Influence of small harmonic terms on eigenvalues of monodromy matrix of piecewise-linear oscillators

In this paper we consider the problem which can appear at the determination of the dynamical stability of the responses of oscillators with discontinuous or steep derivative of the restoring characteristic obtained in the frequency domain. For that purpose, a simple one degree-of-freedom system with piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet-Liapounov theorem. Confirmation of the results obtained in the frequency domain is done by comparing with the results obtained in the time domain by the method of piecing the exact solutions. Determination of the dynamical stability can be made more reliable by using the proposed plot of maximum modulus of the eigenvalues of the monodromy matrix in dependence of non-dimensional frequency and the number of harmonics included in the supposed approximate solution.