Vibration of beams using novel boundary characteristic orthogonal polynomials satisfying all boundary conditions

Boundary characteristic orthogonal polynomials proposed by the author in 1985 have been used in the Rayleigh Ritz method extensively in order to obtain natural frequencies of vibrating plates with different boundary conditions. The method used products of the characteristic orthogonal polynomials along the two directions of the plate. The first member of the boundary characteristic orthogonal polynomials set satisfied all the boundary conditions of the vibrating beam, including the natural conditions. However, the higher members of the set satisfied only the geometry boundary conditions. In this study, a modified Gram-Schmidt orthogonalization method is presented where all the members of the orthogonal set of polynomials satisfy all the boundary conditions including the natural boundary conditions. Furthermore, the exact solution of the beam differential equation is expressed in the form of a generalized Fourier series in terms of the set of new boundary characteristic orthogonal polynomials which forms an eigenvalue problem that can provide the natural frequencies and the corresponding normal modes of the beam more accurately.