Vibrations of a composite shell of hemiellisoidal-cylindrical shell having variable thickness with and without a top opening

Natural frequencies and mode shapes of a composite shell of revolution consisting of a circular cylindrical shell and a deep or shallow hemi-ellipsoidal shell with variable thickness having a circular cylindrical hole or not are determined by the Ritz method using a mathematically three-dimensional analysis instead of two-dimensional thin shell theories or higher order thick shell theories. The present analysis is based upon the circular cylindrical coordinates while in the traditional shell analyses three-dimensional shell coordinates have been commonly used. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the composite shells. The frequencies from the present three-dimensional method are compared with those from three types of two-dimensional thin shell theories (finite element method, finite difference method, and numerical integration) by previous researchers. The present analysis is applicable to very thick shells as well as thin shells; and to shallow shells as well as deep shells.