Approximate Mathematical Model Research Articles

Melting of a finite paraffin slab as applied to phase-change thermalcontrol.

An approximate mathematical model was established for the melting of a finite paraffin slab, a problem of interest for phase-change thermal control devices. The method is based on the numerical computer solution of the two-phase, one-dimensional heat-conduction equations with moving interface and variable thermal properties in both the solid and liquid regions. Results from the theoretical model are compared with experimental results for n-octadecane in a bellows-type test cell, and with information published in the literature. The numerical analysis appears to provide a good rigorous treatment of the unidimensional melting problem in an idealized model.

Study of the Effect of Plane Boundary Location in the Analysis of Characteristic Dispersion

The mathematical analysis of the characteristic dispersive effects of a continuously stratified ocean requires obtaining a solution to the reduced wave equation that simultaneously satisfies the existing boundary conditions. The relative ease with which the dispersion information can be extracted from this boundary value problem is influenced by the location of the boundaries. The desired eigen values for a finite ocean are often embedded in an intractable transcendental equation, the solutions of which are usually obtained through the application of numerical techniques. However, if the physical problem is approximated by an ocean of infinite extent, the characteristic values are given by an exact, explicit, and easily obtained relation. The purpose of this work was to study the effect of this approximate mathematical model upon the characteristic dispersion of the medium. This was accomplished by comparing results from both models for a number of both single- and double-valued sound velocity profiles. In both cases, the exact solution was sought in terms of hypergeometric functions. For the infinite ocean model, further restrictions were applied so that the solution might be expressed in terms of Jacobi polynomials. [Research supported by U. S. Naval Underwater Sound Laboratory, New London, Connecticut.]

On the Application of the Lyapunov Method to Synchronous Machine Stability†

ABSTRACT The Lyapunov stability method is applied to two problems in the dynamic behaviour of round-rotor synchronous machines. In one case analytical results of a sufficient nature are obtained from an approximate mathematical model which explain the dependence of the self-oscillation of a machine on the stator impedance and the rotor excitation, and good agreement is found with experimental results. The other case is the ‘two-machine’ problem of two synchronous machines connected by a transmission line subject to a fault. Using the complete set of Park's equations an analytical expression is obtained for the transient stability surface. In general this is of a fairly weak sufficient nature but in a restricted region of the state-Space it gives a close agreement with the computed region of stability. The dynamic equations of the machines involve trigonometrieal functions of the state variables, and in this study the variable gradient method of Schultz and Gibson has been found useful for generating Lyapu...

An Automated X-Y Dimensional Stability Measuring System for Printed Circuits Substrates

A system is described for measuring length (width) changes in printed-circuit substrates under environmental (temperature/humidity) changes. A fixture for samples 20 inches in length is also described. Data are given for temperature coefficient and humidity coefficient of epoxy-glass substrates 0.010 in and 0.020 in thick, both bare and partially covered with an etched copper pattern. Two approximate mathematical models are discussed. One is static and linear; the other is time-dependent and nonlinear, but treated as a linear expansion. Additional Key Words for Information Retrieval--Circuit boards, circuit stability, dimensional stability, environmental effects, humidity coefficient of expansion, measurement of X-Y coordinates, substrates, temperature coefficient of expansion.

The Theoretical Behavior of a Knitted Fabric Subjected to Biaxial Stresses

Abstract : The theoretical mechanical behavior of a plain knitted fabric subjected to biaxial stresses is derived. An approximate mathematical model of the fabric structure was established from which the stress vs fabric-geometry and stress vs strain relationships were determined. The work was done by considering only the properties of the fabric structure, completely independent of the fiber properties. The results may be used for such applications as predicting the performance of a plain knitted fabric in situations where it will be stressed biaxially. (Author)

A Theoretical Investigation of Pressure Depression in Externally Pressurized Gas-Lubricated Circular Thrust Bearings

A theoretical explanation of the experimentally observed depression of the pressure profile in externally pressurized gas-lubricated circular thrust bearings is presented, based on a concept new to gas lubrication, that of the generation of a shock wave in the bearing clearance space. On the basis of an approximate mathematical model, the bearing pressure distribution and load capacity are determined, and the effects of several bearing parameters are discussed. Good qualitative agreement is found between the analytical results and available experimental data.

A Theoretical Investigation of Pressure Depression in Externally Pressurized Gas-Lubricated Circular Thrust Bearings

A theoretical explanation of the experimentally observed depression of the pressure profile in externally pressurized gas-lubricated circular thrust bearings is presented, based on a concept new to gas lubrication, that of the generation of a shock wave in the bearing clearance space. On the basis of an approximate mathematical model, the bearing pressure distribution and load capacity are determined, and the effects of several bearing parameters are discussed. Good qualitative agreement is found between the analytical results and available experimental data.