Response and Dynamical Stability of Oscillators with Discontinuous or Steep First Derivative of Restoring Characteristic

• Published in: Interdisciplinary description of complex systems Vol. 6; no. 2; p. 117
• Main Authors: Wolf, H, Banić, D, Božić, Ž
• Format: Journal Article
• Language: English
• Published: Zagreb Croatian Interdisciplinary Society 01-01-2008
• Subjects: Bifurcations; Dynamic stability; Eigenvalues; Evaluation; Excitation; Frequency dependence; Frequency ranges; Frequency response; Frequency stability; Incremental harmonic balance method; Oscillators; Time domain analysis
• ISSN: 1334-4684; 1334-4676

Response and dynamical stability of oscillators with discontinuous or steep first derivative of restoring characteristic is considered in this paper. For that purpose, a simple single-degree-of-freedom system with piecewise-linear force-displacement relationship subjected to a harmonic force excitation is analysed by the method of piecing the exact solutions (MPES) in the time domain and by the incremental harmonic balance method (IHBM) in the frequency domain. The stability of the periodic solutions obtained in the frequency domain by IHBM is estimated by the Floquet-Lyapunov theorem. Obtained frequency response characteristic is very complex and includes multi-frequency response for a single frequency excitation, jump phenomenon, multi-valued and non-periodic solutions. Determining of frequency response characteristic in the time domain by MPES is exceptionally time consuming, particularly inside the frequency ranges of co-existence of multiple stable solutions. In the frequency domain, IHBM is very efficient and very well suited for obtaining wide range frequency response characteristics, parametric studies and bifurcation analysis. On the other hand, neglecting of very small harmonic terms (which in-significantly influence the r.m.s. values of the response and are very small in comparison to other terms of the spectrum) can cause very large error in evaluation of the eigenvalues of the monodromy matrix, and so they can lead to incorrect prediction of the dynamical stability of the solution. Moreover, frequency ranges are detected inside which the procedure of evaluation of eigenvalues of the monodromy matrix does not converge with increasing the number of harmonics included in the supposed approximate solution.