Two identical flywheels

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Mathematical Methods for Physics, Ch Eigenvalue problems, diagonalisation and normal modes

Problem

Two identical flywheels, each with moment of inertia I=0.50 kg m2I=0.50\,\mathrm{kg\,m^2}, are each held by a support spring of torsional stiffness κ=8.0 N m rad−1\kappa=8.0\,\mathrm{N\,m\,rad^{-1}} and joined to each other by a much weaker torsional spring of stiffness κ′=0.40 N m rad−1\kappa'=0.40\,\mathrm{N\,m\,rad^{-1}}, arranged exactly as for a weakly coupled pair of oscillators: a support spring on each mass and a single spring joining the two. Bearings are frictionless and twists stay small enough to remain linear. Let θ1,θ2\theta_1,\theta_2 be the angular displacements. (a) Write A\mathsf A for θ¨=−Aθ\ddot{\boldsymbol\theta}=-\mathsf A\boldsymbol\theta, and give its eigenvalues, eigenvectors and angular frequencies. (b) Define normal coordinates η1,η2\eta_1,\eta_2 and give the equation each obeys. (c) At t=0t=0, flywheel 11 is twisted by θ0=0.15 rad\theta_0=0.15\,\mathrm{rad} and released from rest; flywheel 22 is undisturbed and also at rest. Find η1(0),η2(0)\eta_1(0),\eta_2(0) and hence θ1(t),θ2(t)\theta_1(t),\theta_2(t). (d) Using the sum-to-product identity for cosines, write θ1(t)\theta_1(t) and θ2(t)\theta_2(t) each as a carrier times a slowly varying envelope, and find the time at which the envelope of θ1\theta_1 first falls to zero, when flywheel 11 is all but at rest. State roughly how many oscillations of the mean frequency this represents.

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