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Shock and Vibration
Volume 19 (2012), Issue 6, Pages 1341-1357

Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

Seyyed M. Hasheminejad1 and Yaser Mirzaei2

1Acoustics Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
2Department of Mechanical Engineering, Damavand Branch, Islamic Azad University, Damavand, Iran

Received 4 July 2011; Revised 8 January 2012

Copyright © 2012 Hindawi Publishing Corporation. 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.


A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.