An oscillator obeys

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Oscillations and normal modes

Modes, Packets and Media, Ch The anharmonic oscillator: frequency, harmonics and drift

Problem

An oscillator obeys x¨+ω02x+ηx2=0\ddot x + \omega_0^2x + \eta x^2 = 0, with ω0=20.0 rad s−1\omega_0 = 20.0\,\mathrm{rad\,s^{-1}} and η=150 m−1 s−2\eta = 150\,\mathrm{m^{-1}\,s^{-2}}, executing a periodic motion whose fundamental has amplitude A=20.0 mmA = 20.0\,\mathrm{mm} (the coefficient of cos⁡ωt\cos\omega t in the Lindstedt-Poincare solution). ((a)) Apply the Lindstedt-Poincare method to first order: rescale time by s=ωts=\omega t, write x=Acos⁡s+x1(s)x = A\cos s + x_1(s) with x1x_1 of order A2A^2, and require that x1x_1 contain no term in cos⁡s\cos s. Derive the resulting mean-position drift and second-harmonic amplitude, and evaluate both for the numbers given. ((b)) State the turning-point displacement x(0)x(0) implied by part (a), and evaluate it. ((c)) Continuing the method one further order gives the second-order frequency correction ω2=−5η2A2/(12ω03)\omega_2 = -5\eta^2A^2/(12\omega_0^3), independent of the sign of η\eta. Evaluate the fractional frequency shift (ω−ω0)/ω0(\omega-\omega_0)/\omega_0 it produces for the given numbers.

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