理论力学全套解疑10

10-1ωOxyzOx′y′z′ttva=ve+vr12vavevr3va=ve+vrva4va=ve+vrva=ve+vraa=ae+ar1aa=ae+ar2aaaear3aeaaaraaaτaanaarτranrannarreaaaaaa++=+ττ4nnarreaaaaaa++=+ττva=ve+vraa=ae+araaar10-710-7(a)R=100mmOA=100mmn=120(r/min)ϕ=30°BCDA1AO′x′y′OxyAOABCDO1BCD226.130π120100=×=⋅=ωOAvam/sOAvrO1Aveva=ve+vr10-7dve=va=1.26(m/s)BCD3===1.0)26.1(22aaRva15.9m/s2OA10-7bτra===1.0)26.1(22rrRvan15.9m/s2O1ae10-7baa=ae+arnrrraaa+=τnrreaaaaa++=τAξAξτranaaaξξξrea+=naaarea30cos60cos−=DD=+=+=2/39.15219.1530cos60cosraeDDnaaa27.4m/s2aeBCDAητrea60cos30cosaaa+−=−DD)m/s(11.0239.15214.2730cos60cos2aer−=×−×=−=DDaaaττraOAAaaOOAAnaaτaaaa=ae+ar+ak1ak2ak=2ωvrakak=2ωvrsinθθωvr180°akωakvrωvrak345aeaneaτeaaarknrrneenaaaaaaaaa++++=+τττ6va=ve+vrttttddddddrevvva+==tddevtddrvOxyzO′x′y′z′10-91OzωeεeMkjiv′′+′′+′′=tztyxddddddrτ(1)kjia′′+′′+′′=222222rddddddtztytx(2)MMve=ωer(3)ae=εer+ωeve(4)tttddddddreaavvva+==(5)(3)ttttdddd)(ddddeeeerωrωrωv×+×=×=ωeεerreeeaeeeddvωvωrεvωrεv×+×+×=×+×=t(4)aereeeddrωav×+=t(6)veae(5)(1)⎟⎠⎞⎜⎝⎛′⋅′+′⋅′+′⋅′+⎟⎟⎠⎞⎜⎜⎝⎛′′+′′+′′=ttzttyttxtztytxtdddddddddddddddddddd222222rkjikjiv(7)ar[(2)]i′j′k′k′k′ArA10-92ArAωerAAAAtrωrv×==edd2krr′+=′OArO′O′)(ekr′+×′Oddddωkr=′+′OttO′×rOOt′′==ωrveddkωk′×=′eddt(8)iωi′×=′eddt(9)jωj′×=′eddt(10)(7)reetztytxttzttyttxvωkjiωkji×=⎟⎠⎞⎜⎝⎛′′+′′+′′×=⎟⎠⎞⎜⎝⎛′⋅′+′⋅′+′⋅′dddddddddddddddddd(7)rerrddvωav×+=t(11)(6)(11)(5)aa=ae+ar+2ωevrak=2ωevr(12)aa=ae+ar+ak(13)ak(3)(ωe=0)ak=0aa=ae+ar(14)rreeddddavav==ttωevr10-18ak=2ωevr10-101Azωvrak=2(ωevr)ωevrωMωvrθak=2ωvrsinθakωvrakωvrθvrPvr′vr′Pω90°akak=2ωevr1ω=0vr=0ak=02ωvrθ=0ak=03ωvrθ=90°ak=2ωvrvrω90°ak.ak10-111M1ak=0M2ak2ak3ak=04ak5ak6akP7akO2B8ak=010-1112vrar110-121OMM10-12110-12210-122OMωMABMAB2OABCDOuh10-123OAωOABE)1BOA2EBCDBOA(i)BOABBOAOAO(ii)vavevruOE⋅ωOEAOva=ve+vrθθsinsinaeuvv==θθθω2esinsin/sinhuhuOBvOA===EBCD(i)EOABCDEOEBCDEE′EE′BCD(ii)BEτranraROω10-124(a)ABvAB10-124AABAABAAOvavevrOE⋅ωOEAva=ve+vr10-124(b)ωωROEvvvAB=⋅===eaEABE10-124c310-125(a)ϕ=30°ABva10-125CDABCDO1RO1O1O1ABEABO1EveABveO1ABO1OO1ABvavevreωOO1va=ve+vrbωevvvAB2330cosae===DABO1aaaeareω2Oaa=ae+arc2ae21sinωϕeaaaAB===dABαAvABA10-126aBvBBAvrABBAAAx′y′ABAAAx′y′OxyBOBBABBAAvavevr=vAOBABAAx′y′BEAve=vAva=ve+vrαααtgcoscosreAABvvvvv===ABx′y′B10-126b10-126ABBx′y′OxyAABABvavevrvAOBBAva=ve+vrαcoseABvvv==αtgrAvv=′rv′ABvr5ACBDωlAB10-127aMACBDMMACBDACBDvMaMBDBDvMaMMvM(i)ACMACMMACAC(ii)va1ve1vr1AM⋅ω=lωACMAva1=ve1+vr1(1)7a(iii)BDMBDMMBDBD4va2ve2vr2lωBDMBvaz=vez+vrz(2)7(a)M(1)(2)ve1+vr1=ve2+vr2(3)(3)ξ7a2e1r1e30cos30sinvvv=+DD32132)30sin(30cos11e2e1rωωωlllvvv=⎟⎠⎞⎜⎝⎛−=−DDωωωlllvvvM3323222221r21e=+=+=ωωωlllvvvM3134222222e22r=−=−=xy7(a)ωωωlllvvvMx332232360cos30cos1r1e−=1−−=−−=DD02332130cos30sin1r1e=−=−−=ωωllvvvMyDDvMaMMACaa=ae1+ar1+ak1(4)21eωla=ar1AC21r1k32322ωωωωllva===7bBDaa=ae2+ar2+ak2(5)22eωla=ar2BD22r2k32322ωωωlllva===7(b)M(4)(5)ae1+ar1+ak1=ae2+ar2+ak2(6)(6)ξ7b2k1k60cos30cosaa=−DD1r1e30cosaa−D22222k1k1e1r)32213223(32)60cos30cos(30cos1ωωωωllllaaaa−=−−=−−=DDD2424221k21r21e334344)(ωωωlllaaaaM=+=+−=xy7(b)↓−=−−=−−==+−=+−=2221k1e1r221k1e1r334213223230sin30cos)(0233221230cos60cos)(ωωωωωlllaaaallaaaaMyMxDDDDRv0ABϕ=30°AB10-131a10-131ABA1ABAAABAAvavevrv0O′Ava=ve+vrva↑===000ea577.0tgtgvBvvvDϕ2MAB10-131bMABM(1)⎪⎭⎪⎬⎫−=−=)cos(cos)sin(sin00ϕϕϕϕRyRxrrMϕ0ϕϕM220220)cos()sin(RRyRxr+−+−ϕϕ00cos,sinϕϕRyRxrcrc==RABAA(2)⎪⎭⎪⎬⎫===ϕcosRyLxAA(2)tA(3)⎪⎭⎪⎬⎫−===ϕϕsin0RyvxAAAM(4)⎪⎭⎪⎬⎫==+=000cosϕRytvxxMrAxxx+=(1)(2)(4)ϕϕsinsin000RRLtvx−+=+t(5)⎪⎭⎪⎬⎫−=−=ϕϕϕϕcos/cos00RvRv(5)(3)ϕϕϕtgcossin00vRvRvA=⎟⎟⎠⎞⎜⎜⎝⎛−−=ϕ=30°00577.030tgvvvA==D(2)10-1210-132(a)OARωα=45°BD10-132(1)OAABDAOAAOAAavavevrRωOADBva=ve+vrωαRvv22cosae==baaaearRω2AODBaa=ae+ar2ae22sinωαRaa==10-12(2)AA′OAOx′(c)vrvevaωαRvv22cosea==10-132aa=ae+ar+ak=ae+++aτranrak(d)nraρ2rrvn=aρaa(3)A′AAx′y′A′BD(e)ωαRvv22cosea==(f)aa=ae+ary′ωαRaa22cosea==10-132(4)(f)A′tRytRytRRyωωωωωαcossincoscos2−=−===α=ωt=45°22222ωωRyaRyv−====Mω=2trad/sOτea1O2MOA240tOM=OAO1O260°t=1M10-141MOx′y′OxyMMOAO1O22)mm/s(340260sin40)mm/s(80dd)rad/s(2dd)rad/s(232er2eeetttOMvOAttOMvtt=⋅=⋅′======Dωωεω)mm/s(340|22340)60sin()mm/s(380|)2(2340)60sin()mm/s(80dd212ee21222enerrztOMattOMatvatt=⋅=========εωτDD)mm/s(3120|3120)340(dd)260sin40(dd)(dddd21232ee===⋅=⋅′===tttttttOMttvaDωττea40τea3(mm/s2)t=1MOA12|40==ttOM=40(mm)D60sin'OMOM==40sin60)mm/s(340)2320(dd)60sin40dddd2eetttttva=⋅=⋅==ωτD)mm/s(316023802260sin221k=⋅⋅===trettvaDωzxyz356222a==++=azayaxaaaa(mm/s2)ωMMR10-15aMMaaaearakb110-15MMMR2aaaearak02ωR22rωυRR=2r22ωR=×υωOOO0kre=++=aaaaMA150(m)vA=45(km/h)BvB=70(km/h)30(m)1AB2BAvr10-16a10-16ABBA1AB70==BAevv(km/h)aD6.297045tg)km/h(83704511222e21r===+=+=−θAAvvv2BAABvv15030150e+==54km/hbD7.377054tg)km/h(887054122222e2r===+=+=−θBBvvv212r1r,θθ≠≠vvvaaavrar10-7BCDBCDO110-7axO1O1ϕϕϕcos2coscos101RAOOAx=+=ϕωϕϕsin2sin201RRx−=−=ϕωϕϕcos2cos22201RRx−=−=ϕ=30°)()m/s(4.272330π1202.0232)()m/s(26.130π1201.02220101←−=⎟⎠⎞⎜⎝⎛−=−=←−=×−=−=ωωRxRxx10-105DDϕsineRy+=ϕωϕϕcosddcosddetetyv===ϕωsindd2etva−==ϕ=30°)(21)(232↓−=↑=ωωeaevrtr(t)r(t+∆t)r)()(tttrrr−∆+=∆∆r∆r10-181tt+∆trr*[r(t+∆t)](a)(b)Oxyr∆rcOx′y′r∆r∆dr~r∆rtttt∆∆=→∆rr0limdd10-18110-182∆tr~ttt∆∆=→∆rr~limd~d0O′x′y′z′ωrO′x′y′z′i′j′k′O′x′y′z′10