Predicting subharmonic and superharmonic backbone curves in nonlinear mechanical systems

We show that conservative (or primary) backbone curves, defined as amplitude–frequency curves of conservative nonlinear normal modes, have a tight mathematical connection with all subharmonic and superharmonic (or secondary) backbone curves, defined as curves connecting peaks of forced response at higher-order resonances. Specifically, under weak periodic forcing and damping, a single primary or secondary member of a backbone curve family determines all other members of that family via a simple linear scaling relationship that we illustrate in examples. This observation provides a quick theoretical prediction for forced response peaks arising from higher-order external resonances without the need for numerical simulations or experiments.