On the harmonic balance with linearization for asymmetric single degree of freedom non-linear oscillators

This paper deals with analytical approximation of non-linear oscillations of conservative asymmetric single degree of freedom systems, using the method of harmonic balance with linearization. This technique which consists of linearizing the governing equations prior to harmonic balance permits us to avoid solving complicated non-linear algebraic equations. But it could be applied only to symmetric oscillations for which it proves to be very simple and effective. This restriction is due to the fact that the method requires an appropriate initial approximate solution as input. Such a solution could not be readily identified for nonsymmetric oscillations, contrary the symmetric case where the fundamental harmonic works well. For these nonsymmetric oscillations, we propose in this paper to consider an initial approximation which consists of a small bias plus the fundamental harmonic. By expanding the corresponding harmonic balance equations respectively to first and second order in the bias, we are able to easily determine the bias and thus the required initial approximate solution that yields consistent solution at higher order. We use three examples to illustrate the proposed approach and reveal its simplicity and its very good convergence.