The nonlinearity of helical springs: An energy-based approach

A general expression for the elastic potential energy of a helical spring is derived using the basic concepts of elasticity theory and geometry. Both the translational and the rotational displacement of the spring's moving ends are considered. The resulting expression is employed to derive the general relation between force and spring height, and corrections to Hooke's law are discussed. The results can be used to estimate the magnitude of nonlinear effects in realistic situations, to predict how springs unwind under load, and to determine the spring constant when spring ends are allowed to rotate freely. This energy-based approach is suitable for inclusion in advanced mechanics courses. The results are validated via measurements on steel springs using a simple experimental setup appropriate for use in physics laboratories at all levels.