Marian Wiercigroch

Periodic solution finder for an impact oscillator with a drift

In this paper, an efficient semi-analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift, which explains the progression mechanism in vibro-impact systems and can be used to optimize their performance. The method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known. For example, a period may consist of the following sequential phases: (I) contact with progression, (II) contact without progression, (III) no contact and (IV) contact without progression. Using this information, a system of four piecewise linear first order differential equations is transformed to a system of non-linear algebraic equations. The method allows one to accurately predict a range of control parameters for which the best progression rates are obtained.

Engineering Applications of Non-smooth Dynamics

This chapter introduces and discusses practically important concept of non-smooth dynamical systems, which are very common in engineering applications. Mathematically, such systems can be considered as piecewise smooth and therefore their global solutions are obtained by stitching local solutions, which are easy to develop by standard methods. If a dynamical system is piecewise linear then an implicit global analytical solution can be given, however the times when non-smoothness occurs have to be determined first. This leads to a set of nonlinear algebraic equations. To illustrate the non-smooth dynamical systems and the methodology of solving them, three mechanical engineering problems were studied. Firstly, a vibro-impact system in a form of moling device was modelled and analysed to understand how the progression rates can be maximised. For this system, periodic trajectories can be reconstructed as they go through three linear subspaces (no contact, contact with progression and contact without progression), and using combination of analytical and numerical methods the optimal range of the system parameters can be identified. In the second application the influence of opening and closing of a fatigue crack on the system dynamics was investigated. Specifically, a novel apparatus to induce aperiodic loading to a specimen with a fatigue crack was studied. It was shown experimentally that fatigue life can be reduced few times if the sample is loaded aperiodically. The analysis of the developed mathematical model shown that as a crack grows linearly before reaching its critical value, the response of the system remains periodic. When its size exceeds the critical value, the system behaviour becomes chaotic and then the crack growth increases exponentially. This phenomenon can be used in structural health monitoring. The last problem comes from rotordynamics, where nonlinear interactions between the rotor and the snubber ring were studied. The influence of the preloading of the snubber ring on the system behaviour was investigated and the range of the system parameters where chaotic vibrations occur was identified. The results obtained from the developed mathematical model confronted with the experiments shown a good degree of correlation.