Vibrations of Moderately Thick Shallow Spherical Shells at Large Amplitudes

Abstract

A shallow shell theory is presented for the geometrically non-linear analysis of moderately thick isotropic spherical shells. Effects of transverse shear deformation and rotatory inertia are included in the governing equations of motion by means of tracing constants. When these effects are ignored, the governing equations readily reduce to those applicable for thin shallow spherical shells. Solutions to the system of thick shell equations are obtained by means of Galerkin's method and the numerical Runge-Kutta procedure. Numerical results are presented for certain cases of shallow spherical shells considering different geometric shell parameters. Transverse shear and rotatory inertia effects are found to be important in linear as well as non-linear responses of shallow spherical shells. The non-linear frequency-amplitude behavior is of the softening type for shallow spherical shells and of the hardening type for circular plates. Frequency ratios are lower at any given amplitude when the effects of transverse shear and rotatory inertia are included in the analysis.