Free vibration analysis of deep doubly curved open shells using the Ritz method

This paper develops a unified semi-analytical method for the free vibration analysis of moderately thick doubly curved open shells with arbitrary geometry and classic boundary conditions. The only restriction on the shell's geometry is that boundaries are being coincided by the principal curvature lines of the shell. The formulation is based on the first order shear deformation theory by considering effects of curvature in the evaluation of stress resultants. Differential geometry method is used to represent the arbitrary shape of the middle surface of the shell. The Ritz method with algebraic polynomials as trial functions is employed to obtain the natural frequencies and mode shapes of the shell. To demonstrate the efficiency and accuracy of the solution, convergence and comparison studies are carried out for a cantilevered shallow shell and a parabolic panel. Furthermore, a variety of new vibration results including frequencies and mode shapes of ellipsoid and Enneper panels with various boundary conditions are presented which may be used as benchmark results for future studies.