Three-Dimensional Vibration Analysis of Toroidal Sectors With Solid Circular Cross-Sections

This paper studies the free vibration of circular toroidal sectors with circular cross-sections based on the three-dimensional small-strain, linear elasticity theory. A set of orthogonal coordinates, composing the polar coordinate $(r,θ)$ with the origin on the cross-sectional centerline of the sector and the circumferential coordinate $φ$ with the origin at the curvature center of the centerline, is developed to describe the displacements, strains, and stresses in the sector. Each of the displacement components is taken as a product of four functions: a set of Chebyshev polynomials in $φ$ and $r$ coordinates, a set of trigonometric series in $θ$ coordinate, and a boundary function in terms of $φ$⁠. Frequency parameters and mode shapes have been obtained via a displacement-based extremum energy principle. The upper bound convergence of the first eight frequency parameters accurate up to five figures has been achieved. The present results agree with those from the finite element solutions. The effect of the ratio of curvature radius $R$ to the cross-sectional radius $a$ and the subtended angle $φ0$ on the frequency parameters of the sectors are discussed in detail. The three-dimensional vibration mode shapes are also plotted.