Page 493 - Dynamics of Mechanical Systems

0593_C13_fm Page 474 Monday, May 6, 2002 3:21 PM

474 Dynamics of Mechanical Systems

angles θ and θ as shown. Determine the governing equation of motion assuming θ and
1
1
2
θ are sufficiently small that all nonlinear terms in θ , θ , θ ˙ 1 , and θ ˙ 2 may be neglected.
1
2
2
P13.7.9: See Problem P13.7.8. Determine the natural frequencies of vibration for the small-
displacement, double-rod pendulum.
P13.7.10: Repeat Problems P13.7.8 and P13.7.9 for the triple-rod pendulum of Section 8.11
and as shown in Figure P13.7.10. As before, let the rods be identical with each having
length and mass m. Let the joint pins be frictionless. Let the system move in a vertical
plane with the rod orientation being defined by the angles θ , θ , and θ as shown.
3
1
2
θ
1

θ
2

FIGURE P13.7.10 θ 3
A triple-rod pendulum.

Section 13.8 Modes of Vibration
P13.8.1: Consider a vibrating system with two degrees of freedom governed by the fol-
lowing equations:
mx ˙˙ +( k + ) k x = 0
k x −
11 1 2 1 2 2
mx ˙˙ +( k + ) k x = 0
k x −
22 2 3 2 2 1
where m and m are masses; k , k , and k are spring stiffness; and x and x are displace-
1
3
1
2
2
1
2
ments. Determine the natural frequencies for this system.
P13.8.2: See Problem P13.8.1. Let m , m , k , k , and k have the following values: m = 2 kg,
2
3
1
2
1
1
m = 3 kg, k = 200 N/m, k = 150 N/m, and k = 250 N/m. Determine the natural
2
2
3
1
frequencies and the modes of vibration.
P13.8.3: See Problem P13.7.4. Determine the modes of vibration for the system governed
by the equation:
3˙˙ x + 2x − x = 0
1 1 2
3˙˙ x − x + 2x − x = 0
2 1 2 3
3˙˙ x − x + 2x = 0
3 2 3
P13.8.4: See Problem P13.7.5. Determine the modes of vibration for the system governed
by the equations:
4˙˙ x + 3x − 2x = 0
1 1 2
4˙˙ x − 2x + 3x − 2x = 0
2 1 2 3
4˙˙ x − 2x + 3x = 0
3 2 3