Resonant Motions of the Three-dimensional Elastic Pendulum

Peter Lynch


The three-dimensional motion of the elastic pendulum or swinging spring is investigated in this study. The amplitude is assumed to be small, so that the perturbation approach is valid. If the Lagrangian is approximated by keeping terms up to cubic order, the system has three independent constants of motion; it is therefore completely integrable.

The linear normal modes are derived, and some special solutions are considered. For unmodulated motion, with no transfer of energy between vertical and horizontal components, elliptic-parabolic solutions are found, which generalize the solutions first found by Vitt and Gorelik. These solutions are illustrated by numerical integrations. Perturbations about conical motion are then studied, and solutions in terms of elementary functions are found.

When the ratio of the the normal mode frequencies is approximately two to one, an interesting resonance phenomenon occurs, in which energy is transferred periodically between predominantly vertical and predominantly horizontal oscillations. The motion has two distinct characteristic times, that of the oscillations and that of the resonance envelope, and a multiple time-scale analysis is found to be productive. The amplitude of the vertical component may be expressed in terms of Jacobian elliptic functions.

As the oscillations change from horizontal to vertical and back again, it is observed that each horizontal excursion is in a different direction. To study this phenomenon, it is convenient to transform the equations to rotating co-ordinates. Expressions for the precession of the swing-plane are derived. The approximate solutions are compared to numerical integrations of the exact equations, and are found to give a realistic description of the motion.

Keywords: Elastic Pendulum, Swinging Spring, Nonlinear Resonance, Precession.