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0593_C08_fm Page 261 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 261

FIGURE 8.11.1 FIGURE 8.11.2
A triple-rod pendulum. N-rod pendulum.

A little reflection, however, might suggest that, while such procedures are possible, they
require extensive analyses. A comparison of the difference in the analyses of the single-
rod and double-rod pendulums shows a dramatic increase in the analysis required for the
double-rod system. For the triple-rod system of Figure 8.11.1, the analysis required is even
more extensive. Without going into the details, we will simply present here the results
that can be obtained from the analysis. As before, we assume the system is moving in a
vertical plane, that the connecting and support pins are frictionless, and that the rods are
identical, each having length and mass m. After simplification, the governing equations
become:

( 7 3)θ 1 +( 32)θ 2 cos 1 (θ − θ 2) +( 1 2)θ 3 cos 1 (θ − θ 3) +( 32)θ 2 ˙ 2 (θ 1 − θ 2)
˙˙
˙˙
˙˙
sin
(8.11.1)
+( 12)θ 2 ˙ 3 (θ 1 − θ 3) +( 5 2)( )sinθg l 1 = 0
sin
( 32) cosθ 1 2 (θ − θ 1) +( 4 3)θ 2 ( 1 2)θ 3 cos 2 (θ − θ 3)
˙˙
˙˙
+
˙˙
(8.11.2)
sin
+( 32)θ 2 ˙ sin 2 (θ − θ 1) +( 1 2)θ 2 ˙ 3 (θ − θ 3) +( 32)( )sinθg l = 0
1 2 2
( 12) cos θ 3 ( − θ 1) +( 12)θ 2 cos 3 (θ − θ 2) +( 13)θ 3
˙˙
˙˙
˙˙
θ
1
(8.11.3)
sin
sin
+( 12)θ 2 ˙ 1 (θ 3 − θ 1) +( 12)θ 2 ˙ 2 (θ 3 − θ 2) +( 12)( )sinθg l 3 = 0
By examining the patterns in the coefficients of these equations, it can be shown that
the governing equations for the N-rod pendulum may be written in the form:
j]
N
rj [ ∑
˙˙
m θ + n θ +( ) l k sinθ = 0 ( r = 1 ,K , N) (8.11.4)
˙ 2
g
rj
j
rj
j
j=1