On the asymptotic behavior of shells of revolution in free vibration

The present work focuses on shells of revolution in free vibration, in the realm of both Kirchhoff-Love and Reissner-Mindlin small deformation theories. We study the asymptotic behavior of the lowest shell eigenfrequency and of the ratio between bending and total strain energy with respect to decreasing thicknesses. It is shown from a mathematical standpoint that, for fully clamped shells, the basic feature that determines the asymptotic behavior of such physical parameters is given by meridional geometry which may be hyperbolic, parabolic or ellitpic (resp. positive, null or negative Gaussian curvature). A set of numerical results obtained via a ring finite element and a Lagrange collocation method adopting Fourier series decoupling of dependent variables in circumferential direction are presented. These results confirm the theoretical predictions.