Vietnam Joumal of Mechanics, VAST, Vol. 37, No. 1 (2015), pp. 4 3 - 5 6 DOI: 10.15625/0866-7136/37/1 /5508
DYNAMIC STIFFNESS METHOD FOR FREE VIBRATIONS ANALYSIS OF PARTIAL FLUID-FILLED ORTHOTROPIC
CIRCULAR CYLINDRICAL SHELLS
Tran Ich Thinh*, Nguyen Manh Cuong, Vu Quoc Hien Hanoi University of Science and Technology, Vietnam
'E-mail: [email protected] Received November 20, 2014
Abstract. Free vibrations of partial fluid-filled orthotropic circular cylindrical shells are investigated using the Dynamic Stiffness Method (DSM) or Continuous Element Method (CEM) based on the First Order Shear Deformation Theory (FSDT) and non-viscous in- compressible fluid equations. Numerical examples are given for analyzing natural fre- quencies and harmonic responses of cylindrical shells partially and completely filled with fluid under various boundary conditions. The vibrahon frequencies for different filling ratios of cylindrical shells are obtained and compared with existmg experimental and the- oretical results which indicate that the fluid filling can reduce sigmficantly the natural frequencies of studied cylindrical shells. Detailed parametric analysis is carried out to show the effects of some geometrical and material parameters on the natural frequencies of orthotropic cylindrical shells. The advantages of this current solution consist in fast con- vergence, low computational cost and high precision validating for all frequency ranges.
Keywords: Free vibration, continuous element method, dynamic stiffness method, ortho- tropic cylindrical shell, fluid-shell interaction
1. INTRODUCTION
The knowledge on dynamic responses of fluid-filled isottopic and orthottopic cylin- drical shells is of primary importance for the design of pressure vessels, fluid tanks of pro- pellant rockets, seismic effects on liquid storage tanks etc. For coupled fluid-cylindrical shell problems, Jain [1] obtained the natural frequency equations of simply supported orthottopic cylindrical shells, filled completely or partially with an incompressible non- viscous fluid on the basis of a shell theory in which the effects of ttansverse shear defor- mation and rotatory inertia were retained. Warburton and Soni [2] studied the resonant response of orthottopic cylindrical shells, while Bradford and Dong [3] investigated the lateral vibrations of orthottopic cylinders under initial sttess. The ttee vibrations of or- thotropic cylindrical shells based on the three-dimensional elasticity theory considering
© 2015 Vietnam Academy of Science and Technology
44 Tran Ich Thinh, Nguyen Manh Cuong. Vu Quoc Hien
the effect of intemal fluid have analyzed by Chen et al. [4] who obtamed the firequency equation of non-axisymmettic free vibration modes of an ortiiottopic fluid-filled cylin- drical shell with arbittary constant thickness and then compared their results with those based on other sheU theories. Li and Chen [5], basing on the Flugge's linear sheU tiieory and the normal mode flieory, study the dynamic response of orthottopic chcular cylindri- cal sheUs subjected to extemal hydrostatic pressure. The effect of sheU parameters, exter- nal hydrostatic pressure and material properties on the dynamic behavior of the shell has been examined in detafl in their work. Using the Sanders-Koiter non-hnear shell theory, Sehnane and Lakis [6] investigated the influence of non-linearities associated with the shell wall and wifli the fluid flow on the dynamics of elastic tiiin orthottopic cyhndrical shells using a hybrid finite element method. On the basis of the Ritz variation method, Shang and Lei [7] investigated the free vibrations of fluid-conveying orthottopic cylin- drical shell. Sharma et al. [8] have investigated the natural frequency response of vertical cantilevered composite shells containing fluid. They used the Fourier series of ttigono- metric functions to approximate the axial modal dependence. This presents an example of practical applications of cylindrical shells. Recently, Prado et al. [9] studied non-linear vibrations and instabflities of orthotropic cylindrical shells with intemal flowing fluid using the Galerkin approach. Daneshjou et al. [10] have proposed an analytical solution to solve the problem of sound ttansmission through orthotropic cylindrical shells with subsonic external flow. A boundary integral solution has been applied to the hydroelastic analysis of fluid storage tanks. Zhang et al. [11] have employed the wave propagation ap- proach to investigate vibration characteristics of fluid-filled cylindrical shells. This gives relatively more accurate predictions of natural frequencies of cylindrical shells. Further showing the efficiency and efficacy of the wave propagation approach, Li et al. [12] have applied this techruque to study the modal analysis of thin finite cylindrical shells. By the same method, Natsuki et al. [13] have carried out the vibrational analysis of fluid- filled carbon nanotube. This approach seems to be widely appficable to study vibration characteristics of shells involving fluids.
In those studies, low natural frequencies are generally investigated. For medium and high frequency range, the CEM can be applied with many advantages: high preci- sion, rapid calculating speed, reduction of the model size and of the computing time. Nu- merous CEM researches have been performed for isotropic and composite beams: [14,15], plates: [16,17] and sheUs: [18-20]. However, no work based on CEM appears to have been done on the problem of cylindrical shells taking into account the interaction of fluid- sttucture.
The purpose of this article is to present a new continuous element for isotropic and orthotropic cylindrical shells based on the Reissner-Mmdlm shell theory which retains the effects of ttansverse shear deformation and rotatory inertia as well as those of non- viscous incompressible fluid. Numerical results indicate that the fluid fiUiag can reduce significantiy the natural frequencies of cylmdrical shells. Paramettic studies including circumferential wave number, fluid depth, thickness to mean radius ratio, length to mean radius and boundary condition are carried out.
Dynamic s( method for free vibrations analysis of partial fluid-filled orthotropic circular .
2. THEORETICAL FORMULATION OF ORTHOTROPIC FLUID-FILLED CYLINDRICAL SHELLS 2.1. Kinematics of ortrofropic cylindrical shell filled with fluid
Consider an ortrotropic cylindrical shell containing fluid as shown in Fig. 1. R,L and /! are the radius, length and thickness of the cylinder. H is the height of the contained fluid volume.
Fig. 1. Kinematics of orthotropic cylindrical shell filled with fluid
(1)
(2) The material constants Q,^ are defined in terms of the orthotropic material proper- ties by
Q n = £ l / ( l -l^I2l^2l), Ql2 = I'l2£2/{1-I^12l'2l), Q22 = E2/ (1 - ui2i'2i) / Qee = Gu, Qu = G23, Q55 = Gis,
where £/, G,j, Vu, 1^21 • elastic constants of the material and the stiffiiess coefficients (A;,,D,;)are
A,i = Q,jh, D,, = Q,,h'/12 (/,/ = 1,2,6),
A44 - hQu, A55 = hQss. A16 = A26 = A45 = 0, Die = D26 = 0.
FoUowing Reissner-Mindlin assumption, the displacement components are assumed to be
u{x,e,z,t) = UQ{x,e,t)+z(pjc{x,e,t),
v{x,e,z,t) = voix.e.t)-i-zcpe{x,e,t), (3)
w{x,e,z,t) ^wo{x,e,t),
where u,v and lo are the displacement components in the x,6 and z directions respec- tively; iio,Vo and ZOQ are the displacements of the sheU at the neuttal surface in axial, circumferential and radial directions, and (px and ipe are the rotations of the normal to the middle surface of the sheU. The strain-displacement relations of cylindrical shell can be written as
duQ d(px Wo dx dx
1 3 ^ 0 '- - -, V 1 duQ Bvo / 1 0
1 3wt, Up
" R ae R'
(4)
Trail Ich Thinh, Nguyen Manh Cuong, Vu Quoc Hien
2.2. Equations of motion for orthottopic cylindrical shells in contact with fluid The equations of motion of the first-order shear deformation theory for an or- thottopic circular sheU fiUed with fluid hi {x, 6, z) coordinate are written by
(5) aw,
dx dx dx 3 M ,
dx dM,e
dx 1 aWto R ae "
1 aws '^ R de "
1 dQs
^ R ae laiMto '^ R d9 IdMs R de
= Jo"o
= foSo
i«'
-Q, -Qe
+ h9x.
+ 1196, -Pf = he
= J, Uo + h
= I,vo + h
I f'"'"'-
(. = 0,1,2), (6)in which ps is the mass density of shell, Pj is the hydrodynamic pressure of fluid which can be determined from fluid equations.
2.3. Internal force resultants-displacement relationships
Here, the internal force and moments resultants can be expressed in terms of dis- placements as [10]
• N» • Ne Nto Nrf M , Me Me, M,e
All A21
0 0 0
_ D j 2 R
0 0
0 0 Att, Aa, 0 0
/ Q.. 1 _ a A55 0 1 r
1 Qe / - * [ 0 AtiW
0 0 0
R
'S^ + i
0
R
0 0 D12 D22 0 0
0 0
D M R
0 0 0 DM D66
0 0 0 D66
R 0 0
^66
^66
'
r 9"n \ -, "5^
Rde ^ R dvo dx 9i'o 150 'dx Rd0
"57 R5e
= 5/6).
(7)
where k is the shear correction factor {k - 2.4. Fluid equations
The cylindrical shell is partially or completely filled with an incompressible, invis- cid liquid. For the steady-state case, the potential function (r, 9, x, t) satisfies the Laplace
Dynamic stiffness method for free vibrations analysis of partial fluid-filled orthotropic circular .. 47
equation in cylindrical coordinates {r, 6, x)
9^2 r 9J. fl 9^2 3^.2 Then, the BemouUi equation is written
^ + ^ = 0. (9) dt Pf
By linearizing this expression, the pressures on the intemal regions are
Pf=-p4\ (10)
The condition of impermeability of the surface of sheU in contact with fluid may be ex- pressed as
a o l Bwo\
dr I2 dt IJ-
where w is the normal displacement of the sheU, Vf is the velocity of fluid.
The potential function $ may be defined by the following expression
^{r, e, X, t) = Y{r)f(x) cos(m^)e''^^ (12) where T ( r ) is a function to be defined. Substitution (12) into Laplace's equation (8), gives
Eq. (13) may be resolved using modified Bessel functions of first and second kind of order m, yielding
Y(r) = Al J„, (fc„r) -F A2K,,, {k„r). (14) FinaUy, the potential <!> is expressed as
And then the pressure P is written by
Y{r) dhi:
P ; ( r , e , j : , t ) = - p ; ^
dfi (16)
In addition, the pressure Pf must remain finite, which means that the funchon R{r) is defined by: Y(r) = .4iJ„(J:„r) inside the shells.
The hydrodynamic pressure acting on the cylindrical shell is then defined by [21]
1 dhoo
f~^''fm+ k„Rl„,+i {k„R) //„, {k„R) ' W ' ^ ^ This value will be introduced in (5) in order to construct the Dynamic Stiffness Matrix for
the studied structure.
4H Tran Ich Thinh, Nguyen Manh Cuong. Vu Quoc Hien
3. CONTINUOUS ELEMENT FORMULATION FOR ORTHOTROPIC CYLINDRICAL SHELLS COMPLETELY FILLED WITH FLUID
The state-solution vector is y = {uo,vo,ivo,fx>Ve>^x,^xe,Qx>^x>MxeV- This vector can be expressed in the Fourier expansion form, for the symmetric circumferential mode m as
{ UQ{x,e,t) wo{x,e,t] (px{x,e,t) NAx,B,t) Qx{x,e,t) Mx{x,e,t) }^ =
£ { U„,{x) W,n{x) q)xm{x) N:„„{x) Q:,ni{x) M ^ „ , ( x ) j " ^ COS m ^ g " ^ ' ,
'"=1 T (18) {vo{x.e,t) Nxe{x,9,t) ipg{x,e,t) Mxeix,8,t) }'=
E { v,,{x) Nxemix) <pem{x) MxenM Ysinmee'^'.
Similar ttansformations for anti-symmettic modes will easily be obtained by swap- ping cos(m^) and sin{m6) in (18). Further detailed formulations in this research concern only symmettic modes for the sake of simplification.
Applying the CEM procedure presented in [16,20], the following differential equa- tions have been calculated
dl""! 1 ^ I^Vxm 1 », C5 . ,
dx kAss ^ dx Cl c^c^ ^ d(pem B(,(, Aee.. 1
— 7 — = W:r0«j M:,em " fncp^n, - r-fSnu
dx Ce cc R
mcy m I 2 , Cs \ Cam
dx Rci CiR R2 \ " c i c-ij (, 2 , <tf-C7\ 2 mcs / 2 m^cs [l«aj^ + -gr ) -"•: - R N , „ „ + -^9„. + l^hco^ + - ^ iQa, / , 2 2 , kAiim^\ 1 kA„m
- ^ = [io'^' - Pf^^ + ^ ^ j «>». - ^Q». - -^n,..,
dM„„ 1 fcs \ . 1 . 2 Cg\ mcg Ct (19)
(w+ll)..
R2J'^"^ R2dx Rci \ R^ J K2 R -'" ^ ^ ' » +
-I^?.,,,, + ( fcffl' + k4« + - ^ j re„, - -^M.
Dynamic stiffness meihod for free vibrations anaiysis of partial fiuiit-fiiied ortiiotropic circular
with
Cl = -Bf, + AiiD,i,C2 = {AuBn-AnBi2)/cuC3= (BnBn-AnDu)/ci, Ct = (B12B11 - Ai2Dn)/ci,Cs = (BnDi2 - Bi2Dii)/ci,C6 = AuCi + B12C2 + A22, C? = A12C5 + B12C3 + B22,C8 = B12C4 + Di2C2 + B22,C9 = B12C5 + D22 + D12C3, ClO = 8^6 - A66D66,C„ = (A12C3 + D22/R^)/R,C12 = (D12C4 + D22/R^)/R.
Eq. (19) can be expressed in the matrix form for each circumferential mode m dx -- A„,y„,
where A,„ is a 10 x 10 mattix (see Appendix).
The dynamic ttansfer matrix T,„ is evaluated by [16,17]
„A„L , T i 2 T22 And the dynamic stiffness mattix K(a;)„i is [17,18,20]
12 Til -T12 T2:-T22Tf2^Tii T22Tf2^
^i^)m =
(20)
(21)
(22)
4. ASSEMBLY PROCEDURE FOR CONTINUOUS ELEMENTS OF PARTIALLY FLUID-FILLED CYLINDRICAL SHELLS
It is important to note that the proposed Dynamic Stiffness Matrix can be used for both orthottopic cylinders without fluid (by setting p^ = 0) and for cylindrical sheUs completely filled with fluid {pf ^ 0).
Naturally, a partially fluid-filled cylinder is composed by two regions with differ- ent characteristics of material and geometry: an empty zone and a fluid-contained one (see Fig, 2). The problem becomes complex which requires a much more complicated coupled fluid-shell system of equations to solve. In such case, the application of analyt- ical approaches in order to resolve those system is obviously very tedious and difficult.
Moreover, FEM simulation meets also difficulties either in modeling fluid-shell sttuctures with various properties or in choosing an appropriate meshing for obtaining accurate so- lutions.
.fi"«™;,0
%„„
Fig. 2. Assembly of Dynamic Stiffness Matrix for a partially fluid-filled cylinder The assembly technique of continuous elements demonsttates it's performance in this situation. That procedure is a powerful capacity of CEM and has successfully been
50 Tran Ich Thinh, Nguyen Manh Cuong, Vu Quoc Hien
applied in our previous research [17] to study the hard problem of composite plates rest- ing on non-homogenous elastic foundation. The logical development of continuous ele- ment model consists to adapt this assembly algorithm to the problem of partiaUy fluid- fiUed cylindrical sheUs.
Fig. 2 fllusttates the construction of the Dynamic Stiffness Mattix for a partial fluid- filled cylindrical shell using the assembly procedure. Despite the complexity of the prob- lem, the proposed algorithm makes possible a simple and efficient solution. In this case, the different mentioned zones of the sheU are modeled by two Dynamic Stiffness Matii- ces i.e.: KemptyCi^) for the empty region (with pf = 0) and Kfluid(<J^) for the completely fluid-fiUed zone {pf ^ 0). FinaUy, the Dynamic Stiffness Matrix of the partially fluid- filled cylinder wUl be obtained by an assembly of the two above mattices similarly to the method used in our previous researches [16-18,20]. This strong capacity of CEM con- sists an interestmg and powerful approach to deal with complicated structures such as shell partially in contact with inside or outside fluid, combined conical-cylindrical shells, shells with stiffeners ...
5. NUMERICAL RESULTS AND DISCUSSION
A Matlab program has been written using the presented formulation in order to study the vibrational behavior of orthotropic cylindrical shells in contact with fluid and subjected to various boundary conditions. Obtained values will be validated by compar- ing with available results in the literature and with FEM.
5.1. Free vibration of orthotropic cylindrical shell filled with fluid
Now our model will be validated for orthotropic cylindrical shell filled with fluid.
The cylinder is subjected to simply supported-simply supported boimdary condition.
The followmg dfrnensionless natural frequency is used for comparison:
n = ajRy/{l-Vi2V2i)p/E-i.
A parametric analysis for orthottopic cylindrical shells without fluid has been con- ducted using five different cases of orthottopic material with varying E1/E2 relations and Poisson ratios. All stiadied orthottopic cases are resumed in Tab. 1. Only the first longitudinal mode is considered in Tab. 2 and Tab. 3.
Tablel. SheU material properties Case
1 2 3
"12 0.131926 0.131926 0.131926
"21 0.012114
0.04 0.131926
£1 (GPa) 227.350
68.599 20.545
£2 (GPa) 20.876 20.799 20.545
G12 (GPa) 7958 7.958 7.958
Obtained results by CEM comparing to those of Warburton and Soni [2] and Li and Chen [5] basing on theirs analytical solution are illustrated in Tab. 2. For circumferential modes varying from 2 to 7, the high precision of our model is confirmed. The present
Dynamic stiffness method for free vibrations analysis of partial fluid-filled orthotropic circular ...
Table 2. Comparison of natural frequency parameter O for the orthottopic cylindrical shell with pubhshed results. L/R ^ 2.0, h/R ^0.01 and m - 1
Warburton and Soni
[21 Li and Chen [5]
Present Case
1 2 3 1 2 3 1 2 3
n 2
0.119875 0.207798 0.330057 0.119888 0.207822 0.330100 0.1199 0.2078 0.3301
3 0.085272 0.139463 0.198610 0.085277 0.139494 0.198607 0.0852 0.1395 0.1986
4 0.065184 0.100747 0.134684 0.065190 0.100772 0.134742 0.0652 0.1007 0.1346
5 0.054311 0.081915 0.113414 0.054336 0.081952 0.113515 0.0543 0.0819 0.1134
6 0.050975 0.079310 0.122501 0.051014 0.0793S5 0.122652 0.0510 0.0793 0.1225
7 0.054227 0.089331 0.150636 0.054286 0.089433 0.150753 0.0542 0.0893 0.1506 Table 3. Natural frequency parameter O for the fluid-fiUed orthottopic cylindrical shell and
varying « ; L / R - 1.0, ft/R - 0.01 and m = 1
Case
1 2 3
n 5
Prado [9]
0.0628 0.1049 0.1616
Present 0.0629 0.1049 0.1601
6 Prado
[9]
0.0585 0.0954 0.1437
Present 0.0586 0.0953 0.1422
7 Prado
[91 0.0575 0.0929 0.1426
Present 0.0575 0.0926 0.1408
8 Prado
[91 0.0597 0.0975 0.1568
Present 0.0597 0.0971 0.1546
formulation otters good values which are perfectly closed to those of Li and Chen. And this simiUtude is observed for three investigated cases of orthotropic materials.
The last analysis consists of a simply supported orthottopic cylindrical shell filled with water. In Tab. 3, natural frequencies of a cylinder {L/R = 1.0,h/R = 0.01) with five above orthottopic materials are computed and compared to work of Prado et al. [9] using their analytical solutions.
It is obvious to remark very small differences between results calculated by CEM and by Prado et al. [9]. Moreover, the high precision of our model are vahdated through out all five cases of orthottopic materials. Therefore, by using the exact closed form so- lution of the differential equations of the structure, CEM using only two elements is an atttactive approach with high precision and smaU volume of data storage, especially for analyzing vibrations of isottopic and orthottopic cylindrical sheUs partiaUy or fully filled
52 Tran Ich Thinh, Nguyen Manh Cuong, Vu Quoc Hien
with water. Otherwise, these advantages combirung with the possibihty of using the as- sembly procedure of FEM are precious features of CEM which aUows our formulation working weU in aU low, medium and high frequency ranges.
5.2. Effect of filled fluid level and shells parameters on frequency of orthotropic cylin- drical shells
First, in this section, the effect of filled fluid level on frequency of clamped-free orthottopic cylindrical shells is investigated. The dimensions of cylindrical shell and ma- terial properties are: £i = 227.35 GPa; £2 = 20.87 GPa; i;i2 = 0.131926; G12 ^ G13 = G23 = 7.958 GPa; h = 0.0254 m; R = lO/i; L = IK; ps = 1600 kg/m^; pf = 1000 k g / m l Tab. 4 shows the influence of the liquid level on the natural frequencies with correspond- ing modes.
Table 4. Effect of fluid level on natural frequencies (Hz) of a clamped-free orthottopic cylindrical shell
m
1
2
3 n
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
H/L 0
830.97 588.73 663.56 1021.44 1549.77 2090.28 1722.11 1621.99 1782.03 2161.95 3303.99 3233.23 3190.24 3289.25 3539.30
0.25 829.23 587.44 662.34 1019.59 1546.60 2045.59 1682.59 1584.02 1739.55 2108.57 3153.33 3012.01 2971.26 3067.29 3307.88
0.5 784.92 554.96 630.01 968.78 1453.77 1669.42 1388.27 1319.88 1477.01 1838.39 276753 2673.74 2671.77 2798.34 3061.84
0.75 649.29 457.04 530.09 8 2 7 7 4 1258.47 1501.30 1279.81 1237.40 1410.59 1780.29 2162.56 2198.16 2267.92 2435.18 2718.17
1 429.40 241.82 357.04 653.63 1067.59
952.53 728.20 706.47 921.49 1307.25 1183.10 1196.37 1244.81 1432.81 1782.48
The results in Tab. 4 show that, for all the modes of vibration considered, the filled fluid can reduce significantly the nattiral frequency of an orthottopic cylindrical sheU.
For example, for the first mode {m = l,n = 1), the natural frequency is approximately 0.25%, 5.5%, 21.8% and 48.3% correspondmg to fluid filUng levels, H/L = 0.25,0.5,0.75 and 1.0 respectively lower in fluid than in air {H/L = 0), whereas for the mode m = 3,n = 3, the reduction of the natural frequency is approximately 6.8%, 16.3%, 28.9%
and 61% corresponding to fluid filling levels, H/L = 0.25,0.5,0.75 and 1.0 respectively.
Dynar. is method for free vibrations analysis (^partial fluid-filled orthotropic circular...
Consequentiy, the high modes are more sensitive to the effect of the containing fluid than the low modes. From the Tab. 4, it can be seen clearly that the reduction of the natural frequency of the cylindrical shell decreases first slowly then quickly as the fluid height increases.
Next, the effects of length-to radius ratio and thickness-to radius ratio on the fre- quencies of fluid-filled simply-supported orthofropic cylindrical sheU are presented in Figs. 3-4, respectively.
Fig. 3. Variation of natural frequencies (Hz) with various values of L/K for orthottopic fluid-filled simply supported cylindrical shell (m = 1,2,3)
Fig. 4. Variation of natural frequencies (Hz) with various values oih/R for orthottopic fluid-filled simply supported-supported cylindrical sheU (m — 1,2,3)
54 Tran Ich Thinh, Nguyen Manh Cuong. Vu Quoc Hien
Fig. 3 shows variations of natural frequencies of the orthottopic cylindrical wifli sunply supported edges and completely filled with fluid, versus L/R The values of nat- ural frequencies of the cyUndrical shells for all modes are decrease as flie ratio, L/R, increase. For example, as the ratio, L/R, increases from 2 to 10, for the first mode (m = l , n = 1) the natural frequencies are decreased from 921.05 Hz to 144.6 Hz;
for the mode m = 3,n = 3, the natural frequencies are decreased from 2725.76 Hz to 409.48 Hz, etc.
The effect of the variation of the ratio, h/R, on the natural frequencies of orthofropic sheUs with simply supported edges and completely fiUed wifli fluid is iUusfrated in Fig. 4 The values of natural frequencies of orthottopic cyUndrical sheUs are increased, as the ra- tio, h/R, increase. For example, as the ratio, h/R, increases from 0.02 to 0.1, for the first mode {m — l,n = 1) the natural frequencies are increased from 93.36 Hz to 921.05 Hz;
for the third mode (m = 3, n = 3), the natural frequencies are increased from 130.05 Hz to 2725.76 Hz, etc.
6. CONCLUSIONS
A new continuous element for orthotropic cylindrical shell with or without con- tact with fluid has been successfully consttucted in this research. Furthermore, a simple and robust assembly method has been inttoduced for analyzing the vibration charac- teristics of partiaUy fluid-fiUed orthottopic cylindrical shells. Different test cases have been examined which confirm the validity of the proposed formulation. Excellent agree- ments are found between results by Continuous Element Method and by other pubUshed ones demonsttating the high precision of this model. Continuous Element solutions for the mentioned structure reveals strong influences of fluid on vibratoire behaviors of the shell. Fluid effects are expressed by reducing significantly the natural frequencies and by changing the mode shapes of the sttucture. Such phenomenon must be taken into account in designs of isottopic and orthottopic cylinders in contact with fluid. Advan- tages of the presented model are low computational cost and the possibility of application in medium and high frequency range. Those precious characteristics of Continuous Ele- ment Method are high-Ughted in the case of fluid-structure interaction problem while the precision and computational time of Finite Element models are affected by the demand of a huge number of meshing elements.
ACKNOWLEDGEMENTS
This research is funded by Vietnam National Foundation for Science and Technol- ogy Development (NAFOSTED) under grant number: 107.02-2013.25.
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