Page 30 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Free

Vibration

All systems possessing mass and elasticity are capable of free vibration, or
vibration that takes place in the absence of external excitation. Of primary interest
for such a system is its natural frequency of vibration. Our objectives here are to
learn to write its equation of motion and evaluate its natural frequency, which is
mainly a function of the mass and stiffness of the system.
Damping in moderate amounts has little influence on the natural frequency
and may be neglected in its calculation. The system can then be considered to be
conservative, and the principle of conservation of energy offers another approach
to the calculation of the natural frequency. The effect of damping is mainly evident
in the diminishing of the vibration amplitude with time. Although there are many
models of damping, only those that lead to simple analytic procedures are
considered in this chapter.

2.1 VIBRATION MODEL

The basic vibration model of a simple oscillatory system consists of a mass, a
massless spring, and a damper. The mass is considered to be lumped and
measured in the SI system as kilograms. In the English system, the mass is
m = w/g lb • s^/in.
The spring supporting the mass is assumed to be of negligible mass. Its
force-deflection relationship is considered to be linear, following Hooke’s law,
F = kx, where the stiffness k is measured in newtons/meter or pounds/inch.
The viscous damping, generally represented by a dashpot, is described by a
force proportional to the velocity, or F = cx. The damping coefficient c is
measured in newtons/meter/second or pounds/inch/second.

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