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Research Papers

Modes of Wave Propagation and Dispersion Relations in a Cylindrical Shell

[+] Author and Article Information
Yu Cheng Liu

Graduate Institute of the Mechanical and Aeronautical Engineering, Feng Chia University, No. 100 Wenhwa Road, Seatwen, Taichung, Taiwan 40724, R.O.Cp9245215@fcu.edu.tw

Yun Fan Hwang

Electroacoustic Graduate Program, Feng Chia University, No. 100 Wenhwa Road, Seatwen, Taichung, Taiwan 40724, R.O.Cyfhwang@fcu.edu.tw

Jin Huang Huang

Department of Mechanical and Computer-Aided Engineering, Feng Chia University, No. 100 Wenhwa Road, Seatwen, Taichung, Taiwan 40724, R.O.Cjhhuang@fcu.edu.tw

J. Vib. Acoust 131(4), 041011 (Jun 08, 2009) (9 pages) doi:10.1115/1.2981172 History: Received April 29, 2008; Revised June 01, 2008; Published June 08, 2009

Abstract

This paper reinvestigates the classic problem of the dispersion relations of a cylindrical shell by obtaining a complete set of analytical solutions, based on Flügge’s theory, for all orders of circular harmonics, $n=0,1,2,…,∞$. The traditional numerical root search process, which requires considerable computational effort, is no longer needed. Solutions of the modal patterns (eigenvectors) for all propagating (and nonpropagating) modes are particularly emphasized, because a complete set of properly normalized eigenvectors are crucial for solving the vibration problem of a finite shell under various admissible boundary conditions. The dispersion relations and the associated eigenvectors are also the means by which to construct transfer matrices used to analyze the vibroacoustic transmission in cylindrical shell structures or pipe-hose systems. The eigenvectors obtained from the conventional method in shell analysis are not as conveniently normalized as those commonly used in mathematical physics. The present research proposes a new alternative method to find eigenvectors that are normalized such that their norms equal unity. A parallel display of the dispersion curves and the associated modal patterns has been used in the discussion and shown to provide a more insightful understanding of the wave phenomena in a cylindrical shell.

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Figures

Figure 1

Coordinates and displacement orientation.

Figure 2

Comparison between the eigenvectors (of Branch 1 roots) obtained from the conventional and the current methods (n=2). (a) The dispersion curve for Branch 1 roots; (b) the eigenvectors obtained from the current method expressed as [UVW]′; (c) comparison between the eigenvector obtained from the two methods.

Figure 3

Comparison between the eigenvectors (of Branch 2 roots) obtained from the conventional and the current methods (n=2). (a) The dispersion curve for Branch 2 roots; (b) the eigenvectors obtained from the current method expressed as [UVW]′; (c) comparison between the eigenvectors obtained from the two methods.

Figure 6

The dispersion relations for n=1 circular harmonic: (a) real part of the root; (b). imaginary part of the root

Figure 7

The normalized eigenvectors of n=1 circular harmonic: (a) the mode shapes of the Branch 1 roots; (b) the mode shapes of the Branch 2 roots; (c) the mode shapes of the Branch 3 roots; (d) the mode shapes of the Branch 4 roots

Figure 8

The dispersion relations for n=2 circular harmonic: (a) real part of the root; (b) imaginary part of the root; (c) the dispersion curves of Branch 1 and 2 below Ω=0.02

Figure 4

The dispersion relations for n=0 circular harmonic: (a) real parts of the root; (b) imaginary part of the root

Figure 5

The normalized eigenvectors of n=0 circular harmonic (no V component). (a) The mode shapes of the Branch 1 roots; (b) the mode shape of the Branch 2 roots; (c) the modes shape of the Branch 3 roots.

Figure 9

The normalized eigenvectors of n=2 circular harmonic: (a) the mode shapes of the Branch 1 roots; (b) the mode shapes of the Branch 2 roots; (c) the mode shapes of the Branch 3 roots; (d) the mode shapes of the Branch 4 roots

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