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0593_C11_fm Page 381 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 381

n
θ z O
G G
θ 1
B 1 B
n 1
θ
Q G
θ 2
2
n B 2
r
FIGURE 11.10.2 FIGURE 11.10.3
A rod pendulum. A double-rod pendulum.
From Eq. (11.9.6) we find the generalized inertia force to be:

*
F = v G θ ˙ ⋅F * + ωω θ ˙ ⋅T *
θ
=− m(l 4 θ ) − m(l 12 θ ) ˙˙
˙˙
2
2
or
F =− m(l 3 θ ) ˙˙ (11.10.11)
2
*
θ
Example 11.10.3: Double-Rod Pendulum
As an extension of the foregoing example, consider the double-rod pendulum as in Figure
11.10.3. Let the rods, B and B , be identical, each having mass m and length . Let the
2
1
rods be pinned together at Q and supported at O by frictionless pins such that the system
is free to move in a vertical plane as depicted in Figure 11.10.3. The system then has two
degrees of freedom as represented by the angles θ and θ as shown.
1
2
The velocities and accelerations of the mass centers G and G and the angular velocities
2
1
and angular accelerations of the rods themselves are:
v G 1 = (l 2)θ ˙ n , v G 2 = lθ ˙ n +(l 2)θ ˙ n
1 1θ 1 1θ 2 2θ
˙˙
G
a = (l 2)θ n −(l 2)θ 2 ˙ n (11.10.12)
1 1 1θ 1 r 1
˙˙
˙˙
n − lθ
a G 2 = lθ 11θ 2 ˙ 1 n +(l 2)θ 2 n −(l 2)θ 2 ˙ 2 n r 2
2θ
r 1
˙
˙
ωω = θ n , ωω = θ n
B 2
B 1
1 3 2 3
˙˙
˙˙
αα = θ n , αα = θ n
B 1
B 2
1 3 2 3
where the unit vectors are shown in Figure 11.10.4. The unit vectors are related to one
another by the expressions:
n = cos θ 2 ( − θ n ) − sin 2 (θ − θ n ) = cosθ n + sinθ n
1r 1 2r 1 2θ 1 1 1 2
(11.10.13)
n = sin θ 2 ( − θ 1 n ) 2r + cos 2 (θ − θ 1 n ) 2θ =−sinθ 1 1 1 n 2
n +cosθ
1θ