Stiffness Model of Coiled Springs Including the Axial Asymmetry due to the Helical Geometry

Abstract

Abstract The stiffness of a short section of a coiled spring is derived by writing the elastic energy in the spring wire, considering its actual geometric form, and then applying the Castigliano theorem. The resulting linear stiffness model includes the effects of pitch, curvature of the wire and distortion due to normal and transverse forces in the wire. Further, the starting and ending points of the spring wire are taken into consideration. This stiffness model is used to derive the locally linearized stiffness matrix for a complete coiled spring that is laterally loaded. The stiffness matrix is evaluated by first calculating the non-linear deformation for a given external load and then linearizing the analysis about that equilibrium point. The deformation is calculated by summing the contributions from a number of small spring elements for which the deformations are considered to be linear. The natural frequencies of a mechanical system consisting of a rigid body and a laterally loaded spring are calculated and compared with experimental results. These experiments clearly verify the proposed theoretical stiffness matrix for the complete spring and detect the rotational asymmetry of the spring.