Three equal masses between two walls

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch Normal modes of many coupled oscillators: eigenvalues, symmetry and response

Problem

Three equal masses mm between two walls, joined by four equal springs κ\kappa, have the modes (1,2,1)(1,\sqrt2,1) at ω1=0.765 ω0\omega_1=0.765\,\omega_0, (1,0,−1)(1,0,-1) at ω2=2 ω0\omega_2=\sqrt2\,\omega_0 and (1,−2,1)(1,-\sqrt2,1) at ω3=1.848 ω0\omega_3=1.848\,\omega_0, with ω0=κ/m\omega_0=\sqrt{\kappa/m}. The end mass, mass 1, is driven by a force Fcos⁡ωtF\cos\omega t, and the damping is light enough to ignore away from the resonances. ((a)) Scale the three shapes so that anTMan=1\mathbf{a}_n^{\rm T}\mathbf{M}\mathbf{a}_n=1 and find each mode's projection anTF\mathbf{a}_n^{\rm T}\mathbf{F}. ((b)) Write the undamped response of mass 1 as a sum over modes and find the drive frequencies, as multiples of ω0\omega_0, at which mass 1 stands still. ((c)) A sensor on mass 3 replaces the one on mass 1. Decide which resonances it shows and whether its response vanishes at any drive frequency.

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