Deformation and vibration of a spiral spring

In this paper we use elastica theory to study the deformation and natural frequency of a spiral spring. We first consider a spiral spring with circular cross section. Static analysis shows that the spring undergoes planar deformation first and buckles into spatial deformation at a bifurcation point. The load–deflection curve repeats itself after two full turns. Therefore, the spring returns to its initial shape after two full turns. The planar deformation has two types of mode shapes; i.e., in-plane and out-of-plane. If one proceeds from initial shape to the bifurcation point, the natural frequency of one of the out-of-plane modes reduces to zero. This out-of-plane mode shape will be the buckling mode when planar deformation buckles into spatial deformation. Experiments are conducted on a custom made spiral spring with circular cross section. The measured deformations and natural frequencies agree with theoretical predictions very well. We next proceed to consider a spiral spring with rectangular cross section. It is found that the planar deformation is independent of the stiffness ratio between the two principal directions of the cross section. However, the spatial deformation and the bifurcation point depend on the stiffness ratio. Finally, the relation between the critical angle and the stiffness ratio is obtained via vibration method and compared with previous work of others based on equilibrium method.