# Wave-Based Method for Free Vibration Analysis of Orthotropic Cylindrical Shells with Arbitrary Boundary Conditions

Authors: Fuzhen Pang; Ruidong Huo; Haichao Li; Cong Gao; Xuhong Miao; Yi Ren
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The wave-based method (WBM) is a feasible method which investigates the free vibration characteristics of orthotropic cylindrical shells under general boundary conditions. Based on Reissner–Naghid’s shell theory, the governing motion equation is established, and the displacement variables are transformed into wave functions formed to satisfy the governing equations. On the basis of the kinematic relationship between the force resultant and displacement vector, the overall matrix of the shell is established. Comparison studies of this paper with the solutions in the literatures were carried out to validate the accuracy of the present method. Furthermore, by analyzing some numerical examples, the free vibration characteristics of orthogonal anisotropic cylindrical shells under classical boundary conditions, elastic boundary conditions, and their combinations are studied. Also, the effects of the material parameter and geometric constant on the natural frequencies for the orthotropic circular cylindrical shell under general boundary conditions are discussed. The conclusions obtained can be used as data reference for future calculation methods.


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## Body

## 1. Introduction

Orthotropic materials have good material properties, and they are very popular in the engineering application field. As a common engineering geometry, cylindrical shell structure has certain applications in petroleum equipment, coal development, and marine equipment. Because of the development of research in recent years, the free vibration analysis of the orthotropic circular cylindrical shells under general boundary conditions has gradually deepened. With the development of cylindrical shell theory in recent decades, more and more theories are put forward and developed. At present, the main shell theory related to free vibration of cylindrical shell mainly includes three types, namely, classical shell theory [1–4], first-order shear deformation shell theory [5–8], and high-order shear deformation shell theory [9–11].Over the decades, many researchers have put in a lot of time and effort on the vibration analysis of the orthotropic circular cylindrical shells and obtained many excellent research results. Chen et al. [12] analyzed the free vibration characteristics of the fluid-filled FG orthotropic cylindrical shells, and the boundary condition was set as simply supported. The 3D anisotropic elasticity fundamental equations were used to state equations and some numerical examples are presented. Ding et al. [13] extended the nonhomogeneous orthotropic elastic solution for the axisymmetric plane strain cylindrical shell dynamic problems, and the orthogonal expansion technique was adopted to derive the time variable and the solutions were obtained. Najafov et al. [14] conducted the Galerkin method to investigate the vibration and stability characteristics of the FG orthotropic cylindrical shells on elastic foundations. Sofiyev and Kuruoglu [15] proposed the vibration and buckling of FG orthotropic cylindrical shells in the same way. The Donnell shell theory and Galerkin method were adopted to derive the governing equation. Mallon et al. [16] proposed the coupled shaker-structure model to study the dynamic stable problem of the harmonically base-excited thin orthotropic cylindrical shell. The numerical observations were qualitatively confirmed by comparing the results with the experimental solutions. Prado et al. [17] presented the nonlinear vibrations and dynamic instability problem for the orthotropic circular cylindrical shell under simply supported boundary conditions. Donnell’s shell theory and Galerkin method were used to establish the differential equations of the shell. Lakis and Selmane [18] conducted the thin shell theory, fluid theory, and the hybrid FEM to study the influence of large amplitude vibration of orthotropic, circumferentially nonuniform cylindrical shells. Ahmed [19] analyzed the isotropic and orthotropic cylindrical shell with variable circumferentially thickness and complex curvature radius by Flugge’s shell theory. The transfer-matrix method and the Longberg integral method were proposed in the establishment of the motion control equation. Liu et al. [20] proposed the S-DQFEM to investigate the free vibration problem of the orthotropic circular cylindrical shells under classical boundary conditions, and the Donnell–Mushtari shell theory was adopted. Furthermore, the natural frequencies of the closed-form are obtained by the method of variable separation. Sofiyev and Aksogan [21] extended the Galerkin method to investigate the free vibration characteristics of the nonhomogeneous orthotropic thin cylindrical shells with geometric nonlinearity. Fang et al. [22] investigated the vibration characteristics of nanosized piezoelectric double-shell structures under simply supported boundary condition by the Goldenveizer–Novozhilov shell theory, thin plate theory, and electroelastic surface theory. Zhu et al. [23] studied the surface energy effect on the nonlinear free vibration behavior of orthotropic piezoelectric cylindrical nanoshells by the classical shell theory. Also, for the shell and composite structures, some reported literatures have been discussed. Ghassabi et al. [24] presented the solution strategy based on the state vector technique to study the vibroacoustic performance of carbon nanotube- (CNT-) reinforced composite doubly curved thick shells by three-dimensional theory. Talebitooti et al. [25] conducted the nondominated sorting genetic algorithm to optimize sound transmission loss of the laminated composite cylindrical shell by FSDST. Ghassabi et al. [26] investigated the vibroacoustic performance of the doubly curved thick shell by the three-dimensional sound propagation approach and state space solution. Talebitooti et al. [27] analyzed the acoustic characteristics of the doubly curved composite shell with full simply supported by the third-order shear deformation theory (TSDT). Talebitooti and Zarastvand [28] investigated the acoustic transmission of the infinitely long doubly curved shell based on HSDT. Talebitooti et al. [29] discussed the effect of compressing porous material on sound transmission loss of the multilayered cylindrical shell subjected to porous core and air-gap insulation in the presence of external flow based on the three-dimensional elasticity theory. Talebitooti and Zarastvand [30] investigated the wave propagation on infinite doubly curved laminated composite shell which is used in aerospace structures. Talebitooti et al. [31] analyzed the acoustic behavior of laminated composite infinitely long doubly curved shallow shells which is the acoustic behavior of laminated composite infinitely long doubly curved shallow shells. Talebitooti et al. [32] analyzed the acoustic behavior of the laminated composite cylindrical shell which is excited by an oblique plane sound wave by the third-order shear deformation theory.For the wave-based method, it is an unfamiliar semianalytical method to investigate the dynamic characteristics of the engineering structure in some applications such as cylindrical shell structure [33–35], coupled structure [36–40], coupled vibroacoustic problem [41], composite structure [42–45], and so on.


## 2. Theoretical Formulations

### 2.1. Description of the Shell Model

The model of circular cylindrical shell is described in Figure1. The circular cylindrical shell is composed of homogeneous and isotropic materials. h represents the cross section of the uniform circular cylindrical shell with thickness. L represents the length of the circular cylindrical shell. The model is described by curvilinear coordinate system (x, θ, and z), in which x and θ denote the axial and circumferential directions of the shell, respectively. The displacements in the direction of x, θ, and z of the middle surface are denoted as u, v, and w, respectively. For the elastic boundary conditions, there are three pair translational restrained springs (Ku, Kv, and Kw) and one pair rotational restrained spring (Kθ) are set at two ends to simulate the arbitrary elastic boundary conditions.Figure 1
The schematic diagram of the orthotropic circular cylindrical shell.


### 2.2. Kinematic Relations and Stress Resultants

According to Reissner–Naghid’s shell theory [46], the relationship between the strain resultant ε0=εxx0,εθθ0,εxθ0T and curvature change resultant χ0=χxx0,χθθ0,χxθ0T in the middle surface of the shell is shown as follows:(1)εxx0=∂u∂x,εθθ0=∂vR∂θ+wR,εxθ0=∂v∂x+∂uR∂θ,χxx0=−∂2w∂x2,χθθ0=∂vR2∂θ−∂2wR2∂θ2,χxθ0=∂vR∂x−2∂2wR∂x∂θ.Furthermore, the strains of arbitrary point in the shell are given as follows:(2)εxx=εxx0+zχxx0,εθθ=εθθ0+zχθθ0,εxθ=εxθ0+zχxθ0.Also, the stress-strain relationships of the orthotropic shell are obtained by Hooke’s law [47–49]:(3)σxxσθθτxθ=Q11Q120Q21Q22000Q66εxxεθθεxθ,where Qiji,j=1,2,6 are the transform coefficients which are defined as follows:(4)Q11=E11−μ12μ21,Q12=Q21=μ12E21−μ12μ21,Q22=E21−μ12μ21,Q66=G12,where E1 and E2 are the Young’s modulus, μ12 and μ21 are the Poisson’s ratios, and G12 is the shear modulus. The relationship between μ12 and μ21 is determined as μ21E1 = μ12E2. For the isotropic cylindrical shell, the relationship is defined as E1 = E2 and G12=E1/2+2μ12. To obtain the force vector and bending moment resultant of the orthotropic shell, the relationship between the force resultant, bending moment, and strain vector is given as follows:(5)NxNθNxθ=∫zσxxσθθτxθdz,MxMθMxθ=∫zσxxσθθτxθzdz.So, the expression for the resultant forces and moments is as follows:(6)NxNθNxθMxMθMxθ=A11A120000A21A22000000A66000000D11D120000D21D22000000D66εxx0εθθ0εxθ0χxx0χθθ0χxθ0,where Aij and Diji,j=1,2,6 are the coefficients and are defined as follows:(7)Aij=Qijh,Dij=h312Qij.


### 2.3. Governing Equations

Based on Reissner–Naghid’s theory, the governing equation of the orthotropic cylindrical shell in terms of the force resultant, moment vector, and displacement variables are given as follows [50–52]:(8)∂Nx∂x+1R∂Nxθ∂θ=ρh∂2u∂t2,∂Nxθ∂x+1R∂Nθ∂θ+1R∂Mxθ∂x+1R2∂Mθ∂θ=ρh∂2v∂t2,∂2M∂x2+2R∂2Mxθ∂x∂θ+1R2∂2Mθ∂θ2−NθR=ρh∂2w∂t2,where ρ is the destiny of the orthotropic circular cylindrical shell. Submitting the expression of force and moment resultant into equation (8), the governing equation of the orthotropic cylindrical shell is given in the matrix form as follows:(9)T11T12T13T21T22T23T31T32T33uvw=ρh∂2u∂t2∂2v∂t2∂2w∂t2,where Tiji,j=1,2,3 are the transform coefficients, and the expression of them are given as follows:(10)T11=A11∂2∂x2+A66R2∂2∂θ2,T12=A12+A66R∂2∂x∂θ,T13=A12R∂∂x,T21=A66+A21R∂2∂x∂θ,T22=A66+D66R2∂2∂x2+A22R2+D22R4∂2∂θ2,T23=−D22R4∂3∂θ3−D21+2D66R2∂3∂x2∂θ+A22R2∂∂θ,T31=−A21R∂∂x,T32=D22R4∂3∂θ3+D12+2D66R2∂3∂x2∂θ−A22R2∂∂θ,T33=−D11∂4∂x4−D22R4∂4∂θ4−D12+D21+4D66R2∂4∂x2∂θ2−A22R2.


### 2.4. Implementation of the WBM

Using the WBM to analyze the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, the displacement variables should be transformed into the wave functions form as follows [33]:(11)ux,θ,t=U0ejknxcosnθe−jωt,vx,θ,t=V0ejknxsinnθe−jωt,wx,θ,t=W0ejknxcosnθe−jωt,where U0, V0, and W0 are the displacement amplitude variables, kn is the axial wave number, t is the time variable, n is the circumferential number, and ω is the circular frequency. So, submitting equation (11) into equation (9), the governing equation of the orthotropic cylindrical shell in the wave function form is obtained as follows:(12)L11L12L13L21L22L23L31L32L33U0V0W0=−ρhω2000ρhω2000ρhω2U0V0W0,where Liji,j=1,2,3 are the coefficients of equation (12) in the wave functions form, and the expressions are given as follows:(13)L11=−A66n2R2−A11kn2,L12=inknA12+A66R,L13=iknA12R,L21=inknA21+A66R,L22=A66kn2+A22n2R2+D66kn2R2+D22n2R4,L23=D21+2D66nkn2R2+nA22R2+D22n3R4,L31=iknA21R,L32=D12+2D66nkn2R2+nA22R2+D22n3R4,L33=D11kn4+D12+D21+4D66n2kn2R2+A22R2+D22n4R4.To make sure equation (13) has nontrivial solutions, the determinant of the coefficient matrix in equation (12) should be equal to zero. Also, the eighth-order equation of the axial wave number kn is obtained as follows:(14)a8kn8+a6kn6+a4kn4+a2kn2+a0=0,where aii=0,2,4,6,8 are the coefficients of the eighth-order equation, and the solutions of equation (14) are calculated by Reference [53] as follows:(15)kn=±kn1,±kn2,±kn3,±kn4,where knii=1,2,3,4 can be in real, imaginary, and combination form. Furthermore, the displacement variables in the wave function forms are converted as follows:(16)ux,θ,tvx,θ,twx,θ,t=∑n=0∞∑i=18λn,iW0,iejknixcosnθκn,iW0,iejknixsinnθW0,iejknixcosnθe−jωt,where λn,i and κn,i are the wave contribution factors which are expressed as follows:(17)λn,i=L13L12L23L22L12L11L22L21n,i,κn,i=L11L13L21L23L12L11L22L21n,i.Particularly, the expressions for displacement variables are transformed as follows:(18)ux,θ,tvx,θ,twx,θ,t=∑n=0∞∑i=18u¯xWn,icosnθv¯xWn,isinnθw¯xWn,icosnθe−jωt,where(19)u¯x=λn,iejknix,v¯x=κn,iejknix,w¯x=ejknix.On the basis of the relationship between the displacement variables and force and moment resultants, the detailed expression of the force resultant and moment vector is shown as follows:(20)Nxx,θ,tMxx,θ,tNxθx,θ,tQxx,θ,t=∑n=0∞∑i=18Nx¯xWn,icosnθMx¯xWn,icosnθNxθ¯xWn,isinnθQx¯xWn,icosnθe−jωt,where(21)Nx¯x=ikniA11λn,i+nA12Rκn,i+A12Rejknix,Mx¯x=nD12R2κn,i+D12n2R2+D11kni2ejknix,Nxθ¯x=−nA66Rλn,i+ikniA66κn,ieiknix,Qx¯x=inkniD12+D66R2κn,i+iD11kni3+in2kniD12+2D66R2ejkn,ix.For the classical boundary conditions, four type of boundary conditions are considered which are widely used in some engineering applications, as follows [54–56]:Free (F):(22)Nx=Nxθ+MxθR=Qx+∂Mxθ∂θ=Mx=0.Clamped (C):(23)u=v=w=∂w∂x=0.Simply supported (SS):(24)u=v=w=Mx=0.Shear diaphragm (SD):(25)Nx=v=w=Mx=0.For the elastic boundary conditions, the relationship between the force, displacement, and elastic restrained stiffness is discussed. When the elastic restrained stiffness is in the axial direction, the boundary relationship is given as follows:(26)Kuux,θ,t±Nxx,θ,t=0,where Ku is the stiffness constant, and the symbol ± means the elastic restrained spring at the boundary edge x = L and x = 0. Also, for the circumferential displacement, the normal displacement, and the transverse normal rotations of the θ and x axis, the symbol ± has the same meaning for the elastic boundary conditions. For the other elastic restrained states, the boundary relationships can refer to Reference [57]. Related to the introduction of the elastic and classical boundary conditions, the final equation of the orthotropic cylindrical shell is given as follows:(27)KW=F,where K is the overall matrix, F is the external force resultant, and W = {W1; W2} is the wave contribution factor vector. W1 and W2 are the wave contribution factor vectors which are related to the boundary conditions at the two ends. The overall matrix K is shown as follows:(28)K=D10B1LB20D20,where D1 and D2 are the boundary matrixes which depend on the boundary conditions and B1 and B2 are the segment matrixes which are given as follows:(29)B1,2x=±δxfx,where δx and fx are the displacement and force matrix of the orthotropic shell and are expressed as follows:(30)δx=u1¯xu2¯x⋯u8¯xv1¯xv2¯x⋯v8¯xw1¯xw2¯x⋯w8¯x∂w1¯x∂x∂w2¯x∂x⋯∂w8¯x∂x,fx=Nx,1¯xNx,2¯x⋯Nx,8¯xMx,1¯xMx,2¯x⋯Mx,8¯xNxθ,1¯xNxθ,2¯x⋯Nxθ,8¯xQx,1¯xQx,2¯x⋯Qx,8¯x.When analyzing the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, the external force vectorF should be vanished. For each circumferential number n, searching the zero position of the overall matrix K, a series of values are calculated. When the sign occurs, the natural frequencies of the cylindrical shell are obtained.The dichotomy method is a mathematical idea that uses flat partitions and infinite approximations. It is an effective algorithm for avoiding complex calculations and approximations of analytic one-dimensional functions. In this paper, the dichotomy method is adopted to get the natural frequencies of the cylindrical shell, and some numerical examples are established.


## 2.1. Description of the Shell Model

The model of circular cylindrical shell is described in Figure1. The circular cylindrical shell is composed of homogeneous and isotropic materials. h represents the cross section of the uniform circular cylindrical shell with thickness. L represents the length of the circular cylindrical shell. The model is described by curvilinear coordinate system (x, θ, and z), in which x and θ denote the axial and circumferential directions of the shell, respectively. The displacements in the direction of x, θ, and z of the middle surface are denoted as u, v, and w, respectively. For the elastic boundary conditions, there are three pair translational restrained springs (Ku, Kv, and Kw) and one pair rotational restrained spring (Kθ) are set at two ends to simulate the arbitrary elastic boundary conditions.Figure 1
The schematic diagram of the orthotropic circular cylindrical shell.


## 2.2. Kinematic Relations and Stress Resultants

According to Reissner–Naghid’s shell theory [46], the relationship between the strain resultant ε0=εxx0,εθθ0,εxθ0T and curvature change resultant χ0=χxx0,χθθ0,χxθ0T in the middle surface of the shell is shown as follows:(1)εxx0=∂u∂x,εθθ0=∂vR∂θ+wR,εxθ0=∂v∂x+∂uR∂θ,χxx0=−∂2w∂x2,χθθ0=∂vR2∂θ−∂2wR2∂θ2,χxθ0=∂vR∂x−2∂2wR∂x∂θ.Furthermore, the strains of arbitrary point in the shell are given as follows:(2)εxx=εxx0+zχxx0,εθθ=εθθ0+zχθθ0,εxθ=εxθ0+zχxθ0.Also, the stress-strain relationships of the orthotropic shell are obtained by Hooke’s law [47–49]:(3)σxxσθθτxθ=Q11Q120Q21Q22000Q66εxxεθθεxθ,where Qiji,j=1,2,6 are the transform coefficients which are defined as follows:(4)Q11=E11−μ12μ21,Q12=Q21=μ12E21−μ12μ21,Q22=E21−μ12μ21,Q66=G12,where E1 and E2 are the Young’s modulus, μ12 and μ21 are the Poisson’s ratios, and G12 is the shear modulus. The relationship between μ12 and μ21 is determined as μ21E1 = μ12E2. For the isotropic cylindrical shell, the relationship is defined as E1 = E2 and G12=E1/2+2μ12. To obtain the force vector and bending moment resultant of the orthotropic shell, the relationship between the force resultant, bending moment, and strain vector is given as follows:(5)NxNθNxθ=∫zσxxσθθτxθdz,MxMθMxθ=∫zσxxσθθτxθzdz.So, the expression for the resultant forces and moments is as follows:(6)NxNθNxθMxMθMxθ=A11A120000A21A22000000A66000000D11D120000D21D22000000D66εxx0εθθ0εxθ0χxx0χθθ0χxθ0,where Aij and Diji,j=1,2,6 are the coefficients and are defined as follows:(7)Aij=Qijh,Dij=h312Qij.


## 2.3. Governing Equations

Based on Reissner–Naghid’s theory, the governing equation of the orthotropic cylindrical shell in terms of the force resultant, moment vector, and displacement variables are given as follows [50–52]:(8)∂Nx∂x+1R∂Nxθ∂θ=ρh∂2u∂t2,∂Nxθ∂x+1R∂Nθ∂θ+1R∂Mxθ∂x+1R2∂Mθ∂θ=ρh∂2v∂t2,∂2M∂x2+2R∂2Mxθ∂x∂θ+1R2∂2Mθ∂θ2−NθR=ρh∂2w∂t2,where ρ is the destiny of the orthotropic circular cylindrical shell. Submitting the expression of force and moment resultant into equation (8), the governing equation of the orthotropic cylindrical shell is given in the matrix form as follows:(9)T11T12T13T21T22T23T31T32T33uvw=ρh∂2u∂t2∂2v∂t2∂2w∂t2,where Tiji,j=1,2,3 are the transform coefficients, and the expression of them are given as follows:(10)T11=A11∂2∂x2+A66R2∂2∂θ2,T12=A12+A66R∂2∂x∂θ,T13=A12R∂∂x,T21=A66+A21R∂2∂x∂θ,T22=A66+D66R2∂2∂x2+A22R2+D22R4∂2∂θ2,T23=−D22R4∂3∂θ3−D21+2D66R2∂3∂x2∂θ+A22R2∂∂θ,T31=−A21R∂∂x,T32=D22R4∂3∂θ3+D12+2D66R2∂3∂x2∂θ−A22R2∂∂θ,T33=−D11∂4∂x4−D22R4∂4∂θ4−D12+D21+4D66R2∂4∂x2∂θ2−A22R2.


## 2.4. Implementation of the WBM

Using the WBM to analyze the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, the displacement variables should be transformed into the wave functions form as follows [33]:(11)ux,θ,t=U0ejknxcosnθe−jωt,vx,θ,t=V0ejknxsinnθe−jωt,wx,θ,t=W0ejknxcosnθe−jωt,where U0, V0, and W0 are the displacement amplitude variables, kn is the axial wave number, t is the time variable, n is the circumferential number, and ω is the circular frequency. So, submitting equation (11) into equation (9), the governing equation of the orthotropic cylindrical shell in the wave function form is obtained as follows:(12)L11L12L13L21L22L23L31L32L33U0V0W0=−ρhω2000ρhω2000ρhω2U0V0W0,where Liji,j=1,2,3 are the coefficients of equation (12) in the wave functions form, and the expressions are given as follows:(13)L11=−A66n2R2−A11kn2,L12=inknA12+A66R,L13=iknA12R,L21=inknA21+A66R,L22=A66kn2+A22n2R2+D66kn2R2+D22n2R4,L23=D21+2D66nkn2R2+nA22R2+D22n3R4,L31=iknA21R,L32=D12+2D66nkn2R2+nA22R2+D22n3R4,L33=D11kn4+D12+D21+4D66n2kn2R2+A22R2+D22n4R4.To make sure equation (13) has nontrivial solutions, the determinant of the coefficient matrix in equation (12) should be equal to zero. Also, the eighth-order equation of the axial wave number kn is obtained as follows:(14)a8kn8+a6kn6+a4kn4+a2kn2+a0=0,where aii=0,2,4,6,8 are the coefficients of the eighth-order equation, and the solutions of equation (14) are calculated by Reference [53] as follows:(15)kn=±kn1,±kn2,±kn3,±kn4,where knii=1,2,3,4 can be in real, imaginary, and combination form. Furthermore, the displacement variables in the wave function forms are converted as follows:(16)ux,θ,tvx,θ,twx,θ,t=∑n=0∞∑i=18λn,iW0,iejknixcosnθκn,iW0,iejknixsinnθW0,iejknixcosnθe−jωt,where λn,i and κn,i are the wave contribution factors which are expressed as follows:(17)λn,i=L13L12L23L22L12L11L22L21n,i,κn,i=L11L13L21L23L12L11L22L21n,i.Particularly, the expressions for displacement variables are transformed as follows:(18)ux,θ,tvx,θ,twx,θ,t=∑n=0∞∑i=18u¯xWn,icosnθv¯xWn,isinnθw¯xWn,icosnθe−jωt,where(19)u¯x=λn,iejknix,v¯x=κn,iejknix,w¯x=ejknix.On the basis of the relationship between the displacement variables and force and moment resultants, the detailed expression of the force resultant and moment vector is shown as follows:(20)Nxx,θ,tMxx,θ,tNxθx,θ,tQxx,θ,t=∑n=0∞∑i=18Nx¯xWn,icosnθMx¯xWn,icosnθNxθ¯xWn,isinnθQx¯xWn,icosnθe−jωt,where(21)Nx¯x=ikniA11λn,i+nA12Rκn,i+A12Rejknix,Mx¯x=nD12R2κn,i+D12n2R2+D11kni2ejknix,Nxθ¯x=−nA66Rλn,i+ikniA66κn,ieiknix,Qx¯x=inkniD12+D66R2κn,i+iD11kni3+in2kniD12+2D66R2ejkn,ix.For the classical boundary conditions, four type of boundary conditions are considered which are widely used in some engineering applications, as follows [54–56]:Free (F):(22)Nx=Nxθ+MxθR=Qx+∂Mxθ∂θ=Mx=0.Clamped (C):(23)u=v=w=∂w∂x=0.Simply supported (SS):(24)u=v=w=Mx=0.Shear diaphragm (SD):(25)Nx=v=w=Mx=0.For the elastic boundary conditions, the relationship between the force, displacement, and elastic restrained stiffness is discussed. When the elastic restrained stiffness is in the axial direction, the boundary relationship is given as follows:(26)Kuux,θ,t±Nxx,θ,t=0,where Ku is the stiffness constant, and the symbol ± means the elastic restrained spring at the boundary edge x = L and x = 0. Also, for the circumferential displacement, the normal displacement, and the transverse normal rotations of the θ and x axis, the symbol ± has the same meaning for the elastic boundary conditions. For the other elastic restrained states, the boundary relationships can refer to Reference [57]. Related to the introduction of the elastic and classical boundary conditions, the final equation of the orthotropic cylindrical shell is given as follows:(27)KW=F,where K is the overall matrix, F is the external force resultant, and W = {W1; W2} is the wave contribution factor vector. W1 and W2 are the wave contribution factor vectors which are related to the boundary conditions at the two ends. The overall matrix K is shown as follows:(28)K=D10B1LB20D20,where D1 and D2 are the boundary matrixes which depend on the boundary conditions and B1 and B2 are the segment matrixes which are given as follows:(29)B1,2x=±δxfx,where δx and fx are the displacement and force matrix of the orthotropic shell and are expressed as follows:(30)δx=u1¯xu2¯x⋯u8¯xv1¯xv2¯x⋯v8¯xw1¯xw2¯x⋯w8¯x∂w1¯x∂x∂w2¯x∂x⋯∂w8¯x∂x,fx=Nx,1¯xNx,2¯x⋯Nx,8¯xMx,1¯xMx,2¯x⋯Mx,8¯xNxθ,1¯xNxθ,2¯x⋯Nxθ,8¯xQx,1¯xQx,2¯x⋯Qx,8¯x.When analyzing the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, the external force vectorF should be vanished. For each circumferential number n, searching the zero position of the overall matrix K, a series of values are calculated. When the sign occurs, the natural frequencies of the cylindrical shell are obtained.The dichotomy method is a mathematical idea that uses flat partitions and infinite approximations. It is an effective algorithm for avoiding complex calculations and approximations of analytic one-dimensional functions. In this paper, the dichotomy method is adopted to get the natural frequencies of the cylindrical shell, and some numerical examples are established.


## 3. Numerical Examples and Discussion

In this part, some numerical examples are presented to investigate the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions (i.e., classical, elastic, and combinations). Some numerical examples are extended to verify the correctness of the results by the presented method. Furthermore, the effects of the material parameter and geometric constants on the free vibration characteristics of the orthotropic circular cylindrical shell are studied.


### 3.1. Orthotropic Cylindrical Shell with Classical and Elastic Boundary Conditions

First, the comparison of the natural frequencies for the homogeneous cylindrical shell under clamped-clamped boundary condition is presented. The results by the present method are compared with the solutions by Ref. [58] in Table 1. The geometric parameters and material constants are defined as follows: L = 511.2 mm, R = 216.2 mm, h = 1.5 mm, E = 1.83 × 1011 N/m2, μ = 0.3, and ρ = 7492 kg/m3. The range of the circumferential mode number n and longitudinal wave number m is set as 2–6 and 1–3. It can be seen that the results of the two methods are in good agreement, and the maximum error is 5.79% which appears with n = 3 and m = 1. Next, this paper discusses the free vibration characteristics of the orthotropic cylindrical shell under classical boundary conditions. To verify the correctness of the results by the presented method, the natural frequencies of the orthotropic cylindrical shell under several classical boundary conditions are compared with the solutions in the reported literature by Zhao et al. [59]. There are three pair classical boundary conditions which are set as SS-SS, SD-SD, and C-C. The material parameters are given as follows: E1 = 120 GPa, E2 = 10 GPa, G12 = 5.5 GPa, μ12 = 0.27, and ρ = 1700 kg/m3. The geometric constants are set as follows: L = 5 m, h = 0.01 m, and R = 1 m. In Table 2, the natural frequencies of the first two longitudinal modes for the first six circumferential numbers are calculated. By comparing the MRRM in the reference literature and the results by the presented method, we can discover that the errors are small. It is obvious that the free vibration characteristics for the orthotropic circular cylindrical shell under classical boundary conditions calculated by the presented method are correct. From Table 1, we can find that for each circumferential number and longitudinal mode, the minimum natural frequency is associated with the boundary condition SD-SD and the maximum natural frequency is related to the boundary condition C-C. It can be concluded that the classical boundary conditions have a significant effect on the free vibration characteristics for the orthotropic circular cylindrical shell.Table 1
Comparison of the natural frequencies for the homogeneous cylindrical shell under clamped-clamped boundary conditions.
n
m = 1
m = 2
m = 3
Present
Reference [58]
Error (%)
Present
Reference [58]
Error (%)
Present
Reference [58]
Error (%)
2
1281.90
1299
1.32
2593.52
2660
2.50
4245.89
4364
2.71
3
2110.36
2240
5.79
2555.82
2574
0.71
3389.93
3483
2.67
4
3830.74
3872
1.07
4082.66
4043
−0.98
4473.30
4417
−1.27
5
6240.10
6262
0.35
6383.64
6491
1.65
6645.01
6728
1.23
6
9087.86
9125
0.41
9129.98
9276
1.57
9259.20
9540
2.94Table 2
Comparison of the natural frequencies for the orthotropic circular cylindrical shell under classical boundary conditions.
Boundary conditions
Method
n = 1
n = 2
n = 3
n = 4
n = 5
n = 1
n = 2
n = 3
n = 4
n = 5
m = 1
m = 2
C-C
Present
122.7511
76.4642
52.8701
41.4866
39.4239
225.6351
146.3284
103.4431
79.3989
67.0333
Reference [59]
122.75
76.464
52.869
41.486
39.423
225.63
146.33
103.44
79.398
67.033
Error (%)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
SS-SS
Present
122.5910
76.1294
52.4352
41.0422
39.0609
225.2716
145.6195
102.4194
78.1700
65.7589
Reference [59]
122.59
76.129
52.435
41.042
39.061
225.27
145.62
102.42
78.17
65.759
Error (%)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
SD-SD
Present
117.9855
66.1809
40.8206
30.8837
32.3243
225.2414
144.7009
99.8179
74.0717
61.0129
Reference [59]
117.99
66.181
40.821
30.884
32.324
225.24
144.7
99.818
74.072
61.01
Error (%)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00Next, the free vibration characteristics of the orthotropic circular cylindrical shell under classical combination boundary conditions are concerned. The material parameters and geometric constants are equal to the numerical example in Table2. For the classical combination boundary conditions, there are two type boundary conditions, C-F and SS-F, which are considered, and the results by the presented method are compared with the solutions by the MRRM method [59]. From Table 2, the errors between the two numerical methods are small, and the maximum error is 0.95%. So, it can be concluded that the free vibration characteristics which are investigated by the presented method are right. From the two numerical examples in Tables 2 and 3, the free vibration characteristics for orthotropic circular cylindrical shell under classical boundary conditions and their combinations are discussed. Figure 2 shows some modes of orthotropic cylindrical shells under C-C boundary conditions. The purpose is to further study the free vibration characteristics of orthotropic cylindrical shells under classical boundary conditions.Table 3
Comparison of the natural frequencies for the orthotropic circular cylindrical shell under combination classical boundary conditions.
Boundary conditions
Method
n = 1
n = 2
n = 3
n = 4
n = 5
n = 1
n = 2
n = 3
n = 4
n = 5
m = 1
m = 2
C-F
Present
57.2692
29.4248
18.4766
19.1837
27.1424
166.7386
96.3363
62.8738
46.8067
42.1131
Reference [59]
57.272
29.445
18.566
19.365
27.376
166.74
96.347
62.922
46.952
42.414
Error (%)
0.00
−0.07
−0.48
−0.94
−0.86
0.00
−0.01
−0.08
−0.31
−0.71
SS-F
Present
57.2303
29.3770
18.4436
19.1688
27.1366
166.6339
96.0876
62.5356
46.4639
41.8393
Reference [59]
57.233
29.397
18.532
19.35
27.369
166.64
96.098
62.584
46.609
42.137
Error (%)
0.00
−0.07
−0.48
−0.95
−0.86
0.00
−0.01
−0.08
−0.31
−0.71Figure 2
The mode shapes of the orthotropic circular cylindrical shell under C-C boundary condition. (a)n = 1, m = 1, (b) n = 2, m = 1, (c) n = 3, m = 1, (d) n = 1, m = 2, (e) n = 2, m = 2, and (f) n = 3, m = 2.
(a)
(b)
(c)
(d)
(e)
(f)Next, the free vibration characteristics of the orthotropic circular cylindrical shell under elastic boundary conditions and their combinations are concerned. In this paper, there are three type elastic-restrained situations which are considered as follows:First-elastic boundary condition (EBC_1): the axial displacement is under elastic restrained and others are fixed (u ≠ 0, v=w=∂w/∂x=0), and the elastic stiffness value is set as Ku = 109.Second-elastic boundary condition (EBC_2): the circumferential displacement is under elastic restrained and others are fixed (v ≠ 0, u = w = ∂w/∂x = 0), and the elastic stiffness value is set as Kv = 109.Third-elastic boundary condition (EBC_3): the radial displacement is under elastic restrained and others are fixed (w ≠ 0, u = v = ∂w/∂x = 0), and the elastic stiffness value is set as Kw = Kθ = 107.In Table4, there are six type elastic boundary conditions and their combinations (i.e., EBC1-EBC1, EBC2-EBC2, EBC3-EBC3, EBC1-EBC3, EBC2-EBC3, and EBC1-EBC2) are concerned. The first five circumferential numbers (i.e., n = 1–5) and the first four longitudinal modes (i.e., m = 1–4), natural frequencies for the orthotropic circular cylindrical shells under several elastic boundary conditions, are calculated. The material parameters and geometric constants are the same as the numerical example in previous discussion. For various elastic boundary conditions, the natural frequencies are relatively stable within a range and have a small variation range for various circumferential numbers and longitudinal modes. Also, some mode shapes of the orthotropic circular cylindrical shell with EBC1-EBC1 are shown in the Figure 3.Table 4
The natural frequencies for the orthotropic circular cylindrical shell under elastic boundary conditions and their combinations.
n
m
Boundary conditions
EBC1-EBC1
EBC2-EBC2
EBC3-EBC3
EBC1-EBC3
EBC2-EBC3
EBC1-EBC2
1
1
121.211
122.513
122.741
121.859
122.627
120.6801
2
225.634
225.269
225.618
225.626
225.444
223.8509
3
295.019
294.726
294.998
295.008
294.862
293.584
4
332.953
332.765
332.924
332.939
332.845
332.059
2
1
72.788
76.340
76.442
74.491
76.391
73.89489
2
146.075
146.074
146.289
146.194
146.182
145.0127
3
208.553
208.389
208.619
208.581
208.504
207.2762
4
255.979
255.727
255.920
255.950
255.823
254.7795
3
1
48.121
52.807
52.844
50.435
52.826
50.16391
2
102.408
103.281
103.387
102.929
103.334
102.2252
3
154.456
154.656
154.785
154.613
154.720
153.7046
4
199.167
199.040
199.161
199.166
199.100
198.2201
4
1
36.929
41.458
41.463
39.174
41.460
39.06892
2
77.503
79.304
79.336
78.464
79.320
78.08885
3
120.575
121.249
121.294
120.930
121.271
120.3684
4
160.463
160.585
160.623
160.546
160.604
159.9009
5
1
36.110
39.413
39.407
37.717
39.410
37.6851
2
64.670
66.982
66.973
65.868
66.978
65.69338
3
99.855
100.974
100.959
100.409
100.967
100.0973
4
134.854
135.270
135.241
135.051
135.256
134.6571Figure 3
The mode shapes of the orthotropic circular cylindrical shell under EBC1-EBC1 boundary condition. (a)n = 1, m = 1, (b) n = 2, m = 1, (c) n = 3, m = 1, (d) n = 1, m = 2, (e) n = 2, m = 2, and (f) n = 3, m = 2.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.2. Effect of the Material and Geometric Parameters on the Natural Frequencies

In addition, the effect of the material parameter and geometric constants on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions (i.e., C-C, S-F, EBC_1-EBC_2, and EBC_3-EBC_3) is discussed.First, the effect of the modulus ratioE1/E2 for the orthotropic circular cylindrical shell under various boundary conditions is discussed. The material parameter and geometric constants are similar to the numerical example in the previous study. The changing rule of the modulus ratio E1/E2 is from 1 to 12, the natural frequencies for the first two circumferential number (i.e., n = 1 and 2) and the first four longitudinal mode (i.e., m = 1–4) under various boundary conditions are calculated in Table 5. For analyzing the effect of the modulus ratio E1/E2, the changing rule of the natural frequencies under various boundary conditions is shown in the Figure 4 (n = 1). It is obvious that with the changing of the modulus ratio E1/E2, the natural frequencies are generally growing for various longitudinal modes under various boundary conditions. Also, for the boundary condition S-F, the change range of the natural frequencies is more obvious for various longitudinal modes. For other boundary conditions, the change range for the first two longitudinal modes (i.e., m = 1 and 2) are evident, and for the longitudinal mode m = 3, the change range stays in the basic stable range.Table 5
The natural frequencies of the orthotropic cylindrical shell with various modulus ratiosE1/E2 under several boundary conditions.
E
1/E2
Boundary conditions
m
C-C
SS-F
EBC1-EBC2
EBC3-EBC3
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
1
1
110.6992
55.2536
31.2626
11.6005
108.9950
54.0279
110.6879
55.2403
2
211.2475
120.7196
119.1539
58.5856
209.0331
118.8745
211.2434
120.7013
3
292.7848
188.8046
234.6133
132.9303
291.0075
186.7130
292.7825
188.7881
3
1
119.3153
68.6970
45.3992
18.7554
117.2422
66.4857
119.3085
68.6849
2
221.3334
136.5006
144.2771
78.3209
219.3547
134.6394
221.3285
136.4841
3
294.0729
201.8201
253.0665
157.9665
292.4997
200.0412
294.0666
201.8009
5
1
121.1147
72.5290
50.8336
22.7042
118.9951
70.0528
121.1077
72.5153
2
223.5331
140.9972
154.0669
85.6214
221.6405
139.3435
223.5256
140.9765
3
294.4017
205.0738
257.5586
165.3520
292.8816
203.4539
294.3914
205.0469
8
1
122.1522
74.9725
54.7245
26.3878
120.0403
72.3926
122.1439
74.9555
2
224.8418
144.1636
161.5715
91.4974
223.0119
142.7059
224.8302
144.1353
3
294.7082
207.2162
260.3054
170.3087
293.2317
205.7223
294.6917
207.1771
10
1
122.5081
75.8520
56.1950
28.0675
120.4149
73.2671
122.4986
75.8326
2
225.3073
145.4115
164.5223
94.0730
223.5032
144.0357
225.2928
145.3778
3
294.8730
208.0499
261.2583
172.1662
293.4169
206.6100
294.8524
208.0026
12
1
122.7511
76.4642
57.2303
29.3770
120.6801
73.8949
122.7406
76.4423
2
225.6351
146.3284
166.6339
96.0876
223.8509
145.0127
225.6179
146.2894
3
295.0230
208.6741
261.9059
173.4884
293.5837
207.2762
294.9984
208.6188Figure 4
The changing rule of the natural frequencies with various modulus ratiosE1/E2; n = 1. (a) C-C, (b) S-F, (c) EBC1-EBC2, and (d) EBC3-EBC3.
(a)
(b)
(c)
(d)Next, the effect of the geometric constants on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions is discussed. In this paper, the geometric constants are set as thickness to radius ratioh/R and length to radius ratio L/R. For the boundary conditions, geometric and material parameters are same as the numerical example in the previous study for the effect of the modulus ratio E1/E2. In Table 6, the natural frequencies for the orthotropic circular cylindrical shell with the changing of the thickness to radius ratios h/R are calculated. The changing range of the thickness to radius ratios h/R is set as 0.001 to 0.1. The natural frequencies are generally increased for various circumferential number and longitudinal modes for boundary condition C-C, SS-F, and ECB_3-ECB_3. Furthermore, the natural frequencies are first decreased and then increased with the growing of the thickness to radius ratios h/R under EBC_1-EBC_2. In order to reflect the law of natural frequency change more intuitively, the changing rule of the natural frequencies with the increase of the thickness to radius ratios h/R under several boundary conditions is shown in Figure 5. Especially, when the thickness to radius ratios h/R are set from 0.05 to 0.1 and related to the longitudinal wave number m = 3, the growth rates of the natural frequencies are more obvious than the longitudinal wave number m = 1 and 2. Also, for the boundary condition S-F and EBC3-EBC-3, when the longitudinal wave number m = 1 and 2, the natural frequencies have a small range of variation and is basically kept within a certain range. In particular, when the boundary condition is set to EBC1-EBC1 and longitudinal wave number m = 1, as the thickness to radius ratios h/R increases, the natural frequencies tend to decrease, but the variation is small.Table 6
The natural frequencies of the orthotropic cylindrical shell with various thickness to radius ratiosh/R under several boundary conditions.
h
Boundary conditions
m
C-C
SS-F
EBC1-EBC2
EBC3-EBC3
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
0.001
1
122.5913
76.0481
57.2287
29.2184
122.3553
75.6950
122.5913
76.0480
2
225.2229
145.4536
166.6168
95.9782
225.0297
145.2889
225.2229
145.4535
3
294.0861
207.0796
261.7840
173.2139
293.9218
206.8985
294.0861
207.0795
0.005
1
122.6467
76.1854
57.2295
29.2576
121.5370
74.6696
122.6451
76.1818
2
225.3556
145.7348
166.6211
96.0050
224.4220
144.9893
225.3533
145.7292
3
294.3756
207.5760
261.8143
173.2815
293.6008
206.7692
294.3724
207.5683
0.01
1
122.7511
76.4642
57.2303
29.3770
120.6801
73.8949
122.7406
76.4423
2
225.6351
146.3284
166.6339
96.0876
223.8509
145.0127
225.6179
146.2894
3
295.0230
208.6741
261.9059
173.4884
293.5837
207.2762
294.9984
208.6188
0.05
1
124.3256
81.4447
57.2392
32.9221
117.5104
76.1999
123.8317
80.5904
2
230.6216
157.0350
167.0094
98.6036
224.4088
154.2499
229.4403
154.9489
3
308.5174
230.8005
264.5660
179.5628
303.9226
227.8976
305.9156
226.6462
0.1
1
127.4686
92.6207
57.2553
42.0016
118.1959
87.4795
125.6479
89.6293
2
242.0943
181.4167
168.0745
105.8049
233.3506
178.7213
236.1977
171.4342
3
343.1045
283.8211
271.9447
196.1988
335.7079
280.6610
322.5592
255.6987Figure 5
The changing rule of the natural frequencies with various thickness to radius ratiosh/R; n = 1. (a) C-C, (b) S-F, (c) EBC1-EBC2, and (d) EBC3-EBC3.
(a)
(b)
(c)
(d)Furthermore, the influence of the length to radius ratiosL/R on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions is calculated in Table 7. The material constants and geometric properties are the same as the numerical study in previous part, and the changing range of the length to radius ratios L/R are set from 5 to 12. In Figure 6, the changing rule of the natural frequencies with respect to the length to radius ratios L/R under various boundary conditions is shown. It can be seen that with the changing of the length to radius ratios L/R, the natural frequencies are generally decreased for various boundary conditions related to three longitudinal wave number m = 1–3, at the same time, and as the length to radius ratios L/R changes, the attenuation effect of the frequency parameters is more obvious. When the length to radius ratios L/R changes from 5 to 8, the decreased attenuations are larger for four type boundary conditions.Table 7
The natural frequencies of the orthotropic cylindrical shell with various length to radius ratiosL/R under several boundary conditions.
L
Boundary conditions
m
C-C
SS-F
EBC1-EBC2
EBC3-EBC3
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
5
1
122.7511
76.4642
57.2303
29.3770
120.6801
73.8949
122.7406
76.4423
2
225.6351
146.3284
166.6339
96.0876
223.8509
145.0127
225.6179
146.2894
3
295.0230
208.6741
261.9059
173.4884
293.5837
207.2762
294.9984
208.6188
8
1
76.7055
45.4524
31.6944
14.1556
75.2519
43.4839
76.7009
45.4440
2
148.5210
90.2844
99.4478
53.6849
147.5756
89.3387
148.5141
90.2678
3
214.2263
137.1869
178.4795
107.3297
213.0963
136.2748
214.2159
137.1633
10
1
60.6792
34.7071
23.1038
9.8312
59.4140
33.0345
60.6761
34.7020
2
118.8628
70.3232
76.2998
39.8337
118.1806
69.3907
118.8582
70.3123
3
176.4915
109.1948
143.5250
82.7372
175.6285
108.4187
176.4847
109.1790
12
1
49.7780
27.4582
17.5069
7.3573
48.6440
26.0412
49.7758
27.4549
2
98.2672
56.7730
61.0112
30.8062
97.7181
55.8310
98.2638
56.7654
3
148.6260
89.5260
118.5957
65.9236
147.9499
88.8051
148.6211
89.5147Figure 6
The changing rule of the natural frequencies with various length to radius ratiosL/R; n = 1. (a) C-C, (b) S-F, (c) EBC1-EBC2, and (d) EBC3-EBC3.
(a)
(b)
(c)
(d)


## 3.1. Orthotropic Cylindrical Shell with Classical and Elastic Boundary Conditions

First, the comparison of the natural frequencies for the homogeneous cylindrical shell under clamped-clamped boundary condition is presented. The results by the present method are compared with the solutions by Ref. [58] in Table 1. The geometric parameters and material constants are defined as follows: L = 511.2 mm, R = 216.2 mm, h = 1.5 mm, E = 1.83 × 1011 N/m2, μ = 0.3, and ρ = 7492 kg/m3. The range of the circumferential mode number n and longitudinal wave number m is set as 2–6 and 1–3. It can be seen that the results of the two methods are in good agreement, and the maximum error is 5.79% which appears with n = 3 and m = 1. Next, this paper discusses the free vibration characteristics of the orthotropic cylindrical shell under classical boundary conditions. To verify the correctness of the results by the presented method, the natural frequencies of the orthotropic cylindrical shell under several classical boundary conditions are compared with the solutions in the reported literature by Zhao et al. [59]. There are three pair classical boundary conditions which are set as SS-SS, SD-SD, and C-C. The material parameters are given as follows: E1 = 120 GPa, E2 = 10 GPa, G12 = 5.5 GPa, μ12 = 0.27, and ρ = 1700 kg/m3. The geometric constants are set as follows: L = 5 m, h = 0.01 m, and R = 1 m. In Table 2, the natural frequencies of the first two longitudinal modes for the first six circumferential numbers are calculated. By comparing the MRRM in the reference literature and the results by the presented method, we can discover that the errors are small. It is obvious that the free vibration characteristics for the orthotropic circular cylindrical shell under classical boundary conditions calculated by the presented method are correct. From Table 1, we can find that for each circumferential number and longitudinal mode, the minimum natural frequency is associated with the boundary condition SD-SD and the maximum natural frequency is related to the boundary condition C-C. It can be concluded that the classical boundary conditions have a significant effect on the free vibration characteristics for the orthotropic circular cylindrical shell.Table 1
Comparison of the natural frequencies for the homogeneous cylindrical shell under clamped-clamped boundary conditions.
n
m = 1
m = 2
m = 3
Present
Reference [58]
Error (%)
Present
Reference [58]
Error (%)
Present
Reference [58]
Error (%)
2
1281.90
1299
1.32
2593.52
2660
2.50
4245.89
4364
2.71
3
2110.36
2240
5.79
2555.82
2574
0.71
3389.93
3483
2.67
4
3830.74
3872
1.07
4082.66
4043
−0.98
4473.30
4417
−1.27
5
6240.10
6262
0.35
6383.64
6491
1.65
6645.01
6728
1.23
6
9087.86
9125
0.41
9129.98
9276
1.57
9259.20
9540
2.94Table 2
Comparison of the natural frequencies for the orthotropic circular cylindrical shell under classical boundary conditions.
Boundary conditions
Method
n = 1
n = 2
n = 3
n = 4
n = 5
n = 1
n = 2
n = 3
n = 4
n = 5
m = 1
m = 2
C-C
Present
122.7511
76.4642
52.8701
41.4866
39.4239
225.6351
146.3284
103.4431
79.3989
67.0333
Reference [59]
122.75
76.464
52.869
41.486
39.423
225.63
146.33
103.44
79.398
67.033
Error (%)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
SS-SS
Present
122.5910
76.1294
52.4352
41.0422
39.0609
225.2716
145.6195
102.4194
78.1700
65.7589
Reference [59]
122.59
76.129
52.435
41.042
39.061
225.27
145.62
102.42
78.17
65.759
Error (%)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
SD-SD
Present
117.9855
66.1809
40.8206
30.8837
32.3243
225.2414
144.7009
99.8179
74.0717
61.0129
Reference [59]
117.99
66.181
40.821
30.884
32.324
225.24
144.7
99.818
74.072
61.01
Error (%)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00Next, the free vibration characteristics of the orthotropic circular cylindrical shell under classical combination boundary conditions are concerned. The material parameters and geometric constants are equal to the numerical example in Table2. For the classical combination boundary conditions, there are two type boundary conditions, C-F and SS-F, which are considered, and the results by the presented method are compared with the solutions by the MRRM method [59]. From Table 2, the errors between the two numerical methods are small, and the maximum error is 0.95%. So, it can be concluded that the free vibration characteristics which are investigated by the presented method are right. From the two numerical examples in Tables 2 and 3, the free vibration characteristics for orthotropic circular cylindrical shell under classical boundary conditions and their combinations are discussed. Figure 2 shows some modes of orthotropic cylindrical shells under C-C boundary conditions. The purpose is to further study the free vibration characteristics of orthotropic cylindrical shells under classical boundary conditions.Table 3
Comparison of the natural frequencies for the orthotropic circular cylindrical shell under combination classical boundary conditions.
Boundary conditions
Method
n = 1
n = 2
n = 3
n = 4
n = 5
n = 1
n = 2
n = 3
n = 4
n = 5
m = 1
m = 2
C-F
Present
57.2692
29.4248
18.4766
19.1837
27.1424
166.7386
96.3363
62.8738
46.8067
42.1131
Reference [59]
57.272
29.445
18.566
19.365
27.376
166.74
96.347
62.922
46.952
42.414
Error (%)
0.00
−0.07
−0.48
−0.94
−0.86
0.00
−0.01
−0.08
−0.31
−0.71
SS-F
Present
57.2303
29.3770
18.4436
19.1688
27.1366
166.6339
96.0876
62.5356
46.4639
41.8393
Reference [59]
57.233
29.397
18.532
19.35
27.369
166.64
96.098
62.584
46.609
42.137
Error (%)
0.00
−0.07
−0.48
−0.95
−0.86
0.00
−0.01
−0.08
−0.31
−0.71Figure 2
The mode shapes of the orthotropic circular cylindrical shell under C-C boundary condition. (a)n = 1, m = 1, (b) n = 2, m = 1, (c) n = 3, m = 1, (d) n = 1, m = 2, (e) n = 2, m = 2, and (f) n = 3, m = 2.
(a)
(b)
(c)
(d)
(e)
(f)Next, the free vibration characteristics of the orthotropic circular cylindrical shell under elastic boundary conditions and their combinations are concerned. In this paper, there are three type elastic-restrained situations which are considered as follows:First-elastic boundary condition (EBC_1): the axial displacement is under elastic restrained and others are fixed (u ≠ 0, v=w=∂w/∂x=0), and the elastic stiffness value is set as Ku = 109.Second-elastic boundary condition (EBC_2): the circumferential displacement is under elastic restrained and others are fixed (v ≠ 0, u = w = ∂w/∂x = 0), and the elastic stiffness value is set as Kv = 109.Third-elastic boundary condition (EBC_3): the radial displacement is under elastic restrained and others are fixed (w ≠ 0, u = v = ∂w/∂x = 0), and the elastic stiffness value is set as Kw = Kθ = 107.In Table4, there are six type elastic boundary conditions and their combinations (i.e., EBC1-EBC1, EBC2-EBC2, EBC3-EBC3, EBC1-EBC3, EBC2-EBC3, and EBC1-EBC2) are concerned. The first five circumferential numbers (i.e., n = 1–5) and the first four longitudinal modes (i.e., m = 1–4), natural frequencies for the orthotropic circular cylindrical shells under several elastic boundary conditions, are calculated. The material parameters and geometric constants are the same as the numerical example in previous discussion. For various elastic boundary conditions, the natural frequencies are relatively stable within a range and have a small variation range for various circumferential numbers and longitudinal modes. Also, some mode shapes of the orthotropic circular cylindrical shell with EBC1-EBC1 are shown in the Figure 3.Table 4
The natural frequencies for the orthotropic circular cylindrical shell under elastic boundary conditions and their combinations.
n
m
Boundary conditions
EBC1-EBC1
EBC2-EBC2
EBC3-EBC3
EBC1-EBC3
EBC2-EBC3
EBC1-EBC2
1
1
121.211
122.513
122.741
121.859
122.627
120.6801
2
225.634
225.269
225.618
225.626
225.444
223.8509
3
295.019
294.726
294.998
295.008
294.862
293.584
4
332.953
332.765
332.924
332.939
332.845
332.059
2
1
72.788
76.340
76.442
74.491
76.391
73.89489
2
146.075
146.074
146.289
146.194
146.182
145.0127
3
208.553
208.389
208.619
208.581
208.504
207.2762
4
255.979
255.727
255.920
255.950
255.823
254.7795
3
1
48.121
52.807
52.844
50.435
52.826
50.16391
2
102.408
103.281
103.387
102.929
103.334
102.2252
3
154.456
154.656
154.785
154.613
154.720
153.7046
4
199.167
199.040
199.161
199.166
199.100
198.2201
4
1
36.929
41.458
41.463
39.174
41.460
39.06892
2
77.503
79.304
79.336
78.464
79.320
78.08885
3
120.575
121.249
121.294
120.930
121.271
120.3684
4
160.463
160.585
160.623
160.546
160.604
159.9009
5
1
36.110
39.413
39.407
37.717
39.410
37.6851
2
64.670
66.982
66.973
65.868
66.978
65.69338
3
99.855
100.974
100.959
100.409
100.967
100.0973
4
134.854
135.270
135.241
135.051
135.256
134.6571Figure 3
The mode shapes of the orthotropic circular cylindrical shell under EBC1-EBC1 boundary condition. (a)n = 1, m = 1, (b) n = 2, m = 1, (c) n = 3, m = 1, (d) n = 1, m = 2, (e) n = 2, m = 2, and (f) n = 3, m = 2.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.2. Effect of the Material and Geometric Parameters on the Natural Frequencies

In addition, the effect of the material parameter and geometric constants on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions (i.e., C-C, S-F, EBC_1-EBC_2, and EBC_3-EBC_3) is discussed.First, the effect of the modulus ratioE1/E2 for the orthotropic circular cylindrical shell under various boundary conditions is discussed. The material parameter and geometric constants are similar to the numerical example in the previous study. The changing rule of the modulus ratio E1/E2 is from 1 to 12, the natural frequencies for the first two circumferential number (i.e., n = 1 and 2) and the first four longitudinal mode (i.e., m = 1–4) under various boundary conditions are calculated in Table 5. For analyzing the effect of the modulus ratio E1/E2, the changing rule of the natural frequencies under various boundary conditions is shown in the Figure 4 (n = 1). It is obvious that with the changing of the modulus ratio E1/E2, the natural frequencies are generally growing for various longitudinal modes under various boundary conditions. Also, for the boundary condition S-F, the change range of the natural frequencies is more obvious for various longitudinal modes. For other boundary conditions, the change range for the first two longitudinal modes (i.e., m = 1 and 2) are evident, and for the longitudinal mode m = 3, the change range stays in the basic stable range.Table 5
The natural frequencies of the orthotropic cylindrical shell with various modulus ratiosE1/E2 under several boundary conditions.
E
1/E2
Boundary conditions
m
C-C
SS-F
EBC1-EBC2
EBC3-EBC3
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
1
1
110.6992
55.2536
31.2626
11.6005
108.9950
54.0279
110.6879
55.2403
2
211.2475
120.7196
119.1539
58.5856
209.0331
118.8745
211.2434
120.7013
3
292.7848
188.8046
234.6133
132.9303
291.0075
186.7130
292.7825
188.7881
3
1
119.3153
68.6970
45.3992
18.7554
117.2422
66.4857
119.3085
68.6849
2
221.3334
136.5006
144.2771
78.3209
219.3547
134.6394
221.3285
136.4841
3
294.0729
201.8201
253.0665
157.9665
292.4997
200.0412
294.0666
201.8009
5
1
121.1147
72.5290
50.8336
22.7042
118.9951
70.0528
121.1077
72.5153
2
223.5331
140.9972
154.0669
85.6214
221.6405
139.3435
223.5256
140.9765
3
294.4017
205.0738
257.5586
165.3520
292.8816
203.4539
294.3914
205.0469
8
1
122.1522
74.9725
54.7245
26.3878
120.0403
72.3926
122.1439
74.9555
2
224.8418
144.1636
161.5715
91.4974
223.0119
142.7059
224.8302
144.1353
3
294.7082
207.2162
260.3054
170.3087
293.2317
205.7223
294.6917
207.1771
10
1
122.5081
75.8520
56.1950
28.0675
120.4149
73.2671
122.4986
75.8326
2
225.3073
145.4115
164.5223
94.0730
223.5032
144.0357
225.2928
145.3778
3
294.8730
208.0499
261.2583
172.1662
293.4169
206.6100
294.8524
208.0026
12
1
122.7511
76.4642
57.2303
29.3770
120.6801
73.8949
122.7406
76.4423
2
225.6351
146.3284
166.6339
96.0876
223.8509
145.0127
225.6179
146.2894
3
295.0230
208.6741
261.9059
173.4884
293.5837
207.2762
294.9984
208.6188Figure 4
The changing rule of the natural frequencies with various modulus ratiosE1/E2; n = 1. (a) C-C, (b) S-F, (c) EBC1-EBC2, and (d) EBC3-EBC3.
(a)
(b)
(c)
(d)Next, the effect of the geometric constants on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions is discussed. In this paper, the geometric constants are set as thickness to radius ratioh/R and length to radius ratio L/R. For the boundary conditions, geometric and material parameters are same as the numerical example in the previous study for the effect of the modulus ratio E1/E2. In Table 6, the natural frequencies for the orthotropic circular cylindrical shell with the changing of the thickness to radius ratios h/R are calculated. The changing range of the thickness to radius ratios h/R is set as 0.001 to 0.1. The natural frequencies are generally increased for various circumferential number and longitudinal modes for boundary condition C-C, SS-F, and ECB_3-ECB_3. Furthermore, the natural frequencies are first decreased and then increased with the growing of the thickness to radius ratios h/R under EBC_1-EBC_2. In order to reflect the law of natural frequency change more intuitively, the changing rule of the natural frequencies with the increase of the thickness to radius ratios h/R under several boundary conditions is shown in Figure 5. Especially, when the thickness to radius ratios h/R are set from 0.05 to 0.1 and related to the longitudinal wave number m = 3, the growth rates of the natural frequencies are more obvious than the longitudinal wave number m = 1 and 2. Also, for the boundary condition S-F and EBC3-EBC-3, when the longitudinal wave number m = 1 and 2, the natural frequencies have a small range of variation and is basically kept within a certain range. In particular, when the boundary condition is set to EBC1-EBC1 and longitudinal wave number m = 1, as the thickness to radius ratios h/R increases, the natural frequencies tend to decrease, but the variation is small.Table 6
The natural frequencies of the orthotropic cylindrical shell with various thickness to radius ratiosh/R under several boundary conditions.
h
Boundary conditions
m
C-C
SS-F
EBC1-EBC2
EBC3-EBC3
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
0.001
1
122.5913
76.0481
57.2287
29.2184
122.3553
75.6950
122.5913
76.0480
2
225.2229
145.4536
166.6168
95.9782
225.0297
145.2889
225.2229
145.4535
3
294.0861
207.0796
261.7840
173.2139
293.9218
206.8985
294.0861
207.0795
0.005
1
122.6467
76.1854
57.2295
29.2576
121.5370
74.6696
122.6451
76.1818
2
225.3556
145.7348
166.6211
96.0050
224.4220
144.9893
225.3533
145.7292
3
294.3756
207.5760
261.8143
173.2815
293.6008
206.7692
294.3724
207.5683
0.01
1
122.7511
76.4642
57.2303
29.3770
120.6801
73.8949
122.7406
76.4423
2
225.6351
146.3284
166.6339
96.0876
223.8509
145.0127
225.6179
146.2894
3
295.0230
208.6741
261.9059
173.4884
293.5837
207.2762
294.9984
208.6188
0.05
1
124.3256
81.4447
57.2392
32.9221
117.5104
76.1999
123.8317
80.5904
2
230.6216
157.0350
167.0094
98.6036
224.4088
154.2499
229.4403
154.9489
3
308.5174
230.8005
264.5660
179.5628
303.9226
227.8976
305.9156
226.6462
0.1
1
127.4686
92.6207
57.2553
42.0016
118.1959
87.4795
125.6479
89.6293
2
242.0943
181.4167
168.0745
105.8049
233.3506
178.7213
236.1977
171.4342
3
343.1045
283.8211
271.9447
196.1988
335.7079
280.6610
322.5592
255.6987Figure 5
The changing rule of the natural frequencies with various thickness to radius ratiosh/R; n = 1. (a) C-C, (b) S-F, (c) EBC1-EBC2, and (d) EBC3-EBC3.
(a)
(b)
(c)
(d)Furthermore, the influence of the length to radius ratiosL/R on the natural frequencies for the orthotropic circular cylindrical shell under several boundary conditions is calculated in Table 7. The material constants and geometric properties are the same as the numerical study in previous part, and the changing range of the length to radius ratios L/R are set from 5 to 12. In Figure 6, the changing rule of the natural frequencies with respect to the length to radius ratios L/R under various boundary conditions is shown. It can be seen that with the changing of the length to radius ratios L/R, the natural frequencies are generally decreased for various boundary conditions related to three longitudinal wave number m = 1–3, at the same time, and as the length to radius ratios L/R changes, the attenuation effect of the frequency parameters is more obvious. When the length to radius ratios L/R changes from 5 to 8, the decreased attenuations are larger for four type boundary conditions.Table 7
The natural frequencies of the orthotropic cylindrical shell with various length to radius ratiosL/R under several boundary conditions.
L
Boundary conditions
m
C-C
SS-F
EBC1-EBC2
EBC3-EBC3
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
n = 1
n = 2
5
1
122.7511
76.4642
57.2303
29.3770
120.6801
73.8949
122.7406
76.4423
2
225.6351
146.3284
166.6339
96.0876
223.8509
145.0127
225.6179
146.2894
3
295.0230
208.6741
261.9059
173.4884
293.5837
207.2762
294.9984
208.6188
8
1
76.7055
45.4524
31.6944
14.1556
75.2519
43.4839
76.7009
45.4440
2
148.5210
90.2844
99.4478
53.6849
147.5756
89.3387
148.5141
90.2678
3
214.2263
137.1869
178.4795
107.3297
213.0963
136.2748
214.2159
137.1633
10
1
60.6792
34.7071
23.1038
9.8312
59.4140
33.0345
60.6761
34.7020
2
118.8628
70.3232
76.2998
39.8337
118.1806
69.3907
118.8582
70.3123
3
176.4915
109.1948
143.5250
82.7372
175.6285
108.4187
176.4847
109.1790
12
1
49.7780
27.4582
17.5069
7.3573
48.6440
26.0412
49.7758
27.4549
2
98.2672
56.7730
61.0112
30.8062
97.7181
55.8310
98.2638
56.7654
3
148.6260
89.5260
118.5957
65.9236
147.9499
88.8051
148.6211
89.5147Figure 6
The changing rule of the natural frequencies with various length to radius ratiosL/R; n = 1. (a) C-C, (b) S-F, (c) EBC1-EBC2, and (d) EBC3-EBC3.
(a)
(b)
(c)
(d)


## 4. Conclusions

In this paper, a semianalytical method is conducted to investigate the free vibration characteristics of the orthotropic cylindrical shell under general boundary conditions, including the classical boundary conditions, elastic boundary conditions, and their combinations. Reissner–Naghid’s shell theory is utilized to obtain the governing motion equations and the displacement variables are transformed into wave function forms to accurate the motion relationship. According to the motion relationship and boundary conditions, the final equation of the orthotropic circular cylindrical shell is established. Then the natural frequencies can be obtained by solving the zero position of the overall matrix determinant based on the dichotomy method. In the numerical examples and discussion parts, the effect of the material parameter and geometric constants on the free vibration characteristics of the orthotropic circular cylindrical shell is studied and some conclusions are obtained.To verify the correctness of the calculation results by the presented method, some numerical examples are proposed by comparing with the solutions in the reported literatures. The effect of the material parameter modulus ratioE1/E2, geometric constants thickness to radius ratio L/R, and thickness to radius ratio h/R on the free vibration characteristics of the orthotropic cylindrical shell under several boundary conditions are discussed. For the effect of different parameters, various properties have its one influence impact on the free vibration characteristics of the orthotropic circular cylindrical shell. The changing ranges of the natural frequencies are different for various circumferential number and longitudinal modes under several boundary conditions.This paper proposed a new numerical method to research the free vibration characteristics of the orthotropic circular cylindrical shell under general boundary conditions, and it provided a theoretical basis for the development of subsequent numerical studies.



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