Boundary Characteristic Orthogonal Polynomials (BCOPs) in Vibration of Plates
This chapter discusses the method and procedure for generating these polynomials, specific to the vibration of plates. The importance of orthogonal polynomials is well known in numerical approximation problems. Methods of least squares, Gauss quadrature, interpolation, and orthogonal collocation are a few areas in which the advantage of orthogonal polynomials is well documented. Quite a large variety of one-dimensional problems have been solved using orthogonal polynomials. The use of Boundary Characteristic Orthogonal Polynomials in the Rayleigh–Ritz method for the study of vibration problems involves three steps. The first step is the generation of orthogonal polynomials over the domain occupied by the structural member and satisfying the appropriate boundary conditions. The second step is the use of these polynomials in the Rayleigh–Ritz method that renders the problem into a standard eigenvalue problem, rather than as a generalized one. The third and the last step involves the solution of this eigenvalue problem to get the vibration characteristics.