理论力学全套解疑11

O′x′y′O′O′O′O′11-111-1O′M11-2O′xO′yO′O′MOxϕO′ϕt(1)⎪⎪⎭⎪⎪⎬⎫===′′)()()(321tftfytfxOOϕ(1)O′⎪⎪⎭⎪⎪⎬⎫′′==′==)(dd)(dd3223tfttftϕεϕω(2)M⎪⎭⎪⎬⎫′+=′+=′′ϕϕsincosMOyyMOxxOMOM(3)tM⎪⎭⎪⎬⎫′+=′−=′′ϕϕϕϕcossinMOyyMOxxOMOM(4)tM⎪⎭⎪⎬⎫′+′−=′−′−=′′ϕϕϕϕϕϕϕϕcossinsincos22MOMOyyMOMOxxOMOM(5)(4)(5)O′MO11-4tABC∆tA′B′C′1AdAAϕ2CCdCCϕCdCAdACAAC90°ACACdAdC123456711-5(a)O1Aω1O2Dω2vAvBPAD11-511-5(b)ABvAvBACBCCCvAvBPPAB111-6111-6111-622PvPr=0PvPn=0PPPvP=0r11-62CuMϕϕϕcossinrrBCHCHBAMyrrMBMHAHOHOAx−=−===−=−=−==rutMCOHMCMH===ϕϕMrutrryrutrutxcossin−=−=trutuvrutuvyxsincos1=⎟⎠⎞⎜⎝⎛−=Mϕ=2π00==yxvvvP=0OPPEBDu1Ou11-7a(b)(c)11-7022====PDBEvuvvuv2PRu=ω(c)ω=0ωAvAAvAvAω90°APωAvP11-811-811-91211-9ABAB123ABABDv=120(mm/s)OOE11-11aOωOR40(mm)11-111OADPPv=120(mm/s))PvP=v=120(mm/s)POOEvODD30cos60cosOPvv=3402/32112030cos60cos=×==DDPOvv(mm/s)vO2ωOωOvPvOCb2021==RCO(mm)3220340===COvOOω1/sωOPcvO=vP+vOP380260cos===OOOPvvvD(mm/s)OOPRvω=3240380===RvOPOω(1/s)OA=AB=lOAω11-12aBvBABωAB1A(vA=lω)vB=vA+vBABvB(1)DD60cos60cosBAAXvvv−−=B(2)030cos30cos=−=DDBAAYvvvB11-12ωlvvvABAB===ωω==lvBAAB2BvA=vB+vABA11-12b(3)DD60cos60cosABBXAvvv+−=−(4)DD30cos030cosABAvv+=ωlvvvBABA===ωω==lvABABvAvBAvBvABAB11-13(vA)x(vB)xxvA(vA)x=0(vB)x0xAB11-1311-14ABA11-14AAx′y′BAx′y′BBAx′y′BB′B′Ave=vAva=ve+vrvB=vA+vBA11-1512311-1545vAvBvAvB6vAvBvA=vB11-16(a)ABaA=aBaB=2aωεAnBABAABaaaa++=τAxy(b)xnBAAaa−+=0020ωla−=la±=ωωy00=+=τBABaa2a=lεla2=ε11-16AB(1)nBABAABaaaa++=τC*(2)nACACAC***aaaa++=τ11-17(3)AnACACa−=+**aaτ2****ωε⋅=⋅=ACaACanACACy(3)⎪⎪⎭⎪⎪⎬⎫=+=242*tgωεϕωεAaAC(4)(4)AaAεϕ*ACC*11-17(a)C0*=Ca*MnMCMCnMCMCCM*****aaaaaa+=++=ττMCnMCMC**aaτ*2****,ωετ⋅=⋅=MCaMCanMCMC42*ωε+⋅=MCaMaMC*Mϕtgϕ2ωε=1-17(b)1ω=0ε=0C*2ω=0ε0=0(4)nAC*aτAC*aD90*==ϕεAaAC11-17(c)3ε=0ω0=0(4)τAC*anAC*aD02*==ϕωAaAC11-17(d)11-17123RvOaOCC*ωvORvO=ωRaRvttOO=⎟⎠⎞⎜⎝⎛==ddddωεOCnCOCOOCaaaa++=τOCOaR==ετanCOOaa=RvRaOC22==ωCvC=0aC011-19a11-19C*C*11-17O2142*tgωεϕωε−=+=OaOCaOεϕOC*42ωε+OaC*(b)C*(c)C*0*=Ca0*≠Cv11-17(b)Mτ*MCaCC*CC*11-20MvMMCvMMMCMMMMvτ*MCanMC*aMMCMτManMa11-2111-21142536AaABnBABAABaaaa++=τω=002=×=ABanBAωABaBA⊥τABAABBaa][][=11-212nBABAABaaaa++=τBAAB11-23nBAaABnBAABBAABAABB][][][][aaaa++≠τ=0nBAa2ω⋅=ABanBAω=0ABAABB][][aa=11-24(a)1α=30°3101=v(mm/s)3101=a(mm/s2)2B1BO=30mm2v2=50(mm/s)a2=10(mm/s2)3B311-241232B313ω3vBa=vBe+vBr(1)vBa=v2vBrvBeB3E3AvBe=va+vBA=v1+vBA(2)(2)(1)v2=vBr+v1+vBA(3)11-24(b)(3)xyααααcossin)90cos(cos1212vvvvvvBrBA+=+−−=D)mm/s(330)mm/s(10==BABrvv3=====2321/30330sin/3303αωBOBAvBA0.866(rad/s)23B3(4)kreaBBBBaaaa++=aBa=v2aBr3aBkB=×==1023290sin23DBrBkvaω17.3(mm/s2)(c)aBeA(5)nBABAnBABAABaaaaaaa++=++=ττ1e3ετ⋅=ABaBAAB3=⎟⎟⎠⎞⎜⎜⎝⎛×=⋅=2232360ωABanBA45(mm/s2)BA(5)(4)(6)kr12BBnBABAaaaaaa++++=τ3(6)xyr12k12cossinsincosBnBABBAaaaaaaa++=−−+−=−ααααατ)(mm/s65)(mm/s3.172r2−==BBAaaτB3===603.173ABaBAτε0.29(rad/s)OAR=0.2(m)OωO=10(1/s)ABl=1(m)α=β=45°ABB11-25a1ABωABBvBωABvBABvAvBPPARvABOA⋅==ωω===lABPA1(m)212.010=×==PAROABωω(rad/s)28322=×=⋅=PBvABBω(m/s)11-25vB(a)ABvv=αcos=×===2/12.010coscosαωαRvvOAB2.83(m/s)ωABvBABAABvvv+=(b)=×===2/12.010coscosαωαRvvOAB2.83(m/s)=×====12.010lRlvlvOABAABωω2(rad/s)ωABvBOxy(c)BβαsinsinlRyB+=tltRyvBBddcosddcosββαα+==Otωα=ddABtωβ=ddβωαωcoscosABOBlRv+=tABddβω=(c))cos2.024.0cos(arccos2.024.0cos45cos)(cosαβααβ+=+=+−=lRRlD2)cos2.024.0(1ddsin2.0)]cos2.024.0cos(arc[ddddααααβω+−=+==tttABα=45°==Otωαdd10(rad/s)ωAB=2(rad/s))m/s(83.2222122102.0coscos=××+××=+=βωαωABOBlRv2ABεABBaBεABaBAnBABAABaaaa++=τ(C)(d)11-25(m/s20102.022=×==OARaω2)(d)=×=⋅=2221ABnBAABaω4(m/s2)(d)(d)τBAaaB(d)xy(1)DDD45cos45cos45cosnBABAAaaaO++−=τ(2)DDD45cos45cos45cosτBAnBAABaaaa+−−=(1)16(m/s=−=−=420nBAABAaaaτ2)AB===116laBAABτε16(rad/s)(2)−=−=+−−=2422162242220Ba5.66(m/s2)aBβωαωcoscosABOBBlRyv+==tBβωβεαωββωωβααωsincossinddsinddcosddsin22ABABOABABOBBllRtltltRya−+−=−+−==2222222)cos2.0)cos2.024.0(1ddsincos08.0ddsin216.021ddsin2.024.0(1ddcos2.0)cos2.024.0(1)cos2.024.0(1ddsin2.0ddddddαααααααααααααααβωε+⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛+−+−×−⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+−===ttttttttABABα=45°10dd==Otωα(rad/s)=ABε16(rad/s2)22212216122102.022×−××+×−=Ba−=−=245.66(m/s2)aBy11-26aO1O22O2O12O2O1O22O2122b11-26ωaωeωr2ωa=ωe+ωr1O22(a)O2O2x′y′(b)O1x′y′O1O2O1x′y′3222P22P11-27(a)(b)rO2O1O2R1(a)ϕO2θϕ11-27θrRθϕ=ϕθrR=ϕϕϕϕθrrRrR−=−=−ϕω=eϕθωrR==rϕωrrR−=aϕεrrR−=aP2vO2=rωaϕ)(2rRvO−=ϕωrrRrvO)(2a−==ϕωεrrR)(aa−==2(b)ϕO2θϕθRϕ=θrϕθrR=ϕϕϕϕθrrRrR+=+=+ϕω=eϕθωrR==rϕωrrR+=aϕωεrrR+==aaP2a2ωrvO=ϕ)(2rRvO+=ϕωrrRrvO+==2aϕωε⎟⎠⎞⎜⎝⎛+==rrRaa11-28(a)R1ωOεO3RO3R3M(AMOA)vMaM1vM1OA23OAABOOAROAvωω3=⋅=OAOOBROBvωω5.1=⋅=OB12D1ODRvω=OD11-28BD2P2b22DPvBPvDB=vBvD225.1DPBP=RDPBP5.022=+RDPRBP2.0,3.022==2OOBRRBPvωωω53.05.122===23EOOERRREPvωωω45)3.05.0(22=⋅+==BEEA3P3b33APvEPvAE=33APREP+=vAvBAP3=3R3OOARRAPrωωω===33333OOMRRRAPRMPvωωωω10)3(22323233=⋅+=⋅+=⋅=MP3(b)cOAωe=ωO()1ω1a=ωOOAOA1231ω1rOω1a=ω1aωeωO=ω1rωOω1r=2ωOω3r=2ωOω3rωe3ω3a=ω3aωe=2ωOωO=ωO22er===OOAPOPωωωωRMPRAP10,3==ωωRMPvM103a=⋅=MP11-28dva=ve+vr3MOAMOAMOOROMvωω10e=⋅=OMvr=Rω3rω3r=2ωOvr=Rω3r=2ωOdOMRvvω10e==e3AMvM=vA+vMAOOAROAvωω3=⋅=OAOMARvωωω==a3a3OMARvω=eOOOMRRRvωωω10)()(322=+=11-282aMf3AτMAnMAAMaaaa++=aAa322a3)()(3)(23εωωεωττRaRRaRaRaMAOnMAOAOnA=↑==↓=←=OOttεωωε===dddda3a3OMA