1.1 Introduction

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Vibrations can be analyzed both in the time domain and in the frequency domain. Free vibrations and

forced vibrations may have to be analyzed. This chapter provides the basics of the time-domain

representation and analysis of mechanical vibrations.

Free and natural vibrations occur in systems because of the presence of two modes of energy storage.

When the stored energy is repeatedly interchanged between these two forms, the resulting time response

of the system is oscillatory. In a mechanical system, natural vibrations can occur because kinetic energy

(which is manifested as velocities of mass [inertia] elements) may be converted into potential energy

(which has two basic types: elastic potential energy due to the deformation in spring-like elements, and

gravitation potential energy due to the elevation of mass elements under the Earth’s gravitational pull) and

back to kinetic energy, repetitively, during motion. An oscillatory excitation (forcing function) is able to

make a dynamic system respond with an oscillatory motion (at the same frequency as the forcing

excitation) even in the absence of two forms of energy storage. Such motions are forced responses rather

than natural or free responses.

1-1

© 2005 by Taylor & Francis Group, LLC

An analytical model of a mechanical system is a set of equations. These may be developed either by

the Newtonian approach, where Newton’s Second Law is explicitly applied to each inertia element, or

by the Lagrangian or Hamiltonian approach which is based on the concepts of energy (kinetic and

potential energies). A time-domain analytical model is a set of differential equations with respect to

the independent variable time ðtÞ: A frequency-domain model is a set of input – output transfer

functions with respect to the independent variable frequency ðvÞ: The time response will describe how

the system moves (responds) as a function of time. The frequency response will describe the way the

system moves when excited by a harmonic (sinusoidal) forcing input, and is a function of the

frequency of excitation.

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