7.1 Introduction

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Vibration phenomenon, common in mechanical devices and structures [2,9], is undesirable in many

cases, such as machine tools. But this phenomenon is not always unwanted; for example, vibration is

needed in the operation of vibration screens. Thus, reducing or utilizing vibration is among the

challenging tasks that mechanical or structural engineers have to face. Vibration modeling has been used

extensively for a better understanding of vibration phenomena. The vibration modeling here implies a

process of converting an engineering vibration problem into a mathematical model, whereby the major

vibration characteristics of the original problem can be accurately predicted. The mathematical model of

vibration in its general sense consists of four components: a mass (inertia) term; a stiffness term; an

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© 2005 by Taylor & Francis Group, LLC

excitation force term; and a boundary condition term. These four terms are represented in differential

equations of motion for discrete (or, lumped-parameter) systems, or boundary value problems for

continuous systems. A damping term is included if damping effects are of concern. Depending on the

nature of the vibration problem, the complexity of the mathematical model varies from simple spring –

mass systems (see Chapter 1) to multi-degree-of-freedom (DoF) systems (see Chapter 3); from a

continuous system (see Chapter 4) for a single structural member (beam, rod, plate, or shell) to a

combined system for a built-up structure; from a linear system to a nonlinear system. The success of the

mathematical model heavily depends on whether or not the four terms mentioned before can represent

the actual vibration problem. In addition, the mathematical model must be sufficiently simplified in

order to produce an acceptable computational cost. The construction of such a representative and simple

mathematical model requires an in-depth understanding of vibration principles and techniques,

extensive experience in vibration modeling, and ingenuity in using vibration software tools.

Furthermore, it also requires sufficient knowledge of the vibration problem itself in terms of working

conditions and specifications.

Except for few special cases that promise exact and explicit analytical solutions, vibration models have

to be studied by means of approximate numerical methods such as the finite element method. The finite

element method has been very successfully used for vibration modeling for the past two decades. Its

success is attributed to the development of sophisticated software packages and the rapid growth of

computer technology.

In this chapter, several aspects of the construction of mathematical models of linear vibration

problems without damping will be addressed. The capabilities of the available software packages for

vibration analysis are listed and the basic procedure for vibration analysis is summarized. As an

illustration, an engineering example is given.

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