Mathematical models

This thesis describes a method to detect, localize and identify a faulty bearing in a rotating machine using narrow band envelope analysis across an array of accelerometers. This technique is developed as part of the machine monitoring system of an ocean turbine. A rudimentary mathematical model is introduced to provide an understanding of the physics governing the vibrations caused by a bearing with a raceway defect. This method is then used to detect a faulty bearing in two setups : on a lathe and in a dynamometer.

The study of ancient Egyptian monumental art is based on subjective and qualitative analyses by art historians and Egyptologists who use the change in stylistic trends as Dynastic chronological markers. The art of the ancient Egyptians is recognized the world over due to its specific and consistent style that lasted the whole of Dynastic Egypt. This artwork exhibits fractal qualities that support the applicability of applying fractal analysis as a quantitative and statistical tool to be used in this field. In this thesis, I show the fractality of ancient Egyptian monumental art by analyzing black and white line drawings of twenty-eight spearate bas-reliefs with three separate programs : Benoit 1.3, ImageJ, and Fractal3e. After preparing the images with GIMP2 software - used to remove non-original lines - I analyzed each image using the fractal box-counting analysis function in the above programs and calculated their fractal dimension, D. The resulting fractal dimension supported the consistency visually identified in the artwork from ancient Egypt, both chronologically and geographically.

This is a novel attempt to produce a rigorous mathematical model of a complex system. The complex system under study is the relationship between therapists and their clients. The success of psychotherapy depends on the nature of the relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model. ... We describe the emotional state of both the therapist and client with coupled, first order, nonlinear ordinary differential equations (ODE's). The rate of change of the emotional state of the therapist and client is proportional to their previous state, their uninfluenced state when alone, and an influence function which depends on the state of the other person. We formulated influence functions based on the research literature on psychotherapy and the therapeutic alliance. We then determined the critical points from the intersection of the nullclines and used a numerical ODE solver (Matlab ODE113) to compute the trajectories from different initial conditions. ... The results validate this prototypical approach to psychotherapy ; we have shown that human interaction (in the context of psychotherapy) can be quantified and modeled using differential equations.

An ocean turbine extarcts the kinetic energy from ocean currents to generate electricity. Machine Condition Monitoring (MCM) / Prognostic Health Monitoring (PHM) systems allow for self-checking and automated fault detection, and are integral in the construction of a highly reliable ocean turbine. MCM/PHM systems enable real time health assessment, prognostics and advisory generation by interpreting data from sensors installed on the machine being monitored. To effectively utilize sensor readings for determining the health of individual components, macro-components and the overall system, these measurements must somehow be combined or integrated to form a holistic picture. The process used to perform this combination is called data fusion. Data mining and machine learning techniques allow for the analysis of these sensor signals, any maintenance history and other available information (like expert knowledge) to automate decision making and other such processes within MCM/PHM systems. ... This dissertation proposes an MCM/PHM software architecture employing those techniques which were determined from the experiments to be ideal for this application. Our work also offers a data fusion framework applicable to ocean machinery MCM/PHM. Finally, it presents a software tool for monitoring ocean turbines and other submerged vessels, implemented according to industry standards.

In this thesis, a 2D CHebyshev spectral domain decomposition method is developed for simulating the generation and propagation of internal waves over a topography. While the problem of stratified flow over topography is by no means a new one, many aspects of internal wave generation and breaking are still poorly understood. This thesis aims to reproduce certain observed features of internal waves by using a Chebyshev collation method in both spatial directions. The numerical model solves the inviscid, incomprehensible, fully non-linear, non-hydrostatic Boussinesq equations in the vorticity-streamfunction formulation. A number of important features of internal waves over topography are captured with the present model, including the onset of wave-breaking at sub-critical Froude numbers, up to the point of overturning of the pycnoclines. Density contours and wave spectra are presented for different combinations of Froude numbers, stratifications and topographic slope.

Ji and Wang (2010) propose that the dominant source of sound from a forward facing step is the stream wise dipole on the face of the step and that sources acting normal to the flow are negligible. Sound radiation normal to flow of forward facing steps has been measured in wind tunnel experiments previously by Farabee and Casarella (1986, 1991) and Catlett (2010). A method for evaluating sound radiation from surface roughness proposed in Glegg and Devenport (2009) has been adapted and applied to flow over a forward facing step which addresses the sound normal to the flow that was previously unaccounted for. Far-field radiation predictions based on this method have been compared with wind tunnel measurements and show good agreement. A second method which evaluates the forcing from a vortex convected past surface roughness using RANS calculations and potential flow information is also evaluated.

Optimization of compaction in granular material without the use of traditional ground improvement methods may be possible by optimizing the percentage of finer material and the median grain size ratio in binary soil mixtures. In this study, the median grain size ratio D50/d50 was explored as a fundamental parpmeter affecting the compaction characteristics of binary mixes made from natural sands as opposed to singular measurements such as fines content and mean grain size traditionally used to represent granular soils. A total of 18 binary granular mixes were synthetically generated from natural sands obtained from Longboat Key, Florida and evaluated through grain size analysis, laboratory compaction and determination of relative density. Results indicate that the D50/d50 ratio shows promise as a fundamental parameter for compaction optimization in binary mixes with values exceeding six approaching the densest packing configuations.

The main objective of this thesis is to simulate, evaluate and discuss three standard methodologies of calculating Value-at-Risk (VaR) : Historical simulation, the Variance-covariance method and Monte Carlo simulations. Historical simulation is the most common nonparametric method. The Variance-covariance and Monte Carlo simulations are widely used parametric methods. This thesis defines the three aforementioned VaR methodologies, and uses each to calculate 1-day VaR for a hypothetical portfolio through MATLAB simulations. The evaluation of the results shows that historical simulation yields the most reliable 1-day VaR for the hypothetical portfolio under extreme market conditions. Finally, this paper concludes with a suggestion for further studies : a heavy-tail distribution should be used in order to imporve the accuracy of the results for the two parametric methods used in this study.

This research is concerned with algorithmic representation of technoeconomic growth concerning modern and next-generation telecommunications including the Internet service. The goal of this study thereof is to emphasize efforts to establish the associated forecasting and, the envisioned tasks thereof include : (i) Reviewing the technoeconomic considerations prevailing in telecommunication (telco) service industry and their implicating features; (ii) studying relevant aspects of underlying complex system evolution (akin to biological systems), (iii) pursuant co-evolution modeling of competitive business structures using dichotomous (flip-flop) states as seen in predator evolutions ; (iv) conceiving a novel algorithm based on information-theoretic principles toward technoeconomic forecasting on the basis of modified Fisher-Kaysen model consistent with proportional fairness concept of comsumers' willingness-to-pay, and (v) evaluating forecast needs on inter-office facility based congestion sensitive traffics encountered. Commensurate with the topics indicated above, necessary algorithms, analytical derivations and compatible models are proposed. Relevant computational exercises are performed with MatLab[TM] using data gathered from open-literature on the service profiles of telecommunication companies (telco); and ad hoc model verifications are performed on the results. Lastly, discussions and inferences are made with open-questions identified for further research.

This work incorporates previous work done by Guerra and the application of fluid dynamics. The structure attached to the turbine will cause unsteady fluctuations in the flow, and ultimately affect the acoustic pressure. The work of Guerra is based on a lot of assumptions and simplifications to the geometry of the turbine and structure. This work takes the geometry of the actual turbine, and uses computational fluid dynamic software to numerically model the flow around the turbine structure. Varying the angle of the attack altered the results, and as the angle increased the noise levels along with the sound pulse, and unsteady loading increased. Increasing the number of blades and reducing the chord length both reduced the unsteady loading.