Identifying phase-varying periodic behaviour in conservative nonlinear systems

Nonlinear normal modes (NNMs) are a widely used tool for studying nonlinear mechanical systems. The most commonly-observed NNMs are synchronous (i.e. single-mode, in-phase and anti-phase NNMs). Additionally, asynchronous NNMs in the form of out-of-unison motion, where the underlying linear modes have a phase difference of 90◦, have also been observed. This paper extends these concepts to consider general asynchronous NNMs,where the modes exhibit a phase difference that is not necessarily equal to 90◦. A single-mass, two-degree-of-freedom model is firstly used to demonstrate that the out-of-unison NNMs evolve to general asynchronous NNMs with the breaking of the geometrically-orthogonal structure of the system. Analytical analysis further reveals that, along with the breaking of the orthogonality, the out of-unison NNM branches evolve into branches which exhibit amplitude-dependent phase relationships. These NNM branches are introduced here and termed phase-varying backbone curves. To explore this further, a model of a cable, with a support near one end, is used to demonstrate the existence of phase varying backbone curves (and corresponding general asynchronous NNMs) in a common engineering structure.