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Axisymmetric free vibration of moderately thick polar orthotropic hemispherical shells are studied under the various boundary conditions with sliding, guided pin, clamped, and hinged edges. Based on the improved linear elastic shell theory with the transverse shear strain and rotatory inertia taken into account, the dynamic equlibrium equations are formulated and transformed into the displacement form in terms of mid-surface meridian and radial displacements and parallel circle cross-section rotation. These partial differential equations are solved by the Galerkin method using proper Legendre polynomials as admissible displacement functions with the aid of the orthogonality and a number of special integral relations. Natural frequencies and modes found from the eigenproblems are shown with reasonable results.

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This paper presents studies of the axisymmetric free vibration of moderately thick orthotropic complete spherical shells. The mathematical model is formulated in terms of the mid-surface meridian and radial displacements, and cross-section rotation of the meridians by the improved linear elastic shell theory with the effects of transverse shear strain and rotary inertia included. By the use of the Legendre polynomials, the exact solu tion from a set of three governing partial differential equations for the transversely iso tropic case and the series solution from the Ritz method for the polar orthotropic case are obtained. Natural frequencies and modes are found from the eigenproblems by using orthogonality and some special integrals with reasonable results.