Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Free vibrations of a polar body at elastic range


Authors: Gülay Altay and M. Cengiz Dökmeci
Journal: Quart. Appl. Math. 64 (2006), 711-734
MSC (2000): Primary 74H25, 49R50, 74H45
DOI: https://doi.org/10.1090/S0033-569X-06-01042-3
Published electronically: October 31, 2006
MathSciNet review: 2284467
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Abstract: The purpose of this paper is to study certain features of the equations governing the time-harmonic free vibrations of a polar body at elastic range. The governing equations of micropolar elasticity are expressed in differential form, and then, the uniqueness of their solutions is investigated. The conditions sufficient for uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained that leads to all the governing equations of the free vibrations as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh's quotient is given. As an application, the high-frequency vibrations of an elastic plate are treated.


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Additional Information

Gülay Altay
Affiliation: Faculty of Engineering, Boḡaziçi University, Bebek, 34342 Istanbul, Turkey
Email: askarg@boun.edu.tr

M. Cengiz Dökmeci
Affiliation: Istanbul Technical University, P.K. 9, Gümüsuyu, 34430 Istanbul, Turkey
Email: cengiz.dokmeci@itu.edu.tr

DOI: https://doi.org/10.1090/S0033-569X-06-01042-3
Received by editor(s): February 17, 2006
Published electronically: October 31, 2006
Additional Notes: The authors acknowledge the financial support in part by their departments and TUBA, and the second author (M.C.D.) is grateful to Prof. Dr. Oral Büyüköztürk (Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge) for his kind invitation and support to the three-day workshop on engineering materials on June 9-11, 2004, Cambridge, Mass.
Article copyright: © Copyright 2006 Brown University

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