Resonant phase lags of a Duffing oscillator

TL;DR

This paper investigates the phase resonance phenomena in a forced Duffing oscillator, revealing specific phase lag conditions associated with different resonance types using averaging methods.

Contribution

It provides a detailed analysis of phase resonance conditions in the Duffing oscillator, linking phase lag values to amplitude resonance behaviors.

Findings

Phase resonance occurs at phase lag {}2 or 3{}4{ u}.

Different resonance families are characterized by specific phase lag conditions.

The study clarifies the relationship between phase and amplitude resonance in nonlinear oscillators.

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:{\nu}) and ultra-subharmonic (k:{\nu}) 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 {\pi}/2 (phase quadrature) or 3{\pi}/4{\nu}.