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Research Papers

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

[+] Author and Article Information
D. Zhou

College of Civil Engineering, Nanjing University of Technology, Xinmofan Road, Nanjing 210009, People’s Republic of China

Y. K. Cheung, S. H. Lo

Department of Civil Engineering, University of Hong Kong, Porkfulam Road, Hong Kong, People’s Republic of China

J. Appl. Mech 77(4), 041002 (Mar 31, 2010) (8 pages) doi:10.1115/1.4000906 History: Received September 16, 2008; Revised December 10, 2009; Published March 31, 2010; Online March 31, 2010

## Abstract

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.

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## Figures

Figure 1

A circular toroidal sector with circular cross-section. (a) 3D view and (b) coordinate system.

Figure 2

The first three frequency parameters of symmetric and antisymmetric modes about the coordinate φ for the C-C sectors when the sectors make symmetric vibrations about their centerline plane. Solid line: the symmetric mode; dash line: the antisymmetric mode; ◻: φ0=120 deg; ◇:φ0=180 deg; △: φ0=240 deg.

Figure 3

The first three frequency parameters of symmetric and antisymmetric modes about coordinate φ for the C-C sectors when the sectors make antisymmetric vibrations about their centerline plane. Solid line: the symmetric mode; dash line: the antisymmetric mode; ◻: φ0=120 deg; ◇:φ0=180 deg; △: φ0=240 deg.

Figure 4

The first four modes symmetric about the centerline plane for a C-C sector with the subtended angle φ0=180 deg and the radius ratio R/a=2. The superscript s means the symmetric modes in the φ direction and the superscript a means the antisymmetric modes in the φ direction.

Figure 5

The first four modes antisymmetric about the centerline plane for a C-C sector with the subtended angle φ0=180 deg and the radius ratio R/a=2. The superscript s means the symmetric modes in the φ direction and the superscript a means the antisymmetric modes in the φ direction.

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