山东大学学士学位论文
????p?m?r1x??r2x?1r1?mr??m?2r2=m1r???1y???2?2y??r1z????r2z???rM??r1?A1rC1M?r2?A?2rC2M??r??C1Mx??r2x??rC2Mx????r1x?r??A??r?C1My???r??A??11?rC??1y????r1z????r??2y2My?CMz??r???1?2z??r?C2Mz???vM??r1?A1ω?1rC1M?r2?A2ω?2rC2M???r1x????0??1z?1y??rC1Mx?????r1y??A????1??1z0??1x??rC???1My???1y?1x????r1z???0??r?C1Mz?????0??2z?2y??rC2Mx
???r2x??r??A?2??2y?????2z0??2x??rC2My????r?2z??????2y?2x0????r?C2Mz?刚体1相对于其质心C1’的动量矩为
h1??J1A1ω1 其中
?J1xyJ1xz?
J?J1x1r??J1yxJ1yJ?1yz?? ?J1zxJ1zyJ1z??J?AJT111rA1 代入得，
h1??A1J1rω1 同理，刚体2相对于其质心C2’的动量矩为
h?2?J2A2ω2 其中， ?J2xyJ2xz?
J?J2x2r??J2yxJ2yJ?2yz?? ?J2zxJ2zyJ2z??J2?A2J2rAT2
25
(79)
(80)
(81)
山东大学学士学位论文
代入得，
??A2J2rω2 h2刚体1相对于原点O的动量矩为，
? h1?h1??r1?m1v1?AJ11rω1?r1m1r1同理可得，刚体2相对于原点O的动量矩为，
??r2?m2v2?A2J2rω2?r2?m2r2 h2?h2则刚体系统相对于原点O的动量矩为，
h?h1?h2
约束条件亦为刚体1和刚体2在连接点M?的位移和速度一致，
?rM??OM??OC1??C1?M??r1?A1rCM1????C2?M??r2?A2rC2M?OC2? ????v??OC??ω?C?M??OC??ω?C?M?111222?M???r?Aωr?r?AωrC2M?111CM222?1代入得，
26
山东大学学士学位论文
???h?A1J1rω1?r?1m1r1?A2J2rω2?r?2m2r2???1x??0?r?1zr1y??r1x???A1J?1r???????1y????r1z0?r1x?m1?r1y?????1z?????r1yr1x0????r1z??????2x??0?r2zr2y??r2x??A?2J2r????r0?r?m??r???2y????2z2x?2??2z?????r?2y?2yr2x0????r2z????rM??r1?A1rC1M?r2?A2rC2M???r1x?rC1Mx??r2x??rC2Mx?????r??A?r??1??r??A?r??1y?????r1z???C1My??r??2y?1?C2My?C??r????1Mz?2z?rC2Mz???vM??r1?A1ω?1rC1M?r2?A2ω?2rC2M???r1x??0??1z?1y??r?????r?1y?A???C1Mx??1??1z0??1x??rC1???My??r1z????1y?1x0??r??????C1Mz? ????0??2z?2y???r2x????rC2Mx???r?2y??A???2??2z0??2x??rC2My????r2z??????2y?2x0????r?C2Mz?刚体系统的动能为
T?T1m1T1T11?T2?(21vT1v1?2ω1J1rω1)?(2m2v2v2?2ωT2J2rω2)约束条件同样为刚体1和刚体2在连接点M?的位移和速度一致，
??rM??OM??OC1??C1?M??r1?A1rC1M???OC?2?C2?M??r2?A2rC?2M??? ?vM??OC1??ω1?C1?M??OC?2?ω2?C?2M????r1?A1ω?1rC1M?r?2?A2ω2rC2M代入得
27
(82)
山东大学学士学位论文
??T???????????????????????????????1111?(m1r1Tr1?ω1TJ1rω1)?(m2r2Tr2?ω2TJ2rω2)2222??r1x???1x???1?r??1?????J?????=?m1?rrr1y1z?1r?1y???1x1y1z??1y?2?1x2?????r1z????1z??????r2x???2x???1?r??1??????????m2?rrr??J2y2z?2r?2y???2x2y2z??2y?2?2x2?????r2z????2z????rM??r1?A1rC1M?r2?A2rC2M?rC1Mx??r2x??rC2Mx??r1x???A?r???r??A?r???r?1y?1?C1My??2y?1?C2My????r????rr2z?r1z?????C1Mz??C2Mz??vM??r1?A1ω1?rC1M?r2?A2ω2rC2M?0?r1x???A????r1y??1?1z???1y???r1z???0?r2x???A????r2y2?2z?????2y??r2z?????1z?1y??rCMx????0??1x??rCMy????1x0?r??CMz???2z?2y??rCMx????0??2x??rCMy????2x0?r??CMz?111222(83)
对于完整约束系统，有第二类拉格朗日方程，
d?T?T??Qk?k?qkdt?q(k?1,2,?)
将系统拆分成两个独立的刚体1和刚体2，将铰链M分别作用于刚体1和刚体2的力视作刚体1和刚体2所受外力。
刚体1的位置可以由6个独立的坐标r1x、r1y、r1z、?1、?1、?1来确定，则
d?T1?T1??Qk?k?qkdt?q(k?1,2,3,4,5,6)
?T1?0 ?r1x?T1?0 ?r1y 28
山东大学学士学位论文
?T1?0 ?r1z?T1?0 ?ψ1?T11?ω1TJ1rω1 ???12??1?T11?ωT1J1rω1??? 12??1?T1?r?m1r1x 1x?T1?r?m1r1y 1y?T1?r?m1r1z 1z?T11?ωT1J1r???ω1?? 121?T11?ωT1J1rω1??? 12??1?T11?ωT1J1rω1??? 12??11的第二类拉格朗日方程为
??m1r1x?f1x?m1r1y?f1y??m1r1z?f1z? ?1d?ωT1J1rω1?2dt???g1? ?1?1d?ωT1J1rω11?ωT1J1rω??2dt???1?g1?12??1?1d?ωTωT1J1rω1?1?1J1rω1??2dt???g1?12?12的位置也可以由6个独立的坐标r2x、r2y、29
(84)
r2z、?2、?2、?2来确
因此，刚体同理，刚体