Equations of motion for the rigid and elastic double pendulum using Lagrange’s equations

Abstract

The double pendulum is a well-known system exhibiting nonlinear dynamics and chaotic behavior. This study extends the conventional rigid double pendulum by introducing elastic extensions in the links, leading to a system known as the elastic double pendulum. The mathematical model incorporates both rotational and translational motion, accounting for elastic deformations using Hooke’s Law. The governing equations are derived using Lagrangian mechanics, considering both gravitational and spring potential energy contributions. Numerical simulations are performed to compare the motion of the elastic and rigid double pendulums, highlighting differences in phase-space trajectories, energy transfer, and stability characteristics. Results demonstrate that elasticity introduces additional oscillatory components, increases system nonlinearity, and affects the overall predictability of motion. These findings provide insights into elastic multi-body dynamics and have potential applications in flexible robotic arms, soft mechanisms, and bio-inspired locomotion.