A well has only a quadratic force term,

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Modes, Packets and Media, Ch Driven and pumped nonlinear oscillators and the route to chaos

Problem

A well has only a quadratic force term, η=0.90 m−1s−2\eta = 0.90\,\text{m}^{-1}\text{s}^{-2}, no cubic term, natural frequency ω0=2.00 rad s−1\omega_0 = 2.00\,\text{rad s}^{-1} and damping β=0.020 s−1\beta = 0.020\,\text{s}^{-1} (so Q=50Q = 50). It is driven at Ω=ω0/2=1.00 rad s−1\Omega = \omega_0/2 = 1.00\,\text{rad s}^{-1} with F0/m=0.050 m s−2F_0/m = 0.050\,\text{m s}^{-2}. The identity cos⁡2Ωt=12(1+cos⁡2Ωt)\cos^2\Omega t = \tfrac12(1+\cos2\Omega t) shows that a fundamental response Acos⁡ΩtA\cos\Omega t, fed through the quadratic term ηx2\eta x^2, generates a second-harmonic forcing of amplitude ηA2/2\eta A^2/2 at 2Ω=ω02\Omega = \omega_0: this is the secondary resonance mechanism. ((a)) Treating the well as linear and off-resonant at the drive frequency Ω\Omega (damping negligible there because Ω\Omega is far from ω0\omega_0), estimate the fundamental response amplitude AA at Ω\Omega. ((b)) Using AA from (a), find the amplitude of the second-harmonic forcing at ω0\omega_0. ((c)) This forcing drives the well exactly at its own natural frequency ω0\omega_0, where the linear resonant response amplitude is (force amplitude)/(2βω0)/(2\beta\omega_0). Find the resulting secondary-resonance amplitude at ω0\omega_0, and compare it with AA.

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