An alternate state-space algorithm for dynamic solution, sensitivity analysis and parameter identification of dry friction systems

Abstract

Dry friction is one of the most complicated nonlinear phenomena in mechanical systems, due to its discontinuous stick–slip property. This property certainly brings difficulty to the solution and identification of dry friction systems. To overcome the difficulty, an alternate state-space algorithm is proposed in this paper to transform the dry friction system into a piecewise linear system. In doing so, the stick–slip process is firstly modeled by a zero–one representation, with which the stick–slip transition conditions are deduced. Then, the transition points are detected by the transition conditions to break the system into several pure stick or slip phases; each phase corresponds to a linear system. Dynamic solution of the piecewise linear system is quickly acquired by precise time step integration over an arbitrary pure stick or slip phase and the sensitivity analysis, pertaining also to a piecewise linear equation, can proceed exactly in a similar way. As is noteworthy, the sensitivity may admit jumps at the transition points due to the discontinuity of the dry friction system and therefore, the affine relation between the sensitivity before and after a transition point is elaborately derived. Eventually, based on the acquired sensitivity and the Tikhonov regularization, a sensitivity approach is developed to inversely identify the parameters of the dry friction system. Numerical examples and an experimental case are studied to demonstrate the accuracy and efficiency of the proposed alternate state-space algorithm.