Departure from linear mechanical behaviour of a helical spring

We model small deviations from linear mechanical behaviour of a loaded vertically suspended helical spring. A Taylor expansion of the elasticity equations governing the axial extension of the spring is used to determine the relative magnitudes of linear and (quadratic and cubic) nonlinear terms in the force–extension relationship. This relationship is the basis for the derivation of a model for the static extension of a loaded spring, and a wave equation that models small amplitude oscillation. The models account for the natural decline in pitch angle down a suspended spring, and provide accurate fits to measurements of static extension and periods of oscillation that are not adequately represented by equations based on Hooke’s law. The static and dynamic data yield consistent estimates of the spring rate.