Derivative-free arclength control-based continuation for secondary resonances identification

Mechanical structures exhibiting nonlinear dynamical behaviors possess co-existing stable or unstable periodic solutions at given excitation parameters, with possible jumps between solutions when the parameters are varied. In addition to their fundamental resonances, they can resonate at any multiple or fraction of the excitation frequency, a phenomenon known as superharmonic or subharmonic resonance, respectively, which has received less attention so far in the technical literature.
Control-based approaches have been developed for more systematic and reliable testing of nonlinear structures. This work exploits a novel derivative-free method for experimental continuation, termed arclength control-based continuation (ACBC). An extended version of ACBC (x-ACBC) relying on a double-control strategy is proposed herein. We show that this approach can successfully identify superharmonic and subharmonic resonances. Furthermore, x-ACBC can transition effectively to isolated branches of periodic solutions by leveraging a temporarily invasive control. Numerical and experimental demonstrations of x-ACBC carried out on a Duffing oscillator serve to validate the proposed developments.