A driven oscillator obeys with , ( ) and (hardening)

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos · Nonlinear dynamics and chaos

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

Problem

A driven oscillator obeys x¨+2βx˙+ω02x+εx3=(F0/m)cos⁡ωt\ddot x+2\beta\dot x+\omega_0^2x+\varepsilon x^3=(F_0/m)\cos\omega t with ω0=1.000 rad s−1\omega_0=1.000\,\mathrm{rad\,s^{-1}}, β=0.0040 rad s−1\beta=0.0040\,\mathrm{rad\,s^{-1}} (Q=125Q=125) and ε=1.0 s−2 m−2\varepsilon=1.0\,\mathrm{s^{-2}\,m^{-2}} (hardening). It is driven at F0/m=3.0×10−3 m s−2F_0/m=3.0\times10^{-3}\,\mathrm{m\,s^{-2}}. ((a)) Find the critical drive F0c/mF_{0\rm c}/m above which the frequency response becomes triple-valued. ((b)) Find the factor by which the given drive exceeds this critical drive, and state whether a slow frequency sweep at the given drive should show a jump and hysteresis.

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