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공학박사학위논문

공력열탄성 특성을 고려한 다기능 내장

안테나 구조물의 능동 플러터 제어

Active Flutter Control of

Multifunctional Skin Antenna Structures

considering Aerothermoelastic Characteristics

2015년 2월

서울대학교 대학원

기계항공공학부

이 창 열

i

Abstract

Active Flutter Control of

Multifunctional Skin Antenna Structures

considering Aerothermoelastic Characteristics

Chang-Yull Lee

Department of Mechanical and Aerospace Engineering

The Graduate School

Seoul National University

Multifunctional skin antenna structures (MSAS) are studied for aerothermoelastic

analysis and the active flutter control using piezoelectric sensors and actuators

under thermal and aerodynamic loads in this thesis. The structure is multi-layered

composite sandwich panel using carbon/epoxy, glass/epoxy and dielectric polymer

layers. The model is based on the first-order shear deformation theory (FSDT) for

the plate. The von Karman nonlinear strain-displacement relation is based to

consider geometrical nonlinearity. In order to apply the airflow, the first-order

piston theory is adopted to represent the aerodynamic pressures. The governing

equations are obtained from the principle of virtual work. For the nonlinear static

ii

analysis, the nonlinear governing equation is applied as incremental form by using

Newton Raphson iteration method. For the nonlinear flutter analysis, Newmark

time integration method is applied in time domain. To check the validity of results,

numerical model in this study is compared with reported data. Specifically, the

stability regions are obtained for various ranges of temperatures and aerodynamic

pressures. Then, the regions for buckling, post-buckling and limit cycle oscillation

of the structures are clearly discussed. For more analysis, the model is investigated

the different sizes and shapes of dielectric portion within enclosure layer in detail.

The constant gain negative velocity feedback control and linear quadratic regulator

(LQR) control are applied in the flutter regions. The effects of position and size for

piezoelectric patches on the flutter suppression behaviors of the panel also

discussed. The results show that the structures based on the controllers effectively

suppress panel flutter motions.

Keywords: Active flutter control, Multifunctional skin

antenna structure (MSAS), Piezoelectric sensor and actuator,

LQR control

Student Number: 2009-21712

iii

Contents

Page

Abstract..................................................................................................................... i

Contents.................................................................................................................. iii

List of Tables........................................................................................................... vi

List of Figures....................................................................................................... vii

Chapter 1 Introduction........................................................................................... 1

1.1 Background and Motivation........................................................................... 1

1.2 Conformal Load-Bearing Antenna Structure.................................................. 3

1.3 Panel Flutter.................................................................................................... 6

1.4 Smart Structures.............................................................................................. 7

1.5 Thesis Scope and Outline................................................................................ 9

Chapter 2 Composite Plate.................................................................................. 14

2.1 Constitutive Equations.................................................................................. 15

2.2 Governing Equations.................................................................................... 19

2.2.1 Aerodynamic Load................................................................................. 20

2.2.2 Equations of Motion.............................................................................. 22

2.3 Solutions of Nonlinear Equations of Motion................................................ 23

2.3.1 Postbuckling Analysis............................................................................ 24

2.3.2 Flutter Analysis...................................................................................... 25

2.4 Code Verifications......................................................................................... 28

iv

2.5 Design of Active Flutter Control................................................................... 29

2.5.1 Modeling of the Plate with Piezoelectric Layer..................................... 30

2.5.2 Dynamic Equation................................................................................. 33

2.5.3 Control Algorithms................................................................................ 34

2.6 Code Verification.......................................................................................... 36

Chapter 3 Modeling of MSAS.............................................................................. 44

3.1 Concept of Conformal Load-Bearing Antenna Structure............................. 44

3.2 Design of CLAS............................................................................................ 45

3.2.1 Cover...................................................................................................... 45

3.2.2 Radiating Element.................................................................................. 46

3.2.3 Dielectric Substrate................................................................................ 46

3.2.4 Separating Core...................................................................................... 47

3.2.5 Load Bearing Face Sheet....................................................................... 47

3.2.6 Other Extra Components........................................................................ 48

3.3 Code Verifications......................................................................................... 48

Chapter 4 Aerothermoelastic Behaviors of Structures...................................... 56

4.1 Stability Regions........................................................................................... 57

4.2 Postbuckling Behaviors................................................................................ 58

4.3 Flutter Behaviors........................................................................................... 63

4.3.1 Panel Flutter........................................................................................... 64

4.3.2 Limit Cycle Oscillations........................................................................ 65

Chapter 5 Active Flutter Control ........................................................................ 93

v

5.1 Composite Sandwich Plate with Piezoelectric Patches................................ 94

5.2 Design of MSAS with Piezoelectric Patches................................................ 96

5.3 Active Flutter Suppression of MSAS............................................................ 97

Chapter 6 Conclusions........................................................................................ 112

6.1 Summary..................................................................................................... 113

6.2 Future Works............................................................................................... 115

References............................................................................................................ 117

Appendix...............................................................................................................125

Abstract (Korean)................................................................................................130

vi

List of Tables

Page

Table 1.1 Panel flutter analysis categories.............................................................. 11

Table 2.1 Material properties of PZT G1195N piezoceramic and T300/976

graphite-epoxy composite........................................................................ 38

Table 3.1 Material properties of MSAS.................................................................. 51

Table 3.2 Nondimensional natural frequencies of sandwich plates........................ 52

vii

List of Figures

Page

Fig. 1.1 Effects of static deformation and vibrations behaviors on antenna radiation

patterns..................................................................................................... 12

Fig. 1.2 Aircraft with CLAS................................................................................... 13

Fig. 2.1 Limit cycle amplitude of the simply supported isotropic plate................. 39

Fig. 2.2 Static stability boundary of the isotropic plate.......................................... 40

Fig. 2.3 Limit cycle amplitudes of a composite panel............................................ 41

Fig. 2.4 Centerline deflection curve of a plate under piezoelectric load.................42

Fig. 2.5The effect of negative velocity feedback control gain G.............................43

Fig. 3.1 Basic composition of CLAS model........................................................... 53

Fig. 3.2 Modeling of multifunctional skin antenna structure................................. 54

Fig. 3.3 Natural frequencies according to the variation of dielectric layer............. 55

Fig. 4.1 Thermal stability boundaries of clamped model according to the size of the

dielectric region........................................................................................ 70

Fig. 4.2 Nondimensional deflections with respect to temperature increment for the

variation of aspect ratios of the model..................................................... 71

Fig. 4.3 Nondimensional deflections due to aerodynamic loads............................ 72

viii

Fig. 4.4 Deformed shapes of the model as the increase of aerodynamic pressure..73

Fig. 4.5 Nondimensional deflections according to the temperature increment for the

variation of dielectric portion................................................................... 74

Fig. 4.6 Nondimensional deflections with the variations of dielectric portions......75

Fig. 4.7 Deformed shapes of the model according to the increase of temperature

variations.................................................................................................. 76

Fig. 4.8 Nondimensional deflections according to the increase of temperature

variations.................................................................................................. 77

Fig. 4.9 Nondimensional deflections with the variations of parameters for

improvement of structural behaviors........................................................ 78

Fig. 4.10 Shape of the dielectric portion................................................................. 79

Fig. 4.11 Nondimensional deflection with the three types of shape for dielectric

portion......................................................................................................81

Fig. 4.12 Nondimensional deflection due to three types of shape for dielectric layer

under aerodynamic flow........................................................................... 81

Fig. 4.13 Three types of models with various shapes of the dielectric portion.......82

Fig. 4.14 Thermal flutter boundaries with the shapes of the dielectric portion...... 83

Fig. 4.15 Frequency coalescence flutters according to the models......................... 84

Fig. 4.16 Time responses of the model (III) in the flat and stable region ( 600 ,

30T C )............................................................................................... 85

Fig. 4.17 Time responses of the model subjected to the aerodynamic load during

the limit cycle oscillation........................................................................ 86

ix

Fig. 4.18 Time responses of the model for the variation of dielectric portion

( 30T C , 1600 )............................................................................ 87

Fig. 4.19 Time response of the model for the variation of temperature

( 1600 )............................................................................................... 88

Fig. 4.20 Phase plot of the model according to temperature increments

( 1600 )............................................................................................... 89

Fig. 4.21 Time responses of the model according to the shapes of dielectric region

( 30T C )............................................................................................ 90

Fig. 4.22 Deformed shapes at the points in LCO behaviors (Model (III),

30T C , 1600 )........................................................................... 91

Fig. 4.23 Deformed shapes at the points in LCO behaviors (Model (I), 30T C ,

1600 )................................................................................................ 92

Fig. 5.1 Modeling of composite sandwich plate with piezoelectric patches........ 100

Fig. 5.2 The effect of negative velocity feedback control gain G on the responses of

the model.................................................................................................. 101

Fig. 5.3 The effect of locations for sensor/actuator pairs..................................... 102

Fig. 5.4 The effect of sizes for the sensor/actuator pairs...................................... 103

Fig. 5.5 The effect of the positions for sensor/actuator through the thickness..... 104

Fig. 5.6 Stability regions with Model (I) and (II)................................................. 105

Fig. 5.7 Active control model with piezoceramic actuator................................... 106

Fig. 5.8 Modeling of MSAS with distributed piezoelectric patches..................... 107

x

Fig. 5.9 Three types of models with dielectric portion and piezoelectric patches 108

Fig. 5.10 The aerodynamic pressure effect on flutter suppression (Model (I),

0T )..................................................................................................109

Fig. 5.11 The thermal effect on flutter suppression (Model (I), 1200 )......... 110

Fig. 5.12 The flutter suppression control of the designed models........................ 111

1

Chapter 1

Introduction

1.1 Background and Motivation

Development of stealth function for military aircraft has been one of the interesting

topics during 20 years. Specifically, advanced technologies for new material

composition have been proposed, and then many engineers have studied the

conformable load-bearing antenna structures (CLAS). The structure is one type of

multifunctional aircraft structure (MAS). The concept of MAS offers the potential

to alter the capabilities of military air vehicles. Integrating airframe structure with

functional applications such as the structural monitoring, radiofrequency signal

communication systems could eliminate the weight, volume and radar cross section

(RCS) associated with the current approach of designing structures and functional

systems separately. Main advantage of the CLAS makes the antenna to reduce the

radar cross section (RCS) as well as to increase the stealth functions. In this regards,

Varadan and Varadan [1-3] proposed a concept of the smart skin antenna structure

as a CLAS model using composite materials. Based on this work, many engineers

tried to develop and fabricate different types of the structures using various kinds

of material. Also, Locker et al. [4] designed a multifunctional antenna with

fabrication and confirmed the structural integrity.

2

The electromagnetic antenna embedded in the airframe structure has several

advantages. However, conformal and structure integrated antennas are subject to

aerodynamic loads which will cause vibrations and deformations of the antenna

elements. This will lead to severe deviations of the phase information of the

incoming and outgoing signals and affects the degradation of the antenna

performance as shown in Fig. 1.1 [5-7]. Furthermore, the phenomena create phase

errors which will lead to a perturbed far-field pattern. Recently, some researchers

have emphasized the necessity of the active control analysis of the antenna

structures [8-10]. However, little research exists about the structural behaviors of

the CLAS [11]. Though numerous researches on CLAS have been performed

widely up to now, only radiation patterns and performances of antenna function

have been studied in a limited range.

The structures of flight vehicles are deformed due to thermal and aerodynamic

loads, and theses structural deformations are related to the aerodynamic load

distributions. The deformations of the structure are caused by the static and

dynamic behaviors such as buckling, postbuckling, vibration and flutter, etc. These

phenomena may cause an abrupt failure of the structure of lead to a significant

degradation of control performance.

In recent years, the subject of smart materials and structures has become an

important research topic. There are numerous applications of smart structures in

different industries. These include aerospace and aviation, biomedical services,

civil engineering, mechanical systems with various utilization possibilities of the

3

practical examples of control applications.

The piezoelectric materials are the most popular smart materials. The piezoelectric

materials generate an electric charge under the mechanical deformation, and

conversely produce mechanical strain in response to an applied electric field. The

use of piezoelectric materials as actuators and sensors has been successfully

demonstrated by many researchers during the last decade. This material shows

good performance to control the vibration of flutter behaviors of the various

structures.

During these investigations, the design of the antenna structure is performed at first,

and then the various characteristics and behaviors of the model under thermal and

aerodynamic loads are investigated in detail. Afterwards, active flutter control of

designed CLAS model with piezoelectric patches is investigated with various s a

wide range of possibilities.

1.2 Conformal Load-Bearing Antenna Structure

Conformal Load-Bearing Antenna Structure (CLAS) replaces separate aircraft

structure and antennas such as blades, wires and dishes, with electromagnetic

radiators embedded in the structure [12]. It can be replaced the term 'smart skins'.

This is a term for a structure that is very thin and can be mounted on any surface,

no matter the shape [13]. The first publicized CLAS program was 'Smart Skin

Structures Technology Demonstrator' (S3TD) program and it ran from 1993 to

4

1996. The program was the design, manufacture and test of CLAS as called smart

skin panel. The MUSTRAP program commenced in 1997 as a follow-on to the

S3TD program. The first of VHF/UHF antennas was made and was known as the

'Smart Skin Antenna'. It was designed manufactured, installed on the right hand

vertical tail of a NASA F/A-18 and flight tested in February 1997 [14].

Additionally, CLAS is also a part of the load-bearing airframe structure of the F-22

as shown in Fig. 1.2 [15].

There are many benefits of CLAS [12,13,16]. The most commonly quoted benefit

of the structure is drag reduction. Clearly replacing externally mounted antennas

with antennas that are flush to the outer of the aircraft will reduce drag. Large

antenna structures, such as reflecting dishes or planar arrays, are usually housed in

fairings or radomes. While these shield the antennas from the airstream, thereby

reducing the extent of drag, the shape of the vehicle can depart significantly from

the aerodynamic optimum. Secondly, it can enhance the electromagnetic

performance of radar. Thirdly, it can reduce the radar cross section (RCS).

Reducing the number of protruding antenna will certainly reduce the radar cross-

section (RCS) of an aircraft. There are many features besides the antenna that

contribute to the RCS of an aircraft. Fourthly, it can reduce the foreign object

damage (FOD). Protruding antenna could be often damaged when foreign objects

such as birds pass close to the aircraft and contact the antenna. Lastly, it can

enhance structural efficiency. The traditional approaches to install aircraft

antenna/sensors are to drill fastening holes, install the antenna/sensor mounting

5

into the cut-outs, and then fasten the mounting into the holes. This is structurally

inefficient, and this added weight is sometimes beyond that of the original airframe.

In addition, antenna/sensor mountings tend to be relatively massive so they can

retain their dimensional tolerances while being subjected to aerodynamic loading.

The loads transmitted into the airframe arising from the weight of these mountings

and the aerodynamic load can necessitate further reinforcement. On the other hand,

there are some demerits of CLAS. Firstly, the designing CLAS will be far more

complex than separately designing the airframe and antenna. Airframes are

designed by structural engineers using the principles of mechanics and materials

engineering while antennas are designed by electrical engineers using the

principles of radiofrequency photonics and electronics. In designing CLAS the

requirements from each of these fields will impose constraints on the other. Thus

the traditional approaches to design cannot be used. Secondly, it is not easy to

match existing radiation patterns. Different antenna applications require different

radiation patterns. For example target tracking requires tightly focused beams

while direction finding requires uniform coverage over an entire hemisphere. Many

antenna concepts are available and antenna design is a well developed area of

electrical engineering.

Development of stealth function for military aircraft is closely connected safety of

the countries, a lot of engineers in many countries tried to develop and fabricate

different types of the structures using various kinds of material. You and Hwang

[17] widely investigated for the design procedure including the material selection

6

and improving the antenna performance. Further, Yoon et al. [18] designed and

fabricated a simple conformal load-bearing antenna structures, and compared the

experiments and numerical results for unidirectional compression case.

Additionally, Lee et al. [19] performed the design, and experimentally validated the

skin model. Lee and Kim [20,21] also analyzed the thermo-mechanical

characteristics and stability boundaries of the structures in supersonic flows.

Recently, many researches and applications have been studied and tested under

Agency for Defense Development in Korea [22-24].

1.3 Panel Flutter

Panel flutter is the self-excited oscillation of the plate or shell when exposed to

airflow along its surface [25]. This is a dynamic instability phenomenon of only on

one side of a panel in the supersonic/hypersonic speed regime, and is induced by

the aerodynamic loads. Generally, flutter is an oscillatory aeroelastic instability

characterized by the loss of system damping due to the presence of unsteady

aerodynamic loads [26].

The panel can be oscillated with small amplitudes at the aerodynamic pressure

below the critical value, and it will die out with increase of the time. On the other

hand, the amplitude of the structures will be large at the aerodynamic pressure over

the critical value, and it can be a disaster. The earliest reported structural failures

due to the panel flutter were known as the failures of the early German V-2 rockets

7

during World War II [27,28]. And then, many works have been studied on panel

flutter over several decades, and Table 1.1 shows the six categories based on the

structural and aerodynamics theories [29].

Linear panel flutter can be studied using the frequency analysis. The critical

dynamic pressure and flutter boundary are obtained by increasing the aerodynamic

pressure until two linear frequencies coalesce. The panel will be fluttering over the

flutter boundary, and the amplitude of the panel motion diverges. However the

motion is vibrated in a limited value, and this phenomenon is called limit cycle

oscillation (LCO). It is due to the interactions between damping due to the

structural nonlinearities and instability due to aerodynamic pressure effect [30,31].

Additionally, the limit cycle oscillation can be obtained by the numerical time

integration in a time domain. Many researchers also solved the nonlinear panel

flutter using the harmonic balance method and the perturbation method [32-35].

1.4 Smart Structures

Pierre and Jacques Curie [36] discovered that some materials produce charges on

their surfaces when compressed, and this effect is called piezoelectricity.

Piezoelectric materials generate an electric charge when mechanical pressure is

applied (direct piezoelectricity), and conversely produce mechanical strain under

an applied electric field (converse piezoelectricity). In this regard, they can be

usually bonded to or embedded in the surface of a structure as the actuators and

8

sensors. Recently, the most commonly used piezoelectric materials are PZT (Lead

Zirconate Titanate), PVDF (Poly Vinylidene Fluoride), MFC (Macro Fiber

Composites) and AFC (Active Fiber Composites).

Modern structures have been controlled in various shapes and vibration control

applications by the use of smart sensors and actuators. Some studies have been

performed on induced strain actuation for beams and plates [37-42]. Burke and

Hubbard [43] performed the vibration control of a simply supported beam using the

distributed actuator. Additionally, Lee [44, 45] dealt with a laminate theory based

on modal sensors and actuators. Furthermore, Tanaka [46] designed some sensor

patches to measure the response of a number of modes. Anderson et al. [47]

investigated the cantilevered beam using a coupled electromechanical modeling.

Dosch et al. [48] presented a technique for vibration suppression of intelligent

structures using a self-sensing actuator in a closed-loop system.

Though the control of the structures has been demonstrated by many researchers, a

few papers have been reported in the area of panel flutter suppression using

piezoelectric materials [49,50]. Frampton et al. [51] increased the flutter

boundaries using the active control of panel flutter with piezoelectric materials by

implementing direct rate feedback control. Some studies have been performed on

the suppression of panel flutter using piezoelectric actuators using the finite

element method [52-55]. Additionally, both active and passive suppression

methods are studied for nonlinear flutter of composite panel using the linear control

theory [56,57].

9

1.5 Thesis Scope and Outline

The CLAS can reduce substantially weight, volume, drag and signature penalties,

enhanced electromagnetic performance, damage resistance and structural efficiency.

However, the structures are subject to aerodynamic loads which will cause

vibrations and deformations of the antenna elements. This will lead to severe

deviations of the phase information of the signals and affects the degradation of the

antenna performance. In this regard, the design of the antenna structure is

performed at first, and then the various characteristics and behaviors of the model

under thermal and aerodynamic loads are investigated in detail. Afterwards, active

flutter control of CLAS model with piezoelectric patches will be performed. The

various studies will be developed to investigate a wide range of possibilities.

After this introduction, geometrically nonlinear equations of motions under

aerodynamic force and thermal loads based on FSDT of composite plates will be

derived in Chapter 2. Also, the solution procedures of nonlinear equations of

motions will be discussed and validated. Furthermore, the theoretical background

of the active flutter control system will be introduced and applied. The formulation

will be derived and validated.

In Chapter 3, the conformal load-bearing antenna structure (CLAS) model will be

introduced and discussed in detail. The model is designed with basic components

and concepts which are expected to be required and applied. The suggested CLAS

model will be validated.

10

In Chapter 4, the thermal stability regions of the model are discussed, and the static

characteristics of the model will be analyzed in detail. Especially, thermal

postbuckling analysis and the parametric study will be performed in various

conditions. Additionally, the flutter boundaries will be obtained using linear flutter

analysis, and then flutter behaviors will be analyzed using nonlinear flutter analysis

in the stable and flutter regions.

In Chapter 5, active flutter control of designed model with piezoelectric patches

will be performed. The various studies will be developed to investigate a wide

range of possibilities.

In Chapter 6, summarizes the contributions of this research and suggests the future

works.

In addition, additional detailed formulations will be presented in Appendix.

11

Table 1.1 Panel flutter analysis categories [29]

Type Structural theory Aerodynamic theory Mach number

1 Linear Linear piston 2 5M

2 Linear Linearized potential flow 1 5M

3 Nonlinear Linear piston 2 5M

4 Nonlinear Linearized potential flow 1 5M

5 Nonlinear Nonlinear piston 5M

6 Nonlinear Euler of Navier-Stokes equations Transonic,

supersonic,

hypersonic

12

(a) Shape of undeformed antenna array

(b) Shape of deformed antenna array

Fig. 1.1 Effects of static deformation and vibrations behaviors

on antenna radiation patterns [5-7]

13

(a) Smart skin antenna fitted to the NASA F/A-18 research aircraft [14]

(b) F-22 Raptor with CLAS [15]

Fig. 1.2 Aircraft with CLAS

14

Chapter 2

Composite Plate

In this chapter, the governing equations are derived for the composite plate

considering the piezoelectric effect under aerodynamic forces and thermal loads.

The numerical rectangular plate model is based on the first-order shear deformation

theory (FSDT) for the plate. To consider geometrical nonlinearity, the von Karman

nonlinear strain-displacement relation is applied. In order to study the effects of the

airflow, first-order piston theory is adopted to represent the aerodynamic force due

to the supersonic flow. The governing equations are obtained from the principle of

virtual work. To solve the nonlinear differential equation, the solutions are divided

by the static deflection and the dynamic displacement from the equilibrium

equation. Then, equation of motion is divided into two coupled equations such as

static and dynamic problems. For the nonlinear static analysis, the nonlinear

governing equation is applied as incremental form by using Newton Raphson

iteration method. For the nonlinear flutter analysis, Newmark time integration

method is applied in time domain. Additionally, the generalized nonlinear dynamic

equations for panel with piezoelectric layers are presented. A laminated composite

plate with PZT piezoceramic layers embedded on top and bottom surfaces to act as

sensor and actuator is considered. The linear piezoelectric theory is used to derive

the equations of piezoelectric actuation and sensing. And then, active control

15

systems usually has input and output to the structure using sensors and actuators to

activate the application of forces on a structure.

2.1 Constitutive Equations

Based on the first-order shear deformation theory (FSDT), the displacement fields

of the plate are expressed as

0

0

0

, , , , , , ,

, , , , , , ,

, , , , ,

x

y

u x y z t u x y t z x y t

v x y z t v x y t z x y t

w x y z t w x y t

(2.1)

where u , v and w are the mid-plane displacements in the x , y and z

directions, respectively. Also x and

y are the originally perpendicular to the

longitudinal plane. The subscript '0' indicate the mid-plane.

The von Karman nonlinear strain-displacement relations are expressed to consider

the geometric nonlinearity.

2

2

, ,1

, ,2

, , 2 , ,

xx x x

yy y y

xy y x x y

u w

v w

u v w w

e (2.2)

16

where e is the in-plane strain vector.

Substituting Eq. (2.1) into Eq. (2.2), the in-plane strain vector is expressed as

0

m θ

2

0 0

2

0 0

0 0 0 0

, , ,1

, , ,2

, , 2 , , , ,

x x x x

y y y y

y x x y x y y x

z

z

u w

v w z

u v w w

e ε κ

ε ε κ

(2.3)

where 0ε ,

mε , θε and κ are the in-plane strain vector at the mid-plane, the

linear in-plane strain vector, the nonlinear in-plane strain vector and the curvature

strain vector, respectively.

Additionally, transverse shear strains are given as

0

0

,

,

yz y y

xz x x

w

w

γ

(2.4)

where γ is the transverse shear strain vector.

On the other hand, the stress-strain relation of the arbitrary composite model with

temperature rise T is presented as in Ref. [58]: C T .

In the two dimensional domain, the stress of the thk layer is obtained by the

transformation of coordinates as

17

11 12 16

12 22 26

16 26 66

44 45

45 55

xx xx xx

yy yy yy

xy xy xyk k kk

yz yz

zx zxk kk

Q Q Q

Q Q Q T

Q Q Q

Q Q

Q Q

(2.5)

where ijQ is the transformed reduced stiffness.

0T T T is temperature

rise. Also, 0T and are reference temperature and thermal expansion

coefficient, respectively. Additionally, xx , yy , xy are defined as

2 2

1 2

2 2

1 2

1 2

cos sin

= sin cos

2 sin cos

xx

yy

xy

(2.6)

where is the ply angle, and 1 and

2 are the thermal expansion

coefficients in the principal directions.

Finally, constitutive equation of a laminate plate under thermal effect can be

obtained as

18

0b ΔT

b ΔT

s

N NA B ε

M MB D κ

Q A γ

(2.7)

where bN ,

bM and Q stand for the in-plane force, the moment and the

transverse shear force resultant vectors, respectively. Meanwhile, thermal force

ΔTN and moment ΔTM induced by the temperature change derived as

11

, 1,k

k

n z

k kz

k

z Tdz

ΔT ΔTN M Q

(2.8)

Also, A , B , D and sA means matrices that represent extensional, bending-

extension coupling, bending and shear stiffness matrices, respectively.

1

1

2

1

1

, , 1, , ,

,

k

k

k

k

n z

kz

k

n z

p kz

k

z z dz

dz

s

A B D Q

A Q

(2.9)

where p is shear correction factor.

19

2.2 Governing Equations

Using the principle of virtual work, the governing equation of the composite plate

under aerodynamic forces and thermal loads with geometrical nonlinearity can be

derived as

int 0extW W W (2.10)

where intW and

extW represents the internal and external virtual work,

respectively.

At first, the internal virtual work is given by

int

=

1 1

2 3

V

A

W dV

dA

T

T T T

b b

T T

ΔT ΔT

e σ

ε N κ M γ Q

δd K K N1 N2 d d P

(2.11)

In Eq. (2.11), [ ]T

x yu v w d means the displacement vector. Additionally, K ,

ΔTK , N1 and N2 are matrices for the linear elastic, thermal geometric, first-

order non-linear, and second-order non-linear stiffnesses, respectively. Further,

ΔTP means the thermal load vector.

20

On the other hand, the external virtual work is derived as

ext 0 0 0 0 0 0 0

1 0 0 0 0

2

[

( ) ]

A

x x y y

x x y y a

W I u u v v w w

I u u v v

I p w dA

T Tδd Md δd f

(2.12)

where ap , M and f are the aerodynamic pressure, the mass matrix and the

external force vector, respectively. Additionally, the moment of inertia are defined

as /2

2

0 1 2/2

( , , ) (1, , )h

hI I I z z dz

, in here represents the material density.

2.2.1 Aerodynamic Load

The external force in this study is an aerodynamic pressure that is caused by a

supersonic air flow. It can be approximated by the first-order piston theory for the

range of 52 M [59]. The aerodynamic force can be expressed as

2 2

22

110 110

3 4

0

2 1( , , )

11

aa

a

V Mw wp x y t

x M V tM

D g Dw w

a x a t

(2.13)

21

where, V , M and a are the airflow speed, Mach number and air density,

respectively. In addition, non-dimensional aerodynamic pressure is defined as

2 3

110

aV a

D

(2.14)

Additionally, different parameters are defined as

22110

a 03 4

0

( 2), , 1aV M D

g Mh ha

(2.15)

where ag , 110D , 0 and represent the non-dimensional aerodynamic

damping parameter, bending rigidity, convenient reference frequency and

aerodynamic pressure parameter, respectively.

On the other hand, using following approximation is reasonable for 1M , ag

can be obtained as [60]

agM

(2.16)

where is the air-panel mass ratio defined as / a ma h [61].

22

2.2.2 Equations of Motion

The external virtual work term Td f due to aerodynamic force can be

transformed by the first-order piston theory as

aAP w dA

T

T

f d

d f

d A d A d (2.17)

where, fA and

dA are the aerodynamic influence matrix and aerodynamic

damping matrix, respectively. Then, the external virtual work in Eq. (2.12) can be

obtained as

ext

TW f dd Md A d A d (2.18)

Finally, the internal virtual work intW and external virtual work

extW are

derived. Then, the equations of motion are obtained using the principle of virtual

work as

1 1

2 3

d ΔT f ΔTMd A d K K A N1 N2 d P (2.19)

23

2.3 Solutions of Nonlinear Equations of Motion

In order to solve the nonlinear equation derived in previous section, the

displacement d is assumed as a sum of the static solution Sd and a tiem

dependent part td as S td d d . Subscript s and t denote the static and

dynamic terms, respectively.

1 1

2 3

ΔT f s s s ΔTK K A N1 N2 d P (2.20)

0

1 1

2 3

ΔT f s s

t d t t

st t t

K K A N1 N2

Md A d d 0N2 N1 N2

ag

(2.21)

Eq. (2.20) is the equation of motion for static analysis such as an aero-thermal

postbuckling analysis. On the other hand, Eq. (2.21) is the equation of motion for

dynamic problems like the vibration or flutter behaviors. Additionally, the stiffness

matrices are the function of the displacements, and they are coupled. In this regard,

to solve the dynamic governing equation, the static equation should be solved

beforehand.

24

2.3.1 Postbuckling Analysis

In this section, a solution procedure for aero-thermal postbuckling analysis is

presented. To obtain the non-linear behaviors of the model, the Newton-Raphson

iterative method is employed [62]. The tangent stiffness matrix and load vector for

the Newton-Raphson iterative method can be written using the function sψ(Δd )

as

1 1( )

2 3s s

ΔT f s s ΔTψ Δd K K A N1 N2 Δd P (2.22)

For the ith iteration,

tan

( ( ))

( )

1 1( )

2 3

i

i

s

is i

s i s

i

d

ΔT f s s

ΔT f s s ΔT

ψ ΔdK K K A N1 N2

d Δd

ψ Δd K K A N1 N2 Δd P

(2.23)

And then, the incremental force vector and the updated displacement vector can be

written as

tan 1

1 1

( )i i s i

i i i

s

s s s

K d ψ Δd

d d d (2.24)

25

The post-buckling behaviors are calculated repeat until the converged incremental

displacement as

0 s sd d d (2.25)

2.3.2 Flutter Analysis

1) Linear Flutter Analysis

The linear flutter analysis is employed to obtain the critical conditions for the

flutter motion of the model. Firstly, small incremental time dependent solution

td is assumed, and the time dependent nonlinear stiffness matrices are

approximated to zero. Then, the Eq. (2.21) can be linearized as

0

agt d t ΔT f S S tM d A d K K A N1 N2 d 0 (2.26)

To reduce the degree of freedom, Guyan Reduction [62] is applied to above the

equation, and the reduced equation is obtained as

R R RM w C w K w 0 (2.27)

26

where RM ,

RC and RK are the reduced mass, damping and stiffness matrices.

And they are defined as

T

R R RM T MT ,

0

ag

T

R R d RC T A T and T

R R tan R K T K T (2.28)

where

T

srss ms

tan

ms mm

dK KK d

wK K,

tan ΔT f s sK K K A N1 N2 ,

{ , , , }T sr x yd φ φ u v , and

-1 T

ss ms

R

K KT

I.

A small incremental transverse displacement w can be assumed to be a

harmonic motion as

0

te w φ (2.29)

where 0φ is a time independent vector, and the panel motion parameter is a

complex number defined as

27

R Ii (2.30)

Then homogeneous equations for eigenvalue analysis with state variables are

obtained as

0R R

R R R

0 M M 0 Δwω

K C 0 M Δw

(2.31)

The eigenvalue has a positive real part ( 0 R), the deflection will grow

exponentially with time. Especially, when 0 R and 0 I

, the instability is

called ‘divergence’, whereas it is called ‘flutter’ for the condition of 0 R and

0 I.

As monotonically increasing from zero, imaginary parts of two eigenvalues

approach each other until they coalesce at a cr. When the coalescence occurs,

real parts of the two eigenvalues are bifurcated and one of them becomes positive,

then the structure becomes dynamically unstable and flutter occurs.

2) Nonlinear Flutter Analysis

The linear flutter analysis can be obtained the critical conditions for flutter

behaviors. However, it is not sufficient to simulate the motion of flutter behaviors.

28

In this regard, the nonlinear flutter analysis should be performed to obtain the

flutter motion of the model including the geometrical nonlinearity due to

aerodynamic pressures. Thus, the Newmark time integration method is used to

analyze the flutter motions in time domain [63].

The Newmark method is introduced in Appendix, and the routine of time

integration for flutter response is illustrated.

2.4 Code Verifications

To verify the present procedures, three cases are compared with data of the

references.

Firstly, Fig. 2.1 represents the limit cycle amplitude of the simply supported

isotropic model with respect to nondimensional aerodynamic pressures. As shown

in the plot, present results are good agreement with the data in Ref. [64].

Secondly, to verify the program for the thermal buckling analysis with non-

dimensional aerodynamic pressure, Fig. 2.2 depicts the static stability boundary of

isotropic plate for thermal post-buckling analysis. Temperature and aerodynamic

load are compared with the results in Ref. [65], and the data show good agreement

with the previous data.

Lastly, Fig. 2.3 represents the limit cycle amplitudes of the panel to check the

validity of time integration routine. In the analysis, the Newmark method is

adapted with 0.1ms as the time step. Also, deflection is the transverse deformation

29

at x/a=3/4 and y/b=1/2 for maximum magnitude of the LCO. As shown in the plot,

present results are good agreement with the data in Ref. [66].

On the whole, the governing equations of the composite plate considering

geometrical nonlinearity under aerodynamic forces and thermal loads are derived.

The numerical rectangular plate model is based on the first-order shear deformation

of the plates and von-karman nonlinear strain-displacement relation. Also, the first-

order piston theory is adopted to study the aerodynamic pressures. The governing

equations are obtained from the principle of virtual work. Then, equation of motion

is divided into two coupled equations such as static and dynamic problems. For the

nonlinear static analysis, the nonlinear governing equation is applied as

incremental form by using Newton Raphson iteration method. For the nonlinear

flutter analysis, Newmark time integration method is applied in time domain.

2.5 Design of Active Flutter Control

In this section, the generalized nonlinear dynamic equations for panel with

piezoelectric layers are presented. A laminated composite plate with PZT

piezoceramic layers embedded on top and bottom surfaces to act as sensor and

actuator is considered. The piezoelectric materials are in the form of distributed

patches or continuous layers. The linear piezoelectric theory is used to derive the

equations of piezoelectric actuation and sensing. And then, active control systems

30

usually has input and output to the structure using sensors and actuators to activate

the application of forces on a structure. Different control algorithms such as the

constant gain negative velocity feedback control and linear quadratic regulator

(LQR) control are applied.

2.5.1 Modeling of the Plate with Piezoelectric Layer

Linear piezoelectric coupling for thk layer between the elastic field and the

electric field can be expressed by the ‘direct’ and the ‘converse’ piezoelectric

equations, respectively. The transformed equations of a piezoelectric material can

be written as

T

k k k k k

k k k k k

σ Q ε e E

D e ε E (2.32)

where ε , σ , D and E are strain, stress, electric displacement and electric

field vectors. Additionally, e , and Q are piezoelectric constants, permittivity

coefficients and elastic matrices, respectively. The first equation represents the

converse effect as the actuator, and second one governs the direct effect as the

sensor.

On the other hand, the electric field vector E is the negative gradient of the

31

applied electric potential V . The voltage applied in the thickness direction as,

V E (2.33)

The voltage applied to the actuator only in the thickness direction, the electric field

vector E can be expressed as

0 0 1/T

a ah VE (2.34)

where ah and

aV are the thickness of the actuator layer and applied with a

voltage, respectively.

The sensor equation can be derived from the second Eq. (2.32). The electric

displacement in the thickness direction zD can be written as

31zD e ε (2.35)

where 31e is the dominant piezoelectric constant. The total charge ( )q t

developed on the sensor surface is the spatial summation of all the point charges

and can be calculated by integrating the electric displacement over the sensor

surface area as

32

( ) zS

q t D dS (2.36)

where S is the surface area of the sensor. The sensor voltage output SV from the

sensors can be derived as

( ) ( )S cV t G i t (2.37)

where cG is the gain of the current amplifier, which transforms the sensor current

to voltage. The current ( )i t on the sensor is the time derivative of the total charge

and can be expressed as

d ( )( )

d

q ti t

t (2.38)

where ( )q t is the total charge presented in Eq. (2.36).

A piezoelectric patch is either surface bonded or embedded into the substrate

composite plate perfectly. The layered composite plate with piezoelectric layer

modeled is modified to include the transverse piezoelectric parameters.

The constitutive equation of a laminate plate under thermal and piezoelectric

effects can be obtained as

33

0b ΔT P

b ΔT P

s

N N NA B ε

M M MB D κ

Q A γ

(2.39)

All of the quantities with thermoelastic effects appearing in Eq. (2.39) were already

derived earlier in section. The thermal force and moment resultants are also already

calculated. Similarly, the electric force and moment resultants can be obtained as

11

, 1,k

k

npz

k kz

k

z dz

P PN M e E

(2.40)

where np represents the number of piezoelectric layers, and kE is the electric

field vector derived in Eq. (2.34).

2.5.2 Dynamic Equation

The energy principles are developed for a layered composite plate include the

piezoelectric resultants as

0

0 0 0

uu uuu

u

u u

K KM F

K K (2.41)

34

where uuK and K are the elastic and the electric stiffness matrix, respectively.

Additionally, uK and

uK are the coupling matrices. Actuator and sensor

equations can be written as

1

1

uu u A

s u

u

u

K F K

K K

(2.42)

where A and s are electric displacement vectors of actuation and sensing.

Assembling the element equations gives the global dynamic equation

uu uu u u u Au u u M C K K K K F K (2.43)

The stiffness matrix definitions are given in Appendix.

2.5.3 Control Algorithms

A basic control system has an input, a process and an output. The input and the

output represent the desired response and the actual response respectively. In this

section, control gain negative velocity feedback control and linear quadratic

regulator (LQR) control algorithms are applied.

At first, constant gain negative velocity feedback control algorithm is introduced.

35

The sensor voltage output SV from the sensors is already presented as

( ) ( )S cV t G i t in Eq. (2.37). The distributed sensor generates a voltage when the

structure is oscillating, and this signal is fed back into the distributed actuator using

a control algorithm. The actuating voltage under a constant gain control algorithm

can be expressed as

( )a c i i sV G G i t GV (2.44)

where iG means the feedback control gain.

Next, linear quadratic regulator (LQR) control algorithm is introduced. For the

feedback active control system, a linear quadratic regulator (LQR) is one of the

powerful optimization routines. This control algorithm is optimal control law based

on full state feedback. The control gain is obtained which minimizes the

performance index given by

0

1

2

T T

a aJ dt

ξ Qξ R (2.45)

where Q and R are semi-positive definite and positive definite weighting

matrices on the output and control inputs, respectively. The actuating voltage can

be expressed as

36

1( ) T

a c stV t G ξ R B Pξ (2.46)

The MATLAB software has inbuilt functions for estimating the control gains using

LQR method. An optimal feedback controller is designed using the gain obtained

from MATLAB LQR function.

2.6 Code Verifications

To verify the present procedures, two cases are compared with data of the

references. Firstly, Fig. 2.4 shows the Linear static analysis of a cantilevered

composite plate ( 20 20cm cm ) with both the upper and lower surfaces

symmetrically bonded by piezoelectric ceramics. The actuator input voltage is

10V , and the stacking sequence of the composite is [-45/45/-45/45]. Additionally,

the material properties for composite and PZT are presented in Table 6.1. The

present result shows good agreement with the previous work [78].

Next, for the verification of control algorithm, the feedback control gain G on the

response of the plate in Fig. 2.5. The model is simply supported laminated

composite plate with integrated piezoelectric sensor and actuator. The plate

dimensions considered are a=b=400mm and h=0.8mm. The composite plate is

constructed of four layers of T300/976 with [-30/30/30/-30], and the thickness of

PZT G1195N piezoceramic is 0.1mm with upper and lower layer of the plate. The

37

material properties of the model are shown in Table 2.1. The distributed load of

250 /N m . The present result shows good agreement with the previous work [79].

On the whole, the governing equations of the composite plate considering

geometrical nonlinearity under aerodynamic forces and thermal loads are already

derived in section 2.1. In this section, the piezoelectric effect is added to the

dynamic equations of the composite plate. A laminated composite plate with PZT

piezoceramic layers embedded on top and bottom surfaces to act as sensor and

actuator is considered. The linear piezoelectric theory is used to derive the

equations of piezoelectric actuation and sensing. Additionally, the constant gain

negative velocity feedback control and linear quadratic regulator (LQR) control are

applied.

38

Table 2.1 Material properties of PZT G1195N piezoceramic and

T300/976 graphite-epoxy composite [78]

Properties PZT G1195N T300/976

Young’s modulus (GPa) : 11E 63.0 150

22 33E E 63.0 9

Poisson’s ratio : 12 13 0.3 0.3

23 0.3 0.3

Shear modulus (GPa) : 12 13G G 24.2 7.10

23G 24.2 2.50

Density (3/kg m ) : 7600 1600

Piezoelectric constants ( /m V ) : 31 32d d 12254 10 -

Electrical permittivity ( /F m ) : 11 22 915.3 10 -

33 915.0 10 -

39

400 500 600 700 800

0.0

0.2

0.4

0.6

0.8

1.0

wm

ax/h

Non-dimensional aerodynamic presseure

Present

Dixon and Mei [64]

Fig. 2.1 Limit cycle amplitude of the simply supported isotropic plate

40

0.5 1.0 1.5 2.0

0

100

200

Present

Xue [65]

T/Tcr

Fig. 2.2 Static stability boundary of the isotropic plate

41

100 200 300 400 500 600 700

0.00

0.25

0.50

0.75

1.00

ply angle (00)

ply angle (300)

Present

Shiau and Lu [66]

wm

ax/h

Nondimensional aerodynamic presseure

Fig. 2.3 Limit cycle amplitudes of a composite panel

42

0 50 100 150 200

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Present

Lam et al. [78]

De

fle

ctio

n w

(m

m)

Distance x (mm)

Fig. 2.4 Centerline deflection curve of a plate under piezoelectric load

43

0.0 0.5 1.0

-2

-1

0

1

2

Present (G=0)

Present (G=600)

Ref. [79]

w(m

m)

Time [sec]

Fig. 2.5 The effect of negative velocity feedback control gain G

44

Chapter 3

Modeling of MSAS

3.1 Concept of Conformal Load Bearing Antenna Structure

During the past century, a number of breakthroughs have been achieved in various

structural, material and mechanical parts of the aircraft field. The concept of

multifunctional aircraft structure (MAS) offers the potential to alter the capabilities

of military air vehicles as integrating airframe with functional applications. The

concept of Conformal Load Bearing Antenna Structure (CLAS) is the one type of

the MAS. CLAS replaces separate aircraft structure and antennas such as blades,

wires and dishes with electromagnetic systems embedded in the airframe structure.

The structure refers to load bearing aircraft structure, typically exterior skins

manufactured from carbon fiber reinforced polymer (CFRP) composite, that also

contain radiofrequency transmitters and receivers. This structure must be improved

to existing airframes or incorporated in new platforms for the purpose of antenna

structure [12].

As previously stated in Chapter 1, this approach can reduce substantially weight,

volume, drag and signature penalties, enhanced electromagnetic performance,

damage resistance and structural efficiency. However the design, manufacture,

certification and through-life-support of CLAS will be more complex than for its

45

non-integrated counterparts.

In this regard, the essential and basic components of the CLAS are presented, and

the model used in this study is compared and designed with the presented concepts.

3.2 Design of CLAS

Many of the reported CLAS demonstrators and laboratory test specimens have

taken the form of honeycomb stiffened sandwich panels. A typical configuration is

presented in Fig. 3.1 [67]. CLAS must support significant structural load, so ideally

the outer skins of the panel would be manufactured from high stiffness materials

such as carbon fiber reinforced polymer (CFRP) or even high strength aluminum

alloys. Unfortunately these materials are opaque to electromagnetic radiation, thus

a bathtub shaped recess is usually manufactured into the inner skin to support

structural loads and provide a space which the antenna components can be located.

3.2.1 Cover

The outer face of the CLAS lies flush with the aircraft outer mould line to protect

the element with a cover. The cover must be transparent to radiofrequency (RF)

radiation and so is typically constructed from glass fiber reinforced polymer

(GFRP) composites. Transmission losses through this cover are minimized by

46

controlling their thickness and distance from the radiator. The lower stiffness of the

GFRP covers relative to the load bearing CFRP skins must be accounted for

otherwise secondary bending will reduce fatigue life and allow deformation under

load that may impair antenna performance. In this regard, face sheets are made up

of glass/epoxy because the RF radiation penetrates cover layer in this study.

3.2.2 Radiating Element

Radiating elements are usually used to be electrodeposited copper films in the

order of 0.015 mm thickness. Generally, a thin copper plate is attached to the

dielectric layer because of its high conductivity. The shape and dimension of

radiators determine the electromagnetic performance of an antenna. The element

can be changed various shapes to produce the desired radiation pattern for any

specific application. Generally, the copper layer is too thin to consider the strength

and stiffness, the effect of the layer is neglected in mechanical analysis [18].

3.2.3 Dielectric Substrate

The radiating element is attached onto dielectric substrates (dielectric layers). The

dielectric layer is essential part of the CLAS, and thus a few research have been

investigated according to the various shapes of the dielectric region such as a circle

[68], square [69] and rectangular [70] shapes and etc. In this regard, various shape

47

and size of dielectric layer are designed and performed the comparative studies.

3.2.4 Separating Core

This core layer is usually designed as a honeycomb or foam core between the face

sheets. The core component is mechanically very efficient because the skins

support the applied loads as an airframe structure. Most CLAS are used

honeycomb core because of lower density and electromagnetic losses than foam.

And then, the honeycomb cores are used in this study to transmit shear between

sheets and provide the air gap to the antenna.

3.2.5 Load Bearing Face Sheet

The load bearing face sheet bonded to the core is usually made of CFRP. The cover

of structure can be transparent to RF radiation, and then it is made of GFRP

composites. However, CLAS must support significant structural load, so the panel

would be manufactured from high stiffness materials such as CFRP. The sheet is

presented as the bathtub shaped to put the radiating components in the layer. If the

face sheet has sufficient conductivity then it may also act as the ground plane. In

this regard, the dielectric and dielectric enclosure layers are developed using

materials assumed to be perfectly bonded, and the dielectric enclosure layer are

designed to protect the dielectric layer from external disturbances such as

48

aerodynamic loads forces and thermal disturbances. Additionally the dielectric

layer is assumed to consist of Phenol [18].A carbon/epoxy layer was located on

each side of the dielectric layer.

3.2.6 Other Extra Components

1) Absorber

Some antenna designs produce EM energy directed in the backwards direction. If

this back-lobe occurs then a layer of dielectric material could be absorb the

undesirable radiation in the backwards of the model.

2) Absorber Pan

It is design to contain the absorber material with the lightweight and non load-

bearing enclosure layer.

3.3 Code Verifications

First of all, a model in this study is designed with basic components and concepts

which are expected to be required and applied as presented in Fig. 3.2 from Ref.

[69]. The structure is designed of multi-layer sandwich composite plate with the

whole five layers. From the bottom side of the model, the staking sequences are the

49

face sheet, honeycomb core, face sheet, dielectric enclosure layer with dielectric

layer and face sheet. And then, face sheets are made up of Glass/Epoxy. While the

dielectric and dielectric enclosure layers are developed using materials assumed to

be perfectly bonded, and the face sheets are designed to protect the dielectric

enclosure layer from external disturbances. Further, the Carbon/Epoxy layer covers

around the dielectric layer. This model is un-symmetric multi-layer sandwich

model. Thus the total transverse displacement of a sandwich plate is obtained as the

sum of the displacement due to bending of the plate and the displacement due to

shear deformation of the core. Based on the Fig. 3.2, face sheet layer is [θ/-θ]S, and

dielectric enclosure layers are made up of composite layers [0/θ/90]S. And then, the

honeycomb cores transmit shear between sheets, and provide the air gap to the

antenna. Table 3.1 shows the properties of the constituent materials are as in Ref.

[18].

For the code verification of the designed model, two cases are compared with the

reported data. At first, the natural frequencies of multi layered composite sandwich

plates under the thermal effect are calculated. The structure is consisted of [θ/-

θ/…/-θ]10[core][-θ/θ/…/θ]10. Additionally, parameter fh and h denotes the

thickness of face sheets and total thickness of the sandwich plate, respectively.

The ratios of the thickness of honeycomb core and face sheets are chosen as 0.3

and 0.2. The nondimensional natural frequencies are calculated as

50

2/h E . Table 3.2 shows that the obtained nondimensional natural

frequencies agree with the data of Ref. [71].

Next, Fig. 3.3 shows the variation of natural frequencies according to the ratio

(Dielectric region / Total area of dielectric layer) for two kinds of ply angle. In

other words, the parameter denotes the size of the dielectric portion relative to the

full area of the layer. The natural frequencies are decreased as the dielectric portion

is increased, and then the present results are good agreement with data of the Ref.

[72].

On the whole, the essential and basic components of the CLAS are presented using

a typical configuration. CLAS must support structural load, so ideally the outer

skins of the panel would be high stiffness materials such as CFRP. Unfortunately

these materials are opaque to electromagnetic radiation. In this regard, the cover

must be transparent to RF radiation, and thus it is typically constructed using GFRP

composites. The CFRP is used as the dielectric enclosure layers due to the lower

stiffness of the GFRP covers. Finally, a model is designed and presented with basic

components and concepts which are expected to be required and applied. For the

code verification of the designed model, composite sandwich plates and the

dielectric portion are verified with reported data.

51

Table 3.1 Material properties of MSAS [18]

G/E C/E Phenol Honeycomb

E1 24 Gpa 67 Gpa 7.2 Gpa 0.09 Mpa

E2 28 Gpa 57 Gpa 7.2 Gpa 0.08 Mpa

ν12 0.105 0.103 0.3 0.3

G12 4.54 Gpa 5.9 Gpa 2.77 Gpa 0.1 Mpa

G13 1.0 Gpa 1.0 Gpa 2.77 Gpa 19.7 Mpa

G23 1.0 Gpa 1.0 Gpa 2.77 Gpa 11.5 Mpa

α1 9.7-6

/ oC 2.1

-6/ oC 75

-6/ oC 1.5

-6/ oC

α2 17.7-6

/oC 2.1

-6/ oC 75

-6/ oC 1.5

-6/ oC

ρ 2200kg/m3 1450kg/m

3 9000kg/m

3 96.1kg/m

3

52

Table 3.2 Nondimensional natural frequencies of sandwich plates

hf/h θ=30∘ θ=45∘

: Ref. [71] : Present : Ref. [71] : Present

0.3 0.2294-2

0.2198-2

0.2426-2

0.2333-2

0.2 0.2328-2

0.2310-2

0.2462-2

0.2459-2

53

Fig. 3.1 Basic composition of CLAS model [67]

54

Fig. 3.2 Modeling of multifunctional skin antenna structure [69]

55

0 20 40 60 80 100

200

250

300

350

400

450

ply angle (450)

ply angle (00)

Present

Yoo and Kim [72]

Na

tura

l fr

eq

ue

ncy [ra

d/s

ec]

Dielectric region / Total area of dielectric layer (%)

Fig. 3.3 Natural frequencies according to the variation of dielectric layer

56

Chapter 4

Aerothermoelastic Behaviors of Structures

In this chapter, multifunctional skin antenna structures (MSAS) are investigated for

the stability regions, the postbuckling behaviors and the flutter behaviors under

aerodynamic and thermal conditions. First of all, thermal buckling and

postbuckling analyses are performed to investigate the static characteristics of

MSAS. The designed model in this work is already shown in Fig. 3.2, and the

material properties are presented in Table 3.1. The structure is consisted of multi-

layer sandwich composite plate with the whole five layers. From the bottom side of

the model, the staking sequences are the face sheet, honeycomb core, face sheet,

dielectric enclosure layer with dielectric layer and face sheet. And then, face sheets

are made up of glass/epoxy, and the carbon/epoxy layer covers around the

dielectric layer. For numerical analysis, finite element method is using 7 7

meshes for nine-node plate elements. The reference temperature T0 is 27 oC.

Further, ply angle of each layer is selected as 450 and simply-supported boundary

conditions are used except the cases with special comments. Unless otherwise

noted, square panels ( / 1a b ) are considered and the thickness ratio ( /a h ) is

chosen as 100. Additionally, first-order piston theory is used to analyze the model

in the supersonic flow.

57

4.1 Stability Regions

The role of the dielectric layer is important in the MSAS model, thus the analysis

on the layer is necessary to evaluate the performances of the structure. In this point

of view, thermo-mechanical behaviors of the structure are discussed to focus on the

effects for the sizes and shapes of dielectric layer.

First of all, there are generally four types of clamped panel behaviors in the thermal

stability boundaries as shown in Fig. 4.1: (A) flat and stable, (B) statically buckled

but dynamically stable, (C) flutter and (D) chaos. At region (A), the panels remain

flat and statically stable as well as dynamically stable. On the other hand, the

panels are buckled but dynamically stable at region (B) as increasing the

temperature. The region (B) is defined by thermal post-buckling behaviors, and the

boundaries between the regions (A) and (B) indicate critical conditions for

buckling. Furthermore, the dynamic pressure increases, flutter occurs in the region

(C). And the boundaries between the regions (A) and (C) can be determined by

linear flutter analysis. Additionally, chaotic motions can be observed in region (D).

Additionally, as the area of the region (A) is larger, the model is more stable.

In this regard, thermal stability boundaries of the structure with the change of

dielectric portion under thermal and non-dimensional aerodynamic loads are

depicted in Fig. 4.1. The effect of area ratio for the dielectric layer relative to the

enclose layer (Dielectric region / Total area of dielectric layer) are compared. In

other words, it deals with the boundaries of the model according to the size of the

58

dielectric portion relative to the full area. Additionally, the square shape of

dielectric layer is located at the center portion of the model. Primarily, to obtain the

flutter boundaries of the model, linear flutter analysis are preformed. The area

ratios for dielectric layer stand for 18.37 % ( 3 3 mesh), 51.02 % (5 5 mesh) and

100 % ( 7 7 mesh: fully dielectric layer) of the dielectric layer. As the sizes are

increased, critical aerodynamic pressures for flutter and critical temperatures for

buckling behaviors are decreased simultaneously. The reason is due to the

flexibility of the dielectric portion in the layer. That is to say, lower stiffness

characteristics of the layer result in more flexible behaviors, and the model is more

easily to shift the regions for lower temperatures and aerodynamic pressures.

In this regard, the postbuckling behaviors concerned with regions (A) and (B), and

the flutter behaviors related to the regions (A) and (C) are investigated.

4.2 Postbuckling Behaviors

From now on, the dielectric area ratio is fixed 18.37% ( 3 3 mesh) except the cases

with special comment in the postbuckling analysis.

Fig. 4.2 presents the postbuckling characteristics with the variation of aspect ratio

for the face sheets and dielectric enclosure layer according to the ranges between

0.5 and 2.5. As the aspect ratio increases, the non-dimensional deflection increases.

Furthermore, aspect ratio is larger than 2, the temperature increment has negligible

59

effect on the deflection. Similar phenomena are already stated in Ref. [72], and

then the aspect ratio is larger than 1.8, the natural frequencies become very small

amounts of differences.

Fig. 4.3 (a) and (b) depict the effects of airflow on the deformed shapes of model

with temperature variation. Fig. 4.3 (a) shows the deflection with the temperature

difference variation, while Fig. 4.3 (b) presents center deflection shapes along the

x-direction at the point y/b =1/2 with 20T C , respectively. The deflections

decrease as the aerodynamic pressures increase as shown in Fig. 4.3 (a). On the

other hand, the shape of the panel deformed un-symmetrically due to supersonic

airflow in the Fig. 4.3 (b). As the aerodynamic pressure increases, the deflection

reduces and the peak point is moved backward as similarly observed in Ref. [73].

Specially, Fig. 4.3 (b) presents interesting things around the center of model. For

the cases of over than 400 , there are suddenly gone downward in

deformations at the center of the model. It is due to the dielectric layer based in the

center of the model is more ductile than dielectric enclosure face sheets, and thus

center of the structure is more sensitive to aerodynamic pressure at same

temperature increasing. Add to that, this phenomenon is not observed in the case

without aerodynamic pressure.

For more emphasis on the specific characteristic of antenna structure, Fig. 4.4 (a-c)

shows the deformed shapes with variation of non-dimensional aerodynamic

pressures for 200 , 400 and 600. The results show that the peak of the

deflection is moving backwards direction as the aerodynamic load increases.

60

Especially, it is easily to observe that the center of the model is moving downward

direction as in Fig 4.4. (b) and Fig. 4.4 (c) relative to Fig. 4.4 (a). Also, the

aerodynamic pressure increases, center of the structure is more moving downward

direction due to the flexible characteristics of the dielectric layer.

Due to the dielectric portion is essential part of the MSAS model, thus the

performances of the layer is necessary to analyze. Fig. 4.5 shows the thermal post-

buckling behaviors according to the change of area ratio of the dielectric layers. In

this work, full cross section is divided by 7 7 uniform mesh. The area ratios for

dielectric layer stand for 0 (without dielectric layer), 18.37 % ( 3 3 mesh), 51.02 %

( 5 5 mesh) and 100 % ( 7 7 mesh: fully dielectric layer). The deflection is more

developed as the area of the layer increases according to the temperature rises.

Because, lower stiffness characteristics of the layer result in more flexible

behaviors, and finally more sensitive to the temperature variations are occurs.

Fig. 4.6 (a) and (b) show the effects of dielectric portions on the deformed shapes

of model with temperature variation in the case of 400 . Fig. 4.6 (a) shows the

deflection with the temperature variation, while Fig. 4.6 (b) presents non-

dimensional center deflection shapes along the x-direction at the point / 1/ 2y b

with 20T C , respectively. The deflections increases as the dielectric portions

increase as shown in Fig. 4.6 (a). However, the shape of the panel deformed un-

symmetrically due to aerodynamic flow in the Fig. 4.6 (b). And the results show

that the maximum deflection occurs at around / 3 / 4x a [74,75]. As the dielectric

61

portions increase, the maximum deflections increase and deflections at around the

center of the model are more gone downward. This characteristics are also due to

the dielectric portion based in the center of the model is more ductile than dielectric

enclosure face sheets.

Fig. 4.7 shows the effects of temperature variations on the deformed shapes of

model with the case of 400 . The deflections increases as the temperature

variations increase. However, the shape of the panel deformed un-symmetrically

due to aerodynamic flow.

For more detailed comparison study, Fig. 4.8 presents the non-dimensional center

deflection shapes along the x-direction at the point y/b =1/2. As the temperature

increase, the maximum deflections increase and deflections at around the center of

the model are dominantly gone downward with 40T C . This characteristics

are also due to the dielectric portion based in the center of the model is more

ductile than dielectric enclosure face sheets.

These may lead to severe deviations of the phase information of the signals and

affects the degradation of the antenna performance. To improve the structural

performance of the phenomena in Fig. 4.8, the design variables are chosen as two

parameters such as ply angles and the honeycomb core thickness. At first, Fig. 4.9

(a) shows the nondimensional center deflections according to the ply angles. The

results show that the ply angles of each sheet do not affect dominantly on the

deformed shape of the model. It is due to that carbon/epoxy layer is not covered the

dielectric portion, the change of ply angles of the model do not affect the stiffness

62

of the dielectric portion large. Additionally, the shape of the model with between 0o

and 90o are same under no aerodynamic pressure, while those are different patterns

under aerodynamic pressure as shown in Fig. 4.9 (a). Additionally, when the fibers

oriented at 45o, the structure has the largest bending stiffness which results in the

lowest change. Next, Fig. 4.9 (b) shows the nondimensional center deflections with

the variations of honeycomb core thickness. The thickness of the core is chosen as

40% and 80% of total thickness for the model. According to increments of the

honeycomb core thickness, the phenomena dominantly decrease. It is due to that

honeycomb core layer is covered the dielectric portion, the change of honeycomb

core thickness of the model improves the stiffness of the dielectric portion directly.

As a result, structural performance of the model can improve with increase of

honeycomb core thickness than ply angles of layers.

Up to now, the effect of the dielectric layers and the area ratios are discussed.

Further, the effects of shapes of the dielectric region are presented in this work. In

Fig. 4.10, three types of the region are shown as Model (I), (II) and (III) with the

same area (5/49=10.20%) located at the center portion of the model. The Fig. 4.11

shows the post-buckling behaviors of the model without aerodynamic pressure.

Primarily, the deflection of model (I) is smaller than the other cases due to the

dielectric area is concentrated at the center portion of the model than the other

cases. The deflection of models (II) and (III) are almost similar due to the (II) and

(III) are symmetric cases in the x- or y- direction.

On the other hand, Fig. 4.12 (a) and (b) show the post-buckling behaviors for three

63

types of the model with the airflows. The airflow direction is chosen along the x-

direction. On the other hand, models (II) and (III) show equal deformation shapes

irrespective of the direction of the dielectric layer as shown in Fig. 4.11. However

the shapes are different under the airflow as shown in Fig. 4.12. The peak points of

the deflection for the model (III) and the model (II) represent the highest and

lowest, respectively. This means the model (III) is less sensitive than the other

types in the airflow. In other words, the model (III) has the smallest area in the

perpendicular direction of the airflow. Fig. 4.12 (b) presents deformed shapes of

the three types for the model with airflow along the x-direction at 40T C . As

similar to Fig. 4.12 (a), the maximum deflection of model (II) is the lowest, and the

more decreased in deformations at the almost center of the model. It is confirm that

the model (II) is more affected by the airflow due to the largest area in the

perpendicular direction of the airflow.

4.3 Flutter Behaviors

In this section, numerical results are discussed for the panel flutter behavior and

limit cycle oscillations (LCO) of MSAS. The MSAS is modeled as multi-layered

sandwich structure as in Fig. 3.2, and material properties of components are

summarized in Table 3.1. In the analysis, finite element method is using 6 6

64

meshes for nine-node plate elements. Unless otherwise noted, square panels with

thickness ratio (a/h) as 100, and simply-supported boundary conditions are used.

4.3.1 Panel Flutter

The role of the dielectric layer is important in the structure, thus the analysis on the

layer is necessary to evaluate the performances. The thermo-mechanical behaviors

of the structure are discussed to focus on the effects for the sizes and shapes of

dielectric layer. In this regard, three types of the dielectric portions are shown as

Model (I), (II) and (III) with the same area (4/36=11.11%) located at the center

portion as shown in Fig. 4.13. And then, thermal flutter boundaries of the structure

with the change of dielectric portion under thermal and non-dimensional

aerodynamic loads are depicted in Fig. 4.14.

Primarily, to obtain the flutter boundaries between the stable and flutter regions,

linear flutter analysis is preformed. At the 0T , the critical aerodynamic

pressure of Model (III) is the lowest, while the pressure of Model (I) is the highest.

Though the dielectric areas of the three types are same portion of the layer, the

region of Model (III) is widely distributed in the layer. Thus, flexibility of the

dielectric region can affect broadly on the model. However, the region of the

Model (I) is concentrated at the central part of the model than the other types.

Therefore, model (I) is stiffer than the other models at 0T . To obtain the

flutter boundaries in Fig. 4.14, linear flutter analysis at 0T for models (I), (II)

65

and (III) are performed as presented in Fig. 4.15. As monotonically increasing

from zero, real parts of two eigenvalues approach each other until they coalesce at

cr . When the coalescence occurs, imaginary parts of the two eigenvalues are

bifurcated and one of them becomes positive, then the structure becomes

dynamically unstable and flutter occurs. It is noted from the figure that the critical

flutter aerodynamic pressure of the Model (I), (II) and (III) are around at

1000, 880 and 800 , respectively.

Fig. 4.16 depicts time responses of the model (III) in the flat and stable region. The

deflection of the panel converges to zero in a stable region. Additionally, phase plot

is converge to zero point well.

4.3.2 Limit Cycle Oscillations

In this section, limit cycle oscillations (LCOs) are investigated using the Newmark

time integration method. From now on, behaviors of the model are considered in

the LCO region in Fig. 4.14. The dielectric area is chosen as the Model (I) in Figs.

4.17-4.20, and the dielectric part of the Model is located at the center portion of the

layer in the skin. Unless otherwise noted, the dielectric portions are selected for the

model with 2 2 mesh (11.11%).

Figs. 4.17 (a-b) describe the time responses as the non-dimensional aerodynamic

pressures, and aerodynamic load λ is changed from 1200 to 1600. When the

66

aerodynamic loads are increased, the limit cycle amplitudes are increased.

Additionally, periods of the motions are decreased as the pressures are increased.

However, interesting thing is the different shapes of the pattern of the oscillation in

Fig. 4.17 (a). Maximum amplitude in the upper direction and minimum amplitude

in the lower direction show different magnitudes in low aerodynamic pressure

( 1200 ). In other words, maximum points of the wave are almost similar, while

minimum points are different with change of aerodynamic loads. It is due to the un-

symmetrical layers of composite structure. Additionally, the behaviors of

unsymmetrical amplitude are not occurring without thermal effects as shown in Fig.

4.17 (b). Similar results are reported in Ref. [76,77]. The increase of temperature of

simply supported panels can cause the unsymmetrical amplitudes of the vibration

or flutter.

Fig. 4.18 shows the time responses according to the sizes for the dielectric area.

The dielectric portions are selected for the model with 2 2 (11.11%), 4 4

(44.44%) and 6 6 (100%) meshes. As the dielectric areas are increased, the limit

cycle amplitudes and periods are increased simultaneously due to the flexibility of

the dielectric area.

Fig. 4.19 depicts the time responses according to change of the temperature

increment ( T ). The temperature increments are chosen as 0 C and 30 C . The

patterns of two behaviors are almost similar, while LCO behaviors with 30 C

oscillate at a little upward direction.

Fig. 4.20 (a) is phase plot for 0T , and then limit cycle oscillation is observed

67

well. On the other hand, Fig. 4.20 (b) is phase plot for 30T C , and the plot

shows un-symmetric motion in comparison with the Fig. 4.20 (a).

Fig. 4.21 depicts the time responses of the model as the shapes of the dielectric

areas. Increment of temperature 30T C , and the area ratio is fixed as 11.11%

( 2 2 mesh). Additionally, three types of the region are chosen as Model (I), (II),

and (III) as in Fig. 4.13. In here, Figs. 4.21 (a) and (b) are the cases of 1200

and 1600 , respectively. However, interesting thing is the different shapes of

the pattern of the oscillation. Maximum amplitude in the upper direction and

minimum amplitude in the lower direction show different magnitudes in low

aerodynamic pressure ( 1200 ). In other words, maximum points of the wave

are almost similar, while minimum points are different with change of aerodynamic

loads. It may be due to the un-symmetrical layers of composite structure.

Furthermore, period of the Model (III) is shortest than any other models, but the

periods of the Model (I) and (II) are almost equal. In Fig. 4.21 (b), the period of

Model (III) is also shortest. However, the limit cycle amplitudes of the three types

of model are symmetrical behaviors. Figs. 4.21 show that symmetry of amplitude

in the structures is affected by the non-dimensional aerodynamic pressures.

For more emphasis on the specific characteristic of the structure, Figs. 4.22 (a-c)

show the deformed shapes of the Model (III) at three points as marked on Fig. 4.21

(b). In here, the Figs. 4.22 (a), (b) and (c) indicate the deformed shapes at the

lowest, zero and highest amplitudes, respectively. The results show that the peak of

the deflection is moving backwards direction. Furthermore, the specific motions

68

near the corners are slightly observed in Fig. 4.22 (b) and (c) due to the flexible

characteristics of dielectric portions near the corners. In other words, the corners of

model (III) are sensitive to aerodynamic pressure.

Additionally, Figs. 4.23 (a-c) show the deformed shapes of the Model (I) at three

points as marked on Fig. 4.21. As a result of the previous results that the center is

sensitive due to the dielectric portions.

On the whole, the dielectric layer is essential part of the MSAS model, and then the

layer is specially focused in this chapter. The characteristics are analyzed with sizes

and shapes of the dielectric region of the model. The deflection increases as the

area of dielectric layer increases due to low stiffness characteristics of the layer.

Furthermore, sudden decreased of the deformation shapes are observed at the

center of the model due to the increase of flexibility of the dielectric layer. While,

as the antenna structures are usually operated as a part of aircraft in the air,

aerodynamic pressure is important factor to be estimated the performance of the

structure. Also, the aircrafts are possible to move in any direction, thus the shapes

of dielectric layer can develop the different performance of the antenna. Especially,

the difference of the vertical and horizontal shapes of the dielectric portion is not

appear without changing direction of motion, while vertical shape of the layer is

most sensitive due to the largest contact portion in the airflow direction.

Furthermore, the thermal stabilities as well as limit cycle oscillations are deeply

investigated in the supersonic airflow region. The dielectric layer is essential part

69

of the skin, and thus the layer is mainly concerned. Numerical analyses are

preformed for thermal stability boundaries in the three cases of dielectric portions

according to the shapes of dielectric layer. Also, limit cycle oscillations are studied

according to the change of dielectric area, temperature, aerodynamic pressure and

shapes of the dielectric region. Furthermore, abrupt changed of deformed shapes

are observed at the dielectric parts due to the flexibility of the dielectric portion. An

interesting thing is that symmetry of amplitude for un-symmetric model is affected

by the non-dimensional aerodynamic pressures in simply-supported boundary

conditions.

70

0 10 20 30 40 50 60 70 80 90 100

0

200

400

600

800

1000

1200

3X3 mesh (18.37%)

5X5 mesh (51.02%)

7X7 mesh (100%)

oC

C : Flutter (LCO)

A : Stable (flat)

B : Buckled

D : Chaotic

Fig. 4.1 Thermal stability boundaries of clamped model according to

the size of the dielectric region

71

0 4 8 12 16 20

0.0

0.5

1.0

a/b=0.5

a/b=1

a/b=1.5

a/b=2

a/b=2.5

wm

ax/h

oC

Fig. 4.2 Nondimensional deflections with respect to temperature increment

for the variation of aspect ratios of the model

72

0 4 8 12 16 20

0.0

0.2

0.4

0.6

w

ma

x/h

oC

(a) Deformed shapes with temperature variation

0.0 0.5 1.0

0.0

0.2

0.4

0.6

w/h

x/a

(b) Deformed shapes along the x- direction ( 20T C )

Fig. 4.3 Nondimensional deflections due to aerodynamic loads

73

(a) 200

(b) 400

(c) 600

Fig. 4.4 Deformed shapes of the model as the increase of aerodynamic pressure

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

x/ay/b

w/h

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

x/ay/b

w/h

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

x/ay/b

w/h

74

0 2 4 6 8 10 12 14 16 18 20

0.0

0.5

1.0

without dielectric region

3X3 mesh (18.37%)

5X5 mesh (51.02%)

7X7 mesh (100%)

oC

wm

ax/h

Fig. 4.5 Nondimensional deflections according to the temperature increment

for the variation of dielectric portion

75

0 4 8 12 16 20

0.00

0.05

0.10

0.15

1X1 mesh (2.04%)

3X3 mesh (18.37%)

5X5 mesh (51.02%)

wm

ax/h

oC

(a) Deformed shapes with temperature variation

0.0 0.5 1.0

0.00

0.05

0.10

0.15

1X1 mesh (2.04%)

3X3 mesh (18.37%)

5X5 mesh (51.02%)

w/h

x/a

(b) Deformed shapes along the x- direction ( 20T C )

Fig. 4.6 Nondimensional deflections with the variations

of dielectric portions

76

(a) 10T C (b) 20T C

(c) 30T C (d) 40T C

Fig. 4.7 Deformed shapes of the model according to

the increase of temperature variations

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

0.25

x/ay/b

w/h

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

0.25

x/ay/b

w/h

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

0.25

x/ay/b

w/h

0

0.5

1

0

0.5

10

0.05

0.1

0.15

0.2

0.25

x/ay/b

w/h

77

0.0 0.5 1.0

0.0

0.1

0.2

0.3

10oC

20oC

30oC

40oC

w/h

x/a

Fig. 4.8 Nondimensional deflections according to

the increase of temperature variations

78

0.0 0.5 1.0

0.0

0.1

0.2

0.3

0o

45o

90o

w/h

x/a

(a) Deflections with the variations of ply angles

0.0 0.5 1.0

0.0

0.1

0.2

0.3

40%

80%

w/h

x/a

(b) Deflections with the variations of honeycomb core thickness

Fig. 4.9 Nondimensional deflections with the variations of parameters for

improvement of structural behaviors

79

(a) Model (I): Cross shape

(b) Model (II): Vertical shape

(c) Model (III): Horizontal shape

Fig. 4.10 Shape of the dielectric portion

80

0 5 10 15 20 25 30 35 40

0.0

0.5

1.0

1.5

Model (I)

Model (II)

Model (III)

wm

ax/h

oC

Fig. 4.11 Nondimensional deflection

with the three types of shape for dielectric portion

81

0 5 10 15 20 25 30 35 40

0.0

0.1

0.2

Model (I)

Model (II)

Model (III)

wm

ax/h

oC

(a) Nondimensional deflections

0.0 0.5 1.0

0.0

0.1

0.2

Model (I)

Model (II)

Model (III)

w/h

x/a

.

(b) Deformed shapes along the x- direction ( 40T C )

Fig. 4.12 Nondimensional deflection due to three types of shape for dielectric

layer under aerodynamic flow

82

Model (I)

Model (II)

Model (III)

Fig. 4.13 Three types of models with various shapes of the dielectric portion

83

0 10 20 30 40 50 60 70

400

600

800

1000

1200

Stable (flat)

Flutter (LCO)

Model (I)

Model (II)

Model (III)

oC

Fig. 4.14 Thermal flutter boundaries with the shapes of the dielectric portion

84

700 800 900 1000 1100

-2000

0

2000

4000

6000

8000

Model (I)

Model (II)

Model(III)

Eig

en

-Fre

qu

en

cy

Nondimensional dynamic pressure

Fig. 4.15 Frequency coalescence flutters according to the models

85

0.00 0.02 0.04 0.06 0.08 0.10

-0.10

-0.05

0.00

0.05

0.10

w

/h

Time [sec]

(a) Non-dimensional deflection

-0.10 -0.05 0.00 0.05 0.10

-50

0

50

w / h

w/h

(b) Phase plot

Fig. 4.16 Time responses of the model (III)

in the flat and stable region ( 600 , 30T C )

86

0.00 0.01 0.02 0.03 0.04

-0.4

-0.2

0.0

0.2

0.4

w

/h

time [sec]

(a) 30T C

0.00 0.01 0.02 0.03 0.04

-0.4

-0.2

0.0

0.2

0.4

w/h

Time [sec]

(b) 0T C

Fig. 4.17 Time responses of the model subjected to the aerodynamic load

during the limit cycle oscillation

87

0.00 0.01 0.02 0.03 0.04

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

2X2 mesh (11.11%)

4X4 mesh (44.44%)

6X6 mesh (100%)

w/h

Time [sec]

Fig. 4.18 Time responses of the model for the variation of dielectric portion

( 30T C , 1600 )

88

0.00 0.01 0.02 0.03 0.04

-0.4

-0.2

0.0

0.2

0.4 = 0

= 30oC

w/h

Time [sec]

Fig. 4.19 Time response of the model for the variation of temperature ( 1600 )

89

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

-750

-500

-250

0

250

500

750

T = 0

w

/ h

w / h

(a) 0T

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

-750

-500

-250

0

250

500

750

T = 30oC

w / h

w / h

(b) 30T C

Fig. 4.20 Phase plot of the model according to temperature increments ( 1600 )

90

0.00 0.01 0.02 0.03 0.04

-0.4

-0.2

0.0

0.2

0.4

Model (I)

Model (II)

Model (III)

w/h

Time [sec]

(a) 1200

0.00 0.01 0.02 0.03 0.04

-0.4

-0.2

0.0

0.2

0.4 c

b

a Model (I)

Model (II)

Model (III)

w/h

Time [sec]

(b) 1600

Fig. 4.21 Time responses of the model according to the shapes of dielectric region

( 30T C )

91

(a) At the lowest amplitude

(b) At zero

(c) At the highest amplitude

Fig. 4.22 Deformed shapes at the points in LCO behaviors

(Model (III), 30T C , 1600 )

0

0.5

1

0

0.5

1

0.4

0.2

0

-0.2

-0.4

x/ay/b

w/h

0

0.5

1

0

0.5

1-0.4

-0.2

0

0.2

0.4

x/ay/b

w/h

0

0.5

1

0

0.5

1-0.4

-0.2

0

0.2

0.4

x/ay/b

w/h

92

(a) At the lowest amplitude

(b) At zero

(c) At the highest amplitude

Fig. 4.23 Deformed shapes at the points in LCO behaviors

(Model (I), 30T C , 1600 )

93

Chapter 5

Active Flutter Control of MSAS

The location and area of the piezoelectric patches are important for the effective

flutter control. Prior to analyze the MSAS model, the composite sandwich plate is

investigated with the form of distributed patches or continuous layers in various

studies for the effective control. The dielectric portions of the model and

aerodynamic pressures are not considered as shown in Fig. 5.1. With these results

in Figs. 5.2-5.6, the optimal locations of the piezoelectric actuators and sensors for

flutter control for MSAS is selected, and then MSAS models with patches under

aerodynamic and thermal loads are proposed for the effective control. Unless

otherwise noted, piezoelectric layers embedded on top and bottom surfaces to act

as sensor and actuator, respectively. Numerical simulations using the developed

finite element model with linear piezoelectric patches on them. A clamped

composite plate with two additional actuator layers (PZT G1195N) at the top and

bottom of the plate is modeled. The plate dimensions considered are a=b=100mm

and total thickness h=1mm and the thickness of piezoelectric layers is 0.1mm. The

material properties considered are in Table 2.1.

94

5.1 Composite Sandwich Plate with Piezoelectric Patches

The control the free vibration of the plate, the collocated sensors and actuators

should be coupled into sensor/actuator (S/A) pairs through closed control loops.

The load original load is 250 /N m . First of all, a constant gain negative velocity

feedback control algorithm is applied. It is assumed that the composite plate is

vibrating freely due to an initial disturbance. Fig. 5.2 shows the effect of feedback

control gain G on the responses of the plate. It can be seen that with higher control

gain, the vibration of the plate is damped out more quickly. The Increasing

feedback control gains can result in a higher damping matrix in the system

equation. So the vibration of the plate can be suppressed much faster at higher

feedback control gain.

The effect of the position for sensor/actuator pairs on the responses of the model is

investigated in Fig. 5.3. The piezoelectric sensors/actuators are bonded on different

positions of the upper and lower full surfaces of the plate, respectively. The

constant gain negative feedback control is applied. The results show that the

vibration control effect is the better as the sensor/actuator pairs are bonded on the

center of the model. It is due to that the maximum displacement appears at near the

center of the vibration.

However, linear quadratic regulator (LQR) optimal control is much more effective

when compared to the constant negative velocity feedback. In this regard, the LQR

95

method is used for the analysis in remained researches.

To study the relations between the control effects and sizes of piezoelectric patches,

the flutter bounds of structural systems with different sizes of actuator/sensor pairs

are calculated and shown in Fig. 5.4. The sizes are chosen as 2 2 (6.25%) and

8 8 (100%) meshes. The result shows that as the increase of the piezoelectric

patch sizes, the vibration is more quickly damped. In other words, the area of the

piezoelectric material covered on the panel is larger, the control effect is better.

However, the quantity of the piezoelectric materials is limited in practice, so the

optimal location design for the piezoelectric patches is important.

The effect of the piezoelectric layers positions between the inner layer and outer

layer on the vibration suppression is shown in Fig. 5.5. It is observed that as the

smart material layer is moved farther from the mid-plane, the suppression time

decreases. It is due the moment effect by smart layer actuations. In other words, the

surface-bonding case shows the control is more efficient due to the largest moment

arm for the piezoelectric forces about the laminate mid-plane.

Up to now, vibration suppressions of the model with piezoelectric patches are

investigated. Now on, the flutter suppressions of the model are studied.

Fig. 5.6 shows the stable regions of the panels with different actuator and sensor

placements. The distributed patches are located at the center in Model (I), while the

patches are positioned at the 3/4 point of the Model (II). The location of Model (II)

is the peak of the aeroelastic mode of the laminated panel as it is shown in Fig. 5.6.

By comparing the areas of the flat regions of the two types of structural system, it

96

is noted that the aeroelastic stability of the structural system type (II) is better than

that of the model (I). It is due to that the actuator/sensor pairs of the structural

system types (II) is bonded near and even right on the peak of the aeroelastic mode

of the laminated panel, so the piezoelectric patches and their actuations are more

helpful for strengthening the stiffness of the aeroelastic structure than those of the

type (I) whose actuator and sensor pairs are bonded near the center of the

aeroelastic structural system.

LQR is more efficient than negative feedback control. In these regards, the LQR

algorithm, sensor/actuators are located 75% point from the airflow direction, large

size of the piezoelectric patches and location of outer layers conditions are efficient

for the flutter suppression.

5.2 Design of MSAS with Piezoelectric Patches

Recently, some engineers have tried to perform the active vibration control by

integrating sensors and actuators into the antenna structures, and propose the model

as shown in Fig. 5.7 [8-10]. As previously explained in chapter 3, glass fiber

reinforced polymer (GFRP) composites are used due to the cover must be

transparent to radiofrequency (RF) radiation. Additionally, the piezoceramic

actuators are located beside the antenna elements in order not to block the antenna

portion.

Three types of models are proposed in order not to block the antenna portion in Fig.

97

5.9. The model in Fig. 5.7 is designed to control the vibration, so the piezoceramic

actuators are located at the center. However, the focus of this research is the flutter

suppression performance. Therefore, the 3/4 point of maximum displacement

structures appear to have greater effect on flutter control as in previous results in

5.6, the piezoelectric patches are designed. Especially, the piezoelectric patches are

embedded in the model (III). It is due to that the piezoceramics are weak and brittle.

Additionally, noted that the piezoelectric patches are used same area and same

number. As previously mentioned, the quantity of the piezoelectric materials is

limited in practice, so the optimal location designs for the piezoelectric patches are

chosen.

5.3 Active Flutter Suppression of MSAS

Fig. 5.10 shows that the effect of aerodynamic pressure on responses of flutter

suppression. The control is started after a lapse of 0.05sec in order to compare both

the controlled and uncontrolled responses. The aerodynamic pressures ( ) are

chosen as 1200 and 1600. The result shows that the flutter suppression control with

given piezoelectric material and properties with a thickness of suppression is more

difficult under high dynamic pressure loads.

Fig. 5.11 also shows that the thermal effect on responses of flutter suppression. The

control is started after a lapse of 0.05sec in order to compare both the controlled

98

and uncontrolled responses. The thermal variations ( T ) are chosen as 0T

and 30T C . The result shows that the flutter suppression control is more

difficult under high thermal loads.

Fig. 5.12 shows the flutter suppression behaviors of the models using distributed

piezoelectric patches. The result shows that the Model (I) is most efficient, and

Model (II) is the worst design. It is due to that the piezoelectric patches of the

Model (I) are located at maximum displacement point for the suppression of the

flutter behaviors. Additionally, the suppression behaviors of the Model (I) and (III)

are almost similar due to almost equal position of the patches.

It is important to increase the area of antenna elements due to the increase of

performances of antenna structure. However, the quantity of the piezoelectric

materials is limited in practice, and piezoelectric patches are located beside the

antenna elements in order not to block the antenna portion. It can be degraded the

performance of the piezo sensor and actuator. Optimized design for the size of the

antenna for the radiofrequency radiation and the position of the piezo sensor and

actuator for the suppression of flutter are important.

On the whole, prior to analyze the MSAS model, the composite sandwich plate is

investigated with piezoelectric patches in various studies for the effective control.

A laminated composite plate with PZT piezoelectric layers embedded on top and

bottom surfaces to act as sensor and actuator is considered. Additionally, the

constant gain negative velocity feedback control and linear quadratic regulator

99

(LQR) control are applied. With these results, the optimal locations of the

piezoelectric actuators and sensors for flutter control for MSAS is selected, and

then MSAS models with patches under aerodynamic and thermal loads are

proposed for the effective control. The flutter suppression control behaviors of the

structures are analyzed.

100

Fig. 5.1 Modeling of composite sandwich plate with piezoelectric patches

101

0.00 0.03 0.06

-0.4

-0.2

0.0

0.2

0.4

G=0

G=1000

G=2000

w/h

Time [sec]

Fig. 5.2 The effect of negative velocity feedback control gain G

on the responses of the model

102

Model (I) Model (II)

(a) Model (I) and (II)

0.00 0.03 0.06

-0.4

-0.2

0.0

0.2

0.4

Model (I)

Model (II)

w/h

Time [sec]

(b) Response of the locations for sensor/actuator pairs

Fig. 5.3 The effect of locations for sensor/actuator pairs

103

0.00 0.03 0.06

-0.4

-0.2

0.0

0.2

0.4

2X2 mesh (6.25%)

8X8 mesh (100%)

w/h

Time [sec]

Fig. 5.4 The effect of sizes for the sensor/actuator pairs

104

0.00 0.03 0.06

-0.4

-0.2

0.0

0.2

0.4

Inner layer

Outer layer

w/h

Time [sec]

Fig. 5.5 The effect of the positions for sensor/actuator through the thickness

105

Model (I) Model (II)

(a) Model (I) and (II)

0 15 30 45 60 75 90

0

300

600

900

1200

B : Buckled

A : Stable (flat)

C : Futter (LCO)

Model (I)

Model (II)

oC

Fig. 5.6 Stability regions with Model (I) and (II)

106

(a) Active control modeling with piezoceramic actuator

(b) Fabricated active control model

Fig. 5.7 Active control model with piezoceramic actuator [8-10]

107

Fig. 5.8 Modeling of MSAS with distributed piezoelectric patches

108

Model I (Top) Model I (Side)

Model II (Top) Model II (Side)

Model III (Top) Model III (Side)

Fig. 5.9 Three types of models with dielectric portion and piezoelectric patches

109

0.00 0.05 0.10 0.15

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

1200

1600

w/h

Time [sec]

Fig. 5.10 The aerodynamic pressure effect on flutter suppression

(Model (I), 0T )

110

0.00 0.05 0.10 0.15

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0

30oC

w/h

Time [sec]

Fig. 5.11 The thermal effect on flutter suppression (Model (I), 1200 )

111

0.00 0.05 0.10 0.15

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Model (I)

Model (II)

Model (III)

w/h

Time [sec]

Fig. 5.12 The flutter suppression control of the designed models

112

Chapter 6

Conclusions

The conformal load-bearing antenna structure (CLAS) has many merits such as

reductions of weight, drag and RCS. However, the structures under aerodynamic

loads may cause vibrations and deformations of the model. And then, these will

affect the degradation of the antenna performance. However, numerous researches

on CLAS have been performed on only performances of antenna function in a

limited range. Thus, the antenna structure is investigated with aerodynamic flows

and aerodynamic heating that may occur during supersonic flight for more realistic

effects. The results show that severe deformations of antenna element are appeared

due to thermal and aerodynamic loads. To improve the postbuckling behaviors of

the model, the structural performance is dominantly improved according to

increments of the honeycomb core thickness. Additionally, active flutter controls of

the antenna structure are studied using piezoelectric sensors and actuators for

suppressions of panel flutter.

The summary of the present works are given as follows.

113

6.1 Summary

1. The governing equations of composite plates under thermal and aerodynamic

loads are formulated. The first-order shear deformation theory of the plates and von

karman nonlinear strain-displacement relation are applied. The governing equations

are obtained from the principle of virtual work. Then, equation of motion is divided

into two coupled equations such as static and dynamic problems. For the nonlinear

static analysis, the nonlinear governing equation is applied as incremental form by

using Newton Raphson iteration method. For the nonlinear flutter analysis,

Newmark time integration method is applied in time domain. Additionally, the

piezoelectric effect is added to the dynamic equations of the composite plate. The

linear piezoelectric theory is used to derive the equations of piezoelectric actuation

and sensing. Furthermore, the constant gain negative velocity feedback control and

linear quadratic regulator (LQR) control are applied.

2. The essential and basic components of the CLAS are presented using a typical

configuration, and the MSAS model used is designed with the concepts of CLAS.

The cover must be transparent to RF radiation, and thus it is typically constructed

using GFRP composites. The CFRP is used as the dielectric enclosure layers due to

the lower stiffness of the GFRP covers. Finally, a model is design and presented

with basic components and concepts which are expected to be required and applied.

For the code verification of the designed model, composite sandwich plates and the

114

dielectric portion are verified with reported data.

3. The dielectric layer is essential part of the CLAS model, and then the layer is

specially focused. The characteristics are analyzed with sizes and shapes of the

dielectric region of the model. The deflection increases as the area of dielectric

layer increases due to low stiffness characteristics of the layer. Furthermore, sudden

decreased of the deformation shapes are observed at the center of the model due to

the increase of flexibility of the dielectric layer. The aircrafts are possible to move

in any direction, thus the shapes of dielectric layer can develop the different

performance of the antenna. Especially, the difference of the vertical and horizontal

shapes of the dielectric portion is not appear without changing direction of motion,

while vertical shape of the layer is most sensitive due to the largest contact portion

in the airflow direction.

4. The thermal stabilities as well as limit cycle oscillations are deeply investigated

in the supersonic airflow region. Numerical analyses are preformed for thermal

stability boundaries in the three cases of dielectric portions according to the shapes

of dielectric layer. Also, limit cycle oscillations are studied according to the change

of dielectric area, temperature, aerodynamic pressure and shapes of the dielectric

portion. An interesting thing is that symmetry of amplitude for un-symmetric

model is affected by the non-dimensional aerodynamic pressures in simply-

supported boundary conditions.

115

5. Prior to analyze the MSAS model, the composite sandwich plate is investigated

with piezoelectric patches in various studies for the effective control. A laminated

composite plate with PZT piezoceramic layers embedded on top and bottom

surfaces to act as sensor and actuator is considered. Additionally, the constant gain

negative velocity feedback control and linear quadratic regulator (LQR) control are

applied. With these results, the optimal locations of the piezoelectric actuators and

sensors for flutter control for MSAS is selected, and then MSAS models with

patches under aerodynamic and thermal loads are proposed for the effective control.

The results show that the systems based on the controllers effectively suppress

panel flutter motions.

6.2 Future Works

Although this thesis suggest that the characteristics of MSAS model and active

flutter control with piezoelectric patch, there are a few challenging further works as

follows.

1. The present study developed structural model and active flutter control system.

However, only structural analysis is performed with designed MSAS model in this

thesis. Therefore, the antenna structure is finally required to consider the radiation

patterns and performances of antenna model as well as the structural performance

116

of the antenna structure.

2. Modern aircrafts can be freezing during the flight under subsonic flows at high

altitudes. In spite of improvements in design and techniques, icing-related

accidents and operating problems still occur, and thus icing condition is important

issue to the performance and safety of air vehicles. The antenna structures are

required to investigate the capability to operate in the low temperature environment.

3. The present study is investigated under aerodynamic and thermal conditions. For

more practical model, considering the mechanical load would be better used for

realistic various demands and application of the MSAS model.

117

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125

Appendix

Matrix Transformations

The transformed elastic coefficients given in equation (2.5) have the form,

11 12 16

12 22 26

16 26 66

44 45

45 55

0 0

0 0

0 0

0 0 0

0 0 0

Q Q Q

Q Q Q

Q Q Q Q

Q Q

Q Q

where

4 2 2

11 11 12 66

2 2 4 4

12 11 22 66 12

4 2 2 4

22 11 12 66 22

3 3

16 11 12 66 12 22 66

3

26 11 12 66

cos 2( 2 )sin cos

( 4 )sin cos (sin cos )

sin 2( 2 )sin cos cos

( 2 )sin cos ( 2 )sin cos

( 2 )sin cos

Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q Q Q

Q Q Q Q

3

12 22 66

2 2 4 4

66 11 12 12 66 66

2 2

44 44 55

45 55 44

2 2

55 44 55

( 2 )sin cos

( 2 2 )sin cos (sin cos )

cos sin

( )sin cos

sin cos

Q Q Q

Q Q Q Q Q Q

Q Q Q

Q Q Q

Q Q Q

with

126

11 11 12 21

12 12 12 12 21

22 2 12 21

66 12

44 23

55 13

/ (1 )

/ (1 )

/ (1 )

Q E v v

Q E v v

Q E v v

Q G

Q G

Q G

The piezoelectric stress coefficient matrix e has the form as

14 15

24 25

31 32 36

0 0 0

0 0 0

0 0

e e

e e

e e e

e

where

2 2

31 31 32

2 2

32 31 32

36 31 32

14 15 24

2 2

24 24 15

2 2

15 15 24

25 15 24

cos sin

sin cos

( )sin cos

( )sin cos

cos sin

cos sin

( )sin cos

e e e

e e e

e e e

e e e

e e e

e e e

e e e

where [ ]ije are piezoelectric stress coefficients.

127

Sensor and Actuator Equations

The electro-mechanical coupling

. The components iD of electrical displacement vector are related to the

components of strains and electrical field by

1 1 11 1

2 2 22 2

3 31 32 6 33 3

0 0 0 0 0 ε

0 0 0 0 0 ε

0 0 0 ε

k k kD

D

D e e

where ije are piezoelectric stiffness and ij are dielectric coefficients. The

transformed equations are

11 12

21 22

31 32 36 33

0 0 0 0 ε

0 0 0 0 ε

0 0 ε

kk k

x xx x

y yy y

z xy z

D

D

D e e e

Sensor equations

According the Gauss law, the closed circuit charge measured through the electrodes

of the kth layer is

1( ) ( )

1 1( )

2 2I k I k

k

s z zS z z S z z

Q t D dxdy D dxdy

128

where S is the surface electrode. And subscript 's' denotes sensor.

The total charge in the laminate is calculated by summing over the number of

sensor layer as

1( ) ( )1

1( )

2

s

I k I k

Nk

s z zS z z S z z

k

Q t D dxdy D dxdy

The current I(t) on the surface of the sensor is given by

( ) sdQI t

dt

When the sensors are used as strain rate tensors, the current can be converted to the

open circuit sensor voltage output Vs by

( ) ( ) ss c c

dQV t G I t G

dt

where Gc is the gain of the current amplifier.

0

ε 0

/s sV h

129

Here sh denotes the thickness of the sensor layer. The forces and moments will be

determined in terms of the strain rates in the sensor layers.

The Matrix in the global dynamic equation

The Electric field E can be written in terms of electric potential as

0 0 1/ [ ]T

a a ah V B V E

Because it is assumed that the electric potential is constant over an element and

varies linearly through the thickness. Using the total strain definition and the strain

displacement matrices, the linear stiffness matrices of the system can be written as

1

1

1

1

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

n l T l

uu k A z

n l T

u k A z

n T l

u k A z

n T

k A z

B Q B dzdA

B Q B dzdA

B Q B dzdA

B Q B dzdA

K

K

K

K

where lB is the linear strain displacement matrix.

130

국문초록

공력열탄성 특성을 고려한

다기능 내장 안테나 구조물의 능동 플러터 제어

이창열

서울대학교 대학원

기계항공공학부

본 논문은 열과 공기력의 하중에서의 압전 센서와 작동기를 이용한 다기

능 내장 안테나 구조물 (MSAS)의 공력열탄성 해석과 능동 플러터 제

어에 관한 연구를 수행하였다. 구조물은 카본에폭시와 글라스에폭시 그

리고 유전체로 구성된 다층 샌드위치 패널로, 압전재료 층은 구조물의

위 아래에 각각 센서와 작동기로 구성되었다. 모델은 1차 평판 전단 변

형 이론을 사용하였고, 기하학적 비선형을 고려하기 위해 폰 칼만 비선

형 이론을 적용하였다. 공기력을 고려하기 위해 1차 피스톤 이론을 통

하여 공기의 흐름을 나타내었고, 지배 방정식은 가상일의 원리를 이용하

131

여 유도되었다. 비선형 정적 해석을 위해 뉴톤 랩슨 방법이 사용되었으

며, 비선형 동적 해석을 위해 뉴마크 시간 적분법이 시간영역에서 사용

되었다. 이 연구에서 사용된 수치해석 결과를 검증하기 위해 다양한 보

고된 결과들과 비교, 검증을 수행하였다. 특히, 다양한 온도와 공기력

영역에서의 모델의 안정성 영역을 분석하였고, 이에 따라 좌굴, 후좌굴,

제한주기 플러터 거동에 대해서 고찰하였다. 또한 안테나 모델의 핵심

부분인 유전체의 다양한 크기와 모양에 관하여 자세히 해석하였다. 제어

이론으로 Constant gain negative velocity feedback control 과

Linear quadratic regulator (LQR) 를 사용하여 압전 패치의 위치와

크기가 플러터 저감 거동에 미치는 영향을 알아보았다. 본 연구를 통해

압전 재료를 이용하여 다기능 내장 안테나 구조의 패널 플러터를 효과적

으로 제어할 수 있었다.

주제어: 능동 플러터 제어, 다기능 내장 안테나 구조 (MSAS), 압전 센

서와 작동기, LQR 제어

학번: 2009-20712