Forced vibrations of oscillators with a purely nonlinear power-form restoring force

Harmonically excited oscillators with non-negative real-power geometric nonlinearities and no linear term in the restoring force are considered. Perturbation approaches are developed for the cases of weak and strong nonlinearity. Frequency–amplitude equations are derived for an arbitrary value of the non-negative real power of the restoring force as well as analytical expressions for the steady-state response at the frequency of excitation. It is shown that the system response is of a softening type for the powers lower than unity and of a hardening type for the powers higher than unity. Frequency-response curves of the antisymmetric (constant force) oscillator, the restoring force of which has a zero power, are also discussed. Comparisons with numerical results are presented for confirmation of the analytical results obtained.