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Research Papers

Free Vibration Characteristics of a Finite Ring-Stiffened Elliptic Cylindrical Shell

[+] Author and Article Information
Min Fang

School of Naval Architecture
and Ocean Engineering,
Huazhong University of Science and Technology,
Luoyu Road 1037,
Wuhan 430074, China
e-mail: iverson_fm@hust.edu.cn

Xiang Zhu

School of Naval Architecture
and Ocean Engineering,
Huazhong University of Science and Technology,
Luoyu Road 1037,
Wuhan 430074, China
e-mail: zhuxiang@hust.edu.cn

Tianyun Li

School of Naval Architecture
and Ocean Engineering,
Huazhong University of Science and Technology,
Luoyu Road 1037,
Wuhan 430074, China
e-mail: ltyz801@hust.edu.cn

Guanjun Zhang

School of Energy and Power Engineering,
Wuhan University of Technology,
Heping Road 1040,
Wuhan 430063, China
e-mail: gjzhang_hust@126.com

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 9, 2017; final manuscript received May 2, 2017; published online August 2, 2017. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 139(6), 061012 (Aug 02, 2017) (13 pages) Paper No: VIB-17-1052; doi: 10.1115/1.4036870 History: Received February 09, 2017; Revised May 02, 2017

A theoretical method is employed to study the free vibration characteristics of a finite ring-stiffened elliptic cylindrical shell. Vibration equations of the elliptic cylindrical shell are derived based on Flügge shell theory, and the effects of the ring stiffeners are evaluated via “smeared” stiffener theory whereby the properties of the stiffeners are averaged over the shell surface. The displacements of the shell are expanded in double Fourier series in the axial and circumferential directions, and the circumferential curvature is expanded in single Fourier series in the circumferential direction. The partial differential characteristic equations with variable coefficients are converted into a set of linear equations with constant coefficients which couple with each other about the circumferential modal parameters. Then, the natural frequencies of the finite ring-stiffened cylindrical shell are obtained. To verify the accuracy of the present method, the finite ring-stiffened elliptic cylindrical shell is degenerated into two models: one of which is a ring-stiffened circular cylindrical shell and the other of which is an elliptic cylindrical shell without ring stiffeners. The present results of the two degenerated shells show good agreements with available results from the literature. The effects of main parameters, including the ellipticity, the shell length ratio, the stiffener's interval, the stiffener's depth, and the stiffener's eccentricity, on the free vibration of the ring-stiffened elliptic cylindrical shell are examined in detail. The ellipticity makes the difference between the symmetric and antisymmetric modal frequencies of the shell. The stiffeners have a greater influence on the free vibration at relatively higher order circumferential modal parameters. The circumferential modal parameters corresponding to the fundamental frequency are affected by the ellipticity, shell length, stiffeners' interval, and depth. The eccentricity of the ring stiffeners has a weak effect on the vibration of the structure.

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Figures

Grahic Jump Location
Fig. 1

Geometric parameters and coordinate system of the finite ring-stiffened elliptic cylindrical shell

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Fig. 2

The change of the nondimensional natural frequency versus truncated number (m = 1, n = 8): (a) L/mr0 = 3, d/mr0 = 0.6; (b) L/mr0 = 3, d/mr0 = 0.3; (c) L/mr0 = 3, d/mr0 = 0.1; (d) L/mr0 = 1, d/mr0 = 0.1; (e) L/mr0 = 10, d/mr0 = 0.1; and (f) L/mr0 = 20, d/mr0 = 0.1

Grahic Jump Location
Fig. 3

Mode shapes of the ring-stiffened elliptic cylindrical shell

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Fig. 4

The natural frequencies of simply supported internally ring-stiffened elliptic cylindrical shell (ε = 0.6) (— — represents symmetric modes;… … represents antisymmetric modes)

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Fig. 5

The change of the natural frequency of the finite ring-stiffened elliptic cylindrical shell versus ellipticity (— — represents symmetric modes;… … represents antisymmetric modes): (a) L/mr0 = 4.54 without ring; (b) L/mr0 = 4.54, d/mr0 = 0.6; (c) L/mr0 = 4.54, d/mr0 = 0.3; (d) L/mr0 = 4.54, d/mr0 = 0.167; (e) L/mr0 = 2, d/mr0 = 0.167; and (f) L/mr0 = 12, d/mr0 = 0.167

Grahic Jump Location
Fig. 6

The change of the natural frequency of finite ring-stiffened elliptic cylindrical shell versus the shell length ratio L/mr0 (— — represents symmetric modes;… … represents antisymmetric modes): (a) d/mr0 = 0.3, ε = 0; (b) d/mr0 = 0.3, ε = 0.2; (c) d/mr0 = 0.3, ε = 0.6; and (d) d/mr0 = 0.3, ε = 1

Grahic Jump Location
Fig. 7

The change of the natural frequency of finite ring-stiffened elliptic cylindrical shell versus the stiffeners' interval ratio d/mr0 (— — represents symmetric modes;… … represents antisymmetric modes): (a) L/mr0 = 4.54, ε = 0; (b) L/mr0 = 4.54, ε = 0.2; (c) L/mr0 = 4.54, ε = 0.6; and (d) L/mr0 = 4.54, ε = 1

Grahic Jump Location
Fig. 8

The change of the natural frequency of finite ring-stiffened elliptic cylindrical shell versus the stiffeners' depth–width ratio a1/b1 (— — represents symmetric modes;… … represents antisymmetric modes): (a) ε = 0, (b) ε = 0.2, (c) ε = 0.6, and (d) ε = 1

Grahic Jump Location
Fig. 9

The change of the natural frequency of finite ring-stiffened elliptic cylindrical shell versus the stiffeners' eccentricity c1 (— — represents symmetric modes;… … represents antisymmetric modes): (a) ε = 0, (b) ε = 0.2, (c) ε = 0.6, and (d) ε = 1

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