Reduced-order modelling of parameter-dependent systems with invariant manifolds: application to Hopf bifurcations in follower force problems

TL;DR

This paper develops reduced-order models for parameter-dependent systems experiencing Hopf bifurcations, demonstrating improved accuracy in predicting limit cycle amplitudes in follower force problems like the Ziegler pendulum and Beck's column.

Contribution

It introduces a modified parametrisation method considering multiple master modes and a numerical strategy for exceptional bifurcation points, enhancing ROM accuracy near bifurcations.

Findings

ROMs accurately predict limit cycle amplitudes within a certain parameter range.

Using two master modes improves the ROM's accuracy over a single unstable mode.

Computing ROMs after the bifurcation yields better results, especially with the proposed adjustments.

Abstract

The direct parametrisation method for invariant manifolds is adjusted to consider a varying parameter. More specifically, the case of systems experiencing a Hopf bifurcation in the parameter range of interest are investigated, and the ability to predict the amplitudes of the limit cycle oscillations after the bifurcation is demonstrated. The cases of the Ziegler pendulum and Beck's column, both of which have a follower force, are considered for applications. By comparison with the eigenvalue trajectories in the conservative case, it is advocated that using two master modes to derive the ROM, instead of only considering the unstable one, should give more accurate results. Also, in the specific case where an exceptional bifurcation point is met, a numerical strategy enforcing the presence of Jordan blocks in the Jacobian matrix during the procedure, is devised. The ROMs are constructed for the Ziegler pendulum having two and three degrees of freedom, and then Beck's column is investigated, where a finite element procedure is used to space discretize the problem. The numerical results show the ability of the ROMs to correctly predict the amplitude of the limit cycles up to a certain range, and it is shown that computing the ROM after the Hopf bifurcation gives the most satisfactory results. This feature is analyzed in terms of phase space representations, and the two proposed adjustments are shown to improve the validity range of the ROMs.