Page 458 - Dynamics of Mechanical Systems

0593_C13_fm Page 439 Monday, May 6, 2002 3:21 PM

13

Introduction to Vibrations

13.1 Introduction
Vibration is sometimes defined as periodic (repeating) oscillatory movement. It occurs in
virtually all mechanical systems. Vibration produces noise, unwanted wear, and even
catastrophic failure. On the other hand, for many systems vibration is essential for the
proper functioning of the systems. Therefore, analysis and control of vibration are principal
problems of mechanical design.
In this chapter we will develop a brief and elementary introduction to mechanical
vibration. It is only an introduction and is not intended to replace a course or a more
intense study. The reader is referred to the references, which provide a partial listing of
the many books devoted to the subject.
We begin with a brief review of solutions to second-order ordinary differential equations.
We then consider single and multiple degree of freedom systems. We conclude with a
brief discussion of nonlinear vibrations.

13.2 Solutions of Second-Order Differential Equations

Vibration phenomena are often modeled by second-order ordinary differential equations.
Solutions of these equations provide a representation of the movement of vibrating sys-
tems; therefore, to begin our study, it is helpful to review the solution procedures of second-
order ordinary differential equations. The reader is encouraged to also independently
review these procedures. References 13.1 to 13.7 provide a sampling of the many texts
available on the subject.
We will consider first the so-called linear oscillator equation:

˙˙ x + ω 2 x = 0 (13.2.1)

where, as before, the overdot represents differentiation with respect to time, and ω is a
constant.
In Eq. (13.2.1) the time t is the independent variable and x is the dependent variable to
be determined. It is readily verified that the solution of Eq. (13.2.1) may be expressed in
the form:

+
x = Acosω t Bsinω t (13.2.2)

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