Tautochrone in the damped cycloidal pendulum

TL;DR

This paper investigates the motion of a damped cycloidal pendulum under drag force, demonstrating that the tautochrone property persists even with damping, enabling experimental studies in viscous media.

Contribution

It shows that the tautochrone property remains valid for a damped cycloidal pendulum using Lagrangian mechanics, extending classical results to include damping effects.

Findings

Tautochrone property persists with linear damping

Dynamics analyzed using Lagrangian formulation

Potential for experimental studies in viscous fluids

Abstract

The tautochrone on a cycloid curve is usually considered without drag force. In this work, we investigate the motion of a damped cycloidal pendulum under presence of a drag force. Using the Lagrange formulation, and considering linear dependence with velocity for damping force, we found the dynamics of the system to remain tautochrone. This dictates the possibility for studying the tautochrone experimentally, e.g. the cycloidal pendulum in water or oil.