An elliptic averaging method for harmonically excited oscillators with a purely non-linear non-negative real-power restoring force

Harmonically excited oscillators with a purely non-linear non-negative real-power restoring force are considered in this paper. The solution for motion is assumed in the form of a Jacobi cn elliptic function, the frequency and the parameter of which are obtained from the exact period of motion calculated from the energy conservation law. The parameter of the elliptic function is found to be negative for under-linear restoring forces and positive for over-linear restoring forces. By taking into account the time variation of the parameter of the elliptic function, a new elliptic averaging method is developed. The use of the equations derived is illustrated on the examples of oscillators with van der Pol damping and linear viscous damping with various integer and non-integer powers of the restoring force. New insights into dynamics of these oscillators are engendered. Numerical confirmations of analytical results are provided.

► Forced oscillators with power-form geometric nonlinearity are considered.
► A solution for motion is assumed in the form of a Jacobi elliptic function.
► A new elliptic averaging method is developed.
► General differential equations for the amplitude and phase shift are derived.
► New insight into dynamics of van der Pol and viscously damped oscillators is engendered.