Consider a driven undamped spring/mass system described by

Chapter 5, Problem 44E

(choose chapter or problem)

Consider a driven undamped spring/mass system described by the initial-value problem

\(\frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \sin ^{n} \gamma t, \quad x(0)=0, \quad x^{\prime}(0)=0\)

(a) For n = 2, discuss why there is a single frequency \(\gamma_{1} / 2 \pi\) at which the system is in pure resonance.

(b) For n  3, discuss why there are two frequencies \(\gamma_{1} / 2 \pi\) and \(\gamma_{2} / 2 \pi\) at which the system is in pure resonance.

(c) Suppose \(\omega=1\) and \(F_{0}=1\). Use a numerical solver to obtain the graph of the solution of the initial-value problem for n = 2 and \(\gamma=\gamma_{1}\) in part (a). Obtain the

graph of the solution of the initial-value problem for n = 3 corresponding, in turn, to \(\gamma=\gamma_{1}\) and \(\gamma=\gamma_{2}\) in part (b).

Text Transcription:

fracd^2xdt^2+omega^2x=F_0\sin^ngammat,x(0)=0,x^prime(0)=0

gamma_1/2pi

gamma=gamma_1

omega=1

F_0=1