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

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.

Figures (11)

• Figure 1: Two models of Ziegler pendulum. (a) classical 2-DOF system. (b) A 3-DOF version.
• Figure 2: Eigenvalue trajectories for the 2-DOF Ziegler pendulum. Three distinct scenarios are considered: without damping, with mass-proportional damping, and with stiffness-proportional damping. Parameter values set as $m_1=m_2=k_1=k_2=L=1$. (a) undamped system (b) mass-proportional damping with $\xi_m = 0.2$ (c)-(d) cases with stiffness-proportional damping, and $\xi_k=0.01$ or $\xi_k=0.1$. The black, solid line on the top plots indicates zero.
• Figure 3: Eigenvalue trajectories for the 2-DOF Ziegler pendulum with $L=1$ and other parameters as given by \ref{['eq:2DOFZieglerAleParam']}.
• Figure 4: Bifurcation diagrams for the 2-DOF Ziegler pendulum with $L=1$ and other parameters as given by \ref{['eq:2DOFZieglerAleParam']}. The amplitude of the limit cycle for $\theta_2$ is given as a function of $P-P_H$. $P_e$ denotes the parametrisation point, $P_c$ the point of eigenfrequencies coalescence and $P_H$ the Hopf bifurcation point. (a), (c) and (e): solutions for increasing orders when the parametrisation is computed for $P_e=P_H$. (b), (d) and (f): solutions for a fixed order 9 of the parametrisation and different values for the expansion point $P_e$. Each line corresponds to a different damping scenario. Full-order solutions computed by numerical continuation implemented in the package Matcont dhooge2004matcont in red. Dashed line for unstable solutions. Stability is reported for the full-order model only.
• Figure 5: Bifurcation diagrams for the 2-DOF Ziegler pendulum with $m_1=m_2=k_1=k_2=L=1$, $\xi_k = 0$ and $\xi_m = 0.2$. The amplitudes of the limit cycles for $\theta_2$ are shown as a function of $P-P_H$. $P_e$ denotes the parametrisation point, $P_c$ the point of eigenfrequencies coalescence and $P_H$ the Hopf bifurcation point. (a) Solutions for increasing orders when $P_e=P_H$. (b) Solutions for fixed order 9 and different expansion points.