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Advances in Acoustics and Vibration
Volume 2010, Article ID 984361, 11 pages
http://dx.doi.org/10.1155/2010/984361
Research Article

Free and Forced Vibrations of Elastically Connected Structures

Department of Mechanical Engineering, The University of Akron, Akron, Oh 44235, USA

Received 21 October 2010; Accepted 11 November 2010

Academic Editor: K. M. Liew

Copyright © 2010 S. Graham Kelly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A general theory for the free and forced responses of 𝑛 elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for 𝐢 π‘˜ [ 0 , 1 ] , then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on π‘ˆ = 𝑅 𝑛 Γ— 𝐢 π‘˜ [ 0 , 1 ] . This leads to the definition of energy inner products defined on π‘ˆ . When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in π‘Š is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of 𝑛 intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.