Keywords: eigenvalues, Rayleigh-Ritz, dynamic stiffness, superposition, Wittrick-Williams algorithm, Lagrangian multiplier method.

This is a brief review paper which discusses the application of several analytical methods of finding eigenvalues and eigenvectors of continuous systems with particular emphasis on problems involving structural mechanics. The dynamic stiffness method (DSM) would be the first choice for eigenvalue calculations where it is applicable as it gives exact results. However, for problems where DSM is not applicable, which method should one use? How are these methods related? The paper attempts to address these questions. The methods under consideration include the Rayleigh-Ritz method (RRM), Gorman's superposition method (GSM), the exact DSM and the associated computational procedures such as the Wittrick-Williams algorithm. Issues such as the accuracy and boundedness of the results and the rate of convergence are discussed.

The superposition method for eigenvalue problems was advanced by Daniel Gorman. In this method known solutions to building blocks corresponding to different sets of boundary conditions are superimposed to obtain the solution to a new problem under different sets of boundary conditions. This method is compared with the DSM. The main difference between GSM and other variational methods such as the RRM is that in GSM the governing equations are satisfied exactly and it is only the boundary conditions that need to be satisfied through a series approximation. Where the DSM is not applicable, GSM may be the most efficient method due to its rapid rate of convergence.

On the question of versatility, the RRM stands out as the favourite choice. The paper discusses some recent work on overcoming the limitations associated with the admissibility requirements of the RRM. This has been handled in two different ways. One is the Lagrangian multiplier method where individual functions are allowed to violate some or all of the essential boundary conditions, but the series is forced to satisfy these conditions by incorporating the constraint equations. Another approach is the penalty method. In both approaches, recent publications suggest that instead of specially tailored orthogonal functions, extended trigonometric series may prove to be a more reliable and convenient choice.

The paper also raises questions such as whether it is possible to: model a cut-out by embedding negative bodies into larger bodies; use a Poisson's ratio that is slightly higher than 0.5 to model incompressible fluid; or follow Courant's suggestion to sensitize the energy functional with terms including higher derivatives to improve the accuracy of the derivatives of field variables such as stress.

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