Analysis of the forced damped oscillation equation

Chapter , Problem 40

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Analysis of the forced damped oscillation equation Consider the equation my_ + cy_ + ky = F0 cos vt, which describes the motion of a forced damped oscillator. Assume all the parameters in the equation are positive. a. Explain why the solutions of the homogeneous equation decay in time. b. Show that a particular solution is yp = A sin vt + B cos vt, where A = cvF0 1cv22 + 1k - mv222 and B = 1k - mv22F0 1cv22 + 1k - mv222. c. Using the amplitude-phase form of a solution, show that yp = A sin vt + B cos vt = C*sin 1vt + w2, where C* = 2A2 + B2 and tan w = B A . d. Show that C* = F0 2c2v2 + m21v0 2 - v222 , where v0 2 = k m . e. What is the relationship between the forcing frequency v and the natural frequency v0 that produces the largest amplitude C*? Explain why this result is analogous to resonance in the case of forced undamped motion. f. Let m = c = F0 = 1 and v0 = 3. Graph the amplitude C* as a function of v. Describe how C* varies with respect to v.