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

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, theta) at each circular cross section and the circumferential coordinate phi around the ring, is developed. Each of the displacement components u(r), v(theta), and w(phi) in the r, theta, and phi directions, respectively, is taken as a product of the Chebyshev polynomials in the r direction and the trigonometric functions in the theta and phi 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.