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Free vibration analysis of thin rotating cylindrical shells using wave propagation approach

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Abstract

The wave propagation approach is extended to study the frequency characteristics of thin rotating cylindrical shells. Based on Sanders’ shell theory, the governing equations of motion, which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotation, are derived. And, the displacement field is expressed in the form of wave propagation associated with an axial wavenumber k m and circumferential wavenumber n. Using the wavenumber of an equivalent beam with similar boundary conditions as the cylindrical shell, the axial wavenumber k m is determined approximately. Then, the relation between the natural frequency with the axial wavenumber and circumferential wavenumber is established, and the traveling wave frequencies corresponding to a certain rotating speed are calculated numerically. To validate the results, comparisons are carried out with some available results of previous studies, and good agreements are observed. Finally, the relative errors induced by the approximation using the axial wavenumber of an equivalent beam are evaluated with respect to different circumferential wavenumbers, length-to-radius ratios as well as thickness-to-radius ratios, and the conditions under which the analysis presented in this paper will be accurate are discussed.

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Abbreviations

E :

Young’s modulus

H :

Thickness of the cylinder

k m :

Axial wavenumber of traveling mode

K :

Flexural rigidity

L :

Length of the cylinder

m :

Order of the axial mode for a thin cylindrical shell or an equivalent beam with similar boundary conditions as the cylindrical shell

L L :

3-by-3 linear differential operator matrix base on Sanders’ shell theory for a non-rotating thin cylindrical shell

L R :

3-by-3 modifying operator matrix taking into account Coriolis and centrifugal forces as well as the initial hoop tension due to rotation

n :

Circumferential wavenumber of traveling mode

q :

Three-dimensional loading vector

R :

Mean radius of the cylinder

t :

Time

u, v, w :

Axial, circumferential and radial displacement

U :

Three-dimensional displacement vector

x, θ, z :

Axial, circumferential and radial coordinates

γ :

Quantity expressed as \({R\sqrt{\rho \left({1 - \mu^{2}}\right)/E}}\)

\({\bar{\delta}}\) :

Coefficient in the general governing equations of motion

η :

Non-dimensional quantity expressed as H 2/12R 2

μ :

Poisson’s ratio

ρ :

Density of the material

ω :

Natural frequency

Ω :

Angular velocity

ω*:

Non-dimensional frequency parameter

Ω*:

Non-dimensional rotating speed

Subscript b :

Backward wave

Subscript f :

Forward wave

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Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, PO Box 137, Harbin, 150001, People’s Republic of China

    Shupeng Sun, Dengqing Cao & Shiming Chu

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  1. Shupeng Sun
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  2. Dengqing Cao
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  3. Shiming Chu
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Correspondence to Dengqing Cao.

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Sun, S., Cao, D. & Chu, S. Free vibration analysis of thin rotating cylindrical shells using wave propagation approach. Arch Appl Mech 83, 521–531 (2013). https://doi.org/10.1007/s00419-012-0701-x

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