An Eigenvalue Method of Calculation of Stability and Limit Cycles in Nonlinear Systems

Abstract

In the calculation of periodic oscillations of nonlinear systems – so-called limit cycles – approximative and systematic engineering methods of linear system and analysis are known. The techniques, working in the frequency domain, perform a quasi-linearization of the nonlinear system, replacing nonlinearities by amplitude-dependent describing functions. Frequently, the resulting equations for the amplitude and frequency of presumed limit cycles are solved directly by a graphical procedure in a Nyquist plane or by solving the nonlinear equations or a parameter optimisation problem. In this paper, an indirect numerical approach is described which shows that, for a system of nonlinear differential equations, the eigenvalues of the quasi-linear system simply indicate all limit cycles and, additionally, yield stability regions for the linearized case. The method is applicable to systems with multiple nonlinearities which may be static or dynamics. It is demonstrated for an example of aircraft nose gear shimmy dynamics in the presence of different nonlinearitiese and the results are compared with those for simulation.