achapt 12

# achapt 12 - REGULATING STEADY-STATE BEHAVIOR 12 REGULATING...

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REGULATING STEADY-STATE BEHAVIOR 12 REGULATING STEADY-STATE BEHAVIOR This section treats the regulation problem associated with a two degree-of-freedom system acted on by a persistent excitation. For simplicity we’ll assume that the persistent excitation is harmonic, although the material presented in this chapter can be extended to systems acted on by general persistent excitations. The material covered in this chapter can be extended to general persistent excitations by representing the general persistent excitation as a linear superposition of harmonic excitations. Also, we’ll assume in this section that the control force is continuously-acting. When a system is acted on by a persistent excitation, the dominant part of the response is a steady-state response, at least after the transient response has damped out. The regulation problem is essentially about changing steady- state behavior. This is clearly very different than the regulation problems treated earlier in the notes where the focus was on the transient response. MAE 461: DYNAMICS AND CONTROLS

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1. The Single Degree-of-Freedom System Consider a damped single degree-of-freedom system whose base is excited harmonically, as shown in Fig. 12 - 1. Figure 12 - 1 The equation of motion is (12 - 1) f x x c x x k x m + = ) ( ) ( 0 0 where is the base excitation and f is the regulation force. Equation (12 - 1) is rewritten as 0 x (12 - 2) f x c kx kx x c x m + + = + + 0 0 in which the right side of the equation contains the non- homogeneous terms. Assume that the system is subject to the harmonic base excitation (12 - 3) t i e X t x ω 0 0 ) ( = where X 0 is the amplitude of the base excitation. We now develop a regulator that prescribes a harmonic force f that opposes the response in such a way as to reduce the amplitude of the response. Accordingly, the regulating
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achapt 12 - REGULATING STEADY-STATE BEHAVIOR 12 REGULATING...

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