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Research Papers

Wave Based Method for Free and Forced Vibration Analysis of Cylindrical Shells With Discontinuity in Thickness

[+] Author and Article Information
Meixia Chen

School of Naval Architecture
and Ocean Engineering,
Huazhong University of Science and Technology,
Wuhan 430074, China
e-mail: chenmx26@hust.edu.cn

Kun Xie

School of Naval Architecture
and Ocean Engineering,
Huazhong University of Science and Technology,
Wuhan 430074, China
e-mail: xiekun79@163.com

Kun Xu

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

Peng Yu

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

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 29, 2014; final manuscript received March 3, 2015; published online April 27, 2015. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 137(5), 051004 (Apr 27, 2015) (14 pages) Paper No: VIB-14-1409; doi: 10.1115/1.4029995 History: Received October 29, 2014

Wave based method (WBM) is presented to analyze the free and forced vibration of cylindrical shells with discontinuity in thickness. The hull is first divided into multiple segments according to the locations of thickness discontinuity and/or driving points, and then the Flügge theory is adopted to describe the motion of cylindrical segments. The dynamic field variables in each segment are expressed as wave function expansions, which accurately satisfy the equations of motion and can be used to analyze arbitrary boundary conditions, e.g., classical or elastic boundary conditions. Finally, the boundary conditions and interface continuity conditions between adjacent segments are used to assemble the final governing equation to obtain the free and forced vibration results. By comparing with the results existing in open literate and calculated by finite element method (FEM), the present method WBM is verified. Furthermore, the influences of the boundary conditions and the locations of thickness discontinuity on the beam mode frequency and fundamental frequency are discussed. The effects of the direction of external force, location of external point force, and the structural damping on the forced vibration are also analyzed.

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References

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Figures

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Fig. 1

Geometry and global coordinate system for a cylindrical shell with N stepped thicknesses

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Fig. 2

Local coordinate system, displacement and force resultants

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Fig. 3

Interaction displacement and force resultants between adjacent segments

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Fig. 4

Mode shapes of a four-stepped cylindrical shell with m = 1 (h1/R = 0.01, h2/h1 = 2, h3/h1 = 3, h4/h1 = 4, L1/L = 0.25, L2/L = 0.25, L3/L = 0.25, and L4/L = 0.25): (a) L/R = 1 and (b) L/R = 5, —△— n=1, —▽— n=2, —□— n=3, —○— n=4, —◇— n=5, —◁— n=6, —▷— n=7, and —☆— n=8

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Fig. 5

Mode shapes of a four-stepped cylindrical shell with m = 2 (h1/R = 0.01, h2/h1 = 2, h3/h1 = 3, h4/h1 = 4, L1/L = 0.25, L2/L = 0.25, L3/L = 0.25, and L4/L = 0.25): (a) L/R = 1 and (b) L/R = 5, —△— n=1, —▽— n=2, —□— n=3, —○— n=4, —◇— n=5, —◁— n=6, —▷— n=7, and —☆— n=8

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Fig. 6

Beam mode frequency parameters versus the stiffness constant (h1/R = 0.01, h1/h2 = 0.5, and L1/L = 0.5): (a) L/R = 1, (b) L/R = 5, and (c) L/R = 10, —□— Ku, —○— Kv, —△— Kw, and —▽— Kθ

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Fig. 7

Fundamental frequency parameters versus the stiffness constant (h1/R = 0.01, h1/h2 = 0.5, and L1/L = 0.5): (a) L/R = 1, (b) L/R = 5, and (c) L/R = 10, —□— Ku, —○— Kv, —△— Kw, and —▽— Kθ

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Fig. 8

Beam mode frequency parameters versus the thickness variation ratio L1/L for SD–SD (h1/R = 1): (a) L/R = 1, (b) L/R = 5, and (c) L/R = 10, —□— h1/h2=2, —○— h1/h2=0.5, —△— h1/h2=4, and —▽— h1/h2=0.25

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Fig. 9

Fundamental frequency parameters versus the thickness variation ratio L1/L for SD–SD (h1/R = 1): (a) L/R = 1, (b) L/R = 5, and (c) L/R = 10, —□— h1/h2=2, —○— h1/h2=0.5, —△— h1/h2=4, and —▽— h1/h2=0.25

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Fig. 10

Beam mode frequency parameters versus the thickness variation ratio L1/L for C–F (h1/R = 1): (a) L/R = 1, (b) L/R = 5, and (c) L/R = 10, —□— h1/h2=2, —○— h1/h2=0.5, —△— h1/h2=4, and —▽— h1/h2=0.25

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Fig. 11

Fundamental frequency parameters versus the thickness variation ratio L1/L for C–F (h1/R = 1): (a) L/R = 1, (b) L/R = 5, and (c) L/R = 10, —□— h1/h2=2, —○— h1/h2=0.5, —△— h1/h2=4, and —▽— h1/h2=0.25

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Fig. 12

Effect of the truncated circumferential mode number n on the frequency response calculated by WBM at the driving point: (a) axial displacement and (b) radial displacement

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Fig. 13

Convergence of FEM of frequency response at the driving point: (a) axial displacement and (b) radial displacement

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Fig. 14

Comparison of frequency response of WBM and FEM at the driving point: (a) axial displacement and (b) radial displacement

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Fig. 15

Effect of the direction of the external force on the frequency response at driving point: (a) axial displacement and (b) radial displacement

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Fig. 16

Effect of the location of the external force on the frequency response at driving point: (a) axial displacement and (b) radial displacement

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Fig. 17

Effect of structural damping on the frequency response at the driving point: (a) axial displacement and (b) radial displacement

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