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.
Tribology assignment regarding the fundamentals of friction, containing: Introduction
Rules of Friction
Exceptions
Mechanism
Cases of Friction
Friction of common materials
& Conclusion
Tribology assignment regarding the fundamentals of friction, containing: Introduction
Rules of Friction
Exceptions
Mechanism
Cases of Friction
Friction of common materials
& Conclusion
Fracture mechanics CTOD Crack Tip Opening DisplacementDavalsab M.L
Fracture Mechanics .Whilst the Crack Tip Opening Displacement (CTOD) test was developed for the characterisation of metals it has also been used to determine the toughness of non-metallics such as weldable plastics.
The CTOD test is one such fracture toughness test that is used when some plastic deformation can occur prior to failure - this allows the tip of a crack to stretch and open, hence 'tip opening displacement
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
Buckling and Postbuckling Loads Characteristics of All Edges Clamped Thin Rec...theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Fracture mechanics CTOD Crack Tip Opening DisplacementDavalsab M.L
Fracture Mechanics .Whilst the Crack Tip Opening Displacement (CTOD) test was developed for the characterisation of metals it has also been used to determine the toughness of non-metallics such as weldable plastics.
The CTOD test is one such fracture toughness test that is used when some plastic deformation can occur prior to failure - this allows the tip of a crack to stretch and open, hence 'tip opening displacement
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
Buckling and Postbuckling Loads Characteristics of All Edges Clamped Thin Rec...theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Гоман, Загайнов, Храмцовский (1997) - Использование бифуркационных методов дл...Project KRIT
М.Г.Гоман, Г.И.Загайнов, А.В.Храмцовский "Использование бифуркационных методов для исследования задач нелинейной динамики полёта", Prog.Aerospace Sci., том.33, стр.539-586, 1997
В статье рассказывается об использовании методов бифуркационного анализа и анализа глобальной устойчивости для исследования различных задач нелинейной динамики полёта, таких, как инерционное вращение, сваливание, штопор и т.д. Приведены примеры расчетов с использованием моделей реальных самолетов, таких как F-4, F-14, F-15 и модели High Incidence Research Model (HIRM). Также кратко описаны основные понятия и методы теории динамических систем.
M.G.Goman, G.I.Zagainov and A.V.Khramtsovsky "Application of Bifurcation Methods to Nonlinear Flight Dynamics Problems". Prog.Aerospace Sci., Vol.33, pp.539-586, 1997
Applications of global stability and bifurcation analysis’ methods are presented for different nonlinear flight dynamics problems, such as roll-coupling, stall, spin, etc. Based on the results for different real aircraft, F-4, F-14, F-15, High Incidence Research Model (HIRM), the general methods developed by many authors are presented. The outline of basic concepts and methods from dynamical system theory are also introduced.
During the last decades, several studies on suspension bridges under wind actions have been developed in civil engineering and many techniques have been used to approach this structural problem both in time and frequency domain. In this paper, four types of time domain techniques to evaluate the response and the stability of a long span suspension bridge are implemented: nonaeroelastic, steady, quasi steady, modified quasi steady. These techniques are compared considering both nonturbulent and turbulent flow wind modelling. The results show consistent differences both in the amplitude of the response and in the value of critical wind velocity.
Ultrasonic guided wave techniques have great potential for structural health monitoring applications. Appropriate mode and frequency selection is the basis for achieving optimised damage monitoring performance.
In this paper, several important guided wave mode attributes are
introduced in addition to the commonly used phase velocity and group velocity dispersion curves while using the general corrosion problem as an example. We first derive a simple and generic wave excitability function based on the theory of normal mode expansion and the reciprocity theorem. A sensitivity dispersion curve is formulated based on the group velocity dispersion curve. Both excitability and sensitivity dispersion curves are verified with finite element simulations. Finally, a
goodness dispersion curve concept is introduced to evaluate the tradeoffs between multiple mode selection objectives based on the wave velocity, excitability and sensitivity.
Static Aeroelasticity Analysis of Spinning Rocket for Divergence Speed -- Zeu...Abhishek Jain
Above Research Paper can be downloaded from www.zeusnumerix.com
The research paper aims to develop a method to model the spin effects of rocket for Aeroelastic analysis. As the speed of the rocket increases, the structural integrity of the fins becomes more dependent on aeroelastic loads. Methods exist to analyze aeroelasticity of fins for non-spinning missiles. Most software use panel methods for calculation of load distribution. The current research replaces the panel methods to RANS CFD and introduces source terms in equations to model spin. The results of new formulation are validated w.r.t. published data on non-spinning projectile and then the method is used to simulate current projectile. Mode shapes up to 6th mode are delivered as result. Authors - Sanjay Kumar and Prof GR Shevare (Zeus Numerix), Subhash Mukane and PT Rojatkar (ARDE, DRDO)
TRANSIENT ANALYSIS OF PIEZOLAMINATED COMPOSITE PLATES USING HSDTP singh
Piezoelectric materials have excellent sensing and actuating capabilities have made them the most practical smart materials to integrate with laminated structures. Integrated structure system can be called a smart structure because of its ability to perform self-diagnosis and quick adaption to environment changes. An analytical procedure has been developed in the work based on higher order shear deformation theory subjected to electromechanical loading for investigating transient characteristics of smart material plates. For analysis two displacement models are to be considered i.e., model-1 accounts for strain in thickness direction is zero whereas in model-2 in-plane displacements are expanded as cubic functions of the thickness coordinate. Navier’s technique has been adopted for obtaining solutions of anti-symmetric cross–ply and angle-ply laminates of both model-1 and model-2 with simply supported boundary conditions. For obtaining transient response of a laminated composite plate attached with piezoelectric layer Newmark’s method has been used. Effect of thickness coordinate of composite laminated plates attached with piezoelectric layer subjected to electromechanical loadings is studied.
Estimation of Damping Derivative in Pitch of a Supersonic Delta Wing with Cur...iosrjce
In the Present paper effect of angle of incidence on Damping derivative of a delta wing with Curved
leading edges for attached shock case in Supersonic Flow has been studied. A Strip theory is used in which
strips at different span wise location are independent of each other. This combines with similitude to give a
piston theory which gives closed form solutions for damping derivatives at low to high supersonic Mach
numbers. From the results it is seen that with the increase in the Mach number, there is a progressive decrease
in the magnitude of damping derivatives for all the Mach numbers of the present studies; however, the decrease
in the magnitude is variable at different inertia level. It is seen that with the increase in the angle of attack the
damping derivative increases linearly, nevertheless, this linear behavior limit themselves for different Mach
numbers. For Mach number M = 2, this limiting value of validity is fifteen degrees, for Mach 2.5 & 3, it is
twenty five degrees, whereas, for Mach 3.5 & 4 it becomes thirty five degrees, when these stability derivatives
were considered at various pivot positions; namely at h = 0.0, 0.4, 0.6, and 1.0. After scanning the results it is
observed that with the shift of the pivot position from the leading edge to the trailing edge, the magnitude of the
damping derivatives continue to decrease throughout. Results have been obtained for supersonic flow of perfect
gases over a wide range of angle of attack and Mach number. The effect of real gas, leading edge bluntness of
the wing, shock motion, and secondary wave reflections are neglected.
Why would a company hire a trainer? To produce a change. The trainer by default is
an agent for change. Regardless of any results a trainer may accomplish, the bottom line is a
measurable change in employees’ performance.
What is marketing?
How to find out about customers?
How to reach them?
How to get them to know about you?
What is a product life cycle?
How about Marketing strategies?
Learn more ...
http://AcademyOfKnowledge.org
Brief description of current state of drones and some future challenges.
The presentation is prepared for delivery in the "Interact with Today's World" conference held in Bibliotica Alexandria 5-6 August 2016
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Finite Element Analysis of the Aeroelasticity Plates Under Thermal and Aerodynamic Loading
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 1iW
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
References
1 Mei,C.,“AFiniteElementApproachforNon-linear panel-flutter,”AIAA Journal,
Vol. 15, No. 8, 1977, pp. 1107-1110.
2 Zhou,R.C.,Xue,D.Y.,andMei,C.,“OnAnalysisofNonlinearPanel-flutter at
Supersonic Speeds,” Proceedings of the First Industry/Academy Symposium On
Research For Future Supersonic And Hypersonic Vehicles, Vol. 1, Greensboro,
North Carolina, 1994, pp. 343-348.
3 Xue,D.Y.,andMei,C.,“FiniteElement Non-linear Panel-flutter with Arbitrary
TemperatureinSupersonicFlow,”AIAA Journal, Vol. 31, No. 1, 1993, pp. 154-162.
4 Frampton, K. D., Clark, R. L., and Dowell, E. H., “State-Space Modeling For
AeroelasticPanelsWithLinearizedPotentialFlowAerodynamicLoading,”Journal
Of Aircraft, Vol. 33, No. 4, 1996, pp. 816-822.
5 Dowell,E.H.,“PanelFlutter:AReviewofTheAeroelasticStabilityofPlates and
Shells,”AIAA Journal, Vol. 8, No. 3, 1970, pp. 385-399.
6 Bismarck-Nasr, M. N., “Finite Element analysis of Aeroelasticity of Plates and
Shells,”Applied Mechanics Review, Vol. 45, No. 12, 1992, pp. 461-482.
7 Bismarck-Nasr, M. N., “Finite Elements in Aeroelasticity of Plates and Shells,”
Applied Mechanics Review, Vol. 49, No. 10, 1996, pp. S17-S24.
42. 42
8 Mei, C., Abdel-Motagaly,K.,andChen,R.,“ReviewofNonlinearPanelFlutterat
SupersonicandHypersonicSpeeds,”Applied Mechanics Review, Vol. 52, No. 10,
1999, pp. 321-332.
9 Sarma, B. S., and Varadan, T. K. “Non-linear Panel-flutter by Finite Element
Method,”AIAA Journal, Vol. 26, No. 5, 1988, pp. 566-574.
10 Yang,T.Y.,andSung,S.H.“FiniteElementPanel-flutter in Three-Dimensional
Supersonic Unsteady Potential Flow,” AIAA Journal, Vol. 15, No. 12, 1977, pp.
1677-1683.
11 Ashley,H.,andZartarian,G.,“PistonTheory– A New Aerodynamic Tool for the
Aeroelastician,”Journal of Aeronautical Sciences, Vol. 23, No. 12, 1956, pp. 1109-
1118.
12 Dixon, I. R., and Mei, C., “Finite Element Analysis of Large-Amplitude Panel-
flutterofThinLaminates,”AIAA Journal, Vol. 31, No. 4, 1993, pp. 701-707.
13 Abdel-Motagaly,K.,Chen,R.,andMei,C.“NonlinearFlutterofCompositePanels
Under Yawed Supersonic Flow Using Finite Elements,”AIAA Journal, Vol. 37, No
9, 1999, pp. 1025-1032.
14 Zhong,Z.“ReductionofThermalDeflectionAndRandomResponseOfComposite
StructuresWithEmbeddedShapememoryAlloyAtElevatedTemperature”,PhD
Dissertation, 1998, Old Dominion University, Aerospace Department, Norfolk,
Virginia.
43. 43
15 Frampton,KennethD.,Clark,RobertL.,andDowell,EarlH.“ActiveControlOf
Panel-flutterWithLinearizedPotentialFlowAerodynamics”,AIAAPaper95-1079-
CP, February 1995.
16 Gray, C. E., Mei, C., and Shore, C. P.,“FiniteElementMethodforLarge-Amplitude
Two-Dimensional Panel-flutteratHypersonicSpeeds,”AIAA Journal, Vol. 29, No.
2, 1991, pp. 290-298.
17 Benamar,R,Bennouna,M.M.K.,andWhiteR.G.“Theeffectoflargevibration
amplitudes on the mode shapes and natural frequencies of thin elastic structures
PART II: Fully Clamped Rectangular Isotropic Plates”, Journal of Sound and
Vibration, Vol. 164, No. 2, 1993, pp. 295-316.
18 Benamar,R,Bennouna,M.M.K.,andWhiteR.G.“Theeffectoflargevibration
amplitudes on the mode shapes and natural frequencies of thin elastic structures
PARTIII: Fully Clamped Rectangular Isotropic Plates – Measurements of The Mode
Shape Amplitude Dependence And The Spatial Distribution Of Harmonic
Distortion”,Journal of Sound and Vibration, Vol. 175, No. 3, 1994, pp. 377-395.
19 Liu, D. D., Yao, Z. X., Sarhaddi, D., and Chavez, F., “From Piston Theory to
Uniform Hypersonic-SupersonicLiftingSurfaceMethod,”Journal of Aircraft, Vol.
34, No. 3, 1997, pp. 304-312.
20 Lee, I., Lee, D.-M., and Oh, I.-K, “Supersonic Flutter Analysis of Stiffened
LaminatedPlatesSubjecttoThermalLoad,”Journal of Sound and Vibration, Vol.
224, No. 1, 1999, pp. 49-67.
44. 44
21 Surace, G. and Udrescu, R., “Finite-Element Analysis of The Fluttering Panels
Excited byExternalForces,”Journal of Sound and Vibration, Vol. 224, No. 5, 1999,
pp. 917-935.
22 Bismarch-Nasr,M.N.andBones,A.,“DampingEffectsinNonlinearPanelFlutter,”
AIAA Journal, Vol. 38, No. 4, 2000, pp. 711-713.
23 Young, T. H. and Lee, C. W., “Dynamic Stability of Skew Plates Subjected to
Aerodynamic and Random In-PlaneForces,”Journal of Sound and Vibration, Vol.
250, No. 3, 2002, pp. 401-414.
24 Zhou, R.C., Lai, Z., Xue, D. Y., Huang, J. K., and Mei, C., “Suppression Of
Nonlinear Panel-flutter with PiezoelectricActuatorsUsingFiniteElementMethod”,
AIAA Journal, Vol. 33, No. 6, 1995, pp. 1098-1105.
25 Frampton,K.D.,Clark,R.L.,andDowell,E.H.,“ActiveControlOfPanel-flutter
WithPiezoelectricTransducers,”Journal Of Aircraft, Vol. 33, No. 4, 1996, pp. 768-
774.
26 Dongi, F., Dinkler, D., and Kroplin, B. “Active Panel-flutter Suppression Using
Self-SensingPiezoactuators,”AIAA Journal, Vol. 34, No. 6, 1996, pp. 1224-1230.
27 Scott, R. C., and Weisshaar, T. A., “Controlling Panel-flutter Using Adaptive
Materials,”AIAAPaper91-1067-CP, 1991.
28 Suzuki, S. and Degali, T., “Supersonic Panel-flutter Suppression Using Shape
Memory Alloys,” International Journal of Intelligent Mechanics: Design and
Production, Vol. 3, No. 1, 1998, pp. 1-10.
45. 45
29 Tawfik, M., Ro, J. J., and Mei, C., “Thermal post-buckling and aeroelastic
behaviourofshapememoryalloyreinforcedplates,”SmartMaterialsandStructures,
Vol 11, No. 2, 2002, pp. 297-307.
30 Xue,D.Y.,“FiniteElementFrequencyDomainSolutionofNonlinearPanel-flutter
WithTemperatureEffectsAndFatigueLifeAnalysis”,PhDDissertation,1991,Old
Dominion University, Mechanical Engineering Department, Norfolk, Virginia.