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0593_C13_fm Page 475 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 475

P13.8.5: See Problem P13.7.8 and P13.7.9. Determine the modes of vibration of the double-
rod pendulum with small oscillation.
P13.8.6: See Problem P13.7.10. Determine the modes of vibration of the triple-rod pendu-
lum with small oscillation.
P13.8.7: Consider a system with three degrees of freedom whose governing equations are:

2˙˙ x + 6x − 3x = 0
1 1 2
2˙˙ x − 3x + 6x − 3x = 0
2 1 2 3
2˙˙ x − 3x + 6x = 0
3 2 3

where x , x , and x are measured in feet. Determine the natural frequencies and the modes
2
1
3
of vibration.
P13.8.8: See Problem P13.8.7. Determine expressions describing the movement of the
system of Problem P13.8.7 if initially (t = 0) the system is at rest and x , x , and x have
2
3
1
the values:
x 0 () = 0 5 ft , x 0 () =− 2 0 ft , x 0 () = 1 0 ft
.
.
.
1 2 3
P13.8.9: Repeat Problem P13.8.8 if initially x , x , and x are:
2
3
1
x 0 () = 1 2 ft , x 0 () = 2 2 ft , x 0 () = 1 2 ft
2
1
3
P13.8.10 See Problem P13.8.7. Determine expressions describing the movement of the
system of Problem P13.8.7 if initially (t = 0) x , x , and x are zero but ˙ x 1 , ˙ x 2 , and ˙ x 3 have
1
3
2
the values:
˙ x = . 05 ft sec , ˙ x = − . 10 ft sec , ˙ x = . 075 ft sec
1 2 3
Section 13.9 Nonlinear Vibrations
P13.9.1: Consider the rod pendulum depicted in Figure P13.9.1 where the rod has length
and mass m and oscillates in a vertical plane supported by a frictionless pin. Develop
equations analogous to Eqs. (13.9.24) and (13.9.25) for the large-angle oscillations of the
pendulum.

θ

FIGURE P13.9.1
A rod pendulum.