Master's Thesis

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MASTER'S THESIS

Analysis of a Flutter Suppression System for an Efficiently Designed UAV

Niclas Bramstång 2015

Master of Science in Engineering Technology Space Engineering

Luleå University of Technology Department of Computer Science, Electrical and Space Engineering Masters Thesis

Conducted at Aerospace Department at Collage of Engineering University of Illinois at Urbana-Champaign

Analysis of a Flutter Suppression System for an Eﬃciently Designed UAV

Examiner: Associate Professor Lars-G¨oranWesterberg

Author: Supervisor: Niclas B. Bramst˚ang Professor Emeritus Harry H. Hilton September 2, 2015 Master Thesis

Abstract With a market for Unmanned Aerial Aircrafts (UAV) exploding and in an age were composite materials are used to construct light and flexible structures it is of importance to have the knowledge of how the choice of material and model of plane will act together. UAV missions can vary with different flying conditions and loads and with high launching costs for satellites and the increasing amount of space debris, long endurance, high altitude UAVs might come to replace some of the functions that were previously handled by satellites. This thesis will study the influence of aeroelastic effects on an electric flexible tailless flying wing design UAV. A flying wing is chosen due to its efficiency, since less control surfaces means less drag and are therefore suitable for long endurance missions. Endurance of such plane can be enhanced by fitting of equipment such as piezo electric devices and solar panels. Different design options consists of, for example, choosing the optimum location of the center of gravity and the use of morphing control surfaces instead of conventional hinged ones. Due to lack of longitudinal stability compared to a conventional aircraft, aeroelastic effects such as flutter will have large consequences to the whole plane instead of just the wings. Long slender wings means less drag but more flexibility. Therefore, a flutter suppression system is the key to reach higher speeds with more efficient designs. Other aeroelastic effects such as divergence and control reversal also requires investigation since these effects cannot be suppressed and can have critical effects on the plane. Since UAVs are remote controlled, a flight control system (FCS) is often used in order to have it fly autonomous and stable during different maneuvers. Therefore a aero-servo-elastic analysis will be made to test the interaction between a FCS with the aeroelastic effects. The analysis may show whether it would be possible to suppress flutter and how the model handles wind gusts and turbulence.

Keywords: Aeroelasticity, UAV, Flying wing, ﬂutter suppression

Acknowledgments Many thanks to Professor Emeritus Harry H. Hilton who agreed to su- pervise me during my thesis and has put in big eﬀort to give me the opportunity to do my research at the Aerospace Department at the Col- lage of Engineering at University of Illinois at Urbana-Champaign.

Personally I would also like to thank everyone who has supported me from home in my endeavour of learning and exploring new areas within the ﬁeld of aerospace.

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Contents

Title Page i

Abstract i

Acknowledgments i

Contents ii

1 Nomenclature 1

2 Introduction 3 2.1 Motivation ...... 3 2.2 Mission ...... 3 2.3 Design ...... 3

3 Aerodynamic Forces 5 3.1 Atmospheric Model ...... 5 3.2 Unsteady Aerodynamics ...... 6 3.3 Simpliﬁed Unsteady Aerodynamic Model ...... 7 3.4 Eﬃcient Design ...... 8

4 Elastic Forces 13 4.1 Deformation of a Slender Beam ...... 13 4.2 Mode Shapes ...... 13 4.3 Deformation of the Wings ...... 14 4.3.1 Static Load ...... 15 4.3.2 Dynamic Load ...... 16 4.4 Energy Methods ...... 17

5 Flight Mechanics 18 5.1 Deﬁnitions ...... 18 5.2 Rigid Aircraft ...... 18 5.2.1 Dynamic model ...... 19 5.2.2 Longitudinal state-space model ...... 21 5.2.3 Pitch ...... 21 5.2.4 Pitch for ﬂexible wings ...... 23

6 Stability 24 6.1 Static Stability ...... 25 6.2 Dynamic Stability ...... 25 6.2.1 Short Period ...... 26 6.2.2 Phugoid ...... 26 6.3 Pole Placement ...... 26 6.4 Flight Control System ...... 27 6.4.1 PID controller ...... 28 6.4.2 Full state feedback ...... 28

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7 Aeroelasticity 31 7.1 Basic Example ...... 32 7.2 Static Aeroelastic Effects ...... 32 7.2.1 Torsional Divergence ...... 33 7.2.2 Static aeroelastic effects on stability ...... 35 7.2.3 Control reversal ...... 35 7.3 Flutter ...... 37 7.4 Binary Aeroelastic Model ...... 38 7.4.1 Structural damping ...... 40 7.4.2 Moving flexural and mass axis ...... 41 7.5 Entire Aircraft Model ...... 41 7.6 Aeroelastic Model for Simulation ...... 42 7.7 Effects of Swept Wing and Low Aspect Ratio ...... 45 7.8 Viscoelasticity ...... 46 7.9 Maneuvers ...... 46 7.10 Turbulence and Gust Response ...... 46 7.10.1 Sharp edged gust ...... 46 7.10.2 1-cosine gust ...... 47 7.10.3 Turbulence Modeling ...... 47 7.11 Aero-servo-elasticity ...... 48 7.11.1 State-space modeling ...... 48

8 Analysis 50 8.1 Eﬃcient Design ...... 51 8.2 Static Aeroelastic Eﬀects ...... 52 8.3 Flutter Prediction ...... 52 8.4 Simulation ...... 53

9 Results 55 9.1 Efficient Design ...... 55 9.1.1 Airfoil Analysis ...... 55 9.1.2 3D model with hinged elevon ...... 56 9.1.3 3D model with morphed elevon ...... 58 9.2 Aeroelastic Effects ...... 59 9.3 Flutter Prediction ...... 60 9.3.1 Binary aeroelastic model ...... 60 9.3.2 Entire aircraft aeroelastic model ...... 61 9.3.3 3D Plots of entire aircraft model ...... 61 9.4 Simulation ...... 62 9.4.1 Linearized model without control ...... 62 9.4.2 Linearized model with control ...... 63 9.4.3 Simulink response without control ...... 64 9.4.4 Simulink response with LQR control ...... 64 9.4.5 Simulink response climb profile with LQR control . . . . . 65 9.4.6 Control Reversal ...... 66 9.4.7 Manual Tuning ...... 67 9.4.8 Simulink response climb profile with adaptive controller . 68 9.4.9 Poles and zeros plot ...... 69 9.5 Summary ...... 69

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10 Discussion 71 10.1 General Considerations ...... 71 10.2 Simulation ...... 71 10.3 Limitations of the Controller ...... 72

11 Conclusion 73

List of Tables

1 Design properties ...... 4 2 Aeroelastic velocities (m/s) at sea level ...... 69 3 Aeroelastic velocities (m/s) at 10,000 feet ...... 70

List of Figures

1 Swept unmanned flying wing design ...... 4 2 Flying wing airfoil MH 62 ...... 8 3 Weight and lift distribution of conventional aircraft [6] ...... 9 4 Design considerations flying wing [39] ...... 11 5 Hinged Design ...... 12 6 Wing morphing Design ...... 12 7 Coordinates and force vectors [33] ...... 20 8 Simple closed-loop with controller ...... 24 9 Effects of pole position on stability ...... 27 10 Full State Feedback control system ...... 29 11 Properties of the eigenvalues [41] ...... 30 12 Interaction between forces and their effects[1] ...... 31 13 Mechanical spring and damping system ...... 32 14 Divergence ...... 33 15 Flutter 2DOF ...... 37 16 Illustration of the 3 DOF wing-flap flutter model [18] ...... 43 17 Flow chart ...... 50 18 Flying wing 3D model in XFLR 5 with flaps down configuration 51 19 Flying wing 3D model in XFLR 5 with flaps down morphed configuration ...... 52 20 Flying Wing Flutter suppression system ...... 53 21 Lift to drag ratio for different flap settings and different Reynolds number ...... 55 22 Inviscid 3D analysis at different flap settings hinged elevon . . . 56 23 Viscous 3D analysis at different flap settings for hinged elevon . . 57 24 Inviscid 3D analysis at different flap settings for morphed elevon 58 25 Viscous 3D analysis at different flap settings for morphed elevon 59 26 Aeroelastic response due to airspeed ...... 60 27 Flutter prediction of binary model ...... 60 28 Flutter prediction of entire aircraft model ...... 61 29 Damping ratio with respect to velocity and altitude ...... 61 30 Damping ratio with respect to velocity and pitch stiffness . . . . 62 31 Linearized without control ...... 63 32 Step response of the linearized model with LQR control . . . . . 64

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33 Simulink response without control ...... 64 34 Simulink LQR response with control at 10,000 feet with airspeed of Mach 0.172 and 5 m/s gust at 10 seconds ...... 65 35 Simulink response with control at 10,000 feet with airspeed of Mach 0.172 with turbulence ...... 65 36 Climb and acceleration profile response for LQR controller with gusts and turbulence ...... 66 37 Control effectiveness for Simulink model ...... 67 38 Simulink response with manually tuned controller at 10,000 feet with airspeed of Mach 0.3 and 5 m/s gust at 5 seconds with turbulence ...... 68 39 Climb and acceleration profile response for an adaptive controller with gusts and turbulence ...... 68 40 Pole/Zero plot for plunge and pitch at 10,000 feet at Mach 0.15 with LQR controller ...... 69

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1 Nomenclature

Abbreviations

Re - Reynolds number AR - Aspect Ratio IAS - Indicated Airspeed TAS - True Airspeed UAV - Unmanned Aerial Vehicle RSS - Relaxed Static Stability FCS - Flight Control System FBW - Fly-By-Wire SISO - Single-input single-output MIMO - Multiple-input multiple-output PID - Proportional, Integral and Derivative Controller LQR - Linear-quadratic regulator

Variables

α - angle of attack CL - lift coefficient CM - moment coefficient CD - drag coefficient V - air speed a - speed of sound / distance between half chord and flexural axis M - Mach number p - roll rate q - dynamic pressure / pitch rate / generalized coordinate r - yaw rate a0 - zero lift coefficient b0 - zero moment coefficient a1 - lift curve airfoil b1 - moment curve airfoil ac - lift due to elevon deflection bc - moment due to elevon deflection aw - lift curve wing bw - moment curve wing c - chord Φ - Wagner’s function / PSD σ - Turbulence standard deviation Ω - Wavenumber h - plunge / bending deflection θ - pitch and twist / torsion deflection CG - center of gravity E - elastic modulus G - torsional elastic modulus I - moment of inertia J - polar moment of inertia K - shearing stiffness constant Sh - coupling inertia

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Sθ - coupling inertia ω - downwash / frequency ωg - gust speed ψ - mode shape Q - generalized force Kh - spring constant plunge Kθ - spring constant twist ρ - density g - gravity / structural damping coefficient m - mass s - span b - half span S - wing area Λ - sweep e - position of flexural axis δ - elevon deflection

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2 Introduction 2.1 Motivation Motivation for this thesis is to design and analyze a flutter suppression system for an efficiently designed UAV. An efficient design means that all available tools for maximizing the efficiency have been used during the design process. This requires the lift versus drag (L/D) ratio to be as high as possible. How this can be achieved will be discussed later on. Since more efficient wings often means higher flexibility, a flutter suppression system has become vital for efficient designs [13]. The tools for aeroelastic modeling and control have been available for some time [2], but the application of such a control system for small efficient UAV’s is still limited [33]. In order to create an aeroelastic model, the plane must be treated as a whole system. This requires knowledge in areas such as aerodynamics, structural mechanics, flight mechanics and control theory. A theoretical mission and design will be used to emphasize the importance of an aeroelastic model when designing airplanes and especially efficient UAV’s. By modifying and improving existing aeroelastic models to represent a flying tailless wing and to include control theory, the author hopes that the thesis will cover different possible design configurations and flying conditions and demonstrate the possibility to increase both efficiency and flying speed by the use of a flutter suppression system.

2.2 Mission The model will be an electrical powered swept unmanned tailless flying wing with a mission that includes long endurance flying at altitudes about 10 000 feet (∼ 3000 meters) and with a structure that will handle speeds of up to at least 60 m/s. For long endurance missions efficiency is a key role, however if the mission consists of fast in and out response in dangerous areas or search and rescue, the ability to be fast and invisible is of higher importance. For these purposes a flying wing design could be used with advantage.

2.3 Design Tailless Flying Wing Design The tailless flying wing design has been tested and tampered with since the dawn of flight. Many chose the design due to its efficiency but soon gave up because of stability issues [8]. During WWII, the German Horten brothers built the Ho- 229 which was a purely tailless flying wing. Due to its design it had a very small radar cross section and its purpose was to avoid the allies radar. Luckily the war ended before the planes went into mass production. The design got more popular after the first flight of Jack Northrop’s N-1M in 1940 and Northrop was determined that the future of flight lied with the flying wing design. But it took 47 years until technology caught up and in 1987 the B-2 stealth bomber flew for the first time. It had a computerized stability augmentation system and thereby the previous stability problems were fixed. The B-2 was very successful but due to high manufacturing costs, only 21 were built. In the last decade, UAVs have

3 Master Thesis become popular and they come in many different designs. Northrop Grum- man for example, still holds on to the flying wing and are currently testing out there X-47. Lockheed Martin also uses a flying wing for research purposes with there X-56A, known as MUTT, which stands for Multi-Use Technology Testbed.

Design Parameters

Figure 1: Swept unmanned ﬂying wing design

Parameter Value Mass m 10 kg 2 Inertia Iy 0.503 kgm Span s 2 m Wing area S 1 m2 Root chord 0.7 m Tip chord 0.3 Mean aerodynamic chord MAC 0.53 m Aspect ratio AR 4 Taper ratio 0.43 Effective sweep Λ 23◦ Plunge stiffness 1644 kg/m Pitch stiffness 413.2 kg/rad

Table 1: Design properties

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3 Aerodynamic Forces

Some basic knowledge about aerodynamics will be required for the understand- ing of this thesis. The general equations of importance are:

Lift:

L = qCLS (3.1) Moment:

M = qCM Sc (3.2) Drag:

D = qCDA + Di (3.3) where S is the wing surface area, c the wing chord, Di is the induced drag and q is the dynamic pressure which depends on density and velocity squared: 1 q = ρV 2. (3.4) 2 The lift, moment and drag coeﬃcients can be found experimentally by split- ting them into parts that can be analyzed individually. For instance the lift coeﬃcient can be written as:

CL = CL0 + CLαα + CLδδ (3.5) where CL0 is the lift coefficient at zero angle of attack due to the camber of the airfoil, CLα is the coefficient due to angle of attack (assuming that the angle of attack is small and in the linear region) and CLδ is the contribution due to elevon deflection.

For small angles the angle of attack can be deﬁned as:

rigid body twist angle angle z }| { z }| { h˙ (t) α(x, y, t) = α(y, t) + αr(y) + α0(y) + θ(y, t) + . (3.6) | {z } | {z } V angle of zero lift |{z} angle Addition due attack to plunge

The Mach number is deﬁned as the airspeed relative to the speed of sound a, which varies with altitude and temperature: V M = . (3.7) a 3.1 Atmospheric Model For simulation, a standard atmospheric model is used, namely the 1976 COESA model which uses the altitude as argument to calculate the temperature, speed of sound, air pressure and air density. This is then used together with the Mach number to produce the dynamic pressure q.

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3.2 Unsteady Aerodynamics Consider an instant change of angle of attack ∆α, this will develop into a change in lift. A common representation of the lift is by using the Wagner function Φ(τ). 1 ∆L = ρV 2ca ∆αΦ(τ) (3.8) 2 1 where a1 is the lift curve for the airfoil. Replacing V ∆α with downwash velocity w gives: 1 ρV ca wΦ(τ). (3.9) 2 1 The Wagner function can be approximated as [1]: τ + 2 Φ(τ) = (3.10) τ + 4 and represents the step response due to the increase in angle of attack.

Theodorsen’s Function When analyzing ﬂutter, it might be more interesting to analyze the motion at a single frequency than looking at the time domain response. When comparing the unsteady aerodynamics to the quasi-steady it can be shown that the lift decreases and there is an introduction of a phase-lag since the quasi-steady values per deﬁnition always are in phase.

The amplitude and phase-lag of the forces can be expressed as functions of the reduced frequency k which is defined using the semi chord b = c/2 and can be interpreted as the number of oscillations during the time it takes for the air to flow over the semi chord b, times 2π: ωb k = . (3.11) V Theodorsen’s function can then be used to express the amplitude and phase-lag with respect to k. In order to accomplish that, Theodorsen’s function consists of a real part and an imaginary part and is expressed as C(k) = F (k) + iG(k) for constant flight velocity. F and G must be expressed using Bessel functions of second kind, these wont be provided, but can be easily implemented. Theodorsen’s function as a function of Bessel functions K: K (ik) C(k) = 1 . (3.12) K0(ik) + K1(ik) Quasi-steady aerodynamics can be obtained by letting ω = 0 and k → 0 which leads to F = 1 and G = 0.

Oscillating Airfoil Lift and moment for an airfoil in simple harmonic motion can be divided into two diﬀerent parts. One part that will consist of the circulatory terms related to aerodynamic forces and one part that will be the noncirculatory terms related to inertia. The deﬂection due to bending is denoted as h and twisting θ.

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h i 1 L = πρb2 h¨ + V θ˙ − baθ¨ + 2πρV bC(k) h˙ + V θ + b − a θ˙ (3.13) 2

1 1 M = πρb2 bah¨ − V b − a θ˙ − b2 + a2 θ¨ + 2 8 (3.14) 1 1 2πρV b2 a + C(k) h˙ + V θ + b − a θ˙ 2 2 where a is the distance between the half chord b and the ﬂexural axis. These equations can be written in derivative form, using oscillatory derivatives, and by considering simple harmonic motions for a 2 DOF system so that:

n (φ+iω)to ˙ n (φ+iω)to h = Re h0e , h = Re (φ + iω)h0e

n (φ+iω)to ˙ n (φ+iω)to θ = Re θ0e , θ = Re (φ + iω)θ0e where h0 and θ0 are complex numbers, φ is the phase relation and ω the frequency.

The derivatives can be found in the appendix. Resulting derivative form for the oscillating airfoil becomes: ! bh˙ b2θ˙ L = ρV 2 L h + L + L bθ + L (3.15) h h˙ V θ θ˙ V ! b2h˙ b3θ˙ M = ρV 2 M bh + M + M b2θ + M . (3.16) h h˙ V θ θ˙ V This is conveniently expressed using matrices for further calculations:

2 ˙ L bLh˙ b Lθ˙ h 2 Lh bLθ h = ρV 2 3 ˙ + ρV 2 . (3.17) M b Mh˙ b Mθ˙ θ bMh b Mθ θ

3.3 Simplified Unsteady Aerodynamic Model For some of the flutter calculations a more simplified unsteady aerodynamic model will be used, since the results will still be within the required tolerance. [2] First, the position of the flexural axis ec is defined in relation to the position of the aerodynamic center ac and the mid chord ab: c c ac ec = + ab = + . 4 4 2 Then assume quasi-steady by letting k → 0,G → 0,F → 1, effectively removing the derivatives. Then, the lift and pitching moment can be expressed as:

1 h˙ L = ρV 2ca θ + (3.18) 2 w V

1 h˙ M = ρV 2ec2a θ + (3.19) 2 w V

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where aw is the lift curve for the wing. It has been proven that the unsteady aerodynamic term Mθ˙ is of significant importance due to its damping effect, a value of -1.2 has been shown to provide sufficient accuracy for flutter predictions [2]. Therefore, it will be the only remaining unsteady aerodynamic term in the simplified moment equation 1 h˙ θ˙ M = ρV 2c2 ea θ + + M c . (3.20) 2 w V θ˙ 4V

3.4 Eﬃcient Design For an eﬃcient design there are many things to consider. A selection of these will be presented here.

Airfoil As usual when considering an airfoil there are lift and drag coeﬃcients and the L/D ratio that is supposed to be as high as possible at a reasonable angle of attack. But depending on the mission, diﬀerent airfoils might be used along the span of the wing for best performance. To simplify the design process in this case, only one airfoil is used along the entire span.

For tailless ﬂying wings it is important to also consider the moment coeﬃcients. These must be as low as possible considering that all the moment produced by the wing needs to be compensated by the elevons and the lever arm for the elevons is much shorter than for example the lever arm for a conventional airplane with a tailplane. The elastic axis is located between the aerodynamic center and the center of gravity, and the center of gravity is aft of the aerodynamic center.

There are specially designed airfoils for ﬂying wings that have been thoroughly tested [38]. One of these is the MH 62. It has a thickness of 9.30% and a relatively high lift to drag ratio and a vary small moment coeﬃcient.

Figure 2: Flying wing airfoil MH 62

Elliptic lift distribution Lifting line theory implies that the optimum lift distribution for a wing without winglets is accomplished by an elliptic lift distribution [4]. The lift distribution is directly related to the induced drag. Since elliptic wings (such as those on the Spitﬁre) are hard to produce, a taper ratio of 0.4 has proved to give the closest approximation to the elliptic wing.

Washout One of many reasons to consider washout is to ensure that the root of the wing

8 Master Thesis stalls before the tip. It is accomplished by for instance twisting the wing and decreasing the angle of attack along the span. This reduces the lift distribution but lowers the probability of a wingtip stall. It can also improve the stability and static aeroelastic eﬀects by altering of the pitching moment coeﬃcient of the wing.

Wingtip Devices Winglets are commonly used since they are known to increase eﬃciency. The need for winglets increases with the size of the airplane, higher mass increases downwash and incduced drag which can be reduced with winglets by moving the wingtip vortices further away from the wings. The addition of winglets requires further aeroelastic calculations and for small UAV’s the aeroelastic eﬀects might have bigger impact on performance than the gain given by the winglets. Therefore, wingtip devices will not be considered in this thesis.

Engine Placement Placement of the engine has proven to be of great importance for the aerodynamic efficiency. The most obvious reason is the affect of the CG, but if placed in the back on top of the wing, the engine will work as an active suction device that eliminates separation of the air flowing over the center of the wing. It should also increase the velocity of the air over the wing, lowering the pressure on top of the wing according to the Bernoulli’s equation and therefore increasing the lift.

Position of CG Flying wings are considered, due to their lack of tailplane, the most eﬃcient plane design. The tailplane generates both regular drag and induced drag since the down force of the tailplane needs the main wings to produce higher lift. This is made clear by Figure 3:

Figure 3: Weight and lift distribution of conventional aircraft [6]

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In Figure 3, the lift from the main wings needs to compensate for both the weight of the airplane and the downforce of the tailplane. Therefore, by moving the CG aft, the needed lift will be decreased as the CG approaches the aerodynamic center, making the tailplane less important. This also decreases the steady-state induced drag or lift-drag. The induced drag is defined: 1 D = ρV 2SC (3.21) i 2 Di and after entering the parameters for the induced drag coefficient, the latter becomes: C2 C = L . (3.22) Di πeAR This equation clearly states the dependency of lift coefficient squared, which indicates that the induced drag, even if smaller than the form drag, depends of the weight needed to be countered by the lift. The main drawback by mowing the CG aft is stability. The elevator becomes more effective, but the plane will become inherently unstable and have a tendency to begin to oscillate on its own. The equation is also governed by the aspect ratio and e, which in this case, is related to the lift distribution and is equal to 1 for elliptic wings.

By choosing a tailless flying wing design the drag can be reduced significantly, but the drag is still dependent on the position of the CG according to Figure 4. As in previous case the position of CG affects the stability and if the position of CG is aft of the aerodynamic center, the airplane might become impossible to fly without a stability augmentation system. As following figures show the position of CG leads to different flow patterns while maintaining steady flight:

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Figure 4: Design considerations ﬂying wing [39]

Figure 4 shows how a forward CG, while increasing stability, also increases drag due to the shape of the flow and the wake generated behind the wing. Choosing an aft CG, however, gives both better lift performance due to the increased camber and less drag. This shows that it is preferable to have an aft CG combined with an advanced stability augmentation system for best flying efficiency. For a CG located right at the aerodynamic center the stability is governed by the pitching coefficient of the airfoil CM0.

Wing Morphing Testings of an adaptive compliant trailing edge made by NASA and Flexsys Co. has shown that drag forces can be dramatically reduced compared to the

11 Master Thesis conventional hinged design of trailing edge devices. As much as 12% of fuel can be saved for a clean sheet design [35]. This makes the morphing design highly desirable for long endurance UAV’s. Flexsys also claims that the design is both lighter and requires less force to operate than hinged control surfaces [36].

Figure 5: Hinged Design

Figure 6: Wing morphing Design

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4 Elastic Forces

One of the big factors in aeroelasticity are elastic forces which have to be analyzed in order to create an aeroelastic model for a structure. It is important to know the shape and which components are used to make up the structure and what boundary condition need to be met to choose the right approach for the problem. Since the wings make up the whole plane on a ﬂying wing, the fuselage bending will be left out and the focus will lie exclusively on the deformation of the wings. For best precision, a ﬁnite element method would be preferred. This requires time consuming simulations and therefore simpler models will be considered as the results only have to be representative and not exact.

4.1 Deformation of a Slender Beam For larger aspect ratios the wings can be modeled as slender beams. The most common description for the bending of beam with applied load q is described with the Bernoulli-Euler equation:

∂2 ∂2h EI = q (4.1) ∂y2 ∂y2 and the solution for moment yields:

∂2h M = EI (4.2) ∂y2 and the torsion can be described as: ∂θ T = GJ . (4.3) ∂y The potential energy in the beam due to bending and torsion can then be written as: Z l EI ∂2h Z l GJ ∂2h U = 2 ∂y + dy. (4.4) 0 2 ∂y 0 2 ∂y∂x With these equations deﬁned, the elastic forces can be included in the model at a later stage.

4.2 Mode Shapes Mode shapes are the assumed shapes that the wing will take due to loads that satisfy the boundary conditions. In one dimension the mode shape is described as: N X w(y, t) = ψj(y)qj(t) (4.5) j=1 where w is the deﬂection, ψj is the assumed shape and qj is the unknown function. Following are examples of how the expression is used for two simple deformations:

Bending The mode shape for a slender beam can be found analytically by integrating

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Equation 4.2. For a uniform load distribution W(t) the moment at any point of the cantilever wing is given by: W (t) M = (s − y)2. (4.6) 2 Using this expression in Equation 4.2 gives:

∂2h W (t) EI = (s2 − 2sy + y2). (4.7) ∂y2 2 Integration gives:

∂h W (t) y3 EI = s2y − sy2 + + C . (4.8) ∂y 2 3 1 Using boundary conditions where the wing is clamped at y = 0 and free at y = s ∂h(0) gives = 0 which leads to C = 0. ∂y 1 Another integration gives:

W (t) s2y2 sy3 y4 EIh = − + + C . (4.9) 2 2 3 12 2

Boundary condition h(0) = 0 gives C2 = 0, leading to the ﬁnal expression for deﬂection h:

W (t) h = − 6s2y2 − 4sy3 + y4 . (4.10) 24EI

Assuming a dominating quadratic term and using general coordinate qb(t) for bending, following simpliﬁcation can be obtained which satisﬁes the one clamped end and one free end boundary condition of a slender beam [2]:

y 2 h(y, t) = q (t). (4.11) s b

Equation 4.11 describes a quadratic deﬂection with maximum deﬂection q(t)b at the tip of the wing where y = s. This expression will be used for aeroelastic bending since it reduces the workload and still gives a good representative result [2].

Torsion Similar calculations with the same boundary conditions in torsion assuming linear twist and general coordinate qt(t) gives: y θ(y, t) = q (t). (4.12) s t

4.3 Deformation of the Wings This will give a brief description of how to model the deformation of the wings in more detail. However, for best precision, a ﬁnite-element analysis would be required.

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4.3.1 Static Load Assuming a more complex shape of the wings requires the inclusion of inﬂu- ence functions to describe the deformation. The governing equation for the deformation of an elastic wing under a static normal load Z(x,y) can be written as: ZZ ω(x, y) = C(x, y; ξ, η)Z(ξ, η)dξdη. (4.13)

Since the wings can be considered slender enough to assume that the chordwise segments are rigid, the inﬂuence function can be written:

C(x, y; ξ, η) = Czz(y, η) − xCθz(y, η) + ξxCθθ(y, η) − ξCzθ(y, η) (4.14) where Cpq(y, η) is the linear or angular deﬂection in the p-direction at y due to a unit force or a torque in the q-direction at η.

The inﬂuence functions for an elastic swept wing by angle Λ may be written as [1]:

η ~ y ~ Z y/cosΛ − λ − λ Z y/cosΛ d~λ Czz(y, η) = cosΛ cosΛ d~λ + (η ≥ y) 0 EI 0 GK (4.15) η ~ y ~ Z η/cosΛ − λ − λ Z η/cosΛ d~λ Czz(y, η) = cosΛ cosΛ d~λ + (y ≥ η) 0 EI 0 GK (4.16)

Z y/cosΛ cos2Λ sin2Λ Cθθ(y, η) = + (η ≥ y) (4.17) 0 GJ EI

Z η/cosΛ cos2Λ sin2Λ Cθθ(y, η) = + (y ≥ η) (4.18) 0 GJ EI

Z y/cosΛ ( η − ~λ) Czθ = Cθz = sinΛ cosΛ d~λ (η ≥ y) (4.19) 0 EI

Z η/cosΛ ( η − ~λ) Czθ = Cθz = sinΛ cosΛ d~λ (y ≥ η). 0 EI (4.20) By expressing the deﬂection as:

ω(x, y) = h(y) − xθ(y) (4.21) and substituting Equation 4.13 and Equation 4.14 into Equation 4.21 results in:

Z l Z l h(y) = Czz(y, η)Z(η)dη + Czθ(y, η)t(η)dη (4.22) 0 0

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Z l Z l θ(y) = Cθz(y, η)Z(η)dη + Cθθ(y, η)t(η)dη (4.23) 0 0 where the running spanwise load Z and torque τ in η can be written as: Z Z(η) = Z(ξ, η)dξ (4.24) chord Z τ(η) = − ξZ(ξ, η)dξ. (4.25) chord For a swept wing with sweep angle Λ, the running spanwise torque may be expressed as: Z τ(η) = − (ξ − ηtanΛ)Z(ξ, η)dξ. (4.26) chord

4.3.2 Dynamic Load Since ﬂutter is of main concern in this thesis, dynamic loads must be considered. ˙ ¨ Dynamic loads include a time dependent force Fz(ξ, η, z, z,˙ z,¨ θ, θ, θ, t) which represents the lift force created by the unsteady aerodynamics, with the mass per unit area deﬁned as ρ. The load Z is then written as:

Z(ξ, η, t) = −ρ(ξ, η)¨ω(ξ, η, t) + Fz(ξ, η, etc.). (4.27) The deﬂection equation becomes: ZZ h(x, y, t) = C(x, y; ξ, η)[−ρ(ξ, η)¨ω(ξ, η, t) + Fz(ξ, η, etc.)]dξdη. (4.28)

This can be solved using natural shape modes. This is done by putting:

∞ X h(x, y, t) = φi(x, y)ξi(t) (4.29) i=1 where φi(x, y) represents the natural modes of vibration and are the eigenfunc- tions of the integral equation: ZZ 2 φi(x, y) = hi C(x, y; ξ, η)φi(ξ, η)ρ(ξ, η)dξdη. (4.30) S Resulting coupled equations of motion become: Z l zz ¨ h(y, t) = C (y, η)[−m(η)¨ω(η, t) + Sy(η)θ(η, t) + Fz(η, etc.)]dη 0 (4.31) Z l zθ ¨ + C (y, η)[−Iy(η)θ(η, t) + Sy(η)¨ω(η, t) + t(η, t)]dη 0 Z l θz ¨ θ(y, t) = C (y, η)[−m(η)¨ω(η, t) + Sy(η)θ(η, t) + Fz(η, etc.)]dη 0 . (4.32) Z l θθ ¨ + C (y, η)[−Iy(η)θ(η, t) + Sy(η)¨ω(η, t) + t(η, t)]dη 0 Depending on the inﬂuence functions, above equations could be used to describe the deﬂection for swept wings for as long as they are rigid in x-direction.

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4.4 Energy Methods The equations for motion are based on D’Alembert’s principle and give a good picture of what to expect when the wings are subjected to diﬀerent loads. To ﬁnd a solution it is more convenient to adapt an energy method where equilibrium is found with expressions for work and energy. By using the principle of virtual work and including the inertial and external forces into D’Alembert’s principle following expression is obtained:

δWe + δWin = δU. (4.33)

This can then be used in Lagrange’s equation, but ﬁrst the work need to be expressed in generalized coordinates. The external forces become:

n X δWe = Qiδqi (4.34) i=1 where Z ∂u ∂v ∂w Qi = Fx + Fy + Fz dy. (4.35) S ∂qi ∂qi ∂qi The inertial forces are expressed in terms of kinetic energy. This gives the following Lagrange equation:

d ∂T ∂T ∂U ∂(δW ) − + = Qi = . (4.36) dt ∂q˙i ∂qi ∂qi ∂(δqi) This equation is used throughout the report to ﬁnd solutions to the equations of motion.

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5 Flight Mechanics

Flight mechanics covers the interaction between the inertial forces of the airplane and the surrounding aerodynamic forces. As one of the main parts of the aeroelastic triangle, ﬂight mechanics are necessary to include mostly because it gives valuable information of the stability of the airplane.

5.1 Deﬁnitions First we deﬁne the velocity, position, force vectors, the symmetric moment and the inertia matrix.  U   long.vel   V   lat.vel      W   vert.vel  ~v =   =   (5.1)  P   roll rate       Q  pitch rate R yaw rate     XE x − positon YE  y − position      h   Altitude  ~η =   =   (5.2)  φ   roll angle       θ   pitch angle  ψ yaw angle

X   long.force   Y   lat.force       Z   vert.force  ~τ =   =   (5.3)  L   roll moment      M pitch moment N yaw moment   Ix Ixy Ixz I = Iyx Iy Iyz (5.4) Izx Izy Iz Flight mechanics involve the motions of the aircraft around its body axe, such as roll, pitch and yaw. In this section the equations of motion for the airplane are deﬁned and simpliﬁed by making some basic assumptions. The equations of motion will be derived from Newton’s second law of motion and Euler moment equations.

Newton’s second law: d~v ΣF~ = m . (5.5) dt 5.2 Rigid Aircraft Initially the airplane is assumed to be rigid. This can be used to give an approximation of the pitch response for the whole airplane for which the stability can be analyzed. Assuming a rigid airplane leads to moment of inertia around

18 Master Thesis the y-axis to be calculated as just: Z 2 2 Iy = (xr + zr )dm (5.6)

The governing nonlinear force and moment equations are [3]:

X = m(U˙ − RV + QW ) (5.7)

Y = m(V˙ − PW + RU) (5.8) Z = m(W˙ − QU + PV ) (5.9)

L = IxP˙ − (Iy − Iz)QR − Ixz(PQ + R˙ ) (5.10) 2 2 M = IyQ˙ + (Ix − Iz)PR + Ixz(P − R ) (5.11) ˙ N = IzR˙ − (Ix − Iy)PQ + Ixz(QR − P ). (5.12) Due to aircraft symmetry all other products of inertia are zero. If the body axis were aligned with the principal axis of the airplane, then Ixz would be zero as well.

To linearize these equations perturbation theory can be used. By assuming trimmed equilibrium ﬂight the velocity components can be written as the sum of nominal value + perturbation value:

U = Ue + u V = Ve + v W = We + w (5.13) and rotation rates:

P = Pe + p Q = Qe + q R = Re + r. (5.14)

If disturbance velocities are small and small quantities like products of rate can be neglected, then it is possible to divide the equations into longitudinal and lateral motion.

Longitudinal

X = m(u ˙ + Weq) Z = m(w ˙ − Ueq) M = Iyq.˙ (5.15) Lateral

Y = m(v ˙ − Wep + Uer) L = Ixp˙ − Ixzr˙ N = Izr˙ − Ixzp.˙ (5.16)

5.2.1 Dynamic model The following dynamic model is derived from the Euler moment equations and can be used to express a rigid six DOF vehicle. The deﬁning parameters are

19 Master Thesis the external forces (X, Y, Z, L, M, N) which will represent those of an aircraft, namely the aerodynamic and actuator forces.

 X  −gsinθ + rv − qw + m  Y     gcosθsinφ − ru + pw +  u˙  m   Z  v˙   gcosθcosφ + qu − pv +     m  w˙   1     [L + Ixz(r ˙ + pq) + (Iy − Iz)qr]  p˙   Ix     1 2 2  q˙ = [M + Ixz(r − p ) + (Iz − Ix)rp] (5.17)    Iy   r˙       1   ˙   [N + Ixz(p ˙ − qr) + (Ix − Iy)pq]  φ  Iz      θ˙   p + (qsinφ + rcosφ)tanθ    ψ˙    qcosφ − rsinφ    (qsinθ + rcosφ)secθ

Figure 7: Coordinates and force vectors [33]

In order to linearize the model some general assumptions need to be made: • Small angles → α ≈ w/u.

• Small disturbances. • Products of inertia equal zero. • Fast response → neglect derivatives of all velocities.

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• Iy << Ix = Iz for a ﬂying wing. • Uncoupled motion in longitudinal and lateral directions.

These assumptions results in a linear state space model that can be used to analyze the dynamic modes of motion that can later be coupled and implemented in the aeroelastic model.

5.2.2 Longitudinal state-space model General state-space representation:

x˙ = Ax + Bu. (5.18)

The governing variables for longitudinal motion are u, w, q and θ. Converting them into a state-space representation results in:

u˙      u w˙  w = A + B δ (5.19) q˙ e    q  θ˙  θ  where δe is change in elevon angle and the matrices A and B are expressed as:

 Xu Xw  m m 0 −gcosθ0

 Zq +mu0 −mgsinθ   Zu Zw 0   m−Zw˙ m−Zw˙ m−Zw˙ m−Zw˙  A =    1 Mw˙ Zu 1 Mw˙ Zw 1 Mw˙ (zq +muo) 1 −Mw˙ mgsinθ0   Mu + Mw + Mq +   Iy m−Zw˙ Iy m−Zw˙ Iy m−Zw˙ Iy m−Zw˙    0 0 1 0

 Xδe   m   Z   δe   m−Zw˙  B = . 1 Mw˙ Zδe  Mδ +   Iy e m−Zw˙       0  It can be practical to further reduce this model into three DOF by applying α ≈ ω/u for small angles. This would result in following model:

α˙      α q˙ = A q + Bδe. (5.20) θ˙ θ

5.2.3 Pitch Due to the lack of longitudinal stability of a flying wing it is important to analyze the pitch response. Since the phugoid, or long-term oscillation, does not affect the stability and flying characteristics as much as the short-period oscillations do, the terms that corresponds to phugoid can be neglected [33]. This includes

21 Master Thesis the term responsible for the fore and aft motion. Therefore, only two equations of importance remain:

Z = m(w ˙ − Ueq) M = Iyq.˙ (5.21)

Expressing Z and M in terms of aerodynamic stability derivatives:

Z = Zw˙ w˙ + Zq˙q˙ + Zww + Zqq + Zδδ (5.22)

M = Mw˙ w˙ + Mq˙q˙ + Mww + Mqq + Mδδ. (5.23) With small perturbations and lack of tailplane following simpliﬁcations can be made:

Zw˙ = Zq˙ = Mw˙ = Mq˙ = Zq = Mq = 0 The remaining terms are given by: 1 Z = − ρV (S a + S a (1 − k ) + S C ) (5.24) w 2 0 w w T T w D 1 M = ρV [S a l − S a l (1 − k )] (5.25) w 2 0 w w w T T T 1 Z = − ρV 2S a (5.26) δ 2 0 T E 1 M = − ρV 2S a l . (5.27) δ 2 0 T E T The solution to the rigid ﬂight mechanics equations can then be found by ﬁrst writing them in matrix formation:

m 0 w˙ −Z −mU w Z + w E = δ δ (5.28) 0 Iy q˙ −Mw 0 q qδ and applying Laplace transform

sm − Z −mU w(s) Z w E = η η(s). (5.29) −Mw sIy q(s) qη It is then possible to solve the matrix for pitch rate per elevator angle: q smM + M Z − M Z (s) = η w η η w (5.30) η D(s) where D(s) is the determinant of the square matrix:

2 s (Iym) − s(IyZw) − mUeMw. (5.31)

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5.2.4 Pitch for ﬂexible wings Even if a ﬂying wing lacks fuselage, there will be an impact on pitch performance due to the elasticity in bending and torsion of the wings.

Bending mode It is shown with examples of different natural frequencies that the bending mode of the wings have a small impact on elevator effectiveness and stability in pitch and that the results only differ about 2% from the rigid case [2]. There will therefore not be any further calculations on this mode.

Torsion mode The torsion of the wings however results in a significant change in elevon effectiveness as the flying speed increases compared to the bending mode. As the twist of the wing increases due to static aeroelasticity it leads to loss in control effectiveness. This will be described further in the aeroelastic part.

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6 Stability

The relaxed static stability (RSS) design implies that the center of gravity is behind the neutral point. This gives the aircraft a tendency to change its attitude by itself and will also oscillate in a divergent fashion around its trimmed attitude. This behavior is desired for most modern fighters for their increased maneuverability and also for airliners since they require smaller control surfaces to change their attitude which gives less drag and better fuel economy [34]. This design will have to rely on a sophisticated flight control system (FCS) with a stability augmentation system in order to achieve stable flights.

By using the existing FCS to decrease aeroelastic eﬀects it may be possible to increase the general stability of the whole airplane as well as increasing the maximum velocity and decrease the drag forces associated with ﬂutter. Other advantages could be increased life span to structures due to less fatigue, less vibration leading to sharper imaging if a camera is present and increased passenger comfort for airliners.

By adding sensors in order to create a closed-loop control system and to include controllers, flutter can be controlled until a point were the controls become unresponsive. Since flutter frequencies often are too high for a pilot to be able to react, a flutter suppression system is both convenient and vital for a flying wing as it approaches its flutter speed. However, as the wings twist, other effects such as torsional divergence and control reversal may occur which the flutter suppression system cannot handle.

Figure 8: Simple closed-loop with controller

Figure 8 represents a simple closed-loop single-input single-output system (SISO) with input X, output Y, controller K and system G. The transfer function for the Laplace transformed system is found by deﬁning the following:

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Y(s)=G(s)u(s) u(s)=K(s)e(s) e(s)=X(s)-Y(s) which results in a transfer function: Y (s) K(s)G(s) = . (6.1) X(s) 1 + K(s)G(s) This is the main loop. In order to analyze flutter effects, this system will be ex- panded by adding sensors for the plunge, pitch and elevon deflection. It will also include the aerodynamic forces and the flight mechanics of the flying wing model as well as accounting for elastic forces and including a comprehensive controller.

6.1 Static Stability The static stability depends heavily on the placement of the center of gravity. The governing stability term is the change in moment due to change in angle of attack:

∂M ∂L ∂L = x w + qScC − (l − x ) t (6.2) ∂α cg ∂α Mα t cg ∂α | {z } | {z } main wing contribution tailplane contribution

where xcg is the distance of the center of gravity from the aerodynamic center (positive if aft). Lack of tailplane gives:

∂M ∂L = x w + qScC . (6.3) ∂α cg ∂α Mα

For stability this term needs to be negative, but for performance this term is preferably positive. However, this requires a sophisticated FCS to ease the work load of the pilot who would need to make adjustments continuously in order to keep the plane flying. Since such control system exists in most modern airplanes the question then is if it is possible to increase the maximum flight velocity by decreasing the effects of flutter by utilizing the already existing flight computers and sensors.

6.2 Dynamic Stability Dynamic stability includes the modes of oscillation of an airplane as it encoun- ters a disturbance from steady ﬂight. The open-loop performance is analyzed by ﬁnding the eigenvalues of matrix A. There will be two pairs of complex conjugate roots. One low and one high frequency. The low frequency root represents the stability of the so-called phugoid motion, while the high frequency represents the stability of the short period motion.

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6.2.1 Short Period As its name implies it is an oscillation with a period of around a second. This is a critical state for an airplane and the amplitude is often effectively damped. This mode poses a threat to flying wings due to their already inherent instability and its connection to flutter must be thoroughly investigated to ensure stability at the edge of the flying envelope. If the flutter frequency coincides with the short period the airplane is most likely to encounter high angle of attack and stalling or even worse, structural failure of the wings.

6.2.2 Phugoid The longer period oscillation mode with a period of 20-60 seconds is called phugoid. When the airplane tries to retain a speciﬁc altitude or energy level after a disturbance, there is a slow exchange between kinetic and potential energy which results in a slow oscillation. This mode is easily compensated for by the autopilot or the pilot and does not pose a threat to the general stability of the airplane.

6.3 Pole Placement To determine the stability of the system it is useful to analyze the poles of the system. A general expression for the transfer function can be written as:

K(s − z )(s − z ) + ... + ()s − z G(s) = 1 2 m . (6.4) (s − p1)(s − p2) + ... + ()s − pn The roots of the denominator are the poles and the roots of the nominator are the zeroes of the system. For oscillatory systems, the poles form conjugating pairs: p σ ± iθ = −ζω ± iω 1 − ζ2. (6.5) The location of the poles in the Im and Re plane strongly aﬀects the stability of the system. As Figure 9 below suggest, the system becomes unstable if the real part of the poles is positive. The system becomes oscillatory if the imaginary part is nonzero.

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Figure 9: Eﬀects of pole position on stability

6.4 Flight Control System There are two main types of FCS. One that limits outputs of the pilot to ensure the security of the passengers and one which the pilot cannot live without due to the unstable characteristics of the plane. The first one is often found in modern passenger jets and the latter on is used in fighters where maneuverability is crucial. A common abbreviation is FBW which stands for Fly-By-Wire and simply means that the pilot gives commands to the on board flight computer which takes values from sensors, calculates and sends an optimized signal to the actuators controlling the control surfaces. Nowadays it is required for some planes to also have an active flutter control system to ensure stability and passenger comfort at the edge of the flight envelope and to avoid structural failure.

Lockheed-Martin Co. is currently using their unmanned plane X-56A to carry out tests of an active flutter suppression system for the ability to fly through the flutter region with help computers [37]. The program emphasizes the importance of such a system, especially for flying wing designs. It shows that flutter speeds no longer have to be avoided. Instead slender, elastic and lighter designs may be flown beyond the flutter speed with reduced fuel consumption compared to earlier designs.

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6.4.1 PID controller Probably the most common used control system is called PID controller. P stands for proportional, I for integral and D for derivative. It can be written as:

Z t d u(t) = kpe(t) + ki e(τ)dτ + kd e(t). (6.6) 0 dt The Laplace transform of the transfer function K(s) is written as:

k K(s) = k + i + k s. (6.7) p s d This is easy to implement and a powerful solution for simple linear systems. The problem with PID controller is tuning. It has three gains for each computation, and each one of them needs to be tuned separately. These gains will only work for some unique condition and will need to be tuned again if the conditions are changed. This can be solved by enormous look-up tables for each and different flight condition that the system might encounter. Look-up tables can be very ineffective and require large amounts of memory. However, the PID controller can be optimized with the least-square method in order to make it self-adaptable in order to get rid of look-up tables. If the response of the FCS is too slow to react it might end up out of phase and cause serious stability issues, or if the pilot response is out of phase with the FCS it can cause pilot induced oscillations, which have been proven to be very harmful. Therefore another controller would be preferable that can handle a dynamic and non-linear system.

6.4.2 Full state feedback The state-feedback controller beneﬁts from an already derived state-space model by adding an extra matrix K which is known as the feedback matrix. This alters the control system from being open-loop to closed-loop and therefore increasing the stability if required. [29]

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Figure 10: Full State Feedback control system

The control signal to the actuators can be expressed as:

u(t) = −Kx −→ δ(k) = hdes(k) − Kθ(k) (6.8) which gives the following expression for the enhanced state-space model:

x˙ = (A − BK)x(t) + Bu(t) (6.9)

y(t) = Cx(t). (6.10) K can be optimized in order to make a unstable system stable. One method consists of using the Linear-quadratic algorithm (LQR). The algorithm is designed to minimize the quadratic cost function: Z ∞ J(u) = (xT Qx + uT Ru + 2xT Nu)dt. (6.11) 0 The Q and R matrices are typically chosen with respect to desired settling time tsi and control constraints of the system ximax and uimax and the chosen trade oﬀ constant ρ. N is set to zero.     q1 r1  q2   r2      Q =  .  R = ρ  .       .   .  qn rm where: 1 1 qi = 2 ri = 2 ρ > 0. tsi(ximax) uimax

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Eigenvalues of the new state matrix A − BK can then be found for analyzing the stability of the system according to Figure 11.

Figure 11: Properties of the eigenvalues [41]

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7 Aeroelasticity

Aeroelasticity is the interaction between aerodynamic forces, elastic forces and inertial forces. How these relate to each other is best showed with the modiﬁed aeroelastic triangle of forces.

Figure 12: Interaction between forces and their eﬀects[1]

Aeroelastic effects have been present since the dawn of flight. Early attempts of flying often involved structures that were not very rigid and would therefore encounter flutter and torsional divergence very quickly. Torsional divergence might have been one of the main causes for the popularity of bi-plane during World War I [1]. But as planes developed other effects were discovered. During World War II pilots encountered an effect called control reversal when trying to push their planes into super-sonic speeds leading to pilot confusion and crashing.

Improvements to the structures have been made since. These improvements often involve making the wings more rigid, but also heavier, leading to a penalty in fuel needed. Since the arrival of new composite materials and more advanced FCS, the general aeroelastic theory has been outdated and needs to be comple- mented. This new theory is called servo-aero-viscoelasticity [9] and covers both the time dependency of viscoelastic materials such as composites and the use of servo controls to suppress flutter, for instance. Composite materials have the advantage of having lower density than metals and the ability to become stiffer depending on fiber directions etc.

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The realization and prevention of aeroelastic effects are of especially high importance when designing and building a tailless flying wing since they affect the stability of the already inherent unstable plane.

7.1 Basic Example Consider a one degree of freedom mechanical system shown in Figure 12.

Figure 13: Mechanical spring and damping system with governing equation: ( f(t) − open loop mx¨ + cx˙ + kx = . (7.1) f(t, x, x,˙ x¨) − closed loop

This motion can be used for demonstrating aeroelastic eﬀects for one degree of freedom since it involves inertial, damping and elastic terms on the left hand side and the right hand side could include forces such as aerodynamic forces. The system is referred to as a closed loop system if there is any dependency of displacement, velocity or acceleration of the structure on the right hand side. For example aerodynamic forces are heavily dependent on the angle of attack which depends on the velocity in the left hand side.

The inclusion of a force on the right hand side might convert a stable system into an unstable system depending on phase and size of the force. For multiple degrees of freedom with coupling, numerical solutions are often used since it can be diﬃcult to ﬁnd analytical solutions.

7.2 Static Aeroelastic Effects Static aeroelastic effects can be described as deformations of the structure due to aerodynamic loads and are independent of time and inertia. This thesis will put emphasis mainly on dynamic aeroelastic effects, but, since the static effects are just as important, two common effects will be discussed. These are torsional divergence and control reversal.

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7.2.1 Torsional Divergence Torsional divergence, or simply divergence, occurs as the aerodynamic moments overcome the structural stiffness of a wing. The result is a twisting wing that eventually breaks. If flutter is suppressed by some controller then divergence might occur at the same velocity as flutter, suddenly and without warning. It is typically preferable to have both effects at the same speed and therefore it is important to analyze the effect in order to determine what speeds to expect.

Consider a twisting airfoil mounted on a spring at a distance ec from center of lift and with spring constant Kθ. Where θ is the deviation in pitch from its starting position θ0.

Figure 14: Divergence

The moment around the ﬂexural axis is deﬁned as: 1 M = Lec = ρV 2ca (θ + θ) ec = qec2a (θ + theta). (7.2) 2 w 0 w 0

The energy stored in the spring is: 1 U = K θ2. (7.3) 2 θ The generalized force in θ becomes:

2 Qθ = qec aw(θ0 + θ) = Kθθ. (7.4)

Solving for θ yields: 2 qec awθ0 θ = 2 . (7.5) Kθ − qec aw This equation for the elastic angle of twist becomes inﬁnite as q approaches Kθ 2 therefore the divergence speed can be solved from the expression for ec aw qdiv: s 2Kθ Vdiv = 2 . (7.6) ρawec s

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For a flying wing design, the spring is connected to a lump mass which represents inertia around the center of gravity and the flight mechanics limiting the twisting motion. When torsional divergence occurs, the flying wing will pitch with increasing angle of attack, leading to stall, unless the inertia limits its movement. Only then will the wings break as for a conventional airplane.

Finite Wing For more accurate results one must consider a finite wing. To solve the finite wing problem it is convenient to apply strip theory[5]. Strip theory suggests the following: dT T + dy − T + ∆Lec + ∆M = 0 (7.7) dy 0 rewriting gives: dT dL dM + ec + 0 = 0 (7.8) dy dy dy where torque and the incremental change in lift and moment are given by: dθ T = GJ ∆L = qcδ C (α + θ) ∆M = qc2δ C . dy z Lα 0 y M0 The equation then becomes: d2θ qec2C θ −qec2C α qc2C + Lα = Lα − M0 . (7.9) dy2 GJ GJ GJ The angle θ can then be solved by integrating twice: C θ = Asinλy + Bcosλy − M0 + α . (7.10) eCLα dθ By using boundary conditions that θ = 0 at y = 0 and = 0 at y=s, the dy constants A and B becomes: C C A = M0 + α tanλ B = M0 + α eCLα eCLα where: qec2C λ2 = Lα . GJ Effects of sweep Increasing angle of sweep Λ increases the divergence speed according to Equation 7.11 [2]: s 2KθKh Vdiv = 2 2 2 2 2 . (7.11) ρaw[Khsc cos (Λ)/4 − Kθ(cs tan(Λ)/2 + c sin (Λ)/4)]

It shows that the divergence speed decreases for forward swept wing. Therefore planes with forward swept wings, like the X-29, had to use the advantages of composite materials and aeroelastic tailoring to counter the twisting moment of the wing [32]. Aeroelastic tailoring means that the direction of the ﬁbers have been used to tailor the material for desired properties. In this case, increased stiﬀness in torsion.

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7.2.2 Static aeroelastic effects on stability Trim As the wing is twisting as it approaches divergence speed the trim angle needed for level flight increases negatively. This might become critical as the FCS tries to counter the flutter, the plane will eventually run out of trim and effectively run out of control authority to counter the flutter leading to a condition where flutter no longer can be suppressed. Flutter encountered close to divergence speed can be considered as hard flutter, a much more serious condition than the initial soft flutter that would occur at lower speeds if its not being controlled by the FCS.

Longitudinal stability Since the static aeroelastic effects causes twist of the wing, the static stability becomes affected. Following analytic results were given in Bisplinghoff et al. (1955):

dC 2 2q −1 L = b1c[W ] [As] − q[E] + [G]{mg}b1c[W ] {1} (7.12) dα S W

dC 2tanΛ 2q −1 M = − byc[W ] [As] − q[E] + [G]{mg}blc[W ] {1}. (7.13) dα S(MAC) W dC dC These equations imply that as the wing twists, L decreases and M in- dα dα creases, which aﬀects the stability negatively. This occurs due to the forward movement of the center of pressure as the wing twists.

7.2.3 Control reversal As an elastic wing differentially deflects its aileron it changes the lift on either side and creates a rolling motion. For a flying wing, the aileron and elevator are usually combined into what is called elevons. What also happens is that the wing will twist due to the deflection and increasing roll rate. This leads to less responsive controls and finally, after reaching a certain speed, the response will be reversed. For explanation purposes the wing is assumed to be linearly flexible in twist using the expression: y θ = θ (7.14) s 0 where θ0 is the angle of incidence at the tip of the wing.

Lift and moment coeﬃcients can be expressed as:

CL = CL0 + CLα(θ0 + θ) + CLδδ (7.15) CM = CM0 + CMα(θ0 + θ) + CMδδ where CL0 and CM0 are lift and moment coeﬃcients at zero degrees of angle of attack and CLα and CMα are lift and moment curves for wing while CLδ and CMδ are lift and moment curves for the elevon deﬂection.

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The equations for lift and moment for an elevon deﬂection for quasi-steady aerodynamics are:

y θy˙ dL = qcdy C θ + θ − + C δ Lα 0 s 0 V Lδ (7.16) y θy˙ dM = qc2dy C θ + θ − + C δ . Mα 0 s 0 V Mδ

The work done by the wing for incremental roll and twist angles (δφ and δθ) is defined as: Z δW = (dLyδφ + dMδθ). (7.17) wing By using generalized coordinates to define the generalized forces and using the strain energy it is possible to apply Lagrange’s equation and solve the situation where the roll rate φ˙ due to elevon deflection δ equals zero. This gives the reversal speed qrev. Following these steps leads to an expression for the roll velocity p per elevon angle δ and the reversal speed qrev: p 3V [C (µ − e) + C ] = Lδ Mδ (7.18) δ 2µsCLα

3GJCLδ qrev = 2 2 (7.19) c s CLα(eCLδ − CMδ) where µ is: 3GJ µ = 2 2 . qc s CLα Entire aircraft model This approach is better suited for the industry due to the inclusion of the rigid body motion of the aircraft. This approach yields that there is a constraining moment M such that the roll rate becomes zero. This moment can be found by solving the following equations [2]:

 2  −2qsc CLα 0 φ 1 qcs2C  3  + M = Lδ = R (7.20)  2GJ 2qc2sC  θ 0 qc2sC δ con 0 − Mα T Mδ s 3 which results in: 2 CMδ (M/δ)flexible = qcs 2 CLα + CLδ . (7.21) 3GJ/s − qc sCMα

This can be compared to the rigid response by letting G → ∞. The control eﬀectiveness is now easily calculated by dividing (M/δ)flexible by (M/δ)rigid.

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7.3 Flutter The main consideration of this thesis is the dynamic aeroelastic effect known as flutter. Mainly because of the effects on stability but also since flutter can be suppressed by a flight control system. Flutter arises due to coupling between the inertia, the elastic modes of the wing structure and the aerodynamic flow. Flutter includes two or more modes of motion and the flutter velocity is defined as the lowest air speed for which the structure will be able to maintain a simple harmonic motion, i.e the damping becomes positive. By integrating over the span of the wing with, for instance, Galerkin’s method and thus eliminating the span effect, one will end up with a solution that reassembles a rigid wing on a flexible support. This can be done in order to lower the degrees of freedom without loosing any vital information. This leads tpo faster calculations and more time for analysis. Therefore, a two degrees of freedom model will be used to demonstrate flutter in this thesis.

General considerations Consider a rigid airfoil connected to two springs, one bending and one torsional with stiﬀnesses Kh and Kθ.

Figure 15: Flutter 2DOF

The motion in h and θ can be found to be oscillations with frequency ω.

n (φ+iω)to ˙ n (φ+iω)to h = Re h0e , h = Re (φ + iω)h0e

n (φ+iω)to ˙ n (φ+iω)to θ = Re θ0e , θ = Re (φ + iω)θ0e where h0 and θ0 are complex numbers and the phase relation φ governs the ﬂutter condition. Eigenvalue solutions have to be used to determine the ﬂutter velocity.

The ﬂutter condition states the following:

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 < 0 − stable  φ = 0 − neutrally stable . (7.22) > 0 − unstable To find the flutter velocity, state the governing equations for bending and torsion using the unsteady equations for lift and moment and Newtons second law. Now, for this system, the solution is found by solving the determinant for the equation: a b h 0 = . (7.23) c d θ 0 This approach can be adapted into multiple degrees of freedom and includes the displacements and first and second derivatives of the displacements. For 2 DOF, uncoupling of the motions is possible by using Laplace transformation. A more general approach is to use the eigenvalue solution to analyze frequency and phase relations and thereby find the theoretical flutter velocity. If the real part of any of the eigenvalues is positive, it indicates that some mode has caused the system to flutter, i.e. the damping is non-negative. Flutter speed is then found by increasing the velocity until the damping of the system becomes positive.

The general form of second-order N DoF aeroelastic equations with general coordinates q can be written as following:

A ¨q + (ρV B + D) ˙q + (ρV 2C + E) q = 0. (7.24) |{z} | {z } | {z } Inertia Damping Stiffness This equation can be solved as an eigenvalue problem. The solution produces frequencies and damping ratios for speciﬁc ﬂight conditions. Numerical solutions are preferable for more complex systems with higher degrees of freedom.

7.4 Binary Aeroelastic Model From the general principles of flutter, it is possible to consider a more detailed 2 DOF aeroelasic model that can be used for flutter prediction. Classic binary flutter suggest that the wing is unswept and rectangular with span s and chord c, uniform mass distribution and rigid on a flexible support with rigidity EI for plunge and GJ pitch. Spanwise distributions of lift and stiffness are assumed to be constant. The stiffness is uncoupled between the two motions and the elastic axis appears at a distance ec from the aerodynamic center.

The displacement plunge h using general coordinates qb for bending/plunge and qt for torsion/twist is given by: y 2 y h(x, y, t) = q (t) + (x − x )q (t) (7.25) s b s f t and for twist: y θ(y, t) = q (t). (7.26) s t

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Lagrange’s equations can be used to ﬁnd the equations of motion. The kinetic energy exist due to the dynamic motion and the potential energy consist purely of the strain energy of the bending and torsion motion.

Z Z s Z c 2 2 1 2 m y y T = dmz˙ = q˙b + (x − xf )q ˙t dxdy (7.27) wing 2 2 0 0 s s

Z s 2 Z s 2 1 2qb 1 qt U = EI 2 dy + GJ dy. (7.28) 2 0 s 2 0 s Applying Lagrange’s equations:

dT dT sc s c2 = m q¨b + − xf c q¨t (7.29) dt dq˙b 5 4 2

2 3 dT dT s c s c 2 2 = m − cxf q¨b + − c xf + xf c q¨t (7.30) dt dq˙t 4 2 3 3 dU 4EI dU GJ = 3 qb = qt. (7.31) dqb s dqt s The equations without any aerodynamic forces in matrix form then becomes:   msc ms c2 5 4 2 − csf 4EI   q¨b 3 0 qb 0   + s = . 2 3 GJ  ms c ms c 2 2  q¨t 0 s qt 0 4 2 − cxf 3 3 − c xf + cxf (7.32) Before the aerodynamic forces can be included in the model, they need to be transferred into generalized forces by combining the derivatives of the lift and moment equations into the incremental work on the wing for small incremental forces δqb and δqt.

Z y 2 y δW = dL − δqb + dM δqt . (7.33) wing s s Now the ﬁnal step is to include the unsteady aeroelastic forces in the shape of generalized coordinates in order to receive the full aeroelastic equations of motion for a binary aeroelastic model. The forces in general coordinates becomes:

∂(δW ) Qqb = = (7.34) ∂(δqb) " 2 2 2 # 2 s 1 c Lz˙ b s 1 c Lθb Lθ˙b −ρV Lz q + + cx qt + q˙ + + cx q˙t + qt + q˙t 2 b 3 2 f V 2 b 3 2 f 3 3V

∂(δW ) Qqt = = (7.35) ∂(δqt) " 2 2 2 3 # 2 2 1 Mz˙ b 2 1 c Mθb Mθ˙b ρV Mzb q + (x − x )qt + q˙ + + cx q˙t + qt + q˙t . 3 b 2 f V 3 b 2 2 f 2 2V

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The resulting aeroelastic equations become:

  msc ms c2 5 4 2 − csf   q¨b   + (7.36)  ms c2 ms c3 2 2  q¨t 4 2 − cxf 3 3 − c xf + cxf

 2  − 1 L bs − 1 L b c + cx + L b2 2 z˙ 3 z˙ 2 f θ˙ q˙ ρV   b +  2 2 1 2 c2 3 q˙t 3 Mz˙ b 2 Mz˙ b 2 + cxf + Mθ˙b

 2  ( 1 1 c 2 ) − 2 Lz˙ s − 3 (Lz 2 + cxf + Lθb ) 4EI 2 s3 0 qb 0 ρV   + GJ = .  2 M  q 0 2 1 c θb3 0 s t 3 Mzb 2 Mzb 2 + cxf + V

It is the coupling of the aerodynamic forces due to the nonsymmetric matrices that can produce ﬂutter.

7.4.1 Structural damping It has been shown that a small amount of structural damping in some configurations can have a positive effect by increasing the flutter velocity [25]. It can also have a negative effect depending on the phase relations. Structural damping occurs due to forces between the independent elements of the structure, such as friction in joints, etc. A common description of the structural damping is done by combining it with the structural stiffness by adding the term:

E∗ = E(1 + ig) (7.37) where g is not to be mixed up with gravity but is a structural damping coef- ﬁcient. This method requires a rewriting since it involves a complex number in the time domain. The solution will only consist of the real part which is included in the elastic modulus.

Another way of describing the structural damping of the system is by adding a matrix D which is dependent of the mass and stiﬀness matrices A and E.

D = αA + βE. (7.38)

This damping model is called proportional damping or Rayleigh damping where α and β are Rayleigh coeﬃcients. To get α and β for some assumed frequencies ωa and ωb and damping ratios ζa and ζb, they have to be deﬁned according to NAFEMS, 1987 as follows:

2ωaωb(ζaωb − ζbωa) α = 2 2 (7.39) ωaωb

2(ζaωa − ζbωb) β = 2 2 . (7.40) ωb − ωa

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7.4.2 Moving ﬂexural and mass axis For examining diﬀerent CG locations, the following procedure can be applied. Consider a strip mass M per unit length mounted at the leading edge. Mass axis is now given by: mc2 X = . (7.41) cm 2(mc + M) The position of mass axis is now easily varied by changing the value if M. This results in a new expression for the kinetic energy which includes one new term:

m Z s Z c y 2 y 2 M Z s y 2 y 2 T = q˙b + (x − xf )q ˙t dxdy+ q˙b − xf q˙t dy. 2 0 0 s s 2 0 s s (7.42)

7.5 Entire Aircraft Model It is often preferred in the industry to model the entire aircraft instead of just its wings in order to allow for movement of the fuselage. The simplest way of doing this is to add a rigid lump mass M at the elastic axis of the aeroelastic model. This makes it possible to estimate the flutter velocity of the whole aircraft. To accomplish that (at least) three different modes of motion must be considered. These are the bending and twisting of the wing and also the heave of the entire aircraft and the coupling between those motions. This approach works equally well for a flying wing design since the fuselage is considered rigid and the mass can be divided between the wing and the body

The displacement h downwards remains the same with the addition of the body heave displacement qh: y 2 y h = q + (x − x )q + q . (7.43) s b s f t h The twist will remains as: y θ = q . (7.44) s t Total kinetic energy can be written:

Z Z s Z c 2 2 1 ˙ 2 1 2 1 y y T = dmh = Mq˙h+ m q˙b+ (x−xf )q ˙t+q ˙h dxdy. wing 2 2 2 0 0 s s (7.45) Applying Lagrange’s equation using the simpliﬁed unsteady aerodynamics equations (3.18 and 3.20) results in the following aeroelastic equation for the entire aircraft assuming a rigid center body:

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 2  m sc m s c − cx m sc  5 4 2 f 3        q¨b  s c2 s c3 2 2 s c2  m − cxf m − c xf + cx m − cxf  q¨t  +  4 2 3 3 f 2 2    q¨h    sc s c2  m 3 m 2 2 − cxf msc + M

 cs cs   4EI 2 cs  aw 0 aw 3 ρV aw 0 10 6   s 8     q˙b   qb  c2s c3s c2s   GJ 2 c2s  +ρV − bw − M ˙ − bw q˙t  +  0 − ρV bw 0 qt  = 0.  8 24 θ 4   s 6    q˙h   qh cs cs 2 cs 6 aw 0 2 aw 0 ρV 4 aw 0 (7.46)

7.6 Aeroelastic Model for Simulation While the binary model of the wing only has 2 DOF in plunge and pitch and the entire aircraft model included a lump mass. This model will consider springs for heave, pitch and flap motion and will include more details than previous models. The model has been used by NASA for modeling of the Benchmark Active Control Technology (BACT) Flutter Suppression System at Langley Re- search Center, Hampton VA and has been proven to comply well with the real model[18]. A Comparison between the results of the different approaches can then be made to draw a conclusion about the performance of the different systems.

Using a 3 DOF model concerning only plunge, pitch and flap movement makes simulations and calculation much easier instead of handling the many individual degrees of freedom of each span wise part of the wing. Therefore a 3 DOF model is chosen to reassemble the Flying Wing model analyzed in the simulation part. The wing will still be considered rigid with uniform mass distribution on a flexible support. To account for sweep angle and finite wing the aerodynamics of the 3D model are analyzed separately. The aerodynamic derivatives from the analysis are then used to compute the lift and moment coefficients. This approach was shown to work well with the real model at NASA Langley [18]. The figure below illustrates the system:

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Figure 16: Illustration of the 3 DOF wing-ﬂap ﬂutter model [18]

The problem can be solved as previous using Lagrange’s equations. First construct the three generalized coordinates: • Pitch angle: θ • Plunge displacement:h

• Trailing edge control surface angle: δTE = δ The angle of attack will be of big importance and it is deﬁned in this case using the general coordinates as follows (assuming small angles): ˙ ˙ h(t) l(x)θ(t) wg(x, y, t) α(x, y, t) = α0 + θ0 + θ(t) + + − . (7.47) U0 U0 U0

Here α0 is the angle for lift to be equal the weight, θ0 is the trim angle at which the airplane ﬂy, U0 is the freestream velocity, wg is the normal perturbation velocity caused by turbulence and gusts and l(x) is the distance from the origin of body ﬁxed coordinates system to the angle of attack reference where x is positive aft.

Now the kinetic and potential energies need to be deﬁned. The kinetic energy will consist of a heave component and a rotational component for both the wing and the ﬂap. The potential energy consists of a gravity part and two spring parts.

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Kinetic energy 1 1 1 1 T = m (h˙ + e θ˙)2 + I θ˙ + m (h˙ + e θ˙ + e δ˙)2 + I (θ˙ + θ˙)2. (7.48) 2 1 1 2 1 2 2 2 δ 2 2 Potential energy Addition due to gravity:

Ug = −m1g(h + e1sinθ)cosθ0 − m2g(h + e2sinθ + eδsinθ)cosθ0. (7.49) Addition due to strain in springs: 1 1 1 U = K h2 + K θ2 + K (δ − δ)2. (7.50) e 2 h 2 θ 2 δ c Now Lagrange equations can be applied as in previous example but using following formulation: d dT dT dU − + = Qqi (7.51) dt dq˙i dqi dqi where qi are the generalized coordinates and Qqi are generalized forces related to qi. These consists of external forces, nonconservative forces, etc. The result are the following matrix equations:

  ¨     m Shθ Shδ h Kh 0 0 h ¨ Shθ Iθ Sθδ θ +  0 Kθ 0  θ = Shδ Sθδ Iδ δ¨ 0 0 Kδ δ (7.52)       0 m QH =  0  δc + Shθcosθ gcosθ0 +  Qθ  Kδ Shδcosδ Qδ where the generalized parameters are:

m ≡ m1 + m2 2 2 Iθ ≡ I1 + I2 + m1e1 + m2e2 2 Iδ ≡ I2 + m2e2 and the coupling parameters are:

Shθ ≡ m1e1 + m2e2

Shδ ≡ m2eδ

Sθδ ≡ I2 + m2e2eδ. The deformation due to hinge load can be assumed to be insignificant since the control surface stiffness becomes large, leading to some simplifications by eliminating the term Kδ which gives that δ = δc.

m S θ h¨ K 0 h h + h = Shθ Iθ θ¨ 0 Kθ θ (7.53) Shδ ¨ m QH − δc + gcosθ0 + . Sθδ Shθ Qθ

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This leaves only generalized forces QH and Qθ to be determined and to do that the principle of virtual work will be used.

The principle of virtual work done by the system for an inﬁnitesimal displacement is deﬁned as:

∂(δW ) Qqi = . (7.54) ∂(δqi) Aerodynamic forces Qh ≡ −L (Lift) Qθ ≡ Mp (Moment around point Xp).

By expressing the lift coeﬃcient and moment coeﬃcient in terms of aerodynamic derivatives the equations used for simulation become:

c ˙ ˙ L = qSCL = qS[CL0 + CLαα + CLδδ + (CLα˙ α˙ + CLqθ + CLδ˙δ)] (7.55) 2U0 c ˙ ˙ Mp = qScCM = qS[CM0+CMαα+CMδδ+ (CMα˙ α˙ +CMqθ+CMδ˙δ)]. (7.56) 2U0 Non-conservative forces (viscous damping)

nc Qh m Shθ 2ζhωh 0 nc = − (7.57) Qθ Shθ Iθ 0 2ζθωθ where ζh and ζθ has to be obtained experimentally and ωh and ωθ are the vibration frequencies.

The equations for the generalized forces can now be obtained by including the expression for angle of attack. The full aeroelastic model for the system is then obtained by combining equations 7.53 and 7.55-7.57. The result can be expressed in this general form:

e ˙ (Ms−Ma)q¨+(Ds−Da)q˙ +(Ks−Ka)q = Q0+QT θ0+Mggcosθ0+B1δ+B0δ+Ew (7.58) with q as the general coordinates h and θ.

Full expansion of this expression is presented in the appendix. For small systems such as these, software such as Simulink can be used with advantage.

7.7 Effects of Swept Wing and Low Aspect Ratio By decreasing the aspect ratio and increasing the sweep angle Λ flutter speeds generally increase [1]. This is because the normal speed of the wing decreases with the cosine of the angle of sweep and it is the speed which affects the flutter characteristics. The equations for the aerodynamic forces such as lift and moment are simply adapted by multiplying by factor cosΛ [27]. For better precision, the lift and moment coefficients will be obtained from Xfoil simulations of a three dimensional model of the airplane.

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A decrease in aspect ratio means that the wing form can be reassembled more like a plate instead of a slender beam, which has more modes of oscillation due to elasticity in two directions. Therefore there is no general conclusion to be drawn whether an increase in aspect ratio favors ﬂutter or not without a detailed analysis of the wing.

7.8 Viscoelasticity In the age of longer, slender, more flexible and efficient wings made possible by the entering of composite materials, it is important to consider viscoelastic effects as well. New composite materials makes it possible to tailor the structures for optimum strength and stiffness where it is needed the most. But viscoelastic materials such as composites require an extended theory of aerovisocelasticity. A theory that includes time and temperature dependency and take the lay- up and the direction of the fibers into account. This theory considers that each flight will have different conditions of the viscoelastic material, since viscoelastic material have a memory and will have different properties depending on the climate and will creep with respect to time [10].

7.9 Maneuvers For maneuvers to be executed a change in velocity is required. For all previous models derived, maneuvers can be added by using the resultant time dependent ∂V (t) velocity V (t) = V (t) + . This will require re-deriving the governing R ∂t equations for lift and moment. Modiﬁcations of Theodorsen’s function is also required since the reduced frequency concept was mainly deﬁned for constant velocities [13].

7.10 Turbulence and Gust Response One of the things that can trigger ﬂutter is turbulence because ﬂutter might occur if the aircraft is disturbed from its equilibrium position. Therefore, gusts and turbulence need to be included in the model.

7.10.1 Sharp edged gust The easiest way to model the gust is as a step function that goes from zero to ωg0 instantaneous and effectively changing the angle of attack of the aircraft and disturbing the aircraft equilibrium. For a stable system, the aircraft should be able to find its equilibrium position within a required time called the settling time. If the aircraft is inherently unstable, it will not be able to find its equilibrium position by itself but will need help from the FCS. If the FCS is designed to suppress flutter, the gust may also trigger flutter if the airplane flies at or above the structural flutter velocity. This gust model is very easy to implement and give a hint on the stability of the system, but the realism of the model is questionable. There are better models for more realism.

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7.10.2 1-cosine gust To analyze a more realistic gust response, a 1-cosine gust is often used with strength ωg0 at position Xg. The aircraft will be considered rigid for simplicity. The upward velocity due to the gust ωg is expressed as: ωg0 2πXg ωg(Xg) = 1 − cos . (7.59) 2 lg

If the position is written as Xg = V t the equation becomes: ωg0 2πV ωg(t) = 1 − cos t . (7.60) 2 lg

Now the equation depends on the true airspeed (TAS) and time.

For conventional airplanes, gust penetration effects will have bigger impact as the gust will propagate backwards and affect the main wings at a different time than for the tailplane. But since the length of the flying wing is much shorter, the gust will be assumed to affect the whole plane at the same time.

The instant increase in angle of incidence due to the gust is written as:

ω + Z˙ ∆a ≈= g c (7.61) g V leading to a expression for the increase in lift: 1 ∆L = qS a∆a = ρV S a(ω + Z˙ ). (7.62) w g 2 w g c A more realistic approach would involve unsteady aerodynamics and to let the gust and the lift force build up over time and not be instantaneous. To achieve this a Kussner¨ 0s function can be used to allow the gust to build up. The equiva- lent function for lift can be expressed with Wagner’s function (see Aerodynamic Forces).

7.10.3 Turbulence Modeling For turbulence modeling in Simulink, the Dryden model was chosen. It deﬁnes each wind component as a power spectral density (PSD) which are varying as random processes. The PSD functions for each component are deﬁned as:

2 2Lu 1 Φug(Ω) = σu 2 (7.63) π 1 + (LuΩ)

2 2 2Lv 1 + 12(LvΩ) Φvg(Ω) = σv 2 2 (7.64) π 1 + 4(LvΩ) ) 2 2 2Lw 1 + 12(LwΩ) Φwg(Ω) = σw 2 2 (7.65) π 1 + 4(LwΩ) ) where σ is the standard deviation of the turbulence Ω is the wave number and L is the turbulence scale length. Subscripts deﬁne the direction of each component.

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7.11 Aero-servo-elasticity This section demonstrates the mathematical modeling when a control system is included in the aeroelastic model. The control system will help suppress aeroelastic effects by controlling the elevons and their deflection, which is denoted by δ. This can be done up to a point were controls become ineffective due to static effects. A single derivative servo has been chosen for the analysis, but there are more general versions with multiple derivatives and integral available. The simplified equations for lift and moment with the addition of control surface deflection become: 1 y2 y dL = ρV 2cdy a q˙ + q + a δ (7.66) 2 w s2V b s t c 1 y2 y cy˙ dM = ρV 2c2dy ea q˙ + q + M c q˙ + b δ (7.67) 2 w s2V b s t θ˙ 4sV t c where ac and bc are the corresponding lift and moment curves per elevon deflection angle δ.

Now consider the equations derived from the binary aeroelastic model in equation [7.36] expressed in its general form, but adding the contribution of servo control and gust response. The results becomes:  csac   csaw  − − 2 2 A¨q + ρV B ˙q + (ρV C + E)q = ρV cs s 6 δ + ρV s 6 ω . (7.68)  c 2bc  c 2eaw  g | {z } Binary aeroelastic model 4 4 | {z } | {z } g h By adding sensors that give feedback, the control system becomes a closed-loop. This is not to be mixed-up with the aeroelastic model which is already a closed- loop due to the interaction between the aerodynamics and the structure. For a full-state feedback, including the integral of h and θ for elimination of the steady-state error, the command signal sent to the elevons δcom becomes:  h   h˙     θ  δcom = K •   (7.69)  θ˙  R   hdt R θdt where K is the optimal gain matrix found using the Linear Quadratic Regulator (LQR) algorithm (see Stability).

7.11.1 State-space modeling To analyze the stability of the system and to be able to apply optimal control theory, a state-space model will be required. The state-space model for an aero-servo-elastic system will have the following appearance: q¨ 0 I q˙ 0 0 = + δ + ω q˙ −A−1(ρV 2C + E) −A−1(ρV 2B) q A−1g A−1h g (7.70)

48 Master Thesis where the matrices A,B,C, E are defined differently for different aeroelastic models and I is the identity matrix.

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8 Analysis

The analysis is an iterative process that follows the simplified flow chart below. For the thesis to be able to focus on the aeroelastic effects, the design process was shortened and only serves as an example.

Figure 17: Flow chart

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MATLAB and Simulink were the main tools for analyzing the aeroelastic effects. Xfoil [31] analysis was made with XFLR 5 to show the efficiency of the chosen design by calculating lift,drag and moment coefficients for the chosen airfoil at different configurations.

8.1 Efficient Design XFLR 5 XFLR 5 is a free tool that can be used to draw and create 2D airfoils with infinite number of variations. XFLR 5 then lets the user define different testing conditions to establish pressure distribution and lift, moment and drag coefficients. The calculations are based on the program Xfoil which is designed for fast estimations of airfoils at low Reynolds number[31]. The airfoil used is MH62 at Reynolds number varying between 800,000 and 1,500,000 and angle of attack varying from -5 to 20 degrees. For more detailed analysis of the design, XFLR 5 allows the user to build a 3D model of the airplane and test both efficiency and stability.

Initially a 2D analysis was conducted to find the optimal flap setting. Later a 3D analysis was used to find more realistic coefficients and derivatives for different angles of attack and flap settings. The coefficients were taken in the linear region, well before stall occurred. The 3D analysis could be either viscous or inviscid. Obviously the viscous approach should be more accurate, but this requires values from 2D analysis to be interpolated and due to limitations of XFLR 5 all values could not be interpolated. Therefore, the inviscid approach is used as a complement to the viscous. The resulting coefficients could then be used in the Simulink model for flutter analysis.

Figure 18: Flying wing 3D model in XFLR 5 with ﬂaps down conﬁguration

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Figure 19: Flying wing 3D model in XFLR 5 with ﬂaps down morphed conﬁg- uration

8.2 Static Aeroelastic Effects The divergence and control reversal speeds were found analytically (see Aeroe- lasticity) using MATLAB. These effects can be illustrated by plotting the twist angle of the wing and the control effectiveness versus speed.

8.3 Flutter Prediction An analytic approach to determine the flutter velocity is analyzed. The calculations are described in the aeroelastic part. These are implemented in MATLAB to plot the corresponding natural frequencies and damping ratios for the wing twist and plunge for different velocities. By analyzing the plots, flutter occurs as the damping ratio reaches zero and eventually becomes positive and the system becomes unstable.

Brief description of method: - Initialize variables

- Define the inertia matrix A - Define the structural stiffness matrix E ∗ 0 ∗ - Calculate damping matrix C = ρV ∗ ∗ ∗ ∗ 0 ∗

∗ ∗ 0 - Calculate stiﬀness matrix K = ρV 2 0 ∗ 0 + E 0 ∗ 0

zeros(3) eye(3) - Create Eigen matrix = −A \ K −A \ C

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- Calculate the ﬁrst order Eigenvalues λi=1,..,6

- Phase = Re(λi)

p 2 2 - Natural Frequency = Re(λi) + Im(λi)

p 2 2 - Damping ratio = 100Re(λi)/ Re(λi) + Im(λi) - Add to plot 3D Plots Same procedure as previously, except that two parameters where varied, for example speed and altitude. This produced a results matrix which could be plotted in three dimensions.

8.4 Simulation The 2 DOF Simulink model used to simulate flutter is based on an active flutter suppression system developed at NASA Langley Research Center in Hampton, Virginia. It will be used to determine whether it will be possible to suppress flutter effects with the help of computers on a relaxed stability designed flying wing in order to increase the stability close to the flutter region and to improve its top speed. The system is illustrated in fig(19).

Figure 20: Flying Wing Flutter suppression system

The model simulates aeroelastic eﬀects for chosen altitude and Mach number and plots the plunge, pitch and elevon movement in three separate graphs. These graphs are easily analyzed to determine the ﬂutter velocity for the system. For illustrative purpose the model also includes a 3D model of a wing with

53 Master Thesis aileron that animates the response of the system.

Gust Requirement To satisfy the gust requirement set by the FAA, the Dryden turbulence model can be added to the system. This model uses power spectral densities for deﬁn- ing the continuous gusts. The turbulence model requires altitude, speed and the directional cosine matrix as input and creates a velocity vector and an acceleration vector that can be added to the model to simulate turbulence.

Optimal Control In order to ﬁnd the optimum control gain K used by the LQR controller, the Simulink model had to be linearized and analyzed in MATLAB outside of Simulink. Unnecessary states had to be removed and the control gain was found with the LQR command. By changing the input matrices according to the preferred settling time and overshoot, an optimal gain matrix could be found. This gain matrix was then used in Simulink to give the smoothest possible response by the servo control.

Climb and Acceleration To simulate the flight performance throughout the intended flight envelope, a climb and acceleration profile was used as input. The profile started at sea level and at a velocity of around 15 m/s and ended at 3048 m (10,000 feet) and 130 m/s. For more realism, turbulence and regular gusts at 5 m/s was used. The controller was set to achieve the smoothest response at 3048 m (10,000 feet) and at a speed of 55 m/s. This controller showed promising performance up to when reversal speed was met and reached the highest flutter velocity of around 59 m/s. For higher speeds a new controller is needed.

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9 Results 9.1 Efficient Design The section will show results of simulations done in XFLR 5 for different flap configuration to establish the most efficient design option. All 3D simulations are carried out using 5 degrees of twist at the end of the wings for washout.

9.1.1 Airfoil Analysis As Figure 4 suggest, an aft center of gravity needs the flap to be in down position to counteract the moment around the aerodynamic center. For greatest stability the center of gravity should be placed forward of the aerodynamic center. This requires the flap to be in upward position. Following figure shows the lift to drag ratio for these configurations at Reynolds number varying from 500,000 to 2,500,000. Flap variations range from -10 to 10 degrees, where downward position is defined as positive.

Figure 21: Lift to drag ratio for different flap settings and different Reynolds number

Figure 21 shows that the maximum L/D ratio is achieved for the flap in a 10 degrees down position in the 2D airfoil case. This confirms that efficiency is somewhat increased if the flap is in down position.

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9.1.2 3D model with hinged elevon For more precision, a 3D model was analyzed instead of just a 2D airfoil. For maximum efficiency the L/D ratio of the 3D model should be as high as possible. The figures 22-25 consist of four different plots showing lift, drag, moment coefficients and L/D versus angle of attack for different flap configurations, considering different flap designs and both viscous and inviscid flow. Top left plot shows the drag coefficient, top right shows the lift coefficient, bottom left shows the moment coefficients and bottom right shows the lift over drag coefficient. The most interesting figure for efficiency is the L/D ratio in the bottom right corner. The lift and moment coefficients were taken from the linear region at small angles of attack from these plots and put into Matlab and Simulink for aeroelastic calculations.

Figure 22: Inviscid 3D analysis at diﬀerent ﬂap settings hinged elevon

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Figure 23: Viscous 3D analysis at diﬀerent ﬂap settings for hinged elevon

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9.1.3 3D model with morphed elevon

Figure 24: Inviscid 3D analysis at diﬀerent ﬂap settings for morphed elevon

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Figure 25: Viscous 3D analysis at diﬀerent ﬂap settings for morphed elevon

As expected, the viscous plots give higher (more realistic) values of the drag and therefore give much smaller value on the L/D graphs. The results show that the morphed elevon is able to produce higher lift per angle of attack than the hinged one. Otherwise, there is only a small improvement by using morphed elevon instead of the hinged one. The relatively low aspect ratio makes the wings more rigid and allows higher speeds but on the other hand, a low aspect ratio implies more drag and less efficiency. Taken this into account, a taper ratio close to 0.4 was chosen to represent an elliptic lift distribution to minimize the induced drag, even if the aspect ratio is low. For future research, wingtip devices could be added to the model to evaluate how they effect both efficiency and aeroelastic behavior.

9.2 Aeroelastic Effects Most of the calculations were conducted at a presumed altitude of 10,000 feet. For illustrative purposes, a presentation of both static and dynamic aeroelastic effects is shown in Figure 26. All effects are presented as percentage as the air speed increases. Divergence is displayed as the wing twist divided by the maximum twist angle, control effectiveness is displayed as the ratio between a flexible and rigid wing and for flutter, the damping ratio gives the best representation. All speeds are in m/s.

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Figure 26: Aeroelastic response due to airspeed

Figure 26 indicates that for this case, ﬂutter will occur at 56 m/s, divergence at around 90 m/s and reversal at 63 m/s.

9.3 Flutter Prediction For ﬂutter prediction purposes two results are shown for each mode of oscillation, plunge and pitch. One is the natural frequency, the other one is the damping ratio, which is related to the phase relation φ. Flutter is known to occur as the damping ratio of any of the modes becomes positive. Figures 27 and 28 compare the quasi-steady aerodynamic model with the unsteady aerodynamic model for the binary aeroelastic model respective the entire aircraft aeroelastic model. For the entire aircraft model, the body heave motion is also displayed.

9.3.1 Binary aeroelastic model

Figure 27: Flutter prediction of binary model

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9.3.2 Entire aircraft aeroelastic model

Figure 28: Flutter prediction of entire aircraft model

9.3.3 3D Plots of entire aircraft model Here are results from the entire aircraft model presented by varying two parameters. Flutter for this specific model occurs as the damping ratio of one of the modes becomes positive which is indicated by the use of a color scale. For illustrative purposes only the mode causing flutter is presented. Figure 29 illustrates how the flutter speed depends on airspeed and density, which decreases as altitude increases, while Figure 30 illustrates the dependency of stiffness in torsion/pitch.

Figure 29: Damping ratio with respect to velocity and altitude

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Figure 30: Damping ratio with respect to velocity and pitch stiﬀness

The results obtained from a pure structural analysis point of view show that the flutter velocity highly depends on how detailed the model is. As unsteady aerodynamics are introduced in the entire aircraft model, bifurcations starts to occur as the damping becomes positive. Results show that including more aerodynamic terms changes the results and can both increase and decrease the flutter velocity. The entire aircraft model should in reality give a more realistic results due to the contribution of the rigid body motion. However, as unsteady aerodynamics are included in the entire aircraft model the results become un- clear due to the great complexity of the system, making it hard to determine the flutter velocity.

9.4 Simulation Results from the analysis of the ﬂutter suppression model are presented in this section. This includes both the linearized results from MATLAB and the Simulink simulations. In MATLAB, colors indicate which mode of oscillation that is being analyzed and in Simulink each mode is represented in a separate window.

9.4.1 Linearized model without control Figure 31 shows the results of the linearized open-loop Simulink model as it was exposed to a disturbance. This model was later optimized with the LQR algorithm in order to provide feedback and control.

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Figure 31: Linearized without control

As Figure 31 clearly suggests, the model is not capable of stable ﬂight due to its relaxed static stability design. In order for the model to ﬂy stable a controller is needed.

9.4.2 Linearized model with control When the linearized state-space model had been obtained, the optimal gain can be found with the LQR algorithm and applied to the model. This resulted in the optimal step response in Figure 32. Even if the settling time of the linearized model was around 6 seconds, the controller showed suﬃcient performance in the Simulink model. For improved settling time, the weight matrices could be modiﬁed but this never improved this Simulink model.

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Figure 32: Step response of the linearized model with LQR control

9.4.3 Simulink response without control The response of the Simulink model without control in Figure 33 is comparable to the response in Figure 31. It also shows oscillations with increasing amplitude, but for the Simulink model the amplitude is maximum at 180◦ for the pitching motion.

Figure 33: Simulink response without control

9.4.4 Simulink response with LQR control LQR The response of the Simulink model using LQR in Figure 34 is comparable to

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Figure 32 and shows approximately the same settling time. When the model is exposed to turbulence it begins to struggle to maintain control, see Figure 35. However, the amplitude of the plunge motion never exceeds 4 cm upwards and 8 cm downwards and the pitching motion stays within a few degrees. Therefore, the controller appears to handle turbulence well.

Figure 34: Simulink LQR response with control at 10,000 feet with airspeed of Mach 0.172 and 5 m/s gust at 10 seconds

Figure 35: Simulink response with control at 10,000 feet with airspeed of Mach 0.172 with turbulence

9.4.5 Simulink response climb proﬁle with LQR control In an attempt to establish the ﬂight envelope for the model, two ramp functions were used. One for speed and one for altitude. Same plots are used as before

65 Master Thesis but including an altitude and velocity plot to illustrate the ramp functions used to simulate climb and acceleration. Both continuous turbulence and 5 m/s gusts at a regular interval are present.

Figure 36: Climb and acceleration proﬁle response for LQR controller with gusts and turbulence

In Figure 36, the model using the LQR ensures its stability throughout the ﬂight envelope until it reaches a speed of 59 m/s at 3048 m (10,000 feet). After that, ﬂutter occurs with increasing amplitude.

9.4.6 Control Reversal By plotting the moment per elevon angle versus increasing airspeed (Figure 37), control reversal was found to occur at around 54 m/s for the Simulink model. This means that the limit for the LQR will lie around 54 m/s and it will not be able to handle higher speeds. Conclusion is that since the reversal effect occurs at 54 m/s controls begins to amplify the flutter instead of suppressing it. This gives hard flutter where divergence and flutter occurs at the same speed. However, since reversal is not a critical state as divergence or flutter, it can be handled by adapting the controller and reversing the outputs.

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Figure 37: Control eﬀectiveness for Simulink model

Figure 37 shows how moment per elevon angle increases until a maximum point at 46 m/s where the controls have their maximum eﬀectiveness and after that the eﬀectiveness decreases until a point at 54 m/s where they become unresponsive and later reversed.

9.4.7 Manual Tuning As the Simulink model is highly non-linear and due to static aeroelastic eﬀects, the linearized optimal controller could not handle velocities larger than Mach 0.176 at 10,000 feet. By tuning the parameters manually it was possible to make an adaptive controller that worked for much higher velocities (Figure 38). Note that the aileron output has changed sign due to the reversal eﬀect. Since the controller was manually tuned an improved settling time could be achieved after a few iterations.

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Figure 38: Simulink response with manually tuned controller at 10,000 feet with airspeed of Mach 0.3 and 5 m/s gust at 5 seconds with turbulence

9.4.8 Simulink response climb profile with adaptive controller The adaptive controller uses the LQR controller until the controls become unresponsive at around 50 m/s after that a manually tuned controller takes over. This sort of controller makes it possible to push the flight envelope even further. In Figure 39 the maximum flying speed reaches 120 m/s before flutter and divergence occurs at approximately the same speed.

Figure 39: Climb and acceleration proﬁle response for an adaptive controller with gusts and turbulence

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9.4.9 Poles and zeros plot As described in the stability part, the position of the poles highly aﬀects the stability of the system. To prove this, a pole/zero plot is provided for the linearized system using LQR at an altitude of 10,000 feet and at a speed of Mach 0.15. At this condition the system should be stable. The poles are marked with x and the zeroes are marked with circles and Figure 40 conﬁrms that all poles are in the stable region for both modes.

Figure 40: Pole/Zero plot for plunge and pitch at 10,000 feet at Mach 0.15 with LQR controller

9.5 Summary A summary of the flutter, divergence and reversal velocities in m/s for the different models and for two different altitudes are presented in the tables below. These velocities apply only to the specific model analyzed. Since the adaptive controller was deigned for altitudes of 10,000 feet, it did not improve the performance at sea level. Therefore, it is not included in Table 2.

Sea level 2DOF 3DOF Simulink Quasi Unsteady Quasi Unsteady LQR Vf 49 54 49 40 66 Vdiv 81 81 81 81 NA Vrev 55 55 55 55 51

Table 2: Aeroelastic velocities (m/s) at sea level

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10,000 feet 2DOF 3DOF Simulink Quasi Unsteady Quasi Unsteady LQR Adaptive Vf 56 60 55 46 59 >120 Vdiv 94 94 94 94 59 124 Vrev 64 64 64 64 53 53

Table 3: Aeroelastic velocities (m/s) at 10,000 feet

Comparison in Table 2 and 3 clearly shows static effects occur at lower speeds for the Simulink model and that the inclusion of unsteady aerodynamics sometimes increases the flutter speed and sometimes it decreases it. The inclusion of the rigid body motion seems to generally lower the flutter speed. This can be explained by the inclusion of another degree of freedom which can get into resonance with the other two. The inclusion of a LQR increases the flutter speed by at most 13 m/s and compared to one model, it lowers the flutter speed by 1 m/s. However, as the adaptive controller takes reversal effects into account it reaches up to twice the speeds achieved with the LQR. But bear in mind that without any controller, the Simulink model was completely unstable due to its efficient design. For such a unstable design to reach beyond its structural limits is a huge improvement.

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10 Discussion 10.1 General Considerations Since the purpose of this thesis is to provide the reader with the necessary knowledge about aeroelastic calculations and simulations, the design is arbitrary. It is only supposed to provide a concrete example and should not be treated as a plane that could enter manufacturing. For those purposes there are teams of engineers that work on one and each of their respective area.

Electric propulsion has the advantage that there is no liquid fuel that can move around, eﬀectively changing the center of gravity. Otherwise the technology is still limited. Batteries are too heavy, solar panels ineﬀective and engines less powerful than engines running on liquid fuel. Any thrust requirements were not established since this was not a part of the research. As progress is made, sometime in the future electric long endurance UAV’s might become feasible.

If the flutter suppression system would fail then airbrakes need to be used to lower the speed fast and effectively. But since the flutter suppression system in this case is embedded in the flight control system it would mean total loss of control and most probably loss of the airframe (unless the pilot is able to make a controlled descent and landing).

10.2 Simulation With Simulink, the flight dynamics of the model and the aeroelastic effects are both taken into account. Furthermore, the software also solves the equations numerically. Therefore, it gives a better representation of how the model would react. Results show that due to its relaxed static stability design, it is not capable of stable flight without any controller. Therefore, it is hard to draw a conclusion whether the FCS increases the flutter velocity without comparing with the analytic results. Comparison shows that depending on model the flutter velocity increases with up to 28% at 10,000 feet using the LQR controller. However, it seems as static aeroelastic effects generally occur at lower speeds for the Simulink model than for the analytic models. Since the servo controllers are trying to keep the airplane flying at a steady attitude while also suppressing flutter they might cause even more twisting to the wing than expected, leading to loss of control effectiveness before flutter occurs. This effect is highly undesirable since it causes so called hard flutter as the controls suddenly becomes ineffective. A conclusion that also has been made previously [30]. Therefore, it is important to establish an absolute limit for which the aircraft can operate safely. This limit can be estimated with the climb and acceleration plots (Figure 38-39) where both gusts and turbulence is present throughout the intended flight envelope. It shows clearly that the model is not intended to fly faster than 59 m/s at 10,000 feet, and should for safety reasons never fly faster than 56 m/s using a LQR controller. However, as the plane passes the reversal speed, it was found that a manually tuned controller could keep the plane flying for much larger velocities than with the LQR optimized controller. Due to the highly time consuming process of finding the new optimal controller manually, speeds higher than 120

71 Master Thesis m/s were never achieved. This shows at least that flutter can be prevented as long as the controller can handle the non-linear dynamics of the airplane and as long as the twist of the wing does not reach divergence. In reality, static aeroelastic effects could be avoided by adding stiffness or inertia to the wings or trying different airfoils. This is a much safer option then modifying the controller, but requires new calculations for flutter, since higher stiffness doesn’t necessarily mean higher flutter velocity. As speeds become greater than Mach 0.3 one can no longer ignore compressible effects. Therefore, 98.5 m/s is the highest valid speed to be considered for the model at 10,000 feet. One need also to bear in mind that the drag will make it highly inefficient to fly at such speeds.

10.3 Limitations of the Controller Even if the LQR gives good results for a specific altitude and speed, it is not a dynamic controller. This means that it is not adaptable for different flying conditions. To take different flying conditions into account for this specific controller, a lookup table or a least square method would be required since the relations between the control parameters are not linear. This could be done, but for now the performance of the controller throughout the flight envelope is sufficient for this research, using only one set of parameters. Dynamic controllers should be used with high precaution. Switching modes can cause extreme instabilities if the phase relations are wrong even if the flight conditions are favorable and should generally not cause instabilities. To ensure stability of the adaptive controller, a sinus function was used to simulate fluctuations of airspeed around the chosen switch velocity, which was 50 m/s. This caused oscillations, but none were severe enough to be considered dangerous, and thereby the controller proved to be stable even in speeds close to the control reversal speed.

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11 Conclusion

As airplane designs becomes more efficient, the need for a flight control system increases. Flutter can be suppressed using a linear optimized controller embedded in the FCS, but only until the point where controls become ineffective, causing hard flutter. This dangerous state needs to be avoided at all costs, since it will occur almost instantly and will most probably cause structural failure of the wings and total hull loss. One way is to adapt the controller parameters as the plane reaches reversal speed. This was proven to work with the model and the system stability was increased so that it was able to reach much higher flying velocities than before. This opens up for more efficiently designed UAV’s which might in the future be able to perform electric powered long endurance missions.

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[8] E.T. Wooldridge ”The History of Flight: Early flying wings (1870 to 1920)” http://www.century-of-flight.net/Aviation%20history/flying%20wings/ Early%20Flying%20Wings.htm [9] Craig G. Merrett ”Aero-servo-viscoelasticity theory: lifting surfaces, plates, velocity transients, flutter, and instability”, Dissertation, University of Illi- nois at Urbana-Champaign, 2011 [10] Craig G. Merrett, Harry H. Hilton ”Elastic and viscoelastic panel flutter in incompressible, subsonic and supersonic flows”, ASDJournal (2010 ), Vol. 2, No. 1, pp. 53-80, University of Illinois at Urbana-Champaign, 2010 [11] Craig G. Merrett, Harry H. Hilton, ”Generalized Linear Servo-Aero- Viscoelasticity theory and applications”, AIAA Paper 2008-1997, University of Illinois at Urbana-Champaign, 2008 [12] Craig G. Merrett, Harry H. Hilton ”Aero-servo-viscoelastic flutter and torsional divergence alleviation for a wing in subsonic, compressible flow”, AIAA Paper 2011-1716, University of Illinois at Urbana-Champaign, 2011

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[16] Vivek Mukhopadhyay ”A conceptual wing ﬂutter analysis tool for systems analysis and parametric design study”, textitnasa techdocs, NASA Langley Research Center, Hampton VA, 2003

[17] Moses G. Farmer, ”A two-degree-of-freedom ﬂutter mount system with low damping for testing rigid wings at diﬀerent angles of attack”, NASA Technical Memorandum 83302, NASA Langley Research Center, Hampton VA, 1982

[18] Martin R. Waszak, ”Modelling the benchmark active control technology windtunnel model for application to ﬂutter suppression”, AIAA Paper No. 96-3437, NASA Langley Research Center, Hampton, VA, 1996 [19] Martin R. Waszak, Jimmy Fung , ”Parameter estimation and analysis of actuators for the BACT wind-tunnel model”, AIAA Paper 96-3362, NASA Langley Research Center, Hampton, VA, 1996 [20] Jennifer L. Pinkerton, Anna-Maria R. McGowan, Robert W. Moses, Robert C. Scott, Jennifer Heeg ”Controlled aeroelastic response and airfoil shaping using adptive materials and integrated systems”, SPIE’s 1996 Symposium on Smart Structures and Integrated Systems, Aeroelasticity Branch, NASA Langley Research Center, Hampton, VA 23681-0001, 1996 [21] Michael J. Logan, Julio Chu, Mark A. Motter, Dennis L. Carter, Michael Ol, Cale Zeune, ”Small UAV research and evolution in long endurance electric powered ﬂight”, AIAA Paper 2007-2730, NASA Langley Research Cen- ter, Hampton, VA, USAF Air Force Research Laboratory, WPAFB, OH, 2007

[22] Chang-gi Pak, ”Unsteady aerodynamic model tuning for precise flutter prediction”, Journal of Aircraft, Vol. 48, No.6(2011), pp. 2178-2184, NASA Dryden Flight Research Center, Edwards, CA, 2011 [23] Martin J. Brenner, ”Actuator and aerodynamic modeling for high-angle- of-attack aeroservoelasticity”, NASA Technical Memorandum 4493, NASA Dryden Flight Research Center, Edwards, CA, 1993 [24] Jeffry Block, Heather Gilliatt ”Active control of an aeroelastic structure”, AIAA Paper 97-0016, Texas A&M University, 1997 [25] E. G. Broadbent, Margaret Williams ”The Effect of Structural Damping on Binary Flutter”, Reports and Memoranda No. 3169, The deputy controller aircraft (research and development) ministry of aviation, London, August, 1956 [26] E. G. Broadbent, Ola Mansfield, ”Aileron reversal and wing divergence of swept wings”, Reports and Memoranda No. 2817, The deputy controller aircraft (research and development) ministry of aviation, London, September, 1947 [27] W. G. Molyneux, H. Hall ”The Aerodynamic Effects of Aspect Ratio and Sweepback on Wing Flutter”, Reports and Memoranda No. 3011, The principal director of scientific research (air) ministry of supply, London, 1955

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[28] N. C. Lambourne ”An experimental investigation on the ﬂutter characteristics of a model ﬂying wing”, Reports and Memoranda No. 2626, Aerody- namics Division, N.P.L, London, April, 1952

[29] Katalin M.Hangos, Rozalia Lakner, Miklos Gerzson ”Intelligent Control Systems: An introduction with Examples” Kluwer Academic Publishers, 2011, Netherlands [30] Terrence A. Weisshaar, David K. Duke, ”Induced drag reduction using aeroelastic tailoring with adaptive control surfaces”, Journal of Aircraft, Vol. 43, No. 1, Purdue University, Lockheed-Martin Company, January-February, 2006 [31] Mark Drela ”XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils” MIT Dept. of Aeronautics and Astronautics, Cambridge, Massachusetts, Springer-Verlag Berlin Heidelberg, 1989 [32] S. L. Butts, A. D. Hoover ”Flying Qualities Evaluation of the X-29A Re- search Aircraft”, AFFTC-TR-89-08, U.S. Air Force Flight Test Center, 1989 [33] Shael Markin ”Multiple simultaneous specification attitude control of a mini flying-wing unmanned aerial vehicle”, Thesis, Department of Mechan- ical and Industrial Engineering, University of Toronto, Toronto, ON, 2011 [34] http://www.boeing.com/commercial/aeromagazine/aero 02/textonly/fo01txt.html ”The Effect of High Altitude and Center of Gravity on The Handling Char- acteristics of Swept-Wing Commercial Airplanes” The Boeing Company

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76 Master Thesis

Appendix Oscillating Airfoil Derivatives Lift Derivatives

k2 L = 2π − − Gk h 2

Lh˙ = 2πF k2a 1 (.1) L = 2π + F − Gk − a θ 2 2 1 1 G L = 2π + F − a + θ˙ 2 2 k Moment Derivatives

k2a 1 M = 2π − − k a + G h 2 2 1 M = 2π a + F h˙ 2 (.2) k2 1 1 1 1 M = 2π + a2 + F a + − Gk − a − a θ 2 9 2 2 2 k 1 1 1 G 1 M = 2π − + F − a + kF a + − a + a + θ˙ 2 2 2 2 k 2

Full aeroelastic equation for Flying wing model ¨ m Shθ h Kh 0 h Shδ ¨ m + = − δ + gcosθ0 − Shθ Iθ θ¨ 0 Kθ θ Sθδ Shθ | {z } | {z } | {z } | {z } Inertial forces Elastic forces Coupled inertia Gravity m Shθ 2ζhωh 0 −CL0 −CLα − + qs + qs θT + Shθ Iθ 0 2ζθωθ ~cCM0 ~cCMα | {z } | {z } Generalized Damping Aerodynamic forces " ~c # qS~c −C −lC h¨ qS −CLα −lCLα − (CLα˙ + CLq) h˙ Lα˙ Lα˙ 2U0 + + ~c + 2U ~cCMα˙ ~clCMα˙ θ¨ U ~cCMα ~clCMα − (CMα˙ + CMq) θ˙ 0 0 2U0 | {z } Aerodynamic forces " ~c # 0 −C h qS~c −C −C qS − CLα˙ −CLα ω˙ Lα Lδ˙ ˙ Lδ 2U0 g + qS + δ + qS δ − ~c 0 CMα θ 2U ~cC ˙ ~cCMδ U CMα˙ ~cCMα ωg 0 Mδ 0 2U0 | {z } | {z } Aerodynamic forces Disturbance forces (.3)

77 Master Thesis

Linearized Aeroelastic Flying wing State-space model without control Following matrices were obtained from the linearized Simulink model and were used to ﬁnd the optimal control parameters.

−1.0000 0 1.0000 0   0 0 0 0 1.0000  A =   (.4) 100.8774 −205.8916 −553.8032 −26.9256 −25.7326 21.0454 10.5342 30.1238 0.2473 −5.2687

 0   0    B =  0  (.5)   3.4785 0.7257  0 0 0 0 0   0 −1.0000 0 0 0     0 0 0 −1.0000 0    C =  0 0 −1.0000 0 0  (.6)    0 0 0 0 −1.0000   0.5009 −0.2044 −0.1301 −0.0877 −0.1191 0 0 0 0 0  0   0     0    D =  0  (.7)    0    0.0173 0

Linearized Aeroelastic Flying wing State-space model with control −1.0000 0 −1.0000 0 0   0 0 0 1.0000 0    Aenhanced =  0 0 0 0 1.0000  (.8)    70.7138 −217.7978 −616.9943 −27.5706 −34.5308 14.7525 8.0503 16.9407 0.1127 −7.1042 K = 8.6714 3.4228 18.1660 0.1854 2.5293 (.9)