Approximations to limit cycles for a nonlinear multi-degree-of-freedom system with a cubic nonlinearity through combining the harmonic balance method with perturbation techniques

This paper presents an approach to obtaining higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system with a single cubic nonlinearity based on a first approximation involving first and third harmonics obtained with the harmonic balance method. This first approximation, which is similar to one which has previously been reported in the literature, is an analytical solution, except that the frequency has to be obtained numerically from a polynomial equation of degree 16. An improved solution is then obtained in a perturbation procedure based on the refinement of the harmonic balance solution. The stability of the limit cycles obtained is then investigated using Floquet analysis.
The capability of this approach to refine the results obtained by the harmonic balance first approximation is demonstrated, by direct comparison with time domain simulation and frequency components obtained using the Discrete Fourier Transform. The particular case considered was based on an aeroelastic analysis of an all-moving control surface with a nonlinearity in the torsional degree-of-freedom of the root support, and parameters corresponding to air speed, together with linear stiffness and viscous damping of the root support were varied. It is also shown, for the cases considered, how the method can reveal further bifurcational behaviour of the system beyond the initial Hopf bifurcations which first lead to the onset of limit cycle oscillations.