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0593_C12_fm Page 435 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 435

Finally, substituting from Eqs. (12.5.6) and (12.5.9) into Lagrange’s equations in the form
of Eq. (12.3.3), we obtain the governing dynamical equations of the system:

N
rj [ ∑
˙˙
m θ + n θ +( ) l k rj] = 0 (12.5.12)
˙ 2
g
j
rj
j
j=1
where the coefficients m , n , and k are:
rj
rj
rj
[
cos θ
m = ( ) + ( 2 N − p) + ( 2 M m)] ( − θ r)
1 2 1
rj j
(12.5.13)
r ≠ j and p is the larger of i and j
rr [
m = N r +( ) +( M m)] (12.5.14)
−
13
[
sin θ
1 2 1
n =−( ) + ( 2 N − p) + ( 2 M m)] ( − θ r)
rj j
(12.5.15)
r ≠ j and p is the larger of i and j
n = 0 (12.5.16)
rr
k = 0 r ≠ j (12.5.17)
rj
rr [
k = N r +( ) +( M m)] sinθ r (12.5.18)
−
1 2

12.6 Closure
The computational and analytical advantages of Kane’s equations and Lagrange’s equa-
tions are illustrated by the examples. In each case, the effort required to obtain the
governing dynamical equations is significantly less than that with d’Alembert’s principle
or Newton’s laws. As noted earlier, the reason for the reduction in effort is that non-
working constraint forces are automatically eliminated from the analysis with Kane’s
and Lagrange’s equations; hence, an analyst can ignore such forces at the onset. Also,
with Kane’s and Lagrange’s equations, the exact same number of governing equations
are obtained as the degrees of freedom. Finally, Lagrange’s equations offer the additional
advantage of using energy functions, which makes the computation of vector accelera-
tion unnecessary. The disadvantages of Lagrange’s equations are that they are not
applicable with nonholonomic systems, and the differentiation of the energy functions
may be tedious and even unwieldy for large systems.
In the following chapters we will consider applications of these principles in vibra-
tions, stability, balancing, and in the study of mechanical components such as gears
and cams.