Low-Frequency Asymmetric Vibrations of a Thin Shell with a Sign-Changing Curvature

Nhawu, Gerald (2009) Low-Frequency Asymmetric Vibrations of a Thin Shell with a Sign-Changing Curvature. Masters thesis, University of Zimbabwe.

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Abstract

Most problems in Applied Mathematics involving difficulties such as nonlinear governing equations and boundary conditions, variable coefficients and complex boundary shapes preclude exact solutions. Consequently exact solutions are approximated with ones using numerical techniques, analytical techniques or a combination of both. We need to obtain some insight into the character of the solutions and their dependence on certain parameters. Often one or more of the parameters becomes either very large or very small. Typically these are very difficult situations to treat by straight-forward numerical procedure. The analytical method that can provide an accurate approximation is by asymptotic expansions. This thesis is largely influenced by Professor M.B. Petrov and P.E. Tovstik [12] paper, which examines the role of turning points on shells of revolution with low frequency vibrations and how the resulting solutions behave. The thesis focuses on the general idea introduced by Langer, where he realized that any attempt to express the asymptotic expansions of the solutions of turning point problems in terms of elementary functions must fail in regions containing the turning point. A uniformly valid expansion must be expressed in terms of the solution of non-elementary functions which have the same qualitative features as the equation, for example, Airy equations and the exploration of the shell of revolution with emphasis on the negative Gaussian curvature region where the instability occurs due to low frequency vibrations. This thesis consists of 4 Chapters. Historical developments, refinements and definitions are provided in Chapter 1. This Chapter also contains a number of examples and applications illustrating the practical use of shells. The concept of surfaces is introduced and the basic equilibrium and stress-strain relations are derived using the Love-Kirchhoff assumptions. The concept of Airy functions as solutions of the Airy equation is introduced in Chapter 2. Using Bessel functions of the first kind, it is shown that the Airy functions have an oscillatory character for negative values of the argument. The Chapter ends with a discussion of the Liouville’s differential equation and turning points. Chapter 3 deals with a shell of revolution with low frequency vibrations of sign-changing curvature. It begins with a brief account of the derivation of the governing equations and the use of separating variables. Principal sections in this Chapter are asymptotic solutions, zeroth approximation, first approximation, Airy solutions and the variational approach to the boundary conditions. The Chapter ends with the conclusion, where it has been shown that, in the region with negative Gaussian curvature both solutions oscillate and in the remaining region, where the curvature is positive, both Airy functions and the unknown displacements and stresses exponentially increase or decrease. Chapter 4 deals with the numerical and asymptotic aspects of the theory. It begins with an example, where different values of the eigenvalue are found for different shell thickness and the results tabulated and compared.

Item Type:	Thesis (Masters)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Africana
Depositing User:	Tim Khabala
Date Deposited:	15 May 2018 08:04
Last Modified:	15 May 2018 08:04
URI:	http://thesisbank.jhia.ac.ke/id/eprint/4001

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