Two identical discs

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 discs, each of moment of inertia II, are mounted on a shaft. Each disc is joined to the fixed frame by a torsional spring with κ/I=50 s−2\kappa/I = 50\,\mathrm{s^{-2}}, and the two discs are joined to each other by a torsional spring with κ′/I=10 s−2\kappa'/I = 10\,\mathrm{s^{-2}}. Let θ1,θ2\theta_1, \theta_2 be the rotation angles of the discs from equilibrium; the equations of motion assemble into θ¨=−Aθ\ddot{\boldsymbol\theta} = -\mathsf{A}\boldsymbol\theta with A=(60−10−1060) s−2.\mathsf{A} = \begin{pmatrix}60 & -10\\ -10 & 60\end{pmatrix}\,\mathrm{s^{-2}}. (a) Write down A\mathsf{A} in terms of κ/I\kappa/I and κ′/I\kappa'/I, and confirm it matches the numbers given. (b) A motion is observed in which disc 1 rotates clockwise by an angle θ0\theta_0 while disc 2 rotates anticlockwise by the same angle θ0\theta_0, at every instant. Write this motion as an eigenvector, taking clockwise as positive. (c) Verify by substitution that this is indeed an eigenvector of A\mathsf{A}, find its eigenvalue, and state the angular frequency of this mode.

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