Page 440 - Dynamics of Mechanical Systems

0593_C12_fm Page 421 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 421

In Section 11.10, we found the generalized inertia forces to be (see Eqs. (11.10.30) to
(11.10.33)):

1 [
θ
F =− m x ˙˙ −(l + x ) ] (12.2.28)
2 ˙
*
x 1 1
2 [
2 ˙
θ
F =− m x ˙˙ −( l x ) ] (12.2.29)
2 +
*
x 2 2
3 [
F =− m x ˙˙ −( l x ) ] (12.2.30)
θ
3 +
2 ˙
*
x 3 3
and
[
2
2
˙˙
*
F =− (l + x ) +( l + x ) +( l + x ) 2 ] θ − M L ( 2 θ ) 3 ˙˙
2
m
3
θ
2
3
1
(12.2.31)
m ( [ x x ˙ ) x x ˙ ) + ) ] ˙
˙
−2 l + +( l2 + +( l3 x x θ
1 1 2 2 3 3
Hence, from Kane’s equations, the governing dynamical equations are:
mgcosθ + kx − 2 kx − m x ˙˙ −(l + x θ ) ] = 0 (12.2.32)
˙ 2
2
1
[ 1
1
mgcosθ + kx + kx − m x ˙˙ −(2l + x θ ) ] = 0 (12.2.33)
˙ 2
3 1 [ 2 2
˙ 2
mgcosθ − 2 kx + kx − m x ˙˙ −(3l + x θ ) ] = 0 (12.2.34)
3 2 [ 3 3
and
(
( )
−Mg L 2 sinθ − mg 6l + x 1 + x 2 + )sinθ
x
3
[ 2 2 2 ] 2 )
˙˙
˙˙
− (l + ) +(2l + ) +(3l + ) θ − (L 3 θ (12.2.35)
x
m
M
x
x
1
2
3
+ ( [ + ) ]
− m l x ) x ˙ +(2l + ) x ˙ +(3l x x ˙ θ ˙ = 0
2
x
1 1 2 2 3 3
Example 12.2.5: Rolling Circular Disk
As a final example, consider again the rolling circular disk of Figure 12.2.6 (we considered
this system in Sections 4.12, 8.13, 11.3, 11.10, and 11.12). This is a nonholonomic system
with three degrees of freedom represented by the angles as shown.
The applied forces acting on the disk D may be represented by a weight force –mgN 3
passing through the mass center G together with a contact force C passing through the
contact point C. Because the contact point has zero velocity (the rolling condition), C does