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Hybrid Finite Element in Non Linear Structural Dynamics of

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					                                                          2nd WSEAS Int. Conf. On NON-LINEAR ANALYSIS,
                                                  NON-LINEAR SYSTEMS and CHAOS (WSEAS NOLASC 2003)

                                                              Vouliagmeni, Athens, Greece, December 29-31, 2003



         Hybrid Finite Element in Non-Linear Structural Dynamics of
                  Anisotropic Tubes Conveying Axial Flow
                                       M.H. Toorani1 and A.A. Lakis2

                      1. Nuclear Engineering Department, Babcock & Wilcox Canada
                           P.O. Box 310, Cambridge, Ontario, Canada N1R 5V3

                 2. Mechanical Engineering Department, Ecole Polytechnique of Montreal
                     C.P. 6079, Station Centre-ville, Montreal (QC), Canada H3C 3A7
                             Tel.:(514) 340-4711, x4906, Fax: (514) 340-4176



Abstract: A semi-analytical approach has been developed in the present theory to determine the geometrical
non-linearity effects on the natural frequencies of anisotropic cylindrical tubes conveying axial flow.
Particular important in this study is to obtain the natural frequencies of the coupled system of the fluid-
structure, taking the geometrical non-linearity of the structure into consideration and estimating the critical
flow velocity at which the structure loses its stability. The displacement functions, mass and stiffness
matrices, linear and non-linear ones, of the structure are obtained by exact analytical integration over a
hybrid element developed in this work. Linear potential flow theory is applied to describe the fluid effect that
leads to the inertial, centrifugal and Coriolis forces. Stability of tubes subjected to the flowing fluid is also
discussed. Numerical results are given and compared with those of experiment and other theories to
demonstrate the practical application of the present method.

Key-Words: Non-Linear, Dynamics, Flowing Fluid, Anisotropic, Cylindrical Tubes

1         Introduction                                      instability can be illustrated as a feedback
Component failures due to excessive flow-                   mechanism between structural motion and the
induced vibrations continue to affect the                   resulting fluid forces. A small structural
performance      and    reliability   of     nuclear        displacement due to fluid forces or whatever alerts
components, piping system and tube heat                     the flow pattern, inducing a change in the fluid
exchangers. Fluid-elastic vibrations have been              forces; this in turn leads to further displacement,
recognized as a major cause of failure in shell-            and so on. When the flow velocity becomes
and-tube-type heat exchangers. Fluid-elastic                higher, the vibration amplitude becomes larger
vibrations result from coupling between fluid-              and the impact phenomenon occurs that can lead
induced dynamic forces and the motion of                    to unacceptable tube damage due to fatigue and
structure. Depending on the boundary conditions,            /or fretting wear in critical process equipment.
static (buckling) and dynamic (flutter) instabilities       Therefore, the evaluation of complex vibrational
are possible in the structures at sufficiently high         behavior of these structures is highly desirable in
flow velocities. The nature of fluid-elastic                nuclear industry to avoid such problems.

2
    Corresponding author


                                                        1
                                                            2.1          Structure Model
While several studies have been conducted on the            The equations of motion of anisotropic cylindrical
dynamic stability of circular cylindrical shells            shells in terms of U, V and W (axial, tangential
conveying fluid, but in contrast, non-linear studies        and radial displacements of the mean surface of
of shells subjected to either internal or external          the shell),  x and   (the rotations of the
axial flow are few. Particularly interesting, in case
                                                            normal about the coordinates of the reference
of internal axial flow, are the studies of Païdoussis
                                                            surface), see Fig. 1, and in terms of Pij’s elements
&Dennis [1], Lakis & Païdoussis [2, 3], Weaver
                                                            are written as follows:
& Unny [4], Pettigrew [5], Selmane &Lakis [6]
and Amabili et al. [7]. These works are performed
based on the classical shell theory by neglecting
                                                                                             
                                                                  Lm U , V , W ,  x ,   , Pij  0.          m  1,...,5
                                                                                                                              (1)
the shear deformation effect while this later plays                                                     i, j  1,2,3,...,10
a very important role in reducing the effective             where anisotropic elasticity matrix (Pij’s
flexural stiffness of composite shells and also the         elements), and five linear differential operators
moderately thick structures. The present work               (Lm), are fully given in [8].
addresses the question of stability of anisotropic
cylindrical shells, based on a shearable shell              The finite element developed is shown in Fig. 1.
theory, subjected to internal and external axial            It is a cylindrical panel segment defined by two
flow. The non-linearities due to large amplitude            nodal lines i and j. Each node has five degrees of
shell motion are taken into account, by using the           freedom, three displacements and two rotations.
modal coefficient approach, while the amplitude
of shell displacement remains within the linear                              2         i j
                                                                         1
range from the fluid point of view.
                                                                     1       m        N          Wmi
                                                                                               1mi
2       Mathematical Model                                                                 Vmi                  m
                                                                                                    i
The analytical solution involves the following                                              2mi
steps:                                                                                    Umi
                                                                                                                              j

a) The strain-displacement relations expressed in                                                        
an arbitrary orthogonal curvilinear coordinate are                  (A)
inserted into the equations of motion, obtained
based on shearable shell theory, of anisotropic
cylindrical shells. The mass and linear stiffness                                                         (B)
matrices are determined for an empty finite                 Figure 1: (A) Finite element discretization.
element, Fig. 1, and assemble the matrices for the                    (B) Nodal displacements at node i.
complete shell.
                                                            The displacement functions associated with the
b) The coefficients of the modal equations are              axial wave number are assumed to be:
derived using the non-linear part of the kinematics                           m                         m 
relations.                                                  U ( x, )  ACos       xe ;  x ( x, )  DCos     xe
                                                                               L                            L
                                                                             m                        m  (2)
c) A finite fluid element bounded by two nodal              V ( x, )  BSin      xe ;  ( x, )  ESin     xe
                                                                              L                           L
lines, Fig. 1-B, is considered to account the effect                                          m 
of the fluid on the structure.                                               W ( x, )  CSin    xe
                                                                                               L
d) The linear and non-linear natural vibration              where m is the axial mode, and  is a complex
frequencies are then obtained and compared with
the available results.                                      number. A system of five homogeneous linear
                                                            functions is obtained by substituting (2) into
                                                            equations of motion (1). For the solution to be
                                                            non-trivial, the determinant of this system must be


                                                        2
equal to zero. This brings us to the following                                  equation (6). The global matrices [Ms] and [Ks(L)]
characteristic equation (see References [8, 9] for                              may be obtained, respectively, by superimposing
more detail):                                                                   the mass and stiffness matrices for each individual
                                                                                panel finite element [9].
DetH   f10 10  f 8 8  f 6 6  f 4 4  f 2 2  f 0     (3)
                                                                                2.1.2    Non-Linear Stiffness Matrices of
Each roots of this equation yields a solution to the                                     Structure
linear equations of motion (1). The complete                                    The exact Green strain-displacement relations are
solution is obtained by adding the ten solutions                                used in order to describe the non-linear behavior,
independently. After carrying out the some                                      including large displacements and large rotations,
intermediate manipulations, that are not displayed                              of anisotropic cylindrical shells. In common with
here (see Reference [9]), the following equations                               linear theory, it is based on refined shell theory in
are obtained :                                                                  which the shear deformations and rotary inertia
                          U ( x,  )                                          effects are taken into account. The approach
                          V ( x,  )                                          developed by Radwan and Genin [10] is used with
                         
                                       
                                               i                  (4)       particular attention to geometric non-linearities.
                          W ( x,  )   N   
                           ( x,  )          j 
                          x                                                   The coefficients of the modal equations are
                            ( x,  ) 
                                                                              obtained through the Lagrange method. Thus, the
where   is the displacement vector at the                                    non-linear stiffness matrices of second and third
boundaries and [N] represents the displacement                                  order are then calculated by precise analytical
function matrix. The constitutive relation between                              integration and superimposed on the linear part of
the stress and deformation vector of cylindrical                                equations to establish the non-linear modal
shells is given as [9]:                                                         equations. The main steps of this method are as
                                                                                follow:
   N xx N x Qxx N    Nx Q M xx M x M  M x  
                                                                                2.1.2.a Shell displacements are expressed as
             
   P B   i                                                (5)             generalized product of coordinate sums and
             j                                                               spatial functions:
The matrices [P], as a function of geometrical and
mechanical parameters of anisotropic cylindrical
shells, and [B] are given in [9].                                                 u   qi (t ) Ui  x,                 x   qi (t )  xi  x,  
                                                                                      i                                            i                     (7)
                                                                                  v   qi (t ) Vi  x,                   qi (t )  i  x,  
                                                                                      i                                            i
2.1.1 Mass and Linear Stiffness Matrices                                                                 w   qi (t ) Wi  x,  
Using the procedure of the classical finite                                                                  i
element, the mass and stiffness matrices are then
calculated. For one finite element, they may be                                 where the qi (t)’s functions are the generalized
written as follows:                                                             coordinates and the spatial functions U, V, W,  x
                                L
                                                                                and  are given by equation (2).
                  m   s h   N  T N  dA
                                00
                           L
                                                                      (6)       2.1.2.b The deformation vector is written as a
                  k L     B T PB dA                                  function of the generalised coordinates by
                           00
                                                                                separating the linear part from non-linear one:
where s is the density of the shell, h its
                                                                                                         L    NL T                            (8)
thickness, dA a surface element, [P] the elasticity
matrix and the [N] and [B] are derived from
equations (4, 5). The matrices [ms] and [ks(L)] are                             This vector is given in [11]. The subscripts “L”
obtained analytically by carrying out the                                       and “NL” mean “linear” and “non-linear”,
necessary matrix operations over x and  in


                                                                            3
respectively. In general, these terms can be                     carrying out a large number of the intermediate
expressed in the following form:                                 mathematical operations, while are not given here
      x   a j q j    AA jk q j q k
       o                                                         due to the complexity of the manipulations, the
                    j                j       k                   following non-linear modal equations are
      x   b j q j    BB jk q j q k
       o                                                         obtained. These non-linear modal equations are
                    j                j       k                   used to study the dynamic behavior of an empty
                                                                 anisotropic cylindrical shell.
        c j q j    CC jk q j q k
        o
        x                                            (9-a)
                        j            j           k
                                                                             
                                                                        mij j   kij  j    kijk           j k
                                                                                        ( L)            ( NL 2 )
      o   d j q j    DD jk q j q k                               j               j                     j   k                 (12)
                    j                j           k
                                                                           kijks
                                                                                            ( NL 3)
                                                                                                       j k  s  Qi     i  1,2,...
      o   e j q j    EE jk q j q k                                   j   k   s
                    j            j           k                   Where mij, kij(L) are the terms of mass and linear
                                                                 stiffness matrices given by equation (6). The
      o   g j q j    GG jk q j q k                        terms of kijk(NL2) and kijks(NL3) , which represent the
                    j                j           k               second and third-order non-linear stiffness
      x  n j q j    NN jk q j q k                          matrices, are given by the following integrals in
            j                j           k                       the case of anisotropic laminated cylindrical shell
      x   p j q j    PPjk q j q k              (9-b)       based on the refined shell theory in which the
                j                j           k                   shear deformation and rotary inertia effects are
         s j q j    SS jk q j q k                         considered:
                    j            j           k
                                                                                     P AAijk  P22 BBijk  P33CCijk   
         t j q j    TT jk q j q k                                            
                                                                                        11                                      
                j            j           k                                           P44 DDijk  P55 EEijk
                                                                                     
                                                                                                                                
                                                                                                                                   (13-a)
                                                                                                                                 
                                                                                     P66GGijk  P77 NN ijk  P88 PP  ijk 
                                                                                                                             
                                                                                                                               dA
Note: AAij=AAji, BBij=BBji and etc.                               kijk ( NL 2)     P99 SSijk  P
                                                                                                     1010TTijk               
                                                                                                                                 
                                                                                     Pmn  AUX ijk I  AUX ijk J   
                                                                                                                              
2.1.2.c Lagrange’s equations of motion in the                                                                                
                                                                                                                                
generalized coordinates qi (t) is defined as:                                        P36  AUX ijk 57  AUX ijk 58  
                                                                                                                               
                                                                                                                             
                                                                 and
            d  T           T V                                                 P AAijks  P22 BBijks  P33CCijks  
                           
                             q  q  Qi           (10)                          
                                                                                       11                                  
            dt  qi
                             i    i                                               P44 DDijks  P55 EEijks
                                                                                    
                                                                                                                            (13-b)
                                                                                                                          
                                                                                                                               
                                                                                    P66GGijks  P77 NN ijks  P88 PP  
                                                                                                                       ijks 
                                                                                                                            
                                                                  kijks( NL3)     P99 SSijks  P1010TTijks              dA
Where T is the total kinetic energy, V the total                                                                            
                                                                                                                               
elastic strain energy of deformation and the Qi’s                                   Pmn  AUXijksI  AUXijksJ   
                                                                                                                            
                                                                                                                           
are     the    generalized    forces.  Assuming                                                                               
                                                                                           AUX 57  AUX 58  
 NL   1 ,  2 ,..., 10 T , the strain energy V can                          P36 
                                                                                              ijks         ijks  
                                                                                                                    
                                                                                                                                
                                                                                                                                

be defined as follow:                                            Where dA=R dx d  and:
                                                                 m =1,2,...,9                I=1,3,5,...,55 and
                                                                                             m, n3,6
         Pij i  j  Pkl  k  l  R d x d
     a L
V                                                   (11)        n=m+1 to 10                 J=I+1
     2 00                                                        The Pij’s are the terms of the elasticity matrix [P]
Where:                                                           and the terms AAijk, BBijk, ..., AUXijk58 and AAijks,
a =1 if i = j or k = l (i, j =1,2,...,10), (k,l=3,6)             BBijks,..., AUXijks58 represent the coefficients of the
       and i , j 3,6                                            modal equations in step (2.1.2.d). Follow are the
a =2 if i  j or k  l (i ,j=1,2…,10) (k,l=3,6)                  expressions for the coefficients ai, AAij, AAijk and
                                                                 AAjkrs, the others coefficients are obtained in the
2.1.2.d After developing the total kinetic energy                same way, details are given in [11].
and strain energy, using definitions (9), and then
substituting into the Lagrange equation (10) and


                                                             4
       U i                                                        2.2.1 Dynamic Pressure
ai                                                                Based on the previous hypothesis, the potential
        x
                                                                   function must satisfy the Laplace equation. This
         1  U i U j Vi V j Wi W j                          relation is expressed in the cylindrical coordinate
AAij                                 
         2  x x
                       x x    x x                           system by:
                                                                              2 
                                                                                        1
                                                                                          r ,r ,r  12  ,  , xx  0. (16)
AAijk  ai AA jk  a j AAki  ak AAij                  (14)                             r              r

AAijks  2 AAis AA jk                                               is the potential function that represents the
                                                                   velocity potential. The components of the flow
where U,V and W are spatial functions                              velocity are given by:
determined by equations (2). In equation (14), the                                                             1
subscript “i,j”, “i,j,k” and “i,j,k,s” represent the                    Vx  U xu  , x ; V                   , ; Vr  , r    (17)
                                                                                                               R
coupling between two; three and four mode,
respectively. Substituting equation (2) into
                                                                   where Vx ,V and Vr are respectively the axial,
equations (14), we obtain
                                                                   tangential and radial components of the fluid
                                                                   velocity; Uxu is the velocity of the liquid through
ai  Ci i mSinmx e i 
                                                                   the shell section. The Bernoulli equation is given
             m 2 Sin 2 mx                                     by:
           i j                          ( i   j ) (15)
AAij  Ci                          C je
           (1     )m Cos 2 mx 
                          2
                 i   j                                                                      2   
                                                                                     , t  V  P                      0.        (18)
m  m / L                                                                                   2 f 
                                                                                                               r 

where  i ( i =1,...,10) are the roots of                          A full definition of the flow requires that a
characteristic equation (3) and m is the axial mode                condition be applied to the structure-fluid
number. The same definitions, as relation (15), are                interface. The impermeability condition ensures
obtained for other parameters and given in [11].                   contact between the shell and the fluid. This
The constants Ci(i=1,...10) can be determined                      should be:
using ten boundary conditions for each element.
The axial, tangential and radial displacements as                       Vr   r R
                                                                                     , r   r R
                                                                                                     W,t  U xW, x r  R         (19)
well as the rotations have to be specified for each
node.                                                              From the theory of shells, we have:
Substituting these definitions into equation (13)
                                                                                                                               m
and then integrating over x and  , the two
                                                                                        j 1
                                                                                                           
                                                                      W x,  , t    C j exp  j  it sin
                                                                                         10
                                                                                                                               L
                                                                                                                                  x (20)
expressions for the second- and third-order non-
linear matrices are obtained, as given in equation
(12).                                                              Assuming then,


                                                                               x, , r , t    R j (r ) S j x, , t 
                                                                                                      10
                                                                                                                                    (21)
2.2      Fluid Model                                                                                  j 1
Linear potential flow theory is applied to describe
the fluid effects that lead to the inertial,                       The function S j ( x,  , t ) is explicitly determined
centrifugal and Coriolis forces. The mathematical
                                                                   by applying the impermeability condition (19) and
model is based on the following hypothesis: i) the
                                                                   using the radial displacement (20). Substituting
fluid flow is potential; ii) the fluid is irrotational,
                                                                   the assumed function  into equation (16) leads to
incompressible and non-viscous.
                                                                   the following differential Bessel equation:



                                                               5
                                                                           Substituting relation (24) into (18), we obtain the
                     2
                  d R j (r )                 dR j (r )                     equation for the pressure on the shell wall.
             r2             2
                                      r                 
                                                              (22)
                                                                                                                                            
                     dr                        dr
                                                 0.
                                                                                            10
                                                                               Pu    f u  Zuj W j ,tt  2U xuW j , xt  U xuW j , xx (27)
                                                                                                                              2
              R j (r ) i                     i j
                                2                        2
                                     2
                                    mk r 2                                                 j 1

where “i” is the complex number, i2=-1 and  j is
                                                                           By introducing the displacement function (20)
the complex solution of the characteristic equation
                                                                           into pressure expression (27), performing the
for the empty shell. The general solution of
                                                                           matrix operation and thereafter integration over
equation (22) is given by:
                                                                           the fluid element required by the finite element
                                                                           method, the linear matrices (mass [mf], damping
                              im              im 
         R j r   A J i j     r   B Yi j     r       (23)         [cf] and stiffness [kf]) of moving fluid are
                               L               L                       obtained. Finally, the global linear matrices [Mf],
                                                                           [Cf] and [Kf] may be obtained, respectively, by
where Ji j and Yi j are, respectively, the Bessel                        superimposing the different matrices for each
                                                                           individual fluid finite element.
functions of the first and second kind of complex
order “ i j ”. For inside flow, the solution (23)
must be finite on the axis of shell (r=0); this
means we have to set the constant “B” equal to                             3          Non-Linear                                     Differential
zero. For outside flow ( r   ); this means that                                     Relations
the constant “A” is equal to zero. When the shell                          The structural and fluid mass and stiffness
is simultaneously subjected to internal and                                matrices, either linear or non-linear, as well as the
external flow, we have to take the complete                                fluid damping matrix, obtained in the previous
solution (23). We carry the Bessel equation                                sections, are only determined for one element.
solution back into (21) to obtain the final                                The global mass, stiffness and damping matrices
expression of velocity potential evaluated at the                          are obtained by assembling the matrices for each
shell wall:                                                                element. Assembling is done in such way that all
                             imRu
                                   )W j ,t  U xuW j , x 
                    10                                                     the equations of motion and the continuity of
  (r , x, , t )   Z uj (                                (24)
                     j 1                L                                 displacements at each node are satisfied. These
where                                                                      matrices are designated as [M], [K] and [C],
                                                                           respectively.
         mRu
                                                                                          
                                 Ru
              )                                      if u  i (25)
                                                                            (M s   M f )   ( K s  K f )    
Z uj (                                                                                               ( L)
          L               imRu J i j 1 (imRu / L)                                                                                            (28)
                   i j 
                                 J i j (imRu / L)                          C   K
                                                                                            K     0
                                                                                                 ( NL 2 )          2       ( NL 3)   3
                            L                                                     f          s                         s

and
  Z uj (
           mRu
                )
                                   Ru
                                                       if u  e
                                                                (26)       where {  } is the displacement vector and [Ms],
            L               imRu Yi j 1 (imRu / L)                     [KsL], [KsNL2] and [KsNL3] are, respectively the
                     i j 
                              L    Yi j (imRu / L)                       mass, linear and second- and third-order non-
                                                                           linear stiffness matrices of the structure,
where  j ( j  1,...,10)                     are the roots of the
                                                                           respectively, and [Mf], [Kf] and [Cf] are the
characteristic equation of the empty shell;                                inertial, centrifugal and Coriolis forces,
J i j and Yi j are, respectively, the Bessel                             respectively, due to the fluid effect.
functions of the first and second kind of order
                                                                           Setting:
“ i j ”; “m” is the axial mode number; “R” is the
mean radius of the shell; “L” its length; the
subscript “u” is equal to “i” for internal flow and
                                                                                                         q
                                                                                                            (r )

                                                                                                       q j (t )   j j (t )                    (29)
is equal to “e” for external flow.
                                                                                                       j (0)  1 and  j (0)  0.
                                                                                                                       


                                                                       6
where             represents the square matrix for                                    in this work is a hybrid finite element based on a
eigenvectors of the linear system and qis a time                                     combination of refined shell theory, modal
                                                                                       expansion approach and potential flow theory.
related vector. Numerical solution of the coupled                                      This method is capable of obtaining the high as
system (28) is difficult and costly. Here, we limit                                    well as low frequencies with high accuracy. The
ourselves to solving the uncoupled system. In this                                     values of shear correction factors used in the
case, equation (28) is reduced to the following
equation:                                                                              calculation have been taken  2 / 12 .
                                                                                                                   90


      i   i i    i   i (i / h) 
                                                                                                                                                        o

                              2                  2                                                                                            Fibre 90




                                                                                       Non-Dimensional Frequency
                               i                  i                                                                80
                                                          (30)
         i (i / h 2 ) i3  0.
                                                                                                                                                                x
                                                                                                                   70                               y
                                                                                                                                                        o
                                                                                                                                                 Fibre 0
                                                                                                                   60
where                                                                                                                                            Sanders' Theory
                                                                                                                   50                            Present
                                                   NL 2
               C f ii                k ii        k
      i               ;  i2           ; i 
                                                   sii                                                             40
                                                      h                                                                                                             R/h=50
                                                                                                                                                                    L/R=5
               mii                   mii          mii                                                              30
                                                                                                                                           m=4

                   NL 3
                                                                                                                            m=3
               k                                                                                                   20
      i                 h ; mii  msii  m fii
                   sii     2
                                                          (31)
                mii                                                                                                10
                                                                                                                                                                             m=1
      and           k ii  k  k f ii
                               L
                               sii                                                                                 0
                                                                                                                        0     1   2   3    4    5           6        7       8     9   10


where “h” represents the shell thickness. The                                                                                Circumferential Wave Number (n)
square root of coefficient k ii / mii represents the ith
linear vibration frequency of system. The solution                                     Figure 2: Variation of non-dimensional natural
                                                                                       frequencies in conjunction with variation of m.
 i (t ) of the non-linear differential equations (30),
which satisfies the conditions in (29), is calculated                                  a) Linear Vibrations of empty and liquid filled
by a fourth order Runge-Kuta numerical method.                                         isotropic and anisotropic cylindrical shells- It
The linear and non-linear natural frequencies are                                      should be noted that in the two first examples, the
evaluated by a systematic search for the                                               natural frequencies of the structures are also
 i (t ) roots as a function of time. The  NL /  L                                   obtained using Sanders’ theory (non-shearable
ratio of linear and non-linear frequency is                                            shell theory), by authors.
expressed as a function of non-dimensional ratio
 i / h where i is the vibration amplitude.                                           In the first example, the different longitudinal
                                                                                       vibration modes (    o R 2 (  / E2 ) / h ) as a
                                                                                       function of the circumferential wave number are
                                                                                       drawn in Fig. 2. This figure shows the results for
4         Numerical                     Results           and                          four symmetric layers cross-ply (0o/90o/90o/0o)
          Discussions                                                                  laminated shell whose mechanical properties are
                                                                                       given as:
This research work is focused on the shear
deformation and geometrically non-linear effects
                                                                                       E1=25E2; G23=0.2E2; G13=G12=0.5E2, ν12=0.25;
on the dynamic behavior of anisotropic cylindrical
                                                                                       ρ=1
shells conveying fluid. Non-linearity effects
produce either hardening or softening behavior in                                      3
circular cylindrical shells. Considering the shear
deformation effects leads to reducing the flexural                                                                                                      Present
                                                                                       2.5
                                                                 ensional Freque/ncy




stiffness of the structures. The developed method
                                                                                                                                                        Sanders' Theory
                                                                                                                                          m=5
                                                                                             2
                                                                 7
                                                                                       1.5
                                                                                                                      Figure 4: Natural frequencies of a simply
                                                                                                                      supported cylindrical shell.
                                                                                                                                   0.7




                                                                                                                                   0.6




                                                                                          Non-Dimensional Frequency
                                                                                                                                   0.5


                                                                                                                                                                                                         [Ref. 14]
                                                                                                                                   0.4
                                                                                                                                                                                                         Present
                                                                                                                                                                    R/h=20
Figure 3: Variation of non-dimensional natural                                                                                                                                                       m=1
                                                                                                                                   0.3
frequencies in terms of m & n.                                                                                                                                                                       n=1

                                                                                                                                   0.2
In the next example (Fig. 3), the effect of axial                                                                                                             R/h=100
mode number on the non-dimensional natural
                                                                                                                                   0.1
frequencies (    o R(  (1   2 ) / E )1 / 2 ) of an
                                                                                                                                                  R/h=300
isotropic cylindrical shell is studied and the results                                                                                    0
                                                                                                                                              1        2      3         4    5      6        7       8        9      10
are compared with the obtained corresponding
values based on the Sanders’ theory.                                                                                                                        Length-to-Radius Ratio L/R
Fig. 4 shows the natural frequencies computed for                                                                     Figure 5: Frequency distribution of a fluid-filled
closed simply supported, circular cylindrical shell                                                                   cylindrical shell.
for m=2 and compared with the experimental
results, given in [13]. To determine natural                                                                          The fluid depth effect is also studied for the half-
frequencies with the developed program, based on                                                                      filled cylindrical shell in Fig. 4.
the present theory, only 10 elements are required
to provide acceptable accuracy. As can be seen,                                                                       Fig. 5 is carried out for a simply supported,
there is good agreement between the present                                                                           isotropic circular cylindrical shell completely
theoretical results and those of experimental.                                                                        filled with liquid. The frequency parameter,
Dimensions and material properties are given as                                                                       (    o R (  (1   2 ) / E )1 / 2 ), is shown for
follow:                                                                                                               different values of R/h and L/R and is compared
 R  0.175(m)      L  0.664(m) t  1(mm)                                                                             with provided results in Ref. [14].
E  206(GPa)                                   0.3       7680(kg / m3 )
                                                                                                                      b) Stability of the shells subjected to flowing fluid-
                           200
                           0                                                                                          The influence of the flow velocity on the
                                                       Present, Empty
                           180
                           0                           Present, Full
                                                                                                                      frequency parameter of cylindrical shells is
                                                                                                                      studied through Figs 6 and 7 for different values
  Natural Frequency (Hz)




                           160                         Experiment [13],
                           0
                                                       Empty Half-Filled
                                                       Present,                                                       of R/h, L/R, axial and circumferential wave
                           140
                           0                                                                                          numbers. The obtained results, in Fig. 6, are
                           120                                                                                        compared with those of theory [4].
                           0
                           100
                           0
                                                                                                                                              25
                           80
                           0                                                                                                                                                               Present
                                                                                                                      nsional Frequency




                                                                                                                                                                  m=2
                           60
                           0
                                                                                                                                                                                           [Ref.4]
                                                                                                                                              20
                           40
                           0                                               m=2                                                                                                   L/R=2
                           20                                                                                                                                                    R/h=100
                           0                                                          8
                             0
                                                                                                                                                  15                             f/s=0.128
                                 0       2      4       6       8          10    12

                                     Circumferential Wave Number (n)                                                                                                               n=5
                                                                                                                                              10
                                                                                                                       Figure 7: Stability of a cylindrical shell as a
                                                                                                                       function of flow velocity.

                                                                                                                       In Fig. 6, the first frequency becomes negative
                                                                                                                       imaginary at U=2.96, indicating static divergence
                                                                                                                       instability in the first axial mode, and reappeared
                                                                                                                       and coalesced at U=3.36 with that of the second
                                                                                                                       axial mode to produce mode flutter. Fig. 7 shows
                                                                                                                       the divergence instability phenomenon for an
                Figure 6: Stability of a simply supported                                                              isotropic simply supported cylindrical shell.
                cylindrical shell as a function of internal flow
                velocity.                                                                                              c) Linear vibration of submerged cylindrical
                                                                                                                       shells- Fig.        8 shows the non-dimensional,
                In these two figures, the following parameters are                                                      (   o R(  s (1   2 ) / E )1 / 2 ) ,frequency variation
                defined:                                                                                               as a function of circumferential wave number for
                 U  u / u o ;    /  o ; o  u o / L                                                              three different cases, shell in air, fluid-filled shell
                            K 1/ 2   2 Eh3                                                                            and shell immersed in fluid and are compared
                   uo  2 (   ) ;K                                                                                    with those of theory [15]. The two theories give
                       L sh         12(1   2 )                                                                      nearly identical results.

                The u and ω are, respectively, the velocity of the                                                     d) Non-linear vibrations of empty and submerged
                flowing fluid and the natural frequency. As the                                                        cylindrical shells- The influence of non-linearities
                flow velocity increases, Fig. 6, the two theories                                                      on the frequencies of a simply supported
                generate significantly different results. This might                                                   cylindrical shell, along with corresponding values
                be attributed to i) not considering the influence of                                                   given in References [16 and 17] is shown in Fig.
                transverse shear deformation in Ref.[4] and ii)                                                        9. The given results in Ref. [16] were obtained
                limitations of the theory (Ref.[4]) associated with                                                    based on Donnell’s simplified non-linear method.
                the use of too few terms in the application of                                                         Raju and Rao [17] used the finite element method
                Galerkin’s method.                                                                                     based on an energy formulation.



                            30
                                                                L/R=2
                                       n=3                      R/h=100                                                              1
                                                                f/s=.128
Non-Dimensional Frequency




                            25
                                                                                                                                                                        d
                                                                                                                                                                    b
                                                                                                      Non-Dimensional Frequency




                                      n=4

                            20                                                                                                                                                  Present
                                     n=5
                                                                                                                                                                                [Ref.15]
                                                                                                                                   0.1
                                                                                        m=1
                            15                                                   - - - -m=2
                                                                                                                                                              Shell in
                                                                                                                                                              Air
                                     n=3
                                                                                                                                                     Fluid Filled
                            10
                                     n=4

                                     n=5                                                                                                                                              m=1
                                                                                                                                   0.01
                                                                                                                                                                                    L / R=4
                            5
                                                                                                                                                                                   R / h=400
                                                                                                                                             Shell Immersed in                      b /d=1.0
                                                                                                                                             Fluid                                 /=.132
                                                                                                                                                                                    f s

                            0                                                                     9
                                 0           1         2        3            4                5


                                                 Dimensionless Velocity                                                           0.001

                                                                                                                                         0       2        4             6   8         10       12
                                                                                                                               Figure 10: Non-linearity effect on the frequency
                               Figure 8: Frequency variation of empty, fluid-                                                  ratio of a submerged cylindrical shell.
                               filled and immersed in fluid shell with respect to
                               (n).                                                                                            5       Conclusion
                                                                                                                               This paper deals with some of the problems that
                               Fig. 10 shows the non-linearity effect on the                                                   arise when considering geometric non linearities,
                               frequency ratio of a steel open (φ=100o)                                                        shear deformation, rotary inertia and flowing fluid
                               cylindrical shell totally submerged in fluid. The                                               effects in the study of dynamic and stability
                               shell is simply supported. The following data are                                               behavior of elastic, anisotropic and isotropic
                               considered into calculations:                                                                   cylindrical shells. An efficient hybrid finite
                                                                                                                               element method, modal expansion approach,
                               R=450mm; h=1.5mm; L=1350mm, ρf/ρs=.128                                                          shearable shell theory and linear potential flow
                              1.3                                                                                              one have been used to develop the non-linear
                                                                                                  Present
                                                              E=200GPa
                                                                                                  Raju and Rao [17]
                                                                                                                               dynamic equations of the coupled fluid-structure
                                                              =0.3
Relative Frequency NL/L




                                                              =7800kg / m
                                                                           3                      Nowinski [16]                system. The shell equations are used in full for the
                              1.2
                                                              R=2.54                             m=1                           determination of the displacement functions. It is
                                                              L=40
                                                              cm
                                                              h=0.0254
                                                              cm
                                                                                                 n=4                           believed that the refined shear deformation theory
                                                              cm                                                               and effects of geometric non-linearities of the
                                                                                                                               structures presented here are essential for
                              1.1
                                                                                                                               predicting an accurate response for anisotropic
                                                                                                                               shell structures. A full implementation of the non-
                                                                                                                               linear dynamic equations is conducted to show the
                                 1                                                                                             reliability and effectiveness of the present
                                                                                                                               formulation that gives a very good description of
                                                                                                                               geometrical non-linear and shear deformation
                                                                                                                               effects on the dynamic and stability behavior of
                              0.9
                                     0                 0.5           1           1.5         2           2.5          3        the cylindrical shells subjected to flowing fluid.
                                            Amplitude to Thickness Ratio j/h                                                  The natural frequencies of the coupled fluid-
                                                                                                                               structure are lower than the corresponding values
                               Figure 9: Relative frequency of simply supported                                                of empty shells due to increased kinetic energy
                               cylindrical shell versus relative amplitude.                                                    without a corresponding increase in the strain
                                                                                                                               energy. In the case of flowing fluid, the
                                                                                                                               centrifugal and Coriolis terms generate complex
                                             2.1
                                                                                                                               eigenvalue problems, non-self-adjoint differential
                                             1.9
                                                                                                                               equations. Therefore, the system may experience
                                                                                            n=1         n=2                    static (buckling) and dynamic (flutter)
                                             1.7                                                                               instabilities. As long as the effective stiffness of
                                                                                                                               the system remains positive as flow velocity
                            Relative Frequency




                                             1.5
                                                                                                                               increases, the system will oscillate asymptotically
                                             1.3                                                                               about its neutral equilibrium position; otherwise it
                                                                                                                               will diverge to a new equilibrium position,
                                             1.1
                                                                                                                               different from neutral (buckling). As long as the
                            ωNl/ωL




                                             0.9



                                             0.7
                                                                                                                          10
                                             0.5
                                                   0    0.2    0.4       0.6   0.8     1   1.2    1.4    1.6   1.8    2



                                                       Amplitude to Thickness Ratio Γ /h
effective fluid damping of the system remains               Quiescent Fluid, Journal of Fluid and Structures,
positive as flow velocity is increased, vibrations          No. 12, 1998, pp. 883-918.
will be damped; otherwise they will be amplified
(flutter).                                                  [8] M.H. Toorani and A.A. Lakis, General
                                                            Equations of Anisotropic Plates and Shells
It is shown that the non-linearities associated with        Including Transverse Shear Deformations, Rotary
fluid, under no-flow conditions, have no or very            Inertia and Initial Curvature Effects, Journal of
little effect on the natural frequency of a                 Sound and Vibration, 237 (4), 2000, pp. 561-615.
cylindrical shell for amplitudes up to two times
the shell thickness [6]. Under flow conditions,             [9] M.H. Toorani and A.A. Lakis, Shear
non-linear effects were found to increase with              Deformation Theory in Dynamic Analysis of
flow rate increasing but the importance of the              Anisotropic Laminated Open Cylindrical Shells
contribution of flow-non-linearities to this overall        Filled With or Subjected to a Flowing Fluid,
trend has yet to be determined. Attempting to this          Computer Methods in Applied Mechanics and
work is left to future investigations.                      Engineering, 190, 2001, pp. 4929-4966.

                                                            [10] H. Radwan and J. Genin J., Non-Linear
References:                                                 Modal Equations for Thin Elastic Shells,
                                                            International Journal of Non-Linear Mechanics,
[1] M. P. Païdoussis and J.P. Denis, Flutter of             10, 1975, pp. 15-29.
Thin Cylindrical Shells Conveying Fluid, Journal
of Sound and Vibration, No. 20, 1972, pp. 9-26.             [11] M.H. Toorani and A.A. Lakis, Geometrically
                                                            Non-Linear Dynamics of Anisotropic Open
[2] A.A. Lakis and M.P. Païdoussis, Free                    Cylindrical Shells with a Refined Shell Theory,
Vibration of Cylindrical Shells Partially Filled            Technical Report, Polytechnique of Montreal,
with Liquid, Journal of Sound and Vibration, No.            EPM-RT-01-07, 2002.
19, 1971, pp. 1-15.
                                                            [12] J.L. Sanders J.L., Nonlinear Theories for
[3] A.A. Lakis and M.P. Païdoussis, Shell Natural           Thin Shells, Quarterly of Applied Mathematics,
Frequencies of the Pickering Steam Generator,               21, 1963, pp.21-36.
Atomic Energy of Canada Ltd., AECL, Report
No. 4362, 1973.                                             [13] M. Amabili and G. Dalpiaz, Breathing
                                                            Vibrations of a Horizontal Circular Cylindrical
[4] D.S. Weaver and T.E. Unny, On the Dynamic               Tank Shell, Partially Filled with Liquid, Journal
Stability of Fluid-Conveying Pipes, Journal of              of Vibration and Acoustics, 117, 1995, pp. 187-
Applied Mechanics, No. 40, 1973, pp. 48-52.                 191.

[5] M.J. Pettigrew, Flow-Induced Vibration                  [14] A.A. Lakis and M. Sinno, Free Vibration of
Technology: Application to Steam Generators,                Axisymmetric and Beam-Like Cylindrical Shells
Lecture Series Presented at Babcock & Wilcox                Partially Filled with Liquid, Int. J. of Num. Meth.
Canada, November 2000.                                      In Eng., 33, 1992, pp. 235-268.

[6] A. Selmane and A.A. Lakis, Non-Linear                   [15] P.B. Gonçalves and R.C. Batista, Frequency
Dynamic Analysis of Orthotropic Open                        Response of Cylindrical Shells Partially
Cylindrical Shells Subjected to a Flowing Fluid,            Submerged or Filled with Liquid, Journal of
Journal of Sound and Vibration, No. 202, 1997,              Sound and Vibration, 113(1), 1987, pp. 59-70.
pp. 67-93.
                                                            [16] J.L. Nowinski J.L., Non-Linear Transverse
[7] M. Amabili M., F. Pellicano and M.P.                    Vibrations of Orthotropic Cylindrical Shells,
Païdoussis, Non-Linear Vibrations of Simply                 AIAA Journal, 1, 1963, pp. 617-620.
Supported Circular Cylindrical Shells, Coupled to


                                                       11
[17] K.K. Raju and G.V. Rao, Large Amplitude        Shells of Revolution, Journal of Sound and
Asymmetric Vibrations of Some Thin Elastic          Vibration, 44, 1976, pp. 327-333.




                                               12