Dynamic from the general equation of dynamics we have (OA-QB + OB COS a)o,+(CB coS a+Osin a-LBOSB=0 The system has two degree of freedom, we can choose &x, and s as independent virtual displacement. Moreover Q=mg. So we get Ma +ma-ma. cosa=0 macosa+mgsina-ma =0 lB B Solving them we get oxa SB m sin 2a a P 2(M+msin a g
11 From the general equation of dynamics we have (− − + cos ) + ( cos + sin − ) = 0. B r B e A B r B e A B Q Q Q x Q Q Q s The system has two degree of freedom, we can choose as independent virtual displacement. Moreover, . So we get A B x and s Q=mg cos sin 0 cos 0 + − = + − = r r ma mg ma Ma ma ma Solving them we get . 2( sin ) sin 2 2 g M m m a + =
学 由动力学普遍方程: C-OA-OB+OB COSa)a +(@ cosa+Osina-ORSSB=0 系统为二自由度,取互不相关的&,∞S为独立虚位移, 且Q=mg,所以 Ma +ma-ma COsa=0 macosa+mgsina-ma, =0 解得: LB B m sin2a oxa a- 8 SB 2(M+msin 2a P 12
12 由动力学普遍方程： (− − + cos ) +( cos + sin − ) B =0 r B e A B r B e A B Q Q Q x Q Q Q s 系统为二自由度，取互不相关的 为独立虚位移， 且 ，所以 A B x ,s Q=mg cos sin 0 cos 0 + − = + − = r r ma mg ma Ma ma ma 解得： g M m m a 2( sin ) sin2 2 + =
Dynamic 817-2 Lagrange's equations of the second kind Here we will deduce the general equations of dynamics using generalized coordinates Suppose there are n particles in the system and that there are s constraints which all are ideal ones. The number of degrees of freedom is k-3n-s Particle M If we choose a set of generalized coordinate of the system 1>92, qk then we have 7=(q1929k1)(=1,2;m)(a ,; V (i=1,2m)(b) dt j=10q at are called the generalized velocities dt 13
13 §17-2 Lagrange's equations of the second kind Suppose there are n particles in the system and that there are s constraints which all are ideal ones. The number of degrees of freedom is k=3n- s. Here we will deduce the general equations of dynamics using generalized coordinates. q q qk , , 1 2 Particle . If we choose a set of generalized coordinate of the system, , then we have i i i M :m ,r ( 1,2 ) ( ) ( , , , ) ( 1,2, ) ( ) 1 1 2 i n b t r q q r dt dr v r r q q q t i n a k j i j j i i i i i k = = + = = = = are called the generalized velocities. dt dq q j j =
学 §17-2拉格朗日第二类方程 下面推导以广义坐标表示的动力学普遍方程的形式 设质点系有n个质点,受s个完整约束且系统所受的约束是 理想约束,自由度k=3n-s。 质点M1:m1,F。若取系统的一组广义坐标为1,q2,9k,则 =F(q14q2…q2t)(=1,2,m)(a dr k or Vi dt zag (i=1,2mn)(b) 称q=“为广义速度
14 §17-2 拉格朗日第二类方程 设质点系有n个质点，受s个完整约束且系统所受的约束是 理想约束，自由度 k=3n- s 。 下面推导以广义坐标表示的动力学普遍方程的形式。 质点 Mi :mi ,ri 。若取系统的一组广义坐标为 q1 ,q2 , qk ，则 ( 1,2 ) ( ) ( , , , ) ( 1,2, ) ( ) 1 1 2 i n b t r q q r dt dr v r r q q q t i n a k j i j j i i i i i k = = + = = = = 称 为广义速度。 dt dq q j j =
Dynamic ∑xn(i=12,m) (c) iog Substituting equation(c) into the general equation of dynamics of the system of particles, we get ∑(F-ma1)c=∑F·C-∑ma,1=0(d) F=F①)=2(F,m) +rOi =(X001z,02 ∑Q,M 15
15 = = = k j j j i i q i n c q r r 1 ( 1,2, ) ( ) Substituting equation (c) into the general equation of dynamics of the system of particles, we get = = = − = − = n i n i i i i i i n i Fi mi ai ri F r m a r d 1 1 1 ( ) 0 ( ) = = = = = = = = = + + = = = k j j j k j j j i i n i j i i j i i k j n i j j i i k j j j i n i i n i i i Q q q q z Z q y Y q x X q q r q F q r F r F 1 1 1 1 1 1 1 1 [ ( )] ( ) ( )