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PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
The Superposition Method for Computing Free Vibration Eigenvalues of Elastic Structures
D.J. Gorman1 and S.D. Yu2
1Department of Mechanical Engineering, University of Ottawa, Ontario, Canada
D.J. Gorman, S.D. Yu, "The Superposition Method for Computing Free Vibration Eigenvalues of Elastic Structures", in , (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 191, 2010. doi:10.4203/ccp.93.191
Keywords: cylindrical shells, eigenvalues, mode shapes, plates, vibration.
Historically, the most widely accepted analytical method has been the Ritz method. This energy-based method requires the selection, in advance, of admissible displacement functions to represent the vibratory mode shapes of the structure of interest. Recently an alternative analytical method, referred to as the superposition method (SM), was developed by Gorman  for obtaining analytical solutions for plate vibration problems. In the past few decades, SM has been exploited with considerable successes in obtaining accurate free vibration eigenvalues and mode shapes for an array of problems. To solve free vibration problem of a rectangular plate, a set of two or more judiciously selected edge-driven forced vibration problems, termed as the building blocks, are first solved. The exact Levy solution is obtainable for each building block. One edge in each of these building blocks is driven by a distributed harmonic bending moment or vertical edge reaction. The distribution of the amplitudes of the driving quantity is expressed in the same series as the Levy solution with the coefficients to be determined. The next step is to superimpose the building block solutions and to constrain the driving coefficients in such a way that the superimposed solution satisfies the desired boundary conditions along the driven edges. By doing so, a set of homogeneous algebraic equations of the unknown coefficients is obtained. The condition for a non-trivial solution to the algebraic equations is then sought for determining the eigenvalues and mode shapes. Experience has shown that convergence in problem solutions is rapid. The problem of trying to select appropriate functions to represent plate displacement, as required in the Ritz Method, is completely eliminated. Selection of appropriate building blocks for use in the superposition method is straightforward.
Early solutions pertained mainly to thin rectangular plate with classical boundary conditions. Solutions were then extended to handle non-classical boundary conditions such as translational and rotational elastic edge support, point supports, and attached masses. Subsequently, the SM was extended to triangular and trapezoidal plates. A significant milestone was achieved when the SM was employed to handle Mindlin plates, laminated plates and in-plane loaded thin, moderately thick plates. More recently the SM was employed for in-plane vibration of rectangular plates.
Beside its successes in handling vibrations and buckling problems of flat plates, the SM has also been employed to deal with free vibration of open cylindrical shells with combinations of classical boundary conditions. The SM is currently being exploited to handle transient vibration, forced vibrations, and free vibrations of very thick plates in accordance with a third order shear deformable plate theory.
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