Page 441 - Dynamics of Mechanical Systems

0593_C12_fm Page 422 Monday, May 6, 2002 3:11 PM

422 Dynamics of Mechanical Systems

Z
n
3 N 3
θ
D n 2
ψ
G
X
φ Y
N 1
c
FIGURE 12.2.6 n N 2
A rolling circular disk. L 1
not contribute to the generalized active forces. Regarding the weight force, we found in
Section 11.10 that the partial velocities of G relative to θ, φ, and ψ are (see Eq. (11.10.43)):

v =− r n , v = r sinθ n , v = n r 1 (12.2.36)
G
G
G
˙ ψ
˙ φ
˙ θ
2
1
Hence, the generalized active forces are:
F =− ( mg ) ⋅− ( r ) = mgrsin θ (12.2.37)
N
n
θ
3
2
F =− ( mg ) ⋅− ( rsin θ ) (12.2.38)
N
n
φ
1
3
and
F =− ( mg ) ⋅( ) = 0 (12.2.39)
N
n
r
ψ
1
3
Also in Section 11.10 we found that the generalized inertia forces for the angles are (see
Eqs. (11.10.53), (11.10.54), and (11.10.55)):
[
˙ ˙
˙˙
θ
F = mr −( ) +( ) 2 ψ φcos +( ) φ sin cos θ ] (12.2.40)
θ
θ
*
˙ 2
2
3
54
54
θ
[
˙ ˙
θ
F = mr −( ) ψ ˙˙ sin −( ) ˙˙ 2 θ −( ) φθsin cos θ
θ
*
2
32
φsin
32
52
φ
(12.2.41)
−( ) ˙˙ 2 θ +( ) 2 ψθcos θ]
˙ ˙
φcos
14
1
and
[
˙˙
˙ ˙
θ
F =− mr ( ) +( ) φsin +( ) θφcos θ ] (12.2.42)
2
*
ψ ˙˙
52
32
32
ψ
Kane’s equations then lead to the expressions:
F + F = 0 , F + F = 0 , F + F = 0 (12.2.43)
*
*
*
φ
φ
θ
ψ
θ
ψ