Page 391 - Dynamics of Mechanical Systems
P. 391
0593_C11_fm Page 372 Monday, May 6, 2002 2:59 PM
372 Dynamics of Mechanical Systems
By projecting the forces applied to T and the particles along the partial velocities at the
points of application, we obtain the generalized forces as:
⋅
F = O x ( i O y ) ⋅ O Mg j v + C nv Q 1 + C nv Q 2
⋅
+
j v −
⋅
G
˙
˙
x 1 x 1 x 1 ˙ 1 2 x 1 2 2 x 1 ˙
[
2]
2 (
⋅
+ C nv Q 3 + − mg j k x − ) n − kx n − C n ⋅ v P 1
+
x
˙
3 2 x 1 ˙ 1 1 1 1 1 x 1
[
j k x − )
+− mg + ( 3 x 2 n − ( 2 − x 1 1 − C n 2 ] ⋅v P 2
k x − )n
1
˙
2
x 1
[ ]
+−mgj − kx n 1 − ( k x 3 − )n 1 − C n 2 ⋅v P 3
x
3
3
2
x 1 ˙
= − ++++ mgcosθ + kx − kx ++ 0
00000
2
0
2 1
or
F = mgcosθ + kx − 2 kx (11.7.14)
x 1 2 1
and
F = O x ( i O y ) ⋅ O Mg j v + C nv Q 1 + C nv Q 2
⋅
⋅
+
j v −
⋅
G
˙
˙
˙
x 2 x 2 ˙ x 2 1 2 x 2 2 2 x 2
2]
[
2 (
+
⋅
+ C nv Q 3 + − mg j k x − ) n − kx n − C n ⋅ v P 1
x
3 2 x 2 1 1 1 1 1 x 2
˙
˙
[
j k x − )
+− mg + ( 3 x 2 n − ( 2 − x 1 1 − C n 2 ] ⋅v P 2 ˙
k x − )n
2
1
x 2
[ ]
+−mgj − kx n 1 − ( k x 3 − )n 1 − C n 2 ⋅v P 3
x
2
3
3
x 2 ˙
= − ++++ mgcosθ + kx + kx − kx + 0
00000
2
3 1 2
or
Fx = mgcosθ + kx + kx − 2 kx (11.7.15)
2 3 1 2
and
Fx = ( i O ) ⋅ O j v + ⋅ Q 1 + ⋅ Q 2
+
j v −
⋅
G
3 O x y x 3 Mg x 3 C 1 nv ˙ x 3 C 2 nv ˙ x 3
˙
2
˙
2
2]
2 (
n −
+
⋅
+ C 3 nv ˙ Q 3 + − [ mg j k x − ) n − kx 1 1 C 1 n ⋅ v ˙ P 1
x
2
1
1
x 3
x 3
+− [ mg + ( 3 x 2 n − ( 2 − x 1 1 − C n 2 ] ⋅v ˙ P 2
j k x − )
k x − )n
2
1
x 3
+− [ mgj − kx n 1 − ( k x 3 − )n 1 − C n 2 ] ⋅v ˙ P 3
x
2
3
3
x 3
= − ++++++ mgcosθ − kx 3 + kx 2
2
0000000