# Resonant Motions of the Three-dimensional Elastic Pendulum

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm0101.html
**Parent**: https://www.maths.tcd.ie/research/papers/

## Resonant Motions of the Three-dimensional Elastic Pendulum

### Peter Lynch

#### Abstract

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.