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A vibration isolator, sometimes called an isolating mount, is the device situating between the vibration source and the sensitive system preventing the transmission of undesired disturbances. The performance is measured by the force or the displacement transmissibility, both functions of frequency. A good vibration isolation system has three main properties - a low resonant peak, a large isolation range and low transmissibility at non-resonant regions. Unfortunately, these characteristics cannot be achieved simultaneously by a simple linear vibration isolation system. The thesis addresses this problem for single-degree-of-freedom (sdof) vibration isolation systems by introducing nonlinear damping and stiffness devices into the system. First, theoretical studies were carried out to rigorously reveal the benefits of the proposed nonlinear vibration isolation systems over linear ones. Next, the performance of these nonlinear systems were analysed by simulations. Then, experimental studies were conducted to verify the theoretical and simulations results. Finally, a systematic approach was developed to design the parameters of the nonlinear damping and stiffness devices in order to satisfy specific vibration isolation requirements. Many vibration isolators can be modelled as a single-degree-of-freedom mass-spring-damper system. Many researchers have attempted to enhance the vibration isolation performance by designing springs with nonlinear stiffness. Others have focused on different types of damping nonlinearities. The new vibration isolation system proposed in the thesis combines both spring and damping nonlinearities in one system to exploit the advantages of both components while avoiding their undesirable effects. The theoretical properties of this proposed nonlinear vibration isolation system were analysed rigorously using the output frequency response function (OFRF) approach, a novel and unique method recently proposed at Sheeld. The stiffness nonlinearity is already a well researched area and can readily be realised in practice. Therefore, the implementation of the proposed nonlinear vibration suppression system focused on the realisation of the nonlinear damping component using commercially available magneto-rheological (MR) dampers which provide a damping force that is dependent on a control current. With feedback control, the force-velocity relationship of an MR damper can be shaped into a designed function. This implementation has been incorporated first in a vibration isolation system by simulation, then in a physical experimental rig which has a moving mass. The simulation and experimental data not only showed the successful realisation of a damping device with a particular nonlinear damping characteristic, but also confirmed the theoretical findings on the beneficial effects of nonlinear damping on a vibration isolation application. The final part of the thesis is devoted to the practical design of the proposed vibration isolation system. Given specific transmissibility requirements at certain critical frequencies, the values of the linear parameters are first designed, then the OFRF approach is applied to determine the nonlinear parameters. This pragmatic method simplifies the design of a complicated nonlinear system, which was traditionally difficult to work with, into a step-by-step guide and, therefore, has significant potential of industrial applications. The thesis has exploited the special effects of two nonlinear components on the performance of a passive sdof vibration isolation system. With the support of theoretical, simulation and experimental studies, the newly proposed configuration has shown substantial benefits to many vibration isolation problems. The simple yet effective design and implementation has significant implications for a wide range of engineering applications such as car suspension designs and building protection against earthquakes.

Conventional methods of modelling financial data using Generalised AutoRegressive Conditional Heteroskedastic (GARCH) models make many assumptions involving the distribution of the data and the structures of the underlying mean and variance models. This research intends to develop a unified framework for estimating GARCH models using Non-Linear AutoRegressive Moving Average with eXogenous inputs (NARMAX) methodology without making many assumptions about the structures of the mean and variance models. This thesis starts with a review of financial volatility and the different models that have been used to model financial volatility. These models are collectively termed as GARCH-class models. All GARCH-class models attempt to model financial volatility of a given return series by fitting a mean model, obtaining the residuals, and then fitting a variance model using the obtained residuals. Whilst a great deal of research has been done to develop different types of linear and non-linear variance model, researchers have ignored the possibility of the mean model being non-linear in nature, and most GARCH models use a very simple constant mean model to describe the means process. In 2010, Zhao developed a NARMAX based Weighted Orthogonal Forward Regression method to identify non-linear mean models whilst assuming that the structure of the variance model is known. In this thesis, this method is extended to accommodate the case where the variance is unobservable. The working of this method is demonstrated with a simulated example. The method is also used to select and estimate the mean models of two real financial data sets and to demonstrate that a constant mean model is often inadequate and can result in inaccurate variance estimates. A new Weighted Least Squares approach for the estimation of the variance model is also developed. Since the true variance is unobservable and unknown, a linear ARCH estimate of the variance is used as a proxy for the true variance. The results of simulations to demonstrate the working of the new method are also shown. Identification of a non-linear variance model is not possible using this method, since a linear estimate of the variance is used. The thesis goes on to generate a non-linear estimate of the variance without making strong assumptions about the structure of the variance model making use of Radial Basis Function models. These have been used to create a generalised representation of linear and non-linear multivariate functions in other fields. In order to create a generalised non-linear variance estimate, a generalised RBF representation of a linear ARCH variance estimate is first created. The parameters of the obtained RBF model are optimised using Maximum Likelihood to generate a variance estimate that is much more accurate. The proposed method is tested and demonstrated in three simulations and succeeds in creating a non-linear variance estimate that is more accurate than a linear ARCH variance estimate in all the demonstrated simulations. The methods introduced in this thesis build upon the newly developed application of NARMAX methodology to modelling financial volatility using GARCH models and provides a fresh new perspective to estimating financial volatility using GARCH-class models without making many assumptions about the structures of the underlying mean and variance processes.