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.

The Hamiltonian of a stably stratified incompressible fluid in an internal water wave in a deep ocean is constructed. Studying the ocean internal wave field with its full dynamics is formidable (or unsolvable) so we consider a test-wave Hamiltonian to study the dynamical and statistical properties of the internal water wave field in a deep ocean. Chaos is present in the internal test-wave dynamics using actual coupling coefficients. Moreover, there exists a certain separatrix net that fills the phase space and is covered by a thin stochastic layer for a two-triad pure resonant interaction. The stochastic web implies the existence of diffusion of the Arnold type for the minimum dimension of a non-integrable autonomous system. For non-resonant case, stochastic layer is formed where the separatrix from KAM theory is disrupted. However, the stochasticity does not increase monotonically with increasing energy. Also, the problem of relaxation process is studied via microscopic Hamiltonian model of the test-wave interacting nonlinearly with ambient waves. Using the Mori projection technique, the projected trajectory of the test-wave is transformed to a form which corresponds to a generalized Langevin equation. The mean action of the test-wave grows ballistically for a short time regime, and quenches back to the normal diffusion for a intermediate time regime and regresses linearly to a state of statistical equilibrium. Applying the Nakajima-Zwanzig technique on the test-wave system, we get the generalized master equation on the test-wave system which is non-Markovian in nature. From our numerical study, the distribution of the test-wave has non-Gaussian statistics.

A fundamental question in Complexity Science is how numerous dynamic processes coordinate with each other on multiple levels of description to form a complex whole - a multiscale coordinative structure (e.g. a community of interacting people, organs, cells, molecules etc.). This dissertation includes a series of empirical, theoretical and methodological studies of rhythmic coordination between multiple agents to uncover dynamic principles underlying multiscale coordinative structures. First, a new experimental paradigm was developed for studying coordination at multiple levels of description in intermediate-sized (N = 8) ensembles of humans. Based on this paradigm, coordination dynamics in 15 ensembles was examined experimentally, where the diversity of subjects movement frequency was manipulated to induce di erent grouping behavior. Phase coordination between subjects was found to be metastable with inphase and antiphase tendencies. Higher frequency diversity led to segregation between frequency groups, reduced intragroup coordination, and dispersion of dyadic phase relations (i.e. relations at di erent levels of description). Subsequently, a model was developed, successfully capturing these observations. The model reconciles the Kuramoto and the extended Haken-Kelso-Bunz model (for large- and small-scale coordination respectively) by adding the second-order coupling from the latter to the former. The second order coupling is indispensable in capturing experimental observations and connects behavioral complexity (i.e. multistability) of coordinative structures across scales. Both the experimental and theoretical studies revealed multiagent metastable coordination as a powerful mechanism for generating complex spatiotemporal patterns. Coexistence of multiple phase relations gives rise to many topologically distinct metastable patterns with di erent degrees of complexity. Finally, a new data-analytic tool was developed to quantify complex metastable patterns based on their topological features. The recurrence of topological features revealed important structures and transitions in high-dimensional dynamic patterns that eluded its non-topological counterparts. Taken together, the work has paved the way for a deeper understanding of multiscale coordinative structures. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

We consider the flow of an ideal gas with internal friction and heat conduction in a layer between a fixed plane and an upper free boundary. We describe the top free surface as the graph of a time dependent function. This forces us to exclude breaking waves on the surface. For this and other reasons we need to confine ourselves to flow close to a motionless equilibrium state which is fairly easy to compute. The full equations of motion, in contrast to that, are quite difficult to solve. As we are close to an equilibrium, a linear system of equations can be used to approximate the behavior of the nonlinear system. Analytic, strongly continuous semigroups defined on a suitable Banach space X are used to determine the behavior of the linear problem. A strongly continuous semigroup is a family of bounded linear operators {T(t)} on X where 0 ≤ t < infinity satisfying the following conditions. 1. T(s+t)=T(s)T(t) for all s,t ≥ 0 2. T(0)=E, the identity mapping. 3. For each x ∈ X, T(t)x is continuous in t on [0,infinity). Then there exists an operator A known as the infinitesimal generator of such that T(t)=exp (tA). Thus, an analytic semigroup can be viewed as a generalization of the exponential function. Some estimates about the decay rates are derived using this theory. We then prove the existence of long term solutions for small initial values. It ought to be emphasized that the decay is not an exponential one which engenders significant difficulties in the transition to nonlinear stability.

Stability against capsizing is one of the most fundamental requirements to design a ship. In this research, for the first time, CFD is performed to predict main modes of capsizing. CFD first is conducted to predict parametric rolling for a naval ship. Then CFD study of parametric rolling is extended for prediction of broaching both by using CFD as input to NDA model of broaching in replacement of EFD inputs or by using CFD for complete simulation of broaching. The CFD resistance, static heel and drift in calm water and static heel in following wave simulations are conducted to estimate inputs for NDA and 6DOF simulation in following waves are conducted for complete modeling of broaching. CFD parametric rolling simulations show remarkably close agreement with EFD. The CFD stabilized roll angle is very close to those of EFD but CFD predicts larger instability zones. The CFD and EFD results are analyzed with consideration ship theory and compared with NDA. NDA predictions are in qualitative agreement with CFD and EFD. CFD and EFD full Fr curve resistance, static heel and drift in calm water, and static heel in following waves results show fairly close agreement. CFD shows reasonable agreement for static heel and drift linear maneuvering derivatives, whereas large errors are indicated for nonlinear derivatives. The CFD and EFD results are analyzed with consideration ship theory and compared with NDA models. The surge force in following wave is also estimated from Potential Theory and compared with CFD and EFD. It is shown that CFD reproduces the decrease of the surge force near the Fr of 0.2 whereas Potential Theory fails. The CFD broaching simulations are performed for series of heading and Fr and results are compared with the predictions of NDA based on CFD, EFD, and Potential Theory inputs. CFD free model simulations show promising results predicting the instability boundary accurately. CFD calculation of wave and rudders yaw moment explains the processes of surf-riding, broaching, and periodic motion. The NDA simulation using CFD and Potential Flow inputs suggests that CFD/ Potential Flow can be considered as replacement for EFD inputs.

Different theoretical models have tried to investigate the feasibility of recurrent neural mechanisms for achieving direction selectivity in the visual cortex. The mathematical analysis of such models has been restricted so far to the case of purely linear networks. We present an exact analytical solution of the nonlinear dynamics of a class of direction selective recurrent neural models with threshold nonlinearity. Our mathematical analysis shows that such networks have form-stable stimulus-locked traveling pulse solutions that are appropriate for modeling the responses of direction selective cortical neurons. Our analysis shows also that the stability of such solutions can break down giving raise to a different class of solutions ("lurching activity waves") that are characterized by a specific spatio-temporal periodicity. These solutions cannot arise in models for direction selectivity with purely linear spatio-temporal filtering.

During the last few years there was a tremendous growth of scientific activities in the fields related to both Physics and Control theory: nonlinear dynamics, micro- and nanotechnologies, self-organization and complexity, etc. New horizons were opened and new exciting applications emerged. Experts with different backgrounds starting to work together need more opportunities for information exchange to improve mutual understanding and cooperation. The Conference "Physics and Control 2007" is the third international conference focusing on the borderland between Physics and Control with emphasis on both theory and applications. With its 2007 address at Potsdam, Germany, the conference is located for the first time outside of Russia. The major goal of the Conference is to bring together researchers from different scientific communities and to gain some general and unified perspectives in the studies of controlled systems in physics, engineering, chemistry, biology and other natural sciences. We hope that the Conference helps experts in control theory to get acquainted with new interesting problems, and helps experts in physics and related fields to know more about ideas and tools from the modern control theory.

Computational research making use of molecular dynamics (MD) simulations has begun to expand the paradigm of structural biology to include dynamics as the mediator between structure and function. This work aims to expand the utility of MD simulations by developing Principal Component Analysis (PCA) techniques to extract the biologically relevant information in these increasingly complex data sets. Gramicidin is a simple protein with a very clear functional role and a long history of experimental, theoretical and computational study, making it an ideal candidate for detailed quantitative study and the development of new analysis techniques. First we quantify the convergence of our PCA results to underwrite the scope and validity of three 64 ns simulations of gA and two covalently linked analogs (SS and RR) solvated in a glycerol mono-oleate (GMO) membrane. Next we introduce a number of statistical measures for identifying regions of anharmonicity on the free energy landscape and highlight the utility of PCA in identifying functional modes of motion at both long and short wavelengths. We then introduce a simple ansatz for extracting physically meaningful modes of collective dynamics from the results of PCA, through a weighted superposition of eigenvectors. Applied to the gA, SS and RR backbone, this analysis results in a small number of collective modes which relate structural differences among the three analogs to dynamic properties with functional interpretations. Finally, we apply elements of our analysis to the GMO membrane, yielding two simple modes of motion from a large number of noisy and complex eigenvectors. Our results demonstrate that PCA can be used to isolate covariant motions on a number of different length and time scales, and highlight the need for an adequate structural and dynamical account of many more PCs than have been conventionally examined in the analysis of protein motion.