Many oscillations of a rigid rod

Abstract

The many oscillations of a rigid rod—pendular, bifilar, torsional, coupled, nonlinear (including pure x3), and chaotic—provide interesting opportunities for experimental and theoretical investigations in introductory, intermediate, and advanced mechanics courses. Because of the rod’s simple geometry, precise expressions for the periods of its oscillations can be calculated, allowing comparisons between theory and experiment to within 0.5%. At this level of precision, the finite diameter of the pin upon which the rod oscillates has an observable effect on the period. A derivation is given of this effect, and applied to a ring oscillating on rods of various diameters. The large-angle oscillations of the bifilar pendulum are shown to depart by a measurable amount from the large-angle oscillations of the physical pendulum. The unequal-arm bifilar pendulum couples torsional and pendular modes in a calculable way. The vertical bifilar pendulum is a double pendulum with calculable coupled oscillations for small amplitudes and chaotic oscillations for large amplitudes. Other oscillators that can be built from a rigid rod, including an x3 oscillator, are described.