Three identical blocks

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Action and Symmetry, Ch Normal modes as a generalised eigenvalue problem

Problem

Three identical blocks, each of mass m=0.50 kgm=0.50\ \mathrm{kg}, sit in a line on a frictionless track. Each block is joined to its neighbour, and each end block to a fixed wall, by four identical springs of stiffness κ=200 N m−1\kappa=200\ \mathrm{N\,m^{-1}}. Let x1,x2,x3x_1,x_2,x_3 be the displacements of the three blocks from equilibrium. ((a)) Write down T\mathbf T and V\mathbf V. ((b)) Find the three normal-mode angular frequencies and mode shapes. ((c)) A force F(t)=5.0cos⁡ωdt NF(t)=5.0\cos\omega_{\rm d}t\ \mathrm N with ωd=20 rad s−1\omega_{\rm d}=20\ \mathrm{rad\,s^{-1}} is applied to the middle block only. Before calculating any amplitude, say which mode this force cannot excite, and why. Then find the steady-state amplitude of each block.

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