A three-dimensional analysis of the spheroidal and toroidal elastic vibrations of thick-walled spherical bodies of revolution

O. G. Mcgee, S. C. Spry

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This paper addresses the spheroidal (i.e. coupled bending-stretching) and toroidal (i.e. torsional or equivoluminal) elastic vibrations of thick-walled, spherical bodies of revolution by means of the threedimensional theory of elasticity in curvilinear (spherical) co-ordinates. Stationary values of the dynamical energies of the spherical body are obtained by the Ritz method using a complete set of algebraic-trigonometric polynomials to approximate the radial, meridional, and circumferential displacements. Extensive convergence studies of non-dimensional frequencies are presented for the spheroidal and toroidal modes of thin-walled spherical bodies of revolution. Results include all possible 3-D modes, i.e. radial stretching, combined bending-stretching, pure torsion, and shear deformable flexure through the wall thickness (including thickness-shear, thickness-stretch, and thickness-twist). It is shown that the assumed displacement polynomials yield a strictly upper-bound convergence to exact solutions of the title problem, as a sufficient number of terms is retained. Since the effects of transverse shear and rotary inertia are inherent to the present 3-D formulation, an examination is made of the variation of non-dimensional frequencies with non-dimensional wall thickness, h/R ranging from thin-walled (h/R = 0.05) to thick-walled (h/R = 0.5) spherical bodies. The findings confirm that the variation of the spheroidal frequencies increases with increasing h/R and mode number, whereas the variation of the toroidal frequencies decreases with increasing h/R and mode number. This work offers some accurate 3-D reference data for the title problem with which refined solutions drawn from thin and thick shell theories and sophisticated finite element techniques may be compared.

Original languageEnglish
Pages (from-to)1359-1382
Number of pages24
JournalInternational Journal for Numerical Methods in Engineering
Issue number8
StatePublished - 1997


  • Elastokinetics
  • Spherical bodies of revolution
  • Spheroidal
  • Toroidal


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