research article free vibration analysis of circular...

Research ArticleFree Vibration Analysis of Circular CylindricalShells with Arbitrary Boundary Conditions by the Method ofReverberation-Ray Matrix

Dong Tang, Guoxun Wu, Xiongliang Yao, and Chuanlong Wang

College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Guoxun Wu; [email protected]

Received 14 November 2015; Accepted 6 January 2016

Academic Editor: Lorenzo Dozio

Copyright © 2016 Dong Tang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An analytical procedure for free vibration analysis of circular cylindrical shells with arbitrary boundary conditions is developedwith the employment of the method of reverberation-ray matrix. Based on the Flugge thin shell theory, the equations of motion aresolved and exact solutions of the traveling wave form along the axial direction and the standing wave form along the circumferentialdirection are obtained. With such a unidirectional traveling wave form solution, the method of reverberation-ray matrix isintroduced to derive a unified and compact form of equation for natural frequencies of circular cylindrical shells with arbitraryboundary conditions. The exact frequency parameters obtained in this paper are validated by comparing with those given by otherresearchers. The effects of the elastic restraints on the frequency parameters are examined in detail and some novel and usefulconclusions are achieved.

1. Introduction

Circular cylindrical shells are widely used in engineeringstructures, such as rockets, aircrafts, submarines, and pipe-lines. Due to the extensive application, the vibration char-acteristics of circular cylindrical shells have been a topic ofmajor interest to many researchers. A comprehensive reviewof various thin shell theories developed before 1973 waspresented by Leissa [1], in which a few solution techniqueshave been developed for some types of boundary conditionsas well.

A large number of researches on vibration characteris-tics of circular cylindrical shells with various methods areavailable in the literature. Chang and Greif [2] studied thevibrations of amultisegment cylindrical shell with a commonmean radius based on the component mode method coupledwith Fourier series and Lagrange multipliers. Chung [3]presented a general analytical method for evaluating the freevibration characteristics of a circular cylindrical shell withclassical boundary conditions of any type based on Sanders’shell equations. Lam and Loy [4] used a formulation basedon Love’s first approximation theory andwith beam functions

used as axial modal functions in the Ritz procedure to studythe effects of various classical boundary conditions on thefree vibration characteristics for a multilayered cylindricalshell. Mirza and Alizadeh [5] investigated the effects ofdetached base length on the natural frequencies and modalshapes of cylindrical shells. Zhou and Yang [6] proposed adistributed transfer function method for static and dynamicanalysis of general cylindrical shells. Loy et al. [7] presentedthe free vibration analysis of cylindrical shells using animproved version of the differential quadrature method. Yimet al. [8] extended the receptance method to a clamped-free cylindrical shell with a plate attached in the shell atan arbitrary axial position. An analytical procedure to studythe free vibration characteristics of thin circular cylindricalshells was presented by Naeem and Sharma [9], in which Ritzpolynomial functions are assumed to model the axial modaldependence and the Rayleigh-Ritz variational approach isemployed to formulate the general eigenvalue problem. Basedon Love’s equations and an infinite length model, Wang andLai [10] introduced a novel wave approach to predict thenatural frequencies of finite length circular cylindrical shellswith different boundary conditions without simplifying the

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 3814693, 18 pageshttp://dx.doi.org/10.1155/2016/3814693

2 Shock and Vibration

equations of motion. Ghoshal et al. [11] and Zhang et al.[12, 13] presented the vibration analysis of cylindrical shellsusing a wave propagation method. El-Mously [14] presenteda comparative study of three approximate explicit formulaefor estimating the fundamental natural frequency of a thincylindrical shell and its associated fundamental mode num-ber. Karczub [15] presented algebraic expressions for directevaluation of structural wave number in cylindrical shellsbased on the Flugge equations of motion. Mofakhami et al.[16] developed a general solution to analyze the vibration offinite circular cylinders utilizing the infinite circular cylinderssolution based on the technique of variables separation.Sakka et al. [17] presented an analytical investigation to thefree vibration response of moderately thick shear flexibleisotropic cylindrical shells with all edges clamped. Based onthe Flugge thin shell theory, Li [18] obtained the free vibrationfrequencies of a thin circular cylindrical shell using a newmethod which introduces a general displacement represen-tation and develops a type of coupled polynomial eigenvalueproblem. Pellicano [19] presented a method for analyzinglinear and nonlinear vibrations of circular cylindrical shellshaving different boundary conditions based on the Sanders-Koiter theory. Based on the Flugge thin shell theory, Zhangand Xiang [20] presented exact solutions for the vibration ofcircular cylindrical shells with stepwise thickness variationsin the axial direction. Li [21] presented a wave propagationapproach for free vibration analysis of circular cylindricalshell based on Flugge classical thin shell theory.

Recently, Brischetto and Carrera [22] carried out the freevibration analysis of one-layered and two-layered metalliccylindrical shell panels for both thermomechanical and puremechanical problems. Based on the Flugge thin shell theory,Zhou et al. [23] investigated the free vibrations of cylindricalshells with elastic boundary conditions by the method ofwave propagations. Chen et al. [24] presented a wave basedmethod to analyze the free vibration characteristics of ring-stiffened cylindrical shell with intermediate large frame ribsfor arbitrary boundary conditions. Chen et al. [25] investi-gated the free vibrations of cylindrical shell with nonuniformelastic boundary constraints using the improved Fourierseriesmethod. Dai et al. [26] obtained an exact series solutionfor the vibration analysis of circular cylindrical shells witharbitrary boundary conditions by using the elastic equationsbased on the Flugge thin shell theory. Qu et al. [27, 28]presented a domain decomposition technique for solvingvibration problems of uniform and stepped cylindrical shellsas well as isotropic and composite cylindrical shells with arbi-trary boundary conditions. Based on the Donnell-Mushtarishell theory, Xing et al. [29] presented exact solutions of asimple and compact form for free vibration of circular cylin-drical shells with classical boundary conditions. Chen et al.[30] presented an analytic method to analyze free and forcedvibration characteristics of ring-stiffened combined conical-cylindrical shells with arbitrary boundary conditions. Li et al.[31] investigated the influence on modal parameters of thincylindrical shell under bolt looseness boundary. Yao et al.[32] carried out the free vibration analysis of open circularcylindrical shells with either the two straight edges or the twocurved edges simply supported and the remaining two edges

z12 x12

𝜃12

R

h

L

x21

𝜃21 z21

Figure 1: Geometry and the dual local coordinate system for acircular cylindrical shell.

supported by arbitrary classical boundary conditions basedon the Donnell-Mushtari-Vlasov thin shell theory.

As far as the researches of circular cylindrical shells areconcerned, most of the existing works are limited to classicalboundary conditions. Some efforts are recentlymade to studythe vibration characteristics of circular cylindrical shells witharbitrary boundary conditions. Based on the Flugge thin shelltheory, this paper presents a unified and compact formula-tion for free vibration analysis of circular cylindrical shellswith arbitrary boundary conditions including classical onesand nonclassical ones using the method of reverberation-ray matrix (MRRM). With the analytical solution of thetravelingwave form along the axial direction and the standingwave form along the circumferential direction obtainedby Yao et al., MRRM is introduced to derive the unifiedequation for natural frequencies of the circular cylindricalshell with arbitrary boundary conditions. To validate themethod presented in this paper, some results for both classicalboundary conditions and nonclassical boundary conditionsare compared with those found in the published literature.The effects of the elastic restraints on the natural frequenciesare examined in detail and some novel and useful conclusionsare achieved at the end of this paper.

2. Formulation

Consider an isotropic circular cylindrical shell with length𝐿, uniform thickness ℎ, middle surface radius 𝑅, Young’smodulus 𝐸, Poisson’s ratio 𝜇, and mass density 𝜌 as shown inFigure 1. The axial, circumferential, and radial displacementsof the middle surface of the circular cylindrical shell withreference to the coordinate system are denoted as 𝑢(𝑥, 𝜃, 𝑡),V(𝑥, 𝜃, 𝑡), and 𝑤(𝑥, 𝜃, 𝑡). The problem at hand is to determinethe natural frequencies of the circular cylindrical shell witharbitrary boundary conditions.

2.1. Force and Moment Resultants. The force and momentresultants acting on the cross section perpendicular to theaxial direction in a circular cylindrical shell, shown in Fig-ure 2, are expressed in terms of the displacement components𝑢, V, and 𝑤 as

𝑁𝑥= 𝐶[

𝜕𝑢

𝜕𝑥+ 𝜇

1

𝑅

𝜕V𝜕𝜃

− 𝜆𝑅𝜕2

𝑤

𝜕𝑥2+ 𝜇

1

𝑅𝑤] , (1)

Shock and Vibration 3

x

Nxx +𝜕Nxx

𝜕x

Vxz +𝜕Vxz𝜕x

Fx𝜃 +𝜕Fx𝜃𝜕x

Mx𝜃 +𝜕Mx𝜃

𝜕x

Mxx +𝜕Mxx

𝜕x

Mx𝜃

Fx𝜃

Vxz

Nxx

Mxx

𝜃

Figure 2: Force and moment resultants in a circular cylindricalshell.

𝑁𝑥𝜃=1 − 𝜇

2𝐶[

1

𝑅

𝜕𝑢

𝜕𝜃+ (1 + 𝜆)

𝜕V𝜕𝑥

− 𝜆𝜕2

𝑤

𝜕𝑥𝜕𝜃] , (2)

𝑀𝑥= 𝐷[

1

𝑅

𝜕𝑢

𝜕𝑥+ 𝜇

1

𝑅2

𝜕V𝜕𝜃

−𝜕2

𝑤

𝜕𝑥2− 𝜇

1

𝑅2

𝜕2

𝑤

𝜕𝜃2] , (3)

𝑀𝑥𝜃= (1 − 𝜇)𝐷(

1

𝑅

𝜕V𝜕𝑥

−1

𝑅

𝜕2

𝑤

𝜕𝑥𝜕𝜃) , (4)

𝑄𝑥𝑧= 𝐷[

1

𝑅

𝜕2

𝑢

𝜕𝑥2− (1 − 𝜇)

1

2

1

𝑅3

𝜕2

𝑢

𝜕𝜃2+1 + 𝜇

2

1

𝑅2

𝜕2V

𝜕𝑥𝜕𝜃

−𝜕3

𝑤

𝜕𝑥3−1

𝑅2

𝜕3

𝑤

𝜕𝑥𝜕𝜃2] ,

(5)

𝐹𝑥𝜃=1 − 𝜇

2𝐶[

1

𝑅

𝜕𝑢

𝜕𝜃+ (1 + 3𝜆)

𝜕V𝜕𝑥

− 3𝜆𝜕2

𝑤

𝜕𝑥𝜕𝜃] , (6)

𝑉𝑥𝑧= 𝐷[

1

𝑅

𝜕2

𝑢

𝜕𝑥2− (1 − 𝜇)

1

2

1

𝑅3

𝜕2

𝑢

𝜕𝜃2+3 − 𝜇

2

1

𝑅2

𝜕2V

𝜕𝑥𝜕𝜃

−𝜕3

𝑤

𝜕𝑥3− (2 − 𝜇)

1

𝑅2

𝜕3

𝑤

𝜕𝑥𝜕𝜃2] ,

(7)

where 𝑢, V, and 𝑤 denote the displacement components inthe axial (𝑥), circumferential (𝜃), and radial (𝑧) directions.𝐶 = 𝐸ℎ/(1 − 𝜇

2

) and 𝐷 = 𝐸ℎ3

/12(1 − 𝜇2

) represent themiddle surface stiffness and the bending stiffness of the shell.𝑁𝑥and 𝑁

𝑥𝜃denote the in-plane normal force and in-plane

shear force.𝑀𝑥and𝑀

𝑥𝜃represent the bending moment and

torsional moment. 𝑄𝑥𝑧

denotes the out-of-plane shear force.𝐹𝑥𝜃= 𝑁𝑥𝜃+𝑀𝑥𝜃/𝑅 and𝑉

𝑥𝑧= 𝑄𝑥𝑧+𝜕𝑀𝑥𝜃/𝑅𝜕𝜃, respectively,

represent the in-plane andout-of-plane resultant shear forces.

2.2. Governing Differential Equations. Based on the Fluggethin shell theory, the governing differential equations for freevibration of a circular cylindrical shell are written as

𝑅2𝜕2

𝑢

𝜕𝑥2+ (1 + 𝜆)

1 − 𝜇

2

𝜕2

𝑢

𝜕𝜃2+1 + 𝜇

2𝑅𝜕2V

𝜕𝑥𝜕𝜃

− 𝜆𝑅3𝜕3

𝑤

𝜕𝑥3+ 𝜆

1 − 𝜇

2𝑅𝜕3

𝑤

𝜕𝑥𝜕𝜃2+ 𝜇𝑅

𝜕𝑤

𝜕𝑥

− 𝑅2𝜌 (1 − 𝜇

2

)

𝐸

𝜕2

𝑢

𝜕𝑡2= 0,

(8)

1 + 𝜇

2𝑅𝜕2

𝑢

𝜕𝑥𝜕𝜃+ (1 + 3𝜆)

1 − 𝜇

2𝑅2𝜕2V𝜕𝑥2

+𝜕2V𝜕𝜃2

− 𝜆3 − 𝜇

2𝑅2𝜕3

𝑤

𝜕𝑥2𝜕𝜃+𝜕𝑤

𝜕𝜃− 𝑅2𝜌 (1 − 𝜇

2

)

𝐸

𝜕2V𝜕𝑡2

= 0,

(9)

− 𝜆𝑅3𝜕3

𝑢

𝜕𝑥3+ 𝜆

1 − 𝜇

2𝑅𝜕3

𝑢

𝜕𝑥𝜕𝜃2+ 𝜇𝑅

𝜕𝑢

𝜕𝑥

−3 − 𝜇

2𝜆𝑅2𝜕3V

𝜕𝑥2𝜕𝜃+𝜕V𝜕𝜃

+ 𝜆𝑅4𝜕4

𝑤

𝜕𝑥4

+ 2𝜆𝑅2𝜕4

𝑤

𝜕𝑥2𝜕𝜃2+ 𝜆

𝜕4

𝑤

𝜕𝜃4+ 2𝜆

𝜕2

𝑤

𝜕𝜃2+ (1 + 𝜆)𝑤

+ 𝑅2𝜌 (1 − 𝜇

2

)

𝐸

𝜕2

𝑤

𝜕𝑡2= 0,

(10)

where 𝜆 = ℎ2/12𝑅2 is a dimensionless parameter related withthe ratio of the shell thickness to shell radius.

The Fourier transform of an arbitrary physical quantity𝑋(𝑡) is defined by

�� (𝜔) = ∫

+∞

−∞

𝑋 (𝑡) 𝑒−𝑖𝜔𝑡

𝑑𝑡, (11)

where 𝜔 is the circular frequency, and a tilde over a symbolrepresents the corresponding quantity expressed in the fre-quency domain.

Taking the Fourier transforms of (8)∼(10) and elimi-nating the other two displacement components from theFourier transforms, three independent equations of motion,respectively, in terms of the axial, circumferential, and radialdisplacement components are obtained as

[(𝐿4𝐿6− 𝐿3𝐿3) (𝜇1𝐿8− 𝐿5𝐿7)

+ (𝐿1𝐿5− 𝐿4𝐿7) (𝐿2𝐿7− 𝐿5𝐿6)]

𝜕��

𝑅𝜕𝜃= 0,

(12)

[(𝐿4𝐿6− 𝐿3𝐿3) (𝜇1𝐿8− 𝐿5𝐿7)

+ (𝐿1𝐿5− 𝐿4𝐿7) (𝐿2𝐿7− 𝐿5𝐿6)]𝜕V𝜕𝑥

= 0,

(13)

[(𝐿4𝐿6− 𝐿3𝐿3) (𝜇1𝐿8− 𝐿5𝐿7)

+ (𝐿1𝐿5− 𝐿4𝐿7) (𝐿2𝐿7− 𝐿5𝐿6)]

𝜕2

��

𝑅𝜕𝑥𝜕𝜃

= 0,

(14)


where 𝐿𝑗(𝑗 = 1 ∼ 6) are linear differential operators defined

in Appendix A and 𝜇1is a dimensionless parameter defined

in Appendix B.It can be observed from (12)∼(14) that the differences

among the linear differential operators operating on theaxial, circumferential, and radial displacement componentslie in the parts outside the braces, which represent zero wavenumber solutions or, in other words, the rigid-body motions.As normal in free vibration analysis, the zero wave numbersolutions are neglected throughout this paper.

2.3. Solutions for the Circular Cylindrical Shell. Solutions tothe equations of motion of the circular cylindrical shell aregenerally assumed in the following forms:

�� (𝑥, 𝜃) =

∞

∑

𝑛=0

𝑈 (𝑥) cos (𝑛𝜃) ,

V (𝑥, 𝜃) =∞

∑

𝑛=0

𝑉 (𝑥) sin (𝑛𝜃) ,

�� (𝑥, 𝜃) =

∞

∑

𝑛=0

𝑊(𝑥) cos (𝑛𝜃) ,

(15)

where 𝑛 is the circumferential mode number and𝑈(𝑥),𝑉(𝑥),and𝑊(𝑥) are axial, circumferential, and radial displacementcomponent functions of the axial variable. Assuming that theaxial, circumferential, and radial displacement componentscan be expressed by exponential functions of the axial var-iable, a common axial wave number equation for the axial,circumferential, and radial displacement components isobtained as

(𝜆2𝜇2𝑘4

𝑥+ 𝜁2

1𝑘2

𝑥+ 𝜁2

2𝜁2

3) (𝜆3𝑘4

𝑥+ 𝜁2

4𝑘2

𝑥+ 𝜁4

5)

+1

𝜆(2𝜆𝜇2𝑘4

𝑥+ 𝜁2

6𝑘2

𝑥+ 𝜁2

2𝜁2

7) (𝜆𝜆2𝜇2𝑘4

𝑥+ 𝜁2

8𝑘2

𝑥+ 𝜁4

9)

= 0,

(16)

where 𝜇2, 𝜆2, and 𝜆

3are dimensionless parameters and

𝜁𝑗(𝑗 = 1 ∼ 9) are parameters of the same dimension to the

axial wave number. They are presented in detail inAppendix B.

By solving (16), eight axial wave numbers are obtained as

𝑘𝑥= ±𝑘𝑥1, ±𝑘𝑥2, ±𝑘𝑥3, ±𝑘𝑥4. (17)

Therefore, the solutions for the displacement componentsof an isotropic circular cylindrical shell can be explicitlyrewritten as

�� (𝑥, 𝜃) =

∞

∑

𝑛=0

4

∑

𝑗=1

𝛼𝑗(𝑎𝑗𝑒𝑖𝑘𝑥𝑗𝑥

− 𝑑𝑗𝑒−𝑖𝑘𝑥𝑗𝑥

) cos (𝑛𝜃) , (18)

V (𝑥, 𝜃) =∞

∑

𝑛=0

4

∑

𝑗=1

𝛽𝑗(𝑎𝑗𝑒𝑖𝑘𝑥𝑗𝑥

+ 𝑑𝑗𝑒−𝑖𝑘𝑥𝑗𝑥

) sin (𝑛𝜃) , (19)

�� (𝑥, 𝜃) =

∞

∑

𝑛=0

4

∑

𝑗=1

(𝑎𝑗𝑒𝑖𝑘𝑥𝑗𝑥


) cos (𝑛𝜃) , (20)

where 𝛼𝑗and 𝛽

𝑗(𝑗 = 1, 2, 3, 4) are ratios of the axial and

circumferential wave amplitudes to the radial wave ampli-tude. The expressions of 𝛼

𝑗and 𝛽

𝑗are defined as follows:

𝛼𝑗=

𝑖𝑅𝑘𝑥𝑗(𝜆𝜆2𝜇2𝑘4

𝑥𝑗+ 𝜁2

8𝑘2

𝑥𝑗+ 𝜁4

9)

(𝜆2𝜇2𝑘4

𝑥𝑗+ 𝜁2

1𝑘2

𝑥𝑗+ 𝜁2

2𝜁2

3)

,

𝛽𝑗= −

𝑛 (2𝜆𝜇2𝑘4

𝑥𝑗+ 𝜁2

6𝑘2

𝑥𝑗+ 𝜁2

2𝜁2

7)

(𝜆2𝜇2𝑘4

𝑥𝑗+ 𝜁2

1𝑘2

𝑥𝑗+ 𝜁2

2𝜁2

3)

.

(21)

The rotation of the normal to the middle surface aboutthe circumferential direction is defined as

𝜑𝑥= −

𝜕𝑤

𝜕𝑥. (22)

Substituting (20) into the Fourier transform of (22) yieldsthe frequency domain expression of the rotation

��𝑥=

∞

∑

𝑛=0

4

∑

𝑗=1

− 𝑖𝑘𝑥𝑗(𝑎𝑗𝑒𝑖𝑘𝑥𝑗𝑥


) cos (𝑛𝜃) . (23)

Substituting (18)∼(20) into the Fourier transforms of (1),(3), (6), and (7) yields the frequency domain expressions ofthe force and moment resultants of the circular cylindricalshell:

��𝑥=𝐶

𝑅

∞

∑

𝑛=0

4

∑

𝑗=1

(𝜇 + 𝜆𝑅2

𝑘2

𝑥𝑗+ 𝑖𝑅𝑘𝑥𝑗𝛼𝑗+ 𝜇𝑛𝛽

𝑗)

⋅ (𝑎𝑗𝑒𝑖𝑘𝑥𝑗𝑥


) cos (𝑛𝜃) ,

(24)

��𝑥𝜃= 𝜇2

𝐶

𝑅

∞

∑

𝑛=0

4

∑

𝑗=1

(3𝜆𝑛𝑖𝑅𝑘𝑥𝑗− 𝑛𝛼𝑗+ 𝜆2𝑖𝑅𝑘𝑥𝑗𝛽𝑗)



) sin (𝑛𝜃) ,

(25)

��𝑥𝑧=𝐷

𝑅3

∞

∑

𝑛=0

4

∑

𝑗=1

[(2 − 𝜇) 𝑛2

𝑖𝑅𝑘𝑥𝑗+ 𝑖𝑅3

𝑘3

𝑥𝑗

+ (𝜇2𝑛2

− 𝑅2

𝑘2

𝑥𝑗) 𝛼𝑗+ 𝜇3𝑛𝑖𝑅𝑘𝑥𝑗𝛽𝑗] (𝑎𝑗𝑒𝑖𝑘𝑥𝑗𝑥


) cos (𝑛𝜃) ,

(26)

��𝑥=𝐷

𝑅2

∞

∑

𝑛=0

4

∑

𝑗=1

(𝜇𝑛2

+ 𝑅2

𝑘2

𝑥𝑗+ 𝑖𝑅𝑘𝑥𝑗𝛼𝑗+ 𝜇𝑛𝛽

𝑗)



) cos (𝑛𝜃) .

(27)

2.4. Solutions Expressed in Matrix Form. For an arbitrarycircumferential mode number 𝑛, (18)∼(20) and (23) can beexpressed in matrix form as

W𝑑= H𝑛(𝜃)W∗

𝑑, (28)


whereW𝑑denotes the displacement vector of the shell,H

𝑛(𝜃)

represents the circumferential mode matrix, andW∗𝑑denotes

the wave vector of the displacement with respect to the axialvariable. They are presented in detail as follows:

W𝑑= {�� V ��

𝑥}𝑇

, (29)

H𝑛(𝜃)

= diag {cos (𝑛𝜃) sin (𝑛𝜃) cos (𝑛𝜃) cos (𝑛𝜃)} ,(30)

W∗𝑑= A𝑑Pℎ(𝑥) a +D

𝑑Pℎ(−𝑥) d (31)

in which Pℎ(𝑥) denotes the phase matrix, A

𝑑and D

𝑑are

coefficientmatrices of fourth order, and a andd are amplitudevectors of the arriving wave and the departing wave corre-sponding to the displacement vector. They are presented indetail as follows:

Pℎ(𝑥) = diag {𝑒𝑖𝑘𝑥1𝑥 𝑒𝑖𝑘𝑥2𝑥 𝑒𝑖𝑘𝑥3𝑥 𝑒𝑖𝑘𝑥4𝑥} ,

a = {𝑎1𝑎2𝑎3𝑎4}𝑇

,

d = {𝑑1𝑑2𝑑3𝑑4}𝑇

,

A𝑑(1, 𝑗) = −D

𝑑(1, 𝑗) = 𝛼

𝑗,

A𝑑(2, 𝑗) = D

𝑑(2, 𝑗) = 𝛽

𝑗,

A𝑑(3, 𝑗) = D

𝑑(3, 𝑗) = 1,

A𝑑(4, 𝑗) = −D

𝑑(4, 𝑗) = −𝑖𝑘

𝑥𝑗,

(32)

where 𝑗 = 1, 2, 3, 4.In the same manner, (24)∼(27) are expressed in matrix

form as

W𝑓= H𝑛(𝜃)W∗

𝑓, (33)

where H𝑛(𝜃) is defined in (30). W

𝑓denotes the force vector

of the shell, andW∗𝑓denotes the wave vector of the force and

moment resultants with respect to the axial variable.They arepresented in detail as follows:

W𝑓= {��𝑥��𝑥𝜃

��𝑥𝑧

��𝑥}𝑇

,

W∗𝑓= A𝑓Pℎ(𝑥) a +D

𝑓Pℎ(−𝑥) d

(34)

in which the physical significances and expressions of Pℎ(𝑥),

a, and d are the same to those presented in (31). A𝑓and D

𝑓

are coefficientmatrices of the arrivingwave and the departingwave corresponding to the force andmoment resultants of theshell. They are presented in detail as follows:

A𝑓(1, 𝑗) = D

𝑓(1, 𝑗) =

(𝜇 + 𝜆𝑅2

𝑘2

𝑥𝑖+ 𝑖𝑅𝑘𝑥𝑖𝛼𝑖+ 𝜇𝑛𝛽

𝑖) 𝐶

𝑅,

A𝑓(2, 𝑗) = −D

𝑓(2, 𝑗) =

(3𝜆𝑛𝑖𝑅𝑘𝑥𝑖− 𝑛𝛼𝑖+ 𝜆2𝑖𝑅𝑘𝑥𝑖𝛽𝑖) 𝜇2𝐶

𝑅,

A𝑓(3, 𝑗) = −D

𝑓(3, 𝑗) =

[(2 − 𝜇) 𝑛2

𝑖𝑅𝑘𝑥𝑖+ 𝑖𝑅3

𝑘3

𝑥𝑖+ (𝜇2𝑛2

− 𝑅2

𝑘2

𝑥𝑖) 𝛼𝑖+ 𝜇3𝑛𝑖𝑅𝑘𝑥𝑖𝛽𝑖]𝐷

𝑅3,

A𝑓(4, 𝑗) = D

𝑓(4, 𝑗) =

(𝜇𝑛2

+ 𝑅2

𝑘2

𝑥𝑖+ 𝑖𝑅𝑘𝑥𝑖𝛼𝑖+ 𝜇𝑛𝛽

𝑖)𝐷

𝑅2,

(35)

where 𝑗 = 1, 2, 3, 4.

2.5. Equation of Natural Frequencies. Taking advantage ofthe unidirectional traveling wave solutions of the circularcylindrical shell obtained in the preceding subsections, theMRRM is introduced to derive the equation of naturalfrequencies. Since the scatteringmatrix is related to boundaryconditions of the shell, it will be discussed in the first step.Then, the phase matrix and permutation matrix, which areindependent of the boundary conditions, are derived. Finally,the reverberation-ray matrix and the equation for naturalfrequencies of the circular cylindrical shell are obtained. Theformulation mentioned above is presented in detail in thefollowing discussions.

2.5.1. Scattering Matrix for Various Boundary Conditions. Invarious boundary conditions including classical ones andnonclassical ones, for instance, the elastic-support boundaryconditions can be expressed in a unified compact form at thelocal coordinate (𝑜𝑥𝜃𝑧)12 as

K1𝑓W12𝑓= K1𝑑W12𝑑, (36)

whereK1𝑓andK1

𝑑are coefficient matrices of fourth order. For

common boundary conditions, the coefficient matrices K1𝑓

and K1𝑑are presented in Table 1.


Table 1: Coefficient matrices for common boundary conditions.

BCs K1𝑓

K1𝑑

CE diag {0 0 0 0} diag {1 1 1 1}

FE diag {1 1 1 1} diag {0 0 0 0}

SS diag {1 0 0 1} diag {0 1 1 0}

ES diag {1 1 1 1} diag {𝑘𝑢𝑘V 𝑘𝑤𝑘𝜑}

EI diag {1 0 0 0} diag {𝑘𝑢0 0 0}

EII diag {0 1 0 0} diag {0 𝑘V 0 0}

EIII diag {1 1 0 0} diag {𝑘𝑢𝑘V 0 0}

Substituting (28) and (33) into (36) yields

K1𝑓[A𝑓Pℎ(0) a12 +D

𝑓Pℎ(0) d12]

= K1𝑑[A𝑑Pℎ(0) a12 +D

𝑑Pℎ(0) d12] .

(37)

Since the phase matrix turns into a unit matrix at theorigin of the local coordinate, which is obvious from its def-inition, (37) can be rewritten as

d1 = S1a1, (38)

where d1 = d12 = {𝑑12

1𝑑12

2𝑑12

3𝑑12

4}𝑇 and a1 = a12 =

{𝑎12

1𝑎12

2𝑎12

3𝑎12

4}𝑇 are amplitude vectors of the depart-

ing wave and the arriving wave and S1 = −(K1𝑑D𝑑−

K1𝑓D𝑓)−1

(K1𝑑A𝑑− K1𝑓A𝑓) is the scattering matrix at one end

of the circular cylindrical shell.Similarly, the scattering relation for the other end of the

circular cylindrical shell can be obtained as

d2 = S2a2, (39)

where d2 = d21 = {𝑑21

1𝑑21

2𝑑21

3𝑑21

4}𝑇 and a2 = a21 =

{𝑎21

1𝑎21

2𝑎21

3𝑎21

4}𝑇 are amplitude vectors of the depart-

ing wave and the arriving wave and S2 = −(K2𝑑D𝑑−

K2𝑓D𝑓)−1

(K2𝑑A𝑑−K2𝑓A𝑓) is the scattering matrix at the other

end.Assembling the local scattering equations at both ends of

the circular cylindrical shell by stacking d1, d2 and a1, a2 intotwo column vectors d and a, the global scattering equation isobtained as

d = Sa, (40)

where d = {(d1)𝑇 (d2)𝑇}𝑇

and a = {(a1)𝑇 (a2)𝑇}𝑇

are globalamplitude vectors of the departing wave and the arrivingwave and S = diag {S1 S2} is the global scattering matrix.

2.5.2. Phase Matrix and Permutation Matrix. The phase rela-tions of harmonic waves in the dual local coordinate systemof MRRM provide additional equations for solving theunknown amplitude vectors. Note that the departing wavefrom one end of the circular cylindrical shell is exactly thearriving wave to the other end, and vice versa. Therefore, the

amplitudes of the departing wave and the arriving wave differwith each other by a phase factor. The relations between theamplitudes of the departing wave and the arriving wave arepresented as

𝑎12

1= −𝑒−𝑖𝑘𝑥1𝐿

𝑑21

1;

𝑎12


𝑑21

2;

𝑎12


𝑑21

3;

𝑎12


𝑑21

4

𝑎21


𝑑12

1;

𝑎21


𝑑12

2;

𝑎21


𝑑12

3;

𝑎21


𝑑12

4

(41)

which can be rewritten in matrix form as

a1 = −Pℎ(−𝐿) d2,

a2 = −Pℎ(−𝐿) d1.

(42)

Assembling both local phase equations results in theglobal phase equation

a = Pd∗, (43)

where a is defined in (40), d∗ = {(d2)𝑇 (d1)𝑇}𝑇

is a rear-ranged global amplitude vector of the departing wave, andP = −diag {P

ℎ(−𝐿) P

ℎ(−𝐿)} is the global phase matrix.

A comparison of the global amplitude vectors of thedeparting waves d and d∗ indicates that the two amplitudevectors contain the same scalar state variables arranged indifferent sequential orders. The relation between d and d∗ is

d∗ = Ud, (44)

where U = [04I4; I404] is the permutation matrix from d

to d∗, in which 04and I4are, respectively, the zeromatrix and

the unit matrix of fourth order.

2.5.3. Equation for Natural Frequencies. Substituting (43) and(44) into (40) yields

(I − R) d = 0, (45)

where R = SPU is defined as the reverberation-ray matrix.To obtain a nontrivial solution of the global amplitude

vector of the departing wave d, the determinant of (I − R)must be zero; namely,

det (I − R) = 0 (46)

which is the equation of natural frequencies of the circularcylindrical shell.


Table 2: Comparison of the frequency parameter Ω = 𝜔𝑅[(1 − 𝜇2

)𝜌/𝐸]1/2 for a circular cylindrical shell with some classical boundary

conditions (𝑅 = 1m, 𝐿/𝑅 = 20, ℎ/𝑅 = 0.01,𝑚 = 1).

𝑛

SS-SS CE-SS CE-CEReference

[7]Reference

[28]Reference

[13] Present Reference[7]

Reference[28]

Reference[13] Present Reference

[7]Reference

[28]Reference

[13] Present

1 0.016101 0.016103 0.016101 0.016101 0.023974 0.024065 0.024721 0.023930 0.032885 0.033176 0.034879 0.0327922 0.009382 0.009388 0.009382 0.009382 0.011225 0.011238 0.011281 0.011215 0.013932 0.013970 0.014052 0.0139023 0.022105 0.022108 0.022105 0.022105 0.022310 0.022314 0.022335 0.022309 0.022672 0.022677 0.022725 0.0226674 0.042095 0.042097 0.042095 0.042095 0.042139 0.042141 0.042166 0.042139 0.042208 0.042210 0.042271 0.0422085 0.068008 0.068008 0.068008 0.068008 0.068024 0.068026 0.068054 0.068025 0.068046 0.068048 0.068116 0.0680466 0.099730 0.099730 0.099731 0.099731 0.099738 0.099739 0.099771 0.099739 0.099748 0.099749 0.099823 0.0997497 0.137239 0.137239 0.137240 0.137240 0.137244 0.137244 0.137279 0.137245 0.137249 0.137250 0.137328 0.1372518 0.180527 0.180528 0.180527 0.180530 0.180531 0.180531 0.180569 0.180533 0.180535 0.180535 0.180617 0.1805369 0.229594 0.229594 0.229596 0.229596 0.229596 0.229596 0.229636 0.229599 0.229599 0.229600 0.229684 0.22960110 0.284435 0.284436 0.284438 0.284439 0.284437 0.284438 0.284478 0.284440 0.284439 0.284441 0.284526 0.284442

2.6. Mode Shape Calculation. By substituting one of thenatural frequencies obtained by solving (46) in the precedingsubsection into (45), the global amplitude vector of thedeparting wave d is obtained. By normalization, the globalamplitude vector of the departing wave is determinate.Subsequently, by substituting the global amplitude vector ofthe departing wave into (43) and (44), the global amplitudevector of the arriving wave a is determined.

Finally, substituting the global amplitude vector of thedeparting wave and the arriving wave into the expressionsof the displacement components of the circular cylindricalshell, themode shape corresponding to the natural frequencyis obtained.

3. Numerical Results and Discussions

To begin with, the coefficient matrices K1𝑓and K1

𝑑for com-

mon boundary conditions are defined in Table 1, in which𝑘𝑢, 𝑘V, and 𝑘𝑤, respectively, denote the translational stiffness

coefficients for the springs in axial, circumferential, and radialdirections, and 𝑘

𝜑represents the rotational stiffness coef-

ficient for the spring around the circumferential direction.In order to simplify the presentation, CE, FE, SE, and ESrepresent clamped edge, free edge, simply supported edge,and elastic-support edge, respectively. Tomake a comparisonwith the results obtained by Qu et al. [28], three types ofelastic-support edges indicated by symbols EI, EII, and EIII

are introduced. EI type edge is considered to be axially elasticonly (i.e., 𝑁

𝑥= 𝑘𝑢𝑢, V = 𝑤 = 𝜑

𝑥= 0), support type EII

only allows elastically restrained displacement in the circum-ferential direction (i.e., 𝐹

𝑥𝜃= 𝑘VV, 𝑢 = 𝑤 = 𝜑

𝑥= 0), and EIII

type edge indicates both axial and circumferential displace-ment components are elastically restrained (i.e., 𝑁

𝑥= 𝑘𝑢𝑢,

𝐹𝑥𝜃= 𝑘VV, 𝑤 = 𝜑

𝑥= 0).

3.1. Validation of the Present Method. In this subsection, afew well-studied classical boundary conditions and somenonclassical boundary conditions frequently encounteredin practice are taken as calculation examples. The method

proposed in this paper is validated by comparing the presentnumerical results with those previously published in theliterature.

Frequency parameters calculated for the three classicalboundary conditions are presented in Table 2 together withthose previously given by Loy et al. [7], Qu et al. [28], andZhang et al. [13]. It can be observed from Table 2 that thepresent results agree very well with the reference solutions,which indicates that the method presented in this paper issuitable and of high accuracy for free vibration analysis ofcircular cylindrical shells with classical boundary conditions.

Subsequently, a comparison of the frequency parametersobtained by the method presented in this paper and thosepresented by Qu et al. [28] for the combination of the threetypes of elastic-support edges is shown in Table 3, whichshows that the frequency parameters obtained by themethodpresented in this paper agree well with those from Qu et al.[28]. The slight differences in the results may be attributed tothe different shell theories and solution approaches adoptedin the literature and in this paper. It should be reminded thatthe present analysis is based on the Flugge thin shell theory,whereas Qu et al. [28] employed the Reissner-Naghdi thinshell theory.

Tables 2 and 3 also show that the boundary conditionsaffect the frequency parameters of the circular cylindricalshell more significantly for small mode numbers than largemode numbers.Thiswill be further proved in the next discus-sions. Since it is impossible to undertake an all-encompassingsurvey of the free vibrations for every pair of mode numbers,only those mode numbers, for which the frequency parame-ters are strongly influenced by the boundary conditions, arechosen to be investigated for the circular cylindrical shells.Therefore, in the following analysis, the effects of the elastic-support stiffness on frequency parameters of the circularcylindrical shell are to be analyzed for small mode number𝑚 ≤ 3 and 𝑛 ≤ 3.

3.2. Effect of Elastic-Support Stiffness on Frequency Parameters.In what follows, the effects of each of the axial, circumferen-tial, radial, and torsional stiffness of the elastic-support on


Table 3: Comparison of the frequency parameterΩ = 𝜔𝑅[(1 − 𝜇2

)𝜌/𝐸]1/2 for a circular cylindrical shell with some elastic-support boundary

conditions (𝑅 = 1m, 𝐿/𝑅 = 20, ℎ/𝑅 = 0.01,𝑚 = 1).

𝑛

EI-EI EI-EII EI-EIII EII-EII EII-EIII EIII-EIII

Reference[28] Present Reference

[28] Present Reference[28] Present Reference

[28] Present Reference[28] Present Reference

[28] Present

1 0.021812 0.020328 0.027104 0.025721 0.021669 0.020163 0.033157 0.031534 0.026984 0.025467 0.021527 0.0200042 0.011674 0.010301 0.012695 0.011816 0.011471 0.010301 0.013957 0.013881 0.012457 0.011816 0.011272 0.0103013 0.022463 0.022211 0.022558 0.02239 0.02244 0.022209 0.022673 0.022667 0.022531 0.022387 0.022418 0.0222064 0.042178 0.04212 0.042193 0.042156 0.042175 0.042118 0.042209 0.042208 0.04219 0.042154 0.042172 0.0421175 0.068038 0.067936 0.068043 0.068031 0.068037 0.067936 0.068048 0.068046 0.068042 0.06803 0.068037 0.0680166 0.099745 0.099736 0.099747 0.099742 0.099745 0.099735 0.099748 0.099749 0.099746 0.099741 0.099745 0.0997357 0.137248 0.137243 0.137249 0.137247 0.137248 0.137243 0.13725 0.137251 0.137249 0.137246 0.137248 0.1372438 0.180534 0.180532 0.180535 0.180534 0.180534 0.180532 0.180535 0.180536 0.180535 0.180534 0.180534 0.1805319 0.229599 0.229598 0.229599 0.229599 0.229599 0.229598 0.229599 0.229601 0.229599 0.229599 0.229599 0.22959810 0.284439 0.28444 0.284439 0.284441 0.284439 0.28444 0.284441 0.284442 0.284439 0.284441 0.284439 0.284440

the frequency parameters for the clamped, free, and elastic-support boundary conditions are, respectively, investigatedin the following discussions. The clamped, free, and elastic-support boundary conditions, respectively, indicate the otherthree degrees of freedom are clamped, free, and elastic-supported.

3.2.1. The Clamped Boundary Conditions. Firstly, each of theaxial, circumferential, radial, and torsional stiffness is taken asfrom 10 to 1012N/m (orN/rad for torsional stiffness)while theother three degrees of freedom are assumed to be clamped.The frequency parameters against the nondimensional stiff-ness log(𝑘/𝑘

0) formode numbers (𝑚, 𝑛) = (1∼3, 1) and (1, 1∼3)

are plotted in Figure 3, in which 𝑘 is a general appellation ofthe axial, circumferential, radial, and torsional stiffness andthe reference stiffness 𝑘

0is taken as 1N/m (or N/rad).

It can be observed from Figure 3 that, for the clampedboundary conditions, only axial and circumferential stiffnessexert a significant influence on the frequency parameters ofthe circular cylindrical shell. Figures 3(a) and 3(b) indicatethat the axial and circumferential stiffness mainly affect thefrequency parameters in the range of 107 ∼ 1011N/m.

Besides, Figure 3(a) shows that the increase of the axialstiffness affects the frequency parameters in almost the sameextents for different axial mode numbers; however, it exertsa more evident influence for small circumferential modenumbers than it does for large circumferential mode num-bers. Figure 3(b) shows that the effect of the circumferentialstiffness on the frequency parameters increases with theincrease of the axial mode number while decreasing with theincrease of the circumferential mode number.

3.2.2. The Free Boundary Conditions. Subsequently, each ofthe axial, circumferential, radial, and torsional stiffness istaken as from 10 to 1012N/m (or N/rad) while the other threedegrees of freedom are assumed to be free. The frequencyparameters against the nondimensional stiffness log(𝑘/𝑘

0) for

mode numbers (𝑚, 𝑛) = (1∼3, 1) and (1, 1∼3) are plotted inFigure 4.

Figure 4 indicates that, except for the torsional stiffness,the stiffness of all the other three degrees of freedom exertsdramatic influence on the frequency parameters of thecircular cylindrical shell with a sensitive stiffness range of105

∼ 1010N/m.

To put it specifically, Figure 4(a) shows that the increaseof the axial stiffness affects the frequency parameters for thefree boundary conditions in the same manner as it does forthe clamped boundary conditions. However, as shown inFigure 4(b), the effect of the circumferential stiffness on thefrequency parameters becomes complicated. In the sensitivestiffness range, there exists switch stiffness, at which thefrequency parameters switch into another variation curvewhich is quite inconsistent with the original one. The modeshape analysis indicates that the free modes turn into theclamped ones at the switch stiffness. To make it intuitive withthe free mode shapes and the clamped mode shapes, someselectedmode shapes are presented in Figure 5, fromwhich itcan be found that themost remarkable difference between thetwo types ofmode shapes is that the freemode shapes containfree edge modes while the clamped mode shapes do not.

It can be observed from Figure 4(b)-(1) that, for circum-ferential mode number 𝑛 = 1, all the switch stiffness fordifferent axial mode numbers is the same to be 108N/m, atwhich there are both free modes and clamped modes formode numbers 𝑚 = 2 and 3. Figure 4(b)-(2) shows that,as the axial mode number 𝑚 = 1, the free modes turninto the clamped modes with the switch stiffness 108N/mfor circumferential mode number 𝑛 = 1. However, as thecircumferential mode numbers 𝑛 = 2 and 3, the free modesdisappear at circumferential stiffness 𝑘V = 10

7N/m whilethe clamped modes appear at the stiffness of 106N/m, whichindicates that there exist both freemodes and clampedmodesin the stiffness range of 106 ∼ 107N/m.

Figure 4(c)-(1) shows that, as the circular cylindrical shellis free in the axial, circumferential, and torsional directions,the frequency parameters vary against the radial stiffnesswith only free modes for small radial stiffness. However, asthe radial stiffness increases to a certain value, for instance,


1 2 3 4 5 6 7 8 9 10 11 12 Clamped

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

log(ku/k0)1 2 3 4 5 6 7 8 9 10 11 12 Clamped

log(ku/k0)

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1 m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

(2) m = 1, n = 1∼3(1) m = 1∼3, n = 1

0.020.040.060.08

0.10.120.14

0.01

0.015

0.02

0.025

0.03

(a) Frequency parameters versus axial stiffness

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

1 2 3 4 5 6 7 8 9 10 11 12 Clamped

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

1 2 3 4 5 6 7 8 9 10 11 12 Clamped

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

(2) m = 1, n = 1∼3(1) m = 1∼3, n = 1

0.020.040.060.08

0.10.120.14

0.01

0.015

0.02

0.025

0.03

0.035

log(kv /k0) log(kv /k0)

(b) Frequency parameters versus circumferential stiffness

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

1 2 3 4 5 6 7 8 9 10 11 12 Clamped

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

1 2 3 4 5 6 7 8 9 10 11 12 Clamped

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

(2) m = 1, n = 1∼3(1) m = 1∼3, n = 1

0.020.040.060.08

0.10.120.14

0.01

0.015

0.02

0.025

0.03

0.035

log(kw/k0) log(kw/k0)

(c) Frequency parameters versus radial stiffness

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

1 2 3 4 5 6 7 8 9 10 11 12 Clamped

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

1 2 3 4 5 6 7 8 9 10 11 12 Clamped

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

0.01

0.015

0.02

0.025

0.03

0.035

(2) m = 1, n = 1∼3(1) m = 1∼3, n = 1

0.020.040.060.08

0.10.120.14

log(k𝜙/k0) log(k𝜙/k0)

(d) Frequency parameters versus torsional stiffness

Figure 3: Effect of elastic-support stiffness on the frequency parameters for the clamped boundary conditions.


Free 1 2 3 4 5 6 7 8 9 10 11 120.040.060.08

0.10.120.140.160.18

0.2

0.020.030.040.050.060.07

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

log(ku/k0)

(2) m = 1, n = 1∼3(1) m = 1∼3, n = 1

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

Free 1 2 3 4 5 6 7 8 9 10 11 12log(ku/k0)

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3


00.020.040.060.08

0.10.120.140.160.18

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

log(k�/k0)Free 1 2 3 4 5 6 7 8 9 10 11 12

(1) m = 1∼3, n = 1

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

(2) m = 1, n = 1∼3

Free 1 2 3 4 5 6 7 8 9 10 11 12log(kv /k0)

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

0.0050.01

0.0150.02

0.0250.03

0.0350.04

0.0450.05


00.020.040.060.08

0.10.120.140.160.18

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

log(kw/k0)Free 1 2 3 4 5 6 7 8 9 10 11 12

(1) m = 1∼3, n = 1

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

log(kw/k0)

(2) m = 1, n = 1∼3

Free 1 2 3 4 5 6 7 8 9 10 11 12

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

00.010.020.030.040.050.060.070.080.09


0.020.040.060.08

0.10.120.140.16

0.010.015

0.020.025

0.030.035

0.04

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

log(k𝜙/k0)Free 1 2 3 4 5 6 7 8 9 10 11 12

(1) m = 1∼3, n = 1

m = 1, n = 1

m = 2, n = 1

m = 3, n = 1

log(k𝜙/k0)

(2) m = 1, n = 1∼3

Free 1 2 3 4 5 6 7 8 9 10 11 12

m = 1, n = 1

m = 1, n = 2

m = 1, n = 3


Figure 4: Effect of elastic-support stiffness on the frequency parameters for the free boundary conditions.


1.51

0.50

−0.5−1

1.5

1

0.5

0

−0.5

−1

1

0.5

0

−0.5

−1

−1.5

1.51

0.50

−0.5−1

−1.5

1.51

0.50

−0.5−1

−1.5

1.5 1 0.5 0 −0.5 −1−1.5

1.51 0.5 0

−0.5 −1 −1.5

21

0−1

−2 0 5 10 15 20

05

1015

20

05

1015

20

0 510 15

20

0 5 10 1520

−2−1

01

2

−2−1

01

2

(1) (m, n) = (1, 1)

(2) (m, n) = (1, 2)

(3) (m, n) = (1, 3)

(4) (m, n) = (2, 1)

(5) (m, n) = (3, 1)

(a) Clamped mode shapes

0 5 1015 20

05

1015

20

05

1015

20

0 510 15 20

0 5 10 1520

−2

−2

−1

−1

0

0

1

1

2

−2−1

01

2

2

−2

−1

0

1

2

1.51

0.50

−0.5−1

1.5 10.5 0 −0.5 −1 −1.5

−1.5

−2

−2

−1

−1

0

0

2

2

1

−2−1

01

1

−2

−1

0

2

2

1

(1) (m, n) = (1, 1)

(2) (m, n) = (1, 2)

(3) (m, n) = (1, 3)

(4) (m, n) = (2, 1)

(5) (m, n) = (3, 1)

(b) Free mode shapes

Figure 5: Some selected free mode shapes and clamped mode shapes.


107N/m for (𝑚, 𝑛) = (1, 1) and 108N/m for (𝑚, 𝑛) = (2, 1),the clamped modes appear with no disappearance of the freemodes, which indicates that the free modes and the clampedmodes coexist with each other for large radial stiffness.However, this only applies to small mode numbers like (𝑚, 𝑛)= (1, 1) and (2, 1). Figure 4(c)-(2) shows that there are only freemodes for small radial stiffness and only clamped modes forlarge radial stiffness. The free modes disappear at the radialstiffness of 106N/m while the clamped modes appear at theradial stiffness of 105N/m.

Besides, it can be observed from Figure 4(d) that thereis hardly any influence of the torsional stiffness on the fre-quency parameters for the free boundary conditions.

3.2.3. The Elastic-Support Boundary Conditions. Then, withthe other three degrees of freedom elastic-supported, thecurves of the frequency parameters against the axial, cir-cumferential, radial, and torsional stiffness are, respectively,presented in Figures 6(a)∼6(d).

It can be found from Figure 6(a) that the sensitive rangeof the axial stiffness is 106 ∼ 1011N/m for the elastic-supportboundary conditions.There are both freemodes and clampedmodes formode numbers (𝑚, 𝑛) = (1, 1) and (2, 1).The effect ofthe axial stiffness on the frequency parameters correspondingto the clamped modes decreases with the increase of theaxial mode numbers, while it increases with the increase ofthe axial mode number on those corresponding to the freemodes. However, only free modes exist for the circumferen-tial mode number 𝑛 = 1 and axial mode numbers 𝑚 ≥ 3,and only clamped modes appear for the axial mode number𝑚 = 1 and circumferential mode numbers 𝑛 ≥ 2. In thisinstance, the effect of the axial stiffness on the frequencyparameters decreases with the increase of the circumferentialmode number.

Figure 6(b) shows that the circumferential stiffnessmainly affects the frequency parameters in the range of 105 ∼1010N/m. Formode numbers (𝑚, 𝑛) = (1, 1) and (2, 1), the free

modes and the clamped modes coexist as the circumferentialstiffness is small. However, with the increase of the circum-ferential stiffness, for instance, to 107N/m for mode numbers(𝑚, 𝑛) = (1, 1) and 108N/m for mode numbers (𝑚, 𝑛) = (2,1), the free modes disappear while the clamped modes arereserved. For the circumferential mode number 𝑛 = 1 andaxial mode numbers 𝑚 ≥ 3, there is only one type of modesfor each circumferential stiffness except for 𝑘V = 10

8N/m, atwhich the free modes turn into the clamped modes with theincrease of the circumferential stiffness. For the axial modenumber 𝑚 = 1 and circumferential mode numbers 𝑛 ≥ 2,only clamped modes exist and the circumferential stiffnessexerts little influence on the frequency parameters.

It can be observed from Figure 6(c) that the radialstiffness mainly affects the frequency parameters in the rangeof 105 ∼ 1010N/m. For mode number (𝑚, 𝑛) = (1, 1), both thefree modes and the clamped modes are present for smallradial stiffness; however, the free modes disappear as 𝑘

𝑤≥

108N/m. The frequency parameters corresponding to the

free modes increase rapidly while those corresponding tothe clamped modes increase slightly with the increase of

the radial stiffness in the sensitive stiffness range. For modenumber (𝑚, 𝑛) = (2, 1), the free modes and the clampedmodes coexist for all the radial stiffness. The curves of thefrequency parameters corresponding to the two types ofmodes are of the same pattern while in different heights.As for mode number (𝑚, 𝑛) = (3, 1), only free modes arepresent for small radial stiffness 𝑘

𝑤≤ 108N/m; however, as

it increases to be larger than 109N/m, both the free modesand the clamped modes are present. Besides, the effect of theradial stiffness on the frequency parameters corresponding tothe clamped modes increases with the increase of the axialmode numbers while it decreases with the increase of theaxialmode numbers with respect to the freemodes. However,for the axial mode number 𝑚 = 1 and circumferential modenumbers 𝑛 ≥ 2, both the free modes and the clamped modesare present for small radial stiffness; as it increases to be largerthan 106N/m, only the clamped modes are reserved and thefree modes disappear. The increase of the radial stiffness canhardly affect the frequency parameters corresponding to theclampedmodes; however, it increases those corresponding tothe free modes in the sensitive stiffness range.

Figure 6(d) indicates that, for the elastic-support bound-ary conditions, the torsional stiffness can hardly affect thefrequency parameters of the circular cylindrical shell.

3.2.4. Comparison of the Three Types of Boundary Conditions.Finally, curves of frequency parameters varying with theaxial, circumferential, radial, and torsional stiffness for differ-ent mode numbers of the circular cylindrical shell with thefree, clamped, and elastic-support boundary conditions areshown in Figure 7.

It can be observed from Figure 7(a) that the increase ofthe axial stiffness will not cause mode switch for all of thethree types of boundary conditions. For mode number (𝑚,𝑛) = (1, 1), the frequency parameters for the free boundaryconditions are much greater than those for the clampedboundary conditions. There exist both the free modes andthe clamped modes for the elastic-support boundary con-ditions. The frequency parameters corresponding to thefree modes of the elastic-support boundary conditions aregreater than those for the free boundary conditions whilethose corresponding to the clamped modes are smaller thanthose for the clamped boundary conditions. The increaseof the axial stiffness enlarges the disparity of the frequencyparameters corresponding to the two clamped modes whilereducing that of the two free modes. For the axial modenumber 𝑚 = 1 and circumferential mode numbers 𝑛 ≥ 2,only the clamped modes are present for the elastic-supportboundary conditions, and the frequency parameters arevery close to those for the clamped boundary conditions.With the increase of the axial stiffness, the disparity of thefrequency parameters corresponding to the two clampedmodes increases. However, for axial mode numbers 𝑚 ≥ 2

and the circumferential mode number 𝑛 = 1, the frequencyparameters for the elastic-support boundary conditions areclose to those for the free boundary conditions, and the dis-parity of the frequency parameters corresponding to the twofree modes decreases with the increase of the axial stiffness.


m = 1, n = 1

m = 2, n = 1

m = 3, n = 1 m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

00.020.040.060.08

0.10.120.140.160.18

0.2

00.010.020.030.040.050.060.07

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

(1) m = 1∼3, n = 1 (2) m = 1, n = 1∼3


m = 1, n = 1

m = 2, n = 1

m = 3, n = 1 m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

0 00.020.040.060.08

0.10.120.140.160.18

0.010.020.030.040.050.06

(1) m = 1∼3, n = 1 (2) m = 1, n = 1∼3


m = 1, n = 1

m = 2, n = 1

m = 3, n = 1 m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

0 00.020.040.060.08

0.10.120.140.160.18

0.2

0.010.020.030.040.050.060.070.080.09

(1) m = 1∼3, n = 1 (2) m = 1, n = 1∼3


m = 1, n = 1

m = 2, n = 1

m = 3, n = 1 m = 1, n = 1

m = 1, n = 2

m = 1, n = 3

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

2 3 4 5 6 7 8 9 10 11 121log(k𝜙/k0)

2 3 4 5 6 7 8 9 10 11 121log(k𝜙/k0)

00.020.040.060.08

0.10.120.140.16

00.010.020.030.040.050.06

(1) m = 1∼3, n = 1 (2) m = 1, n = 1∼3


Figure 6: Effect of elastic-support stiffness on the frequency parameters for the elastic-support boundary conditions (elastic-support stiffness𝑘 = 1 × 10

7N/m or N/rad).


0.010.020.030.040.050.060.07

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

2 3 4 5 6 7 8 9 10 11 121log(ku/k0)

0.0080.01

0.0120.0140.0160.018

0.020.0220.024

0.022

0.0225

0.023

0.0235

0.024

0.0245

0.030.040.050.060.070.080.09

0.10.110.120.13

0.110.120.130.140.150.160.170.180.19

FreeClamped

Elastic-supported

(1) (m, n) = (1, 1)

(2) (m, n) = (1, 2)

(3) (m, n) = (1, 3)

(4) (m, n) = (2, 1)

(5) (m, n) = (3, 1)


Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

0.010.015

0.020.025

0.030.035

0.040.045

0.050.055

0.0080.009

0.010.0110.0120.0130.0140.0150.016

0.02180.022

0.02220.02240.02260.0228

0.0230.0232

0.020.040.060.08

0.10.120.14

0.050.060.070.080.09

0.10.110.120.130.14

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

2 3 4 5 6 7 8 9 10 11 121log(k�/k0)

FreeClamped

Elastic-supported

(1) (m, n) = (1, 1)

(2) (m, n) = (1, 2)

(3) (m, n) = (1, 3)

(4) (m, n) = (2, 1)

(5) (m, n) = (3, 1)


Figure 7: Continued.


FreeClamped

Elastic-supported

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

0.010.020.030.040.050.060.070.080.09

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

0.0080.01

0.0120.0140.0160.018

0.020.0220.0240.026

0.0220.0230.0240.0250.0260.0270.028

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

0.020.040.060.08

0.10.120.14

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

0.1

0.12

0.14

0.16

0.18

0.2

2 3 4 5 6 7 8 9 10 11 121log(kw/k0)

(1) (m, n) = (1, 1)

(2) (m, n) = (1, 2)

(3) (m, n) = (1, 3)

(4) (m, n) = (2, 1)

(5) (m, n) = (3, 1)


FreeClamped

Elastic-supported

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

Freq

uenc

y pa

ram

eter

(Ω)

0.0150.02

0.0250.03

0.0350.04

0.0450.05

0.0550.06

2 3 4 5 6 7 8 9 10 11 121log(k𝜙/k0)

0.0090.01

0.0110.0120.0130.014

2 3 4 5 6 7 8 9 10 11 121log(k𝜙/k0)

2 3 4 5 6 7 8 9 10 11 121log(k𝜙/k0)

0.0220.02210.02220.02230.02240.02250.02260.02270.02280.0229

2 3 4 5 6 7 8 9 10 11 121log(k𝜙/k0)

0.030.040.050.060.070.080.09

0.1

113 4 5 6 7 8 9 10 121 2log(k𝜙/k0)

0.135

0.14

0.145

0.15

0.155

0.16

(1) (m, n) = (1, 1)

(2) (m, n) = (1, 2)

(3) (m, n) = (1, 3)

(4) (m, n) = (2, 1)

(5) (m, n) = (3, 1)


Figure 7: Effect of elastic-support stiffness on the frequency parameters for the free, the clamped, and the elastic-support boundary conditions(elastic-support stiffness 𝑘 = 1 × 107 N/m or N/rad).


What is particular is that there are also clamped modes formode number (𝑚, 𝑛) = (2, 1) and the frequency param-eters are almost independent of the axial stiffness.

Figure 7(b) shows that, with the increase of the cir-cumferential stiffness, mode switch occurs for all modenumbers for the free boundary conditions while appearingonly for mode numbers (𝑚, 𝑛) = (1∼3, 1) for the elastic-support boundary conditions.However, as for elastic-supportboundary conditions, only the clampedmodes are present forthe axial mode number 𝑚 = 1 and circumferential modenumbers 𝑛 ≥ 2 and the frequency parameters are almostindependent of the circumferential stiffness. There is nomode switch for the clamped boundary conditions. For thefree boundary conditions, the free modes are present forsmall circumferential stiffness while the clamped modes arepresent for large circumferential stiffness. The frequencyparameters corresponding to the free modes of the elastic-support boundary conditions are greater than those of thefree boundary conditions. However, the frequency param-eters corresponding to the clamped modes for the elastic-support boundary conditions are very close to those of thefree boundary conditions. Both of them are much smallerthan those of the clamped boundary conditions.

Figure 7(c) indicates that the effects of the boundaryconditions of the other three degrees of freedom on thefrequency parameters depend on mode numbers and theradial stiffness. For small radial stiffness (e.g., 𝑘

𝑤≤ 108N/m

for (𝑚, 𝑛) = (1, 1) and (2, 1), 𝑘𝑤≤ 106N/m for (𝑚, 𝑛) = (1, 2)

and (1, 3)), both free modes and clamped modes are presentfor the elastic-support boundary conditions, while only freemodes are present for the free boundary conditions andonly clamped modes are present for the clamped boundaryconditions. The frequency parameters corresponding to thefree modes of the elastic-support boundary conditions aregreater than those of the free boundary conditions and thefrequency parameters corresponding to the clamped modesof the elastic-support boundary conditions are smaller thanthose of the clamped boundary conditions. As for axial modenumbers𝑚 ≥ 3 and the circumferential mode number 𝑛 = 1,only free modes are reserved for small radial stiffness.

However, for large radial stiffness, as for mode number(𝑚, 𝑛) = (1, 1), the free modes of the elastic-support bound-ary conditions disappear and the clamped modes of freeboundary conditions appear. The frequency parameters cor-responding to the freemodes of the free boundary conditionsare much greater than those of the clamped boundary con-ditions and subsequently greater than those correspondingto the clamped modes of the free boundary conditions andthe elastic-support boundary conditions, and the latter twoare very close to each other. For mode numbers (𝑚, 𝑛) = (1,2) and (1, 3), only the clamped modes are present for thethree types of boundary conditions. The frequency param-eters for the clamped boundary conditions are greater thanthose for the free boundary conditions and the elastic-support boundary conditions, and the latter two are veryclose to each other. For mode number (𝑚, 𝑛) = (2, 1), boththe free modes and the clamped modes are present for thefree boundary conditions and the elastic-support boundary

conditions. The frequency parameters corresponding to thefree modes of the elastic-support boundary conditions aregreater than those of the free boundary conditions, and thoseof the clamped boundary conditions are greater than thosecorresponding to the clamped modes of the elastic-supportboundary conditions and subsequently greater than thosecorresponding to the clamped modes of the free boundaryconditions. For mode number (𝑚, 𝑛) = (3, 1), the free modesare present for the free boundary conditions and the elastic-support boundary conditions, while the clamped modes arepresent for the clamped boundary conditions and the elastic-support boundary conditions.The frequency parameters cor-responding to the freemodes of the elastic-support boundaryconditions are much greater than those of the free boundaryconditions, and those corresponding to the clamped modesof the elastic-support boundary conditions are much smallerthan those of the clamped boundary conditions.

Figure 7(d) shows that, for mode numbers (𝑚, 𝑛) = (1, 1)and (2, 1), both the free modes and the clamped modes arepresent for the elastic-support boundary conditions, whileonly the free modes are present for the free boundary con-ditions and the clamped modes are present for the clampedboundary conditions.The frequency parameters correspond-ing to the free modes of the elastic-support boundary condi-tions are greater than those of the free boundary conditions,and those corresponding to the clampedmodes of the elastic-support boundary conditions are smaller than those of theclamped boundary conditions. Formode numbers (𝑚, 𝑛) = (1,2) and (1, 3), there is only one type of modes for the free, theclamped, and the elastic-support boundary conditions. Thefrequency parameters corresponding to the free boundaryconditions are greater than those of the clamped boundaryconditions and subsequently much greater than those of theelastic-support boundary conditions. For mode number (𝑚,𝑛) = (3, 1), there is also only one type of modes for each of thethree types of boundary conditions. However, the frequencyparameters corresponding to the three types of boundaryconditions are arranged from the highest to the lowest orderof elastic-support, free, and clamped boundary conditions.

4. Conclusions

An analytical procedure is developed to analyze the freevibration characteristics of circular cylindrical shells witharbitrary boundary conditions. Based on the Flugge thin shelltheory, exact solutions of the traveling wave form along theaxial direction and the standing wave form along the cir-cumferential direction are obtained. MRRM is introduced toderive the equation of the natural frequencies of a unified andcompact form. The agreement of the comparisons in the fre-quency parameters obtained by MRRM and those presentedin the published literature proves the suitability and accuracyof applying MRRM to the study of free vibration of circularcylindrical shells with arbitrary boundary conditions.

Free vibration characteristics of circular cylindrical shellswith elastic-support boundary conditions are investigated byMRRM. Results show that the elastic-support stiffness effec-tively affects the frequency parameters in the range of 105 ∼1011N/m, which is comparable to the bending rigidity of a


shell. The effects of the elastic-support stiffness on the fre-quency parameters vary with different mode numbers. Theaxial and circumferential stiffness of the elastic-support havea dramatic influence on the frequency parameters in thesensitive stiffness range. The radial stiffness affects the fre-quency parameters in certain circumstances. However, thetorsional stiffness can hardly affect the frequency parametersof the circular cylindrical shell. The free-clamped modeswitch frequently occurs for the free boundary conditions andthe free-clamped mode coexistence is usually present for theelastic-support boundary conditions at small mode numbers.Generally, the elastic-support can effectively increase thefrequency parameters corresponding to the free modes anddecrease those corresponding to the clamped modes.

Appendices

A. Linear Differential Operators 𝐿𝑗(𝑗 = 1 ∼ 6)

Consider the following:

𝐿1= 𝜇1𝑅2𝜕2

𝜕𝑥2,

𝐿2= 𝜇1

𝜕2

𝜕𝜃2,

𝐿3= 𝜇1𝑅

𝜕2

𝜕𝑥𝜕𝜃,

𝐿4= 𝑅2𝜕2

𝜕𝑥2+ 𝜆1𝜇2

𝜕2

𝜕𝜃2+ 𝑅2

𝑘2

𝐿,

𝐿5= −𝜆𝑅

2𝜕2

𝜕𝑥2+ 𝜆𝜇2

𝜕2

𝜕𝜃2+ 𝜇,

𝐿6= 𝜆2𝜇2𝑅2𝜕2

𝜕𝑥2+𝜕2

𝜕𝜃2+ 𝑅2

𝑘2

𝐿,

𝐿7= 1 − 𝜆𝜇

3𝑅2𝜕2

𝜕𝑥2,

𝐿8= 𝜆(𝑅

2𝜕2

𝜕𝑥2+𝜕2

𝜕𝜃2+ 1)

2

− (2𝜆𝑅2𝜕2

𝜕𝑥2+ 𝑅2

𝑘2

𝐿− 1) ,

(A.1)

where 𝑘𝐿= 𝜔[(1 − 𝜇

2

)𝜌/𝐸]1/2 is the longitudinal wave num-

ber and 𝛾 = {[𝑅2𝑘2𝐿− (1 − 𝜇)/(1 + 𝜇)]/𝜆𝑅

4

}1/4 is a parameter

of the same dimension with the axial wave number.

B. Dimensional Parameters 𝜁𝑗(𝑗 = 1 ∼ 6)

Consider the following:

𝜁2

1=

(1 − 𝜇2

1+ 𝜆1𝜆2𝜇2

2) 𝑛2

𝑅2− (1 + 𝜆

2𝜇2) 𝑘2

𝐿,

𝜁2

2=𝜆1𝜇2𝑛2

𝑅2− 𝑘2

𝐿,

𝜁2

3=𝑛2

𝑅2− 𝑘2

𝐿,

𝜁2

4=

[(2𝜇1+ 𝜆𝜇2𝜇3) 𝑛2

− (1 + 𝜇𝜇3)]

𝑅2,

𝜁4

5=

[𝜇1(𝑛2

− 1)2

+ 𝜇2𝑛2

+ (𝜇2− 𝜇1𝑅2

𝑘2

𝐿) /𝜆]

𝑅4,

𝜁2

6=

[(1 − 𝜇𝜇1) + 𝜆𝜇

2(𝜇1+ 𝜆1𝜇3) 𝑛2

− 𝜆𝜇3𝑅2

𝑘2

𝐿]

𝑅2,

𝜁2

7=1

𝑅2,

𝜁2

8=

[𝜆2𝜇2𝜇 + 𝜆 (1 − 𝜆

2𝜇2

2− 𝜇1𝜇3) 𝑛2

− 𝜆𝑅2

𝑘2

𝐿]

𝑅2,

𝜁4

9=

(𝜆𝜇2𝑛2

𝑅2

𝑘2

𝐿− 𝜇𝑅2

𝑘2

𝐿− 𝜇2𝑛2

− 𝜆𝜇2𝑛4

)

𝑅4,

(B.1)

where 𝜇1= (1+𝜇)/2, 𝜇

2= (1−𝜇)/2, 𝜇

3= (3−𝜇)/2, and 𝜆

1=

1 + 𝜆, 𝜆2= 1 + 3𝜆, 𝜆

3= 𝜇1− 𝜇3𝜆 are dimensionless param-

eters.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is financially supported by National Natural Sci-ence Foundation of China (Grants nos. 51279038 and51479041). The authors would like to express their profoundthanks for the financial support. The first author would liketo sincerely thank Miss Jingjing Yu and Mr. Qingshan Wangfor the scientific discussions and suggestions.

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