Vibration of Hemispherical-Cylindrical-Hemispherical Shells and Complete Hollow Spherical Shells with Variable Thickness

The natural frequencies and mode shapes of enclosed shell typed structures with variable thickness (hemispherical-cylindrical-hemispherical shells and complete hollow spherical shells) are determined by the Ritz method using a three-dimensional (3D) analysis. However, in the conventional shell analysis, mathematically two-dimensional (2D) thin shell theories or higher order thick shell theories are often employed, which adopt limiting assumptions about the displacement variation through the shell thickness. While most researchers have adopted the 3D shell coordinates that are normal and tangential to the shell mid-surface, the present analysis is based upon the circular cylindrical coordinates. By the Ritz method, the Legendre polynomials, which are mathematically orthonormal and minimal, are used as the admissible functions, instead of the ordinary algebraic polynomials. The strain and kinetic energies of the combined shell structures are formulated, and upper bound solutions of the frequencies are obtained by minimizing the solution for frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. The frequencies from the present 3D method are compared with those from other 3D approach and 2D thin and thick shell theories existing in the literature. The present 3D analysis is applicable to both very thick shells and very thin shells.