A doubly clamped micro-electromechanical (MEMS) beam oscillating about its…

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

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

Problem

A doubly clamped micro-electromechanical (MEMS) beam oscillating about its equilibrium obeys the symmetric Duffing equation x¨+ω02x+εx3=0\ddot x + \omega_0^2 x + \varepsilon x^3 = 0 (stretching of the beam produces no quadratic term, so η=0\eta=0), with ω0=2π×50.0×103 rad s−1=3.1416×105 rad s−1\omega_0 = 2\pi\times50.0\times10^3\ \mathrm{rad\,s^{-1}} = 3.1416\times10^5\ \mathrm{rad\,s^{-1}} and ε=5.00×1021 m−2 s−2\varepsilon = 5.00\times10^{21}\ \mathrm{m^{-2}\,s^{-2}}. It is set oscillating freely with amplitude A=5.00×10−7 mA = 5.00\times10^{-7}\ \mathrm{m} (the coefficient of the fundamental cos⁡ωt\cos\omega t). ((a)) Using the Lindstedt-Poincare result for the Duffing equation, find the fractional frequency shift Δω/ω0\Delta\omega/\omega_0 to first order, and state whether the beam hardens or softens. ((b)) Find the amplitude of the third harmonic generated in the motion.

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