The two-spring oscillator of the hardening worked example has

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

The two-spring oscillator of the hardening worked example has ω0=14.14 rad s−1\omega_0=14.14\,\mathrm{rad\,s^{-1}}, η=0\eta=0 and ε=6400 m−2 s−2\varepsilon=6400\,\mathrm{m^{-2}\,s^{-2}}, and swings freely with amplitude A=0.050 mA=0.050\,\mathrm{m}. ((a)) In the naive series at fixed frequency, the part of the forcing on the first correction that sits at the oscillator's own frequency is −34εA3cos⁡ω0t-\tfrac34\varepsilon A^3\cos\omega_0t. Using the resonant response (F/2ω0) tsin⁡ω0t(F/2\omega_0)\,t\sin\omega_0t to a forcing Fcos⁡ω0tF\cos\omega_0t, find the time t∗t^* at which the secular term has grown as large as the leading term Acos⁡ω0tA\cos\omega_0t, in seconds and in small-swing periods. ((b)) The Lindstedt-Poincare solution runs at ω=ω0+3εA2/8ω0\omega=\omega_0+3\varepsilon A^2/8\omega_0. Find the phase (ω−ω0)t∗(\omega-\omega_0)t^* by which Acos⁡ωtA\cos\omega t has pulled ahead of Acos⁡ω0tA\cos\omega_0t at the time t∗t^*, and say in a sentence what this tells you about the secular term.

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