A two-harmonic homotopy method to connect a primary resonance to its secondary resonances

Source abstract

Besides the well-known primary resonances, forced nonlinear systems can exhibit secondary (namely, superharmonic, subharmonic and ultrasubharmonic) resonances the frequencies of which are rationally related to the forcing frequency. Some of these secondary resonances can appear as isolated branches of solutions, challenging their characterization. This work leverages two-harmonic forcing to transition from a primary resonance to a specific secondary resonance. A homotopy problem is formulated, the limit cases of which correspond to these resonances. The proposed method is shown to be able to uncover isolated responses in a deterministic and reliable way. It is illustrated on a Duffing oscillator, a two-degree-of-freedom system and a beam with contact.