A systematic approach to the free vibration analysis of thin, flat, non-rectangular quadrilateral plates with combinations of simple and clamped supports is presented, using the parallelogram plate as an example. Modifications that Saliba made for the right-triangular plate to the building block superposition method developed by Gorman are implemented. The superposition method is an analytical solution. No simplifications are made to take advantage of the point symmetry of the parallelogram plate, keeping the solution general. The whole plate is divided into two right-triangular and one rectangular segment. Rectangular building blocks are superimposed for each of the segments to meet the required net boundary conditions and the conditions of continuity along the segment interfaces. The Levy-type solution to the eight building blocks used are given, along with the necessary Fourier expansions and a comprehensive guide to assembling the eigenvalue matrix. Numerical results for a wide variety of plates are presented, including numerous mesh and contour plots. Comparison is made to previously published data. The results for the fully clamped parallelogram plate are supported by experimental results. Six aluminum plates were tested for the first six resonance frequencies and the first three mode shapes. Details of the simple, yet highly accurate experimental method are included.

This study evaluated the ability of two different methods of measuring mechanical work, absolute power (AP) and absolute work (AW), to distinguish between normal and impaired gait. The relation between these two measures was examined as well as their relation to oxygen consumption. Global work measures from all three were compared and, in the case of the absolute power method individual joint power curves of the ankle, knee and hip, for normal and two impaired conditions were examined. Four subjects of each gender were examined for one full gait cycle, by three video cameras, over two AMTI force platforms, under three conditions; normal, locked knee and locked ankle. Oxygen consumption was measured with a TEEM 100 unit carried by the subject in a "fanny" pack. Five normal gait and one trial of each impaired condition were analysed. (Abstract shortened by UMI.)

In this thesis we consider the following model for a three-dimensional r ˣ s ˣ t contingency table: [formula omitted]. A dot indicates summation over the replaced subscript. The f[formula omitted]’s represent the frequencies and the P[formula omitted]'s represent the proportions. The problem we are concerned with is testing the hypothesis H₀: [formula omitted] = 0 for all i, j, k. i.e. no second order interaction is present. We then seek to extend the model and problem to a w-way table. We use the method of the likelihood ratio . To assist us in determining the numerator of the likelihood ratio we reformulate a theorem about constrained extrema and Lagrange multipliers and prove this reformulation. Some general conclusions we draw are: there are two extensions to our 3-way model; results we obtain using our model and methods are in close agreement with results obtained using the models and methods of other statisticians. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis describes an investigation of the propagation of a pressure wave, through a long straight pipe, in a rarefied atmosphere of Argon. This phenomenon is approximately analogous to thermal wave propagation in the boundary scattering limit. Casimir and Ziman studied this latter phenomenon for the steady state case. P.W. Matthews extended their theory to the non-steady state and found a diffusion coefficient D depending on L. Smoluchowski studied the corresponding phenomenon in gases for the steady state and again P.W. Matthews extended the theory to the non-steady state, and again found a dependence of D on L. Therefore at the low frequency limit a wave pressure velocity would depend on L. An experiment was set up in order to produce a wave pressure and measure its velocity as a function of the frequency and the pressure and hence L. Results have been obtained and no dependence on the pressure has been observed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

An investigation is made into some aspects of the analysis and design of high order systems. The problems treated are system stabilization, parameter optimization, computation of an optimal controller and parameter sensitivity. The methods developed for solving these problems are applied to a 9ᵗʰ order linearized power system. To stabilize the system, an eigenvalue shift technique is used. Eigensystem sensitivity analysis is applied to determine both the parameter change required and the new eigensystem after the change has been made. A correction method is applied to the new eigensystem for improving accuracy in order that large steps in parameter change may be taken. This method is subsequently used in an optimization procedure for parameter setting to minimize a cost functional of quadratic form. For the computation of an optimal controller, Puri and Gruver's successive approximation method is used in conjunction with a fast recursive method developed for solving each approximation of the Ricatti matrix. The calculation can be initiated by the eigenvalue shifting method to ensure that the system is initially stable. Finally, a time response sensitivity study is made using a method developed for simultaneous sensitivity function determination. This method reduces computation time significantly over the conventional method ⁽⁹⁾ thus enabling the investigation of time response sensitivity to a large number of parameters. The results of the sensitivity study are then applied to the design of a suboptimal controller. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

This thesis is a study of some of the financially important and technically interesting situations which appear in the economic operation of power systems. Static type optimization problems, which arise when a given plant demand is distributed among the running units of that plant, and in thermal power systems, are first considered. For this case, scheduling equations which are valid in the presence of bounds on the plant operating range are derived through the application of Kuhn-Tucker conditions. A dynamic type short-range hydro-thermal problem based essentially according to the currently prevalent formulation of assuming a specified water usage, is extended to include operating range bounds and unit commitment aspects. Pontryagin's maximum principle is found to be useful in obtaining suitable solutions to the extended formulations. For the above dynamic type short-range hydro-thermal problems to be meaningful, the water usage specification should usually be obtained through a careful analysis of long term resource management. With the above view, the overall hydro-thermal economic optimization problem is formulated and the long-range and short-range problems are deduced as its components. This results in a dynamic type long-range component while allowing a static type short-range problem. Among other practical aspects, the stochastic nature of the river flows and the very long and indefinite nature of the period of optimization are taken into account. A mathematical basis for the intuitively appealing concept of water values is developed. As a solution to the long-range problem, an equation which gives the interrelationship of water value functions is derived through the application of the principle of optimality. This equation is shown to be useful in computing water values as a function of reservoir state and time. Finally, complications which arise due to the variation of water head, flow interconnected reservoirs and interruptible consumers are analysed. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

In this thesis, a stiffness matrix which includes the non-linear effects of principal plane shears, moments and axial loads on lateral and torsional deflections is developed for a doubly symmetric wide flange section. Initially, an exact eight by eight linear matrix is developed for an element of constant section properties. The eight allowable deflections allows the independent representation of the deflections of either flange at either end. The non-linear effects are included in the differential equations by considering the effect of the primary stresses on the equilibrium of a dis-placed element. Two approximations are then introduced. The first consists of a numerical technique for solving the differential equations. The second consists of a simplification of the boundary conditions in solving the differential equations. Using these two approximations, the non-linear portion of the matrix is then built. Several structures are then analyzed. Each structure is divided into several elements. This allows beams of non-constant section properties to be analyzed, and increases the accuracy of the results of the approximate matrices. The results of these analyses are then compared to theoretical results and tabulated. It is seen that the matrix gives good agreement for all cases tested. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

The general area of investigation was first suggested by the problem of designing a plug to block off a tunnel that was to be filled with water. The specific purpose of this project was to determine if the shear stress along the edge of a plug in a circular hole could be considered uniform when the hole on one side of the plug was subjected to hydrostatic pressure. Three mathematical solutions were attempted using the theory of elasticity, but none yielded a simple solution. The problem was then attacked experimentally by plane photoelasticity and 32 configurations of six plane models were examined. The results showed that the shear stress was not uniform, but rose to a high peak and then declined rapidly. Three-dimensional photoelastic techniques were also used and the results of five "stress-freezing" models confirmed this conclusion. Other significant conclusions concern the variation of stresses with plug thickness; the use of fillets to strengthen the plug; and the dissipation of shear stress with distance from the plug. Some discussion is also made of the optimum design for a plug. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

The deformation behaviour of polycrystalline compacts of Mg(OH)₂ during dehydroxylation has been studied in an attempt to evaluate the nature of strain that can be introduced into the compact during the reaction. A study of neck-growth between tips of single crystals of Ca(OH)₂, and between two hemispherical tips of Mg(OH)₂, compacts showed both deformation and interaction at the contact point during the dehydroxylation reaction. Load-dependent deformation of the compacts gave a total strain proportional to the one-third power of the applied stress. The creep deformation of Mg(OH)₂ compacts during dehydroxylation was also studied under isothermal conditions. The overall creep behaviour can be divided into three stages. The initial stage is initiated by the dehydroxylation reaction. During the second or steady state creep stage the highest creep rate was obtained. The steady state creep rate was determined as a function of temperature pressure, and relative density of the green compact. The results are represented by: [formula omitted] Particle sliding was considered to be the most probable mechanism for creep during the second stage. / Applied Science, Faculty of / Materials Engineering, Department of / Graduate

Methods of determining the deformation characteristic physical properties and dynamic response of cohesive soils were evaluated in order to obtain design parameters for soil-machine systems. The study was limited to simple scaled soil-machine systems on Haney clay and on mixed Haney clay and Ottawa sand in the semi-solid to plastic range. A special moire method was successfully developed to study large deformation and translation paths of soil as a function of tool shape and position. Compression, direct shear, stress wave and forced vibration methods were used to measure mechanical properties of soils. Advantages and disadvantages of each method were evaluated. Quasi-static stress-strain relationships were established from improved unconfined compression tests to produce unixial compression. Yield stress and strain hardening effects could be observed from these tests. Rate dependency of cohesive soil was verified by observing stress wave propagation in soil. It was observed that stress wave propagation velocity was more sensitive to soil particle size than to soil strength. Forced vibration methods were used to evaluate elastic constants such as Young's modulus and shear modulus which are useful in determining the contribution of elastic strains to the total force required to deform a soil. The theory of plasticity was successfully used in conjunction with experimental observations to establish stress-strain relationships in the soil on the assumption that strain hardening was linear and elastic strains were negligible. The maximum difference between theoretical forces deduced on the above basis and measured forces was less than twenty percent. The use of gelatin as a simulated soil was investigated to determine whether its use could provide a useful qualitative aid to design of soil-machine systems. It was found that the gelatin study gave stress trajectories and slip lines which resembled the results observed by the moire method in actual soil. Results obtained from the application of soil-machine systems on simulated soil and prototype soil were compared. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate