(формат)Тарг

123): uaeaAbflbl.MU Ha3bl8Q/OmcH.

oeAflHJtt{UX tWJJO.JICeHue cucme.Mbl, coBnaaaem c <tucAO.M ee cmeneHeu CBo·

6oabt. fiOolTOMY y T3KOH CHCTeMhl lJHCJIO CTeneHeH CB060,ll.bl MOJKHO onpe- ,!l.eJIHTb KaK IIO lJHCJIY He3aBHCHMbiX B03MOJKHbiX nepeMeru.eHHH, TaK It no lJHCJIY HesaBHCHMhiX Koop,!l.HHaT. TaK, y KPHBOIIIHnHo·nOJI3YHHoro MeX3HI13M3 (eM. HHJKe puc. 356) O,!l.H3 CTeiTeHb CB060,!l.bl (y Hero O,ll.HO He33BHCHMOe B03MOJKHOe nepeMeru.eHHe, HanpHMep IIOBOpOT KpHBOIIIHlla

OA, H O,ll.Ha HesaBHCHMaH Koop,!l.HHara, HanpnMep yroJI fP). Y cBo6o)l,- HOro TBep,ll.OrO TeJia IIIeCTb CTeiieHefi CB060)l.bl (He3aBHCHMbiX nepeMeru.e· Hnii - TpH nocTyrraTeJibHhiX B,!l.OJih Koop.u.mJaTHbiX oceii H TPH noBopoTa BoKpyr 3THX oceU:, a HesaBHCHMbiX Koop,ll.HHaTTPH Koop,!l.HHaTbl nomoca H TPH yrJia 3iiJiepa).

§ 139. UPHHU.HU B03MO)I(Hb1X nEPEMEU.tEHHA

flepeii,!l.e.M K paCC.MOTpeHHIO eru.e O)l.HOro npHHIJ,Hfia MexaHHKH, KOTOpblll ycTaH3BJIHBaeT o6w,ee yCJIOBHe paBHOBeCMH MeXaHHlJeCKOH CllCTeMbl. flo,ll. paBHOBeClfeM (eM.§ 1) Mbl IIOHHMaeM TO COCTOHHHe CHCTeMbl, TipH KOTOpoM BCe ee TOlJKI:l IIO,ll. ,ll.eHCTBHeM npHJIOJKeHHb!X CHJI H3XO)l.5ITCH B IIOKOe no OTHOIIIeHHIO K HHepii,HaJibHOH CHCTeMe OTClJeTa (paccMaTpHBaeM TaK Ha3hiBaeMoe «a6co:rnoTHoe» paBHoBecue). O_uHoBpeMeHHO 6y- ,!l.eM ClJHTaTb BCe HaJIOJKeHHbie Ha CHCTeMy CB.H3H CTal'HJHapHbiMH II crreurraJihHO 3To B ,!l.aJihHei'riiieMKaJK.U.hiH pas oroBapuBaTh He 6y,!l.eM.

BBe,!l.eM noii.HTHe o B03MOJKHoli pa6ore, KaK o6 3JieMeHTapHoii pa6oTe, KOTopyiO ,ll.eHCTBYIOili.3H Ha MaTepHaJibH}'IOTOtiKY CHJia MOrJia 6bi COBepIIIHTh Ha nepeMeru.eHHH, COBTia)l.aiOill.eM C B03MOJKHbiM nepe:\1elll.eHHeM

~noli TOlJKH. Ey,!l.eM B03MOJKHYIO pa6oTy aKTHBHoii CHJibi f• o6osHalJaTb CHMBOJIOM tJA a (6A •=p• ·6r), a B03MOJKHY!O pa6oTy peaKIJ,lUI N CBH3HCHMBOJIOM tJA" (6A" =N ·6{) .

.H,a.u.nM Terrepb o6lll,ee onpe,!l.e.1er-me ITOHHTHH o6 n,u:eaJihHbiX CBH3HX, KOTOpb!M Mbl Y}Ke ITOJib30BaJIHCb (eM. §

C6H.3U, 0/lH. JWmOpblX cyM.MQ B/leM.ehmapHblX pa6om UX peaKt{UU Ha AI060M B03MOOICHOAt nepeM.et.Uf!HUu cucme.Mbl paeHa HYAJO, T. e.

~6A~=0. (98)

flpuBe,ll.eHHoe B § 123 H BbipaJKeHHOe paseHCTBoM (52) ycJioBne H,!l.e- aJihHOCTH CB.H3eii, KOr)l.a OHH O)l.HOBpeMeHHO .HBJI.HIOTCH CTaiJ,HOH3pHbiMH, COOTBeTCTByeT onpe,!l.eJieHH!O (98), TaK KaK npH CTaiJ,HOHapHb!X

CBH3HX KaJK)l.Oe ,!l.eHCTBHTeJihHOe nepeMeru.eHHe COBIIa)l.aeT C O,!l.HHM H3 B03MOJKHhiX. ilo3TOMY npHMepaMH H.n_eaJibHb!X CBH3ell 6y,!l.yT BCe npH· Mephi, rrpHBe)l.eHHble B § 123.

.UJIH onpe.u.eJieHHH Heo6xo,!l.HMoro ycJioBHH paBHoBeCHH .u.oKaJKeM,

'ITO eCJIH	MeXaHHlJeCK3.H CHCTeMa	C H)l.eaJibHh!MH	CBH3HMH	H3XO)l.HTCH
non; ,ll.ellCTBHeM np~JIO.}KeHHbiX CHJI B paBHOBecHH,			TO npH	JII060M B03·
MOJKHOM	nepeMelll.eHHH CHCTeMhl	,ll.OJIJKHO BblllOJIHHTbCH paBeHCTBO

~6A~= 0	(99)
HJII1
~F~·brk = ~F~ 6sk cos ak = 0,	(99)

r)l.e ak - yroJI MeJK)l.y CHJIOH H B03MOJKHhiM nepeMelll.eHHeM.

360

0603H8'liiMpaBHO)l.eHCTBYIOill,IIe BCeX (!I BHeWHHX, II BHyTpeHHH"'S) aKTIIBHbiX CHJI II peaKIJ.liH CBH3eH, )l.eHCTBY!OliJ.IIX Ha KaKy!O-HII6y)l.h TO'l- Ky cHcTeMbi Bk, cooTBeTCTBeHHo qepe3 F't_ II Nk. Tor.na, nocKOJihKY Ka}l{)l.aH II3 TO'leKCHCTeMhr Haxo)l.IITCH B paaHoBeCHII, Fk +Nk=O, a c.rre-

.noaaTeJibHO, II CyMMa pa60T 3TIIX CII,TJ npii JI!060M nepeMelll,eHIIII TO'lKH

Bk 6y.neT To}l{e pasHa HYJI!O, T. e.	6A%+oAk =0. CocTaBIIB TaKHe pa-
BeHcTBa )l.JIH BCeX TO'leK CHCTeMbi	II CJIO}I{HB HX llO'lJieHHO, llOJIY'liiM
~<'>A~ +~<'>Ak = 0.

Ho TaK KaK CBH3H II.neaJibHhle, a <'>rk npe.ncTaBJIHIOT co6oi1: B03MO}I{Hhre nepeMelll,eHIIH TO'leKCHCTeMhi, TO BTOpaH CYMMa ITO yCJIOBIIIO (98) 6y,neT paBHa HYJIIO. Tor.na pasHa HYJIIO II nepBaH cyMMa, T. e. BhinOJIHHeTcH paBeHCTBO (99). TaKHM o6pa3oM, .noKa3aHo, 'lTOpaBeHcTBo (99) Bbrpa- }1{8eT HeOOXO)l.IIMOe yCJIOBHe paBHOBeCIIH CIICTeMbi.

DoKa}l{eM, 'lTO3TO ycJioarre HBJIHeTCH II .nocTaTO'lHhiM,T. e. 'lTOecJIII K TO'lK8MMeXaHII'leCKOH CHCTeMhl, HaXO!J.HIII.eHCH B llOKOe, npiiJIQ)!H!Tb aKTIIBHhie CHJibi tj,, y)l.OBJieTBOpH!OIIJ.He paBeHCTBy (99), TO CIICTeMa

OCTaHeTCH B TIOKOe. Ope)l.nO.TJO}I{II'\1o6paTHOe, T. e. 'lTOCHCTeMa npll 3T0\1

I1pii)l.eT B )l.BII}I{eHIIe !I HeKOTOpbie ee TO'lKHCOBeprnaT )l.eHCTBHTeJibHbie nepeMem.eHHH drn. Tor.na cnJihi PC coBeprnaT Ha 3THX nepeMeiiJ.eHIIHX pa60Ty H no TeopeMe 06 II3MeHeHHH KHHeTH'leCKOH3Hepnm 6y.neT:

dT = ~ dAk (dM = fi· drk),

r.ne, oqeBII)l,HO, dT>0, TaK KaK BHaqaJie CHCTeMa 6biJI8 B fiOKoe; CJie- )l.OBaTeJibHO, H ~dA~>O. Ho rrpii cTauHoHapHhiX CBH3HX .nefrcTBHTe.'lh-

Hbre nepeMeiiJ.eHHH drk COBll8,!1.8IOT C KaKHMH-TO H3 B03MO>I<Hb!X nepe:vleIIJ.eHHll 6rk II Ha 3TliX nepeMelll,eHIIHX TO}I{e )l.OJI/!{HO 6biTb ~OA~>O,

'lTOnpoTIIBopeqHT ycJIOBHIO (99). TaKHM o6pa30M, Kor.na npiiJIO}I{eHHhie CHJibi y)l.OBJieTBOpH!OT ycJIOBII!O (99), CHCTeMa II3 COCTOHHHH llOKOH BbiHTII He MO}I{eT II 3TO yCJIOBIIe HBJIHeTCH )l,OCT8TOqHhiM yc.TJOBHeM

paBHOBeCHH.

Ib .noKa3aHHoro BbiTeKaeT cJie.nyrom.nii: n p II H u II n B o 3 M o }1{-

H hi x n e p e M e m. e H II il *: OAJl paeHoeecuR .MexaHu<teawu cucmeMbt

c uoeaAbHbl.MU C6Jl3RMU He06XOOUM0 u oocmamO'iHO, 'im06bt CYMMa 3Ae-

.MeHmapHblX pa6om 8CeX OCUCmBYIOUI,UX Ha Hee aKmU6HbtX CUA ttpU A1060.M eos.MOJJCHO!rt nepe.MeU{eHuu cucme~rtbl 6btAa paeHa HYAIO. MaTeMaTII'leCKH

apopMyJIHpOB8HHOe yCJIOBIIe paBHOBeCIIH	Bbipa}l{aeTCH	paBeHCTBOM
(99), KOTOpoe H83biB8IOT T8K}I{e ypaBHemreM	B03M0}1{HblX	pa60T. 3TO

paBeHCTBO MO}I{HO em.e npe)l.CT8Bl1Tb B aHaJIIITH'leCKOHcpopMe (eM. § 87):

~ (F~x <'>xk + F~u 6yk + F~z ozk) = 0.

(100)

* B cpopMe, 6JH!3KOH K COBpeMeHHOH, HO 6e3 ,1\0Ka3aTeJibCTBa .9TOT npHHUKll :BDlCKa3aJI 3HaMeHHTDI'HMareMaTHK H M€XaHHK (IIIBeftuapeU 110 npOHCXOJK,1\€HHIO) l1oraHH BepuyJIJIH (1667-1748). B o6III.eM BHAe npHHUHTI BrrepBhie ccpopMyJiuposaJI H ,1\0KasaJI' )K. JlarpaHJK (1788 r.) 06o6JII.eHue npuHuuna Ha cJiyqafi Hey,nepJKuBaiOIII.HX c:sl!seji: DblJIO ,naHo M.B. OcTporpa.l\CKHM B pa6oTax 1838-1842 rr.

361

TipHHUHn B03MO)I{Hh!X nepeMeuJ;eHHR ycraHaBJIHBaeT 06~ee YCJIOBHe

paBHOBeCH51 MeXaHIIlleCKOH CIICTeMhl, He Tpe6yiOW.ee paCCMOTpeHIIH paBHOBeCHH OT,ll.eJibHbiX 118CTeH (TeJI) :nofi CliCTeMhl II n03~0JI210lll,ee npii

H)J.eaJihHblX cBH3HX HCKJIIOllHTb us paccr.mrpenHH see IIanepe.n. HeHsBecrHhle peaKI.J.HH CBHseii:.

§ 141. 0611\-EE YPABHEHUE ,l.l.UHAMUKU

flpHHUIIIT B03MOJKHhiX nepeMerueHIIH )J.aer o6ruHil MeTO)J. perneHHH 3a)J.a"' CTaTHKll. C )J.pyroif CTOpOHhi, npiiHUIIII llaJiaM6epa II03BOJIHeT ucrroJib30BaTh MeTO)I.bl cTaTIIKH )J.JIH perneHHH 3a)J.all )J.HHaMHKII. CJie-

)J.OBaTeJihHo, npHMeH.HH 3TH )J.Ba npHHUlma Q)J.HOBpeMeHHO, Mbl MOJKeM rroJiytiiiTh o6ruiiil MeTO)J. perneHIIH 3a)J.all )J.HHaMHKH.

PaCCMOTpliM CIICTeMy MaTepHaJibHhiX TOIJeK, Ha KOTopyiO HaJIOJKeHbl H)J.eaJihHbie CBH3II. ECJIH KO BCeM TOI!KaM CHCTeMbi KpoMe )J.eHcTBYIOIII,IIX Ha HHX aKTHBHbiX ciiJI F~ n peaKUHH cBH3eil Fik npn6aBIITb coOTBeTcrByiOIII.IIe CliJibl HHepUHII F~=-mkah., TO corJiaCHO npiiHUIIIIy naJiaM6epa noJiytieHHaH ciicTeMa CHJI 6y)J.eT Haxo)J.IIThCH B paaHoBecHII. Tor)J.a, npHMeHHH K 3THM CHJiaM npiiHUHII B03MOIKHblX nepeMeUJ.eHHH, IIOJIYilliM

~ 6A~ +~ bA~ +~ 6A~ = 0.

Ho nocJie)J.UHfl cyMMa no ycJioBIIIO (98) pasHa HYJIIO II OKOHIIaTeJibHO

6y)J.eT:		~ BA~ +~ 6A~ = 0.
				(102)
Hs	noJiyiieHHOro	pe3yJihTaTa BhiTeKaeT	cJie)J.yiOIII,IIii	n p II H u II n
ll a JI a	M 6 e p a -	J1 a r p a H 1K a: npu	06U:J/Cet-tUU	.Mexat-tUt.teCIWU

cucmeMbt c uoeallbt-tbtMU cBR3RMU 8 Ka:JICObtii MOM.eHm epeMeHu cyMMa alle.Met-tmapHbtX pa6om Bcex npullo:J/CeHHbtX aKmUBHbtX cull u scex CUll uHep- t{UU H.a Jl1060M. sos.MO:JICHO.M nepeMetu;;HUU cucme.Mbt 6yoem pasHa HYll/0.

YpaaHeHIIe ( 102), BhrpaJKaiOruee 3TOT np IIHUIIrr, Ha3biBaiOT o 6 ru H M

y p	a B H e H II e M )I. II H a M II K II. B aHaJIIITHIIeCKOH cpopMe ypaBHe-
HIIe	(102) IIMeeT BII)J.
	~ [(F~x+ F~x) 6xk +(F);y +F~y) 6yk +(F~z+ F~z) <'3zk] = 0. (103)
YpaaueHIIH (102) HJIH (103) rro3BOJUIIOT cocraarrTh )J.wcpcpepeHuHaJib-
Hhie ypaBHeHHH )J.BHJKeHIIH		MeXaHIIqecKOH CHCTeMbi.
	EcJIH npii 3TOM cncreMa	npe)J.CTaBJIHeT co6ofi coaoKyrrHocrb KaKIIX-

HII6y)J.h TBep)J.hiX TeJI, TO )J.JIH COCTaBJieHIIH ypaBHeHIIH Hy:tKHO K )J.eii:crBYIOIUHM Ha KaJK)J.Oe TeJIO aKTHBHbiM CHJiaM I1pH6aBIITb npiiJIO/KeHHYIO B JII060M UeHTpe CHJIY, paBHYIO rJiaBHOMY BeKTopy CIIJI HHepUHH, II rrapy C MOMeHTOM, paBHhlM rJiaBHOMY MOMeHTY CHJI HHepUIIH OTHOCIITeJibHO ::noro uenrpa (rum O)J.Hy It3 3TIIX aeJIIIIIHH, CM. § 134), a saTeM rrpHMelliiTb ITpHHUHII B03MO>KHbiX nepeMeruemfii.

367

UIIH nparuaiOIUlU-.Cli TC,1 (eM.§ 134). 3TH BCJIII'!Hllbl TIO \IO.!IYJIIO paBHbl:

F~=(Q/g) as, IM~ I= (PI!g) p~e1, IM~ I= (Pz/g) P~82·

HanpanJienHll ncex neJIHqHH noKasaHhi Ha •IepreMe. Coo6ruall cncre~1e B03MOiKHoe nepe~IeruenHe H COCT3BJillll )p3Bt1Clll!C (102), DOJIY'IH\1

- (Q +F;) os3 +(M -M~) 6cp1-M~6<p2=0.

BhlpaMal! nee ncpeMew,eBHll qepe8 6<p 2, nafl.ne'.I,qro

6ss = r&p2,	O<Jll	(>)J	fz
	;;--=-=- ll
	ucp2	Wz	f1

fJlaBa XXIX
YCJIOBIHI PABHOBECH51
H YPABHEHH51 ,li.BH)I(EHH51 CHCTEMbl
B 060Bll.\EHHbiX			I(OOP,li.HHATAX
§ 142. 0606lll.EHHblE KOOP)J.HHATbl				H 0606W.EHHbiE CKOPOCTH
4HCJIO I<OOp,!!.llH8T		(rrapaMeTpOB), orrpe)l.eJIHIOIII.llX			IIOJIOllieHHe Mexa-
HH'-JeCKOtlCllCTeMbl, 38BIICHT OT KOJill'-leCTBaTO'-leK(liJIH TeJI),						BXO)l.Hll.I,HX
B CHCTeMy, II OT '-JHCJia ll xapaKTepa				H8JIOMeHHb!X	CBJI3eii.	oy)l.eM B
,l/,8JlbH€MI11€M paCCM8TpHB8Tb TOJibKO				CHCTeMbi C	T€0'-'feTpii<reCKHMH
cBH3Hl\IH	(To'-IHee TOJibKO roJIOHOMHhre			ciiCTeMbi). KaK ycTaHoBJieHo B
§ 138, y	T8KOll CHCTeMhl		'-lliCJIO He38B!ICliMbiX KOOp)l.HHaT, onpe)l.eJIHlQ-
lll.llX IIOJIOMeHHe CliCTeMbi, COBI18)l.8eT				C '-IHCJIOM ee CTerreHeii		CBOOO)l.hl.
B Ka'-leCTBe 3THX K00p,!!.HH8T MOMHO Bh!6Hp8Tb rrapaMeTpbT,						HMeiOrn;He
JII06yro	pasMepHOCTb	H	JII060fi reoMeTpli'-IeCKIIll (HJill <\JH3II'-leCKHH)
CMbiCJI, B '-l8CTHOCTH0Tpe3KH IIpHv!b!X HJIH )l.yf, yrJihl,					!1JIOUJ;8)l.II H T. II.
He3aBHCHMhre MeM.U.Y co6oii rrapaMeTphr Jiro6oii pa3MepHOCTH, '-IHCJIO
KOTOpb!X	paBHO '-IHCJIY CTerreHeii CB060)l.hl CHCTeMhl				II KOTOpb!e O)l.H0-

3H8'-IHOorrpe)l.eJIH!OT ee IIOJIO)!{eHHe, H83b!BaiOT o606U{eHI-tbUtU KOopauHa- maMU cnCTeMbr. By.u.eM o603Ha'-laTbo6o6rn.eHHhre Koop.n.HHaThr 6yKsoii q.

Tor.u.a rroJioMeHHe CHCTeMhi, HMerolll,eil s cTerreHeil cBo6o.n.hl, 6y.u.eT onpe-

369

u=				=		g1*			ё
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Ё AЕЁA$*ЕЁЕзЕ									в вх 5
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-

liqЕfli?

Еil}ЁЁeвsвЁgЕ

BhiHOCH 06ill,IIH MHOMHTeJib {)q1 38 CK06KII, HaH,ll.eM OKOHqareJibHO

	6A1=Q16q17	(108)
r11..e o6o3Ha'!eHo
		(1 09)
no aHaJIOriiH c	paBeHCTBOM tSA = F,6s, onpe.n;emiiOill,HM 3JieMeHrap·
HYIO pa6ory cHJibi	F, BeJII!'IHHy Q1 Ha3biBaiOr	o6o6wpuwtl cuJWtl,
COOTBeTCTBYIOm.eii	Koop.n;HHaTe ql.
Coo6m.aH ciiCTeMe .n:pyroe He3aBIICIIMoe B03MOMHoe nepeMem.eHHe,
npH KOTOpOM H3MeHHeTCH TOJibKO KOOp.IJ:IIHaTa q2 ,		IIOJIY'IIIM .IJ:JIH 3Jie•

MeHrapHOH pa60Tbl BCeX .n;eficTBYIOIJJ.HX CliJI Ha 3TOM nepeMelll,eHIIH

BbrpaJKeHIIe

(110)

(111)

BeJIIItJHna Q2 npe.n;cTaBJIHeT co6oii: o6o6m.eHHyiO CIIJiy, cooTBeTcTBYIO· my10 Koop.n;uHare q2. H T . .n:.

QqeBH)l.HO, liTO eCJill CHCTeMe C006lll,IITh T8KOe B03MO}I{HOe nepe· Melll,eHHe, npH KOTOpOM O.IJ:HOBpeMeHHO H3MeHHIOTCH Bee ee 0606lll,eHHble KOOp_nHH8Tbl, TO CyMMa 3JieMeHTapHbiX pa60T npiiJIO}I{eHHhiX CHJI Ha 3TOM: nepeMem.emrn orrpe)l.eJIIITCH pasenCTBOM

(112)

ci>opMyJia (112) .n;aeT BbtpaJICet-tue noAHoii &AeMeHmapt-toii pa6ombl ecex oeiicmey10uwx Ha cucme;.ty cuA e o6o6t«eHHIJtX Koopout-tamax. Yh

:noro paBeHCTBa BH.IJ:HO, 'ITOo6o6~Hiibte CUAbl - 9tn0 eeAWiUHbl, paeHbte IWfX/JC/Jtu{ltet-tmaM npu npupau{eHUflX o6o6u{eHHbtx KoopouHam o 8btpa31CeHuu lWAHou &AeMet-tmapHoii pa6ombt iJeiicmey10wpx Ha cucmeMy CUA.

ECJIH Bee H8JIO}I{eHHhie Ha CHCTe~IY CBH3H HBJIHIOTCff H.n;eaJihHb!MH, TO pa6oTy npn B03MOMHbiX nepeMeill,eHHHX COBep!llaiOT TOJibKO 8KTHBHhie CHJibl II BeJIH'IIIHbr Qll Q2, ... , Qs 6y.n;yT npe.n;cTaBJIHTh co6oii o6o6- U{eHHble GKtnU81ibte CUAbl CHCTeMbl.

Pa3MepHOCTh o6o6m.ennoii ciiJihi saBIICIIT or pa3MepHoCTII cooTBer-

CTByiOm.eii: o6o6III.eHHoii: KoOpJI.HHaThl.	TaK KaK npoH3Be.n;eHIIe	Q6q,
a CJieJl..OB8TeJibHO, ll Qq IIMeeT pa3MepHOCTb pa60Tbl, TO
[Q]=~~1	,	(113)

T. e. pa3MepHocmb o6o6u{eHHoii Cll.!lbt paetta pasMepHocmu pa6ombl, oeAeHHOU Ha pa3MepHOCtnb COOtn8emcm8yiOLJ{eU 0606U{eHHOU KOOpOUHatnbl.

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312