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0593_C11_fm Page 385 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 385

Finally, from Eq. (11.9.6), the generalized inertia forces are:

F = v ⋅ F + v ⋅ F + v ⋅ F + v ⋅ F + ω ⋅ T *
*
*
*
*
*
*
P 2
P 3
P 1
˙
˙
x 1 x 1 P 1 x 1 ˙ P 2 x 1 ˙ P 3 x 1 ˙ T x 1 T (11.10.30)
1 [
2 ˙
=− mx ˙˙ −(l + x ) ]
θ
1
F = v ⋅ F + v P 2 F ⋅ * + v ⋅ F + v ⋅ F + ω ⋅ T *
*
*
G
*
*
P 1
P 3
˙
˙
˙
x 2 x 2 P 1 x 2 P 2 ˙ x 2 P 3 x 2 ˙ T x 2 T (11.10.31)
2 [
2 ˙
2 +
θ
=− mx ˙˙ −( l x ) ]
2
F = v ⋅ F + v P 2 F ⋅ * + v ⋅ F + v ⋅ F + ω ⋅ T *
*
G
*
*
*
P 3
P 1
˙
˙
˙
x 3 x 3 P 1 x 3 P 2 ˙ x 3 P 3 x 3 ˙ T x 3 T (11.10.32)
3 [
=− mx ˙˙ −( 3 +l x ) ]
2 ˙
θ
3
*
*
F = v P 1 ⋅F P 1 + v P 2 ⋅F P 2 * + v P 3 ⋅F P 3 * + v G ⋅F T * + ω ⋅T T *
θ
θ ˙
θ ˙
θ ˙
θ ˙
θ ˙
]
˙˙
θ
˙˙
=− m(l + ) ( [ l + x θ ) + 2 θ ˙ x ˙ − m(2l + x ) ( [ 2l + x ) + 2 θ ˙ x ˙ ]
x ˙
1 1 1 2 2 2
)[
˙˙
˙
θ
˙˙
− m(3l + x ) (3l + ) + 2 x ˙ ] − ( ) 2 2 θ ˙˙ − ( 2 12 ) θ
θ
x
ML
ML
3 3 3
or
[
2
2
˙˙
*
F =− (l + x ) +( l + x ) +( l + x ) 2 ] θ − M L ( 2 ˙˙
3
2
m
θ ) 3
θ
2
3
1
(11.10.33)
m ( [ + ) + ) + ) ] ˙
˙
− 2 l x x ˙ +( l x x ˙ +( l x x θ
3
2
1 1 2 2 3 3
Example 11.10.5: Rolling Circular Disk
As a final example consider again the rolling circular disk D with mass m and radius r of
Figure 11.10.7. (We first considered this system in Section 4.12 and later in Sections 8.13
and 11.3.) As we observed in Section 11.3, this is a nonholonomic system having three
Z
n
3 N 3
θ
D n 2
ψ
G
X
φ Y
FIGURE 11.10.7 N 1 c
A circular disk rolling on a horizontal N 2
surface. L n 1