The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force

The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple scales with a new expansion parameter is presented in order to cover the cases when the nonlinearity is not necessarily small. Amplitude-frequency equations and approximate solutions for the steady-state response at the frequency of excitation are obtained and compared with numerical results, showing good agreement.

► Oscillators with power-form geometric nonlinearity and different linear terms are analyzed.
► They are under the action of harmonic excitation and an arbitrary power-form damping force.
► A perturbation approach is presented with a new expansion parameter.
► General expressions for frequency-amplitude equations are derived.
► Boundaries for the steady-state response obtained are emphasized.