LEARNING OBJECTIVES FOR THIS CHAPTER

1. 4–1 To use analytical solution methods for ODEs to predict the response of first- and second-order systems to nonzero initial conditions and typical input signals.

2. 4–2 To estimate key parameters (i.e., time constant, natural frequency, and damping ratio) in system responses.

3. 4–3 To use solution methods for ODEs to derive the relationship between the complex roots of underdamped second-order systems and the natural frequency and damping ratio.

4. 4–4 To use the concept of “dominant poles” to estimate the response of higher-order systems when one or two poles dominate the system's dynamic behavior.

INTRODUCTION

In Chap. 3, the state representation of system dynamics was introduced and the derivation of state equations was shown to be a relatively simple and straightforward process. Moreover, the state models take the form of sets of first-order differential equations that can be readily solved by use of one of many available computer programs. Having these unquestionable advantages of state-variable models in mind, one might wonder whether devoting an entire chapter to the methods for solving the old-fashioned input–output model equations is justified. Despite all its limitations, the classical input–output approach still plays an important role in analysis of dynamic systems because many of the systems to be analyzed are neither very complex nor nonlinear. Such systems can be adequately described by low-order linear differential equations. Also, even in those cases in which a low-order linear model is too crude to produce an accurate solution and a computer-based method is necessary, an analytical solution of an approximate linearized input–output equation can be used to verify the computer solution.