Classical Jacobi polynomials, closed-form solutions for transverse vibrations

This paper deals with transverse vibrations of nonuniform homogeneous beams and plates. Classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations, respectively, can be reduced to an eigenvalue singular problem (singularities occur at both ends) of orthogonal polynomials, are reported. Exact natural frequencies and Jacobi polynomials as exact mode shapes, which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials, are reported for these classes. The above classes of beams and plates hereafter called Jacobi classes are given by geometry and boundary conditions. The geometry consists of parabolic thickness variation, with respect to the axial coordinate for beams, and with respect to the radius for plates. Beams belonging to this class have either one or two sharp ends (singularities) along with certain boundary conditions. Plates have zero thickness at zero and outer radii. The boundary value problems associated with plates, and beams of two sharp ends, are free boundary problems. Two other boundary value problems, hinged-free and sliding-free, are reported for beams with one sharp end. Also, exact natural frequencies and mode shapes for uniformly rotating beams with hinged-free boundary are reported.