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 rst 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 pr...

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