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0593_C12_fm Page 427 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 427

In Section 11.12, we found the kinetic energy of the system to be (see Eq. (11.12.13)):

K = ( ) ml θ 2 +(1 m ) 2 l θ θ cos θ − )
2 ˙
θ
2 ˙ ˙
23
1 1 2 ( 2 1
(12.3.19)
2 ˙
+(16 m ) l θ 2
2
As with the single-rod pendulum, gravitational forces are the only forces doing work
on the system. In Section 11.11, we found the potential energy P of the gravitational forces
to be (see Eq. (11.11.5)):
P = mgh + mgh (12.3.20)
1 2
where h and h are the elevations of the rod mass centers G and G above an arbitrary
2
1
1
2
but fixed reference level. If, as before, we take the reference level through the support O
we find h and h to be (see Figure 12.3.3):
2
1
h =−(l 2)cosθ and h =−lcosθ −(l 2)cosθ (12.3.21)
1 1 2 1 2
where, as before, each rod has length and mass m.
Using Eqs. (12.3.19) and (12.3.20), the Lagrangian L becomes:

−
θ
2 ˙ ˙
2 ˙
L = K P = ( ) ml θ 2 +(1 m ) 2 l θ θ cos θ − )
23
1 1 2 ( 2 1
2 ˙
l
+(16 m ) l θ 2 = ( mg ) 2 cosθ (12.3.22)
2 1
+ mglcosθ +( mg ) 2 cosθ
l
1 2
Substituting L into Lagrange’s equations, Eq. (12.3.5) then produces the expressions:
( 4 3)θ 1 +( 12)θ 2 cos 2 (θ − θ 1) −( 12)θ 2 ˙ 2 sin 2 (θ − θ 1) + 3 ( g 2 )sinθ 1 = 0 (12.3.23)
˙˙
˙˙
l
( 12)cos θ 2 ( − θ 1)θ ˙˙ 1 +( 13)θ 2 −( 12)θ 2 1 sin 1 (θ − θ 2) +(g 2 ) sinθ 2 = 0 (12.3.24)
˙˙
l
Eqs. (12.3.23) and (12.3.24) are identical to Eqs. (8.10.12) and (8.10.13) obtained using
d’Alembert’s principle, and Eqs. (12.2.22) and (12.2.23) using Kane’s equations. Comparing
the three approaches it should be clear that Lagrange’s equations produce the governing
equations with the least amount of analysis effort.

Example 12.3.4: Spring-Supported Particles in a Rotating Tube
Finally, consider the system of spring-supported particles in a rotating tube as in Figure
12.3.4. We considered this system in Sections 11.7, 11.10, 11.11, 11.12, and 12.12. Recall that
the system has four degrees of freedom represented by the angle θ and the coordinates
x , x , and x as in Figure 12.3.5.
3
1
2