Analytical study of the amplitude and phase resonances of a Duffing oscillator

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) and subharmonic (1:ν) resonances are carefully studied using first and higher-order averaging. When the phase lag is calculated between the k-th 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.