## Mathematics, Engineering and Computer Science Undergraduate Theses

Spring 1979

Thesis

#### Department

Mathematics, Engineering & Computer Science

Ronald Knoshaug

Alfred Murray

John Semmens

#### Abstract

The modern world has experienced an increase in the applications of mathematics to physical activity. Many physical conditions can be described in terms of partial differential equations, which result in industrial applications. One such application deals with the problem of imbedding metal bolts into a metal frame, achieving a secure fit. The problem is in determining the factors which affect the security of the seal. There are two ways by which the bolt is secured. The first involves heating the frame and inserting the bolt snugly into it. When the frame cools it contracts tightly around the bolt. The second involves cooling the bolt and fitting it snugly into the frame. Then a tight seal forms when the bolt warms up to the temperature of the frame. The problem that the engineer faces is in determining how hot or how cold the bolt should be in relation to the temperature of the frame. This problem is generally referred to as the problem of shrunken-fittings. Both methods can be described mathematically with the use of the heat equation. The intent of this thesis is to indicate how this problem can be solved mathematically. Specifically it is to create a mathematical model of the process of heat conduction in the cylindrical coordinate system. Included in the study will be several basic requirements for the mathematical model. A description of the physical aspect of
heat conduction will be introduced to derive the heat equation. The heat equation, in the Cartesian coordinate system, will be translated to the cylindrical coordinate system. A look at the method of separation-ofvariables will show how it applies to the heat equation. A compact review of Bessel function theory and its use in the study of the Sturm-Liouville problem will be included in this work. This body of knowledge will be used to solve a particular problem in heat conduction. The problem will involve a rod of infinite length, made up of isotropic mate rial, and initially at a constant temperature A. Slipped over the rod will be a sleeve of the same material and length, but initially with a different constant temperature B. Also, at the outside boundary of the sleeve the temperature will always be B. What I intend to do is describe the heat distribution of this system for anytime forward of time zero. To conclude the thesis I will discuss briefly the limitations of this mathematical model.

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