Bifurcations in impact oscillators : theoretical and experimental studies
One of the most important features of nonlinear dynamicai systems is that, as system parameters are varied, qualitative changes in the overall behaviour of the system can occur at a bifurcation . For smooth systems, the local bifurcations which occur under the change of one parameter are well understood. Non-smooth dynamical systems, which frequently arise due to the way certain physical processes are modelled, undergo bifurcations which have not been widely studied. We examine a particular type of bifurcation arising in a commonly occurring class of non-smooth dynamical system, combining theoretical and experimental results. In this thesis we are concerned with the study of the important class of dynamical system we call impact oscillators, which undergo oscillations under the influence of some forcing, and additionally can undergo impacts at rigid stops. Such systems are of interest because a large number of physical and engineering systems display behaviour which can be classified as impacting, where it is important to use a dynamical analysis to identify and thus avoid the noise, wear or failure which could be caused by repeated impacts producing unacceptably large loads. Recent interest in such systems has concentrated on the unusual bifurcational behaviour which occurs when part of an orbit begins to undergo low velocity impacts. Using analytical methods to locate particular simple steady state solutions of an impact oscillator these grazing bjfurcations are investigated. Comparisons are made between the behaviour of these special bifurcations, which arise because of the instantaneous reversal of velocity in the mathematical model of the impact process, and the standard bifurcations of smooth dynamical systems. An experimental study of an electromagnetically forced metal beam impacting against a stop is used to show that the overall qualitative behaviour displayed by a simple theoretical model is also displayed in a physical impact oscillator. Finally the theoretical studies are related to a particular problem of offshore engineering and it is shown how a very simple model can be used to explain some unusual observed behaviour.