Bifurcation analysis of rotating machinery in non-stationary operations: an angular harmonic balance approach

In this paper, we introduce an approach that extends the applicability of numerical continuation methods to the study of rotating machinery where the hypothesis of constant instantaneous angular speed is relaxed. Besides having a profound impact on the system’s dynamics, in particular close to bifurcations, this choice has the practical consequence of forcing the state of the system to include angular coordinates which increase without bounds over time, thus preventing the straightforward use of typical numerical continuation methods. We describe a series of transformations, inspired by the so-called angular approach used in the field of condition monitoring, which systematically recast the system in an equivalent form, in such a way as to render periodic solutions admissible and within the grasp of the harmonic balance method. Furthermore, we detail a practical implementation of this method which reduces the number of back-and-forth transformations between angle and angle-frequency domains to a minimum, an approach which we call the Angular Harmonic Balance Method (AHBM). For the selected test case, we report numerous numerical results in order to showcase the versatility, robustness, and performance of AHBM-based continuation on the simplest possible models which remain representative of the phenomenology expected from more complex rotating machines. This includes the prediction of bifurcations during a ramp response, bifurcation tracking to determine the validity of the simplified model, and the use of continuation as an aid for the study of defects in condition monitoring.