Modeling and Analytical Approaches

Introduction

The analytical modeling of vibro-impact systems is very crucial in predicting their dynamical behavior. The system can be linear or weakly nonlinear in the absence of impact. However, in the presence of impact, or friction, or both, it becomes “strongly” nonlinear. This strong nonlinearity owes it origin to the fact that the velocity before and after impact experiences a sudden change in its direction, and thus resulting in what is known as “non-smooth dynamics.” Three particular techniques have been developed over the years in order to transform the non-smooth models into smooth ones. These include the power-law phenomenological modeling, the Zhuravlev and Ivanov non-smooth coordinate transformations, and the Hertzian contact law. Vibro-impact dynamics of linear systems, known as piecewise linear systems, have been treated in the literature using point-wise mapping. This approach solves the linear differential equation in two stages. The initial conditions of each stage are taken as the values of the solution of the previous stage at the end of its period. Other techniques include the saw-tooth-time transformation and the Lie group transformation. The basic principles of these approaches are presented in this chapter. The analysis of different models of vibro-impact systems was considered by Babitsky and Krupenin ([59], [61]). Analytical models approximating purely elastic and inelastic impact using smooth functions have been reviewed by Manevich and Gendelman [636].