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0593_C12_fm Page 417 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 417

clever in their choice of points to take moments about or in the choice of directions to add
forces. Finally, as noted before, Kane’s equations are applicable with both holonomic and
nonholonomic systems. We can illustrate the use of Kane’s equations with the examples
of the previous chapter.

Example 12.2.1: The Simple Pendulum
Consider again the simple pendulum as in Figure 12.2.1 (we considered this system in
Sections 11.5, 11.10, and 11.12). This system has one degree of freedom represented by the
angle θ. In Section 11.5, Eq. (11.5.9), we found that the generalized active force F was:
θ
F =− mglsin θ (12.2.5)
θ
In Sections 11.10 and 11.12, Eqs. (11.10.5) and (11.12.8), we found the generalized inertia
force F θ * to be:

˙˙
F =− ml 2 θ (12.2.6)
*
θ
Hence, from Kane’s equations (Eq. (12.2.1)), we have the dynamical equation of motion:

F + F = 0 (12.2.7)
*
θ
θ
or

−mglsinθ − ml θ = 0 (12.2.8)
˙˙
2
or

˙˙ g sinθ = 0 (12.2.9)
θ +( ) l

Equation (12.2.9) is identical to Eq. (8.4.4) obtained using d’Alembert’s principle.

k n z
θ θ
n G
θ
B
n
θ

n n r
r

FIGURE 12.2.1 FIGURE 12.2.2
A simple pendulum. A rod pendulum.