Wilberforce pendulum: modelling linearly damped coupled oscillations of a spring-mass system

Abstract

The Wilberforce pendulum is a coupled spring-mass system, where a mass with adjustable moment of inertia is suspended from a helical spring. Energy is converted between the translational and torsional modes, and this energy conversion is most clearly observed at resonance, which occurs when the damped natural frequencies of the two oscillation modes are equal. A theoretical model—with energy losses due to viscous damping proportional to velocity accounted for—was formulated using the Lagrangian formalism to predict the pendulum mass’ trajectory. Theoretical predictions were compared with experimental data, showing good agreement. Fourier analysis of both theoretical predictions and experimental data further corroborate the validity of our quantitative model. The dependence of oscillation features like beat frequency and maximum conversion amplitude on relevant parameters such as the initial vertical displacement, initial angular displacement and moment of inertia was also investigated and experimentally verified.