Page 415 - Dynamics of Mechanical Systems

0593_C11_fm Page 396 Monday, May 6, 2002 2:59 PM

396 Dynamics of Mechanical Systems

FIGURE 11.12.2
The rod pendulum.

Example 11.12.2: Rod Pendulum
Consider next the rod pendulum of Figure 11.12.2, using the same notation as before.
Recall that the mass center velocity v and the angular velocity of the rod are (see Eqs.
G
(11.10.6) and (11.10.7)):

v = ( ) 2 θ ˙ n and ωω = θ ˙ n (11.12.9)
l
G θ z
Hence, the kinetic energy K of the rod is:
K = ( ) mv 2 +( )I ωω 2
12
12
G G
2
2 ˙
l
1 2
= (12 m ) ( ) 2 θ 2 ˙ +( )(1 12 m ) l θ 2 (11.12.10)
2 ˙
= (16 m ) l θ 2
Then, from Eq. (11.12.5), the generalized inertia force F θ * is:

˙˙
*
F =−( )ml 2 θ (11.12.11)
13
θ
This result is the same as in Eq. (11.10.11).
Example 11.12.3: Double-Rod Pendulum
To extend this last example, consider again the double-rod pendulum of Figure 11.12.3.
Unlike the two above examples, this system has two degrees of freedom, as represented
by the angles θ and θ .
1
2

O
θ G 1
1
B
1
Q G
θ 2
FIGURE 11.12.3 2 B
A double-rod pendulum. 2