2A.1 Introduction

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Many people use “transforms” without even knowing it. A transform is simply a number, variable, or

function in a different form. For example, since 102 ¼ 100, one can use the exponent (2) to represent the

number 100. Doing this for all numbers (i.e., using their exponent to the base 10) results in a “table of

logarithms.” One can perform mathematical computations using only logarithms. The logarithm

transforms all numbers into their exponential equivalents; a table of such transforms (i.e., a log table)

enables a user to quickly transform any number into its exponent, do the computations using exponents

(where a product becomes an addition and a division becomes a subtraction), and transform the result

back into the original form (i.e., by an inverse logarithm). It is seen that the computations become simpler

by using logarithms, but at the cost of the time and effort for transformation and inverse transformation.

Other common transforms include the Laplace transform, Fourier transform, and Z transform. In

particular, the Laplace transform provides a simple, algebraic way to solve (i.e., integrate) a linear

differential equation. Most functions that we use are of the form tn; sin vt; or et ; or some combination of

them. Thus, in the expression

y ¼ f ðtÞ

the function y is quite likely to be a power, a sine, or an exponential function. Also, often, we must work

with derivatives and integrals of these functions and differential equations containing these functions.

These tasks can be greatly simplified by the use of the Laplace transform.

Concepts of frequency-response analysis originate from the nature of the response of a dynamic system

to a sinusoidal (i.e., harmonic) excitation. These concepts can be generalized because the time-domain

analysis, where the independent variable is time (t), and the frequency-domain analysis, where the

independent variable is frequency (v), are linked through the Fourier transformation. Analytically, it is

more general and versatile to use the Laplace transformation, where the independent variable is the

Laplace variable (s), which is complex (nonreal). This is true because analytical Laplace transforms may

exist even for time functions that do not have “analytical” Fourier transforms. However, with compatible

definitions, the Fourier transform results can be obtained from the Laplace transform results simply by

setting s ¼ jv: In the present appendix, we will formally introduce the Laplace transformation and the

2-40 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Fourier transformation, and will illustrate how these techniques are useful in the analysis of mechatronic

systems. The preference of one domain over another will depend on such factors as the nature of the

excitation input, the type of the analytical model available, the time duration of interest, and the

quantities that need to be determined.

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