Limit cycle computation of self‐excited dynamic systems using nonlinear modes

Abstract

A self-excited dynamic system is able to oscillate periodically by itself. Corresponding solutions of the autonomous differential equation are called limit cycles or periodic attractors. To find these solutions, a simple approach would be brute-force search for the corresponding basins of attraction. However, grid searching might become unfeasible with increasing number of degrees of freedom. Instead, solution path continuation techniques are often used to keep computational costs low. As the continuation of solution branches and their bifurcations provides only solutions which are connected to each other, isolas and detached branches are missed out. We present a method for fast limit cycle detection of self-excited systems with isolas based on nonlinear modes. A nonlinear mode, often referred to as nonlinear normal mode, is defined as a periodic motion of the undamped and unforced mechanical system. For nonconservative systems however, e.g. with friction nonlinearity, damping cannot be neglected as it is characteristic for the oscillators nonlinear dynamics. Therefore, the Extended Periodic Motion Concept (E-PMC) was proposed recently to find periodic solutions of nonconservative nonlinear systems. In this work, the E-PMC is applied to self-excited dynamic systems in order to find periodic attractors along its nonlinear modes. Zero crossings of the nonlinear damping curve indicate autonomous solutions which can be used as starting points for single parameter continuation. Thus, solutions corresponding to the main branch and detached curves in the solution space are connected by nonlinear modes. The proposed method is applied to a frictional oscillator with cubic stiffness and proves to be robust in the search for isolated periodic solutions that are already known from literature.