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Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation

Part of: Equations of mathematical physics and other areas of application Hyperbolic equations and systems

Published online by Cambridge University Press:  27 January 2016

H. M. Tenkam ,
M. Shatalov ,
I. Fedotov  and
R. Anguelov
H. M. Tenkam*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Pretoria 0028, South Africa
M. Shatalov
Affiliation:
Department of Mathematics and Statistics, Tshwane University of Technology, Private Bag X680 Pretoria, 0001, South Africa Materials Sciences and Manufacturing, Council for Scientific and Industrial Research, Private Bag X395, Pretoria 0001, South Africa
I. Fedotov
Affiliation:
Department of Mathematics and Statistics, Tshwane University of Technology, Private Bag X680 Pretoria, 0001, South Africa
R. Anguelov
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Pretoria 0028, South Africa
*
*Corresponding author. Email:[email protected] (H. M. Tenkam)
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Abstract

In this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.

Keywords

Longitudinal vibration of rodsvariable cross-sectionexact solutionmethod of lineshypergeometric functions

MSC classification

Secondary: 33C90: Applications 35Q74: PDEs in connection with mechanics of deformable solids 35L35: Initial-boundary value problems for higher-order hyperbolic equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 2 , April 2016 , pp. 257 - 270
Copyright
Copyright © Global-Science Press 2016 

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