A survey of studies dealing with vibrating structural elements using simple polynomial approximations in connection with Rayleigh-Ritz or Galerkin-type methods is presented.
The classical use of polynomials when solving dynamic problems of deformable bodies consists of constructing a set of coordinate functions in such a way that they satisfy at least the essential boundary conditions and that they represent “reasonably well” the deformation field of the structural element under study.
An alternative and more rational procedure has been developed and used in recent years whereby orthogonal polynomials are used. A “base function” is constructed and then one generates a set of orthogonal polynomials using the Gram-Schmidt or equivalent procedure.
The present paper presents comparisons of numerical results in the case of different types of vibrating structural elements Special emphasis is placed on Rayleigh’s optimization procedure which consists of taking one of the exponents of the polynomial coordinate functions as an optimization parameter “γ”. Since the calculated eigenvalues constitute upper bounds, by minimizing them with respect to “γ” one is able to optimize the eigenvalues.