### UNIVERSITA’ DI PISA

### FACOLTA’ DI INGEGNERIA

### Corso di Laurea Magistrale in Ingegneria Aerospaziale

### Analysis of the effects of nonlinear actuator

### dynamics on the flutter of wings with

### electromechanical flight controls

### Relatore:

**Prof. Ing. G. Di Rito **

**Prof. Ing. R. Galatolo **

**Ing. F. Schettini **

**Prof. Ing. E. Denti **

### Candidato:

**Federico Betti **

**Abstract **

This thesis has as objective to study, via nonlinear modelling and simulation, the effects of electro-mechanical flight controls dynamics on wing flutter.

The basic step for performing the analysis is to develop a generic aeroservoelastic model of a wing with a control surface moved by an electro-mechanical actuator, and the case study has been referred to the actuation system of a subsonic small-scale plane. Starting from an aeroelastic model derived from literature, the actuator dynamics has been included, by introducing the non-linear behaviour caused by freeplay in the actuator hinges. The variation of flutter speed is thus characterised as function of freeplay amplitude, by performing simulations of gust responses.

Behaviour in case of failure of the actuator control system has been also addressed: by assuming that failure is detected and the motor of the actuator is shut-off, the compensator damper has been designed in order to assure safe operations with respect to aeroelastic issues.

All simulations have been performed using a complete Matlab® - Simulink model of the aeroservoelastic system.

Analysis of the effects of nonlinear actuator dynamics on the flutter of wings with electromechanical flight controls

**Sommario **

Questa tesi ha come obiettivo di studiare, tramite modellazione non lineare e simulazione numerica, gli effetti della dinamica dei controlli elettro-meccanici sul flutter delle ali.

Il passo fondamentale per eseguire l'analisi è stato quello di sviluppare un generico modello aeroservoelastico di un'ala con una superficie di controllo mossa da un attuatore elettro-meccanico. Lo studio si è concentrato sul sistema di attuazione di un piccolo aereo subsonico. Partendo da un modello aeroelastico della letteratura, la dinamica dell’attuatore è stata inclusa introducendo inoltre il comportamento non lineare causato dal gioco nella cerniera dell’attuatore. La variazione della velocità di flutter è stata così caratterizzata come funzione dell’ampiezza del gioco, effettuando delle simulazioni alla risposta alla raffica.

É stato affrontato anche il comportamento in caso di guasto del sistema di controllo dell’attuatore: assumendo che l’avaria sia rilevata e che il motore dell’attuatore sia spento, è stato progettato un compensatore al fine di garantire un funzionamento sicuro per quanto riguarda i problemi aeroelastici.

Tutte le simulazioni sono state eseguite utilizzando un modello aeroservoelastico completo Matlab® - Simulink.

Analysis of the effects of nonlinear actuator dynamics on the flutter of wings with electromechanical flight controls

### CONTENTS

**Abstract ... ii **

**Sommario ... iii **

**List of figures ... viii **

**List of tables ... xii **

**List of acronyms ... xiii **

**List of symbols ... xiv **

**Introduction ... 2 **

**1. Effects of flight actuator dynamics on wing flutter ... 4 **

**1.1.** **General overview ... 4**

**1.2.** **EMA actuation for flight control system ... 12**

1.2.1. Advantages of EMA actuation ... 15

1.2.2. EMA applications... 17

**2. Reference model for control surface flutter ... 21 **

**2.1.** **Basic concepts for dynamic aeroelasticity ... 21**

2.1.1. Hamilton’s principle ... 21

2.1.2. Lagrange’s Equations ... 23

2.1.3. Simplified Unsteady Aerodynamics ... 25

**2.2.** **Control surface flutter ... 27**

2.2.1. Baseline system parameters ... 30

**3. Electromechanical actuator dynamic model with hinge freeplay 35 **
**3.1.** **Electro-Mechanical Actuator model ... 35**

3.1.1. Architectures and components ... 35

3.1.2. Electro-Mechanical Actuator dynamics with speed and position closed loop controls ... 38

3.1.3. Matlab – Simulink actuator implementation ... 46

3.1.4. Actuator dynamic stiffness ... 47

3.1.5. Effects of inner structural modes ... 51

3.1.6. Effects of freeplay ... 54

**3.2.** **Servo-hydraulic actuator model ... 59**

3.2.1. Actuator transfer functions ... 62

**4. Complete aeroservoelastic model ... 67 **

**4.1.** **Integration of the Electro-Mechanical actuator in the aeroelastic **
**model. ... 67**

**4.2.** **Inclusion of Gust Terms ... 68**

**4.3.** **Governing equations ... 70**

**4.4.** **Matlab - Simulink Implementation ... 72**

**4.5.** **Flutter analysis ... 73**

4.5.1. LTI aeroservoelastic model ... 73

4.5.2. Nonlinear aeroservoelastic model ... 78

4.5.3. Without freeplay ... 80

**4.6.** **Effects of actuator technological solution on flutter behaviour ... 90**

**4.7.** **Aeroservoelastic model with actuator failure ... 94**

4.7.1. Actuator fault modes ... 94

4.7.2. Motor failure simulation and compensator damper design ... 96

4.7.3. Fail-Safe mode implementation issues ... 103

**Conclusions And Future Works ... 106 **

**Appendix ... 109 **

**Appendix A ... 109**

A.1 Static aeroelasticity ... 109

**Appendix B ... 113**

**B.1 Binary aeroelastic model ... 113**

B.1.1 Aeroelastic equations of motions ... 113

Analysis of the effects of nonlinear actuator dynamics on the flutter of wings with electromechanical flight controls

**References ... 121 **
**Ringraziamenti ... 126 **

### LIST OF FIGURES

Figure 1.1: Block diagram of a typical aeroservoelastic system [1] ______________________ 5

Figure 1.2: Aeroservoelastic pyramid [2] ____________________________________ 5

Figure 1.3: The aeroelastic triangle of forces [6] _______________________________ 7

Figure 1.4: Types of power supply of aircraft systems [9] __________________________ 12

Figure 1.5: Estimated weight of onboard systems in the transition to All electric aircraft [9] ______ 13

Figure 1.6: Example of Power-by-wire flight actuators [9] _________________________ 15

Figure 1.7: EMA actuator [12] _________________________________________ 17

Figure 1.8:Scenario of the EMA introduction in aircraft flight control systems [13] ___________ 18

Figure 1.9: Types of actuators used on the Boeing 787 [14] _________________________ 19

Figure 1.10: Actuation system on the Boeing 787 [14] ____________________________ 19

Figure 1.11 Trends in aeronautical design [15] ________________________________ 19

Figure 2.1: Two-dimensional aerofoil [2] ____________________________________ 25

Figure 2.2: Aeroelastic model including a full span control surface _____________________ 27

Figure 2.3: V! and Vg plot for low control stiffness _____________________________ 31

Figure 2.4: V! and Vg plot for medium control stiffness __________________________ 32

Figure 2.5: V! and Vg plot for high control stiffness ____________________________ 32

Figure 2.6: Literature model response to gust (𝑘𝛽 = 103) _________________________ 33

Figure 2.7: Literature model response to gust (𝑘𝛽 = 104) _________________________ 33

Figure 2.8: Literature model response to gust (𝑘𝛽 = 105) _________________________ 33

Figure 2.9: Literature model response to gust (𝑘𝛽 = 105) at Flutter speed _______________ 34

Figure 3.1: EMA's typologies [19] _______________________________________ 36

Figure 3.2: Exploded view of a brushless motor [19] _____________________________ 36

Figure 3.3 Example of a planetary gearbox [19] _______________________________ 37

Figure 3.4: Three-phase motor connected to a 6-switches power electronic [19] _____________ 38

Figure 3.5: Root locus of Speed Loop _____________________________________ 43

Figure 3.6: Bode diagram of Speed loop transfer function __________________________ 43

Figure 3.7: Root locus of Position Loop (High Frequency) _________________________ 44

Figure 3.8: Root locus of Position Loop (Low Frequency) _________________________ 45

Figure 3.11: Simulink EMA Speed loop ____________________________________ 46

Figure 3.12: Simulink EMA Position loop ___________________________________ 46

Figure 3.13: Static and dynamic stiffness model of EMA [22] _______________________ 47

Figure 3.14: Bode diagram of "/$ _______________________________________ 48

Figure 3.15: Bode diagram of T/β _______________________________________ 50

Figure 3.16: Position response to unit step torque disturbance _______________________ 50

Figure 3.17: Position response to unit command _______________________________ 51

Figure 3.18: Mechanical transmission scheme [23] ______________________________ 51

Figure 3.19: Bode Diagram of EMA actuator stiffness with the effect on vibrational mode _______ 52

Figure 3.20: Position response to unit step torque disturbance _______________________ 53

Figure 3.21: Position response to unit command _______________________________ 53

Figure 3.22: Backlash model [25] ________________________________________ 55

Figure 3.23: Deadband [26] ___________________________________________ 55

Figure 3.24: Bode of 𝛽/𝑇 without freeplay __________________________________ 56

Figure 3.25: Bode of 𝛽/𝑇 with 0.3° of freeplay ________________________________ 57

Figure 3.26: FFT analysis at 3.46 Hz with freeplay = 0.3° _________________________ 57

Figure 3.27: Zoom of FFT analysis at 3.46 Hz with freeplay = 0.3° ____________________ 58

Figure 3.28: Hydraulic scheme of the tested flight actuator _________________________ 59

Figure 3.29: Flight actuator closed-loop control _______________________________ 60

Figure 3.30: Actuator flowrate absorption (simplified scheme) _______________________ 60

Figure 3.31: Servo hydraulic actuator dynamic stiffness ___________________________ 63

Figure 3.32: Comparison between EMA dynamic stiffness and SA dynamic stiffness __________ 65

Figure 3.33: Comparison between EMA dynamic stiffness (with and without freeplay) and SA dynamic stiffness ______________________________________________________ 65

Figure 4.1: System with a control surface with bending and torsion modes [2] ______________ 67

Figure 4.2: Effective angle of incidence due to the vertical gust [2] ____________________ 69

Figure 4.3: Example of gust input 1-cosine __________________________________ 69

Figure 4.4: Complete Aeroservoelastic model _________________________________ 72

Figure 4.5: Aeroelastic subsystem _______________________________________ 73

Figure 4.6: Real part of the poles vs velocity _________________________________ 76

Figure 4.7: Root locus ______________________________________________ 76

Figure 4.8: Damping Ratio vs velocity ____________________________________ 77

Figure 4.9: Frequency plot ___________________________________________ 78

Figure 4.11: Simulation point position _____________________________________ 79

Figure 4.12: FFT of z signal @ variable speed with no-freeplay model __________________ 79

Figure 4.13: Zoom of FFT of z signal @ variable speed with no-freeplay model _____________ 80

Figure 4.14: Vertical deflection [gust=10m/s & V=172m/s] ________________________ 80

Figure 4.15: Nose up twist [gust=10m/s & V=172m/s] ___________________________ 80

Figure 4.16: 𝛽 Response [gust=10m/s & V=172m/s] ____________________________ 81

Figure 4.17: 𝛽 Response [gust=10m/s & various speeds] __________________________ 82

Figure 4.18: Vertical deflection [gust=10m/s & V=190.5m/s] with the zoom ______________ 83

Figure 4.19: 𝛽 Response [gust=10m/s & V=190.5m/s] ___________________________ 83

Figure 4.20: 𝛽 Response [gust=10m/s & Over Flutter Speed] _______________________ 83

Figure 4.21: : 𝛽 Response – Variable Freeplay [gust=10m/s & V=160 m/s] _______________ 84

Figure 4.22: Vertical deflection – Variable freeplay [gust=10m/s & V=160m/s] _____________ 84

Figure 4.23: Vertical deflection – Variable freeplay [gust=10m/s & V=172m/s] _____________ 85

Figure 4.24: 𝛽 Response – Variable Freeplay [gust=10m/s & V=172 m/s] ________________ 85

Figure 4.25: Vertical deflection – 0.3° Freeplay [gust= 10m/s & V=184.5 m/s] _____________ 86

Figure 4.26: 𝛽 Response – 0.3° Freeplay [gust= 10m/s & V=184.5 m/s] _________________ 86

Figure 4.27: Vertical deflection – 0.6° Freeplay [gust= 10m/s & V=180 m/s] ______________ 86

Figure 4.28: 𝛽 Response – 0.6° Freeplay [gust= 10m/s & V=180 m/s] __________________ 87

Figure 4.29: FFT of z and z @192 m/s – Freeplay 0° ____________________________ 87

Figure 4.30: FFT of z and z @184.5 m/s – Freeplay 0.3° __________________________ 87

Figure 4.31:FFT of z and z @179m/s – Freeplay 0.6° ____________________________ 88

Figure 4.32: FFT of z and z @172m/s – Freeplay 1° ____________________________ 88

Figure 4.33: FFT of z signal with variable freeplay _____________________________ 89

Figure 4.34: Zoom of FFT of z signal with variable freeplay ________________________ 89

Figure 4.35: Vertical deflection –Literature model vs EMA model [gust=10m/s & V=181 m/s] ____ 89

Figure 4.36: 𝛽 Response – Literature model vs EMA model [gust= 10m/s & V=181 m/s] _______ 90

Figure 4.37: FFT of z signal with variable freeplay and speeds _______________________ 90

Figure 4.38: ASE model with Servohydraulic actuation system ______________________ 91

Figure 4.39: 𝛽 Response – Variable Actuator [gust=10m/s & V=183 m/s] ________________ 91

Figure 4.40: Vertical deflection – Variable actuator [gust= 10m/s & V=183 m/s] ____________ 92

Figure 4.41: 𝛽 Response – Variable Actuator and speed [gust=10m/s] __________________ 92

Figure 4.42: Vertical deflection – Variable actuator and speed [gust= 10m/s] ______________ 92

Figure 4.43: 𝛽 Response – Variable Actuator [gust=10m/s] ________________________ 93

Figure 4.45: Simulink approach of ASE system with actuator in failure __________________ 97

Figure 4.46: EMA Simulink block with failure and damping device ____________________ 97

Figure 4.47: 𝐷 ∗ vs flutter velocity ______________________________________ 100

Figure 4.48: Zoom of Figure 4.47 ________________________________________ 100

Figure 4.49: EMA failure - 𝛽 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑤𝑖𝑡ℎ 𝐷 ∗= 1𝑁𝑚/𝑟𝑎𝑑/𝑠 @ 𝑣 = 129 𝑚/𝑠 ___________ 101

Figure 4.50: EMA failure - 𝛽 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑤𝑖𝑡ℎ 𝐷 ∗= 140𝑁𝑚/𝑟𝑎𝑑/𝑠 𝑎𝑛𝑑 𝐷 ∗= 180𝑁𝑚/𝑟𝑎𝑑/𝑠 @ 𝑣 = 184 𝑚/𝑠 _____________________________________________________ 101 Figure 4.51: FFT of z @184m/s with variable 𝜁 _______________________________ 102

Figure 4.52: EMA failure - 𝛽 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑤𝑖𝑡ℎ 𝐷 ∗= 180𝑁𝑚/𝑟𝑎𝑑/𝑠 @ 𝑣 = 170 𝑚/𝑠 _________ 102

Figure 4.53: EMA failure - 𝛽 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑤𝑖𝑡ℎ 𝐷 ∗= 60𝑁𝑚/𝑟𝑎𝑑/𝑠, 𝐷 ∗= 120𝑁𝑚/𝑟𝑎𝑑/𝑠, 𝐷 ∗=

180𝑁𝑚/𝑟𝑎𝑑/𝑠 @ 𝑣 = 170 𝑚/𝑠 ________________________________________ 102 Figure 4.54: EMA failure - 𝛽 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑤𝑖𝑡ℎ 𝐷 ∗= 180𝑁𝑚/𝑟𝑎𝑑/𝑠, 𝐷 ∗= 1000𝑁𝑚/𝑟𝑎𝑑/𝑠 𝑎𝑛𝑑 𝐷 ∗= 2000 𝑁𝑚/𝑟𝑎𝑑/𝑠 @𝑣 = 190.5 𝑚/𝑠 ______________________________________ 103 Figura 4.55: Hydraulic scheme of a Servohydraulic actuator ________________________ 104

Figure 4.56: Scheme of a Fail-Safe EMA Actuator ______________________________ 104

Figure A.0.1: Geometry of the typical section ________________________________ 109

Figure A.0.2: Adding of the control surface __________________________________ 111

Figure B.0.3: Binary aeroelastic model showing bending and torsion modes _______________ 113

Figure B.0.4: Frequency and damping trends for the baseline system with zero aerodynamic and

structural damping _______________________________________________ 119

Figure B.0.5: Frequency and damping trends for the baseline system with quasi steady aerodynamic damping included (%& = 0) __________________________________________ 119

Figure B.0.6: Frequency and damping trends for the baseline system with unsteady aerodynamic

### LIST OF TABLES

Table 1.1: Features of EHA and EMA systems _______________________________ 16

Table 2.1: Aerodynamic parameters ______________________________________ 29

Table 2.2: Baseline parameters for the model _________________________________ 30

Table 3.1: Loops design bandwidths ______________________________________ 39

Table 3.2: Actuator parameters ________________________________________ 41

Table 3.3: Zeros/Poles values _________________________________________ 42

Table 3.4: First mode TF parameters _____________________________________ 52

Table 3.5: Servo hydraulic TF parameters __________________________________ 62

Table 4.1: Variation of flutter speed ______________________________________ 85

Table 4.2: Common EMA Fault Modes [31] _________________________________ 95

### LIST OF ACRONYMS

**Acronym ** **Definition **
AEA All-Electric aircraft

AFW Active Flexible Wing

ASE Aeroservoelastic / Aeroservoelasticy CFD Computational fluid dynamics DAST Drones for Aeroelastic Testing

DDV Direct drive valve

DOF Degree of Freedom

ECS Environmental control system EHA Electro hydrostatic actuator EMA Electromechanical actuator

FBW Fly by wire

FCS Flight control system

FFT Fast Fourier Transform

LAMS Load Alleviation and Mode Stabilization

LTI Linear time invariant

MEA More Electric aircraft

PI Proportional integral

PMSM Permanent Magnet Synchronous Motor

SA Servohydraulic actuator

SHA Servohydraulic actuator

TF Transfer function

### LIST OF SYMBOLS

**Symbol ** **Definition **

α Angle

- _{External force }

𝑘_{𝛿0.3°}** ** _{Dynamic Stiffness with 0.3° of freeplay }
𝑋_{𝑎} Velocity of the actuator piston
𝜃_{𝑚}_{𝑖} Reference speed signal

𝜃_{𝑚} Speed

./ Actuator pushing area

01 Aerodynamic parameter
02 Aerodynamic parameter
31 Aerodynamic parameter
32 Aerodynamic parameter
41 Aerodynamic parameter
42 Aerodynamic parameter
𝑫∗ _{Damping factor }
𝑫∗ _{Damping factor }
𝑓_{𝑒𝑥𝑡} External load
5678 External load

591 Non conservative forces

:_{;} _{Open loop TF }
<= Current
>/ System inertia
>= Motor inertia
>8?8 Equivalent inertia
𝐾𝑎

Actuator stiffness transfer functions

A= Motor torque constant

𝐾_{𝑜𝑖𝑙} Stiffness of the oil entrapped in the actuator’s chambers
BCD Distributed torsional spring of stiffness

𝐿𝑎 Position response transfer functions

EF, %F… Non dimensional oscillatory aerodynamic derivatives

𝑀_{𝛽} Control aerodynamic damping derivative
𝑀_{𝜃} Pitch damping term

𝑀𝑎 Piston mass

I1 Mass per unit area for the control surface

I2 Mass per unit area for the wing

𝑃_{𝑎𝑒} Fluid pressure in the extraction chamber
𝑃_{𝑎𝑟} Pressure in the retraction chamber

J/ Flowrate entering the extraction chamber

K; i-th generalized coordinates

J_{;} _{Generalized forces }
𝑆_{𝑎} Actuator stroke

$; Closed loop TF

LM Gust term

N_{9O} _{Non conservative work }
𝑥𝑎 Displacement

𝑋_{𝑎} Displacement of the actuator piston
𝑥_{𝑎𝑖} Command Displacement

PQ Elastic axis

RQ Downwards displacement at the elastic axis

𝛽_{𝑎𝑡𝑡} Actuator Output
𝛽_{𝑖} Reference position signal
ST; Uniform strain components

𝜻𝒂 Servohydraulic TF parameter: damping

𝜁_{𝑚} Damping poles First mode vibration
𝜻_{𝒗} Servohydraulic TF parameter: damping
𝜁_{𝑧𝑚} Damping zeros First mode vibration

𝜃_{𝑚} Rotation

UT; Components of stress tensor

VM Reduction ratio

𝝎_{𝒂} Servohydraulic TF parameter: frequency
𝜔_{𝑚} Frequency poles First mode vibration
𝝎_{𝒗} Servohydraulic TF parameter: frequency

𝜔𝑧𝑚 Frequency zeros First mode vibration

3 _{Semi-chord }

4 _{Chord }

𝑫 Non-dimensional damping device term

W _{Error }
XY _{Bending rigidity }
Z _{Mass forces }
:> _{Torsional rigidity }
[ _{Hinge moment }
< _{Current }
𝑰𝒎 Immaginary
E _{Lift }
I _{Mass }
% _{Moment }

𝑀 First mechanical vibration mode

\ _{Pressure }

] _{Displacement }

𝑹𝒆 Real

^ _{Semi-span }

_ _{Time }

$ _{Virtual kinetic energy }

` _{Elastic potential (or strain) energy }
𝑼 Velocity simulation

N _{Virtual work }

a _{Heaviside function }
R _{Vertical deflection }
𝛽 Control rotation
" _{Fluid bulk modulus }
𝜹 Freeplay value
& _{Nose up twist }

b _{Density }

**INTRODUCTION **

The present thesis lies in what it is a major trend in aircraft systems design
*philosophy, the so-called “More-Electric” and “All-Electric Aircraft”. This approach *
implies the extensive use of electrically-powered systems and the gradual elimination of
hydraulic or pneumatic power sources. In this context, there is the necessity for
developing, analysing and testing the applicability of the electromechanical technologies,
especially for safety-critical applications.

The basic objective of this thesis is to analyse flutter phenomena in wings with control surfaces moved by Electromechanical Actuator (EMA) actuator. We consider the presence of nonlinearities as the freeplay on actuator hinge. Later we develop a model with a failure of the actuator. We also consider the comparison with more conventional technology solutions, servohydraulic actuator, pointing out advantages and drawbacks in terms of performances and reliability.

In Chapter 1 we present a general overview of the fields of aeroservoelasticity, introducing the problems and effects of an actuator on an aeroelastic model. First of all we present the aeroservoelasticity, then we describe the actuator technology used in this thesis.

In Chapters 2,3,4, the approach to modelling the aeroservoelastic (ASE) system is discussed. The mathematical models used for the structural, the aerodynamic and the actuation system simulation are described. In Chapter 2, the model is developed based on patterns of literature for reference. In Chapter 3, we analyse the isolated EMA model including a comparison with the Servohydraulic actuator (SHA) model. In Chapter 4, we provide the integration between Chapter 2 and 3 models to form a full LTI system that lead to the complete analysis of flutter behaviour. In the paragraph 4.7, we consider the a major failure of the actuator and we present a possible solution (damping device) to prevent catastrophic events.

General overview

The final comments on thesis are made within the chapter Conclusions. Also, the alternatives for future work are discussed.

## Chapter 1

**1. EFFECTS OF FLIGHT ACTUATOR **

**DYNAMICS ON WING FLUTTER **

**1.1. General overview **

Aeroservoelasticity (ASE) is the interface of unsteady aerodynamics, structural dynamics, and control systems, and is an important interdisciplinary topic in aerospace engineering. It is a study of dynamic interactions among air loads, structural deformations, and automatic flight control systems commonly experienced by the modern aircraft. The relevance of ASE to modern airplane design has increased considerably with the advent of flexible, lightweight structures, increased airspeeds, and closed-loop automatic flight control. Since aeroservoelasticity lies at the interface of aerodynamics, structures, and control, its impact on aircraft design and operation requires a thorough understanding of these core areas as far as they contribute to building an accurate mathematical model.

The interaction between control dynamics and aeroelastic properties of an aircraft naturally inherits transient dynamics, therefore aeroservoelasticity is a topic of dynamic aeroelasticity. Studies on aeroservoelasticity mainly focuses on modelling, analysis and control of aeroelastic responses.

General overview

*Figure 1.1: Block diagram of a typical aeroservoelastic system [1] *

The science of (ASE) extends the aeroelastic interactions between aerodynamic forces and a flexible structure, to include a control system. The classic Collar aeroelastic triangle can be extended to form the aeroservoelastic pyramid shown in Figure 1.2, where there are now forces resulting from the control system as well as the aerodynamic, elastic and inertial forces.

*Figure 1.2: Aeroservoelastic pyramid [2] *

ASE effects [3] [4] [5] are becoming of increasing importance in modern aircraft design as it is usual nowadays to employ some form of Flight Control System (FCS) to improve the handling and stability, flight performance and ride quality throughout the flight envelope, and also to reduce loads and improve service life. For commercial aircraft, the FCS might include a Gust and/or Maneuver Load Alleviation System in addition to a control system that meets the basic handling requirements. All control implementations

involve the use of sensors, usually accelerometers and rate gyros placed at the aircraft centre of mass and air data sensors (e.g. angle of incidence, air speed). It is feasible to develop flutter suppression systems that enable aircraft to fly beyond the flutter speed; however, such an approach has a very high risk and would not be contemplated for a commercial aircraft.

ASE effects, sometimes referred to as “structural coupling”, can potentially cause a major structural failure due to a flutter mechanism involving coupling of the aeroelastic and control systems. However, there is also the possibility of causing fatigue damage and reducing control surface actuator performance.

The problems of aeroelasticity tend to be summarized in four major categories:
• *static instability, under the action of constant forces: divergence. *

• *dynamic instability, under the action of non-stationary forces: flutter. *
• *responses to forcing the static system: static in nature. *

• *responses to forcing the dynamic system: purely non-stationary. *

The aeroelasticity is a discipline that combines within it the interaction of aerodynamic forces, elastic forces, due to structural deformation, and aerodynamic forces. The term was coined for the first time in 1930 by Alfred Pugsley and Harold Cox, two engineers of the British Royal Aircraft Establishment. To clarify how these three forces of fields can interact between them, it is reported in Figure 1.3 the aeroelastic triangle:

General overview

A: Aerodynamic forces E: Elastic forces

I: Inertia forces

Static Aeroelastic Phenomena L: Load distribution

D: Divergence

R: Control system reversal Related Fields

V: Mechanical vibrations DS: Dynamic stability

Dynamic Aeroelastic Phenomena F: Flutter

B: Buffeting

Z: Dynamic response
*Figure 1.3: The aeroelastic triangle of forces [6]*

We can distinguish numerous phenomena depending on the forces that are involved with each other. For example:

• (L) load distribution: the study of the influence of pressure on the structural deformation of the structures;

• (D) divergence: the study of the static instability of the lifting surfaces;

• (R) inversion of the control surfaces: the study of the modification of the response of the aircraft's control surfaces.

• (F) flutter: the study of the instability dynamic of the lifting surfaces; • (B) buffeting: study of transient vibration due to aerodynamic pulses;

• (Z) dynamic responses: study of the dynamics response due to external forcings as the gusts.

Aeroservoelastic studies were performed since 1960’s. Aircraft wings are mainly composed of a fixed lifting surface and a trailing control surface (flap). Flutter suppression of the aircraft wing can be achieved by controlling the motion of the flap. The same flap is moved due to an actuator.

While aeroelastic interactions have been studied for nearly a century, the impact of an active control system on dynamic aeroelasticity is a relatively new topic, and has come into focus with the advent of modern fly-by-wire designs. In such aircraft, the controller bandwidth can encroach the upon aeroelastic modal spectrum, thereby leading to resonance-like behavior in certain flight conditions. The most common example of such an interaction between the control and aeroelastic systems is the closed-loop flutter—a catastrophic dynamic coupling between the elastic motion, the unsteady aerodynamic loading, and a controller-actuated surface. Many airplane accidents (such as Taiwan IDF fighter and Lockheed YF-22 prototypes) have been blamed on unforeseen and unstable ASE couplings. In order to understand how such a phenomenon can remain unforeseen in the modern technical era, let us consider a well-designed car with the best engine, chassis, and electronics, and thoroughly tested for the most adverse road conditions that can be expected. However, when put into production, the same car could experience poor performance and even engine stalling due to a minor feature such as cable routing. In such a case, engine vibration at a certain speed can interact with the natural frequency of one of the spark plug cables, thereby leading to its coming loose and causing an even greater engine vibration. The engine controller would detect the poor combustion as a lean or cold mixture condition, and try to correct it by increasing the fuel volume injection. The result would be an even rougher idle with black smoke, fouled spark plugs and injectors, and possibly an engine failure. Troubleshooting such a condition would be a nightmare, and the fix is either an expensive redesign of the engine, a reprogramming of the fuel controller, or an identification and change of the culprit natural frequency by merely redesigning the cable routings and clamps. If a mathematical model is constructed of such a dynamic interaction between electromechanical connectors, fuel control system, and engine dynamics, such a model is likely to be a formidable interdisciplinary exercise.

ASE is a fledgling discipline when compared to the other traditional aerospace areas of aerodynamics, structures, propulsion, and flight mechanics. Its formal beginning can be traced to the early 1970s, when ways of addressing the problem of flutter were being

General overview

investigated in earnest due to the several new aircraft designs evolving in that era. Highly maneuverable fighters such as the Lockheed F-16 and the McDonnell Douglas F/A-18, as well as efficient passenger transports such as the Boeing-767 and the Airbus A-320 that were being developed, had inbuilt automatic flight control systems, which could be programmed relatively easily to achieve secondary tasks, such as active flutter suppression and maneuver/gust load alleviation. Prior to that era, a passive redesign of the structural components was the only way to avoid flutter, whose analysis often required thousands of hours of painstaking and dangerous flight flutter testing, and wind-tunnel tests of aeroelastically scaled models, thereby increasing the already high costs of prototype development. Consequently, flutter analysis and prevention was a stumbling block in developing novel aircraft configurations.

In order to overcome the inadequacy of passive techniques, and to fly at a velocity greater than the open-loop flutter velocity for greater speed and efficiency, the concept of active flutter suppression was developed in the 1970s [7] [8] [1].

Active flutter suppression requires accurate knowledge of the aeroelastic modes that cause flutter, which are then actively changed in such a way that closed-loop flutter occurs at a higher flight velocity. Although the classical flutter of a high aspect-ratio wing—such as that of a Boeing 747 or an Airbus A-380—is caused by an interaction between the primary bending and torsion aeroelastic modes, the flutter mechanism of a low aspect-ratio wing, such as that of an F/A-18 (or F- 22) fighter airplane is rather more complicated, comprising a coupling of several higher aeroelastic modes. In order to actively suppress flutter, it is necessary that an accurate aeroelastic model based on modeling of the unsteady aerodynamic forces as a transfer matrix be derived. After the transfer matrix is derived, a linear, time-invariant, state-space model for the aeroelastic system, including the control surface actuators, can be obtained. The multivariable feedback controller for active flutter suppression can then be designed by standard closed-loop techniques, such as eigenstructure assignment and linear optimal control [1].

The first practical demonstration of active flutter suppression was carried out by the US Air Force in 1973 in their Load Alleviation and Mode Stabilization (LAMS) program, which resulted in a Boeing B-52 bomber flying 10 knots faster than its open-loop flutter velocity. This was accompanied by flight flutter testing of aeroelastic drones under NASA-Langley’s Drones for Aeroelastic Testing (DAST) program. These pioneering developments in active flutter suppression received an impetus at NASA-Langley and Ames laboratories [1] with novel control laws developed at Ames being tested and further developed in Langley’s transonic dynamics wind tunnel. These developments in the 1970s were greatly enabled by the optimal control theory advancements [1] of that era. ASE design and analysis efforts continued in the 1980s and 1990s [1], which were given a further boost by the newly developed robust multivariable control theory [1]. Boosted by their early achievements in active flutter suppression, the US Air Force and Rockwell, initiated the ambitious Active Flexible Wing (AFW) program, wherein the objective was to utilize favorable aeroservoelastic interactions to produce performance, stability, and control improvements on a highly flexible and overly instrumented wind-tunnel wing model, employing multiple control surfaces and gain scheduled control laws.

The main challenge in ASE mathematical analysis and design is in deriving a suitable unsteady aerodynamic model of aircraft wings and tails (or canards). The aeroelastic plant for flutter suppression of a thin wing-like surface is derived at subsonic and supersonic speeds using small-disturbance, potential aerodynamic models [1] with a harmonic (frequency response) theory. Such a model is linear, and can be directly employed in developing an aerodynamic transfer matrix, and finally a linear, time-invariant state-space model through analytic continuation in the Laplace domain. However, there are important flow regimes where such a linearized model is inapplicable. The ASE applications which involve unsteady separated flows and transonic shock-induced flows, are inherently nonlinear in nature and require advanced computational fluid dynamics (CFD) modeling techniques [1]. An example of nonlinear aeroelasticity is

General overview

the post-stall buffet arising due to a sudden and large increase in the angle of attack, either by an abrupt maneuver, or a vertical gust. The ASE plant for such a case is further complicated by the separated wake and/or leading-edge vortex from the wing interacting with the tail, resulting in an irregular and sometimes catastrophic deformation of the tail— either on its own or driven by rapid and large deflections of the elevator. Such a wing-tail-elevator coupling of a post-stall buffet, or a shock-vortex interaction requires a fully viscous flow modeling that is only possible by a Navier–Stokes method.

Despite the early successes in demonstrating active flutter suppression/load alleviation, ASE has remained largely an experimental area and has still not reached operational status on any aircraft. This remarkable failure is mainly due to the difficulty of designing a multivariable control system, which is sufficiently robust to the parametric uncertainties in the underlying unsteady aerodynamic model. Clearly, the aircraft designers and operators are reluctant to take risks until (what might be considered) suitably reliable ASE modeling and analysis methods become prevalent.

Due to the inherent uncertainty of an unsteady aerodynamic model, a closed-loop controller for ASE application must be quite robust to modeling errors. Furthermore, such a controller must also adapt to changing flight conditions. Hence, an ASE control law must not only be robust, but also self-adaptive, which renders it mathematically nonlinear even for a linear aeroelastic plant of the subsonic and supersonic regimes. Furthermore, designing a control law based upon nonlinear aeroelastic iterative models can be a very cumbersome and computationally intensive process. Instead, adaptive control techniques can be used for extending the subsonic and supersonic linear feedback designs to predict and suppress the transonic flutter. Adaptive control has been an area of active research in the past few decades [1], and many useful design techniques have emerged that can be applied to ASE. However, these remain “application specific” (rather than general), if not completely ad hoc in many cases. Thus, ASE control-law derivation for a particular case is as challenging as the problem of aeroelastic modeling. For this reason, ASE has remained a formidable technological problem.

**1.2. EMA actuation for flight control system **

In the design of aircraft there is a strong push towards integration and energy optimization of the subsystems. From this need comes a design trend known as All-Electric Aircraft according to which all the on-board systems can be designed as electrical utilities.

*Figure 1.4: Types of power supply of aircraft systems [9] *

The All-Electric Aircraft concept has been established for many years describing an aircraft with all systems fully electrically powered and no hydrostatic or pneumatic systems remaining. The envisioned benefits are:

**- Weight benefit: the removal of the hydraulic and pneumatic systems and replacing **

them with electric ones influence weight reduction and the clutter of the system as a whole. A real advantage in weight is achieved by sophisticated electronics to realize the Power Management System, able to divide the electricity to the systems (like the Environmental control system ECS) that can withstand temporary reductions of energy. In contrast, the electrical generators would be enormous and heavy.

By analyzing the weight of onboard systems in their entirety we can see the overall weight reduction in switching from a conventional to an All-Electric aircraft, Figure 1.5.

EMA actuation for flight control system

*Figure 1.5: Estimated weight of onboard systems in the transition to All electric aircraft [9] *

Considering a civil transport aircraft of 300 passengers, in the transition from conventional to All-Electric where the control surfaces are moved by a total of 32 actuators, it is estimated that a penalty of about 214 kg due to the use of electromechanical actuators than hydraulic ones, there will be a weight gain of about 2077kg in total system, for a total savings of about 1863kg.

**- Cost reduction: another fundamental aspect is the reduction of the overall costs of **

the aircraft. Connected to the overall weight reduction is to reduce fuel consumption. Specifically, it is estimated that you can potentially reduce 25% peak power demand to the propeller of the on-board systems, resulting in a reduction of 5% of the fuel consumed.

**- Eco-compatible: The introduction of new technologies, aimed at improving **

performance and reducing costs, must still take into account the effects that can have on the environment and in this perspective is directed the design of aircraft with fully electric systems. In traditional systems, hydraulic fluid must meet and maintain certain standards and this will require frequent maintenance and replacement of the fluid it-self. Remove the plumbing, and then related fluids, eliminates the problem of disposing of the hydraulic fluid. The shift to a single source of energy (electricity), entails a simplification of realization and installation of the entire system (absence of rigid pipes) and therefore to a reduction of maintenance costs; you don’t need more skilled personnel for different types of systems and do not require periodic intensive maintenance as for hydraulic systems.

**- Energy Optimization: plumbing elimination means eliminate the need to feed the **

hydraulic actuators with streams of leakage, resulting in energy dissipation. In the electrical system, the generator delivers power only when there is a real need to move the control surfaces. In addition, eliminating pressurized conduits with flammable fluids at high temperature decreases the risk of fire and allow the introduction of new structural solutions using lower heat protection systems. Pneumatic system removal affect on air conditioning and cabin pressurization system. Currently the air in the cabin come from the engine compressor through the Environmental Control System (ECS). An ECS electrically would reduce quality issues of the air, pressure changes for passengers during thrust variation and would result in savings of approximately 1% of the fuel consumed.

**- Better diagnosability. **
**- Better actuator dynamics. **

Realizing this vision has proven very challenging especially in the area of flight control actuation where the trend towards the all-electric aircraft causes a strong need for novel optimized electrical actuators. Although for many years there has been substantial research effort in the area there still is no large commercial aircraft on the market or even in development that fully implements the concept. However, first steps ahead have been taken by the Airbus A380 (stand by electro-hydrostatic flight control actuation system, 2 instead of 3 hydraulic systems) and the Boeing B787 with electromechanically actuated airbrakes.

Main challenges in developing solutions for electromechanical flight control actuation are:

**- Weight optimized design of power electronics and the electromechanical drive train to be **

competitive with hitherto used and mature hydraulic actuation technology;

**- Electromechanical piezoelectric actuator and electronics designs offering low failure rates **

under prevailing adverse environmental conditions (temperature, pressure variations, humidity, contamination, radiation, vibration);

EMA actuation for flight control system

As there is little experience with the relevant failure mechanisms and failure probabilities of EMAs for failure modes such as actuator jamming, the first generation of electrically powered flight control actuators introduced on the A380 transport aircraft recently uses EHA (electro hydrostatic actuator) technology. The EHA concept comprises an electric motor and a pump acting as a local source of hydraulic power and a conventional hydraulic cylinder for linear actuation.

To get to all electric solutions the above-mentioned challenges have to be addressed e.g. by novel gear technologies to exclude jamming and/or freewheeling of linear drives, encapsulation technologies to protect actuators from moisture or by system topologies sufficiently tolerant against the occurrence of these failure modes in individual actuators.

**1.2.1. Advantages of EMA actuation **

*According to More-Electric Aircraft (MEA) concept, a large set of actuators has *
been studied, among which Hydrostatic Actuators (EHA) and
Electro-Mechanical Actuators (EMA). In the EHA solution, a distributed hydraulic system is
used whereas in the EMA solution the hydraulics is replaced by an electrical machine, a
gearbox, and/or a screw mechanism, as shown in Figure 1.6.

*Figure 1.6: Example of Power-by-wire flight actuators [9] *

The EMA replaces the electrical signalling and power actuation of the electro-hydraulic actuator with an electric motor and gearbox assembly applying the motive force to move the ram. The three main technology advancements that have improved the EMA to the point where it may be viable for flight control applications are: the use

of rare earth magnetic materials in 270 VDC motors; high power solid-state switching devices; and microprocessors for lightweight control of the actuator motor.

In new aircraft, such as Airbus A380 and Boeing B787, one or more hydraulic systems have been replaced by EHA networks, which are used as a backup for other hydraulic systems [10]. This is an electrical subsystem that provides locally hydraulic actuation capability. However, a drawback linked to the use of these actuators is their initial and maintenance cost, which is higher than for totally electromechanical actuators. EMA is an appealing alternative to EHA, since it allows the elimination of local hydraulic circuits, implying a significant maintenance cost reduction due to the absence of wearing parts such as seals. In the following table, we observe a trade-off summary of both systems.

*Table 1.1: Features of EHA and EMA systems *

*Electro-Hydrostatic Actuator *
**(EHA) **

*Electro-Mechanical Actuator *
**(EMA) **

ü The reliability of the EHA driven by pump has been well studied and it is sufficiently guaranteed.

ü The reliability seems to be competitive with conventional actuation, but additional research must be done. ü Traditional Active Standby and Active

configuration have been developed and validated.

ü Possible usage as a Standby Actuator on a flight critical surface, or on a spoiler-type surface-rotary or linear design are practical.

û Hydraulic maintenance is required. û Flutter concerns due to mechanical transmission free-play.

û EHA cost and weight baseline need improvement.

û Jam susceptibility.

A great deal of attention will be paid to the safety issues introduced by these new design concepts. In hydraulic systems, an actuator leak that could result in jeopardizing the flight safety is usually isolated. The same concept is applied to electrical systems, that is, a critical actuator failure can easily be isolated. On the other hand, hydraulic system is a continuous load on the engine if the hydraulic power is used for actuation or not, while in EMA actuators the electric load is demanded just when it is needed. So, compared to an EHA, the EMA has certain advantages. It is lighter, smaller, and less

EMA actuation for flight control system

complex than an equivalent EHA; since, there are no hydraulic fluid in the load path, the EMA tends to be stiffer than an equivalent EHA. The EMA tends to be more efficient because there are no winding losses or pump inefficiencies. Finally, since there is no leak potential with an EMA, it is better suited to long-term storage or space applications.

**1.2.2. EMA applications **

EMA technologies are already being used in aeronautics, but for safety reason they
*are limited to secondary flight controls or used in military aircraft, where they were *
proposed and validated during the 1990s. In its basic form, the EMA is susceptible to
single-point failure that can lead to a mechanical jam. Additional devices can be used to
mitigate against this failure mode, but in so doing complexity, cost, and weight are
increased. For these reasons, the basic EMA is not suited for primary flight control
application [11].

*Figure 1.7: EMA actuator [12] *

*To make a distinction, we can define basic commands like: *

**- Ailerons: they are placed at the trailing edge of the outer part of the two wing and **

they control the aircraft’s rotation around the longitudinal axis (roll axis).

**- Rudder: it is positioned on the vertical tail surface and it controls the aircraft’s rotation **

around the vertical axis (yaw axis).

**- Elevator: it is located on the horizontal tail surface and it controls the rotation of the **

aircraft around the pitch axis.
*While “minor” commands include: *

**- Spoilers: they increase the resistance of the wing forcing flow separation through a **

strong geometric variation of the profile.

**- Flap: they increase the curvature of the profile and increase the wing area producing **

an increase in lift coefficient.

**- Slats: they increase the curvature of the profile and used coupled with the flaps, they **

increase the curvature of the airfoil to work in ideal conditions of incidence.

The limited use of electromechanical actuation systems is fundamentally tied to the safety aspect. Being relatively new in the aerospace field, they have not been used for a sufficiently high time that they have stored reliable failure statistics. However, newer generation aircraft such as the Airbus A380, Boeing 787 use EMA in roles traditionally reserved for hydraulic systems.

In Figure 1.8 is illustrated the scenario of the EMA introduction in aircraft flight control systems; power source types are located in the vertical axis on the left, actuator type on the right. For example, on the Boeing 787 (Figure 1.9 and Figure 1.10) are located also in the landing gear, brakes and into the air conditioning system.

EMA actuation for flight control system

*Figure 1.9: Types of actuators used on the *
*Boeing 787 [14] *

*Figure 1.10: Actuation system on the Boeing 787 *
*[14] *

*Figure 1.11 Trends in aeronautical design [15] *

In this thesis we have been considering the application of an EMA for a primary control actuator (aileron). The use of EMAs in primary flight-control systems has been investigated for several years [16], but some central issues have still not been solved satisfactorily. For example, EMAs do not currently reach the necessary lifetime required for primary flight control actuators. Wear of the incorporated gears may further result in an unacceptable amount of backlash, which affects flight control quality and could lead to flutter.

A major concern regarding the flight-control architecture is the probability of EMAs jamming resulting from numerous mechanical contacts of the incorporated gear teeth and ball bearings. Simply introducing redundancy by actuator duplication even increases the probability of control surface jamming, because each actuator has the ability to jam the entire control surface. Conservative solutions might be shear pins or couplers, but they are either irreversible and reduce the availability of the aircraft in combination with high maintenance costs, or increase system weight, making them an undesired solution for jamming concerns.

Basic concepts for dynamic aeroelasticity

## Chapter 2

**2. REFERENCE MODEL FOR CONTROL **

**SURFACE FLUTTER **

**2.1. Basic concepts for dynamic aeroelasticity **

In this section, we gradually enter in dynamic range. The first part describes the established theory of the dynamics of deformable bodies and then describe the various models used in the dynamical theory of Aeroelasticity.

**2.1.1. Hamilton’s principle **

Newton's second law or law of proportionality is:

𝐹 = 𝑚𝑑_{𝑑𝑡}2_{2}𝑟 ( 2.1 )

where F is the external force, r the displacement, and m the mass.

If the configuration changes from 𝑟 to 𝑟 + 𝛿𝑟 and if 𝑡1 → 𝑡2 then we can assume that 𝑟

is null. By applying now to the ( 2.1 ), the product of the first and of the second member with 𝛿𝑟, and integrating between the moments of time 𝑡1 and 𝑡2, bringing all at first

𝑚𝑑2𝑟

𝑑𝑡2𝛿𝑟 − 𝐹𝛿𝑟 𝑑𝑡
𝑡_{2}

𝑡_{1}

= 0 ( 2.2 )

where the second term of ( 2.2 ) can be identified as virtual work. It is assumed that the force remains fixed or equivalently that the virtual displacement is instantaneous, meaning that 𝛿𝑡 = 0. The first term inside the integral in ( 2.2 ) has also the size of a work or an energy and can be revised to as follows:

𝑚 𝑑2𝑟
𝑑𝑡2𝛿𝑟𝑑𝑡
𝑡_{2}
𝑡_{1}
= −𝑚
2 𝛿
𝑑𝑟
𝑑𝑡
𝑑𝑟
𝑑𝑡 𝑑𝑡
𝑡_{2}
𝑡_{1} ( 2.3 )

so the ( 2.2 ) can be rewritten as follows:
1
2𝑚𝛿
𝑑𝑟
𝑑𝑡
𝑑𝑟
𝑑𝑡 + 𝐹𝛿𝑟 𝑑𝑡 = 0
𝑡2
𝑡1
( 2.4 )
𝛿 𝑇 + 𝑊 𝑑𝑡 = 0
𝑡_{2}
𝑡_{1}
( 2.5 )
where the quantity 𝛿𝑇 and 𝛿𝑊 are virtual kinetic energy and virtual work respectively.
The discussion of Newton and Hamilton lead to the same results even though the second
is based on a time interval of interest to us, while the first is still valid.

Obviously the newly developed theory for a single particle is also valid for a distribution of masses in space. Extending the discussion to continues bodies theory, the kinematic quantities and the work quantities must be calculated in the following manner:

𝛿𝑇 = 𝜌 2𝛿 𝑉 𝑑𝑟 𝑑𝑡 𝑑𝑟 𝑑𝑡 𝑑𝑉 ( 2.6 ) 𝛿𝑊 = 𝑓𝛿𝑟𝑑𝑉 𝑉 + 𝑝𝛿𝑟𝑑𝐴 𝑆 ( 2.7 )

where in ( 2.6 ) 𝜌 indicates the density, 𝑝 pressure forces and with 𝑓 mass forces. In dealing with a continuous body, which deforms in the elastic way, the handling of reference is the one developed by Hooke. We show that the work done by the internal elastic forces is equal, but with sign changed, to the virtual elastic potential energy. All this can be summed up in the following equation:

Basic concepts for dynamic aeroelasticity

𝑈 =1_{2} [𝜎𝑥𝑥𝜖𝑥𝑥 + 𝜎𝑦𝑦𝜖𝑦𝑦+ 𝜎𝑧𝑧𝜖𝑧𝑧… ]𝑑𝑉

𝑉 ( 2.8 )

where 𝜎_{𝑗𝑖} are the components of the stress tensor 𝑗𝑖 and 𝜖_{𝑗𝑖} are the uniform strain
components.

With regard to non-conservative forces that do not allow the discussion with potential, Hamilton's principle can be reformulated in the following way:

𝛿𝑇 − 𝛿𝑈 + 𝐹_{𝑁𝐶}𝛿𝑟 𝑑𝑡 = 0 ( 2.9 )

where the term 𝐹𝑁𝐶 only contains non-conservative forces. In the aeroelastic

analysis, an example of these forces components are represented by the aerodynamic loads, which in practical terms are the contributions of non-stationary pressure acting on the structural element taken into account.

**2.1.2. Lagrange’s Equations **

The treatment developed by Hamilton uses an infinite number of degrees of freedom, because the speeds, forces, mass deployments and all parameters that come into the equations are described by continuous functions. Engineering purposes, the system is shown through the use of a particular set of coordinates and using the equations of motion developed by Lagrange. The equations can be obtained by tracing back the way we did to describe those of Hamilton.

The coordinates are called generalized coordinates or Lagrange’s coordinates: these are an arbitrary and independent group of degrees of freedom, which can sufficiently describe the motion of a dynamic system.

For any physical system, there are endless combinations of generalized coordinates. Given a source and then a shift, we can write the radius vector function through the use of generalized coordinates:

where in ( 2.10 ), 𝑞𝑖 are the i-th generalized coordinates. The kinetic and potential

energy for as are defined, depend on displacement, velocity, and time, so we can write:

𝑇 = 𝑇 (𝑞_{𝑖}, 𝑞_{𝑖}, 𝑡) ( 2.11 )

𝑈 = 𝑈(𝑞𝑖, 𝑞𝑖, 𝑡) ( 2.12 )

and using the principle of Hamilton we can connect all writing: 𝛿𝑇 − 𝛿𝑈 + 𝛿𝑊𝑁𝐶 𝑑𝑡

𝑡_{2}
𝑡_{1}

= 0 ( 2.13 )

where using ( 2.11 ) and ( 2.12 ) and inserting them in ( 2.13 ) gives: 𝜕(𝑇 − 𝑈) 𝜕𝑞𝑖 𝛿𝑞𝑖+ 𝜕(𝑇 − 𝑈) 𝜕𝑞𝑖 𝛿𝑞𝑖+ 𝑄𝑖𝛿𝑞𝑖 𝑑𝑡 𝑡2 𝑡1 𝑖 = 0 ( 2.14 )

in the ( 2.14 ) generalized forces 𝑄𝑖 form the non-conservative work:

𝛿𝑊𝑁𝐶 = 𝑄𝑖𝛿𝑞𝑖

𝑖 ( 2.15 )

Integrating by parts the first term of ( 2.14 ) and noting that 𝑞𝑖 is null at the start

and at the end, we obtain:
𝜕 𝑇 − 𝑈
𝜕𝑞_{𝑖} 𝛿𝑞𝑖
𝑖 _{𝑡}_{1}
𝑡_{2}
+ − 𝑑
𝑑𝑡
𝜕(𝑇 − 𝑈)
𝜕𝑞𝑖 𝛿𝑞𝑖+
𝜕(𝑇 − 𝑈)
𝜕𝑞𝑖 𝛿𝑞𝑖
𝑡_{2}
𝑡_{1}
+ 𝑄𝑖𝛿𝑞𝑖 𝑑𝑡 = 0
( 2.16 )

where the first term of ( 2.16 ) vanishes for all the above mentioned explicitly. So now we can collect all contributions in the integral. Analyzing the whole thing, and using any arbitrary displacement 𝛿𝑞𝑖, to satisfy the equality, the sufficient condition that checks

( 2.16 ), is that the integrand is nil. Writing this gives:
−_{𝑑𝑡}𝑑 𝜕 𝑇 − 𝑈_{𝜕𝑞}

𝑖 +

𝜕 𝑇 − 𝑈

𝜕𝑞_{𝑖} + 𝑄𝑖 = 0 ( 2.17 )

Basic concepts for dynamic aeroelasticity

**2.1.3. Simplified Unsteady Aerodynamics **

Flutter is arguably the most important of all the aeroelastic phenomena and is the most difficult to predict. It is an unstable self-excited vibration in which the structure extracts energy from the air stream and often results in catastrophic structural failure. Classical binary flutter occurs when the aerodynamic forces associated with motion in two modes of vibration cause the modes to couple in an un-favourable manner, although there are cases where more than two modes have combined to cause flutter and in industry the mathematical models employ many modes.

At some critical speed, known as the flutter speed, the structure sustains oscillations following some initial disturbance. Below this speed, the oscillations are damped, whereas above it one of the modes becomes negatively damped and unstable (often violent) oscillations occur. Flutter can take various forms involving different pairs of interacting modes, e.g. wing bending/torsion, wing torsion/control surface, wing/engine, T-tail etc. In this thesis, a simple flutter model is developed, making use of strip theory with simplified unsteady aerodynamic terms.

*Figure 2.1: Two-dimensional aerofoil [2] *

A simplified unsteady aerodynamic model will be introduced. Consider the
*two-dimensional aerofoil shown in Figure 2.1 with the elastic axis positioned a distance ec aft *
*of the aerodynamic center and ab aft of the mid chord, so *

𝑒𝑐 =𝑐_{4}+ 𝑎𝑏 =_{4}𝑐+𝑎𝑐_{2} ( 2.18 )

The unsteady lift and moment per unit span for an aerofoil may be expressed, for a particular reduced frequency, as:

𝐿 = 𝜌𝑉2 _{𝐿}
𝑧𝑧 + 𝐿𝑧𝑏𝑧 _{𝑉} + 𝐿𝜃𝑏𝜃 + 𝐿𝜃𝑏
2_{𝜃}
𝑉 ( 2.19 )
𝑀 = 𝜌𝑉2 _{𝑀}
𝑧𝑏𝑧 + 𝑀𝑧𝑏
2_{𝑧 }
𝑉 + 𝑀𝜃𝑏2𝜃 + 𝑀𝜃𝑏
3_{𝜃}
𝑉 ( 2.20 )

where 𝐿𝑧, 𝑀𝑧* etc., are the non-dimensional oscillatory aerodynamic derivatives. *

Taking the quasi-steady assumption (𝑘 → 0, 𝐹 → 1, 𝐺 → 0) and the quasi-steady
derivatives [15], then the lift and pitching moment per unit span about the elastic axis
become:
𝐿 = 𝜌𝑉2 _{𝐿}
𝑧𝑏𝑧 _{𝑉} + 𝐿𝜃𝑏𝜃 =1_{2}𝜌𝑉2𝑐𝑎1 𝜃 +_{𝑉}𝑧 ( 2.21 )
𝑀 = 𝜌𝑉2 _{𝑀}
𝑧𝑏
2_{𝑧 }
𝑉 + 𝑀𝜃𝑏2𝜃 =
1
2𝜌𝑉2𝑒𝑐2𝑎1 𝜃 +
𝑧
𝑉 ( 2.22 )

An extra term in each expression due to the effective incidence associated with the aerofoil moving downwards with constant heave velocity 𝑧̇, so causing an effective “up wash”. The quasi-steady assumption implies that the aerodynamic loads acting on an aerofoil undergoing variable heave and pitch motions are equal, at any moment in time, to the characteristics of the same aerofoil with constant position and rate values.

The major drawback in using quasi-steady aerodynamics is that no account is made for the time that it takes for changes in the wake associated with the aerofoil motion to develop (as defined by the Wagner function) and this can lead to serious aeroelastic modelling errors. Consequently, the 𝑀𝜃 unsteady aerodynamic derivative term in

Equation ( 2.20 ) will be retained in the flutter analysis as it has been shown [17] that this has an important effect on the unsteady aerodynamic behaviour. It adds a pitch damping term to the pitching moment (Equation ( 2.22 )) and so the model then becomes:

𝐿 =1_{2}𝜌𝑉2_{𝑐𝑎}

1 𝜃 +_{𝑉}𝑧 ( 2.23 )

𝑀 =1

2𝜌𝑉2𝑐2 𝑒𝑎1 𝜃 +_{𝑉}𝑧 + 𝑀𝜃_{4𝑉}𝜃𝑐 ( 2.24 )

where 𝑀_{𝜃} is negative and is assumed to have a constant value. This “simplified unsteady
aerodynamic” model will now be used to develop a binary aeroelastic model.

Control surface flutter

Note that the pitch damping term here differs numerically from that in [17] by a factor of four, which occurs because the unsteady aerodynamic derivatives are derived here in terms of semi-chord b instead of the chord c.

**2.2. Control surface flutter **

Historically, flutter involving the control surfaces has occurred more frequently than classical wing bending/torsion flutter. This has often resulted in the loss of control surfaces and/or part of the wing/tail structure but in many cases the aircraft has survived. Usually the flutter mechanism still occurs due to the interaction of two modes. To illustrate some of the characteristics of control surface flutter, consider the 3DoF aeroelastic system shown in Figure 2.2.

*Figure 2.2: Aeroelastic model including a full span control surface *

The binary bending/torsion aeroelastic model (Appendix B) has been altered so
that a full span control surface is attached to the wing by a distributed torsional spring
of stiffness 𝑘_{𝛽}_{𝑠𝑡𝑟𝑢𝑐𝑡} per unit span. As before, the non-dimensional pitch damping
derivative 𝑀_{𝜃 } is included in order to approximate the unsteady aerodynamic behaviour,
but now a control aerodynamic damping derivative 𝑀_{𝛽} is also included [18].

The vertical deflection (positive downwards) is expressed as:

where 𝛽 is the control rotation (trailing edge down and the same at every spanwise position) and [𝑋] is the Heaviside function defined by 𝑋 = 0 𝑖𝑓 𝑋 < 0 𝑎𝑛𝑑 𝑋 = 𝑋 𝑖𝑓 𝑋 ≥ 0.

The downwards displacement at the elastic axis and the nose up twist are given by:
𝑧𝑓 = 𝑦_{𝑠}

2

𝑞𝑏 ( 2.26 )

𝜃 = 𝑦

𝑠 𝑞𝑡 ( 2.27 )

The kinetic energy terms for the wing (w) and control (c) are given by:
𝑇𝑤 =𝑚_{2}𝑤 𝑦_{𝑠} 𝑞𝑏+ 𝑥 − 𝑥𝑓 𝑦_{𝑠} 𝑞𝑡
2
𝑠 𝑥_{ℎ}
0 𝑥_{ℎ}
𝑑𝑥𝑑𝑦 ( 2.28 )
and
𝑇_{𝑐} =𝑚𝑐
2
𝑦
𝑠
2
𝑞_{𝑏}+ 𝑥 − 𝑥_{𝑓} 𝑦
𝑠 𝑞𝑡+ 𝑥 − 𝑥ℎ 𝛽
2
𝑠 𝑐
0 𝑥ℎ
𝑑𝑥𝑑𝑦 _{( 2.29 ) }

where 𝑚𝑤, 𝑚𝑐 are the mass per unit area for the wing and control surface. The elastic

potential (or strain) energy terms are the same for bending and torsion of Appendix B but with an additional term due to the distributed control spring, so

𝑈 = 1_{2} 𝐸𝐼 2𝑞𝑏
𝑠2
2
𝑠
0
𝑑𝑦 + 1_{2} 𝐺𝐽 𝑞𝑡
𝑠
2
𝑠
0
𝑑𝑦 +1_{2}𝑘𝛽_{𝑠𝑡𝑟𝑢𝑐}𝛽2 ( 2.30 )

The lift, pitching moment about the elastic axis and the hinge moment about the hinge
*line are written for a strip of width dy as follows: *

𝑑𝐿 =1
2𝜌𝑉2𝑐𝑑𝑦 𝑎𝑤 𝑦
2_{ 𝑞}
𝑏
𝑠2_{𝑉} +
𝑦
𝑠𝑞𝑡 + 𝑎𝑐𝛽 ( 2.31 )
𝑑𝑀 = 1_{2}𝜌𝑉2_{𝑐}2_{𝑑𝑦 𝑏}
𝑤 𝑦
2_{ 𝑞}
𝑏
𝑠2_{𝑉} +
𝑦
𝑠𝑞𝑡 +𝑀𝜃_{4𝑠𝑉}𝑐 𝑦 𝑞𝑡 + 𝑏𝑐𝛽 ( 2.32 )
𝑑𝐻 =1
2𝜌𝑉2𝑐2𝑑𝑦 𝑐𝑤 𝑦
2_{ 𝑞}
𝑏
𝑠2_{𝑉} +
𝑦
𝑠𝑞𝑡 + 𝑀𝛽
𝑐 𝛽
4𝑉 + 𝑐𝑐𝛽
( 2.33 )
Here the lift, pitching moment and hinge moment coefficients may be estimated as in
“Scanlan et al.” [2] namely

Control surface flutter

*Table 2.1: Aerodynamic parameters *

**Parameter ** **Value **
𝑎𝑤 2𝜋
𝑎𝑐
𝑎𝑤
2𝜋 cos−1 1 − 2𝐸 + 2 𝐸(1 − 𝐸)
𝑏𝑤 𝑒𝑎𝑤
𝑏𝑐 −
𝑎𝑤
𝜋 1 − 𝐸 𝐸(1 − 𝐸)
𝑐𝑤 −𝑇_{2}12
𝑐𝑐 −𝑇12_{2𝜋}𝑇10
𝑑 2𝑥ℎ
𝑐 − 1
𝐸𝑐 𝑐 − 𝑥_{ℎ}

The incremental work done by the aerodynamic force and moments is: 𝛿𝑊 = − 𝑑𝐿𝛿𝑧𝑓+ 𝑠 0 𝑑𝑀𝛿𝜃 + 𝑠 0 𝑑𝐻𝛿𝛽 𝑠 0 ( 2.34 ) Using Lagrange’s energy equations then yields the 3DoF aeroelastic equations that involve interaction of the wing bending, torsion and control rotation, namely

𝐴_{𝑏𝑏} 𝐴_{𝑏𝑡} 𝐴_{𝑏𝛽}
𝐴_{𝑡𝑏} 𝐴_{𝑡𝑡} 𝐴_{𝑡𝛽}
𝐴_{𝛽𝑏} 𝐴_{𝛽𝑡} 𝐴_{𝛽𝛽}
𝑞𝑏
𝑞_{𝑡}
𝛽
+ 𝜌𝑉
𝑐𝑠
10𝑎𝑤 0 0
−𝑐_{8}2𝑠𝑏𝑤 −𝑐
3_{𝑠}
24 𝑀𝜃 0
−𝑐_{6}2𝑠𝑐𝑤 0 −𝑐
3_{𝑠}
8 𝑀𝛽
𝑞𝑏
𝑞_{𝑡}
𝛽
+ 𝜌𝑉2
0 𝑐𝑠
8 𝑎𝑤
𝑐𝑠
6 𝑎𝑐
0 −𝑐2_{6}𝑠𝑏𝑤 𝑐
2_{𝑠}
4 𝑏𝑐
0 𝑐_{4}2𝑠𝑐𝑤 −𝑐
2_{𝑠}
2 𝑐𝑐
+
4𝐸𝐼
𝑠3 0 0
0 𝐺𝐽_{𝑠} 0
0 0 𝑘𝛽𝑠
𝑞_{𝑏}
𝑞𝑡
𝛽 =
0
0
0
( 2.35 )

𝐴𝑏𝑏 = 𝑚𝑠𝑐5 ( 2.36 )
𝐴𝑡𝑡 = 𝑚𝑠𝑐3 𝑐
3
3 − 𝑥𝑓𝑐2+ 𝑥𝑓2𝑐 ( 2.37 )
𝐴_{𝑏𝛽 } = 𝑚𝑠
3 𝑐
2_{−𝑥}
ℎ
2
2 − 𝑥ℎ(𝑐 − 𝑥ℎ) ( 2.38 )
𝐴𝑏𝑡 = 𝑚𝑠4 𝑐
2
2 − 𝑥𝑓𝑐 ( 2.39 )
𝐴𝑡𝛽 = 𝑚𝑠2
𝑐3_{−𝑥}
ℎ
3
3 − (𝑥𝑓 + 𝑥ℎ)𝑐
2_{−𝑥}
ℎ
2
2 + 𝑥𝑓𝑥ℎ(𝑐 − 𝑥ℎ) ( 2.40 )
𝐴𝛽𝛽 = 𝑚𝑠 𝑐
3_{−𝑥}
ℎ
3
3 − 𝑥ℎ𝑐2+ 𝑥ℎ2𝑐 ( 2.41 )

Flutter characteristics of this system can be obtained using the same methodology for the bending/torsion flutter of Appendix B.

**2.2.1. Baseline system parameters **

The baseline system parameters considered are shown in Table 2.2, noting that the mass axis is at the semi-chord (𝑥𝑚 = 0.5𝑐) and the elastic axis is at 𝑥𝑓 = 0.48𝑐 from

the leading edge.

*Table 2.2: Baseline parameters for the model *

**Symbol ****Description ****Value ****Unit **

s Semi - span 7.5 [m]

c Chord 2 [m]

𝒙_{𝒇} Elastic axis 0.48c [m]

Mass axis 0.5c [m]

Mass per unit area 100 [kg/m2_{] }

𝑬𝑰 Bending rigidity 2 x107 _{[Nm}2_{] }

𝑮𝑱 Torsional rigidity 2 x106 _{[Nm}2_{] }

𝒂𝒘 Lift curve slope 2𝜋

𝑴𝜽 Pitch damping derivative -1.2

𝝆 Air density 1.225 [kg/m3_{] }

𝑀_{𝛽} Control aerodynamic

damping derivative