Forced oscillators with non-linear spring: A simple analytical approach

An analytical method of finding the amplitude frequency and phase frequency characteristics of a forced oscillator with non-linear restoring force is proposed. The non-linear restoring force is simply fitted by a linear one with varying stiffness. The analytical expression of amplitude frequency characteristics predicts the jump or bifurcation points; and keeps highly consistent with the numerical results. This method can be used to both smooth and piecewise linear restoring force.