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JOURNAL OFSOUND ANDVIBRATION

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Journal of Sound and Vibration 314 (2008) 330–342

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A study on free vibration of a ring-stiffened thin circularcylindrical shell with arbitrary boundary conditions

Zhi Pan�, Xuebin Li, Janjun Ma

Wuhan 2nd Ship Design & Research Institute, Hubei Province, PR China

Received 1 July 2007; received in revised form 18 December 2007; accepted 4 January 2008

Handling Editor: L.G. Tham

Available online 4 March 2008

Abstract

The vibration of ring-stiffened cylinders associated with arbitrary boundary conditions is investigated. Displacements of

cylinders can be easily described by trigonometric functions when the cylinders are shear diaphragms supported at both

ends. As to other boundary conditions, exponential functions are used and axial factors are introduced. An eighth-order

algebraic equation for this axial factor is derived. The physical meaning of the axial factor is studied. Both analytical and

numerical studies prove that, when the axial factor is a pure imaginary number, the cylinder appears to have a certain

length with shear diaphragm boundary conditions. The effects of shell parameters and hydrostatic pressure on the axial

factor are determined in the analysis.

r 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The use of cylindrical shells is common in many fields, such as aerospace, mechanical, civil and marineengineering structures. To effectively enhance the flexural and axial stiffness, stiffeners are frequently used.Many engineers have considerable interest in the free vibration of thin circular cylindrical shells and numerousinvestigations have been devoted to this, e.g. Ref. [1–6]. The methods used to study the vibrational behavior ofthin shells range from energy methods based on the Rayleigh–Ritz procedure to analytical methods in which,respectively, closed–form solutions of the governing equations and iterative solution approaches were used.The circular cylindrical shell supported at both ends by shear diaphragms (SD–SD) has received the mostattention in the literatures. This is due to the fact that one simple form of the solutions to the eighth-orderdifferential equations of motion is capable of satisfying the SD–SD boundary conditions exactly. Thefrequency equation is a polynomial of order six in the shell frequency parameter. When solving problems witharbitrary boundary conditions, many authors [7–11] select general solutions of shell equations. They havechosen exponential functions for the modal displacements along the axial direction, substituted them into theequations of motion and then enforced the eighth-order frequency determinants which are coupled together.

ee front matter r 2008 Elsevier Ltd. All rights reserved.

v.2008.01.008

ing author.

ess: pan_zhi@yahoo.com.cn (Z. Pan).

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Nomenclature

A2 sectional area of the stiffenerB stretching stiffness of the shell, B ¼ Eh/

1�m2

br width of the rectangular stiffenerD bending stiffness of the shell, D ¼ Eh3/

12(1�m2)d2 stiffener spacingE modulus of Young’s elasticitye2 eccentricity of the centroid of the ring

stiffener sectionG shear modulush thickness of the shellh h+A2/d2hr height of the rectangular stiffenerI0 sectional moment of inertia of the stiffen-

er about the centroid of the stiffenerI2 sectional moment of inertia of the stiffen-

er about the middle surface of the shell

iffiffiffiffiffiffiffi�1p

J2 torsion constant of stiffener cross-sectionk h2/12R2

L length of the shell between endsm axial mode numbern number of circumferential wavesQ hydrostatic pressureR radius of the mean surface of the shellt timex, y, z cylindrical coordinates (Fig. 1)Un, Vn, Wn amplitudes of the displacement in the

x, y and z directionsu, v, w components of the displacement in the x,

y and z directionsl axial factorm Poisson’s ratior mass densityo circular frequency

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 331

Recently, many new approaches were developed for cylinder shell for free vibration analysis ofcylindrical and conical shells. Xiang et al. [12] use the state-space technique to derive the homogenousdifferential equation system for a shell segment and a domain decomposition approach is developed tocater to the continuity requirements between shell segments. Bingen Yang and Jianping Zhout [13] haveused the distributed transfer functions of the structural components, and then various static and dynamicproblems of stiffened shells have been systematically formulated. With this transfer function formulation,the static and dynamic response, natural frequencies and mode shapes and buckling loads of generalstiffened cylindrical shells under arbitrary external excitations and boundary conditions can bedetermined in exact and closed forms. Civalek Omer [14] proposes a discrete singular convolution methodfor the free vibration analysis of rotating conical shells. Frequency parameters of the forward modesare obtained for different types of boundary conditions, rotating velocity and geometric parameters.A wave propagation approach [15–18] is also developed to predict the natural frequency of thecylinders. This method combines an exact frequency wavenumber characteristic formula with appropriatebeam functions in the axial direction to give predictions of natural frequencies of circular cylindricalshells.

In the general solutions of shell equations, axial factors are usually introduced. But few literatureshave discussed the physical meanings of the axial factors. It has been numerically proved that the modesassociated with pure imaginary axial factors conform to the shear diaphragm end conditions of a finiteshell of length [19]. The analysis in this paper complements the author’s previous work [19] in the sensethat it investigates the physical meaning of the axial factors both numerically and analytically, and the effectsof the shells’ parameters and hydrostatic pressure are also studied. In this paper, a general solution usingexponential functions for the modal displacements along both the axial and the circumferential directions ispresented.

The investigation into the problem of the cylinder stiffened by ring stiffeners uses a ‘‘smeared frame’’approach, and represents the discretely reinforced cylinder by a uniformly orthotropic cylinder. When thestiffener spacing increases or becomes irregular or if the wavelength of the eigenmode becomes smaller thanthe stiffener spacing, this approach is no longer useful. The classical linear shell theory of Flugge [3] is adoptedin the present analysis.

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2. Exact solution of motion equations

The stiffeners are not considered as discrete members, but their effects are averaged or ‘‘smearedout’’ over the shell. Shell equations of motion for a thin-walled circular cylinder, stiffened by evenlyspaced uniform rings (Fig. 1) under hydrostatic pressure, are given below according to Flugge’s shelltheory:

qNx

qxþ

qNyx

Rqy�Q

q2u

Rqy2þ

qw

qx

� ��

QR

2

q2u

qx2¼ rh

q2u

qt2,

qNxy

qxþ

qNy

Rqy�

1

R

qMy

Rqyþ

qMxy

qx

� ��Q

q2v

Rqy2�

qw

Rqy

� ��

QR

2

q2v

qx2¼ rh

q2vqt2

,

q2Mx

qx2þ

q2Mxy

Rqxqyþ

q2Myx

Rqxqyþ

q2My

R2qy2þ

Ny

RþQ

qu

qx�

qv

Rqy�

q2w

Rqy2

� ��

QR

2

q2w

qx2

� �¼ rh

q2w

qt2. (1)

The forces and moments are given in the Appendix.The general solution for the equations can be written in the following form:

ðunðx; yÞ; vnðx; yÞ; wnðx; yÞÞ ¼ ðUn; V n; W nÞelx=Reinyeiot; (2)

where the axial factor l is a complex number.Substituting Eq. (2) into Eq. (1), it can be written as

½Ljk�fUn; V n; W ngT ¼ f0g; j; k ¼ 1; 2; 3. (3)

The elements of the coefficient matrix Ljk are given in the Appendix.The nondimensional circular frequency and hydrostatic pressure used in Eq. (3) are given as

O2 ¼r 1� m2� �

R2

Eo2, (4)

Q0 ¼QR 1� m2� �Eh

. (5)

For the unstiffened cylinder case, the coefficient matrix of Eq. (3) is symmetrical. The coefficient matrix thatcontains the influence of hydrostatic pressure is always antisymmetric.

For nontrivial solutions of Eq. (3), one requires

detðLjkÞ ¼ 0; j; k ¼ 1; 2; 3. (6)

The determinant can be expressed as

F ðo; n; l; Q0Þ ¼ 0. (7)

Assuming a known vibration frequency and a number of circumferential waves n, we can obtain an eighth-order algebraic equation for l:

P8l8þ P6l

6þ P4l

4þ P2l

2þ P0 ¼ 0. (8)

hr

br

Rh

e2

d2

z

x

z yx R

L

Fig. 1. Ring-stiffened circular cylindrical shell.

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The roots of Eq. (8) usually have the form [6]:

l ¼ �k1 � ik2; �k3;�ik4 (9)

where kj (j ¼ 1, 2, 3, 4) are real quantities.Analytical solutions for Eq. (8) can be written as

ðl1;2Þ2¼ G16 �

ffiffiffiffiffiffiL1

p

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �

L3

4ffiffiffiffiffiffiL1

p

s;

ðl3;4Þ2¼ G16 �

ffiffiffiffiffiffiL1

p

2þ

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �

L3

4ffiffiffiffiffiffiL1

p

s,

ðl5;6Þ2¼ G16 þ

ffiffiffiffiffiffiL1

p

2�

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ

L3

4ffiffiffiffiffiffiL1

p

s;

ðl7;8Þ2¼ G16 þ

ffiffiffiffiffiffiL1

p

2þ

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ

L3

4ffiffiffiffiffiffiL1

p

s, (10)

where

G1 ¼ P24 � 3P2P6 þ 12P0P8,

G2 ¼ 2P34 � 9P2P4P6 þ 27P0P

26 þ 27P2

2P8 � 72P0P4P8,

G3 ¼ ð�4G31 þ G2

2Þ1=2; G4 ¼ G2 þ G3,

G5 ¼ G1=34 ; G6 ¼ 21=3G1,

G7 ¼ P26=4P2

8; G8 ¼ G5=3P821=3,

G9 ¼ G6=3G1=34 P8; G10 ¼ �8P2=P8,

G11 ¼ P36=P3

8; G12 ¼ 4P4P6=P28,

G13 ¼ � 2P4=3P8; G14 ¼ P26=2P2

8,

G15 ¼ � 1=3P821=3; G16 ¼ �P6=4P8,

G17 ¼ � 4P4=3P8; G18 ¼ �G6=3G1=34 P8,

L1 ¼ G13 þ G7 þ G8 þ G9; L2 ¼ G14 þ G17 þ G18 þ G15G5,

L3 ¼ G10 � G11 þ G12.

3. Numerical examples

An examples is studied in the present analysis. The assumptions for the basic parameters of shells andstiffeners are [20]: h ¼ 0.119� 10�2m, R ¼ 10.37� 10�2m, d2 ¼ 3.14� 10�2m, br ¼ 0.218� 10�2m, hr ¼

0.291� 10�2m, and m ¼ 0.3, E ¼ 2.06� 1011N/m2.The roots of Eq. (8) are computed for the following five cases:

(1)

an unstiffened shell with O variation for Q0 ¼ 0 and n ¼ 3. (2) an unstiffened shell with O variation for Q0 ¼ 0 and n ¼ 9. (3) an internal-ring stiffened shell with O variation for Q0 ¼ 0.008 and n ¼ 3. (4) an unstiffened shell with O variation for Q0 ¼ 0.008 and n ¼ 9. (5) an internal-ring stiffened shell with O variation for Q0 ¼ 0.008 and n ¼ 9.

The results from Eq. (10) are shown in Figs. 2–6 for nondimensional frequencies. At any frequency, fourdistinctly different values of l are seen to exist, although only two may be seen on the figures. This is due to

ARTICLE IN PRESS

41,2

Ω2

-3

3

6

9

12

15

18

21

0-π/2

1,21

2

3

4

3

1,2,4

A

B

Am

plitu

deP

hase

1 2 3 4 5 6

Fig. 3. Roots of Eq. (8) for unstiffened cylinders (n ¼ 9, Q ¼ 0).

0.5

Am

plitu

de

4

31,2

-3

3

6

9

12

15

18

0-π/2

Pha

se

A

1,2

1

23

4

1,2,4

Ω2

1.0 1.5 2.0

Fig. 2. Roots of Eq. (8) for unstiffened cylinders (n ¼ 3, Q ¼ 0).

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342334

curves 1, 2 and 3, 4 representing coincident complex conjugate pairs. Each curve has its negative counterpart,meaning that eight l exist altogether at any frequency. The phrases are �p/2 and 0 indicate that l is pureimaginary or real, respectively.

It can be seen in the figures that curves 1 and 2 always translate to two curves representing pure real roots atpoint A from coincident complex conjugate pairs. Curves 3 and 4 always represent pure imaginary roots andreal roots, respectively, over the frequency ranges calculated, except in Fig. 3. Fig. 3 shows that curves 3 and 4represent coincident complex conjugate pairs in the very low frequency range, and translate to curves 3 and 4at point B.

A further comparison made between Figs. 3 and 5 shows that hydrostatic pressure delays the appearance ofbifucation point A, and increases the amplitude of the pure imaginary roots markedly. Hydrostatic pressuredoes not effect curves 1, 2, 4 very much. Examination of Figs. 2 and 3 shows that point A appears much laterin high-order vibration than in low-order vibration. The stiffeners affect mainly the initial values of the pureimaginary roots on comparing Fig. 5 with 6.

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1,2 1,23

4

-3

3

6

9

12

15

18

0-π/2

A1,2

1

23

4

1,2,4

Am

plitu

deP

hase

0.5

Ω2

1.0 1.5 2.0

Fig. 4. Roots of Eq. (8) for internal-ring cylinders (n ¼ 3, Q0 ¼ 0.008).

Am

plitu

de

1,2,44

31,2

A

-3

3

6

9

12

15

18

21

0-π/2

Pha

se

1,2

1

2

3

4

Ω2

1 2 3 4 5 6

Fig. 5. Roots of Eq. (8) for unstiffened cylinders (n ¼ 9, Q0 ¼ 0.008).

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 335

3.1. The physical meaning of pure imaginary roots

The above calculations show that Eq. (8) has the pure imaginary roots as 7ils for certain combinations of(o, n, Q0). Substituting l ¼7ils into Eq. (8), Eq. (8) can be rewritten as

P8l8s � P6l

6s þ P4l

4s � P2l

2s þ P0 ¼ 0. (11)

In order to investigate the physical meaning of the pure imaginary roots, the classical separation of variables,method for determining the natural frequencies of cylinders with shear diaphragm boundary conditions isdeduced here. A comparison will be made between two approaches. Suppose that the cylinders supported atboth ends by shear diaphragms (SD–SD) have a certain length L for classical analysis. The shear diaphragmboundary conditions can be closely approximated in physical application simply by means of rigidly attachinga thin, flat, circular cover plate at each end [6].

ARTICLE IN PRESS

4

1,2

A

-3

3

6

9

12

15

18

21

0-π/2

Pha

se

1,2

1

2

4

3

1,2,4

3

Am

plitu

de

Ω2

1 2 3 4 5 6

Fig. 6. Roots of Eq. (8) for internal-ring cylinders (n ¼ 9, Q0 ¼ 0.008).

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342336

Consider the closed circular cylindrical shell of finite length L which satisfies the boundary conditions

v ¼ w ¼ Nx ¼Mx ¼ 0 at ends. (12)

Generally, deformation functions for the boundary conditions are given as follows:

umn x; yð Þ ¼ Umn cos nyð Þcos lclxð Þcosoclt,

vmn x; yð Þ ¼ V mn sin nyð Þsin lclxð Þcosoclt,

wmn x; yð Þ ¼W mn cos nyð Þsin lclxð Þcosoclt, (13)

where lcl ¼ mpR/L.

Substituting Eq. (13) into Eq. (1), a set of equations similar to Eq. (3) is obtained. For nontrivial solution ofequations, the determinant of the coefficient matrix must be zero. This leads to an equation with variablesðocl;m; n; lcl;Q0clÞ as follows:

F 2 ocl;m; n; lcl;Q0cl� �

¼ 0. (14)

Eq. (14) is a cubic equation with variable vibration frequency o2cl when lcl ¼ mpR/L is given. Each

combination of m and n has three roots of frequency. The minimum root among the roots for all combinationsof m and n is namely the basic frequency of the cylinder.

Eq. (14) can also be written as an eighth-order algebraic equation for lcl:

P08l8cl þ P06l

6cl þ P04l

4cl þ P02l

2cl þ P00 ¼ 0. (15)

A comparison was made between fP8;P6;P4;P2;P0g and P08;P06;P04;P02;P00

� �; it is interesting to note that:

P08 ¼ P8; P06 ¼ �P6; P04 ¼ P4; P02 ¼ P2; P00 ¼ P0. (16)

These findings appear to prove that Eqs. (11) and (15) are the same. This reasoning certainly seems to offer amathematical explanation for the conclusion that, when the axial factor is a pure imaginary number, thegeneral solution, Eq. (2), for the shell equations appears to describe the vibration mode of a cylinder with acertain length and shear diaphragm boundary conditions. The length of the cylinder can be obtained from

lsj j ¼mpR

L. (17)

After a pure imaginary root is obtained from Eq. (8), the length (L/R) of the cylinder with shear diaphragmssupported at ends is determined according to Eq. (17). Then the, corresponding natural frequency ocl fromEq. (14) is compared with the given frequency o, which is given before Eq. (8), and is resolved. Incidentally,

ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 337

if the cylinder is shear diaphragm supported, the length of the cylinder equals to m multiplies the half-wavelength in the axial direction; hence, only m ¼ 1 is discussed in the paper.

A numerical example is given for comparison in Table 1. Good agreement can be seen between the approachpresented and the classical method for cylinders with shear diaphragm boundary conditions on comparing thesecond column with the fifth column.

3.2. Effects of Stiffeners on ls

The effects of stiffeners on pure imaginary roots are studied here. The symmetrical-ring stiffened cylinder isintroduced here to make a comparison. The eccentricity of the centroid of ring stiffener section e2 equals tozero when the ring is symmetrical.

Fig. 7 shows that the values of |ls| increase along with an increase of frequency. The starting points of thepure imaginary roots in Fig. 7 are the bifurcation points of the complex conjugate pairs. In order to comparethe difference of the curves in Fig. 7 clearly, curves of differences based on internal-ring cylinder are shown inFig. 8. The difference is defined as

Diffll2;3s

� l1s

l1s

� 100%, (18)

Table 1

Comparison with classical method

Present approach (Eq. (7)) Classical method (Eq. (14))

O |ls| Lcl(� 10�2) Ocl

n ¼ 3

0.1 1.0362 31.44018 0.10000

0.5 3.2444 10.04137 0.50000

1.0 11.3558 2.86886 1.00000

1.5 18.6236 1.75129 1.49792

Unstiffened shell, Q0 ¼ 0, h ¼ 0.119� 10�2m, R ¼ 10.37� 10�2m, E ¼ 2.06� 1011N/m2, m ¼ 0.3.

0.20

2

4

6

8

10

hr = 0.582×10-2mn = 2

Ω

internal-ringexternal-ring symmetrical-ring

| λ s

|

0.4 0.6 0.8 1.0

Fig. 7. Effects of stiffeners’ locations (Q ¼ 0).

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-30

-20

-10

0

10

20external-ring symmetrical-ring

hr = 0.582×10-2mn = 2

Diff

λ

0.2Ω

0.4 0.6 0.8 1.0

Fig. 8. Curves of difference in stiffeners’ locations.

0

2

4

6

8

| λ s

|

hr= 0.291×10-2mhr= 0.582×10-2m

external-ringn=2

0.2Ω

0.4 0.6 0.8 1.0

Fig. 9. Effects of stiffeners’ parameters (Q ¼ 0).

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342338

where l1s corresponds to the internal ring; l2;3s corresponds to the external ring and the symmetrical ring,respectively.

It can be seen in Fig. 8 that |ls| obtained from the external-ring cylinder are greater than thoseobtained from the internal-ring cylinder when Oo0.235, but smaller when O40.235. |ls| obtained from thesymmetrical-ring cylinder are always the smallest in the frequency ranges. This indicates that the naturalfrequency of the symmetrical-ring cylinder is always the highest in the three location forms.

The effects on |ls| of stiffeners’ parameters with no initial pressure(Q0 ¼ 0) are shown in Fig. 9. It can beseen that the stronger the ring, the smaller the |ls|, meaning that the increase of stiffeners’ stiffness heightensthe cylinder’s natural frequency. Fig. 10 shows the difference between |ls| obtained from the stiffeners of twodifferent sizes, and the difference is defined as Eq. (18). It can be seen that the difference fluctuates from largeto small along with a reduction in shell length.

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-20

-15

-10

-5

0

Diff

λ external-ringn=2

0.2Ω

0.4 0.6 0.8 1.0

Fig. 10. Curves of difference in stiffeners’ parameters.

0

5

10

15

20hr= 0.291×10-2mhr= 0.582×10-2m

external-ringn=2

| λ s

|

0.2Ω

0.4 0.6 0.8 1.0

Fig. 11. Effects of stiffeners’ parameters(Q0 ¼ 0.008).

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 339

When the hydrostatic pressure exists(Q0 ¼ 0.008), the effects of stiffeners’ parameters are presented inFig. 11. The hydrostatic pressure increases the value of |ls|, and the increase of stiffeners’ stiffness increase thecylinder’s natural frequency as we found with no hydrostatic pressure. As multiple values appear withincreasing frequency, we cannot obtain the difference line between the two lines in the whole frequency range.Before multiple values appear, the hydrostatic pressure reduces the difference between the two lines in the low-frequency range, but heighten the difference in the high-frequency range.

3.3. Effects of initial stress on ls

In this paper, hydrostatic pressure is considered as the initial stress. Figs. 12 and 13 show theeffects of hydrostatic pressure on the pure imaginary roots. It can be seen that hydrostatic pressure reducesthe natural frequency of cylinders with a certain length. Hydrostatic pressure has much more effectson the vibration behavior when the vibration order is higher. Curves start at zero frequency, which

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0

2

4

6

8

10

| λ s

|

Q′= 0Q′=1.0×10-3

n=3internal-ring

0.2Ω

0.4 0.6 0.8 1.0

Fig. 12. Effects on high-order vibration of hydrostatic pressure (n ¼ 3) (internal-ring, br ¼ 0.218� 10�2m, hr ¼ 0.291� 10�2m).

0.0

2

4

6

8

10

| λ s

|

Ω

Q′= 0Q′=1.0×10-3

n=9internal-ring

0.2 0.4 0.6 0.8 1.0

Fig. 13. Effects on low-order vibration of hydrostatic pressure (n ¼ 9) (internal-ring, br ¼ 0.218� 10�2m, hr ¼ 0.291� 10�2m).

Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342340

indicates that the hydrostatic pressure is just the buckling pressure of the cylinder with shear diaphragmends.

4. Conclusion

When solving the vibration problem of ring-stiffened cylinders associated with arbitrary boundaryconditions, exponential functions are used and axial factors are introduced here. An eighth-order algebraicequation for this axial factor is derived. The physical meaning of the axial factor is studied. Both analyticaland numerical researches prove that when the axial factor is a pure imaginary number, the cylinder appears tohave a certain length with shear diaphragm boundary conditions. The effects of stiffeners’ parameters andhydrostatic pressure on the axial factor are studied.

Hydrostatic pressure increases the module of the pure imaginary roots markedly, but does not affect theother curves very much. The stiffeners affect mainly the initial value of the pure imaginary roots. The natural

ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 341

frequency of the symmetrical-ring stiffened cylinder is always the highest in the three disposal forms.Increasing the stiffeners’ stiffness can increase the cylinder’s natural frequency.

Appendix A. Forces and moments

Eight internal forces including membrane forces Nx, Ny, Nxy and Nyx, bending moments Mx and My andtorsioning moments Mxy and Myx are given

Nx ¼ Bqu

qxþ m

qv

Rqy�

w

R

� � �þ

D

R

q2w

qx2,

Ny ¼ B 1þ m2� � qv

Rqy�

w

R

� �þ m

qu

qx� w2

w

Rþ

q2w

Rqy2

� � ��

D

R

q2w

R2qy2þ

w

R2

� �,

Nyx ¼1� m2

Bqu

Rqyþ

qv

qx

� �þð1� mÞD

2R

qu

R2qy�

q2wRqxqy

�,

Nxy ¼1� m2

Bqu

Rqyþ

qv

qx

� �þð1� mÞD

2R

q2wRqxqy

þqv

Rqx

�,

Mx ¼ D �q2w

qx2� m

q2w

R2qy2�

qu

Rqx� m

qv

R2qy

�,

My ¼ D � 1þ Z2� � w

R2þ

q2w

R2qy2

� �� m

q2wqx2þ z2

qv

R2qy�

w

R2

� � �,

Mxy ¼ 1� mð ÞD �q2w

Rqxqy�

qv

Rqx

� �,

Myx ¼ D 1� mð Þ �q2w

Rqxqyþ

1

2R

qu

Rqy�

qv

qx

� �� �� Zt2

q2wRqxqy

�.

Appendix B. The elements of the coefficient matrix

L11 ¼ l2 þð1� mÞ

2ð1þ kÞðinÞ2 �Q0 ðinÞ2 þ

l2

2

� �þ O2,

L12 ¼ð1� mÞ

2lin,

L13 ¼ � lmþ kl3 �ð1� mÞk

2lðinÞ2 �Q0l,

L21 ¼ð1� mÞ

2lin,

L22 ¼ ð1þ m2ÞðinÞ2� kz2ðinÞ

2þ

1� m2ð1þ 3kÞl2 �Q0 ðinÞ2 þ

l2

2

� �þ O2,

L23 ¼ � ð1þ m2Þinþð3� mÞ

2l2ink þ Z2 inþ ðinÞ3

� k � z2ðinÞ

3k þQ0ðinÞ,

L31 ¼ lm� l3k þð1� mÞ

2lðinÞ2k þQ0l,

L32 ¼ ð1þ m2Þin�ð3� mÞ

2l2ink þ z2ðinÞ

3k �Q0in,

L33 ¼ k �l4 � ð2þ Zt2Þl2ðinÞ2 � ð1þ Z2ÞðinÞ

4� ð2þ 2z2 þ Z2ÞðinÞ

2� ð1þ z2Þ

� ��ð1þ m2Þ �Q0 ðinÞ2 þ

l2

2

� �þ O2,

ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342342

where

m2 ¼EA2

Bd2; w2 ¼

EA2e2

Rd2B; Z2 ¼

EI2

Dd2; Zt2

¼GJ2

Dd2; z2 ¼

12R2w2h2

.

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