《理论力学》 第九章 点的复合运动

1、coscosABvvnBABAaaaaAB8-1 xCOyMxyxyOMABxyOABCxyxyaaa,vrra,veea,vOM1MrveveaOMve2OMae8-2 动系上与动点重合动系上与动点重合 的点的绝对轨迹的点的绝对轨迹zxyOM绝对运动轨迹绝对运动轨迹相对运动轨迹相对运动轨迹 M1 M2MM M1 M2MtMMtlim0avavtMMtlim10evevtMMtlim20rvrvMMMMMM11tMMtMMtMMttt10100limlimlimreavvvreavvvM M1 M2MavevrvCOABavevrvreavvvlvacoscosaelvvvvevavrreav

2、vvvvevvvea330cotOMAB2C已知：已知：R， =t (为常数为常数)求：求：(1) 小环小环M的速度的速度(2) 小环小环M相对于相对于AB杆的速度杆的速度vavevrcos2RAMveRvvea2cos/reavvvsin2sinRvvarOO1ABvavevrreavvvrv asinsinrvvae222211sin/sin/rlrrvAOveeOO1ABCvBavBevBrvAavAevArArAeAavvvBrBeBavvvOABhvavevrreavvvhv etantanhvveaCO1O3rA2vavrvereavvvrv a)90sin(2sinaevv51/

3、11COveAOrvv510sin2aeyO1O2ve1vr1vr2ve211arevvv22arevvv2211rerevvvv21145cos45coserevvv22112;2rvrvee112rvr1114rvvveraCO1O3rACOABM8-2OyxzijkyxzOrrOrOrrrdtddtddtdOrrrdtdarvdtddtzddtyddtxdrrkjiveOdtdvvrOreavvvkjir zyx8-2reavvvMrOyxzrijkyxzOOrdtddtddtdreavvvdtddtzddtyddtxdrr222222vkjiadtdaavaeeaavvOOdtddtd

4、reaaaakjivdtzddtyddtxdrreaaaanrrneenaaaaaaaaCOABearaaaAreaaaa2laa22130sinlaaaeOO1ARBCvavrvereavvvs /m4 . 0OAvam/s4 . 0arevvvm/s4 . 0eBCvv=30OO1ARBCaeranraaaAnrreaaaaa22s /m6 . 1OAaa22s /m6 . 1/Rvarnrynreaaaa30cos60cos2m/s771. 2eBCaaOO1ARBC=30 xcos21RxOs /m4 . 0sin2sin211rdtdRdtdxvoO2211s /m7712. 2co

5、s2rdtdvaoOOO1ABCDvAvavrvereavvvrvvAervvvarervvaCDOO1ABCD2raaAe其其中中reaaaaaaaraeaAy60cos30coseaaa23330tanraaaeaCD8-3OrreavRvvvr2rraavRvRRvRRva2)(222MreaaaaOyxzrAijkyxzOOrAOrkrAAedtdrrvAAOeOOdtdrrv)(krkrOeOdtddtdkkedtddtddtddtdAOrkrkk, jj, iieeedtddtddtdkjivdtzddtyddtxdrdtddtzddtddtyddtddtxddtzddtyddtx

6、ddtdkjikjiv222222rdtddtzddtyddtxdrr222222vkjiareeeedtzddtyddtxddtddtzddtddtyddtddtxdvkjikji)()()(rrrdtdvaverveeOrrMrijkyxzOOyxz dtddtddtdeeerrvrevraeereedtdvave)(reeeedtdvvrveedtdreadtdvvvrrrrdtdvavereedtdvavedtddtddtdreaavvvaCreaaaaa科科氏氏加加速速度度rvae2CaCvrreCva 2sin2reCv aCreaaaaaOMAB2C已知：已知：R， =t(为常数

7、为常数）求：求：(1)小环小环M的速度、加速度的速度、加速度 (2)小环小环M相对于相对于AB杆的速度、加速度杆的速度、加速度vavevrcos2RAMveRvvea2cos/reavvvsin2sinRvvaraearnaaaaaCOMAB2Cvavevrsin2,2cos/RvRvvreaCreaaaaaanasin42cos24/2222RvaRaRRvarCeanacossincosrCenaaaaacos22RarOO1ABvavevrreavvvrv asinsinrvvae222211sin/sin/rlrrvAOveeOO1ABaCaraaneaeaCraaaaaaene222

8、/322241212/cos)(rlrlvvrlrAOaraarnea2/32321)(290sin2rllrvarCCeaaaacos22/32222)()(rlrlrlae2222)22)(rlrlrlOABhvavevrreavvvhv etantanhvveaseccos/hvverOABhaaaeaCnraraCeaaaaaarnr/)sec(/222hvaharnresec290sin221hvarCCeraaaaacoscosn) 1secsec2(322ahhaCO1O3rAreavvvrv a)90sin(2sinaevv2vavrve51/11COveAOrvv510sin

9、2aeCO1O3rAaaarneaeaCaaaaaarene22122510;1rCOaraAOnea25290sin21rvarAOCCneeaaaaasincos2cos2107511rae2175111COaeAOaCOABvAvevrreavvvsincosArAvvvvelvOAvAeecos/BOAarneaeaaCCaaaaaarene222cos0Aeneavllaa2sin90sin222lvvaAreCCeaa 02sin2lvaAe2sin22lvlaAeeOABvAreavvv0evlvACvAAABecosvrraOvvvCsinAABrOvOCvvOABvACreO

10、aaaa0eaar2sin90sin22lvvaAreCaCOAnOAaaaaAOnOAaOAa222cosAABnOAvlla0AaOAnOAaaaaCrxCOAaalaOAABe2sin22lvAO1OO2ABCDvCvBEvA0332BEvB0332aavvAC02332COvCCOEO1OO2ABCDvCvBvA02130cosavvAAr06330sinavvAAe063OAvAeODvAevArArAeAvvvO1OO2ABCD20)31 (aaCaBaBnCaCanCBaCBaCBnCBCnCaaaaaB20aaB202312aaaCOnC20234aaanCBnCBBnCCaa

11、aa30cos60cos30cos202)31 (2COaCCOO1OO2ABCD20332aaCBaBaBnCaCanCBaCBaCBnCBCnCaaaaaB20aaB202312aaaCOnC20234aaanCB60cos30cosnCBCBnCaaa20332BCaCBABnABaaaaBAO1OO2ABCDaB20aaB20332aaaaAB20234aaanABnABaABaCrAaaaaaeneaCarneaea202121aaaODne206390sin2avaArODCCrBaaaaaaaeneABnABaBO1OO2ABCDaBnABaABaaCaraBxCeABnABBa

12、aaaa60cos30cos60cos206373aae206373OAaeODneaeaCrBaaaaaaaeneABnABvAA BD C vBPvCevCrvCarad/s1PAvAABsmPCvvABCCe/2 .0CrCeCavvvsmvvvCeCrCa/15330tanvAA BD C nBAaBAaaBBAnBAaaaaAB22m/s4 . 0ABnBAABa0Aa2m/s34 . 030cotnBABAaa2rad/s3ABaBAABxvAA BD C CAnCAaaaaACnCAaCAaaCraCaaCCaaaaCeCrCaCCrCAnCAaaaaaCa2m/s32.0ABCAACa2Cm/s153290sin2CrABvaC30cosaaaCACa2m/s32CaCDaavAOABEDCvBPrv2ArvvAB20ABOBvBOBEvevrvarvvveaEDreavvv