A xylophone bar of mass behaves as an undamped simple harmonic oscillator obeying

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Delta functions, distributions and generalised functions

Problem

A xylophone bar of mass m=50 gm = 50\,\mathrm{g} behaves as an undamped simple harmonic oscillator obeying mx¨+mω02x=F(t)m\ddot x + m\omega_0^2 x = F(t), with natural angular frequency ω0=754 rad s−1\omega_0 = 754\,\mathrm{rad\,s^{-1}} (natural frequency 120 Hz120\,\mathrm{Hz}). Its impulse response, for the bar at rest before t=0t=0, is h(t)=Θ(t)sin⁡(ω0t)/(mω0)h(t) = \Theta(t)\sin(\omega_0 t)/(m\omega_0). A mallet strikes it with a rectangular force of magnitude F0=45 NF_0 = 45\,\mathrm{N} acting from t=0t=0 to t=τt=\tau, and zero force outside this interval. ((a)) For τ=0.40 ms\tau = 0.40\,\mathrm{ms}, find the amplitude of the ringing for t≥τt\ge\tau, and the relative difference between it and the ideal-impulse amplitude J/(mω0)J/(m\omega_0), with J=F0τJ=F_0\tau, as a fraction of the ideal. ((b)) The bar also rings in a second mode at 3ω03\omega_0, which responds to the same force as an independent undamped oscillator. What is the shortest blow duration τ\tau that leaves this mode silent after the blow? ((c)) For that τ\tau, what fraction of the ideal-impulse amplitude F0τ/(mω0)F_0\tau/(m\omega_0) does the fundamental receive?

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