Behavior of Complex Propagation Roots in Thick and Thin Cylindrical Shells

Abstract

On the basis of linear elasticity analysis, the set of modes, which may exist in an infinitely long cylindrical shell, have real, pure imaginary, and complex propagation numbers. Previous investigations, except for a recent paper by McNiven et al. [J. Acoust. Soc. Am. 40, 1073–1976 (1966)], have concentrated almost entirely on the real branches, but the complex and imaginary branches are important when general boundary conditions of a finite shell are considered. The complex branches of the mode dispersion curves for elastic curves in an isotropic cylindrical shell occur at lower frequencies than in a solid cylinder of the same dimensions. The modes describe the end distortion displacement component in the characteristic vibrations of a finite cylinder. The change from thin to thick shell behavior is traced. The complex branch intersects on the imaginary plane for thin shells and on the real plane for thick shells. The exact transition from thin to thick shell behavior is a function of Poisson's ratio. For a Poisson's ratio of 0.3, it occurs at a thickness/mean radius ratio of 0.65 for the axially symmetric case. The results of approximate analysis based on thin-shell theory (Flügge) and thick-shell theory (Mirsky-Herrmann) are compared to three-dimensional linear elasticity analysis. For thin shells, the approximate analysis reproduces the full low-order three-dimensional branches in detail. [This investigation was performed at the Ordnance Research Laboratory under U. S. Naval contract with the Naval Ordnance Systems Command (NOSC) and the Office of Naval Research (ONR).]