The magnitude gain of the response of a circuit or other oscillatory system to a

Question

The magnitude gain of the response of a circuit or other oscillatory system to an input of frequency omega ("omega") is described by the function Both (the natural frequency of the system) and D (the damping factor) are positive constants. The graph of Phi is called a resonance curve, and the positive frequency omega r > 0, where Phi takes its maximum value, if it exists, is called the resonant frequency. Show that and that no resonant frequency exists otherwise (Figure 21).

Explanation / Answer

This is a very cool problem. The idea is to find out what forcing frequencies w will produce the biggest response in the steady state solution. The response comes from what you call phi(w). You can maximize the response by minimizing the denominator. Since the denominator is a square root, you can minimize the denominator by minimizing the stuff under the square root. Now you have a calc I min/max problem. Find the critical points of f(w) = (w0^2 - w^2) ^2 - 4D^2 w^2 You have no critical points if D > w0/sqrt(2) and the critical point is a minimum if 0

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