Paper Title:

Hyperbolic Models Arising in the Theory of Longitudinal Vibration of Elastic Bars

Author(s):

¹I. Fedotov, ¹J. Marais, ^1,2M. Shatalov and ¹H.M. Tenkam

¹Department of Mathematics and Statistics,
Tshwane University of Technology
 Private Bag X6680, Pretoria 0001
South Africa.

fedotovi@tut.ac.za, julian.marais@gmail.com, djouosseutenkamhm@tut.ac.za.

 ²Manufacturing and Materials
Council of Scientific and Industrial Research (CSIR)
P.O. Box 395, Pretoria, 0001
South Africa.
mshatlov@csir.co.za

Abstract:

In this paper a unified approach to the derivation of families of one
dimensional hyperbolic differential equations and boundary conditions describing
the longitudinal vibration of elastic bars is outlined. The longitudinal and
lateral displacements are expressed in the form of a power series expansion in
the lateral coordinate. Equations of motion and boundary conditions are derived
using Hamilton's variational principle. Most of the well known models in this
field fall within the frames of the proposed theory, including the classical
model, and the more elaborated models proposed by by Rayleigh, Love, Bishop,
Mindlin, Herrmann and McNiven. The exact solution is presented for the
Mindlin-Herrmann case in terms of Green functions. Finally, deductions regarding
the accuracy of the models are made by comparison with the exact
Pochhammer-Chree solution for an isotropic cylinder.

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