Classical Mechanics (Kibble & Berkshire) · Problem 8.14

The energy principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · super-harmonic-resonance-duffing

Problem

Super-harmonic resonance A driven non-linear oscillator satisfies the equation x¨+9x+ϵx3=cos⁡t,\ddot{x} + 9x + \epsilon x^3 = \cos t, where ϵ\epsilon is a small parameter. Use perturbation theory to investigate the possible existence of a superharmonic resonance. Show that the zero order solution is x0=18(cos⁡t+a0cos⁡3t),x_0 = \frac{1}{8}(\cos t + a_0 \cos 3t), where the constant a0a_0 is a constant that is not known at the zero order stage. By proceeding to the first order stage, show that a0a_0 is the unique real root of the cubic equation 3a03+6a0+1=0,3a_0^3 + 6a_0 + 1 = 0, which is about −0.164-0.164. Thus, when driving the oscillator at this sub-harmonic frequency, the non-linear correction appears in the zero order solution. However, there are no infinities found in the perturbation scheme at this (or any other) stage. Plot the graph of x0(t)x_0(t) and the path of the phase point (x0(t),x0′(t))(x_0(t), x_0'(t)).

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