Frequency of free vibration in systems with power-law restoring force

To model the nonlinear behaviour of vibrating systems, Taylor expansion with integer powers is often used. Some systems, however, are inherently nonlinear. In the case of a non-integer real exponent, the power-law system cannot be linearised around the equilibrium position using Taylor expansion. The approach presented here provides a simple estimate of the principal frequency of free vibration in systems with power-law restoring force. Without seeking the precise mathematical form of the output waveform, we only focus on the principal frequency. The first step is the use of dimensional analysis to reduce the number of parameters. Two independent non-dimensional groups are formed and functional dependence between them is sought using numerical simulations. Once this dependence is known, the principal frequency of free vibration can be readily determined for any system properties and any initial conditions. Finally, we compare the numerical results to analytical expressions for a few restoring force exponents.