An eigenvalue method for calculation of stability and limit cycles in nonlinear systems (Q5959336)

The author considers nonlinear homogeneous systems which can be quasilinearized through the use of sinusoidal describing functions, under the assumption that higher frequencies are well damped due to low pass filtering. A numerical eigenvalue method is presented to calculate stability and limit cycles, using the same unified procedures and avoiding to solve nonlinear equations or optimization problems by parameter sweeps. The method is demonstrated for a third-order system of shimmy oscillations of an aircraft nose gear, where three different static and one dynamic nonlinearity are taken into account simultaneously. The author finds regions in parameter and amplitude space, where instability or stability, as well as stable, unstable and semi-stable limit cycles occur. The results are compared with those of numerical simulations and, for an algebraic approximation of the function describing the tire, with analytical solutions of harmonic balance equations. The calculation times for limit cycle from the eigenvalue method are in general much shorter than those from simulation, and require only a few seconds for a complete bifurcation diagram.