Simulated Analysis of the Friction-Governed Dynamics of the Looping Pendulum

Overview

The looping pendulum — a rigid pendulum driven through a full vertical loop — exhibits rich nonlinear behaviour governed by friction, angular momentum, and the transition between oscillatory and rotational regimes. Despite its apparent simplicity, the system produces hysteresis, period-doubling, and energy-dependent mode switching that make it an instructive testbed for computational mechanics.

Methods

We derived coupled nonlinear ordinary differential equations incorporating bearing friction, air drag, and variable driving torque. A real-time Web-VPython simulation platform was built to integrate these equations and render the pendulum state interactively, allowing parameter sweeps across friction coefficients and drive amplitudes.

Experimental validation used 240 fps motion tracking to extract angular position and velocity time series. The simulation was calibrated against these trajectories, achieving 98.5% predictive accuracy across the tested parameter range — defined as the fraction of variance in angular position explained by the model.

Key results

The model successfully captured the critical driving amplitude at which the pendulum transitions from partial oscillation to complete looping, including the friction-dependent hysteresis loop where the transition threshold differs for increasing versus decreasing drive. The Web-VPython platform enabled real-time visualisation of phase-space trajectories, making the connection between friction dissipation and stability margins accessible to students.

Publication

Simulated analysis of the friction governed dynamics of the looping pendulum
Avighna Daruka, Gyaneshwaran Gomathinayagam, and Aneesh Agarwal
Published 9 January 2026 · © 2026 European Physical Society
European Journal of Physics, Volume 47, Number 1
Citation: Eur. J. Phys. 47 015012 · DOI 10.1088/1361-6404/ae257a