Incremental harmonic balance method with multiple time variables for dynamical systems with cubic non-linearities

The incremental harmonic balance method with multiple time variables is developed for analysis of almost periodic oscillations in multi-degree-of-freedom dynamical systems with cubic non-linearities, subjected to the external multi-tone excitation. The method is formulated to treat non-autonomous as well as autonomous dynamical systems. The almost periodic oscillations, which coexist with periodic oscillations in a rotating system model with cubic restoring force and an electromagnetic eddy-current damper are analysed. The closed form solutions based on generalized Fourier series containing two incommensurate frequencies are obtained in the case of small non-dimensional sti ness ratio. Almost periodic oscillations of a rotating system model in dependence on variable parameters are also analysed, where solutions are computed through an augmentation process including a greater number of harmonics and combination frequencies involved