Many engineering problems involve modelling of systems th a t exhibit non-smooth dynamics. It is therefore essential th at the phenomena associated with such dy namical behaviour are investigated and understood, in as much detail as possi ble. Sound knowledge about the behaviour of such systems can greatly enhance the effectiveness of their modelling, and consequently the forecasting of their be haviour. This would directly benefit their design. Vibro-impact systems are prob ably the most common examples of physical systems exhibiting non-smooth dy namics. They are encountered in many engineering problems, a representative selection of which has been presented in chapter 1.
In scientific and engineering practice, monitoring and analysis of dynamical systems is at least as im portant as their modelling. Meaningful results often have to be produced by monitoring systems whose underlying dynamics are not known in advance. Furthermore, the experimentally observable quantities may not be the ones used in their models, such as the variables of a system. Methods th at enable this kind of analysis are constantly sought by engineers and academics for the obvious benefits they carry. Moreover, the attem p t to extract the necessary information from expressions of the result of a dynamical process, instead of the process itself, is very challenging.
In this thesis, we have tried to enhance our understanding of vibro-impact sys tems through experiments and simulations of their behaviour, using single degree
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of freedom models. At the same time, we have attem pted to address certain is sues relating to the modelling and use of the impact process. The goal of these investigations was to exploit the impacts as a useful source of information for the behaviour of vibro-impact systems, and to dem onstrate the use of control for the regulation of the impact process.
Our source of experimental d ata is a harmonically forced beam, fixed at the bottom and free to oscillate laterally at the top. The beam impacts to a rigid constraint for a certain range of frequencies. The benefit of using this particu lar system is th at its vibro-impacting motion has been studied in the past, and therefore its dynamics are known. Prior knowledge of its behaviour would assist in interpreting our results.
By monitoring the time of the impacts, the dynamics of the beam would es sentially become a point process, from an analysis point of view. The dynamics of point processes is a very active field of research, due to its immediate applications to neuron activity studies. A specially constructed impact load cell was mounted at the tip of the impact stop. We used the cell to obtain time series of the force ex erted on the impact stop upon impact. These time series had the form of impulse spike trains.
We calculated the interspike intervals of the vibro-impact motion for selected forcing frequencies, using impulse spike trains thresholded in real-time. This was necessary in order to avoid recording useless d ata and to obtain enough impact timings for our results to be representative and valid. Using the interspike intervals, we calculated the correlation dimension for two periodic orbits of the system. In this type of data, the correlation dimension is directly linked to the embedding dimension, should a reconstruction be attem pted. The correlation dimension was found to be zero, which is the value we expected for a fixed point attractor.
Since the correlation dimension was zero, the embedding dimension for these data sets had to be equal to or greater than 1. We therefore reconstructed the dynamics using delay plots in IR^. This suggests th a t the dynamics of the system can essentially be reduced to a simple one-dimensional mapping, th at captures the characteristic features of the motion. In the interpretation of the results we found
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th at the probability distribution of the interspike intervals was a very convenient tool.
Encouraged by the results obtained from the impulse spike trains, we sought a more remote way of obtaining appropriate information for reconstruction of the dynamics. We used recordings of the sound of the impacts as a possible suitable source of information. As we demonstrated in the analyses, although sound carries similar d ata about the impact process, this information is much more difficult to extract from the sound times series, than in the case of the impulse spikes. Nevertheless, we successfully estimated the interspike intervals from these time series with enough accuracy to reconstruct the dynamics of the system.
Previous research, involving the same experimental system, has suggested the use of a single degree of freedom model to simulate the observed behaviour. The model treats the system as linear, away from the impact stop. Impacts are assumed to be instantaneous and they are modelled using a coefficient of restitution rule. We used d ata from the impact load cell to assess the impact durations, for which the assumption of being instantaneous is valid. We formulated this quantification with the introduction of a contact time measure. We futher showed th at for the stiff beam, used in our experiments, the instantaneous im pact assumption does not sacrifice the accuracy of the results. These findings can be potentially very useful for other research, involving similar vibro-impact systems which are treated as single degree of freedom.
During the course of our investigations we found th a t the d ata acquisition process — and essentially the thresholding of the d ata — induced artefacts in the results. We numerically simulated the data acquisition process and isolated its effects on the results, as well as their cause. This enabled us to interpret the results in a more accurate manner and to draw conclusions th a t were not affected by lim itations in our method.
In order to address the issue of noise, and especially how it affects the interspike intervals (Longtin et al. 1991), we simulated a vibro-impact system th at operates in a ‘noisy’ environment. This was done by adding levels of Gaussian noise to one of the system variables during integration. We found th a t noise affects the
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interspike intervals in qualitatively the same manner as the thresholding performed on the experimental time series. High noise levels can seriously affect an otherwise periodic process and totally disguise any indication of periodicity. We have shown th at the im pact of noise is much greater for impacting orbits of higher periodicity.
Our attention was then focused on exploiting impacting itself, via control tech niques. A self-adaptive, proportional feedback method was chosen because of its suitability for experimental systems, as well as because of its proven effectiveness in the control of other numerical and experimental systems. We showed th a t tiny carefully planned perturbations can achieve stabilisation of many different periodic impacting orbits, while the system operates in its chaotic regime. We also tested the ability of the method to switch between any of the controlled orbits at will.
Although this particular control method had not been previously used on a vibro-impact system, the results were encouraging. In chapter 1 we gave a brief account of engineering vibro-impact systems, th a t actually make use of impacting. A control scheme th a t allows precise customisation of the impacting could therefore find many applications in such systems.