Nonlinear Dynamics

, Volume 49, Issue 3, pp 401–424 | Cite as

An analytical prediction of sliding motions along discontinuous boundary in non-smooth dynamical systems

  • Albert C. J. Luo
  • Brandon C. Gegg
Original Article


This paper presents a method for the analytical prediction of sliding motions along discontinuous boundaries in non-smooth dynamical systems. The methodology is demonstrated through investigation of a periodically forced linear oscillator with dry friction. The switching conditions for sliding motions in non-smooth dynamical systems are given. The generic mappings for the friction-induced oscillator are introduced. From the generic mappings, the corresponding criteria for the sliding motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the sliding motions is illustrated. Finally, numerical simulations of sliding motions are carried out to verify the analytical prediction. This analytical prediction provides an accurate prediction of sliding motions in non-smooth dynamical systems. The switching conditions developed in this paper are expressed by the total force of the oscillator, and the nonlinearity and linearity of the spring and viscous damping forces in the oscillator cannot change such switching conditions. Therefore, the achieved force criteria can be applied to the other dynamical systems with nonlinear friction forces processing a C 0-discontinuity.


Sliding motion Force product Non-smooth dynamical systems Sliding fragmentation 


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringSouthern Illinois University, EdwardsvilleEdwardsvilleUSA

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