Formula of Cylindrical Spring Stiffness for Nonlinear Large Deformation and Its FEM Verification

Springs are fundamental components in mechanical systems, crucial for ensuring the safety and functionality of mechanisms. Timoshenko’s stiffness formula accounts for both bending and torsional energy effects, providing accurate results for small deformations. However, when the deformation becomes large, the spring stiffness becomes a nonlinear problem due to the changing inclination angle and radius during deformation. In this study, we derive a formula for the cylindrical spring stiffness under nonlinear large deformation by considering two assumptions: the invariability of the polar angle at any point and spring wire length during deformation. This formula incorporates the effects of inclination and radius changes on the spring wire. We analyze the stiffness of the cylindrical spring with different initial inclinations using the finite element method (FEM). FEM results were compared with those obtained from Timoshenko’s formula, Hiroyuki’s formula, and the derived formula. For small deformations, the FEM results matched well with all three formulas. However, for nonlinear large deformations, the calculated results from Timoshenko’s formula showed a discrepancy of up to 32.58% compared to the FEM results. The modified Hiroyuki formula also exhibited slightly poorer agreement with the FEM results than the formula proposed in this paper. On the other hand, our derived formula demonstrated excellent agreement with the FEM results for nonlinear large deformations. Therefore, our stiffness formula for cylindrical springs is recommended for mechanical engineering spring design applications involving nonlinear large deformations.