Backbone analysis for nonlinear vibrations in rotor dynamics

Backbone curves, a form of nonlinear normal mode analysis, are a well established practice for understanding the vibrations of nonlinear structures, by charting the frequency-amplitude relations for the underlying conservative system. However, in their typical form they are ineffective in tracing many of the phenomena seen in the vibrations of isotropic rotating systems, including both periodic responses related to the main resonance and isolated quasiperiodic responses. This is because in the fully conservative and isotropic system there are no mechanisms to drive the mode locking that is an essential part of these responses. However, to include mechanisms such as out of balance forcing that can induce these behaviours would reduce the generality of the analysis, and may also require knowledge of parameters that can be hard to control or measure. This work produces backbones curves with additional constraints that enforce synchronisation with the out of balance forcing and therefore act in the place of the physical causes of mode locking. These curves provide a skeleton that sits underneath the bifurcation diagrams of a wide range of nonconservative and also weakly anisotropic rotating systems, despite being calculated with just the underlying conservative and isotropic parts of the system. This allows a systematic means of exploring the complex response space of rotating systems, enabling continuation approaches to more efficiently find isolated response regions that previously required brute force simulation approaches to discover. The approach also provides some commonality to the analysis of a diverse range of responses. The method is demonstrated on an isotropic 2 degree of freedom overhung rotor with a smooth radial stiffness nonlinearity, but is shown to have relevance to harsher nonlinearity systems and weakly anisotropic systems. An experimental comparison is also given.