Wave based method for vibration analysis of elastically coupled annular plate and cylindrical shell structures

Applied Acoustics 123 (2017) 107–122

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Wave based method for vibration analysis of elastically coupled annular plate and cylindrical shell structures Kun Xie, Meixia Chen ⇑, Lei Zhang, De Xie School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Received 14 December 2016 Received in revised form 1 March 2017 Accepted 8 March 2017

Keywords: Wave based method Elastically coupled plate-shell structures Elastic boundary conditions Vibration analysis

a b s t r a c t Free and forced vibrations of elastically coupled thin annular plate and cylindrical shell structures under elastic boundary conditions are studied through wave based method. The method is involved in dividing the coupled structure into shell segments and annular plates. Flügge shell theory and thin plate theory are utilized to describe motion equations of segments and plates, respectively. Regardless of boundary and continuity conditions, displacements of individual members are expressed as different forms of wave functions, rather than polynomials or trigonometric functions. With the aid of artificial springs, continuity conditions between segments and plates are readily obtained and corresponding governing equation can be established by assembling these continuity conditions. To test accuracy of present method, vibration results of some coupled structures subjected to different boundary and coupling conditions are firstly examined. As expected, results of present method are in excellent agreement with the ones in literature and calculated by finite element method (FEM). Moreover, effects of annular plates, elastic coupling and boundary conditions, excitation and damping are also studied. Results show that normal displacement of annular plate mainly affects free vibrations of the coupled structures, while tangential displacement has the greatest effect on forced vibrations as meridinoal or normal excitation forced on the annular plate. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Plates and shells are widely used in engineering and vibration characteristics of these independent members have been deeply investigated by lots of researchers. Most of these investigations were well summarized by Leissa [1] and Qatu [2,3]. On the other hand, due to the complexity involved in modeling and solution process, literature about coupled annular plate and cylindrical shell structures is rare, whereas the coupled structures also play an importance role in industrial vessels, missiles, submarines and so forth. Since vibrations of the coupled structures have significant effect on their performance, knowing vibration behaviors of the coupled structures is important in analysis and design process. Of course, FEM is a powerful tool and commercial FEM programs, such as ANSYS, ABAQUS and NASTRAN, have been well developed. Nevertheless, FEM has inherent disadvantages in efficiency. To identify the mode shape of a certain frequency, the solution must be extracted for each mode shape and classified one by one, which is a time-consuming and tiresome work. In addition, to meet requirements of convergence, the number of elements rapidly ⇑ Corresponding author. E-mail address: [email protected] (M. Chen). http://dx.doi.org/10.1016/j.apacoust.2017.03.012 0003-682X/Ó 2017 Elsevier Ltd. All rights reserved.

increases as the range of analysis frequency increases, which seriously reduces computation efficiency and increases storage space in return. To this end, proposing an accurate and efficient method to evaluate vibrations of coupled plate-shell structures is considerable. To the authors’ best knowledge, there are some but not many papers studying vibrations of coupled plate-shell systems. In the early research, Smith and Haft [4] and Takahashi and Hirano [5,6] were the typical researchers investigating vibrations of coupled plate-shell systems. Neglecting in-plane motions of the circular plate, Smith and Haft [4] analytically determined natural frequencies of a cylindrical shell closed by a circular plate at one end and clamped at the other end. Takahashi and Hirano [5,6] studied vibrations of cylindrical shells with circular plates at ends and intermediate section. By using boundary and continuity conditions, the Lagrangian in terms of unknown boundary values of displacements was formulated, and corresponding frequency equations were obtained by minimizing the Lagrangian. Soon afterwards, more and more researchers began to study vibrations of the coupled structures. In those studies, the transfer matrix method [7–9], state space method [10,11], receptance method [12–14], Rayleigh-Ritz approach [15–18] and finite element method [19–21] were usually adopted.

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Irie et al. [7] used the transfer matrix method to discuss free vibrations of joined conical-cylindrical shells. Natural frequencies and mode shapes of an annular plate-cylindrical shell system were presented as the special case. Then, the same method was adopted by Yamada et al. [8] to analyze free vibrations of circular cylindrical double-shell systems closed by end plates. Liang and Chen [9] combined the vibration theory with the transfer matrix method to calculate natural frequencies and mode shapes of a conical shell with an annular or a round end plate. State space method (SSM), which is similar with the transfer matrix method, was proposed by Tavakoli and Singh [10] for the eigensolution of axisymmetric joined/ hermitic thin shells. Then they adopted SSM to investigate free vibrations of a hermetic can consisting of one cylindrical shell and two end circular plates [11]. Since distinct boundaries required in SSM, a ‘‘pinhole” with free edge was additionally introduced at the center of end circular plates. Huang and Soedel [12,13] used the receptance method for free vibrations of a simply supported cylindrical shell welded one or more circular plates at arbitrary axial positions. Then, the receptance method was extended by Yim et al. [14] to analyze vibrations of a clamped-free cylindrical shell with a plate attached to the shell at arbitrary axial position. Rayleigh-Ritz method is another extensively adopted approach. With the help of artificial springs, Cheng and Nicolas [15] used the variational principle to establish analytic formulation for free vibration analysis of a cylindrical shell-circular plate system with elastic coupling and boundary conditions. In the analysis, circumferential and radial displacements at two ends of the shell were assigned as 0, which simplified admissible functions of cylindrical shells. Correspondingly, eigenfunctions of shear diaphragm supported shells were utilized. In addition, in-plane motions of circular plates were not considered to meet the prior boundary conditions. Yuan and Dickinson [16] applied the Rayleigh-Ritz approach for free vibrations of cylindrical shell and plate systems, which were coupled by artificial springs. General orthogonal polynomials were selected as trial functions of separate components regardless of boundary and continuity conditions. The FourierRitz method, which were widely used for vibration analysis of conical shells, cylindrical shells, annular plates and so forth [22–25], was extended by Ma et al. [18] to analyze free and forced vibrations of coupled cylindrical shell and annular plate systems. In theoretical formulation, artificial springs were also utilized to restrain displacements at boundaries and to combine plates and shells. Differing from the displacement functions adopted by Yuan and Dickinson [16], the modified Fourier series composed of standard Fourier series and auxiliary functions were selected. However, only classic boundaries were taken in account in the results analysis, and annular plates with pinhole, rather than circular plates, were adopted to model circular plates. In vibration analysis of the coupled structures, transmission of structure-borne vibrations between the coupled shell and plate is important and difficult. Tso and Hansen [26] emphatically studied transmission characteristics of vibration waves through the junction of two semi-infinite cylindrical shells and one annular plate. Motions of the cylindrical shells were described by DonnellMushtari equations while motions of the plate were represented by Bessel functions. The results showed that wave transmission properties of a cylindrical shell were approximately same with those of a flat plate after the cut-on of the plate longitudinal wave. From above review it is clearly known that the cited literature mainly was focused on free vibrations of plate-shell systems with rigid coupling and classic boundary conditions. However, the weld defects or bolted connections can lead to incompletely coupling conditions. In addition, more complex boundary conditions, rather than classic ones, may appear in engineering applications. To compensate for these shortcomings, the paper presents an accurately

and efficiently analytic method for both free and forced vibration analysis of elastically coupled annular plate and cylindrical shell structures with elastic boundary conditions. Wave based method, which has been adopted by the research group of the authors to study vibrations of cylindrical shells [27–29], is extended to establish the governing equation of the coupled thin structures. The method is involved in dividing the coupled structures into shell segments and annular plates. Flügge shell theory and thin plate theory are respectively utilized to describe motion equations of these thin segments and plates. Regardless of boundary and continuity conditions, displacements of individual members are expressed as different forms of wave functions. In addition, artificial springs are employed to restrain displacements at edges and to couple annular plates with the cylindrical shell at any axial locations. Since Flügge shell theory and thin plate theory are adopted, the present method is applicable to analyze linear vibrations of coupled thin cylindrical shell and annular plate structures. Correspondingly, only thin shells and plates are considered in present paper. On the other hand, in practical engineering applications, e.g. submarines, missiles, rockets and so on, the coupled structures mainly consist of thin shells and plates. In the circumstances, present method can be utilized for vibration analysis of these engineering structures. Although adopting the first shear deformation shell theory, high shear deformation shell theory or 3-D elastic theory can lead to more extensive applications, utilizing thin shell and plate theories to establish the governing equation is much more concise in theoretical derivation process and the computation efficiency is higher in some degree. Additionally, as the coupled structures consist of thin shells and plates, high accurate results can be also predicted, which can be readily observed from above cited references. Besides, Lee and Kwak [30] pointed out that DonnellMushtari theory was not sufficiently accurate to calculate natural frequencies of cylindrical shells while there was no discernible difference among other shell theories, including Sanders theory, Love-Timoshenko theory, Reissner theory, Flügge theory and Vlasov theory. On the whole, adopting Flügge shell theory and thin plate theory to investigate the titled problem can predict accurate results and the accuracy will be emphatically discussed in the following analysis.

2. Theory equations Schematic diagram of a cylindrical shell with P-1 uniform annular plates is shown in Fig. 1(a). h, R and L are the thickness, radius and length of the shell. hp and R1 are the thickness and inner radius of annular plates. Li is the axial length between the ith and (i + 1)th annular plate. The global cylindrical coordinate system ðr; h; XÞ is also presented. In the figure, end plates are not taken into account, but the case can be easily considered.

2.1. Description of wave based method As adopting wave based method to establish governing equation of the coupled structure, the first step is to decompose the coupled structure into separate shell segments and annular plates according to the locations of annular plates. In Fig. 1(b), local coordinate systems of a shell segment and a plate are presented. Meanwhile, positive directions of displacement and force resultants at edges are also given. After decomposition, to solve motion equations of separate substructures, appropriate wave functions, rather than polynomials or trigonometric functions used in the literature [15–25], are adopted to express displacement functions of individual substructures, which is the essence of wave based method. In addition, it should be emphasized that the wave functions can

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Fig. 1. Schematic diagram of a coupled annular plate-cylindrical shell structure, and local coordinate systems of substructures: (a) coupled structure, (b) substructures.

accurately satisfy the equations of motions since the derivation of these specific wave functions is based on the motion equations. 2.2. Cylindrical shell segments The local cylindrical coordinate system ðr; h; xÞ, directions of displacement and force resultants of a shell segment are presented in Fig. 1(b). u, v , and w are displacements in axial, circumferential and radial directions, and b ¼ @w=@x is slop. On the basis of Flügge shell theory, motion equations of a cylindrical segment are [31] @ u 1  t @ u qð1  t Þ @ u 1 þ t @ v t @w @ w 1t @ w þ ¼0  kR 3 þ k þ ð1 þ kÞ  þ 2R @x@h R @x 2R @x2 @h @x2 E @x @t2 2R2 @h2 2 2 2 2 3 2 1þt @ u 1  t @ v 1 @ v qð1  t Þ @ v 1 @w 3t @ w ¼0 þ  þ þ ð1 þ 3kÞ k 2R @x@h 2 @x2 R2 @/2 2 @x2 @h E @t 2 R2 @h 2

2

2

2

2

3

2

ð1Þ

where r ¼ r r , r ¼ R@ =@x þ ð1=RÞ@ =@h . k ¼ h =12R is the thickness parameter. q, E and t are the density, Young’s modulus and Poisson’s ratio, respectively. x is angular frequency and t is time. For a given circumferential mode number n, wave functions are adopted to expand the displacements in Eq. (1) as

u¼

2

2

2

2

2

2

2

8 8 X X U ci Wcui cosðnhÞejxt ¼ ni W ci Wcui cosðnhÞejxt i¼1

v

2

ð2Þ

i¼1

8 8 X X w¼ W ci Wcwi ðxÞ cosðnhÞejxt ¼ W ci Wcwi ðxÞ cosðnhÞejxt i¼1

i¼1

where j is the imaginary unit, and the structure wave functions, Wcui , Wcv i and Wcwi , are

Wcui ¼ Wcv i ¼ Wcwi ¼ ejki x ;

i¼1:8

v

 c T D T w bjN T S M ¼ c  fwc g F

ð4Þ

where

2

u1 ðxÞ . . . u8 ðxÞ

3

6 v ðxÞ . . . v ðxÞ 7 8 6 1 7 Dc ¼ 6 7 4 w1 ðxÞ . . . w8 ðxÞ 5 b1 ðxÞ . . . b8 ðxÞ 2

N1 ðxÞ . . . N8 ðxÞ 6 T ðxÞ . . . T ðxÞ 8 6 1 Fc ¼ 6  4 S1 ðxÞ . . . S8 ðxÞ M 1 ðxÞ . . . M 8 ðxÞ w ¼ c

fW c1

W c2

W c3

W c4

ð5Þ 48

3 7 7 7 5

ð6Þ

48

W c5 W c6 W c7 W c8 g c

ð7Þ

c

Elements in matrixes D and F are given in Appendix A. 2.3. Annular plates

i¼1

8 8 X X ¼ V ci Wcv i sinðnhÞejxt ¼ gi W ci Wcv i sinðnhÞejxt i¼1

 u

3

t @u @3 u 1  t @3 u 1 @v 3  t @3 v k @2 w 1 þ k  kR 3 þ k þ k þ kr4 w þ 2 þ 2 w @x R @x 2R @x2 @h R2 @h 2 @x2 @h R @h2 R qð1  t2 Þ @ 2 w ¼ 0 þ E @t 2 4

ni ¼ U ci =W ci and gi ¼ V ci =W ci are ratios of axial and circumferential wave contribution coefficients. Expressions of ki , ni and gi can be found in authors’ opened paper [29]. Substituting Eq. (3) into Eq. (2) into expressions of force resultants, which are given in Appendix A, displacement and force resultants at cross-section of cylindrical segments can be expressed as

ð3Þ

In Eqs. (2) and (3), U ci , V ci and W ci are unknown wave contribution coefficients of segments. ki is the axial wave number, and it can be obtained by substituting the middle part of Eq. (2) into Eq. (1).

Directions of displacement and force resultants of an annular plate in polar coordinate system ðr; hÞ are presented in Fig. 1(b). In the figure, positive directions of four displacement resultants, up ; v p ; wp ; bp , and four force resultants, N px ; N pr ; N ph ; M p , are also presented. In Appendix B, the relationships of force resultants and displacement resultants are given. For thin annular plates, out-plane motions are decoupled with in-plane motions, and motion equations of annular plates are [26]

r4 wp þ @ @r 1 r





¼0    q ð1t2 Þ 2 1t @ @ v p v @u @ u þ þ þ rp  1r @hp ¼ p Ep p @t2p  2r p @h @r    q ð1t2 Þ @ 2 v @up u @v 1t @v v @u þ rp þ 1r @hp þ 2 p @r@ @rp þ rp  1r @hp ¼ p Ep p @t2p @r

@up @r

@ @h

qp hp @ 2 wp Dp

up r

@t 2

1 @v p r @h

ð8Þ

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where r4 ¼ r2 r2 , r2 ¼ @ 2 =@r2 þ ð1=rÞ@=@r þ ð1=r 2 Þ@ 2 =@h2 . qp , hp , tp and Ep are the density, thickness, Poisson’s ratio and Young’s 3

modulus, respectively. Dp ¼ Ep hp =12ð1  t2p Þ is the bending stiffness. Displacements in Eq. (8) can be expressed in terms of wave functions [26]

up ðr; h; tÞ ¼

8 X W ai Waui ðrÞ cosðnhÞejxt

2.4. Elastic boundary and coupling conditions 2.4.1. Elastic boundary conditions Artificial springs are employed to restrain displacements at two ends of cylindrical shells, as shown in Fig. 2. K cu ; K cv ; K cw and K cb are stiffness constants of artificial springs restraining axial, circumferential and radial displacements and slop, respectively. The corresponding boundary conditions are

i¼1 8 X v p ðr; h; tÞ ¼ W ai Wav i ðrÞ sinðnhÞejxt

ð9Þ

i¼1

wp ðr; h; tÞ ¼

8 X W ai Wawi ðrÞ cosðnhÞejxt i¼1

where W ai are unknown wave contribution coefficients of annular plates, and the wave functions Waui , Wav i and Wawi are dJ ðk rÞ

nJ ðk rÞ

Wau1 ¼ n drpL ; Wau2 ¼ n r pT ; Wau3 ¼ Wau5 ¼ Wau6 ¼ Wau7 ¼ Wau8 ¼ 0

Wv 1 Wav 5 Waw1 Waw5 a

dY n ðkpL rÞ ; dr

Wau4 ¼

nY n ðkpT rÞ r

nJ ðk rÞ dJ ðk rÞ nY ðk rÞ dY ðk rÞ ¼  n r pL ; av 2 ¼  n drpT ; av 3 ¼  n r pL ; av 4 ¼  n drpT a a a ¼ v6 ¼ v7 ¼ v8 ¼ 0 ¼ aw2 ¼ aw3 ¼ aw4 ¼ 0 ¼ Jn ðkpB rÞ; aw6 ¼ In ðkpB rÞ; aw7 ¼ Y n ðkpB rÞ; aw8 ¼ K n ðkpB rÞ

W

W W

W W

W

W

W W

W

W

W

ð10Þ In Eq. (10), kpL

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x qp ð1  t2p Þ=Ep and kpT ¼ x 2qp ð1 þ tp Þ=Ep are

the in-plane longitudinal and tangential wave numbers, and 1=4

is the out-plane bending wave number. Jn kpB ¼ ðqp x2 hp =Dp Þ and Y n are Bessel functions of the first kind and second kind, respectively. In and K n are modified Bessel functions of the first kind and second kind. Similarly, displacements and forces of annular plates are also expressed in matrix form as

 wp

vp

up  bp jNpx Nph Npr Mp

T

 a D T ¼ a  fwa g F

ð11Þ

where

wp1 ðrÞ . . . wp8 ðrÞ 6 v p1 ðrÞ . . . v p8 ðrÞ 6 Da ¼ 6 4 up1 ðrÞ . . . up8 ðrÞ

7 7 7 5

bp1 ðrÞ . . .  bp8 ðrÞ

w

K cv v  T ¼ 0

K cb b  M ¼ 0

ð15Þ

In Eq. (15), ‘þ’ indicates boundary conditions at X ¼ 0 while ‘’ indicates boundary conditions at X ¼ L. By assigning appropriate stiffness constants of artificial springs, both classic and elastic boundary conditions can be analyzed. In the following analysis, the stiffness constant is set as 0 if corresponding displacement is free or a very large value (1016 N=m) if corresponding displacement is fixed, e.g. K cu ¼ K cv ¼ K cw ¼ K cb ¼ 0 for free boundaries and K cu ¼ K cv ¼ K cw ¼ K cb ¼ 1016 N=m for clamped boundaries. 2.4.2. Elastic coupling conditions Elastic coupling conditions at the ith annular plate are analyzed as an example. Fig. 3 shows displacement and force resultants at the junction of the ith annular plate. Four artificial springs to couple the annular plate with adjacent segments are also shown in the figure, and the stiffness constants are K au ; K av ; K aw and K ab . At the outer edge of the ith annular plate (r ¼ R), displacement continuity conditions of adjacent two segments are

uLi ¼ uRi ;

v Li ¼ v Ri ;

wLi ¼ wRi ; bLi ¼ bRi

ð16Þ

where superscripts L and R denote edges of adjacent segments at left and right side of the ith annular plate. Equilibrium equations of forces of adjacent segments and plates are

NLi  NRi þ Npx;i r¼R ¼ 0; T Li  T Ri þ Nph;i r¼R ¼ 0 SL  SR þ Npr;i ¼ 0; M L  M R þ M p;i ¼ 0 i i i i r¼R r¼R

ð17Þ

where the forces of the ith annular can be further expressed in terms of stiffness constants and displacements as

3

2

K cu u  N ¼ 0; K c w  S ¼ 0;

ð12Þ

Npx;i r¼R ¼ K aw ðwp;i  uLi Þ; Nph;i r¼R ¼ K av ðv p;i  v Li Þ Npr;i r¼R ¼ K au ðup;i  wLi Þ; M p;i r¼R ¼ K ab ðbp;i þ bLi Þ

ð18Þ

or

48

3

2

Npx1 ðrÞ . . . Npx8 ðrÞ 6 N ðrÞ . . . N ðrÞ 7 ph8 7 6 ph1 Fa ¼ 6 7 4 Npr1 ðrÞ . . . Npr8 ðrÞ 5 M p1 ðrÞ . . . Mp8 ðrÞ

ð13Þ 48

wa ¼ fW a1 W a2 W a3 W a4 W a5 W a6 W a7 W a8 g

ð14Þ

In Eqs. (12) and (13), expressions of upi ; v pi ; wpi ; bpi ; N pxi ; Nphi ; N pri and Mpi are given in Appendix B. By setting W a3 ¼ W a4 ¼ W a7 ¼ W a8 ¼ 0, Eq. (9) denotes displacement functions of circular plates, and all the others are the same for annular and circular plates. On the other hand, as the inner radius of the annular plate is small enough, the annular plate plays the same role of circular plate [11]. Under this circumstance, results of two approaches to model circular plate will be compared in the following analysis. In addition, as adopting the annular plate to analyze the circular plate, the inner radius is assigned as 1% of the outer radius and the corresponding boundary conditions are free.

Fig. 2. Schematic diagram of artificial springs distributed on boundaries.

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

Fig. 3. Interaction displacements and forces of adjacent substructures.

Npx;i r¼R ¼ K aw ðwp;i  uRi Þ; Nph;i r¼R ¼ K av ðv p;i  v Ri Þ Npr;i r¼R ¼ K au ðup;i  wRi Þ; Mp;i r¼R ¼ K ab ðbp;i þ bRi Þ

ð19Þ

When annular plates located at middle part of the shell, Eqs. (18) and (19) are the same in essence. However, for the annular plate located at one end of the shell, only one group of equations can be adopted. As similar with boundary conditions, different coupling conditions, such as no coupling, elastic coupling or rigid coupling conditions, can be easily obtained by selecting appropriate stiffness constants. Besides elastic coupling conditions, boundary conditions at the inner edge of annular plates cannot be neglected. Although elastic boundaries can be also considered at the inner edge of annular plates (r ¼ R1 ), only free boundary conditions are analyzed in present paper, and corresponding equations are

Npx;i r¼R ¼ 0; 1 Npr;i r¼R ¼ 0; 1

Nph;i r¼R ¼ 0 1 M p;i r¼R ¼ 0

ð20Þ

1

2.5. External point force In theory deduction, one point excitation forced at the free boundary of one annular plate is emphatically discussed, and one or more point excitations forced on any position of the shell and/ or plate can be easily dealt. The external excitation with amplitude F 0 is expressed in terms of Dirac delta functions as

F ¼ F 0 dðX  X 0 Þdðh  h0 Þ=R1

ð21Þ

where ðR1 ; h0 ; X 0 Þ is the global coordinate of the excitation. For radial force, F results in the modification of the third equation in Eq. (20) and it becomes

Npr;i r¼R ¼ F 0 dðX  X 0 Þdðh  h0 Þ=R1 1

ð22Þ

Multiplying above equation by cosðnhÞ and taking the integral from 0 to 2p, Eq. (22) further becomes

Npr;i r¼R ¼ eF 0 cosðnh0 Þ 1

where

e ¼ 1=ð2pR1 Þ if n ¼ 0 and e ¼ 1=ðpR1 Þ if n P 1.

ð23Þ

2.6. Governing equation Assembling all elastic coupling and boundary conditions, Eqs. (16)–(23), the governing equation of the coupled structure is established.

c a T K jK  fwc jwa g ¼ fFg c

ð24Þ a

where W ¼ fwc1 ; wc2 ; . . . ; wcP g and W ¼ fwa1 ; wa2 ; . . . ; waP1 g are 1  8P and 1  8ðP  1Þ vectors of wave contribution coefficients of segments and annular plates, respectively. F is a ð16P-8Þ  1 vector of external force. Sub-matrixes Kc and Ka are given in Appendix C. For Eq. (24), only a particular circumferential mode number is taken into account. As studying free vibration characteristics of the coupled structures, natural frequencies can be solved by vanishing the determinant of coefficient matrix, and mode shapes are obtained by substituting the natural frequencies back into Eq. (24). By setting different circumferential mode numbers, all natural frequencies can be obtained. However, to analyze forced vibrations of the coupled structures, Eq. (24) cannot give accurate results since contributions of one circumferential mode number are considered. On the other hand, accurate vibration responses are obtained by taking into account circumferential mode numbers from 0 to infinite, but it is mathematically impossible. In the following analysis, influences of the truncation will be discussed. 3. Results and discussions 3.1. Free vibration analysis 3.1.1. Validity analysis for free vibrations To test the validity of present method, a cylindrical shell closed by two circular plates is firstly employed. The model was adopted by Yuan and Dickinson [16] to analyze free vibrations of circular cylindrical shell and plate systems. In theoretical derivation of Ref. [16], the maximum strain and kinetic energy of cylindrical shells and circular plates were firstly expressed in terms of displacements, respectively, and displacements were expanded as orthogonal polynomials regardless of boundary conditions. Then the shells and plates were coupled by artificial springs, and the Rayleigh-Ritz minimization procedure was employed to establish

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

governing equations. Although the artificial spring technique simplified the selection of polynomials, the adopted polynomials strongly depended on mode shapes. Correspondingly, the governing equation was not uniform for all modes. The detailed dimensions of the coupled structure are L ¼ 0:347 m, R ¼ 0:1143 m and h ¼ hp ¼ 0:002 m. The shell and plates are made of the same mate3

rial with E ¼ 207 GPa, q ¼ 7800 kg=m , t ¼ 0:3. Natural frequencies of present method and Ref. [16] are tabulated in Table 1. In the table, m indicates the order that modes appear for a particular circumferential mode number. Two kinds of modes, symmetric and antisymmetric modes, are accounted for. The former modes mean radial and tangential displacements of the shell are symmetric about the centerline and axial displacement is antisymmetric, while the later modes denote axial displacement of the shell is symmetric about the centerline and radial and tangential displacements of the shell are antisymmetric. As stated in Section 2.3, both annular plates and circular plates can be dealt by present method. Herein, annular plates with pinhole and circular plates are adopted to model the end plates, and corresponding nature frequencies are compared in Table 1. It can be observed that the discrepancy of natural frequencies of two models is negligible. That is to say, adopting annular plates with pinhole to model circular plates is an accurate approach to analyze circular plates. More importantly, natural frequencies of present method show excellent agreement with the ones in literature. In above analysis, only rigid coupling conditions are analyzed, and elastic ones are emphatically investigated now. The model discussed by Ma et al. [18] is adopted to illustrate accuracy of present method. In Ref. [18], a unified analytical method was presented for free and forced vibrations of coupled cylindrical shell and annular plate systems. The theoretical derivation in Ref. [18] is similar with the one in Ref. [16], e.g. the artificial spring technique and the Rayleigh-Ritz method, but the most striking difference is the selection of trail displacement functions. Modified Fourier series consisting of standard Fourier series and additional auxiliary functions were adopted in Ref. [18] to uniformly express displacements of shells and plates. Nevertheless, annular plates with pinhole, rather than circular plates, were directly adopted to model circular plates as analyzing coupled cylindrical shell and circular plate systems. Besides, only classic boundaries were studies. The dimension and material parameters of the coupled structure are: the length of shell L ¼ 0:5 m, the radius of shell R ¼ 0:1045 m, the inner radius of annular plate R1 ¼ 0:03 m, the thickness of shell

and plate h ¼ hp ¼ 0:003 m, Young’s modulus E ¼ 206 GPa, density

q ¼ 7850 kg=m3 and Poisson’s ratio t ¼ 0:3. The shell is clamped at the left end X ¼ 0 and elastically coupled with an annular plate at the right end X ¼ L. Without otherwise specified, the geometry and material parameters keep unchanged in the following analysis. Meanwhile, a finite element model, which is developed in ANSYS, is employed to further illustrate the validity of present method. The shell and plate consist of 160  160 and 160  20 Shell63 elements (in circumferential and meridional directions, respectively). At X = 0, displacements of all nodes are completely restrained. At X = L, namely the junction of the shell and plate, the mesh of shell is the same with the one of plate. By utilization of Combin14 elements, corresponding nodes of the shell and plate are connected one by one, and arbitrary coupling conditions are achieved by selecting different stiffness constants of Combin14 elements. Furthermore, the elements will be approved to satisfy the requirements of convergence in the following analysis. Comparisons of natural frequencies of present method, literature and ANSYS for different coupling conditions are tabulated in Tables 2 and 3. In Table 2, one group of displacements of the shell and plate is elastically coupled by selecting appropriate stiffness constant and the others are not coupled by setting corresponding stiffness constants to 0. It is observed that differences of natural frequencies of three methods are slight, which demonstrates the validity of both present method and the finite element model. That is to say, the developed finite element model satisfies the requirements of convergence. It is further observed that the difference of natural frequencies of present method and ANSYS is smaller than the difference of ANSYS and literature, especially for the modes of n = 2 and m = 2. Consequently, present method is of higher accuracy for free vibration analysis of elastic coupling plate-shell structures. Differing from the elastic coupling conditions in Table 2, the conditions in Table 3 are that one group of displacements is elastically coupled and the other three are rigidly coupled. Once again excellent agreement is observed as comparing natural frequencies of present method and ANSYS. From Tables 2 and 3 it is further found that the increase of some stiffness constants can significantly increase or reduce natural frequencies, while effects of other ones are negligible. In addition, the increase of stiffness constant cannot certainly lead to the increase of natural frequency. In the following, effects of elastic coupling conditions will be investigated in detail.

Table 1 Natural frequencies of a cylindrical shell closed by two circular plates. n

m

Symmetric modes Ref. [16]

Antisymmetric modes Present

Ref. [16]

Present

Annular plates

Circular plates

Annular plates

Circular plates

0

1 2 3

353.4 1389 3120

353.372 1388.200 3119.282

353.415 1388.738 3121.008

381.7 1428 3175

381.664 1427.567 3174.274

381.719 1428.122 3176.053

1

1 2 3

735.4 2121 3763

735.364 2121.207 3762.999

735.365 2121.210 3762.891

741 2124 4087

740.962 2124.314 4086.259

740.962 2124.215 4086.248

2

1 2 3 4

1191 1254 2940 4312

1190.932 1254.451 2939.832 4312.468

1191.017 1254.489 2940.252 4312.535

1206 2825 3075 5113

1205.659 2824.808 3075.096 5110.826

1205.779 2825.010 3075.337 5111.254

3

1 2 3 4

776.5 1772 3372 3916

777.049 1771.467 3372.613 3915.228

777.049 1771.467 3372.613 3915.228

1759 2130 3843 4492

1759.464 2130.945 3842.959 4492.597

1759.464 2130.945 3842.960 4492.597

113

K. Xie et al. / Applied Acoustics 123 (2017) 107–122 Table 2 Natural frequencies of the coupled structure with one group of displacements elastically coupled and the others uncoupled. n

m

Methods

K au 1  105

1  109

1  105

1  109

1  105

1  109

1  105

1  109

0

1

Ref. [18] Present ANSYS

565.706 565.745 566.273

565.706 565.745 566.273

565.620 565.745 566.273

565.620 565.745 566.273

569.9791 570.0036 570.533

315.20212 315.2025 315.365

916.548 914.512 915.244

950.649 948.126 948.897

1

1

Ref. [18] Present ANSYS Ref. [18] Present ANSYS

667.411 667.102 666.995 1252.091 1252.226 1254.300

574.680 574.003 573.947 1252.091 1252.226 1254.300

667.668 667.098 666.991 1252.091 1252.227 1254.300

580.492 580.254 580.173 1252.091 1252.227 1254.300

667.155 666.855 666.747 1253.715 1253.814 1255.900

664.933 664.660 664.557 871.421 871.908 872.704

171.618 172.538 172.620 667.497 666.924 666.816

178.797 179.668 179.762 667.240 666.931 666.823

Ref. [18] Present ANSYS Ref. [18] Present ANSYS

314.091 313.931 313.959 327.338 334.886 334.952

327.424 334.886 334.952 830.568 826.915 826.675

313.322 313.149 313.177 330.415 334.886 334.952

330.415 334.886 334.952 697.239 802.530 802.111

313.065 312.896 312.924 336.483 342.828 342.897

313.065 312.904 312.933 1229.271 1229.079 1228.500

313.151 313.027 313.055 483.316 488.455 488.510

313.151 313.028 313.057 498.101 502.968 503.020

2

2

1

2

K av

K aw

K ab

Table 3 Natural frequencies of the coupled structure with one group of displacements elastically coupled and the others rigidly coupled. n

m

Methods

K au

K av

K aw

K ab

1  105

1  109

1  105

1  109

1  105

1  109

1  105

1  109

0

1

Present ANSYS

607.322 607.744

644.605 644.658

661.005 660.901

661.005 660.902

1003.589 1003.800

657.248 657.143

616.762 616.720

661.000 660.896

1

1

Present ANSYS Present ANSYS

581.405 581.322 1135.614 1137.100

581.464 581.381 1174.331 1175.500

576.810 576.737 1118.867 1120.400

580.270 580.189 1171.503 1172.700

196.808 196.821 582.582 582.500

581.473 581.390 1176.259 1177.200

581.458 581.375 1139.391 1140.400

581.485 581.402 1189.573 1190.600

Present ANSYS Present ANSYS

898.302 897.769 1929.984 1931.100

902.587 902.068 1963.679 1964.400

865.105 864.679 1777.048 1777.900

881.791 881.330 1864.288 1865.100

522.732 522.602 905.109 904.594

903.973 903.458 1931.394 1931.900

903.615 903.100 1915.811 1916.300

903.990 903.475 1972.329 1972.900

Present ANSYS Present ANSYS

725.144 724.386 1447.443 1446.000

727.718 726.959 1460.601 1459.200

726.817 726.064 1436.305 1435.000

727.265 726.511 1443.869 1442.500

726.534 725.789 1056.451 1056.100

728.510 724.749 1464.525 1463.100

728.027 727.270 1463.314 1461.900

728.534 727.772 1464.575 1463.200

Present ANSYS Present ANSYS

1057.564 1057.100 1389.682 1388.000

1059.357 1058.900 1398.798 1391.100

1059.873 1059.400 1398.854 1397.200

1059.874 1059.400 1399.319 1397.600

1059.010 1058.500 1396.105 1394.500

1059.851 1059.400 1401.310 1399.600

1059.476 1059.000 1400.101 1398.400

1059.877 1059.400 1401.404 1399.700

Present ANSYS Present ANSYS

1636.330 1636.200 1806.244 1804.900

1638.014 1637.800 1814.083 1812.700

1638.427 1638.300 1816.016 1814.700

1638.430 1638.300 1816.031 1814.700

1637.951 1637.800 1813.813 1812.500

1638.424 1638.300 1816.021 1814.700

1638.165 1638.000 1815.060 1813.700

1638.450 1638.300 1816.126 1814.800

2 2

1 2

3

1 2

4

1 2

5

1 2

Fig. 4. Effects of the axial location of annular plate on natural frequencies: (a) m = 1, (b) m = 2.

3.1.2. Effects of annular plates on free vibrations Fig. 4 shows effects of the axial location of annular plate on natural frequencies. As the annular plate shifts from the clamped end

to the free end, natural frequencies of n = 0 and n = 1 slightly decrease for both m = 1 and m = 2. However, for n = 2 and n = 3, tendencies of natural frequencies versus axial locations depend

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

on m. To be specific, one peak around 0.38 appears for m = 1 while two peaks around 0.25 and 0.42 appear for m = 2. In addition, two peaks of n = 2 and m = 2 are very flat. Comparing curves in Fig. 4 (a) and (b), it can be further observed natural frequencies of the modes of n = 2, m = 1 and n = 3, m = 1 reach the peaks while the natural frequencies of the modes of n = 2, m = 2 and n = 3, m = 2 are around the valleys. As a result, selecting an appropriate axial location to make all natural frequencies increased is impossible. Fig. 5 presents effects of the inner radius of annular plate on natural frequencies. In the figure, the inner radius of annular plate increases from 0 to R, which means the end circular plate gradually disappears. It is found that increasing the inner radius can make natural frequencies of n = 0 and n = 1 increased, while natural frequencies of n = 2 and n = 3 overall decrease as the inner radius increases. However, for n = 2 and m = 2, small fluctuations appear around R1 = 0.65, which may result from the strong coupling effects of the plate and shell. It is further observed that variations of natural frequencies of n = 0 are significantly larger than the others since the modes of n = 0 are dominated by the end plate. Fig. 6 shows effects of the number of annular plates on natural frequencies. Besides the end annular plate, the other annular plates are uniformly spaced along axial direction. As the number of annular plates varies from 1 to 2, the increase of natural frequencies is obvious for n = 2–5 and negligible for other circumferential mode numbers, especially for the modes of m = 2. On the other hand, effects of annular plates become more significant as 4 annular plates are accounted for. On the whole, increasing the number of annular plates can make the natural frequencies increased.

3.1.3. Effects of elastic coupling conditions on free vibrations It is pointed out in Section 3.1.1 that elastic coupling conditions have significant effect on free vibrations of the coupled structure,

and the effect is further investigated in this subsection. Fig. 7 presents curves of natural frequencies versus stiffness constants. In the figure, one group of displacements is elastically coupled and the others are rigidly coupled, which is the same with the coupling conditions in Table 3. Except for some natural frequencies of n = 0, the increase of stiffness constant leads to the increase of natural frequencies as the values of stiffness constants are in appropriate range. However, for n = 0, curves of natural frequencies versus stiffness constant K aw fluctuate around 107 , and the overall tendencies are downward. To analyze the reason why the tendencies of K aw of n = 0 are obviously different from others, some mode shapes are shown in Fig. 8. For these mode shapes, deformations of two meridional lines are considered, whereas the variations in circumferential direction are not given. Circumferential interval of the two meridional lines is 180°, and appropriate circumferential coordinates are selected, especially for n = 1, so that the maximum deformations of the coupled structure can be shown. The dashed lines stand for original shapes of the coupled structure and the solid lines are deformed shapes. In addition, only normal deformations are considered, which results in obvious discontinuities at the junction between the shell and plate. It can be clearly found that the increase of stiffness constant results in variations of mode shapes. In Fig. 8, some mode shapes of n = 1 are also shown. It can be further observed that the mode shapes of m = 2 with small stiffness constants are similar with the mode shapes of m = 1 with large stiffness constants, which leads to the result that the curve of m = 2 of K aw almost coincides with some curves of m = 1 as stiffness constant is small. Generally, K aw has the greatest influence on free vibration characteristics while effects of K au and K av become obvious as n and m increase. As a result, assigning in-plane displacements of the plate at the junction as 0 can lead to obvious errors of natural frequencies, however, the method was used in Refs. [4,15].

Fig. 5. Effects of the inner radius of annular plate on natural frequencies: (a) m = 1, (b) m = 2.

Fig. 6. Effects of the number of annular plates on frequencies: (a) m = 1, (b) m = 2.

K. Xie et al. / Applied Acoustics 123 (2017) 107–122

115

Fig. 7. Effects of coupling stiffness constants on frequencies: (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 3.

Fig. 8. Mode shapes (normal displacements) of different stiffness constant K aw .

3.1.4. Effects of elastic boundary conditions on free vibrations Effects of elastic boundary conditions on natural frequencies are presented in Fig. 9. For each curve, only one displacement is elastically restrained and the other three are fixed. In general, tendencies of the curves are similar with elastic coupling conditions.

and it is not repeated. It is further observed that natural frequencies are mainly affected by K cu .

Furthermore, obvious fluctuations appear around 108 for curves of K cu in Fig. 9(a), the reason of which is the variations of mode shapes of different stiffness constants, as shown in Fig. 10. The detailed meaning of these mode shapes is the same with Fig. 8

In current section, forced vibrations of the coupled structure are investigated. Based on the model discussed above, one more annular plate located at the intermediate section ðX ¼ L=2Þ is considered. The annular plates are rigidly coupled with shell. By

3.2. Forced vibration analysis

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

Fig. 9. Effects of boundary stiffness constants on frequencies: (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 3.

introducing complex Young’s modulus, E ¼ Eð1  jgs Þ, structure damping gs is taken into account and it equals 0.01. Two external point forces in meridional directions are located at ðR1 ; 0; LÞ and ðR1 ; 0; L=2Þ, respectively, and they are defined as F1 and F2. Normal displacement responses of three points on the shell, Point A ðR; 0; 3L=4Þ, Point B ðR; 0; L=2Þ and Point C ðR; 0; L=4Þ, are examined to discuss forced vibration characteristics. Amplitudes of the

external excitations are 1 N, range of analysis frequency varies from 500 Hz to 4000 Hz and the corresponding step is 5 Hz. 3.2.1. Convergence and validity analysis for forced vibrations First of all, convergence of present method for forced vibrations is discussed. Fig. 11 compares frequency responses of the driving point of three different truncated circumferential mode numbers

Fig. 10. Mode shapes (normal displacements) of different stiffness constant K cu .

Fig. 11. Convergence analysis of frequency responses of the driving point as F1 considered: (a) Meridional displacement, (b) normal displacement.

K. Xie et al. / Applied Acoustics 123 (2017) 107–122

as F1 considered. It is observed that the meridional displacement responses converge rapidly as the truncated number increases, while the increase of the truncated number from 5 to 10 to 15 has no influence on the normal displacement responses. Correspondingly, the truncated circumferential number is assigned as 10 in the following analysis. To validate the accuracy of present method for forced vibrations, a new finite element model is developed in ANSYS. Based on the finite element model in Section 3.1.1, an annular plate located at X = L/2 is added, and the elements of the additional plate are the same with the ones of end annular plate. That is to say, the shell and two annular plates are still composed of 160  160 and 160  20 Shell63 elements, respectively. The connection type between the shell and plates is the same with the one stated in Section 3.1.1 and it is not repeated. Structural damping and corresponding excitation are considered before calculation. Since the mesh generation method of two finite element models is consistent, the convergence of the new model can be guaranteed. Figs. 12 and 13 show comparisons of frequency responses of present method and ANSYS as F1 and F2 are considered, respectively. It is observed that, for the two excitations, the curves of present method and ANSYS almost completely overlap, which demonstrates high accuracy of present method for forced vibration analysis.

117

3.2.2. Effects of external force on forced vibrations From Figs. 12 and 13 it can be found that the difference of frequency responses of two excitations is obvious. To clearly discuss influences of external forces, frequency responses of the coupled structure under F1 and F2 are plotted in Fig. 14. It is observed that responses of the coupled structure under F1 are greater than those of F2 at low frequencies, which is attributed to the fact that the excitation is forced at free edge of the end annular plate and more power can be generated by the external force as the beam mode (n = m = 1) is excited. At high frequencies, responses of the coupled structure under F2 become larger. It is further observed that some minor resonant peaks appear as F2 is forced on the structure, which may result from natural frequencies dominated by the intermediate annular plate. Effects of the direction of F1, in meriondional and normal direction, on forced vibrations are presented in Fig. 15. It is found that, excepting the first resonant peak, frequency responses of the normal force are greater than those of the meridional force. For the first peak, beam mode (n = m = 1) is excited and the coupled structure vibrates like a cantilever beam, which explains the reason why responses of the meridional force are larger. As frequency increases, due to the smaller out-plane bending stiffness, bending modes of the annular plate can be easily excited by the normal force. Consequently, more vibration energy can be transmitted to the shell through the plate-shell junction.

Fig. 12. Comparisons of frequency responses of present method and ANSYS as F1 considered: (a) Point A, (b) Point B, (c) Point C.

Fig. 13. Comparisons of frequency responses of present method and ANSYS as F2 considered: (a) Point A, (b) Point B, (c) Point C.

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

Fig. 14. Effects of the location of external force on frequency responses: (a) Point A, (b) Point B, (c) Point C.

Fig. 15. Effects of directions of F1 on frequency responses: (a) Point A, (b) Point B, (c) Point C.

Fig. 16. Effects of coupling stiffness constants Kau and Kav on frequency responses as F1 in meridional direction: (a) Point A, (b) Point B, (c) Point C.

3.2.3. Effects of elastic coupling conditions on forced vibrations The limit values of some stiffness constants are selected to investigate effects of elastic coupling conditions on forced vibrations. As F1 is in meridional direction, comparisons of vibration responses are presented in Figs. 16 and 17. In the figures, only one stiffness constant is set as 0 and the others are very large (1016 ), which means that three groups of displacements are rigidly coupled and the other one is uncoupled. It is observed that setting one group of displacements uncoupled has negligible effect on

vibration responses at very low frequencies, but the effect becomes obvious as frequency increases. In contrast with the completely coupled structure, more resonant peaks are excited if one group of displacements is not coupled. It is further observed that effects of meridional and tangential displacements of the annular plate are obviously larger than the others. In addition, if the tangential displacement of the annular is uncoupled with the shell, frequency responses of the shell are larger than the other cases at middle and high frequencies.

K. Xie et al. / Applied Acoustics 123 (2017) 107–122

119

Fig. 17. Effects of coupling stiffness constants Kaw and Kab on frequency responses as F1 in meridional direction: (a) Point A, (b) Point B, (c) Point C.

As the direction of F1 is changed from meridional direction to normal direction, influences of elastic coupling conditions are shown in Figs. 18 and 19. It is found that the tendencies of curves in the figures become more complex and irregular, which are obviously different from those of the meridional excitation. That is to say, effects of coupling conditions are more obvious as normal excitation forced on the annular plate. When normal and tangential displacements of the plate are uncoupled with the shell, larger

amplitudes of resonant peaks of the shell can be obtained. Furthermore, vibration responses of the structure with tangential displacement uncoupled are the largest at high frequencies. On the basis of the above results about coupling conditions on forced vibrations, it can be concluded that setting the tangential displacements at the plate-shell junction uncoupled can significantly increase vibration responses of the shell as the excitation forced on the annular plate.

Fig. 18. Effects of coupling stiffness constants Kau and Kav on frequency responses as F1 in normal direction: (a) Point A, (b) Point B, (c) Point C.

Fig. 19. Effects of coupling stiffness constants Kaw and Kab on frequency responses as F1 in normal direction: (a) Point A, (b) Point B, (c) Point C.

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

Fig. 20. Effects of structure damping on frequency responses as external force F1 considered: (a) Point A, (b) Point B, (c) Point C.

3.2.4. Effects of structure damping on forced vibrations Fig. 20 presents influences of structure damping on responses. It is observed that amplitudes of resonant peaks are significantly reduced by the damping while frequencies of those peaks keep unchanged, which is attributed to the small damping. Nevertheless, the influences are negligible in the non-resonant region. 4. Conclusions An analytic method (wave based method) is presented to investigate free and forced vibrations of elastically coupled thin annular plate and cylindrical shell structures with arbitrary boundary conditions. To establish the governing equation, the method is involved in dividing the coupled structure into cylindrical shell segments and annular plates. Flügge shell theory and thin plate theory are respectively utilized to describe motion equations of segments and plates. Regardless of boundary and continuity conditions, displacements of individual members are expressed as different forms of wave functions. Meanwhile, artificial springs are employed to restrain displacements at edges and to combine annular plates with the shell at any axial locations. Through elastic boundary and coupling conditions, the governing equation for free and forced vibration analysis of the coupled structure are established. Before adopting present method to investigate vibration characteristics of the coupled structures, many comparisons of present method and literature and ANSYS are presented, and they demonstrate high accuracy of present method for free and forced vibrations. Meanwhile, the comparative results also illustrate that adopting annular plates with pinhole to model circular plates is an accurate method to deal with circular plates. The axial location of annular plate has significant effect on natural frequencies as circumferential mode number is larger than 1. As the inner radius of annular plate increases, natural frequencies may increase or decrease, which depends on circumferential mode number. Increasing the number of annular plates is an efficient approach to elevate natural frequencies of the coupled structure. Variations of stiffness constants of elastic coupling and boundary conditions affect both mode shapes and corresponding natural frequencies. For elastic coupling conditions, the normal displacement of annular plate has the greatest effect on natural frequencies, while tangential displacement significantly affects frequency responses of the shell as the excitation forced on the annular plate. As a result, neglecting in-plane motions of the plates can lead to big errors. For elastic boundary conditions, meridional (axial) displacement at the edge of shell is the main factor affecting free vibration characteristics. Small structure damping can efficiently reduce amplitudes of resonant peaks whereas frequencies of resonant peaks and ampli-

tudes at non-resonant range are almost not affected by the damping. In contrast with FEM, present method has higher computation efficiency, whereas high accurate free and forced vibration results can be obtained. As compared with analytic approaches in literature, adopting wave functions to express displacements of shells and plates has more definite physical meaning, which can be conveniently used to analyze the wave transmission in the coupled structures. Appendix A At the cross-section of cylindrical shells, force resultants are [31]

M¼

D

"

@2w @2w @v R þ t  @x2 @h2 @h 2

R2

!

# @u R @x

ðA:1Þ

" # 3 3 2 2 2 S ¼  D R3 @ w þ ð2  tÞR @ w  3  t R @ v  R2 @ u þ 1  t @ u @x3 @x2 2 @x@h 2 @h2 @x@h2 R3 ðA:2Þ " # D 1  t @u @v @2w þ ð1 þ 3kÞR T ¼ 3  3kR @x@h 2k @h @x R

ðA:3Þ

" #

2 D 1 @u @v 2@ w N¼ 3 R þt þ w  kR @x @x2 @h R k

ðA:4Þ

3

where D ¼ Eh =12ð1  t2 Þ is the bending stiffness of shell. Detailed expressions of ui ; v i ; wi ; bi ; N i ; T i ;  Si and ði ¼ 1 : 8Þ are

Mi

ui ¼ ni ejki x

ðA:5Þ

v i ¼ gi ejk x

ðA:6Þ

wi ¼ ejki x

ðA:7Þ

bi ¼ jki ejki x

ðA:8Þ

i

i 2 jRgi ki þ tni n þ t þ kR k2i ejki x

ðA:9Þ

D 1t ½gi n þ jð1 þ 3kÞRni ki þ j3kRki nejki x T i ¼ 3 R 2k

ðA:10Þ

Ni ¼

D h kR

2

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122



 Si ¼  D jR3 k3  jRð2  tÞki n2  j 3  t Rni ki n þ g R2 k2  1  t g n ejki x i i i i 2 2 R3

ðA:11Þ

Mi ¼

Dh R

3

i R2 k2i  tn2  tni n  jRgi ki ejki x

ðA:12Þ

Force resultants of annular plates are [26] 3

2

3

2

@ wp 1 @ wp 2  tp @ wp 1 @wp 3  tp @ wp þ þ   r @r2 @r 3 r2 @r@h2 r 3 @r r3 @h2

!

ðB:1Þ

Ep hp @up tp @v p þ þ u Npr ¼ p 1  t2p @r r @h

ðB:2Þ



Ep hp @up @v p þr  vp ¼ 2rð1 þ tp Þ @h @r

ðB:3Þ

Nph

Mp7 ¼ Dp ½C1 In ðkpB rÞ  kpB v2 Inþ1 ðkpB rÞ;

ðB:12Þ

Mp8 ¼ Dp ½C1 K n ðkpB rÞ  kpB v2 K nþ1 ðkpB rÞ where

C1 ¼ v1 þ k2pB ; C2 ¼ v1  k2pB ; C3 ¼ kpB ðrk2pB þ n2 v2 Þ;

Appendix B

Npx ¼ Dp

Mp5 ¼ Dp ½C2 J n ðkpB rÞ  kpB v2 J nþ1 ðkpB rÞ; Mp6 ¼ Dp ½C2 Y n ðkpB rÞ  kpB v2 Y nþ1 ðkpB rÞ

" !# @ 2 wp tp @wp @ 2 wp M p ¼ Dp þ 2 r þ @r 2 r @r @r@h Non-zero expressions of upi ; Mp;i are

v pi ;

C4 ¼ kpB ðrk2pB  n2 v2 Þ

ðB:13Þ

C5 ¼ v3 þ k2pT r; C6 ¼ v1  k2pL r 2

ðB:14Þ

v1 ¼ ð1  tp Þnðn  1Þ=r2 ; v2 ¼ ð1  tp Þ=r; v3 ¼ 2nðn  1Þ=r

ðB:15Þ

Da ¼ Ep hp =½2rð1 þ tp Þ; Db ¼ Ep hp =ð1  t2p Þ

ðB:16Þ

Appendix C Sub-matrixes Kc and Ka in Eq. (24) are

ðB:4Þ wpi ; bpi ; N pxi ; N phi ; N pri and

up1 ¼ nJn ðkpL rÞ=r  kpL J nþ1 ðkpL rÞ; up2 ¼ J n ðkpT rÞ

ðB:5Þ

up3 ¼ nY n ðkpL rÞ=r  kpL Y nþ1 ðkpL rÞ; up4 ¼ Y n ðkpT rÞ

ðC:1Þ

v p1 ¼ Jn ðkpL rÞ; v p2 ¼ nJn ðkpT rÞ=r þ kpT Jnþ1 ðkpT rÞ v p3 ¼ Y n ðkpL rÞ; v p4 ¼ nY n ðkpT rÞ=r þ kpT Y nþ1 ðkpT rÞ

ðB:6Þ

wp5 ¼ J n ðkpB rÞ; wp6 ¼ Y n ðkpB rÞ; wp7 ¼ In ðkpB rÞ; wp8 ¼ K n ðkpB rÞ ðB:7Þ and

bp5 ¼ nJn ðkpB rÞ=r  kpB J nþ1 ðkpB rÞ; bp6 ¼ nY n ðkpB rÞ=r  kpB Y nþ1 ðkpB rÞ bp7 ¼ nIn ðkpB rÞ=r þ kpB Inþ1 ðkpB rÞ; bp8 ¼ nK n ðkpB rÞ=r  kpB K nþ1 ðkpB rÞ ðB:8Þ Npx5 ¼ Dp =r½nC1 J n ðkpB rÞ  C3 J nþ1 ðkpB rÞ; Npx6 ¼ Dp =r½nC1 Y n ðkpB rÞ  C3 Y nþ1 ðkpB rÞ Npx7 ¼ Dp =r½nC2 In ðkpB rÞ  C4 Inþ1 ðkpB rÞ;

ðB:9Þ

Npx8 ¼ Dp =r½nC2 K n ðkpB rÞ þ C4 K nþ1 ðkpB rÞ Nph1 ¼ Da ½v3 J n ðkpL rÞ þ 2nkpL J nþ1 ðkpL rÞ; Nph2 ¼ Da ½C5 J n ðkpT rÞ  2kpT J nþ1 ðkpT rÞ Nph3 ¼ Da ½v3 Y n ðkpL rÞ þ 2nkpL Y nþ1 ðkpL rÞ;

ðB:10Þ

Nph4 ¼ Da ½C5 Y n ðkpT rÞ  2kpT Y nþ1 ðkpT rÞ

ðC:2Þ In Eq. (C.1) and Eq. (C.2),

Npr1 ¼ Db ½C6 J n ðkpL rÞ þ kpL v2 J nþ1 ðkpL rÞ; Npr2 ¼ Db ½v1 J n ðkpT rÞ  kpT nv2 J nþ1 ðkpT rÞ Npr3 ¼ Db ½C6 Y n ðkpL rÞ þ kpL v2 Y nþ1 ðkpL rÞ; Npr4 ¼ Db ½v1 Y n ðkpT rÞ  kpT nv2 Y nþ1 ðkpT rÞ

T ¼ diagðK aw ; K av ; K au ; K ab Þ ðB:11Þ

ðC:3Þ

Sub-matrices of boundary conditions in (C.1) are

Bci

¼ Tc  Dci  Fci

i ¼ 1; P

ðC:4Þ

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K. Xie et al. / Applied Acoustics 123 (2017) 107–122

where c

T ¼

diagðK cu ;

c

Kv ;

K cw ;

K cb Þ

ðC:5Þ

References [1] Leissa AW. Vibration of shells. OH, US: Acoustical Society of America; 1993. [2] Qatu MS. Recent research advances in the dynamic behavior of shells: 1989– 2000, part 1: laminated composite shells. Appl Mech Rev 2002;55(5):325–49. [3] Qatu MS. Recent research advances in the dynamic behavior of shells: 1989– 2000, part 2: homogeneous shells. Appl Mech Rev 2002;55(5):415–34. [4] Smith BL, Haft EE. Vibration of a circular cylindrical shell closed by an elastic plate. AIAA J 1967;5(11):2080–2. [5] Takahashi S, Hirano Y. Vibration of a combination of circular plates and cylindrical shells (1st report, a cylindrical shell with circular plates at ends). Bull Jpn Soc Mech Eng 1970;13(56):240–7. [6] Takahashi S, Hirano Y. Vibration of a combination of circular plates and cylindrical shells (2nd report, a cylindrical shell with circular plate in the intermediate section). Bull Jpn Soc Mech Eng 1971;14(67):20–8. [7] Irie T, Yamada G, Muramoto Y. Free vibration of joined conical-cylindrical shells. J Sound Vib 1984;95(1):31–9. [8] Yamada G, Irie T, Tamiya T. Free vibration of a circular cylindrical shell doubleshell system closed by end plates. J Sound Vib 1986;108(2):297–304. [9] Liang S, Chen HL. The natural vibration of a conical shell with an annular end plates. J Sound Vib 2006;294:927–43. [10] Tavakoli MS, Singh R. Eigensolutions of joined/hermetic shell structures using the state space method. J Sound Vib 1989;100(1):97–123. [11] Tavakoli MS, Singh R. Modal analysis of a hermertic can. J Sound Vib 1990;136 (1):141–51. [12] Huang DT, Soedel W. Natural frequencies and modes of a circular plate welded to a circular cylindrical shell at arbitrary axial positions. J Sound Vib 1993;162 (3):403–27. [13] Huang DT, Soedel W. On the free vibrations of multiple plates welded to a cylindrical shell with special attention to mode pairs. J Sound Vib 1993;162 (2):315–39. [14] Yim JS, Sohn DS, Lee YS. Free vibration of clamped-free circular cylindrical shell with a plate attached at arbitrary axial position. J Sound Vib 1998;213 (1):75–88. [15] Cheng L, Nicolas J. Free vibration analysis of a cylindrical shell-circular plate system with general coupling and various boundary conditions. J Sound Vib 1992;155(2):231–47.

[16] Yuan J, Dickinson SM. The free vibration of circularly cylindrical shell and plate systems. J Sound Vib 1994;175(2):241–63. [17] Shakouri M, Kouchakzadeh MA. Free vibration analysis of joined conical shells: analytical and experimental study. Thin-Walled Struct 2014;85:350–8. [18] Ma XL, Jin GY, Shi SX, Ye TG, Liu ZG. An analytic method for vibration analysis of cylindrical shells coupled with annular plate under general elastic boundary and coupling conditions. J Vib Control 2017;23(2):305–28. [19] Joseph A, Ganesan N. Frequency response of shell-plate combinations. Comput Struct 1996;59(6):1083–94. [20] Ozakca M, Hinton E. Free vibration analysis and optimization of axisymmetric plates and shells-I. Finite element formulation. Comput Struct 1994;52(6). 1181-1107. [21] Benjeddou A. Vibrations of complex shells of revolution using B-spline finite elements. Comput Struct 2000;74:429–40. [22] Jin GY, Ma XL, Shi SX, Ye TG, Liu ZG. A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions. Appl Acoust 2014;85:82–96. [23] Su Z, Jin GY, Shi SX, Ye TG, Jia XZ. A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions. Int J Mech Sci 2014;80:62–80. [24] Jin GY, Yang CM, Liu ZG, Gao SY, Zhang CY. A unified method for the vibration and damping analysis of constrained layer damping cylindrical shells with arbitrary boundary conditions. Compos Struct 2015;130:124–42. [25] Zhang CY, Jin GY, Ma XL, Ye TG. Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions. Appl Acoust 2016;110:176–93. [26] Tso YK, Hansen CH. Wave propagation through cylindrical/plate junctions. J Sound Vib 1995;186(3):447–61. [27] Wei JH, Chen MX, Hou GX, Xie K, Deng NQ. Wave based method for free vibration analysis of cylindrical shell with non-uniform stiffener distribution. J Vib Acoust 2013;135(6):061011. [28] Chen MX, Wei JH, Xie K, Deng NQ. Wave based method for free vibration analysis of ring stiffened cylindrical shell with intermediate large frame ribs. Shock Vib 2013;20:459–79. [29] Chen MX, Xie K, Xu K, Yu P. Wave based method for free and forced vibration analysis of cylindrical shells with discontinuity in thickness. J Vib Acoust 2015;137(5):051004. [30] Lee HK, Kwak MK. Free vibration analysis of a circular cylindrical shell using the Rayleigh-Ritz method and comparison of different shell theories. J Sound Vib 2015;353:344–77. [31] Flügge W. Stress in shells. Berlin, Germany: Springer; 1973.