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A THREE‐DIMENSIONAL ANALYSIS OF THE SPHEROIDAL AND TOROIDAL ELASTIC VIBRATIONS OF THICK‐WALLED SPHERICAL BODIES OF REVOLUTION
Author(s) -
MCGEE O. G.,
SPRY S. C.
Publication year - 1997
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19970430)40:8<1359::aid-nme14>3.0.co;2-j
Subject(s) - toroid , ritz method , spherical shell , torsion (gastropod) , physics , geometry , normal mode , vibration , materials science , shell (structure) , mathematical analysis , classical mechanics , boundary value problem , mathematics , composite material , plasma , medicine , surgery , quantum mechanics
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 three‐dimensional 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. © 1997 by John Wiley & Sons, Ltd.