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0593_C12_fm Page 418 Monday, May 6, 2002 3:11 PM

418 Dynamics of Mechanical Systems

Example 12.2.2: The Rod Pendulum
Consider next the rod pendulum of Figure 12.2.2. (We also considered this system in
Sections 11.5, 11.10, and 11.12.) As with the simple pendulum, this system also has one
degree of freedom represented by the angle θ. In Section 11.5, Eq. (11.5.14), we found that
the generalized active force F was (neglecting the torsion spring):
θ
F =− mg( ) 2 sin θ (12.2.10)
l
θ
In Sections 11.10 and 11.12, Eqs. (11.10.11) and (11.12.11), we found the generalized inertia
force F θ * to be:

F =− m(l 3 θ ) ˙˙ (12.2.11)
2
*
θ
Hence, from Kane’s equations [Eq. (12.2.1)], we have:

F + F = 0 (12.2.12)
*
θ
θ
or
)
− ( ) 2 sinθ − (l 3 θ ˙˙ = 0 (12.2.13)
2
mg l
m
or
˙˙ 2l ) sinθ (12.2.14)
θ +(3g

Equation (12.2.14) is seen to be equivalent to Eq. (8.9.9), obtained using d’Alembert’s
principle.

Example 12.2.3: Double-Rod Pendulum
As an extension of the rod pendulum, consider again the double-rod pendulum of Figure
12.2.3 (we previously considered this system in Sections 11.10 and 11.12). This system has
two degrees of freedom as represented by the angles θ and θ . In Section 11.10, Eq.
2
1
(11.10.19), we found that the partial velocities of the mass centers G and G are:
2
1
G 1 =−(l 2) sinθ 1 n +(l 2) cosθ 1 n 2
1
v ˙ θ 1
G 2 =−l sinθ 11 cosθ 1 n 2
n + l
v ˙ θ 1
(12.2.15)
G 2 = 0
v ˙ θ 1
G 1 =−(l 2) sinθ n +(l 2) cosθ
2 1 2 n 2
v ˙ θ 2
The only applied forces contributing to the generalized active forces are the weight, or
gravity, forces. These may be represented by single vertical forces mgn passing through
1
G and G . Hence, the generalized active forces are:
1
2
⋅( ⋅( mg )
G 1 G 2
n
F = v ˙ mg ) + v ˙ n 1
1
θ 1
θ 1 θ 1