Abstract
The azimuthal-radial pendulum is a coupled pendulum-beam system where a pendulum and a supporting string are attached to the free end of a cantilever beam with its other end clamped. Conversion between radial (parallel to cantilever) and azimuthal (perpendicular to cantilever) modes of oscillations is observed when the radial frequency is twice the azimuthal frequency. A theoretical Lagrangian formulation of the system is presented and numerical predictions are compared to experimental data, showing good agreement. A non-linear coupling term arising from the inextensibility of the cantilever beam is shown, via simplifying assumptions that reduce the equations of motion to the Mathieu equation for parametric oscillators, which is responsible for the parametric resonance observed in the system. Experimental work involved measurements of the dynamic trajectory of the pendulum bob in the lab frame, and systematic investigation into the maximum conversion amplitude, beat frequency, and parametric resonance regime boundaries as a function of bob mass, string length, cantilever length, initial release amplitude, and other relevant parameters, shows good agreement with numerical results.
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