While nonlinear behavior in mechanical systems typically degrades the behavior and performance the devices, the presence of system nonlinearities can sometimes improve the quality of the system. A reason for avoiding nonlinearities within a device is the difficulty in controlling the device due to the effects of the nonlinearities on system behavior. However, careful analysis of nonlinear systems can allow for one to take advantage of the nonlinear behavior to improve system performance. The objective of this thesis is to exploit the use of nonlinearities to enhance system performance, specifically the sensitivity of a micromachined magnetic sensor. A device design will be presented that is similar to a prototype that has been fabricated by a student within the Electrical and Computer Engineering Department at Georgia Tech. The operating principle of the device is that changes in the orientation and the strength of an external magnetic field will result in changes in the dynamic behavior of the sensor. While previous device provided a proof of the design concept, it was unable to achieve a sensitivity that would allow for its use as a compass. Improvements in the sensitivity of the sensor are achieved through the modeling and optimization of the magnetic sensor. The optimization and redesign of the magnetic sensor will improve the quality of the device and provide another step towards sensor commercialization. A new design that incorporates the use of variable force comb drives will be proposed that will further improve the sensitivity of the device by modifying the dynamic behavior of the sensor. Another approach that is presented to exploit the nonlinear behavior of the magnetic sensor involves a frequency detection scheme that uses nonlinear vibrations to characterize sensor behavior. Some benefits of this detection technique are that it is insensitive to noise in the vibration of the sensor and is also independent of the damping present within the system. In addition, the implementation of this sensing technique can be readily applied to variety of sensors types without the redesign of a system or the addition of complex components such as vacuum packaging or signal processing electronics.

We carry out a preliminary group classification of the following family of non-linear wave equations u_tt =f(u_x)u_xx+g(u_x)+x. We first re-obtain the principal Lie algebra obtained by Ibragimov et al[3) and then construct the equivalence Lie algebra. In order to partially classify this family of wave equations, optimal systems of one-dimensional sub-algebras of the equivalence Lie algebra are constructed and in so doing, two distinct equations are obtained. We furthermore determine some invariant solutions of these equations. / Thesis (MSc. Mathematics) North-West University, Mafikeng Campus, 2003

Rotor-dynamic stability is a very important subject impacting the design, control, maintenance, and operating safety and reliability of rotary mechanical systems. As rotor-dynamic nonlinearities are significantly more prominent at higher rotary speeds, the demand for better and improved performance achievable through higher speeds has rendered the use of a linear approach for rotor-dynamic analysis both inadequate and ineffective. To establish the fundamental knowledge base necessary for addressing the need, it is essential that nonlinear rotor-dynamic responses indicative of the causes of nonlinearity, along with the bifurcated dynamic states of instability, be fully characterized. The objectives of the research are to study the various rotor-dynamic instabilities induced by crack breathing and bearing fluid film forces using a model rotor-bearing system and to investigate the applicability of the fundamental concept of instantaneous frequency for characterizing rotor-dynamic nonlinear responses. A comprehensive finite element model incorporating translational and rotational inertia, bending stiffness and gyroscopic moment is developed. The intrinsic modes extracted using the Empirical Mode Decomposition along with their instantaneous frequencies resolved using the Hilbert transform are applied to characterize the inception and progression of bifurcations suggestive of the changing rotor-dynamic state and impending instability. The dissertation presents and demonstrates an effective approach that integrates nonlinear rotor-dynamics, instantaneous time-frequency analysis, advanced notions of dynamic system diagnostics and numerical modeling applied to the detection and identification of sensitive variations indicative of a bifurcated dynamic state. All presented studies on rotor response subjected to various system configurations and ranges of parameters show good agreements with published results. Under the influence of crack opening, the rotor-bearing model system displays transitional behaviors typical of a nonlinear dynamic system, going from periodic to period-doubling, chaotic to eventual failure. When film forces are also considered, the model system demonstrates very different behaviors and failures from different settings and ranges of control parameters. As a result, a dynamic failure curve differentiating zones of stability and bifurcated instability from zones of dynamic failure is constructed and proposed as an alternative to the traditional stability chart. Observations and results such as these have important practical implications on the design and safe operation of high performance rotary machinery.

A new system identification algorithm, multiple autoregressive exogenous (ARX) inputs-based Takagi-Sugeno (TS) fuzzy model, is developed to identify nonlinear behavior of structure-magnetorheological (MR) damper systems. It integrates a set of ARX models, clustering algorithms, and weighted least squares algorithm with a TS fuzzy model. Based on a set of input-output data that is generated from building structures equipped with MR dampers, premise parameters of the ARX-TS fuzzy model are determined by clustering algorithms. Once the premise part is constructed, consequent parameters of the ARX-TS fuzzy model are optimized by the weighted least squares algorithm. To demonstrate the effectiveness of the proposed ARX-TS fuzzy model, it is applied to a three-, an eight-, a twenty-story building structures. It is demonstrated from the numerical simulation that the proposed ARX-TS fuzzy algorithm is effective to identify nonlinear behavior of seismically excited building structures equipped with MR dampers. A new semiactive nonlinear fuzzy control (SNFC) algorithm is developed through integration of multiple Lyapunov-based state feedback gains, a Kalman filter, and a converting algorithm with TS fuzzy interpolation method. First, the nonlinear ARX-TS fuzzy model is decomposed into a set of linear dynamic models that are operated in only a local linear operating region. Based on the decomposed models, multiple Lyapunov-based state feedback controllers are formulated in terms of linear matrix inequalities (LMIs) such that the structure-MR damper system is globally asymptotically stable and the performance on transient responses is guaranteed. Then, the state feedback controllers are integrated with a Kalman filter and a converting algorithm using a TS fuzzy interpolation method to construct semiactive output feedback controllers. To demonstrate the effectiveness of the proposed SNFC algorithm, it is applied to a three-, an eight-, and a twenty-story building structures. It is demonstrated from the numerical simulation that the proposed SNFC algorithm is effective to control responses of seismically excited building structures equipped with MR dampers. In addition, it is shown that the proposed SNFC system is better than a traditional optimal algorithm, H2/linear quadratic Gaussian-based semiactive control strategy.

Rotor-dynamic stability is a very important subject impacting the design, control, maintenance, and operating safety and reliability of rotary mechanical systems. As rotor-dynamic nonlinearities are significantly more prominent at higher rotary speeds, the demand for better and improved performance achievable through higher speeds has rendered the use of a linear approach for rotor-dynamic analysis both inadequate and ineffective. To establish the fundamental knowledge base necessary for addressing the need, it is essential that nonlinear rotor-dynamic responses indicative of the causes of nonlinearity, along with the bifurcated dynamic states of instability, be fully characterized. The objectives of the research are to study the various rotor-dynamic instabilities induced by crack breathing and bearing fluid film forces using a model rotor-bearing system and to investigate the applicability of the fundamental concept of instantaneous frequency for characterizing rotor-dynamic nonlinear responses. A comprehensive finite element model incorporating translational and rotational inertia, bending stiffness and gyroscopic moment is developed. The intrinsic modes extracted using the Empirical Mode Decomposition along with their instantaneous frequencies resolved using the Hilbert transform are applied to characterize the inception and progression of bifurcations suggestive of the changing rotor-dynamic state and impending instability. The dissertation presents and demonstrates an effective approach that integrates nonlinear rotor-dynamics, instantaneous time-frequency analysis, advanced notions of dynamic system diagnostics and numerical modeling applied to the detection and identification of sensitive variations indicative of a bifurcated dynamic state. All presented studies on rotor response subjected to various system configurations and ranges of parameters show good agreements with published results. Under the influence of crack opening, the rotor-bearing model system displays transitional behaviors typical of a nonlinear dynamic system, going from periodic to period-doubling, chaotic to eventual failure. When film forces are also considered, the model system demonstrates very different behaviors and failures from different settings and ranges of control parameters. As a result, a dynamic failure curve differentiating zones of stability and bifurcated instability from zones of dynamic failure is constructed and proposed as an alternative to the traditional stability chart. Observations and results such as these have important practical implications on the design and safe operation of high performance rotary machinery.

This dissertation is a continuation of the work by Zabolotskaya (Sov. Phys. Acoust. 32, 296-299 (1986)) and Wochner et al. (J. Acoust. Soc. Am. 125, 2488-2495 (2008)) on the nonlinear propagation of shear wave beams in an isotropic solid. In those works, a coupled pair of nonlinear parabolic equations was derived for the transverse components of the particle motion in a collimated shear wave beam, accounting consistently for the effects of diffraction, viscosity and nonlinearity. The nonlinearity includes a cubic nonlinear term that is equivalent to the nonlinearity present in plane shear waves, as well as a quadratic nonlinear term that is unique to diffracting beams. The purpose of this work is to investigate the quadratic nonlinear term by considering second-harmonic generation in Gaussian beams as a second-order nonlinear effect using standard perturbation theory. Since shear wave beams with translational polarizations (linear, elliptical, and circular) do not exhibit any second-order nonlinear effects, we broaden the class of source polarizations considered by including higher-order polarizations that account for stretching, shearing and rotation of the transverse plane. We find that the polarization of the second harmonic generated by the quadratic nonlinearity is not necessarily the same as the polarization of the source-frequency beam, and we are able to derive a general analytic solution for second-harmonic generation that gives explicitly the relationship between the polarization of the source-frequency beam and the polarization of the second harmonic. Additionally, we consider the focusing of shear wave beams with this broader class of source polarizations, and find that a tightly-focused, radially-polarized shear wave beam contains a highly-localized region of longitudinal motion at the focal spot. When the focal distance of the beam becomes sufficiently short, the amplitude of the longitudinal motion becomes equal to the amplitude of the transverse motion. This phenomenon has a direct analogy in the focusing properties of radially-polarized optical beams, which was investigated experimentally by Dorn et al. (Phys. Rev. Lett. 91, 233901 (2003)). / text

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The complexity of large industrial engineering systems such as chemical plants has continued to increase over the years. As a result, flexible control systems are required to handle variation in the operating conditions. In the classical approach, first the plant model should be linearized at the nominal operating point and then, a robust controller should be designed for the resulting linear system. However, the performance of a controller designed by this method deteriorates when operation deviates from the nominal point. When the distance between the operating region and the nominal operating point increases, this performance degradation may lead to instability. In the context of traditional linear control, one method to solve this problem is to consider the impact of nonlinearity as “uncertainty” around the nominal model and design a controller such that the desired performance is satisfied for all possible systems in the uncertainty set. As the size of uncertainty increases, conservatism occurs and at some point, it becomes impossible to design a controller that can provide satisfactory performance. One of the methods proposed to overcome the aforementioned shortcomings is the so-called Multiple Model approach. Using Multi-Models, local designs are performed for various operating regions and membership functions or a supervisory switching scheme is used to interpolate or switch among the controllers as the operating point moves among local regions. Since the Multiple Model method is a natural extension of the linear control method, it inherits some benefits of linear control such as simplicity of analysis and implementation. However, all these benefits are valid locally. For example, the multiple model method may be vulnerable when global stability is taken into account. The core objective of this thesis is to develop new tools to study stability of closed-loop nonlinear systems controlled by local controllers in order to improve design of multiple model control systems. For example, one of the aims of this work is to investigate how to determine the region where closed loop system is stable. A secondary objective is to study the effects of the exogenous signals on stability of such systems. / Controls

Thesis (doctoral)--Université de Rennes I, 1984. / "Série: 17. No d'ordre: 386. No de série: 57." Includes bibliographical references.