Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 86594, 13 pages

Regular and chaotic motion of a bush-shaft system with tribological processes

Jan Awrejcewicz and Yuriy Pyryev

Department of Automatics and Biomechanics (K-16), Technical University of Łódź, 1/15 Stefanowskiego Street, Łódź 90-924, Poland

Received 19 July 2004; Revised 16 August 2005; Accepted 23 August 2005

Copyright © 2006 Jan Awrejcewicz and Yuriy Pyryev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The methods of both analysis and modeling of contact bush-shaft systems exhibiting heat generation and wear due to friction are presented [3–5]. From the mathematical point of view, the considered problem is reduced to the analysis of ordinary differential equations governing the change of velocities of the contacting bodies, and to the integral Volterra-type equation governing contact pressure behavior. In the case where tribological processes are neglected, thresholds of chaos are detected using bifurcation diagrams and Lyapunov exponents identification tools. In addition, analytical Mel'nikov's method is applied to predict chaos. It is shown, among the others, that tribological processes play a stabilizing role. The following theoretical background has been used in the analysis: perturbation methods, Mel'nikov's techniques [7,8], Laplace transformations, the theory of integral equations, and various variants of numerical analysis.