Lagrange's method applied to a Slinky

Abstract

The Lagrange method is applied to two dynamic models of a Slinky, one based on point masses and linear springs and a second where the Slinky is represented as a sequence of half hoops connected by torsion springs. For the first time, the use of Lagrange's method applied to a Slinky has produced a multi-body dynamic model that can potentially with minor modification reproduce all the interesting behaviour Slinkies are well known for; descending stairs, pseudo levitation, transmission of longitudinal and transverse waves. In this paper, the models are derived and a limited exploration of the two dynamic models’ behaviour is considered. For unforced oscillation, the point mass model and torsion spring model produce a similar amplitude and frequency. When considering forced oscillations, the point mass model has a very different spectrum of natural frequencies than the torsion spring model. The torsion spring model is considered for different forcing conditions.