Page 449 - Dynamics of Mechanical Systems

0593_C12_fm Page 430 Monday, May 6, 2002 3:11 PM

430 Dynamics of Mechanical Systems

points on the pin axes, and B , B , and B are the rods themselves. Using this notation, the
3
2
1
angular velocities of the rods are:
˙
˙
˙
B 3
ωω = θ n , ωω = θ n , ωω = θ n (12.4.1)
B 1
B 2
1 3 2 3 3 3
Also with this notation, the mass center and pin velocities are:
n
v G 1 = (l 2)θ ˙ 11θ
n +(l
v G 2 = lθ ˙ 11θ 2)θ ˙ 2 n 2θ (12.4.2)
n + lθ
v G 3 = lθ ˙ 1 1θ ˙ 2 n +(l 2)θ ˙ 3 n 3θ
2θ
and

v Q 1 = lθ ˙ 11θ
n
n + lθ
v Q 2 = lθ ˙ 11θ ˙ 2 n 2θ (12.4.3)
n + lθ
v Q 3 = lθ ˙ 1 1θ ˙ 2 n + lθ ˙ 3 n 3θ
2θ
The partial angular velocities of the rods are then

1 B
1 B
1 B
ωω = n , ωω = 0 , ωω = 0
˙ θ 1 3 ˙ θ 2 ˙ θ 3
ωω = 0 , ωω = n , ωω = 0 (12.4.4)
B
B
B
2
2
2
˙ θ 1 ˙ θ 2 3 ˙ θ 3
ωω = 0 , ωω = 0 , ωω = n
B
B
B
3
3
3
˙ θ 1 ˙ θ 2 ˙ θ 3 3
Similarly, the partial velocities of the mass centers and of point Q are:
v G 1 = (l 2) n , v G 1 = 0 , v G 1 = 0
˙ θ 1 θ 1 ˙ θ 2 ˙ θ 3
v G 2 = l n , v G 2 = (l 2) n , v G 2 = 0 (12.4.5)
˙ θ 1 θ 1 ˙ θ 2 2 θ ˙ θ 3
v G 3 = l n , v G 3 = l n , v G 3 = (l 2) n
˙ θ 1 θ 1 ˙ θ 2 2 θ ˙ θ 2 3 θ
and
Q
v = l n , v Q = l n , v Q = l n (12.4.6)
˙ θ 1 θ 1 ˙ θ 2 2 θ ˙ θ 3 3 θ
The applied forces that contribute to the generalized forces are weight forces mgn 1
through the mass centers, the weight force Mgn through Q, and the moments at the pin
1
joints. Hence, the generalized forces become:
⋅
⋅
F = mgnv G 1 + mgnv G 2 + mgnv G 3
⋅
˙
˙
˙
θ 1 1 θ 1 1 θ 1 1 θ 1
⋅
+ Mgnv Q +( M − M )n ωω B 1 (12.4.7)
⋅
˙
1 ˙ θ 1 1 2 3 θ 1
+( M − M )n ωω B 2 + M n ωω B 3
⋅
⋅
2 3 3 ˙ 3 3 ˙
θ 1 θ 1