Abstract

Wave based method (WBM) is presented to analysis the free vibration characteristics of cylindrical shells with nonuniform stiffener distributions for arbitrary boundary conditions. The stiffeners are treated as discrete elements. The equations of motion of annular circular plate are used to describe the motion of stiffeners. Instead of expanding the dynamic field variables in terms of polynomial approximation in element based method (finite element method etc), the ring-stiffened cylindrical shell is divided into several substructures and the dynamic field variables in each substructure are expressed as wave function expansions. Boundary conditions and continuity conditions between adjacent substructures are used to form the final matrix to be solved. Natural frequencies of cylindrical shells with uniform rings spacing and eccentricity distributions for shear diaphragm-shear diaphragm boundary conditions have been calculated by WBM model which shows good agreement with the experimental results and the analytical results of other researchers. Natural frequencies of cylindrical shells with other boundary conditions have also been calculated and the results are compared with the finite element method which also shows good agreement. Effects of the nonuniform rings spacing and nonuniform eccentricity and effects of boundary conditions on the fundamental frequencies and the beam mode frequencies have been studied. Different stiffener distributions are needed to increase the fundamental frequencies and beam mode frequencies for different boundary conditions. WBM model presented in this paper can be recognized as a semianalytical and seminumerical method which is quite useful in analyzing the vibration characteristics of cylindrical shells with nonuniform rings spacing and eccentricity distributions.

Introduction

Ring stiffened cylindrical shells are widely used in many structure applications such as aeronautic, aerospace, underwater vessel and so on. The free vibration characteristics of cylindrical shells are quite important for use of on-board equipments and so on. Most research focus on cylindrical shells with uniform stiffener distribution because of the convenience of the establishment of mathematical model of the problem, especially convenient for dealing with stiffeners. But in practical engineering applications, stiffeners are sometimes unequally spaced and often with different sizes, such as the existence of intermediate large frame ribs, so a more general model is needed to deal with cylindrical shells with nonuniform stiffener distribution, such as nonuniform eccentricity, unequally spaced and different sizes for ring stiffeners.

Many investigations have been developed to analyze the vibration characteristics of cylindrical shells, e.g., Refs. [1–10]. Energy methods based on the Rayleigh–Ritz procedure and analytical methods are most widely used. Most often cylindrical shells with shear diaphragm-shear diaphragm boundary condition have been investigated because the solution to the governing equations which satisfies boundary conditions can be easily expressed by trigonometric functions. Exponential functions have been chosen for the modal displacements along the axial direction, which are substituted into the equations of motions and then the eight specified boundary conditions are enforced to calculate the natural frequency of cylindrical shells with arbitrary boundary conditions.

In order to effectively enhance the flexural and axial stiffness of cylindrical shells, stiffeners are frequently used. The even spaced rings or stringers are either “smeared out” over the surface of the cylindrical shells or treated as discrete members [11–28]. With rings or stringers “smeared out,” the cylindrical shells are treated as orthotropic cylindrical shells [14–18]. The stiffeners must be equally and closely distributed and also with the same depth and width and large errors will be introduced if the stiffeners are large. And also this method is not applicable at mid and high frequencies if the wavelength and the rib spacing are in the same order for neglecting wave transmission and reflection at the discontinuities. Analytical method for the determination of the vibration characteristics of even spaced ring-stiffened shells with shear diaphragm–shear diaphragm (SD-SD) was presented in Refs. [19–20], where rings or stringers are treated as discrete numbers and equations of motions of beams are used to describe the motion of stiffeners. Assuming that the stiffeners uniformly spaced, the displacement at the position of stiffeners are expanded as trigonometric functions which satisfy simply supported boundary conditions to calculate the natural frequencies in Ref. [19]. The method is confined to solve the problem of even spaced ring-stiffened cylindrical shells and simply supported boundary condition. C.H. Hodges et al. [21,22] studied the vibration characteristics of periodic ring-stiffened cylindrical shells with circular T-section ribs according to Bloch's or Floquet's theorem [23]. Recently, Wave propagation method is given in Refs. [24–27] to study the vibration characteristics of cylindrical shell, with the effects of ring stiffeners “smeared out” over the surface of the shell in Refs. [25,27]. Beam function with shear diaphragm-shear diaphragm (SD-SD), clamped-clamped (C-C) and clamped-shear diaphragm (C-SD) boundary condition is used to approximate the axial mode function of ring-stiffened cylindrical shell. With the stiffeners treated as discrete elements, Ritz method is applied to analyze the free vibration characteristics of simply supported cylindrical shells with nonuniform rings spacing and eccentricity in Ref. [28] and some conclusions have been made.

A wave based method (WBM) was first proposed in Ref. [29] for prediction of the steady-state dynamics analysis of coupled vibro-acoustic systems which enables accurate predictions in the midfrequency range. Wave based method can be understood as a semianalytical and seminumerical method which is computationally less demanding than corresponding element based models. In contrast with the finite element method (FEM), in which the dynamic field variables within each element are expanded in terms of local, nonexact shape functions, usually polynomial approximation, the dynamic field variables in WBM are expressed as wave function expansions. Modeling of the vibro-acoustic coupling between the pressure field in an acoustic cavity with arbitrary shape and the out-of-plane displacement of a flat plate with arbitrary shape was discussed in Ref. [30] and the unbounded domain was discussed in Ref. [31]. The application of the wave based method for the particular case where stress singularities appear in one or more corners of a polygonal plate domain is discussed in Ref. [32].

In this paper, WBM method is extended to analyze free vibration characteristics of cylindrical shells with nonuniform rings spacing and eccentricity distributions for arbitrary boundary conditions. The cylindrical shell is divided into several substructures the motions of which are described by the equations of Donnell–Mushtari theory and the stiffeners are treated as separate substructures the motions of which are described by the equations of annular circular plates. The displacements within each substructure of cylindrical shells are expanded as a set of wave functions obtained in Ref. [19], and those within the stiffener substructure are expanded as a set of wave functions in Ref. [33]. Then boundary conditions and continuity conditions between substructures are used to form the final matrix to calculate the natural frequencies. Compared with traditional wave propagation method in analyzing free vibration characteristics of ring stiffened cylindrical shells, WBM method has a great advantage in dealing with stiffener of arbitrary sizes and arbitrary distributions for arbitrary boundary conditions. Compared with finite element method possessing a significantly greater generality, a better convergence can be obtained by WBM method and also the size of the final matrix formed to calculate the natural frequencies is much smaller than that formed by finite element method. As a semianalytical and seminumerical method, the shape function chosen in WBM method is more convenient to explain the mechanism of wave propagation including vibration transmission and reflection. The first part of this paper reviews the work already done about free vibration of cylindrical shells. WBM model of cylindrical shells with nonuniform rings spacing and eccentricity distribution is described in the second part. Numerical results are discussed in the third part and some conclusions are made in the fourth part.

Basic Concept of Wave Based Model

Wave based method (WBM) is developed for steady-state dynamic analysis of cylindrical shells with nonuniform rings spacing and stiffener distribution. In contrast with the finite element method (FEM), in which the dynamic field variables within each element are expanded in terms of local, nonexact shape functions, usually polynomial approximation, the dynamic field variables within each substructure in WBM model are expressed as wave function expansions, which exactly satisfy the governing dynamic equations of the substructure, and then boundary conditions and continuity conditions are utilized to form the final matrix to calculate the natural frequencies. After that, wave function contribution factors used for postprocessing can be determined.

Similar to the finite element method, several steps are needed to construct a WBM model as follows:

1. (1)

Divide the whole model into different substructures. Different governing equations are used for different types of structure, such as beam, plate and shell. Also different physical properties and the positions of the discontinuities where coupling effects occur need to be considered.

2. (2)

Select suitable wave functions for each substructure. The dynamic field variables, such as the displacement, the velocity and so on, can be expanded by the wave functions which can exactly satisfy the governing equations of the substructures.

3. (3)

Form the final matrix to be solved according to boundary conditions and continuity conditions. Eight specified boundary conditions and eight continuity conditions between different substructures, including forces and displacements, are used to form the final matrix.

4. (4)

Solve the final matrix to get the natural frequencies and the wave function contribution factors. The size of the matrix depends on the total number of substructures and the number of wave function contribution factors in each substructure. Compared with traditional element based model, more unknowns exist in the shape functions but the number of substructures is far more less than the number of elements so the final size of matrix formed is much smaller than FEM.

5. (5)

Postprocessing. The dynamic field variables can be obtained with the wave function contribution factors, such as the displacement, stress, strain and so on.

Substructure Division of Ring-Stiffened Cylindrical Shells.

Cylindrical shells and annular circular plates are main types of structures composing the ring-stiffened cylindrical shells and the stiffeners appear as a discontinuity dividing the cylindrical shell into different substructures. As a ring-stiffened cylindrical shell shown in Fig. 1, the total number of stiffeners is N, and the cylindrical shell is divided into (N + 1) bays. The cylindrical shell and the stiffeners are made of the same material, so the total number of substructures is (2N + 1) with N substructures describing the motions of stiffeners and (N + 1) substructures describing the motions of cylindrical shells. Compared with element based method, this method of substructure division has much less elements.

Fig. 1
Fig. 1
Close modal
According to Donnell–Mushtari theory, the following equations describe the motion of thin cylindrical shells [1,2]:
$L11u+L12v+L13w=0L21u+L22v+L23w=0L31u+L32v+L33w=0}$
(1)

Differential operators $Lij(i,j=1,2,3)$ are given in Appendix  A. u, v, w are the axial, circumferential and outward normal displacements.

The stiffeners are treated as discrete members the motions of which are described by the equations of motions of annular circular plates. Internal stiffening type is considered here which can be applied to external stiffening type easily. The vibrations of annular circular plates with inner radius a1 and outer radius a (also the radius of the cylindrical shell) with bending and in-plane motions are described in Eq. (2). The axial displacement wp, radial displacement up and circumferential displacement vp of the plate in polar coordinates are shown in Fig. 2. $θp=∂wp/∂r$ is the twist angle.

Fig. 2
Fig. 2
Close modal
${∂∂r(∂up∂r+upr+1r∂vp∂θ)-1-νp2r∂∂θ(∂vp∂r+vpr-1r∂up∂θ)=ρp(1-νp2)Ep∂2up∂t21r∂∂θ(∂up∂r+upr+1r∂vp∂θ)+1-νp2∂∂r(∂vp∂r+vpr-1r∂up∂θ)=ρp(1-νp2)Ep∂2vp∂t2∇4wp-ρpω2hpDpwp=0$
(2)

hp is the plate thickness. Ep, rp and $νp$ are respectively the Young's modulus, density and Poisson's ratio of the annular circular plate. r is the radial coordinate. ω is the angular frequency. $Dp=Ephp3/12(1-νp2)$ is the flexural rigidity.

Selection of Wave Functions.

For a modal vibration, the axial, circumferential and outward normal displacements of the cylindrical shells and the annular circular plates are usually expressed as a solution expansion
$w=∑i=1nsAiΨwi(x,φ)sin(nφ)e-jωtv=∑i=1nsBiΨvi(x,φ)cos(nφ)e-jωtu=∑i=1nsCiΨui(x,φ)sin(nφ)e-jωt}$
(3)

$Ψwi(x,φ),Ψvi(x,φ),Ψui(x,φ)$ are the structure wave functions which satisfy Eqs. (1) and (2) for a specified circumferential wave number n, Ai, Bi, Ci are the wave contribution factors. ns is the number of wave functions designating the wave propagation in the axial direction for the cylindrical shell and the radial direction for the annular circular plate.

To insure the WBM model to converge towards the exact solution, the selection of suitable wave function is quite important. According to analysis in Ref. [19], a set of wave functions describing the free vibration of cylindrical shell is selected as follows:
$Ψw1=Ψv1=Ψu1=eλ1x,Ψw2=Ψv2=Ψu2=e-λ1xΨw3=Ψv3=Ψu3=cosλ2x,Ψw4=Ψv4=Ψu4=sinλ2xΨw5=Ψv5=Ψu5eλ3xcosλ4x,Ψw6=Ψv6=Ψu6=eλ3xsinλ4xΨw7=Ψv7=Ψu7=e-λ3xcosλ4x,Ψw8=Ψv8=Ψu8=e-λ3xsinλ4x$
(4)
The wave contribution factors are as follows:
$B1=ξ1A1,B2=ξ1A2,B3=ξ2A3,B4=ξ2A4B5=ξ3A5+ξ4A6,B6=-ξ4A5+ξ3A6B7=ξ3A7-ξ4A8,B8=ξ4A7+ξ3A8$
(5)
$C1=η1A1,C2=-η1A2,C3=-η2A3,C4=η2A4C5=η3A5+η4A6,C6=-η4A5+η3A6C7=-η3A7+η4A8,C8=-η4A7-η3A8$
(6)

In Eqs. (4)–(6), $λ1,±iλ1,±(λ3±iλ4)$ are character roots to be determined which are given in Appendix  B. ξ1 ∼ ξ4 and η1 ∼ η4 are constant coefficients related to character roots which are also given in Appendix  B.

According to Ref. [33], a set of wave functions describing the free vibration of annular circular plate are selected as follows:
$ψwp1=Jn(kpBr), ψwp2=Yn(kpBr),ψwp3=In(kpBr), ψwp4=Kn(kpBr)ψvp1=nJn(kpLr)/r, ψvp2=dJn(kpTr)/dr,ψvp3=nYn(kpLr)/r, ψvp4=dYn(kpTr)/drψup1=dJn(kpLr)/dr, ψup2=nJn(kpTr)/r,ψup3=dYn(kpLr)/dr, ψup4=nYn(kpTr)/r$
(7)
The wave contribution factors are as follows:
$A1=B1=B1n,A2=B2=B2n,A3=B3=B3n,A4=B4=B4nC1=A1n,C2=A2n,C3=A3n,C4=A4n$
(8)

ns = 8 is the number of wave functions. Jn, Yn are, respectively, Bessel functions of the first and the second kind. In, Kn are, respectively, modified Bessel functions of the first and second kind. $kpB=(ρpω2hp/Dp)1/4$ is the plate bending wave number, $kpL=ω[ρp(1-υp2)/Ep]1/2,kpT=ω[2ρp(1+υp)/Ep]1/2$ are the wave numbers for in-plane waves in the circular plate. The coefficients $Ai,n,Bi,n(i=1,2,3,4)$ are determined from the boundary conditions of the annular circular plates.

Matrix Formed for Calculating Natural Frequencies.

As shown in Fig. 3, the cylindrical shells with arbitrary boundary conditions have four displacement constraints (u, v, w, θ) and four force or moment constraints (M, S, T, N), i.e.,

Fig. 3
Fig. 3
Close modal
$u=0orM=0v=0orS=0w=0orT=0θ=0orN=0}$
(9)

where θ designate the twisting angle and M, S, T, N designate bending moment, transverse shear, tangential shear and axial force per unit length of the cylindrical shell and the expressions for M, S, T, N are expressed by Eq. (C6) in Appendix  C. Combination of these eight boundary conditions can present arbitrary boundary conditions.

The cylindrical shell with inside stiffeners is divided into two bays by the stiffeners as shown in Fig. 4 where continuity equations must be satisfied.

Fig. 4
Fig. 4
Close modal
The displacements of the connection of the adjacent cylindrical shells must be equal, which can lead to the following relationship:
$ukL=ukR,vkL=vkR,wkL=wkR,θkL=θkR$
(10)
At the outer radius of the annular circular plate which is shown in Fig. 4, the continuity conditions of displacements and forces can be expressed as follows:
$up,k|r=a=wkR,vp,k|r=a=vkR,wp,k|r=a=ukR,θp,k|r=a=-θkR$
(11)
${NkL=Npx,k+NkRSkL=Npr,k+SkRTkL=TkR-Npθ,kMkL=MkR-Mp,k$
(12)
The annular circular plates have one free edge at the inner radius (outer radius for the external stiffening type) where boundary conditions are applied as follows:
$Npx|r=a1=0,Npr|r=a1=0,Npθ|r=a1=0,Mp|r=a1=0$
(13)

$ukL,vkL,wkL,θkL$ and $ukR,vkR,wkR,θkR$ denote the axial, circumferential, radial displacement and the twist angle of the left bay and the right bay of the cylindrical shell of the kth stiffener respectively. $up,k|r=a,vp,k|r=a,wp,k|r=a,θp,k|r=a$ denote the radial, circumferential, axial displacement and the twist angle of the kth stiffener at the outer radius of the circular plate. $MkL,SkL,TkL,NkL$ and $MkR,SkR,TkR,NkR$ designate the bending moment, transverse shear, tangential shear and axial force per unit length of the right and the left bay of the cylindrical shell, respectively. $MkL,Npr,k,TkL,Npx,k$ and $Mp|r=a1,Npr|r=a1,Npθ|r=a1, Npx|r=a1$ designate the bending moment, transverse shear, tangential shear and axial force per unit length of the kth annular circular plate at the outer and inner radius of the cylindrical shell, respectively.

Combing Eqs. (11) and (13), the eight wave function contributors of the annular circular plate can be expressed by the eight wave function contributors of the adjacent cylindrical shells. The cylindrical shell is divided into (N + 1) subsystems which means that there are totally 8(N + 1) wave contribution factors to be solved. The boundary conditions at both ends of the cylindrical shell and the continuity conditions between the stiffeners and adjacent bays of cylindrical shells can be written and assembled in matrix form for each circumferential mode number n. Omitting the subindex n, the assembled matrix is
$[K]{A}=0$
(14)
where {A} is the 8(N+1) unknown coefficient vectors and
$[K]=[[B1(0)][D1(b1)]-[D2(0)][F1(b1)]-[F2(0)][D2(b2)]-[D3(0)][F2(b2)]-[F3(0)]············[DN(bN)]-[DN+1(0)][FN(bN)]-[FN+1(0)][BN+1(bN+1)]]$
(15)
For the kth substructure of the cylindrical shell, the 4 × 8 matrix blocks [Dk] and [Fk] designating displacement and force continuity conditions used in Eq. (15) are as follows. For the continuity equations between stiffeners and adjacent bays of cylindrical shells, blocks [Dk] and blocks [Fk] are given by
$[Dk(x)]4×8=[wk,1(x)···wk,8(x)vk,1(x)···vk,8(x)uk,1(x)···uk,8(x)θk,1(x)···θk,8(x)], k=1,2,…,N+1$
(16)
$[Fk(x)]4×8=[Sk,1(x)···Sk,8(x)Tk,1(x)···Tk,8(x)Nk,1(x)···Nk,8(x)Mk,1(x)···Mk,8(x)], k=1,2,…,N+1$
(17)

The initial and final blocks [B1] and [BN+1] are expressed in terms of displacement and/or forces, depending on the boundary conditions at each end of the cylindrical shell. Combination of eight boundary conditions in Eq. (9) can present arbitrary boundary conditions. Free, clamped and shear-diaphragm boundary conditions are considered here and are given by

Free end (F)
$[Bk(x)]4×8=[Sk,1(x)···Sk,8(x)Tk,1(x)···Tk,8(x)Nk,1(x)···Nk,8(x)Mk,1(x)···Mk,8(x)], k=1,N+1$
(18)
Clamped end (C)
$[Bk(x)]4×8=[wk,1(x)···wk,8(x)vk,1(x)···vk,8(x)uk,1(x)···uk,8(x)θk,1(x)···θk,8(x)], k=1,N+1$
(19)
Shear-diaphragm (SD)
$[Bk(x)]4×8=[wk,1(x)···wk,8(x)vk,1(x)···vk,8(x)Nk,1(x)···Nk,8(x)Mk,1(x)···Mk,8(x)], k=1,N+1$
(20)

$[B1],[B+1],[Dk],[Fk](k=1~N+1)$ are given in Appendix  C. $Φ,1(Φ=w,v,u,θ,S,T,N,M)$ designate the values of field variables at the position x of the kth bay of cylindrical shell. bk designates the length of the kth bay.

Solution of the Matrix and Postprocessing.

When analyzing the free vibration characteristic of ring-stiffened cylindrical shell, the value of the determinant of [K] is calculated for a sequence of trial values of frequency until a sign change is met, then the zero of the determinant is located by iterative interpolation. After a natural frequency has been accurately calculated, the solution to the homogeneous equations (Eq. (14)), normalized by taking $A+1,8=1$, can be calculated and the mode shape can be obtained.

Numerical Results and Discussion

Cylindrical Shells With Uniform Ring Spacing and Eccentricity.

The WBM model is first utilized to calculate the natural frequencies of two ring-stiffened cylindrical shells with C-C, C-SD, SD-SD, C-F, SD-F, F-F boundary conditions. The geometry and material properties of the two shells denoted as model M1 and model M2 which are given in Table 1. Both of the two shells are with external stiffeners of rectangular cross section. Three kinds of shells with stiffeners of different depths are calculated in model M1. The cylindrical shells and the stiffeners have the same materials in both model M1 and model M2. The WBM results are compared with the analytical results already done in relevant references and also FEM results.

Table 1

Geometry dimensions and material properties of stiffened cylindrical shells

Geometry and material properties
CharacteristicsM1 ModelM2 Model
Length, l(m)0.47090.3945
Thickness, h(m)0.001190.001651
Density, ρ(kg/m3)78502762
Young's modulus, E(Pa)2.06 × 10116.895×1010
Poission's ratio, ν0.30.3
Ring width, bR(m)0.002180.003175
Ring depth, dR(m)0.00291,0.00582,0.008730.005334
Number of rings, N1419
Stiffening typeExternalExternal
Geometry and material properties
CharacteristicsM1 ModelM2 Model
Length, l(m)0.47090.3945
Thickness, h(m)0.001190.001651
Density, ρ(kg/m3)78502762
Young's modulus, E(Pa)2.06 × 10116.895×1010
Poission's ratio, ν0.30.3
Ring width, bR(m)0.002180.003175
Ring depth, dR(m)0.00291,0.00582,0.008730.005334
Number of rings, N1419
Stiffening typeExternalExternal

The finite element package ANSYS is used to calculate natural frequencies of model M1 and model M2. Both the cylindrical shells and the stiffeners are modeled with shell63 element. In order to ensure the convergence of results calculated by FEM, M1 model with dR = 0.00291 is modeled with three different meshes as shown in Table 2 and the natural frequencies of vibrations modes m = 1, n = 1∼6 (m, n denote the axial half wave number and the circumferential wave number respectively) with SD-SD boundary conditions have been calculated with the results shown in Table 3. The two numbers in Table 2 for cylindrical shell denote the number of elements in the circumferential direction and the axial direction respectively and the three numbers for stiffeners denote the number of elements in the circumferential direction, the radial direction and the number of stiffeners in the axial direction, respectively. 10,500, 21,600 and 33,000 nodes are used in mesh 1, mesh 2 and mesh 3, respectively, with each node having six degrees of freedom UX, UY, UZ, ROTX, ROTY and ROTZ designating three translation degrees and three rotational degrees of freedom. Mesh 2 which can achieve both high computation efficiency and adequate converged results is used in the following analysis.

Table 2

FEM models with different meshes

Number of elements
Mesh 1Mesh 2Mesh 3
Cylindrical shell100 × 90160 × 120200 × 150
Stiffeners100 × 1 × 14160 × 1 × 14200 × 1 × 14
Total number104002144032800

Number of elements
Mesh 1Mesh 2Mesh 3
Cylindrical shell100 × 90160 × 120200 × 150
Stiffeners100 × 1 × 14160 × 1 × 14200 × 1 × 14
Total number104002144032800
Table 3

Natural frequencies of M1 model calculated by three different meshes

Natural frequencies (Hz)
Mode numberMesh 1Mesh 2Mesh 3
n = 116101610.31610
n = 2690.87690.85690.84
n = 3555.15555.04555.04
n = 4875.46874.94874.91
n = 5137213711370
n = 619711967.51967

Natural frequencies (Hz)
Mode numberMesh 1Mesh 2Mesh 3
n = 116101610.31610
n = 2690.87690.85690.84
n = 3555.15555.04555.04
n = 4875.46874.94874.91
n = 5137213711370
n = 619711967.51967

Table 4 shows the comparison of natural frequencies of model M1 between WBM results and the analytical results of Basdekas and Chi [20], Galletly [14], Wah and Hu [19], Bosor [12] and L. Gan and X. B. Li [27] and also with FEM results. From the table we can see that good agreement can be achieved between different methods for most cases while for other cases large differences have been introduced. The reason why large differences exist is mainly due to some simplifications dealing with the stiffeners in different methods. In method (a), a modified variation method is adopted and the motions of the stiffeners are described by the equations of motions of beams. In method (b), the stiffeners are treated as discrete members but the inter-ring displacement is neglected. In method (c), the stiffeners are also treated as discrete members but the eccentricity of the stiffeners with respect to the shell midsurface is neglected. In method (d), the stiffeners are treated by “smeared method” which can only present an accurate solution when the wavelength is small and the stiffeners are with small sizes. In method (e), the stiffeners are also treated by “smeared method” while beam function is used to approximate the axial mode function of ring stiffened cylindrical shells.

Table 4

Comparison of circular frequencies of M1 model (rad/s). (a) Basdekas and Chi [20]; (b) Galletly [14]; (c) Wah and Hu [19]; (d) Bosor (smeared rings) [12] and (e) Gan and Li [27].

dR(m)n(a)(b)(c)(d)(e)FEMWBM
0.0029124550447043144420440943414370
33870365531733680367434873560
46550595045656000600054975590
510000951070589620960486148715
0.005822458044504235448143574386
3671062354615649257175782
412830117907982121491015410248
5201201902012660196941563915767
0.008712504048854378495446594689
31033095006651987381938269.8
4202001801012154188841460814735
5318002557019221306062180321992
dR(m)n(a)(b)(c)(d)(e)FEMWBM
0.0029124550447043144420440943414370
33870365531733680367434873560
46550595045656000600054975590
510000951070589620960486148715
0.005822458044504235448143574386
3671062354615649257175782
412830117907982121491015410248
5201201902012660196941563915767
0.008712504048854378495446594689
31033095006651987381938269.8
4202001801012154188841460814735
5318002557019221306062180321992

Table 5 shows the comparison of natural frequencies of model M2 between WBM results and the experimental results and the analytical results of Hoppmann [11], the analytical results of Mustafa and Ali [13] and the analytical results of Jafari and Bagheri [28] for various modes of vibrations with SD-SD boundary conditions. As we can see from the tables, when the wave number is small, quite good agreement can be achieved between WBM methods and other methods. The differences increase with the increment of the wave number, while in all the cases WBM results and the experimental results agree quite well. The reason is that the wave length decreases with the increment of the wave-number and either the stiffener is treated with “smeared method” or treated as discrete members with the motions of stiffener described by the equations of motions of beams can both introduced errors while WBM model is much more exact in such cases.

Table 5

Comparison of natural frequencies of M2 model (Hz). (e) Gan and Li [27]; (f) Jafari and Bagheri [28]; (g) Mustafa and Ali [13]; (h) Hoppmann [11]; and (i) Hoppmann [11].

nm(e)(f)(g)(h)Experiment (i)WBM
1112161199.5812041154.67
235363493.5934983325.82
359075839.8958445533.53
47538.00
59249.70
2116351564.471587153015301585.07
221762113.842129210020402112.62
334303378.173386333032003298.39
4486044404712.78
5648062006121.56
3145784387.594462423040804156.73
245734400.584437432040904174.51
347884595.794627450045204407.83
4504050004937.64
5576057005707.83
4187818377.75855981007424.87
287318392.63848281007403.14
387288449.898438819075207435.08
4828078007580.76
579207873.03
511417213490.7137801305011059.28
21411913508.91369513100
31406913555.41359513140
4132301140011087.96
511207.63
nm(e)(f)(g)(h)Experiment (i)WBM
1112161199.5812041154.67
235363493.5934983325.82
359075839.8958445533.53
47538.00
59249.70
2116351564.471587153015301585.07
221762113.842129210020402112.62
334303378.173386333032003298.39
4486044404712.78
5648062006121.56
3145784387.594462423040804156.73
245734400.584437432040904174.51
347884595.794627450045204407.83
4504050004937.64
5576057005707.83
4187818377.75855981007424.87
287318392.63848281007403.14
387288449.898438819075207435.08
4828078007580.76
579207873.03
511417213490.7137801305011059.28
21411913508.91369513100
31406913555.41359513140
4132301140011087.96
511207.63

Table 6 shows comparison between WBM results of M1 model for dR = 0.00291m with C-C, C-SD, SD-SD, C-F, SD-F, F-F boundary conditions. Good agreements have been achieved which shows that WBM model has adequate accuracy to determine the natural frequencies of ring stiffened cylindrical shells with arbitrary boundary conditions.

Table 6

Comparison of natural frequencies of M1 model

C-C(Hz)

SD-SD(Hz)

C-SD(Hz)

Mode
FEMWBMErrorFEMWBMErrorFEMWBMError
m=1n=12005.82010.30.22%1610.31614.10.24%1786.21790.20.22%
n=21105.51109.80.39%690.9695.50.67%900.8905.20.49%
n=3784.7793.71.15%555.0566.52.07%658.9669.01.53%
n=4950.6964.51.46%874.9889.61.68%904.3918.71.59%
n=51396.01412.11.15%1371.01387.11.17%1379.71396.11.19%
n=61977.91995.70.90%1967.51985.30.90%1971.11989.00.91%

C-C(Hz)

SD-SD(Hz)

C-SD(Hz)

Mode
FEMWBMErrorFEMWBMErrorFEMWBMError
m=1n=12005.82010.30.22%1610.31614.10.24%1786.21790.20.22%
n=21105.51109.80.39%690.9695.50.67%900.8905.20.49%
n=3784.7793.71.15%555.0566.52.07%658.9669.01.53%
n=4950.6964.51.46%874.9889.61.68%904.3918.71.59%
n=51396.01412.11.15%1371.01387.11.17%1379.71396.11.19%
n=61977.91995.70.90%1967.51985.30.90%1971.11989.00.91%

F-F(Hz)

SD-F(Hz)

C-F(Hz)

Mode
FEMWBMErrorFEMWBMErrorFEMWBMError
m=1n=13155.03122.21.04%2249.12254.20.23%2282.02287.40.24%
n=21483.01488.60.38%1044.01048.80.46%1221.81226.50.38%
n=3882.7892.71.13%682.7693.51.58%818.7820.20.18%
n=4925.5941.11.69%888.8903.81.69%935.6950.31.57%
n=51374.01390.21.18%1371.81388.31.20%1383.01400.01.23%
n=61968.71986.50.90%1968.01985.90.91%1972.41990.20.90%

F-F(Hz)

SD-F(Hz)

C-F(Hz)

Mode
FEMWBMErrorFEMWBMErrorFEMWBMError
m=1n=13155.03122.21.04%2249.12254.20.23%2282.02287.40.24%
n=21483.01488.60.38%1044.01048.80.46%1221.81226.50.38%
n=3882.7892.71.13%682.7693.51.58%818.7820.20.18%
n=4925.5941.11.69%888.8903.81.69%935.6950.31.57%
n=51374.01390.21.18%1371.81388.31.20%1383.01400.01.23%
n=61968.71986.50.90%1968.01985.90.91%1972.41990.20.90%

Cylindrical Shells With Nonuniform Ring Spacing and Eccentricity.

Numerical calculations have been performed here to study the effects of nonuniform rings spacing and nonuniform stiffeners eccentricity distribution, separately and simultaneously. M1 model with dR = 0.00291 m with uniform stiffener distribution is considered here, whose properties are shown in Table 1. Some cases of nonuniform rings spacing and nonuniform stiffener distribution are shown in Fig. 5, for which cases the total stiffener mass maintain constant with that of M1 model with evenly rings spacing and equally depth for all stiffeners.

Fig. 5
Fig. 5
Close modal
Equating the mass of stiffeners in uniform and nonuniform distributions, a second-order equation corresponding to dRk' can be obtained as follows [28]:
$1N(1a+dR1+h/2)2∑k=1NdRk'2+2N(1a+dR1+h/2)∑k=1NdRk'-2(a+h/2)(d0-dR1)(a+dR1+h/2)2-(d02-dR12)(a+dR1+h/2)2=0$
(21)
Here the difference $dR'$ between the depths of adjacent stiffeners is considered to maintain constant and the nonuniform eccentricity is symmetrically distributed about the midsection of the shell. Select a value of dR1 and a distribution function of dRk, $dRk'$ can be determined by solving Eq. (21). The depth of each stiffener can be obtained as follows:
$dRk'={(k-1)dR'k≤N/2(N-k)dR'k>N/2$
(22)
$dRk=dR1+dRk'$
(23)
Similarly, the nonuniform spacing is considered to be symmetrical distributed about the midsection of the shell and the differences between the lengths of adjacent bays of cylindrical shell maintain constant, the nonuniform distribution can be obtained by selecting a value of d1. The spacing of the kth bay of cylindrical shell can be obtained as follows:
$d'=L-(N+1)*d1N2/4$
(24)
$dk={d1+(k-1)d'k≤N/2d1+(N+1-k)d'k>N/2$
(25)

In the above equations, d1 denotes the first ring spacing at the left end and dR1 denotes the depth of the first stiffener at the left end. d′ denotes the differences between the lengths of adjacent bays of cylindrical shells and $dR'$ denotes the differences between the depths of adjacent stiffeners. $dRk'$ denotes the differences between the depths of the kth stiffener and the first stiffener. dRk and dk denote the depth of the kth stiffener and the length of kth bay of cylindrical shell, respectively. d0 denotes the depth of the stiffeners of uniform eccentricity distribution.

For d1 < 0.00314, the rings are closely spaced at the two ends of the cylindrical shell and on the other hand for d1 > 0.00314, the rings are compressed in the midsection of the cylindrical shell. For dR1 > 0.00291, the eccentricities of the stiffeners at the two ends are larger than those in the midsection and for dR1 < 0.00291, the eccentricities of the stiffeners at the two ends are smaller than those in the midsection. For d1 = 0.00314 and dR1 = 0.00291, it denotes uniformly rings spacing and eccentricity distribution. In the following calculation, $d1=0.00114~0.00514$ and $dR1=0.00091~0.00491$.

Figures 6(a)–6(c) shows the variations of natural frequencies $(m=1,n=1~6)$ of cylindrical shells with respect to the depth (dR1) of the first stiffener with SD-SD, F-F, SD-F boundary conditions. Here the rings spacing is uniform and only the effect of nonuniform eccentricity is considered. It should be noted that decrement of the depth of the first stiffener increase the mass and stiffness in the midsection of the cylindrical shell and decrease them at the two ends.

Fig. 6
Fig. 6
Close modal

For the three kinds of boundary conditions, the fundamental frequencies all occur at mode m = 1, n = 3 except when dR1 = 0.00491 for SD-F boundary condition for which case the fundament frequency mode changes from m = 1, n = 3 to m = 1, n = 4. Increment of the depth of the first stiffener lowers the fundamental frequencies for SD-SD boundary conditions and for other two boundary conditions the fundament frequencies first decrease and then increase with the increment of the depth of the first stiffener and the lowest fundamental frequencies both occur when d1 = 0.00314 and dR1 = 0.00291, which denotes uniformly rings spacing and eccentricity distribution.

From Fig. 6, we can see that for SD-SD and SD-F boundary conditions, increment of the depth of the first stiffener leads to increment of natural frequencies of the beam mode (m = 1, n = 1) while for F-F boundary conditions it leads to the decrement of natural frequencies of the beam mode. Also the natural frequencies of beam mode with F-F boundary conditions are the highest among the three kinds of boundary conditions and those with SD-SD boundary conditions are the lowest.

Figures 7(a)–7(c) shows the variations of fundamental frequencies with respect to the depth of the first stiffener for given d1 with SD-SD, F-F, SD-F boundary conditions. For SD-SD boundary condition, increment of the depth of the first stiffener lower the fundamental frequencies and increment of the first ring spacing increase the fundamental frequencies from which we can make a conclusion that the cylindrical shells get the highest fundamental frequency when dR1 = 0.00091 and d1 = 0.0514, in another word, more mass concentrated in the midsection of the cylindrical shells, the higher the fundamental frequencies are. For the other two kinds of boundary conditions, the fundamental frequencies first increase and then decrease with respect to the increment of the depth of the first stiffener but the fundamental frequencies not always get the lowest value when rings spacing and eccentricity are uniformly distributed which depends on the value of d1. Natural frequency curve crossing is observed which shows that the cylindrical shells get the lowest fundamental frequencies when d1 = 0.0114 before the intersection point and get the lowest fundamental frequencies when d1 = 0.0514 after the intersection point.

Fig. 7
Fig. 7
Close modal

Figures 8(a)–8(c) shows the variations of beam mode frequencies with respect to the depth of the first stiffener for given d1 with SD-SD, F-F, SD-F boundary conditions. We can see from Fig. 7 that for SD-SD boundary conditions the beam mode frequencies increase with the increment of depth of the first stiffener and decrease with the increment of the first ring spacing from which we can make a conclusion that the cylindrical shells get the highest beam mode frequency when dR1 = 0.00491 and d1 = 0.0114 which means that the less the mass distributed in the midsection of the cylindrical shell, the higher the beam mode frequencies are, which is opposite to the variations of the fundamental frequencies with respect to the mass distribution. This conclusion is the same with Jafari and Bagheri [28] where the reason is explained by energy theory. In the beam mode, the shape of stiffeners remain circular and the strain energy of stiffeners does not affect the total energy of the system and only the kinetic energy of stiffeners contributes in the total energy; therefore decrement of mass distribution in the midsection of the shell reduces the kinetic energy and raises the beam mode natural frequency. For F-F boundary condition, the effects of nonuniform ring spacing and eccentricity distribution is rather complicated but a conclusion can be made that the cylindrical shells get the lowest beam mode frequency when dR1 = 0.00491 and d1 = 0.0114 and get the highest beam mode frequency when dR1 = 0.00491 and d1 = 0.0514. For SD-F boundary condition, beam mode frequencies increase with the increment of the depth of the first stiffener for given d1 and before the first intersection point dR1 = 0.00241 the beam mode frequencies decrease with the increment of the first spacing and after that point the beam mode frequencies first decrease then increase. The cylindrical shells get the highest beam mode frequency when dR1 = 0.00491 and d1 = 0.0214 and get the lowest beam mode frequency when dR1 = 0.00091 and d1 = 0.0414.

Fig. 8
Fig. 8
Close modal

Effects of Boundary Conditions on Cylindrical Shells With Nonuniform Rings Spacing and Eccentricity.

From the above analysis we can see that boundary conditions have an important effect on the beam mode frequencies and fundamental frequencies of cylindrical shells with nonuniform rings spacing and eccentricity distribution. Here C-C boundary condition is first considered, and then the boundary conditions are released simultaneously at the two ends gradually to consider the effects of boundary conditions on the vibration characteristics of cylindrical shells. The natural frequencies of three models are calculated here which is shown in Fig. 5 and they are denoted as M1 Model 1, M1 Model 2 and M1 Model 3, respectively. The fundamental frequencies are shown in Table 7 and the beam mode frequencies are shown in Table 8. The two numbers in the bracket in Table 7 denote the axial half wave number and the circumferential wave number, respectively.

Table 7

Fundamental frequencies with different boundary conditions

Fundamental frequency (Hz)

Boundary condition
M1 model 1M1 model 2M1 model 3
C-Cu = v = w = θ = 0710.369(1,4)869.084(1,3)793.693(1,3)
One boundary condition releasedN = v = w = θ = 0489.112(1,3)674.216(1,3)571.517(1,3)
u = T = w = θ = 0710.160(1,4)842.905(1,3)770.544(1,3)
u = v = S = θ = 0709.458(1,4)865.559(1,3)789.784(1,3)
u = v = w = M = 0709.833(1,4)866.113(1,3)790.518(1,3)
Two boundary conditions releasedN = T = w = θ = 0488.939(1,3)672.183(1,3)570.852(1,3)
N = v = S = θ = 0485.590(1,3)671.791(1,3)568.789(1,3)
N = v = w = M = 0 (SD-SD)483.323(1,3)669.668(1,3)566.477(1,3)
u = T = S = θ = 0701.333(1,4)1305.736(1,3)852.117(1,4)
u = T = w = M = 0690.939(1,3)798.868(1,3)730.671(1,3)
u = v = S = M = 0709.452(1,4)865.205(1,3)789.447(1,3)
Three boundary conditions releasedN = T = S = θ = 0624.969(1,4)928.643(1,3)849.354(1,4)
N = T = w = M = 0476.919(1,3)655.691(1,3)557.706(1,3)
N = v = S = M = 0483.280(1,3)570.729(1,2)566.456(1,3)
u = T = S = M = 01309.268(1,3)1305.412(1,3)1121.270(1,4)
Four boundary conditions releasedN = T = S = M = 0 (F-F)622.777(1,4)921.218(1,3)847.124(1,4)

Fundamental frequency (Hz)

Boundary condition
M1 model 1M1 model 2M1 model 3
C-Cu = v = w = θ = 0710.369(1,4)869.084(1,3)793.693(1,3)
One boundary condition releasedN = v = w = θ = 0489.112(1,3)674.216(1,3)571.517(1,3)
u = T = w = θ = 0710.160(1,4)842.905(1,3)770.544(1,3)
u = v = S = θ = 0709.458(1,4)865.559(1,3)789.784(1,3)
u = v = w = M = 0709.833(1,4)866.113(1,3)790.518(1,3)
Two boundary conditions releasedN = T = w = θ = 0488.939(1,3)672.183(1,3)570.852(1,3)
N = v = S = θ = 0485.590(1,3)671.791(1,3)568.789(1,3)
N = v = w = M = 0 (SD-SD)483.323(1,3)669.668(1,3)566.477(1,3)
u = T = S = θ = 0701.333(1,4)1305.736(1,3)852.117(1,4)
u = T = w = M = 0690.939(1,3)798.868(1,3)730.671(1,3)
u = v = S = M = 0709.452(1,4)865.205(1,3)789.447(1,3)
Three boundary conditions releasedN = T = S = θ = 0624.969(1,4)928.643(1,3)849.354(1,4)
N = T = w = M = 0476.919(1,3)655.691(1,3)557.706(1,3)
N = v = S = M = 0483.280(1,3)570.729(1,2)566.456(1,3)
u = T = S = M = 01309.268(1,3)1305.412(1,3)1121.270(1,4)
Four boundary conditions releasedN = T = S = M = 0 (F-F)622.777(1,4)921.218(1,3)847.124(1,4)
Table 8

Beam mode frequencies with different boundary conditions

Beam mode natural frequency (Hz)

Boundary condition
M1 model 1M1 model 2M1 model 3
C-Cu = v = w = θ = 02109.4051923.4362010.241
One boundary condition releasedN = v = w = θ = 01685.6491553.8401614.918
u = T = w = θ = 04053.7644114.1674100.927
u = v = S = θ = 02106.3211920.2702006.712
u = v = w = M = 02107.0661920.9012007.418
Two boundary conditions releasedN = T = w = θ = 03398.0493520.2053497.146
N = v = S = θ = 01685.1431553.4341614.483
N = v = w = M = 0 (SD-SD)1684.6131553.0381614.050
u = T = S = θ = 03823.1963804.4433879.559
u = T = w = M = 03925.9623931.6173991.735
u = v = S = M = 02106.1921920.0452006.462
Three boundary conditions releasedN = T = S = θ = 03027.1033106.5423122.503
N = T = w = M = 03207.8363325.4363320.550
N = v = S = M = 01684.6101553.0381614.050
u = T = S = M = 03823.1783804.4283879.551
Four boundary conditions releasedN = T = S = M = 0 (F-F)3026.6073106.2113122.168

Beam mode natural frequency (Hz)

Boundary condition
M1 model 1M1 model 2M1 model 3
C-Cu = v = w = θ = 02109.4051923.4362010.241
One boundary condition releasedN = v = w = θ = 01685.6491553.8401614.918
u = T = w = θ = 04053.7644114.1674100.927
u = v = S = θ = 02106.3211920.2702006.712
u = v = w = M = 02107.0661920.9012007.418
Two boundary conditions releasedN = T = w = θ = 03398.0493520.2053497.146
N = v = S = θ = 01685.1431553.4341614.483
N = v = w = M = 0 (SD-SD)1684.6131553.0381614.050
u = T = S = θ = 03823.1963804.4433879.559
u = T = w = M = 03925.9623931.6173991.735
u = v = S = M = 02106.1921920.0452006.462
Three boundary conditions releasedN = T = S = θ = 03027.1033106.5423122.503
N = T = w = M = 03207.8363325.4363320.550
N = v = S = M = 01684.6101553.0381614.050
u = T = S = M = 03823.1783804.4283879.551
Four boundary conditions releasedN = T = S = M = 0 (F-F)3026.6073106.2113122.168

We can see from Table 7 that M1 Model 2 always gets the highest fundamental frequency and M1 Model 1 always gets the lowest fundamental frequency except when u = T = S = M = 0. For M1 Model 2, m = 1, n = 3 always appears as the fundamental frequency mode except when N = v = S = M = 0 and m = 1, n = 4 appears as the fundamental frequency mode in some cases for M1 Model 1.

The axial displacement constraint has the largest effects on the fundamental frequencies. Release of the axial displacement constraint decreases the fundamental frequencies sharply and does not change the vibration modes most of which still appear as m = 1, n = 3. Release of other three kinds of boundary conditions decrease the fundamental frequencies much less and lead to the fundamental frequency mode switching from m = 1, n = 3 to m = 1, n = 2 or m = 1, n = 4 for some cases.

From Table 8 we can see that the effects of boundary conditions on beam mode frequencies are rather complicated. Different rings spacing and eccentricity distributions are needed to increase the beam mode frequency for different boundary conditions. M1 Model 1 always gets the highest beam mode frequency when the circumferential displacement constraint is imposed except for C-C boundary condition.

Release of the circumferential displacement constraint can increase the beam mode frequencies rapidly. Release of the other three kinds of displacement constraints will decrease the beam mode and release of the axial displacement constraint lower the beam mode frequency more than the other two. A conclusion can be made that the circumferential displacement constraint has the largest effects on the beam mode frequencies of the cylindrical shells, and the effects of the axial displacement constraint is larger than the other two.

Conclusions

Wave based method (WBM) which can be recognized as a semianalytical and seminumerical method presented in this paper is quite useful in analyzing the free vibration characteristics of cylindrical shells with nonuniform rings spacing and stiffener distribution for arbitrary boundary conditions. The ring-stiffened cylindrical shells can be divided into different substructures according to the type of the structure. The motion of each bay of cylindrical shell is described by the equations of Donnell–Mushtari theory and the motions of the stiffeners are described by the equations of motion of annular circular plate. In contrast with the finite element method (FEM), in which the dynamic field variables within each element are expanded in terms of local, nonexact shape functions, usually polynomial approximation, the dynamic field variables within each substructure in WBM are expressed as wave function expansions, which exactly satisfy the governing dynamic equations of the substructure. Numerical results show good agreement with experimental results and analytical results of other researchers for shear diaphragm-shear diaphragm boundary condition and also show good agreement with FEM results for other boundary conditions.

Effects of the nonuniform rings spacing and eccentricity distribution on fundamental frequencies and beam mode frequencies of ring-stiffened cylindrical shells have been studied. For SD-SD boundary conditions, more mass concentrated in the midsection of the cylindrical shells, the higher the fundamental frequencies are while the lower the beam mode frequencies are. For F-F and SD-F boundary conditions, the effects are a little complicated. Numerical results of the effects of boundary conditions show that the axial displacement constraint has the largest effects on fundamental frequencies and the circumferential displacement constraint has the largest effects on beam mode frequencies. Release of the axial displacement constraint decreases the fundamental frequencies sharply and release of the circumferential displacement constraint increases the beam mode frequencies rapidly.

Acknowledgment

All the work in this paper obtains great support from the National Natural Science Foundation of China (51179071) and the Fundamental Research Funds for the Central Universities, HUST: 2012QN056.

Appendix A

The differential operators in Eq. (1) are as follows:
$L11=∂2∂x2+1-ν2∂2∂φ2-ρ(1-ν2)a2E∂2∂t2$
$L22=1-ν2∂2∂x2+∂2∂φ2-ρ(1-ν2)a2E∂2∂t2$
$L33=1+β(∂4∂x4+2∂4∂x2∂φ2+∂4∂φ4)+ρ(1-ν2)a2E∂2∂t2$
$L12=L21=1+ν2∂2∂x∂θ, L13=L31=ν∂∂x, L23=L32=∂∂φ$

The radius of the cylindrical shell is designated by a, and the thickness by h. $b=h2/12a2$, The axial and circumferential coordinates are $x,φ$, $x=x¯/a$. The mass density of the shell's material is designated by ρ, Young's modulus by E and Poission's ratio by $ν$.

Appendix B

For modal vibration of cylindrical shell with a specified circumferential wave number n, the general solution to Eq. (1) can be written as
$w=Aeλxsinnφcosωtv=Beλxcosnφcosωtu=Ceλxsinnφcosωt}$
(B1)

where λ is a characteristic root to be determined, ω is circular frequency and $φ$ is circumferential coordinate angle.

Substituting Eq. (B1) to Eq. (1) leads to three homogenous equations for the three constants A, B, C. For nontrivial solution, the determinant of their coefficients must vanish, which give the following fourth-degree equation in λ2.
$λ8+g6λ6+g4λ4+g2λ2+g0=0$
(B2)
where
$g6=(3-ν1-ν)Ω2-4n2$
$g4=6n4-3(3-ν)1-νn2Ω2+21-vΩ4+1β(1-ν2-Ω2)$
$g2=n2Ω21-ν[3(3-ν)n2-4Ω2]-4n6 +Ω2β(3+2ν+2n2-3-ν1-νΩ2)$
$g0=[1/β(1-ν)][(1-ν)n2-2Ω2][βn6-Ω2(1+βn4+n2-Ω2)]$
where Ω is a dimensionless frequency parameter:
$Ω2=ω2a2(1-ν2)(ρ/E)$
(B3)
For the usual range of parameters and n ≥ 1, the roots of Eq. (B2) have the form
$λ=±λ1,±iλ2,±(λ3±λ4)$
(B4)
where λ1, λ2, λ3, λ4 are real quantities. For each root, the ratios C/A, B/A can be determined by Eq. (1), so the general solution of u, v, w can be expressed by eight real constants A1 ∼ A8.
$w={A1eλ1x+A2e-λ1x+A3cosλ2x+A4sinλ2x+A5eλ3xcosλ4x+A6eλ3xsinλ4x+A7e-λ3xcosλ4x+A8e-λ3xsinλ4x}sinnφcosωt$
(B5)
$v={A1ξ1eλ1x+A2ξ1e-λ1x+A3ξ2cosλ2x+A4ξ2sinλ2x+A5eλ3x(ξ3cosλ4x-ξ4sinλ4x)+A6eλ3x(ξ4cosλ4x+ξ3sinλ4x)+A7e-λ3x(ξ3cosλ4x+ξ4sinλ4x)-A8e-λ3x(ξ4cosλ4x-ξ3sinλ4x)}cosnφcosωt$
(B6)
$u={A1η1eλ1x-A2η1e-λ1x-A3η2sinλ2x+A4η2cosλ2x+A5eλ3x(η3cosλ4x-η4sinλ4x)+A6eλ3x(η4cosλ4x+η3sinλ4x)-A7e-λ3x(η3cosλ4x+η4sinλ4x)+A8e-λ3x(η4cosλ4x-η3sinλ4x)}sinnφcosωt$
(B7)
where
$ξ1=G1/D1,η1=H1/D1ξ2=G2/D2,η2=H2/D2ξ3=R1Q1+R2Q2Q12+Q22, η3=S1Q1+S2Q2Q12+Q22ξ4=R2Q1-R1Q2Q12+Q22, η4=S2Q1-S1Q2Q12+Q22$
and
$D1=(1-ν)λ14+λ12[2n2(ν-1)+(3-ν)Ω2] +(n2-Ω2)[(1-ν)n2-2Ω2]$
$G1=n[λ12(ν2+ν-2)+(1-ν)n2-2Ω2]$
$H1=-λ1[λ12ν(1-ν)+2ν(Ω2-n2)+n2(1+ν)]$
$D2=(1-ν)λ24-λ22[2n2(ν-1)+(3-ν)Ω2] +(n2-Ω2)[(1-ν)n2-2Ω2]$
$G2=n[-λ22(ν2+ν-2)+(1-ν)n2-2Ω2]$
$H2=λ2[λ22ν(1-ν)-2ν(Ω2-n2)-n2(1+ν)]$
$Q1=(1-ν){(λ32-λ42)2-4λ32λ42}+(λ32-λ42){2n2(ν-1) +(3-ν)Ω2}+(n2-Ω2){(1-ν)n2-2Ω2}$
$Q2=4λ3λ4(λ32-λ42)(1-ν)+2λ3λ4{2n2(ν-1)+(3-ν)Ω2}$
$R1=n{(λ32-λ42)(ν2+ν-2)+(1-ν)n2-2Ω2}$
$R2=2nλ3λ4(ν2+ν-2)$
$S1=-λ3{ν(1-ν)(λ32-3λ42)+2ν(Ω2-n2)+n2(1+ν)}$
$S2=-λ4{ν(1-ν)(3λ32-λ42)+2ν(Ω2-n2)+n2(1+ν)}$

Appendix C

Equations (B5)–(B7) can be written as follows:
$w(x)=w(x)·A,v(x)=v(x)·A,u(x)=u(x)·A$
(C1)
where
$w(x)=[w1(x)w2(x)w7(x)w8(x)]$
(C2)
$v(x)=[v1(x)v2(x)···v7(x)v8(x)]$
(C3)
$u(x)=[u1(x)u2(x)···u7(x)u8(x)]$
(C4)
$A=[A1A2···A7A8]$
(C5)
The twisting angle, forces and moments are given in Ref. [1],
${θ=∂w∂xM=Da2(∂2w∂x2+ν∂2w∂φ2)S=Da2[∂3w∂x3+(2-ν)∂3w∂x∂φ2]T=Eh2a(1+υ)(∂u∂φ+∂v∂x)N=Eha(1-υ2)[∂u∂x+ν(∂v∂φ+w)]$
(C6)
Substituting Eqs. (B5)–(B7) to Eq. (C6)
$w1=eλ1x, w2=e-λ1x, w3=cosλ2x, w4=sinλ2xw5=eλ3xcosλ4x, w6=eλ3xsinλ4x, w7=e-λ3xcosλ4x, w8=e-λ3xsinλ4xv1=ξ1eλ1x, v2=ξ1e-λ1x, v3=ξ2cosλ2x, v4=ξ2sinλ2xv5=G1eλ3x, v6=G2eλ3x, v7=G3e-λ3x, v8=G4e-λ3xu1=η1eλ1x, u2=-η1e-λ1x, u3=-η2sinλ2x, u4=η2cosλ2xu5=F1eλ3x, u6=F2eλ3x, u7=-F3e-λ3x, u8=F4e-λ3xθ1=λ1eλ1x, θ2=-λ1e-λ1x, θ3=-λ2sinλ2x, θ4=λ2cosλ2xθ5=H1eλ3x, θ6=H2eλ3x, θ7=-H3e-λ3x, θ8=H4e-λ3x$
$M1=Da2δ1eλ1x,M2=Da2δ1e-λ1x,M3=-Da2δ2cosλ2x,M4=-Da2δ2sinλ2xM5=-Da2eλ3x(δ4sinλ4x-δ3cosλ4x),M6=Da2eλ3x(δ3sinλ4x+δ4cosλ4x)M7=Da2e-λ3x(δ3cosλ4x+δ4sinλ4x),M8=-Da2e-λ3x(δ4cosλ4x-δ3sinλ4x)$
$S1=Da3γ1eλ1x,S2=-Da3γ1e-λ1x,S3=Da3γ2sinλ2x,S4=-Da3γ2cosλ2xS5=Da3eλ3x(γ3cosλ4x-γ4sinλ4x),S6=Da3eλ3x(γ3sinλ4x+γ4cosλ4x)S7=-Da3e-λ3x(γ3cosλ4x+γ4sinλ4x),S8=-Da3e-λ3x(γ3sinλ4x-γ4cosλ4x)$
$T1=Da3χ1eλ1x,T2=-Da3χ1e-λ1x,T3=-Da3χ2sinλ2x,T4=Da3χ2cosλ2xT5=-Da3eλ3x(χ4sinλ4x-χ3cosλ4x),T6=Da3eλ3x(χ4cosλ4x+χ3sinλ4x)T7=-Da3e-λ3x(χ3cosλ4x+χ4sinλ4x),T8=-Da3e-λ3x(χ3sinλ4x-χ4cosλ4x)$
$N1=Da3μ1eλ1x,N2=Da3μ1e-λ1x,N3=Da3μ2cosλ2x,N4=Da3μ2sinλ2xN5=-Da3eλ3x(μ4sinλ4x-μ3cosλ4x),N6=Da3eλ3x(μ4cosλ4x+μ3sinλ4x)N7=Da3e-λ3x(μ3cosλ4x+μ4sinλ4x),N8=-Da3e-λ3x(μ4cosλ4x-μ3sinλ4x)$
$F1=η3cosλ4x-η4sinλ4x,F2=η4cosλ4x+η3sinλ4xF3=η3cosλ4x+η4sinλ4x,F4=η4cosλ4x-η3sinλ4x$
$G1=ξ3cosλ4x-ξ4sinλ4x,G2=ξ4cosλ4x+ξ3sinλ4xG3=ξ3cosλ4x+ξ4sinλ4x,G4=-ξ4cosλ4x+ξ3sinλ4x$
$H1=λ3cosλ4x-λ4sinλ4x,H2=λ4cosλ4x+λ3sinλ4xH3=λ3cosλ4x+λ4sinλ4x,H4=λ4cosλ4x-λ3sinλ4x$
$δ1=λ12-νn2,δ2=λ22+νn2,δ3=λ32-λ42-νn2,δ4=2λ3λ4$
$γ1=λ1{λ12-(2-ν)n2},γ2=λ2{λ22+(2-ν)n2}γ3=λ3{λ32-3λ42-(2-ν)n2},γ4=λ4{3λ32-λ42-(2-ν)n2}$
$χ1=[(1-ν)/2β](nη1+ξ1λ1),χ2=[(1-ν)/2k](nη2+ξ2λ2)χ3=[(1-ν)/2β](nη3+ξ3λ3-ξ4λ4),χ4=[(1-ν)/2k](nη4+ξ4λ3+ξ3λ4)$
$μ1=(1/β){η1λ1+ν(1-nξ1)},μ2=(1/β){-η2λ2+ν(1-nξ2)}μ3=(1/β){η3λ3-η4λ4+ν(1-nξ3)},μ4=(1/β)(η4λ3+η3λ4-νnξ4)$

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