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FIGURE 4.2 (a) Spring-mass-damper system. (b) Free-body diagram.
To illustrate the usefulness of the Laplace transformation and the steps involved in the system analysis, reconsider the spring-mass-damper system being studied by you earlier described by Equation (3.1), which is
We wish to obtain the response, y, as a function of time. The Laplace transform of Equation (4.16) is
Left and Right Limits: Left Hand Limit: Right Hand Limit: When we have
Solving for Y(s), we obtain
The denominator polynomial q(s), when set equal to zero, is called the characteristic equation because the roots of this equation determine the character of the time response. The roots of this characteristic equation are also called the poles of the system. The roots of the numerator polynomial p(s) are called the zeros of the system; for example, At the poles, the function Y(s) becomes infinite, whereas at the zeros, the function becomes zero. The complex frequency s -plane plot of the poles and zeros graphically portrays the character of the natural transient response of the system. For a specific case, consider the system when k/M = 2 and b/M = 3. Then Equation (4.19) becomes
FIGURE 4.3 An S -plane pole and zero plot.
The poles and zeros of Y(s) are shown on the s-plane in Figure 2. Expanding Equation (4.20) in a partial fraction expansion, we obtain
where
and
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