Page 279 - Dynamics of Mechanical Systems

0593_C08_fm Page 260 Monday, May 6, 2002 2:45 PM

260 Dynamics of Mechanical Systems

Setting moments of these force systems about O and Q equal to zero leads to the equations:

l
l
−( ) 2 n ×− ( mg n ) −( ) 2 n × F * + T *
12 2 12 1 1
[ +( ) ] ( )
− ln 12 l 2 n 22 ×−mg n 2 (8.10.9)
[ +( ) ] * *
− ln 12 l 2 n 22 × F 2 + T 2 = 0
and

l
l
−( ) 2 n ×− ( mg )n −( ) 2 n × F * + T * = 0 (8.10.10)
22 2 22 2 2
By substituting from Eq. (8.10.4) to (8.10.8), these equations become (after simplification):
( 4 3)θ 1 +( 12) cosθ 1 2 (θ − θ 1) +( 13)θ 2 +( 12)θ 2 cos 2 (θ − θ 1)
˙˙
˙˙
˙˙
˙˙
2 )sinθ
sin
−( 12)θ 2 ˙ 2 (θ 2 − θ 1) +( 12)θ 1 2 ˙ sin 2 (θ − θ 1) + 3 ( g l 1 (8.10.11)
+(g 2 )sinθl = 0
2
and

( 12) (θ 2 − θ 1)θ 1 +( 13)θ 2 −( 12)θ 1 2 ˙ sin 1 (θ − θ 2) +(g 2 )sinθ 2 = 0 (8.10.12)
˙˙
˙˙
l
cos
Observe that all of the terms of Eq. (8.10.12) are also contained in Eq. (8.10.11). Therefore,
by eliminating these terms from Eq. (8.10.11), as their sum is zero, Eq. (8.10.11) may be
written as:

( 4 3)θ 1 +( 12)θ 2 cos 2 (θ − θ 1) −( 12)θ 2 (θ 2 − θ 1) + 3 ( g 2 )sinθ 1 = 0 (8.10.13)
˙˙
2 ˙
˙˙
l
sin
Equations (8.10.12) and (8.10.13) are the desired governing equations for the two-rod
system. They form a system of coupled ordinary differential equations. We will explore
the solutions of such systems in Chapter 12.

8.11 The Triple-Rod and N-Rod Pendulums

As a further extension of the pendulum problems consider the three-rod and N-rod
systems shown in Figures 8.11.1 and 8.11.2. In principle, these systems may be studied by
following the same procedures as in the foregoing problems: that is, we can study the
kinematics and from that develop expressions for the inertia forces. Then, by considering
the applied forces, we can construct free-body diagrams. Finally, by using d’Alembert’s
principle we can assert that the systems of applied and inertia forces are zero systems and
thus obtain the governing equations of motion.