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IMA Journal of Applied Mathematics Advance Access originally published online on December 15, 2008
IMA Journal of Applied Mathematics 2009 74(1):149-162; doi:10.1093/imamat/hxn043
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Solvability of a Reissner–Mindlin–Timoshenko plate–beam vibration model

N. F. J. van Rensburg

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

L. Zietsman

Department of Mathematics, Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24060, USA

A. J. van der Merwe{dagger}

Department of Mechanical Engineering, Cape Peninsula University of Technology, Box 652, Cape Town 8000, South Africa

{dagger} Email: vandermerwea{at}cput.ac.za

Received on December 20, 2007; Revision received October 27, 2008. Accepted on November 13, 2008

In this paper, we consider the vibration of a plate–beam system consisting of a Reissner–Mindlin plate and a Timoshenko beam. A variational form is derived directly from the equations of motion and the constitutive equations. We show how an existence result for a general linear vibration problem in variational form may be applied to the weak variational problem for this system.

Keywords: vibration; plate–beam system; existence of solutions; semigroups; variational form; Reissner–Mindlin plate; Timoshenko beam.


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