Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid

ARTICLE IN PRESS

JID: APM

[m3Gsc;July 6, 2016;11:41]

Applied Mathematical Modelling 0 0 0 (2016) 1–16

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid Tran Ich Thinh∗, Manh Cuong Nguyen Dept. Mechanical Engineering, Hanoi University of Science and Technology, 1 Dai Co Viet Str., Hai Ba Trung Distr., Hanoi, Vietnam

a r t i c l e

i n f o

Article history: Received 12 June 2015 Revised 27 May 2016 Accepted 16 June 2016 Available online xxx Keywords: Free vibration Composite cylindrical shell Dynamic Stiffness Method Fluid–shell interaction

a b s t r a c t The present work deals with a theoretical investigation on free vibration of composite circular cylindrical shells containing fluid. A new precise analytical model using the Dynamic Stiffness Method (DSM) or Continuous Elements (CEM) based on the Reissner–Mindlin theory and non-viscous incompressible fluid equations has been proposed for the studied structures. Numerical examples are given for analyzing natural frequencies and harmonic responses of clamped-free cylindrical shells partially and completely filled with fluid. To compare with the theoretical results, some experimental results have been obtained on the free vibration of a clamped-free glass fiber/polyester cylindrical shells partially filled with water by using a multi-vibration measuring machine (DEWEBOOK-DASYLab 5.61.10). Results calculated by the proposed computational model for studied composite cylindrical shells are in good agreement with experiments. © 2016 Published by Elsevier Inc.

1. Introduction In recent years, the use of partially fluid-filled laminated composite circular cylindrical shells in engineering industry has been steadily increasing. In the design of such structure, dynamic response is still a major concern. The free vibration analysis of the partially fluid-filled composite shells is very useful indeed to better study and understand of the dynamic behavior. Owing to the significance of the problem, a few investigators have carried out the vibration analyses of partially fluid-filled laminated composite circular cylindrical shells. Xi et al. [1,2] analyzed the free vibration of a laminated composite circular cylindrical shell partially filled with fluid using a semi-analytical finite element technique based on the Reissner– Mindlin theory and compressible fluid equations. Sharma et al. [3] have presented analytical solutions to free vibrations of fluid-filled vertical cantilever composite cylindrical shells. Chen and Ding [4] determined the natural frequencies of nonaxisymmetric vibrations of an anisotropic elastic spherical shell filled in a compressible fluid medium. Katsutoshi et al. [5] analyzed the free vibrations of a laminated composite circular cylindrical shell partially filled with liquid. Toorani and Lakis [6] studied the effect of shear deformation in the dynamic behavior of anisotropic laminated open cylindrical shells filled with fluid and later they investigated the dynamic behavior of axisymmetric and beam-like anisotropic cylindrical fluid-filled shells (Toorani and Lakis [7]). Using linearized boundary conditions of fluid free surface, Pal et al. [8] made a study on the sloshing dynamics in a fluid-filled laminated composite open cylindrical tank and later accomplished their work assuming nonlinear free surface boundary conditions using Finite Element Method (Pal et al. [9]). Amabili [10,11] has performed a non-linear vibration analysis of fluid filled cylindrical shells. A semi-analytical approach has been utilized by

∗

Corresponding author. Tel.: +84 4 38692775; fax: +84 4 38681655. E-mail address: [email protected] (T.I. Thinh).

http://dx.doi.org/10.1016/j.apm.2016.06.015 0307-904X/© 2016 Published by Elsevier Inc.

Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

JID: APM 2

ARTICLE IN PRESS

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

Toorani and Lakis [12] to determine the swelling effect on the dynamic behavior of composite cylindrical shells conveying fluid and recently Larbi et al. [13] presented the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid. Senthil and Ganesan [14] performed a dynamic analysis on composite conical shells filled with fluid. Kuo et al. [15] investigated acoustic-structure interaction of sound by considering various fluid filled composite shells of different stacking sequences. In the study of Firouz-Abadi et al. [16], a modal based finite element model of structural dynamics in combination with a boundary element formulation of fluid dynamics has been used to determine the free vibration frequencies of the composite tanks partially filled with fluid. In those studies, low natural frequencies are generally investigated. For medium and high frequency range, the CEM can be applied with many advantages: high precision, rapid calculating speed, reduction of the model size and of the computing time. Numerous CEM researches have been performed for isotropic and composite beams: Lunden and Akesson [17], Banerjee [18], plates: Nguyen [19], Fazzolari et al. [20], Thinh et al. [21] and shells: Casimir et al. [22], Khadimallah et al. [23], Thinh and Nguyen [24], Fazzolari [25]. It should be noted that the majority of the conducted investigations were associated with only theoretical studies; however, not many experimental studies on free vibrations of empty and fluid-filled cylindrical shells are available in the literature. The main reason for this is that the measurement is extremely difficult to perform. Mixson and Herr [26] have determined natural frequencies and mode shapes of the clamped cylindrical shell made of aluminum and steel. Water was used as the contained liquid. Similar studies were conducted by Lindholm et al. [27] for a steel test cylinder. Chiba [28] tested two empty cantilevered circular cylindrical shells made of polyester sheets. Chiba [29] has also studied experimentally large-amplitude vibrations of two vertical cantilevered circular cylindrical shells made of polyester sheets, partially filled to four different levels of water with a free surface. Koval’chuk and Lakiza [30] experimentally investigated forced vibrations of large amplitudes in empty fiberglass cantilevered shells of revolution. In the work of Amabili et al. [31], the response of two water-filled circular cylindrical shells made of steel has been investigated in the neighborhood of the fundamental mode. The boundary conditions at the shell edges approximate the simple support type. Recently, a few studies can be found dealing with experimental modal analysis of composite cylindrical shells. Hosokawa et al. [32] have studied numerically and experimentally the free vibrations of angle-ply laminated carbon fiber reinforced plastic cylindrical shells with clamped edges. Okazaki et al. [33,34] experimentally determined the modal characteristics of a cross-ply composite cylindrical shell partially filled with fluid and followed the study by analyzing the free vibrations using finite elements. Wu et al. [35] performed an experimental study as well as a finite element analysis on the dynamic response of filament wound pressure vessels filled with liquid. They also provided the primary information which can be used for vibration-based damage detection of composite vessels. In the present study, free vibrations of fluid-filled composite circular vertical cylindrical shells are investigated. For the theoretical study, a new Dynamic Stiffness model for partially fluid-filled composite cylinders based on the Reissner–Mindlin theory and incompressible fluid equations are constructed. This continuous element provides high precision results, saves computing time and can be applicable for high frequency range. For the experimental study, the natural frequencies of the composite specimens are determined by using a multi-vibration measuring machine (DEWEBOOK-DASYLab 5.61.10). Theoretical solutions are compared to experimental results for analyzing free vibrations of partially and completely fluid-filled cross-ply composite circular cylindrical shells with clamped-free boundary condition. Emphasis will be placed on the advantages of the DSM and on the effects of the fluid filling on the natural frequencies of glass fiber/polyester composite circular cylindrical shells. 2. Formulation of cross-ply laminated composite circular cylindrical shells filled with fluid 2.1. Kinematics of cylindrical shells with fluid Investigate a thick composite circular cylindrical shell of length L, thickness h and mean radius R; H is the height of the contained fluid volume (Fig. 1). The shell consists of a finite number of layers which are perfectly bonded together. Following Reissner–Mindlin theory, the displacement components are:

u(x, θ , z, t ) = u0 (x, θ , t ) + zφx (x, θ , t ),

v(x, θ , z, t ) = v0 (x, θ , t ) + zφθ (x, θ , t ), w(x, θ , z, t ) = w0 (x, θ , t ),

(1)

where u, v and w are the displacement components in the x, θ and z directions, u0 and v0 are the in-plane displacements of the shell in the mid-plane, and φ x and φ θ are the rotations of the normal to the middle surface of the shell. The strain– displacement relations of cylindrical shells are expressed as:

  ∂ u0 ∂ φx 1 ∂ v0 z ∂ φθ w0 1 ∂ u0 ∂ v0 1 ∂ φx ∂ φθ ∂ w0 εx = +z , εθ = + + , γxθ = + +z + , γxz = φx + , ∂x ∂x R ∂θ R ∂θ R R ∂θ ∂x R ∂θ ∂x ∂x 1 ∂ w0 v0 γθ z = φθ + − . R ∂θ R

(2)

Consider a composite shell composed of N orthotropic layers of uniform thickness with the principal material axis of the kth layer is oriented at an angle α with the x axis. The stress-strain relations of the kth layer by neglecting the transverse Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

3

Fig. 1. Laminated composite cylindrical shell containing fluid.

normal strain and stress, are written as:

⎧ (k ) ⎫ ⎡ (k ) Q 11 ⎪ ⎪σx(k) ⎪ ⎪ ⎪ ⎪ ⎢ (k ) ⎪ ⎪ ⎨σθ ⎬ ⎢Q 12 ⎢ (k ) τx(θk) = ⎢Q 16 ⎪ ⎪ ⎢ ⎪ τ (k ) ⎪ ⎪ ⎪ ⎣0 ⎪ ⎩ θ(zk) ⎪ ⎭ τxz 0

(k )

(k )

Q 12

Q 16

0

Q 22

Q 26

0

Q 26

Q 66

0

0

0

Q 44

0

0

Q 45

(k )

(k )

(k )

(k )

⎤⎧ ⎫ (k ) ⎪ ⎪εx(k) ⎪ ⎪ ⎥⎪ ⎪ 0 ⎥⎪ ⎨ε ⎪ ⎬ ⎥ θ(k) , 0 ⎥ γxθ (k ) ⎪ (k ) ⎥⎪ ⎪ ⎪ γ ⎪ ⎪ ⎦ Q 45 ⎪ ⎩ θ(zk) ⎪ ⎭ (k ) γxz Q 0

(k ) (k )

(3)

55

(k )

where Q i j are the transformed stiffness and Qi j are the lamina stiffness referred to principal material coordinates of the kth lamina. The stress and moment resultants are calculated by:

(Nx , Nθ , Nxθ , Qx , Qθ ) = ( M x , M θ , M xθ ) =





(σx , σθ , τxθ , τxz , τθ z )dz,

(4)

z

(σx , σθ , τxθ )zdz.

(5)

z

The laminate constitutive relations become:



  {N} = [A] {M} [B]

     Qθ A {ε } and : = k 44 Qx A45 {γ }

[B ] [D ]

A45 A55



 γθ z , γxz

(6)

in which the laminate stiffness coefficients (Aij , Bij , Dij ) are defined by:

Ai j =

N 

Q¯ ikj (zk+1 − zk )

(i, j = 1, 2, 4, 5, 6 ),

k=1

Bi j =

N 1 k 2 Q¯ i j (zk+1 − zk2 ) 2

(i, j = 1, 2, 6 ),

N 1 k 3 Q¯ i j (zk+1 − zk3 ) 3

(i, j = 1, 2, 6 ),

k=1

Di j =

(7)

k=1

with k = 5/6: the shear correction factor, zk −1 and zk are boundaries of the kth layer. Internal force and moment resultants for general cross-ply composite (A16 = A26 = A45 = B16 = B26 = D16 = D26 = 0) are computed as [36]:

∂ u0 ∂ v0 w0 ∂φ ∂φ + A12 ( + ) + B11 x + B12 θ , ∂x R∂θ R ∂x R∂θ ∂ u0 ∂ v0 w0 ∂φ ∂φ Nθ = A12 + A22 ( + ) + B12 x + B22 θ , ∂x R∂θ R ∂x R∂θ ∂ v0 ∂ u0 ∂φ ∂ φx Nxθ = A66 ( + ) + B66 ( θ + ), ∂ x R∂θ ∂x R∂θ ∂ u0 ∂ v0 w0 ∂φ ∂φ Mx = B11 + B12 ( + ) + D11 x + D12 θ , ∂x R∂θ R ∂x R∂θ

laminated

cylindrical

shells

Nx = A11

(8)

Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM 4

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

∂ u0 ∂ v0 w0 ∂φ ∂φ + B22 ( + ) + D12 x + D22 θ , ∂x R∂θ R ∂x R∂θ ∂ v0 ∂ u0 ∂φ ∂ φx Mxθ = B66 ( + ) + D66 ( θ + ), ∂ x R∂θ ∂x R∂θ ∂ w0 Qx = kA55 (φx + ), ∂x ∂ w0 v0 Qθ = kA44 (φθ + − ). R∂θ R Mθ = B12

The equations of motions based on the first-order shear deformation shell theory for a laminated circular cylindrical shell filled with fluid taking to account the hydrodynamic fluid pressure P are:

 1 ∂ Nx 1 ∂  ∂ 2 u0 ∂ 2 φx + Nxθ − Mxθ = I0 + I1 , 2 ∂x R ∂θ 2R ∂t ∂t2  ∂N ∂  1 Q ∂ 2 v0 ∂ 2 φθ θ Nxθ + M xθ + + θ = I0 + I1 , 2 ∂x 2R R∂θ R ∂t ∂t2 ∂ Qx ∂ Qθ Nθ ∂ 2 w0 + − − P = I0 , ∂x R∂θ R ∂t2 2 ∂ M x ∂ M xθ ∂ u0 ∂ 2 φx + − Qx = I1 + I2 , 2 ∂x R∂θ ∂t ∂t2 ∂ M xθ ∂ M θ ∂ 2 v0 ∂ 2 φθ + − Qθ = I1 + I2 , 2 ∂x R∂θ ∂t ∂t2

(9)

where:

Ii =

z N k+1 

ρ (k) zi dz (i = 0, 1, 2 ),

k=1 zk

in which ρ ( k ) is the material mass density of the kth layer 2.2. Fluid equations The composite cylindrical shell is partially or completely filled with an incompressible, inviscid liquid. The potential function Ф(x,θ ,z,t) satisfies the Laplace equation in cylindrical coordinates (x,θ ,z):

∂ 2 1 ∂  1 ∂ 2 ∂ 2 + + 2 + = 0. z ∂z ∂ z2 z ∂θ2 ∂ x2

(10)

Then, the Bernoulli equation is written:

∂ P + = 0. ∂t ρf

(11)

By linearizing this expression, the pressures on the internal regions are:

P = −ρ f

 ∂   , ∂ t 

(12)

where is the portion of the structure’s surface in contact with fluid. The condition of impermeability of the surface of shell in contact with fluid can be expressed as:

vf =

  ∂   ∂ w0  = , ∂ z  ∂ t 

(13)

where w0 is the normal displacement of the shell, vf is the velocity of fluid. Regarding the equilibrium equations of the problem, the general equations of axisymmetric shells will be used. The effect of fluid on the shell manifests itself in terms of pressure P(x,θ ,z,t) occurring in the system of equations as external efforts. Then, the fluid-structure interaction is reflected by the condition of impermeability of the shell surface in contact with fluid. The potential function  may be defined by the following expression:

(x, θ , z, t ) = (z ) f (x ) cos(mθ )eiωt ,

(14)

where (z) is a function to be defined. The substitution of (14) into Laplace’s Eq. (10) gives:



 m2 1 1 ∂ 2 f (x ) ∂ 2 (z ) 1 ∂ (z ) + − + = 0. z ∂z

(z ) z2 f ( x ) ∂ x2 ∂ z2

(15)

Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

5

This equation leads to the following system:



1 ∂ 2 f (x ) = −k2n , f ( x ) ∂ x2



 m2 1 ∂ 2 (z ) 1 ∂ (z ) + − 2 = k2n . 2 z ∂z

(z ) z ∂z

(16a)

(16b)

From (16a), we have:

∂ 2 f (x ) + k2n f (x ) = 0. ∂ x2

(17a)

General solution of Eq. (17a) is:

f (x ) = B1 cos (kn x ) + B2 sin (kn x ).

(17b)

The term kn will be determined based on the fluid boundary condition. If free surface fluid modes of are not considered and by neglecting gravity, the following boundary conditions at x = 0 and x = L must be satisfied: (a) Clamped – Clamped at x = 0 and x = L: the fluid velocities is zero at x = 0 and x = L, thus:

kn = nπ /L.

(18)

(a) Clamped at x = 0 and free at x = L: the fluid velocities is zero at x = 0 and the fluid pressure must be zero at x = L. In this case:

kn = (2n − 1 )π /2L.

(19)

From (16b), we have:

 2  ∂ 2 1 ∂ m 2 + − + kn = 0. z ∂z ∂ z2 z2

(20)

Eq. (20) may be resolved using modified Bessel functions of first and second kind of order m, yielding:

(z ) = A1 Im (kn z ) + A2 Km (kn z ).

(21)

Finally, the potential  is expressed as:



(z ) ∂ w  (x, θ , z, t ) =  .

(z ) ∂ t z=R

(22)

And then the pressure P is written by:

P (x, θ , z, t ) = −ρ f



(z ) ∂ 2 w  .

 (z ) ∂ t 2 z=R

(23)

In addition, the pressure P must remain finite, which means that the function (z) is defined by:

(z) = A1 Im (kn z) inside the shells. The hydrodynamic pressure acting on the cylindrical shell is then defined by Paidoussis and Denis [37]:

P = −ρ f

1 ∂ 2 w0 ∂ 2 w0 = m∗ , m + kn RIm+1 (kn R )/Im (kn R ) ∂ t 2 ∂t2

(24)

where m∗ is the added mass which will be determined as:

m∗ = −ρ f

1 . m + kn RIm+1 (kn R )/Im (kn R )

(25)

This value will be introduced in (9) in order to establish the Dynamic Stiffness Matrix for the studied structure. 3. Continuous element method for vibration analysis of fluid-filled thick laminated composite cylindrical shells 3.1. Strong formulation The state-solution vector is {y} = {u0 , v0 , w0 , φ x , φ θ , Nx , Nxθ , Qx , Mx , Mxθ }T . For natural vibration of the cylindrical shell, the displacements and internal force resultants can be expressed in the Fourier expansion form, for the symmetric circumferential mode m as (Casimir et al. [22], Thinh and Nguyen [24]): Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM 6

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

⎧ ⎫ ⎧ ⎫ u0 (x, θ , t ) ⎪ um (x ) cos(mθ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨v0 (x, θ , t ) ⎬  ∞ ⎨vm (x ) sin (mθ ) ⎬ w0 (x, θ , t ) = wm (x ) cos(mθ ) eiωt ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ⎪ ⎩φx (x, θ , t ) ⎪ ⎭ ⎩φxm (x ) sin(mθ ) ⎪ ⎭ φθ (x, θ , t ) φθ m (x ) cos(mθ )     ∞ Nθ (x, θ , t ) Nθ m (x ) cos(mθ )  Mθ (x, θ , t ) = Mθ m (x ) cos(mθ ) e jωt . Qθ (x, θ , t ) m=1 Qθ m (x ) sin (mθ )

⎧ ⎫ ⎧ ⎫ Nx (x, θ , t ) ⎪ Nxm (x ) cos(mθ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Nxθ (x, θ , t ) ⎬  ∞ ⎨Nxθ m (x ) sin (mθ ) ⎬ Mx (x, θ , t ) Mxm (x ) cos(mθ ) eiωt = ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ⎪ ⎩Mxθ (x, θ , t )⎪ ⎭ ⎩Mxθ m (x ) sin(mθ )⎪ ⎭ Qx (x, θ , t ) Qxm (x ) cos(mθ ) (26)

Similar transformations for anti-symmetric modes can be obtained by swapping cos(mθ ) and sin(mθ ) win (26 ). For the sake of simplification, further developments in this work concern only symmetric modes. Note that the fluid pressure P on the shell is not an independent variable because it can be calculated from other state variables, especially from w. So the same state vector {y}m = {um , vm , wm , φ xm , φ θ m , Nxm , Nxθ m, Qxm , Mxm , Mxθ m }T will be used to solve the problem of fluid–shell interaction. It is necessary to establish the relation between P and other state variables by solving equations of fluid and then introduce the added mass m∗ into the system of equations. Applying the CEM procedure presented in (Casimir et al. [22], Thinh et al. [21], Thinh and Nguyen [24]), the following differential expressions have been calculated:

d um dx d vm dx d wm dx dφxm dx d φθ m dx dNxm dx dNxθ m dx

= m c 4 v m + c 4 w m + m c 5 φθ m +

B11 D11 Nxm − Mxm c1 c1

D66 m B66 um − N + M R c10 xθ m c10 xθ m 1 = −φxm + Qxm kA55 B11 A11 = m c 2 v m + c 2 w m + m c 3 φθ m − Nxm + Mxm c1 c1 A m B = φxm + 66 Nxθ m − 66 Mxθ m R c10 c10 m = −I0 ω2 um − I1 ω2 φxm − Nxθ m  R     kA44 k A kA44 44 2 2 2 2 = m c6 + 2 − I0 ω vm + m c6 + 2 wm + m c7 − − I1 ω φθ m − mc4 Nxm − mc2 Mxm R R R =



dQxm kA44 = m c6 + 2 dx R

     kA44 m2 kA44 ∗ 2 vm + c6 − (I0 + m )ω + φθ m w m + m c7 − 2 R

R

− c4 Nxm − c2 Mxm m dMxm = −I1 ω2 um − I2 ω2 φxm + Qxm − Mxθ m dx   R    kA44 A d M xθ m 2 2 vm + mc8 − 44 wm + m2 c9 − I2 ω2 + kA44 φθ m = m c8 − I1 ω − dx R R − mc5 Nxm − mc4 Mxm ,

(27)

with:

c3 = (B11 B12 − A11 D12 )/Rc1 ,

c4 = (B11 B12 − A12 D11 )/Rc1 ,

c5 = (B11 D12 − B12 D11 )/Rc1 ;

c6 = (A12 c4 + B12 c2 + A22 /R )/R,

c7 = (A12 c5 + B12 c3 + B22 /R )/R;

c8 = (B12 c4 + D12 c2 + B22 /R )/R;

c9 = (B12 c5 + D12 c3 + D22 /R )/R;

c10 = B266 − A66 D66 .

Eq. (27) can be expressed in the matrix form for each circumferential mode m:

dym /dx = Am ym

(28)

where Am is a 10 × 10 matrix (see Appendix). This system will help us to build a continuous element of thick cylindrical shell in contact with an internal fluid. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

7

Fig. 2. Assembly of Dynamic Stiffness Matrix for a partially fluid-filled cylinder.

3.2. Dynamic transfer matrix, Dynamic Stiffness Matrix K(ω) The dynamic transfer matrix Tm is calculated as:



Tm = e

Am L

T11 = T21

T12 T22



.

(29)

m

The Dynamic Stiffness Matrix K(ω)



K (ω )m =

T−1 T 12 11 T21 − T22 T−1 T 12 11

m

is then determined by Casimir et al. [22] and Thinh et al. [24]:

−T−1 12 T22 T−1 12



.

(30)

m

Natural frequencies will be extracted from the harmonic responses of the structure. 4. Assembly procedure for continuous element of partially fluid-filled cylindrical shells It is interesting to remark that the proposed Dynamic Stiffness Matrix can be used for both composite cylinders without fluid (with ρ f = 0) and for cylindrical shells completely filled with fluid (ρ f = 0). A partially fluid-filled cylinder can be divided into two different regions: an empty zone and a fluid-contained one (see Fig. 2). The whole problem becomes complex requiring a much more complicated coupled fluid–shell system of equations to solve. In such case, finite element simulation meets also difficulties either in modeling fluid–shell structures with various properties or in choosing an appropriate meshing for obtaining accurate solutions. The assembly technique of continuous elements offers a powerful approach to deal with this situation. This procedure is a strong capacity of CEM which has successfully been used in our previous research (Thinh et al. [21]) to analyze the complicated problem of composite plates resting on non-homogenous elastic foundation. The natural development of continuous element method consists to adapt this assembly algorithm to the problem of partially fluid-filled composite cylindrical shells. The construction of the Dynamic Stiffness Matrix for a partial fluid-filled cylindrical shell using the assembly procedure has been introduced in Fig. 2. In detailed, different mentioned zones of the shell are modeled by two Dynamic Stiffness Matrices i.e. : Kempty (ω) for the empty region (with ρ f = 0) and Kfluid (ω) for the completely fluid-filled zone (ρ f = 0). At last, the Dynamic Stiffness Matrix of the partially fluid-filled cylinder will be obtained by an assembly of the two above matrices similarly to our previous researches [21]. In conclusion, the proposed algorithm makes possible a simple and efficient solution, despite the complexity of the problem. This special capacity of CEM can be efficiently applied to solve complicated structures such as shell partially in contact with inside or outside fluid, combined conical-cylindrical shells, shells with stiffeners. 5. Validation studies A computer program based on Matlab is developed using the present formulation to solve a number of numerical examples on free vibration of composite cylindrical shells (emptyand filled with fluid) with various types of boundary conditions. First, the frequency parameters are defined as ω = ωL2 ρ /E2 /h and are calculated for cross-ply simply supported laminated cylindrical shells. In Table 1, the present solutions are compared to the corresponding exact analytical results given by Reddy and Liu [38] using Navier solution based on first shear deformation shell theory (FSDT). Now the present results are compared with those obtained by using Lèvy-type closed form exact solution based on FSDT and thin classical shell theory (CST) of Khdeir et al. [39]. Table 2 presents nondimensionalized fundamental frequencies of two, three and ten-layer cross-ply cylindrical shells for a radius to thickness ratio R/h = 5 and for various boundary conditions and length to radius ratios. Both methods (DSM and analytical) based on the FSDT give almost the same values for all cases, with the largest difference occurring for the clamped boundary conditions. The CST overpredicts the shell frequencies while the FSDT gives lower frequencies than those computed by CST. The excellent agreement of DSM results with the other published results based on FSDT given in Tables 1 and 2 indicates that the present analysis is accurate. It is interesting to note that, if the term (Mxө /2R) in the first two equations of (9) is neglected, by our formulation, we obtained the same results to the last digit presented in two above Tables. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM 8

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16 Table 1  Nondimensionalized fundamental frequencies ω = ωL2 ρ /E2 /h of cross-ply simply supported circular cylindrical shells (E1 /E2 = 25, G12 = G13 = 0.5 E2 , G23 = 0.2 E2 , υ 12 = 0.25, [38]). 0°/90°/00

0°/90°

0°/90°/90°/0°

R/L

Theory

L/h = 100

L/h = 10

L/h = 100

L/R = 10

L/R = 100

L/R = 10

10

FSDT Present FSDT Present FSDT Present

11.831 11.829 10.265 10.259 9.7816 9.7809

8.8879 8.8865 8.8900 8.8878 8.8951 8.8943

16.625 16.618 15.556 15.551 15.244 15.239

12.173 12.169 12.166 12.157 12.163 12.160

16.634 16.627 16.559 16.548 15.245 15.241

12.236 12.228 12.230 12.218 12.228 12.224

20 50

Table 2  Comparison of the dimensionless minimum frequencies ω = ωL2 ρ /E2 /(100h )of cross-ply circular cylindrical shells for various boundary conditions (R/h = 5, E1 /E2 = 40, G12 = 0.6 E2 , G13 = G23 = 0.5 E2 , υ 12 = 0.25, ρ = 1500 kg/m3 , [39]). SS

SC

CC

FC

Lamination

Theory

L/R = 1

L/R = 2

L/R = 1

L/R = 2

L/R = 1

L/R = 2

L/R = 1

L/R = 2

0°/90°

FSDT CST Present FSDT CST Present FSDT CST Present

0.0791 0.0866 0.0766 0.1004 0.1479 0.0996 0.0982 0.1235 0.0949

0.1552 0.1630 0.1519 0.1779 0.2073 0.1722 0.1899 0.1958 0.1821

0.0893 0.1152 0.0823 0.1036 0.1850 0.1025 0.1037 0.1605 0.1039

0.1697 0.1841 0.1661 0.1945 0.2662 0.1950 0.2012 0.2304 0.2012

0.1002 0.1048 0.9822 0.1093 0.2049 0.1083 0.1110 0.1963 0.1086

0.1876 0.2120 0.1737 0.2129 0.3338 0.2083 0.2137 0.2752 0.2013

0.0435 0.0480 0.0396 0.0495 0.0669 0.0483 0.0513 0.0598 0.0504

0.0914 0.0938 0.0872 0.0988 0.1099 0.0914 0.1004 0.1077 0.0940

0°/90°/0°

0°/90°/ 10 layers

S: simply-supported, C: clamped and F: free.

Next, the free vibration of composite cylindrical shells containing fluid with clamped-free boundary condition is considered. Both fundamental frequencies and harmonic responses will be validated with respect to those of literature and to the results obtained by Finite Element Method (FEM). Finite element models (ANSYS) are constructed using different mesh sizes on the same structure to ensure the convergence of responses from FEM to themselves and to the proposed analytical model. The SHELL181 element taking into account the effect of transverse shear is chosen in order to achieve a consistent comparison with our continuous element. The fluid is modeled by the 3D fluid element FLUID80. The composite circular cylindrical shells has following properties (Xi et al. [1]): E1 = 206.9 GPa; E2 = 18.62 GPa; ν 12 = 0.28, G12 = 4.48 GPa; G13 = 4.48 GPa; G23 = 2.24 GPa; ρ = 2048 kg/m3 ; h = 9.525 mm; R = 0.1905 m; L = 0.381 m; layer scheme: [0°/90°]; the water characteristics are: ρ f = 10 0 0 kg/m3 , c = 150 0 m/s. The shells is partially filled with fluid with various fluid level (H/L = 0, 0.25, 0.5, 0.75 and 1). First, natural frequencies computed by CEM are shown in Fig. 3 and compared with FEM solutions and with Xi et al. [1] results which used a modified finite element taking into account the influence of fluid in composite shell model of FEM. The compared results concern circumferential waves number n = 1÷5 and longitudinal mode m = 1. It is seen from these graphics that the agreement among the present results and those of FEM and Xi et al. [1] is very good which confirms the exactness of our formulation. At low frequencies, CEM offers excellent solutions which are very closed to those of FEM with different levels of contained fluid. Next, the high precision of the present analytical formulation will be highlighted through the comparison of harmonic responses of composite cylindrical shell drawn by CEM and by FEM. Responses of bending displacements are evaluated at the point subjected to a concentrated force located at the free edge of the structure. Harmonic responses by FEM using ANSYS SHELL181 element with different meshes (60 × 20 × 6 and 90 × 30 × 10) are compared to those computed by CEM. Figs. 4 and 5 illustrate the harmonic response obtained by CEM and by FEM of a clamped-free [0° /90° ] composite cylindrical shell filled with liquid for H/L= 0.5 (Fig. 4) and H/L= 1 (Fig. 5) with following properties: E1 = 206.9 GPa, E2 = 18.62 GPa, G12 = 4.48 GPa, G13 = 4.48 GPa, G23 = 2.24 GPa, ρ = 1824 kg/m3 , υ 12 = 0.26, h = 0.0254 m, R = 0.254 m, L = 2R. It is easy to note from these figures that in low range of frequencies, two methods give almost identical values until 820 Hz (when H/L= 0.5) and 575 Hz (when H/L=1). The discrepancies between two models are clearly noticed from 946 Hz (when H/L= 0.5) and 757 Hz (when H/L=1) because the divergence of the three curves becomes more and more important. From these limits, due to the strong dependence on the size of meshing elements, FEM provides less precise results compared to our formulation which assures high exact solutions in medium and high frequencies. Huge important discrepancies between FEM and CEM curves occurring in high frequencies which emphasizes the advantage of the presented formulation. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

JID: APM

ARTICLE IN PRESS

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

9

Fig. 3. Fundamental natural frequency of a clamped-free laminated composite circular cylindrical shell partially filled with a fluid.

Fig. 4. Harmonic responses of a clamped-free composite cylindrical shell containing fluid computed by FEM and by CEM (H/L = 0.5).

Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

JID: APM 10

ARTICLE IN PRESS

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

Fig. 5. Harmonic responses of a clamped-free composite cylindrical shell fully filled with fluid calculated by FEM and by CEM (H/L = 1). Table 3 Laminated cylinder specifications. Given name

L/R ratio

Dimension (L x R x h), mm

C1 C2 C3

1.5 2 3

427.5 × 285 × 2 570 × 285 × 2 855 × 285 × 2

Fig. 6. Photographs of the tested glass fiber/polyester composite shells.

In addition, more meshing finite elements are required to get the same precision of CEM results but in this case more computing time and data storage volume are also needed. Therefore CEM saves computational time compared to the traditional approximate method. For a full-fluid filled composite cylinder (H/L= 1), our model with 3 elements required 18 min to plot the harmonic response while FEM models needed much more calculating time: 83 min with 60 × 20× 6 and 250 min for 90 × 30 × 10 mesh. The reason for these differences is that CEM uses a minimum meshing of the structure yielding to a very small size of system of equations to solve. 6. Experimental study An experimental procedure for determining natural frequencies of partially fluid-filled composite cylindrical shells will be presented and validated here. Let us consider three composite cylinders made of glass fiber in Woven Roving (denote WR8) form with its weight per unit area is 800 g/m2 . The unsaturated polyester 3210 is used as resin. The first specimen is labeled “C1”, the second is “C2” and the third “C3”. The detailed dimensions of these samples are presented in Table 3. The photograph of the tested composite shells is shown in Fig. 6. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

JID: APM

ARTICLE IN PRESS T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

[m3Gsc;July 6, 2016;11:41] 11

Fig. 7. Schematic diagram of experimental set-up for a frequency measurement.

Fig. 8. Frequency response curves of the test composite cylinder C1, H/L = 1.

The measured composite cylinder was glued at the base to a 5 mm thick composite circular plate of radius R = 335 mm which was in turn bolted to a heavy steel base block (see Fig. 6). The shell was filled with water up to the level H. The filling ratio of the contained water H/L are 0, 0.25, 0.50, 0.75 and 1 successively. The boundary condition of the shell assures theoretically the clamped-free type. The schematic diagram for the testing experimental frequency is shown in Fig. 7. First five natural frequencies of each clamped-free composite cylindrical shell containing various levels of water were measured by using a multi-vibration measuring machine (DEWE BOOK-DASYLab 5.61.10). Fig. 8 shows an example of the frequency response curves for the test composite cylinder C1 with H/L = 1. At least five measures for each frequency were carried out and the average and standard deviations are reported. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM 12

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16 Table 4 Comparison of natural frequencies (Hz) of C1 specimen by experiment, FEM and by CEM. H/L

Mode

Experiment

FEM

CEM

CEM-Exp. errors (%)

H/L = 0

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

77.9 84 92.8 107.9 115.6 73.5 79.4 88.1 106.7 112.5 47.8 52.7 63.7 66.3 69.2 26.4 32 35.5 39.3 46.6 19.3 20.1 25.9 26.4 32.1

77.59 85.65 88.53 112.02 117.97 76.86 83.79 88.27 109.47 113.19 50.49 53.56 54.54 60.45 67.03 31.60 32.55 36.56 40.45 44.43 21.1 21.53 25.79 27.17 34.54

76.24 84.42 87.03 111.04 117.89 75.93 83.23 86.61 106.64 113.1 49.37 53.13 53.22 58.97 66.51 30.14 31.42 34.97 40.19 42.85 19.68 20.34 23.89 26.28 31.91

2.13 0.50 6.22 2.91 1.98 3.31 4.82 1.69 0.06 0.53 3.28 0.82 16.45 11.06 3.89 14.17 1.81 1.49 2.26 8.05 1.97 1.19 7.76 0.45 0.59

H/L = 0.25

H/L = 0.5

H/L=0.75

H/L=1

%Errors = 100∗ abs (ωExp −ωCem )/ωExp . Table 5 Comparison of natural frequencies (Hz) of C2 specimen by experiment, FEM and by CEM. H/L

Mode

Experiment

FEM

CEM

CEM-Exp. errors (%)

H/L = 0

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

57.9 67.9 75.3 85.6 100.3 57.6 61.3 74.9 79.6 84.6 35.7 40.8 41.5 45.6 46.1 20.5 25.2 26.1 29.2 35.4 14.5 15.1 18.9 19.8 25.4

59.68 63.18 78.84 81.56 108.52 58.53 62.57 75.73 78.76 87.32 37.36 38.7 42.89 45.02 48.72 22.51 24.84 26.76 30.81 38.4 14.58 16.77 17.51 23.18 25.92

59.29 62.76 77.11 80.32 103.41 58.06 61.72 75.59 77.21 85.51 36.43 37.57 41.77 44.9 47.33 21.39 23.66 25.89 29.73 37.26 13.64 15.61 16.6 21.56 25.37

2.40 7.57 2.40 6.17 3.10 0.80 0.69 0.92 3.00 1.08 2.04 7.92 0.65 1.54 2.67 4.34 6.11 0.80 1.82 5.25 0.55 3.37 7.35 8.89 0.12

H/L = 0.25

H/L = 0.5

H/L = 0.75

H/L = 1

The next research consists to validate the experiment model and measurement by comparing experimental results with those obtained by CEM and by Finite Element Method (ANSYS) for the same specimens. The natural frequencies of all clamped-free test cylinders C1, C2 and C3 were calculated accurately by CEM and by FEM, with various filling ratio of the containing water. It is necessary to note that, in our calculation, one-WR8 layer with twr8 thickness, can be considered as two unidirectional layers, which have the following characteristics: Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

13

Table 6 Comparison of natural frequencies (Hz) of C3 specimen by experiment, FEM and by CEM. H/L

Mode

Experiment

FEM

CEM

CEM-Exp. errors (%)

H/L = 0

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

39.8 44.7 54.4 76.7 78.1 38.1 42.5 53.7 59.3 62.1 24.1 25.2 28.8 31.5 35.2 14.5 15.1 17.3 22.1 29.5 10.3 11.2 12.7 14.4 21.0

40.79 43.30 55.20 72.57 78.20 40.64 42.18 53.24 58.20 61.76 24.84 25.73 27.98 32.63 36.74 14.84 15.17 19.76 21.46 25.92 9.32 10.81 13.95 14.59 22.38

40.05 42.21 54.85 71.34 77.62 39.69 41.72 50.95 54.75 59.43 23.53 24.69 26.48 30.59 36.63 13.94 14.17 18.64 20.69 24.38 8.68 10.05 13.2 13.36 20.34

0.62 5.57 0.83 6.99 0.61 4.17 1.84 5.12 7.67 4.30 2.37 2.02 8.06 2.89 4.06 3.86 6.16 7.75 6.38 17.36 15,73 10.27 3.94 7.22 3.14

H/L = 0.25

H/L = 0.5

H/L = 0.75

H/L = 1

- The thickness of each unidirectional layer is: t1 = t2 = twr8 /2 = 0.5 mm - The angles, which is made by direction of fiber in each layer and x-direction, are respectively: θ 1 = 0°; θ 2 = 90°. As a result, the cylinder’s configuration is: [0°/90°/0°/90°] and the mechanical properties are (Thinh et al. [40]): E1 = 10.58 GPa, E2 = 2.64 GPa, G12 = G13 = G23 = 1.02 GPa, υ 12 = 0.19, ρ = 1600 kg/m3 , ρ water = 10 0 0 kg/m3 . High precision results (CEM), approximate numerical solutions (FEM) and measured natural frequencies of the first five modes are reported separately for each specimen in Tables 4–6, respectively. In our study, the 3D ANSYS model is built using an elastic composite shell element (SHELL181) and a 3D contained fluid element (FLUI80). The FE solutions were obtained using a 100 × 40 ×12 mesh for C1 and C2 and a 60 × 32 ×10 mesh for C3. The natural frequencies of first three modes calculated by CEM are shown in Fig. 9 and compared with our experimental results. It can be seen that the agreement between the theoretical and experimental results is very good (the numerical results’ error compared with experimental results almost remain nearly constant at 7% in air (dry) and about 10% in water. The relatively large difference (up to 17%) between the theoretical and measured natural frequencies of the C3 for certain water depths was probably due to a variation of the mode shapes of vibration of the shell with the height of the liquid level. The consistence between high precision CEM solutions and experimental results confirms the accuracy and the reliability of both CEM model and the experimental measurement. Obtained results in Tables 4–6 show that, for all the investigated cases, the filled fluid can reduce significantly the natural frequency of a glass fiber/polyester composite cylindrical shell. For example, for cylinder C1, the natural frequency of first mode is approximately 6%, 39%, 66% and 75% corresponding to water filling levels, H/L = 0.25, 0.5, 0.75 and 1 respectively lower in water than in air. From these figures it can be seen clearly that the reduction of the natural frequency of the shell decreases first slowly and then quickly as the water height increases. Fig. 10 shows the effect of the ratio of length to radius, L/R = 1.5, 2, 3 on the fundamental frequency of glass fiber/polyester composite cylindrical shells. From this figure, it can be seen that the longer a shell, the more remarkable the effect of the filled fluid on its natural frequency. This is due to the fact that a longer shell becomes more flexible, and it is thus susceptible to the filled fluid.

7. Conclusions Free vibrations of partially and completely fluid-filled laminated glass-fiber/polyester composite circular vertical cylindrical shells are studied theoretically and experimentally. Through the above examples, we note that the Dynamic Stiffness Method is an attractive possibility for solving the problem of fluid-composite shell with a simple model of the fluid and mesh minimized. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

JID: APM 14

ARTICLE IN PRESS

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

Fig. 9. Comparison of natural frequencies by experiment and by CEM.

Based on the numerical results presented in this paper, the following conclusions may be drawn: + The presented new continuous element for composite cylindrical shells containing fluid provides excellent precision results in low, medium and high frequencies and by this method the number of meshing elements has been reduced strongly. Therefore, our model allows fast calculation and saves data storage. + Experimental data agree well with the theoretical results obtained by using the Dynamic Stiffness Method of Continuous Elements based on the Reissner–Mindlin theory and non-viscous incompressible fluid equations which validates our proposed formulation as well as the experimental procedure. Both the theoretical and experimental results showed that: - The fluid filling can reduce significantly the natural frequencies of cross-ply glass-fiber/polyester composite circular cylindrical shells. Frequency reduction is shown to increase with liquid depth. - As the ratio of length to radius increases, the effect of the fluid filling on the natural frequencies of cross-ply laminated composite circular cylindrical shells becomes strong. Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

ARTICLE IN PRESS

JID: APM

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

15

Fig. 10. Effects of length on the fundamental frequency of clamped-free glass fiber/polyester composite cylindrical shells partially filled with a fluid.

Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number: 107.02-2015.14. Appendix Matrix [A(ω)]10 × 10 : ⎡

0

⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 Am = ⎢ ⎢−I0 ω2 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣−I ω2 m R

1

0

mc4 0 0 mc2 0 0  2 m c6 +



m c6 +

0

kA44 R2 kA44 2 R

− I0 ω2

kA44 R

− I1 ω2

m2 c8 −







c4 0 0 c2 0 0 m c6 +



c6 +

0 0 −1 0

0

m c7 −

−I2 ω2 0

0

m R



kA44 R2 m2 kA44 R2

0  mc8 −

−I1 ω2 0

mc5 0 0 mc3 0 0  2 m c7 −

kA44 R



− (I0 + m )ω ∗

2





D11 c1

kA44 R kA44 R



− I1 ω2



m2 c9 + kA44 − I2 ω2



0 0 − Bc111 0 0 −mc4

0 − Dc1066 0 0

0 0 1 kA55

− Bc111 0 0

0

⎥ ⎥

⎥ ⎥ ⎥ ⎦

0 0 −mc2

⎥ 0 ⎥ ⎥ 0 A66 ⎥ − c10 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎥

A11 c1

0

0 0 0 0

−c4

0

0

−c2

0

0 −mc5

0 0

1 0

0 −mc4

−m R 0

B66 c10 −m R

⎤

B66 c10

References [1] Z.C. Xi, L.H. Yam, T.P. Leung, Free vibration of a laminated composite circular cylindrical shell partially filled with fluid, Compos. Part B Eng. 28 (4) (1997) 359–374. [2] Z.C. Xi, L.H. Yam, T.P. Leung, Free vibration of a partially fluid-filled cross-ply laminated composite circular cylindrical shell, J. Acoust. Soc. Am. 101 (2) (1997) 909–917. [3] C.B. Sharma, M. Darvizeh, A. Darvizeh, Natural frequency response of vertical cantilever composite shells containing fluid, Eng. Struct. 8 (20) (1998) 732–737. [4] W.Q. Chen, H.J. Ding, Natural frequencies of a fluid-filled anisotropic spherical shell, J. Acoust. Soc. Am. 105 (1) (1999) 174–182. [5] O. Katsutoshi, T. Unji, S. Mikio, Free vibrations of a laminated composite circular cylindrical shell partially filled with liquid, Trans. Jpn. Soc. Mech. Eng. C 65 (640) (1999) 4597–4604. [6] M.H. Toorani, A.A. Lakis, Shear deformation in dynamic analysis of anisotropic laminated open cylindrical shells filled with or subjected to a flowing fluid, Comput. Methods Appl. Mech. Eng. 190 (37) (2001) 4929–4966. [7] M.H. Toorani, A.A. Lakis, Dynamic behavior of axisymmetric and beam-like anisotropic cylindrical shells conveying fluid, J. Sound Vib. 259 (2003) 265–298. [8] N.C. Pal, S.K. Bhattacharyya, P.K. Sinha, Coupled slosh dynamics of liquid- filled, composite cylindrical tanks, J. Eng. Mech. 125 (4) (1999) 491–495. [9] N.C. Pal, S.K. Bhattacharyya, P.K. Sinha, Non-linear coupled slosh dynamics of liquid-filled laminated composite containers: a two dimensional finite element approach, J. Sound Vib. 261 (2003) 729–749. [10] M. Amabili, Comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach, J. Sound Vib. 264 (2003) 1091–1125. [11] M. Amabili, Nonlinear vibrations of circular cylindrical shells with different boundary conditions, AIAA J. 41 (2003) 1119–1130. [12] M.H. Toorani, A.A. Lakis, Swelling effect on the dynamic behaviour of composite cylindrical shells conveying fluid, Int. J. Numer. Methods Fluids 50 (2006) 397–420. [13] W. Larbi, J.F. Deu, R. Ohayon, Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid, Int. J. Numer. Methods Eng. 71 (12) (2007) 1412–1435. [14] K.D. Senthil, N. Ganesan, Dynamic analysis of conical shells conveying fluid, J. Sound Vib. 310 (2008) 38–57. [15] Y.M. Kuo, H.J. Lin, C.N. Wang, W.C. Wang, C.L. Chang, Study of the interaction of two acoustical fields and one FRP structure with FEM-BEM coupling analysis, J Reinf. Plast. Compos. 28 (2009) 2083–2096.

Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015

JID: APM 16

ARTICLE IN PRESS

[m3Gsc;July 6, 2016;11:41]

T.I. Thinh, M.C. Nguyen / Applied Mathematical Modelling 000 (2016) 1–16

[16] R.D. Firouz-Abadi, H. Haddadpour, M.A. Kouchakzadeh, Free vibrations of composite tanks partially filled with fluid, Thin Walled Struct. 47 (2009) 1567–1574. [17] R.B. Lunden, B. Akesson, Damped second order Rayleigh Timoshenko beam vibration in space-an exact complex dynamic member stiffness matrix, Int. J. Numer. Methods Eng. 19 (1983) 431–449. [18] J.R. Banerjee, Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam, J. Sound Vib. 270 (2004) 379–401. [19] M.C. Nguyen, Eléments Continus de plaques et coques avec prise en compte du cisaillement transverse. Application à l’interaction fluide-structure, Thèse de Doctorat, Université Paris VI, 2003. [20] F.A. Fazzolari, M. Boscolo, J.R. Banerjee, An exact dynamic stiffness element using a higher order shear deformation theory for free vibration analysis of composite plate assemblies, Compos. Struct. 96 (2013) 262–278. [21] T.I. Thinh, M.C. Nguyen, D.G. Ninh, Dynamic stiffness formulation for vibration analysis of thick composite plates resting on non-homogenous foundations, Compos. Struct. 108 (2014) 684–695. [22] J.B. Casimir, M.C. Nguyen, I. Tawfig, Thick shells of revolution: Derivation of the dynamic stiffness matrix of continuous elements and application to a tested cylinder, Comput. Struct. 85 (23-24) (2007) 1845–1857. [23] M.A. Khadimallah, J.B. Casimir, M. Chafra, H. Smaoui, Dynamic stiffness matrix of an axisymmetric shell and response to harmonic distributed loads, Comput. Struct. 89 (2011) 467–475. [24] T.I. Thinh, MC Nguyen, Dynamic stiffness matrix of continuous element for vibration of thick cross-ply laminated composite cylindrical shells, Compos. Struct. 98 (2013) 93–102. [25] F.A. Fazzolari, A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells, Compos. Part B Eng. 62 (2014) 143–158. [26] J.S. Mixson, R.W. Herr, An investigation of the vibration characteristics of pressurized thin-walled circular cylinders partly filled with liquid, Techn. Rept. NASA 1962 (1963). [27] U.S. Lindholm, D.D. Kana, H.N. Abramson, Breathing vibration of a circular cylindrical shell with an internal liquid, J. Aerosp. Sci. 29 (1962) 1052–1059. [28] M. Chiba, Non-linear hydroelastic vibration of a cantilever cylindrical tank. I. Experiment (empty case), Int. J. Nonlinear Mech. 28 (1993) 591–599. [29] M. Chiba, Non-linear hydroelastic vibration of a cantilever cylindrical tank. II. Experiment (liquid-filled case), Int. J. Nonlinear Mech. 28 (1993) 601–612. [30] P.S. Koval’chuk, V.D. Lakiza, Experimental study of induced oscillations with large deflections of fiberglass shells of revolution, Int. Appl. Mech. 31 (1995) 923–927. [31] M. Amabili, R. Garzier, A. Negri, Experimental study on large-amplitude vibrations of water-filled circular cylindrical shells, J. Fluids Struct. 16 (2) (2002) 213–227. [32] K. Hosokawa, M. Murayama, T. Sakata, Free vibration analysis of angle-ply laminated circular cylindrical shells with clamped edges, Sci. Eng. Compos Mater. 9 (2) (20 0 0) 75–82. [33] K. Okazaki, J. Tani, M. Sugano, Free vibrations of a laminated composite coaxial circular cylindrical shell partially filled with liquid, Nippon Kikai Gakkai Ronbunshu, C Hen/Trans. Jpn. Soc. Mech. Eng. Part C 68 (2002) 1942–1949. [34] K. Okazaki, J. Tani, J. Qiu, K. Kosugo, Vibration test of a cross-ply laminated composite circular cylindrical shell partially filled with liquid, Nippon Kikai Gakkai Ronbunshu, C Hen/Trans. Jpn. Soc. Mech. Eng. Part C 73 (2007) 724–731. [35] Z. Wu, W. Zhou, H. Li, Modal analysis for filament wound pressure vessels filled with fluid, Compos. Struct. 92 (2010) 1994–1998. [36] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells. Theory and Analysis, CRC Press, 2004. [37] M.P. Paidoussis, J.P. Denis, Flutter of thin cylindrical shell conveying fluid, J. Sound Vib. 20 (1) (1972) 9–26. [38] J.N. Reddy, C.F. Liu, A higher-order shear deformation theory of laminated elastic shells, Int. J. Eng. Sci. 23 (3) (1985) 319–330. [39] A.A. Khdeir, J.N. Reddy, D. Fredrick, A study of bending, vibration and buckling of cross-ply circular cylindrical shells with various shells theories, Int. J. Eng. Sci. 27 (11) (1989) 1337–1351. [40] T.I. Thinh, T.H. Quoc, Finite element modeling and experimental study on bending and vibration of laminated stiffened glass fiber/polyester composite plates, Comput. Mater. Sci. 49 (2010) S383–S389.

Please cite this article as: T.I. Thinh, M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.015