TY - JOUR

T1 - A three-dimensional analysis of the spheroidal and toroidal elastic vibrations of thick-walled spherical bodies of revolution

AU - Mcgee, O. G.

AU - Spry, S. C.

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

KW - Elastokinetics

KW - Spherical bodies of revolution

KW - Spheroidal

KW - Toroidal

UR - http://www.scopus.com/inward/record.url?scp=0031118983&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1097-0207(19970430)40:8<1359::AID-NME14>3.0.CO;2-J

DO - 10.1002/(SICI)1097-0207(19970430)40:8<1359::AID-NME14>3.0.CO;2-J

M3 - Article

AN - SCOPUS:0031118983

SN - 0029-5981

VL - 40

SP - 1359

EP - 1382

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

IS - 8

ER -