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Accurate Approximations for Nonlinear Vibrations

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As global issues such as climate change and overpopulation continue to grow, the role of the engineer is forced to adapt. The general population now places an emphasis not only on the performance of a mechanical system, but also the efficiency with which this can be achieved. In pushing these structures to their maximum efficiency, a number of design and engineering challenges can arise. In particular, the occurrence of geometric nonlinearities can lead to failures in the linear modelling techniques that have traditionally been used. The aim of this thesis is to increase the understanding of a number of widely-used, nonlinear methods, so that they may eventually be used with the same ease and confidence as traditional linear techniques. A key theme throughout this work is the notion that nonlinear behaviour is typically approximated in some way, rather than finding exact solutions. This is not to say that exact solutions cannot be found, but rather that the process of doing so, or the solutions themselves, can be prohibitively complicated. Across the techniques considered, there is a desire to accurately predict the frequency-amplitude relationship, whether this be for the free or forced response of the system. Analytical techniques can be used to produce insight that may be inaccessible through the use of numerical methods, though they require assumptions to be made about the structure. In this thesis, a number of these methods are compared in terms of their accuracy and their usability, so that the influence of the aforementioned assumptions can be understood. Frequency tuning is then used to bring the solutions from three prominent methods in line with one another. The Galerkin method is used to project a continuous beam model into a discrete set of modal equations, as is the traditional method for treating such a system. Motivated by microscale beam structures, an updated approach for incorporating nonlinear boundary conditions is developed. This methodology is then applied to two example structures to demonstrate the importance of this procedure in developing accurate solutions. The discussion is expanded to consider non-intrusive reduced-order modelling techniques, which are typically applied to systems developed with commercial finite element software. By instead applying these methods to an analytical nonlinear system, it is possible to compare the approximated results with exact analytical solutions. This allows a number of observations to be made regarding their application to real structures, noting a number of situations in which the static cases applied or the software itself may influence the solution accuracy.
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