Text
H3HKH H TeXHHKH
P.ll. T
M
HrC
CnpaBO"lHL4K
L4H>KeHepa -
MeXaHL4Ka
nepeBOA c aHrnMMCKOrO
nOA PeAaK4MeM LI1.IO. WKaAMHOM
TEXHOCCI>EPA
MOCKBa
2008
THMHHrc P.JI.
CnpaBOqHHK HIDKeHepa-MeXaHHKa
MOCKBa:
TexHoc epa, 2008. - 632 c. ISBN 978-5-94836-183-3
CnpaBOQHHK no,nrOTOBJIeH B nOMOm;b HH)I{eHepaM-MexaHHKaM, 3aH TbIM
npoeKTHpOBaHHeM H H3rOTOBJIeHHeM TeXHHtJeCKHX H3,neJIHH H o6opy,noBaHH .
M3,naHHe yqHTbIBaeT H3MeHeHH B TeXHOJIOrH X H CTaH,napTax HHxeHepHoro
npoeKTHpOBaHH H co,nepXHT BCIO caMYIO Heo6xo,nHMYIO HH<popMaUHIO no
pa3pa6oTKe npoeKTOB, TeopHH, MaTepHarraM H HX CBOHCTBaM.
KHHra npe,nHa3HaqeHa ,nJI HH)I{eHepOB-MexaHHKOB, cTy,neHTOB, a TaIOKe
Bcex, KOMY Tpe6yeTc qaCTO 06palI(aTbC K TeXHHqeCKOH HH<popMaUHH.
Newnes Mechanical
Engineer's Pocket Book
Third edition
Roger L. Timings
. Awl.1tIIO*. BmfON. HtatMfIc;.1.QNDOH . NfwY()IIIl.
()XJON) . PAM . SNlOIrGO . SAIl fttAHclSCO . s..cwo.
. SVOMY . TOitYO
ELSEVIBR Newnes is MI impflnt of EbMIr .......
@ 2006, Elsevier Ltd.
@ 2008, 3AO «PMIJ; «TexHoc<pepa», nepeBO,n Ha PYCCKHH 3bIK,
OpHrHHaJI - MaKeT, o<popMJIeHHe
ISBN 978-5-94836-183-3
ISBN 978-0-75066-508-7 (anrJl.)
COAEP>KAHME
BBe.neHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
TIpe.nHCJIOBHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
BJIaro.napHOCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
fJlaBa t. HlDKeHepHaJl MaTeMaTHKa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1. fpeqeCKHH aJIcpaBHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2. MaTeMaTHqeCKHe CHMBOJIbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3. E.nHHHnbI H3MepeHIDI: CM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1. OCHoBHbIe H .nonOJIHHTeJIbHbIe BeJIHqHHbI. . . . . . . . . . . . . . . . . . . . . . . 23
1.3.2. TIpoH3Bo.nHbIe BeJIHqHHbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.3. E.nHHHnbI H3MepeHIDI, He BXO,nHW;He B CHCTeMY CM. . . . . . . . . . . . . . . . 24
1.3.4. TIpHMeqaHIDI OTHOCHTeJIbHO 3anHCH pa3MepHOCTeH . . . . . . . . . . . . . . . . 25
1.3.5. ,LJ;OJIbHbIe H :KpaTHbIe e.nHHHnbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4. K03cpcpHnHeHTbI npeo6pa30BaHIDI ,WUI e.nHHHn H3MepeHIDI <I><I>C H CM . . . . . 26
1.5. TIpe.nnOqTHTeJIbHbIe qHCJIa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6. Onpe.neJIeHHe pa3MepOB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6.1. TI.1IocKHe CPHrypbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6.2. 06DeMHbIe cpHrypbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7. CTeneHH, KOpHH H o6paTHbIe BeJIIftIHHbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.8. TIporpeccHH................................................ 34
1.8.1. ApHcpMeTHqeCKaH nporpeCCIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8.2. feoMe1]>H'IeCKaH nporpeCCIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8.3. fapMOHHqeCKaH nporpeCCIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.9. TpHroHoM e1]>HqeCKHe CPOPMYJIbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.9.1. OCHoBHbIe onpe.neJIeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.9.2. T ox,n eCTBa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.9.3. COCTaBHbIe cpOPMYJIbI H cpOPMYJIbI .nBoHHoro yrJIa . . . . . . . . . . . . . . . . . 36
1.9.4. <l>OPMYJIbI «npOH3Be.neHHe-CYMMa» . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.9.5. <l>OPMYJIbI 1]>eyrOJIbHHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.10. Kpyr H OKpyxHOCTb: HeKOTopbIe onpe.neJIeHIDI H CBOHCTBa. . . . . . . . . . . . . . . 36
1.10.1. Kpyr: TIJIOW;a,nb H ,WIHHa O:KpyxHOCTH. . . . . . . . . . . . . . . . . . . . . . . . .37
1.11. Kaa,npaTHbIe ypaBHeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.12. HaTYPaJIbHbIe JIOrapHcpMbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.13. CTaTHcTHKa: BBe.neHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.13.1. OCHoBHbIe nOIDITIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.13.2. BepO.SlTHOCTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.13.3. BHHOMHaJIbHOe pacnpe.neJIeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.13.4. Pacnpe.neJIeHHe nyaccoHa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.13.5. HOpMaJIbHOe pacnpe.neJIeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.14. ,LJ;HcpcpepeHnHaJIbHOe HCqHCJIeHHe (npoH3Bo.nHbIe) . . . . . . . . . . . . . . . . . . . . . 42
1.15. I1HTerpaJIbHoe HCqHCJIeHHe (Heonpe.neJIeHHbIe HHTerpaJIbI). . . . . . . . . . . . . . . 43
1.15.1. MHTerpHpoBaHHe no qaCT.SIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.15.2. Onpe.neJIeHHbIe HHTerpaJIbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
CoiJep:JICaHUe
1.16. BHHOM HbIOToHa (6HHOMHaJIbHaH TeopeMa) . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.17. Pa3JIOXeHHe B PM MaKJIOpeHa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.18. Pa3JIOXeHHe B PM TeKrropa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
fJlaBa 2. HlDKeHepaaJl CTaTHKa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1. HIDKeHepHaH CTaTHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2. Macca, CHJIa H Bec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1. Macca. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2. CHJIa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.3. BeKTOpbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.4. Bec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.5. Macca Ha e,lJ;HHH 06'beMa (nrroTHocTb) . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.6. Bec B e,lJ;HHHnax OObeMa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.7. OTHOCHTeJIbHaH nrrOTHOCTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.8. ,LJ;aBJIeHHe (:>KH,l.(KOCTH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3. BeKTOpHbIe ,lJ;HarpaMMbI CHJI: rpacpH'IeCKOe pemeHHe . . . . . . . . . . . . . . . . . . . 50
2.3.1. PaBHO,lJ;e:HCTByIOInHe CHJIbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
2.3.2. TIapaJIJIeJIorpaMM CHJI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.3. YpaBHoBemHBaIOInaH CHJIa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.4. Pa3JIOXeHHe CHJI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.5. PaBHoBecHe 1]>ex CHJI (1]>eyrOJIbHHK CHJI) . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.6. MHoroyrOJIbHHK CHJI: nOC1]>oeHHe ,lJ;HarpaMMbI MaKCBeJIJIa-KpeMoHbI . . 54
2.3.7. HenepeCeKaIOInHeCg KonrraHapHbIe CHJIbI (BepeBOqHbI:H
MHoroyrOJIbHHK). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4. MOMeHTbI CHJI, neH1]> TIDKeCTH H neH1]>bI TIDKeCTH nrroInane:H . . . . . . . . . . . . 56
2.4.1. MOMeHTbI CHJI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.2. TIpHHnHn MOMeHTOB (HcnOJIb3yeMaH TepMHHOJIOrIDI). . . . . . . . . . . . . . . 57
2.4.3. TIpHHnHn MOMeHTOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.4. PaBHoBecHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5. Pa3HOBHJJ;HOCTH pbftlarOB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5.1. PbfllarH nepBoro pO,lJ;a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5.2. PbfllarH BToporo pO,lJ;a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.5.3. PbfllarH 1]>eTbero pO,lJ;a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.6. QeH1]> TIDKeCTH, reOMe1]>HqeCKH:H neH1]> cpHrypbI H paBHOBeCHe . . . . . . . . . . . 63
2.6.1. QeH1]> TIDKeCTH (TBep,lJ;Oe TeJIo) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.6.2. QeH1]> TIDKeCTH HeO,lJ;HOpO,lJ;HbIX H COCTaBHbIX TBep,lJ;bIX TeJI. . . . . . . . . . . 64
2.6.3. QeH1]> TIDKeCTH (TOHKaH nrracTHHa) . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6.4. feOMe1]>HqeCKHe neH1]>bI CPHryp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.6.5. PaBHoBecHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.7. TpeHHe................................................... 68
2.7.1. CMa3Ka. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7.2. 3aKoHbI 1]>eHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7.3. K03CPCPHnHeHT 1]>eHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2. 7 .4. Y rOJI 1]>e HIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.7.5. TpeHHe Ha HaKJIOHHO:H nrrOCKOCTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.7.6. YroJI 1]>eHIDI nOKOg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8. HanpIDKeHHe H ,lJ;ecpopManIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.8.1. HOpMaJIbHOe HanpIDKeHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.8.2. HanpIDKeHHe C,lJ;BHra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.8.3. OTHOCHTeJIbHaH JIHHe:HHaH ,lJ;ecpopManIDI. . . . . . . . . . . . . . . . . . . . . . . . 73
2.8.4. ,LJ;ecpopManIDI C,lJ;BHra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.8.5. M O,zryJIb ynpyrOCTH (3aKoH fYKa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.8.6. MO,zryJIb C,lJ;BHra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.8.7. HanpIDKeHHe KpyqeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
COOep:JICllHUe
2.8.8. KOJIbIJ;eBOe HanpIDKeHHe B TOHKHX IJ;HJIHHJ(pH'IeCKHX 060JIOQKax. . . . . . . 76
2.8.9. TIpO,lJ;OJIbHOe HanpIDKeHHe B TOHKHX IJ;HJIHHJ(pH'IeCKHX 060JIOQKax . . . . . 77
2.9. BaJIKH.................................................... 77
2.9.1. YCHJIHe C,lJ;BHra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
2.9.2. H3rH6aIOInHH MOMeHT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.9.3. )J;HarpaMMbI YCHJIIDI C,lJ;BHra H H3rH6aIOInerO MOMeHTa . . . . . . . . . . . . . . 79
2.9.4. BaJIKH (KOHCOJIb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.10. HanpIDKeHHe, ,lJ;ecpopMaIJ;IDI H H3m6 6aJIOK. . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.10.1. H3rH6aIOInee HanpIDKeHHe H HeH1]>aJIbHaH JIHHIDI . . . . . . . . . . . . . . . . 86
2.11. <l>epMbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.11.1. MeTO,lJ; CeqeHHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.12. fHJJ;pOCTaTH'IeCKOe ,lJ;aBJIeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.12.1. )J;aBJIeHHe Ha nOBepXHOCTb, HaxO,nHIUYIOCg B XH,l.(KOCTH . . . . . . . . . . . . 94
2.12.2. 3aKoH TIaCKaIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
fJlaBa 3. HlDKeHepHu ,lOIHaMHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1. HIDKeHepHaH ,lJ;HHaMHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2. Pa60Ta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3. 3HepfHH.................................................. 98
3.3.1. CoxpaHeHHe 3HeprHH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4. MOInHOCTb................................................. 99
3.5. K03cpcpHIJ;HeHT nOJIe3HOrO ,lJ;eHCTBIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6. CKOpOCTb H YCKopeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6.1. MO,zryJIb CKOpOCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6.2. CKOpOCTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6.3. Y CKopeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6.4. YpaBHeHIDI, OTHOCgInHeCg K CKOpOCTH H YCKopeHHIO . . . . . . . . . . . . . 101
3.6.5. HMnYJIbc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1
3.6.6. 3aKOHbI MexaHHKH HbIOTOHa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.6.7. fpaBHTaIJ;IDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102
3.6.8. CoxpaHeHHe HMnyJIbCa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6.9. B03,lJ;eHCTBHe C1]>YH XH,l.(KOCTH Ha HenO,lJ;BIDKHOe TeJIO . . . . . . . . . . . . . 103
3 .6.10. HH ep IJ;IDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.6.11. ConpOTHBJIeHHe ,lJ;BIDKeHHIO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7. BpaInaTeJIbHOe ,lJ;BIDKeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.7.1. Pa,nHaHbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3. 7 . 2. Y rOJI no BO pOTa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7.3. YrJIOBaH CKOpOCTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7.4. CooTHomeHHe Mex,ny yrJIOBOH H JIHHeHHOH CKOPOCTgMH. . . . . . . . . . . 108
3.7.5. YrJIOBOe YCKopeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.7.6. MOMeHT BpaIneHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.7.7. Pa60Ta MOMeHTa BpaIneHIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.7.8. [(eH1]>oc1]>eMHTeJIbHOe YCKopeHHe H IJ;eH1]>OC1]>eMHTeJIbHaH CHJIa. . . . . 110
3.7.9. [(eH1]>o6e:>KHaH CHJIa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
3.8. YpaBHoBemHBaHHe BpaInaIOInHXCg Macc . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.8.1. YpaBHoBemHBaHHe KOMTIJIaHapHbIX Macc (cTaTHqeCKOe paBHoBecHe). . . 112
3.8.2. YpaBHoBemHBaHHe KOMTIJIaHapHbIX Macc (,lJ;HHaMH'IeCKOe paBHoBecHe) 114
3.9. OTHOCHTeJIbHbIe CKOpOCTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.9.1. OTHOCHTeJIbHbIe CKOpOCTH (HeCOe,lJ;HHeHHbIe TeJIa). . . . . . . . . . . . . . . . 115
3.9.2. OTHOCHTeJIbHbIe CKOpOCTH (cOe,lJ;HHeHHbIe TeJIa) . . . . . . . . . . . . . . . . . 117
3.10. KHHeMaTHKa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.10.1. BaJIJIHCTHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.11. KHHeTHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.11.1. BceMHpHoe TgrOTeHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
CoiJep:JICaHUe
3.11.2. JIHHe:HHOe nepeMeIneHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.11.3. TIepeMeIneHHe no KpHBOJIHHe:HHO:H 1]>aeKTopHH. . . . . . . . . . . . . . . . . 121
3.11.4. KOHHqeCKH:H MaJITHHK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
3.11.5. BpaIneHHe TeJIa BOKpyr HenO,lJ;BIDKHO:H OCH . . . . . . . . . . . . . . . . . . . . 122
3. 11.6. Pa,nHYC KpHBH3HbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.11.7. [(eH1]> CTOJIKHOBeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.11.8. MOMeHT HMnYJIbCa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.12. TIpocToe rapMOHHqeCKOe ,lJ;BIDKeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.12.1. MaTeMaTHqeCKH:H MagTHHK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.12.2. Co6cTBeHHbIe KOJIe6aHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.13. fH,lJ;pO,lJ;HHaMHKa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.13.1. CKOPOCTb TeqeHHg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
3.13.2. HenpepbIBHoCTb TeqeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.13.3. 3HepIWI ,lJ;B e:HCg Te e:H Cpe,lJ;bI (ypaBHeHHe BepHYJIJIH) . . . . . . 130
3.13.4. TeqeHHe qepe3 OTBepCTHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.13.5. Bg3KOCTb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.13.6. <l>oPMYJIa TIya3eitrnI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.13.7. 3aKoH CToKca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
fJlaBa 4. KpenelKllbie ,neTaJlH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1. BHHToBbIe Kpene)I{HbIe H3,lJ;eJIIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.1. Pa3MepbI H COOTHomeHIDI Ha qepTexax . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.2. Pa3JIHqHbIe rOJIOBKH BHHTOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.1.3. Pa3JIHQHbIe KOHnbI BHHTOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.1.4. BHHTbI C meCTHrpaHHbIM yrJIy6JIeHHeM B rOJIOBKe . . . . . . . . . . . . . . . . 135
4.1.5. TIpaKTHQeCKOe npHMeHeHHe BHHTOBbIX Kpene)I{HbIX H3,lJ;eJIH:H . . . . . . . . 135
4.1.6. TpaneneH,L(aJIbHaJI pe3b6a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.1.7. TIpgMOyrOJIbHWI pe3b6a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.1.8. YnopHaJI pe3b6a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.1.9. OC1]>OyrOJIbHaJI pe3b6a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.1.10. OCHOBHaJI ,lJ;IO:HMOBWI pe3b6a (55°): nHJIHHJ(pHQeCKaJI pe3b6a. . . . . . . . 137
4.1.11. <l>OPMbI ISO Me1]>HQeCKO:H H ISO 60° YHHcpHnHpoBaHHO:H pe3b6bI . . . . 137
4.1.12. BBe,lJ;eHHe B BHHTOBbIe Kpene)I{HbIe H3,lJ;eJIHH . . . . . . . . . . . . . . . . . . . 137
4.1.13. BS EN ISO 4014: 2001. DOJITbI C mecTHrpaHHO:H rOJIOBKO:H -
KJIaCCbI H3,lJ;eJIIDI A H B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.1.14. BS BS EN ISO 4016: 2001. BOJITbI C mecTHrpaHHO:H rOJIOBKO:H -
KJIacc H3,lJ;eJIIDI C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1.15. BS EN ISO 4017: 2001. BHHTbI C mecTHrpaHHO:H rOJIOBKO:H -
KJIaCCbI H3,lJ;eJIIDI A H B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.1.16. BS EN ISO 4018: 2001. BHHTbI c mecTHrpaHHO:H rOJIOBKO:H -
KJIaCC H3,lJ;eJIHH C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.1.17. BS EN ISO 4032: 2001. illecTHrpaHHbIe ra:HKH HCnOJIHeHIDI 1 -
KJIaCCbI H3,lJ;eJIIDI A H B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.1.18. BS. EN ISO 4033: 2001. illecTHrpaHHbIe ra:HKH HCnOJIHeHIDI 2 -
KJIaCCbI H3,lJ;eJIIDI A H B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.1.19. BS EN ISO 4034: 2001. illecTHrpaHHbIe ra:HKH HCnOJIHeHIDI 1 -
KJIacc H3,lJ;eJIIDI C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.1.20. BS EN ISO 4035: 2001. illecTHrpaHHbIe HH3KHe ra:HKH
(c cpJIaHneM) - KJIaCCbI H3,lJ;eJIHH A H B . . . . . . . . . . . . . . . . . . . . . . 167
4.1.21. BS EN ISO 4036: 2001. illecTHrpaHHbIe HH3KHe ra:HKH
(6e3 cpJIaHQa) - KJIacc H3,lJ;eJIHg B . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.1.22. BS EN ISO 8765: 2001. BOJITbI C mecTHrpaHHO:H rOJIOBKO:H
C Me1]>HQeCKOH MeJIKO:H pe3b6o:H - KJIaCCbI H3,lJ;eJIIDI A H B . . . . . . . . 169
CoiJepxaHue
4.1.23. BS EN ISO 8676: 2001. BHHTbI C mecTHrpaHHO:H rOJIOBKO:H
C MeT})H'IeCKO:H MeJIKO:H pe3b6o:H - KJIaCCbI H3,l1;eJIIDI A H B. . . . . . . . . . . . 123
4.1.24. BS EN ISO 8673: 2001. illecTHrpaHHbIe ra:HKH HCnOJIHeHIDI 1
C Me1]>H'IeCKO:H MeJIKO:H pe3b6o:H - KJIaCCbI H3,l1;eJIIDI A H B. . . . . . . . . . . . 181
4.1.25. BS EN ISO 8674: 2001. illeCTHyrOJIbHbIe ra:HKH HCnOJIHeHIDI 2
C Me1]>H'IeCKO:H MeJIKO:H pe3b6o:H - KJIaCCbI H3,l1;eJIIDI A H B. . . . . . . . . . . . 183
4.1.26. BS EN ISO 8675: 2001. illecTHrpaHHbIe HH3KHe ra:HKH
C Me1]>H'IeCKO:H MeJIKO:H pe3b6o:H - KJIaCCbI H3,l1;eJIIDI A H B. . . . . . . . . . . . 184
4.1.27. BS 7764: 1994. illecTHrpaHHbIe npope3HbIe H KOpOHqaTbIe ra:HKH. . . . . . . . 186
4.1.28. BS EN ISO 898-1: 1999. TIpOqHOCTHbIe CBO:HCTBa Kpene:>KHbIX H3,l1;eJIH:H:
60JITbI, BHHTbI H mnHJIbKH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.1.29. BS EN ISO 898-1: 1999. MapKHpoBKa. . . . . . . . . . . . . . . . . . . . . . . . . . .197
4.1.30. BS EN 20898-2: 1994. MeXaHHqeCKHe CBO:HCTBa Kpene:>KHbIX H3,l1;eJIH:H:
ra:HKH C YCTaHOBJIeHHbIMH 3HaqeHIDIMH np06HO:H HarpY3KH -
KPynHaR pe3b6a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.1.31. BS EN ISO 898-6: 1996. MeXaHHqeCKHe CBO:HCTBa KPene:>KHbIX H3,l1;eJIH:H:
ra:HKH C YCTaHOBJIeHHbIMH 3HaqeHIDIMH np06HO:H HarpY3KH -
MeJIKaH pe3b6a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.1.32. BS EN 20898-7: 1995. MeXaHH'IeCKHe CBO:HCTBa Kpene:>KHbIX H3,l1;eJIH:H:
HcnbITaHHe Ha KpyrgW;HH MOMeHT H MHHHMaJIbHbIe KpyrgW;He MOMeHTbI
AIDI 60JITOB H BHHTOB C HOMHHaJIbHbIMH ,lI;HaMe1]>aMH 1-10 MM . . . . . . . . .208
4.1.33. BS EN ISO 4762: 2004. Me1]>HqeCKH:H BHHT C nHJIHHJ(pHqeCKO:H rOJIOBKO:H
H meCTHrpaHbIM yr.rry6JIeHHeM no,ll; KJIIOq . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.1.34. BS EN ISO 10642: 2004. BHHTbI C nOTa:HHO:H rOJIOBKO:H H meCTHrpaHbIM
yr.rry6JIeHHeM no,ll; KJIIOq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.1.35. BS4827 ISO Me1]>H'IeCKHe BHHTOBbIe pe3b6bI, MHHHaTIOpHaR CepHg . . . . . . 217
4.1.36. BS3643-1j2 ISO MeTpH'IeCKHe BHHTOBbIe pe3b6bI: cepIDI C nOCTOgHHbIM
marOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.1.37. BS EN ISO 228-1: 2003. Tpy6HbIe pe3b6bI He 06eCneqHBaIOW;He
repMeTH'IHOCTb COe,ll;HHeHHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.1.38. ISO. Tpy6HaR KOHlf1IeCKaR pe3b6a: OCHOBHbIe pa3MepbI . . . . . . . . . . . . . . . 226
4.1.39. BpHTaHCKaR ,lI;IO:HMOBaR pe3b6a (BSW), 60JITbI H ra:HKH . . . . . . . . . . . . . . . 228
4.1.40. BpHTaHCKaR MeJIKaR ,lI;IO:HMOBaR pe3b6a (BSF), 60JITbI H ra:HKH. . . . . . . . . . 229
4.1.41. ISO YHHcpHnHpOBaHHaR TOqHOCTb BHyTpeHHe:H pe3b6bI: PM C KpynHO:H
pe3b6o:H (UN C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
4.1.42. ISO YHHcpHnHpOBaHHaR TOqHOCTb HapyxHO:H pe3b6bI: PM C KPynHO:H
pe3b6o:H (UN C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.1.43. ISO YHHcpHnHpOBaHHaR TOqHOCTb BHyTpeHHe:H pe3b6bI: PM C MeJIKO:H
pe3b6o:H (UNF). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
4.1.44. ISO YHHcpHnHpOBaHHaR TOqHOCTb HapyxHO:H pe3b6bI: PM C MeJIKO:H
pe3b6o:H (UN F). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
4.1.45. BA: npocpHJIb 6pHTaHCKO:H CTaHJ(apTHO:H pe3b6bI . . . . . . . . . . . . . . . . . . . . 234
4.1.46. BA: 6pHTaHcKHe CTaHJ(apTHbIe BHyrpeHHHe H HapyxHbIe BHHTOBbIe pe3b6bI 234
4.1.47. Pa3MepbI CBepJI AIDI oTBepCTH:H no,ll; BA pe3b6y H AIDI CKB03HbIX oTBepCTH:H
C 3a30pOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.1.48. Pa3MepbI CBepJI AIDI oTBepCTH:H no,ll; ISO Me1]>JItIeCKYIO pe3b6y H
AIDI CKB03HbIX oTBepCTH:H C 3a30pOM: pe3b6a C KpynHbIM marOM. . . . . . . . . 235
4.1.49. Pa3MepbI CBepJI AIDI oTBepCTH:H no,ll; ISO MeTpHqecKYIO pe3b6y H
AIDI CKB03HbIX oTBepCTH:H C 3a30poM: PM C KpynHO:H pe3b6o:H . . . . . . . . . . 236
4.1.50. Pa3MepbI CBepJI AIDI oTBepCTH:H no,ll; ISO YHHcpHnHpOBaHHyK> pe3b6y H
AIDI CKB03HbIX oTBepCTH:H C 3a30pOM: pe3b6a C MeJIKHM marOM. . . . . . . . . . 236
4.1.51. Pa3MepbI CBepJI AIDI oTBepCTH:H no,ll; ISO YHHcpHnHpOBaHHyIO pe3b6y H
AIDI CKB03HbIX oTBepCTH:H C 3a30pOM: pe3b6a C MeJIKHM marOM. . . . . . . . . . 237
CoiJep:JICaHUe
4.1.52. Pa3MepbI CBepJI AIDI oTBepcTHH no,ll; ISO MeTPHqeCKYIO pe3b6y H AIDI
CKB03HbIX oTBepcTHH C 3a30pOM: MHHHaTIOpHaH cepIDI . . . . . . . . . . . . 238
4.1.53. Pa3MepbI CBepJI AIDI oTBepcTHH no,ll; BSW pe3b6y H AIDI CKB03HbIX
oTBepcTHH C 3a30pOM pe3b6bI, Hape3HbIe H npOqHw;aIOW;He MeT'lHKH . . 238
4.1.54. Pa3MepbI CBepJI AIDI oTBepcTHH no,ll; BSF pe3b6y H AIDI CKB03HbIX
oTBepcTHH C 3a30pOM pe3b6bI, Hape3HbIe H npOqHw;aIOW;He MeT'lHKH . . 238
4.1.55. llJIocKHe maH6bI, qHCTbIe: MeT})H'IeCKHH PM . . . . . . . . . . . . . . . . . . . 239
4.1.56. llJIocKHe maH6bI, qepHbIe: MeTPHqeCKHH PM . . . . . . . . . . . . . . . . . . . 240
4.1.57. YCTPOHCTBa AIDI CTonopeHIDI, OCHOBaHHbIe Ha TPeHHH . . . . . . . . . . . . 241
4.1.58. YCTPoHcTBa AIDI CTonopeHIDI, OCHOBaHHbIe Ha MeXaHHqeCKOH
4>HKCaIJ;HH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
4.1.59. TIpyxHHHaH maH6a C KBa,npaTHbIM nOnepeqHbIM CeqeHHeM:
MeTPHqeCKHH PM, THn A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4.1.60. TIPyxHHHaH maH6a C npgMoyrOJIbHbIMH KOHIJ;aMH: MeTPHqeCKaH
cepIDI, THnbI B H BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
4.1. 1. TIpyxHHHaH ,lI;ByxBHTKOBaH maH6a C npgMoyrOJIbHbIM nOnepeqHbIM
CeqeHHeM: MeTPHqeCKHH PM, HCnOJIHeHHe D. . . . . . . . . . . . . . . . . . . 245
4.1.62. TIpyxHHHbIe maH6bI C 3y6bgMH, MeTPHqeCKHe . . . . . . . . . . . . . . . . . . 247
4.1.63. CTOnOpHaH maH6bI C 3y6bgMH, MeTPHqeCKHe . . . . . . . . . . . . . . . . . . . 248
4.1.64. ISO MeTPHqeCKaH npyxHHHaH BOJIHHCTaH maH6a: o6w;ee
MaIDHHocTPoeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
4.1.65. TIP04>HJIb na30B no,ll; T -06pa3HYIO rOJIOBKY 60JITa . . . . . . . . . . . . . . . . 251
4.1.66. Pa3MepbI 60JITOB H raeK K T -o6pa3HbIM na3aM . . . . . . . . . . . . . . . . . . 252
4.1.67. Pa3MepbI mnOHOK AIDI T -o6pa3HbIX na30B . . . . . . . . . . . . . . . . . . . . . 253
4.2. 3aKJIenOqHbIe COe,ll;HHeHIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
4.2.1. THnH'IHbIe rOJIOBKH H CTep)l(}lH 3aKJIenOK . . . . . . . . . . . . . . . . . . . . . . 255
4.2.2. THnHqHbIe 3aKJIenOqHbIe COe,ll;HHeHIDI BHaXJIeCTKY . . . . . . . . . . . . . . . . 256
4.2.3. THnHqHbIe CTbIKOBbIe 3aKJIenOqHbIe COe,ll;HHeHIDI . . . . . . . . . . . . . . . . . 257
4.2.4. CooTHomeHIDI Me)l{):(y ,lI;HaMeTPOM OTBepCTIDI H ,WIHHOH 3aKJIenKH . . . . . 258
4.2.5. 3aKJIenKH C nOJIYKPyrJIOH 3aKJIa,nHoH rOJIOBKOH, H3rOTOBJIeHHbIe
XOJIO,lUlOH mTaMnOBKOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
4.2.6. 3aKJIenKH C nOJIyKpyrJIOH 3aKJIa,nHoH rOJIOBKOH, H3rOTOBJIeHHbIe
rOpgqeH mTaMnOBKOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
4.2.7. TIpe,ll;nOJIaraeMbIH ,lI;Hana30H HOMHHaJIbHbIX ,WIHH, CBg3aHHbIX
C ,lI;HaMeT})OM CTep)l(}leH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.2.8. POP@ 3aKJIenKH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.2.9. 0630p BH,L(OB POP@ 3aKJIenOK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4.2.10. TIpaKTHQeCKHe peKOMeHJ(aIJ;HH no HCnOJIb30BaHHIO :KpenrreHHH . . . . . . 263
4.2.11. BbI60p POP@ (HJIH BbITIDKHbIX) 3aKJIenOK. . . . . . . . . . . . . . . . . . . . . . 264
4.2.12. PeKOMeHJ(aIJ;HH no npOeKTHpOBaHHIO . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.2.13. POP@ «F» cepIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
4.2.14. CTaHJ(apTHbIe 3aKJIenKH (c OTKpbITbIM TOPIJ;OM) H3 cnrraBa aJIIOMHHHH-
MarHHH 3,5 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4.2.15. CTaHJ(apTHbIe 3aKJIenKH (c OT:KpbITbIM TOpIJ;OM) H3 yrJIepO,ll;HCTOH CTaJIH 274
4.2.16. 3aKpbITbIe 3aKJIenKH H3 cnrraBa aJIIOMHHHH-MarHHH 5%. . . . . . . . . . . . 276
4.2.17. 3aKJIenOQHbIe raHKH (<<raeqHbIe 3aKJIenKH») . . . . . . . . . . . . . . . . . . . . 277
4.2.18. POP@ pe3b6oBbIe 3aKJIenKH: npHMeHeHHe . . . . . . . . . . . . . . . . . . . . . 277
4.2.19. POP@ pe3b6oBbIe 3aKJIenKH: YCTaHOBKa. . . . . . . . . . . . . . . . . . . . . . .278
4.2.20. POP@ faHKa: CTaJIb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.3. <l>aJIbIJ;eBbIe COe,ll;HHeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
4.3.1. <l>aJIbIJ;eBbIe COe,ll;HHeHIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281
4.3.2. TIPHnyCKH Ha 4>aJIbIJ;eBbIe COe,ll;HHeHIDI . . . . . . . . . . . . . . . . . . . . . . . . 282
4.4. Pa3JIHqHbIe Kpene)l(}lbIe H3,l1;eJIIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
4.4.1. KOHHqeCKHe mTH4>TbI, He3aKaJIeHHbIe . . . . . . . . . . . . . . . . . . . . . . . . . 283
COOep:JICaHue
4.4.2. TIpyxHHHbIe HapyxHbIe KOJIbna: Me1]>HqeCKaH cepIDI . . . . . . . . . . . . . 285
4.4.3. CTonopHbIe KOJIbna, BHYTPeHHHe: Me1]>HqeCKaH cepIDI . . . . . . . . . . . . 288
4.5. CKJIeHBaHHe MeTaJIJIOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
4.5.1. AHa3po6HbIe KJIeH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
4.5.2. KneH Y<I>-oTBep)l{)];eHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
4.5.3. [(HaHOaKpHJIaTHbIe KJIeH - aHHOHHaH nOJIHMepH3anIDI. . . . . . . . . . . . 293
4.5.4. Mo,n;HcpHnHpoBaHHbIe aKpHJIOBbIe KJIeH - OTBep)l{)];eHHe
C HCnOJIb30BaHHeM aKTHBaTOpOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
4.5.5. KneH, OTBep)l{)];eHHe KOTOpbIX npOHcxo,n;HT no,n; B03,n;eHcTBHeM
BJIaxHOCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
4.5.6. 3nOKCH,L(HbIe KJIeH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
4.5.7. BoccTaHOBHTeJIbHbIH nponecc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
4.5.8. KneeBoe coe,n;HHeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
fJlaBa 5. MeXalfll1leCKHe nepe,ZJ;a1JH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.1. MeXaHH'IeCKHe nepe,n;aqH: 3y6qaTbIe nepe,n;aqH. . . . . . . . . . . . . . . . . . . . . . . . 299
5.1.1. HeKoTopbIe THnH'IHbIe 3y6qaTbIe nepe,n;aqH. . . . . . . . . . . . . . . . . . . . . 299
5.1.2. TIpocTbIe npgMo3y6ble nHJIHHJ(pHqeCKHe nepe,n;aqH . . . . . . . . . . . . . . . 301
5.1.3. CTyIIeHqaTaH npgMo3y6aH 3y6qaTag nepe,n;aqa. . . . . . . . . . . . . . . . . . . 302
5.1.4. 3BOJIbBeHTa KpHBOH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
5.1.5. OCHOBbI reOMe1]>HH 3y6qaTOrO KOJIeCa . . . . . . . . . . . . . . . . . . . . . . . . 304
5.1.6. illar 3anenrreHIDI 3y6beB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
5.1.7. BbIcoTa 3y6beB KOJIeCa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.1.8. 3JIeMeHTbI CTaHJ(apTHOro 3y6qaTOrO KOJIeCa (,n;IOHMbI) . . . . . . . . . . . . . 309
5.1.9. 3JIeMeHTbI MeJIKOMO,LJ;yJIbHOrO 3y6qaTOrO KOJIeCa (,n;IOHMbI) . . . . . . . . . . 309
5.1.10. 3JIeMeHTbI CTaHJ(apTHOro 3y6qaTOrO KOJIeCa C YKopOqeHHbIMH 3y6bgMH
(,n;IO HMbI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
5.1.11. 3JIeMeHTbI CTaHJ(apTHOro 3y6qaTOrO KOJIeCa (Me1]>H'IeCKaH cHcTeMa). . . 311
5.1.12. ByJ(BeHHbIe 0603HaqeHIDI ,lJ;.IDI pa3MepHocTeH 3y6qaTOrO KOJIeCa H
BbJ1IHCJIeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
5.1.13. OCHoBHbIe cpOPMYJIbI ,lJ;.IDI npgMo3y6oro nHJIHHJ(pH'IeCKOrO 3y6qaTOrO
KOJIeCa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
5.1.14. OCHoBHbIe ypaBHeHIDI Koco3y6oro nHJIHHJ(pH'IeCKOrO 3y6QaTOrO KOJIeCa 314
5.1.15. BcnoMOraTeJIbHbIe ypaBHeHIDI A1llI 3y6QaTOrO KOJIeca. . . . . . . . . . . . . . 315
5.1.16. TepMHHOJIOrIDI npgMo3y6oH KOHHQeCKOH 3y6QaTOH nepe,n;aqH . . . . . . .316
5.1.17. TepMHHOJIOrIDI QepBgtIHOrO 3anenrreHIDI (QepBgK H QepBHQHOe KOJIeco) 316
5.2. MeXaHHQeCKHe nepe,n;aQH: peMeHHbIe nepe,n;aQH. . . . . . . . . . . . . . . . . . . . . . . 317
5.2.1. npOCTbIe nrrOCKopeMeHHbIe nepe,n;aQH . . . . . . . . . . . . . . . . . . . . . . . .317
5.2.2. CTyIIeHQaTbIe nrrOCKopeMeHHbIe nepe,n;aQH . . . . . . . . . . . . . . . . . . . . . 318
5.2.3. THnH'IHbIe HaTIDKHbIe YC1]>OHCTBa . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
5.2.4. THnHQHble npHMeHeHIDI KJIHHOBbIX H CHHXpOHHbIX peMeHHbIX nepe,n;aQ 320
5.2.5. ULTRAFLEX@ KJIHHOBbIe peMHH C o6epTKoH 60KOBbIX rpaHeH H
Y3KHM CeQeHHeM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
5.2.6. FO@-Z MOW;HbIe 3y6QaTble KJIHHOBbIe peMHH 6e3 o6epTKH 60KOBbIX
rpaHeH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
5.2.7. MULTIFLEX@ KJIHHOBbIe peMHH C o6epTKoH 60KOBbIX rpaHeH
H KJIaCCHQeCKHM CeQeHHeM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
5.2.8. MULTIBELT MHoropyqbeBbIe KJIHHOBbIe peMHH . . . . . . . . . . . . . . . . 326
5.2.9. illKHBbI KJIHHOBbIX peMHeH, COOTBeTCTByIOW;He CTaHJ(apTaM BS 3790
H DIN 2211 ,lJ;.IDI peMHeH FO@-Z H ULTRAFLEX@. Rz xx
0603HaQaeT mepOXOBaTOCTb nOBepXHOCTH .... . . . . . . . . . . . . . . . . . 327
5.2.10. illKHBbI KJIHHOBbIX peMHeH, COOTBeTCTByIOW;He CTaHJ(apTY DIN 2217
qacTb 1 A1llI peMHeH FO@-Z H MULTIFLEX@ . . . . . . . . . . . . . . . . . 328
5.2.11. illKHBbI C r.rry6oKHMH KaHaBKaMH . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
(:;, COOep:JICaHUe
5.2.12. CHHXpoHHbIe peMeHHbIe nepe,n;aqH: BBe,n;eHHe. . . . . . . . . . . . . . . . . . . 330
5.2.13. CHHXpoHHbIe peMeHHbIe nepe,n;aqH: THnbI H pa3MepbI peMHeH . . . . . . . 334
5.2.14. CHHXpoHHbIe peMeHHbIe nepe,n;aqH: mKHBbI . . . . . . . . . . . . . . . . . . . . 339
5.2.15. SYNCHROBELT@ HTD KOHCTPYKD:IDI . . . . . . . . . . . . . . . . . . . . . . . 342
5.2.16. SYNCHROBELT@ HTD CHHXpOHHbIe (3y6qaTbIe) peMHH: npocpHJIH
pacnpe,n;eJIeHIDI 3y6beB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
5.2.17. CHHXpoHHbIe (3y6qaTbIe) peMHH: H3MepeHHe ,WIHHbI. . . . . . . . . . . . . .346
5.2.18. SYNCHROBELT@ HTD 3y6qaTbIe mKHBbI: npe,n;nOqTHTeJIbHbIe
pa3MepbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
5.3. MeXaHHqeCKHe nepe,n;aqH: n;enHbIe nepe,n;aqH. . . . . . . . . . . . . . . . . . . . . . . . . 350
5.3.1. Pa60qHe xapaKTepHCTHKH n;eneH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
5.3.2. K03cpcpHn;HeHT H3Hoca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
5.3.3. THnbI I.J;eneH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
5.3.4. Me)l{):(YHapo,n;HbIe CTaH,n;apTbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
5.3.5. HopMaTHBHbIe CCbillKH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
5.3.6. TIpeH ecTBa n;enHbIX nepe,n;aq. . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
5.3.7. TIpeH ecTBa I.J;enHbIX npHBo,n;OB. . . . . . . . . . . . . . . . . . . . . . . . . . .358
5.3.8. COBMecTHMocTb 3Be3,n;OqKH H n;enH . . . . . . . . . . . . . . . . . . . . . . . . . . 360
5.3.9. PacnOJIOXeHHe npHBo,n;a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
5.3.10. MeTo,n; BbI60pa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
5.3.11. On;eHKa xapaKTepHCTHK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
5.3.12. OI.J;eHKa xapaKTepHCTHK I.J;eneH, H3rOTOBJIeHHbIX no
EBponeHCKOM)' CTaH,n;apTY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
5.3.13. On;eHKa xapaKTepHCTHK n;eneH, H3rOTOBJIeHHbIX no CTaH,L:(apTY ANSI . . 373
5.3.14. TIpoBHcaHHe I.J;enH..................................... .374
5.3.15. CMa3bIBaHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
5.3.16. MeTo,n;bI CMa3bIBaHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
5.3.17. nO,n;'beMHbIe YCTPoHCTBa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
5.3.18. ANSI X-,n;Hana30H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
5.3.19. <l>aKTopbI, BJIIDIIOIUHe Ha CpOK CJl)')K6bI I.J;enH . . . . . . . . . . . . . . . . . . . 379
5.3.20. YATIHHeHHe n;enH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
5.3.21. COrJlaCOBaHHe (BbIpaBHHBaHHe) n;eneH. . . . . . . . . . . . . . . . . . . . . . . . 382
5.3.22. 113MepeHHe H3Hoca I.J;enH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
5.3.23. PeMoHT H 3aMeHa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
5.3.24. PeryJIHpOBKa I.J;enH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
5.3.25. KOHCTPYKTOpCKHe H,L(eH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
5.3.26. Ta6JIHI.J;a K03cpcpHI.J;HeHTOB ,n;HaMeTPa HaqaJIbHOH OKpyxHOCTH . . . . . . . 388
5.3.27. npHBO,n;bI mar 3a marOM: npHMep nepBbIH . . . . . . . . . . . . . . . . . . . . . 389
5.3.28. npHBO,n;bI mar 3a marOM: npHMep BTOpOH . . . . . . . . . . . . . . . . . . . . . 391
5.3.29. npHBO,n;bI mar 3a marOM: npHMep TPeTHH. . . . . . . . . . . . . . . . . . . . . . 394
5.3.30. npe,n;ocTepexeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
5.4. MeXaHHqeCKaH nepe,n;aqa: BaJIbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
5.4.1. Kaa,npaTHbIe H npHMoyrOJIbHble npH3MaTHqeCKHe IIInOHKH,
MeTPHqeCKaH cepIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
5.4.2. Pa3MepbI H ,n;onYCKH ,WIH KBa,npaTHbIX H npSlMoyrOJIbHbIX
npH3MaTHqeCKHX mnOHOK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
5.4.3. Kaa,npaTHbIe H npSlMoyrOJIbHbIe KJIHHOBbIe mnOHKH, MeTPHqeCKag
CepHg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
5.4.4. Pa3MepbI H ,n;onYCKH ,lI;illI KBa,npaTHbIX H npHMoyrOJIbHbIX KJIHHOBbIX
mnOHOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
5.4.5. CerMeHTHbIe mnOHKH H na3bI ,lI;illI mnOHOK, MeTPHqeCKaH cepIDI . . . . . 406
5.4.6. Pa3MepbI H ,l(OnyCKH ATIH cerMeHTHbIX mnOHOK. . . . . . . . . . . . . . . . . . 408
5.4.7. THnbI KOHI.J;OB BaJIOB: OCHOBHbIe COOTHomeHIDI. . . . . . . . . . . . . . . . . . 409
CoiJep:JICaHUe
5.4.8. Pa3MepbI H ,n;OrryCKH IJ;HJIHH,lJ;pHqeCKHX KOHIJ;OB BaJIOB, ,LUIHHHOe
H KopOTKoe HCnOJIHeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
5.4.9. Pa3MepbI KOHHqeCKHX KOHIJ;OB BaJIOB C npH3MaTHqeCKHMH mnOHKaMH,
,LUIHHHOe HCnOJIHeHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
5.4.10. Pa3MepbI KOHHqeCKHX KOHIJ;OB BaJIOB C ,n;HaMe1'paMH CBbIme 220 MM
C na3aMH AIDI mnOHOK, napaJIJIeJIbHbIMH nOBepXHOCTH BaJIa, ,LUIHHHOe
HCnOJIHeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
5.4.11. Pa3MepbI KOHH1IeCKHX KOHIJ;OB BaJIOB C npH3MaTHqeCKHMH mnOHKaMH,
KopOTKoe HCnOJIHeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
5.4.12. ,lJ;orrycKaeMbIe BpaInalOInHe MOMeHTbI. . . . . . . . . . . . . . . . . . . . . . . . . 417
5.4.13. llpHM060qHbIe lIIJIHIJ;bI AIDI IJ;HJIHH,lJ;pH1IeCKHX BaJIOB, Me1'pHqeCKHe
e,n;HHHIJ;bI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
5.5. KOHYCbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
5.5.1. CaMo3IDKHMHbIe Mop3e H Me1'pHqeCKHe 5% KOHYCbI . . . . . . . . . . . . . .421
5.5.2. KOH)'CHbIe KOHIJ;bI mnHH,lJ;eJIeH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
5.5.3. KOH)'CbI AIDI XBOCTOBHKOB HHCTPYMeHTOB. . . . . . . . . . . . . . . . . . . . . . 435
5.5.4. <l>JIaHIJ;bI HHCTPYMeHTaJIbHOH onpaBKH . . . . . . . . . . . . . . . . . . . . . . . . 437
5.5.5. KOHYC BpH,l()KlIOpT R8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
5.6. CHcTeMbI 06'beMHOH nepe,n;aqH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
5.6.1. THnHqHaH nHeBMaTHqeCKaH CHCTeMa . . . . . . . . . . . . . . . . . . . . . . . . . 439
5.6.2. THnH1IHaH fH,ll;paBJIHqeCKaH CHCTeMa . . . . . . . . . . . . . . . . . . . . . . . . . 441
5.6.3. BH,lI;bI B03,[(yuIHbIX KOMnpeccopOB . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
5.6.4. fH,ll;paBJIHqeCKHe HaCOCbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
5.6.5. I1CnOJIHHTeJIbHbIe MexaHH3MbI (JIHHeHHbIe) . . . . . . . . . . . . . . . . . . . . 450
5.6.6. I1CnOJIHHTeJIbHbIe MexaHH3MbI (BpaInaTeJIbHbIe) . . . . . . . . . . . . . . . . . 452
5.6.7. I1CnOJIHHTeJIbHbIe MexaHH3MbI CMemaHHOfO THna . . . . . . . . . . . . . . . .454
5.6.8. 0603HaqeHIDI AIDI fH,ll;paBJIH1IeCKHX CHCTeM . . . . . . . . . . . . . . . . . . . . 454
5.6.9. ,lJ;aHHbIe AIDI paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (OCHoBHbIe
<POPMYJIbI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
5.6.10. llpoeKTHbIe ,L(aHHbIe 06'beMHOH fH,ll;pOnepe,n;aqH (rH,ll;paBJIHqeCKHe
IJ;HJIHH,n;p bI ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
5.6.11. ,lJ;aHHbIe ,L(JUI paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (rH,ll;paBJIHqeCKHe
Tpy6bI H lIIJIaHfH) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
5.6.12. ,lJ;aHHbIe ,WUI paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (pa60qHe XH,l(KOCTH,
fepMeTHKH H KOH1'pOJIb 3arpH3HeHHocTH). . . . . . . . . . . . . . . . . . . . . . 460
5.6.13. ,lJ;aHHbIe AIDI paCqeTOB 06'beMHOH rH,ll;ponepe,n;aqH (fH,ll;paBJIHqeCKHe
aKKYMYJIHTOpbI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
5.6.14. ,lJ;aHHbIe ,LUIH paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (fH,ll;paBJIH1IeCKOe
OXJIIDK,LJ;eHHe H HarpeB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
5.6.15. ,lJ;aHHbIe AIDI paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (KJIanaHbI) . . . . . . . . 465
5.6.16. ,lJ;aHHbIe ,L(JUI paCqeTOB 06'beMHOH rH,ll;ponepe,n;aqH (nHeBMaTH1IeCKHe
IJ;HJIHH,lJ;pbI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
5.6.17. llpoeKTHbIe ,n;aHHbIe 06'beMHOH fH,ll;pOnepe,n;aqH (yruIOTHeHHe,
<pHJIb1'paIJ;IDl H cMa3Ka). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
5.6.18. ,lJ;aHHbIe AIDI paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (B03,n;ymHbIe
KOMnpeccopbI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
5.6.19. ,lJ;aHHbIe AIDI paCqeTOB 06'beMHOH fH,ll;pOnepe,n;aqH (Ta6JIHIJ;bI
H K03<P<PHIJ;HeHTbI nepeCqeTa B nHeBMaTHKe) . . . . . . . . . . . . . . . . . . . 472
5.6.20. HopMaTHBHbIe CCbIDKH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
fJIaBa 6. MamHHocTpOHTeJIbHbIe MaTepHaJIbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
6.1. MeXaHHqeCKHe CBOHCTBa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
6.1.1. llpOqHOCTb Ha paCTIDKeHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
6.1.2. llpOqHOCTb Ha C)KaTHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
COOep:JICaHUe
6.1. 3 . II po qH OCTb Ha c,n;BHr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
6.1.4. llpOqHOCTb npH y,n;ape: y,n;apHaH BH3KOCTb . . . . . . . . . . . . . . . . . . . . . 478
6.1.5. Y npyrocTb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
6.1.6. llrracTHqHOCTb......................................... 479
6.1.7. BH3KOCTb............................................. 479
6.1.8. KOBKOCTb............................................ 480
6.1.9. TBep,lJ;OCTb........................................... .480
6.1.10. McnbITaHHe Ha paCTIDKeHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
6.1.11. MHTepnpeTanIDI HcnbITaHIDI Ha paCTIDKeHHe: MaTepHaJIa
C BbIpIDKeHHbIM npe,lJ;eJIOM Tex:yqeCTH. . . . . . . . . . . . . . . . . . . . . . . . . 481
6.1.12. MHTepnpeTanHH HcnbITaHIDI Ha paCTIDKeHHe: YCJIOBHbIH npe,lJ;eJI
Tex:yqeCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
6.1.13. MHTepnpeTanIDI HcnbITaHIDI Ha paCTIDKeHHe: CeKYIIJ;HH MO.nyJIb . . . . . . 483
6.1.14. McnbITaHHe Ha y,n;apHYIO BH3KOCTb: no M30.ny. . . . . . . . . . . . . . . . . . . 484
6.1.15. McnbITaHHe Ha y,n;apHYIO BH3KOCTb: no IIlapnH. . . . . . . . . . . . . . . . . .484
6.1.16. MHTepnpeTanIDI pe3YJIbTaTOB HcnbITaHIDI Ha y,lJ;apHYIO BH3KOCTb. . . . . . 485
6.1.17. Onpe,n;eJIeHHe TBep,n;OCTH no BpHHeJIIO . . . . . . . . . . . . . . . . . . . . . . . . 485
6.1.18. Onpe,lJ;eJIeHHe TBep,n;OCTH no BHKKepcy. . . . . . . . . . . . . . . . . . . . . . . . 486
6.1.19. Onpe,n;eJIeHHe TBep,n;OCTH no POKBeJIJIy . . . . . . . . . . . . . . . . . . . . . . . . 486
6.1.20. Onpe,lJ;eJIeHHe TBep,n;OCTH no POKBeJIJIy (npH )'MeHbmeHHbIX
no cpaBHeHHIO C 06bNHbIMH HarpY3KaMH) . . . . . . . . . . . . . . . . . . . . . 487
6.1.21. CpaBHHTeJIbHaH mKaJIa TBep,n;OCTH . . . . . . . . . . . . . . . . . . . . . . . . . . .488
6.2. qepHbIe MeTaJIJIbI H cn.rraBbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
6.2.1. qepHbIe MeTaJIJIbI: HeJIerHpOBaHHaH yrJIepO,lJ;HCTaH CTaJIb . . . . . . . . . . . 489
6.2.2. BJIIDIHHe co,n;ep)KaHIDI yrJIepo,n;a Ha C1'pyKTypy, CBOHCTBa
H npHMeHeHHe HeJIerHpOBaHHbIX yrJIepo,n;HCTbIX CTaJIeH. . . . . . . . . . . . 490
6.2.3. qepHbIe MeTaJIJIbI: JIerHPYIOInHe 3JIeMeHTbI. . . . . . . . . . . . . . . . . . . . . 491
6.2.4. BpHTaHcKHe CTaH,ll;apTbI, OTHOCHInHeCH K qepHbIM MeTaJIJIaM . . . . . . . . 492
6.2.5. HeKoTopbIe THnHqHbIe CTaJIH H HX npHMeHeHHe . . . . . . . . . . . . . . . . . 495
6.2.6. HeKoTopbIe THnHqHbIe HHC1'p)'MeHTaJIbHbIe CTaJIH . . . . . . . . . . . . . . . . 502
6.2.7. qyryH (cepbIH) C n.rraCTHHqaTbIM rpa<J;>HToM . . . . . . . . . . . . . . . . . . . . 504
6.2.8. BS EN 1561: 1997 CepbIe qyryHbI . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
6.2.9. KOBKHe qyryHbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
6.2.10. qyryH C mapOBH,lJ;HbIM rpa<J;>HToM. . . . . . . . . . . . . . . . . . . . . . . . . . . .510
6.2.11. JIerHpoBaHHbIe qyryHbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
6.2.12. CocTaB, CBOHCTBa H npHMeHeHHe HeKOTopbIX THnWlHbIX qyryHOB. . . . . 514
6.3. I(BeTHbIe MeTaJIJIbI H cn.rraBbI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
6.3.1. I(BeTHbIe MeTaJIJIbI H cn.rraBbI. BBe,lJ;eHHe. . . . . . . . . . . . . . . . . . . . . . .515
6.3.2. Cn.rraBbI C BbICOKHM co,n;ep)KaHHeM Me,n;H . . . . . . . . . . . . . . . . . . . . . . 515
6.3.3. KOBaHaH Me,lJ;b H Me,lJ;HbIe cn.rraBbI: KO,n; COCTOHHIDI. . . . . . . . . . . . . . . . 516
6.3.4. BpHTaHcKHe CTaH,lJ;apTbI, OTHOCHInHeCH K Me,lJ;H H Me,lJ;HbIM cn.rraBaM . . . 517
6.3.5. llpyTKH H npo<J;>HJIH H3 Me,n;H H Me,n;HbIX cn.rraBOB . . . . . . . . . . . . . . . . 518
6.3.6. ,lJ;e<J;>opMHpyeMaH Me,lJ;b H Me,n;HbIe cn.rraBbI. . . . . . . . . . . . . . . . . . . . . . 522
6.3.7. ,lJ;e<J;>opMHpyeMaH Me,n;b H Me,lJ;HbIe cn.rraBbI. . . . . . . . . . . . . . . . . . . . . . 526
6.3.8. nHCTOBaH, nOJIOCOBaH Me,n;b H Me,lJ;HaH <J;>OJIbra. . . . . . . . . . . . . . . . . . . 528
6.3.9. naT)'HHbIe JIHCTbI, nOJIOCbI H <J;>OJIbra: ,n;ByxKOMnOHeHTHbIe (6HHapHbIe)
cn.rraBbI Me,n;H H nHHKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · · . . . 529
6.3.10. naT)'HHbIe JIHCTbI, nOJIOCbI H <J;>OJIbra: CnenHaJIbHbIe cn.rraBbI
H JIerHpOBaHHaH JIaT)'Hb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · . . . . 531
6.3.11. nHCTbI, nOJIOCbI H <J;>OJIbra H3 <J;>OC<J;>OpHCTOH 6pOH3bI . . . . . . . . . . . . . . 532
6.3.12. Cn.rraBbI aJIlOMHHHeBoH 6pOH3bI. BBe,n;eHHe. . . . . . . . . . . . . . . . . . . . . 534
6.3.13. nHCTbI, nOJIOCbI H <J;>OJIbra H3 aJIlOMHHHeBOH 6POH3bI. . . . . . . . . . . . . . 534
6.3.14. Me,n;Ho-HHKeJIeBbIe (KYJIPO-HHKeJIeBbIe) JIHCTbI, nOJIOCbI H <J;>OJIbra . . . . 535
6.3.15. HHKeJIb-Cepe6pHHbIe JIHCTbI, nOJIOCbI H <J;>OJIbra. . . . . . . . . . . . . . . . . . 536
CoiJep:JICaHUe
6.3.16. CMemaHHbIe ,L(e<J;>opMHpyeMbIe Me,L(HbIe CnrraBbI. . . . . . . . . . . . . . . . . .537
6.3.17. Me,L(HbIe cnrraBbI AIDI JIHTbH: rpynna A. . . . . . . . . . . . . . . . . . . . . . . .540
6.3.18. Me,L(HbIe cnrraBbI AIDI JIHTbH: rpynna B. . . . . . . . . . . . . . . . . . . . . . . . 542
6.3.19. Me,L(HbIe CnJIaBbI AIDI JIHTbH: rpynna C. . . . . . . . . . . . . . . . . . . . . . . . 544
6.3.20. Me,L(HbIe cnrraBbI AIDI JIHTbH: THnHqHbIe CBOHCTBa H 3HaqeHIDI
xeCTKOCTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
6.3.21. .AJIIOMHHHH H aJIlOMHHHeBbIe cnrraBbI. . . . . . . . . . . . . . . . . . . . . . . . . 548
6.3.22. BpHTaHCKMe CTaH,napTbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
6.3.23. llrrHTbI, JIHCTbI H nOJIOCbI H3 qHCTOrO aJIlOMHHIDI . . . . . . . . . . . . . . . . 549
6.3.24. TIJIHTbI, JIHCTbI H nOJIOCbI H3 aJIlOMHHHeBoro cnrraBa:
6e3 TepMHqeCKOH o6pa6oTKM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
6.3.25. llrrHTbI, JIHCTbI H nOJIOCbI H3 aJIlOMHHHeBoro cnrraBa:
TepMHqeCKaH o6pa6oTKa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
6.3.26. TIpyrKM, mTaMnOBaHHbIe 1py6bI H npo<J;>MJIH H3 aJIlOMHHIDI
H aJIlOMHHHeBoro cnrraBa AIDI o6w;ero MamHHOC1pOeHIDI:
6e3 TepMHqeCKOH o6pa6oTKM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
6.3.27. TIpyrKM, mTaMnOBaHHbIe 1py6bI H npo<J;>MJIH H3 aJIlOMHHHeBoro
cnrraBa ,L(. UI o6w;ero MamHHOC1poeHIDI: TepMlNeCKaH o6pa6oTKa . . . . . 555
6.3.28. nHTbIe aJIlOMHHHeBbIe cnrraBbI, rpynna A: o6w;ee Ha3HaqeHHe . . . . . . . 557
6.3.29. nHTbIe aJIlOMHHHeBbIe cnrraBbI, rpynna B: CneI(HaJIbHOe Ha3HaqeHHe . . 558
6.3.30. nHTbIe aJIlOMHHHeBbIe cnrraBbI, rpynna C: CneIJ;HaJIbHOe Ha3HaqeHHe
H orpaHHqeHHaH npHMeHHMOCTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
6.3.31. nHTbIe aJIlOMHHHeBbIe cnrraBbI: MeXaHHqeCKMe CBOHCTBa . . . . . . . . . . . 560
6.3.32. BS EN 29453: 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
6.3.33. THnHqHbIe npHMeHeHIDI MHrKOrO npHnOH . . . . . . . . . . . . . . . . . . . . . 563
6.3.34. TIaHKa cepe6pHHbIM npHnoeM (naHKa TBep,L(bIM npHnoeM) . . . . . . . . . . 563
6.3.35. fpynna AG: naHKa cepe6pHHbIM TBep,L(bIM npHnoeM . . . . . . . . . . . . . . 564
6.4. Pa3MepbI MeTaJIJIHqeCKMX H3,L(eJIHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
6.4.1. Pa3MepbI MeTaJIJIHqeCKMX H3,L(eJIHH: BBe,L(eHHe B BS 6722: 1986 . . . . . . . 565
6.4.2. PeKOMeH.nYeMbIe ,L(HaMe1pbI npOBOJIOKM, Me1plNeCKMe. . . . . . . . . . . . . 566
6.4.3. P KOMeH.nYeMbIe pa3MepbI npyrKoB H nrrocKMX H3,L(eJIHH . . . . . . . . . . . 567
6.4.4. PeKOMeH.nYeMbIe mHpHHa H ,L(JIHHa nrrOCKMX H3,L(eJIHH . . . . . . . . . . . . . 567
6.4.5. Macca Me1plNeCKMX npyrKoB KpyrJIOrO H KBan;paTHoro CeqeHIDI . . . . . 568
6.4.6. Pa3MepbI npyrKoB meCTMyrOJIbHOrO CeqeHIDI AIDI BHHTOBbIX KpenexeH,
Me1pHqeCKMe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
6.4.7. Pa3MepbI KaJIH6pOB H MX 3KBHBaJIeHTbI . . . . . . . . . . . . . . . . . . . . . . . 570
6.5. TIOJIHMepHbIe MaTepHaJIbI (nrracTMaccbI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
6.5.1. llOJIHMepHbIe MaTepHaJIbI (nrracTMaccbI). BBe,L(eHHe. . . . . . . . . . . . . . .571
6.5.2. HeKoTopbIe BIDKHbIe TepMOpeaKTHBHbIe nOJIHMepbI . . . . . . . . . . . . . . . 573
6.5.3. HeKoTopbIe BIDKHbIe TepMOnrraCTINHbIe nOJIHMepbI . . . . . . . . . . . . . . . 574
--
----
fJIaBa 7. Ha3Ha1JeHHe JIHHeHHbIX H reOMeTpH1JeCKHX JJ;OnyCKOB. . . . . . . . . . . . . . . . . 575
7.1. Ha3HaqeHHe JIHHeHHbIX ,L(OnYCKOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
7 .1.1. onycKM H nOCan;KM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
7.1.2. THnbI nOCan;OK. . .A--O-- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
7.1.3. T OqHOCTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
7.2. CTaH,napTHaH CHCTeMa ,nOnyCKOB H nOCan;OK. BBe,L(eHHe . . . . . . . . . . . . . . . . . 578
7.2.1. llpHMeHeHHe Ta6JIHIJ; ,L(OnYCKOB H nOCan;OK. .. . . . . . . . . . . . . . . . . . . . 579
7.2.2. BbI60p TOqHOCTH ,L(onycKa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
7.3. Ha3HaqeHHe ,L(OnYCKOB <J;>OpMbI H pacnOJIOXeHIDI nOBepXHocTeH . . . . . . . . . . . 583
7.3.1. Ha3HaqeHHe ,L(OnyCKOB <J;>OpMbI (OCHoBHbIe nOIDITHH). . . . . . . . . . . . . . 584
7.3.2. PaMKa ,L(OnYCKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
7 .3.3. OnyCK <J;>OpMbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
7.3.4. TIoJIe ,L(OnYCKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
CoiJep:JICaHUe
7.3.5. 3aBHcHMbIH ,Z:(OrryCK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
7.3.6. 06JIaCTH npHMeHeHIDI ,Z:(OrryCKOB <pOpMbI H pacnOJIOXeHIDI
nOBepXHocTeH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
7.4. ,lI;eHcTBHTeJIbHbIH pa3Mep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
7.5. 3KOHOMHqeCKHH 3<p<peKT ,Z:(onYCKa <pOpMbI . . . . . . . . . . . . . . . . . . . . . . . . . . 591
fJIaBa 8. ABToMaTH3HpoBaHHoe KOHCTPYHPOBaHHe. . . . . . . . . . . . . . . . . . . . . . . . . . 593
8.1. qllYTHna CNC............................................ .593
8.1.1. THnHqHbIe 06JIaCTH npHMeHeHIDI qHCJIOBOrO nporpaMMHoro
ynpaBJIeHIDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
8.1.2. llpeHMYJl(eCTBa H orpaHHQeHIDI qllY . . . . . . . . . . . . . . . . . . . . . . . . 593
8.1.3. YnpaBlliIeMbIe Koop,Z:(HHaTbI CTaHKOB . . . . . . . . . . . . . . . . . . . . . . . . . 595
8.1.4. CHCTeMbI ynpaBJIeHIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
8.1.5. <l>opMaT H TepMHHOJIOrIDI nporpaMMbI . . . . . . . . . . . . . . . . . . . . . . . . 597
8.1.6. <l>opMaT C an;peCOM B KaJK,LJ;OM CJIOBe . . . . . . . . . . . . . . . . . . . . . . . . . 598
8.1.7. KO,Z:(HpoBaHHaH HH<popManIDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
8.1.8. Bxo,Z:(HbIe ,Z:(aHHbIe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
8.1.9. KoppeKUIDI Ha ,Z:(JIHHY HHCTPYMeHTa: <ppe3epoBaHHe . . . . . . . . . . . . . . 602
8.1.10. KOMneHcanIDI Ha ,Z:(HaMe1p <ppe3bI: <ppe3epoBaHHe. . . . . . . . . . . . . . . .603
8.1.11. MeTo,Z:(bI nporpaMMHpOBaHIDI: <ppe3epoBaHHe H CBepJIeHHe. . . . . . . . . . 605
8.1.12. llpHMep nporpaMMbI: <ppe3epoBaHHe . . . . . . . . . . . . . . . . . . . . . . . . .606
8.1.13. KoppeKUIDI Ha HHC1pYMeHT: o6pa6oTKa Ha TOKapHOM CTaHKe. . . . . . . .608
8.1.14. KoppeKUIDI Ha pan;HYC BepmHHbI pe3na: TOKapHbIH CTaHOK . . . . . . . . . 609
8.1.15. MeTo,Z:(bI nporpaMMHpOBaHIDI: o6pa6oTKa Ha TOKapHOM CTaHKe. . . . . . . 610
8.1.16. llpHMep nporpaMMbI: o6pa6oTKa Ha TOKapHOM CTaHKe . . . . . . . . . . . . 612
8.1.17. CJIOBapb TepMHHOB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
8.2. ABToMaTH3HpoBaHHoe npOeKTHpOBaHHe (CAD) . . . . . . . . . . . . . . . . . . . . . . . 617
8.2.1. BBe,Z:(eHHe B aBTOMaTH3HpOBaHHoe npoeKTHpOBaHHe . . . . . . . . . . . . . . . 617
8.2.2. AnnapaTHoe 06eCneQeHHe CHCTeMbI CAD . . . . . . . . . . . . . . . . . . . . . . 619
8.2.3. llporpaMMHoe o6eCneQeHHe CHCTeMbI CAD. . . . . . . . . . . . . . . . . . . . . 622
8.2.4. ABToMaTH3HpoBaHHoe npoeKTHpOBaHHe H npOH3BO,Z:(CTBO. . . . . . . . . . . . 623
8.2.5. llpeHM ecTBa H orpaHHQeHHH CAD . . . . . . . . . . . . . . . . . . . . . . . . . 624
8.3. llpOMbIIIIJIeHHbIe P060TbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
8.3.1. BBe,Z:(eHHe B po6oTOTeXHHKY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .624
8.3.2. YnpaBJIeHHe P060TOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
8.3.3. feoMeTpIDI PYKH po6oTa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
BBE.QEHME
IIpOIIIJIO yxe 14 JIeT C Tex lIOp, KaK lIepBoe H3,l(aHHe ClIpaBOqHHKa HIDKeHepa-Me-
xaHHKa 6 bill 0 BbIrryI.l(eHO B ero CylI(eCTByrolI(eM <popMaTe. XOT.H B TeKCT BToporo H3-
,l(aHIDI, orry6JIHKOBaHHoro lIpH6JIH3HTeJIbHO 7 JIeT Ha3M, 6billO BKJIlOqeHO MHoxe-
CTBO 06HOBJIeHHH 6pHTaHcKHX CTaH)];apTOB, CTPYKTYPa KHHrH He H3MeHHJIaCb.
B TeqeHHe 14 JIeT, C MOMeHTa BbIxo,n;a lIepBoro H3,n;aHIDI CnpaBOqHHKa HIDKeHe-
pa-MexaHHKa, 6pHTaHcKoe MaIIIHHOCTPoeHHe lIpeTeplIeJIO PM cylI(ecTBeHHbIX H3-
MeHeHHH C nepeMeII eHHeM aKUeHTa C lIpOH3Bo,n;cTBa Ha npOeKTHpOBaHHe H pa3BH-
THe. B TO xe caMoe BpeM.H lIpOH3BO,n;CTBO B 3HaqHTeJIbHOH CTeneHH lIepeMeCTHJIOCb
B BOCTOqHylO EBpony, A3HIO H Ha ,lJ;aJIbHHH BOCTOK C ueJIblO YMeHbIIIeHIDI 3KCnrry-
aTaUHOHHbIX pacxo,n;OB B YCJIOBIDIX Bce 60JIbIIIeH KOHKYPeHUHH Ha rJI06aJIbHOM
pbIHKe.
IIo3ToMY lIepe,l( BbIrryCKOM 3Toro TpeTbero H3,l(aHIDI ClIpaBOQHHKa H3,n;aTeJIH H
aBTOp npe,l(npHIDIJIH IIIHpOKoe H3yqeHHe pbIHKa. fJIaBHbIe pe3YJIbTaTbI 3Toro HCCJIe-
,l(OBaHHR nOKa3aJIH, qTO:
. nOTpe6HOCTb B ClIpaBOqHHKe no meXHOJl02UU npou38oiJcm8a CMeHHJIaCb no-
TPe6HOCTblO B CnpaBOqHHKe no UH'JICeHepHOMY npoeKmup08aHUlO;
. 60JIbIIIe HH<popMaUHH TPe6yeTcH no TaKHM 06JIaCTHM, KaK pOJIHKOBbIe uelI-
HbIe lIpHBO,n;bI, lIHeBMaTHqeCKHe H rH,ll;paBJIHqeCKHe CHCTeMbI B pa3,n;eJIe, lIO-
CBHlI(eHHOM MeXaHHqeCKHM nepe,n;aqaM;
. 60JIbIIIe HH<popMaUHH TPe6yeTcH lIO BonpocaM UH'JICeHepHOU cmamUKU, iJuHa-
MUKU H MameMamUKU;
. pa3,n;eJIbI, HHTepec K KOTOpbIM CHH3HJICH, KacalOTCR pe Hx HHCTPYMeHTOB.
B HaCTOHI..I.J;ee BpeM.H ,l(aHHbIe 0 pe Hx HHCTPYMeHTax npe,n;CTaBJI.HIOTCH Me-
Hee Ba)l{HbIMH, nOCKOJIbKY aKUeHT CMeCTHJICH OT npOH3Bo,n;CTBa K npoeKTHpo-
BaHHIO. KpOMe Toro, ,n;aHHbIe 0 pe Hx HHCTPYMeHTax Tenepb IIIHpOKO
npe,n;CTaBJIeHbI Ha HHTepHeT-CTPaHHuax lIpOH3Bo,n;HTeJIeH pe HX HHCTPy-
MeHTOB. IIepeqeHb lIOJIe3HbIX Be6-caHTOB BKJIlOqeH B IIpHJIoxeHHe 4.
PoiJ'JICep JI. TUMUH2C (2005)
npE.QMCnOBME
KaK CKa3aHO BO BBe,n;eHHH, 3TO HOBOe H3,n;aHHe ClIpaBOqHHKa HIDKeHepa-MeXaHHKa
0TPIDKaeT H3MeIDIlOmyIOCH lIpHpO,n;y MaIIIHHOCTPOeHIDI H CB.H3aHHbIe C HeH TeXHO-
JIOrHH B BeJIHK06pHTaHHH. MHOrHe H3 6pHTaHCKHX CTaH,ll;apTOB (BS), UHTHpyeMbIe
B lIpe,n;bLl(ylI(HX H3,n;aHIDIX, 6billH HClIpaBJIeHbI HJIH y,n;aJIeHbI H 3aMeHeHbI Ha CTaH-
,n;apTbI BS EN H BS EN ISO. KaTaJIOr 6pHTaHcKHX CTaH,ll;apTOB 60JIbIIIe He ,n;OCT}'-
neH B 6)'Ma)l{HOM BapHaHTe, O,L(HaKO H3,n;aH Ha KOMnaKT-,n;HCKe.
CxeMLI YCJIOBHLIX 0603HaqeHHH CTaH,ZJ;apTOB
CTaH,ll;apTbI B KaTaJIOre 3aKo,n;HpOBaHbI, KaK lIOKa3aHO Ha CJIe,n;yrolI(eM npHMepe.
Pa3JIHqHbIe 3JIeMeHTbI lIOMeqeHbI 6YKBaMH A-J. KrrlOq 06'b.HCIDIeT K3.)I()J;bIH 3JIe-
MeHT. He Bce 3JIeMeHTbI BCTpeqaIOTC.H B K3.)I()J;OM CTaH,ll;apTe.
ABC
I I I
BS 1361: 1971 (1986)
D
I
"* IEC 269-1
npe,n;oxpaHHTeJIH nJIaBKHe ,ZJ;JIB D;eneii nepeMeHHoro TOKa B 6LITOBLIX nOMem;eHHJIX.
Tpe6oBaHIDI, KJIaCCbI H HClIbITaHIDI ,LUI.H IIJIaBKHX BCTaBOK, OCHOBaHHH npe,n;o-
xpaHHTeJI.H H ,n;epxaTeJIeH IIJIaBKHX BCTaBOK. Pa3MepbI H BpeM.HTOKOBbIe 30HbI ,LUI.H
IIJIaBKHX BCTaBOK. THII I-KJIacc 240B H 5-45A,LUI.H 3aMeHbI lIoTPe6HTeJI.HMH B 6bITO-
BbIX YCJIOBIDIX; THn II - KJIacc 415 B H 60, 80 HJIH 100A,LUI.H HClIOJIb30BaHHR npe,n;-
CTaBHTeJI.HMH lIOCTaBII(HKOB 3JIeKTp03HeprHH BO BXO,L(HbIX 6JIOKaX )I{HJIbIX HJIH aHa-
JIOrHQHbIX lIOMelI(eHHH.
G
I
AMD 4171, HHBapb 1983 (Gr 0) R 00000821
AMD 4795, HHBapb 1985 (Gr 2) R 00000833 - E
AMD 6692, HHBapb 1991 (FOC) 00000845
J - A5, 44 CTpaHHUbI
00000818
I
I
Gr7
I
H
PEL/32
I
F
llpeOUCIl08ue
A - l1.neHTH<pHKaTOp H3,n;eJIIDI.
B - ,lJ;aTa lIepBOH rry6JIHKaUHH.
C - ITO,n;TBep)l{)];eH B 1986 r., YKa3bIBaeT Ha rrpo,n;OJDKeHHe ,n;eHcTBIDI CTaH,ll;apTa
6e3 lIOJIHOH nepepa60TKH.
D - = - l1.neHTWlHbIH CTaH,ll;apr: rry6JIHKaUIDI BSI H,lI;eHTWlHa BO Bcex lIo,n;p06-
HOCTRX COOTBeTCTByrolI(eM)' eBporreHCKOM)' H/HJIH MeX,ZJyHapo,n;HOM)' CTaH,ll;apTY.
HJIH = - (TeXHWlecKH) 3KBHBaJIeHTHbIH CTaH,ll;apT: rry6JIHKaUIDI BSI BO Bcex
TeXHWlecKHX OTHOIIIeHIDIX aHaJIOrWlHa COOTBeTCTByromeMY eBpolIeHcKOM)' H/HJIH
MeX,ZJyHapo,n;HOM)' CTaH,ll;apTY, XOT.H <POPMYJIHpOBKa H lIpe,n;CTaBJIeHHe MOryT OTJIH-
qaTbCH B 3HaqHTeJIbHOH CTerreHH.
HJIH '* - OTHOC.HII(HHCH K TOM)' xe BOlIPOCY, HO He 3KBHBaJIeHTHbIH CTaH,ll;apT:
rry6JIHKaUIDI BSI, KOTOpasI OlIHCbIBaeT lIpe,n;MeTbI, aHaJIOrHqHbIe onHcaHHbIM B eB-
pOlIeHCKOM H/HJIH MeX,ZJyHapo,n;HOM CTaH,ll;apTe. Co,n;epxaHHe, o,n;HaKO, HCKJIlOqaeT
nOJIHyIO H,lI;eHTWlHOCTb HJIH TeXHHqecKYIO 3KBHBaJIeHTHOCTb.
E - YHHKaJIbHbIH KO,n; H3,L(eJIIDI ,LUI.H lIonpaBOK.
F - TeXHWlecKHH KOMHTeT BSI HeceT OTBeTCTBeHHOCTb 3a 3TY rry6JIHKaUHIO.
G - ITolIpaBKH BKJIlOqeHbI B lIepeH3,n;aHHbIH TeKCT. OTcyrcTBHe R 03HaqaeT,
qTO lIOlIpaBKH He .HBJI.HIOTC.H qaCTblO TeKCTa.
H - :QeHa rpynnbI: 06paTHTeCb K KJIanaHY Ha BHyrpeHHeH CTopOHe 3MHeM 06-
JIO)l{lGf KaTaJIOra.
I - YHHKaJIbHbIH KO,n; H3,n;eJIIDI.
J - BOJIbIIIHHCTBO HOBbIX H lIepepa60TaHHbIX CTaH,ll;apTOB ny6JIHKYIOTC.H B <pop-
MaTe A4. Pa3MepbI, OTJIWlHbIe OT A4, BKJIlOqeHbI B ClIHCOK.
nonpaBKH
Bce OT,n;eJIbHbIe lIorrpaBKH K MOMeHTY BbInYCKa BKJIlOqeHbI BMeCTe C JII06bIMH
OCHOBHbIMH rry6JIHKaUIDIMH. :QeHbI YKa3aHbI B lIpHJIOXeHHH. ECJIH lIOlIpaBKa
BKJIlOqeHa B Oqepe,n;Hoe H3,n;aHHe, TeKCT co,n;ep)I{HT YKa3aHHe, npHBJIeKalOlI(ee K HeM
BHHMaHHe, H lIOMeTKH Ha noIDIX B cooTBeTcTByrolI(HX MecTax Ha HCnpaBJIeHHbIX
cTPaHHUax.
0630P
IIoJIHTHKa BSI ,LUI.H K3.)I()];orO CTaH,ll;apTa 3aKJIIOQaeTC.H B lIepecMoTPe ero OTBeT-
CTBeHHbIM TeXHHqeCKHM KOMHTeTOM, He 60JIee qeM qepe3 5 JIeT nOCJIe rry6JIHKa-
UHH, qT06bI yCTaHoBHTb, .HBJI.HeTC.H JIH OH Bce elI(e ,n;eHCTByroII(HM, H, eCJIH HeT,
OlIpe,n;eJIHTb H npe,n;lIpHIDITb lIO,n; KOHTpOJIeM COOTBeTCTByrolI(He MepbI. 06CTO.HTeJIb-
CTBa MOryT npHBeCTH K 60JIee paHHeMY lIepeCMOTPY.
IIpH paCCMOTPeHHH CTaH,ll;apTa KOMHTeT MoxeT BbI6paTb O,n;HH H3 QeTbIpex Ba-
pHaHTOB:
. Om038aHue: YKa3bIBaeT, QTO CTaH,ll;apT 60JIee He HBJI.HeTCR ,n;eHCTByroII(HM.
. 06ofl8JleHUe 06 ycmape8aHUU: YKa3bIBaeT B lIonpaBKe, qTO CTaH,ll;apT He peKO-
MeH,D;yeTC.H ,LUI.H HClIOJIb30BaHIDI B HOBOM o6opy,n;OBaHHH, HO ,n;OJDKeH 6MTb
coxpaHeH lIPH 06CJI}T)KHBaHHH cyru;eCTByrolI(ero o6opy,n;OBaHIDI C MHTeJIbHbIM
O)l{H,ll;aeMbIM CpOKOM CJIy:>K6bI.
llpeducno8ue
. llepeCMOmp: BOBJIeqeHHe npOue,n;ypbI HOBbIX npOeKTOB.
. lloiJm8ep'JICiJeHUe: YKa3bIBaeT, qTO CTaH,lJ;apT lIpO,n;OJl)l{aeT ,n;eHCTBOBaTb 6e3
nOJIHOrO lIepeCMOTPa. IIocJIe lIO,n;TBep)l{)];eHIDI lIy6JIHKaUHH Ha KOnHH CTa-
BHTCH HOBaR lIeqaTb C MeCRueM H rO,n;OM lIO,L(TBep)l{)];eHIDI.
IIpH pa3pa60TKe HOBbIX lIpOeKTOB H3,n;eJIHH H 060py,n;OBaHIDI BCer,n;a ,n;OJDKHbI
HClIOJIb30BaTbC.H nOCJIe,n;HHe BbIrryCKH CTaH,lJ;apTOB. O,n;HaKO MHOrHe H3,n;eJIIDI BCe
eI..I.J;e H3rOTaBJIHBalOTCH lIO YCTapeBalOlI(HM H YCTapeBIIIHM CTaH,lJ;apTaM, qT06bI y,n;OB-
JIeTBOpHTb BCe elI(e YCTOHqHBbIH ClIpOC. 3TO ,n;eJIaeTC.H He TOJIbKO B ueJI.HX TeXHWle-
CKoro 06CJI}')KHBaHIDI, HO TaIOKe H M.H TeKYIJ,J;ero lIpOH3Bo,n;CTBa TaM, r,n;e PbIHOq-
HbIe CHJIbI elI(e He nOTPe60BaJIH 06HOBJIeHIDI pa3pa60TKH. 3TO oc06eHHO KaCaeTC.H
BHHTOBbIX KpelIe)I{HbIX H3,n;eJIHH. IIo 3TOH lIpHqHHe 6billH coxpaHeHbI Ta6JIHUbI
BHHTOBbIX pe3b6 H3 lIpe,n;bI,lJ;yru;HX H3,n;aHHH HapMY C HOBbIMH TPe60BaHIDIMH
BE EN ISO.
HOBbIe CTaH,lJ;apTbI, KaK lIpaBHJIO, HBJI.HIOTC.H 60JIee ,n;eTaJIbHbIMH H OlIHCaTeJIb-
HbIMH, qeM HX lIpe,n;IIIeCTBeHHHKH, n03TOM)' B 3TOT ClIpaBOqHHK MOryT 6bITb BKJIIO-
qeHbI TOJIbKO HaH60JIee cyru;eCTBeHHbIe HH<popMaUHOHHbIe Ta6JIHUbI. IIpH He06xo-
,n;HMOCTH 3a ,n;aJIbHeHIIIeH HH<popMaUHeH Bcer,n;a CJIe,n;yeT 06palI(aTbCR K nOJIHbIM
CTaH,lJ;apTaM, H ClIHCOK 6H6JIHOTeK, no,n,n;ep)I{HBaIOII(HX lIOJIHbIe Ha60pbI COBpeMeH-
HbIe CTaH,L(apTOB, BKJIlOqeH B lIpHJIOXeHHe B KOHue 3TOH KHHrH. CTaH,lJ;apTbI, YKa-
3aHHbIe B ClIpaBOqHHKe, 6billH nOCJIe,n;HHMH Ha MOMeHT lIy6JIHKaUHH, HO, BBH,lJ;Y
nOJIHTHKH peryJIHpHbIX lIepeCMOTpOB BSI, HeKOTopbIe H3 HHX MOryT CTaTb npe,n;Me-
TOM nepeCMOTpa B TeqeHHe BpeMeHH cyru;eCTBOBaHIDI 3TOH KHHrH. IIo 3TOM npHqH-
He CTaTYC YKa3aHHbIX CTaH,lJ;apTOB Bcer,n;a CJIe,n;yeT npOBep.HTb. B03MO)I{HbI KOHCYJIb-
TaUHH TeJIe<poHY ceTH no,n,n;ep)l{lGf BSI.
3TOT CnpaBOqHHK 6bill no,n;rOTOBJIeH B nOMOlI(b HIDKeHepaM-MexaHHKaM, 3aH.H-
TbIM B npoeKTHpOBaHHH, pa3pa60TKe H npOH3Bo,n;CTBe TeXHHQeCKHX H3,n;eJIHH H 060-
PY,L(OBaHH.H. OH TaIOKe lIOJIe3eH B KaqeCTBe HCTOqHHKa HH<popMaUHH BceM TeM HH-
xeHepaM, KOM)' 6bICTPO H exe,n;HeBHO TPe6yroTC.H HopMaTHBHO-TeXHHqeCKHe ,n;aH-
HbIe. ,lJ;JIR y,n;06cTBa CCbillOK 3Ta KHHra pa3,n;eJIeHa Ha 8 OCHOBHbIX pa3,L(eJIOB:
1. HIDKeHepHaR MaTeMaTHKa
2. HIDKeHepHaR CTaTHKa
3. HIDKeHepHaJI ,L(HHaMHKa
4. KpelIe)I{HbIe ,n;eTaJIH
5. MeXaHHqeCKHe lIepe,n;aqH
6. MalIIHHocTpOHTeJIbHbIe MaTepHaJIbI
7. Ha3HaqeHHe JIHHeHHbIX H reOMeTPHqeCKHX ,n;OrryCKOB
8. ABToMaTH3HpoBaHHoe KOHCTpYHPOBaHHe
B paMKax 3THX OCHOBHbI 3,n;eJIOB MaTepHaJI c06paH B JIOrlftleCKOM lIOCJIe,n;o-
BaTeJIbHOCTH M.H y,n;06cTBa 0<:. " " eHHM H COOTBeTCTBeHHO lIpOHYMepOBaH. 3TO lIO-
3BOJIReT qHTaTeJIIO HanpaBJIRTbC.H Henocpe,n;CTBeHHO K TPe6yeMOM)' IIYHIcry npH no-
MOI..I.J;H HCqepnbIBalOlI(e nOJIHOrO co,n;epxaHIDI H no,n;p06Horo aJI<paBHTHoro YKa3a-
TeJIR.
3TOT CnpaBOqHHK He yqe6HHK, a C60PHHK lIOJIe3HOH HH<popMaUHH. II03TOM)' B
pa3,n;eJIbI, CBR3aHHbIe C MaTeMaTHKOH, CTaTHKOM H ,L(HHaMHKOH, pa60qHe npHMepbI
BKJIlOqeHbI TOJIbKO TaM, r,n;e HHaqe MOrJIH 6bI B03HHKJIyTb He,n;opa3)'MeHIDI. ABTOp
IIpeouCJl()(Jue
lIpH3HaTeJIeH BpHTaHCKOM)' HHCTH1)'TY CTaH,ll;apTOB H BCeM HH,nyCTPHaJIbHbIM H
KOMMepqeCKHM KOMlIaHIDIM B BeJIHK06pHTaHHH H 3a py6exoM, KOTOpbIe lIOMOrJIH
06eCneqHTb COBpeMeHHbIe ,n;aHHbIe lIO TaKOM)' 60JIbIIIOM)' KOJIHqeCTBY 06JIaCTeH
TeXHHKH. K COXaJIeHHIO, OrpaHWleHHOe lIpOCTPaHCTBO 3TOH KHHrH n03BOJIHJIO
BKJIlOqHTb TOJIbKO KpaTKHe 0630pbI H3 BCerO pa3H006pa3IDI npe,n;OCTaBJIeHHOrO Ma-
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PoiJ'JICep JI. TUMUH2C (2005)
SnArO.QAPHOCTM
MbI XOTeJIH 6bI lI06JIaro,n;apHTb BCe KOMlIaHHH, JII06e3HO lIpe,n;OCTaBHBIIIHe aBTOp-
CKHe MaTepHaJIbI, HClIOJIb3yeMbIe B ,n;aHHOM H3,n;aHHH:
British Fluid Power Association (BFPA) (Pa3,n;eJIbI 5.6.9-5.6.18, 5.6.20).
ContiTech United Kingdom Ltd. (Pa3,n;eJIbI 5.2.4-5.2.18).
David Brown Engineering Ltd. (Pa3,n;eJI 5.1.1).
Henkel Loctite Adhesives Ltd. (Pa3,neJIbI 4.5.1-4.5.9).
IMI Norgren Ltd. (Pa3,n;eJI 5.6.19).
National Broach & Machine Co. (Pa3,n;eJIbI 5.1.4-5.1.17).
Renold pIc. (Pa3,n;eJIbI 5.3-5.4,5.3.10-5.3.13,5.3.16,5.3.18-5.3.19,5.3.21-
5.3.22, 5.3.27-5.3.30).
Emhart Teknologies (Tucker Fasteners Ltd.) (Pa3,n;eJIbI 4.2.8-4.2.18).
ButteIWorth Heinemann 3a pa3peIIIeHHe BOClIpOH3BeCTH ,n;aHHbIe H3 KHHrH Hig-
gins, R.A., «CBoHcTBa MaTepHaJIOB» B pa3,neJIax 6.2.2 H 6.2.3.
ButteIWorth Heinemann 3a pa3peIIIeHHe BOClIpOH3BeCTH ,n;aHHbIe H3 KHHrH Parr,
Andrew, «fH,ll;paBJIHKa H lIHeBMaTHKa» B pa3,n;eJIaX 5.6.1-5.6.4.
Newnes 3a pa3peIIIeHHe BOClIpOH3BeCTH ,n;aHHbIe H3 KHHfH Stacey, Chris,
«IIpaKTWlecKasI lIHeBMaTHKa» B pa3,n;eJIe 5.6.3.
Pearson Education Ltd. 3a pa3peIIIeHHe BocnpOH3BeCTH ,n;aHHbIe H3 KHHfH Ti-
mings, R. L., «MaTepHaJIoBe,n;eHHe», TOMa 1 H 2, B pa3,n;eJIax 6.1.1-6.1.21 H 7.1-7.5.
BbI,ll;ep)l{lGf H3 6pHTaHcKHx CTaH,ll;apTOB, BOClIpOH3Be,n;eHHbIe C pa3peIIIeHIDI BSI.
IIOJIHbIe KonHH 3THX ,n;OKYMeHTOB MOryr 6bITb nOJIyqeHbI lIO lIOQTe OT BSI Sales,
Linford Wood, Milton Keynes, Bucks., MK14 6LE.
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YfJIOBaH CKOpOCTb
1 060pOT/MHH 1=0,10472 panic 11 pa.n!c 1=9,5493 060pOT/MHH.
IlrroIUanb
1 ,ruoHM =6,4516 CM 1 CM =0,15500 ,ruoHM
I<pYT =0,092903 M 1 M =10,764 <PYT
1 Hp,n =0,83613M 1M =1,1960 Hp,n
1 aKp =0,40469 fa 1 fa =2,4711 aIq)
II.JJomOCTh
1 <PYHT/<PYT 1=16,018 Kr/M 11 Kr/M 1=0,062428 <PYHT/<PYT
3uepl1lJl
1 <PYT nayH,ZJ;aJIb =0,042140,lQK 1 ,lQK =23,730 <PYT nayH,ZJ;aJIb
1 <PYT <PYHT-CHJIa =1,355 82 ,lQK 1 ,lQK =0,73756 <PYT <PYHT-CHJIa
1 KBT 'lac =3,6000 M,[(x 1 M,[(x =0,27778 KBT 'lac
1 TepM =0,10551 f,lQK 1 f,lQK =9,4781 TepM
CHJla
1 nayH,ZJ;aJIb =0,13826 H IH =7,2330 nayH,ZJ;aJIb
1 <PYHT-CHJIa =4,4482 H 1H =0,22481 <PYHT-cHJIa
,nmma
1 ,nIOHM =2,5400 CM 1 CM =0,39370,lUOHM
I<pYT = 0,30480 M 1 M =3,2808 <PYT
1 Hp,n = 0,91440 M 1 M =1,0936 Hp,n
1 MHlliI = 1,6093 KM 1 KM =0,62137 MHlliI
Macca
1 YHUIDI =28,350 f 1 f =0,035274 YHUIDI
1 <PYHT =0,453 59 Kr lKr =2,2046 <PYHT
1 aHfJI. ueHTHep =50,802 Kr lKr =0,019684 aHfJI. ueHTHep
1 aHfJI. TOHHa =1,0161 TOHHa 1 TOHHa =0,98421 aHfJI. TOHHa
MOMeHT CHJIbI
1 <PYHT-CHJIa <PYT 1= 1,3558 H M 11 H M 1=0,73756 <PYHT-CHJIa <PYT
II.JJOCKHH yroJi
1 0 =0,01745 pan 1 pan =57,296 0
MOIIUIOCTh
1 <PYT <PYHT-CHJIa/ C =1,3558 BT 1 BT =0,73756 <PYT <PYHT-CHJIa/c
1 JI.C. =0,745 70 KBT 1 KBT =1,3410 JI.C.
,J.J;aBJIeHHe H uanpJDKeUHe
1 ,nIOHM pT.CT. =33,864 M6ap 1 M6ap =0,02953 ,nIOHM pT.CT.
1 <PYHT-CHJIa/,nIOHM =6,8948 Kna 1 KIla =0,14504 <PYHT-CHJIa/,nIOHM
1 aHfJI. TOHHa-CHJIa/,nIOHM =15,444 H/MM 1 H/MM =0,064749 aHfJI. TOHHa-CH-
JIa/,nIOHM
1.5. IIpeono'lmume/lbHble 'IuCIla
O"OH"taHue
E.zunutnw H3MepelllDl cJ}cJ}C 8 e,lUllllllOtl CII E.zunutnw H3MepelllDl CII 8 e,lUllllllOtl cJ}cJ}C
Y.n;eJlhHaJI TelIJIoeMKOCTh
1 BpHTaHCKaH TennOBaH 1=4,1868 /(KI' 0c) 11 /(KI' 0c) 1=0,23885 BplrnUlcKaH Ten-
e,lJ;HHHua/ (<pYUT OF) JIOBaH e,lJ;HHHua/( <PYHT OF)
CKOpoCTh
1 <PYT/c =0,30480 M/e 1 M/e =3,2808 <PYT/e
1 MHlliI/q =1,6093 KM/q 1 KM/q =0,62137 MHJIb/q
Ooo.eM
1 ,lJ;IOHM =16,387 eM lcM =O,061024,lJ;IOHM
1 <PYT =0,028317 M 1M =35,315 <PYT
hlp,lJ; =0,76456 M 1M =1,3080 Hp,lJ;
1 nHHTa =0,56826 JI 1 JI =1,7598 nHHTa
1 raJIJIOH =4,5461 JI 1 JI =0,21997 raJIJIOH
1.5. npeAnO'lTMTenbHble 'IMCna
Kor,na qeJIOBeK nOKYIIaeT, HanpHMep, 3JIeKTpHqecKYIO JIaMrry .rum HCnOJIb30BaHIDI B
6bITY, 06bIqHO B HaJIHqHH HMeIOTC.H JIaMnbI MOII(HOCTblO 15, 25, 40, 60, 100 BT H
T.,n. 3TH 3HaqeHIDI MOII(HOCTH CJIe,n)'lOT O,nHO 3a ,npyrHM npHMepHO B reOMeTpHqe-
CKOH lIporpeCCHH, np eM OTHOCHTeJIbHbIe H3MeHeHIDI CBeTOBoro H3.rryqeHIDI Me)l{-
l1Y coce,nHHMH 3HaqeHIDIMH MOII(HOCTeH B 3TOH nOCJIe,nOBaTeJIbHOCTH OCTaIOTC.H
npH6JIH3HTeJIbHO lIOCTO.HHHbIMH. KaK npaBHJIO, COOTHorneHIDI Me:>I<nY napaMeTpaMH
H3,neJIHH He c.rryqaHHbI, a OCHOBaHbI Ha CHCTeMe npeono"tmumeAbHblX "tuceA. (napa-
MeTp npO,nyKUHH - 3TO KOJIHqeCTBeHHIDI xapaKTepHCTHKa ee CBOHCTB. HaH60JIee
BIDKHbIMH napaMeTpaMH .HBJUIIOTC.H pa3MepHbIe, BeCOBbIe, 3HepreT eCKHe xapaKTe-
pHCTHKH npO,nyKUHH. - llpUMe"t. nep.)
IIpe,nnOqTHTeJIbHbIe qHCJIa 6a3Hp)'lOTC.H Ha R-"tUCAax, H306peTeHHbIX nOJIKOB-
HHKOM IllapJIeM PeHapOM. B OCHOBHOM HCnOJIb3)'1OTC.H p.H,LJ;bI - R5, RIO, R20, R40
H R80 H no,nrpynnbI 3THX p.H,LJ;OB. 3HaqeHIDI B KIDK,lJ;OM P.H.nY npe,nCTaBJUIIOT C060H
npH6JIH3HTeJIbHbIe reOMeTpHqeCKHe nporpeCCHH co 3HaMeHaTelliIMH ViQ, 1\¥iQ,
2\¥iQ, 4\¥iQ H 8\¥iQ, npH 3TOM 3HaqeHIDI ByrpH p.H,LJ;OB H3MeIDIIOTC.H Ha 58 % .rum p.H,LJ;a
R5, Ha 26% .rum p.H,LJ;a RIO, Ha 12% .rum R20, Ha 6% .rum R40 H Ha 3% .rum p.H,LJ;a
R80.
bOJIee no,np06HyID HH<popMaUHIO 0 3HaqeHIDIX H HCnOJIb30BaHHH npe,nnOqTHTe-
JIbHbIX qHCeJI MO)I{HO HaHTH B BS (British Standards) 2045:1965. (B HaCTO.H ee Bpe-
M.H ,neHcTByeT fOCT 8032-84 «IIpe,nnOqTHTeJIbHbIe qHCJIa H p.H,LJ;bI npe,nnOqTHTeJIb-
HbIX qHCeJI». YTBep)l(,L(eH: fOCCTaH,L(apT CCCP, 09.08.1984. B HeM npe,nYCMOTpeHbI
pR,nbI - R5, RIO, R20, R40. - llpuMe"t. nep.) AmI p.H,LJ;a R5 oKpyrJIeHHbIe 3HaqeHIDI
paBHbI 1,00, 1,60, 2,50, 4,00, 6,30 H 10,00; TaKHM 06pa30M, npHBe,neHHbIe BbIrne
3HaqeHIDI HOMHHaJIbHbIX MOII(HOcTeH 3JIeKTP eCKHX JIaMn 6a3HpyroTC.H Ha p.H,LJ;e
R5. bOJIbIIIHHCTBO HCnOJIb3yeMbIx CTaH,L(apTOB OCHOBaHO Ha p.H,LJ;ax lIpe,nnOqTHTeJIb-
HbIX qHCeJI; B TOM qHCJIe CTaH,L(apTbI Ha TOJIII(HHY JIHCTOBOro MeTaJIJIa H KaJIH6p
npOBOJIOKH, pa3MepbI raeK H 60JITOB, CTaH,L(apT 3JIeKTP eCKoro TOKa (A) H CKOpO-
CTH BpalI(eHH.H HHCTpyMeHTaJIbHbIX rnnHH,L(eJIeH B cTaHKax.
F/la8a 1. lfH:JICeHepHaJl MameMamUlCO
1.6. OnpeAeneHMe pa3MepoB
1.6. 1. nnOCKJfe cpJfryP&1
KBaLJ;paT
IIp.HMoyrOJIbHHK
.,
"
"
"
"
b
"
h
"
"
"
"
"
I"
I
I-
-I
b
CTopoHa b
IIJIolI(aLJ;b = b 2
):LrrHHa ,lJ;HarOHaJIH = ..fi x b
b
CTopoHa b, BbICOTa h
nJIOII(aLJ;b = b x h
):LrrHHa ,lJ;HarOHaJIH = .J b 2 + h 2
napaJIJIeJIorpaMM
TpalIeUIDI
a
I"
b
"I
I..
b
..I
CTopOHa b, BbICOTa h
IIJIolI(aLJ;b = b x h
OCHOBaHIDI a H b, BbICOTa h
1
llrrOII(aLJ;b = - x (a + b) x h
2
TpeyroJIbHHK
Kpyr
I.
b
I
CTopOHa b, BbICOTa h
1
nJIOII(aLJ;b = - x b x h
2
PaLJ;HYC r
nJIOII(aLJ;b = 1t X r 2
nepHMeTp = 2 x 1t x r
1.6. OnpeOe/leHUe pQ3Mep08
CeKTOp Kpyra
s
3JIJIHnC
i
I.
a
..I
P8)J;HYC r, ueHTpaJIbHbIH yrOJI 9, ,nyra s
1
llrrOII(an;b = - x r 2 x 9
2
):lrrHHa ,nyrH S = r x 9 (9 B pan;HaHax)
Pan;HYCbI a H b
nJIOII(an;b = 1t x a x b
nepHMeTP = 1t X (a + b)
HenpaBHJIbHbIH MHoroyrOJIbHHK
PI
I
I
I
I
t
YpoaeHb
OTC"IeTa
X 1
d
CylI(eCTByeT HeCKOJIbKO MeTo,nOB onpe,neJIeHIDI IIJIOII(an;H 3allITpHXOBaHHoH 06-
JIaCTH. ,[lJrn 3Toro HCnOJIb3yroTC.H lIpaBHJIO cpe,nHHX np.HMoyrOJIbHHKOB, npaBHJIO
TpaneUHH H npaBHJIO CHMncoHa. B KaqeCTBe lIpHMepa npHBe,neM <popMY.1IY CHMn-
COHa. Pa3,neJIHM OTpe30K XtX2 Ha qeTHOe KOJIJI1IeCTBO paBHbIX OTPe3KOB ,lUIHHOH d
K3.}K)J;bIH. nYCTb p, q, r, ... - ,lUIHHbI BepTHKaJIbHbIX JIHHHH, H3MepeHHbIe OTHOCH-
TeJIbHO HeKOToporo ypOBIDI OTCqeTa H nyCTb A - lIpH6JIH3HTeJIbHasI IIJIOII(8)J;b He-
lIpaBHJIbHOrO MHoroyrOJIbHHKa, BbI,l(eJIeHHOrO llITpHXOBKOH. B TaKOM c.rryqae:
d d
A = -[(p + t) + 4(q + s) + 2r] - -[(p + t) + 4(u + w) + 2v].
3 3
B 061I(eM, npaBHJIO CHMncoHa <POPMYJIHPyeTC.H CJIe,nyroII(HM 06pa30M:
.
lIpH6JIH3HTeJIbHasI IIJIOII(an;b =
= (%) x [(lIepBasI + nOCJIe,nHIDI) + 4(cyMMa qeTHbIX) + 2(cyMMa HeqeTHbIx)],
r,ne nepBasI, lIOCJIe,nHIDI, qeTHbIe H HeqeTHbIe - 3TO ,lUIHHbI COOTBeTCTB)'IOlI(HX Bep-
THKaJIbHbIX JIHHHH, a d - ,lUIHHa paBHbIX OTpe3KOB Ha ypOBHe JIHHHH OTCqeTa.
F/la8a 1. HH:JICeHepHaR MameMamU1CO
1.6.2. 06.eMH&le cpJfrYP&1
np.HMoyrOJIbHIDI npH3Ma
QHJIHH,L(P
hI
h ·
-- --,
,
06'beM = bhl
nJIOII(aLJ;b nOBepXHOCTH = 2(bh + hi + Ib)
06'beM = 1tr 2 h
nJIOII(aLJ;b nOBepXHOCTH = 21tr(r + h)
KoHYc
Y CeqeHHbIH KOHYC
h
06'beM = ! 1tr 2 h
3 1
lliolI(a,nb IIOBepXHOCTH = Ttr(1 + r) 06'beM = 3 rrh(R2 + Rr + r 2 )
lliolI(a,nb nOBepXHOCTH = 1t/(R + r) + 1t(R2 + r 2 )
C<pepa
nO.HC c<pepbI
\
\
4
06'beM = - 1tr 3
3
nJIOII(aLJ;b nOBepXHOCTH = 41tr 2
1th
06'beM = -(h 2 + 3R2 + 3 2)
6 1
lliOII(8)J;b lIOBepXHOCTH = 21trh + 1t( R 2 + 'i 2 ),
r,ne r - PaLJ;HYC C<pepbI
1.6. OnpeOeJleHUe p03Mep08
IIHpaMH)];a
TBep,nbIe TeJIa lIpaBHJIbHOH <pOpMbI
h
1
06'beM = - Ah
3 '
r,ne A - IIJIOII(an:b OCHOBaHH.H,
h - BbICOTa lIHpaMH)];bI
06'beM JII06oro TBep,noro TeJIa npaBHJIbHOH
<pOpMbI MO)I{HO HaHTH lIPH lIOMOII(H <POPMY-
JIbI CHMncoHa. YqHTbIBaIOTC.H TPH lIapaJI-
JIeJIbHbIe IIJIOCKOCTH C IIJIOII(a)J;.HMH AI' A3 H
A 2 , COOTBeTCTBeHHO pacnOJIOXeHHbIe Ha Kpa-
.HX H B n;eHTpe TBep,noro TeJIa. T or,na:
I
06'beM = -(AI + 4A 2 + A 3 ),
6
r,ne I - MHHa TBep,noro TeJIa.
TBep,nbIe TeJIa HelIpaBHJIbHOH <pOpMbI
I
,\ I
I \ I
I \ I \ ,
I \, \'
, 'I , ,
I .4_ , I .4 _ II .4
1'""'511 II
, , I , I
, I, , I
\ I , "
\' \ I ,
\' \
\
\
):lIDI OlIpe,neJIeHIDI 06'beMa TeJIa HelIpaBHJIbHOH <pOpMbI MO)I{HO HClIOJIb30BaTb
pa3JIJI1IHbIe MeTo,nbI; O,nHH H3 HHX OCHOBaH Ha npHMeHeHHH <pOPMYJIbI CHMlICOHa
(CMOTPH paHee B 3TOM pa3,neJIe). TBep,noe TeJIO CJIe,nyeT pa3,neJIHTb Ha qeTHOe KOJIH-
qeCTBO ceKUHH lIapaJIJIeJIbHbIMH IIJIOCKOCT.HMH, paClIOJIOXeHHbIMH Ha paBHbIX pac-
CTO.HHIDIX d o,nHa OT ,npyrOH H HMelOlI(HMH IIJIOII(an:H AI' A 2 , A 3 , ... . ECJIH HClIOJIb-
3yeTC.H, HalIpHMep, ceMb TaKHX IIJIOCKOCTeH, TO:
IIpH6JIH3HTeJIbHbIH 06'beM = d [(AI + A 7 ) + 4(A 2 + At + ) + 2(A 3 + As)].
3
F/la8a 1. HH:JICeHepHaJI MameMamUlCO
1.7. CTeneHM, KOpHM M o6paTHbie BenM'IMHbl
n n 2 .In .JIOn n 3 (n)x (IOn )X (lOOn )X
1 1 1,000 3,162 1 1,000 2,154 4,642 1,00000
2 4 1,414 4,472 8 1,260 2,714 5,848 0,50000
3 9 1,732 5,477 27 1,442 3,107 6,694 0,33333
4 16 2,000 6,325 64 1,587 3,420 7,368 0,25000
5 25 2,236 7,071 125 1,710 3,684 7,937 0,20000
6 36 2,449 7,746 216 1,817 3,915 8,434 0,16667
7 49 2,646 8,367 343 1,913 4,121 8,879 0,14286
8 64 2,828 8,944 512 2,000 4,309 9,283 0,12500
9 81 3,000 9,487 729 2,080 4,481 9,655 0,11111
10 100 3,162 10,000 1000 2,154 4,642 10,000 0,10000
11 121 3,317 10,488 1331 2,224 4,791 10,323 0,09091
12 144 3,464 10,954 1738 2,289 4,932 10,627 0,08333
13 169 3,606 11,402 2197 2,351 5,066 10,914 0,07692
14 196 3,742 11,832 2,744 2,410 5,192 11,187 0,07143
15 225 3,873 12,247 3375 2,466 5,313 11,447 0,06667
16 256 4,000 12,649 4096 2,520 5,429 11,696 0,06 50
17 289 4,123 13,038 4913 2,571 5,540 11,935 0,05882
18 324 4,243 13416 5832 2,621 5,646 12,164 0,05556
19 361 4,359 13784 6859 2,668 5,749 12,386 0,05263
20 400 4,472 14,142 8000 2,714 5,848 12,599 0,05000
21 441 4,583 14,491 9261 2,759 5,944 12,806 0,04762
22 484 4,690 14832 10648 2,802 6,037 13,006 0,04545
23 529 4,796 15, 166 12167 2,844 6,127 13,200 0,04348
24 576 4,899 15492 13824 2,884 6,214 13,389 0,04167
25 625 5,000 15,811 15625 2,924 6,300 13,572 0,04000
26 676 5,099 16,125 17576 2,962 6,383 13,751 0,03846
27 729 5,196 16,432 19683 3,000 6,463 13,925 0,03704
28 784 5,292 16,733 21952 3,037 6,542 14,095 0,03571
29 841 5,385 17,029 24389 3,072 6,619 14,260 0,03448
30 900 5,477 17,321 27000 3,107 6,694 14,422 0,03333
31 961 5,568 17,607 29791 3,141 6,768 14,581 0,03226
32 1024 5,657 17,889 32768 3,175 6,840 14,736 0,03125
33 1089 5,745 18,166 35937 3,208 6910 14,888 0,03030
34 1156 5,831 18,439 39304 3,240 6,980 15,037 0,02941
35 1225 5,916 18,708 42875 3,271 7,047 15,183 0,02857
36 1296 6,000 18,974 46656 3,302 7,114 15,326 0,02778
37 1369 6,083 19,235 50653 3,332 7,179 15,467 0,02703
38 1444 6,164 19,494 54872 3,362 7,243 15,605 0,02632
39 1521 6,245 19,748 59319 3,391 7,306 15,741 0,02564
40 1600 6,325 20,000 64000 3,420 7,368 15,874 0,02500
41 1681 6,430 20,248 68921 3,448 7,429 16,005 0,02439
42 1764 6,481 20,494 74088 3,476 7,489 16,134 0,02381
43 1849 6,557 20,736 79,507 3,503 7,548 16,261 0,02326
44 1936 6,633 20,976 85184 3,530 7,606 16,386 0,02273
45 2025 6,708 21,213 91125 3,557 7,663 16,510 0,02222
46 2116 6,782 21,448 97,336 3,583 7,719 16,631 0,02174
47 2209 6,856 21,679 103823 3,609 7,775 16,751 0,02128
48 2304 6,928 21,909 110592 3,634 7,830 16,869 0,02083
49 2401 7,000 22,136 117649 3,659 7,884 16,985 0,02041
1. 7. CmeneHu, KOpHU U o6pamHble 8eJlU'IUHbl
O"OH"taHue
n ,;z .In oJlOn n 3 (n)X (lOn)X (lOOn)X
50 2500 7,071 22,361 125000 3,684 7,937 17,100 0,02000
51 2601 7,141 22,583 132651 3,708 7,990 17,213 0,01961
52 2704 7,211 22,804 140608 3,733 8,041 17,325 0,01923
53 2809 7,280 23,022 148877 3,756 8,093 17,435 0,01887
54 2916 7,348 23,238 157464 3,780 8,143 17,544 0,01852
55 3025 7,416 23,452 166375 3,803 8,193 17,652 0,01818
56 3136 7,483 23,664 175616 3,826 8,243 17,758 0,01786
57 3249 7,550 23,875 185193 3,849 8,291 17,863 0,01754
58 3364 7,616 24,083 195112 3,871 8,340 17,967 0,01724
59 3481 7,681 24,290 205379 3,893 8,387 18,070 0,01695
60 3600 7,746 24,495 216000 3,915 8,434 18,171 0,01667
61 3721 7,810 24,698 226981 3,936 8,481 18,272 0,01639
62 3844 7,874 24,900 238328 3,958 8,527 18,371 0,01613
63 3969 7,937 25,100 250047 3,979 8,573 18,469 0,01587
64 4096 8,000 25,298 262144 4,000 8,618 18,566 0,01562
65 4225 8,062 25,495 274625 4,021 8,662 18,663 0,01538
66 4356 8,124 25,690 287496 4,041 8,707 18,758 0,01515
67 4489 8,185 25,884 300763 4,062 8,750 18,852 0,01493
68 4624 8,246 26,077 314432 4,082 8,794 18,945 0,01471
69 4761 8,307 26,268 328,509 4,102 8,837 19,038 0,01449
70 4900 8367 26,458 343000 4,121 8,879 19,129 0,01429
71 5041 8,426 26,646 357911 4,141 8,921 19,220 0,01408
72 5184 8,485 26,833 373,248 4,160 8,963 19,310 0,01389
73 5329 8,544 27,019 389017 4,179 9,004 19,399 0,01370
74 5476 8,602 27,203 405224 4,198 9,045 19,487 0,01351
75 5625 8,660 27,386 421875 4,217 9,086 19,574 0,01333
76 5776 8,718 27,568 438976 4,236 9,126 19,661 0,01316
77 5929 8,775 27,749 456,533 4,254 9,166 19,747 0,01299
78 6084 8,832 27,928 474552 4,273 9,205 19,832 0,01282
79 6241 8,888 28, 107 493039 4,291 9,244 19,916 0,01266
80 6400 8,944 28,284 512,000 4,309 9,283 20,000 0,01250
81 6561 9,000 28,460 531441 4,327 9,322 20,083 0,01235
82 6724 9,055 28,636 551368 4,344 9,360 20,165 0,01220
83 6889 9,110 28,810 571787 4,362 9,398 20,247 0,01205
84 7056 9,165 28,983 592,704 4,380 9,435 20,328 0,01190
85 7225 9,220 29,155 614125 4,397 9,473 20,408 0,01176
86 7396 9,274 29,326 636056 4,414 9,510 20,488 0,01163
87 7569 9,327 29,496 658503 4,431 9,546 20,567 0,01149
88 7744 9,381 29,665 681472 4,448 9,583 20,646 0,01136
89 7921 9,434 29,833 704969 4,465 9,619 20,724 0,01124
90 8100 9,487 30,000 729000 4,481 9,655 20,801 0,01111
91 8281 9,539 30,166 753,571 4,498 9,691 20,878 0,01099
92 8464 9,592 30,332 778688 4,514 9,726 20,954 0,01087
93 8649 9,644 30,496 804357 4,531 9,761 21,029 0,01075
94 8836 9,695 30,659 830584 4,547 9,796 21 , 105 0,01064
95 9025 9,747 30,822 857375 4,563 9,830 21,179 0,01053
96 9216 9,798 30,984 884736 4,579 9,865 21,253 0,01042
97 9409 9,849 31, 145 912673 4,595 9,899 21,327 0,01031
98 9604 9,899 31,305 941192 4,610 9,933 21,400 0,01020
99 9801 9,950 31,464 970299 4,626 9,967 21,472 0,01010
2-3255
F/la8a 1. HH:JICeHepHaJI MameMamUlCO
1 .8. nporpeccMM
IIocJIe,noBaTeJIbHOCTb qHCeJI, B KOTOpOH KIDK)];Oe qHCJIO CB.H3aHO co lIOCJIe,nyroII(HM
qHCJIOM B COOTBeTCTBHH C KaKHM-JIH60 3aKOHOM, Ha3bIBaeTC.H npozpeccueu.
1.8. 1. ApJfcpMeTJflfeCKSfI nporpeccJffl
IIOCJIe,nOBaTeJIbHbIe qHCJIa apH<pMeTJI1IeCKOH lIporpeCCHH CB.H3aHbI o6lI(eit pa3HO-
CTblO. KIDK)];bIH H3 qrreHOB lIOCJIe,nOBaTeJIbHOCTH 3, 6, 9, 12, 15, ... lIo.rryqeH lIpH6aB-
JIeHHeM qHCJIa 3 K lIpe,nbl.nYlI(eM)' qrreHy lIOCJIe,nOBaTeJIbHOCTH, T .e. pa3HOCTb lIpO-
rpeCCHH paBHa 3. B o6IUeM c.rryqae, eCJIH a - lIepBbIH qrreH lIporpeCCHH, a d - ee
pa3HOCTb, apH<pMeT}lqeCKa.H lIporpeCCIDI HMeeT CJIe,nyIOIUHH BIf,l(:
qJIeH nporpeCCHH
3HaqeHHe
I-H
a
2-H
a+d
3-H
a + 2d
4-H
a + 3d, ...
lIOCJIe,nHHH
a + (n - I)d
r,ne n - KOJIHqeCTBO qrreHOB lIporpeCCHH.
CYMMa Sn Bcex qrreHOB lIporpeCCHH onpe,neillIeTC.H KaK cpe,nHee 3HaqeHHe qrre-
HOB, YMHoxeHHoe Ha KOJIJI1IeCTBO qrreHOB nporpeCCHH, T.e.:
Sn = [(lIepBbIH + lIOCJIe,nHHH) / 2] x (qHCJIO qrreHOB) =
= [(a + a + (n - I )d) / 2] x n = (n / 2) [2a + (n - I )d].
1.8.2. reOMeTpJflfeCKSfI nporpeCCJffl
IIOCJIe,nOBaTeJIbHbIe qHCJIa reOMe1pHqeCKOH lIporpeCCHH CBR3aHbI 06W,UM 3HaMeHa-
melleM. Kax,nbIH H3 qrreHOB nOCJIe,nOBaTeJIbHOCTH 3, 6, 12, 24, 48, ... lIo.rryqeH YM-
HoxeHHeM lIpe,nbl.nYlI(ero qrreHa nOCJIe,nOBaTeJIbHOCTH Ha 2. B 06lI(eM CJIyqae, eCJIH
a - lIepBbIH qrreH lIporpeCCHH, a r - 3HaMeHaTeJIb, TO reOMe1pHqeCKa.H lIporpec-
CIDI HMeeT BIf,l(:
qJIeH lIporpeCCHH I-H
3HaqeHHe a
2-H
ar
3-H
ar 2
4-H
ar3, ...
lIOCJIe,nHHH
ar"-I
r,ne n - KOJI}lqeCTBO qrreHOB lIporpeCCHH.
CYMM)' Sn Bcex qrreHOB lIporpeCCHH MO)I{HO HaHTH CJIe,nyIOII(HM 06pa30M:
Sn = a + ar + ar 2 + ar 3 + ... + ar n -I . ( I)
YMHOXa.H KIDK)];bIH qrreH ypaBHeHIDI (I) Ha r, lIo.rryqaeM
rS n = ar + ar 2 + ar 3 + ... + ar n - 1 + ar n . (2)
BbIqHTaHHe ypaBHeHIDI (2) H3 ypaBHeHIDI (I) lIpHBO,nHT K pe3YJIbTaTY:
Sn(1 - r) = a - ar n ,
Sn = [a(1 - r n )] / (I - r). (3)
fuH xe, YMHOXa.H qHCJIHTeJIb H 3HaMeHaTeJIb lIOCJIe,nHero ypaBHeHIDI Ha -I,
lIOJIyqaeM:
Sn = [a(r n - I)] / (r -I).
(4)
1.9. TpuzoHoMempu'teclCue fjJopMY/lbl
06bIqHO HClIOJIb3yeTC.H ypaBHeHHe (3), Kor,n;a r < 1, H (4), Kor,n;a r > 1.
Kor,n;a -1 < r < 1, KIDK,D;bIH qrreH reOMe1pHqeCKOH nporpeCCHH MeHbIIIe, qeM
lIpe,n;bI):(ylI(HH; B 3TOM c.rryqae rOBOp.HT, qTO lIOCJIe,n;OBaTeJIbHOCTb cxooumcJl. MO)I{HO
HaHTH CYMM)' Bcex qrreHOB CXO,lUIIUeHC.H lIOCJIe,n;OBaTeJIbHOCTH. TaKYK> CYMM)' Ha3bI-
BalOT CYMMOH 6eCKOHeqHOH reOMe1p eCKOH lIporpeCCHH. qJIeH a(l - r n ) / 1 - r
MO)I{HO nepelIHcaTb B Blf,l(e [a / (1 - r)] - [ar n / 1 - r]. IIOCKOJIbKY r MeHbIIIe 1, r n
CTaHOBHTC.H Bce MeHbIIIe H MeHbIIIe lIPH B03paCTaHHH n. Kor,n;a n CTaHOBHTC.H OqeHb
60JIbIIIHM, 3HaqeHHe r n lIpaKT eCKH paBHO HyJIIO, H TaKHM o6pa30M [ar n / 1 - r]
CTaHOBHTC.H paBHbIM HyJIIO. OTclO,n;a CJIe,nyeT, qTO CYMMa 6eC"OHeltItOU zeOMempUlte-
CKOU npozpeccuu paBHa a / (1 - r) wrn c.rryqa.H -1 < r < 1.
1.8.3. rSpMOHJflfeCKSfI nporpeCCJffl
fapMOHHqeCKYK> lIporpeCCHIO o6pa3yroT qHCJIa, o6paTHbIe lIOCJIe,n;OBaTeJIbHbIM qrre-
HaM apH<pMeT eCKOH lIporpeCCHH. TaK, wrn apH<pMeT eCKOH lIporpeCCHH 1, 2, 3,
4, 5, ... COOTBeTCTByrolI(a.H rapMOH eCKa.H lIporpeCCIDI HMeeT BIf,l(: 1, 1/2, 1/3, 1/4,
1/5, ... .
1.9. TpMrOHOMeTpM'IeCKMe cl>opMynbl
1.9. 1. OCHOSH&le onpeAeneHJffl
B np.HMoyrOJIbHOM 1peyrOJIbHHKe, lIOKa3aHHOM HIDKe, a - CTopOHa, npomU80lleJICa-
U4aR yr.rry A (oPP), b - rHlIOTeHY3a 1peyrOJIbHHKa (hyp), a C - CTopOHa, npU/leJICa-
U4aR yr.rry A (adj). IIo OlIpe,n;eJIeHHIO:
sin A = opp / hyp = a / b
cos A = adj / hyp = C / b
tgA = opp / adj = a / C
cosecA = hyp / opp = b / a = 1 / sin A
secA = hyp / adj = b / C = 1 / cos A
ctgA = adj / opp = c / a = 1 / tgA,
c
A
80pp
r,n;e opp - CTopOHa, lIpOTHBOJIeXalI(a.H
yrJIY A, adj - CTopOHa, lIpHJIeXalI(a.H K
yr.rry A, hyp - rHnOTeHy3a.
B
Cadi
1.9.2. TOJK,qeCTSS
sin 2 A +cos 2 A =1
1 +tg 2 A =sec 2 A
1 + ctg 2 A = cosec 2 A
sin(-A) = -sin A
cos( -A) = cos A
tg( -A) = -tgA
F/la8a 1. HH:JICeHepHaJI MameMamUlCO
1.9.3. COCTSSH&le cpopMyn&1 Jf cpopMyn&1 AsoMHoro yrns
sin( A + B) = sin A cos B + cos A sin B
sin( A - B) = sin A cos B - cos A sin B
cos(A + B) = cos AcosB - sin A sinB
cos( A - B) = cos A cos B + sin A sin B
tg(A + B) = (tgA + tgB) / (1 - tgA . tgB)
tg(A - B) = (tgA - tgB) / (1 + tgA . tgB)
sin2A = 2 sin A cos A
cos 2A = cos 2 A - sin 2 A = 2 cos 2 A-I = 1 - 2 sin 2 A
tg2A = (2tgA) / (1 - tg 2 A)
1.9.4. tPopMynw «npOJf3seAeHJfe-cYMMS»
sinAcosB = l[sin(A + B) + sin(A - B)]
2
1
cos A cos B = - [cos( A + B) + cos( A - B)]
2
cos A sinB = l[sin(A + B) - sin(A - B)]
2
sinAsinB =l[cos(A + B) -cos(A -B)]
2
1.9.5. tPopMyn&1 TpeyrOn&HJfKS
C yqeToM 0603HaqeHHH, YKa3aHHbIX Ha pHCYHKe:
TeopeMa CHHyOB:
a / sin A =b / sinB =e / sinC.
c
TeopeMa KOCHHYCOB:
a 2 =b 2 +e 2 -2becosA
b 2 = a 2 + e 2 - 2aecosB
e 2 = a 2 + b 2 - 2abcosC.
8
c
IlrrOlI(aab:
IIJIolI(aab = lab sinC = lbe sin A = lea sinB
222
TaIOKe:
IIJIolI(aab = .J s(s - a)(s - b)(s - e),
r,n;e s - lIOJIyrrepHMe1'p, T.e. s = (a + b + e) / 2.
1 .10. Kpyr M OKPY>KHOCTb: HeKOTopble onpeAeneHMSI
-
M CBOMCTBa
):LIDI Kpyra ,n;HaMe1'pOM d II paaHYCOM r.
,l/;AuHa O"PY:>ICHOemU paBHa 1td HJIH 21tr.
llllOUl,aOb "pyza paBHa 1td 2 / 4 HJIH 1tr 2 .
1.9. TpuzoHoMempu'IecICue rjJOPMY/lbl
)];yra OKpyxHOCTH - 3TO ee qaCTb.
KacamellbHou K OIq)yxHOCTH Ha3bIBalOT np.HMyIO JIH-
HHIO, cOlIpHKaCalOlI(yroC.H C OIq)yxHOCTblO TOJIbKO B o,n-
HOH TOqKe. Pan;HYc, lIpoBe,neHHbIH B TOqKY KacaHIDI, 06-
pa3yeT C KaCaTeJIbHOH IIp.HMOH yrOJI.
CeKmopoM Iq)yra Ha3bIBalOT 06JIaCTb, orpaH eHHyIO
.nyfOH OIq)yxHOCTH H .D:BywI PMHYCaMH. IIrrOII(a,nL ceKTopa
1
paBHa - r 29, r,ne 9 - yrOJI Me:>K:,ny pan;HYcaMH B pan;HaHax.
2
XopiJou Ha3bIBalOT IIp.HMyIO JIHHHIO, coe,nHlUllOlI(yro
,nBe TOqKH OKpyxHOCTH. IIpH lIepeCeqeHHH ,nBYX Xop,n
lIpOH3Be,neHHe MHH qaCTeH O,nHOH XOp,nbI paBHO lIpOH3-
Be,neHHIO MHH qaCTeH BTOpOH XOp,nbI. Ha pHC. 1
AE x BE = CE x ED.
CeZMeHmoM Iq)yra Ha3bIBalOT 06JIaCTb, orpaH eH-
HyIO XOp,nOH H ,nyr<>H OIq)yxHOCTH. YrJIbI, lIOC1'poeHHbIe
Ha O,nHOM H TOM xe cerMeHTe, paBHbI: Ha pHC. 2 yroJI A =
= yroJI B.
A
1
1. 10. 1. Kpyr: nnOIl.4SA& Jf ,qJIJfHS OKpyJlCHOCTJf
,lI;HUeTp fiJIOIna.zn. ,lI;Jnma ,lI;HaMeTp fiJIouuw. ,lI;Jnma ,lI;HUeTp fiJIouuw. ,lI;Jnma
Olq)ymlOCTH Olq)ymlOCTH Olq)ymlOCTH
1 0,7854 3,142 24 452,39 75,40 47 1734,9 147,7
2 3,1416 6,283 25 490,87 78,54 48 1809,6 150,8
3 7,0686 9,425 26 530,93 81,68 49 1885,7 153,9
4 12,566 12,57 27 572,56 84,82 50 1963,5 157,1
5 19,635 15,71 28 616,75 87,96 51 2042,8 160,2
6 28,274 18,85 29 660,52 91,11 52 2123,7 163,4
7 38,485 21,99 30 706,86 94,25 53 2206,2 166,5
8 50,265 25,13 31 754,77 97,39 54 2290,2 169,6
9 63,617 28,27 32 804,25 100,5 55 2375,8 172,8
10 78,540 31,42 33 855,30 103,7 56 2463,0 175,9
11 95,033 34,56 34 907,92 106,8 57 2551,8 179,1
12 113,10 37,70 35 962,11 110,0 58 2642, 1 182,2
13 132,73 40,84 36 1017,9 113,1 59 2734,0 185,4
14 153,94 43,98 37 1075,2 116,2 60 2827,4 188,4
15 176,71 47,12 38 1134, 1 119,4 61 2922,5 191,6
16 201,06 50,27 39 1194,6 122,5 62 3019, 1 194,8
17 226,98 53,41 40 1256,6 125,7 63 3117,2 197,9
18 254,47 56,55 41 1320,3 128,8 64 3217,0 201 , 1
19 283,53 59,69 42 1385,4 131,9 65 3318,3 204,2
20 314,16 62,83 43 1452,2 135,1 66 3421,2 207,3
21 346,36 65,97 44 1520,5 138,2 67 3525,7 210,5
22 380,13 69, 11 45 1590,4 141,4 68 3631,7 213,6
23 415,48 72,26 46 1661,9 144,5 69 3739,3 216,8
F/la8a 1. HH:JICeHepHaJI MameMamU1Ca
O"OH"taHue
,lI;HaMeTp fiJIOIQa.zn. ,lI;Jnma ,n;HaMeTp fiJIOIQa.zn. ,lI;Jnma ,n;HaMeTp fiJIOIQa.zn. ,lI;Jnma
OKpymtOCTH Olq)ymtOCTH Olq)ymtOCTH
70 3848,5 219,9 80 5026,5 251,3 90 6361,7 282,7
71 3959,2 223,1 81 5153,0 254,5 91 6503,9 285,9
72 4071,5 226,2 82 5381,0 257,6 92 6647,6 289,0
73 4185,4 229,3 83 5410,6 260,8 93 6792,9 292,2
74 4300,8 232,5 84 5541,8 263,9 94 6939,8 295,3
75 4417,9 235,6 85 5674,5 267,0 95 7088,2 298,5
76 4536,5 238,8 86 5808,8 270,2 96 7238,2 301,6
77 4656,6 241,9 87 5944,7 273,3 97 7389,8 304,7
78 4778,4 245,0 88 6082, 1 276,5 98 7543,0 307,9
79 4901,7 248,2 89 6221,1 279,6 99 7697,7 311,0
1 . 11. KBaApaTHbie ypaBHeHMSI
PeIIIeHHeM (KopHeM) KBaapaTHoro ypaBHeHIDI:
ax 2 + bx + c = 0
.HBillI eTC.H:
-b + .J b 2 - 4ac
x=
2a
1 . 12. HaTYpanbHble norapMcI>Mbl
HaTYPaJIbHbIH JIOrapH<pM ,neHCTBHTeJIbHOrO qHCJIa x 0603HaqaeTC.H KaK In x. IIo
OlIpe,neJIeHHIO, OH paBeH TaKOM)' qHC.rry, qTO:
e 1n x = X
,
r,ne e = 2,1782 .HBillIeTC.H OCHOBaHHeM HaTYPaJIbHOrO JIOrapH<pMa.
HaTYPaJIbHbIH JIOrapH<pM 06JIaaaeT CJIe,nyroII(HMH CBOHCTBaMH:
In(xy) = In x + lny
In(x / y) = In x -lny
In(yx) = xlny
1. 13. CTaTMCTMKa: BBeAeHMe
1. 13. 1. OCHOSH&le nOHflTJffl
B 60JIbIIIHHCTBe PYKOBO,nCTB, HMelOlI(HX OTHOIIIeHHe K BOlIpocaM KOHTpOillI KaqeCT-
Ba, npe,nlIOJIaraeTC.H HeKoTopbIH 6a30BbIH ypOBeHb 3HaHHH no CTaTHCTHKe, He06xo-
,nHMbIH .n.rnI ,nOJDKHoro nOHHMaHHH yr.rry6JIeHHOH CTaTHCTHKH, JIeXalI(eH B OCHOBe
3Toro lIpe,nMeTa. B KpaTKOM BBO,nHOM Kypce, lIpeMaraeMOM ,naJIee, co,nepxaTC.H
CBe,neHH.H, KOTopbIe lIoTpe6yroTC.H qHTaTeJI.HM npH lIOCJIe,nyIO eM H3yqeHHH pa3-
JIHqHbIX PYKOBO,nCTB, CB.H3aHHbIX C KOHTpOJIeM KaqeCTBa.
1.13. CmamucmU1CO: 88eoeHue
CpeiJHee apurjJMemu"teCKoe HJIH npOCTO cpeiJHee - 3TO cpe,nHee 3HaqeHHe Ha60pa
,naHHbIX. QT06bI ero HaHTH, JIy}KH0 CJIO)I{HTb BMeCTe 3HaqeHIDI Bcex qrreHOB Ha60pa
,naHHbIX H 3aTeM pa3,neJIHTb CYMM)' Ha KOJIHqeCTBO qrreHOB B Ha60pe. MaTeMaTHqe-
CKH 3TO MO)I{HO BbIpa3HTb CJIe,nyroII(HM 06pa30M:
X=(X 1 +X 2 +...+XN)/N
TaKHM 06pa30M, cpe,nHee apH<pMeTHqeCKOe Ha60pa qllceJI 4, 6, 9, 3 II 8 6y,neT
(4 + 6 + 9 + 3 + 8)/5 = 6.
MeiJuaHa - 3TO RTIH cepe,nIlHHoe 3HaqeHlle, HJIII cpe,nHee ,nBYX cepe,nIlHHbIX
3HaqeHHH lIOCJIe,nOBaTeJIbHOCTII qHCeJI, paClIOJIOXeHHbIX B lIOp.H,Z:l:Ke B03paCTaHIDI.
TaK, Me,nHaHa lIOCJIe,nOBaTeJIbHOCTII 3, 4, 5, 6, 8, 9, 13 II 15 paBHa (6 + 8)/2 = 7, a
Me,nHaHa lIOCJIe,nOBaTeJIbHOCTII 4, 5, 7, 9, 10, 11, 15, 17 II 19 paBHa 10.
MoiJa - 3TO 3HaqeHHe, KOTopoe Hall60JIee qaCTO BC1'peqaeTC.H B Ha60pe qllceJI.
TaKHM 06pa30M, Mo,na Ha60pa 2, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9 II 9 paBHa 6.
Pa3Max Ha60pa qHCeJI - 3TO pa3HOCTb Mex,ny MaKCIIMaJIbHbIM II MIIHHMaJIb-
HbIM 3HaqeHIDIMH B Ha60pe. TaK, pa3Max Ha60pa qllceJI 3, 2,9, 7,4, 1, 12, 3, 17 II
4paBeH 17-1 = 16.
CmaHiJapmHoe omKIlOHeHue, IIHor,na Ha3bIBaeMoe cpeiJHeK6aiJpamu"tecKUM omKIlO-
HeHueM, OlIpe,neillIeTC.H KaK:
S= [(XI -X)2 +(X 2 -X)2 +",+(X N -X)2]/ N.
TaK, ,nJUI qHCeJI 2, 5 II 11 cpe,nHee 3HaqeHHe paBHO (2 + 5 + 11 )/3 = 6. CJIe,noBa-
TeJIbHO, CT aH,l(apTHOe OTKJIOHeHlle:
s = [(2 - 6)2 + (5 - 6)2 + (11 - 6)2] / 3 = (16 + 1 + 25) / 3 = Jl4 3,74.
06WqHO S HClIOJIb3yeTC.H ,nJUI 0603HaqeHIDI CTaH,l(apTHOro OTKnOHeHIDI reHe-
paJIbHOH COBOKYIIHOCTII (lIOJIHOrO Ha60pa qHCeJI), a 0 IIClIOJIb3yeTC.H ,nJUI 0603Haqe-
HIDI CTaH,l(apTHOro OTKJIOHeHIDI BbI60pKH.
1.13.2. BepoflTHOCT&
IIycTb x - KOJI eCTBO 3JIeMeHTapHbIX C06bITIIH, 6JIarOlIpIDITHbIX ,nJUI HeKOToporo
C06bITIDI A, a n - lIOJIHOe KOJI eCTBO Bcex B03MO)I{HbIX II paBHOBepO.HTHbIX 3JIeMeH-
TapHbIX C06bITIIH; Tor,na BepO.HTHOCTb C06bITIDI A OlIpe,neillIeTC.H KaK p = x/no BepO.HT-
HOCTb C06bITHH eCTb qHCJIO OT 0 ,no 1. ECJIH q - BepO.HTHOCTb Toro, qTO ,naHHoe co-
6bITlle He IIp 0113 oi1,z:(eT , TO H3 CKa3aHHoro paHee CJIe,nyeT, qTO p + q = 1. TaK, lIpll
6pocaHHII lIpaBHJIbHOH IIIecrn:rpaHHoH IIrpaJIbHOH KOCTH (T.e. KY6HKa C H<ppaMII),
BepO.HTHOCTb BbIrran;eHIDI orrpe,neJIeHHOrO qHCJIa, HalIpHMep 1'pex, paBHa 1/6, TaK KaK
y KY6HKa IIIeCTb rpaHeH H qHCJIO 3 lIpllcyrcTByeT TOJIbKO Ha O,nHOH H3 HHX.
1.13.3. 5JfHOMJfSn&HOe pscnpeAeneHJfe
DUHOMUaIlbHOe pacnpeiJelleHue lIpHMeHHTeJIbHO K KOH1'pOJIIO KaqeCTBa MoxeT 6bITb
OlIpe,neJIeHO CJIe,l(ylOII(IIM 06pa30M. BepO.HTHOCTb lIO.rryqHTb 0, 1, 2, 3, ..., n ,ne<peKT-
HbIX H3,neJIHH B BbI60pKe H3 n H3,neJIHH, lIo.rryqeHHoH c.rryqaHHbIM 06pa30M H3 reHe-
paJIbHOH COBOKYIIHOCTH, ,l(JIH KOTOpOH BepOHTHOCTb lIO.HBJIeHIDI ,ne<peKTHOrO H3,ne-
JIIDI paBHa p, a He,ne<peKTHoro - q, OlIpe,neillIeTC.H lIOCJIe,nOBaTeJIbHbIMH qrreHaMH
pa3JIOXeHIDI (q + p)n HaqllHa.H C lIpaBoro qrreHa pa3JIOXeHIDI.
F/la8a 1. HH:JICeHepHaJI MameMamU1Ca
TaK, eCJIH BbI6opKa, COCTOIIT, HalIpHMep, H3 qeTbIpex ,neTaJIeH, BbI6paHHbIX c.rry-
qaHHbIM 06pa30M H3 rrpO,LJyKIJ;HH CTaHKa, lIpOH3BO,lUI ero B Cpe,lUleM 5 % ,ne<peKTHbIX
,neTaJIeH, TO BepOHTHOCTb lIOnyqlITb 0, 1, 2, 3 H.JIH 4 ,ne<peKTHbIe ,neTaJIH B BbI60pKe MO-
xeT 6bITb orrpe,neJIeHa CJIe,nyIOII(HM 06Pa30M. IIOBTOpHbIM YMHoxeHHeM lIonyqaeM:
(q + p)4 = q4 + 4q3 p + 6q2 p2 + 4 qp 3 + p4.
3HaqeHIDI q H P paBHbI COOTBeTCTBeHHO 0,95 H 0,05. Tor,na:
(0,95 + 0,05)4 = 0,95 4 + 4 x 0,95 3 x 0,05 + 6 x 0,95 2 x 0,05 2 + 4 x 0,95 x 0,05 3 +
+ 0,05 4 = 0,8145 + 0,1715 + 0,011354 + ...
3TO 03HaqaeT, qTO:
1) 81 % BbI6opOK, BepO.HTHO, He co,nep)I{HT ,ne<peKTHbIX ,neTaJIeH;
2) 17% BbI6opOK, BepO.HTHO, co,nep)I{HT O,nHY ,ne<peKTHyro ,neTaJIb;
3) 1 % BbI6opOK, BepO.HTHO, co,nep)I{HT ,nBe ,ne<peKTHbIe ,neTaJIH;
4) MaJIOBepO.HTHO, tffO B BbI60pKe lIOlIa,nyTC.H 1'pH HJIH qeTbIpe ,ne<peKTHbIe ,neTaJIH.
qTO KaCaeTC.H KOH1'pOillI KaqeCTBa, TO, eCJIH lIPH lIOBTOpHbIX BbI60pKax TaKoe
lIpOn;eHTHoe COOTHOIIIeHHe lIpH6JIH3HTeJIbHO coxpalUleTC.H, KOH1'pOJIep MoxeT c,ne-
JIaTb BbIBO,n, qTO CTaHOK lIpo,nOJDKaeT lIpOH3BO,nHTb OKOJIO 5 % ,ne<peKTHbIX ,neTaJIeH.
O,nHaKo eCJIH lIpOn;eHTHoe COOTHOIIIeHHe H3MelUleTC.H, TO, BepO.HTHO, lIpOn;eHT 6pa-
Ka TaIOKe H3MelUleTC.H. T OqHO TaK xe lIOKYIIaTeJIb, lIo.rryqalOlI(HH 60JIbmyIO lIapTHIO
H3,neJIHH, MoxeT, HClIOJIb3Y.H c.rryqaHHbIe BbI6oPKH, no,nCqHTaTb KOJIJI1IeCTBO ,ne<peK-
THbIX H3,neJIHH B BbI60PKax H C nOMOlI(blO 6HHOMHaJIbHOrO paclIpe,neJIeHIDI on;eHHTb
BepO.HTHOe KOJI}lqeCTBO ,ne<peKTHbIX H3,neJIHH BO BceH lIapTHH.
1.13.4. PscnpeAeneHJfe nYSCCOHS
BbIqHCJIeHIDI C HClIOJIb30BaHHeM 6HHOMHaJIbHOrO paClIpe,neJIeHIDI MOryT OKa3aTbC.H
OqeHb 1'py,noeMKHMH, eCJIH KOJIHqeCTBO n ,neTaJIeH B BbI60pKe 60JIbIIIe IIIeCTH HJIH
ceMH. B 3TOM c.rryqae pacnpeiJe.n.eHue llyaccoHa .HBlliIeTC.H XOpOIIIHM npH6JIIDKeHHeM
6HHOMHaJIbHOrO paClIpe,neJIeHIDI. OHO OlIpe,neJIeHO CJIe,nyroII(HM o6pa30M.
ECJIH BepO.HTHOCTb HaCTYTIJIeHIDI C06bITIDI B JII060H MOMeHT lIOCTO.HHHa H MaTe-
MaTHqeCKOe OXll,lJ;aHHe np HaCTYTIJIeHIDI C06bITIDI paBHO A, Tor,na BepO.HTHOCTb Ha-
CTYTIJIeHIDI C06blTIDI 0, 1, 2, 3, 4, ... pa3 OlIpe,neillIeTC.H KaK:
e-A., ')..,e-A., ')..,2e-A. / 2!, ')..,3e-A. / 3!, ')..,4e-A. / 4!,... ,
r,ne e - KOHCTaHTa 2,71828... H 2! = 2 x 1, 3! = 3 x 2 x 1, 4! = 4 x 3 x 2 x 1, H T.,n.
(3,necb 4! qHTaeTC.H KaK «qeTbIpe <paKTOpHaJI»).
IIpHMeHeHHe paClIpe,neJIeHIDI IIyaccoHa K HCnOJIb30BaHHOM)' BbIIIIe .n.rnI HJIJIIO-
C1'pan;HH lIpHMeHeHIDI 6HHOMHaJIbHOrO paClIpe,neJIeHIDI npHMepy co CTaHKOM, lIpO-
H3BO,lUI HM 5 % ,ne<peKTHbIX ,neTaJIeH, ,naeT:
MaTeMaT}lqeCKOe OXll,lJ;aHHe np = 4 x 0,05 = 0,2
BepO.HTHOCTb OTCyrCTBIDI ,ne<peKTHbIX ,neTaJIeH paBHa e-A. = e- O ,2 = 0,8187
BepOHTHOCTb HaJI}lqIDI O,nHOH ,ne<peKTHOH ,neTaJIH paBHa Ae-A. = 0,2e- O ,2 = 0,1637
BepOHTHOCTb Ha.rrwnm,nBYX ,ne<peKTHbIX ,neTaJIeH paBHa ')..,2e-A. j2! = 0,2 2 e-O,2 /2 = 0,0164.
MO)I{HO Blf,l(eTb, qTO BepO.HTHOCTH, lIpH6JIH3HTeJIbHO paBHbIe 82, 16 H 2 %, xo-
pOllIO COrJIacyroTC.H C pe3YJIbTaTaMH, no.rryqeHHbIMH paHee.
1.13. CmamucmuKa: 88eoeHue
1.13.5. HOpMSn&HOe pscnpeAeneHJfe
,n;aHHbIe, CBR3aHHbIe C H3MepeHHeM TaKHX BeJlHqHH, KaK Macca ,nJIHHa, BpeM.H H
TeMlIepaTYPa, Ha3bIBalOTCR HelIpepbIBHbIMH, TaK KaK OHH MOryr lIpHHHMaTb JlI06bIe
3HaqeHIDI B 3a,naHHOM ,nHalIa30He. npe,nlIOJlO)I{HM, qTO paClIpe,neJleHHe ,nJIHH ,ne-
TaJIeH, lIpOH3Be,neHHbIX HeKOTopbIM CTaH-
KOM, 6y,neT lIpe,nCTaBJIeHO B BH,[(e rpa<pH-
Ka, KaK lIOKa3aHO Ha pHCyHKe; Tor,na <poPMY
,naHHoro rpa<pHKa MO)I{HO OlIpe,neJlHTb MaTe-
MaTWleCKH. OHa OlIpe,neJI.HeTCR <POPMYJlOH
y = (1/ cr)e Z , r,ne Z = -x 2 / 2cr 2 , cr - CTaH-
,napTHoe OTKJIOHeHHe Ha60pa ,naHHbIX; x -
qaCTOTa, C KOTOpOH ,naHHOH 3HaqeHHe BC1'pe-
qaeTCR. TaKa.H KPHBa.H Ha3bIBaeTCR 2parjJUKOM
6epORmHocmu HOpMaJlbH020 pacnpeOe/leHUR
HJIH KPU80U HOpMaJlbH020 pacnpeOe/leHUR.
HaH60Jlee BIDKHbIMH ,[(.IDI KOH1'pOJI.H Kaqe-
CTBa RB.IDIIOTCR CJle,nyroII(He CBOHCTBa ,naHHOH KpHBOH.
1. nJlOII(a,nb 06J1aCTH, orpaHWleHHoH KpHBOH H ,nBywI BepTHKaJIbHbIMH JlHHIDI-
MH, OTCTORII(HMH Ha + 1 CTaH,[(apTHOe OTKJIOHeHHe OT cpe,nHero 3HaqeHIDI, COCTaB-
JI.HeT lIpHMepHO 67 % 06lI(eH IIJIOma,nH.
2. nJlOII(a,nb 06J1aCTH, orpaHWleHHoH KpHBOH H BepTHKaJIbHbIMH JlHHIDIMH, OT-
CTORII(HMH Ha + 2 CTaH,[(apTHbIX OTKJIOHeHIDI OT cpe,nHero 3HaqeHIDI, COCTaB.IDIeT
lIpHMepHO 95 % 06lI(eH IIJIOII(a,nH.
3. lliOII(a,nb 06J1aCTH, orpaHHqeHHOH KpHBOH H BepTHKaJIbHbIMH JlHHIDIMH, OT-
CTORII(HMH Ha + 3 CTaH,[(apTHbIX OTKJIOHeHH:SI OT cpe,nHero 3HaqeHIDI, COCTaBJI.HeT
lIpHMepHO 99,75 % 06meH IIJIOII(a,nH.
4. 06J1aCTb, orpaHWleHHa.H KpHBOH, lIpOlIOPUHOHaJIbHa IIJIOTHOCTH BepORTHO-
CTH B reHepaJIbHOH COBOKyIIHOCTH
,n;JI.H HJIJIIOC1'paUHH HClIOJlb30BaHIDI ,naHHbIX CBOHCTB paCCM01'pHM BbI60PKY, co-
COTORI.l(yID H3 30 KpyrJlbIX ,neTaJIeH, lIo.rryqeHHyID c.rryqaHHbIM 06pa30M H3 lIapTHH B
1000 ,neTaJIeH, lIpOH3Be,neHHbIX CTaHKOM. 11:3MepeHHeM 6blJI0 YCTaHOBJleHO, qTO
cpe,nHHH ,nHaMe1'p 06pa3uoB B BbI60pKe paBeH 0,503 CM, a CTaH,[(apTHOe OTKJIOHe-
HHe BbI60pKH COCTaBJI.HeT 0,0005 CM.
ECJlH ,nOrryCTHMbIMH RB.IDIIOTCR, HalIpHMep, TOJlbKO ,neTaJIH ,nHaMe1'pOM
0,502-0,504 CM, TO TeopeTWlecJ<yIO KpHByro HOpMaJIbHOrO paClIpe,neJleHIDI MO)I{HO
HClIOJlb30BaTb ,[(.IDI lIpe,nCKa3aHIDI lIpoueHTa 6paKa. ,n;OrryCTHMbIH pa36poc ,nHaMeTPoB
,neTaJIeH COCTaB.IDIeT 0,504-0,502 = 0,002 CM. TaK KaK CTaH,[(apTHOe OTKJIOHeHHe paB-
HO 0,0005 CM, ,naHHbIH pa36poc COOTBeTcTByeT + 2 CTaH,[(apTHbIM OTKJIOHeHIDIM. 11:3
CBoHcTBa (2), lIpHBe,neHHoro BbIIIIe, CJle,nyeT, qTO 95 % ,neTaJIeH yKJIaLJ:bIBaIOTCR B ,no-
rryCTHMbIH ,nHalIa30H, T.e. B BbI60pKe co,nep)I{HTCR 28 HJIH 29 ,nOrryCTHMbIX ,neTaJIeH, a
B lIapTHH, BepORTHO, co,nep)I{HTCR 95 % OT 1000, T. e. 950 ,nOrryCTHMbIX ,neTaJIeH.
,n;aHHbIH lIpHMep 6bIJI BbI6paH TaK, qTO pa36poc 6bIJI paBeH TOqHO + 2 CTaH,[(apT-
HbIM OTKJIOHeHIDIM. O,nHaKo CylI(eCTByroT Ta6J1HUbI qaCTHqHbIX IIJIOII(a,neH lIO,n KPH-
BOH HOpMaJIbHOrO paClIpe,neJIeHIDI, KOTopbIe lI03BOJI.HIOT CBR3aTb JlI060e CTaH,[(apT-
Hoe OTKJIOHeHHe C IIJIOII(a,nblO lIO,n KpHBOH. (06bJqHO TaKaR Ta6J1Hua Ha3bIBaeTCR
«<I>yHKUIDI paClIpe,neJleHIDI HOpMHpOBaHHoro H ueHTPHpOBaHHoro HOpMaJIbHOrO
paClIpe,neJleHIDI». - IlpuMe"Lf,. nep.)
F/la8a 1. HH:JICeHepHaJI MameMamUKa
1. 14. AMcI>cI>epeHu.ManbHoe MC'IMCneHMe (npoM3BoAHble)
ECJlH y = xn
ECJlH y = ax n
ECJlH f(x) = ax n
ECJlH y = sin e
ECJlH Y = cos e
ECJlH Y = tge
ECJlH Y = ctge
ECJlH Y = seee
ECJlH Y = eosece
ECJlH Y = sin -1 X
a
X
ECJlH Y = COS-1 -
a
ECJlH Y = tg-1 X
a
X
ECJlH Y = ctg- 1 -
a
X
ECJlH Y = see- 1 -
a
X
ECJlH Y = cosec -1 -
a
ECJlH Y = ex
ECJlH Y = e ax
ECJlH Y = ax
ECJlH Y = In X
TO dy = nx n - 1
dx
TO dy = anxn-1
dx
TO f'(x) = anx n - 1
TO dy = cose
de
TO dy = -sine
de
TO dy = see 2 e
de
TO dy = cosec 2e
de
dy sin e
TO - = tgeseee =
de COS2 e
dy Cose
TO - = -ctgecosece = -
de sin 2 e
dy 1
TO -
dx .J a 2 - x 2
dy 1
TO -
dx .J a 2 - x 2
dy a
TO -
dx a 2 + x 2
dy a
TO -
dx a 2 + x 2
dy a
TO -
dx x .J x 2 -a 2
dy a
TO -
dx x .J x 2 -a 2
TO dy = ex
dx
TO dy = ae ax
dx
TO dy =a X lna
dx
TO dy =l
dx X
IIpoH3Bo,n;uaB npOH3Be eHHB
dy dv du
ECJlH Y = UV, r,ae U H V - <PYllKUHH OT X, Tor a - = U - + v -
dx dx dx
1.14. ,4ufjJfjJepeHlJ,UOJlbHOe UC"tUCileHue (npoU36ooHble) 43
IIpoH3Bo,n;uaB qaCTHOro
du dv
v- -u-
ECJIH y = u / V, r.n;e U H V - cPyHKIJ;HH OT x, Tor.n;a dy = dx dx
dx v 2
IIpoH3Bo,n;uaB CJIOlKHOH fj)YUKD;llH
dy dy du
ECJlH y eCTb <PYllKUIDI OT x, Tor,na - = - x -
dx du dx
IIocJIe,n;oBaTeJILHoe ,n;Hfj)fj)epeHD;HpoBaHHe
ECJlH y = f(x), Tor,na ee lIepBa.H lIpOH3Bo,nHaR 3alIHCbIBaeTCR B BH)];e dy HJIH f'(x).
dx
ECJlH 3TO BbIpIDKeHHe lIpo,nH<p<pepeHUHpoBaTb BTOpOH pa3, TO lIOJlyqHTCR BTOpaR
d 2 y
lIpOH3BO,nHa.H, KOTOpaH 3aIIHCbIBaeTCR B BH)];e - HJIH f"(x).
dx 2
1 . 15. MHTerpanbHoe MC'IMCneHMe
(HeonpeAeneHHble MHTerpanbl)
3,neCb C - lIOCTORHHa.H HHTerpHpOBaHIDI; n * -1.
J xn+l
xndx = + C
n+l
J sin9d9 = -cos9 + c
J sec 2 9d9 = tg9 + c
J tg9sec9d9 = sec9 + c
J dx . I x
= sm - - + C
-J a 2 - x 2 a
J adx =tg- 1 X + C
a 2 + x 2 a
J adx = sec- 1 X + C
x -J x 2 - 0 2 a
J exdx = ex + s
J axn+l
axndx = + C
n+l
J cos9d9 = sin 9 + c
J cosec9d9 = -<:tg9 + c
J ctg9cosec9d9 = -<:osec9 + c
J -dx =cos- 1 X + C
-J a 2 - x 2 a
J -adx = ctg- 1 X + C
a 2 + x 2 a
J -adx = cosec-l X + C
x -J x 2 - 0 2 a
J eaxdx = eax + s
a .
J dx =lnx+c
X
J coshxdx = sinh x + s
f "
)
J axdx = ax + s
lna
J sinh xdx = cosh x + s
J sech 2 xdx = tghx + s
J dx = sinh-I x +c'
-J a 2 + x 2 a
J = COSh-I x + c
.JX2 a 2 a
J dx =!tgh- 1 X +c
a 2 - x 2 a a
HJIH
In[ x + .Jx a 2 + a 2 ] + c
[ X + -J x 2 -0 2 ]
In + C
a
1 1 (a + x)
- n +c
2a (a - x)
HJIH
HJIH
(::, F/la8a 1. HH:JICeHepHaJI MameMamUKa
1. 15. 1. HHTerpJfpOSSHJfe no IfSCTflM
J udv = uv - J vdu + c
1.15.2. OnpeAeneHH&le JfHTerpSn&1
BbIIIIeOlIHcaHHbIe HHTerpaJIbI co,n;epxaT npOH3BOJlbHyID KOHCTaHTY C H Ha3bIBalOTCR
HeOnpeOe/leHHblMU UHmeZpaJlaMu. OlIpe,n;eJleHHbIMH HHTerpaJIaMH Ha3bIBalOT HHTer-
paJIbI, ,lI;illI KOTOpbIX lIpe,n;eJlbI 3a.naHbI TaK: [x] = (b) - (a), CJle,n;OBaTeJlbHO:
3 [ 3 ] 3 [ 33 ] [ 13 ] ( 1 ) 2
y = f x 2 dx = x 3 + c 1 = 3 + c - 3 + c = (9 + c) - 3 + c = 8 3 ,
OTMeTHM, qTO lIOCTORHHbIe HHTerpHpOBaHH.H C B OlIpe,n;eJleHHOM HHTerpaJIe
B3aHMHO yHHqTOXaIOTCR.
1 . 16. &MHOM HblOTOHa (6MHOMManbHaR TeopeMa)
n(n -1) n(n -1)(n - 2)
(a + x)n =a n + nan-Ix + a n - 2 x 2 + a n - 3 x 3 +... + x n
2! 3! '
r,n;e 3! eCTb <t>aKTOpHaJI qHCJla 3 H paBH.HeTCR 1 x 2 x 3.
1 . 17. Pa3nO)l(eHMe B PRA MaKnOpeHa
x 2
f(x) = f(O) + xf'(O) + - f"(O) + ... .
2!
1.18. Pa3nO)l(eHMe B PRA TeMnopa
h 2
f(x + h) = f(x) + hf'(x) + - f"(x) + ... ·
2!
,n;OIIOJlHHTeJlbHyro HH<popMaUHIO lIO HIDKeHepHoH MaTeMaTHKe qHTaTeJlb MoxeT
lIOJIyqHTb, 06paTHBIIIHcb K cJle,n;yIOlI(eMY ClIpaBOqHHKY: ,lbKOH Bep,n; «ClIpaBOqHHK
lIO HIDKeHepHoH MaTeMaTHKe», «Newnes», TpeTbe H3,n;aHHe (Newnes Engineering
Mathematics Pocket Book, Third Edition, John Bird, 0750649925).
rnABA 2
MH>KEHEPHASI
CTATMKA
2.1. MH)I(eHepHaR CTaTMKa
HIDKeHepHYIO MexaHHKY MO)I{HO pa3,neJlHTb Ha cmamUKY H iJUHaMUKY. CTaTHKa B
OCHOBHOM H3yqaeT CHJIbI H pe3YJlbTaTbI HX B03,neHCTBIDI B COCTORHHH paBHOBeCIDI,
Tor,na KaK ,nHHaMHK:l H3yqaeT 06'beKTbI B ,nBIDKeHHH.
2.2. Macca, CMna M Bee
2.2. 1. Msccs
. Macca - 3TO Mepa KOJlWleCTBa BelI(eCTBa B TeJle; 061I(a.H C)'MMa Macc Bcex
cy6aToMHbIX qaCTHU, H3 KOTOpbIX COCTOHT ,naHHoe TeJlO. .
. BelI(eCTBO 3alIOJlHReT lIpOC1'paHCTBO H MoxeT HaxO,nHTbCR B TBep,nOM, )I{H)J;-
KOM HJIH ra3006pa3HoM COCTORHHH.
. ECJlH BelI(eCTBO He ,n06aBlliIeTCH K Te.rry HJIH He y,naJI.HeTCR H3 Hero, Macca
3Toro TeJla He MeH.HeTCR. OHa OCTaeTCR lIOCTORHHOH lIPH JII06bIX 06CTORTeJlb-
CTBax. B O,nHOM KHJIorpaMMe, HarrpHMep, MeTaJIJIa Ha JIyHe co,aep)I{HTCR CTO-
JlbKO xe aTOMOB, CKOJlbKO H B KHJIorpaMMe TaKoro xe MeTaJIJIa Ha f3eMJIe.
O,nHaKO KHJIorpaMM MeTaJIJIa Ha JIyHe 6y,neT BeCHTb MeHbIIIe, qeM Ha 3eMJIe
(CM. pa3,neJl 2.2.4).
. OCHOBHa.H e,nHHHua H3MepeHIDI MaCCbl - KWl02paMM (Kr). CaMa.H paClIpOCTPa-
HeHHa.H KpaTHa.H e,nHHHua,lJ)Ul 3TOH OCHOBHOH e,nHHHUbI - TOHHa (1000 Kr);
CaMa.H paClIpOCTPaHeHHa.H ,nOJlbHasI e,nHHHua - rpaMM (0.001 Kr).
(TaKoe OlIpe,aeJleHHe 6billO lIpHH.HTO B KJIaCCHqeCKOH <pH3HKe H HClIOJlb30Ba-
JlOCb, HalIpHMep, HbIOTOHOM. OHO HMeeT CMhICJI TOJlbKO AJUI O,nHOpO,nHbIX TeJl H
lI03BOJI.HeT BBeCTH rrOHRTHe IIJIOTHOCTH. Macca - 3TO <pH3W1eCKa.H BeJlHqHHa, o,nHa
H3 xapaKTepHCTHK MaTepHH, OlIpe,neJI.HlOlI(a.H ee HHepUHOHHbIe H rpaBHTaUHOHHbIe
CBOHCTBa. OlIbIT lIOKa3aJI, qTO HHepTHa.H H rpaBHTaUHOHHa.H MaCCbI lIPOlIopUHQHa-
JlbHbI o,nHa ,npyrOH. 3TOT <pYJI,LI;aMeHTaJIbHbIH 3aKOH lIpHpO,nbI Ha3bIBaeTCR lIPHHUH-
lIOM 3KBHBaJIeHTHOCTH.
npHpo,na MaCChI - o,nHa H3 BIDKHeHIIIHX elI(e He pelneHHbIX 3a,naq <pH3HKH.
npHHRTO CqHTaTb, qTO Macca 3J1eMeHTapHOH «<cy6aToMHOH») qaCTHUbI OlIpe,neJI.H-
eTCH lIOJI.HMH, KOTopbIe C HeH CBR3aHbI (3J1eK1'pOMarHHTHbIM, .H,L(epHbIM H IIp.). O,n-
HaKO KOJlHqeCTBeHHasI TeopIDI MaCCbI elI(e He C03,naHa.
11:3 CTaTbH Macca, <l>H3W1ecKHH 3HUHKJIOlIe,nHqeCKHH CJlOBapb. - IlpUMe"Lf,. nep.)
F/la8a 2. HHJlCeHepHaJI cmamU1CO
2.2.2. CJfns
QT06bI lIOH.HTb, KaK CBR3aHbI Macca H 8ec TeJla, He06xo,n;HMO paCCM01'peTb lIOH.HTHe
CUJlbl. CaMY lIO ce6e CHJIy HJIH CHCTeMY CHJI YBHJJ:eTb HeJlb3R; MO)I{HO 06Hap'YXHTb
TOJlbKO B03,n;eHcTBHe CHJIbI HJIH CHCTeMbI CHJI. HalIpHMep:
. CHJIa MoxeT H3MeHHTb HJIH lIOlIbITaTbCR H3MeHHTb <poPMY 06'beKTa.
. CHJIa MoxeT lIepeMeCTHTb HJIH lIOlIbITaTbCR lIepeMeCTHTb lIOKORlI(eeCR TeJlO.
ECJlH BeJlHqHHa CHJIbI ,aOCTaTOqHO 60JlbIIIa.H, TO 3TO lIpHBe,n;eT K lIepeMelI(e-
HHIO TeJla B HalIpaBJleHHH lIpHJIOXeHIDI CHJIbI. ECJlH BeJlWlHHa CHJIa He,n;OCTa-
TOqHa, qT06bI lIpeo,n;OJleTb COlIpOTHBJleHHe ,n;BIDKeHHIO, CHJIa Bce xe nonblma-
emCR C,n;BHHYTb TeJlO, XOTH H 6e3YClIeIIIHO.
. CHJIa MoxeT H3MeHHTb HJIH rrorrbITaTbCR H3MeHHTb ,n;BIDKeHHe TeJla, KOTopoe
yxe lIepeMelI(aeTCR. HalIpHMep:
- CHJIa BC1'peqHOrO Be1'pa MoxeT )'MeHbIIIHTb CKOpOCTb CaMOJleTa OTHOCH-
TeJlbHO 3eMJIH, Tor,n;a KaK CHJIa lIolIyrHoro Be1'pa MoxeT YBeJlWlHTb CKO-
pOCTb CaMOJleTa OTHOCHTeJlbHO 3eMJIH;
- CHJIa 60KOBoro Be1'pa MoxeT 3aCTaBHTb aBTOM06HJIb OTKJIOH.HTbCR OT KYPca.
. B03,l(eHCTBHe JlI060H CHJIbI 3aBHCHT OT:
- BeJlWlHHbI (pa3Mepa) CHJIbI, H3MepReMOH B HblOTOHax (H);
- HalIpaBJleHIDI CHJIbI;
- TOqKH lIpHJIOXeHIDI CHJIbI;
- clIoc06HOCTH TeJla COlIpOTHBJI.HTbCR B03,n;eHcTBHIO CHJIbI.
. KaK yxe 6billO CKa3aHO, CHJIY HeJlb3R YBHJJ:eTb; o,n;HaKO OHa MoxeT 6bITb
rrpe,aCTaBJleHa BeKTOpOM.
2.21'3. BeKTop&1
<l>H3HqeCKHe BeJlHqHHbI MO)I{HO pa3,aeJlHTb Ha ,n;Be KaTeropHH:
. CKaJI.HpHbIe BeJlHqHHbI;
. BeKTopHbIe BeJlHqHHbI.
CKaJIHpHbIe BeJlWlHHbI OlIpe,n;eJlRIOTCR TOJlbKO qHCJlOBbIM 3HaqeHHeM (pa3-
MepoM). HanpHMep:
. CKOpOCTb aBTOM06HJI.H MO)I{HO OlIpe,n;eJlHTb KaK 50 KHJIOMe1'pOB B qac (KM/q).
. MOII(HOCTb ,aBHraTeJI.H MoxeT 6bITb oueHeHa KaK 150 KHJIOBaTT (KBT).
. TOK, TeKYIIJ:HH B 3J1eKlpWleCKOH uelIH, MoxeT 6bITb paBeH 12 aMlIepaM (A).
BeKTopHbIe BeJlHqHHbI OlIpe,n;eJl.HIOTCH KaK qHCJlOBbIM 3HaqeHHeM, TaK H Ha-
rrpaBJleHHeM. HalIpHMep, CKOpOCTb aBTOM06HJIR - 50 KM/q TOqHO Ha ceBep H3
JIOH.nOHa. CKOpOCTb - BeKTOpHa.H' BeJlHqHHa, TaK KaK Y Hee eCTb H qHCJlOBOe 3Ha-
qeHHe (CKOpOCTb); H HarrpaBJleHHe (TOqHO Ha ceBep).
TOqHO TaK xe BeKTOpbI MO)I{HO HCrrOJlb30BaTb ,lJ;illI lIpe,n;CTaBJIeHHH CM H CHC-
TeM CHJI. Ha pHCyHKe 2.1 lIOKa3aHO, KaK MO)I{HO HCnOJlb30BaTb BeKTOp ,lJ;illI rrpe,n;-
CTaBJleHIDI CHJIbI (F) BeJlHqHHOH 150 HblOTOHOB (H), ,n;eHcTByrolI(eH Ha TOqKY (P) B
HarrpaBJIeHHH, COCTaBJlHlOlI(eM 30° C ropH30HTaJIblO. BeKTopHbIe ,n;HarpaMMbI 6y,ayr
no,ap06HO paCCMOTpeHbI B pa3,aeJle 2.3.
QT06bI lIpe,nCTaBHTb CHJIy BeKTO-
pOM, He06xo,nHMa cJle,nyrolI(a.H HH<pOP-
MaUIDI.
. T OqKa lIpHJIOXeHIDI CHJIbI.
. BeJl HHa (pa3Mep) CHJIbI.
. JIHHIDI B03,neHCTBIDI CHJIbI.
. HalIpaBJleHHe, B KOTOpOM ,neH-
cTByeT CHJIa (YKa3aHO cTpeJl-
KOH). OHa MoxeT HaxO,nHTbCR B
KOHue BeKTopa HJIH B cepe,nHHe
BeKTOpa, KaK 60Jlee y,n06HO.
2.2.4. Bec
2.2. Maeea, eU/la u 8ee
F = 150 H
p
Puc. 2.1. BeKTopHoe npe,n;CTaBJIeHHe CHJIbI
TBep,noe TeJlO, XH,llKOCTb HJIH ra3 MaCCOH 1 Kr 6y.nyr BeCHTb 9,81 H 6J1aro,napR rpa-
BHTaUHOHHOMY lIpHTIDKeHHIO IIJIaHeTbI 3eMJUI. TaK KaK IIJIaHeTa 3eMJIR He H)];eaJIb-
Ha.H c<pepa, TeJlO MaCCOH 1 Kr 6y,neT BeCHTb HeMHoro 60JlbIIIe Ha lIOJlIOCaX H HeMHO-
ro MeHbIIIe Ha 3KBaTope. O,nHaKo AJUI lIpaKT eCKHX ueJleH cpeoHuu 8ee MaCCbI 1 Kr
Ha IIJIaHeTe 3eMJUI CqHTaeTCR paBHbIM 9,81 H:
. rpaBHTaUHOHHa.H CHJIa rrpHTRrHBaeT Maccy JlI060ro TeJla (TBep,noro, )l{lf)]:KOrO
HJIH ra3006pa3Horo) K ueHTpy IIJIaHeTbI 3eMJUI;
. Bec TeJla - 3TO rpaBHTaUHOHHa.H CHJIa (cHJIa TIDKecTH), ,neHcTByrolI(a.H Ha 3TO
TeJIO;
. lI03TOMY Bec - 3TO CHJIa, H3MepHeMa.H B HbIOTOHax;
. K03<P<PHUHeHT 9,81 - 3TO YCKopeHHe cB060,nHoro lIa,neHIDI TeJla, 06YCJlOB-
JleHHOe CHJIOH TIDKeCTH, OHO paBHo 9,81 MeTPa Ha ceKJH,ll;Y B KBa,npaTe
(M/C 2 ). YCKopeHHe, 06YCJlOBJleHHOe CHJIOH TIDKeCTH, 06Cy)I(,l(aeTCR ,naJIee B
pa3,aeJle 3.6. 7; (
. XOTR Bec TeJla MaCCOH 1 Kr paBeH lIpH6J1H3HTeJlbHO 9,81 H (cpe,nHee 3Haqe-
HHe) Ha IIJIaHeTe 3eMJUI, Bec ,naHHoro TeJla H3MeHHTCR Ha IIJIaHeTe, HMelO-
lI(eH HHYIO CHJIy rpaBHTaUHOHHoro rrpHTRxeHHR, HJIH lIPH rrepeMelI(eHHH B
KOCMoce. HarrpHMep, rpaBHTaUHOHHoe lIpHTIDKeHHe JIYHbI COCTaBlliIeT lIPH-
6J1H3HTeJlbHO 1/6 lIO cpaBHeHHIO C IIJIaHeTOH 3eMJUI. n03TOMY Macca 1 Kr 6y-
,aeT BeCHTb Ha JIYHe )i x 9,81 = 1,64 H.
2.2.5. Msccs HS eAJfHJfqy 06"heMS (nnoTHOCT&)
I1YCTb TBep,aoe TeJlO HMeeT 06'beM 60 CM 3 . ECJlH Macca 3Toro TeJla COCTaBJlHeT
600 rpaMMOB (r), TO MO)I{HO BbIqHCJlHTb
Macca 1 CM 3 = 600 r/60 CM 3 = 10 rpaMMOB B CM 3 (10 r/cM 3 )
TaK KaK 1 CM 3 B 3TOM cJlyqae - e,nHHHQHbIH 06'beM, TO Macca e,nHHHUbI 06'beMa
,[(aHHoro TeJla COCTaBJI.HeT 10 r/cM 3 . Maccy e,nHHHUbI 06'beMa JlI060ro TeJla TaIOKe
F/la8a 2. HH:JICeHepHaJI cmamUl€a
Ha3bIBaIOT IIJIOTHOCTblO 3Toro TeJla. I1JIOTHOCTb 0603HaqaeTCR rpeqeCKOH 6yKBOH
«PO» (p). 061I(a.H <poPMYJla AJUI MaCCbI e,nHHHUbI 06'beMa (IIJIOTHOCTH):
p = m/V,
r,ne: p = IIJIOTHOCTb, m = Macca, V = 06'beM.
I1JIOTHOCTH HeKoTopbIX paclIpoc1'paHHbIX TBep,nbIX TeJl, )l{H)J;KOCTeH H ra30B, HC-
lIoJlb3yeMbIX B TeXHHKe, lIpHBe,aeHbI B Ta6J1Hue 2.1.
Ta6J1. 2.1. llJIOTHOCTb pacnpOCTpaHHbIX BerneCTB.
II.11OTHOCTh II.11OTHOCTh
BeIQeCTBO Kr/M 3 r/cM 3 BeIQeCTBO Kr /M 3 r/cM 3
AAIOMHHHH 2720 2,72 CnHPT 800 0,80
JIaTyHb 8480 8,48 PryTb 13590 13,59
Me,n;b 8790 8,79 llapa<p HH 800 0,80
qyryH 7200 7,20 BeH3HH 720 0,72
CBHHeI( 11350 11,35 Bo,n;a (qHCTaH npH 4 °C) 1000 1,10
HeHJIOH 1120 1,12 Au;eTHJIeH 1,17 0,1017
llOJIHBHHHJIXJIOPH.n (llBX) 1360 1,36 B03,ZJ;yX (CYXOH) 1,30 0,1013
Pe3HHa 960 0,96 YrJIeKHCJIbIH ra3 1,98 0,10198
CTaJIb 7820 7,82 Bo,n;opo,n; 0,19 0,10009
OJIOBO 7280 7,28 A30T 1,25 0,10125
QHHK 7120 7,12 KHcJIopo,n; 1,43 0,10143
2.2.6. Bec s eAJfHJfqsx 06"heMS
Bec e,nHHHUbI 06'beMa BelI(eCTBa MoxeT 6bITb OlIpe,neJleH:
a) lIpRMbIM H3MepeHHeM,
6) lIpe06pa30BaHHeM IIJIOTHOCTH BelI(eCTBa.
a) Orrpe,neJleHHe Beca e,nHHHUbI 06'beMa lIpRMbIM H3MepeHHeM:
Bec e,nHHHUbI 06'beMa (H/M 3 ) = Bec (H) / 06'beM (M 3 ) = W/
6) OlIpe,neJleHHe Beca e,nHHHUbI 06'beMa lIpe06pa30BaHHeM:
Bec e,nHHHUbI 06'beMa = W/V
nJlOTHOCTb (p) = m/V
Bec = m x 9,81
nO,acTaBJI.H.H ypaBHeHHe (3) B ypaBHeHHe (2):
Bec e,nHHHUbI 06'beMa = (m x 9,81) / V
m/V=p
Bec e,nHHHUbI 06'beMa = p x 9,81
(1)
(2)
(3)
Ho:
n03TOMY:
IlpuMettanue:
. 3TO lIpe06pa30BaHHe lIpHMeHHMO TOJlbKO Ha IIJIaHeTe 3eMJI.H. B ,npyrHX Mec-
Tax KOCMoca CJle,nyeT HClIOJlb30BaTb JlOKaJIbHOe 3HaqeHHe rpaBHTaUHOHHoro
YCKopeHH.H g.
. 3HaqeHHe IIJIOTHOCTH ,n0JI)I{H0 6bITb BbIpIDKeHO B KHJIorpaMMax Ha e,nHHHIfj'
06'beMa.
2.2. Maeea, eU/la u 8ee
2.2.7. OTHOCJfTen&HSfI nnOTHOCT&
OTHOCHTeJlbHa.H IIJIOTHOCTb BelI(eCTBa MoxeT 6bITb olIpe,n;eJleHa KaK OTHOIIIeHHe:
(nrrOTHOCTb BelI(eCTBa) / (IIJIOTHOCTb qHCTOH BO,n;bI lIpH 4°C)
HJIH
(Macca BelI(eCTBa) / (Macca paBHoro 06'beMa qHCTOH BO,n;bI lIpH 4°C)
TaK KaK OTHOCHTeJlbHa.H IIJIOTHOCTb lIpe,n;CTaBJI.HeT C060H OTHOIIIeHHe, OHa RB-
JI.HeTCR 6e3pa3MepHOH BeJlWlHHOH.
OTHOCHTeJlbHbIe IIJIOTHOCTH HeKOTopbIX paClIpOC1'paHeHHbIX BelI(eCTB, HClIOJlb-
3yeMbIX B TeXHHKe, lIpHBe,n;eHbI B Ta6J1Hue 2.2.
Ta6J1. 2.2. OTHOCHTeJIbHbIe llJIOTHOCTH HeKOTopbIX pacnpOCTpaHeHHbIX BeIIJ;eCTB.
BeIQecTBO OmOCHTeJlhHaJI BeIQeCTBO OmOCHTeJlhHaJI
WlOTBOCTh (d) WlOTBOCTh (d)
ArrIOMHHHH 2,72 CnHpT 0,80
JIaTYHb 848 Pryrb 13,95
KanMHH 8,57 llapa<pH H 0,80
XpOM 7,13 BeH3HH 0,72
Me,n;b 8,79 Bo,n;a (qHCTaH npH 4 °C) 1,10
qyryH 7,20 Bo,n;a (MOpCKaH) 1,12
CBHHeI( 11,35
HHKeJIb 8,73 Au;eTHJIeH 0,1017
HeHJIOH 1,12 B03,ZJ;yX (CYXOH) 0,1013
llOJIHBHHHJIXJIOPH.n (llBX) 1,36 YrJIeKHCJIbIH ra3 0,10198
Pe3HHa 0,96 YrapHbIH ra3 0,10126
CTaJIb 7,82 Bo,n;opo,n; 0,10009
OJIOBO 7,28 A30T 0,10125
QHHK 7,12 KHcJIopo,n; ( 0,10143
2.2.8. p,SSneHJfe ()lCJfAKOCTJf)
,n;aBJIeHHe, ,n;eHcTByrolI(ee Ha IIJIOII(a,nb lIOBepXHOCTH ,n;aHHoro TeJla, OlIpe,n;eJI.HeTCR
KaK HOpMaJIbHa.H CHJIa Ha e,n;HHHIfj' IIJIOII(a,nH, ,n;eHcTByrolI(a.H Ha lIOBepXHOCTb. Hop-
MaJIbHaR CHJIa 03HaqaeT CHJIY, ,n;eHcTByrolI(yro lIO,n; lIpRMbIM yrJlOM K lIOBepXHOCTH.
B CM e,n;HHHueH ,aaBJleHIDI KOCTH RBJI.HeTCR lIaCKaJIb (Ila). 1 Ila paBeH 1 H/M 2 .
,n;aBJIeHHe (P) = F/ A,
r,n;e: p - ,n;aBJleHHe B rraCKaJUIX; F - CHJIa B HbIOTOHax, ,n;eHcTByrolI(a.H lIO,n; rrpHMbIM
yrJlOM (IIO HOpMaJIH) K lIOBepXHOCTH; A - IIJIOII(a;J:b lIOBepXHOCTH B KBa,npaTHbIX
Me1'pax.
TaK xe:
1 MIla = 1 MH/M2 = 1 H/MM 2 .
TaK KaK naCKaJIb OqeHb MaJIaH e,n;HHHua, Ha lIpaKTHKe qalI(e HClIOJlb3yeTcR 6ap:
1 6ap = 10 5 lIaCKaJIeH.
F/la8a 2. HH:JICeHepHaJI cmamwco
llpUMe"taHue.
. ATMoc<t>epHoe ,naBJleHHe 06bJqHO BbIpIDKalOT B MHJIJIH6apax.
. MaHoMe'rpHqeCKOe ,naBJleHHe - 3TO ,naBJleHHe )lQf)];KOCTH, H3MepReMoe 'rpy6-
qaTbIM MaHoMe'rpoM, U-06pa3HbIM MaHoMe'rpoM HJIH aHaJIOrHqHbIM YC'rpOH-
CTBOM, TaK qTO H3MepReMoe ,naBJleHHe RBJlReTCR ,n06aBOqHbIM K aTMoc<t>epHo-
MY ,naBJleHHIO.
. A6coJIIOTHOe ,naBJleHHe, HClIOJlb3yeMoe B <pH3HKe H TepMo,nHHaMHKe, - 3TO
,naBJleHHe OTHOCHTeJlbHO a6COJllOTHOrO BaKY}'Ma, lI03TOMY:
a6coJllOTHoe ,aaBJIeHHe = aTMoc<PePHoe ,naBJIeHHe + MaHOMerp ecKoe ,naBJIeHHe.
2.3. BeKTOpHbie AMarpaMMbl CMn:
rpacl>M'IeCKOe peweHMe
KaK yxe 6billO OTMeqeHO, CHJIbI HeJlb3R }'BHJJ:eTb, lI03TOMY HX HeJlb3R HapHCOBaTb;
o,nHaKO B03,neHCTBHe CHJI H CHCTeM CHJI MO)I{HO rrpe,nCTaBHTb BeKTopaMH. BeKTopbI
6billH OlIpe,neJleHbI B pa3,neJle 2.2.2.
2.3. 1. PssHoAejfcTsYIO Jfe CJfn&1
CHJIa, KOTopaR MoxeT 3aMeHHTb ,nBe
HJIH 60Jlee CHJI H OKa3aTb TaKoe xe
B03,neHcTBHe, Ha3bIBaeTCR pe3YJlbTHPy-
IOlI(eH HJIH paBHo,neHcTByromeH CHJIOH.
HarrpHMep, CHJIbI Fl H F 2 , ,neHcTByro-
II(He B O,nHOM H TOM )l(e HalIpaBJleHHH
Ha TOqKY P B,nOJlb O,nHOH H TOH xe JlH-
HHH ,neHCTBH.H, MO)I{HO 3aMeHHTb e,nHH-
CTBeHHOH CHJIOH, oKa3bIBalOlI(eH Ta-
Koe )l(e B03,neHCTBHe, KaK rrOKa3aHO Ha
pHC. 2.2, a.
I10CKOJlbKY 06e CHJIbI Fl H F 2 HMe-
lOT 06lI.(yIO JlHHHIO ,aei1cTBHH H ,aeHcT-
BYIOT B O,nHOM H TOM )l(e HalIpaBJleHHH,
qT06bI lIOJlyqHTb BeJlHqHJIy BeKTopa
e,nHHCTBeHHOH CHJIbI, KOTopaR MoxeT
3aMeHHTb HX, ,nJIHHbI BeKTOpOB 3THX
CHJI MO)I{HO lIpOCTO CJlO)I{HTb. I10.rry-
qHBIIIaRCR CHJIa F R rrOKa3aHa Ha pHC.
2.2, 6. Ee Ha3bIBalOT paBHo,aeHcTByro-
meH CHJIOH. OTMeTHM, qTO BeKTOp paB-
Ho,neHcTBYlOmeH CHJlbI HMeeT ,aBOHHyID
pa3MepHYlO c'rpeJIKY, QTO rr03BOJlHeT
OTJlHqHTb ero OT ,apyrHX CHJI, ,aeHcTBY-
IOmHX B CMCTeMe.
B MaclIlTa6e 1 CM = 5 H BeKTOp,
0603HaqalOlI(HH F 1 , 6y,aeT HMeTb ,nJIHHY
4 CM, a BeKTOp, 0603HaqalOlI(HH F 2 , -
4 CM 5 CM
r F 1 = 20 H -r F 2 = 25 H -I
P . . .
a
9 CM
- -
F R = 45 H
P
6
4 CM
[ F 1 = 20 H T
P
I
B .
5 CM
F 2 = 25 H :I
F R = 5 H
..
Puc. 2.2. PaBHo,n;eHcTBYIOll(He CHJIbI:
a - CM.JIbI Fl H F 2 , ,n;eHCTByIOIl(He Ha TOtIKy P;
6 - paBHo,n;eHcTByrornaH CHJI Fl H F 2 ;
B - paBHo,n;eHcTBYIOInaH CHJI Fl H F 2 , ,n;eH-
CTBYIOInHX B npOTHBOnOJIO)l(HbIX HanpaB-
JIeHHHX.
MacllITa6: 1 CM = 5 H
2.3. BelCmopHble OUaZpaMMbl CUll: zparjJu'teclCoe peUleHue
AJIHHY 5 CM. B pe3YJIbTaTe CJIOXeHIDI, BbIlIOJIIDIeMOrO, qT06bI lIO.JIyqHTb ,nJIHHY BeK-
Topa pe3YJIbTHpyrolI(eH CHJIbI (F R ), lIOJIyqaeTC.H e,l(HHCTBeHHbIH BeKTOp ,nJIHHOH
4 CM + 5 CM = 9 CM, lIpe,l(CTaBIDIIOII(HH CHJIy B 45 H, ,l(eHcTByrolI(yID Ha TOqKy P.
ECJIH 6bI CHJIbI ,l(eHCTBOBaJIH B lIpOTHBOlIOJIO)I{HbIX HalIpaBJIeHIDIX B,l(OJIb TOH xe
JIHHHH ,l(eHCTBIDI, TO paBHO,l(eHCTByrolI(yID CHJIy (F R ) MO)I{HO 6 bill 0 6bI OlIpe,l(eJIHTb,
BbIqHTasI ,nJIHH)' BeKTOpa FI H3 ,nJIHHbI BeKTOpa F 2 , KaK lIOKa3aHO Ha pHC. 2.2, B.
2.3.2. nspsnnenorpsMM CJfn
Kor,l(a Ha 0,l(H)' TOqKy P ,l(eHcTByroT ,l(Be HJIH 60JIee CHJI, JIHHHH ,l(eHCTBIDI KOTOpbIX
JIeXaT lIO,l( yrJIOM ,l(pyr K ,l(pyry, me pe3YJIbTHpyrolI(yro CHJIy HeJIb3R orrpe,l(eJIHTb
rrpOCTbIM apH<pMeTHqeCKHM CJIOXeHHeM HJIH BbIqHTaHHeM. ECJIH ,l(eHcTByroT TOJIbKO
,l(Be CHJIbI, KaK lIOKa3aHO Ha pHC. 2.3, BeJIWlHHY H HalIpaBJIeHHe paBHO,l(eHcTByro-
eH CHJIbI MO)I{HO OlIpe,l(eJIHTb lIOC1'poeHHeM B MacIIITa6e lIapaJIJIeJIorpaMMa CHJI.
Ha pHC. 2.3, a lIOKa3aHbI BeKTOpbI ,lI)UI ,l(BYX CHJI, ,l(eHcTByroII(HX iIa TOqKy P,
rro,l( yrJIOM 60° O,l(Ha K ,l(pyrOH. Ha pHC. 2.3, 63TH BeKTOpbI 06pa3yroT ,l(Be CMe)I{HbIe
CropOHbI lIapaJIJIeJIorpaMMa P ABC. BeKTop P A COOTBeTcTByeT CHJIe F I , a BeKTOp
PC COOTBeTcTByeT CHJIe F 2 . QT06bI ,l(OCTpOHTb lIapaJIJIeJIorpaMM, lIPRMasI AB rrpOBe-
,neHa lIapaJIJIeJIbHO BeKTOpy PC, a rrpRMasI CB lIpOBe,l(eHa lIapaJIJIeJIbHO BeKTOpy
PA. TOqKa B JIeXHT Ha lIepeCeqeHHH IIp.HMbIX AB H CB. Pe3YJIbTHpyrolI(a.H CRJIa F R ,
,neHcTByrolI(aR Ha TOqKy P, RBJI.HeTCR ,l(HarOHaJIblO lIapaJIJIeJIorpaMMa PABC. BeJIH-
qHHY pe3YJIbTHpyrolI(eH CHJIbI F R MO)I{HO OlIpe,l(eJIHTb, H3MepHB ,nJIHHY ,l(HarOHaJIH
PB H YMHOXHB ee Ha MacIIITa6 ,l(HarpaMMbI (1 CM = 10 H B ,l(aHHOM c.JIyqae). YroJI a
MO)I{HO OlIpe,l(eJIHTb lIpH rrOMOII(H 1'paHClIopTHpa. ,n:JIR pHC. 2.3 TIOJIyq HOTCR CJIe.ny-
IO He 3HaqeHIDI : F R = 87 H H a = 23° (6JIl:OKaHIIIee ueJIoe qHCJIO).
B
'I
I
(
I
I
I
R = 87 HI
I
I
I
I
I
p
a
F 2 = 50 H
p
6
F 2 = 50 H
c
Pac. 2.3. naparrJIeJIOrpaMM CHJI.
MacllITa6: 1 CM = 10 H
2.3.3. YpSSHOSeWJfSSIOUCSfI CJfns
CRJIa, KOTOpasI ypaBHOBeIIIHBaeT ,l(eHCTBHe ,l(pyrOH CRJIbI HJIH CHCTeMbI CRJI, Ha3bI-
BaeTCR ypaBHOBeIIIHBalOlI(eH CHJIOH (FE)' YpaBHOBeIIIHBalOlI(aR CHJIa:
. HMeeT 1)' xe caMyro BeJIHqHHY, qTO H paBHO,l(eHCTByrolI(aR cRJIa;
. HarrpaBJIeHa B,l(OJIb TOH xe caMoH JIHHHH ,l(eHCTBIDI, qTO H paBHO,l(eHCTByro-
lI(aR cRJIa;
. ,l(eHcTByeT B HalIpaBJIeHHH, rrpoTHBOlIOJIO)I{HOM paBHO,l(eHCTByrolI(eH cl1JIe.
F/la8a 2. HH:JICeHepHaJl cmamUlCa
Ha pHC. 2.4 lIOK83aHa ypaBHOBeIIIHBalOlI(asI CHJIa (FE)' paBHasI lIO BeJI}fqHHe H
lIpOTHBOlIOJIO)I{HasI lIO HalIpaBJIeHHIO paBHO,l(eHcTByrolI(eH CHJIe (F R)' OlIpe,l(eJIeHHOH
Ha pHC. 2.3, lI03TOMY OHa ypaBHOBeIIIHBaeT B03,l(eHCTBHe CHJI FI H F 2 Ha TOqKy P.
A
B
FE = 87 H
Pac. 2.4. YpaBHoBellIHBaIOlI(aH CHJIa.
MacllITa6: 1 CM = 10 H
2.3.4. PS3nOJICeHJfe CJfn
Pa3JI0XeHHe CHJI - 3TO 06paTHasI OlIepaUIDI K HaxO)l{)J;eHHIO paBHO,l(eHcTByrolI(eH
CHJIbI. To eCTb pa3JIOXeHHe CHJIbI - 3TO 3aMeHa ,l(aHHOH e,l(HHCTBeHHoH CHJIbI ,l(BY-
M.H HJIH 60JIee CHJIaMH, ,l(eHcTByroII(HMH B TOH xe caMOH TOqKe B yxa3aHHbIX Ha-
lIpaBJIeHIDIX.
PaCCM01'pHM e,l(HHCTBeHHyIO CHJIy Fs, paBHyIO 50 H, ,l(eHcTByrolI(yID lIO,l( yrJIOM
30° K ropH30HTaJIH, KaK lIOKa3aHO Ha pHC. 2.5, a. QT06bI pa3JIO)I{HTb 31)' CHJIy Ha ee
ropH30HTaJIbHyIO H BepTHKaJIbHyIO COCTaBJI.HIOII(He, BHaqaJIe HapHcyeM JIHHHH ,l(ei1:-
CTBIDI 3THX CHJI, KaK lIOKa3aHO Ha pHC. 2.5, 6. 3aTeM ,l(OC1'pOHM lIapaJIJIeJIorpaMM
CRJI, KaK lIOKa3aHO Ha pHC. 2.5, 6. BeJIHqHHbI CHJI Fv H F H MO)I{HO lIOJIyqHTb H3Me-
peHHeM pHCYHKa B MacIIITa6e (BeKTOpHasI ,l(HarpaMMa) HJIH lIpH lIOMOII(H 1'pHrOHO-
Me1'pHqeCKHX <PYHKUHH.
Fv = 50 H x sin 30° = 25 H
F H = 50 H x cos 30° = 43,3 H
Fs = 50 H
p
a
=50H
------------- 5
I
Fv I
I
I
I
I
I
p
6
F H
Pac. 2.5. Pa3JIOXeHHe CHJI.
MacllITa6: 1 CM = 10 H
2.3. BelCmopHble OUaZpaMMbl CUll: zparjJu'teclCoe peUleHue
2.3.5. PSSHOSeCJfe Tpex CJfn (TpeyronbHJfK cJfn)
. BeKTOpbI 1'pex CHJI, ,l(eHcTByroII(HX Ha TeJIO B O,l(HOH H TOH xe TIJIOCKOCTH,
MOXHO HapHCOBaTb Ha JIHCTKe 6YMarH.
. CHJIbI, KOTopbIe ,l(eHcTByroT B O,l(HOH TIJIOCKOCTH, H83bIBaIOTCR KOMTIJIaHap-
HbIMH.
., . ECJIH 1'pH CHJIbI, ,l(eHcTByroII(He Ha TeJIO, HaxO,lUlTC.H B paBHOBeCHH, JII06asI
CHJIa ypaBHOBeIIIHBaeT B03,l(eHCTBHe ,l(BYX ,l(pyrHX CHJI. TpH CHJIbI MOryr 6bITb
rrpe,l(CTaBJIeHbI 1'peyrOJIbHHKOM, CTOpOHbI KOToporo RBJI.HIOTC.H BeKTOpaMH
3THX CHJI.
. ECJIH JIHHHH ,l(eHCTBIDI 1'pex CHJI lIepeceKalOTCR B O,l(HOH TOqKe, TO OHH 06pa-
3yroT CHCTeMY CXO,lUlII(HXCR CHJI.
Ha pHC. 2.6 lIOKa3aHbI 1'pH KOMTIJIaHapHbIe CHJIbI, ,l(eHcTByroII(He Ha TeJIO. Mx
JIHHHH ,l(eHCTBIDI lIpO,l(OJDKeHbI Ha38)J;, ,l(0 lIepeCeqeHIDI B TOqKe P. ToqKy P Ha3bIBa-
lOT TOqKOH lIepeCeqeHIDI CHJI. (Ee TaIOKe Ha3bIBalOT 06lI(eH TOqKOH lIpHJIOXeHIDI
CRJI. K HeH BCer,l(a MO)I{HO lIpHBR3aTb cHcTeMY KOOp,l(HHaT. - llpUMe"t. nep.) ECJIH
CRJIbI HaxO,lUlTC.H B paBHoBecHH:
. HX B03,l(eHCTBIDI ypaBHoBeIIIHBaIOTcR;
. TeJIO, Ha KOTopoe OHB ,l(eHCTByroT, OCTaHeTCR HelIO,l(BIDKHbIM;
. BeKTOpbI CHJI 06pa3yroT 3aMKJIyTbIH 1'peyrOJIbHHK;
. BeKTOpbI CJIe,l(ylOT ,l(pyr 3a ,l(pyrOM lIO 3aMKJIyT0MY KOHTYPY.
Ha pHC. 2.7 lIOKa3aH ,l(OlIOJIHHTeJIbHbIH lIpHMep C peIIIeHHeM. ToqHOCTb OTBe-
TOB 3aBHCHT OT TOqHOCTH pHCYHKa. ToqHOCTb pHCYHKa ,l(OJDKHa 6bITb KaK MOXHO
BbIIIIe, qT06bI lIOMOqb H3MepeHHIO.
F 2 = 20 H
F3 = 35 H
6
a
Pac. 2.6. TpeyrOJIbHHK CHJI:
a - cxeMa KOHC'rpYKI(HH;
6 - ,n;HarpaMMa YCHJIHH.
MacllITa6: 1 CM = 5 H
CVlJ1a B OTT5I)I(-
K = 20 KH
(
Ha-
rpY3Ka
20 KH
\
CVlJ1a B CTpeJ1e
KpaHa = 34,6
6
Pac. 2.7. CHJIbI B npOCTOM CTpeJIOBOM KpaHe.
a - cxeMa KOHC'rpYKI(HH;
6 - ,n;HarpaMMa YCHJIHH.
MacllITa6: 1 CM = 5 KH
F/la8a 2. HH:JICeHepHf1Jl cmamUica
2.3.6. MHoroyronbHJfK CJfn: nOCTpOeHJfe
AJfSrpSMMbl MaKcsenns-KpeMOH&1
MHoroyrOJIbHHK CHJI HClIOJIb3yeTC.H JVUI HaxO)l{)J;eHIDI paBHO,l(eHcTByrolI(eH JII060ro
KOJII-NeCTBa KOruIaHapHbIX CRJI C nepeceKalOmHMHC.H JIHHIDIMH .n.eHCTBIDI, B03.n.eHCT-
ByrolI(HX Ha TOqKy P. 3TH CHJIbI RBJI.HIOTCR lIepeCeKaIOII(HMHC.H, lIOCKOJIbKy ,l(eHCTBY-
lOT Ha e,l(HHCTBeHHyIO TOqKy P. OHH RBIDIIOTCR KOTIJIaHapHbIMH, TaK KaK JIeXaT B O -
HOH H TOH xe TIJIOCKOCTH. Ha pHC. 2.8, a lIOKa3aHa cxeMa KOHC1'pYKUHH, lIpWleM 06-
JIaCTH, orpaHWleHHbIe JIHHIDIMH ,l(eHCTBHH CHJI, 0603HaqeHbI 3arJIaBHbIMH 6yJ(BaMH.
CooTBeTcTByrolI(asI ,l(HarpaMMa YCHJIHH (,l(HarpaMMa MaKCBeJIJIa-KpeMoHbI) lIOKa3a-
Ha Ha pHC. 2.8, 6. CRJIbI 0603HaqeHbI C1'pOqHbIMH 6yJ(BaMH. 3TO rpa<pWlecKHH cno-
c06 paCqeTa YCHJIHH lIOC1'poeHHeM ,l(HarpaMMbI MaKCBeJIJIa- KpeMoHbI:
F 1 = 40 H
A
F 2 = 50 H
b
c
D
B
p
F3 = 30 H
c
d'
a
6 d
Pac. 2.8. MHoroyrOJIbHHK CHJI (HaKJIOHHbIe 0603HaQeHHH):
a - cxeMa KOHCTpYKI(HH (He B MacllITa6e);
6 - ,n;HarpaMMa YCHJIHH (MacllITa6: 1 CM = 10 H)
. CRJIbI, ,l(eHcTByroII(He Ha TOqKy P, CJIe,l(yeT lIpe,l(CTaBHTb BeKTOpaMH, ,l(JIHHbI
H yrJIbI KOTOpbIX ,l(OJDKHbI 6bITb Hap:ucoBaHbI C C06JIIO,l(eHHeM MacIIITa6a Ha-
CTOJIbKO TOqHO, HaCKOJIbKO 3TO B03MO)I{H0. 06JIaCTH, orpaHHqeHHble JIHHIDI-
MH ,l(eHCTBHH CHJI, 0603Haqa1OTC.H 3arJIaBHbIMH 6yJ(BaMH. OTlIpaBHasI TOqKa
He BaxHa, HO 6yJ(BbI ,l(OJDKHa CJIe,l(OBaTb BOKpyr lIO nop5JJJ;Ky, 06bIqHO B Ha-
npaBJIeHHH lIO qaCOBOH C1'peJIKe.
. ,n:HarpaMMa YCRJIHH ,l(O]I)I{Ha 6bITb HapHCOBaHa c1'poro B MaCIIITa6e, HaqHHasI C
rrO,l(XO,l(RlI(ero BeKTOpa. HanpHMep, MO)I{HO HaqaTh C BeKTopa BepTIlKaJIbHOH
CRJIbI Fl. TaK KaK 3TOT BeKTOp HaxO,l(HTCR Me)K,l(y 06JIaCT.HMH D H A, OH Map-
KHpOBaH da. TIoMHHTe, qTO 06JIaCTH npHH.HTO 0603HaqaTb 3arJIaBHbIMH 6yJ(Ba-
MH, TOr,l(a KaK CHJIbI 0603HaqalOT COOTBeTCTByrolI(HMH C1'pOqHbIMH 6yJ(BaMH.
,n:06aBbTe c1'peJIKy, qT06bI yKa3aTb, B KaKOM HalIpaBJIeHHH ,l(eHcTByeT CHJIa.
. CJIe,l(yIDlI(aR CRJIa (F 2 ) paClIOJIOXeHa Me)K,l(y 06JIaCTRMH A H B. OHa HapHCO-
BaHa B MaCIIITa6e, HaqHHaR OT KOHua lIepBoro BeKTOpa, H 0603HaqeHa ab.
CHOBa ,l(06aBbTe c1'peJIKy, qT06bI yKa3aTb, B KaKOM HalIpaBJIeHHH ,l(eHcTByeT
CRJIa.
. AHaJIOrHqHbIM ClIoc060M ,l(06aBbTe OCTaJIbHbIe CRJIbI TaK, qT06bI OHH CJIe,l(O-
BaJIH no nopMKy B COOTBeTCTBHH C 6yJ(BeHHbIMH 0603HaqeHIDIMH.
. HaKOHeu, pe3YJIbTHpyroII(yID CRJIY (FR) MO)I{HO onpe,l(eJIHTb, COe,l(HHeHIDI
TOqKH d H d' Ha ,l(HarpaMMe YCRJIHH. 06paTHTe BHHMaHHe, qTO pe3YJIbTHpyro-
2.3. BeICmopHble OUll2pflMMbl CUll: zparjJu'tecICoe peUleHue
IUasI CHJIa HalIpaBJIeHa rrpoTHB 06lI(ero HalIpaBJIeHIDI CHJI, 3an;aHHbIX Ha ,[(Ha-
rpaMMe YCHJIHH. llpH He06xo,[(HMOCTH pe3YJIbTHpyroII(yID CHJIY Terrepb MOXHO
rrepeHecTH Ha cxeMY KOHCTPYKUHH, qT06bI lIOKa3aTb ee BeJIHqHIfY H HarrpaB-
JIeHHe OTHOCHTeJIbHO TOqKH P.
. lloMHHTe, qTO pe3YJIbTHpyrolI(asI CHJIa MoxeT 3aMeHHTb Bce 3an;aHHbIe CHJIbI
H Bce xe OKa3bIBaTb TO xe caMoe B03,[(eHCTBHe Ha TOqKy P.
. ECJIH 6bI 1'pe60BaJIOCb HaHTH ypaBHOBeIIIHBalO1I(yID CHJIy, TO ee BeJIHqHHa
paBH.HJIaCb 6bI BeJIHqHHe pe3YJIbTHpyrolI(eH CHJIbI, HO ,[(eHCTBOBaJIa 6bI 3Ta
CHJIa B lIpOTHBOlIOJIO)I{HOM HalIpaBJIeHHH. OHa 6bma 6bI HalIpaBJIeHa B 06-
lI(eM HalIpaBJIeHHH Bcex 3an;aHHbIX CHJI Ha ,[(HarpaMMe YCHJIHH.
2.3.7. HenepeCeKSIOUCJfeCfI KonnSHspHble CJfnbl
(sepesolfHblH MHoroyrOn&HJfK)
KaK CJIe.nyeT H3 Ha3BaHIDI, HelIepeCeKalOlI(HeCR KOIIJIaHapHbIe CHJIbI JIeXaT B O,[(HOH
IIJIOCKOCTH, HO B03,[(eHCTByroT He Ha 0,[(1fY H 1)' xe TOqKY (T.e. HeT 06lI(eH TOqKH rrpH-
JIOXeHIDI CHJI). llpHMep lIOKa3aH Ha pHC. 2.9, a. ECJIH lIpHMeHHTb MeTO,[( lIOC1'poeHIDI
,[(HarpaMMbI MaKCBeJIJIa- KpeMOHbI, TO Ha OCHOBaHHH cxeMbI KOHC1'pYKUHH pe3YJIb-
mpyrolI(asI CHJIa (FR) MoxeT 6bITb OlIpe,[(eJIeHa 06bIqHbIM ClIoc060M H3 ,[(HarpaMMbI
YCHJIHH, KaK lIOKa3aHO Ha pHC. 2.9, 6. O,[(HaKO,LVIR Toro, qT06bI OlIpe,[(eJIHTb lIOJIOXe-
HHe H JIHHHIO ,[(eHcTBIDI pe3YJIbTHpyrolI(eH CHJIbI, He06xo,[(HMO rrOC1'pOHTh BepeBOq-
HbIH MHoroyrOJIbHHK, KaK lIOKa3aHO Ha pHC. 2.9, B, CJIe,ayroII(HM 06pa30M:
F3
d
c
c
A
F3
b
a
D
F R - pe3YJ1bTVlpYIOw.a51 CVI-
J1a. Ee BeJ1V1"1V1Ha OnpeAeJ1f1-
eTCfI Vl3 AViarpaMMbl YCVI-
J1V1 , a ee n0J10)l(eHVle VI
J1V1HVl5I Ae CTBVl5I OnpeAeJ1f1-
IOTC5I Vl3 BepeBO"lHOrO MHO-
royrOJ1 bH VI Ka.
a
w ------
\
\
(, \
\
\ I
\ / ' y
\ /
\ //
\ /
\ /
F3
.
B
Pac. 2.9. BepeBOqHbIH MHoroyrOJIbHHK:
a - cxeMa KOHCTPYKUHH;
6 - ,n;HarpaMMa YCHJIHH (HapHcoBaHa B MacllITa6e);
B - BepeBOqHbIH MHoroyrOJIbHHK (HapHcoBaH B MacllITa6e)
. HapHCyHTe ,[(HarpaMMY CHJI (pHC. 2.9, 6) 06bIqHbIM crroc060M H HaH.nHTe Be-
JIHqHHY H HarrpaBJIeHHe pe3YJIbTHpyroI.l..(eH CHJIbI (F R ), orrpe,[(eJIReMbIe BeKTO-
pOM ad.
. BbI6epHTe TOqKy rrOJIIOCa 0 TaK, qT06bI OHa HaXO,[(HJIaCb BHYTPH HJIH cHapy-
)l(H ,[(HarpaMMbI YCHJIHH, H coe,[(HHHTe 31)' TOqKy rrOJIlOca 0 C KIDK.n;bIM H3 yr-
F/la8a 2. HH:JICeHepHaJl cmamU1ca
JIOB ,l(HarpaMMbI YCHJIHH, KaK lIOKa3aHO Ha pHC. 2.9, 6. TIOJIOXeHHe TOqKH
lIOJIIOCa 0 He HMeeT lIpHHUHlIHaJIbHOrO 3HaqeHIDI, H OHa MoxeT 6bITb pa3Me-
lI(eHa TaM, r,l(e 3TO HaH60JIee Y,l(06HO.
. 113 JII060H TOqKH W, JIeXalI(eH Ha JIHHHH ,l(eHCTBIDI CHJIbI F 1 , lIpOBe,l(HTe lIP.H-
MyIO qepe3 06JIaCTb A, lIapaJIJIeJIbHO IIp.HMOH Oa Ha ,l(HarpaMMe YCHJIHH. 3Ta
IIp.HMasI lIOKa3aHa Ha pHC. 2.9, B.
. 113 TOqKH W qepe3 06JIaCTb B lIpOBe,l(HTe IIp.HMyIO WX, lIapaJIJIeJIbHyIO IIp.HMOH
Ob Ha ,l(HarpaMMe YCHJIHH H lIepeceKalO1I(yID JIHHHIO ,l(eHCTBIDI CHJIbI F 2 B
TOqKe X. 3Ta JIHHIDI lIOKa3aHa Ha pHC. 2.9, B.
. TIpO,l(OJDKaHTe lIOC1'poeHHe, lIOKa IIp.HMbIe He 6y,l(yr lIpOBe,l(eHbI qepe3 Bce 06-
JIaCTH; lIO.JIyqHBIIIasIC.H 06JIaCTb ,l(OJDKHa TelIepb HalIOMHHaTb pHC. 2.9, B.
TIpRMbIe, lIapaJIJIeJIbHbIe Oa HOd, TelIepb MO)I{HO lIpO,l(JIHTb ,l(0 me lIepeCeqe-
HIDI B TOqKe Z. TIpoBe,l(HTe qepe3 TOqKy Z IIp.HMyIO, lIapaJIJIeJIbHyIO ad Ha ,l(Ha-
rpaMMe YCHJIHH, qT06bI lIpe,l(CTaBHTb JIHHHIO ,l(eHCTBIDI BeKTOpa pe3YJIbTHpy-
IOlI(eH CHJIbl FR'
CJIe.nyeT oTMeTHTb, qTO, XOT.H p83JIOXeHHe CHJI H CHCTeM CHJI rpa<pWleCKHMH
MeTO,l(aMH .HBJI.HeTCR lIpOCTbIM H lIpRMbIM ClIoc060M, TOqHOCTb 3Toro clIoc06a 3aBH-
CHT OT MacIIITa6a pHcYHKa H KaqeCTBa lIoc1'poeHIDI. ECJIH 1'pe6yroTCR 60JIee TOqHbIe
pe3YJIbTaTbI, me MO)I{HO lIO.JIyqHTb MaTeMaTWleCKHMH BbNHCJIeHIDIMH, HClIOJIb3YR
TpHroHoMe1'pHIO.
2.4. MOMeHTbl CMn, u.eHTp TR>KeCTM
M u.eHTpbl TR>KeCTM nno aAeM
2.4. 1. MOMeHTbl CJfn
Ha pHC. 2.10, a lIOKa3aHO, KaK raeqHbIH KJIlOq HClIOJIb3yeTC.H ,l(JUI lIOBopOTa raHKH.
CHJIa (F) He ,l(eHcTByeT HelIOCpe,l(CTBeHHO Ha raHKy, BMeCTO 3Toro OHa ,l(eHcTByeT
Ha paCCTO.HHHH (d) OT OCH raHKH. TIPH 3TOM C03,l(aeTCR BpalI(alOlI(ee YCHJIHe, KOTO-
poe Ha3bIBaeTCR MOMeHTOM CHJIbI OTHOCHTeJIbHO OCH raHKH. Ero TaIOKe Ha3bIBalOT
BpalI(alOlI(HM MOMeHTOM HJIH lIapoH CHJI, HO Bce 3TH TepMHHbI 03HaqalOT O,l(HO H TO
xe. PaCCTORHHe (d) Ha3bIBaeTCR TIJIeqOM CHJIbI HJIH ,l(JIHHOH pbIqara, 06a TepMHHa
RBJI.HIOTCR B3aHM03aMeH.HeMbIMH. MaTeMaTHqeCKH 3TO MO)I{HO BbIpa3HTb CJIe,l(yID-
II(HM 06pa30M:
M = F x d,
r,l(e: M - MOMeHT CHJIbI (HM); F - lIpHJIOXeHHasI CHJIa (H); d - TIJIeqo CHJIbI
(,l(JIHHa pbNara) (M).
llpUMe"laHue.
CYIUeCTBeHHo, qTO TIJIeqo CHJIbI (d) H3MepReTC.H lIeplIeH;:I;HKyJI.HpHO JIHHHH ,l(eii-
CTBIDI CRJIbI. ECJIH CHJIa HarrpaBJIeHa lIO,l( yrJIOM, KaK lIOKa3aHO Ha pHC. 2.10, 6, 3<p-
<peKTHBHasI ,l(JIHHa raeqHOrO KJIlOqa YMeHbIIIaeTCR. B 3TOM c.JIyqae ypaBHeHHe 6Y,l(eT
BbIr.JUl)]:eTb CJIe,l(yIDII(HM 06p830M:
M = F x de'
r,l(e: M - MOMeHT CHJIbI (HM); F - lIpHJIOXeHHasI CHJIa (H); de - 3<p<peKTHBHOe
IIJIeqo CHJIbI (,l(JIHHa pbIqara) (M).
a
6
2.4. MOMeHmbl CWl, eHmp mJI:JICecmu u eHmpbl mJI:JICecmu fl./lOU4aOeU
F
rae"lHblVi KJ11O"I
I
I
d
I ·
AnVlHa (d)
rae"lHblH KJ11O"I
I
/
Puc. 2.10. nOBOpaQHBaIOII(He MOMeHTbI
(,
.
2.4.2. npJfHqJfn MOMeHTOB (JfCnOn&3yeMsfI TepMJfHOnOrJffl)
Toqu onopld
UJIeqo CHJILI
MOMeHT no
qaCOBOH CTJ)eJIKe
MOMeHT npOTHB
qaCOBOH CTpeJiKH
Pe3YJILTHPYJOIIUlH
MOMeHT
ToqKa (ueH1'p) HJIH OCb lIOBopOTa, BOKpyr KOTOpOH lIpOHCXO,l(HT
HJIH MoxeT lIpOHCXO,l(HTb BpalI(eHHe.
PaCCTORHHe Me)K,[{y JIHHHeH ,l(eHCTBIDI lIpHJIOXeHHOH CHJIbI H
TOqKOH OlIOpbI, H3MepeHHoe lIO,l( lIpRMbIM yrJIOM K JIHHHH ,l(eH-
CTBIDI CHJIbI. ( \
JII060H MOMeHT CHJIbI, KOTOpblli IIOBOpaqllBaeT HJIII CT:p(¥IITCSI
lIOBepHYTb TeJIO BOKpyr ero TOqKH OlIOpbI B HalIpaBJIeHHH lIO qa-
COBOH C1'peJIKe. MOMeHTbI lIO qaCOBOH C1'peJIKe lIpHH.HTO CqHTaTb
lIOJIO)I{HTeJIbHbIMH (+), KaK lIOKa3aHO Ha pHC. 2.11.
JII060H MOMeHT CHJIbI, KOTOpbIH lIOBOpaqHBaeT HJIH C1'peMHTCR
lIOBepHYTb TeJIO BOKpyr ero TOqKH onopbI B HanpaBJIeHHH lIpO-
THB qaCOBOH C1'peJIKH. MOMeHTbI lIpOTHB qaCOBOH C1'peJIKH lIPH-
H.HTO CqHTaTb 01'pHuaTeJIbHbIMH (-), KaK lIOKa3aHO Ha pHC. 2.11.
Pa3HocTb no BeJIHqHHe Me)l{)J;Y CYMMapHbIM MOMeHTOM lIO qaco-
BOH C1'peJIKe H CYMMapHbIM MOMeHTOM lIpOTHB qaCOBOH C1'peJIKH.
F/la8a 2. HH:JICeHepHaJl cmamUKa
I
A- TO'lKa onopbl
npOTVlB "IaCOBO CTpeJ1KVI
(-)
no "IaCOBO CTpeJ1Ke
(+)
Pac. 2.11. MOMeHTbI no qaCOBOH H npOTHB qaCOBOH CTpeJIKH
2.4.3. npJfHqJfn MOMeHTOB
Arrre6paWleCKa.H CYMMa MOMeHTOB HeCKOJIbKHX CHJI, ,neHcTByroII(HX OTHOCHTeJIbHO
JII060H TOqKH, paBHa pe3YJIbTHpyrolI(eMY MOMeH1)' CHJIbI OTHOCHTeJIbHO 3TOH TOqKH.
MaTeMaTWleCKH 3TO MO)I{HO BbIpa3HTb CJIe,l(yIDI.I..(HM 06pa30M:
M R =Ml +M 2 +(-M3)+...+Mn'
r,ne: M R - pe3YJIbTHpyrolI(HH MOMeHT CHJIbI; M I , ..., M n - JII060e KOJIHqeCTBO MO-
MeHTOB, ,neHcTByroII(HX OTHOCHTeJIbHO O,nHOH H TOH xe TOqKH OlIOpbI.
IIoMHume. TIOJIO)I{HTeJIbHbIe MOMeHTbI 6y,nyr ,neHCTBOBaTb B HalIpaBJIeHHH lIO
qaCOBOH C1'peJIKe, Tor,na KaK OTpHuaTeJIbHbIe MOMeHTbI (TaKHe KaK -M 3 ) 6y.nyr ,neH-
CTBOBaTb B HalIpaBJIeHHH lIpOTHB qaCOBOH CTpeJIKH.
TIpHMeH.H.H BbIIIIeynOM.HHYTbIe lIpHHU;HlIbI K pbIqary Ha pHC. 2.12, a, BbIqHCJIHM:
1. BeJIHqHHY pe3YJIbTHpyrolI(eH CHJIbI.
2. BeJIHqHHY pe3YJIbTHpyrolI(erO MOMeHTa.
3. TIOJIOXeHHe JIHHHH ,neHcTBIDI pe3YJIbTHpyrolI(eH CHJIbI.
1. TIOCKOJIbKy JIHHHH ,neHcTBIDI lIpHJIOXeH-
HbIX CHJI, ,neHcTByroII(HX Ha pbIqar, lIOKa3aHHbIH
Ha pHC. 2.12, a, lIapaJIJIeJIbHbI o,nHa ,npyrOH, pe-
3YJIbTHpyrolI(asI CHJIa (F R ) MoxeT 6bITb lIo.rryqeHa
lIpOCTbIM CJIOXeHHeM H BbJqHTaHHeM:
2M
2M
F 1 = 25 H
F 2 = 40 H
. CHJIbI FI H F 2 CTpeM.HTCR lIepeMeCTHTb
pbIqar BBepx. OHH ,neHCTByroT B O,nHOM H
TOM xe HalIpaBJIeHHH, rr03TOM)' HX MOXHO
CJIO)I{HTb.
. CHJIa F3 ,neHcTByeT B lIpOTHBOlIOJIO)I{HOM
HalIpaBJIeHHH, CJIe,nOBaTeJIbHO, ee CJIe.nyeT a
BbIqeCTb H3 CYMMbI FI H F 2 .
TI03TOMY:
F R = FI + F 2 + F3 = 25 H + 40 H + 50 H = 15 H.
2. HaitneM BeJIHqHHY pe3YJIbTHpyrolI(erO MO-
MeHTa (M R ).
MOMeHTbI lIO qaCOBOH CTpeJIKe (+):
TO"lKa onopbl .
F 2 = 50 H
3 M
2M
6
F R = 15 H
M I = FI x (2 M + 2 M) = 25 H x 4 M = 100 HM.
Pac. 2.12. npHMeHeHHe npHHUH-
na MOMeHTOB
2.4. MOMeHmbl CUll, Z4eHmp mJI:JICecmu u Z4eHmpbl mJI:JICecmu n./loU(aoeu
M 2 = F 2 x 2 M = 40 H x 2 M = 80 HM.
MOMeHT lIpOTHB qaCOBOH CTpeJIKH (-):
M3 = F3 x 3 M = 50 H x 3 M = 150 HM.
Pe3YJIbTHpyrolI(HH MOMeHT:
M R = M I + M 2 - M3 = 100 HM + 80 HM - 150 HM = 30 HM.
3. QT06bI HaHTH lIOJIOXeHHe JIHHHH ,l(eHCTBIDI pe3YJIbTHpyrolI(eH CHJIbI (F R ),
06'be,l(HHHM pe3YJIbTaTbI 1 H 2. 06paTHMcR K pHC. 2.12, 6.
M R = F R x d = 30 HM.
II03TOMY:
30 HM = 15 H x d,
d = 30 HM/15 H = 2 M.
TaKHM 06pa30M, CHJIbI F I , F 2 H F3 MOryr 6bITb 3aMeHeHbI O,l(HOH CHJIOH F R ,
.neHcTByrolI(eH Ha paCCTO.HHHH 2 M OT TOqKH OlIOpbI. TIOCKOJIbKy M R lIOJIO)I{HTeJIeH,
CHJIa F R 6Y,l(eT lIepeMelI(aTbCR HJIH C1'peMHTbC.H lIepeMeCTHTb pbIqar B HalIpaBJIeHHH
no qaCOBOH C1'peJIKe, TaKHM 06pa30M, CHJIa F R 6Y,l(eT ,l(eHCTBOBaTb B HalIpaBJIeHHH
BBepx, KaK lIOKa3aHO Ha pHCYHKe.
B lIpHMepe H3 lIpe,l(bl,l(ylI(ero pa3,l(eJIa MOMeHTbI, ,l(eHcTByroII(He B lIpOTHBOlIOJIOX-
HbIX HanpaBJIeHIDIX, He 6billH paBHbI. B HTore 3TO lIpHBO,l(HJIO K TOMY, qTO pe3YJIb-
THpyrolI(asI CRJIa CTaHOBHJIaCb HJIH C1'peMHJIaCb CTaTb lIPHqHHOH B lI(eHIDI OTHOCH-
TeJIbHO ueH1'pa BpalI(eHIDI (TOqKH OlIOpbI).
Ha pHC. 2.13 lIOKa3aHbI lIpOCTbIe BeCbI, B qaIIIKax KOTOpbIX MaCCbI M I H M 2
paBHbI, TaK qTO BeCbI HaxO,l(HTCR B COCTORHHH paBHOBeCIDI, KOpOMbICJIO BeCOB - B
ropH30HTaJIbHOM lIOJIOXeHHH. TIOCKOJIbKy CHJIa TIDKeCTH ,l(eHcTByeT O,l(HHaKOBO Ha
M I H M 2 , CHJIbI FI H F 2 TaIOKe paBHbI. CHJIa peaKUHH OlIOpbI (R) paBHa CYMMe Ha-
npaBJIeHHbIX BHH3 CHJI.
CHJIbI paBHOY,l(aJIeHbI OT TOqKH OlIOpbI, TaK qTO lIOBOpaqHBalOlI(He MOMeHTbI TaK-
xe paBHbI. TIOCKOJIbKy lIOBOpaqHBalOlI(He MOMeHTbI ,l(eHcTByroT B lIpOTHBOlIOJIO)I{HbIX
HanpaBJIeHIDIX, 3TH MOMeHTbI TaIOKe HaxO,l(HTC.H B paBHOBeCHH. OHH ypaBHOBeIIIHBa-
lOT ,lij)yr ,l(pyra, H KOpOMbICJIO BeCOB He C1'peMHTCR lIoBepHyrbCR. 3TO MO)I{HO c<pop-
MyJIHpOBaTb CJIe,ayrolI(HM 06pa30M.
ECJIH TeJIO no,n; B03,n;eicTBHeM MHOrux CHJI
HaxO,ZJ;HTCJI B COCTOJlHHH paBHOBeCHJI (c6aJIaHCH-
pOBaHHoM COCTOJlHHH), TO OTHOCHTeJILHO JII060i
TO'lKH MOMeHTLI no 'IaCOBOH CTpeJiKe paBHLI MO-
MeHTaM npoTHB qacoBoi CTpeJiKH OTHOCHTeJILHO
TOH )Ke caMOH TO'lKH.
DOJIee KpaTKO MO)I{HO CKa3aTb TaK.
AJIre6paH'leCKaB cYMMa MOMeHTOB OTHOCH-
TeJlLHO JII060H TOqKH paBHa HyJIIO.
2.4.4. PSSHOSeCJfe
"
F 1
F 2
F R
Pac. 2.13. npOCTbIe BeCbI
(::, F/la8a 2. HH:JICeHepHf1Jl cmamUlCa
,lVrn TeJIa, HaxO,lUllI(eroc.H B COCTORHHH paBHoBecIDI, ,l(OJDKHbI BbIlIOJIH.HTbC.H CJIe-
,l(yK>II(He YCJIOBIDI.
. Arrre6paWlecKasI cYMMa ropH30HTaJIbHbIX CHJI, ,l(eHcTByroII(HX Ha TeJIO,
,l(OJDKHa paBH.HTbC.H H)'JIIO, L F H = O.
. Arrre6pawIecKasI cYMMa BepTHKaJIbHbIX CHJI, ,l(eHcTByroII(HX Ha TeJIO, ,l(OJDKHa
paBH.HTbCR H)'JIIO, L Fv = O.
. Arrre6paWlecKasI cYMMa MOMeHTOB CHJI, ,l(eHcTByroII(HX Ha TeJIO, ,l(OJDKHa paB-
H.HTbC.H H)'JIIO, LM = O.
Ha pHC. 2.14 lIOKa3aH pbJqar, 3a-
KpeTIJIeHHbIH B ueH1'pe Ha OCH. JIHHIDI
,l(eHCTBIDI CHJIbI F 1 CMelI(eHa OT ueH1'pa
Ha 2 M BJIeBO H C1'peMHTCR C006lI(HTb
Teny MOMeHT lIpOTHB qaCOBOH c1'peJI-
KH. JIHHIDI ,l(eHCTBIDI CHJIbI F 2 CMelI(e-
Ha OT UeH1'pa Ha 3 M BlIpaBO H c1'pe-
MHTC.H C006lI(HTb TeJIY MOMeHT lIO qa-
COBOH C1'peJIKe. ECJIH BeJIWlHHa F 2
paBHa 20 H, BbIqHCJIHM:
F 1
F 2 = 20 H
F R
Puc. 2.14. CHJIbI B paBHOBeCHH
1. BeJIWlHHY CHJIbI F 1 , He06xo,l(HMyID ,l(JUI Toro, qT06bI pbJqar HaxO,l(HJIC.H B co-
CTO.HHHH paBHOBeCIDI.
2. BeJIHqHH)' CHJIbI peaKUHH (F R ) B TOqKe OlIOpbI.
,lVrn paBHOBeCIDI:
MOMeHTbI lIO qaCOBOH C1'peJIKe = MOMeHTbI lIpOTHB qaCOBOH C1'peJIKH
(20 H x 3 M) = (F) x 2 M)
F 1 = (20 H x 3 M)/2 M = 30 H
CHJIa peaKUHH F R , ,l(eHcTByrolI(asI BBepx, paBHa lIO BeJIWlHHe H lIPOTHBOlIOJIO)l{-
Ha lIO HalIpaBJIeHHIO CYMMe CHJI F 1 H F 2 , ,l(eHcTByroII(HX BHH3. TI03TOMY:
R = F 1 + F 2 = 20 H + 30 H = 50 H
Ha pHC. 2.15 lIOKa3aHa lIpOCTasI 6aJIKa Ha ,l(BYX OlIOpax, paClIOJIOXeHHbIX C ,l(BYX
KOHUOB. TIOCKOJIbKy OHa HaxO,l(HTCR B paBHOBeCHH, BpalI(alOlI(HH MOMeHT OTCyrCT-
ByeT H CHJIbI peaKUHH OlIOp R A H R B MOryr 6bITb BbJqHCJIeHbI C lIOMOlI(blO lIPHHUH-
lIa MOMeHTOB. 3TO MOXHO BbIlIOJIHHTb, BbJqHCJI.H.H MOMeHTbI OTHOCHTeJIbHO KIDK;J;OH
H3 OlIOp lIOOqepe,l(Ho.
3M
4M
3 M
600 H
500 H
R A
R B
Puc. 2.15. BbIqHCJIeHHe CHJI peaKI(HH onopbI
2.5. Pa3H06uonocmu pbl'laZ06
QT06bI HaHTH BeJIHtlHHY R A , onpe,n;eJIHM MOMeHTbI OTHOCHTeJIbHO TOqKH npHJIo-
xeHIDI R B , C TeM qT06bI YC1paHHTb 31)' CHJIy H3 paCCM01peHIDI npH nepBOHaqaJIb-
HbIX BbIqHCJIeHIDIX. nOCJIe 3Toro OCTaeTC.H TOJIbKO o,n;Ha HeH3BeCTHasI BeJIHtlHHa,
KOTOpyro MO)I{HO BbJqHCJIHTb CJIe,n;yroII(HM 06pa30M:
MOMeHTbI no qaCOBOH C1peJIKe = MOMeHTbI npOTHB qaCOBOH C1peJIKH
R A x 10 M = (600 H x 7 M) + (500 H x 3 M)
R A x 10 M = 4200 HM + 1500 HM
R A x 10 M = 5700 HM
R A = 5700 HM/I0 M
R A = 570 H
AHaJIOrHtlHO, qT06bI HaRTH BeJIHtlHHY R B , onpe,n;eJIHM MOMeHTbI OTHOCHTeJIbHO
TO'IKH npHJIOXeHIDI R A , C TeM qT06bI YC1paHHTb 31)' CHJIy H3 paCCM01peHIDI npH
nepBOHaqaJIbHhIX BbIqHCJIeHIDIX. nOCJIe 3Toro OCTaeTC.H TOJIbKO o,n;Ha HeH3BeCTHasI
BeJIH1IHHa, KOTOpyro MO)I{HO BbJqHCJIHTb CJIe,n;yrom:HM 06pa30M:
MOMeHTbI no qaCOBOH c1peJIKe = MOMeHTbI npOTHB qaCOBOR C1peJIKH
(600 H x 3 M) + (500 H x 7 M) = R B x 10 M
1800 HM + 3500 HM = R B x 10 M
5300 HM = R B x 10 M
R B = 5300 HM/I0 M
R B = 530 H
nOCKOJIbKY CHCTeMa CHJI HaxO,n;HTC.H B paBHOBeCHH, CyMMa CHJI, HanpaBJIeHHbIX
BHH3, ,n;OJDKHbI 6bITb paBHa CyMMe CHJI, HanpaBJIeHHbIX BBepx.
CJIe,n;oBaTeJIbHo:
R A + R B = 600 H + 500 H
570 H + 530 H = 600 H + 500 H
1100 H = 1100 H
()
nOJIyqHBIIIeeC.H TO)l{,l(eCTBO no,n;TBep)l{,l(aeT KOppeKTHOCTb BbIqHCJIeHHH R A H R B . He
COIq>am:aHTe BbfqHCJIeHIDI, onpe,n;e.rnrn CHJIy pea HH TOJIbKO O,n;HOH onopbI H BbIqH-
TasI ee H3 CyMMbI CHJI, HanpaBJIeHHbIX BHH3. JII06asI oIIIH6Ka, BHeceHHasI B nepBO-
Haq8JIbHbIe BbIqHCJIeHIDI, nOBJIIDIeT Ha Bce ,n;aJIbHeHIIIHe paCqeTbI, H BbI.HBHTb ee 6y-
.neT HeB03MO)I{HO.
B BbIIIIenpHBe,n;eHHbIX npHMepax He ytlHTbIBaJIC.H Bec KOpOMbICJIa BeCOB H 6aJIKH.
Bec O,ZJ;HOpO,ZJ;HOH 6aJIK11 (KopOMLlCJIa) M01KllO paCCMaTpHBaTL KaK cocpe,n;OTO'leH-
nyIO CBJIY, JIHHHJI ,n;eacTBHJI KOTopoa npe,znIOJIaraeTCJI npoxo,ZJ;JIInea '1epe3 cepe,ZJ;HHy
,ZJ;JlllllLI 6UD.
2.5. Pa3HOBMAHOCTM pbl'larOB
PbIqarH no,n;pa3,n;eillIIOTC.H Ha pbJqarH nepBoro, BToporo H 1peTbero po,n;a (pHC. 2.16).
(QacTo BbI,ll;eillIlOT TOJIbKO pbJqarH nepBoro H BToporo po,n;a, npHHHMasI BO BHHMa-
HHe JIHIIIb B3aHMHoe pacnOJIOXeHHe TOqKH onopbI H TOqeK npHJIOXeHIDI CHJI. -
IIpUMe"t. nep.) B COOTBeTCTBHH C npHHnHnOM MOMeHTOB COOTHOIIIeHHe Me)l{){y Ha-
rpY3KOR (W), paCCTO.HHHeM OT HarpY3KH .no TOqKH onopbI (d 1 ), npHJIOXeHHOH CH-
JIOR (F) H paCCTO.HHHeM OT TOqKH npHJIOXeHIDI CHJIbI,n;O TOqKH onopbI (d 2 ) o,n;HHa-
F/la8a 2. HH:JICeHepHaR cmamU1ca
d 1
_1 4
.1
1 4
TO"lKa onopbl n pVlJ1araeMa51
HarpY3Ka (w) CVlJ1a (F)
a
I: .1
d 1 -I
npVlJ1araeMa51
CVlJ1a (F)
TO"lKa onopbl
6 HarpY3Ka (w)
d 1
npVlJ1araeMa51 CVlJ1a (F)
TO"lKa onopbl
HarPY3Ka (w)
B
PRC. 2.16. PblqarH:
a - nepBoro po,n;a;
6 - BToporo po,n;a;
B - TPeTbero po,n;a
KOBO ,l(.IDI Bcex pO.n;OB pbIqarOB H MoxeT 6bITb BbIpaxeHo MaTeMaTHqeCKH CJIe.nyro-
IUHM 06pa30M:
Wx d J = Fx d 2 .
CJIe.n;OBaTeJIbHO:
W=Fxd 2 /d l
H:
F= Wx d l /d 2 .
2.5. 1. P&llfSrJf nepsoro POAS
. TOtIKa orropbI HaxO,ll;HTC.H Mex.ay TOqKOH rrpHJIOXeHIDI CHJIbI H HarpY3KoH.
. I1pHJIaraeMasI CHJIa MeHbIIIe, qeM HarpY3Ka.
. TOqKa lIpHJIOXeHIDI CHJIbI rrpOXO,ll;HT 60JIbIIIHH rryTb, qeM HarpY3Ka.
. Pb:Nar lIepBoro pO,ll;a MO)I{HO paCCMa1pHBaTb KaK YBeJIHqHTeJIb CHJIbI.
2.6. I(eHmp tn.R:JICecmu, zeOMempU'IeCKUU 14eHmp tjJUZYPbl U pa8H08ecue
2.5.2. P&llfSrJf SToporo POAS
. TOtIKa rrpHJIOXeHIDI CHJIbI H HarpY3Ka pacrrOJIOXeHbI rro o,n;HY H TY xe CTOpO-
HY OT TOqKH orropbI, HO CHJIa ,n;eHcTByeT B HarrpaBJIeHHH, rrpOTHBOrrOJIO)I{HOM
HarpY3Ke.
. HarpY3Ka pacrrOJIOXeHa Me)l{){y TOqKOH rrpHJIOXeHIDI CHJIbI H TOtIKOM OrrOpbI.
. I1pHJIaraeMa.H CHJIa MeHbIIIe, qeM HarpY3Ka.
. TOqKa rrpHJIOXeHIDI CHJIbI rrpoxo,nHT 60JIbIIIHM rryrb, tIeM HarpY3Ka.
. PbItIar BToporo po,n;a MO)I{HO paCCMaTpHBaTb KaK YBeJIHqHTeJIb CHJIbI.
2.5.3. P&llfSrJf TpftT&erO POAS
. TOqKa rrpHJIO}l(eHIDI CHJIbI pacnOJIOXeHa Me)l(,l(y HarpY3KoM H TOtIKOH OnOpbI.
. I1pHJIaraeMaH CHJIa 60JIbIIIe, qeM HarpY3Ka.
. HarpY3Ka rrpoxo,n;HT 60JIbIIIHH rryTb, qeM TOqKa rrpHJIOXeHIDI CHJIbI.
. PbIqar 1'peTbero po,n;a MO)I{HO paCCMa1'pHBaTb KaK YBeJIHqHTeJIb paCCTO.HHIDI.
2.6. LJ.eHTp TSI>KeCTM, reOMeTpM'IeCKMM LJ.eHTp cl>Mrypbl
M paBHOBeCMe
2.6.1. LleHTp TflJlCeCTJf (TsepAoe Teno)
fpaBHTan;HOHHoe rrpHTIDKeHHe ,n;eHcTByeT Ha Bce qaCTHn;bI TBep,n0F TeJIa, )I{H,l(KO-
cm HJIH ra3a, KaK rrOKa3aHO Ha pHC. 2.17. I1pe,n;rrOJIaraeTCH, qTO JIHHHH ,neHcTBIDI
CRJI rrpHTIDKeHIDI OT,n;eJIbHbIX qaCTHn; rrapaJIJIeJIbHbI. ,l(aJIee rrpe.n;rrOJIaraeTC.H, qTO
3TH OT,n;eJIbHbIe CHJIbI rrpHTIDKeHIDI rrpH BbJqHCJIeHIDIX MO)I{HO 3aMeHHTb O,n;HOH pe-
3YJIbTHpyrom:eH CHJIOH, paBHoM CyMMe CHJI rrpHTIDKeHIDI OT,neJIbHbIX qaCTHn;. U;eH1'p
TIDKeCTH (G) TBep,n;oro TeJIa - 3TO TOqKa rrpHJIOXeHIDI pe3YJIbTHpyrom:eH rpaBHTa-
n;HOHHOH CHJIbI, T.e. TOtIKa rrpHJIOXeHIDI Beca
TeJIa. HarrpHMep, n;eH1p TIDKeCTH n;HJIHH,lI;pa C
rrOCTOHHHbIM rrOrrepeqHbIM CeqeHHeM JIe)I{HT
Ha ero OCH rrocepe,n;HHe Me)l(,l(y ,n;ByM.H OCHO-
BaHIDIMH. 3TOT H HeKOTopbIe ,n;pyrHe rrpHMe-
pbI rrpe,n;CTaBJIeHbI Ha pHC. 2.18. (nOIDITHe
n;eH1pa TIDKeCTH HMeeT CMbICJI TOJIbKO ,l(.IDI
TBep,n;oro TeJIa, HaXO,lUIm:eroc.H B O,n;HOpO,n;HOM
rrOJIe TIDKeCTH; ,l(.IDI rrpOH3BOJIbHOH MeXaHHqe-
CKOH CHCTeMbI BBO,n;HTC.H rrOIDITHe l(eH1pa
HHepn;HH, He CBH3aHHoe C CHJIOBbIM rrOJIeM.
,lLJm TBep,n;oro TeJIa rrOJIOXeHIDI n;eH1pa TIDKe-
cm H n;eHTpa HHepn;HH COBrran;aIOT. - IIpu- PRC. 2.17. l(eHTP TH)I{eCTH
Me . nep.)
\. \
JlIiIHIiISI, CTBIiISI_
pe3YJ1 bTVI PYIO w.e VI
CVlJ1b1 (Bec W)
CVlJ1a, AeVl-
CTBYIOw.a51
Ha "IaCTVlu.bl
w
F/la8a 2. HH:JICeHepHaR cmamU1ca
opMa TBep oro TeJIa
nMo.eDe a (
(pacCTOJlHHe x, n0K838HHoe
Ha pHCyuKe)
061.eM
IJ;HJIHH.D:P
h(2
1tr 2 h
C<pepa
R
41tr 3 /3
lloJIyc<pepa
3r,/8
21tr 3 /3
KoHYc
h
h/4
1tr 2 h /3
llHpaMH,lI;a
h/4
abh /3
Pac. 2.18. llOJIOXeHHe I(eHTPa TH)I{eCTH (G) ,lI;JIH HeKOTopbIX 06w;epacnpOCTPaHeH-
HbIX TBep,n;bIX TeJI
2.6.2. LleHTp TflJlCeCTJf HeOAHOpOAH&IX
Jf COCTSSH&IX TSepA&IX Ten
Ha pHC. 2.19 rrOKa3aH n;HJIHH,D;pWIeCKHH BaJI C ,n;BywI ,n;HaMe1paMH:
. G} - n;eH1p TIDKeCTH n;HJIHH,D;pa C 60JIbIIIHM ,n;HaMe1pOM, OH pacrrOJIOXeH Ha
OCH rrocepe,n;HHe Me)l(,l(Y OCHOBaHIDIMH 3Toro n;HJIHH,D;pa;
. G 2 - n;eH1p TIDKeCTH n;HJIHH,lI;pa C MeHbIIIHM ,n;HaMe1pOM; OH pacrrOJIOXeH Ha
OCH rrocepe,n;HHe Me)l(,l(Y OCHOBaHIDIMH 3Toro n;HJIHH,lI;pa;
. G - n;eH1p TIDKeCTH COCTaBHoro TBep,n;oro TeJIa KaK e,n;HHoro IJ;eJIoro; OH pac-
rrOJIOXeH Ha paCCTOHHHH d OT 6a30Boro TOpn;a BaJIa. Ero rrOJIOXeHHe MO)I{HO
orrpe,n;eJIHTb, rrpHMemrn rrpHHn;Hrr MOMeHTOB.
1. CYMMapHbIH Bec BaJIa paBeH CyMMe BeCOB ,n;BYX IJ;HJIHH,lI;pOB. To eCTh 10 KH +
+ 5 KH = 15 KH.
2.6. I(eHmp mH:JICecmu, zeOMempU'IeCKUU u,eHmp rjJUrypbl U pa8H08ecue
2M 2M
G1 G
--4----.- ----4---
1 M
10 KH
d
( 1 0 + 5) KH
5 KH
3M
PRC. 2.19. Onpe,n;eJIeHHe I(eH1'pa TH)I{eCTH Heo,n;Hopo,n;Horo TBep,n;oro TeJIa
2. CYMMapHbIH Bec B 15 KH 6y,n;eT npHJIOXeH B n;eH1pe TIDKeCTH G BaJIa, KaK
e)];HHOrO n;eJIoro, Ha HeKOTopOM paCCTOHHHH d OT TOtIKH OTCtIeTa P Ha 6a30BOM
TOpn;e B8JIa, OTHOCHTeJIbHO KOTOpOH onpe,n;eillIIOTC.H MOMeHTbI.
3. B 3TOM BbIqHCJIeHHH YCJIOBHO npe,n;nOJIaraeTCH, qTO MOMeHT G OTHOCHTeJIbHO
roqKH P npOTHBOnOJIOXeH no HanpaBJIeHHIO MOMeHTaM G l H G 2 , XOT.H Ha caMOM
)];eJIe Bce OHH ,n;eHCTByroT no qaCOBOH C1peJIKe.
MOMeHTbI no qaCOBOH C1peJIKe = MOMeHTbI npOTHB qaCOBOH C1peJIKH
(G} x 1 M) + (G 2 x 3 M) = (G x d)
(10 KH x 1 M) + (5 KH x 3 M) = (15 KH x d)
10 KHM + 15 KHM = 15 KH x d
25 KHM = 15 KH x d.
(',
II03ToMY:
d = 25 KHM/15 KH = 1,67 M.
TaKOH xe MeTO,n; MO)I{HO HCnOJIb30BaTh ,l(.IDI COCTaBHbIX TBep,n;bIX TeJI, eCJIH
TIJIOTHOCTH HX pa3JIHqHbIX 3JIeMeHTOB OTJIH'laIOTCH.
2.6.3. LleHTp TflJlCeCTJf (TOHKSfI nnSCTJfHs)
ECJIH 3JIeMeHTbI H3 TOHKoro JIHCTOBOro MeTaJIJIa HJIH xeCTKoro JIHCTOBOro IIJIaCTH-
Ka HMelOT nOCTOHHH)'IO TOJIIUHHY (ToHKHe IIJIaCTHHbI) H 3Ta TOJIIUHHa He3HatIH-
TeJIbHa no cpaBHeHHIO C IIJIOIUaL(blO, TO MO)I{HO nOJIaraTb, qTO HX n;eH1p TIDKeCTH
pacnOJIOXeH B n;eH1paJIbHOH TOqKe IIJIOII(aL(H.
. U;eH1p TIDKeCTH G np.HMoyrOJIbHOH TOHKOH IIJIaCTHHbI pacnOJIOXeH B TOqKe
nepeCeqeHIDI ,n;HarOHaJIeH IIJIaCTHHbI.
. U;eHTp TIDKeCTH G KpyrJIOM TOHKOH IIJIaCTHHbI pacnOJIOXeH B n;eH1pe Kpyra.
3-3255
F/la8a 2. HH:JICeHepHaJI cmamU1ca
2.6.4. reOMeTpJflfeCKJfe qeHTp&1 cpJfryp
I10IDITHe reOMe1pHqeCKOrO ueH1'pa OTHOCHTC.H K ,n;ByMepHOH IIJIOCKOH <pHrype, He
HMelOIUeti TOJIII(HHbI. Y Hee eCTb TOJIbKO IIJIOIUaLJ:b. TaK KaK IIJIOIUaLJ:b He HMeeT
TOJIIUHHbI, Y Hee HeT MaCCbI, Ha KOTOpyro MOrJIa 6bI ,n;eHCTBOBaTb CHJIa rpaBHTan;HH.
CJIe,n;OBaTeJlbHO, Y Hee HeT Beca. TepMHH «reOMe1pHtleCKHH n;eHTp» MO)I{HO HC-
rrOJIb30BaTb rrpHMeHHTeJIbHO K IIJIOIUaLJ:H ,n;ByMepHOH IIJIOCKOti <pHrypbI, aHaJIOrHtlHO
TOMY, KaK rrOIDITHe n;eH1'pa TIDKeCTH HCrrOJIb3yeTC.H ,l(.IDI TBep,nbIX IIJIaCTHH. I103TOMY
reOMe1pHqeCKHH n;eH1p <pHrypbI - 3TO TOqKa Ha IIJIOCKOH <pHrype, B KOTOpOti, KaK
rrpe,n;rrOJIaraeTCH, cocpe,n;OTOqeHa BC.H IIJIOIUaLJ:b. KaK H n;eH1p TIDKeCTH, reOMe1pHtle-
CKHH n;eH1'p <pHrypbI TaIOKe 0603HaqaeTC.H 6YKBOH G. I1pHMepbI rrOJIOXeHHH reo-
Me1pHqeCKHX n;eH1'pOB ,l(.IDI HeKOTopbIX <pHryp rrpHBe,n;eHbI Ha pHC. 2.20.
opMa WIOUWUI
a
KBa,npaT
a
IIpHMoyrOJIbHHK
Kpyr
IIoJIyKpyr
.c::
IIPHMOyrOJIbHbIH TpeyrOJlbHHK
b
PRC. 2.20. feOMeTpHqeCKHe I(eHTpbI <pHryp
2.6.5. PSSHOSeCJfe
PaCCTOJlHHe x PaCCTOJlHHe y lIJIoma,zu.
a/2
a 2
a/2
a/2
b/2
ab
r
1tr 2
r
4r/3n
nr 2 /2
r
b/3
h/3
bh/2
nOH.HTHe paBHOBeCIDI 6billO BBe,n;eHO B pa3,n;eJIe 2.4.4. Ha pHC. 2.21, a rrOKa3aH Me-
TaJIJIHqeCKHH CTepxeHb C rrOCTO.HHHbIM CeqeHHeM, 3aKpeIIJIeHHbIH Ha OCH, rrpoxo-
,lUIIUeH qepe3 ero n;eH1p, TaK, qTO CTepxeHb MoxeT cBo6o.ll;HO BpaIUaTbC.H, a TaIOKe
2.6. I(eHmp mR:JICecmu, zeOMempU'IeCKUU u,eHmp rjJU Pbl u pa8H08ecue
1 4
_I. 4
p
d 1
-I
o
/......
/; / )
I I
I I
I I
I I
Y I I
/ f ll
I I
II G
P
o
o
OCb Bpa-
w.eHVlS1
M 1
M 2
G
a
6
PRC. 2.21. PaBHoBecHe:
a - 6e3pa3JIHqHOe HJIH YCTOHqHBOe paBHoBecHe;
6 - HeycTOHqHBOe paBHoBecHe;
B - YCTOHqHBOe paBHOBeCHe
/
/
/ /1
/ /
// 1 //
/ /
\ //G
/ /
/0/
l.. /
....j
B
nOKa3aHO, qTO MOMeHTbI Bpam:eHIDI, ,n;eHcTByrom:He Ha K oe IIJIeqo CTep)l{ID(
BCJIe,n;CTBHe CHJIbI TIDKecTH, paBHbI rro BeJIHtlHHe H rrpoTHBOrrOJIO)I{HbI rro HarrpaBJIe-
HHIO. I103TOMY H,ZJ;eaJIbHO ypaBHoBeIIIeHHbIH cTepxeHb OCTaHeTCH B JII060M rrOJIO-
xeHHH, B KOTopoe 6y,n;eT rroBepHYT. B TaKOM cJIyqae rOBopHT, qTO cTepxeHb Haxo-
)];HTCH B COCTO.HHHH 6e3pa3JIHtlHOrO paBHoBecIDI.
Ha pHC. 2.21, 6 TaKoH xe CTepxeHb 3aKpeIIJIeH Ha OCH, He rrpOXO,lUIm:eH qepe3
ero n;eH1p, TaK, qTO O,n;HO IIJIeqo LUIHHHee, qeM ,n;pyroe. npH BepTHKaJIbHOM rrOJIO-
xeHHH CTep)l{ID(, KaK rrOKa3aHO Ha pHC. 2.21, 6, JIHHIDI ,n;eifcTBIDI BeKTO rrpe,n;-
CTaBJUIIOm:ero CHJIy TIDKeCTH, rrpoxo,n;HT qepe3 TOqKY orropbI H MOMeHT BpalI(eHIDI
OTcyrcTByeT. O,n;HaKo Terrepb CTepxeHb HaxO,n;HTC.H B COCTO.HHHH HeycTOHqHBOrO
paBHOBeCIDI, TaK KaK MaJIeHIIIHH rrOBopOT CTep)l{ID( rrpHBe,n;eT K CMem:eHHIO n;eH1'pa
TIDKeCTH B CTOPOHY OT BepTHKaJIH, KaK rrOKa3aHO Ha pHCyHKe. B pe3YJIbTaTe B03-
HHKHeT MOMeHT Bpam:eHIDI H CTepxeHb rrpH,ZJ;eT B ,n;BIDKeHHe, KOTopoe 6y,n;eT rrpo-
)];OJl)l{aTbC.H ,n;o Tex rrop, rrOKa n;eH1p TIDKeCTH 60JIee LUIHHHoro IIJIetIa He OKIDKeTC.H
Herrocpe,n;CTBeHHO rro,n; TOqKOH OrrOpbI. Tor,n;a MOMeHT Bpam:eHIDI HCqe3HeT H 6aJIKa
OCTaHOBHTC.H.
Ha pHC. 2.21, B rrOKa3aH CTepxeHb, y KOToporo n;eH1'p TIDKeCTH 60JIee LUIHHHO-
ro TIJIeqa pacrrOJIOXeH TOqHO rro,n; TOtIKOM OrrOpbI. CJIe,n;oBaTeJIbHO, MOMeHT BltClm:e-
u \ /
HID! OTcyrcTByeT, CTepxeHb He rro,n;BepraeTCH ero B03,n;eHCTBHIO H OCTaeTC.H a no-
)];BH)I{HbIM. Ha pHCyHKe TaIOKe rrOKa3aHO, qTO rrpH JII060M OTKJIOHeHHH CTep)l{ID( OT
BepTHKaJIH C03,n;aeTCH MOMeHT, rrpHBO,lUIm:HH CTepxeHb B ,n;BIDKeHHe, KOTopoe 6y,n;eT
npo,n;OJl)l{aTbCH ,n;o Tex rrop, rrOKa n;eH1p TIDKeCTH 60JIee LUIHHHoro IIJIeqa CHOBa He
OKIDKeTCH Herrocpe,n;CTBeHHO rro,n; TOqKOH orropbI. I103TOMY B TaKOM CJIyqae
rOBopHT, tITO CTepxeHb Haxo,n;HTCH B COCTO.HHHH YCTOHqHBOrO paBHOBeCIDI.
IIpuMe"taHue.
I1pH MaJIOM B03MYlI(eHHH TeJIO Bcer,n;a rrepexo,n;HT HJIH C1'peMHTCH lIepeHTH 113
COCTOHHHH HeYCTOHqHBOrO paBHOBeCIDI B COCTOHHHe YCTOHqHBOrO paBHOBeCID!.
F/la8a 2. HH:JICeHepHaR cmamU1ca
2.7. TpeHMe
BC.HKHH pa3, Kor,n;a O,n;HO TeJIO CKOJIb3HT HJIH KaTHTC.H rro ,n;pyrOM)', eM)' rrpOTHBO-
,n;eHcTByeT CHJIa COrrpOTHBJIeHIDI, KOTOpasI Ha3bIBaeTC.H CHJIOH 1peHIDI CKOJIbXeHIDI
(,n;BIDKeHIDI). ,l(axe Kor,n;a ,n;BIDKeHIDI He rrpOHcxo,n;HT, Ha TeJIO ,n;eHcTByeT CKpbITasI
CHJIa corrpOTHBJIeHIDI, CHJIa 1peHIDI rrOKOH. TpeHHe rrOKO.H, KaK rrpaBHJIO, 60JIbIIIe,
qeM 1peHHe CKOJIbXeHIDI. ,l(.mI rrpeo,n;OJIeHIDI 1peHIDI rrOKOH Me)l{){y ,n;BywI ,n;aHHbIMH
MaTepHaJIaMH H HaqaJIa ,n;BIDKeHIDI H}')I{Ha 60JIbIIIasI CHJIa, qeM ,l(.IDI rrO,LUJ;epXaHIDI
,n;BIDKeHIDI C rrOCTO.HHHOH CKOpOCTblO.
2.7. 1. CMS3KS
TpeHHe Me)l{){y rrOBepXHOCT.HMH CKOJIbXeHIDI HJIH KaqeHIDI H H3HOC rroBepXHocTeH
MO)I{HO cyw;eCTBeHHO yMeHbIIIHTb, eCJIH o,n;HY rrOBepXHOCTb MaTepHaJIa OT,n;eJIHTb OT
,n;pyrOH cooTBeTcTByroIUeH IIJIeHKOH CMa3KH. ,l(.mI CMa3KH 6bIC1p0 ,n;BIDKYID:HXC.H rro-
BepXHocTeH, rro,n;BepraIOIUHXC.H OTHOCHTeJIbHO MaJIbIM HarpY3KaM, 06bIqHO HCrrOJIb-
3yroTC.H MaCJIa, a ,l(.IDI CMa3KH Me,n;.rreHHO ,n;BIDKYID:HXC.H rroBepXHocTeH, rro,n;BepralO-
IUHXCH 60JIbIIIHM HarpY3KaM, KaK rrpaBHJIO, HCrrOJIb3yroTC.H )l{HpbI. ECJIH 3a30p Mex-
l1.Y rrOBepXHOCT.HMH 3arrOJIHeH, TO cyxoe 1peHHe YC1paHeHO, H rrpeo,n;OJIeBaTb rrpHXO-
,n;HTCH TOJIbKO )l{H)];KOCTHoe 1peHHe, T.e. BHYTPeHHee 1peHHe Te eH cpe,n;bI, HJIH
B.H3KOCTb CMa3KH. CMa3Ka He TOJIbKO yMeHbIIIaeT H3HOC H 1peHHe, HO TaIOKe 3aIUH- I
IUaeT CKOJIb3.HIUHe rrOBepXHOCTH OT KOpp03HH H yMeHbIIIaeT TeIIJIOBbI,lI;eJIeHHe, a
3HaqHT, rrOTepH 3HeprHH. JII060e BbI,ll;eillIlOIUeeC.H TeIIJIO pacceHBaeTC.H CMa3KOH.
2. 7.2. 3SKOH&1 TpeHlffl
Ha BeJIHqHHY 1peHIDI Me)l{){y ,n;BywI rrOBepXHOCT.HMH CKOJIbXeHIDI HJIH KaqeHIDI
BJIIDIIOT CJIe,n;yroIUHe <paKTOpbI.
. C1pYKTYPa corrpHKaCalOIUHXC.H rroBepXHocTeH. fJI8)];KOCTb rroBepXHocTeH
yMeHbIIIaeT 1peHHe; HepOBHOCTH YBeJIHtlHBalOT 1peHHe.
. BH,lI; MaTepHaJIOB, HaxO,lUIIUHXC.H B KOHTaKTe (HarrpHMep, CTaJIb CKOJIb3HT rro
HeHJIOHY HaMHoro JIerqe, qeM rro pe3HHe).
. BeJIHqHHa HOpMaJIbHbIX (HarrpaBJIeHHbIX rro,n; rrp.HMbIM yrJIOM) CHJI rrpOTHBO-
,n;eHCTBIDI Me)l{){y rrOBepXHOCT.HMH.
. TpeHHe He 3aBHCHT OT IIJIOIU8)];H.
2.7.3. KoacpcpJfq1AeHT TpeHJffl
K03<P<P H D;HeHT 1peHIDI (J..1) ,l(.IDI rrapbI rroBepXHocTeH, HaxO,lUIII(HXC.H B KOHTaKTe,
orrpe,n;eillIeTC.H BbIpaxeHHeM:
J..1 = (cHJIa 1peHIDI Fj)/(cHJIa HOpMaJIbHOH peaKUHH orropbI Fn).
3TO rrOKa3aHO Ha pHC. 2.22. nOCKOJIbKY Fj H Fn BbIpaxalOTC.H B O,n;HHX H Tex xe
e,n;HHHD;ax H3MepeHIDI, HX OTHOIIIeHHe J..1 OKa3bIBaeTC.H 6e3pa3MepHOH BeJIHtlHHOH.
2. Z TpeHue 69'
CJIY'laH 1.
BJIOK BeCOM W T.HHyr no ropH30H-
TaJIbHOH nOBepXHOCTH, npHKJIaL(bIBasI
CHJIy BeJIHtlHHOH Fa, KaK nOKa3aHO Ha
pHC. 2.22. ECJIH 6JIOK ,n;BIDKeTC.H C no-
CTOHHHOH CKOpOCTblO HJIH Heno,n;BIDKeH,
TO Ha Hero ,n;eHcTByeT CHJIa conpOTHBJIe-
HID!, paBHasI no BeJIHqHHe H npOTHBO-
nOJIO)I{HasI no HanpaBJIeHHIO npHJIOXeH-
HOH CHJIe Fa. 3Ta CHJIa conpOTHBJIeHIDI
H eCTb CHJIa 1peHIDI Fj. CHJIa 1peHIDI
Bcer,n;a HanpaBJIeHa B CTOpOHY, npOTH-
BOnOJIO)I{Hyro ,n;BIDKeHHIO. TaKHM 06pa-
30M, B 3TOM c.rryqae:
Bec (W)
CVlJ1a Tpe-
HVl5I (F,)
npVlJ10)l(eH-
Ha51 CVlJ1a (Fa)
CVlJ1a HopMaJ1bHOVI
peaKu.VIVI onopbl (F n )
PRC. 2.22. TpeHHe
J.1 = FJFn = Fa/
. ECJIH CHJIa 1peHIDI paBHa npHJIOXeHHOH cHJIe, 6JIOK Heno,n;BIDKeH HJIH paBHo-
MepHo ,n;BIDKeTC.H H ,n;eHcTByroIUHe Ha Hero CHJIbI HaxO,ll;HTCH B paBHOBeCHH.
. ECJIH 6JIOK ,n;BIDKeTC.H C YCKopeHHeM, TO npHJIOXeHHasI CHJIa 60JIbIIIe, qeM
CHJIa 1peHIDI.
. CHJIa 1peHIDI MoxeT 6bITb MeHbIIIe HJIH paBHa npHJIOXeHHOH CHJIe. OHa HH-
Kor,n;a He MoxeT 6bITb 60JIbIIIe npHJIOXeHHOH CHJIbI. To eCTb CHJIa Fj paBHa
HJIH MeHbIIIe Fa H Fn paBHa HJIH MeHbIIIe
C.nyqaH 2.
ECJIH CHJIa (Fa) npHJIOXeHa K CKOJIb3.HIUeMy 6JIOKY no,n; yrJIOM 0, KaK nOKa3aHO
Ha pHC. 2.23, Tor,n;a CHJIa 1peHIDI Fj 6y,n;eT paBHa no BeJIHqHHe H npoTHBOnoJI6XHa
no HanpaBJIeHHIO ropH30HTaJIbHOH cocTaBillIlOIUeH CHJIbI (F h ), r,n;e Fh = Fa cosO.
BepTHKaJIbHasI COCTaBillIIOIUa.H CHJIbI Fv = Fa sinO. 3Ta CHJIa HanpaBJIeHa npOTHBO-
nOJIO)I{HO Becy W H, CJIe,n;OBaTeJIbHO, yMeHbIIIaeT HOpMaJIbH)'lO CHJIy pea HH ono-
pbI Fn. n03TOMY:
J.1 = Ff / Fn = (Fa cosO) / (W - Fa sinO).
TaK, eCJIH npHJIOXeHHasI CHJIa Fa ,n;eHcTByeT BHH3 no OTHOIIIeHHIO K ropH30HTa-
JIbHOH nOBepXHOCTH, Tor,n;a Fh 6y.neT YBeJIHtlHBaTb CHJIy HOpMaJIbHOH pea HH ono-
phI Fn. n03TOMy B 3TOM CJIyqae:
J.1 = Ff / Fn = (Fa cosO) / (W + Fa sinO).
Bec (W)
CVlJ1a TpeHVl5I (F,)
npVlJ10)l(eHHa51
CVlJ1a (Fa)
(
"- /;
BepTVlKaJ1bHa51 COCTaBJ151-
lOw.a51 CVlJ1bl Fv = Fa sine
Fn
PRC. 2.23. TpeHHe - CHJIa npHJIOXeHa no,n; yrJIOM
F/la8a 2. HH:JICeHepHaR cmamU1ca
B Ta6JIHD;e 2.3 rrpHBe,n;eHbI HeKOTopbIe THrrHqHbIe 3HaqeHIDI K03<P<PHD;HeHTa
1peHIDI ,lUIH pa3JIHqHbIX COqeTaHHM rroBepxHocTeH CKOJIbXeHIDI.
Ta6JI. 2.3. THrrHqHbIe 3HaqeHIDI K03<P<PHD;HeHToB TpeHIDI.
MarrepHaJIhI (cyxHe) p. MarrepH8JIhI (cyxHe) p.
CTaJIb no qyryHy 0,20 T OpM03HasI KOJIo,n;Ka no qyryHy 0,60
CTaJIb no JIa1)THH 0,15 Pe3HHa no aC<paJIbTy 0,65
qyryH no JIa1)THH 0,15 Pe3HHa no 6eTOHY 0,70
Koxa no qyryHy 0,55
2.7.4. Yron TpeHJffl
I10CKOJIbKy 06e CHJIbI Fj H Fn rrpHJIOXeHbI K CKOJIb3HIUeMY
6JIoKY, KOTOpbIH HaxO,n;HTCH B KOHTaKTe C ropH30HT8JIb-
HOM rrOBepXHOCTblO, B pe3YJIbTaTe BeKTOpHOrO CJIOXeHIDI
3THX CHJI, KaK rrOKa3aHO Ha pHC. 2.24, MO)I{HO rro.rrytIHTb
pe3YJIbTHpyrolI(ylO CHJIy peaKIJ;HH rrOBepXHOCTH Fro
TaK KaK CHJIbI Fj H Fn HarrpaBJIeHbI rro,n; rrp.HMbIM yr-
JIOM o,n;Ha K ,n;pyrOH:
tgq> = Ff / Fn,
HO
fl = Ff / Fn,
CJIe,n;OBaTeJIbHO:
tgq> = fl
YroJI <I> ,n;OCTHraeT MaKCHMaJIbHOrO 3HaqeHIDI rrpH
MaKCHMaJIbHOM 3HaqeHHH Fj. 3TO MaKCHMaJIbHOe 3Haqe-
HHe <I> Ha3bIBaeTCH yrJIOM 1peHIDI.
2.7.5. TpeHJfe HS HSKnOHHOH nnOCKOCTJf
Orrpe,n;eJIHM CHJIY Fa, lIpHJIOXeHH)'IO K TeJIY
BeCOM W, KOTOpas! T.HHeT ero BBepx rro IIJIOC-
KOCTH, HaKJIOHeHHOH rro,n; yrJIOM 0 K ropH-
30HT8JIH, eCJlH Fa ,n;eHcTByeT rro,n; yrJIOM <I> K
HaKJIOHHOH IIJIOCKOCTH, KaK rrOKa3aHO Ha
pHC. 2.25.
CHJIa Fa ,n;eHcTByeT rro,n; yrJIOM <I> K IIJIOC-
KOCTH, KaK rrOKa3aHO Ha pHC. 2.25, H Fn -
HOpMaJIbHas! peaKIJ;ID1 IIJIOCKOCTH, ,n;eMCTByro-
IUas! Ha TeJIO. I1pHpaBHHBaH KOMrrOHeHTbI
CHJI, lIeprreH,ll;HKyJI.HpHbIe HaKJIOHHOM IIJIOC-
KOCTH, rrOJIyqHM:
Fn + Fa sinq> = W cosO
Fn = W cosO - Fa sinq>
Fr
Fn
F,
PRC. 2.24. YroJI 1'peHHH
w
"
PRC. 2.25. TpeHHe Ha HaKJIOHHOH
nJIOCKOCTH:
Fa - npHJIOXeHHasI CHJIa; Fn - HOp-
MaJIbHaH peaKI(HH onopbI; Fj - npo-
THBo,n;eHCTBYIOw;aH CHJIa TpeHHH; W-
Bec TeJIa; fl - K03<P<PHI(HeHT 1'peHHH
2. 7. TpeHue 71
CHJIa TPeHIDI, lIpoTHBo,n;eHCTBYIOIUa.H ,n;BJDKeHHIO TeJIa, paBHa:
Ff = JJFn = J.l{W cose - Fa sin<p) (1)
IIpHpaBHHBa.H KOMlIOHeHTbI CHJI, lIapaJIJIeJIbHbIe HaKJIOHHOH IIJIOCKOCTH, lIO.rry-
qHM:
Fa COs<p = Ff + W sine
IIocJIe lIo,n;CTaHOBKH Fj H3 ypaBHeHIDI (1) lIo.rryqaeM:
Fa cos<p = J.l{W cose - Fa sin<p) + W sine
Fa cos<p = W(f.lCOSe + sin<p) - JJFa sin<p
Fa(f.l sin<p + cos <p) = W(f.lCOS e + sin<p)
Fa = [W(f.lCOSe + sin<p)] / [(f.lsin<p + cos<p)]
(2)
1. ECJIH Fa lIpHJIOXeHa lIapaJIJIeJIbHO IIJIOCKOCTH, TO <p = O. CJIe,n;OBaTeJIbHO,
sin <p = 0 H COS <p = 1. IIo,n;cTaBJIIDI 3TH 3HaqeHIDI B ypaBHeHHe (2), lIOJIyqHM:
Fa = [W(f.lCOS e + sine)] / (1 + 0)
Fa = W(f.lCOSe + sine)
2. ECJIH Fa lIpHJIOXeHa ropH30HTaJIbHO, TaK qTO OHa lIOMoraeT TeJIy,n;BHraTbC.H
BBepx lIO IIJIOCKOCTH, TO <p = -e. CJIe,n;OBaTeJIbHO:
cos <p = cos( -e) = cos e H
sin<p = sin(-e) = -sine
IIo,n;cTaBJI.H.H B ypaBHeHHe (2):
Fa = [W(f.lCOSe + sine)] / (cose + sine)
)
2.7.6. Yron TpeHJffl nOKOfl
TeJIo, HaxO,lUIIUeeC.H Ha IIIepoxoBaToH IIJIOCKOCTH, paClIOJIOXeHHOH lIO,n; yrJIOM e
K ropH30HTaJIH, KaK lIOKa3aHO Ha pHC 2.26, Haxo,n;HTC.H B COCTO.HHHH paBHOBeCIDI,
eCJIH rpaBHTaUHOHHa.H CHJIa, BbIffY)K,lI;alOIUa.H TeJIO CKOJIb3HTb BHH3 lIO HaKJIOHHOH
w
,
..
,# .. ..
W sin 8
/
" I\.
PRC. 2.26. YrOJI TpeHHH IIOKOH
F/la8a 2. HH:JICeHepHQJl cmamUica
IIJIOCKOCTH, ypaBHoBeIIIHBaeTC.H paBHoH lIO BeJIHqHHe H lIpOTHBOlIOJIO)I{HOH lIO Ha-
lIpaBJIeHHIO CHJIOH TPeHIDI, ,n;eHcTByrolI(eH BBepx lIO HaKJIOHHOH IIJIOCKOCTH. YroJI
HaKJIOHa IIJIOCKOCTH, lIPH KOTOpOM BbIlIOJIIDIIOTC.H YCJIOBIDI paBHOBeC:mI, olIpe,n;eIDI-
eTC.H Y2.ll0M mpeHUJl nOKOJl.
fpaBHTaUHOHHyIO cHJIY, ,n;eHCTBYIO Ha TeJIO (W), MO)I{HO pa3JIOXHTb Ha co-
cTaBIDIIOII(He; lIPH 3TOM COCTaBJI.HIOII(a.H, lIeplIeH,nHKYIDIpHa.H HaKJIOHHOH IIJIOCKO-
CTH, 6y,n;eT paBHa W cos 9, H COCTaBIDIIOII(a.H, lIapaJIJIeJIbHa.H HaKJIOHHOH IIJIOCKOCTH,
6y,n;eT W sin 9. B YCJIOBIDIX paBHOBeCIDI, lIPH KOTOpOM TeJIO yxe lIOqTH COCKaJIb3bI-
BaeT C HaKJIOHHOH IIJIOCKOCTH:
Jl = Ff / Fn = (W sin 9) / (W cos 9) = tg{}
B pa3,n;eJIe 2.7.4 yrOJI TPeHIDI OlIpe,n;eJIeH KaK Jl = tg<p; lIO 3TO MY , qT06bI TeJIO Ha-
XO,n;HJIOCb B COCTO.HHHH lIOK05.l, He06xo,n;HMO 9 = <po ECJIH 9 > <p, TeJIO 6y,n;eT CKOJIb-
3HTb BHH3 lIO HaKJIOHHOH IIJIOCKOCTH. ECJIH yrOJI 9 < <p, TeJIO 6y,n;eT OCTaBaTbC.H B
COCTO.HHHH lIOKO.H, nOKa BHeIIIH.IDI CHJIa He lIpHBe,n;eT ero B ,n;BJDKeHHe.
2.8. HanpR>KeHMe M Aecl>opMaLJ.MR
2.8. 1. HOpMan&HOe HanpflJICeHJfe
Ha pHC. 2.27, a lIOKa3aH UHJIHH,npHqeCKHH CTepxeHb C IIJIOIUMblO lIOlIepeqHOrO
CeqeH:mI A lIPH paCmJl:JICeHUU, a Ha pHC. 2.27, 6 lIOKa3aH TOT xe CTepxeHb lIPH c:JICa-
muu. IIpHJIoxeHHa.H CHJIa F HalIpaBJIeHa lIO OCH CTep)I{IDI H HOpMaJIbHa (lIeplIeH,nH-
KYIDIpHa) ero lIOlIepeqHOMY CeqeHHIO. B 3TOM CJIyqae rOBOp.HT, qTO CTepxeHb lIO,n;-
BepraeTC.H HOpMaJIbHOMY HalIp.HXeHHIO. HOpMaJIbHOe HalIp.HXeHHe 0603HaqaeTC.H
CHMBOJIOM cr (rpeqeCKa.H 6yxBa CHrMa).
HOpMaJIbHOe HalIp.HXeHHe cr =
= lIpHJIOXeHHa.H CHJIa F /IIJIOIUMb lIOlIepeqHOrO CeqeHIDI A.
A
F
F
a
F
+ =----.:_-=+
I
I
F
6
PRC. 2.27. HOpMaJIbHOe HarrpHxeHHe:
a - CTepxeHb rrpH paCTH)I{eHHH - HarrpHxeHHe paCTHxeHHH;
6 - CTepxeHb rrpH cxaTHH - HarrpH)l{eHHe cxaTHH
2.8. HanpJl:JlCeHUe u iJerjJopMal(UR
2.8.2. HSnpflJICeHJfe CASJfrS
Ha pHC. 2.28 lIOKa3aHO 33KJIelIOtffiOe coe,n;HHeHHe. IIoCKOJIbKY lIpHJIOXeHHbIe CHJIbI Q
CMelI(eHbI OTHOCHTeJIbHO OCH, B 3TOM c.rryqae rOBOp.HT, qTO 3aKJIelIKa lIo,n;BepraeTC.H
HarrpIDKeHHIO c,n;BHra, KOTopoe 0603HaqaeTC.H CHMBOJIOM 't (rpeqeCKa.H 6yxBa Tay).
HalIpIDKeHHe c,n;BHra 't = c,n;BHraIOIUa.H CHJIa Q/IIJIOII(a;:r;b c,n;BHra A.
HOpMaJIbHOe HalIpIDKeHHe H HalIpIDKeHHe c,n;BHra 06bIqHO ,n;OCTaTOqHO BeJIHKH
H n03TOMY H3Mep.HIOTC.H B MH/M2.
Q
Q
PRC. 2.28. HarrpH)l{eHHe c,n;BHra
2.8.3. OTHOCJfTen&HSfI nJfHeHHSfI AecpopMsqJffl
CR1IbI, ,n;eHcTByroII(He Ha TeJIO TaK, qTO B03HHKaeT HOpMaJIbHOe HalIpIDKeHHe, TaK-
xe RBIDIIOTC.H lIpWlHHOH H3MeHeHHH pa3MepOB TeJIa. HalIpHMep, paCT.HrHBaIOIUa.H
HarpY3Ka C03,n;aCT HalIpIDKeHHe paCTIDKeHIDI, a TaIOKe lIpHBe,n;eT K y,nJIHHeHHIO-- teJIa,
KaK lIOKa3aHO Ha pHC. 2.29, a. AHaJIOrHqHO, C)l{HMaIOIUa.H HarpY3Ka lIpHBe,n;eT K
)'KopOqeHHIO TeJIa, KaK lIOKa3aHO Ha pHC. 2.29, 6. OTHOCHTeJIbHa.H JIHHeHHaR ,n;e-
<l>opMaUIDI 0603HaqaeTC.H CHMBOJIOM E (rpeqeCKa.H 6yxBa 3lICHJIOH).
OTHOCHTeJIbHa.H JIHHeHHa.H ,n;e<popMaUIDI E = H3MeHeHHe ,nJIHHbI X/HCXO,n;Ha.H ,nJIHHa /.
F ..
. I .. F
I_ x .1- I .1 -J
;; )
,)
a
F
..
I: x .1
F
I
I-
.1
6
PRC. 2.29. OTHOCHTeJIbHaH JIHHeHHaH ,n;e<p0pMaI(HH:
a - B03HHKaIOInaH rrpH paCTHrHBaIOIneM HarrpH)l{eHHH;
6 - B03HHKaIOInaH rrpH C)l{HMaIOIneM HarrpHxeHHH
F/la8a 2. HH:JICeHepHQR cmamU1ca
IIOCKOJIbKY H3MeHeHHe ,1J)IHHbI H HCXO,n;Ha.H ,1J)IHHa H3Mep.HIOTC.H B o,n;HHaKOBbIX
e,n;HHHuax, TO OTHOCHTeJIbHa.H JIHHeHHa.H ,n;e<popMaUIDI, OlIpe,n;eIDIeMa.H KaK HX OTHO-
IIIeHHe, .HBillIeTC.H 6e3pa3MepHOH BeJIHqHHOH.
2.8.4. p,ecpopM8qJffl CASJfrS
,lJ:e<l>opMaUIDI c,n;BHra lIOKa3aHa Ha pHC. 2.30. IIpHJIoxeHHa.H CHJIa Q cMerueHa H
lIpHBO,n;HT K ,n;e<l>opMaUHH MaTepHaJIa. ,lJ:e<l>opMaUIDI c,n;BHra 0603HaqaeTC.H CHMBO-
JIOM Y (rpeqeCKa.H 6yxBa raMMa).
,lJ:e<l>opMaUH.H c,n;BHra y = BeJIHqHHa ,n;e<l>opMaUH.H X/HCXO,n;HbIH pa3Mep I.
3TO OII.HTb OTHOIIIeHHe, a CJIe,n;OBaTeJIbHO, 6e3pa3MepHaR BeJIHqHHa.
Q
n
\
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Q
PRC. 2.30. ,LJ;e<popMaI(HH c,n;BHra
2.8.5. MOAyn& ynpyroCTJf (3SKOH rYKS)
3aKOH fyJ(a YCTaHaBJIHBaeT, qTO B 06JIaCTH yrrpyrHX ,n;e<p0pMaUHH ,n;aHHoro MaTepH-
aJIa ,n;e<l>opMaUIDI lIpOlIOPUHOHaJIbHa lIpHJIOXeHHOH CHJIe, C03,n;aIOrueH 3TY ,n;e<l>op-
MaUHIO. ECJIH rrpHMeHHTb 3TOT 3aKOH K rrpyxHHe:
)l(ecTKocTb lIpyxHHbI =
= HOpMaJIbHOe HalIpIDKeHHe cr/OTHoCHTeJIbHa.H JIHHeHHa.H ,n;e<p0pMaUIDI E.
MO,LJ;yJIb yrrpyrOCTH E MO)I{HO rrOJIyqHTb H3 3aKOHa fyJ(a. HalIpIDKeHHe IIp.HMO
lIporrOpUHOHaJIbHO Y,1J)IHHeHHIO, lIOKa MaTepHaJI lIo,n;BepraeTC.H HalIpIDKeHHIO B 06-
JIaCTH yrrpyrHX ,n;e<l>opMaUHH, rr03TOMY:
HalIpIDKeHHe - ,n;e<l>opMaUIDI.
IIo3TOMY:
HalIp.H)I{eHHe = ,n;e<l>opMaUH.H x KOHCTaHTa.
CJIe,n;OBaTeJIbHO:
HalIpIDKeHHe/,n;e<l>opMaUIDI = KOHCTaHTa.
3Ta KOHCTaHTa (0603HaqaeMaR E) Ha3bIBaeTC.H Mo,n;YJIeM yrrpyrOCTH (KJIH MO,n;y-
JIeM IOHra) H H3Mep.HeTCR B fH/M2.
E = HalIpIDKeHHe cr/,n;e<l>opMaUIDI E.
2.8. HanpJl:JICeHUe u iJetjJoPMlll(UJI
2.8.6. MOAyn& CASJfra
MO.nyJIb c,n;BHra G CB.H3bIBaeT ,n;e<l>opMaUHIO c,n;BHra H HalIpIDKeHHe c,n;BHra CJIe,l(ylO-
lI(HM 06pa30M:
MO,n;YJIb c,n;BHra G = HalIpIDKeHHe c,n;BHra 't/,n;e<l>opMaUIDI c,n;BHra y.
MO.nyJIb c,n;BHra TaIOKe H3Mep eTC.H B fH/M2. B Ta6JIHue 2.4 lIpHBe,n;eHbI MO.nyJIb
c,n;BHra H MO,n;YJIb yrrpyrOCTH ,lJ;JUI HeKOTopbIX THlIHqHbIX MaTepHaJIOB.
Ta6JI. 2.4. MO,ZJ;yJIb ynpyrocm H MO,ZJ;yJIb C,lI;BHra ,lI)UI HeKoropbIX 'fHIIHqJIbIX MaTepHaJIOB
MO,ZJyJlh Mo,ZJy JIb Mo,ZJy JIb Mo,ZJy JIb
MaTepHaJI ynp E, c.ura G, MaTepHaJI ynpyroCTH E, c.ura G,
fH/M fH/M2 fH/M 2 fH/M2
ArrIOMHHHH 70 26 CBHHeu 16 6
JIaTyHb (70/30) 101 37 CTaJIb (MHrKaH, HH3- 212 82
KoyrJIepo,n;HCTaH)
KanMHH 50 19 OJIOBO 50 18
XpOM 279 115 THTaH 116 44
Me,n;b 130 48 BOJIb<ppaM 411 161
)l(eJIe30 ('IyryH) 152 60 QHHK 108 43
( '1
2.8.7. HSnpflJlCeHJfe KpYlfeHJffl
Ha pHC. 2.31, a MO)I{HO 3aMeTHTb, qTO, KOr,lIa Ha BaJI KpyroBoro rrOlIepeqHOrO ceqe-
HID! ,n;eHcTByeT MOMeHT KpyqeHIDI (3aKpyqHBaIOIUHH MOMeHT) T:
. Bce CeqeHIDI Barra OCTaIOTC.H KpyrOBbIMH H HX ,n;HaMeTP He H3MeIDIeTC.H;
. IIJIOCKOCTb lIOlIepeqHOrO CeqeHIDI OCTaeTC.H IIJIOCKOCTbID (TOJIbKO KpyroBa.H)
lIPH YCJIOBHH MaJIOCTH yrJIa lIOBopOTa.
TopueBbIe lIOBepXHOCTH H30JIHpOBaHHoro UHJIHH,lJ;pHqeCKOrO 3JIeMeHTa BaJIa,
nOKa3aHHoro Ha pHC. 2.31, 6, OCTaIOTC.H IIJIOCKHMH, H JII060H rrp.HMoyrOJIbHbIH 3JIe-
MeHT lIOBepXHOCTH abcd 6y,n;eT Haxo,n;HTbC.H B COCTO.HHHH qHCTOrO c,n;BHra. jIalIp.H-
xeHIDI C,l(BHra B,n;OJIb ad H bc BbI30Byr ,n;OlIOJIHHTeJIbHbIe HalIpIDKeHHe c,n;BIf B,n;OJIb
ab H cd. IIpoHcxo,n;HT lIpO,n;OJIbHOe cKpyqHBaHHe BaJIa TaK, qTO oTPe3KH\")Gib H cd
CMeIUaIDTC.H Ha yrOJI OTHOCHTeJIbHO JIHHHH AB, KaK lIOKa3aHO Ha pHC. 2.31, B.
IIpH qHCTOM c,n;BHre H MaJIbIX yrJIOB cKpyqHBaHIDI yrOJI lIpe,n;CTaBIDleT yrJIOBYIO
,n;e<l>opMaUHIO, a -r - KaCaTeJIbHOe HalIpIDKeHHe Ha lIOBepXHOCTH BaJIa. II03ToMY:
MO,n;YJIb c,n;BHra G = KaCaTeJIbHOe HalIpIDKeHHe -r/yrJIOBa.H ,n;e<l>opMaUIDI ; (1)
,l];JIHHa n;yrH BBI = I , lIPH YCJIOBHH, qTO ,n;OCTaTOqHO MaJIO;
,l];yra BB 1 = re,
r,n;e r - Pa)J;HYC BaJIa, e - yrOJI cKpyqHBaHIDI Ha BceH ,1J)IHHe BaJIa I.
II03TOMY:
I = ,lLrrHHa ,[(yrH BB I = re,
<p = re / I.
F/la8a 2. HH:JICeHepHQJl cmamU1CO
d
a
6
A
PRC. 2.31. HarrpHxeHHe H ,n;e<popMaI(HH
rrpH KpyqeHHH:
a - cKpyqHBaHHe KpyroBoro Bana;
6 - H30JIHpOBaHHbIH 3JIeMeHT abcd;
B - ,n;e<popMaI(HH KpyroBoro Bana
BCJIe,n;CTBHecKpyqHBaHHH
B
IIo,lJ;cTaBJI.H.H B ypaBHeHHe (1), lIo.rryqaeM:
G = 't / (re / I) HJIH 't / r = Ge / I.
IIpUMettaHue. IIoCKOJIbKY G H I .HBJI.HIOTC.H KOHCTaHTaMH AIDI JII060ro ,lJ;aHHOrO
BaJIa H e - KOHCTaHTa AIDI Bcex PMHYCOB JII060ro KOHKpeTHoro lIOlIepeqHOrO ce-
qeHIDI, TO:
't/r = KOHCTaHTa.
TaKHM 06pa30M, KaCaTeJIbHOe HalIpIDKeHHe B JII060H TOqKe ,lJ;aHHOrO lIOlIepeq-
Horo CeqeHIDI lIpOlIOPUHOHaJIbHO PMHYCY, H HalIpIDKeHHe YBeJIHqHBaeTC.H paBHO-
MepHO OT HYJIR Ha OCH BaJIa ,lJ;0 MaKCHMYMa Ha ero BHeIIIHeM PMHyce. II03TOMY
OqeBH,LI;HO, qTO lIOJIbIH BaJI MoxeT lIepe,lJ;aBaTb 60JIbIIIHH MOMeHT KpyqeHIDI, qeM
CTIJIOIIIHOH BaJI C TaKOH xe TIJIOII(MbIO lIOlIepeqHOrO CeqeHIDI.
2.8.8. Kon&qesoe HanpflJICeHJfe s TOHKJfX
qJfnJfHAPJflfeCKJfX 060nOlfKax
113 pHC. 2.32 lIOIDITHO, qTO ,lJ;aBJIeHHe P CTPeMHTC.H YBeJIHqHTb ,lJ;HaMeTP UHJIHH,lJ;pa.
OHO paCTRrHBaeT CTeHKH UHJIHH,lJ;pa lIO BceH ,lJ)IHHe OKpy:>KHOCTH H C03,lJ;aeT paCT.Hrn-
BaIOlI(ee HanpIDKeHHe, KOTopoe Ha3bmaeTCR KOJIbueBbIM (OKpyxHbIM) HanpIDKeHHeM crh'
CHJIa, B03HHKalOlI(a.H BCJIe,lJ;CTBHe BHyrpeHHero ,lJ;aBJIeHIDI P, ypaBHOBellIHBaeTCR
KOJIbueBbIM HalIpIDKeHHeM crh' OTCIO,lJ;a:
KOJIbueBOe HarrpIDKeHHe x TIJIOruMb = ,lJ;aBJIeHHe x lIpoeKUIDI lIOBepXHOCTH.
crh x 21t = Pdl.
crh=Pd/2t.
2.9. DOJIICU
npOeKu.Vl5I nOBepXHOCTVI
PRC. 2.32. KOJIbI(eBOe HarrpH)l{eHHe B TOHKHX I(HJIHH,lI;pHqeCKHX 060JIOqKax.
P - ,n;aBJIeHHe, ,n;eHcTBYKnnee PaLI.HaJIbHO Ha TOHKYIO 060JIoqKY
2.8.9. npO.ROn&HOe HanpflJICeHJfe S TOHKJfX
qJfnJfHAPJflfeCKJfX 060nOlfKax
Ha pHC. 2.33 MO)I{HO 3aMeTHTb, qTO BHyTpeHHee ,n;aBJIeHHe P C03,n;aeT paCT.HrHBalO-
mee HalIp.HXeHHe TaIOKe B lIpO,n;OJIbHOM HalIpaBJIeHHH, 3TO HalIpIDKeHHe 0603Haqa- (
eTC.H a[. CJIe,n;oBaTeJIbHO, ,n;aBJIeHHe P, ,n;eHcT-
BYIOlI(ee Ha IIJIOIIJ;a;:r;b nd 2 /4, ypaBHOBeIIIHBa-
eTC.H lIpO,n;OJIbHbIM (oceBbIM) HalIpIDKeHHeM
CJb ,n;eHCTBYIOIIJ;HM Ha IIJIOIIJ;a;:r;H ndt. (CTPoro
roBOp.H, ,n;OJDKeH HCnOJIb30BaTbC.H cpe,n;HHH
,n;HaMeTP, o,n;HaKO lIPH YCJIOBHH, qTO TOJIIIJ;H-
Ha CTeHKH t MaJIa lIO cpaBHeHHIO C ,n;HaMeT-
pOM, ,n;aHHoe lIpH6JIJDKeHHe .HBIDleTC.H KOp-
peKTHbIM.) TaKHM 06pa30M:
P x (1td 2 / 4) = a[ x 1tdt.
a[ = (nPd 2 ) / (41tdt).
a[ = Pd / 4t.
PRC. 2.33. npO,n;OJIbHOe HarrpH)l{eHHe B
TOHKHX I(HJIHH,lI;pHqeCKHX 060JIOqKax
IIpUMettaHue.
lIpaKTHqeCKHX ueJIeH B ypaBHeHIDIx ,LJ;.IDI a h Hal HalIpIDKeHIDI IIp e,n;lIOJIara -
IOTC.H lIOCTORHHbIMH lIO BceH TOJIIIJ;HHe CTeHOK. 3TO ClIpaBe,1J)lHBO TOJIbKO B TOM
c.rryqae, Kor,n;a OTHOIIIeHHe TOJIII(HHbI K BHyTpeHHeMY ,n;HaMeTPY MeHbIIIe 1:20.
2.9. &anKM
2.9.1. YCJfnJfe CASJfra
,lLJrn paBHOBeCIDI 6aJIKH CHJIbI, ,n;eHCTBYIOII(He CJIeBa OT JII060ro CeqeHIDI X, KaK rro-
Ka3aHO Ha pHC. 2.34, a, ,n;OJDKHbI ypaBHOBeIIIHBaTbC.H CHJIaMI1, ,n;eHCTBYIOIIJ;HMH
F/la8a 2. HH:JICeHepHQJl cmamU1ca
nOAAep)I(Ka
nOAAep)l(Ka
, F
" F
'F
HarPY3Ka
HarPY3Ka
Mx
M'
x
" F
a
6
B
PRC. 2.34. CHJIbI, ,n;eHcTByK>InHe Ha 6aJIKY:
a - CHJIbI, ,n;eHcTByK>InHe Ha 6aJIKY;
6 - rrOJIO)I{HTeJIbHaH rrOrrepeqHaH CHJIa;
B - OTpHI(aTeJIbHaH rrOrrepeqHaH CHJIa
ClIpaBa OT CeqeHIDI X. MOMeHTbI CHJI OTHOCHTeJIbHO CeqeHIDI X CJIeBa TaIOKe ,n;OJI-
)l{HbI ypaBHoBeIIIHBaTbC.H MOMeHTaMH CHJI OTHOCHTeJIbHO CeqeHIDI X clIpaBa.
HeCMOTp.H Ha TO qTO B COCTO.HHHH paBHOBeCIDI, CHJIbI H HX MOMeHTbI B3aHMHO
KOMlIeHCHpYIOTC.H, BeJIHqHHa H lIpHpo,n;a 3THX CHJI H MOMeHTOB HMelOT 3HaqeHHe,
lIOCKOJIbKY OHH OlIpe,n;eIDIIOT HalIpIDKeHIDI B CeqeHHH X, a TaIOKe KPHBH3HY H lIpO-
fH6 6aJIKH. Pe3YJIbTHPYIOIUa.H CHJIa CJIeBa OT X H pe3YJIbTHpYIOIUa.H CHJIa ClIpaBa OT
X (CHJIbI HJIH KOMlIOHeHTbI CHJI, rreplIeH,lJ;HKYillIpHbIe 6aJIKe) COCTaBillIlOT lIapy CHJI,
cTPeM.HIUYIOC.H C03,n;aTb C,n;BHf B 3TOM CeqeHHH 6aJIKH. ITolIepeqHa.H CHJIa OlIpe,n;eilll-
eTC.H KaK CHJIa, lIeplIeH,lJ;HKYIDIpHa.H K 6aJIKe H ,lJ;.IDI JII060fO 3aL(aHHOfO CeqeHIDI
lIpHBO,n;.HIUa.H K C,n;BHry B 3TOM CeqeHHH.
ECJIH C,n;BHf lIpOHCXO,n;HT, KaK lIOKa3aHO Ha pHC. 2.34, 6, TO lIOlIepeqHa.H CHJIa
YCJIOBHO CqHTaeTC.H lIOJIOXHTeJIbHOH (T.e. +F); eCJIH c,n;BHr lIpOHCXO,n;HT, KaK lIOKa-
3aHO Ha pHC. 2.34, B, CHJIa CqHTaeTC.H 0TPHuaTeJIbHOH (T.e. -F).
2.9.2. 1f3rJf6alOUlJfH MOMeHT
H3fH6aIOIUHH MOMeHT B ,n;aHHOM CeqeHHH 6aJIKH OlIpe,n;eIDIeTC.H KaK pe3YJIbTHPYIO-
IUHH MOMeHT OTHOCHTeJIbHO 3Toro CeqeHH.H HJIH Bcex CHJI, lIpHJIOXeHHbIX CJIeBa OT
CeqeHIDI, HJIH Bcex CHJI, lIpHJIOXeHHbIX ClIpaBa OT CeqeHIDI. Ha pHC. 2.34, a H3fH-
6alOIUHH MOMeHT - 3TO HJIH M x, HJIH M . 3TH MOMeHTbI 6y,n;yr HalIpaBJIeHbI lIO
qaCOBOH CTPeJIKe CJIeBa OT CeqeHIDI H lIpOTHB qaCOBOH CTpeJIKH - ClIpaBa OT ceqe-
HIDI. H3-3a HHX 6aJIKa 6y,n;eT lIporH6aTbC.H. Y CJIOBHO TaKOH lIpOfH6 CqHTaeTC.H lIO-
JIO)l{HTeJIbHbIM lIPOfH60M, T. e. lIOJIOXHTeJIbHbIe H3fH6alOIUHe MOMeHTbI C03,n;aIOT
lIOJIOXHTeJIbHbIH H3fH6 (lIpOfH6). AHaJIOfHqHO, OTpHuaTeJIbHbIe H3rH6aIOIUHe MO-
MeHTM CTaHOB.HTC.H lIPHqHHOH H3fH6a 6aJIKH B lIpOTHBOlIOJIO)I{HOM HalIpaBJIeHHH,
T. e. 0TPHuaTeJIbHbIe H3rH6alOIUHe MOMeHTbI C03,n;a1OT OTpHuaTeJIbHbIH H3fH6 (BbI-
rH6). Pa3HHua Me)l{)];y lIpOfH60M H BbIfH60M lIOKa3aHa Ha pHC. 2.35.
ITepefH6 B03HHKaeT, KOf,n;a BbIfH6 H lIporH6 lIpO.HBJIRIOTC.H B 6aJIKe o,n;HOBpe-
MeHHO, KaK lIOKa3aHO Ha pHC. 2.35, B. ITo,n; B03,n;eHcTBHeM HarpY3KH, lIOKa3aHHOH
Ha pHC. 2.35, a, 6aJIKa lIpOfHeTC.H BHH3 Me)l{,lJ;Y TOqKaMH B H C, a lIO,n; B03,n;eHCTBH-
eM HarpY3KH, lIOKa3aHHOH Ha pHC. 2.35, 6, 6aJIKa BbIfHeTC.H BBepx Me)l{,lJ;Y TOqKaMH
AH B.
2.9. DOJIICU
F L
Fs
Fs
a
6
Fs
Fs
B
PRC. 2.35. BbIrH6, rrporH6 H rreperH6:
F L - HarpY3Ka, Fs - orropa;
a - rrporH6;
6 - BbIrH6;
B - rreperH6
BeJIWlHHa lIOlIepeqHOH CHJIbI H BeJIWlHHa H3fH6alOIUefo MOMeHTa 06bIqHO H3-
MeH.HIOTC.H B,n;OJIb JII060H 6aJIKH. fpa<l>HKH, lIOKa3bIBalOIUHe lIOlIepeqHbIe CHJIbI H
H3fH6alOlI(He MOMeHTbI ,lJ;.IDI Bcex CeqeHHH 6aJIKH, Ha3bIBaIOTC.H 3lIIOpOH lIOlIepeq-
HbIX CHJI H 3lIIOpOH H3fH6alOIUHx MOMeHTOB COOTBeTCTBeHHO.
IIOlIepeqHbIe CHJIbI H 3lIIOpbI lIOlIepeqHbIX CHJI HMelOT MeHbIIIee 3HaqeHHe, qeM
H3fH6alOIUHe MOMeHTbI H 3lIIOpbI H3fH6alOIUHx MOMeHToB; o,n;HaKO OHH lIOJIe3HbI,
1l0CKOJIbKY yJ(33bIBaIOT Ha caMbIe Ba)l{HbIe aClIeKTbI 3lIIOpOB H3fH6alOIUHX MOMeH-
TOB HalIpHMep, TaM, f,n;e lIOlIepeqHa.H CHJIa paBHa HYJIIO, H3fH6alOIUHH MOMeHT
6y,n;eT MaKCHMaJIbHbIM HJIH MHHHMaJIbHbIM.
2.9.3. p,JfSrpSMM&1 YCJfnJffl CASJfrS
Jf Jf3rJf6slOIJ.4erO MOMeHTS
( ,
((J
)
//
PaCCMOTPHM 311lOpbI lIOlIepeqHOH CRJIbI H H3fH6alOIUefo MOMeHTa .l(JI.H CHCTeMbI
CM, ,n;eHcTByroIUHX Ha 6aJIKY, Ha pHC. 2.36. ,lI;JI.H MOMeHTa 6y,n;eT paCCMOTpeHa TOJIb-
KO lIpOCTa.H CHCTeMa H3 Tpex cocpe,n;OTOqeHHbIX HarpY30K.
IIpe)l{,lJ;e BcefO, He06xo,n;HMO paCCqHTaTb peaKUHH OlIOp B TOqKaX A H B, KaK
6bIJIO OlIHcaHO paHee B pa3,n;eJIe 2.4.4. OlIOpbI B TOqKaX A H B lIpOCTO lIO,LJJJ;ep)J{HBa-
lOT 6aJIKY. 3TO 03HaqaeT, qTO OHa rrOKOHTC.H Ha orropax H B 3THX TOqKaX ,n;eHcTByeT
24 KH
C"
A
B
5M
3 M
-
R A
R B
PRC. 2.36. CHJIbI, ,n;eHcTBYIOIUHe Ha 6aJIKY
F/la8a 2. HH:JICeHepHQJl cmamwco
BepTHKaJIbHa.H peaKII;lUl OlIOp. ,[LrnI BbIlIOJIHeHIDI YCJIOBHH paBHoBecIDI TPe6yeTC.H,
qT06bI pe3YJIbTHPYIOIUHH MOMeHT OTHOCHTeJIbHO JII060H TOqKH paBHIDIC.H HYJIIO H
cYMMa CHJI, ,n;eHcTByroIUHX BBepx, paBHIDIacb cYMMe CHJI, ,n;eHcTBYIOIUHX BHH3. IIo-
3TOMY, eCJIH olIpe,n;eIDITb MOMeHTbI OTHOCHTeJIbHO TOqKH A, MOMeHT, c03,n;aBaeMbIH
CHJIOH peaKII;HH RB, ,n;OJDKeH ypaBHOBeIIIHBaTb MOMeHT, c03,n;aBaeMbIH HarpY3KoH B
TOqKe C:
R B x 8 M = 24 KH x 5 M.
R B = (24 KH x 5 M) /8 M.
R B = 15 KH.
AHaJIOfWlHO:
R A = 24 KH - 15 KH = 9 KH.
HelIocpe,n;CTBeHHO ClIpaBa OT TOqKH A lIOlIepeqHa.H CHJIa OlIpe,n;eIDIeTC.H peaK-
UHeH R A H, CJIe,n;OBaTeJIbHO, paBHa 9KH. IIoCKOJIbKY 3Ta CHJIa lIpHJIOXeHa CJIeBa OT
paCCMaTpHBaeMOfO CeqeHIDI H HalIpaBJIeHa BBepx, OHa .HBIDIeTC.H lIOJIOXHTeJIbHoit.
IIOlIepeqHa.H CHJIa 6y,n;eT lIOCTO.HHHOH ,LJ;.IDI Bcex TOqeK Me)l{,LJ;y A H C, lIOCKOJIbKY
Me)l{,LJ;y 3THMH TOqKaMH Ha 6aJIKY He ,n;eHCTBYIOT HHKaKHe ,n;pyrHe CHJIbI.
IIpH paCCMOTPeHHH TOqKH ClIpaBa OT C CJIe,n;yeT lIpHHHMaTb BO BHHMaHHe Ha-
rpY3KY B TOqKe C, TaK xe KaK H CHJIY peaKII;HH R B . B KaqeCTBe aJIbTepHaTHBbI peaK-
l.(HIO R B MO)I{HO paCCMOTPeTb caMY lIO ce6e. IIOlIepeqHa.H CHJIa paBHa 15 KH. 3TO
3HaqeHHe R B BbNHCJIeHO paHee; OHO TaIOKe MoxeT 6bITb lIo.rryqeHO ,LJ;.IDI ,n;aHHOH Ha-
rpY3KH C - R A = 24 KH - 9 KH = 15 KH. ,[LrnI JII060H TOqKH Me)l{,LJ;y C H B CHJIa
lIpHJIOXeHa ClIpaBa H HalIpaBJIeHa BBepx, lI03TOMY lIOlIepeqHa.H CHJIa 0TPHuaTeJIb-
Ha, KaK 6bIJIO lIOKa3aHO paHee Ha pHC. 2.34. Hy:>KHo oTMeTHTb, qTO BeJIWlHHa lIO-
lIepeqHOH CHJIbI pe3KO H3MeIDIeTC.H B TOqKe C. 3TO lIOKa3aHO Ha pHC. 2.37, a.
CooTBeTCTBYIOIUHe H3rH6alOIUHe MOMeHTbI lIOK33aHbI Ha pHC. 2.37, 6. M3fH6a-
IOIUHH MOMeHT B TOqKe A paBeH HYJIIO, lIOCKOJIbKY HeT CHJI, lIpHJIO)l{eHHbIX CJIeBa
OT TOqKH A. B TOqKe, cMeIUeHHoH BlIpaBO Ha 1 M OTHOCHTeJIbHO TOqKH A, ,n;eHcTBY-
eT MOMeHT TOJIbKO O,n;HOH CHJIbI R A , lIPH-
JIO)l{eHHOH CJIeBa OT 3TOH TOqKH; OH paBeH
R A x 1 M = 9 KHM. IIoCKOJIbKY 3TOT MO- a
MeHT OTHOCHTeJIbHO TOqKH A HalIpaBJIeH
lIO qaCOBOH CTPeJIKe, OH lIOJIO)I{HTeJIeH
(+9 KHM). B TOqKax, CMeIUeHHbIX Ha 2, 3,
4 H 5 M BlIpaBO OT TOQKH A, H3fH6alOIUHe
MOMeHTbI COOTBeTCTBeHHO paBHbI:
A
9 KH
C
B
R A x 2 M = 9 KH x 2 M = 18 KHM.
R A x 3 M = 9 KH x 3 M = 27 KHM.
R A x 4 M = 9 KH x 4 M = 36 KHM.
R A x 5 M = 9 KH x 5 M = 45 KHM.
-15 KH
I 45 KH
6
IIOCKOJIbKY Bce 3TH MOMeHTbI HalIpaB-
JIeHbI lIO qaCOBOH CTPeJIKe, Bce OHH .HB.IDI-
IOTC.H lIOJIO)I{HTeJIbHbIMH H3fH6alOIUHMH
MOMeHTaMH. ,l);ID1 TOqeK lIpaBee C CJIe,n;yeT
yqHTbIBaTb HarpY3KY B TOqKe C, TaK xe
PRC. 2.37. 3IIIOpbI IIOIIepeqHOH CHJIbI H
H3rH6aIOInerO MOMeHTa.
a - 3IIIOpa IIOIIepeqHOH CHJIbI;
6 - 3IIIOpa H3rH6aIOInerO MOMeHTa
2.9. IiOJl1(,u
KaK H CIDIy peaKIJ;HH R B , HJIH, lIpOlI(e, KaK 6billO lIOKa3aHO paHee, MO)I{HO HClIOJIb-
30BaTb B BbIqHCJIeHlUIX TOJIbKO R B . B TOqKax, CMelI(eHHbIX Ha 5, 6 H 7 M OT TOq-
KH A, H3rH6alOlI(He MOMeHTbI COOTBeTCTBeHHO paBHbI:
R B x 3 M = 15 KH x 3 M = 45 KHM.
R B x 2 M = 15 KH x 2 M = 30 KHM.
R B x 1 M = 15 KH x 1 M = 15 KHM.
TaK KaK 3TH MOMeHTbI C03,n;aHbI CHJIOH, lIpHJIOXeHHOH ClIpaBa OT paCCMa1pHBa-
eMbIX TOqeK, H HalIpaBJIeHbI lIpOTHB qaCOBOH C1peJIKH, BCe OHH .HBJUIIOTCH nOJIO)I{H-
TeJIbHbIMH. B TOqKe B H3rH6alOlI(HH MOMeHT paBeH HYJIIO, TaK KaK HeT CHJI, lIpHJIO-
XeHHbIX ClIpaBa OT Hee. Pe3YJIbTaTbI CBe,n;eHbI B Ta6JIHUY 2.5.
Ta6J1. 2.5. 3HaqeHHH nOnepeqHOH CHJIbI H H3rH6aIOIIJ;erO MOMeHTa AJlH pHC. 2.37.
PaCCTOHHHe OT TOqI{H A (M) 0 1 2 3 4 5 6 7 8
Y CHJIHe c,n;BHra (KH) +9 +9 +9 +9 +9 +9 -15 -15 -15
H3m6aIOIIJ;HH MOMeHT (KHM) 0 +9 +18 +27 +36 +45 +30 +15 +15
WCnOJIb3YH pe3YJIbTaTbI H3 Ta6JIHUbI 2.5, MO)I{HO HapHCOBaTb 3lIIOpbI lIOlIepeq-
nOH CHJIbI H H3rH6alOlI(erO MOMeHTa, KaK lIOKa3aHO Ha pHC. 2.37. Kor,n;a 6aJIKa He-
ceT Ha ce6e TOJIbKO CKOHueH1pHpOBaHHyIO B TOqKax HarpY3KY, CTYIIe aTaH 3lIIOpa
nOnepeqHOH CHJIbI COCTOHT TOJIbKO H3 ropH30HTaJIbHbIX H BepTHKaJIbHbIX lIPHMbIX:-)
CKaqK006pa3Hoe H3MeHeHHe lIOlIepeqHOH CHJIbI lIpOHCXO,n;HT B Tex TOqKaX, r,n;e co-
cpe,n;OTOqeHa HarpY3Ka, B TOM qHCJIe B TOqKax lIpHJIOXeH:IDI peaKIJ;HH onop. ,lLJrn
3TOro npOCToro BH,l(a HarpY3KH 3lIIOpa H3rH6alOlI(erO MOMeHTa TaIOKe COCTOHT H3
npHMbIX JIHHHH, 06bIqHO HaKJIOHHbIX. Pe3KHX H3MeHeHHH H3rH6aJplI(ero MOMeHTa
06bflIHO He lIpOHCXO,n;HT, HCKJIlOqaH He06bIqHbIe YCJIOB:IDI, Kor,n;a K 6)mKe lIpHJIOXeH
... /
MOMeHT, OTJIHqHbIH OT HarpY3KH.
Ha caMOM ,n;eJIe, B peaJIbHOCTH He MoxeT cym;ecTBoBaTb 6aJIKa, Hecym;aH e,n;HH-
CTBeHHyIO HarpY3KY, cocpe,n;OTOqeHHyIO B O,n;HOH TOqKe. DaJIKa caMa lIO ce6e HMeeT
Maccy H, CJIe,n;OBaTeJIbHO, Bec. ECJIH lIOlIepeqHOe CeqeHHe 6aJIKH lIOCTOHHHO, Ha-
rpY3Ka paClIpe,n;eJIeHa paBHOMepHO lIO BceH A1IHHe 6aJIKH. PaCCM01pHM TelIepb
B03,n;eHCTBHe paBHOMepHO paClIpe,n;eJIeHHOH HarpY3KH.
Ha pHC. 2.38, a lIOKa3aHa paBHOMepHO HarpyxeHHaH 6aJIKa A1IHHOH 1 H Be-
COM Cocpe,n;OTOqeHHaH HarpY3Ka 3,n;eCb lIpHcyrcTByeT TOJIbKO B BH,l(e peaKIJ;HH
onop R A H R B . TaK KaK HarpY3Ka CHMMe1pHqHa, H R A , H R B paBHbI W12. M3 3nlOpbI
nOnepeqHOH CHJIbI BH,l(HO, qTO MaKCHMaJIbHbIe 3HaqeH:IDI ,n;OCTHraIOTCH B TOqKax
onopbI, a lIocepe,n;HHe YCHJIHe c,n;BHra paBHO HYJIIO. O,n;HaKO Ha 3TOT pa3 lIPHMaH,
onpe,n;eJI.HlOlI(aH BeJIINHHY lIOlIepeqHOH CHJIbI, HBJI.HeTCH HaKJIOHHOH JIHHHeH, npo-
XO,lUI eH qepe3 cepe,n;HHY.
PaCCMOTpHM CeqeHHe XX, paClIOJIOXeHHOe Ha paCCTOHHHH a OT JIeBOH OlIOpbI,
B03,n;eHCTByIOlI(eH Ha 6aJIKY C CHJIOH peaKIJ;HH R A :
IIOlIepeqHaH CHJIa, ,n;eHCTByrolI(aH Ha CeqeHHe .IT" = (WI I 2) - (W x a).
F/la8a 2. HH:JICeHepHOR cmamUlCO
a
x
PaBHOMepHa51 HarpY3Ka (WjL)
WI
R A =-
2
x
Rs= WI
2
I
a
WI
-
2 B
A
WI
-
2
6
A
B
B
PRC. 2.38. oarrKa c paBHOMepHO pacnpe,n;eJIeHHOH HarpY3KoH (UDL):
a - 3nIOpa pacnpe,n;eJIeHHH HarpY3KH;
6 - 3nIOpa YCHJIHH c,n;BHra;
B - 3nIOpa H3rH6aIOm;ero MOMeHTa
3TO ypaBHeHHe lIpe,n;cTaBJISIeT lIpSIMyro JIHHHIO, H 3lIIOpa lIOlIepeqHOH CIDIbI lIO-
Ka3aHa Ha pHC. 2.38, 6. B OTJIHqHe OT lIpe,nbl)];yw;ero lIpHMepa, Ha 3TOT pa3 3lIIOpa
H3rH6alOm;ero MOMeHTa COCTOHT He H3 lIpSIMbIX JIHHHH, a lIpe,nCTaBJI.HeT C060H He-
npepbIBHyIO KPHByIO C MaKCHMYMOM lIocepe,n;HHe.
W3rH6alOlI(HH MOMeHT Mx B CeqeHHH XX = [(wi / 2) x a] - [W x a x (a / 2)].
fI03ToMY:
Mx = W / 2(la - a 2 ).
(1)
3TO ypaBHeHHe lIapa60JIbI, MaKCHM)'M KOTOpOH MO)I{HO HaRTH, HClIOJIb3Y.H lIep-
BYIO npOH3BO,LJ;HyIO, H3 YCJIOBIDI:
dM x / da = O.
fIpo,nH<p<pepeHUHpyeM ypaBHeHHe (1):
dM x / da = W / 2(1 - 2a) = O.
OTKY,na:
a = I / 2.
2.9. EaIllCU
W3rH6alOlI(HH MOMeHT MaKCHMaJIeH B ueH1pe lIapa60JIbI, KaK H CJIe,n;OBaJIO
O)l{H,lJ;aTb lIPH cHMMeTpHqHOH HarpY3Ke. CJIe,n;oBaTeJIbHO, lIo,n;CTaB.IDI.H a = 1/2 B
ypaBHeHHe (1):
M max = W /2[(/ 2 /2) -(1 2 / 4)] = Wl2 /8.
3lIIOpa H3rH6alOlI(erO MOMeHTa lIOKa3aHa Ha pHC. 2.38, B. KpoMe CJIyqaH, lIOKa-
3aHHoro Ha pHC. 2.38, a, Kor,n;a paBHOMepHaH HarpY3Ka B03HHKaeT B pe3YJIbTaTe
.neiicTBIDI rpaBHTaUHH Ha Maccy caMo 6aJIKH, e,n;HHcTBeHHbIH CJIyqaH, Kor,n;a 6aJIKa
raIOKe paBHOMepHO HarpyxeHa, 3TO eCJIH OHa SIBJI.HeTCSI qaCTblO KapKaca H HeceT Ha
j
ce6e TIDKeCTb KHPlIHqHOH KJI3)(KU.f
EOJIee qaCTO, BC1peqaeTCSI COqeTaHHe cocpe,n;OTOqeHHbIX HarpY30K C paBHOMep-
HO paclIpe,n;eJIeHHOH HarpY3KoH, KaK lIOKa3aHO Ha ,n;HarpaMMe paClIpe,n;eJIeHIDI Ha-
rpY3KH, lIpHBe,n;eHHoH Ha pHC. 2.39, a.
45 KH
24 KH
UDL = 125 KHM
A B
2M
9M
R A
a
81 KH
D
3 M
Rs
A
24 KH
24 KH
c
D
6
- 72 KH
144 KHM
(' '\
\" )
/
!
c
B
3 M
T 0"1 Ka
neper 6a
7,9 M
-72 KHM
B
PRC. 2.39. EarrKa c cocpe,n;oTOqeHHOH H paBHoMepHoH HarpY3KoH:
UDL - paBHOMepHO pacnpe,n;eJIeHHaH HarpY3Ka;
a - ,n;HarpaMMa pacnpe,n;eJIeHHH HarpY3KH;
6 - 3nIOpa nOnepeqHOH CHJIbI;
B - 3nIOpa H3rH6aIOw;ero MOMeHTa
F/la8a 2. HH:JICeHepHtlH cmamU1ca
1. PaCCqHTaeM peaK«HH OlIOp R A H R B , OlIpe,n;e.IDI.H MOMeHTbI OTHOCHTeJIbHO A:
(45 x 2) + (12 x 9) x 4,5 + (24 x 12) = 9Rc
TaKHM 06pa30M: Rc = 96 kN H R A = 81 KH.
2. fIOlIepeqHbIe CHJIbI (eCJIH x 6y,n;eT paCCTO.HHHe OT TOqKH A):
,lLJrn 0 x 2, lIOlIepeqHaH CHJIa = (81 - 12x) KH (1)
,lLJrn 2 x 9, lIOlIepeqHaH CHJIa = (81 - 12x - 45) = (36 - 12x) KH
,lLJrn 9 x 12, lIOlIepeqHaH CHJIa = (81 - 12 x 9 - 45 + 96) = +24 KH
3TH pe3YJIbTaTbI MO)I{HO lIpe,n;CTaBHTb IIp.HMbIMH JIHHIDIMH, KaK lIOKa3aHO Ha
pHC. 2.39, 6.
3. W3rH6alOlI(He MOMeHTbI (eCJIH x - paCCTO.HHHe OT TOqKH A):
,lLJrn 0 x 2, M = 81x -12(x 2 / 2) = 81x - 6x 2 KHM.
fI03ToMY: B TOqKe A x = 0 H M A = 0
B TOqKe B x = 2 H M B = 162 - 24 = 138 KHM
,lLJrn 2 x 9, M = 81x - (12x 2 / 2) - 45(x - 2) = 36x - 6x 2 + 90 KHM. (2)
fIpH x = 2 H M B = 72 - 24 + 90 = 138 KHM (KaK BbIIIIe)
fIpH x = 9 H Me = 324 - 486 + 90 = -72 KHM
113rH6alOlI(HH MOMeHT MaKCHMaJIeH Me TOqKaMH B H C, TaM r,n;e dM/dx = O.
OTMeTHM, qTO 3TO Ta xe CaMaH TOqKa, B KOTOpOH lIOlIepeqHaH CHJIa paBHa JIy-
JIIO. 113 ypaBHeHIDI (2):
dM/dx = 36 - 12x = O.
CJIe,n;OBaTeJIbHO:
x = 3 M.
fIo,n;cTaBJUISI x = 3 M B ypaBHeHHe (2):
(36 x 3) - (6 x 3 2 ) + 90 = 144 KHM.
TaKHM 06pa30M, MaKCHMYM H3rH6alOlI(erO MOMeHTa paBeH 144 KHM H OH ,no-
CTHraeTCSI B TOqKe, CMelI(eHHOH Ha 3 M OT TOqKH A.
fIeperH6 lIpOHCXO,n;HT B TOqKe, r,n;e H3rH6alOIUHH MOMeHT paBeH JIyJIIO. qT06bI
HaHTH 311' TOqKY, ypaBHeHHe (2) CJIe,nyeT lIpHpaBHSITb HYJIIO H HaHTH ero lIOJIOXH-
TeJIbHbIH KopeHb.
36x - 6x 2 + 90 = O.
OTCIO,na:
x = 7,9 M (e,n;HHcTBeHHoe lIOJIO)I{HTeJIbHOe peIIIeHHe)
fI03ToMY lIeperH6 lIpOHCXO,n;HT B TOqKe, paClIOJIOXeHHOH Ha paCCTOSIHHH 7,9 M OT
TOqKH A.
2.9.4. BSnKJf (KoHCOn&)
KpOMe lIpOCTbIX 6aJIOK, cB060,n;HO JIeXalI(HX Ha Ollopax, HClIOJIb3yroTC.H 6aJIKH, BbI-
CTYIIalOlI(He B BH,l(e KOHCOJIH. B 3TOM c.rryqae TOJIbKO O,n;HH KOHeu 6aJIKH HMeeT
OlIOPy, B TO BpeM.H KaK ,naJIbHHH OT OlIOpbI KOHeu 6aJIKH OCTaeTCSI He3aKpeIIJIeH-
2.9. IiOJlKU
HbIM, KaK lIOKa3aHO Ha pHC. 2.40, a. 3aKpeIIJIeHHbIH KOHe 6aJIKH MoxeT 6bITb
BCTPoeH B KHplI HyID KJIMKY HJIH lIpOCTO lIpe,n;CTaB.IDITb C060H BbICTYIIalOlI(yIO
qaCTb 6aJIKH, JIeXalI(eH Ha Ollopax. THlIHqHbIe 3lIIOpa lIOlIepeqHOH CHJIbI H 3lIIOpa
H3rH6aIOlI(erO MOMeHTa ,nmI KOHCOJIbHOH 6aJIKH C cocpe,n;OTOqeHHbIMH TOqeqHbIMH
HarpY3KaMH lIOKa3aHbI Ha pHC. 2.40, 6, B.
W 1 W 2 w 3
I I
a
W 1
(W 1 + W 2 )
-(W 1 + w 2 + w:J
6
( '\
j'J
(
-51W 1 - 41W 2 - 31W 3
B
PRC. 2.40. KOHCOJIb - TOJIbKO cocpe,n;OTOqeHHble HarpY3KH:
a - ,n;HarpaMMa pacnpe,n;eJIeHHH HarpY3KH;
6 - 3nIOpa nOnepeqHOH CHJIbI;
B - 3nIOpa H3rH6aIOm;ero MOMeHTa
Ha pHC. 2.41 lIOKa3aHbI ,n;HarpaMMa paClIpe,neJIeHIDI HarpY3KH, 3lIIOpa lIOlIe-
peqHOH CHJIbI H 3lIIOpa H3rH6alOlI(erO MOMeHTa ,nmI KOHCOJIbHOH 6aJIKH lIPH 60JIee
peaJIbHOM COqeTaHHH paBHOMepHO paclIpe,n;eJIeHHOH H TOqeqHOH HarpY30K.
F/la8a 2. HH:JICeHepHtlH cmamUK.a
40 KH
3 M
UDL = 125 KHM
2M
a
I
-90 KH
-130 KH
6
-190 KH
-455 KHM
Pac. 2.41. KOHCOJIb - COqeTaHHe cocpe.aoTOqeHHOH H pacnpe,n;eJIeHHOH HarpY30K: '
a - ,n;HarpaMMa pacnpe,n;eJIeHHH HarpY3KH;
6 - 3nIOpa nOnepeqHOH CHJIbI;
B - 3nIOpa H3rH6aIOw;ero MOMeHTa
2.10. HanpSI>KeHMe, Aecj)opMau.MSI M M3rM6 6anoK
2. 10. 1. H3rJf6slO ee HSnpflJICeHJfe Jf HeHTpSn&HBfI nJfHJffl
I
Ha pHC. 2.42, a lIOKa3aHO, qro lIpOHCXO,n;HT, Kor,n;a cB060,n;HO OlIepTaSI 6aJIKa, HMe-1
IOlI(aSI lIpSIMoyrOJIbHOe CeqeHHe, H3rH6aeTcSI (lIporH6aeTcSI) lIO,n; B03,n;eHcTBHeM I
lIpHJIOXeHHOH HarpY3KH. BHyrpeHHHe CJIOH 6aJIKH C)l{HMaIOTCSI, a BHeIIIHHe - pac-
TSIrHBaIOTCSI. B HeKoTopoH TOqKe Me)l()];Y 3THMH lIpOTHBOlIOJIO)I{HOCTSIMH paClIOJIO-
xeHa HeH1paJIbHaH JIHHIDI, KOTOpaH He HalIpIDKeHa, TaK KaK He cxaTa H He paCTH- I
Hyra. 3lIIOpa HalIpIDKeHHM A1ISI 6aJIKH lIOKa3aHa Ha pHC. 2.42, 6. MO)I{Ho BH,l(eTh'l
qTO HalIpIDKeHHe MaKCHMaJIbHO BBepxy H BHH3Y 6aJIKH H paBHO HYJIIO Ha HeH1paJIh.:
HOM JIHHHH.
2.10. HanpJl:JICeHUe, oetjJopMa1.4UJl U U3zu6 60Jlo"
F
R A 'B Ra
,
a PaCT5I HYTbl t1 MeTaJ1J1
MaKc MYM C)l( MalO-
w.ero Hanp5l)l(eH 5I
- Het1TpaJ1bHa51 J1 H 5I (J = 0 5 PRC. 2.42. H3rH6alOw;ee HanpHxeHHe
H HeHTpaJIbHaH JIHHH:
a - ,n;HarpaMMa pacnpe,n;e-
MaKc MYM paCT5Ir Ba- JIeHHH HarpY3KH;
6 O't B lOw.ero Hanp5l)l(eH 5I 6 - 3nlOpa HanpHxeHHH
fI03TOMY TBep,l(a.H IIp.HMOyrOJIbHa.H 6aJIKa MeHee 3<p<peKTHBHa, qeM THlI Ha.H
.n;ByraBpOBa.H 6aJIKa, Y KOTOpOH OTHOIIIeHHe lIPOqHOCTH K Becy HaMHoro 60JIbIIIe.
B .n;ByraBpoBoH 6aJIKe MaKcHM)'M HalIpIDKeHIDI HClIbITbIBaIOT BepXHIDI H HIDKIDI.H lIOJI-
KH 6aJIKH, a 06JIerqeHHa.H CTeHKa Me)l(,l(y lIOJIKaMH 6aJIKH lIo.n;BepraeTC.H TOJIbKO He-
60JIbIIIOMY HalIpIDKeHHIO. <l>aKT eCKH, qT06bI eme 60JIbIIIe YMeHbIIIHTb Maccy 6aJIKH
6e3 OII(yTHMOH lIoTepH lIPOqHOCTH, 3aqacTYIO B lIOJIKaX MO)I{HO lIpO,l(eJIaTb oTBepcTIDI.
B lIpaKTHqeCKHX YCJIOBIDIX 06bIqHO HClIOJIb3yIOTC.H CJIe.n;yrolI(He .n;orryI.l(eHIDI.
. 113rH6 MaJI lIO cpaBHeHHIO C pa3MepaMH lIOlIepeqHOrO CeqeHIDI 6aJIKH, T. e.
PMJIyC H3rH6a BeJIHK lIO cpaBHeHHIO C pa3MepaMH lIOlIepeqHOrO CeqeHIDI
6aJIKH.
. fIpe,l(lIOJIaraeTCSI, tITO 6aJIKa CJIaraeTCSI H3 6eCKOHe1ffioro KOJIH1IeCTBa lIpO.n;OJIbHbIX
CJIOeB, KOTOpbIe COrrpHKaCaIOTCSI rrpH H3rn6e, HO He3aBHCHMbI ,npyr OT .n;pyra.
. fIaJIKa no.n;BepraeTcSI TOJIbKO lIpO.n;OJIbHOMY HalIpIDKeHHIO. He cym;ecTByeT pe-
3YJIbTHpyrolI(eH lIpO.n;OJIbHOH CIDIbI, .n;eHCTByIOlI(eH Ha KaKOe-JIH60 CeqeHHe
6aJIKH.
. HarpY3Ka Ha 6aJIKY Ha,n,e)I{HO Haxo.n;HTCSI B lIpe.n;eJIax ynpyrOCTH MaTepHaJIa,
TaK qTO HalIpSI)J{eHHe lIpOlIOPUHOHaJIbHO .n;e<popMaUHH.
. MO,l(YJIb ynpyrOCTH MaTepHaJIa 6aJIKH OCTaeTCSI lIOCTOSIHHbIM KaK lIPH paCTSI-
)J{eHHH, TaK H lIPH C)J{aTHH. L \ )
. fIOlIepeqHbIe CeqeHIDI, IIJIOCKHe .n;o H3rn6a, OCTaIOTCSI IIJIOCKHMH lIOCJI€t -:tI3rH6a.
. HeHTpaJIbHa.H JIHHIDI lIpOXO.n;HT qepe3 reOMeTpHqeCKHH ueHTp CeqeHIDI He3a-
BHCHMO OT <pOpMbI lIOlIepeqHOrO CeqeHHSI 6aJIKH.
OCHoBHoe ypaBHeHHe ,l(JUI H3rH6a 6aJIKH:
(J / y = M / I = E / R = Mil = EIR,
r.n;e (J - HalIpIDKeHHe, 06YCJIOBJIeHHOe H3rH60M, Ha paCCTOSIHHH Y OT HeHTpaJIbHOH
OCH; y - paCCTO.HHHe OT HeHTpaJIbHOH OCH; M - H3rH6alOlI(HH MOMeHT; I - MO-
MeHT HHepUHH ,l(JISI CeqeHHSI 6aJIKH; E - MO.n;YJIb ynpyrOCTH (MO.n;YJIb IOHra); R -
pan;HYC KpHBH3HbI 6aJIKH.
<l>OPMYJIbI ,l(JUI CeqeHHH, BKJIlOqaSI MOMeHT HHepUHH, MOMeHT COlIpOTHBJIeHIDI
CeqeHHSI H PMHYC HHepUHH, lIpe,l(CTaBJIeHbI Ha pHC. 2.43 BMeCTe C lIOlIepeqHbIMH
CeqeHIDIMH HaH60JIee paClIpOCTpaHeHHbIX 6aJIOK.
F/la8a 2. HH:JICeHepHtlH cmamUico
Ce1leHHe
MOMeHT
WIOIIWUI
lu
qf : d!
bd 3
12
y
d
bd 3
3
x x
; A-=r t
bd 3
36
!l !l
2 2
BD3 -bd 3
12
D
2
D
2
D
reD 4
32
y
nD4
64
D
y
d
D
re(D4 - d 4 )
64
PRC. 2.43. <l>OPMYJIbI CeKI(HH
MOMeHT cOllpOTllBJlelllDl
Ce1lelllDl
z = I/y
bd 2
6
bd 2
3
bd 2
Zl=-
24
bd 2
Z2=-
12
BD3 -bd 3
6D
reD 3
16
nD3
32
re(D4 - d 4 )
32D
P8)OO'C
HHepQIIII
k = Ji7A
d
Jfi
d
J3
d
J18
BD3 -bd 3
12(BD - bd)
D
J8
D
4
.J D2 -d 2
4
2.11. (/JePMbl
2. 11. cDepMbl
Pa3pa60TKa TaKHX KOHC1'pyxI.J;HH, KaK C1'pOlIHJIbHa.H <pepMa HJIH lIpOC1'paHCTBeHHa.H
KOHCTPyxI.J;IDI, COCTOSIII(HX H3 HeCKOJIbKHX 3JIeMeHTOB, coe,n;HHeHHbIX CBOHMH KOHua-
MH, BKJIlOqaeT B ce6SI OlIpe,n;eJIeHHe YCHJIHH B KIDK,l(OM 3JIeMeHTe lIPH ,n;aHHOH CHCTe-
Me HarpY30K. B 3TOM pa3,n;eJIe 6y,nyT paCCM01'peHbI TOJIbKO «lIpOCTbIe <pepMbI», T.e.
<pepMbI, 3JIeMeHTbI KqTOpbIX coe,n;HHeHbI IIIapHHpHO. Ha pHC. 2.44, a lIOKa3aHbI xeCT-
KO coe,n;HHeHHbIe 3JIeMeHTbI, Tor,n;a KaK 3JIeMeHTbI, lIOKa3aHHbIe Ha pHC. 2.44, 6, co-
e.nHHeHbI IIIapHHpHO H MOryI' cB060,n;HO BpalI(aTbC.H OTHOCHTeJIbHO TOqKH P. ,L(.mI
npocToH <pepMbI lIpe,n;lIOJIaraeTC.H, qTO:
)
. Bce 3JIeMeHTbI coe,n;HHeHbI Me C060H IIIapHHpaMH;
. lIOCKOJIbK)' 3JIeMeHTbI MOryI' cB060,n;HO BpalI(aTbCSI, B IIIapHHpax He 6y,n;eT MO-
MeHTOB H He 6y,n;eT H3rH60B 3JIeMeHTOB;
. BHeIIIHHe HarpY3KH lIpHJIOXeHbI TOJIbKO K IIIapHHpHbIM coe,n;HHeHIDIM. IIo-
3TOMY OT,n;eJIbHbIe 3JIeMeHTbI <pepMbI lIo,n;BepraIOTC.H TOJIbKO paCTSIrHBalOlI(HM
HJIH C)l{HMalOlI(HM HarpY3KaM;
)
o 0 0 0 0,'
p
WapH pHoe
coeA HeH e
F 1
,
,
,
,
,
,
,
,
,
,
,
- - --,
,
,
,
,
,
,
,
,
" F 2
..
a
6
B
F 2
R A
Ra
..
r
A
(' " I
(/ ->.
I
PRC. 2.44. <l>epMbI:
a - xeCTKoe 3aKJIenOqHOe coe,n;HHeHHe;
6 - no,n;BIDKHoe mapHHpHoe coe,n;HHeHHe;
B - coe,n;HHeHHbIH mapHHpaMH npHMoyrOJIbHbIH MO,n;YJIb <pepMbI He 06-
JIa,n;aeT BHYTpeHHe npHcyw;eH xeCTKOCTbIO;
r - coe,n;HHeHHbIH mapHHpaMH TPeyrOJIbHbIH MO,l(yJIb <pepMbI HBJIHeTCH
xeCTKHM no caMOH CBoeH npHpo,n;e;
,n; - <pepMa Y oppeHa, COCTOHm;aH H3 CBH3aHHbIX napa.rr.n:eJIbHbIMH nOHca-
MH TPeyrOJIbHbIX Mo,n;YJIeH
F/la8a 2. HH:JICeHepHOR cmamUlCO
. 3aTIDKKa - 3TO 3JIeMeHT, pa60TalOlI(HH Ha paCTIDKeHHe B KIDK,l(OM KOHue, T.e.
BOClIpHHHMalOlI(HH paCT.HrHBalOlI(yIO HarpY3KY, KOTopaSI CTpeMHTCSI YA1IHHHTb
3JIeMeHT;
. paClIopKa - 3TO 3JIeMeHT, pa60TalOlI(HH Ha cxaTHe B KIDK,l(OM KOHue, T.e.
BOClIpHHHMalOlI(HH C)l{HMalOlI(yro HarpY3KY, KOTOpaH C1peMHTCSI YKOpOTHTb
3JIeMeHT.
DOJIbIIIHHCTBO <pepM OCHOBaHO Ha 1peyrOJIbHHKe HJIH 1peyrOJIbHOH peIIIeTKe.
fIpHqHHY 3Toro MO)I{HO lIOIDITb H3 pHC. 2.44, B, r. MO)I{Ho BH,l(eTb, qTO eCJIH coe,n;H-
HeHHbIH IIIapHHpaMH lIpSIMOyrOJIbHHK lIo,n;BeprHeTCSI B03,n;eHcTBHIO CHJI FI H F 2 , TO
OH lIpOH3BOJIbHO ,n;e<popMHpyeTCSI. O,n;HaKO, eCJIH coe,n;HHeHHbIH IIIapHHpaMH 'rpe-
yrOJIbHHK lIo,n;BeprHeTC.H B03,n;eHCTBHIO Tex xe CHJI, ero <popMa He H3MeHHTC.H ,n;o Tex
nop, lIOKa He H3MeHHTCSI A1IHHa XOT.H 6bI o,n;Horo H3 ero 3JIeMeHTOB. TpeyroJIbH3j{
peIIIeTKa SIBJI.HeTCSI xeCTKOH lIO caMOH CBoeH lIpHpo,n;e. Ha pHC.. 2.44, ,n; lIOKa3aHa
THlIHqHaH CKB03HaH <pepMa, COCTO.HlI(aH H3 1peyrOJIbHHKOB. 3Ta KOHKpeTHaH CKB03-
HaH 6aJIOqHaSI <pepMa Ha3bIBaeTC.H <pepMoH C lIapaJIJIeJIbHbIMH lIO.HCaMH H 1peyrOJIb-
HOH peIIIeTKoH (<pepMoH YoppeHa).
2.11.1. MeToA CelleHJfH
OCHOBHoe lIpeHMym;ecTBo MeTo,n;a CeK«HH 3aKJIIOqaeTCSI B TOM, qTO YCHJIHe lIOqTH B
JII060M 3JIeMeHTe 06bIqHO MO)I{HO lIOJIyqHTb, pacceKaSI <pepMY lIO 3TOMY 3JIeMeHTY If I
lIpHMeH.H.H lIpHHUHlI MOMeHTOB. HalIpHMep, paCCMOTpHM, KaK, HClIOJIb3Y.H MeTO,n;
CeqeHHH, MO)I{HO OlIpe,n;eJIHTb YCHJIHSI B 1pex 3JIeMeHTax, paCCeqeHHbIX Bo06paxae- I
MOH IIJIOCKOCTblO XX Ha pHC. 2.45, a.
CeKyIUaH IIJIOCKOCTb XX ,n;eJIHT <pepMY Ha ,n;Be qaCTH. fIpe,n;lIOJIO)I{HM, qTO lIpaBWI
qaCTb <pepMbI y,n;aJIeHa. 3TO lIpHBe,n;eT K TOMY, qTO JIeBaH qaCTb lIOTepSIeT YCTOHqH-
BOCTb OTHOCHTeJIbHO coe,n;HHeHIDI Rl, eCJ1H Ha paCCeqeHHbIe 3JIeMeHTbI He 6y,n;yr
,n;eHCTBOBaTb BHeIIIHHe CHJIbI F BF , FFE H FED' KaK lIOKa3aHO Ha pHC. 2.45, 6. TaKHM I
06pa30M, 3TH BHeIIIHHe CHJIbI, lIOKa3aHHbIe Ha pHC. 2.45, B, Ha caMOM ,n;eJIe lIpe,n;- '
CTaB.1UIIOT C060H BHYTPeHHHe CHJIbI, KOTopbIe HYXHO OlIpe,n;eJIHTb B 3TOM lIpHMepe.
PaCCMo1pHM YCJIOBIDI paBHOBeCIDI JIeBOH qaCTH <pepMbI. BHeIIIHHe CHJIbI, lIpH-
JIOXeHHbIe K OTCeqeHHOMY <pparMeHTY, ,n;OJDKHbI HJIH TOJIKaTb, HJIH TSIH)'Tb IIIapHHp-
HbIe coe,n;HHeHIDI, paClIOJIOXeHHbIe CJIeBa OT CeqeHIDI XX. fIpHIDITo lIpe,n;lIOJIaraTh I
3apaHee, qTO Bce OTCeqeHHbIe 3JIeMeHTbI HaXO,lJ;.HTCSI B COCTO.HHHH paCTIDKeHIDI, TOf- I
,n;a lIOJIO)I{HTeJIbHbIH OKOHqaTeJIbHbIH pe3YJIbTaT lIo,n;TBep)l()];aeT, qTO 3JIeMeHT pa60-
TaeT Ha paCTIDKeHHe (3aTIDKKa), B TO BpeM.H KaK 01pHuaTeJIbHbIH pe3YJIbTaT:
lIo,n;pa3YMeBaeT, qTO 3JIeMeHT pa60TaeT Ha cxaTHe (paClIopKa).
JIeBaH qaCTb <pepMbI TelIepb HaxO,n;HTCSI B COCTOSIHHH paBHOBeCIDI lIO,n; ,n;eHCTBH- I
eM CHJIbI peaK«HH OlIOpbI R I , HarpY3KH BeJIWIHHOH 20 KH H 1pex HeH3BeCTHbIX CKJI I
F BF , FFE H FED' OlIpe,n;e.IDI.H MOMeHTbI OTHOCHTeJIbHO JIeBOH OlIOpbI (TOqKH lIpHJIO- I
xeHIDI Rl H o,n;HOBpeMeHHO TOqKH lIepeCeqeHIDI ,n;BYX HeH3BeCTHbIX CHJI FFE H FED)'
MO)I{HO JIerKO HaHTH 3HaqeHHe F BF . AHaJIOrWIHO, BeJIHqHHY FED MO)I{HO HaHTH,
OlIpe,n;e.IDI.H MOMeHTbI OTHOCHTeJIbHO TOqKH (1) (pHC. 2.46, 6). HaKoHeu, OlIpe,n;eJIe- I
HHe MOMeHTOB OTHOCHTeJIbHO TaKOH TOqKH, KaK P, lI03BOJIHT HaHTH BeJIHqHHY YCH- I
JIIDI B FE. !
2.11. tPepMbl
2,4 M 3 M
20 KH 30 KH
B
x
9 M D
R 1
fJ
a
20 KH
x
F BF
------.....--.
I ",
I "" F.
,,' FE
p
,,------------
,
,
,
\Q
,
,
,
FeD
. - - - - - - ..,- - -
Puc. 2.45. MeTo,n; CeqeHHH:
a - THnHqHaH <pepMa )];JIH
aHaJIH3a MeTo,n;OM CeqeHHH;
6 - JIeBaH qaCTb - paBHO-
BeCHe no,n;,n;epXHBaeTCH CH-
JIaMH F BF , FFE H FED
x
R 1
6
ITpe)l{)J;e qeM lIpHCTYIIaTb K HaxO)l{)J;eHHIO BbIIIIeyrrOM.HHYTbIX CHJI, He06xo,nHMO
paCCqHTaTb BeJIHqHHbI Rt H R 2 , HCnOJIb3YR npHHUHn MOMeHTOB, 06cyx,n;aBIIIHHCR
paHee.
Onpe,neJIHM MOMeHTbI OTHOCHTeJIbHO Rt:
(20 x 2,4) + (30 x 5,4) = 9R 2 .
CJIe,nOBaTeJIbHO:
Rz = 23,33 KH.
R 1 = 26,67 KH.
YCHJIHe B DF (pHC. 2.46, a)
CeKyI.I.J;a.H IIJIOCKOCTb XX:
Onpe,neJUIeM MOMeHTbI OTHOCHTeJIbHO TOqKH lIpHJIOXeHIDI R t :
C \ \ I
(
20 x 2,4 + FBFx 2,4 = O.
II03TOMY:
F BF = -20 KH.
ITOCKOJIbKY 3HaqeHHe F BF RBJlReTCR OTpHuaTeJIbHbIM, BF - pacnopKa.
F/la8a 2. HH:JICeHepHaJI cmamUica
2,4 M 20 KH
20 KH
3M
20 KH
F ED = 35 KH
p
r---------
"
, \1#
Q
, \
,
,
I
I
,
.
-...
xF ED = 35 KH
x F BF = -20 KH
----..
x
D
26,67 KH
R 1
-------,-,-,-(1)
"
x
26,67 KH
R 1
x
a
6
B
F AF = -28,3 KH
Y
p
:'; ::A5; - - - -, FFE = -16,4 KH
,R 1/
, \ '
: Q\ I
: FED = 35 KH
..
2,4 M
..
..
..
..
..
..
...."...... h EC = -4 2 KH
J!!!I z '
,
,
,
,'8
,
,
,
,
,
,
,
-------- .
z
36M
D
r
A
R 2 = -23,33 KH
Puc. 2.46. Y CHJIHH B 3JIeMeHTax <pepMbI:
a - YCHJIHe B BF; 6 - YCHJIHe BED; B - YCHJIHe B FE;
r - YCHJIHe B AF; ,n; - YCHJIHe B EC
YCHJIHe B ED (pHC. 2.46, 6)
CeK}'IUa.H IIJIOCKOCTb XX:
Onpe.neJUIeM MOMeHTbI OTHOCHTeJIbHO TOqKH (1):
(26,67 x 5,4) - (20 x 3) - FED x 2,4 = O.
IT03TOMY:
FED = 35 KH.
ITOCKOJIbKY 3HaqeHHe FED .HBJUIeTC.H nOJIO)I{HTeJIbHbIM, ED - 3aTIDKKa.
YCHJlHe B FE (pHC. 2.46, B)
CeK}'IUa.H IIJIOCKOCTb XX:
Onpe.neJUIeM MOMeHTbI OTHOCHTeJIbHO TOqKH P:
(20 x 2,4) - FEE x XX - FED x 2,4 = O.
48 - FEE x 2,4cos9 - 35 x 2,4 = o.
r.ne:
9 = 24°.
II03ToMY:
FEE = -16,4 KH.
IIoCKOJIbKY 3HaqeHHe FEE .HBJUIeTC.H OTpHuaTeJIbHbIM, FE - paClIopKa.
YCHJIHe B AF (pHC. 2.46, r)
CeK}'IUa.H IIJIOCKOCTb XX:
Onpe.neJUIeM MOMeHTbI OTHOCHTeJIbHO TOqKH P:
16,4 xx - FAExy - 35 x 2,4 = O.
2.11. (/)ePMbl
II03ToMY:
F AF = -28,3 KH
ITOCKOJIbKY 3HaqeHHe F AF RBJUIeTCR OTpHuaTeJIbHbIM, AF - paClIopKa.
YCHJIHe B EC (pHC. 2.46, ,n;)
Ce a.H IIJIOCKOCTb zz:
Onpe,neJUIeM MOMeItTbI OTHOCHTeJIbHO TOqKH (2):
23,33 x 3,6 + F EC x Z = 0
II03TOMY:
F EC = -42KH
ITOCKOJIbKY 3HaqeHHe FK RBJUIeTC.H OTpHuaTeJIbHbIM, EC - pacnopKa.
OTMeTHM, qTO BO Bcex paCCMOTpeHHb.x BbIIIIe c.rryqa.HX ,no Tex nop, nOKa YCH-
J
JIID! B 3JIeMeHTax HeH3BeCTHbI, npe,nnOJIaraJIOCb, qTO 3JIeMeHTbI HaxO,lUITCR B COCTO-
HHHH paCTIDKeHIDI.
2. 12. rJfAPOCTBTJflfeCKOe ABSneHJfe
)KH,nKOCTb B cocy,ne, lIOKa3aHHa.H Ha pHC. 2.47, a, OKa3bIBaeT ,naBJIeHHe Ha CTeHKH H
,ZJ;HO cocy,na. aBJIeHHe, OKa3bIBaeMoe )l{H)J;KOCTblO, 3aBHCHT OT:
. llJIOTHOCTH )l{H)J;KOCTH (p);
. BbICOTbI cToJI6a )l{H)J;KOCTH, ,neHCTByrolI(eH Ha TeJIO (h);
. rpaBHTaUHoHHoH lIOCTORHHOH (g).
- BbiCOTa
rViAPOCTaTVI"Ie-
CKoro Hanopa
a
6
Puc. 2.47. rH.D:pOCTaT ecKoe ,n;aBJIeHHe:
a - ,n;aBJIeHHe B cocy,n;e, co,n;epXalI(eM )f{H)J;KOCTb;
6 - ,n;aBJIeHHe PI = P 2 H 3aBHCHT OT h. ,n;aBJIeHHe He 3aBHCHT OT llJIOII(3)J;H.
Ha pHC. 2.47, 6 lIOKa3aHbI ,nBa cocy,na A H B, B KOTOpbIX BbICOTa cToJI6a )l{H)J;KO-
CTH (h) o,nHHaKOBa. IIpH YCJIOBHH, qTO IIJIOTHOCTb )l{H)J;KocTeH B 060Hx C.rrYqa.HX
O,ZJ;HHaKOBa, H nOCKOJIbKY rpaBHTaUHOHHa.H nOCTORHHa.H B 060HX C.rrYqa.HX o,nHHaKO-
Ba, ,naBJIeHIDI PI H P2 6y.nyT o,nHHaKOBbI, HeCMOTpR Ha TOT <paKT, qTO IIJIOII(a,zJ;H
OCHOBaHIDI KOHTeHHepOB pa3JIWIHbI. BbICOTa cToJI6a )l{H)J;KOCTH h OTHOCHTe!lpf!O
( -"
,ZJ;aHHOH TOqKH TaIOKe Ha3bIBaeTC.H: \
. BbICOTa rH;:J;pOCTaTWIeCKoro Hanopa;
. BbICOTa ,naBJIeHIDI.
F/la8a 2. HH:JICeHepHaJI cmamUica
aBJIeHHe (P) B )I{H,l(KOCTH onpe,n;eJUIeTCR CJIe,nyIOII(HM BbIpaxeHHeM:
P = P x g x h,
r,ne P - ,naBJIeHHe (ITa); h - r.rry6HHa, Ha KOTOpOH H3MepReTcR ,naBJIeHHe (M);
p _ IIJIOTHOCTb )I{H,l(KOCTH (Kr/M 3 ); g - rpaBHTaUHOHHa.H nOCTORHHa.H (9,81 M/C 2 ).
llpHMep
Ha pHC. 2.48 nOKa3aH KOHTeHHep, 3alIOJIHeHHbIH napa<pHHoM ,no BbICOTbI 2 M.
OlIpe,neJIHTb ,naBJIeHIDI PI H P2, eCJIH H3BeCTHO, qTO IIJIOTHOCTb napa<p HHa paBHa
800 Kr/M 3 H g = 9,81 M/C 2 .
PI = hI x P x g = 2 M x 800 Kr / M 3 x 9,81 M / c 2 = 15696 ITa HJIH 15,696 KITa.
P2 = h 2 x P x g = 1 M x 800 Kr / M 3 x 9,81 M / C 2 = 7848 lla HJIH 7,848 KITa.
B TOqKe Ha nOJIOBHHe r.rry6HHbI ,naBJIeHHe paBHO
nOJIOBHHe OT 3HaqeHIDI ,naBJIeHIDI Ha ,nHe cocy,na. Ha
nOBepXHOCTH ,naBJIeHHe paBHO HYJIIO, nOCKOJIbK}' r.rry-
6HHa OTCyTCTByeT. CJIe,noBaTeJIbHO, ,naBJIeHHe npo-
lIOPUHOHaJIbHO r.rry6HHe, Ha KOTOpOH OHO H3MepR-
eTCR.
2M
1 M
Puc. 2.48. <l>aKTopbI, BJIHSI-
IOIJJ;He Ha ,n;aBJIeHHe
2. 12. 1. p,SSneHJfe HS nosepXHOCT&,
HSXOAR YIOCR S JlCJfAKOCTJf
PaccMoTpHM npRMoyrOJIbHbIH KOHTeHHep, nOKa3aHHbIH Ha pHC. 2.49. B npe,nbI,ny-
lI(eM npHMepe 6bmo nOKa3aHO, qTO lIOJIOBHHa ,naBJIeHIDI ,nOCTHraeTC.H Ha lIOJIOBHHe
rJIy6HHbI. ITOCKOJIbKY ,naBJIeHHe H3MeIDIeTCR OT HYJUI Ha lIOBepXHOCTH ,no MaKCHMa-
JIbHOrO 3HaqeHIDI Ha lIOJIHOH r.rry6HHe, ,naBJIeHHe Ha nOJIOBHHHOH r.rry6HHe RBJUIeT-
CR cpe,nHHM ,naBJIeHHeM (PA). TaK:
PA =(PI + P2) /2 = P2 /2 (TaK KaKPI = 0)
2 M
Puc. 2.49. ,n;aBJIeHHe Ha nOBepXHOCTb, Haxo,n;}JIJJ;yroCSI B XH,lI;KOCTH
2.11. f/>epMbl
n03TOM)' CHJIa (Hanop), ,neHcTBYIOII(a.H Ha cTeHKY KOHTeHHepa 6y,neT paBHa
rrpoH3Be,neHHIO IIJIOII(a,zJ;H Haxo,lUIlI(eHCR B )l{H)J;KOCTH CTeHKH (As) Ha cpe,nHee ,naB-
JIeHHe (p A) . To eCTb:
Fs = PAX As ·
r,n;e FS - CHJIa (HanOp), ,neHCTByrolI(aR Ha CTeHKY KOHTeHHepa (H); PA - cpe,n;Hee
,n;aBJIeHHe (ITa); As - llJIOII(a,zJ;b HaxO,lUIlI(eHCR B )l{H)J;KOCTH CTeHKH KOHTeHHepa (M 2 ).
IIpuMe"taHue.
1. ITpe,nnOJIaraeTCR, qTO CHJIa npHJIOXeHa B ueHTpe llJIOII(a,zJ;H CTeHKH KOHTeH-
Hepa.
2. CHJIa, ,neHcTByrolI(a.H Ha ,nHO KOHTeHHepa, BbIqHCJUIeTCR qepe3 MaKCHMaJIb-
Hoe, a He qepe3 cpe,nHee ,naBJIeHHe.
IIpHMep , (
Ha pHC. 2.49 nOKa3aH npRMoyrOJIbHbIH pe3epByap C IIJIOma,zJ;blO ,nHa 9 M 2 . OH
HanOJIHeH BO,nOH ,no BbICOTbI 2 M. MCnOJIb3YR g = 9,81 M/C 2 , BbJqHCJIHM:
1) ,naBJIeHHe, ,neHcTByrolI(ee Ha ,nHO pe3epByapa;
2) CHJIY, ,neHcTByroll(yIO Ha ,nHO pe3epByapa;
3) cpe,nHee ,naBJIeHHe, ,neHcTByrolI(ee Ha CTeHKH pe3epByapa;
4) CHJIY (F s ), ,neHcTByroll(yIO Ha CTeHKY pe3epByapa;
5) BbICOTY (h) OT OCHOBaHIDI pe3epByapa, Ha KOTOpOH ,neHCTByeT CHJIa (Fs);
6) P B = h x p x g = 2 M x 1000 Kr / M 3 x 9,81 M/ C 2 = 19220 ITa = 19,22 KIT a;
7) F B = PB x A B = 19220 ITa x 9 M2 = 172980 H = 172,98 KH;
8) P A = (Ps + PB) / 2 = (0 ITa + 19220 ITa) / 2 = 9600 ITa = 9,6 KITa;
9) Fs =PA +As =9600ITax6M2 =57600H=57,6KH (r,ne As - IIJIOII(a,zJ;b
HaxO,lUIlI(eHCR B )l{H)J;KOCTH cTeHKH);
10) CHJIa Fs ,neHcTByeT Ha ueHTp IIJIOII(a,zJ;H Haxo,nRlI(eHCR B )l{H)J;KOCTH CTeHKH
pe3epByapa, T. e. Ha >i x 2 M = 1 M BblIIIe OCHOBaHIDI pe3epByapa.
2.12.2. 3SKOH nSCKsnR
3aKOH ITaCKaJI.H YTBep)K)];aeT, qTO:
,n:aBJIeHHe, OKa3LIBaeMoe Ha 1KII,ZJ;KOCTL B KOHTeHHepe, nepe,n;aeTCJI 1KII,ZJ;K0CTLIO
O,lUlHaKOBO BO Bcex HanpaBJIeHHJIX.
3TO lIOKa3aHO Ha pHC. 2.50, a. aBJIeHHe (P), KOTopoe RBJUIeTCR o,nHHaKOBbIM B
rrpe,neJIax 3aMKH)'Toro npOCTpaHCTBa, MoxeT 6bITb BbJqHCJIeHO CJIe,nyroII(HM 06pa-
30M:
,D:aBJIeHHe (P) = CHJIa (F)/IIJIOII(a,zJ;b (A) = 100 H/0,5 M 2 = 200 H/M2 = 200 ITa.
B COOTBeTCTBHH C 3aKOHOM ITaCKaJI.H ,naBJIeHHe B 200 ITa ,neHcTByeT o,nHHaKOBO
Ha Bce lIOBepXHOCTH. OHO ,neHcTByeT Ha lIopIIIeHb, Ha CTeHKY UHJIHH,l(pa H Ha
KpbIIIIKY UHJIHH,l(pa. ITOCKOJIbKY Macca )l{H)J;KOCTH MaJIa, B 3TOM npHMepe rHJJ;pOCTa-
TWleCKHM ,naBJIeHHeM H COOTBeTCTByrolI(HMH CHJIaMH MO)I{HO npeHe6peqb.
Ha pHC. 2.50, 6 nOKa3aHbI ,nBa UHJIHH,l(pa pa3JIHqHOH nJIOII(a,zJ;H, coe,nHHeHHbIe B
o,n;Hy CHCTeM)' H HanOJIHeHHbIe BO,nOH HJIH rHJJ;paBJIHqeCKOH )l{H)J;KOCTblO. aBJIeHHe
F/la8a 2. HH:JICeHepHaJI cmamUlCO
.. CVlJ1a (F) = 100 H
)KVlAKOCTb
nJ10W.aAb (A) = 0,5 M 2
a
HarpY3-
Ka
CVlJ1a (F 1 ) = 200 H
t C..m8 (F 2 )
nJ1ow.aAb (A 1 ) = 0,5 M 2
nJ10W.aAb (A 1 ) = 0,1 M 2 I
6
CoeAVlHVlTeJ1bHa51 )KVlAKOCTb
Tpy6a
Puc. 2.50. 3aKoH llaCKaJIH:
a - ,n;aBJIeHHe, ,n;eitcTByK>lI(ee Ha )f{H)J;KOCTb B 3aMKH)'TOM 06'beMe;
6 - CHCTeMa C0061I(aIOII(HXCH COCy,n;OB
B CHCTeMe BbIqHCJUIeTCR npH nOMOIl(H paHee npHBe,neHHoH <pOPMYJIbI p = FlA.
AmI MeHbIIIero UHJIHH,l(pa ,naBJIeHHe paBHO:
p = Fi I At = 200 N 10,1 M 2 = 2000 HI M 2 = 2000 ITa HJIH 2 KITa.
O,nHaKo ,nBa UHJIHH,l(pa coe,nHHeHbI BMeCTe, 06pa3YR o,nHY CHCTeMY, CJIe,nOBa-
TeJIbHO, B COOTBeTCTBHH C 3aKOHOM ITaCKaJI.H cyw:eCTByeT 06lI(ee ,naBJIeHHe B 2 KITa,
,neHcTByrolI(ee BO BceH CHCTeMe. IT03TOMY CHJIa, ,neHcTByrolI(a.H co CTOpOHbI 60JIb- i
IIIero nopIIIIDI Ha HarpY3KY, 6y,neT paBHa:
F 2 = P x A 2 = 2 KITa x 0,5 M2 = 1000 H HJIH 1 KH.
CHJIa F 2 B IDITb pa3 60JIbIIIe, qeM CHJIa Ft, lIOCKOJIbKY IIJIOII(a;:J;b lIOpIIIIDI A 2 B I
nRTb pa3 60JIbIIIe IIJIOII(a,zJ;H lIOpIIIIDI At. TaKa.H B03MO)I{HOCTb YBeJIWIHBaTb CHJIbI,
npHMemrn rHJJ;pOCTaTHqeCKHe lIpoueCCbI, HCnOJIb3yeTC.H B TaKHX YCTpOHCTBax, KaK
rHJJ;paBJIWIeCKHe npeCCbI, rH;ipaBJIWIeCKHe ,nOMKpaTbI H fHJJ;paBJIHqeCKHe TOpM03-
HbIe CHCTeMbI cpe,nCTB nepe,nBIDKeHID!. O,nHaKo CJIe.nyeT OTMeTHTb, qTO, XOT.H 60JIb-
IIIHH nopIIIeHb ,neHcTByeT C CHJIOH, B IDITb pa3 60JIbWeH, qeM CHJIa, ,neilcTByrolI(asI ,
Ha MeHbIIIHH nopIIIeHb, 60JIbIIIHH lIopweHb npoxo,nHT TOJIbKO IDITyID qaCTb paCCTO-
RHID!, npeo,nOJIeBaeMOro MeHbIIIHM nopIIIHeM.
rnABA 3
MH>KEHEPHASI
AMHAMMKA
3.1. MH>KeHepHaSi AMHaMMKa
KaK CKa3aHO B HaqaJIe BTOpOH fJIaBbI, HIDKeHepHa.H MexaHHKa MO:>KeT 6bITb pa3,ne-
JIeHa Ha CTaTHKY H ,nHHaMHKY. fJIaBa 3 onlfcbIBaeT ,nHHaMHKY, B KOTOpOH H3yqa1OT-
CR 06'bCKTbI B ,nBIDKeHHH. '
3.2. Pa60Ta
Kor,na npHJIOXeHHa.H K TeJIY CHJIa npHBo,nHT ero B ,nBIDKeHHe, rOBOp.HT, qTO CHJIa
COBepIIIaeT pa60TY. BbInOJIHeHHaR pa60Ta onpe,neJIReTCR KaK npOH3Be,neHHe npOH-
.n;eHHoro paCCTO.HHIDI B MeTpax (M) Ha BeJIHqHHY npHJIOXeHHOH CHJIbI B HblOTOHax
(H), H3MepeHHoH B HanpaBJIeHHH ,nBIDKeHHR. II03ToMY BbIlIOJIHeHHa.H pa60Ta
H3MepReTC.H B HbIOTOHaX-MeTpax, a e,nHHHueH pa60TbI RBJUIeTCR ,lCKOYJIb (,l(:>K),
1 H M = 1 ,[bK.
Ha pHC. 3.1, a lIOKa3aHO TeJIO, ,nBIDKYJI.(eeCR no,n ,neHcTBHeM CHJIbI F no fOpH-
30HTaJIbHOH IIJIOCKOCTH. IIoCKOJIbKY CHJIa ,neHcTByeT Ha TeJIO napaJIJIeJIbHO fOpH-
30HTaJIbHOH IIJIOCKOCTH, BbInOJIHeHHa.H pa60Ta paBHa npOH3Be,neHHIO CHJIbI F (H)
F = 50 H
.- - - - - - --.
. .
. .
----- . .
. .
a
1 4
S=3M
I
.- - - - - - - -
.
.
I
.
F H = 50 cos 30 0
6
1 4
S=3M
.1
Puc. 3.1. Pa60Ta:
a - BbInOJIHeHHaH CHJIOH F;
6 - BbInOJIHeHHaH CHJIOH F H
4.3255
(:;, F/la8a 3. HH:JICeHepHlUl OUHaMUlCa
Ha paccToRHHe S (M). ECJIH CHJIa paBHa 50 H, a paCCTORHHe 3 M, TO BbIlIOJIHeHHasI
pa60Ta paBHa:
50 H x 3 M = 150 H · M = 150 ,L(x.
Ha pHC. 3.1, 6 CHJIa HalIpaBJIeHa no,n yrJIOM K HanpaBJIeHHIO ,nBIDKeHHR TeJIa.
B 3TOM c.rryqae BeKTOp CHJIbI CJIe.nyeT pa3JIO)I(HTb Ha BepTHKaJIbHyro H ropH30HTaJIb-
HyID COCTaBIDIIOII(He, KaK nOKa3aHO Ha pHCyJlKe. COCTaBJUIIOII(a.H CHJIbI, HanpaBJIeH-
Ha.H lIapaJIJIeJIbHO ,nBIDKeHHIO TeJIa, paBHa F cos a.. TI03TOMY BbInOJIHeHHa.H pa60Ta
paBHa S x F cos a. ,l(x. ECJIH CHJIa (F) BHOBb paBHa 50 H, paCCTORHHe (s) CHOBa paB-
HO 3 M H yrOJI a. paBeH 30°, BbInOJIHeHHa.H pa60Ta 6y,neT paBHa:
W = 3 M x 50 H x 0,866 = 129,9 H · M = 129,9 ,L(x.
3.3. 3HeprMSI
ECJIH TeJIO cnoc06HO COBepIIIaTb pa60TY, rOBopRT, qTO OHO 06JIa,naeT 3HeprHeH.
3HeprIDI cyw:ecTByeT B pa3JIHqHbIX <popMax, H ee MO)I{HO paccMaTpHBaTb KaK «3a-
naceHHyID pa60TY», o)l{H)J;alOmyIO CBoero HCnOJIb30BaHIDI. HanpHMep:
. XHMWIeCKaR 3HeprIDI norJIolI(aeTCR HJIH BbICB060)K)];aeTCR B npouecce peaK-
UHH (HalIpHMep, lIpH cropaHHH HCKonaeMoro TOIIJIHBa);
. aTOMHa.H 3HeprIDI BbICB060)K)];aeTCR npH paClI(eIIJIeHHH HJIH coe,nHHeHHH
aTOMHbIX qaCTHU. PaClI(eIIJIeHHe aTOMHbIX qaCTHI( Ha3bIBaeTCR ,neJIeHHeM, a
coe,nHHeHHe aTOMHbIX qaCTHU Ha3bIBaeTCR CHHTe30M. TIpH 3THX peaKI(IDIX
BbI,neJUIeTCR 60JIbIIIOe KOJIHqeCTBO TeIIJIOBOH 3HeprHH;
. TellJIOByro 3HeprHIO MO)I{HO pa3,neJIHTb Ha RBHyID (OlI(yTHMyIO) H CKpbITYIO
TeTIJIoTY. .HBHa.H TeIIJIOTa onpe,neJUIeTCR KOJIWIeCTBOM 3HeprHH, oT,naBaeMoH
HJIH nOJIyqaeMoH npH HarpeBe HJIH OCTbIBaHHH TBep,noro TeJIa, )l{H)J;KOCTH HJIH
ra3a. CKpbITa.H TeTIJIOTa - 3HeprHR, He06xo,nHMa.H .nIDI H3MeHeHIDI arperaT-
Horo COCTORHIDI TBep,noro TeJIa, )l{H)J;KOCTH HJIH ra3a lIPH lIOCTORHHOH TeMne-
paTYPe;
. MeXaHlNeCKa.H 3HeprIDI MoxeT 6bITb no,npa3,neJIeHa Ha nOTeHUHaJIbHyID 3Hep-
rHIO, KHHeTHqecKyID 3HeprHIO H 3HeprHIO ,ne<p0pMaUHH.
1. TIoTeHUHaJIbHa.H 3HeprIDI (B KOHcepBaTHBHoH cHcTeMe) - 3TO 3HeprIDI,
KOTOpOH 06JIa,naeT TeJIO BCJIe,nCTBHe CBoero nOJIOXeHIDI B npOCTpaHCTBe.
HanpHMep, lIOCJIe BbInOJIHeHIDI CBaHHbIM KOlIpOM pa60TbI no no,nIDITHIO
MOJIOTa, 3Ta pa60Ta coxpaIDIeTCR B MOJIOTe KaK lIOTeHUHaJIbHa.H 3HeprIDI.
2. KHHeTHqeCKa.H 3HeprIDI - 3TO 3HeprIDI, KOTOpOH 06JIa,naeT TeJIO 6JIaro,na-
pR CBoeH CKOpOCTH. HanpHMep, lIpH na,neHHH MOJIOTa CBaHHoro Konpa ero
nOTeHUHaJIbHa.H 3HeprIDI YMeHbwaeTCR lIpRMO lIponOpUHOHaJIbHO lIOTepe
BbICOTbI. KHHeTHqeCKa.H 3HeprIDI npH 3TOM YBeJIWIHBaeTCR npRMO npo-
nOpUHOHaJIbHO B03paCTaHHIO CKOpOCTH. TIoTeHUHaJIbHa.H 3HeprIDI npe06-
pa3yeTcR B KHHeTHqecKyID 3HeprHIO BCJIe,nCTBHe ,nBIDKeHIDI MOJIOTa. B MO-
MeHT y,napa MOJIOTa 0 CBalO BCR lIOTeHUHaJIbHaR 3HeprIDI yxe npe06pa30-
BaHa B KHHeTHQecKyID 3HeprHIO, KOTOpa.H H HClIOJIb3yeTcR .nIDI COBepIIIe-
HIDI pa60TbI, a KOHKpeTHO .nIDI Toro, qT06bI 3a6HTb CBalO B 3eMJIIO.
KHHeTHqeCKa.H 3HeprIDI paBHa .!. m V 2 .
2
3.4. MOll/llocmfJ
3. 3HeprIDI ,ne<popMaUHH (B KOHcepBaTHBHoH cHcTeMe) - 3TO 3HeprIDI, Ha-
KOIIJIeHHa.H B HanpIDKeHHOM yrrpyrOM MaTepHaJIe. HanpHMep, eCJIH MeTaJI-
JIHqeCKHH CTepxeHb y.n.rrHHeH pacTRrHBalOlI(eH CHJIOH, Han CTep)I{HeM co-
BepIIIeHa pa60Ta. IIpH YCJIOBHH, qTO ,ne<popMaUIDI He BbIIIIJIa 3a npe,neJIbI
06JIaCTH yrrpyrHX ,ne<popMaUHH .nIDI ,naHHoro MaTepHaJIa, BbInOJIHeHHaB pa-
60Ta coxpaIDIeTCR B paCT.HHYTOM CTep)I{He B BH)];e 3HeprHH ,ne<popMaUHH.
3Ta 3HeprIDI MoxeT 6bITb BbICB060)l{)];eHa nOCJIe YCTpaHeHIDI ,ne<popMHPY-
IOlI(eH CHJIbI. TaKHe xe paccyx,neHIDI npHMeHHMbI H B TOM c.rryqae, Kor,na
CTepxeHb 3aKpyqeH, KaK TOpCHOHHbIH BaJI.
3.3. 1. COXpSHeHJfe 3HeprJfJf
3aKOH coxpaHeHIDI 3HeprHH 3aKJIIOqaeTCR B CJIe.nyIOlI(eM: 3HeprIDI He MoxeT 6bITb
C03,naHa HJIH yJlHqTOXeHa, OHa TOJIbKO npe06pa30BbIBaeTC.H H3 O,nHOH <pOpMbI B
)
,n;pyryIO. 5
HanpHMep, Ha 3JIeKTpOCTaHUHH XHMHqeCKa.H 3HeprIDI roplOqero BbICB060)l{)];a-
eTCH npH C)l{HraHHH H npe06pa3yeTC.H B TeIIJIOByro 3HeprHIO, KOTOpa.H, B CBOIO oqe-
pe,n;b, C03,naeT nap B KOTJIe. IIap BpalI(aeT TYP6HHY, KOTOpa.H npe06pa3yeT 3HeprHIO
napa B MeXaHHqecKYIO 3HeprHIO. 3Ta MeXaHHqeCKa.H 3HeprIDI npe06pa3Y TCR B
3JIeKTIlWlecKYIO 3HeprHIO reHepaTopOM nepeMeHHoro TOKa, npHBo,nHMbIM B ,nBIDKe-
HHe Typ6HHOH. Bo Bcex 3THX npoueccax npe06pa30BaHIDI npOHCXO,lUIT nOTepH
3HeprHH BCJIe,nCTBHe HeCOBepIIIeHCTBa HClIOJIb3yeMoro 060Py,nOBaHHR. «IIOTepRH-
HaR» 3HeprIDI He HCqe3aeT (3aKOH coxpaHeHIDI 3HeprHH), HO OHa npe06pa3yeTcR B
Tame <pOpMbI 3HeprHH, KOTopbIe He BHOCRT BKJIan B nOJIe3HyID pa60TY. HanpHMep,
B pe3YJIbTaTe pa60TbI, BbInOJIIDIeMOH .nIDI npeo,nOJIeHIDI TpeHIDI, HarpeBalOTCR
pa60qHe nOBepXHOCTH H CMa30qHbIe MaTepHaJIbI.
3.4. MO HOCTb
MOII(HOCTb - 3TO CKOpOCTb BbInOJIHeHIDI pa60TbI, H MaTeMaTHqeCKH ee MO)I{HO
BbIpa3HTb CJIe,nyroII(HM o6pa30M:
MOII(HOCTb (BT) = BbIlIOJIHeHHa.H pa60Ta (,L(x)/3aTpaqeHHOe BpeM.H (c) =
= CHJIa (H) x CKOpOCTb (M/C).
E,nHHHua MOII(HOCTH BaIT (BT), r,ne 1 BT paBeH pa60Te B 1 ,L(x, BbInOJIHeHHOH
3a 1 c.
npHMep.
TeJIo BeCOM 5 KH HyxHO no,nIDITb Ha BbICOTY 15 M 3a 10 c. Onpe,neJIHTb Tpe6ye-
MyIO MOII(HOCTb.
BbInOJIHeHHa.H pa60Ta = CHJIa (Bec TeJIa) x npOH,neHHOe paCCTORHHe =
= 5 KH x 15 M = 75 KH M = 75 .
Tpe6yeMaR MOII(HOCTb = BbInOJIHeHHa.H pa60Ta/BpeMR = 75 K,[(x/l0 C =
= 7,5 KBT (lIpH 100% K03<P<PHUHeHTe nOJIe3HOrO ,neHcTBIDI).
(::: F/la8a 3. HH:JICeHepHaJI OUHaMUlCa
3.5. KOsct>ct>MLJ.MeHT none3Horo AeMcTBMSI
K03<P<PHUHeHT nOJIe3HOrO ,neHCTBIDI (KII ) - 3TO Mepa «nOJIe3HOCTH» MaIIIHHbI,
lIpouecca HJIH onepaUHH.
KII,LJ; (,,) = [BbIXO,nHasI pa60Ta (HJIH MOII(HOCTb)/BXO,nHasI pa60Ta (HJIH MOII(HOCTb)].
KII,LJ; MO)I{HO BbIpaxaTb B lIpoueHTax (%) HJIH B ,neCRTWIHbIX ,nOIDIX OT 1. Ha
npaKTHKe Krr YCTpoHCTBa HHKor,na He MoxeT ,nOCTHraTb 100% HJIH 1.
npHMep.
ECJIH Krr lIo,n'beMHHKa B npe,nbI.D;ylI(eM lIpHMepe paBeH 80%, Tpe6yeTcR BbI-
qHCJIHTb <paKTWIeCKYIO Tpe6yeMyID MOII(HOCTb.
KII,LJ; (%) = (BbIXO,nHasI MOII(HOCTb x 100)/Bxo,nHasI MOII(HOCTb.
CJIe,noBaTeJIbHO 80% = 7,5 KBT/BXO,nHasI MOII(HOCTb
OTclO,na BXO,nHasI MOII(HOCTb = (7,5 x 100)/80 = 9,375 KBT.
3.6. CKOPOCTb M YCKopeHMe
. MO.D;yJIb CKOpOCTH - 3TO CKaJI.HpHasI BeJIHqHHa, lIOCKOJIbK}' HMeeT TOJIbKO
qHCJIOBOe 3HaqeHHe (HalIpHMep, 30 KM/q).
. CKOpOCTb - 3TO BeKTOpHasI BeJIWIHHa, nOCKOJIbKY HMeeT H qHCJIOBOe 3Haqe-
HHe, H HalIpaBJIeHHe (HanpHMep, 30 KM/q Ha ceBep).
. Y CKopeHHe - 3TO 6bICTpoTa H3MeHeHIDI CKOpOCTH.
3.6.1. Mo,qyn& CKOpOCTJf
MO.D;yJIb CKOpOCTH - 3TO 6bICTpOTa H3MeHeHIDI npOH,neHHOrO paCCTORHIDI. To eCTb
MO.D;yJIb CKOpOCTH = paccToRHHe/BpeM.H. 06bIqHO MO.D;yJIb CKOpOCTH H3MepReTCR B
KHJIOMeTpax B qac (KM/q) HJIH B MeTpax B ceKYH.D;Y (M/C).
3.6.2. CKOPOCT&
CKOpOCTb - 3TO 6bICTpOTa H3MeHeHIDI paCCTORHIDI, npoH,neHHoro B onpe,neJIeHHOM
HanpaBJIeHHH. 0 TeJIe, ,nBIDKyI.I:(eMCR C nOCTORHHOH CKOpOCTblO lIO lIpRMOH JIHHHH,
rOBopRT, qTO OHO ,nBIDKeTC.H paBHOMepHO H np.HMOJIHHeHHO. TeJIo ,nBIDKeTCR C no-
CTORHHOH CKOpOCTblO, eCJIH 3a paBHbIe npOMexyrKH BpeMeHH OHO lIpOXO,nHT
paBHbIe paCCTORHIDI B,nOJIb lIpRMOH JIHHHH.
CKOpOCTb = ds/dt B 3a,naHHOM HanpaBJIeHHH.
3.6.3. YCKopeHJfe
YCKopeHHe - 3TO 6bIcTpoTa H3MeHeHIDI CKOpOCTH TeJIa:
. TeJIO ,nBIDKeTCR paBHOYCKopeHHO, T.e. C nOCTORHHbIM YCKopeHHeM, eCJIH 3a
paBHbIe lIpOMexyrKH BpeMeHH CKOpOCTb TeJIa H3MeIDIeTCR Ha O,nHY H 11' :>Ke
BeJIWIHHY. YCKopeHHe H3MepReTCR B M/C 2 ;
. TopMoxeHHe HJIH 3aMe.n.rreHHe ,nBIDKeHIDI - lIpouecc 06paTHbIH YCKopeHHIO.
Ero qaCTO onpe,neJUIIOT KaK OTpHuaTeJIbHOe YCKopeHHe.
3.6. C1copocmb u YCICopeHue
3.6.4. YpssHeHJffl, OTHOCfI JfeCfI K CKOpOCTJf Jf YCKopeHJfIO
B npHBe,n;eHHbIX HIDKe ypaBHeHIDIX HCnOJIb3yroTC.H CJIe,n;yrolI(He CHMBOJIbI:
VI - HaqaJIbHa.H CKOpOCTb B Me1pax B ceKYH.nY (M/c);
V2 - KOHeqHa.H CKOpOCTb B MeTpaX B ceKYH.nY (M/c);
a - YCKopeHHe B Me1pax Ha ceKYH.nY B KBMpaTe (M/c 2 );
s - npoaneHHbIH rryTb B Me1pax (M);
t - BpeWl, B3.HTOe B ceKYJl,l(ax (c).
(Bce npHBe,n;eHHbIe ,n;aJIee <pOPMYJIbI OTHOC.HTC.H HCKJIIOqHTeJIbHO K paBHOYCKO-
peHHOMY ,n;BIDKeHHIO. - llpUMett. nep.)
1. Cpe,n;HIDI CKOpOCTb = (VI + V 2) I 2
2. KOHeqHa.H CKOpOCTb V2 = VI + at
3. IIpoaneHHbIH lIYTb S = (VI + V2)t I 2
4. llpoaneHHbIH rryTb S = Vit + at 2 I 2
5. YCKopeHHe a = (V2 - VI) I t
6. YCKopeHHe a = (vi - vl) I 2s
)
)
3.6.5. HMnyn&C
KOJIWleCTBO ,n;BIDKeHIDI HJIH HMrryJIbC TeJIa OlIpe,n;eJUIeTC.H CJIe,l(yIOII(HM 06pa30M:
HMrryJIbC = Macca x CKOpOCTb = mv.
OCHOBHa.H e,n;HHHQa H3MepeHIDI HMrryJIbCa Kr M/c.
3.6.6. 3SKOH&1 MeXSHJfKJf H&IOTOHS
IIepBbIH 3aKOH HbIOTOHa YTBep)l{)];aeT, qTO TeJIO coxpaIDIeT COCTO.HHHe nOKO.H IDIH
paBHOMepHoro np.HMOJIHHeHHOro ,n;BIDKeHIDI ,n;o Tex lIOp, nOKa pe3YJIbTHpyrolI(aR
Bcex CHJI, ,n;eHcTByroII(HX Ha TeJIO, paBHa HYJIIO. 3TO 03HaqaeT, qTO TeJIO He HaqHeT
,lJ;BIDKeHHe, He H3MeHHT HanpaBJIeHHe HJIH CKOpOCTb CBoero ,n;BIDKeHIDI ,n;o Tex lIOp,
nOKa ero He lIpHHY,n;HT K 3TOMY HeKOTOpa.H BHeIIIHIDI CHJIa. ll03TOMY CHJIY MO)I{HO
onpe,n;eJIHTb KaK HeqTO H3MeIDIIOlI(ee HJIH lIbITaIOlI(eeC.H H3MeHHTb ,n;BIDKeHHe TeJIa
nyTeM B03,n;eHCTBIDI Ha Hero.
BTopOH 3aKOH HbIOToHa YTBep)l{)];aeT, qTO CKOpOCTb H3MeHeHIDI HMrryJIbCa TeJIa
nponOPQHOHaJIbHa pe3YJIbTHpyrolI(eH CIDIe, ,n;eHcTByrolI(eH Ha 3TO TeJIO, H H3MeHe-
HHe lIpOHCXO,n;HT B TOM HalIpaBJIeHHH, B KOTOpOM npIDIoxeHa CHJIa.
TpeTHH 3aKOH HbIOTOHa YTBep)l{)];aeT, qTO ,lI;IDI KIDK,lI;OH ,n;eHCTByrolI(eH CHJIbI CY-
m;ecTByeT paBHa.H no BeJIHqHHe H lIpOTHBOnOJIO)I{Ha.H lIO HanpaBJIeHHIO CHJIa npOTH-
Bo,n;eHCTBH.H (CM. pa3,n;eJI 3.6.8).
H3 BToporo 3aKOHa ,n;BIDKeHIDI HbIOToHa CJIe,n;yeT, qTO:
llpHJIOXeHHaR CHJIa ,., 6bIC1poTa H3MeHeHIDI HMrryJIbca.
llpHJIOXeHHa.H CHJIa ,., H3MeHeHHe B HMnYJIbC/BpeWl.
CJIe,n;oBaTeJIbHO, eCJIH CHJIa F ,n;eHcTByeT Ha TeJIO MaCCOH m, TaK qTO ero CKO-
pOCTb H3MeIDIeTC.H OT VI ,n;o V2 3a BpeWl t, TO:
F ,., (mV2 - mVI) I t
F ,., m(V2 - VI) I t
F'" ma, r,n;e a =(V2 - VI) I t
F/la8a 3. HH:JICeHepHaJI OUHllMUICa
ll03TOMY
F = ma x const
llpH BbI60pe B KaqeCTBe e,n;HHHU;bI H3MepeHIDI CHJIbI (H), e,n;HHHQbI H3MepeHIDI
MaCCbI (Kr) H e,n;HHHQbI H3MepeHIDI YCKopeHIDI (M/C 2 ), const CTaHOBHTC.H paBHoH 1:
F=ma
DpHMep
CKOpOCTb JlerKOBOro aBTOM06HJI.H MaCCOH 4000 Kr 3a 10 C YBeJlHqHBaeTC.H OT 10
,n;o 60 KM/q. Onpe,n;eJlHTb BeJlHqHHY CHJIY, KOTOpyro TPe6yeTC.H npHJIOXHTb, qT06bI
06eClIeqHTb TaKoe YCKopeHHe.
V2 = VI + at (ypaBHeHHe (2), pa3,n;eJl 3.6.4),
r,n;e: V2 = 60 KM/q = 16,67 M/C, VI = 10 KM/q = 2,78 M/C, t = 10 C
T or,n;a:
a = (V2 - VI) / t = (16,67 - 2,78) / 10 = 1,39 M/C 2
CJle,n;OBaTeJlbHO:
F = ma = 4000 Kf x 1,39 M/C 2 = 5,56 KH.
.
3.6. 7. rpSSJfTsqJffl
Bec - 3TO CHJIa. HMeHHO lIO,n; BJIIDIHHeM CHJIbI T.HXeCTH, ,n;eHcTByrolI(eH Ha Maccy,
TeJlO ,n;BIDKeTC.H HJIH CTPeMHTC.H ,n;BHraTbC.H C YCKopeHHeM. OTMeTHM, qTO BeJlHqHHa
YCKopeHIDI 9,81 M/C 2 .HBJUIeTC.H cpe,n;HHM 3HaqeHHeM ,lI;IDI HaIIIeH nnaHeTbI. OHa
CJlerKa MeIDIeTC.H B 3aBHCHMOCTH OT MeCTOlIOJlOXeHH.H, o,n;HaKO 3HaqeHHe 9,81 M/C 2
lIpHMeHHMO ,lI;JI.H JlI06bIX lIpaKTHqeCKHX u;eJleH. YCKopeHHe, 06ycJloBJleHHoe CIDIOH
T.HXeCTH, 0603HaqaeTC.H CHMBOJlOM g, n03TOMY BbIpIDKeHHe F = ma npe06pa3yeTC.H B
F = mg npH BbIqHCJleHIDIX, CB.H3aHHbIX C CIDIOH T.HXeCTH.
(Ha nnaHeTe 3eMJISI CHJIa F ,n;eHcTByeT Ha Maccy 5 Kf = 5 x 9,81 = 49,05 KH. llo-
CKOJlbKY CHJIa TIDKeCTH Ha JIYRe COCTaBJUleT lIpH6J1H3HTeJlbHO 1/6 CHJIbI T.HXeCTH Ha
3eMJIe, Ha JIYRe CIDIa F ,n;eHcTByeT Ha Maccy 5 Kf = 5 x (9,81/6) = 8,175 KH.)
3.6.8. COXpSHeHJfe JfMnyn&CS
3aKOH coxpaHeHIDI KOJlHqeCTBa ,n;BIDKeHIDI HJlH HMrryJlbCa CJle,n;yeT H3 TPeTbero 3a-
KOHa HbIOTOHa: ,lI;IDI K3.)I{,lI;0H ,n;eHcTByrolI(eH CHJIbI CylI(eCTByeT paBHa.H lIO BeJlWlHHe
H npOTHBOnOJlO)I{HaSI no HanpaBJleHHIO CHJIa lIpoTHBo,n;eHcTBH.H. 3aKOH coxpaHeHIDI
HMrryJlbCa YTBep)l{)];aeT, qTO lIOJlHblH HMnYJlbC CHCTeMbI B ,n;aHHOM HalIpaBJIeHHH
OCTaeTCSI nOCTO.HHHbIM ,n;o Tex nop, rtoKa HHKaKHe BHeIIIHHe CIDIbI He ,n;eHCTByroT Ha
CHCTeMY B 3TOM HalIpaBJleHHH. 3TOT 3aKOH lI03BOJl.HeT JlerKO peIIIaTb 3a;:J;aqH, B KO-
TOpbIX ytIHTbIBaIOTC.H TOJlbKO BHyrpeHHHe CHJIbI.
PaCCMOTPHM cHapM, BbmeTalOlI(HH H3 opy,n;IDI. PaBHbIe H lIpOTHBOlIOJlOXHO
HanpaBJleHHbIe CHJIbI, ,n;eHCTByro He Ha CHapM H Ha opy,n;He, npHBO,nRT K TOM}',
qTO CHapM ,n;BIDKeTC.H BlIepe,n;, a opy,n;He ,n;BIDKeTC.H Ha3a;:J;. llOCKOJlbKY BHeIIIHHe
CHJIbI OTCYTCTByroT, eCJlH 6bI opy,n;He H CHapM HMeJlH paBHbIe MaCCbI, OHH ,n;BHra-
3.6. C1copocmb u YCICopeHue
JlHCb 6bI C o,n;HHaKOBbIMH CKOpOCT.HMH. K CqaCTbIO, Ha rrpaKTHKe Macca opy,n;IDI Ha-
MHoro 60JIbIIIe MaCCbI CHapg,na, rr03TOMY CKOpOCTb OTKaTa HaMHoro MeHbIIIe, qeM
CKOpOCTb ,n;BIDKeHIDI cHapg,na Brrepe,n;.
DpHMep.
TeJIo P MaCCOH 15 Kr, ,n;BIDKYIUeeC.H co CKOpOCTbIO 30 M/c, CTaJIKHBaeTC.H co
BTOpbIM TeJIOM Q MaCCOH 5 Kr, ,n;BIDKYIUHMC.H B TOM xe HarrpaBJIeHHH CO CKOpOCTbIO
5 M/c. Orrpe,n;eJIHTb 06lI(y1O CKOpOCTb ,n;BYX TeJI lIOCJIe CTOJIKHOBeHIDI.
COrJIaCHO 3aKOHY coxpaHeHIDI HMrrYJIbca:
HMrryJIbC ,n;o CTOJIKHOBeHIDI = HMrrYJIbC nOCJIe CTOJIKHOBeHIDI
Macca x CKOpOCTb ,n;o CTOJIKHOBeHIDI = Macca x CKOpOCTb lIOCJIe CTOJIKHOBeHIDI
15 Kr x 30 M/c + 5 Kr x 5 M/c = (15 Kr + 5 Kr)v,
r,ne v - CKOpOCTb rrOCJIe coy,n;apeHIDI.
v = 23,75 M/c.
Ha pHC. 3.10, a lIOKa3aHbI ,n;Ba TeJIa, ,n;BIDKYIUHeC.H ,n;o CTOJIKHOBeHIDI B lIpOTHBO-
rrOJIO)I{HbIX HalIpaBJIeHIDIX. ll03TOMY CKOpOCTb BToporo TeJIa Q CqHTaeTC.H 0TPHIJ;a-
TeJIbHOH BeJIHtlHHOH. BbIqHCJIeHIDI BbIlIOJIIDIIOTC.H CJIe,n;yroII(HM 06pa30M:
HMrryJIbC ,n;o CTOJIKHOBeHIDI = HMrryJIbC lIOCJIe CTOJIKHOBeHIDI
15 Kr (+ 30 M/c) + 5 Kr (-5 M/c) = (15 Kr + 5 Kr)v.
v = 21,25 M/c.
3.6.9. B03,qeiicTSJfe CTPYJf JlCH,qKOCTJf HS HenOASJfJICHOe Teno
. ECJIH CTPY.H OTKJIOIDIeTC.H OT CBoeH JIHHHH ,n;BIDKeHIDI, TO H3 rrepBoro 3aKOHa
HbIOTOHa CJIe,n;yeT, qTO Ha Hee ,n;eHcTByeT CHJIa.
. ECJIH CHJIa, ,n;eHCTByrolI(a.H Ha CTPyro, rrporrOPQHOHaJIbHa 6bICTPOTe H3MeHe-
HIDI HMlIYJIbCa H ,n;eHcTByeT B HarrpaBJIeHHH H3MepeHIDI HMrrYJIbCa, TO lIpHMe-
HHM BTOpOH 3aKOH HbIOTOHa.
. ECJIH CHJIa, C KOTOpOH TeJIO ,n;eHcTByeT Ha CTPyro, H3MeIDIeT ee HMrrYJIbC, TO
3Ta CHJIa ,n;OJDKHa 6bITb paBHa lIO BeJIHqHHe H lIpOTHBOrrOJIO)I{Ha lIO HarrpaB-
JIeHHIO CHJIe, C KOTOpOH CTPY.H ,n;eHcTByeT Ha TeJIO.
IIOCKOJIbKY ,n;aHHbIH rrpHMep BKJIIOqaeT Harrop CTPYII )l{Jl)];KOCTH, TO OH CBSl3aH
CO CKOpOCTbIO rrOTOKa CTPYII, H BTOpOH 3aKOH HbIOTOHa MO)I{HO 3arrHcaTb KaK:
CHJIa = CKOpOCTb H3MeHeHIDI HMIIyJIbCa =
= (Macca x H3MeHeHHe CKOpOCTH)/3a1paQeHHOe BpeWI =
= MaCCOBaH CKOpOCTb nOTOKa x H3MeHeHHe CKOpOCTH B HanpaBJIeHHH CHJIbI.
npHMep
CTPY.H Bo,n;bI,n;HaMeTpOM 50 MM co CKOpOCTbIO 75 M/c rreplIeH)J;HKYJI.HpHO y,n;ap.H-
OT B Herro,n;BIDKHoe TeJIO. OlIpe,n;eJIHTb CHJIY, ,n;eHcTByrolI(ylO Ha TeJIO.
IIycTb CTPY.H BO,n;bI pa36pbI3rHBaeTC.H lIpH CTOJIKHOBeHHH C TeJIOM pan;HaJIbHO,
-T.e. OHa OTKJIOIDIeTC.H OT lIepBOHaqaJIbHOrO HalIpaBJIeHIDI ,n;BIDKeHIDI Ha yrOJI 90°.
CJIe,noBaTeJIbHO, CTPY.H Tep.HeT BeCb CBOH HMrrYJIbC B HalIpaBJIeHHH, rreprreH)J;HKY-
IDIpHOM TeJIy, KaK lIOKa3aHO Ha pHC. 3.2.
F.na8a 3. HH:JICeHepHaJI OUHllMUlCa
050 MM
)l(ecTKa51
nJ1aCTVlHK8
Puc. 3.2. Hanop XH,lI;KOCTH Ha Heno-
,n;BHXHoe TeJIO
75 M/C
llJIOII(Mb CTPYII = (1t I 4) x (50 I 1000)2 M2 = 6,251t X 10-4 M 2 .
061>eMHaR CKOpOCTb lIOTOKa = 6,251t X 10-4 M 2 x 75 MI C = 0,0471t M3 I c.
MaCCOBa.H CKOpOCTb lIOTOKa (Bo,n;a) = 061>eMHaR CKOpOCTb lIOTOKa x
x IIJIOTHOCTb BO,n;bI = 0,0471t M 3 I s x 10 3 Kr/M 3 = 471t MKrI C.
CHJIa, ,n;eHCTByrolI(aR Ha TeJIO = MaCCOBaR CKOpOCTb lIOTOKa x H3MeHeHHe CKOpOCTH.
llpUMettaHue.
llpH CTOJIKHOBeHHH CTPYII )I{H,lI;KOCTH C TeJIOM rrpOHcxo,n;HT MrHOBeHHoe H3Me-
HeHHe CKOpOCTH OT 75 ,n;o 0 M/c.
ll03TOMY:
CIDIa, ,n;eHcTByrolI(a.H Ha TeJIO = 471t Kr/c X 75 M/c = 3525 H = 3,525 KH.
3.6. 10. HHepqJffl
llepBbIH 3aKOH ,n;BIDKeHH.H HbIOTOHa YTBep)l{)];aeT, qTO TeJIO coxpaIDIeT COCTO.HHHe
rrOKO.H HJIH paBHOMepHoro rrp.HMOJIHHeHHOro ,n;BIDKeHIDI, rrOKa Ha Hero He ,n;eHCTBY-
eT BHeIIIIDI.H CHJIa. 3TO corrpOTHBJIeHHe H3MeHeHHIO COCTO.HHIDI Ha3bIBaeTC.H HHep-
QHeH TeJIa, H BeJIHqHHa HHepIJ;HH 3aBHCHT OT MaCCbI TeJIa. ll03TOMY, qT06bI lIPH-
,n;aTb YCKopeHHe HaXO,nRlI(eMYcSI B COCTO.HHHH lIOKO.H lIOJIHOCTbIO 3arpyxeHHoMY T.H-
XeJIbIM 6arIDKoM aBToM06IDIIO, TPe6yeTC.H HaMHoro 60JIbIIIaR cHJIa, qeM cIDIa, He-
06XO,n;HMaR ,lI;IDI C006lI(eHIDI TaKoro xe YCKopeHIDI HaXo,nRlI(eMYc.H B COCTO.HHHH
rrOKO.H BeJIOCHne,n;y.
3.6. 11. COnpOTJfSneHJfe ASJfJICeHJfIO
MCrrOJIb3Y.H BbIpIDKeHIDI F = ma, CJIe,n;yeT 06palI(aTb BHHMaHHe Ha CJIe,l(yIOlI(ee.
. F MoxeT 6bITb KaK YCKOp.HIOlI(eH CHJIOH, TaK H 3aMe,ll;IDIIOlI(eH (TOpM03.HlI(eH).
. llpH BbIlIOJIHeHHH HH)I{eHepHbIX paCqeTOB qaCTO lIpHXO,n;HTC.H OlIpe,n;eJUITb Be-
JIHqHJIy YCKopeHIDI HJIH 3aMe,ll;JIeHIDI, HCrrOJIb3Y.H ypaBHeHIDI ,n;BIDKeHIDI pac-
CMOTPeHHbIe BbIIIIe.
. Kor,n;a CIDIe, BbI3bIBaIOlI(eH YCKopeHHe TeJIa, rrpoTHBo,n;eHcTByeT ,n;pyraR CHJIa,
cTPeMSllI(a.HC.H 3aMe,ll;JIHTb (3aTopM03HTb) ,n;BIDKeHHe TeJIa, TO rOBOp.HT, qTO
,n;BIDKeHHe TeJIa BCTpeqaeT COlIpOTHBJIeHHe.
Ha pHC. 3.3, a nOKa3aHa CIDIa P, ,n;eHCTByro aR Ha TeJIO MaCCOH m, TaK qTO OHO
,n;BIDKeTC.H C YCKopeHHeM a, rrpeo,n;OJIeBaR rrpOTHBOrrOJIO)I{HO HarrpaBJIeHHOe COlIPOTHB-
JIeHHe R. YCKOp.HIOlI(aR CIDIa Fa = (P - R) H Fa = ma, CJIe,n;OBaTeJIbHO, P - R = ma.
Ha pHC. 3.3, 6 rrOKa3aHO lIo,n;BeIIIeHHoe TeJIO MaCCOH m. llYCTb P - HaT.HXeHHe
BepeBKH, Ha KOTOpOH no,n;BeIIIeHO TeJIO.
3.6. C"opocmb u yc"opeHue I 05
HaT5I)I(eHVle (P) H
YCKopeHVle (a) M/C 2
C na conpOTVlB-
neHVl5I (R) H
Macca (m) Kr
YCKopeHVle
(a) M/C 2
f 1
Macca (m) Kr
CVlJ1a(P)H
Bec W = mg H
I
a
6
YCKopeHlile (a) M/C 2 \
)
'\
/
Bec W = mg H
B
Puc. 3.3. ConpOTHBJIeHHe ,n;BH)I{eHHIO:
a - Ha ropH30HTaJIbHOH llJIOCKOCTH;
6 - no,n;BemeHHoro TeJIa;
B - Ha HaKJIOHHOH llJIOCKOCTH
1. llpH ,n;BIDKeHHH BBepx HJIH BHH3 C IIOCTO.HHHOH CKOpOCTbIO HJIH eCJIH TeJIO
HaxO,n;HTC.H B COCTO.HHHH IIOKO.H YCKopeHHe paBHO HYJIIO; II03TOMY YCKOp.HIOlI(aR
CHJIa (F) TaIOKe paBHa HYJIIO.
2. llpH ,n;BIDKeHHH BHH3 C YCKopeHHeM a W 60JIbIIIe, qeM P, r,n;e W = mg. Tor,n;a
YCKOp.HIOlI(aR CHJIa F = (W - P), lI03TOMY (W - P) = ma.
3. llpH ,n;BIDKeHIDI BBepx C YCKopeHHeM a F = (P - W), II03TOMY (P - W) = ma.
Ha pHC. 3.3, B IIOKa3aHO TeJIO, paCIIOJIOXeHHOe Ha rJIa;:J;KoH HaKJIOHHOH IIJIOCKO-
CTH. B 3TOM lIpHMepe JII06bIM TPeHHeM IIpeHe6peraeM. KOMIIoHeHTa Beca TeJIa W;
.neHCTByIOlI(aR BHH3 B,n;OJIb IIJIOCKOCTH, paBHa Wsin8, r,n;e W = mg (H):
1. ECJIH P 60JIbIIIe W sin 8, TeJIO ,n;BIDKeTC.H BBepx lIO IIJIOCKOCTH C YCKopeHH-
eM a. YCKOP.HIOlI(aR CHJIa F = P - W sin 8, II03TOMY P - W sin 8 = ma.
2. ECJIH W sin 8 60JIbIIIe P, TeJIO ,n;BIDKeTC.H BHH3 lIO IIJIOCKOCTH C YCKopeHHeM a.
Tor,n;a YCKOp.HIOlI(aR CHJIa F = W sin 8 - P, II03TOMY W sin 8 - P = ma.
(::: F/la8a 3. HH:JICeHepHaJl OUHaMUlCa
DpHMep
Ha TeJIO MaCCOH 600 Kf ,n;eHcTByeT CHJIa corrpOTHBJIeHH.H 400 H. OlIpe,n;eJIHTb,
KaKYJO lIOJIHyIO CHJIY He06xo,n;HMO rrpHJIOXHTb K Te.rry, qT06bI OHO ,n;BHraJIOCb C rro-
CTO.HHHbIM YCKopeHHeM 3 M/c 2 .
llYCTb HCKOMaR rrOJIHa.H CIDIa = P.
Tor,n;a: YCKOp.HIOlI(aR CHJIa F = (P - 400).
TaK KaK F = ma, TO: (P - 400 H) = 600 Kr x 3 M/c.
ll03TOMY: P = 400 H + 1800 H = 2200 H.
DpHMep
llaccIDKHpcKHH JIH<pT MaCCOH 1500 Kr HaqHHaeT ,n;BIDKeHHe H3 COCTO.HHIDI rro-
KOSI H, ,n;BHraSiCb BBepx, Qepe3 10 M ,n;OCTHraeT CKOpOCTH 5 M/c. Onpe,n;eJIHTb HaTSI-
)KeHHe TPoca JIH<pTa, eCJIH YCKopeHHe nOCTO.HHHO. llpHMHTe g = 9,81 M/c 2 .
QT06bI HaHTH YCKopeHHe, HClIOJIb3yeM BbIpIDKeHHe (V 2 )2 = (Vi)2 + 2as.
ll03TOMY: (5 M 2 /c 2 )2 = 0 + 2 x a x 10 M.
TaKHM o6pa30M: a = 25/(2 x 10) = 1,25 M/C 2 .
lloKa JIH<pT Haxo,n;HTC.H B COCTO.HHHH rroKo , HaT.HXeHHe B TPoce 06YCJIOBJIeHO
TOJIbKO ,n;eHcTByromeH Ha JIH<pT CHJIOH T.HXeCTH (mg), paBHoH 1500 Kr x 9,81 M/c 2 =
= 14715 H. llOCKOJIbKy JIH<pT ,n;BIDKeTC.H C YCKopeHHeM BBepx, HaT.HXeHHe P 60JIbme
14715 H.
llYCTb P = HaTIDKeHHe B rro,n;'beMHOM Tpoce.
Tor,n;a YCKOpSiIOmaR CIDIa F = P - 14715 = ma = 1500 x 1,25 M/c 2 .
ll03TOMY: P = 14715 + 1500 x 1,25 = 16590 H = 16,59 KH.
3.7. Bpaw.aTenbHoe ABM>KeHMe
Kor,n;a xeCTKoe TeJIO BpamaeTC.H BOKpyr <pHKcHpoBaHHoH OCH, Bce TOqKH 3TOro
TeJIe BbIHyx,n;eHbI ,n;BHraTbCSI lIO KpyrOBbIM TPaeKTopIDIM. ll03TOMY 3a JII060H rrepH-
o,n; BpeMeHH Bce TOqKH TeJIa COBepIIIaT O,n;HO H TO )Ke KOJIHqeCTBO 060pOTOB BOKpyr
OCH BpaIl(eHIDI; TaKHM 06pa30M, CKOpOCTb BpaII(eHIDI MO)KHO orrpe,n;eJIHTb qepe3
060POTbI B MHHyry (06/MHH). O,n;HaKO ,lI;IDI MHorHX HH)I{eHepHbIX 3Maq qaCTO 6bI-
BaeT He06xo,n;HMO BbIpa3HTb CKOpOCTb BpaII(eHIDI Qepe3 yrOJI rrOBopOTa B PMHaHax
B e,n;HHHn;y BpeMeHH.
3. 7. 1. PSAJfSH&1
PMHaH - 3TO n;eHTpaJIbHbIH yrOJI, CTSlrHBaeMbIH ,n;yrOH, ,lI;JIHHa KOTOpOH paBHa ee
PMH)'CY. 3TO lIOKa3aHO Ha pHC. 3.4, a. CooTHoIIIeHHe Mex,n;y ,lI;JIHItOH ,l(yrH, PMHY-
COM H yrJIOM B PMHaHax rrOKa3aHO Ha pHC. 3.4, 6.
M3 orrpe,n;eJIeHIDI PMHaHa:
rr03TOMY
sir = 8,
s = reo
3.7. Bpaw,ame.tlbHOe 06UJlCeHUe 107"
r
s
a
Puc. 3.4. a - PMHaH (r - PMHYC Kpyra); 6 - COOTHOIIIeHHe Me)l{,ll;Y ,lI;JIHHOH (s),
PMHYCOM (r) H yrJIOM (9) B PMHaHax;
S - ,lI;JIHHa ,n;yrH; - yrOJI B PMHaHax; r - pa,n;HYC OKpyxHOCTH
(
/
ECJIH B BbIpIDKeHHe s = re 1I0,n;CTaBHTb s, paBHoe ,lI;JIHHe OKpyxHOCTH Kpyra, Tor,n;a:
21tr = re,
1I03TOMY 21t = e.
To eCTb 1 060POT = 21t panHaH (= 360°)
AHaJIOrWlHO 1 panHaH = 360/21t = 57,3° (lIpH6JIH3HTeJIbHo).
3. 7.2. Yron nosopOTS
IIpH 1I0CTYIIaTeJIbHOM ,n;BIDKeHHH CHMBOJIOM S 0603HaqaeTC.H paCCTO.HHHe, lIpOMeH-
Hoe 110 IIp.HMOH JIHHHH, T.e. JIHHeHHOe lIepeMelI(eHHe. llpH BpalI(aTeJIbHOM ,n;BIDKe-
HHH yrJIOBOe lIepeMelI(eHHe (HJIH yrOJI 1I0BopOTa), H3MepeHHoe B panHaHax, 060-
3HaqaeTC.H rpeqeCKOH 6)'KBOH e.
3.7.3. YrnossfI CKOPOCT&
TBep,n;oe TeJIO BpalI(aIOlI(eeC.H BOKpyr <pHKcHpoBaHHoH OCH 0 C lIOCTO.HHHOH CKOpO-
CTbIO n 06./c, 1I0BOpaqHBaeTC.H Ha 21t panHaH (pan) 3a K3.)I{,lI;bIH 060pOT. ll03TOMY
yrJIOBaR CKOpOCTb ro (rpeqeCKa.H 6)'KBa oMera) Ollpe,n;eJI.HeTC.H BbIpIDKeHHeM:
ro = 21tn pan/c
,lJ;aJIee, eCJIH TeJIO 1I0BOpaqHBaeT Ha yrOJI e panHaH 3a BpeWl t (c) H eCJIH ,n;BH-
xeHHe paBHOMepHoe, Tor,n;a
YrJIOBa.H CKOpOCTb ro = e / t pan/c.
OTMeTHM, qTO yrJIOBaR CKOpOCTb He HMeeT JIHHeHHbIX pa3MepHOCTeii.
F/la8a 3. HH:JICeHepHaJI OUHllMUICa
3. 7.4. CooTHOWeHJfe MeJICAY yrnosoH Jf nJfHeHHOH
CKOpOCTflMJf
PaCCMOTPHM TOqKY A Ha HapyxHOM 060,n;e TBep,n;o-
ro TeJIa, rrOKa3aHHoro Ha pHC. 3.5.
llYCTb TeJIO nOBOpaqHBaeTC.H Ha yrOJI e (pan) 3a
BpeWl t (c). HCrrOJIb3yeM <poPMY.rry: ,lI;JIHHa ,l(yrH =
= yrOJI B panHaHax x panH)'C, T.e.
s = reo
T errepb, qT06bI rrOJIyqHTb cpe,n;HIOIO CKOpOCTb,
rro,n;eJIHM 06e qaCTH BbIpIDKeHIDI Ha t Tax:, qTO:
sit = r(e It),
HO
s It=v He/t=OO.
v
s
Puc. 3.5. COOTHOIIIeHHe Me)l{,ll;Y
yrJIOBOH H JIHHeHHOH CKOpO-
CTHMH
n03TOMY:
V = roo HJIH JIHHeHHaR CKOpOCTb = yrJIOBaR CKOpOCTb x panH)'c.
llpUMettaHue.
1. ToqKa TeJIa B HMeeT TaKYIO xe yrJIOByro CKOpOCTb, KaK H TOqKa A, HO ee !lH-
HeHHaR CKOpOCTb MeHbIIIe, rrOCKOJIbKY eH COOTBeTcTByeT MeHbIIIHH panH)'C.
2. ECJIH panH)'C r BbIpIDKaeTC.H B MeTPax H yrJIOBaR CKOpOCTb BbIpIDKaeTC.H B
panic, Tor,n;a JIHHeHHaR CKOpOCTb v BbIpIDKaeTC.H B M/c. HHor,n;a yrJIOBa.H CKO-
pOCTb BbIpIDKaeTC.H B 06./MHH. Tor,n;a He06xo,n;HMO lIpe06pa30BaTb 06./MHH.
B panic ,n;o lIo,n;CTaHOBKH B TaKHe BbIpIDKeHIDI, KaK v = roo.
N (06/MHH) = 21tN (pan/MHH) = 21tN I 60(pan/c).
3. 7.5. Yrnosoe YCKopeHJfe
COOTHOIIIeHHe, aHaJIOrHqHOe TOMY, KOTopoe CB.H3bIBaeT JIHHeHHyro H yrJIOByro CKO-
pOCTH, CylI(eCTByeT ,lI;IDI JIHHeHHOro H yrJIOBOrO YCKopeHHH. TaKHM 06pa30M:
a = ra,
r,n;e a - JIHHeHHOe YCKopeHHe, r - panH)'C, a (rpeqeCKaR 6)'KBa aJIb<pa) - yrJIOBOe
YCKopeHHe.
B pa3,n;eJIe 3.6.4 rrepeqHCJIeHbI BbIpIDKeHIDI, CB.H3bIBaIOII(He JIHHeHHbIe CKOpOCTb
H YCKopeHHe. AHaJIOrHqHO, CylI(eCTByeT Ha60p BbIpIDKeHHH, CB.H3bIBaIOII(HX yrJIO-
BbIe CKOpOCTb H YCKopeHHe:
002 = 001 + at
e =[(001 + (02) I 2] x t
e = 001 t + at 2
00 2 = 00 2 + 2ae
21,
e = 002 t - 11 at 2
001 = HaqaJIbHaR yrJIOBaR CKOpOCTb
002 = KOHeqHaR yrJIOBaR CKOpOCTb
a = yrJIOBOe YCKopeHHe
e = yrOJI rrOBopOTa
t = 3aTpaqeHHOe BpeMSI
3. 7. Bpaw,ame.tlbHOe 06UJlCeHUe I 09'
3. 7.6. MOMeHT SpS eHJffl
MOMeHT BpaII(eHIDI - 3TO BpalI(aIOIUHH MOMeHT CHJIbI. ECJIH CHJIa npHBo,n;HT K
BpalI(eHHIO TeJIa lIPH OTCYTCTBHH nOCTYITaTeJIbHOrO ,n;BIDKeHIDI, ,n;OJDKHa CylI(eCTBO-
BaTh BTOpaR CHJIa, ,n;eHCTByrolI(aR Ha TeJIO, paBHaR no BeJIWlHHe H lIpOTHBOlIOJIOX-
Hag lIO HanpaBJIeHHIO nepBOH, TaK qT06bI pe3YJIbTHpyrolI(a.H CHJIa paBHIDIaCb HYJIIO.
Tame CHJIbI HaxO,nRTC.H B paBHOBeCHH. PaCCMOTPHM raeqHbID KJIIOq H raHKY, lIOKa-
3aHHbIe Ha pHC. 3.6, a. Kor,n;a CHJIa Fl npHJIOXeHa K KOH:Qy raeqHOrO KJIIOqa, paB-
Hag no BeJIHqHHe H lIpOTHBOnOJIO)I{HaR no HalIpaBJIeHHIO CHJIa F 2 B03HHKaeT B raD-
Ke. ECJIH 6bI 3TO 6billO He TaK, HaqaJIOCb 6bI nOCTYITaTeJIbHOe ,n;BIDKeHHe B HanpaB-
JleHHH CHJIbI F I _ Kor,n;a ,n;Be paBHbIe lIO BeJIWlHHe H lIpOTHBOnOJIO)I{HO HalIpaBJIeH-
HbIe CHJIbI ,n;eHcTByroT Ha TeJIO, rOBOp.HT, qTO OHH 06pa3yroT napy.
a
Puc. 3.6. a - MOMeHT Bpam;eHHH (Bpam;aIOm;HH MOMeHT);
6 - pa60Ta, BbInOJIHeHHaH MOMeHTOM Bpam;eHHH
o
s
6
MOMeHT BpaII(eHIDI (1) = BeJIHqHHa npHJIOXeHHOH CHJIbI (F I ) x ,lI;JIHHa IIJIeqa (d),
r,ne T - MOMeHT Bpa eHH.H (HM), F - CIDIa (H), d - IIJIeqo CHJIbI (M).
13. 7. 7. PS60TS MOMeHTS SpS eHJffl
PaCCMOTPHM pa60TY, BbInOJIHeHHyIO CHJIOH F (H), ,n;eHcTByrolI(eH lIO,n; np.HMbIM yr-
JlOM K IIJIeqy CHJIbI OA ,lI;JIHHOH r (M) H cOBepIIIHBIIIeH lIOBOpOT Ha yrOJI e pan, KaK
nOKa3aHO Ha pHC. 3.6, 6.
BbmOJIHeHHaR pa60Ta = JIHHeHHOe paCCTO.HHHe, lIpOH,n;eHHoe lIO,n; ,n;eHCTBHeM CHJIbI =
= s x F (HM HJIH ,l];x).
Ho: s = re (M).
II03TOMY:
BbInOJIHeHHaR pa60Ta = re x F = Fr x e.
F/la8a 3. HH:JICeHepHaJI OUHaMUICa
Ho:
ll03TOMY:
MOMeHT BpaII(eHIDI T = Fr.
BbIlIOJIHeHHaR pa60Ta = 10 (HM HJIH )];)K:).
Pa6oTa, BLlnOJIHeHHaJi Bpam:alOlIUIM MOMeHTOM =
= Bpam:alOlIUIH MOMeHT x yroJi nOBopOTa B pa.zntaHax.
npHMep
MaxOBHK BpalI(aeTC.H co CKOpOCTbIO 2400 06/MHH. B pe3YJIbTaTe HeO)l{H)];aHHoro
B03,n;eHcTBIDI TOpM03Horo MOMeHTa B03HHKaeT yrJIOBOe TopMoxeHHe (3aMe,ll;JIeHHe)
paBHoe 10 pan/c 2 . BbIqHCJIHTe:
1) BpeWl, 3a KOTopoe CKOpOCTb MaxoBHKa }'MHbIIIHTC.H ,n;o 600 06/MHH;
2) yrJIOByro CKOpOCTb MaxoBHKa qepe3 300 060pOTOB C MOMeHTa HaqaJIa TOpMO-
xeHIDI.
1. llpHMeHHM ypaBHeHHe:
002 = 001 + at (pa3,n;eJI 3.7.5),
r,n;e 001 = HaqaJIbHaR yrJIOBaR CKOpOCTb = (2400 06/MHH x 21t)/60 panic;
002 = yrJIOBaR CKOpOCTb qepe3 BpeWl t = (600 x 21t)/60 panic;
a = yrJIOBOe YCKopeHHe = -10 pan/c 2 (3HaK MHHYC yKa3bIBaeT Ha TopMoxeHHe).
ll03TOMY:
CJIe,n;OBaTeJIbHO:
2. npHMeHHM ypaBHeHHe:
oo = 001 + 2aa (pa3,n;eJI 3.7.5),
r,n;e 002 = yrJIOBaR CKOpOCTb qepe3 300 060pOTOB;
001 = (2400 06/MHH x 21t)/60 = 251,33 panic; a = -10 pan/c 2 ; a = 300 x 21t pan.
T or,n;a:
oo = 251,332 + 2 x (-10) X 6001t = 63166,8 - 37699,1 = 159,6 pan/c.
oo = (159,6 X 60)/21t = 1524 06/MHH.
(600 x 21t)/60 = (2400 06/MHH x 21t)/60 + (-10)t.
lOt = (1800 x 21t)/60.
t= 18,8 c.
HJIH
3.7.8. LleHTpOCTpeMJfTen&HOe YCKopeHJfe
Jf qeHTpoCTpeMJfTen&HSfI CJfns
COrJIaCHO CJIOBapIO, QeHTPOCTPeMHTeJIbHOe YCKopeHHe - 3TO YCKopeHHe, HanpaB-
JIeHHOe K n;eHTPY KpHBH3HbI TPaeKTOpHH TeJIa, ,lI;BIDKYIUerOC.H no TPaeKTOpHH C no-
CTO.HHHOH no MO,IJ;yJIIO CKOpOCTbIO. qHCJIeHHO OHO paBHO v 2 /r, r,n;e v - CKOpOCTb,
r - panHYC KpHBH3HbI, ero TaIOKe MO)I{HO BbIpa3HTb qepe3 yrJIOByro CKOpOCTb 00,
KaK 00 2 r.
Ha pHC. 3.7, a lIOKa3aHO TeJIO MaCCOH m, ,n;BIDKYIUeeC.H C nOCTO.HHHOH yrJIOBOH
CKOpOCTbIO 00 lIO KpyrOBOH TPaeKTOpHH panHYCOM r C n;eHTPOM B TOqKe O. Kor.na
TeJIO Haxo,n;HTC.H B TOqKe P, ero MrHOBeHHaR JIHHeHHaR CKOpOCTb v HanpaBJIeHa no
KaCaTeJIbHOH K OKpyxHOCTH, B,n;OJIb KOTOpOH OHO ,n;BIDKeTC.H. 3a npOMexyrOK Bpe-
3. Z Bpa14ameJlbHOe 08U:JICeHUe II
o
p
q
r
o
a
)
)
6
Puc. 3.7. a - JJ;eHTpOCTPeMHTeJIbHOe YCKopeHHe;
6 - nHarpaMMa H3MeHeHHH CKOpOCTH npH ,n;BH)I{eHHH no OKp)')KHOCTH C
I(eHTpOCTpeMHTeJIbHbIM YCKopeHHeM
MeHH ot TeJIO lIepeMemaeTcH no OKpyxHOCTH H3 TOqKH P B TOqKY Q. CooTBeTcTBY-
IOm;asI 3TOMY nepeMemeHHIO ,nyra PQ CT.HrHBaeT yrOJI oe. XOT.H TeJIO ,nBH)I{eTC.H C
nOCTO.HHHOH no MO,!{yJIIO CKOpOCTblO, HanpaBJIeHHe ero ,nBH)I{eHIDI He OCTaeTC.H no-
CTO.HHHbIM. TI03TOMY BeKTOp CKOpOCTH TeJIa H3MeH.HeTC.H; a nOCKOJIbKY BeKTOp CKO-
pOCTH H3MeH.HeTC.H co BpeMeHeM, TeJIO, no onpe,neJIeHHIO, ,nBH)I{eTC.H C YCKopeHHeM.
Ha pHC. 3.7, 6 nOKa3aHa BeKTOpHa.H ,nHarpaMMa CKopocTeit, r,ne:
. CKOpOCTb B TOqKe P (MO,!{yJIb CKOpOCTH H HanpaBJIeHHe) npe,nCTaBJIeHa BeKTO-
pOM Op;
. CKOpOCTb B TOqKe Q npe,nCTaBJIeHa BeKTopOM Oq;
. pa,nHaJIbHOe H3MeHeHHe CKOpOCTH npH nepeMemeHHH TeJIa H3 TOqKH P B TOq-
KY Q npe,nCTaBJIeHO BeKTopOM pq.
Pa,nHaJIbHOe H3MeHeHHe CKOpOCTH = pq = v x oe.
O,nHaKO:
e = rot.
TI03TOMY:
oe = ro x ot = (v / r) x ot.
TaKHM 06pa30M:
M3MeHeHHe BeKTOpa CKOpOCTH = (v 2 x ot) / (r x ot) = v 2 / r = ro 2 r.
,lI0 CHX nop MbI paCCMa1pHBaJIH Te.rry MaCCOH m, L(BIDICylI(eeC.H C ueH1pOC1peMH-
TeJlLHbIM YCKopeHHeM. 1f3 BToporo 3aKOHa MexaHHKH HblOTOHa CJIe,nyeT, qTO TeJIO
He MoxeT ,nBHraTbC.H C YCKopeHHeM, eCJIH Ha Hero He ,neitcTByeT BHeIIIHIDI CHJIa.
IT03TOMY C03,nalOma.H YCKopeHHe ueH1pOC1peMHTeJIbHa.H CHJIa = Macca x YCKope-
HHe. MaTeMaTHqeCKH 3TO MO)I{HO BbIpa3HTb:
o;eHTpOCTpeMHTeJILHaJi CHJIa = m x (V 2 / r) = mm 2 r.
F/la8a 3. HH:JICeHepHaJI OUHaMUKa
3. 7. 9. eHTpo6eJKHafl CHna
TpeTHH 3aKOH HblOTOHa )'TBep)l{)J;aeT, qTO ,l(lliI KaJK,l(OH ,neHcTByroIUeH CHJIbI cym;ecr-
ByeT paBHa.H no BeJIHqHHe H lIpOTHBOnOJIO)I{Ha.H lIO HalIpaBJIeHHIO CHJIa lIpOTHBO-
,neHCTBIDI. n03TOMY ,nOJDKHa cym;ecTBoBaTb CHJIa lIpoTHBo,neHcTBIDI, paBHa.H lIO Be-
JIWIHHe H npOTHBOnOJIO)I{Ha.H lIO HalIpaBJIeHHIO ueH1pOc1peMHTeJIbHOH CHJIe. 3ra
CHJIa npoTHBo,neHcTBIDI HalIpaBJIeHa lIO pa,nHYCY OT ueH1pa KpHBH3HbI 1paeKTOpHH
H Ha3bIBaeTC.H n;eH1p06e)I{HOH CHJIOH. PaccM01pHM noe3,n, ,nBIDICylI(HHC.H no 3aKpyr-
JIeHHOMY yqaCTKY rryTH. Pe60p,nbI KOJIeC noe3,na ,naB.HT Ha peJIbCbI B HanpaBJIeHHH
OT ueH1pa KpHBH3HbI 3aKpyrJIeHIDI (ueH1p06e)I{Ha.H cHJIa); B TO xe BpeM.H peJIbCbI e
CHJIOH, paBHOH lIO BeJIWIHHe, B03,neHCTByroT Ha KOJIeCa noe3,na B HalIpaBJIeHHH K
ueH1py KpHBH3HbI 3aKpyrJIeHIDI (ueH1pOC1peMHTeJIbHa.H CHJIa). ToqHO TaK xe, Kor-
,na rHpbKa, y,nep)I(HBaeMa.H IIIHypOM, BepTHTC.H BOKpyr OCH, ueH1p06e)I{Ha.H CHJIa TH-
HeT rHpbKY Hap)')I{Y, B TO BpeM.H KaK ueH1pOC1peMHTeJIbHa.H CHJIa T.HHeT ee BHYTPb;
npH 3TOM IIIHyp OCTaeTC.H HaT.HHYTbIM. (HeJIb3.H )'TBep)l{)J;aTb, qTO «ueH1p06e)I{HaH
CHJIa T.HHeT rHpbKY», lIOCKOJIbKY ueH1p06e)I{Ha.H CHJIa Bcer,na ,neHcTByeT Ha CBH3H, a
He Ha ,nBIDICylI(eeC.H TeJIO. npaBHJIbHO CKa3aTb MO)I{HO TaK: «ToqHO TaK xe, Kor,na
rHpbKa y,nep)I(HBaeMa.H IIIHypOM, BepTHTC.H BOKpyr OCH, ueH1p06e)I{Ha.H CHJIa ,neHcr-
ByeT Ha IIIHyp, HaT.HrHBa.H ero, B TO BpeM.H KaK ueH1pOC1peMHTeJIbHa.H CHJIa T.HHer
rHpbKY BHYTPb». - llpUMe"t. nep.)
3.8. YpaBHOBewMBaHMe BpaLQalOLQMXCR Mace
QT06bI MexaHH3MbI MOrJIH pa60TaTb nrraBHo, 6e3 BH6paUHH, He06xo,nHMO, qT06bI
Bce BpaIUaIOIUHeC.H MaCCbI HaxO,nHJIHCb B COCTO.HHHH paBHoBecIDI.
(MexaHH3M CqHTaeTC.H nOJIHOCTblO ypaBHoBeIIIeHHbIM, eCJIH CHJIa ,naBJIeHIDI Ha
onopbI OCTaeTC.H lIOCTO.HHHOH npH ,nBlUKeHHH 3BeHbeB. Y CJIOBHeM CTaTWIeCKoro
paBHoBecIDI .HBJI.HeTC.H nOCTO.HHHOe nOJIOXeHHe ueH1pa Mace MexaHH3Ma. ,[(HHaMH-
qeCKOe ypaBHOBeIIIHBaHHe - 3TO ypaBHOBeIIIHBaHHe peaKUHH, B03HHKalOIUHX npH
BpaIUeHHH TeJIa, T.e. o,nHOBpeMeHHO CTaTWIeCKoe H MOMeHTHoe ypaBHOBeIIIeHHe.
ECJIH OCb BpaIUeHIDI npoxo,nHT qepe3 ueH1p Macc TeJIa H COBna,naeT C W(HOH H3 ero
rJIaBHbIX oceH HHepUHH, TO rOBOp.HT, qTO BpaIUalOIUeeC.H TeJIO ,nHHaMWIeCKH ypaB-
HOBeIIIeHO Ha OCH BpaIUeHIDI. - llpUMe"t. nep.)
3.8. 1. YpasHosewHsaHHe KOMnnSHapH&IX Macc
(CTaTHlfeCKoe pasHoseCHe)
KOMnrraHapHbIMH 6y,neM Ha3bIBaTb MaCCbI, paClIOJIOXeHHbIe B O,nHOH nrrOCKOCTH.
QT06bI TeJIO HaxO,nHJIOCb B COCTO.HHHH CTaTWIeCKoro paBHOBeCIDI, pe3YJIbTHpyroIUaH
Bcex CHJI, ,neHcTByroIUHX Ha TeJIO, H aJIre6paWIeCKa.H CYMMa MOMeHTOB ,nOJDKHbI
paBH.HTbC.H HYJIIO. Ha pHC. 3.8, a nOKa3aHbI 1pH KOMnrraHapHbIe MaCCbI, paClIOJIO-
xeHHbIe BOKpyr TOqKH O. Bo Bcex CJIyqasIX ,l(lliI Bcex lIpaKTHqeCKHX ueJIeH Ha 3eM-
JIe MO)I{HO CqHTaTb, qTO rpaBHTan;HOHHa.H CHJIa, ,neHcTByroIUa.H Ha Maccy, paBHa
9,81 x Macca.
QT06bI CHCTeMa Macc, lIOKa3aHHa.H Ha pHCYHKe, 6bma ypaBHoBeIIIeHHoH, OHa
,nOJDKHa HaxO,nHTbC.H B COCTO.HHHH CTaTWIeCKoro paBHOBeCIDI H, CJIe,nOBaTeJIbHO, aJI-
3.8. Ypa6Ho6eUlu6aHue 6paUl,alOUI,UXCJl Mace 113
c
a
b
9,81 Ins
a
c
a
9,81 m 2
6
Puc. 3.8. CTaTHqeCKOe paBHOBeCHe KOMnJIaHapHbIX Macc:
a - KOMnJIaHapHbIe MaCCbI B nepBOM yrJIOBOM nOJIOXeHHH;
6 - KOMnJIaHapHbIe MaCCbI nOCJIe nOBopOTa Ha 90 0 BO BTOpOM yrJIOBOM
nOJIOXeHHH
re6palNeCKa.H C)'MMa MOMeHTOB OTHOCHTeJIbHO UeH1pa BpameHIDI 0 ,nOJDKHa paB-
HHThC.H HYJIIO. MaTeMaTWIecKH 3TO MO)I{HO BbIpa3HTb CJIe,nyromHM 06pa30M:
9,81mlrl COSel + 9,81m2r2 Cose 2 + 9,81m3r3 Cose 3 = o.
ECJIH pa3,neJIHTb KIDK,ZJ;Oe H3 CJIaraeMbIX Ha 9,81, lIOJIyqHTC.H:
mlrl COSel + m2 r 2 Cose 2 + m3 r 3 Cose 3 = O.
F/la8a 3. HH:JICeHepHaJI OUHaMUICll
O,nHaKO 3Toro e,nHHCTBeHHoro YCJIOBIDI He,nOCTaTOqHO, qT06bI CHCTeMa Macc
6bma ypaBHoBeIIIeHHoH npH Bcex yrJIOBbIX nOJIOXeHIDIX. ECJIH Tenepb nOBepJIyTb CH-
cTeMy Ha 90°, KaK nOKa3aHO Ha pHC. 3.8, 6, aJIre6paHqeCKa.H C)'MMa MOMeHTOB OTHO-
CHTeJIbHO TOqKH 0 CHOBa ,nOJDKHa paBH.HTbC.H HYJIIO.
9, 81m lrl sin9 1 + 9,81m2r2 sin9 2 + 9,81m3r3 sin9 3 = o.
CHOBa pa3,neJIHM KIDK,ZJ;Oe H3 CJIaraeMbIX Ha 9,81 H nOJIy1IHM:
mlrl sin9 1 + m2 r 2 sin9 2 + m3 r 3 sin9 3 = o.
ECJIH CHCTeMa ypaBHOBeIIIeHa B JII06bIX ,nBYX yrJIOBbIX nOJIOXeHIDIX, COCTaBJI.HIO-
IUHX Me)l{)J;Y C060H np.HMOH yrOJI, TO OHa TaIOKe 6y,neT ypaBHOBeIIIeHa B JII06bIX npo-
MexyTOqHbIX nOJIOXeHIDIX. CJIe,nOBaTeJIbHO, npHBe,neHHbIX BbIIIIe ypaBHeHHH ,no-
CTaTOqHO ,l(lliI npOBepKH ypaBHOBeIIIeHHOCTH CHCTeMbI. ECJIH CHCTeMa He ypaBHOBe-
IIIeHa, Tor,na Te xe caMbIe ypaBHeHHH n03BOJI.HT onpe,neJIHTb BeJIHqHHY H nOJIoxe-
HHe ypaBHOBeIIIHBalOIUeH MaCCbI.
3.8.2. YpasHosewHsaHHe KOMnnaHapH&IX Macc
(AHHaMHlfecKoe pasHosecHe)
BonpOCbI ,nHHaMHqeCKOrO ypaBHOBeIIIHBaHH.H OqeHb CJIO)I{HbI, H eCJIH nonbITaTbCjI
HCCJIe,nOBaTb HX ,nOCTaTOqHO rJIY60KO, TO B 3TOM CnpaBOqHHKe npOCTO He XBaTHT
MeCTa. O,nHaKO paCCMOTpHM npocTyro KOMllJIaHapHylO CHCTeMY CHJI, ,neHCTByIO-
IUHX B CHCTeMe npH BpaIUeHHH Macc OTHOCHTeJIbHO TOqKH 0, KaK nOKa3aHO Ha
pHC. 3.9, a. B npouecce BpaIUeHH.H KIDK,ZJ;OH H3 Macc Ha TOqKY 0 (OCb HJIH BaJI) ,neH-
cTByeT ueH1p06e)I{HaH CHJIa mr0)2. CKOpOCTb BpalI(eHH.H 0) BbIpaxeHa 3,neCb B pa,n/c.
KaK H paHee, aJIre6paHqeCKaH C)'MMa MOMeHTOB OTHOCHTeJIbHO TOqKH 0 ,noJI)I{Ha
paBH.HTbC.H HYJIIO B ,nBYX HanpaBJIeHH.HX, COCTaBJIHIOIUHX Me)l{)J;Y C060H np.HMOH yrOJI
(x H y), KaK nOKa3aHO Ha pHC. 3.9, a.
CJIe,noBaTeJIbHO, YCJIOBIDI ,nHHaMHqeCKOrO paBHOBeCIDI onHCbIBaIOTC.H ,nBywI
ypaBHeHIDIMH:
mlrl0)2 cos9 1 + m2r20)2 cos9 2 + m3r30)2 cos9 3 = o.
ml1i0)2 sin9 1 + m2r20)2 sin9 2 + m3r30)2 sin9 3 = o.
nOCne ,neJIeHH.H Ha 0)2 (nOCTOHHHOH BeJIHqHHbI ,l(lliI JII060H 3a,naHHOH CKOpOCTIi :
BpaIUeHIDI) ypaBHeHH.H CTaHOB.HTC.H H,neHTHqHbIMH npe,nbI,nylI(HM ypaBHeHIDIM, onH-
CbIBalOIUHM CTaTHqeCKOe paBHOBeCHe. CJIe,nOBaTeJIbHO, ,l(lliI JII060H KOMllJIaHapHOH
CHCTeMbI CM YCJIOBH.H CTaTHqeCKOrO H ,nHHaMHqeCKOrO paBHOBeCIDI o,nHHaKOBbI Ii
He 3aBHC.HT OT CKOpOCTH BpaIUeHIDI. Ha pHC. 3.9, 6 nOKa3aHa BeKTOpHa.H ,nHarpaM-
Ma CHJI, HJIJIIOC1pHpyroIUa.H npe,nbI,LJ;ylI(He ypaBHeHH.H.
. ECJIH MHoroyrOJIbHHK 3aMKHYT, KaK nOKa3aHO Ha pHCYHKe, TO MaCCbI ypaB-
HOBeIIIeHbI.
. ECJIH MHoroyrOJIbHHK He 3aMKHYT, TO MaCCbI He ypaBHOBeIIIeHbI, H 3aMbIKa-
IOIUHH BeKTOp nOKa3bIBaeT yrJIOBOe nOJIOXeHHe H 3HaqeHHe mr ,nJI.H MaCCbI,
He06xo,nHMOH, qT06bI ypaBHOBeCHTb CHCTeMy.
3.9. OmHocume.nbHble cICopocmu
x
.-------
IY
.
I
.
I
.
x
------
.
I
.
I
.
S Iy
a 9,81 I7Js
6
PRC. 3.9. a - ,n;HHaMHqeCKOe (B npOI(eCce Bpam;eHHH) paBHOBeCHe KOMnJIaHapHbIX
Macc;
6 - BeKTopHaH ,n;HarpaMMa CHJI ,lI;JIH KOMnJIaHapHbIX Macc
3.9. OTHOCMTenbHble CKOpOCTM
3.9.1. OTHOCHTen&H&le CKOpOCTH (HecoeAHHeHH&le Tena)
YTBep)f{)];eHHe, qTO TeJIO ,nBH)I{eTC.H co CKOpOCTblO 50 KM/q, 03HaqaeT, qTO TeJIO ,nBH-
)KerCH co CKOpOCTblO 50 KM/q OTHOCHTeJIbHO KaKorO-TO ,npyroro TeJIa, TaKoro, Ha-
npHMep, KaK nOBepXHOCTb nrraHeTbI 3eMJI.H. Ho Kor,na rOBOp.HT, qTO CKOpOCTb caMO-
JIera B B03,nyxe paBHa 300 KM/q, Ha caMOM ,neJIe yKa3bIBalOT CKOpOCTb CaMOJIeTa OT-
HOCHreJIbHO CKOpOCTH B03,nynIHbIX lIOTOKOB, CKB03b KOTopbIe OH JIeTHT. MCTHHHa.H
CKOpOCTb CaMOJIeTa OTHOCHTeJIbHO 3eMJIH MoxeT 6bITb 60JIbIIIe, qeM ero CKOpOCTb
F/la8a 3. HH:JICeHepHaJI OUHaMU1Ca
OTHOCHTeJIbHO B03,nyxa, eCJIH BeTep nonyrHbIH, HJIH MeHbIIIe, eCJIH BeTep
BC1peqHbIH.
Ha pHC. 3.10, a nOKa3aHbI ,nBa Hecoe,nHHeHHbIX TeJIa, 0603HaqeHHbIe TOqKaMH P
H Q. 3TH TeJIa ,nBIDKyrC.H B npOTHBOnOJIO)I{HbIX HanpaBJIeHIDIX co CKOpOCT.HMH 3 If
5 M/c COOTBeTCTBeHHO. By,neM CqHTaTb HanpaBJIeHHe ,nBH)I{eHIDI CJIeBa HanpaBO no-
JIO)I(HTeJIbHbIM, a cnpaBa HaJIeBO - 01pHuaTeJIbHbIM.
P 3 M/C
. ..
(+)
....
5 M/C a
.
(-)
P 3 M/C
.
(+)
..
. 5 M/C
(+)
a
6
;/ p a7
P a " "
" _ Vi p ',- Vp
, ,
, ,
, ,
, ,
, ,
Va
a
,
,
,
,
,
, /"
, ,
\ /"
'",
-Vp
R
B
r
A
Puc. 3.10. OTHOCHTeJIbHbIe CKOpOCTH Hecoe,n;HHeHHbIX TeJI:
a - OTHOCHTeJIbHbIe CKOpOCTH ,n;BYX Hecoe,n;HHeHHbIX TeJI, ,n;BIDJCYII(HXC$I B
npOTHBOnOJIOXHbIX HanpaBJIeHHHX;
6 - OTHOCHTeJIbHbIe CKOpOCTH ,n;BYX Hecoe,n;HHeHHbIX TeJI, ,n;BHXYIQHXC$I B
O,n;HOM HanpaBJIeHHH;
B - OTHOCHTeJIbHbIe CKOpOCTH ,n;BYX Hecoe,n;HHeHHbIX TeJI, ,n;BHxyIIJ;HXCH
no pa3JIHqHbIM TpaeKTopHHM;
r - MeTO,n; OCTaHOBKH P H BJIHHHHe Ha Q;
,n; - CKOpOCTb Q OTHOCHTeJIbHO P (pe3YJIbTHPYlOIQaH CKOpOCTb QR)
CKOpOCTb Q OTHOCHTeJIbHO P = CKOpOCTb Q - CKOpOCTb P =
= (-5 M/c) - (+3 M/c) = -8 M/c.
CKOpOCTb Q OTHOCHTeJIbHO P 01pHuaTeJIbHa, nOTOMy qTO TeJIO Q ,nBH)I{eTCjI
cnpaBa HaJIeBO.
AHaJIOrHqHo:
CKOpOCTb P OTHOCHTeJIbHO Q = (+3 M/c) - (-5 M/c) = +8 M/c.
CKOpOCTb P OTHOCHTeJIbHO Q nOJIO)I(HTeJIbHa, nOCKOJIbKY TeJIO P ,nBH)I{eTC.H CJIe-
Ba HanpaBO. B ,naHHOM CJIyqae BbI60p nOJIO)I(HTeJIbHOrO H 01pHuaTeJIbHOrO HanpaB-
JIeHHH 6bill c,neJIaH npOH3BOJIbHO. OTHOCHTeJIbHbIe HanpaBJIeHH.H H pacnOJIOXeHlfe
nOJIO)I(HTeJIbHbIX H 01pHuaTeJIbHbIX 3HaqeHHH ,nOJDKHbI YCTaHaBJIHBaTbC.H nepe,n;
HaqaJIOM peIIIeHIDI KIDKJJ:OH KOHKpeTHoH 3a,n;aqH.
ECJIH TeJIa ,nBIDKyrC.H B npOTHBOnOJIO)I{HbIX HanpaBJIeHH.HX HaBC1peqy ,npyr ,npy-
ry, HX OTHOCHTeJIbHa.H CKOpOCTb - 3TO CKOpOCTb C6JIH)I{eHIDI. nYCTb Tenepb 3TH
,nBa TeJIa ,nBIDKyrC.H B O,nHOM H TOM xe HanpaBJIeHHH, KaK nOKa3aHO Ha pHC. 3.10,6.
CKOpOCTb Q OTHOCHTeJIbHO P = +5 M/c - (+3 M/c) = +2 M/c.
3.9. OmHOCUmetlbHble Clwpocmu
ECJIH ,nBa TeJIa ,nBIDKyTC.H He no O,nHOH H TOH xe 1paeKTOpHH, HX OTHOCHTeJIb-
Hble CKOpOCTH HeJIb3.H onpe,neJIHTb npOCTbIM CJIOXeHHeM HJIH BbIqHTaHHeM. Ha
pHC. 3.10, B nOKa3aHbI ,nBa TeJIa P H Q, ,nBIDICylI(HeC.H B pa3JIHqHbIX HanpaBJIeHIDIX
IIO pa3JIINHbIM 1paeKTOpIDIM B O,nHOH llJIOCKOCTH. CKOpOCTb TeJIa P paBHa V p , a
CKOpOCTb TeJIa Q - V Q .
QT06bI HaHTH CKOpOCTb TeJIa Q OTHOCHTeJIbHO P, He06xo,nHMO HaHTH Kaxyw;y-
IOC.H CKOpOCTb Q, MbICJIeHHO OCTaHOBHB TeJIO P. ,l:LrnI 3Toro lfY)KH0 npH6aBHTb CKO-
pOCTb -VP K CKOpOCT.HM 060HX TeJI, KaK nOKa3aHO Ha pHC. 3.10, r, B pe3YJIbTaTe
qero CKOpOCTb P CTaHeT paBHoH HYJIIO. 113 BeKTOpHOH ,nHarpaMMbI (pHC. 3.10, ,n)
MO>KHO BH,neTb, qTO Tenepb CKOpOCTb TeJIa Q COCTOHT H3 ero c06CTBeHHOH CKOpO-
CTH VQ H nepeHocHoH CKOpOCTH (-VP). (nepeHocHoH Ha3bIBalOT CKOpOCTb no-
,lJ;BIDKHOH CHCTeMbI OTCqeTa no OTHOIIIeHHIO K OCHOBHOH CHCTeMe OTCqeTa. -
llpUMe"t. nep.) Pe3YJIbTHpyromasI CKOpOCTb TeJIa Q npe,nCTaBJIeHa BeKTOpOM QR, KO-
TOpbIH COOTBeTcTByeT CKOpOCTH Q OTHOCHTeJIbH P. I1MeHHo TaK BbIrJIR,nHT ,nBlUKe-
HHe TeJIa Q ,l(lliI Ha6JIIO,naTeJI.H, CB.H3aHHOro C TeJIOM P. Ha6JIIO,naTeJIb, CB.H3aHHbIH C
TeJIOM Q, 6y,neT BH,neTb TeJIO P, ,nBIDICylI(HMC.H B npOTHBOnOJIO)I{HOM HalIpaBJIeHHH,
HO C TaKOH xe CKOpOCTblO. TaKHM 06pa30M, 3HaK pe3YJIbTHpyromero BeKTOpa CKO-
pOCTH CJIe,nyeT H3MeHHTb Ha npOTHBOnOJIO)I{HbIH.
3.9.2. OTHOCHTen&H&le CKOpOCTH (coeAJfHeHH&le Tena)
B OTJIINHe OT npe,nbI,nylI(erO pa3,neJIa, B 3TOM pa3,neJIe 6y,neM HMeTb ,neJIO C OTHOCH-
TeJIbHbIMH CKOpOCT.HMH TeJI, coe,nHHeHHbIX Me)l{)J;y C060H. Ha pHC. 3.11, a nOKa3aHbI
,lJ;Ba TeJIa P H Q, coe,nHHeHHbIe xeCTKHM 3BeHOM ,nJIHHbI I. CKOpOCTH coe,nHHeH-
HbIX TeJI P H Q paBHbI COOTBeTCTBeHHO v;, H V q H HanpaBJIeHbI, KaK nOKa3aHO Ha
pHCyJIKe.
Va
a
o
p
6
Puc. 3.11. OTHOCHTeJIbHbIe CKOpOCTH (coe,n;HHeHHbIe TeJIa):
a - ,n;Ba TeJIa (P H Q), coe,n;HHeHHbIe xeCTKHM 3BeHOM;
6 - BeKTopHaH ,n;HarpaMMa CKopocTeH
F/la8a 3. HH:JICeHepHaJI OUHaMUICa
nOCKOJIbKY paccToHHHe / Mex.n.y TeJIaMH P H Q <pHKCHpOBaHO, He MoxeT c}'Ine-
CTBOBaTb KOMlIOHeHTa CKOpOCTH, HanpaBJIeHHa.H B,nOJIb 3BeHa, coe,nHH.HlOmerO P Ii
Q. TaKHM 06pa30M, JI106a.H OTHOCHTeJIbHa.H CKOpOCTb 6y,neT neplIeH;J;HKYJI.HpHa 3Be-
HY, coe,nHH.HlOmeMY P H Q, nOCKOJIbKY 3TO e,nHHCTBeHHoe HalIpaBJIeHHe, B KOTOpOM
OTcyrcTByeT KOMnOHeHTa, HalIpaBJIeHHa.H B,nOJIb 3BeHa, coe,nHH.HlOmerO P H Q. H3
BeKTOpHOH ,nHarpaMMbI CKopOCTeH, lIOKa3aHHOH Ha pHC. 3.11, 6, MO)I{HO BH,Z:(eTb,
qTO BeKTOp pq nepneH;J;HKYJI.HpeH 3BeHY, coe,nHH.HlOmeMY TeJIa P H Q. n03TOMY
OqeBH,Z:(HO, qTO P BpamaeTC.H C TaHreHUHaJIbHOH CKOpOCTblO pq OTHOCHTeJIbHO Q.
CJIe,noBaTeJIbHo:
YrJIOBa.H CKOpOCTb 3BeHa = (TaHreHUHaJIbHa.H CKOpOCTb)/(Pa,nHYc).
ro = (BeKTOp pq)//.
BbNHCJIeHHe CKOpOCTH Q OTHOCHTeJIbHO P npHBeJIO 6bI K TOMY xe pe3YJIbTaTY, Tal{
KaK 3BeHO, coe,nHIDIlOmee TeJIa P H Q, MoxeT HMeTh TOJIbKO 0,l(Hy yrJIOByro CKOpOCTh.
3.10. KMHeM8TMK8
KHHeMaTHKa H3yqaeT ,nBIDKeHHe TeJI, He paCCMa1pHBa.H CHJIbI, HBJI.HlOmHeC.H nplNH-
HOH ,nBIDKeHID!. JIHHeHHa.H CKOpOCTb - 3TO 6bIC1pOTa H3MeHeHID! MO,Z:(yJI.H BeKTOpa
nepeMemeHID! no OTHOIIIeHHIO KO BpeMeHH, H ee MO)I{HO BbIpa3HTb KaK:
CKOpOCTb = ds / dt.
(BeKTOp lIepeMemeHIDI - 3TO BeKTOp, HaqaJIbHa.H TOqKa KOToporo COBna,naeT C
HaqaJIbHOH TOqKOH ,nBIDKeHID!, a KOHeu - C KOHeqHOH. Cpe,nHIDI CKOpOCTb - OT-
HOIIIeHHe MO,nyJI.H BeKTOpa lIepeMemeHIDI K lIpOMexyrKY BpeMeHH, B TeqeHHe KOTO-
poro 3TO lIepeMemeHHe lIpOH30IIIJIO. MrHOBeHHa.H CKOpOCTb (CKOpOCTb, 0 KOTOpOH
3,neCb rOBOpHTC.H) - lIpe,neJI OTHOIIIeHIDI BeKTOpa lIepeMemeHID! K npOMexyrKY
BpeMeHH, 3a KOTOpbIH 3TO lIepeMemeHHe lIpOH30IIIJIO, Kor,na MHTeJIbHOCTb lIpOMe-
xyrKa BpeMeHH C1peMHTCH K HYJIIO. - llpUMe"t. nep.)
nOMHHTe, qTO CKOpOCTb TeJIa xapaKTepH3yeTC.H KaK Mo,nYJIeM CKOpOCTH, TaK Ii
HalIpaBJIeHHeM. QHCJIeHHOe 3HaqeHHe CKOpOCTH paBHO ee MO,nYJIIO. nepeMemeHHe
paBHO llJIOma,nH no,n rpa<pHKOM 3aBHCHMOCTH CKOpOCTH OT BpeMeHH. TaK KaK:
v = ds / dt,
TOr.n;a f tis = f vdt.
AHaJIOr HO, JIHHeHHOe YCKopeHHe - 3TO 6bIC1pOTa H3MeHeHIDI CKOpOCTH:
Y CKopeHHe = H3MeHeHHe CKOpOCTH/BpeM.H = dv / dt HJIH d 2 S / dt 2
OTMeTHM, qTO YCKopeHHe - 3TO, KpOMe Toro, BTOpa.H npOH3Bo,nHaH nepeMem;e.
HID! no BpeMeHH. CKOpOCTb paBHa llJIOma,nH no,n rpa<pHKOM 3aBHCHMOCTH YCKope.
HID! OT BpeMeHH. TaK KaK:
Tor,na
a = dv / dt,
f da = f adt.
AHaJIOrINHO, yrJIOBOe YCKopeHHe - 3TO 6bIC1pOTa H3MeHeHIDI yrJIOBOH CKOpOCTII
a. = dq,/ dt HJIH a. = d 2 e / dt 2 .
3.10. KUHeMamUlCO
3. 10. 1. 6annHcTHKa
Ha pHC. 3.12 nOKa3aHa 1paeKTOpIDI CHap.H,na, BbIrryI.l(eHHoro C HaqaJIbHOH CKOpO-
CTblO v, M/C, npH TOM qTO yrOJI B03BbIIIIeHIDI Opy,nIDI paBeH e. Tor,na BepTHKaJIbHa.H
COCTaBIDIIOIUa.H CKOpOCTH V v = vlsine H ropH30HTaJIbHa.H COCTaBlliIIOIUa.H CKOpOCTH
VB = VIcose. ECJIH npeHe6peqb conpOTHBJIeHHeM B03,nyxa, e,nHHCTBeHHOH CHJIOH,
,lI;eHCTByroIUeH Ha CHapM, 6y,neT CHJIa TIDKeCTH, co06IUalOIUa.H Macce cHapMa rpa-
BHTaUHOHHoe YCKopeHHe (9,81 M/C 2 ) B HanpaBJIeHHH BHH3. nOCKOJIbKY He CYIneCT-
ByeT ropH30HTaJIbHO HanpaBJIeHHOH CHJIbI, ,neHCTByroIUeH Ha CHapM, ropH30HTaJIb-
HWI COCTaBJUlIOIUa.H CKOpOCTH OCTaHeTC.H nOCTO.HHHOH Ha npOTIDKeHHH Bcero nOJIe-
Ta. (B peaJIbHOCTH OHa 6y,neT nOCTeneHHO )'MeHbIIIaTbCH 3a CqeT COnpOTHBJIeHIDI
B03,nyxa. )
v sin 8
MaKCH bHaR BbiCOTa nOAbeMa h
I.
TpaeKTopVl51 CHap5lAa
s
I
· v cos 8
Puc. 3.12. TpaeKTopHH CHapH,n;a
COrJIaCHO CooTHoIIIeHHIO, npHBe,neHHoMy B pa3,neJIe 3.6.4, nOJIHOe BpeMH nOJIe-
Ta MOXHO nOJIyqHTb, HCnOJIb3Y.H BbIpaxeHHe s = VI t + 1 / 2at 2 . nOCKOJIbKY npe,nno-
JIaraeTC.H, qTO cHapM CTapTYeT C ypOBH.H 3eMJIH, ,nOCTHraeT MaKCHMaJIbHOH BbICO-
TbI h, KaK nOKa3aHO Ha pHCYHKe, H CHOBa B03BpaIUaeTC.H Ha ypOBeHb 3eMJIH, TO pe-
3YJIbTHpyro:w;ee paCCTO.HHHe, npOH,neHHOe CHapMOM B BepTHKaJIbHOM HanpaBJIe-
HHH, 6y,neT paBHO HYJIIO. n03TOMy:
s = 0, V v = VI sin e H a = -g.
TIOCKOJIbKY:
o = (VI sine)t - gt 2 / 2.
t = (2VI sin e) / g.
Tor,na
TaK KaK ropH30HTaJIbHa.H COCTaBJI.HIOIUa.H CKOpOCTH cHapMa OCTaeTCH nOCTO.HH-
HOH, paCCTO.HHHe s, npOH,neHHOe CHapMOM B ropH30HTaJIbHOM HanpaBJIeHHH (,na-
JIbHOCTb), MO)I{HO paCCqHTaTb CJIe,nyroIUHM 06pa30M:
TIpoi!f,neHHoe paCCTO.HHHe HJIH ,naJIbHOCTb (s) =
= ropH30HTaJIbHa.H KOMnOHeHTa CKOpOCTH x BpeM.H nOJIeTa.
S = (VI cos e) x (2VI sin e) / g = [(VI)2 sin 2e] / g (TaK KaK sin 28 = 2 sin ecos e).
MaKCHMaJIbHyro BbICOTY (h), ,nOCTHraeMyID CHapMOM B nOJIeTe, MO)I{HO HaHTH
H3 BbIpaxeHIDI:
h = [(VI)2 sin 2 e] / 2g.
F/la8a 3. HH:JICeHepHaJI OUHaMU1Ca
3.11. KMHeTMKa
KHHeTHKa HCCJIe,nyeT npWIHHbI B03HHKHOBeHIDI ,nBH)I{eHIDI, B OTJIHqHe OT KHHeMa-
THKH, KOTOpa.H H3yqaeT TOJIbKO reOMe1pHIO ,nBH)I{eHIDI. CJIe,noBaTeJIbHO, B KHHeMa-
THKe lIpHMeH.HIOTC.H 3aKOHbI MexaHHKH HbIOTOHa.
3. 11. 1. BceMHpHoe TflrOTeHHe
3aKOH, onHcbIBalOIUHH B3aHMHoe npHTIDKeHHe TeJI, 06JIa,naIOIUHX MaccoH, Ha3bIBa-
eTCH 3aKOHOM BceMHpHoro T.HrOTeHIDI. 3TOT 3aKOH 6bill C<POpMyJIHpOBaH C3pOM
McaaKOM HbIOTOHOM, H OH yrBepx.n;aeT, qTO KIDIQ:J:Oe TeJIO BO BCeJIeHHOH lIpHT.HrH-
BaeTC.H K JII060MY ,npyroMy TeJIY C CHJIOH, KOTopaH IIp.HMO nponOpUHOHaJIbHa npo-
H3Be,neHHIO Macc 3THX TeJI H 06paTHO lIpOlIOPUHOHaJIbHa KBa,npaT}' paCCTO.HHIDI
Mex.n;y HHMH. MaTeMaTWIeCKH 3TO MO)I{HO C<POpMyJIHpOBaTb CJIe,nyIOIUHM 06pa30M:
F =( Gm lm2) / d 2 ,
r,ne F - B3aHMHa.H CHJIa npHTIDKeHIDI Mex.n;y MaccaMH ml H m2; d - paCCTO.HHHe
Mex.n;y MaccaMH; G - rpaBHTaUHOHHa.H nOCTO.HHHa.H (He 3aBHCHT OT B3aHMo,neHCT-
ByroIUHX Macc)
,l:LrnI JII06bIX ,nBYX Macc Ha JII060H nrraHeTe, BKJIlOqa.H nrraHeT}' 3eMJI.H, CHJIa B3a-
HMHoro npHTIDKeHIDI Mex.n;y MaccaMH F npeHe6pe)I(HMO MaJIa lIO cpaBHeHHIO C CH-
JIOH npHTIDKeHIDI Mex.n;y KIDIQ:J:OH H3 Macc H nrraHeTOH, Ha KOTOpOH 3TH MaCCbI Ha-
XO,lUITC.H. TIOCKOJIbKY CHJIa npHTIDKeHIDI nJIaHeTbI 3eMJI.H ,neHcTByeT Ha Bce OCTaJIb-
HbIe TeJIa, TO H3 BToporo 3aKOHa MexaHHKH HblOTOHa CJIe,nyeT, qTO 3Ta CHJIa coo6-
IUaeT HM YCKopeHHe g (M/C 2 ). TaK KaK
(GMoM)/ r 2 = F = Mg.
TI03TOMy:
g = (GM 0)/ r 2 = 9,81m / s 2 ,
r,ne G - rpaBHTaUHOHHa.H nOCTO.HHHa.H (6,668 x 10- 11 M 3 /KrC 2 ); Mo - Macca nrraHe-
TbI 3eMJI.H (5,98 x 10 24 Kr) npe,nlIOJIaraerc.H cocpe,nOTOqeHHOH B ueH1pe nrraHeTbI;
r - cpe,nHHH pa,nHYC nJIaHeTbI 3eMJI.H (6,375xl0 6 M); g - cpe,nHee YCKopeHHe CBO-
60,nHoro na,neHIDI Ha nJIaHeTe 3eMJI.H; F = B3aHMHa.H CHJIa npHTIDKeHIDI.
3. 11.2. nJfHeHHOe nepeMe eHHe
JII060e TeJIO, Ha KOTopoe ,neHcTByeT HeCKOJIbKO BHeIIIHHX CHJI TaK, qTO pe3YJIbTHPY-
IOIUaH 3THX CHJI npoxo,nHT qepe3 n;eH1p Macc TeJIa, 6y,neT ,nBHraTbC.H nOCTYIIaTeJIbHO.
KpoMe Toro, lIOCKOJIbKY pe3YJIbTHpyroIUa.H CHJIa npHJIOXeHa K ueH1py Macc TeJIa,
MOMeHT BpaIUeHIDI He B03HHKHeT, If TeJIO He 6y,neT BpaIUaTbC.H. TI03TOMy JII060e
TaKoe ,nBH)I{eHHe 6y,ner nOCTYIIaTeJIbHbIM, H TeJIO 6y,neT nepeMeIUaTbC.H HJIH c1pe-
MHTbC.H nepeMeCTHTbC.H C o,nHoro MeCTa B ,npyroe no np.HMOH JIHHHH. (TIpH nocTY-
naTeJIbHOM ,nBH)I{eHlfH Bce TOqKH TeJIa ,nBIDKyTC.H o,nHHaKOBO, Hero MO)I{HO pac-
CMa1pHBaTb KaK MaTepHaJIbHyro TOqKY. - llpUMe"t. nep.)
11. KUHemu a
3. 11.3. nepeMeU,f,eHJfe no KpJfSOnJfHeHHOH TpSeKTOpJfJf
Kor,na TeJIO lIepeMelI(aeTCR lIO KpHBOJIHHeHHOH 1paeKTOpHH PMHYca r C lIOCTORH-
HOD lIO MO.l(yJIIO CKOpOCTblO v, 3TO 03HaqaeT, qTO OHO ,nBIDKeTcR C YCKopeHHeM v 2 /r,
HanpaBJIeHHbIM lIO PMHYCY K n;eH1pY KpHBH3HbI TpaeKTOpHH, KaK 06Cy:>K,ZJ;aJIOCb B
pa3,neJIe 3.7.8. 3TO YCKopeHHe Ha3bIBaeTCR n;eH1pOC1peMHTeJIbHbIM. COrJIaCHO BTO-
poMy 3aKOH)' MexaHHKH HblOToHa, TeJIO He MoxeT ,nBHraTbCR C YCKopeHHeM, eCJIH
OHO He rro,nBepraeTCR B03,neHCTBHIO BHeIIIHeH CHJIbI. TaKHM 06pa30M:
:QeH1pOCTpeMHTeJIbHaR CHJIa = Macca x n;eH1pOC1peMHTeJIbHOe YCKopeHHe =
= (mv 2) / r.
ITOCKOJIbKY rrpH paBHOMepHOM )J;BIDKeHHH rro, OI(P}')KHOCTH v = !to, n;eHTPOCTPe-
MIITeJIbHaR CHJIa = mro 2 r.
:QeHlpOC1peMHTeJIbHaR CHJIa ,nOJDKHa 6bITb lIpHJIOXeHa K Te.rry H3BHe; HalIpHMep,
B c.rryqae aBTOM06HJI.H 3T0 CHJIa 1peHIDI IIIHH, COlIpHKaCalOlI(HXCR C ,noporOH, HJIH B
cJIy'lae lIoe3,na - 60KOBoe ,naBJIeHHe peJIbC Ha pe60p,nbI KOJIeC lIo,nBIDKHoro COCTaBa.
COrJIaCHO 1peTbeMY 3aKOH)' HbIOTOHa, AJUI KIDK.nOH aKTHBHOH CHJIbI CylI(eCTByeT CHJIa
peaKIJ;HH, paBHaR eH lIO BeJIHqHHe H lIpOTHBOlIOJIO)I{HasI lIO HalIpaBJIeHHIO. TI03TOMY
,ZJ;OJDKHa CylI(eCTBOBaTb CHJIa, paBHaR lIO BeJIINHHe H lIpOTHBOlIOJIO)I{HaR lIO HalIpaB-
JIeHllIO n;eH1pOC1peMHTeJIbHOH CHJIe. 3TO - n;eH1p06e)I{HaR CHJIa, H OHa ,neHcTByeT
no PMHYCY B HalIpaBJIeHHH OT n;eH1pa Iq)HBH3HbI 1paeKTOpHH. B c.rryqae aBTOM06HJI.H
ero WHHbI TOJIKalOT ,nopo)I{Hoe lIoIqJbITHe OT n;eH1pa KpHBH3HbI (n;eIITp06e)I{HaR
CHJIa), a ,nopora TOJIKaeT IIIHHbI aBTOM06HJI.H B lIpOTHBOlIOJIO)KHOM HalIpaBJIeHHH
(u;eIITpOCTPeMHTeJIbHaR cHJIa) ,no Tex lIOp, lIOKa CHJIa 1peHIDI Me)l{,[(y IIIHHaMH H ,no-
porOD OCTaeTCR ,nocTaTOqHOH, qT06bI aBTOM06HJIb He HaqaJI CKOJIb3HTb.
3.11.4. KOHJflfeCKJfH MSflTHJfK
ECJIII TBep,noe TeJIO MaCCOH m lIo,nBeIIIeHO K TOq-
Ke P Ha HHTH (JIerKOM CTep)I{He) .n.rrHHOH I TaK,
QTO OHO OlIHCbIBaeT OKpyxHOCTH PMHYca r C
nOCTORHHOH CKOpOCTblO <0, KaK lIOKa3aHO Ha
pllC. 3.13, TO AJUI JII060H 3MaHHOH CKOpOCTH
BpaIneHIDI yrOJI OTKJIOHeHIDI OT BepTHKaJIbHOH
OCII, rrpOXO,nRlI(eH qepe3 TOqKY lIo,nBeca, e H pa-
,ZJ;HYC BpalI(eHIDI r 6y,nyr BlIOJIHe OlIpe,neJIeHHbIMH
II B3aHMOCBR3aHHbIMH BeJIHqHHaMH. MMeHHO
npllHn;HlI KOHlNeCKOrO MaRTHHKa, TaK xe KaK H
npllHn;HlI peryJUlTOpa, HClIOJIb3yroTCR AJUI ynpaB-
JIeHIDI CKOpOCTblO ,nBHraTeJIeH. B lIOCJIe,nHeM
cJIy'lae, eCJIH HarpY3Ka Ha ,nBHraTeJIe YMeHbIIIaeT-
CH, TaK qTO CKOpOCTb ,nBHraTeJUI HaqHHaeT B03pa-
CTaTb, BeJIHqHHbI r H e TaIOKe YBeJIHqHBaIOTCR, H
PbIqIDKHbIH MexaHH3M, K KOTOpOMY lIpHcoe,nHHe-
Ha Macca m, 3aKpbIBaeT ,npOCCeJIbHyID 3aCJIOHKY
TaK, qT06bI CKOpOCTb ,nBHraTeJUI CHOBa CHH3H-
h
,
I ,
.... ""
I .........__...-'
I
,
9,81 m
Puc. 3.13. KOHHqeCKHH MaHTHHK
j
F/la8a 3. HH:JICeHepHaJI OUHaMUlCa
JIaCb ,no HOpMaJIbHOH (6e30rracHoH) BeJIHqHHbI. Ha060poT, eCJIH HarpY3Ka YBeJIINII-
BaeTCR TaK, qTO CKOpOCTb ,nBHraTelliI HaqHHaeT YMeHbIIIaTbcR, Tor,na r H e YMeHb-
IIIaIOTCR, H PbJqIDKHbIH MexaHH3M, K KOTOpOMY rrpHcoe,nHHeHa Macca m, OTKpbIBa-
eT ,npoCCeJIb H BOCCTaHaBJIHBaeT rrpaBHJIbHyID pa60qyro CKOpOCTb ,nBHraTelliI. TaKHM
06pa30M 06eCrreqHBaeTCR rrOCTORHHaR CKOpOCTb pa60TbI ,nBHraTelliI.
TIYCTb T - CHJIa HaT.HXeHIDI HHTH, Ha KOTOpOH rro,nBeIIIeHa Macca m; Tor,na
IJ;eH1poc1peMHTeJIbHaR CHJIa, rro.z:r.n;ep)I(HBalOlI(aR ,nBH)I{eHHe lIO O:KpyxHOCTH, 06ec-
lIeqHBaeTCR ropH30HTaJIbHOH cOCTaBlliIlOlI(eH CHJIbI T.
fOpH30HTaJIbHaR COCTaBlliIlOlI(aR CHJIbI T paBHa:
T sin a = mrro 2
TIOCKOJIbKY CHCTeMa Haxo,nHTCR B paBHoBecHH, BepTHKaJIbHaR COCTaBlliIlOlI(WI
CHJIbI T AJUI paBHoBecIDI paBHa:
TI03TOMY:
T cos a = 9,81m.
sin a / cos a = tg(}.
tg(} = (rro 2 ) / 9.81.
tg(} = (rro 2 ) / 9,81 H tg(} = r / h,
r / h = (rro 2 ) / 9,81.
h = 9,81 / 00 2 .
r = I sin a.
I = r / sin a.
T = m / 00 2 .
TIOCKOJIbKY:
To:
TaK KaK
TO
3. 11.5. BpSUf,eHJfe Tens sOKpyr HenOASJfJICHOH OCJf
Kor,na pe3YJIbTHpyrolI( BHeIIIHHX CHJI, ,neHCTByroII(HX Ha TeJIO, lIpHJIOXeHa K ero
IJ;eH1pY Macc, TeJIO ,nBH)I{eTCR TOJIbKO lIOCTYIIaTeJIbHO, H BpalI(eHIDI He B03HHKaeT (CM.
p33,neJI 3.11.4). O,nHaKo Kor,na pe3YJIbTHpyrolI(aR CHJIa lIpHJIOXeHa He K IJ;eHTPY Mace
TeJIa, B03HHKHeT MOMeHT BpalI(eHIDI, H TeJIO HaqHeT BpalI(aTbCR. Ha pHC. 3.14 lIOKa-
3aHO TeJIO, BpalI(alOlI(eeCR BOKpyr OCH P lIO,n B03,neHcTBHeM MOMeHTa BpalI(eHIDI T.
MrHOBeHHaR yrJIOBaR CKOpOCTb TeJIa paBHa 00, a ero yrJIOBOe YCKopeHHe - a.
YCKopeHHe, C KOTOpbIM ,nBIDKYTCR MaTepHaJIbHbIe TOqKH, HMeIOII(He MaccbIom,
8m! H 8m2, MO)I{HO pa3JI0)I(HTb Ha ,nBe cOCTaBlliIIOII(He: TaHreHIJ;HaJIbHyID COCTaBJUI-
IOIUyIO ar H IJ;eH1poc1peMHTeJIbHyID COCTaBlliIlOlI(yro 00 2 r, KOTopaR HalIpaBJIeHa no
PMHYCY KOCH P, BOKpyr KOTOpOH BpalI(aeTCR TeJIO. TIOCKOJIbKY IJ;eHTPOCTPeMllTe-
JIbHbIe COCTaBlliIlOlI(He AJUI KIDK,ZJ;OH H3 MaTepHaJIbHbIX TOqeK HalIpaBJIeHbI K OCII P,
COOTBeTCTByrolI(He CHJIbI He C03,na1OT MOMeHTOB OTHOCHTeJIbHO 3TOH OCH. O,nHaKO
CHJIbI, 06eClIeqHBalOlI(He TaHreHIJ;HaJIbHbIe COCTaBlliIlOlI(He AJUI KIDK,ZJ;OH H3 3JIeMeH-
TapHbIX MaCC (8mar 2 ), ,neHCTBHTeJIbHO C03,na1OT BpalI(alOlI(HH MOMeHT OTHOCHTeJIL-
HO OCH P. TI03TOMY lIOJIHbIH MOMeHT BpalI(eHIDI OTHOCHTeJIbHO OCH P paBeH:
21.8mar 2) = alJ..8mr 2 ),
TaK KaK Bce MaTCpHaJIbHbIe TOqKH BpalI(alOTCR BOKpyr P C O,nHHaKOBOH CKOpOCTblO.
3.11. KUHemUK,a
\00 -yrnoBaSi CKOpOCTb
) IX - yrnoBoe YCKopeHHe
TaHreHu.VlaJ1bHa51 (KacaTeJ1bHa5l)
COCTaBJ151IOw.a51 YCKopeHVI
J
PRC. 3.14. BpalI(eHHe TeJIa BOKpyr <pHKcHpoBaHHoro I(eHTpa
CYMMa lIpoH3Be,neHHH Macc Bcex MaTepHaJIbHbIX TOqeK TeJIa Ha KBan;paTbI HX
paccToRHHH ,no paccMaTpHBaeMoH OCH P lJ...omr 2 ) Ha3bIBaeTCR MOMeHTOM HHepIJ;HH.
TaKHM 06pa30M:
MOMeHT HHepIJ;HH I p = lJ...omr 2 )
MOMeHT BpalI(eHIDI T OTHOCHTeJIbHO P = I pa.
E,nHHHIJ;bI H3MepeHIDI: T (HM), Ip (KrM 2 ) H a (pan;/c 2 ).
3. 11.6. PSAJfYC KpJfSJf3H&1
TIOCKOJIbKY C)'MMa 3JIeMeHTapHbIX Macc paBHa 06lI(eii Macce TeJIa (m = Lom), eCJIH
k - 3TO paCCTORHHe OT BbI6paHHoH OCH ,no TOqKH, B KOTOpOH, KaK lIpe,ZJ;lIOJIaraeT-
CH, cocpe,nOTOqeHa C)'MMapHaR Macca, TaK qTO lIPH 3TOM MOMeHT HHepIJ;HH OTHOCH-
reJIbHO 3TOH OCH paBeH MOMeHTY HHepIJ;HH TeJIa mk 2 = lJ...omr 2 ), Tor,na MOMeHT
HHepUHH MO)I{HO BbIpa3HTb KaK:
Ip == mk 2 ,
r,n;e k - pan;HYC HHepIJ;HH OTHOCHTeJIbHO KOHKpeTHoH OCH BpalI(eHIDI.
3. 11. 7. LleHTp CTonKHOSeHJffl
TIpH peIIIeHHH 3an;aq, CBR3aHHbIX C BpalI(eHHeM, qaCTO 6bIBaeT He06xo,nHMO OlIpe-
,n;eJIHTb JIHHHIO ,neiicTBIDI pe3YJIbTHpyrolI(eH CHJIbI F (KaK lIOKa3aHO Ha pHC. 3.15}, I
r,n;e F paBIDIeTCR Macce TeJIa, )'MHOXeHHoii Ha YCKopeHHe IJ;eH1'pa Macc. TIpe,nlIOJIO-
)l{HM, qTO 3Ta JIHHHR ,neHcTBIDI pe3YJIbTHpyrolI(eii CHJIbI F lIpOXO,nHT qepe3 HeKOTO-
PYlO TOqKY P Ha JIHHHH RGP, r,ne R - OCb BpalI(eHIDI H G - IJ;eH1'p Macc. KaK 06-
Cyx,zJ;aJIOCb B pa3,neJIe 3.11.6, TOJIbKO TaHreHIJ;HaJIbHaR COCTaBIDIlOlI(aR CHJIbI F MO-
xeT C03,naBaTb MOMeHT BpalI(eHHR OTHOCHTeJIbHO R. P - 3TO TOqKa TeJIa, r,ne MO-
MeHTbI Bcex BHeIIIHHX CHJI ypaBHOBeIIIeHbI, H HX C)'MMa paBHa H)'JIIO. TOqKa P
Ha3bIBaeTCR IJ;eH1pOM y,napa. ,[L1rn Tex CJIyqaeB, Kor,na TeJIO BpalI(aeTCR BOKpyr CBoe-
ro n;eH1'pa Macc (R H G cOBlIan;aIOT), MO)I{HO 3alIHcaTb:
F/la8a 3. HH:JICeHepHaR OUHaMUlCa
(0 - yrJ10Ba51 CKOpOCTb
T aHreHu.VlaJ1bHa51 (KacaTeJ1 bHa5l)
COCTaBJ151IOw.a51 F = ma.t
- yrnoBoe YCKopeHloIe
PRC. 3.15. QeHTp Y,lJ;apa
MOMeHT BpalI(eHIDI T = m(kG)2a = IGa,
r,ne kG - panHYC HHepn;HH OTHOCHTeJIbHO G.
Kor,na TeJIO BpalI(aeTCR BOKpyr OCH, lIpOXO,nRlI(eH qepe3 ero n;eH1p Macc, panHYC
HHepn;HH MHHHMaJIeH H, CJIe,nOBaTeJIbHO, MOMeHT BpalI(eHIDI, He06xo,nHMbIH ,nmI
BpalI(eHIDI TeJIa, TaIOKe MHHHMaJIeH.
(EcJIH TeJIO, 3a:KpeIIJIeHHOe Ha OCH BpalI(eHIDI, HClIbITbIBaeT y,nap, TO ,neHCTBHe
y,napa B 06lI(eM c.rryqae lIepe,naeTcR H Ha OCb. Ho eCJIH y,nap HaHeceH CTporo B
n;eH1p y,napa, TO OCb He HClIbITbIBaeT HHKaKHX ,nOlIOJIHHTeJIbHbIX HarpY30K, CBR3aH-
HbIX C y,napOM. B 3TOM c.rryqae CKOpOCTb lIOCTYIIaTeJIbHOrO ,nBH)I{eHIDI, lIpH06peTae-
Moro TOqKOH BMeCTe C n;eH1pOM Macc, KOMlIeHcHpyeTcR JIHHeHHOH CKOpOCTblO Bpa-
lI(aTeJIbHOrO ,nBH)I{eHIDI BOKpyr n;eH1pa Macc (06a 3TH ,nBH)I{eHIDI HHHIJ;HHpyroTCR
CHJIOH F H lIpOHCXO,nRT o,nHOBpeMeHHO). - llpUMett. nep.)
3. 11.8 MOMeHT JfMnyn&CS
M3 pa3,neJIa 3.6.8 H3BeCTHO, qTO HMlIYJIbC TeJIa paBeH Macce TeJIa, YMHOxeHHoH Ha
ero JIHHeHHyID CKOpOCTb. MOMeHT HMrryJIbCa BpalI(alOlI(erOCR TeJIa - 3TO MOMeHT
ero HMrryJIbCa OTHOCHTeJIbHO TOqKH BpalI(eHIDI. Ha pHC. 3.16 lIOKa3aHO TeJIO, Bpa-
lI(alOlI(eeCR BOKpyr OCH P C yrJIOBOH CKOpOCTblO ro (pan/c). PaCCM01pHM MaTepH-
aJIbHyro TOqKY MaCCOH 8m, paClIOJIOXeHHyID Ha paCCTORHHH r OT OCH P, CKOpOCTb
KOTOpOH paBHa v M\C.
MMrryJIbC MaTepHaJIbHOH TOqKH = 8mv = 8mrro.
MOMeHT HMrryJIbCa OTHOCHTeJIbHO P = r(8mrro) = 8mr 2 ro.
MOMeHT HMrryJIbCa OTHOCHTeJIbHO P AJUI Bcero TeJIa = L8mr 2 ro.
L8mr 2 = I,
Ho:
r,ne I - MOMeHT HHepn;HH (CM01pH pa3,neJI 3.11.5).
TaKHM 06pa30M:
MOMeHT HMrryJIbCa OTHOCHTeJIbHO P = lro.
3.12. IIpocmoe zapMOHU'teCK,Oe 08U:JICeHUe
(0 - yrJ10Ba51 CKOpOCTb
,
,
,
,
,
", V - reo- J1V1HeVlHa51 CKOpOCTb
,
)
)
PRC. 3.16. MOMeHT HMnyJIbCa
ECJIH rrpoHcxo,nHT H3MeHeHHe yrJIOBOH CKOpOCTH, BbIpaxeHHe, lIpHBe,neHHoe
BbIwe, lIpe06pa3yeTcR KaK:
M3MeHeHHe MOMeHTa HMrryJIbCa = 1(002 - (01)'
CJIe,nOBaTeJIbHO: BpalI(aTeJIbHbIH HMrryJIbC Tt = 1(002 - (01),
r,ne T - MOMeHT BpalI(eHIDI (HM); t - BpeM.H (ATIHTeJIbHOCTb HMrryJIhCa B ceKYJI,ZJ;ax).
KHHeTlNeCKaR 3HeprIDI, KOTOpOH 06JIMaeT BpalI(alOlI(eeCR TeJIO = 1 p002 ,
r,n;e Ip - MOMeHT HHepIJ;HH OTHOCHTeJIbHO OlIOpHOH TOqKH P.
KHHeTHqeCKaR 3HeprIDI TeJIa, KOTopoe o,nHOBpeMeHHO H BpalI(aeTCR, H lIepeMe-
m;aeTCR = KHHeTHqeCKaR 3HeprIDI BpalI(eHIDI + KHHeTlNeCKaR 3HeprIDI lIepeMelI(e-
HID! = /002 + mv2.
3.12. npOCToe rapMOHMl.leCKOe ABM>KeHMe
)];BIDKeHHe, KOTopoe lIOBTopReTCR qepe3 HeKOTopbIe HHTepBaJIbI BpeMeHH, Ha3bIBaeT-
/
CH nepHo,nlNeCKHM. HalIpHMep, eCJIH ,nepHYTb TaKoe ynpyrde TeJIO, KaK cTPYHa My-
3bIKaJIbHOrO HHC1pyMeHTa, HJIH y,napHTb lIO HeH, TO OHa 6y,neT CB060,nHO KOJIe6aTbCR
OKOJIO cpe,nHero lIOJIOXeHIDI. ECJIH B03HHKIIIHe 3ByKOBbIe KOJIe6aHIDI HaxO,nRTCR B
npe,neJIax 06JIaCTH CJIbIIIIHMOCTH qeJIOBeqeCKOrO yxa, pO)l{)];aeTCR My3bIKaJIbHbIH 3ByK;
n03TOMy TaKHe KOJIe6aHIDI xapaKTepH3yroTCR KaK rapMOHlNeCKHe. TaKoe lIepHo,nH-
qeCKOe ,nBH)I{eHHe, lIOpO)l{)];alOlI(ee 3ByKOBbIe KOJIe6aHIDI, He lIpO.ZJ;OJDKaeTCR 6ecKo-
HeqHO, a B KOHIJ;e KOHIJ;OB 3aT}'XaeT. Ero «raCHT» BHYTPeHHee COlIpOTHBJIeHHe C1pYHbI
Ii BHeWHee COlIpOTHBJIeHHe B03,nyxa, oIqJ)')I{alOlI(ero CTPYHY.
TIpocToe rapMOHlNeCKOe .ZJ;BH)I{eHHe olIpe,nelliIeTcR KaK ,nBH)I{eHHe TOqKH B,nOJIb
npRMoH JIHHHH, TaK qTO YCKopeHHe TOqKH Bcer,na HalIpaBJIeHO K <PHKcHpOBaHHOH
TOQKe Ha 3TOH JIHHHH H lIpOlIOpIJ;HOHaJIbHO ee CMelI(eHHIO OT 3TOH TOqKH. Ha
pliC. 3.17 lIOKa3aHa Bo06paxaeMaR TOqKa P, BpalI(alOlI(aRCR lIO OKpyxHOCTH C PMHY-
COM r H IJ;eH1pOM B TOqKe 0 C lIOCTORHHOH yrJIOBOH CKOpOCTblO (0. ECJIH ClIPO UHpO-
BaTh TOqKy P Ha ,nHaMeTP OIqJyxHOCTH AB, TO ,nBH)I{eHHe TOqKH Q BlIepe,n-Ha3M lIO
I
,n;HaMeTpY AB 6y,neT lIpOCTbIM rapMOHlNeCKHM ,nBH)I{eHHeM. To eCTb, Kor,na T01JKa P
F/la8a 3. lfH:JICeHepHaJl OUHaMUlCa
AMnJ1V11)'Aa
AMnJ1V11)'Aa
PRC. 3.17. npOcToe rapMOHHqeCKOe ,nBH-
xeHHe.
COBepIIIHT O,nHH lIOJIHbIH 060POT, TOqKa Q
cOBepIIIHT lIOJIHOe KOJIe6aHHe OT A ,no B Ii
06paTHo K A. TI03TOMY BpeM.H o,nHoro
lIOJIHOrO 060poTa P paBHO BpeMeHH O,nHO-
ro lIOJIHOrO KOJIe6aHIDI Q. PaccToRHHe AO
(HJIH BO) Ha3bIBaeTCR aMIIJIHTY,noH KOJIe-
6aHHH.
1. qacTOTY KOJIe6aHHH B repn;ax (fu)
MO)I{HO HaHTH H3 BbIpaxeHIDI:
f = 1 / T = ro / 2TC,
r,ne f - qaCTOTa KOJIe6aHHH, T - lIepHO,lJ;,
ro - yrJIOBaR CKOpOCTb.
2. MaKCHMaJIbHyID CKOpOCTb Q ,no-
CTHraeT B cpe,nHeH TOqKe, Kor,na P HaxO-
,nHTCR HelIocpe,nCTBeHHO Han HJIH lIO,n
n;eH1pOM 0 H YCKopeHHe P paBHO H)'JIIO.
TI03TOMY:
V max = (J)T.
3. MaKCHMaJIbHOe YCKopeHHe Q paBHO YCKopeHHIO P H ,nOCTHraeTCR B KpaHHHX
TOqKax A H B, Kor,na CKOpOCTb Q paBHa HYJIIO. TI03TOMY:
a max = ro 2 r.
npHMep
TeJIo COBepIIIaeT lIpOCToe rapMOHlNeCKOe ,nBIDKeHHe C aMIIJIHTY,noH 50 MM H qac-
TOTOH 2 fn;.
TIOCKOJIbKy qaCTOTa f = ro / 2TC, Tor,na ro = f X 2TC = 2 X 2TC = 12,566 pan / c.
a) MaKCHMaJIbHaR CKOpOCTb = ror = 12,566 x 50 = 0,628 M/C B cpe,nHeit TOqKe.
6) MaKCHMaJIbHOe YCKopeHHe = ro 2 r = 12,566 2 x 50 = 7,895 M/C 2 B IqJaHHHX TOqKax.
3. 12. 1. MSTeMSTJflfeCKJfH MSflTHJfK
rpY3
mg
PRC. 3.18. MaTeMaTHqeCKHH MaHTHHK
MaTeMaTHqeCKHH MaRTHHK COCTOHT H3 He-
60JIbIIIOrO MaCCHBHoro rpY3a, lIo,nBeIIIeHHO-
ro Ha HIDKHeM KOHn;e JIerKOrO CTep)J{}UI.
BepXHHH KOHen; CTep)I{H.H 3aKpeIIJIeH TaK,
qTO CTepxeHb H rpY3 MOryr KOJIe6aTbCR B
BepTHKaJIbHOH IIJIOCKOCTH, KaK lIOKa3aHO
Ha pHC. 3.18.
PaCcM01pHM lIpOCTOH MaRTHHK co
CTep)I{HeM ATIHHOH I H rpY30M MaCCOH m,
OTKJIOHeHHbIH Ha yrOJI 9. TaK KaK rpY3 Mac-
COH m ,nBIDKeTCR lIO KpyrOBOH 1paeKTOpHH
panHYca I, e,nHHCTBeHHbIe CHJIbI, ,neHcTByK>-
II(He Ha Hero, 3TO CHJIa T.HXeCTH mg H CHJIa
HaT.HXeHIDI CTep)I{H.H T. TIpH OTKJIOHeHHIi,
3.12. llpocmoe zapMOHU'teCK,oe 08UJICeHUe
nOKa3aHHOM Ha pHCYHKe, CHJIa T.HXeCTH C03.n;aeT MOMeHT BpalI(eHIDI OTHOCHTeJIbHO
TOqKH P, C1peM.HII(HHCR B03BpaTHTb rpY3 B ero paBHoBecHoe lIOJIOXeHHe, T.e. B lIO-
JIO:>KeHHe, Kor.n;a MaRTHHK BHCHT BepTHKaJIbHO H rpY3 HaxO.n;HTCR HelIocpe.n;cTBeHHo
no,lJ; TOqKOH lIo.n;Beca (P). CJIe.n;oBaTeJIbHo:
B03BpalI(alOlI(HH MOMeHT = -mgl sin 9 = -mgl9 (LtIDI MaJIbIX 3HaqeHHH 9).
,[lJm rpY3a MaCCOH m MOMeHT HHepIJ;HH OTHOCHTeJIbHO P paBeH I p = ml 2 .
TIpHMeHHM BTOpOH 3aKOH MexaHHKH HblOToHa K BpalI(eHHIO MasITHHKa BOKpyr P:
T = I pa,
r,lJ;e T - lIepHO.n; \)
./
TaKHM 06pa30M: -mgl9 = ml 2 a r.n;e a = (-g(}) / I.
TI03TOMY YCKopeHHe = CMelI(eHHe x KOHCTaHTa H, lIPH YCJIOBHH, qTO CMelI(eHHe
MaJIO, KOJIe6aHIDI MaRTHHKa 6y.n;yr rapMOHlNeCKH M .n;B H)I{eHHeM. TI03TOMY:
1. TIepHo.n; T = 21t / ro = 21t ..J( 1 / g) ce KYH.ZJ;.
2. qacToTa f = 1 / T = (1 / 21t) ..J( 1 / g) fIJ;.
TaK KaK lIepHO.n; KOJIe6aHHH MaRTHHKa (H, COOTBeTCTBeHHO, qaCTOTa ero KOJIe6a-
HHH) 3aBHCHT TOJIbKO OT ero ATIHHbI H MeCTHoro YCKopeHIDI cB060.n;Horo lIan;eHIDI,
OTCIO,na CJIe.nyeT, qTO MaRTHHK - H,lJ;eaJIbHOe cpe.n;CTBO ynpaBJIeHIDI MeXaHlNeCKHMH
qaCaMH, lIPH YCJIOBHH, qTO CTepxeHb, Ha KOTOpOM lIo.n;BeIIIeH rpY3, H3rOTOBJIeH H3
MaTepHaJIa C OqeHb HH3KHM K03<P<PHIJ;HeHTOM paCIIIHpeHIDI lIpH H3MeHeHIDIX TeMne-
paTYPbI.
3.12.2. Co6CTSeHH&le Kone6sHJffl
C06CTBeHHbIe (cB060.n;HbIe) KOJIe6aHIDI lIPH-
6JIIDKalOTCR K lIpOCTOMY rapMOHlNeCKOMY
,lJ;BIDKeHHIO, Kor.n;a OHH B03HHKalOT B TeJIaX,
H3rOTOBJIeHHbIX H3 ynpyrHX MaTepHaJIOB,
nO,ZJ;TIHIDIIOII(HXCR 3aKOH)' fyKa. Ha pHC. 3.19
nOKa3aHa lIpyxHHa paCT.HXeHIDI, K KOTOpOH
no,nBeIIIeH rpY3 MaCCOH m. B lIOKoe lIOJIOXe-
HHe rpY3a 6y.n;eT 3aBHceTb OT xeCTKOCTH lIPY-
:>KHHbI H CHJIbI T.HXeCTH, .n;eHcTByrolI(eH Ha
rpY3. 3TO 03HaqaeT, qTO .n;Be 3TH CHJIbI lIpO-
THBo,neHCTByroT o.n;Ha .n;pyrOH, H rpY3 Haxo-
,lJ;HTCR B COCTORHHH paBHOBeCIDI. ECJIH CMec-
THTb rpY3 BHH3 Ha paCCTORHHe x H OTrryC-
THTb, TO no.n; B03.n;eHCTBHeM lIpyxHHbI OH
HaqHeT KOJIe6aTbCR C qaCTOTOH U0 B BepTH-
KaJIbHOH IIJIOCKOCTH.
TIYCTb S - xeCTKOCTb lIpyxHHbI (CHJIa
Ha e,nHHHIJ;y H3MeHeHIDI ATIHHbI), a x - Be-
npy)I(VlHa
Macca (paBHOBecVle VI
n0J10)l(eHVle «nOK05l») ,
x
, ,
, m
'T
PRC. 3.19. C06cTBeHHbIe KOJIe6aHHH
F/la8a 3. HH:JICeHepHaJI OUHaMUlCa
JIINHHa CMelI(eHIDI OT lIOJIOXeHIDI paBHoBecIDI; Tor,na H3MeHeHHe CHJIbI B lIpyxHHe
lIOCJIe paCT.HXeHIDI paBHo sx. 3TO H eCTb YCKOpRlOlI(IDI CHJIa F, ,neHcTByroII(IDI Ha
Maccy m, TaK qTO F = sx. CJIe,noBaTeJIbHo:
Y cKopeHHe Q = f I m = sx I m.
3HaqHT, YCKopeHHe lIpRMO lIpOlIOpU;HOHaJIbHO BeJIHqHHe CMelI(eHIDI MaCCbI m
OT lIOJIOXeHIDI paBHoBecIDI, H ,lJiBH)I{eHHe RBlliIeTCR lIpOCTbIM rapMOHHqeCKHM ,nBH-
xeHHeM. TIepHo,n KOJIe6aHHH T B ceKYJI,na x olIpe ,nelliIeTcR BbIpaxeHHeM:
T = 21t .J( m I s)
qacToTa KOJIe6aHHH B repu;ax = 1 I T
npHMep
K lIpyxHHe, pa60TalOlI(eH Ha paCT.HXeHHe, lIo,nBeIIIeH rpY3 MaCCOH 5 Kr, B co-
CTORHHH paBHOBeCIDI YATIHIDIIOII(HH ee Ha 12 MM. fpY3 CMeCTHJIH BHH3 elI(e Ha
15 MM H 3aTeM OTrryCTHJIH. TIpeHe6peraR lIOTepRMH Ha COlIpOTHBJIeHHe B03,nyxa Ii
lIOTepRMH Ha rHCTepe3HC B MaTepHaJIe lIPyxHHbI, OlIpe,neJIHTb: a) qaCTOTY KOJIe6a-
HHH, 6) MaKCHMaJIbHyID CKOpOCTb rpY3a, B) MaKCHMaJIbHOe YCKopeHHe rpY3a.
a) Bec rpY3a paBeH mg = 5 Kr x 9,81 M/c 2 = 49,05 H.
)l(ecTKocTb lIpyxHHbI S = CHJIa/YATIHHeHHe = 49,05 H/15 MM = 3,27 H/MM.
llpUMettQHue. 3,27 H/MM = 3,27 KH/M.
qacToTa KOJIe6aHHH f = 1 I T = 1 I 21t .J( s 1m).
TI03TOMY:
f = 1 I 21t .J [(3,27 x 103) I 5] = 4,07 fu;.
6) MaKCHMaJIbHaH CKOpOCTb HarpY3KH =
= <OX = 21tfx = 21t x 4,07 x 15 = 383,59 MMI c.
B) MaKCHMaJIbHOe YCKopeHHe HarpY3KH =
= ro 2 x = (21Cf) 2 X = (21C x 4,07)2 x 15 = 9809,3 MMI c 2 HJIH 9,81 MI c 2 .
3. 13. rMAPOAMH8MMK8
BOlIpOCbI rH,lJ;pOCTaTHKH 06Cy:>K,ZJ;aJIHCb B pa3,neJIe 2.13. B ,naHHOM pa3,neJIe 06cy:>K,ZJ;a-
IOTCR Te He cpe,nbI ()l{H)J:KOCTH HJIH ra3bI) B ,nBH)I{eHHH (,nHHaMHKa )l{H)J:KOCTeH Ii
ra30B). TIo CylI(eCTBY, lIpe,nlIOJIaraeTCR, qTO B YCTaHOBHBIIIeMCR paBHOMepHOM lIO-
TOKe MOJIeKyJIbI )l{H)J:KOCTH ,nBIDKYTCR lIO lIapaJIJIeJIbHbIM TPaeKTOpIDIM, a ,naBJIeHHe
H CKOpOCTb B cpe,ne He H3MeIDIIOTCR B 3aBHCHMOCTH OT BpeMeHH. TaKHe YCJIOBIDI
06bIqHO lIpHMeHHMbI K MeATIeHHOMY lIOTOKy B rJIMKOCTeHHbIX U;HJIHH,npHqeCKHX
TPy6ax C ,nOCTaTOqHO 60JIbIIIHM lIOlIepeqHbIM CeqeHHeM.
YCTaHoBHBIIIHHCR HepaBHoMepHbIH lIOTOK B03HHKaeT, Kor,na )l{H)J:KOCTb (HJIIi
ra3) TeqeT lIO KOHHqeCKOH TPy6e. TIo Mepe Toro KaK lIOlIepeqHOe CeqeHHe TPy6bI
CTaHOBHTCR MeHbIIIe, CKOpOCTb MOJIeKyJI )l{H)J:KOCTH HJIH ra3a YBeJIHqHBaeTCR TaK,
qT06bI TOT xe caMbIH 06'beM )l{H)J:KOCTH HJIH ra3a Mor lIpOHTH qepe3 )'MeHbIIIeHHoe
3.13. ruiJpoiJuHllMU"a 129
IIOrrepeqHOe CeqeHHe 3a ,naHHOe BpeM.H. O,nHaKo lIOCKOJIbKY CKOpOCTb H ,naBJIeHHe
HBJUIIOTCR lIOCTORHHbIMH B JII060M KOHKpeTHOM CeqeHHH Tpy6bl, 3TOT c.rryqaH Bce
:>Ke paCCMaTpHBaeTCR KaK YCTOHqHBbIH lIOTOK, HeCM01'pR Ha TO, qTO lIOTOK HepaB-
HOMepeH lIO .n.rrHHe KOHHqeCKOH TPy6bI.
(06 H3MeHeHIDIX CKOpOCTH MOJIeKYJI ra3a MO)I{HO Cy,nHTb, TOJIbKO ytlHTbIBaR
TaIOKe H ,naBJIeHIDI B CeqeHIDIX, KOTopbIe MOryr CylI(eCTBeHHO pa3JIlNaTbCR. -
l1pUMeli. nep.)
3. 13. 1. CKOPOCT& TelfeHJffl
f
/"
CKOPOCTb TeqeHIDI HJIH pacxo,n )l{H)J:KOCTH, lIpOXO,nRlI(eH qepe3 TPy6y, MO)I{HO OlIpe-
,n;eJIHTb H3 CJIe,nyrolI(ero BbIpaxeHIDI:
Q = Av,
r,ne Q - pacxo,n )l{H)J:KOCTH (M 3 /C); A - IIJIOII(Mb lIOlIepeqHOrO CeqeHIDI TPy6bI;
v - CKOpOCTb lIOTOKa.
Y,neJIbHbIH MaCCOBbIH pacxo,n MO)I{HO OlIpe,neJIHTb CJIe,nyrolI(HM 06pa30M:
pQ = pAv,
r,ne p - IIJIOTHOCTb )l{H)J:KOCTH; pQ - y,neJIbHbIH MaCCOBbIH lIOTOK (Kr/c).
3.13.2. Henpep&ISHOCT& TelfeHJffl
BbIwe lIpHBe,neHHbIe <pOPMYJIbI OTHOCRTCR K YCTaHOBHBIIIeMYCR lIOTOKY B Tpy6ax C
IIOCTORHHbIM CeqeHHeM. TIpH YCTaHOBHBIIIeMCR lIOTOKe B KOHlNeCKOH (cyxalOlI(eH-
CR) 1'py6e y,neJIbHbIH MaCCOBbIH pacxo,n, lIPOXO,nRII(HH qepe3 JII060e ,naHHoe lIOlIe-
peqHOe CeqeHHe, OCTaeTCR lIOCTORHHbIM. TI03TOMY:
pQ=PIAIvi =P2 A 2 V 2,
r,ne pQ - y,neJIbHbIH MaCCOBbIH pacxo,n; Al H A 2 - MaKCHMaJIbHaR H MHHHMaJIbHaR
llJIOInMH lIOlIepeqHOrO CeqeHIDI (M 2 ) COOTBeTCTBeHHO; VI - CKOpOCTb lIOTOKa )I{H)J:-
KOCTH B lIOlIepeqHOM CeqeHHH Al (M/C); v2 - CKOpOCTb lIOTOKa )l{H)J:KOCTH B lIOlIe-
peQHOM CeqeHHH A 2 (M/C); PI - IIJIOTHOCTb )l{H)J:KOCTH B lIOlIepeqHOM CeqeHHH AI;
P2 - IIJIOTHOCTb )l{H)J:KOCTH B lIOlIepeqHOM CeqeHHH A 2 .
3T0 BbIpaxeHHe Ha3bIBaeTCR ypaBHeHHeM Hepa3pbIBHOCTH, H OHO lIpHMeHHMo
KO BceM TeKyqHM cpe,naM, KaK K )l{H)J:KOCTRM, TaK H K ra3aM.
,[lJm HeC)l(HMaeMbIX TeKyqHx cpe,n ()l{H)J:KOCTeH) P RBlliIeTCR KOHCTa )0ID,nmI
Bcex lIOlIepeqHbIX CeqeHHH, lI03TOMY:
Q=AIvi =A 2 V 2.
5-3255
F/la8a 3. HH:JICeHepHaJI OUHaMUlCa
3.13.3. 3HeprJffl ASJfJKYU,f,eHCfI TeKYlfeH CpeA&1
(YPssHeHJfe 6epHynnJf)
HeC)l(HMaeMbIe Te He cpe,nbI ()l{H)J:KOCTH) 06JIan;a1OT 'rpeM.H BH,lJ;aMH 3HeprHH, a
HMeHHO:
. lIOTeHIJ;HaJIbHOH 3HeprHeH: mgZ, r,ne m - Macca (Kr), g - 9,81M/c 2 H Z -
BbICOTa )l{H)J:KOCTH Han; 3an;aHHbIM ypOBHeM OTCqeTa. TIoTeHIJ;HaJIbHIDI 3Hep-
rIDI B e,lJ;HHHIJ;y MaCCbI = gZ ( /Kr);
. KHHeTlNecKOH 3HepmeH: mv2, r,ne m - Macca (Kr) H V - lIOCTO.HHHasI CKOpocTb
)l{H)J:KOCTH (M/C). KHHeTlNeCKIDI 3HeprIDI Ha e,nHHHIJ;Y MaCCbI = v2 UJ:x/Kr);
. 3HeprHeH ,naBJIeHIDI: Kor,na )l{H)J:KOCTb TeqeT lIO Tpy6e lIO,n ,lJ;aBJIeHHeM, lIPIl
,nBIDKeHHH )l{H)J:KOCTH qepe3 KIDK,ZJ;Oe CeqeHHe TPy6bI BbIlIOJIIDIeTCR pa60Ta no
lIpeo,nOJIeHHIO rH,lJ;pOCTaTlNeCKOrO ,naBJIeHIDI, lIpHcyrcTByrolI(erO B 3TOM ce-
qeHHH. CJIe,noBaTeJIbHo:
BbIlIOJIHeHHIDI pa60Ta Ha e,nHHHIJ;Y MaCCbI = pQ / pQ = p / P ( /Kr),
r,lJ;e p - ,lJ;aBJIeHHe, p - IIJIOTHOCTb )l{H)J:KOCTH, Q - CKOpOCTb lIOTOKa, a p / p -
3HeprIDI ,naBJIeHIDI Ha e,nHHHIJ;y MaCChI )l{H)J:KOCTH.
CJIe,nOBaTeJIbHO, C)'MMapHIDI 3HeprIDI, KOTOpOH 06JIan;aeT e,nHHHIJ;a MaCCbI )I{Jf)];-
KOCTH, 6y,neT C)'MMOH ee lIOTeHIJ;HaJIbHOH 3HeprHH, KHHeTHqeCKOH 3HeprHH II
3HeprHH ,naBJIeHIDI.
C)'MMapHIDI 3HeprIDI Ha e,lJ;HHHIJ;y MaCCbI UJ:X/Kr) = gZ + v 2 / 2 + p / p.
B YCTaHoBHBIIIeMcR nOTOKe HeC)l(HMaeMOH Te eH cpe,nbI ()l{H)J:KOCTH) lIOTeH-
IJ;HaJIbHIDI, KHHeTlNeCKIDI H 3HeprIDI ,naBJIeHIDI MOryr pa3JIINaTbCR B pa3HbIX Mec-
Tax B,nOJIb lIOTOKa, o,nHaKO COrJIaCHO lIpHHIJ;Hrry coxpaHeHIDI 3HeprHH lIOJIHWI
3HeprIDI 6y,neT OCTaBaTbCR lIOCTORHHOH. TI03TOMY COrJIaCHO ypaBHeHHIO BepHYJIJIH:
gZ + v 2 / 2 + p / p = KO,HCTaHTa(K) HJIH Z + v 2 / 2g + p / W = KOHCTaHTa (K),
r,ne W - Bec e,lJ;HHHIJ;bI 06beMa (y,neJIbHbIH Bec).
3. 13.4. TelfeHJfe lfepe3 oTSepCTJfe
C60KY H3 6aKa Bo,na BbITeKaeT B BH,lJ;e ropH30HTaJIbHOH C'rpYH lIO,n HalIopOM; CKO-
pOCTHOH HallOp HJIH ,nHHaMHqeCKHH HallOp - pa3JIHqHbIe Ha3BaHIDI o,nHoro H Toro
xe lIOIDITIDI. BO,l(a B pe3epByape HMeeT HaqaJIbHbIH lIOTeHIJ;HaJIbHbIH HalIOp, 3aBH-
CRII(HH TOJIbKO OT BbICOTbI h OTHOCHTeJIbHO ypOBIDI OTCqeTa HJIH cpe,nHero ypOBIDI
OTBepCTIDI. TI03TOMY:
V 2 / 2g = h .
v = ,J( 2gh).
TIOCKOJIbKY HeKOTopoe KOJIlNeCTBO 3HeprHH TepReTCR H3-3a TPeHIDI, CTPYR
)l{H)J:KOCTH, BbITeKalOlI(IDI H3 OTBepCTIDI, 6y,neT HMeTb HeMHoro MeHbIIIyro CKOpOCTb,
qeM TeOpeTHqeCKHH MaKCHMyM.
,l(eHcTBHTeJIbHIDI CKOpOCTb CTPYH/TeopeTlNeCKIDI CKOpOCTb CTPYH =
= K03<P<P H IJ;HeHT (lIoTepH) CKOpOCTH (C v )'
3.13. FuopoiJuHaMUICa
h
.....
---.. -....
"... ....--..........:-:.:-...- -.
.......... ---....-.
- - - - - - ..... .... - -- - -t- -
- - - - - - - - - - - - -4.-.
.. :..-....-.,.,. .- --.....-..-...- :---; -:r:- -:
:...-::........- I '
....'
,
I
, ,
: I
. ,
L.. CY>I(eHVle (AVIaMeTp J
CTPYVI MeHbwe) "IeM
AViaMeTp 0:raepCTVl5I)
Pac. 3.20. TeqeHHe qepe3 OTBepCTHe
,lJ;HaMeTP cTPYH :>KH,ZJ;KOCTH, BbITeKalOlI(eH H3 oTBepcTIDI C OCTPOH ,nHa<pparMoH,
Ha npaKTHKe 6y,neT MeHbIIIe, qeM ,neHcTBHTeJIbHbIH ,nHaMeTP oTBepcTIDI, KaK nOKa-
3aHO Ha pHC. 3.20. 3TO TaIOKe Ha3bIBaeTC.H cyxeHHeM.
IIno a,nb nonepetffioro CeqeHIDI cTPYH/nrrOII(a,n;b nonepetffioro CeqeHIDI OTBepCTIDI =
= K03<p<pmnIeHT cyxeHIDI (C c )'
AHaJIOIWIHO:
,n;eHCTBHTeJIbHbIH pacxo,n :>KH,ZJ;KOcTH/TeopeTHqeCKHH pacxo,n =
= K03<P<PH:QHeHT pacxo,na (Cd)'
II03TOMy:
Cd = C v x C c .
3.13.5. Bfl3KOCT&
ConpHKaCaBCb C nOBepXHOCTblO, :>KH,ZJ;KOCTb KaK 6bI npHJIHnaeT K HeH. Y,n06HO pac-
CMaTpHBaTb :>KH,ZJ;KOCTb COCTO.HlI(eH H3 CJIOeB TaK, qTO CJIOH, conpHKaCalOII(HHC.H C
rrOBepXHOCTblO TPy6bI, Haxo,nHTC.H B nOKoe, a OCTaJIbHbIe CJIOH CKOJIb3.HT O,nHH no
,n;pyroMy C nOCTeneHHO YBeJIWIHBalOlI(eHC.H CKOpOCTblO, BIIJIOTb ,no HaH60JIee y,na-
JleHHOrO OT nOBepXHOCTH TPy6bI CJIO.H, ,nBIDKyIIJ;erOC.H C MaKCHMaJIbHOH CKOpOCTblO.
TaKOH nOTOK Ha3bIBaeTC.H JIaMHHapHbIM, H TaKoe npH6JIIDKeHHe npHMeIDIeTC.H, Kor-
,n;a )J{Jf)(KOCTb no,nBepraeTC.H B03,neHCTBHIO nOnepeqHOH CHJIbI. HanpHMep, CMa3Ka B
rro,n;mHnHHKe, OT,neJI.HlOlI(aB BpalI(alOII(HHC.H BaJI OT BKJIa,n;bIIIIeH no,n;IIIHnHHKa. TIpo-
THBo,n;eHcTByrolI(aB CHJIa - 3TO B.H3KOCTb :>KH,ZJ;KOCTH, H3-3a KOTOpOH B03HHKaeT co-
rrpOTHBJIeHHe cpe,nbI, HanpHMep eCJIH ,nBHraTeJIb TPaHcnopTHoro cpe,nCTBa 3aBo,nHT-
CH B XOJIO,nHyID noro,ny, Kor,na B.H3KOCTb CMa3KH YBeJIHqHBaeTC.H. ECJIH CKOpOCTb
rrpeBbImaeT HeKOTopoe KpHTHqeCKOe 3HaqeHHe, JIaMHHapHbIH nOTOK pa3pyrnaeTC.H
H CTaHOBHTC.H TYP6YJIeHTHbIM. AHaJIOrIDI co CJIO.HMH Tor,na yxe 60JIbIIIe He npHMe-
HHMa.
F/la8a 3. HH:JICeHepHaR OUHaMUKa
B JIaMHHapHoM nOTOKe KaCaTeJIbHOe HanpIDKeHHe Mex.ny JII06bIMH ,nBYM.H co-
ce,nHHMH CJIO.HMH paBHo FIA, r,ne A - nJIOII(a,nb COlIpHKOCHOBeHIDI Mex.ny CJIO.HMH
H Av / Ay - rpa,nHeHT CKOpOCTH. CJIe,noBaTeJIbHO, nOCKOJIbKY KaCaTeJIbHOe Hanp.H-
xeHHe H3MeIDIeTC.H nponop:QHOHaJIbHO rpa,nHeHTY CKOpOCTH:
F / A =(Av / AY)l1,
r,ne 11 - KOHCTaHTa, Ha3bIBaeMaB K03<P<PH:QHeHToM B.H3KOCTH.
TaKHM 06pa30M:
11 =(F / A) / (A v / Ay),
r,ne e,nHHH:QbI H3MepeHH.H B.H3KOCTH paBHbI H/CM 2 (TIac) HJIH Kr/MC, TaK KaK 1 H =
= 1 KrM/C 2 .
3.13.6. tPopMyns nys3eRnfl
,[lJrn JIaMHHapHoro TeqeHIDI B rJIa,nKOCTeHHOH KpyrJIOH TPy6e 06'beM npoTeKalOlI(eH
:>KH,ZJ;KOCTH (V) B ceK}'H,nY onpe,neJI.HeTC.H <pOPMYJIOH TIya3eHmI:
v = (1tpr 4 ) / (811/),
r,ne p - pa3HOCTb ,naBJIeHHH Ha KOH:Qax TPy6bI; r - BHyrpeHHHH pa,nHYC TPy6bI (M);
1 - ,nJIHHa TPy6bI (M); 11 - B.H3KOCTb :>KH,ZJ;KOCTH, TeKYlI(eH no TPy6e.
3. 13. 7. 3SKOH CTOKCS
3aKOH CTOKca HCnOJIb3yeTC.H ,l(IDI onpe,neJIeHIDI CHJIbI F, B03HHKalOIUeH H3-3a B.H3-
KOCTH :>KH,ZJ;KOCTH; C TaKOH CHJIOH np.HMOJIHHeHHO ,nBIDKyIUHHC.H nOTOK B03,neHcTByeT
Ha rJIa,nKYIO c<pepy. <l>oPMYJIa npHMeIDIeTC.H, JIH60 Kor,na c<pepa ,nBIDKeTC.H CKB03b
:>KH,ZJ;KOCTb, JIH60 Kor,na :>KH,ZJ;KOCTb 06TeKaeT c<pepy.
CHJIa F = 61t11rv,
r,ne r - pa,nHYC C<pePbI (M).
rnABA 4
KPEnE>KHbIE
AETAnM
, X /
4. 1. BMHToBble Kpene)l(Hble M3AenMSI
4. 1. 1. Pa3Mep&1 Jf COOTHOWeHJffl Ha lfepTeJICaX
BOJITY H BHHTLI
1,5D Me)I(AY
napaJ1J1eJ1bHbIMVI
rpaH5IMVI
AnVlHa pe3b6bl
( 60J1T)
5D
HOMVlHaJ1bHa51 AJ1V1Ha
illUHJILKH
\
/
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nOCaAO
KOH
- -
-
D ,
- - .
\ J
"IHbIVi 1,5D - 2,5D CT5I
eu. - . -
-
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HOMVlHaJ1bHa51 AJ1V1Ha
D
)l(HOVI (rae"lHbIVl)
KOHeu.
r J1aAKa51 "IaCTb
CTeP)l(H51
D
F/la8a 4. Kpene:JICHble iJemOJlu
CTaH)l;apTHaJi raiiKa
1,50
napaJ1J1eJ1bHble
nJ10CKOCTVI
HH3KaJI raio (KoHTpraio)
.1 O,5D I.
.0
"
)
,
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J
"
-
-
0,150
0.80
npOCTaJI mal6a
--I I... O,15 D
.D
4.1.2. PS3nJflfH&le rOnOSKJf SJfHTOS
nOTaVlHa51
r0J10BKa
yrOJ1 90 0 ,
co WJ1V1u.eM
n0J1YKpyrJ1a51
r0J10BKa co
WJ1V1u.eM
nJ10CKa51 U.VI- KBaApaTHa51
J1V1HAPVl"leCKa51 r0J10BKa
r0J10BKa co
WJ1V1u.eM
KpyrJ1a51 U.VI-
J1 VlHAPVl"le-
CKa51 r0J10BKa
co WJ1V1u.eM
nOJ1ynoTa .
Ha51 nJ1OC-
KOBblnYKJ1a
r0J10BKa co
WJ1 VI u.eM
o
@
r 0J10BKa C
KpecTo06pa3-
HblM WJ1V1u.eM
r0J10BKa co
WJ1 VI u.eM
@
LJ.VlJ1 VI HAPVl"leCKa51
r0J10BKa C weCTVI-
rpaHHblM OTBepCTVleM
4.1. BUHm08ble "pene:JICHble U3iJeJlWl
4.1.3. PS3nJflfH&le KOHq&1 SJfHTOS
-a- -a=t 1- - -a- -a-
nJ10CKVlVI
LJ.VlJ1V1HA- KOHVI"IecKVlVI Cct>epVl-
pVl"leCKVlVI "IeCKVlVI
3acBepJ1eHHblVi
4. 1.4. BJfHT&1 C WeCTJfrpSHH&IM yrny6neHJfeM S ronOSKe
Yr J1y6J1eHVle
nOA KJ11O"I
AViaMeTp
CTep)l(H5I
AViaMeTp
r0J10BKVI
Pa3Mep weCTVI-
yrOJ1 bHoro
OTBepCTVl5I
J
4. 1.5. npSKTJflfeCKOe npJfMeHeHJfe SJfHTOS&IX
KpeneJICH&IX Jf3AenJfH
60nT AOJ1)1(eH 6blTb BCTaBJ1eH
TaK, "IT06b1 OH He Mor BblnaCTb,
ecn raViKa He3aT5IHYTa
nOBepXHOCTb CTblKa AOJ1)1(Ha
npVlxOAVlTbC5I Ha rJ1aAKYIO "IaCTb
CTep)l(H5I 60J1Ta VI HViKorAa He
panOJ1araTbC5I nonepeK pe3b6bl
DOJ1T AOJ1)1(eH npoxo-
AVlTb "Iepe3 OTBepCTVle
C rapaHTVlpOBaHHblM
3a30pOM
l.IT06b1 06eCne"lVlTb nJ10CKYIO
nOCSAKY B Hecyw.VlX CTaJ1bHbIX
KOHCTPYKU.Vl5IX, CJ1eAyeT VlC-
n0J1b30BaTb KOCble waVl6bl
LlT06bl rapaHTVlpOBaTb nOJ1HOe
3au.enJ1eHVle 3aT5IHYTOVI raVlKVI,
3a ee npeAeJ1aMVI AOJ1)1(Hbl 6blTb
BVlAHbl no KpaVlHeVi pe ABa
BVlTKa pe3b6bl. )
4. 1.6. TpsneqeJfASn&HSfI pe3&6s
0, 3707 P I
l ·
War pe3b6b1 P "I
f
0,5P + 0,01
I. .1
0.3707 P
F/la8a 4. Kpene:JICHble iJemOJlu
4. 1. 7. npflMoyrOn&HSfI pe3&6s
I'" P/2 "I" War pe3b6bl P ..I
P/2
4. 1.8. YnopHSfI pe3&6s
O 7 __
War pe3b6bl P
O,245P
P/2
4. 1.9. OCTpoyrOn&HSfI pe3&6s
YrOJ1 npoct>VlJ151 pe3b6bl
O,1395P
BblCOTa npoct>VlJ151 pe3b6bl
OCb pe3b6bl
J5
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Q) a.
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::I:
4.1. BUHm06ble "peneJICHble U30e.tlUR I 37
4. 1. 10. OCHOSHSfI AIOHMOSSfI pe3&6s (55°):
qJfnJfHAPJflfeCKSfI pe3&6s
3TO OCHOBHMI <popMa pe3b6bI ,l(IDI BSW (6pHTaHCKMI ,nIOHMOBMI pe3b6a), BSF
(6pHTaHCKMI MeJIKMI ,nIOHMOBMI pe3b6a) H BSP (6pHTaHCKMI TPy6HMI KOHWIeCKM
pe3b6a) BHHTOBbIX pe3b6.
Hie
H h
War P
H = 0,960 491 P
h = 0,640 327 P
r = 0,137 329P
H'6
4. 1. 11. tPOPM&1 ISO MeTpJflfeCKOH Jf ISO 60°
YHJfcpJfqJfPOSSHHOH pe3&6&1
War pe3b6bl P
PaAVlYC BnaAVlHbl npoct>VlJ151
pe3b6bl raVlKVI
a..
,...
o
J5
\0
.0
(t)
Q)
a.
a.
j5
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co )s .0
S (t)
S :I: Q)
C[ :I: a.
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:I: S Q)
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m co
a. s
co a. C[
::I: U
a..
o
a..
<0
<0
co
o
0,217P
II BHYTpeHH5I5I pe3b6a (raVlKa)
HapY>KHa51 pe3b6a (60J1T)
4. 1. 12. BseAeHJfe S SJfHTOS&le KpeneJICH&le Jf3AenJffl
XOT.H B CJIe.nyro:m;HX Ta6JIH:Qax BHHTOBbIX pe3b6 pa3MepbI npHBe,neHbI B ,nIOHMax
(pa3,n;eJIbI 4.1.38-4.1.43), OHH He nOTepIDIH CBoero 3HaqeHIDI B KaqeCTBe ,naHHbIX
wm TeXHHqeCKOrO 06CJIy)l{HBaHIDI H aHaJIOrWIHbIX npHJIOXeHHH. BHHTOBbIe Kpe-
ne)I{HbIe H3,neJIIDI COrJIaCHO 3THM cne:QH<pHKa:QIDIM ,no CHX nop npOH3BOMTC.H:
4.1.39. opHTaHCKMI ,nIOHMOBMI pe3b6a (BSW) 60JITbI H raHKH
4.1.40. opHTaHCKMI MeJIKa.H ,nIOHMOBMI pe3b6a (BSF) 60JITbI H raHKH
F/la8a 4. Kpene:JICHble oemOJlu
4.1.41 ISO YHH<pH:QHpoBaHHMI TOqHOCTb BHyrpeHHeH pe3b6bI: PM C KpynHOH
pe3b60H (UNC) (aMepHKaHCKMI YHH<pH:QHpOBaHHMI KpynHMI pe3b6a)
4.1.42 ISO YHH<pH:QHpoBaHHaH TOqHOCTb HapyxHoH pe3b6bI: PM C KpynHOH pe-
3b60H (UNC) (aMepHKaHCKa.H YHH<PH:QHpOBaHHM KpynHMI pe3b6a)
4.1.43 ISO YHH<pH:QHpoBaHHMI TOqHOCTb BHyrpeHHeH pe3b6bI: PM C MeJIKOH
pe3b60H (UNF) (aMepHKaHCKa.H YHH<pH:QHpOBaHHMI TOHKa.H pe3b6a)
4.1.44 ISO YHH<pH:QHpoBaHHMI TOqHOCTb HapyxHoH pe3b6bI: PM C MeJIKOH pe-
3b60H (UNF) (aMepHKaHCKMI YHH<PH:QHpOBaHHMI TOHKMI pe3b6a)
XOT.H cJIe,nyrolI(MI Ta6JIH:Qa BHHTOBbIX pe3b6 (pa3,neJI 4.1.45) H YCTapeJIa, OHa CO-
xpaHeHa no TeM xe npHqHHaM, qTO lIpHBe,neHbI BbIIIIe:
4.1.45 DpHTaHcKM aCCO:QHa:QIDI TeXHHqeCKHX CTaH,napTOB (BA): 6pHTaHcKHe
CTaH,napTHbIe BHyTpeHHHe H HapyxHbIe BHHTOBbIe pe3b6bI
Ta6JIH:QbI, OCHOBaHHbIe Ha CTaH,napTax PMa BS EN 24 000, 3aMeHeHbI TeKY-
HM PMOM BS EN ISO 4000 ,lW.H BHHTOBbIX Kpene)I{HbIX H3,neJIHH C MeTPHqeCKHMH
pa3MepaMH. TIpHBe,neHbI CJIe,nyro He HOBbIe CTaH,napTbI (pa3,neJIbI 4.1.13-4.1.21):
4.1.13 BS EN ISO 4014: 2001 DOJITbI C IIIecTHrpaHHoH rOJIOBKOH - KJIaCCbI H3-
,neJIHR A H B
4.1.14 BS EN ISO 4016: 2001 DOJITbI C mecTHrpaHHoH rOJIOBKOH - KJIacc H3-
,neJIHR C
4.1.15 BS EN ISO 4017: 2001 BHHTbI C mecTHrpaHHoH rOJIOBKOH - KJIaCCbI H3-
,neJIHR A H B
4.1.16 BS EN ISO 4018: 2001 BHHTbI C IIIecTHrpaHHoH rOJIOBKOH - KJIacc H3-
,neJIIDI C
4.1.17 BS EN ISO 4032: 2001 IllecTHrpaHHbIe raHKH HCnOJIHeHHH 1 - KJIaCCbI
H3,neJIIDIAH B
4.1.18 BS EN ISO 4033: 2001 IllecTHrpaHHbIe raHKH HCnOJIHeHIDI 2 - KJIaCCbI
H3,neJIIDIAH B
4.1.19 BS EN ISO 4034: 2001 IllecTHrpaHHbIe raHKH HCrrOJIHeHIDI 1 - KJIacc
H3,neJIIDI C
4.1.20 BS EN ISO 4035: 2001 IllecTHrpaHHbIe HH3KHe raHKH (c <pJIaH:QeM) -
KJIaCCbI H3,neJIHH A H B
4.1.21 BS EN ISO 4036: 2001 IllecTHrpaHHbIe HH3KHe raHKH (6e3 <pJIaH:Qa) -
KJIacc H3,neJIIDI B
Bce BbImeyrrOM.HHyrbIe CTaH,napTbI OTHOC.HTCH K BHHTOBbIM Kperre)I{HbIM H3,ne-
JIHRM C MeTpHQeCKHMH pa3MepaMH H pe3b6e C KpynHbIM marOM. CTaH,napTbI ,[(JHI
MeJIKOrO IIlara pe3b6bI 6y,nyr npHBe,neHbI ,naJIee.
IIpUMeltaHue.
a) BS EN ISO 4015 KacaeTCR 60JITOB C IIIecTHrpaHHoH rOJIOBKOH, CTep)I{HH KO-
TOpbIX YMeHbIIleHbI ,no cpe,nHero ,nHaMeTpa HX pe3b6bI. OHH rrpe,nHa3HaQeHbI
LUIR CneUHaJIbHbIX rrpHJIOXeHHH H He BKJIlOqeHbI B 3TOT ClIpaBOqHHK.
6) CTaH,napTbI MeXaHHqeCKHX CBOHCTB, KOTopbIe co,nepXaJIHCb B rrpe,neJIax BS
4190 H BS 3692, MO)I{HO Tenepb HaH,nTH B BS EN 20898 qacTb 1 (60JITbI) H
qacTb 2 (raHKH).
Krracc H3,[(eJIIDI 03HaqaeT CJIe,nyro ee:
. TIpoaHaJIH3HpyHTe Ta6JIHUY H3 pa3,neJIa 4.1.13.
. ,lJ:HaMeTPY CTep)I{H.H (d s ) COOTBeTCTBYIOT MaKCHMaJIbHbIH ,nHaMeTp, paBHbIH
80 MM, HOMHHaJIbHbIH ,nHaMeTp, a TaIOKe MHHHMaJIbHbIH ,[(HaMeTp.
4.1. BUHmo6ble "peneJICHble U30e.tlUR 139
. Y MHHHMaJIbHOrO ,nHaMel'pa MoxeT 6bITb ,nOrryCK H3,neJIIDI KJIacca A HJIH ,no-
rryCK H3,neJIIDI KJIacca B. ,nOrryCK KJIacca A 60JIee xeCTKHH (60JIee TOqHbIH),
qeM ,nOrryCK KJIacca B.
. ,nOrryCKH H3,neJIHH KJIacca A OTHOC.HTC.H K KpelIe)I{HbIM H3,neJIIDIM C pa3Mepa-
MH B ,nHana30He OT Ml.6 ,no M24 BKJIlOqHTeJIbHO.
. ,nOrryCKH H3,neJIHH KJIacca B OTHOC.HTC.H K Iq>elIe)I{HbIM H3,neJIIDIM C pa3Mepa-
MH B ,nHana30He OT M16 ,no M64.
. II03ToMY pa3MepbI OT M16 ,no M24 BKJIlOqHTeJIbHO MOryr HMeTb ,nOrryCKH H3-
,neJIIDI KJIacca A HJIH B H CJIe.nyeT onpe,neJIHTb l'pe6yeMbIH ,nOrryCK.
. 06paTHTe BHHMaHHe Ha TO, qTO KJIaCC H3,neJIIDI 3aBHCHT He TOJIbKO OT ,nHa-
Mel'pa, HO TaIOKe H OT ,nJIHHbI.
npHMep 1
BOJIT C IIIecTHrpaHHoH rOJIOBKOH M5 npHaaMe)I{HT KJIaccy TOqHOCTH A H ,nHa-
MeTp ero CTep)I{H.H 6y,neT Haxo,nHTbC.H B ,nHana30He Mex.ny 5,00 H 4,82 MM BKJIlOqH-
TeJlbHO.
npHMep 2
BOJIT C IIIecTHrpaHHoH rOJIOBKOH M36 npHH3A1Ie)I{HT KJIaccy TOqHOCTH B H ,nHa-
MeTp ero CTep)I{H.H 6y,neT HaxO,nHTbC.H B ,nHana30He Mex.ny 36,00 H 35,38 MM BKJIIO-
QHTeJIbHO.
npHMep 3
Y 60JITa C IIIecTHrpaHHoH rOJIOBKOH M16 ,nHaMel'p CTep)I{IDI 6y,neT JIeXaTb B
,ZJ;Hana30He Mex.ny 16 H 15,73 MM BKJIlOqHTeJIbHO, eCJIH 3TO H3,neJIHe KJIacca TOqHO-
CTH A. ECJIH 3TO H3,neJIHe KJIacca TOqHOCTH B, TO ,nHaMel'p CTep)I{IDI 6y,neT JIeXaTb B
,ZJ;Hana30He Mex.ny 16 H 15,57 MM BKJIlOqHTeJIbHO.
llpUMettaHue.
BbIIIIeH3JIOXeHHaB CHCTeMa ,nOrryCKOB OTHOCHTCR H KO BceM ,npyrHM pa3MepaM
:Kpene)I{Hb!X H3,neJIHH B 3THX Ta6JIH:Qax.
. TIpoaHaJIH3HpyHTe Ta6JIHU}' B pa3,neJIe 4.1.14. Bce pa3MepbI B 3TOH Ta6JIHQe
OTHOC.HTC.H K BHHTOBbIM Kpene)I{HbIM H3,neJIIDIM KJIacca TOqHOCTH C. CpaBHe-
HHe 3TOH Ta6JIH:QbI C lIpe,nbmyIl(HMH nOKa3bIBaeT, qTO Kpene)I{HbIe H3,neJIIDI,
H3rOTOBJIeHHbIe COrJIaCHO Te \ecKHM YCJIOBIDIM KJIacca TOqHOCTH C, HMe-
lOT HaMHoro 60JIee rpy6bIe ,nOrryCKH, qeM H3,neJIIDI, H3rOTOBJIeHHbIe B COOT-
BeTCTBHH C l'pe60BaHIDIMH KJIaCCOB TOqHOCTH A H B.
. Y 60JITa M12 KJIacca TOqHOCTH A ,nHaMel'p CTep)I{H.H (ds) MoxeT HaxO,nHTbC.H B
,nHana30He Mex.ny 12 H 11,73 MM (,nOrryCK COCTaBJI.HeT 0,27 MM); Tor,na KaK y
60JITa M12, H3rOTOBJIeHHOrO B COOTBeTCTBHH C l'pe60BaHIDIMH KJIacca TOqHO-
CTH C, ,nHaMel'p CTep)I{H.H (ds) MoxeT Haxo,nHTbC.H B ,nHana30He Mex.ny 12,77 H
11,3 MM (,nOrryCK - 1,4 MM). .n:OrryCKH H3,neJIHH KJIacca B He npHMeIDIIOTC.H K
3TOMY pa3Mepy 60JITa.
CTapa.H TepMHHOJIOrIDI «qepHbIe 60JITbI» (rop.HqaB IIITaMnoBKa) H «qHCTbIe/HOp-
MaJIbHOH TOqHOCTH 60JITbI» (XOJIO,nHaB BbIca,nKa HJIH MamHHHaB 06pa60TKa) K 60JI-
TaM H raHKaM 60JIbIIIe He lIpHMeIDIeTC.H. O,nHaKO 60JITbI H raHKH, H3rOTOBJIeHHbIe
no cne:QHaJIbHOMY 3aKa3Y MeTo,nOM rOp.HqeH IIITaMnOBKH (qepHbIe, HeOT,neJIaHHbIe),
MOryT OTHOCHTbC.H TOJIbKO K KJIaccy TOqHOCTH C.
Bce npe,nbmyIl(He npHMeqaHIDI OTHOC.HTC.H K 60JITaMH H BHHTaM C IIIeCTHrpaH-
BOH rOJIOBKOH H K IIIeCTHrpaHHbIM raHKaM C KpynHbIM IIIarOM pe3b6bI. Y 60JITOB
F/la8a 4. Kpene:JICHble oemallu
qaCTb CTep)I{IDI Mex.ny pe3b60H H rOJIOBKOH OCTaeTC.H rJIa,nKoH, Tor,na KaK Ha BHHTe
pe3b6a Hape3aeTC.H <paKTWIeCKH no BceH ,nJIHHe BIIJIOTb ,no rOJIOBKH. CJIe,nyroI.U;He
Ta6JIH:QbI H CTaH,napTbI (pa3,neJIbI 4.1.22-4.1.26) OTHOC.HTC.H K COOTBeTCTByrolI(HM
p.SI,ll;aM BHHTOBbIX Kpene)I{HbIX H3,neJIHH C MeJIKHM marOM pe3b6bI. 3TH p.SI,ll;bI BHHTO-
BbIX Kpene)I{HbIX H3,neJIHH C MeJIKHM IIIarOM pe3b6bI H3rOTaBJIHBaIOTC.H TOJIbKO B
COOTBeTCTBHH C l'pe60BaHIDIMH KJIaCCOB TOqHOCTH A H B.
4.1.22 BS EN ISO 8765: 2001 DOJITbI C mecTHrpaHHoH rOJIOBKOH C Mel'pHqe-
CKOH pe3b60Hc MeJIKHM IIIarOM - KJIaCCbI H3,neJIIDI A H B
4 1.23 BS EN ISO 8676: 2001 BHHTbI C IIIecTHrpaHHoH rOJIOBKOH C Mel'pHqe-
CKOH pe3b60Hc MeJIKHM IIIarOM - KJIaCCbI H3,neJIIDI A H B
4.1.24 BS EN ISO 8673: 2001 IllecTHrpaHHbIe raHKH HCnOJIHeHIDI 1 C Mel'pHqe-
CKOH pe3b60H C MeJIKHM IIIarOM - KJIaCCbI H3,neJIH.H A H B
4.1.25 BS EN ISO 8674: 2001 IllecTHrpaHHbIe raHKH HCnOJIHeHIDI 2 C Mel'pHqe-
CKOH pe3b60H C MeJIKHM IIIarOM - KJIaCCbI H3,neJIIDI A H B
4.1.26 BS EN ISO 8675: 2001 IllecTHrpaHHbIe HH3KHe raHKH C MeTPHqeCKOH
pe3b60H C MeJIKHM marOM - KJIaCCbI H3,neJIIDI A H B
4. 1. 13. BS EN ISO 4014: 2001. 60nT&1 C WeCTJfrpSHHOH
ronOBKOH - KnSCC&1 Jf3AenJffl A Jf B
Pa3MepLI
llpUMettaHue. CHMBOJIbI H 0603HaqeHIDI pa3MepOB ,naHbI B COOTBeTCTBHH C ISO 225.
KOHeu. AOJ1)1(eH 6blTb C
ct>acKoVi VlJ1V1 AJ151 pe3b6
M4 MO)l(eT 6blTb 6e3
X ct>acKVI (cpe3HoVi 60J1T)
(CM. ISO 4753)
Pa3Mepbl B MVlJ1J1V1MeTpax
Is U(1) S
l (b)
k
X
k' I, MaKCVlMa11bHOe YTOJ1w.eHVle
nOA rOJ1oBKoVi BVlHTa
"tf-d
(1) HelIOJIHa.H pe3b6a (c6er pe3b6bI) u 2P
TIpHMep 0603HaqeHIDI 60JITa C IIIecTHrpaHHoH rOJIOBKOH C pe3b60H M12, HO.
MHHaJIbHOH ,nJIHHOH 1 = 80 MM H KJIaCCOM npOqHOCTH 8.8:
60JIT C mecTllrpaHHOH rOJIOBKOH ISO 4014 - M12 x 80 - 8,8.
4.1. BUHm06ble "peneJICHble U30eJl!J.R 14.
\0 \0 \0 V)
00 V) V) 00
("r') ("r') ....-4 N... ....-4 ("r') 0 ... ... \0... V) N'"
\0 , , V) ... ....-4 I ... , , , 0\ 0 , N ,
::a 0 t"-- \0 ("r') 00 ....-4 t"-- 0\
....-4 0 ("r') 0\
\0 00 ....-4 N
'oC \0 \0 \0 V) V"l V) 00
V) V"l t"-- ("r') ("r') \0 N \0... V) N V) ... ....-4 V) N
, , ("r') ... ....-4 ... , , ... , , , N ,
::a V) 0 \0 V) V) 00 ("r') ....-4 ("r') V) 00
....-4 ("r') ("r') 00
V) t"-- 0\ N
00 \0 00 V) \0 00 N ....-4 ....-4
00 ....-4 ("r') N'" 00 ("r') ... N 0 0 V) t"-- V) ...
V) , 0 N ... ....-4 ... , , , , ... , ... I ("r')
::a 0 t"-- 0\ ....-4 ("r') 0\ 0 0 ....-4 t"--
....-4 ....-4 V) 00 t"--
\0 N ("r') N
N \0... 00 V) ("r') 00 N ....-4 ....-4
V) \0 0\ N ("r') ... \0 V) ... V) ...
... , 0 ....-4 ... , , 00 I , ... , , ("r')
0\ 0 V) ....-4 0\ ....-4 N V) \0 t"-- ....-4 \0
....-4 t"-- \0
V) N N ....-4
'oC ... 00 ....-4 0\ V) 00 N \0
ff') t"-- N 00 \0 ("r') ....-4 t"--... N'" 0 0\ V)
, ... ... ... , ... , \0 , , N'" N'" , ....-4 I
::a 00 0\ 0 0 0\ ("r') V) ....-4 0 V) V) ("r')
("r') ("r') V) \0 N N N ....-4 V)
= ... 00 V) V) t"--... 00 N 00
ff') V"l \0 N V) N 00 0 ... t"-- 00... N... ....-4 N'" \0 V)
... 0 , N'" , \0 00 , , ... , ....-4 ,
::a ("r') \0 t"-- 00 0 ("r') ("r') 0\ 0 00 0\
("r') N V) ....-4 ....-4 ....-4 ....-4
t"-- 00 ....-4 V) 00 V) J V) V) V) V) V) \0 00
... 00 ....-4
0 ("r') N 00 \0... ... N V)- V) t"-- N \0 ("r') ("r') N... 00... \0 ("r') V)
("r') 0'" ... ... ... ... ... ...
::a V) \0 t"-- 0 \0 N ("r') ("r') ("r') ("r') 0\ 0\ ....-4 ... ... V) 0 0 0 ("r') V) ("r')
N N N ("r') ("r') ("r') ("r') V) ....-4 ....-4 ....-4 ....-4 ("r')
....-4 ....-4
t"-- 00 0\ ("r') V) V) V) V) V) t"-- \0
= t"-- V) 00 ....-4 ....-4
N V) \0 N N 00 N'" 0 ....-4 ... V) N N t"-- ....-4 00 \0... V) 00 0 \0... ....-4
N , ... ... t"-- ... N'" N 0 a\
::a V) 0 0 N 0\ 0\ 00 ("r') N N N'" 00 ... ("r') 0\
N ....-4 ....-4 N N ("r') ("r') ....-4 ....-4 ....-4 00 N N
....-4 ....-4
'oC t"-- ("r') t"-- 0\ V) t"-- N 00 ....-4 0\ t"-- t"-- \0
.,.,.,.. 00 N 00 ... \0 t"-- V) N t"--... ....-4 0 00 ....-4 t"--... N 00 00 \0 ....-4
N , ... ... N ("r') ... ... ... 0 ...
::a ("r') 0 0 t"-- ....-4 V) V) N \0 \0 ....-4 ... 0 0 ... \0 N ("r') ("r')
....-4 ....-4 ....-4 N N N 0\ ....-4 0\ ....-4 \0 N N
N V) V) t"--... ("r') ("r') ("r') N 00 N ("r')
.,.,.,.. t"--... 0 ....-4 \0... N t"--... \0... 0... ("r') V) ("r') \0... ....-4 \0... 00 t"--
I , , , , ... , , , ,
::a ("r') 0 0 ("r') ....-4 ....-4 \0 0 t"-- ... ... 0 ....-4 t"--
....-4 ....-4 ....-4 ....-4 N t"-- t"-- V) ....-4
= V) N 00 ("r') t"-- N 00 V) ("r')
.,.,.,.. V"l \0 ....-4 \0... ... 0 t"--... \0 t"--... N N V) ("r') \0 t"--...
, , ... I , \0'" , , , 0 ,
::a ....-4 N ... 0 ....-4 ....-4 t"-- ... ... ... ....-4 V)
0 ....-4 0\ ....-4 ....-4 \0 \0 ....-4
V) V) 00 ("r') 00 V) V) ....-4 ("r')
00 \0 N \0... ("r') ("r') t"--
::a N N , , ....-4 0'" ... 00 t"--... , I , N ... ....-4 ... , , \0... , ("r') N I
... N ... 0\ ....-4 V) ... 0 ....-4
....-4 0 t"-- ....-4 ....-4 V) V) ("r') ....-4
'oC V) N 00 V) V) V) V) 00
V) 00 0 ... t"-- 0
::a ....-4 00 I I ....-4 ... ... \0 00 , 00 , , 00 ....-4 I , N , N... t"--... ,
....-4 ... 0 \0 ... ... ....-4 ....-4 ... ....-4
0 V) 00 ....-4 ("r') 0 0\
V) 00... V) V) t"--... N 00 0\ N V) V) V) V) N 00
::a \0 , , ....-4 0 V) 00 , 00 , t"--... I ... ... ("r') \0... , , ("r') , ... 00 t"--... ,
0 ....-4 ... V) ... ... ....-4 ("r') ... ... 0
0 \0 00 ("r') ("r') N t"--
V) N 00 \0 V) V) t"-- 00
t"--... ... t"-- N 00 t"-- N N
::a , , ....-4 ... 00 , 00 I \0... , ... ... \0 0\ , , 00 , 0'" t"-- t"--... I
0 ....-4 ... 0 ... ... ....-4 N ...
0 ("r') V) t"-- N'" N'" ....-4 \0
V) \0 \ ....-4 V) V) ....-4 N
i V) ... \0... t"-- N ....-4 V)
... N , I ....-4 ("r') 00 L , 0 ....-4 N 00 ....-4 , I ("r') , ... ... ("r') I
0 ....-4 ... 0 ("r') N'" \0'" 0 V) ...
0 ... N ....-4 V)
....-4
V) V) V) \0 t"-- V) V) V) N
., ....-4 ....-4 V) t"--... t"-- N ....-4 ....-4
N ... , , ... N ... N'" ("r') 0 , ... , ....-4 V"l 00 , I ... I ... V) 00 ,
::a 0 0 0'" ("r') N ... V) ....-4 ... ....-4 0 ...
....-4 ....-4
V) \0 t"-- N V) V) 0\ N
N 0 ....-4 \0 00 t"-- N ....-4
::a I , 0'" N N'" N 00 I q I I ... V) I I 00 I ... 00 ,
0 ....-4 ... ... 0 ....-4 ... 0 ...
0 ....-4 ("r') ... 0 ("r')
....-4 ....-4
'oC V) V) \0 t"-- ....-4 V) V) 00 N
...;' ("r') ....-4 \0... N ... \0 ....-4 t"-- N \0 ....-4 0...
0\ , I ... N , , , 0 ... N... , , I 0 I
::a 0 0 0 ....-4 ....-4 N'" ("r') ....-4 0 ("r') ("r')
0 ....-4
.0 1 (,.) = = = :i = = = =
I I == == == 0 == == == ==
=
:i II I II
1 <1j:Q <1j:Q C"') « <1j:Q <1j:Q
:i == :i
0 <1j:Q 0
= (,.) (,.) (,.) (,.) (,.) = (,.)
t (,.) 5 (,.) 5 (,.) 5 (,.) 5 (,.) 5 (,.) 5
I g
== == == == == ==
c...;
"'r:;jr:::J "'r:;jl",,) ...c:u
..c::) c:J <u "" ,
F/la8a 4. Kpene:JICHble oemOJlu
P83Mep pe3h6w, d Ml,6 M2 M2,S M3 M4 MS M6 M8
KJIacc H3 eJIIUI I H I f, g
A B s g
ROM. MHH. MHC. MHH. MaKe. . . . . . . . .
:I :I :I :I :I :I :I :I
_fIJ ...... tIC _fIJ ...... tIC _fIJ ...... tIC _fIJ ...... tIC _fIJ ...... tIC _fIJ ...... tIC _fIJ ...... tIC _fIJ .....'C
12 11,65 12,35 - - 1,2 3 ,l:lrnI pa3MepOB BbIme cTYIIelf1la-
16 .15,65 16,35 - - ,.7. .7. 4 6 2,75 5 TOB JIHHHH, 0603HaQeHHOB KaK
20 '19,58 20,42 - - :8 10 6,75 9 5,5 8 _, peKoMeaayeTCH ISO 4017
1 . . . . . .
25 24,58 25,42 - - : \ t ,.7.5. .\4 10,5 13 7,5 11 5 9
30 29,58 30,42 - - : \ ,.5. .\& 12,5 16 10 14 7 12
35 34.5 35.5 - - : 17.5 21 15 19 12 17
.40 39,5 40,5 1 . 20 24 22 18
- - 122 5 17 11,75
. .'. .
45 44,5 45,5 - - : 25 29 22 27 16,75 23
50 49.5 50.5 - - :30 34 27 32 21.75 28
55 54,4 55,6 - - : 32 37 26,75 33
60 59,4 60,6 - - : 37 42 31,75 38
65 64.4 65.6 - - : 36.75 43
70 69,4 70,0 - - :41,75 48
80 79,4 80,6 - - :51 75 .
..'...
90 89.3 90.7 - -
100 99,3 100,7 - -
110 109,3 110,7 108,25 111,75
120 119.3 120.7 118.25 121..75
130 129,2 130,8 128 132
140 139,2 140,8 138 142
150 149.2 150.8 148 152
160 - - 158 162
180 - - 178 182
200 - - 197.7 202.3
220 - - 217,7 222,3
240 - - 237,7 242,3
260 - - 257.4 262.6
280 - - 277,4 292,6
300 - - 297,4 302,6
320 - - 317.15 322.85
340 - - 337,15 342,85
360 - - 357,15 362,85
380 - - 377.15 382.85
400 - - 397,15 402,85
420 - - 416,85 423,15
440 - - 436.85 443.15
460 - - 456,85 463,15
480 - - 476,85 483,15
500 - - 496.85 503.15
ap - mar pe3b6bI; b ,lI;IDI ,nJIHH I HOM . 125 MM;
C ,lI;IDI ,nJIHH 125 MM < I HOM . 200 MM; d ,nJIH ,nJIHH IHoM. > 200 MM;
e k'min = 0,7 k min ; f I gmaks = I HOM - b; I smin = I gmaks - 5P;
gig - MHHHMaJIbHaH ,nJIHHa ,nJIHHa rJI3):(KoH QaCTH CTep)I{IUI.
4.1. BUHmo6ble "peneJICHble U30eJlUR 143:::
OK,OH"taHue
MIO M12 M16 M20 M24 M30 M36 M42 M48 M56 M64
I H I I, g
s g
i . . . . . . . . . .
.
:I :I :I :I :I :I :I :I :I :I :I
..... GI'J ..... ..... GI'J ..... b. ..... GI'J ..... tJ( .....(0) ..... .....(0) ..... .....(0) ..... .....GI'J ..... .....(0) ..... tJ( .....(0) ..... tJ(I .....(0) ..... tJ( .....(0)
,n;m pa3MepOB BbIllie
warOBOH JIHHHH, 060-
3HaQeHHbIX TaK _, pe-
KOMeHn VeTCH ISO 4017
J
(
)
11,5 19
16 5 24 11 25 20
21,5 29 16,25 25
26,5 34 21,25 30
31,5 39 26 25 35 17 27
36,5 44 31,25 40 22 32
46,5 54 41,25 50 32 42 21,5 34
56.5 64 51.25 60 42 52 31.5 44 21 36
6.6., 7 61,25 70 52 62 41,5 54 31 46
:71,25 80 62 72 51,5 64 41 56 : 26,5 44
1 61.5 I
181.25 90 72 82 74 51 66 1 36.5 54
: 76 86 65,5 78 55 70 : 40,5 58
:86 1
96 75,5 88 65 80 : 50,5 68 36 56
1
: 96 106 98 I 78 46 66
85.5 75 90 I 60.5
106 116 95,5 108 85 100 70,5 88 56 76 41,5 64
115,5 128 105 120 90,5 108 76 96 61,5 84 47 72
135.5 148 125 140 110.5 128 96 116 81.5 104 67 92
132 147 117,5 135 103 123 88,5 III 74 99 55,5 83
152 , \137,5 155 123 143 108,5 131 94 119 75,5 103
\ 157.5 175 143 163 128.5 151 114 139 95.5 123 77 107
177,5 195 163 183 148,5 171 134 159 115,5 143 97 127
197,5 215 183 203 168,5 191 154 179 135,5 163 117 147
203 223 188.5 211 174 199 155.5 183 137 167
223 243 208,5 231 194 219 175,5 203 157 187
243 263 228,5 251 214 239 195 5 223 177 207
248.5 271 234 259 215.5 243 197 227
268,5 291 254 279 235,5 263 217 247
288,5 311 274 299 255,5 283 237 267
308.5 331 294 319 275.5 303 257 287
314 339 295,5 323 277 307
334 359 315,5 343 297 327
335.5 363 317 347
llpUMe"taHue. PacnpocTpaHeHHbIe ,nJIHHbI 0603HaQeHbI Is H I g:
113,n;eJIHH KJlacca A BbIllIe CTYIIeHqaTOH JIHHHH, 0603HaqeHHOH KaK ----
113,n;eJIHH KJlacca B HHxe 3TOH CTYIIewraToH JIHHHH.
F/la8a 4. Kpene:JICHble oemOJlu
::a
C"')
= \0 ....c ....c V') V') V') 00
\0 V) V') ("f"') t- o N .. N N 00 .. .. N 0 ..
::a .. I I 0" ....c \0 \0 I .. I I 0\ ....c ("f"') I I t- oo I \0" N 0\ I t-
V') ....c 0\ ("f"')
V') 00 0\ ("f"') ("f"') N 00
N \0 0\ \0 \0 N V') V') V') V') .-4
It') ("f"') \0" N N .. N 0 ("f"') .. .. t- \0.. 0 ..
V') I ....c N 0" ....c I .. I I .. I I N ("f"') I .. I 00
....c .-4 V') .-4 00 ....c ("f"') N ....c 00
V') V') t- oo ("f"') ("f"') N t-
It') \0.. 00 t- V') 00 .-4 ....c
V') N V') ("t") \0 ("f"') .. 0\ 00 V') .. ("f"') N 0 ..
::a .. I 0 .-4 .. ....c 00 I .. I I .. 00 N I I .. I .. .. t- I 00
....c ....c 0 \0 \0 t- oo 0\ ....c \0
t- N N ....c
CJ\ .. 00 \0 .. 00 N ....c 00
ff') 0 ("f"') ("f"') 0\ ("f"') 00 V') V') .. N 0 ..
I 0 .-4 N I .. I .. I \0 I I .. I .. ....c I 00
::a 0\ 0 ("f"') 00 V') \0 N V') t- \0
....c V')
("f"') V') \0 N N ....c
ff') 00 V') t- oo N ....c
ff') V') 00 ....c N 00 \0" ("f"') ("f"') V') ("f"') \0 .-4 V') .. 0 0\
::a .. I t- 0\ 0" 0" ("f"') I .. I \0" I .. N I I 0" I .. ....c V') I
("f"') ("f"') N \0 ....c
("f"') V') N N ....c
I' 00 N V') V') \0
N 0 \0 0\ N 00 0" t- 00 .. \0 t- \0 ("f"') \0 ....c 0
::a ("f"') \0 \0 t- o" 0" N I \0" I ("f"') I V') ....c I I \0" .. I .. ....c I
("f"') t- ....c
N ....c ....c ....c
t- oo ....c V') N 0\ V') V') V') V') 00
N V') N 00 \0 .. t- ("f"') t- N 00 ....c \0.. ("f"') V') \0 00 ("f"')
N .. 0 \0 0\ .. .. .. N .. .. .. .. .. t- N .. \0 V') .. .. ("f"')
::a N V') V') \0 0 0 N ....c ....c ....c ....c t- t- ....c .. .. ("f"') .. .. 0 ("f"') ("f"') ("f"')
N N N ("f"') ("f"') ("f"') ("f"') ("f"') ....c ....c 0\ 0\ ("f"')
....c ....c
("f"') t- V') \0 V') V') V') V') t- \0
00 N V') 00 t- ....c
.,.,.,.. V') N 00 N 00 .. 00 t- V') ("f"') 00 ....c V') .. N ....c .-4 00 0\ 00 \0.. t- \0 ....c
::a .. I 0" 0" 0 ....-4 .. .. .. .. 0" 0\ ("f"') ....c .. .. .. .. .. .. N \0" \0"
N N t- t- V') ....c ....c ....c ....c ....c t- t- o
.-4 ....c N N ("f"') N ....c ....c ....c ....c N N
V') t- ("f"') t- \0 N 00 ("f"') t-
.,.,.,.. N 0 ....c \0.. .. t- ("f"') ("f"') ("f"') 00 \0.. 0\ 0.. \0.. ....c \0
::a ("f"') I .. V') ....c .. I .. I .. I .. .. I I I N 0" I
0 0 ....c ("f"') 0\ ("f"') 00 00 00 \0 0
....c ....c N N
It') V') N t- oo V') V') 0\ N
\0 ("f"') ....c V') .. t- N ....c
0" I I ....c 0" .. .. ("f"') I 0.. I V') I ....c N V') I I V') I 0" \0 00 I
::a ....c 0" ("f"') .. .. N .. ..
("f"') V') \0 .. .. .-4 V')
N N
,.01 0 1 "'C) == == :i == ==
=: 0 =: =:
«S =
II < II
:i <IIXI <IIXI <IIXI <IIXI :i <IIXI
.,
:3 0 0
= 5 5 5 5 = 5
i
t =: =: == =: =: =: =:
I 0 0 0 0 0 0 0
0 0 0 0 0 0 0
«S «S «S «S «S «S «S
1-4 ...c:u
..t::) c:J &... CI'J
::a
\0
..Q
C"')
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=
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z
CJ'.)
4. 1. BuHm08ble Kpene:JICHble U30eJlUR
-- V') V')
0\ ....c
I ....c
:I
-- . V') V')
.. ..
t"- t"-
\0 00
-- ....c ....c ....c
00 0\ ....c ("f"')
:I ....c .-4
- . 0\ \0 \0 \0
V') \0 00 0
....c
-- t"- V') V')
....c 00 00 V')
0 t"- 0\ 0 M
:I .. ....c .-4 ....c
- . O V') \0.. V') V') V')
5a .. M" M" M"
r 1/ V') V') 0 M
V') t"- oo
t::5 .-4 ....c
-- ):s: 0 0 0 0 t"- t"- t"-
oG) \0 t"- 0'\ ....c ....c ("f"') V')
:I t; ....c ....c .-4 ....c
-- . G) 0 0 0 0 t"- t"- t-
G) V') t- 0\ 0\ ....c ("f"')
....c ....c
-- N N 0'\ 0\ 0\
() N N N N
V') \0 t- oo 0 N M \0
G) 1 ....c ....c .-4 ....c ....c
:I
. -- . V') V') V') V') V') V') V') V') V')
.. .. .. ..
...: .. .. .. .. .. ....c ....c ....c
..... ("f"') V') \0 00 0 ....c ("f"') V')
II & ....c ....c .-4 ....c
....rIj -- 0 0 0 .-4 ....c ....c
G)):s: V') \0 \0 t- oo 0\ ....c ("f"') \0 00
:I o ....c ....c ....c ....c ....c
=
-- . = V') V') V') 0\ 0\ 0\ 0\ 0\ 0\ \0 \0 \0
j N ("f"') V') \0 t"- 0'\ ....c N \0
.-4 ....c ....c .-4
-.......- ------...
-- C"') 0 0 0 0 : ....c
0 N V')
\0 V') \0 t- t- oo 0\.0 ....c ....c ....-4
:I 0 . .-4
-- . V') V') V') V') V') V') V').V') V') V') V')
.. .. ..
.. .. .. .. .. .. ..' .. V') ....c 00
t- t- t- t- ....c ....c ....c. ....c
N ("f"') V') \0 t- 00:0\ ....c ("f"') ("f"')
....-4 ....c ....c
U 00 00 00 00 00 00 N N N:N M
__I.\C)= N ("f"') V') \0 t- oo 0\ o.....c ("f"')
:I ....c. ....c ....c
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. . . . . . . . .
HOM. MHH. MaKC.
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60 58,5 61,5 16 26 ,[(JIR R-a epOB BbIllie CTYTIeHqaTOM JIHHHH,
65 63,5 66,5 21 31 0603HaQeHHOM KaK ,
peKOMeH,[(yeTCR ISO 4018
70 68,5 71,5 26 36
80 78,5 81,5 36 46 25,5 38
90 88,25 91,75 46 56 35,5 48 27,5 40
100 98,25 101,75 56 66 45,5 58 37,5 50
110 108,25 111,75 66 76 55,5 68 47,5 60 35 50
120 118,25 121,75 76 86 65,5 78 57,5 70 45 60
130 128 132 80 90 69,5 82 61,5 74 49 64 34,5 52
140 138 142 90 100 79,5 92 71,5 84 59 74 44,5 62
150 148 152 89,5 102 81,5 94 69 84 54,5 72 40 60
160 156 164 99,5 112 91,5 104 79 94 64,5 82 50 70
180 176 184 119,5 132 111,5 124 99 114 84,5 102 70 90 55,5 78
200 195,4 204,6 131,5 144 119 134 104,5 122 90 110 75,5 98 59 84
220 215,4 224,6 138,5 151 126 141 111,5 129 97 117 82,5 105 66 91
F/la8a 4. Kpene:JICHble oemOJlu
O"OHllaHue
Pa3Mep pe3L6hI, d MI4 MI8 M22 M27 M33 M39 M45 M52 M60
K.JJacc npo,ZJyKT8 I Hlf,g
A B s g
HOM. MHH. MaKe.
:I :I :I :I :I :I :I :I :I
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240 235,4 244,6 146 161 131,5 149 117 137 102,5 125 86 111 67,5 95
260 254,8 265,2 166 181 151,5 167 137 157 122,5 145 106 131 87,5 115
280 274,8 285,2 171,5 189 157 177 142,5 165 126 151 107,5 135
300 294,8 305,2 191,5 209 177 197 162,5 185 146 171 127,5 155
320 314,3 325,7 211,5 229 197 217 182,5 205 166 191 147,5 175
340 334,3 345,7 217 237 202,5 225 186 211 167,5 195
360 354,3 365,7 237 257 222,5 245 206 231 187,5 215
380 374,3 385,7 257 277 242,5 265 226 251 207,5 235
400 394,3 405,7 277 297 262,5 285 246 271 227,5 255
420 413,7 426,3 282,5 305 266 291 247,5 275
440 433,7 446,3 302,5 325 286 311 267,5 295
460 453,7 466,3 306 331 287,5 315
480 473,7 486,3 326 351 307,5 335
500 493,7 506,3 346 371 327,5 355
a P - mar pe3b6bI.
b ,LUUI ATlHH I HOM . :::; 125 MM.
C ,LUUI ATlHH 125 MM < I HOM :::; 200 MM.
d ATlH ATlHH 1 HoM . > 200 MM.
e k ' . - 0 7k .
mm- . rom
f/pnaks =/ HOM -b; I smin =Igmaks -5P.
gig - MHHHMaJIbHaH ATlHHa rJIa,nKoH qaCTH cTepXHH
llpUMellaHue. PacnpocTPaHeHHbIe ATlHHbI 0603HaqeHbI JIHHHeH Is H I g'
4.1. BUHmo8ble "pene:JICHble U30e.tlWl
4.1.15. BS EN ISO 4017: 2001. BJfHT&I C WeCTJfrpSHHOii
ronoBKoii - KnSCC&1 Jf3AenJffl A Jf B
Pa3MepLl
llpUMellaHue. CHMBOJlbI H 0603HaqeHH.H pa3MepOB ,naHbI B COOTBeTCTBHH C
ISO 225.
k
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ct>acKoVi VlJ1V1 AJ151 pe3b6
M4 MO)l(eT 6blTb 6e3
ct>acKVI (cpe3HoVi 60J1T)
(CM. ISO 4753)
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MHHaJIbHOH ,nJIHHOH I = 80 M H KJIaCCOM npOqHOCTH 8.8:
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F/la8a 4. Kpene:JICHble oemOJlu
BS EN ISO 4017: 2001. Ta6JI. 2. CneI(HaJIbHbIe pe3b6bI. Pa3MepbI B MM
P83Mep pe3L6hI, d M3,5 MI4 MI8 M22 M27 M33 M39 M45 M52 M60
pi! 0,6 2 2,5 2,5 3 3,5 4 4,5 5 5,5
MaKc. b 1,8 6 7,5 7,5 9 10,5 12 13,5 15 16,5
a
MHH. 0,6 2 2,5 2,5 3 3,5 4 4,5 5 5,5
MHH. 0,15 0,15 0,2 0,2 0,2 0,2 0,3 0,3 0,3 0,3
c
MaKC. 0,4 0,6 0,8 0,8 0,8 0,8 1 1 1 1
d a MaKC. 4,1 15,7 20,2 24,4 30,4 36,4 42,4 48,6 56,6 67
KJIacc 5,07 19,37 25,34 31,71 - - - - - -
d w H3,n;eJIHH MHH.
- - 24,85 31,35 38 46,55 55,86 64,7 74,2 83,41
KJIacc 6,58 23,36 30,14 37,72 - - - - - -
e H3,n;eJIHH MHH.
- - 29,56 37,29 45,2 56,37 66,44 76,95 88,25 99,21
HOM. 2,4 8,8 11,5 14 17 21 25 28 33 38
KJIacc MHH. 2,275 8,62 11,285 13,785 - - - - - -
k H3,n;eJIHHA MaKC. 2,525 8,98 11,175 14,215 - - - - - -
KJIacc MHH. - - 11,15 13,65 16,65 20,58 24,58 27,58 32,5 37,5
H3,n;eJIHH B MaKC. - - 11,85 14,35 17,35 21,42 25,42 28,42 33,5 38,5
KJIacc 1,59 6,03 7,9 9,65 - - - - - -
k'e H3,n;eJIHH MHH.
- - 7,81 9,56 11,66 14,41 17,21 19,31 22,75 26,25
r MHH. 0,1 0,6 0,6 0,8 1 1 1 1,2 1,6 2
HOM. = MaKC. 6 21 27 34 41 50 60 70 80 90
s KJIacc 5,82 20,67 26,67 33,38 - - - - - -
H3,n;eJIHH MHH. 26,16 33 40 49 58,8 68,1 78,1 87,8
- -
4.1. BUHmo8ble "pene:JICHble U30e.tlWl
O"OHliaHue
P83Mep pe3hOhI, d M3,5 MI4 MI8 M22 M27 M33 M39 M45 M52 M60
K.JJacc H eJJHJI
A B
[d
80M. MHB. MaKe. MHH. MaKe.
r ./
-'
8 7,71 8,29 - -
10 9,71 10,29 - -
12 11,65 12,35 - -
16 15,65 16,35 - -
20 19,58 20,42 - -
25 24,58 25,42 - -
30 29,58 30,2 - -
35 34,5 35,5 - -
-----
40 39,5 40,5 .
- - .
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45 44,5 45,5 .
- - .
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65 64,4 65,6 63,5 66,5 . .
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150 149,2 150,8 148 152 . .
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160 - - 158 162
180 - - 178 182
200 - - 197,7 202,3
a P - war pe3b6bI.
b 3HaqeHHH, COOTBeTCTByJOll(He a MaKc ., HOpMaJIbHOrO pH,ZJ;a, B ISO 3508.
C k:nm = 0,7 k min .
d )];Hana30H pacnpOC'rpaHeHHbIX .lUIHH pacnOJIOXeH Mex.ny CTYTIeHqaTbIMH JIHHHHMH:
Krracc H3,n;eJIHH A BbIwe cTYTIeHqaTOH JIHHHH, 0603HaqeHHOH KaK
Krracc H3,n;eJIHH B HHXe 3TOH cTYTIe aToH JIHHHH.
F/la8a 4. Kpene:JICHble oemOJlu
4. 1. 16. BS EN ISO 401 B: 2001. BJfHT&1 C WeCTJfrpSHHOH
ronoBKoii - KnSCC Jf3AenJffl C
Pa3MepLl
llpUMellaHue. CHMBOJIbI H 0603HaqeHH.H pa3MepOB ,naHbI B COOTBeTCTBHH C
ISO 225.
Cneu.VlanbHble Tpe-
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MHHaJIbHWI ,lUIHHa I = 80 MM H KJIacc lIPOqHOCTH 4.6:
BHHT C meCTHrpaHHOH rOJIOBKOH ISO 4018 - M12 X 80 - 4,6.
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oV')V') t'-t'-t'-NNN
' V') 00 V')V')V')V')V')V')V')V')
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NNNNN V')
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.. .. N .. N .. - 00 00 - - ." ._ ._ ....; - t'- 0- -- -
- V')t'- , , ,
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_ , 00 _ 0 ........ _, _, _, -, \J' , \J' , \J'
\J' ........ N
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= NNNNN V')
.
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4.1. BUHmo6ble "peneJICHble U30eJlUR 161
BS EN ISO 4018: 2001. Ta6JI. 2. CneI(HaJIbHbIe pe3b6bI. Pa3MepbI B MM
Pa3Mep pe3hOhl, d MI4 MIS M22 M27 M33 M39 M45 M52 M60
pa 2 2.5 2.5 3 3.5 4 4.5 5 5.5
a MaKC. 6 7,5 7,5 9 10,5 12 13,5 15 16,5
MHH. 2 2.5 2.5 3 3.5 4 4.5 5 5.5
c MaKC. 0.6 0.8 0.8 0.8 0.8 1 1 1 1
d.. MaKC. 16.7 21.2 26.4 32.4 38.4 45.4 52.6 62.6 71
d... MHH. 19.15 24.85 31.35 38 46.55 55.86 64.7 74.2 83.41
e MHH. 22.78 29.56 37.29 45.2 55.37 66.44 76.95 88.25 99.21
HOM. 8,8 11,5 14 17 21 25 28 33 38
k MHH. 8,35 10,6 13,1 16,1 19,95 23,95 26,95 31,75 36,75
MaKC. 9.25 12.4 14.9 17.9 22.05 26.05 29.05 34.25 39.25
k'e MHH. 5.85 7.42 9.17 11.27 13.97 16.77 18.87 22.23 25.73
r MHH. 0.6 0.6 0.8 1 1 1 1.2 1.6 2
HOM. = MaKC. 21 27 34 41 50 60 70 80 90
S MHH. 20,16 26,16 33 40 49 58,8 68,1 78,1 87,8
IC
HOM. MHH. MaKC.
30 28,95 31,05
35 33,75 36,25
40 38.75 41.25
45 43,75 46,25
50 48,75 51,25
55 53.5 56.5
60 58,5 61,5
65 63,5 66,5
70 68.5 71.5
80 78,5 81,5
90 88,25 91,75
100 98.25 101.75
110 108,25 111,75
120 118,25 121,75
130 128 132 /
140 138 142 \.._-
150 148 152 I
\
160 156 164
180 176 184
200 195,4 204,6
220 215.4 224.6
240 235,4 244,6
260 254,8 265,2
280 274.8 285.2
300 294,8 305,2
320 314,3 325,7
340 334.3 345.7
360 354,3 365,7
380 374,3 385,7
400 394.3 405.7
420 413,7 426,3
440 433,7 446,3
460 453.7 466.3
480 473,7 486,3
500 493,7 506,3
a P - mar pe3b6bI. b ktMHHo = 0,7 k MHHo
c )];Hana30H pacnpOC'rpaHeHHbIX,lI)IHH Mex.ny cTYIIe aTbIMH JIHHIDIMH, 0603HaQeHHbIMH KaK -.
6-3255
F/la8a 4. Kpene:JICHble OemOJlu
4. 1. 17. BS EN ISO 4032: 2001. WeCTJfrpSHH&le rSHKJf
JfCnOnHeHJffl 1 - KnSCC&1 Jf3AenJffl A Jf B
Pa3MepLI
30 0
7S o
...:
e
m'
c
CbOPMY BblCTYna
nOA wat16y CJ1e-
AyeT orOBap -
BaTb OTAeJ1bHO
o 0
i
.....
"tJ "tJ t
\
mil
m
- .
- -
CD
s
m
IIpHMep 0603HaqeHIDI IIIecTHrpaHHoH raHKH HCrrOJIHeHIDI I, C pa3MepOM pe3b-
6bI d = MI2 H KJIaCCOM rrpOqHOCTH 8:
lliecTHrpaHHaJi raHKa ISO 4032 - MI2 - 8.
BS EN ISO 4032: 2001. Ta6JI. 1. CTaH,l(apTHbIe pa3MepbI. Pa3MepbI B MM
Pa3Mep pe3hOhl, d MI,6 M2 M2,5 M3 M4 M5 M6 M8 MIO MIl
pa 0,35 0,4 0,45 0,5 0,7 0,8 1 1,25 1,5 1,75
MaKC. 0,2 0,2 0,3 0,4 0,4 0,5 0,5 0,6 0,6 0,6
c
MIIH. 0,1 0,1 0,1 0,15 0,15 0,15 0,15 0,15 0,15 0,15
MHH. 1,6 2 2,5 3 4 5 6 8 10 12
d a
MaKC. 1,84 2,3 2,9 3,45 4,6 5,75 6,75 8,75 10,8 13
d w MHH. 2,4 3,1 4,1 4,6 5,9 6,9 8,9 11,6 14,6 16,6
e MHH. 3,41 4,32 5,45 6,01 7,66 8,79 11,05 14,38 17,77 20,03
MaKC. 1,3 1,6 2 2,4 3,2 4,7 5,2 6,8 8,4 10,8
m 6,44
MHH. 1,05 1,35 1,75 2,15 2,9 4,4 4,9 8,04 10,37
m' MHH. 0,8 1,1 1,4 1,7 2,3 3,5 3,9 5,2 6,4 8,3
m" MHH. 0,7 1 1,2 1,5 2 3,1 3,4 4,5 5,6 7,3
HOM. = MaKC. 3,2 4 5 5,5 7 8 10 13 16 18
s 3,02 3,82 4,82 5,32 6,78 7,78 9,78 12,73 15,73 17,73
MHH.
4.1. BUHmo6ble "peneJICHble U30eJlUR 163"
O"OH'taHue
P83Mep Pe3hOhl, d MI6 M20 M24 M30 M36 M42 M48 M56 M64
pa 2 2,5 3 3,5 4 4,5 5 5,5 6
MaKC. 0,8 0,8 0,8 0,8 0,8 1 1 1 1
C
MHH. 0,2 0,2 0,2 0,2 0,2 0,3 0,3 0,3 0,3
d a MHH. 16 20 24 30 36 42 48 56 64
MaKC. 17,3 21,6 25,9 32,4 38,9 45,4 51,8 60,5 69,1
d w MHH. 22,5 27,7 33,3 42,8 51,1 60 69,5 78,7 88,2
e MHH. 26,75 32,95 39,55 50,85 60,79 71,3 82,6 93,56 104,86
MaKC. 14,8 18 21,5 25,6 31 34 38 45 51
m
MHH. 14,1 16,9 20,2 24,3 29,4 32,4 36,4 43,4 49,1
m' MHH. 11,3 13,5 10 -1 19,4 23,5 25,9 29,1 34,7 39,3
m" MHH. 9,9 11,8 14,1 17 20,6 22,7 25,5 30,4 34,4
HOM. = MaKC. 24 30 36 46 55 65 75 85 95
s 23,67 29,16 35 45 53,8 63,1 73,1 82,8 92,8
MHH.
a P - mar pe3b6bI.
BS EN ISO 4032: 2001. Ta6JI. 2. CneI(HaJIbHbIe pa3MepbI. Pa3MepbI B MM
P83Mep pe3hOhl, d M3,5 MI4 MI8 M22 M27 M33 M39 M45 M52 M60
pa 0,6 2 2,5 2,5 3 3,5 4 4,5 5 5,5
MaKC. 0,4 0,6 0,8 0,8 0,8 0,8 1 1 1 1
C
MHH. 0,15 0,15 0,2 0,2 0,2 0,2 0,3 0,3 0,3 0,3
d a MHH. 3,5 14 18 22 27 33 39 45 52 60
4 15,1 (19,5 23,7 29,1 35,6 42,1 48,6 56,2 64,8
MaKC.
d w MHH. 5 19,6 ( .4,9 31,4 38 46,6 55,9 64,7 74,2 83,4
e MIIH. 6,58 23,35 29,56 37,29 45,2 55,37 66,44 76,95 88,25 99,21
MaKC. 2,8 12,8 15,8 19,4 23,8 28,7 33,4 36 42 48
m
MHH. 2,55 12,1 15,1 18,1 22,5 27,4 31,8 34,4 40,4 46,4
m' MHH. 2 9,7 12,1 14,5 18 21,9 25,4 27,5 32,3 37,1
m" MHH. 1,8 8,5 10,6 12,7 15,8 19,2 22,3 24,1 28,3 32,5
HOM. = MaKC. 6 21 27 34 41 50 60 70 80 90
s 5,82 20,67 26,16 33 40 49 58,8 68,1 78,1 87,8
MHH.
a P - mar pe3b6bI.
F/la8a 4. Kpene:JICHble oemOJlu
4. 1. 18. BS EN ISO 4033: 2001. WeCTJfrpSHH&le rSHKJf
JfCnOnHeHJffl 2 - KnSCC&1 Jf3AenJffl A Jf B
Pa3MepLI
...:
i!!
b
C\I
g
o
c
AonYCT Ma
aJ1bTepHaT BHa51
ct>opMa BblCTYna
nOA wat16y
CD
1:) -d
3t
1:)
s
m
m
IIpHMep 0603HaqeHIDI IIIecTHrpaHHoH raHKH C pa3MepOM MeTPHqeCKOH pe3b6bI
d = MI2 H KJIaCCOM rrpOqHOCTH 9:
lliecTHrpaHHaJi raHKa ISO 4033 - MI2 - 9.
Pa3MepHOCTH B MM
Pa3Mep pe3hohl, d M5 M6 M8 MIO Ml2 (MI4) Ml6 M20 M24 M30 M36
pa 0,8 1 1,25 1,5 1,75 2 2 2,5 3 3,5 4
c MaKC. 0,5 0,5 0,6 0,6 0,6 0,6 0,8 0,8 0,8 0,8 0,8
d a MHH. 5 6 8 10 12 14 16 20 24 30 36
MaKC. 5,75 6,75 8,75 10,8 13 15,1 17,3 21,6 25,9 32,4 38,9
d w MHH. 6,9 8,9 11,6 14,6 16,6 19,6 22,5 27,7 33,2 42,7 51,1
e MHH. 8,79 11,05 14,38 17,77 20,03 23,35 26,75 32,95 39,55 50,85 60,79
MaKC. 5,1 5,7 7,5 9,3 12 14,1 16,4 20,3 23,9 28,6 34,7
m
MHH. 4,8 5,4 7,14 8,94 11,57 13,4 15,7 19 22,6 27,3 33,1
m' MHH. 3,84 4,32 5,71 7,15 9,26 10,7 12,6 15,2 18,1 21,8 26,5
HOM. = MaKC. 8 10 13 16 18 21 24 30 36 46 55
s 7,78 9,78 12,73 15,73 17,73 20,67 23,67 29,16 35 45 53,8
MHH.
Pa3MepOB B cK06Kax CJIe,n;yeT no B03MO)I{HOCTH H36eraTb.
a p - mar pe3b6bI.
4.1. BUHmo6ble "peneJICHble U30eJlUR 165"
BS EN ISO 4033: 2001. Ta6JI. 1.
HOMHH8JIh1lhlii ,lUIaMeTp Pa3Mep raiiKH llJIom;a,zu. KOJlhneBOii ropneaoii onopHoii nOBepXHOCTH/
pe3hOW (MM) no KJII01f (MM) HOMHH8JIhHaJI DJI0m;a,zu. Ce1felllDl no pe3..oe *
5 8 1,08
6 10 1,44
8 13 1,23
15 0,90
10 16 1,30
17 1,73
18 0,91
12
19 1,16
21 0,96
14 /
22 - 1,24
.J I
16 24 1,02
20 30 0,95
24 36 0,86
30 46 1,02
36 55 1,04
* BbIQHCJIeHHH OCHOBaHbI Ha 3HaqeHHHX .LJ)UI oTBepcTHH C rapaHTHpOBaHHbIM 3a30pOM B
COOTBeTCTBHH C ISO 273 (HcnpaBJIeHHbIM), cpe,n;HHH cepHH.
BS EN ISO 4033: 2001. Ta6JI. 2.
P83Mep pe3how, d MIO MI2 MI4
pa 1,5 1,75 2
d w MHH. 13,6 (15,6 17,4 20,4
e MHH. 16,64 90 21,10 24,49
MaKC. 10 8,8 11,3 13,13
m
MHH. 9,64 8,44 10,87 12,7
m' MHH. 7,7 6,75 8,7 10,2
MaKC. 15 17 19 22
s 14,73 16,73 18,67 21,67
MHH.
a P - mar pe3b6bI.
r--
F/la8a 4. Kpene:JICHble iJemOJlu
4. 1. 19. BS EN ISO 4034: 2001. WeCTJfrpSHH&le rSHKJf
JfCnOnHeHJffl 1 - KnSCC Jf3AenJffl C
Pa3MepLI
30°
15°
.,J
AonYCT Ma
3eHKOBKa s
B Ha"laJ1e
pe3b6bl
IIpHMep 0603HaqeHIDI IIIecTHrpaHHO:H ra:HKH C pa3MepOM pe3b6bI d = MI2 II
KJIaCCOM rrpOCqHOCTH 5:
illecTHrpaHHaJi raHKa ISO 4034 - MI2 - 5.
BS EN ISO 4034: 2001. Ta6JI. 1. CTaH,n;apTHbIe pa3MepbI. Pa3MepbI B MM
Pa3Mep pe3hohl, d M5 M6 M8 MIO MI2 MI6 M20
pa 0,8 1 1,25 1,5 1,75 2 2,5
d w MHH. 6,7 8,7 11,5 14,5 16,5 22 27,7
e MHH. 8,63 10,89 14,20 17,59 19,15 26,17 32,95
MaKC. 5,6 6,1 7,9 95 12,2 15,9 19
m 4,6 6,4 16,9
MHH. 4,4 8 10,4 14,1
m' MHH. 3,5 3,7 5,1 6,4 8,3 11,3 13,5
s HOM. = MaKC. 8 10 13 16 18 24 30
MHH. 7,64 9,64 12,57 15,57 17,57 23,16 29, 16
Pa3Mep pe3how, d M24 M30 M36 M42 M48 M56 M64
pa 3 3,5 4 4,5 5 5,5 6
d w MHH. 33,3 42,8 51,1 60 69,5 78,7 88,2
e MHH. 39,55 50,85 60,79 72,02 82,6 93,56 104,86
MaKC. 22,3 26,4 31,5 34,9 38,9 45,9 52,4
m 36,4 49,4
MHH. 20,2 24,3 28 32,4 43,4
m' MHH. 16,2 19,5 22,4 25,9 29,1 34,7 39,5
HOM. = MaKC. 36 46 55 65 75 85 95
s 35 45 53,8 63,1 73,1 82,8 92,8
MHH.
a p - mar pe3b6bI.
4. 1. BUHm08ble 1(;pene:JICHble U3Oe.nUR
BS EN ISO 4034: 2001. Ta6JI. 2. CneI(HaJIbHbIe pa3MepbI. Pa3MepbI B MM
Pa3Mep pe3hOhl, d MI4 MI8 M22 M27 M33 M39 M45 M52 M60
pa 2 2,5 2,5 3 3J 4 4,5 5 5,5
d w MHH. 19,2 24,9 31,4 38 46,6 55,9 64,7 74,7 83,4
e MHH. 22,78 29,56 37,29 45,7 55,37 66,44 76,95 88,25 99,21
MaKC. 13,9 16,9 20,2 24,7 29,5 34,3 36,9 42,9 48,9
m 12,1 15,1 18,1 22,6 27,4 31,8 34,4 40,4 46,4
MHH.
m' MHH. 9,7 12,1 14,5 18,1 21,9 25,4 27,5 32,3 37,1
s HOM. = MaKC. 21 27 34 41 50 60 70 80 90
MHH. 20,16 26,16 33 40 49 58,8 68,1 78,1 87,8
a P - mar pe3b6bI.
4.1.20. BS EN ISO 4035: 2001. WeCTJfrpaHH&le HJf3KJfe
raHKJf (c cpnaHqeM) - lUJaCC&1 Jf3AenJffl A Jf B
Pa3MepLi
ai
...
fa
.....
s
IIpHMep 0603HaqeHIDI HH3KO:H raHKH C <pJIaHueM C pa3MepOM pe3b6bI d = MI2
H KJIaCCOM rrpOqHOCTH 05: (-
illeCTHrpaHHaJI HH3KaJl raHKa ISO 4035 - MI2 - 00.5
BS EN ISO 4035: 2001. Ta6JI. 1. CTaH,l(apTHbIe pa3MepbI. Pa3MepbI B MM
P83Mep pe3hohl, d MI,6 M2 M2,5 M3 M4 M5 M6 M8 MIO MI2
pa 0,35 0.4 0,45 0,5 0,7 0.8 1 1,25 1,5 1,75
d a MHH. 1,6 2 2,3 3 4 5 6 8 10 12
1,84 2,3 2,9 3,45 4,6 5,75 6,75 8,75 10,8 13
MaKC.
d w MHH. 2,4 3,1 4,1 4.6 5,9 6,9 8,9 11,6 14,6 16,6
e MHH. 3,41 4,32 5,45 6,01 7,66 8,79 11,05 14,38 17,77 20,03
MaKC. 1 1,2 1,6 1,8 2,2 2,7 3,2 4 5 6
m 0,75 0,95 1,35 1,55 1,95 2,45 2,9 3,7 4,7 5,7
MHH.
m' MHH. 0,6 0,8 1,1 1,2 1,6 2 2,3 3 3,8 4,6
HOM. = MaKC. 3,2 4 5 5,5 7 8 10 13 16 18
s 3,02 3,82 4,82 5,32 6,78 7,78 9,78 12,73 15,73 17,73
MHH.
F/la8a 4. Kpene:JICHble OemOJlu
O"OH"taHue
Pa3Mep pe3hohl, d MI6 M20 M24 M30 M36 M42 M48 M56 M64
pa 2 2,5 3 3,5 4 4,5 5 5,5 6
d a MHH. 16 20 24 30 36 42 48 56 64
17,3 21,6 25,9 32,4 38,9 45,4 51,8 60,5 69,1
MaKC.
d w MHH. 22,5 27,7 33,2 42,8 51,1 60 69,5 78,7 88,2
e MHH. 26,75 32,95 39,55 50,85 60,79 71,3 82,6 93,56 104,86
MaKC. 8 10 12 15 18 21 24 28 32
m 7,42 9,10 10,9 13,9 16,9 19,7 22,7 26,7 30,4
MHH.
m' MHH. 5,9 7,3 8,7 11,1 13,5 15,8 18,2 21,4 24,3
HOM. = MaKC. 24 30 36 46 55 65 75 85 95
s MHH. 23,67 29, 16 35 45 53,8 63,1 73,1 82,8 92,8
a p - mar pe3b6bI.
BS EN ISO 4035: 2001. Ta6JI. 2. CneI(HaJIbHbIe pa3MepbI. Pa3MepbI B MM
Pa3Mep pe3how, d M3,5 MI4 MI8 M22 M27 M33 M39 M45 M52 M60
pa 0,6 2 2,5 2,5 3 3,5 4 4,5 5 5,5
d a MHH. 3,5 14 18 22 27 33 39 45 52 60
4 15,1 19,5 23,7 29,1 35,6 42,1 48,6 56,2 64,8
MaKC.
d w MHH. 5,1 19,6 24,9 31,4 38 46,6 55,9 64,7 74,2 83,4
e MHH. 6,58 23,35 29,56 37,29 45,2 55,37 66,44 76,95 88,25 99,21
MaKC. 2 7 9 11 13,5 16,5 19,5 22,5 26 30
m 1,75 6,42 8,42 9,9 12,4 15,4 18,2 21,2 24,7 28,7
MHH.
m' MHH. 1,4 5,1 6,7 7,9 9,9 12,3 14,6 17 19,8 23
HOM. = MaKC. 6 21 27 34 41 50 60 70 80 90
S MHH. 5,82 20,67 26,16 33 40 49 58,8 68,1 78,1 87,8
a p - mar pe3b6bI.
4.1.21. BS EN ISO 4036: 2001. WeCTJfrpaHH&le HJf3KJfe
raHKJf (6e3 cpnaHqa) - IUJBCC Jf3AenJffl B
Pa3MepLI
Pa3Mepbl B M J1J1 MeTpax
1:)
m
s
IIpHMep 0603HaQeHIDI IIIecTHrpaHHoH
HH3KOH raHKH C MeTPHQeCKOH pe3b6oH
d = M6, H3rOTOBJIeHHOH H3 CTaJIH C TBep-
,nOCTblO IIO BHKKepcy MHH. 110 HV (CT.):
lliecmrpaHHaJI raiiKa ISO 4036 - M6 CT.
4.1. BUHmo6ble "peneJICHble U30eJlUR 169
BS EN ISO 4036: 2001. Ta6JI. 1.
HOMHH8JIh1lhli ,lUIaMeTp Pa3Mep raiKH llJIOIQa,ZU. KOJlhneBOH TOpu;eBOH onopHOH nOBepxeOCTH/
pe3hO'" (MM) no KJII01f (MM) HOMHH8JIhHaJI DJI0m;a,zu. Ce1felllDl no pe3hoe *
5 8 1,08
6 10 1,44
8 13 1,23
15 0,90
10 16 1,30
17 1,73
* BbIQHCJIeHHH OCHOBaHbI Ha 3HaQeHHHX ,L(JIH oTBepcTHH C rapaHTHpOBaHHbIM 3a30pOM B
COOTBeTCTBHH C ISO 273 (HcnpaBJIeHHbIM), cpe,n;HHH) cepHH.
BS EN ISO 4036: 2001. Ta6JI. 2.
Pa3Mep Pe3hOW, d MIO
pa 1,5
e MHH. 16,46 18,73
MaKC. 5 5
m 4,52 4,52
MHH.
MaKC. 15 17
s MHH. 14,57 16,57
a p - mar pe3b6bI.
4. 1.22. BS EN ISO 8765:r.. OO 1. 50RThI C WeCTlffrpaHHOH
ronosKoif C MeTp...;:ecKoif MenKoH pe3&6oif -
KnSCC&1 Jf3AenJffl A Jf B
Pa3MepLI
llpUMe"taHue. CHMBOJIbI H 0603HaqeHIDI pa3MepOB .naHbI B COOTBeTCTBHH C
ISO 225.
KOHeu. AOJ1)1(eH Pa3Mep B M J1J1 MeTpax X MaKC MaJ1bHOe
6blTb C ct>acKot1 YTOJ1w.eH e
(CM. ISO 4753) k' I, nOA rOJ10BKot1
X c B HTa
-- Q)
I. U(1) S
Ig (b)
k I
(1) HerrOJIHa.H pe3b6a (c6er pe3b6bI) u 2P.
F/la8a 4. Kpene:JICHble OemOJlu
:a
Q)
M
\0 \0 \0 V)
x 00 V) V) 00
N w.. 0 w.. ow.. \Ow.. V) w..
I I V) ow.. I w.. I w.. I I I I N I
\C \0 00 0\ 0\
\0 00 0 N 0\
\0 \0 \0 V)
x V) V) 00
\C \0 N \Ow.. V) N V) w.. w.. V) w..
tn I I w.. \0 V) I w.. I I w.. I I V) I w.. N 00 I
0 V) 00
V) 0\ N 00
tf') 00 V) 00 N
X 00 \Ow.. w.. \Ow.. V) w.. \Ow.. ,....c
QO 00 0 0 V) ...
I 0 N ow.. N I w.. I I N I I w.. I ow.. I
0\ 0\ 0 M
V) \0 00 N N ['...
tf') 00 V) 00 N
X 0\ \Ow.. ,....c
N \0 N w.. \0 V) w.. N V) ...
I 0\ 0 ow.. V) I w.. I I 00 N I I w.. I w.. \0 I M
0\ V) \0
V) N N \0
tf') 0\ 00 N \0
x 00 V) 00
\C N 00 0\ \0 w.. w.. Ow.. w.. V) ...
tf') I 00 0\ ow.. ow.. I V) I w.. I ow.. \0 N I I I V) I M
N N V)
V) \0 N N N t/")
N 00 V) V) 00 N
X ... w.. 00
= \0 N V) N 00 0 w.. 00 N w.. \0 t/")
tf') \0 00 ... w.. I 0\ I I w.. \0 00 I I w.. w.. I N I
0 0 N 0 00 0\
N V)
N 00 V) 00 V) V) V) V) V) V) \0 00
x w.. 00
0 N 00 \0 \Ow.. N 0\ V) V) N \0 N 00 \0 t/")
w.. ow.. \0 w.. w.. w.. w.. w.. w.. w.. ow.. ow.. w..
V) \0 0 N 0\ 0\ w.. w.. V) 0 V) M
N N N V)
tn V) V)
00 0\ V) V) V) \0
...-4 w.. V) 00
x \0 N N 00 0 \Ow.. w.. w.. V) w.. N 00 \Ow.. V) 00 0 \Ow.. ......-I
= V) I w.. w.. N N w.. w.. N w.. w.. ow.. ...
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171
u
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00 0
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X / V) \0 t"-- 0\ ....... N
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ff') --- i :I: 1-4 M M
:S:t:J:: \0 \0 \0 \0 \0 0 N
u M V) t"-- 0\ ....... .......
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F/la8a 4. Kpene:JICHble oemOJlu
::s
:t:
:t:
C
U t"-- t"-- t"-- t"-- t"-- t"-- t"-- t"-- t"-- t"-- t"-- t"-- t"--
X --- = 0 N \0 00 0 N \0 00 0 N
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t:C
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CIS
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= 'I') I"- 0 N... ... N N 00 ... N 0
'..c I I ... t-4 \0 \0 I I I ... I I I"- 00 I ... N 0\ I I"-
t-4 0 0\ 0\ t-4 \0
'I') 00 0\ N 00
\0 'I') 'I')
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N ... N N ... N 0 ... ... I"- \0 0 ...
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t-4 t-4 0 t-4 00 N t-4
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ff') 00 'I') 00 N t-4
X \0... 1"-... t-4
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I 0 t-4 ... t-4 00 I I I 00 M I I ... ... I ... ... l"- I 00
t-4 t-4 0 \0 \0 I"- 00 0\ t-4 \0
l"- N N t-4
ff') 00 \0 00 N t-4
X ... 00
0'\ 0 0\ 00 'I') 'I') ... N 0 ...
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t-4 t-4
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0 t-4 0\ t-4 t-4 \0 \0 t-4
I .D 1 0 1 := 0 :i := 0 := 0 := := 0 :=
:= 0 := := := := :=
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x 00 0\
N ...
In
0\ \0
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M . 0 0 0 0 t---
0\0
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N :s:"'o N N N N 0 N N
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o 0 .......
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0 ....... N M V) \0 00 0 N
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V) V) V) V) V) V) V) V) V) V) V)
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V) V) V) V) V) V) V) V) V) V) V)
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KnSCC&1 Jf3AenJffl A Jf B
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c
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d a 32,4
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d w MHH. 11,63 14,63 16,63 22,49 27,7 33,25 42,75 51,11
e MHH. 14,38 17,77 20,03 26,75 32,95 39,55 50,85 60,79
MaKC. 7,5 9,3 12 16,4 20,3 23,9 28,6 34,7
m
MHH. 7,14 8,94 11,57 15,7 19 22,6 27,3 33,1
m' MHH. 5,71 7,15 9,26 12,56 15,2 18,08 21,84 26,48
m" MHH. 5 6,26 8,1 10,99 13,3 15,82 19,11 23,17
HOM. = MaKC. 13 16 18 24 30 36 46 55
s 12,73 15,73 17,73 23,67 29,16 35 45 53,8
MHH.
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BS EN ISO 8674: 2001. TaoJI. 2. CneI(HaJIbHble pe3b6bI. Pa3MepbI B MM
P83Mep pe3how, dxP MI0xl,25 M12xl,25 M14x 1,5 M18xl,5 M20x2 M22xl,5 M27x2 M33x2
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c
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MHH. 10 12 14 18 20 22 27 33
d a 21,6 35,6
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d w MHH. 14,63 16,63 19,64 24,85 27,7 31,35 38 46,55
e MHH. 17,77 20,03 23,36 29,56 32,95 37,29 45,2 55,37
MaKC. 9,3 12 14,1 17,6 20,3 21,8 26,7 32,5
m
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m' MHH. 7,15 9,26 10,72 13,52 15,2 16,4 20,32 24,72
m" MHH. 6,26 8,1 9,38 11,83 13,3 14,35 17,78 21,63
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s
MHH. 15,73 17,73 20,67 26,16 29,16 33 40 49
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PHC. 2. Pa3MepbI KOpOHqaTbIX raeK
4.1. BUHmo8ble K.pene:JICHble U30eJlUR
BS 7764:1994. TaoJI. 1. Pa3MepHocTH meCTHrpaHHbIX rrpope3HbIX H KOpOHqaTbIX raeK.
Pa3MepbI B MM
d M4 M5 M6 M8 MIO MI2 (MI4) MI6 (MI8) M20 (M22) M24
p 0,7 0,8 1 1,25 1,5 1,75 2 2 2,5 2,5 2,5 3
c MaKC. 0,4 0,5 0,5 0,6 0,6 0,6 0,6 0,8 0,8 0,8 0,8 0,8
d a MHH. 4 5 6 8 10 12 14 16 18 20 22 24
MaKC. 4,6 5,75 6,75 8,75 10,8 13 15,1 17,3 19,5 21,6 23,7 25,9
de MaKC. - - - - - l?J 19 22 25 28 30 34
( 24,16 27, 16 29,16
MHH. - - - - - 15,57 18,48 21 ,48 33
d w MHH. 5,9 6,9 8,9 11,6 14,6 16,6 19,6 22,5 24,8 27,7 31,4 33,2
e MHH. 7,66 8,79 11,05 14,38 17,77 20,03 23,36 26,75 29,58 32,95 37,29 39,55
MaKC. 5 6 7,5 9,5 12 15 16 19 21 22 26 27
m 4,7 5,7 7,14 9,14 11,57. 14,57 15,57 18,48 21,48 21,48 25,48 25,48
MHH.
m' MHH. 2,32 3,52 3,92 5,15 6,43 8,3 9,68 11,28 12,08 13,52 14,48 16,16
MHH. 1,2 1,4 2 2,5 2,8 3,5 3,5 4,5 4,5 4,5 5,5 5,5
n 1,45 1,65 2,25 2,75 3,05 3,8 3,8 4,8 4,8 4,8 5,8 5,8
MaKC.
MaKC. 7 8 10 13 16 18 21 24 27 30 34 36
s 6,78 7,78 9,78 12,73 15,73 17,73 20,67 23,67 26,16 29, 16 33 35
MHH.
MaKC. 2,9 4 5 6,5 8 10 11 13 15 16 18 19
w 3,2 3,7 4,7 6,14 7,64 9,64 10,57 12,59 14,57 15,57 17,57 18,48
MHH.
KJIacc H3,n;eJIHH A KJIacc H3,n;eJIHH B
CMO'rpH B Ta6JI. 2 CMO'rpH B Ta6JI. 2
d (M27) M30 (M33) M36 (M39) 42 (M45) M48 (M52) M56 (M60) M64
TJ 3 3,5 3,5 4 4 -4,5 4,5 5 5 5,5 5,5 6
c MaKC. 0,8 0,8 0,8 0,8 1 1 1 1 1 1 1 1
d a MHH. 27 30 33 36 39 42 45 48 52 56 60 64
MaKC. 29,1 32,4 35,6 38,9 42,1 45,4 48,6 51,8 56,2 60,5 64,8 69,1
de MaKC. 38 42 46 50 55 58 62 65 70 75 80 85
MHH. 37 41 45 49 53,8 56,8 60,8 63,8 68,8 73,8 78,8 83,8
d w MHH. 38 42,7 46,6 51,1 55,9 59,9 64,7 69,4 74,2 78,7 83,4 88,2
e MHH. 45,2 50,85 55,37 60,79 66,44 71,3/ ,.16,95 82,6 88,25 93,56 99,21 104,86
MaKC. 30 33 35 38 40 46 48 50 54 57 63 66
m .
MHH. 29,48 32,38 34,38 37,38 39,38 45,38 47,38 49,38 53,26 56,26 62,26 65,26
m' MHH. 18 19,44 21,92 23,52 25,44 25,92 27,52 29, 12 32,32 34,72 37,12 39,3
MIIH. 5,5 7 7 7 7 9 9 9 9 9 11 11
n 5,8 7,36 7,36 7,36 7,36 9,36 9,36 9,36 9,36 9,36 11,43 11,43
MaKC.
MaKC. 41 46 50 55 60 65 70 75 80 85 90 95
s 40 45 49 53,8 58,8 63,1 68,1 73,1 78,1 82,8 87,8 92,8
MHH.
MaKC. 22 24 26 29 31 34 36 38 42 45 48 51
w 21 ,48 23,48 25,48 28,48 30,38 33,38 35,38 37,38 41,38 44,38 47,38 50,26
MHH.
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llpUMe"taHue.
1. CneHaJIbHble pa3MepbI rrOKa3aHbI B cKo6Kax.
(2\KopOHqaTbIe raHKH HaqHHaIOTCH C pa3Mepa M12.
\ 3. KOpOHqaTbIe raHKH 60JIbme M39 HMeIOT BoceMb rpaHeH.
F/la8a 4. Kpene:JICHble oemOJlu
BS 7764:1994 TaoJI. 2. XapmcrepHCTHKH mecrnrpaHHbIX npOpe3HbIX H KOpO aThIX raeK
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TIOJIe ,n;onycKa pe3b6bI CooTBeTCTBYI<>IUee KJIaccy 6H B BS 3643: qacTb 1:1981 H BS 3643:
qacTb 2: 1981
npOqHOCTHble CooTBeTCTBYI<>IUHe CooTBeTCTBYI<>IUHe CooTBeTCTBYI<>I.U;He
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TepHaJIa 6, 8 HJIH 10 B MaTepHaJIa A2- 70
BS EN 20898-2: 1992 B BS 6105: 1981
B03MO)I{Hble OTKJIOHeHHH B COOTBeTCTBHH C KJIaCCOM H3,n;eJIHH A B BS 6322: qaCTh 1: 1982
pa3MepOB H <l>opMbI ,lI)UI pa3MepOB pe3b6bI M16
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4. 1.29. BS EN ISO 898-1: 1999. MSpKJfPOSKS
38am
3HaKH MapKHpOBKH npHBe,n;eHbI B BS EN ISO 898-1: 1999. Ta6JI. 1.
MapmpOBKa
J
\
bOAmbl U 8UHmbi c lUecmuzpaHHoii ZOA08KOii
BOJITbI H BHHTbI C IllecTHrpaHHoH rOJIOBKOH ,n;OJI)I{HbI MapKHpOBaTbCH 3HaKaMH
KJIacca npOqHOCTH, onHcaHHbIMH B nYHKTe 3.
f' """'" ,,-- ---... ""'"
8.8
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PHC. 1. npHMepbI MapKHpOBKH 60JITOB H BHHTOB
C mecTHrpaHHoH rOJIOBKOH
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apKHpOBKa 06H3aTeJIbHa
Bcex KJIaCCOB npOqHOCTH; npe,n;noq-
THTeJIbHO HaHOCHTb yr.rry6JIeHHbIe
HJIH BbInyKJIbIe 3HaKH MapKHpoBKH
Ha TOpueByID nOBepXHocTb rOJIOBKH
HJIH yr.rry6JIeHHbIe - Ha 60KOByID
lIoBepXHocTb rOJIOBKH (CM. pHC. 1).
apKHpoBKa TPe6yeTcH
60JITOB H BHHTOB C IllecTHrpaHHoH
rOJIOBKOH C HOMHHaJIbHbIM ,n;Ha-
MeTpOM d 5 MM, TaM, r,n;e <popMa
H3,n;eJIIDI 3TO n03BOillIeT, npe,n;noq-
THTeJIbHO Ha rOJIOBKe.
BUHmbl c lUecmUZpaHHbLM YZJly6AeHUeM 8 ZOA08Ke
BHHTbI C IlleCTHrpaHHbIM yr.rry6JIeHHeM B rOJIOBKe ,n;OJI)I{HbI MapKHpOBaTbCH 3Ha-
KaMH KJIaCca npOqHOCTH, onHcaHHbIMH B rryHKTe 3.
MapKHpoBKa 06H3aTeJIbHa ,lJ;IDI KJIaCCOB npOqHOCTH, paBHbIX HJIH 60JIbIllHX 8.8;
npe,nnOQTHTeJIbHO HaHOCHTb yrJIy6JIeHHbIe 3HaKH MapKHpOBKH Ha 60KOByID no-
BepXHOCTb rOJIOBKH HJIH yr.rry6JIeHHbIe HJIH BbInyKJIbIe - Ha TopueByID nOBepXHOCTb
rOJIOBKH (CM. pHC. 2).
I I i I I
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12.9
PHC. 2. llpHMepbI MapKHpOBKH BHHTOB C lUec-
THrpaHHbIM yrJIy6JIeHHeM B rOJIOBKe
apKHpOBKa TPe6yeTcH ,lLIDI
BHHTOB C IllecTHrpaHHbIM yr.rry6JIe-
HHeM B rOJIOBKe C HOMHHaJIbHbIM
,n;HaMeTPoM d 5 MM, r,n;e <popMa H3-
\
,n;eJIIDI 3TO n03BOillIeT, npe,n;nOqTH-
TeJIbHO Ha rOJIOBKe.
,l:LJrn MaJIeHbKHX BHHTOB C Illec-
THrpaHHbIM OTBepCTHeM B rOJIOBKe B
KaqeCTBe aJIbTepHaTHBbI ,n;orrycKaeTcH
HCnOJIb30BaHHe UH<pep6JIaTHOH Map-
KHpOBKH, peKoMeH,l(yeMoH ,lLIDI raeK
B ISO 898- 2.
F/la8a 4. Kpene:JICHble oemOJlu
BS EN ISO 898-1:1999. Ta6J1. 1. MapKHpOBOqHbIe 3HaKH
Krracc npOqHOCTH 3.6 4.6 4.8 5.6 5.8 6.8 8.8 9.8 10.9 12.9
MapKHpOBOqHbIH 3HaK a,b 3.6 4.6 4.8 5.6 5.8 6.8 8.8 9.8 10.9 12.9
a TOqKa B CHMBOJIe MapKHpOBKH MoxeT 6bITb onymeHa.
b Kor,n;a ,lI)UI KJIacca npOqHOCTH 10.9 (CMO'rpH BS EN ISO 898-1: 1999. Ta6JIHI(a 2) HC-
nOJIb3yK>TCH HH3KoyrJIepo,n;HCTbIe MapTeHCHTHbIe CTaJIH, 3HaK 10.9 ,n;OJDKeH 6bITb no,n;qepK-
JIyT: .
BS EN ISO 898-1: 1999. Ta6J1. 2. H,n;eHTH<pHKaI(HOHHbIe 3HaKH )];JIH illnHJIeK
Krracc npOqHOCTH 8.8 9.8 10.9 12.9
H,neHTH<pHKaI(HOHHaH MeTKa 0 + 0
mnUllbKU
IllnHJIbKH ,lI;OJI)I{HbI MapKHpOBaTbCR 3HaKaMH KJIacca npOqHOCTH, onHcaHHbIMH
B nyHKTe 3.
MapKHpOBKa 06R3aTeJIbHa ,lI;IDI KJIaCCOB npOqHO-
CTH, paBHbIX HJIH 60JIbIllHX 8.8; npe,ll;nOqTHTeJIbHO
HaHOCHTb yr.rry6JIeHHbIe 3HaKH MapKHpOBKH Ha TOp-
ue C pe3b60H (CM. pHC. 3). Y IllnHJIeK C HenO,ll;BIDK-
HOH nOC8)];KOH MapKHpOBKa ,lI;OJDKHa pacnOJIaraTbCg
Ha CTIDKHOM (raeqHOM) KOHue.
MapKHpOBKa TPe6yeTcR ,lI;IDI IllnHJIeK C HOMH-
HaJIbHbIM ,lI;HaMeTPOM paBHbIM HJIH 60JIbIllHM 5 MM.
CHMBOJIbI BS EN ISO 898-1: 1999. Ta6JIHua 2
Pac. 3. MapKHpOBKa illnHJIbKH ,lI;onYCTHMbI ,lI;IDI H,lJ;eHTH<pHKaUHH B KaqeCTBe aJIbTep-
HaTHBbI.
J(pyzue munbl 60/lm08 U 8UHm08
CHCTeMa MapKHpOBKH, aHaJIOrHqHa.H onHcaHHOH B pa3,l1;eJIaX «BOJITbI H BHHTbI C
IlleCTHrpaHHOH rOJIOBKOH» H «BHHTbI C IlleCTHrpaHHbIM yr.rry6JIeHHeeM B rOJIOBKe»,
,lI;OJDKHa HCnOJIb30BaTbCR ,lI;IDI ,lI;pyrHX THnOB 60JITOB H BHHTOB KJIaCCOB npOqHOCTH
4.6, 5.6 H Bcex KJIaCCOB, paBHbIX HJIH 60JIbIllHX 8.8, KaK onHcaHO B COOTBeTCTByro-
m;HX Me)l{J(YHapO,ll;HbIX CTaH)l;apTaX (ISO) HJIH ,lI;IDI CneUHaJIbHbIX KOMnOHeHTOB no
COrJIaCOBaHHIO Me)l{J(Y 3aHHTepeCOBaHHbIMH CTopOHaMH.
MapKHpoBKa JIeBOH pe3L6LI
BOJITbI H BHHTbI C JIeBOH pe3b60H ,lI;OJDKHbI MapKHpOBaTbCR CHMBOJIOM, nOKa3aH-
HbIM Ha pHC. 4, HaHeceHHbIM Ha TopueByIO nOBepXHOCTb rOJIOBKH HJIH KOHua.
MapKHpoBKa TPe6yeTcR ,lLIDI 60JITOB H BHHTOB C HOMHHaJIbHbIM ,lI;HaMeTPOM pe3b-
6bI d 5 MM.
,[{JIR 60JITOB H BHHTOB C IlleCTHrpaHHOH rOJIOBKOH ,lI;orrycKaeTC.H HCnOJIb30BaHHe
aJIbTepHaTHBHOH MapKHpOBKH JIeBOH pe3b6bI, KaK nOKa3aHO Ha pHC. 5.
4.1. BUHmo8ble Kpene:JICHble U30e.tlUR
k
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J
PHC. 4. MapKHpoBKa JIeBOH pe3b6bI Puc. 5. A.n:bTepHaTHBHaH MapKHpOBKa JIeBOH pe3b6bI:
S - illHpHHa Me)K)J;Y napaJIJIeJIbHbIMH rpaHHMH;.
k - BbICOTa rOJIOBKH
AJILTepHaTHBHaJiMapKHpOBKa
HaHeceHHe aJIbTepHaTHBHoH HJIH He06R3aTeJIbHOH ,lI;OrryCTHMOH MapKHpoBKH,
KaK 3a.HBJIeHO B pa3,l1;eJIaX «3HaKH», «MapKHpoBKa» H «0603HaqeHHe JIeBOH pe3b-
6bl», MoxeT 6bITb OCTaBJIeHO Ha YCMoTPeHHe H3rOTOBHTeillI.
ToprOBaJi (H,ZJ;eHTH4JHKaD;HJI) MapKHpOBKa
HaHeceHHe KJIeHMa (ToBapHoro 3HaKa H3foToBHTeillI) RBillIeTCR 06R3aTeJIbHbIM
,lUUI Bcex npo,[(yKToB, MapKHpoBaHHbIx KJIaCCOM npOqHOcTH.
4.1.30. BS EN 20898-2: 1994. MeXSHJflfeCKJfe CSOHCTSS
KAeneJICH&IX Jf3AenJfH: rs C YCTSHOSneHH&IMJf
'aHslfeHJfflMJf npo6HOH HS 3KJf - KpynHsfI pe3&6s
BS EN 20898-2:1994. Ta6J1. 1. MeXaHHqeCKHe CBOHCTBa
KJIacc np01JHOCTH
04 05
Pe3hoa
)= TBep ocTh T8ep OCTh )= TBep oCTh
e e
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Ii! Kepcy HV i! Kepcy HV Kepcy HV
S S
rI1 HC- rI1 HC- rI1 HC-
I COCTO- nOJl- I COCTO- nOJl- I COCTO- nOJl-
OT o = MHH. MaKC. JIHIIe He- = MHH. MaKe. JIHIIe He- = MHH. MaKe. JIHIIe He-
r! r! r! \
HHe HHe HUe
== == ==
- M4
M4 M7
M7 MI0 6e3 TOH- 353 3/0 b TOH- - - - - -
380 188 302 3/0 a KaH 500 272 KaH
M10 M16
M16 M39 510 117 302 6e3 1
3/0 a
F/la8a 4. Kpene:JICHble oemOJlu
llpoiJollJICeHUe
KJIacc np01JHOCTH
Pe3hoa 5 c 6
HanpJDKeHHe TBep oCTh no faiiKa HanpJDKeHHe TBep ocTh no faiiu
BHKKepcy HV BHKKepcy HV
OT npooHoii OT npooHoii
HarpY3KH Sp COCTO- HCnOJl- HarpY3KH Sp COCTO- HCnOJl-
OT o (H/MM2) MHH. MaKC. (H/MM2) MHH. MaKe.
JIHIIe HeHHe JIHIIe HeHHe
- M4 520 600
M4 M7 580 670
130 6e3 150 6e3
M7 MI0 590 302 3/0 a 1 680 302 3/0 a 1
MI0 M16 610 700
M16 M38 630 146 720 170
KJIacc npo1JHOCTH
Pe3hoa 8
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BHKKepcy HV BHKKepcy HV
OT npooHoii OT npooHoii
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OT o (H/MM2) MHH. MaKe. (H/MM2) MHH. MaKe.
JIHIIe HeHHe JIHIIe HeHHe
- M4 800 180
M4 M7 855 6e3
302 3/0 a - - - - -
M7 MI0 870 200 1
MI0 M16 880
M16 M38 920 233 353 3/0 b 990 180 302 6e3 2
3/0 a
KJIacc np01JHOCTH
Pe3hoa 9 10
HanpJDKeHHe TBep oCTh no faiiu HanpJDKeHHe T8ep OCTh no faHKa
BHKKepcy HV BHKKepcy HV
OT npooHoii OT npooHoii
Harp)'3KH Sp COCTO- HCnOJl- Harpy:JKH Sp COCTO- HcnWl-
OT (H/MM2) MaKe. (H/MM2) MHH. MaKe.
O MHH. JIHIIe HeHHe JIHIIe HeHHe
- M4 900 170 1040
M4 M7 915 1040
M7 MI0 940 302 6e3 2 1040 272 353 3/0 b 1
188 3/0 a
MI0 M16 950 1050
M16 M38 920 1060
4.1. BUHmo6ble "peneJICHble U30eJlUR 20 I
OKOHltaHUe
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Pe3b6a 12
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OT DpoOHoii OT Jt'6HOi
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JlHHe HeHHe BIllIe HeHHe
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M4 M7 1140 1150
295 353 3/0 b 1
M7 MI0 1140 1160 272 353 3/0 b 2
M10 M16 1170 1 190
M16 M38 - - - - - 1200
a Be3 30 - 6e3 3aKaJIKH H oTnycKa.
b 30 - 3aKaJIeHHble H oTnyrn;eHHble.
C MaKCHMaJIbHaSl TBep,n;OCTb 60JITOB KJlaCCOB npOQHOCTH 5.6 H 5.8 6y,neT H3MeHeHa Ha
3HaQeHHe 220 HV B CJIe,nyroIUeH pe,naKI(HH ISO 896-1: 1988. 3TO - MaKCHMaJIbHCUI TBep-
,l(OCTb 60JITa B 06JIaCTSlX 3aI(enJIeHHH pe3b6bI, Tor,na KaK TOJIbKO KOHIJ.bI pe3b6bI HJIH rOJIOBKa
MOryr HMeTb MaKCHMaJIbHyro TBep,nOCTb 250 HV. n03TOMY BeJIHQHHbI HanpH)l{eHHH OT np06-
HOB HarpY3KH OCHOBaHbI Ha MaKCHMaJIbHOH TBep,nOCTH 60JITa 220HV.
llpUMe'taHUe. "\
MHHHMaJIbHble 3Ha IJ:jlSI TBep,nOCTH YCTaHOBJIeHbI TOJIbKO J:wSl TepMHqeCKH 06pa6oTaH-
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):(Jrn Bcex OCTaJIbHbIX raeK MHHHMaJIbHCUI TBep,nOCTb He 06S13aTeJIbHa, a npHBe,neHa npOCTO
,l(IDI cnpaBKH.
):Lrrn raeK 6e3 3aKaJIKH H OTnYCKa, BbI,LJ;epXaBllIHX HCITbITaHHe np06HOH HarpY3KOH, MH-
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BeJIIf1IHHLI npo6HOH HarpY3KH
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\
BeJIMqMHbI lIp06HO:H HarpY3KH lIpMBe,ll;eHbI B BS EN 20898-2: 1994. Ta6JI. 2.
HOMMHaJIbHa5I IIJIOrn;anb CeqeHH.H lIO pe3b6e As BbJt:lMClliIeTCH lIO <popMyJIe:
2
A = 1t ( d 2 + d 3 )
s 4 2 '
f,ne d 2 - HOMMHaJIbHbI:H Cpe,ll;HM:H ,lI;MaMeTp HapyxHO:H pe3b6bI (CMOTpM ISO 724);
d 3 - BHYTpeHHM:H ,lI;MaMeTP HapyxHO:H pe3b6bI lIO ,lI;HY BlIanMHbI:
H
d 3 = d l - 6'
f,ne d 1 - BHYTpeHHM:H ,lI;MaMeTP HapyxHO:H pe3b6bI; H - BbICOTa MCXO,ll;HOrO TPe-
yrOJIbHMKa pe3b6bI.
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4.1. BUHmo6ble "peneJICHble U30eJlUR 203
BS EN 20898-2:1994. TaoJl. 3. 3HaKH MapKHpOBKH )].JHI raeK C KJIaCCaMH npOqHOCTH
B COOTBeTCTBHH C BS EN 20898-2 nYHKT 3.1
KJIacc npo1fHOCTH
= JII060e 0603HaqeHHe CHMBOJIa
==
!
&1 i H.rrH 3HaK MapKHpOBKH, RJIH KO,nO-
f c=.. BbIH 3HaK (I( H <pep6JIaTHCUI CHCTeMa
I MapKHpoBKH)
4
4
5
5
6
6
8
8
KJIacc Dp01fHOCTH
= JII060e 0603HaQeHHe CHMBOJIa
==
!
&1 i H.rrH 3HaK MapKHpOBKH, HJIH KO,nO-
c=.. BbIH 3HaK (QH<pep6JIaTHCUI CHCTeMa
I MapKHpoBKH)
9
9
10
10
12 8
12
a TOQeqHaH MapKHpOBKa He MO>KeT 6bITb 3aMeHeHa MapKHpOBKOH H3rOTOBHTeJISI.
BS EN 20898-2:1994 TaoJl. 4. 3HaKH MapKHpOBKH )J)ISI raeK C KJIaCCaMH npOQHOCTH
B COOTBeTCTBHH C BS EN 20898-2 nyHKT 3.2
KJIacc Dp01fHOCTH
04
05
)
MapKHpOBKa
MapmpOBKa JIeBOH pe3LoLI
faMKH C JIeBOM pe3b60M ,nOJDKHbI MapKHpoBaTbcH, KaK lIOKa3aHO Ha pMC. 1, yr-
JIy6JIeHHbIM 3HaKOM Ha OlIOpHOH IIOBepXHOCTM raMKH.
MapKHpoBKa TPe6yeTC.H ,lLJUI raeK C pe3b60M M5. /
,lJ;orrycKaeTcH TaIOKe MClIOJIb30BaHMe aJIbTepHaTMBHoM MapKHpOBKH JIeBOM pe3b-
6bI, KaK lIOKa3aHO Ha pRC. 2.
CI)
1\
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pe3b6bI. s - paCCTOSlHHe Me:>K.LJ:Y rrapaJIJIeJIb-
HbIMH rpaHSlMH
F/la8a 4. Kpene:JICHble oemOJlu
AHLTepHaTHBHaJi MapKHpoBKa
HaHeceHMe aJIbTepHaTMBHoH IDIM He06H3aTeJIbHOH ,nOrryCTMMOH MapKHpoBKH,
KaK 3aHBJIeHO B pa3,neJIaX «3HaKH», «MapKHpOBKa» M «0603HaqeHHe JIeBOH pe3b-
6bI», MoxeT 6bITb OCTaBJIeHO Ha YCMoTPeHMe M3rOTOBMTelliI.
ToprOBLIe (If)J;eHTH HKanaoHHLIe) 0603Ha'leHIUI
HaHeceHMe KJIeHMa (ToBapHoro 3HaKa M3roToBMTelliI) HBlliIeTCH 06H3aTeJIbHbIM
,LUISI Bcex lIpO,nyJ(TOB, MapKHpOBaHHbIX KJIaCCOM lIPOqHOCTM, npM HaJIWIMM TeXHlflIe-
CKOH B03MO)I(HOCTH. YnaKOBKH, TeM He MeHee, ,nOJDKHbI 6bITb npOMapKHpOBaHbI B
JII060M CJIyqae.
4.1.31. BS EN ISO 898-6: 1996. MeXSHJflfeCKJfe CSOHCTsa
KpeneJICH&IX Jf3AenJfH: rSHKJf C YCTSHOSneHH&IMJf
3HSlfeHJfflMJf npo6HoH HsrPY3KJf - MenKSfI pe3&6a
BS EN ISO 898-6:1996. TaoJl. 1. CHCTeMa 0603HaQeHHH ,l(JISI raeK C HOMHHaJIbHOH
BbICOTOH 0,8 d
faiKII
KJIacc CODpSlfaeMhIH 60JIT
Dp01fHOCTH HCDOJIHeHHe 1 HCDOJIHeHHe 2
raiiKH
KJIacc 1Ip01JH0CI1I ,lI;HaD83oH HOMHHaJIbHhIX aMeTp08 pe3bOhl (MM)
5 3.6, 4.6, 4.8 d 39 d39 -
5.6, 5.8
6 6.8 d 39 d 39 -
8 8.8 d 39 d 39 d 16
10 10.9 d 39 d 16 d 39
12 12.9 d 16 - d 16
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KH 60JIee HH3Koro KJIacca npOqHOcTH. TaKaH 3aMeHa peKOMeH,nyeTcH ,lLJUI coe,nMHe-
HMH 60JIT/raHKa, HalIpIDKeHMe B KOTOpbIX 6y,neT BbIme, qeM H:IDKHMH npe,neJI TeKY-
qeCTH HJIM HanpIDKeHMe 60JITa OT lIp06HOH HarpY3KM.
ECJIM xe coe,nMHeHMe 6y,neT 3aTHHyro BbIme lIp06HOH HarpY3KH 60JITa, TO KOH-
CTpyKUMH raHKH ,nOJDKHa 6bITb paCCqMTaHa TaK, qT06bI B pe3YJIbTaTe nepe3aTIDKKH
06eClIeqMTb no KpaHHeH Mepe 10 % nOBpe)l(,[{eHMH no CTep)I(H1O 60JITa ,lLJUI lIpe,ay-
npe)l(,[{eHHSI nOTpe6MTeJISI 0 HenpaBHJIbHOH YCTaHOBKe Kpene)I(HOro coe,nMHeHMSI.
IIpuMettaHue.
BOJIee nO,lI,p06HYIO MH<p0pMaUMIO 0 ,norrycKaeMbIX HarpY3Kax B pe3b60BbIX coe-
,nMHeHMSlX, a TaIOKe 0 BapMaHTax MClIOJIHeHIDI raeK CMOTpM B ISO 898-2: 1992,
TIpHJIOXeHMe A.
4.1. BUHmo8ble ICpene:JICHble U30e.tlUJI
faiiKH C HOMHHaJILHOH BLICoroH O,5d H < O,8d (a4KIJeKTHBHaJI AJIHHa pe3LOLI O,4d
H <O,6d)
faHKH C HOMMPIaJIbHOH BbICOTOH O,5d M <O,8d (3<p<peKTMBHaH ,[(JIMHa pe3b-
6bI O,4d M <O,6d) 0603HaqalOTCH KOM6MHaUMeH ,nBYX UM<PP: BTopaH YKa3bIBaeT
HOMMHaJIbHOe HalIp.HXeHMe OT lIp06HOH HarpY3KM B 3aKaJIeHHOH MClIbITaTeJIbHOM
onpaBKe, B TO BpeMH KaK lIepBaH YKa3bIBaeT Ha TO, qTO HarpY30QHaH ClIoc06HOCTb
COe,[(MHeHMH raHKM C 60JITOM HIDKe, qeM C 3aKaJIeHHOH OlIpaBKOH, M HIDKe, qeM Y
COe,n:MHeHMH, OlIMCaHHbIX B pa3,neJIe 3.1. <l>aKTMqeCKaH HeCYIIJ;aH clIoc06HOCTb
raRKH orrpe,l(eJISleTCSI He TOJIbKO TBep,lI,OCTblO raH -lI 3<p<peKTMBHOH ,[(JIMHOH pe3b-
6bI, HO TaIOKe lIPOqHOCTblO Ha paCT.HXeHHe 60JITa, C KOTOpbIM COlIpHraeTCH raHKa.
Ta6JIMue 2 BS EN ISO 898-6: 1996 lIpMBe,neHa CMCTeMa 0603HaqeHMH M HalIpSl-
:>KeHIDI OT lIp06HOH HarpY3KM. TIp06HbIe HarpY3KM ,naHbI B BS EN ISO 898-6:
1996 Ta6JIMua 3. B BS EN ISO 898-6: 1996 Ta6JIHua 4 lIpMBe,neHbI ,lI,illI ClIpaBOK
MHHMMaJIbHbIe pa3pymalOIIJ;Me (lIpMBo,nHrn;Me K CPbIBY pe3b6bI) HarpY3KH ,[(JISI coe-
,[(HHeHMH raeK C 60..1TaMH pa3JIMQHbIX KJIaCCOB lIPOQHOCTM.
BS EN ISO 898-6:1996. TaoJl. 2. CHcTeMa 0603HaqeHHH H HanpH)l{eHHSI OT npo6-
HOH HarpY3KH ,lI;JIH raeK C HOMHHaJIbHOH BbICOTOH 0,5d H < 0,8d
KJIacc HOMHHaJIbHOe eHHe AKTyJlbHOe aBJIeHHe
CBOHCTB raiiKH DO TeCTOBOH HarpY3KoH (H/MM 2 ) DO TeCTOBOH HarpY3KoH (H/MM 2 )
04 -, 400 380
05 U 500 500
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4.1. BUHmo8ble Kpene:JICHble U30e.tlUR
BS EN ISO 898-6:1996 TaoJl. 4. MHHHMaJIbHbIe pa3pymaIOm;He (npHBo,nSlIUHe K
CPbIBY pe3b6bI) HarpY3KH,lI;JIH coe,n;HHeHHH raeK C 60JITaMH B % np06-
HOH HarpY3KH 60JITOB
KJIacc MHHHMaJlbHOe HanpSDKeHHe B CTepmte 60JITa DpH cphme pe3bOW,
% DpofiHOH HarpY3KH 6oJlTOB KJlaCCOB Dp01fHOCTH
Dp01fllOCTH raeK 6,8 8,8 10,9 12,9
04 85 65 45 40
05 100 85 60 50
MaTepHaJILI
faHKH KJIaCCOB lIPOqHOCTM 05, 8 (MclIOJIHeHMe 1), 10 H 12 CJIe,nyeT lIo,nBepraTb
3aKaJIKe M OTrryCKY.
faHKH CJIe,nyeT M3rOTOBlliITb M3 CTaJIeH, XHMMqeCKHH COCTaB KOTOpbIX COOTBeT-
cTByeT BS EN ISO 898-6: 1996 Ta6JIMua 5. XMMMqeCKHH COCTaB He06xo,nMMO aHa-
JIH3MpOBaTb B COOTBeTCTBMM C cymecTByroIIJ;HMM Me)l()];YHapo,nHbIMM CTaH,napTaMM.
BS EN ISO 898-6:1996. TaoJl. 5. XHMHqeCKHH COCTaB CTaJIeH
KJIacc npe eJIbHOe CO ep]K3HHe 3J1eMeHTOB (KOIITpOJlhHWH aH8JlH3) %
Dp01fHOCTH C MaKC. Mn MHH. P MaKC. S MaKC.
5 a : 6 - 0 50 - o 060 O 150
8 b 04 a 0..58 0..25 0..060 0..150
lOb 05 b 0..58 0..30 0..048 0..058
12 b - 0..58 0..45 0..048 0..058
a faHKH 3THX KJIaCCOB npOqHOCTH ,nonycKaeTcSI H3rOTaBJIHBaTb H3 aBTOMaTHOH CTaJIH,
eCJIH HeT HHoro COrJIallIeHHSI Mex.ny nOTpe6HTeJIeM H H3rOTOBHTeJIeM. B TaKHX CJIyqaSlx ,n;o-
nycKaeTcH CJIe,l(yIOm;ee MaKCHMaJIbHOe co,nepxaHHe cepbI, <poc<popa H CBHHI(a: cepa 0,34 %;
4>OC<pOP 0,11 %; CBHHeQ 0,35 %.
b TIpH He06xo,nHMOCTH YJIyqmeHHH MeXaHHqeCKHX CBOHCTB raeK B03MO)I(H0 npHMeHeHHe
CTaJIH C ,n06aBKaMH JIerHpyrom;HX 3JIeMeHTOB.
MeXaHHlJeCKHe CBOHCTBa
TIpM lIpOBepKe MeTo,naMM, OIIMcaHHbIMM B rryHKTe 8, MeXaHMqeCKHe CBOHCTBa
raeK ,nOJDKHbI COOTBeTCTBOBaTb BS EN ISO 898-6: 1996 Ta6JIMua 1 C yqeToM pa3,ne-
JIa 4.1.30.
BeJIHlJHHLI npOOHOH HarpY3KH
BeJIMqMHbI lIp06HOH HarpY3KH lIpMBe,neHbI B BS EN 898-6: 1996. Ta6JI. 3.
HOMMHaJIbHaH IIJIOII(Mb CeqeHMH lIO pe3b6e As BbIqMClliIeTCH lIO <pOpMYJIe:
2
A = 1t ( d 2 + d 3 )
s 4 2 '
r,ne d 2 - HOMMHaJIbHbIM cpe,nHMH ,nMaMeTP Hap)')KHOH pe3b6bI (CMOTpM ISO 724);
d 3 - BHyrpeHHMH ,nMaMeTP Hap)')KHOH pe3b6bI lIO ,nHY BlIMMHbI:
H
d 3 = d} --,
6
r,ne d l - BHYTpeHHMH ,nMaMeTp Hap)')KHOH pe3b6bI; H - BbICOTa Mcxo,nHero Tpe-
yrOJIbHMKa pe3b6bI.
F/la8a 4. Kpene:JICHble oemOJlu
4.1.32. BS EN 20898-7: 1995. MeXSHJflfeCKJfe CSOHCTSS
KpeneJICH&IX Jf3AenJfH: JfCn&ITSHJfe HS KPYTflUCJfH
MOMeHT Jf MJfHJfMSn&H&le KPYTflUCJfe MOMeHT&1
Anfl 6onTos Jf SJfHTOS C HOMJfHSn&H&IMJf
AJfSMeTpSMJf 1-10 MM
BS EN 20898-7:1995. TaoJl. 1. K03<P<PHI(HeHT 3anaca npOqHOCTH X
Krracc 3anaca npOQHOCTH 8,8 9,8 10,9 12,9
K03<P<PHUHeHT X 0,84 0,815 0,79 0,75
BS EN 20898-7:1995. TaoJl. 2. MHHHMaJIbHbIH pa3pymaIOIUHH KpyrSlIUHH MOMeHT
MlmHMaJlhllhlii p83pymaIOlIUIii KPYTJIIIUIH MOMe...ra M B miD (11M)
Pe3boa mar (MM) KJIacc Dp01fHOCTH
8,8 9,8 10,9 12,9
M1 0,25 0,033 0,036 0,040 0,045
M1,2 0,25 0,075 0,082 0,092 0,10
M1,4 0,3 0,12 0,13 0,14 0,16
M1,6 0,35 0,16 0,18 0,20 0,22
M2 0,4 0,37 0,40 0,45 0,50
M2,5 0,45 0,82 0,90 1,0 1,1
M3 0,5 1,5 1,7 1,9 2,1
M3,5 0,6 2,4 2,7 3,0 3,3
M4 0,7 3,6 3,9 4,4 4,9
M5 0,8 7,6 8,3 9,3 10
M6 1 13 14 16 17
M7 1 23 25 28 31
M8 1,25 33 36 40 44
M8xl - 38 42 46 52
M10 1,5 66 72 81 90
M 10xl - 84 92 102 114
M10x1,25 - 75 82 91 102
a 3TH MHHHMaJIbHbIe 3HaQeHHSI pa3pymaIOIUHx KpyrSlIUHX MOMeHTOB OTHOCSlTCSI K 60JI-
TaM H BHHTaM C ,nonYCKaMH pe3b6bI 6g, 6f H 6e.
4. 1.33. BS EN ISO 4762: 2004. MeTpJflfeCKHH SJfHT
C qJfnHHAPJflfeCKOH ronosKOH Jf WeCTJfrpSH&IM
yrny6neHJfeM nOA KnIOIf
ISO 10683, KpeIIe)I(HbIe M3,neJIIDI - He3JIe:KTpOJIMTMQeCKHe UMHKOBbIe IIOKpbITIDI.
ISO.23429, KaJIM6pOBKa meCTMrpaHHbIX yrJIy6JIeHMH.
CMOTpM pMC. 1 M BS EN ISO 4762: 2004. Ta6JI. 1.
4.1. BUHmo8ble "pene:JICHble U30e.nUR
Pa3MepLI
IIpUMettaHue. CMMBOJIbI M 0603HaQeHHH pa3MepoB ,naHbI B ISO 225.
t
b
C\I
'r'"
AI
x
(1 )
1J
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4
CD
'T!
CI) -e-
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U(3)
Is
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(b)
J
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)
orrYCTHMaH aJIbTepHaTHBHaH <pOpMa yrJIy6JIeHHH
t
BepXHHH H HIDKHHH KPaH rOJIOBKH
o
o
C\I
AI
ri = t r tl
(5)
V
612
An nOJ1Y"leHHbIX paaBepTKO yrJ1y6J1eHVI
Ha 60J1bWerO npeAellbHOrO paaMepa aapea,
nOJlY"l BWVlVlC51 npVl CSepJ1eHVlVI, He AOJ1)1(eH
npeBblwaTb 1/3 AJ1V1Hbl J1t060 rpaHVI yrJ1y6-
neH , KOTOpa51 paBHa e/2.
( 612)/3
Puc. 1.
(1) ,lJ;OrryCTHMO He3HaQHTeJIbHOe cKpyrJIeHHe HJIH 3eHKOB Bxo,!:{a yrJIy6JIe-
HIDI.
(2) KOHeu ,nOJDKeH 6bITb C <paCKO:H HJIM ,lLJUI pe3b6 M4 MO)l(eT 6bITb 6e3 <pac-
KIf B COOTBeTCTBMM C ISO 4753.
(3) HeIIOJIHa5I pe3b6a u 2P.
(4) d s MCIIOJIb3yeTcH, eCJIM 3HaQeHHH Is min OIIpe,neJIeHbI.
,
(5) BepXHM:H KpaH rOJIOBKH MO)l(eT 6bITb cKpyrJIeH HJIM MMeTb <paCKY, KaK IIOKa-
3aHO, no BbI60py M3rOTOBMTelliI.
(6) HIDKHM:H Kpa:H rOJIOBKH MO)l(eT 6bITb cKpyrJIeHHbIM IDIM CKomeHHbIM ,no d w ,
HO B JII060M c.rryqae He ,nOJDKHO 6bITb 3ayceHueB.
MaKCMMaJIbHOe yrOJIIIJ;eHMe IIO,n rOJIOBKO:H BMHTa:
If,max =1,7r max '
da,max - ds,max
r nlax =
2
'min CM01'pH BS EN ISO 4762: 2004. Ta6JI. 1, pMC. 1.
F/la8a 4. Kpene:JICHble oemOJlu
BS EN ISO 4762: 2004. TaoJl. 1. Pa3MepbI B MM
P83Mep pe3hOW, d MI,6 M2 M2,5 M3 M4 M5 M6 M8 MIO MI2
pa 0,35 0,4 0,45 0,5 0,7 0,8 1 1,25 1,5 1,75
bb 15 16 17 18 20 22 24 28 32 36
MaKCC 3,00 3,80 4,50 5,50 7,00 8,50 10,00 13,00 16,00 18,00
dk MaKcd 3,14 3,98 4,68 5,68 7,22 8,72 10,22 13,27 16,27 18,27
MHH. 2,86 3,62 4,32 5,32 6,78 8,28 9,78 12,73 15,73 17,73
d a MaKC. 2 2,6 3,1 3,6 4,7 5,7 6,8 9,2 11,2 13,7
d s MaKC. 1,60 2,00 2,50 3,00 4,00 5,00 6,00 8,00 10,00 12,00
MHH. 1,46 1,86 2,36 2,86 3,82 4,82 5,82 7,78 9,78 11,73
ee,f MHH. 1,733 1,733 2,303 2,873 3,443 4,583 5,723 6,863 9, 149 11,429
If MaKC. 0,34 0,51 0,51 0,51 0,6 0,6 0,68 1,02 1,02 1,45
k MaKC. 1,60 2,00 2,50 3,00 4,00 5,00 6,0 8,00 10,00 12,00
MHH. 1,46 1,86 2,36 2,86 3,82 4,82 5,7 7,64 9,64 11,57
r MHH. 0,1 0,1 0,1 0,1 0,2 0,2 0,25 0,4 0,4 0,6
HOM. 1,5 1,5 2 2,5 3 4 5 6 8 10
sf MaKC. 1,58 1,58 2,08 2,58 3,08 4,095 5,14 6,14 8,175 10,175
MHH. 1,52 1,52 2,02 2,52 3,02 4,020 5,02 6,02 8,025 10,025
t MHH. 0,7 1 1,1 1,3 2 2,5 3 4 5 6
v MaKC. 0,16 0,2 0,25 0,3 0,4 0,5 0,6 0,8 1 1,2
d w MHH. 2,72 3,48 4,18 5,07 6,53 8,03 9,38 12,33 15,33 17,23
w MHH. 0,55 0,55 0,85 1,15 1,4 1,9 2,3 3,3 4 4,8
Ig Aruma CTep1KllJl/ s H ,lI,JIHHa rJla,nKoii 1faCTH CTep1KllJl Ig
HOM. MUH. MaKC. - - - - - - - - - -
_fIJ J -1 J J J J J J J -1 J J J fIJ "..,p. fIJ J J J
- -
2,5 2,3 2,7
3 2,8 3,2
4 3,76 4,24
5 4,76 5,24
6 5,76 6,24
8 7,71 8,29
10 9,71 10,29
12 11,65 12,35
16 15,65 16,35
20 19,58 20,42 2 4
25 24,58 25,42 5,75 8 4,5 7
30 29,58 30,42 9,5 12 6,5 10 4 8
35 34,5 35,5 11,5 15 9 13 6 11
40 39,5 40,5 16,5 20 14 18 11 16 5,75 12
45 44,5 45,5 19 23 16 21 10,75 17 5,5 13
50 49,5 50,5 24 28 21 26 15,75 22 10,5 18
55 54,4 55,6 26 31 20,75 27 15,5 23 10,25 19
60 59,4 60,6 31 36 25,75 32 20,5 28 15,25 24
65 64,4 65,6 30,75 37 25,5 33 20,25 29
4.1. BUHmo6ble "peneJICHble U3iJeJlUR 211'
llpoiJollxeHue
P83Mep pe3b6w, d MI,6 I M2 I M2,S I M3 I M4 I MS M6 I M8 I MIO I MI2
Ig ,lI;JIHHa CTep1KllJl/ s H .zuoma rJla,nKoA 1faCTH CTep1KllJlI g
HOM. MHH. MaKe. - - - - - - - - - -
J J -1 ""., J ...I J ...I J J J J J J. J ,.,JI. J J. J "",p.
70 69,4 70,6 35,75 42 30,5 38 25,25 34
80 79,4 80,6 45,75 52 40,5 48 35,25 44
90 89,3 90,7 -- 50,5 58 45,25 54
100 99,3 100,7 ,. 60,5 68 55,25 64
110 109,3 110,7 65,25 74
120 119,3 120,7 75,25 84
130 129,2 130,8
140 139,2 140,8
150 149,2 150,8
160 159,2 160,8
180 179,2 180,8
200 199,075 200,925
220 219,075 220,925
240 239,075 240,925
260 258,95 261,05
280 278,95 281,05
300 298,95 301,05
P83Mep pe3b6bI, d (MI4)h MI6 M20 M24 M30 M36 M42 M48 MS6 M64
PI 2 2 2,5 3 3,5 4 4,5 5 5,5 6
bb CCbillKa 40 44 52 60 72 84 96 108 124 140
MaKCC 21 ,00 24,00 30,00 36,00 45,00 54,00 63,00 72,00 84, 3,6,00
/
dk MaKcd 21,33 24,33 30,33 36,39 45,39 54,46 63,46 72,46 84,54 96,54
MHH. 20,67 23,67 29,67 35,61 44,61 53,54 62,54 71 ,54 83,46 95,46
d a MaKC. 15,7 17,7 22,4 26,4 33,4 39,4 45,6 52,6 63 71
d s MaKC. 14,00 16,00 20,00 24,00 30,00 36,00 42,00 48,00 56,00 64,00
13,73 15,73 19,67 23,67 29,67 35,61 41,61 47,61 55,54 63,54
MHH.
ee,f MHH. 13,716 15,996 19,437 21,734 25,154 30,854 36,571 41,131 46,831 52,531
If MaKC. 1,45 1,45 2,04 2,04 2,89 2,89 3,06 3,91 5,95 5,95
k MaKC. 14,00 16,00 20,00 24,00 30,00 36,00 42,00 48,00 56,00 64,00
MHH. 13,57 15,57 19,48 23,48 29,48 35,38 41,38 47,38 55,26 63,26
r MHH. 0,6 0,6 0,8 0,8 1 1 1,2 1,6 2 2
HOM. 12 14 17 19 22 27 32 36 41 46
sf MaKC. 12,212 14,212 17,23 19,275 22,275 27,275 32,33 36,33 41,33 46,33
MHH. 12,032 14,032 17,05 19,065 22,065 27,065 32,08 36,08 41,08 46,08
t MHH. 7 8 10 12 15,5 19 24 28 34 38
v MaKC. 1,4 1,6 2 2,4 3 3,6 4,2 4,8 5,6 6,4
d w MHH. 20, 17 23,17 28,87 34,81 43,61 52,54 61,34 70,34 82,26 94,26
w MHH. 5,8 6,8 8,6 10,4 13,1 15,3 16,3 17,5 19 22
F/la8a 4. Kpene:JICHble oemOJlu
O"OHltaHUe
P83Mep pe3b6w, d (M14)h M16 M20 M24 M30 M36 M42 M48 M56 M64
Ig ,lI;JntHa XBOCTOBHKa Is H .zuoma 3aXBaTa Ig
HOM. MMH. MaKe. - - - - - - - - - -
J J J J J ,..",. J ,..",. J ,..",. J J. J J J J J J J ,..II-
2..5 2..3 2..7
3 2..8 3..2
4 3,76 4,24
5 4,76 5,24
6 5..76 6..24
8 7..71 8..29
10 9,71 10,29
12 11,65 12,35
16 15,65 16,35
20 19..58 20..42
25 24,58 25..42
30 29,58 30,42
35 34,5 35,5
40 39..5 40..5
45 44,5 45,5
50 49,5 50,5
55 54,4 55,6
60 59..4 60..6 10 20
65 64.4 65..6 15 25 11 21
70 69,4 70,6 20 30 16 26
80 79,4 80,6 30 40 26 36 155 28
90 89,3 90,7 40 50 36 46 255 38 15 30
100 99..3 100.7 50 60 46 56 35..5 48 25 40
110 109.3 110..7 60 70 56 66 45..5 58 35 50 .5 38
120 119,3 120,7 70 80 66 76 55.5 68 45 60 30..5 48 16 36
130 129,2 130,8 80 90 76 86 65,5 78 55 70 40,5 58 26 46
140 139,2 140,8 90 100 86 96 75,5 88 65 80 50,5 68 36 56 215 44
150 149.2 150..8 96 106 855 98 75 90 ffi..5 78 46 66 31.5 54
160 159,2 160,8 106 116 95,5 108 85 100 70.5 88 56 76 41,5 64 27 52
180 179,2 180,8 1155 128 105 120 ,5 108 76 96 61,5 84 47 72 28,5 56
200 199,075 200,925 1355 148 125 140 110..5 128 96 116 81,5 104 67 92 48,5 76 30 60
220 219.075 220.925 101,5 124 87 112 68,5 96 50 80
240 239..075 240,925 121.5 155 107 132 88..5 116 70 100
260 258,95 261,05 141.5 164 127 152 108.5 136 90 120
280 278.,95 281,05 161,5 184 147 172 128..5 156 110 140
300 298.95 301,05 181,5 204 167 192 148,5 176 130 160
a P - mar pe3b6bI. b ,lJ.JHI ,lLJIHH Mex.ny )l(HpHbIMH cTYIIeHqaTbIMH JIHHH$lMH B He3aTeHeH-
HOH 06JIaCTH. C ,lLrrn rJIa,nKHX rOJIOBOK. d ,lJ..rrH rOJIOBOK C pH<pJIeHHeM. e e min = 1,14 smin
f npOBe,L(eHHe COBMeCTHOH KaJIH6pOBKH pa3MepOB yrJIy6JIeHH$I e H S CM. B ISO 23429.
g ,lJ..rrHHbI H3,n;eJIHH CTaH,ZJ;apTHOro HCnOJIHeHH$I pacnOJIaraIOTC$I Me)l(,l(y )l(HpHbIMH CTYTIeH-
qaTbIMH JIHHH$lMH. Pe3b6a Ha H3,n;eJIIDIX, ,lLJIHHbI KOTOpbIX pacnOJIaraIOTC$I B 3aTeHeHHOH 06-
JIaCTH, ,n;OXO,n;HT ,n;o rOJIOBKH B npe,n;eJIax 3P. ,lJ..rrH ,lLJIHH, pacnOJIOXeHHbIX HHXe 3aTeHeHHoH
06JIaCTH, Ig H Is npHHHMaIOT 3HaqeHHH B COOTBeTCTBHH co CJIe,n;yIOI.QHMH <popMYJIaMH:
I g max = I HOM - b; Is nun = I g max - 5P.
, "
h Pa3MepoB B cK06Kax CJIe,n;yeT no B03MO)l(HOCTH H36eraTb.
4.1. BuHmo8ble "pene:JICHble U3iJe/lUR
4. 1.34. BS EN ISO 10642: 2004. BJfHT&1 C nOTsHHoH
ronosKoH Jf WeCTJfrpSH&IM yrny6neHJfeM
nOA KnIOIf
Pa3MepLI H KaJIHOpOBKa rOJIOBKH
CMOTPH pHC. 1 H 2 H BS EN ISO 10642: 2004. Ta6JI. 1.
Pa3MepLI
IIpUMe'1.aHue. CHMBOJIbI H 0603HaQeHIDI pa3MepOB ,naHbI B ISO 225.
i J
/(
t w
AonYCTLflMa aJ1bTepHaTLflB-
Ha51 ct>opMa yrJ1y6J1eHLfl5I
t
o
o
N
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(2)
Q)
o
o
C\I
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AI
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's
Ig
U(5)
s
(b)
l
- /
e/2
(e/2)/3
An51 nOJ1Y"leHHbIX pa3BepTKOVI yrJ1y6J1eHLflVI HaLfl60J1bWerO npeAeJ1bHOrO
pa3Mepa 3ape3, nOJ1Y"lLflBWLflVlC5I npLfl CBepJ1eHLflLfI, He AOJ1)1(eH npeBblwaTb
1/3 AJ1Lf1Hbl J11060Vl rpaHLfi yrJ1y6J1eHLfl5I, KOTOpa51 paBHa e/2.
PRC. 1. BHHT C nOTaHHOH rOJIOBKOH H H llIeCTHrpaHHbIM yrJIy6JIeHHeM no,n; KJIIOq
(1) ,l(OrryCTHMO He3HaQHTeJIbHOe cKpyrJIeHHe HJIH 3eHKOBKa y Bxo,na yrJIy6JIe-
HHR.
(2) KOHeu ,nOIDKeH 6bITb C <pacKoH HJlH ,lUIR pe3b6 M4 MO)l{eT 6bITb 6e3 <pac-
KH B COOTBeTCTBHH C ISO 4753.
(3) KpaH rOJIOBKH MO)l{eT 6bITb cpe3aHHbIM HJlH cKpyrJIeHHbIM.
(4) a = 90° - 92°
(5) HeIIOJIHaR pe3b6a u 2P.
(6) d s HCIIOJIb3yeTcR, eCJIH 3HaQeHHR Is min OIIpe,neJIeHbI.
,
F/la8a 4. Kpene:JICHble iJemOJlu
KaJIHopoBKa roJIOBKH
BepXHIDI IIoBepXHocTb BHHTa ,nOJDKHa 6bITb paCIIOJIO)l{eHa Me)l{)];Y KOHTaKTHbIMH
IIOBepXHOCTRMH A H B.
90°-0°30 1
AonYCKLfI B MM
A
o a
; D-0,05
B
;d a
PUC. 2. npOBepKa 1lJI0CKOCTHOCTH
a D - dk.Teop. max (CMOTPH BS EN ISO 10642: 2004 Ta6JIHua 1).
b F - ,nOIIYCK Ha IIJIOCKOCTHOCTb rOJIOBKH (CMOTpH BS EN ISO 10642: 2004
Ta6JIHua 1).
BS EN ISO 10642: 2004. Ta6JI. 1. Pa3MepbI B MM
Pa3Mep pe3b6w, d M3 M4 MS M6 M8 MtO Mt2 (Mt4 Mt6 M20
pa 0,5 0,7 0,8 1 1,25 1,5 1,75 2 2 2,5
bb 18 20 22 24 28 32 36 40 44 52
d a MaKC. 3,3 4,4 5,5 6,6 8,54 10,62 13,5 15,5 17,5 22
Teope'I'JNe- 6,72 8,96 11,20 13,44 17,92 22,40 26,88 30,8 33,60 40,32
dk CKHH MaKC.
Cl>aKTJf1Ie- 5,54 7,53 9,43 11,34 15,24 19,22 23,12 26,52 29,01 36,05
CKHH MHH.
d s MaKC. 3,00 4,00 5,00 6,00 8,00 10,00 12,00 14,00 16,00 20,00
2,86 3,82 4,82 5,82 7,78 9,78 11,73 13,73 15,73 19,67
MHH.
ec,d MHH. 2,303 2,873 3,443 4,583 5,723 6,863 9,149 11,429 11,429 13,716
k MaKC. 1,86 2,48 3,1 3,72 4,96 6,2 7,44 8,4 8,8 10,16
Fe MaKC. 0,25 0,25 0,3 0,35 0,4 0,4 0,45 0,5 0,6 0,75
MHH. 0,1 0,2 0,2 0,25 0,4 0,4 0,6 0,6 0,6 0,8
r 6
HOM. 2 2,5 3 4 5 8 10 10 12
sd MaKC. 2,08 2,58 3,08 4,095 5,14 6, 140 8,175 10,175 10,175 12,212
MHH. 2,02 2,52 3,02 4,020 5,02 6,020 8,025 10,025 10,025 12,032
t MHH. 1,1 1,5 1,9 2,2 3 3,6 4,3 4,5 4,8 5,6
w MHH. 0,25 0,45 0,66 0,7 1,16 1,62 1,8 1,62 2,2 2,2
4.1. BUHmo6ble "peneJICHble u3deJlUR 215:::
O"OHttaHUe
Pa3Mep pe3b6w, d M3 M4 M5 M6 M8 MtO Mt2 (Mt4 Mt6 M20
If ,lI;JIHHa cTep1KllJl/ s H .zuoma rJla,nKOH 1faCTH CTep1KllJl Ig
HOM. MHH. MaKe. - - - - - - - - - -
fI.I J J J J J J J J J J J fI.I J J J J J J J
- -
8 7,71 8,29
10 9,71 10,29
12 11,65 12,35 (J
)
16 15,65 16,35
20 19,58 20,42
25 24,58 25,42
30 29,58 30,42 95 12 6,5 10
35 34,5 35,5 11,5 15 9 13
40 39,5 40,5 16,5 20 14 18 11 16
45 44,5 45,5 19 23 16 21
50 49,5 50,5 24 28 21 26 15,75 22
55 54,4 55,6 26 31 20,75 27 15,5 23
60 59,4 60,6 31 36 25,75 32 20,5 28
65 64,4 65,6 30,75 37 25,5 33 20,25 29
70 69,4 70,6 35,75 42 30,5 38 25,25 34 20 30
80 79,4 80,6 45,75 52 40,5 48 35,25 44 30 40 26 36
90 89,3 90,7 50,5 58 45,25 54 40 50 36 46
100 99,3 100, 7 60,5 68 55,25 64 50 60 46 56 35,5 48
a P - mar pe3b6bI.
b ,D;JIH ,lLJIHH Me)l()]:Y )l(HpHbIMH cTyrreHqaTbIMH JIHHHSlMH B He3aTeHeHHoH 06JIaCTH.
C e min = 1, 14s min .
d npoBe,n;eHHe COBMeCTHOH KaJIH6pOBKH pa3MepOB yrJIy6JIeHIDI e H S CMOTpH B
ISO 23429.
e F - nJIOCKOCTHOCTb rOJIOBKH, CMOTpH pHC. 2. llIa6JIoH ,lI;JUI H3MepeHHH F HMeeT ,n;o-
nYCK O.
f ,D;JIHHbI H3,n;eJIHH CTaH,n;apTHoro HCnOJIHeHHSI pacnOJIaraIOTCH Me)l(,l(y )l(HpHbIMH C1)'neH-
qaTbIMH JIHHHSlMH. Pe3b6a Ha H3,n;eJIHHX, ,lLJIHHbI KOTOpbIX pacnOJIaraIOTCH B 3aTeHeHHOH 06-
JIaCTH, ,n;OXO,nHT ,n;o rOJIOBKH B npe,n;eJIax 3P. ,D;JIH ,lLJIHH, pacnOJIO)l(eHHbIX HH)I(e 3aTeHeHHOH
06JIaCTH, Ig H Is npHHHMaIOT 3HaQeHHSI B COOTBeTCTBHH co CJIe,n;yIDI.QHMH <popMYJIaMH:
I g,max = I HOM - b; Is ,nun = I g,max - 5P.
g Pa3MepoB B cK06Kax CJIe eT no B03MO)I{HOCTH H36eraTb.
TpeOOBaHHB H COOTBeTCTBHe Me}K,ll;yllapo,n;HLIM CTaH,ZJ;apTaM
CMOTpH BS EN ISO 10642: 2004 Ta6JIHUbI 2 H 3.
F/la8a 4. Kpene:JICHble oemOJlu
BS EN ISO 10642: 2004. Ta6JI. 2. Tpe60BaHHH H COOTBeTCTBHe Me)l()];)'Hapo,n;HbIM
CTaH,ZJ;apTaM
MaTepHaJI
OCHoBHble T e60BaHIDI Me a
Pe3b6a )].OnyCK
CT8JIh
IIpHeMKa
ISO 8992
6g ,lLJIH KJIaCCOB npOqHOCTH 8.8 H 10.9;
5 6 ,lI,JUI KJIacca n OQHOCTH 12.9
I ISO 261 ISO 965-2 ISO 965-3
8 8 10 9 12 9
ISO 898-1
A
ISO 4759-1
Be3 CneI(HaJIbHOH 06pa60TKH
Tpe60BaHHH K raJIbBaHOnOKpbITIDIM
onpe,n;eJIeHbI B ISO 4042
Tpe60BaHIDI K He3JIeKTPOJIHTH'IeCKHM
I(HHKOBbIM nOKpbITHHM onpe,n;eJIeHbI B
ISO 10683
)].onycKaeMble npe,n;eJIbHble BeJIWlHHbI
HeCllJIOlliHOCTH nOBepXHOCTH onpe.ne-
JIeHbI B ISO 6157-1 H ISO 6157-3 ,[(JUI
KJIacca n OqHOCTH 12.9
IIpoI(e.nypa npHeMKH onHcaHa B 180
3269
MexaHWleCKHe
CBOHCTBa
)].OnyCKH
HeCllJIOlliHOCTH nOBepXHOCTH
a J13-3a KOH<pHrypaI(HH rOJIOBKH 3TH BHHTbI MOryr He y,n;OBJIeTBOpHTb Tpe60BaHIDIM no
MHHHMaJIbHbIM pa3pynIaIOI.QHM HarpY3KaM ,lI,JUI KJIaCCOB npOqHOCTH 8.8, 10.9 H 12.9, onpe.ne-
JIeHHbIX B ISO 898-1, npH TeCTHpOBaHHH B COOTBeTCTBHH C nporpaMMoH HcnbITaHHH B. OHM
,n;OJDKHbI, TeM He MeHee, y,n;OBJIeTBOpHTb ,n;pyrHM TPe60BaHHHM K MaTepHaJIaM H npOqHOCTM
,lLJIH KJIaCCOB npOqHOCTH 8.8, 10.9 H 12.9, onpe,n;eJIeHHbIX B ISO 898-1. KpoMe Toro, npH Ha-
rpY3Ke ueJIbIX BHHTOB, Kor,n;a rOJIOBKa nOMep)I(HBaeTCH Ha no,U,Xo,n;HIUeM xOMyre (c KOHHqe-
CKOH onopHoH nOBepxHocTbIO) C nOMOIUbIO npHcnoc06JIeHHH ,lUll[ HcnbITaHHH, onHcaHHoro
B ISO 898-1, OHH ,n;OJDKHbI BbI,n;ep)I(HBaTb 6e3 pa3pynIeHHH MHHHMaJIbHyro pa3pynIaIOI.QyIO
HarpY3K)' B COOTBeTCBHH C BS EN ISO 10642: 2004 Ta6JIHI(a 3. IIpH HcnbITaHHH ,n;o pa3py-
llIeHIDI pa3pymeHHe MO)l(eT npoH30HTH B pe3b60BoH qaCTH, rOJIOBKe, CTepXHe HJIH B MeCTe
coe,n;HHeHIDI rOJIOBKH co CTepXHeM.
BS EN ISO 10642: 2004. Ta6JI. 3. MHHHMaJIbHble pa3pynIaIOI.QHe HarpY3KH ,lI,JUI BHH-
TOB C nOTaHHOH rOJIOBKOH H llIeCTHrpaHHbIM yrJIy6JIeHHeM no,n; KJIlOq
KJIacc Dp01fHOCTH
P83Mep pe3hOW, d 8,8 10,9 12,9
MHHHM8JIhHaJI p83pymalOInaJI HarpY3K8 (H)
M3 3220 4180 4190
M4 5620 7300 856
M5 9080 11800 13800
M6 12900 16700 19600
M8 23400 30500 35700
M10 37100 48200 56600
M12 53900 70200 82400
M14 73600 96000 112000
M16 100000 130000 154000
M20 162000 204000 239000
80% 3HaqeHHH COOTBeTCTByroT ISO 898-1.
4.1. BUHmo8ble "pene:JICHble U3iJeJlUR
0003HaQeHHJI
llpHMep. BHHT C IIoTaHHoH rOJIOBKOH C IIIecTHrpaHHbIM yrJIy6JIeHHeM IIO,n; KJIIOq
C pe3b60H M12, HOMHHaJIbHOH ,lUIHHOH I = 40 MM H KJIaCCOM IIPOqHOCTH 12.9 060-
3HaQaeTCR CJIe,n;yroII(HM 06pa30M:
BHHT C IIoTaHHoH rOJIOBKOH C IIIeCTHrpaHHbIM yr.rry6JIeHHeM IIO,n; KJIIOq
ISO 10642-M12 x 40-12,9
HH<popMaUHIO, KacaIOlI(yIOCR BHHTOB C IIOTaHHOH rOJIOBKOH C IIIeCTHrpaHHbIM
yrny6neHHeM IIO,n; KJIIOq C pa3MepaMH B ,lI;IOHMaX, MO)I{HO HaHTH B BS2470.
4.1.35. 854827 ISO MeTpJflfeCKJfe SJfHTOS&le pe3&6&I,
MJfHJfSTlOpHSfI cepJffl
War e3b6bl P :z: /
It)
0,125P C\I
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0
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(1) ('f) co
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co II
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s:
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co ct 0
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a. OCb pe3b6bl ('f)
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Pa3MepbI B MM
HOMHllaJlhllhli p83Mep Hap)')KHhli Cpe.znmi Buyrpellllllii
mar pe3b6w P (HOMHH8JIhIlhIii)
1-1 PBA 2-ii PBA .1nfaMeTp .1nfaMeTp .1nfaMeTp
8-0 3 0,080 0,300000 0,248038 0,223200
,
8-0,35 0,090 0,350000 0,291543 0,263600
8-0 4 0, 100 0,400000 0,335048 0,304000
,
8-0,45 0,100 0,450000 0,385048 0,354000
8-0 5 0,125 0,500000 0,418810 0,380000
,
8-0,55 0,125 0,550000 0,468810 0,430000
8-0 6 0, 150 0,600000 0,502572 0,456000
,
8-0,7 0,175 0,700000 0,586334 0,532000
8-0,8 0,200 0,800000 0,670096 0,608000
8-0,9 0,225 0,900000 0,753858 0,684000
8-1 0,250 1 ,000000 0,837620 0,760000
8-1 1 0,250 1,100000 0,937620 0,860000
,
8-1 2 0,250 1 ,200000 1,037620 0,960000
,
8-1 4 0,300 1,400000 1,205144 1,112000
,
Donee IIOJIHyro HH<popMaUHIO CMOTPH B BS 4827.
F/la8a 4. Kpene:JICHble iJemOJlu
4.1.36. 853643-1/2 ISO MeTpJflfeCKJfe SJfHTOS&le pe3&6b':
cepJffl C nOCTOflHH&lM wsrOM
Pa3MepbI B MM
HapymrLli (HOMHH8JIhHLIi) .1nfaMeTp Cpe,lUflli BuyrpellllHii .1nfaMeTp
mar pe3b6w .1nfaMeTp
I-ii p 2-ii p 3-ii p DO .lUIY BD8,lUIIIbI BuyrpH
0,25 2,0 - - 1,84 1,69 1,73
0,25 - 2,2 - 2,04 1,89 1,93
0,35 2,5 - - 2,27 2,07 2,12
0,35 3,0 - - 2,77 2,57 2,62
0,35 - 3,5 - 3,27 3,07 3,12
0,50 4,0 - - 3,68 3,39 3,46
0,50 - 4,5 - 4,18 3,86 3,96
0,50 5,0 - - 4,68 4,39 4,46
0,50 - - 5,5 5,18 4,86 4,96
0,75 6,0 - - 5,51 5,08 5,19
0,75 - - 7,0 6,51 6,08 6,19
0,75 8,0 - - 7,51 7,08 7,19
0,75 - - 9,0 8,51 8,08 8,19
0,75 10,0 - - 9,51 9,08 9,19
0,75 - - 11,0 10,51 10,08 10,19
1,0 8,0 - - 7,35 6,77 6,92
1,0 - - 9,0 8,35 7,77 7,92
1,0 10,0 - - 9,35 8,77 8,92
1,0 - - 11,0 10,35 9,77 9,92
1,0 12,0 - - 11,35 10,77 10,92
1,0 - 14,0 - 13,35 12,77 12,92
1,0 - - 15,0 14,35 13,77 13,92
1,0 16,0 - - 15,35 14,77 14,92
1,0 - - 17,0 16,35 15,77 15,92
1,0 - 18,0 - 17,35 16,77 16,92
1,0 20,0 - - 19,35 18,77 18,92
1,0 - 22,0 - 21,35 21,77 21,92
1,0 24,0 - - 23,35 22,77 22,92
1,0 - - 25,0 24,35 23,77 23,92
1,0 - 27,0 - 26,35 25,77 25,92
1,0 - - 28,0 27,35 26,77 26,92
1,0 30,0 - - 29,35 28,77 28,92
1,25 10,0 - - 9,19 8,47 8,65
1,25 12,0 - - 11,19 10,47 10,65
1 25 a - 140 a - 13,19 12,47 12,65
, ,
4.1. BUHlTl08ble "pene:JICHble U3iJe.tlUR
llpoiJollxeHue
HapYJKllhIi (HOMHH8JIhHhIi) .1nfaMeTp Cpe,lUflli BuyrpellllHii .1nfaMeTp
mar pe3hOW
I-ii PBA 2-ii PBA 3-ii PBA .1nfaMeTp DO .lUIY BD8,ZUIIIW BllyrpH
1,5 12,0 - - 11,03 10,16 10,38
1,5 - 14,0 - 13,03 12,16 12,38
1,5 - - 15,0 14,03 13,16 13,38
1,5 16,0 - - 15,03 14,16 14,38
1,5 - - 17,0 16,03 15,16 15,38
1,5 - 18,0 - 17,03 16,16 16,38
1,5 20,0 - - 19,03 18,16 18,38
1,5 - 22,0 - 21,03 20,16 20,38
1,5 24,0 - - 23,03 22,16 'I 22,38
1,5 25,0 24,03 23,16 I 23,38
- -
1,5 - - 26,0 25,03 24,16 24,38
1,5 - 27,0 - 26,03 25,16 25,38
1,5 - - 28,0 27,03 26,16 26,38
1,5 30,0 - - 29,03 28,16 28,38
1,5 - - 32,0 31,03 30,16 30,38
1,5 - 33,0 - 32,03 31,16 31,38
1,5 - - 35,0 34,03 33,16 33,38
CepUJl C UlaZOM 1,5 MM npoiJollxaemCJl iJo MaKCUMOJlbHOZO iJullMempa 80 MM
2,0 18,0 16,70 15,55 15,84
2,0 20,0 - - 18,70 17,55 17,84
2,0 - 22,0 - 20,70 19,55 19,84
2,0 24,0 - - 22,70 21,55 21 ,84
2,0 - - 25,0 23,70 22,55 22,84
2,0 - - 26,0 24,70 23,55 23,84
2,0 - 27,0 - 25,70 24,55 24,84
2,0 - - 28,0 26,70 25,55 25,84
2,0 30,0 - - 28,70 27,55 27,84
2,0 - - 32,0 30,70 29,55 29,84
2,0 - 33,0 - 31,70 30,55 30,84
2,0 - - 35,0 33,70 32,55 32,84
CepUJl C UlaZOM 2 MM npoiJollxaemcJl iJo MaKCUMaJlbHOZO iJullMempa 150 MM
3,0 30,0 - - 28,05 26,32 26,75
3,0 - 33,0 - 31,05 29,32 29,75
3,0 36,0 - - 34,05 32,32 32,75
3,0 - - 38,0 36,05 34,32 34,75
3,0 - 39,0 - 37,05 35,32 35,75
3,0 - - 40,0 38,05 36,32 36,75
F/la8a 4. Kpene:JICHble iJemOJlu
OKOHttaHUe
HapYJKllhIi (HOMHH8JIhHhIii) .1nfaMeTp Cpe.znmii BuyrpeHHHii .1nfaMeTp
mar pe3b6w .1nfaMeTp
I-ii p 2-ii p 3-ii p DO .lnIY BD8,lUIIIbI BIfYTPH
3,0 42,0 - - 40,05 38,32 38,75
3,0 - 45,0 - 43,05 41,32 41,75
3,0 48,0 - - 46,05 44,32 44,75
3,0 - - 50,0 48,05 46,32 46,75
3,0 - 52,0 - 50,05 48,32 48,75
3,0 - - 55,0 53,05 51,32 51,75
CepUR C UlaZO.M 3 JAM npOaOJl:JICaemCJl ao .MaKCUMaJlbHOZO aUa.Mempa 250.M.M
4,0 42,0 - - 39,40 37,09 37,67
4,0 - 45,0 - 42,40 40,09 40,67
4,0 48,0 - - 45,40 43,09 43,67
4,0 - - 50,0 47,40 45,09 45,67
4,0 - 52,0 - 49,40 47,09 47,67
4,0 - - 55,0 52,40 50,09 50,67
4,0 56,0 - - 53,40 51 ,09 51,67
4,0 - - 58,0 55,40 53,09 53,67
4,0 - 60,0 - 57,40 55,09 55,67
4,0 - - 62,0 59,40 57,09 57,67
4,0 64,0 - - 61 ,40 59,09 59,67
4,0 - - 65,0 62,40 60,09 60,67
CepUR C UlaZO.M 4,0.M.M npOaOJl:JICaemcJl ao .MaKCUMaJlbHOZO aUa.Mempa 300 .M.M
6,0 - - 70,0 66,10 62,64 63,50
6,0 72,0 - - 68, 10 64,64 65,50
6,0 - 76,0 - 72,10 68,64 69,50
6,0 80,0 - - 76,10 72,64 73,50
6,0 - 85,0 - 81,10 77,64 78,50
6,0 90,0 - - 86,10 82,64 83,50
6,0 - 95,0 - 91,10 87,64 88,50
6,0 100,0 - - 96,10 92,64 93,50
6,0 - 105,0 - 101,10 97,64 98,50
6,0 110,0 - - 106, 10 102,64 103,50
6,0 - 115,0 - 111,10 107,64 108,50
6,0 - 120,0 ' - 116,10 112,64 113,50
6,0 125,0 - - 121,10 117,64 118,50
CepUR C UlaZO.M 6,0 JAM npOaOJl:JICaemCJl ao .MaKCUMaJlbHOZO aUa.Mempa 300 JAM
a 3TOT pa3Mep npHMeHHeTCH TOJIbKO .lUIH CBeqeH 3aJKHraHHH.
,[(aJIbHeHmyIO HH<popMaUHIO CMOTPH B BS 3643.
4.1. BUHm06ble "peneJICHble U30eJlUJl 221
4. 1.37. BS EN ISO 228-1: 2003. Tpy6H&le pe3&6&1,
He o6ecnelfJfSSIO Jfe repMeTJflfHOCT& COeAJfHeHlAfI
Pa3MepLI, ,n;OnyCKH H 0003Ha1JeHHJI
06JlaCmb npUMenenUR
3Ta qaCTb ISO 228 onpe,neJUIeT TPe60BaHlUI K npo<pHJIIO, pa3MepaM, ,norrycKaM
H 0603HaqeHlUIM TPy6HOH pe3b6bI ,l(JUI pe3b60BbIX coe,nHHeHHH C pa3MepaMH pe3b-
6bIOT 1/16 ,no 6 BKJIlOqHTeJIbHO. H BHyrpeHHHe, H Hap)')KHbIe [(HJIHH):(pHqeCKHe pe-
3b6bI npe,nHa3HaqeHbI ,l(JUI MexaHWIeCKHX coe,nHHeHHH ,neTaJIeH <pHTHHroB, KpaHOB
H BeHTHJIeH, BcnOMOraTeJIbHOrO 060py,noBaHIDI H T.,n.
3TH pe3b6bI He IIO,nXO,n.HT B KaqeCTBe coe,nHIDIIOIUHX, eCJIH pe3b60Boe coe,nH-
HeHHe ,nOJDKHO 6bITb repMeTHqHbIM. ECJIH coe,nHHeHHe C HCnOJIb30BaHHeM Ta-
KHX pe3b6 ,nOJDKHO 6bITb repMeTHqHbIM, CJIe,nyeT YCTaHOBHTb ,nBe CTonopHbIe IIJIac-
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/
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ISO 7 -1.
. B ISO 228-2 no,np06HO OIIHcaHbI MeTo,nbI npOBepKH pa3MepOB H npo<pIDIeH
Kpene)I{HOH pe3b6bI, a TaIOKe peKOMe eMbIe KaJIH6pbI.
HOpMaTHBHLIe CCLIJIKH
HIDKeCJIe,ayroIUHH HopMaTHBHbIH ,nOJ<yMeHT co,nep)I{HT nOJIOXeHlUI, KOTopbIe,
Qepe3 CCbillKH B TeKCTe, .HBJUlIOTC.H IIOJIOXeHlUIMH cooTBeTcTByroIUeH qaCTH ISO
228. LVrn ,naTHpOBaHHbIX CCbillOK He npHMeHHMa JII06aH H3 rry6JIHKa[(HH C nOCJIe.ny-
IOInHMH nonpaBKaMH HJIH B ,npyrOH pe,naKUHH. O,nHaKO yqacTHHKH COrJIaIIIeHlUI,
OCHOBaHHoro Ha 3TOH qaCTH ISO 228, nO,lI,l(ep)I{HBaIOT HCCJIe,nOBaHlUI, HanpaBJIeH-
HbIe Ha B03MO)I{HOCTb npHMeHeHlUI HOBeHllIHX BbIrryCKOB HopMaTHBHoro ,nOJ<yMeH-
Ta, YKa3aHHoro HIDKe. He,naTHpOBaHHbIX CCbIJIOK npHMeIDIeTC.H IIOCJIe,nHHH BbI-
rrYCK HopMaTHBHoro ,nOJ<yMeHTa, 0 KOTOpOM eT peqb. qJIeHbI ISO (MelK.UYHapo,n-
Hag opraHH3a[(IDI no CTaH):(apTH3a[(HH) H IEC (MelK.UYHapo,nHaH 3JIeKTpOTeXHlNe-
CKaH KOMHCClUI, M3K) IIO,lI,l(ep)I{HBaIOT peeCTpbI ,neHCTBHTeJIbHbIX B HaCTO.HIUee
BpeWI MelK.UYHapo,nHbIX CTaH):(apTOB.
ISO 7-1 1994, Tpy6HbIe pe3b6bI, 06eCneqHBalOIUHe repMeTHqHOCTb coe,nHHe-
HIDI. qacTb 1. Pa3MepbI, ,nOrryCKH H 0603HaqeHHe.
CHMBOJILI
B 3TOH qaCTH ISO 228 HCnOJIb3yroTC.H CJIe,ayroIUHe 0603HaqeHlUI:
A Krracc C xeCTKHMH ,norrycKaMH Ha Hap)')KHbIe TPy6HbIe pe3b6bI, He 06ecne-
qHBalOIUHe repMeTHqHOCTb coe,nHHeHlUI.
B Krracc C 60JIee IIIHpOKHMH ,norrycKaMH Ha Hap)')KHbIe Tpy6HbIe pe3b6bI, He
06eCneqHBalOIUHe repMeTHqHOCTb coe,nHHeHlUI.
F/la8a 4. Kpene:JICHble oemOJlu
D = d; Hap)')KHbIH JJ;HaMeTP BHYTPeHHeH pe3b6bI.
D} = D - 1,280 654 P = d}; BHYTPeHHHH JJ;HaMeTP BHYTpeHHeH pe3b6bI.
D 2 = D - 0,640 327 P = d 2 ; cpeJJ;HHH JJ;HaMeTPa BHYTPeHHeH pe3b6bI.
d Hap)')KHbIH JJ;HaMeTP Hap)')KHOH pe3b6bI.
d} = d - 1,280 654 P; BHYTPeHHHH JJ;HaMeTP Hap)')KHoH pe3b6bI.
d 2 = d - 0,640 327 P; CpeJJ;HHH JJ;HaMeTPa Hap)')KHOH pe3b6bI.
G Tpy6Ha.H pe3b6a, He 06eCIIeqHBaIOIUa.H repMeTINHOCTb coeJJ;HHeHlUI.
H BbICOTa HCXOJJ;Horo TPeyrOJIbHHKa pe3b6bI.
h Pa60qa.H BbICOTa IIPO<pHJI.H pe3b6bI (c 3aKpyrJIeHHbIMH BepIIIHHaMH H Bna-
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P IIIar.
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T Dt ,lJ;OrryCK Ha BHYTPeHHHH JJ;HaMeTP BHYTPeHHeH pe3b6bI.
T D2 ,lJ;OrryCK Ha CpeJJ;HHH JJ;HaMeTPa BHYTpeHHeH pe3b6bI.
Td ,lJ;OrryCK Ha Hap)')KHbIH ,ZJ;HaMeTPa Hap)')KHOH pe3b6bI.
T d2 ,lJ;OrryCK Ha cpeJJ;HHH JJ;HaMeTPa Hap)')KHoH pe3b6bI.
Pa3MepLI
IIpo<pHJIb 3THX pe3b6 H,LJ;eHTlNeH IIpO<pHJIIO :QHJIHH):(pHqeCKOH pe3b6bI, OIIpe,D;e-
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peHHHe, TaK H Hap)')KHbIe pe3b6bI.
ECJIH He orOBopeHO HHoe, pe3b6a, COOTBeTCTByroIUa.H 3TOH qaCTH ISO 228, SIB-
JUIeTC.H IIpaBoH pe3b60H (CMOTpH TaIOKe rryHKT 5).
BepIIIHHbI pe3b6bI 06bIqHO BbIIIOJIIDIIOTC.H C IIJIOCKHM cpe30M B IIpeJJ;eJIax JJ;ony-
CKa, KaK yx:a3aHO B KOJIOHKax 14 H 15 BS EN ISO 228-1: 2003 Ta6JIH[(a 1. MCKJlIO-
qeHHe COCTaBJUIIOT BHYTPeHHHe pe3b6bI, eCJIH CylI(eCTyeT B03MO)I{HOCTb HX coe,D;H-
HeHlUI C BHeIIIHHMH pe3b6aMH B COOTBeTCTBHH C ISO 7-1, H B 3TOM c.rryqae ,lJ)IHHa
pe3b6bI JJ;OJDKHa 6bITb paBHoH HJIH 60JIbIIIeH, qeM onpeJJ;eJIeHa B ISO 7-1.
,lJ;OrryCK Ha CpeJJ;HHH JJ;HaMeTP BHYTPeHHHX pe3b6 COOTBeTcTByeT nOJIO)I{HTeJIbHO-
MY OTKJIOHeHHIO JJ;OrryCKOB JJ;HaMeTPa B ISO 7-1, 3a HCKJIlOqeHHeM Tex pe3b6 C pa3-
MepaMH 1/16, 1/8, 1/4 H 3/8, ,l(JUI KOTOpbIX onpeJJ;eJIeHbI HeMHoro 60JIee BbICOKHe
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,lJ;.mI Hap)')KHbIX pe3b6 OIIpeJJ;eJIeHbI JJ;Ba KJIacca JJ;OrryCKOB Ha CpeJJ;HHH JJ;HaMerp
(CMOTPH BS EN ISO 228-1: 2003 Ta6JIH[(a 1).
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KIDK;:(bIH H3 KOTOpbIX 3KBHBaJIeHTeH COOTBeTCTByroIUeMY JJ;orrycKy Ha BHYT-'
peHHIOIO pe3b6y.
. Krracc B (KOJIOHKa 11) COCTOHT H3 nOJIHOCTbIO OTPH[(aTeJIbHbIX JJ;OrryCKOB,
KIDK;:(bIH H3 KOTOpbIX paBeH yJJ;BoeHHOMY 3HaqeHHIO COOTBeTCTyIDIUero ,ZJ;onyc-
Ky Ha BHYTPeHHIOIO pe3b6y.
BbI60p Me KJIaCCOM A H KJIaCCOM B 3aBHCHT OT YCJIOBHH npHMeHeHIDI Ii
JJ;OJDKeH 6bITb CJJ;eJIaH B CTaH):(apTaX Ha npO,nyK[(HIO, ,l(JUI KOTOpOH pe3b6bI onpeJJ;eJIe-
HbI B COOTBeTCTBHH C 3TOH qaCTbIO ISO 228.
Pa3MepbI TPy6HbIX pe3b6, B MHJIJIHMeTPax, JJ;aHbI B BS EN ISO 228-1: 2003
Ta6JIH[(a 1.
4.1. BUHmo6ble "peneJICHble U30eJlUJl
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I ...-4 ...-4 V) V) 00 00 00 00 \0 \0 \0 \0 \0 \0 M M M M M M M M M M
N N N N C'! N N C'! M M M M M M v
= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
I I I I I I I I I I I I I I I I I I I I I I I I
It N N V) V) ...-4 ...-4 ...-4 ...-4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S- 00 00 v v v v v v v v v v v v v v v v v v v v v
i i :;: N N V) V; V; V) d d \0 \0 d d ; d \0 d d d d d d
0 0 0 0 0 0 0 0 0 0 0 0
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t = v v 0 0 v v v v 0 0 0 0 0 0 v v v v v v v v v
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i ! ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 C'! C'! C'! C'! C'! C'! C'! C'! C'! C'!
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S- ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 ...-4 C'! C'! C'! C'! N N C'! C'!
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Ha pRC. 1 nOKa3aHa KpeIIe)I{H8j{ pe3b6a C nOJIHbIM npo<pHJIeM R noJUI ,norrycKa.
Ha pRC. 2 nOKa3aH npo<pHJIb Kpene)I{HOH pe3b6bI C IIJIOCKOCpe3aHHbIMH BepmH-
HaMR H noJUI ,norrycKa.
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(1) H = 0,960491P.
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4.1. BUHmo8ble "pene:JICHble U30eJlUJI
0003Ha1JeHHJI
0603HaqeHHe pe3b6bI ,nOJDKHO COCTO.HTb H3 CJIe,ayroIUHx 3JIeMeHTOB B ,u;aHHoH
rrocne,nOBaTeJIbHOCTH:
1. OnHCaTeJIbHbIH 6JIOK: Tpy6HaH pe3b6a.
2. DJIOK, yx:a3bIBaIOIUHH Ha HOMep Me)l()];yJIapo,nHoro CTaH):(apTa: ISO 228.
3. DJIOK, oIIpe,neJUIIOIUHH KOHKpeTHyIO pe3b6y (O,nHH H3 CJIe,n;yIOIUHX, B 3aBHCH-
MOCTH OT CJIytIaH):
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BS EN ISO 228-1: 2003 Ta6JIH[(a 1 ,l(JUI BHyrpeHHHX pe3b6 (TOJIbKO O,nHH
KJIacc TOqHOCTH); HJIH
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H3 BS EN ISO 228-1: 2003 Ta6JIH[(a 1 H 6yxBa A,l(JUI KJIacca TOqHOCTH A
Hap)')KHbIX pe3b6; HJIH
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H3 BS EN ISO 228-1: 2003 Ta6JIH[(a 1 H 6YKBa B ,ZJ;JI.H KJIacca TOqHOCTH B
Hap)')KHbIX pe3b6.
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Krracc TOqHOCTH B
Tpy6HbIe pe3b6bI ISO 228 - G 1 1/2
Hap)')KHaH pe3b6a
Tpy6HbIe pe3b6bI ISO 228 - G 1 1/2 A
Tpy6HbIe pe3b6bI ISO 228 - G 1 1/2 B
C01JeTaHHJI c COe,ZJ;llHHTeJILHLIMH pe3LoaMH
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COOTBeTcTByroIUeH C ISO 228-1, H BHYTpeHHeH [(HJIHH):(pHqeCKOH pe3b6bI Rp, COOT-
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8-3255
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OCHOSH&le pS3Mep&l
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(AI1 Ha cnonb3yeMoH pe3b6bl 3a OCHOBHOH
nnOCKOCTblO, He06xOA Ma , "IT06bl 06ecne-
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HOMHH8JIb1lhIi p83Mep 1'py6hI a qHCJIO mar Pa601faJI ,lI;HaMeTp pe3h6hI B OCHOBHOA DJIOCKOCTH (0)
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25,4 MM (MM)
1/8 6 28 0_907 0_581 9_728 9_147 8_566
1/4 &- 19 1_337 0_856 13_157 12_301 11_445
3/8 10 19 1_337 0_856 16_662 15_806 14.950
1/2 15 14 1.814 1.162 20.955 19.793 18.631
3/4 20 14 1.814 1.162 26.441 25.279 24.117
1 25 11 2.309 1.479 33.249 31.770 30_291
1 1/8 32 11 2_309 1_479 41_910 40_431 38_952
1 1/8 40 11 2_ 309 1_479 47 _803 46_324 44.845
2 50 11 2309 1_479 59_614 58.135 56.656
2 1/8 65 11 2.309 1_479 75.184 73.705 72.226
3 80 11 2309 1.479 87.884 86.405 84.926
4 100 11 2.309 1.479 113.030 111.551 110_072
5 125 11 2_309 1_479 138_430 136_951 135_472
6 150 11 2_309 1_479 163_830 162_351 160_872
a 3KBHBaJIeHTbI HOMHHaJIbHbIX ,lI;HaMeTpOB 'rpy6, a He pe3YJIbTaTbI npe06pa30BaHHH.
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Ta6JI. 2.
HOMH- Aruma HapymtoA pe3h6hI Aruma ,n;OnyCK
B8Jlb1lhdi Pa601faJI ,lUIIIIIa pe3h6hI Ha npa- Ha DO-
p83Mep OT ropna .no OCHOBHOA (MHH.) c60PKY nyCK JI01Ke- ,n;OnyCK
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1/8 6 4 3/8 1 5 3/8 3 3/8 7 1/8 8 1/8 6 1/8 2 1/2 1 1/2 1 1/4 0,071
(4,0) (0,9) (4,9) (3,1) (6,5) (7,4) (5,6) (2,5) (1,4) (1,1)
1/4 8 4 1/2 1 5 1/2 3 1/2 7 1/4 8 1/4 6 1/4 2 1/2 1 1/2 1 1/4 0,104
(5,0) (1,3) (7,3) (4,7) (9,7) (11,0) (8,4) (3,7) (2,0) ( 1 , 7)
3/8 10 4 1/2 1 5 1/2 3 1/2 7 1/2 8 1/2 6 1/2 2 1/2 1 1/2 1 1/4 0,104
(6,4) (1,3) (7,7) (5,1) (10,1) (11,4) (8,8) (3,7) (2,0) ( 1 , 7)
1/2 15 4 1/2 1 5 1/2 3 1/2 7 1/4 8 1/4 6 1/4 2 1/2 1 1/2 1 1/4 0,142
(8,2) (1,8) (10,0) (6,4) ( 13,2) (15,0) (11,4) (5,0) (2,7) (2,3)
1/2 20 5 1/4 1 6 1/4 4 1/4 8 9 7 2 1/4 1 1/2 1 1/4 0,142
(9,5) (1,8) (11,3) (7,7) (14,5) ( 16,3) (12,7) (5,0) (2,7) (2,3)
1 25 4 1/2 1 5 1/2 3 1/2 7 1/4 8 1/2 6 1/4 2 1/2 1 1/2 1 1/4 0, 180
(10,4) (2,3) (12,7) (8,1) (16,8) (19,1) (14,5) (6,4) (3,5) (2,9)
11/4 32 5 1/2 1 6 1/2 4 1/2 8 1/4 9 1/4 7 1/4 2 1/2 1 1/2 1 1/4 0, 180
(12,7) (2,3) (15,0) (10,4) (19,1) (21,4) (16,8) (6,4) (3,5) (2,9)
11/2 40 5 1/2 1 6 1/2 4 1/2 8 1/4 9 1/4 7 1/4 2 3/4 1 1/2 1 1/4 0,180
(12,7) (2,3) ( 15,0) (10,4) (19,1) (21,4) (16,8) (6,4) (3,5) (2,9)
2 50 6 7/8 1 7 7/8 5 7/8 10 1/8 11 1/8 9 1/8 3 1/4 2 1 1/4 0, 180
(15,9) (2,3) (18,2) (15,6) (23,4) (25,7) (21,1) (7,5) (4,6) (2,9)
2 1/2 65 7 9/16 1 1/2 9 1/16 6 1/16 11 9/16 13 1/16 10 1/16 4 2 1/2 1 1/2 0,216
(17,5) (3,5) (21,0) (14,0) (26,7) (30,2) (23,2) (9,2) (5,8) (3,5)
3 80 8 15/16 1 1/2 10 7/16 7 7/16 12 15/16 14 7/16 11 7/16 4 2 1/2 1 1/2 0,216
(20,6) (3,5) (24,1) (17,1) (29,8) (33,3) (26,3) (9,2) (5,8) (3,5)
4 100 11 1 1/2 12 1/2 9 1/2 15 1/2 17 14 4 1/2 3 1 1/2 0,216
(25,4) (3,5) (28,9) (21,9) (35,8) (19,3) (32,3) (10,4) (6,9) (3,5)
5 125 12 3/8 1 1/2 13 7 /8 10 7 /8 17 3/8 18 7/8 15 7/8 5 3 1/2 1 1/2 0,216
(28,6) (3,5) (32,1) (25,1) (40,1) (43,6) (36,6) (11,5) (8,1) (3,5)
6 150 12 3/8 1 1/2 13 7 /8 10 7 /8 17 7 /8 18 7/8 15 7 /8 5 3 1/2 1 1/2 0,216
(28,6) (3,5) (32,1) (25,1) (40,1) (43,6) (36,6) (11,5) (8,1) (3,5)
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M48 43,00 - 50,0 52,0 56,0
M52 4 7 00 - 54_0 56_0 62 0
F/la8a 4. Kpene:»cHble oemOJlu
4. 1.49. PS3Mep&1 csepn Anfl oTSepCTJfH nOA
ISO MeTpJflfeCKYIO pe3&6y Jf Anfl CKS03H&IX
oTSepCTJfH C 3S30pOM: PflA C KpynHoH pe3&6oif
HOMHHaJIh.....H Pa3Mep CBepJla DO pe3WY (MM) P83Mep CBepJla .zum CKB03HhIX OTBepcTHii (MM)
p83Mep PeKOMeu.nyeMoe AJIhTepHaTHBHoe nOC8,lI,K8 C nepexo.lUlaJI noc8,ll,K8c
80% 3aQeDJIeHHe 70% 3aQeDJIeHHe HaTJlroM (1 PB.lO DOC8,ll,K8 (2 PW 3a30pOM (3 pJiJQ
M6 5,20 5,30 6,4 6,6 7,0
M8 7,00 7,10 8,4 9,0 10,0
MI0 8,80 8,90 10,5 11,0 12,0
M12 10,80 10,90 13,0 14,0 15,0
M14 12,50 12,70 15,0 16,0 17,0
M16 14,50 14,75 17,0 18,0 19,0
M18 16,50 16,75 19,0 20,0 21,0
M20 18,50 18,75 21,0 22,0 24,0
M22 20,50 20,75 23,0 24,0 26,0
M24 22,00 22,25 25,0 26,0 28,0
M27 25,00 25,25 28,0 30,0 32,0
M30 28,00 28,25 31,0 33,0 35,0
M33 31,00 31,25 34,0 36,0 38,0
M36 33,00 - 37,0 39,0 42,0
M39 36,00 - 40,0 42,0 45,0
M42 39 00 - 43 0 45 0 48 0
4. 1.50. PS3Mep&1 csepn p,nfl oTSepCTJfH nOA
ISO YHJfcpJfqJfPOSSHHYIO pe3&6y Jf Anfl CKS03H&IX
oTSepCTJfH C 3S30pOM: pe3&6s C MenKJfM wsrOM
HOMHHaJIhllhlii P83Mep CBepJla DO pe3hOY Pa3Mep CBepJla .zum CKB03....1X OTBepcTHii
p83Mep, ,lJ.IOHM MM ,lJ.IOHM MM oyua HJIH ,lJ.IOHM
1 5,20 13 6,50 17
-x20 - A HJIH -
4 64 64
5 6,60 17 8,00 21
-x18 - o HJIH -
16 64 64
3 8,00 5 9,80 25
-x16 - W HJIH -
4 16 64
7 9,40 3 11,30 33
-x14 - -
16 8 64
1 10,80 27 13,00 33
-x13 - -
2 64 64
9 12,20 31 14,75 37
-x12 - -
16 64 64
5 13,50 17 16,25 41
-xlI - -
8 32 64
3 16,50 21 19,50 47
-x10 - -
4 32 64
7 19,25 49 20,25 51
-x9 - -
8 64 64
4.1. BUHmo8ble Kpene:JICHble U30e.tlUJI
OKoH"taHue
HOMHH8JIh1lldi Pa3Mep CBepJIa DO pe3hoy P83Mep CBepJIa .n;JUI CKB0311h1X OTBepcTHii
p83Mep, ,lJ.IOHM MM ,lJ.IOHM MM oyua HJIH ,lJ.IOHM
1x8 22,25 7 25,75 1-.L
-
8 64
1 25,00 63 26,00 1
1-x7 -
8 64 64
1 28,25 a 1 28,25 1 17
1-x7
4 64 64
3 30,50 a III 30,75 1 25
l-x6
8 64 64
1 34,00a 1 21 34,00 1 33
1-x6
2 64 64
3 39,50 a 1 35 45,00 1 49
1-x5
4 64 64
1 45,50 a 1 25 52,00 2-.L
2x4-
2 64 64
a DJIHXaHIlIHH CTaH,lJ;apTHbIH MeTpHQeCKHH pa3Mep: npH6JIH3HTeJIbHO Ha 0,25 MM 60JIb-
me peKOMeH,l(yeMoro ,lI;IOHMoBoro pa3Mepa.
4.1.51. PS3Mep&1 csepn Anfl OTSepCTJfH nOA
ISO YHJfcpJfqJfpOSSHHYIO pe3&6y Jf Anfl CKS03H&IX
OTSepCTJfH C 3S30pOM: pe3&6s C MenKJfM warOM
HOMHHaJIhHhlH p83Mep Pa3Mep CBepJIa DO pe3hOY Pa3Mep CBepJIa .n;JUI CKB0311h1X oTBepCTHii
,lJ.IOHM MM oyua HJIH ,lJ.IOHM MM oyua HJIH ,lJ.IOHM
)
1/4x28 5,50 7/32 6,50 F
5/16x24 6,90 I 8,00 0
3/8x24 8,50 R 9,80 W
7/16x20 9,90 25/64 11 ,30 29/64
1/2x20 11,50 29/64 13,00 33/64
9/16x18 12,90 33/64 14,75 37/64
5/8x18 14,50 37/64 16,50 41/64
1/2x 16 17,50 11/16 19,50 49/64
7/8x14 20,50 13/16 22,75 57/64
1x12 23,25 59/64 25,80 1 1/64
1 1/8x 12 26,50 1 3/64 29,00 1 9/64
1 1/4x12 29,50 ] 11/64 32,50 1 17/64
1 3/8x 12 33,00 1 19/64 35,50 1 25/64
] 1/2x 12 36,00 1 27 /64 38,50 1 33/64
F/la8a 4. Kpene:JICHble oemOJlu
4. 1.52. PS3Mep&1 csepn p,nfl oTSepCTJfH nOA
ISO MeTpJflfeCKYIO pe3&6y Jf Anfl CKS03H&IX
oTSepCTJfH C 3S30pOM: MJfHJfSTlOpHSfI cepJffl
HOMHH8JIhHhIii p83Mep mar Pa3Mep CBepJIa DO pe3hOY P83Mep CBepJIa JUUI
BHTKH CKB03HhIX OTBepcTHii
Ha ,lJ.IOHM
ISO MM ASA Bl.10 MM MM MM 1fHCJIO HJIH OOh MM 1fHCJIO HJIH p06h
S-O 3 0,30 unm 0,080 318 0,25 - 0,32 -
,
(S-0,35) (0,35 unm) 0,090 282 0,28 - 0,38 79
S-O 4 0,40 unm 0,100 254 0,35 80 0,45 77
,
(S-0,45) (0,45 unm) 0,100 254 0,38 79 0,50 76
S-O 5 0,50 unm 0,125 203 0,42 78 0,55 75,74
,
(S-0,55) (0,55 unm) 0,125 203 0,45 77 0,60 73
s-o 6 0,60 unm 0,150 169 0,50 76 0,65 72
,
(S-0,7) (0,70 unm) 0,175 145 0,58 74 0,78 1/32 ,lI;IOHMa
S-O 8 0,80 unm 0,200 127 0,65 72 0,88 66,65
,
(S-0,9) (0,90 unm) 0,225 113 0,72 70 0,98 62
S-10 1,00 unm 0,250 102 0,80 1/32 ,lI;IOHMa 1,10 57
,
(5-1,1) (1,10 unm) 0,250 102 0,90 65 1,20 3/64 ,lI;IOHMa
S-12 1,20 unm 0,250 102 1,00 61 1,30 55
,
(S-1,4) (1,40 unm) 0,300 85 1,15 3/64 ,lI;IOHMa 1,50 53
4. 1.53. PS3Mep&1 csepn p,nfl OTSepCTJfH nOA SSW pe3&6y
Jf p,nfl CKS03H&IX OTSepCTJfH C 3S30pOM pe3&6&I,
HSpe3H&le Jf npOlfJf SIO Jfe MeTlfJfKJf
Pa3Mep BHTKH no ,lI;mI CKB03HhIX Pa3Mep BUTKH no ,lI;mI CKB03HhIX
Ha ,lJ.IOHM pe3hOY OTBepcTHii Ha ,lJ.IOHM pe3hOY OTBepcTHii
1/16 60 58 49 1/4 20 7 G
3/32 48 49 36 5/16 18 F P
1/8 40 38 29 3/8 16 0 W
5/32 32 31 19 7/16 14 U 29/64
3/16 24 26 9 1/2 12 27/64 33/64
7/32 24 15 1
4. 1.54. PS3Mep&1 csepn p,nfl OTSepCTJfH nOA SSF pe3&6y
Jf p,nfl CKS03H&IX OTSepCTJfH C 3S30pOM pe3&6&I,
HSpe3H&le Jf npOlfJf SIO Jfe MeTlfJfKJf
Pa3Mep BUTKH no ,lI;mI CKB03HhIX Pa3Mep BHTKH no ,lI;mI CKB03HhIX
Ha ,lJ.IOHM pe3hOY OTBepcTHii Ha ,lJ.IOHM pe3hOY OTBepCTHii:
7/32 28 14 1 3/8 20 21/64 W
1/4 26 3 G 7/16 18 W 29/64
9/32 26 C M 1/2 16 7/16 33/64
5/16 22 H P
4.1. BUHmo6ble "peneJICHble U30eJlUR 23 9
4.1.55. nnOCKJfe WSH6&I, IfJfCT&le: MeTpJflfeCKJfH PflA
Pa3MepbI B MM
d 1
d 2
-!s
BuyrpeHHHH BHeIlIllllii TOJIDUllla s
0603Ha'leHHe aM d t aMeTp d z
WtaMeTp TOJiCThie (HopMaJIhHLIe) TOHKHe
Pe3hOhl)8 I
MaKe. MHH. MaKe. MHH. MaKe. MHH. MaKe. MHH.
M1..0 1..25 1.1 2.5 2.3 0.4 0.2 - -
M1.2 1..45 1.3 3.0 2.8 0.4 0.2 - -
(M1.4) 1..65 1.5 3.0 2.8 0.4 0.2 - -
M1.6 1.85 1..7 4.0 3.7 0.4 0.2 - -
M2,0 2.35 2.2 5.0 4.7 0.4 0.2 - -
(M2,2) 2.55 2.4 5.0 4.7 0.6 0.4 - -
M2,5 2,85 2,7 6.5 6.2 0.6 0.4 - -
M3 3,4 3,2 7,0 6,7 0.6 0.4 - -
(M3,5) 3,9 3,7 7,0 6,7 0,6 0,4 - -
M4 4.5 4.3 9,0 8,7 0,9 0,7 - -
(M4.5) 5.0 4.8 9,0 18,7 0,9 0,7 - -
M5 5.5 5.3 10.0 9.7 1.1 0.9 - -
M6 6.7 6.4 12.5 12.1 1.8 1.4 0.9 0.7
(M7) 7,7 7,4 14.0 13.6 1.8 1.4 0.9 0.7
M8 8,7 8.4 17.0 16.6 1..8 1..4 1.1 0.9
MI0 10,9 10,5 21,0 20.5 2.2 1.8 1.45 1.05
M12 13,4 13,0 24,0 23.5 2.7 2.3 1..8 1.4
(M14) 15,4 15,0 28,0 27,5 2,7 2,3 1..8 1,4
M16 17.4 17.0 30,0 29,5 3,3 2,7 2,2 1,8
(M18) 19.5 19.0 34,0 33,2 3,3 2,7 2,2 1,8
M20 21.5 21.0 37.0 36.2 3,3 2,7 2,2 1,8
(M22) 23.5 23.0 39.0 38.2 3,3 2,7 2,2 1,8
M24 25.5 25.0 44.0 43.2 4.3 3.7 2.7 2.3
(M27) 28.5 28.0 50.0 49.2 4.3 3.7 2.7 2.3
M30 31,6 31.0 56.0 55.0 4.3 3.7 2.7 2.3
(M33) 34,6 34.0 60.0 59.0 5.6 4.4 3.3 2.7
M36 37,6 37,0 66,0 65.0 5.6 4.4 3.3 2.7
(M39) 40,6 40,0 72,0 71,0 6.6 5.4 3.3 2.7
a CneI(HaJIbHbIe pa3MepbI yJ<a3aHbI B cK06Kax ().
DOJIee nOJIHyID HH<popMaI(HIO CMOTpH B BS 4320.
F/la8a 4. Kpene:JICHble oemOJlu
4. 1.56. nnOCKJfe waH6&I, lfepH&le: MeTpJflfeCKJfH PflA
d 1
Pa3MepbI B MM
d 2
s s
0003Ha'leHHe Buyrpellllllii: )UlaMeTp d 1 BHeIlIllllii )UlaMeTp d z TOJIDUllla s
()UlaMeTp
pe3hohl)8 MaKe. MHH. MaKC. MUH. MaKe. MHH.
M5 5,8 5,5 10,0 9,2 1,2 0,8
M6 7,0 6,6 12,5 11,7 1,9 1,3
(M7) 8,0 7,6 14,0 13,2 1,9 1,3
M8 9,4 9,0 17,0 16,2 1,9 1,3
MI0 11,5 11,0 21,0 20,2 2,3 1,7
M12 14,5 14,0 24,0 23,2 2,8 2,2
(M14) 16,5 16,0 28,0 27,2 2,8 2,2
M16 18,5 18,0 30,0 29,2 3,6 2,4
(M18) 20,6 20,0 34,0 32,8 3,6 2,4
M20 22,6 22,0 370 35,8 3,6 2,4
(M22) 24,6 24,0 39,0 37,8 3,6 2,4
M24 26,6 26,0 44,0 42,8 4,6 3,4
(M27) 30,6 30,0 50,0 48,8 4,6 3,4
M30 33,8 33,0 56,0 54,5 4,6 3,4
(M33) 36,8 36,0 60,0 58,5 6,0 4,0
M36 39,8 39,0 66,0 64,5 6,0 4,0
(M39) 42,8 42,0 72,0 70,5 7,0 5,0
M42 45,8 45,0 78,0 76,5 8,2 5,8
(M45) 48,8 48,0 85,0 83,0 8,2 5,8
M48 53,0 52,0 92,0 90,0 9,2 6,8
(M52) 57,0 56,0 98,0 96,0 9,2 6,8
M56 63,0 62,0 105,0 103,0 10,2 7,8
(M60) 67,0 66,0 110,0 108,0 10,2 7,8
M64 71,0 70,0 115,0 113,0 10,2 7,8
(M68) 75,0 74,0 120,0 118,0 11,2 8,8
a CneUHaJIbHbIe pa3MepbI yJ<a3aHbI B cK06Kax ().
Donee nOJIHyID HH<popMaUHIO CMOTpH B BS 4320.
4.1. BUHmo6ble "peneJICHble U30eflUR 241
4.1.57. YCTpOHCTBS Anfl cTOnOpeHJffl,
OCHOBSHH&le HS TpeHJfJf
CTOnOpHaJi raiKa
CaMOCTOnOpJlInaJlCJI raiiKa
(c KOJILD;eBOi BCTaBKoi)
rat1Ka nOJ1-
HOt1 BblCOTbl
BCTaBKa 3 Het1J10-
Ha J1 nJ1aCT Ka
CTonopHa J1
3a)l( MHa rat1Ka
nJ10CKa wat16a
nJ10CKa
wat16a
CaMOCTOnOpJlInaJlCJI raHKa (IneJIeBaJi rOJIOBKa)
nJ10CKa
wat16a
eJ1eBa51 r0J10BKa
(c AByxcerMeHTHblM
npOn J10M)
eJ1eBa r0J10BKa
(c KOJ1bu.eBbIM npOn J10M)
nJ10CKa
wat16a
3YO'laTan nPY1KHHHaJi maHoa
/
Cn0J1b30BaH e 3y6"1aTot1
np HHot1 (3aw. w.eHHot1
OT caMOOTB H"I BaH 5I np
COTp5lCeH X) wat16bl
CMOTPH npo CTOIIopHbIe IIIai1:6bI:
IIp)')KHHHa.H maH6a 4.1.59,4.1.60,4.1.61
IIp)')KHHHa.H maH6a C 3y6bHMH 4.1.62
CTOIIOpHa.H maH6a C 3y6bHMH 4.1.63
IIp)')KHHHIDI BOJIHHCTaSI IIIaH6a 4.1.64
F/la8a 4. Kpene:JICHble oemOJlu
4. 1.58. YCTpOHCTBS,lInfl CTOnOpeHJffl, OCHOBSHH&le
HS MeXSHJflfeCKOH cpJfKCSqJfJf
npOpe3HaJi raHKa
CTOnOpHaJi
DJIaCTHHa
BJl3Ka
npOBOJIOKOH
KOpOJl1laTaJI ranKa
CTOnOpHaJI manoa
(c JIanKaMH HJIH HOCKOM)
KOHLJ.bl npOBOAa
CKPY"IeHbl BMeCTe
aaKpenJ1eHbl
4.1. BUHmo8ble Kpene:JICHble U30e.tlUR
4. 1.59. npyJKJfHHSfI wsii6s C KBSAPSTH&IM nonepelfH&lM
CelfeHJfeM: MeTpJflfeCKJfii PflA, TJfn A
1 s r-
,
I
Ce"leHVle xx
h = 2s :!:: 15%
Pa3MepbI B MM
HOMHHaJILHai BuyTpeHHHi BHemHHi P8,lUIycr
p83Mep H ,lUIaMeTp ,lUIaMeTp d t TOJIIIUIHa ,lUIaMeTp d 2
pe3L6Ll a d H IIIIIpHHa s
MaKC. MHH. MaKC. MaKC.
M3 3,3 3,1 1 :to, 1 5,5 0,3
(M3,5) 3,8 3,6 1 :to, 1 6,0 0,3
M4 4,35 4,1 1 2 :to, 1 6,95 0,4
M5 5,35 5,1 1,5 :to, 1 8,55 0,5
M6 6,4 6,1 1,5 :to, 1 9,6 0,5
M8 8,55 8,2 2 IO, 1 12,75 0,65
MIO 10,6 10,2 2,5 :to, 15 15,9 0,8
M12 12,6 12,2 2,5 :to, 15 17,9 0,8
(M 14) 14,7 14,2 3 :to, 2 21,1 1,0
M16 16,9 16,3 3,5 :t 0,2 24,3 1,15
(M 18) 19,0 18,3 3,5 :to,2 26,4 1,15
M20 21,1 20,3 4,5 :t 0,2 30,5 1,5
(M22) 23,3 22,4 4,5 :t 0,2 32,7 1,5
M24 25,3 24,4 5 :to, 2 35,7 1,65
(M27) 28,5 27,5 5 :to, 2 38,9 1,65
M30 31,5 30,5 6 :to, 2 43,9 2,0
(M33) 34,6 33,5 6 :to, 2 47,0 2,0
M36 37,6 36,5 7 :t 0,25 52,1 2,3
(M39) 40,8 39,6 7 :t 0,25 55,3 2,3
M42 43,8 42,6 8 :t 0,25 60,3 2,65
(M45) 46,8 45,6 8 :t 0,25 63,3 2,65
M48 50,0 48,8 8 :t 0,25 66,5 2,65
a B cK06Kax )'Ka3aHbI cneQHaJIbHbIe, pe,ll;KO BCTpeqaIOID;HeCH pa3MepbI.
,lJ.aJIbHeHrnyIO HH<popMaQHIO CMOTpH B BS 4464.
F/la8a 4. Kpene:JICHble oemOJlu
4. 1.60. npyJKJfHHSfI wsif6s C npflMoyrOn&H&IMJf KOHqSMJf:
MeTpJflfeCKSfI CepJffl, TJfn&1 B Jf BP
k
WTpVlx-nYHKTVlpOM
nOKaaaHa np VlHHa
wat16a C nJ10CKVlM
KOHU.OM
YBeJ1V1"1eHHblt1 ct>parMeHT A
McnonHeHMe BP
h 1 = (28 + 2k) :t 15%
McnonHeHMe B
Ce"leHVle XX
h 2 = 28 :t 15%
Pa3MepbI B MM
HOMHHaJlbHLIH BuyrpelUllfii BHeomHii P8,lUIyC , k (TOJlbKO
pa3Mep ,lUI8MeTp d t illHpHHa b TOJlllUllla s ,lUI8MeTp d 2 .lJ,JUI THDa
H ,lUIaMeTp BP)
pe3L6 y a d MaKC. MHH. MaKC. MaKC.
Ml,6 1,9 1,7 0,7:t 0,1 0,4 :t 0,1 3,5 0,15 -
M2 2,3 2,1 0,9 :t 0,1 0,5 :t 0,1 4,3 0,15 -
(M2,2) 2,5 2,3 1,0 :t 0,1 0,6 :t 0,1 4,7 0,2 -
M2,5 2,8 2,6 1,0 :t 0,1 0,6 :t 0,1 5,0 0,2 -
M3 3,3 3,1 1,3 :t 0,1 0,8 :t 0,1 6,1 0,25 -
(M3,5) 3,8 3,6 1,3 :t 0,1 0,8 :t 0,1 6,6 0,25 0,15
M4 4,35 4,1 1,5 :t 0,1 0,9 :t 0,1 7,55 0,3 0,15
M5 5,35 5,1 1,8 :t 0,1 1,2:t 0,1 9,15 0,4 0,15
4.1. BUHmo6ble "peneJICHble U30eJlUR 245
O"OHltaHUe
HOMHHaJlbllhlii BuyrpelUllfi BHeDIHHH P8,lUIyC , k (TOJlhKO
p83Mep ,lUIaMeTp d t illupHHa b TOJlllUllla s ,lUIaMeTp d 2 .lJ,JUI TUDa
H 8MeTp BP)
pe3L6 y a d MaKC. MHH. MaKC. MaKC.
M6 6,4 6,1 2,5 :t 0,15 1,6 :t 0,1 11,7 0,5 0,2
M8 8,55 8,2 3 :t 0,15 2 :t 0,1 14,85 0,65 0,3
MI0 10,6 10,2 3,5 :t 0,2 2,2 :t 0,15 18,0 0,7 0,3
M12 12,6 12,2 4 :t 0,2 2 5 :t 0,15 21,0 0,8 0,4
(M14) 14,7 14,2 4,5 :t 0,2 3 :t 0,15 24,1 1,0 0,4
M16 16,9 16,3 5 :t 0,2 3,5 :t 0,2 27,3 1,15 0,4
(M18) 19,0 18,3 5 :t 0,2 3,5 :t 0,2 29,4/ I 1,15 0,4
M20 21,1 20,3 6 :t 0,2 4 :t 0,2 33,5 1,3 0,4
(M22) 23,3 22,4 6 :t 0,2 4 :t 0,2 35,7 1,3 0,4
M24 25,3 24,4 7 :t 0,25 5 :t 0,2 39,8 1,65 0,5
(M27) 28,5 27,5 7 :t 0,25 5 :t 0,2 43,0 1,65 0,5
M30 31,5 30,5 8 :t 0,25 6 :t 0,25 48,0 2,0 0,8
(M33) 34,6 33,5 10 :t 0,25 6 :t 0,25 55,1 2,0 0,8
M36 37,6 36,5 10 :t 0,25 6 :t 0,25 58,1 2,0 0,8
(M39) 40,8 39,6 10 :t 0,25 6 :t 0,25 61,3 2,0 0,8
M42 43,8 42,6 12 :t 0,25 7 :t 0,25 68,3 2,3 0,8
(M45) 46,8 45,6 12 :t 0,25 7 :t 0,25 71,3 2,3 0,8
M48 50,0 48,8 12 :t 0,25 7 :t 0,25 74,5 2,3 0,8
(M52) 54,1 52,8 14 :t 0,25 8 :t 0,25 82,6 2,65 1,0
M56 58,1 56,8 14 :t 0,25 8 :t 0,25 86,6 2,65 1,0
(M60) 62,3 60,9 14 :t 0,25 8 :t 0,25 90,8 2,65 1,0
M64 66,3 64,9 14 :t 0,25 8 :t 0,25 93,8 2,65 1,0
(M68) 705 69,0 14 :t 0,25 8 :t 0,25 99,0 2,65 1,0
a B cK06Kax )'Ka3aHbI cneQHaJIbHble, pe,ll;KO BC'rpeqaIOID;HeCH pa3MepbI.
,lJ.aJIbHeHrnyIO HH<popMaQHIO CMO'rpH B BS 4464.
4. 1.61. npyJKJfHHSfI ASYXSJfTKOSSfI WSM6s
C npflMoyrOn&H&IM nonepelfH&IM CelfeHJfeM:
MeTpJflfeCKJfM PflA, JfCnOnHeHJfe D
'j'
r r
Ce"leH Vie XX
F/1080 4. Kpene:JICHble oemOJlu
Pa3MepbI B MM
HOMHHaJlbllhlii BuyrpellllHii ,lUIaMeTp BHelllllHii P8,lUIycr
p83Mep IIIHpHHa b TOJlllUllla s ,lUIaMeTp d 2
H,lUIaMeTp
pe3L6 y a d MaKC. MHH. MaKC. MaKC.
M2 2,4 2,1 0,9 :t 0,1 0,5 :t 0,05 4,4 0,15
(M2,2) 2,6 2,3 1,0 :t 0,1 0,6 :t 0,05 4,8 0,2
M2,5 2,9 2,6 1,2:t 0,1 0,7 :t 0,1 5,5 0,23.
M3,0 3,6 3,3 1,2:t 0,1 0,8 :t 0,1 6,2 0,25
(M3,5) 4,1 3,8 1,6 :t 0,1 0,8 :t 0,1 7,5 0,25
M4 4,6 4,3 1,6 :t 0,1 0,8 :t 0,1 8,0 0,25
M5 5,6 5,3 2 :t 0,1 0,9 :t 0,1 9,8 0,3
M6 6,6 6,3 3 :t 0,15 1 :t 0,1 12,9 0,33
M8 8,8 8,4 3 :t 0,15 1,2 :t 0,1 15,1 0,4
MI0 10,8 10,4 3,5 :t 0,20 1,2 :t 0,1 18,2 0,4
M12 12,8 12,4 3,5 :t 0,2 1,6 :t 0,1 20,2 0,5
(M 14) 15,0 14,5 5 :t 0,2 1,6 :t 0,1 25,4 0,5
M16 17,0 16,5 5 :t 0,2 2 :t 0,1 27,4 0,65
(M18) 19,0 18,5 5 :t 0,2 2 :t 0,1 29,4 0,65
M20 21,5 20,8 5 :t 0,2 2 :t 0,1 31,9 0,65
(M22) 23,5 22,8 6 :t 0,2 2,5 :t 0,15 35,9 0,8
M24 26,0 25,0 6 5 :t 0,2 3,25 :t 0,15 39,4 1,1
(M27) 29,5 28,0 7 :t 0,25 3,25 :t 0,15 44,0 1,1
M30 33,0 31,5 8 :t 0,25 3,25 :t 0,15 49,5 1,1
(M33) 36,0 34,5 8 :t 0,25 3,25 :t 0,15 52,5 1,1
M36 40,0 38,0 10 :t 0,25 3,25 :t 0,15 60,5 1,1
(M39) 43,0 41,0 10 :t 0,25 3,25 :t 0,15 63,5 1,1
M42 46,0 44,0 10 :t 0,25 4,5 :t 0,2 66,5 1,5
M48 52,0 50,0 10 :t 0,25 4,5 :t 0,2 72,5 1,5
M56 60,0 58,0 12 :t 0,25 4,5 :t 0,2 84,5 1,5
M64 70,0 67,0 12 :t 0,25 4,5 :t 0,2 94,5 1,5
a B cK06Kax )'Ka3aHbI cneQHaJIbHbIe, pe,ll;KO BCTpeqaIOID;HeCH pa3MepbI.
IIpuMettaHue. BbICOTa ,L(ByxBHTKOBOH maH6bI B CB060,L(HOM COCTO.HHHH rrepe,n
cxaTHeM 06blqHO rrpH6JIH3HTeJIbHO B II.HTb pa3 60JIbllIe ee TOJIII(HHbI, HO eCJIH rpe-
6yroTCH llIai1:6bI C ,L(pyrOH BbICOTOH B CB060,L(HOM COCTO.HHHH, OHH MOryr 6bITb rrony-
qeHbI B COOTBeTCTBHH C ,L(orOBopeHHOCTbIO Me)l{,L(Y rroTpe6HTeJIeM H H3rOTOBHTeJIeM.
)];aJIbHeHmyIO HH<popMaUHIO CMOTPH B BS 4464.
4.1. BUHmo8ble "pene:JICHble U30e.tlUR
4.1.62. npyJKJfHH&le wsii6&1 C 3y6&flMJf, MeTpJflfeCKJfe
HCnOJIHeHHe A C Hap)T)Kllh.MH ay6hSlMH
HCnOJIHeHHe J C BHyrpeHHHMH ay6hBMH
81
min. 2B.t
min. 28 1
HCnOJIHeHHe V C ay6hBMH noon BHHTh. C nOTaHHOH rOJIOBKOH
90°
IlapaMeTpbI, He orrpe,L(eJIeHHble TeXHHqeCKHMH YCJIOBH.HMH, ,L(OJDKHbI 6bITb rrpo-
pa60TaHbI COOTBeTCTByrolI(HM 06pa30M. )
0603HaqeHHe 3y6qaTOH rrp)')KHHHOH rnaH6bI THna J C ,L(HaMeTpOM OTBepCTIDI
d] = 6,4 MM, H3rOTOBJIeHHotf H3 rrp)')KHHHoH CTaJIH, C <poc<paTHbIM nOKpbITHeM ,L(JI.H
3a HTbI OT KOpp03HH: 3y6qaTaH IIp)')KHHHIDI maH6a J 6.4 DIN 6797 - XHM. <l>oc.
npM.
0603HaqeHHe 3y6qaTbIX rrpYXHHHbIX IIIatf6 ,L(JIH 60JITOB C JIeBotf pe3b60tf qHTa-
eTCH: 3y6qaTIDI rrp)')KHHHa.H rnatf6a J 6.4 JIeBa.H DIN 6797- XHM. <l>oc. rrpM.
F/la8a 4. Kpene:JICHble oemOJlu
Pa3MepbI B MM
qHCJIO 3yOheB Bee (7,85 Kr/JJ.M 3 ) ,lI;mI
d d d3 sl s2 MHH. Kr/lOOO DJTYK a-
(HI3) (1\14) MeTpa
AHJ V A J V pe3bOw
1..7 3,6 - 0,3 - 6 - 0,01 - - 1..6
1,8 3,8 - 0,3 - 6 - 0,015 - - 1,7
1,9 4 - 0,3 - 6 - 0,02 0,03 - 1,8
2,2 4,5 4,2 0,3 0,2 6 6 0,025 0,04 0,02 2
2,5 5 - 0,4 0,2 6 6 0,03 0,025 - 2,3
2,7 5,5 5,1 0,4 0,2 6 6 0,04 0,045 0,025 2,5
2,8 5,5 - 0,4 0,2 6 6 0,04 0,045 - 2,6
3,2 6 6 0,4 0,2 6 6 0,045 0,045 0,025 3
3,7 7 7 0,5 0,25 6 6 0,075 0,085 0,04 3,5
4,3 8 8 0,5 0,25 8 8 0,095 0,1 0,05 4
5,l a 9 - 0,5 - 8 - 0,14 0,15 - 5
5,3 10 98 0,6 0,3 8 8 0,18 0,2 0,12 5
6,4 11 11,8 0,7 0,4 8 10 0,22 0,25 0,2 6
7,4 12,5 - 0,8 - 8 - 0,3 0,35 - 7
8,2 a 14 - 0,8 - 8 - 0,4 0,45 - 8
8,4 15 15,3 0,8 0,4 8 10 0,45 0,55 0,4 8
10,5 18 19 0,9 0,5 9 10 0,8 0,9 0,7 10
12,5 20,5 23 1 0,5 10 10 1..1 1,3 1..2 12
14,5 24 26,2 1 0,6 10 12 1,7 2 1,4 14
16,5 26 30,2 1,2 0,6 12 12 2,1 2,5 1,4 16
19 30 - 1,4 - 12 - 3,5 3,7 - 18
21 33 - 1,4 - 12 - 3,8 4,1 - 20
23 36 - 1,5 - 14 - 5 6 - 22
25 38 - 1,5 - 14 - 6 6,5 - 24
28 44 - 1,6 - 14 - 8 8,5 - 27
31 48 - 1,6 - 14 - 9 9,5 - 30
a TOJIbKO ,lI)UI 60JITOB C llieCTHyrOJIbHOH rOJIOBKOH.
,lJ.JIH ,lI;aJIbHeHlliHX ,lI;eTaJIeH CMOTpH DIN 6797.
4. 1.63. CTonopHSfI Wsif6&1 C 3y6&flMJf, MeTpJflfeCKJfe
HCnOJllleHHe A c Hap)T)KllhlMH 3y6hBMH
HCnOJlHeHHHe J C BuyrpeHHHMH 3y6bBMH
51
3S1
:==38 1
4.1. BUHmo6ble "peneJICHble U30eJlUR 249
HCnOJlHeHHHe V C 3y6hBMH
DO)); BHHTh. C nOTaHHOH
rOJIOBKOH
d:3
90°
IlapaMeTPbI, He orrpe,L(eJIeHHble TeXHHqeCKHMH YCJIOBH.HMH, ,L(OJDKHbI 6bITb rrpo-
pa60TaHbI COOTBeTCTByrolI(HM 06pa30M.
0603HaqeHHe CTonopHOH maif6bI C 3y6b.HMH HCHOJlHeHH.H J C ,L(HaMeTPOM OT-
BepCTIDI d l = 6,4 MM, H3rOTOBJIeHHOH H3 rrpyxHHHOH CTaJIH, C <poc<paTHbIM rrOKpbI-
THeM AIDI 3alUHTbI OT KOpp03HH: CTOrrOpHa.H C 3y6b.HMH maH6a J 6.4 DIN 6797 -
XHM. <Doc. rrpM.
0603HaqeHHe CTOIIOpHbIX maH6 C 3y6b.HMH AIDI 60JITOB C JIeBoif pe3b60if qHTa-
eTCH: 3y6qaTIDI rrpyxHHHIDI rnaH6a J 6.4 JIeBaH DIN 6797 - XHM. <Doc. rrpM.
Pa3MepbI B MM
qHCJIO 3yOheB MHH. Bec (7,85 KI'/JUl3) ,lI;mI
d d d KI' /1000 DJTYK ,lUI8 -
(HI3) (1\14) 3 Sl S2 MeTpa
A J V AHJ V pe3L6w
1,7 3,6 0,3 - 9 7 - 0,02 - 1,6
1,8 3,8 0,3 - 9 7 - 0,02 - 1,7
1,9 4 0,3 - 9 7 - 0,025 - -
2,2 4,5 4,2 0,3 0,2 9 7 10 0,03 0,025 2
2,5 5 0,4 0,2 9 7 10 0,04 - 2,3
2,7 5,5 5,1 0,4 0,2 9 7 10 0,045 0,03 2,5
2,8 5,5 0,4 0,2 9 7 10 0,05 - 2,6
3,2 6 6 0,4 0,2 9 7 12 0,06 0,04 3
3,7 7 7 0,5 0,25 10 8 12 0,11 0,075 35/
,
4,3 8 8 0,5 0,25 11 8 14 0,14 0,1 4
5 l a 9 0,5 - 11 8 - 0,22 - 5
,
5,3 10 9,8 0,6 0,3 11 8 14 0,28 0,2 5
6,4 11 11,8 0,7 0,4 12 9 16 0,36 0,3 6
7,4 12,5 0,8 - 14 10 - 0,5 - 7
82 a 14 0,8 - 14 10 - 0,75 - 8
,
8,4 15 15,3 0,8 0,4 14 10 18 0,8 0,5 8
10,5 18 19 0,9 0,5 16 12 20 1,25 1 10
12,5 20,5 23 1 0,5 16 12 26 1,7 1,5 12
14,5 24 26,2 1 0,6 18 14 28 2,4 2 14
16,5 26 30,2 1,2 0,6 18 14 30 3 2,4 16
19 30 - 1,4 - 18 14 - 5 - 18
F/la8a 4. Kpene:JICHble oemOJlu
OKOH"taHue
qHCJIO 3y6heB MHH. Bee (7,85 KI' / 3)
d d d3 KI' /1000 DJTYK ,lUIa-
(HI3) (1\14) St s2 MeTpa
A J V AHJ V pe3L6w
21 33 - 1,4 - 20 16 - 6 - 20
23 36 - 1,5 - 20 16 - 7,5 - 22
25 38 - 1,5 - 20 16 - 8 - 24
28 44 - 1,6 - 22 18 - 12 - 27
31 48 - 1,6 - 22 18 - 14 - 30
a TOJIbKO ,lI)UI 60JITOB C meCTHrpaHHOH rOJIOBKOH. DOJIee nO,ll;p06HO CMOTpH B DIN 6797.
4. 1.64. ISO MeTpJflfeCKSfI npyJKJfHHSfI BOnHJfCTSfI wsii6a:
o6tqee MSWJfHOCTpOeHJfe
d 2 d 1
--L
h
f
Pa3MepbI B MM
HOMHHaJlbHhIii Buyrpelllllli aMeTp d t BHelIllllli aMeTp d 2 BhlCOTa h
(pe3hOOBoii) TOJIDUDlas
aMeTpa MaKC. MHH. MaKC. MHH. MaKC. MHH.
MI,6 1,8 1,7 3,7 3,52 0,51 0,36 0,16
M2 2,3 2,2 4,6 4,42 0,53 0,38 0,16
M2,5 2,8 2,7 5,8 5,62 0,53 0,38 0,16
M3 3,32 3,2 6,4 6,18 0,61 0,46 0,16
M4 4,42 4,3 8,1 7,88 0,84 0,69 0,28
M5 5,42 5,3 9,2 8,98 0,89 0,74 0,30
M6 6,55 6,4 11,5 11,23 1,14 0,99 0,40
M8 8,55 8,4 15,0 14,73 1,40 1,25 0,40
MI0 10,68 10,5 19,6 19,27 1,70 1,55 0,55
M12 13,18 13 0 22,0 21,67 1,90 1,65 0,55
,
(M14) 15,18 15,0 25,5 25, 17 2,06 1,80 0,55
M16 17,18 17,0 27,8 27,47 2,41 2,16 0,70
(M18) 19,21 19,0 31,3 30,91 2,41 2,16 0,70
M20 21.21 21.0 34.7 34.31 2.66 2.16 0.70
a BTOpOH PH.D: pa3MepOB B cK06Kax.
DOJIee nOJIJIyIO HH<popMaQHIO CMOTpH BS 4463.
4.1. BUHmo6ble "peneJICHble U30eJlUJl 25 I
4.1,,65. npocpJfn& nS30B nOA T-06pS3HYIO ronoBKY 60nTs
r
YAaJU1Tb OCTpbl
yrOJ1
0.3 x 45° max.
K
H
G
Pa3MepbI B MM
mllpHHa fnyfiHHa nOJIHaJI 4>acKa x 45 0 PacCTOJllllle Me
mllpHHa ropJlOBHHhI At D83a Bt Da3a C t rJIyOHHa HJIH P8.ZUIYC D838MH P (H36eraiiTe
0603ua- T-D83a H 3Ha1feHHii mara B
qeHHe ,lI;JuI HCDOJll»30- CKoOK8X, DOCKOJlh&y
T-0838 Ho- 00""- BaIllUl B Ka1fe- OHH MOryr CTaTh
MH- HhIii . . . :I :I :I Dpll'llllloii
HaJJ (H 12) CTBe IIIIIIIJIhKH t.::) Henp01lllOCTH)
(HB) :I :I :I
M4 5 +0,12 +0,018 10 11 3,5 4,5 8 10 20 25 32
M5 6 0 0 11 12,5 6,7 5 11 13 25 32 40
M6 8 +0,15 +0,022 14,5 16 7 8 15 18 1,0 0,6 1,0 32 40 50
M8 10 0 0 16 18 7 8 17 21 40 50 63
MI0 12 +0,18 +0,027 19 21 8 9 20 25 (40) 50 63 80
- -
M12 14 0 0 23 25 9 11 23 28 (50) 63 80 100
-
M16 18 30 32 12 14 30 36 1,6 1,6 (63) 80 100 125
M20 22 +0,21 +0,033 37 40 16 18 38 45 (80) 100 125 160
1,0 2,5
M24 28 0 46 50 20 22 48 56 100 125 160 200
-
M30 36 +0,25 +0,039 56 60 25 28 61 71 125 160 200 250
M36 42 0 0 68 72 32 35 74 85 2,5 1,6 4,0 160 200 250 320
M42 48 80 85 36 40 84 95 200 250 320 400
M48 54 +0,30 +0,046 90 95 40 44 94 106 2,0 6,0 250 320 400 500
0 0
,lJ;OnyCKH Ha paCCTOJlHHJI Me1K,ZJ;y oa3aMB p ,ZJ;JIJI T -oa30B
mar (MM) ,n;OnyCK (MM)
Or 20 ,lI;0 25 :to,2
OT 32 ,lI;0 100 :to, 3
OT 125 ,lI;0 250 :to,5
OT 320 ,lI;0 500 :to,8
)];a.rrbHeHIIIyro HH<popMaUHIO CM01'pH B BS 2485.
F/la8a 4. Kpene:JICHble oemOJlu
4.1.66. PS3Mep&1 6onToB Jf rseK K T-06pS3H&IM nS3SM
T -raMKa
T -6onT
K A aMeTp
pe3b6bl
K A aMeTp
pe3b6bl
-1
B.2
M
L
PaBHbie
paCCT05lH 5I
A A a-
MeTp
C 2
F
t
f
N
F
R
E
8 2
E
C 2
B.2
Ta6JI. 1.
0603Ha- IIIHpHHa BWCOTa ,n;Ha- <l>aCKa Pa;ntyc illHpHHa BWCOTa <l>acu
1feHHe I'OJlOBKH BWCOTa CTYDeHH MeTp E r T -raiiKII G T -raiiKH F
T -ooJlTa H (KBa.npaT) B 2 I'OJlOBKH c CTyIIeHH H
;ntaMeTpa ,lI;oD.:tO,2 CTep1KllJl CTep1KllJl
Pe3hOW K HOM. ,lI;OD. N A 2 MaKe. MaKe. HOM. ,lI;OD. MaKe. MaKC.
M4 9 2 5 - 4 5 6
M5 10 4 - 5 6 -0 3 8
M6 13 6 - 6 8 -0 5 10
M8 15 6 8 1,0 10 12
-
MI0 18 0 7 - 10 12 16
M12 22 -0 5 8 - 12 14 -0 3 19
M16 28 10 - 16 18 -0 6 25
M20 34 14 20 1,0 22 30 0,3x45°
-
M24 43 18 20 26 1,5 28 40
M30 53 23 20 33 36 -0 4 45
M36 64 28 25 39 42 -0,7 55
M42 75 32 25 46 48 65
M48 85 36 30 52 2,5 54 -0,4 75
-0,8
aJIbHeHmyIO HH<p0pMaUHIO CMOTpH B BS 2485.
4.1. BUHm06ble "peneJICHble U30eJlUJl 253
Ta6JI. 2.
Pa3MepbI B MM
,[(HaMeTp PeKOMeH,ZJ;OBaHHaJI .zunma CTep1KllJl OOJITa L Aruma Pe3hOhI Ha
pe3hOhI K CTep1Klle 60JITa M
M4 30 40 50 60 70 80 100 ,[(IDI L 100 M = 0,5L
M5 30 40 50 60 70 80 100
M6 60 70 80 100
M8 60 70 80 100
MI0 60 70 80 100 125 160 180 / I
M12 70 80 100 125 160 180 ,[(IDI L > 100 M = 0,3L
M16 70 80 100 125 160 180 200 250 300
M20 70 80 100 125 160 180 200 250 300
M24
M30
M36 100 125 160 180 200 250 300
M42
M48
)];aJIbHeiimyIO HH<popMaUHIO CM01'pH B BS 2485.
4. 1.67. PS3Mep&1 wnOHOK AJlfl T-06pS3H&IX nS30B /
M- Ad!a.
B dia.
1
L
j
w
E
p
Q
1
D
B.2
E
F/la8a 4. Kpene:JICHble iJemOJlu
=
::is
c.
Q)
M
t'<S
Q...
I =;
1- t "! V) V) V) V)
c:t.. 8u «I e= .., .., .., ..,
c:t.. + N \0 00
t!1I ):I1u
i
I ::II
81 0.., 0
c:t.. I Q == 0... ...
i i ! ... \0 ......c
......c ......c
i= =&i
8
i = i
1°-1= ... ... \0 0
it = \0'" 0\
e 5 i; = N
I f'11!!0
= u
iill;li N \0 00
i....fl..
::I IIQ
i" ..
!!I iE! !!I
U IIQ c:t..:11 IIQ
i \0 0... V)
... ...
0 ......c ......c
::I
! V) 0 0 0
N \0
i Q, 0 N 0 0
eue ......c ......c
=
«I ;
S eO V) V) V) V)
... ......c N
a V)
.-..
«I roo- 00 ......c V) 0
..c:: ......c N N
I --- 0 0... 0 0... 0 0... 0 0...
i; =1 0 0 0 0
I I I I
=e
ia :i \0 0 0 0
e e ......c V)
= =
.-..
a roo- N V) 00 ......c V) 0
..c:: ......c ......c ......c N N
--- 0 0 0 0 q 0 0... 0 0... 0 0...
0
i 0'" 0'" 0 0 0 0
I I I I I I
, :i V) \0 00 0 N 00 N 00 \0 N 00
i ......c ......c ......c ......c N N V)
I
ia
'r 0 N \0 0 0 \0 N 00
o E-4 V) \0 00 ......c ......c ......c N N
.
f'.
00
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00
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:=
o
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:=
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:=
:=
i
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4.2. 3a1(,Jleno"tHble coeOUHeHUJl 255"
4.2. 3aKnenO'lHble coeAMHeHMH
4.2.1. TJfnJflfH&le rOnOBKJf Jf CTepJICHJf 3SKJlenOK
<l>D
2D
D
HHa [
3 D
2
f
nOJ1YKpyr J1a51
J1 n0J1YKpyrJ1a51
3aKJ1aAHa51 r0J10BKa
U. J1 HAPVl"leCKa51
cKpyrJ1eHHa51
r0J10BKa
H 3Ka51
nOJ1YKpyr J1 a51
r0J10BKa
2D
KOHycoo6pa3Ha51
r0J10BKa
nJ10CKa51
r0J10BKa
nOTaVlHa51
r0J10BKa
yrOJ1 90°
U. J1 HAP "IeCK V1
J1 cnJ10WHOVi
CTep)l(eH b
nOJ1ynycTOTeJ1bIVi
CTep)l(eHb
nYCTOTeJ1blVI
CTep)l(eH b
J1 n CTOH
Pa3ABoeHHblVi
(B J1b"laTbIVl)
CTep)l(eHb
CTe )l(eHb
AJ151 M5IrK X
MaTep aJ10B
F/la8a 4. Kpene:JICHble iJemOJlu
4.2.2. TJfnJflfH&le 3aKJIenOIfH&le COeAJfHeHJffl BHaxnecTKY
O,ZJ;llopJl,ZJ;Hoe COe,ZJ;HHeHHe BHaxJIeCTKY
M. r-I'" 3D _I. 3D _I D - AlllaMeTp CTep)l(HSI 3aKJ1enKIII
W
, , , ,
BeJ1 "I Ha Ha-
XJleCTK 30 Q
..-
,[(BYXPJl,ZJ;HOe COe,ZJ;HHeHHe BHaxJIecTKY (,ZJ;Be D;enO'lKH)
M. r -I'" 3D _I. 3D -I
PaCCT05lH e Me)I(AY WBa ,
p5lAaM 3aKJ1enOK ' , ,
- -- --- -----
BeJ1 "I Ha Ha- Q
XJleCTK 60 c-)
Q
..-
,[(BYXPJl,ZJ;HOe coe,n;HHeHHe BHaXJIecTKY (3Hr3ar)
1.5D
3D
ar
3D
3D
PaCCT05lH e Me)I(AY
p5lAaM 3aKJ1enOK
BeJ1 "I Ha Ha-
XJleCTK 60
Q
c-)
Q
....
3D 3D
4.2. 3a1(,Jleno"tHble coeOUHeHUJl 257
4.2.3. TJfnJflfH&le CT&IK08&le 3SKhenOlfH&le COeAJfHeHJffl
3aKJIenO'IHOe CTLIKOBoe COe,ZJ;HHeHHe C O,ZJ;llOH HaKJIa;ncOH (o,ZJ;llop.H,ZJ;IIoe)
1.5D 3D 3D 3D D - A aMeTp CTep)l(H5I 3aKJ1enK
War
W p Ha
HaKJ1aAK
6D
i
(CTbIKOBa5l)
HaKJ1aAKa
IlpUMettanue. 3TO coe,nHHeHHe TaIOKe MoxeT 6bITb BbInOJIHeHO ,nBOHHbIM
pH,Il;OM 3aKJIenOK, uenOqKaMH HJIH 3Hr3arOM. IllHpHHa HaKJIa,nKH = 12D npH
,IJ;BOHHOM pR,LI.e 3aKJIenOK (IIIar Me)l{)(y pR,LI.aMH = 3D).
3aKJIenO'IHOe CTLIKOBoe COe,ZJ;HHeHHe C ,ZJ;BYMJI HaKJIa)Q(aMH (,ZJ;BYXPJl)l;Hoe)
1.5D 3D
3D
3D
W p Ha
(CTbIKOBO )
HaKJ1aAK
12D
Q
.....
PaCCT05lH e Me)I(AY
p AaM 3aKJ1enOK
IIpuMettanue. 3TO coe,nHHeHHe TaIOKe MOxeT 6bITb BbInOJIHeHO ,nBOHHbIM pR,LI.OM
3aKJIenOK, pacnOJIOXeHHbIX 3Hr3arOM (CM01'pH pa3,neJI 4.2.2), MH o,nHHapHbIM P.H-
,aOM 3aKJIenOK, CM01'pH BbIIIIe.
9-3255
F/la8a 4. Kpene:JICHble OemOJlu
4.2.4. CooTHOWeHJffl MeJKAY AJfaMeTpOM oTSepCTJffl
Jf A/lJfHoii 3aKJIenKJf
1.0625D
1 D r
4.2.5. 3aKJIenKJf C nonYKPyrnoif 3aKJIaAHOif ronosKoif,
Jf3rOTOSneHH&le xonoAHoif wTaMnosKoif
D
I-
I
-I
K
npH d = 16 MM HJlH MeHbme
D=1,75d
K = 0,6 d
L = ,lI;JIHHa
L
LJ
p
a3Mepbl B MM
HOMHH8JIhllblH ,n;OnyCK Ha HOMHH8JIh1lblH ,n;OnyCK Ha HOMHH8JIhHaJI ,n;OnyCK Ha ,n;OnyCK HI
;ntaMeTp ;ntaMeTp d ;ntaMeTp ;ntaMeTp D BMCOTa BMCary ,zuumy L
CTep1KllJl a d I'OJIOBKH D I'OJlOBKH K I'OJlOBKH K
1 1..8 0..6 +0,2
1..2 2..1 :f:0,2 0..7
-0,0
1..6 2..8 1..0
2..0 :f:0,07 3..5 1..2
2..5 4..4 :f:0,24 1..5 +0,24 +0,5
-00
3..0 5..3 1..8 , -0,0
(3..5) 6..1 2..1 +0,29
4 7..0 :f:0,29 2..4
:f:0,09 -0,0
5 8..8 3..0
6 10..5 3..6 + 0,35
(7) 12..3 :f:0,35 4..2
-0,0 +0,8
8 :f:0,11 14..0 4..8
-0,0
10 18..0 6..0
12 21..0 :f:0,42 7..2 + 0,42 + 1,0
(14) :f:0,14 25..0 8..4 -00
, -00
16 28..0 9..6 ,
a B cKo6Kax y:Ka3aHbI CneUHaJIbHbIe pa3MepbI 3aKJIenOK.
,LJ;aJIbHeHmyIO HH<popMaI(HIO CMOTpH B BS 4620: 1970.
4.2. 3aK./leno"tHble coeiJuHeHUR 259
4.2.6. 3aKJIenKJf C nonYKpyrnoH 3aKJIaAHOif ronoBKoif,
Jf3rOTOSneHH&le ropfllfeif wTaMnosKoif
D
K
L
ITpH d = 14 MM HJlH 60JIbme
D = 1,6 d
K = 0,65 d
L = )];JIHHa
Pa3MepbI B MM
HOMHllaJlbllblH ,n;OnyCK Ha HOMHHaJlbHWi ,n;OnyCK Ha HOMHHaJlbHaJI ,n;OnyCK Ha ,n;OnytK
ABaMeTp ;ntaMeTp d ;ntaMeTp ;ntaMeTp D BMCOTa BMCary Ha MIlBY L
CTeJUKlUla d I'OJlOBKH D I'OJlOBKH K I'OJlOBKH K
(14) 22 9 + 1,0
16 :1:043 25 :I: 1,25 10 + 1,00 -0,0
-00
,
18 28 11,5
20 32 13 + 1,6
(22) :f:0,52 36 + 1,8 14 + 15 -00
,
-00
,
24 40 16
(27) 43 17
30 48 :f:2,5 19 + 2,0
-0,0
(33) :f:0,62 53 21 + 3,0
-0,0
36 58 23 + 2,5
62 :f:3 ,0 25 -00
39 ,
a B cKo6Kax y:Ka3aHbI CneI(HaJIbHbIe pa3MepbI 3aKJIenOK.
)];a.rrbHeiimyIO HH<popMaUHIO CM01'pH B BS 4620: 1970.
F/la8a 4. Kpene:JICHble oemOJlu
4.2. 7. npeAnonaraeM&IH Alfana30H HOMJfHan&H&IX ,qJJJfH,
CBfl3aHH&IX C AJfaMeTpOM CTepJICHeH
Pa3MepbI B MM
HOMHHaJlbHWi HOMHHaJlbHaJI MIIIIa a L
,lUIaMeTp
CTep1KllJl* d 3 4 5 6 8 10 12 14 16 (18) 20 (22) 25 (28) 30 (32) 35 (38) 40 45 50 55
1,0 x x x x x x x x x x
1,2 x x x x x x x x x x
1,6 x x x x x x x x x x x x
2,0 x x x x x x x x x x x x
2,5 x x x x x x x x x x x X
I
3,0 x x x x x x x x x x x X
I
(3,5) I
4,0 x x x x x x x x x
5,0 x x x x x x x x x x x x x x
6,0 x x x x x x x x x x x x x x x
a Pa3MepbI H ,lI;JIHHbI, y:Ka3aHHbIe B cKo6Kax, HBIDIIOTCH CneI(HaJIbHbIMH H HX cne.nyeT no'
B03MO)l{}{OCTH H36eraTb. BKJIlOqeHHe B Ta6nHI(bI ,lI;aHHbIX 0 pa3Mepax He 03HaQaeT, QTO Bce
onHcaHHbIe H3,l1;eJIHH BXO,ll;HT B accopTHMeHT BbInYCKaeMOH npO,n;yKUHH. nOTpe6HTeJIIO CJIe-
,n;yeT npOKOHCYJIbTHpOBaTbCH C H3rOTOBHTeJIeM OTHOCHTeJIbHO cnHCKa HMeIO HXCH B HaJIH-
QHH pa3MepOB H3,l1;eJIHH.
nOJIHbIH ,lI;Hana30H THnOB H pa3MepOB rOJIOBOK, BKJIIOQ3jl ,lI;HaMe'rp 39 MM npH ,nJIHHe
CTepXHH 60 MM, CMO'rpH B BS 4620: 1970.
4.2.8. POFJf8J 3aKJIenKJf
POP@ HJIH O,nHOCTOpOHHIDI KJIenKa - 3TO MeTO,n, KOTOpbIH n03BOJUIeT BbInOJIIDITb
MeXaHlNeCKHe coe,nHHeHHH, Kor,na ,nOC1)'I1 orpaHlNeH TOJIbKO O,nHOH CTOpOHOH co-
e,nHH.HeMbIX qaCTeH; HO 6JIaro,nap.H Ha,ne)I{JIOCTH, npe,nCKa3yeMocTH, HH3KOH CTOH-
MOCTH C60PKH H npOCTOTe npHMeHeHHH O,nHOCTOpOHHAA KJIenKa IIIHpOKO HCnOJ!bool
3yeTC.H TaIOKe H Tor,na, Kor,na ,nOC1)'I1HbI 06e CTOpOHbI coe,nHHeHHH. PO P@, KaK If,
,npyrHe BH.l(bI CHCTeM O,nHOCTopoHHeH KJIeIIKH, onpe,neJUIeTC.H ,nBywI 3JIeMeHTaMH -
BbITIDKHOH 3aKJIenKOH H BbI6paHHbIM YCTPOHCTBOM ,l.(JUI YCTaHOBKH 3aKJIenOK (3a-
KJIenOqHHKoM).
BbITIDKHa.H 3aKJIenKa - 3TO MeXaHHqeCKOe Kpene)I{JIOe cpe,nCTBO, COCTO.Hru;ee H3!
,nBYX qaCTeH. OHa npe,nCTaBJUIeT C060H TPy6qaTbIH KOprryC C 60PTHKOM, 3apaHee 3a-
Be,neHHbIH B cTepxeHb, KOTOpbIH 06JIa,naeT CBOHMH OC06bIMH CBoHcTBaMH. Kor,n;a B
npouecce YCTaHoBKH ero BTHrHBaIOT B TeJIO 3aKJIenKH, npoHcxo,nHT pacIIIHpeHHe
Tpy6qaTOrO Kopnyca H <pOpMHpOBaHHe 3aMbIKaIOII(eH rOJIOBKH, qTO npHBo,nHT K CO-
e,nHHeHHIO cKpeIIIDIeMbIx ,neTaJIeH. 3aKJIenOqHHK 06bIqHO BKJIIOqaeT B ce6.H ynop,.
KOTOpbIH y,nep)I{HBaeT 60pTHK 3aKJIenKH, H 3IDKHM, KOTOpbIH 3axBaTbIBaeT H BhITjf-
rHBaeT cTepxeHb, B pe3YJIbTaTe qero 3aKJIenKa YCTaHaBJIHBaeTC.H, a 3aTeM, npH'
3apaHee onpe,neJIeHHOH HarpY3Ke, cTepxeHb pa3pbIBaeTC.H.
CyIUeCTByeT MHoxeCTBO pa3JIHqllbIX BHJ(OB BbIT.H)l{JIbIX 3aKJIenOK, HO HaH6o-
JIee IIIHpOKO pacnpocTpaHeHbI CTaH.napTHbIe (c OTKpbITbIM TOpUOM) 3aKJIenKH.
4.2. 3aKlleno"tHble coeOUHeHUJl 261
B BS EN ISO 14588: 2000 TaKaH 3aKJIenKa onpe,neJIeHa TaK: «BbIT.H)I{HaH 3aKJIenKa,
nycTOTeJIaH no BceH ,nJIHHe, C KOTOpOH MO)I{HO HCnOJIb30BaTb CTaH,[(apTHbIH CTep-
xeHb». IIpHHIJ;HI1 O,nHOCTopoHHeH KJIenKH npH HOMOII(H 3aKJIenOK C OTKpbITbIM
KOHn;OM I1pe,nCTaBJIeH Ha CJIe.nyrolI(eM pHCYHKe.
CTepxeHb 3aKpeIlIDIIOT B 3aKJIenOqHHKe H TeJIO 3aKJIenKH BCTaBJI.HIOT B npe,nBa-
pHTeJIbHO np06HToe HJIH npOCBepJIeHHOe OTBepCTHe B coe,nHH.HeMbIX ,neTaJI.HX. 3a-
KJrenOqHHK THHeT CTepxeHb, BTHrHBaH ero rOJIOBKY B TeJIO 3aKJIenKH, qTO BbIH)'X-
,naeT ee paCIIIHpHTbCH Ha HIDKHeH CTopOHe coe,nHHeHIDI, B TO xe BpeM.H B I1pOn;eCce
YCTaHOBKH CTHrHBaH ,neTaJIH BMeCTe H YCTPaH.HH JII06bIe 3a30pbI Me)l()(y HHMH.
B onpe,neJIeHHbIH MOMeHT, Kor,na 3aMbIKalOIUaH rOJIOBKa Ha HIDKHeH CTopOHe nOJI-
HOCTblO C<P0PMHpOBaJIaCb, ,naJIbHeHIIIee BbITHrHBaHHe CTep)I{IDI 3aKJIenOqHHKOM
HpHBO,nHT K pa3pbIBY CTep)I{IDI. OTPa60TaHHyID qaCTb CTep)I{IDI BbIHHMaIOT, H YCTa-
HOBKa 3aKJIenKH 3aBepIIIeHa.
3aKpbITaH 3aKJIenKa (c 3aKpbITbIM Topn;OM) onpe,neJIeHa B BS EN ISO 14588:
2000 TaK: «BbIT.H)I{HaH 3aKJIeI1Ka, KOTOpa.H 3aKpbITa H OCTaeTCH 3aKpbIToH nOCJIe
YCTaHOBKH». 3TOT BH,l( 3aKJIenOK TaIOKe IIIHpOKO H3BeCTeH KaK «repMeTHqHbIH». 3a-
KpbITWI 3aKJIenKa npe,nOTBpalI(aeT nonanaHHe napa HJIH BJIarH qepe3 OTBepCTHe B
YCTaHOBJIeHHOH 3aKJIenKe, a TaIOKe rapaHTHpyeT Y).J;epxaHHe roJIOBKH CTep)I{IDI, qTO
oco6eHHO BaxHO, HanpHMep, B 3JIeKTp eCKOM 060Py,noBaHHH.
3aKJIaLJ;HbIe 3aKJIenKa POP@ ,nOCTYIIHbI BO MHoxeCTBe MaTepHaJIOB, <pOPM KY30-
B8. H B03rJIaBJI.HIOT <pOPMbI, qT06bI 06eCneqHTb BapHaHTbI 3aKpeIIJIeHIDI ,lLillI IIIHpO-
Koro CneKTPa c06paHIDI H 3KOJIOrHqeCKHX TPe60BaHHH OT JIOMKHX H XpynKHX MaTe-
pHaJIOB, TaKHX KaK aKpHJIOBbIe IIJIaCTMaCCbI K HepXaBelOlI(eH CTaJIH. Pe310Me nOKa-
3bIBalOT B pa3,neJIe 4.2.9. 3TOT pHCyJlOK H CJIe.nyrolI(He Ta6JIHn;bI B3HTbI H3 ny6JIH-
IGlI HH Emhart Teknologies (Tucker Fasteners Ltd.). A,npec 3TOH KOMnaHHH ,naH B
IIpHJIoxeHHH 3.
F/la8a 4. Kpene:JICHble oemOJlu
4.2.9. 0630P SMAOS POpIJ 3alUJenOK
CTaHAapTHa51 (OTKpbITa5l) aaK11enKa
aK
3aKpblTa51 (repMeTH"IHa5l) 3aK11enKa
JlenecTKOBa51 3aK11enKa
I(
c:.u III po Kill A Alllana30HQaKJ1enOK' nplllroAH blX AIlS! nplllMe-
HeHVl5I B Tex npHnO)l(eHH5IX, rAe He npeAb5lBJ151IOTC5I BbI-
COKVle Tpe6oBaHVl5I K pa60"lHM HarpY3KaM
S! nplllnO)KeHIIIA, BQOTOPblX Tp e6YIOTCS! BOAOHenpOHM-
u.aeMble, repMeTH"IHble coeAHHeHH5I HJ1V1 B KOTOpblX
Tpe6yeTc5I YAep)I(aHHe CTep)I(H5I
riPIIIMeHIIIMa AIlS! CO IIIHeHIIIS! nn aCTMaCCbI, pe3l11HbI,
ApeBeCHHbI, CTeK110nJ1aCTHKa HJ1V1 J1aMHHaTOB
MHor03IDKVlMHa51 3aK11enKa
tc
IIIPOKIIIA Alllana30H p eKPaCHO nOAXOAIIIT ATlSI
HCn0J1b30BaHH5I npH HeCOBnaAalO HX OTBepCTVl5IX
PaCnOpHa51 aJ1IOMHHHeBa51 3aK11enKa
0tC
aKJ1enKa C pacnpeineHllleM H arpY3KIII nOAXOAIIIT ATlSI
HCn0J1b30BaHVl5I C M5IrKHMH, J10MKHMH HJ1H xpynKHMVI
MaTepHaJ1aMH
Cl)OPMLI 60p'rHKOB
3aKJIenKH C HH3KHM K:yIIOJI006pa3HbIM 60pTHKOM (CTaH,[(apTHbIe) no,nXO)UT JI.IDI
npHMeHeHIDI B 60JIbIIIHHCTBe npHJIOXeHHH, o,nHaKO, eCJIH M.HrKHe HJIH xpynKHe
MaTepHaJIbI 3a:KpeII.IDIIOTCH Ha xeCTKOH OCHOBe, CJIe,nyeT paCCMOTPeTb B03MO)I{HOCTb
HCnOJIb30BaHIDI 3aKJIenOK C YBeJIHqeHHbIMH 60pTHKaMH. 3aKJIenKH C nOTaHHbIM
(yrOJI 120°) 60pTHKOM CJIe,nyeT HCnOJIb30BaTb B Tex CnytlaHX, Kor,na TPe6yeTcH POB-
HaH nOBepXHOCTb.
B I cTeplKHei
CTaH,[(apTHbIe POP@ 3aKJIenKH (c OTKpbITbIM TopnoM) 06bIqHO cHa6xalOTcB
CTep)I{IUIMH, 0603HaqaeMbIMH BS «<pa3JIaMbIBaIOII(HHCH cTepxeHb») H pa3pa60TaH-
HbIMH TaK, qT06bI rOJIOBKa CTep)l{}ffi y,nep)J{HBaJIaCb npH YCTaHOBJIeHHOH 3aKJIenKe.
CTep)l{}ffiMH C 0603HaqeHHeM BH «<Pa3JIaMbIBalOlI(aHCH rOJIOBKa»), pa3pa60TaHHbI-
MH TaK, qTO rOJIOBKa H3BJIeKaeTCH H3 TeJIa 3aKJIenKH, MoxeT 6bITb cHa6xeHo 60Jlb-
IIIHHCTBO OTKpbITbIX 3aKJIenOK, HO oc06eHHo TaKHe CTep)KHH nOJIe3HbI I1PH npe,nBa-
pHTeJIbHOH c60pKe 3JIeIcrpHqeCKHX MOHTaxHbIX IIJIaT.
nOKpLlTHe
. CmaHoapmHoe nOKpblmue Kopnyca 3aK./lenKU. CTaJIbHbIe H HHKeJIb-Me,nHbIe
(MoHeJIb) 3aKJIenKH 06biQHO nOCTaBJI.HIOTCH OUHHKOBaHHbIMH.
. OKpawu6aHue U opyzue 6UObl o6pa6omKu n06epXHocmu. 110 TPe60BaHHIO MOryT
I10CTaBJI.HTbCH 3aKJIenKH C pa3JIHqHbIMH BH,l(aMH nOKpbITHH nOBepXHOCTH H
OKpaIIIeHHbIe B pa3JIHqHbIe nBeTa. 3aKJIenKH H3 aJIlOMHHHeBbIX CIIJIaBOB MO-
ryT 6bITb aHo,nHpOBaHHbIMH H OKpaIIIeHHbIMH, MaTOBbIMH HJIH 6JIeCTHlI(HMH B
3aBHCHMOCTH OT 3CTeTHqeCKHX Tpe60BaHHH H npe,nnOJIaraeMbIX BHeIIIHID
YCJIOBHH npH 3Kcrr.rryaTaUHH.
4.2. 3aKlleno"tHble coeouHeHUR 26
4.2.10. npaKTHlfeCKHe peKoMeHAaqHH
no HCnOn&30SaHHIO KpenneHHH
O,lJ;HOCTOPOHH.H.H KJIenKa - 3TO BbICOKOHan;e)f{HbIH H npoBepeHHbIH MeTO,n BbII1OJI'
HeHIDI Hepa3'beMHbIX coe,nHHeHHH. QT06bI nonyqHTb HallJI)"lIIIee coe,nHHeHHe, CJIe.
,nyeT npH,ll;ep)I{HBaTbCH CJIe,nyroII(HX npHHUHI1OB.
MaTepHaJILi COe,ZJ;HHJleMLIX ,n;eTaJIei
IIpH coe,nHHeHHH MarepHaJIOB pa3JIHqHOH TOJIII(HHbI H npOqHOCTH 60JIee npoq.
HbIH MaTepHaJI no B03MO)I{HOCTH CJIe,nyeT pacnOJIaraTb C HIDKHeH CTOpOHbI. Hal1pH'
Mep, npH coe,nHHeHHH IIJIaCTHKa H MeTaJIJIa IIJIaCTHKOBbIH JIHCT ,nOJl)l{eH panOJIaraTh'
CR I1O,n 60pTHKOM 3aKJIe11KH, a MeTaJIJI eCKaH ,neTaJIb ,noJI)I{Ha 6bITh CHH3Y.
Pa3Mep OTBepCTHJI H no,n;rOTOBKa
IIoJIyqeHHe xopOIIIero coe,nHHeHIDI 3aBHCHT OT xopoIIIeH no,nrOTOBKH OTBep'
CTIDI, npe,nnOqTHTeJIbHO np06HToro H npH He06xo,nHMOCTH paCTOqeHHOrO ,no pa3'
MepOB, peKOMeH,nyeMbIX B Ta6JIHUax ,naHHbIX 110 BbITIDKHbIM 3aKJIenKaM.
,l);HaMeTP 3aKJIenKH
COrJIaCHO CnpaBOqHHKY no Hecyru;HM HarpY3KY coe,nHHeHIDIM, ,nHaMelp 3a-
KJIenKH ,nOJl)l{eH 6bITb no KpaHHe Mepe paBHbIM TOJIII(HHe caMoro TOJICTOrO JIHCTa I'
He 60JIee qeM B TPH I1peBbIIIIaTb TOJIIUHHy JIHCTa, Haxo,nHlI(erOCH Henocpe,nCTBeaH(
no,lJ; 60pTHKOM 3aKJIenKH. 06paTHTecb K Ta6JIHn;aM ,naHHbIX 110 xapaKTepHCTHKa
npOqHOCTH 3aKJIenOK.
PaCCTOJlHHe OT 3aKJIenOK ,n;o KpaJi 3JIeMeHTa
OTBepcTIDI ,l.(.illI 3aKJIenOK ,nOJDKHbI 6bITb npOCBepJIeHHbI HJIH np06HTbI Ha pac-
CTOHHHH He MeHbIIIe ,nBYX ,nHaMeTPOB OT KpaH, HO He ,naJIee qeM B 24 ,nHaMeTpax 01
3TOro KpaH.
mar mBa
COrJIaCHO CnpaBOqHHKY paCCTOHHHe MelK,l(y 3aKJIenKaMH B cnyqae Hecyru;HX Ha-
rpY3KY coe,nHHeHHH HHKor,na He ,nOJl)l{O npeBbIIIIaTb TPex ,nHaMeTPOB 3aKJIenKH
IIpH pa3pa60TKe CTbIKOBbIX coe,nHHeHHH ueJIeC006pa3HO BKJIlOqHTb YCHJIeHHyID Ha-
KJIMKY, npHKpenHB ee K JIexalI(eMy BHH3Y JIHCTY 3Hr3aro06pa3HbIM 3aKJIeIIOqHbI
mBOM.
MaTepHaJI 3aKJIenOK
BbI60p MaTepHaJIa 3aKJIenOK 06bIqHO 3aBHCHT OT Tpe6yeMoH npOqHOCTH 3aKJIe-
nOQHOrO coe,nHHeHIDI. ECJIH npHxo,nHTCH HCnOJIb30BaTb 3aKJIel1KH H3 MaTepHaJIa
OTJIHqalOlI(erOCH OT MaTepHaJIa coe,nHH.HeMbIX JIHCTOB, BaxHO 3HaTb, qTO 3JIeIcrpO-
JIHTlNeCKOe B03,neHCTBHe MoxeT CTaTb I1p HHOH KOpp03HH (CMOTpH pa3,neJ
4.2.12).
Y CTaHOBKa H 6e30nacHocTL
3aKJIenOqHHK 06bNHO BbI6HpalOT B COOTBeTCTBHH C YCJIOBIDIMH npOH3Bo,nCTBa
IIepe,n YCTaHoBKoH CJIe,nyrolI(eH 3aKJIenKH HHCTpYMeHT ,nOJl)l{eH 6bITb OqHlI(eH 01
OTpa60TaHHbIX MaTepHaJIOB. 3JIeIcrpHqeCKHH 3aKJIenOqHHK He ,nOJl)l{eH HCnOJIb30Ba-
!bCH 6e3 ,ne<pJIeKTOpa CTep)KHeH HJIH YCTaHOBJIeHHOrO KOHTeHHepa ,l.(.illI c60pa OTpa-
60TaHHbIX 3aKJIenOK. Bcer,na CJIe,nyeT nOJIb30BaTbCH 3aIUHTHbIMH OQKaMH.
F/la8a 4. Kpene:JICHble oemOJlu
4.2.11. B&160p POpIJ (HnH B&ITflJlCH&IX) 3alUJenOK
np01JHOCTL COe,ZJ;HHeHHJI
IIpe)l{,L(e Bcero CJIe,nyeT On;eHHTb TPe6yeMYlO npOqHOCTb coe,nHHeHIDI Ha pa3pbIB
H Ha c,nBHr. QT06bI y,nOBJIeTBOpHTb 3THM TPe60BaHIDIM, He06xo,nHMO npaBHJIbHO
BbI6paTb KOJIHqeCTBO KpenexeH H paCCTOHHHe Me HHMH, a TaKXe MaTepHaJI 3a-
KJIenOK H HX ,nHaMeTp. ,lI;aHHbIe 0 npOqHOCTH, npHBe,neHHbIe Ha COOTBeTCTByroIIJ;:HX
CTPaHHn;ax, n03BOJI.HIOT BbI6paTb 3aKJIenKH TPe6yeMoH npOqHOCTH.
TOJIDJ;HHa COe,lUfHeHHJI
Ha CJIe,nyIOlI(eH CTa,rJ;HH He06xOHMO onpe,neJIHTb CYMMaplfYIO TOJIIIJ;HJIy coe,nHIDIe-
MbIX MaTepHaJIOB, He 3a6bIBaH yqHTbIBaTb JII06bIe B03,l(yll1HbIe 3a30pbI HJIH npOMe-
xyrOqHbIe CJIOH, TaKHe KaK ynnoTHHTeJIH (repMeTHKH). Tenepb onpe,neJIHTe p33Mep
BbI6paHHOH 3aKJIenKH C He06xo,nHMOH ,l.(JIHHOH 33XBaTa, 06paTHBIIIHCb K COOTBeTCT-
ByIOII(eH CTPaHHl(e C ,naHHbIMH. 3TO BaxHO c,neJIaTb, nOCKOJIbKY npH HenpaBHJIbHOM
BbI60pe ,l.(JIHHbI 33XBaTa 3aKJIenKa He CMoxeT y,nOBJIeTBOpHTeJIbHO 3aXBaTHTb 06paT-
IfYIO CTOpOJIy 3arOTOBKH HJIH naKeTa.
YCHJIeHHe KOpp03HH (CBOHCTBa MaTepHaJIOB)
HaKOHen;, CJIe,nyHTe 06lI(eMy npaBHJIY, qTO BbI6paHHaH 3aKJIenKa ,noJDKHa HMeTb
TaKHe xe <pH3 eCKHe H MeXaHHqeCKHe CBOHcTBa, KaK H 3arOTOBKa. 3aMeTHoe pa3-
JIHqHe B CBOHCTBax MoxeT CTaTb np HHOH pa3pyrneHIDI coe,nHHeHIDI B pe3YJIbTaTe
B03HHKHOBeHIDI YCTaJIOCTH MeTaJIJIa HJIH KOHTaKTHOH KOpp03HH. KOPp03IDI YCKO-
pHeTcH npH onpe).J;eJIeHHbIX COqeTaHH.HX MaTepHaJIOB H B03,neHcTBIDI OKpyxalOru;eit
cpe,nbI. KaK npaBHJIO, CJIe,r(yeT H36eraTb KOHTaKTa Me)l{,L(y pa3Hopo,nHbIMH MeTaJIJIa-
MH. By-KBbI A, B, C H D B Ta6JIHue 03HaqalOT CJIe.nyrolI(ee:
A KOpp03IDI paCCMaTpHBaeMoro MeTaJIJIa He YCHJIHBaeTCH npH KOHTaKTe C ,npy-
rHM MeTaJIJIOM.
B KOpp03IDI paCCMaTPHBaeMoro MeTaJIJIa MoxeT CJIerKa YCHJIHTbCH npH KOH-
TaKTe C ,npyrHM MeTaJIJIOM.
C KOpp03IDI paCCMaTPHBaeMoro MeTaJIJIa MO)l{eT 3aMeTHO YCHJIHTbCH npH KOH-
TaKTe C ,npyrHM MeTaJIJIOM.
D B YCJIOBIDIX BJIaxHOCTH KOM6HHan;IDI paCCMaTPHBaeMoro MeTaJIJIa C KOHTaK-
THpyrolI(HM MeTaJIJIOM Hen;eJIeco06pa3Ha; ,nIDKe B 60JIee M.HrKHX YCJIOBIDIX He-
06XO,nHMbI COOTBeTCTByIOlI(He 3aIIJ;HTHbIe MepbI.
ECJIH npHBe,neHbI ,nBa CHMBOJIa (HanpHMep, B HJIH C), TO 3TO 03HaqaeT, qTO YCH-
JIeHHe, BepOHTHO, H3MeHHTCH npH H3MeHeHHH BHeIIIHHX YCJIOBHH HJIH COCTOHHIDI
MeTaJIJIa.
KOHTaKTHPYJOIQHi MeTaJJJI
MaTepH8JI
3aKJIeDKH HHKeJJh- Hep1K88elO- Me CT8JIh AmoMHIIIfii IJ;HHK
Me,lJ,llhlii CDJIU OUUI CT8JIh H CDJlULI
HHKeJIb- Me,ll;HbIH CllJIaB - A A A A A
HepXaBeIOIIJ;aH CTaTlb A - A A A A
Me,ll;b B HJIH C B HJIH C - A A A
CTaJIb C C C - B A
ArrIOMHHHH H CllJIaBbI C B HJIH C D B HJIH C - A
IJ;HHK C C C C C -
4.2. 3aKlleno"tHble coeOUHeHUR 265
4.2. 12. PeKoMeHAaqHH no npOeKTHpOSaHHIO
npHCOe,ZJ;HHeHHe MJlfKoro MaTepHaJIa K TBep,n;oM)'
ECJIH 60pTHK 3aKJIenKH HaxO,nHTCH co CTOpOHbI WIrKoro MaTepHaJIa, MO)I{HO
HCnOJIb30BaTb 3aKJIenKY C YBeJIHqeHHbIM 60pTHKOM. Ii Ha060poT, paCnOpHaH 3a-
KJIenKa pacnpe,neJI.HeT C)J{HMalOlI(yID HarpY3KY no 60JIbIIIeH 06JIaCTH, qTO n03BOJI.HeT
H36exaTb nOBpe)K,l(eHIDI M.HrKOrO MaTepHaJIa.
YBeJ1V1"1eHHblt1 60pTiI1K
PacnopHa aa KnenK a
TBePA b t It
MflrK1II1II -
nJiaCTHKH H xpynKHe MaTepHaJILI
,[LrnI JIOMKHX TIJIaCTMaCC H XpynKHX MaTepHaJIOB POP@ KJIenKa npe,l(JlaraeT MHO-
xeCTBO pa3Ho06pa3HbIx peIIIeHHH. JIerKo YCTaHaBJIHBaeMbIe aJIlOMHHHeBbIe 3aKJIeI1-
KH ,norrycKalOT HH3KHe HarpY3KH I1PH YCTaHOBKe, B TO BpeM.H KaK H JIeneCTKOBbIe, H
pacnopHbIe 3aKJIenKH 06eCI1eqHBa1OT YBeJIHqeHHe TIJIOIIJ;Cl,l.(H 0I10pbI Ha coe,aHIDIe..
MbIe MaTepHaJIbI. ,[LITH 60JIee npOqHbIX TIJIaCTMaCC MO)I{HO HCI10JIb30BaTb CTaH,l(apT-
HbIe BbITH)l{HbIe 3aKJIeIIKH - C OTKpbITbIM HJIH 3aKpbITbIM TOpUOM.
JlerKa YCTaHoBKa (PAD) YBeJ1i11"1eH- JlenecTKoBa PacnopHa
AAIOMiI1HiI1V1 (AD/ASS) Hblt1 60pTiI1K aaKnenKa aaKJ1enKa
MaTepHaJI c Ce1JeHHeM, HanOMHHalOIIUlM 6YKBY «n» (mBeJlJIep)
QT06bI ,nOCTHqb ,nHa Y3Koro CeqeHIUI, MO)f{HO HCnOJIb30BaTb YMHHeHHyID Ha-
Caru(y (A) (CMOTPH pHCYHOK HHXe). IIpH 3TOM CJIe,nyeT HCI10JIb30BaTb 60JIee ,l(JIHH-
HbIH CTepxeHb 3aKJIeI1KH, H MaKCHMaJIbllbIH ,nHaMeTp HaCan;KH ,nOJl)l{eH 6bITb paBeH
.D;HaMeTPY 60pTHKa 3aKJIeilKH. B KaqeCTBe aJIbTepHaTHBbI 3aKJIeilKY MO)I{HO YCTaHO-
BHTb C ,npyrOH CTOpOHbI (B) IDIH paCIIIHpHTb KaHaJI, QT06bI npHMeHHTb CTaH,[(apT-
HyIO 3aKJIeilKY H 3aKJIeI10qHHK(C).
YAJ1i11HeHHa51
HaCSAKa
A
B
c
F/la8a 4. Kpene:JICHble oemOJlu
TOJICTLIe/TOHKHe JIHCTLI
IIpH coe,nHHeHHH MaTepHaJIOB pa3JIWlHOH TOJIII(HHbI JIyqIIIe paCnOJIO)J{HTb 60-
JIee TOJICTYIO TIJIaCTHJIy co CTOpOHbI 3aMbIKalOlI(eH fOJIOBKH (A) (CMOTPH pHCyJlOK
HIDKe). ECJIH ,nHaMeTp OTBepCTIDI B 60JIee TOHKOH TIJIaCTHHe BeJIHK, CJIe,nyeT HC-
nOJIb30BaTb 3aKJIenKY C YBeJIHqeHHbIM 60PTHKOM (B). ECJIH 60JIee TOHKaH IIJIaCTHHa
pacnOJIO)l{eHa co CTOpOHbI 3aMbIKalOlI(eH rOJIOBKH, Hy)KH0 JIH60 HCnOJIb30BaTb
onopHyro IIIaH6y (C), JIH60 06eCneqHTb, qT06bI ,nHaMeTP OTBepCTIDI B 60JIee TOJI-
CTOH TIJIaCTHHe 6bill MeHbIIIe, qeM B 60JIee TOHKOH TIJIaCTHHe (D).
A
B
c
D
fJIYXlle OTBepCTHJI H na3LI
YCTaHoBKa POP@ HJIH BbITIDKHOH 3aKJIenKH B rJIyxoe OTBepCTHe HJIH B npo<ppe-
3epoBaHHbIH na3, nepeCeKalOlI(HH OTBepCTHe HJIH BHYTpeHHlO1O nOJIOCTb, B03MOlKHa
6JIaro,napR paCIIIHpeHHIO TeJIa 3aKJIenKH npH YCTaHOBKe.
A
B
c
illapHHpHoe COe,ZJ;HHeHHe
HCnOJIb30BaHHe Cnen;HaJIbHOH HaCan;KH 06eCneqHBaeT He60JIbIIIOH 3a30p Mex,ny
60pTHKOM 3aKJIenKH H naKeTOM, TaKHM 06pa30M 06eCneqHBaR no,nBIDKHOCTb co-
e,nHHeHIDI. HacaAKa, o6ec-
ne"lil1BalOw.a
t wapHVlpHoe
coeAVlHeHiI1e
AeTaJ1b
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XOTR CTaH,[(apTHbIe ,nHaMeTpbI oTBepcTHH paBHbI ,nHaMeTpy Koprryca 3aKJIenKH
IIJIIOC 0, 1 MM, pa3MepbI oTBepcTHH B coe)J;HH.HeMbIX ,neTaJI.HX He Bcer,na MOryT 6bITb B
TOQHOCTH paBHbIMH 3an;aHHOMY 3HaQeHHIO, HanpHMep eCJIH OTBepCTHR np06HTbI B
,neTaJUIX 3apaHee. B Tex CJIyqaRx, KOf,na OTBepCTHe Ha HIDKHeH CTopOHe (co CTOpO-
HbI 3aMbIKalOIIJ;eH rOJIOBKH) 60JIbIIIe, CJIe,nyeT BbI6paTb MHor03IDKHMHyIO 3aKJIenKY
4.2. 3aKlleno"tHble coeOUHeHUR 267'
H3-3a ee HCKJIlOqHTeJIbHOrO CBOHCTBa 3anOJI-
IDITb OTBepCTHe. JIenecTKoBbIe H pacnopHbIe 3a-
KJIeilKH TaIOKe MOryT CTaTb aJIbTepHaTHBHbIMH
perneHIDIMH npH 3THX 06CTOHTeJIbCTBax, oco- MHOr03a)l(VlM-
6eHHo npH pa60Te C JIOMKHMH HJIH XpynKHMH Ha 3aK11enKa
MaTepHaJIaMH. Ii Ha060poT, eCJIH OTBepCTHe 60-
JIbme co CTOpOHbI 60pTHKa, CJIe,r(yeT BbI6paTb 3aKJIeilKY C
)'BeJIWleHHbIM 60pTHKOM.
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50 100 150 200 °C
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4.2.17. 3aKnenOlfH&le raNKJf (<<raelfH&le 3aKnenKJf»)
3aKJIeIIOqHbIe raHKH pa3pa60TaHbI CIIeI..J;HaJIbHO, qT06bI 06eCIIeqHTb 60JIee IIpOqHOe
pe3b60Boe coe,nHHeHHe JIHCTOB (CTaJIH) H ,npyrHX MaTepHaJIOB, aHaJIOrHqHO BbITIDK-
HbIM 3aKJIeIIKaM, TaM, r,ne ,[(JI.H YCTaHOBKH B03MoxeH ,nOCTYII TOJIbKO K O,nHOH CTO-
pOHe H3,neJIIDI. OHH He TOJIbKO <pOpMHpyroT HaKH)];Ilyro raHI{)', HO, B OTJIWIHe OT He-
KOTOpbIX ,npyrHX BH,nOB, TaIOKe ,noIIycKalOT CKJIeIIbIBaHHe KOMIIoHeHToB, 3ao,nHO
06eCIIeQHBa.H 3aKpeIIJIeHHe BHHTOBOH pe3b6bI. IIpHHUHII HX HCIIOJIb30BaHIDI nOKa-
3aH HIDKe.
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YCTaHOBOQHOrO HHc1]JYMeHTa ( «3aKJIenOQHHKa ,[(lliI 3aKJIeIIOQHbIX
raeK», «pe3b60Boro 3aKJIenOqHHKa»), KOTOpbIH 3aTeM BCTaBJI.HeTC.H B
npOCBepJIeHHOe HJIH IIp06HToe oTBepcTHe.
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BasI HX BMeCTe.
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or M3 ,no M12, IIPH 3TOM B03MO)KHbI caMbIe pa3Ho06pa3HbIe COqeTa-
HIUI <pOPMbI 60pTHKa, <pOpMbI KopIIyca H MaTepHarra. BbI,l(eJI.HIOT TPH
THna 60pTHKOB - CTaH,[(apTHbIH, IIOTaHHOH (yrOJI 90°) H ,[(lliI TOHKHX
MaTepHaJIOB YMeHbIIIeHHbIH 60PTHK, KOTOpbIH 06eCIIeQHBaeT YCTa-
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CTaH,[(apTHbIH KOPIIYC - KpyrJIbIH OTKpbITbIH HJIH 3aKpbITbIH. 3a-
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rpaHHbIe KopIIyca (YCTaHOBJIeHHbIe B IIIeCTl1yrOJIbHbIe OTBepcTIDI) HJIH Koprryca co
IllJIHl(aMH (YCTaHOBJIeHHbIe B KpyrJIOe oTBepCTHe).
4.2. 18. POP@ pe3&6os&le 3aKnenKJf: npJfMeHeHJfe
POP@ pe3b60BbIe 3aKJIenKH 06eCIIeqHBalOT npocToH H
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3<p<peKTHBHbIH CIIoco6 coe,nHHeHHR aTepHaJIOB, n03B( f '
JUIIOmHH B pa3JIHqHbIX IIpHJIO)f(eHHRX H IIOJIb30BaTb npe-
HMyIneCTBO BHyrpeHHeH pe3b6bI.
POP@ pe3b6oBbIe 3aKJIenKH .HBJl.HIOTC.H IIpeKpaCHbIM
pemeHl1eM ,[(.JlSI <pOpMHpoBaHIDI BbICOKOKaqeCTBeHHbIX, Ha-
rpyxeHHbIX pe3b60BbIX coe,nHHeHHH B pa3JIHqHbIX MaTepH-
arrax TaM, r,ne B ,npyrHe coe,nHHeHIDI, He C03,na101I(He Bpa-
m;aIOIIJ;erO MOMeHTa, He MOryT y,nepxaTb BbInOJI3alOII(yro
HarpY31{)'. POP pe3b60BbIe 3aKJIenKH IIO,nxOMT ,nIDKe ,[(.JlSI
o,nHHapHbIX JIHCTOB TOJIII(HHOH He 60JIee 0,5 MM.
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IIpHJIOXeHHH, B KOTOpbIX ,nocTYIIHa TOJIbKO o,nHa CTOpO-
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IIOBpe)l(,[(a.H IIpe,nBapHTeJIbHO o6pa6oTaHHyro OKpyxalO1I(yIO IIOBepXHOCTb HJIH TOH-
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B COBpeMeHHbIX YCJIOBH-RX IIpOH3Bo,nCTBa:
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. IllHpoKoe pa3Hoo6pa3He BH,l.(OB
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KJIeIIKH C)l{HMaeTC.H, <POpMHPY.H 6ypT Ha HIDKHeH CTopOHe IIaKeTa, H CHJIbHO CXH-
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4.2. 3aK./leno"tHble coeOUHeHUJl 279"
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HOrO 3JIeMeHTa KpeIIJIeHH.H rrpH rrOCJIe,nyroII(HX C60POqHbIX pa6oTax.
4.2.20. PO raHKa: CTan&
OTKpLITaJl, CTaH,ZJ;apTHLIH OOpTHK (c pH4JJIeHHeM)
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d 0003Ha1feHHe (MM) DaKeTa e OTBepCTHJI CTep1KllJl OOPTHKa OOpTHKa BO B KOH- CTBO B
L (MM) (MM) (MM) D (MM) B (MM) S TeiHepe KopoOKe
M4 PSFON430 10,0 0,3-3,0 10000 500
PSFON440 11,5 3,1-4,0 6,0 5,9 9,0 1,0 10000 500
M5 PSFON530 12,0 0,3-3,0 8000 500
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M6 PSFON630 14,5 0,5-3,0 4000 500
PSFON645 16,0 3,1-4,5 9,0 8,9 12,0 1,5 4000 500
M8 PSFON830 16,0 0,5-3,0 2000 250
PSFON855 18,5 3,1-5,5 11,0 10,9 15,0 1,5 2000 250
MI0 PSFONI030 17,0 0,5-3,0 1500 200
PSFONI060 22,0 3,0-6,0 12,0 11,9 16,0 2,0 1500 200
M12 PSFON1240 23,0 1,0-4,0 16,0 15,9 22,0 2,0 1500 200
F/laBa 4. Kpene:JICHble oemOJlu
3aKpLITaJI, cTau,napTHLIH OOPTHK (c pH4JJIeHHeM)
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d 0003Ha1feHHe (MM) DHeTa e OTBepCTHJI cTep1KllJl oOpTHKa 60pTHKa 80 8 KOH- CTBO B
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M5 PSFCN530 18,0 0,3-3,0 7,0 6.,9 10,0 1,0 5000 500
M6 PSFCN630 20,5 0,5-3,0 9,0 8,9 12,0 1,5 2000 500
M8 PSFCN830 25,0 0,5-3,0 11,0 10,9 15,0 1,5 1500 250
OTKPLITaJI, CTaH,ZJ;apTHLlH OOPTHK, meCTHrpaHHHK
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d 0003Ha1feHHe (MM) DaKeTa e OTBepCTHJI CTep1KllJl OOp1'HKa 6op1'HKa 80 8 KOH- CTBO B
L (MM) (MM) (MM) D (MM) B (MM) S TeiHepe Kopo6Ke
M4 PSFOH430 11,5 0,5-3,0 6,1 6,0 9,3 1,0 10000 500
MS PSFOH530 13,5 0,5-3,0 7,1 7,0 10,3 1,0 8000 500
M6 PSFOH630 15,5 0,5-3,0 9,1 9,0 12,3 1,5 4000 500
MB PSFOH830 17,5 0,5-30 11,1 11,0 14,3 1,5 2000 250
MI0 PSFOHI040 22,0 1,0-4,0 13,1 13,0 16,3 2,0 1500 200
4.3. (/JQJlbu,e6ble coeouHeHUR 281
OTKpLITaJl, nOTaHHoH OOpTHK (c pHfj)JIeHHeM)
goo
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d 0003H81feHHe (MM) DHeTa e OTBepCTHJI crrep1KllJl oOpTllKa 60pTllKa 80 8 KOH- cTB08
L (MM) (MM) (MM) D (MM) B (MM) S rreiHepe KopOOKe
M4 PSKON435 11,5 2,0-3,5 6,0 59 9,0 1,5 10000 500
M5 PSKON540 13,5 2,0-40 7,0 69 10,0 1,5 8000 500
M6 PSKON645 16,0 2,0-4,5 9,0 89 12,0 1,5 4000 500
MS PSKON845 19,0 2.CM,5 11,0 10,9 14,0 1,5 2000 250
MI0 PSKONI045 21,0 2,0-4,5 12,0 11,9 14,7 1,5 1500 200
,D;aJIbHeHmyIO HH<popMaUHIO MO)I{HO HaHTH BO MHoxeCTBe rry6JIHKaUHH, rrOCB.H-
m;eHHbIX POP KJIerrKe, BbIrryI.l(eHHbIX Emhart Teknologies (Tucker Fasteners Ltd) -
CMOTpH rrpHJIOXeHHe 3.
4.3. Cbanbu.eBble coeAMHeHMSI
4.3. 1. C/)sn&qes&le COeAJfHeHJffl
Jle)KaqHH 4JaJILD;
/j-
/
,[(BOHHOH JIe)Ka'lHH 4JaJILD;
Pila8a 4. Kpene:JICHble oemOJlu
npocToi yrJIOBoi 4JaJILD;
KOMOHHHpOBaHHLIH yrJIOBoi 4JaJILn
nOCJIe,n;OBaTeJILHOCTL 4JopMHpOBaHHJI JIe1Ka'lero 4JaJILna
np cnOC06J1eH e c
KaHaBKoVi AJ151 06-
)I( Ma coeA HeH
) (
OTr 6 KpOMK AJ151 ct>OpM pOBaH 5I
OA HapHoro J1 e)l(a"lero ct>aJ1bu.a
)
CoeA HeHVle J1 CTOB OA HapHbIM
J1e)l(a"lViM ct>aJ1bu.eM
CTau.VlOHapHa51 J1 ne-
peHOCHa51 HaKOB J1bH
OKOH"IaTeJ1bHOe ynJ10THeH e coeA HeH 5I np
nOMow. np cnOC06J1eH 5I AJ151 06)1( Ma C Ka-
HaBKoVi cooTBeTcTBYIOw.eVl W pVlHbl
4.3.2. npJfnycKJf HS cpsn&qes&le COeAJfHeHJffl
Jle)Ka'lHH 4JaJILD;
1
w
T
w
w
4.4. PQ3Jlu"tHble "peneJICHble U30eJlUR 283
npOCTOH yrJIOBOH 4JaJILD;
KOMOHHHpOBaHHLIH yrJIOBOH 4JaJILn
T
W - IIIHpHHa OTOrHYTOH KpOMKH
G - IIIHpHHa JIexaqero <paJIbua
L - IIIHpHHa <pHKcHpyrolI(eH rrOJIOCKH
P - IIIHpHHa rrpOCToro yrJIOBOrO <paJIbua
K - IIIHpHHa KOM6HHHpoBaHHoro yrJIOBOrO <paJIbua
T - TOJIII(HHa MeTaJIJIa
THo coe,znmelllDl
npHMepHlde
onyIQelllDl
npHMepHbIe npHnycKH
CYMMapHbIH npHnycK = 3G-4T paCnpe,ll;eIDIeTCH:
(a) nOpOBHY Mex.ny ,lI;BYMH KOHI(aMH 1 H 2; HJIH
(b) ,lI;Be 'rpern - KOHeI( 1 H O,ll;Ha 'rpeTb - KOHeI( 2, eCJIH BIDKHO
nOJIOXeHHe eH a coe HHeHHHeHH n03H HH
BOHHOH JIeXaqHH <PaJIbI( ,LJ;o6aBHTb W - T K Iq>OMKe KIDK,ZJ;OH H3 COe,ll;HIDIeMbIX 3arOTOBOK.
llIH HHa HKCH eH nOJIOCKH = 4W + 4T Tor a L = 2W + 4T
ITpOCTOH yrJIOBOH <paJIbI( ,LJ;o6aBHTb W ,lI)UI O,ll;HHapHOH Iq>OMKH K KOH
3.
,LJ;o6aBHTb 2W + T ,lI)UI ,lI;BOHHOH Iq>OMKH K KOH
4.
P = 2W + 2T
,LJ;o6aBHTh W ,lI)UI O,ll;HHapHOH Iq>OMKH K KOH
5.
,LJ;o6aBHTb 2W+ T ,lI)UI ,lI;BOHHOH Iq>OMKH K KOH
6.
K = 2W + 3T
JIe:>KaqHH <PaJIbI(
KOM6HHHpOBaHHbIH
yrJIOBOH <paJIbI(
4.4. Pa3nM'IHble Kpene>KHble M3AenMSI
4.4. 1. KOHJflfeCKJfe WTJfcpT&I, He3SKSneHH&le
Pa3MepLI
Krracc A (IIIJIH<p0BaHHbIe IIITH<pTbI): IllepoxoBaTocTb Ra = 0,8 MKM
Krracc B (o6TOqeHHbIe IIITH<pTbI): IllepoxoBaTocTb Ra = 3,2 MKM
1:50
r2
a + d + (0,02/)2
2 8a
1J
a
a
FII060 4. KpeneJICHble iJemOllu
Pa3MepbI B MM
d hl0 a 0,6 0,8 1 1,2 1,5 2 2,5 3 4 5 6 8 10 12 16 20 25 30 40 50
a :=::: 0,08 0,1 0,12 0,16 0,2 0,25 0,3 0,4 0,5 0,63 0,8 1 1,2 1,6 2 2,5 3 4 5 6,3
lb
HOM. MHH. MaKC.
2 1,75 2,25
3 2,75 3,25
4 3,75 4,25
5 4,75 5,25
6 5,75 6,25
-
8 7,75 8,25
10 9,75 10,25
12 11,5 12,5 -
--
14 13,5 14,5
16 15,5 16,5
- -
18 17,5 18,5 ,L(Hana30H
20 19,5 20,5
22 21,5 22,5
24 23,5 24,5 -
-
26 25,5 26,5
28 27,5 28,5
30 29,5 30,5 npOMbIlliJIeHHbIX
-
32 31,5 32,5
35 34,5 35,5
-
40 39,5 40,5
-
45 44,5 45,5
50 49,5 50,5
-
55 54,25 55,75 ,l(JIHH -
-
60 59,25 60,75
65 64,25 65,75
70 69,25 70,75
75 74,25 75,75
80 79,25 80,75
85 84,25 85,75
90 89,25 90,75
95 94,25 95,75
100 99,25 100,75
120 119,25 120,75
140 139,25 140,75
160 159,25 160,75
180 179,25 180,75
-
200 199,25 200,75
a ,L(pyrHe ,n:OnyeKH, HanpHMep all, ell, f8, B eOOTeTeTBHH e eOfJIaWeHHeM Me
110-
Tpe6HTeJIeM H 110eTaBIl(HKOM
b ,L(JHI HOMHHaJIbHbIX ,l(JIHH eBbIwe 200 MM mafH 20 MM.
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3:
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FII060 4. KpeneJICHble iJemOllu
r.Q
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p,.
I .-..
)1 Q. t 8Cu N 00 \C .-4 0 N \C'" \C'" \C'" \C'" 00 0 N \C'" 00 00 0", N '"
0'" '" '" '" '" '" '" '" '" '" '"
!ei '" '" N V") \C f' 00 0'\ 0 N V") \C f' 0'\ 0 .-4 N
I f' 00 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N N N N N N M
=
11 Q,) \J
I= \C N 00 .-4 0;, 0 N N .-4 .-4 N '" V") \C'" 00 00 00 0", N '" \0
'" '" '" '" '" '" '" '" '" '" '" '" 0
= '" '" '" .-4 N V") \C f' 00 0'\ 0 .-4 N V") f' 00 0'\
\C 00 0'\ .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N N N N N N N M
=
I 00 0", 0", V") N N N V") V") f'", f' f' f' f' f' 0", 0", 0 0 0 q
I ii '" .-4 '" '" '" '" '" '" '" '" '" '" '" '" '"
0 .-4 .-4 '" .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N N N N
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'" '" M '" '" '" '" '" '" '" '" '" '" '" '"
.-4 N N
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c i i ii 00 0;, '" V") f' 00 00 00 0", .-4 N N '" V") \C'" f' 00 0\
'" '" '" '" '" '" '" '" '" '" M '" '" '" '" '"
0 0 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N N N N N N N N
= = '-"
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0 0",
0 0'" 0 0
I I
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= '" '" \C'" f' 00 0 0", 0", 0", 0", 0 0 0 0", N N N", N N
'" '" '" '" '" '" '" '" '" '"
0 0 0 0 0 0 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4
\C V") 0'\ 00 00 \C
0 .-4 0 .-4 .-4
'" '" <;'" '" '" '"
0 0 0 <; 0
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t \C V") V") V") V") 0 0", 0;, 0;, 00 f' f' V") V'l V") V") V) "1
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N V") \C f' 00 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N
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! fl '" '" \C'" \C \C'" f' f'", 0'\ 0;, .-4 N N V") V") V") V") "1- 'i')
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0 0 0 0 0 0 0 0 0 0 0 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4
i .-4
<;'" 0
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\C V") 0'\
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t 00 00 00 f' f' \C'" \C'" \C'" V") V") '" '" N", N", 0", 0", 0", 0", q
I\:) '" '" '"
'" '" '" '" '" 0 .-4 N V") \C f' 00 0'\ 0 .-4 N
N V") \C f' 00 0'\ .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N
t 0 N 0 N M
I- V") \C f' 00 0'\ .-4 V") \C f' 00 0'\ .-4
i .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N N
&rill
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C U 0 0 0 0 0 0 0 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N N
u
= te I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
rJ'.) rJ'.) rJ'.) rJ'.) rJ'.) rJ'.) rJ'.) rJ'.) rJ'.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) 00 00
4.4. P03IIulfHble Kpene:JICHble U3iJeJ1UJ/
00 00 0 N '" \C 0", N '" 00 N N '" \C'" 00 q N 0", N N '" '" \C.. 00 00 00 0",
M '" '" '" '" '" '" v-) '" '" '" '" .. '" N '" '"
\C f' 00 0'\ .-4 N f' 00 0'\ 0 .-4 \C f' 00 0'\ 0 .-4 f'
M V") V") V") V") V") V") V") V") \C \C \C \C \C \C
'- " /
t--. f' 0'\ 00 0", N \C'" V") f'", 0;, .-4 N N \C'" 00 V") f' V") f' f' 0'\ 0'\ 00 0",
'" '" '" '" '" '" '" '" '" '" '" '" '" '" '" '" '" '" 0 ..
N \C f' 00 0'\ 0 .-4 V") \C f' 00 0'\ 0 N V") \C 00 0'\ 0
M V") V") V") V") V") V") V") V") \C \C \C
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N N N N N N N N N N N N N N N N N N N N N N N N N N N N
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P""4 N V") 0 0'\ 00
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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N N N N N N V") V")
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
00 00 00 C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) C/.) 00 C/.) C/.) C/.) C/.) C/.)
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4.4. P03IIU'IHble ICpene:JICHble U3iJeIIUJ/
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N V") V") \C f' 00 00 0'\ 0 N N V") f' 00 0 .-4 N V") V") \C f' f'
. N N N N N N N N
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. . . . . . . . . . . . . . . t; f' f' f' f' f' f' f' f' f' N N N
1"'"4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 . . . . . . . . .
.-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4
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. . .r) ...0 . . . N .r) ...0 r..: 00 0\ d N .r) ...0 r..: 00 0\ d . N .r) \0
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.-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 .-4 N N N
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N 0 N N
d d 0 d 0 d 0
+ + + +
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0 N .r) \0 r---: N .r) . N .r) \0 r---: 00 0\ d .
00 0'\ 0 .-4 f' 00 0'\ 0 .-4 .-4
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0 .-4 N V") \C f' 00 0'\ 0 .-4 N V") \C f' 00 0'\ 0 .-4 N V") \C f' 00 0'\ 0 .-4 N
N N N N N N N N N N V") V") V")
::E ::E ::E ::E ::E ::E ::E ::E ::E ::E ::E ::E ::E
0\ 0 .-4 N V") \C f' 00 0'\ 0 .-4 N V") \C f' 00 0'\ 0 .-4 N V") \C f' 00 0'\ 0 .-4 N
N N N N N N N N N N V") V") V")
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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FII060 4. Kpene:JICHble iJemOllu
4.5. CKneMBaHMe MeTannOB
4.5. 1. AHBapo6Hble KneJf
CyrueCTByIOT BbICOKOnpOqHble KJIeH, mHpOKO HCnOJIb3yeMble B MamHHOCTpoeHHH.
AHa3po6Hble KJIeH - O,n;HOKOMnOHeHTHble MaTepHaJIbI, KOTopble OCTalOTCR HeaK-
THBHbIMH B YCJIOBHRX KOHTaKTa C KHCJIOpO,n;OM. ,[LJrn OTBepX,ll;eHHR 3THM KJleRM He-
06XO,n;HMO BblnOJIHeHHe o,n;HOBpeMeHHO ,n;BYX YCJIOBHH: OTC)'TCTBHe KHCJIOpo,n;a H Ha-
JIJ{qHe MeTaJIJIJ{qeCKHX ,n;eTaJIeH. TaKHe KJIeH mHpOKO HCnOJIb3yIOTCR ,nJIR <pHKca:U;HH
H repMeTH3aU;HH pe3b6oBbIX coe,n;HHeHHH, yruIOTHeHIDI H y,n;ep:>KaHIDI (IUIH coe,n;H-
HeHIDI) ,n;eTaJIeH U;IUIHH,lJ;p:uqecKoH <pOpMbI npH KOMHaTHOH TeMnepaType, Kor,n;a cy-
IUeCTByIOT :u,n;eaJIbHbIe YCJIOBIDI ,LLJUI OTBepX,ll;eHHR KJIeR, KaK nOKa3aHO Ha pHC. 1.
PUC. 1. K.1IeH 3aTeKaeT B 3a30pbI Me)K,[(Y rneCTepHeH H BaJIOM, r,n:e cyru;eCTByroT H,ZJ;e-
aJIbHbIe YCJIOBHH ,l(JUI ero OTBep)K,[(eHHH (OTcyrcTBHe KHCJIOpo,n:a H aKTHBHbIe
MeTaJlJIHQeCKHe ,n:eTaJIH)
Me,n;b, JIaTYHb H HeJIerHpOBaHHaR yrJIepo,n;HCTaR CTaJIb «aKTHBHbI» no OTHome-
HHIO K aHa3po6HbIM KJIeRM, nOCKOJIbKy B HX npHc)'TcTBHH KJIeH 6bICTPO oTBep,n;eBaIOT
npH KOMHaTHoH TeMnepaType. Hep:>KaBelOIUaR CTaJIb, aJIlOMHHHH H aJIlOMHHHeBbIe
CTIJIaBbI, a TaIOKe MeTaJIJIbI C nOIqJbITHRMH - «naCCHBHbI» H, qTo6bI o6eCneqHTb nOJI-
Hoe OTBep,n;eHHe, TPe6yIOT HCnOJIb30BaHHR aKTHBaTopa. TaK KaK KJIeH :>KJf,l:I;KHH, OH
3aTeKaeT B MeJIbqaHmHe IUeJIH Me BCTaBJIeHHbIMH o,n;Ha B ,n;pyryro KOMnOHeHTa-
MH, r,n;e, B OTC)'TCTBHH aTMoc<pepHoro KHCJIOpo,n;a, 3aTBep,n;eBaeT H OKa3bIBaeTCR «3a-
nepTbIM» Me HepOBHOCTRMH nOBepXHOCTH. TIpo:Qeccy OTBepX,ll;eHIDI TaIOKe cno-
co6cTByeT KOHTaKT KJIeR C MeTaJIJI:uqeCKOH nOBepXHOCTblO, KOTOpaR ,n;eHcTByeT KaK
KaTaJIH3aTOp B TOM cJIyqae, eCJIH MeTaJIJI «aKTHBeH». ECJIH nOCJIe OTBepX,ll;eHIDI a,n;re-
3HB ,n;OCTHraeT BR3KOCTH TepMOpeaKTHBHOH TIJIaCTMaCcbI, OH o6eCneqHBaeT BTIJIOTb ,n;o
60% npOqHOCTH coe,n;HHeHIDI npH IqJyqeHHH, KaK nOKa3aHO Ha pHC. 2. QTo6bI rapaH-
THpOBaTb npOqHOCTb coe,n;HHeHIDI, conpHKacalOIUHeCR nOBepXHOCTH CJIe,n;yeT Mexa-
HJ{qeCKH H XHMJ{qeCKH OqHCTHTb nepe,n; HaHeceHHeM KJIeR H JII06oro Heo6xo,n;HMoro
aKTHBaTOpa - ,LLJUI y,n;o6cTBa ero 06b}qHO HaHOCRT Ha OXBaTbIBaeMYIO qaCTb coe,n;HHe-
HHR. BpeWi <pHKcaU;HH COCTaBillIeT 10-20 MHH)'T, a OKO aTeJIbHaR npOqHOCTb ,n;o-
CTHraeTCR 3a 4-24 qaca B 3aBHCHMOCTH OT BeJIJ{qHHbI 3a30pa Me ,n;eTaJIRMH, Ma-
TepHaJIOB conpHKacalOIUHXCR nOBepXHocTeH H TeMnepaTypbI. OTBep,n;eBmHe a,n;re3H-
BbI cnoco6HbI BbI,lJ;ep:>KHBaTb TeMnepaTypy BTIJIOTb ,n;o 150°C, a. TaIOKe RBJlRIOTCR
CTOHKHMH K 60JIbmHHcTBY XHMJ{qecKHX B03,n;eHcTBHH, o,n;HaKO cyrueCTByIOT a,n;re3H-
4.5. CKJleU60HUe MemOJlll06
- 60 DonT M10
J: 50
-
as
CD
Ji 40
a.
0
30
:r:
Q)
20
0
10
0
40 HM YCVlnVle
OTBOpat.lVlBaHVlS1
B OT-
CYTCT-
BVIVI
KJ1 eS1
Kne Kne BbICOKO-
HVl3KO cpeAHe npOt.lHbl
npot.l- npot.l- KJ1e
HOCTVI HOCTVI
PUC. 2. fpa<pHK, nOKa3bIBalO HH MOMeHT cpbIBa ,l(JUI 60JITa M lOB 3aBHCHMOCTH OT
KJIaCCa KJIeH, npH YCHJIHH OTBOpaQHBaHHH paBHOM 40HM
BbI, CnOC06Hble BbI,lJ;ep:>KaTb H 60JIee BbICOKHe TeMnepaTypbI. THnJ{qHble aHa3po6Hble
KJIeH H3 aCCOpTHMeHTa Loctite H HeKOTOpble HX npHMeHeHHR nepeqHCJIeHbI B Ta6JI. 1.
Ta6J1. 1. )];aHHbIe 0 npO,n:yKUHH Loctite (JIoKTaHT)
ARas o611Ye Meu
KJIeu ,lUIJI
660
HaMe aJIbHbIH 3a30
II 01JHOCTb BbICOKCUI BbICOKCUI C e,l(IDUI
BpeWi <pHKca HH npH 23 °C (MHH.) 10 10 30 10
Te MOCTOHKOCTb °C
150
150
150
175
KJIeu ,lUIJI
e3hOOBbIX coe.znmeHHii
243 2701
)];0 0,15 )];0 0,15
3a30p B
HH3KCUI C e,l(IDUI BbICOKCUI
15 10 10
150 150 150
577
542
C e,l(IDUI
30
150
napa BTIJIOTb ,l(0 ,l(aBJIeHIDI
pa3pbIBa TPy6bI I1PH
fH,lJ;paBJIWleCKHX coe-
,l(HHeHIDIX
FII060 4. Kpene:JICHble iJemOllu
4.5.2. KneJf YcP-oTSepJlCAeHJffJ
CyrueCTByIOT KJIeH, KOTopble OTBepX,ll;alOTCR no,n; B03,n;eHcTBHeM YJIbTPa<pHOJIeToBoro
(Y<I» H3JIyqeHHR. BpeMR OTBepX,ll;eHHR 3aBHCHT OT HHTeHCHBHOCTH H ,nJIHHbI BOJI-
HbI Y<I>-H3JIyqeHHR, KOTopoe HHH:QHHpyeT nOJIHMepH3a:QHIO.
IIpuMel.taHue. TIPH pa60Te C Y<I>-H3JIyqeHHeM Heo6xo,n;HMO Bcer,n;a HCnOJIb30BaTb
COOTBeTCTByIOIUHe cpe,n;CTBa 3aIUHTbI rJIa3.
K.JIeH, KOTopble OTBepX,ll;alOTCR no,n; B03,n;eHcTBHeM Y<I>- H3JIyqeHHR, RBillIlOTCR
BbICOKOnpOqHbIMH, 06JIa,n;a1OT xopomeH 3anOJIIDIIOIUeH cnoco6HoCTblO, xapaKTepH-
3yIOTCR He60JIbmHM BpeMeHeM <pHKcaO;HH, npORBillIlOT BbICOKyIO YCTO:HqHBOCTb K
B03,n;eHCTBHIO BHemHHX YCJIOBHH H JIerKO ,n;03HpyIOTCR aBTOMaTHqeCKHMH CHCTeMa-
MH, nOCKOJIbKy RBillIlOTCR O,n;HOKOMnOHeHTHbIMH.
MCTOqHHK Y<I>-H3JIyqeHHR RBillIeTCR BIDKHOH COCTaBillIlOIUeH npoo;ecca. Hana-
30H 3THX YCTPOHCTB npOCTHpaeTCR OT npOCTbIX pyqHbIX JIaMn ,n;o nOJIHOCTblO aBTO-
MaTH3HpOBaHHbIX KOHBeHepoB C HeCKOJIbKHMH Y<I>-npo:>KeKTOpaMH, BKJIlOqalOII(HX
B ce6R cne:QHaJIbHble KpeTIJIeHHR, y,n;ep:>KHBaIOIUHe KOMnOHeHTbI BMeCTe B npoo;ecce
OTBepX,ll;eHHR. ,[LITR MaJIeHbKHX KOMnOHeHTOB peKOMeH,lJ;yeTcR HCnOJIb30BaTb CBeTO-
BO,n;bI, TaK KaK 3TH YCTPOHCTBa 06eCneqHBalOT BbICOKyIO HHTeHCHBHOCTb H3JIyqeHIDI
Ha TIJIOIUa,n;H ,n;HaMeTPOM OKOJIO 10 MM. TaKHe CBeTOBO,n;bI rapaHTHpyIOT, qTO Y<I>-
H3JIyqeHHe HanpBJIeHO TOqHO Ha 06JIaCTb nOJIHMe-
pH3aO;HH, qTO n03BOJIReT MHHHMH3HpOBaTb MOm;-
HOCTb Y<I>-H3JIyqeHHR HerO pacceRHHe. CBeTOBO,l(
nOKa3aH Ha pHC. 1.
B HeKOTopbIX CJIyqaRX He Bcer,n;a MO:>KHO rapaH-
THpOBaTb, qTO BeCb KJleH no,n;BeprCR Y<I>-o6.rryqe-
HHIO, n03TOMY pa3pa6oTaHbI KJleH, KOTopble OTBep-
,n;eBalOT nOCJIe O;HKJla Y<I>-06JIyqeHHR. CyrueCTByeT
HeCKOJIbKO BapHaHTOB BTopJ{qHoro OTBepX,ll;eHIUI.
nO)l;orpeB
BTopJ{qHoe OTBepX,ll;eHHe no,n; B03,n;eHCTBHeM
TeTIJIa rapaHTHpyeT nOJIHOe OTBepX,ll;eHHe ,LLJUI MHO-
rHX BH,lJ;OB Y<I>-KJIeeB. TeMnepaTypbI OTBepX,ll;eHIDI
06bIQHO JIe:>KaT B ,n;Hana30He 100-150°C.
OTCYTCTBHe KHCJIOpO)l;a
Y<I> aHa3po6Hble copTa npHMeIDIIOTCR B Tex
npIUlO:>KeHHRX, r,n;e Tpe6yeTcR OTBepX,ll;eHHe B 06-
JIaCTH TeHH. TaKHe aHa3po6Hble KJIeH OTBep,n;eBaIOT
PUC. 1. rH6KHH Y<I>-cBeTOBO,n: Ha rJIy6HHe 0,2 MM BCJIe,n;CTBHe KOHTaKTa C MeTaJI-
,l(JUI OTBep)K,[(eHHH Y<I>-KJIeH.
JIOM H OTC)'TCTBIDI KHCJIOpo,n;a.
BJImKHOCTb
Y<I> CIUIHKOHOBble KJleH CXBaTbIBalOTCR B pe3YJIbTaTe Y<I>-nOJIHMepH3aO;HH, H
3aTeM OTBepX,ll;eHHe npo,n;OJI:>KaeTCR B TeqeHHe 24 qaCOB 6JIaro,n;apR BJIIDKHOCTH
OKpyxalOIUero B03,n;yxa.
nOBepxnocTHaB BJImKHOCTb
Y<I> O;HaHoaKplUlaTHble KJIeH OTBepX,ll;alOTCR Ha HeCKOJIbKO MHJIJIHMeTPOB no,n
B03,n;eHCTBHeM Y<I>-06JIyqeHHR, HO OHH TaIOKe OTBepX,ll;alOTCR 6JIaro,n;apR nOBepXHO-
CTHOH BJIIDKHOCTH B 3aTeHeHHbIX 06JIaCTRX.
4.5. C1(;JleU8aHUe Mematll108 2.
O,n;HOH H3 OCHOBHbIX npJ{qHH HCnOJIb30BaHHR Y<I>-KJIeeB RBJlReTCR npeHMYIUec
BO «OTBepX,ll;eHHR no TPe60BaHHIO» (T. e. cnoco6HoCTb OTBepX,ll;aTbCR TOqHO B JIy)
HbIH MOMeHT, TOJIbKO nOCJIe Toro KaK KOMnOHeHTbI 6y,n;yT nOJIHOCTblO BblpoBHeHbI)
4.5.3. LlJfBHOBKpJfnBTHble KneJf -
BHJfOHHBfJ nOnJfMepJf3BLlJffJ
O,n;HOKOMnOHeHTHble U;HaHoaKplUlOBble KJIeH nOJIHMepH3HpyIOTCR B KOHTaKTe c
CJIerKa IUeJIOqHbIMH nOBepXHOCTRMH. KaK npaBIUIO, BJIIDKHOCTH oKpyxalOIUeH Cpt
,[(bI H BJIIDKHOCTH nOBepXHOCTH coe,n;HHeHHR ,n;OCTaTOqHO, qTo6bI HaqaJIOCb OTBep)J
,[(eHHe H B TeqeHHe HeCKOJIbKHX ceKyH,lJ; 6bma ,n;OCTHrHYTa <pHKCaU;HR. HalUlyqml1
pe3YJIbTaTbI MO:>KHO nOJIyqHTb npH OTHOCHTeJIbHOH BJIIDKHOCTH B03,n;yxa 40-60
rrpH KOMHaTHOH TeMnepaType B pa60qeM nOMeIUeHHH. EOJIee HH3KaR BJIIDKHOCl
rrpHBo,n;HT K 60JIee Me,nJIeHHOMY OTBepX,ll;eHHIO, 60JIee BbICOKaR BJIIDKHOCTb YCKOp5.1
eT OTBepX,ll;eHHe, HO B KpaHHHX CJIyqaRX MO:>KeT OCJIa6HTb OKOHqaTeJIbHYIO npotJ
HOCTb coe,n;HHeHHR.
IToCJIe HaHeCeHHR KJIeR qaCTH CJIe,n;yeT 6bICTPO coe,n;HHHTb, TaK KaK nOJIHMepH
3a:U;HR HaqHHaeTCR yxe qepe3 HeCKOJIbKO ceKyH,lJ;. BpeMR ,n;o HaqaJIa nOJIHMepH3a
nHH 3aBHCHT OT OTHOCHTeJIbHOH BJIIDKHOCTH B03,n;yxa, BJIIDKHOCTH coe,n;HIDIeMbIX no
BepXHocTeH H TeMnepaTypbI oKpyxalOIUeH cpe,n;bI. M3-3a 6bICTPOTbI OTBepX,ll;eHID
nHaHoaKplUlaTHble KJIeH oco6eHHO nO,llXOMT ,lI)IR CKJleHBaHHR MeJIK.HX ,n;eTaJIeH
TalOKe cyrueCTByIOT BH,lJ;bI KJIeR Y<I>-oTBepX,ll;eHHR, n03BOillllOIUHe YJIyqmHTb OTBep.
:>K.LJ;eHHe no BceH rJIy6HHe KJIeR, a TaIOKe cnoC06CTByIOIUHe OTBepX,ll;eHHIO JII06bD
H3JIHmKOB KJIeR. l(HaHoaKplUlaTHble KJleH CJIe,n;yeT HaHOCHTb TOJIbKO Ha o,n;HY no-
BepXHOCTb, npHqeM JIyqmee coe,n;HHeHHe no.rryqaeTCR, TOJIbKO eCJIH HaHeCTH ,n;OCTa-
TOqHO KJIeR, qTo6bI 3anOJIHHTb Bce 3a30pbI B coe,n;HHeHHH. AIcrHBaTopbI MOryr HC-
rrOJIb30BaTbCR ,LLJUI YCKopeHHR npou;ecca OTBepX,ll;eHHR, a TaIOKe ,LLJUI OTBepX,ll;eHID]
JII06bIX H3JIHmKOB KlIeR. ,LJ;onOJIHHTeJIbHOe npeHMyrueCTBO U;HaHoaIqJlUlaTHbIX KJleeE
3aKJIIOqaeTCR B TOM, qTO HX MO:>KHO HCnOJIb30BaTb KaK rpYHTOBKy (npaHMep) npl1
coe,n;HHeHHH TaKHX MRrKHX TIJIaCTHKOB, KaK nOJIHnpOnlUleH H nOJIH3TlUleH.
,LJ;OCToHHCTBa :QHaHoaIqJlUlaTHbIX KlIeeB:
. OqeHb BbICOKaR npOqHOCTb Ha paCTR:>KeHHe H KpyqeHHe;
. OqeHb BbICOKaR CKOpOCTb OTBepX,ll;eHHR (CKopee ceKyH,lJ;bI, qeM MHHYTbI);
. MHHHMaJIbHbIH pacxo,n; KJIeR;
. coe,n;HHeHHe nOqTH Bcex BH,lJ;OB MaTepHaJIOB;
. JIerKaR ,n;03HpOBKa 6JIar,n;apR xopomeMY COnpOTHBJIeHHIO CTapeHHIO O,n;HOKOM-
nOHeHTHbIX KJIeeB;
. o,n;HOBpeMeHHbIH 3<p<peKT repMeTH3aU;HH.
4.5.4. MOAJfCPJfLlJfPOSBHHble BKpJfnOSble KneJf -
OTSep eHJfe C JfCnOnb30SBHJfeM BKTHSBTOpOS
OTBepX,ll;eHHe 3THX KlIeeB npOHcxo,n;HT npH KOMHaTHOH TeMnepaType B npHC)'TCT-
BHH aKTHBaTopOB. K.JIeH H aKTHBaTOp OT,n;eJIbHO HaHOCRTCR Ha coe,n;HIDIeMbIe no-
BepXHOCTH. KOMnOHeHTbI 3TOH CKJleHBalOIUeH CHCTeMbI 3apaHee He nepeMemHBa-
IOTCR, n03TOMY HeT Heo6xo,n;HMOCTH 6ecnOKOHTbCR 0 «:>KH3HeCnoco6HOCTH» KJIeR.
rl1060 4. Kpene:JICHble iJemOllu
Mo,n;H<pHnHpOBaHHble aKplUloBble KJIeH 06JI8JJ:a1OT CJIe,n;yIOIUHMH xapaKTepHbIMH
CBoHcTBaMH:
. OqeHb BblCOKaR npOqHOCTb Ha pacTIDKeHHe H Ha c,n;BHr;
. xopomaR y,n;aponpOqHOCTb H cOnpOTHBJIeHHe OTCJIaHBaHHIO;
. mHpoKHH ,n;Hana30H pa60qHX TeMnepaTYP (-55 ... +120°C);
. coe,n;HHeHHe nOqTH Bcex MaTepHaJIOB;
. BbICOKaR 3anOJIIDIIOIUaR cnoco6HoCTb;
. XopomaR YCTOHqHBOCTb K B03,n;eHcTBHIO oKpyxalOIUeH cpe,n;bI.
4.5.5. KneJf, OTsepJK,qeHJfe KOTOpblX npoJfCXOAJfT
nOA s03AeiicTSJfeM SnSJlCHOCTJf
3TH KJIeH/repMeTHKH nOJIHMepH3HpyIOTCR (B 60JIbmHHCTBe CJIyqaeB) 3a CqeT KOH-
,n;eHcanHH, KOTOpaR BKJIlOqaeT peaKI.J;HIO C BJIarOH H3 B03,n;yxa. B 3TOH KaTeropHH
cyrueCTByeT TpH THna KJIeeB, a HMeHHO:
. CIUIHKOHOBble;
. nOJIHypeTaHOBble;
. Mo,n;H<pHnHpOBaHHble ClUlaHOBble.
CHJlHKOHOBble
BYJIKaHH3anIDI (OTBep)l{,lJ;eHHe) npOHcxo,n;HT npH KOMHaTHOH TeMnepaTYPe B pe-
3YJIbTaTe peaKI.J;HH C BJIarOH H3 B03,n;yxa. MOHOJIHTHaR pe3HHa H3 CIUIHKOHa OTJI:uqa-
eTCR CJIe,n;yIOIUHMH CBOHCTBaMH:
. npeBocxo,n;HaR TepMOCTOHKOCTb;
. 3JIaCTHqHOCTb, npOqHOCTb, HH3KHH MO,n;yJIb ynpyrOCTH H 60JIbmOe OTHOCH-
TeJIbHOe y,nJIHHeHHe;
. 3<p<peKTHBHbIH repMeTHK,LLJUI pa3JI:uqHblx BH,lJ;OB )I{H,lJ;KOCTH.
nOJluypeTaHOBble
MexaHH3M 06pa30BaHIDI nOJIHypeTaHOB OCHOBaH Ha peaKI.J;HH BO,n;bI (B 60JIb-
mHHCTBe CJIyqaeB) C BeIUeCTBaMH, co,n;ep:>KaIUHMH rpynnbI H30IJ;HaHaTa. 3TH KJIeH
OTJIHqalOTCR CJIe,n;yIOIUHMH CBOHCTBaMH:
. npeBocxo,n;HaR npOqHOCTb H 3JIaCTHqHOCTb;
. npeBocxo,n;Hoe 3anOJIHeHHe 3a30pOB (,n;o 5 MM);
. nOCJIe OTBep)l{,lJ;eHIDI nOMalOTCR OKpaCKe;
. MO:>KeT HCn0JIb30BaThCR C npaHMepaMH (rpYHTOBKOH) ,nmI yJIyqmeHHR anre3HH.
MO.znt4»HIUlpoBaHHble CHJlaHOBhIe
OTBep)l{,lJ;eHHe MO,n;H<pHIJ;HpOBaHHblx ClUlaHOB TaIOKe npOHcxo,n;HT qepe3 peaK-
IJ;HIO C BJIarOH H3 B03,n;yxa. OHH 06JI8JJ:a1OT CJIe,n;yIOIUHMH CBOHCTBaMH:
. nOCJIe OTBepX,ll;eHIDI nOMalOTCR OKpaCKe;
. xopomo coe,n;HHRIOTCR C mHpOKHM ,n;Hana30HOM oCHoBaHHH;
. OTCYTCTBHe H30nHaHaTOB;
. npeBocxo,n;HaR npOqHOCTb H 3JIaCT:uqHOCTb.
3,n;ecb npe,n;CTaBJIeH Bcero JIHmb KpaTKHH 0630P OCHOBHbIX BH,lJ;OB 3THX cnenHa-
JIH3HpOBaHHblX KJIeeB, H qHTaTeJIb MO:>KeT o6paTHTbcR K TeXHHqeCKOH JIHTepaTYPe,
H3,n;aHHoH Henkel Loctite Adhesives Limited, qTo6bI nOJIyqHTb 60JIee no,n;po6JIYIO
HH<popManHIO 0 BH,lJ;ax KJIeeB H HX npHMeHeHHH.
4.5. CKJleU60HUe MemOl/l106
4.5.6. 3nOKCJfAHble KneJf
3nOKCH,lJ;Hble KJIeH OCHOBaHbI Ha CMOJIax, no eHHbIX H3 3nHXJIOprH,lJ;pHHa H 6HC-
<l>eHoJI'il-AJ OHH xapaKTepH3yIOTCR HH3KOH YC8JJ:KOH npH nOJIHMep 3anHH, BbICOKOH
a):{re3HeH H xopomeH MexaH:uqeCKOH npOqHOCTblO. 06bIHO 3TH KJIeH ,n;ByxKOMnO-
HeHTHble, H OTBep)l{,lJ;eHHe HaCTYIIaeT B pe3YJIbTaTe CMemHBaHIDI KJIeR C KaTaJIH3a-
TOpOM (OTBep,n;HTeJIeM) HenOCpe,n;CTBeHHO nepe,n; HCnOJIb30BaHHeM. OTBepX,ll;eHHe
HaqHHaeTCR C MOMeHTa ,n;06aBJIeHIDI KaTaJIH3aTOpa. O,n;HaKO B HeKOTOpbIX CJIyqaRX
OTBep,n;HTeJIb CMemHBaeTCR C KJIeeM B npOnecce H3rOTOBJIeHIDI, a aKTHBH3HpyeTcR
COCTaB HarpeBaHHeM (06blqHO) ,n;o 150°C B TeqeHHe 30 MHHYT. B accopTHMeHe
Loctite@ Hysol@ npHCYTCTByIOT Bce THn:uqHble 3nOKCH,lJ;Hble KJIeH. PyxOBO,n;CTBO no
BbI60py KJIeR B 3aBHCHMOCTH OT OCHOBaHIDI npe,n;CTaBJIeHO Ha pHC. 1.
CyruecTByIOT CJIe.l(ylOIUHe THn:uqHble Hysol@ 3nOKCH,lJ;Hble KJIeH:
1. 3nOKCH,lJ;ble KJIeH o6IUero Ha3HaqeHIDI:
. YHHBepCaJIbHble KJIeH,LLJUI coe,n;HHeHHH o6IUero Ha3HaqeHIDI.
. YJIbTpanp03paqHble KJIeH ,LLJUI np03paqHbIX mBOB.
. 3JIaCTHqHble KJIeH ,LLJUI CJIa60HanpIDKeHHbIX coe,n;HHeHHH, TIJIaCTMaCC H Heo,n;-
HOpO,n;HbIX OCHOBaHHH
2. DbICTPonOJIHMepH3yIOIUHeCR 3nOKCH,lJ;Hble KJIeH:
. DbICTPOe OTBep)l{,lJ;eHHe B TeqeHHe KopOTKoro BpeMeHH <pHKcanHH.
. YJIbTpanp03paqHble KJIeH,LLJUI 6bICTPOro np03paQHOrO coe,n;HHeHIDI.
. CpOQHbIH peMOHT MeTaJIJI:uqecKHX QaCTeH, TaKHX KaK TPy6bI H 6aKH.
3. BblcOKOnpOQHble 3nOKCH,lJ;Hble CMOJIbI:
. BbICOKaR npOQHOCTb Ha c,n;BHr H OTCJIaHBaHHe.
. TIpeBocxo,n;HaR y,n;aponpOQHOCTb.
. CKJIeHBaHHe CJIO:>KHbIX KOMn03HTOB, TaKHX KaK CTeKJIOTIJIaCTHK H yrJIeTIJIac-
THK (TIJIaCTHK, apMHpoBaHHbIH yrJIepo,n;HbIM BOJIOKHOM).
4. TepMocToHKHe 3nOKCH,lJ;Hble CMOJIbI:
. Y CTOHqHBOCTb npH BbICOKHX pa60QHX TeMnepaTYPax.
. CooTBeTcBHe UL94- VO ,LLJUI orHeynopHbIX ,n;eTaJIeH.
. BblcoKaR TeTIJIOnpOBo,n;HOCTb.
5. MeTaJIJIOHanOJIHeHHble 3nOKCH,lJ;Hble CMOJIbI:
. PeMoHT H BOCCTaHOBJIeHHe H3HomeHHbIX H nOBpe)l{,lJ;eHHbIX MeTaJIJI:uqecKHX
,n;eTaJIeH.
. PeMoHT MeTaJIJI:uqecKHX TPy6 H OTJIHBOK.
. M3rOTOBJIeHHe <pOPM, HHCTPYMeHTa H apMaTYPbI.
. CHIDKeHHe H3Hoca ,n;BIDKYIUHXCR QaCTeH npH TpeHHH CKOJIb:>KeHIDI.
THnHQHbIe pa60QHe xapaKTepHCTHKH npHBe,n;eHbI Ha pHC. 2. qHTaTeJIb MO:>KeT
o6paTHTbcR 3a HCQepnbIBalOIUeH HH<popManHeH K ,n;aHHbIM, ony6JIHKOBaHHbIM Hen-
kel Loctite Adhesives Limited, CBR3aTbCR C KOTOpOH MO:>KHO no 8JJ:pecy, npHBe,n;eH-
HOM)' B npIUlO:>KeHHH 3.
4.5.7. BOCCTSHOSJfTenbHblH npOLf,eCC
MeTo,n;, pa3pa6oTaHHbIH ,LLJUI coe,n;HHeHHR HCXO,n;HbIX JIHCTOB MeTaJIJIa B CaMOJIeTOCT-
poeHHH C nOMOIUblO ,n;ByxKOMnOHeHTHoro KJIeR, npH TIUaTeJIbHOM KOHTPOJIe 3a
TeMnepaTYPOH H ,n;aBJIeHHeM.
FII060 4. Kpene:JICHble iJemOllu
@)
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o
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@)
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o t;:(I:S r.Q:= - ............ 0
e a
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sag s g1 C!Sg gJ
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0 ..,Q Q $Q
cd cd:= cd E-t':' p,. o M
o ogog 20S2 O
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4.5. CKJleU60HUe MemO/lJl06 2 -1
a,)g e
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ci o
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FII060 4. Kpene:JICHble iJemOllu
4.5.8. Kneesoe COeAJfHeHJfe
KaK yxe 6billO CKa3aHO, ,nJIR OnTHMaJIbHOrO coe,n;HHeHHR Heo6xo,n;HMa npaBlUlbHasI
no,n;rOTOBKa nOBepXHOCTH. TIpOqHOCTb coe,n;HHeHHR B 3HaqHTeJIbHOH CTeneHH onpe-
,n;eJIReTCR CneTIJIeHHeM MeX,ll;Y KJIeeM H OCHOBaHHeM. TI03TOMY npH no,n;rOTOBKe c
MeCTa coe,n;HHeHHR CJIe,n;yeT y,n;aJIHTb MaCJIO, CMa3Ky, OKHCHble TIJIeHKH H rpR3L,
qTo6bI 06eCneqHTb XHMHqecKyIO H MexaH:uqecKyIO qHCTOTY. MeXaHHqeCKaR o6pa-
60TKa nOBepXHOCTH 06,n;YBKOH CTaJIbHOH KpomKoH IUIH npOBOJIOqHOH IUeTKOH ra-
paHTHpyeT xopomee CneTIJIeHHe.
KpoMe Toro, Heo6xo,n;HMO npaBlUlbHO CnpOeKTHpOBaTb coe,n;HHeHHe. Conpo-
THBJIeHHe KJIeeB paCTIDKeHHIO If C,n;BHry OTHOHTeJIbHO BeJIHKO no cpaBHeHHIO C HX
conpOTHBJIeHHeM CKaJIbIBaHHIO H OTCJIaHBaHHIO. 3TO Heo6xo,n;HMO yqecTb npH npo-
eKTHpOBaHHH coe,n;HHeHHR, a TaIOKe npe,n;yCMOTPeTb ,n;OCTaTOqHYIO nOIU8JJ:b nOBepx-
HOCTH Me coe,n;HIDIeMbIMH qaCTRMH, qTo6bI 06eCneqHTb TPe6yeMYIO npOqHOCTb.
TaKHM o6pa30M, coe,n;HHeHHe C caMoro HaqaJIa CJIe,n;yeT npOeKTHpOBaTb C yqeToM
HCnOJIb30BaIDI KJIeR, a He npOCTO npHcnoca6JIHBaTb cyrueCTByIOIUee MexaH:uqe-
CKoe/CBapHoe coe,n;HHeHHe. Ha pHC. 1 nOKa3aHO HeCKOJIbKO npaBlUlbHO H Henpa-
BlUlbHO CnpOeKTHpOBaHHblX coe,n;HHeHHH, no,n;xoAAIUHX,LLJUI CKJleHBaHHR.
ABTOP 6JIaro,n;apHT MHCTepa Bo6a rocca, CTapmero CnenHaJIHCTa no TeXHOJIO-
rHH B Henkel Loctite Adhesives Limited 3a ero nOMomb B no,n;rOTOBKe BblmeH3JlO-
:>KeHHOH HH<popManHH 0 KJleeBbIX coe,n;HHeHIDIX.
npaBMnbHO
AnbTepHaTHBHble BapHaHTbl npOCToro
COe,D,HHeHHR BHaxnecTKY
HenpaBMnbHO
nepeKawHBaHHe H pa3pyweHHe
npOCToro COe,D,HHeHHR BHaxnecTKY
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c -.
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B03HVlKHOBeHVle pa3pyweHVlS1 OTpblBOM
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1 HarPY3Ka
HarpY3Ka
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arpY3Ka
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CoeAVlHe-
HVie npVI
C)f(aTVIVI
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PUC. 1. npeKTHpOBaHHe KJIeeBbIX coe,n:HHeHHH
rnABA 5
MEXAHM'tECKME
nEPEAA'tM
5.1. MeXaHM"IeCKMe nepeAa"lM: 3y6"1aTbie nepeAa"lM
5. 1. 1. HeKoTopble TJfnJfI.fHble 3y61.fsTble nepeASI.fJf
npRMo3y6aR
MnMHAPM'IeCKaR 3y6'1aTaR nepeAa'la
Pee'tHaR nepeAa'la: npeo6pa30BaHMe Bpal1.\aTenbHoro ABM)I(eHMR B nOCTYnaTenbHoe
3y6t.1aToe Koneco
I
/
3y6t.1aTas:a pe
Ka
Dla8a 5. MexaHulteclCue nepeiJaltu
Koco3y6aR 3y6'taTaR nepeAa'ta
WeBpOHHaR 3y6'taTaR nepeAa'ta
npRMo3y6aR KOHM'teCKaR 3y6'taTaR nepe.qa'l8
'fepBRK M 'tepBR'tHoe Koneco
5.1. MexaHUlteC1(,Ue nepeoaltu: 3y6ltamble nepeoaltu 30 I:::
5.1.2. npOCTble npflMo3y6ble LfJfnJfHAPJflleCKJfe nepe,q8l1Jf
BeAy a weCTepH
npOMe)K}'TOYHOe KOl1eco
BeAOMoe KOl1eco
IIpocTaJl nepe,ZJ;a1fa
1. Be,n; (UI rneCTepIDI H Be,n;OMoe KOJIeCO BpalI..(alOTCR B rrpOTHBOrrOJIO)I(HbIX
HanpaBJIeHHRX.
2. OTHollIeHHe qaCTOTbI BpalI..(eHHR Be,n; eit llIeCTepHH K qaCTOTe BpalI..(eHHR
Be,nOMoro KOJIeCa HJIH rrepe,n;aTOqHOe OTHOllIeHHe (oTHollIeHHe yrJIOBbIX CKopocTeH
Be,nyIUeH llIeCTepHH H Be,n;OMoro KOJIeca) BbIqHCIDIeTCR no CJIe.nyIDlI..(eH <popMYJIe:
06/MHH Be,LJylI(ee _ qHCJlO 3y6beB B Be,nOMOM
06/MHH Be,nOMoe - qHCJlO 3y6beB B Be,LJylI(eM.
(BbIpaxeHHe crrpaBa OT 3HaKa paBeHCTBa Ha3bIBaeTCR rrepe,n;aTOqHbIM qHCJIOM. -
llpUMett. nep.)
IIpHMep.
PaCCqHTaTb qaCTOTY BpalI..(eHIDI Be,n;OMoro KOJIeCa, eCJIH qaCTOTa BpalI..(eHIDI Be.ny-
meH llIeCTepHH COCTaBJUleT 120 06/MHH. Be,n;oMoe KOJIeCO HMeeT 150 3y6beB, a Be.ny-
mWI llIeCTepIDI - 50 3y6beB:
120 150
06/MHH Be,nOMoe - 50 '
120 x 50
06/MHH Be,nOMoe = = 40 06/MHH.
150
Dla8a 5. MexaHulteCK,Ue nepeiJaltu
npOCTaJI nepe,ZJ;a1fa C npOMe1K)'T01fHbIM KOJleCOM
1. Be,l YIIUUl meCTepIDI II Be,ZJ;OMoe KOJIeCO BpalI..(alOTCR B O,n;HOM HanpaBJIeHHH,
eCJIII KOJIlftleCTBO rrpOMexyrOqHbIX KOJIeC HeqeTHOe, H B npOTIIBOrrOJIO)KHbIX Ha-
rrpaBJIeHIDIX, eCJIII HX KOJIlftleCTBO qeTHO.
2. npoMexyrOqHbIe KOJIeCa HCrrOJIb3yroTCR .nIDI 113MeHeHIDI HarrpaBJIeHIDI Bpa-
lI..(eHIDI II/R1I11 .nIDI YBeJIllqeHIDI MexoceBoro paCCTORHIDI Me y Be.nyIneit
mecTepHeH Be,n;OMbIM KOJIeCOM.
II'
3. KOJIlftleCTBO rrpOMexyrOqHbIX KOJIeC II qHCJIO 3y6beB Ha rrpOMexyrOqHbIX KO-
JIecax He BJIIDIIOT Ha o61I..(ee nepe,n;aTOqHOe OTHomeHlle.
4. 061I..(ee nepe,n;aTOqHOe OTHomeHlle MO)KHO BbIqllCJIIITb no <poPMYJIe:
06/MHH Be,nyInHH _ qHCJIO 3y6beB B Be,nOMOM
06/MHH Be,nOMbIH - qHCJIO 3y6beB B Be,nyIneM'
5. 1.3. CTyneHII8T8f1 npflMo3y68f1 3y6118T8f1 nepeA8118
npOMe)l(YTOYHOe KOl1eco '1
npOMe)l(YTOYHOe KOl1eco '2
BeAy aS1 weCTepHS1
1. B OTJIllqlle OT npOMexyrOqHOrO KOJIeCa npocToH rrepe,n;aqll, rrpOMexyrOqHbIe
KOJIeCa cTYIIeHqaTOH nepe,n;aqll BJIIDIIOT Ha o61I..(ee rrepe,n;aTOqHOe OTHOllIeHlle (nOJI-
Hoe OTHOllIeHlle yrJIOBbIX CKopOCTeH Be,n;yII..(eH llIeCTepHII II Be,n;OMoro KOJIeca).
2. 06a npOMexyrOqHbIX KOJIeCa (It II 1 2 ) YCTaHOBJIeHbI Ha O,n;HOM II TOM xe BaJIe
II IIMelOT o,n;HY II TY xe qaCTOTY BpalI..(eHIDI.
3. Be.nyIn(UI llIeCTeplUl II Be,n;OMoe KOJIeCO BpalI..(alOTCR B O,n;HOM HanpaBJIeHIIII. qTO-
6bI 113MeHIITb HanpaBJIeHlle BpalI..(eHIDl, Heo6xo,n;IIMO YCTaHOBIITb rrpOMexyrOqHOe KO-
JIeco JIII60 Me Be.nyIneH llIecrepHeH II II' JIII60 Me Be,n;oMbIM KOJIeCOM II 1 2 ,
5. 1. MexaHUlteCKUe nepeiJaltu: 3y6ltamble nepeiJaltu 3
4. 061I..(ee rrepe,n;aTOqHOe OTHOllIeHHe MO)J(}lO BbIqHCJIHTb no <popMyJIe (T.e. KaK
rrpoH3Be,n;eHHe rrepe,n;aTOqHbIX qHCeJI):
06/MHH Be,LJyI.QHH _ qHCJlO 3y6beB B /} qHCJlO 3y6beB B Be,nOMOM
- x .
06/MHH Be,nOMbIH qHCJlO 3y6beB B Be,LJyI.QeM qHCJlO 3y6beB B /2
npHMep
PaCCqHTaTb qaCTOTY BpalI..(eHIDI Be,n;OMoro KOJIeCa, eCJIH H3BeCTHO: Be.nyIUIDI llIe-
CTepIDI BpalI..(aeTCR C qaCTOTOH BpalI..(eHIDI 600 06/MHH H HMeeT 30 3y6beB; II HMeeT
60 3y6beB; 1 2 HMeeT 40 3y6beB; a Be,n;OMoe KOJIeCO HMeeT 80 3y6beB:
06/MHH Be,LJyI.QHH = 60 x 80 = i
06/MHH Be,noMbIH 30 x 40 1
Ho qaCTOTa BpalI..(eHIDI Be,n;yII..(eH llIecTepHH = 600 06/MHH
TI03TOMy
600 06/MHH Be,LJyI.QHH = i
06/MHH Be,noMbIH 1
600 x 1
qacToTa Bpa eHHH Be,nOMoro = = 150 06/MHH.
4
5. 1.4. 3sonbseHT8 KpJfsoif
C
/
/
/
/
I
OSOl1bseHTa ---,
I
I
A I //
--f--
rV16KV1 HepaCT )I(V1MblVi WHYP
T(TOYKa KacaHV1 )
OCHOSHa OKpY)I(HOCTb
B
I
3BOJIbBeHTa HCnOJIb3yeTcR ,L(JIR npo<pHJIeH 3y6beB 60JIbllIHHCTBa cepHHHo H3ro-
TaBJIHBaeMbIX npRMo3y6bIX n;HJIHH,L(pHqeCICHX 3y6qaTbIX nepe,n;aq. Ha pHCYHKe nOKa-
3aHa 3BOJIbBeHTa, no.rryqeHHIDI npH CMaTbIBaHHH Boo6paxaeMoro TYI'0 HaTRHyroro
illHypa BTC C OCHOBHOH OKpyxHOCTH C n;eHl'pOM B TOqKe 0, Ha'lHHaR C TOqKH A.
KpHBIDI AC, KOTOpyro onHCbIBaeT KOHen; llIHypa C, HMeeT <POPMY 3BOJIbBeHTbI.
KpHBH3Ha C 3aBHCHT OT pa.nHYca OCHOBHOH OKpyxHOCTH. ECJIH pa.nHYC
OCHOBHOH OKpyxHOCTH YBeJIHqHBaeTCR, KpHBH3Ha 3BOJIbBeHTbI YMeHbllIaeTCR. ,lJ;JIR
OCHOBHOH OKpyxHOCTH 6eCKOHeqHOrO p.a.nHYca JIHHIDI AC nepeCTaeT 6bITb KpHBOH
H CTaHOBHTCR npRMOH JIHHHeH. TaKHM 06pa30M, peeqHIDI nepe,n;aqa HMeeT 3y6bR C
Dla8a 5. MexaHulteclCue nepeiJaltu
npRMbIMH cTopoHaMH, nOCKOJIbKy 3y6qa1)1O peHKy MO:>KHO paccMal'pHBaTb, KaK
qaCTb OKpyxHOCTH 3y6qaTOrO KOJIeCa 6eCKOHeQHOrO pa.n:M)'ca. n03TOMY 3y6qaTWI
peHKa C npRMbIMH cTopoHaMH 3y6beB Bcer,n;a y,n;oBJIeTBOpHTeJIbHO BXO,n;HT B 3a en-
JIeHHe C 3y6qaTbIM KOJIeCOM C npRMbIMH 3y6bRMH B <popMe 3BOJIbBeHTbI.
<l>opMa 3BOJIbBeHTbI ,L(IDI 3y6beB npRMo3y6oH :U;R1IHH,L(pHtleCKOH 3y6qaTOH nepe-
,n;aqH HMeeT CJIe.nyIDII..(He npeHMYII..(eCTBa.
1. 3BOJIbBeHTHbIe 3y6bR npe,n;OCTaBIDIIOT rrepe,n;aTOqHOMY MexaHH3MY B03MO)l{-
HOCTb I1JIaBHO nepe,n;aBaTb 60JIbllIHe HarpY3KH rrpH rrOCTORHHOH CKOpOCTH.
2. npH corrpIDKeHHbIX 3y6bRX C npo<pR1IeM, OqepqeHHbIM no 3BOJIbBeHTe, B03-
HHKaeT npeHMYII..(eCTBeHHO KOHTaKT KaqeHIDI Mex.ny HO)l{l(aMH 3y6beB, Kor,n;a OHH
BXO,nRT H BbIXO,nRT H3 3a:u;eIIJIeHIDI. nOCKOJIbKy KOHTaKT CKOJIbXeHIDI MHHHMaJIeH,
CTHpaHHe H H3HOC HoxeK 3y6beB TalOKe MHHHMaJIbHbI.
3. oJIaro,n;apR COOTHOllIeHHIO Mex.ny 3BOJIbBeHTHbIMH 3y6bRMH H 3y6bRMH C
npRMbIMH CTopOHaMH 3y6qaTOH peHKH, 3y6bR 3y6qaTOrO KOJIeCa MO:>KHO 6bICTPO H
aKKYPaTHO Hape3aTb C nOMOlI..(blO <ppe3bI B BH.l(e peHKH C JIerKO H3rOTaBJIHBaeMbIMH
pexyII.J;HMH 3y6bRMH C npRMbIMH CTopOHaMH. OT 3Toro, OqeBH.l(HO, BbIHrpbIBaer
CTOHMOCTb HHCl'pYMeHTOB.
5. 1.5. OCHOBbl reOMeTpJfJf 3y6118Toro KoneC8
QTo6bI Ha npaKTHKe nOCl'pOHTb npo<pR1Ib 3y6a 3y6qaTOrO KOJIeCa Ha OCHOBaHHH 6a-
30BOH 3BOJIbBeHTbI, onHcaHHOH B pa3,n;eJIe 5.1.4, Heo6xo,n;HMO paCCMOl'peTb ,n;Ba CJIe-
,n;YIOII..(HX reOMel'pHqeCKHX COOTHOllIeHHR.
1. KaCaTeJIbHaR K 3BOJIbBeHTe B JII060H TOqKe Bcer,n;a rrepneH,L(HKyIDIpHa K Kaca-
TeJIbHOH K OCHOBHOH OKpyxHOCTH, npoBe,n;eHHoH H3 TOH xe caMOH TOqKH. B paJ,n;e-
JIe 5.1.4 nOKa3aHO, qTO npRMIDI CT RBIDIeTCR KaCaTeJIbHOH K OCHOBHOH OKpyxHO-
CTH H qTO 3Ta KaCaTeJIbHaH nepneH,L(HKyIDIpHa KaCaTeJIbHOH, npoBe,n;eHHoH K 3BOJIb-
BeHTe B TOqKe C. oy,n;eT nOKa3aHO, qTO Ka-
CaTeJIbHaR CT RBIDIeTCR TalOKe JIHHHeH
3a:u;eIIJIeHIDI 3y6beB KOJIeca.
2. ,LLrrHHa npRMoH CT TalOKe paBHa .nnH-
He ,L(yrH AT.
PaCCMOl'pHM ,n;Be OKpyxHOCTH, KacalOlI(H-
eCR B TOqKe P, KaK nOKa3aHO Ha pHCyHKe.
T OqKa P Ha3bIBaeTCR nOJIIOCOM 3a ellJIeHIDI,
a OKpyxHOCTH Ha3bIBalOTCR HaqaJIbHbIMK
OKpyxHOCTRMH. ECJIH 6bI HaqaJIbHbIe OIq)YJK-
HOCTH 6billH <ppHK:QHOHHbIMH ,n;HCKaMH, T0
OHH KaTHJIHCb 6bI O,ZJ;HH no ,n;pyrOMY C TPe6ye-
MbIM OTHOllIeHHeM CKopocTeH B 3aBHCHMocm
OT COOTHOllieHIDI HX ,n;HaMel'pOB. O,n;HaKo Boa-
MO:>KHOCTH MexaHH3MOB, B KOTOpbIX nepe aqa
MOIIJ;HOCTH 3aBHCeJIa 6bI HCKJIlOqHTeJIbHO OT
CR1IbI l'peHIDI, 6billH 6bI CR1IbHO orpaHWleHLI.
Ha npaKTHKe, qTo6bI npe,n;OTBpaTHTb rrpOCKa-
Ha al1bHaS1 JIb3bIBaHHe, ,n;HCKH coe,n;HIDIIOT 3a:u;eIIIDIIOII(H-
OKPY>KHOCTb 2 MHCR 3y6bRMH. ECJIH HCnOJIb3yroTCR 3BOJIb-
Ha al1bHaS1
OKPY?KHOCTb 1
/
5.1. MexaHulteclCue nepeiJaltu: 3y6ltamble nepeiJaltu 3
BeHTHbIe 3y6bR, TO Ha pHCYHKe rrpH.l(eTCR ,n;06aBHTb ,n;Be ,n;OrrOJIHHTeJIbHbIe OKpyxHO-
cm. 3TH ,n;orrOJIHHTeJIbHbIe OKpyxHOCTH RBIDIIOTCR OCHOBHbIMH OKpyxHOCTRMH, ,L(JUI
KOTOpbIX Cl'pORTCR 3BOJIbBeHTbI (p83,n;eJI 5.1.4). Terrepb 3BOJIbBeHTbI Cl'pORTCR, HCXO,nR
H3 3THX OCHOBHbIX OKpyxHOcTeH, TaK, qTo6bI OHM KaCaJIHCb Ha JIHHHH 3a:u;ellJIeHIDI.
3m JIHHIDI 3aueIIJIeHIDI - 061I..(aR KaCaTeJIbHaR K OCHOBHbIM OKpyxHOCTRM H RBIDleT-
CH npo,n;OIDKeHHeM rrpRMoH CT (pa3,n;eJI 5.1.4).
YroJI 3an;eIIJIeHIDI , rrOKa3aHHbIH Ha pHCYHKe, 3TO yrOJI Mex.ny JIHHHeH 3a:u;eI1JIe-
HID! H 061I..(eH KaCaTeJIbHOH K OCHOBHbIM OKpyxHOCTRM, rrpOBe,n;eHHbIM Qepe3 TOqKY
P. OrrbIT rrOKa3aJI, qTO = 20° 06eCrreQHBaeT OrrTHMaJIbHYK> CR1IY B COQeTaHHH C
IDIaBHbIM XO,n;OM. 3Ta BeJIHtlHHa rrpHHRTa B 60JIbllIHHCTBe COBpeMeHHbIX MexaHH3-
MOB.
Ha al1bHaS1
OKPY>KHOCTb 1
0 1
ConpVlKOCHO-
BeHHe ay6beB
Ha l1V1HVIVI aa-
u,enl1eHVlS1
I
nMHHS1 aau,enl1eHVlS1
Ha al1bHaS1
OKPY>KHOCTb 2
,
OCHOBHa
I OKPY>K-
HOCTb 2
°2
BblCOTa
ay6a
.........
.........
.........
.........
.........
--- .........
-
Add. - BbiCOTa rOl1OBKVI ay6a
Oed. - BblCOTa HO)l(KVI ay6a
. - yrol1 aau,enl1eHVlS1
P - nOl1lOC aau,enl1eHVlS1
T - TO'iKa KacaHVlS1
Dla8a 5. MexaHUlteCKUe nepeiJaltu
QT06bI CrrpOeKTHpOBaTb pa60qHH rrpo<pHJIb 3y6a Ha OCHOBaHHH a6Cl'paKTHOH
3BOJIbBeHTbI, He06xo,n;HMbI CJIe,nyIOmHe ,n;orrOJIHHTeJIbHbIe rrocl'poeHIDI.
OKPYJKHOCTb roJlOBOK 3y6beB
Y CTaHaBJIHBaeT BbICOTY 3y6a, TaK qT06bI rrpH 3a:QeIIJIeHHH rrpe,n;oTBpaTHTb ero
CTOJIKHOBeHHe C Brr3.,L(HHOH 3y6a rrapHoH rnecTepHH.
OKPYJKHOCTb BDa;ntH 3y6beB
Y CTaHaBJIHBaeT rrOJIOXeHHe ,n;Ha HJIH Brr3.,L(HHbI 3y6a H TaKHM 06pa30M YCTaHaB-
JIHBaeT rJIy6HHY, ,n;o KOTOpOH rrpo<pHJIb 3y6a BbIpe3aeTcR H3 3arOTOBKa ,nmI
3y6qaTOrO KOJIeca.
BblcoTa roJlOBKH 3y6a
P3.,L(HaJIbHOe paccToRHHe Me OKp OCTbID rOJIOBOK 3y6beB H HaqarrbHOH
OKp OCTbID.
BblcoTa HO)KKH 3y6a
P3.,L(HaJIbHOe paccToRHHe Me y OKp OCTbID Brr3.,L(HH 3y6a H HaqarrbHOH
OKp OCTbID.
----
---
---
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
, .
. .
. .
. .
. .
. .
. .
. I .
f 0 2
OBOl1bBeHT-
_ .. .. Ha 4>opMa
.... T. OCHOBHO
o KpY)I(HOCTVI
AO OKPY>KHO-
oCTVI rOl1OBOK
ay6beB
3aKPyrl1eHHa
4>opMa OT
OCHOBHO
OKPY>KHOCTVI AO
OKPY>KHOCTVI
Bnap.VlH ay6beB
Add. - BblCOTa
rOl1OBKVI ay6a
Oed. - BblCOTa
HO)l(KVI ay6a
BblCOTa ay6a
Add.
Ded.
BblCOTa 3y6a
P3.,L(HarrbHOe paCCTORHHe Mex.ny
OKp OCTbID rOJIOBOK 3y6beB H OIq)YX-
HOCTbID BII3.,L(HH 3y6beB. II03TOMY' BbICO-
Ta 3y6a = BbICOTa rOJIOBKH 3y6a + BbICO-
Ta HOXKH 3y6a.
OTMeTHM, qTO rrpO<pHJIb 3BOJIbBeHT-
Horo 3y6a 3y6qaTOrO KOJIeCa OqepqeH no
3BOJIbBeHTe OT OCHOBHOH OKp OCTH
,n;o OKp OCTH rOJIOBOK 3y6beB, o,n;HaKO
HMeeT 3aKpyrJIeHHyIO <POPMY' OT OKPYX-
HOCTH Brr3.,L(HH 3y6beB ,n;o OCHOBHOH OK-
p OCTH, KaK rrOKa3aHO Ha pHCYHKe.
npocIJHJlb 3y6beB peRKH
Ha pHCYHKe rrOKa3aHO, KaK COOTHO-
rneHIDI 3BOJIbBeHTbI rrpHMeIDIIDTCR ,lQUI
3y6beB peHKH. PeHKa RBJIReTCR OCHOBOH
JIID60H CHCTeMbI 3y6qaTbIX rrepe,n;aq II
rr03TOMY CJIy)KHT HaqarrbHOH TOqKOH, OT
KOTOpOH Cl'pOHTCR rrOCJIe,n;OBaTeJIbHOCTb
3y6qaTbIX rrepe,n;aq.
....
HaYal1bHa np Ma
Add. - BblCOTa rOl1OBKVI ay6a
TeOpeTVlYeCKVI npo4>Vll1b ay6a pe KVI Oed. - BblCOTa HO)l(KVI ay6a
5.1. MexaHUlteC1(,Ue nepeoaltu: 3y6ltamble nepeoaltu 307
1 ,5708
1 ,5708
.
0,02 max. } AOn Y CTHMaS1 MOAH4>HKa-
0,6 max. HS1 npo4>HnS1 rOl1OBKH
1
\.L}
_ HC! anbHaS1 (Ael1V1-
Tel1bHaS1) l1V1HHS1
1,25
"
1,00
} Pap.VlYC
0,39 max. OKPY>KHOCTH
0,25 max. Bnap.HH ay6a
BpHTaHCKHH CTaH,ll;apT rrpO<pHJIR 3y6a peHKH
(pa3MepbI npHBe,n;eHbI ,L(JIR MO,n;YIDI 3y6beB, paBHOrO e,n;HHHue)
KaK nOKa3aHO Ha pHCYHKe, TeOpeTlftleCKHH rrpO<pHJIb 3y6a peHKH HMeeT npR-
Mble 60Ka H qeTKO 0603HaqeHHbIe yrJIbI. H3 npaKTHqeCKHX co06paxeHHH TaKOH
rrpO<pHJIb 3y6a MO,n;H<pHUHpyeTcR rryreM cKpyrJIeHIDI y BepllIHHbI 3y6a H 3aMeHbI
qeTKO 0603HaQeHHbIX yrJIOB BO Brra,nHHe 3aKpyrJIeHHeM.
PeaJIbHbIH rrpO<pHJIb 3y6a peHKH COrJIaCHO BS 436 (Mel'plftleCKHe e,n;HHHUbI) no-
Ka3aH BbIllie. 3T0 OCHOBHOH rrpo<pHJIb 3y6beB peHKH, cooTBeTCTByIDI.l(HH 6pHTaHcKo-
MY CTaH,ll;apTY; pa3MepbI yJ(a3aHbI ,L(JUI MO,L(yIDI 3y6beB paBHoro e,n;HHHue. AmI JII060ro
3HaqeHIDI MO,L(yIDI, OTJIIftIHOrO OT e,n;HHHUbI, p83MepbI 3y6a onpe,n;eIDIIOTCR YMHoxe-
HHeM yJ(83aHHbIX QHCeJI Ha BeJIIftIHHY MO,L(yJUl. H3 rrpaK1'J{qeCKHX co06paxeHHH ,n;aH-
HbIH OCHOBHOH npo<pHJIb MoxeT 6bITb ,n;aJIee MO,n;H<pHUHpoBaH CJI