeMeIl.\eHHi1 B P,D; (pHC. 3.51),
BKJIIO'leHHbIM B LleITb 06paTHoli CBH31l npH-
BO)J,OB. TaK KaK YCKopeHMe KOpHOJIHCa npo-
nOpLlHOHaJIbHO JIlIHel'IHoH CKOpOCTH dx/dt
11M, npHHLlHnnaJIbHO Ba)KHO nO)J,)J,ep)KaHHe
nocToSlHHOM aMIImnY)J,bI :noM CKOpOCTM.
,D;ml KOHTpOJISI CKOpOCTll 11M B P,D; MO)KeT
6bITb llCIIOJIb30BaH TOK CMellJ,eHIUI, paBHbIH
._ elC U = f>Cflx U
I - elf on fix Of on '
(3.123)
V on
i
-
t y .
I
-
V on
x
-
PIIC. 3.51. K B03HHKHOBeHilIO TOKa Cl\lell\eHIIH
me C - CMKOCTb rpe6eH'laTol'1 CTpYKTYpbI
H3MepI1TeJIH nepeMellJ,eHMH B P,D;.
rapMOHI1'lCCKOe J).BM)Kemle 11M MO)KeT
6bITb 06eCne<IeHO cxeMOM C YCHJIMTCJIeM, Ha
BbIXO)J,e KOToporo C03)J,aeTCSI ITepeMeHHoe
Hanp1DKeHMe C aMIIJIHT]')J,ol'l, nO)J,)J,ep)KHBae-
MOl'l nOCTOSlHHOH, B COOTBeTCTBllM C TOKOM
CMellJ,eHHH. [CM. OPMYJIY (3.123)], npoTe-
KaIOIl.\I1M B LlenM 06paTHol'1 CBSl3M cxeMbl,
oCYIl.\eCTBJIJUOIl.\el'1 P,D;.
Ha pMC. 3.52 npMBe)J,eHa YHK-
l{1l0HaJIbHa}1 cxeMa l{eIIH 06paTHoH CBH3l1,
y
pqt
. P.ll
x
.0". .....
2
PIIC. 3.52. CPYHK IIOHaJJbllall CXCMa neDIl ofipaTHoii CBH3U B PA:
J H 2 - HenO)J,BIDKHble 3JIeKTpo)J.bI 06panloH Cml3H H 3JIeKTpoCTIITK'IeCKOrO DpHBo,na cooTBeTCTBeHHO
206 Duma 3. 3JIEMEHTHA.sJ. EA3A MI1.KPOMEXAHHlfECKI1X rIPHEOPOB
BKJ1lO'IalOll..laJI B ce6 HenOnBJ1:>KHbIe 1 II
nO.ll.BIDKHbJe 3JIeKTpo.nbl, SlBIDIIOll..lUeCSI 'la-
CTblO CTpYKrypbl 11M, H3MepllTCJIH. nepeMe-
meHHH B P L{ H CXCMY pe3llCTopHoro YCHJIH-
reIDI Y p (R-pC3J1CTOpbl), BKJ1IO'leHHOrO B
lIenH npHBOJ1.0B C HenO)J,BHiKHbIMIi 2 M no-
.lI.BIDKHb1MH 3JIeKTpO.ll.a (1l Ha cTpYKTYPe 11M.
KOJIe6aTeJlbHO .!1BIDl\:eHHe lIM o6eCne'lI1Ba-
eTCJI nepCMeHHbIM HanpjDKeHHeM, KOTopoe
npm(.11a.;:u,IBaeTCJI MC>KD;)' nO.D.BIf>KHbIMH J1
HenO.!IBHiKHLlMlf 3JIeKTpo.naMH npIfBO.IlOB
(HCTO'IHMK nepeMeHHoro HanpJliKeHIfJl He
nOKaJaH ).
B COOTBeTCTBHH C OPMYJIOH (3.123)
38.IIHIlleM BblpaiKeHIDI .nml TOKOB B JIeBofl H
npaBo" BeTBXX cxeMbI, o6CCnCqHBalOIlJ,HX
,llBIDKCHMe lIM:
J+} _ oC . ( ) .
I -Uon-XO<.o"Sill<.oz!,
Ox
.(-) - -U oC X . ( ) .
I - on Ox 0<.0" Sill <.o,,! ,
r.ne Xo, w.. - aMIIJIll'I)'.iJ.a H 'IaCTOTa lIM B PL{,
oC
- - H3MeHeHIDI eMKOCTH rpe6CH'laTblX
Ox
CTpYKfYp.
Ha BbIXO.!1C YCH.'IHTeJIR
U{-rJ = U on + U" sin(<.oxt};
U(-} =U -U sin ( <.o t ) . (3.124)
011 X r ,
r.ne
oC
U x = RUon-XOw" .
Ox
(3.125)
B COOTBeTCTBHH C BbIpaiKeHHeM (3.91)
Ii c yqCTOM .popMynbl (3.125) nOJIY1fllM BbI-
paxeHHe .Q)lJI CHJIbI, 06ecneqHBalOllJ,eH pC-
JfCHM .llBIDKeHIDI:
F = 4 OC U onU" sin(w"t) =
ox
2 (3.126)
= { U on ) RX oW" sin(w"J).
BancHO OTMCTHTb, qTO o6ecne'IHBaeTCH
pe3OH8HCHLIH pe1KHM )1BIDKeHIDI. TalC KaJ(
CHJI8. H CJ(opOCTh HaXO}VITCJI B tPa3C. Cyllte-
CTBCHHO T8.K)!(C 06CTOHTe lbCTBO, '11'0 CIIJIa
ner.m l1pOBaHJ1S1 II 3JIeKTpOCTaTII4eCKaSi CII-
JI80 nponOpIlIfOH8JIbHbl CKOpOCTIi .nBl1iKeHiISI
11M. CJIe)J,OBaTeJIbHO, eCJlII YCI1JICHHe B cxe-
Me He,llOCT8.T04HO. TO LloMmmpyeT CIIJla
.D.eMn J1pOBaHIiSi H KOJIe6aHHSI 11M nOCTe-
neHHO 3aTyxalOT. ECJUI me YCNJ1eHUe BeJ1H-
KO, TO Kone6aHliSi 6Yllyr B03paCTaTb J10 Tex
nop, nOKa He orp8.HI1'laTCSI ynopaMH. K03 )-
1iUl1eHT YCI1:1eHHSI (MOIltHOcm) B cxeMe
(pl1c. 3.52) onpe)J,eJISleTCSI BeJII1'1IJHOH R, KO-
TOpylO MOiKHO HaHTli 113 YCJ10BI1S1 6RJlaHCa
CHJl 3J1CKTpOCTaTli'leCKIiX 11 .lI:eMn HpOB8.HlUI.
ITpHp80BHSleM MO)J,YJ1H :nHX CI1JI:
4 ( OC U;n ) RlX 0<.0,,)= bAx o<.oy),
OX
OTKY)J.8.
R= b x
4 ( BC U ) 2'
s: on
uX
(3.127)
r.ne b x - a6COJIIOTHbll\ K03q) HUHCHT )J,eMn-
llpOBaHllSi.
3anllllieM 113BCCTHoe COOTHOllieHHe
b" / m = 2l;x<.o" ( - OTHOCHTeJIbHbIH K03 -
Hl{l1eHT .lI:eMn JHpOBaHllS1) M, MMeSi B BM.lI:y
BbIpaiKeHHe )J,JISI )J,06pOTHOCTM no 3aTyxa-
HHIO Qx = 1/2l;x , nOJIY'illM BCJIH'IHHY
b x = (m<.o,,)1 Q", nOCJ1C nO)J,CTaHOBKM KOTO-
pOH B OPMYny (3.127) HaH.D.eM K03 MUH-
eHT YCHJIeHHSI, OM:
R= ( : ) 2'
4Qx n ---2.. U on
OX
(3.128)
oc.
r.ne ----L - 1l3MeHeHl1e eMKOCTH Ha i-M
Ox
n8JIbl{e rpC6eH'IaTOH CTpYKTYPbl; n - 4HCJIO
naJIb eB.
B. PeiJCHM "YBCTBHTeJIbHOCTH
113MepHTenb nepeMellJ,eHHH JI)1S1 pq,
KaK OTMe'IaJ10Cb, BbmOJIH eT YHKUHH M3-
MepeHml ITepCMeIUeHI1H 11M, KOMnCHcaUHH
KBa,l.tpa-rypHOH OlliH6KH H HaCTpOHKM 'I8oCTD-
Tbl Kone6aHHH 11M. Ha pHC. 3.53 nOKa3aH8o
'7JIEKTPOHHhlE CPEJJ:CTRA O(jPAEOTKYI Cl1fHAJIOB
207
r '""' .""' ,,, ,,,," :,, ..=-
. ... . .. .". .. .... ...... . 1 /2
::(::::.:. -.:: ':::: :..::':; :' -.:=;";.:;:::"-; , ':"::::':::;;':-::;::::::' :;::-: :':=r
: i: y i .:'<. .......... :...,:--. .,;:-, ::5
7,::':.X ill
_ x
PJJ:
8
\
<J......
+
"<J
:::,0 IE::
:5
9
3
4
:: :'..-:., :.. .;:0.:.... :::,.
"0" ;.: : :-;:'::::::. .....::{
.,
:.-:.::...-..z.:..:.:...-.;.... :-;.
JJ
6
7
f:
....
.'
. 5 '.
_.::=:.:;.. ..:.;....:.;:-;:;v::.;-:':-;::.;::::::-:-;::.'::;..==::.:;::-.: ::"::::::.:::;=::::::::
. ::::;:;:: ::.! :: :.:-;.; :; ; i: .;:-..:: -:x ;:;r: :::
PIIC. 3.53. CPYHKl1IIOH3J1bH3H CXCJlla BKJlIO':ICHHH JJlCKTpOJtOB eMKOCTHoro
npeo6pmoBaTc:ul ncpeJlleUleliHii:
3,4.6. 7 - HerrO,/J.BI1JKHhle 3JIeKTpO.lIhl: 5 - nO.lIBI1)KHble 3JIeKTpO.lIbl;
8-12 HCTO'lHHKH HanpIDKeHlliI
YHKI!110HaJIbHaSi cxeMa BKJUO'leHllSi :meK-
TpO,llOB eMKOCTHoro npe06pa30BaTeJISI nepe-
MellleHHi'I, 06eCne'lllBarorn.aH. BbtnOJIHeHMe
nepe'lHCJIeHHblX YHKI..{I1i1 (01U1 poBKa 1IB-
JISleTCSI npo.nOJI)!(eHl1eM pllc. 3.52).
11CTOqHI1K HanpSl)!(eHHH. 12 npHKJIa)J,bJ-
BaeT nepeMeHHoe HanpSl)!(eHMe Me:>K)J,y 3JIeK-
TpO)J,aMM 3, 4 H 6, 7 C aMmlMTY oi'1 0,1...
0,5 B M QaCTOTOH, 3Ha'll1TeJIbHO 60JIbIIIeH,
'leM 'laCTOTa P,I:{, Hanpl1Mep, 1 MrQ. 11C-TO'l-
HllKH HanpSl)!(eHI1H 8-11 C03)J,aIOT KOM6uHa-
QI1M no ITpI1HQHny, KOTOpbli'1 I1J1JIIOCTpwpO-
BaH pHC. 3.50.
TaKHM 06p830M MoryT 6bITb C03.D.aHbI
3JIeKTpOCTaT1l4eCKlle Cl1JIbI )!JISI KOMneHca-
QllM OIlIM60K, B TOM 4HCJIe KBa)J,paTypHoi'l, 11
)!JISI 1l3MeHeHMSI 3JIeKTpOCTaTll'leCKOH )!(eCT-
KOCTI1 C l..{eJIblO KOppeKItm1 'laCTOTbl KOJIe6a-
HI1H 11M B pq.
3JIeKTpo)J,bI 5 H 3, 7 C03)J,a1OT 1l3Mepl1-
TCJIbHYIO eMKOCTb C" a 3JICKTpO)J,bI 5 H 4,6-
113MepllTeJIbHYIO eMKOCTb C 2 .
BapllaHTbI BKJIIO'lCHH KOH,LIeHCaTOpoB
C eMKOCTSlMM C 1 I1 C 2 nOKa3aHbl Ha pl1c. 3.54.
OQeBM)J,HO, qro BBM)J,Y MaJIblX 3Ha'lellliH
eMKocTei'1 KOH)J,eHcaTopoB B Mr KpHTH'iC-
CKoe 3HaQeHlle HMelOT pa.3JIH'lHble napa3HT-
HbIe eMKOCTll, 06pa30BaHHbIe mIHH.H.MI1 CB -
311, MeTaJIJI11311pOBaHHbIMH IIOBepXHOCTHMM
Kopnyca, M T.)J,.
B KaqeCTBe O)J,Horo H3 B03MO:>KHblX Ba-
pl1aHTOB Ha pMC. 3.55 npl1Be eHa CXCMa
BKJIIO'leHHSI nOJIYMocTa C uHTerpaTOpoM B
Ka4eC-TBe YCHJIllTeJISI 3apSl.D.OB KOH eHca
pOB. TIo BH)I(Hble 3JIeKTpO.D.bI (naJIbl..{bI) 5
(CM. pllc. 3.53) Qepe3 nO)J,BH)!(HYIO cTpYKTY-
py 11M I1MelOT 3JIeKTpM'lecKylO CBSl3b C OT-
pHl!aTeJIbHbIM BXO.nOM YCllJ1HTeJIH..
aJ
6)
Puc. 3.54. KOHlIJHrypal{IISI nOJlYMocT081oIX
cxeM BKJlIO'IeIIllSi KOH.!'eHCaTopOB:
a - c 06mefl TO'UWH; 6 - .IU1clJrpepeHlUlaJIbHWI
208 rJIaBa 3. 3flEMEHTHAJI EA3A MIIKPOMEXAHUqECKIIX rIPHI>OPOB
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1 v.
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Pec. 3.55. IUlTerpaTop B pO.1H )'CHJIIITe.1H
33pHltOB:
Cl, C2 - KOlUeHcaropLI !1.aT'lm:a nepeMemeHllii;
C p - napa3IfTHIDI eMKOCTb; CII - eMKOCTb
KO!UleHCaTOpa HHTerpaTopa; VII - Hanpjj)KeHl!e
IIHTerparopa
I1ocKmIbl\.-y YCHillfTe..'Ib. 06CCne"llfBalO-
lUHM peJKHM roUDKeHIDI, H UHTerpaTOp 06"be-
llHHeHbI B CI1CTeM)' 'lepC3 .t!BH)h"ylUHCCSl rpe-
6eH'laTbIe CTPY"-LYpbI Mr. HMCeTC1I 3JIeK-
"l]>OHHax CBX3b MeiK!IY HHMH. KOTOpax npn-
BO.1HT K nOSiBJIeHHIO .t!onO.1HHTeJlbHOI"0 CMr-
11a.:1a Ha BbIXO!l.e I1HTerpaTopa.
PaCCMOTpI1M OpMMpOBaHHe 3Toro )J,D-
no..'IHJITc"'IbHOro CHrHa.'1a. B peilmMC )J,BIDKe-
HIDI Ha BbIXO,3e YCH.iIHTe.JUl OPMHPYeTCSI Ha-
npSDKeHHe [CM. <}>OpM)'JIY (3.124)]. EMKOCTH
B3aMMO;IeHCTBYlOlllHX na.'IbUeB I-pe6eH"IaTbIX
CTpytcryp onpelleillllOTcR 3aBHCl1MOCTRMH
CH:;:::C o + O C Xo cos(w x I);
Ox
C(-) = Co - oC Xo cos(w x I), (3.129)
Ox
r.ne Co - eMKOCTb B HeMTpa.1bHOM II0J10)Ke-
HHH 11M.
e yqe-rOM Bblpan<eHHH (3.124) H (3.129)
3aIlHIDeM Be.JJI1qJ1H)' 3apBJla, HaKanJ1HBaJO-
ntylOCx Ha HHTerpaTope:
Q = C<-r}U H + CHU<-) =
oC . (3.130)
= W mCa + U xX 0 -s1D(2ro x f).
Ox
TaKHM 06Pa30M, HapR.IlY C nOCT01lHHOH
COCTaBIDIIOllleM 3apRIDl, KOTOpaR MO:HreT
6L1Tb OTBe,neRa, CYlUecmye-r COCTaBmIlOlUaR
Ha .nBOHHOH 1JacTOTe pe'30HaHcHoro pe-ACHMa
,llBIDKeHIDI. KOTOpaR H.a BbIXOJ],e HHTerpaTOpa
clJopMHpyer .nonOJIHHTeJIbHbIH Cl1maJI:
U1J. =l.- BC X o U x sin(2ro x f). (3.131)
Cft Bx
r10)J,CTaBIlM B Bblpa)l(CHlle (3.131) BC-
J1It'lHHY ll.T 113 OPMYJIbl (3.125) C Y'leTOM
COOTHOlllCHIfH (3.128) 11 nOJ1Y'IIIM OCTaTO'l-
Hblfl CHrRan Ha )J,BOHHOH 'laCTOTe P)J, np"
paBeHCTBe CI1JI 3JIeKTpOCTaTI1'leCKIIX 11
lleMn l1pOBaHnSl:
V - I m(w,Xof . (2 )
.:! - SIn W,!. (3.132)
4 Q,UonC Il
)J,BofIHaH. 'laCTOTa lweT B03MO:>KHOCTb
HCnOJIb30BaHHSI HJIbTpaUI1H CHrH8.,1a. 3aMe-
TI1M TaIOKe, 'ITO, 113MeHSIH eMKOCTb HHTerpH-
PylO ero KOH)l.CHcaTopa. MO)KHO KoppeK-
THpOBaTb nOJIOcy nponycKaHHSI 'laCTOT PLI.
r. 06pa6oTKa CIII"1I8JIOB
HTorOBofI npoue)J,ypOH 3JIeKTpOHuKH
Mr SlBJIHeTCSI M3BJIe'leHllC Im opMauMH 06
H3MepSieMoll yrnoBoH CKOpOCTI1. 3JIeKTpO-
HMKa Mr BKJ1IO"IaeT B ce6S1 yCHJIHTeJIb, op-
MUPYIOIlJ,I1H CMHYCOI1J1.anbHOC nIITaHHe u.c-
nell rl1pOCKOna, 11 llHTerpaTOp, KOHUCHT-
pI1PYIO I1H BCIO llH OpMaUHIO 0 )J,BM)KeHHHX
fHpOCKona B Bl1)J,e 3apSinOB, HaKamIMBalO-
IlJ,HXCg B KOH)l.eHCaTOpax rpe6eH'laTblX
crpyKT)'p. qacTOTa BblXO.l\HOro CMrHana
YCllilHTeJIH. )J,OCTaTO'lHO CTa6HnbHa M MaJJO
3aBHCHT OT TeMnepaTypbl, a aMnJIliTY,l\a KOH-
TpOJIMpyeTc1I CMCTeMOH aBTOMaTHKM Ii TaK:>KC
OTHOCHTeJIbHO nOCTOSlHHa. CneKTp 'laCTOT
Y3KIiM:, II era nMK 6JIM30K K pe30HaHCHOH
'laCTOTe KOHCTPYXUMM.
Y:>Ke OTMe'la..1l0Cb, 'lTO CYllJ,eCTByeT
B3aHMOCBR3b MC)K,l\y YCHJIMTCJIeM M MHTerpa-
TOpOM, CJIC.nCTBllCM 'lero SlBJIH.eTCSI B03HMK-
HOBeHMe CllfHana Ha ,l\BOHHOfI pe30HaHCHofI
IJaCTOTC w X ' ,[{BI1.MeHMe mpOCKona B pq eCTb
CJIe.nCTBl1e yCKopeHIDI KOp1l0JIMCa, KOTOpOC
Ope,l\CTaBJJger C060H aMIIJIHTY.nHO MO,l\YJIM-
pOBaHHbIH cHrHaJJ. KpOMe TOro, lie06xo.l\l1-
MO BcnOMHHTb 0 KBa.l\pa'I)'pHOH COCTaBJISlIO-
llJ,eM BblxonHoro CHrHaJIa,
TaKHM 06pa30M, aMOJII1'I)'lI.Y Ii a3Y
KOJIe6aHHH mpOCKona B P,[{ MO)KliO "H3-
BJIe"lb" H3 BblXO):\HOrO CHrHana YCI1JIHTCJIB. C
Bblxo,na HHTerpaTopa MO:>KHO nOJIY'lI1Tb llH-
4JopMaltHIO 06 yCKopeHHH KOpHOJIHC8, T.e.
3JIEKTPOHHhIE CPE.LJ:CTBA OEPAEOTKJ1 CHrHAJIOB
209
06 H3MepSieMoM YfJIOBOH cKOpOeTI1 H KBa,ll-
paTypHoH 01llH6Ke. l'hBJIe'leHlle nonC3Horo
CllrHaJIa 3TO Hl:>lepeHl1c aMn:mTY}J,L! CHf-
!laJIOI! Ha pa3J111<IHbIX [IaCTOTax. ,lJ.aHHaH. 3a-
naIra aHaJIOrl1'llia TOi1, KOTopaSi HMeCT MeCTO
B JII1HWIX CB lH, f)J,e TIHOKe H3BJICKatOTCSI
ClIrHaJIbl. COCpC,llOTO'leHHblC Ha pa3JII1'lHbIX
1.JaCTOTax TeKYmero TOKa.
I1pc.nCTaBJISlCTCSI, 1.JTO HaH60JIcc npl1-
CMJIe\1bIM )J,:ISI Mr MeTo.n;m.l H3B,le1.JCHH}{
nOJIe3Hoi1 MH OpMaL\llM HBJISlCTCSJ CllHXPOli-
HaH }J,eMon:YJISlUHSI. OCHOBHaH CJIO:>KHOCTb
(OC06CHflOCTb) KOTOpOH 3aKJIEOIlaCTCSI B TOM,
1.JTO Tpe6yeTcSI nocToSlHHblfl fIO aMfIJIMTYnc
CHfHa.l, KOTOpblH llMeCT TC >Ke 1.JacTOTY 11
a3Y, [ITO I1 HY>KHblfl (BbIAeJISIeMbIfl) CHfHaJI.
B Mr :no YCJIOI!He MO>KHO 06eCne'lmb pC-
>KHMaMH HaCTpOHKl1.
,lJ.JI ITOSlCHeHI1S1 MCTona ITpeI(nOJIO>KI1M.
1.JTO CCTb CllrHaJI
U(f) = Uo(t )COS(Wf),
KOTOpbIH CMClllHBaeTC}{ (YMHO>KaeTcH) C :na-
JIOHHbIM CMrHaJIOM C)J,HHI1'lHOfl aMITJIHTY)J,bI
H IlaCTOTOH W x :
U(f )COS(Wxf) = Uo(t )cos(wt )COS(Wxf) =
= U 0(( ) [cos(w - wx)t + cos(w + wx)1] .
ECJIH w=w x ' TO U(t)=!Uo(t)x
2
x [I + cos(2wt)], T.e. nOSlBJIJlCTCJI nOCTOSlHHaH.
COCTaBJISlEOmaH. CHrHaJIa, KOTopmr (aMITJIHTY-
Aa) MO>KeT 6blTb BbIAcneHa H1l3KO'laCTOTHbIM
HJIbTpOM.
TIpH H3MCpCHl11l ymoBoH CKOpOCTU
nz(t) HMCEOT MCCTO KOJIc6aHH1I 11M rl1po-
CKona B pq C 'laCTOTOfl w Y ' T.e. BbIXO,l\HOH
Cl1fHaJI Mr MO>KHO npe.ncTaBHTb B BHne
U(t) = QAt )cos(wxt )COS(Wyf),
rAe QAt)= / (w y ); Wx.W y - IlaCTOTbI P,lJ.
H pq COOTBeTCTBeHHO. ECJIH 3TOT CHrHaJI
CMelllaeM C Y'fCTBepeHHbIM 3TaJIOHHblM CHI'-
HaJIOM, TO nony'll1M
U>. (t) = 4U(t )cos(Wxt )COS(Wi)=
= 4!1 z (t )COS 2 (W./ )cos 2 (Wi) =
= !1 z {t )[1 + COS(2w J ( )][1 + cos(2w\t)].
Oc06eHHOCTb 3JIeKTpOHHbIX cxeM CMC-
CHTenCH 3aKJIEO'iaCTCSI B TOM, 'iTO BXO)J,HOll
CHHYCOH.naJIbHbIH Cl1rHaJI YMHO:>KaeTCSI Ha
3TaJIOHHbIH CHmaJI, OPMHPYCMblfl L\HqJpo-
BOH CXCMOH B l!H)J,e ITOCJIe)J,OBaTeJIbHOCTU
npSlMoyronbHbIx HMrryJIbCOB H3BCCTHOH 'lac-
TOTbl.
C nOMOllibEO HI13KO'iaCTOTHOrO lmbT-
pa MO>KeT 6b!Tb BbI)J,eJIeHa aMnJImY)J,a .0= (t),
co.ncp>KarnaH. H3MepSlCMYEO BCJIH'iHHY .0=.
,lJ.mr HJ1J1IOCTpaQllll OTMe'leHHbIX OCO-
6eHHOCTeH B3aI1MO)J,eHCTBlUI 3JIeKTpH'leCKnX
CMfHaJIOB Ha pllc. 3.56 npHBe)J,eHa YHKL\Ho-
HaJIbHaH. CXCMa Mr, MllKpOCTpYKTypa KOTO-
pom COOTBeTCTByeT pMC. 3.47, a CXCMa
BKJIEO'lCHllS1' 3JICKTpO.nOB rpC6eHQaTbIX
CTPYKTYP, 06CCnC'lllBaIOIlJ,l1X P.LJ: II pq, OTBC-
'laeT pMC. 3.52 M 3.53.
P,lJ. o6CCnCQHBaeTCSI npHBo)J,aMM, H3Me-
pMTCJIeM ncpCMClllCHI-rn B P.LJ:, BKJIEOQCHHbIM
B L\enb 06paTHOH CBJl3H 'iCpC3 YCHJIMTeJIb Y p'
OJIOKOM qJa30BOH aBTOno)J,cTpoflIill 'laCTOT
({J)AJ1q), CMCCI1TeJICM /8 H I1CTOQHHKOM
ncpeMcHHoro HanpJDKCHIUI 12. HenOABI1:>K-
Hble 3JIeKTpOAbl 1, 2 B L\CnH 06parnofl CB1I3M
coenMHCHbI aHaJIOrH'lHO pllc. 3.52. pq 06cc-
nC'lHBaeTCSI ll'3MCpHTCJIeM nepcMellJ,eHHH,
llHTerpaTopoM H C YCJiJlmeJIeM Y, MCTO'IHI1-
KaMH HanpSln<eHMH 8-12 M .n;eMo.D.)'JISITOpa-
MI1-CMCCI1TCJI1IMll 13-17. 3nCKTpoAbI 5 11 3,
7, a T8K:>KC 5 II 4, 6 BKJIlOI.JCHbI TaK >Ke, KaK Ha
pMC. 3.53. CMCTCMa {J)AJILI rCHepl1pyeT TO'l-
Hble UH pOBbIC CHrHaJIbI, KOTOpbIe J1.1VI p,o;
JIe)KaT B .nl1ana30HC W x = 7... 100 KIL\ 11 co-
OTBCTCTB}'lOT nOJIO:>KeHHIO 11M, 11 Bblpa6a-
TbIBaeT onpe,l\eJIeHHbrn Cl1rHaJI (JIHHIDI CBJl3H
22), a TaIOKe CKOpOCTHOH Cl1rHaJI (nHHHSI
CBH311 21), C.n;BI1HYTbIH Ha 90° no OTHome-
HI1IO K cHrHaJJY nonO:>KCHl-UI.
CHfHaJJbI nOJIO>KeHHSJ Ha JIHHIUIX CBSI'3H
19, 20 MMelOT nponmOnOJIOJKHble aMnmrry-
,l\bI H nocrynalOT H8 nOJIO:>KHTeJ1bHbIH H OT-
210 fJlaaa 3. 3nEMEHTHMI 6A3A MI1KPOMEXAHl1"4ECKJ1X rIPI160P013
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PIIC. 3.56. cJ»YHKUBOHSJ1bH3H cxeJ\la OJ(HOMSCCOBOro Mf LL-Tnns:
1 - HenOJ(BlDKHhIe 3JIeKTpO.!lb1 o6paTHoH CBII3H; 2 - TO n<e, 3JIeKTpO-
CTaTWleCKoro npHBO.l!a; 3, 4, 6, 7 - TO n<e, eMKOCTHlolX H3Mepl1Tenei1
nepeMeIQeHHH; 5 - nOJ1BH)K}{hIH 3neKTpo.n; 8-12- I1CTO<JHHKH HarrpJl)J(eHl1H.;
13-18 - CMeCl1TeJIH; 19-22 - 3JIeKTpH<JecKHe ]JHHliH CBH.3H
x
pHIlaTeJIbHblH BbIX01U>I YCMJIHTeIDI Y p. C KO-
ropJ,lMJ1 CBjD(UJbI T8lOKe HenO,l\BM:>KHbIC
3!1eIcrpO,lIbI 2 npHBOJl.a. Heno}J.BH:>KHble 3JIeK-
Tp01U>I 1 06paTHOH CBg3H CoeJtHHeHbI C no-
J10:ilCHTeJIbHblM H O1pHuaTeJIbHbIM BXO,l\aMI1
YCHJIHTeJISI y p'
o.ZUiH 113 BblXO}J.OB YCHJIHTCIDI Y p 'lepe3
cMecHTeJIb J 8 06"be.JJ.HHeH co CKOpOCTHblM
CHJ1laJIOM <1>AJI1I, H KOM6HHHpOBaHHblH
CHI1IaJI nOC1)'JlaeT Ha Y p .nIDI 06ecne1JeHIDI
l1ellH aBTOMaTH'JecKOH peryJ1JipOBKH YCHJIe-
HIDI 'ITO n03BOIDIeT ynpaBIDITb aMnJIM'T)',l\OH
KOlIe6aUHH HM. I1CTO'lHHJC HanpJDKeHIDI 8
BLIpa6aTbIBaeT HTOrOBOe HanpJDKeHMe CMe-
meHWI U oo - I!U - U C , HCTO'iHMJ( 9 - Ha-
npoceHHe U on + AU - U e . HCTO'iHHK 10 -
HaDpJDKeHHe U oo -AU + U e H HCTO'iHHK J 1 -
HaIIpJDICeHHe CMellteHIDI UOI! + AU + U e . Ha-
60p 3THX HanpJDICeHHH o6UYIDIeT KBaJl.pa-
TYPH)'IO OWWKY, 06ecne'IHB8eT BbI60p He-
o6xo.amfOH qaCTOTbI KOJle6aHHH 11M B pq H
KOMneHCal\J1I0 u:eHTp06e>KHbIX I1JIH m06bIX
HH3KOqaCTOTHbIX CllJ1.
B03MO)l<Ha II )J,pyraSi KOM61lHaUHSI I1C-
TOQHI1KOB HanpSliKCHHH.
BblXO,llHOii CHrHan HHTerpaTopa 11 no-
cJle YCHJIllTeIDI Y MOiKCT 6bITb CMelliaH C
CHrHanOM OT I1CTO'lHllKa J 2, a TalOKe C CMr-
HanaMI1 OT <DAJ11.l. I1cTOQHI1K 12 Bblpa6a-
TblBaeT HanpIDKcHHc BbICOKOii QaCTOTbl
(-1 MfI.\), KOTOpWi j{BJISleTCSI HecYllI.ei1: 'lac.
TOTOH .nIDI BCCX CHruaJIOB M nOCTYnaCT Ha
CMCCI1TeJIH 13-17. Ha 3TOH HecYllI.eH 'laCTO-
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CHTeJJb 13). UI< (cMecMTeJIH 14, 17), UI<U
(cMeCHTeJlI1 J 5, 16), nponopU:HOHaJIbHble
nepeMelllcHHJO 11M, YCKopemHo KOpHOJlI1Ca,
T.C. H3Mepj{eMoH ymoBoH CKOpOCTI1. M KBa).l.-
pa-rypHoMY YCKopeHHEO. 3TH CMfHaJIbI MO}K-
HO npOnyCTI1Tb Qepe3 HI13KOQaCTOTHblC
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211
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Horo CHrHaJIa B PII H CHrHaJIOB, 06ecITe"lH-
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pHC. 1.43. 11M 1 B P.D: cOBeprnaIOT npOTMBo-
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CHJIa, pa3BMBaeMaH. 3JIeKTpoCTaTJ1qe-
CKHM npHBO)l,OM, ITO OTHorneHHID K ynpaB-
JIHIDllJ,eMY ClITHany H3MeHSlCTCSI no 3aKoHY,
MH3KOMy K KBa,apaTH"IHoMy [CM. OPMY1IY
(3.85)]. TIpM pe30HaHcHoH HJIH 6JIH3KOH K
HeH HacTpoHKe Mr He06xo.ll,HMO, 'lT06bl
CHJIa ("cHrHaJI" CHJIbI) 06na,aaJIa HaH6oJIb-
meH ::meprCTMKOH Ha pe30HaHcHoH "IaCTOTe,
T.e. B MaTeMaTH"IeCKOM BblpIDKeHHH 3JIeK-
TpOCTaTH"IeCKaS! CHJIa )l,on:>KHa MMeTb HaH-
60nbIIIYID COCTaBAAIOrn.yID Ha pe30HaHCHOH
"IaCTOTe.
3JIeKTpO,lJ,bI H3MepMTeJISI ITepeMellJ,eHHH
3 HaxO)J,H.TCSI no.u. Hanp1l)j{eHHeM U o , a 3JIeK-
TpO,nb1 M3MepHTeJIH nepeMellleHHii 5 - no)l,
HaITpj{)!(eHlleM U on . 3TM onopHble HanpIDKe-
HH1I C03,l\a1OT Ha 3JIeKTpo,l\ax eMKOCTHbIX
H3MepMTeJIeH 3apH;::t;bI, npOnOpl.\HOH8JIbHble
M3MeHeHlUIM eMKOCTeH Me)Kl{}' COOTBeTCT-
BYlOlllMMH 3neKTpO)l,aMM. CMfHaJI C 1l3MepH-
Teml 3 nocT)'nacT Ha llHTerpaTop Ml, BhI-
nOJIHHIOIl.\HH ponb YCHJIMTeJISI 3apMoB, Ha
BbIXO)l,e KOToporo OPMHPYCTCj{ CHrHaJI,
C,l\BHfaeMbIH 3aTeM B a30BpalllaTeJIe 41 Ha
90° no OTHomeHHID K CHrHany Ha BbIXO,l\e
HHTerpaTopa 112, B KOTOpoM npC06paJylOTCSI
3ap1lnOBble CHrHaJIhI npe06pa30BaTeneii 5.
212 rJIaB8 3. 3JIEMEHTHA..5I EA3A MI1KPOMEXAHUt.IECKI1X DPHI10POB
Ha BLIXO,i1e HHTerpaTopa 112 opMnpy-
eTClI a..1\UlJllrry.i1HO-MO.nYJIUpOBaHHbIH U3Me-
pller.tOM yrnoBoH CI<OpOCTbIO CHfHan Ha "Iac-
TOTe P)]., KOTOpblH COBMeCTHO C CUfHaJIOM
06paTHOH CB1I3M U OC Ha TOM >Ke "IaCTOTe, HO
co C.i1BMrOM no 83e Ha 90 c , nOC1)'naeT Ha
.neMo.nyJIHTOp )]., a 3aTeM Ha 4JUJ1bTp. BLl-
XO.!UWH CHrHan H1l3I<0'laCTOTHOrO UJIbTpa
npe.!J,CTaBJIj[eT HallpIDKeHUe, nponoPQuo-
HfulbHoe H3MepSieMoH yr.noBoH CKOpOCTH.
B38HMHOe BJIIDIHMe BbIXO.nHbIX cnrHa-
.nOB C mrrerparopoB 111 H 112 SlBJIXeTCg
C.JIe.aCTBHeM nap83HTHOH eMI<OCTH. B03HM-
KaJOmeH npH H3rOTOBJIeHUH MHI\.-p03JIel\.1p0-
HHKH. T aK KaK 3m CnrHaJILI HMelOT onH)'
QaCT01)'. B38HMOB.1JIDIHHe KaHaJIOB YCTpaIDl-
eTCg C TpY.!J,OM.
ITpoXO>K.LleHl1e CI1I"Hana U oc 'lepC3 MOK
4JOpMMpOBaHIDI ynpaBJ1SIIOmerO CHrHaJIa
npHBO.n;HT I< H3MeHeHHIO (nepeHOCY) "IaCTO-
TbI CHniana U oc no CpaBHeHllIO C pe30HaHC-
HOH QaCTOTOH, I<OTopaH. 1IB.!1JIeTCg HecyrueH
.a.'UI CHf'HanOB Ha BLIXo.n,e MHTerpaTopoB 111
HID.
Ha.'Jj{I!He npeo6p83oBaTCJl}I Ha BXO)J,e
CJITHaJIa Uoc; Hco6B3aTCJIloHO. TIoCJIeJtHMH
MO)JCe1' 6bITb npeJ1.CTa.BJIeH B BHlI,e
U oc = Acosroxl.
r.ne A - Ko34}l$HlmeHT YCHJ1eHIDI uenH,
BKJIJOQaJOm.eH B ce6g npc06p830BaTeJIb 3,
HHTerpaTOp HI H <}Ia3oBpamaTCJIb «1>; ro x -
q,acr()1'3. nepeMeIl!eHHH HM.
B cxeMY aBTOMaTH'leCKOrO ynpaBJIeHHg
YCKJ'IeHHeM BXO.IVIT YMHOIKJiTeJIb YMHl,
lCoMnaparop (6JIOK cpaBHeHIDl) K H I1HTer-
pHpytonudf YCHJIHTeJIb Y. Y MHO)KJ1TeJIb 11
KOMnapaTOp CJIY>KaT )J,JISI )J,BYXITOJIynepl1o.L\-
HOro BbmpSlMJIeHU5! Cl1rHana V oc . Ha BbIXO;l,e
YClillUTeJISI Y 06pa3yeTcSI pa31i1lua l\le>K; Y
cpe)J,HuM ypOBHeM BbmpllMJ1eHHOrO CMrH8..18
Ii noporoBblM HanpH>KeHHeM VII C ueJIblO
nOJIY<ieHHlI nOCTOHHHoro Cl1fHana CMelileHI1S1
U CM ' KOTOpblll HBJISleTClI MepoH aMnJIM1)'llHoH
norpelliHOCTH CHrHana V ec -
CnrHanbI U ec Ii V CM nocTynalOT Ha
CYMMaTOp L, Ha BbIX0.i1e KOToporo 06-
pa3yeTclI cKoppeKTHponaHHblii no aMnJII1-
TY.i1e M CMell\eHHbIH no a3e curHan, KOTO-
pLIH nO.i1aeTClI Ha YMHO>KMTeJIb YMH2, Ky.n,a
npuxo.n,UT TaJ.-JKC curRan OT reHepaTopa
KOMMyrauuoHHoro CMrHana, ynpaBJISIeMOrO
CHrHaJIOM U ec . Cl1rHan KOMMYTaUUH MOn<eT
ITocTynaTb Ha YMHO>KMTeJIb Yr.1H2 11 OT He-
3aBUCHMOrO I1CTO"lHUKa. Ha BLIXOlle Yl\1H2
opMHpyeTcSI ynpaBJISIIOll\MH CH rRaJI , no-
C1)'naIOIl!MH K 3JIeKrpOCTaTU'leCKUM npH-
Bo.n,aM 2 H 4.
B CJIy<iae, ecnH CHfHaJI KOMMYTaunu
llMeCT CllH)'COJi.n,aJIbHYIO OPMY, yrrpaB-
JIHIOIl!HM CMrHaJI BblrJIH.,llMT TaK:
U y = (U CM + Acosroxt)acosroJ(c, (3.133)
r.n,e a, ro" aMllJIM1)'na H a3a CMrHaJIa KOM-
MyraU HH .
KaK CJIe.nyeT H3 OPMYJIbl (3.133), CHr-
Han U y H,l\eT Ha qaCTOTaX, OTJIM'lHbIX OT ro x , H
B3allMOBJIIDIHHe uenu 06paTHoH CBSl3M H
QenH HHTerpaTopa 112 .n,M>KHO OCJIa6JISlTbcSI.
CHJ10BOH CHfHaJI, pa3BHBaeMLIH 3JIeK-
TpoCTaTJ11{eCKHMM npHBo.n,aMM, HMeeT KBa)J,-
pa1)'pHYIO 3aBMCMMOCTb no OTHOllleHI1IO K
YITpaBJISIIOIl!eMY CHfHaJIY:
tV)=Krl-l; = Kn o2 (U c lil. + Acosror/f cos 2 rokl =
= Kn o2 (U';' +2AU(;M cosrorl + A 2 cos 2 wi ) (1 + cos2wJ(/) =
1 [ A2 (, ) ]( ) (3.134)
='2K n c? U:)4 +T +cos 2 ro x ' -sin 2 roi + 2UcldAcosroxi 1 + cos2roJ(1 =
I [ ]
='1. Kn c? U:III. +TWcIII.ACOSWxl+TcoS2rolCl (l+cos2roJ(/),
3JlEKTPOHHbIE CPE.lJ:CTBA OEPAEOTKH CHrHAJlOB
213
r)J,e K" - K03(})(IH1L{HCH1 nCpe)J,a'lH, 3aBHCR-
IllHH OT l-eO:'le I pHH 3:IeKTpOCTaTM'ICCKOrO
npHBOJ1a.
113 BblpIDKCHI1S1 (3.134) Cne.D.YCT, 'lTO
CI1JIOBOH CHrHaJl I1MeeT nOCTOSlHHYID H oc-
HOBHYID COCTaBJIRIDIllYID Ha pe30HaHcHoH
1.JaCTO re npHBOJIa, KOTopble 11 oITpe)leJIRIDT
ero lHepreTHllecKHe B03MO)KHOCTl1. Boc-
npHI1M'lHBOCTb 11M. SlBJISIIOll(HXCH 1.JaCTbID
nO,D.BI1>KHbIX CTpYKTYP npl1BOAa, K COCTaB-
JlSlIDIllMM Ha ,llBOHHbIX 'laCTOTaX 3Ha1.JMTeJ1b-
HO MCHblllC.
Ecm1 CJlrHaJI KOMMyraL{HH npH:'w-
yrOJIbHbIM. npe,llCTaBJ1HeMbIH B BHne pRna
S ( ) _ 4a ( . sin 3x sin 5x )
q x - smx+ + +... ,
n 3 3
(3.135)
rAe x = ffiKt; ffi K = 2n1T; a, T - aMllJIMTYAa H
nepl1011 npRMO}TOnbHOH nOCJIe)J,OBaTeJIbHO-
CTH, TO ynpaBJIRIOIlJ,I1H CHrHan npMHHMaCT
BH,D.
V y = (V CM + ACOSffi.l)sq(ffiKt).
AHanOrM'lHO npe,llblAYIl\cMY, MO)l(HO
nOJIY'lMTb Bblpa»<eHHe AJUI CHJIOBOrO cur-
Hana:
( 2 A 2
f(t)=Kn U CM +2+2Uc IACOSffixt+
A 2 )
+Tcos2ffixt .
B :.nOM c.rryqae CMJIOBOH CHrHaJJ TaK:>Ke
o6JIa.L\aeT 3Ha'll1TeJIbHOH COCTaBJIRlOllleH Ha
'laCTOTe pe30HaHca, HO C Y'feTOM OPMYJlbI
(3.135) AO TeX nop, ITOKa 'laCTOTa KOMMyra-
L{HH He RBJIHeTCR He'leTHOH rapMOHMKOH
pe30HaHCHOH 'laCTOTbI.
3alla'la npe06pa30BaTeJISI Ha BbIXOlle
c}Ja30BpallJ,aTeJISI 3aKJII0'laeTCH B npeBpallJ,e-
HHM CHrHana V ec B npRMoyronbHbIH:
U ec = Bsq( roxt),
rAe B - aMnJlHl)'Aa BbIXonHOro CHrHana
npeo6pa30BaTeJISI.
TaKl1M 06pa30M, eCJII1 Cl1rHan KOMMY-
TaL{MH no-npe:>KIiCMY npHMO}'f'OJlbHbIM, TO Ha
Bblxolle YMH2 (POpMMpyeTcR ynpaBJUIIDlllMH
Cl1rHaJI
V y = [Uc\! + Bsq(ffixt)}sq(ffiKt).
KBa)J,paTM'lHlliI 3aBI1CHMOCTb CMJIbl
npMBOna OT ynpaBJUlIDIllero Cl1rHana npl1BO-
.aHI K CHJIOBOMY Cl1rHany. 06ecnelll1BalOIIle-
My P,[.I, 11M;
f(t) = Kn[V;M + B 2 + 2U CM Bsq(ffi x t )).
I1cnoJlb30BaHHe B 6JIOKe opMHpoBa-
HlUI ynpaBJIHIOll(ero CIITHana npe06pa30Ba-
TeJIH C 60JIblliOH aMnJIHrynoH CMrHana Ha
BbIXO)J,e o6eCne'llffiaCT 60J1ee BbICOKHH KO-
3 c}JI1lrneHT YCMJIeHmr 3ar.fKHYTOH lI.enH H
B03MO>KHOCTb npMMeHeHMSI lI.H pOBOH cxe-
MbJ 06pa6OTKH npH OpMI1pOBaHl1ll ynpaB-
JUIIDl1lero CMrHana.
,[.I,onOJIHHTCJIbliaH. lI.enb 06H) eHHH
KBa.L\pal)'pHoro CMrHana (KBaAparypbJ) MO-
:>KeT 6blTb BKJIID'leHa Me>K.L\)' BbIXOAOM MHTe-
rpaTopa 111 I1 3JIeI\.'TpOnaMH npHBona.
TeMLIAJIHCaMOKOHTpOJIH
1. )l(ecTKocTb ynpymx 3JICMeHTOB M
I1X 3aBHCllMOCTb OT cxeMbI HarpY:>KeHIDI.
2. )l(ecTKOCTb H 'laCTOTbI c06CTBeH-
HbIX KOJIe6aHHH.
3. Bm1H.HMe MaCCbI ynPYTHX 3JleMeHTOB
Ha 'laCTOTY co6CTBeHHbIX KOJIe6aHMH q3.
4. BJIHIDIMe paCTSlrHBalOIl\I1X CHJI Ha
:>KeCTKOCTb ynpyrux 3JIeMeHTOB H tJaCTOTY
KOJIe6aHHH q3.
5. fJIaBHble OpMbI H 'laCTOTbl KOJle-
6aHHH q3.
6. Oc06eHHoCTH lle opMHpOBaHMH
MeM6paHbI nocToIDIHOH TOJIIlJ,HHbI H CBH3aH-
HOH C HHMH TOnOJIorHH TCH30pe3HcTopOB.
7. Oc06eHHOCTH Ae OpMI1pOBaHHSI
MeM6paHbi C :>KeCTKHM UeHTpoM H cBH3aHHoH
C HMMH TOITOnOfUH TeH30pe3MCTopoB.
8. CxeMa, npHHlI.Hn pa60TbI, OCHOB-
Hble nOCTOMHCTBa 11 HellOCTaTKH eMKocTHOro
npeo6pa30BaTeJIH. nepeMellJ,eHHH.
214 rJIasa 3. 3nEMEHTHAJI 6A3A MI1KPOMEXAHWlECKI1X rIPI1EiOPOB
9. Cnoco6bI H3roTOBJIeHWI 11 oc06eH-
HOCTH pa60Tbl TeH30pe3HCTopHbIX npe06pa-
30BaTeJIeH ,l\eI}>OpMaJ.{Mii.
10. TCITJ10BOH lllYM H none3HblH CllrHaJJ
TeH30pe3HCTOpa. AHH30Tponll51 3neKTpU'le-
CKOro cOnpOTH8JlelUUl nonynpoBo.n,HHKOBblX
TeH30pe3HCTOpoB.
1 L llbe30pe3HCTMBHble, HOpMHpOBaH-
HLIe nbe3ope3MCTMBHble K03 HUHeHn,1 H
HX H3M'ieCKHH CMhICJI.
12. qysCTBHTeJIbHOCTb TeH30pe3HCTHB-
HbIX cxeM.
13. TonoJIorHSI TeH30pe3HCTopOB Ha
KpyrnoH MeM6paHe nOCTOHHHoH TOJIIltHHbI,
06ecne'lHBaJOlI.tWl MaKCHMaJIbHYIO qyBCTBH-
Te.JIbHOCTb cxeM.
14. llpeo6pa30BaTenb )J,e opMaUH:H Ha
IIAB. CxeMa, npHHUMn pa60TbI, CTaTH'le-
CKaJI xapaA"TepllCnfKa.
15. llpe06pa30BaTeJIb .n,e4JoPMaUHH Ha
CTp)'He. CxeMa, npllHUHn pa60Tbl, CTaTH'le-
CKMl xapaA"TepHcTliKa.
16. <IJH3M'iecKHe OCHOBbI pa60TbI :meK-
TpOC'I'aTH'IeCKI1X npe06pa30BaTeJ1eH CHJIbI.
17. 3J1eKTpOCTaTH'IeCKHH J1HHeHHLIH
MHKpO.nBHraTeJIb. CxeMa H nOCJIenOBaTeJIb-
HOCTb pac'JeTa.
18. 3!1eKI'pOCT'aTU'IeCKHH BpalI.taTeJ1b-
HblH MHKpO.nBHraTeJ1b. CxeMa H nOCJIenOBa-
TeJIbHOCTb pac'JeTa.
19. MarnHT03JIeKI'pH'ieCKHH npe06pa-
3OB8TeJIb CHJIbJ C CHJ10BOH (Bo3Bp a lI.t alO ll\eH)
06MOTKOH. CxeMbl H OCHOBHaH paC'1eTHilll
3aBlfCMMOCTb.
20. MarHliT03JIeKTpH'lecKnH npe06pa-
30BaTenb CHJIbI KOJIbUeBoro pe30HaTopa.
CxeMa, npl1HUHn pa60TbI. OCHOBHillJ pac-
'leTHWI 38BI1ClfMOCTb.
21. 3neKTpoMarHHTHble npc06pa30Ba-
TeJUI CHJIbl co B3aJlMo)J,eHcTBYI01llI1MH npo-
BO)J,HMKaMIi C TOKOM. CxeMbI H: OCHOBHilll
pactJeTHaH. 3aBliCHMOCTb.
22.3neKTpoMarHHTHblH npe06pa30Ba-
Tenb CHJ1bI C HallmrliHtJl1BaeMbIM HCnOJIHlI-
TeJIbHblM 3neMeHTOM. CxeMali OCHOBHilll
paC'leTHaH. 3aBHCI1MOCTb.
23. AHaJJH:3 pa60TbI cxeMbI BKJIIO'1eHlIH
eMKOCTHoro npc06pa30BaTeJ151 nepeMellle-
HHH C aHaJJorOBbIM BbIXO.n,OM no pHC. 3.41.
24. AHaJJm pa60TbI cxeMbI BKJIIO'leHI1H
TeH30.n,aT'lH:KOB C 'laCTOTHbIM BblXo.n,OM no
pHC. 3.42.
25. TIpMHUHn pa60Tbl BbIXOnHblX YCT-
pOHCTB C 1lIl1M Ha BbIXOne CXCMbI.
26. Oco6eliHoCTH 3JIeKTpoHHblx cpe.n,CTB.
06eCne'lHBaIOIlJ,MX pa60ry Mr.
27. MeTo.n,bI ycrpaHeHHSI 3neKTpOHHbI-
MH cpe.n,cTBaMH: ocHOBHblX nOrpellJHO-
CTeH Mr.
28. 06ecnetJeHHe 3JIeKTpoHHblMH cpen-
CTBaMH P)J( Mr.
29.06ecne'leIiHe 3neKTpoHHbIMH cpen-
CTBaMH P.z:I: Mr.
30. 3neKTpoHHble MeTonbI Bbl,lleJIeHMSI
nOJIe3HOrO CHrHaJJa Mr.
r JIaBa 4
,[(I1HAMHKA q3 MHKPOMEXAHIflIECKHX
nPIIEOPOB
B COOTBCTCTBHH C onpe)J,eneHHHMM, BBe)l,eHHbIMH B rn. 1, qYBCTBHTeJIbHbIM '3JIeMeHTOM
(43) MHKpoaKCeJIepOMeTpa (MA) RBJIHeTCH HHepUMOHHaH. Macca (HM), 06bl'lHO BLInOJIHSle-
MaH. B BH.ne nJIaCTHHKM C ynpymM nonBeCOM. 3 MHKpO)l,aT'lHKOB naBJIeHllH (MM,[() npen-
CTaBJIHeT C060H MeM6paHY, ITOCTORHHYJO no TOJIll\HHe JIH60 C :>KeCTKMM ueHTpoM. 3 MHKpO-
rHpOCKOnoB (Mf) :no OnHa HJIH HeCKOJIbKO HM Ha ynpyroM nO)J,Bece H nepBH'lHb1e npe06-
pa30BaTeJIM, o6eCnC'lHBalOllme pe>KHM )J,BH>KeHHSI (P,ilJ HM. TIo)l, ,lI,MHaMHKoi1 q3 MMKpoMe-
XaHH'leCKI1X ITpH60pOB 6Y)I,eM nOHMMaTb KaK c06cTBeHHoe, TaK H BbIHY))()J.eHHoe - no,ll, neHcT-
BHeM 113MepSieMblx M BpeL\HbIX (nap8.3HTHbIX) B03,l1,eHCTBHi1 ,lI,BM>KeHlle 11M C Y'feTOM oco-
6eHHocTeH KOHCTPYKUHl1 Q3. [10 TeopMH aBTOMaTll'leCKoro ynpaBJIeHIDI q3 MHKpoMexaHM-
qeCKHX npH60poB SlBJIRIOTCH KOJIe6aTeJIbHLIMH 3BeHbHMM, ITapaMeTpLI KOTOpbIX H xapaKTCpH-
3YJOT ,l\HHaMMKY Q3.
4.1. YP ABHEHIDI,l{BHJKEHIDI "tJ3 MlIKPOAKCEJIEPOMETPOB
4.1.1, OceBoii MHKpOaKCeJIepOMeTp
npennOJIaraeTCSI, "TO ueHTp Macc (QM, TO<iKa C) [lJ]aCTHHbJ CMellJ,eli B nJIOCKOCTH XZ
OTHOCllTeJIbHO reOMeTpH'leCKoro uempa (TO'lKa 0) Ha BenH'lHHLI lx, lz (pHC. 4.1, a). QeHTp
Macc HaXOnHTCR B ITJIOCKOCTH cHMMeTpHH nJIaCTI1HLl (PHC. 4.1, 6). TIpH )J,eikTBHH CHJI may M
mg (m - Macca ITJIaCTMHLI), 06YCJIOBJIeHHbIX H3MepHeMbIM YCKopeliMeM aJ' M YCKopeHMeM g
CHJILI TH>KeCTM, nepeMellJ,eHHe ITJIacTHHbl onpe,l\CJUIeTCR JIHHeHHOH Y M ymoBoH a KOop,ll,HHa-
TaMH (PHC. 4.1, 6).
TeKYllJ,ee ITOnO>KeHHe HM (ITJIaCTMHbI) onpeneneHO CJIe.D.)'IOIlJ,HM 06p8.30M (pHC. 4.1, 2). C
l<opnYCOM aKcenepOMeTpa eBH3aHa CHCTeMa KOOPnHHaT XYZ, Ha'lanO KOTOpOH: COBna.naeT C
UeHTpOM CMMMeTpHH ITJIaCTHHLl.
BL\OJIb oceH X, Y, Z HaITpaBJIeHbI BeKTOpbl BH6ponepeMellJ,eHMH X B , h, ZB, KOTopbIe SlBJIH.-
JOTCH ITpOeK1lHHMH BH6ponepeMell\eHllR TO'lKH 0, M BeKTOpLI JIHHeHHbIX YCKopeHllH ax, a. ., a z ,
SlBJIHlOllJ,HeCSI npoeKUHRMH JIHHeHHOro YCKopeHHJI. TO'lKH lJ. BCJIC)I,CTBHe ,lI,eHCTBMJI. o6061lJ,eH-
HLIX CM H KOHe'lHOH :>KeCTKOCTH nOnBeca UeHTp CHMMCTpHH 11M ITepeMellJ,aeTCH B TO'lKY 0(,
onpeneJISleMYJO KOOp,ll,HHaTaMH x, y, z ynpyroro CMell\eHMJI..
OCH CHCTeMLI KOOpnHHaT XIYIZ' napaJIJIenbHLl COOTBeTCTBYJOll\HM OCHM CllCTeMLI KOOp-
,lI,HHaT XYZ. YrJIOBOe nOJIO>KeHlle [lJ]aCTHHbI 3a,l\aHO yrJIaMH 13 H a, KOTOpbIM OTBeqaIOT CHC-
TCMbI KOOp,ll,MHaT X2)'2Z2 I1 XVJ3Z3. TIJIOCKOCTb nJIaCTMHbI onpe,ll,eneHa OCHMM 0IX3 H 0tZ3, r,l\e
Ha paccToHHMH Ix B,lI,OJIb OCH 0'X3 H /z B,l\OJ1b OCH 0lZ3 HaxO,ll,HTC}[ QM (TO'lKa C). C TOqKOH C
CBSI3aHa CMCTeMa Koop,ll,HHaT xyz, OCH KOTOpOH napaJIJIeJIbHbI COOTBCTCTBYJOIltHM OCRM CMCTe-
MbI KOOp,ll,HHaT XVJ3Z3. TaKJ1M 06P8.30M, nOJIO:>KeHHe HM aKcenepoMeTpa onpe.neJIeHO TpeMH
nHHeHHbIMH (x, y, z) II )I,B}'MH yrJIOBLIMM KOOp,l\HHaTaMH (a, 13) (nHTb 0606ll.{eHHbIX KOOp-
n HHaT ).
B npHHSlTOH nOCTaHOBKe 3a)l,aQH B03MO>KHLIH yrOJI pa3BOpOTa nJIaCTHHbI BOKpyr OCM
OIY3 He paCCMaTpHBaCTCH, TaK KaK npe,ll,nOJIaraerCH, qTO :>KeCTKOCTL no.nBeca BOKpyr 3TOH OCR
3Ha'lHTeJIbHO 60JIbllIe :>KeCTKOCTeH no yrJIOBLlM KOOp,ll,HHaTaM a Ii 13.
216
rnasa 4. LUffiAMHKA q3 MHKPOMEXAHW1ECKI1X rIPHEOPOB
aJ
............,.
.
..
6) A-A
Y
.,... -_..."" "="' .
_Y'. .....1'..... ..."Io.e, ...
4
.B : .,.-;
, ". -..:X
, "' "-- I II; -"," I ' :J
LJ " " ",""",:".
l ...
...-. ..."... .
\ . ;.. "": r
... .. '.>
". " :::=\ ,> " '"
" ..." ..
T I"
., I.,
.-'- __n--.J __ _ ____
_ ",_ ", _h'
Z
e)
mal
;:- t a
mg
Z2,ZJ
Pac. 4.1. K BbIBO Y ypaBHeHHu l1BUal:eHHH q3 oceBoeo MA:
a, 6 - nOnO"IKeHHSI L{M fiJIacTHHLI; 6 - nOJIOjj{eHMe fiJIaCTMHbl npl1l1eilcTBI1M YCKOpCHI1S1;
2 - CMCTeMbI KOOP.lI.HHBT
YpmmeHIDI .nBIDKeHIDI q3 nonyqHM C nOMOWblO ypaBHeHMH flarpaHJKa BToporo po.na:
( a ) _ aT =Qj' j=1...5,
dt aqj aqj
(4.1)
r..n.e T - KHHCTH'IecKIDI 3HeprIDI IIJIaCTHHbI; Q} - 0606ll\eHHLle CMJIbl no COOTBeTCTByIOll{I1M
0606m:eHHLlM KOOp.rornaT8M; qj, q} - o606weHHble Koop,nJiHaTbI II CKOpOCTI1 (ql = X,
QJ=x,q2.=y,qz=Y' q'j=z, q'j=i., q4=a, q4=a,qS=p,qs=(3).
Bhlpa7ll:euue, onpe.nemllomee KHHe'IWjecKYIO 3HeprHIO ITJIaCTUHbI, HMeeT BI1.Q
T I (J p'2 J .2 ) m ( .2 .2 .2 )
=- x.... + z(J. +- Xc + Yc +Zc '
2 2
r,Ae J x , Jz - rJIaBHhle lteH1paJIbHble MOMeHTbI HHCpLtHH fiJIacTMHbJ OTHOCHTeJIbHO oceM Cx 11 Cy;
,xc. Pc' it - nHHeif:H1.Ie CKOpocni l!M DJJaCTHHbI B HanpaBJIeHHI1 oceu --y. Y, Z COOTBeTCT-
(4.2)
BeHHO.
YP ABHEHHJI BYDKEHIDI q3 MMKPOAKCEJIEPOMETPOB
217
B COOTBeTCTBHl1 C pHC. 4.1
Xc = x + XII + Ixcoso.:
Yc y + Yo + l.xsinacos -l z sinl3; Zc = Z + ZB + Izcosl3.
J1HHeHHble CKOpOCHI UM IDIaCTHHbl
Xe :::C:X+X" -Ixasina;
.l'L = jq j'" + I., (acosacosl3 -(3sin sina )-lz(3cosl3 .v + YB + Ixacosacosl3 -lz(3cosl3
(TaKKaK acosacos » sinl3sina): zc=i+il!-I= sin .
B Ka'leCTBe npHMepa nOJIY'lMM ypaBHelUle .u,BH:>KeHllJI 1:13 no KOOpnHHaTe y. Bbl'lHCJIHM
npoH3Bo)J,Hble:
T T T .
o 0 0 0 eye . t.. I . 113 ' n )
--;:;-= ;-:;:-= '. 8j =mYC=m\Y+YB+ xacosacosl3- z cos...;
r;,. cv Uy ,
J . _ C
: = m{v + YII + lJicoso.cosp - a. (a. sin acosJ3 + psin J3cosa)]-I= cosl3-(32 sin (3)}.
HMCH B BH.lJ:Y, '-ITO 06061lJ,eHHble KOOpnHHaTbI a, 13, y - MaJIble BeJIH'lHHLl, 6yneM nOJIa-
raTb, liTO sino. R< a, sinl3 R< 13, coso. R< cosl3 1.
KpoMe TOro, CTaHeM npeHe6peraTb npoH3Be.u,eHHKMM M KBa)J,paTa.MM MaJILIX BenH'IMH, a
TaK:>Ke BellH'lHHaMH, HMeIOll.\HMH ell.\e 60J1ee BLICOKHH ITOp'L'J,OK MaJIOCTH.
C y'leTOM CKa3aHHoro, ypaBHeHMe ,llBH:>KeIDIH q3 no JIMHeHHOH KOOpmmaTe Y HMeeT BH,ll
m (y + lxii + lzi3) = Qy - mYB'
(4.3)
TIoCT)'naH. aHaJIOrH'lHbIM 06p830M, nOJ1y<rnM ypaBHeHMJI ,QBH:>KeHMH q3 no OCTaJIbHLIM
Koop)J,H:HaTaM:
m'i = Qx - mX B ; mz = Qz - mzo ;
(J x + m!;)j3 + mlzCv + lx.o.) = Qp -mlz (yo - ZBI3);
(J z + m'- )a+ml.,.(y+lz )= Qa -mix (yB -xBo.}
(4.4)
0606ll.\eHHble CHJIbI BKJIIO'laIOT B ce6S1 CMJILI, 06ycnOBJIeHHble :>KeCTKOCTbIO nO,l],Beca, yc-
KopeHMeM nepeHOCHoro .u,BM:>KeHH:R H neMn HpOBaHHeM.
B 06rueM cJIY'lae nepeMeIlleHHe ITJIaCTMHbl (MM) MO:>KeT 6bITL ITpenCTaMeHO KaK pe-
3YJIbTaT nBYX nBJ1)KeHHH: ITapaJIJIeJIbHorO nepeMellJ,eHMR no KoopnMHaTe y 11 p83BOpoTa Ha
yrOJI a (aHaJIOfH'lHO - Ha yrOJI (3) (pHC. 4.1, 6). TIpH 3TOM 06a KOHl..{a KIDK)J,Oro ynpyroro 3JIe-
MeHTa no.nBeca :>KeCTKO 3aIl\eMJIeHbI.
MaTpH:l..{a CYMMapHLlX CMJI H: MOMeHTOB, ,l],eHCTByroIlJ,HX Ha IDIaCTMHY Co CTOpOHbl OJtHoro
ynpyroro 3JIeMeHTa, onpc.u,eJIJleTCJI OPMYJIOH (3.2), B COOTBeTCTBHH: C KOTOpOH 0606meHHbIe
CJ1JIa H MOMeHT, 06YCJlOBJIeHHbIe :>KeCTKOCTblO o,ltHorO ynpyroro 3JIeMeHTa, onpe.u,eJ1JIlOTCJI
3aBH:CH:MOCTHMH (3.3), r.u,e K03 I1[.1MeHTbI :>KeCTKOCTM onpe,l\CJIJlIOTCH PMYJI3.M" (3.1).
218
rnaBa 4. LU1HAMHKA q3 MI1KPOMEXAHI1 ECKI1X rIPI1EOPOB
B 06IlteM cJ1)"iae, nOJIaraH. l 1= II HMe}! B Bli.ny MaJIbIe )TJ1bl "3aKpYTKIi" ynpyrnx 3JIe-
MeHTOB OTHOCHTeJIbHO oceH x H Z, MO:>KHO HCITOJIb30BaTb 3aBI1CHMOCTli (3.3), )J,OMHO:>KaS! HX lia
'iHCJIO napaJIJ1enbHO pa60TalOIlJ,MX 6aJIOK no.n;Bcca.
3anHmeM BblpIDKeHIU! CHJIbl, 06YCJIOBJICHHbIC YCKopeHl1cM nepcHocHoro BI1:>KeHI1J1, H
CHJIbI TH:>KeCHI JlJI}! COOTBeTcTBYlOllllfX Koop.n;llHaT:
Qx = ma x ; Qy = mea,. - g); Qz = ma z ;
Qu = m(/" + -"o)(a.,. - g);
Qp =m(/= +zo)(g-a,.).
(4.5)
0606111eHHble CIUlLI .aeMn lIpoBaHMSI nponOpUHOHaJIbHbl J1J1HeHHOH .i l 11 yrnoBoH a
(a TalOKe (3) CKOpOCTSIM JIBH)!\eHliSi ITJIaCHIHLI M Moryr 6bITb 3anncaHbl B MaTpH'lHOH opMe:
FIl
M;1,
k;t,\" k;1,ya y
kJJ.\.a k:l u a
,
(4.6)
r.n;e k1!,'" k:1o. - K03 HI.{HeHTLI .n;eMn MpoBaHH}! no JIHHeHHOH H YfJIOBOH KOOpmmaTaM, OITpe-
.n;eJIeHHe KOTOpblX 6y.n;eT paccMoTpeHo B nOnpa3n. 4.4; a - JIHlieiiHaH. KOOpnHHaTa HeHTpa yr-
J10BbIX KOJ1e6aHllH WlaCTHHbl, KOTOPYlO MO:>KHO C'lHTaTb paBHoH nOJIOBHHe )J,JIHHbI ITJIaCTHHbI.
npH I13Y<IeHI1I1 !l,HHaMllKH q3 B ITepBOM npM6JIH:>KeHllH MO:>KHO He )"-IHTbIBaTb B3aUMO-
CB$I3b Me)f()IY 0606IlleHHbIMH KOOpJ1.HHaTaMH npH BLI'lMCJIeHHH 06061lleHHLIX CHJI :>KeCTKOCTM Ii
CHJI .neMn HpoBaHIDI.
YpaBHeHID! !lBH:>KeHIDI (4.3), (4.4) C )"-IeTOM 1l3JIO)!\eHHoro npHHHMalOT BM.n;
nii+k:u:x+Gxx =m(a" -xo);
mZ+kIlZz+Grz=m(a z -zo);
my+k yY+ GyY +m«(..o. +lzi3) = mea}' - g- Yo);
Jjj +k + Gp13 +m{l" +z)(y +lxa.) = m(/z +z)(g -ay - Yo)+
+m(g-a).)(z" +z)+m(lz +z)zo13;
Juo. +k;w.a. + Ga.a+ m(lx +xxY+lz )= m(lx +x)(a y - g - Yo)+
+ m(a)' - g)(x o + x) + m(lx + x)xoa;
(4.7)
ypaBHeHIDI cBJJ:3eH
Yc = Y+/xo.+/,,13+ Y,,;
xc=x+lx+x,,;
Zc =z+l z +ZB'
(4.8)
T.n;e J jj =J x +m/;; Ja. =J z +m/; - MOMeHT HHepUHll llJIacTHHbI (11M) OTHOCHTeJIbHO DceM
0IZI H OIXJ COOTBeTCTSeHHO; G", Go., G p onpe.n;ellilJOTC1J no 4>opMYJIe (3.4, 3.5); G x , G z - :>Ke-
CTKOCTH no.nseca no KOOp,lUiHaTaM x H z, KOTopble MOTyr 6bITb BbI'jHCneHbI no 4JOpMYJ1e (3.3 I)
(HH.n;eKc y cne.nyer 3aMeHHTio Ha HHJ1eKC z).
YPAHHEI-IJ151 JJ:UH)KEHM5I 43 MJ1KPOAKCEJIEPOMETPOB
219
}J.JJH aKCeJICpOMeTpoB C OCblO lfYBcTBHTeJIbHOCTM no KOOpnHHaTe Y lie06xonHMo KOHCT-
PYKTIHUIO o6CCne(lI1Tb q,:::: G:» G 1 .. Ha :nOM OCHOBaHHH MO)!(HO C'lHTaTb x:::: z:::: O. CMcTeMa
ypaBHcliHM (4.7) Ii ypanHeHI1H. CBSl3eH (4.8) npumlMalOT BH,lI,
mY + k;J.".i' + Gl,y -I- m(lla + I:B) = mea" - g - Yo);
Jd 3 +k](IIB + G(1 +l}llzCy +I"ii) = mlz<g- a. v - j\,) +m(g-al,)zo +ml zB(3;
Jari -"-k]((1Ct + Gua +mlx<Y + lz ) = m1xC a y - g - jiB) +m(a v - g)x B +mIXXBa;
,
(4.9)
Y c = y+I.,.a+IA3+ YD'
(4.10)
TIpe)J,IIOJIQ)KI1M Tenepb, <ITO Iz = 0,1\.::1; 0, Y'iTeM OPMYJ1bl (3.2), (4.6) H 3aITHllleM ypaBHe-
flllSi (4.3), (4.4), Clfl1TaSl, <ITO Bce 6CL1KH pa60TalOT Ha H3nl6 H x = Z = 0, B BHnC
k
m(Y+lxii)+n(kl1y+kI2a)+kljvj'+ Ct =m(a l . -g- YB);
a
Jn + n(k 2t y + kd 3 ) + kn,ray + k.npP = mzo(g-a y );
Jaii + mlxji + nk 21 J' + (nk n -mlxxo)a + k)1.vo,V +k.na6. = mlAa y - g- yo)+mx,,(a y - g);
Yc=y+lxa+YII'
(4.11)
r,Qe 11 - 'lMCJIO yrrpymx 3J1eMemOB (6aJIOK) nO)J,Beca.
npH IICCJIenOBaHI1H CUCTeMbl (4.11) Ha,ll,O HMeTb B BH)J,y, liTO MO>KHO pewaTb cHCTeMY H3
nepBoro H TpeTbero ypaBHeHllj:j, TaK KaK B HHX He BXOnHT KOOp,ll,MHaTa [3, TaK :>Ke, KaK H B
ypaBHcHHe CBSl3H.
1-13 ypaBHeHHH (4.9) )J,JISI yeTaHoBHBweroc1I pe)!\MMa (6. = J3 = j; = ii = P = y = 0) eJIe,ll,y-
lOT 3aBI1CHMOCTH )J,JI1I Bbl'lHCJIeHH1I CTaTH'leCKHX CMell\eHHH llJIaCTHHbl no 06061lJ,eHHblM KO-
Op,ll,HHaTaM:
In ( .. )
Y CT =c ay-g-YB ;
y
(3CT - Ill.. [cg-aY)ZB+(g-ay-Ye)l,J;
G p - mlzz B
(4.12)
m
a. =
CT
Go. -mlxxo[(a y - g)x B +(a y - g - Yo) d
flplwep 4.1
Bbl(IJ1CnliM CTaTWleCKJ1e CMemeHHH OJJaCTHHbl .n;nJ! HCXO.n;HbIX .n;aHHbIX: QJ' = 100 M/C 2 ;
.\'" = x,,"sinro/, Yo = Y"asinro/, z" =: zuasinro/, X.a == Yua= Z.o = 10- 3 M, ro = 27t.100 IIc; Lx = Lz = 5 . 10- 5 M.
BOCnOJIb'3yeMcJI Pe3)'JIbT8TaMH BblIlHCJIeHI1H lI3 npHMepa 3.1: m =: m n =: 7,5 . 10- 5 Kr, G y = 525 HIM;
G n = G p = 47,72 . 10- 5 H'M. BbtIlI1CJII1M ImLzzol =lmlxxlIl =0,148.10- 5 H'M II no 4JopMyne (4.12) noJIY-
'111M Yo:r = 10- 5 (1,288 + 5,63sinrot) M, Ct. cr = Pc-r = 10-4 (7,08 + 30sinro/) pa,n.
220
rJlaBa 4. JUIHAMIIKA 43 MIlKPOMEX..\HIlt.JECKHX rIPII60POB
113 BbI'iHc."1eHUH c,"1e.JyeT, 'ITO .1aA-:e npll HeOO:IblllOU qaCTOTe Bllopallllll (100 r u) nOCTOJJllIIOe
CMeweWfe lThl no.1 .J.eHCTBHeM H3MepJJe fOrO YCI;:opeHIIH. a, !\leHbWe a m.lllT)".J.bl BliopaulioHliblX ne-
pe!otemeHHii. JlItHeliH(U! BHopaUIU! npltBo.J.11T I;: 3Haq]lTe::IbHbl!\1 Ha"-lOHa f 11M. H3 KOTOpblC T3KiKe H3-
K..l3,3blBaIOTCJI yr.:JoBbie I;:O:IeOaHlU! Ha qaCTOTe BH6paUIIII. JllOOIDI 113 COCTaB!ilIlOWIiX 'lOiKeT OblTb OT-
qUL"llirpoBaHa cpe.J.CTBa...m 3:IeKTpOHHKlI.
flpu.\Iep 4_1
BbI'lUC.1H!ot CTaTII<JecKoe C!\ICllleHIIC n_laCTIIHbl -' cr' IICnO:Ib3ySl CIlCTe'IY (4.11 J. npll OTCYTCTBlIII
BuopaUHH. Ilo.'laraJI npOH3BO:tHbIC OT KOOp.1I1HaT Hy.leBb] m. nO:J)"imf
11( I1Y + k aCT) = m(a,r g): }
l1(k l.lCT k "2acr)=mlx(ay g).
BblpaA-.'CHHe ;L'"VI -' cr H!\!eeT BII.J.
m(a, -g)(k"22 -/Tkr:.J
l' - -
. CT ,
11(k\lk 2 -k l -:!)
lICXO.J.Hue .J.aHHhle 113 npH!\!epOB 3_1 H 4.1: m = 7.5 . IO- Kr. Ix = 5 . 10- 5 !\f, a, = 100 !\UC!. 11 = 4.
k ll = 13125 H;M. k: 1 = 6.56 . 10-; H. k = 17.5 . 10- 5 H '!\f.
CrarnqecKoe c !ellleHHe ll.laCTHHb1:
1" = 7.5-IO-.5(100-9.8)(17,5-10- -5.10- 5 -6.56.10- 2 )
- CT 4{I3L25.17.5-10-5 -(6.56.10- 2 )2]
1 _- 10 - 5
,))- M.
B DpH.",epe 4.1 nO:J)"leHo)'C'T = 1,288. 10- 5 M. OnmCHTe,lbH(U! norpelliHOCTb Bbl'UlC::JeHlliI no OT-
HOmeHHJO K no.'J)rqeHHOMY pe3y;u,TIll)' 17.4 %.
4.1.2. MaJrnmKOBblH MBh-poaKce.'1epoMeTp
II') MA npe)J,CTaIL"UleT co60H 11M (n;JaCTHHY) Ha ynpyrOM nO,11Bece, KOTOpblH r.lOiKer
6bITL o4>opM.leH B BH;J.e KOHCOJlLHblX 6anol(, pa60TalOlUHX Ha mm6, .'lH60 KaK YITpyrHe 3JIe-
MeHTbl nlll<i TOpcI10HOB, pa60TalOIlUiX Ha KpY'leHHe (CM. pHC. 1.11).
OqeBH;IHO, 'ITO ecJ1M B cxeMe no pl1c. 4.1, a YCTpaHMTb ynpyme CBSl3M B!1.0J1b JIlO60H OCIi
(x H.."1H z). TO DO.;JyqaeTCg cxeMa MWlTHllKOBOro MA C nO}1.BeCOM B BH.lI,e 6a.'10K, pa60TalOlUHX
H3. KpyqeHHe (TOpcHOHOB). B 3TOM c.lyqac MOn<HO nOJIb30BaTbCSI ypaBHeHIDIMH (4.7}-{4.9),
HMe:ll B BH.!I)', 'ITO BblXO.!IHOH KOOp.il:l1HaTOH 6y.neT OnHH M3 yrnoB (a. llJJH J3).
no.:I.BeC 11M H3. KOHCOJIbHbIX 6anKax .lI,O!J:iKeH HCKJ1IO'laTb B03MO:>KHOCTb cJ.:>OPM KO-
ne6aHHH no pHC. 3.5. nroTOM)' BblBO.a ypaBHeHHH J1.BffiKeHH:II BLJnOJJHHM JtJIH. nepcMelUcHHii
11M B IL1OCKOCTH (pHC. 4.2), .!I.OnyCKalOI.l.lHX ncpByIO H BTOPYIO cJ.:>OPMbl KOJIe6aHHH MaHTHHKa
(11!IacTHH:bI 11M).
npe.lUI OnaraeTCg . liTO llM (TOIJKa C) He.ne4JopMHpyeMoH ITJ1aCTMlibI (HM) pacnoJIaraeT-
C3I B ee reoMe1pUQecKoM ueJfI]>e Ha pacCTORRl111 a OT ee KpaeB. HM CBR3aHa C KopnYCOM aK-
cenepoNerpa m6KHM 3J1eNeHTOM 6anOQHOrO THna A-1HHOH AB = J.
}1wI onpe}leJJeHIDI TeK)'wero nOJJOiICeHH:II HM BBe,l\eHbl CJ1e):tYIOIllHe CI1CTeMbI KOOp)J,H-
ua:r: OXY - aenOJIBK"AOiWl CIiCTeMa; OXQ}'o - CHCTCMa KOOp.l1HHaT. CB:II3aHHWI C OCHOBaHl1eM,
Ha KOTOpoM yCTaHOBJIeH MA; O\X"yll - TO ilCe. C KopnyCOM MA (TOtUm B H 0 1 COBna):taIOT).
YPABHEHIUI L\B DKEHHJI lJ3 MHKPOAKCEJIEPOMETPOB
221
musinp
J'
Xo
x
....,J
YB
Xo
I
.
x
x
PIIC. 4.2. Ie BblBony ypaBHeHuii llBlIm:eHUH q3 laHTHlIROBOm MA
[IOJIOiKeHHe CIICTeMbl (kQVO OTHOCIITeJIbHO OCII OX onpe,QeJIeHO YfJIOM y, KOTOpbIH MO-
iKeT npOII3B0J1bHO M3MeHHTbCSI BO BpeMeHH. OeHoBaHIIe, a BMeCTe C HI1M I1 CIICTeMa KOOP)J,Jf-
HaT O\"QVO MOIyr nepeMemaTbCH C YCKOpeHJleM 1I, HanpaB!leHl1e Be""TOpa KOToporo 3aBHCIIT OT
yrna p. KpOMe Toro, OCHOBaHHe MOjf(eT COBeplIIaTb "KOCYlO" BlI6pal!HlO, 3a)J,aHH}'ID Bllopone-
peMemeHmlMll X n I1 YB' Ha'laJIo CIICTeMbI Koop,QHHaT O\XkYk B)J,OJIb OCH 00 onpeJ1.eJIeHO KaK
00 1 = L, a ee nOnOiKeHlle OTHOCllTeJIbHO CllCTeMbI OXQVO - npOH3BOJIbHbIM, HO HKCHpOBaH-
HbIM B npe,Qe.nax 0.. .360° YfJIOM Yo. MrHoBeHHOc nOJIOiKeHlle 1 (TO'lKa C) onpeJ1.eJIeHO KO-
op,QIIHaTa m x II Y B CIICTeMe oceH OXY. TIonoiKeHl1e ocn, npoxo!UlmeH 'lepe3 TO'IKH A M C B
CllCTeMe KOOp!lIIHaT OIX Yk. oITpeneJIeHO YTJlOM , a nonO)!\eHne TOl{KII A - KoopAHHaToH y"
ITapaJIJleJIbHOH OCll OU'k. YroJI Men<.rIY OCblO 0 1 _,-", H KaCaTeJIbHOH K 1130rnyrOH OCH ynpyroro
3JIeMeH"ra B ToqKe B B CHJIY em MaJIOCTn: a yrl/. B TO'lKe C ITpHJ10jf(eHbI BHClliHHC CIDThl,
06YCJIOBJIeHHble YCKopemlcM II II g, a TaIOKe B1l6poycKopeHHeM.
YpaBHeHHH. J1.BIIiKCHHSI ITOnyqllM BTOpbIM MeTOJJ;OM JIarpaH:>Ka, HMeH B B , 'ITO 0606-
meHHble Koop)J,IIHaTbI q. = ,q2 = Yr'
KJlHeTH'lCCKWI 3HeprHH. 11M 111 aKCeJICpOMeTpa
T I J n2 111 ( .2 .2 )
=2" c'i:1 +"2 x + Y ,
(4.13)
r,!{e J c - fJIaBHbIH l!CHTpaJIbHbIH MOMCHT HHepl.(HH 11M BOKpyr OCII, nCpnCH.LIHK)'IDIpHOH K
nnOCKOCrn XY.
3anHlliCM BbIpIDKCHHH. ,11)151 KOOpAI1HaT TO'IKH C:
X=/cos(a +Yo +y) +acos( +yo +y) -Lsiny+x B COSY- YB Siny; }
y = Isin(a +Yo +y) +asin( +yo +Y) +Lcosy+xBsiny+ Y B cosy.
(4.14)
222
rmrna 4. J],HHA11JiKA tI3 MHKPOMEXAHMlJECKMX rIPHEOPOB
Y'IHThIBID!, '!:TO yrJIbI a 11 MaJlbl (sin :::: , sina:::: a. cosa = cos ::::; 1) 11 npH y:l:- 0 HMe-
lOT MecTO HcpascHCTBa a «y, «Y, 113 CI1CTeMbI (4.14) nOJI)"IItM CJIC.n.YJOIlJ,IiC paBcHcTBa:
Yo=oo
x = -(L + a + Yr)sin Y +(/ +a)cosy + x" cos)' -)'B sin )'; }
y = (I + a) sin )' + (L + a + y,)cosy + x B sin y +,vB cosy:
yo = 90°
x = -(L +1 +a)sin y -(.1', + a )cosy + x B cosy -)'B sin y; }
Y = (L + 1 + a)cosy -(.1', +a )sin y + X B sin y + y" cosy;
(4.15)
yo = 180 0
x=(-L +a + y,)siny -(I +a)cosy+x B cosy- YB sin y ; }
y =-(1 + a) sin y +(L - a - y,)cosy +x B sin y + YB cOS)';
Yo = 270°
x = (-L + 1 + a) sin y + (y, +a )cos)' + x B cosy - Y B sin y; }
y = (L -I-a) cosy + (}', + a )sin y + X B sin y + .vB cOS)'.
B KaqeCTBC npMMepa nOny<illM ypasHCHHH. }1.BH:JKCHUH. J]JIJf cny4ID! Yo = 0°. I1MCSI B BllJ1.Y
paseHCTBa (4.15) (Yo = 0°), BbItJMCJIHM X. Y II B COOTBeTCTBHH C BbIpa:>KCHHCM (4.13) 3anll111eM
T = Jc 2 + m { ((I +a+xB + a + jlr + .YBf +[(L +a + Y r + yJY-XBF}.
2 2
HaitrleM npoH3BO.1lHbIC OT nOJI)"IcHHoro BbIpa:>KCHlUr:
iJT = ma [ (L+a + Yr + YBXif -Xai]; a = Jc +ma[ +a+xB +a + Y, + .YB];
o
d O ::Yr .. [( )y .. ]
- =Jc +ma I +a+x B .. +xBi+a + Yr + YB ;
dt03
T =m[(L+a +Yr+YB}(yr-XBY]; =m[ +a+xa)y+a +Yr+.YB];
QVr V/'r
doT r, / \: .. 0" .. ]
--:::;:-=mL +a+xe/T +xej'+aQ'+ Yr + Y B .
dt QVr
B COOTBeTCTBHH C ypaBHeHHeM (4.1) nonY'lI1M ypaBHeHlUI JtBl1iKCHI1R q3 MA ,l\JUI CJIY-
'IWlYo = 0°:
c +ma 2 )9 + ma.Yr -ma(yr +a }(Yf =
=-ma{L+a+xeh+ma(L+ Ye)(-rf -mays -2amx"y+Qs;
(4.16)
m{Yr +aa-(yr +as)(yf]=-m(t +a+xs}y+m + Ys)(-rf -my" -2mx s y+Q),.
r,ne Qs, Qy- 0606meHHbIe CHJIbI no COOTBeTCTBYlOlllHM Koop.nHHaTaM.
YPABHEHlliI D:BI1)KEHlliI q3 MI1KPOAKCEJIEPOMETPOB
223
IlocTymm aHaJIOrH'lHbJM 06Pa30M, ITOJIY1JHM ypasHCHlliI BH:>KCHIDI LI3 MA .l1.iLlI cne-
JJ.YIOll.{HX cnY'laeB:
Yo = 90°
(Jc +m0 2 )&+ majir -ma(Yr +a9)(1')2 =
= -ma( L + I + a + YI,)? - amr: O (yi + am o - 2mayo y + Qs;
m[ii r + a& - (Yr + a9)('y)2 ]=
=-m(L +1 +a+ Yo)? - mx o (y)2 + m o -2myoy +Qy;
(4.17)
Yo = I 80°
(Jc +ma 2 )&+ma)ir -ma(Yr +a9)(if =
=-ma(1 +a -xB)y-ma(L+ Yo)('yf +amyo +2amx B y+Qs;
m&i r +aB-(Yr -t-a9)(y)2]=
= -m(l +a-x B )? -m(L + Yo)(y)2 +mYB + 2mXoy+Qy;
(4.18)
Yo = 270°
(J +ma 2 )&+maYr -ma(Yr +a9)(y)2 =
= ma(L -1- a - Y B )? - amx o (y)2 + ami o + 2maYo Y + Qs;
m r +a&-(Yr +a9)(y)2]=
= m(L-I-a- Yo>y+mx o (y)2 -m. o + 2mYBY+Qy.
(4.19)
0606ll.{CHHb1C CHJIbI Qs, Qy B ypaBHeHI1Hx (4.16)-{ 4.\ 9) BKJIIO'laIOT B CC6H CHJIbI, 06y-
CJIOBJIeHHble iKeCTKOCTbIO ynpyrOH 6aJIKH nO Beca, YCKOPCHHCM g H It, eMmlmpoBaHHcM.
0606ll.{eHHbIC CI1JIbJ, 06YCJIOBnCHHbIe iKecTKOCTbIO nOJ1.Beca, BbNHClliIIOTCSI MaTpM'lHOH
cjJOpMOH (3.11), B KOTOpOH K03cjJclm HeHTbI iKeCTKOCTI1 oITpCJ1.eJlHIOTC.II ITO clJopMymlM (3.10).
0606ll.{eHHbIe CHJIbJ, CBSl3aHHblC cMnclJllpOBaHHcM, HaxO,l(.llTCH aHaJIOfH1JHO (4.6):
FJJ.
MJJ.
kJ1Y k!1J./a
,
k J1y a kJJ.s
(4.20)
mC kJJ.v , k{J.s - K03 m. lleHTbJ .ncMn HpoBaHlliI, COOTBCTCTBYIOll.{HC JIHHCHHOH H YTJ1OBOH KO-
op)J,I1HaTaM; a - reoMeTpH'IeCKllH napaMeTp (CM. pl1c. 4.2).
0606ll.{eHHbJC Cl1JIbI, 06YCJ10BJlCHHbJC YCKopcHHeM U H g, B COOTBCTCTBHU C pHC. 4.2 .l1.iLlI
0606ll.{eHHbIX KOOp llHaT Yr H 9 OnpCJ1.ellil!OTCH no opMYJlaM
o = m {- gcosho +(9+y)]+ usin(j3-(9+y + YO)]; }
M = o(l+a).
(4.21)
TaKI1M 06Pa30M, 06061lJ,CHHbJC CHJIbI Qs M Qy onpeJ1.eJIH!OTcSI CYMMHpoBaHHcM COOTBeT-
CTBYIOll.{HX Bblpa:>KCHHH (3.11), (4.20), (4.21). 3aMeTliM, 'lTO 06061l1.CHHblC CHJILl, 06ycn0BJ1eH-
HbJC iKecTKOCTbIO nO,l\Beca H J1.CMn HpOBaHHeM, HC 3aBI1CHT OT yrJIa Yo YCTaHOBKH KOp-
nyca MA.
224
rnaBa 4. ,l{HHAMUKA 1I3 MI1KPOMEXAHUlIECKHX rIPI160POB
C )"leToM (4.16}-{4.19) H (3.11), (4.20). (4.21) 3anllweM ypaBHeHHSI ,llBlDKeHIISI l13 MA
)J,J1H. 'leTblpCx BapHaHTOB YCTaHOBKIi ero Kopnyca Ha 06"bel(Te B CJIenYlOIlJ,eM BI1)J,e:
mb'r + a&-(Yr + aB)(y)2]+ ll(k ll Yr + kJ:!B) + k.'t,.j',. + k;J.9 B = F + Fro:
a
(Jc+ma2 ) &+maYr-mab'r+aB )( y ) 2+1l ( k I I' +k ?B ) +k .ai' +k 9 9=
__r ..._ ;1.1." .J
= Fa+ Fyo(1 +a),
(4.22)
rl1e Il - 'lHCIIO 6fu"10K; F Ii FfJ BbltmCIHllOTC}! no 3aBHCI1MOCT}!M
Yo = 0 0
F =-m(1 + a +xBh+(L + J'B)(rf mYB -2mi B y:
FrO = m[- gcos(B+y)+usin( - B-Y)];
Yo = 90"
F=-m(L +1 +a + YB)y-mx B hr +nt B -2mYB i';
Fro =m[- gsin(B+y)-usio( -B-y)]:
Yo = 180"
F =-111(1 +a -xB)y-m(L + YB)(i'f +mY B +2mi B i';
(4.23)
Fro = m[gcos(B +y )-usin( - B -y)];
Yo = 270 0
F =m(L -1-a- YBh+mx B Ci'f -mx B +2mY B i' ;
Fro =m[- gsin(B+y)+ucos( - B-y)].
YPaBHCHH» ABH)f(CHIDI (4.22) n03BOJIHlOT npoaHaJJH3HpOBaTb JtHHaMHK)' q3 MIDITHHKOBOro
MA)J.IDI pa3.'IH1lHbIX BapHaHTOB B03Myn1CHHH.
YpaBHeHIDI .z:\BM)f(CHIDI q3 MA cnpasCJtJ1HBbl H )J,JISI aKcenepoMCTpoB B MaKpOHCnOJIHe-
Him.
OueHHM BJIIDIHMe YCTaHOBKH Kopnyca MA Ha 'laCTOTbl C06CTBCHHbIX, HCJ1.CMn HpoBaH-
HblX Kone6aHHH MaHTHHKOBOH 11M (MaHTHHKa).
)1.m:I yrnoB yo = 0 M 180 0 YCTaHoBKH Kopnyca ornOCHTCJIbHO HCnO,l\BHiKHOro OCHOBaHHSI
ypaBHeHHe 'iacTOT (3. I 9) BbIrIDI,llI1T CJJe.z:\YIOIl1.HM 06pa30M:
4 ( ) 2 2 2
Jcmp -n JAk ll +mk 22 -2mak l2 'P +n (k ll k 22 -k I2 )=0.
(4.24)
. 080 HMeeT peweHHe BM,M (3.20) C K031}>I}IHllHeHTaMH
a=Jcm;
b=JAk ll +mk 22 -2mak I2 ;
c=k ll k I2 -k'A.
(4.25)
YPABHEHl1JI JJ:BI;JJKEHIDI 43 MI1KPOAKCEJIEPOMETPOB
225
,AnR ymoB Yo = 90 Ii 270 0 YCTaHOBKH Koprryca OTHOCMTeJIbHO HCITO)J,BIDKHOrO OCHOBaHHH.
ypaBHcHIie '-IaCTOT 3aI1llCblBaeTCR TaK:
Jcn1p4 - [n(J A k l1 + mk 22 - 2mak 12 )+ m 2 gl]p2 + n{n(k ll k 12 - kl ) + mg(kJl (I + a)-k I2 J}= 0,
(4.26)
r.ll,e 3HaK "_" nepe.rr <InCHaMll, CO[(Cpn(8lUI1MI1 BCJIIi'-lliHY g, cooTBeTCTByeT ymy Yo = 90°, a
31JaK "+" - yrJIY ycraHoBKH Yo = 270°.
PerneHllC ypaBHeHW[ (4.26) COOTBeTCTBYeT ypaBHeHIilO (3.20), B KOTOpOM
a = J m'
c ,
b = J AkJl + mk 22 + 2mak 12 + m 2 gl;
(4.27)
c == k l [k 22 -k I 2 2 + mg(k I1 (1 +a)-k 1Z l
3aMeTIiM, '-ITa B BbIpa)Kemm illUI K03 HlmeHTa b '-IJICH m 2 g1 Ha HecKOJIbKO nOpSl,llKOB
MeHbllle OCTaJIbHbiX II cro MO:>KHO He yqllTbIBaTb npll paC'-IeTax.
IlpuMep 4.3
Pacc'unaeM 'IaCTOTbI co6cTBeHHbIx, He.QeMmlmpoBaHHLIx KOJIe6aHIDi MaJlTHlIKa no npHMepy 3.2 C
napaMeTpar.m: m = 0,29 . 10- 3 Kf: J c == 1,775 . 10- 9 Kf'M 2 ; J A == 7,093 . 10-9 Kf'M 2 ; kJ] == 2,036 . 103 HIM,
k 21 == k l2 = -0,819 H, k zz = 4,398 . 10-4 H'M 2 ; a == 2,28 . 10- 3 M, I + a == 5 . 10- 3 M.
K03!jJ!jJI1I.\lIeHTbI no 3aBHClIMOCTHM (4.25) paBHL!:
a == 0 514 . 10- 12 Kr'M 2 . b == 16 601 - 10-6 Kr.rl/c 2 . c == 0 224 r<? 'M 2 /c 4 .
, " " ,
KOJ!jJ!jJHJ lIeHTbI a Ii b no 3aBlICHMocniM (4.27) llMelOT Te )Ke 3fIa'IeHIDl, a K03<P4>HIU1eHT
c = (0,224 + 0,031) r<?rl/c 4 . PeJynbTIlTLI paC'IerOB no !jJopMyne (3.20) npHBe eHbl BTa6miI.\e.
4:aCTOTa, KOJI-BO Yron YCTaHoBKH Kopnyca aKcenepoMeTpa 'Yo. 0
lIc (fl.\) ynpymx
WOK o lII80 90 0 270
1 116,81 (18,6) 112,56 (] 7.92) 124,87 (19,88)
PI
3 202,32 (32,22) 194,96 (31,04) 216,28 (34,44)
I 5687.0 (905,57) 5687,52 (905,65) 5687,26 (905,62)
P2
3 9850,17 (1568,49) 985],07 (1568,64) 9850,62 (1568,57)
113 nonY'ICHHbIX peJyJ1LTIlTOB CJIe.Qyer, 'ITa nepBaJI (HH3IllaJI) 'laCTOTIi c06CTBeHHbIX KOJIe6RHHH
MaJlTHHKa 3aBHCJIT OT era OpHeHTIll.\HIi OTHOCHTeJILHO OCHOBRHIDl. I1pH 3TOM HaH60JIbillee 3H8'leHHe
'laCTOTbI COOTBeTCTByer yrny 'Yo = 270 0 ("npHMoii" MaJlTHI:IK), a HaHMeHLIllee 3H8'leHHe - yrny Yo = 90 0
("nepeBepHyrblH" MWlTHHK).
226
fJIaBa 4. ,lJ,I1HAMI1KA l:l3 MHKPOMEXAHM4ECKl1X IlPI1EOPOB
4.2. ITEPE.lJ.AT01J.HbIE Cl>YHKU.1llI ':1.3 MnKPOAKCEJIEPOMETPOB
4.2,1. OcesoH l\UlKpOaKce.llepol\leTp
EY.!lcM nonaran, 'lTO >KCCTKOCTb no,nBeca oceBoro MA (CM. pnc. 4.1, a) ITO Koop.nIlHaTaM
x, Z 3Ha'llfTeJIbHO 60JIbllle, lJ.eM )KeCTKOCTb no KOOpJUlHaTaM )', 0., p. J],JISI :noro Cny'laSl OCb Y SlBJISI-
eTCH OCbIO 'I)'BCTBHTeJlbHOCTU. nCpemIlllCM ypaBHeHHR (4.9), (4.10):
mj;+k::!"j' +GyY +m(lxa + l=i3) = F - FBI';
Jj)P + k;$p + GDP + m(,li; + 1 xa) = -(F + Fey)l= - FZe + F 8 i=P;
Jaa +k:laCx. + Gao.+ mlx(i;+ lzp) = (F - Fe,,)l,. + F"\:e + Fa.J,.o.;
(4.28)
Yc = Y + lxo. + l=P + )'8'
(4.29)
r,ncm{ay-g) =F, m-,\, = Fe-,"' OOe =FBJ:' mZe = Fe::.
Ha pHC. 4.3 B COOTBeTCTBHH C ypaBHeHIDIMH (4.28) npUBe,neHa CTPYKTypHaJI cxeMa
(F..... = 0) q3, H!l..'IIOcrpHp)'lOlJ.{aSI CJIO:>KHbJH xapaKTCp nepCl\.-pCCTHbJX CBSl3eH, 06YCJIOBnCHHbIX
CMemeHHCM l.I.M q3 OTHOCHTeJlbHO reOMeTpHlJ.CCKOrO UCHTpa no,nBcca. EcnH B CHCTCMC
(4.28) npeHe6pcQb ITepCKpeCTHbIMll CB}l3J1MH, TO OHa pacna)J,eTCR Ha Tpl1 HC3aBHCllMbIX ypaa-
BeHID! C nepeMCHHblMll y, 0., P, KOTOpblM COOTBeTCTBYIOT KaHaJIbI npOXO:>K)J,CHlliI H3McpRCMoro
YCKOpeHIDI, nOKa3aHH"bIe Ha pHC. 4.3 :>Kl1pHbIMH JIHHIDIMH [22].
EY.!leM nOJIaraTb, 'iTO napaMCTpbl q3 H JIllHCHHOH BH6paUHI1 TaKOBbI, 'lTO B J1.BYX ITO-
CJIeJI:HHX YPaBHCHIUIX CHCTeMbJ (4.28) B npaBbIX lJ.aCTSIX ncpBblc COCTaBJISllOll{He, no Kpai-1HeH
Mepe, Ha ,llB8 nopAAKa 60JIbillC OCTaJIbHbIX, KOTOpbJC Mbl HCKJIIO'lUM 113 ,lJ,aJIbHCHlliCro pac-
CMOTpeHIDI.
.{. s2+ k;Ja:HGa
u
I
mil+kg.yS+G y
Yc
pes)
Pac. 4.3. CrpYKIYPa8J1 cxeM8 q3 ocesoro MHKpo8KceJJepOMeTp8
IlEPE/J.A TO HbIE <DYHKQI1I1 43 MYIKPOAKCEJIEPOMETPOB
227
3amiWCM cHCTeM}' (4.28) B oncpaTopHoH <popMe ( = S) :
ns2 +ktys + G y )y(s )+ml z s 2 13(s)+ mlxs20.(s) = F - Fay;
ml z s 2 y(s)+ Jps2 + kAps + Gp)p(s)+ ml x l z s 2 0.(s) = -(F + Fey)lz;
mlxs2y(s)+ ml x l z s 2 13(s) + (Ju.S2 + kJJ.u.s + Gu.)o.(s)= (F - FBJ1x;
(4.30)
TIpC,QCTaBHM nepC,l\aT01.JHYIO YHl(ltHIO q3 ITO OTHOIlICHMJO K CHJIC F:
ffj.c(s) = Yc(s) IF(s),
rn.e Yc(s) - ypaBHcHHC (4.29) B onepaTOpHOH <popMe,
Yc(S) = yes) ..L lxo.(s) + lzl3(s) ,
(4.31)
(4.32)
rn.c
y(s) = l1y , l3(s) = 1113 , o.(s)= 11u. .
11 11 11
(4.33)
B BbIpa)!{eHHJ!X (4.33) oITpen.eJJMTeJIMI1. l1y, 11 13 , 11u. nony'lalOTCR H3 CHCTeMbI (4.30) B co-
OTBCTCTBHI1 C npaBllnOM KpaMepa.
IlO1I.CTaBI1M paBCHCTBa (4.33) B <popMyny (4.32) H nOJ1YQHM
Yc(s) = (I1). + 'x l1 u. +lzl1p)= W}'C(s)F(s) ,
11
OTKY,l\a
( ) _ l1y + 'xl1u. + I z l1 p
W1'C s - .
- M'
(4.34)
3aITMllICM BbIpaiKCHllR ,llJIH onpc,QCJII1TeJICH:
., ml z s 2 I 2
ms- + kJJJ,s + G y m xS
ml z s 2 ., ml)zs2
11= Jps- + kAfJs + G p .
,
ml x s 2 ml,Jzs2 Ju. s2 + kJJ.rJ.s + Gu.
F
ml z s 2 mlxs2
J p s 2 +kAPs+G p m Jzs2 ; 1113 =
m1xZzs2 Ju. s2 +kAas+Gu.
ms 2 +kllJ,s+G y
ml z s 2
mlxs2
F ml s2
X
- Flz ml x l z s 2
FIx Jo.s 2 +kJJ.a s + Gu.
.
,
l1y = -Flz
Fix
ms 2 +k]JJ.s+G y mlxzs2 F
11 - ml z s 2 J p s 2 +kJJ.ps+G p -Fl
a- z
ml x S 2 tnl x l z s 2 Fix
PaCKpoeM onpCJ1.enHTenll 11 nOJIylJ.HM
65432
l1=a68 +a5s +a 4 s +a3s +a2s +a\s+ao,
(4.35)
228
rmma 4. ,lUlliAMI1KA q3 MI1KPOMEXAHWlECKl1X IlPI1EOPOB
r.ne
a6 =mJZJ X ;
05 = m(k;JaJx +kAJIJJ+ K.o1\.(JaJ -m 2 (;1;);
a4 '= m(GaJ x + G Jz + kAJIkna)+Gy(JaJ - m 2 1;1; )+ki1.,.(J kna +Jnki! );
a3 = G a (mk;X13 +k;J.,. J I3)+G p (mk;J,a +ki1.,.Ja)+G,.(J kna +J n ki.\p)+k;J..,. k ;J.l3 k ila;
a2 =GAJ 13 G a +JaGI3+kn(J.knP}+k.1JJ.(kJlJ3Go. +k!lo. G a)+mGJ3G a ;
aI :=: k!J,\.G o. GI3 + Gy(k;{IPa + k;J.a G I3); a o = GyGr,ca;
I 4 3 2 )
/),\. =F\b 4 S +b 3 s +b 2 s +b(s+b O ,
(4.36)
r.ne
b 4 = Jz{Jp + m/;);
hJ = k;J.j!J;: + (JJ3 + ml; )k na ;
b 2 = Gr. Jz + G a {JJ3 + mi; )+knakAJI;
hi = k;J.a G J3 + kJlJ3 Go.;
ho = G a Gj3 ;
(4.37)
/),13 = Flz(e 4 s4 + e3 s3 + e2 s2 + els + eo) ,
r.ne
e4 =2m 2 i;; e3 =-2mk Jla -k.1JJ,(Ja +ml;);
e2 =-2mG a -Gy(Ja +mi;)-kJJyk na ;
eI = -k.1JJ,G o. -Gyk na ;
eo = -GyG a .
/),a = F1A c 3 s3 + C2 s2 + cIs + co) ,
(4.38)
r.ne
C3 = k]J"V<JJ3 +ml;); c 2 = Gy(Jr. + mi;) +knykAJI;
c, =k ny GJ3 + GykAJI; Co = G y GI3'
3ammJeM BbIpaJKeHHe ,WUI nepe.naTo'moH YHKwm:
4 3 2
( ) n4s +n3s +n2 s +n,s+no
Wyc s 6 S 4 3 2 '
06s +oss +a4s +a3 s +a2 s +als+ao
r.ne
n4 = Jr.J +m/;(Ja +ml;);
n3 = k1S.aJx +kJJfJJ + knyl;(J13 +m/;) -kny/;(Ja + mi;);
'= Gy(Jf>/; -J a l;) +kny(kJJfJi; -k na l;) + GJ3Jz + GaJ x + knakJlj!;
II) ,=kJl(lq +kJJfJG a + (kl!)'Gr. + GykJJfJ) /; -(k!JyG a + Gyk na ) 1;;
110 = G y( GJ31; - Gal;) + Gr,Ga.
rIEPE,lJ,A TOllHbIE C1JYHKI.U111 113 MI1KPOAKCEJIEPOMETPOB
229
}1h nCpC.D,aTO'lHOH YHKl(l1l1 'C(s) npM S = 0 cnC)J,yeT BbIpmKCHHC )J,JIH '1}'BCTBMTCJIbHO-
cm K JIMHeHHOMY YCKOPCHMIO:
Y c
ay-g
[G . (GjJ/; - Gal;)+ GjJGa]m
GvG p G a
(4.39)
O'leBll,IJ:HO, 'lysCTBllTenbHOCTb no OTHOlliCHMIO K B1l6poYCKOPCHHIO YB TaK)f(C MO:>KCT
6bITb paCC<JHTaHa no BbIpa:>KCHHIO (4.39), H3 KOTOporo CJle.nyeT, 'lTO ccnH BbInOJIHHIOTCH pa-
BeHCTBa G a = GfJ, Ix = I z , TO yrJIOBblC )f(CCTKOCTH Ha <JysCTBI1TeJIbHOCTb K JIHHCHHbJM YCKOPC-
HHSlM HC BnmIIOT.
y CTOH1.JHBOCTb ') MO)f(eT 6bITb HCC.1CnOBaHa C nOMOlUblO KpllTepIDI rypBHlla. J\:ID! xa-
paKTepllCTI1'lCCKOro ypaBHCHllSi /). = 0 [CM. Bblpa:>KeHHe (4.35)] COCTaBMM KBa.n.paTH}'1O MaTpH-
uy (Ta6nHUY) K03 <pMUHCHTOB:
a5 a3 al 0 0 0
a6 a4 a 2 aD 0 0
0 a5 a3 a. 0 0
OnpeneneHHC rypBHua
0 a6 a 4 a 2 aD 0
0 0 a5 a3 a. 0
0 0 a6 a4 a2 aD
KpHTCpllH YCTOH<mBOCTH CBOJ1.11TCH K TOMY, 'lTO npH a6 > 0 .D,OJI:>KHbI 6bITb 60JIblIIC HYJ1X
BCC lllCCTb onpCJ1.eJIllTeJICH rypBll11a:
/)., > 0 (i = 1...5), /).6 = aD . /).5 > O.
TIonY<ICHMC KOHKpCTHblX BbIpIDKCHHH Ltml COOTHOIIICHMH K03 HUHeHToB, xapaKTepll-
3YIOIllHX YCTOHlJ.HBOCTb, HC npC.D,CTaBJIHeT TPYJ1.HOCTCH.
npw,Jep 4.4
Bbl'lHCnHM 1JYBCTBHTenbHoCTb H CTIlTlIQCCKHe CMCmeHJiJI 11M MA npH a}. = 0, a)' = 109 ,lIID! napa-
MerpOB 113 no npHMcpy 3.1: m n = m = 7,5 . IO- s Kf, Gu. = GfJ = 47,72 . 10- 5 H . M, GJ' = 525 HIM. IlpH-
HHMaeM /x = 5. 10- 5 M, /z= 25. 10- 5 M. IlpH Gu. = G p HMeCM
2 2 7,5.1O- S { 525 [ (5.IO- S [ - '1O-5f J +47,72'1O-5 }
Yc m[Gy(lx -/z )+G a l _
= := -5
a-g GyG a 525.47,72.10
= 7,5.10- 5 (0,11+47,72) M/(Mlc 2 ).
525.47,72
neHHii, npaKTH1JeCKH He BJlMJUOT Ha pC3ynbTaT Bbl1JHCnCHHH, T.e.
OTCIO)J,a BHJ1HO, <!TO CMemeHJiJI l.J;M n1U1 npHJiJlThIX 3HaQeHHH, HMCJI B BHJ1Y norpemHOCTH BbNHC-
Yc m 7,5,10- 5
--=
a-g G y 525
= 0,14 MI(Mlc 2 ).
IlpH a y = 0 CMemeHlle Yc = 1,37.10-6 M; npH o}' = 109 CMemeHlle Yc = 12,35 . IO-6 M .
230
rnaaa 4. ,lJ.HHAMHKA 3 MI1KPOMEXAHI1 ECKI1X IlPHEOPOB
3aMCTUM, 'ITO 'I)'BcTBHTeJ1bHOCTb MA K CMcmCHHHM UM MO)KHO OCJIa6UTb n. BH1U1 1O.
npaKTI1'lecKM YCTpaHl1Tb, eCJ1H B ee rCOMCTpU'leCKOM UCHTpe pa3MccTHTb ,nonOJIHlneJILHYIO
Maccy, npcBbIlllaIOUJyIO Maccy Ca!\WH nJIaCTHHbl. O<ICBUlJ,HO, <ITO npn 3TOM YBeJ1H<IHBaeTc}J 11
'I)'BCTBUTeJIbHOCTh aKCeJ1CpOMCTpa.
TIpHMCHMM OPMY 3anHCH (d I df = s), BBClJ,eM 0603Ha<IeHUH
m(a,. - g - J;B}= F,,, - m Bla,. - g}= Al x'
- mxB(a y g)= M z .
H nOJ1y<IHM}J.JlSI Ka)KJ1.0rO H3 Tpcx ypaBHcHHH (4.28) (l,,= 1== 0)
K
Wq(s) = ,") 9
Tg-s- + 2c,qT g s + 1
( 4.40)
rlle Tq - nOCTOHHHaJI BpCMCHH q3; c,q - OTHOCllTeJ1bHbli1 K03 uulieHT lJ,CMncjmpoBaHHblx
II3; CDgo = 11 Tg - C06CTBCHHaJI qaCTOTa HClJ,CMn HpOBaHHbIX KOJIe6aHHi1 43).
,lI,.'VI Ka)KJ1.0H 0606mcHHoH KOOpJ1.HHaTbI HMeeM CJ1CJJ.YIOIlJ,I1C BbIpa)KCHI1H:
y(S) /" /" 1 N; I klJ,\' kIJ,V
W,.(s)=-; Kq = K,. =-. Ty = -, CD,-£) =-; c,\. = C:;::: - .
- F".,. . G y G y - T." - 2V mG ) 2mw y o
Jt (S)= a(s) ; Kg=Ka= .Ta= JZ .Wao= ; /;a= ./lla = kJJ.a .
M= G a G a Ta 2 J=G a 2J z w a o
. P(s) I IE 1 . k llP k.np
"'13(s) = Mx ; Kg = Kp = G p , Tp =:: V G p , (J}po = Tp · /;p = 2 JxGp = 2J x w/JO .
4.2.2. MaHTIlHKOBLIH MHKpoaKceJJepoMeTp
TIOJIyqMM nepC,llaTO'iHbIC I}>YHKUHH II3, HMCSI: B Bl1,lly, qTO Hal160JIblliCH 'lYBCTBllTCJIbHO-
CTbIO K H3MeHCHHIO yrJIa y 06JIa,llaeT ncpeBepHYTbIH MWlTHHK, a np})t..IOH - HCCKonbKO MCHb-
meiL I103TOMY H3MepeHHH yrJIOBbIX KOJIe6aHHH (yrnoBoH Bu6paUlil1) OCHOBaHHSI JIY<IlllC Bbl-
nOJIIDITb npH ycraHoBKe MWlTHMKa, xapaKTep1l3yeMoH yrnaMH 'Yo = 90 H 270°.
K J1HHeHHOH BM6pauHH MaJlTHHK qysCTBHTeJIeH ITPH mo60M BapllaHTC yCTaHOBKH OTHO-
CHTeJJ1>HO OCHOB8HIDI. XOTSI KOJIH'leCTBeHHaSI Mepa I.JYBCTBHTenbHOCTH pa3HaJI.
,lL'UI cnyqaH JIMHeHHOH BH6pallHH nono:>KHM B yPaBHCHH})X (4.22), (4.23) Y = 0, a JJ.J1S1: yr-
JIOBbIX KOJIe6amdf Xs = Y.. = 0, a T810Ke (yr +a&)y2 :::: O. BBe,!lCM 0603HaQCHI1S1: J A = J c + ma 2 ,
x B :;: ax(t), YB :;: Oy(t), y = T(f), L, = £ - / - a, £2 = L + / + a H ypaBHcHJ-lSI (4.22) npCJ1.CTa-
BHM B M8TpJ1qHOH .popMe JJ,.JUI npSl:MOrO H nepcBcpHyroro MaSlTHHKOB (n = 1):
m
ma
.. k
+ l\Y
& kl\Yo
k]J.s / 1 Yr k ll
.+
k1J..S & k 2J
k J2 + mg Yr
k 22 + mag &
F
(4.41)
ma
J A
,
F.a
nEPEflA roLlHblE <DYHKlU1M lJ:3 MI1KPOAKCEJIEPOMETPOB
231
me
F + ma (t)
-
F.a + 111aaAt)
(J1I1HCHHaJI BM6pa1U1S1) ;
(4.42)
+ JIlL, }
r(S)
F -mL
_ 2
F.a - +maL 1 }
res)
- maL 2
(yrnoBble Kone6aHI1S1) .
(4.43)
B ypaBHeHmlx (4.41) 11 Bblpa)l{eHMSIX (4.42) BepXHlle 3HaKl1 COOTBeTCTBYIOT npHMoMY
CYo= 270°), a HMiKHI1C - ncpcBepHYToMY (Yo = 90°) MWlTHl1KaM.
B BblpaiKeHMSIx (4.43) npSlMOMY Ma}lTHMKY OTBe'iaIOT Ko!>mOHeHTbI, CO)J,epiKall.lHe napa-
MCTp L 1> a ncpcBepuYToMY napaMeTp L 2 .
YpaBHeHliSi ,iJ,BMiKeHI1S1 :\!aSlTHMKa. COOTBCTCTBYIOll1l1C BepTMKaJJbHOH BM6paUI111 OCHOBa-
Hl1S1, l1MeWT BII.11:
m ma Y r kJJ,>
.. +
ma J A 8 kl!v a
k;!.,,/a
kJJ.9
Y r k ll
. +
9 k 2I
k 12 Y r
k 22 9
+ m[a/t)+g]
-
+ ma[av(t) + g)
(4.44)
3HaK "_" B npaBblX 'lacn!x ypaBHeHI1H (4.44) COOTBeTCTByerymy 1'0= 0°, a 3HaK (+) - YfJIY
YCTaHoBKM Yo = 180 0 .
BBe)J,eM 0603Ha'leHl1e s "" Ii ITepenmncM CMcTeMY (4.41) B onepaTopHoH opMe:
dt
?
I1lS- + k!l,vs + k ll
mas 2 + kJJ..l,as + k 2I
2 k& ( )
mas + -E...- S + k 12 + mg
a
J A ). + knss + (k 22 + mag)
yAs) F(s)
9(s) - F(s)a ,
(4.45)
me Ha OCHOBaHl111 MaTpMU (4.42) Ii (4.43) I1MeeM
F(s) _ + max(s) .
- ,
F(s)a + maax(s)
+mL 1 }
f(s)
F(s) _ -mL2
F(s)a - + maL. } r(s)
- maL 2
(4.46)
113 ypaBHcHI1H (4.45) c yl.JeToM (4.46) nOJIY'll1M nCpe,llaTO'iHbIe CPYHKUHI1
( ) + LI } m(akll - k 2I )
) & S -L 2
Ws,r(s "" r(s) = l ;
+ L. } ( ? )
( ) m\Jcs- +k Z2 -akJ2
y s -L 2
W...,r(s)= s) - I '
(4.47)
Z32
rnaB84. ,lU1HAMHKA q3 MMKPOMEXAHMQECKI1X nPMEOPOB
r.n.e
6 1 = J ems 4 + J ek lJ,.1.s3 + (J Akll + mk 22 - 2mak 12 ) s2 +
+(k,!l\'k 22 + k.n klI -k::J,\.ak I2 -k I2 k s } + k ll k 12 -kf2 + mg(akl1 -k 12 ).
IIepe)],ftTOqHble CPYHKUHH. (4.47) n03BOJISIIOT onpe.neJIl1Tb peaKl.\111O 1.J3 Ha B03MyrnaWll.\ec
B03 eHCTBHe B BHlle yrnoBblx KOJIe6aHHH OCHOBaHHSI. TIepell.aTO'lHble YHKl.\HI1, OITpe;:J,eJISlW-
ll{He peaKUHlO 43 Ha JIl1HeHHbIe BH6pal.\HOHHble B03MYll1eHI1S1 a", l1MClOT Bl1)J,
( 4.48)
J¥, ( s ) = &(s) _ + m(ak ll - k 2 J .
S.D. ( ) A '
ax S 01
W ( S ) = Yr(S) = +m(J e s 2 +k 22 -ak 21 )
Y,.D. ax(s) 6 1 .
I1epenHllleM ypaBHeHIDI (4.44) B onepaTOpHoH cpopMe
(4.49)
')
ms- +k . +k ll
mas 2 +k,!ll'as +k 21
., k n
mas- + .J,,, s+k
- 12
a
J A S 2 +k.ns s + k 22
Yr(S) _ + m[aAs) + g]
9(s) + ma[a .1' (s) + g]
H noJI)"ilfM nepellaTO'IHbIe CPYHKIlHH 43, onpe)J,eJUlIOIl.\He ero peaKUllW HB BepnlKaJIbHYW
nHReHHOrO BH6pal.\HOHHOro B03 eHHSI COBMeCTHO C YCKopcHHeM CllJIbI
TIDKecTH [a,(I) + g]:
J¥, ( ) = 9(s) = + m(aklI - k 21 ) .
S D +g S ( ) ,
. ) a y S + g 6 2
W ()= Yr(S) +m(J e s 2 +k 22 -ak 12 )
>".D) +g S ay(s)+ g 6 2 '
(4.50)
rJ1.e
6 2 = 6 1 - [+ mg(ak J I - k I2 )].
(4.51)
IIepe.naTO'IHLIe CPyHKIJ:HM (4.47), (4.49), (4.50) n03BOlUUOT onpe.neJ1HTb peaKl.\MIO q3 HB
B03M)'IUWOlUHe B0311eHCTBWI B BH)],e yrnoBoro H nHHeHHoro YCKOpCHHSI.
Twc I(aK HaJf60JIbmeH If)'BCTBllTeJ1bHOCTblO no OTHorneHHlO K yrJ1Y HaKJ10Ha Y(/) OCHOBa-
HID[ 06JIa,ll3.lOT npgMoH b'o = 270°) H nepeBCpHYTblH (Yo = 90°) MaJlTHHKll, TO llMeHHO MJI HMX
H onpe.neJJHM nepe,l\aTO'lHbIe CPYHKUHH q3.
IIOJIaraJI. 'lTO CPYHKl.\.IDI y(/) B onepaTOpHOH 4JopMe HMeeT 3anHcb y(s), M npeHe6perWl,
JCa.K H npejf()(e. qneHOM (y" + a&)(i'f ' a TaK:>Ke C<JHTaH. X. = Y. = 0, cHCTeMY (4.22) 3anHrneM B
MaTpH'lHOH $opMe:
ms 2 + k11J' + k 11
m£lS2 + kwos + k 21
k
m£lS2 +...1!!.s +(k}2 :t mg)
a
J As2 + k/1ss + (kn :t mag
+ my (s ) { ( S2 - g)
Yr(s) - (Lzs2-g)
&(s) = ( ){ ( S2 - g)
:t may s
(Lzs2 - g)
(4.52)
rIEPE)J,A T01.J:HbIE YHKQ1111 1.J:3 MI1KPOAKCEJIEPOMETPOB
233
W 4
,9
W 3
a a
W 2
WI Yr Yr +ait
PHC. 4.4. CTPYKTYpHaH cxeMa "iJ3 MaHTHHKOBoro MUKpOaKCeJ1epOMeTpa
(Yo = 270 c , Yo = 90 C )
B ypaBHCHl1HX (4.52) 3HaK "+" B CI1CTeMC ,l!BOHHbIX 3HaKOB I1 COMHO:>KI1TCnH, co.nep:>Ka-
IlJ,l1e napaMeTp L], OTHOCHTCH K npHMOMY MaJlTHHKY (yo = 270°), a 3HaK ,,_It Ii COMHO:>KMTeJIH C
napaMCTpaMll L 2 - K nepcBepHYToMy MlliITHI1I<)' (yo = 90°).
113 ypaBHeHI1H (4.52) nOJIY<JHM nCpc,l\aTo<JHblc YHKUI1H 1.l3 no YTJIY y{t))1.JUI JtByx BapH-
aHTOB YCTaHOBKI1 aKCCJIepoMeTpa:
- np"-MoH MaJlTHHK (y = 270°)
W a (s) = 9(s) = m(ak ll -k 21 )(lt s2 - g) ;
,r y(s) 6 1
- ITepeBepHYTbIH MWlTHHK (y = 90°)
W s (s) = 9(s) = m(ak ll -k2I)(- s2 + g) ;
,r y(s) I
(4.53)
- npHMoH MWiTHIiK
W" r (s) = Yr(s) = Jcs4 +i [ (k22 -ak 12 )-J c g]- g(k 22 -ak I2 )};
m y(s) 6 1
- nepeBepHYTblH MaJlTHHK
W (s) = Yr(s) = Jcs4 +s2[ (ak12 -k 22 )+J c g]+ g(k 22 -ak I2 )J
y,;y y(s) 6 1
(4.54)
TIepe,l!aTO<JHble <lIYHKUHH (4.53),(4.54) n03BOJIRIOT onpe.neJ1HTb peaKUMIO q3 no KOOp-
AI1HaTaM 8- H Yr B OTBeT Ha Bxo.nHoe B03AeHcTBMe B BH.!J.e yrnoBbIx KOJIe6aHl1H OCHOBaHIDI. -
234
rJIBBB 4. AHHAMI1KA t.J.3 MHKPOMEXAHH ECKHX rIPHEOPOB
4.1. IIepenaToqRl>le CPYH ItIIll 1.13
Yo, 0 W1(s) Wl(s) JfTM) 11'4('\ )
0; 90; 1 ')
mas +k ,a+
180' .., - -
, ms- + k "s + k)l
270 +k::!1
O. ') k;J.ss 1
,
180 ')
mas- +-s +k 12 JAs- + k;J.9' + k 22
idem a idem
k s
270' mas"!. + s + ku + mg 1
,
90 a .,
JAS + k:J.s s +k 22 + mag
B 06I11.eM cJI)"!ae npM 11 "* 1 nepc.l! KIDK.i1.bIM K03 I11.\ueHTOM :>KeCTKOCTll B 3aBI1Cl1MOCTSIX
(4.47H4.S4) He06xoJ1.l1t.1O nOCTBBMTb B KalJeCTBe COMHO)KI1TeJISI HY)KHOe 'mCJIO YITpyrl1x :me-
MeHTOB n = 2, 3,.__
B COOTBeTCTBHH C ypaBHeHHSlMH (4.44), (4.45), (4.52) Ha pnc. 4.4 ITpl1Be)J,cHa CTPYKTYP-
HWI cxeMa q3 MasITHMKOBOro MA, r.ne ncpe)J,aTO'lHble YHKl.{lU1 W,{s) (i = 1...4) onpe.n eJISI -
IOTC'i B COOTBeTCTBMH C -ra6j}. 4.1, a I}>YHKUHM K03 I1UneHTa "ncpe,iJ,a'iu" tD - no Ta6JI. 4.2, B
KOTOpOH .aml BapHBHTOB yrJIOB Yo YCTaHOBKH aKCeJIepOMeTpa, He peKOMeH)J,yeMblx ,lJ)UI JiC-
nOJIl,30BaHH}!. HMeeTC}! <1> = O.
C nOMOI1l.b1O crpYKTYPHoii cxeMbl H Ta6JI. 4.1 11 4.2 MOryr 6blTb nony4eHbi nepe.n.aT04Hble
q.)'HKllHH 9{s)fB(s). y,(s)/B(s); [y,(s) + a9(s)]/B(s) ,!l..!U! JI1Q60i1. BbIXO)J,HOH Koop)J,I1HaTbl no OT-
HOlIJeHHIO K mo60MY BH.!lY B03MYlllaIQI1I.erO B B03.l!eHCTBH}!. llITSi 3Toro Heo6xo.l!I1MO, JiCnOJIb-
3YJI pHC. 4.4. 3aITHCaTb CJIeJ1.YIOI1I.He anre6pal1'ieCKl1C ypaBHemU!:
[-(BmtD-W29)U W3 + BmtDa]W 4 = 9;
(4.55)
[- (BmtDa - W 3 Yr )W 2 W 4 + Bm<1>]Jt) = Yr ;
(4.56)
[-(Bm<I>a-W 3 Yr)W 2 W 4 + BmtD] WI +a[-(Bm<1>-W29)U W3 + Bm<1>a]W 4 = Y r +a9. (4.57)
4.2. «PyHKItUH K03cpcpUItMeHTa 1)3
<Dym<f..lI1j! "nepeM'm" <D
B03M)'WeHHe Yo = 270 0
1'0=0 "(0 = 90° Yo= 180 0
xB(t) 0 +1 0 -1
YB(t) + g -I 0 +J 0
yet) -£2 +L,
0 0
yet) -L 2 i + g Lli-g
11 EPE.D:A T04H hIE <1JYHKltI111 43 MI1KPOAKCEJlEPOMETPOB
235
H3 ypaBHeHl'IH (4.55) II (4.56) nonyqHM
&(s) m<D(a - U f )11"4
B(s) = 1- U 11"211"3W4 ;
(4.58)
Yr(S) _ mcD(l-aW 2 11" 4 )W;
B(s) - 1 - Wi W 2 W 3 11" 4
(4.59)
.AJIH JlJmeHHOH BI16paJ..\HH -"a (t) r!JYHKI.{l1SI B03MYll1CHHSI B(s) = a...(s). <D}'HKUUSI K03r!Jr!JI1-
Iluema "nepe,l\a'lll" <D onpe)J,eJlJreTCH 113 nepBOH CTpOKH Ta6JI. 4.2; nepe)J,aTO'IHbre }'HlQ{l1H
WI I1 U':1 H3 ITepBOH cTpOlm, a W 2 11 11"4 - 113 TpeTbeH CTpOKI1 Ta6JI. 4.1. TIocJle ux nO,llCTa-
HOBKll B Bhlpa>KemU! (4.58) H (4.59) rIOJIyqalOTCSI ncpe)J,aTo<lHble YHKI.{l1Ii (4.49)..lJ:mr B03-
MYll1 eHI1H j:'a(t) + g ITepe)J,aTOqHbIe r!JYHKUI1I1 WI Ii W 3 onpe)J,eJIHIOTCH. 113 nepBoH CTpOKH,
r!JYHKUIHi 11"2 U W 4 , a TaK>Ke YHKI..\HSI <D - 113 BTOpOH crpOKH m6n. 4.2. TIocJJe npe06pa30Ba-
HI1H Bbrpa)!(eHI1S1 (4.58), (4.59) TpaHC OpMI1PYIOTCH B nepC,l\aTO'lHble }'HKI.{l1Ii (4.50).
.AJISI CJIy<raSl yrJIOBOH Bl16paul111 OCHOBaHI1S1 yet) r!JYHKI..\HH <D onpe)J,emeTcSI TpeTbeH
C'I]JOKOi1 TaOO. 4.2, YHKI..\HH WI 11 W 3 - nepBoH CTpOKOi1, a I}>YHKUI1I1 W 2 11 W 4 - TpeTbeH
CTpOKOH Ta6JI. 4.1. .AJJSI :noro cJ1yqaSl Bblpa)!(CHMH (4.58) H (4.59) npe06pa3YIOTCH K BI1)J,y
(4.47) H (4.50). ,l].mI yrnoBbIx KOJIe6aHI1H OCHOBaHIiSi yet) YHKUl1S1 <D onpe)J,eJJHeTCH <ieTBep-
TOH CTpOKOH Ta6JI. 4.2, a I}>YHKUHI1 W;{s) - aHaJIOrHqHO npenbI)J,YUleMY cnY'/aIO.
TIpe06pa30BaB COOTBeTCTBYlOll1l1M 06pa30M Bblpa)!(CHHSI (4.58) 11 (4.59), MO:>KHO nony-
'iI1Tb ncpC.lIaTO'iHbIe YHKI..\I1I1 (4.53) 11 (4.54). 113 ypaBHeHI1S1 (4.57) HaH)J,cM ncpc.lIaTo mylO
YHKUI1IO
(yr +o&)(s) _ m<D[U (I-a11"3 W 4 )+aW 4 (a-W I W 2 )]
B(s) I-WI 11"2 W 3 W 4
(4.60)
.AJISI CJIY'iaeB Yo = 90 !1 270 0 BbIpa)!(CHI1C (4.60) npe06pa3YeTcSI K B Y:
(yr +a&)(s) m<D 2. 2
B(s) =--;;;(Jc s +k 22 +kLIa -2ak(2)'
(4.61)
.AJISI nImeHHOH Bl16paUl1H Xa(s) B BbIpa)!(eHIiI1 (4.61) <D = +1 .lIJIH
cD = -I)J,JIH )'0 = 270 0 (CM. Ta6JI. 4.2). .AJIH yrJIOBOi1 BH6paUl1l1 OCHOBaHMH
cD = -L 2 Ii + LI COOTBCTCTBeHHO ,l\JJH ymoB)'o = 90 H 270°.
ECJII1 B Ka<JeCTBe B03MYllleHHH paccMaTpHBaTh yrJIbI HaKJloHa )'(t) [B(s) = y(s)], TO nepc-
)J,aTO<JHaJI YHKUl1SI ,lInSi )'0 = 270 0 (npHMoH MaJlTHI1K) 3aITMIlleTCH CJJe)J,)'IDIlJ,HM 06pa3oM:
)'0 = 90° u
yet) HMeeM
(yr +a&)(s)
Y(8)
= m {L(J c s 4 +[L.(k 22 +k Il a 2 -2ak I2 )-J c g]s2 - g(k 22 + kJla 2 -2ak J2 )}.
6 1
(4.62)
236
rmma 4. }J,I1HAMl1KA q3 Ml1KPOMEXAHW.JECKI1X IlPHEOPOB
,lJ,.rrSI nepeBepHYToro MaJlTHIiKa (Yo = 90°) B Bblpa:>KemUl (4.62) Heo6xo)J,l1MO BeJIIi'-iHHY L 1
3aMeHHTb Ha L 2 M g Ha g.
,lJ,.rrSI B03M}'IUeHIDI .vs(t) + g nepc)J,aTO'IHWI YHK MR (4.60) npllHHMaeT BH)J,
(v r + a&)(s ) _ + 171 (J 2 k k :>. - 2 k )
B(s} - l c S + 22 + n a a 12 '
(4.63)
r)J,e 3HaK "-" COOTBcTcTByeT yrny YCTaHoBKH Yo = 0°, a 3HaK "+" - yrJIY Yo = 180°.
nOJIyqUM Bblpa:>KeHHSI JI5[ qyBCTBJ1TeJIbHOCTeH MWiTHIiKa, KOTOpbIe onpe)J,eJU!IOTc OT-
HomeHHeM BbIXO,!{HOH Koop.nuHaTbI (S; Yr;)'r + as) K BXO.nHOMY B03)J,eHcTBUl() (B03Mym: eHIi l()
B) B CTaIlMOF'apHOM pe/lmMe tdJdt = s = 0).
A'll! JIHHeHHOH BH6pa1um OCHOBaHIDI is (B -)! ax)' nOJIO:>KUB s = 0 B BbIpa)KeHHHX
(4.49), (4.61), HMeeM
.
S _ =+:m(ak ll -k 21 } .
9 o.J: - /1. ,
1
S = =+: m(k 22 - ak 12 } .
h D '
S = =+: m(k l Ia 2 - 2ak l2 )
Yr+ a9 ,OJ: ; ,
(4.64)
r;t.e S3.a, - <i)'BCTBHTenbHOCTb ITO yrny, C 2 /M; S"..a, - qYBCTBHTeJIbHOCTb no ncpeMellJ,eHMlO,
c 2 ; Sy. aa,a, - c)'MMapHaJI qyBCTBHTeJIbHOCTb, c 2 ; ; = k ll k 22 -kl2 + mg( ak ll -k I2 ), H2;
aepXHHe 3HaKH nepe,ll napaMeTpoM m cooTBeTCTBY!OT yrJIY YCTaHOBKH 'Yo = 270°, a HllJKHl1e
'Yo = 90°.
AHanOfI11JHO J1.J"U1 B03M)'lll.eHHR jilJ + g (B -)! a y + g) H3 Bblpa)KeHHH (4.50) Ii (4.63) no-
JI}"'lHM
S = =+: m(akIl -k 21 ) .
S,a) +g . ,
2
S _ =+:m(k 22 -ak I2 ) .
y,.uy+g - ; ,
S _ + m(k 22 + k J la 2 - 2ak 12 )
Y. +uS,o) +g - /1; ,
(4.65)
r.nc b; = kJ J 2 - k 2' H2; 3HaK "-" nepe.n napaMeTpOM m COOTBeTCTByeT yrny YCTaHOBKI1
10= 0°, a 3Hax "+" - yrny 'Yo = 180°.
llytlCTBHTeJIbHOCTH ]JJUJ cnyqWl yrnoBoH BH6paWiH OCHOBaHH5[ y (B -)! f) B COOTBeTCT-
BUH C Bblpa:JICeHIDIMH (4.47), (4.61) Bbl'iHCJUlIOTC5f no 3aBHCHMOCTj{M
nEPE)lA TOLIHhIE <» YHKlIJU1 1.J:3 MI1KPOAKCEJIEPOMETPOB
237
& } m (aklI k 21 )
-
S ,r = fj,* ;
I
+ LI } m (k22 - ak I2 )
-L
S ) ,rcc:: 2. ;
n fj,]
h } m (kit a 2 2ak J2 )
L2
+ = '
I
rAe S .r - QYBCTBI1TenbHOCTb no yrny, c 2 ; SYr,r - qYBCTBI1TeJIbHOCTb no ITepeMelll.eHHlO,
2 S 2
M'C; Yr+a .r - CYMMapHa>I QyBcTBI1TeJIbHOCTb, M'C; napaMeTp LI COOTBeTCTBYCT yrny
Yo = 270°, a napaMeTp L 2 - yrJ1Y YCTaHoBKH Yo = 90°.
ECJ111 B KaQeCTBe B03MYlll.eHlUI paccMaTpHBaCTcSI yroJI HaKJIOH8 OCHOBaHI1>I
yet) (B y) , TO 113 BbIpa>KeHllH (4.53), (4.54), (4.61) cnCAYIOT opMynbI L\JIH. BbIqHCneHI1H
(4.66)
'lYBCTBl1TeJIbHOCTeH:
S _ +mg(ak lI -k 2I ) .
9,y - b '
S - +mg(k 22 -ak I2 ) .
y"y fj,* ,
1
+ mg ( 2 )
SYr+ o9 ,y - fj,* k22 +klIa -2ak I2 ,
1
.
(4.67)
fIle S 'Y - 'lYBCTBI1TeJIbHOCTb no yrny, 6e3p83MepHWI BenWllm8; Sy"y - 'lysCTBHTeJIIoHOCTb
no nepeMe eHllIO, M; SYr+a ,y - CYMMapHaJI 'lYBCTBHTenbHOCTb, M; BepXHHC 3HaKll COOTBCT-
CTayIOT yrny Yo = 270°, a HI1:>KHHe - yrJIY 'Yo = 90°.
flpll.\1ep 4.5
BbI'IHCJIHM 'IYBCTBHTenbHoCTH MIDITHHKa K pa3HbIM B03MYI.QeHHJlM. HCxo.n;Hble .n;aHHble:
m = 0,29 . 10- 3 Kr; / + a = 5 . 10- 3 M; L = 0; a = 4,281 . 10- 3 M; g = 9,81 Mfc 2 ; k ll = 2,036. loJ HIM;
k 12 = k 21 = 0,879 H; k 22 = 4,398 . W- 4 H . M. IlpH Yo = 2700 nOJIyqHM fj, = 0,239 H 2 , a llpH Yo = 90 0 -
3lia'leHHe fj, = 0,225 H 2 . JUI yrnoB 1'0= 0 1I 180 0 3Ha'leHHe b = 0,224 H 2 [BeJ1H'1HHL1 fj, , fj,; npu-
Be.n;eHbI B nOSiCHCHH)!X K q>opMynaM (4.64)-{4.65)].
PC3ynbTaTbI BbI'IHCJIeHHH no qx,pMynaM (4.64)-{4.67) npe.n;CTaBHM B Ta6JIHI-,ax.
qYIJCTIJHTeJlbHOCTb M8HTHHK8 no OTHOWCHHIO K YCKOPCHHIO JIHHCHHOH BH6paJ(HH .fa = ax
Yo, 0 SSa ,1.10- 3 r:-/M SYr.o ,I'IO-6 c 2 Sy +090 .1-10-6 c 2
. x r , "
270 2,992 1.238 13,91
90 3,178 1,316 14,77
238
rmma 4. ,D;HHAMI1KA I.J:3 MHKPOMEXAHllLj ECKHX IlPIIEOPOB
qYI5CTlIHTe.,ILHOCTh M8SnH" 8 no OTHomCHIflO h 'ChOpCHUIO in + g =u" + g
0 S9.a, +g,I.1O-3 C /M -6 : S j-6 ,
'YO, S"".a. .I']O C ",+as,a, +g' .10 C-
O; 180 3.193 1,321 14,9R9
q 'I5CTBHTe."LI{oCTh 1It8HTHH a no OTHOUlCHIlIO h T."OBO:\l)' YChOPCIUlIO Y = r
0 S9,r,I.]0-6 , 1 10- 9 2 Y ,
Yo, c- S" r,' C S",+os,r,I.IO M'C'
, ,.
270 14,961 12,401 69,549
90 15,892 13,173 73,876
q '8CTBHTe.:'ILHOCTb lItaHTHH 8 no OTHomCHHIO )TJ'IY H8K.'IOH8 Y
Yo. 0 59.r ].]0- 3 Sy"y' 1.10--6 M S",+09;(, 1.]0' <} M
270 29,353 24,333 13 7 ,809
90 31,178 25,844 146,382
ITepc;IaTOqHble <}JYHJ<Ium (4.58), (4.59) I1 Bce BbITeKa1QIllHe I13 HI1X 'i8CTHbIe CJIY'lall ,llJIH.
pa:i-"lI1'llibIX BapUaJITOB B03MyuI.eHJili MOryr 6bITb CyuI.eCTBeHHO ynpOIlJ,CHbI, CCJII1 Benl1'Uma Yr Ha
HecKOJIbKO nopj!JU(OB MeHbwe, 'IeM as, a .neMIT HpoB8Hl1e 06YCJI0BJ1eHO B OCHOBHOM yrJIOBbIM
HA\CHMeM MaSlTIfHKa B onpell,eneHHOM 'IaCTOTHOM .lIMan830HC. BMeCTe c TeM, HMeJl B Bl1J1.Y op-
MhI KOJIe6a.HHH MIDITHHKa, MOjKHO yrnepiK,!l.aTb, 'ITO cYI..IJ:eCTBYJOT 'laCTOTHbIe ,l\Hana30Hbl, B KOTO-
pblX po.'IH nocrynareJIbHoro II yrnoBoro nepeMern.eHHll MaSlTHI1J<a npaKTH'leCKH paBHOl.{eHHbI.
3amnne ynpollleHHbIH BapmUIT ypaBHCHI1H (4.41):
+ } -
(k 12 + mg)S(s)+kIIYr(s)= r(s )+maAs);
-
( 2 ) +ma4 }
JAs +k!l.ss+k 22 + mag S(S)+k 21 Yr(S)= r(s)+maax(s).
-ma
(4.68)
B CHCTeMe ypaBHeHJ1i:i (4.68), TaR :>Ke, KaK H B opMynax (4.45), (4.46), BepXHHe 3HaKl1, TaM,
me OHM JJ;BOHm.Ie, a TaIGKe napaMe1p LI COOTBe'TCTBytOT np MoM)' MaSlTJillK}' (Yo = 270°), a HJ1:>KHI1C
3HaKH H napaMeTp Lz - nepeBepHyroM)' (yo = 90°). 3aJIHWeM peIlJeHHC CllCTeMbI (4,68) OTHOCI1-
TeJIbHO napaMeTpa S(8):
Knr f(s)+Kna a(8)
9(8) = ' ;
T 2 s2 + 2 Ts + 1
(4.69)
nEPb)J.A T04HbIE <1>YHKl{I1M 3 MI1KPOAKCEJIEPOMETPOB
239
=
T= JAk lJ
n(kllk n - k 2) + mg(k 21 - ak ll ) ,
kTJ.9 k Jl
,
2 JAkI! l (kllk n -kI22) + mg(k21 -ak ll )]
r)J,c
+ m'ak -k ) } 0.<-YO =270°)
11 21 L 2 (yo =90°)
Knr = ? _ ;
n(k ll k 22 -k t - 2 )+ mg(k 2I -ak ll )
(4.70)
K = +m(ak)]-k 21 )
no, n(k ll k 22 -k 1) + mg(k2I -ak ll )
B COOTBeTCTBJfH C Bblpa)KeHHeM (4.69) 3anHllJeM nepe)J,aTO'IHYIO YHKUI1IO MWlTHHKa no
OTHOllleHIUO K ynIOBon Y(t) 11 .1I1HeHHoH io (t) B1l6paumIM:
K
W ( s ) = n\i
9: r, 0. T 2 S2 + 2 Ts + 1 '
(4.71)
[LIe Kn" = Knr HJIIi KnM = Kna B 3aBI1CI1MOCTM OT BMLIa B1l6paul1l1.
O'IeBHLlHO, 'ITO K03 qJl1UHCHTbI K nf , Kn , - :ITO 'IysCTBHTenbHOCTI1 MaJlTHMKa [(CM. Bbl-
paJKeHIie (4.64) II (4.66)].
CoxpaHSISI ITpe)KHl1e npCLlITOnO)KeHIiSl, 113 Cl1CTeMbI (4.44) lln cny'IaH. JIImCnHOn B1l6pa-
UllH Yo + g nOJIy<II1M
kI2B(s)+ k 1 \Yr(s )= + m [ay(s)+ g]; }
(J AS2 + k n9 s + k22JB(s)+ k 21 Yr(S) = + ma [aAs) + g].
(4.72)
flpUMep 4.6
BbI'iHCJ1HM napaMeTpbI Kmu T nepeJ1aTO'ilJOM cPYHKl(HH (4.71) J1m! 3I1a'ieHHH (CM. npHMep 3.2):
In = 0,29 . 10- 3 Kr; J A = 7,093 . 10- 9 Kr'M 2 ; k ll = 2,036 . 1& HIM; k 22 = 4,398 . 10-4 H'M; k l2 = k 21 =
= --08 79 H' g = 9 8 1 M /c2. a = 4 281 ' Hf 3 M' 1+ Q:::: 5 . lO J M' n = I' 3
, , , " , " .
npHMeM L = 0; TOfJ1a Lt = L 2 = I + Q. Pe3ynbTaTbI BbI'IHCneI1HM no cPopMyn8M (4.70) npe)1CT8BJIe-
Hbl B Ta6nHl(e:
YrOJI YCT8HOBKlI, c. H 'iHCnO rIapaMeTp
ynpyrnx 3neMel1'I"OB n Kna , 1.10- 3 c 2 /M --6 2 T,I.lO- J C
Knr: ,1.10 C
.
1 11,016 55,079 7,585
270
3 3,956 19,778 4,545
1 14,035 70,177 8,562
90
3 4,287 21,434 4,732
r:JI,e 3HaK "_" COOTBCTCTByeT ymy /,0= 0°, a 3HaK "+" - ymy /'0 = 180° [CM. (4.72)].
240
fnaBa 4. Ln1HAMHKA 43 MI1KPOMEXAHU4ECKl1X rIPI1EOPOB
P83peUIMB CMCTeMY (4.72) OTHOCIITeJIbHO napaMeTpa &(. ), Ha")J,e 1 3aTeM nepenaW'iHYfO
YHKUmo MWlTHI1Ka no OTHorneHllfO K nlmCllHOll BepTI1KaJIbHOH BH6paUl1H OCHOBaHHSI C Y'ie-
TOM YCKopeHIDI CHJIbl TH:>KCCTH:
K
IV ( s ) - no,
9.0, - T'!.S2 + 2 Ts + 1 '
(4.73)
r,lle
KfJD
..
_ + m(aklI - k 2l J.
- n(k Il k'!.2 - k l '!.'!.) .
T=
J A k l1
l1(k ll k 22 -k\:'.2) ,
= J :I9kll
2 IlJ Ak\ ] (k 11 k 22 - k 2)
. (4.74)
K034>cl>Mu,IeHT nepe.Ja'iIi Kna -:nO 'lYBCTBHTeJIbHOCTb MaRTHI1Ka K B03MymeHl1fO
.
YB + g [(CM.4>OpM)'.'1bI4.65)].
Tlpu\tep 4.7
BbI'illc.:TIlM napaMeTpbl nepellaTO'iHoM I}>YHKlUIH (4.73) Jl1151 lICXOJ1HbIX J1aHHbIX 1I3 npeJ1blJ1YlL1ero
npHMepa. Pe3)'"ThTaTbI BbI'lHcneHHM npHBe,!J;eHbI B Ta6nHue.
f1apaMeTp
tlHC.'10 ynpyrHX 3.'1eMeHTOB, 11 Kna' ].1O-3 c 2/M T, ].1O- 3 c
-'
I 12,344 8,029
3 4,1 ]5 4,636
3aMeTRM, qro TaX KaK MaJITHHK B .naHHOM npHMepe ropH30HTaJ1bHbIH, TO MO.nynll Bbl'illCneHHblX
napaMC1J>OB O.mrnaKOBbI ,!I.lliI 060HX yrJIOB ycraHOBKH (yo = ° 11 180°).
aMHKa MWlTHllKa KaK H3MepHTeJISI yrJIOB HaKJ}OHa OCHOBaHl1H MO:>KeT 6bITb IiCCJIellO-
B8Ha H2 683e ynpoweHHOH CHCTeMbI ypaBHeHI1H, nOJIyqeHHbIx 113 ypaBHeHMH (4.52):
(k 12 + mg) &(S) + kIIYr(S) = + m { S: - g } yes);
L 2 s - g
(J A s2 +kJlss+k u + mag}&(s) + k2IYr(S)= + ma { s: - g } yes).
Lzs - g
(4.75)
.KaK B B ypaBHeHHj{)( (4.52), 3HaK "+" B rpynne .z:\BOHHbIX 3HaJ<OB H COMHO:>KHTeJIH C na-
paMeTpOM L. COOTBeTCTBYJOT yrny YCTaHOBKH Yo = 270°, a 3HaJ< "_" Ii COMHO:>KHTeJIH c napa-
MeTpOM 0. - yrJIY YCTaHOBKH MWlTHMKa Yo = 90°.
AHanorn'iHO npeJtb1)J,yrneM)' nonYQHM nepeJ.{aTO'iH)'KJ cl>YHK1.\HKJ MaJlTHHKa no OTHOIlIe-
HMID K yrny HaJ(J)OHa OCHOBaHIDI:
K (1.2s2 -1)
W; ( s ) - y
S.y T 2 s2 +2STs+l '
(4.76)
YPABHEHH5I BJ;DKEHI15l H rIEPE ATOqHNI II>YHKl{HJI 43 MHKPO,[(AT4I1KA 241
r,Uc
JAk lI
,
n(J(ll k 22 - kt2) + mg(k 21 - ak ll )
T",-,
11 "'-' LII g, (Yo "'-' 270°); 11 = L2 / g, (Yo = 90°);
K = =tmg(k 21 -ak ll ) .
Y n( ku k 22 - k;;) + mg( k 21 - ak ll ) ,
kJJ.BklI
£;=
2 JAkll[n(klIk22 -k I 2 2 ) + mg(k 21 -ak ll )]
(4.77)
(BcpXHHe 3HaKH B rpynnc ,UBOHHbIX 3HaKOB COOTBeTCTBYlOT yrny YCTaHOBKU Yo = 270°, a HM:>K-
HHe Yo = 90°).
ITp/l.\lep 4)i
BbI'IHCJlHM napaMCTpbI Ky, T, T I nepe)1aTO'IHOH iPYHKQHH (4.76) ,l\JIH. MaH.THHK8 C HCXO)1HbIMH )1aH-
HbIMH no npllMepy 4.6.
3aMCTI1M, 'ITO napaMeTjJ T nepe,naTO'lHOH iPYHKl\HlI (4.76) COBnll)1ae-r C aHaJlOrH'IHhIM napaMeTjJOM
iPYHI<LtHH (4.69) 1I 6hlJl BbI'IHCneH paHee. ECJlH L =' 0, Tor,na ,l1JlJI 060HX ymoB YCTaHOBKH Yo (270 H 90°)
nonY'lHM T. = 2,258 . 10 2 C, 'iTO Ha nOpH)10K 60nbllIe nOCTo HHOH BpeMeHH T. HaH)1eM TaK)!(e cneJ.IY1O-
I.I.1He 3Ha'leHH K03iPiPHuHeHTa nepeL\a'lH ('IYBCTBHTenbHoCTh MWlTHHKa, 6e3pa3MepHllil BenH'IHHa):
npH Yo = 270 0
{ 0,1 08(n = 1);
K -
y - 0,038(11 =3),
npH Yo = 90°
{ 0,138(n = 1);
K =
y 0,042(n = 3).
4.3. YP ABHEHIDl )J.BlDKEHIDI II IIEPE)J.ATOQHAH «I>YHK:QJUI Q3
MHKPO)J.A T\fiIKA )J.ABJIEIIHiI
YpaBHCHMR nBH:>KeHMSI q3 M nepenaTOqHaJI YHI<I.(IDI MHKpo.naT'lMKa .naBJICHIDI, BbInon-
HCHHorO B BH,Ue MeM6paHbl C :>KeCTKUM I.(CHTpOM, onHCblBalOTCSI ypaBHeHHRMH (4.9) ,UBM:>KeHIDI
q3 mmeHHoro MA, B KOTOpbIX Hc06xo,l\HMO yqeCTb .ncHCTBHe ,l\aBJIeHIDI.
TIonaraeM, '1TO Ha :>KeCTKHH (He,l\e OpMHPyeMbIH) I.(eHTp 1 C ynPYTOH nepcMbIQl(OH 2,
COC,l\HHSIlOllleH cro C KopnycHoH nJIaCTHHOH 3, CHM3Y I1 CBepxy .neHCTBYIOT AaBJIeHHH. PI H P2,
p83HOCTb KOTOpbIX, pacnpc.u.eJIeHHYIO ITO nJ10IlJ,a.u.H MCM6paHbl, MO:>KHO 3aMCHHTb CHJIOH FIJ.'
npHJ10:>KCHHOH K I.(CHTpY CHMMeTpHH MeM6paHbI (PHC. 4.5). B COOTBCTCTBHll C npHHJITbIM Ha-
npaBJIeHHeM CI1JIbI FIJ. no nOJIO:>KHTCJIbHOMY HanpaBJIeHHlO OCH OY npH6aBMM ee K npaBOH
'laCTH nepBoro ypaBHeHHlI (4.9) H, OCTaBJIlIR OCTaJIbHbIC 6e3 H3MCHCHIDI, 3anHllIeM ypaBHeHIDI
)J,BH:>KeHHR q3 MM:
my + kJJ.YY + GyY+ mVii+IA3)= m(u- g- YB)+FJ].;
Jpp + kJJ.pP + Gpp + mIz (y + Ix ii )::: mIAg -u - Yo)+ mZ B (g-u)+ mIZzBP;
(4.78)
J aii+kJ].IlP + Gila + mlA.Y + lzp)= mlAu - g- YB)+mxB(U - g)+m( zBa;
Yc = y+Ixa+Izp+ Y B .
YP ABHEHlliJ: ,lJ,BI1)KEHlliJ: H rIEPE,lJ,A TOt.fHA5I (})YHKW151 43 MI1KPO.D,A T411KA 243
Llmr CTaLlI1OHapHoro pe)I<HMa (s = 0) aHaJJOflf1.JHO ypaBHeHHIO (4.39) MO>I<HO ITOny1.JHTb
cnc'/xYlol1(ce Bblpa)KeHHe, onpe.nemllDllJ,ee 1JYBcTBMTeJIbHOCTb .naT'lHKa .aaBJIeHHH K B03MYllJ,e-
/UUlM:
(2 2 )
Y c _ Gy\GI',lx -GuI;; +GrPa
m(u - g - )i,,)+ FJI GyGpG a
(4.80)
OqeBHlXliO, 1.JTO 1JYBCTBMTeJIbHOCTb
ICM6paHbI C )I<eCTKUM ueHTpOM no OTHOllIeHHID K
CI1Jl 8.M" , 06YC.:IOBJIeHHbIM J1HHeHHbIM (u - g) YCKopeHHeM, BM6poYCKopeHHeM YB M CHnOH )J,aB-
neImSl, OITpe,llefl}leTCR BblpIDI<eHHeM (4.80).
B COOTlJeTCTBHH C nepBbIM ypaBHeHHeM CHCTeMbI (4.78) 3aJIHmeM COOTHOllIeHHe, onpe-
)J,eJIHlDllJ,ee OTHOCllTeJIbHYID 01llH6KY H3MepeHllSi naBJIeHHH, 06YCJIOBJIeHHyIO J1HHeHHbIM YCKO-
pCIiHeM:
8 = (a.A,cM)p (u - g - )iO) 100 % = [(u - g - )iB) pc" / t1Pmax] 100 %,
t1Prnax a.,b,!
(4.81)
me a", b... c" - pa3MepbI :>KeCTKOro L(eHTpa (CM. pHC. 4.5); t1pmax - nHall830H M3MepSieMbIX
naBJ1eHHH.
HeTpY,llHO BM)J,eTb, 'ITO BbIpa)KeHlle (4.80) AJIH 06Il(ero CJIY'lWi Ix =F 0 Ii l. =F 0 C T01JHO-
CTblO no 3HaqeHHR BeJ1H1.JHHbI Gy{l; IGa -I; IG p ) paBHo
Y c I(m(u - g- )io)+ F,II]= I/G)' (4.82)
flpll.\fep 4.8
BbIqllCJIHM norpelllHocTb, BHOClIMyro mmeHHoM BH6pal..\HeH OCHOBaHJ.UI B 1I3MepeHHH. nepeMeli\e-
lIHH 43 MM .lI.JISI HCXO.ll.HbIX .lI.aHHbIX: I1pm.x = 0...10 KIla = lO't HIM:!; P = 2,33 . 10 3 Kr/M J ; e.. = 0,5 x
x 10- 3 M' Y = ) ' cosco(' (y = 10- 3 M' CD = 100 r.. = 6 28 . 10 2 IIc )
., B sO , DO 'J ..... 'J .
BbjqHCmlM IYol = U?Yoo = (6,28.10 2 )2.10- 3 ::::: 394,38 wc 2 .
rIo .popM)'ne (4.81) rrOJIyqHM
39438.05.10- 3 .233.10 3
s: " ,
u= x
10 4
xl00 % :::::4,59%.
flpll.\fep 4.9
IloJI)"IHM npe.neJIbHYIO oueHKY norpelllHoCTeM BbI'IHCneHHM no .popMyne (4.82). TaK KaK rrpe-
JIenbHaJI 01llH6Ka HMeeT MeCTO npH Ix = 0 HJIH I
= 0, nOJIO)j{JfM, HanpHMep, Iz = O. OqeBHllHO TaJOKe (CM.
pllC. 4.5), 'ITO MaKCHMaJlLHOe 3HaqeHHe Ix = a.l2. IlorpelllHoCTH BblqHCneHHH no .popMYJIe (4.82) no
cpaBHeHHIo C .popMynoH (4.80) B 3aBHCHMOCTlI OT OTHOllleHHj[ 0 y /Oa .lI.JIH. 3HaqeHHM a.. = 0,5 . 10- 2 H
10- 2 M npHBe.llCHbI B Ta6mure.
Oy/Oa
a..,M
10 4 leY 10 1
0,5 . 10- 2 6,25% 625.10- 2 % 6 20 . 10- 3 % 6,25.10-4 %
, ,
10- 2 25% 25 . 10- 2 % 25 . 10- 3 % 25.10-4%
244
fnaBa 4. .n:HHAMI1KA 1.J:3 MHKPOMEXAHI11.J:ECKHX IlPHEOPOB
YCTOllqHBOCTb MeM6paHbl onpenemleTC}I TaK )I(e, KaK 11 YCTOH1.JHBOCTb q3 JIMHei1Horo
MA. ECJIll pa60Ta MeM6paHbl npOMCXOnHT B npeneJIax 30Hbl ynpyrocTH ee MaTepl1aJIa (t<peM-
HHii) H l = 1= = 0 (6e3JJ:e eI,"lHbIH 1.I3), Tof,tIa MeM6paua a6COJ11OTHO YCTOH'lHBa H eH He rpo-
3HT }lBJIeHHe CXJIOnbI8aHIDI [25].
4.4. r A30BOE ){El\ffi<I>HPOBAHHE 1J3 MIIKPOAKCEJIEPOMETPOB II
l\filKPO,l(A PIIIKOB ,l(ABJIEHIDI
B 06w:eM cJIY'!ae K03 HI.tHeHT lI,er.m lipoBaHHSI q3 onpeneJI}leTCSI ITO opMYJIe
k = Fill Y ,
(4.83)
r,ae F:J., v - .ne mqJHpYK>W:WI ClL1m H CKOpOCTb ITepeMew:eHIDI ITO.nBH>KHorO Y3JIa q3.
TaKMM 06pa30M, .ilID1 onpe.neJIeHIDI K03 HlIMeHTa ,l:teMn IfpOBaHH5I He06xo,l:tHMO oITpe-
J1.eJIHTb CIL"I}' lI,eMnI}>HpoBaHH5I, KOTOpaJI eCTb pe3ynbTaT ,l:teHCTBIDI pacnpeJ1.CJIeHHOrO no nJIO-
w:anu S )J,aB.:'1eHIDI p ra30BOH CpCJJ:bI.
FlJ, = HpdS.
(4.84)
5
B H3BCCTHbIX pa60TaX lI,aBJIeHHe onpe.neJI5lIOT no ,l\aHHbIM HHrerpHpoBaHH51 C Pa3JIM1.J-
HLIMH ynpoW:aJOIIlHMH nonymeHHHMH npH6JIH:>KeHHOH CHCTCMbl ypaBHeHHM HaBbe-CToKca
}1...'llI BX3KOH HeC:>Kl1MaeMOH cpe.nbI H ypaBHeHIDI ee Hepa3pbIBHOCTH:
{ .., .., ..... .., ) ( ? 2 2 )
au au au au vp v-u v U v u
-+u-+v-+w- =--+Jl -+-+- .
at ax oy Bz & ax 2 oy2 Bz2 ·
{ Bv Bv fJv Bv ) op ( 02 v 02y 02 v )
-+u-+v-+w- =--+Jl -+-+- .
Ot ax oy Bz oy ox 2 oy2 oz2 '
(4.85)
J&w +U &w +V &w +W &w ) =_ +Jl ( 02W + 02W + 02W ) .
1. ot Ox By Bz Bz &2 oy2 Bz2 '
op + o(pu) + o(Pv) + a(pw) = 0,
at ax oy Bz
(4.86)
rJ1.e u, v, w - COCTaBJI5llOlWfe CKDpoCTH TeqeHIDI ra3a no KOOp,l\HHaTaM x, y, Z COOTBeTCTBeHHO;
p - IDIOTIIOCTh cpe]1.l>I; }.t - K03 H1.(HeHT .z:\HHaMM1.JeCKOH BX3KOCTH; p - ,l\aBJIeHHe Cpe,l\bI; t-
BpeMjI.
B pa60re [6] BLInOJIHeHO HHTerpHpOBaHHe CHCTCMLI (4.85) H ypaBHeHIDI (4.86).nnx CJle-
JIYlOII1HX J1.onyrn.eHHH:
. ]) TeqeHHe ma CTa1.(HOHapHOe, Hec>KHMaeMOe, BH3Koe (BbIITOJIH}leTCSI YCJIOBHe npHJIM-
naHIDI ra38 Ha CTeHKaX), 3a3Op ho MeHbwe reoMerpHqeCKHX JIMHeHHblx pa3MepOB aM' b M no.n-
B,IDKHoro)'3JIa B JIJIaHe Ha TpH-<reTblpC nopll,ll,Ka (pIiC. 4.6);
r A30BOE ,LlEMO(}>HPOBAHHF 43 MHKPOAKCEJIEPOMETPOB
245
A
A-A
..c
1tf:
: *f:
;{f:t.
Erli
tr: ;3
x
v
aM
A
ho
PIfC. 4.6. K onpe eJJeHlllo K03Ij1r!ull.\lIeHTa JIlIHeUHOro Ael\mIjlHpoB8HHH
2) HOpMaJIbHaJl COCTaBmUOllJ,aJl cKopocnl v y 3aBHCHT TOJIbKO OT HOpMaJIbHOH KOMno-
HeHTbly,coCTaBJISIfOll.\He vx,v z -OTBcexTpexKoop.nHHaT,npM3ToM v y « v x ' v y « v z ,
Te1.JeHHe B nJIOCKOCTH x, Z SlBmleTCSI nOTeHlll1aJIbHblM, T.e. Te1.JeHHeM 6e3 TpeHIDI. rpa)J,HeHT
op
naBJIeHHSI - = 0 .
ay
8 pe3yJIbTaTe HaM)J,eHa CHJIa (4.84), H K03 HllHeHT ,l\eMll HpoB8HHSI (4.83) B HanpaB-
.1eHHH OCM y
k = 2I-la b?
l\.v 3 2 2'
hO(a..+b M )
(4.87)
K03 £l>HLlHeHTbl )J,eMn HpOBaHillI JUIJI nO)J,BM)KHbIX Y3JIOB, MMelOll1HX B nmllie OPMY
KBa)J,paTa (aM = b M ), Kpyra Pa)J,HYCOM R H YJUIHHeHHoro npJlMoyrOJIbHHKa - nOJIOCbI
(aM» b..), onpeneJUlfOTCSI B COOTBeTCTBHH C ypaBHeHHeM (4.87) ITO OPMYJI8M
4
k = J.la.. .
)\I' 3'
ho
k = 3J.lnR 4 . k :=: 2llaMb .
l1,V h3' J\I' h3'
o 0
(4.88)
a)
6)
y
{......
I ......
,........... Vo ..................
.....
v
PHC. 4.7. K onpe.ll.eJIenHIO K031j11j1Hl{HeHT8 ymoBoro J1.eMnIjlHpoB8HHR:
a - nO.l\BeC 11M Ha KOHconbHblX ynpymx 3neMeHTaX; 6 - nO.l\Bec 11M ua ynpyrHx :meMeHTaX
KpY'ieHHH(TOpCHoHax)
246
rnaBa 4. )JJ1HAMMKA q3 MI1KPOMEXAHlfllECKI1X IlPI1EOPOB
IIpw.cep 4.10
PaCC'lHTaeM K03rprpH HeHTbl lleMnrpnpoBaHHH. Q3, 11M KOTOpbIX JlMelOT npHMepHo paBHbIe nno-
ma,aH, HO pa3H)'IO rpOPMY. LI,eMnrpnp)'IOWlli! cpe a - B03 ; npll T= +20 °C = 1,7 . 10- 5 Kr/(M'C).
3a30p = 18. 1O-{) M. Ilo rpopMYJIaM (4.87), (4.88) I10n)"lHM:
- npmrO)TOITbHHK a" = 9 . 10- 3 M, b" = 10- 2 M
k:>".=23,48 H'c!M;
- KBa,apaT a.. = b" = 9,48.10- 3 M
k:>".=23,56 H'C/M;
- J...-pyr R = 535. 10- 3 M
k:>".=22,41 H'C/M;
- I1OJlOCY aM = 29.98.10- 3 M. b.. = 2.99. 10- 3 M
k;J,)" = 4,67 H,c/M.
IIo.rry<leHHbIe pe3ynbTaThI .neMoHcTpHPYJOT 3aBHCHMOCTb K03rprpH HeHTa .ner.mrpHpOB8HHH OT
cpopMbl 1Th1. CymecTBeHHa em 3aBHCHMOCTb H OT pa3MepOB 1UUI orumM rpOpMbl 11M. IlpHMeM 1UUI npH-
MO)TO!ThHOH B mIme 11M pa3MepbJ: a.. = 8,56 . 10- 3 M, b" = 9,1 . 10- 3 M (m npHMepa 3.2) H non)"lMM
2.1,7.10- 5 (8,56.10- 3 Y(9,1'10- 3 Y = 17,65 H'C/M.
8.10-6 r [(8,56,10- 3 J + ,1.1O-3 r]
kJJ,l
OnpeJ1.eJIHM K03 HUHeHTbl ,l:\eMn HpOBaHJU! q3 M8SlTHMKOBbIX MA, nOMHj!, 'ITO 11M
!m60 HMeeT ynpyrne 3neMeHTbI nO,l\BeCa, )J,JIMHbI KOTOpbIX 3HaLJHTeJlbHO MeHb!1lC ):\JIHHbI aM
nJIaCTHHbI 11M (pHC. 4.7, a), JlH60 cc nO)J,Bec BbInOnHeH B Blme TOpCHOHOB, 06eCneLJHBalOI.L{I1x
BpaIUeHHe JIM BOI<pYT OCR, npOXO!1.H.ll{eH LJepe3 TOLJKY 0 (pHC. 4.7, 6).
a
B npeJ1.ll0n01KeHHH, 'ITO neHTp ,!I;aBJIeHIDI pacITOJIO:>KeH Ha pacCToSlHHR Iu = (pRc.4.7,
2
a
a), MOMeHT D.eMnl}>RpOBaHIDI paBeH MIJ. = FIJ.IU = Fj1 , a KO HeHT YfJIOBOrO neMn-
2
I}>HpoBaHIDI onpe,l\emreTcj! CJ1eD.)'!Ol!J,HM 06p830M:
k]J.a. = M 11. 16. = FAa/>! 126. .
(4.89)
1
lli pHC. 4.7, a CJIeD.)'eT v 0 = '2 v, 6. = 2v 0 I aId H B COOTBeTCTBMH C BbIpa)KeHIDIMH
(4.89), (4.83), (4.87) n01l)'LJHM
k AU
F 2 5 3
IJ.aM _ fJO)db M
4vo - 2h 3 (a 2 +b 2 )
o M )d
(4.90)
.l(mr 11M no pHC. 4.7, 6 BeJIHLJHHa klJf]. TaJQKe onpc.neJISleTCB <l>opMynoH (4.90), HMej! B BH-
JI.Y, 'ITO /JJ. = 0../4, a B .n.eMnl}>HpoBaHHH npHHRMalOT yqacrHe leaK 61>1 ,l\Be IDIaCTHHbI, Ka:>K,l\aj!
,lJ,1IHHOH 0..12.
r A30BOE ,[{EMn(}JI1POBAHYlE 43 MYlKPOAKCEJIEPOMETPOB
247
(I)
6)
;:;; -
...Q
6)
c)
. ...". o': .,...
.
aM
PHC. 4.8. llepl»opaUHoHHble OTBCpCTHH:
a-? - BapllaHTLI pacnOnO)f(eHHjj no
O'leBl1,nH0 ,UJUI 11M, BbIITOJIHeHHOH B BH,Qe KBa.apaTHoH IIJIaCTliHbI, H3 ypaBHeHHH. (4.90)
c;Ic.J.yeT:
k J'.a I-W 1( 4h 6 ).
(4.91)
<DOPMYJIbl (4.90), (4.91) MorYT 6bITb 3arrHcaHbI B Bl1):\e
kJla. = k Il )' ll .
(4.92)
3 eKTHBHbIM KOHCTPYKTHBHbIM npHeMoM H3MeHeHHSI CTeneHH ,QeMn MpOBaHHSI ITO-
MMMO M3MeHeHHJI )J,eMrr Hp}'lOllJ,eH cpe)J,bI (B03,l\YX, a3OT, BO)J,Opo)J" MaCJlO If T.,Q.), RBlliIeTC
BblIIOJIHeHHe B rrJIaCTHHe CneLlMaJIbHbIX OTBepcTHH, KOTopbIe MorYT 6bITb Ha3BaHbI nep opa-
LlHOHHbIMI1 - no (pHC. 4.8).
<DOPMYJIbI ,QJIJI BbIQHCJleHIDI K03 HLllleHTOB )J,eMmllHpoBaHml nnaCTHHbI C Y<IeTOM no
I1MelOT CTPYKTYPY, HneHTH<IHYlO opMYJIe (4.87), H, HanpMMep, Ko3C}J HLlMeHT J1llHeliHoro
neMII HpOBaHI1JI ,QJISI ITJIaCTIIHbI C LleHTpaJIbHbIM no (pHC. 4.8, a) MO:>KeT 6bITb BbI<IHCJleH aHa-
J"IOrH<IHO opMYJIe (4.87):
k = 2J.l ( a;tb
J\}' h 3 a 2 + b2
o M M
a?bl )
a +b J 2 .
(4.93)
Y'lHTbIBaJI npH6JIl1:>KeHHbIH xapaKTep 3aBHcHMocTeH (4.87)-(4.93), MO>KHO npe,l\nOJIO-
>KHTb, 'ITO YMeHbIlIeHHe K03 HUMeHTOB )J,eMIT HpoBaHHSI ITPH HaJIHIJHM 110 B nJJaCTHHax
nponOpLlHOHaJIbHO nJIOllJ,a,l\H 110.
flpwtep 4.11
Bb)qHCJ1HM K03t}>t}>HlIHeHT nHHeHHoro ,LleMmlJHpoBaHID! 11M, KOTOpaJI HMeeT t}>OPMY nnaCTI1HbI
(PHC. 4.8, a) C pa3MepaMH QM = 9 . 10- 3 M, b M = 10- 2 M; QI = 4,5 . 10- 3 M, b l = 0,5 . 10- 2 M. OCTaIIbHbre
napaMeTpbI B3J1TbI H3 npHMepa 4.10.
no t}>opMyne (4.87) nOJly<lHM kBJ' = 5,69 H'c/M. CpaBHHBM 3TOT pe3YJIbTaT C BeJIH'lHHOH k11Jl =
= 23,48 H'cfM llJlJl nJlaCTHHbl 6e3 rIO, 3aKJII<J'laCM, 'ITO cOKpameHHe ,LleMnt}>HJJYJOIl\eH nnOIl\a;:rH nnaCTH-
RbI Ha 25 % npHBeJlO K YMeHbweHHlo K03t}>ilJHlXHeHTa ,LleMnilJMpoBaHID! B 4,12 pa:3a. 8bl'rnCJlHM TenCpb
K03t}>ilJHI. HeHT ,LleMnt}>HpOBaHWI C y<leTOM IlO, 3aHHMal<JIl\HX 1,4 nJIOmauH nJIaCTHHbI, CJIe.nyroUUIM o6pa-
30M: kBJ' = 23,48/4 = 5,87 H'c/M, 'ITO no cpaBHeHHIo C BbJ'iHCJleHHeM no ilJopMyne (4.93) ,nae-r OlliH6rcy
3,16%.
248
fnaBa 4. J],UHAMIIKA q3 MnKPOMEXAHII ECKHX IlPHEOPOB
PaCCMOTpH f J.e m upOBaHlie KO.iIe6aHUll le:'16paHbl B l{aT'H1KaX .'laB.leHI!SI OT C>KaTI1S1
(KoMnpecCMM) r83oBoll n:leHKU npu ;:tBU>KCHlut MCM6paHbI. -'m60 3a CLJeT nepeTeKaHI1S1 BSl3KOH
cpe.!Ibl (r83 ILilIl :>KIO,KOCTb) Qepe3 cneUI1a.TJbHble KammJUlpHblC oTBepcnul. 8 nepBoM cJIY'-Iae
AeMn HpoBaHHe cYllJ,eCTBCHHO 3aBlICHT OT )J,aBJICHUSI B nOJIOCTlf, B KOTOPO" npoucxo.IUIT C>Ka-
TIle r83a. nOCKOJIbt-."Y BSI3KOCTb ra3a l\.-pUTU'lHa K naBJIeHHIO.
,[[,ml onpe.Je.neHIDI BR3KOCTH r83a DonbUMaH npe,lJ,JIO>KUJI OPMYJlY
= 0.3502pVA.
r.:le p - l1.10THOCTh r83a; \'. i. - cpe.:lWlSI CKOpOCTb H )J,JHlHa cB060nHoro np06era MOJICKy.lbI
COOTBeTCTBeHHO.
.1J...'llIHOH cB06o.:lHOrO np06era H83blBaeTCSI cpe)J,HSI}1 IVlHHa nym, npoxo.nllMoro MOJIeKY-
10H Me)IG)' COY.TI.apeHlliIMH C .:IpyrMMu MO..'Iel\."yJIaMM. ,l:LnS! r83a OHa onpe,lJ,enSleTCSI 3aBI1Cl1MD-
cn.1O
kT
t. =
find"}. P ,
r.:le k= 1.38066 . 10-:: 3 ,Il:>KiK (ITOCTOSlHHWI DO.:IbUMaHa); T - TCMITCparypa no lIlKane KeJIbBIf-
Ha (K = °C + 273); p - .!laBJleHHe; d - .!lUaMeTp MoneKYJIbl r83a.
TaK KaK Be.1H'lliHa i. 3aBMCMT OT .!laBJIeHH}1. TO OHO CK83bIBaeTCH H Ha BSl3KOCTH raJa.
llpu"-ep 4.12
Pacc<IHTaeM ;:L1HH)' cB06o!1Horo np06era MOl1eI-.")'JIbl a:30Ta npll t = 20°C II ;J;aBJ1eHlIli p = 1 H
0,1 aTa (1 Topp = 1/760 aTa).
l1MeeM d = 3,7 . 10- 10 M; I lIOpM. aTa = 760 MM pT. CT. = 1.013 . 10 5 HJM 2 .
JJ,.u P = laTa:
A
1;38006.10- 23 .293
J2 '3,14.(3,7.10-10)2.1,013.10 5
65,68.10- 9 M.
lLu p = 0.1 aT8 3HaqeHHe 'J.. = 0,656 . 10-{; M, a nn.ll p = 1 T opp A = 49,916 - 10-{; M <; 50 MKM.
H3 llpHMepa c!le.:tYeT. 'ITO cpemiHH npo6er MOJIeK)'JIhI a30Ta npll CTaHllapTHoH TeMneparype 1I
.:taB.:leHHH p = 1 aTa npHMepHo COCTaBJ1JleT 70 lIM. Ilpll na,!1eHHlI )J.aBneHH.II,lJ,O 0,1 aTa ero 3l1aqeHlIe npll-
6.mDKaercx K 1 MKM, a!l.o I Topp np06er B03paCTaeT 1I0 50 MKM.
B JIHTepa-rype, B qaCTHOC'rH B [22], JllliI raJa, HaxOJ1.H.ll\erOCSI MCiK,llY napaJIJ1eJIbHbIMH
IL'1aCTHHaMH, nepeMeLUafOIllHMMcR HaBcrpeqy ,lJ,pyr ,llpyry (C)KaTHe ra3a). JllliI Manoro .naBJIe-
HHJt (p « 0,1 am) npMBC,l\eHa 3aBHCHMOCTb JllliI BbI'lliCJIeHlliI K03 MUHeHTa ,l\eMn HpoBa-
HHJi:
J.l p a 3 b
K JJY = 7 2 p,
hO
(4.94)
r,nc J.lp = 2,776 . 1O-fi elM (JJ.IDI a3OTHOH H B03)J.YWHOH cpe,l\); a, b - lIIMpHHa H JI.JIHHa MeM6pa-
RbI nOCTOmHOH TO.lW.tHHbl HJIH JlCeCTKOrO ueHTpa MeM6paHbI; ho - pa3Mep 3a30pa, B KOTOpOM
npoHCXO)tHT CJlC8TI1e raJa.
()qeBHJ(HO. IJTO npH ,necl>oPMMpoBamfH (nporn6e) MeM6paHbI nOCTOSlHHOH TOJIIlJ,HHbI
npoHCXOlIHT HeKOTopoe )'MeHbllieHHe o61.eMa, a CJ1e,l\OBaTenbHO, yseJIHQeHlle ,l\aBJleHIDI B
06pa.3U OBOH KaMepe. OJUl3KO 3TH H3MeHeHIDI CTOm. He3Ha':lHTeJIbHbI, qTO HMH MO:>KHO npe-
He6pe%.
r A30BOE ,llEMIl<1>HPOBAHHE 43 MHKPOAKCEJIEPOMETPOB
249
J/plwep 4./3
PaCC'IlITaeM KOJi!JI}JI1L\l1eHT .!J.cMni!JHpOBaHlISi MeM6paHLI (mraCTI1HLI) C >KeCTKHM ueHTpoM C napa-
MeTpa !I1 A = 5 10- 3 :\t; a = b = 2,5 - 10- 3 M. IlpH:\!eM ho = 2 . 10- 5 M, P = 1 Topp = 1/760 aTa =
== 1.013 10 5/760 = 133 H/:l1 2 1I no 1}J0pMyne (4.94) nonyqHM
K = 7.2,776.\0-6(2,5.10- 3 )4 133
- 2,53 . \0-4 H/(M'C).
.1\)' (2.5 .10- 5 )2
1-13 npm.(epa cnellYeT, '1TO npH 3Ha'lI1TeJlbHOM paJpe>KeHHH .!J.aBJIeHJiH B 06paJLlOBOM KaMepe raJO-
Boe .!J.eMnl}JllpoB8HHe BeCbMa ManeHbKoe.
,Amma cB060iJ,HorO rrp06era MonerynbI 06paTHo rrponopLlHOHaJIbHa )J,aBJ1eHHIO H, B qa-
CTHOCTH, KaK CJIe)J,yeT H3 npMMepa 4.12, npM P = 1 an. OHa MeHbllle 1 MKM.
TaK KaK B )J.aT'lHKaX naBJIeHJHI paccToRHMe Me)K)J,Y nJIaCTHHaMH, rne HaXOAMTCJl C)fGf-
MaeMblH fa3, 3Ha<IHTeJIbHO 60JIbllle cB060)J,Horo npo6era MoneKYJI ra3a, MO:>KHO nOJIaraTb, 'ITO
BH.3KOCTb fa3a He 3aBl1CHT OT )J,aBJIeHHSI. ):(JIH. :noro cnyqaJI K034>4>I1ID1eHTbl lIeMn4>HpoBaHIDI
MO}[{HO paCC<IHTaTb ITO OPMYJIa.\f (4.87), (4.88), (4.90), (4.91).
B naPlMKaX naBJIeHIDI C )[(eCTKHM UeHTpOM .neMIT HpoBaHHe MeM6paHbI MO)l(eT oC)'IUe-
CTBJIH.TbCSI 3a C'IeT ITepereKaHHH BSl3KOH Cpenbl (ra3, :>KHJJ:KOCTb) 113 nonMeM6paHHOH nOJIOCTH
'lepe3 KaJI116pOBaHHble OTBepCTI1S1. Ha pRC. 4.9 nOKa3aHO O)J,HO H3 TaKliX OTBepCTHH C pa3Me-
par.1H ro M I. I1MeH B BliJJ.Y JIaMHHapHbIH pe}[{I1M Te<IeHWI :>KH)J,KOCTH, nOJIaraeM, qTO CKOpOCTb B
ITpOlIOJIbHOM CeqeHMH OTBepcTIDI M3MeHJ1eTCSI no H3BecTHoMY napa60JIH1.JeCKoMY 3aKoHY:
?
V = vo[I-(rl ro)-],
(4.95)
rne Vo - CKOpOCTb no OCH oTBepcTIDI.
.D:eMn HpyIOIlJ,WI clUIa FD. paBHa CHJIe TpeHlUl F-rp BR3KOii cpe)J,bl 0 CTeHKH OTBepcTIDI:
FfJ. = F TP = -kD, v'v, (4.96)
f)J.e v y - CKOpoCTb nepeMellJ,eHH5I )l(eCTKoro LleHTpa MeM6paHbL
K03 I1LlHeHT neMn4>HpoBaHIDI
kIJ.. =-FTp/vy-
CI1JIa TpeHHR BSl3KOH cpe)J,bl oITpenCJISleTCSI cne)),ylOlIJ,HM 06pa30M:
FTp = 21trohTp'
r,l\e L Tp - KacaTenbHoe HanpR)l(eHRe -rpeHIDI BR3KOH cpCAbI Ha CTeHKaX OTBepCTIDI M oITp eJIH.-
eTCR H3BecTHoH opMynoH:
(4.97)
'tIp = J.l(dv I dr) Ir=ro .
+ + , , , , ,p, , , , , + ,
,. . ".-
_.. \\"H' .' __L
V o
Pile. 4.9. K onpe eJIeHHIO K034141HltHeHTS AeMn4tHpoB8HRR
250
fnaBa 4. ,l]J1HAMHKA q3 MI1KPOMEXAHUQECKHX npl1EOPOA
I1MeR B BIUJ;)' BblpIDKeHlfe (4.95), nOny<iHM (dv I dr) IJ' 'o = -2\10/10 Ii, CJIelIOBaTeJIbHO:
F Tp = -4rrvo/ll.
(4.98)
Cl;:OpOCTb Vo HalJileM 1f3 ycnoBIiR paBeHcTBa 06"beMHbIX pacXOnoB BSl3KOH cpetJ,bI: voSz ==
= V_,SI (Sz, Sj - COOTBeTCTBeHHO ITnOl a)J,1i nOnepC1.JHbIX CC1.JeHHH KaJIH6pOBaIJHOrO 0'1 BepcTIUI
H nOllMeM6paHHoH nOJIOCTIi). on'-Ylla \'0 = (\'ySI)/S z 11 C yqeTOM Bblpa)KCIiIiSi (4.98) B COOT-
BeTCTBHIi c 4JopMynoH (4.97) nOJIY'-IHM
k::J. = (4rr!li S I)/(n O S2) .
(4.99)
r.n.e no > 1 - 1.JHCJIO KaJIH6pOBaHHbIx oTBepcTHll.
llplL\tep 4.14
Pacc'urraeM K03iPiPm.{HeHT ner.miPlfpoBaHIDI llJllI napaMeTpoB: S. = 10-4 M 2 , S2 == 10-6 M 2 .
1=0,5. 10-: M, no = I. JJ:eMTIiPup)'lOUlM cpe.na - 3niJIOBbIH cnHpT /l = 0,122 . 1O 2 Kr/lM'c) 11)111 aJOT /l =
= 1,67 - 10- 5 Kf/(M'C).
Ilo iPopMyne (4.99) nO!I)"lliM: 1{.IDI 3TIUIOBoro CTInpTa k:J. = 7,7 . 10- 3 H'c/M, ,n:ml aJOTa k:J. == 1,05 x
x 10-4 H-CiM.
B MHOrO'lI1CJ1eHHbIX pa60Tax, rlIe paCCMaTpHBalOTcR 3alIa'lH Te<IemUI r83a, B 1.JaCTHOCTH
[21J, H cnelUia.'IhHbIX pa6oTax, HanpID.lep D IJ, npl1Be,l:\eHbI 4JOPMYJ1bI 11JIR BbI<IHCJIeHM}J ,l\eMn-
Hp}'lOlllHX CH.Jl, KOTopble TO:>Ke HMelOT onpeneJIeHHbIe orpaHH1.JeHIDI Ha npHMeHeHHe, a HX
,1{OCTOBepHOCTb, TaK :iKc. KaI< M opMyn, paccMoTpeHHbIx BbIllle, He nOnTBep)K)J,eHa PC3YJ1bTa-
Ta.\4H npIDdhIX 3KcnepHMeHTOB.
4.5. YP ABHEHIDI )(BIDKEHIDI MHKPOrHPOCKOIIOB
4.5.1. MHKpOrHpocKoDbl LL- H LR-nmoB
A. 0606I.QeHHble ypaBHeHHJI
YpaBHeHIDI JtBM:iKeHIDI COCT8BHM npHMeHMTeJIbHO K cxeMe no pHC. 4. I 0, KOTOpaH. RBJI5I-
eTCR 06061I1.eHHoH cxeMoH MMKpoMeXaHH1.JeCKJ1X rHpocKoIlOB LL- H LR-TMnoB.
Mr npelICTaB.1llleT C060H JtBe 11M 5 H 9, KOTopbIe C nOMOll(blO ynpyrHx 3neMeHTOB 3, J J
CMOHTHpOBaHbI B paMe 2, a OHa, B CBOlO 01.Jepe.nb, nOCpe,D.CTBOM ynpyrHX 3JIeMeHTOB 1 CKpeIT-
JIeHa c KopnycoM..I1M 3J1eKTpoCTaTJf1.JeCKHMH BH6ponpMBO,l:\aMH 4 (neBbIM), 7 (ueHTpanbHbIM)
H 10 (npaBbIM), HMeJOIlU1MH rpe6CH'IaTYlO crpYKTYPY, C03.nalOIWIMI1 BbIHY)K,lI.aIOIWIe CMJIbI F,
npMBo.wrrc5I B Kone6aTeJIbHble npoTHBotP83HbIe mUf:iKeHIDI B CBoeH flJ10CKOCTM (XZ).
TIpH H8J1H'iIrn nepeHocHoH YTnOBOH CKOpOCTH !2 z B03HHKalOT YCKopeHl1e KopHonl1ca H
COOTBeTCTB)'lOIlU1e eMY CHJIbI HHeplUiH, KOTOpbJe BbI3bIBalOT JIH60 BH6palUioHHble nocTyna-
Te.JI&Hble JtBH:iKeHIDI qyBCTBMTeJIbHbIX Mace B HanpaBJJeHHH OCH OY, nH60 Kone6aTeJIbHOe
JtBffiKeHHe paMbI 2 BOKpyT OCH OZ B 3aBHCMMOCTH OT BapHaHTa HcnOflHeHHB ynpyrHX nO,l\Be-
COB. Y J{ "IaH Hhle KOJJe5aHID1 co.nep:>KaT HHl}JopMalUiJO 06 H3MepaeMoH yrnoBOH CKOpOCTH !2 z
H H3MepmOTCB C nOMOl1U>lO eMKOCTHLlX Jl,ar<IllKOB nepeMell1eHHH, 06p83oBaHHblx IlO,lI.BM:>KHbl-
MH 3neK'IpO.naMH, pacnOJIO)l(eHHblMH Ha q3 5, 9, H HenO,l\BH:>KHbIMH 3neKTpO,llaMH 6,8, Pa3-
MeIqeHHbJMH Ha lCopnyce.
YPABHEHI1.H ):{BI-DKEHI1.H MI1KPOrHPOCKOIIOB
251
z
]
2
.'
'.
. . ......
".-"..
. .,....
. -.'
.. ."' 0'; :.".:.:.. .".. '". : ....-.:.:......: ...'.:-:. . '.";-. _
x
3
n z
]]
4
()
" ;.!/J;}f&!i ) r!:k.:
.. : ::{:;<! fii ;t lifir ' .....
5
,
....
.'
. ".-.".' ,". '. . ,." - . "-" ..... A:' . -'.. :.".", .' .: . ;.:. .' ...._....: :.}' ". ';':-.':':" ," .....:-:-:..",... ....: A:' ."
y
:=:
PIIC. 4.10. 0606Il\eHHHA cxeMB Mr LL- II LR-TunoB:
1,3, /1- ynpyme 3J1eMeHTbJ; 2 - paMa; 4, 7, /0 Bu6poIIpHBO,l{bJ; 5, 9 -lfHep HoHHble MaccbI;
6, 8 - Hen01!BmKHbJe 3neKTpO,D.bJ ,LIaT'lHKOB IIepeMeIIIemdi
ITpH BbIBO.n.e ypaBHeHHH ,l\BM:>KeHHSI 6y.neM rrOJIaraTb, 'lTO paMa 2 COBMeCTHO C ynpyrnMH
'3neMeHTaMH I1MeeT KOHe'lHYlO :>KCCTKOCTb B HaITpaBneHHH oceH Ox, OY, a 3JIeMeHTbI J - KO-
He<lHYlO :>KCCTKOCTb Ha Kpy'leHHe BOKpyr OCll OZ. 3JIeMeHTbl 3 H 1 J 06JIallalOT B 06IlIeM CJ1y-
<Iae pa.3HbIMI1 :>KCCTKOCUIMll Ha H3fH6 B HanpaBJIeHHH oceM OX H OY. Cql1TaeM, 'ITO :>KeCTKO-
CTH BCeX 3JIeMeHTOB (KpOMe 1) BOKpyr OZ 6CCKOHe'lHO 60nbllll1e, 11M M paMbI HMelOT CTaTll-
'lecrylO Hec6anaHCHpOBaHHOCTb, a OCHOBaHl1e rrepeMelll.aeTC5I rrocTyrraTeJIbHO C IIHHeHHbIM
YCKopeHl1eM. BBe.neM CI1CTeMbl KOOp,l\llHaT I1 0603Ha'leHHSI (pl1c. 4.11).
C KoprrycoM CBSl3aHa CI1CTeMa Oxyz. OH :>KCCTKO COeL\HHeH C OCHOBaHl1eM, KOTopoe no-
cTyrrareJIbHO nepcMeIl\aeTC5I C YCKopeHl1eM, HMelOll.{I1M Ha OCI1 CI1CTeMbI KOOp,l\HHaT OXYZ
npOeKl.{HI1 ax, a y , az. OCHoBaHHe, KpoMe Toro, Bpall.{aeTCR BOKpyr oceH CHCTeMbI KOOp)J,I1HaT
OXYZ C YfJIOBblMI1 CKOpOCTSIMH nX,ny,n z ' BeKTop H3MepReMoH CKOpOCTH n z HallpaBJIeH
BL\OJIb OCI1 OZ, cOBrra)J,alOIl\eH C OCblO KpY'leHIDI 3JIeMeHTOB J (CM. pHC. 4.10).
CI1CTeMa KOOp,l\HHaT Ox'y'z' 0603HaQaeT pa3BOpOT paMbI Ha yroJI ex 6naro.n.apSi KOHCQ-
HOH :>KCCTI<OCTH :meMeHTOB 1 Ha KpY'leHlle. C reOMCTpH'leCKllM I..{eHTpOM paMbI Cml3aHa CHC-
TeMa KoopL\I1HaT O.,xW'.,xp. COOTBeTCTBYIOll.{He OCI1 KOTOpOH napanJIeJIbHbI OCRM CHCTeMbI KO-
Op,l\llHaT Ox'y'z' . IIOIIO:>KeHl1e reoMeTpl1'leCKOrO I..{eHTpa paMbI (TOQKa O ) OnpelleJIeHO KOOp-
L\I1HaTaMH x, y B CMCTeMe Oxyz, KOTOpble 06YCJIOBJIeHbI KOHeQHOH :>KeCTKOCTblO Ha H3fH6 pa-
MbI I1 3J1eMeHTOB 1.
C 4M (TOQKa O ) paMbl CBJl3aHa CI1CTCMa Koop.nI1HaT O x y z , OCH KOTOpOH nap an-
JIeJIbHbI COOTBeTCTBYIOIl\I1M OCRM CI1CTeMbI KOOp,l\I1HaT OpXpYpZ p , a nOJIO:>KeHHe TOIfl<I{ O
onpe.n.eneHO Koop,l\llHaraMI1 0xp, ow' 0zp' KOTopble MorYT I1MeTb c)IH3J1qeCl\YIO npHpO.l\)'
TeXHOJIoruQeCKllX HJII1 TeMnepaTYpHbIX CMeIl\eHI1H.
252
rmma 4. .lJ,I1HAMI1KA q3 MJ1KPOMEXAHHQECKHX OPHEOPOB
C reoMeTI)HqecKHM QeHTpOM (TO'lKa 0\) nepBoll HM CB1I3aHa cnCTeMa Koop;umaT
O]XLY\X), nOJIOJKeHlle KOTOpoil B cnCTeMe OpXpYpZ p onpe):\eJIeHO KOOp)J,liHaTaMI1 xLVI. a COOT-
BeTCTBytOll!lle OCH )'Ka.3aHHbIX CliCTCM Koop.!UlHaT napa.TmenbHbI. KOOpl1liHaraXI 06YCJIOBJleHa
ynpyrnMH CBOHCTBaMIf KOHCTpYKliHH, a TaIOKe rrepeMCll.\eHU1IMH, BbI3BaHHbIMI1 Im6ponpl1BO-
.n.a.\tH, a JCoopllHHarn,v\ - TOJ1bKO ynpyrHMI1 cBoiicTBaMH Mr.
C l{M (TO'lKa 0;). nOJIO:>KeHMe KOToporo B CI1CTeMe OI.\WIZ\ onpe)J,eJIeHO Koop)J,lmaTaMM
Oxp'O)'P' 0:p, CBH3aHa CllCTeMa KoopllHHaT O;x;)';z;. COOTBeTCTBYIOll.\l1e OCM CMCTCM KOOp-
O 0 ""
J1.HHaT \XlrIZ\ I1 \XI)'\ZI napaJme.rlbHbI.
C reOMeTpHqeCKHM QeHTpOM (TO'lKa O 2 ) BTOpOH 11M CBSl3aHa CI1CTeMa KOOp,l\I1HaT
02X!V2Z2, OCI1 KOTOpoH napa,i1JIeJIbHbl cooTBeTcTBytOll!HM OCRM CI1CTeMbI OpxpYpZl" a rrOJIO:>Ke-
HHe TO'IK:' O 2 orrpC1leJIeHO KOOplll1HaTaMH X2 (ynpyrne 11 Bn6ponCpCMCll.\eHI1R) H )'2 (ynpyme
rrepeMeIQeHHH). no npJiHUHrry pa60TbI Mf KoopMHaTbI XI. X2,)'h)'2 HaXO)J,SlTCSI B npOH!Boqm-
3e. JTHM o6cTmrre.TIbCTBOM I1 06"bJiCIDIeTCR B3anMHoe rrOJIO:>KeHl1e CI1CTeM Koop):\HHaT OIXlVIZI
H 02X2.1'2Z2 OTHOCHTeJ1bHO cnCTCMbI Op\'JlVpZp.
TIOJIO:>KeHHe l{M (TOqKa 0;) BTOpOH 11M B CI1CTeMe 02X])'2=2 onpe)J,eJIeHO Koop)J,llHaTaMM
0x2,Oy2,O=2' a ocn CHCTeMbI O;x;y;z; napa.1IJIeJIbHbl cooTBeTcTBylOll.\HM OCSIM CHCTCMbI
Koop.:urnaT OiXV'2Z2'
CMeIQeHIDI I..IM (roqlm 0;. 0;) OTHOCI1TeJIbHO reoMeTpI1QeCKHX QeHTpOB (TO'IKI1 0 1 ,
O 2 ) 11M MOryr 6bITb BbI3BaHbI H3MCHeHMeM TeMneparypLI HJII1 TeXHOJIOrH<ICCKHMll npllQll-
HaMH.
YpaBHeHIDI .D.BH:>KCHIDI Mf rrOJIyqHM Ha OCHOBaHHM ypaBHeHMH flarpaIDKa BTOpOro pO-
;la (4.1).
B KaqecTBC 0606IQeHHbIX KOOp.nHHaT ,lI)UI. paMbI BbI6epeM yrnoBYIO a. H JII1HeHHble x, y;
lL'VI nepBOH H BTOpoH 11M COOTBeTCTBeHHO XI'}" H X2'}'2' TaKHM 06pa.30M, ,lI)UI. paMbI
j=o.,x,y (qa. =0., qx =X, q}. =y); ,lI)UI.l1Mj=XI,YI(qx] = x), %'1 = YI); i =X2,Y2(Qx2=XZ, qy2 = Y2).
3anHweM BbapaJKeHHe ,lI)UI. KHHeTH'ieCKOH '3HeprnH:
T = RA" +AI +A2)p2 +(B p +B, +B2)q2 + (C p +C I +C 2 )r 2 +mp(v +v +v )+
+ml(V;1 +V;J +v;J+m 2 (v;2 +V '2 +v;J, (4.100)
r.ne A". Bp' C p - rJIaBHbIe MOMeHTbl HHcpwm paMLI OTHOCHTeJIbHO oceH OX p , Oyp, OZp co-
OTBeTCTBeHHO; AI ,B). C,; A 2 . B 2 . C 2 - TO :>KC, 11M OTHOCHTeJIbHO oceH OX p , Oyp, OZp COOT
Be'I'CTBeHHO; mp' ml' m2 - MaCCbI paMbI, nepBOH M BTOpoH 11M; p.q,r - rrpOeKltHH a6COJIIOT-
HHX yrnoBLlX CKOpOCTeH Ha OCH OX p , OyP' OZp COO1BCTCTBeHHO;
v .xp' VoW' v zp; v I' v yl' v zl; V x2, V y2' v z2 - IIpoeKllliH JIMHeHHbIX CKOpOCTCH l!M paMbl, rrepBoii
H BTOPOH JiM H8 COOTBCTCTByt011(He OCH.
Eyp.eM nOJIaran.. lITO )KecTJ(OCTb paM.bl no COOTBCTCTBYlOlWiM 06061l.leHHbiM Koop.nHHa-
TaM onpe.neJIJleTCR BeJllf'DlliaMH Ga.,GJC,G y , a :>KeCTKOCTb nO.D.BeCOB 11M - BCnH'lHHaMH
Gxt.G y ) H G JC2 ,Gy2 nOCOOTBeTCTBYI011(HM 0606meHHLIM KOop,l\MHarnM.
YPABHEHI15I J],BIDKEHIDI MI1KPOrl1POCKOITOB
253
TIOTCHW1aJIblla5! 3HepmSl Mr 3aBI1CI1T OT ynpymx )J,eQ>OpMaI..(I1H -:lJ1eMeHTOB nO)J,Beca I1
3apR,lla Q. HaKanJII1BalOll.\erOCSI Ha KOH.aeHCaTopax eMKOCTblO C rpe6eHQaTbIX crpyKl)'P
( W = QZ Ie):
I (G 2 G 2 G 2 G Z 2 G Z
TI = '2 u a + xX + yY + xI XI + +G x2 Xz + Y2 Y2 ) + W.
(4.101)
,lJI1CCl1naTI1BHYIO Q>YHKl\lllO paCCeSlHHH. 3Hepnm Mr npe)J,CTaBI1M B Bl1)J,e
cD = [cb a a 2 + (b x x 2 +b),j}Z) + (b;t.lxf +byryf )+(bxJ + bY2Y )],
(4.102)
r.ae ha. h" by - KO'1<pQ>I1l\I1eHTbl )J,CMIIQ>llPOBaHllSi JlJ1S1 paMbI npl1 ee l\BI1:>KeHHH.X no o606meH-
HbIM Koop)J,I1UaTaM a, X, y; b xh b. ,,, b xz , b Y2 - K03 Hl\HeHTbI llcMnQ>HpoBaHHR AM rrepBoH 11
BropOH 11M IIpll HX ,llBll)J(eHHRX B HanpaBJIeHl111 COOTBCTCTBYIOII.U1X 0606lll.eHHbIX KOOp,l\I1HaT.
OnpClI.eJII1:'w1 0606ll.\eHHble CI1JIbI.
,lJJISI paMbI:
Qa = - - ; - : + M a; Qx = - - : - : + Fx ;
aI1 aw a<D
QJ' =-ay- (7y - ay +Fy.
.lJnR IIepBOH I1 BTOpOH 11M
an aw a<l> .
Q ::::------+F +F. o sm p r
XI :::c.. :::c.. . XI '
VAl VAl IJX I
aI1 aw a<D
QYI =- ayl - ayI - a.Y[ + FYI ;
aIT aw a<t> .
Q . =------+F -F. o sm p t"
x ax z ax z axz -"2 '
aI1 aw a<t>
QY2 = - ayz - (7yz - a.Yz + F. V 2
(4.103)
(Fo H P - aMnJIHry,l\a I1 'laCTOTa CHJI, reHepupyeMblx BI16pOIIpI1BO,QaMH).
.lJnR MaJIbIX 3HaQeHl1H ymoB a HMelOT Mecro paBeHCTBa (pl1c. 4.11):
Ma =aAmp(x+oxp)+m[(x+XI +0-"1 )+m Z (x-x2 +OX2 )]-
-ax[mp(y+oyp)+mt(Y+ Y[ + O}'I)+ mz(y- Yz +Oy )];
Fx =(m p +mt +mz)a x ; Fy =(m p +m[ +mz)a y ;
FXI ;:, mtax; FYI = mI a }.; Fx = mza x ; F Y2 = m2 a }..
(4.104)
254
rJlaBa 4. ,lJ.I1HAMHKA q3 MHKPOMEXAHI1QECKI1X rIPI1EOPOB
a)
.
Z=
6)
Zp
Z2
,
z2
.D"
Xp
y
0.
Zp
a"
y'
xp
PHC. 4.11. CHcTelllbl KOOP.llHH3T:
a - .IlAA paMbI; 6 - .IlAA 11M
3anHllleM 0606111eHHbIe CH.i1bI [CM. OPMYJIY (4.103)] C )"IeTOM CPYHKlU1H (4.101) H
(4.102):
oW . oW
Qa =-Gao.---;:::--bao.+Ma; Qx =-Gxx---bxx+Fx;
00. fu
Qy = -GyY - 0; - byY + Fy;
Qx =-G x,- oW -b x XI +F;x +Fosinpt;
] .1: 1 Ox I I
1
Q ) ' =-G ) . YI- oW -b v YI +Fv ;
I I By, _I _I
QX1 =-G x ,X2 - W -b x ,x2 +F;x, -Fosinpt;
- - OX2 - -
QY2 =-G y2 Y2 - :: -b y2 Y2 +F y2 '
(4.105)
3arrnrneM npOCKl1HM a6COJlIOTHOH yrnoBoH CKOpOCTI1 Ha OCH XpYpZ p :
p = .D;xcoso. +Qysino.;
(4.106)
q =!1 y coso.-.D x sino.;
r =.D z +0..
C )"IeTOM Toro '1TO npoeKl1HH a6COJIlOTHbIX yrnoBblx cKopoCTeH 11M Ha OCH CI1CTeM KD-
opromaT O;X;yjzj H 02xiY2z2 TaJOICe onpe,l\CmrlOTCJI BblpaiKeHHJlMI1 (4.106), 3aOHweM MaT-
YPABHEHH5I ,lJ,B¥DKEHI15I MMKPOrMPOCK0I10B
255
pmtbl ,ll,JISI Bbl<II1CJIemrn JIl1HeihlblX CKOpOCTeH DM paMbI II 11M OTHOCHTCJIbHO TO'lKH 0
(CM. pHC. 4.11).
ip jp kp
vO p = P q r .
.
xcosa+o x vcos a +oy Oz
p - p p
Ip J p kp (4.107)
VOl = P q r .
,
xcosa +xI +8xl ycosa+ YI +0>'1 O.
-I
Ip J p kp
V 02 = P q r ,
xcosa-x2 +OX2 ycosa - Y2 +8 Y2 OZ2
r;J.e ip' jp, kp - e;J.llIDI'iHble BeKTOpbI B HanpaBJIeHl111 oceH xp' Y p . zp COOTBCTCTBeHHO.
Jlh MaTpHl.{ (4.107) CJIe)J,YlOT BbIpa)KeHllS1:
V O p = ip [qozP -rGcosa + 0Y P )]+ jp (xcosa+ ox p )- pOz p ]+
+fp Gcosa + 0Y P )-q(xcosa +ox p )];
V Ol = ip [qozi -r(ycosa + Yl +OYI )1+ jp '(xcosa+ Xl +OX 1 )- pozJ+
+ kp [p(ycosa + Yl +0)'1 )-q(xcosa +XI +OX 1 )];
(4.108)
V 02 = Ip [qOz2 -r(ycosa - Y2 + OyJ] + jp (xcosa-x2 +OX2)- p8 Z2 1+
+ kp [p(ycosa - Yz +OY2 )-q(xcosa-x2 +ox 2 )].
3anl1111eM npOCKUI1H a6COJIlOTHbiX JIHHeHHhIX CKOpOCTCH l.{M paMbI, nepBOH H BTOpOH
11M c yqeTOM (4.108) Ha COOTBCTCTBYlOllJ,Me OCM:
V x =x+qoz -/ cosa+o y ) ;
p p \Y p
v YP =y+r(xcosa+o x )-p8 zp ;
v Zp = pGcosa+o},J-q{xcosa+oxJ;
(4.109)
256 rJIaB3 4. JJ;HHAMI1KA q3 MHKPOMEXAHI1QECKI1X rIPI1EOPOB
V XI = i + x. + qO::1 - ,.(vcosa + .1'\ + 0\'1 );
v,. = i'+ i'l +r ( xcosa+x l +0 \ ) - [ :>0. ;
, I - . 1-'
V=I = p(vcosa + ."1 +0 1 '1)- q(\'cosa + xI +0\1):
V x, = .\-- x:! + qO;:. - ,.(\'cosa - y:! + 0 1 , );
V ,. = j' - .i'::! + ,(\,'cos a - x::: + 0 \, )- PO . ;
v;: " p(vcosa-y::! +0\'1)-Q(,cos'a-x:! O'\'2)'
(4.110)
(4.11] )
TIo;1arWl yro 1 a MaJIblM (sina a. cosa 1). nepenllweM Bblpa:>KeHIUI (4.109)... (4.111 )
c yqeToM paaeHCTB (4.106):
"x, = X+ (nl' -nxa)o::p - (n:: + a)ll'+ 0,.)
"y, =jl+(n;: +a)(x+oxJ-(n +nl.a}ozp:
v"p = (n x +nya)(v + 0,\' )-(n)' -n\a}( :+o p);
V XI =x+xI +(n)' -nXa ;:, -(n;: + a.}(v + YI +0,1");
VI'I = .i'+ j'l +(n;: +a)(x+x, +0'\1 )-(n x +n,la}o::\; (4.112)
"ZI = (n x + nya)(v+)', + 0."1 )-(n... -nxa)(x+ XI + OXI);
V 1' , = X-X::: +(n,. -np)o;:, -(n;: +a}(v- Y2 +0)',);
V y: = y - jr2 + (n + a}(x - x + OX c )- (n x + n)'a )0=: ;
V: 2 = (n x +nya)(v- Y2 +0"2 )-(n)' -n x a)(x-X2 +OX2)'
BbJ1IHCJIHM CJIellYIOlIDIe npoH3BO)J,Hble (n x = const, ny = const, n z "" const):
aT ( ) Op ( ) oq ( Ovx Ov}' Ov. ]
-::::- = A, + AI + A 2 p - + Bp + B I + B 2 q - + mp v x p + v y p + v:: 'p +
m fu fu pfu pfu pfu
( Ov"1 OV I' Ov z J ( Ov]{, Ov y , Ov", )
+ mJ v x + V } ' . , + v _ --L + m., V:x . + v v . + v - - =
1 fu ) fu ., fu - 2 fu . 2 fu -2 oa
= A, +AI +A )(n x +nya)n y - Bp +B, +B2)(n y -n r a)n x +(n z +a)x
+mz -)'2 + yJnr -(x-XZ +OX2 )nJOZ2 }+m p { +(ny -nxa)ozp -n xp OzJ+
+ -(ox +QiX zp](-ny Ozp)+[(nx +np) +OYp)-(ny -nxa)(x + oX p )]x
X l ny + yJ+nx + "P)} }+ml {[x+x, +(ny -nxa)ozJ nxoJ+(Y+ y, -(nx +n}.a)ozJx
x -ny J+[(Ox +Qya + YI +OyJ-(oy -nxa) +x, +OXI )Hny + y, +Oy)+
+ [(ox +n y a)G-Y2 + ,J-(Oy -n X a)(x-X2 +O J. [n Y &-)'2 + >;.J+n1' x+XZ +01'2 ]);
(4.113)
YPABHEHHJI ):{ByDKEHJ1JI MJ1KPOrJ1POCKOTIOB
257
oT ( , ) or [ av , av 1', av = ]
= C p +C 1 +(,2 p-+m p v.. --:'2-+ VI' :' +V z - . 'p +
ca 2a p CO .p ca P ca
+mJ ( V tl aV:'1 +V,. (::'1 +V= i)V I ) +m2 ( VX' () > + V", av' '2 +V=, av 2 ) =
Ga oa 1 Ga. 'ca.'" aa. . ad.
= (CI' + C I + C 2 )(0" + 0.)+ I1lp { f + (0.1' - 0xa. )0" - (Oz + o.) + 0 I'p)] [- (v + 0)'0)
+ [i 1 + (0= +u)(,+oJ-(O, +Oya.)oJ(x+o.JJ+ In, {[i-+.i l +(0)' -Oxa)o=l-
-(0= + a)G' + Yl + \J][- G'+)'1 +0.1'.)]+ [i,+ J't +(0= +0.)',+ XI + 0....)- (ox +O..a.)oJx
x (x -r XI -r o.,.)}-r m2 {[-r-.i-2 +(0), -O,.a)o;;, (O= + 0.)- G,- 12 + oJ][- (v- Y2 -OIJ+
+[V .V2+(O=+0.)(X-X2+0.J- (01:+0) a)oJ(X-X2+0.J}' " "
(4.1 14)
; : = {cp + lnp [(v +oYr.f + +Ox) ]+C I +lnl[(y-r YI +Oy} + ,,"+x\ + Ox} ]+
+C 2 + ln 2 [G,- )'2 +01'1 f +(X-X2 +OX2 f Dii+
+mp [(x+o xp ) ji- +OJp ).t ]+ml [(y+ Yl +0)'1 )(- X-X\)+(X+X I +0.1: 1 )(y+ j\)]+
+m2[(Y-)'2 +0,1'2 )(-X+X2)+(X-X 2 +OX 2 )(y- h) ]+2(Oz +o.){mJy +oyp)+
+x(x+oxJ]+mJ [(j!+ YI)G,+ YI +0,1'1 )+(.f+.fj)',"+xl +OXI )]+
+ 1Il2 [(y - h )(y- Y2 +0)', )+(x - '\-2) - X2 +Ox,)]}+ M(n", 0 v' 0),
. -'
(4.115)
rJ1.C
M(Ox,O)" 0)= np [oJv+ 0Y p )-O)x+ oxJ]ozr + ml[ Ox &+ YI +0,1'1)-
-O)'(X+X I +0-'1 )]OZI +111 2 [Ox&-)'2 +OyJ-0,)X-X2 +OxJ]OZ2 }a-
-lnp[(Oy -Oxa)y+(Ox +Oya)xJoz -mj[(Oy -O...a)(y+ y\)+(Ox +Oya)(x+XI)]Ozl -
-ln 2 [(0). -Oxa.)(y - h)+(Ox +Oya)(x- x 2)]OZl'
(4.116)
TaKllM 06pa30M, B COOTBeTCTBHI1 C 06ll.\eH OpMOH (4.1), rll.e npOH3BOll.HbIe onpe.nen11-
IOTC11 Bblpa:>KCHI111MH (4.113)...(4.115), nOJIyqcHo ypwmeHHe ,l\BH:>KeHHJI Mf no KOOp.nu-
HaTe a..
EcnH npe)J,nOJIO)KHTb, '1TO MMbl CMell.\eHIDI 0 (c mo6bIM l1H.z:\eKCOM) I1 yrnoBb\e cKopocm
Ox' OAOx <Oz,Oy <oz), TO MO)KHO B BbIpa:>KCHHH.X (4.113)...(4.116) npeHe6pe% BeJ1H'Il1-
HaMH nopSl,!l,Ka HX npOl13Se,l\eHHH (8 mo6blX CO'leTaHHH.X). B 3TOM c.rry<iae aT lOa I::: 0,
M(O x' n 1'0 ) 0 II ypaBHeHlle .nBH)KeHlUI Mf no Koop.n,maTe a npHHHMaeT BH)J.
258 rJIaBa 4. .lU1HAMHKA 43 MI1KPOMEXAHH"lJECKHX IlPI1EOPOB
{C p +mp[(v+o."p f +(x+oxpf ]+C, + 111\[(y+ )'1 +0)'1 f +(X+X\ +OXI f]+
+C 2 +m 2 Kv- Y2 +0_"2 f + &-X 2 +OX f]}a +
+ mp [(X + OXJy-(V + OyJ\'J+ 11l\[-(V+ Yl + 0"'1 )(.\,+i\)+ (X +X I +0-"1 )C1 i + y,)]+ (4.117)
+m 2 [(1'- Y2 +o\" )(-x+x 2 )+ (X-X2 + ox, )(;;- h)]+2(0" +a)x
x {m p j{l' + 0.\1' r:.i(X + oX p )+ m I [Ci'+ .i'\)(\ +)'\ +OYI )+(x+x, )' +x, + OXI )]+
+ m 2 [(j,- J'2)(V- J'2 + \. )+(.i-X2)( -X2 + Ox ) ]}= Qa.
nOCTynWl 8HaJIOni1fHO, B COOTBeTCTBI1H C 06rn:eH <jJOpMOH (4.1) ITOJ1YQI1M ypaBHeHHe
.nBH:>KeHHH. Mr no KOOp,IJ:MHaTe x 6C3 .nonyrn:eHI1H 0 MaJIOCTH yrJIOBblX cKopoCTeH Ox, Oy no
CpaBHeHlUO C H3MepJleMOH n.=:
(m p +m, +m 2 )x+m I x\ -m2x2 - [m p (v + 0.\1' )+mI(V+)'\ +0-"1 )+m2&-)'2 + Oy )]a-
-2(0= + 6.)[m p y+m,(j1+ y\}+m2(j,- Yz)]-Ox6.(m p o zp +111,OZ! + m20z2 )-(0= +6.)x
x {mJ(O= +6.) +OxJ-(Ox +Oya)ozp ]+mI[(O= +6.Xx+ x, +o.J-(Ox +Oya)oz.J+
+m 2 [(0. +6.Xx- x 2 +OxJ-(O +O)'a)OzJ+111I[(OX +o)'a)&+)'1 +OyJ-(Oy -Oxa)x
x (X+XI + OXI) m2[(OX +Oya)(y-)'2 +O)' )-(0)' -Oxa)(X-X2 +OX 2 )]}= Qx' (4.118)
B npe.rmonO:>KeHHH MaJIOCTH yrJIOBbIX CKopocTeii 0x« Oz, Oy« n" H BeJIHql1H 0
(c !Il06bIMH HlUleKcaMH), eCJ1H npeHe6peQb CJ1araeMIJMH nOpH,l\Ka HX npOH3Be CHllH H Bb\IIIe,
ypaBHeHHe (4.118) 3anMllIeTCJI CJ1eJ1.Y1OIl.\HM 06pa30M:
(m p + m\ + m2)x + mlxJ - m2 x 2 - [m p & + OyJ+ m, & + )', + 0YI )+ m2 (y - )'2 + 0Y2 )]a-
-2(n z + 6.}[m p y+ mlv+ y\)+m2(Y- Y2) ]-(oz +6.? [mp(x+oxJ+mI(X+XI +Ox l )+
+m2 -x2 +o,,J]=Qx'
YpaBHeHMe J1.BMJKeHIDI M:r no KOOp.lUfHaTe)' HMeeT BHll
(4.119)
(m p +m\ + )y+mIYI -m 2 Y2 -[mp +OXp)+mI(x+X\ +OXI )+m2(x-x 2 +OX2)]a-
-2(oz +a)[').i+ml(x+.i\)+m2(.i-X2)]-0},a ! mpozp +mloZI +m 2 o z J-
-(!1z +a pKOz + a)), + o)'J-(ny -nxa)<\ +m\[(Oz + 6.){y + )'1 +OYJ )-(o}' -oxa)ozJ+
+ Roz +a)(v- Y2 +Oy )-(Oy +ora)oz.]}-(g" -nya)x
x {mpl(ox +Oya)(y + l)yJ-(Oy -nxa)(x oxJJ+ml[(ox +O)'a)&+ YI +o)'J)-
-(Oy -nxa}{x+x\ +oJ]+ [(ox +n y a)(v-)'2 +OyJ-(o)' -Oxa)(X-X2 +OxJ]}=Qy-
(4.120)
YPABHEHIDI ,[{BIDKEHMJI MHKPOfHPOCKOITOB
259
)],JISI MaJIbIX 'ma'leHMM BeJIl14l1H n x , ny, 0 aHaJIOrllQHO COOTHOllleHl1lO (4. 119) nOJIY'lI1M
ypaBHcHHe
lIp +m\ + mJji +m\j\ -m2Yz -[mp(x+o>J+mI(x+x 1 +oxJ+m2(x-x2 + OxJ] a +
+2(0.... +a.) npx+ml(x+Xl)+mAx-X2)]-(nZ +a.)2[m p &+OyJ+mI(y+ YI +Oy,)+ (4.121)
+m2(v- Y2 +oJ]= Qy.
)],m1 ncpBoH (Macca ml) I1 BTOpOH (Macca m2) HM B COOTBeTCTBl111 C 06ll.leH <pOpMOH
(4.1) nOJIyql1M ypaBHeHIDI )J,B11:>KeHllS1:
ml{ +xl -nxoz,a.-(v+ Yl +ozJa-[2(Y+ YI)+(n Z +a.)(x+x] +oxJ-
-(n x +nya)ozJn z -ra.)+[(n T +nya)(y+ YI +OyJ-(ny -nxa)(x+xl +OxJ]x
x(o.y -nxa)}= Qx,;
(4.122)
m l + Y] -nyo=,a.- +Xl +ox 1 )a- [2(x+x 1 )-(n Z + a.)(y + Yl +OYI)+
-(ny -nxa)oZ! ](n z +a.)-[(n x + nya)(y + Yl +OYI )-(n y -nxa)(x+x] + OXI )]x
x(nx +nxa)}= QYI'
m z x+xz -n X O Z2 a. +(v- Y2 + oyJa -[2(Y- Y2)+(n Z + a.)(X-X2 +OxJ-
-(n x +n y a)oZ2 ](n z +a.)- [(n x +nya)G'- Y2 +OY2 )-(n y -n x a)(x-x2 +OX2 )]x
x(n y -n X a)}= QT 2;
m z {- Y+ YI -n v o z ,a.-(X-X2 +0:<. )a-[2(X-X 2 )-(n Z +a.)(y- Y2 +OY2)+
-(ny -nxa)ozJ(oz + a.) + [(n x +:Oya)(y- Y2 +OyJ-(ny -nxa)(x-Xz +o}J]x
x(nx+n y a)}=QY2'
(4.123)
)],JIH MaJIbIX 3HaQeHllH BCJII1'lI1H n x ' ny. 0 H3 BbIpIDKeHHH (4.122) I1 (4.123) nOJIyqMM
ml {i+x\ -(y+ Yl +0)'1 )a- (y+ Yl)+(n z +a.)(X+Xl +OXI)] (n z + a.) }= QXI;
m2 + YI +(X+Xl +OXI )a+ (X+XI)-(nZ +a.)(y+ Yl +OYI)] (n z +a.)}=Qy..
(4.124)
m2 { -i+i 2 +'- Y2 +o } ' ) a+r2(y- h)+(nz +a.)(X-X2 + ox. )](n z + a.) }= Qx.;
\)' 2 - - (4.125)
m2{ Y+ Y2 -(X-X2 +OX2 )a- (X-X2)-(nZ +a.)(y- Y2 +OyJ](n z +a.)}= QY2'
260 rJlaBa 4. )]JIHAMHKA q3 MI1KPOMEXAHI1QECKI1X rIPHEOPOB
B nOJI)"-leHHbIX ypaBHeHIUlX 0606ll\eHHbIe CIIJ1bl onpe..'J.c.1S110TC B COOTBeTCTBIUI C cpop-
MYJIaMI1 (4.105).
O':lCBIt.rJ:HO, ':ITO cnCTeMbI ypaBHeHnii (4.117), (4.119), (4.12]), (4.124), (4.]25) Onl1CbI-
BalOT .w!H:JKeHl1e Mr He TOJIbKO )J,JIH. MaJIblX 3Ha':leHI1M BeJIl1'II1H 0" 0, ' 8, HO 11 1V1 1Ha'1e-
HIm. Ox::: 0,. = 0, B - HeMaJIOe.
3arIMllleM. IIcnOJlb3)'1I BbIpa:>KeHIH! (4.113)-{4.116). (4.I1R), (4.120). (4.]22), (4.123),
cnCTc fY ypaBHeHIli1 .1BIDKCHIiSi Mr .'VISI CJIY<IaSI 8 = 0 (OTC)'] CTB)'lOT .1lO6bIe C fell\eHlISi UM
paMbI II HM):
{C p +mp(1" +X )+C I +1111 [CrT .1'1 Y + (x + x l f]+ C 2 +1Jl2lCv- )'2)2 +(X-X2Y J}o. +
+mp( ;- .1'1')+ 111I[-(V+ YI)(X+XI)+(X+X,)(V+ YI)]+
+m 2 [(y- .1'2)(-'1' +1'2 )+(x-x 2 )(V - .1;2)]+
+ 2(0;: +a){mp(j '+iX)+ 111 1 [(\' + ,i'I)(Y + .1'1 )+(i+ il)(x+xl )]+
+ m [(i'- ,i"2)(Y- )'2)+ (i- X 2 )(X- X2)]}-( +AI + A 2 )(O, -r o,.a)Oy +
+(B p +B I +BJ(O.,. -oxa)n.. -mp[(Ox +O)ah'-(O\, -Oxa)x]{OyY+O,x)-
- ml[(Ox +O,.a)(y+ YI)-{OI' -Oxa)(x+x[)] [0,. G1+ YI)+Ox (X+XI)]-
-m2[(Ox +Oya)(y- Y2)-( Y -oxa)(x-xJ[Oy (y- )'2)+0", (X-X2)]= Qa.'
(4.126)
(m p + m I + m 2 )x + mix i - m2x2 - [(m p + 111 1 + m2)Y + 111IYI - m 2 )'2]0. -
-2(0= +a)[(m p +ml +m2).>,'+m l )1 1 -m 2 >'2]-(Oz Tar [(m p +1111 +m2)x+mIX[ -111 2 X 2]+
+(o)' -Oxa)«Ox +o)'a)[(m p +m l +m2))'+ml)'1 -m2)'2]-
-(0)' -o.-a) [(m p +mI +mJx+mIXI -m2 X 2]}=Qx.
(4.127)
(m p +m l + m:Jy + m l Yl -m 2 Y2 -[(m p +m I +mJx+mlxl -m 2 X 2 ]o.+
+2{Oz +a)[(m p +m I +m2)x+m I X I -m2X2]-(Oz +oy [(m p +ml +m2))'+mIYI -1112Yz]-
-(Qx -o}'a)«Ox +o)'a)[(m p +ml + m2))' + ml)'1 -m2)'2]-
-(0)' -nxa) [(m p +mI + mJx + mixi -m2 x 2])= Qy-
(4.128)
m){x+xl -(y+ YI)o.-[2(5'+ .YI)+(Oz + a)(x + XI)](Oz +a)+
+ [(ox + O}'a}(y + YI)-(O)' -Ox a )(x + xI )](O)' -Ox a )}= QXI;
(4.129)
ml{Y+ YI +(x+xl)o.+[2(x+x,)-(Oz +a)(y-r )'1)](Oz +a)-
- [(Ox +Oya)(y + )'!)-(O)' -OXa)(x + XI)](Ox + o}'a)} = QYI;
{-X+X2 -(y- Y2)a + [2(y- .Y2)+(Oz +a)(X-X2)](Oz +a)-
-[(Ox +Oya)(y- Y2)-(n)' -n x a)(X-X2)](0}' -Ox«)}::: QX2;
(4.130)
(4.131)
YPABHEHHH ,L(BJlDKEHJ15I MI1KPOfI1POCKOrrOB
261
1I/2{- Y+ h -(X-x 2 )a-[2(x-X2)-(n z +6.)(y- Y2)](n z +6.)+
+ [(n x + nl'o.)(y - Y2)-(n y -np)(X X2)](nX +nl'o.)}= QY2'
(4.132)
CHCTe Ja ypaBHeHHH (4.126)--{4.132) OITHCblBaeT l HHaMHKY Mf LL- H LR-THnOB npl1
neCll!>1MeTpHJ1 MaCCOHHepUHOHHbIX H ynpymx xapaKTepllCTllK I1X 3JIeMeHTOB npH H3MepeHI11l
yr.rrOBOM CKOpOCTH n z OCHOBaHHSI, KOTopoe, KpOMe TOro, ITepeMellJ,aeTCH C nOCTOHHHblM YC-
KOpeHl1e H Bpall(aeTCSI co CKOpOCTSlMH n x ' n)' OTHOCI1TeJIbHO nepeKpeCTHblX oceH.
HaJm lHe pa BHToro nporpaMMHoro 06ecne ' leHHSI DJUI '3BM neJIaeT ,llOCTYITHbIM 'lHC-
.1eHHoe liCC"lClJ,OBaIiHe JIl060H 113 ITOnY leHHblx CMCTeM ypaBHeHI1H.
O<JeBHlJ,HO TaK)Ke, <no H HCCnelJ,OBaHHe Ha :JBM ITpelJ,ITOJIaraCT pa3YMHoe YITpOllJ,eHHe
MaTeMaTH'-IeCKHX OlJ.C:JeH, HMeSi B BHlJ.Y UeJIbIH pHlJ. nOITymeHlIH MaTeMaTJi<JeCKOrO H 1l3WJe-
CJ{{WO xapaKTepa. KOTOphiC H3naQ1lJlbHO ,'].eJIaIOT HCXO)J,HYIO MaTeMaTHqecK)'IO MO,'].eJIb JIlO60H
InOJIHOTbI" npIi6.'1H:>KeHHOH.
E. qaCTHbie ypaBHeHIUI
YnpOCTHM MaTeMaTHtJeCKHe MOlJ,enH Ha npHMepe CMCTeM ypaBHeHJll! (4.126)-(4.132).
IlpHMeM CJIe,'].}'1OIlJ,He lJ,onymemISl, HMex B BHJ1.Y TO'lHOCTb TeXHOJIOrH'lCCKHX npoueccoB 1I3ro-
rOBJIeHHH Mf:
In l = 111 2 = In; XI = x2 = Xo ; YI "" Y2 "" Yo ; AI = A 2 ; BI = B 2 ;
C I = C 2 ; b XI = b X2 = b xo ; b y, = b Y2 = byo ; G XI = G X2 "" G xo ;
GI' =G y =G y ; Fx =Fx =F" ; F y =FI' =F y ,
_I 2 0 1 2 .0 1 .2 0
H nOJIy<mM CHCTeMY ypaBHeHHH
[C p + (m p + 2m)(\"2 + y2 )+2C, + 211l(\" + y )]a+ba6.+{Ga +( + 2AI)n;-
-(B p +2BI)n; -Inp(nyy+nxxr -21ll[n &,2 + Y5)+2n x nAxy+xoYo)+n;(-r 2 +x )]}a.+
+ 2(n z +6.) &np{;.j. + xi") + 2m(w +i:x + YoYo +xoxo)]+ mp(xji - yX)+
+ 2m(xji- yx+xoYo - YoXo) = Ma -oW lOa.;
(4.133)
(In +2m) { X+b x X+ [ Gx (n z + 6. f ] X- ya-2(nz +6.)y+
p Inp+2m
+(n 1' -nxo.)[(nx +nya.)y-(ny -nxo.)x])= Fx -oW lox;
(4.134)
n +2m) { Y+b Y+ [ Gy (n z + 6. f ] y+xa+2(n z +6.).x-
P y III +2m
P
-(nx -nya.)[(nx +n1'a.)y-(n1' -nxo.}x]}=Fy -oW loy;
262 rJIaBa 4. JIHHAMHKA q3 MHKPOMEXAHJ1QECKHX OPJ1EOPOB
m{. +xo - (y+ YO)a - [2(.i'+ .VO)+ (n= + 6.)(x + xo)](n, +6.)+
+ [(n" +nyo.)(y + .1'o)-(n.,. -n"o.)(x+xo)](n,. '-Oxo.)}=
=-G...ox o -b"'o..io -oW 18.'1: 0 +1\0;
"{i;+ Yo +(x + xo)a +[2(x+xo)- (n= +6.)(.1' + Yo)](O= + a)-
-[(0... + 0) o.)(y+ )'o)-(n)' -nxo.)(x+xo} J(n... +nyo.)}=
(4.135)
= -GyoYo - b)oJ'o - oW 10)'0 + Fro;
m{-x+_ o + (v- yo}a + [2(i.- Yo)+(O= +a)(x-xo)](n= +a)-
-[(0... +0".0.)(1'- Yo)-(Oy -n...o.)(x-xo)] (Oy -Op)}=
=-G... Xo -b x io -oW 10.;-':0 +F... ;
m{- jt+ Po -lx-xo)a- [2(.i-,i )-(n:: +6.)0, - )'0)](0= +a)+
+[(0... -rOp)(y- Yo)-(O) -n...o.)(x-xo)] (n... +n,.o.)}=
=-G,. yo-b, j'o-oWloyo+F r .
_ 0 _ 0 _ 0
(4.136)
EY.IleM ITo.rraraTb, qTO J1.Jlj{ Mr no pHC. 4.10 pq - :no nOCT)'naTeJIbHOe lJ:BH)!(eHl1e 11M no
Koop;umaTe y(o. = 0) (mpocKon LL-THna).
.lJ..'Di paccp,mlpHBaeMoro CJI)"IW! ypaI!HeHHe (4.133) npHHl1Macr BHlJ:
2n.J(mp + 2m)Gj1+.u)+ 2mV'oyo +ioxo) ]+(m p + 2m)(J.ji- yX)+
+ 2m (xo.vo - )'o- o)= Ma -oW loa.
)],aHHOe ypaBHeHHe MO:>KeT 6blTb HCITOnb30BaHO )J,nSl BhI'IllCJIeHllSi .nHHaMI1'leCKHX
peaKl.uiii B CBSl3S1X, oCYIlteCTBJIHIOIlU1X KpeITJIeHHe paMKH C KopnycoM, ITPH paC<IeTe HX
Ha npO'lHOCTb.
CllCTeMa ypaBHeHHH (4.134) H CHCTeMbI (4.135), (4.136) npHHHMaJQT BH.n
(m p + 2m) [ x +b...i + ( Gx 0 ; ] x-20zy+oAOxy-oyx)1 =Fx -oW lox;
mp + 2m J
(4.137)
(m p +2m) [ Y+b y Y + ( Gy 0 ; ] Y+2n z x-n...{0...y-oyx)1=F y -oW loy;
mp +2m J
i + x+ "'0 +[ 2(yx Yo)+ o.,(x + xo)]o., + [o.,(y + yo)-o.y(X + Xo)Py}
= -GXjjXO -bXflxo -oW I 8xo + F Xjj;
(4.138)
m{:ty+ Yo + [2(x + xo}-Oz(y + yo}]Oz + [n...(y + yo)-ny(x + xo)]n...}=
=-GyGYo -byoYo -oW loyo +Fyo.
YPABHEHI15I .n:mDKEHIDI MI1KPOf11POCKOrrOB
263
B ClfCTeMe ypaBHeHJfH (4.138) BepXHMe 3HaKH COOTBeTCTByIOT JIeBOH, a HIDKHHe - npa-
BOH HM.
O'leBJfJl,Ho, tlTO H3 CI1CTeM ypaBHeHllH (4.137), (4.138) KaK qaCTHbIH CJIyqaH MOryr 6bITb
IIOJ1YtJeHbI ypaBHeHllH. D,Bll:>KeHl1S1 mpocKon8 C O.lmOH 11M (CM. puc. 3.47):
m[xo -(2yo +.ozxo).o z + (.oxYo -.oyXo).oy]= -Gxoxo -bxoxo + F:xo -oW 10.,\"0;
(4.139)
m o + (2x o -.ozYo).o z - (.oxYo -.oyXo).oJ= -G v Yo -by Yo + Fy -oW n%.
_ _ 0 . 0 . 0
TIOJIYtJlfM ynpomeHHbIH Bap1l8HT ypaBHeHMH nBU:>KeHWI rMpocKona LR-Tllna nOJIafID!,
'ITO ynpyme :meMeHTbI nOD,BeC8 11M D,OnycKalOT MX .nBU)I(eHHe TOJIbKO B HanpaBJIeHMH OCM X,
T.e. xI ::c: Xz '" xO;YI = Y2 = O. CucTeMa ypaBHeHuH (4.134) OCT8eTCSI 6C3 u3MeHcHl1H, a ypaB-
HeHue (4.133) U CJfCTeMbI (4.135) JI (4.136) npllHllMalOT BH1\:
[ C p + (m p + 2m) (x 2 + y2 )+ 2{c1 + mx )]a + b a 0.+ { G a + ( + 2AI)X
2 ( '} ) 2 ( )2 [ 2 2 2 ( 2 2 ) }
X.oy - Bp +_B, .ox -mp\.oyy+.oxX) -2mL.oyy +2.o x .o y XY+.o x X +XO a+
+ 2(.o z +0.) [mplrY + xx) + 2m(yy+.u+xoxo)]+ (m p +2m) (xji- )IX) = Ma -oW loa;
(4.140)
m {+ x+xO + y&. + [2y+(.o z + a) (x + xo)](.o: + a):!:
+ [(.ox +Oya)y-(.o y -.oxa)(x + xo) ](.oy .op)}=
'" -G xo Xo - b xo Xo + Fxo - oW I oXo;
(4.141)
m[:!: ji + (x + xo)&' + [2(x + xo)-(.o z +o.)y](.o z +0.) +
+ [(.ox +.oya)y-(Oy -.o"a)(x + Xo)](Ox +.oya)}=Fyo'
B CUCTeMe ypaBl-leHHH (4.141) BepXHHe 3HaKll COOTBeTCTBYKJT JIeBOH 11M (CM.
puc. 4.10), 8 HH:>KHlle - np8BOH. IJ:tUlee MO)l(HO npe.nnOJIO:>KHTb, 'ITO COOTBeTCTByIOllJ,HM BbI60-
pOM p83MepOB paMKI1 H :meMeHTa nOABcca, pa60TalOruero Ha KJ>yqeHHe, 06CCneqHBa1OTC
p8BeHCTBa x = y = O. B T8KOM CJIyqae CMCTeMa ypaBHeHMH (4.134) 06UYJIHeTcH. a ypaBHeHIDI
(4.140), (4.141) 38ITl1CblBaKJTCR CJIe.nyKJIUHM 06p830M:
[C p +2(C I +mx )]&.+bao.+{Ga +( + 2AI)O -[B p +2(B 1 +mx ]O; }a+
+4m(Oz +o.)ioxo =-oW loa;
m{xo - [(.0: +0.)2 +(Oy -.o:xar ]XO}= -Gxoxo -bxoxo + Fxo -oW I o.xo.
(4.142)
TIOJIyqeHHbIe ypaBHeHHH. .nBH)I(eHHR C y'lCTOM 3HeprnH W. HaKallJIMBaeMOH Ha 3JIeKIpO-
CTaTIf'lCCKI1X npe06pa30BaTeJIH.X, onHCbIBaIOT .nHHaMHKY rHpocKonoB LL- H LR-THIIOB npH PaJ-
HOH CTeneHH yqeTa KOHCTpYKTHBHb]X H B03MyWalOIlJ,HX aKTOpOB. YpaBHeHID£ CnpaBe)J,J1l1BbI H
Am, ,l\pyrnx wnOB npeo6pa30B8TeJIeH (MarHHT03JIeKTpHqCCKUX, 3JIeKTpOM8fHJITHb1X), J])UI KOTO-
pbIX 3HeprnR W.noJDI<Ha Bbl'lHCJIH.TbCH. B COOTBeTCTBHH C I}IH3H'leCKOH npHpO,l\OH I1X }'1iKQHOHH-
pOBaHl1Jl.
264
rnaB8 4. ,Z:lIlHAMIlKA 3 MIIKPOMEXAHII4ECKHX 111'1 [l)OPOB
4.5.2. MIII ponlpoc..on"l RR-nma
YpaBHeHlU1 llBlllKeHItSi Mr RR-Tlma pa3JI11'1II0rO KOHCTPYKTItBiIOro IICIlO!IIICHIIR
(paMO'lHbJe, OllHOpOTOpHblC, lIB 'XpOTopHbIe C KIIHCManl'lccKoi1 CBSI%IO MCIK,3,Y poropar.m II
6e3 Hce II 1'..:1.), a TalVKC npll He06xOllIIMOCTII C Y'leTOIII napal\lCTpOB, I\mOpble MmKHO rpaKTO-
BaTb KaK TeXHOJIoru'leCKlle norpeWHOCTII IIX 113mTOBJ1CIHUI, Moryr 6hITh 6e3 npIHII ltrIllaJlb-
HbIX 3aTpY)J,HCHllii ITOJIY'leHbl C IICn0J1b30B3HlteM M3Tepllana, 113J1U>KCHHOro B Ilo,llpa'm. 4.5.1,
B pO.1H MeTO:J,It<JCCKoro YK33aHIISI.
YpaBHcHlH1 .!1.BIIIKCHI!SI (MaTeMaTII<JccKIiC MO]J,eJIIt) ,!J.JISI pa3:J1I'lHhIX CXCM Mr, B TOM 411C-
Ile H pOTOpHbIX, np1me.n-eHbl B paoOTax [8,22.23], a TaK>KC B MaTCpn3.1aX .npyrnx aBTOpoB.
DO.1Y<JllM ypaBHeHIU! ,!lBIt>KeHliSl pOTopHom Mr npllMeHllTeJlbBO K cxeMe. nOKa'381111011
H8 pnc. 3.9.
A. 0606meHHLle ypaBUetUlJl
I1pll BhIBo.n-e ypaBHeHIIH llBlliKeHiUl Y<JTeM KOHe<JHYJO )KCCTKOCTh ynpyrux ')JlCMCHTOB
(TOpCIIOHOB) Ha KPY<JeHuc M n3ru6, CTaTII<JecKYKJ Hcc6anaHCllpOBaHHOCTb pOTopa II 6Y,3,eM
nO:JaraTb, <JTO OCHOBaHlle Hap}i)J,y C nepeHOCHhJM BpaUlcHlleM no.nBepraeTCSI JIllHeiiuoMY YC-
KOpeHHIO BI16paUlfli 11 ITOCT)'naTeJIhHOm nepeMellJ,eHII}i.
BBelleM C!1ell)'101llI1e elfCTCMbI KOOp,!lI1HaT (pllc. 4.12). C KopnycoM CBSl3aHa ClfCTCMa KO-
Op,l\HHaT 0),17. B HatJ8J1bHblH MOMeHT npo,,\OJIbHble oell TOpCIIOHOB COBIl8)J,aKJT C OCSIMII OX,
OY. OCHOBaHI1C BpalllaeTC}i c yrnoBbIMn CKOpoCTSlMlt .o.y, Q.1 nocTynaTeJIbHO nepeMelllaCTcR
C ,1I1HeHHbIMI1 YCKopeHlll1MIt a;r;, a... a, (B COCTaB a; BKJIlotJeHO YCKopcHne CIiJIbl TH.iKecTIi) Ii
nO.:.lBepiKeHO !HfHeHHOH B116paLHm. oTo6paiKeHHoH B Bn,'\e BI16ponCpeMeIlleHlfH Xu, )'.1> Zo no
COOTBeTCTBYfOlllUM OC}iM. Bcne.n-CTBne ynpyrl1x CMellleHI1H nO,,\Beca em reoMeTPHtJecKllH
ueHTp H3 TO'lKH 0 CMeCTHJICSI B TO'll{)' 0 1 , Koop,!U1HaTbI KOTOpOH x, y. Z. OCM CI1CTeMbI KOOp-
J1.HHaT qX},'=' nap<L1JIeJIbHbI COOTBeTcTByKJLI.U1M OCSIM 0...\'Yz. I1onO)KCHne CnCTeMh. KOOp-
O v 0 ' II
;rnH8T ]X)'Z, CB1I3aHHOH C pOTOpOM, OTHOClfTeJIbHO enCTeMbJ KOop,'\llHaT IX Y Z onpe.neJIeHO
oj
z
6)
x
z
}j.Y
y
PRe.. 4.12. K BWBony ypaBJleJlll1i nelf:ilCelfJlfI Mr RR-THna:
a - JCHHeMaTH ecKajj cxeMa; 6 - CHCTeMhl KOOPJ.\HU8T
YPA.BIIEHI151 BJ.1)j{EHH5I MHKPOrHPOCKOnOB
265
yrnaMIl y, r1. [3. nOCJICJ]OBaTe.1hHble ITOBOpOThi Ha KOTopble onpeD,eJlSlfOT CHCTeMbl KOOpJUI-
lIaT OJ '"}Y 2 . 0 1 '"1.1'121 II O I 'YZ. 110JIO)KcIme I {M pOTopa B TO'IKe 0 0 onpe,1eJIeHo BeJIl14HHaMit
ii.\", OJ' B n;IOCKOC m 'y. OCII CIiCTeMLI Koop,11maT OU.\"Qvo:-o ITapaJIJIeJIbHbI COOTBeTCTBYfOlllJlM
OCJI eHCTCMbl KOOpD,IIH8T 0 1 ,'9'2'.
Yp8BIICHIHI Jl.BU}lccmm rupOCKOWI ITOJlY'IHM aHaJIOrH4HO npeD,bI)),yllleMY, Bbl6Hprur B Ka'Ie-
CTBe o606utCHHhiX /mop,1I1HaT B ypanHellHSIX J1arpamKa nCJIH4J1Hbl a, , y ,x, y, z.
!>YD,eM nOJIaraTh, 'ITO BBH,1Y MaJIhIX 3Ua4eHJlH Ox, by no cpaBHCHHfO C reOMeTpl1'.JCCKI1MH
pa3MCp8MH paropa n nO.l1BCCa rJIaBHble (arHOCIlTeJIbHO oceH CHCTeMb/ KOOp.l1HHaT 0 1 _\1>::) H
rnaBHhle l{eHTpaJ/hHblC (arHocJrrcnbl/o oceM CMCTeMbl /<OOP,111HaT OOXQ1'oZo) MOMeHTbJ tHep-
L(IIII rupOCKona CORIT8,lJ,aIOT C rO<I!-IOCTbfO ,10 3Ha4eHHJI BeJUI4HHbl mB: II mB lm - Macca
pOTOpa).
[3blpa}KeHHC )J,JlSl KHlie'l H4CCKOH :mepmH Mr IIMeeT BHD,
I [ 2. 2 2. 2 2 2. ( 2 2 2. )
T=2 A 2 P2 +B'1H2 +C 2 1"::! + Atp/ +BJqJ +Ct'i +m Vy+v.y+vz '
(4.]43)
rD,C AI. BI. C I /"J1aBHble MOMeHTbll1Hepl.(lUl pOTopa; A 2 , B 2 , C 2 - rnaBHbIe MOMeHTbII1Hepu.l1J1
3neMeHTOlJ ITO.l1BeCa, I(oTopbre nepeMelLlaIOTCSI BMeCTe C pOTOpOM no KOOp)JJIHaTaM a, , y; P2.
lf2, 1'2; Pl. q" 1", - npOeKI{HIl a6COJIfOTHbiX YfJIOBblX CKopOCTeH Ha OCII CI1CTeM KOOP,1HHaT
OIXI)'I=1 H OIXYZ COOrBeTCl'BeHHO; v" V Y ' v, - npOeKl.(l111 a6COJIlOTHOH mlHeHHOM CKOpOCTlI
Ll,M Ha COOTBeTCTBY101ll1le ocn.
l1oTeHUlmnbH8J1 3Hepn1H Mr 11 1II1CCHnaTMBHaJi tPYHKUMSI paeCeHHI1S1 ::)Heprl1ll BbJfJISI)J,JlT
TaK:
11 = (G a a 2 +G[I 2 +Gyl + G x x2 + G y y2 + G z z 2 )+W;
" I ( b . 2. b A2 b . 2 b ,,2 b .2. b . 2 )
'J)=2 aa + [II-' + yY + xX + }'Y + zZ ,
(4.144)
r,1e G a , G(1, G y , G.y, G.I" OZ - >KeeTKOCTlI nOJ\Beca no COOTBeTCTBYfOIUIiM 0606IUeHHbIM KOOp-
NlHaTaM.
npOCKUlI11 a6coJ!fOTHbIX yrnoBblx cKopoCTeH Mf HMefOT BI1,l1.:
P2 = 6.+!1 x COSI' + !1)' sin 1'; q2 =!1ycosy-.o..ysiny; 1'2 =y;
PI = 6.cos +n.,. cosycos +.o. y sinycos -ycosasin ;
ql = P +ysin a +.0.)' cosycosa -.o. x sin ycosCl.;
'i = ycosacosp + 6.sin +!1 x cosysin + .o.y sinysin p .
(4.145)
KoopJ\ltHaTbl UM onpe).lenJl)OTCH. BblpmKeHIHIMH
x = x+x u +ox cos cosY-Oy cosasiny;
y =)'+ YII +Oy cosacosy+ox cospsiny;
Z = z+zu +0)' sin a-ox sinpcosCl.,
266
rnaBa 4. ,lUIHAMI1KA t.I3 MIlKPOMEXAHlillECKHX IlPII!)OPOB
a a6COJIfOTHbIC J1HHdIHbIe clwpocn1 UM no KOOpll,l1113T8M 33BIICHT OT I1X npoll3BOJUlhlX no
BpeMeHH:
v..=.X:,
v y = };,
v 7 = Z.
(4.146)
Bbl'l11CJlItB KIiHent'lecKYfO 3Hepnno no )OpMYJIe (4.143) C Y'ICTOM IJhlpa)J(cHHii (4.145),
(4.146). BbmOJIHHB onepawUt )J,1t CpeHu.ltpOBaHI1S1 B COOTBeTCTBlI1I C ypaBlJellllHMIl J1arpall-
)f(a (4.1) no 0606meHHb1M Koop.lH1HanlM It CKOpOCTSIM n onpeD,cJIl1B 0606mcIIIlhie Cl1JlhI C Y'le-
TOM COOTHoweHliM (4.144) Ii )J,eifcTBYfOlll.JfX Ha fltpOCKOn YCKOpCHllrl, nOJIaraH ,'.I,8ncc. 'ITO BC-
.'lH'lHHbl a, p, 'Y. x, y. Z ManbIe Ii y» 6,., .0.,. . ypaBHeHIlSi ,'.I,BJ1)J(e IIISI pOTOpllOro Mr MO>KHO
npHBeCTH K BIt.ny
(A:! + A1)a+baa + [Go -(B 1 + B:! -C 1 _C:!)y2] a -(A, + BI -cJ)yp ==
= -(c:! + B 2 - A 2 + C I + BI - A. »)11.., - (AI + A:! )6.\. - m(z + =" )0.1' + mal).., - : ;
BIP + bj3P + [G 13 +(C I - AJ)f] p+(A I + B, - CI)ya = -(C I +Bl - A,)yn x -
B A ( .... ) s: s: oW
- J:!'ly +m Z+Z8 U x -ma:u: - 0f3 ;
(4.147)
C{y + (by + AJO",a + Bln.l.p h + G y I' == M 0 sin pf + m [(x + xI! )0-" - (y + YII ) &:t'] +
+m(a.J.o", -a",oJ;
.. b . (G . 2 ) 2 . . ( " s: " ) aw
mx + xX + ;r -11T'{ x - myy = m ax - XII + u 1'1' - -;
. ax
.. b . (G . 2 ) 2 . . ( .. s: " ) aw
my+ 1'1'+ \.-11T'{ 1'+ mxy=mal,-YJj-u r --;
.. . 81'
.. b . G ( .. S:" s: Ii ) aw
mz+ ::z+ zz=ma::-zJj-u)'a+u x .... - f)z '
(4.148)
r.ne bu. bl'>' by. b:.. b)., b z - K031}>I}>HUHeHTbJ ,l\eMnl}>HpoBaHwl no COOTBeTCTBYIOUlliM KOOpJlIiUa-
TaM; Mo, p - aMnnMry,l\a H 'iaCTOTa MOMeHTa .lUmraTeml (BH6ponpliBo.na) no KoopJJ.llUaTe y.
YpaBHeHIDI (4.147). (4.148) OnJ-iCbIBaJOT MHaMHKY pOTOpHoro Mf no llJCCTH 0606111CH-
HbiM KOOpMHaTaM C )"IeTOM MaccoHHepUHoHHoi1 HecHMMeTpHH pOTopa H nOD,Beca Ii B03MY-
meHHH, 06ycnOBJleHHblX nHHei1HbIM H yrnoBLIM nepeHocHblM ,l.J,BH:>KeHlieM OCHOBaHHH.
I). l.faCTHble ypaBHeHHH
B M3BeC1HbJX KOHCTpYKUHJJX pOTOpHbJX Mr Macca pOTopa Ha neCKonhKO nOpJl,l.J,KoB
60JIbwe MaCCbI no,naeca, nOOToMY MO)J(JiO nonaraTbA 2 = B 2 = C 2 = O. ,lJ,JUI CJIyqaSl Ox = Oy = 0
ypaoHeHHJI ,llBH)f(eHHJI npHHHMalOT BMA
YPABHEI.11151 )J,BI1)I<EI'JI.151 MI1KPOfl1POCKOnOB 267
Ala + h"u.. + [G u + (C I - 81)r2 ]a - (AI + 1J 1 - C 1 hr3 = -(C, + 8( - AI ha l . - Aln. r _ oW ;
. oa
BIB+h'IJ3"'[GrI +(C I -Alh 2 ]J3+(A I + 81- C lho.=-(c l +81 - Alhnx -Bin.. - ; (4.149)
('It + (hy + A,nra + Bln,J3 h + G.( y = Mo sin pl.
" b . (G .2 ) 2 .. ( .. aw
mx -I- xX -I- x - my x - my; = In a,,' - XII) - ax ;
(4.150)
n b . (G .2 ) 2 . . ( .. ) aw
my + \.y + I' - my y + IIIXY = m a y - Y" - -;
. 8y
.. b . G ( .. ) aw (4 I 51 )
mz + "z + "z = In (1;: - ZI' - az . .
CMCTeMbl YP8BHCHI1H (4.149), (4.150) II yp8BHeHMc (4.151) HBJISIKITCSI HC3aBMCI1MblMH.
BocnoJlb3yeMcR yp8BHeHHSlMM (4.149), B KOTOpbIX npeHe6pe>KeM 'lJIeHaMH
(AI + 81 - C 1 h P 11 (AI + B, - Cl)yo., TaR KaK fiX BJIMSlHHe ITpOSlBJISleTCSI B B03HHI\HOBeHI1H
I(OJIC6mmH Ha Yl BoeHHoH 'IaCTOTe no OTHOWeHl11O K llaCTOTe nepBl1'IHblX KOJle6aIIHH C 8M-
nnHTYlI.OH, Ha Hec,wnbKO nopSl.nKOB MeHbweH aMnJIMTY.nbl Kone6aHllH, 06ycnoBneHHbiX H3Me-
pHeMblMI1 C'WpOCTRMH. 6YJ.leM C ' Il1TaTb, 'ITO rupocKOnll ' ICCKHe MOMeHTbl B npaBbIX qaCTJlX
ypaBlIeImH (4.149) 60JlbWe MOMemOB A,o.x H B1o. y , ITpeHe6pe>KeM )J,eiicTBMeM 3JIeKTpoCT8-
TH'IecKHX CM, BBe,'J,eM 0603Ha'IeIH1SI J a = AI' I n = B I , J y = C 1 , J o = -AI + BI +C. H 3anM-
UICM cI1cTeMY (4.149) B BHne He38BHCI1MblX ypaBHeHllH:
J a a+b a 6.+G;a=J o yn y ; (4.152)
. . .
Jpp + bjJP + Gpp = JoY n x ;
(4.153)
J./v +byY + Gy'Y = M,,(t),
(4.154)
r,'J,e
Mn(/)= Mo sin(p/); G: = G a +(C I - Blh 2 ;
G; = G p + (C I - A()y2.
(4.155)
. .
B ypaBHeHI1HX (4.152), (4.153) :>KeCTKOCTH G a , G p B COOTBeTCTBHH C ypaBHeHlUlMH
(4.149) COCTOSIT 113 >KCCTKOCTeH ynpymx 3J1eMeHTOB G a , Gj) H TaK Ha3bIB8eMbIX KB83Hynpy-
rJiX )I(eCTKOCTCH, nponOpLl1l0HaJIbHbIX BeJIHqHHe 1 2 . C Y'IeTOM KBa3Hynpymx >KeCTKOCTCH
lIaC-r6Tbl c06cTBeHHb'x Kone6aHMH Mf no BblXO.l.\HblM Koop.nHHaTaM onpe.n.emIIOTCJI OPMY-
1I8MH
())aO = G: I AI;
())pO = G; I BI .
(4.156)
JJ,JUI 60JIblliHX Y qaCTOTbl (4.156) MorYT 3HaqHTeJIbHO OTnHQaTbCSI OT HK 3Ha'leHH!:\ 6e3
Y'leTa KBa3HynpyrMx :>KeCTKOCTeH..
268
rmrna 4. LJ,UHAMHKA 1J3 MHKPOMEXAHHT.J:ECKHX rrpl1EOPOB
flpu.\tep 4.15
BOCnOJIh3yeMCSllIaHHhlMII npuMepa 3.4: AI = BI = 10-'13 Kr'M 2 , C 1 = 2,5. 10- 13 Kr.1lf2, Go = 23,8 x
x 10-6 H'M, Gf> = 29.76. LO-6 H-M, ,!VlR KOTOpblX 6e3 yqeTa neml'lIlHhI i' '13CTOTbl roaO = 1.54.10 4 lie,
CI)\>O = 1,73 . 10 4 IIc. IlpuMeM i' = (OrO = 1.32.10 4 I/c II no cpopMYJIaM (4.156) nonY'1HM lOaO = 2,23 x
x 1O IIe, WIIO = 2,36 . 10 4 lIc.
Tal\.'ulII 06p330M. yqeT KBa3If)'npyroH lKeCTKOCTIi npIlBeJI K YBeJIH'1eHlllO 'f3CTOT KOJIe6aHHH npH-
MepHO Ha 40 % no o6euM OCRM.
4.6. 3JIEKTPOCT A TJIqECKIIE CIiJIhI II qACTOTHMI HACTPOHKA
PE'JKll.MOB KOJIEEAHmI MIIKPOrliPOCKOnOB
3JIClrrpOCTaTl{'leCKHe CU.JlbI B3auMo.neikTBHSI Me ' 3J1el\.Tpo,'-\aMH rpe6CH'iaTbIX CTpYK-
Typ, KaK nOKa3aHO B no pa3 . 3.4.2, HBJUlJOTCH npU'iHHOH TaK Ha3bIBaeMOH >KeCTKOCTI1 :meK-
TpOCTaTH'leCKHX CUJ) [CM. OPMYJ)Y (3.121)], KOTOpaH. npUBO HT K YMeHbllIeHHJO 'JaCTOTbl
KOJIe6a.IIl1H lfM B pq [CM. tP0PMYJIY (3.122)]. nOKa3aHO TaKiKe, 'JTO no.n60POM BeJUl'lHH no-
TeHltHfu'OB [CM. cpOpM)'J1Y (3.119)] Ha 3J1el\.Tpo,,-\ax rpe6eH'iaTbIX crpYKTYP pq MO:>KHO 06ecne-
'lHTb KOMneHCaIlUJO KBa.!l:paT)'PHOH norpelliHocTH, CillJ nonepetlHoro TIDKeHlUI 3JIeKTpo,,-\OB, a
TaIVKe BbIllOJIHllTb qaCTOTHYIO HacTpOHKY peiKJ1MOB Kone6aHlUI Mr. PaCcMOTpHM 3ne"TpoCTa-
TH'lecKHe CHJThI 60JIee nOLlp06HO.
Ha pHC. 4.13 npHBe.neHa paCqeTHaH. cxeMa Mr LL-TllITa C o,'-\HHapHbIMH rpe6eH'laTblMl1
3..'1e TpoCTaTHqecKliMM CTpYKT)'PaMll, 3JIeMeHTbI KOTOpblX BKJItO"leH.bI B )J,H epeHQHanbHbIe
uenll C HCTO"lHHKaMn HanpSl:iKeHHH VI> V 3 H V:!, V 4 . TIpe.nITOnafaeTcH, 'iTO H3MepHeTcSI yrno-
BID! CKOpOCTb 4, a nepeMellleHIDI q3 B P.D: H pq npoHcxo)J,JIT COOTBCTCTBeHHO B HaTIpaBJIeHUH
oce" x,y.
z
"
J'
x
:.-;'.. / ..
. ""' '.,. :' .-
, - : w / ,::
." , , . :," , .
.,-;' .
. /.
".':
..,
.;;...,
. .....
.:."
.. "'...:
"."
,
.r
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,
J
VI V 2
U 3 V4
Pae:. 4.13. CxeM3 Mr LL-THD3 C JIHcJlcJlepcH H3JJbJll>IM BKJlID<JCIUteM 3JJeMeHTOB
3JJeKTpoCT8TII<JecKllx rpe6ell<J3TbJX CTPYKTYP
3J1EKTPOCT A TI1lJ:ECKHE CI1J1bI H lJ:ACTOTHA5£ HACTPOI1KA
269
Pa60Ta JJIeKrpOeTaTlltJecKlix CHJI Ha B03MO>KHbIX nepeMellleHlffix I1x Ii y aHaJlOn1tJBO
JOpMyne (3.89) MmKCT 6bITb npe)J.CTaBJIeHa cnenYlOlllHM Bhlpa)KeHI1CM:
A= ELOC [ I (vJ-vi)+ I (vi-v I 2 )111x-
2 vo + v Yo -.6.y J
CLoe [ a+.6.x V 2 a-fu V 2 a+L'.,.-c 2 a-fu 2 ]
- ,+ 4 - 2 VI - V 2 y =
2 (.6.Yo + y)- ( y() + y) (.6.Yo - y) ( Yo - y)-
[ I"J 2 2 2 ) I 2 2 "J )]
ELOC }o \V 3 +V I V4 V2 + y\V3 -V t - -V 4 +Vi
"J L'.,.-c+
2 tWo - y
-I- LEoc [ a( Y +.6.y2) (vi -U +v; -vi )-2a yO Y (vi +vt +uJ +ui) +
2 I 2 ) 2
\ Yo - y
(2 2 ) { 2 - 2 ) (2 2 "J 2 )]
fu\.6.Yo ....6.y \U 3 -U 1 - -U 4 +Ui -2a yo Y \V 3 +V I -V 4 -U 2
+ I ) '" y.
\ y _.6.y2 -
(4.157)
HanpH>Kemm U j ...V 4 MorYT npe.n.CT<IBJISlTb cnC,DylOlllY!O KOM61lHaUM!O (CM. pl1c. 3.56):
VI =U on +U r -U q -U y ;
V 2 = Von -V x +U q -V y ;
U 3 =U on +V x +V q +U y ;
U 4 =V on -V x -U q -U y ,
(4.158)
r,':(e U x - ITepeMeHHaJI COCTaBnHlOlllaH. (aHaJlOrI1'lHa v Ha pHC. 3.56); U q - COCTaBJIHIOll.lM 11JHl
KOMneHCaUI1H KBa)J.parypHoM 01ll116KM; U y - COCTaBJIHlOlllaH. )J.JUI YCTpaHeHl1jJ nonepe(lHoro
TSl:>KeHIDI 3JIeKTpO)J.OB.
C Y<IeTOM BbIpa)KeHHH (4.158) 3anl1111eM TaKHe paBeHCTBa:
V ' 2 V '2 V '2 V .2 8 1 U V . V . V . ) .
3 - 1 + 4 - 2 = on y+ q x'
U 2 V '2 V .2 V .2 - 8 ' V ' V . V V ) .
3-1-4+2- onq+xy'
U -t-V t 2 -vl-vi = 8 (UonV r +VqU y );
V} +V? +UJ +ui = 4 (U;n +u: +V; +U;).
Pa60TY 3JIeKTpOCTaTM<IeCKI1X CJ1JI [CM. ypaBHeHHjJ (4.157)] C )"IeTOM nony"leHHblX pa-
BeHCTB ,IJ)lJI CJIy<IWI Y « Yo MO:>KHO npe)J.cTaBJ1Tb B ynpollJ,eHHOM BH.n.e:
4EE o C(V on V x +VqV y ) 4EEoca(UonUy +UqU x )
A = tu- 2 y. (4.159)
Yo Yo
270
rJUma 4. JU1HAMI1KA q3 MI1KPOMEXAHI1 ECKHX rIPl>lEOPOB
3amnneM 3J1eMeHTbl MaTpI1l{bl :>KeCTKOCTH, BbI'lI1CJlSleMble B COOTBeTCTBl1l1 C tPOPMYJIOH
(4.157):
e =0'
n ,
.3 _ [ (t!.yJ + t!.../ )(U onU q + U xU}') - 2t!.y o t.y(U onU + U qU y ) ] .
k X1 : - 4EE o C ( \2 '
t!.Yo -t!.y- J
k;'. = -4c:Eoc[2at.)'(3t!.y + t.y2 )(UonU x +UqU,.)- at!.yo(t!.y +3t!. y 2 )(U n + V: +U; +u;)+
f 2 ,2 ) {. . ) 2xt!.yO(t!.y + 3t!.i )(UonU + U qU ,. ) ]
+2xt!.}\3t!.yo+t!.} \UoPq+UxU y ( 2 2'\3 . (4.160)
t!.Yo - t!.y J
,lL'lJI CJJ)"-IaJI t.y« t!.Yo Bblpa:>KeHlUl (4.160) yrrpollJ,alOTCSI Ii MaTpliua K03tPtPl1L\HeHTOB
)KeCTKOCTH npMHHMaeT BH)],
o
UonU q +UxU y
a (2 U 2 U .2 2 ) .
--\U on + q + x +U,.
t!.yo
(4.161)
K = 4£EOC
'3 t. 2
o
UonU q +UxU y
3neMeHTbI MaTpML\bJ :>KeCTKOCTH 3JIeKTpOCTaTllQCC"HX CliJI (4.160) SlBnSllOTCSI HeJIl1HeH-
HHMH $yma1HH.MH ITepeMellJ,eHHH t!.y H CJIO:>KHbIM, TpY./:lHO npofH03HpyeMbJM 06p830M BllliSl-
lOT Ha !U1HaMHK)' Mr. Onpe.neJIHM xapalITep HeJIHHeHHOCTeH 3neMeHTOB MaTpHL\bI )!(eCTKOCnJ
(4.160), nOJIaraJI /),y« t.Yo .I1MeeM CJIeJ1.Y1OJl.llie paBeHcTBa:
k . = 4££oc
Ar.[I+ (:;' )}ooU, + up,)
AY [l- (:;' )']
1 ( /),Y ) 2
'3 + t!.Yo
= k:c;'(6y=O) 2 .
1- ( t!.y )
t!.Yo
(4.162)
AHanorn"lUO MO:IKeT 6bITb nOR)"-IeHO Bblpa:>KeHHe
( ) 2
1 3 /),Y
k =k ' ) [l:( l
(4.163)
B BbJp87lCeHIDIX (4.162), (4.163) k (l>Y=o) H k (l>y=oJ jlBnJJIOTCH COOTBeTCTB)'lOlllHMM
3JleMeHTaMH MaTpHUbJ (4.160), KOTOpaJI OTBeqaeT YCJlOBHIO /),y:=; O. Hanl1QHe HeJIHHeHHOcTeH
3J1EKTPOCTA TI14ECKI1E CI1Jlhl 11 4ACTOTHA5I HACTPOI1KA
271
B COtieTaHl1I1 C 60JIbllll1MH nepeMellJ,eHHJIMI1 3JIeMeHTOB Mr MO:>KeT ITpl1BeCTM K HX CTOJIKHO-
lJeHHlO.
Cle,iJ,yeT OTMeTl1Tb, 'lTO HaCTpOHKa 1JaCTOT pe30HaHCHbIX peJKHMOB HanpSlMYlO CBSl3aHa C
OrpaHH1JeHI1S1MH Ha B03MOJKHbIe nepeMeI.QeHMH 3JIeMeHTOB Mr. EcnH, HanpHMep, )J,BIDKeHHe
no KoopJJ:l1HaTe y (npll H3MepeHH11 yrnoBoH CKOpOCTH Oz) npOHCXO)J,HT C aMrurnry)J,OH BcerO
!lI1llJb B 1 % OT a30pa, T.e. l1y = O,Oll1yo' 11 C 1JaCTOTOH ffiy ,llJISI pq, TO aMillU11)',D,a YCKopeHHH.
3Tom )J,BI1JKeHI1S1 a y = 0,0 1 l1yo ffi; Ii IIPH ffiy = 10 5 pa)J,/c, l1yo = 10-6 M OHa pasHa
,
a j . = 100 M/C- ;:;: lOR. 3TO yCKopeHl1e )J,OCTaT01JHO 60JIblliOe H MO:>KeT 6bITb cOH3MepHMo C
YCKopeHlleM, )J,eHcTBYlOllJ,llM Ha Kopnyc Mr. Pe30HaHCHaJl1JaCTOTa pq H3 ycnom!i:j HacrpoHKM
Mr He OTJIl1tJaeTCSI CYllJ,eCTBeHHO OT pe30HaHCHOH 'laCTOTbI P,ll;, T.e. OHa JIBJISlCTCSI 60J1bllJOH M
HMeeT nOpSl,llOK YKa3aHHOH BbIllJe BeJIl1 t ll1HbI.
CJIe,iJ,OBaTenbHO, no YCJ10BI1S1M MI1HMMI13al\I1H HeJIl1HeHHOCTeH 3neMeHTOB MaTpI1I.J;bI JKe-
CTKOCTl1 11 OrpaHl1tJeHI1S1 YCI<OpeHHSI ,D,BI1:>KeHHSI crpYKTYP Mr nepeMeIlJ;eHllH l1y (H3MepeHlle
Oz) HJIH fu (J13MepeHl1e 0 J') JJ:OJI:>KHbI HaXO)J,l1TbCSI B npe)J,eJIax 1 % OT nCpBOHa1JaJIbHbIX
3a30pOB. 3TO OrpaHl1tJeHl1e pacnpocTpaIDIeTcSI H Ha poTopHbIe mpOCKOnbJ RR-THITa Ii Ha
J1.Pyme.
06paTl1M BHHMaHl1e TaK:>Ke Ha TO, tJTO )J,eMn HpOBaHl1e )J,OJI:>KHO 06eCne1.iMTb ycrpaHeHHe
(crnaJKl1BaHlie) HeJIl1HeHHOCTeH B K03 HlJ;HeHTaX k [CM. OPMYJ1Y (4.162)]. O'leBM.L\HO H
TO, tJTO K03 HlJ;HeHT k;'Y [CM. BbIpa)«eHHe (4.163)J 06J1a,n;aCT60JIbllleH HeJIHHeHHOCTblO, 1JeM
k , 11, KpOMe TOro, onpe,II,eJIHeMaJI HM )«eCTKOCTb 3JIeKTpOCTaTHtJeCKHX CMJI HanpaBJIeHa npo-
111B MeXaHH'leCI<OH )«eCTKOCTH C.V nO)J,Bcca. HCXOM H3 3Toro, He06xo)J,HMO C06JIKI)J,aTb YCJIO-
BHe
Gy>k .
(4.164)
OtJeBH)J,HO, ammorH1JHoe YCJ10BHe MO:>KeT 6bITb 3anHCaHO H ,D,J1S1 pq B HanpaBJ1eHHH OCM Z
(H3MepeHl1e fly):
G z > k; .
(4.165)
HepaBeHCTBa (4.164), (4.165) MorYT 6bITb HCnOnb30BaHbl )J,JIH onpe,D,eJIeHMH npe,D,CnbHbIX
(KpHTlltJecKHx) CMeI.QeHI1H, npH KOTOpbIX He npoHcxo,nuT 3neKTpOCTaTM'leCKOrO "3axBaTa"
3neMeHTOB C HX B03MO:>KHbIM CTOJIKHOBeHHeM. 0603Ha1JHM k (f1y=o) = k;y(o)H nepenHllleM
YCJIOBHe (4.164) C yqeTOM paBeHCTBa(4.l63):
( J 2
1+3 l1y
l1yo
GY>k O)H ::,r r
(4.166)
272
r1I3B3 4. ,DJIHAMHKA l.J.3 MHKPOMEXAHI14ECK LX rIPl1bOPOB
H3 Hero, 3ID.1eHliB 3HaK HepaaeHCTBa 3HaKOM paBCHCTBa. HaHJ1.CM npe)J,eJIbH}'IO (KpUTH'ICCI\)'IO)
/1J' = L1Yl>l' BeJIll'IUH)' nepeMC CHlIiI:
,
9 ( k )(o) ] - + 16 k ,.(o)
G, G,.
(4.167)
1
lI.)' = lI.\' o
. "p - ,
' k 3
-' "n t 'O )
- +2-
G,
PaCCMOTpIlM BJIlUlHlIe :iKeCTKOCTI1 3JleKTpocTaTli'leCKI1X elm H3 ncpcrpY10'lHYIO cnoc06-
HOCTb Mr no OTHOlliCHlUO K BHCllIHCMY YCKopeHIuo. B YCTaHoBIIBllieMcSI pe:iKHMe CIL1a HHCp-
UHU BHelllHero YCKopeHlUl a" ypaBHoBelliHB3cTCSI CIIJ10H MeX3IiH'leCKOH :>KeCTKOCTI1 nOllBeca.
ITOJIOiKHB ;J,.'Dl npc)J,CJ1bHOrO CJl)"!851 v = vo , nOJIY'UlM mal' = G,./1yo. 0603Ha'lHM MaKCH-
Ma.ibHOC YCKopeHIfC. KOTOpOC MO:iKeT BbIlIep:iKaTb Mr C Y<iCTOM "0TplluaTe;JhHOH" 3JlCKTpo-
CT3TII'leCKOll :iKCCTKOCTI1, a,1:J 11 3am111IeM paBeHCTBO CIf:!: ma':J = (G)' - k. . '\}'1<jJ . Pa3,L1enl1M
BTopoe If3 npHBC;reHHbIX paBCHCTB Ha ncpBOC II, npeHe6pefaJI B BbIpmKcHHH (4.163) BTOphlM
C:18raeMbIM 'lHC.'lIfTC:ISI no cpaBHeHI1IO C e)J,HHHl\eil, nOJIY'lHM OTIiOllieHl1e, xapaKTepH3Ylo ee
ncperpY30'lH)'IO cnoco6HoCTb Mr no OTHOlliCHIfIO K BHelliHeMY yCKopeHuIO:
a.1"3
a y
1- k_ ,(o) 1
G. [1-( %: JJ
/1y,,'P
L1yo
(4.168)
113 BblpIDKeHIDI (4.168) C.Tle.!IYeT, 1JTO neperpY301JHaJI cnOC06HOCTh Mr YMeHblliaeTcSI
npH H8:iM'lMH CHJ1bl 3ncKTpocTaTH'leCKOH :iKeCTKOCTH.
OTMeTHM TaIOKe. 'lTO npH pac1Jcre 3JICKTpOCTaTHIJeCKMX CHJ1 B HanpaBJICHHH OCH y HC
Y'II1TbIBa."IHCb pa3.'lH'lHbIe napa3MTHbIe eMKOCTH H B03MO:;KHOCTb nOSlBJIeHHSI CHJl B HanpaBJIe-
HHH OCll Z. '3Tu ynpomeHM5I, BHJ1.MMO, )J,OnyCTHMbI, nOCKOJIbKY lllHpHHa nJIaCTHH KOH}1.eHCaTO-
POB Ha rpe6eHIJaTblX crpYKTYpax COH3MepHMa C 3a30paMH MCn<.llY HHMH.
HanOMHMM, 1JTO npH1JHHOH KBa.llparypHoH norpelliHOcn1 Mr SlBJI5IeTC5I HeCOBna,l\CHHe
BeKTOpa CKOpoCTH HM C BeKTOpOM 3JIeKTpOCTaTHIJeeKI1X CHJI, BbI3bIBaIOlUHX P,lI,. cDl1311'lecKOH
npH'IIUIOH TIoro MO)i(eT 6bITb HaJ1H'lHe nepeKpeCTHblX iKeCTKOCTeH G XY YITPyrofO nO)J,Beca.
JI.06aBHM K ,meMeHTaM MaTpHLlbI (4.161) nreCTKOCTb 3JIeKTpOCTaTH1JeCKOH CHJIbI 3JIeMeHTa
M8-rpHUbI MexaHM'lecKoH iKeCTKOCTH nO)J,Beca H nonyqHM MaTpHUY, KOTOpyIO Ha30BCM 3 eK-
mBHoH Ma-rpHueH :>KeCTKOCTeH Mr:
G x
Co
GX). + UOJ,uq
a/1yo
K =
,
(4.169)
Co
GX)' + UonU q
aL1yo
G y - C02 (U n+U;+U;)
L1yo
r.ne Co - HOMHHaJIbHaJI eMKOCTb 3JIeMeHTOB rpe6eHIJaTbIX crpYKTYP; a - nHHeHHoe nepeKpbI-
THe 3JJeMeHTOB rpe6eHqaTOH CT)JYKTYPbl.
3JIEKTPOCT A THIfECKI1E Cl11Ibl H LIACTOTHMI HACTPOI1KA
273
ylAy
_90'
W
W X < (j)y
W x < (j)y
PIIC. 4.14. K nOHCIlCllll1O pe'"JKHJ\fOB '1aCTOTlfOii HacTpoiiKu Mr
OtjeBI1):\HO, eCJ1l1 nO)J,06paTb Hanp )Ke}Ule U q =
G ,allvo
X) - , TO MaTpMQa (4.169) npHHU-
COU on
MaeT BHJJ.
G r 0
K,cp = 0 G CO I U 2 U 2 U 2 ) ,
y- on+ q+ X
llyo
(4.170)
B COOTBeTCTBIUl C KOTOpOH npHQl1Ha (G Xy), npHBO,llHlllWi K KBMpaT)'pHOH 01I1H6Ke rHpocKo-
na, YCTpaHeHa.
C yqeTOM MaTpl1l..{bI (4.170) opMyna (3.122) npH06peTaeT BH)J,
1 [ Co I 2 2 2 )
W y = m G)' - dY ,U on + U q +U x J .
(4.171)
HCnOJIb3YSl H3JJOjKeHHbIH ITOJJ.XO,ll, MO:>KHO Bb!qHCJIl1Tb MaTpHI..{bI 3 eKTHBHbIX :>KeCTKO-
cTei1 nO)J,8eCOB H COOTBeTCTBYJOlI.{Ue 11M tjaCTOTb! H JJ,JUI )J,pymx THnOB Mr.
TaK KaK YCKOpeHHC KOpHOJIHCa - 3TO CHrH8JI Y3KOH nOnOCb! tjaCTOT C QeHTpOM Ha qac-
TOTe B036Y)I{)J,eHHSI KOJIe6aHMH P,n:, MexaHHtjeCKHH K03 J1[.{J1eHT YCHJIeHIDI pq MO:>KeT 6LlTb
HaCTpoeH ITYTeM COOTBeTCTBMSI HnH npH6JIH31lTeJIbHOfO COOTBeTCTBMH pe30HaHCHblX tjaCTOT
PC)KlfMOB B036Y)I{)J,eHHSI H pq.
Ha pHC. 4.14 nOKa3aHbI 3aBHCHMOCTM OTHOrneHllH aMruIHT)')J,bI Kone6aHHH 11M B pq K
aMmIHT)',l\e yCKopeHJ1}[ KOpHOJIHCa H cjJa3bI 3THX KOJIe6aHHH OT QacTOTbI, a TaIOKe B03MO:>KHble
pe:>KHMbI HaCTpOHKH Mr.
274
rmma 4.1-U1HAMHKA ll") MIIKI J OMEXAl-II,!lIECKHX l1PIII,OI'OB
KOf)J,a pe30HaHCllh1C <mCTOThl connU.rW.10T «(0, = II),. ), MCXi1IlIl'lCCKlll1 1(O')(IIIIUII'IiCIiT YCII-
J1CIHUI Mr BMCOK, 110 nOJ1oca '!aCTOl" nxoJ\lIoro cllm:um I1c60JlhIlIali II cyu,ccmyci 111:\'10111.111 CJUIIU
Ha BbIXOJ1.C. ECJ111 'IaCTOTa BOJ6)')KJI.3ClI1bIX KOJIe6mlllii 1131\11101'0 MCllblllC 'inCH)) hI KOJlc6mll1ii Pl!
( (!)x « w...), rllpOCKOll IIMCCT 6OJU,IIIYIO llOJIOC)' npon)'CKalilUI 'I:1CHH, M:I:I<=lIhI<IIli Ilm'JOllblii
CllUM!" It HC3Ua<IltTCJ1hHhllt K(nlj)(jIllIUICIlT MCX:l1H1'ICCKnl"O )'CII1ICIIIIII. 1-luKOIiCII. CCJIII pC'IO-
naHCHhlC <IaCT01'J.1 HC COBnanalO1 IHI IIc60111.1II01l1 OTPC I(C 'I:1C1'01' (l1all)1I1I\1C)1. 1\ 10 'x,), Mcxa-
UW'JCCKIIH KO'J q)liU.IICHT YCItJ1CIIIUI 1\11" OCT:1Cl"CH 6OJ1blllJlM, a IIOJIOCa 1I)101l)'CI(l1II1U1 'I:IC"IIH,
no cpaBUelHtIO C ncpBhlM CJ1)"JaCIII yncJ1I1'IItBaCTCH.
HaCTpoihi:a '!aCTOTbl (lI, UbmY>KJ\Cll1IbIX KOJIC6allllH ocymcCTIlJUICTCJI IIOJ,60p0I\1 'I:1CTO-
TbI HanpS!>KCUlUI nIlTalUtSi npIIBOJ OB. aCTOTa KOJIc6mlllii Pll IlaCTpnllllaCTCJI II'IMCIICIIIICM
>KCCTKOCTIi nO.!lBCca B HanpaBJICHIHt :nllx KOJIc6a1lltii lIyrcM ItCIIOJlb'lOn:lllllJ! ')J[CICI )10CT:lTII'IC-
CI(HX CIIJ1.
KaK C.iICJJ.)'eT 113 OPMYJIbI (4.171) [CM. TaKJKC QlOpll1YJlY 3.122)], yncJIIl'lclilic lIallpmKc-
HUSI CMCIlJ,CHIUI (.,'"" npItBOmtT K YMCllhWClIlIIO <I:\CTOThi 10,.. l-IacTpoiiKY 'WCTOT lIaJIO 11:1'11111:1'1'''
npn w." > WI \I 3aTCM, )'Bemi'!IiBaH. U nn , 06CCIlC'IIIT" IIc06xoJ.IIMOC OTIIOIllCIIlIC 'mC'rOT (1),.1(0),.
HY>KUO OTMeTltTb, '11'0 )J,J1S! !l.lUiaMIIKI1 Mr 60JICC IJa>KIILIM JlIUIHCTCJI OTIIOIllCIIIIC 'UICTOT. 'ICM
I1X a6COJIlOTHhlC 3Ha'lCIH1S!.
TOIJHaJi HacTpoiil\a <IaCTOT B03MO>Klia npu C06J110)J,CIIIHI ['COMC'I pl1'ICCKltX pa'iMCpOJI '»IC-
MeHTOB ynpyroro nOJ1.0eca. <ITO nplt I1X MaJIbIX pa'lMCpaX nCCJ,Ma ' :1Tp)'JI.m1TCJ[hIlO. rIpl1 :>TOM
c.'JC!l.YCT HMCTb B BIUIY, 'lTO )J,J1lma ynpyrux 3JICMCIITOU OllpCJ CJIHCTCH TOJlhlW JIIITOI-palimci-i It
nOTIOMY c0611IO.J.acTcJI C 60JIbWOH TO<IliOCTLIO. IUUpllJl3 11 TOJIlI.lma ynpyrlix :>JICMCIITOU 01[-
pc.ae!U!IOTCS! He TO;)bKO n1nOI-pa4mCH, no 11 TpanJICIII1CM. OTiIOCU rCJII>IILIC pa'!MCplol Yllpyrllx
'J11eMCliTOB COOTHOCJlTCJI MeJK)I,y c0601i cnc)J,YIOUt\1M 06pa30M:
tJ.', tJJ Il . J\c"
-«-,-,
'1 b n ell
r.ae IYJ" M n , Ac.. - nOI-pCWIlOCTI1 113rOTOBJ[CIUlSI JVII1Ht.I ynpyn)['o 'JJICMC[1T3, Ilmpllllhi 11
TOJIIllHHhl ero nonepcLJlioro pa3Mcpa COOTBCTCTBCHHO.
PaccMOTpHM 38BHCHMOCTM OTHOlliClUiJ! <IaCTOT I1HQJopMaTHlIIIblx KOJIC6alll1H K 'laCTOTC
BblHy"AC.D.eIiHblX K0J1c6aHMH 11M OT reOMCTpH'leCKHX pa3Mepou ynpyrl1x 3JICMCIITOU. [JPOCJIC-
J1HM BHa'la.1e cJlY'la.i1 H3Mepem-lSi yrnoBoH CKOpOCTll !l: C nO)J,ueCOM 11M 110 CXCMC, IIpOUCJ\CII-
HOH Ha pHC. 3.7. 3anHwcM OTHowelmc pC30H811ClibIX 'faCTOl' mullOpMaTI1BIIOm (PY) 11 UMlIy-
1tl)1CHIiOrO (PJJ.) Kone6auHH COOTueTCl'Bel1HO B]J,OJlb oceM Y H x:
"',< - i G ,. - i Go,
- - ,
wxo G x G2J.-
(4.172)
flle
q.v = Ebnl ( "1 ) 3; G 2x = Eb"2 ( n2 ) 3,
11 'r2
(4.173)
r.ne h ni , C nlo If; - COO1'BCTCTBeHHO WHpHHa, TOJlIlJ,HHa Ii MHII8 ITCpDOrO (i = 1) 11 DTOpOro (i "" 2)
ynpyniX :meMeIiTOB.
')JIEK'r!.OCTATWIECKHE OUlhl '" lJA('TOTIlMI IIACTJ>Oj,'IKA
275
IloCHC 1!O/(CTalIOUKIIIIJ.lIXI)ICCIII-UI (4.173) n lllOPMYJIY (4, J 72) I10nY'II1M
( ) 3
(Oyll bill ('IIII. J2
(°. 111 = b ll2 c n 2 / d '
{)l 'K YJ ' , 'I , IIOJIlIr:U I b 1 1 ( ' C ' IIMCC "
, III = n2' 'III = 112' ...
3
(11\'11 ( I? ) 2
- = -.!:-
{r}xll 1. 11
(4. J 74)
P"CCMOTPUM TCIICp" lI:lpI1HII'I' InMcpcIIIHI YI'Jlolloii CKOpOCTlI fl.... B :)TOM cJly'mc HlJ(I)op-
M:lTlIIIIIOC JUIIJ)KCIIIIC I-IM IIpOlICXOJ(I1T II mlllpallllClIlI11 OCII Z 11 OTIIOIIICIUlC pC'WllrUICllblX 'laC-
TOT palm (I
{J)z(l =
(0"'11
CIZG h
G 2 ".(C lz + G 2z ) ,
(4.175)
I'j{C
( ) 3
c
- " --1!.L.
G 1z - 1.11 111 '
I'll
G = J ' C ' ( bll2 ) 3.
? 11 ? ,
-- - 11"2
2
G L l ( ell? )
2x = J II 2 --=-
11'2
(4.176)
3aMCUIM, 'ITO (IJOPMYJlhI (4.176) :lUnIlC:UII,1 II npC/{1I0ll0>l\cmm MCllhllldi )KCCTKOCTU :)lleMCH-
'lOll IlIlIanpmmclIlIIl DCII Z II 3JICMCIITOB 2 II lIanpmlJlClIlI11 OCII X (CM. pliC. 3.7).
nOCllC IIOHC'nIllOltKll nCJlU'JUIi (4,176) II OTIIOIIICIII1C (4.175) nOJIY'lIIM
( ) 3
2 ell I
b lll b ll2 --,
1'1'1
[ 3 3 ] ,
c 2 ? b 1( : lt ) .1. e 2( b 12 )
11- n I II I
11 'J2
(0'0
:::
W x ()
OTKYJ(U IIlIpCJUlOnO)I\CIIHU, 'ITO h nl b 112 ::::; bll' enl £.'112 ell' cnCAYCT
Wz(J _
--
I
(4.177)
(O"n
1
[( e'l ) 2 + ( l:rr. ) 3 ] 2
b ll /'1'2
113 cpaBIIHTCJlbHOrO alUlIlH3a OTIJOllleHHM (4.174) 11 (4.177) 3aKJlIO'laeM. '1'1'0 DBu.ny Tex-
c
nonOI'H'lCCKOH CJIO:>KIIOCTl1 TO'lHOrO c06moAcHHJI Q1'UOWeHMM -1L npH H3MepcmU1 yrnoBoit
h n
276
rnaBa 4. .lU1HAMlfKA LI3 MI1KPOMEXAHHLIECKI1X rIPHEOPOB
CKOpOCTH 6Y)J,eT I1MeTb MeCTO 60nblllaSl BapnaTIIBHOCTb B OTHOCnTeJIbHOM pacno;lO}I{cHfm
pe30HaHcHblx <IaCTOT no cpaBHeHlUo C n3MepeHueM yr.ilOBOH CKOpOCTI1 n .
AHanOrH<IHO MO>KHO npoamlJUnnpOBaTb BJIIliIHHe pa3MepOB YITpyrnx 3.iIeMeHTOB Ha TO<I-
HOCTb HacrpOHKH pe30HaHcHbIx pe>KIiMOBIi JlJISi )J,pyrHX cxeM nO)J,BecoB.
J{Ju nOllBeca Mr no pnc. 3.11, rrOJIO}l{HB. liTO BbIXOlIHOii KooplIHHaTOH SlBnSlCTcSl yroJI a.,
MMeJl B BH.!!;)' BLlpa)KeHIUI (3.28), (3.29), 3aITHllleM OTHOllleHHe 'laCTOT:
( 2G(n)k'bnIC + E(n)b ]
C I T1 I T2
I
A ( 3 )
I E b I b n2
(n) - +-
IT] I T2
(4.178)
(OaO
-=
(01'0
I1cnO.::IL3YSl rrOJl)"'IeHHble 1}>0pM)'nbI If, B lIaCTHOCTH, (4.177) 11 (4.178), MO:>KHO OLleHHTb
B."IIDIHHe TeXH0J10rH'IeCKUX )J,OnycKOB H3fOTOBJIeHIDI ynpyrHx 3neMeHTOB nO Beca Mf Ha H3-
MeHeHlle HacrpoHKH 1JaCTOT no OTHOllleHI1({) K HacrpoHKe. paCCLJHTaHHOH ,llJUI HOMHHaJIhHbIX
3Ha1JeHHH 1JaCTOT.
lJptL"Iep4./6
"UI HCXO!!.HbIX lI.BHHbIX no npHMepy 3.4, nonarM, 'ITO pa3MepbI ITj, IT2 Ii napBMe-rpbI C" AI He Me.
HJUOTCjI, Bbl'IHCJ"lHM H3MeHeHHe pacc-rpoHKH lIacroT, o6YCJJOBJIeHHOe TeXIiOnOrnQeCKl1MII ,nonYCKBMI1 Ha
llIHpHHY Ii BbIC01)' TOpCHOHOB. IIpHMeM bnl = I1b n2 = I1c n = II 0- 8 M (IO- 2 MKM) 11 no l}J0pMyne (4.178)
HaHlleM:
2.0,79.10 11 . 0,2(20 .10- 6 :!:: 10- 8 Y + 1,29.10 11 (15 . 10- 6 :!:: 10- 8 r
2 5 0,8.10- 3 0,5.10- 3
, 1,295 '10 11 ( (20 .10-6 :!:: 10- 8 r + (15.10- 6 :!:: 10- 8 ) )
0 8.10- 3 05.10-3
, ,
= 1,17 L
(OaO
-=
(OyO
liMes B BHllY, 'ITO <.0]'0 = 1,32. 10 4 IIc, nonytIHM (J)QQ = 1,171 . 1,32. 10 4 = 1,55 . 10 4 11c. PBCCTpOHKB
'iaCTOT hro = 0,23.10 4 lie = 17,42%.
B npHMepe 3.4 nOJJyqeHB 'lacTOTa <DaD = 1,54.10 4 lIe H paecrpoHKB 'iBcrOT COCTaBIDIeT
hro = 0.22.10 4 Ilc = 16,66 %. TBKHM 06PaJOM, TeXHOnOrnQeCKl1e ,nonycKH B 10- 2 MKM Ha llJHpHHy 11
BLlCO'I)' ynpyrHX 3JJeMeHTOB Bbl3bIBafOT H3MeHeHHe paCCTpOiiKH QBCTar Ha 0,76 %.
.nmr 06ecne1JeHIliI He06xo)J,I1MbIX xapaKTCpHCTHK Mf Tpe60BaHHSI K T01JHOCTH 1JaCTOT-
Hoif HacrpDifKl1 01JeHb BLICOKH. HanpHMep, ,l\onyCTHMaSI paccrpoHKa 1JaCTOT MO)KeT COCTaB-
IDITb COTble (H MeHee) ,l\OJIH npOl{eHTa. HCTpYJtHO nO,l\C<IUT8Tb, <ITO npH 3TOM ,/.J,onycKI1 Ha
feoMCTpJ1qecKHe pa3Mepbl ynpYfl'lX :meMeHTOB 6y,l\yr 10- 3 ...10-6 MKM, LJTO npH COBpeMeHHoM
ypoBHe TeXHOJIOrHU npoH'3BO.nCTB8 Mr rrp06J1eM8TH'fHO. TI03TOMY MeTO,/.J,bI 3neKTpOHHOH H8-
CTpOifKH npH06peTI1lOT oc06oe 3H8'JeHHe.
.l1:EMfI<lJJ.1POBAHJ.1E 113 MJ.1KPOrJ.1POCKOIlOB
277
4.7. ,L{EMn<I>MPOBAIDIE ...3 MMKPOfMPOCKOnOB
B Mf pacceSlHMC 3Hepnm Kone6aTeJIbHblX nBM)i(eHMH npom:xomIT 3a CqeT CJ1JI BSl3Koro
TpeHIfSi ra mBDH CPC,i.(bl (ra30Boe ,lleMm}mpoBaHHe) H Cl1JI BHyrpHKpHcTaJJnHlIecKoro TpeHHSI B
MaTcpMaJJe YITpYnIX JneMeHTOB nO)J.Beca (KOHCTPYKl.\HOHHOe )J,eMn HpoBaHHe). Pac 'leT )J,eMn-
ejmpoBaHl1ll KOJIe6aHI1H OTJlI1'laeTCSI 6OJlblliOH HeonpeneneHHOCTblO. H npOBO)JHMbIe paCqeTbI
06)'I'lHO HOC}!T Ol.\eHO<JHbIM xapaKTep.
YHMBepCaJJbHOH MeTpOJlOfMlIeCKoli xapaKTepJ1CTHKOH )J,JIH. KOJIe6aTeJIbHblX CJ1CTeM (OC-
LU1JIJ1HTOpOB) SJBJIHeTCSI )J,06p01liOCTb no )J,eMnc}JHpOBaHHlO. rIpH HaJJM'lMJ1 3KcnepHMeHTaJIbHO
onpeneJIeHHI.IX 1l06pOTHOCTeH, ym13aHHbIX C KOHCTpYKTl1BHbIM HCnOJIHeHl1eM Mr, B03MO:>KHO
mJJJYlIeHMc K03 cJH1l.\HeHTOB BJIH51HHSI )J,JI51 yqeTa B paC'-leTHblX opMYJIaX pa3nH'lHbJX aKTO-
pas: CTeneHM BaKYYMHpOBaHHSJ, cpOpMbl 11M OCl.\HJIJIH.TOPa, reOMCTpJ1QeCKMX napaMCTpoB yn-
PYI MX 3JIeMeHTOB H .iip. 3TOT CJIO)i(HbIM BOrrpOC Tpe6YeT CITeUJ1aJJbHbIX HccnenOBaHJ1H, O)J,HaKO
MO)i(HO OTMeTHTb, 'lTO ,l\06pOTHOCTb > 10 4 MO:>KCT 6b1Tb )J,OCTMfuyra B 06'beMe, B KOTOpOM
naBlleHl1e 0,1 Topp.
4.7.1. ra30Boe )J,eMm}mpoBaHHe
ra30BOe )J.eMmjmpoBaHMe KOJIe6aHMM q3 rrpOHCXO,llHT 1l3-3a pa3JIl1'-lHbIX MeXaHM3MOB
B3aI1MOneMCTBll51 ra3a H rrepeMellJ,alOll\MxCSI 3JIeMeHTOB Mr. 3TO B3aHMO,/.J,ei1eTBJ1e npOHcxo-
)J.lIT maBHblM 06pa30M B )J,ByX pe>KllMax: nepeTeKaHHSI ra3a BnOJIb nepeMell\alOllJ,MXCH 3J1eMeH-
TOB Mr J1 c)J,aBJIllBaHHH. (KOMnpeCCl1H) raJa Me)i(,/.J,Y )J,BH:>KYll{HMHCSI HaBcrpe'lY .npyr )J.pyry
3JIeMeHTaMll KOHCTPYKUl1l1.
113 OCHOBHoro 3aKOHa HblOTOHa BSl3Koro TeQeHJ1S1 )i(I1,/.J,KOCTJ1 (ra3a) Me>K.D.Y ITapaJJJleJI.b-
HbIMJ1 nJlaCTHHKaMH cne.tt:YeT Bblpa)i(eHHe ,/.J,JISI TaHfeHl.\HaJJbHOM CMJIbI, BbI3bIBalOll{eif C)J,BHr
CJIOeB >KlI,/.J,KOCTI1:
I1v
F=J.!-S=bl1v,
h
r)J.e I1v - pa3HOCTb CKOpOCTeH cnoeB B 3a30pe; S - nnOllla.D.b cnoSi (nnacrnHKU), no KOTOPOMY
npOHCXO)J,HT C)J,Bl1r (nepeTeKaHJ1e); h - 3a30p MeiK1I.Y rrnaCTHHKaMH; b - K03 J1eHT BH3Koro
TpeHHSI ()JeMn HpOBaHJ1H), paBHbrn, H'c!M,
b=,J.Slh.
(4.179)
B no.npa3,/.J,. 4.4 ITpHBe,/.J,eHbI 3aBllCHMOCTH EOJIbl.\MaHa .om! BbI'"IHCJIeHIli! B5l3KOCTH ra3a J1
)J.JIHHbI cB06o)J,Horo npo6era MoneKYJIbI, KOTOpaJI 3aBHel1T OT .naBJIeHIDI. a CJIe)J.OBaTeJIbHO, OT
)J.aBJIeHMSI 3aBHCHT H BSl3KOCTb ra3a. TaM )i(e Ha npJ1Mepe 6bIJIO nOKa3aHO, liTO 3HaQeHHe 'J... rrpH
Hl13Kl1X )J,aBneHHSlX COH3MepHMO C pa3MepaMJ1 MHKpOCTpYrcryp. Kor)J.a 3HalleHHe 'J... COJ13MepH-
MO C pa3MepaMl1 h, BSl3KOCTb MO:>KHO ITpe.oCTaBMTb B Bune
J.! =J.1o{l+a }
f.oe a - KOHCTaHTa; J.10 = J.1 rrpJ1 'J... --). 0 ('J... «h).
278
Duma 4. JU1HAl\1I1KA '-13 MI1KPOMEXAHI1'-1ECKI1X rIPI1EOPOB
TaK KaK BeJIH'lHHa A 06paTHo npOnOPQHOHaJIbHa AaBJIeHlUO p Cpell.bI, MO)!(HO 3anHcaTb
npH6JIH:>KeHHOe BbIpIDKeHife
j..l j..lpph.
nOCJIe nO)J,CTaHOBKU KOToporo B COOTHOllIeHl1e (4.179) nOJIYQHM
b fJ.pSp ,
(4.180)
r)J,e fJ.p - napaMeTp BSl3KOCTIt, cooTBeTCTB}'IOlliItH HH3KOMY (p « 0.1 aTa) )J,aBJIeHHlO ()J,JISI
a30THOH cpe)J,bl j..lp = 2.776. 10-6 elM).
<1JopMY_lla (4.180) MO:>KCT 6bITb HCnOJIb30BaHa npH oueHKe BepxHeH rpaHHUbl a6coJ1lOT-
Horo K03 HuHeHTa lleMmlmpoBaHHJI )J,JI 3JIeMeHTOB KOHcrpYKUHH Mr, nepeMell.{alOllIHXCSI
napa.J1J1eJIbHO llpyr APYI}'. B pOTOpHbIX Mr RR-Tl1na rrpH6JIH)!(eHHYIO OueHKY BSl3Koro Tpe-
HIDI, MMelOll\erO aHa.110nf'lHYIO npHpo.ny, MO)!(HO onpe)J,eJ1HTb, C'IlfT85I, '1TO
.,
r- (J)
M = Fr = II ....L- S
C ,.. h '
r.J.e r(' = ("0 + r )/2; S = 1t - r ) 3)J,eCb ro - pa.L\I1Yc OKPY:>KHOCTH, B KOTOpOH pa3Mell.{aeTCSI
aHKep pOTopa (CM. pHC. 3.11); r - HapY'AiliblH pa.L\HYC pOTopa.
C )"'IeTOM Bblpa)!(eHIDI (4.180) nony<IHM Opl\f)'JIY JtJ]SI Bbl'lHCJIeHM K03 HuHeHTa
.iJ.er-fiulmpoBaHIDI poTopa no KOOp,llHHaTC y, HMeJl B BH.ny ero )J,BycTopoHHee o6TeKaHHe ra30M:
bys = knkp 1tfJ.p (r - ro)(r + roY p,
2
(4.181)
r;re k n - )(03.p MUllCHT, yqHThIBalOlll.HH .L\eHCTBHTeJIbHYIO nJIOlll.anb pOTOpa, 06TeKaeMYlO ra-
30M; kp - K03 MUHeHT, )"'IHTbIBaJOUU1H pa.L\HYC npHJIO)!(eHIDI .neMn HPYIOll.{eH CHJIbl, B )J,OJISIX
OT (r + ro)/2.
JLu 3JleMeHTOB Mr nma nnaCTHHOK .u.rrHHOH aM H WHpHHOH b", nepeMemalOllJ,HXCSI Ha-
Bcrpeary .iJ.pyr .iJ.Pyry, ,l\eMn4JHpOBaHHe npOHCXO)J,HT B OCHOBHOM 3a C'IeT C)J,aBJIHBaHHSI raJa.
nOJIa.J"'WI cnpaBe.nnHBOH 3aBHCMMOCTb (4.87), 3anHllIeM Bblpa)!(eHHe a6COJIIOTHOrO K034J HQH-
eHTa )J,eMn4JHpOBaHIDI J].JUI mo6oH KOOpJ1.HHaTbI B HanpaBJIeHHH BCTpe'lHOrO )J,BH)!(eHHH nJ1a-
CTHHOK:
3 b 3
j..l paM MP
k =2
1l h2(a +b )
(4.182)
K034J4>ID.(.HCHT ,CI.eMn4JHpOB8HIDI 3a ClJer KOMnpcccHH raJa TopuaMH naJIbQeB rpe6eH'la-
TOM CTp)'KTYpbl npHBo,CJ.a C Y<IeTOM paBeHCTBa (4.182) MO:>KHO oueHHTb no 3aBHCHMOCTH
3 3 ( n ] 2
fJ. pcnb n "
byr. 2 L}i
h 2 (b 2 + c ) ,,,,I
(4.183)
r,CI.e b - WHpHHa naJIhua .nBHfaTeIDI; h - Tonlluma c.naBJIHBaCMOrO CJIOSI raJa; n - lJHCJJO naJIh-
QeB; L r; - C)'MMa Beex paJUiycoB, Ha KOTOpbIX npHJIO:>KeHbl CMJIbI KOMnpeccHH f83a.
.LJ:EMrI<l>MPOBAHME q3 Ml1KPOfMPOCKOrIOB
279
K03 ( m..{lleHT ;J,eMn l1pOBaHI1S1, 06YCJIOBJIeHHbIM BSl3K11M TpeHl1eM fa3a B)J,OJIb nOBepX-
HOCTeH naJIb eB, ITJIOIl{a):\11 KOTOpblX onpe)J,emIlOTC5I pa3MepOM C II Ii ,Wl11H0i1 B3aHMHOfO nepe-
KpbITllSi naJJb eB, MmKCT 6bITh Bbl'lHCJIeH aRaJJOHlliHO no <popMYJIaM (4.181), (4.183). 06Il{l1H
Km HUlleHT ;J,eMn HpOBaHI1S1 pOTopa onpe)J,cnSlereSi CYMMl1pOBaHl-!eM 3Tl1X Bblpa)l(eH11H:
by = by" +b YK .
HCnOJIb3YSI 113JIO)l(eHHbIH no)J,xo)J" MO)l(HO nOJIytIl1Tb OPMYJ1bl )J,Jl51 Bbl'll1CneHH5I K03 -
<pl1I l1eHTOB )J,eMIT HpOBaH11S1 3 Mr mo6bIX KOHCTpYKTI1BHbIX I1CnOJIHeH11H.
4,7.2. KOHcTpYKnHoHHoe ,I:\el\-mc)mpoBalllle
Ilpl1'll1HOH KOHCTpYKU110HHOro eMn HpOBaHH5I 51Bn5leTCSI BHyrpeHHee TpeHl1e, KOTopoe
noH11MaeTCSI KaK nOTepSi 3Heprm1 np" ITepMo)J,HlIecKoM )J,e OpM11pOBaH11H He3aBHCHMO OT pC-
JIaKCaL(110HHoro MeXaH113Ma 11 xapaKTepa 3aTyxaHHSI: penaKcaUMoHHoro, pe30HaHcHoro I1JI11
mCTepe3HcHoro. BHYTpeHHee TpeHHe lIaIl{e Bcero 0603Ha'lalOT BeJIHlIHHOH, 06paTHoH Mexa-
Hl1lJeCKOH )J,06poTHocrn C11CTeMbI g-I. no aHaJJOfMH C )J,06poTHOCTbI0 KOHT}'pa B 3neKTpoTex-
HHKe AJUI BHYTpeHHero TpeHH}! npMHHMaeTC5J
Q-I =MVI2nW,
f,l\e W - MaKCliMaJJbHaSI 3HeprHSI, 3anaceHHaSI 06pa3 OM 3a nepHo)J,; t.W - 3Heprml paCCe5lH-
HWI, 3a nepllo)J, B TOM )l(e 06pa3 e.
PaCcMOTp11M KOHcrpYK 110HHOe )J,eMn l1pOBaHl1e Ha npHMepe Mr RR-TMna no
pHC. 3.11. O'leBH)J,Ho, 'lTO B P)], pacceSlHHe 3HepfMll npoHcxo)J,HT H3-3a 113rn6HbIx ne opMa-
I..{l1H Bcex 'leTblpeX ynpyrI1X 3JIeMeHTOB. B pq ,WlSi JIlo60H 113 Koop)J,MHaT 0., J3 pacceSlHHe 3Hep-
fMll H)J.eT B pe3YJIbTaTC 113rH6HbIX lle OpMalll1H J1.BYX H3 'leThIpeX ynpyrI1X 3J1eMeHTOB Ii
BCJIe)J,CTBHe )J,ecpopMa 11H KpYQeHH5I OCTaIDIl{I1XCSI )J,BYX ynpymx 3J1eMeHTOB. OCTaROBHMCSI Ha
BTOpOM, KaK HaH60nee 06Il{eM, cnY'lae. Bocnoflb3yeMcSI ypaBHeHlieM (4.153) H nepenMllleM
ero B BH)J.e
n b". G p J o
J3 + J3+-J3 =-y!lx'
J p J 13 J p
(4.184)
C06CTBeHHbIe Kone6aHI1S1 pOTopa on11CblBaIOTCSI JIeBOH lIaCTblO ypaBHeHlUI (4.184), KOp-
Hli xapaKTep11CTHQeCKoro ypaBHeHH5I KOToporo paBHbl
'i.2 = -b + b2 - O) o '
r)J,e b = b 13 /(2J p ); 0)130 = Gr31 J p
n pH O) o > b 2 )J,B11:>KCHHe pOTopa HOCHT xapaKTep 3aT)'XalOllJ,I1X Kone6aHHH C 'laCTOTOH
O)r = O) o - b 2 , a HHTeHCHBHOCTb YMeHbllIeHHSI aMnJIMT)')J, Kone6aRHH onpe.n e n5leTCH BcnH-
'iHHOH b (pHC. 4.15).
,[{eKpeMcHT 3aTyxaHHSI npeACTaBJISIeT C060H HaTypaJIbHblH norapH M OTHOllieHIDI aM-
nnl1T)',ll JtJ1SI ,l\BYX cne.u.ytOlUHx OAHH 3a APyrMM nepHo)J.oB:
= InU3, 1J32)'
(4.185)
,aEMf1<1>HPOBAHHE 43 MI1KPOrUPOCKOnOB
281
)l{CCTKOCTb YITpyrux ,L,CMeHTOB Ha Kpy<JCHI1C B
opMYJIe (4.188) 3aITHCaHa B COOTBeT-
CTBI1I1 C Bblpa>KeHI1CM (3.6) [.J K = k'bIl2C
,llJISI npHMoyroJ1bHOm Ce'leHMH ynpyroro 3JIeMeH-
Ta; .J K = 0,14Ib
- ..'!JISI KBai
paTHoro CCQCHl1SI; J K = O,1d 4 - ..'!JIH Kpymoro ce<JeHIDI, r.D:e
J. - B /,,/].
,[{JIH MaKCI1MaJIbHbIX aMITJIH1)')J, 3aTyxalOIl{JiX KOJIe6aHHH, CJIC.D:yroIl{HX B TC<JCHHC ncpHo-
Ita OJHiH 3a iJ:pyrI1M, HMeeM
ITmax(I) =
GI3
oe'bt f;
IT - 1 In -b(I+T) \2
max(l) --G 13 'Po e J .
2
BbI<JllCJII1M pa31111UY
LlW = ITma!t(I) - IT max (2) ,
H nOJIY<JHM OTHOllleHHe paCCCSlHIiOH: 3Heprl1H K 3anaCeHHOH::
LlW 1 -2bt 1 -ZB
- -e --e
W - - .
,[{ml MaJIbIX 3Ha<JCHHH .D:CKpCMCHTa 3aTyxaHIDI (e -2B ;::;: 1- 2$)
$
LlW 1(2W)= nQ-t .
(4.189)
06'be.uHHI1B Bblpa:>KeHl1SI (4.189) I1 (4.186), nOJIY'lHM
OPMYJ1Y )J,JIJl Bbl'lHCJIeHIDI K03I}>-
11u.HCHTa KOHCTPYKl\110HHOrO .neMnI}>HpOBaHHH YITPYfHX 3JIeMCHTOB:
b 13 = J 13 $ ro r In.
(4.190)
ITpe06pa3yeM
opr,f)'.rry (4.187), BBC)J,H. B Hee napaMe1pbI KOHC1pYKl\HH Mr. 1-13 ypaBHe-
HI1S1 (4.184), npHHSlB l' = p , BbIpa3HM MO
JIb MaKCHMaJIbHOrO yrJ1a nOBopoTa pompa:
! /301 = J o P0. x I GJ)'
(4.191)
<DopMYJIa (4.187) C )"IeTOM BbIpa:>KeHIDI (4.191) npHHllMaeT BH)J;
W = (Jop0. x )2/(2QJ.
(4.192)
ITpH HCO.nHOpoJtHOM pacnpe,l\eJIeHHM .ne
opMauHi! nOTepSi 3HeprHM BBJIHeTCH BeJIH'lH-
HOH, llHTerpaJIbHO 3aBHCmlIei! OT pacnpe,ll,eJIeHIDI HanpIDKeHHSI B yrrpyroM :,meMeHTe npH KO-
ne6aHJiHX, TaK :>Ke, KaK H BenH1JMHa ee 06rn:ero 3anaca 3HepfHH, T.e. .wUI nepeMCHHoro HOp-
MaJIbHOrO Hanp
DKeHIDI
MV., = J cer m dV ;
v
(4.193)
282
r.!]aBa 4. .lU1HAMHKA 1..I3 MI1KPOMEXAHI11.IECKI1X IlPI160POB
- .lI.JUl nepCMcHHoro KaCaTeJIbHoro HanpIDKeHHJI
w" = JC TmdV,
'"
( 4.194)
r,l\e c, C" 111 - K03 I}>UUIfCHTbI, xapa....-repH3YIOll{He MaTCpUaJI ynpynlx :meMeHToB; T - ITcpe-
MeHHoe KacaTe.,1bHoe HanpR)KCHlfe: 0 - nepeMeHHoe HOpMa.1bHoe HanpJl>KeHne: V - 06'beM
YITpyroro 3JIeMeHTa.
IIpcl\noJIo)KllB 6J11l30CTb xapaKrepl1eTI1K KpeMHIHI 11 CTaJJl1 KaK KOHCTPYKUl10HHblX MaTC-
pHa."lOB. BbmO HnM .1I.aJ1bHeHllll111 aHaJ1lf3 D,.illl 06bI'lHO Ifcnonb3yeMoro 3Ha'leHI1S1 m = 3 .
H3BeCTHo. <ITO MaKCI1MaJ"ll:.HOe KaCaTeJIbHOC HaITpIDKCHI1C B CC4CHI111 ynpyroro 3JICMCHTa
onpe.'J.eillleTcH l}>0pM)'noH
Tmax = Afl,p IW",
(4.195)
rlle W" = k'b cn -.:L"UI ITpHMoyronbHoro Ce'leHJi5I ynpyroro 3neMCHTa (bn/c n = 11 > 1; k'-
B3HTO 1'13 Ta6JI. 3.2): W" = 0.208b -)J,JUI ynpyroro 3neMeliTa C KBa)J,paTHbIM CC'lCHI1CM;
7f.d3
W" = 16:::: O,2d'" - C t,..pyrnbIM CeqeHl1eM; "',1".. -l\.-pyrSlll{I111 MOMCHT.
YnpyrI1e 3:iCMeHTbI l\."pyrJIoro Ce4eHJi5I I1MCIOT Tmax Ha cro Ol\."PY:>KHOCTI1 pa)J,l1yca
ry = d /2. a C npSlMoyrOJIbHbIM II KB8.il:paTHbIM Ce'leHWIMI1 t max Ha6JIIOAaIOTCH B CCpC)J,HHe
K:DK,1I.OH CTOpOHbl cro Kom)'pa (B yrJI8X t..-om)'pa t = 0).
IIepeMcHHoe KaCaTeJIbHoe HanpHn\:eHI1e .rVTH I\."pyrnoro B Ce4eHl111 3JICMCHTa OnpCl\eJISleT-
cx H3BeCTHOM l}>opMynoH:
r
T = Lmax - .
ry
(4.196)
,L{.nx )'ITpoll{eHIDI M3JIO)KCHIUI 11 HMeH B BJ1..UY. liTO B yrJ1ax npHMO)'T'OJIbHOrO 11 KBal\paTHO-
ro ceqeHHM L = O. 6Y,l\CM nonaraTb, 'lTO I1 }1,lliI. HI1X cnpaBe.nnHBa 1}>0pMYJIa (4.196), HO Benl1-
'lMHY ry Bblpa3HM 'iepe3 pa.3Mep b n 113 YCJlOBHH paBeHcTBa Lmax .lIlliI pa3JIH4HbIX q,OpM ce4e-
HMH. IIonmKHM O,208b = 0,2d 3 M HaH.neM
d
-=r T =1,OI6bnl2 bI1/2.
2
AHanOni'iHO )J,JlSI npSlMoyrOJIbHOrO ceqeHIDI B opMYJle (4.196) HY:>KHO nOJIaraTb
r. :;: .!- V k'b .
T 2 O,2lI
3anMlUeM Bl>lpIDKeHHe .llJUI 06'beMa l<OJIbueBOro 3JIeMeHTa (pHC. 4.16, a) dV = (2nrdr)111
c yqeTOM $OpM)'JIbI (4.196) BI>I'IUCnHM B COOTBeTCTBHH C '3aBI1CHMOCTblO (4.194) nOTepIO
3HeprnM B ynpyroM '3JIeMeHTe, no.nsepralOll\eMCH .ne4>0pMaUI1M KpyqeHHH:
AW. = 1+- J 2_ =2nlC.( ' )}dr = nlc, =r;
,l{EMOcDI1POBAHHE 1.J3 MHKPOrl1POCKOOOB
283
.:.
. ==0
b
(j
x
'C-o
t=O
ll)
6)
PMC. 4.16. K onpe eJleHMI{) nOTepM JMeprMM B ynpyrHx JJle leHTax:
a - npM Kp)"IeHMH; 6 - npH 1I3rH6e
,QJISI ,llBYX YITpyrl1x 3JIeMeHTOB 1= 2/ T2 I1 nOJIY'UlM
4 3 2
llWI( = -1[I T2 C, trnaxr T .
5
(4.197)
AHaJIOn1'lHbIC BbIKJ1a;:J;KIi BbITIOJIHliM .lI)UI ynpyrHX 3JIeMeHTOB, B KOTOpbIX pacceHHHe
3Hepn111 npOHCXO).(I1T 3a C'lCT ,l\C OpMaQJ111 113rl16a I1 MaKCliMaJIbHOC HOpMaJIbHOC HanpIDKe-
Hlie onpe).(eJISleTCSI ITO 1}>0pMYJIe
O'rnax = MIl IW K ,
r,l\e WI( = bnlC /6; M H - 113m6alOIlJ,l111: MOMCHT.
TIcpeMcHHoe HOpMaJIbHOe HanpSl>KeHHe B CeqeHMH (pllc. 4.16, 6) HaXOllMTCH no 3aBMCH-
MOCTI1
z
cr=cr
rnax a,5c n
(4.198)
3anlilllCM BbIpIDKCHllC )J;JISI 3J1eMeHTapHoro 0610eMa dV = Ibndz H C yqeTOM ypaBHeHIDI
(4.198) BbI'lliCJIllM no 3aBliCJ1MOCTM (4.193) (m = 3) nOTcplO 3HepfHll B ynpyroM 3JIeMeHTe,
nO).(BepralOll{eMCSI ,l\e OpMaQI1I1 H3f116a:
c"t2 ( ) 3
llW H = 2 J c cr rnax z Ibndz = !ccr ax Ibncn.
0,5c n 4
o
HMCH B BH,l\y, 'lTO I = 2/ T [, b n = b nh nOJIY'"lliM
1 3
ll =-ccr max I T1 b nl cn.
2
(4.199)
TaKllM 06prooM, CYMMapHbJe ITOTepH ::lHeprnH paBHEJ
llW = llWI( + llW Il .
(4.200)
284
r!I8B8 4. .!lIIHAMHKA '-:13 l\1HKPOMEXAHH'-:IECKHX IlPI1EOPOB
Q, HOot
2.0
1.5
LO '---'-'-'-'---.-.-.---.-.
0.5
-10"'
.'
-10'
-10 1
_10 0
p. Topp
Puc. 4.17. 38BIfCIf IOCTL Q = f(P)
B COOTBeTCTBMJ1 C OPMY:JOH (4.189) lJ.CKpeMeHT 3a1)'xaHuSi
2Gj3 (l'. WI( + l'. W u )
B= Vop!1 x f '
(4.201)
II no OPMYJ1C (4.190) Mo",eT 6hITb HafuJ.eH K03 HQHeHT KOHCTPYKQI10HHOro ,l\eMn l1pOBa-
HIDI.
Bonpoc 0 QHCJIeHHbIX 3Ha'IeHIDIX K03il>iPI1ltHeHToB C, C t B il>0pMynax (4.199), (4.197) oc-
TaeTC OTh-pbITblM, TaK KaK .'l;TII I\.-pCMHIDI M Bo06we ).UIH nOJIynpOBO)lHI1KOB nOA06HbIC HCCJ1C-
.:rOBaHHH.. BH.:IHMO. ewe He npOBO.il:l1JUlCb. TIpaBoMepHO ITOCTaBlfTb Bonpoc H 0 caMOH He06xo-
;IHMOCTH aHa."IHTWleCKOrO BhIIffiCJICHn K03 I1UHeHTOB )J,CMn l1pOBaHn , 06YCJ10BJIeHHbIX
BH),-peHHHM TpeHHeM, HMeH B B 1J1.Y, 'ITO ,l\OCTOBepHble pC3ynhTaTbI 6ynyr nOJl}"-leHbI TOJIbKO
noc::re BbInO:!IHeHHH. cneUHa.'IbHbIX 3KcnepHMCHTOB.
Cae.:reHIDI 0 nO!I06HbIX 3KcnepHMeHTaX l\.-paHHe CKY.l{HbI, B 'laCTHOCTI1 Ha pHC. 4.17 npc)J,-
CTaB..1eHbI pe'3y:rbTaTbI M3MepeHHH. A06poTHocrn 06pa:ma H3 I\.-pCMHI1S1 B 3aBMCl1MOCTH OT lJ.aB-
.!JeHIDl, no:ryqeHHbJe MeTOJ1.0M l\.-pyrHJIbHbIX KOJIe6aHMH, 113 KOTOphIX cncnYeT. qro pCaJIl13a-
UHj:J BbICOKOH CTeneHH !I06pOTHOCTH Q B03MOiKHa TOJIhKO npH BaK)'YMl1pOBaHI1H 06'beMa
(p = 10- 2 ...10- 3 MM pT.CT.), B KOTOpOM HaXOJJ.l1TCH Mr.
CJIeJlYeT BCC 1Ke 06paTHTb BHHMaHHe Ha 6e3YCJIOBHYlO none3HocTb aHaJIHTH'leCKOrO BbJ-
'lHC.'IeHIDI K03il> HUHeHTOB .l{eMnI}>HpOBaHHH. XOTJI 6bI C n03MQHH aHaJIH3a Bnl1S1HllH Ha HI1X
pa'3.!IH'iHbIX KOHcrpyKTHBHbIX napaMeTpoB Mr.
11, HaKoHeu, MMeH )IOCTOBepHbJe 3KCnepl1MeHTaJIbHbJe pe3ynbTaThl no onpc,l\eJ1eHm{) Q
,lL1J1 KOHKpeTHOH KOHcrpyKliHH Mr, MO)I(HO, Hcnonb3YH npHBe.l{eHHbre 3aBHCHMOCTH, pellll1Tb
06paTH}'IO 3a!Iaqy no OnpelleJIeHHIO qliCJIOBbIX 3Ha'IeHHH K03 HQHeHToB C Ii Ct.
llpu.wep 4. J 7
PaccllHTaeM ll06poTHOCTh, 06ycJloBJleHuylO BHyrpeHHuM 1peHueM B ynpyrux ::meMCtlTaX nOABeca
JLIDI Mf no pltc. 3.11, nplt KOJle6aHWix poTopa no t<oopJ]JiH8Te p. J1MeR B BItAY puc. 4.17, nOJlaraeM, '1TO
poTop HaxO.mrrCJI B BaK)'YMltpoBaHHoM I<opnyce.
)J,EMn<1>I1POBAHI1E 43 MI1KPOrl1POCKOrIOB
285
I1pI1HHMael\1 e "" c, '" 10- 24 M 4 /H2, Qx = 1 1/c. BocnOm,'3yeMcH MCXOD,HbIMM D,aHHbIMI1 H pe3ynbTa-
TaMil pactjeTOB npHMcpa 3.4; J o = C 1 = 2,5-10- 13 Kf'M 1 , J p = 10,13 Kf'M 2 , b nl = 2.10- 5 M, IT] = 0,8.10- 3 M,
b,t2 = 1,5 . 10 5 M, 'T2 = 0,5.\0-3 M, en = 2-10- 5 M. E(1()(]) = 1,3.10" HlM 2 . G(lOO) = 0,79.10 11 HlM 2 , (Of30 =
4 4
= 1,73.\0 IIc, p = (OyO = 1,32.10 lIc.
PaC'leTb! BbInOJlHI1M IJ TaKOH nOCJIe.L(OBaTeJlbHOCHI:
I. Bbl'U1C,lHM BCnJ1tjHHY G p 110 iPOp!tIYJ1C (4.188).
BHa'HIJlC paCC'IlITaeM J k = k'b n2 e = = 0,18. 1,5 . 10- 5 (2. 10'5)3 = 2,16. 10- 10 M . rD,e K03iPiPHl.\H-
CHT k' = 0,18 onpeJ.\errcH 113 Ta6J1. 3.2.
1l.anee nOJlY'U!M
079.10 11 .216.10- 20 13.10 1 I. 2.1O-S {2 .10- 5 \J
G - ' , + ' L 29.41.10- 6 H'M.
13 05.10- 3 08.10-3
, ,
2. HaHJ.\eM MaKC IMaJ1bHOe HopManbHoe HanpIDKeHHe C1 max B ynpyrllx 3J1eMeHTax WII1HOH IT" KOTO-
pble nO;J,BepfalOTCH H3rn6HbIM D,ciPopMaI.\HHM. I1cno:Jb3ycM MaTpHI.\Y (3.11) 11 3anHIIIeM H3rn6alOlQHfi
OJ k R k 4£(100)1 0 I _ _ bI1le _ 2.1O- 5 (2.IO- S f - 133.10 - 20 M 4 .
MOMeHT M" = 221-'00 fLle '22 = 'TI . npe.neJIHM 12 12 - ,
R 1 0 OJ C
ITpHMCM PO = 0, = --=-- pal!. neD,OBaTenbHO,
57,3
4 0 11 -20 0 1
M = .1,3-1 .1,33-10 ., 1,5.1O-9H'M.
II 08.10- 3 .573
, ,
MaKCHManbHoe HOpMaJll>HOe HarrpJl)KeHHe C1 max = M HI W K , r.ne
w: = 2.1O- 5 (2.1O- 5 Y - 133.1O- IS M 3
K 6 '
1,5.10- 9 6 HI z
11,3Ha'lHT, a max = 15 - 1,13-10 M.
1,33.10-
3. OnpeD,eJIlIM no iPopM)'J1e (4.199) nOTepw 3HeprHH H3-3a H3rn6HbIX D,ciPopMa!.\HH ynpynIx 3JIe-
MeHTOB:
A W =! 10- 24 1113.10 6 \3 0 8.10- 3 .2.10- 5 .2.10- 5 =0,23.10- 18 H'M.
u II 2 , J,
4. BLI'lHCJIHM MaKCHMaJlbHOe KacarenbHoe HanpH)KeHHe 't max B ynpyrnx 3JIeMeHTax WIliHOii lTI,
KOTOpble no,nBepralOTCJI ,neiPopMaI.\ffH KpY'leHIDI. KpyrSllQIiH MOMeHT B ynpyrOM 3JIeMeHTe
M "J! = G( O) J K Po, rl1.e J K = 2,16 . 10- 20 M 4 (CM. n. 1), a 130 = 0,1157,3 p . TaKHM 06pa30M,
T2
II -20 1
M = 0,79.10 .2.16.10 .0, =2,3.1O- 8 H'M
KP 2.0,5.10-3.57,3
11
_ M KP _ 2,3.10-8 - 243 . 10 7 HI 2
't --- r ' M .
max W 5 5
K 0,21.1,5.10- .2.10-
286
fnaBa 4. .zu.rnAMHKA 113 MI1KPOMEXAHULJECKHX IlPHEOPOB
5. HaiilleM no cpopM)'ne (4.197) nOTeplO 3Hepnm OT KPYTUJIbHblX D.ecpopMa(!uii ynpynix 3JIeMeH-
TOB:
w" = : .3,14.0,5.10- 3 .1O- Z4 (2,43.10 7 Y(0.75.1O- s r =1O- 15 H'M.
6. BbI'lliCJlH?>1 Beml'IJIH)' (Jopn,i=(2,5' 10. 13 .1,32 - 10 4 . li=27,22' 10- 18 H 2 'M 2 .
7. Orrpe.!lemlM .!leKpeMeHT 3al)'XaHlUl no cpopM)'JIe (4.201)
B
2.29,41.10-6(0,23.10- 18 + 10- 15 ) :::: 2,16,10- 3
27,22 _10- 18
H no cpopMy.rre (4.189) .!106pOTHOCTb Mf no KOOp.1l.IiHaTe 13;
0= 'iC1 B = 3, 14 10 3 =1,45.10 3 "
- 2,16
1-13 npnBe.::xeHHoro npuMepa CJ1e.n.yeT, '{TO lIeMncpHpoBaHl1e KOJ1e6aHHH npOHCXOD.JiT nO'ITIi HCKJIIO-
'iJITe.1JbHO 3a ct.zeT pacceRHlliI 3Hepnm B ynpyrHx 3neMeHTax, lICnbITbIBa!Ol.QHX KpYTlIJIbHble Kone6aHHJI
(Be.1:UQUHa l'.W" Ha TpH nopAAKa 60nbllIe BenHqHHbll'.W ).
4.8. JI:IlliAMHKA r APMOIIlPIECIillX OCIUUIJUITOPOB
q3 MRJ".-P0MeXaHJPleCKHX npn60poB CO,l\ep:>KaT 11M C ynpyruM no.nBeCOM M He3aBHCMMO
OT YHKliHOHaJThHOrO H83Ha'lCHHH. MHKpOITpn60pa (MA, MJJ.l1, Mr) HBIDIIOTCH KOJIe6aTenb-
HhlMH 3BeHbHMll, JUIll MHa'ie - rapMOHH'1eCKllMI1 OCl.{liJIJISlTOpaMH. B03My eHHH., .neHCTBYIO-
lIlHe Ha q3, - 3TO KOM6MHaI{I1I1 nOCTOHHHbIX H CllHYCOH.D:aJIbHbIX CHJI HIm MOMeHTOB. I1paK-
TH'ICCKMH HHTepcc ITPH HCCJIe.D:OBaHHM )J,HHaMMKH q3 npc)J,CTaBlliIlOT XapaKTepMcn1Kll BbIHY-
jKJleHHbIX. KOJIe6aHlrn, KOTOpbIC npoHcxoJUIT OTHOCHTeJIbHO CMellleHHH. 11M, 06YCJIOBneHHoro
nOCToHHHoH COCTaBJIHlOllleii B03My eHIDI.
4.8.1. JIHHeiiHbIe OCQHJlJlJlTOpbI
BbIH)"JK.!leHHble KOJIe6aHHH. rapMOHM'ieCKoro OCQI1JIJISITOpa orrnCbIBalOTCH JIMHeMHbIM
)J,H«p<PepeHUMaJThHbIM ypaBHeHMeM BTOporo nopR,l\Ka C nepeMeHHoH npaBoii 'iaCTbIO. Hac HH-
Tepecyer CJI)"JaH, Kor.na npaBaH. 'i8CTb CHHYCOHJl.aJIbHa, T.e. ynpOllJ,eHHble ,llH epeHQl1aJIb-
HbIe ypaBHeHIDI q3 MA, MM JUIH Mr no O)lHOH nlO60H 0606111eHHoH Koop,l\HHaTe npHBo-
,lUITCH K ypa8HeHl1IO BH)la
x+2g+ro x=EcoSIDt ,
(4.202)
r.ne x - MeXaHH1JCCKOC nepeMell.leHHe 11M (JIHHeiiHOe MOO yrnoBoe); E = Eolm (Eo - aMllJIHT)'-
.na M qacrOTa BHeomeH CHJIl>I, m - Macca 11M) MOO E = MoIJ (Mo - aMllJIMTY,l\a BHelliHero MO-
MeHTa, J - MOMeHT HHep11.HH 11M); 2 = k m (kJW - oceBoii K03 I1I.lH eHT AeM n HpOBaHHH.)
HJIM 2 = k;rjJ (k JJy - YfJIOBoii K 03 H.UHeHT ,l\eMn«pHpoBaHHH); IDa = Go 1m (Go - oceBaJI
nceCTKOCTb no.naeca) JUIll IDa = .J G y I J (G y - yrnoBaJI xeCTKOCTb no.nseca).
,lJ,HHAMI1KA fAPMOHHlffiCKJ1X OCQJ1JIJUlTOPOB
287
LfaCTO lI.JIJI pellleHIDI ypaBHeHUSI (4.202) nOJIb3YlOTCSI KOMITJICKCHbIMH BbIpIDKeHI1S1MI1.
npaBYIO <JaCTb 3aMCliSiI0T BbIpa:>KCHl1eM Ee JW ' , I1 nony<JacTcH Hoeoe ypaBHeHl1e:
X + 217 + w = Ee jrJJ ' .
(4.203)
,[{eikTBI1TCJIbHaJi 'laCTb pellJeHUSI ypaBHeHI1H (4.203) HBlliI.eTCH pCllleHl1eM ypaBHeHIDI
(4.202).
YpaBHCHl1e (4.203) pelllacTcH noncTaHoBKOM
x = CeJ((J/ .
(4.204)
IlOllCTaBJIHH BbIpa:>KCHHC (4.204) B ypaBHeHHe (4.203), IToJI)"laeM
(- w 2 + 29w + w6 )Ce JC01 = Ee Jw1 ,
OTKY)J,a KOMIT;IeKCHaH. a:.!n:mry.!J.a paBHa
E
C= ? 2 .
Wjj -w +29W
(4.205)
Hac I1HTepecyeT lI.e fcTBI1TeJlbHaH 'laCTb Bblpa:>KeHI1H (4.204), KOTopaH HMCCT BI1.n
x = X cos(wt-Ij»,
(4.206)
f)J,e
E
X- .
- (w6 _w 2 J +4 2W2 '
2 w
tgcp= 2 ') .
W o - w-
(4.207)
ECJII1 w - w 2 > 0, TO Hal1MeHblllee nOJIOJKI1TeJIbHOe 3HaqeHMC cp Haxo.nHTCH BO BTOpOM
KBa,llpaHre.
Mbl HallJJlJ1 O,ll.HO 113 <JaCTHbIX pellleHI1H HCO,llHOpO)J,Horo ypaBHeHlUI. 06II1CC pelllemle
ITOnY'lI1M, npl16aBMB K IiCMY 061lJ,cc pellleHHe HeO,l\HOpO,l\HOrO ypaBHeHIDI:
x = X cos{wt - cp)+ Ae --1;/ cos(pt - \jI),
(4.208)
f,lle p2 = w6 - 2 ; A, 'II - np0113BOJIbHble nOCTOHHHble.
TIo I1CTeqeHl111 HeKOToporo BpeMeHH c06CTBeHHbIC Kone6aHIDI ,lI.eJIalOTCSI CKonb yro)J,HO
6JIH3KI1MH K HYJIIO I1 HaCTYITaeT YCTaHOBHBlllHHCH npouecc. )],0 3TOro MOMeHTa roBOpMTb 0
pe30HaHce 6eCCMbICJIeHHO.
BbTHY:>K.L\eHHOe nepHO,l\I1<JecKoe KOJIe6aHHe HMeeT qaCTOTY p, paBHYIO qaCTOTe .neHCT-
BYIOll.(eH CI1JIbl H, Bo061lJ,e rOBOpH., OTJIH'IHyIO OT W.{{aCTOTa, C KOTOpOH KOJlc6JICTCH OCL\HJIIDI-
TOp B YCTaHOBllBllleMCSI pC:>KllMe npH BbIHY)K)J,eHHbIX KOJle6aHIDIX. COBepllleHHO He 3aBHCHT OT
era c06CTBeHHOH 'laCTOTbl.
TIpH npO<JHX paBHblX YCJIOBmIX aMnJIHry.na BbIHY)K)J,eHHb1X KOJIe6amiH X nponOPQHO-
HaJJbHa aMnJIH1)',l\e ,lIeHCTBYIOll.(eH Cl1JIbI E. AMnJIHTY.na H l}>83a CHJIbHO 3aBHCJIT OT TOro. B
KaKOM OTHOllleHHI1 HaXO,l\UTCH ITepHO,l\ BHeIllHeH CHJIbl K nepHO.ny co6CTBeHHhIX KOJIe6amrn.
288
fnaBa 4. ,lJJlHAMIlKA 43 MIIKPOfvlEXAHI14ECKliX nPIIEOPOB
PaCCMOTpUM, npH KaKOM COOTHOlueHHM 'laCTOTbI BHCIllHCll CIiJIbl 11 C06CTBeHHOll 'JaCTO-
TbI HaCT)'naeT pC30HaHC (nOHUMaH. ITO,a. pe30HaHCOM MaKCIIMYM aMnJIlITYllbI BbIHY>K)J,eHHbIX
Kone6aHllti ).
IlyCTb Wo = const. a U3MeIDIeTCH (0. Ilpollll epeHUlfpOBaB X no (0. Hailnet.!. 'lTO :'>laKCIIMY\1
aMruuU)'.J.bI CMellJ,eHIUI 6YlI.eT npn '1aCTOTe
W= J (O -2E,2.
(4.209)
TaK.H f OOP830M. pe30HaHCHM qaCTOTa BHeUIHen CII;lbl H e paBHa co6CTBeHHoil 'laCTOTe
roo II 'laCTOTe 3aryxaIOlllero OCUli!l:JI5ITOpHOrO npouecca ro = (O6 - S2 . PC30HaHC HaCTynae1
He npM n30xpOHH3Me. a npH ro. HeCKO.ITbKO MCHblliCii. qeM WOo
TIYCTb no-npe:iKHeM)/ roo = const. a 113 leHSleTCSI (0. Ho TeITepb !\Ibl 6yp:eM I1HTepeCOBaTbCH
MaKCnM) IOM a!t.UUHT)'.J.bI CKOpOCTll X. H)')KHO. CJIenOBaTeJIbHO, UCKaTb MaKcm.IYM BbIpa>Ke-
HlUi
. roE
X=wX=
(ro6 - ro 2 J + 41;2(02
E
[ ' ) 2
4 2 + (00 (0 ro-
(4.210)
CHO_ qTO MaKcMM)'M 6YlI.eT npl1
., .,
roo - (0- = 0 ,
(4.211)
T.e. pe30HaHc HaCl)'naeT Tor.na, KorlI.a ncpMo.n BHClllHCH CHJ1bI paBeH nepHo.ny He3aTyx8IOIl(e-
ro Ko.ae6aHIDI.
JL-ui MaKCHMaJIbHbIX 3HaIJeHMti aMIU1HT}'Abl H CKOpOCTH I1MeeM
X = E .
= 2S(Oo'
. E
X max =-.
2
(4.212)
HTaX, pC30HaHc 8MlmHl)')J,bI CMemCHIDI I1 pe30HaHc aMnJIHT}').I,bI CKOpOCTI1 3Toro CMeIl(e-
HIDI HaCTYflaIDT npH p83JlHIJHbIX 3HaQeHIDIX ro. PaCCMoTpHM, HaCKOJIbKO BeJIHKa p83HI1Qa Me-
JK,!I)' HHMH. BbIllle 6bUIO nOKa38HO, IJTO MaKCHMYM aMIU1HT}'AbI CMeIl(CHIDI COOTBeTCTBYeT BbI-
pIDKcmno (4.209), a MaKCHMYM aMnJIH1)')J,bI CKopOcrn cMerueHMlI opMYJIe (4.211).
llepenHweM yp8.BHeHHe (4.209) B BH)J,e
(02 = (O [I- 2 ;)
H..'IH, nOCKOJIbK}' I Wo = 3/21t (3 - JIOrapHI}>MMIJCCKMH ,l\el\.-peMeHT 3a1)'xaHHSI (CM. nO,l\p83,l\.
4.7.2), nonyqMM
2 2 [ 32 )
(0 = roo 1 - 2 41t 2 .
,lJ,HHAMI1KA r APMOHH4ECKI1X OCltHJU15ITOPOB
289
npH MaJlblX 3Ha'ICHI1S1X S, 'ITO HMCeT MeCTO B MHKpOMeXaHI1'leCKHX npl160pax, MO>KHO
C'IIn:nb, 'l TO 06a pe30HaHca (CMeUlCIII1H I1 CKOpOCTH) HacryrraIOT npH p = (0 11, 3Ha t UIT,
2 2
(1)0 '" CO = Wo - .
l1TaK, CCJII1 'JaTyxaHHe OCQHJIJISITOpa MaJIO, MO>KHO 3aKJIIO'lliTb, tITO pe30HaHC B MllKpO-
npu60pax HaCTYITaeT npH 1130XpOH113Me.
PaCC IOTpI1 1 TCITCpb Bonpoc 06 OCTpOTC pe30HaHcHoH KpI1BOH, 'lTO I1MceT ITCpBOCTCnCH-
Hoe 3l1a'IeHHe JJ.JTSI MI1KpOnpI160pOB, TaK KaK, HaITpl1Mep, Mr pa60TalOT B pe:>KI1Me pe30HaHC-
Ho i HaCTpOHKI1 HJIH 6ml3KOM K HeMY.
113 CooTHoIlleHHIl (4.207) .'ICrKO no.rry'Il1Tb BblpaJKeHl1e
. 2 (O
SIn cp = J (w _ (1)2) + 4 2(1)2 '
OTKy.LI.a C Y'leTOM OPMYJI (4.207) H (4.210) nonY'lHM
(4.213)
x
E sin cp
21;w '
j( = E sin cp .
2
(4.214)
,[{OITYCHIM, 'ITO Hac I1HTepecyeT 3aBHCHMOCTb OT paCCTpOHKH MaKCI1MaJII.HOro 3HaQCHHSI
KHHeTI1'IeCKOii :mcpnm, T. e. BenH'II1HbI
T= m j(l..
2
(4.215)
MI.l 3HaeM, 'ITO T = T,nax npH (1)0 = w, T.e. npl1 simp = 1. TI03TOMY Ha OCHOBaHHH (4.214),
(4.215) H (4.207) ITOJIYQaeM
T -T
max
T
( ",2 ",2 ) 2
2 '""0 - '""
ctg cp= ,
2 w
(4.216)
OTKY,lla
T
1
( 2 ? ) 2 .
1 + <0 0 - (0-
21;<0
(4.217)
Tmax
Pa3).(eJIHM QMCJIHTeJIb M 3HaMCHaTenb BblpaJKeHIDI (4.217) Ha (OoW, BBe.neM "cTeneHb 1130-
XpoHM3Ma", MJIM paccTpoHKY 'laCTOT
(I)
-=11
Wo
(4.218)
Ii ncpenHIllcM ypaBHeHHe (4.217) B BH).(e
T
T II1IlX
1
1+( -11r/( : r'
(4.219)
113 paBeHcTBa (4.219) BH)J,HO, 'iTO OTHOIlIeHMe rlT max 3aBHCHT He OT Tpex OT,l\eJIbHbIX ne-
peMeHHbIX (0, (00, , a OT .nsyx I1X KOM6HHaQMH (1111 B). B 3TOM 3aKJIIOIJaeTCX o.nHa H3 rrpH'-IHH
TOro, nO'leMY TaK CYI.l.I.eCTBeH JIOrapH MHqeCKHH ).(eKpeMeHT 3aTYXaHHx.
290
rnasa 4. HHAMI1KA 3 MI1KPOMEXAHl14ECKl1X IlPHEOPOB
B03bMeM. HanpMMep. Bin = 0,01. 11 = 0,8 M nOJI)"II1M
T 1
Trnax 1+2.10 3 '
on'-YI\a CTaHOBHTC}l }lCHO, n01JCMY pe30HaHc HMceT TaKoe orpOJl.1Hoe 3Ha<IeHl1e. 3.necb ,l\OCTa-
T01JHO 20%-Hol\ paCcTpoHKJi, 1JT06bl 3HeprmJ ynaJIa B 2000 pa3.
IIpM Ma.,lbIX paCCTpoHKaX, Le. KOrI\a ffio MaJIO OTJII1'laCTC}l OT ffi, I1MeeT MeCTO np1l6JII1-
:>KeHHOe paBeHCTBo
(ffi O + <.0 )(ffi O - ro) 2ro(ro o - ffi) ,
H opMYJIa (4.216) npMHMMaCT BIUI
( ) 2
roo - ro
Tmax - T _ (<.0 0 - ro f _ <.0 0
T 2 - ( 2 r
,
OTh')'.n.a C.ile.n.yeT, 'lTO 3HeprHSl na.rraCT B,l\Boe no cpaBHeHHIO C pC30HaHCOM npH
ffiO - <.0
roo
B
(4.220)
2n
Onpe.ileJIHB paccrpoHI>.')'. npM KOTOpOH op,l\MHaTa pe30HaHcHoH KpHBOH paBHa nOJIOBJ1He
op.romaTbI B MaKCHM)'Me, MO:>KHO BbJ'IMCJIHTh no opMYJIe (4.220) JIOrapH MM'leCKl1H ,l\el>.-pe-
MeHT 3aTYXaHIDI.
OCTJ)()TOH HacrpoHKl1 Ha3bIBaeTCH 06paTHaJI BeJIH'IMHa OTHOCHTeJIbHOH paCCTpOHKl1
I roo I !J.ro I. KOTOpaJI HYiKHa .D.JLII Toro, 'lT06bI opJumarn pe30HaHcHoH I>.-pHBOH YMeHblllHJIaCb
B.nsoe. 113 1}>0PMYJIbI JI)UI {Trnax - T)I T CJIeAyeT, 'ITO
I roo - 21t
/).ro - B '
T.e. 0CTp0Ta HaCTpOHKl1 06paTHo nponOpUHOHaJIbHa JIOrapHI}>MH'leCKoMY I\CI>.-peMeHT}' 3aT}'-
xaHIDI OCUHJIIDJTOpa.
BBe,l\eM B paccMmpeHMe TIlK Ha3blBaeMble paBHOBeCHbJe aMrIJIllTYI\bl
Xo ",li- (WIH li-)
Go G y
H nepenHUIeM BbJpaiKeHIDI (4.207) B BMAe
v _ X{ro) _ 1 .
- X(O) J 6-.,2Y +4E/.,2 '
(j> := arctg 21;'11 2 '
1-'11
(4.221)
r,l\e., - OTHOCmenbJWI llacTOTII [CM. 4IopMyny (4.218)].
,AI1HAMJ1KA r APMOHJ1{{ECKI1X OCW1JIJI.HTOPOB
291
0)
,.-.,.
S'i
- ':;-1-;-
---- ->
,.-.,.
4 q:
2
--- - - - -
,
6)
tp
-r - -
--+++-
.-- +---0--
:j .
n
<;=0,1
I; 0,2
0,3
,...
<;=0.
v
JlI
4
n/2
1
o
1
TJ
-
1
2
3 TJ=ro/roo
PHC. 4.18. HOp lHpo8aHHble 'iaCTOTHbie xapaKfepHCTHKH KOJle6aTeJJbHoro 38eHa:
a - aMnJIlIT)'.l1HbIe; 6 -1}J83oBbIe
K03 HUl1eHT V xapaKTepH3yeT J:\HHaMH'lCcKYIO BoCrrpHMM'lllBOCTb KOJIc6aTCJIbHOH
CHCTCMbI (OCUllJIJIHTopa) H qaCTO H83bIBaeTCH K03tP HUHCHTOM MHaMH'lHOCTH. 3Ha'lCHMC V
npH TJ = 1 ,L\aeT xapaKTepHCTI1KY ,l\eMIT HpOBaHHJI, HMCHYCMYlO ,L\06pOTHOCTblO (no ,L\eMIT M-
pOBaHHI{) ):
Q=1I2 .
(4.222)
Ha..n:o 06paTHTh BHHMaHl1e Ha TO, 'lTO BbIpa)l(CHIDI (4.221) HBJUllOTCH HOpMHpOBaHHbIMH
8MIIJIHT)',l\HOH H 830BOH qaCTOTHblMH xapaKTCpl1CTHKaMI1 KOJIc6aTCJlbHOrO 3BCHa (pHC. 4.18).
113 pHC. 4.18, a CJ1e,l\yeT, B qaCTHOCTM, qTO MaKcHM}'M aMnJ1HT)',L\ CMCJ:I{CHllJ1 nepCMeIlta-
eTCH B 38BHCHMOCTH OT ,l\CI<pCMCHT8 3aTYXaHHJI. Jh pHC. 4.18,6 BH):(HO, qTO npH MaJIOH qaCTO-
Te BHClllHCH CHJIhl IIpaKTH1.JCCKM tgtp = 0, T.e. CMCllJ,eHlle H3MCHJJCTCH B tP83C C BHCIIIHCH CHJIOH.
ECJIH CiI 4 00, TO tgtp TO:>KC CTpCMHTCH K HYJ1lO, HO 6yJtY'lH OTpMuaTCJ1bHbIM. CJ1C,L\OBaTCJIbHO,
C)J,BHr 83 CTpCMHTCH K n. B TOT MOMCHT, Kor,L\a BHClllHHH CHJIa ,l:(CHcTByeT BnpaBO, CMCllJ,eHMe
HanpaBJ1CHO BnCBO. TIpl1 pe30HaHce (TJ = 1) C.[\BHr tPa3 paBeH n/2. qCM MCHblliC , TCM MeHblue
HY)l(HO y,l\aJIJlTbCH OT pC30HaHca, IJT06hl C)J,BMf CP83 CTaJI npaKTJ1qCCJ(}f pasCH HYJ1lO HJIH n.
Ba:>KHOH ITpaKTMIJCCKOH XapaKTCpHCTHKOH KOJIe6aTCJIbHOrO 3BCHa HBJUlCTClI cro nOJIOca
nponycKaHHH 'laCTOT, KOTOpa1l B COOTBCTCTBMH C pHC. 4.19 onpe)J,eJUleTCH CJIC):IyI{)lI{HM 06pa-
30M:
ro 2 - ro l
ro = Cil2 - (01 = 2 TJroo = 2 (00 .
2(00
(4.223)
<DH3HIJCCKH 3TO 03HalJaeT, qTO npH HeKOTopoM CHJ1)KeHI1l1 YCHncHIDI )J,JIH. 6JIH3KHX qaCTOT
Ha n-l{) qaCTb MaKCHMaJIbHOH BCJIH1JHHbI V rnax = Q, OCUHJIJIHTOp HC pearHpyeT Ha OTHOCH-
TeJ1hHblC lJaCTOThl, KOTOpblC OTJ1H1JaI{)TCH OT qaCTOTbI CilO Ha BeJIH'IHH}' TJ .
292
rJIaBa 4.lIHHAMI1KA 43 MHKPOMEXAHUI.J:ECKHX IlPHEOPOB
v
V rnax
V rnax
n
____L_____
, ,
, ,
, ,
I !i ll '
:.. . '
, I
1
l 1
lQL
roo
1
(02
roo
11
PUC. 4.19. K oopelle.'leHHIO nO.'lOCbl nponycKaHHH 'iaCTOT
npM CHIDKCHMH yCJUICHJUI KOJ:Ic6aTcIlbHoro 3BCHa B n pa3 CnpaBe,llJIMBbI paBcHCTBa
Vw(l+ 11)= Q = = 1 ,
n 2/;11 (1-112)2 +4eTJ 2
Oncyn.a C;"IcJ1.YeT KBa,llpaTHoe ypasHeHHC,llJIj{ 11 2 :
TJ4 -2(1-2 2)TJ2 +I-4n2 2 = 0,
KOTOpoe MMeeT pelllCHHC
TJI,2 = J I-2 2 + 2 n2 -1+1;2
(4.224)
TaK Kal( « 1, H3 BLIpa:>KeHIDI (4.224) nOJIyqaeM npH6JIH:>KCHHOC pClllcHMe:
111.2 :::: 1 + 2e; .J n 2 -1 :::: 1 + .J n 2 -I .
3Ha'IJIT, HMeeT MecTO npH6JJIDKCHHOe pascHCTBO
ATJ=II-TJI,21::::1; .J n 2 -I,
M. B COOTBetCTBHH C rpOPMYJIOH (4.223), nOJJoca nponYCKaHIDI 1.JaCTOT paBHa
Aro = ro2 - rol :::: roo .J n 2 -1 = roo .J n 2 -1 .
2Q
(4.225)
)JI1HAMI1KA r APMOHI14ECKl1X OCI...lI1JIJU1TOPOB
293
TaImM 06pa30M, paCllll1pCHI1C nonOCbJ nponYCKaHI1J1 1.JaCTOT B03MO:>KHO npl1 COXpaHeHI111
Tpe6yeMoro YCI1JIelUlSI (V rnax In) 3a C1.JeT YMCHbllIeHI1S1 )J,06pOTHOCTM Q.
3aMeTUM. 4TO nOJIOca nponYCKaHlUi 1.JaCTOT OnpC eJISleTCSI no pC30HaHCHOH KpUBOH rrpl1
paBeHCTBC [X«(J)/ X(0)]2 = 0,5, r)J,e X(O), X(O) - aMITJII1TY)J,b1 I(QJIc6aHUH Ha pC30HaHcHoH
llaCTOTe (J)o U Ha 4aCTOTe (J).
Cle OBaTC.:IbHO, CHU:>KeHUC YCHJICHIDI I(oJJe6aTCJIbHOrO 3BCHa n = X(O) I X(w) =.J2 11
H3 <pOpMy.lbI (4.225) nOJIy<1aeM BeJIl1ql1HY ITOJIOCbI nponycI(aHI111 4aCTOT
110) = (J)o /2Q . (4.226)
TOqKH pe30HaHcHoH KpI1BOH, no I<OTOpblM onpC CJIH.eTCSI ITOJ1oca nponycI<aHIUI <IaCTOT
[CM. <POPMYJIY (4.226)], Ha3bIBalOTCR TO'H<8MH nOnOBI1HHOH MOIl!HOCTl1.
4.8.2. BmfHHHe HeJlHHeHHOii ynpyroCTII nO,l\Beca
HeJIHHCHHbIC 3<p<}JCKTbl B MHI<poMeXaHI1'lCCImX npH60pax 06ycnOBJICHhl BYMSI OCHOB-
HbIMH npl1qHHaMU: 3JICKrpOCTaTU4CCImMH CI1JIaMH B3aHMo eHCTBH» B rpC6eHIJaTbIX CTpYKTY-
pax nCpB1l4HbIX 11 BTOPU1.JHbIX npe06pa3oBaTencH H HeJIHHCHHOH :>KCCTI(QCTbIO nO)J,Bcca.
BJII1S1HI1CM 3JICKTpOCTaTHIJCCKI1X CI1JI npH YCJIOBMH, 1.JTO nepeMCIl!CHIDI 11M HaxO,l\KTC5! B rrpe-
CJIax 10 % OT nCpBOHaqaJIbHOrO 3a30pa. MO:>KHO npeHc6pc'lb.
KpoMc Toro, 3 eKTHBHbIM MeTO OM YCTpaHCHIDI BJIH»HIUi :noro BHM HCJIHHCHHOCTH
SlBJUl.eTC» COOTBeTCTBYIOIl!I1H no 60p ynpaBmlIOIl!HX H onopHoro HarrpIDKeHHH (CM. no,l\-
pa3 . 4.6).
OL{CHHM BJII1S1HHC HCJIHHCHHOH yrrpyrOCTI1 nO Bcca (CM. nO,llpa3,l\. 3.1.2, B) Ha MHaMHKY
OCL{HJIJI1ITOpa Ha npl1MCpe Mr RR-THITa ITO Koop I1HaTe nCpBH1.JHblX KOJIe68HI1H.
3anMlIIcM ypaBHeHHc BH:>KCHWI q3 MHKpOMCXaHH1.JeCI(oro rl1pOCKOITa RR-rnna OTHOCH-
TeJIbHO OCI1 nePB1l4HblX K0J1c6aHHH B npc OJIO:>KCHMI1, 1.JTO ynpYT BOCCT8HaBJIMBaIOlL{
CMa SlBillIeTCSl HeJ1HHCHHOH [CM. BbJpa)KeHI1C (3.39)]:
Jy'y+b/I+G J y+G 3 y 3 =Mocospt,
(4.227)
r)J,e J y - MOMeHT I1HepL{HH Mr OTHOCHTeJIJ)HO OCM nCpBH'lHhJX KOJIc6aHHH; by - K03<}JI}J HI .lH-
eHT cMnI}JHpoBaHI1»; G 1 , G 3 - K03I}J<}JHL{HCHThl JII1HCHHOH M Ky6H'lCCKOH :>KCCTKOCTI1 coorneT-
CTBCHHO; .N/ o , P - aMnJIHTy a I1 qaCTOTa MOMCHTa ITpl1Bo,lla B036Y)f()J.CHM1I nePBl1qHbIX KOJIc6a-
HUH.
I1cpcnHllICM ypaBHCHI1C (4.227) B Bll,l\C
Y + 2 wyot + w;o(Y + A/)= M cos pt , (4.228)
r,l\C w 0 = G 1 - KpyrOBaJI qaCTOTa C06CTBCHHbIX HC3aryxaIOIl!HX
'I J
'I
by
OCL{I1JIJI»TOpa;
2J yWyO
M Mo v
= - - npHBc cHHbIH MOMCHT rrpHBo a.
Joy
KOJIc6aHHH nHHcHHoro
OTHOCIfl'CJIbHbIH K03 l1UHeHT ,llCMnI}JHpOBaHH5!;
G
A. = ---1. .
G 1 '
294
rJ1aBa4. LU1HAMHKA lJ3 MHKPOMEXAHl1lJECKHX rIPM\)OPOB
BBe.neM 6C3p83MepHoe BpeMH: "C;:::: (J).yot . I1MCIOT MeCTO TaKHC paaCHCTBa:
., .,
d-y 2 d-y
---=;- = wyO ;
dr dt
dy -w dy
dt - yO dt
(4.229)
C }''leTOM OPMYJI (4.229) ypaBHcHuc (4.228) OTHOCHTeJIbHO 6e3pa3MepHoro BpeMcHH t
npMMeT CJIeA)'lOIltHH BH.!I.:
d 2 dy
-++2 -+y+Al =M.cosTJ"C,
dt- dt
(4.230)
M M P
me lv!. = ----;;- ;:::: ; TJ ;:::: - .
w:;o G, wyo
llrrSi perneHHH. ypaImeHml (4.230) BOCnOJIb3ycMcH MeTO):{OM rapMOHH'lCCKoro 6anaHca
[12], KOTOpblH npHMeHMM,llJ1SI HeqeTHbIX BOCCTaHaBJII1BalOll\HX CMJI (MOMCHTOB):
f(x) = - f(- x).
(4.23 I)
OCHOBHoe npe.nnOJIO:>KCHHe MeTO,lla rapMOHI1'leCKOrO 6anaHca 3aKJIIO'laCTCSI B TOM, qro
Kone6aHIDI C'llITaKJTCH 6JIH3KHMH K rapMOHHqCCKHM:
x ;:::: A cos wt .
(4.232)
ECJII1 nO)J,CTaBHTb Bblpa)l(CHI1C (4.232) B OPMYJIY (4.231). TO HCJII1HeHHWI BOCCTaHaBJII1-
BaIOIUIDI CMJIa f(x) TaK:>KC CTaHeT nepHO)J,H'lCCKOH ym(l.{HCH BpCMCHH 11 6Y,lleT I1MCTh I1MCH-
HO TaK)'KJ )l(e KpyrOBYIO qaCTOTY w, KaK 11 x. 3Ty nCp(10):{H'lCCK)'1O YHKUHIO MO:>KHO pa3Jlo-
)K}fTb B pB.:J. <1>ypbC:
x
f(x);:::: t(A cos wt);:::: ao + L (ak cos krot + b k sin krot ),
k='
r.ne ao. ak. b k - K03$q,HUHCHTbI pH):{a <DYPbC:
1 T
ao;:::: T Jf(Acoswt)dt;
o
(4.233)
2 T
ak ;:::: T Jt(Acosrot)coskwtdt;
o
T
b k ;:::: : Jt(Acoswt)sinkwtdt
o
( T ;:::: 2n _ nepl1o,l\ Kone6aHHH).
(j)
B cHJl)' npe,nnOJIO:>KeHIDI (4.231) Bce K03cl11}>HUI1CHTbI b k , a TaK)I(C nOCTOHHHbIH K03£l!qJl1-
[(l1eHT ao paBHbI Hymo. BTOpoe npe.nnonO:>KeHMe MeTO,l\a rapMOHM'leCKOrO 6anaHca COCTOHT B
TOM, <ITO MOjKHO npeHe6pe'lb BbIClliHMH rapMOHHKaMH pH):\a (4.233), T.e. IjJleHaMI1 C k:> 1, Ii
yqJfTbIBaTb TOJIbKO OCHOBHYIO rapMOHMK)' C KpyroBoH 'laCTOTOH w. Tor):\a C yqeTOM BblpIDKC-
HHH (4.232) nonyqaeM
f(x) = t(Acosrot) aJ coswt =5-x;:::: ex.
A
(4.234)
J],HHAMI1KA rAPMOHWlECKI1X OCL.{I1J1JU1TOPOB
295
TaKI1:'l 06pa30M, MeTOD,OM rapMOHH'leClmrO 6anaHca HCJIHHCHHaH tPYHKUWI f(x) npH-
BCnCHa K mmCHHOMY npH6mDKeHHOMY BblpIDKCHHlO ex. O HaKO I(03tPtPHl.{HCHT npOnOpl.{HO-
HanbHOCTH e HC SlB:ISlCTCSI nOCTOHHHbIM H 3aBHCHT OT aMnJIHTY):\bI A. nOCJIC ITO,llCTaHOBKH B
cpopMyny (4.234) K03cptPHl.{I1CHT8 <Dypbe OJ, 3aITMIIICM
T
e = e(A) = =...!... J f(Acosrot )cosrotd(rot).
A 1tA
o
(4.235)
HHrerpaJIbHOC COOTHOlliCHHe (4.235) n03BOJISIeT npC06p83oBaTb tPYHKl.{HlO f OT ITepc-
MeHHoH x B YHKl.{HIO e OT ncpcMeHHoH A.
CnC,l\oBaTeJIbHO, Ky6H'lCCI<HH BOCCTaHaBJIUBalOllJ,MH MOMCHT no.nBcca Mr MO:>KHO 3aMe-
HHTb nl1HCHHbIM BbIpIDKCHHCM C 1(Q3 I1l.{HeHTOM, 3aBHCHIlJ,HM OT OTKJIOHCHHSI:
Ax 3 J.,,:x,
2:<
r;:J,c J...." = JA3 cos 4 TJ1:"d(r)'t).
o
Y'lHTbIBWI,4TO
J 4 dx 3x sin2x sin4x
cos x =-+ + ,
8 4 32
ITOnY'1aCM
. 31..,e
A =
4
(4.236)
BBC,l\CM OTHOCHTenbHYlO c06CTBCHH}'IO qacTOTY TJA ' T8IOKC 3aBHCSlIlJ,ytO OT OTKJIOHCHlUI:
. 3 2 2
1+1.. =l+-A,A =TJA'
4
(4.237)
H YpaBHCHI1C (4.230) npHBe)J,cM I( BI1JJ:Y
d 2 y dy 2
- + 2 - + TJAY = Jvf. COS TJ'C .
dr. 2 dt
(4.238)
nCpHO,lJ,HqeCI(Oe pClllCHHC nHHCHHoro ypaBHCHHH (4.238) HMeeT Bll
'Y = Acos(TJ'C -tp),
r,l\e
A
M.
(TJ - TJ2 j + 41;2TJ2 '
(4.239)
21;TJ
tp=arctg 2 2 '
TJA - TJ
(4.240)
296
rJIaBa 4.11HHAMHKA lJ3 MI1KPOMEXAHl1lJECKI1X rIPHEOPOB
a) y,O
0,2
o
0,98
6) y,O
0,4
0,4
1
1,02
T}
0,2
o
0,98
1
1,02
T}
PHC. 4.20. Pe30uRHcHble KpHBbie Mr C ..y6H'iCC"HM BOCCT3H3B.rIllB3WU{lt!\1 !\IO!\ICHTO!\I:
a - nplf A > 0; 6 npH A < 0
Or3.J1qHCM BbIpaJKeillJ.H (4.239) H (4.240) OT aMITJ1I1TY)J,bI 11 c}Ja3bI BbIHY:>K.z:\eHHbIX Kone6a-
HHH Mr B cjJyqae JIMHeHHOro ynpYToro IIOJ:lBCCa HBJISlCTCSI 3aBMCHMOCTb BCJIH'1I1Hbl TJ OT
aMILTHIT)'.'1bl A. TaK 'ITO paBeHCTBO (4.239) cnc!!,yeT paccr>1aTpI1Ban KaI( ypaBHCHlle nml A:
') [( ') ') \:! ') ') ] ')
A- 117.1 - TJ-} + 41;-TJ- = M;.
(4.241)
Hafu[eM 3aBHCHMOCTb qaCTOThI TJ OT aMnJIHTYJ:lbI 11 = TJ(A)
(4.237) B ypaBHeHI1C (4.241). B pe3ynbTaTe KOTOpOH nony'lI1M
ITO,l\CTaHOBKOH paBCHCTBa
'-{I+ 31 ' 2<' ) ' +[(1+ 3 ' r <l o'
(4.242)
oTK)'J1.a C!.Ic.<I}'eT
') ( 3/...42
TJI2 = 1 +
, 4
21;2 ) + M; _41;2 ( 1+ 3M2
A 2 4
1;2 ).
(4.243)
B 3aBI1CHMOCTH OT BeJ1H<:IMHbJ BXOWlllUJ.X B ypaBHCHHC (4.243) ITapaMeTpoB MOryT 6blTb
,!IBa., OJ1.HO HJIH iKe illJ. OJ1.Horo )J,CHCTBI1TeJlbHoro pClllCHI1J1 JI,.IDI TJ. Hc 06cY:>K,llaJI B03MO:>KHOCTH
pemeBJUI 3aMeTHM, qTO ITOCTpoeHHLIe camacHo COOTHOllleHMIO (4.243) pC30HaHCHblC Kpl1BblC
OICa3bIBaIOTC.!I ropa3,l\O pa3Hoo6pa3Hec, <:ICM B CJIyqac nl1HCHHbIX CI1CTCM. 3J:lCCb KpOMC K03 -
«}JHl{I1CHTa J1.eMnc}JHpOBaHIDI 60nbllioe BJlIDIHI1C OKa3bIBaIOT BCJ1I1'lHHbl A H Nf.. AMnmITY,l\a
B03M)'IIJ.eHHH. M. npaKTllQeCKH He BJIHMa Ha ITOBC,l\CHHe JIMHCHHbIX CHCTCM. B TO :>KC BpeMJI
IIOBe,lleHHe HeJII1HeHHbIX CHCTeM caMblM cyweCTBeHHblM 06pa30M 3aBHCHT OT aMnJIMTYJ:lbl
M. . PaccMOTpHM 3TY 3aBHCHMOCTb.
OCTaHOBHMCJI Ha HeKOTopLIx xapalCTepHLIx cBoHCTBax pe30HaHcHblx KpHBblX HCJIHHeH-
HbIX CHCTeM. Ha pMC. 4.20 npHBe eHbl pe30HaHcHble KpHBble ,l\JUJ CJlc,l\yromHX IIapaMeTpoB
MI': G 1 = 2. ](J-5 H.M!pa,l(; G 3 = 1 . ](J-2 H.Mlpd I; = ](J-4Jy = 3 . 10- 13 Kr.M 2 ; M = 10- 10 H'M.
HCJIHHeHMOCTb, BblpIDKCHHaJI K03 I1I.(HeHTOM A, Bbl3blBaCT H3m6 nHKOB pC30HaHCHbIX
KpHBbIX. ITpM A. > 0 M A. < 0 JIHKH H3J'H6aJOTc.!I B HarrpaBnCHHH 66JIblllHX H MCHblIlHX 3Ha'lCHI1H 11
,AI1HAMI-IKA rAPMOHHlJ:ECKI1X OCQHJIIDlTOPOB
297
a) 6) y,O
1,5
1
0,5
0 <
-0,1 -0,5
-0,2 -1
-0,3 -1,5 0
0 0,5 I 1,5 2 0,5 1 1,5 2
t, c t, C
PHC. 4.21. Pe3YJ]bT3TLI "lUCJ]eHuoro uHTerpHpoB3HUU:
a - C yqeTOM Ky6H<Jecl<Oro BOCCTaHaBJIHBalOll\ero MOMeHTa; 6 - 6e3 Y<JeTa Ky6H<JecKoro
BOCCTaHaBJIHBaJOll\erO MOMeHTa
COOTBeTCTBeHHO. Cne,l\CTBHeM 3THX H3rH60B SlBJJHeTCH cyrneCTBOBaHHe 06nacTeH qaCTOT TJ, B
KOTOpbIX HeKOTOPOMY tPHKcHpoBaHHoMY 3HaQemnO TjoTBeqalOT TpH 3Ha'leHIDI 8MITJIHTYllbI A,
T.e. TpI1 B03MO>KHbIX pcrneHIDI YpaBHeHIDI (4.243).
qT06bI onpe.QenHTb MaKCHMYMbI H30fUYTbIX pe30HaHCHbIX KpHBbIX, °HY)KHO HaHTH KpaT-
HbIM ImpeHb TJ2 ypaBHeHHH (4.243), T.e. YClIOBHe, rrpH KoropOM nO.QKopeHHoe Bblpa:>KeHHe 06-
parnaCTCR B uynb. 3TO YCJ10BMe .QaeT KB8J],paTHoe ypaaHeHHe.QIDI A 2 , pellleHHe KOToporo HMe-
eT BH.Q [12]:
A2 =_ 2(1 1;2 t
max 3A.-
4(1- 2 f + /vi:
9'). 3A1;2 '
OTKY,l\a B COOTBeTCTBI1H C OPMYJIOH (4.243)
2 _ 1 3A.A
TJrnax - +
4
21;2.
(4.244)
I1peHe6perWl B paaeHcTBe (4.244) 'illeHOM 2 2 no cpOO3HeHlUO C e,l\HHMUeH, nOJIyqaeM
2 3 G] 2
TJ :::::1+--1'0'
4 G I
(4.245)
r.Qe 1'0 - aMnJII1TY.Qa nepBH<iHbIX KOJIe6aHMH.
113 BbIpa>KeHHH (4.245) CJIe.nyeT
G] 4 (2 1) - 4 P-(f)yO ( 1)
-:::::- TJ - - TJ+ .
G I 31' 31' O)yO
P-(f)O
BBe.QeM 0600HaqeHl1e oW y = y (.QorryCTMMaJI paccTpoHKa qaCTOTbl, 06YCJIOBJIeH-
O)yO
HWI HenHHeHHOCTbIO ITO,l\Beca) H, HMeH B BH , 'ITO 1') ::::: 1 , ITOny'lHM
G 3 800)y
A.=-::::: 2 '
G 1 3yo
(4.246)
298
DIaBa 4. .lUlliAMHKA q3 MI1KPOMEXAHI11ffiCKUX IlPI-IEOPOA
aJ y,O 6) y,"
1,5
1,5
1
1 0,5
0
0,5 -0,5
-1
0 ....1;'-- -1,5 0 0,5 1 1,5 2
0,98 1 1,02 11 t, C
PHC. 4.22. Pe30HauCHaH KpusaH (a) H nepeXO,!l.Hblii nponecc (6) Mr npH A = 1
BbIpaJKeHI1C (4.246) onpe emlcT ITpe,lleJIbHYIO BeJIH'IHHY A, npH KOTOpOH He npml8J1HeT-
ClI 3 clJet..l HeJIHHCHHOH }!(eCTKOCTH.
O'IeBIUl.HO, 'ITO MaKCHMaJIbHWI aMllJIMl)'.LIa K0J1e6aHI1H 6Y.LIeT He npH 11 = I, KaK B CJI}"Iae
Mr C J1HHeHHbIM BOCCTaHaB.ilMBalOll{I1M MOMeHTOM ynpYToro nO.LIBeca. B COOTBeTCTBMI1 C
pHC. 4.20 npMl1 = 1 aMllJIHTY.LIa BbIHY:iKJ1.eHHbIX K0J1e6aHHH COCTaBJIHeT Bcero 0,13°. B CJIyqae
llHHeHHoro BOCCTaHalLll1BalOJ.IJ.ero MOMeHTa YCTaHOBHBIlleeCH 3Ha'leHHC aMnJIHTY.LIbI K0J1e6a-
HMH Mr paBIDUlocb 6bI 1,4°. OueHKH aMnJ1Hl)')J,bI nepBH'lHbIX KOJIe6aHI1H Mr cOrJ1acylOTCSI C
pe3YJIbTaTaMH 'lI1CJIeHHoro HHTerpl1pOBaHWI ypaBHeHIDI (4.227), I<OTOpbIe npe,l\CTaBJ1eHbI Ha
pRC. 4.21.
AHaJIH3 nOJIyqeHHbIx pe3YJ1bTaTOB ITOKa3bIBaeT, If TO K)'611'IeCKaH. :>KeCTKOCTb nOJ1.Beca
3Ha<JHTelli>HO CHH:>KaeT aMnJIMl)'.LIY nepBM'IHbIX K0J1e6aHHH pOTopa Mr H, IqJOMe TOro, npl1-
BO.!UfT K 6HCHIDIM, KOTopbIe CBOHCTBeHHbI JI,mI JIHHeHHOro OCL{I1J1JUITOpa npH HeKOTopOH pac-
CTpOHKe <JaCTOT.
B COOTBeTCTBHH C OPMYJIOH (4.246) npH A < 1 HeJ1HHeHHble 3 eKTbl He OK83bIBa!OT
3aMeTHOrO BJIIDIHIDI Ha .LIMHaMHK)' Mf (PHC. 4.22).
3<pl}>eIITHBHhIM MeTO.nOM ycrpaHeHllSi BJIIDIHHH K)'611'leCKOH HeJIMHeHHOcTH nO.LIBeCa, TaK
jf(e, KaK 11 KBallpaTH'1HOH, 3aBI1CHll{eH OT 3J1eKTpoCTaTI1'leClmX CHJI, HBJIlIeTCH orpaHl1'IeHl1e
aMrurnl)'J1.bI Kone6aHI1H OCItHJl1IHTopa.
4.9..l(BHAMHKA 1 13 MAHfHlIKOBbIX MllKPOAKCEJIEPOMETPOB
4.9.1. MHKp08KCeJlepOMeTp H8 BH6pHpYlOmeM OCHOBaHMM
I1pM H113KOlfaCTOTHblX KOJIe6aHWIX OCHOBaHIDI, MeHblllHX H C0l13Mepl1MblX C nepBoH
rJIaBHOH <J.aCTOTOH, OCHOBHOH «}IOpMOH Kone6aHHH HBMeTClI nepBWI rnaBHWI clJoPMa no JCOOp-
.wrnare 9. CJIe,l\oBaTeJIbHO, B03MO:>KeH npHGJIM:>KeHHbIH aHaJIH3 )J,MHaMI1KH MallTHI1Ka Ha oc-
HORe ypaBHeHHSI, nOJI}"IeHHoro 113 CHCTeMbI (4.22) 6e3 yqem nepeKpCcTHbIX CBH3eH Me:>K.llY
KOOp)J,MHaTaMH SHY,:
J A 9+k ns 9+ 22 -m(aYY]9 = Fa+ F.(oV + a),
(4.247)
r.ne «}IYHKItHH F H F-IJ BbI'iHCMJOTCH no «}IopMYJIaM (4.23).
.DJ1HAMHKA 43 MA5ITHHKOBbIX Ml1KPOAKCEJIEPOMETPOB
299
B 3aBHCI1MOCTI1 OT Bapl1aHTa YCTaHOBKI1 aI(CCJICpOMCTpa OTHOCI1TCJIbHO OCHOBaHM B co-
OT8eTCTBHI1 C BbIpa:>KCHI1}IMH (4.23) pcaK1lliH. Ma THHKa 6Y)1eT p83J1H'lHOH B OTBeT Ha yrJ10BYIO
I1 mmeHHYIO BI16pal.\l1lO. Bonpoc 0 npc)1ITO':lTHTeJIbHOM Bapl1aHTe yCTaHOBKI1 aKCCJICpoMeTpa
J]11S1 lf3McpemUI napaMCTpoB yrnoBoH H nl1HCHHOH BH6p8lll1H OCHOBaHl1R MO:>KHO pClinlTb Ha
OCHOBC aHaJJI13a CHCTeMbI, nOJI)"ICHHOH 113 CHCTeMbI (4.22) .11m! 'lYBcTBHTeJIbHOCTI1 aKceJIepo-
Merpa:
k) IYr + k)2 B == F + Fyo;
(4.248)
k 21 Y r + k 22 B == Fa + Fro(l + a).
CI1CTCMY (4.248) p83pCWHM OTHOCI1TCJIbHO I1Hl}JopMaTHBHoro napaMc-rpa B, Bblpa)l(CHM
}:iJISI KOToporo 3anHlllCM ,I.(JISI yrnoBoH I1 JIMHCHHOH BI16paQMH 11 ,llJISI p83JII1':1HbIX Bapl1aHTOB
YCTaHOBKI1 aI(CeJICpOMCTp8, BbI':II1CJIRR F 11 FfJ no Bb]pa:>KCHH M (4.23).
,[{nH yrnoBoH Bl16pa[um OCHOBaHllH. ('Y"* 0, Xu = Yu = 0):
B ::!::m[k I2 -(l +a)kIlJf(l +a}y-L(yr + gL .
k) (kl2 - kl '
(4.249)
B= m[k 21 -(l + a)k..][::!::(L +l +aN+ gy]
[k 22 + mg(l + a )] -k 2 [(k 12 + mg)
(4.250)
.l0S1 nl1HeHHOH BI16paUHH OCHOBaHH (xu"* 0, Y8"* 0; 'Y = 0):
f\ = ::!:: m[k I2 -(I + a)kllJ(:YB + g) .
\7 .."
k lI k 22 - ki2
(4.251 )
B= ::!::m[(l + a)klI -k21]Xs
k ll [k 22 + mg(l +a)-k 21 (k 12 + mg)]
(4.252)
B Bblpa:>KeHHSIX (4.249), (4.251) 3HaK "+" B ':IHCJIHTeJIC COOTBCTCTByeT yrny YCTaHOBKH
Yo= 0°, a 3HaK ,,_It - yrny 'Yo = 180°, a B BbIpIDKCHlUIX (4.250), (4.252) BCpXHl1e 3HaKH "+" HJIll
"_" COOTBCTCTBYIOT yrny Yo = 90°, a HH)KHMe - yrJIY 270 0 .
113 BbIpa)l(eHHH (4.249)-( 4.251) CJIe)1yeT: )1J1H H3MCpeHH HH3KOQaCTOTHbIX 113McHeHHH
yrnoB y npe,l\nO'ITHTeJICH yroJI YCTaHoBKI1 'Yo = 90° (ncpcBcpHYTblH MiUl.11II1K); npH H3MepeHHH
BepTl1KaJJbHOH BH6pa1.tI1M yrnbI YCTaHoBKH aKCCJIepoMeTpa yo = 0 H 180° p8.BH03Ha'llibI; npH
113MepeHI1H rOp113OHTaJIbHOH BH6paUHH (Yo = 90 H 270°) HaH60JIbllll1C 3Ha<ICHHH yrnoB 6y-
.z:\YT OTBe<IaTb nepcBcpHYTOMY MWlTHI1KY ('Yo = 90°).
TIpe,l\nOJ1O:>KMM, liTO MaJlTHllK nO,l\BCp:>KCH yrJIOBblM Kone6aHlUIM OCHOBaHHJI, H3MeIDIl0-
LQI1MCH no 3aKoHY y = 'YMsinco/ C'YM' co - COOTBCTCTBCHHO aMnJIH1)')1a H 'laCTOTa KOJIe6aHHH oc-
300
rJIaBa 4.III1HAMHKA lJ3 MI1KPOMEXAHH4ECKI1X nNIEOP013
HOBaHHjl). BOCnOJIb3yeMcH 3aBJIClIMOCTJIMlf (4.23) (Yo = 27010) H 3anMIlleM ypaBHeHHe (4.247) H
BIUI.e
.. k;J.s' 1 r ( ) ( ) ]
+JS+Jlk::!2 +mg l+a -ma- Yow -COS-(J)( s=
.4 .4
- '7 M [o(I -1-a}u2 + g(l + a)] sin (Ot.
A
1 ( ) I
DY.J.eM nO.laraTb. 'lTO cos-(Ot = - 1 + cos 2wt -. l! ypaBHeHHe KOJIe6aHl!H MWlTHl1I<a
- . 2 2
npe.J,CTaBl1M B clJoPMe
.. . M .
S+ 2 rooS+N roo 9 = Y", smwt, (4.253)
J A
r.J.e (00 =
k 22 + mg(l + 2
J.4
- c06cTBeHHCUI <IaCTOTa K0J1e6aHI1H MCUlTHHKa; E = k llS 1(2J AWO)
OTHOCHTej]bHblH
K03clJ l1uHeHT
,llCMn l1pOBaHMjl;
N = 1- ma 2 (y",wY .
2[k 22 + mg(l + a )] ,
[ ( ) ]
a L-l-a (O-
M 0 = -mg(/ + a) 1 + g(l + a) .
B ypaBHeHHll (4.253) !lJU! <IaCTOT (00 > W J(03 MUl1eHT N 1 H -sinrot = cos«(Ot + 90°).
YpaBHeHl1e (4.253) nepenl1lllCM B BI1.ae
9- + 2 rooS + (O B- = M 0 'Y M cos(rot + 90° ).
J A
(4.254)
TaR KaK c06CTBeHHoe .aBJ1)KeHHe MWlTHHKa npH HaJlH<IMH .D.eMn l1pOBaHI1H 6blCTpO 3aTY-
xaeT, HafuIeM BbIH)'"A<.D.eHHOe ,llBI1:>KCHl1e MaHTHMKa. OITMCblBaeMoe <IaCTHbIM pellleHl1eM ypaB-
HeHIDI (4.254) aHaJlOrH<IHO nO.!l.p83ll. 4.8.1:
B- = Acos[«(Ot- q»+ 90°]= -Asin(rot - q» .
r.!:{e
A _ MoYo
- J A ((O - 00 2 r + 4 2oo2oo '
21;00000
q> = arctg 2 2
(00 - ro
(4.255)
OrHoweHHe BeIDi":lMHbI A K TaK H83blBaeMoH paBHOBeCHoH aMruImY,lle
Ao = (MoY M )/[k 22 + mg(/ + a)], KOTOpaJI B pacCMaTpHBaeMoM cJIY'lae npe,nCTaBJIHeT C060H
CTaTIf<lecxylO ,l\e4>OpMallHIO ynpyroro nO,l\BeCa MaJlTHI1Ka no.l1 J{eHCTBl1eM MaKCI1MaJIbHOrO
MOMeHTa Mo "PH HaKJIOHe OCHoBaHHR Ha yroJI 1.., onpe,lleJIHeT K034>QmJJ.HeHT )J,I1HaMH<IHocm:
A
13=-
Ao
I
(I- TJ2 r + 4 2T}2 '
00
T) --
- .
(00
(4.256)
.lU1HAMI1KA q3 MA5lTHI1KOBblX MI1KPOAKCEJIEPOMETPOB
301
(fJa30BbIH C.LI.Bl1r KOJIc6aHHH MaHTHI1Ka OTHOCI1TeJIbHO KOJIe6aH11H OCHOBaHMSI B <PYHKUHH
OT OTHOUlCHllH tIaCTOT:
(jJ = arctg 2 11 .
1- TJ2
(4.257)
AMnJlll1y.n.a KOJIc6aH11H MaHTHHKa B YHKl\HM OT 6C3pa3MepHOH 'IaCTOTbI HaXO)JJITClI no
BbIpa)KeH11I0:
-mg(1 + all + a(L -1- a)(roOTJY ]y
t g(l + a) M
A=
J A ro J (1-TJ2 } +4 2TJ2
(4.258)
nplwep 4. J 8
Bbl'JI1C.JI1M aMIUII1l)'.l1bl Ko.rre6aHI1H QM MaJITHHKa 11 era 4J!I30BOe 3an!l3.nbIBaHHe .D:JUl 3Ha'JeHHH
L = 0 H ] M npH 'IaCTOTax ymoBbIx KOJIe6aHHH OCHOBaHI1H (0 = 1 (6,28 lIc), 2 (12,56 I/c) 1I 3 rQ
(18,84 lIc) 11 a:.m.rrmy.l1e Y.. = 8 0 = 0,139 pa.n.
napaMeT]Jbl MaJITIlI1Ka: In = 0,29 . 10- 3 Kr, 1 + a = 5 . 10- 3 M, g = 9,81 Mlc 2 ; k 22 = 4,4 . 10-4 H'M;
J A = 7,093 . 10- 9 Kr.M 2 ; k!1.f} = 49,93 . 10 H.M'C, tjl1CJIO ynpyrux 6anoK 3.
IlonY"lI1M
(00 =
IIc (68,67 rQ);
s = klJ.s 1(2J A(()O) = 8,16.
Ina ( l + a )y
Crarn"lecKM aMIU1I1I)'.l1a Ao = 0 2 M = 5,15'.
JA(Oo
Pe3y.rrbT<iTbl BbI'IlICJIeHlIH no 4JopMynaM (4.257), (4.258) npHBe.l1eHbI B Ta6nl1Qe.
IlapaMeT]J L,M (0 = 6,28 IIc (0 = 12,56 lIc (0 = 18,84 IIc
(I ru) (2 rQ) (3 ru)
0 4,96 4,40 4,15
AMnnHl)'.na A, ,
I 22,17 68,55 133,61
<1>!I3a!p, 0 0...1 13,3 25,3 35,3
113 nonY"leHHbIX pe3ynbTaTOB CJIe,!1}'eT, 'ITO MWlTHHK npH L = 0 He ycneB8eT oTcneJKHB8Tb KOJIe6a-
Hml OCHOBaHIUI H aMnnllTY,l\bI cra KOJIef)!uIHH MeHbwe CTaTH'IeCKOH 8MmmTY,l\bI (A < AD)' TIpH L *" 0
HHepLlHoHHbIe CUJIb!, .neHCTBylOunie H8 MWITHHK, Bcne.nCTBHe Kone68HHH OCHOB8.HUSl, npHBO):\H.T K pe3-
KOMY B03p8craHlIIo aMnnHTy.nb! era KOJIe6aHHH (A » AD)'
TIocTynaJI aHaJIOrJ1<iHblM 06pa30M, ol\eHllM BJUUlHl1e JIHHeHHOH BH6paltMH OCHOBaHWI Ha
MaJlTHHK. npe.n.fiOJIO)l(llM, '1TO, KaK II B paCCMOTpeHHbIX CJIyqaJIX, yo. = 270°, a yrJIOBaH BH6pa-
QHSI OTcyrcTByeT (y = 0). YpaBHeHHe (4.247) c yqeToM Bl:JpWKeHHH (4.23) npliHHMaeT BH)J,
J) +klJ.se.+[k22 +mg(Z +a)l =-mxB(Z +a).
(4.259)
302
rJIaaa 4.)],MHAMMKA "43 MI1KPOMEXAHU1.JECKHX IlPHEOPOB
IIpe.nnonO>KJiM, qTO nHHeHHaJI Blf6paQHSI onlfCbIBaeTCSI YHI(QlfeH Xu = xuosin(rot)
(x"o, ro - aMllJ1HTy.na II 'laCTOTa BH6paQHu), II ypaBHeHHe (4.259) npe)J,cTaBHM CJ1e,l\YlOllJ,llM
06p830M:
n . F ( )
a + 2 rooa + ro(ja = - xBo COS\rot - 90° ,
J A
(4.260)
r e F = mro 2 (I + a), a c06cTBeHHaH. 'IaCTOTa KOJIe6aHHH H OTHOCI1TeJIbHbIH K03 m.iMeHT
eMn HpOBaHIDI Te )Ke, qTO H B ypaBHeHliII (4.253).
KaK M B npe.nbUJYI.U.eM cJIY"lae, BbIH)')J()I,eHHoC )J,BH:>KeHlle npe}J.cTaBmleT C060H rapMoHIi-
qeCKIie KOJIe6aHIDI C a30BbIM 3an83.nbIBaHl1eM, fIle aMnnHTy)J,a Ii a38 onpe}J.emllOTCSI Bblpa-
)KeHIDIMII
A = m(l +a}, 2x uo
JA (I-t/J +4Z;2YJ '
<p = arctg 2 YJ ,
1- YJ2
ro
YJ=-.
roo
(4.261)
liMeJl B BI1,!1)', qTO paBHOBCCHaH. aMnJIHTY)J,a BbIQllCJISleTCJl no opMyne
Ao = m(YJro f (I + a )x uo
k 22 + mg(l + a) ,
(4.262)
K03<l><l>HIUleHT .nHHaMH'IHOCTH HaxO,lJ.I1TC.H. no opMyne (4.256).
flJ7lL'-fep 4.19
Bbr'iUC.WM aMnJIIUYJIY Kone6aHHH QeHTpa Mace MaJlTHHKa c napaMeTpar.m H3 npe)l,bI,nYlQerO
npHMepa npH lleikTBMH JIHHeHHOH BH6pamm: XBO = 5 . 10-6 M, (0 = 200 . 6,28 lie (200 rQ).
HMeeM 1'\ = 200/68,87 = 2,91, no $opMYJIe (4.262) JJ,JL<ITpex ynpymx 6arrOK nO.ll.BeCa HaM)l.eM
Ao = 0.85' 10- 2 M. no $opM)'JIe (4.256) onpe.!l;eJIMM K03cpcpHQHeHT .ll.HHaMHQHOCTH 13 = 0,021 H aMnmn)'-
1l)' KOJIe6aHHH IJ,M A = Aof3 = 17,8 . 10- 5 M.
B COOTBeTCTBHH C 1l3BeCTHb1Mll pe3YJ1bTaTaMll TeOpl1ll Kone6aHHH, npH O}J.HOBpeMeHHOM
B03 eHCTBHH Ha MaJlTHHK yrJ10BOH Ii nlfHeHHOH BH6paull1i JJBl1jKeHHe MWlTHllKa 6Y,lleT npe.ll.-
CTa.BJUITb co60H HH3KO<JaCTOTHble KOJIe6aHWI, H8 KOTOpb1e HaJIO)KeHbI BbICOKO<JaCTOTHbIe MJIll,
MHaqe, BbICOKOqaCTOTHbIe KOJIe6aHIDI MOJJYJIllPYlOTCSI HIf3KO<JaCTOTHbiMll. ):(aHHoe nOJIO>KeHMe
HJIJIIOC'IpHpyercR pHC. 4.23, Ha KOTOpOM npHBe.neHb1 pe3YJIbT8Tb1 HHTerpHpOBaHIDI CMCTeMLI
ypaJmeHHH (4.22) )J..IDl napaMCTpoB WlTHHKa no npHMepy 4.3: m = 0,29 . 10- 3 Kr, J c = 1,775 x
x 10-9 Kr. .l= 805. 10-6 M, a = 4281,25. 10-6 M,L = 0, kJJJl = kJJ.fJ = 0; k lI = 2,036.10 3 HIM,
k J2 = k 21 = --0,819 H, k 22 = 4,389 . 10-4 H'M; IJHCJIO ynpymx 6aJIOK 3; yrOJJ YCTaHOBKJI aKCeJIe-
poMCTpa ""'0 = 90"; ymoBasI BH6paUIDI, rpa,l\, Y - 8 sin(6,28 t); J1IfHeHHaSI B1l6paUllS1, M,
/' 57,3.
X. = 10- 3 sin(50. 6,28 t) .
,AI1HAMI1KA lJ3 MA5ITHI1KOBbIX MI1KPOAKCEJIEPOMETPOB
303
'''r.r,.",...F''J''- .r __,.,
. .", ,, -r;:w-'-""" ....... "'.. -r:-r '"'.....
{
.
<
:x
.
;.:
;)
.
:,. '-. :: -t\
: .
--. --
,
.; ':.._".1__
,
" "". -= _., ,
!.' <.: 5>:: '<: "0:':"'- ...
. y;.,,"; a.
t .
-- - ...
PHC. 4.23. 3aBHcHMoCTH & = f(t) H Jl r = f(t) AJIH MaHTHHKa, YCTItHOBJlCHHoro
l1a OCI10BaHHH, COBcpw8IOweM HHJIW'IaCTOTHbiC yrJloBbic H BbiCOKO'laCTOTHbiC JlHHeHHLle H
BH6paLlHoHHblC KOJle6aHHH (OIUl4JpoBKa BepTHKaJlbHOn OCH .llill! rpa4JI1Ka B03M)'IUeHHH. .llaHa
B rpaLIycax, JIlIH. rpacpHKa yrnoBoro nepeMeIlleHIDI - B yrnoBbIx MHHyrax,
MJI rpa4JHKa mmellHoro nepeMemeHIU! - B MHKpoMeTpax)
KaK cne)J,YeT M3 pMC. 4.23, JlHHCiiHaH. BH6pa ml npHBena K nm£BJIemnO cOCTaBmUOIl{eH,
KOTOpaJI HaJ10:>KeHa Ha HI13KOtIaCTOTHbIH npo ecc yrJIOBbIX KOJIe6amt:H OCHoBaHIDl. EcnH H3-
Mepj{IOTCSI I1MeHHO yrJIOBble KOJIe6aHll}1 OCHOBaHHj{, TO B116pa HOHHWI COCTaBJIHIOIlJ,WI I1H-
opMaTHBHoro napaMeTpa 9- j{BJIHeTCj{ "rnyMOM", KOTOpbIH .llOJI>KeH 6bITb OTltmJIbTpOBaH B
61IOKe 31IeKTpOHHKH. ECJ1M :>Ke 113Mepj{eTCj{ JIl1HeHHaJI BH6paQH1I, TO nOMeXOH j{BJIHeTCSI CO-
CTaBJISlIOIl{WI, KOTOpWi B .naHHOM CJIyqae AOJI>KHa 6bITb YCTpaHeHa B 6JIOKe HJIbTpaQHH.
4.9.2. MUKp08KCeJIepOMeTp U8 nocTYn8TeJIbHO nepeMeIQ810IQeMCH OCHOB8HHH
B npeAnOJlO:>KeHI1M, QTO OCHOBaHlle, Ha KOTOpOM YCTaHOBJleH aKCeJIepOMeTp, ncpeMeIl{a-
eTCj{ nocTynaTCJIbHO C nOCTOSlHHbIM YCKopeHlleM u, lltJTO B ypaBHeHIDIx (4.22) H BbIpa:>KeHHJIX
(4.23) x B = Y B = 0, nOJIytJllM
m'A +ma +klIYr +k 12 9-+k w Yr + kJJS = m(u + g);
a
(4.263)
J A +majir +k 21 Yr + k22B+kJU'aYr +k/.\s =m{u + g)(l + a);
304
r.rraBa 4, )1HHAMHKA 43 MlIKPOMEXAHl14ECKIIX IlPIIEOPOU
"" \0 k :1 .
m.i r + maB + k 11 J'r + (kJ: + mg JU + k:1\'.i' r + B = mil ;
. a
(4.264)
J)}'+maj r +k 2l Yr + [k 22 9- + mg(1 + a)p+k.:<Ai'r +knBB = mu(1 + a).
B ypaBHeHIISIx (4.263) 3HaK "-" B cK06Kax cooTBeTcTByeT ymaM)'o = 0 0 , p = 90°. a 3HaK
"+" - )'f.'IaM Yo = 180°, P = 270°, B ypaBHeHI1S1x (4.264) 3H8K "-" B cK06Kax OTBetJaCT yrJJaM
Yo = 90°. P = 180°,8 3HaK "+" -)'f.il.aM)'o = 270°, P = 0°,
ITPllHS1TbIH BbI60p 3HaKoB cooTBeTCTByeT H3MeHeHI1K> yrJIa B no)J, ,l\eikTBI1CM CHIIbl mIl B
nO.'lO)K1fTeJ1LHOM (npoTlIa qaCOBOH cTpe.TIKu) HanpaBJIeHiUl (CM. pl1c. 4.2),
IToC]lC 3aT)'XamUi C06CTBCHHbIX Konc6aHI1ll MaJlTHllKa 3HaQCHllSi YCTaHOBl1BlIIl1XCSI yrnoB
& MOryr 6bITb onpe,aeneHbI 113 CI1CTeM:
kllYr + k l2 B = m(u + g); }
k 2J J'r + k 22 & = m(u + g )(1 + a); (4.265)
kIIJ'r + (k 12 + mg)& =mu; }
k 21 Yr + [k22 + mg(l + a )]B = mu (I + a).
(4.266)
113 CHCTeMbI (4.265) C.'leJ1.YIOT BblpIDKemUi .1I.ID1 yrJ1a OTKJIOHeHI1H MaHTHllKa npl1 )J,eHcT-
BHH BepTMKaJIbHOfO YCKopeHIDI:
& _ m(u + g)[kIl(1 + )- k 21 ] ,
k 11 k 22 - k 12
(4.267)
I1pH .neHCTBHH ropH30HTaJIbHblX YCKOpeHMH YCTaHoB1IBIlll1ecx ymbI OTKJIOHeHHH MaHT-
HHKa, nonyqeHHlde H3 CHCTeMhI (4.266), onpe.neJISlK>TCSI B1dpIDKeHHeM
B = mu[kll(1 + a)-k 21 ] ,
k lI k 22 -kf2 +mg [+ k 2J + k J1 (I + a)]
(4.268)
f..!l.C 8CpXHHe 3HaKH B pambIx CK06Kax COOTBeTCTBytOT ymaM 1'0 = 90°, P = 180°, a HI1:>K-
Hlie - yrJIaM 'Yo = 270°, f3 = 0°.
Ec.'lH H3BeCTHO npeJ1.CJ1bHOC 3HaqeHHe yrna ITOBOpOTa MWiTHIIKa 9rnax, orpaHH1JeHHoe
nporrnOCTLIO ynpyroro nO)J,Beca li.JlH KOHCTpYKTMBHbIMll oco6eHHOCTXMI1, TO MaKCHMaJIbHO
H3MepxeMoe YCKopeHHe MO:;KHO OITpC..!I.eJIHTb no 3aBllCllMOCTXM, KOTopLle CJIeAytOT 113 opMyn
(4.267) Ii (4.268):
U max = KBrnax + g,
(4.269)
U max = Bmax{K + g),
(4.270)
r.ne K
n (k 11 k 22 - kl 2 )
m[kll(l +a)-k 2J J
; n - qHCJIO ynpyrnx 6aJIOK nOJJ,Beca.
,D,MHAMMKA 43 MMITHHKOBblX MI1KPOAKCEJIEPOMETPOB
305
B qJOpMYJIe (4.269) 3HaK "+" nepel! BeJIWUmOH g COOTBeTcTByeT ymaM 'Yo = 0°, p = 90°.
a 3HaK "-" - yr.laM Yo = 180°, P = 270°. B opMYJ1e (4.270) 3HaK "+" COOTBeTCTBYeT yrJIaM
Yo = 90°, P = 180°, a 3HaK "-" -ymaM Yo = 270°, P = 0°.
lIpll.1fep 4.20
PaCCqHTaeM MaKCI1MaJlbHble 3HaIJeHmi YCKopemdf, KOTopbIe MOryr 6bITb H3MepeHbI aKCeJIepOMeT-
pml ,U.:IJI yrnoB yo = 0, 90, 180 H 270° C 43. HMelOllieM napaMeTpbI: a.. = 9.10- 3 M, b.. = 10- 2 M.
e" = 0.35 10- 3 M, P = 2.33 . 10 3 Kr /M 3 , m = 0 , 075 . 10- 3 Kr I + a = 5 . 10- 3 M g = 981 M/c 2
-6 " , ,
II =- (20.. .21)10 M, 11 = 3; klJ = 2,036 . 10 3 HIM, k l2 = k z ! = -D,819 H, k 22 = 4,398 . 10-4 H"M.
h
BbIIJI1CJIHM &ma = - = 2,32. 10- 3 pM (8').
aM
Pe3YJIbTaTbI paC'-IeTOB no cpopMYJIaM (4.269), (4.270) CBe,UeM B Ta6JIHl\Y:
Ilo.:Jo>KeHHe MaJITHHK8 H BeKTOpOB U H g
rIapaMe1p
Yo = 0°, p = 90° Yo = 180°, P = 270° Yo = 90°, P = 180° yo=270°,p=0°
IIm.IX' MI c 2 11,70 7,92 8,36 8,50
011eBHl!HO, npH H3MeHeHHlI HanpaBJIeHHJI BeKTopa U ,!UIJi ClI)'IJaeB 'Yo = 90 H 270 0 Mo.nym, BbIXO,n-
Horo CHrHaJla, npOnOpl "OHam.Horo lln1l1..\' He 1I3MeHlITCJ!.
4.9.3. nepo3H H BTOp3H r1JOPMLI KOJIe63HHH M3HTHHKOBoro q3 H3
BH6PUPYI01l(eM OCHOBaHHH
B pa60Te [22] npHBe)J,eHbl YCJIOBIDI, npH KOTOpbIX H3MeHJ!eTCJ! opMa KOJIe6aHHH (eM.
nO.z:qJa3.n. 3.1.1, E) MaHTHliKa Ha yrrpyroM nO,nBece. Ha pHC. 4.24 nOKa3aHa pacl1eTHWI cxeMa
Q3, me 0603Hal1eHbI: I..{M (TOl1IGI C), KpaHHHe TOl1IGI A H B nJIaCTllHbI H npOH3BOJIbHaH TOq-
Ka E, Koop.l\I1HaTa I<OTOpOH onpe.l\CJIneTCSI paBeHCTBoM
YE = YA + e& .
(4.271)
Bocno.TIb3yeMcn Cl1CTeMOH ypaBHeHHH (4.22) II nOJIO:>KI1M )J,JIX onpe,l\eJIeHHOCTH 'Yo =
= 270 0 (npxMoH MaHTHHK). B :ITOM cJlY'lae ,l:\JIX 1l3MepSieMoH ropH30HTaJIbHOH BM6paumi
x. = A cos rot (A, (i) - aMnJII1TY.l\a H qaCT0T8 BH6paullll), llMCSI B BH,I:\y, qTO Yr = YA (nepe06o-
3HaqeHHe KOOp.l\HHaTbI), CllCTeMa (4.22) npllHHMaeT BH.n (n = 1)
y
e
x
I
PRC. 4,24. P8C'IeTH8H cxeMB
366
rJIaBa 4. AlffiAMHKA t.[3 r-.1l1KPOMEXAHI1t.IECKI1X npl1EOPOB
.. {\ k (k ) n k . k Jlf}
mYA+ma-cr+ 11)'A+ 12+mg..:7+ -n.vYA+---;;&:::::b coswt ;
a
(4.272)
J A &+ maj'A +k 2I )'A + (k 22 + mga)B + kn.\.aj'A + kJJ.f}&::::: bcoswt,
rne b = maAw 1 .
PemeHHe CllCTeMhl (4.272) ,llJUI TOqKI1 A 3anHIlIeM B BH.ne BbIpa:>KeHH H
Y A::::: C A cos(wt + tpA -n/2); ::::: Cf} cos(wr + (jJa -n/ 2),
(4.273)
C A ::::: + l\ + l\ / l\; C a ::::: + /).22 + /). / l\;
l\ /).
(jJ A ::::: arctg /).3 ;tps ::::: arctg I\. 4 ;
1 °2
r)J,e
l\= - f+ - f; :::::b -
(4.274)
/).2 =b(a,a2as -a a6); /).3 :::::b(a,a2a4 -aia 3 ); l\4 :::::b(a a4 -al a 2 a 3);
aJ :::::k l1 a-k I2 ; a2 :::::k I2 a-k n + Je w2 ; a3 :::::k21-maw2;
a4 == mga - J e w 2 + k n ; as == -kJlyaw; a6 = -k}lf} w.
qaCTOTa, npH KOTOpOH aMnJIH1)'.na K0J1e6aHllH TOqKI1 A paBHa HYJ1lO, onpenemleTCH BbI-
paJKeHHeM
WA == (k 22 - k I2 a ) / J c .
(4.275)
Tm< K.aK /). > 0 npH m060H qaCTOTe w > 0, TO C yqeTOM Bblpa:>KeHMH (4.273) MO:>KHO 3a-
KJ1IOqHTb. 'lTO npH nepexon e qaCTOTbI qepe3 3HaqeHlle WA aMnJIl1Ty.D.b I Ay, By MeWllOT CBOH
3HaK. nocKOJ1blC)' BhlpaJKellID1 /)... /).3 TaIOKe ero MeWllOT. TaKUM 06pa30M, nOJlaraH C:4 0,
npH mo6oM CiI > 0 nonyqeHHble BbIpa:>KeHIDI 3anHIlIeM B BH,l\e
l\
tp A == arctg -1. npH
/).,
W<WA;
(4.276)
C A == t?j + /). I/), npH CiI> 0;
l\
tp A == arctg.-1. + n npll CiI > w A .
/).1
paccy-A(J).aJI aHaJJomqHO. 38KJIJOqaeM, QTO aMnJIJl1yna yrJJOBOro ABH:>KeHllH C s :;:' 0 npl1
J1J060H 'laCTaTe w > O. TaK KaK ypaBHeHWI /).2 = 0 H l::..t = 0 He HMelOT 06UJ.HX KopHeH. OTCIO.n a
cJ1ep;yeT. qTO cpa3a yrJJOBOro .nBH eHWI H3MeWleTCJI 6C3 cKaqKOB. CJle,l\OBaTenbHO, cnpaBen-
JIHBb1 CJ1e.nylOlIUie BbIP3)KeHIDI;
b. 4
tpA =arctg-.
/).2
(4.277)
C s == J b. + b. Il\;
.z:U1HAMI1KA 43 MNITHI1KOBblX Ml1KPOAKCFJIEPOMETPOB
307
nOJJ:CTaBllM )J,anee BbIpa)KeHlUI: (4.273) B paBeHCTBO (4.271) II nOJIY'lllM paBeHCTBo
YE == C A sin(mt + ({JA)+eCgsin(mt + ({Jg)= C E sin(mt+ ({JE)= C E cos( mt + ({JE - ;).
(4.278)
BOCnOJIb30BaBllll1Cb H3BeCTHOH OPMYJIOH ,llJIJI CJIO:>KeHMJI K0J1e68HHH, 3aIIHllleM Bblpa-
)KeHn C ,llJIJI aMIDII1TY)J,bI:
C E = (eCgY + C + 2eC AC 9 cos(({J A -({JS),
TaKKaK ({JA =({Jg npl1 m<m A 11 ({JA =({Jg -1t lIpll m>mA, nOJIyqHM
C E = C A +eC s lIpl1 m<m A ; C E =IC A -eC g I lIpH m>m A ;
(4.279)
({JE = ({Js lIpll C A > eC s ; ({JE =({Js +1t lIpll C A < eC g .
ITOJIyql1M BeJII1'lI1HY 'laCTOTbI mE, lIpl1 KOTOpOH aMnJJllry,lla K0J1e6aHHH TO'lKH E paBHa O.
HMeeM paBeHCTBa:
C E = C A -eC g
I1JII1
D. +D. I A =e A +D. I A,
OTKYAa
m E =
(4.280)
l'13 OPMYJIbI (4.280) lIpH e = 0 cne.nYeT clIopMYJIa (4.275).
fh Bblpa)KeHIDI (4.280) HaH,l\eM Koop,l\HHary e q TO'lKH Ew aMnJIHry,l\a KOTOpOH paBHa 11)'-
JIlO npH 'laCTOTe B03MyllJ,eHIDI m:
2
k l2 q - k 22 + Jem
e =
11. k Il q-k I2
(4.281)
To'lKY E q MO:>KHO Ha3BaTb QeHTpoM K0J1e6aHHH, OTHOCHTeJIl>HO KOToporo MWlTHHK co-
BCplllaeT yrJIOBbIe )J,BH)KeHWI. BIDKeHHe TO'lKH B (CM. pHC. 4.24) onHCbIBaeTCJI Bblpa:>KeHHeM
YB ::C B sin(mt + ({JB)= CBCO{ mt+({JB - ;).
(4.282)
BOCnOJIb3YCMCJI paBeHcTBaMH (4.279), (4.280) H ITPH e :: 2a nony'iHM
308
fmma4. ,DJ1HAMHKA t.{3 MHKPOMEXAHlft.{ECKHX rIPHEOP013
C B = C A + 2aC & npn ID > ID A : C B = jc 4 - 2aC & I npn W < (04 ;
(jJB =(jJa npn m<w B : (jJB =(jJ& +71: npn m<m B :
(4.2S3)
W B =
.,
2a-k ll + k:!.2 - 3k l2 q
J c
I101I0:>KUB e = a, aHaJIOflI'iHO nonY<IllM Bblpa:>KeHI1S1 )J)UI Onl1CamlSi )J,BI1)1(eHIISI llM TO'lKH
C ll.laCnrnbl:
Yc = C c sm(COI + (jJJ= C C cos( mt +(jJc - ; ),
r;J.e
C c = C A +aC a ITpH ID > mA; C C =IC A -aCsl npli m< IDA;
(jJc = <i>a npM m < COc ; (jJc = (jJa + 1t npM m > me ;
(4.284)
co =
c
.,
a klI + k 22 - 2k)2a
J c
fJpILwep 4.21
BbI'lHCJlHM 3HaqeHHjj qaCTOT. .llIDI KOTOpblX aMIUlI1TY a KOJIe6aHHH TOqeK A, C, B (CM. pHC. 4.24)
n!lacTIiHbl p,uma 0, a 4>a3a MeHHeT 3Ha'IeHHe Ha 11/2 .llIDI napaMeIpoB aKCeJIepoMe1pa no npJIMepy 4.3.
Bocno.llb3yeMCJJ 4>oPMy.'1aMH (4.275), (4.283) H (4.284) H nonY'iHM: .llIDI TO'IKH A (e = 0) (J)A = 413 fu,
;l!U1 TO'iKH C (e = 0) (J)c = 1380 fIt, J],illI TOqKH B (e = 20) (J) = 1917 fu. BOCnOJIb3yeMcJI tfJOPMYJIOi1
(4.281) H nOCTpOHM rpa lmK (PHC. 4.25) 3aBHClIMOCTH nOJIO)l{eHI1J1 ueHTpa KOJIe6aHIDI ruraCTHHbI OT qac-
TOTbJ B03MYlIleHIDI (B.llOm, op.nHHaT YCJIOBHO H306pa)!(eHa IUlaCTHHa-MWlTHHK).
B
8
6
C
4
0 ......
.-<
2 t: rr. 00
M
- - .-<
'<3"
400 600 800 1000 1200 1400 1600 1800 2000
m,n
PHC. 4.25. K onpe.ne.JJeHHJO nOJlmKeHHH QeHTpa KOJlef)auHit
)]J1HAMHKA q3 MMITl-II1KOBbIX MI1KPOAKCFJIEPOMETPOB
309
a)
r-i2
_' . .. . -.' ' ' _' - ....-... ....-.-. ......-.- ..-_....-........r.....-... -..-. -..
.rImtJ«(lI-I e WUITH Ka
- - Jtfl'-fanJ j 14M
--II. JMM
.. :-t.
-,- ' 1 ;-'
.. i "
-, '
. ,
.
rDP H3C<ij18I1bHaH
f!IiI OpaltM M
-:-::: .;It:
t
6)
t
PHC. 4.26. TIepBble I'Jl8BHble lJIopMbI KOJIe68HHH M8HTHHKa
(BapnaHTbI a H 6 OTJIHQalOTCH napaMeIpI1MlI BH6pa HH)
Ha pHC. 4.26.. .4.28 npllBeAeHLI pe3YJ1bTaTLI HCJIeHHOrO HHTerpHpoBaHIDI ypaBHeHHH
(4.22) AnSI YCTaHoBKH al\."CeJ1epOMeTpa, COOTBeTCTBYlOllJ,eH ymy Yo = 270 0 npH OTCyrCTBHH
yrJIOBOH BH6paQHl1 (y = 0). Ha MaHTHllK AeHCTBYeT TOJIbKO ropH30HTaJJbHWI BM6paQHA
(xo ::f:. 0). aMnJIl11y.na KOropoH 5 . 10-{; M Ha BCeM ,l\Hana30He 'laCTOT. napaMCTpbI aKcenepoMcT-
pa B3jITbI H3 npMMepa 4.3. TIpe.nnonaraeTCH, '1TO nnaCTHHa HMeeT 50 % nep<}Jopar.urn OT 06meH
nJIOllJ,aAll.
Ha pllc. 4.26.. .4.28 B HM:>KHeH QaCTll TOJICTOH JIHHHeH nOKa3aHbI JIHHeHHb1e nepeMeIU.e-
HH TO'lKM A (CM. pMC. 4.24), Yr :::: Y A' a TOHKOH HHHeH - J1HHeHHhIe nepeMeIU.eHH KpallHeH
TO'iKH l1j]aCTHHL1 Ya = aM 3- :::: Yo .
310
fmilia 4. ):{l1HAMI1KA 3 MHKPOMEXAHI1QECKHX f1PHEOPOB
a)
"
-"-_.+"- - -- -===-
f
..
-.
;:
-.}
:.
,;
::
t
.
I - d ' r ' " -'- '- '._u_"" ,- ,
-:-- + - " -='L::_-_ - --- :.:_-- t! : - f: ' t
-. .., , ---- --- - "------ -!-. . -- - ---- . .
- . . .- --':-T: ' T!--- . -
. .
'-"U",." cI..
.... ":;,H'
. .- . .'
., 'J'.. 4ki
. '. "
.."
, .
6) In.- __..I
1t ==== I
-;1 ,. I
I
I
"' IUI
SM
: I a '917("u
725,35
,
. ,
.
I.
. I". I ::
-. -- -. ...,..-...--- ._-_.- -
, .
I . ; :
t
I
. '
, "
- -1--
; . - - -
: J I
.. - ..,.
r
. no I,
.
i - -- - -
-- -- ---.- ,-
. ,
. _! 1 L
. .
t .
Pac. 4.27. BTopLie rnaBHLle 4»OpMLI KOJle68HHH M8J1THHK8
(aapHaHThi Q H 60TJUf'iIUOTCR napaMe11HIMH BH6pallliH)
B sepXHeH IJaCTH pHcyHKOB cBe'rnbIH 4>OH - 3T0 IVIOIlla,l\b. "OMeTaeMWI" nJIaCTHHOH Ma-
JlTHHKa, TeMHbIH - IVIO II l\J> . "OMeTaCMWI" IVIaCTHHOH MWlTHllKa nocne 3aryxaHHSI C06CTBeH-
HbIX Kone6aHIDf.
JU1HAMMKA \.)3 MAIDl-IHKOBbIX Ml1KPOAKCEJIEPOMETPOB
3)]
t
I
n. .."
1:l l.Q e"';;HH8 MaRTH KII
- - I,""""", ......
--lI:cttet.l MP(M
m m=.* '
.(.?;
I"
{ ...
u <.
,
t
t
. : 1
i ! J
! : t
,
I' '
. :!
. : j
, ,
-......
: '
. ' j '
,I ,-
I I
. ,
: .
:-t-
: f
: !
I,
. ;'7
{/L.A:-"" ,,'
. It'
, ' I ':
, ,
, .
. ,
}....; t
(
. .'
. '
". ..{
.....--. --f--"'--+ "'-' ....;. .........- - .....
.-, 1 !.. .,
, v'
.,
, "
I- I I q ,
j . ' j -<
, ' \ ,
4"' " ,
: . '.j,
, . ..
., ; f ,
- .
I
;
PHC. 4.28. BTopaH rnaBHaH tjlopMa KOJle6aHHH MaHTHHKa
Ha OCHOBaHH1-1 nOJIyqCHHbIX pe3ynbTaTOB cJIe.z:lYlOT TaKllC BbIBO.nbl:
. }l,BHiKCHHe IIJIaCTHHbI-MaJl.THHKa MO:>KHO paccMaTpHBaTb KaK yrnoBoc ,lI,Bll:>KeHMe OTHO-
CHTCJIbHO [{eHTpa KOJIe6aHllH [<lJoPMyna (4.281 )];
. )J,JIH. JIl060H TO'lKH Ha nJIaCTHHe c ecTByeT '1aCTOTa B03MymcHIDI [ oPMYna (4.280)],
npH KOTOpOH aMnnMTY,l\a nOH TOqIill paBHa 0, a 83a 1l3MeIDIeTCJJ: Ha 180 0 (ecJI1-1 npOTHB 3TOH
TOqKH paciionaraeTcJJ: ,l\aT'lllK nepeMell\eHHH, TO CHrHaJI C Hero 6Y,Il,eT 6J11130K K HYneBoMY);
. Ha 'laCTOTax B03MymeHIDI. MeHblllHX (1)A, ueHTp KOJle6aHHJI HaXO,l\llTCJI: BHC nnaCTHHbI,
co CTOpOHbI ynpyroro nO,l\BeCa (CM. p1-1c. 4.25 1-14.26, a), a tl>83bI nllHeHHbIX M yrnoBbIX ,l\BH-
iKCH1-1H COBnaL\aIOT;
. Ha ':IaCTOTe B03MymcHMJI (1) = (1)A [{eHTp KOJIe6aHHH COBnaL\aeT C TOqKOH A, aMnnllTY,l\a
ee KOJlc6aHHH HyneBaJl, a l}>83a }l,BMiKCH1-1JJ: TO'lIill A nperepncBaeT CKa':lOK Ha 180 0 (CM.
pHC. 4.25 1-14.26, 6);
. Ha qaCTOT8X B03M}1l1eHIDI OT (1)A [l}>oPMYJ1a (4.275)] ,l\0 (1)B [$oPMYJILI (4.283)] [{CHTp
Kone6aHllH pacnOJIO>KeH Ha nJIaCT1-1He. Ha :nllX qaCTOTax TO'lK1-1 nnaCTHHbI Me)l()J,y ynpYTllM
no.nBeCOM II [{eHTpoM K0J1e6aH1-1H .nBH:>KYTCJI B npOT1-1BO 83e C yrnoBbIM .nBHiKcHHeM, a TO'lKa
MC>K,II,y [{eHTpoM KOJIe6aHllH 1-1 KOH[{OM nnaCTHHbI (TOqKa B) - B O,Il,HOH I}>83C C yrnoBbIM .nBH-
>KCHHeM (pHC. 4.25 1-1 4.27);
. Ha "laCTOTe B03M}'ll{eHIDI (1) = (1)B [{eHTp K0J1e6aHHH COBna)J,ReT C TO'lKOH B (CM.
pHC. 4.27. 6). Ee aMnnHTY,II,a paBHa HYmo, a 83a }l,Bll:>KCHIDIl13MeIDIeTCJJ: Ha 180°;
. Ha '1aCTOTax B03Myrn.eHIDI (1) > roB (CM. pllc. 4.28) l!CHTp Kone6aHHH HaxOAHTCJJ: sa CBo-
6o.nHbIM KOHl!OM IIJIaCTHHbI. Ha 3THX qaCToTax Bee TOIJIill rrJlaCTHHbI Kone6J1lOTcJJ: B npOTHBo-
!j>a3e C ee yrnoBbIM ,lI,Bll:>KCHllCM.
312
f.i1aBa 4. .illIHAMllK.o\ q3 r..mKPOMEXA.Hl ECKILX rIPl160POB
4,10. ):{ImAMURA 1.I3 I\UIKPO,JAT1.IJIKA lIAB.TJEHIUI
OueHlP..l 8.\m:J1fIY.!lHb1e H 1}>830Bb1e xapanepJlCTIum M)J)1 no .1J1HeIlHOIl KOop.:UlHaTC
npH /x= 1:= 0 Ha OCHOBe BHaml3a nepBofo ypaBHeHJUi CnCTC Ib1 (4.78). no.laraSl. <ITO .1J1HeiiHaSI
BH6pauHH. 11 .u:aB..lIeHUe onpe.u:eJUUOTCH <l>YHKillUiMH
Y II = .1'110 COS (011; F::J.:::: F COS <02' ;
(-1..285 )
rile V"o. F::JO - 8..\flLlIHT)'Lmble 3Ha'leHRJI BH6pauMlf 1111as"leHJUI; ro I. ro - IJaCTOTbI.
.. C )"'Ie<rOM ypromemrn (4285) nepennme 1 nepBoe ypaBHeHHe CIlCTeMbI (4.78):
n6; + k:J,)..j' + Gy.1'::Om(u- g)+ mYBOro cosroI'+ F:Jo cosro:!.t,
(4.286)
KOTOpoe npeliCTaBHM B BH.Je
y + 2!;;n'Cl)0.i'+ro y:::: (u - g)+ o COS( lt -+- F:Jo COS<02',
.. m m
(4.287)
r.!{e o =myIlOro;;(OO = G)' Im;s:ry = k:J,)."/(2 mG)' ) .
OIJeBlUIHo, 'IT{) npH F::JO = 0 JpaBHemle (4.286) OITHCbIBaeT .:ufHa.'m "y q3 oceBoro MA.
DTMeTIIM BHaqaIie, 'ITO CH."la m(u - g) BhI3hIBaer craTlf'leCKOe CMememle )f\eCTKoro QeH-
1]>a MeM6paHbI M,[l[{:
Ycr = m(u - g)/G.\:;
(4.288)
OTHOCHTCJIbHO Hero H 6y.:I)'T npOHCXO.;:uITb ero BbIHyJK.!{eHHhIe KOJ1e6aIilUl, 06YCJ10B.ileHHble
rapMOHJl1lCCKHM H3MeHeHHeM H3MepReMoro ;IaB..lIeHIDi Ha 1}>0He BH6panHOHHJ,lX KoneOaHlfli
OCHOBa.HID{.
PemeHHe ypasHeHIDi (4.287) C }"IeTOM I}>OPMY!lhI (4.288), onpe.u:eJUleMOe ero npaBOH qa-
CTblO, HMeeT BHlJ.
y = m(u - g)/G]" - [AI sine rolt - CPI) + A 2 sine rozl - CP2)]'
(4.289)
IJ1e
F-
A - 110 .
1 - ,
m J (ro -wi f + .roiro
2!; .rol<OO
CPI = arctg 2 2 '
roo - rol
(4.290)
-
F:JfJ
(22\1 222'
m \roo - rol J + a).ro2roO
21;;n. ro 2(00
CP2 = arctg i 2
(00 - (02
B COOTB ell,,;LB HH C pemeHHeM ypBBHeHIDi (4.289) BLlH)-'iI<,neHHoe .u:B I]KeHHe q3 ,a,aT'IUKa
.n8BJleHHJJ npellCTaBIUler co6oii rapMOHHqCCKHe Ko.1Je6aHIDI C '1BCTOTOii H3MepReMoro ,a,aBJIe-
HHJJ ID:2 ua 4>oHe BH6PaII.HOHHOii nOMeXH 'IaCTOTbI rol OTHOCHTem.HO CMemeHHoro nOnO:>KeHmt
:aKeCTJ{oro uelllpa MeM6p8HhI.
LUIHA mKA 4"3 HlKP0.1A1'-I1IK.\ "'B.1EHlfJf
3B
BBe!J.e 1 B pacc IOTpeHJle pasHoBeCHb[e a m.1J1T)'.J.hl
A ifl = F 30 G.,:
A:., = F:;.rJ G.
.'
If 3aITUWe 1 OTHOllleHHSI a m.1HT}.J.
j 1 ' ( .
'1. I .., ..._
P = = I , i 1 - "1'\-' - .E:. n-
I A 'I, . '_' 'I
,0
'- ) ..,
1- ._.
. . r' l
I.... _'1 I
f.:Ie 1')[ = (01" (00' 1'):: = 0)::.0)0 - OTHOCHTe.1bHbIe (6e3pa:mepHbIe) qaCToTbL
06paTII 1 BHlmaHlIe Ha TO. qTO Bblpa:-KeHUe (4,291) eCTb He 'iTO UHoe. KaK Hop mposaH-
HWI a"n.1Hl}'.J.H<UI 'laCTOTHaJI xapa",epUCTm:a q') .J.aPnIKa .J.aB..1eHIUL
OlleHU 1 norpelliHOCTIl, Bbl3BaHHbIe nOCTOSlHHbD.f YCKopeHUe\! II Bu6poYCKopemle\!.
O'leBlt1HO, 'ITO nlellIeHHe (4.288) n.;eCTKOro lleHTpa no.J. B..ln5!HHe { ClUbI mt.Z1 - g 1 paSHD-
ueHHO .J.eHCTBlllo CII.lbI JlKTIIBHOrO .:IaB.leHH5! .1fJ Q Ha n:lOlIIa.JH n;:eCTKoro ueHTpa:
m(lI- g) = a'lb,!!¥J q ILlU a"b"c"p(u - g) = a,Af.-¥J Q . OT"'Y.J.a -'¥J? = C"P(U - g I. O'iemoHo.
norpelliHocTb no OTHomeHlIlO K :\laKc e.{a.lbHD:\()' n3 lepJJ:e IO I}' .1Uarr830HY .J.aB..lemrn L"
onpe.:Ie.LlleTcH Bblpan\eHlle 1
r ] 00 0 , "9"'"
0" = LC,!p(ll- g) :¥J=-x I /0. (...._ _)
IJplL\lep 4.22 _
Bbl'illc.:TIIM oIHoclrre.ThH)'lO norpelIIHOCTb. BHocml)'lO YCKopelille!>! ZI = I 00 g !>L c- npli u1!>lep.;--
.
Him .:I<iB.!leHJUI -¥IIl.3..X = 10 5 Da. DQ;IO:iK!{}! C" = 0.35 '1O- \1. P = 2.33.10- I-.T \C If no q>0p!>!y_le
(4.292) no.'I)"ilI}1
O - 10 - 3 .., 10 3 9 8 99
- ,,);). '';'..).)' '.' 100 0 . _ 0 _ 9 °'
b - u,- ,/ '0.
II - 10 5
Onpe!1. eJnlM norpelliHoCTb, BHOCIIM)'IO BJl6pauIlOHHbl 1 )TKOpeHne}( lBn6POI.Il} o 1 oc-
HOBaHiUl. npH u3 lepemUl nOCTOKHHOfO .JaB.leHlUl B BIDe OTHOllieHIUl
0 9 = (AI A ) 100 'o.
B COOTBeTCTBlUl C paBeHCTBO 1 (4.191) U lee (
Al = AIO I (1- 1') r +4 ;\.1') ;
A::! = Azo (ill::! = 0).
Cne,r{OBaT 1bHO, OTHOCHTe.!1bHaJI norpelliHOCTb palma
o = AJO
B A:!o
100
,
'BO(Oj, PI' IOO ';o.
<¥Jnu., .-1.-
l-C Q 31
f: \! ..'
I- 1') J + -t;;)'TJj
314
rmlBa 4. .lU1HAMI1KA 1:13 MI1KPOMEXAHI11:1ECKl1X IlPl160POB
rAe K03 H[UleHT lUlHaMH<IHOCTH PI onpelleJISleTCJI no pHC. 4.18, CYllSi no KOTOPOMY MaKCH
MaJIbHWI norpelliHocTb H3MepeHIDI nOCTOJlHHOrO )J,aBJIeHHJI HB iPoHe BH6pornYMa HMeeT MeCTO
npH '11 1 H, 3Ha'lHT. MBKCHMaJIbHWI OTHOCHTeJIbH<U! norpelliHoCTb MJI q3 MM onpe,lleJIJI-
eTCH OpMYnOH
2
s: = mYBOO)O 1 00 0/
uBma.x 2 J'O.
2S.i\\.fo.pma.x A
(4.294)
llplL\fep 4.23
BLI'IHCJillM OTHOCHTeJlbH)'lO norpeIllHoCTb H3MepeHHH. nOCTOHHHoro llaBJ1eHlIJi Ha 1j)0He BH6powy-
Ma lL'VI c.:.e;I)'lOlIIHX HCXO.:!HbIX .lI.aHHbIX: Pmax = 10 5 Ila, A = 2.10- 2 M , m = 2,08.10--4 Kr, YBO = 10- 3 M,
ffiJ =6,28.102 lIc, 0\) =2,73.104 l/c. TaK KaK rol/roO = 2,3 .10- 1 , TO no pHC. 4.18 !)::: 1 H no 1j)0pMyne
(4.293) nO.1IY'lHM:
B = 2,08.1O '1O-3 .6/8.10 4 100 %=0,21 %.
10 .4.10-
MaKCHMWIbHylO OTHOCHTeJIbH}'lO norpeIllHoCTb BbI'lHCJ1HM JVUI CJ1)"1aH., KOr.ll.a H3 YCJ10BHSI npo<J-
HOCTIi KpeMHHeBOH DJ1acTHHKH BH6poycKopeHHe OCHOBaHHH. orpaHHqeHO BCIIWlliHOH YBOro? = 1000g .
Ec.'1R npellnO!IO)fQITb, 'ITO ecTb HCTO'IHHK BH6paUHH c qaCTOTOH rot = roo (pe30HaHcHbIH cJ1)"1ai1), TO
aMIIJIHl)'lla BH6pauHH orpBHH'lelia 3Ha'IeHHeMYBO= (1000.9,8)/(2,73. 10 4 f= 1,35 . IO- s M. Ilo ljJopMY-
ne (4.294) DO.'I)"'lHM
Bmax
--4 0 3
2,8.10 .;,8,1 4 100 %=4,85 %.
2.0,707.10 +10-
OrnOCHTeJlbHaJI norpelliHoCTb H3MepeHIDI nepeMeHHoro no qaCTOTe (J)2 ABBJIeHIDI Ha
.poHe BH6polll)'Ma C qaCTOTOH (J)I C )"IeTOM iPOPMYJIbI (4.290) OnpellCJISleTCJI CJIe)J,YlOll.{HM 06-
p830M:
2
o = AI 100% = mYB.cCJ)I
B.n A An A 2
2 '-'i"'max
II 2\2 +4):2 2
-'112) ':>lJ,YTJ2 100 %.
(1-11;) + 4l; TJ;
(4.295)
OtIeBMJtHO, MaKCHM8J1bHWI norpemHOCTb HMeeT MeCTO npM (J)I = 0)2 :
2
o = mYe.oCJ)1 100 %.
B.n max f¥Jrnax A 2
(4.296)
):VvI 3H8IJeHHH '111 «I H 112« I , <ITO COOTBeTCTBYeT peaJIbHOCTH, BeJIH'II1HbI nOfpeW-
HOCTeH, BbI<fMCJUleMblX no .popMynaM (4.295) H (4.296), COBna.n.alOT.
.AHa.Jrn3 .wUIaMMKH q3 oceBoro MA MO)KeT 6bITb BbInOJIHeH HB OCHOB8HHH peweHlul
(4.289) C ycieTOM coOTHomeHHH (4.290), HMeg B BH,l\Y, 'ITO A 2 = 0, <pz = O.
lU1HAMI1KA 1I3 MI1KPOrl1POCKOIlA
315
4.11. )J.HHAMHKA lJ:3 MHKPOfHPOCKOfiA
.D:HHaMHKY 3 paccMOTpHM Ha npHMepe pOTOpHoro Mr RR-THna, HCnOJIb3YH ypaBHe-
HUH (CM. nO)J,pa3)J,. 4.5.2, E):
Juii + bua + Gua. = JoYo. y ;
(4.297)
Jp + b13f3 + Gpp = JoYo.x;
J/y+byy+G../y = Mu(t),
(4.298)
(4.299)
[)J,e M 8 (I) - BH6paUMOHHbIH MOMeHT B036Y)f()J,eHIDI nepBH<IHblX Kone6aHHH pOTOpa.
PaCCMOTpHM BHa'lane ypaBHeHHe (4.299) B npenn0J10:>KeHHH, 'lTO MOMeHT BH6ponpHBOna
onpe,lleJIHCTCH BbIpa:>KeHHeM
Mu(/) = Mo sin(pt),
(4.300)
r,ne Mo, P - aMnmny,na H 'laCTOTa MOMCHTa B036Y)f()J,CHHSI KOJIe6aHHH pOTOpa.
YpaBHeHHe (4.299) C }"IeToM paaeHcTBa (4.300) 3amUlIeM B 1}>0pMC
.. 2 ): . 2 Mo. ( )
Y+ ':IyWyoY+(i)yoy=-sm pI ,
J y
(4.301)
r)J,e (i)yO = Gy / J y ; /;y = by 1(2J y (i)yo).
qaCTHoe perneHHe ypaBHeHIDI (4.301) HMCeT BH)J,
Y = Esin(pt +q»,
(4.302)
r,ne
E = Mo{Jy ((i):O - p2) +4!; p2(i);o); tgcp= -21; y p(i)yo I((i) o - p 2 ). (4.303)
Y CJIOBHMCH, 'lTO
(i)yO =6P,
(4.304)
r.n.e 6 - K03I}>I}>Hl(HCHT, onpe.n.eJIHIOll.{MH paccTpoifKy <IaCTOTbl co6CTBeHHbIx K0J1e6aHHH pOTopa
OTHOCHTCJIbHO '1aCTOTbl B036Y)f()J,eHHH pOTOpa reHepaTOpOM K0J1e6aHHH.
C }"IeToM paBcHcTBa (4.304) BblproKeHHH. (4.303) npHHllMaIOT BH)J,
E = Mo - MOKyQy;
G y ( 1 _ J.... ) 2 + 4 ;
6 2 6 2
2 6
tgq>= ,
6 2 -I
(4.305)
316
IJ1aBa 4. ,[{HHAMHKA 43 MliKPOMEXAHI14ECKHX rIPHEOPOB
rite
K = .
y .
G y
Qy =
1
( 1 ) 2 4/;;
1-- +-
, ,
C C
(4.30/1)
TIpl1 He60JIblllOH paccTpOHKe (£ 1) MMeeM
cp -90 e ; Qy = 112/;'1'
( 4.307)
11 pellleHl1e (4.302) nplf06peTaeT OPMY
l' ::::: Yo sin{pr - 90° )::::: -Yo cos(OJypr),
(4.308)
r.:le
Yo = MoKyQy;
°\0
OJ yp = P = - .
£
(4.309)
Y'lliTblBaJI H;J.eHTH'iHOCTb ypaBHeHHH (4.297) H (4.298), paccMoTpMM O)J,HO 113 HMX,
(4.298). KOTopoe C yqeTOM BbIpIDKeHJU! (4.208) nepenl1111eM B mUte
.. .., Ho . ( )
P + 2!;I3 OJ I30P + copoP = -Ox Slll\co'fPt ,
J 13
(4.310)
f-'l:e OJ130 = GrdJ[J; Ho = Joyoco YP - MO.D.YJIb KHHeTl1QCCKOrO MOMCHTa po'ropa;
13 = b p I(2J I3 co [30)'
Mo.nyJlb KHHeTIfqecKOfO MOMeHTa MO:>KeT 6bITb npC,llCniB1JCH HHa'le. ITOMHJI, 'lTO DJUI po-
Topa B cpopMe TOHKoro ,llHCKa J 0 ::::: C l = J r' 11 nOJIarWl, liTO co ru = coyp (E 1). C yqeTOM
ypaBHeHMH (4.309) nOJIyql1M
Ho = Joyoco YP = JyYoCOyO = Jy M oKIQ y CO y O =JyMo QyCOi'O =
G y
1 MoQr
= JyMO 2 QyCOyO =
JyCOyO OJrO
(4.311)
Ih BbIpIDKCHIDI (4.311) cJ1e.nyeT, liTO KI1HeTl1qCCKHH MOMeHT YBeJ1I1'lI1BaeTCH npH
}'Mem.lIIeHHH co6crBeHHOH ljacTOTbI Kone6aHllH rnpocKona no Koop,l\MHaTC y.
tJacrnoe pellJeMI1C ypaBHeHIDI (4.310) HMeeT BH)J,
13 ::: E, sin(coypt + <p,),
(4.312)
r.ne
E _ Hof2x .
,- JJJ J (ro o -w r +4c;pw w o '
tgCPI :::
2C;13 w po w yp
2 2
wpo - COyp
(4.313)
)J,J1HAMI1KA q3 MI1KPOB1POCKOIlA
317
A(ro)
_...V.I. V . 1
I I .
I I I
I I I
I I I
I I I
,
,
-----r-----------------
I
I
I
I
I
r
r
I
I
co
<p(ro)
0"_
I
rol royp ro2 IO) O
A O)po-royp
"
-90
W
D
-180-
I
I
I
I
I
I
I
I
I
---------------- -----
I
I
PIIC. 4.29. CooTHoweHHe 'IaCTOT paccTpoiilm pe:iKIIJl.I08 KOJIe6aHHU
npH pe30HaHcHoH HaCTpOHKC w 130 :::: W yp ' II H3 opMynb[ (4.313) cnc)J,yeT
E,:::: H 0 Ox K 13 Q p H oOx npH <PI = -90 0 ,
J r> 21;r>w o
(4.314)
K 1 Q 1
r.u.e p ::::-; J3 ::::-.
G 13 21;13
3anMllleM perneHlle ypaBHeHIDI (4.312) C )"IeToM paBCHCTBa (4.314):
13 = -KpQpH oflx COs(wypt ).
(4.315)
B COOTBeTCTBHH C pCrnCHHeM (4.315) Kone6aHMH pOTOpa B pq HOCHT MOHorapMOHH'le-
CKItH xapaKTCp C 'laCTOTOH w yp H a30BbIM c.nBHrOM Q), = -n12 no OTHOrneHllIO K MOMCHT}'
ClUl HHCpltHH KopHonHca Mx = HoOx sin(wypt) [CM. COOTHorneHMe (4.410)], Bbl3LIBaIOIlJ,eMY
3T11 Kone6aHHJI. KOJIe6aHHJI pOTOpa B P.LI: H pq CHH a3HLI.
113 pClllCHIDI ypaBHeHHJI (4.315) CJIe.z:\YeT BLIpa:>KeHlle .u.JIH. CTaTHQeCKOrO K03c}JqmWfema
ncpe,l\a'm no CKOpOCTH:
13 Ho
K =-;::;KpHoQp;::; .
CT Ox ' . 21; p G p
(4.316)
318
rnullu 4. J.\HUAMHKA In MHKI'OMEXAllI-I l lE('KI1X III'W;OI>(}B
Ami ncpC30naBCHOI'O PC>KItM:t KOJlcfilllmii IIMCCT MCCTO pucnpoiilCll 'IUCTOT (pIlC. 4.29):
6 = roll o - (l)w .
AJ1R :nol"o CJlY'IaR pClIIcnltc paUCBC1"Ua (4.312) npltllllMUCT 1\11)'
J3 = E\ sin[(1J 1III - 1\)( + tpl] = 1 1 [sm(wl\O - I\ OStpl -to COS(II)IIn - l\ iIHPI].
(4,317)
r,llC
£1'"
J I1
UuD.,
... ..."' ,.. 2
[{ (0rP ... A)" - (J)TI»- ... 4!; ('ITI> «(o)YI' ... /\)
tgtpl =
2 1\ (lI)yp + 1\ )(I)yp
2 2'
«(Oyp + 1\) - (\))'1'
(4.3IX)
3anHIlJeM CJ1C.llYIQUUiC pauencnm:
sm<P1 "'-
2S l1 «(Oyp -+- 1\ )(J)YI'
r .,., ] ., ., .,
l«(I)yp + M- - (J) , + 4S «(J)yp + 1\)-(0;1'
,
.
(4319)
COS<P\ '"
., .,
(row +.6 r - LO yp
] 2
.,") ") ") ")
(LO yp + h)' - (I)yl' + 4sji (LO yp +.6). roYI'
nepemuueM peweuHe (4.317) C YI.JCTOM BhlpmKclIHH (4.318) H (4.319):
J3 = HoQx )(
J { "'w + 1»' - ", ]'+ 4l; ("'w + 6)' "':. }
)( {[(royp +f1)2 -LO ]sjn(L01'P -.6)t-2S p (LO yp +.6)LOy/,COS(roy!' -.6)/).
( 4.320)
J'b BbIpwKeHl1J1 (4.320) cne,l\yer, q'ro Kone6aHHH no BblXO.llHOH OCM Mr I1MeIO'f ,llnc co
CTaBJIJlJOIllHe, nepBaJI H3 KO'fOpbIX HaXO.llHTCX B JlpOTHBO )a'JC c Bbllf}?K.llCIUlI>IMU KOJlc6amUI-
MM pOTopa, a BTOptul - B )a1e C HHMH. Ha nOM OCHOBaHHH ncpoYJO COCnmJIHIOI1!YIO KOJIC6n-
HHH MO)l(HO Ha3S8Tb KBa.aparypHoH, a B'fOpyJO - CHII Ja:JHOH.
"IYSCTBHTeJJbHOCTb Mf K H3MepHeMOH yrnoooH CKOpOCTH Onpe.llC1UICTCSI MBCIJITB6UhlM
K03clJcJlH1!.HeHTOM:
KN =J3/0 y ,
KOTOpblli pasCH:
-.l1JIX CHHcJla3HoJi COCTaBnJlJOllJ,eJi
K = 2t;'ICJ)'YP(CJ)YP +f1);
A
(4.321)
;U'IiIAMHKA tn MHI<POrHPOCKOriA
319
)'-!UI ImH)tpll'l'YpIlOii COCTHIlJUIIOII\di
KK _ K r 2 2 ]
M - - L«(!)yl' + /\) - (Ow '
II
(4.322)
mC
K == I/o !.III' Jl == [(O)yp + 1\)2 - W;IJ -I- 4 «(J.)Yr -I- 6.)2C1) II'
(4.323)
B coornCTCTIIllII c ypUlIIICIIHCM (4.317) Ii C Y'IC'I'OM (I)OPMYJI (4.321), (4.322) M3cllna6-
IIhlM Iw')(IJllmI\IfCIlT 110 06utcii HMIIJIH'I'YHC H (11U1011101ii cl\Imr OIlPC)lClUIIOTCH BblpU)I(CllllJ!MH
KM == (K Y -I' (K:Y :
KK
(PI == urctg K .
M
(4.324)
BIoI1Il1J11IUM IIpI16JIU)f{CIIIIYI0 OllCIl KY ']UBHCHMOCTCH (4.321 )...(4.324). 3UI1H1l1CM:
_ ( 2 \:! 2{ 2 2 \:! 2 2 2 2 4 2 ( 2 2 2 )
/l- 2(I).YII/\'" /). / -I- 1;11 \(I)YII -I- 2(1)'1'1 16 + 6. J ffi Y11 :::: 4(j)yp6. .1. 41;(lffiyl' = 4ffi YI1 t:i. + 1;llffiyp .
(I>opMym,1 (4.321), (4.322) IIplllll1MatOT IUm
/C K . 2I';rl(U yp (Cl)yp + 6.) K1;11
K I == 2 2 2 2 :::: 2 2 2
4(1)Yl'(6 -I- 1;11(1)'1'1 1 ) 2(6. + Sllffi yp )
.!,,'YOWypS!1 .
2 .2 2 '
2J11( -I- I';J}W yll )
(4.325)
rK K(2<oyp6.+ 6. 2 ) Kf). _ .I0'YOro yp f).
Jl. M == :1 :1 ,22:::S 2 2:1 - ( 2 1=22 ) '
4(Oyp(6 +I';II(I)y[1) 2o).yp(6. + llro'Yl') 2.!fJ(J)yp IJ "' IJWyp
ECJIJ1 KOIIC" I 1JyKIlHH JlOnYCKUC'f 1I1>e,l\IIOnO)f{eIlHC .I() J y Up, TO H3 BblPU)I(CHHM (4.325)
CJICJlYJo'J' "I>IP3)f{CIIHII:
K c _ 'Yo W YI I S rl .
III -.. 2 2 '
6.. + 1;,I(J)yp
K I< 'Yo6.
M 2 22'
... Sllro yp
(4.326)
<DOI>MYJlbl (4.324) npHIIHMaloT DHJ1
K - 'Yo .
M - f).2... 1; ro .
11
!PI = urctg
Sproyp
(4.327)
Jib cOOTJt011JeJtH (4.326) JlCHO, 'ITO MnKCHM8JlbHoe 3HU'leHHe K HMceT MecTO npH
f). a 0:
K IIII1X = 'Yo/(l;pwyl')' (4.328)
320
IJIaBa 4. ,WiHAMI1KA q3 MI1KPOMEXAHH1.IECKHX rIPH60POB
BbI'IllCJUlM np01l3BOJIHYlO d(K=)/ d/1, npllpaBHReM ee HYJIlO H nonY'iHM paccTpOHKY
/1 = 1;jjCO yp , npn KOTOpOH 3Ha'leHI1e K:' MaKCI1Ma..TlbHO:
K max ;="Yo /(2 p(()'YP)'
(4.329)
113 nonyqeHHhIX PC3YJIbTaTOB CJIe.nyeT, 'iTO YBeJIl1'lliTb MaCIIITa6HblH K03 IIUlIeHT, T.e.
'I)'BCTBIITCJIbHOCTh. MO:>KHO )'BeJIM'leHlIeM allfllmuy.nbl nepBII'iHbIX KOJIe6aHllH pOTopa (P )J,).
YBeJIllqeHl1e paCCTpOHKll (napaMeTp /1) I{ OTHOCHTeJIbHOro K03cpcpnUI1eHTa ,lleMncpl1pO-
BaHHSI YMeHblIlaeT qyBCTBIITeJIbHOCTb m:pOCKona.
llpU\fep 4.24
-, -4 4
JLu napaMe1pOB rnpOCKona: Yo = 1,5°= 2,62 . 10 . pa.n,t;j3 == 10 . (J).fP == 3 KIU = L88 . 10 pawc
pacC'iKTaeM Macnrra6Hble K03cpqJHIlHeHTbi H cpa30Bb1H c.J1Bnr npH pa31UI'IHbIX lHa'ieHII5!X paCCTjJOHKl1 /'1.
Pe3YJIhTaThI paC'IeTOB no cpopMYnaM (4.426). (4.427) npe.llCTaBJIeHbI B Ta6J1l1ue, B KOTOpOH npuBe.lleHbI
TaIOKe OTHOCHTe.;"IbHa5! paccrpoHKa /) = bJ(J)yp II OTHOlIJeHHe K / K :
b., Il1 (pa.alc) 0 1(6,28) 2( 12,56) 3(18,84) 4(25,120) 5(31,4)
o = f./ro.,p, % 0 0,033 0,066 0,095 0,132 0,165
K , 1 .10""pCUIl(p3,!l/c) 139,0 11,0 3,0 1,4 0,8 0,5
K=, 1 . 1 pa.n/(P3,!l/c) 0 38,0 20,0 13,7 10,3 8,3
K... ' I . 10-4 pa.nl(p3,!l/c) 139,0 39,56 20,22 13,77 10,33 8,32
K I I K C 0 3,45 6,66 9,78 12,87 16,6
... ...
«Ph rpal{ 0 73,34 81,49 84,3 85,73 86,58
Ha pue. 4.30 npHBe.neHbI rpa4JH ecKHe 3aBHCHMOCTH MacIIJTa6HbiX K03cpcpHl1HemOB OT 3Ha'leHHH
paccrpoiixH.
fu npm.cepa cneJIYeT, 'ITO }"ICe npH paccrpoHKe A = 5 rl1 (0 = 0,165 %) cpa30BblH cnBlir
«PJ = 86,58°, 'ITO npHBOJJ.I1T K npeBaJIHpoBaHHJO KBa.npa'I)'pHoH cocraBn"JOLUeH Ha,n, cIlHcpa3HoH, T.e. BbI-
xO./UIoli CHJ'HaJI rnpocKona npaKnf ecKH onpe.neIDIeTCR ero KBa,n,parypHoH cocTaBIDIIOLUeH.
lhno-AreHHLlH no)1Xo.n MOiKeT 6bfTh I1CnOJIb30BaH A/VI I1CCJIe,l\OBaHI1J1 ,l\HHaMHKH 43 Mr
.npyrnx THnOB.
)J,I1HAMVIKA 43 MJ1KPOrJ1POCKOrIA
32]
(K, .K .KM), 1'10- 4 pa;v'(pa,[{/c)
----r----r---'----,----,----,----,----,----,-----
I I I I I I I I I I
____ ____ ___ ____J____ ____ ____ ____ ____ ____
I V I I I I I I I I
I fi\{1 I I I I I I I I
-- ---- ---- ---- ---- ---- - -- ----
I I I I I I I I I
100 - r--c- --- ---- ---- ---- ---- ---- ---- ----
I KM I I I I I I I I :
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I I I I I I I I I
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\ KK, , , I , , , , I
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: ". I I I I I I . I I
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: I I., I I I I I t I I
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2
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3
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4
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2. Cl1CreMbl Koop)J,MHaT, oITpe)J,eJISlIOIlJ,He nOJIO>KeHHe q3 MaHTHIIKOBoro aKCeJIepo-
MeTpa.
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4. CHcTeMbI Koop)J,l1HaT, onpe)l;emnOIUl1e nOJIO>KeHUe q3 Mf RR-nma.
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6. Oco6eHHocTH paC'leTa KOHCTpYKllMoHHoro LIeMn HpOBaHH}{ Mr.
7. 3neKTpOCTaTMQeCKaJl >KeCTKOCTb nO)J,Beca.
8. qaCToTHaJi HacTpoHKa Mr.
9. Oc06eHHOCTII )J,I1HaMHKH nHHeHHblx OCllJlJJmITOpoB.
10. BJIHSlHMe Ky611'leCKOM :>KeCTKOCTl1 noJtBeca Ha )J,HHaMMKY OCl.\HJIID1TOpa.
11. Oc06eHHocTH )l;IiHaMHKH MWlTHI1KOBOrO MA, YCTaHOBJICHHOrO Ha BH6pHpyIOIl\CM
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13. HOpMHpoBaHHbIc 1J:aCTOTHbIe xapaKTepl1CTl1KH OCl.\HJImITOpOB, K03 Hl.\HCHT )l;I1Ha-
MH'lHOCTI1, nonoca nponycKaHHH. 'laCTOT, ,ll06pOTHOCTb.
14. OC06CHHOCTH )l;I1HaMI1Kll MM, pa6oTaIOIUcfo Ha BH6plipyIOIl\eM OCHOBaHllll.
15. MaCIIITa6HbIe K03<tJ l1IweHTb] CHH 83HOH H KBMparypHoH COCTaBlliIIOIUHX BTOPHq-
HbIX Konc6aHl1H Mr.
16. 3aBHCliMOCTb MacIlITa6Hofo K03 I}>Hl.\l1eHTa OT KOHCTpyKTHBHbIX napaMCTpoB Mr.
r.TJaBa 5
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MHKPOMEXAHWIECKHX llPHEOPOB
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lllTa6HbJM K03 I1UHCHTOM, linI1 K034)4JHUH
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H3 Bapl1aHTOB 113MepMTeJIbHOH uenl1 nOKa3aH
Ha puc. 5.2.
KOH.l\eHcaTOpbI C{ M C 2 , BKJIIO<IeHHble
nOeJIe)J,OBaTeJIbHO, COCTaBJJj{IOT nBa nJIe'la
MOCTOBOH cxeMbl, a pOJIb )J,BYX npYTux nne'l
BblDOJIHIDOT }J,BynOllilpHbIC I1CTO'lllI1KI1 mrra-
HHH. + u cn' OnopHoe Hanp.Sl))(CHI1C K eMKOCT-
HOMY MOCT}' nocT}'naeT 'lepe3 KJIIO'lCBym
u-g _I m l F .1 W y 1 y= /1h -I W n I /1U -I W4> 1 U BbIX -
PHC. 5.1. CxeMII H3MepmeJ1bHOii LleOH ocemoro MA
R5
Knl Cl
+U an R2 C3
KJI3 R4
.r KJI4 OY2
C4
-Uan T + j,U ObIX
Kn2
PHC. 5.2. If3MepmenbH3H LleBb ocemoro MA
324 rmma5.1BMEPHTEJIbHblE CBOI1:CTBA MHKPOMEXAHI14ECKHX IlPI160POB
<J
a)
<J
6)
PHC. 5.3. IbMepHTe.lJhHl.lii IIIOCT:
a - cxeMa BKJ11Q'ieHHll; 6 - 3KBHBanCHTH<UI cxeMa
cxeM)' K.,"1-K'14, ynpaBmleMYIO CneUI1aJ1b-
HblM 1'ak"TOBblM feHcpaTopoM. BbIXO,lIHOe
Hanpj{)f{emfe AU C M3MepHTeJIbHOrO MOCTa
DO.llaeTCR Ha HHBepTHPYIOIIJ,JfH ITOBTopHTeJIb
C 60JIbwHM BXO)J,HbIM cOnpOTHBJ1eHHeM Ha
ODepaJJ.HOHHOM YCH..'1MTCJIC OY 1.
llOJIY'lHM BbIpIDKeHHe IDHI OITpeneJle-
HID!. flU. Ha pHC. 53, a nOKa3aHa CXCMa
H3MepHTe.JIbHoro MOCTa, a Ha pl1C. 5.3, 6 -
3KBHBaJIeHTHWI CXCMa, Ha KOTOpOH 0603Ha-
qeHhI: R - BHyrpeHHee COITpOTHBJ1eHlle
HCTO'IHI1Ka onopHoro HanpIDKCHIDI;
XCI =lI(C I ro), X C2 = I/(C 2 ro) - COITpO-
TIfBJleHIDI eMI\OCTeH (ro - IJacrOT8 TaKTOBOrO
reHepaTopa); /" h - TOKH B BeTBRX MOCTO-
BOH CXCMbl.
EcJIH Bb16paTb pe:UfCTopbI RI = R 2 = 1
MOM (CM. pHC. 5.2), TO MO:>KHO C'iHTaTb, 'ITO
H3MepHTeJIbHb1H MOCT pa60TaeT 6e3 HarpY3-
KIf. B 3TOM cJl)"iae nCpe3ap'i,l\ KOH,lI,eHCaTO-
POB CI H C2 6YJJ.eT OC}meCTBJISlTbCR 'iepC3
MaJILle BHyrpeHHHe COnpOTl1BJJeHH'i HCTOIJ-
HJfJCOB mmUflUI R (CM. pMC. 5.3). qacroTa
H8IIl"DKeHIHI nHTaHID!. In 113MepHTeJIbHoro
MOCTa BJ>l6Hpae-rCg T3KHM 06Pa30M, IJT06bT
BLUlOJlHJJJ1OCb YCJJOBHe
J
1m «- .
2fn
r.ne 'm - speMJl DepC3apsma KOHJJ:eHcaTOpOB.
Tor)J,a BbIXO)J,HOC HanpSl>KCHHC 6Y)J,CT
HMeTb OPMY. 6mi3KYIO K MeaH)J,PY.
B COOTBeTCTBHI1 C pHC. 5.3, 6 BeJ1H'lliHa
t-.U 06pa3yeTcSI KaK pa3HOCTb nOTeHUHaJ10B
Me)K)J,Y TO'iKaMH A H B:
!1U = IIR - 1 2 XCI =
_ 2U 011 R 2U on X _
- - Cl-
2R XCI +X C2
=U X C2 -XCI =U C 2 -C I
on X on'
CI + X C2 C I + C 2
C Y'leTOM Bblpa>KeHHH (5.1), nonY'lHM
!1U=(UonM)lho- (5.2)
B COOTBeTCTBl1ll C OPMYJ10ij (5.2) Ha-
npR:>KeHl1e B 113Mepl1TeJJbHOH )J,HarOHaJ1H
MOCTa He 3aBHCHT OT 'laCTOTbI reHepaTopa, a
nepe,l\aTOqHaH I}>YHKLlH'i npC06pa30B8Teml
ITepeMellJ,eHHH HMceT BH,l\
w. = t-.U = U on ( 5.3)
n!1h ho'
KaK OTMe'iaJ10Cb, HanpR>KeHHC t-.U
DocrynaeT Ha HHsepnlPYlOllllfH nOBTOpl1-
TeJJb, K BbIXO.ny KOToporo no,nCOC,lI,lmeH
KJIIO'i Kn5 CHHXpOHHOro ,lI,eTCKTopa. YnpaB-
neHHe KJIIO'iOM Kn5 oCYllJ,eCTB.lUJCTCR np'i-
MHKPOAKCEJIEPOMETPbI np.HMOrO rIPEOEPA30BAHIUI
325
Mbt:\! CnrHajIo"t C 4aCTOToH HaITpSl>KeHlUt
mnaHI1S1 MOCTa. Ilpe06pa30BaHHe ncpeMeH-
Horo HallpSl)l(eHl1H nOCJIe CHHxpOHlloro ,lIe-
TeKTopa B CUrHajJ U,".OK nOCTOSlHHoro HanpSl-
)!(eHlljI pca:UUYC'I CSI C JIOMOlUblO aKTHBHoro
l)mnbTpa HH:>KHJ1X 'laCTOT BToporo nOpJl,lIKa.
IlOlJ,06I1bIe <pHJlbTphI MorYT 6blTb pca-
I1U30BaHhI pa3HbI:>'UI cxer-.raMH. O)J,Ha 113 HHX,
nOKa:mHHaSI Ha pHC. 5.2. nOCTpoeHa Ha OnC-
paunoHHoM YCHJIHTeJIe OY2 11 IToJIY'lI1JIa
Ha3BaHHC "CTpYKl)'pa Payxa". C ec nOMO-
IUblO :\(0)1(110 peaJI1130BaTb <pHJIbTp C laJIbIM
3Ha4CHHeM )J,06pOTHOCTH. YBeJIH'ieHHe )J,06-
pOTHOCTH HJIbTpa nOBbIlliaeT era H3611pa-
renbHbIC cBoi1cTBa. IlpH 3TOM CY>Kae-rCH
,lIHana30H <IaCTOT, B KOTOpOM oCYllJ,ecTBnHeT-
CH ITcpexolJ, OT nOJIOCbl nponYCKaHI1S1 qaCTOT
K n0J10Ce 3alJ,ep>KllBaHHSI.
O)J,HaKo C YBeJII1'leHl1CM )J,06pOTHOCTl1
ITOBbllllaCTCH K0J1e6aTeJ1bHOCTb ITepexo)J,Horo
npOl(ecca B HJIbTpe ITpl1 CKa'lK006pa3HOM
B03,l1,eHcTBHH. TIO,ll,06HaH. CI11)'auHSI MO>KeT
B03HHKHYTb, HanpHMep, npM 113MCpCHBl1
YCKopeHlUI KaTaITYJ1bTHpyeMoro 06'beKTa,
npl1 pe3KoM (YlJ,ap) TOpMO>KeHHI1 aBToM06H-
JISI H T.ll. B 3THX cJlY'laH.X HY>KHO Hcnmlb30-
BaTh HJIbTp C MaJIOM )J,06POTHOCThIO.
TIepe)J,aTO'lHaJl YHKUI1H <pHIIbTpa BTO-
paro ITOpHllKa HMeeT H3BeCTHbIM BH)J,:
W 4J (S) = K j(Tgs2 + 2ToJ>1;rj>s + I), (5.4)
mc
Krj> =-Rs/R3; Trj> = .J R4R5C3C4 ;
c{ R4 +Rs + R :s J
1;rj> (5.5)
2 .J R 4 R s C 3 C 4
3am11l1eM aMnmITy)J,HO-'laCTOTHYID xa-
paKTCpl1CTHKY HJIbTpa, nonaraSl S = jro, r,nc
ro - KpyroBaSI 'laCTOTa:
A(ro)=KfjJ/ (1-Tgro 2 ) +4T 1; ro2.
(5.6)
B COOTBCTCTBHM C <popMynoH (5.6) aM-
nJ1l1Ty,na nynbcaQl1H BbIXO,l\HOfO CHrHana
onpe,l\enHeTCH BbIpa>KeHI1CM
/ If 2 2 ) 2:!"2
!1U n = K4JU on 'V ,1- T4J ror + 4Trj> 1;4J(J)r .
(5.7)
r)J,e ro = ro l - KpyroBaJl 'laCTOTa TaKTOBoro
reHepaTOpa.
Hcxo.llSl 113 pl1c. 5.1 11 C Y'leTOM BbIpa-
>KCHl1M (4.31), (5.3), (5.4), ncpCnaTO'iHaH.
<PYHKUHSI MA HMeCT BHn
v
W(s) = BLIX =
a v -g
_ mUonK)'KoJ>
- 22 2 .
ho(TI'S + 21;J)'s + 1)(Ti s + 21;4JToJ>s + 1)
(5.8)
H3 ncpc)J,aToqHOM YHKUHH (5.8) npH
s = 0 cne,l\yeT MaCllITa6HbIM K034> HuHeHT
OCCBoro MA, B/(Mlc 2 ):
U BLfX _ mUonKyKoJ>
aJ' - g 110
,
(5.9)
OTKY)J,a MO>KHO onpe)J,CJIHTb Tpe6ycM)'1O BC-
JIH'll1HY K03 m.\l1eHTa nepella'll1 HJIbTpa:
K - UBLIXho
4J mgUonKy
(5.10)
flpu.Mep 5.1
PaCC<IHTaeM napaMeTpbI 4>HJILTpa, o6ecne-
4HBalOll\He KpyrH3HY BbIXOAHOH xapmcrepHCTHKH
U BLIX / U = 1 B / g, H 3HaqeHHe nyJIhCaUI1H BbI-
XO,D,HOrO CHrHana tJ.U n < 10- 5 B ,II)IJI napaMeT-
pOB MA:
m=O,2.10- 3 Kr,'10 =20.10-6M ,Ky =6,34-10-4 M/H.
Ty =3,55.10-4 C, }' =15,28. IloJIaraJI u on =5 B,
no l}>opMYJIe (5.10) nOJIyqaeM:
K = 1.20.10- 6 =322
fjJ 98.02.10- 3 .5.634.10- 4 ' .
" ,
IIpHMCM Krp = 3,5 , TOr,na B COOTBeTCT-
BHB C opMYJIaMI1 (5.5) Rs = 3,5R 3 .
YcnOBHMcH, 'ITO R3 =14, Ii nOJIy'iI1M
3aBliCHMOCTb l;rj> = 2,14 .J C3/C4' TIpl1ID1B
326 fllaBa 5. H3MEPHTEJlbHbIE CBOi1CTBA MI1KPOMEXAHII4ECKID" npl1£OPOB
n - J
pll.\fep J._
Pacc'lIlTaeM lJaCTOTilLIC xapaKTcpHcl Him
MA C napaMeTpaMIl IB npC!lLIlI.ywero npIJMCpa:
m =0.2.\0-3 Kr ,U on = 5 U. K I =6,34-\0-4 M/H.
Krj> = 3,5, "0 = 20.10- 6 M. T I = 3,55 -I 0-4 e.
y=15.28. Trj>=5,61']0 3c. 1;4>=0.707.
rj>=0,707, npliMeM C 3 =10-6<D II onpe)J,e-
ID1M C 4 "" 9.10-6 <D. EY.QeM CqI1TaTb
. l
R3 = = 10-' OM, Tor.u.a R5 = 3.5 .10' OM II
T rj> = 5,61.10- 3 e (O)rj> = 0.17-10 3 lIe"" 28,3Rflt).
TI0J10:>KHM, 'ITO 0).. = 6.28 .1051/c, II nOJI)":IIIM
BOenOJIb3yelllcJI nepe!lRTO'LHOIi 4JYHKIlllcii
I1U n =0,157.10- 5 B < 10- 5 B - (5.8) H 3amf.weM
W(s) 0,2,]0- 3 -5,6,34-\0-4 -3,5
20.10- 6 [(3.55.10- 4 r s2 + 2 -15,28- 3.55 .\0-4 s+ 1][(5,61. IO- 3 r sl + 2-0.707 - 5,6]-10- 3 s+ I J
'J 3 Ig (I)
- IIOO;;:r:\ - -.-1: IO (J) I/c
---- ------ --:...: -'I'\(J)J - --I---r----r- -+-r- '
- "-
In.rco ) b6"':. ++-
_ - . 'I1p'. -
aMes B ma)', 'lT0 Bb1paJKeHHe B nepBoi1:
KBa.:I:parnOll CKo6KC 3HaMeHaTe;"U! MMeeT )lBa )lci1:-
CTBHTe.ThHbIX Iwpm. nepenllllleM nepClIaTO'iH)'IQ
4J)'HKUHIO CJle.JylOIll.HM 06pa30M:
W(s)
K
., ., ,
(7js+I)(T2S+I)(T s- +2Trj> ljJs+l)
r;J.e K = 0,11(20 IgK = -19,172 lIb); T. = 0,0 I 08 C,
]
0)1 =r,-=92,59 lIe (lgO)I = 1.966); T 2 = 1,162 x
J
x lO-5 C,
I
0)., = -:::: 86058,5
- T.,
l/c (lg0)2 =
o
L(CO), cp(oo).
;iD
I
_ =.:::::IO .-
180
- A(oo)
-2
-_..
-40
= 4,93); Trj> = 5,61.] 0- 3 c,
I
corj> =- = 178,25
T4>
lie,
(Ig (iJ1jJ = 2,25) .
B !laHHOM npllMepe IB-3a 6onbworo lICMn-
cpHpoBaHIUI 43 cooTBeTcTBYlOwee eMY KOlle6a-
TenbHoe 3BeHO npe06pa3yeTC5! B !lsa anepHolIM-
lleCKliX. B COOTBeTCTBIni C BHllOM Mo.n.H )Hm!po-
BaHHOi1: nepC.!1aTOlLHoi1: CPYHKl.llm II Bbl<IHC.!1eHlUI-
!\Ill H3 pllC. 5.4 npHBe.aeHbl 'iaCTOTHbIe xapaKTc-
pHCTHKIt B npelIe1l3X H3MeHCHlUI lJaCTOTbI 110
1000 IIc, TaK KaK .!I..rul npHHHTbIX napaMeTpoB MA
nO.!1oca nponycKaHlUI orpaHHlleHa 3Ha'ieHlleM
'iaCTOThl 0) '" 100 IIc "" 16 f u, II npH 3TOM 3ana3-
lIblBaHllC no tPa3C lIOCmraeT 'P = 90°.
T ,
-+-,, - -:.;:.....:- ...;..... --++-:.
"",""' t=::;- .-- . -" --- - :-";""f':;
-.--.. -,- '-'\. -1--+-,-...,.-
.....-.=. --,-.- -r-+-o :.-...... 'IT' ' -'::'''' :ti:rt
---r-V _ -0_ +
'" ---> +---+ -.::, - -=:= --q.:; :-+tl:i-;--I (jJ1{OO) 1>",.- ,-
_ .._.. -r- ..L. --.
=---h. _-, , , ---,-,--" t'" .n r=;:
..J ..J':" _ __1 _p.. -=1-: - . H-
-60-
- . -.-
-80 -
-----4 ...... -
- j.';'IV, .
. ..J jj m, -- _ l- - I-' '
A _ _ ...... :_._ _......._. . _ '\._.L
--.-'-._ -,_+.L1_ ,__-.-' v.,..; './. _j, 1./--;1 -;., .- , j- rr
4 a.-.-.-1-...----y.- --......... - H.....-... -r-..,-.,...-', I -;-1-r-1'I
-T---=t -, -1- =:- t .. ,-I-, - i-t-
-, ---'--L . h--'-'; " . 1 _, -+-"- -, ' II ,..,
.....-.,..-.. "_-i-.+ _,.--..-++-H---4---i. .. --;.,...... --...--..
..;....
o
Poco 5.4. CTOTHLle xapaKTepHCTHICH MA:
A(ro) -aMnnmy,/lHaJI xapaxrepHCTHl<a (nYHKTHpHOH nHHHei1: nOl<a3aHa aCUMnTOTll'leCKaJI
norapHIjJMH'lecKaJI aMIIJlHl)'.IlHo-'iaCTOTHaJI xapaKTepHC11fl<a); 'PI (0) - cpa30BaJl xapaKTepHCTHKa,
COOTBeTCTBYJOI1l3JI nocroSlHHoH BpeMeHH 1j; 'P1jJ(0) -ljJa30BCUI XapaKTepHcTHKa tPHJlbTpa;
q>( ro) - cYMMapHaJI 4Ja30BCUI XapaKTepHCTHKa
MI1KP01\KCEflEPOMETPbl rIP5IMOrO DPEOEPA30BAHI1JI
327
5.1.2. MaHTHHKOBb.ii MIIKpoaKcMepol\le. p
CXCM8 H3MCpHTCJIhHOM ueml Ma»TIIH-
KOllom MA, oro6pa)!(alo[] a» nOCJIe OBa-
TcnbHOC fb IIpe06pa1013aHHSI BXOll.HO i Be:m-
<JllIIbI (mlHdiHoc H yrnoBoe YCKopemle oc-
H0I38HHSI) fI BbIXO,LtHyIO BCJIJ1tJI1HY (Hanp}l)!(e-
Hue BhIXo! a 3:le!\.lpl1<reCKOM cxeMbI). ITOKa-
'3aIla H3 pHC. 5.5. llepBbIM 6JIOK C lIepella-
TOtlHoii <pYIIKul1ei-i W M oTo6pIDKaeT npe06-
pa'JOl3aIme YCKopeHHH B KOJIc6am!}I MaHTHli-
Ka. BTOpOH 6JIOK C nepellaT04HoM <PYIiKuHei1
I + a oT06pa)!(8eT npe06pa30BaHI1e Yf'J10BbIX
Kone6aHl1H 8 MaSlTIIl1Ka B JHlHeMHOC nepe-
1ellleHl1e /'..11 nnaCTHH KOH;'{CHcaTopOB npe-
06pa30BaTeJJSI IICpCMClllCHHM. nepellaTOqllaJl
<PYHKUHH KOTopom W n , a HanpSl)!(eHlle flU ,
,..
oopa30BaHlloc Ha cro I3bIXOllC, nocTynaeT B
CXCMY <jHiJIbTpa C lIepe)J,aTO'lHOH YHKuHeM
WtjJ, KOTOpbdi C03)J,aeT BbIXO)J,HOC Hanpll)!(e-
HIIC U aMX .
B npe)J,nOJIO)!(eHIH1, 'lT03J1eKTpH'leCKaJl
uenh MA aHaJlOnl'lHa cxe!>1e Ha pl1C. 5.2,
nepe11aTotJHaSi <PYHKUl1}1 W M onpe)J,eJI}leTC}I
Bhlpa)!(eHIISlMI1 (4.71) IIJIH (4.73), a nepe)J,a-
TO'lHhIC <PYHKUUH W n H W<\> - COOTBCTCT-
I3CHHO I3hlpIDKCHM}lMH (5.3) H (5.4), 3anl1WeM
nepellaTO'iHYJ<) <PYHKUHJ<) MA:
Kn(/ +a)UonKr:p
W(s) = ') 2 2 ') ,
(T-s + 2Tf;5 + 1)(TtjJs- + 2Tr:p1;r:ps + 1)
(5.11)
r.n.e K03<p<pI1Ul1eJ-IT nepe.na'lH K" npHHI1MaCT
3HatJCHHH. Knr, Knu [(eM. OPMYJIbI (4.70)]
x
I1nl1 Kna [CM. Bblpa)!(eHHH (4.74)] B 3aBH-
)
CI1MOCTl1 OT )J,eHCTI3YIOlL\erO B03Mym eHHlI .
TIOJIaraJl 13 <PYHKIUIH (5.11) s=O, nO-
;JYtJaeM :\JaClll ra6HbIe K03 HlUieIiTbI MA
no OTHOWCHI1ID K JIHIIej:'IHbIM. B/(M/c 2 ). Ii
')
ym0I30MY, B/( I/c-), YCKopeHHIO:
U m,IX KK
= nu;
x
U BblX = KK
n 1Il1\
}i .
. "
X B
[) IIbl» - KK
... nr,
Yo
(5.12)
f)J,e K = [(I + a)U onKr:pJI 110'
')
MaKcHMaJlbHOC yCKopeHl1e, M/C-, 113Me-
pSieMoe MA, onpe)J,e.JSleTcSI MaKCHMaJIbHblM
r> '" hma.,
YI'JIOM n0130pOTa MaHTHI1Ka \Jmax--
2a
( h max - )J,onycTI1MblH p83Max KOHua nJ1a-
CTI1HhI MaSlTHMKa) H K03 nuHeHTaMH K n :
XII" = 8max / K'I<' x ; YaM = 8max / Kna, ;
Y M = 8max I Knl .
(5.13)
KpYTI13Ha CTaTH'ieCKOH xapaKTepl1CTI1-
KH l1:ISI Bcex l3apnaHTOB B03MymeHlUl c yqe-
TOM I}JOPMYJI (5.12), (5.13) Bbl'lI1CJISlCTCSI 113
U (I + a)U onK<\>
paBeHCTBa B"'); - , OTK)')J,a
8 max 110
K - UBblXI10
<\>-
Uon(l + a)8max
(5.14)
llpll.\1ep 5.3
PaCC<JHTaeM napaMeTpb! tPI1JlbTj)a, o6ecne-
<JHBalOWerO U BblX =] B npH MaKCUMaJIbHOM
3Ha<JeHHH H3MepJleMOH JIMHeiiHOH BH6pa[!Hl! 1I
nYJ1bCaLtI1H BbIXO./I.HOro CHI'HaJIa flU (1 <] 0-6 B ,
,IVI» aKCenepoMeTpa co c]]eJ{YJO HMH MCXOJ{HbIMH
J{aHHbIMI1:
Xn, Yo, Yo_I WM I 9 -1/+a 18h 1 Wn 18U. Wr:p I UDb
PHe 5.5. CXCM3 H31\1CpHTCJlbHOii LtCnH I\IRHTHHKOBOro MA
328 fJlaBa 5.113MEPHTEJIbHbIE CB0I1cTBA MI1KPOMEXAHIJl1ECKIIX npI1EOPOR
-t' I ___-I ... ,_ J. , __ L_ .,. : _... .1 V. I -L1 k_ t
I I' -J-' . _
A(co) _ . -.t-- .H _.;:;- .::j +-+= =;::t i ,- ((O . -1-,--; i-:=j
8.4 . ::j:::. - --' L' :tit { :-' ":::' iV " J: 1- jj 'i- t -; L;: J '
-180 ' - . ' -,-+-, . -" 0 . -.,-. j . T L T'-- 1
- ... <I . .. -- --! f- --,' " . ,-,-" ... '-'--1 r
o . .lfli/" ' . '. -r-..---->-
- (''';. ---. t---'--+ "'...(ro) et::'
4 ._,.-.-- "t'1JI"-:- -
- . _ -. --- _L .. CJ, -, .-1.' I h --'"....- I. .
, _ - . .; ---: !-"M-_-=- .i-"!=-
, -I- -_. - - i - .-;.- """" - --'''' -++ , .,.-. j. I
. ' _.
-+-r- L _.-j j-_..... I i _ r-../! ' I ""'-\. /--1 i . -t--- I.. I
----f- !-:- .: -+
--,-t----r- - '---+- -- 1 - : -:' .,.
I I +-:--f _
. -. ..-. . . I=--=-;..!' _ -= . .,-,.:-. : ;1.1 I \..t:t -I '-. - .J ....;....:_
-40 -90- -. ,- (Mro) " ,
-., -----'"-
. C,,",---:- :,-rj-:t: -.--'-- 21:: :.- - : '-, 'i-:p:. <fh:;. T t-:
V7""----'-- - ,-+--- ._--' - :t\v ..r-. . - .
60 4 5-:- _.-:;: - - 7' e+--.---.- . = =i" t:. !- :' t:--i
- I I --: fU. . "=-t-T
_ _' _ --'_ _ ' ,, , -1 '
I " --;---r' v. ,
----!-- .1-...---, _"\,- -1_
_--'- ____ - 21=2,18::.q>1(ro) . ' 3 '_"- Igro
L(co),
AE
2U
!P(co), C
.135
-20 ,
-80
'
:u,85
1
10
100
1000
co,l!c
PHC. 5.6. qacToTHLle xapah."TepHCTHKlI MA
KQ(J = 3,956.10- 3 c 2 /M , T == 4,545 .10- 3 C,
%
1;==16,052, U on ==5B. 0:;5.1O- 3 M, 1+0:;0,
hmax =15.10-6 M, Ita =20.10-6 M, n = 3.
liMeeM Smax = hma.» 20 =(15.10-6)1
(10.10- 3 ) = 1,5. ]0- 3 pall..
B COOTBeTCTBHH C BLIpIDKeHHeM (5.13):
.is.. ==(1,5.10- 3 )1 (3,956.10- 3 ) = 0,38 wc 2 .
no q.opMyne (5.14) BLI'1HCIDIeM K03rl>tPHlU1-
em nepe;J.a'lH tPH.!lbTpa:
K = 1.20.10-6 =0,53.
4> 5.5.10-3.1,5 .10- 3
O6pamaJlCL K pHC. 5.2, nOlJaraeM, 'iTO
R3 = R4 . HM B BHJ1Y q.oP MYJU.I (5 .5), HaH.neM:
I4 ==0,53R 3 H 4J =],4I J C 3 /C 4 npHMeM
l;cjJ == 0,707 , C 3 = 10-9 4> H nony'iHM
C 4 =3.98.10-94>.
Onpe.QeJJJB( R3 ==R4 ==TIC 3 =4,5.10 6 OM
H I4 == 2,38.10 6 0..- .
B COOTBeTCTBnlf C tPopMYJIaMli (5.5) YCTI\-
HOBlIM, 'ITO:
Tq, == 4,5.l06 .2,38.10 6 .10- 9 .3,98.1O 9 =
=6,53.10- 3 c.
npHlI1eM <Jacnrry TaKTOBoro reHepaTopa
(Or =6,28,10 5 l/c H no tP0pMYJle (5.7) HaH.neM
/:!;.U n = 1,57 .} 0- 7 B < 10-6 8, <JTO OTBe<JaeT npH-
WITOMY orpaHH'iel-ll11O.
C )"ieTOM <JHCJ10BLIX :ma'1eHHH napaMeTpoB,
3anHllleM nepe.naTO'IHYIO CPYHKl1HIO (5. I I),
?
B/(wc"):
W(s) = 2,62
(T 2 s2 +219"+ 1)(T s2 + 2TcjJ rl>S+ 1) ,
r.ne T=4,545.10- 3 c, =]6,052, TcjJ=6,53.1O- 3 c,
Scp = 0,707.
IlepBLIH COMHO)!(HTeJ1L B 3HaMeHaTeJ1e ne-
pe.naTo<JHoH tPYHKWiH HMeeT ,l1Ba .neiiCTBHTeJ1L-
HhiX KOpln1 sl =-7127,18, s2 = -7, KOTO-
pLIe OnpeAeJ1 1OT ABC nepBLle lJaCTOTbI
0» == 7 ] 27,] 8 I/c, co2 = 7 ] Ic H, CJ1ellOBaTeJ1b-
HO,
T Z S2 +219"+ 1 ==(1jp +1)(T 2 P+I) ,
MHKPOAKCEITEPOMETPbJ KOMrIEHCAI.{J10HHOrO rIPE06PA30BAHlliI 329
I J
me 7j =-=O,OO0l4e, T 2 =-=0,I4285e.
wI 002
TaKHM 06pa30M, nepe.n;aTO'iHIDI I}>YHKUIUI
npHHHMaer BH
W(s) = 2,62
(O,OO0l4s+ I)(O,I4285s+ 1)(0.0065 2 ..1'2 + 2.0,707.0,0065..1' + I)
B COOTBeTCTBlIH e nepe aTO<iHoj,j I}>YHK1.1H-
eN, n0C1poH!'>1 'iaCTOTHble xapaKTepHCTHKIf.
Bbf'illCIIHM 20 Ig 2,62 = 8,4 E; Ig wI = 3,85 ;
Igw2 =0,85; m] =I1T<j> =: 153.8 IIc, Igw] =2,18.
ITorapul}>MH<lccKHe aMnnH1)' HIDI H ljIa10BIDI <iac-
TOTlible XapaK'l'epHCTHKH MA npMBe eHbl Ha
pue. 5.6.
113 nOIIY<leHHblx xapaKTepHCTHK clle,!!YeT,
'ITO o 'iaCTOTbl H3lIOMa 002 = 7 I/c ("" I,ll f11)
3ana3,nblBaHHe 110 rjJa3e KOJIe6aHlIH MaJlTHHKa no
mHOllJeHHIO K KOJIe6aHHHM OCHOBaHHH He npe-
BuwaeT -47°.
113MepHTenbHaSI [tenb MA ,l!.illI BXO)J,HO-
ro B03,lleHCTBmI B Bl1L\e ymoB Y HaKJ10Ha
OCHOBaHIUI COOTBeTCTBYCT puc. 5.5 C yqeTOM
TOro, liTO nepeL\aTOqHRj{ YHK[tUSI
W.'I == WS,y H BbI'lHCJISIeTCR no Bblpa>KeHHIO
(4.76). TIepeL\aTO'lHRSI YHIo..\l1j{ aKcenepo-
MeTpa L\JHI 3Toro CJIY'lRj{ onpc)J,emleTCR op-
MYJIOH
W(s) ==
?
_ K y (l+a)Uon K cj>(7i s -- l )
- ?") 2 ") ,.
(T- s- + 21;Ts + 1)(Tcj>s- + 21;cj>Tcj>s + 1)h o
(5.15)
rne Benl1<JHHbJ Ky, T, Ii, I; HaXOL\H.TCSI B co-
OTBeTCTBHH C BblpIDKeHHRMU (4.77). TIOJIO-
f
2U ou
Cl
C2
>KliB B opMYJIe (5.15) s = 0, nOJIY1fHM Mac-
lIITa6HbiM K03 H[tI1CHT MA, B/pa)J,:
UB IX Iy ==[Ky(l + a)UonKcj>]I ho.
O'leBHL\HO, qTO MaKCHMaJIbHbIM yrOJI
y max HaKJ10Ha OCHOBaHllH orrpe)l,CJUleTcjI
pasCHCTBOM Ymnx == 9max I Ky (9mux ::::
:::: h max / 2a), a KpYTl13Ha CTaTU'leCKOM xa-
paKTCpHCTl1KU MA BblpIDKCHHCM
UB IX /Ymax ==[Ky(/ +a)UonK<j>]/ ho.
(5.16)
KpYTH3Ha HJ1bTpa nO-npe:>KHeMY on-
pe)l,eJIHCTCR OPMYJ10H (5.14).
5,2. MHKPOAKCEJIEPOMETPbI
KOMTIEHCA HOHHOrO
nPEORPA30BAHIUI
5.2.1. MIIKp03KCeJIepOMeTp C
3J1eKTpOCTaTlIlJecKoii 06paTHoii CBH3blO
CxeMa, llJIJIIOCTPHPYIOI1.1;Rj{ C03.D.aHHC
pa3HOCTHOH 3JICKTpOCTaTU'leCKOH CHJIbI Ha
ITO.nBU:>KHOM nJIaCTUHe (MRSlTHl1K) MA, SlB-
JIjIlOlllCHCSI nO)l,Bll>KHbIM 3J1CKTpO)l,OM )J,11 -
epeHLll1aJIbHOrO KOH)I,eHCaTOpa, npu ee OT-
KJ10HeHHH OT HeHTpaJJbHoro nOJIO>KCHUH
nOKa3aHa Ha pHC. 5.7.
R
UOId)(
R
PHC. 5.7. CxeMa cJlOpMHpOBaHHH :)JJelCTpOCT8TulfecKoii o6paTHoii CBB3H
330 f.naBa 5. 113MEPHTEJIbHbiE CBOI1CTBA MHKPOMEXAHH4ECKIIX rIPI1EOPOB
aj
-- -
on t
on
- '-1---1....- -- ---
u
u
0)
--- -- ,...
e u.."
I
c
--
Uon+U o
Von-U o
Puc. 5.8. Bu.Q MCaUiJ.POB:
a - cpopMl1pyeMbIx renepaTopoM; 6 - CMellleHHbIX OTHOCI1TeJIbHO o6111cro npoBolla
IhMepnre:lbHblH MOCT 06pa30BaH
..1H<t>l}>epeHUua:lbHbIMU KOH.leHCaTOpa m CI
H C2, Y KOTOpbIX 06lLll1H npoBo)], ("3eMJIH")
COe,.!lHHeH C 3JIC""Tp0,l\OM, pacnOnO>KeHRbIM
Ha ITO..JBHjJQiOij n.:laCTHHe MA. H pe3HcTopa-
Ml1 R. leHepaTop r BbJpa6aTblBaeT onopHoe
HanpX)KeHHe C a!>H1..'IH1J'.!I.OH MeaH,.1pOB U on '
KOTopoe nMTaeT H3MepMTCJIbHbIH MOCT (PUC.
5.8.0).
HanpH:iKeHl1C f1U , CHHMaeMoe C H3Me-
pHTe..,ThHOrO MOCTa, qepe3 YCHJIMTeJIb Y.
CHHXpOHHbIH .'1eTe"-rop CJJ: 11 HJIbTp tlI Hl13-
KHX qacroT, HMetOlLlHX COOTBeTCTBeHHO KO-
3 JlUl1eRTbl nepe.uaqH K I _ K 2 H K 3 , noCTY-
naeT Ha BhIXO.n. (U mA) H no !1HHHl1 06paT-
HOH CBX3H - Ha cyMMaTOp (U oe) , Ha KOTOpbIH
no.;xaerCH TaIOKe onopHoe Harrp}[)KeHMC rCHC-
paTopa. OqeBHllHo. V BLIX = -V oe = KI1U
(K = K 1 K 2 K 3 ). HanpIDKeHI1e 06paTHoH
CB1I3H CMelIlaeT onopHoe HaITpSDKeHHe OTHO-
CHTeJILHO 06mero npoBo,!J,a Ha Bem1'lI1HY
V OC (PHC. 5.8, 6), HO BeJIH'mHa nHTaHI1H
H3MepHTeJIbHOrO Mocra no aMnJIHT)',lI,e OCTa-
eTCH HeH3MeHHOH (2U on) H, CnC.nOBaTenbHO,
HanpH:JKeHHC I1U onpe)J,ellileTCH no <t>OPMY-
ne (5.2). BCJIe.nCTBI1e CMemeHIDI onopHofo
HaOpSDKeHH1I KOH)1eHCaTOpbI Cl H C2 OKa-
3bIBaJOTC5I no.n pa3HbIM HanpIDKeHI1eM:
V oo +V oc H V OR -Vex;.
J1...{esJ B BHJIY pa3HOCTUYlO 3neKTpoCTa-
TMqecK)'IO CHJIY, .neHCTBYIOIlJ,YIO Ha nO):tBH:>K-
HbIH 3neKTpO,lI, - MaH.THHK, a TaIOKe paBeHCT-
80 VBw. =-W oc , nOJI)"fHM nepe.naTOqHYlO
<t>YHKItIDO KaHaJIa 06paTHOH CBH3H:
F
Jr = p
oe
U Db!>
_ ££osU on
- ?
h()
l5.17)
B TOM CJIY'lae, eCJIU 113MCpnTCJlbHbli:i
MOCT 06pa30BaH nBynOJIJlpHbIM IlCTOllHl1KOM
HarrpH>KCHI1H I1 113MCpHTCJlbHblM11 KOHlleHCa-
TOpaMI1 aHaJIOfMqHO pUC. 5.2, 3J1CKTpOCTaTI1-
'leCKUM naT'lHK Cl1JIbI MO>KeT 6bITb pcamno-
BaH, KaK nOKa3aHO Ha pl'lC. 5.9, nYTeM 1I0)J,a-
'111 CMrHaJIa C Bblxo.na (ImJlbTpa HH3KHX 'laC-
TOT Ha CYMMaTOpbI, BbInOJ1HCHHbIC lIa one-
paUI10HHbIX YCHJIHTeJISiX OYI H OY2. B pe-
3ynbTaTe 113MepHTeJlbHbIC KOH,lleHcaTOpbI
ITI1TalOTCH HCCI1MMCTpJ1IJHbIM HanpJl>KeHHeM,
liTO npHBO,ll,I1T K B03HHKHOBCHI11O 3neKTpO-
CTaTI1'lCCKOM CHJIbI. ,lI,eHCTBYlOIllei1 Ha q3.
OnepaUHOHHblH ycnnHTeJ1b OY3. BKJItOlJeH-
RbIM no CXCMC YCHJIlrreJJSI 3ap}J)J,a, peafHpyeT
Ha npl1pameHHC BXO)J,Horo HaITpSl>KeHHSI 11
06eCne'll1BaeT nepe3apSl)J, H3MCpl1TeJIbHbiX
KOH)J,CHcaTopOB.
CXCMbI 113Mep11TeJ1bHbJX uenCM aKCeJIC-
pOMeTpOB nOKa3aHbJ Ha pnc. 5.10.
f1cpc)J,aTOllHa YHK I1Sl W y BbI<IHCJISI-
CTCH no opMYJIe (4.3 I), a YHKUI1S1 W no
opMync (4.71). TIepe)J,aTO'lHbIe t1JYHKLlHI1
W n H WIj> onpe,ll,enSltOTCSI COOTBCTCTBCHHO
BLlpa:>KeHMJlMI1 (5.3) H (5.4), a YHKLll1Sl Woe -
BbIpIDKCHHCM (5.17). BBc.neHHe 06paTHoH
CB5I311 YBeJIHIJI1BaeT :>KCCTKOCTb 3neKTpoMe-
XaHHlJeCKOH CHCTCMbI "nnaCTI1Ha Ha YITpy-
rOM no.uBece C 3JIeKTpOCTaTI1'lCCKI1M ,lI,aTlJl1-
KOM CHJIbI".
MYIKPOAKCFJII:-POMETPbl KOMfTEHCA 110HHOrO fTPEOEiPA30BAHJ151 331
R2
R1 R8
OYI C3 C5
+U R3 + KJI3
on
R5 u m ,,,,
1
R6 OY2 KJI2 1
+
PIIC. 5.9. I1JMepllTCJlbHaH uenb MA C ,lByno.1HpHbJ}IIICTO<JIJIIKO:\l Il3npHiKcHIIH:
R I... R9 - pe.mcTOPbl; C I.. .C5 - KOHIleHcaTopbI; OY I.. .OY5 - Onepal\HOHHbIe ycuIDITeml;
KIIL. .K.1I5 - KJIlOtJlI
a)
YCI<O
U OhIX
F p
6)
Y CI<OpeIilie
-L
m
Fp
W
U ObJX
Pllc.5.10. CxeMbl1l31\IepHTCJIbllbiX l(cneii MA:
a - oceBoro; 0 - MMTHUKOBoro
TIepe aTO'lHble <!JYHl<l.{HI1
l..{eneH MA no OTHOllleHHID K
UMeIDT BH)J,:
oceBoH MA -
$;::: mWyWnWIjJ ;
1+ WyWnWIjJW oc
3aMKHYTbIX
yCKopem(ID
(5.18)
MaHTHH«OBblH MA -
W(l + a)WnWIjJ
$=
1
1+ W(l+a)WnWcI>Woc
m
(5.19)
3anHlueM BbIpIDKeHl1e (5.19) JJSI CTa-
QHoHapHofo pe>KI1Ma (s = 0):
$s-o = Kn K , (5.20)
1 + Kn K EEOS on
mho
r)J,e Kn HaxO)J,JITCD no opMYJIaM (4.70) I1JIM
(4.77), a K03I}> Hl.\HeHT K onpe.neJIeH B BbI-
paJKeHHSlX (5.12).
CpaBHHBaH. BbIpaJKeHHSI (5.20) M (5.12),
3aKJIID'laeM, 'lTO C BBe.neHlleM 06paTHOM CBX-
3M MaKCMMaJIbHO 113MepHeMoe YCI<OpeHHe
I
YBCfll1'lllBaeTCSI B I + - KnKWoc pB.3.
m
I1pH HCCne OBaHHH MA MO:>KeT OKa-
3aTbCSI IIOJIe3HOH nepe,l\aTO'lHWI YHKl.\IiSi C
BblXO OM He no HanpXmeHI1ID, a IIO nepeMe-
332 fnaBa 5. H3MEPHTEJIbHbIE CBOllCTBA MHKPOMEXAHH4ECKIIX rIPHEOPOB
meHHlO. ITepe,!J.aTO'iHaSl c}JYHI(l.(JU! 38MKH)'Toii
l{ellH no pl1c. 5.10, 6 C BbIXO)J,OM no yrny S
HMeeT BH..I!
8 W
<D - .
1 + 'YWocHTIjJWn(l + a)
m
OTl\.')',!J.a CJIe eT BbIpIDKeHHe .!tJ1S1 CT8TH'le-
CKOTO pe:>KHMa:
<D -o = 1 ? SU -
- 1 K (I ) on K EEo on
+- n +a - IjJ .,
m '10 ho
(5.21 )
EC:lH 3allaH MaKCHMa.1JbHO ,lJ.OrrYCTHMbIH
}TOJI Smax nOBopOTa ll.'I8CTHHbI, TO MaKCH-
Ma.'IbHOe 3Ha'leHHe H3MepReMOro YCKOpeHlli!
onpe.aem!eTCR ITO c}JopM)7JIe
X max = 3ma.x 1c1> =O-
flpu.uep 5.4
PaCC'lHTaeM MaKCHMa.'lbH}'IQ aMI1nHl)'.l!Y
ropH30HTa..'ILHOH BH6panHH, H3MepReMOH aKCeJJe-
pOMeTpOM C 3JleKTpOCTa-Il1<JCCKOii o6paTHoii l:B}I-
3MO.
HCXO.nl1bIC .naHHhle: m = 0.075 .10- 3 Kr ;
l+a::::5.1O- 3 M;h o =20.IO-{iM: 11=3. ¥o=270o;
klI=2,036.10 3 HIM: k I2 =k:!I=-O,819H; k n .=
= 4,398. I 0-4 H'M; E =],00058: EO =8.85.] 0- 1 2 I])I\!,
S = 6.10- 5 M 2 : KIjJ = 1, U ull = 5 B .
B COOTBeTCTBlflf C Q:JopMynaMII (4.70) IIMC-
eM: Kn = Kna "" 3,956.] 0- 3 c 2 /M. rIYCTb h max '=
=] 5.1 O-{i II! (!!.llmax = IImax) If, CJle.nOBaTC!1bHO,
MaKCHJ\!8J]bHO .nonYCTlIlI!blii yron nOBopoT8 Iwa-
CTHHbl 3max = IImax /2a = 1,5. I 0- 3 p8i1. MaKCI1-
J\!8J]bHCUI aMll.T1I1l)'.na YCKopeHlU! nJmellHoii B116-
pawm, KOTOPYIO II!O)!\CT U3MCPUTb MA 6e3 06-
paTHoi1 CBJl3H, pasHa
X smax = 3max I Kno = 1,5 .10- 3 /3,956.10- 3 =
A
1
= 0,38M1c- .
K03Q:JQ:JHlllieHT nepenaQH MA C 06paTIIOH
CBR3blO onpe.neJ1J1CM no Q:JopMync (5.21):
3,956.10- 3
1+ 1 .3956.10-3.5.10-3. 5 .1. 1,00058.8,85.10- 12 .6.10- 5 .5
0,075.10- 3 ' 20.10--6 20 2 .10- 12
_ 3,956.10- 3 - 2,75.10- 3 , 2
1+0,437 C 1M.
8
c1>.s=O
Cne.llOaaTe.iThHO, MaKCHMaJ1bHCUI aMJ1JJH1}',lIa
YCKOpeHIDl, H3MCJ"IeMCUI MA c 06paTHOH CBJl3blO,
paBHa:
X max = Bmax 1<JJ =o
1,5.10- 3
2,75.10- 3
0,55 Mlc 2
H B 1,437 pa3 60JIbwe MaKCHMaJ1bHOH aMI1.J1Jf1)',lIbI
YCkopeHID!. H3MepS!eMOH aJ<CeJJ:epoMeT]>oM 6e3
06pamoH CBfI.3Jf.
IIac-ronw:e xapaKTepMCTHKM 8Kccne-
POMeTpOB MOryr 61>JTb nonyqeHLI Ha OCHOBe
nepe,naTOIJHLJX rp}'HIClUiH (5.18), (5.19) }.IJVI
3aMKHyT:bIX no JCaHaJIy 06paTHoH CBR3H ID-
MepJffeJlLHLlX neneH, a TaJOKC Ha OCHOBe
nepe,naTO'IHLlX .p}'Hl'Wlif. HalfAeHHLlX H3
ypaBHeHMH ,llBH>KeHWI 11M (nJIaCTHHbI) aK-
CeJIepoMeYpa, B I<OTOpble He06xo,lJ.HMO BBe-
CTH BbIp8)1(eHHR .D.lliI CMJI, ,lleHCTBYIOll\l1X Ha
llJIRCTMHY co CTOpOHbI ,lIaT<IHKa CMJIbI. Pac-
CMUlpHM BHa1.Ja.TJe aKCeJIepoMeTp C 3JIeKTpo-
CTaTH<IeCKI1M ,lJ.aT'lHKOM CMJIbI.
BOCnOJIh3YeMCH., HaIIpHMep, pHC. 5.10, 6
H, HMeR B BM,!{Y, 'lTO YHKl{HM W n II Woe
orrpe,lJ.enRlOTCSI BbIpa)l(eHlli!MM (5.3) I1 (5.17),
8 TaJ(:>Ke nOlIaraSI WIjJ = KIjJ' nOJIY<IaeM BbI-
pa)l(eHl1e CHlIbl, C OpMHpOBaHHOH 31IeKTpo-
CTaTH<IeCI<HM )J,aT<IHKOM:
Fp = MiK'J = K 3 (Yr +a9),
f,l\e K'J = (KIjJESOSU: n ) I ht (3,l\ecb - B
Hl>lOTOHaX Ha MeYp).
MHKPOAKCEflEPOMETPbI KOMrIEHCAltI10HHOrO ITPE06PA30BAHJ15J 333
LJ:06aBlIl-t K JIcnol! 'raCTl1 nepBoro ypaB-
HeHUSI CHCTeMbl (4.68) BeJIl1'lHHY Fp, a K ne-
Boii LJaCTI1 BTOpOro ypanHclIlISI BC."lH'4HHY
Fp (l + a) 1-1, nOJImKI1B, qTO HMeeTCR 17 ynpy-
nIX 3JIeMCHTOB nOlIBeca 11M, nOJIy'lJ1M
(lIk 12 :t mg + K}G )S(s)+ (nk ll + K})Y,.(s)= + mL I } r(S) + mat (s);
m
k1S2 +k;J,ss+ nk 22 :tmag+ K}a(l + a)]s(s) +
[ ] + maLl } \
+ nk 21 + K}(l +a) Yr(s)= r(s) + maaJ f
- maL 2
npaBHJIO HCITOflb30BaHHSI 3HaKOB B
npaBbIx LJaCTSIX ypaBHeHMH TO :>KC, liTO H B
CHCTeMC (4.68).
PeweHHc CHCTCMbI (5.22) OTHOClfreJ1b-
HO napaMcTpa 9(s) aHaJIOrl1qHO BbIpa:>KeHHIO
(4.69), a HMeHHO:
+m(ak Il -k 21 )}Zl
K} - 2 .
nr- /1 '
}
T =
3
J.,{k ll + : K}) .
/1}
(5.22)
K rf(s)+K aAs)
S ( ) = na,
S 2 2 '
T} s + 2't,}T}s + ]
(5.23)
r.ne
K 3
flax
_ +m{ak ll -k 21 ) .
/1 '
}
kiJ,S
} == 2
I
kfl +-K 3
n
J A/1 3
.
,
(5.24)
/1} = n(k Il k 22 -kl 2 )+mg(k 21 -akIl)+K)R.
3.neCb R = -[(I +a)(k12 -ak l d-k 22 +ak I2 J;
- B HblOTOHax B KBa)J,paTe; R - B HbIOTO-
Hax Ha MeTp.
B CHCTeMax ,/J,BOHHbIX 3HaKOB Y Benl1-
'lMH B <lJopM)lJ1ax (5.24) BCpXHI1C 3HaKll H
ITapaMeTp COOTBeTCTBYIOT yrJIY yCTaHOB-
KH Yo = 270 0 , a Hl1:>KHI1C H ITapaMeTp L 2
yrny Yo = 90° .
113 BbIpa)KeHHSI (5.23) cne.nyIOT nepe-
.D,aTO<IHbIC <lJYHKQI1H q3 MA C yqeTOM 06-
paTHOl! CBR3H:
W 3 _ K M
- 2 2 '
1; s + 2c,}T}s + 1
r,l\c K M = K r .llJ1S1 yrnoBoH BH6paQHH oc-
HOBaHH H K 3 = K n 3 -,Wl JIHHCHHOH
nt.t ax
(5.25)
BH6p a I.U1 H .
nepC,/J,aTO'lHaJI I}>YHK HR q3 MA no
B03Mym. eH HIO (jiB + g) B COOTBeTCTBHH C
BbIpa:>KCHHRMH (4.73), (4.74) 6y.neT TaK:>Ke
l1MeTh BH)J, (5.25), r,/J,e napaMcTpbI T == 1; ,
c, = c,} BbI'lMCJIH.lOTCSI no Bb1pa)KCHHSl:M
(5.24), B KOTOpbIX CJIe.nycT nOJIO:>KHTb g = 0,
I1 npM 3TOM K M = K:W" = K a,. .
ITep c .n aTO 'lHa5I <lJYHKQIDI q3 MA no
B03Mym.CHI1IO B BM)J,e yrna HaKJIOHa OCHOBa-
HIDI HaxO,/J,IITCR no Bblpa)KCHHIO (4.76), B
KOTOpOM ITapaMeTp 11 BbI'iHCJI eTCSl B COOT-
BeTCTBHI1 C opMYJIaMM (4.77), napaMeTpbI
T = 1;, c, =;3 - no BbIpa)KeHWIM (5.24), a
K03<lJ Hl{HCHT nepe.na'iH
Ky = K; = [ + mg(k 21 - ak I 1)] I /1} . (5.26)
334 fJIaBa 5. H3MEPIiTEJIbHblE CBOMCTBA MHKPOMEXAHJ1lIECKHX rIPHEOPOB
Hcxo,WI U3 pHC_ 5_10.6. 3arnru..ICM, HarIpH-
Mep, nepe.naTO'lH)'lO YHKUlno MA no OTHO-
llleHIUO KJllUleiiHoii Bu6paulU1 ax<s) TaK:
K 3 K
n a \
<D(s) = .,.,' ,
T 3 -s- + 2 3T3S + 1
(5.27)
r.ne K = [(I + a)U onK<j>]1 110 [CM. I}>OPMYJIY
(5.12)].
flpu\fep 5.5
PaCCqHTaeM napaMerpw nepeJaTOqHOfi
CPYHKlUUI (5.25) Ii nocrpoliM qaCTOTHl.le xapaKI'e-
pliCTIUm !\'IA lJ..!UI IICXOJHhIX ).IaHHbIX:
a =9-1O- 3 !1L b" =1O- 2 M. C" = 0.35-IO- J M,
a=45-1O- 3 M, 1=0.8.W- 3 M. p=2.33-10 3 Kr/M 3 ,
L! =L =1+a=53-1O- 3 M. k ll =2.036-10 3 HIM_
k l2 =k 21 =-o,819H. k2 =4,398.10-4 H'M.
k;J.3 =49,93.IO-{;H.M'c, n=3. 110 =20-10- 6 M,
E=100058, EO =8,85.10- 12 <D!M. S=60.IO-{;M 2 ;
U on =5B: K<j>=l.
BbIQMCJ1IIM
11M
m=aMhMcMP-
:= 0,075 .10- 3 Kr \I ee MOMeHThI I1HeplUlI1:
m .,., -9"
JC= 12 (aM+c )=0.51'1() KI r'; .f..1
., -'I 2
= J c + ma- = 2.03.10 Kr'M.
HaiU.l,eM K03.p.pMUl1eHTbI.
-I -{; 2
K..= I,00058.8,85.10 .60.10 .5 - 1.66 HIM,
(20_10- 6 )3
R = -[5.3.10- 3 (-0.819-4,5.10 -3 .2.036-10 3 )-
-4,391\-10-4 -4,5.10- 3 .0,819]=0,057 H-M.
BbJnOJ1HI1M BblQIlCJ1eHHjI B COOTBeTeTBlU1 C
Bblpa:iKeHIUIMM (5.24) M pe3ynbTaTbI npe.l1CTaBlIM
B Ta6m1ue ,!IJ1j1 ).IBy>; BapHaHTOB YCTaHoBKI1 MA:
'/0 = 270 (npjlMOM MaJlTHHK) H 90° (o6paTHblii
MaHTHMK) n .!lBy>; 3HaqeHI1H K03.p.pnUHeHTa:
K3 = 1,66 HIM (HWmQne 06paTHOH CBjl311) \I 0
(OTcyrCTBHe 06paTHoii cml3u).
K 3 , ilapaMerp
Yo- 0
HIM K 3 , 1.10- 3 C 2 !M K 3 I.1O-{; c 2 Tp 1.10- 3 C 1;3,6e3pa3MepHbIH
na, nr,
1,66 0,963 5,103 2,305 28,369
270
0 1,097 5,815 2,461 30,281
],66 0,980 5,194 2,326 28,6 I 6
90
0 1,119 5,932 2,486 30,583
113 nO;I)"!eHHLlx pe3}'!JLTaTOB CJIe}:(yeT, '1TO
;L'lJt npHIDlThlX HCXO.!IHLlX ):laHHbIX :meKTpOCTaTH-
'iecKaB 06parnaJI CBg3b He3Ha'iHTe!JbHO BJIHjleT Ha
DapaWe1pY nepe.a ITOtJHoH 4JyHKWfH.
3anHweM oepe):laTO'iH)'1O tPynKUHlO (5.27)
]t.!11l Yo = 270 0 :
0963.10 - 3 .5,3 .10- 3 .5
C1J - 20 JO-6(r; s2 +21;3 T '}s+I J =
1.28
- (0.OO231f s 2 +2.0.00231'28,369s+1'
KOpHH 3HaMeuaTeIDI nepe,naTo<JHoH tPYHK-
Ulm: 5) = -8,44; s2 = -24580,67, II, CJIe,nOBa-
TeJIbHO, nepe.llaTO'iHa1l tPYHKUHjI npHHHMae-r BH.l1
'1> _ 1,28
(1)3S + 1)(T 23 s + 1)
I1MeeM ):lanee:
1
1)3 =-=0,118c; (JJI =8,44 lIe; Igwl =0,93;
sJ
T 23 = = 4,1.10- 5 c; (JJ2 = 24580,67 IIc;
s2
19 (JJ2 = 4,39; 20 19 1,28 = 2,144.
MMKPOAKCEJIEPOMFTPbI KOMOEHCAUl10HHOrO nPE06PA30BAHJ15I 335
. ; t;::::::: -.:- --: :++- 'rl--;:: _1 +- f
! , i : I-i;-r.-,- -. 1'- :--\it.: -R=-h -t -j +r4--'. '
j I --L .. -f.... I " I
h-t- - .H.l. - A(w} -i- -'. --H+ t+. '
o I r-- I .
-180 -t. .; j-t -" -,'-!-j Xl - l_ - - -J I -.- -'-1---
- " +, -I -:- : - I :":'> <OH LI-I- - -1- - -+ - - ++ - -
-.. T. 'T .-+-t-'--_ £v, -
---I +; . i- ' I '.:Q( : I_--+- + ' .
-.-. .- L h-; -, . ;+-{- -:-+ - "':"i), _ " . _
-135 - i -:+ -Lo__ -- - H--t_ - -v.(q:' " : r
--"'T"+- - r-:- ---...A.. r :t - -.- I, I
......t, _, _..__ _ l_ _ _
if j -::{:j:: --t- -- -i-=!-=i.:f-. . -I- . ---- , -
-l tLt-_ -t.t.:.: 1: --f " -1tt . , . +=' - ' -L-
-90 -- " --.--J-+ -H-"" -r-I-- . -1-P. --
_ JJ___ . 'j ..:.J __'; - (j)(wl. 1-1 + t- I' I-I '-I-
....--.......;+ _-:.-. _ --1_ +- _
. . .. +-..->---:- "'..;.- 4 ..J....,_ ...++ I-t-f- -......... ++-r-
L( W),
nE
20
n
<p( w).
-20
-40
-60
__'__j 'I ---!:.i:::- .. I
--..-.-. . ---- - " l ' 4- -:--:MT V' I I
-45 i.- . _:--+--t rJ-'.- - tL-
"1+.:I:i=' '. ,-' '+i.
Tr!-f ttt:t:.t Vr 1-: ' ': ,
j I. .,--I ' :i-' "
,-:- I I -.- -=L --t-r- I _ ' . _
1-'1"""'1 ....J - _& L_, -I .- -1 -, "f"
=4-+...- -h-+-r-
-RO
10
'___ I I I I
, ,.
-;- -, L I
-1--' -- -1-'-
I !- -f- -,-i- -
- - -..... -
. .
., ,
- .
, !-- 2
100
-H-;- L..H-i-
-I; I:T f+ - -3 -H Ig W
1000 w,l/e
PIIC. 5.11. qacToTilble XapaKTepllCTlIl\:l1 MA C )JJel..:TpOCTaTlI'ieCl\:oii
o6paTHoii CBH3blO
4aCTOTHble xapaKTepHCTliKH 111m orpaHli-
lfeHHoro .D,liana30Ha qaCTOT npliBeneHbi Ha
pllC. 5.1 I:
O'leBH.D.HO, 'lTO, BbmOJJHHB npc06p830-
BaHl1S1 B COOTBeTCTBliM C 1}>0PMYJIOlI (5.19),
nOJIYT-lMM BbIpa:>KeHI1C (5.27). npn 3TOM no-
JIe3HO npHBeCTlI OPMYJIbI, nOKa3bIBalOIlJ,HC
xapaKTcp 3aBMCl1MOCTH napaMCTpOB ITepena-
TO<IHOH clJYHKltHH (5.27) OT K03 l1eHTa
K , xapaKTepli3YJOll\ero o6paTHYJO CillI3h
I13MepHTeJIbHoi1u enH :
K 1 = K 1 ( 1 + Q) ' T 3 = T I l + Q .
nox nux' ,
3 = /(I+Q)'
(5.28)
I
r)J,e Q=-Kna (/+a)K 3 .
111 x
. HanOMHltM, 'ITO norapH4JMH'ICCKIJC a mmrryt1ltble
xapaKTCpllCTHKH Moryr 6LITb nOCTpOCHbl ,!InJi 6C3pa3-
MCpHLIX nepeJ\aTo'IHbJX 4JYHKI1HiI. ECJIH nepeAIITO'IHiIJI
.
4JYIIKLUUI UMCeT pa3MeplloCTb, mmpHMep B/(M/c-), TO B
:nOM CJJY'Iae nOA MOJ\YJICM nOHlfMaeM OTHOllleHl1e "10-
JlynH '1aCTOTIIOIt nepeJ\aTO'lHOil 4JYHKUIIII K pa3MepHoii
.
CJUlIlIIue 1 B/(M/C-).
3a\IeTI1M, 'lTO Bce MA MpMa Analog
Devices (CIlIA) I1MelOT 3JIeKTpOCTaTI1QeCKI1e
nepBll<IHbIe npe06p830BaTeJII1I1 3JIeKTpocTa-
TMQeCK)'ID 06paTHYID CBR3b.
5.2.2. MIIKpOaKCeJlepOMeTp
C (arHUT03JleKTpu..ecKoii o6paTHoii
CBH3blO
CxeMa, HJIJIIOCTpMPYl{)llJ,aJI
BaRHe MarHfiT03JIeKTpll'leCKOH
CIDI3I1, ITpHBe.D.eHa Ha pl1c. 5.12.
)J;aTQMK CHJlbI BKJIIO'laCTCSI B KaHaJI 06-
paTHOH CBR3H. I13Mepl1TenhHbIe KOH.UeHCa-
TOphI Cl H C2 BXOMT BO BpeM>I3an83)],bI-
BalOll\He l.{enH rCHcpaTopa r, KOTOpbIH op-
Ml1pyeT Ha BbIxo.ne Hanp}J)KeHl1e VI' noCTY-
ITaIDll\ee Ha HJIbTpbI $1 H $2, KOTopbIe
BbI)J,eJIHIOT ITOCTOHHHYIO COCTaBJlHlOlllYI{)
Cl1maJIa. HanpSI:>KeHIDI V 11>1 I1 V 11>2 npo-
nOpUl10HaJIbHbI OTHOllleHHHM lilT H 1 2 1T ,
r)J,e /1 I1 12 - BpeMH, B TeqeHHe KOToporo Ha-
np}J)KeHl1e Ha KOH,l\eHCaTOpax Cl H C2 )],oc-
THrnCT noporoBhIX 3Ha<IamH. a BCJlWIHHa
T = I, +/2 onpe):\emICT ITepl10,l\ HMnyIlhCHOH
nOCJIe):\OBaTeJIbHOCTI1 reHepaTopa.
OpMHpO-
06paTHOH
336 lJIaaa 5. H3MEPHTEJIbHbIE CBOHCTBA tvU1KPOMEXAHW1ECKI1X rIPHEOP013
R H
U 8N "
1
PHC. 5.12. CxeMs fIIOpMHpOBSHHH MSmHT03!lel..'7pIl"ieCIWii 06paTHoH CBHJ)!
DAI +15 B
cd). R8
Rl C3
R5
R3 .I. DA3 flTl
a:JA 12 R L U 8b t>.
DDI R6
VT2 H
R7 -15 B
R9
RlI
HarrpX:>KCHHe V 4>1 H U 4>2 nocrynaeT
Ha M8cmra6Hble YCHJlJITCJIH Y 1 11 Y2, BbI-
xOJ1.Hoe HaIIpX:>KCHl1e KOTOpbIX HBJUlCTCR
BXO,llaMll ,lI.JUI YCHJlHTeJUJ: Y3: U 8x1 H U 8>2 .
Harrp$DKCHHC Ha BbIXO,l\e YCI1JlHTeJUJ: Y3,
paBHoe U B hV<3 = V 8 >.::! -V Bxl ' BKJIlOqeHO B
nOC,le){OBareJIbH)'lO Qerr:b, 06pa30BaHH)'IO
Ka1)'UJJCaMU .!XaT<lHKa CHJIbI C 06lIlHM conpo-
THB..TICHHeM R L 11 conpOTHBJIeHMeM HarpY3Kll
. ITo KaHany 06paTHOH CBR3H npOTeKaeT
TOK 1, nponopLumHaJIbHb1H ncpcMemeHHIO
HM H C03.!laromHH CHJl)' II MOMeHT, .neHCT-
Byronrne 3aollHO C ynpyrnM MOMeHTOM no,ll-
Beca 11M. TOK I II Hanpjj)KeHHC U RblX npo-
nOpLlHOHaJIhHbl YCKOpeHHIO, .n.CMcTBYlOIlleMY
Ha 11M lnJIaCTHHY) MA.
Ha pHC. 5.13 npHBe.n.cHa 3JIeKTpH'IC-
CKWI cxeMa H3Mepl1TeJIbHOM l!enl1 aKceJlCpO-
MCTpa AT-l 104. OHa COCTOHT M3 reHcpaTOpa
HeCYIllCM qaCTOTbI Ha MHKpocxeMe DD1,
11J1bT]JOB HH3KHX 'laCTOT C3, R3 I1 C4, R4,
YCHJlMTCJ1eH HanpSlJKCHHR Ha MMKpocxcMax
DA 1 H DA2, .n.11 epeHLll1aJIbHoro yCHJlHTe-
ill! Ha MHKpocxeMc DA3, yCHJlHTenSi MolUHO-
CTH Ha T]JaH3l1CTopax VTl I1 vn, KaTYlliKH
MarHI1T03JIeKTpHqCCKOrO )J,aTqHKa MOMeHTa
R L H COITpOTHBJIeHHSI HarpY3Kl1 R H .
PRe. 5.13. 3neJCJ])H'IecJCaH CXCM8 MA C MsrHHTo:meJCJ])H"iCCKOH 06pSTHOH CBH3blO
MHKPOAKCEJlEPOMETPbI KOMIlEHCAllHOHHOrO IlPEOEPA30BAHlliI 337
------- --- ------- --- --
I I
, I
I I
I I
I I
I
I I . I
I I . I
------- --- ------- --- --
I !
I I
I I
I I
I I
U CI
aJ U nop
U C2
U nop
6) lh
Q
I
I
I I I I
I I I I
--- I I I-
II 12
1
U 1
Q
t
PHC. 5.14. <I>opMHpoBauHe HanpmKeuHiI Ha H3J\1epIlTCJlbHbIX el'itKOCTHX (a)
H HlI1ny.lJbCHblX nOCJIellOB8TeJIbHOCTeii ua BblXOllax reHcpaTopa (6)
reHepaTOp Hecy([(eH 'laCTOTbI nOCTpoeH
H8 MHKpOcxeMe D-TpHfrepa DDI. pa6o-
TaJ{)IltHH B pe)!(HMe acMHXpOHHOro RS-
TpMrrepa. OC06eHHoCTbJ{) )J,aHHoH cxeMbI
reHepaTOpa $IBJISleTCR TO, qTO H3Mepl1TeJIb-
Hble KOH)JJ],eHCaTOpbI Cl H C2 BKJIlO1.JeHb1 BO
BpeMR3Rna3.nbIBaIOIltl1e uenH reHeparopa.
flpM M3MeHeHHH eMKOCTeH KOHJ(eHCaTOpOB
Cl M C2 BCJIe.n.CTBMe nepeMe([(eHMSI ITnacTH-
HbI C LleHTpaJIbHbIM 3neKTpO)J,OM Ha npRMOM
H MHBepcHOM BbIXO)J,ax MHKpOCxeMbI DDI
OpMI1PYlOTCR HMnynbcbI, .D,JIMTeJIbHOCTb
KOTOpbIX npOITOpUHOHaJIbHa eMKOCTRM KOH-
.n.eHcaTOpOB Cl I1 C2.
Ha piIC. 5.14 nOKa3aHbI 3aBl1CMMOCTM
OT BpeMeHI1 BeJIHqIiH HanpR:>KeHWI U CI H
U C2 Ha M3MepHTeJIbHbIX eMKOCTHX, a TaK:>Ke
HMnynbcHbIe nocne.nOBaTeJIbHOCTI1, OpMI1-
pYJ{)lUl1eCR Ha BbIXO)J,ax reHepaTOpa Q M Q .
ECJIH npH BKJIlO'leHI1H ITl1TaHHSI Ha BbI-
XO.D,e Q YCTaHOBMJIacb J10rn<IecKWI "1 ", TO
KOH.D,eHCaTOp Cl 'lepe3 pe:mcTOp Rl (CM.
pMC. 5.13) Haql1Haer 3apmKaTbCR 3THM Ha-
ITpSl)!(eHHeM. B MOMeHT BpCMeHI1 lit KOr.D,a
HaITp}JJKeHHe Ha KOH.D,eHCaTope Cl ,llOCTHf-
HeT nopofOBoH BeJIl1ql1HbI HanpRJKeHHSI no-
fM<JecKoH "1" U;10P' RS-TpHrrep nepexoAHT B
pe)!(HM c6poca, Ha BbIXO.D,e Q YCTaHaBJIMBa-
CTCSI 3HaqeHl1e JIOrMQeCKOrO "0", a Ha BbIXO-
)J,e Q - JIOfH'!ecKoH "1". TIO.D, .D,eHCTBI1CM
:noro Hanp}JJKeHIDI QepC3 pe:mcTOp R2 Ha-
'lI1HaeTCSI 3ap}J)l KOH)J,eHcaTopa C2, a KOH-
,l\cHcaTOp Cl 6bICTpO pa3pR)KaeTCSI qepe3
BHyrpeHHee COITpOTMBJIeHHe MMKpOCXeMbl
DDI. 3a BpeMSI 1'2 HaITpSlJKeHHe Ha BXO.D,e S
RS-TpHfrepa )J,ocTMraeT noporOBOH BeJII1'!ll-
HbI Hanp}JJKeHHR nOrHqeCKOH "1" M TpHrrep
nepexo.nMT B pe)KMM YCTIlHOBKM "1" Ha BbI-
xo.ne Q.
TaKHM 06Pa30M, )J,JIHTeJIbHOCTM MM-
nYJIbCOB t[ H 12 3aBHCSlT OT nOCTORHHbIX Bpe-
MeHH CIR I Ii C 2 R 2 . ECJIH R[ = R'2 M YCKope-
Hlle, ,l\eHCTBytOllJ,ee Ha 11M, paBHo UYJIlO
(C I = C 2 ), TO Ha BbIXO.D,aX MHKpocxeMbI DDI
c}JOpMHpYJ{)TCR HMnYJIbCbI O.n.HHaKOBOM JtJIM-
TenbHOCTH.
nonyqHM 3aBl1CHMOCTM lIillI tI M t2 OT
3Ha'leHI1H C l I1 C 2 .
3anHllIeM BbIpa)KeHIDI .n.JIH. HanpH:>Ke-
HHM:
U CI =U{l-' .:G } U C2 =U{l-' :;" J.
(5.29)
338 rnaBa 5. H3MEPHTEJlbHbiE CBOl1cTBA MHKPOMEXAHl1'4ECKI1X nPHEOPOR
r.ne U I - 3Ha'leHUe Hanp}[)KCHlUI JIOrM'leCKOH
"1" Ha BbIXO.!le MHl\.-pOCXeMbI DD 1.
Y'lJITbIBWI. 'lTO nepeKJIIO'lCHIISI TpUfrc-
pa OCYllI.eCTBJlSlIOTCR ITpll )J,OCTIDKeHMIi Ha-
npIDKeHIDIMll U Cl 11 U c:! BeJIlf'lUHbl V nop ,
Bblpa3HM li3 c}J0pM)7JI (5.29) 11 If 12:
_ VI -U nop .
I) --R)C I In ,
V)
V) -V nop
I.. = -R..C.. In
- U
I
(5.30)
H:lbTpbl HU3KJfX qaCTOT C3, R3 Ii C4.
R4 npe.::rna3Ha'leHbl .J.,JUI BbIlle.1JCHli5! ITOCTO-
RHHbLX COCTaB:ISlIOI..UUX CMfHaJIOB Ha BbIXO-
;1ax Tplirrepa DDI. K03c}Jc}JHuMelITbJ nepe-
;1a<IH RC-ueneH no nOCTORHHOMY TO}"")' B
.!laHHOM CJIyqae paBHbI elIHHHue, Tal< KaK
HarpY3KOH J1.JISI HHX RB1]SlIOTCR OITepaUHOH-
Hble YCH.'lHTeJIM DA 1 M DA2. BKJIJOqeHHble
no cxeMe HeMHBepTHpYIOWI1X YCH;1MTe.1JeH I1
MMelOlllHe O'leHb OOJlbliJOe BXOJ1.HOC conpo-
THB.:leHHe. IT03TOM)7 HanpSl)KCHHe Ha BbIXO-
lIax M.:lbTpOB Hlf3KHX 'laCTOT MO:>KHO 3anM-
caTb B BH..'le
V -V I).
ct>I - ) T '
TT - u 12
u ct>2 - IT'
r.ne T = I) + 1 2 - nepl1o.!l HMnYJIbCHOH no-
CJIe.n:OBaTeJlbHOCTH reHepaTOpa.
MMKpOcxeMbI DA I Ii DA2 BKJIlO'leHbl
no CTaH,J.aprnoH cxeMe HeMHBepTHp)'IOlllero
YCIL1HTeIDI H Bbffi011ID!lOT YHKl{HM MaCll1Ta-
6HPYJOIIlI1X YCHJIHTeJIeH. K03 I}>JfU.MeHTbl
YCHJIeHIDI KOropblx BbIqMCJISIIOTCR no I}>op-
M)'naM
K ) =I+ Rs .
R6 '
R7
K 2 =1+-.
,lLmJ Toro qT06bI K034> HLlMeHTbI yCH-
JIeHIDI 6blJlH paBHbI MC"JKJIY co60H, BbJ6lipa-
lOT R5 =R7 MTor.na KJ = K 2 = Ky'
MMKpOCxeMa DA3 1}>0pMMpyer CHfHaJI,
nponopLlMOHanbHbIH pa3HOCTH HanpR>KeHIDI
Ha BbIXO,nax YCHlIHTCJ1eH DA 1 H DA2, a TaK-
A<e BbmOJIIDIeT YHK1!H1O ,l\onOJIHMTeJIbHOrO
HJIbTpa HH3KMX 'lacTOT C nOCToRHHOH Bpe-
MeHM T", = RIDC S , npe)J,Ha3HaqeHHOrO ,llJIjj
no.naBJ1CHliSi nYJIbcauui-i HecyweH LJaCTOTbJ.
HaITpSl:>KeHl1e Ha Bblxone DA3 oITpellemIeTCjj
no c}Jop:\IYJ1e
RII ( RIO ) RII1
V B 1>I;\) = 1 +- V Bx2 --V axl ,
R9 +RII Rs Rs
(5.31)
r)J,e V Bxl Ii V Bx2 - HanpSl/KeHlUI Ha BblXO.Qax
YCHnMTCJIeH DA 1 If DA2 COOTBCTCTBeHHO
ECJIM BbI6paTb pe3l1CTOpbI R8-Rl1
paBHbIMH. TO K03 I1UllelITbI YCIIJICHIISI no
KIDK.lI.OMY M3 BXO.QOB 6ynyr paBHbI enMHlIUC
Ii opMYJIa (5.31) ITpl1MCT BI1.n
U B I,Jx3 = U Bx2 - V BXI .
(5.32)
TpaH3HCTOpbl VTI Ii VT2 BKJ)lO'-IeHbI
no cxeMe .nBYXTaKTHOrO 3MMTTepHoro no-
BTOpliTeJJSI II BblITOJIHSllOT YHKUMIO YCliJIM-
Te.iUI TOKa C K03c}Jc}JI1UMeHTOM YCI1JIeHMSI no
HanpJDKCHI1IO. paBHblM eJJ:l1HMl{e. [lo3TOMY,
ITOliCTIlBMB B Bblpa/KeHMe (5.32) JHaqeHMR
U Bx2 I1 V BxI' nony'lMM HanpRJKeHMe. npl1-
JIUiKeHHOe K 06MOTKe R L )J,aTqliKa MOMeHTa M
cOnpOTMBJ1eHHlO HarpY3KM RH' C KOToporo
CHHMaerCR BbIXOLI.HOH CMrHaJI MA. YqHTbI-
BWI, 'lTO R H » R L , MmKHO 3anMcaTb
V = K V C 2 - C)
BbIX y) C C
2 + )
TaKliM 06Pa30M, HanpRJKeHlie Ha BbI-
xOJJ:e aKCeJlepOMeTpa ITpOnOpUMOHaJIbHO
OTHOCHTeJIbHoMY npMpaIlleHl110 BeJIM<IHHbl
113MepMTeJIbHOH eMKOCTH M, CJIe.nOBaTeJIbHO,
,l\eHCTBYIOWeMY Ha Hero YCKopeHMlO. MMeR B
BH)J,y, 'lTO C) = EEoS /(h o + I1h) M
C 2 = EEOS I(h o -l1h). B COOTBeTCTBI1H C
opMynoH (5.33) nOJ1Y<IllM
U sbrx = (K y U)l1h) I ho. (5.34)
(5.33)
TOK, npoTeKalOIllHH B KaHane 06paTHOH
CBJl3M, paBeH (CM. pMC. 5.13),
1::= USJ.Jx I(Ru + R L ), (5.35)
f,l\e R H , R L - aKTHBHble conpoTHBneHMSI
HarpY3KM M 06MOTOK KaryliJeK.
1HKPOAKCEJIEPOMETPhI KOMrIEHCAQI10HHOrO IlPEOEPA30BAHWI 339
..L
m
1\0
Pnc. 5.15. W3MepHTeJIbH3)1 uenb MA c MarHHT03JIeKTpn'lecKOH o6paTHoH CBH3b10
<l>opMy y (3.97) IL1Si BblYllC1CmUl CM-
:lbI, pa3Bl1BaeMOH MarHMT03.'ICKTpH ' leCKMM
,lI,aTYHKOM, Y ' Il1ThlBaSl, 'lTO V = V BblX '
L :::: 111 . 3anMlllCM CJICl{yfOlllMM 06pa30M:
F.'I) :::: IB3 L = K,n/l h ,
(5.36)
r)J,e K.", KrlC 2 ,
K MJ = (KvUIBJL)/[ho(RH + RrJ]. (5.37)
113:\1CpHTc.lbHaSi UCITb MA nOKa3aHa Ha
pHc.5.15.
I1epe.n:aTO'lHaH. YHKUH5I W Bbl'lI1CJI5IeT-
CR no Bblpa:>KemnO (4.71) MJIM (4.73). ECJIM
B03My afO I1M B03)J,cikTBl1eM SlBlUIeTCSI yrOJI
HaKJ10Ha OCHOBaHMR, TO nepe.n:aTOqHytO YHK-
l(I1K) W CJIClI.YeT Bbl'lHCJII1Tb no BbIpIDKeHillO
(4.76). I1epe)J,aTO'lHaH. YHKl.{lli! W y no Ha-
npSl)l(CHHfO onpe)J,eJISleTcSI H3 OPMYJIbI (5.34):
W y = V DLLX 1 fl..h:::: (K 'ul)1 110' (5.38)
I1cpc)J,aTO'lHaR YHKUlUl o6paTHoH CBR-
3M B COOTBCTCTBlIM C opMYJIaMH (5.35) M
(5.36) onpe.neJI5ICTCSI BblpaJKeHHeM
Woe:::: F;.n 1 V DLI!\. :::: BJL I(R H + R[). (5.39)
I1ep e .n aTO 'lHWI YHKUH5I (5.39) C yqe-
TOM 3aBMCHMOCTH (3.98) MOJKeT 6blTb npe.n-
CTaBJ1eHa TaK)J{e B BH.nC
Woe = [L(Jl3 crH M B M V M I V 3 )O,5]I(R H + R L ).
(5.40)
I1epe.n:aTO'lHaSI YHKUMSI
ucnl1 MMeeT OPMY
3aMKHYTOH
W(I + a)W y
$=
1
1 + - W(I + a) .Woc
m
W(I + a)W y
I
I+-W(/+a)K M )
m
(5.41 )
.D:lliI CTaTHllccKOro pC)J{H m (s = 0) no-
,
JIY'lHM, B/(M/C-),
Krr{/ + a)W v
$ s=o :::: 1 ' (5.42)
l+-Kn(l + a)K'rJ
m
r.ne Kn onpe,l\CmleTCjj no opMYJlaM (4.70)
MJIM (4.74).
I1YCTh, HanpMMep, K n = K na ' Tor)J,a
BblpaiKCHHe (5.42), KOTOpoe no M3I1'leCKO-
MY CMbICJIY COOTBeTCTBYCT Kp YT H3He pa60-
'leH XapaKTCpMCTl1KH MA, MO)J{eT 6bITb npM-
BC)J,eHO K BH.ny
$s=o = q = K x (l + a)W y , (5.43)
r.ne
K :::: [+ m(ak ll - k ZI )]1 /lM3;
11M3 ::::n(k ll k 22 kI22} + mg(k2I -akll)+kmR;
OITpe)J,eJIemfe R LlaHO B opMYJle (5.24).
(5.44)
AHaJIOrM'lHO MOryT 6bITb nOJIyqeHbI
BbIpa)J{eHwl .nJUI Bbl'lMCJ1eHHR KPYT1l3HbI pa-
6o'lMX xapaKTepMCTI1K IIPH B03M)'1II.eHHRx B
Bl1)J,e BepTHKaJIbHOH BH6paul1H OCHOBaHH5I,
ymoBoH BM6p HH JIH60 ymoB HaKJ10Ha
OCHOBamUl.
flpUMep 5.6
BblQllCJ1IIM KpyrM3H)' pa60Qeii xapall.,epHC-
TlIKM MA .IUlSI CJIe.ayIOll\HX HCXO,llHbIX ,llaHHbIX;
m=O,29.1O- 3 Kr; a:::: 4,5.10- 3 M; I +a = 5,3 x
x 10- 3 M; n = 3; k ll = 2,036.10- 3 HIM;
k 12 =k 2l =-O,819H; k 22 =4,398.10-4 H-M;
h o =20.1O- 6 M; U,=2,7B; Ky=8,5;
RH::::lOOO OM; R L ::::160 OM; B 3 ::::O.25 Tn;
L=lOM.
340 rnaBa 5. H3MEPHTEJIbHbiE CBOHCTBA MlIKPOMEXAHIJ1.IECKHX nPHEOPOB
Haiiu.eM:
W,. <=(K ,.udi ho - 8,5.2,7 6 = 1,147 .10 6 BiM :
- :!O.IO-
BL 1,147.10 6 .0,25.10
K)on = W,. =
- R H +R L 1000+160
= :!.475 .10 3 KY/e 2 ; R = 0.057 H'M:
/)",.3 =3(2,036.10 3 .4,398.10-4 -0.819 2 h:
+ 0.29.10- 2 .9.8(-0.819-4.5.10- 3 .2.036.10 3 )+
+ 2,472 .10 3 -0.057. on.')'lla .'UBI Yo = 270 0 no-
.JJ)"iHM il M3 = 141.70, a.:\,""U! Yo = 90°
LI."n :: 141,645. Ilo:J "ieHHbJe 3Ha'ieHI'ISI OTJTl'I'ia-
KJTCJi HecymeCTBeHHo, n03ToM)' nplfMeM
/)",.3 = 141,7 W.
Bbf'l.HC."'JHM
K M3 =
no.
+0,29.10- 3 (4.5.10- 3 '2,036.10 3 + 0,819)
-
141,7
=2,0.10- 5 e 2 !M.
no cpopM)'.ne (5.43) paCCQI'lTaeM Kpym3HY
xapaKrepuCTInrn
q = 2,0.10- 5 .5,3.10- 3 . 1,147 .10 6 = 0,121 B/(Mlc 2 ).
npH .neHCTBHH YCKopeHIDI ax =9,81 MJC 2
ua BbIXOl1e nOJIyqH."I HanpJDKelUle:
U swx ::9,8Jq = 1,185 B
I _1,185 - 1 02.10- 3 A.
BWX 1160 '
llilH
TOK
B MA C 06paTHoH CBj!3blO BTOpoe CJIa-
raeMoe B 3HaMeHaTene nepC,l\aTO'lHOH YHK-
llHH (5.41) 06LlIJHO 3Ha'lI1TeJIbHO 60JIblIIe
eJtHHHl!bI. liMej! :no B BHJty, H3 nepC,llaTO<I-
nOH .p)'HKWfH (5.42) aaiUIeM
U mW
_ _ BYX_ Y
""',,:() -
o%.)J KII.)
KyU J
m
ho
:;::
K y U J B 3 L
ho(R" +R L )
m
B '
Ru + R L
OTKY)J,a CJIe,llyeT
U Db1X
R H + RJ
B L = may,\' ' T.e. JB /, = ma,.I"
(5.45)
PaseHcTBo (5.45) SlBJlj!eTCSI o6Il{enpu-
H})TblM paC'leTHbiM COOTHOlIIeHl1e:\1. KOTopoe
MMeeT npoCToll 1}>113Jf'leCKHH CMbICJI: Clma,
p83BlIBaeMaH. MarHI1T03neKTpU ' ICCKllM IlaT-
'lllKOM B Llenn 06paTHoii CB})3I1, ypaBHoRc-
lIIHBaeT HHepL\MOHHYIO CHJJy MiUlTHHKOBOro
q3. o6yc.l0BJIeHHYIO M3MepSieMbl"1 YCKope-
HMeM (a x' a,,). lh paBeHCTBa (5.45) MO>KHO
ITOJIyqHTb, liaITpl1MCp, Tpe6ye lOe 3Ha'leHMe
HHnYKLlHH MarHHTHOH L\cnu
B3 = (ma_T.!") I(IL) npl1 3a)J,aHHblX 3HaqeHH-
}IX OCTaJJbHblX napaMeTpOB.
.z:Lrnr nOHI1MaHllSi M3J1'leCKnx npouec-
COB, npllBOJ1.H.Il{l1X B HTore K COOTHOlIIeHHIO
(5.45), BIDKHO nO)J,qepKHyrb, qTO CHrHaJJ B
H3MepHTeJIbHOif L\em1 npH ,lleHCTBHI1 YCKO-
peHMSI B03HHKaeT BCJ1e)J,CTBMe nOBopoTa Ma-
}ITHI1Ka Ha yroJI B. TIoKa:>KeM CB 3b Me:>KJJY
yrnoM B M YCKopeHHcM.
B COOTBeTCTBHll C pHC. 5.15 3alll1111eM
paBeHCTBo
]
B;:=(ma x -F M3 )-W,
m
KOTopoe ,D,JISI YCTaHoBMBlllerocSi pe:>KMMa
(W = K;:;x) npl1HHMaeT Bl1n
9 = ( ax - F M3 ) KI.O .
m nUr
liMeSi B BH.ny BLlpa)KeHIDI (5.37) M
(5.43), npeo6pa3yeM nOCJIe,llHee paBeHCTBO:
9 _ a [ 1 BJLq ]
K M3 x m(R H +Rd '
I1U.
OTKY.lla, npHI1H.B B = Bmax M ax::: a max . no-
JI)"IHM
8 = ["- l
K"o I B)Lq
/lllx meR" + Rd
(5.46)
MI1KP01\KCEJIEPOMETPbl KOMnEHCALLI10HHOrO IIPEOEPA30BAHI151 341
f.ae B l11a - r.IaKCI1:1oHLrIhllhIH yro.l nOBopoTa
rlJlaCTl1IIhi aKcencpoMCTpa.
Tlpll.llep 5.7
PaCC'll1TaCM MaKCI1MaJlbHoe 3Ha'JeHHe yc-
KopeHJn! lIa OCHOlJallI1U CPOPMY:Ibl (5.46), KOTopoe
MOiKeT 6blTb IIJMCpeHO MA C napaMeTp8!\1H 113
IIpe)l.bl.!lYlIlcro npl1Mepa: K'n = 2.0 _\0-5 c 2 /H ;
nu);
BJ = a.25T:!; L = 10 \1; q = 0,]21 B/(M/c 2 );
,
In = 0,29.\0 Kr; R" +R L = 1160 OM
rIYCTl, nl1Hel1110e nepCMemCHHe KOHl!a nJla-
CTllHbI OrpaHI1<JeHO 1Ha<JeHI1CM t:,.h max = 5-10- 6 t,
Tor}J:a
9 = M max
max 2a
5,10- 6
- 0,55x
2.45.]0-3
,
xl 0- 3 p . rIo <jJopMyne (5.46) nOJIy'IHM
0.55-]0-3
a ma " = 2'°'10- 5 ( 1- 0,25.]0.0,121 J -
0,29-10- 3 -1160
) ?
= 275 M/C- = 28 g M/e
TIepc.aaTOqHaSi }'HKl!JHI MA C MafHl1-
T03JIeKTpl1QeCKOH 06paTHOll CB}I3bIO MOiKeT
6bITb nonYQeHa Ha OCHOBaHHI1 ypaBHCHI1H
l{Bl1iKeHIDI (5.22), B KOTOpbIX BMeCTO BenM-
<IHHbI K) HYiKHO npI1MCH}ITb Benl1QHHY K",),
BbI<mcllileMYIO no opMYJ1e (5.37). npH
3TOM Bblpa:>KeHH}I (5.25), (5.26) cnpaBC1V1I1-
Bbl H .il:Jl}I MA C :\tafHHT03neKTpl1qeCKOH 06-
paTHoll CB}I3bIO, HO npH 3TOM HY)l(HO nona-
K K A A K = K M3 [
faTh ) = j)' Ll) = Ll M3 , nox no, CM.
cpOPMYnbl (5.44)] 11, .aanee
!-,'D _L ( k k )} LI J 'A - T.
"nf -l+m a 11- 21 L 2 ILl,n' M) =
J A( k II + : K\t3 ); D.\I);
(5.47)
kl1.9 ( 1 ) ( )
1;M3 =2 k ll +-;;K 13 / J A t:,.m .
3am1111eM nepe.aaTO'lHYIO cp}'HKl\HIO
MA ITO OTHOllIeHHIO K nl1Heillioll B116pal!I1H
?
ax (CM. pHC. 5.15), B/(M/C-):
<Du
KJ'u1
K"') (l +a) -
na r 110
? 2
Tr.7)S + 2 M31;.I3S + I
(5.48)
q
? ? .
1;.7)8- + 2 M:J1;.I3S + 1
Ecnl1 B KalfeCTBe BbIXO)J,HOll BCnJ1qI1Hbl
MA CQI1TaTb He HanpSl>KeHl1e, a TOK, nepe)J,a-
?
TO'lHaSi cpYHKl\IDI ITpl1MeT BI1,l\, AI(M/c-),
<D _ q/(R H +Rr)
1- ? 2
T.73S + 2 M3TM3S + I
ECJ1H B Ka'leCTBe B03MYIl(eHHH, .nCHCT-
BYIOIlJ,HX Ha MA, paccMaTpMBaTb HaKJIOHbl
OCHOBaHI1}1, TO nepe.aaTOQHWI <PYHKl\llS1 q3
MA BbI'lI1CJISleTCSI no Bblpa>KeHHIO (4.76),
r.ne ITapaMeTp 1) He06xo.D.HMo onpe,l\eJIHTb B
COOTBeTCTBl111 C BbIpa>KCHHSlMH (4.77), ITapa-
MeTpbI T = T MJ H l; = M3 - no 1}>0PMYJIaM
(5.47), a K03c}Jc}JHQl1eHT ITepe a'll1 CJIe.nylO-
Il.{MM 06proOM:
(5.49)
.
,
Ky = K;"J = [ + mg(k 2I - ak ll )] / Ll M3 . (5.50)
L(HHaMH'leCKl1e XapaKTepHCTI1KI1 MA:
HOpMl1pOBaHHYIO aMllJIMTy.aHO-qaCToTHYIO
/(m)/ /(0), cproO<IaCToTHYIO <p(m) H cpYHK-
WIIO nepeXO)J,Horo ITpouecca h(t) MO:>KHO
ITpe,l\CTaBHTb JI3BeCTHOH CHCTeMOH COOTHO-
llICHI1H. eCJII1 B Bblpa>KeHI1RX (5.48) 11 (5.49)
n0J10)l(HTb s = jm 11 BbIIIOJIHHTb COOTBeTCT-
BYIOllJ,l1e npe06prooBaHIDI:
A(o»
Afo)"-
I
,
l 2 2 \2 11):.2 2 2
t 1 -0> 1;,13} +-'-'M3™ 3 0>
..f' 2):.T.0> 1
'l'\O>J = -arct g M3 t JI.lliI 0> s; - ;
1- 0> 1;,13 1;,\3
..! ) 2):. T.o> I
'1'\0> =--1t-arct g M;} 2M )J.JUI 0»- ;
1-0> T M3 T M3
h(t) = Jl_ exl -l;M3t ) sin(At+ )1;
l 'AT M3 1 T M3 J
I r.-:2 'AT
'A=-vl-l;;13; =arct&-,
T M3 l;r.o
(5.51)
342 rllaBa 5. 113MEPIITEJIbHhiE CB0l1cTBA MIIKPOMEXAHH'IECKI1.X r1PHEOPOB
q>( (0), <>
.
.
, .
,
. . ---
.
- . I --==
-180 . . .
A(oo) -.?O- . , , . +--
-.J/, . ,
-135. . ., .
.
q .
. .
.. '. .
(/11(00)
-90
. . , ,
00 YO -r
, , ,
-45 :-'
,
, q ...
.,. . - Ig w
3 -.-
+++ -+ - -+
f,!1,e A(re), .4(0) - COOTBeTCTBeHHO BblXOllHoe
HanpSDf\eHlfe (U.;llf BbIXO,J:HOH TOK) aKce.:,e-
pOMeTpa Ha npOl!3BO.'lbHOii aCTOTe 11 npl1
nOCTOSlHHO!-1 BbIXO.J.HO!-i ClffHa.le. Hanplmep
,
ax = 9.81 !.{/c K = q .:Il100 K = q/(R" ..... R L ).
HaKOHeu. OT.MeTlL'.i, 'ITO cnpaBe.1.mIBbI
If 3a.\iC'ialDU!, BblpaJoKeHHble B opMe pa-
BeHCTB (5.28). B KOTOphLX .J.."lSI pacc!-18Tpu-
Bae!\mfO BapmUfTa :'larHlfT03.1cKlpll'leCKOH
06parnoii CBH311 CJIe.J)'eT nOJlaraTb K3 = K'M3'
r:r fL\fep 5.8
Pacc'llrr:J.e'l "IaCTOTHhIe xapahtepJlcTium
MA C ManiIIT03.:Jelo:.plf"JeCKlnl .J.aT'lHKO!\I CII3:bl B
l>.-aHa:Je 06panJOH CBJl3H.
.1.u Man1iIlKa (n."JaCTIIHbl) In!eeM:
.J.{ = 7.093.10-9 Kr. r. k;:;.s = 49.93-10-" H-J\!'e XJJI
ynp TOro no,J.BeCl: n = 3. k ll = 2.036-100 H-M.
UCnO.Th3yeM .:JaHHble. nO:J)"'IeHHble B npH-
Mepe 5.6:
- ,
Kw,= 2.475. 10' H-M ,..,= 141.7 H-
.,
q = 0.121 B.(:.! c-) = 1.04. 10- A/(J\!'e-).
BhI"IHc.."'JJIM napaMeTpbl nepe.J.aTo'-/HOH
.pymanrn (5.481:
T.., = J J_4( k l1 -;- : K"'3). n =
4(0).
.w
20
o
-20
-40
-60
-80
I <y WI
= 7,093 10-9( 2,036-10' + -2.472.103 )f.t1.7 =
= 3.78 . 10 -I e.
f,,,, = k;s ,( k ll + Km }(JA n) =
2,036.10 3 + .2.472 .10 3
3
7,093.10- 9 -141,7
=
49.93 -10- 6
2 ,
- I ....
- .-'-'
HaltlJ.eM e06CTBeHH)'1O "IaCTOl)':
CO,, ] = = 2645,5 lie = 421.26 fl.!.
.. T
,,
TaK Io.(lK C. > I. nepel1aTO"lHa CPYHKI.!JU!
(5.48) npliHHMaeT BIlA
c];J = q
(J:"DI.I + 1 )(T"132S + I)'
r;J.e T.. ol = 1/s 1 = 8.34 . 10 4 e; T"'2 = I/s:, =
= 1.71'10'-4 e; Sl = -1198,91; S2 =-5838,46 - KOp-
HlI 3HaMeHaTe!lS! nepe!{aTO<iHOH CPYHKI.!HIJ (5.48)_
HMeeM )laJIee: 20 Ig (0,121) = -18.34n.E
0)1 = lIT,,]] = 1198.91 lIc, Igwl = 3,08; W2 =
= lIT,02 = 5838,46 lIc, Igwz = 3,76. Ha pHC. 5.16
Bl>ITIOJIHeHO nOCTpoeHMe a mJIInyn.HoM 'iaCTOT-
HOM II cpa30BoH <iaCTOTHOM xapaKTepl1CTHK.
J(f
10 5 w,l/c
Pae. 5.1 6. OTBlole xapanepHCTHKH MA C M8rHHT03JJeKTpHtJeclCoH
06paTHoii CBH3l>1O
KOPPEKlU1JI 4ACTOTHhlX XAPAKTEPI1CTl1K
343
L(ro), <p(W)."
Jll)
20 _ _.
" - _ t_T ' . . t . - ....... -'-"--,
...L.. -=-- : -:-:-=-: _; - !. --
.- - ..:..:;..;-. -, -.-.;.-----
._ -:- A(ru ..;..L . + .:._,--!_' I ".; !...t ".-+ .'
r -180 .d;, Ara) .L,...,.... i-1 _._Lj, 1_ '-:- ::.-,., - Ij\{ w)
- --'- ---. i"- ..
_ n ---. ---. . '_ -:s::;..... -. -.: '. ..0.--. '------,_ __--'-
--- - - -..."... -- -,--.,- I ........-.- --r-- ..........-
.> .... <__If- -""-- - ---I- rn,_..-
-"i35f' H"' - -d' '-:fr- i:t ,-- . n+=r ". ..+ ::' =-=1=
-20
- - :-;.. =J L- - -:-1- T L' _, 'I----;-'"
__ ---:: _:':::'4___ ==-:-'-- -.-0---- ...... _._ -' __ --... . . ' :
. .. ..._ _;:'_::':":::...::.J; -.- _ v ---- - -.:.- .:t:. _t. --- -. - ..L-
-90--- -.; 1- r - ---r- . -=- ---+--"----: -< , r .- -.--. ...--+
-4() , " _ , '.;_ U.J_ ......... , _,....,; I - . ' _L_ , , ' ---'---'--J- ',.'
I , _.:.; ++ . _.__ -- J J I tT+" -. - ;: t :t..-....L::.:::::::::
::. : I ! -_ - ' tr i.: -;- 1 L 3-'-: K I -t
- 'n' . . - -- m ::Z::::-: .. - - .. _
-45 -. -. --. . :- / .. - 0- --r-- -+-----'--
-60 '--" '. j. ."' -.-.. <---f-- -+ ,i -, - I. ,.
"_. J_; i ' r 't-- .c:;:::1:'-"";- " n "u- - ..t + '+--+>-:"..,... J-l:::1::
L__:'_ __.._ ..... L. IT _.1 ..;.... -:---+: , -t-r-- . . - '!'t. --I
- -- . ,-- +--'-t--!-... --: -:-- -+--'-1-....- y---'--;-- --;- .
= -- - _ 3 .. - -:..1.: :::.. _. 4 .-r-::::-:t- -":::':h- ---I-, _:-- 10 co
-RO --- '- . ' - T"-",\-- - -..-..1- - "'-->-.;- ---. -, -,) -1--_ . 1"
1O 10' co,lie
10'
PIIC. 5.17. HOp.'\.lIfpOB3HH3H 3-"'m:IHTy,!lH3H II 1tt330BaH qaCTOTHLIe
xapat,,"TepliCTHlm MA C 'fannlTo'),leKTpHqeC OH o6paTHOH CBH3blO
npu."ep 5.9
PacCQIUaeM HOpMHpoBaHHble 'iaCTOTHb1e
xapaICrepHCTlIKH}VUI MA c m:XO]1HbIMH )J,aHHh1MH
1-13 npe.nbl]1ymero npl-lMepa.
PeJy.1JbTaTb1 paC'ieTOB npl1Be]1eHbI B Ta6:1II-
L1e. B KompOR 3HaQeHlUI [.04(<0)/.04(0)] ]1aHh1 B 6e3-
pa3MepHb1X L1I1cppax, nepeBe.neHHb1X TalGKe B ]1e-
L1H6eJ1bI, lIMe)! B BI-IJ1Y, 'iTO H3MeHeHI:Ie B 3 .aD B
mlHeuHoM MacllITa6e COOTBeTcTByeT OTHOlIIeIDfl{)
npl-lMepHO B [J2 = 1.41] pa3a.
XapaKTepHCTHKa 4aeTOTa w. lIc
6e3pa3- 100 500 1000 2000 2645,5 5000
.04(<0)/.04(0) MepHIDl
]111 0,996 0.919 0,757 0,282 0,177
2,13 -2,31 -2,8J -7,5 -12,0
<jJ, c
-5,7 - --49,4 -77,9 -89,99
no .naHHbIM Ta6J1uL1b1 Ha pHe. 5.17 no-
c-rpoeHh1 '-IacrollIh1e Xapah'TepI-lCTl-!KH.
OQeBn.nHo, ecnl-l aMnnnTy,!:tHYIO xapaKTe-
pllCTI-IKY Ha pile. 5.16 "nOlllillTh" ,!:to OCH a6e-
tiliee, TO nOJIYQHM HOpMl1pOBaHHYIO xapaKTepH-
CTHKY, l-I,!:teHTH'-IHYIO xapaKTepl-lCTHKe ro = <0.,) =
= 42645,5 lie (lg<o.., = 3,42) <jJ = -90°.
5.3. KOPPEKI.(IDI qACTOTHhIX
XAPAKTEPHCTHK
ra30,llI1HaMl1lfeCKOe, a TeM 60JIee :>KH,!J.-
KocTHoe, ,l\eMIT HpOBaHHc ITpHBO,l1,l-1T K nepc-
,l\eMm)mpoBaHHIO 43 .najKe npll HaJIHlfHH
nep Opal11l0HHbIX OTBepCTMH. BMecTe C TeM.
npli BaKYYMHpoBaHHH Kopnyca ,!I,eMnI}>Mpo-
BaHlle MO:>KeT OKa3aTbCSI He,!I,OCTaTOlfHbIM.
KpoMe TOro, )J,JlH. KOHcTp)TKmm MA xa-
paKTepHbI MaJIWI Macca q3 H 60nbllIWi )l\e-
CTKOCTb ITO)J,Beca, 06YCJ10BneHHWI HaJIl1qHeM
)J,aT<JHKa CHJIbI B l1CIIH 06paTHoH CBK311
(3JIctcrpHlfeCKaJI npY:>KHHa), If TO npHBO,!I,UT K
yseJIM'IeHHIO c06cTBeHHoH qaCTOTbI Kone6a-
HHH "tJ3. IIpH ::ITOM YSeJIwu1BaeTc}l BocnpH-
HMlfMBOCTb MA K BbICOI<OlfaCTOTHbIM BU6-
pOIlI)'MaM.
344 fJIaBa 5. H3MEPHTEJIbHblE CBOI1:CTBA MMKPOMEXAHHlIECKHX IlPI1EOPOB
UBiill'.
PUC. 5.18. CxeMa H3 lepuTe.rJbHOii nenH MA c oppeKTllpY)QLUIIIII YCTpoiicTBOI\I
Hcnpaa.:"leHlle (KoppeKlI.m!) OTHOCII-
Te!lbHOrO K03 Ull.UeHTa .!J.CMIT HpOBaHllSi B
H)')KH)'JO CTOpOH)' (YMeHbllleHue liJIH YBenu-
<Jemie), a Tal0Ke nOCTOjlHHOll BpeMeHH J\1A
06ecne'lHBaeTCSI cnCUHa.TJbHbIMH KOppe""TI1-
PYlOl1U{!oiU YCTpOllCTBa.\m. BKmO<JaeMbIMU B
ueru, 06parnOll CBSl31l. PaCCMOTpHM KoppeK-
lUilO OTHOCHTeJlbHOrO K03 HUl1eHTa t1eMn-
MpOBaHIDI H nOCT05ulHoi1 BpeMeHH Ha npH-
Mepe oceBoro MA, CXCMa U3MepHTeJIbHOll
uemi KOToporo, BbInOJIHeHHID! B COOTBeTCT-
BHll C pHC. 5.10. a. npHBCt1eHa Ha pl1c. 5.18.
nepe.1aTO'iHID! YHKI.{ID! 3aMKH}'TOll
n.eIIH MA MMeeT BH,!I
<1>=
m",:" JJF n W $
1 + W)JVnWcpWocWI<)'
(5.52)
THJIOBble 3JIeKIpH'leCKHC CXCMbI KOp-
peJcrHpyJOUlHX YCTpOHCTB npllBe,lleHbI Ha
pHC. 5.19.
aJ
C
R2
8)
C2
R3
KoppCKmpYIOlllHC YCTjJOI1CTBa BbIlIOJI-
HCHbl Ha oITCpauUOHHbJX YCHJIHTeJlHX (OY) C
BHelliHUMl1 RC-ueml:'\H1 H H leIOT CJIeP:YIO-
lll.HC ncpc.naTo<JHble YHKUHIf:
cxeMa (5.19, a) - TV"y = R /RI + R C ;
cxeMa (5.19, 6) - W"y = R /RI -R)Cs;
cxeMa (5.19, 6) -
,
W "Y = R2R Ct C2S +
( R4 R2 ) R2R4
+ -R 2 C I +-R 4 C Z s+ .
R3 HI Rt R 3
,l],mJ Toro 'lT06bl CTaTH'leCKUC xapaKTe-
pl1CTHKJI MA He 3aBHCe!Hf OT BBC)J,CHHbIX YCT-
pOHCTB H COXpaIDlJIaCb B03MO:>KHOCTb KOp-
peKI.{l1l1 JJ:HHaMM<IeCKI1X xapaKTCpUCTHK, He-
06XO)J,HMO C06JIIOCTH pSl,LJ, YCJJOBHH: 1) K03 -
HUl1eHTbI nepe.na'ln OY )J,OJI:>KHbI 6bITb
eJJ:HHHQHbIMU; 2) pC3HCTOpbl ,llOmKHbl 6bITb
BbmonHCHbl B O.n:HOM TeXHOJIOmQeCKOM
UHKJIe H HMeTb OMH HOMnHaJI, T.C.
RI = Rz = R3 = 10. = Rs = R6 = R.
6)
R2
Rl
R4
R5
R6
PRC. 5.19. 3neKlpHifecKHe cxeMLlICOppeKTHp)'lOJUllx YCTpoiiCTB:
a, 6 -1J]JJf yBe1IJfQeHJfJI Ii yMeHl.meHHH KmclJclJHUHeHTa ,AeMncpHpOBaHHJI; 6 - .D.JUI KOppel<lu rn
K034J4)J(llHeHTa ner.mclJHpOBaHJfJI Ii nOCTORHHOH BpeMcnH
KOPPEKl.{I151 4ACTOTHhIX XAPAKTEPI1CTI1K
345
TIpl1 BbIrIOJIHeHHli YKa3aHHbJX YCJIOBHH
nepe.n.aTO'IHbIC CPYHKl I1H KOppeKTHpYK)[l1HX
YCTPOi1c fB ITpHHHMaKJ I BHn
cxeMa (5.19, a)
W Ky = t + RCs;
cxe:\1a (5.19, 6) -
W KY = 1 - RCs; (5.54)
cxeMa (5. t 9, e)
WI<)' = R 2 C l c z i + R(C I + C 2 )s + 1. (5.55)
(5.53)
TIO)J,roHKa TpC6yeMbIX HOMHHaJIOB 1(0-
3 qJIU l1eHTOB .n.cMn HpoBaHHSI H C06CTBCH-
HOH qaCTOTbl 06eCnC'lHBaeTCH ITO)J,60pOM
KOHD.eHCaTOpOB Cl, C2.
B BbIpa)!(emm (5.52) YHKI..{HJO Uj,(s)
OrrpC,l\CJIHM 3aBI1CHMOCTbJO (4.31), BeJIH'lI1HY
W n - cpopMynoH (5.3), nepc.n.aTO'lHYJO YHK-
UHJO cPl1JIbTpa H3 BblpaJKCHHSI (5.4) 6YD.eM
CqHTaTb W<jJ = KfjI, CT8HeM ITOJIaraTb, 'ITO
cPH.J"IbTp BbrnOJIWleT yHKI..{HH MaCllITa6Horo
YCHJII1Temr. TIepe)J,aTO'lHYIO YHKl..{l1IO 06-
paTHoH CBH3H B npeD.IT0J10:>KCHHI1, 'ITO
Woe = Koc 6Y)J,eM Bbl'lHCJISlTb J1H60 ITO op-
MYJIe (5.17) )J,JIH 3JIeKTpOCTaTI1'ICCI<OH 06-
paTHOH CBSl3H, JIH60 ITO opMYJ1e (5.39) .n;ml
MarHHT03JICKTpH'leCI<OH 06paTHOH CBH3H
TIpu 3Tl1X YCJIOBIDIX ncpCllaTO'lHM CPYHKl!HH
(5.52) npl1HHMaCT BH)]:
<D = mK. ( 5 -6 )
2 ' .
ms +KJJ,vs+Cy+K3KOCWKY
r;J,e K, = UoJ(<jJ Iho.
TIo.n.CTaBllM B BbJpa)!(CHlle (5.56) no-
CJ1e.n.OBaTeJIbHO CPOPMYJIbJ (5.53) H (5.54) II
ITOJ1)"lHM
<D = mK,
.,
ms- + (KlJ,lI J:K 3 K oc RC)s +C y +KCJKoc
(5.57)
II3 BbIpa)!(CHI15I (5.57) CJIe.nyeT, 'ITO npH
HCn0J1b30BaHHI1 KOppCKTHpYJOll{Cro YCT-
pOHcTBa C nepe,l\aTO'lHOH YHKl\HCH (5.53)
YBeJIJ1qI1BaeTC5I a6COJIJOTHhIH K031}> HUHeHT
)J,eMIT HpOBaHHSI (3Hal< "+" B cKo6Kax 3Ha-
MeHareJ1S1), a 1<0ppeKmpYJOll{ero YCTpoHcTBa
C nepe.n.aTOqHOH cpYHKl1HeH (5.54) YMeHb-
lllaeTCSI K03 HUHeHT ,lleMn HpOBam1Jl q3
MA.
I10cJIe nO,l\CTaHOBKH ncpCnaTO'lHOH
YHI<l!HH (5.55) B cpYHKl\HJO (5.56) onpc).(c-
JIHM
mK3
(m+K3KocR2C,C2)s2 + [k.ll Y +KCJKocR(C t +C 2 )}+G y +K,Koc
Kcr , (5.58)
T 2 s2 + 21;Ts + 1
$=
r,l\e
[ K K R 2 CC ) 0.5 T( k ,+R ( C +C., ) KK
T= 111 + ;) oc I 2 . K _ m '3 . = 1 - CJ oc . ( 5.59 )
G KK 'Co G KV'':! 2J
y+ 3 oc .v+ y' c (m+K.3KocR2CIC2)(Gy+K3Koc)
B COOTBeTCTBHH C BbIpa:>KeHJ1JIMH
(5.58), (5.59) l<oppeKTHpyIO(ljCe YCTpOHCTBO
C nepe.n.aTO'lHOH YHKUHeH (5.55) npHBo,l\llT
K H3MeHeHHJO 1<03 m.mcHTa .n.eMn HpoBa-
HHH H nocTOHHHOH BpcMeHH MA.
$YHI<Ul1S1 (5.57) T8K)!(e MO)!(eT 6bITb
ITpHBe.n.eHa K BH).(Y (5.58), a ee napaMeTpbI
6y)J,YT paBHbl
K - mK3 . T =
Co ,
G y + K3 K OC
m
,
G y +K3 K OC
K:-... + K3KocRC
; = VI
2 m(Gy +K3KOC)
(5.60)
IIp,wep 5.10
PacC'BrraeM eMKOCTb lUUl nepelJaTO'lHOH
4>YHKf..\lIH (5.54), KOTOpWi 06ecne'ulT OTHOCHTeJTh-
HbIH K034>4>1I UelIT ,QeMn4>HpoBaHIDIl; = 0,707 AIDI
oceBoro KOMneHCal.{MOHHOrO MA C M8fHIIT03J1eK-
1p f'lecKoii o6paTHoH CB»3bJO co CJIe1JYlOlltHMlI
HCXO,lI,HbIMH ):\aHHbIMH: m = 0,29 . 10- 3 Kf, KJI;y =
= 17,166 H'c/M, GJ'= 1575 HIM. B COOmeTCTBHH C
Bb'pIDKellHeM (5.37) K$oc = K",) Ii "3 npHMepa
(5.6) HMeeM K",) = 2,475'10 3 Kr/c 2 . Y CJIOBHMC», 'iTO
R = 1000 OM H H3 4>op MYJI (5.60»)J,JIJIl; nOJIY'lHM
C = 17,1l6-2'0,707 0,29'1O-3(l575+2475) =
1000.2,475.10 3
= 6,29. 1O...{,<I> = 6,29 M<I>.
346 rnaBa 5. H.3MEPI1TEJIbHbIE CB0I1CTBA MI1KPOMEXAHIII1ECKIIX IWHIJOPOA
S.4. OllilibKII H3MEPEHIUl
MHKPOAKCEJIEPOMETPAMH
ECJIH npll60p COCTOItT 113 3BCHbCB. npe-
06pa3YIOInI1X BXO.nHOH CUfH;l.J"I X B BbIXOJ!-
Hble CHrHaJ1bl 3BeHbeB .1'1. .1'1. .__, y", TO
BCJIeJ!CTBUC ITorpeWHOCTeH 3BeHbeB 6.1'1.
.v:!, ..., Ay" norpelliHOCTb npn60pa
n
Y = L(I3,Y,),
,:\
(5.61 )
r e Y = tw/}' Y =tw / .; f.I. - K03rt..rt..UUI1CHT
.... , _ I.. I. PI '¥'t-'
B.,'IWIHHJI ;-1'0 3BeHa Ha CYMMapHYIO norpelll-
HOCTb.
lLu onpC.1 eJIeH IiJi 1', npe.nnOJIO/KHM.
liTO Bce 3BeHbSl, KpOMe i-ro, He HMelOT no-
rpellIHOCTeH, H Tor a 113 <}>OPMYJ1b1 (5.61)
no.aYtlHM
/).1'1 .
1'. = . . ,
1', I y,
a TaK KaK /).y/y = /J.S/S (S, /J.S - COOTBeTCT-
BeHHO "'])yn13Ha H npUpallJ,eHHe Kpyni3HbI
xapall."TepllCTHICH npH6opa); /).y/y, = /J.S/Sj
(S,. /J.S, - KpYT1l3Ha H npHpalJ..{eHl1e "-pyrH3Hbl
i-ro 3BeHa). TO 1', = (/J.SI/J.S;)/(S,IS). TIpHHSlB
l:!.S1/J.S,"= cSlcS" Haj:j eM:
f.I. = as
p, as. s'
,
(5.62)
JLu CXCMbl npH60pa C nOCJJe,l\OBaTeJIb-
HLIM coe.!UiHeHHeM 3BeHbeB (CM. pl1c. 5.1)
"
HMeeM S = n s, H nOJI MM as las, = SIS"
;=1
YCKOpeHHC
m
T.e. 13, = I, II. C.:1Cn.OB3TCJIhHO, CYMMapH3S1
norpcLlIHocTb paBHa CYMMC nm'peUJHOCTeH
BCC}. 3BCHbCB.
)J.nSl cxeMbl npll60pa C napaJlJ1eJIbHO-
BCTpeqHbIM CO,lJ,IIHeHlleM 3BCHbCB (CM. pliC.
5 .15) 1f leeM S = k,,/(\ + I3k"r) (kill" 13 -
COOTBeTCTBeHHO K03<!HIHlUIICHTbl ITepen.atll1
npSlMoil II 06panlOil ucml), 3 <laCTHbIC
ITpOll3BOD.libiC
as
,
as k;,p
-
"} "10 ") .
ok np (I + I3knr >- fl3 (I + I3kllr)-
B COOTBeTCTBIIH C Bblpa/KelllleM (5.62)
K03 qJll1mCHTbI BJlliSiHIlSi npJlMOH 131 H 06-
paTHoii 1'1 ueneil:
I k 111'13
1'1= ; 131 = , (5.63)
1+ k l1p l' 1+ k np l3
a 06waSi norpelliHOCTb cornaCHO opMYJIe
(5.61) paBHa
I
y = y,1'1 + Y21'1;
(5.64)
r.n.e Ylo Y2 - nOrpCIlJHOcTH. BHOCHMble ITpSl-
MOH H 06paTHoH uenJlMH.
Ha pHC. 5.20 nOKa3aHa H3McpHTeJIbHaSi
ueITb aKCeJIepOMeTpa, 3KBHBaJIeHTHaSi cxeMe
Ha pl1c. 5.15 )1JISI YCTaHoBI1BIlJerOCSI pe)l(UM3
c yqeTOM OIlJH60K npSlMoH BliP H 06paTHoH
1
Boe l.[eITel1, r,lle kill' = k(l + a) .; 13 =-W oc '
m
B COOTBCTCTBHH C cpopMy.rraMH (5.63),
(5.64) H puc. 5.20 cYMMapHaSi norpelliHocTb
MA onpC eJISleTCJI paBeHcTBoM:
B[ = BIIPI'\ + oocl32'
O"p
K np
UU""X
J3
PHC. 5.20. CrpYKTYpa8H CXeM8 MA
OUIHEKi'1 H3!v1EPEHI151 MI1KPOAKCEJIEPOMETPAM 1
347
lIpll_IIep 5.//
Pacc'IIITacM Im3 )(JmI1l1e"Tbl [JJllIHHlU1 OWH-
60K 0 11 1' HOne ,1ml MA c napaMCTpaMH (In npH-
5 6) ' K - K MJ - -5 2 .
Mcpa .). ,,- liD - 2,0.10 C /M, I + a =
,
""5.3.10 J M: W,. = 1,147.10 6 R/M; W no = 2,15 x
x IO- J T.1/0 1. .
UbI'IIfCJII1M
K"p = KI1(/- a)W,.=
=?10 s.5 3.10- 3 .1 147.1 0 6 = 01 2 2 8 .C2 / M'
.... , , , ,
A = W /m=215.1 0 -3/ 0 29.1O-3=74 1 TJI'M
I.J ne , , ,
OM'Kr
rIo )OpMYJlaM (5.63)nOJIY'lHM PI = 1/(1 +
+ 0,122.7,41) = 0,53: Pz=0,53.0,122-7,41 = 048.
OCHOBHblMH ITpH'mHaMH, BbI3b1BalO-
11(I1MH ITOrpeWHocTb H3MCpCHHH MA, SlBJlSI-
IOTCR Te mepaTypa, Bl16palll1S1 II ITepeKpeCT-
Hoe YCKopeHHe. M3MeHeHHe TeMnepaTypM
OKpY)f(alOllleH cpellbI rrpHBOIUJT K H3MeHe-
HHIO 3Ha'leHHSI )J,H3J1eKTpH'JeCKOH ITPOHH-
l\aCMOCTH E, 3a30pa h H JIHHeHHblX pa3MepOB
MaSlTHI1Ka H npOBO,D.a Ka'J)'weK MarHHTO-
3.rreKTpH'IeCKoro AaT'IHKa MOMema. IToSlc-
HI1M nOCJIc.nHce YTBep:>K,lleHHe.
DOMHSI, <ITO C 60JIblllOH TOtJHOCTblO
HMelOT MeCTO ITpH6JIH:>KCHHbIe paBCHCTBa:
t. MJ KM:fi, R (I + a)(akJl - k 12 ) [CM. op-
MYJIY (5.44)], Bblpa)f(eHHe (5.43) AJISI KPY-
TI13HbI xapaKrepHCTI1KH MA C BbIXO,l\OM no
HanpSl)f(eHHIO 3aITI11lleM B CJIe.n.ylOl.lteM BlUIe,
B/(M/c 2 ):
q ::::m(Rn + Rt>/(B 3 L). (5.65)
Bhlpa)f(eHHe JI)lSl KpYTH3HbI xapaKTe-
pHcnlKH c BbIXO.nOM no TOKY:
q/ = m/(B 3 L) = (pa..b..c.JI(B3L). (5.66)
f1pllA1ep 5./2
Bbl'lHCJlHM norpewHOCTb pac"IeTa no npH-
6JIH)J(eIIHOH <p0pMync (5.66). I1CXO.l\HbIC J.jaHHble:
m = 0,29.]0-3 Kf, BJ = 0,25 Tn, L = 10 M. 3Ha"lc-
HlIe KPYfH3UbI, BbI"IHClIeHHoe no " TO "lHblM" <pop-
MynaM (CM. npHMep 5.6), q/ = 1,04.]0-4 N(wc 2 ).
BbIl./HCJlHM KPYfI13UY xapaKTepllCTlIKH no <pop-
Myne (5.66), N(M/c 2 ):
(. = 0,29.10- 3 - 1.16.10- 4
71 0,25-]0
OTHOCIITCnbHal1 norpewHocTb Bbl"lIlCJICHUH
11,54%.
Ha OCIIOBaHlIl1 paC'leTa, BbinOJlHeHHoro B
npHMcpe, 3aKlIlO'laCM, 'ITO Bblpan<eHHc (5.66)
MO)J(eT 6bl1'b HCnOJIb30Bm!O .!J:JI!I aHaJIma norpew-
HOCTeiJ: MA.
y CTaHOBI1M BJII1S1HHe Tel\1nCpaTypbl Ha
ITOrpCWHocTb H3MepeHHH MA, nOJIaraSi
B3 = const H p = const.
Bbll.H1CJIHM <JaCTHbIe npoH3Bo)J,HbIe ITO
TeMnepaType 01' BeJIH'lHH, BXO,D.SII.ltIfX B q,
ITepeHAeM K KOHC<IHOH pa3HOCTH H pe3YJIbTa-
TbI pa3AeJIHM Ha HCXOAHoe Bblpa)f(eHI1C
(5.66). B HTOfC ITOJIY<IHM Bblpa)f(eHHe }lJISI
oTHOCHTeJIbHOH TCMnepaTypHoH norpcuJHO-
CTH MA:
8L= ( oaM +J... ab M +J... oC M _J... dL ) I1T.
a 1 aT b" aT c" aT L dT
(5.67)
OTMeTI1M, 'ITO KOHCTpYKTHBHblMH
npHeMaMH OTHOCHTeJIbHaSi oIlJl16Ka KpyrH3-
HbI OT TeMnepaTypHoro paCllll1peHHSI MOHO-
KpHCT 1.JIH'ICCKOro MWITHI1Ka MO)f(eT 6b1Tb
CKOMneHCHpOBaHa. 3TO OTHOCHTCSI H K 3a30-
PY, eCJIH MaSlTHHK H KpbIWKH, Me:>K,llY KOTO-
pblMH OH HaXOlIHTCSI, BbIITOJlHeHbI 113 O.nHOfO
MaTepJia.na. B cnYtJac, eCJ1H KpbIIllKH H3ro-
TOBneHbl H3 )J,Pyroro MaTCpl1aJIa, HaITpHMep
OnTH'leCKOfO CTeKJIa, nOJIHOH cTa6HJI1l3aUl1H
3a30pOB ,D.OCTHtJb He Y,llaCTCSI.
TaKHM 06pa30M, reMITepaTypHaJi no-
rpeWHOCTb MA 06ycnOBJICHa 113MeHCHl1eM
3a30pOB Me)J(}:{Y ITJIaCTHHOH MWlTHMKa 11
KpbIIllKaMH, AJlI1HOH npOBO,lla KaTYWCK H,
CJIC,l\OBaTeJIbHO, H3MeHeHHeM HOpMllpOBaH-
Horo TOKa. CJle)J,OBaTCJIbHO, Tpe6YCTCSI TIlta-
TCJIbHIDI TeMnepaTypHWI KaJI1l6pOBKa MA.
DOrpCIlIHOCTb MA, 06YCJ10BneHHIDI He-
CHMMeTplleH MarHHT03JIeKTpH'ieCKOH CHC-
TeMbl ,l\aTtlHKa MOMCHTa, paCCMOTpeHB B pa-
60Te [11]. ,[(OJISI :noH COCTBBJISl.IOll(eJi no-
rpelliHocTH He3Ha'iHTeJIbHa.
348 rnaBa 5. H3MEPlfTEJ1bHbIE CBOI1CTBA MHKPOMEXAHI'I ECKHX TIPHEOP013
I1pH CHCTBHIf ncpcKpecTHoro YCKope-
HM B03HliKaIDT .iIOnOJIHllTeJIbHaH. e OpMa-
lI.HH ynpyrHX 3JleMCHTOB no.nBcca If COOTBCT-
CTByIDII!liC 11M nepeMe eHml MaJlTHMKa. De-
peMe eHH BllOJIb ocn Y COBna,naIDT C Ha-
npaBneHHeM oeli 'l)'BCTBllTeJIbHOCTlt H KOM-
neHCHpyroTC .iJ:aT1!HKOM MOMeHT8, T.e.
olllH6KH He BHOC5!T. I1epeMemeHIDI MaJiTHU-
Ka B3,OJIb oen = OTHOCHTeJIbHO HenO,llBIDKHblX
3JleKIpOllOB .uaT1!HKa nepeMellleHlIll 1I3Me-
fU!IOT 3 eKrnBH)'1O m1O a.!Ib ITepeKpbITItSi
3JlCI\.1pOllOB H 6C3 npHfU!TH5! KOHCTpYl\.iIlB-
HbIX Mep MOf)'T npHBeCTH K CjlY'-IaHHOif
01IIH6Ke. Bep05!THOCTb nOSlBJIeHHj{ )'K83aH-
HOH 0lIIH6KH npe,l\OTBp8lllaeTC}I yseJIHQCm1-
eM II.!lOIl!aJJ.H 3JIeKTpO,l\OB Ha l\."])bIIlIKSX.
3aMeTHM, 'lTO H3J10IKeHHbIH MaTepHful
no norpelllHOCTJlM IL"fCeT B OCHOBHOM 03Ha-
KOMUTe!lbHbIH Xapal\.iep, TaK KaK 3Ta npo-
6J1eMa HOCHT CaMOCT05!TeJIbHOe 3Ha"IJeHHe. B
qaCTHOCTH, BOITpOCbI BH6poYCTOWrnBOCH!
aKce.'lepoMeTpOB. B TOM QI1C.JIe C MOHO-
I\.-pHCTIL"L'lH"lJeCKI1M MaJlTHllKOM. paCCMOTpe-
HbI B pa60Te [11].
5.5. MBKPO)lA T1fiIK )lABJIEHIDI
llPRMOrO IIPE06P A30BAHIDI
R3MepHTe.JIbHaJI uenb M)J.[( Mom:eT
6blTh BblDO:IHCHa no pHC. 5.2. B COOTBCTCT-
BUH C HUM M C }"IeTOM nepBoro ypaBHeHHH
CHCTeMhI (4.78) npe1l,.<10iKCH8 CTpYKTYpHaJI
cxeM8 113MepHTCJlbHOH uenll M)J)]., ITpell-
craSJIeHHWI Ha pMC. 5.21.
liMeSi B BH.UY. 'ITO 1IJl0 aJU> KBa.np8T-
HO" I1JIaCTI1HbI (MeM6paHbl) A 2 , ncpe.naTO'l-
HWI CP}'lflCUH'llIJlaCTHHbI C m:eCTKliM lI.eHTpOM
W y onpe,l\eIDIeTcj{ BbIpIDKclllfeM (4.40). 8
nepe.naroqHble CPYHKUHH )J,8TtII1Ka nepeMe-
u-
/).
A 2
llleHIIiI 1J.Tn II WIj> BJlbTpa - COOTBeTCrBCH-
HO BblpalKeHI1S1MII (5.3) II (5.4), '3amUlJCM
ncpe.naTO'lHyIO YHKUIlIO MM npJlMOIO
ITpe06pa30B8H1tSl no OTHoweHll1O K .naBJIC-
HIIIO:
W(s) = U AMX -
- !>.jJ
A:!KyK UQn
2 :2 2 2 '
ho(T.,.s +2Ti'.1Iys+1)(Tlj>s +2TojJ/;ojJs+l)
(5.68)
f.iJ,e K03 li ll.lleHT K}' = 11 G" ; BpCMSI
T,. =lIwo = mIG\,; /;.DJ' =kJJ.,J(2 mG)' );
110 - 3a30p MCIK.llY HcnO)J,BHIKHbIMI1 :meK-
TpO,l\a m II 1l0.nBIIIKHbIMII, paCn0J101KeHHbIMl1
Ha MCM6paHC; U on - onopHoe HanpJllKe-
HHC m1TaHM I13MepllTeJIbHOH Uellll;
Toj>, KIj>' /;Ij> - napaMeTpbJ HJIbTp8, OllpC-
,lleJISleMble Bblp8IKCHIISIMli (5.5).
flpliAlep 5.13
PaCC'-IlITaeM napaMeIpbl I}JHJIbTpa (CM. pHC.
5.2) H3MepmeJIbHOM UenH MM, pa6oTalOlUero B
.lmanaJOHe H3MepeHHH aBJIeHml 10 5 ...
10 6 ITa (l aTa = 10 5 ITa). Yfcxo,UHbJe AaHHbJe:
U on = 5 B, Tpe6yeMaSi Kpym3Ha xapaKTepHCTI-!-
U
KH BbIXO,lIHOro CHrH8JJa 8L1X - I B/aTa = ] BI
P
(10 5 Ila) = 10- 5 B/(H/M 2 ); aMnJIHT)'Aa nYJIbCawm
BbIXO,llHOrO CHrH8JJ8 Un < lOB.
I1apaMeIphl G3 MM: In = 2,08'10-4 Kr,
A = 2.10- 2 M, .I\J' = 0,22, G y =15,55.10 4 HIM,
roo = 2,73-10 4 lIc, ho = 20'1O M,
WcjJ
U eLIX
PRe:. 5.21. CxeM8 H3MepHTeJJbHOii l.(eOH MM
MHKPOL/.A TLlHK L{ABJIEHHJI nPJIMOrO nPEOEP A30BAHlliI
349
L(ro),' (jJ(ro),"
.QE
-100
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--;- -H- -, I-t- __ ',--,.-,. t- _ __ "_ _
=-t 1t:J >- :J:
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t - -' '_11_ '-
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J
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:q(.> . .
q ifF.. '
o _;
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1 +1-1-1--1 '- -- -
.
.
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=P.i=' E:r ,
F P::: -i. , , j "Slgco
:2 3 . 4
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-300 0 10 100 2 138 1000 10 000 4.437 (O,l/c
PIIC. 5.22. qacToTHMe xapaKTepllCTHKl1 MM
BbllJHCJ1HM K), = I/G y = 6,4.10- 6 MIH ;
r;. = \fCiJo = 3,66 .10- 5 C. B Bblpa)KeHHH (5.68)
nOJIO)[illM S = 0 H non}"lHM CTaTH<IecKyro xapaK-
TepHCTHKY UBWJb.p= A2KyK ,uoti/ho' oTKy,!!.a
K U BL1X 110 1.20.10--6
Ij> pA2 KyU on 10 5 .4.10- 4 .6,4.10- 6 .5
=< 1,56. 10- 2 .
B COOTBeTCTBHH C 3aBHCHMOCTHMH (5.5)
R5 = 1,56.10- 2 R3' rIpHMeM R3 = R4 = 10 4 OM .
1
TOr a Rs = 1,56.10- OM. p;anee [CM. iPoPMYJIbl
(5.5)] ,lUIJi nOJI}"I eHHblx 3Ha'leHHH COnpOTHBne-
HHll 1j>=4,13 .J C3/C4' YCJIOBHMCH, 'ITO
C 3 = 10- 6 <}) H Haii.u,cM,!I.JISi Ij> = 0, 707 3H8lJeH He
C 4 =34.1O--{; <IJ. BbllJlfCJ1IfM TIjJ = .J R 4 R s C 3 C 4 =
== 104 .1,56.10 2 .10- 6 .34.10--6 = 7,28 . 10- 3 C
(CiJIj> == 1,37.10 2 I/c). IlpHMeM CiJ r =6,28.10 5 llc
H no iPopMYJIe (5.7) nOJI}"IHM U tI < IO--{; B .
3amulleM nepe.[(aTOQHYIO I!JYHlG.jHIO (5.68) JIJ]j[
BbI'lHCJ1eHHbIX napaMeT1xm:
W(s)
10- 5
? ? 1 2 '
(Tis. + n:v1;.n,vS+I)(T.j;s +2TIj> lj>s+l)
r e
Ty ;:,3,66.1O- 5 c, ,!I,)' ::00,22, CiJo =2,7'I0 4 l1c
(lgCiJo = 4,047), TIj> == 7,28.10- 3 C, Ij> ::0 0,707 ,
CiJIj> ::0 1,37.10 2 lie (lg CiJIj> = 2,024).
Bbl<rnCJ1HM 201glO- 5 ::0-99/65 ,l],E. AM-
nJIHTY,I],HaJI A(CiJ) H 1!Ja30BaJI cp(CiJ} 'l8CTOTRhIe
Xap8I<TepHCTHKH npHBe.n;eHbl Ha pHC. 5.22.
113 'IaCTOTHbIX Xap8I<TepHCTHK cne,!!.}'er. 'ITO
q3 (MeM6paHa), o61Ia,!!.IDI 6olIbrnoH co6CTBeHHOH
'UlCTOTOH CiJo, He BJIIi»er Ha xapalCfepHCTHKH
,IJ.JlJI BXO,l],HblX B03)J.eHCTBHH B ,l],lIan830He
'IaCTOT ,1],0 1 if lIc.
350 rJIasa S. H3MEPl1TEJIbHbIE CBOI1.CTBA MIlKPOMEXAHl1l.IECKHX nPI'U;OPOB
5.6. MHKPO)I,A TqHKII )].ABJIEHIUI
KOMnEHcAu,UOHHOrO
nPEOEP A30BAHIUI
5.6.1. MHKpO.n8T'IHK .n8B.TJeHHH C
::mel..,pocT8TH'IeCKOii 06p8THOH CBSl3blO
IIpHHLl.HnHaJlbHaH. cxeMa JIaTtH1Ka )l.aB-
JIeHIISI, 06"beJUfHeHHWI C 3JIeJ-.IpIfqeCKOil
cXeMoH. npHBe.neHa Ha plfC. 5.23. q3 3TOro
,!taT'lHKa lIaB.rleHrUl BblnOJIHeH 1f3 ,l\BYX 0)1.11-
HaKOBbIX J-.-pcMHlieBblx MeM6paH C >KeCTKIfM
Ll.eHTpoM, CBOHMJi KpoMKaMIf npIfBapcHHbIX
K OCHOBaHHIO If3 60pocmmKaTHofo CTeKJIa.
I1p e .!l.nOJIaraeTcSI. '1TO nJroCKOCTb K8>K1I.OH
MeM6paHbI COBMellleHa C J-.-pHCTaJlJIorpa lf-
qeCKOH nJlOCKOCTblO (100). OCH KOOp)J.ItHaT
COBna.:raJOT C HanpaBJIeHHeM [110). Y ITpyra51
nepeMbI'lKa no nepHMeTpY nJIaCTlfHKIf
C OpMHpoBaHa CHMMeTplf'lHOH JIYHKOH C
HaKJIOHOM rpaHeH 54°44' B ceMCHCTBe tJeTbl-
pex IL"JOCKOCTCH [I I I]. ,lLrJSI CHIi:>KeHIDI Mac-
Cbl ncCCTKOfO ueHTpa C BHCn1Heif CTOpOHbl
KaJfQOH eM6paHbl QaCTb MaTepl1t1Jla Bbl-
TpaB.leHa. .rreMn lfpoBaHHc MeM6paH oCY-
1lleCTBJIjICTCSI 3a CQeT nepCKa'lKll >KHnKOCTH,
HaXO)J,5IllleI1CJI B nO)J,MeM6paHHbIx 1I0JIOCTSlX,
tJcpe3 KMlI6poBaHHoe oTBepCTHe.
IbMCplneJIbHblC KOIIJICHCaTOpf,1 C, II
C;.! 06pa30naHbl HenO)J,BII:>KHbIMII JJICKTpO-
)J,al\m, BblnOJIHCHHblMII H3 CTeKJISlHHOM oc-
HOnaHItJI. It nO,l\BHiKHbIMII, QJYHKUIIII KOTO-
pbIX BblnOJIHSlJOT >KeCTKlte ueHTpbl npoBo-
;urnmx Kpel\lHIICBblX MeM6paH. Pa1110cTHoe
.n,aBJJeHlle f¥J = PI - P2 Bbl3bIBaeT nepcMe-
llJ,eHrIC MeM6paH OTHOCI1TeJIbHO HenO)J,BJi)I<-
HOro 3JIeKTpO)l.a. KOTopoe eMKOCTHblM MOC-
TOM npe06pa3yeTcSI B 3J1eKTpH'leCKOC Ha-
npJlIKCHHC, IImerpllpyeTcSI (OITepaIUiOHHbIM
YClfJIHTCneM OYl), MaCIllTa6lipyeTCJI (OITe-
paQl10HHbIM YCliJIlfTCJICM OY2), Ii BbIXO)J.Hoe
HanpJllKeHlle U OblX B ITpOTJIBO a3e cMew.c-
HIfIO MCM6paH rrocTynaeT Ha nO;J.Bl1iKHble
3JIel\.IpOLIbJ, peaJIlI3YSI npHHQI1IT CI1JIOBofi
06paTHOH CBSl31t.
'3JICKTpl1qeCKaSi CXCMa BKJIJOl.laeT n ce-
6S1 TaK)Ke reHcpaTOp BblCOKOH 'laCTOTbl C
TplfrrepHblM lIeJIl1TeJ1CM (T), JIOrHqeCKHe
3JICMeHTbJ, KJIIOQeBOe YCTPOHCTBO K'lI. ..
K'l4 ,llJISi nHTaHHSI eMKocTHOro npc06pa30Ba-
11
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OY2
C
PHC. 5.23. 3neKTpHliecK8H CxeM8 BKJlIOIiCIIHSJ MM
C :JneKrpoCTanllfecKoH OfiP8Tuoii CBSJ1LIO
MMKPO,lJ,A P-II1K )J.ABJIEHI151 KOMrIEHCAL.{I10HHOrO rIPEOEPA30BAHMJI 351
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p
PIIC. 5.24. CxeM3 I13MCpIITCJlhllOii lI,eml M;J)J. C J.1CKTpOCT3TWICCKOii 06P3Tlloii CBJl3h10
Te.'1S1 H IOllo<Jenoe YCTPOHCTBO K1S, Kn6 ,aJISI
nepeioll04eHlfSi cxeMLI C pe)l(l1Ma H3MepeHHSI
B pe:>KHM Cl1JIOBOH KOMnenCaI..\HH. CXeJl.18
lI3:'<1epl1Te.lbHoH uenH MJU( npHBe,aeHa Ha
pHC. 5.24
B COOTBeTCTBI1H C pl1c. 5.24 Pa3HOCT-
Hoe p:aBJIeHl1e I1p BocnpHHHMaeTCSI KBak
paTHoH MeM6paHOH ITJIOllla,l\t.Jo A 2 . Ha 3
(MeM6paHY) C nepe)J,aT04HoH cjJyHKLlHeH
w-L- 1
- - ?
/1F ms + kJl,vs + G y
(5.69)
.aeHCTByeT pa3HocTH8SI Cl1JIa /1F = FiJ. - Fp .
3JIeKTpoCTaTI1'leCKaH. CI1JIa LlenH 06paTHoH
CBSl3U Fp = U nblxWoc' TIepe,aaTO'/H8SI YHK-
LlI1S1 Woe onpe,IJ.eJISlCTCSI Bblpa)l(eHUeM (5.17).
nepeMellleHl1e MeM6paHbI y = /1h 113MepSl-
eTCSI npe06p830BaTeJIeM ITepeMellleHHH (eM-
KOCTHblH MOCT), nepe,aaTO</HaH. cjJYHKQI1S1 W n
KOTOpOro HaXO,l\I1TCSI no cjJopMYJIe (5.3). Ha-
npSl)l(eHl1e /1U C eMKOCTHoro MOCTa npe06-
p83yeTcSI B 3JIeKTpoHHoM 6J10Ke B HanpSl)l(e-
Hl1e UUbIX' nocrynalOI.LJ.ee TaK)I(e B Llenb 06-
paTHoH CBSl311. TIoJIaraH., <ITO ,lJ.l1HaMHKa
3JIeKTpoHHoro 6JIOKa onpe,IJ.emleTC HHTe-
rpaTopoM, 3aITI1IUeM ero nepe,IJ.aTo</HYIO
cjJYHKLll1l0:
w] = (1 + 1:2S)/(1:IS),
(5.70)
r,ae 1:1 = R I C 3 ; 1:2 = R I C 4 ; S - nJIOI.LJ.a,D.b
B3al1MHOrO nepeKpbITl1S1 3JIeKTpo,aOB eMKO-
CTHoro npe06pa30BaTeJISl.
6Y,lJ.eM nOnafaTb, "nu B Bblpa)l(eHl1l1
(5.70) 1: 1 = 1:2 = 1:, 11 3anl1IUeM nepe,IJ.a'IU'I-
HYIO 4)YHKLlI1IO 3RMKHYTOJ;J QenH ITO pMC.
5.24 C Y'ICTOM oPMYn (5.69), (5.17) 11 (5.3):
<1>(s) = Um..,Js) - K(1:TS) ,(5.71)
l1p(s) as 3 + ps- + ys + I
r,llC
11 2 A 2
K - .
- ,
2EEuU on S
kJl",/131:
13 = ? ;
2EE O U nS
1111131:
a= ? '
2EE O U;;n S
J ?
1:(/1 G)' + 2EEOU;;nS)
y= ?
2EE o U;;n S
,
(5.72)
r)J,e S - nJIOllJ,a)J,b B3aHMHoro nepeKpblTMSI
3JIeKTpO)J,OB eMKocTHOro ITpe06pa30BaTeJISI.
PaC'leT 'JaCTOTHLIX XapaKTepl1CTI1K
MM no nepe,IJ.aTO'lHOH yHKLlHI1 (5.71) npl1
HaJIl1<Jl111 3BM He npe,l\CTaBJI eT 3aTpY)J,He-
HUH. BITpO'IeM, B03MO)l(eH H aHaJIHTI1<ieCKl1H
pac'IeT <JaCTOTHblX XapaKTepHCTI1K, TaK KaK
pelUeHl1e Ky611<lecKoro ypaBHemUI B 3HaMe-
HaTeJIe BbIpIDKeHH (5.71), Hanpl1Mep, C ITO-
MOlllblO OPMYJI Kap,IJ.aHa He CBSl3aH C
60JIblUlIMI1 BLI'II1CllI1TeJIbl1bIMH TpY,l\HOCTSI-
MH. TIOKIDKeM, 'ITO nepe,IJ.aTO</HYIO YHKLlHlO
MM MO)l(HO npe,IJ.CTaBHTb B BH,lJ.e KOJIe6a-
TeJIbHoro 3BeHs, BBe)J,H. HeKOTopble orpaHl1-
'JeHHSI Ha ero napaMeTpbl.
TIapaMeTpbl MLI).J, TaKOBbI, 'JTO 3HaMe-
HaTeJIb nepe.n:aTO'/HOH YHKLlHI1 I1MeeT O,IJ.I1H
,lJ.ei1cTBI1TCllbl1blM I1 ,llBa KOMnJIeKCHO-
cOnpJl:>KeHHbIX KOpHfl, 11, cne.n:OBaTenbHO,
YHKLlI1IO (5.71) MO:>KHO npe,IJ.CTaBIITb B BI1,IJ.e
KlI +TS)
OS3 +pS2 +ys+1E
K(1 + TS )
(1 + 1;s) (Tls2 + 2l;T 2 s + I)
K(1 + TS)
1;T2S3 +(Tl +2l;1JT2 2 +(1; +2l;T 2 )s+1
352 r.ilaBa 5. H3MEPIITETIbHbiE CBoflCTBA l\lJIKPOMEX.A.l.llflfECKIIX IIPIIEOPOB
nOTpeoyeM BbillO':IHeHlUI YC:IOBIISI
Ii = '( H U3 paBeHCTBa a = IiT-:.. IIMeSi B
BRiIY coornOllIeH H.Sl (5.72). no.:J Y'lJl 1
/ mh 3
y., = ., -
- ., TT
-€€ov nS
(5.73)
.,
lla':Iee. y'nlTblBaSI. 'ITO T -« 2;T TI'
ll3 paBeHCTBa 2:;.T 2 Ii = J3 Harue t:
K
4'
--
(5.14)
., ££oC';n S
- h 3
Bblp:DKeHHe (5.74) 3aIIHIlle 1 B CTaH-
.J.aprnoH cpop:'o1e:
_ K.:!1'
:;;::;: 2 .Jm'G .
(5.75)
r.J.c
') TT"' S
GT = _€€o ;n
h"
(5.76)
113 EepaseHCTBa 1j» 2ST I1 pa-
BeHCTBa "( = 1) c.'1e.J.yeT, 'ITO BbillO':IHeHHe
YC.'10BIDl 1j ;::;: 't B03MOiKHO, ec.:rn
.,
O 2££oU;!]S ' 0 G )
\_ « ':t ,« . T.e. *eCT-
. h J .
KOCTb MeM6pam.1.J.oJDKHa OblTb 3Ha<rnTe.JlbHO
MeHhme EecTKOCTH, c03;1a.BaeMOH 3.1eIITpO-
1<11}f'q ecKOH CH.loit
MO&EO Ha.:IOEiTh T31GKe orpaHH'IeHHe
Ha 3aaqeHHe h, HCXO)UI H3 Toro, 'ITO npH
f¥J = Pm:n BblXO;utoe HanpJDl(eHHe .i1.0JDKHO
6I:.lIb waKCH.\43..'ThHhlM U &lX = U ou' nO.1D-
mm B (5.71, s =0, nO:JY'lHM
U m.a. = U 00 = Kpmv. = 2 pmaxV(2€€oU 011)'
onyJta
h = J (2uu?mY Pmn -
(5.77)
TaKHM OOpa3oM. AIDI npHIDll1>lX .nony-
lI(eHHii nepe.ztaTo'IIWI tIJ)'HKltIDI M.IJ.lt C
3.JIerrpoc-nrrH'IecKOH ICOMlleHcaUHeH MO)KeT
6I:.lIb npe.i1.CTaB.lleHa B BHJ1.e
K
<D(s) = .,.,
T-:.- s- 2;;T-:.s-<-}
f.1e K Onpe.1eHSIeTCH In <POPMY-' (5.72), a T.,
II - no OP'IY':Ia:\1 (5.73), (5.74).
(5.78)
nplL\lCp. 5././
Bh!'1I1C.'II 1 napa:\leTpbI nepe.:taTO'iHOii
.pYHKllIIH (5.78) M.IJJJ: I1noLIHble LIaHHbie
Pm;!., = 50 Ila. Con = 5 B . EO = 8.85 ,10- 12 <I:>! I .
1\-pe IHHeBaJI [maCTIiHa opneHTupOBaHa B nJ1OCKO-
CTII (100) II mleeT napa le-rpbl
E(l(J()} = 1.69-10 11 H' I-:'. ,,= 0,358. p = 2,33 "
. . .,
x 10' IIT!:\I', Pa3 lepbI J:L1aCTI!HbI: A=2.10- \1.
a=05..1:
JKeCTKoro
ueHTPa
ro,lUlI!Ha
C" = 05 .10- 3 'I : TO:IJl.UIHa )'rrpyroH nepe lbl'iKn
C n = 10- 5 I: paHee 6bl.'1 onpe.J.eJ1CH KmtlJ.pllun-
eRT aOCo.'IOTHOro .J.e m.pHpOBaHml
k;J.y=0.624 H.c.'!.
rIo tlJop:\I)'.1e (5.77) pacc<nlTae 1 3a30p leJK-
.J.y 'XICI-"'IpO.:ra.,tII:
h= (2.8.85.10- 12 .25V50 =3.1O-6 1.
Ilo .pop !)'.'Ie (5.76) BbI'lHCJIJ!M JKeCTKOCTb,
06ecne'lHBae l)lO 3.'leKTpOCTanJ'IeCKHMI! CHna:\1I!
UenH 06paTHoH CB)J:3H:
Gr. =(2.8.85.IO- I2 .25.4.1O-4}127 ,10- 18 =
= 6.5 _10 3 HIM.
PaCC'iHTac!>! JKCCTKOCTb ynp THX nepe lbl-
'1eK. BHa'l3JIC onpC.!Ie.'IHM llIJJIHH.lIIHNecK)'1O
;KCCTKOCTh:
D= E(IOOIC _ 1,69.10 11 .10- 15
12(I-v 2 ) 12(1-0,358 2 )
= 1,62.10- 5 H. M.
Ilo .popMj"11C (3.50) nOJ1}"'1HM:
8-3,14 4 '1,62.10 . - 5 - 5 .10 2 HIM.
G y 4
4.10- (I-O,5f
3HaQeHHC senH'Ujj{j,J G y Ha nopH,aOK
MeHbwe '3HaQCHIDI G:r., 'frO o-rBeQaer O.llHOMY H3
ycnOBHH. npHIDITbIX .IUlJI nonyqcIDIJ! .pYHKUIIII
(5.78). BhI'illCnHM Macey MCMopaHhI
m=1,23.IO-4 KJ" H no .popMyne (5.73) HaHneM
nOCTOHHH}'iO BpeMeHH:
MIIKPO.lJ.APUIK .lJ.A.BJlI::HH5I KOMnEHCA UOHHOrO IlPEOEPA30BAHWI 353
l.n .10-4 .3 1 -10 18 , -of
_ _p _ --4 =1..J7-10 C
.SJb.1O - 2)'4.10
(m2 = 0.73.10-4 lic) .
rIo .popMynae (5.75) pacc<JHTael\! OTHOCiI-
TC_lbHbIH K03.pcplIwleH [ ;J.e m.pupoBaIiHJI:
T, =
t = 0.624 = 0.35 .
. 2 1.23'10--4 .6.5 10 3
B COOTBeTCTBHJI C .pOp Iy.laMu (5.72) no.lY-
<JIIM
K - 9.10- 12 - 01 BfITa
2-8.85 10- 12 .5 . .
(h npHBe.J.eHlioro I1pmlepa cnellYcr. 'ITO
B03MOJKeH no.IIoop KOHCTpy",IffiHbIX napaMerpOB
MLIJJ:, ;JJU! KOTOpbIX Cl1paBe.wuma l1epe)l.aTO'lHID!
.pYBKUHSI BU.::!3 (5.78).
AHaJlli3UPY}l UH.ppoBble )l.aHHble I1PH BbI-
I1O IHemlil I1pH Iepa, Herpy.!lHO 6blJIO 3aMeTIlTb.
'-ITO 06ecne<JIlTb G!; > G y WU1 )l.aBJIeIUUi
PIT!i'{ > 100 Ila YJKe '3a-rpYllHIlTeJIbHo.
M.lJ.[{ C 3JIe"lpOCTaTli'leCKOll 06paTHoii
,-
CBSl3blO OOJIa,LIalOT npllei\.lJleMblMU )JJ1HaMH-
'leCKUMU xapa"-repuCTIIKaMII npu H3Mepemm
.!laBJIeHlUl, He npeBblllialOIUero 100...200 lla.
5.6.2. lVhu'-P0,lJ.aTQIlK ,lJ.aBJ1emUI
-, -
C :\lanIIIT03JIel\.--rpIlQecl\:on oopaTHon CBH3hlQ
M.l1.D. C MafHllT03JIe"lplI'ieCKOii uenblO
KOMneHCalll1H nMelOT YBeml'leHHblH Ha ,LIBa-
Tpn ITOpH):\Ka !lllana:30H l13MepeHIIH ITO cpaB-
HeHMlO C M,l],[{, HMelOIlUIMIi 3.']e"-rpoCTaTII-
lJ:eCKyto uenb Km.meHCaWII!. Ha pliC. 5.25
np nBe ll.eHa cxeMa 43 M.lJ.[{ B rn6pHJ.Hml
ncnonHeHlrn.
3 co,aep>KHT TpH Kpe 1HneBble \le l-
6paHbT: pa60lJ:YIO 1 H J1.Be npOMe:iK}'TOlJ:Hble -
3 n 8. KOTopbIe HenocpellCTBeHHo Bocnpn-
HHMalOT )J,aBjIeHile PI 11 P2' ra3. 3aIlO_l-
HHlOllUlll ITO,10CTb q3. no):\ ,.1ellCTBlleM paJ-
HOCTHOro )J,aBITeHlUi I1p = PI - P2 'iepe3
, --
KannopOlillHHoe oTBepCTne / B03,ileIlCTBYeT
Ha pa6o'l}W MeM6paH)' 1, KOTOpWl IIMeeT
BHenn-me pa3MepbI A x A H :.KeCTKlrn ueHTp
c pa3Mepa:\IJ[ a x a. Ha :iKeCTKOM ueH1pe
pa3MeIUeHbI KaT}'lllKH 5. Ha CTeKmiHHbIX
"1JblilIKax. 2 B MafHlITOnpoBo,aa.X npe):\)'-
cMoTpeHbl nOCTOHHHbIe MarHliTbI 4. ll3 le-
.... "..... -
peHue nepe:\-!eIUeHJIH paoo'len Me:\lOpaHbl
MO:i[(eT 6blTb BbInOJIHeHO C ITOMOIUblO TeH30-
uml MarHIITOpe311CTOpOB, IL"\m.rlaIITUpOBaH-
- ,-
HbIX B paoo'l}W MeMopaH)'.
B Ka<IeCTBe B03MO&:Horo BapnalITa Ha
pnc. 5.25 nOKaJaH 3JIeKTpOCTaTIf'ICCKHii
,- -
npeOOpa30BaTe.;1b ITepe)lelIleHUU, nO;IBJDK-
Hble 3JIeKTpo)J,bI KOToporo pacnOJIo&:eHbI Ha
>KeCTKOM ueHTpe Me?>16paHbT, a HenOJ1.BIDK-
Hble - Ha TOpU8.X MarHUTOB. B Hcxo,:rnml
ITOJIO:iKeHUn !\IeJK,lly napoii 3JIeKTpo,aoB 9
3a30p O,allHaKOB (h o ). llpnHI.{HIT OpMllpoBa-
-
HJUI MafHHT03JIe"lpU'leCKOH OOpaTHOH CB}I-
311 1l3JlO:iKeH B no.u.paJlI.. 5.2.2, a cxeMa H3Me-
pl1TeJIbHOii uenu MO:iKeT 6bITb BbIIlOJIHeHa
aHanOfHQHO puc. 5.13.
PI
J !!!!!!!!!!I!!I
J
," ,.:..' p", .. :. " ' :. . . '. . :.-,": " . '". ... .
.. -
::=.-, .- "-" -..........-. .
'..-:
.....-.
._...-.......- _.._....., ..,"'''A
I
-'.-.-;,.
:.,,"%.:
9
5 )f(ecn:Hii
lIemp
6
7
8
t t t t t t t t t t t t t t
P 2
Pnc. 5.25. q3 MM c P,18rUHT03JIeKTpH'IeCKH1'll CHJlOBLlP,1 npeo6pa30B8TeJIeM
B rH6ptmHOP,1 UCDOJlUeHUU:
1 - pa6o<JaJ! MeM6paHa; 1- creKJUlHHaJI KpLlIlIKa 3, 8 - npoMeJKYfO'iHhle l\leM6paHhI; 4 - nocrOJlHHhlii
Maf1lHT 5 - KaT}'lllKa; 6 - MarHIITOnpoBo,n; 7 - KaJIlI6poBaHHoe OTBepcTHe; 9 - 3JIe,,'PO.nhI
354 fnaBa 5. H3MEPHTERbHbIE CBoi.JcTBA MI'IKPOMEXAHlllIITKHX rIPHI>OI'OB
/j,
F.
MJ
PMC. 5.26. CXCM8 M1!\ICpIITC!H,IIOii IlCnll !\1.l1J1. C III an III I () J:ll'lnpll'Il'CIWii
o6poTllOii cnn:n,1O
Cxe!\1a 1I3McplHcnblJOii Henll MM
npltBelleHa Ha pliC. 5.26, ua KOTOpoii nepc-
.naTO'lHaJI YHKUlUI W(s) 1.J3 COOTBCTCTBYCT
opMync (5.69). a nepe aTO'JHh[C )YHKI lm
6J10Ka :meJ...'P0Hlum 11 3BCHa 06paTHoH CBSl311
onpe.i1emHOTC H3 Bblpa:>KCH1Iii (5.34) II
(5.37):
Jf'3 = (K,.U.) I 11 0 ; WI', = BL I (R II + Rd.
(5.79)
f;xe K,-. VI - napaMeTpbl 1neKTpH'leCKoi1 cxc-
MbI, onpeJ1.eneHHble B nOllpa3J:l. 5.2.2; B -
HHaYKllIUI, C03J:laBaeMaJI MarHifTaMIf; L -
06111aJI ;).;1HHa npoBo,!J.a K8T)'WCK; R II , R L -
aA-rHBHoe H HH.nYlnHBHoc conpOTHBJ1eUH
06MOTOK Ka"I)'weK.
B COOTBeTCTBHH C pHC. 5.26 nonY'IIfM
ncpc.lJ.aTO'lH)'1O YHKlll'110 33MKHyrOH uenH:
<1>(s)= IBu\ = ., K . (5.80)
!J.p T- 52 + 2l;Ts + I
f;xe
T = J m(G). +K,o); f,,=kn.l./2 J (G y +K",. );
K = A 2 K WJ I[BL(G,. + KN- )];
K tlJ = (KyU.BL)/[ho(R H +R L )].
(5.81)
3aMeTHM. '1TO BenH'H1Ha K.n BLI'IHCJUI-
eTCJI no opMyne (5.37). 3 BenH'IHH3 G). no
opMYJJe (3.50).
llpUMep5./5
Pacc'urraeM napllMeTpLi nepeJlaro'iHoli
cjJyHKUHH (5.80) ,WIJI HCXO.i1HLlX ,IlaHI1b1X: nonara-
eM, 'ITO npOM 'IHLle MeM6paHLI nepe.ll8/OT
,tIasneuHe Ha pa60'fYIO MeM6paHY c e.i1HHH'IHLlM
Ko) ) IIIIUIClnOI\l l1epC)l;I'lII; l\IIllJlpa 11Il1J1 pafiotla)1
MCM6pallL\ InroroliJICllli WI KpCMIII UI n IIJIOCICOC'11i
,
(IOO) II IIMCCT p;nMcp'" A 2.10 - 1\1, a O,S A,
e.. '" 0,5.10- 1 M, c'" 0.: I 0 M (II IIpCJlhlJlYIUCM
npllMcpc 6blJIa Ubl t lllCJ1Cli1i )I\Cl.1I\OC n, MCM6pllllhi
G, = 5. I Ol HIM): II" ""' S -I 0--(' 1\1; Macca MCM-
6pallhl IIMCCl C C J<UTYIIIKa 1II - 111 = 2 .( (1.4 1<1".
rIpltHl1MtlCM: /J = 0.25 TJ1, R" = I (100 OM.
RI. = 160 OM (I. = 10 M); J JIH mClnplltlCCKOj.j
CXCMbl no pile. 5.13 6h1J1O 1I0JIY"ICIIO VI = 2,7 B,
K,.= 8,5.
nOJlaraCM, till) 110JlOCTh ll,) 31111OJIIICII3 lno-
TOM III = 1.67 . 1 0 Krl (CM'C)], K:tJIIIGpOllllllllhiC
,
OTIICpl.1JUI I1MCIO'I I = n,5 . 10' M II JmaMCTp
d = 2 . 10 -I M. rIo 1II0PMYliC (4.99) II....IIICJIIIM
KO') Jll>HltIICJlT J CMI1 )Jfp()Ballll}j k." = 2.6H )(
.' .
x 10. H.c/M.
rIo COOTIJOIIJCHlUIM (5.H I) Uhl'IIICJII1M:
K . = 11.5-2,7 -0,25.10 ::::: 10 4 I-11M;
M) (1000+160)5.10- 6
2.1{)-4
5_10 2 +10 4
T=
1,3H.IO 4 c;
t;=
2,68.10- 2 -001
2 J 2.1O- 4 (5.10 2 + 10 4) -. ,
4.10-4 _10 4
K= 1,52-10- 4 A/(I-I'M 2 ).
0,25.10(5.10 2 + 1 0 4 )
HanOMHHM. '/TO .u.aBJ]CUI1C Hl31ll1eTCJI
B3)f(Hol1 qJH3H'ICCKOH xapaKTCpHCTHKOH MHO-
mx npoLleCCOB. )J,aT'iI1KH .u.aBJJeHHH npl1MC-
Jf IOTCH B aBHaUHOHllblX npH6opax: H3MCpH-
TemJX B03.nYWHof:i CKOpOCTI1 H CKOpOC111
no.n"beMa (BapHOMCTpbI), BblCOTOMepax 11 .up.
((IOPMHPOBAIU1E BhIXOrU-lhIX nlfHAJIOB MI1KPOnWOCKOI!{)B 3SS
5.7. (I)(WM M ()OHAI-I M E HbIXO)J,HhIX
CMrt-IA,110H MMI{POfMPOCI{OnOH
5.7.1. nplll\lOC IIpc(61)RJOII:IIIIIC
A. I\hllq)()nf})OCICOIII,1 LR- If LL-nlllolJ
P<lCCMm pll:\.1 11<1 nplI lcpax f'lIpOCKOIIOIi
LR- II LL-Tllllon IIpJIMOC npco6pLBlmam!C
lI'IMcpslcMoi.j l!m'!II'lccKoi-i BCJIl1'llIlIhl - YrJlO-
BuH CimpOCTl1 n Bblxu;moM CI1f'IHIJI Mr -
IIHIlP1l}!(CIII1C lIa nbIXOj.\C H3MCpU ['CJ1bIlOii
CXCMbl.
BOCIIOJlb3YCMCJI BII<I'IUJIC ypmJIICIII1S1MU
CIICTCMhl (4.142) IIpl1MCIIl1TCJlbllO K CXCMC
11<1 pl1c. 4.10 Ami napllmrra rHpocKolia LR-
"!'111m (P.Ll - 111MCIICIII1C KOOp.LU1IIUTbi x; P4 -
II'IMCIICIIHC KOopJ.\UHaTbi a). OllpC/ eJJI1M
')JICI{TpOCTaTl1'ICCKIIC Cl1Jlhl B npaBblx <IaCTSIX
yptIBIIClIlIli CI1CTCMbI (4.142).
B Mf no pllC. 4.10 }W I)KCIII1C no KOOp-
}(IIIHlTC a IIC Il'iMelHlCT 3a30pa MC)K}(Y IlJJa
CTMllaMIl rpc6cII'mTblx C'rpYKTYP npl1IJo.LloB,
R PIt 110 KOop/\HHaTC X npmICXOj.\I1T 113Me-
IIClme IIJ1Olua)J,ci.j B3al1MII01'O ncpCKpblTMJI
rulaCTIIlI, 110 IJo:illHKalOLLH1e 3neKTpOCTaTl1-
'ICCKHC nIJII,I MC)/()J,Y HI1MI1 HanpaBneHbl
lJ,Qom, OCl1 Z. TaKIIM 06pa30M, B3aI1MOACH-
CTBl1C MC)I()J,Y nnaC'fHllaMI1 rpC6CH'IaTbIX
CTPYKTYP npl1BoJlOB HC npIlBO,QHT K n03HI1K-
IIOIJelll11O MOMCIITa Cl1n BOKPYI' OCII Z.
nYCTh paCCT01lHHe 0'1' OCII Z ,QO LlM
HM Xu = 1 + fl., (/ - nOCTOSIlltiblii rCOMeTpH-
'1CCKl1ii pa3Mep, fl.r. - CMCluelmc 113-3a ynpy-
!'OM .LlcllmpMaUIHI nO)J,Bcca). npl1 yrnoBblX
KOJlc6alll1JIX no KOOpl\I1IJaTC a H3MCHSlCTCJ1
'!a30p flyo Me>I{JJ.Y IIJlaCnmaMU KOIID.CHCtITO-
pOB C1 11 C2 C'beMa IIlIIj)QpMaul1l1 (Pl1c. 4.10
11 5.27). ECJII1 flJIOll\aAh D3auMIIOro ITepc-
KpblTI111 rlJlaCHm S lie MCIU!eTCJI, TO B COOT-
F:JY
/),x
1
Cl
a
IICTcnmu C (IJOpMynoi-i (3. 5) 1I0nY'IHM 'HIeK-
TpocTaTu<lccKHC CHJII,I
F I' = -(EEoSV;,,)/ 2(6,vo + 6,V)2,
I"')\C Villi - onoplloe lIanpSl)I(CmlC Ha rumCTl1llaX
KOIIJICIiCaTopOB; 3113K "+" j.\.IISi KOII,QcllcaTopa
CI, a 311aK "_" 1.J;I1I KOllllcllca'ropa C2.
MOMCIIT :melrrpOCTaTI1'ICCKI1X CI1JI
' f - 8W _ c';n SU ,7" (I A_ )
Iv. --- -1-0.\ X
))1" ')
uf.i _
x[ (t.Yo - /I,Yf (t.Yo J\vY ] =
_ 21;COSU' 1I (1-1- 6.:0: )L\ynl\y
- (t.yo + t.y)2 (t.Yo - 4v f .
HMeSi B I3U.ny, '11'0 t!.y « t!.Yo H fly
(/ + fl.r.)a, nOJ1Y'U!M
'J
2EEnS ( 1I (I +' x)L\v = -2c:.. ( V IlII ) - x
llyo 4 v o
x(1 +fl.'"f a,
BW
-
Ba
(5.82)
mc C,. = (I:CoS)/ t.Yo - CMKOCTb KOHlleHcaTO-
pOD npH t.y = 0 .
nonaraJI umpllHY nanbueB rpe6eH1Ja-
TblX CTPYlrryp npHBO,QOB TaKOH, <1T06bI npll
IUVII1 1 11111 ymoBblx ,QBH}I(eHl1i.i no KoopJJ.I1Ha-
Te a H3MeHeHl1e nnOllJ,a.ueH B3aHMHoro ne-
peKpblTI1f1 IlJIaCTI1H npOHCXOAHJIO TOJIbKO 3a
C'ICT KOOpAI1HaTbI Xu =: I + 6.:0:, B COOTBeTCT-
BI1H C )OPMYJlOH. (3.126) 3anllweM
F,? =: -oW / SXo = 4n oCt UoU x sin pI , (5.83)
&\'0
fAe p = (i), ,l\nfl pe30HaHcHoro CJIYlJRfI P JJ,.
I
tu
c
.6.Yo
x
z
.6.y
y
Puc. 5.27. K onpc"-CJlCIIHIO 3J1CKfpOCT8TH'ICCl«JrO MOMellT8
356 rJIaBa 5. H3MEPHTEJIbHbiE CBOI1CTBA MHKPOMEXAHH4ECKHX flPl1EOPOB
C yqeToM :me""TpocTaTli'leCKUX CHn If.. nHweM ypaBHeHlU1 CI1CTeMbI (4.142) B CJIe-
HMe}1 B BU.ny, 'lTO Xo = I + /).X II I» /),x, 3a- .aYlOllJ,eii opMe:
[cf + 2(C 1 +111/2 )!o.+ aa + {G a - G 1 + (A p + 2A 1 )O -
-lBp + 2(B 1 + 1111- )] 0; }a = -4m(Oz + a )f1:i-/;
(5.84)
111 -[(OZ +aT + (ny -Oxaf] I}+bxoA\" + GxoA\'= ma\ + Fr.) sin pt.
,
G - 1 c ( Uon ) - / 2. BC,
rD,e 3 -- S - . Fo = 411-U on U x .
l1yo (5xO
113 nepBoro ypaBHeHIUI CI1CTeMbl (5.84)
c;Ie.J.)'eT, 'ITO MOMeHT. 06YCJIOBJleHHbIH yc-
KopeHHeM KOpHo.'Ulca, npHBo)J,I1T K l\one6a-
ffiU[M Mr no Koop;lImare a. 3JIell."Tpocrnru-
QCCKWI )KecTKOCTb G 3 YMeHblliaeT 3 eKnfB-
H)'lO )KeCTKOCTb nOD,Beca. HaJIH'me nepe"-pe-
CTHhIX yrJJOBbIX cKopoCTei1 Ox, OJ HemmeH-
HbIM 06pa3oM BIIH5IeT Ha 3 e,,-rif.BH)'lO )Ke-
CTKOCTb nOJ1.Beca. 113 BToporo ypaBHeHHH.
CMCTeMbI (5.84) BHAHO, 'ITO CKOpOCTI1 Ox, .0..,
[l" C03D,a1OT ueHTpo6e)KHOe YCKopeHl1e 11
COOTBCTCTBYlOllJ,l1e 11M CMnbl HHepIUIH.
BbInOnHI1M npI16m1>l(eHHOe aHaJIl1Tl1-
'leCKoe uCCjJeD,OBaHHe ypaBHeHI1M CHCTeMbI
(5.84), lIMe}! B BI1.ay, 'lTO yron a - fanWl
Bemf'iHHa 11 0= » 0" .oj'" llpM TaKl1X
ycnoBIISIX Ha6nlO,naCTCSI npI16J1H:>KeHHOe pa-
BeHCTBO: (0., +o.f +( , -o.a n; + , +
+ (20z + o.)a. - If. CHCTeMa (5.84) ITpI1Hl1-
MaeT BH,ll
C:rCi + baa + G!.a = -4mOiM - 4mo./M; }
mlli+bxoM+ Gxotu =mo!. +Fo sinpt +m(20= + 0.)0./,
r,i1e CI: =C p +2(C 1 +mP); G:r. =G a -G 3 +
+(Ap +2 )n;. -[B p +2(B 1 +mP )p;; 0I: =Ox +
+I{o; +0;.).
IIOCJle;IHee CJIaraeMOe B npaBoH <{aCTH
rrepBOfo ypaBHeHIDI MOiKHO paccMaTpHBan
Ka.K rrorpewHOCTb, BHOCHMyro B H3MepeHHe
ymoBOH CKOpocru [l" OTHOCHTCJ1bHOH CKo-
J>OCThIO BpallleHIDI paMKH 0.. I1ocne,llHee
cnaraeMoe B npaBOH 'laCTH BToporo ypaBHe-
MIDI JIBJIJIeTCjJ HHepl.{HOHHOH CHJIOH, TalOKe
06ycJIoBJ1eHHoH yrnoBOH CKOpOCTbIO 0.. 1\0-
TOpM HecICOJU,KO HCKIDKaeT Pt(. MO:>KHO
rrpe.nnoJ1o:iKJfTb. 'no BJIIDIHHe 3THX COCTaB-
mnoIIIHX Y.naeTCJi MHHHMH3HpoBaTb COOT-
Be'IClIs yIOIltHM no.n60poM napaMCTpoB Mf H
pemIMOB ero Hac1pOHKH. B mo60M cJIy'tae
1i)"KH0 BHaq8J1e pe11IHTb CHCTeMY (5.85) 6e3
}"Iera 3THX cnaraeMIoIX, a 3aTCM OneHHTb HX
BJIIDlHHe. nOH,neM rro :rr0M)' nyrH. H B TaKOM
cnyqae BTopoe ypaBHeHlie CTaHOBMTClI He3a-
BHCHMI>IM OT rrepBOfo M MO:>KeT 6bITb 3anH-
calIO B opMe:
(5.85)
A:: 'J!' . 2 A_ Fo .
ilA+ _':>xAX+ffixoilA = o!. +-smpt ,
m
(5.86)
f.n.e 2/;x = b xo 1m; ffi;O = G xO 1111.
PewcHHe ypaBHeHM5I (5.86) HMeeT H3-
BeCTHbIH BH.n:
l\x =e -{'1 { - [!;x(ESiny+ )+ Epcosy] sinffixt-
ffix
-(Esiny+ )COSffixt } +Esin(Pt+Y)+ .
(5.87)
r.ne
ro x = ffi;O -/;; = G xo [ I b;o ) ;
m 4mG xO
E=FO{ m J (ffi;o-p2) +4 ;p2}
E I = [ax +l(O +n;)yro;o; tgy= ;/;xp 2 '
roxO - P
(5.88)
ct>OPMIIfPOBAI [ME BbIXO.[l;HbIX CHrHAJIOB MHKPOrHPOCKOIlOB 357
- D. z 4F O lp COS(pl + y)
CI, [{(O O - p2 r + 4 p2 J[ {(O O - p2 r + 4 p2 ]
I1MCSI B 13l1l1.Y JaryxaHl1e c06cTBeHHbix
KOJIe6aHI1H Mr, a TaK:>KC fIo):{60p napaMCT-
pOB 3J1eKTpOHl1KJ1 11 npI1Bo):{a., 06ccnc'll1-
BaIOLl{l1X HeHTpa..1113aLlHIO CI1:I lI.C mq)l1poBa-
111111 B P,LI" MO)l"IO Orpam1<IHTbCH <IaCTHbJM
pCWeHl1eM
r:= Esin(pl+y)+E I .
(5.89)
CY;lH no )OpMY,le (5. 9), 11M B P,L( co-
BepllialOT rapMOll114eCKI1e K0J1e6aHHH. OTHO-
CHTcnhHO nOnO)l(CHI1H paBHOBeCIHl E I co
ClI.Bl1roM ITO cjJa3C Y ITO OTHOllleHl11O K KO.iIC-
6aHWIM, rCHcpMpYCMhlM I1CTOLJHI1KOM ITCpC-
a=
r.l1e tg(y - 8) = 2 OoP /((O o _ p2 ).
TaR KaK npOI13Be):{eHl1e a i,: B ncpBoM
ypaBHelll1l1 CHCTeMbl (5.85) npl1BO,llIiT K no-
HBJIeHl110 B03MYllJ,aIOLl{ero MOMeHTa Ha
,ltBOHHOM 'laCTOTC p, TO COOTBeTCTBytollJ,8JI
CMY COCTaBnHIOIU8JI CIirHaJIa no yrny a Mo-
)I(eT 6blTb OT HnbTpOBaHa Cpe,llCTBaMl1 3IIeK-
TpOHllKI1.
Y'II1TbIBM MaJIhlH nopMOK BeJIH4J1H
x, a' 3aITHllICM ITpl16JIlI>KeHHOC BbIpa)l{eHl1e
4KM.yFo/D.= COS pI ( 5.92 )
a ] '
CI,P
r):{e K M . y = 1/[(1- (0;0 I p2 ){1- (O o I p2 )]
MexaHH'leCKHM K03 l1I.ll1eHT "ycMJIeHI1J1"
mpOCKOITa.
B COOTBCTCTBHI1 C opMynoH (5.92)
.nnSi nepBoro :nana npe06pa30BaHHH BXO.n-
HorO CI1I1uwa nony'IHM K03 q)l1IJ.HeHT nepe-
.na'IH:
a I D. z = (4FolKM.y )/(C1: p J ).
113 pHC. 5.27 cne.nycT reot.urrpHLJeCKHH
K03 iPHLll1eHT nepe,l\a<m:
D.y I a. = I .
MeHHoro HanpIDKeHI1H U x .
AHaJIOrM'IHO ypaBHcHMIO (5.86) Ii npl1
Tex :>Ke )J,onYLl{eHI1S1X nepenHweM nepBoe
ypaBHCHIie CMCTCMbI (5.85) C y'leTOM cjJop-
MYJIbl (5.89):
ci+2 c,cH(o 0 = D cos(pt+y), (5.90)
C},;
')
fJl,e 2 a = bOo I GI,; (O o = GI, / CI,; D =
= -4mD.)pE.
11MCSI B BI1,lty 3aryXaHHC C06CTBCHHbIX
Kone6aHHH, 3alIl1111CM LJaCTHOe pellleHl1e
ypanHeHI1J1 (5.90):
,
(5.91 )
Ecnl1 npl1HHTb, LJTO KOH.ncHcaTophl
Ct, C2 BKJIIOLJeHbl B MOCTOBYIO cxeMY, Ha-
ITpl1 Mep aHaJIOrWIHO pl1c. 3.18, TO, I1cnOJIb-
3YSl OPMYJIY (3.57), nonyqHM
tJ.U I tJ.y = U on I 24vo .
C BbIXo,lta 3JJeKTpOHHOM cxeMbl C K03 -
)J1Lll1eHTOM YCHJIeHHH Ky 6y):{CT nonyqcHo
BbLXO,ltHOe HanpSI:>KeHl1e UgJ.lx = KyD.U .
06'bell.11HHM npl1Be,lteHHbIe 3BBHCMMO-
CTM I1 nOJI)"II1M MacIllTa6HhIH K03 qlHIJ.HeHT
ITepe,[laQI1 Mf, B/(I1c):
U gblX lD. z = (2FoPUonKM.yKy )/(C1: p J D.Yo).
(5.93)
OLJeBH,[lHO, QTO KpyrH3Ha xapaKTCpHC-
THKH HeJIllHeHHbIM 06pa30M 3aBHCHT OT pe-
:>KI1Ma HaCTpOHm rHpOCKOITa (CM. BbIpa:>Ke-
HHe.l{JlSl K M . y )'
PacCMOTpHM Tenepb cxeM)' Mf no pHC.
4.10, B KOTOpOM pq - 3TO nOC'I)'IlaTeJIbHOC
,l\BI1:>KeHHe 11M nepneH,llMKymlpHo K IIJIOCKO-
cm, B KOTOpOH ocyllJ,eCTBlliIeTCH P,L\, a ne-
pCMellJ,eHHe no Koop.nHHaTe a. OTC}'TCTByeT
(a. = 0). YpaBHeHl1e (4.133) .ZJ.IDI :flora cny-
LJWI npHHI1MBe-r BI1)J,
358 fJIasa 5. H3MEPHTEflbHbIE CBOHCTBA MHKPOMEXAHI14ECKHX flPHEOPOR
20J l1p +2m)(1:i'+x.t)+2m(.i'OYO +iOxO)] +
+ ( + 2m)(1);- .\i') + 2m(\"oYo - Yoi'o)=
aw
=Ma--'
00.
)IaHHOe ypaBHeHHe MO:>KeT 6bITb nc-
nO_1J.b30BaHO Ai"U! BbI'/HCJIeHliSi .lJ;liHaMu'IecKUX
peaxuuH B 3JlCMCHTax l\.-penj}eHItSi paMKH C
KopnycoM npll pac'/eTe HX Ha npO'lHOcTb.
nOJIanUI, 'ITO MaJJhle npHpamelHHI
x, y 3a C'leT ynpyrocTH paMKI1 3Ha'lIlTCJIhHO
MeHbllle TeKYllJ,lfX KOOpII,liHaT xo, Yo 11M
Jm60 paMKa BbInOJIHCHa raKoH )l(eCTKOCTM,
'lTO \" = J' = O. cnCTCMY (4.137) HCKJIlO<IMM
113 paCCMOTpeHI1S1.
I1MeSi B BH)J,)'. 'lTO Xo = I + t'\,X n
/ » I1x, a Yo = 11)', 3anMllleM CI1CTCM)'
(4.138) B BI1):{e
mo.\' +b...oAi + Gxoo.\" = m[a x + /(0; + O . ) + o..n,.t.y]+
+ Fo sin pi + 2m!lj>D. z ;
(5.94)
m!J..) +byo41' +G.,oL'1.y = mk + (0; -;- o; V-OxO)'l X]+
+F -, + 2m&[1",
r):{e Fo OnpelleJIHeTC5I B nORCHeHHII K CHCTeMe
(5.84), a 3!1elrrpoCTalwlecKWI CIU1a npMTIDKe-
HIDI JLlacnrn KOH.1leHcaTOpoB - B COOTBCTCT-
BIm C $oPMYJIOH (3.85): F 3 )' "" (£C:oSU n)1
(24V ) , (l1yo- paccTOSIlllie MeiK1IY nnaCTMHaMl1
113Meplfre.iIbHhIX KOH.1leHcaTopoB, t.Yo»L'1.y).
ECJIH, KaK I1 B npe.nhI.nyrn:eM cJI)"Iae,
HMeTb B BU):!)', 'ITO npHMeHeHlle Mr npe):{-
nOJIaraeT n" » Ox, Po. H C'IHTaTb, "ITO npH-
BOJ1 06ecne'IHBaeT BLInOJIHeHlle YCJIOBHR
!Fol » 2mL'1.j>D.::, TO nepBoe ypaBHeHnC CI1C-
TCMbI (5.94) CTaHOBMTCSI l1)J,eHTI1'IHbIM YPaB-
HeHHIO (5.86) H )J,JlR Hero cnpaBe.nnl1BbI pC-
llleHH5I (5.87), (5.89). I1CITOJIb3yeM peweHue
(5.89) 11 nepeITl1111eM BTopoe ypaBHeHl1e CHC-
TeMbI (5.94) B BI1):{e
4v + 2C; yo L'1.y + w;,t.y =
(5.95)
= (Jr.)' - EL sin [(pt + y ) + v],
r):{e
2 ,)o{) = byo 1m; <0; = w;o + (0; + 0;); w;o = G yO 1m;
aIy =a y - Q...OyE I +F'3J' 1m; E ,El;y -CM. )OPMYJIbI (5.88);
L = 4(O PY- + (O...Oyr ; tgv = (20 p)/(OxOJ
3anHweM fJacTHoe pemeHHe ypaBHeHIDI
(5.95).!J,IDJ H)'JIeBhIX, HafJaJIbHbIX ycnOBH":
a}; ELcosS. r/_ ) ]
y = --f + 2 2 SlDLV't + 'Y + V , (5.97)
my P - w y
rJ:(e tgS = 21;yo 1(P2 -<0;).
(5.96)
I1MeSi B BH.a)', 1.JTO yO 0, v 90°,
90° C y'IeTOM )OPMYJ] (5.87), (5.96) pe-
weHHe (5.97) npHHHMaCT BHn
aIy + 2F o Oz cos(pt + 'Y )
Ay <0; mp'(l- : )(1- : J . (5.98)
ctJOPMHP013AIIHE BbIXO,lJ,HbIX CIITHAJIOB MI1KPorl1POCKOflOB 359
O'leBH)J,HO, 'lTO rapMOHH<IeCKaSI co-
CTaBJ1H.lOll\aSI BblpIDICCHHSI (5.98) cjJI1311<ICCKl1
II.neHTl1'lHa BbIpIDIceHHIO (5.91). <1>opMaJIbHOe
ux COBnaACHl1C MO)l(HO nOJIY'lI1Tb, CCJII1 no-
JIO:>KI1Tb C = 2mP H fly = 1/ a. (-13 )OPMY-
JIbl (5.98) CJICAyeT, <ITO I1MCCT MCCTO 1l0CTO-
SlHHoe CMellJ,eHI1C 11M, HC 3aBI1CSlll\Ce OT H3-
MepSieMoM yrnoBoH CKOpOCTlI, KOTopoe ITpl1-
Be):{CT K nOSlBJ1eHlUO JIO)l(Horo Cl1rHaJIa Ha
BbIxo,ne 3JIeKTpOHHOH CXCMhI. O',eBH)J,Ho,
'lTO TIOT Cl1rHaJI MO)lCCT 6bITb OTcjJl1JIbTpOBaH
OT nOJIe3Horo CllrH8.'Ia.
AHaJIorH<IHo npe;':\bI)J,yllJ,cMY MO)l(HO
ITOJIY<Il1Tb BbIpa)KemlC ,lL'1S1 MaClllTa6HOrO
K03 cjJHUI1CHTa Mr:
UB IX / n z = (FoKMVK),U 0" )/( mp 3 flyo),
(5.99)
r,ne B BbIpa:>KeHHI1 lIJI K M )' BMeCTO BeJ1l1<m-
Hbl waD HYiKHO 3aITl1CaTb W y .
YpaBHeHH CI1CTeMbI (4.138) npllro.nHbl
)J,JI cxeMbI Mr no pl1C. 3.8, I1MeSi B BJ1,I\)',
liTO CTpYK'rypa Mf pacnOJIO)lCeHa B nJJOCKO-
CTl1 xy. JISI Tex )Ice YCJIOBl1H, npn KOTOpbIX
3allnCaHa CHCTeMa (5.94), nepBoe ypaBHeHHe
:nOH cnCTeMbI, Onl1CblBaIOll\ee P , oCTaeTCSI
6e3 113MeHeHHH H )J,JISI paCcMaTplofBaeMOH
cxeMbl Mr. '3JIeKTpOCTaTl1',eCKHe CIDThI Jt1ISI
3TOM CXCMbI pacCMOTpCHbJ B no):{pa3.n. 4.6.
TIYCTb BbIITOJIHeHO YCJIOBl1e, npH KOTOpOM
KOMneHCl1pyCTCSI KBa,llpaTYpHaJI 01ll116Ka.
BOCnOJlb3yeMcSI MaTpI1I(eH K03 ) MI HeHTOB
:>KeCTKOCTI1 (4.170) I1 BTOpoe ypaBHeHl1e CI1C-
reMbI (4.138) .nJ1S1 P4 :mITl1111eM B BH,lle
mflJ' + hyoflY + G y =
= mk + (0; + O )L\y - OxOyL\x]- 2m&Oz'
(5.100)
r,ne G) = G yO - C O2 (U6 + U; + U;) - 3 -
flyo
eKTHBHaJI :>KeCTKOCTb nO):{Beca.
YpaBHeHl1e (5.100) nepenMIlleM B
cjJopMe
L\Y + 2 yOLly + W Lly =
= aE)' - EL sin[(pt + y )+ v],
(5.101)
r):{e
W; = w;o - (n; + n;)- Co(U + V + v; )/ 1ILlY );
a )' = a y - nxnyEI; BeJIll'/HHbI wyo, E, E" L, y, v
(5.102)
OITpe,lleJISIIOTCSI no 3aBHCHMOCTSIM (5.88), (5.96).
YpaBHeHl1e (5.101) ITO )OpMe H):{eH-
TI1</HO ypaBHeHl1lO (5.95), 11, cne.nOBaTCJIbHO,
,nJJSI Hero cnpaBe)J,JIl1BO perneHl1e (5.98).
nOJIY'/HM BbIpa)l(eHl1e ,nJl MacIllTa6HOrO
K03 ) J1l.lJfCHTa Mr, nonarWl, 'ITO 113Mepl1-
TeJIbHble KOH,l\eHCaTOpbI BKJIIO<JeHbl no cxeMC
3.55 (napa3I1THble eMKOCTI1 yql1TbIBaTb He
6y.neM). 113Mepl1TcJlbHaJI UeITb Mr BKJII01.JaCT
B ce6H TaKytO nOCJ1e):{OBaTeJIbHOCTb ITpe06pa-
30BaHI1H: Oz Lly flC V" VB IX .
BeJIH1.JHHa Lly Ollpe,l\eJIllM H3 Bblpax<eHIDI
(5.98). BbI1.Jl1CJTHM 1l3MeHeHHe eMKOCTH:
LlC=C t -C 2 =
BBOS
Llyo - Lly
l1pH S = const.
HMeSi B BH,ny, qTO Llyo» fly, nony-
<JHM LlC - 2C o Lly , r.ne Co - Ha1JaJIhHoe 3Ha-
Llyo
1JeHlle 113MepHTeJ1bHbIX eMKOCTeH.
Hanp}J:>KeHHe Ha BbIxo.ne HHTerpaTopa:
VI! =Von(CI-C2)/CH =VonLlCICH"
BEOS
Llyo + L\y
EE o S2Lly
Lly - Lly2
360 rnaBa 5. H3MEPIHEJIbHbIE CBOHCTBA MHKPOMEXAHI14ECKHX flPHEOPOB
Ecnll HMceTCSI )J,onOJIHMTeJIbHOe YClfne-
HHe, TO U 8m = K yU 1\' TaKHM 06p830M.
MaclllT86HbIH K03 ) I1UlfeHT Mr, B/(l/c),
onpe,l\CJISleTCR BLlpa:>KeHHeM
U Bbl..... _ tJ.y tJ.C U 1\ U BbI....
Q_ Q= tJ.y tJ.C U 1\
4FoKM}.K,D on Co
mp3CI\tJ.yo
,
(5.103)
I".l!.e B K03 q)lfU:HeHTC K 9 BeJ1H'lHHY CO aD
H)'JKHO 3aMeHlITb BeJIlJ1IHHOH co, lt3 Cl1CTeMbI
(5.102).
AHan.orlt'iHO 1f.3.J10)KeHHOM)' MOryr 6bITb
onpe;J.enelIbl K03 HUlleHTbI nepep:a'/ 11 1f3-
MepMTe..'lbHbIX u:eneii npmlOro npCo6p830Ba-
HIDI "Ul m06LIX nmOB Mr.
flpu.uep 5.16
Pacc<nrraeM napaMerpbl Ii xapaI..-repHCTI1KH
Mf c O;lHOH If.M..
CxeMa Mf c c06mo.neHHeM Macrrrra6a OT-
HOCHTe.'U.HbIX pa3MepOB 11M Ii 3JIeMeHTOB ynPY-
roro no.D.Beca DpHBelleHa Ha pliC. 5.28.
y
c ..
I (j"3M
Mr npe.nHama1./eH ;:I.nSl 113"lepeHlISJ YI'J/OIJOU
eKopoCTIi Q., )].1I3na30H H3MeHeHHJI KOTOPOU
6y.neT onpe.neneH B npoLlecee paC1./eTa. J Jonepe'l-
HbJe yrJIOBble cKopoeTII H nocrynarCl1bHhle yeKo-
peHI1H. OCHOBaHlUl oTcyreTBYIOT.
l.J:3 11M I BblnOnHeH B BJ1,ne eTep)lmeBOll
CTpY1'-"T)'pbl. KOTOpIDI BKJlIO'l3er B ce6S! TaK)I(e
BoceMb n&lbLleB KOH.neHcaTopoB 'lYBcTBIITej]bHo-
eTIi 6. l.J:3 B P4 nepcMewaeTcH B H3npaB!IeHlIII
OCII Y no.n .neHcTBHeM cun I1HepLllll1 KOpHol1l1ca
Ha 1./eTblpeX TOpCHOHax 1 OTHOCHTeJlbliO )I(eCTKO-
ro 3.!leMeHTa 4 II Kopnyca npu6opa. SCSI CTep>KHe-
BM CTpY1'-"T)'pa, B TOM 'lHCJle 3neMeHTbl 4, H3 'le-
nlpex TOpCHOHax 3 B P.D: nepeMeW3ercSI B Ha-
npaBJ1eHHH OCH X c nOMOWblO KOH.lleHCaTOpoB
.nBIDKeHIDI 5. f\1r 1t3rOTOBneH H3 KpeMHHS! C na-
paMerpaMH
EIIIO] = 1,68.10 11 HlM 2 ;
G[110] = 6,17 .1010 HlM 2 ; P = 2,33 . 10 3 Kr/M 3
Pa3Mepbl
Beex
CTep>KHeBbIX 3neMeH-
10- 3 .2.10- 5 x
TOB npl1ID1TbI O.nl1HaKOBhIM I:
x 2 .10- 5 M, TaK >Ke, KaK Ii pa3MepbI naJIbLleB KOlI-
3
fpaHHLlbI
npH60pa
Cl
x
.. .
oc.
ox
PRe. 5.28. PaC'IeTlf&H cxeMR Mr:
1- crepJKH eBIUI 11M; 2. 3 - ynpyrne :meMeHThI (TOpCHOHIoI); 4 - CTep>KHeBOH :meMeHT;
5 - KOH,D;eHcaTophI ABH7ICeHIDI; 6 - KOH,IleHCaTOplol 'lYBCTBHTenbHoCTH
{J)OPMHPOBAHHE BbIXOLLHblX CHfHAJIOB MHKPOfHPOCKOnOB 36)
)J.eHcaTopOB 'IYIICTBHTeJ1bHOcTH: 23.10- 5 x
x 2.10- 5 x 2.] 0 5 M. Pa3 1CpLI nanbucn 1<0Hl\eH-
caTOpOB JIBHiKCIIWI 0,5 . 10 3 X 2 . 10- 5 X 2 . 10- 5 M.
3a30p MeiK)J.Y llJ1aCTHH8MI1 KOH)J.eHcaTopOB
Yo == 2.] 0- 6 M. AMnmrrYl\B I<OJIe6aHlII1 11M B
P X () == 10- 5 M. 4BCTOTa reHepBTOpa, o6ecne-
5
<JHBalOlliero P,lJ" P == 10 pa,L\/c.. HanpH.>KcHHe Ha
1<0H.iJ.CHCaTOpBx J1BI1>KeHHH U UII ==] 0 B. B HB KOH-
)J.eHcaTopax <JYBCTBHTCJIbHOCTlI U 0 ==] B. Macca
43 m'l == 5 - ] 0-'1 Kr. B P K Macce q3 )J.06BBJ1HeT-
CH Macca :3lIeMCHTOB 4, T.e. m;! = 6.5 . 10 'I KL
Pac<JeT BbmO.JHH 1 B C:JclIYlQllIeH o<Jepe.u-
HOCTl!.
I. OnpeJJ.eJIHM p83Mepbl TOpcHOHOB, HCXO,WI
H3 pC30HBHCHbJX <JaCTOT: P,lJ, w xu = 0,9 p ==
= 0,9.] 0 5 pa,L\/c: P4 Wyo == I.Iwxo.
Pe30HBH CHaJ! <JBCTOTB Jf.JU! P (j) xO ==
== 2 G xO I 1n;:J , oTKy.ua >KeCTKOCTb onHoro TOp-
CHOlla B HBnpaBjlellHlI ocn X
2 -9 { 5 \2
_ mllwxo _ 6,5-10 t o ,9.1O } = ]3 2 HI
G x3 - 4 - 4 ' M.
TaK KaK G x3 =E[IIO]b n (c n Il3Y' TO, nOJIB-
r<UI nIU! Bcex TOpCHOHOB nmpHHY b n == 2.10- 5 M,
a TOJIIlIHHY c n == 10- 5 M, Bbl<JHCJIHM n!JHH)' TOp-
CHOHa 3:
13 ==cn (E[1I0]bn)/Gx 3 =
==1O- 5 V (1,68.10 11 .2-10- 5 )/13,2 =0,64.10- 3 M.
AHaJIom 'lHO .uJ1J! P4 HMeeM wyo ==
== 2 GY2 I m'l , OTKy.ua >KeCTKOCTb onHoro TOp-
CHOHa 2 B HBnpBBJ1eHHH OCH Y
_ m'lw;o _ 5.10- 9 (1 ,1.0,9.10 5 f _
G 2 - -
.v 4 4
== 11,75 HIM.
BbI<JHCJ1HM .D.JIlIHY OAHoro TopcHOHa 2:
1 2 =c n V(E[1I0]b n )/G Y2 ==
=1O- 5 V (1,68.10 11 .2.10- 5 )/11,75 = 0,66.10- 3 M.
2. PaCC'IHTaeM H3MeHeHHe eMKOCTI1 KOH-
neHcBTopOB 'IYBcIBHTeJ1bHOCTH npH nepeMellle-
HHH II3 B HanpaBJIeHlIH OCH Y.
IlpH OTCyrCTBHlI nBlI",eHH)! eMKOCTb KBj!{-
noro 1I3 KOH,LIeHCaTOpoB 'IYBCTBHTeJIbHOCTH
(0=21.10- 5 M, c==2.10- 5 ,E== I):
.p 5 5
C == EnS == 8,85.10 -.21.10 .2.10
0.'1 Yo 2.10- 6
=0,02.10- 12 {J)(O,02n{J)).
KmK)\<UI rpe6eH'IBT<UI crpYKTypa KOHlleHca-
TOpOB <JYBcTBlITeJ1bHOCTH CO.uePiKHT <JeTblpe eM-
KOCTH (no nBe e IKOCTH C I H C 2 ). Bcc eMKOCTIf
C I BKJIlQ<JeHbI naparrJ1eJ1bHO llpyr C npyroM (Bce-
ro HX 8). EMKOCTH C 2 TaJOKe coenHHeHbJ MeJKlIy
c060i1 napaJIJIeJIbHo. TaKHM 06p830M, upn nepe-
MellleHHH 43 B HanpaBJ1eHlIH OCH Y H3MeH)!eTCH
eMKOCTh, {J), KOH,LIeHCaTOpoB '1)'BcTBHTeJIbHOcTH:
C==8(C1-C2)=8toS ( I _ I )
.6.Yo -.6.y .6.)'0 + y
16ES.6.y _ 16C oq .6.y= 16.0,02.10- 12 .6.v=
.6.Y5 .6.yo 2 .10- 6 .
= (16.10- 8 )t,y.
OrnOllleHHe .6.C l.6.y = 16 .10- 8 {J)/M 6yneT
HCnOJIb30BaHO npH BbI'IHCJJeHlIH MaCIlITa6Horo
K03cpQJHltHeHTa Mr.
3. OnpelIeJIllM YMeHbllleHHe co6CTBeHHOH
<JaCTOTbl Mr Jf.JU! pq 1f3-3a 3JJeKTpOCTaTH<JeCKoro
npHT)!)!(eHlI)! Me>K.L\y nJ1aCTHHaMlI KOH,LIeHCaTOpOB
'1YBCTBHTCJJhHOCTlI. BOCnOJIb3yeMcH cpOPMYJIOH
(3.122) H nOJ1yqHM:
O>yo:: {2G Y2 -8 ;; UJ )tm q =
8.002.10- 12 2
- 2 2.11,75 ' 1 15.10- 9 =
(2.10--6 r
= 0,97 . IO s pa.Wc,
T.e. Ha<JaJIhHIUJ <JaCTOTa wyo = 0,95.loS pa,n/c
YMeHbiliaeTCH Ha 2 %.
4. Bbl<JHCJIHM .Q06pOTHOCTb Mr ,/J,IDI P,l:( H
pq.
362 rnaBa 5. 113MEPI1TEJIbHbIE CBoflCTBA MI1KPOMEXAHl1lIECKMX flPI1EOP013
BHa'laJIe HafuJ.eM K03cpqnuIHeHT ,lleMncpHpo-
BaMIDI CTpYl\.1YPbI Mr. ):{el\mcpnpoBaHlIe oC)'lQe-
CTBJUIeTCH. nyreM Bjj3Koro 1peHHjj r3.3a, Ha.XOllJl-
werOCjj B 3a3opax Me>I\.llY IqJbIlllKaMIl Kopnyca Ii
nOBepXHOCTI!MIi nO.!U!Ii»rnbIX 3J1eMeHTOB, II n)'TeM
CIKaTIUI r3.33 MeJl\.i1Y TopuaMH naJIbUeB KOH,lleHCa-
TOpOB H 3J1eMeHTaMH KOHCTpYKUlIli. CTpy"typa
Mf CO.nepIKHT ceMb CTepIKHeBblx 3neMeHTOB.
BoceMb naJIbUeB KOH1l,eHcaropoB 'i)'BCTBHTe!1bHO-
CTH H 'leTblpe na.'lbUa KOH1l,eHcaropoB ,llBHIKeH1U!.
CKO.ilbIKeHHe npoHcxonHT C nB)'X cropOH CTpYK-
1)'pbI. OowW! nnOW3,IJb CKOJ1b)KeHIiSi
S = 2[7(10- 3 . 2 . 10- 5 ) +
+ 8(23 . 10- 5 .2. 10- 5 ) -:- 4(0,5 . 10- 3 . 2 . 10- 5 )] :>:
:>:4.10- 7 !>f2.
JJ,JIJI .naBJ1ellitJ1 ra30BOM Cpe.nbl B Kopnyce
p = I Topp = 1,33 . 1 0 2 H/ HafuJ.eM
b = pSp= 2,776.10-6. 40 . 10 - 8 '1,33.10 2 =
= 14,76 . lOll IJ/C.
Bbf'iHCilHM lleMncpHpOBaHHe 3a C'IeT C)KaTIDI
H nQC.;"Iell)'lOwero nepereKaHlUI ra3a B TOPUax
na:IbueB. npe.zlO0!10xmM, tfTO MHHHMaJIb-
HbIH 3a30p B pai10He TopueB h = 2 .10-6 M. 06-
uaaR nnoru8nb Bcex neCjjTH TOpUeB
S""10(2.1O- 5 '2.1O-5)=40.IO-JO . Ilo cpop-
M)'!Ie (4.94) DOJI)"lHM
k:r. =( pS2 p)/ h 2 =[2,776.10--6(40.10-10)2 x
x 1,33 .10 2 ]t(2 _10- 6 )2 = 1,48.10- 9 Kr/c.
()fu.e;ufHHM 06a pe3y.:n.TaTa H HafuJ.eM a6-
CO.IIJQTHbIH K0341qJHUHeHT ,lleMnqIHpoBaHIDI .IIJljI
P):{:
b x =(14,76+148)10- 11 ",,162,76.10- 11 Kr/c.
BbI'IHCJIHM 2S x =bx/(m.!lOJxO)= 162,76 x
xIO- Il / (6,5.10-9 .O,9.lO S )=2,78.1O...{; H nony-
'lHM JIo6porHOCTh B P JJ.:
Qx = 1I2S x = 1,8-105 .
Pa.conrraeM JI06pornOCTh JIJJjI PlI. Ilno-
1I1a.r:tb nOBepXHOCTH .llByx CTOpoH crepJKHeBbIX
3JIeMeBTOB q3 Ii BOCIoMH DaJlbUeB KOHneHcaTopoB
'i}'BCTBJITeJIbHOCTH:
2[5.10- 3 .2.\0-5 + 8(23 .10- 5 .2 _10- 5 )]:>:
-? 73 10 -7 2
........... t .
flOllY'II1M b = 2,776 . 1O-{; . 2,73 10 7 X
x 1,33 . 10 2 :>: 10- 10 lIT/C. B BoebMH 3a30pax. Me)K-
Ay 60KOBbIMH nOBepXHoenlMH nanbl.leB 06lueH
IlJIOUl3ll.blO, pillmon S = 8(23 . 10- 5 . 2 . 10- 5 ) =
= 366 . 10- 10 M 2 , npOlICXOllRT CjJ(aTlle \-I nocne-
A)'lOwee nepeTeKaJme raJa 'Jepe3 3a30pbI
11 = lI.)'2 = 10-6 M. flOllY'II1M K03cpcpHI\HeHT
AeMncpHpOBaHlIJI:
k;1 == (2,776 .10-6.(366 f .10- 20 .},33.1 O )I
(1O-6)2 = 4,95 _10- 7 Kr/c.
CnenOBaTeJIbHO,
by =b+k ll :>:4,95.10- 7 Kr/e
AHaJlOm'JHO npenbl.l\yweMY BbI'JHClIlIM
2 \. = b.\./(m",royo) = (4,95 .10- 7 )/
(5-10- 9 .0,95.10 5 )=10- 3
H, 3Ha'JHT,
Qy =1/2 y =10 3 «Q".
5. OnpeneJ1l1M K0341cplllll1eHTbI nepena'JlI
Mf .IIJljI pa3nll'JHblx peiKl1MOB HaCTpOHKlI. HMejj B
BHlIY, 'no X = X 0 sin pt 1I, ClIenOBaTeJIhHO,
Fo = mX = mX op2, BOCnOllb'3yeMCg 3aBHClIMD-
CTblO (5.98) H 3aIDfilleM Bellll'JnHY aMnllHTYlI bI
11M B PlI:
A 2F o Oz
LJ.y-
np'('- : )(1- J
2X o O z
p(1-e 2 ) [1-e 2 (OJ y o /OJ"oY]
nOnO)KHM e = OJ ,,0 I P = 0,95 H nOllY'JlI M
BbIpIDKeHHe JlJIg K03CPclJHuHelffa nepelIa'lH:
c1JOPMHPOBAHJ1E BbIXOL(HbIX CJ1rHAJ10B MI1KPOrJ1POCKOIlOB 363
i\.\
D.
2 10- 5
10 5 (1 - 0,9) [1- 0,9(CJ)I'0 / wro f ]
O'iCBI'IlIHO, MaKcHMa:lbHbIH KO:J.p.pmmeHT
ncpe"a'-m 6y.ueT npl'l wvo'" 1,054w x o. 3H8'ieHIUI
I\y / D. z )J.ml pa3J1H'-lHbIX pe>KHMOB HacTpoiiKIl
npHBe)J.cHbI B Ta6J1HIW.
WyO" cO\(J 0.\ 0.3 0.5 0.7 0.9 1.0 \,05 1,054 \,08 1.1 \.3 1.5
v
IO
n" ' 0.206 0,223 0.264 0,366 0.756 2,05 26,45 2329.5 75.09 -2.3 -0,29 -0,\64
!'.I/(I/c)
HCJ1l1'iI'lHa ((JyO = I,08m y Cl' KOTOpOH COOT-
BC rCTH)icT ,IlOCT8TO'iHO 60!ll,lUOii K03cpQJllUUCHT
YCIIJ1CmUI H B K01'OpOH He U3MCHSIerCR cpa3a Bbl-
XO,IlHOfO Cl'lfHaIJa, 6mnKa K peKOMeH,!.I)'eMOMY
pe>KIIMY HaCTpOHKH: ((J vO = 1,lw...o .
DOJ1bIllOH K03cpcpl1UlICHT YCI1J1CHI1JJ MO>Ker
npl1BCCTl1 K nCpeMCl11eHHJlM y, C0l13MCpllMblM C
3a30pOM Yo. 1.J1'O He,IlOnYCTlIMO. TaK K8K 6y)J.yr
npOJlBJ1HTbCSI HCJ1HHeHHOCTH B 3neKrpOCTaTlI1.Je-
CKHX crpYKrypax.
6. PaCC1.JHTaeM Kpym3HY pa601.JeH xapal-..'Te-
pHCTlIKlI Mr. nOJI8rM, 'no ICn0J1b3yercR CXCMa C
HlITerpaTopoM. C1.JHTaii eMKOCTb CII '" 10- 10 <IJ,
nOJ1YfJaCM
UII/ C=Von/CIl =1 B/IO- 12 <IJ= W1ZB/<IJ.
flYCTb o6ecnC'IlIB8erCJI K03cpcpHIUleHT YClI-
nemnl V BW( 1 VII = 1 0. B bl6epCM pe>KHM H8-
CTpOHKH
)J.JU!
KOTOpOro
Wyo=Wxu,
ylo.z =2,05.10- 8 MI(I/c) H B COOTBeTCTBlIH C
cpOPMYJ10H (5.103) BbI1.JHCnHM:
VB!.IX = y C VII V OLJX _
o.z D. z y I:1C VII
=(2,05.1O 8) (16.10- 8 )10 12 .10=32,6,10- 3 B.
Onpe.neJIlIM M8KClIMaIJbHYJO CKOpOCTh o.z.
npll KOTOpOi1 l:1y=O,ll:1yo =2.10- 7 M. 11301'Ho-
llIeHHH ylD.z =2.10- 7 Io. z =2,052,10- 8 HaxO-
)J.HM o.z mnK ,., 10 pa.nfc.
113 npHBe)J.eHHoro MaTepMaJIa CJ1e)J.yeT
onpe)J.emllOllJ,ee BJ1HRHHe Ha BblXOIlHOH Cllr-
HaJI Mr K03c}Jc}JMI..\HeHTa nepe,llaqM nepBoro
3BeHa 113MepllTeJIbHOH QenH, KOTOpbIH onpe-
)J.eJ1S1eT peaKUlUO 3 Ha BXO}::{HOH ClirHaJI -
H3MepMeMYlO yrJIOBYIO CKOpOCTb. PaCCMOT-
pHM 'laCTOTHbIe xapal<TeplICTllKH Mr. 06pa-
TllMCM, HanpMMep, K ypaBHeHMlO (5.101).
I10JIO>KllM .or I1 (HJIll) .0)' paBHbIMI1
HyJIIO, O)xO = p, B COOTBeTCTBIiM C CMCTeMOH
(5.96) BbIqlICJIMM 'Y 90°, L = 2.o"p, BBe-
)J.eM onepaTOp S = d I dt M 3anHilleM ypaB-
HeHHe (5.101) B onepaTopHOH c}JopMe:
( ) 2 ( .....!... 2 21;)'0 IJ \ _ 2F o o.As)
!y S ffiy 2 s + 2 S + - ( ) ,
W v W y "II?
- mp- ---,S- + 1
p-
OTKY,lla nOJIyqHM qaCTOTHYI{) nepep:aTOqHYI{)
c}JYHKUllIO
w( ) _ l:1y(s) _
S - o.z(s) -
KO
2 "I ? 2 '
(T[ s- +1)(TiS +2I;T 2 s+1)
(5.104)
"I
rp:e KO = 2Fo I mpO)j,; 1\ = 1/ P; T 2 = 1/ <u)';
I; = l;yO 1 W .
ITpn HaJIM'lHJi )J,a)l(e O'leHb MaJIOro OT-
HOCmeJIbHOrO K03c}Jc}JHQlIeHTa AeMnc}JHpOBa-
HJiM 1;..,0 '* ° KOHCepBaTMBHOe 3BeHO
I I(T./ S2 + I) CT8HOBHTCR KOJIe6aTeJIbHbIM.
364 rnaBa 5. H3MEPHTEJIbHhiE CBOHCTBA MHKPOMEXAHH ECKHX rIrHEOPOB
L(ro)
20 19 Ko
o
o
-90
-18d'
q>(ro }
PRC. 5.29. 'laCToTHble xapaK'TepHCTUKIl Mr
06llU1H BHJ1 a mmrry11Ho- H <lJa3o'la-
CTOTHblX X8paKTepHCTHK, COOTBeTCTBYIOJ.l.lHX
I}>YHKl1.HI1 (5.104), nOKa3aH Ha pHC. 5.29.
CJ1e.nycr OTMeTllTh, 'iTO 011HHM H3 nep-
BblX HCCJIe,!tOBaHHe ,!lliHaMHKH pOTOpHblX Mf
RR-rnna B OTC'lecTBeHHOH JlIITepaTYPe 6br-
!IO, BHJUlMO, BbInO.:IHeHO B pa60Te [23].
E. MHk-porHpocKoDbI RR-THD3
PaccMOTpHM 061IUfi1 cny'lai1 I13MepeHIDI
nepeMeHHoH CKOpOCTH, nOJIaraH. HOpMHpO-
BaHHYJO no CKOpOCTb n;r:::: COS(V/) )J,JISI
Mr RR-TMna no pMC. 3.11. YpaBHeHHe
(4.110) B 3TOM cJIY'fae npHHHM8eT BH,l\
.. . 2
P + 2;pCl)jWl3 + WIIO ::::
== H 0 Ox sin (PI }cos( vt):::: Kw;,
J II
r..ne K = Ho ; CI) ::::OJt sin(pt)cos(vt)
J p
MO,n}'JIHpoBaHHbIH BXO,lUfOH CHfH8J1.
(5.105)
YpaBHeHHIO (5.105) cooTBeTcTByeT ne-
pe.naTO'lHaH <lJYHKllHJI
W(s)= l3(s) :::: ., K ., . (5.106)
ro;! (s) s- + 21;propos + roj3o
TIepenHllleM ypaBHeHlIe (5.105) B BlUle
.. .?
13 + 21;propoP + roj3o ::::
:::: K no [sin(p-v)t +sin(p + v)t].
2
(5.107)
COCTaBJIHlOllJ,aH pellleHHSI ,llJ1S1 nepBoH
60KOBOH 'laCTOTbJ ro) = p - v
13. :::: E. [sin (ro)1 )COSqJl + cos(rorl )sin qJl],
(5.1 08)
a )J,JISI BTOpOH w2 = P + v
132::::: Ez[sin(ro 2 t)cOSqJ2 + COS( ro 2 / )sinqJ2] ,
(5.109)
ctJOPMI1POBAHYIE BbIXO,[{HbIX CHfHAJIOB MHKPOfl1POCKOflOB 365
l.i1.e
E=
I
2
KD.o
'
2 2 222
rorJO - roi r + 4 fJro, ro pO
SIn<p, =
21;pro po ro, .
,
(m O - ro7) + 41; ro ro 0
COs <p . =-
. ,
2 2
ropO - ro, .
,
(ro O - rof) + 41;pro ro o
(i = 1, 2).
(5.110)
TaKHM o6pa30 1, peWCHml (5.108),
(5.109) npllHllMaJOT BM,/:\
PI = no [VI sin(rolt)+ VI cos(rolt )];
2
P2 = o [V 2 sin(ro 2 t)+ V 2 cos(ro 2 t)];
(5.111)
r,/:\e
_ K[ropo -(P-vJ] .
VI - [ ] ,
ropo - (p - v J - + 41;propo(p -v r
K21;fJ rofJo(p-v) .
VI - [ ropO -(p-vJ ] \41;propo(p-vJ'
K[ropo -(P+vr] .
V 2 = [ro po -(p+vJ ] 2 +41;p ro po(p+vJ'
K21;fJW po & + v)
V 2 = l ropo - (p + v r ]\ 41; p W po (P + v J .
(5.112)
CJIO:>KI1M pellieHIDI (5.111) M, HMeH B BH-
.ny, qTO sin(wI,2t) = sin(p =F v)t. cOS(Wt,21) =
= cos(p + v)t , nonyqHM perneHMe Ha '1aCTO-
Te MO)J,ymll{Mll (or1l6aI{JII eH):
p = o [(U 2 -U))sin{vr)+
+ (VI + V 2 )cos{vt )Jcos(Pt)+ o [(VI + vJx
x cos{vt)+ (VI - V 2 )sin{vt )]sin{pt)
(5.113)
TIepBa5! COCTaBnSlIOI..I.\aH. peweHMSI
(5.113) cMHc}Ja3HaH., TaK K8K OHa COBITa)J,aCT
no a3e c ITepBM'IHbIMI1 KOJIe6aHI15!MM pOTO-
pa, a BTopaH. cOCTaBJUJIOI..I.\aJI KBa'/:\parypHa5!.
BbInOnHHM B nepe)J,aTO'lHOH YHKl{I1H
(5.106) 3aMeHY s=i(p-v) (nepBaH. 60KO-
BaSI. qaCTOTa) H IT03JY4HM
W;(S)=U I +i .
(5.114)
AHaJIOm'IHO npH 3aMeHe s = i(p + v)
(BTOpaH. 60KOBWI 'IaCTOTa) nepe,/:\aTO'IHaJI
c}JYHKl.{H5! (5.1 06) npMMeT BI1):l
W 2 (s)= U 2 +iV 2 '
(5.115)
B COOTBeTCTBlU1 C Bblpa){(eHlleM (5.113)
3anl1111eM aMITmrry}1b) orH6alOI..I.\eH BbIXO,nHO-
ro CMrHaJIa ,l:(JI5! CMHc}J83HOH II KBa.r\Pa1)'pHOH
COCTaBJI5!IOI..I.\MX:
A = 0 0 [(U 2 -U))sin{vt)+(v( +V 2 )cos{vt)];
2
A = 0 0 [(U 2 -VI)sin{vt)+(v( +VJcos(vt)].
2
(5.116)
PacCMaTpMBaJI B CMCTeMe (5.116) MHO-
:>KMTeJIH npH rapMOHHqeCKHX COCTaBlliIlOIl.{HX
KOJ1e6aHMH aMnJIHry.n KaK BellJ,eCTBeHHbIe 11
MHHMble qaCTH qaCTOTHbIX xapalCTcpHCTHK
no Orn6alOll{eH BblXO Horo CHrHana, MO:>KHO
3anHcaTb, HMCH B BH.ny 3aMeHY s = iv , CJIe-
JlYIOll{He BbIpa:>KeHHH.:
366
rJIaBa 5. H3MEPH.TEJIbHbIE CBOtiCTBA MHKPOMEXAHW-lECKI1X flPH60POB
A,nE
-40 --T-T.,.."Tl"1'r--r-r' ,"Tf"\T'J---r -,.. "1"'"1'1'1')- - - T-r',"rl''r'--- y- r,.."r-n.
I I IIIIIII 'I 1.11". I I 111'111 I I I 1IIII I I I 111'1.
I 1111'111 J 11111111 1 IT IIII I I I .11.' I 11111,1.
I IIIIIILI I I 111111 I 11 II11 I I I 1IIII I I '11.11
I 1,111111 I I II III I II I11I I I I 11111 I I 1.1111
I L J 1,1,11 I I II II, I II I1II I I I 11111 I I I1II1I
-60 I I I 1 I 1 I 1 I I I II tit I II .1. I I I 11 1 11 I I 111111
__ _ - __ _ U __ _ _ _ J L__ _ J I
I I I 1,1111 I I I II II I I I III I I I 1.1.. I I I U II
I 1.I.h'l I I I II I I II It I I I I 11III I I 11111,
A . O ' r ' tllill I I I I 1 I I ,I .1. I I I I1I1I I I Itlill
LJ.. I I.1 Itll I 1I1I I II ... I I I I1I1I I I 111.11
11I1III I II .1. I I I 11III I I 11111,
I I' .11 I I III.. I I .1. I I I 1.111 I I I IIIII
I 11111.11 1.11111 I I III I I I 11111 I I III1II
_ 80 I 1111111) I 11111111 I II I 11111111 I I II1II
--"",-..-.--.'tli--T-T"'rrr,--- '1 '-T'"')'Tnr--l-T "TO I
f1=50rUII11 11111111 I 11111 I IIIII1I1 I I 11111:
I I II I I I1IIIII I I 111111
f1-100 [nil' , I 11 1 1111 I I II I I I I1IIII I I III1II
4 I I I I I11I1I1 I I II1III
A- 200r II I 11111 I I1II I I I I I II1I I I
100 tJ..- III I I 1111111 III I I1IIIII I I II I I I I
_ L -t"""""t- +.J........1.iL__ _.&. -I.LtJ,
I 1I11III 1111111 I IIIII1I 1111111, I I Pili
II1I1III 11'1111 1II11I1I 11'1111, I IIIIII
1111111.. IIIIUI II11IIII 111111 1I1I1IIII
I II1III1I 1111111 1II11I1I 11III I I1 I I 1 I I I
I IIIIIIII 1141111 I II11III I III I 1111111
I 111,1111 1111111 I 11111,11 I ,11 I 1111111
_ 120 1II11IIII III"IIUI I 1 I I 1 I 1 1 1 II I 1 I 1 I 1 1II I I I I
--,- '- r--T-T ' r"r--r- , , pr-- -T ' Th - , ' rtl
I II11I1I1 '11111111 I I II1I11I I I II1III1 I I IIIII
I I 1111111 I I 1111111 I I 1111111 I I I111III I I IIII I
I I 1111.11 I I 1111111 I I , IIIIII 'I 11'1111 I I I IIII
I '111 11I1 I I I IIIUI I I I I 1IIII I I I 1II1I1 I I I III I
I 'I II1III I I 1IIIIh I I II ,lilt I I I1IIIII 1 I I II III
I I 1'111'1 I 11111111 I I1I1II1I I 11111111 I I III II
1 4 0 .J"""'J.. 'J..'J"'!L J..'J'J'I!L 'J..'J'1'J' .1'.1'.1'
- -- I-"-I-r;l'i'li --T-I , l'iYTl - - I - r I' ITil'1 --'1- I' 1'1 fir--- .-T ,I. ITI I
I 11'11111 I 11II11II I I II11I1I I I III1III I 1111111,
I. I I I Jill, I I I I 11111 I I I J 1IIII I I I I I1III I I I I Illt l
, I I I I I 111 I I I I I 1111 I I I I I I III I I 1 I I II1I ,I, I ,I I I
I 1111'lh I 1111,111 I I II1I11I I 11111111 I IIIIII1I
I I III1III I 11111111 I Iltlllll I I I PIIII I III1II1I
I I I I111I1 I I 1111111 I I I I1I111 I I 1,11111 I I 1IIIIII
-160
o
I
2 3
Ig ro (ro, c l )
4
5
Puc. 5.30. A qx nJIH pa3.1U'IUblX 3Ua'leHHH 'IacroT (no orufialOweii)
Hr.:0(s)=K 2p[s+ p(p+ )]
[s2 + 2 p(P + A)s + (p + t.f - p2 j + 4p2[1;p(p + t.)+ sr
,
(5.1I7)
S2 + 2S p (p + t.)s + (p + t.f _ p2
W:(s)=K
2 + 2 p(p + t.)s + (p + t.f - p2 j + 4p2[l;p(p + A)+ sr
Ha pHC. 5.30 npMBe.neHbl PC3YJIbTaTbl
BbltIHCJIeHHH Ha 3BM nO cpopMYJIaM (5.117)
pe3ymilHpyIOIWIX aMI1JJHT}',!Uio-liaCTOTHbIX
xapaKIq)HC=niK (A 1fX) ,!JJVI pa3JlH'IHbIX 3Ha'le-
HHH pacCTpOHKH H CJIe.ll)'lOIUHX MCXO,ll.HbIX
JU!HHhOC J.,-;::; 2111 = 3 . 10- 13 Kf'M 2 ; P = 1,88 x
x t(t pa.nlc; Yo , 1,5°; Qu = Qp :=:: Qr = 10 4 .
TIpe.nnOJIO:>KMM, liTO BbInOJ1HeHa pe30-
HaHCHa HacrpOHKa pe)KHMOB KOJIe6aHHH
(A = 0), C yqeTOM TOro, qTO 1;2« I, a TaK)Ke
rorp = 2...5 JGl{ II HMeS! B Blmy, qTO ,llMana-
30H liaCTOT H3MepSieMOH CKOpOCTM He ITpe-
Bblll1aeT, KaK npaBHJ10, 50 fl{ H3 (5.117)
nOJIY1JHM
wo )-;::; K '2Pls+l;p p) = K = Kcr
c 4p2 + l3p r 2p2 jJ (T s+l ) Ts+l'
o() K 2+21;JlPs)
WI( s 4p2 {s + !;pp r
(5.118)
&(lIs + 1)
2p31;f3(Ts + 1)2
Kcrs(lIs + 1)
p (T s + 1)
,
(5.119)
cPOPMI1POBAHI1E BblXOf\HblX CI1rHAJIOB MHKPOrYIPOCKOflOB
367
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. :.:-::r:.......:o-.....................« .....{....-... »'1-"/.'-" . .. jo __;r . r'oI' .:. . "":-:'.V!i ..., .. .... .... T.-,. '.' .
..::::.:::.::.i-$::. ;:;Y.!.:: :::: : x::.::..:..:::;:;:-" .. " g :e;J :.' :. .. : :O;;: . j; '
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o ;:. :o::::::t ::::;!: :-:I: : .: : .:.::: .: :... ACi.i ....:...:.7 .':$:wN J :. . :-;.r......
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I ...:...: : ).:.p.:<;:)....t:;:::a:-: .;:- ..y r ..,.-:- "" ":YJ.'. f:;C .ili- < . ...
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I .....,::.:;:t;::>I:: ::::>.::<::;: ["'@$. I;;;::... .,.. m>4-:-..i:t '::k ;:r« :;. ;;:z,"<
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l I : I .., .... :::: :f( <; Z4 mm $ . ;;:-*-
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t I I f I I *. . ..:.: ...:. . .:?- ::::'-:V$
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,. I I I r I I I I I I I
t I J f I I I I I I I
I I I I I I I I I , :
o
0,2 0.4
0.6 0,8
1
1.2
1.6 1,8
2
1,4
t, c
PIIC. 5.31. Oru6alOwue BLlXO,nHOI"O ClIl"HaJla Mr
r c T = 11 J3P; 1( = 1I2 J3P; KL"T = K /
')
(2 I3P-) .
KaK CJIe yeT H3 Bblpa)!(eHJUI (5. Il 8),
npn YCJIomm pe30HaHcHoH HaCTpOHKl! )J,JUI
Mf', pa60TalOlllero B pe)!(MMe ITpJlMOro npe-
06pa30BaHHSI, ncpe aTOlfHaJI tPYHKl{IU! ITO
cMH a3HoH cocTaBIDllOllleH ITpM6JIH)!(aeTCJI K
npM60py MHTerpllpylOlllero ullla C nepe,lla-
TOqHOH YHKlllleH aITepHO lI'lecKoro 3BeHa.
KBa.n.paT)'pHaJI COCTaBIDllOll\aH. BbIXO HOro
CMrHaJIa, KaK BM HO m OPMYJIbI (5.119),
MeHbille CHH aJHOH cocTaBIDllOIIJ;eH.
TIpHBe.n.eHHble pe3YJ1bTaTbI nOK83aJIH,
<ITO pa6oqaH. nOJIoca qaCTOT rHpocKona on-
pe)J,eJ1J1eTCJI B OCHOBHOM napaMeTpOM pac-
crpOHKM 1:1 H cJ1a60 3aBHCIIT OT ITapaMeTpa
3aryxaHlUI /;13'
floJ1HaH. nepeMTOlfHaH. tPyHKllHJI Mr H
ero MaclllTa6HbIH K03 tPlll{HeHT Moryr 6bITb
nonylfeHbl aHaJIomlfHO )J,JIJI Mr LL- H LR-
nIllOB.
5.7.2. KOl\meHcaQuoHHoe npeo6p830B8Hue
JJ:HHaMHlfCCKue xapaKTepHCTMKH Mr
MOryT 6blTb ynYlfIlIeHbl, eCllH 06ecnelIHTb
KOHl)'P KOMneHCal{HM MOMeHTa CHJI HHep-
l{IUI KOpHOJIHCa, T.e. rnpOCKon B 3TOM CJ1y-
Ifae 6y eT pa60TaTb B pe)!(HMe KOMITeHCallH-
OHHoro ITpe06paJoBaHHSI. Bocnonb3yeMcJI
nepe.naTolfHoH YHKl{HeH (5.118), KOTOpaJI
XOpOiliO nO,QXO,QHT ,llml OnHCaHI1S1 BbIXO,QHO-
ro CHfHaJIa Mr no om6alOllleH. JJ:JIJI MnnlO-
CTpamm 3TOro 06CTOSlTeJIbCTBa Ha pHC. 5.31
npHBeJIeH pe3YJIbTaT pCllleHIDI Ha 3BM
ypaBHeHlliI (4.310).
11:3 nepeMTOqHOH tPYHKUllH (5.118)
cJIeltYeT, lITO )J,JIJI yaeJIHqCHHJI nonOCbl npo-
ITYCKaHlIJI qaCTOT Heo6xo.Ql1MO YBeJIU'IMBaTb
napaMeTp J3' lITO llJInlOCTpHpyeTcJI
pHC. 5.32.
BMeCTe C pacllinpeHUeM n0J10CbI npo-
nycKaHHSI lIaCTOT nponOPllHOHaJIbHO ysemf-
lIeHHIO J3 YMeHbiliaeTcJI K03I}> HllHeHT ne-
pe,l\a'IH [CM. OPMYJIbI (4.328), (4.329)].
)];JIJI CHJlTHJI :noro npoTHBOpe'IllSi He06xo-
,l\HMO peaJIH30BaTh pa60l)' Mr B pe)!(IfMe
3aMKHyroro KOHrypa KOMneHCallHll MOMeH-
TOB no HH OpMaI1)fn 0 BbIXO)J,HOH KOOp.Ql1Ha-
Te p C HCnOJIb30B:UmeM Ha no.D,JlO:>KKC Mr
::meKTpo.noB MT'lHKa MOMeHTa (JJ:M).
KOHryp KOMnenCallHll (llem, 06paTHoH
CBSI'3H) BKJIlOlIaeT ,n:M, OpMllPYIOlIUiH MO-
368
fJIaBa S.I13MEPH.TEJ1bHbIE CBOI1CTBA MI1KPOMEXAHI1 ECKI1X flPI1F>OPOB
A,nE
-20
--T-rT r T--I- - i ----r ' r --,- , rrr--r -r n --r- r r 1
I 11 1 1 1 ,1, I I 1'1 1 1 1 , I I 11111t, I J 1 1 1 1 ,1, I I I I111II I 1,1 IIII
I '1 1 ,11 1 , I I "1 1 1 1 , I I I1 I I 1I 1 I ,111,'. I 11111111 I I II I_II
, I "",1, I' ,'llrll I 1111111, I,ll I 11'11111 J 1,1 I,ll
I 11',11 1 1 I 1,11 1 1 1 , I 11"111, 1111't I I II1IIII I I II 1111
I 111 I I I I I I I II '1'11111 I I 111)111 I I II IIII
I . I II I I I II I .1'1 1 1 1 1 I 11 1 11 1 ,1 I I II III1
r 11 1 ,',11 I 1,1,11 1 1 I 1 1 1'1 11 , I '11111 I 11'11111 I I II 1.11
-40 -_...I_ , .-LI..J. r lili..._-L. , '_Lt' r l LiL_-!- - r ' L.I..I n IlL__J._'!-J.. .....!1__.J_.!..... , I.n..I.&.I_.....!- T '... " .! r ll!J
I ,I, ili I ,I, 1'. '" 1'1 I I' I II I 2 ' 111,1 I If,
I 1,1,',11 I 11111,11 I I I 'III, I 111,1 1111111 J I I 1III
, ',1,',1, ,1,',1"1 ',' ',II, 1'11,11 11111 I I I IIII
, ""'ll, ,1 1 '1 1 ,'1 'I' ',lIr 1'111 1 I' ,lilli' I I '1111
, ""'ll, I 11'111'1 ',I ',III I'll II ',111111 I I I1III
I I I I' ,111111 I I I 1111
, "I' 1 I I I' , 'I I I I I 11 1 .11 1 I I , 11'1
I "',II', I' ,'L'II, '1 I 11111 I I 1111,11 '11111' I I '11111
O __ _L+L __+_L __ _ L __ _ L __ __ _L L I
1 1 1 ',1 1 ', I' ""1" 'I I 'III, I I 1 1 ,,11' III I I '11,11
, '1',111. I' ,I,ll" 'I I 'III, I' '1',1 1 , I I I' '111'1
, '1',1,'.. I' ,1,1,', 'I I 'III, ,I 11',11 , III I I '11111
, ',',',IL I' ,1,1"1 'I' 'III, I I 11',1, . LII'I ,'1 1 1,11
, I'll 1'11,1"1 I" 'JII, I' '1',la , '11'11 "11,11
, '1 1 1'1
, ,,11.'11, ,1,1,1"1 I I' ',n, I' 'I 'I ,I 1"'11 '11,'1
80 "llt'.I!J. . 1.1',1"1 I I '.1' '1 I' ", I I "I'll I 1.1,11
- --T-rTT(1""JTT--1-r"TI11'J--"-r,.,.T" ---I-"-'''''T --r-r-rrn"f.T--r- 11l
I '1',',1, "1',1"1 ',' r""1 I' '1',1 1 I '11 1 ,1 I I, I,ll
I 1.',1,1, ,11'1 1 1" "'.1",11 1 I' ',111, , 3 ,,1 I I .11 II
I '.',1 1 1, ,11'11.11 '.' .',11 1 I' '[I,ll , ',I I 1,111,
I 1""1" ""111 1 . " """1 I' ',I,ll I , 1111,11
I . 1,,1 1 ', ""1 1 1 1 , ',1",111 I I ',1111 '111,1 I 1 1 1,1,1,
I ',','.', I I J'IIII, ',I,I,lIt I' 1,1111 I "11,1 I I "II,'J
I ,,',I,', ,. "1111, 'I "I,ll. I I 1,1111 I '..1,1 I I 1111,11
_ 100 --'-. r ' '-}<-'-t:'---.!- r ' .q-!{-! __ , '_L , I"}L' i! ' -_J- , 'J- , ''''!. L__L_L r ' u.!" --L- r ' L r ' !-J ,I I '
, ,p" 1 "" I' 1'1 I I IIII I I, III,T I " II
, "",,1. I' ,',1,1, '11,1,'1, I 111,1111 I "111,1 I 11111,11
, ' ,1,',1, 1',1,1"1 'I 1,1 1 11, ,I I11II1I I ,I.. IlL I I 1111.h
I . ,'.',11 I 1,1,1"1 'I ','III, 1'111'111 , ,IIIIJ' I IIIIIIIJ
, 1,1"111 I 1,1,1"1 '11,11'1, ,I II1IIII I ,1111,1 I 11111111
I 1",',1, ,1,',1"1 I I "'111, I I I11I111 I ,111111 I 1IIIII1I
, 11'",1, "1'11"1 "'1'1'1, 1'1'1"'1 , I 11111' I 11 ' 1'.'1
, J,'" I ' . I I' ,I ,II I" I , III, I I, I ,I III I ,I II I' I I I' I ' 111
-120
-3
-2
-I
o
19 co (co, C- I )
1
2
3
Puc. 5.32. AtIX .nnH OTHOCHTe.TJbUMX o31j!4)11..ueHToB Aer.mlj!upoB3HUH:
J - I! = W 4 ; 2 - I! = 10-3; 3 - I! = 10- 2
MeHT Mm.!, npOTHBOn0J10)KHbIH MOMeHTY CHJI
HHeplUlli KOpMOJIHCa M K . T .e. CllBHHYTbIH no
«pa3e Ha yrOJI 1t Ha HeCYIIleM qaCTOTe p. ,Z:{M
.;J.OmKeH KOMneHcHpoBaTb aMllJIHTYJtY orll-
6a1OII1eH KOJIe6aHHli Ha qaCTOTe v M3MepjJe-
MOM yrnoBoH CKOpoCTH. TaK KaK ynpaBJ1e-
HMe MeT Ha 'iaCTOTe OTH6a1OIUeif, pC30-
HaHCHaJI HacTpOHKa Mr He HapywaeTcjJ.
Crp}'K1YPHa5I cxeMa MMr c KOHT)'pOM
KOMneHcaI.{HH M K npHBe,l\eHa Ha pHC. 5.33.
Ilepe.wnoqHWI l}>}'HKl.tlUl no CKOpOcrn
Mr C KOmypoM 06paTHOH CB1I3H no MOMeH-
TJ HMeeT BM
K*
W(s) = *
T s+1
,
(5.120)
r.ne
T.
T
QpKJJ.yKyK)J,M ,
1+
G 13
K" -
Ho
( ] '
G
K 1+ 13
)J,M QI3 K py K yK)J,M
(5.121)
..L a". K. 13 KJJ.Y U BbIX (Qot
Ho K y ,
J
MIJ)t1 KJJ.M
-
Pac. 5.33. CIPYJOypHaSI cxeM8 MT C KOHTYpOM 06P8Tuoii CBR)H no MOMeHTY:
KW. Ky, Kp,.! - Kro4>QnfilJfeUThI nepe)18.<JJf J18.TQHKa yrna ()J;Y), YCHJIHTeWi J{ .QM
lABHOIMOCTb 1'!3MEPHTEJIbHblX CBOHCTB Mr
369
A,.lI.E
-20
-40
-60
-80
-100
-120
-140
-2
--,- T"" r-......r - "T -r-r.,..,r - -r-.,.." '1 !) .... --,-"'" "rrr-"'- r- r'T"'- -r .....".,....,... - "T-r- rr"1'.,....
I (111,111 I III, lilt I (1"JI(1 I 2 1,1111 1,',1,1 1 1 111.1,',1 1.,1(1111
I 11(1(111 I I IIII lit I 11,1,111 I 1.11 I ,',1,1 1 1 I "1111(1 I '11(1.11
I 11(1,11, I 1(1,1111 I 11,',t,1 ",1, I ,',1 1 1,1 I 1 11 1,1 1 1 I 1(1(1(1'
I 1111.1f, I 1(1(1111 ',1,1"1 1 I '.1,1,11 I ,',11111 I 1111.1,1 I 1,1,1(11
I I I II I '11.111 1'111.1,1 I 1,',1,1 1
--'_J.'J....."L_..L_' L'-'....J.! ..L ..L -" --'-.1 ""L L '-,-,'-0',' ..L ' L....!.J"
I I "T.r.rT, I r. rjr'iI -r., I'ITjr- I y."tili1( --'-. rill I , I I Tj"T,r- .-r, rjr,lI
I 1.,'.111 I ',1(1111 1.11"'11 . IIIIIIIS I "11('" I (1(1,' I "',1,11
II',IIII( I '1 1 ,1 11 1 '.1,1.',' I 1,1111" I ,',1(1,. I'. 1(',' I 1"1',11
I I '.11111 I a 1 1 1 1 11 1 I IIIIIIII I 1111111, I IIIIIIII I III I I 1111',11
11 1 1 1 1 11 1 I II 1 I I II 1 I I I1 I 1 I 1 I I 1IIIIIIr I III1I1II I I II 1 1 I 11111,11
I 1 1 1 1 1 11 1 I II 1 I I II I I I1I1III1 I 1111I1II I 11I111I1 1IIIIIIII 1 I I 1 I I 11
- -,- T "T'i"r-,.->-r', .,,,- -...,.-<., !jfJl---,- ...,,,, i1r--<- r" I"'T'i'- - r -<" >j...... - ... r"""
I I 1 1 1 1 11 1 1'1 1 1 1 11 1 I I IIIIIII I 11111111 I 111111,1 1111111,1 1 1 1 1 1 1
I I 1 I I I all I 11 ' 1 1 11 1 I 11,11111 , 1 111111 I 111111,1 I 111111,1 I 1'111
I 1 1 1 1 1 11 1 I 1 1 11 1 11 1 I I II1111I I 1,1 1 11 I 1I111II1 I III111II I II ,II
I I 1 1 1 1 '11 I 1II,IU l I I 1,1 1 1 1 1 J I III I 11111'1' I I 11111,1 I I III I
I 1 1 1 1 1"1 I 1 1 1,1,,1 I I 1,11111 111,1111 I IIIIIIII I I II1I1II I IIIIII
, '" " " , "I' "1' _ I_L"I I :!i -_L 'JJ....','_..L_, L I-!H '"
I I I I ,.. t I -, I Til, I I -, I ITI'Tlr I r I I I"
1 1 11 11 1 I II ,Inl I J ,11111 111,1111 1 1 1,1 1 1 I I I lit III I I IIIII1I
11 1 1 11 1 1IIIIn i 1 1 ,1 1 1 1 1 '11,1111 I I 1,111 I I1111III I I I I 1 'II I
I 1 1 1 1 1 11 1 I 1 1 1 ,111 1 '1 1 ,11 1 1 1 I II a,I.11 I I' III I 1111,111 I 1I I 1 I I I1
I 1 1 1 1 ,11 1 I 11 1 ,111 1 '11,11111 I 111,lt l , I 1'111 I 1 1 1 1 ,1 1 1 I 1 I 1 I 1 I 1 I
I 1 1 1 1 ,11 1 I 111111(1 'I JIIIIII I 1 1 1,1,11 I I 'II rill 111 1 ,11 1 J 1 1 1 1 1 1 ,1
---'-T T Ur-"-"r r--"" " "-'- ""'rrr- -TLrT -- ," "'-"-"'r
I 1 1 1 1 ,111 I 1,11111 1 '1 1 , 1 1 1 1 1 I 11',1(11 I 1'111111 I I1I1II I 1 1 1 1 1 1 1'
I 1 1 1 1 ,11 1 I 11111111 'I tlllill I 111,1,11 I III1IIII I I 1 1 1 1 1 I a 111111'
I I 1 1 1,11 1 I 111(11(' ',11 1 1 1 1 1 I I I lilLI, I 1 1 ,11 1 1 1 I I I I 1,1 I' 111111'
I 1'1,,111 I 11111111 a I I I 1 I I I 1 I III,ILII I I ',1 1 1 1 1 I I II I I J '1'1111 1
I 1'1'1 11 1 I 111111(1 'I I I I I II 1 I 11'lljll I 11111I1I I I 11 1 1 ' I I 1 1 1 1 1 1 1'
__'_J.'J."IIL_.1_' L'-'....J.!__'..L '.1.....11' __L '.1 '''''L_ '_L' " ' :!i -_L 'JJ'-o',' _ 'L'-'Lo"
I I "T I r'irfl I r, rjTiil r 1'1 TjTlr I T 1"1'111 , I rill I I' IITiTlr I Il)r l 1t
I 11 1 1,11 1 I 1111111' 'I LIIII,I I II Illfll I 1I,IIIIf I I I1 1 1 1 I I I '11 1 1 1
I 1 1 1 1 ,11 1 I 111(111' a I fl'llll I (1 1 ,1 1 1 1 I I II I I lit 1111111,1 I I III'
I 11111 11 1 I 1111 1 1(' I I jlllill I 11 1 ,1 1 1 1 I 1 I 1 I 1 I 1 I IIIIIIIII I 1'1 II
I I I I I , III I I I I I 1 11 1 ',j I II 1,1 I I I I r I I, I I I I II 'I I I I I 11III I I a II I
11 1 1 1 ,11 1 I 111 1 111 1 I IIIIIIII I 111,IJI, I IIIIIIIL IIIIIIIII I l'llllt
1 2
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-I
o
3
4
5
PIIC. 5.34. 1.facToTHble xapaKTepHCTII'-U Mr no OrIl6alO1l\eii:
1 Ii 2 - peiKliMbI njUIMOrO Ii KOMneHcal\HOHHoro npe06p8.30B8HIDI COOTBeTCTBeHHO
TIpM 60nLllloM K03c}Jc}JMQl1eHTe nepe,lI,a-
qll KOHTypa 06paTHOH CBS!3H napaMeTpbI
(5.119) npHHHMalOT BI1,lI,
T*
TG p
QpK D.yKyK D.M
K. == Ho
KJ],M
(5.122)
Ha pllc. 5.34 npllBe)J,eHbl 'laCrOTHble
xapaKTepMCTIIKH Mr .lI.ml pe:>KHMOB npS!-
_1
MOrO (1; == 10 -) H KOMneHcawrOHHoro npe-
o6p83oBaHHH (K)J,Y = 1 B/pa.n.; Ky ==100;
KJ1.M = 10-9 H-MlB; T. = 7,5'1O c; Qp = 104).
K03c}Jc}JHQlleHT nepe.na1.JH, KOTOpblH B
pe:>KHMe KOMneHcaQHOHHoro npeo6p830Ba-
HHH onpe.ll.eJIHeTCSI B OCHOBHOM 3HaqeHHeM
KJJN, He 3aBHCHT OT CTeneHll .neMnI}>HpOBa-
HHH. EOJ1bIllOe 3Ha'leHHe nOCTOSlHHOH Bpe-
MeHM T BCJIe.ll.CTBHe MaJIOro ,l\eMn4>llpoBaHHSI
MO:>KeT 6bITb }'MeHbIlleHO Bbl60pOM COOTBeT-
CTBYlOllJ,erO 3HalleHHSI K03 4>Hl\HeHTa Ky.
TaKHM 06P830M, BBe,l\eHHe KOHT}'Pa
KOMneHCaLl.HM MOMeHTa CHJ1 HHepLl.HH Ko-
pHonHca paCIllHpSleT nonocy nponycKaHIDI
qaCTOT Mr, H npM 3TOM MO:>KeT 6bITb o6ec-
neqeHa Tpe6yeMaJI 'iYBCTBIlTeJIbHOcTb.
5.8. 3ABHCHMOCTb
H3MEPHTEJIbHbIX CBOHCTB
MI' OT KOHCTPYKTHBHbIX
IIAPAMETPOB If BHEIIIHllX
B03MYIQEHHH
5.8.1. MacIDTa6H1.Iii K034»tJmu,HeHT H
lIaCTOTHaH HacTpoHKa
HarrOMHHM, qTO B03MO:>KHbl cJIe.nyro-
Il.lHe PC)KHMbI qaCTOTHOH HacrpollKH, T.e.
coornollleHHH Me:>KA)' 'laCTOTaMll nepBJ1q-
HbIX (P.z:o Ii BTOpH'lHbIX KOJIe6aHHH (Pq")
(CM. pHC. 4.14). TIepe'lI1CJII1M HX.
1. qacTOTbI P)]; H pq COBna.n.aIOT (pe-
30HaHCHRH HacrpoHKa). l1Meer MeCTO pe30-
HaHCHOe YCHJ1eHHe BTOPHqHbIX KOJIe6auHH, a
nOJIoca nponYCKaHHH 'iaCTOT onpe,l\eJIHeTCX
pe30HaHcHoH KpHBOH.
2. qaCTOTbl P)]; H pq 3Ha'IJ1TeJIbHO
p83HeceHbI. Ha6mo,u,aeTcSI 60JIblllWi nOJIoca
nponycK8HlUI qaCTOT H oTcyrCTBHe pe3o-
HaHCHoro YCHJIeHHX BTOpH'lHbIX KOJIe6amrii.
370 rJIaBa 5. H3MEPHTEJIbHbiE CBof£cTBA MHKPOMEXAHH ECKIIX flPHEOPOB
3. I1MeeT MeCTO paCCTpOHKa qaCTOT
P.D: 11 pq, npll KOTOpOH o6eCneQUBalOTCSI
38,!1,aHHble nOJ1oca ITponycKaHI1JI QaCTOT II
MacwTaoHblH K03 lIuueHT.
MaKclma.11bHaJI LI)'BcTBIiTenbHOCTb Mr
K 1i3MepSieMoii ymoBoH CKOpOCTI! 06ecneqll-
BaeTCR B pe)fmMe pe30HaHcHoii HaCTpoHKlI.
TIpMMemrreJ1bHO K IT RR-Tlma. lIMeR B BlI-
::I.Y MaTCpHa.1.. H '10iKeHHblii B nO,!lpa3.!l. 4.11
H nOl1araH., 'ITO CITpaBe.J;1.IfBbl paBeHcTBa
J y :::: 2J p _ Cl)yo:::: Cl)po ::::: CI). 3amfllleM npl1-
O.ilIDf\eHH)"lO, 6e3 yqeTa norepb 1l3-3a CBR-
3aHHOCTIi KOJIe6aHlfH, OP iYJl)' "UI mc-
wTa6Horo K03clJI}>uUl1eHTa, KOTOpbIll OITpe-
.!le.:u!eT 'I)'BCTBHTeJIbHOCTb Mr ITPli pe30-
HaHCHOH HaCTpOHKe:
K=
lot "
£".
MoQyQj)
U)yoCl)l' o J p
2M o Q(Qj) _ 2YoQj)
J y CfJ r f1 3 CfJ
(5.123)
r;J.e "(0' p - yrnoBble OTKJlOHeHMSI pOTopa
OTHOCHTeJbHO oceH nepBll'lHb H BTOPHq-
HbIX KO.:Je6aHHH; Q - H3MepSieMaH. yrJIOBaH.
CKOpocTb; M 0 - MOMeHT )J,BMraTemI .1UUI
C03,!laHIDI nCpBH'lHbIX KOJIe6aHHH; Qy, QI' -
,!l06poTHOCTb OTHOCMTeJIbHO oceH ncp-
BJ1qHhlX II BTOPHQHbIX Kone6aHHH; CI) - qac-
TOTa, paBHIDI 'IaCTOTaM nepBH'lHbIX M BTO-
pll'ffibIX KOJIe6aHHH; J Y' J I' - oceBoH H
3KBaTOpHa;IbHl>IH MOMeHTbl HHeplUfH po-
TOpa.
lli BbIpaJICeHIDI (5.123) CJ1e,l\yeT, ':lTO
DpM pe3OHaHCHOH H8.CTpOHKe MaKCHMaJ1bHaJI
'J)'BCTBHTe."IbHOCTb M.r o6eCne'lHBaeTCR
nOJU{epA<aHHeM yrna nCpBM"iHbIX Kone6aHHH
pcrropa H .no6poTHOCTH no OCH BTOpH'lHbIX
K0J1e6aHHH Ha MaKCHMaJIbHO B03MO)I(HOM
ypoBHe npH MHHMMaJ1bHO B03MO)I(HOH 'lac-
TOTe nepB){q11LlX KOJIe6aHMH.
IlaJ:{0 O11deTHTb, 'ITO npH }'MeHbweHMH
1JaCT0TI0I nepBHQHbIX KOJIe6aHHH YBeJ1M'lHBa-
eTCJI KHHeTH'lecKHH MOMeHT Mr. B caMOM
nene, KMHeTH'leCKI{}i MOMeHT onpe.llemleTCR
3aBHCHMOCThIO (4.31 I):
Ho =JryoCl)yo,
(5.124)
TIpJ1 yCTaHOBI1BWeMCR pe)KIIMe Kone-
6aHHll 113 ypaBHeHI1S1 (4.301) CJIe,llyeT
Mosinpt
1'0 - ,
J y(OyO -
(5.125)
T.e. BCJUI<IIIHa yrJIa ITepBlIqHbIX KOJIc6aHHH
npll nOMepiKaHlfl1 ITOCTOSlHHon a mmny.abl
B036YjK)J,alOllJ,ero MOMeHTa o6paTHo ITpO-
nopUHOHaJIbHa KBa.apaTY <IaCTOTbl nepBI1<I-
HbIX Kone6amlit I10cne rIOJJ:CTaHOBKH pa-
BeHCTBa (5.125) B clJOPMYJIY (5.124) llollyqa-
e i UCKOMbIll pe3YJIbTaT [CM. TaK)Ke BblpaJKe-
Hue (4.311)]:
H Mo.
o =-srnpt, rJJ:e p::::: CI)"(o. (5.126)
CfJ.,o
B CBR3H C nOnOiKl1TenbHbIM 3clJclJeKTOM
YBemiqeHIISi aMITJlHTYJJ:bI nepBl14HblX KOJIe-
6aHllH CJIe.nyeT 3aMeTHTb, 4TO OrpaHH'-IeHM-
eM Ha ee pOCT SlBnSleTCSI COnpOBO)!(LlalOlllee
B03paCTaHUe HaITpR)KeHHSI B 3JIeMeHTax yn-
pyroro nO)J,Beca.
OTMeTHM )J,aJIee, 'lTO VBCJIHqeHMlO
MaCIIITa6Horo K03clJclJHIllieHTa, T.e. 4YBCTBH-
TenbHOCTH Mr, KaK BHJJ:HO H3 clJopMynbl
(4.316), CITOC06CTByeT YMeHbweHHe C06CT-
BeRHoH qaCTOTbI Mr:
K =l= Ho = MoQ.(QI' _
cr n 2 3
Hy JI'2S,pCD JpCl)
MoQyQp
GpCD
)J..rrSi YMeHbllIeHlljJ co6CTBeHHOH qaCTO-
Tbl He06xo)J,HMO CHH)KaTb )KeCTKOCTb ITo.nBe-
ca H YBCJIHQHBaTb MOMeHT HHepUHI1 pOTopa
nH60 nYTeM YBeJJH'leHI151 ero )J.l1aMeTpa HJU1
TOJIIltI1HLI. .lJ:mr yMeHbweHHR qaCTOT ueJ]e-
coo6pa3HO YBeJJH'lHBaTb .nuaMeTp pOTopa, a
He era TOJ1lltllHY. B TO :>Ke BpeMjJ CJIe)J.yeT
MMeTb B BH.ny, 'lTO YMeHbllleHHe TOnllJ,HHbI
pOTopa ,l\eJJaeT ero MeHee )KeCTKI1M M MorYT
B03HHKaTb B03MYllteHI1R, BbI3BaHHbie BH6pa-
IUieH OCHOBaHIDI.
3ABI1CI1MOCTb H3MEPI1TEJIbHblX CBOtfCTB Mr
371
HanOMIfl1M, I{TO MaCIIITa6HbIH K03cp-
cpHUUeHT Mr RR-nma Onpe)J,eJIHeTCSI 3aml-
CIIMOCTbJD (4.327), KmopaSi ITpH w.,p = (0"(0
npllHuMaeT BH)J,
K = 'Yo
M 2 2 ") ,
I). + JJW;O
(5.127)
r)J,e /). = (0130 - (])yO a6COmOTHa>l paccTpo i-
Ka MI.
TIpii OTCYTCTBHH paccrpoHIm (I). = 0) H3
ypaBHeHIDI: (5.127) CJIeJJ:yeT
'\J
K - 1 0
MO -
l3wyO
(5.128)
C y<reToM BbIpaJKeHHH (5.127) H (5.128)
3amlUIeM OTHOCHTenbHoe H3MeHeHHe Mac-
IllTa6HOrO K03 IUHeHTa, 06ycnoBneHHoe
pacCTpoHKOH <raCTOT:
8K13 = 1- K\f = 1
K MO
1
] + 4Q13 1 ( (0130 -I J 2
wyo
=1
1
1 + 4Q13 2 ( J 2
wyo
,
I
r,l\eQI3=-'
2 JJ
I1MeSi B BH)J,y, 'ITO
1+4QI32 ( J 2 R:l+2QI32 ( J 2, nOJIY-
wyo (Oro
1
IHIM oK 13 = 1 ") , OTKY.ll.a
1 + 2Qp 2 ( J -
(Oyo
BbIQMCnSleTCSI BenM'lHHa paCCTpOHKll B 3aBH-
CHMOCTU OT OTHOCHTenbHOro 113MeHeHIDI
MaCIIJTa6HOrO K03<p<pHI. HeHTa:
(0 oK" (J)"o 8K"
I). = --...!L ,, -----.!:.. ( 5.129)
Qp 2(1- 8K p ) Qp 2
II0J10JKJ1M B (4.226) (00 = wpo. Q = Qp M
3anHllleM BbIpaJKeHHe )J,JUI ITOJIOCbI nponyc-
KaHHH l{aCTOT:
1).(0 = (0130 .
2Qp
(5.130)
06'be.ll.HHHM cpOPMYJIbl (5.129), (5.130)
H nOJIY'1HM
I). = 1,41t!(O OKp .
(5.131)
IJplUlep 5.17
IlYCTb -rpe6YeTCR nOIloca nponycKaH}\R
l{aCTOT l!.ro=20 r , a ropo =3000 r . Ilo
ljJopMYIle (5.130) HafuJeM lIo6pOTHOCTb
3000
Qp =-=75. IIplfMeM .l1aJIee BKp =0.1%
2.20
(0,001) H no q,opMYIle (5.129) nonyqlfM .l1onyc-
TMMoe 3HaqeHlfe pacCTpomm
3000 0,001
fj, = 0,9 r , MIll! B OTHOCHTeJIb-
75 2
HblX e.lUIHH ax 100%= 1O0=0.03%.
ropo 3000
IlpelInOIlmKHM Tenepb, '!TO o6eCneqeHa lIo6pOT-
HOCTb Qp = 10 4 . Ilo q,opM)'IIe (5.129) pacc<nrraeM
3000 0.001
fj, = = 0,0067 rl{, H1lli B OTHOCIf-
10 2
TeIlhHbIX e.l1MHM ax: 100 % = 0,0067 100 =
ropo 3000
-4
=2,2.10 %.
113 npMBe.neHHoro npHMepa CJ1e.nyeT, 'ITO
.l1JUI Mr KaK C HIDKOH .D.06poTHOCThIO, TIIK If C
BblCOKOH .lI.onycKalOTcR OQeHb He3Ha'lHTeJIhHble
3HaQeHHJI paCCTpoeK.
06eCneQHTb Tpe6yeMytO paCCTpOHK)' H,
TCM caMbIM, ITOJIOCY nponycKaHIDI qaCTOT
MO:>KHO TOJ1bKO np" OQeHb :>KeCTKHX .lI.onyc-
K8X Ha p83MepbI ynpyrHX 3JIeMeHTOB ITO,ll,Be-
ca B ITpe,llenax 10- 3 (H113KWI.lI.06pOTHOCTb) ...
10- 5 MKM (BbICOl<8JI .lI.06pOTHOCTb). OQeBH)J,-
HO, I{TO norpeIIIHOCTb B paCQCTe 'IaCTOT He
372 [naBa 5. H3MEPHTEJIbHbIE CBoHcTBA MI1KPOMEXAHWIECKIIX nPHEOPOB
W p + ..1.ro T
..1.ro H
..1.roJ3()
.&0:1
- ----------------------- -
PIIC. 5.35. COOTIIOWCHIIS! C06CTBCHHblX 'IaCTOT II n.on 'CKOB 118 IIIIX
.J.o.'l:iKHa npeBbIlllaTb TbICSI'lHbIX !IOlle" npo-
ueHTa OT HOMliHa.,lbHoro 3Ha'leHHSI qaCTOTbI.
TIpH CYllleCTBYlO111eM ypOBHe TeXHOJIO-
fHH BbIllO:IHeHHe TaKUX Tpe60BaHllH npo-
6neMaTH'lHO. 11 J1.JIH. Mr C pe30HaliCHofi Ha-
CTpOHKOH He06xo.!l.HMa 3neKTpH'leCK8X CHC-
Te!>!a ITO.1CTpOHKI1 'IaCTOT. KOTOpWi CQ3)J,aeT
OTpHuaTeJlbHYIO n;:eCTKOCTb npn nO]1a'le no-
CTOJlHHbIX nOTeHUHaJIOB Ha 3JIeKrpO)J,bI J1.aT-
'illKa CllCTeMLl nO.1lCTpOHKH. TIoCJIe)J.JUIJI 'lac-
TOTa llOJI:iKHa 6bITb paCC'lllTaHa TaK. qT06bl
'laCTOTa BTOpH'lHbIX KOJIe6aHHH HeCKOJIbKO
npeBblliIaJIa 'laCTOT)' nepBI1'lHbiX (CM. rn. 3).
)J,:UI Mr RR-Tllna CllCTeMa nO)J,CTpoHKI1
'l8CTOT npeJ1.CTaBIDIeT C060H eMKOCTb, 3neK-
TpO.:lbI KOTOpoH nnOlllaJ1.bJO S pacno.'1onceHbl
Ha poTope H nO!lJIOncKe C 3a30pOM 17 0 MelKllY
HHMH. npH pacCTOSlHHH r OT ueHTpa nO,llBeca
.1l0 ueHTpa 3JICKTpO,llOB H Hanpj\:>KeHHH nO]1-
CTpOHKM U 3neKTpoCTaTH'leCKWI :>KeCTKOCTb,
C031laB8CMWI cHCTeMoH nO)J,CTpofiKH, MO:>Ke1'
6bITb onpe.neJJeH8 3aBHCHMOCTSlMH
K = 2uor2S . (5.132)
110 2
G = KU 2 .
:) ,
Co6cTBeHHaJI 'IaCTOTa ynpyroro no.n-
Beca C yqeroM BbfpIDKeHI1H (5.132) paEHa
JG. G,
CJ>PI := .
Jf3
r,l\e G n - HOMHHaJJhHWI :iKeCTKOCTb ynpyro-
ro no.nseca; J p - MOMeHT HHepLUiH pOTopa
OTHocwrenhHo BblXOJUfoJi OCM.
(5.133)
113 opMynbI (5.133) CJIe;:J,yeT npn6:m-
:m:eHHoe paneHcTno
, "
G.. = G n - J,\(OPI = J IJ (wiio - (Oiil ) =
= JI\«(O o -(O I)«O O +W l)::::: 2J p f1(O,w(\().
(5.134)
r)J,e f1(O" = (OJ!o - (O l - 'laCTOTa. C03;:J,anaeMWI
CI1CTeMOH nO)J,CTpOI1KI1.
HanpR)KeHUe, KOTopoe Tpe6yeTcR nJISI
H3MeHeHHSI 'IaCTOTbl (00 Ha f1(O.., onpe,lleJISI-
eTCSI C Y'leTOM Bblpa)!(eHHH (5.132) H (5.134)
no opMYJIe
u=
2J J!L'1ro"ropo
K
(5.135)
113 JIJ1TeparypbI H3BeCTHO, 'ITO nonyCK
Ha paCCTpOHKY 'IaCTOT nepBH'lHblX 11 B1'O-
pH'lHbIX KOJIe6aHHH npenCTaBJISICT C060H
CYMMY .QonYCKOB Ha HeTO'lHOCTb paC'leTa
C06CTBeHHOH 'laCTOTbI ..1.w p H 1'eXHOJIOfH'le-
CKoro .QOITycKa ..1.(OT' 06YCJIOBJIeHHoro He-
TO'lHOCTbJO H3rOTOBJIeHHSI. CHCTeMa no.Q-
CTJX>HKH 'laCTOTbI HCnOJIb3yeTcSI TOJIbKO JJ,JISI
yMeHbllJeHIDI 'IaCTOTbI BTOpH'lHbIX KOJIe6a-
HHM. n03TOMY MO:>KHO C'lMTaTb, '11'0 Haqanb-
Hoe npeBbIweHHe ..1.ro lt 'laCTOTbI BTOpl1'iHbIX
Kone6aHI1H Ha.Q nepBI1'iHbIMH onpe.QeJ1S1eTCSI
HepaBeHCTBOM (pHC. 5.35)
..1.w lI > ..1.ro p + ..1.ro T .
(5.136)
1ABlK'(-IMOCTb Yl3MEPHTEJIbl-lbIX CBOMCTB Mr
373
H"J pllC. 5.35 CJICjlycTHuoKe 01 paIll1<le-
HlIC lIa BCJIlI'IIIHY pacCTpoHKH:
''''.0), > O)II + 2( (JJp + (o)1 ) . (5.137)
Ilpll paC'lCTe CHCTCMbl "JJIeKTpll<recKoH
nO;J.CTpoiiKII C,lC)J,YCT 11\ICTb B IJII)J.Y ce orpa-
HII'ICIIIIYIO MOLllHOCTh II, CJIenOBaTCJIbHO.
3aTpY.llHeHIUI C o6CCnC'lelll1eM HepaneliCTna
(5.137).
T1plwep 5.18
flpc.unO:IOiKliM, 4'10 IIconpClXc.IJCIJHOCTb B
MCTonax paC<ICTa npl1BOlll1T K 01llH6KaM B onpe-
nCJICHl111 <mCTO'f 2 %. 'ITO IIPI! 3000 fu COC-faa-
n CT Amp 60 fu. f)YlXCM C'IlITaTb, <JTO nplt ,D.O-
nycKc ..a ypOBHC 0,2 MKM, 'ITa l1OCTII>I(lfMO, m-
MellCHIiC <mCTO'fLI IJC.'1CnCTBUC norpclllHocTCii
113rOTOBneHlIJ! TaK,KC HaXO,D.I1TC Ha ypoBHe 2 %.
TaKIIM o6pmoM, !J.m T !J.m p =< 60 ru, II no iPop-
:\1y.1C (5.136) !J.m" > 120 ru (4 %). Ha OCHOB8-
HIli! iPOpMy.'lb1 (5.1 37) CIlCTCMa nOl1cTpoiiKlI <rac-
TOTbl l1on>Ima KOMnCHCJ1pOBaTL 1I3MeHCHJle 'mc-
TO'fbl B npc 'lenax !J.ro = 360 [u.
AnJ! napaMeIpOB rupocKona E = I (BaKYYM),
10(1 =8,85.\0-12 <])/M, r = IO- 3 M, S= 3.\0-7 Ml, h()=-
=2.10-(' M, JfJ = \0.13 Kf'M 2 , mpo = 3040 ru IIMeCM
K = 6, 6.10- 7 AiK!B 2 1I no cjJopMyne (5.135) nOJJ)'<laeM
2.\O-3.2n.360.27t.3040
U = 1 ::::: 3,6 B::::: 4 B .
6,6.\0-
YlT8K, ,lI,n 113MClleHlIH 'laCTO'fbl B 360 ru B ClICTC-
MC nOJlc-rpofiKI1 Tpe6ye-rcH Hanpll>I{CHJ!e U = 4 B.
5.8.2. 3aBlfClfl'tlOCTb 'IaCTOT, paccTpouKIf
'IaCTOT If l'tIacwTa6Hblx KO'JI)I)HulfeHTOB
OT Il3l't1eHeHlIH reOl'tleTplf'lecKlfx pS31't1epOB
pOTopa If ynpYTlfx 'JJJel'tleHTOB no,nBeca
3aBllCIIMOCTM OCHOBHblX xapaKTepM-
CTI1K Mr OT reOMeTpH'leCKI1X napaMeTpoB
3 He SlBJIIIIOTCSI YHHBepCaJIbHblMM II onpe-
,neJISIIOTCSI TllnOM Mf H KOHCTpYKu l1 eH ynPY-
roro ITO,l\Beca.
PaCCMOTpl1M YK83aHHble 3aBI1CHMOCTM
Jl)lSl Mf RR-Tl1na C no)WeCOM no pHC. 3.11,
BbtnOJIHeHHoro H3 KpeMHHSI (CM. npHMep
3.4) C p83Meparvm, M: R = 1,5.10-'. 1'1 =
= 0,8-10- 3 , b nl = 20.1O-{1. Id = 0.4.10- 3 . bll =
= 18,4.10-(', ell = 20.W- . ITo opMynaM
(3.28) II (3.29) Bbl'lIlCJIeHbl <JaCTOTbI nep-
BH<rBbIX (0)1 II BTOplltlHbIX no O)J,HO" KOOP)J,I1-
HaTe I<One6aHIII1 COp B 3aBIICIIMOCHI OT reo-
MeTpHqeCKHX pa3MepOB pOTopa II ynpyrHx
3neMeHTOB (TOpCHOHOB) ITO)J,Beca. Pe3YJlbTa-
Tbl Bbl<JHCJJeHlli1 npHBC)J,eHbl Ha pHC. 5.36.
Onpe)J,eJII1M K03 qnH.!l1eHTbI BJIIISlHIIII
K..2 H3MeHeHI1S1 reo eTpIl1.JeCKIIX pa3MepOB
Mr Ha ero tIaCTOTbI no IjJopMYJIe
K _ (o)mwd.2 -cominl.2
! ., -
.- p ,
r)J,e (o)m:lx 1.2' CO mm 1.2 - COOTBeTCTBeHHO MaK-
CHMaJIbHaSI II MHHI1MaJIbHWI tlaCTOTbl )J,JISI
nepBHqHblX (I) II BTOpH'IHbIX (2) KOJIe6amlH
npll H3MeHeHHH reOMeTpH'leCKOrO napaMeT-
pa Ha BCJlUqHHY p.
B COOTBeTCTBIIII C 3aBHCIIMOCTSlMH no
pllC. 5.36 HMeeM:
pHC. 5.36, a
p = 100 MKM ; coma" 1 = 3468,9 fl.!;
CO mm 1 = 2653.1 fu;
fu
KI =8.158-;
MKM
COmin 2 = 2692 fu:
(o)m;\x 2 = 3519,8 fu ;
fu
K 2 =8,278-.
MKM
pllC. 5.36, 6
p = 3 MKM; (o)max I ::: 3000 fLl ;
fLl
COmin 1 = 3000 fu; KJ = 0 - ;
MKM
(o)ItulX 2 = 3100,4 fu: (o)mm 2 = 2990,7 fu:
fu
K., ::: 36,567 - .
- MI<M
pl1C. 5.36, e
p=5MKM ;
(J}mm I = 2547,3fl..\;
(o)max I = 3249,3 n.\ ;
fl{
K, =140,4-;
MKM
WOlin 2 = 2975,6 fQ ;
(o)Olax 2 = 3075,7 fl\;
fu
K., =20,02-.
- MKM
374 fnaBa 5. 113MEPHTEJIbHhiE CBOI1CTBA MHKPOMEXAHH4ECKI1X nPH60POB
a)
3600
340
::r
,
'-:.320
c
t-
o
t 3000
cI
::r
2800
.....
26op450 14671483 1500 1517 1533 1555
}111a.\leTp, MK.\I
6)
340
32
-
C
0: 300
.-:28
-.
..
26
2400
15
16 17 18 19
1IlHpHHa, MKM
6)
3120
3100
::r308
L.
'" 306
304
cI
-.
... 302
300
298
18,5
'"
0 I.'"
'"
'"
0 '"
'"
0 ",'"
W
0 ,I'...
",' ;oy
Ii-" '"
0
19 19,5 20 20,5 21,5 22
romUl-IHa, MKM
z)
3600
'"
C?13 ;'
'"
i-/ ---- --
"'- V'
W.J - -
- .... .
.......
3400
L.
g 3200
l-
t)
3000
2800
2600
20 18 18,5 19 19,5 20 20,5 21 21,5 22
lllHpHHa, MKM
PilC. 5.36. 3aBilcilMOCTiI qaCTOT Mr OT HJllleUeUHH:
a uapy;KHoro pa.zrnyca pOTOpa; 6 - TOJllIUiHbI CTpYKrypbI; 6 -llUlpl1l1bJ KOpOTKHX TOpCI10HOB;
Z - I1IMpMUbJ }J..'llIHHbIX TOpClIOHOB
pHC. 5.36.2
/1p:::::4 MKM; wma>:1 =3188,2ru;
wmin I = 2836,6ru; KJ =87,9 fu ;
MKM
WJD3Jt 2::::: 3405,2ru; Wmin 2::::: 2712,7 fu;
K 2 ::::: 173,125 ru
MICM
Pe'3}'JIbT8Tb1 pac'leTa npHBe.neHhl Ha
pRC. 5.37.
lli nony'leHHblX 3HaIJeHMH K03<!><!>HllH-
emOB BJIIDlHHJ[ cJIe,llYeT, 'lTO KpHTH'lHhJMH
reDMerpHtJecKHMH napaMerpaMM }.UUI pac-
CMOTpCHHOH KOHCTpYJCWfH Mr lIBJISlIOTCJI
lliHpHHa ICOpoTKHX H JVJHHHblX TOpCMOHOB.
Cne,l\OBaTeJIbHO, ,l\onycK Ha 3TH pa3MepbI
,nOJI:>KeH 6bITb MI1HHMaJIbHO B03MO:tKHbIM.
Cpe,nHHe KBaJJ.paTM'leCKJ1e BeJIH'lHHbI
K03 Hl.lHeHTOB BJIWIHHSI onpe,l\eJISIIOTCSI no
opMyne
K J ,2 = l I(K I ,2)2 .
I1cnoJIb3YSl nOJIY'1eHHbIe BbIwe <mcno-
Bble )J,aHHble, Bbl'1HCnHM:
K I = 8,1582+02 +140.4 2 +87,92 =
= 165,847 ru ;
MKM
3ABHCI1MOCTb H3MEPHTEJIbHbIX CBoBcTB Mr
375
100
200
150
50
o
K1
PHC. 5.37. rIlCTo.-paW\la B.lIIHHIIH H3!11eHeHHH .-eO!lleTpH<JecKHx panlepoB Mr
lIa co6cTBeHHble 'laCTOTbI:
Q 1'8L1nyc 1'OT01'8: 6 - TOJIlUHH8 CTpYKTypbI: 8 - IlUl1'HH8 )1J1HHHhIX TOpCHOHOB;
? - lIU!1'HH8 K01'OTKlIX T01'CI10HOB
oj
52
-
-
50
f-.:'
t 48
'"
:- 46
d
:.:
':s: 44
o
c..
t; 42
(J
d
I'. 40
38
]4501460 ]480 ]5()0 1520
,!:(HaMe"Ip, MKl\1
6)
250
;:!
'-' 20
]5
d
:T 10
'"
:.:
,:s: 5
o
t
(J
-5
-lOq7
18 19 20 21 22 23
TOJllIJ.lIHR, MKM
.. V
0
./
0
/'" yo
0 /"
0/ V
1540
6) 4()0
,::1 300
200
100
0
':s:
o
1;-100
g -20
c..
-30q7] 1 3
lllHpliHa .D./IHHHbIX ropCHOHOB, MKM
)
600
;:!
'-'
400
'"
:r 200
d
:.:
,::;
d-2
c..
-400
15 17 18 I
lllHpHHB KOpOTKHX ropCHOHOB, MKM
PUC. 5.38. 3aBltCH!IIOCTit pacCTpoiiKIl <JaCTOT OT H3MeHeIlHH:
a - HapyJKHoro ,nH8MeTpa 1'OTopa; 6 - TOnlI.{HHbJ CTPYKTYPhI; 8 - IliHpHHhl ,nJlHHHbIX TOpCUOHOB;
Z - IliHpHHbI KOpOTKHX TOpCHOHOB
376 rnana 5. IBMEPHTEJIbHblE CBOI1CTBA MIIKPOMEXAHW1ECKIIX nrHEOPOB
0)
" 1
c..
0,95
c..
v
.
o
- 09
!: t
<;
'"
0,85
-go
"
\5
..
0,8
1450 1460 1470 1480 1490 1500 1510
,[(lIaMe11', MKM
1520
ti
..
e)
.:
..
!:
<;
'"
c..
::::-
0,015
c..
<;
0,0
.e.
'"
0,005
6)
0.02
-5
0,015
c..
t:
"
5 0,01
'"
-eo
!i1 0,005
'X
:;:
.c
"
1:
u
"
2)
0,02
O!
.8:
0,015
c..
t:
u
0,01
.a.
-eo
'"
0,00
'=
\5
..
t:
g
o 15 16 17 18 19 20 21 22
JlW.p"lIa KOpoTKH)( TOpcHOHDB, MKM
PRe. 5.39. 3aBHcHMoCTH MaCWTa6ltbiX KO) 4IHIIHeuToB OT II3MeHCHIIH:
a - Hapy"ACHoro pa.!lHyca pOTopa; 6 - TOJlI.l.\l1HbJ CTPYKTYPbI; e - WlIpl1HbJ )1J1l1HHblX TOpCHOHOB;
Z - WHpMUW KOpOnQiX TOpCIIOHOB
K 2 = 8 ,2782 +36567 2 +20,022 +173.1252 =
= 178,266 fl.l
MKM
ECrrH npIDUlTb ,l\OnyCK Ha H3roTOBJIe-
HHe H3 ypoBHe 0,2 MKM, 'no flBJUJeTC}I ,l\OC-
TKiKJfMbIM, TO H3MeHeHHe 'lliCTOTbI BCJJe,llCT-
BHe norpemHOCTeH H3rOTOBJIeHIDI COCTaBHT
AO)TI =O,2 K) =0,2-165,847 = 33,169f1l;
AO)T2 = 0,2 K 2 = 0,2 -178,266 = 35,653 fu.
MoJICHO c'urraTb, 'no paccIJHTaHHbJe
norpeIllHOCTH npHBOJtSlT Ie OWH6leaM npH
BbI'lHCneHI1I1 'lacTOT < 2 %, 'iTO npll
O)y = 3000 f1l COCTaBmJeT 60 fLl, T.e.
1
BC/)TI == -Bc.o'"(.
2
3aBHCHMOCTl1 paccTpoHKH 'laCTOT
A = 0)13 - 0),( OT pa3MepOB pOTopa H ynpyrMx
3JIeMeHTOB nO Beca (TOpCHOHOB) AnSI npH-
H}l1bIX BbIwe HCXO,l\HbIX aHHbIX npHBe,l\eHbI
Ha pHC. 5.38 (paCCTpoHKa AnSI HOMHHaJJbHbIX
pa3MepOB COCTaBJIHeT A = 3044 - 3000 =
=44 rl.l).
113 rpaqmKoB H3 pHC. 5.38 CJJe)J,yeT, qTO
paccrpoHKa qaCTOT B MeHbilieH CTeneHH 3a-
BllCHT OT H3MeHeHHSI pa3MepOB l1aMeTpa
pOTOpa H B 60JIbIlleH OT 1l3MeHeHHSI TOJIUU1-
HbI CTpYKTYpbI H IllHpHHbJ )UIJ1HHbIX I1 KO-
pOTKHX 'fOpCI10HOB.
MacWTa6HbJe K03 HuHeHTbI ,llJISi CHU-
Ij)a3HOH H KBMparypHoH COCTaBJJSlIOIlJ,I1X ITPH
YCJIOBHH, 'lTO ,llJIfI pOTopa J y :::: 2J p , onpe-
,l\eJIHIOTCfI no opMynaM (4.326), a cyMMap-
3ABHCMMOCTb Yl3MEPMTEJIbHbJX CBOHCTB Mf
377
Hblii MaCllJTa6Hbli1 Km q)JjUHeHT no qJOpMY-
JIe (4.224). )J.nSl npHHSlTblX HOMIfHaJJbHblX
pa1MepoB mleel\l: tJ. = 211:.44:::; 276,6 palLlc;
<0,( = 3000. 211: = I,H85. ] if pa11lc H. npHHHMa51
Yo = 1,5" - 11: / 180 = 0,026 pa,L1, J3 = 10-4 , no
(IJOpMyna:>1 (4.326) H (4.224) ITOJIY'lHM
K = 6.45 . \0-7 pa)J,/(pamc), K = 9,46 x
xlO- pa11l(pa.n/c), K" =9.46.10- 5 pam
(pam c ).
Pe3ynbTaTbl Bbl'-UlcneHHSI cYMMapHoro
MacwTa6Horo K03 q)JjUHeHTa B 1anHCHMO-
CHI 01 H3MCHeHHSI reOMeTIJH<lecKHx pa3Me-
pOB Mr npl1BeIl.CHbl Ha pEC. 5.39.
1-13 pIJc. 5.39 CJIe,nyeT CHnbHa5! 3anllCIf-
MOCTb MacwTa6Horo KO::I(p HUHeHTa OT reo-
MeTpH<lecKHx pa3Mepon ynpyrHx 3JIeMeHTOB
(TOpCIJOHon) ITO,L1BeCa, qTO nOHSITHO H OTMe-
'laIIOCb paHee, TaK KaK OTKnOHeHl1e p83Me-
pon OT HOMIIHaJJbHblX ()J,onYCK) BbI3bIBaeT
H3MeHeHHe 1.JaCTOTHOH paCCTpOHKH.
AHaJJOrH'lHO IJ3nOIKeHHoMy MOIKeT 6blTb
npoaHaJJH3I1pOBaHO B.'JIDIHHe IJ3MeHeliHSI reo-
MeTpH1.JeCKIJX pa3MepOB Ha xapaKTepHCTHKH
Mr JIl06bIX ,l\pyrHx mnOB.
5.8.3. BJlHHHHe JlIfHeUHOu If
ymoBou BIf6pm..um OCHOB3HlfH
OueHHM BHa'laJJe BJIWIHHe nHHeHlioH
nIJ6pauml OCHonaHIDI Ha Mr C HepaBHO>KeCT-
101M ynpymM no,nneCOM.
Ha pHC. 5.40 nOKa3aHa cxeMa Mf C
nepBHqliblMIJ KOJIe6aHl1S1MH OTHOCHTeJIbliO
OCIJ OZ, (Koop,nHHaTa y) H BTOPHqHblMH
OTHOCHTeJIbHO OCIf OX (BbIXO,nHWI KOOP;1H-
HaTa a). H3MepSieMasJ yrnoBa5! CKOpOCTb
OJ'"
O'leBH,l\HO, qTO HaH60JJbUlee BJIl1SIHHe
Ha norpelllHocTb Mf B IJ3MepeHHH yrJIOBOH
CKOpOCTM 0 y 6Y,lleT OK83blB8Tb JIHHeHHaSI
BH6paUHSI B ITnOCKOCTH YZ, TaK K8K MOMeHT
M;, 06YCJIOBJJeHHblH HepaBHO:>KeCTKOCThIO
no,nBeca If HHepuHOHHblMI1 Cl1naMH, ,l\eHCT-
BYIOIlJ,HMI1 Ha pOTOp BCJIenCTBHe BI16poYCKO-
peHM W, BJIHSleT TaK :>Ke, KaK H rnpOCKonM-
z
w
H
}'
PIIC. 5.4U. K onpe eJleHIIIQ norpeWHOCTl1 Mr
qeCKMH MOMeHT M.. = HO.!. BOKpyr BbIXOIl,-
HOH OCH Ox. B COOTBeTCTBMM C pHC. 5.40
HMeeM
M_ = F=J' - Fyz ,
(5.138)
r)1e Fy = -mf .; Fz = -mW= ;
y=mWyIG y ; z=mWzIG z ; m - Macca
pOTOpa; f ., W= - nH6poycKopeHHSI B Ha-
ITpaBJIeHllM COOTBeTCTBYIO llX oceH; G...,
G z - )KeCTKOCTH ynpyroro no,nBeca B Ha-
npaBJIeHHH COOTBeTCTByro HX oceu.
Ilpe06p83yeM paBeHcTBo (5.138) CJIe-
)1YIOIlJ,HM 06p830M:
L 1 11 2 1 1
m x =-m WyWA G - G )=
y :!
=-mww [ - J - mwyw= [ I_ ro;o J '
v z 2 2 ., .,
- ro yO ro zD ro;o <0;'0
(5.139)
r,l\e royo, OOzo - 'illCTOTbI C06CTBeHHbIX KO-
ne6aHHH Mr B HanpanJIeHlfH COOTBeTCT-
BylOllJ,l1X DceH.
BBe,neM B paccMOTpeHHe K03 HUHeHT
HepaBHO:>KeCTKOCTI1 no,nneca:
K;" = 1- OOzo I ffiyO
(5.140)
378 rnaBa 5_ 113MEPMTEJIbHbIE CB0I1cTBA MHKPOMEXAHYl4ECKHX OPI1EOPOR
H nepeITI1IlleM BblpIDKeHl1e (5.139) B BHJJ.e
M =(K zmW)'W:)/ro;o'
(5.141)
OqeBHJJ.HO. 'ITO npl1 paaeHcTBe
ro=o = royo (nO.!IBeC paBHO)KeCTIGIM) M = 0 -
npe)J,llOJIO>KHM. 'ITO COCTaBJlSlIOIl{He
Bl16poycKopeHHH I1MeIQT O)J,HHaKOBbIe 'lac-
TOTbI (()o. HO CMeIIleHhI no 83e B03,l\eHCT-
BmI \jJ :
W, = W o sin root; W= = JI- o sine root + \jJ) ,
flle W o = W yO = W=o - aMITJIlIT)'lla Bu6po-
yCKopeHIDI.
JJ;anee 3anllllleM
,",V= =
= W o 2[ (l-cos 2 roBt)cos\jJ + sin(2ro o t) sin \jJ J
(5.142)
OTl\. lla B COOTBeTCTBMH C paBeHcTBoM
(5.141) c..'IeJIyeT, 'ITO HMeIQT MeCTO noCTO-
jlHHWI COCTIlBJVlIOIllWl B03MYllI8101I1ero MO-
MeHTa II COCTaBillIIQIl{WI Ha Y,llBOeHHOH qac-
TOTe BH6paWiH.
lh BhIpIDKeHIDI (5.142) BI1.!UiO, 'ITO ITpl1
roll = O,5ro u o (ro aO - qaCTOTa C06CTBeHHblX
KO!le6amrli Mf OTHOCMTeJIbHO BbIXO.D.HOH
OCH) Ha6J1JO.naeTCj{ pe30H8HCHOe YCHJIeHHe
MOMeHT8 (5.141), nOCKOJIbK)' qaCTOTa ero
R3MeHeHIDi nplf 3TOM paBHa <iaCTOTe roao.
OqeBHllHO IDJJI TOrO <iT06bI BJIIDIHHe HepaB-
HO:tKecTKOCTU 6bUlO MHHHMaJIbHbIM. He06xo-
AliMO co6mO.neHHe HepaBeHCTBa
M r =HrP y »(K;'.zmW o 2 )/ro;o, (5.143)
OTK)'.na CJIeJIYlOT OfpaHliqeHIDI Ha K03 H-
IUfeHT HepaBHO:tKecTKOCTM:
2
H H rPyCJ)zO
Kyz < 2 ' (5.144)
mW o
Ha OCHOBaHHH opMynbJ (5.144) 3a-
KJIIO'IaeM, 'lTO K03 Hl{HeHT HepaBHo,..eCT-
KOCTU TeM 6oJIbme, lleM 60JlbIlle 'laCTOTa
COzO - o.WUUCO B CHJJY M8J10H TOJlll.UiHbl pOTO-
pa no cpaBHeHHIO C ero ,lUi8MeIpOM 3a-rpy,l\-
HltTeJlbHO nonY'lHTb BbICOKoe 'ma1JeHHSI
(()=o . TaK KaK pOTOp npH 3Ha'lHTeJIbHbIX BH6-
pauHoHHblX B03MymeHI1H.X cnoc06eH ,lle-
OpMUpOBaTbCSI.
113 Bblpa)KeHIiR (5.143) MO)f(CT 6bITb
nOJI) eHO OfpaHIf'IeHlfe Ha MaKCIIMaJIbHYIO
BeJIU1JUHya!\tnJIlIT)T)J,bI BH6pauuu:
Ron,.
J1' o < ro zo
(5.]45)
K H
111 F
flplUlep 5.19
PaCC'HITaeM K031jJlpl1III1eHT HepaRHoiKecT-
KOCTI1 .lI.JlH. cne lOllU!X HCXOf\HbIX ,UaHHbIX:
Q}.=100 c /'I",5.10-4 l1c , m=2.10- 7 Kr.
1
W o = 1 00 M/c- (- 10 g)_ vh npl1Mepa 3.4 I1MeCM
C 1 =J o =2,5-10- 13 Kf-M 2 , rozo = 645Sru =
=4,06.10 4 I1c, royo =2100 ru = [,32.10 4 I/c
c 1,5
Ilpl1HI1MaeM Yo = 1,5 = - Pilll. H nony-
57,3
'JaeM Ho = Jo"foroyO =2,5.1O-13 ],32.104 =
57,3
?
10 Kf.r[
= 0,86 -10- . Ilo .popMYHe (5.144) Haxo-
c
,UI1M
-10 0 -4 (4 4 ) 2
KH < 0,86.10 .5.1 ,06-10 <0034
y;: - 2.10- 7 (100)2 -,.
.z:LnH. peaJIbHbIX napaMeTpoB Mr pa3HH-
118 B 'laCTOTaX ffiyO 11 rozo cYll\eCTBeHHa 11
3Ha'leHl1e K03 I1l.{HeHTa HepaBHO)f(eCTKO-
CTH npH6JIM:>KaeTCSI K e)J,I1HHue, BCJJe.ncTBHe
'JefO YCHJII1BaeTCSI BJlHSlHHe JIMHeHHOH BH6-
pal.{llH Ha norpeIllHOCTH Mr.
flpu-\fep 5.20
PaCC'IHTaeM B03MYIllaIOlJ.1I1H MOMeHT, o6y-
cnoBneHublH nl1HeHHOH BI16paI.\HeH, H COnOCTaBI1M
ero C rl1pOCKOnH'leCKHM MOMeHTOM .lI.IDI napaMeT-
pOB Mr 113 npl1Mepa 3.4: (t)zO =4,06.10 4 lIc;
(0)'0 = 8,17 .10 4 lie.
BbI'JI1CJII1M
K03.p.pI1l.1l1eHf
K H -
)'z -
= 1_ ( 4,06 ) 2 = 0 998 H nOJI)"IHM
88,17 '
= 0,998.2.10-\100)2 - 1,2.10- 12 H'M.
(4,06.10 4 )2
MIi-
x-
3ARI1CHMOCTh H3MEPHTEJIhHhIX CBOi1CTB Mr
379
310T :\1O:\feHT. paBHhlH no Be.1H'IHHe mpo-
CKOnM'lCCKOMY l'vl ! = AIr' B01H11K 6bI npll neH-
CTSUIi yr :IOBOH CKOpOCTI1 D.J = M.:', H(J.
H3
npellbI.L!yweI 0
Ho -0,86.\0-10 Kr ':\1-
C
npHMepa
11,
C:Ie.l10BaTCJIbHO,
1,2.10- 12 1
D.; - JO =0.014-=0,0]4.57,3-3600=
. 086.10- C
,
=2878,3 0/'-1. 3aMe111M, 'ITO }J,JUI 0..1' =100 0/'1
11IpOCKonll<leCK IH MOMeHT ""lr = 0,86.10- lO x
x5.]0-4 = 4,3.10 14 H':\f, '1TO npaKTI1'-1eCKI1 Ha l1Ba
nOpll,iJ,Ka MeHbwe 1Ha4eHI111 M. = 1,2.10- 12 H.!\1.
113 npl1Mepa CJIeJJ,ye"l, 4TO 1II1HeHHaH. Bl16pa-
1!1I11 npn K03(!JIjJHI\lIeHTe nepaBHOiKeCTKOCTH
K;= 1 MOiKe"l npl1BO.l1l1Tb K CYIlleCTBeHHbIM
I1CKaiKeHUJ! BTOpl1'lHLIX (BbIXOlIHbIX) K01le6aHHH
ML flpu 3TOM HYiKHO I1MeTb B BI1.l1Y, '-ITO B03MY-
WalOlU11ii MOMeHT I1MeeT 'laCT01Y BTOpl1'lHbIX
KO;Ic6amli1: 11 nOJTOMY ero He:Jb3H YCTpaJJliTb C
nOMOlIlblO IjJU1IbTpaQml BbIXOl1Horo Cl1rHana Mr.
[Jpll.llep 5.21
Bbl'lI1CJIHM .l10nYCT !MYIO aMnJIl1ry.l1Y 1111-
HeHHoro YCKopeHlliI Bu6paUHlI npl1 1i3MepeHHI1
yrnoBoH CKOpOCTH D.y = 100 °/q = 5.10- 4 pa,n,
C
DJIlI napaMeTpOB Mf 113 npeJJ,bIl{)'IUHX npHMepOB:
2
ro,O =4,06.10 4 .!.; Ho =0.86.10- 10 Kr'M ,
- C c
m=2.10- 7 Kr; K;" =0,997. no q,opM}'ne
(5.145) HMeeM
086.10- 10 .5.10-4
W o 4,06.104 ,
2.10- 7 .0,997
= 18,85 M/C- = 1,92g (- 2g).
PaCC<IHTaeM aMnJIMTY.l1Y 1II1HeHHOH BH6pa-
4 1 2
l!11M npM roao = 1,73 . I 0 - Ao = W o / roao =
c
= 18,85/(1,73.10 4 )2 =63 MKM.
113 npHBe.n.eHHoro MaTepHaJIa CJIe eT,
<JTO npl1 00 0 < 0,5w a o pC30HaHcHoe yCI1J1e-
HHe B03Mywal0llJ,ero MOMeHTa, 06YCJIOBJJeH-
Hoe HepaBHO:H<eCTKOCTbIO nO,l\BeCa, OTcyTCT-
BYCT, T.e. OTna.n:aeT Tpe60BaHue K paBHOiKe-
CTKOCTH ITOJJ:Beca. O<IeBH,llHO, B 3TOM CJIY<Jae
BbIBO)1 0 BJ1HSlHHH efO HepaBHOiKeCTKOCTH
Ha,no ,QeJIaTb Ha OCHOBaHHH paBeHCTBa
M. = HOQJ,Qa,
me 1v1. onpeJ1eJISleTCSI no opMYJIe (5.141).
B ,l\aHHOM cJIY':Iae 'laCTOTbI BH6pamIH H
BTOp1l 1 IHblX Kone6aHHH Mr He COBna.n:alOT H
MOiKeT 6bIn npHMeHeHa cpl1JIbTpalUiSi .D.JUI
Bbl)J,eneHIDI none3Horo Cl1fHaJIa.
(5.146)
npllMep 5.22
CpaBHI1M MOMeHTbI HepaBHOiKeCTKOCTIJ II
rupocKonM4ecKoro. 113 npHMepoB 5.19 II 5.20
MMeeM
M; =1,2.10- 12 H'M, Ho =0,86,10- 10 Kr'M- ,
C
fly = 100 e /4 =5 .10-4 I/c '" nonOiKMM Oa = 75.
BbI'JMCJIUM 11l r = 0,86 . 10- 10 . 5 . 10-4. 75 =
= 3,2 .10- 12 H . M , T.e. rllpOCKOnU4eCK11H MOMeHT
no aMnnHryl1e COH3Mepl1M C B03MYlUaIOlUI1M MO-
MeHTOM, HO era MOiKHO YC1paHI1Tb q,MnbTpanllei:i
Cl1rHanOB, TaK KaK OllH (B03MYTI(alOlUHH 1I nOJIe3-
Hblll) pa3HO'laCTOTHbl.
Ol\eHMM BJIHSlHl1e HepaBHOiKeCTKOCTH
nO)J,Beca B nJIOCKOCTH XY Ha B036Y:H<,lleHHe
nepBM'lHbIX KOJIe6aHHH poTopa BOKpyr OCH
Oz. MOMeHT BOKpyr OCH OZ MOiKeT 6bITb
3aITl1CaH aHaJIOrH'lHO opMYJIaM (5.138),
(5.139) B BHAe
H K IJ I 2
M z = ( . .m W,.W y ) (J)xO,
(5.147)
H 1. I 2
f.n.e K X)' = 1- (J)xo ffiyO'
AMnJII1Ty.na MOMeHTa .n.BHrareJIJ'I B03-
6y:>K)J,eHHSI nepBI1'lHblX Kone6aHHH C Y'leTOM
(4.31 I) onpe,QeJIJ'leTCSI BbIpIDKeHHeM
?
Jo(J)yo'Yo
lv/ o = ,
Qy
(5.148)
fAe 'Yo - aMnml1y.n.a nepBH'lHbIX KOJIe6a-
HI1H; J o - MOMeHT I1HepllHH pOTopa OTHOCH-
TeJIbHO OCll OZ; wro, Qr - qaCTOTa C06CT-
380 fnaDa 5. H3MEPI1TEJ1bHbIE CBOlicTBA MItKPOIvIEXAHl\ l lEl'I(I1X IIPIII;OPOB
BeHHl,lX KOJIe6aHI1i1 11 )J,06pOTIJOCTb pOTopa
OTHOCItTeJIbHO OCI1 OZ COOTBe1'C1'BeHHO.
nplL1Iep 5.23
BbInOnHllM CpaBHllTc.TJbU)'K1 OUCHK)' 1\(1]-
MYIl!BlOll!cro MOMeUTa. o6ycnoB.fIcllllOn1 IIcpan-
HO)!\eCTKOCTblO IIOJIBCC3 OTllOCltTCnbliO {lCU OZ. II
MOMCHT3 .l1BUI'aTcml BOJ6YJK.1elllHl nCpml'lllblX
KO!le6aHHH pOTopa .1JJU! C!IC.1lYIOll!II:\ I1C:\O!l.JthI:\
.l13HHbIX:
J o =2.5.\0-13 Rr'M". m = 2.10 7 Kr, lOrn =
= 3000f11 = 18.R4.10 3 ..!.. Yo -=15°=2,62.10- 2 pa..'1.
C
441
Qy = 10 , (!),'{) = 8Q5.1O -
C
(1),-0 = RR.l4. I 0 4 .!.
- c
If' = f J' = \ 00 M/ C
x y -
Bbl'iIlC:IHM K , = 0.17 Ii no QJopM)'nc
(5.145) nOJl}"'IlIM:
M = 0.17. 2 .10- 7 . (1 00)2 :: 5.2.\ O-H,
. (80.5.1 0 4 )
H-M.
no cpopMY 1e (5.148) Bbl'iIlCJUIM Mo,nynb
MOMetrra ;tBHraTe.'UI BOJ6YJ!UleHIUI:
M _ 2.5.\0-13 .(18.84.10 3 )2 .2.62.10- 2 =
(J - 10 4
=2,37.]0-10 H'M.
1-13 npHMcpa cne.i1ye-r. <iTO B03Myrn3101l!1IH
MOM em OT HepaoHO)!\CCTKOCTH no,nBCC3 HB l1e-
CKOJlbKO nopRllKOB MeHLlliC MOMeHTa llBl1raTenSi
BO'36y;KlleHHR H era MO*HO He Y'lHTbIBaTh.
TIpoBe)J,eHHblH aHaJIH3 nOJBOJISIeT c,l\e-
JJ3Tb CJIellYJOJ..U.He 3aKJ110'leHHSI 0 MeTOllaX
CHlf"ACeHIDI BnHflHIUI Ha BblXO.nHOH CHffiaJI
Mr nHHeHHOH BH6paIUUI:
. YMeHhWeHHe K03$Ij>HWieHTa HepaB-
HO:>KecTf(OCTH nO..l1Beca B nnoCKOCTH nepneH-
,lUUCYlUIPHOH K OCM BTOpH<JHblX (BbIXOJ.lHbIX)
Kone6aHldi pOTOpa;
· ynenHIJeHHe <JaCTOTbi co6cTBeHHhIX
K01Je6aHIfH pOTopa (npH :nOM B03MOiKJfO
npHMeHeHHe r"aCTOTffbJX Mero.nOB Ij>HJJJirpa-
IIHH AJUJ BbJ..!:teJIeJJHJJ none:JHOrO CHrH8J18);
· 1(0ll1'pOJIh 'taC roTM nnGpUllIlollllhlX
B03MymcIIII ii. KO'l"OP,1I1 J\OJl)J{IUI GMTh MCllh-
UlC nOJlOUllllht 'mcTOTIoI C06CTIICllllhL\ KOJIC-
6mwii Mr OTIiOCIITCJlhllO nhlXOJll1oii OCJI
(BO: MmKml 'I <lCTOTI/ml (jlllJlhTpallml nh'XO/\-
Horo Cl1nIaJla).
OlleJlIIM TCHCph HJIIUJIIIIC yrJl(1J!oii Im6-
paUlm OClioumlllH OTlmC1ITCJIJ.110 IIbiXO/lIIOii
OCI1 11O)J,Bcca Jla 113MCpllTCJlhllbiC cnoi-icl'lla
Mr. Ou.clII<y l\I>1Il0JIIUlI\1, CpaBJlllllaSi MaKCH-
MaJlhHble ar-.1nmnY)J,bI Yl"JlOBhlX OTI<JIOIICIIlfi1
pOTopa OTiIOCIITCJIhllO BhIXO,llIlOi-i OCII. IJhI-
:maHHbiX IIIICPUl10!lllhll\1 1\10MCIITOM IJCJIC.n-
C1'RUC ynloBoro YCKOpCll1UI OCIiOBallllH 11
rllpocKODll'lCCKIlM MOMCIITOM B PC'3YJIhTaTC
I13MCpSiCMOii yrnoBorr CKOpOCTII.
AMnJIlfT)')J,a yrJIOBOrO OTKJ]OHCIIIUI po-
Topa ItJ-3a yrJIOBOH Bu6paltJ-II1 OCIiOBallIISI
on pC)J,eJIHCTCJI cpOPMYJ10H
"
am:!"
= J,r. _ E
G
u
(5.149)
.. ,
w;;'o
r)J,e J x , G u . (!)aO - MOMCHT llHCPQMlI p01'O-
pa. iKeCTKOCTb no.n;Beca II <mC1'OTa c06c-fBeH-
HblX KOJIe6aHUM OTHOCMTeJlbHO BhIXO.n.HOH
OCM COOTBeTCTBeHHO; E - aMnJll1TY)J,a yrJIo-
BblX YCKOPCHUH OCHOBn/HiSI.
AMruJl1TY,l\a yrnoBoro OTI<JIOHCUI1S1 po-
Topa 110.n. .n.CHCTBIICM fHpOCKOnll'lCCKOI'O
MOMeHTa Bcne)J,CTBlIC H3MepSicMofl yrnoBoH
CKOpOCTI1 H8X0J1.I11'CSlI10 Bhlpa)J(enIllO:
r
am:."
_ M rQo _ Hun J.Qu
- -..
G u w;;'oJ x
Huny
2IJ.CJ)J x CJ)aU
(5.150)
f,l\e Ho - KHHeTH<JeCKIIM MOMCH1' po'ropa;
Oy - H3Mep1leMaSI yrJIOJ3aS1 CKOpOC1'b; /).CJ) -
JlOJIoca nponYCKaHHSI <JaC1'OT.
flpUJ.tep 5.24
OUCHHM BJ1HRHHC yrnoBoH IIHOpaL\HJoI OCIIO-
BaHHJI C aMIIJ1Jo1T)'AOH 5° H 'laC1'O'fOH 5 fu npH
1f3MepCHHH yrnoBoH CKOpOCJ'JoI fly = 100 o/IJ ""
]AJUICHMO{'Tb H3MEPI-ITEJlbHhIX CImFtcTB Mr
3HI
- 5 . 1 I) ,I 1 Ie II IIOJlocoil IIPOIIYCJ{lImUI 'lac-llH
A(I) - 20 rll 125.6 I/c JUIJI lIapaMcrpoll mpocl(o-
,
1 1' ,, - Ilrl -. (), " (! . I() In 1\1, M" IJ 2
() ./ \ = 1 (). I{I"M ,
C
'C
(II,,(J 2700 I'll ' 16.l).IO I/e.
BhI'lIICJIIIM aMIIJlIlTYJlY YIJIOIIUl 0 YCIW-
5 ( ) ? ,
pC1I1II1 OCIJUlIiIllIUI !:=- 5.21t. =H6 I/c. II
57.3 .
110 (ImIJMY;IC (5.149) 1I0JlY'III M
1\6 7
-= , =3.10- pall'" 0,06".
(16.9,\03 r
MYJlC (5,150) pacC.I....11ICM
.11
a ,"ux =
no (1IOp-
a: tulX =
O.X6.10 III .5.10-- 1
10- 7 pall = 0.02",
2,125,6, 10 13 .16,9.10 3
Tal\l1M ofiplrlOM, a.\lI1JIIITYll<l J<0llc6mlllii pO-
TOptl 110)\ Jtciie'l1JUCM YI'JIOIJOro yel(opCHlUI OCIIO-
lIallllll OI<:\"ntllaCl> GOJlhlllC lIMIIJIIITy.abl 1(0JlcGamJi1
pOTOpll, ()(jYCJIOIIJICllllhiX U'!MCPIICMOM yrJIOIJOH
CJ<OpOC I 1010.
npll allaJllnc B0'3MYUlcllui.i, llliOCI1MhlX
YrJlOHoii BII6paUI1\ i.j OCIIOUaJlII lCM. <llop-
MYJIY (5.149)]. CJlC)),YCT IIMCTI> B BH)),Y, <11'0
BhIllY)!(jtCIHlI>IC KOJIc6a1I1H/ po'ropa OTIIOCI1-
TeJll>lIO HI>IXO)J,llOii OCH I1MCIOT 'laCTOl'Y yrJlo-
BLIX KOJIC6m1l1i.j, a II1Mcpm CJlbmU! IImllOp-
MaUHS! rCM. Bl>lpa)!(CIII1C (5.150).1 HllCT ..a
'IaCTOTC C06CTUCl!lIbiX KOIIC6aJII1ii pm'opa
OTlIoclrrcnbllO BI>IXOt\llOi.i OC I Mr.
1.1.3 npuMcpa 5.24 CJlC,llYCT TaK)f(C. 'ITO
)),aT<II1K ncpeMClltCHl1ii Mr ,llOmKCIi 06HaJlaT"
O'ICIiI> 60JlLlJJOH 'IYUCTBUTCllbIIOCTI>IO. B ca-
MOM ,llCJIC. lllllidiHOC ncpCMCIlJ,CIHlC pOl'opa,
COOTBCTCTBYIOIUCC 111MCpHCMO i YI JIOBoii
CKopocnl (JOO °/(1). COCTaBJ1HCT Bccro
I1z = Ra: n3x (R - pa,llHYC pm'opa) 11 pamlO
k = 1,5 .10- 3 . \0-7 = ].5.\0-111 M '= 1,5.10-4 MKM.
TaKHM 06pa30M. CCll11 HOMI1I1aJJblmH
CMKOCTb )),aT<II1Ka nCpCMClltCI1I1H COCT3BJUICT
Ct\n"HI hl TlI1Ko(IJapaj], (10- 12 cD). Jl,aT<lIIK
llOJI:>J(CII pCarl1pOB3Th Ha H3MCHCIIHH CMKO-
CTI1 n COTLIC )),01111 nl1KOIIJapa)),a [()leMTO( apa)),
(JO 15 cD)].
5.8.4. BmUIIIIIC opllcllnuum ynpynlx
')JICMCII'r08 1IO.IJ,8CCn 118 IIJI8C'rUIiC
IIoUIliOKplleTHJJJm',CC Korn KpCMlUlJI
AIIH30'fpOrHIH cnoi1cTB KpeMIIHH pac
CMOTpCI18 B nO,llpa3Jt. 2.1.2, mc B Ta611. 2.2
nOKa3al1bl H3MCIlCI11111 MOJJ.Yl(}f fQI1ra. MO)),Y-
JI c)J,IJllra II Ko:J( q)HUI1CIlTa ITyaccol1a B 33-
BI1CI1MOCTI1 01' KpI1CTaJ1J10rpaCpWICCKHX I1a-
npaBJICllI1H ,lUl TpeX llall60JIcc 'l8CTO I1C-
nOllh3YCMhiX npH np0I13BO)),CTBC MI1KpOMC-
XaHI1<ICCI<I1X npl160poB nJIOCKOeTCH: (100),
(110) 11 (I I 1 ).
PaCCMOTpl1M BJIMmll1C al11130TpOrll1l1
KpeMHI1 na xapmnCpl1CTHKH Mr, BhIflOJI-
HCIlHoro no pnc, 3.11. Ha pl1c. 5.41 yrOJI 9
OnpCllCJIlICT Opl1CHTaUl1l0 TOnOJlOrl1J1 ynpy-
rllx :JIICMCHTOB. COBnaj],aIOlltIIX e OC11MI1 X, Y
{)'I IIOClrrCllbllO KpI1CTaJ11Iorpa H<JCCKI1X I1a-
npllBJ1CIHlH XI. Y I .
3umHIICM ()JOpMYJ1hl (3.28), (3.29) C
Y'ICTOM yrna 9 opnCHTaLt1111 TonOllonm Mr
OTIIOCI1TCJ1hIIO KpHcTruulorp3 11<leCI<I1X 118-
npaBJ1CHI1H:
Gy(n)::; CII [ E,,(9)br 1 + E" 9+90" br 2 ] ; roy(n)::; @Jfil;
3
Gu(n)= 2 GI1(9)krbnIC + 1 E" 9+90" hIl2(' ; rou(n)::; G",(TI);
/"1 3 /..2 J",
G ,i (n)=2 G " 9+9(t k'bn2C + J E,,(9)bllc ; roJ}(n) =
/"2 3 /11 T'
r,l\e E,,(9), G n (9) - MO,l\YJlI1 IOHra 11 ClI.BHra,
33IJHCj(Ll.I,I1C 0'1' I<p"CTa.rIJIOrp8 ""ecKoi1
(5.151)
nJlOCKOCTI1 (n) II KpnCTaJIJlOrpa$H"leCKOrO
HanpaBJJCHI1Jl (9).
382 fmum 5. J13MEPHTEJIbHbIE CBOI1CTBA MI1KPOMEXAHH1.J:ECKI1X flPI1EOPOB
IY'
. .
x.
"." ... .
.:'.;0 ":..:(" ...;.... ..;
- . ._ .. ."-"_ I.
PHe. 5.41. TOnO;JOrHH Mr Ha
KpHCTIL:'L'Iorpac)JH'IeCKOH D.110CKOCTH (ll)
I1pl1MeM cJIe YJOIl.\He 3Ha eHl1SI napa-
MeTpOB Mr: J[ = 0,2. J., = 2,5.10- 13 Kf'M 1 ,
J a = Jf\ =10- 13 Kr'M 1 (CM: npl1Mep 3.4), b n1 =
= b n2 = 27.1O M, lTI = IT"!. = 0,8.10- 3 M,
C n = 20.1O M, onpe.neJIHM H3 Ta6JI. 2.2 3Ha-
eHIDI En, Go ..!1.IDI pa3.JlI1'lHbIX KpHCTaJIJIO-
rpa<]>J1qeCKHX nJIOCKoCTei1 11 HanpaBJIeHHH H
no $opM)'21aM (5.151) pacct.lHTaeM 3aBHCH-
MOCTH qaCTOT nepBH'lHbIX royO(l1) H BTOpl1 -
Hl>IX CJ)aO(TI) = ro TI) KO.'le6aHHH Mr, rpa-
$I1I0i KOTOpLIX npHBeJ1.eHbI Ha pHC. 5.42.
0)
AHa.r1H3 3aBucHMoCTCH Ha pHC. 5.42, a
n03BOJIJ'leT C)J,CJIaTb CJlCl\yI{)Il\I1C UbIBO)J,bI.
1. I\1aKCIIM3.!lbI1aJl <mCTOT<t nCpBI1<lIibIX
KOJlc6aHI1i1 COOTBCTCTBYC'I' pacnollQ)I(CHIU{)
CTPY"l)'PbI Mf B f1110CKOCTI1 (110) C OpI1CH-
Tal{l1eH TOpCIIOHOU no):! ymoM 54.7° K Ha-
npaBJlCHIHO [100] (CM. Ta6J1. 2.2). 3TOMY
HanpaUnCHH1{) OTBe'laeT CI1MBOll [110].
2. TIpH OpMl1pOUaHHI1 Mf B nJIOCKO-
cm (111) 'laCTOTa Kone6aHHiI 11e 3aBI1CI1T OT
Opl1CHTaUm1 TOpCI10HOB, 'lTO 06'bHCH eTc
M30TponHOCTbl{) KpeMHH B :noii nJIOCKoeTH.
3. MaKCI1Ma.rlbHIDI 'laCTOTa nCpUI1<lUbiX
KOJlc6aHlii1 llJ] Mr B nIlOCKOCTH (100) co-
OTBcTCTByeT HanpaBJlCHIUO, J1C)KaIlJ,CMY no):!
Yf110M 45° K OCI1 [100]. '3TOMY HanpaUJIeHl1I{)
cooTBeTcTByeT cm\1BOJI [110].
113 pIfC. 5.42, 6 CJICnyCT, TO MaKeH-
Ma.nbHble 3Ha<ICHHH qaeTOT BTOpH<lHbIX KO-
J1e6aHHH COOTueTCTUYI{)T TCM )Ke HanpaBJIC-
HHHM, 'ITO H ,WIJ'I nepBH'lHbIX K(>1Ie6aHI1M.
BblnonHHM oueHKY 'f)'BcTUl1TeJ1bHOCTH
Mr K H3McpHeMoH YfnouoH CKOpOCTI1 )J,JIJ'I
p33JIHQHbIX KpI1CTaJIJIOfpa l1qeCKHX IDIOC-
KOCTeH 11 HanpaBnel1l1H, OITpe,llCJIHCMYI{) CTa-
THtJeCKUM K03 Hl{HeHTOM ITepe,lla<m
(5.123). npHHJ'lTbIX napaMeTpoB Mr,
C'lHTIDI 'Yo = 1°, Qp == 10 4 , nOJlYQm.1 3aUI1CI1-
MOCTl1, npl1Be):!eHHbIc Ha pl1c. 5.43.
6)
195 0
210
225 0
o
240 255 0
PIfC. 5.42. 3aBlfcHMoCTH '1aCT01' KOJle6aHHH Mr 01'
KpIfCTILILIJOrp a 4J If 'lecKlfx DJIOCKoCTeji If HanpaBJleHHii:
a - nepBH'flIhIe KOne6aHIDI {P.lO; 6 - BTOpHtJf:lble Kone6aHHH (P1.J)
3ARI1CHMOCTb H3MEPI1TEJIbHbiX CBOI1CTB Mr
383
195 0
210 0
225 0
240 0 0
255 270 0
PIIC. 5.43. 3aBliCIIl'lIOCTII CTallllJeCKOrO K03!JIIIJlII.\lielna nepe)\alJlI
01 KpIiCl"aJIJIOrpa$IiIJCCKIIX n.10cKocnii Ii lianpaBJIeHIiH
Auaml3 pHC. 5.43 n03BOJUleT 1a10IlO-
<IHTL, 'lTO MaKeHMaJILHaH. qYBCTUHTCJlLHOCTh
Mr K Ii3MCpJlCMOM ymoBoH CKOpoCTM 11a-
6nlDJ(acTcJI B cnY<lac, KOf)J,a CTpYKTypa Mf
UbIflOJ1HCHa B nJIOCKOCTH (100), npIiqeM TOp-
CHOHbl ynpyrOfo nO,llBCCa )J,omKHbl pac-
nonaraTLC51 no HaITpaUJlCmUlM [100] H [010].
H3rOTOBJ1elH1C Mr B nnOCKOCTI1 (110)
IiMceT HCKOTOpbIC oe06cHHoCTIi, CBJ13aHHble
c 113MCUCliIieM MO)J,yn51 fOufa U 3aBIiCI1MOCTM
OT UanpaBJICHl151 (CM. Ta6J1I1I.(y 2.2). ECJI 11 ,
Uanpl1MCp, BbIllOJIHHTL O)J,HY napy TOpCIiO-
HOB B,llOJIb HanpaBJICUI1J1 [100], a ,llpyrylO
napy no yrnoM 90° - B,llOJlb HanpaBJIeHIiJI
a)
195 0
210 0
225 0
240 0
255 0
[OIIJ, TO 3TIiM HanpaBJIeHIiJlM 6Y)J,YT COOT-
ueTCTUOBaTb pa3JIIi'lHbIe 3Ha'leHHH. MO)),yml
fOHm (168,9 Ii 130.2 rI1a). TaKaH. efO acm.l-
MeTpIi51 npIiBc.n:eT K pa3JII1'lHbIM 3HaqellliJiM
co6cTueliHoM qaCTOTbI 11 lJyuCTuHTeJIbHOCTI1
MY'ITO KaHaJIaM a Ii p (pHC. 5.44).
KaK CJ1e)J,YC1' 113 pIiC. 5.44, a, paBcHCTBo
C06CTBCUI1bIX qaCTOT BTOpH'lHbIX Kone6aHIiH
OTHOCmeJIbHO oceM X Ii Y (eM. pHC. 5.41)
HMee-r MCCTO ITPH opIicm<U.(HIi TOpCIiOHOB
no)J, yrnoM 45° K HanpaBJIeHIiIO [100], HO
npl1 3TOM MIiHHMaJIbHa qYBCTBIiTeJIbHOCTb
Mf (eM. pIlC. 5.44,6).
6)
0 0 pa)l
D.D25 (paule)
345 0
30 0
315 0
PHC. 5.44. 3aBHCIII'IIOCTH co6cTBelllfLlx lJaCTOT KOJle6aHHH (a) H
CTaTHlJeCKOrO KOO$4tHl\IiCIiT8 nepe.aa'JH (6) OT HanpaBJleHIUI B DJIOCKOCTH (110)
384 rnaBa 5. H3MEPI1TEJIbHbiE CBOIlcTBA MI1KPOMEXAHI14ECKl1X flPHfjOPOB
TaKHM 06pa30M. MOjKHO C)J,eJlaTb Bbl-
BOll. 'ITO npl1 npOCh.-mpOBaHIIlI Mr B03MO-
iKeH BbI60p OnTHMfulbHOH OplIeHTaUHII CfO
CTp}'l\.--rypbI B onpeneJICIll1OH Kpl1CTaJIJlorpa-
q,J1qeCKOH llnOCKOCTII. KplITepll1i OnTHMaJIb-
HOCTH MOryr 6bITb p8.3JIII'lHbIMII: MaKCHMYM
c06CTBeHHhlX qaCTOT. !\.laKCHM)'M 'lYUCTBU-
TeJIbHOCTI1 H .1lp.
5.9. IIIYM B !\J]ll(POMEXAHIf1IECKUX
llPlffiOP AX
lllYMOM MHh.-pOMeXaHlf'leCKOfO npl16o-
pa MO:>KHO H8.3BaTb 1106ylO cOCTaBillIlOI.Uyr<J B
BblXO.aHO!\.1 cnfHa.:1e. KOTOpylO ,;J,onycTHMo
npe.aCTaBI1Tb KaK pe3y.1lhTaT B03.iJ,eikTBWJ
H3MepjleMOH q,1I31{qeCKOH BemiQUHbI (YCKO-
peHHe. ;J.aB3eHl1e, ynlOBIDI CKOpOCTb 11 T.,ll.),
B ,l\eHCTB lbHOCTl1 j1R lOI.U C CJIellCT-
Bl1eM BHelllHHX H BH)rrpeHHI1X nOMex (I1C-
TOqjlliKOM IIIyMa). HanpHMep, npl1 Ii3Mepe-
HIDIX B YCJ10BIDIX BHoparuil1 B BbIXO,IDlOM
CHrHa.,1e MHl\.-ponpM6opa B03HHKaeT COCTaB-
!:unOI.UaJI. KOTOpylO MO:>KHO H8.3BaTb Bl16pa-
lUiOHHhIM IIIyMOM.
MexaHHqCCKHe MHKpOCTpYKrypbJ 11
3:IeK11'oHHbIe cpe.rrcTBa m3ID1lOTCSI BH)'TpeH-
HHMH (KOHcrpytcTMBHblMH) I1CTO'lUI1KaMH
IIIyMa., HMelOlllHMH pa3JIH'lHYIO 11311'1eeKYIO
npHpOllY. TaK Ha3bIBaeMbiM 6pOyHOBCKHH
mYM npoHCXO,llHT OT CTOJIKHOBeHMH Me:>K)J,Y
MOjJeJ.')'.rraMH ra38 M MeXaHl1<ieCKHMH Ml1KpO-
CTp}'l\.--rypaMH. MMeeT TennoBYIO npHpo)J,Y H
06YCJIOBJlHBaeT TeOpeTH'lCCK)'lO npelleJIbHO
B03MOjf(H)'lO pa3pemwolltYJO cnoc06HoCTh
(qyBCTBHTeJIbHOCTb) MHKpOnpH6opa. I1pe-
06pa30BaTeJIH H 3JIeKTpoHHbIe Cpe.rrcTBa
(ycHJIHTeJIM, HHTerpaTOpbl, q,HJIbTpbJ 11 T.ll.)
C1IYJKaT HCT01JHHKaMM (reHepaTOpaMH) Ten-
JIOBOrO UIyMa. lllYMoBOH cneKTp BhIXO,l\HOro
CMrnaJIa pallen cneKTpY UIyMa Ha Bxo.ne,
o
G
I I
R
YMUO:>KeIUiOMY ua 'laCToTHylD xapm<Tepnc-
TUKY CUCTeMbl, T.e. 3aBlICHT OT cxeMOTeXIHI-
KII KOHCTPYKUUM. 3JIeKTpOlHlhlii WYM 3aBM-
CUT 01' KOHKpeTHoi1 cxeMbl 3JIeKTpOmlKI1 II
3,'!eCh He paCC!\taTpIlBaeTCJI.
5.9.1. nponCXOifQeHlle Ten.TIOBoro mYMa
HarmQl1c pe311cTopa B 3JIeKTpOHHOH
cxeMe accoUl1IfpyeTC C HeKoTopbIM rel1epa-
TOpOM mYMa. HCTO'lHI1KOM Uaflp.H:>KeHI1J1 KO-
1
TOpOfO JlBJU1eTCJI TerJnOBIDI 3Hcpnul -kT
2
(Ii - nOeTOJlHHaSi bOJIbllMaHa; T - TeMnepa-
Typa). Ha pilC. 5.45 flpnBcn,eua CXCMa
RLC-pe30HaTOpa (R - pe3McTop; L HU)J,YK-
TlIBHOCTb; C - KOH,iJ.eHCaTop) C feHepaTOp{) 1
lllYMa UUJ'
O'leBI1)J,Ho, <iTO TCflJIOBIDI 3Heprl1.H 3K-
BHBaJIeHTHa 3HCpnil1, 3anaceuHoH B I1H)J,YK-
TI1BHOCTH 11 K0I1)J,eHcan>pc. YpaBHeHl1e Ha-
np.H:iKeHI1M pe30HaTopa I1MeeT BI1.1l
d 2 I dI 1
L-+R-+-I=U (5.152 )
dt'!. dt C IlJ '
H.TII1
d 2 I dI '1. U III
---;- + 2/;w o - + woI = - ,
m- m L
fne W o = .J 1I LC - C06CTBCHHIDI QaCTOTa
pe30HaTopa; I; = R 1(2w o L) - OTHOCI1TeJIb-
HbIH K03 HUl1eHT lleMnq,HpoBaul1 .
,lJ,06pOTHOCTb pC30HaTopa OnpClleJISleT-
e.H BblpIDKeHHeM
Q = 1/2/; = woLi R .
(5.153)
II
c
fYY'\
L
o
PHC- 5.45. RLC-pe30H8TOp c reoepaTOpoM wYMa
llNM B Ml1KPOMEXAflli4ECKMX OPHEOPAX
385
OcpenmueM ypallHcHMe (5.152) II BH,llC
dl I J
L d/ + RI + C hIt = I(t) = VIII' (5.154)
nCpei1.1CM K onepaTopHoH (l)op le 18m1CI1
( .\ = d / dt ) 1 3 mI1IlICM BblpmKcHHc )J,J1S1 TOKa
B I..{CnH 113-3a IlIY lOnofO l-lanpmKCHHH. II
<J)(}PMC
1= Urn
I .
R + sL +-
sC
OOJ1(»KI1M n opMyne (5.155) s = iw 11
BbmOJUHiM CJIe YIOlL!HC npeo6pa30B8HmI:
(5.155)
1= R+{:- J
wC
UUI[ R-{ WL- )J _
1 ( 1 ) :!
W+ wL--
wC
iU rn ( WL- )
R2 + ( WL_ ) 2'
wC
V'Il R
., ( 1 ) 2
W+ wI--
wC
38nl1WeM CpC.D.HIOIO KBa.D.p8THqeCKYIO
BeJIH'II1HY TOKa CJIC)J,YIOIltI1M 06pa30M:
1 ( I ) 2
2 1 U wL --
1 UrnR- + wC _ UUJ . (5.156)
2 [R' + ( roL - role rJ [ R' + ( roL - rolcJ' L { R' + ( roL - rolcJ' ]
1=
Bblpa)KCHHe )J,nSl c06CTneHHOM :mcpnm
10Ka B pe30HaTOpc npeo6p83yeM C yqeTOM
opMyn (5.153), (5.156) H, HMeSI B BI1) Y, 'ITO
L = RQ I Wo , nonyqMM
? ?
E= LJ' = 4[R'+(: role ]'] 4R'[I+; ( - n
U Q
- 4R%[I+Q'( :' - n
Opl1paBHSICM TeITJIOBYIO H 3JIeKTpH<Jec-
1
KYIO :mc p fHIO -kT = E ,11 H3 3Toro paBCH-
- 2
eTna H8HJ]:CM
WO [ 1 +Q'[ :' - )']
V 2 =2kTR . (5.157)
ill Q
OOJIOC8 nponyCKaHIUI <JaCTOT onpeJ]:e-
JISIeTCSI BbIpa)KeHl1eM
I!1w = 03 0 12Q,
(5.158)
113 KOTOpOfO Bblp83HM Wo H nOnCTaBl1M B
OPMY1IY (5.157):
U = 4k m1 I + Q '[ :' - n (5_l591
3HepfIDI pe30HaTOpa cocpeJ]:oTO'leHa B
Y3KOM ,lluan83oHe qaCTOT B paHoHe pC3oHaH-
ca. MO:>KHO C'lHTaTb, liTO w / Wo 1 11
Wo I W 1 H, KpoMe TOfO, 3HeprHH. paBHo-
MCpHO pacnpe.nemlCTCJI BO BceM qaCTOTHOM
.nmina30He H He 3aBI1CIIT OT qaCTOTbl. TaKUM
386 rJIaBa 5. H3MEPIHEJ1bHbIE CBOI1CTBA MHKPOMEXAHI14ECKI1X npl1EOPOB
06pa30M. cpe.aHSlSl KBa!J,paTl1'leeKWI BeJIlf'nI-
Ha TennOBOfO wYMa JlBJUleTCR BH)'TpeHHen
nOMexoH Tl1na 6eJIoro wYMa, cneKTpa.J1bHWI
nnOTHOCTb KOTOporo 113 <POPMYJ1bI (5.159)
onpe,l\eJUleTCR npMMmKeHHoH 3aBHCHMO-
CTblO
-1
SR(ro)=[U / ro] 4kTR. (5.160)
5.9.2. IIpoHcxml\lteHHe MeXaHnqeCKOrO
mYM8
PaCCMOTpMM MeXaHHQeCKMH pe30HaTop
(PMC. 5.46), KoropbJn COCTOMT M3 llM m,
IIpY)KI1HhI C :>KCCTKOCTbIO G x Ii .i1eMn epa C
K03 mmefITOM .J.eMn<pHpOBaHIDI hx'
G b I
ml j]
PRe. 5.46. MeX8HH'IeCli:HH pe30n8Top
YpaBHeHl1e )J,BIDKeHIUI 11M I1MeeT BIi.'!:
d 2 x b dt G ( 6 )
m-----;;-+ -+ \x=F. 5.1 1
dr dt
r.ne F - BHelllHRR B03Myu al{)maJI CHJla.
nepeHLleM K onepaTopHoH opMe 3a-
mfCI1 (s = d / dt) Ii, HMeR B OH.ny, 'lTO
d... I dt = v = sx , nepemmJeM ypaBHeHl1e
(5.161) B BIi)J,e
b G:o: F
smv+ xV+-v= .
s
OTK)'.na, nOJ1araJI s = iro, nony<mM
F
v=
b x +{ rom- )
1bx -{ rom- )]
2 .
h}+(rom- )
.3anHllleM Cpe.nHI01{) KBaLlpaTI1QecKYJ{)
BeJ1HQHHY CKOpOCTI1:
b/ (rom- r F
[h/ +( rom- ; rr + [h.' +( rom- ; rr {b/ +( rom- ; J]
v =F 1
2
HMeJl B BH.z:\Y. 'ITO co6CTBeHHaJI 'lac-roTa
H .n06poTHOCTb MexaHHllecKoro pe30-
HaTOpa onpe )J,eJUlIOTCJl Bblpa:>KeHIDIMl1
COo = J Gxlm. Qx =ffiomlhx, 3anl1WeM
BeJ1H'lI1HY
-2
F
.2 = +/+( rom- ; rr
-2
F
- 2b/[1 + Q/ ( .:: - 7:n
(5.162)
C YQeTOM opMyn (5.158) 11 (5.162)
nonyqHM 3aBHCHMOCTb )J,J1R BblqHCneHHR
KHHeTl1'leCKOH 3HeprHH MeXaHI1'leCKOfO pe-
3OHaTopa:
E= mv2 =
2 4b x
FQx
.++Q/[ - 7: n
-2
F
8b [] +Q/[ - :: n
UlYM B MHKPOIvIEXAHWIECKI1X npl1f}OPAX
387
1
I-b paUeHCTBa -kT = E flonY'II1M
2
[ ( ) 2 ]
-2 2 ro roo
F =4kT Arol+Q roo - .
(5.163)
)J,JI51 pC30HaUCHOH 06JlaCUI
( ro I roo I, roo I ro I ) Bblql1CJII1M H3 Op-
Mynbl (5.163) cpe)J,HI{)1O KBa)J,paUI1JCcKYI{)
BCfll11Jl1IfY MeXaUI1'leCKOrO rnYMa, KOTOpblH
TaK)Ke 51BJ1S1C1'CSI BH}'TpeHHdi nOMexoH 1'Hna
6C1I0fO rnYMa co cneKTpaJIbHOH nIIOTHOCTblO
So(ro):::: F 2f Aro 4kTb x ' (5.164)
DOKa)((eM aHaJIOfl11O lllYMOB 3JIeKTpl1-
1JeCKOfO 11 MeXaHH'leCKOfO pe30Ha1'opoB.
B ypaBHeHI1H (5.161) BBe)J,eM o603Ha-
'leHHe d-.; I dt = J (311eKTpH'leCKHH 1'OK-
<maJIOf MeXaHI1':leCKOH CKOpOCTH) H nepe-
fll1111eM ero B Bl1,lle
dJ J
m-+b:J+G x Idt=f(t)=U w .(5.165)
dt
CpaBHHBaJI 3KBHBaneHTHbIe ypaBHeHHH
(5.154) 1 (5.165), 3anl1111eM paBeHcTBa
L =m, C=I/G x , R=b x = Gxm IQx'
B onepaTopHoH opMe ypaBHeHlUI
(5.154), (5.165) I1MelOT Bl1,ll
J
sLI +Rl +-=U w ' (5.166)
sC
OTKy)J,a CJle.nYeT BeJIl1qHHa J, 3KBHBaneHTHaJI
Ublpa:>KeHHIO (5.155), H, CJIe)J,OBaTeJIbHO, )J,JIH
paBeHCTBa (5.166) cnpaBenJJHBO BbIpa)((eHHe
(5.160). nOMI1S1, 1JTO f = U w' R = b x ' no-
nY<IHM
]2 I Aro 1'= 4kTb x .
(5.167)
TaKI1M 06pa'30M, 113 cpaBHeHH5I Bblpa-
)KCHI1i1 (5.160), (5.167) CJ1e,llycT 3aKJIIO<leHI1C
06 3KBI1BaJIelITHOC1'H lllYMoBoro HanpHji(e-
HHSI 11 McxaHMtJeCKOfO lllYMa.
5.9.3. Mex8HH..eCKJIH WYM
MHKpoaKCeJIepOl\leTpOB H
MHKpo)J,a.....HKoB naBJIeHHJI
MaTcMamtJccKWI MO)J,enb OCCBOfO MA
B J1I1HeHHOH nOCTaHOBKC onHCbIBaeTC51 ypaB-
HCHI1CM (5.161), B KOTOpOM F = ma (a - M3-
MCp5IeMOe YCKopeHHc). Ma1'eManl1JeeKaH.
MO)J,enb MaHTHHKOBOfO (yrnoBofo) MA no
crpYKTYpc aHaJIOfl11JHa ypaBHcHl110 (5.161).
llIYMoBoH cncKTp BbIXO,IDIOfO Cl1fHana
paBeH cncKTpY BXO,l\a, YMHOJKeHHOMY Ha
qaC1'OTHYIO xapaKTepl1C1'l1KY C ICTeMbI, B
.naHHOM CJIY'lac - MA. I1Me.sr B BH)J,Y BbIpa-
)((CHHH. (5.161) H (5.164), nOJifIHM
-2 4kTb x
X m = b x 2ro 2 +(G x _mro 2 )2 .
(5.168)
Cpe.ll.HHH KBanpani'lCCKHH rnyM,
If?
M/(pa)J,/c) -, onpc.aeJIlIeTC51 no q,opMyne
<X m """ x;, 2.jk'ib; .
b;ro 2 + (G x - mro 2 )2
(5.169)
Ha pl1C. 5.47 .D:JISI HCXO)J.HbIX ,l\aHHbDG
m = 7.5.10- 5 Kf; k = 1,38.10- 23 ; T =
= 293 K; G x = 1575 HIM H BeJll1tJHHe
b x = (room) I Q (roo = 2800 lIc), onpe.ll.e-
nCHHoH .nml Q,.= 1; 2; 5; 10, npHBe,ueHa rny-
MOBWI xapmcrepl1CTHKa, paCC'lHTaHHaJI no
q,opMyJle (5.169).
MA HCnOJ1b3yeTc5I )J,JIH H3McpeHHH Ha
qaC1'o1'ax, MeHblllHX qaCTOTbi pe30HaHca, T.e.
113 PMYJIbl (5.169) CJIe.nyeT npH6JIH:>KeH-
Hoe paBeHCTBO:
-2 2.jkTb;
Xtu .
G x
(5.170)
388 rnaBa 5. 113MEPI1TEJlbHbIE CBoi1CTBA MHKPOMEXAHH4ECKHX flPI1EOPOB
<X m > ,
M
-J paJJ.!c
-13
10
-13 S
10'
-14
10
l(r.
10 2
10 3
=10
W, PaWc
10 4
10 S
PHC. 5.47. PaC'leTHbIH cne 'Tp UJ)'l\IB MA
(3HaQeHWI 'lacrOTbJ npuBe;J.eHbl B JJOrapu.pMW1ecKoM MacwTa6e)
,[(."llI YCTaHoBHBllIeroc51 (cTaTIf'leCKofo)
pe:>KHMa H3MepeHJIH Ha 'laCTOTe HHjf(e pe30-
HaHca x = ma I G x' 3HaqI1T. MeXaHHqeCKMH
m:yM oceBoro MA, (M'C-2)/(pa.n/C) I 12, C }"Ie-
TOM BLIpaiKeHIDI (5.170) onpe.ueIDIeTc51 CJIe-
.!!)'JOIDHM 06pa3oM:
_ G x 2 JkTb x 2 J kT2t"ro o m
<a >= - -
m m G x m
=2 JkT WO .
OKOH'faTeJlLHO nOJIyqaeM
- 2 J kTwo
<a w >= .
mQ
(5.171)
113 aHaJIH3a BbJpIDKeHIDI (5.171) CJIe.ny-
CT, lITO OTHoweHHe cJtrnanlw)'M MO:>KeT
61>lTb y-BeJII11:JeHO cJIe.ny101JUfM 06p83oM: }'Be-
JIM"IeHHeM Maccbl, YJIyqllleHHeM ,l\06poTHo-
CTH, }'Mem.weHHeM nOJIOCbJ nponycKaHIDI
IUlCTOT. Qpe3MepHoe nOBbIllleHI1e ,l\06pOTHo-
CTH npH OTcyrCTBHH 06paTHoH CBDM MOjf(CT
npHBeCTH K CJII1lllKOM 6oJILIDoMY UJYMY H8
pe3OHaHCHOH IJaCTOTe. T8K KaK nOJIoca npo-
nyeKaHHSI 'laCTOT 51BJUJeTC51 O,llHI1M 113 Bajf(-
HblX 3KCnJIyaTaUl1oHHbIX napaMeTpOB, e,l\JIH-
CTBeHHaJI Mepa YBeJIM<leIUHI xapaKTepl1CTMKlI
CHfHarr/lllYM - yBeJIl1QeHl1e MaCCbl.
PaCcMOTpl1M Tenepb MaJlTHI1KOBbIH MA
Ha nOCT)'llaTeJIbHO nepeMellJ,aJOllJ,eMCSI OCHO-
BaHKH. OOJIOiKl1M B q,opMYJIaX (4.263) Yr = 0
H nony'lHM ypaBI1elme llBI1>KelmH MA:
J AS + k.as B +k 22 & = m(u + g)(1 +a),
file J A - MOMeHT I1HepUl111 MaJlTHI1Ka OTHO-
eHTeJIbHO TOIJKH 3aKpelUleHI1}J ynpyrofo
3JIeMeHTa; m - Macca MaJlTHHKa; kJJ.3 - K03 -
cpHlll1eHT ymoBofo AeMn l1pOBaHI151; k 22 -
YTJIOBaJI )KeCTKOCTb nO,l\Beea; I + a - pae-
CT051Hl1e 01' TO'lKH 3aKpenJIeHI151 ynpyforo
3JIeMeHTa .nO I..{M MaJlTHHK.'t; u - 113Mep51eMoe
YCKopeHHe.
llI}'MOBOH cneKTp BbIXO,l\HOrO CHfHaJ1a
onpeJ1.CIDIeTCSI BbIpa)KeHJleM, aHaJ10fl1'lHbIM
paBeHCTBY (5.168):
-2 4kTkIJ.S
& -
IU - k;sw 2 + (k 22 - J AW 2 r '
a CpeLJ:HHH KBa.llpaTH<ICCKI1H 1ll}'M,
pa):(!(pa):(!c)1I2, HaXOLJ:HTCSI no opMyJIe
WYM B MI1KPOMEXAHl14ECKl1X rIPl1£OP AX
389
< &111 >=
2 kTk
.n3 : (5.172)
,2 2 [ 2\1
AJJ.3(O +\k 22 -.fAro J
)J.ml CTan1'lCCKOro pC>K IMa I1'3MepCmrn
CpCJJ,HHH KBaJ.{panl<rCCKI1H my:'.! onpc.u,e.:!HCT-
CH npI16m1>Kel1HOH OPMYJIOi1
- 2 kTk. 1B
< & >= .
III k '
22
(5.173)
TaK KaK B CTaTH'ICCKOM PC>KI1MC yrOJI
nOBopOTa MaH.THI1Ka COOTBCTCTBYCT BhIpa-
>KeHlllO ,9 = m(u + g )(1 + a)/ k 22 , C Y<ICTOM
paneHcTBa (5.173) 3arll1rnCM OPMY1IY )J,JIH.
BblqIlCJICHH.H MexaHI1'lCCKOfO wYMa
MaH.THI1KOBOro MA, (M'c - 2 )1(pa.rrl c)ll2, B BII,ne
_ k 2 ., 2 kTkJJ.fj
< 1I + g >= ( - )
ml+a k 22
_ 2 kTJAc.oO
- m(1 +a) Qfj ,
(5.174)
rJle Qfj = 1I2Sfj = J Ac.oO I k.nfj .
I1Mc.H B BII,llY npllMcp 3.2, 3anllWCM
J A = .f c + m(l + a Y ;::!, m(l + a Y M nocne
nOJJ,CTaHOBKIl :nOM BeJIl1'lI1HbI B cpopMyny
(5.174) ITOJIY<II1M
- 2 kTc.oo
<lI+g>;:::; .
mQfj
(5.175)
<})0PMYJIbI (5.175) H (5.171) cOBna,l\a-
lOT, H, CJICJJ,OBaTeJIbHO, peKOMeH,llaI.{HH no
yBeJ1H<ICHHIO OTHOrneHWI Cl1fHanlllI)'M ,l\JIH.
ocCBoro MA cnpaBc,nJIlInbI 11 ,llJUI MaJlTHHKO-
BOfO MA.
HaH)J,CM ruYMOBble xapaKTCpHC11I:KH
MM C >KeCTKHM QCHTpoM, nOJIafaJl, qTO Ha
HefO JJ,CMCTByeT CHJIa F.n = pS (p - H3MepSlC-
MOC JJ,aBJICHHC; S - nJIOIlJ,a,I:\b >KCCTKOfO QeH-
Tpa). 113 CHCTeMbl (4.78) npl1 Ix = I" = 0 no-
J1Y'lIlM ypaBHeHHe ,l\BI1>KeHHSI >KCCTKOrO QeH-
Tpa:
mji+kl\.V}'+Cry=pS,
rnc 111 - Macca :>KCCTKOrO QCHTpa; k:lJ' - KO-
3 HLtI1CH'I ,lleMn l1pOnamIH MCM6paHbl;
G y - >KeCTKOCTb MCM6paHbI.
AHMOfl1<IHO npellbI.u,yweMY CpC,aHHM
KBalipaTII'leClmM WYM, M/(pa.rrlc) 1/2, onpe,ae-
JUleTCH (}JOPMYJIOi1
2 kTk.,,1'
< V >=
.w '
2 2 2 '
kTIJ,c.o + G y - mc.o ')
OTKY,lla )J,J1S1 CTaTHqCCKOfO pC:>KHMa 1I3Mepe-
HIlM CJIe)J,YeT npll6mDKcHHWI 3aBI1CHMOCTb
- 2 kTkl1,v
< Y w >;;:; .
G y
(5. I 76)
B CTaTHqeCKOM pe>KHMe nepCMCllJ,eHHe
:>KeCTKOfO I..{CHTpa MeM6paHbI y = pSIG.,. 11 B
COOTBeTCTBIIH C OPMYJIOH (5.176) MexaHH-
qCCKMH WYM MM, (H'M. 2 )/(pa,rVc)'I2, e >Ke-
CTKHM l..{eHTpoM onpe,neJI.HeTC.H 3aBHCIIMO-
CTblO
- G 2 kTkl1,l' 2
<p>= y =-
S G y S
kTc.oom
QJ'
, (5.177)
mc.oo
f,llC Qy = 1I2S y = .
k.ny
113 BblpmKCHHH. (5.177) CJIe.u,yeT, qTO
OTHOlliCHHe cHfHanllllYM MO>KeT 6bITb yne-
In!'leHO cne)),yIOIltHM 06p830M: paCnmpeHl1-
eM nJIOllJ,a,a1l JKeCTKOfO QeHTpa H YMeHhllle-
HHeM efO MaCCbI, 'lTO HBJIH.eTC.H npeJJ,ITO'lTl1-
TeJIbHbIM, a TalOKe nOBblrneHHeM JJ,06pOTHO-
CTH II CY>KeHl1eM nOJIOCbI nponYCKaHH.H qac-
TOT.
5.9.4. MeX8HnQeCKJlu mYl\f
MHKpornpocKonoB
PaccMOTpHM Bbl'iHCneHHC MeXaHJ11le-
CKoro myMa Mr RR-Tl1na Ha npHMepe OJ(-
HOfO 1I3Mep1lTeJ1bHOfO KaHMa, MaTeMarnqe-
CKaJI Mo,neJ1b KOToporo B COOTBeTCTBHH C
paseHcTBoM (4.153) HMeeT Bl1,ll
390 rllaBa 5. I13MEPHTEJIbHbIE CBOHCTBA MMKPOMEXAHJ11.JECKHX flPI.U;OPOB
Jpp +bpP+ G 13 P = HoD.x.
rJJ;e J p - 3KBaTOpHfu1.hHbIH MOMCHT HHcplum
pOTOpa; b p , G p - K03 Q>HUHeHT ncr.mQ>HpO-
BaHWI 11 :>KeCTKOCTb ynpyrOfO nOJI,Beca pOTO-
pa no OCR BTOpHlffibIX KOJ1e6aHHH; Or - H3-
MepjleMWI yrJIOBaJI CKOpOCTb; Ho - MO.nyJIb
KHHeru'leCKOrO MOMeHTa pOTOpa.
qaCTOTHaJI nCpe)J,aTO<IHWI Q>YHKIll1Sl Mr
3anHCbIBaeTCjI TalC
l3(w) =W(w)= Ho .
D.x(w) J bgw 2 +(G p -J 13 ( 2 )2
(5.178)
AHanorl1<JHo OPMYJIC (5.164) nony-
'lHM Be3J1qJ1HY
[ ] 4kTh,.
(5.179)
on.-y.na c.:Ie.nYeT cpe)J,HHH KB paTl1qeCKI1H
B03M)'IIlaIOII!l1H MOM em, 06YCJlOBJ1eHHbIH
TennOBblMH Q>lI)'KryaL{IDlMH H 3KBHBaneHT-
HbIH HCKOTOPOMY rl1pOCKOJIH'leCKOMY MO-
Memy:
<13m>,
p
"J pa;J./C
-9
-95
,
-]0
Q j!=5 0 Q {i ]00
-10,5
-11
-11,5
Q{i=200
-12
1-10 3
Mo = HoO, = 4kTbl\l'.w. (5.180)
C yqCTOM BbJpa>KeHufi (5.178), (5.180)
nOJIY<il1M OPMYJIY )J,JISI BbI<iI1CJlCHHSI CpC)J,-
In
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JIa B Mr LL-nma.
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na B Mr LR-THna.
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JIa B Mr RR-nma.
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poBaHHH MMKpOMeXaHH<IeC1<l1X ITpH60pOB.
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HbIe ny6JIl1KaLtMM no 3THM BonpocaM liMe-
IDTC , 'lTO OTpa:>KCHO B cnMCKe JIHTeparypbI.
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I. Am )1(. I1 .iJ.p. ,D.aT<U1KH It3MepH-
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nPE.D.METHbIH YKA3A TEJIb
A
AKCC:ICpOMeTp (MHKpOaKCCJICpo-
MeTp) 10
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HI1S! 370. 371, 372, 378,379
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17,18,20,255
- MaHTHI1KOBbIM 10, 21. 23, 24. 25.
26,27,28.29,32.260
- npSlMOro ITpe06pa30BaHI1S1 363, 367
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AHI130TpOITI1S1 100, 101, 102,435
r
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- BOJIHOBOM 90, 91
- KaMepToHHblH 87, 88, 89
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63,64.65,66,248,250,310
- LR-THITa 67, 70, 71, 72
- RR-nma 72, 79, 80, 82, 83, 84, 85
,Ll,
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34,45,46
- KOMneHCa1\ 1OHHoro npc06pa3oBa-
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- ITpSlMOfO ITpeo6pa30BaHHx 394
)J;CKpCMCHT 3aTyxaHlU! 322, 325,
327,331
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:>K
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179, 181, 182, 183
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- 3JIcKTpocmmtIeCKaH 3 I 2, 314
- 3<p<peKTl1BHaH 315
3
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359,368,405,407,408,420,437,438
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106, 107, 203
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JI
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- KpeMHl1eBbIe KOMIla}'H,!l.b1112, 113, 114
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MaTpH1\a
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- 3 CKTl1BHOH :>KCCTKOCTH 315
MCM6paHa
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398
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- B KOHCTp)'KI..I;lUlX 35, 36, 39,40,41,
42,43,44,47
MMl\.-po,nBHraTeJIb (3JIcKTpocTan1tIe-
CKHH)
- Bparn.aTcJIbHbnl 218
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MID..-poo6pa6oTKa
- KpeMHl1eBWI 06"beMHaJI 137, 138,
141, 142, 143
- 'PeMHl1eBWI nOBepxHOCTHaH. 143.
144, 145
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MOMCHT
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- KHHeTJ1qeCKHH 74. 355, 420
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H
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316,317,318,356
o
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391,392,393,429,434
- KBa.nparypHWI 240, 241,315
II
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IInOCKOCTb (lqmCTaJIJlorpaq)HqCCKaH.)
96, 97, 105
lloJIoca (nponycKaHIDJ qacTOT) 11,332,
333,422
rrpe06pa30BaTeJIb (npjIMoii)
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- nepeMem.eHHH 192, 197, 245
IIpe06pa30BaTem. (o6paTHbIH)
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225
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- 3JIeKTpOCTaTHIJ:ecKHH 212
IIpen eccIDI (rnpocKona) 73, 75, 76
p
Pa6oT8 (3JlCKTpOCTaTI1tJecKlIX clin)
215.311
PaCCTpOHKa (qaCToT) 330, 331
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242,244
C
C60pKa 149. 152
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CUJIa
- HHCpLU1H KOpHOJIHCa 74, 87, 88, 91
- MafHMT03JIeKTpHtIeCKaH. 222, 226
- 3JIeKTpoMarIiHTHaH. 229, 230
- 3JIcKTpOCTaTH'leCKaH. 193,213,214,
215,244
CTpYKTYpa(KpHCTaJIJIa) 98,99
T
TeH30pe3HcTop 41, 47, 48, 201, 202,
203,204,206
TpaBJIeHHe 129
- aH113oTponHoe 131, 132
- 130TponHoe 130
y
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- aKcenepoMeTpoB 13, 22, 257, 258,
259,263,264,265
- rnpocKonoB 53, 54, 55, 65, 68, 69, 77,
80,299,300,301,302,303,304,305,308,309
- .naTtJHKa ):\aBJJCHMM 37, 282
- narpaIDKa BTOpOro po,na 256
YCKopeHHc KOpHOJJHCa 73, 87, 91
YCTOHql1BOCTb 270
cp
<I>HJJbTp 364, 365, 368
<I>opMa (KoJIc6aHHH) 171, 174, ) 75,
346,349,350
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<I>yHK HM (nepe)J,aTO'lHWI)
- aKccnepoMCTpoB 267, 269, 271,
272,273,274,275,276,279,280,281,366,
370,372,373
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- ,D,aT'rnKOB )18BJJeHIDI 282, 283, 397,
401