Page 416 - Dynamics of Mechanical Systems

0593_C11_fm Page 397 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 397

n
1 n
3
n 2θ
θ
n
1r
n
2
FIGURE 11.12.4 θ n 2r
Unit vector geometry for the double-
rod pendulum. n 1

In our earlier analysis of this system we found that the mass center velocities and the
angular velocities of the rods were (see Eqs. (11.10.12)):

v G 1 = (l 2)θ ˙ n , v G 2 = lθ ˙ n +(l 2)θ ˙ n
1 1θ 1 1θ 2 2θ
(11.12.12)
ωω B 1 = θ ˙ n , ωω B 2 = θ ˙ n
1 3 2 3

where, as before, the unit vectors are as shown in Figure 11.12.4.
The kinetic energy of the system is then:

K = ( ) ( ) +( ) ( ) 2
2
ωω
G
1 B
v
12 I
12 m
1
G
1
v ( ) +( ) ( ) 2
2
G
B
12
+( )m 2 12 I G 2 ωω 2
)
2 ˙
= ( ) (l 2 ) 4 θ 2 ˙ 1 +( )(1 12 ml θ l 2 1
1 2
12 m
[
2 ˙
2 ˙ ˙
+( )m l θ 1 2 + l θ θ 2 cos θ − ) + lθ 1 ( 2 4 4)θ 2 ˙ 2 (11.12.13)
12
1
( 2
+( 12) ( 1 12)ml 2 ˙ θ 2
m
2
= ( 23)ml 2 ˙ θ 1 ( 1 2)ml 2 ˙ ˙ 2 cos 2 (θ − θ 1)
θ θ
+
2
1
+( 16)ml 2 ˙ θ 2 2
Then, from Eq. (11.12.5), the generalized inertia forces F * and F * are:
θ 1 θ 2
θ cos(
θ −(
F =−( 4 3)ml 2 ˙˙ 12)ml 2 ˙˙ θ − ) +( 12)ml 2 ˙ θ sin( θ − ) (11.12.14)
θ
θ
*
2
θ 1 1 2 2 1 2 2 1
and
θ −(
F =−( 13)ml 2 ˙˙ 12)ml 2 ˙˙ θ cos( θ − ) −( 12)ml 2 ˙ θ sin( θ − ) (11.12.15)
θ
θ
2
*
θ 2 2 1 2 1 1 2 1
These results are identical with the Eqs. (11.10.21) and (11.10.22).