Three-dimensional vibration analysis of paraboloidal shells

A method of analysis is presented for determining the free vibration frequencies and mode shapes of open paraboloidal shells of revolution having arbitrary thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell, as well as the inner and outer curved surfaces, may be free or may be subjected to any degree of constraint. The strain energy of deformation, as well as the kinetic energy of motion, are formulated in terms of three displacement components which are tangent or normal to the shell middle surface. The displacements are taken as periodic in the circumferential coordinate and in time, and as polynomials of arbitrary degree in the other two coordinates, and the Ritz method is used to formulate the eigenvalue problem. Convergence studies are presented, and frequencies are given for moderately thick and thick, moderately deep and deep, paraboloidal shells of uniform and variable thickness.