Two identical inverted pendulums stand side by side

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Linear algebra and vector spaces

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

Problem

Two identical inverted pendulums stand side by side, each a light rod of length l=0.981 ml = 0.981\,\mathrm{m} pivoted at its foot with a bob of mass m=0.50 kgm = 0.50\,\mathrm{kg} at its top. Each bob is braced towards the upright by a horizontal spring of stiffness k=3.0 N m−1k = 3.0\,\mathrm{N\,m^{-1}} running to a fixed wall on its own side, and the two bobs are joined to each other by a horizontal spring of stiffness k′=2.5 N m−1k' = 2.5\,\mathrm{N\,m^{-1}}, all springs at their natural lengths when both rods are upright. For small tilts, gravity pushes each bob away from the upright with a horizontal force +(mg/l)x+(mg/l)x, where xx is that bob's horizontal displacement. Take g=9.81 m s−2g = 9.81\,\mathrm{m\,s^{-2}} and measure x1x_1, x2x_2 positive in the same direction. (a) Write the two equations of motion and assemble them as x¨=−Ax\ddot{\mathbf{x}} = -\mathsf{A}\mathbf{x}, giving the entries of A\mathsf{A} in s−2\mathrm{s^{-2}}. (b) With bob 2 clamped at x2=0x_2 = 0, is bob 1 on its own stable? Then decide, without finding the eigenvalues, whether the free pair is stable. (c) Find the eigenvalues and eigenvectors of A\mathsf{A}. Say which motion runs away, and find the time in which a small displacement in that motion grows by a factor ee. (d) An engineer proposes to cure the problem by fitting a much stiffer spring between the bobs. Will that work? Find the smallest wall-spring stiffness kk for which the pair is stable, with k′k' as given.

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