This document presents a finite element analysis of the aeroelastic behavior of plates under combined thermal and aerodynamic loading. It derives the finite element model using a 16-term polynomial displacement function and Bogner-Fox-Schmidt quadrilateral elements. The model accounts for large deflections using von Karman strain-displacement relations and models the aerodynamic loading using quasi-steady piston theory. It will use this model to analyze the flutter boundary, limit cycle oscillations, and effects of thermal loading on the plate behavior.

1. Finite Element Analysis of the
Aeroelasticity Plates Under Thermal and
Aerodynamic Loading
Mohammad Tawfik, PhD
Aerospace Engineering Department
Cairo University

2. 2
Table of Contents
1. Introduction and Literature Review.....................................................................3
1.1. Panel-Flutter Analysis ....................................................................................5
1.2. Panel-Flutter Control......................................................................................9
2. Derivation of the Finite Element Model............................................................11
2.1. The Displacement Functions ........................................................................11
2.2. Displacement Function in terms of Nodal Displacement.............................12
2.3. Nonlinear Strain Displacement Relation......................................................15
2.4. Inplane Forces and Bending Moments in terms of Nodal Displacements ...18
2.5. Deriving the Element Matrices Using Principal of Virtual Work................19
2.5.1. Virtual work done by external forces .................................................23
3. Solution Procedures and Results of Panel Subjected to Thermal Loading .......25
4. Derivation of the Finite Element Model for Panel-flutter Problem...................28
4.1. Deriving the Element Matrices Using Principal of Virtual Works ..............28
4.2. 1st
-Order Piston Theory ................................................................................29
5. Solution Procedure and Results of the Combined Aerodynamic-Thermal
Loading Problem ..........................................................................................31
5.1. The Flutter Boundary ...................................................................................31
5.2. The Combined Loading Problem .................................................................34
5.3. The Limit Cycle Amplitude..........................................................................36
6. Concluding Remarks .........................................................................................40
References .............................................................................................................41

3. 3
1. Introduction and Literature Review
Panel-flutter is defined as the self-excited oscillation of the external skin of a flight
vehicle when exposed to airflow along its surface ‎1
.‎As‎early‎as‎late‎1950’s,‎the‎problem‎
of panel-flutter has drawn‎the‎researchers’‎attention‎‎1
, but it has not been of high interest
until the evolution of the National Aerospace Plane, the High Speed Civil Transport
(HSCT), the Advanced Tactical Fighter projects, and F-22 fighter ‎2-‎4
among other
projects of high-speed vehicles.
At high speed maneuvering of flight vehicles, the external skin might undergo self-
excited vibration due to the aerodynamic loading. This phenomenon is known as panel-
flutter. Panel-flutter is characterized by having higher vibration amplitude in the third
quarter length of the panel (Figure ‎1.1). This phenomena causes the skin panels of the
flying vehicle to vibrate laterally with high amplitudes that cause high inplane
oscillatory stresses; in turn, those stresses cause the panel to be a subject to fatigue
failure.
Figure ‎1.1 Sketch of panel-flutter phenomenon

4. 4
It is thus required to determine the panel-flutter boundary (critical dynamic
pressure) as well as the amplitude of vibration at which the panel will be oscillating in
post-flutter conditions. Linear analysis of the panel structure could be used to predict the
critical dynamic pressure of flutter, but is limited to that, as post flutter vibration is
characterized by being of high amplitude which needs the application of nonlinear
modeling techniques to describe.
The linear analysis though, predicts exponential growth of the amplitude of
vibration with the increase of the dynamic pressure in a post flutter condition. However,
it is worth noting that under those conditions, the vibration of the panel will be
influenced by inplane as well as bending stresses that lead to the limit cycle oscillation.
Thus, failure of the panel does not occur at post-flutter dynamic pressures, but extended
exposure to the panel-flutter decreases the fatigue life of the panel.
One of the earliest reviews that covered the topic of panel flutter was that presented
by Dowell in 1970. It covered the variety of available literature dealing with the
problem. Analytical and numerical methods predicting the flutter boundary as well as
the limit cycle amplitude for panels (beam models) were presented as well as the effect
of presence of in-plane loading. Later, Bismarck-Nasr presented reviews of the finite
element analysis of the aeroelastic problems of plates and shells. The study also
included the cases of in-plane loading and its effects on the problem. Lately, Mei et al.
presented another review that covered the topics of analytical and numerical analysis of
panel flutter together with the topics involving the flutter control and delay.

5. 5
Different methods were used to predict the post-flutter (limit cycle) attitude, which
is a nonlinear phenomenon by nature, of the panel; modal transformation approach with
direct numerical integration, harmonic balance, perturbation method, and nonlinear
finite element method ‎1,‎9
were used for that purpose.
The aerodynamic loading on the panel was also predicted using different
approaches; unsteady supersonic potential flow ‎10
, linearized potential flow ‎1,‎4
, and
quasi-steady piston theory. The most popular of which is the first order quasi-steady
piston theory that was introduced by Ashley and Zartarian ‎11
. That approximate theory
gives high accuracy results at high Mach numbers (M ∞>1.6).
At the flight condition of panel-flutter (usually supersonic flight conditions), the
phenomenon is associated with elevated temperatures, produced from the aerodynamic
heating through the boundary layer friction and the presence of shock waves. This
heating adds to the complexity of the problem by introducing panel stiffness reduction
and thermal loading, which might also be associated with post-buckling deflection. In
the following, a literature review of panel-flutter analysis and control topics is
presented.
1.1. Panel-Flutter Analysis
The non-linear finite element formulation, introduced by Mei ‎1
, was the basis on
which Dixon and Mei ‎12
, Xue and Mei ‎3
, and Abdel-Motagalay et al. ‎13
, build their finite
element models to analyze the flutter boundary, limit cycle, and the thermal problems,
with the extension to random loading and SMA embedding introduced by Zhong ‎14
.

6. 6
Different finite element models were developed to analyze the behavior of panels
subject to aerodynamic loading. Mei ‎1
introduced the use of nonlinear finite element
methods to predict the behavior of isotropic panels in the limit-cycle oscillations (LOC).
In his work, he used the quasi-steady first order piston theory to predict the
aerodynamic loading for M∞>1.6. He also presented a comparison of the effect of
different structural boundary conditions on the critical dynamic pressure and on
oscillation amplitude. Dixon and Mei ‎12
extended the use of finite element nonlinear
analysis to composite panels. von Karman strain-displacement relations were used to
represent the large deflections and the aerodynamic load was modeled using quasi-
steady first-order piston theory. They solved the equations of motion using the linear-
updated mode with a nonlinear time function (LUM/NTF) approximation. Results were
also presented for different boundary conditions.
Model enhancements were also developed to include the thermal effects as well as
the flow direction. Xue and Mei ‎3
presented a very good study on the combined effect of
aerodynamic forces and thermal loads on panel-flutter problems. They studied the effect
of the temperature elevation on the critical dynamic pressure as well as the buckling
temperature variation under different dynamic pressure conditions. The study also
showed the effect of the different boundary conditions on the amplitude of the limit
cycle oscillations (LOC). Abdel-Motagalay et al.‎13
studied the effect of flow direction
on the panel-flutter behavior using first order shear deformation theory for laminated
composite panels. They formulated the finite element nonlinear equations in structural
node degrees of freedom, and then reduced the number of equations using a modal

7. 7
transformation. The resulting reduced equations were then solved using the LUM/NTF
approximation.
Sarma and Varadan ‎9
adopted two methods to solve the nonlinear panel-flutter
problem, the starting point of the first solution was calculated using nonlinear vibration
mode and in the second they used linear mode as their starting point. They derived the
equations from energy‎considerations‎using‎Lagrange’s‎equations‎of‎motion,‎and‎then‎
reduced the equations to nonlinear algebraic equations to solve a double eigenvalue
problem.
Frampton et al. ‎4,‎15
applied the linearized potential flow aerodynamics to the
prediction and control of the flutter boundary using discrete infinite impulse response
(IIR) filters with conventional ‎15
and modern ‎4
control theories. To get their solutions,
they increased the non-dimensional dynamic pressure and calculated the coupled system
eigenvalues until the point of coalescence at which the first complex eigenvalue
appears. They studied the linear panel-flutter problem, thus, only presenting a prediction
of the flutter boundary.
Gray et al. ‎16
introduced the approximation of the third order unsteady piston theory
aerodynamics for the flow over a 2-D panel. Both nonlinear aerodynamic and structure
terms were considered in their finite element formulation. They also presented results
for different support conditions. They concluded that the third order piston theory
introduces a destabilizing effect as compared to the first order quasi-steady theory.

8. 8
Benamar et al. ‎17,‎18
formulated the large amplitude plate vibration problem and
developed the numerical model to apply the analysis to fully clamped plates. They
claimed that the assumption of the space-time solution w(x,y,t) can be presented in the
form w(x,y,t)=q(t)*f(x,y) may be inaccurate for nonlinear deflections. They suggested
that for high amplitudes and low aspect ratios, the effect of nonlinear (plastic) material
properties must be taken into consideration as well. They also presented a set of
experimental results conducted to investigate the dynamic response characteristics of
fully clamped plates at large vibration amplitudes ‎18
.
Different aerodynamic models were introduced to the solution of the panel-flutter
problem to enhance the results of the finite element model or introduce new ranges of
analysis. Yang and Sung ‎10
introduced the unsteady aerodynamic model in their
research on panel-flutter in low supersonic flow fields where the quasi-steady piston
theory fails to produce accurate results. Liu et al. ‎19
introduced a new approach for the
aerodynamic modeling of wings and panel in supersonic-hypersonic flight regimes.
Their model was a generalization of the piston theory. Their model also accounts for the
effects of wing thickness.
Lately, studies of panel flutter were directed to enhancing the models and
introducing different realistic parameters into the problem. Lee et al. modeled a panel
with sheer deformable model as well as applying boundary conditions in the form a
Timoshenko beams. Surace and Udrescu studied the panel flutter problem with higher
order finite element model with the effect of external static pressure. While Bismarch-
Nasr and Bones studied the effect of the aerodynamic damping on the panel flutter

9. 9
attitude. Most recently, Young and Lee presented a study of the flutter of the plates
subject to in-plane random loading. Those studies introduced more practical models for
the problem of panel flutter which was and still is very hard to study experimentally.
1.2. Panel-Flutter Control
Many researchers studied the problem of panel-flutter, but not as much studied the
control of the flutter problem. The studies of the panel-flutter control were mostly
conducted to increase the flutter boundaries (increase flutter Mach number). The main
aim of controlling the panel-flutter is to increase the life of the panels subjected to
fatigue stresses by delaying the flutter and/or decreasing the flutter amplitude.
Different control algorithms as well as the application of smart material were
studied to determine the feasibility of the application in the panel-flutter suppression
problem. Zhou et al. ‎24
presented an optimal control design to actively suppress large-
amplitude limit-cycle flutter motion of rectangular isotropic panels. They developed an
optimal controller based on the linearized modal equations, and the norms of feedback
control gain were employed to provide the optimal shape and position of the
piezoelectric actuator. They concluded that the in-plane force induced by piezoelectric
layers is insignificant in flutter suppression; on the other hand, the results obtained
verified the effectiveness of the piezoelectric materials in the panel-flutter suppression
especially for simply supported plates for which the critical dynamic pressure could be
increased by about four times. Frampton et al. ‎15,‎25
presented a study of the effect of
adding a self-sensing piezoelectric material to the panel, and using direct rate feedback.
They concluded that the use of their scheme increased the flutter non-dimensional
dynamic pressure significantly.

10. 10
Dongi et al.‎26
used self-sensing piezoactuators as a part of a dynamic feedback
control system to suppress flutter. They concluded that a linear observer-based-state
feedback control system fails in the face of structural nonlinearity. They accomplished
the required control using output feedback from a pair of collocated or self-sensing
piezoactuators. They concluded that this technique possesses good robustness properties
regarding nonlinearity, flight parameter variations, and pressure differentials. They also
concluded that it has the advantage of being a simple feedback scheme that can be
implemented with an analogue circuit.
Scott and Weisshaar ‎27
introduced the use of adaptive material in the control of
panel-flutter. They investigated the use of both piezoelectric material and Shape
Memory Alloys in modifying the flutter characteristics of a simply supported panel.
They concluded that for the piezoelectric material used in their study, no substantial
improvement was accomplished; on the other hand, they concluded that the SMA had
the potential for significantly increasing panel-flutter velocities, and that it is a possible
solution for the panel-flutter problems associated with aerodynamic heating.
The use of SMA in the delay of buckling and panel flutter was of interest to
different studies. Suzuki and Degaki ‎28
investigated the use of SMA into the suppression
of panel-flutter through the optimization of SMA thickness distribution into a 2-D
panel. While Tawfik et al. presented a study that involved the plate model with
combined thermal and aerodynamic loading. Their study extended to the investigation
of partial embedding of SMA in the panels as well as a partial study of the effect of the
fibers direction.

11. 11
2. Derivation of the Finite Element Model
In this section, the equation of motion with the consideration of large deflection are
derived for a plate subject to external forces and thermal loading. The thermal loading
is accounted for as a constant temperature distribution. The element used in this study is
the rectangular 4-node Bogner-Fox-Schmidt (BFS) C1
conforming element (for the
bending‎ DOF’s).‎ The‎ C1
type of elements conserves the continuity of all first
derivatives between elements.
2.1. The Displacement Functions
The displacement vector at each node for FE model is
T
vu
yx
w
y
w
x
w
w











 2
(‎2.1)
The above displacement vector includes the membrane inplane displacement vector
 T
vu and transverse displacement vector
T
yx
w
y
w
x
w
w











 2
.
The 16-term polynomial for the transverse displacement function is assumed in the
form
33
16
32
15
23
14
22
13
3
12
3
11
3
10
2
9
2
8
3
7
2
65
2
4321),(
yxayxayxayxaxyayxa
yaxyayxaxayaxyaxayaxaayxw


(‎2.2)
or in matrix form
  
  }{
xxyxyxyx1 3332232233322322
aH
ayxyxyxyxxyyxyxyyw
w

(‎2.3)

12. 12
where  T
aaaaaaaaaaaaaaaaa 16151413121110987654321}{  is
the transverse displacement coefficient vector. In addition, the two 4-term polynomials
for the inplane displacement functions can be written as
xybybxbbyxu 4321),(  (‎2.4)
xybybxbbyxv 8765),(  (‎2.5)
or in matrix form
  
  }{
00001
bH
bxyyxu
u

and   
  }{
10000
bH
bxyyxv
v

(‎2.6)
where    T
bbbbbbbbb 87654321 is the inplane displacement coefficient
vector. The coordinates and connection order of a unit 4-node rectangular plate element
are shown in Figure ‎2.1.
Figure ‎2.1. Node Numbering Scheme
2.2. Displacement Function in terms of Nodal Displacement
The transverse displacement vector at a node of the panel can be expressed by

13. 13





















































16
15
1
222222
2322322322
3232223222
3332232233322322
2
9664330220010000
3322332020100
3232302302010
1
a
a
a
yxxyyxxyyxyx
yxyxyxyxxyxyxyxyx
yxxyyxxyyyxyxyxyx
yxyxyxyxxyyxyxyyxxyxyxyx
yx
w
y
w
x
w
w

(‎2.7)
Substituting the nodal coordinates into equation (2.7), we obtain the nodal bending
displacement vector {wb} in terms of {a} as follows,







































































































































































































16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
2
2
32
32
222222
2322322322
3232223222
3332232233322322
2
32
2
32
4
2
4
4
4
3
2
3
3
3
2
2
2
2
2
1
2
1
1
1
0000300200010000
0000003000200100
0000000000010
0000000000001
9664330220010000
3322332020100
3232302302010
1
0000030020010000
0000000000100
0000000003002010
0000000000001
0000000000010000
0000000000000100
0000000000000010
0000000000000001
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
bb
bb
bbb
bbb
baabbaabbaba
babababaababababa
baabbaabbbabababa
babababaabbababbaabababa
aa
aaa
aa
aaa
yx
w
y
w
x
w
w
yx
w
y
w
x
w
w
yx
w
y
w
x
w
w
yx
w
y
w
x
w
w
(‎2.8)
or
    aTw bb  (‎2.9)
From equation (2.9), we can obtain
     bb wTa
1
 (‎2.10)
Substituting equation (2.10) into equation (2.3) then
       bwbbw wNwTHw 
1
(‎2.11)
where the shape function for bending is

14. 14
     1
 bww THN (‎2.12)
Similarly, the inplane displacement {u, v} can be expressed by
 b
xyyx
xyyx
v
u













10000
00001
(‎2.13)
Substituting the nodal coordinates into the equation (2.13), we can obtain the
inplane nodal displacement {wm} of the panel















































































8
7
6
5
4
3
2
1
4
4
3
3
2
2
1
1
0010000
0000001
10000
00001
0010000
0000001
00010000
00000001
b
b
b
b
b
b
b
b
b
b
abba
abba
a
a
v
u
v
u
v
u
v
u
(‎2.14)
or
    bTw mm  (‎2.15)
From (2.15),
     mm wTb
1
 (‎2.16)
Substituting equation (2.16) into equation (2.6) gives
       mummu wNwTHu 
1
(‎2.17)
       mvmmv wNwTHv 
1
(‎2.18)
where the inplane shape functions are
     1
 muu THN (‎2.19)
     1
 mvv THN (‎2.20)

15. 15
2.3. Nonlinear Strain Displacement Relation
The von Karman large deflection strain-displacement relation for the deflections u,
v, and w can be written as follows











































































































yx
w
y
w
x
w
z
y
w
x
w
y
w
x
w
x
v
y
u
y
v
x
u
xy
y
x
2
2
2
2
2
2
2
2
2
1
2
1



(‎2.21)
or
         zm  (‎2.22)
where
 m = membrane inplane linear strain vector,
  = membrane inplane nonlinear strain vector,
 z = bending strain vector.
The inplane linear strain can be written in terms of the nodal displacements as
follows
              mmmmmm
vu
v
u
m wBwTCbCb
x
H
y
H
y
H
x
H
x
v
y
u
y
v
x
u

























































1

(‎2.23)
where

16. 16
 







































yx
x
y
x
H
y
H
y
H
x
H
C
vu
v
u
m
010100
1000000
0000010
(‎2.24)
and
     1
 mmm TCB (‎2.25)
The inplane nonlinear strain can be written as follows
       
              bbb
w
w
wBwTCaC
a
y
H
x
H
G
y
w
x
w
x
w
y
w
y
w
x
w




2
1
][
2
1
}{
2
1
}{
2
1
2
1
0
0
2
1
1






























































(‎2.26)
where the slope matrix and slope vector are
 



























x
w
y
w
y
w
x
w
0
0
 (‎2.27)
 

















y
w
x
w
G (‎2.28)
and

17. 17
  






















 2322322322
3232223222
3322332020100
3232302302010
yxyxyxyxxyxyxyxyx
yxxyyxxyyyxyxyxyx
y
H
x
H
C
w
w

(‎2.29)
     1
 bTCB  (‎2.30)
Combining equations (2.23) and (2.26), the inplane strain can be written as follows
          bmmm ww   BB
2
1
 (‎2.31)
The strain due to bending can be written in terms of curvatures as follows
         }{}{}{
2
1
2
2
2
2
2
bbbbbb wBwTCaC
yx
w
y
w
x
w






























 (‎2.32)
where
 







































222222
3232
3322
2
2
2
2
2
1812128660440020000
6622606200200000
6262060026002000
2
yxxyyxxyyxyx
yxyxxxxyyx
xyyxyyxyyx
yx
w
y
w
x
w
Cb
(‎2.33)
and
     1
 bbb TCB (‎2.34)
Thus, the nonlinear strain-nodal displacement relation can be written as
     
         bbbmm
m
wzww
z
BBB 





2
1
}{
(‎2.35)

18. 18
2.4. Inplane Forces and Bending Moments in terms of Nodal Displacements
In this section, the derivation of the relation presenting the inplane forces {N} and
bending moments {M} in terms of nodal displacements for global equilibrium will be
derived. Constitutive equation can be written in the form


























T
T
M
N
D
A
M
N


0
0
(‎2.36)
where ‎14
,
QhA  extensional matrix (‎2.37) (a)
Q
h
D
12
3
 flexural matrix (c)
  

2/
2/
),,(
h
h
T dzzyxTQN  inplane thermal loads (d)
  

2/
2/
),,(
h
h
T zdzzyxTQM  thermal bending moment (e)
and
h thickness of the panel,
{} thermal expansion coefficient vector,
T(x,y,z) temperature increase distribution above the ambient temperature
For constant temperature distribution in the Z-direction, the inplane and bending loading
due to temperature can be written in the following form
  

2/
2/
h
h
T dzQTN 
   0
2/
2/
 
h
h
T zdzQTM  for isotropic plate.
with

19. 19














2
1
00
01
01
1
][ 2




E
Q (‎2.38)
Expanding equation (2.36) gives
     
          
     Tm
Tbmm
Tm
NNN
NwAwA
NAN











BB
2
1
][
(‎2.39)
    bb wDDM B][}]{[   (‎2.40)
2.5. Deriving the Element Matrices Using Principal of Virtual Work
Principal of virtual work states that
  0int  extWWW  (‎2.41)
Virtual work done by internal stresses can be written as
      
V A
TT
ijij dAMNdVW }{}{int  (‎2.42)
where
     
         TTT
b
T
m
T
m
TT
m
T
ww 



BB 

(‎2.43)
and
     T
b
T
b
T
w B  (‎2.44)
Note that
     GG  





2
1

20. 20
Substituting equations (2.39), (2.40), (2.43), and (2.44) into equation (2.42), the
virtual work done by internal stresses can be expressed as follows
          
          
         
dA
wDw
NwAwA
ww
W
A
bb
T
b
T
b
Tbmm
TTT
b
T
m
T
m


































 
BB
BB
BB
][
2
1
*
int



 

(‎2.45)
The terms of the expansion of equation (2.45) are listed as follows
       mm
T
m
T
m wAw BB (‎2.46) (a)
        b
T
m
T
m wAw  BB
2
1
(b)
     T
T
m
T
m Nw  B (c)
         mm
TTT
b wAw BB   (d)
            b
TTT
b wAw   BB
2
1
(e)
       T
TTT
b Nw   B (f)
      bb
T
b
T
b wDw BB ][ (g)
Terms (a) and (g) of equation (2.46) can be written in the matrix form as
 












m
b
m
b
mb
w
w
k
k
ww
0
0
 (‎2.47)
Where the linear stiffness matrices are
   dADk
A
b
T
bb  BB ][][ (‎2.48)
    dAAk
A
m
T
mm  BB][ (‎2.49)

21. 21
While terms (b) + (d) of equation (2.46) can be written as
 
           
                  
                   
          
                     
        



















































A
bm
TT
b
mm
TTT
bb
T
m
T
m
A
mm
TTT
b
mm
TTT
bb
T
m
T
m
A
mm
TTT
bb
T
m
T
m
mbmbbmbmbnmb
m
b
mb
bmnm
mb
dA
wBNw
wAwwAw
dA
wAw
wAwwAw
dAwAwwAw
wnwwnwwnw
w
w
n
nn
ww












B
BBBB
BB
BBBB
BBBB
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
2
1
1
2
1
1
2
1
01
11
2
1
Note that
         
          bmmm
ymxym
xymxm
xymym
xymxm
xym
ym
xm
m
T
mm
T
wBNbCNGN
y
w
x
w
NN
NN
N
x
w
N
y
w
N
y
w
N
x
w
N
N
N
x
w
y
w
y
w
x
w
NwA
















































































0
0
B
where
     }{ mm
xym
ym
xm
m wA
N
N
N
N B










 and   






ymxym
xymxm
m
NN
NN
N (‎2.50)
Thus,
      dAAnn
A
T
m
T
bmmb   BB]1[]1[ (‎2.51)
       dABNn
A
m
T
nm  B1 (‎2.52)

22. 22
The first order nonlinear stiffness matrices,    nmmb nn 1&1 , are linearly dependent on
the node DOF {wm}([Nm]) and {wb}([]).
The second order nonlinear stiffness can be derived from term (e):
               b
TTT
bb
T
b wAwwnw   BB
2
1
2
3
1

Thus,
        dAAn
A
TT
   BB
2
3
]2[ (‎2.53)
Also
           
       
                      bT
TT
bT
TT
bT
TT
b
TyTxy
TxyTxTT
b
TxyTy
TxyTx
TT
b
Txy
Ty
Tx
TT
bT
TTT
b
wBNwbCNwGNw
y
w
x
w
NN
NN
w
N
x
w
N
y
w
N
y
w
N
x
w
w
N
N
N
x
w
y
w
y
w
x
w
wNw





























































































BBB
BB
BB
0
0
Thus,
     dABNk
A
T
T
TN    B][ (‎2.54)
where
  









TyTxy
TxyTx
T
NN
NN
N (‎2.55)
Term (f) of equation (2.46) can be written in matrix form as follows
      dANp
A
T
T
mTm    B (‎2.56)

23. 23
2.5.1. Virtual work done by external forces
For the static problem, we may write:
 



A
Surface
iiext
dAyxpw
dSuTW
),(

(‎2.57)
where T is the surface traction per unit area and p(x,y,t) is the external load vector. The
right hand side of equation (2.57) can be rewritten as   b
T
b pw where
    dAyxpNp
A
T
wb  ),( (‎2.58)
Finally, we may write
 
 
 
   
 
 
 
 











































 
m
b
mb
bmnm
TN
m
b
W
W
N
N
NN
K
K
K
00
02
3
1
01
11
2
1
00
0
0
0
 
 













Tm
b
P
P 0
0
(‎2.59)
Equation (2.59) presents the static nonlinear deflection of a panel with thermal
loading, which can be written in the form
             TTN PPWNNKK  





 2
3
1
1
2
1
(‎2.60)
Where
 K is the linear stiffness matrix,
 TNK  is the thermal geometric stiffness matrix,
 1N is the first order nonlinear stiffness matrix,
 2N is the second order nonlinear stiffness matrix,

24. 24
 P is the external load vector,
 TP is the thermal load vector,

25. 25
3. Solution Procedures and Results of Panel Subjected to
Thermal Loading
The solution of the thermal loading problem of the panel involves the solution of the
thermal-buckling problem and the post-buckling deflection. In this chapter, the solution
procedure for predicting the behavior of panel will be presented.
For the case of constant temperature distribution, the linear part of equation 2.60 can
be written as follows
      0  WKTK TN (‎3.1)
Which is an Eigenvalue problem in the critical temperature crT .
Equation (2.60) that describes the nonlinear relation between the deflections and the
applied loads can be also utilized for the solution of the post-buckling deflection. Recall
           TTN PWNNKK  





 2
3
1
1
2
1
(2.60)
Introducing the error function   W as follows
               02
3
1
1
2
1






  TTN PWNNKKW (‎3.2)
which can be written using truncated Taylor expansion as follows
         W
dW
Wd
WWW 

 (‎3.3)

26. 26
where
             tan21 KNNKK
dW
Wd
TN 

 (‎3.4)
Thus, the iterative procedures for the determination of the post-buckling displacement
can be expressed as follows
              TiiiTNi PWNNKKW  





 2
3
1
1
2
1
(‎3.5)
     iii
WWK 1tan  (‎3.6)
      ii WKW i



1
tan1 (‎3.7)
      11   iii WWW  (‎3.8)
Convergence occur in the above procedure, when the maximum value of the   1iW
becomes less than a given tolerance tol ; i.e.   toliW  1max .
Figure ‎3.1 presents the variation of the maximum transverse displacement of the
panel when heated beyond the buckling temperature. Notice that the rate of increase of
the buckling deformation is very high just after buckling, then it decreases as the
temperature increases indicating the increase in stiffness due to the increasing influence
of the nonlinear terms.

27. 27
0
0.5
1
1.5
2
2.5
6 11 16 21 26
Temperature Increase (C)
Wmax/Thickness
Figure ‎3.1. Variation of maximum deflection of the plate with temperature increase.

28. 28
4. Derivation of the Finite Element Model for Panel-flutter
Problem
In this section, the formulation presented in section 2 will be extended to include the
dynamic (time dependent) terms. The aerodynamic formulation will be derived using
first order quasi-steady piston theory which gives quite accurate aerodynamic model at
high speed regimes.
4.1. Deriving the Element Matrices Using Principal of Virtual Works
Extending the principle of virtual work to include the inertial terms, we may write,



































A
V
iiext
dA
t
v
hv
t
u
hutyxp
t
w
hw
dVuBW
2
2
2
2
2
2
),,( 

(‎4.1)
where B is the body (inertial) forces per unit volume, T is the surface traction per unit
area,  is the mass density per unit thickness, and h is the panel thickness.
The first term of (4.1) can be used to derive the bending mass matrix from
    bb
T
b wmw  , such that the bending mass matrix can be obtained by
    dANNhm
A
w
T
wb  ][ (‎4.2)
Similarly, the inplane mass matrix can be obtained by
        dANNNNhm
A
v
T
vu
T
um   ][ (‎4.3)
The equation of motions of the system can be written as

29. 29
 
 
 
 
 
 
     
 
   
 



















































m
b
mb
bmnmTN
m
b
m
b
m
b
W
WN
N
NNK
K
K
W
W
M
M
00
02
3
1
01
11
2
1
00
0
0
0
0
0


 






TmP
0 (‎4.4)
4.2. 1st
-Order Piston Theory
The fist order quasi-steady piston theory for supersonic flow, states that

































x
w
a
D
t
w
a
Dg
x
w
t
w
vM
Mq
P a
a 3
110
4
110
0
1
1
22


(‎4.5)
where
Pa is the aerodynamic loading,
v the velocity of air-flow,
M Mach number,
q dynamic pressure =av2
/2,
a air mass density,
 12
M
,
ga non-dimensional aerodynamic damping
 
3
0
2
2


h
Mva 
 
 non-dimensional aerodynamic pressure 110
3
2
D
qa


o
2
1
4
110







ha
D

D110 is the first entry of the laminate bending D(1,1) when all the fibers of the
composite layers are aligned in the airflow x-direction.
a is the panel length

30. 30
The virtual work of the quasi-steady 1st
-order piston theory aerodynamic loading is
            
        ba
T
bb
aT
b
b
w
A
T
w
T
bbw
a
A
T
w
T
b
A A
a
aext
wa
a
wwg
g
w
dAw
x
N
a
D
NwdAwN
a
Dg
Nw
dA
x
w
a
D
t
w
a
Dg
wdAwPW
3
0
3
110
4
110
0
3
110
4
110
0
.
][

































 


(‎4.6)
wheres the aerodynamic damping matrix [g] is defined by,
         bb
A
w
T
w MM
ha
D
dAN
a
D
Ng 2
04
110
4
110


  (‎4.7)
and the aerodynamic influence matrix [aa] is defined by,
    dA
x
N
DNa w
A
T
wa 







  110 (‎4.8)
Finally, we get the equation of motion in the form
                 
   )(
2
3
1
1
2
1
3
0
tPP
WA
a
NNKKWG
g
WM
T
aTN
a













(‎4.9)
where
 
 
 
 
 
 
 
 
 




























0
)(
)(
00
0
00
0
0
0
tP
tP
A
A
G
G
M
M
M
b
a
a
m
b

31. 31
5. Solution Procedure and Results of the Combined
Aerodynamic-Thermal Loading Problem
In this section, the procedure of determining the critical non-dimensional dynamic
pressure and the combined aerodynamic-thermal loading on the panel will be presented.
The mass terms, as well as the time dependent terms, will be introduced to the equation
of motion. The aerodynamic stiffness and damping terms are introduced by the quasi-
steady piston theory.
5.1. The Flutter Boundary
Recall from the previous chapter
                 
   )(
2
3
1
1
2
1
3
0
tPP
WA
a
NNKKWG
g
WM
T
aTN
a













(4.9)
Which can be reduced for the solution of the linear (pre-buckling and pre-flutter)
problem to the following equation
                   )(1
2
1
3
0
tPPWNA
a
KKWG
g
WM TaTN
a






 



(‎5.1)
Separating equations (5.1) into lateral and transverse directions, we obtain the following
two equations
     
             )(1
2
1
3
0
tPPWNA
a
KK
WG
g
WM
bTbbnmaTNb
b
a
bb










(‎5.2)

32. 32
and
       Tmmmmm PWKWM  (‎5.3)
It can be assumed that the inplane mass term  mM is negligible ‎12
. Thus, equation (5.2)
can be written in the form
     Tmmm PKW 


1
(‎5.4)
Note that the terms related to N2 and N1mb are dropped as they depend on Wb which is
essentially zero before buckling or flutter, while N1nm terms are kept as they depend in
Wm which might have non-zero values depending on the boundary conditions.
Substituting equation (5.4) into equation (5.2), we get
     
         
       )(
1
2
1
1
3
0
tPPPK
WNA
a
KK
WG
g
WM
bTbTmm
bnmaTNb
b
a
bb















(‎5.5)
Solving the homogeneous form of equation (5.5) reduces to
                 01
2
1
3
0




  bnmaTNbb
a
bb WNA
a
KKWG
g
WM


 (‎5.6)
Now, assuming the deflection function of the transverse displacement  bW to be in
the form of
    t
bb ecW 

(‎5.7)
where  i is the complex panel motion parameter ( is the damping ratio and 
is the frequency), c is the amplitude of vibration, and  b is the mode shape.

33. 33
Substituting equation (5.7) into equation (5.6) we get,
       0 t
bb eKMc  (‎5.8)
where    bob MM 2
 ,  is the non-dimensional eigenvalue; given by
o
a
o
g








 
 2
2
(‎5.9)
and
         nmaTNb NA
a
KKK 1
2
1
3
 

(‎5.10)
From equation (5.8) we can write the generalized eigenvalue problem
       0 bb KM (‎5.11)
where  is the eigenvalue, and b is the mode shape, with the charactaristic equation
written as,
     0 KMb
(‎5.12)
Given that the values of are real for all values of  below the critical value, the
iterative solution can be utilized to determine the critical non-dimensional dynamic
pressure cr for temperatures less than the critical buckling temperature.

34. 34
5.2. The Combined Loading Problem
In this section, the procedures used to determine the buckling temperature under the
influence of aerodynamic loading as well as the flutter boundary variation with elevated
temperatures is presented. Recall the stiffness term from equation (5.10)
         anmTNb A
a
NKKK 3
1
2
1 
  (5.10)
This equation contains the combined effect of thermal and aerodynamic loading. In
other words, the aerodynamic stiffness term (  aA
a3

) can be added to the procedure of
section 3.2 to obtain the critical buckling temperature to investigate the effect of
changing the dynamic pressure on the buckling temperature.
Figure ‎5.1 presents the effect of changing the value dynamic pressure on the post-
buckling deflection of the panel. It is seen clearly that the presence of flow increases the
stiffness of the panel.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
6 8 10 12 14 16 18
Temperature Increase (C)
Wmax/Thickness
Lamda=0
Lamda=150
Lamda=225

35. 35
Figure ‎5.1. Variation of the maximum post-buckling deflection for different values of the dynamic
pressure.
Figure ‎5.2 presents a combined chart of the different regions of the combined
loading‎ problem.‎ The‎ “Flat”‎ region‎ indicates‎ the‎ different‎ combinations‎ of‎ dynamic‎
pressure and temperature increase through which the panel will neither undergo flutter
nor‎buckling.‎The‎“Buckled”‎region indicates the combination of dynamic pressure and
temperature increase under which the panel undergoes static deflection due to buckling.
Note that the curve indicating the change of the buckling temperature with dynamic
pressure reaches an asymptotic value‎at‎dynamic‎pressure‎of‎242.‎The‎“Panel‎Flutter”‎
region indicates the combination of temperature and dynamic pressure under which the
panel undergoes pure panel flutter; in other words, the panel becomes stiff enough, due
to the aerodynamic pressure, that‎is‎does‎not‎suffer‎any‎buckling.‎Finally,‎the‎“Chaotic”‎
region indicates the combination of temperature increase and dynamic pressure under
which the plate undergoes limit cycle flutter about a static deflection position due to
thermal buckling. This final region is not going to be investigated in this project.

36. 36
0
100
200
300
400
500
600
700
800
900
0 2 4 6 8 10 12
Temperature (C)
DynamicPressure
Flat Panel
Buckled (Static)
Panel Flutter (Dynamic)
Chaotic
Figure ‎5.2. The different regions of the combined loading problem.
5.3. The Limit Cycle Amplitude
Following the same procedure outlined in the previous section with the only
difference that we include all the nonlinear stiffness terms, we will end up with an
equation similar to equation (5.8) in the form
                 
    )(1
2
1
2
3
1
1
2
1
3
0
tPWN
WA
a
NNKKWG
g
WM
mbm
banmTNbb
a
bb







 



(‎5.13)
          Tmbmbmmmm PWNWKWM  1
2
1 (‎5.14)
Ignoring the inertial terms of the membrane vibration, we may write,
          bmbmTmmm WNKPKW 1
2
1 11 


 (‎5.15)

37. 37
Substituting into the bending displacement equation we get,
                 
            )(1
2
1
11
4
1
2
3
1
1
2
1
11
3
0
tPPKNWNKN
WA
a
NNKKWG
g
WM
Tmmbmbmbmbm
banmTNbb
a
bb














(‎5.16)
It can be shown that,
        bmbmbmbnm WNKNWN 11
2
1
1
1

Which can be used to write the bending equation in its final form to be,
     
     
      
 
      )(1
2
1
2
3
1
11
2
1
1
1
3
0
tPPKN
W
NNKN
A
a
KK
WG
g
WM
Tmmbm
b
mbmbm
aTNb
b
a
bb























(‎5.17)
Procedure similar to those described earlier can be used to write down the equation
of motion in the form,
       0 t
bb eKMc  (‎5.18)
Where,
              2
3
1
11
2
1 1
3
NNKNA
a
KKK mbmbmaTNb 



Since the nonlinear stiffness terms of the above equation depend on the amplitude of the
vibration, an iterative scheme should be used. The following algorithm outlines the
steps used in the iterative procedure.
1- Normalize the Eigenvector {}, obtained at the flutter point, using the
maximum displacement.

38. 38
2- Select a value for the amplitude c
3- Evaluate the linear and nonlinear stiffness terms.
4- Change the value of the nondimensional aerodynamic pressure .
5- Solve the eigenvalue problem for .
6- If coelacence occurs proceed, else goto step 4
7- Check the differences between the obtained eigenvector and the initial, if small,
proceed, else normaize the eigenvector as described in step 1 and go to step 3.
8- The obtained dynamic pressure corresponds to the initially given amplitude
9- Go to step 2
It have to be noted that the above mentioned procedure is valid for the case when
panel flutter occurs while the plate is not buckled or when the dynamic pressure is high
enough that the buckled plate flat again; which does not cover the region of chaotic
vibration.
Figure ‎5.3 Presents and extended version of Figure ‎5.2. In this figure, we can notice
the linear variaiotn of the dynamic pressure associated with different limit cycle
amplitudes in the panel flutter region.
Figure ‎5.4 presents a full map of the variation of both the limit cycle amplitude as
well as the post-buckling deflection with the dynamic pressure for different values of
the temperature increase. Note the distinction between the static and the dynamic
regions.

39. 39
0
200
400
600
800
1000
1200
0 2 4 6 8 10
Temperature (C)
DynamicPressure
Flutter Limit (C=0)
C=1C=0.8C=0.5
Buckling Limit (Wmax=0)
Figure ‎5.3. Variation of the dynamic pressure associated with different limit cycle amplitudes with
temperature.
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200
Non-Dimensional Dynamic Pressure
Wmax/Thickness
DT=0DT=3.4DT=6.4=TcrDT=9.4
DT=6.9
Buckling Flutter
DT=9.4
DT=10.4
DT=7.4
DT=8.4
Figure ‎5.4. Variation of the limit cycle amplitude with dynamic pressure as well as the variation of the
post-buckling amplitude with dynamic pressure for different values of the temperature increase.

40. 40
6. Concluding Remarks
This project report presents an extension of an earlier study that involved the post
buckling deflection and the investigation of the effect of temperature on the flutter
boundary of composite plate embedded with shape memory alloy fibers‎29
. The
extension presented here is the investigation of the limit cycle vibration of the panel
which is an essentially nonlinear dynamics problem.
The other aspect that is not presented in this study is the investigation of the chaotic
vibration region. This region requires the separation of the solution into two distinct
values, namely the static deflection, which is due to the thermal loading, and the
dynamic deflection, which is due to the flutter. The expansion involves extra nonlinear
terms but is essentially straight forward from the point of stand of the current study‎30
.

41. 41
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