The Free and Forced Vibrations of a Closed Elastic Spherical Shell Fixed to an Equatorial Beam—Part II: Perturbation Approximations

The two small parameters that appear in the final equations developed in Part I (Simmonds and Hosseinbor, 2010, “The Free and Forced Vibrations of a Closed Elastic Spherical Shell Fixed to an Equatorial Beam—Part I: The Governing Equations and Special Solutions,” ASME J. Appl. Mech., 77, p. 021017), namely, $h/R$⁠, the ratio of the constant shell thickness to the radius of curvature of the shell’s reference surface and $H/R$⁠, where $H$ is the depth (or width) of the equatorial beam, are exploited using perturbation techniques (including the WKB method). The natural frequencies depend not only on these parameters, but also on the ratio of the mass densities of the shell and beam, the ratio of the Young’s moduli of the shell and beam, Poisson’s ratio, and the circumferential wave number $m$⁠. Short tables for typical parameter values are given for those cases where the frequency equation is not explicit.