Modal properties and eigenvalue veering phenomena in the axisymmetric vibration of spheroidal shells

Abstract

The axisymmetric free vibration properties of arbitrarily slender thin spheroidal shells are investigated by implementing the method of assumed modes in conjunction with energy functionals derived from classical linear bending theory. Loci depicting the dependence of the natural frequencies on aspect ratio are constructed for aspect ratios ranging from a sphere to a prolate spheroid whose length is seven times its diameter. At certain aspect ratios, loci associated with different eigensolutions come close, then veer away without intersecting. Modal properties, such as the number of nodes in the mode associated with increasing root number for the eigenvalue, are found to change irregularly when veering occurs. An analysis of the partitioning of strain energy between membrane and bending effects is performed in conjunction with an earlier general study of eigenvalue veering phenomena. The analysis demonstrates that veering results in mixing of membrane and bending effects, such that it no longer is meaningful to classify the natural modes by their stored energy. A corollary of the analysis is the recognition that membrane shell theory ceases to be useful for spheroidal shells beyond an aspect ratio of 1.5. Further investigation shows that, in addition to physical causes, two eigenvalue loci might veer as the result of approximation error. The modal properties in such cases are interchanged between the nonintersecting loci, such that when viewed macroscopically all properties seem to change gradually along intersecting loci.