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0593_C12_fm Page 425 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 425

The only forces doing work on the pendulum are gravitational. In Section 11.11, we
found (see Eqs. (11.11.5)) that the potential energy of the gravitational forces on the
pendulum bob may be expressed as:

P = mgh (12.3.8)

where h is the elevation of the bob above an arbitrary but fixed reference level. If we take
the reference level through the support O, we find h to be:

h =−lcosθ (12.3.9)

The Lagrangian L of the pendulum is then

2 ˙
L = K P = ( ) ml θ 2 + mglcosθ (12.3.10)
−
12
Substituting L into Lagrange’s equations, Eq. (12.3.5), then produces the expression:

2 ˙˙
ml θ + mglsinθ = 0 (12.3.11)

or

˙˙ g sinθ = 0 (12.3.12)
θ +( ) l

Equation (12.3.12) is seen to be identical to Eqs. (8.4.5) and (12.2.9) obtained using
d’Alembert’s principle and Kane’s equations.

Example 12.3.2: The Rod Pendulum
Consider next the rod pendulum of Figure 12.3.2 (we previously considered this system
in Sections 11.5, 11.10, 11.12, and 12.2). As with the simple pendulum, this system has one
degree of freedom represented by the angle θ.

k n
θ θ z
n
θ G
B
n

n
r n
r
FIGURE 12.3.1 FIGURE 12.3.2
A simple pendulum. A rod pendulum.