Three-dimensional vibration analysis of a torus with circular cross section

Abstract

The free vibration characteristics of a torus with a circular cross section are studied by using the three-dimensional, small-strain, elasticity theory. A set of three-dimensional orthogonal coordinates system, comprising the polar coordinate (r, θ) at each circular cross section and the circumferential coordinate φ around the ring, is developed. Each of the displacement components ur, vτ, and wφ in the r, θ, and φ directions, respectively, is taken as a product of the Chebyshev polynomials in the r direction and the trigonometric functions in the θ and φ directions. Eigenfrequencies and vibration mode shapes have been obtained via a three-dimensional displacement-based extremum energy principle. Upper bound convergence of the first seven eigenfrequencies accurate to at least six significant figures is obtained by using only a few terms of the admissible functions. The eigenfrequency responses due to variation of the ratio of the radius of the ring centroidal axis to the cross-sectional radius are investigated in detail. Very accurate eigenfrequencies and deformed mode shapes of the three-dimensional vibration are presented. All major modes such as flexural thickness-shear modes, in-plane stretching modes, and torsional modes are included in the analysis. The results may serve as a benchmark reference for validating other computational techniques for the problem. © 2002 Acoustical Society of America.