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 fluidstructure, 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
instability can be illustrated as a feedback
mechanism between structural motion and the
resulting fluid forces. A small structural
displacement due to fluid forces or whatever alerts
the flow pattern, inducing a change in the fluid
forces; this in turn leads to further displacement,
and so on. When the flow velocity becomes
higher, the vibration amplitude becomes larger
and the impact phenomenon occurs that can lead
to unacceptable tube damage due to fatigue and
/or fretting wear in critical process equipment.
Therefore, the evaluation of complex vibrational
behavior of these structures is highly desirable in
nuclear industry to avoid such problems.
Introduction
Component failures due to excessive flowinduced vibrations continue to affect the
performance
and
reliability
of
nuclear
components, piping system and tube heat
exchangers. Fluid-elastic vibrations have been
recognized as a major cause of failure in shelland-tube-type heat exchangers. Fluid-elastic
vibrations result from coupling between fluidinduced dynamic forces and the motion of
structure. Depending on the boundary conditions,
static (buckling) and dynamic (flutter) instabilities
are possible in the structures at sufficiently high
flow velocities. The nature of fluid-elastic
2
Corresponding author
1
2.1
While several studies have been conducted on the
dynamic stability of circular cylindrical shells
conveying fluid, but in contrast, non-linear studies
of shells subjected to either internal or external
axial flow are few. Particularly interesting, in case
of internal axial flow, are the studies of Païdoussis
&Dennis [1], Lakis & Païdoussis [2, 3], Weaver
& Unny [4], Pettigrew [5], Selmane &Lakis [6]
and Amabili et al. [7]. These works are performed
based on the classical shell theory by neglecting
the shear deformation effect while this later plays
a very important role in reducing the effective
flexural stiffness of composite shells and also the
moderately thick structures. The present work
addresses the question of stability of anisotropic
cylindrical shells, based on a shearable shell
theory, subjected to internal and external axial
flow. The non-linearities due to large amplitude
shell motion are taken into account, by using the
modal coefficient approach, while the amplitude
of shell displacement remains within the linear
range from the fluid point of view.
Structure Model
The equations of motion of anisotropic cylindrical
shells in terms of U, V and W (axial, tangential
and radial displacements of the mean surface of
the shell),  x and 
(the rotations of the

normal about the coordinates of the reference
surface), see Fig. 1, and in terms of Pij’s elements
are written as follows:


Lm U , V , W ,  x ,   , Pij  0.
i, j  1,2,3,...,10
(1)
where anisotropic elasticity matrix (Pij’s
elements), and five linear differential operators
(Lm), are fully given in [8].
The finite element developed is shown in Fig. 1.
It is a cylindrical panel segment defined by two
nodal lines i and j. Each node has five degrees of
freedom, three displacements and two rotations.
1
1
2
m  1,..., 5
2
m
Mathematical Model
The analytical solution involves the following
steps:

a) The strain-displacement relations expressed in
an arbitrary orthogonal curvilinear coordinate are
inserted into the equations of motion, obtained
based on shearable shell theory, of anisotropic
cylindrical shells. The mass and linear stiffness
matrices are determined for an empty finite
element, Fig. 1, and assemble the matrices for the
complete shell.
(A)
i j
N
Wmi
1mi
Vmi
i
2mi
Umi
m
j

(B)
Figure 1: (A) Finite element discretization.
(B) Nodal displacements at node i.
The displacement functions associated with the
axial wave number are assumed to be:
b) The coefficients of the modal equations are
derived using the non-linear part of the kinematics
relations.
m 
m 
xe ;  x ( x, )  DCos
xe
L
L
m 
m 
(2)
V ( x, )  BSin
xe ;  ( x,  )  ESin
xe
L
L
m 
W ( x, )  CSin
xe
L
U ( x,  )  ACos
c) A finite fluid element bounded by two nodal
lines, Fig. 1-B, is considered to account the effect
of the fluid on the structure.
d) The linear and non-linear natural vibration
frequencies are then obtained and compared with
the available results.
where m is the axial mode, and  is a complex
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
equation (6). The global matrices [Ms] and [Ks(L)]
may be obtained, respectively, by superimposing
the mass and stiffness matrices for each individual
panel finite element [9].
equal to zero. This brings us to the following
characteristic equation (see References [8, 9] for
more detail):
Det H   f 10 10  f 8 8  f 6 6  f 4 4  f 2 2  f 0
(3)
2.1.2
Non-Linear Stiffness Matrices of
Structure
The exact Green strain-displacement relations are
used in order to describe the non-linear behavior,
including large displacements and large rotations,
of anisotropic cylindrical shells. In common with
linear theory, it is based on refined shell theory in
which the shear deformations and rotary inertia
effects are taken into account. The approach
developed by Radwan and Genin [10] is used with
particular attention to geometric non-linearities.
Each roots of this equation yields a solution to the
linear equations of motion (1). The complete
solution is obtained by adding the ten solutions
independently. After carrying out the some
intermediate manipulations, that are not displayed
here (see Reference [9]), the following equations
are obtained :
 U ( x,  ) 
 V ( x,  ) 


 i 
 W ( x,  )   N   
 j 
  ( x,  ) 
 x

   ( x,  )
(4)
The coefficients of the modal equations are
obtained through the Lagrange method. Thus, the
non-linear stiffness matrices of second and third
order are then calculated by precise analytical
integration and superimposed on the linear part of
equations to establish the non-linear modal
equations. The main steps of this method are as
follow:
where  is the displacement vector at the
boundaries and [N] represents the displacement
function matrix. The constitutive relation between
the stress and deformation vector of cylindrical
shells is given as [9]:
N xx N x Qxx N
Nx Q M xx M x M  M x  
 
P B   i 
 j 
2.1.2.a Shell displacements are expressed as
generalized product of coordinate sums and
spatial functions:
(5)
The matrices [P], as a function of geometrical and
mechanical parameters of anisotropic cylindrical
shells, and [B] are given in [9].
u   qi (t ) U i  x,  
x   qi (t )  xi  x,  
i
i
v   qi (t ) Vi  x,  
   qi (t )  i  x,  
i
i
w   qi (t ) Wi  x,  
i
2.1.1 Mass and Linear Stiffness Matrices
Using the procedure of the classical finite
element, the mass and stiffness matrices are then
calculated. For one finite element, they may be
written as follows:
where the qi (t)’s functions are the generalized
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
k L     B T PB dA
(7)
(6)
2.1.2.b The deformation vector is written as a
function of the generalised coordinates by
separating the linear part from non-linear one:
00
where  s is the density of the shell, h its
    L    NL T
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
obtained analytically by carrying out the
necessary matrix operations over x and  in
(8)
This vector is given in [11]. The subscripts “L”
and “NL” mean “linear” and “non-linear”,
3
respectively. In general, these terms can be
expressed in the following form:
carrying out a large number of the intermediate
mathematical operations, while are not given here
due to the complexity of the manipulations, the
following non-linear modal equations are
obtained. These non-linear modal equations are
used to study the dynamic behavior of an empty
anisotropic cylindrical shell.
 xo   a j q j    AA jk q j q k
j
j
k
 xo   b j q j    BB jk q j q k
j
j
k
   c j q j    CC jk q j q k
o
x
j
j
(9-a)
k
 mijj   kij  j    kijk
( L)
 o   d j q j    DD jk q j q k
j
j
j
k
    kijks
 o   e j q j    EE jk q j q k
j
j
j
j
k
 x  n j q j    NN jk q j q k
j
j
k
 x   p j q j    PPjk q j q k
j
j
(9-b)
k
    s j q j    SS jk q j q k
j
j
j
k
2.1.2.c Lagrange’s equations of motion in the
generalized coordinates qi (t) is defined as:
 T V
 

 Qi
 qi qi
 j k
(12)
i  1,2,...
and
 P11 AAijks  P22 BBijks  P33CCijks  
 

  (13-b)
  P44 DDijks  P55 EEijks



 P66GGijks  P77 NNijks  P88 PPijks  




  dA
kijks( NL3)     P99 SSijks  P1010TTijks
 


 Pmn  AUX ijks I  AUX ijks J   








57
58


 P36  AUX ijks  AUX ijks  





(10)
Where T is the total kinetic energy, V the total
elastic strain energy of deformation and the Qi’s
are
the
generalized
forces.
Assuming
 NL    1 ,  2 ,...,  10 T , the strain energy V can
Where dA=R dx d  and:
m =1,2,...,9
I=1,3,5,...,55 and
m, n3,6
n=m+1 to 10
J=I+1
The Pij’s are the terms of the elasticity matrix [P]
and the terms AAijk, BBijk, ..., AUXijk58 and AAijks,
BBijks,..., AUXijks58 represent the coefficients of the
modal equations in step (2.1.2.d). Follow are the
expressions for the coefficients ai, AAij, AAijk and
AAjkrs, the others coefficients are obtained in the
same way, details are given in [11].
be defined as follow:
V
 j k  s  Qi
( NL 3)
( NL 2 )
 P11 AAijk  P22 BBijk  P33CCijk   
 

 
 P44 DDijk  P55 EEijk
(13-a)



P
GG

P
NN

P
PP


 66 ijk
77
ijk
88 ijk  


  dA
kijk ( NL2)     P99 SS ijk  P1010TTijk
 


 Pmn  AUX ijk I  AUX ijk J   








57
58


P
AUX

AUX
 36 

ijk
ijk  




Note: AAij=AAji, BBij=BBji and etc.
d  T

dt  q i
s
k
k
    t j q j    TT jk q j q k
j
k
j
Where mij, kij(L) are the terms of mass and linear
stiffness matrices given by equation (6). The
terms of kijk(NL2) and kijks(NL3) , which represent the
second and third-order non-linear stiffness
matrices, are given by the following integrals in
the case of anisotropic laminated cylindrical shell
based on the refined shell theory in which the
shear deformation and rotary inertia effects are
considered:
k
 o   g j q j    GG jk q j q k
j
j
a L
  Pij  i  j  Pkl  k  l  R d x d
2 00
(11)
Where:
a =1 if i = j or k = l (i, j =1,2,...,10), (k,l=3,6)
and i , j 3,6
a =2 if i  j or k  l (i ,j=1,2…,10) (k,l=3,6)
2.1.2.d After developing the total kinetic energy
and strain energy, using definitions (9), and then
substituting into the Lagrange equation (10) and
4
ai 
U i
x
AAij 
2.2.1 Dynamic Pressure
Based on the previous hypothesis, the potential
function must satisfy the Laplace equation. This
relation is expressed in the cylindrical coordinate
system by:
1  U i U j Vi V j Wi W j 




2  x x
x x
x x 
AAijk  ai AA jk  a j AAki  ak AAij
AAijks  2 AAis AA jk
 2 
(14)
1
r ,r ,r  12  ,  , xx  0. (16)
r
r
 is the potential function that represents the
velocity potential. The components of the flow
velocity are given by:
where U,V and W are spatial functions
determined by equations (2). In equation (14), the
subscript “i,j”, “i,j,k” and “i,j,k,s” represent the
coupling between two; three and four mode,
respectively. Substituting equation (2) into
equations (14), we obtain
Vx  U xu  , x ; V 
1
, ; Vr  , r
R
(17)
where Vx ,V and Vr are respectively the axial,
tangential and radial components of the fluid
velocity; Uxu is the velocity of the liquid through
the shell section. The Bernoulli equation is given
by:
ai  C i i mSin mx e i 
   m 2 Sin 2 mx 

i j


( i   j ) (15)
AAij  C i 
 C je
2
2
 (1     )m Cos mx 
i
j


2


 , t  V  P 

2  f 

m  m / L
where i ( i =1,...,10) are the roots of
characteristic equation (3) and m is the axial mode
number. The same definitions, as relation (15), are
obtained for other parameters and given in [11].
The constants Ci(i=1,...10) can be determined
using ten boundary conditions for each element.
The axial, tangential and radial displacements as
well as the rotations have to be specified for each
node.
 0.
(18)
r 
A full definition of the flow requires that a
condition be applied to the structure-fluid
interface. The impermeability condition ensures
contact between the shell and the fluid. This
should be:
Vr
rR
 , r
rR
 W,t  U xW, x r  R
(19)
From the theory of shells, we have:
Substituting these definitions into equation (13)
and then integrating over x and  , the two
expressions for the second- and third-order nonlinear matrices are obtained, as given in equation
(12).


W x,  , t    C j exp  j  it sin
10
j 1
m
x (20)
L
Assuming then,
 x, , r , t    R j (r ) S j x, , t 
10
2.2
Fluid Model
j 1
Linear potential flow theory is applied to describe
the fluid effects that lead to the inertial,
centrifugal and Coriolis forces. The mathematical
model is based on the following hypothesis: i) the
fluid flow is potential; ii) the fluid is irrotational,
incompressible and non-viscous.
(21)
The function S j ( x,  , t ) is explicitly determined
by applying the impermeability condition (19) and
using the radial displacement (20). Substituting
the assumed function  into equation (16) leads to
the following differential Bessel equation:
5
2
r2
d R j (r )
dr
2

 R j (r ) i
2
r
m k2 r 2
dR j ( r )
dr
Substituting relation (24) into (18), we obtain the
equation for the pressure on the shell wall.

(22)
    0.
 i j
j 1
where “i” is the complex number, i2=-1 and  j is

 im
r   B Yi j 

 L

r

(23)
where J i j and Yi j are, respectively, the Bessel
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
zero. For outside flow ( r   ); this means that
the constant “A” is equal to zero. When the shell
is simultaneously subjected to internal and
external flow, we have to take the complete
solution (23). We carry the Bessel equation
solution back into (21) to obtain the final
expression of velocity potential evaluated at the
shell wall:
10
imRu
(24)
 (r , x, , t )   Z uj (
)W j , t  U xuW j , x 
j 1
3
mRu
)
L
and
Z uj (
L
( L)
(M s   M f )   ( K s  K f )    
( NL 2 )
f
( NL 3)
2
s
(28)
3
s
where {  } is the displacement vector and [Ms],
[KsL], [KsNL2] and [KsNL3] are, respectively the
mass, linear and second- and third-order nonlinear stiffness matrices of the structure,
respectively, and [Mf], [Kf] and [Cf] are the
inertial, centrifugal and Coriolis forces,
respectively, due to the fluid effect.
Ru
(26)
if u  e
imRu Yi j 1 (imRu / L)
i j 
L
Yi j (imRu / L)
where  j ( j  1,...,10)
Differential
      
  K     0
 C  K
Ru
if u  i (25)
J
imRu i j 1 (imRu / L)
i j 
L
J i j (imRu / L)
mRu
)
L
Non-Linear
Relations
The structural and fluid mass and stiffness
matrices, either linear or non-linear, as well as the
fluid damping matrix, obtained in the previous
sections, are only determined for one element.
The global mass, stiffness and damping matrices
are obtained by assembling the matrices for each
element. Assembling is done in such way that all
the equations of motion and the continuity of
displacements at each node are satisfied. These
matrices are designated as [M], [K] and [C],
respectively.
where
Z uj (

By introducing the displacement function (20)
into pressure expression (27), performing the
matrix operation and thereafter integration over
the fluid element required by the finite element
method, the linear matrices (mass [mf], damping
[cf] and stiffness [kf]) of moving fluid are
obtained. Finally, the global linear matrices [Mf],
[Cf] and [Kf] may be obtained, respectively, by
superimposing the different matrices for each
individual fluid finite element.
the complex solution of the characteristic equation
for the empty shell. The general solution of
equation (22) is given by:
 im
R j r   A J i j 
 L

10
2
Pu    f u  Z uj W j ,tt  2U xuW j , xt  U xu
W j , xx (27)
2
are the roots of the
characteristic equation of the empty shell;
J i j and Yi j are, respectively, the Bessel
functions of the first and second kind of order
“ i j ”; “m” is the axial mode number; “R” is the
Setting:
  q
mean radius of the shell; “L” its length; the
subscript “u” is equal to “i” for internal flow and
is equal to “e” for external flow.
(r )
q j (t )   j j (t )
 j (0)  1 and  j (0)  0.
6
(29)
where

in this work is a hybrid finite element based on a
combination of refined shell theory, modal
expansion approach and potential flow theory.
This method is capable of obtaining the high as
well as low frequencies with high accuracy. The
values of shear correction factors used in the
calculation have been taken  2 / 12 .
represents the square matrix for
eigenvectors of the linear system and qis a time
related vector. Numerical solution of the coupled
system (28) is difficult and costly. Here, we limit
ourselves to solving the uncoupled system. In this
case, equation (28) is reduced to the following
equation:
i   i i    i   i (i / h) 
2
i
Non-Dimensional Frequency
90
2
i
(30)
 i (i / h 2 ) i3  0.
where
i 
i 
and
C f ii
mii
k
;  i2 
NL 3
sii
mii
NL 2
sii
k
k ii
; i 
h
mii
mii
h ; mii  msii  m f ii
2
(31)
y
o
Fibre 0
60
Sanders' Theory
50
Present
40
R/h=50
L/R=5
m=4
30
m=3
20
10
0
0
Results
2
3
4
5
6
7
8
9
10
Figure 2: Variation of non-dimensional natural
frequencies in conjunction with variation of m.
a) Linear Vibrations of empty and liquid filled
isotropic and anisotropic cylindrical shells- It
should be noted that in the two first examples, the
natural frequencies of the structures are also
obtained using Sanders’ theory (non-shearable
shell theory), by authors.
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
four symmetric layers cross-ply (0o/90o/90o/0o)
laminated shell whose mechanical properties are
given as:
and
This research work is focused on the shear
deformation and geometrically non-linear effects
on the dynamic behavior of anisotropic cylindrical
shells conveying fluid. Non-linearity effects
produce either hardening or softening behavior in
circular cylindrical shells. Considering the shear
deformation effects leads to reducing the flexural
stiffness of the structures. The developed method
1
Circumferential Wave Number (n)
E1=25E2; G23=0.2E2; G13=G12=0.5E2, ν12=0.25;
ρ=1
3
ensional Freque/ncy
Numerical
Discussions
x
70
m=1
k ii  k  k f ii
L
sii
where “h” represents the shell thickness. The
square root of coefficient k ii / mii represents the ith
linear vibration frequency of system. The solution
 i (t ) of the non-linear differential equations (30),
which satisfies the conditions in (29), is calculated
by a fourth order Runge-Kuta numerical method.
The linear and non-linear natural frequencies are
evaluated by a systematic search for the
 i (t ) roots as a function of time. The  NL /  L
ratio of linear and non-linear frequency is
expressed as a function of non-dimensional ratio
i / h where i is the vibration amplitude.
4
o
Fibre 90
80
Present
2.5
Sanders' Theory
2
7
1.5
m=5
Figure 4: Natural frequencies of a simply
supported cylindrical shell.
Non-Dimensional Frequency
0.7
Figure 3: Variation of non-dimensional natural
frequencies in terms of m & n.
In the next example (Fig. 3), the effect of axial
mode number on the non-dimensional natural
frequencies (    o R(  (1   2 ) / E )1 / 2 ) of an
isotropic cylindrical shell is studied and the results
are compared with the obtained corresponding
values based on the Sanders’ theory.
  0.3
200
0
140
0
Present
R/h=20
m=1
n=1
0.3
0.2
R/h=100
0.1
1
2
3
4
5
6
7
The fluid depth effect is also studied for the halffilled cylindrical shell in Fig. 4.
Fig. 5 is carried out for a simply supported,
isotropic circular cylindrical shell completely
filled with liquid. The frequency parameter,
(    o R(  (1   2 ) / E )1 / 2 ), is shown for
different values of R/h and L/R and is compared
with provided results in Ref. [14].
b) Stability of the shells subjected to flowing fluidThe influence of the flow velocity on the
frequency parameter of cylindrical shells is
studied through Figs 6 and 7 for different values
of R/h, L/R, axial and circumferential wave
numbers. The obtained results, in Fig. 6, are
compared with those of theory [4].
120
0
100
0
25
80
0
60
0
m=2
20
0
8
0
0
2
4
6
8
10
Circumferential Wave Number (n)
9
Figure 5: Frequency distribution of a fluid-filled
cylindrical shell.
  7680(kg / m3 )
40
0
8
Length-to-Radius Ratio L/R
12
nsional Frequency
Natural Frequency (Hz)
160
0
[Ref. 14]
0.4
R/h=300
Present, Empty
Present, Full
Experiment [13],
Empty
Present, Half-Filled
180
0
0.5
0
Fig. 4 shows the natural frequencies computed for
closed simply supported, circular cylindrical shell
for m=2 and compared with the experimental
results, given in [13]. To determine natural
frequencies with the developed program, based on
the present theory, only 10 elements are required
to provide acceptable accuracy. As can be seen,
there is good agreement between the present
theoretical results and those of experimental.
Dimensions and material properties are given as
follow:
R  0.175(m)
L  0.664(m) t  1(mm)
E  206(GPa)
0.6
Present
m=2
[Ref.4]
20
L/R=2
R/h=100
15
f/s=0.128
n=5
10
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
isotropic simply supported cylindrical shell.
Figure 6: Stability of a simply supported
cylindrical shell as a function of internal flow
velocity.
K 1/ 2
Eh 3
uo  2 (
) ;K 
L sh
12(1   2 )
c) Linear vibration of submerged cylindrical
shells- Fig.
8 shows the non-dimensional,
(   o R(  s (1   2 ) / E )1 / 2 ) ,frequency variation
as a function of circumferential wave number for
three different cases, shell in air, fluid-filled shell
and shell immersed in fluid and are compared
with those of theory [15]. The two theories give
nearly identical results.
The u and ω are, respectively, the velocity of the
flowing fluid and the natural frequency. As the
flow velocity increases, Fig. 6, the two theories
generate significantly different results. This might
be attributed to i) not considering the influence of
transverse shear deformation in Ref.[4] and ii)
limitations of the theory (Ref.[4]) associated with
the use of too few terms in the application of
Galerkin’s method.
d) Non-linear vibrations of empty and submerged
cylindrical shells- The influence of non-linearities
on the frequencies of a simply supported
cylindrical shell, along with corresponding values
given in References [16 and 17] is shown in Fig.
9. The given results in Ref. [16] were obtained
based on Donnell’s simplified non-linear method.
Raju and Rao [17] used the finite element method
based on an energy formulation.
In these two figures, the following parameters are
defined:
U  u / u o ;    /  o ; o  u o / L
2
L/R=2
R/h=100
f/s=.128
n=3
1
25
d
n=4
20
n=5
m=1
- - - -m=2
15
n=3
10
n=4
n=5
5
9
0
0
1
2
3
Dimensionless Velocity
4
Non-Dimensional Frequency
Non-Dimensional Frequency
30
b
Present
[Ref.15]
0.1
Shell in
Air
Fluid Filled
m=1
L / R=4
R / h=400
b /d=1.0
f /s=.132
0.01
Shell Immersed in
Fluid
5
0.001
0
2
4
6
8
10
12
Figure 10: Non-linearity effect on the frequency
ratio of a submerged cylindrical shell.
Figure 8: Frequency variation of empty, fluidfilled and immersed in fluid shell with respect to
(n).
5
This paper deals with some of the problems that
arise when considering geometric non linearities,
shear deformation, rotary inertia and flowing fluid
effects in the study of dynamic and stability
behavior of elastic, anisotropic and isotropic
cylindrical shells. An efficient hybrid finite
element method, modal expansion approach,
shearable shell theory and linear potential flow
one have been used to develop the non-linear
dynamic equations of the coupled fluid-structure
system. The shell equations are used in full for the
determination of the displacement functions. It is
believed that the refined shear deformation theory
and effects of geometric non-linearities of the
structures presented here are essential for
predicting an accurate response for anisotropic
shell structures. A full implementation of the nonlinear dynamic equations is conducted to show the
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
the cylindrical shells subjected to flowing fluid.
Fig. 10 shows the non-linearity effect on the
frequency ratio of a steel open (φ=100o)
cylindrical shell totally submerged in fluid. The
shell is simply supported. The following data are
considered into calculations:
R=450mm; h=1.5mm; L=1350mm, ρf/ρs=.128
1.2
Present
Raju and Rao [17]
Nowinski [16]
m=1
n=4
1.1
1
0.9
0
0.5
1
1.5
2
2.5
3
Amplitude to Thickness Ratio j/h
The natural frequencies of the coupled fluidstructure are lower than the corresponding values
of empty shells due to increased kinetic energy
without a corresponding increase in the strain
energy. In the case of flowing fluid, the
centrifugal and Coriolis terms generate complex
eigenvalue problems, non-self-adjoint differential
equations. Therefore, the system may experience
static (buckling) and dynamic (flutter)
instabilities. As long as the effective stiffness of
the system remains positive as flow velocity
increases, the system will oscillate asymptotically
about its neutral equilibrium position; otherwise it
will diverge to a new equilibrium position,
different from neutral (buckling). As long as the
Figure 9: Relative frequency of simply supported
cylindrical shell versus relative amplitude.
2.1
1.9
n=1
n=2
1.7
Relative Frequency
ωNl/ωL
Relative Frequency NL/L
1.3
E=200GPa
=0.3
=7800kg / m3
R=2.54
L=40
cm
h=0.0254
cm
cm
1.5
1.3
1.1
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
Amplitude to Thickness Ratio Γi/h
2
Conclusion
effective fluid damping of the system remains
positive as flow velocity is increased, vibrations
will be damped; otherwise they will be amplified
(flutter).
Quiescent Fluid, Journal of Fluid and Structures,
No. 12, 1998, pp. 883-918.
[8] M.H. Toorani and A.A. Lakis, General
Equations of Anisotropic Plates and Shells
Including Transverse Shear Deformations, Rotary
Inertia and Initial Curvature Effects, Journal of
Sound and Vibration, 237 (4), 2000, pp. 561-615.
It is shown that the non-linearities associated with
fluid, under no-flow conditions, have no or very
little effect on the natural frequency of a
cylindrical shell for amplitudes up to two times
the shell thickness [6]. Under flow conditions,
non-linear effects were found to increase with
flow rate increasing but the importance of the
contribution of flow-non-linearities to this overall
trend has yet to be determined. Attempting to this
work is left to future investigations.
[9] M.H. Toorani and A.A. Lakis, Shear
Deformation Theory in Dynamic Analysis of
Anisotropic Laminated Open Cylindrical Shells
Filled With or Subjected to a Flowing Fluid,
Computer Methods in Applied Mechanics and
Engineering, 190, 2001, pp. 4929-4966.
[10] H. Radwan and J. Genin J., Non-Linear
Modal Equations for Thin Elastic Shells,
International Journal of Non-Linear Mechanics,
10, 1975, pp. 15-29.
References:
[1] M. P. Païdoussis and J.P. Denis, Flutter of
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
Cylindrical Shells with a Refined Shell Theory,
Technical Report, Polytechnique of Montreal,
EPM-RT-01-07, 2002.
[2] A.A. Lakis and M.P. Païdoussis, Free
Vibration of Cylindrical Shells Partially Filled
with Liquid, Journal of Sound and Vibration, No.
19, 1971, pp. 1-15.
[12] J.L. Sanders J.L., Nonlinear Theories for
Thin Shells, Quarterly of Applied Mathematics,
21, 1963, pp.21-36.
[3] A.A. Lakis and M.P. Païdoussis, Shell Natural
Frequencies of the Pickering Steam Generator,
Atomic Energy of Canada Ltd., AECL, Report
No. 4362, 1973.
[4] D.S. Weaver and T.E. Unny, On the Dynamic
Stability of Fluid-Conveying Pipes, Journal of
Applied Mechanics, No. 40, 1973, pp. 48-52.
[13] M. Amabili and G. Dalpiaz, Breathing
Vibrations of a Horizontal Circular Cylindrical
Tank Shell, Partially Filled with Liquid, Journal
of Vibration and Acoustics, 117, 1995, pp. 187191.
[5] M.J. Pettigrew, Flow-Induced Vibration
Technology: Application to Steam Generators,
Lecture Series Presented at Babcock & Wilcox
Canada, November 2000.
[14] A.A. Lakis and M. Sinno, Free Vibration of
Axisymmetric and Beam-Like Cylindrical Shells
Partially Filled with Liquid, Int. J. of Num. Meth.
In Eng., 33, 1992, pp. 235-268.
[6] A. Selmane and A.A. Lakis, Non-Linear
Dynamic Analysis of Orthotropic Open
Cylindrical Shells Subjected to a Flowing Fluid,
Journal of Sound and Vibration, No. 202, 1997,
pp. 67-93.
[15] P.B. Gonçalves and R.C. Batista, Frequency
Response of Cylindrical Shells Partially
Submerged or Filled with Liquid, Journal of
Sound and Vibration, 113(1), 1987, pp. 59-70.
[16] J.L. Nowinski J.L., Non-Linear Transverse
Vibrations of Orthotropic Cylindrical Shells,
AIAA Journal, 1, 1963, pp. 617-620.
[7] M. Amabili M., F. Pellicano and M.P.
Païdoussis, Non-Linear Vibrations of Simply
Supported Circular Cylindrical Shells, Coupled to
11
[17] K.K. Raju and G.V. Rao, Large Amplitude
Asymmetric Vibrations of Some Thin Elastic
Shells of Revolution, Journal of Sound and
Vibration, 44, 1976, pp. 327-333.
12