上海交通大学：《复杂系统动力学计算机辅助分析》课程教学资源_Chapter 6_Char6.6 Constraint reaction forces

6.6 Constraint reaction forces Variational equation ,(M,+西，7-0)=0 i=l virtual work of all the constraint forces acted on body i o,=西…l=立a"天 =[2…水…m] Consider the k th joint -,'水— virtual work of the constraint forces of the kth constraint acted on body i

6.6 Constraint reaction forces ( ) 0 1      n i A i T i i T i i q M q  q  Q Variational equation     T k nh nh l l T T l i T nh Tk T T i T T i i i i i i            1 1 1         q   q  q  q   q q q q Consider the k th joint   T T i i q  q virtual work of all the constraint forces acted on body i virtual work of the constraint forces of the kth constraint acted on body i k T T k i i  q  q 

Virtual work of the constraint forces of the k th constraint acted on body i 面Q}女ook r=r+As”,m,”=+B,s'6动 > m=6m-B,s6冲， m=-6,更'+动，更h =-m,更，+60，sB,水-64，Φ

    k T k i T T k i k T k Tk i T i k T T k i i i i i W i                              r r q q r r Virtual work of the constraint forces of the k th constraint acted on body i i P i i i P i P i i i P ri  r  A s , r  r  B s    k T k i k T T k i T P i i k Tk T P i k T k i T T k i i i i W i i                          r r r r s B r i P i i P ri  ri  B s 

y PP X ER Tk Constraint force vector and torque with respect to o-xy Virtual work of the constraint forces of the k th constraint acted on body i can be written as δW=(6r,P)TF+60,T equal to w=-m2+δ动，spB,更-60，Φ' Fk =-AI T=yPB,更-西2

k i i k i P T W  (ri ) F   T Virtual work of the constraint forces of the k th constraint acted on body i can be written as k i k Fi , T Constraint force vector and torque with respect to o-xy P Fik riP riP i Tik x y k T k i k T T k i T P i i k Tk T P i W i i i   r  r    s B  r      equal to k T k k T T k i T P i k i k T k k i i i i T            r r s B F

Constraint reaction forces in different coordinate system Ek Tk Constraint force vector and torque with respect to o-xy r Xj F=-,九 T=sPB,西2-Φ X Ek Tk Constraint force vector and torque with respect to body-fixed frame o-xiyi E"R=ATFR Tk =Tk

k T k k T T k i T P i k i k T k k i i i i T             r r s B F k i k Fi , T Constraint force vector and torque with respect to o-xy k T k i i i k k Ti Ti     F A F Constraint reaction forces in different coordinate system k i k Fi  , T  x y Constraint force vector and torque with respect to body-fixed frame o-xiyi xi yi

Example 6.12 Driving constraint 中-ot=0 Get the relation between the Lagrange multipliers and the constraint forces Solution: o-m- g Φ2=中-0t=0,2=元3 。ey-lgeg- Length 21 =[]=0,=1 Mass m g6-

Example 6.12                        2 1 1 1 2 1 , sin cos ,          y l x l Fx F Fy ’x F’y L x1 y1 C x y  g Length 2l Mass m P Driving constraint  t  0       0 1 l P s 1 1 1 1 1 1 1 1 2 2 2 2 1 0 sin , 0 1 cos , 1 x y x y l l                               r r        0                                    cos sin sin cos 0 cos sin 1 0 0 1 1 1 l l l P B s 3 2         2 t 0, Solution: Get the relation between the Lagrange multipliers and the constraint forces

-。6-及 X T=sPB,'Φ，2-D -ew及知-o经]-0 F2=-2=0 T2=s"B =-22=-23 X1 r-卧[]-o T2=T2=-3

  sin cos  0 0 1 1 0 sin cos 2 1 2 1 1 1 1 1 1 1 1 1 1 1                                   l l T T T T T P s B  r                                   2 1 2 1 1 1 1 0 1 1 0 1       T y x F F F r Fx F Fy ’x F’y L x1 y1 C x y  g P     0 T 2 2 1 1 F r 3 2 2 2 2 2 1 1 2 1 1 1             T T T T P T s B  r                                                        sin cos cos sin sin cos cos sin 1 2 1 2 2 1 1 1 1 y T x y x F F F F F A 3 2 1 2 T1   T  