In the first part of this dissertation we explore the consequences of high frequency operation of Josephson junction arrays. At high frequencies these systems are no longer well modeled by Kirchhoffs laws, and new dynamical equations are derived directly from Maxwells equations. From these equations we derive a reduced set of averaged equations which greatly simplify the analysis of high frequency arrays. The averaged equations allow us to examine experimental strategies for obtaining higher power outputs from arrays. These strategies rely on resonant architectures that place the junctions near antinodes of a desired standing wave mode of the fluctuating current. Simple, heuristic rules are derived for the proper placement of junctions. The second part of the dissertation is devoted to stochastic resonance. A new theory is proposed to explain both two-state and excitable stochastic resonance. Previous theories explaining the two types of stochastic resonance yield similar results while using different analytic strategies. A constrained asymmetric rate model is derived that in one limit produces the proper result for the two-state system, while in another limit models the excitable system. The result that the constrained asymmetric rate model gives in the excitable limit is off by a factor of two, and this discrepancy is examined. Furthermore, we study the consequences of adding a colored noise source to the classic two-state model of stochastic resonance. We will find that when both white and colored noise sources are present, stochastic resonance will occur as a function of colored noise strength only if the correlation time of the colored noise source is small enough. Two theories are proposed to explain this phenomenon and both are examined in detail.

This dissertation presents three inter-related studies. Chapter 2 presents a study of scaling of crack propagation in rubber sheets. Two different scaling laws for supersonic and subsonic cracks were discovered. Experiments and numerical simulations have been conducted to investigate subsonic and supersonic cracks. The experiments are performed at 85 °C to suppress strain-induced crystallites that complicate experiments at lower temperature. Calibration experiments were performed to obtain the parameters needed to compare with a theory including viscous dissipation. Both experiments and numerical simulations support supersonic cracks, and a transition from subsonic to supersonic is discovered in the plot of experimental crack speed curves versus extension ratio for different sized samples. Both experiments and simulations show two different scaling regimes: the speed of subsonic cracks scales with the elastic energy density while the speed of supersonic cracks scales with the extension ratio. Crack openings have qualitatively different shapes in the two scaling regimes. Chapter 3 describes a theory of oscillating cracks. Oscillating cracks are not seen very widely, but observed in rubber and gels. A theory has been proposed for the onset of oscillation in gels, but the oscillation of cracks in rubber has not been explained. This study provides a theory able to describe both rubber and gels and recover the experimental phase diagram for oscillating cracks in rubber. The main new idea is that the oscillations of cracks follow from basic features of fracture mechanics and are independent of details of the crack equation of motion. From the fact that oscillations exist, one can deduce some conditions on forms that equations of motion can take. A discrete model of hydraulic fracture is mentioned in Chapter 4. Hydraulic fracturing is a stimulation treatment wherein fluids are injected into reservoirs under high pressure to generate fractures in reservoirs. In this study, a lattice-based pseduo-3D model is developed to simulate hydraulic fracturing. This mode has been validated via a comparison with the KGD model. A series of pilot simulations was systematically tested for complex geometries under more realistic operation conditions, including flexible boundary conditions, randomness in elastic properties of shales and perforations. The simulation results confirm that perforation is likely to increase the complexity of fracture networks; the results also suggest that the interference between neighboring fractures is key to fracture network formation. / text

No description available.

The framework for the Hall magnetohydrodynamic (MHD) model for plasma physics is built up from kinetic theory and used to analytically solve problems of interest in the field. The Hall MHD model describes fast magnetic reconnection processes in space and laboratory plasmas. Specifically, the magnetic reconnection process at an X-type neutral point, where current sheets form and store enormous amounts of magnetic energy which is later released as magnetic storms when the sheets break up, is investigated. The phenomena of magnetic flux pile-up driving the merging of antiparallel magnetic fields at an ion stagnation-point flow in a thin current sheet, called the Parker problem, also receives rigorous mathematical analysis.

We created an analytical reduced-order model (macromodel) for an electrically actuated circular plate with an in-plane residual stress for applications in capacitive micromachined ultrasonic transducers (CMUTs). After establishing the equations governing the plate, we discretized the system by using a Galerkin approach. The distributed-parameter equations were then reduced to a finite system of ordinary-differential equations in time. We solved these equations for the equilibrium states due to a general electric potential and determined the natural frequencies of the axisymmetric modes for the stable deflected position. As expected, the fundamental natural frequency generally decreases as the electric forcing increases, reaching a value of zero at pull-in. However, strain-hardening effects can cause the frequencies to increase with voltage. The macromodel was validated by using data from experiments and simulations performed on silicon-based microelectromechanical systems (MEMS). For example, the pull-in voltages differed by about 1% from values produced by full 3-D MEMS simulations. The macromodel was then used to investigate the response of an electrostatically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode. The method of multiple scales was used to derive a semi-analytical expression for the equilibrium amplitude of vibration. The plate was found to always transition from a hardening-type to a softening-type behavior as the DC voltage increases towards pull-in. Because the response of CMUTs is highly influenced by the boundary conditions, an updated reduced-order model was created to account for more realistic boundary conditions. The electrode was still considered to be infinitesimally thin, but the electrode was allowed to have general inner and outer radii. The updated reduced-order model was used to show how sensitive the pull-in voltage is with respect to the boundary conditions. The boundary parameters were extracted by matching the pull-in voltages from the macromodel to those from finite element method (FEM) simulations for CMUTs with varying outer and inner radii. The static behavior of the updated macromodel was validated because the pull-in voltages for the macromodel and FEM simulations were very close to each other and the extracted boundary parameters were physically realistic. A macromodel for CMUTs was then created that includes both the boundary effects and an electrode of finite thickness. Matching conditions ensured the continuity of displacements, slopes, forces, and moments from the composite to the non-composite regime of the CMUT. We attempted to validate this model with results from FEM simulations. In general, the center deflections from the macromodel fell below those from the FEM simulation, especially for relatively high residual stresses, but the first natural frequencies that accompany the deflections were very close to those from the FEM simulations. Furthermore, the forced vibration characteristics also compared well with the macromodel predictions for an experimental case in which the primary resonance curve bends to the right because the CMUT is a hardening-type system. The reduced-order model accounts for geometric nonlinear hardening, residual stresses, and boundary conditions related to the CMUT post, allows for general design variables, and is robust up to the pull-in instability. However, even more general boundary conditions need to be incorporated into the model for it to be a more effective design tool for capacitive micromachined ultrasonic transducers. / Ph. D.

No description available.

Linear vibration absorbers are a valuable tool used to suppress vibrations due to harmonic excitation in structural systems. Limited evaluation of the performance of nonlinear vibration absorbers for nonlinear structures exists in the current literature. The state of the art is extended in this work to vibration absorbers in their three major physical implementations: the mechanical vibration absorber, the inductive-resistive shunted piezoelectric vibration absorber, and the electronic vibration absorber (also denoted a positive position feedback controller). A single, consistent, physically similar model capable of examining the response of all three devices is developed. The performance of vibration absorbers attached to single-degree-of-freedom structures is next examined for performance, robustness, and stability. Perturbation techniques and numerical analysis combine to yield insight into the tuning of nonlinear vibration absorbers for both linear and nonlinear structures. The results both clarify and validate the existing literature on mechanical vibration absorbers. Several new results, including an analytical expression for the suppression region's location and bandwidth and requirements for its robust performance, are derived. Nonlinear multiple-degree-of-freedom structures are next evaluated. The theory of Nonlinear Normal Modes is extended to include consideration of modal damping, excitation, and small linear coupling, allowing estimation of vibration absorber performance. The dynamics of the N+1-degree-of-freedom system reduce to those of a two-degree-of-freedom system on a four-dimensional nonlinear modal manifold, thereby simplifying the analysis. Quantitative agreement is shown to require a higher order model which is recommended for future investigation. Finally, experimental investigation on both single and multi-degree-of-freedom systems is performed since few experiments on this topic are reported in the literature. The experimental results qualitatively verify the analytical models derived in this work. The dissertation concludes with a discussion of future work which remains to allow nonlinear vibration absorbers, in all three physical implementations, to enter the engineer's toolbox. / Ph. D.

This work investigates the dynamics and vibration in gear systems, including spur and helical gear pairs, idler gear trains, and planetary gears. The spur gear pairs are analyzed using a finite element/contact mechanics (FE/CM) model. A hybrid analytical-computational (HAC) model is proposed for nonlinear gear dynamics. The HAC predictions are compared with FE/CM results and available experimental data for validation. Chapter 2 investigates the static and dynamic tooth root strains in spur gear pairs using a finite element/contact mechanics approach. Extensive comparisons with experiments, including those from the literature and new ones, confirm that the finite element/contact mechanics formulation accurately predicts the tooth root strains. The model is then used to investigate the features of the tooth root strain curves as the gears rotate kinematically and the tooth contact conditions change. Tooth profile modifications are shown to strongly affect the shape of the strain curve. The effects of strain gage location on the shape of the static strain curves are investigated. At non-resonant speeds the dynamic tooth root strain curves have similar shapes as the static strain curves. At resonant speeds, however, the dynamic tooth root strain curves are drastically different because large amplitude vibration causes tooth contact loss. There are three types of contact loss nonlinearities: incomplete tooth contact, total contact loss, and tooth skipping, and each of these has a unique strain curve. Results show that different operating speeds with the same dynamic transmission error can have much different dynamic tooth strain. Chapters 3, 4, and 5 develops a hybrid-analytical-computational (HAC) method for nonlinear dynamic response in gear systems. Chapter 3 describes the basic assumptions and procedures of the method, and implemented the method on two-dimensional vibrations in spur gear pairs. Chapters 4 and 5 extends the method to two-dimensional multi-mesh systems and three-dimensional single-mesh systems. Chapter 3 develops a hybrid analytical-computational (HAC) model for nonlinear dynamic response in spur gear pairs. The HAC model is based on an underlying finite element code. The gear translational and rotational vibrations are calculated analytically using a lumped parameter model, while the crucial dynamic mesh force is calculated using a force-deflection function that is generated from a series of static finite element analyses before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses, and included in the force-deflection function. Elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Extensive comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including near resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy. Chapter 4 investigates the in-plane motions in multi-mesh systems, including the idler chain systems and planetary gear systems, using the HAC method that introduced in Chap. 3. The details of how to implement the HAC method into those systems are explained. The force-deflection function for each mesh is generated individually from a series of static finite element analyses before the dynamic calculations. These functions are used to calculated the dynamic mesh force in the analytical dynamic analyses. The good agreement between the FE/CM and HAC results for both the idler chain and planetary gear systems confirms the capability of the HAC model in predicting the in-plane dynamic response for multi-mesh systems. Conventional softening type contact loss nonlinearities are accurately predicted by HAC method for these multi-mesh systems. Chapter 5 investigates the three-dimensional nonlinear dynamic response in helical gear pairs. The gear translational and rotational vibrations in the three-dimensional space are calculated using an analytical model, while the force due to contact is calculated using the force-deflection. The force-deflection is generated individually from a series of static finite element analyses before the dynamic calculations. The effect of twist angle on the gear tooth contact condition and dynamic response are included. The elastic deformations of the gear teeth along the face-width direction are calculated, and validated by comparing with the FE/CM results. / Ph. D. / Gears are widely used in power transmission systems. The dynamics and vibrations of the gears causes system noise because those vibrations are transmitted to the gear housing through the supporting bearings and shafts. The tooth root strains and stresses are directly related to the system failure. These effect becomes significantly important when the system is operating near resonances that high amplitude vibrations are excited and contact loss nonlinearity occurs. We want a fast, accurate, and reliable model to analyze the nonlinear dynamics in those gear and bearing systems. This work investigates the dynamics and vibration in gear systems, including spur and helical gear pairs, idler gear chains, and planetary gears. The static and dynamic tooth root strains in spur gear pairs are studied using a finite element/contact mechanics (FE/CM) approach. Extensive comparisons with experiments, including those from the literature and new ones, validates the accuracy of the FE/CM formulation. The model is then used to investigate the features of the tooth root strain curves as the gears rotate kinematically and the tooth contact condition changes. The three types of contact loss nonlinearities are investigated and explained. A hybrid analytical-computational (HAC) method is developed for nonlinear gear dynamics. This model takes advantage of the good features of the different traditional models, and is available for fast and accurate nonlinear gear dynamic analysis. The HAC method is validated by comparing with the FE/CM results, including near resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC method is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy.

We associate learning in living systems with the shaping of the velocity vector field of a dynamical system in response to external, generally random, stimuli. We consider various approaches to implement a system that is able to adapt the whole vector field, rather than just parts of it - a drawback of the most common current learning systems: artificial neural networks. This leads us to propose the mathematical concept of self-shaping dynamical systems. To begin, there is an empty phase space with no attractors, and thus a zero velocity vector field. Upon receiving the random stimulus, the vector field deforms and eventually becomes smooth and deterministic, despite the random nature of the applied force, while the phase space develops various geometrical objects. We consider the simplest of these - gradient self-shaping systems, whose vector field is the gradient of some energy function, which under certain conditions develops into the multi-dimensional probability density distribution of the input. We explain how self-shaping systems are relevant to artificial neural networks. Firstly, we show that they can potentially perform pattern recognition tasks typically implemented by Hopfield neural networks, but without any supervision and on-line, and without developing spurious minima in the phase space. Secondly, they can reconstruct the probability density distribution of input signals, like probabilistic neural networks, but without the need for new training patterns to have to enter the network as new hardware units. We therefore regard self-shaping systems as a generalisation of the neural network concept, achieved by abandoning the "rigid units - flexible couplings'' paradigm and making the vector field fully flexible and amenable to external force. It is not clear how such systems could be implemented in hardware, and so this new concept presents an engineering challenge. It could also become an alternative paradigm for the modelling of both living and learning systems. Mathematically it is interesting to find how a self shaping system could develop non-trivial objects in the phase space such as periodic orbits or chaotic attractors. We investigate how a delayed vector field could form such objects. We show that this method produces chaos in a class systems which have very simple dynamics in the non-delayed case. We also demonstrate the coexistence of bounded and unbounded solutions dependent on the initial conditions and the value of the delay. Finally, we speculate about how such a method could be used in global optimization.

Columnar jointing is a fracture pattern, best known from locations such as the Giant's Causeway, or Fingal's Cave, in which cracks self-organize into a nearly hexagonal arrangement, leaving behind an ordered colonnade. In this thesis observations of columnar jointing are reported from both a controlled laboratory setting, and in cooled lava flows. Experiments were performed in slurries of corn starch and water, which form columnar joints when dried. This drying process is examined in detail, and it is shown how desiccation leads to the propagation of a sharp shrinkage front. In general, but with some significant exceptions, the size of columnar joints is inversely dependent on the speed of this shrinkage front during their formation. The exceptions, which include sudden jumps in column scale, show that hysteresis is also important in choosing the column scale. Novel observations of the 3D structure of joints in starch show that columnar joints do not settle down to a perfect hexagonal pattern, but rather mature into a continuously evolving dynamic pattern. This pattern is scale invariant, and the same statistical distribution of column shapes applies equally to joints in both starch and lava. Field work was performed to study columnar jointing in the basalts of the Columbia River Basalt Group and the island of Staffa, and the more heterogeneous lava flows of Southwestern British Columbia. The widths of columns and the heights of striae (chisel-like markings that record details of cooling) were examined in detail, and these length scales are shown to be inversely proportional to each other. An additional length scale, that of wavy columns, is also first reported here. Based on these measurements, empirical advective-diffusive models are developed to describe the transport of water in a drying starch-cake, and the transport of heat in a cooling lava flow. These models have only a single scaling parameter, the Péclet number, which relates the fracture front velocity times the column size to the (thermal or hydraulic) diffusivity. In both cases, the formation of columnar joints occurs at a Péclet number of about 0.2. This model explains the hundred-fold differences in scale between columnar joints in starches and lavas, and can be used as a tool for the interpretation of joint patterns in the field.