Physics Cup – TalTech 2021 – Problem 3

by Jaan Kalda (TalTech).

Find all non-trivial natural oscillation frequencies for a regular octagon made from eight homogeneous bars of mass m and length l. While the bars are rigid, the connectors connecting two neighboring bars are such that the angle \varphi between the bars can be changed without any friction, but a returning torque T=k(\varphi-\frac 34\pi) will appear at the joint as soon as the angle departs from its equilibrium value \frac 34\pi. Indicate how many linearly independent oscillation modes correspond to each of these frequencies. Consider only planar oscillation modes, i.e. modes by which the bars move only in the plane of the octagon.

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