Resonant phase lags of a Duffing oscillator

Abstract

This paper revisits the resonant behavior of a harmonically-forced Duffing oscillator with a specific attention to phase resonance and to its relation with amplitude resonance. To this end, the different families of resonances, namely primary (1:1), superharmonic (k:1), subharmonic (1:ν) and ultra-subharmonic (k:ν) resonances are carefully studied using first and higher-order averaging. When the phase lag is calculated between the kth harmonic of the displacement and the harmonic forcing, this study evidences that phase resonance occurs when the phase lag is equal to either π/2 (phase quadrature) or 3π/4ν.