INCORPORATION OF PHYSICS-BASED CONTROLLABILITY ANALYSIS IN AIRCRAFT
MULTI-FIDELITY MADO FRAMEWORK
Dissertation
Submitted to
The School of Engineering of the
UNIVERSITY OF DAYTON
In Partial Fulfillment of the Requirements for
The Degree of
Doctor of Philosophy in Engineering
By
Christopher Meckstroth, M.S.
Dayton, Ohio
December 2019
INCORPORATION OF PHYSICS-BASED CONTROLLABILITY ANALYSIS IN AIRCRAFT
MULTI-FIDELITY MADO FRAMEWORK
Name: Meckstroth, Christopher Michael
APPROVED BY:
______Raúl Ordóñez, Ph.D. Raymond Kolonay, Ph.D. Advisory Committee Chairman Committee Member Associate Professor Director Electrical and Computer Engineering Multidisciplinary Science and University of Dayton Technology Center AFRL/RQVC
______Eric Balster, Ph.D. Keigo Hirakawa, Ph.D. Committee Member Committee Member Associate Professor Associate Professor Electrical and Computer Engineering Electrical and Computer Engineering University of Dayton University of Dayton
______Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean, School of Engineering Professor School of Engineering
ii ABSTRACT
INCORPORATION OF PHYSICS-BASED CONTROLLABILITY ANALYSIS IN AIRCRAFT
MULTI-FIDELITY MADO FRAMEWORK
Name: Meckstroth, Christopher Michael University of Dayton
Advisor: Dr. Raúl Ordóñez
A method is presented to incorporate physics-based controllability assessment in an aircraft
Multi-disciplinary Analysis and Design Optimization environment with a target fidelity
representing the traditional preliminary aircraft design phase. This method was designed with
specific intended application to innovative vehicle concepts such as the Efficient Supersonic Air
Vehicle, a tailless fighter-type aircraft which requires the use of innovative control effectors to
achieve yaw control requirements. Typically, the layout of an aircraft is determined primarily
through empirical design methods with minimal physical evaluation influencing the shape. As a
result, the evaluation of new technologies such as these innovative control effectors in the past has
been limited to placement and testing of them within existing free real estate on an otherwise
complete vehicle design. The hypothesis of this dissertation is that inclusion of such technology in
earliest stages of the design process has a greater chance of leading to optimal benefit and
potentially a closed design for a tailless fighter-type aircraft.
However, incorporation of technology that does not have a strong statistical basis through prior work requires some form of physical analysis to be performed in the design iteration. An aerodynamic study was performed to determine the optimal combination of fidelity and computation time for analyzing these types of configurations for the controls analysis in the MADO environment, resulting in the use of a multi-fidelity approach to aerodynamic analysis. This
iii approach in turn requires a multi-fidelity, parameterized geometric model of the aircraft with automated generation of analysis mesh.
In traditional aircraft design, the disciplines involved are isolated from each other in a linear manner such that one finishes prior to another beginning. Multidisciplinary approaches attempt to merge these. However, in open literature the fidelity level of various disciplines tends to have an inverse relationship. For example, the most complicated controllers explored in MADO tend to use the lowest fidelity aerodynamics and those that use higher fidelity aerodynamics tend to incorporate only a basic controllability assessment if any at all. In fact, this is the first known aircraft MADO effort to incorporate at least preliminary design levels of fidelity into both the aerodynamics and controls disciplines simultaneously.
The approach of this dissertation was to test the implications of this by executing the
MADO framework with both the low-fidelity aerodynamics and the multi-fidelity aerodynamics approach that is tailored to the controllability method desired. Using only the low-fidelity aerodynamics, the MADO result led to an infeasible aircraft configuration. However, the multi- fidelity approach resulted in an aircraft configuration markedly similar to previous real-life designs and with computed controllability in both the pitch and yaw axes optimized to be slightly above input design requirements.
iv
Dedicated to my father, Steve Meckstroth, whose mantra of “get good grades, go to college, and
get good job” resonated with me from a young age and has led to my continued success in all
three of these endeavors.
v ACKNOWLEDGEMENTS
This work would not be possible without the support of the University of Dayton Research
Institute in allowing me to pursue this effort in parallel to my professional research duties. I would like thank my advisor, Dr. Raúl Ordóñez for accepting me as a student and being patient with me as progress on this work was often slow. I want to thank Dr. Raymond Kolonay for his support and guidance throughout this process and for pushing me to continue and ultimately finish when my motivation was at the lowest points. I would also to thank my other committee members, Dr. Eric
Balster and Keigo Hirakawa for their time and effort in reviewing my work and providing their feedback.
To my wife, Manda, thank you for your continued encouragement throughout these past few years. I could not have finished this without your dedication in taking care of our three children,
Jackson, Corwin, and Alivia, allowing me to focus on this work.
vi TABLE OF CONTENTS
ABSTRACT ...... iii
ACKNOWLEDGEMENTS ...... vi
LIST OF FIGURES ...... xi
LIST OF TABLES ...... xvii
LIST OF SYMBOLS ...... xviii
CHAPTER I INTRODUCTION ...... 1
CHAPTER II LITERATURE REVIEW...... 5
2.1 Background and Problem Statement ...... 5
2.2 Aircraft Multidisciplinary Design Optimization ...... 7
2.3 Controls in Conceptual Design and Preliminary Design ...... 9
2.4 The Control Configured Vehicle (CCV) ...... 11
2.5 Aeroservoelasticity ...... 14
2.6 Controls in MADO ...... 15
CHAPTER III CONTROL POWER REQUIRED METHODS ...... 19
3.1 Controllability Requirements Definition ...... 20
3.2 Control Power Required Background ...... 24
3.3 Linearized Aircraft Model ...... 26
3.4 Control Power Required ...... 29
3.5 Longitudinal CPR ...... 31
3.5.1 Takeoff Rotation ...... 32
3.5.2 Trim CPR ...... 32
3.5.3 Roll Coordination ...... 32
vii 3.5.4 Longitudinal Dynamic Response - Short Period ...... 33
3.6 Lateral-Directional CPR ...... 39
3.6.1 Roll Performance ...... 39
3.6.2 Yaw CPR for Roll Initiation ...... 41
3.6.3 Yaw CPR for Roll Coordination ...... 42
3.6.4 Crosswind CPR ...... 42
3.6.5 Lateral-Directional Dynamic Response - Dutch Roll ...... 43
3.7 Control Power Required Application ...... 47
3.7.1 Numerical Example ...... 53
3.7.2 Control Power Available ...... 61
3.8 Control Power Summary ...... 62
CHAPTER IV AERODYNAMICS IN THE MADO PROCESS ...... 63
4.1.1 DATCOM ...... 64
4.1.2 AVL ...... 65
4.1.3 ZAERO ...... 66
4.1.4 ZEUS...... 67
4.1.5 Cart3D ...... 68
4.2 Aerodynamic Analysis Results ...... 70
4.2.1 Static Analysis ...... 70
4.2.2 Dynamic Derivatives ...... 73
4.2.3 Fuselage Shape ...... 75
4.2.4 Control Derivatives – Innovative Control Effectors ...... 79
4.2.5 Summary of Computation Time, Capability, and Fidelity ...... 81
4.3 Aerodynamic Analysis Selection ...... 84
CHAPTER V MADO GEOMETRY ...... 86
viii 5.1 Computational Aircraft Prototype Synthesis ...... 86
5.2 Baseline Geometry—ESAV ...... 88
5.2.1 Wing Geometry ...... 89
5.2.2 Fuselage Geometry ...... 91
5.3 Multi-fidelity Geometric Representation in CAPS ...... 97
5.4 Parameterization ...... 98
5.4.1 Wing Parameterization ...... 100
5.4.2 Fuselage Parameterization ...... 110
5.4.3 Control Effectors ...... 112
CHAPTER VI THE MADO PROCESS ...... 115
6.1 Mass Properties ...... 116
6.2 Multi-fidelity Aerodynamic Approach for MADO...... 126
6.2.1 Convergence Study ...... 126
6.2.2 Superposition Study ...... 128
6.2.3 Multi-Fidelity Aerodynamics ...... 136
6.3 Cost Function ...... 139
6.4 Distributed Computing Environment and Optimization ...... 140
CHAPTER VII MADO RESULTS AND DISCUSSION ...... 142
7.1 Parameter Sweeps ...... 142
7.2 Results of Low-fidelity MADO with AVL ...... 150
7.3 Results of Multi-fidelity MADO with Cart3D and AVL ...... 153
7.4 Discussion of low-fidelity and multi-fidelity optimization results ...... 157
CHAPTER VIII CONCLUSIONS AND FUTURE WORK ...... 159
8.1 Conclusions ...... 159
8.2 Future work ...... 163
ix REFERENCES ...... 165
x LIST OF FIGURES
Figure 2-1. Northrup YB-49 [30] (left) and Grumman X-29 [31] (right) ...... 12
Figure 3-1. ESAV Planform with horizontal and vertical tails (red) from conceptual tail volume coefficients...... 20
Figure 3-2. Control Margin Requirements [27] ...... 31
Figure 3-3. Computed short period poles for F-16 example ...... 37
Figure 3-4. Simulation of F-16 short period response with original and updated CPR methods ...... 38
Figure 3-5. Computed Dutch Roll poles for F-16 example ...... 46
Figure 3-6. Simulation of F-16 Dutch Roll response with original and updated CPR methods ...... 46
Figure 3-7. CAD geometry of initial vehicle design ...... 48
Figure 3-8. Fuselage geometry and effectors tested in wind tunnel ...... 49
Figure 3-9. Spoiler/deflector in wind tunnel ...... 49
Figure 3-10. Rhino Horn control effector ...... 50
Figure 3-11. Flow visualization of vortex shedding in wind tunnel ...... 50
Figure 3-12. Flight conditions evaluated for ICE Program [12] ...... 51
Figure 3-13. ESAV longitudinal poles during high-angle of attack condition ...... 58
Figure 3-14. Simulation of short period response to vertical gust ...... 58
Figure 3-15. ESAV lateral/directional poles during high-angle of attack condition ...... 59
Figure 3-16. Simulation of Dutch Roll response to lateral gust ...... 60
Figure 4-1. ESAV AVL analysis geometry ...... 65
Figure 4-2. ZAERO and ZEUS analysis mesh ...... 67
Figure 4-3. Triangulation mesh input to Cart3D ...... 69
Figure 4-4. Volume mesh for Cart3D computation ...... 69
xi Figure 4-5. Comparison of longitudinal coefficients: Lift (top), Drag (middle), and Pitch Moment (bottom) ...... 71
Figure 4-6. Comparison of lateral coefficients: Side Force (top left), Roll Moment (top right), and Yaw Moment (bottom) ...... 72
Figure 4-7. Comparison of dynamic derivatives between AVL and published wind tunnel data for the F-16 ...... 75
Figure 4-8. Baseline ESAV vehicle geometry ...... 76
Figure 4-9. Fuselage nose shapes ...... 76
Figure 4-10. Side force and yaw moment coefficients from ZEUS at zero (blue) and five degrees (red) sideslip angles ...... 77
Figure 4-11. Clean configuration aerodynamic coefficients for all fuselage nose shapes...... 78
Figure 4-12. Side force and sideslip angle for each fuselage nose shape ...... 78
Figure 4-13. Comparison of Cart3D and wind tunnel data in roll and yaw moment for combinations of spoiler, deflector and spoiler+deflector ...... 80
Figure 4-14. Rhino Horn effectiveness in roll and yaw ...... 81
Figure 4-15. Average computation time per run ...... 82
Figure 4-16. Computation time for one flight condition ...... 83
Figure 5-1. Examples of CAPS geometries in ESP ...... 87
Figure 5-2. Cross-sections used for dimensioning of ESAV fuselage and wing ...... 88
Figure 5-3. Example of dimensioned cross-sections from ESAV CAD model ...... 89
Figure 5-4. Comparison of measured airfoil and nearest standard airfoil definition for root airfoil shape ...... 89
Figure 5-5. Standard NACA airfoil cross-section syntax in ESP ...... 90
Figure 5-6. Cross-sections used to create wing planform ...... 90
Figure 5-7. ESP syntax for creating ruled body for wing ...... 91
Figure 5-8. Baseline ESAV planform in ESP ...... 91
Figure 5-9. Reduced number of ESAV cross-sections...... 92
Figure 5-10. Fuselage cross-section created with Bézier curves ...... 93
xii Figure 5-11. ESP definition of Bezier cross-section made of four individual Bezier curves ...... 94
Figure 5-12. ESP syntax for creating blended body for fuselage ...... 94
Figure 5-13. Baseline vehicle geometry with fuselage cross-sections defined via Bézier curves ...... 95
Figure 5-14. Comparison of elliptical cross-sections (blue) to dimensioned cross- sections (black) ...... 96
Figure 5-15. ESP definition of elliptical cross-section made of top and bottom ellipses of different minor axes...... 96
Figure 5-16. Final version of baseline ESAV vehicle in ESP...... 97
Figure 5-17. Example wing leading edge deflection ...... 97
Figure 5-18. ESP and Analysis geometries represented for AVL and Cart3D ...... 98
Figure 5-19. ESAV design parameters accessible in ESP ...... 100
Figure 5-20. Lamda wing planform parameters ...... 101
Figure 5-21. Demonstration of wing break factor ...... 102
Figure 5-22. Lamda wing with wing-fuselage intersection parameters ...... 103
Figure 5-23. Geometric changes to the wing for each design parameter; the red outline is the baseline ESAV shape, the left side of the plot shows parameter values lower than baseline and right side shows parameters higher than baseline...... 109
Figure 5-24. Scaling of fuselage in the x-direction with wing placement ...... 111
Figure 5-25. Real examples of extreme forward wing placement (B-2, top) [61] and rear wing placement (B-1, bottom) [62] ...... 111
Figure 5-26. Example scaling of fuselage width ...... 112
Figure 5-27. Planform layout of parameterized ESAV model with control effectors ...... 113
Figure 5-28. Spoiler and deflector on ESP geometry ...... 114
Figure 5-29. Rhino Horn control effector on ESP geometry ...... 114
Figure 6-1. MADO N2 diagram ...... 116
Figure 6-2. Mission Profile for conceptual weight computation ...... 118
Figure 6-3. Mission weight segment analysis summary for baseline ESAV vehicle ...... 119
xiii Figure 6-4. Estimated component weights for baseline ESAV configuration ...... 121
Figure 6-5. CG locations of baseline ESAV vehicle components ...... 122
Figure 6-6. Example planforms generated, component CG locations, and resulting gross weight for Monte Carlo analysis ...... 124
Figure 6-7. Histogram of vehicle gross weight for Monte Carlo analysis with 100,000 samples ...... 124
Figure 6-8. Minimum and maximum weight configurations achieved through Monte Carlo analysis ...... 125
Figure 6-9. Convergence of clean configuration data (21 total runs) ...... 127
Figure 6-10. Analysis of superposition of Spoiler and Deflector. The red line shows the summation of individual analysis of the spoiler and deflector. The black line shows the analysis performed with both fully deflected...... 129
Figure 6-11. Analysis of superposition of trailing edge flap surfaces deflected down. The red line shows the summation of individual analysis of the trailing edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected...... 130
Figure 6-12. Analysis of superposition of trailing edge flap surfaces deflected up. The red line shows the summation of individual analysis of the trailing edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected...... 131
Figure 6-13. Analysis of superposition of leading edge flap surfaces deflected down. The red line shows the summation of individual analysis of the leading edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected...... 132
Figure 6-14. Analysis of superposition of leading edge flap surfaces deflected up. The red line shows the summation of individual analysis of the leading edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected...... 133
Figure 6-15. Analysis of superposition of all surfaces fully extended and/or deflected down. The red line shows the summation of individual analysis of the surfaces. The black line shows the analysis performed with all deflected...... 134
Figure 6-16. Analysis of superposition of all surfaces fully extended and/or deflected up. The red line shows the summation of individual analysis of the surfaces. The black line shows the analysis performed with all deflected...... 135
Figure 6-17. Low pressure area (blue color) shows rear wing flap surface shielded by spoiler/deflector, reducing effectiveness of the flap ...... 136
xiv Figure 6-18. Comparison of Cart3D and AVL analysis to ESAV wind tunnel data after correcting for neutral point ...... 137
Figure 6-19. Comparison of Cart3D and AVL analysis to ESAV wind tunnel data for trailing edge outboard flap left after correcting for neutral point ...... 138
Figure 6-20. MADO aerodynamic analysis process ...... 139
Figure 7-1. Parameter Sweep – Thrust-to-Weight Ratio. MADO cost function(s) and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 144
Figure 7-2. Parameter Sweep – Wing Loading. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 145
Figure 7-3. Parameter Sweep - Aspect Ratio. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 145
Figure 7-4. Parameter Sweep - Sweep. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 146
Figure 7-5. Parameter Sweep – Wing Leading Edge Location. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 147
Figure 7-6. Parameter Sweep - Taper Ratio. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 148
Figure 7-7. Low-Fidelity Parameter Sweep – Wing Break Location. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 148
Figure 7-8. Low-Fidelity Parameter Sweep – Wing Break Factor. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right) ...... 149
Figure 7-9. Vehicle configurations resulting from minimum cost of each parameter sweep excursion from ESAV baseline ...... 150
Figure 7-10. Results of low-fidelity MADO with AVL, iterations 1 through 3 ...... 151
Figure 7-11. Results of low-fidelity MADO with AVL, iterations 4 through 6 ...... 152
Figure 7-12. Final Result of low-fidelity MADO with AVL ...... 153
Figure 7-13. Final vehicle configuration result of low-fidelity MADO with AVL ...... 153
Figure 7-14. Results of multi-fidelity MADO with AVL and Cart3D, iterations 1 and 2 ...... 154
Figure 7-15. Results of multi-fidelity MADO with AVL and Cart3D, iterations 3 through 6 ...... 155
Figure 7-16. Final Result of multi-fidelity MADO with AVL and Cart3D ...... 156
xv Figure 7-17. Final vehicle configuration result of multi-fidelity MADO with AVL and Cart3D ...... 156
Figure 7-18. Pareto front of low-fidelity (left) and multi-fidelity (right) MADO results ...... 157
Figure 7-19. Comparison of X-4 Bantam [66] and multi-fidelity MADO result ...... 158
xvi LIST OF TABLES
Table 3-1. Roll control effectiveness requirements ...... 22
Table 3-2. Lateral-directional control in crosswinds ...... 22
Table 3-3. Short period dynamic requirements ...... 23
Table 3-4. Minimum Dutch Roll frequency and damping ...... 23
Table 3-5. Gust magnitudes ...... 23
Table 3-6. Body-Axes 6-DOF Aircraft Equations of Motion ...... 25
Table 3-7. Mass properties and reference geometry of example ESAV configuration ...... 53
Table 3-8. Computed lift and pitching moment coefficients and derivatives ...... 54
Table 3-9. Tabulated dimensional derivatives of ESAV in power-on departure stall condition ...... 56
Table 3-10. Control Power Required for ESAV in high angle of attack condition ...... 61
Table 4-1. Comparison of Static Stability Derivatives ...... 72
Table 4-2. Dynamic derivatives ...... 73
Table 4-3. Computation time, capability, and fidelity of each program ...... 84
Table 6-1. Monte Carlo design variables with upper and lower bounds ...... 123
Table 6-2. Run Matrix of aerodynamic study in CAPS for MADO ...... 127
xvii LIST OF SYMBOLS
wing span, ft.
𝑏𝑏 wing mean aerodynamic chord, ft.
𝑐𝑐C̅ non-dimensional coefficient with respect to x
x gravitational acceleration, ft/sec.2
𝐷𝐷 𝑔𝑔 , , aircraft moments of inertia, slug-ft2/sec.
𝑥𝑥 𝑦𝑦 𝑧𝑧 𝐼𝐼 𝐼𝐼 𝐼𝐼 aircraft product of inertia, slug-ft2/sec.
𝑥𝑥𝑥𝑥 𝐼𝐼, , dimensional body-axis moments about x, y, and z axes, ft-lb.
𝑙𝑙 𝑚𝑚 𝑛𝑛 lift force, lb.
𝐿𝐿 aircraft mass, slug.
𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 dimensional pitching moment, ft-lb.
𝑀𝑀 load factor
𝑔𝑔 𝑛𝑛, , aircraft body-axis roll, yaw, and pitch rates, rad/sec.
𝑃𝑃 𝑄𝑄 𝑅𝑅 dynamic pressure, lb/in2.
2 𝑞𝑞�, , wing, horizontal tail, and vertical tail planform areas, ft .
ℎ 𝑣𝑣 𝑆𝑆 ,𝑆𝑆 , 𝑆𝑆 aircraft body axis forward, lateral, and vertical velocities, ft/sec.
𝑈𝑈 ,𝑉𝑉 𝑊𝑊 horizontal and vertical tail volume coefficients
ℎ 𝑣𝑣 𝑉𝑉� 𝑉𝑉� aircraft total velocity, ft/sec.
𝑇𝑇 𝑉𝑉 weight, lb.
𝑊𝑊 location of aerodynamic center along the x-axis, ft.
𝑎𝑎𝑎𝑎 𝑥𝑥 , location of center of gravity along the x and z axes, ft.
𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑥𝑥 , 𝑧𝑧 horizontal distance from center of gravity to tail aerodynamic centers, ft.
ℎ 𝑣𝑣 𝑥𝑥 𝑥𝑥, horizontal and vertical distance from center of gravity to main landing gear, ft.
𝑥𝑥𝑚𝑚𝑚𝑚 𝑧𝑧𝑚𝑚𝑚𝑚 xviii , , body-axis aerodynamic forces, lb.
𝐴𝐴 𝐴𝐴 𝐴𝐴 𝑋𝑋 ,𝑌𝑌 ,𝑍𝑍 body-axis thrust forces, lb.
𝑇𝑇 𝑇𝑇 𝑇𝑇 𝑋𝑋 𝑌𝑌 𝑍𝑍 dimensional vertical force, lb.
𝑍𝑍
angle of attack, deg.
𝛼𝛼 angle of sideslip, deg.
𝛽𝛽 , input aileron, elevator deflections, deg.
𝑎𝑎 𝑒𝑒 𝛿𝛿 𝛿𝛿 friction coefficient
𝜇𝜇 density of air, slug/ft3.
𝜌𝜌 , , roll, pitch, and yaw Euler angles, rad.
𝜙𝜙 𝜃𝜃, 𝜓𝜓 short period frequency (rad/sec.) and damping
𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝜔𝜔 ,𝜁𝜁 Dutch roll frequency (rad/sec.) and damping
𝑛𝑛𝑑𝑑 𝑑𝑑 𝜔𝜔 𝜁𝜁
xix CHAPTER I
INTRODUCTION
Traditionally, aircraft design has been based in a large part on previous designs. The initial layout of an aircraft from design textbooks [1] begins with a rough estimate of the aircraft weight, generated from several empirical factors, which are based on many previous aircraft designs. With the exception of aerodynamic panel codes for estimation of drag polars, physical analysis is typically reserved for the preliminary design phase, where the Outer Mold Line (OML) is often limited to only a couple of potential configurations. However, according to the National Research
Council, aircraft life cycle costs are heavily impacted from decisions made at this point in the design phase and propagate throughout the life of the aircraft. In fact, it is estimated that when the design reaches Milestone A of the DoD acquisition process—when the initial preliminary layout is finished—70-75% of the entire life-cycle cost decisions have been made [2].
Over the past several years, there has been significant research into aircraft optimization, particularly within the Air Force Research Laboratory’s Multidisciplinary Science and Technology
Center (MSTC). In MSTC, the focus is to reduce the effect of these early design decisions on the life cycle cost by enabling rapid generation and analysis of aircraft configurations with preliminary design levels of fidelity across multiple disciplines. The higher fidelity analysis methods incorporate the physics of the aircraft, allowing the designers and ultimately the Multidisciplinary
Analysis and Design Optimization (MADO) process to experiment with configurations never
before considered. One such potential configuration is the Efficient Supersonic Air Vehicle
(ESAV). These vehicles will have objectives for supercruise, supersonic dash, maneuverability,
and extended range [3] and it is believed that meeting these objectives will lead to tailless airplane
configurations [4]. In order for the tailless aircraft to meet the maneuvering requirements of a
fighter, it will require the use of various innovative control effectors such as spoilers, clamshells,
1 strakes, or pop-up control effectors placed around the body. With low fidelity tools, these types of
effectors cannot be accurately modeled. As a result, using current design methods, their
consideration is only after a layout has been chosen, limiting their placement to existing free real
estate on the aircraft. The MADO methods being developed will correct this, allowing their
consideration concurrently with the aircraft planform layout.
The desired outcome of this effort is to allow future designs to be based primarily on
physical analysis early in the design process rather than trusted previous designs, thus enabling
exploration of innovative aircraft concepts incorporating new technologies which diverge
significantly from traditional planform shapes. One successful aircraft design example that required
incorporation of physics of new technology from the very beginning is the F-117 Nighthawk. After
the success of the F-117, Dr. Zbigniew Brzezinski, President Carter’s National Security Council
Chief, asked: “Could stealth be applied to a conventional airplane without having to start from
scratch?” The answer was a resounding no. The F-117 was designed from the ground up with stealth
technology as a leading driver in the OML creation. However, it did not fit within the traditional
design process. As a result, the head of the F-117 program in Lockheed’s Skunk Works division,
Ben Rich, did not receive a positive response from his predecessor. When Kelly Johnson first saw
the design, he is said to have literally kicked Ben in the butt. Then he crumpled up the proposal,
threw it at his feet and said “Ben Rich, you dumb ****, have you lost your goddam mind? This
crap will never get off the ground [5].” The physics proved otherwise, with the F-117 playing a
major role in the Gulf War in Iraq and the proven stealth technology being incorporated into many
designs since.
By incorporating physical analysis early in the design process, Lockheed was able to develop a major innovation in aircraft technology. Within MSTC, researchers believe this will be key to future aircraft development as well. With the help of MADO, a multitude of aircraft geometries can be rapidly generated and analyzed early in the conceptual design to a fidelity level typically strictly reserved for the preliminary design phase, where only a couple of ‘locked’ in
2 geometries are chosen to be analyzed in detail. This gives the designer the ability to base the designs
on the latest knowledge of physics rather than solely on previous experience, allowing an
exploration of design space combinations never considered before.
The primary goal of the work presented here is the development and incorporation of an automated, physics-based controllability analysis tool for use in the MADO process with intended applicability directly to the ESAV class of vehicle. However, performing a valid controllability analysis on these vehicles with innovative control effectors requires accurate control derivatives.
A parallel effort related to these ESAV vehicles performed by Lockheed Martin [6] provided wind tunnel data for several of these potential control effectors. This data proved the initial aerodynamic analysis of this work to be inadequate. In turn, this knowledge led to a major undertaking, evaluating various aerodynamic analysis software programs for computing required control derivatives in the MADO process, ultimately resulting in a decision to use a multi-fidelity approach incorporating two separate analysis programs. However, the higher-fidelity analysis required a 3- dimensional representation of the vehicle, which for optimization purposes also had to be both parameterized and automatically generated. In addition to the aerodynamics, the controllability analysis required vehicle mass properties. For this, traditional conceptual and preliminary design methods based on a selected mission profile and assumed component locations were used.
Although the controllability analysis tool was successfully developed and implemented in an MADO framework, the implications related to fidelity level of the associated analysis could be considered the largest contribution related to this effort. In Reference [7], in relation to MADO, Dr.
Cramer states “processes and tools must be matched to the level of analysis and fidelity and be used in the right trade space at the appropriate program milestone.” Indeed, in this effort, this comment is shown to be true when testing the controllability analysis with multiple levels of aerodynamic fidelity in the MADO loop. When a low-fidelity approach is used, the result is clearly an infeasible solution. However, when a multi-fidelity approach is used in which both the controllability and aerodynamic analysis are performed with similar levels of fidelity in the MADO process, an aircraft
3 configuration is generated that resembles a real-life planform. It is believed that this is the first
example of an aircraft MADO process with preliminary design levels of fidelity in both of these
disciplines simultaneously.
This dissertation is organized as follows. Chapter II presents a brief overview of control design in conceptual and preliminary design phases, followed by a literature review of attempts to modify the standard design practice for multidisciplinary design and ultimately optimization. The result of this chapter is the selection of the Control Power Required (CPR) approach for controllability analysis in the MADO framework. Chapter III presents the derivation and application of the CPR method to evaluate aircraft using physics-based criteria derived primarily from military flying qualities requirements. Chapter IV describes an aerodynamic study to determine which analysis codes are capable of modeling the required physics of the ESAV-class vehicle. Chapter V details the development of a parameterized, multi-fidelity geometry model required for the selected aerodynamic analysis. In Chapter VI, the remaining components required for the MADO process are described and the overall process is completed. Chapter VII presents results of the MADO process and difference in vehicle designs achieved using only low-fidelity tools vs. the multi-fidelity approach. In Chapter VIII, the potential implications of these results are discussed along with suggestions for future work.
4 CHAPTER II
LITERATURE REVIEW
2.1 Background and Problem Statement
In recent years, the Air Force has become interested in aircraft with small or completely eliminated vertical tails [4]. The potential benefits of this; namely, lower weight/higher efficiency from decreased drag, have fueled continual research through the present day into realization of these aircraft. This research has been focused primarily on overcoming the stability and control challenges. Conventional aircraft would be highly unstable in yaw without a vertical tail and using vectored thrust as a primary source of yaw control is unsafe for engine-out and near-idle conditions
[8]. The most notable unconventional airplane that successfully flies without a vertical tail, the B-
2, has a large wing span that is conducive to producing significant yaw moments using differential drag devices near the wing tips [8]. These devices, also known as clamshell flaps and split ailerons, are insufficient by themselves to provide adequate yaw control for highly maneuverable fighter
type aircraft, resulting in a need for alternate effectors.
A plethora of studies are available in the open literature that summarize analyses and tests
of various yaw control effectors for fighter type aircraft. Simon et. al. [9] present results for
clamshells, wing mounted spoilers, all moving wing tips and differential trailing edge flaps. Fears
[10] presents results for nose strakes and forebody porosity. Dorsett and Mehl [11] provide results
for clamshells, spoilers, all moving wing tips, differential leading edge flaps, deployable rudders
and spoiler-slot deflectors. Although a spoiler-slot deflector is a roll device, it has very favorable
features including minimal to no additional yaw required for roll maneuvers. Roetman et. al. [12]
present results for forebody strakes, clamshells, and a novel variable dihedral horizontal tail which
can remain flat when desired and deploy either up or down when needed to provide additional yaw
5 stability. Filipone [13] surveys a series of wing mounted pneumatic spoilers. Taken together, these studies indicate that no single device can replace the vertical tail/rudder for a tactical aircraft.
Actuation requirements are important to the overall integration process. Some of the controls described above engender higher hinge moments than typical controls, which results in weight, volume, power and thermal management issues. A slotless spoiler deflector, described by Blake
[14], is an effective yaw device with very low hinge moment but has gaps/reflecting surfaces on the vehicle underside that could impact survivability. In contrast, the differential rearward hinged upper spoilers described by Clark [15] could have very good survivability characteristics at the cost of higher actuation requirements.
As mentioned above, traditional aircraft conceptual design layouts are generated from a combination of empirical comparisons and intuition based on previous designs. With the exception of aerodynamic panel codes for estimation of drag polars, detailed analysis is typically reserved for the preliminary design phase, where the OML is often limited to only a couple of potential configurations. This method also fails when applied to non-traditional designs such as the tailless fighter, or when trying to use innovative control effectors that exhibit complex flow physics, such as those above, where there is no previous data and likewise, a lack of intuition.
The question this dissertation seeks to answer is: how can these aircraft of a priori unknown, non-traditional planforms incorporating innovative control effectors be analyzed with minimal human interaction such that the analysis can be incorporated into an overall MADO framework which accurately predicts the performance of the configurations in conceptual design.
In the next Section, this question is further refined with a description of current aircraft design practices in conceptual and preliminary design phases, with a focus on design practices related to the controls discipline. The last Section of this chapter contains a literature review of previous aircraft MADO efforts.
6 2.2 Aircraft Multidisciplinary Design Optimization
Aircraft are designed in a series of distinct phases: Conceptual Design, Preliminary Design, and Detailed Design. The goal of the Conceptual Design Phase is to determine feasible vehicle designs with a high likelihood of meeting the mission requirements. During this phase, many trade studies are performed to make decisions such as the overall configuration layout, sizing of the vehicle (including wings and tails), and the potential to include new technologies. In order to facilitate the large number of computations required for comparative analyses, designers rely heavily on past experiences with known vehicles—using a combination of intuition and empirical comparisons to make decisions on the layout of these initial configurations. A few (usually no more than two or three) of these potential configurations will move forward into the Preliminary Design
Phase.
At this point, each of the selected aircraft are analyzed in more detail. In many cases, this is the first physical analysis and/or experimental testing associated with the candidate configurations. The planforms are adjusted as needed to update for any identified deficiencies, such as refined tail or control surface sizing, but the OML is generally limited to one of the selected configurations. The best of these designs is selected to move into the Detailed Design Phase, marking Milestone A of the Department of Defense (DoD) Acquisition Cycle [2] mentioned in
Chapter I. At this point, the goal is to finalize the aircraft detailed design and begin low rate production. The effort spent on Detailed Design is immense relative to the previous two phases, making any sort of major change requiring re-design efforts extremely costly. In order to avoid such costly mistakes, Ref. [2] also recommends that at least two alternative concepts should be evaluated prior to Milestone A. This is a large portion of the overall mission within MSTC—to enable not just two, but tens of concepts to be evaluated with preliminary design techniques for the same level of current conceptual design effort.
However, as fidelity increases, so too does computational costs. Specifically, in the case of aerodynamics, only the highest levels of fidelity can capture the physics of some innovative control
7 effectors. For this reason, research is being performed into using a combination of programs of
differing fidelity to capture different aspects of the airplane [16]. Other research focuses on
applying this work in optimization algorithms by generating aerodynamic sensitivities with high-
fidelity codes through various methods, particularly the adjoint because it is considered more cost
effective when the number of design variables is greater than the number of constraints/responses
of interest [17].
With advances in computers, what is considered reasonable analysis in early design is
changing rapidly. Whereas in the aircraft design efforts of just a few decades ago, physics-based
linear panel-type aerodynamic programs were considered too computationally costly [18], these
tasks are accomplished in much less than a second with modern computing power and are often
incorporated in conceptual design processes. Today, much of the focus of MADO efforts within
MSTC is on automating tools to perform preliminary design level of analysis. Allison [19] presents
a design process including of the types of analysis and tools desired in the MADO of an ESAV-
type vehicle. These analyses span several disciplines not traditionally included in conceptual design
such as high-fidelity propulsion, external and internal geometry, high-fidelity aerodynamics, engine exhaust-wash structures, weights, mission performance, stability and control, and aeroacoustics. Incorporating these analyses with a preliminary design level of fidelity has been a huge undertaking with many contributors attempting to simplify processes that normally take entire teams several days to accomplish and turning them into tools that individuals can use from their desk – albeit with the help of distributed computing.
To this end, Burton [20], [21] has worked to create a distributed computing environment to enable this analysis and demonstrated an MADO framework that includes geometry generation, aerodynamic analysis, aeroelastic analysis and performance analysis. Haimes [22] has created the
Engineering SketchPad (ESP) for rapidly generating parameterized, multi-fidelity geometry for
MADO. Alyanak [3] focused on creating aircraft structural models early in conceptual design and later applied this to an MADO incorporating aeroelastic analysis with weight optimization [23].
8 Bryson [24] recently implemented a similar approach to aeroelasticity, but with high-fidelity
parameterized CAD-based structural geometry created in ESP. As Morris notes in [25], what these
and many other MADO efforts like them have in common is that they “routinely neglect aircraft
stability and control analyses beyond simple static assessments.”
This research is focused on developing such methods to automate the design and analysis techniques that are required for physics-based analysis of aircraft utilizing innovative control effectors during conceptual design. Achieving this goal would have an immediate impact on aircraft design in its current state, but ultimately, the goal is to apply these techniques to aircraft MADO.
This dissertation specifically focuses on one aspect of this work within MSTC: how to incorporate physics to analyze the stability and control characteristics of potentially non-traditional, aircraft configurations within an optimization loop.
2.3 Controls in Conceptual Design and Preliminary Design
Traditionally, aircraft controllability analysis in conceptual design is limited to little more than tail volume coefficients based on empirical data from previous aircraft designs and computation of static margins [1], [26]. The volume coefficients are based on the relative size and location of the tails and the wing. For the horizontal and vertical tails, respectively, they are given by:
= , and (2.1) 𝑆𝑆ℎ ℎ ℎ 𝑉𝑉 =𝑥𝑥 𝑆𝑆 𝑐𝑐̅ , (2.2) 𝑆𝑆𝑣𝑣 𝑣𝑣 𝑣𝑣 where, and is the distance from the𝑉𝑉 wing𝑥𝑥 quarter𝑆𝑆 𝑐𝑐̅ -chord to the horizontal and vertical tail
ℎ 𝑣𝑣 quarter-𝑥𝑥chord locations,𝑥𝑥 respectively, and are the tail areas, is the wing area, and is the wing mean aerodynamic chord. With the𝑆𝑆ℎ volume𝑆𝑆𝑣𝑣 coefficient selected𝑆𝑆 from previous empirical𝑐𝑐̅ data and the wing parameters determined from weight estimations, the tail planform design is mostly limited to a tradeoff between location and size.
9 As designers move toward the preliminary design phase, they begin to gather enough data
to consider dynamic stability analyses and aircraft flying qualities. Requirements for the flying
qualities are typically found in MIL-STD-1797 [27] for military aircraft. These requirements focus on the performance of the aircraft in various flight conditions and the response to disturbances such as gusts. The dynamics of the aircraft are considered to be linearized and decoupled so that the longitudinal—pitch axis—and lateral/directional—roll and yaw axes—can be evaluated independently. This leads to five distinct modes typical of the aircraft; two in the longitudinal axis and three in the lateral/directional axes. In the longitudinal axis, limits are placed on both the frequency and damping of the fast, heavily damped short period mode and on the damping of the slow, lightly damped Phugoid mode. In the lateral/directional axes, limits are placed on the
frequency and damping of the highly coupled Dutch Roll, the time constant of the roll mode, and
the time to double of the typically unstable spiral mode. In addition to the limits on the various
aircraft modes, the military flying qualities requirements also place limits on the aircraft roll
performance—the ability of the aircraft to roll a specified angle within a limited time frame—
crosswind landings, maneuvers, and many others.
The typical stability and control analysis in preliminary design uses the algebraic derivation of
these modes computed from the roots of the characteristic equations of the transfer functions of the
aircraft response due to elevator, rudder, or aileron inputs. Roskam [28] claims that the analysis of
dynamic stability with the controls held free is beyond the scope of preliminary design since it
requires detailed design data on the flight control system. He therefore stops short of using physics
in his preliminary assessment of controllability, instead offering empirical factors for limitations of
required control augmentation. In this method, the de-facto or augmented control moments required
are computed and subtracted from the bare airframe moments. This value is divided by the
effectiveness of each (traditional) control effector for the respective rotational axes, both statically
and dynamically. For instance, to achieve de-facto static pitch stability,
10 = (2.3) 𝑚𝑚 𝑚𝑚 𝐶𝐶 𝛼𝛼𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 −𝐶𝐶 𝛼𝛼 𝛼𝛼 𝑚𝑚 𝑘𝑘 𝐶𝐶 𝛿𝛿𝑒𝑒 is computed. In this equation, is the non-dimensional pitch moment derivative with respect to
𝑚𝑚𝛼𝛼 angle of attack (or pitch stiffness),𝐶𝐶 is the pitch moment derivative with respect to elevator
𝑚𝑚𝛿𝛿𝑒𝑒 deflection, and is the desired𝐶𝐶 pitch stiffness. To achieve this stability while meeting
𝐶𝐶𝑚𝑚𝛼𝛼𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 desired short-period frequency and damping requirements,
= 𝑐𝑐� (2.4) 𝑚𝑚 𝑚𝑚 �𝐶𝐶 𝑞𝑞𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 −𝐶𝐶 𝑞𝑞��2𝑈𝑈� 𝑞𝑞 𝑚𝑚 𝑘𝑘 𝐶𝐶 𝛿𝛿𝑒𝑒 must also be computed, where, is the pitch damping derivative—the pitch moment with respect
𝑚𝑚𝑞𝑞 to pitch rate, is the aircraft 𝐶𝐶velocity, and is the desired pitch damping derivative,
𝑈𝑈 𝐶𝐶𝑚𝑚𝑞𝑞𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 which is computed by:
= 𝑍𝑍𝛼𝛼 . (2.5) 𝑛𝑛 𝑠𝑠𝑠𝑠 ( ) 𝛼𝛼 �−2𝜔𝜔 𝑠𝑠𝑠𝑠𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑜𝑜 𝜁𝜁 𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓−� 𝑈𝑈 �−𝑀𝑀 � 𝑞𝑞 2 𝑚𝑚 𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 2𝐼𝐼𝑦𝑦𝑦𝑦𝑈𝑈 ⁄ 𝑞𝑞�𝑆𝑆𝑐𝑐̅ In this equation, 𝐶𝐶 and are �the desired� short period frequency and damping
𝜔𝜔𝑛𝑛𝑠𝑠𝑠𝑠𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝜁𝜁𝑠𝑠𝑠𝑠𝑑𝑑𝑑𝑑−𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 values, respectively, is the dimensional Z-axis derivative with respect to angle of attack, is
𝛼𝛼 𝛼𝛼 the dimensional pitch𝑍𝑍 stiffness, is the y-axis moment of inertia, is the dynamic pressure,𝑀𝑀 and
𝑦𝑦𝑦𝑦 the remaining terms have been defined𝐼𝐼 above. Roskam uses this same𝑞𝑞� concept to meet other flying
qualities as well. For each k value, he gives an empirical rule of thumb for allowable limits. Since a major focus of this work is to bring physics into the MADO controllability analysis, these equations in their current form are not acceptable. As part of the search for a method of performing the controllability analysis in the loop with physics involved in the process, a review of flight controls in modern aircraft design is presented below.
2.4 The Control Configured Vehicle (CCV)
In the mid-1970’s, flight control research was focused on the concept of a Control
Configured Vehicle (CCV) [18]. CCV has been defined in multiple ways and it could be said that
11 perhaps the first CCV was the YB-49 Flying Wing, which was also likely the first aircraft
successfully flown with stability augmentation using a quasi-fly-by-wire configuration [29]. In fact,
the term ‘stability augmenter’ stems from this aircraft. However, the focus of this dissertation is on
these modern designs that began incorporating computers from the 1970’s until now—e.g. what
tools enabled vehicles such as the X-29 to fly with a 35% unstable static margin [18].
Figure 2-1. Northrup YB-49 [30] (left) and Grumman X-29 [31] (right)
For many, CCV is considered synonymous with fly-by-wire technology, as it was enabled by computerized flight. However, it is much more than that. In fact, fly-by-wire technology could be applied to any aircraft as in the Fly-By-Wire F-4 program [32], whereas CCV technology has enabled the development of new aircraft designs. “The original [and present] Air Force interpretation is that it is a design concept. The most significant payoffs in terms of performance and mission effectiveness will be realized with the inclusion of advanced flight control technology in the initial design and configuration definition of new aircraft” [32]. These CCV designs were the first attempts at incorporating physics into controls analysis in a multidisciplinary environment early in the aircraft design cycle.
With computers incorporated in aircraft flight control, pilots no longer have to directly
control each effector. Taking advantage of this, the CCV efforts investigated new concepts such as
Relaxed Static Stability (RSS)—using an inherently unstable aircraft planform to enhance
12 maneuverability while requiring a computer to maintain safe flight—and direct force control— using an overdetermined set of actuators to directly control the forces on the aircraft in all 6 degrees of freedom [32]. In the following paragraphs, a review of several of these design efforts and the methods incorporated in the conceptual and preliminary design phase are presented.
In the CCV program described in [32], the recently enabled technologies listed above were tested on an experimental version of the F-16. Although the work was to start with a production aircraft, it was divided into four phases, beginning with a preliminary design phase, moving through detail design and ultimately ending in a flight test. The preliminary design phase was initiated with two potential concepts: one with a twin vertical canard added below the forward inlet and one with horizontal canards mounted on the upper forward fuselage. These extra control surfaces were primarily used to test direct lift and direct side force techniques, along with further reducing the F-
16’s already unstable static margin. With safety and cost as primary drivers in this program however, a major constraint was a requirement to retain the original flight control system so that the CCV technologies could be ‘turned off’ and the vehicle could safely return to normal flight modes.
Rynaski [33] investigated several other control concepts such as maneuver load control— controlling the force distribution along the wing during a maneuver, relaxed static stability, and incorporating flying qualities by forcing a CCV with various combinations of control surface deflections to follow a defined reference model. According to Rynaski, “If sufficient control power is available, the vehicle can have almost any shape and stability augmentation, within the present state of the art, can alter the flying qualities to the desirable Level I behavior.” Although his work was applied to an existing T-33 aircraft, the authors acknowledge again that the more effective use of CCV concepts would come from incorporating them into the preliminary design phase of aircraft design.
In the 1990’s, Anderson [18] suggested that very few if any flight control design techniques were automated to the point that they could be embedded in an automated design procedure that
13 does not allow human intervention. This fact combined with lack of computing resources at the
time for analyzing aircraft led him to create a fuzzy-logic based flight control system risk
assessment algorithm. In this algorithm, the complexity of the controller was assessed against the
vehicle mission in order to assign a risk in conceptual design rather than actually defining a
controller. This lack of computing power was a common theme through all of the CCV efforts.
Livne [34] backed up this claim in 1999. In a special issue of AIAA’s Journal of Aircraft dedicated entirely to Multidisciplinary Design Optimization of Aerospace Systems, Livne said the number of papers related to structural optimization were purposefully limited due to the relative maturity of the field. Whereas, even with solicitation, an in-depth discussion of how to integrate a control system optimization with an optimization of other disciplines was missing. He writes that the framework until this time of examining multidisciplinary interactions between aerodynamics, loads, structure, weights, control, flight mechanics, and even propulsion was aeroservoelasticity.
2.5 Aeroservoelasticity
Aeroservoelasticity is defined as “a multidisciplinary technology dealing with the interaction of the aircraft's flexible structure, the steady and unsteady aerodynamic forces resulting from the aircraft motion, and the flight control systems” [35]. In recent years, designers of almost every aircraft with high-authority, active control systems have had to consider this phenomenon, whether as part of the design to utilize potential benefits—i.e. active flutter suppression or gust load alleviation—or to avoid undesirable interactions [36]. In Ref. [35], a discussion of several research topics covering these potential benefits are discussed. One topic that is discussed is an integrated structure and control law design methodology based on Linear Quadratic Gaussian
(LQG) control. However, incorporating the structural dynamics of the vehicle into the controller required a high-order controller that was then reduced with potential risk to robustness that required an extra optimization loop to improve performance and stability.
14 Of the modern papers surveyed on this subject (2000 to present), the majority focused on
development of advanced controllers to utilize aeroservoelastic benefits [37][38][39]. However,
several papers surveyed on this subject did incorporate MADO techniques. Brown [40] attempted
to determine optimal size, location, and number of control surfaces placed on a wing to suppress
flutter. Stanford [41] used optimization to reduce the weight of a wing box design subject to flutter
constraints. He also investigated the use of active flutter control to relax the open loop constraints
using a simple LQR control algorithm, allowing the optimization to meet the open loop flutter
constraints without the need to stiffen the structure. Haghighat [42] was able to achieve a 41.5% reduction in wing mass through an optimization that included wing shape, structural thicknesses and an LQR control algorithm that was used to reduce wing loading during maneuvers.
2.6 Controls in MADO
Ozoroski [43] presents an MADO framework developed by NASA specifically for the
Supersonics (SUP) Project of the Fundamental Aeronautics Program (FAP) where they intend to
leverage several disciplines in the design optimization of a supersonic cruise vehicle. This work
led to the development of the Matlab™ Stability and Control Toolbox (MaSCoT) for analyzing
aircraft trim and static stability analysis in the optimization loop. Morris [44] says through private
communication with a NASA researcher, he discovered MaSCoT was updated with dynamic
analysis, but it was later found to be unsatisfactory and removed.
Mieloszyk [45] created an aircraft optimization framework to design a box-wing type
aircraft. This framework included panel-method aerodynamics, basic structural, and flight dynamic
stability analyses. The design variables included 24 wing and 183 structural parameters. The
optimization algorithm used Particle Swarm Optimization (PSO) with parallel computing to reduce
the time needed to compute the analysis required for each particle in the swarm. The handling
qualities of each configuration were computed and compared to MIL-STD-8785C and CS-23
airworthiness requirements, incorporating constraints on the Phugoid, short period, Dutch Roll, and
15 spiral modes. This work led to a RC-class configuration that met all constraints and was
successfully built and flown. Mieloszyk claims to his knowledge “it was the first attempt to
incorporate full flight dynamics constraints in the optimization process of a complete aircraft.”
Cosenze [46] used a modified Routh-Hurwitz Criterion in an MADO process to constrain the design space to only those configurations that meet the flying qualities. This led to a two-step optimization algorithm where the first step consisted of finding feasible configurations in the design space and the second step attempted to optimize performance parameters—horizontal tail weight and drag. In the loop, the optimizer considered both inherently stable platforms and unstable configurations. Two methods of stability augmentation were applied and tested: full-state feedback/pole placement and LQR. Using this optimization on a planform similar to the Airbus
A320-200, the authors found the tail size required for the inherently stable case was 43% larger than estimated from traditional conceptual volume method techniques. Both augmentation techniques in the optimization loop led to the same improvements in performance parameters with a 22% reduction in tail size relative to the baseline.
Perez [47] presents another MADO of a 130 passenger narrow-body aircraft with a goal of maximizing range while meeting individual requirements of various disciplines including weights, aerodynamics, propulsion, performance, and dynamics and control. Perez utilized a state feedback controller and choose gains as part of a lower-level optimization loop relative to the overall optimization. The MADO was performed both with and without the stability augmentation in the loop. The vehicle with augmentation was able to fly with a smaller tail, leading to a 500nm increase in the range.
Although not technically an MADO process, Davies [6] used multidisciplinary analysis to explore MDO-based improvements to the fighter aircraft conceptual design process. This work incorporated low, medium, and high-fidelity phases with a goal of capturing coupled physics early in the design process. This effort was applied to an ESAV-type vehicle through design space exploration with 12 design variables. The focus was on bringing higher fidelity analyses in
16 aeroservoelasticity and variable cycle engines with the analyses attempting to optimize the engine
cycle ‘in the loop’. The resultant configuration was able to achieve a 6.9% increase in subsonic
radius compared to the baseline.
Morris [44] developed an MADO framework incorporating both vehicle design and flying
qualities constraints in the stability and control analysis based on an controller. Since it only
∞ involves solving two Riccati equations, synthesis has a low complexity𝐻𝐻 compared to Linear
∞ Quadratic Gaussian (LQG) synthesis [48]𝐻𝐻. However, traditional control required a tradeoff between performance and robustness. With design performed mostly𝐻𝐻∞ in the frequency domain, there was little control over closed-loop pole locations. However, Chilali presented a method of incorporating Linear Matrix Inequalities (LMI) as constraints on the controller. Morris later
adopted this method in designing his MADO framework for aircraft design.𝐻𝐻∞ His method included
a sub-optimization resulting in feasible solutions that use the aircraft controller in each loop of
the overall optimization which relies on LMI’s for guaranteeing𝐻𝐻∞ that the aircraft meets flying
qualities.
The methods of analyzing aircraft stability and control presented above ranging from CCV
programs to aeroservoelasticity research to modern MADO each have their own benefits. However,
one common trend is an attempt develop fully realizable control algorithms within the optimization
loop. This typically requires either significant human intervention or a secondary inner-loop gain
tuning optimization scheme. While this would be perhaps the most complete analysis possible,
typically, this level of controller detail is not included in the preliminary design phase [28]. For the
work of this dissertation, a simpler S&C analysis is desired to incorporate rapid physical analysis
with preliminary design level of fidelity. Although the method presented by Roskam cannot directly
be applied to non-traditional aircraft with innovative control effectors, it can be modified to
compute actual required aerodynamic moments instead of empirical factors. In fact, a similar
Control Power Required (CPR) approach was cited as being used in the Innovative Control Effector
(ICE) program [11]. Additionally, due to the computational cost of the high-fidelity analysis
17 required to model the potential innovative control effectors of an ESAV vehicle, it is imperative to
keep data requirements to a minimum. For this reason, the CPR approach is implemented for the
MADO framework of this dissertation and is discussed further in Chapter III.
18 CHAPTER III
CONTROL POWER REQUIRED METHODS
In modern control theory, a system is considered controllable if all of the closed-loop poles can be assigned to specified locations [49]. The traditional controllability analysis is performed by computing the rank of the controllability matrix, with full rank meaning the system is controllable.
The goal of the controllability analysis of this work is to use physics to rapidly and robustly evaluate the flying characteristics of the aircraft relative to desired requirements. This means the control system must not only be able to mathematically place the poles as desired, but do so subject to constraints on control deflections. In traditional aircraft design this is an iterative process with the initial analysis based primarily on the layout of the aircraft, assuming that both the aircraft and its control surfaces behave similar to other aircraft within the empirical design space. This means that for most airplanes, the wing and tails are initially placed according to the established volume coefficient parameters of equations 2.1 and 2.2.
For a fighter-type aircraft these values range from 0.2 to 0.51 for a horizontal tail and from
0.041 to 0.13 for a vertical tail. The average of these parameters, applied to the ESAV vehicle could lead to a planform that looks like Error! Reference source not found.. Considering this planform appears to fit well within the traditional design space, there would be a high likelihood it could be controllable.
However as discussed in the previous chapter, there is no established method for situations where the design space leaves the range of empirical data that is available from historical aircraft.
These such cases like the desired tailless ESAV that is a focus of this dissertation are the reason behind the push to bring preliminary design level of fidelity into conceptual design phase, where the aircraft themselves can be directly analyzed using physics. The Control Power Required (CPR) method discussed and derived in this chapter does just this by expanding on well-established
19 Figure 3-1. ESAV Planform with horizontal and vertical tails (red) from conceptual tail volume coefficients
control design principles and applying physical constraints and analyses to them. This requires
collecting a large amount of data through aerodynamic analysis that is not traditionally available in
conceptual design. This data and the methods behind capturing it in the MADO loop will be
discussed further in the subsequent chapters.
As mentioned in the previous chapter, published descriptions of the CPR methods stop
short of incorporating physics, relying instead on the use of empirical factors relating to de-facto stability [28]. Reference [11] suggests an extension to the use of physics in the methods, but has little description of the method. This chapter presents the derivation of the CPR method and its application to planforms which are unstable in both the pitch and lateral/directional axes. The derivation is followed by an academic example where the CPR method is applied the ESAV planform using data from wind tunnel testing.
3.1 Controllability Requirements Definition
Before the CPR method is derived, first a list of desired requirements are made available.
For this work, the requirements are based primarily on the Flying Qualities of MIL-STD-1797B
[27]. They are separated into a series of static and dynamic requirements and are separated further into requirements related to the longitudinal and lateral/directional axes. The static requirements in the longitudinal (or pitch) axis that were selected for this work include the pitch moment required
20 for takeoff rotation, the pitch moment required for trim, and pitch moment required to coordinate roll maneuvers. The Phugoid mode is ignored during this work, but it is shown that the CPR controller has little effect on it. In the lateral/directional axis, the aircraft must be able to maintain heading in a crosswind during takeoff or landing and perform coordinated roll maneuvers within specified time periods. Dynamically, in the longitudinal axis, the aircraft must be capable of rejecting disturbances due to gusts or reacting to commanded pitch maneuvers with limitations on the short period mode frequency and damping. Similarly, limitations are placed on the Dutch Roll mode frequency and damping for lateral gust responses. For this work and initial assessment of the
CPR method, the limitations on the roll and spiral modes are neglected. The specific requirements selected are described below.
The flying qualities are separated by type of vehicle, flight condition, and flying quality level. The ESAV vehicle is a considered a Class IV, fighter type vehicle. The flight conditions requirements are defined by; Category A – “nonterminal Flight Phases that require rapid maneuvering, precision tracking, or precise flight-path control”; Category B – “nonterminal Flight
Phases that are normally accomplished using gradual maneuvers and without precision tracking, although accurate flight-path control may be required”; and Category C – “terminal Flight Phases normally accomplished using gradual maneuvers and usually required accurate flight-path control.”
Each of these categories contains various Flight Phases such as takeoff, landing, air combat, etc. and occasionally there are specific requirements for individual Flight Phases listed, but generally they are separated only by Category. In this work, the aircraft is analyzed in Flight Phases that represent all three Categories. The Level of flying quality is defined in the range of 1 to 3 and correlates to Satisfactory, Tolerable, and Controllable flying qualities. The goal is to achieve Level
1 flying qualities for each requirement.
The roll control effectiveness requirement is defined by the aircraft’s ability to achieve a desired bank angle within a specified time (Table 3-1), which in this case are additionally separated by flight speed within the individual Categories. During takeoff and landing phases, the aircraft
21 may experience crosswinds and still be required to maintain accurate heading (Table 3-2). The short period and Dutch Roll are limited in frequency and damping of these modes (Table 3-3 and
Table 3-4). Finally, the aircraft must be capable of rejecting gusts with magnitudes that vary based on speed an altitude (Table 3-5).
Table 3-1. Roll control effectiveness requirements
Time to Achieve Stated Bank Angle (seconds) Level Speed Range Category A Cat. B Cat. C 𝝓𝝓𝒕𝒕 ° ° ° ° ° + 20 1.1 2.0 1.1 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟗𝟗𝟗𝟗 𝟗𝟗𝟗𝟗 𝟑𝟑𝟑𝟑 𝑉𝑉0𝑚𝑚𝑚𝑚+𝑚𝑚 ≤20𝑉𝑉 ≤ 𝑉𝑉0𝑚𝑚𝑚𝑚𝑚𝑚 1.4 𝑘𝑘𝑘𝑘𝑘𝑘 1.1 1.7 1.1 1 𝑉𝑉0𝑚𝑚𝑚𝑚1.𝑚𝑚4 𝑘𝑘𝑘𝑘𝑘𝑘 ≤ 𝑉𝑉0≤.7 𝑉𝑉0𝑚𝑚𝑚𝑚 𝑚𝑚 1.3 1.7 1.1 0.7𝑉𝑉 0𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑉𝑉 ≤ 𝑉𝑉0𝑚𝑚𝑚𝑚 𝑚𝑚 1.1 1.7 1.1 𝑉𝑉0𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑉𝑉 ≤+𝑉𝑉200𝑚𝑚𝑚𝑚 𝑚𝑚 1.6 2.8 1.3 𝑉𝑉𝑚𝑚𝑚𝑚+𝑚𝑚 ≤20𝑉𝑉 ≤ 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 1.4𝑘𝑘𝑘𝑘 𝑘𝑘 1.5 2.5 1.3 2 𝑉𝑉𝑚𝑚𝑚𝑚1𝑚𝑚.4 𝑘𝑘𝑘𝑘𝑘𝑘 ≤ 𝑉𝑉0≤.7 𝑉𝑉𝑚𝑚𝑚𝑚 𝑚𝑚 1.7 2.5 1.3 0.7𝑉𝑉𝑚𝑚𝑚𝑚 𝑚𝑚 ≤ 𝑉𝑉 ≤ 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 1.3 2.5 1.3 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑉𝑉 ≤+𝑉𝑉𝑚𝑚20𝑚𝑚𝑚𝑚 2.6 3.7 2.0 𝑉𝑉𝑚𝑚𝑚𝑚+𝑚𝑚 ≤20𝑉𝑉 ≤ 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 1.4𝑘𝑘𝑘𝑘 𝑘𝑘 2.0 3.4 2.0 3 𝑉𝑉𝑚𝑚𝑚𝑚1𝑚𝑚.4 𝑘𝑘𝑘𝑘𝑘𝑘 ≤ 𝑉𝑉0≤.7 𝑉𝑉𝑚𝑚𝑚𝑚 𝑚𝑚 2.6 3.4 2.0 0.7𝑉𝑉𝑚𝑚𝑚𝑚 𝑚𝑚 ≤ 𝑉𝑉 ≤ 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 2.6 3.4 2.0 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑉𝑉 ≤ 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚
Table 3-2. Lateral-directional control in crosswinds
Level Crosswind 1 and 2 30 kts 3 15 kts
22 Table 3-3. Short period dynamic requirements
Equivalent Flight Phase Short-period Level 1 Level 2 Level 3 Category Dynamics ( ) 0.28 3.6 0.16 10 A − 1 −2 0.35 1.3 0.25 2 0.15 𝑒𝑒 𝑒𝑒 𝑒𝑒 𝐶𝐶𝐶𝐶𝐶𝐶 ( 𝑔𝑔 /𝑠𝑠𝑠𝑠𝑠𝑠 ) ≤ 𝐶𝐶𝐶𝐶𝐶𝐶1.0 ≤ ≤ 𝐶𝐶𝐶𝐶0𝐶𝐶.6 ≤ 𝜁𝜁 0.085≤ 𝜁𝜁 ≤ 0.038≤ 𝜁𝜁 ≤ 𝜁𝜁 ≥ ( ) 𝜔𝜔 𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠 3𝜔𝜔.6 ≥ 10𝜔𝜔 ≥ B −1 −2 𝑒𝑒 𝑒𝑒 𝑒𝑒 ≤ 𝐶𝐶𝐶𝐶𝐶𝐶 ≤ 𝐶𝐶𝐶𝐶𝐶𝐶 6 , or 𝐶𝐶𝐶𝐶𝐶𝐶 𝑔𝑔 𝑠𝑠𝑠𝑠𝑠𝑠 0.3 2 0.2 2 ≤ ≤ 0.15 𝑇𝑇2 ≥ 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜁𝜁 ) 0.16 ≤ 𝜁𝜁 ≤ 3.6 0.16 ≤ 𝜁𝜁 ≤ 10 C − 1 −2 0.35 1.3 0.25 2 𝜁𝜁 ≥ 0.15 𝑒𝑒 𝑒𝑒 𝑒𝑒 𝐶𝐶𝐶𝐶𝐶𝐶 ( 𝑔𝑔 /𝑠𝑠𝑠𝑠𝑠𝑠 ) ≤ 𝐶𝐶𝐶𝐶0𝐶𝐶.87≤ ≤ 𝐶𝐶𝐶𝐶0𝐶𝐶.6 ≤ 0.6 𝜁𝜁 ≤ 𝜁𝜁 ≤ ≤ 𝜁𝜁 ≤ 𝜁𝜁 ≥ 𝜔𝜔 𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 ≥ 𝜔𝜔 ≥ 𝜔𝜔 ≥
Table 3-4. Minimum Dutch Roll frequency and damping
Flight Phase Level Minimum Minimum Category A (CO, GA, RR, TF, 0.40 1.00 RC, FF, and AS) 𝜻𝜻 𝝎𝝎 1 A 0.19 1.00 B 0.08 0.40 C 0.08 1.00 2 All 0.02 0.40 3 All 0.00 0.40
Table 3-5. Gust magnitudes
Speed/Altitude At or below 50,000k ft 20,000 ft VG 66 ft/sec 38 ft/sec
Vo,max 50 ft/sec 25 ft/sec
Vmax 25 ft/sec 12.5 ft/sec
Up to Vmax(PA) with landing gear 50 ft/sec N/A and other devices
23 3.2 Control Power Required Background
In traditional aircraft design, near the end of preliminary design phase, controls engineers will begin to design actual flight controllers and develop them in a simulation environment where they can tune gains and test responses. For several years now many engineers have been trying to automate this process with marginal success. In order to meet the flying qualities requirements, there is a lot of human in the loop interaction of the controller design and tuning. In some instances such as with Morris [44], there has been success in developing a controller that includes some sort of optimization of the controller itself along with a concurrent analysis of the aircraft planform.
However, in this case, the controller development and optimization are separate from the overall configuration optimization and results in a binary yes or no output of whether the controller optimization algorithm was able to converge on a viable solution. What this does not give as information to the overall algorithm is whether the sub-optimization simply failed, whether the aircraft is truly controllable despite the algorithm, or most importantly how close the controller is to being able to successfully control the airplane. This is the primary advantage of the CPR method when applied in an optimization sense. The math of the algorithm, although a lower fidelity than some of the other methods using LQR or H∞, gives a finite measure of control moments that would be needed to be produced in order to control the aircraft in a desired manner. Comparing these moments to the values the aircraft is able to produce returns a finite mathematical and measureable representation of how much the aircraft needs to change to become controllable. The CPR method itself is based on a dynamic inversion controller. The derivation of the controller starts with the aircraft equations of motion, taken from Reference [49] and listed in Table 3-6.
24
Table 3-6. Body-Axes 6-DOF Aircraft Equations of Motion
= sin + ( + )
𝐷𝐷 𝐴𝐴 𝑇𝑇 Force Equations =𝑈𝑈̇ 𝑅𝑅𝑅𝑅+− 𝑄𝑄𝑄𝑄+− 𝑔𝑔sin 𝜃𝜃cos 𝑋𝑋+ ( 𝑋𝑋+ ⁄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚)
𝐷𝐷 𝐴𝐴 𝑇𝑇 𝑉𝑉̇ =−𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃+ 𝑔𝑔 cos 𝜙𝜙cos 𝜃𝜃+ ( 𝑌𝑌 + 𝑌𝑌 ) ⁄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
𝐷𝐷 𝐴𝐴 𝑇𝑇 𝑊𝑊̇ 𝑄𝑄𝑄𝑄 − 𝑃𝑃𝑃𝑃 𝑔𝑔 𝜙𝜙 𝜃𝜃 𝑍𝑍 𝑍𝑍 ⁄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
= + tan ( sin + cos )
𝜙𝜙̇ 𝑃𝑃 = cos𝜃𝜃 𝑄𝑄 𝜙𝜙sin𝑅𝑅 𝜙𝜙 Kinematic Equations =𝜃𝜃(̇ sin𝑄𝑄 +𝜙𝜙 −cos𝑅𝑅 ) 𝜙𝜙cos
𝜓𝜓̇ 𝑄𝑄 𝜙𝜙 𝑅𝑅 𝜙𝜙 ⁄ 𝜃𝜃
= [ + ] [ ( ) + ] + + 2 𝑋𝑋𝑋𝑋 𝑋𝑋 𝑌𝑌 𝑍𝑍 𝑍𝑍 𝑍𝑍 𝑌𝑌 𝑋𝑋𝑋𝑋 𝑍𝑍 𝑋𝑋𝑋𝑋 Γ𝑃𝑃̇ 𝐼𝐼 𝐼𝐼 − 𝐼𝐼= (𝐼𝐼 𝑃𝑃𝑃𝑃 −) 𝐼𝐼 𝐼𝐼 −(𝐼𝐼 𝐼𝐼 ) 𝑄𝑄+𝑄𝑄 𝐼𝐼 𝑙𝑙 𝐼𝐼 𝑛𝑛 2 2 Moment Equations 𝑌𝑌 𝑍𝑍 𝑋𝑋 𝑋𝑋𝑋𝑋 = [( 𝐼𝐼 𝑄𝑄)̇ +𝐼𝐼 −]𝐼𝐼 𝑃𝑃𝑃𝑃 − 𝐼𝐼[ 𝑃𝑃 −+𝑅𝑅 ] 𝑚𝑚+ + 2 𝑋𝑋 𝑌𝑌 𝑋𝑋 𝑋𝑋𝑋𝑋 𝑋𝑋𝑋𝑋 𝑋𝑋 𝑌𝑌 𝑍𝑍 𝑋𝑋𝑋𝑋 𝑋𝑋 Γ𝑅𝑅̇ 𝐼𝐼 − 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝑃𝑃𝑃𝑃=− 𝐼𝐼 𝐼𝐼 − 𝐼𝐼 𝐼𝐼 𝑄𝑄𝑄𝑄 𝐼𝐼 𝑙𝑙 𝐼𝐼 𝑛𝑛 2 𝑋𝑋 𝑍𝑍 𝑋𝑋𝑋𝑋 Γ 𝐼𝐼 𝐼𝐼 − 𝐼𝐼
= + ( + ) + ( + )
𝑁𝑁 𝑝𝑝̇ = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 + 𝑉𝑉(−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐+ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) + 𝑊𝑊( 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) Navigation Equations 𝐸𝐸 𝑝𝑝̇ 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑉𝑉 𝑐𝑐=𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑊𝑊 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
ℎ̇ 𝑈𝑈𝑈𝑈𝑈𝑈 − 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 − 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊
25 3.3 Linearized Aircraft Model
The approach to modern aircraft flight control is still mostly based on the works of G.H.
Bryan [50], published in 1911. In his book Bryan introduced the application of small perturbation theory to stability and control of aircraft. Studying the linearized motions of airplanes, he discovered the separation of longitudinal and lateral motions and went on to invent stability derivatives as they are known today. Since that time, the only major change in mainstream flight controls has been to the orientation of his axis system in relation to what is used for linear aircraft theory and flying qualities today [29]. Typically, the algebraic derivation is performed in the modern definition of the aircraft stability axis. However, the “linear equations needed for control system design will mostly be derived by numerical methods from the nonlinear computer model”
[49] which is taken directly from the equations of motion above—typically in the body axis. The resulting stability characteristics and ultimately the controller should be equivalent in either case, but it is important to maintain consistency throughout the process.
In Reference [49], Stevens suggests that the derivation of the linearized aircraft model in the stability axis has advantages for both linearization and decoupling. The primary reasoning for this is that states of concern for stability and control include the total velocity and the aerodynamic terms relating to angle of attack and sideslip. These terms are inherently included in the stability axis equations of motion rather than requiring additional work to include them as part of the partial differentiation process. The force equations of motion in the stability axis are:
= ( + ) + , (3.1)
=𝑚𝑚𝑉𝑉̇𝑇𝑇 𝐹𝐹(𝑇𝑇 𝑐𝑐𝑐𝑐+𝑐𝑐 𝛼𝛼 ) 𝛼𝛼𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽+− 𝐷𝐷 𝑚𝑚𝑔𝑔1 , and (3.2)
𝑚𝑚𝛽𝛽̇𝑉𝑉𝑇𝑇= −𝐹𝐹𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐( 𝛼𝛼+ 𝛼𝛼𝑇𝑇) 𝑠𝑠𝑠𝑠𝑠𝑠+𝛽𝛽 − 𝐶𝐶 +𝑚𝑚𝑔𝑔2 −( 𝑚𝑚𝑉𝑉𝑇𝑇𝑅𝑅𝑠𝑠 ), (3.3) where, 𝑚𝑚𝛼𝛼̇𝑉𝑉𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽 −𝐹𝐹𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠 𝛼𝛼 𝛼𝛼𝑇𝑇 − 𝐿𝐿 𝑚𝑚𝑔𝑔3 𝑚𝑚𝑉𝑉𝑇𝑇 𝑄𝑄 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽 − 𝑃𝑃𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 𝛽𝛽
= ( + + ) = ,
𝑔𝑔1 𝑔𝑔𝐷𝐷=−𝑐𝑐𝑐𝑐(𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 )−, and𝑔𝑔𝐷𝐷 𝑠𝑠 𝑠𝑠𝑠𝑠 𝛾𝛾
𝑔𝑔2 𝑔𝑔𝐷𝐷 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐= ( 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐+𝑐𝑐𝑐𝑐 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠).
𝑔𝑔3 𝑔𝑔𝐷𝐷 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 26 The moment equations of motion in the stability axis are:
= + ( [ + ] [ ( ) + ] + + ), (3.4) 1 ′ ′ ′ ′ ′ ′ ′ 2 ′ ′ 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑃𝑃̇ 𝛼𝛼̇𝑅𝑅 𝛤𝛤 𝐽𝐽𝑋𝑋𝑋𝑋 𝐽𝐽=𝑋𝑋 −( 𝐽𝐽𝑌𝑌 𝐽𝐽𝑍𝑍) 𝑃𝑃 𝑄𝑄 − 𝐽𝐽𝑍𝑍 (𝐽𝐽𝑍𝑍 − 𝐽𝐽𝑌𝑌 ) +𝐽𝐽′𝑋𝑋𝑋𝑋, 𝑄𝑄and𝑅𝑅 𝐽𝐽𝑍𝑍𝑙𝑙 𝐽𝐽𝑋𝑋𝑋𝑋𝑛𝑛 (3.5) ′ ′ ′ ′ 2 2 = + (𝐽𝐽[𝑌𝑌(𝑄𝑄̇ 𝐽𝐽𝑍𝑍)− 𝐽𝐽+𝑋𝑋 𝑃𝑃𝑠𝑠𝑅𝑅]𝑠𝑠 − 𝐽𝐽𝑋𝑋𝑋𝑋 𝑃𝑃𝑠𝑠 [− 𝑅𝑅𝑠𝑠 +𝑚𝑚 ] + + ), (3.6) 1 ′ ′ ′ 2 ′ ′ ′ ′ ′ ′ where, 𝑅𝑅̇𝑠𝑠 −𝛼𝛼̇𝑃𝑃𝑠𝑠 𝛤𝛤 𝐽𝐽𝑋𝑋 − 𝐽𝐽𝑌𝑌 𝐽𝐽𝑋𝑋 𝐽𝐽′𝑋𝑋𝑋𝑋 𝑃𝑃𝑠𝑠𝑄𝑄 − 𝐽𝐽𝑋𝑋𝑋𝑋 𝐽𝐽𝑋𝑋 − 𝐽𝐽𝑌𝑌 𝐽𝐽𝑍𝑍 𝑄𝑄𝑅𝑅𝑠𝑠 𝐽𝐽𝑋𝑋𝑋𝑋𝑙𝑙 𝐽𝐽𝑋𝑋𝑛𝑛
= . ′ ′ 2 The subscript ‘s’ denotes stability axis forΓ the𝐽𝐽 𝑋𝑋roll𝐽𝐽𝑍𝑍 −and𝐽𝐽′ 𝑋𝑋𝑋𝑋yaw rates.
For stability and control, the navigation equations are ignored in the linear analysis. The remaining states in the stability axis are:
= [ ] , 𝑇𝑇 with the control vector: 𝑥𝑥 𝑉𝑉𝑇𝑇 𝛽𝛽 𝛼𝛼 𝜙𝜙 𝜃𝜃 𝜓𝜓 𝑃𝑃𝑠𝑠 𝑄𝑄 𝑅𝑅𝑠𝑠
= [ ] 𝑇𝑇 representing the input from thrust and 𝑢𝑢the three𝛿𝛿𝑡𝑡 𝛿𝛿traditional𝑒𝑒 𝛿𝛿𝑎𝑎 𝛿𝛿 𝑟𝑟aircraft control surfaces: elevator,
aileron, and rudder, respectively. The linearized model is based on small perturbations to these
equations of motion, assuming that the aircraft is initially in a steady-state condition ( = =
𝑠𝑠 ). A Taylor series expansion is performed on the equations of motion about the equilibrium𝑃𝑃 point𝑄𝑄 to𝑅𝑅𝑠𝑠 determine the state-space representation of the system:
= + , (3.7)
where, 𝐸𝐸𝑥𝑥̇ 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵
= = = , 𝛻𝛻𝑥𝑥̇ 𝑓𝑓1 𝛻𝛻𝑥𝑥𝑓𝑓1 𝛻𝛻𝑢𝑢𝑓𝑓1
𝐸𝐸 − � ⋮ � 𝑒𝑒 𝐴𝐴 � ⋮ � 𝑒𝑒 𝐵𝐵 � ⋮ � 𝑒𝑒 𝑥𝑥̇ 9 𝑈𝑈=𝑈𝑈 𝑥𝑥 9 𝑈𝑈=𝑈𝑈 𝑢𝑢 9 𝑈𝑈=𝑈𝑈 𝛻𝛻 𝑓𝑓 𝑒𝑒 𝛻𝛻 𝑓𝑓 𝑒𝑒 𝛻𝛻 𝑓𝑓 𝑒𝑒 and the subscript ‘e’ denotes equilibrium,𝑋𝑋=𝑋𝑋 or steady state𝑋𝑋=𝑋𝑋 conditions. 𝑋𝑋=𝑋𝑋
It is assumed that the system can be decoupled into longitudinal and lateral/directional
equations. In the Force Equations, this requires an assumption that the lateral/directional controls—
traditionally rudders and ailerons—do not produce drag or cross forces which would lead to an
27 aerodynamic force in the body x-axis equation. Reference [49] acknowledges that this assumption is not true in practice but claims it does not significantly affect the linearized dynamics.
To linearize and decouple the moment equations, it is assumed the engine can contribute
only a pitching moment to the aircraft. This eliminates the application to propeller aircraft which
can have a strong yaw moment due to the propeller at high angles of attack. However, this is hardly
a concern for modern fighter aircraft of interest. Derivatives in the roll and yaw axes with respect
to angle of attack and derivatives in the pitch axis with respect to sideslip are eliminated.
Additionally, all of the angle of attack rate terms are expected to be small and set to zero. These
assumptions lead to the decoupled longitudinal and lateral/directional state space equations. In the
longitudinal axis, the state matrices are:
0 0 0 1 0 0 = 𝑇𝑇𝑒𝑒 𝛼𝛼̇ , (3.8) 𝑉𝑉 0− 𝑍𝑍 0 1 0 𝛼𝛼 𝛼𝛼̇ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 −𝑀𝑀 𝑞𝑞 𝐸𝐸 � 0 0 0 1� 𝑇𝑇 𝑣𝑣 𝜃𝜃 + ( + ) 𝑻𝑻 = +𝜶𝜶 𝒒𝒒 𝒗𝒗 + 0𝜽𝜽 , and (3.9) 𝑍𝑍𝛼𝛼 𝑉𝑉𝑇𝑇𝑒𝑒 𝑍𝑍𝑞𝑞 𝑍𝑍𝑉𝑉 − 𝑋𝑋𝑇𝑇𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝛼𝛼𝑒𝑒 𝛼𝛼𝑇𝑇 −𝑔𝑔𝐷𝐷 𝑠𝑠𝑠𝑠𝑠𝑠 𝛾𝛾𝑒𝑒 ⎡ 0 + ( + ) ⎤ 𝐴𝐴𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑀𝑀𝛼𝛼 𝑀𝑀𝑇𝑇𝛼𝛼 𝑀𝑀𝑞𝑞 𝑀𝑀𝑉𝑉 𝑀𝑀𝑇𝑇𝑉𝑉 ⎢ 0 1 0 0 ⎥ ⎢ 𝑋𝑋𝛼𝛼 𝑋𝑋𝑉𝑉 𝑋𝑋𝑇𝑇𝑉𝑉 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼𝑒𝑒 𝛼𝛼𝑇𝑇 −𝑔𝑔𝐷𝐷 𝑐𝑐𝑐𝑐𝑐𝑐 𝛾𝛾𝑒𝑒⎥ ⎣ ⎦ ( + ) 𝒆𝒆 𝒕𝒕 = 𝜹𝜹 𝜹𝜹 . (3.10) 𝑍𝑍𝛿𝛿𝑒𝑒 −𝑋𝑋𝛿𝛿𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠 𝛼𝛼𝑒𝑒 𝛼𝛼𝑇𝑇 ⎡ ( + ) ⎤ 𝐵𝐵𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑀𝑀𝛿𝛿𝑒𝑒 𝑀𝑀𝛿𝛿𝑡𝑡 ⎢ 0 0 ⎥ ⎢𝑋𝑋𝛿𝛿𝑒𝑒 𝑋𝑋𝛿𝛿𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼𝑒𝑒 𝛼𝛼𝑇𝑇 ⎥ In the lateral/directional axes, the state matrices⎣ are: ⎦
0 0 0 0 1 0 0 = 𝑇𝑇𝑒𝑒 , (3.11) 𝑉𝑉0 0 1 0 𝛽𝛽 𝜙𝜙 𝐸𝐸𝑙𝑙𝑙𝑙𝑙𝑙 � 0 0 0 1� 𝑝𝑝𝑠𝑠 𝑟𝑟𝑠𝑠
𝜷𝜷 𝝓𝝓 𝒑𝒑𝒔𝒔 𝒓𝒓𝒔𝒔 = 0 0 𝑒𝑒 , (3.12) 𝑌𝑌𝛽𝛽 𝑔𝑔𝐷𝐷 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝑒𝑒 𝑌𝑌𝑝𝑝 𝑌𝑌𝑟𝑟 − 𝑉𝑉𝑇𝑇 and ⎡ 0 ⎤ 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐 𝛾𝛾𝑒𝑒⁄𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠 𝛾𝛾𝑒𝑒⁄𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝑒𝑒 𝐴𝐴 ⎢ ′ 0 ′ ′ ⎥ ⎢𝐿𝐿𝛽𝛽 𝐿𝐿𝑝𝑝 𝐿𝐿𝑟𝑟 ⎥ ′ ′ ′ ⎢ 𝛽𝛽 𝑝𝑝 𝑟𝑟 ⎥ ⎣𝑁𝑁 𝑁𝑁 𝑁𝑁 ⎦
28
𝒂𝒂 𝒓𝒓 = 𝜹𝜹0 𝜹𝜹0 . (3.13) 𝑌𝑌𝛿𝛿𝑎𝑎 𝑌𝑌𝛿𝛿𝑟𝑟 𝑙𝑙𝑙𝑙𝑙𝑙 ⎡ ⎤ 𝐵𝐵 ⎢ ′ ′ ⎥ ⎢ 𝛿𝛿𝑎𝑎 𝛿𝛿𝑟𝑟 ⎥ 𝐿𝐿 ′ 𝐿𝐿 ′ ⎢ 𝛿𝛿𝑎𝑎 𝛿𝛿𝑟𝑟 ⎥ In these matrices, the ‘primed derivatives’ represent⎣𝑁𝑁 𝑁𝑁coupling⎦ between the roll and yaw axes and
are given by:
= + , = + , = + , ′ ′ ′ 𝐿𝐿𝛽𝛽 =𝜇𝜇𝐿𝐿𝛽𝛽 +𝜎𝜎1𝑁𝑁𝛽𝛽 , 𝐿𝐿𝑝𝑝 =𝜇𝜇𝐿𝐿𝑝𝑝 + 𝜎𝜎1𝑁𝑁𝑝𝑝, 𝐿𝐿𝑟𝑟 = 𝜇𝜇𝐿𝐿𝑟𝑟 + 𝜎𝜎1𝑁𝑁𝑟𝑟, ′ ′ ′ 𝑁𝑁𝛽𝛽 𝜇𝜇𝑁𝑁𝛽𝛽 =𝜎𝜎2𝐿𝐿𝛽𝛽 +𝑁𝑁𝑝𝑝 𝑁𝑁, 𝑝𝑝 𝜎𝜎2𝐿𝐿=𝑝𝑝 𝑁𝑁𝑟𝑟+ 𝜇𝜇𝑁𝑁𝑟𝑟, 𝜎𝜎2𝐿𝐿𝑟𝑟 ′ ′ 𝐿𝐿𝛿𝛿𝑎𝑎 = 𝜇𝜇𝐿𝐿𝛿𝛿𝑎𝑎 + 𝜎𝜎1𝑁𝑁𝛿𝛿𝑎𝑎, 𝐿𝐿𝛿𝛿𝑟𝑟 = 𝜇𝜇𝐿𝐿𝛿𝛿𝑟𝑟 +𝜎𝜎1𝑁𝑁𝛿𝛿𝑟𝑟 ,. ′ ′ 𝑎𝑎 𝑎𝑎 𝑟𝑟 𝑟𝑟 where, 𝑁𝑁𝛿𝛿𝑎𝑎 𝜇𝜇𝑁𝑁𝛿𝛿 𝜎𝜎2𝐿𝐿𝛿𝛿 𝑁𝑁𝛿𝛿𝑟𝑟 𝜇𝜇𝑁𝑁𝛿𝛿 𝜎𝜎2𝐿𝐿𝛿𝛿
= ( ) , = ( ) , = ( ) , and
𝜇𝜇 𝐼𝐼𝑍𝑍𝐼𝐼𝑋𝑋 ⁄𝛤𝛤 𝜎𝜎1 =𝐼𝐼𝑍𝑍𝐼𝐼𝑋𝑋𝑋𝑋 ⁄𝛤𝛤 . 𝜎𝜎2 𝐼𝐼𝑋𝑋𝐼𝐼𝑋𝑋𝑋𝑋 ⁄𝛤𝛤 2 The B matrix in each system assumes𝛤𝛤 𝐼𝐼𝑋𝑋𝐼𝐼 𝑍𝑍traditional− 𝐼𝐼𝑋𝑋𝑋𝑋 aircraft control effectors and thrust parameters. The pitch axis uses only an elevator and thrust, and the lateral/directional axes use only ailerons and rudders. This is a possible source of contention when innovative control effectors are
used in combination with controllability analyses that may not be able to be decoupled into two
separate B matrices. However, the assumption will be made not that individual innovative control
effectors will decouple, but instead multiple effectors will be combined to result in an
overdetermined system, where various combinations of effectors can be utilized to realize a single
axis moment or combination of desired moments.
3.4 Control Power Required
From the A matrices of Equation 3.7, represented by Equation 3.9 for longitudinal motion
and Equation 3.12 for lateral/directional motion, the uncontrolled aircraft handling characteristics
for each flight condition of interest can be analyzed. However, in modern airplanes these
characteristics are not always desirable, especially in high-performance aircraft with unstable static
29 margins. In these cases, a controller is developed to force the aircraft to behave as desired. The
simplest of these controllers is full state-feedback or pole placement. However, this method is best applied to Single-Input Single-Output (SISO) systems. SISO control works for traditional aircraft designs with the major mode shapes decoupled and directly controlled by elevator, rudder, and aileron. However, modern aircraft have more complicated control effector arrangements. The F-18 for instance, actively uses both flaps and ailerons, along with leading edge devices and differential elevators. A potential ESAV vehicle may have many more control effectors. In cases like these, control methodologies based on Linear Quadratic Regulators (LQR), H , or dynamic inversion are
used to determine the control allocation for the overdetermined systems.∞ Typically, the controllers
rely on model following cost functions that attempt to track the response of various linear models
representing the mode shapes desired. Regardless of the control methodology, if the vehicle
successfully follows the linear model, it can be said to meet the required flying qualities metrics.
In fact, MIL-STD-1797B [27] defines a method of comparing the closed loop non-linear characteristics to a Low Order Equivalent System (LOES) specifically for these cases.
While the solution exists for handling overdetermined systems in practice, most of the stability and control evaluations of Chapter II relied on bringing the entire controller development into the early vehicle analysis. For this work, the CPR method was selected for use in the MADO loop to avoid this additional complexity. This method includes a combination of static and dynamic control requirements. The dynamic requirements are based on the concept of dynamic inversion— e.g. the undesired aircraft dynamics are subtracted and replaced with desired dynamics. In
Equations 3.9 and 3.12, this means solving for the desired, or augmented, moments required to achieve the desired dynamics. The control power required to meet the frequency and damping limits on the short period and Dutch Roll is the difference in the augmented moments and the uncontrolled aircraft moment. The specific requirements for longitudinal and lateral/directional CPR are discussed further below.
30 3.5 Longitudinal CPR
In the longitudinal axis, the controllability assessment is concerned with the aircraft’s ability to control pitch, both statically and dynamically. In the static sense, the airplane must be able to produce a large enough pitch moment to initiate the takeoff rotation before the aircraft even has enough speed to lift off, trim the system at required angles of attack once in the air, and coordinate roll maneuvers around the stability axis. Dynamically, the pitch controller must be able to perform pitch maneuvers and reject stochastic disturbances (e.g. vertical gusts), with responses to these inputs falling within the short period requirements. Figure 3-2 shows how the control power might be allocated relative to the available travel of a particular effector. In this figure, a control margin is allocated to account for gusts. For this work, the control power required for a vertical gust is specifically computed. The ability to include this margin for unknown or unmodeled parameters is maintained throughout this work, but the margin itself is set to zero. Equation 3.14 shows the computation of the overall control power required in the pitch axis. The following sections describe each of the components in this equation.
Figure 3-2. Control Margin Requirements [27]
= + + + + + (3.14)
𝐶𝐶𝑚𝑚𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐶𝐶𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐶𝐶𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐 𝐶𝐶𝑚𝑚𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐶𝐶𝑚𝑚𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚𝑟𝑟𝑔𝑔
31 3.5.1 Takeoff Rotation
The pitch moment required to perform the takeoff rotation is computed from the summation of moments about the center of gravity, assuming the nose wheel has lifted off and the main gear is still on the ground. The reaction force on the main gear is equal to the difference between lift and weight. Both this upward force behind the c.g. and the rearward friction force below the c.g. tend to create downward moments. If the c.g. is forward of the aerodynamic center, the lift also creates a downward moment, all of which must be overcome by the control system. However, for the unstable aircraft of interest, the c.g. is rearward, helping the aircraft to lift the nose. Summing these forces, the pitching moment coefficient required to perform the takeoff rotation is:
( ) = . (3.15) 𝑊𝑊−𝐿𝐿 ��𝑥𝑥𝑚𝑚𝑚𝑚−𝑥𝑥𝑐𝑐𝑐𝑐�+𝜇𝜇�𝑧𝑧𝑐𝑐𝑐𝑐−𝑧𝑧𝑚𝑚𝑚𝑚��−𝐿𝐿�𝑥𝑥𝑎𝑎𝑎𝑎−𝑥𝑥𝑐𝑐𝑐𝑐� 𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑚𝑚0 𝐶𝐶 𝑞𝑞�𝑆𝑆𝑐𝑐̅ − 𝐶𝐶
3.5.2 Trim CPR
Once the aircraft is airborne, it can be trimmed in straight and level flight or with a steady- state climb or turn rate. The required lift in trim condition is the weight multiplied by the desired load factor. In coefficient form, the approximate trim lift is:
= , (3.16) 𝑛𝑛𝑛𝑛 𝐿𝐿 and the angle of attack required to achieve𝐶𝐶 this𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 lift is:𝑞𝑞 𝑞𝑞
= , (3.17) 𝐶𝐶𝐿𝐿𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇−𝐶𝐶𝐿𝐿0 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐿𝐿𝛼𝛼 assuming a linear lift curve slope, .𝛼𝛼 The pitch moment𝐶𝐶 required to maintain this aircraft angle
𝐿𝐿 of attack is: 𝐶𝐶 𝛼𝛼
= . (3.18)
𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐶𝐶 𝑅𝑅𝑅𝑅𝑅𝑅 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 −𝐶𝐶 0 − 𝐶𝐶 𝛼𝛼𝛼𝛼
3.5.3 Roll Coordination
The derivation of longitudinal control power required to coordinate a roll comes directly from the equations of motion for lateral acceleration,
32 = + + + ( + ) , (3.19) and pitch acceleration𝑉𝑉,̇ −𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃 𝑔𝑔𝐷𝐷 𝑠𝑠𝑠𝑠𝑠𝑠 𝜙𝜙 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑌𝑌𝐴𝐴 𝑌𝑌𝑇𝑇 ⁄𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
= ( ) ( ) + . (3.20) 2 2 A coordinated roll requires that𝐼𝐼𝑦𝑦𝑄𝑄 ̇ both𝐼𝐼 of𝑧𝑧 − these𝐼𝐼𝑥𝑥 𝑃𝑃 accelerations𝑃𝑃 − 𝐼𝐼𝑥𝑥𝑥𝑥 𝑃𝑃 − are𝑅𝑅 zero.𝑚𝑚 If the product of inertia about x-z axes is negligible, the two equations can be solved for the pitch moment, m. Assuming there are no axial forces or initial bank angle ( = = = 0), Equation 3.19 reduces to:
0𝑌𝑌𝐴𝐴= 𝑌𝑌𝑇𝑇 +𝜙𝜙 .
Solving for and substituting the definitions−𝑅𝑅𝑅𝑅 for 𝑃𝑃body𝑃𝑃 axis velocities, and , gives the relationship: 𝑅𝑅 𝑈𝑈 𝑊𝑊
= = = . 𝑊𝑊 𝑉𝑉𝑇𝑇 𝑠𝑠𝑠𝑠𝑠𝑠 𝛼𝛼 Equation 3.20 with zero pitch 𝑅𝑅acceleration𝑃𝑃 𝑈𝑈 𝑃𝑃 and𝑉𝑉𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐 negli𝛼𝛼 gible𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 cross𝛼𝛼 -product of inertia gives the relationship:
0 = ( ) + .
Substituting then gives: 𝐼𝐼𝑧𝑧 − 𝐼𝐼𝑥𝑥 𝑃𝑃𝑃𝑃 𝑚𝑚
𝑅𝑅 0 = ( ) tan + . 2 Finally, solving for gives the required 𝐼𝐼pitch𝑧𝑧 − 𝐼𝐼 𝑥𝑥moment𝑃𝑃 to𝛼𝛼 coordinate𝑚𝑚 a roll with a pitch rate, :
𝑚𝑚 = ( ) . 𝑃𝑃 2 The solution in non-dimensional coefficient𝑚𝑚 − 𝐼𝐼form𝑧𝑧 − 𝐼𝐼 is:𝑥𝑥 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝛼𝛼
( ) = . (3.21) 𝐼𝐼𝑧𝑧−𝐼𝐼𝑥𝑥 2 𝑚𝑚 𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑞𝑞 𝑞𝑞𝑞𝑞 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝛼𝛼
3.5.4 Longitudinal Dynamic Response - Short Period
The Flying Qualities standards of MIL-STD-1797B [27] recommend limitations on the mode shapes of the two sets of state equations above. In the longitudinal axis, there are two standard modes: short period and Phugoid. In practice, it has been shown that the short period and Phugoid modes can be excited independently [49]. With this assumption, the short period mode is derived
33 from only the alpha and pitch rate equations in the matrix and with only the elevator as an
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 input in the matrix. Additionally, small angle of𝐴𝐴 attack, small flight path angle, and low Mach
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 number are 𝐵𝐵assumed in the standard definition. The reduced state equations for the short period
mode are:
0 + = + . (3.22) 1 𝑇𝑇𝑒𝑒 𝛼𝛼̇ 𝛼𝛼 𝑇𝑇𝑒𝑒 𝑞𝑞 𝛿𝛿𝑒𝑒 𝑉𝑉 − 𝑍𝑍 𝛼𝛼̇ 𝑍𝑍 𝑉𝑉 𝑍𝑍 𝛼𝛼 𝑍𝑍 𝑒𝑒 � � � � � � � � � 𝑒𝑒� 𝛿𝛿 −𝑀𝑀𝛼𝛼̇ 𝑞𝑞̇ 𝑀𝑀𝛼𝛼 𝑀𝑀𝑞𝑞 𝑞𝑞 𝑀𝑀𝛿𝛿
Assuming a SISO system, the transfer function from elevator input to angle of attack and pitch rate are computed from the transfer function matrix:
+ + ( ) = , ( + ) + −1 𝐶𝐶 �𝑠𝑠 − 𝑀𝑀𝑞𝑞�𝑍𝑍𝛿𝛿𝑒𝑒 �𝑉𝑉𝑇𝑇𝑒𝑒 𝑍𝑍𝑞𝑞�𝑀𝑀𝛿𝛿𝑒𝑒 𝐶𝐶 𝑠𝑠𝑠𝑠 − 𝐴𝐴 𝐵𝐵 ∆𝑠𝑠𝑠𝑠 � � where, is the short period characteristic𝑠𝑠𝑀𝑀𝛼𝛼̇ equation,𝑀𝑀𝛼𝛼 𝑍𝑍𝛿𝛿 𝑒𝑒 �𝑠𝑠�𝑉𝑉𝑇𝑇𝑒𝑒 − 𝑍𝑍𝛼𝛼̇ � − 𝑍𝑍𝛼𝛼�𝑀𝑀𝛿𝛿𝑒𝑒
𝑠𝑠𝑠𝑠 ∆= + + + + + . 2 In standard∆𝑠𝑠𝑠𝑠 � 𝑉𝑉form,𝑇𝑇𝑒𝑒 − 𝑍𝑍a𝛼𝛼 seconḋ �𝑠𝑠 − �order𝑍𝑍𝛼𝛼 transfer�𝑉𝑉𝑇𝑇𝑒𝑒 − 𝑍𝑍 function𝛼𝛼̇ �𝑀𝑀𝑞𝑞 takes�𝑉𝑉𝑇𝑇𝑒𝑒 the𝑍𝑍 𝑞𝑞form:�𝑀𝑀𝛼𝛼̇ � 𝑠𝑠 𝑀𝑀𝑞𝑞𝑍𝑍𝛼𝛼 − �𝑉𝑉𝑇𝑇𝑒𝑒 𝑍𝑍𝑞𝑞�𝑀𝑀𝛼𝛼
( ) = . 2 𝜔𝜔𝑛𝑛 2 2 𝑛𝑛 Relating this to the equations above, the𝐺𝐺 characteristic𝑠𝑠 𝑠𝑠 +2𝜁𝜁𝜔𝜔 𝑠𝑠equation+𝜔𝜔𝑛𝑛 becomes:
= + . 𝛼𝛼 𝑇𝑇 𝛼𝛼̇ 𝑞𝑞 𝑇𝑇 𝑞𝑞 𝛼𝛼̇ 𝑞𝑞 𝛼𝛼 𝑇𝑇 𝑞𝑞 𝛼𝛼 2 �𝑍𝑍 +�𝑉𝑉 𝑒𝑒−𝑍𝑍 �𝑀𝑀 +�𝑉𝑉 𝑒𝑒+𝑍𝑍 �𝑀𝑀 � 𝑀𝑀 𝑍𝑍 −�𝑉𝑉 𝑒𝑒+𝑍𝑍 �𝑀𝑀 𝑠𝑠𝑠𝑠 𝑇𝑇 𝛼𝛼̇ 𝑇𝑇 𝛼𝛼̇ Combining terms ∆and comparing𝑠𝑠 − to standard�𝑉𝑉 𝑒𝑒−𝑍𝑍 form,� the frequency𝑠𝑠 and�𝑉𝑉 𝑒𝑒 damping−𝑍𝑍 � of the short period
mode are given by:
= and 𝑞𝑞 𝛼𝛼 𝛼𝛼 𝑇𝑇 𝑞𝑞 2 𝑀𝑀 𝑍𝑍 −𝑀𝑀 �𝑉𝑉 𝑒𝑒+𝑍𝑍 � 𝑠𝑠𝑠𝑠 𝜔𝜔𝑛𝑛 𝑉𝑉𝑇𝑇𝑒𝑒−𝑍𝑍𝛼𝛼̇ 2 = + , 𝑀𝑀𝛼𝛼̇ �𝑉𝑉𝑇𝑇𝑒𝑒+𝑍𝑍𝑞𝑞�+𝑍𝑍𝛼𝛼 𝑠𝑠𝑠𝑠 𝑛𝑛𝑠𝑠𝑠𝑠 𝑞𝑞 𝑇𝑇 𝛼𝛼̇ respectively. − 𝜁𝜁 𝜔𝜔 𝑀𝑀 𝑉𝑉 𝑒𝑒−𝑍𝑍
However, and are normally small compared to the velocity and they are ignored.
𝑞𝑞 𝛼𝛼̇ Also, it is normally𝑍𝑍 assumed𝑍𝑍 that 0. In the baseline ESAV configuration with DATCOM
𝑀𝑀𝛼𝛼̇ ≈ 34 data, these assumptions accounted for less than 3% variation in the estimates of short period
frequency and damping in the flight conditions examined. The resulting algebraic estimate of short
period frequency and damping that is used for this work is computed by the two equations:
= , and (3.23) 𝑞𝑞 𝛼𝛼 2 𝑀𝑀 𝑍𝑍 𝑠𝑠𝑠𝑠 𝑉𝑉𝑇𝑇 𝛼𝛼 𝜔𝜔2 =− 𝑀𝑀 + . (3.24) 𝑍𝑍𝛼𝛼 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑞𝑞 For desired short period frequency− 𝜁𝜁 𝜔𝜔 and 𝑀𝑀damping𝑉𝑉𝑇𝑇 requirements, the augmented pitch
stiffness that must be achieved by deflecting a control surface is computed from Equations 3.23
and 3.24. As mentioned above, it is assumed that a dynamic inversion type of controller is employed
such that the baseline vehicle dynamics can be subtracted in lieu of desired dynamics. From 3.23,
the augmented pitch moment required with respect to the angle of attack for a desired short period
frequency is:
= + . (3.25) 𝑞𝑞 𝛼𝛼 2 𝑀𝑀 𝑍𝑍 𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑇𝑇 In previous work, this equation𝑀𝑀 was combined−𝜔𝜔 with𝑉𝑉 Equation 3.24 by solving for in one
𝑞𝑞 equation and substituting into the other. Then, solving for the pitch stiffness, , 𝑀𝑀gives the
augmented pitch stiffness required to meet the designer’s specified short period 𝑀𝑀frequency𝛼𝛼 and damping parameters,
= 2 . (3.26) 2 2 𝑍𝑍𝛼𝛼 𝑍𝑍𝛼𝛼 𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 However, is not a dependent𝑀𝑀 variable−𝜔𝜔 in −this𝜁𝜁 SISO𝜔𝜔 transfer�𝑉𝑉𝑇𝑇� − �function𝑉𝑉𝑇𝑇� and cannot be varied as a
𝑞𝑞 function of𝑀𝑀 the designer’s arbitrary choice of parameters. Instead, it is a fixed value that is a function
of the aircraft geometry and flight condition. Some tests have shown errors in required control
deflection of up to four times larger than based on this mistake.
Anecdotally, we have been told these equations are still used in practice, but the only
reference to it found in open literature comes from the analysis of innovative control effectors in
Reference [11]. However, as mentioned in Chapter II, Roskam does not combine these equations
35 in the same manner. Instead, he computes separate augmentation requirements for angle of attack
and pitch rate derivatives and relates them to the elevator effectiveness. This allows independent
augmentation of both and . Roskam stops short of computing actual control power though,
𝛼𝛼 𝑞𝑞 instead opting to relate𝑀𝑀 this to𝑀𝑀 an empirical factor. For this work, Roskam’s algebra is used, but
instead of applying empirical factors, the actual moments required to achieve various flight conditions and maneuvers are computed from Equations 3.25 and
= 2 . (3.27) 𝑍𝑍𝛼𝛼 𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 The difference in the initial𝑀𝑀 method −of 𝜁𝜁Equation𝜔𝜔 − 𝑉𝑉3𝑇𝑇.26 and the updated method can be significant in regard to the response that is desired compared to what is actually achieved. As an example, the F-16 model of Reference [49] was trimmed and linearized in a low-speed, high angle of attack flight condition and resulted in the reduced state matrix representing the short period:
0.4912 0.9376 = . 0.6405 0.4778 − 𝐴𝐴𝑠𝑠𝑠𝑠 � � The eigenvalues of this matrix are: = 1.26, =−0.29, meaning the short period is unstable, 1 2 and the second row represents the𝜆𝜆 pitch− stiffness𝜆𝜆 and damping terms, = 0.6405, =
𝛼𝛼 𝑞𝑞 0.4778, respectively. If the designer chooses to stabilize this mode with𝑀𝑀 the frequency𝑀𝑀 and
damping− parameters selected to be
= 2.0 = 0.7, 𝑟𝑟𝑟𝑟𝑟𝑟 𝜔𝜔𝑠𝑠𝑝𝑝𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝜁𝜁𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 the original CPR method yields the required𝑠𝑠 augmentation, = 3.7667. In contrast, the 𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 updated method results in the two augmentation values are𝑀𝑀 =− 2.8713 and =
𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 2.3117. Substituting both methods into the short period matrix,𝑀𝑀 the resulting− poles are computed𝑀𝑀
and− plotted in Figure 3-3. The unstable open loop poles are plotted as black circles on the real axis,
while the desired closed loop poles are plotted as black squares. Using the original CPR method yields the red ‘x’s, which do not match the desired poles and in this case fall outside the allowable range of pole locations (shaded in blue), despite the selected poles being well within the boundaries.
36
Figure 3-3. Computed short period poles for F-16 example
In both cases, the response was tested in the full non-linear simulation of the F-16. The vehicle was trimmed at approximately eight degrees angle of attack in this condition and subjected to a vertical gust represented by an initial angle of attack condition 3.5 degrees higher. In Figure 3-4, the comparison of the linear response in dashed lines and non-linear response in solid lines shows that this method is a good approximation to the actual vehicle dynamics response for both cases. The difference in dynamic response in the original CPR method in red and the updated CPR method in blue is significant. It is possible if the selected dynamics were closer the right-half plane, the original method could result in an unstable system.
37
Figure 3-4. Simulation of F-16 short period response with original and updated CPR methods
Using these equations for a commanded maneuver with a specified g-loading, the angle of attack is computed from:
= ( 1) . (3.28) 𝑊𝑊 𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐 𝐿𝐿 In coefficient form, the control power𝛼𝛼 required𝑔𝑔 −to achieve�𝑞𝑞𝑞𝑞𝐶𝐶 𝛼𝛼 �this maneuver with the prescribed
dynamics is:
= . (3.29)
𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐 Similarly, for a discrete gust, the𝐶𝐶 control𝑐𝑐𝑐𝑐𝑐𝑐 power�𝐶𝐶 𝛼𝛼 𝑎𝑎required𝑎𝑎𝑎𝑎 − 𝐶𝐶 is𝛼𝛼 �calculated𝛼𝛼 from the change in angle of
attack,
= . (3.30) 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 −1 𝑉𝑉 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 In this case, it is assumed the angle of𝛼𝛼 attack is𝑡𝑡𝑡𝑡𝑡𝑡 changed� 𝑉𝑉 by𝑇𝑇 �the tangent of the vertical gust velocity
relative to the aircraft forward velocity. The control power required to reject the vertical gust is in
turn computed by:
= . (3.31)
𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐶𝐶 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 �𝐶𝐶 𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 − 𝐶𝐶 𝛼𝛼� 𝛼𝛼
38 3.6 Lateral-Directional CPR
The motions of concern in the lateral-directional axes are highly coupled in roll and yaw.
The static requirements for control power required includes yaw and roll moments required to initiate and coordinate a roll of a specified angle within a specified time (roll performance) and to takeoff and land in a crosswind condition. Dynamically, for this work the primary concern is the vehicle’s ability to reject a lateral gusts subject to Dutch Roll response requirements. Similar to the longitudinal case, a margin is available for both roll and yaw requirements to account for unknown or unmodeled quantities, but is set to zero. These requirements are summarized by the equations:
= + + + and (3.32)
𝐶𝐶𝑙𝑙𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶𝑙𝑙𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶𝑙𝑙𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐶𝐶𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐶𝐶𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = + + + + , (3.33) and each is discussed𝐶𝐶𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟 further𝐶𝐶𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 below.𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐶𝐶 𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐶𝐶𝑛𝑛𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐶𝐶𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐶𝐶𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
3.6.1 Roll Performance
The roll performance is specified in terms of the aircraft’s ability to achieve a specified roll angle within a prescribed time. These specifications change with the type of aircraft and phase of flight. In order to compute roll as a function of time, the 1-dimensional differential equation for roll,
+ = , (3.34) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑎𝑎 𝑎𝑎 𝑥𝑥 ̈ is solved. Assuming pure roll ( = ) and𝜕𝜕𝛿𝛿 rearranging,𝛿𝛿 𝜕𝜕𝜕𝜕 𝑝𝑝 the𝐼𝐼 𝜙𝜙 differential equation of interest becomes ̈ 𝜙𝜙 𝑝𝑝̇ + = . 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑥𝑥 𝑎𝑎 By substituting the standard definitions−𝐼𝐼 of𝑝𝑝̇ the𝜕𝜕 𝜕𝜕control𝑝𝑝 − deriv𝜕𝜕𝛿𝛿𝑎𝑎 𝛿𝛿atives,
= and = , 𝜕𝜕𝐿𝐿⁄𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝛿𝛿𝑎𝑎 𝑝𝑝 𝛿𝛿 this equation becomes: 𝐿𝐿 𝐼𝐼𝑥𝑥 𝐿𝐿 𝑎𝑎 𝐼𝐼𝑥𝑥
+ = .
−𝑝𝑝̇ 𝐿𝐿𝑝𝑝𝑝𝑝 −𝐿𝐿𝛿𝛿𝑎𝑎𝛿𝛿𝑎𝑎
39 Using the relationship,
1 =
𝜏𝜏 − 𝑝𝑝 and dividing both sides by yields the pure roll rate𝐿𝐿 differential equation,
𝐿𝐿𝑝𝑝 + = . 𝐿𝐿𝛿𝛿𝑎𝑎𝛿𝛿𝑎𝑎 𝑝𝑝 The solution to this equation is: 𝜏𝜏𝑝𝑝̇ 𝑝𝑝 − 𝐿𝐿
= 1 = ( ) 1 , 𝛿𝛿 𝑎𝑎 𝐿𝐿 𝑎𝑎𝛿𝛿 −𝑡𝑡⁄𝜏𝜏 −𝑡𝑡⁄𝜏𝜏 𝑝𝑝 𝛿𝛿𝑎𝑎 𝑎𝑎 and integrating this roll𝑝𝑝 rate− from𝐿𝐿 time� −t =𝑒𝑒 0 to �time� t,𝐿𝐿 the𝛿𝛿 solution� 𝜏𝜏 � −is 𝑒𝑒the roll� angle equation:
= ( ) 1 . −𝑡𝑡⁄𝜏𝜏 The steady state roll rate is defined𝜙𝜙 as�𝐿𝐿 𝛿𝛿the𝑎𝑎𝛿𝛿 limit𝑎𝑎� 𝜏𝜏 of�𝑡𝑡 p− as𝜏𝜏� time− 𝑒𝑒goes to�� infinity,
= ( ) 1 = ( ). −∞⁄𝜏𝜏 Using this relationship in 𝑝𝑝the𝑠𝑠𝑠𝑠 above�𝐿𝐿𝛿𝛿 𝑎𝑎equation,𝛿𝛿𝑎𝑎� 𝜏𝜏 � the− 𝑒𝑒desired� steady�𝐿𝐿𝛿𝛿𝑎𝑎 state𝛿𝛿𝑎𝑎� roll𝜏𝜏 rate for a specified bank
angle change in time t is
= . 𝜙𝜙 𝑠𝑠𝑠𝑠 −𝑡𝑡⁄𝜏𝜏 Then, substituting this back into the original𝑝𝑝 roll�𝑡𝑡− 𝜏𝜏rate�1−𝑒𝑒 equation�� gives:
= 1 = = . −𝑡𝑡⁄𝜏𝜏 −𝑡𝑡⁄𝜏𝜏 −𝑡𝑡⁄𝜏𝜏 1−𝑒𝑒 1−𝑒𝑒 𝑠𝑠𝑠𝑠 −𝑡𝑡⁄𝜏𝜏 1 −𝑡𝑡⁄𝜏𝜏 𝑝𝑝 𝑝𝑝 � − 𝑒𝑒 � 𝜙𝜙 𝑡𝑡−𝜏𝜏�1−𝑒𝑒 � 𝜙𝜙 𝑡𝑡+𝐿𝐿𝑝𝑝�1−𝑒𝑒 � Multiplying the numerator and denominator by , the roll rate required to achieve the specified
𝑝𝑝 bank angle, , at a specified time, t, is: 𝐿𝐿
𝜙𝜙 = (3.35) −𝑡𝑡⁄𝜏𝜏 𝐿𝐿𝑝𝑝�𝑒𝑒 −1� −𝑡𝑡⁄𝜏𝜏 𝑝𝑝 This is the roll rate that is used in the roll𝑝𝑝 coordination𝜙𝜙 𝑒𝑒 −𝐿𝐿 𝑡𝑡− equations1 for longitudinal control power
required. The required roll moment to perform this maneuver is
= ,
and from above, this is equal to 𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟 𝐿𝐿𝛿𝛿𝑎𝑎𝛿𝛿𝑎𝑎
40 = = . 2 −𝜙𝜙𝐿𝐿𝑝𝑝 −𝜙𝜙𝐿𝐿𝑝𝑝 1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝 𝑒𝑒 −𝐿𝐿𝑝𝑝𝑡𝑡−1 𝐿𝐿 �𝑡𝑡+𝐿𝐿𝑝𝑝�1−𝑒𝑒 �� Finally, in non-dimensional form, the moment required to achieve the desired roll performance is:
= (3.36) 2 . −𝜙𝜙𝐿𝐿𝑝𝑝 𝐼𝐼𝑥𝑥 𝑡𝑡𝑡𝑡𝑝𝑝 𝑙𝑙𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝 𝐶𝐶 �𝑒𝑒 −𝐿𝐿 𝑡𝑡−1� �𝑞𝑞𝑞𝑞𝑏𝑏�
3.6.2 Yaw CPR for Roll Initiation
In order to initiate a coordinated roll about the stability axis, the control system must generate a yaw moment proportional to the required roll moment to keep the yaw acceleration equal to zero. Similar to the roll coordination equations for longitudinal control power required, this derivation begins with the equations of motion for roll and yaw acceleration,
= + + + + (3.37) 2 and 𝛤𝛤𝑃𝑃̇ 𝐼𝐼𝑥𝑥𝑥𝑥�𝐼𝐼𝑥𝑥 − 𝐼𝐼𝑦𝑦 𝐼𝐼𝑧𝑧�𝑃𝑃𝑃𝑃 − �𝐼𝐼𝑧𝑧�𝐼𝐼𝑧𝑧 − 𝐼𝐼𝑦𝑦� 𝐼𝐼𝑥𝑥𝑥𝑥�𝑄𝑄𝑄𝑄 𝐼𝐼𝑧𝑧𝑙𝑙 𝐼𝐼𝑥𝑥𝑥𝑥𝑛𝑛
= + + + + , (3.38) 2 respectively, where,𝛤𝛤𝑅𝑅̇ �𝐼𝐼𝑥𝑥�𝐼𝐼𝑥𝑥 − 𝐼𝐼𝑦𝑦� 𝐼𝐼𝑥𝑥𝑥𝑥�𝑃𝑃𝑃𝑃 − 𝐼𝐼𝑥𝑥𝑥𝑥�𝐼𝐼𝑥𝑥 − 𝐼𝐼𝑦𝑦 𝐼𝐼𝑧𝑧�𝑄𝑄𝑄𝑄 𝐼𝐼𝑥𝑥𝑥𝑥𝑙𝑙 𝐼𝐼𝑥𝑥𝑛𝑛
= . 2 Again, it is assumed that is negligible,𝛤𝛤 and prior𝐼𝐼𝑥𝑥𝐼𝐼𝑧𝑧 − to𝐼𝐼 𝑥𝑥initiating𝑥𝑥 the roll maneuver, the vehicle is in
𝑥𝑥𝑥𝑥 steady state conditions with𝐼𝐼 no roll or yaw rates ( = = 0). This reduces these equations of motion to 𝑃𝑃 𝑅𝑅
= and = . (3.39) 𝑙𝑙 𝑛𝑛 ̇ 𝑥𝑥 ̇ 𝑧𝑧 Since the vehicle is required to roll about𝑃𝑃 the𝐼𝐼 stability𝑅𝑅 axis𝐼𝐼 instead of the body axis, the next step is to convert the roll and yaw rates to the stability axis using the transformations
= + and 𝑃𝑃𝑠𝑠 𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 𝑅𝑅 𝑠𝑠𝑠𝑠𝑠𝑠 𝛼𝛼
= + .
From the second equation with yaw 𝑅𝑅rate𝑠𝑠 in− the𝑃𝑃 𝑠𝑠 𝑠𝑠stability𝑠𝑠 𝛼𝛼 𝑅𝑅 axis𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼set to zero ( = 0),
𝑅𝑅𝑠𝑠
41 =
and, since a coordinated roll results in no change𝑅𝑅 𝑃𝑃 to𝑡𝑡𝑡𝑡𝑡𝑡 angle𝛼𝛼 of attack, integrating gives
= .
Substituting Equation 3.39 for the roll and𝑅𝑅 ̇ yaw𝑃𝑃 ̇ accelerations𝑡𝑡𝑡𝑡𝑡𝑡 𝛼𝛼 gives the relationship between roll moment and yaw moment required to generate the coordinated roll,
= . 𝑛𝑛 𝑙𝑙 Finally, solving for the yaw moment, n, 𝐼𝐼and𝑧𝑧 computing𝐼𝐼𝑥𝑥 𝑡𝑡𝑡𝑡𝑡𝑡 𝛼𝛼 the non-dimensional form, the control power required to initiate a coordinated roll is
= . (3.40) 𝐼𝐼𝑧𝑧 𝑛𝑛 𝑙𝑙 𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑥𝑥 𝑡𝑡𝑡𝑡𝑡𝑡 𝛼𝛼
3.6.3 Yaw CPR for Roll Coordination
After the roll maneuver is initiated, the steady-state roll and yaw accelerations (Equations
3.39) are equal to zero for the rest of the maneuver. It follows that the total roll and yaw moments
on the aircraft must also be equal to zero, or
= 0 = + 𝑃𝑃𝑃𝑃 𝑙𝑙 𝑙𝑙 and 𝑙𝑙 �𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶 𝑝𝑝 2𝑉𝑉𝑇𝑇�
= 0 = + . 𝑃𝑃𝑃𝑃 𝑛𝑛 𝑛𝑛 This means that throughout the maneuver,𝑛𝑛 the𝐶𝐶 control𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶 effectors𝑝𝑝 2𝑉𝑉𝑇𝑇 must overcome damping in both
the roll and yaw axes. Solving these equations for the yaw moment required yields
= . (3.41) 𝐶𝐶𝑙𝑙𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐶𝐶𝑛𝑛𝑝𝑝 𝐶𝐶𝑙𝑙𝑝𝑝
3.6.4 Crosswind CPR
During takeoff and landing, the vehicle must be capable of maintaining heading while
flying in a crosswind. Similar to the vertical gust above, the crosswind creates a sideslip angle that
is the tangent of the crosswind velocity relative to the vehicle forward velocity,
42 = . −1 𝑉𝑉𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 However, since the heading must𝛽𝛽 be maintained,𝑡𝑡𝑡𝑡𝑡𝑡 this crosswind𝑉𝑉𝑇𝑇 condition is static and does not
excite the Dutch Roll mode and the required yaw control power is simply:
= . (3.42)
𝑛𝑛 𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 Since the vehicle dynamics are not𝐶𝐶 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐used𝑐𝑐 to𝑐𝑐𝑐𝑐𝑐𝑐 return−𝐶𝐶 the𝛽𝛽 𝛽𝛽vehicle to level, the roll moment caused by
the crosswind must also be rejected. This moment is given by:
= . (3.43)
𝑙𝑙 𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐶𝐶 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −𝐶𝐶 𝛽𝛽𝛽𝛽
3.6.5 Lateral-Directional Dynamic Response - Dutch Roll
The lateral directional modes are more complex than the longitudinal modes. In this axis,
three modes exist: the Dutch Roll mode, the roll subsidence mode, and the spiral mode. Each of
these are coupled to the others and in practice cannot truly be separated in the way that the short period mode can be separated from the Phugoid mode. In order to simplify the analysis, multiple assumptions are made. First, it is assumed the vehicle is trimmed in straight and level flight, e.g. the flight path angle, , is zero. Next, the sideforce and yaw effects of the aileron are neglected
( = = 0). The 𝛾𝛾gravity term in the first row of the second column of Alat is neglected. The ′ 𝛿𝛿𝑎𝑎 𝛿𝛿𝑎𝑎 side𝑌𝑌 force𝑁𝑁 derivative with respect to the roll rate is assumed to be zero and the side force derivative with respect to the yaw rate is assumed to be much smaller than the total velocity. Finally, the yaw moment derivative with respect to the roll rate is assumed to be zero.
Applying these assumptions and following Cramer’s rule, the characteristic equation becomes
| | = + + + + ′ . 𝛽𝛽 𝛽𝛽 𝑟𝑟 ′ 2 ′ 𝑌𝑌 ′ 𝑌𝑌 𝑁𝑁 𝑠𝑠𝑠𝑠 − 𝐴𝐴 𝑠𝑠�𝑠𝑠 − 𝐿𝐿𝛽𝛽� �𝑠𝑠 − 𝑠𝑠 �𝑁𝑁𝑟𝑟 𝑉𝑉𝑇𝑇𝑒𝑒� �𝑁𝑁𝛽𝛽 𝑉𝑉𝑇𝑇𝑒𝑒 �� In this equation, , approximates both the spiral and roll subsidence modes and the ′ 𝛽𝛽 remainder of the equation𝑠𝑠�𝑠𝑠 − 𝐿𝐿 approximates� the Dutch Roll mode. Traditionally, the aircraft planform
is designed such that the limitations on all three of these modes are met without feedback. For this
43 work it is assumed that the spiral mode will not be of concern due to the lack of a vertical tail
causing coupling in roll and yaw and the roll subsidence m0ode will also be neglected in the
controller due to the similarity in traditional wing shapes and comparatively low control power
required to manage it relative to roll performance requirements. Therefore, it is assumed that only
the Dutch Roll will need specific constraints relative to CPR analysis.
Similar to the analysis for the short period, the 2nd order Dutch Roll equation can be applied
to standard form to extract frequency and damping parameters:
= + 𝛽𝛽 2 ′ 𝑌𝑌 ′ 𝑛𝑛𝑑𝑑 𝛽𝛽 𝑇𝑇 𝑟𝑟 and 𝜔𝜔 𝑁𝑁 �𝑉𝑉 𝑒𝑒� 𝑁𝑁
2 = + . 𝛽𝛽 ′ 𝑌𝑌 𝑑𝑑 𝑛𝑛𝑑𝑑 𝑟𝑟 𝑇𝑇 Likewise, the yaw damping term has− been𝜁𝜁 𝜔𝜔used in𝑁𝑁 the past𝑉𝑉 𝑒𝑒 to combine these equations into one
approximation for required weathercock stability,
= + 2 + , 𝛽𝛽 𝛽𝛽 2 ′ 2 𝑌𝑌 𝑌𝑌 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑑𝑑 𝑑𝑑 𝑛𝑛𝑑𝑑 𝑇𝑇 𝑇𝑇 which when separated becomes𝑁𝑁 𝜔𝜔 𝜁𝜁 𝜔𝜔 �𝑉𝑉 𝑒𝑒� �𝑉𝑉 𝑒𝑒�
= 2 (3.44) 𝛽𝛽 ′ 𝑌𝑌 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑 𝑛𝑛𝑑𝑑 𝑇𝑇 and 𝑁𝑁 − 𝜁𝜁 𝜔𝜔 − 𝑉𝑉 𝑒𝑒
= . (3.45) 𝛽𝛽 ′ 2 𝑌𝑌 ′ 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑑𝑑 𝑇𝑇 𝑟𝑟𝑎𝑎𝑢𝑢𝑔𝑔 Using the same flight conditions that𝑁𝑁 were used𝜔𝜔 −for� 𝑉𝑉the𝑒𝑒� s𝑁𝑁hort period case, the reduced Dutch Roll approximation matrix of the F-16 is
0.1550 0.9852 = . 2.9516 0.2141 − − 𝐴𝐴𝑑𝑑𝑑𝑑 � � In this case, the eigenvalues are: , = 0.19 ± 1.71− , meaning the F-16, like most traditional 1 2 aircraft, was designed to be stable 𝜆𝜆in the Dutch− Roll mode.𝑗𝑗 The second row of the matrix gives the
44 open loop parameters, = 2.9516 and = 0.2141. Although the vehicle is stable, in order ′ ′ 𝛽𝛽 𝑟𝑟 to test the methodology,𝑁𝑁 assume the designers𝑁𝑁 wanted− to move the poles such that
= 3.0 = 0.7. 𝑟𝑟𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑎𝑎𝑎𝑎𝑎𝑎 Using the original CPR method𝜔𝜔 leads to an augmentation𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎 𝜁𝜁 in the yaw axis of = 9.0332. The ′ 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎 updated method, however, leads to = 8.3730 and = 4.0450𝑁𝑁. The computation of ′ ′ 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎 these poles is plotted in Figure 3-5 with𝑁𝑁 the minimum limit𝑁𝑁 s shown− in the shaded blue region— there are no limitations to maximum frequency or damping in the requirements for Dutch Roll. As mentioned, the open loop poles (shown in black circles) are within the boundary by design. The desired poles are placed well within the allowable region and the updated CPR method is able to achieve these poles. However, the original CPR method fails and instead makes the response fall outside the allowable region.
Figure 3-6 shows the Dutch Roll response to a lateral gust, represented by a sideslip deviation of 3.5 degrees in the initial condition. Comparing the linear and non-linear response here, it is clear that the approximation to the Dutch Roll is not as accurate as it was for the short period mode due to the additional interaction with roll and spiral modes. It appears the non-linear response has both a higher frequency and damping than the linear response in this case. It would be ideal to extend the CPR methods to place the poles of all three modes simultaneously in the future.
45
Figure 3-5. Computed Dutch Roll poles for F-16 example
Figure 3-6. Simulation of F-16 Dutch Roll response with original and updated CPR methods
There are also other, more complicated (and potentially more accurate) methods for the
Dutch Roll approximation in Reference [49], but they do not lend themselves well to the simple feedback controller design methods required of the CPR approach. Alternatively, for more accuracy the entire state space model could be used in the analysis in combination with a pole placement approach. For the F-16 planform, pole placement can be used easily to do this with the CPR
46 required still taken from the initial response with the computed augmentation. This approach was
explored for this work, but was abandoned due to complications with mode switching between the
Dutch Roll, spiral, and roll modes in the MADO environment when vehicle shape changed
dramatically and the focus returned to only controlling the Dutch Roll.
As with the short period, the limits of the response to the Dutch Roll are primarily considered
when attempting to reject a gust in the lateral axis. In this case, the augmentation in the yaw axis is
computed in the same manner as in the vertical axis. First, the sideslip angle is computed as the tangent of the gust velocity relative to the total vehicle velocity,
= , 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 −1 𝑉𝑉 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 then the yaw moment required to reject𝛽𝛽 the gust𝑡𝑡𝑡𝑡𝑡𝑡 is computed𝑉𝑉𝑇𝑇 by
= . (3.46)
𝑛𝑛𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑛𝑛𝛽𝛽 𝑛𝑛𝛽𝛽 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 There is no dynamic response input𝐶𝐶 from the�𝐶𝐶 roll𝑎𝑎 𝑎𝑎𝑎𝑎axis,− 𝐶𝐶but the� 𝛽𝛽 roll moment required for coupling is computed as an additional requirement:
= . (3.47)
𝐶𝐶𝑙𝑙𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐿𝐿𝛽𝛽𝛽𝛽𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
3.7 Control Power Required Application
The CPR method is now demonstrated via numerical example on a vehicle representative
of a modern fighter aircraft with tails removed. The configuration selected for this work is based
on that of a previous ESAV research effort performed by Lockheed Martin under the Efficient
Supersonic Air Vehicle Exploration (ESAVE) program [6]. In this program, AFRL tasked
Lockheed Martin with performing multiple design studies that resulted in a selected planform that
was tested in a wind tunnel with several potential innovative control effectors placed around the
body. A CAD file of this geometry was provided and is shown in Figure 3-7, after being imported
into Solidworks. It has a total length from nose to tail of 64 ft., wingspan of 63 ft. with aspect ratio
4, and an estimated take-off gross weight of 69,000 lb. It utilizes both conventional and non-
47 conventional control effectors to maintain stability. Without a tail, the conventional control
effectors are limited to two leading edge and two trailing edge surfaces per wing side. The surfaces
are divided at the wing break with both leading and trailing edge surfaces having a chord of
approximately 20 percent of the total local wing chord. The non-conventional control effectors may include, but are not limited to: split ailerons, spoilers/deflectors, all moving wing tips, and strakes
or various tabs on the fore and aft body.
Figure 3-7. CAD geometry of initial vehicle design
As part of this previous research effort, this vehicle was tested in a wind tunnel. During this testing, several control effectors were evaluated including wing surfaces, spoilers and/or deflectors, pop up rudders, and various control tabs placed around the fuselage. Since all of the available aerodynamic analysis programs are capable of computing the effectiveness of traditional control effectors such as flaps/ailerons, rudders, and elevators, the focus of the wind tunnel testing was on the potential innovative control effectors (Figure 3-8). During the wind tunnel testing, there were two of these innovative control effectors that stood out as potential yaw effectors and were selected to be examined for this work. The first is a combination of spoiler and deflector on the wing. The second is a rectangular surface on the nose that has been called the ‘Rhino Horn’.
48
Figure 3-8. Fuselage geometry and effectors tested in wind tunnel
The combination of spoiler and deflector isn’t a new concept, but is not a traditional control
effector and is difficult for many CFD programs to model. In the wind tunnel, the spoiler and
deflector were tested both individually and in combination (Figure 3-9). After testing many combinations of size and location of control tabs placed around the vehicle, the Rhino Horn was a potential control effector that stood out from the rest as being exceptionally effective. A picture of it on the wind tunnel model is shown in Figure 3-10. This is of particular interest to this analysis due to the fact that it is a difficult effector to predict due to its strong interaction with the fuselage through vortex shedding, evidenced by smoke in the wind tunnel test (Figure 3-11).
Figure 3-9. Spoiler/deflector in wind tunnel
49
Figure 3-10. Rhino Horn control effector
Figure 3-11. Flow visualization of vortex shedding in wind tunnel
The data from the wind tunnel testing was collected and formatted into lookup tables for the static derivatives required in a Matlab simulation environment incorporating the equations of motion. Linear panel methods were used to determine the remaining dynamic derivatives, rounding out the required simulation data.
The controllability analysis must be performed at various points within the flight envelope.
In the conceptual design phase, even with preliminary design level of fidelity that is desired of this
50 work, it is not practical to analyze the controllability of the aircraft at every potential flight condition. Instead, the designers must choose a set of flight conditions that represent a portion of the most stringent points of the flight envelope to give the design a high-probability of being able to fly throughout the entire range of operation. The number of points does not need to be too large because experience has shown which flight regimes are typically critical. Figure 3-12 shows the flight envelope and control evaluation test points used in the Innovative Control Effectors program of Reference [12]. These six test points cover a broad spectrum of flying qualities representing all three Flight Phase Categories in MIL-STD-1797B.
Figure 3-12. Flight conditions evaluated for ICE Program [12]
The flight conditions shown are separated by flight phase. Air-to-Ground Attack, Air
Combat Maneuver (ACM) corner speed, and maximum sustained load factor conditions are
evaluated according to Category A Flight Phase—requiring rapid maneuvering, precision tracking,
or precise flight-path control. For this work, the maximum sustained load factor is assumed to occur during a 9-g turn at Mach 0.9 and altitude of 30,000 ft., and the ACM is a 4-g turn at Mach 0.6 and
15,000 ft. altitude. The Air-to-Ground Attack condition is evaluated at Mach 0.9 at Sea Level with
51 only a 2-g turn requirement. The maximum sustained load factor is commonly the active constraint
in the pitch effector design, but in this work, all three of these conditions have been seen to be
active in both the pitch and yaw axes.
The cruise condition is the less stringent, Category B Flight Phase. Although it would be
ideal for a Supersonic condition to be evaluated, the data provided does not extend to this region.
For this reason, the cruise condition is instead evaluated at Mach 0.9 and 35,000 ft. altitude. This
provides access to this condition for future work when supersonic data is included in the analysis,
but does not lead to a limiting case for the analysis that is presented here.
The two low-speed conditions are Category C Flight Phases—representing gradual
maneuvers with accurate flight-path control. Landing and takeoff are not evaluated separately in
this analysis. Instead, Takeoff is given priority and the vehicle is analyzed in a full-fuel condition where takeoff rotation often becomes the limiting constraint in the pitch axis. This is the point where the vehicle reaches enough speed to achieve flight and must pitch up to generate the required lift. In the initial design iterations, the takeoff velocity, , is stall speed,
𝑉𝑉𝑇𝑇𝑇𝑇 = , 2𝑊𝑊 𝑉𝑉𝑠𝑠 𝜌𝜌𝜌𝜌𝐶𝐶𝐿𝐿𝑚𝑚𝑚𝑚𝑚𝑚 where, is the maximum lift coefficient the� vehicle is capable of achieving. The aircraft must
𝐿𝐿𝑚𝑚𝑚𝑚𝑚𝑚 also be 𝐶𝐶able to take off and land in a crosswind condition at this speed. The final condition is Power- on departure stall, where the aircraft is in the air at 15,000 ft. altitude, but at the lower limits of the speed range in the flight envelope and potentially pitched up well above the stall angle of attack.
This may be a Powered Approach (PA) condition, on the way in for a landing which also requires application of crosswind conditions. In this low speed range, the aerodynamic control effectors have little effectiveness and in pitch, the ability to nose down can be a limiting factor. In modern aircraft, the aerodynamic effectors can be augmented by thrust vectoring to help the aircraft pitch down. Thrust vectoring has not traditionally been used in the yaw axis however. From inspection of Equation 3.40, the yaw moment required to initiate a roll is directly proportional to the tangent
52 of the angle of attack. Therefore, the higher the angle of attack, the higher the yaw moment that is
required. Coupling this with the lower effectiveness of the control surfaces in this condition makes
it generally the prime concern for control power in the yaw axis.
3.7.1 Numerical Example
Of the 6 flight conditions presented above, the Power-on departure stall condition was found to be the most stringent in yaw and is the condition that is presented numerically. Although the ESAVE
program provided estimates for the gross weight of the vehicle, the inertias and center of gravity
were not readily available. Instead, a traditional mass properties analysis based on an assumed
mission that resulted in similar gross takeoff weight was used in conjunction with individual
component weights and locations. This method is automated for the parameterized planform of the
MADO and is discussed in Chapter V. Using the method, the baseline ESAV gross takeoff weight
came to be 72,456 lbs., requiring 20,578 lbs. of fuel to complete the mission. In the Power-on
departure stall condition, it is assumed that the vehicle has expended 40% of the fuel and the mass
properties are updated to reflect this. These mass properties and the vehicle reference geometry
properties are listed below in Table 3-7.
Table 3-7. Mass properties and reference geometry of example ESAV configuration
Weight (lbs) 64,224.7
𝑊𝑊 1.2956e05
𝑥𝑥𝑥𝑥 𝐽𝐽 2.8768e05 Moments of inertia (slug-ft2) 𝑥𝑥𝑥𝑥 𝐽𝐽 4.1542e05
𝑥𝑥𝑥𝑥 𝐽𝐽 671.4
𝑥𝑥𝑥𝑥 Wing area (ft2) 𝐽𝐽 938.5
Wing span (ft) 𝑆𝑆 61.3
Mean aerodynamic chord (ft) 𝑏𝑏 20.2
𝑐𝑐̅ 53 In this flight condition, the vehicle is flying at 15,000 ft. altitude at what could be a very
high angle of attack and still must be able to maneuver. The requirements for this conditions are
taken primarily from Table 3-1 through Table 3-5 for Category C flight condition at Level I Flying
Qualities. Due to the limitations in available aerodynamic analysis fidelity discussed in the next
Chapter, the angle of attack is assumed to be 10 degrees and the coefficients are assumed to be linear in this range, leading to the following values for lift and pitch moment coefficients and derivatives.
Since this is not a takeoff condition and it is already pitched up, in the longitudinal axis,
the vehicle must only be able to achieve trim, coordinate stability axis rolls, and reject gusts while
achieving short period performance within the limitations:
0.16 3.6, 0.35 1.3, and 0.87.
𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 For this flight condition,≤ the𝐶𝐶𝐶𝐶 𝐶𝐶CAP≤ limitation≤ translates𝜁𝜁 ≤ to an 𝜔𝜔allowable≥ frequency range of:
0.9515 4.5134 . In the lateral/directional axis, the vehicle must be able to initiate and
𝑠𝑠𝑠𝑠 �coordinate≤ rolls,𝜔𝜔 ≤ reject lateral� gusts, and maintain heading in a crosswind. The roll performance specification desired is 30 degrees in 1.1 seconds. The crosswind velocity is 30 knots. The Dutch
Roll limitations are:
0.08 and 1.0 .
The first step to computing CPR≤ is𝜁𝜁 𝑑𝑑to𝑑𝑑 determine≤ 𝜔𝜔the𝑑𝑑𝑑𝑑 approximate linear trim conditions.
The lookup tables are evaluated at zero and one degree angle of attack and finite difference is used to compute the slope of the lift and pitching moment coefficients with angle of attack. The results are given in Table 3-8.
Table 3-8. Computed lift and pitching moment coefficients and derivatives
0.0071 3.243 −1 𝐿𝐿0 𝐿𝐿𝛼𝛼 𝐶𝐶 0.0067 𝐶𝐶 0.1754𝑟𝑟 𝑟𝑟𝑟𝑟 −1 𝑚𝑚0 𝑚𝑚𝛼𝛼 𝐶𝐶 𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟
54 The lift coefficient at 10 degrees angle of attack is 0.5731. The required moment coefficient for trim is computed as:
= = 0.0067 0.1754 10 = 0.0373.
𝜋𝜋 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 Next, the stall𝐶𝐶 velocity𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − at𝐶𝐶 this0 −condition𝐶𝐶 𝛼𝛼𝛼𝛼 is computed− as: − � 180 �
( , . ) = = = 399.6 . . ( . )( . ) 2𝑊𝑊 2 64 224 7 𝑙𝑙𝑙𝑙 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠 𝐿𝐿 2 𝑉𝑉 �𝜌𝜌𝜌𝜌𝐶𝐶 𝑚𝑚𝑚𝑚𝑚𝑚 ��0 0015 𝑓𝑓𝑓𝑓3 � 938 5 𝑓𝑓𝑓𝑓 0 5731 𝑠𝑠 In order to dimensionalize the coefficients, the dynamic pressure is also required, which is:
2 2 = = 0.0015 3 399.6 = 119.4 . 1 1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓 𝑞𝑞� 2 2 � � � 𝑠𝑠 � The remaining static coefficient𝜌𝜌𝑉𝑉 derivatives are𝑓𝑓𝑓𝑓 computed by finite difference𝑝𝑝𝑝𝑝𝑝𝑝 of the sideslip angle and the dynamic derivatives are computed using AVL [51]. For CPR analysis, the dimensional derivatives are required at the flight condition and they are they are tabulated below.
55 Table 3-9. Tabulated dimensional derivatives of ESAV in power-on departure stall condition
Longitudinal Derivatives Lateral/Directional Derivatives
= 2 + -0.0050 = -14.9659 −𝑞𝑞�𝑆𝑆 𝑉𝑉 𝐷𝐷𝑒𝑒 𝐷𝐷𝑉𝑉 𝛽𝛽 𝑞𝑞�𝑆𝑆 𝑌𝑌𝛽𝛽 𝑋𝑋 𝑇𝑇 � 𝐶𝐶 𝐶𝐶 � 𝑌𝑌 𝐶𝐶 𝑚𝑚𝑉𝑉 𝑚𝑚 = 2 + = 0.0000 2 0.0330 𝑞𝑞�𝑆𝑆 𝑞𝑞�𝑆𝑆𝑆𝑆 𝑋𝑋𝑇𝑇𝑉𝑉 � 𝐶𝐶𝑇𝑇𝑒𝑒 𝐶𝐶𝑇𝑇𝑉𝑉 � 𝑌𝑌𝑝𝑝 𝐶𝐶𝑌𝑌𝑝𝑝 𝑚𝑚𝑉𝑉𝑇𝑇 𝑚𝑚𝑉𝑉𝑇𝑇 = = -0.4814 2 -1.205e-04 𝑞𝑞�𝑆𝑆𝑆𝑆 𝛼𝛼 𝑞𝑞�𝑆𝑆 𝐿𝐿 𝐷𝐷𝛼𝛼 𝑟𝑟 𝑌𝑌𝑟𝑟 𝑋𝑋 �𝐶𝐶 − 𝐶𝐶 � 𝑌𝑌 𝑇𝑇 𝐶𝐶 𝑚𝑚 𝑚𝑚𝑉𝑉 = 2 + = -0.0020 -2.3612 −𝑞𝑞�𝑆𝑆 𝑞𝑞�𝑆𝑆𝑆𝑆 𝛽𝛽 𝑍𝑍𝑉𝑉 � 𝐶𝐶𝐿𝐿𝑒𝑒 𝐶𝐶𝐿𝐿𝑉𝑉� 𝐿𝐿𝛽𝛽 ′ 𝐶𝐶𝑙𝑙 𝑚𝑚𝑉𝑉𝑇𝑇 𝐽𝐽𝑋𝑋 = = + -183.0291 2 -1.0641 𝑞𝑞�𝑆𝑆𝑆𝑆 𝑏𝑏 𝑝𝑝 𝛼𝛼 −𝑞𝑞�𝑆𝑆 𝐷𝐷𝑒𝑒 𝐿𝐿𝛼𝛼 𝑝𝑝 ′ 𝑙𝑙 𝑍𝑍 �𝐶𝐶 𝐶𝐶 � 𝐿𝐿 𝑋𝑋 𝑇𝑇 𝐶𝐶 𝑚𝑚 𝐽𝐽 𝑉𝑉 = = 2 0.0000 2 0.0160 −𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑞𝑞�𝑆𝑆𝑆𝑆 𝑏𝑏 𝑍𝑍𝛼𝛼̇ 𝐶𝐶𝐿𝐿𝛼𝛼̇ 𝐿𝐿𝑟𝑟 ′ 𝐶𝐶𝑙𝑙𝑟𝑟 𝑚𝑚𝑉𝑉𝑇𝑇 𝐽𝐽𝑋𝑋 𝑉𝑉𝑇𝑇 = = 2 -3.5249 -1.9899 −𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑞𝑞�𝑆𝑆𝑆𝑆 𝑍𝑍𝑞𝑞 𝐶𝐶𝐿𝐿𝑞𝑞 𝑁𝑁𝛽𝛽 ′ 𝐶𝐶𝑛𝑛𝛽𝛽 𝑚𝑚𝑉𝑉𝑇𝑇 𝐽𝐽𝑍𝑍 = 2 + = 2.639e-04 2 -0.0052 𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑞𝑞�𝑆𝑆𝑆𝑆 𝑏𝑏 𝑝𝑝 𝑀𝑀𝑉𝑉 � 𝐶𝐶𝑚𝑚𝑒𝑒 𝐶𝐶𝑚𝑚𝑉𝑉� 𝑁𝑁𝑝𝑝 ′ 𝐶𝐶𝑛𝑛 𝐽𝐽𝑌𝑌𝑉𝑉𝑇𝑇 𝐽𝐽𝑍𝑍 𝑉𝑉𝑇𝑇 = 2 + = 0.0000 2 5.761e-05 𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑞𝑞�𝑆𝑆𝑆𝑆 𝑏𝑏 𝑀𝑀𝑇𝑇𝑉𝑉 � 𝐶𝐶𝑚𝑚𝑇𝑇 𝐶𝐶𝑚𝑚𝑇𝑇𝑉𝑉 � 𝑁𝑁𝑟𝑟 ′ 𝐶𝐶𝑛𝑛𝑟𝑟 𝐽𝐽𝑌𝑌𝑉𝑉𝑇𝑇 𝐽𝐽𝑍𝑍 𝑉𝑉𝑇𝑇 = 1.3788 𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑀𝑀𝛼𝛼 𝐶𝐶𝑚𝑚𝛼𝛼 𝐽𝐽𝑌𝑌 = 2 0.0000 𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑐𝑐̅ 𝑀𝑀𝛼𝛼̇ 𝐶𝐶𝑚𝑚𝛼𝛼̇ 𝐽𝐽𝑌𝑌 𝑉𝑉𝑇𝑇 = 2 -0.1175 𝑞𝑞 𝑞𝑞�𝑆𝑆𝑐𝑐̅ 𝑐𝑐̅ 𝑚𝑚𝑞𝑞 𝑀𝑀 𝑌𝑌 𝑇𝑇 𝐶𝐶 𝐽𝐽 𝑉𝑉
56 Since is positive, an increase to angle of attack will result in an increased pitching
𝛼𝛼 moment and thus𝑀𝑀 the vehicle is unstable in pitch. Likewise, since the is negative, a positive
𝛽𝛽 sideslip will cause a negative yawing moment, increasing sideslip and𝑁𝑁 resulting in a positive
feedback loop that makes the vehicle unstable in yaw as well. In both axes, the minimum CPR to
achieve the desired flying qualities is achieved by selecting the minimum frequency and damping
for control augmentation. Following the CPR description, Equations 3.31 and 3.46 yield a required pitch moment coefficient of 0.0486 and yaw moment coefficient of 0.0113 to reject vertical and lateral gusts. In order to test the validity of this, a simulation of the ESAV was created with longitudinal and lateral/directional controllers using the CPR augmentation as feedback gains implemented into the full longitudinal and lateral/directional state matrices.
Figure 3-13 shows the longitudinal poles. The black circles show the open loop modes with the slow Phugoid poles near the origin and the unstable short period poles near -1.5 and +1.0 on the real axis. The estimate of the short period poles using Equations 3.23 and 3.24 are shown in green and are very accurate. The colored blue area is the range of acceptable pole locations based on the above frequency and damping limitations and the minimum allowable frequency and damping or desired pole is marked by the blue square. Applying the feedback gains to the short period equations, the closed loop poles are computed to match the desired and are shown in red.
Finally, when the same gains are applied to the full longitudinal system, the short period poles remain at the desired location without changing the Phugoid poles which remain near the origin.
Figure 3-14 shows the simulation of the short period control law when rejecting a gust, assumed to be created using an initial angle of attack condition. In the plot on the left, the black line shows the uncontrolled and unstable system with angle of attack rapidly going toward infinity. The red and blue lines show the reduced, short period only and full longitudinal system, respectively, with little change between the two. In the plot on the right, the pitch moment coefficient generated by the controller is plotted as a function of time with the CPR predicted maximum required coefficient
57 shown in the black dotted line. As described above, the maximum simulated moment required
matches the CPR prediction and is related to the initial condition caused by the gust.
Figure 3-13. ESAV longitudinal poles during high-angle of attack condition
Figure 3-14. Simulation of short period response to vertical gust
In the lateral/directional axes, the Dutch Roll has more interaction with the roll and spiral
modes than the short period does with the Phugoid. As a result, the prediction of the open loop poles are not quite as accurate (but still close), as shown in Figure 3-15 below. The resulting desired
58 Dutch Roll poles (blue) and reduced Dutch Roll simulated poles (red) match equally well, but the
Dutch Roll poles of the full system (black ‘+’) shows the Dutch Roll poles slightly to the right and higher than the desired, with the roll mode farther to the left on the real axis than the uncontrolled modes. With the allowable Dutch Roll poles so close to the real axis, this coupling can in some cases introduce enough error to cause the CPR method to fail as a controller in simulation, allowing the Dutch Roll to go unstable. However, the difference in computed yaw moment required to move the Dutch Roll back to the small distance back to the left half plane is minimal and is considered to be within the desired accuracy of the methods in this dissertation. Similar to the short period case, Figure 3-16 shows the simulation of the Dutch Roll response of the vehicle with unstable open loop. The full lateral system response shown in blue has a slightly faster frequency with a slightly lower damping, but generally within the expected fidelity of the input data this comparison is considered acceptable.
Figure 3-15. ESAV lateral/directional poles during high-angle of attack condition
59
Figure 3-16. Simulation of Dutch Roll response to lateral gust
The remaining control power assessment is tabulated below. In the longitudinal axis, the pitch up maneuver is the worst case. Although the requirement selected was only a 1-g pitch up, at low speed, this requires a lot of control authority. The next worst case is simply trimming the vehicle at such a high angle of attack. The gust and roll coordination have similar magnitudes with a gust in the approach region only requiring a velocity of 25 ft/s. The roll requirement due to performing the roll maneuver is a full order of magnitude higher than the gust or crosswind requirements. This is not surprising, but when evaluating traditional planforms with a vertical tail, it would be expected that both of these numbers would increase significantly. In yaw, without a tail for directional stability, the control required to maintain heading in a crosswind during landing is the defining case with both roll initiation and gust rejection close behind. Once the roll is initiated, it takes a relatively small amount of yaw power to coordinate the maneuver.
60 Table 3-10. Control Power Required for ESAV in high angle of attack condition
0.0 0.0233 0.0132 𝑚𝑚𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑙𝑙 𝐶𝐶 𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 0.0373 𝐶𝐶 0.0028 0.0002 𝑚𝑚 𝑙𝑙𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐶𝐶 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐶𝐶 𝐶𝐶𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 0.0486 0.0056 0.0113 𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛 𝐶𝐶 𝐶𝐶 𝐶𝐶 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 0.0174 0.0 0.0151 𝑚𝑚𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶 𝐶𝐶 𝐶𝐶𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 0.0142 0.0 𝐶𝐶𝑚𝑚𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶𝑛𝑛𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0.0 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
. . . 𝒎𝒎 𝑪𝑪 𝒓𝒓𝒓𝒓𝒓𝒓 𝒍𝒍𝒓𝒓𝒓𝒓𝒓𝒓 𝒏𝒏𝒓𝒓𝒓𝒓𝒓𝒓 𝟎𝟎 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝑪𝑪 𝟎𝟎 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 𝑪𝑪 𝟎𝟎 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
3.7.2 Control Power Available
The CPR computation described above gives a designer or MADO algorithm a numerical measure of the moments that must be generated in order to control the vehicle with the imposed constraints. However, this is only part of the evaluation. The Control Power Available (CPA) analysis sums the maximum available moments that can be created through any combination of control effector deployment. It is assumed that if sufficient moments can be generated, a dynamic inversion-based controller could force the aircraft to behave in the desired manner. With both the
CPR and CPA computed in numerical form, it is easy to measure how close the vehicle is to being controllable, allowing easy development of a cost function for optimization routines or a direct intuitive measure for designers. This is described further in Chapter VI.
Ideally, the CPA analysis would be performed as part of a sub-optimization where the maximum moment in each axis is found subject to each moment in the other axes being equal to zero. Ultimately, developing a controller would require a control mixer to determine the combinations of effector deployment which give the required moments in each axis at any given flight condition. In practice, this would get increasingly complex when many different types of effectors are combined that are able to deploy at different rates with different degrees of coupling
61 in each axis. In traditional aircraft, this complexity is eliminated by assigning specific control
effectors to certain tasks and ignoring the resulting coupling in the initial assessment. E.g. ailerons
are used for roll, rudders for yaw, and elevators to pitch. For the desired fidelity of this work, this
concept will be partially adopted. The traditional wing control effectors will be assumed to be used
exclusively to control pitch and roll while the non-traditional control effectors will be used to control yaw, with all coupling effects ignored.
The ESAV example from above has 8 wing control effectors with 4 leading edge devices assumed to be capable of deflecting up or down by 30 degrees and the 4 trailing edge devices with deflection ranges of up or down 45 degrees. From the wind tunnel data, this sums to a total pitch moment of coefficient of 0.2385 or 203% of the required pitch moment. In roll, with one side fully deflected down and the other side fully deflected up, the vehicle can produce a roll moment coefficient of 0.1378 or 436% of the required roll control power. In yaw however, with a full deflection of the spoiler, deflector, and rhino horn control effectors, the vehicle can only produce a yaw moment coefficient of 0.0178 or 45% of the total required yaw control power.
3.8 Control Power Summary
In this chapter, the derivation of the CPR method was presented and the method was applied to the ESAV vehicle using wind tunnel data with supplementation from AVL. The next three chapters describe how this method is extended into an MADO environment. Chapter IV discusses the options that were explored for aerodynamic analysis required to rapidly and robustly generate the required derivatives of Table 3-9. After selecting the aerodynamic analysis, Chapter V discusses how the ESAV geometry was parameterized for use in the MADO in a manner that can rapidly create 3-dimensional analysis geometry along with inertia properties for the vehicle. Then Chapter
VI discusses how all of this is brought together into the MADO environment.
62 CHAPTER IV
AERODYNAMICS IN THE MADO PROCESS
In order to implement the CPR analysis of the previous chapter, the required control derivatives must be generated accurately, autonomously, and robustly. This became an enormous hurdle in developing the MADO process of this work. As described previously, a major goal of this work is the application of the MADO process to the ESAV vehicle—a tailless supersonic aircraft configuration. However, these configurations have proven to be challenging with respect to the selection of aerodynamic analysis tools that can be used. Such tools must be capable of accurately modeling certain innovative control effectors with computational costs such that analyses can rapidly be generated on several aircraft configurations and at multiple flight conditions. In an effort to determine the usefulness of available aerodynamic tools for use within an MADO process, several of these tools were evaluated. The level of fidelity of multiple tools was assessed and in particular, the ability to capture various trends was leveraged against the computation time required for each tool. With the ability to pick the type of analysis and autonomously generate required input, this chapter discusses the effort to determine not a single best code for all testing, but a best combination of analysis codes in the overall process. For instance, one code may be accurate when modeling traditional aircraft configurations but incapable of modeling certain innovative control effectors that are being considered for tailless aircraft configurations. It may be the most computationally efficient to use these codes to model the wing control effectors while using a different code to model the innovative control effectors.
Several aerodynamic analysis programs are available for use within MSTC. These range in fidelity level from empirical calculations to full Navier-Stokes solutions such as with AFRL’s
Air Vehicles Unstructured Solver (AVUS) [52]. In this work, the focus is on the codes that may be suitable to being integrated into an MADO environment. This means they must be capable of
63 running with little human interaction and with computational costs conducive to large numbers of
runs. This is complicated for codes that only analyze static flight conditions since the derivatives
must be obtained via finite differencing. They were evaluated not just on the ability to match static
wind tunnel results, but to output all required control parameters for the CPR analysis.
For in-the-loop aerodynamic analysis, five codes have been evaluated; USAF Digital
DATCOM [53], Athena Vortex Lattice (AVL) [51], ZAERO [54], ZONA’s Euler Unsteady Solver
(ZEUS) [55], and Cart3D [56]. DATCOM is based on a collection of equations and charts empirically derived from vast amounts of Air Force and industry data. AVL is a vortex lattice code that was based on several prior NASA codes and VORLAX, among others. Both ZAERO and
ZEUS are aeroelastic analysis codes developed by Zona Technology, Inc. ZAERO is a linear panel code, while ZEUS is a higher fidelity Euler solver with similar inputs to ZAERO, making the transition between the two relatively painless. Both codes have the capability to output control derivatives directly. Cart3D is another Euler solver developed by NASA. It is considered to be higher fidelity than ZEUS and is capable of modeling more complicated geometries, but is limited to static analysis. With all five codes having potential to be integrated into the MADO process as options, the primary investigation of this work is the tradeoff between computation time, capability, and fidelity. Each analysis tool is described below.
4.1.1 DATCOM
The USAF Stability and Control DATCOM [53], or Data Compendium, is a collection of empirical methods of determining aircraft stability and control derivatives derived from vast quantities of collected aircraft data. It is run from the command line with a simple text input file and outputs results instantaneously on modern computers. The results are well suited for preliminary design of traditional aircraft. DATCOM is limited primarily to bodies of revolution with straight, cranked, or delta wing planforms and traditional control surfaces, making it unfit for
MADO that attempts to change body shape or use non-traditional control effectors as is the case
64 with the desired ESAV vehicle. However, it is one of only a few available programs capable of
outputting dynamic derivatives—the moments created by aircraft rates—which are required for
controls analyses, and it is currently the only program which gives data for these derivatives with
respect to angle of attack rates.
4.1.2 AVL
Figure 4-1. ESAV AVL analysis geometry
Although not empirical, AVL is still best suited to analyze traditional aircraft designs that can be considered to consist mainly of wings and tails. The body shape can be represented as circular cross-sections consisting of equivalent area to that of the baseline fuselage, however the manual suggests that there is little experience with the slender-body theory that is used to model it and that it should be used with discretion. For this reason, the body is neglected in this work. The
ESAV AVL analysis model is shown in Figure 4-1. AVL is capable of outputting all required data with the exception of the derivatives with respect to angle of attack rates. If desired, it can also perform modal analysis internally with these derivatives to output Eigenmodes for the controllability analysis. However, for the current process, these are computed externally. AVL is executed through a command line call with text file input, making it easy to integrate into MADO tools in Matlab or python.
65
4.1.3 ZAERO
ZAERO is a software system that integrates the essential disciplines required for
aeroelastic design/analysis [54]. It couples linear panel method aerodynamics to free vibration
mode shapes determined using a structural finite element method. The transferal of displacements
and aerodynamic forces between the structural and aerodynamic grids in ZAERO is accomplished
by a 3D spline module. The transferred displacement of the aerodynamic grids is used to define the
unsteady motion using a transpiration boundary condition technique. Therefore, there is no moving
mesh involved in the ZAERO unsteady aerodynamic computation which further reduces the burden
on computational resources.
In the work presented here, all aerodynamic calculations are performed on rigid
configurations and the aeroelastic capabilities of ZAERO are not used. The input to ZAERO is text based, but significantly more complicated than AVL. However, this is still well suited to integration into an optimization loop, and has been automated through previous work in MSTCGeom [3]. Like all panel codes, the number of panels used can affect both the fidelity and computation time. Figure
4-2 shows the analysis mesh that was used for ZAERO. In previous work, when the number of panels were doubled, the change in results of the ZAERO aerodynamic analyses was negligible.
Since the computational cost of the refined mesh was three times that of the initial mesh, the initial mesh spacing was used thereafter. Traditional control effectors such as ailerons, elevators, flaps and rudders are easily modeled as rotating control surfaces in ZAERO. However, ZAERO is not capable of modeling non-traditional effectors. Like AVL, ZAERO is capable of outputting all required stability derivatives except those due to angle of attack rates.
66
Figure 4-2. ZAERO and ZEUS analysis mesh
4.1.4 ZEUS
ZEUS [55] is ZONA’s higher fidelity core solver product that integrates the essential disciplines required for aeroelastic design/analysis. It uses an Euler equation solver with optional viscous effects as the underlying aerodynamic force generator, coupled with the structural finite element modal solution to solve various aeroelastic problems such as flutter, maneuver loads, store ejection loads, gust loads, and static aeroelastic/trim analysis. Structural nonlinearities can also be included to perform a nonlinear aeroelastic analysis.
ZEUS uses the bulk data input format that is very similar to that of NASTRAN and
ZAERO. In fact, the majority of the input bulk data cards of ZEUS are identical to those of ZAERO, minimizing the learning curve for an experienced ZAERO user to effectively utilize higher fidelity flow physics such as transonic shocks. The major difference in input between ZEUS and ZAERO is associated with mesh generation, since ZEUS requires the mesh of the entire flow field domain, whereas ZAERO only needs the surface mesh. However, ZEUS employs an automated mesh generation scheme which grows a flow field mesh from the surface mesh. Additionally, ZEUS has an overset mesh capability to handle very complex configurations such as whole aircraft with
67 external stores. The geometric surface mesh used for ZEUS is the same as the ZAERO mesh in this
case.
In order to solve the Euler equations while retaining the ease in setting up a computational mesh, ZEUS employs approximate boundary conditions applied on a Cartesian grid where the full
Euler boundary condition is replaced by its first-order expansion on the mean plane of the lifting surface. For bodies such as fuselages and stores, ZEUS approximates the exact surface geometry of the body by a rectangular box. Using a slender body theory, the exact surface geometry is mapped onto the surface of the rectangular box where the boundary condition of the bodies is applied. This approximate boundary condition for lifting surfaces and bodies significantly reduces the grid generation effort of ZEUS because no conformal mesh is required. It is this approximate boundary condition that enables the development of an automated mesh generation scheme in
ZEUS.
4.1.5 Cart3D
Cart3D is an inviscid Euler aerodynamic analysis code developed by NASA [56]. It contains tools to rapidly generate a volume mesh from a water-tight geometry in triangulation file format generated in a typical CAD program. It can handle complex geometries like those of this project, while returning results significantly faster than a full Navier Stokes solution. Additional inputs are required in the form of text files that govern the flight conditions, meshing parameters, and boundary conditions. Since all of the previous programs assume empirical results or slender body theory, Cart3D is the only program that is capable of modeling small changes to the fuselage.
It is also the only program capable of modeling the expected innovative control effectors. However, all inputs to the previous programs were in the form of text files and could reasonably be automated by an experienced programmer using Matlab or python. In the case of Cart3D, the input requires a surface triangulation that until recently required hands-on effort in a CAD program that would be translated to STL or similar format before being made use of in Cart3D. The development of the
68 Engineering SketchPad (ESP) [22] and incorporation into the Computational Aircraft Prototype
Synthesis (CAPS) [57] tools has alleviated this downfall, helping to automate the geometry
generation process and allowing Cart3D to be considered in this MADO framework.
An example input triangulation mesh generated via ESP is shown in Figure 4-3. Cart3D
uses this mesh to generate the computational volume mesh shown in Figure 4-4. The volume mesh has multiple levels of refinement so that the cells near the body are very small to capture flow characteristics around the surfaces while the cells near the outer edges of the mesh are much larger to decrease computational effort.
Figure 4-3. Triangulation mesh input to Cart3D
Figure 4-4. Volume mesh for Cart3D computation
69 4.2 Aerodynamic Analysis Results
The ESAV vehicle was analyzed in each of the above aerodynamic analysis programs with the goal of obtaining all required parameters for a linear control power analysis. In this section, the results of each are compared to applicable wind tunnel data and where experimental data is not available, they are compared to each other. It should be noted that there is a certain subjectivity to the results in that different modelling practices may produce different results. The selection of parameters defining the mesh, flow characteristics, and number of iterations greatly affects both the fidelity and computation time of the results in the analysis codes where these are options. The results are separated into four sections: (1) Static Analysis, comparing results at steady values of angle of attack and sideslip and the resulting stability derivatives, (2) Dynamic Derivatives, comparing the output of dynamic derivatives from the codes capable of outputting them, (3)
Fuselage Shape, comparing results of derivatives with respect to fuselage shape changes, and (4)
Control Derivatives, comparing analysis of two of the more promising innovative control effectors with available wind tunnel results.
4.2.1 Static Analysis
A static analysis was performed for angles of attack ranging from -10 to 25 degrees and sideslip angles between -10 and 10 degrees in each of the programs. In the case of DATCOM, data at sideslip is not directly output, but is instead computed from the sideslip derivatives. Figure 4-5 shows the longitudinal coefficients – Lift (CL), Drag (CD), and Pitching Moment (Cm).
Unsurprisingly, all programs predict lift with reasonable accuracy, especially in the linear region.
ZEUS fails to converge above 15 degrees angle of attack and is missing data beyond this point. In drag, ZAERO predicts nearly double the value of the wind tunnel results while the rest of the programs are noticeably closer. The pitching moment is where both AVL and DATCOM show large discrepancies. This is likely due to both programs not fully modeling the fuselage shape, leading to an error in the computed aerodynamic center.
70
Figure 4-5. Comparison of longitudinal coefficients: Lift (top), Drag (middle), and Pitch Moment (bottom)
The lateral coefficients – Sideforce (CY), Roll Moment (Cl) and Yaw Moment (Cn) – are shown in Figure 4-6. These values are compared at a sideslip angle of five degrees. Surprisingly, the two highest fidelity codes produce the largest discrepancy in side force relative to the wind tunnel at higher angles of attack. However, they are not far off within a typical operating angle of attack range (<10 degrees). The other three codes show no change with angle of attack. In roll, all but ZAERO predicts the trend in wind tunnel data from positive to negative roll moment as angle of attack increases.
71
Figure 4-6. Comparison of lateral coefficients: Side Force (top left), Roll Moment (top right), and Yaw Moment (bottom)
Table 4-1. Comparison of Static Stability Derivatives
Wind DATCOM AVL ZAERO ZEUS Cart3D Tunnel
3.3862 4.970 3.4807 3.0796 3.6209 3.2604
𝐿𝐿𝛼𝛼 𝐶𝐶 -0.4773 0.4845 0.2544 0.2677 0.2272 0.2322
𝑚𝑚𝛼𝛼 𝐶𝐶 -0.0400 -0.0920 -0.0149 -0.0317 -0.0138 -0.0619
𝐶𝐶𝑌𝑌𝛽𝛽 -0.0211 -0.1381 -0.0298 -0.0322 -0.0199 -0.0199
𝑛𝑛𝛽𝛽 𝐶𝐶 -0.0556 -0.1219 -0.0160 -0.0007 -0.0482 -0.0614
𝐶𝐶𝑙𝑙𝛽𝛽
72 The important stability derivatives that can be determined from the static analysis are
shown in Table 4-1 at five degrees angle of attack. For Cart3D and wind tunnel data, these are
found via finite difference. For all other programs, they are output as part of the analysis. The lift
curve slope is over-predicted by AVL but otherwise fairly consistent across programs. As
mentioned above, the pitch stiffness is accurately predicted by ZAERO and higher fidelity codes,
but is opposite sign from DATCOM output due to fuselage interaction most likely. AVL again
over-predicts the weathercock stability, or yaw moment due to sideslip, but the other codes are very accurate with Cart3D matching the wind tunnel exactly. The Dihedral Effect, or roll moment due to sideslip angle, is most accurately represented by DATCOM with Cart3D close behind and the others far off.
4.2.2 Dynamic Derivatives
Table 4-2 shows the dynamic derivatives that were computed. Neither Cart3D nor the wind
tunnel were capable of determining any of these values and only DATCOM was capable of
outputting derivatives pertaining to the angle of attack rate term. For this reason, it is difficult to
speak to the fidelity level of these results. Compounding this difficulty, only the roll damping term
( ) shows relative consistency across the four programs capable of computing these terms.
𝑙𝑙 𝐶𝐶 𝑝𝑝
Table 4-2. Dynamic derivatives
Wind DATCOM AVL ZAERO ZEUS Cart3D Tunnel
0.0015 X X X X X
𝐿𝐿𝛼𝛼̇ 𝐶𝐶 -0.2726 -1.5135 -1.1380 -0.4490 X X
𝑚𝑚𝑞𝑞 𝐶𝐶 0.0021 X X X X X
𝑚𝑚𝛼𝛼̇ 𝐶𝐶 0.0244 -0.0669 -0.0015 0.0687 X X
𝑛𝑛𝑝𝑝 𝐶𝐶 -0.0114 -0.0679 -0.0102 -0.0075 X X
𝑛𝑛𝑟𝑟 𝐶𝐶 -0.3098 -0.2639 -0.2811 -0.3967 X X
𝐶𝐶𝑙𝑙𝑝𝑝 73 In a related effort, an artistic model was created in CAPS of the F-16 and analyzed in both
AVL and Cart3D [58]. There is a significant amount of publicly available aerodynamic data
pertaining to the F-16. With the multi-fidelity geometry in CAPS linked to AVL and Cart3D, an automated sweep of analysis parameters was performed to compare wind tunnel data published in
Ref. [49]. This data contains all required parameters used to generate the full 6-dof non-linear simulation with control deflections and dynamic derivatives of the previous chapter. Controlled through a simple python script, the analysis was performed in both Cart3D and AVL across seven angles of attack (-10 to 20 degrees), five sideslip angles (-10 to 10 degrees), four elevator deflections (-25 to 25 degrees), three aileron deflections (0 to 21.5 degrees), and three rudder deflections (0 to 30 degrees). This data is not repeated here, but shows a similarly good match in the linear range until 10 degrees angle of attack for all six forces and moments with Cart3D output.
AVL was used to compute the dynamic derivatives for comparison, and these results are shown in
Figure 4-7. This shows that AVL is much less accurate in determining dynamic derivatives, but there are presently few other options for collecting these parameters with more confidence and it was deemed acceptable for this work moving forward.
74
Figure 4-7. Comparison of dynamic derivatives between AVL and published wind tunnel data for the F-16
4.2.3 Fuselage Shape
In order to test the sensitivity to changes in the fuselage geometry, a study was performed in which the nose of the ESAV fuselage was varied in height-to-width ratio. This was done by dividing each x-coordinate and multiplying each y-coordinate of the geometry in the nose region by the h/w ratio and blending the variation via 3rd order polynomial interpolation into the rear portion of the fuselage. Five fuselage nose shapes were chosen for this study and are shown in
Figure 4-9.
75
Figure 4-8. Baseline ESAV vehicle geometry
Figure 4-9. Fuselage nose shapes
The vehicle was initially analyzed in ZEUS with all of the fuselage nose shapes. However,
the analysis was found to be unresponsive to the changes in the fuselage, specifically in terms of
the lateral coefficients. Figure 4-10 shows the side force (left) and yaw moment (right) output for
each height-to-width ratio at sideslip angles of zero and 5 degrees. In both plots, the coefficients
change very little in magnitude and show no discernable trend with the increase in height-to-width ratio.
76
Figure 4-10. Side force and yaw moment coefficients from ZEUS at zero (blue) and five degrees (red) sideslip angles
The same analysis was then performed in Cart3D. The data for the five clean vehicle
configurations is shown in Figure 4-11 at four combinations of angle of attack and sideslip angle;
1) 0 degrees α, 0 degrees β, 2) 10 degrees α, 0 degree β, 3) 0 degree α, 10 degrees β, and 4) 10 degrees α, 10 degrees β. The data shown is plotted against height-to-width ratio of the fuselage nose. The longitudinal coefficients show very little change with respect to the fuselage configuration, suggesting the wing dominates the forces and moments in this axis. However, as expected, the lateral coefficients show a large change across the different nose shapes in both the side force and yaw moment with respect to a sideslip angle. These two parameters are plotted as a function of sideslip angle in Figure 4-12 for each fuselage nose shape. Here, a well-defined trend can easily be seen. Both the side force and yaw moment coefficients show a nearly linear trend with respect to the height-to-width ratios from 0.55 to 1.5. At a height-to-width ratio of 2.0, the data shows a larger increase. At this point, the nose geometry has a height significantly greater than the fuselage height and it is beginning to act similar to what would be expected of a vertical fin if it were placed on the nose.
77
Figure 4-11. Clean configuration aerodynamic coefficients for all fuselage nose shapes
Figure 4-12. Side force and sideslip angle for each fuselage nose shape
78 4.2.4 Control Derivatives – Innovative Control Effectors
Since the wing control effectors are a fairly standard computation, only the
Spoiler/Deflector combination and Rhino Horn effectors were compared across the five analysis programs. The spoiler alone and a Spoiler-Slot-Deflector (SSD) are modelled in DATCOM for comparison, but only Cart3D is able to model the deflector alone, the Spoiler/Deflector without a slot, or the Rhino Horn. The results of the spoiler, deflector, and combination of spoiler plus deflector are all shown in Figure 4-13 for the roll and yaw moments. Neither program performs very well for these combinations. Cart3D is only marginally capable of capturing general trends relative to angle of attack, but is often far off in magnitude and in some cases even the wrong sign.
DATCOM only outputs a single value for change in roll or yaw due to the spoiler or SSD deflection and, therefore, shows no trend with angle of attack. The slot in the SSD is expected to eliminate the effectiveness in roll relative to not having a slot which appears to be reflected in the DATCOM output, but is not modelled in Cart3D. Overall, these results suggest that if these effectors are to be analyzed, a higher fidelity computation may be desirable in the future.
79
Figure 4-13. Comparison of Cart3D and wind tunnel data in roll and yaw moment for combinations of spoiler, deflector and spoiler+deflector
Figure 4-14 shows the same results for the Rhino Horn effector. In this case, Cart3D does a superb job of matching the wind tunnel in both roll and yaw for angles of attack between -5 and
10 degrees. These types of effectors may be very useful in yaw and as this suggests, they could accurately be included in the MADO process when modelled in Cart3D.
80
Figure 4-14. Rhino Horn effectiveness in roll and yaw
4.2.5 Summary of Computation Time, Capability, and Fidelity
The average computation time for each program to complete one run is shown in Figure
4-15 with all computations performed on the same computer. Here, Cart3D appears to be the favorite solution considering it runs faster and with results that more accurately represent the wind tunnel data. However, the computation time of each software program is a difficult and somewhat subjective comparison when taken in the MADO context. Ignoring the selection of number of iterations and mesh parameters, the next thing to consider is that Cart3D is run in parallel on all 40 cores available on the computer. ZEUS is also run in parallel, but is limited to 8 cores since it has been shown to have diminishing returns on computation time when using more than this. However, with multiple licenses, both ZAERO and ZEUS can run several cases simultaneously leading to up to 5 cases on 40 cores with ZEUS.
81
Figure 4-15. Average computation time per run
The controllability analysis is usually performed at multiple flight conditions such as takeoff/landing and cruise, each consisting of different Mach numbers and altitudes requiring a new set of aerodynamic analysis data. In order to trim the aircraft in these conditions, the programs that do not produce linear results must be evaluated at multiple angles of attack. Assuming the flight conditions result in trim angle of attack in the linear range of the lift curve, approximate trim
(neglecting control surfaces) can be found from the lift curve slope and zero angle of attack lift coefficient. This requires two runs for Cart3D and only one run for each of the other programs.
After finding trim, each program should be computed at that angle of attack. In addition, Cart3D also requires evaluation at a minimum of one sideslip angle for four total runs per flight condition and two for each of the other programs to compute all data needed for the CPR portion of the controllability analysis. The CPA portion of the analysis requires evaluation of each control surface.
The effectiveness of each is determined internally for all programs except Cart3D which must again use finite difference. For the 8 traditional wing control surfaces on the ESAV vehicle, this requires at least four more finite difference evaluations with a symmetric assumption. This leads to a total of eight runs in Cart3D, two in ZEUS and ZAERO, and just one in DATCOM or AVL per flight condition with a total computation time shown in Figure 4-16. As mentioned above, ZAERO and
ZEUS can run up to five cases simultaneously on the computer used, so if there were five flight
82 conditions, the Cart3D computation time would increase 5x while the others would remain
unchanged.
Figure 4-16. Computation time for one flight condition
ZEUS has been shown to have comparable results to Cart3D for the traditional vehicle with
significantly less computational effort. However, Cart3D is the only program of those investigated
that can analyze the innovative control effectors, and none of the other programs could be expected
to give good results for trends due to small changes in fuselage shape. Also of note, ZAERO and
ZEUS are the only programs currently capable of performing aeroelastic analysis if that is desired
in the future. Table 4-3 shows a summary of computation time for one MADO iteration, capabilities, and fidelity of each program. The fidelity is defined between level 0 and 4: 0-empirical data, 1-simple calculations, 2-Euler, 3-Navier Stokes, and 4-experimental data.
83 Table 4-3. Computation time, capability, and fidelity of each program
)
Shape Iteration)
Effectors Rate Terms Fidelity Level Aeroelasticity Fuselage Computation Time Innovative Control Control Innovative (One MDO MDO (One Dynamic Derivatives Dynamic (
DATCOM 0 Seconds X X X
AVL 1 Seconds X X X
ZAERO 1 Minutes X X
ZEUS 2 Hours X X
Cart3D 2 Day X X
4.3 Aerodynamic Analysis Selection
It is clear from the aerodynamic analyses presented that of the five programs compared, none are capable of solely outputting all required data. Of the data that can be output, there is a difficult tradeoff between accuracy, time, and capability. In terms of capability, only DATCOM can output control derivatives related to angle of attack rates (although they are normally
neglected). ZAERO and ZEUS are the only programs that can perform aeroelastic analysis, and
only Cart3D can model innovative control effectors like the Rhino Horn or small changes to the
fuselage shape. None of the programs evaluated give great results for combinations of
spoilers/deflectors. In terms of computation time, if one program is used to attempt to gather all
data for one MADO iteration, DATCOM and AVL can give results in a few seconds, ZAERO in
less than an hour, ZEUS in a few hours, and Cart3D in about a day. .
For MADO of the ESAV class of vehicle, Cart3D (or a similar program) is required at least
for analysis of innovative control effectors and fuselage shape changes and DATCOM could be
used solely for angle of attack rate derivatives (if desired) since its computational cost is negligible.
84 However, DATCOM results are not suited to configurations outside the empirical design space since it does not use physics to determine these values. For this reason, the decision was made for the MADO process of this work to utilize a multi-fidelity aerodynamic approach incorporating both
Cart3D and AVL in the loop to optimize the tradeoff between computational cost and accuracy of resulting data. The optimization of this combination of analyses and additional assumptions required is investigated in more detail in later chapters. The next chapter discusses the next complication in this process—rapidly generating analysis geometry for Cart3D.
85 CHAPTER V
MADO GEOMETRY
As discussed in Chapter III, the initial study vehicle used as a baseline was taken from a previous ESAV research effort performed at Lockheed Martin [6]. The selection of aerodynamic analysis performed in the previous chapter excluded all analysis programs that can be run solely with text-based input files. For Cart3D analysis, the typical process involves developing geometry in CAD and meshing it individually for each analysis case. For MADO studies, a full 3-dimensional representation of the vehicle must be generated parametrically and converted to a surface triangulation for Cart3D without additional user input. This chapter details the methods used to achieve this. It begins with a discussion of the Computational Aircraft Prototype Synthesis (CAPS) program. Then, the specific techniques are discussed that were used to convert the baseline ESAV geometry into CAPS geometry, create a multi-fidelity representation for AVL and Cart3D, and parameterize it for MADO studies.
5.1 Computational Aircraft Prototype Synthesis
In order to evaluate aircraft MADO configurations in the physical realm (e.g. perform real analysis vs. use of human intuition), the CAPS program is currently under development [57]. This program is designed to create parameterized CAD-like geometry, specifically for use in MADO applications. At the core of the CAPS program is the geometry generation tool, Engineering
SketchPad (ESP). This program is easy to use, easy to parameterize, and capable of outputting computational grids and sensitivities to the grids. Although currently primarily used for aircraft
OML applications, ESP (and CAPS as a whole) is not designed to be specific to any application, but instead designed generically such that all geometry is created with primitives (similar to a typical CAD program), but with access to underlying geometry definitions that can seamlessly be
86 passed downstream to analysis software. Analysis Interface Modules (AIMs) allow the user to
select the desired analysis and format the geometry and/or mesh accordingly. For multi-fidelity
aerodynamic analysis, the user can select AIMs ranging in fidelity from linear panel codes – where
the geometry is converted into simple text files – to full Navier Stokes programs. Structural analysis
AIMs have been developed as well for both NASTRAN and Abaqus and AIMs continue to be
developed for additional disciplines. All required tools for CAPS and ESP are open source and
freely available for download at https://acdl.mit.edu/ESP/.
Figure 5-1. Examples of CAPS geometries in ESP
Figure 5-1 shows several geometries that have been created and/or used in ESP and CAPS.
The Porsche Carrera, the turbine engine fan assembly, and the moon lander are geometry-only entities. All aircraft vehicles shown have been incorporated into CAPS with linked analyses. Bhagat et. al. [59] used ESP to create a component-based, parameterized aircraft geometry with various control surfaces placed on the body. Using their script, various components were muted (not
87 shown/output) so that the user could dynamically select which components were desired for a
particular analysis. The components of concern were wing, fuselage, canard, tails, and inlets. In
addition, there were two control effectors on the wing; ailerons and Spoiler-Slot Deflectors (SSDs).
Various combinations of these components were analyzed automatically to determine stability
derivatives. Bryson [24] performed aeroelastic analysis with a single parameter set defining linked structural and aerodynamic models. Heath [60] used CAPS to perform an inlet trade study on a low-boom supersonic aircraft.
5.2 Baseline Geometry—ESAV
The first step to inputting the ESAV geometry into the MADO process is to produce dimensioned cross-sections of the wing and fuselage from the initial CAD representation. For the fuselage, cross- sectional slices are taken approximately every 24 inches with extra slices added near the inlets and nozzle to distinctly capture these features (Figure 5-2). Each cross-section is then dimensioned with enough points to accurately represent the geometry. A sample of the cross-sections used are shown in Figure 5-3. Taking the x and y locations relating each dimension to the origin, the resulting coordinates are collected and manually typed into Matlab and saved as a .mat file to be loaded and processed at a future time.
Figure 5-2. Cross-sections used for dimensioning of ESAV fuselage and wing
88
Figure 5-3. Example of dimensioned cross-sections from ESAV CAD model
5.2.1 Wing Geometry
The wing geometry is directly measured in overall planform using Solidworks. The airfoils
at the wing root, break point, and tip are dimensioned and the coordinates are copied into Matlab.
These coordinates are compared to standard NACA airfoils through a least squares optimization to determine the best match of a standard airfoil to the provided geometry. This comparison is shown in Figure 5-4 for the root airfoil.
Figure 5-4. Comparison of measured airfoil and nearest standard airfoil definition for root airfoil shape
89 With the section defined in standard NACA notation, it is created in ESP using the syntax of Figure
5-5. In this syntax, udprim signifies a function call to an internal primitive function that is built into
ESP and naca is the name of the function that is called. The allowable inputs to the function are called by name—in this case, the Series is input as a local variable, series_w, which is the NACA
4-digit airfoil found through the optimization above. The result is a unit-length airfoil cross-section which is then scaled by the chord length, then rotated and/or translated into correct 3-dimensional position.
Figure 5-5. Standard NACA airfoil cross-section syntax in ESP
To generate the full wing geometry, the cross-section at the fuselage centerline is found by linear
interpolation between the two inboard wing sections. This leads to a total of seven wing cross-
sections used to represent the basic planform of the wing (Figure 5-6).
Figure 5-6. Cross-sections used to create wing planform
To create the wing planform, these sections are ‘ruled’ together, meaning the geometry is linearly
interpolated between them. In ESP, the syntax for this is below in Figure 5-7. The command mark
signifies the start of the sections that will be collected together as one body and after each section
is created, rule joins them together.
90
Figure 5-7. ESP syntax for creating ruled body for wing
After ruling the sections together, the baseline ESAV wing planform is shown in Figure 5-8. This geometry can be output for analysis or combined with other solid geometries through Boolean operations such as unions, subtractions, and intersections.
Figure 5-8. Baseline ESAV planform in ESP
5.2.2 Fuselage Geometry
In order to create a test case for the MADO, the notional vehicle fuselage geometry is first simplified. This decision is based on a desire to make the analysis run smoother, parameterize the fuselage shape, and to allow easy placement of different types of control surfaces. In order to do this, a series of iterations are performed that remove detail from the fuselage that is deemed unnecessary to accomplish the level of analysis fidelity desired. The less detail included in the model, the more robust it is to geometry generation errors. As an additional concern, with less detail, a coarser mesh is used for analysis, reducing overall computation time. The first reduction
of this manner is in the number of cross-sections used to represent the fuselage. Instead of using
91 cross-sections every 24 inches, a subset of the above cross-sections are selected to represent the major outline of the fuselage (Figure 5-9).
Figure 5-9. Reduced number of ESAV cross-sections
The next simplification is performed on the individual cross-sections. This version of the
fuselage is created using four cubic Bézier curves to represent each cross-section (symmetric about the vertical plane). Bézier curves are commonly used in many computer graphics applications to represent smooth shapes in a computationally inexpensive manner, based on the equation:
( ) = , ( ), 𝑛𝑛 𝑖𝑖=0 𝒊𝒊 𝒊𝒊 𝒏𝒏 where, in two dimensions, each vector,𝑪𝑪 P𝑡𝑡 i, is an∑ (x,y)𝑷𝑷 coordinate𝑩𝑩 𝑡𝑡 pair of control points determining
the shape of the curve, t ranges from 0 to 1 along the curve, and Bi,n is a set of basis functions of
the form:
, ( ) = (1 ) 𝑛𝑛 𝑖𝑖 𝑛𝑛−𝑖𝑖 𝑩𝑩𝒊𝒊 𝒏𝒏 𝑡𝑡 � � 𝑡𝑡 − 𝑡𝑡 For the cubic curve with four control points, the𝑖𝑖 expanded Bézier equation is: ( ) = (1 ) + 3(1 ) + 3(1 ) + . (5.1) 3 2 2 3 𝑪𝑪 𝑡𝑡 − 𝑡𝑡 𝑷𝑷𝟎𝟎 − 𝑡𝑡 𝑡𝑡𝑷𝑷𝟏𝟏 − 𝑡𝑡 𝑡𝑡 𝑷𝑷𝟐𝟐 𝑡𝑡 𝑷𝑷𝟑𝟑
92 In this case, P0 and P3 are the end points with P1 and P2 controlling the direction and magnitude of
the curve at these points. To apply this curve to the cross-sectional shapes, P1 and P2 are instead calculated as a vector with both magnitude and direction (Figure 5-10). The direction of P1 is fixed horizontally to match the tangent of the fuselage cross-section at the top. The direction of P2 and the magnitudes of both vectors are optimized using the Matlab function ‘fmincon’ to achieve the correct area at each cross-section.
P0 P1
P2 P3
P2 P1 P0
Figure 5-10. Fuselage cross-section created with Bézier curves
ESP allows the user to create a Bézier curve using any number of points. In order to break
the cross-section into four curves, a zero-length line segment is used between them. Figure 5-11
shows the programming used to create a cross-section in this manner in ESP. In this case, the cross-
sections are created in a pattern similar to a standard programming ‘for loop’ where the coordinates
for all sections are collected as arrays of data and the loop iterates over them for each section. In
the future, the control points that define the curves at each cross-section could be used as parameters
in the MADO process for topology optimization of the fuselage, but they are not selected as such
in this work.
93
Figure 5-11. ESP definition of Bezier cross-section made of four individual Bezier curves
Unlike the wing planform, the fuselage sections are ‘blended’ together, meaning the
geometry sections are interpolated using a spline function similar to lofting in standard CAD
packages. In ESP, the syntax for this is shown below in Figure 5-12. Here, the blend command simply replaces the rule command from above. With the fuselage and wing created, the two are unioned together to create the full baseline vehicle, shown in Figure 5-13.
Figure 5-12. ESP syntax for creating blended body for fuselage
94
Figure 5-13. Baseline vehicle geometry with fuselage cross-sections defined via Bézier curves
The Bézier curve representation of the ESAV fuselage does a superb job of matching the major contours of the vehicle. However, when parameterization is added to the wing and fuselage, this version often fails to union the wing and fuselage together, introducing robustness difficulties into the overall MADO framework. For this reason, the fuselage complexity are reduced one final time for this work. In the final version of the fuselage, the cross-sections are represented by a composite ellipse where the top half and bottom half of the fuselage are each represented by half- ellipses with the same major axes but different minor axes. The major axes match the width of the cross-section and the minor axes match the height of the section, with the axes centered at the widest point in the cross-section. For cross-section i along the fuselage span, the equations for the upper and lower z-coordinates of the cross-section are:
= + 1 (5.2) , , 2 𝑦𝑦 2 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑖𝑖 𝑤𝑤𝑖𝑖 𝑧𝑧 𝑧𝑧 ℎ � − � �2� for the upper portion of the cross-section, and:
= 1 (5.3) , , 2 𝑦𝑦 2 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑖𝑖 𝑤𝑤𝑖𝑖 𝑧𝑧 𝑧𝑧 − ℎ � − � �2� for the lower portion. Figure 5-14 shows an example of two cross-sections computed in this manner.
This representation of the fuselage, though quick to generate and robust to variations required of the MADO, is not as accurate as the Bézier version. As fidelity increases in the desired aerodynamic
95 analysis, it would be expected that this would be improved and is a continued research effort within
MSTC.
Figure 5-14. Comparison of elliptical cross-sections (blue) to dimensioned cross-sections (black)
ESP allows the user to create a cross-section by a superellipse with varying powers and different properties in each quadrant. For the simple version of this described above, the ESP input is shown in Figure 5-15. The inputs, ry, rx_w, and rx_e represent the major axis in the y-direction and the minor axes in the East and West x-directions (prior to rotating the section to y-z coordinates). The final version of the baseline ESAV vehicle used in this work is displayed in
Figure 5-16.
Figure 5-15. ESP definition of elliptical cross-section made of top and bottom ellipses of different minor axes.
96
Figure 5-16. Final version of baseline ESAV vehicle in ESP
The traditional wing control surfaces can be created by cutting them out of the wing using Boolean operations or approximated by rotating the coordinates of the wing sections prior to ruling the wing together. The latter option is utilized for greater robustness to changes in the wing planform. Figure
5-17 shows an example of this with the leading edge deflected. This method is expected to be acceptable for small deflection angles needed for finite difference, but is not likely to be accurate for larger deflections of the wing surfaces.
Figure 5-17. Example wing leading edge deflection
5.3 Multi-fidelity Geometric Representation in CAPS
Since the required aerodynamic analysis uses both AVL and Cart3D, a multi-fidelity representation of this vehicle must be created that is able to output the required analysis geometry. For the desired
MADO studies, the geometry is represented in three different ways. For a quick look at the
97 planform, only the wing is created. Otherwise, the geometry is output according to the desired
analysis. For low-fidelity vortex lattice analysis (e.g. AVL), only the cross-sections of the lifting
surfaces are created. In the case of the clean ESAV vehicle, this representation simply neglects the
fuselage and the rule command on the wing along with any control surface definitions because
AVL defines control surfaces only by percent chord input. For higher fidelity analysis (e.g.
Cart3D), the full geometry is generated. These geometries are shown below with both ESP and
analysis versions for AVL and Cart3D.
Figure 5-18. ESP and Analysis geometries represented for AVL and Cart3D
5.4 Parameterization
With the multi-fidelity geometry represented in CAPS, next step is to parameterize it for
MADO. In ESP, virtually any dimension used to create the geometry can be incorporated as an accessible design parameter. For this project, a total of 22 parameters are accessible in the geometry
98 creation (Figure 5-19). These parameters are primarily related to the wing design and control
surface selection, with a few parameters used for user control over the output. The first three
parameters control the output. OutType determines which of the above representations are to be created. It can be set to ‘1’ for the planform only, ‘2’ for AVL, or ‘3’ for the full geometric representation in Cart3D. dSpan allows the user to select approximate spanwise spacing of vortex lattice boxes in AVL. If set to ‘1’, saveSTL will cause the script to output an STL triangulation file of the geometry.
The wing planform is defined using seven planform variables. The wing planform area,
Area, aspect ratio, AR, leading edge sweep angle, sweep, x-location of the wing leading edge at the wing-fuselage intersection, XleWF, wing taper ratio, taper, wing break location, WBL, and wing break factor, WBF. Eight parameters are used to define the traditional wing control surface deflections; left trailing edge outboard flap – DTEFOL, left trailing edge inboard flap – DTEFIL, left leading edge outboard flap – DLEFOL, left leading edge inboard flap – DLEFIL, right trailing edge outboard flap – DTEFOR, right trailing edge inboard flap – DTEFIR, right leading edge outboard flap – DLEFOR, and right leading edge inboard flap – DLEFIR. The non-traditional control surfaces are the spoiler, controlled by DSpoiler, deflector, controlled by DDeflector, and the Rhino Horn, controlled by DRhinoHorn.
The only accessible direct fuselage design parameter, wf, is the defined fuselage width. It
controls the scaling of height and width of the fuselage, accounting for growth in the engine through
rubber engine sizing techniques. However, the fuselage cross-sections are linked to the wing
surface so that it shrinks or stretches with its sizing and placement. If the wing is moved forward
or rearward, the nose sections adjust accordingly. These parameters are discussed further in the next sections.
99
Figure 5-19. ESAV design parameters accessible in ESP
5.4.1 Wing Parameterization
The wing design parameters described above allow a wide variability in the shape of the wing. It can range from a straight, tapered wing to the lamda shaped wing of the baseline ESAV configuration. In the case of a lamda wing (or other non-straight wings), the traditional planform parameters such as taper ratio, sweep, and aspect ratio can be considered to be a weighted average of these parameters over each individual section of the wing, e.g.
= (5.4) 𝑛𝑛 . ∑𝑖𝑖=1 𝜆𝜆𝑖𝑖𝑆𝑆𝑖𝑖 𝜆𝜆 𝑆𝑆
100 The basic wing shape is defined using three chord sections; one at the root or fuselage centerline, one in the middle at the wing break, and one at the wing tip. The planform is defined using 8 total parameters – 3 chord lengths, 3 leading edge x-locations, and 2 spanwise y-locations.
However, using these coordinates directly as design parameters leads to many configurations that are unlikely to be viable. When choosing a more appropriate parameterization, the first thing to consider is overall wing area and aspect ratio. These two parameters are very influential in aircraft design and aerodynamic performance and are typically selected early in the design process. The leading edge sweep is typically constant along the span and can be used to eliminate two of the leading edge x-coordinate requirements, thus reducing the number of design variables representing the leading edge by one. Figure 5-20 shows these basic parameters defining the planform.
Figure 5-20. Lamda wing planform parameters
An additional consideration of the lamda wing design is how much crank is in the wing. If the middle chord were allowed to vary independently of the other two chords, it would potentially lead
101 to planforms where the outboard chord is larger than the inboard chord, which is likely not structurally viable. Instead, for this work the Wing Break Location (WBL) and Wing Break Factor
(WBF) terms are adopted. The WBL is defined as the location of the wing break in fraction of exposed span from 0 to 1. The WBF determines how much break the wing has. A WBF value of zero leads to a straight, tapered wing, whereas, a WBF factor of 1 leads to a wing in which the outboard section has a taper ratio of 1, or the middle chord is equal to the tip chord length. Figure
5-21 shows a demonstration of the WBF range from 0 to 1.
Figure 5-21. Demonstration of wing break factor
Another desire of the parameterization scheme is the ability to explicitly define the location of the wing in terms of the leading edge at the wing-fuselage intersection. This relationship requires treating the wing as three sections instead of only two. The additional parameters required for this and the WBF definition are shown in red in Figure 5-22.
102
Figure 5-22. Lamda wing with wing-fuselage intersection parameters
5.4.1.1 Building the Lamda Wing
After selecting the desired parameters, the mathematical equations required to build the wing are derived based on the aircraft geometry. Regardless of the wing shape, the overall span is determined by the definition of the Aspect Ratio, using the relationship:
= = ( )( ). 2 𝑏𝑏 The exposed span and the intermediate𝐴𝐴𝐴𝐴 span𝑆𝑆 →values𝑏𝑏 of� the𝐴𝐴𝐴𝐴 two𝑆𝑆 sections are
= , (5.5)
=𝑏𝑏𝑒𝑒𝑒𝑒𝑒𝑒 𝑏𝑏 − 𝑤𝑤+𝑓𝑓 , (5.6)
𝑏𝑏1 𝑏𝑏𝑒𝑒=𝑒𝑒𝑒𝑒𝑊𝑊𝑊𝑊𝑊𝑊 , 𝑤𝑤𝑓𝑓 (5.7)
respectively. 𝑏𝑏2 𝑏𝑏 − 𝑏𝑏1
When starting to determine the wing coordinates, no chord lengths are known. Instead,
taper ratios of the wing and sections are used to determine the chord lengths as a function of the
103 design parameters. The taper ratio defined as an input variable relates the wing tip chord length to
the wing-fuselage intersection chord length:
= . (5.8) 𝑐𝑐,𝑡𝑡 𝑟𝑟 𝑤𝑤𝑤𝑤 The value of the wing break factor determines𝜆𝜆 how𝑐𝑐 the taper ratios relate the other chord lengths to
each other. Since the WBF ranges from straight tapered wing ( = 0) to outboard section taper ratio of 1 ( = 1), computing the lambda wing chord 𝑊𝑊lengths𝑊𝑊𝑊𝑊 begins with computing the straight, tapered𝑊𝑊𝑊𝑊𝑊𝑊 wing chord. The chord length as a function of spanwise location can be linearly interpolated from wing-fuselage intersection to wing tip for a straight, tapered wing planform using the equation:
, ( ) = , + . (5.9) 𝑐𝑐𝑡𝑡−𝑐𝑐𝑟𝑟 𝑤𝑤𝑤𝑤 𝑆𝑆𝑆𝑆 𝑟𝑟 𝑤𝑤𝑤𝑤 𝑏𝑏𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐 𝑦𝑦 𝑐𝑐 𝑦𝑦 � � � At the break location, the chord is: 2
, , = , + 2 , 𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑡𝑡−𝑐𝑐𝑟𝑟 𝑤𝑤𝑤𝑤 𝑚𝑚 𝑆𝑆𝑆𝑆 𝑟𝑟 𝑤𝑤𝑤𝑤 𝑏𝑏 𝑏𝑏𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐 𝑐𝑐 �𝑊𝑊𝑊𝑊𝑊𝑊 � � � � � which reduces to: 2
, = , + , . (5.10)
𝑚𝑚 𝑆𝑆𝑆𝑆 𝑟𝑟 𝑤𝑤𝑓𝑓 𝑡𝑡 𝑟𝑟 𝑤𝑤𝑤𝑤 From the definition for taper from𝑐𝑐 Equation𝑐𝑐 5.8, 𝑊𝑊the𝑊𝑊𝑊𝑊 tip� 𝑐𝑐chord− 𝑐𝑐 is: � = , . Substituting this into
𝑡𝑡 𝑟𝑟 𝑤𝑤𝑤𝑤 Equation 5.10, the middle chord for the equivalent straight, tapered𝑐𝑐 wing𝜆𝜆𝑐𝑐 is:
, = , + , , . (5.11)
Next, define the outboard taper𝑐𝑐𝑚𝑚 of𝑆𝑆𝑆𝑆 the 𝑐𝑐straight,𝑟𝑟 𝑤𝑤𝑤𝑤 𝑊𝑊 tapered𝑊𝑊𝑊𝑊�𝜆𝜆𝑐𝑐 wing𝑟𝑟 𝑤𝑤𝑤𝑤 − as𝑐𝑐:𝑟𝑟 𝑤𝑤𝑤𝑤�
, , = = 𝑡𝑡, 𝑟𝑟,𝑤𝑤𝑤𝑤 2 𝑆𝑆𝑆𝑆 𝑐𝑐 𝜆𝜆𝑐𝑐 𝜆𝜆 𝑚𝑚 𝑆𝑆𝑆𝑆 𝑚𝑚 𝑆𝑆𝑆𝑆 and divide both sides of Equation 5.11 by , 𝑐𝑐 . Then invert𝑐𝑐 them to give this equation in terms of
𝑟𝑟 𝑤𝑤𝑤𝑤 the input design variables representing taper𝑐𝑐 of exposed section and the wing break location:
= (5.12) , ( ) . 𝜆𝜆 2 𝑆𝑆𝑆𝑆 𝜆𝜆 �1+𝑊𝑊𝑊𝑊𝑊𝑊 𝜆𝜆−1 �
104 From this equation, the definition of WBF is used to interpolate the actual taper ratio of outboard section, leading to:
= , + 1 , . (5.13)
This equation gives the relationship𝜆𝜆2 between𝜆𝜆2 𝑆𝑆𝑆𝑆 the𝑊𝑊 𝑊𝑊𝑊𝑊tip� chord− 𝜆𝜆 2and𝑆𝑆𝑆𝑆� middle chord of the actual lamda wing. The next step is to determine the taper ratio of inboard section. Since the inboard section is split into two sections from wing root to wing-fuselage intersection and wing-fuselage intersection to the break point or middle chord, define the ratio of the middle chord to the wing-fuselage intersection chord as:
= . (5.14) 𝑐𝑐,𝑚𝑚 𝑚𝑚 𝑟𝑟 𝑤𝑤𝑤𝑤 The exposed wing taper relates to this taper𝜆𝜆 ratio through𝑐𝑐 the relationship:
= = = , 𝑐𝑐,𝑚𝑚 𝑐𝑐𝑡𝑡 𝑐𝑐,𝑡𝑡 𝑚𝑚 2 𝑟𝑟 𝑤𝑤𝑤𝑤 𝑚𝑚 𝑟𝑟 𝑤𝑤𝑤𝑤 or, 𝜆𝜆 𝜆𝜆 𝜆𝜆 �𝑐𝑐 � �𝑐𝑐 � 𝑐𝑐
= , 𝜆𝜆 𝑚𝑚 and this relates to the inboard taper ratio through𝜆𝜆 𝜆𝜆2
= = . (5.15) 𝑐𝑐,𝑟𝑟 𝑐𝑐𝑚𝑚 𝑚𝑚 𝑅𝑅 1 𝑟𝑟 𝑤𝑤𝑤𝑤 𝑟𝑟 The wing chord length is linear between𝜆𝜆 the𝜆𝜆 𝜆𝜆root and�𝑐𝑐 break� � 𝑐𝑐 point� and can be interpolated between
, and using the equation:
𝑟𝑟 𝑤𝑤𝑤𝑤 𝑚𝑚 𝑐𝑐 𝑐𝑐 , ( ) = , + . (5.16) 𝑤𝑤𝑓𝑓 �𝑐𝑐𝑚𝑚−𝑐𝑐𝑟𝑟 𝑤𝑤𝑤𝑤� 𝑤𝑤 𝑟𝑟 𝑤𝑤𝑤𝑤 𝑏𝑏1 𝑓𝑓 𝑐𝑐 𝑦𝑦 𝑐𝑐 �𝑦𝑦 − 2 � � − � Setting the y-coordinate to zero, the root chord is computed2 by:2
, = , + . 𝑤𝑤𝑓𝑓 �𝑐𝑐𝑚𝑚−𝑐𝑐𝑟𝑟 𝑤𝑤𝑤𝑤� 𝑤𝑤 𝑟𝑟 𝑟𝑟 𝑤𝑤𝑤𝑤 𝑏𝑏1 𝑓𝑓 𝑐𝑐 𝑐𝑐 �− 2 � � 2 − 2 � Substituting = , and dividing through by , yields:
𝑐𝑐𝑚𝑚 𝜆𝜆𝑚𝑚𝑐𝑐𝑟𝑟 𝑤𝑤𝑤𝑤 (𝑐𝑐𝑟𝑟 𝑤𝑤𝑤𝑤 ) = 1 . (5.17) 𝜆𝜆𝑚𝑚−1 𝑅𝑅 𝑓𝑓 1 𝑓𝑓 𝜆𝜆 − 𝑤𝑤 �𝑏𝑏 −𝑤𝑤 �
105 These equations now describe the relationship between each chord in terms of the input wing design
variables. The derived parameters for the inboard and outboard sections are used in combination
with the input wing area to compute the root chord using the equation:
= (5.18) ( ) ( ), 2𝑆𝑆 𝑟𝑟 and the computation of the other chord𝑐𝑐 lengths𝑏𝑏1 1+ 𝜆𝜆follow1 +𝑏𝑏2 1from+𝜆𝜆2 intermediate section taper ratios that
were determined above:
= , = , and , = . (5.19) 𝑐𝑐𝑚𝑚 𝑚𝑚 1 𝑟𝑟 𝑡𝑡 2 𝑚𝑚 𝑟𝑟 𝑤𝑤𝑓𝑓 Since the leading edge sweep𝑐𝑐 is constant𝜆𝜆 𝑐𝑐 𝑐𝑐across𝜆𝜆 the𝑐𝑐 wing 𝑐𝑐span, the𝜆𝜆 𝑚𝑚leading edge x-location is:
( ) = , + , (5.20) 𝑤𝑤𝑓𝑓 𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿 𝑤𝑤𝑤𝑤 and the trailing edge x-location𝑥𝑥 at any𝑦𝑦 spanwise𝑥𝑥 point�𝑦𝑦 is− given2 � 𝑡𝑡𝑡𝑡𝑡𝑡 by:𝛬𝛬
( ) = ( ) + ( ), (5.21)
where ( ) is piecewise continuous 𝑥𝑥in𝑇𝑇 𝑇𝑇inboard𝑦𝑦 𝑥𝑥 and𝐿𝐿𝐿𝐿 𝑦𝑦outboard𝑐𝑐 𝑦𝑦 sections.
𝑐𝑐 𝑦𝑦
5.4.1.2 Additional Wing Aerodynamic Parameters
At this point, all of the information required to create the planform is available. However, there are a few additional parameters that are required for computing aerodynamic and geometric properties in the next chapter. The mean aerodynamic chord, , is computed as a weighted average of the inboard and outboard mean aerodynamic chords. The 𝑐𝑐definition̅ of the mean aerodynamic chord for each trapezoidal wing section is:
= = (5.22) ( )2 , and ( )2 . 2 �1+𝜆𝜆1+𝜆𝜆1� 2 �1+𝜆𝜆2+𝜆𝜆2� 1 𝑟𝑟 2 𝑚𝑚 In order to determine the weighted𝑐𝑐̅ 3 𝑐𝑐 average,1+𝜆𝜆1 the planform𝑐𝑐̅ 3 𝑐𝑐 area 1of+𝜆𝜆 2each section is computed as a
trapezoid,
= 2 , and = 2 , (5.23) 𝑐𝑐𝑟𝑟+𝑐𝑐𝑚𝑚 𝑏𝑏1 𝑐𝑐𝑚𝑚+𝑐𝑐𝑡𝑡 𝑏𝑏2 1 2 and the overall wing aerodynamic𝑆𝑆 � chord2 � is:� 2 � 𝑆𝑆 � 2 � � 2 �
106 = . (5.24) 𝑐𝑐1̅ 𝑆𝑆1+𝑐𝑐2̅ 𝑆𝑆2 The spanwise location of the mean aerodynamic𝑐𝑐̅ 𝑆𝑆chord is computed similarly for the individual
sections, except the outboard section in this case must be related to the distance from the wing root
by adding the half-span of the inboard section:
( ) ( ) = = + (5.25) ( ) , ( ) , 𝑏𝑏1 1+2𝜆𝜆1 𝑏𝑏1 𝑏𝑏2 1+2𝜆𝜆2 𝑀𝑀𝑀𝑀𝑀𝑀1 𝑀𝑀𝑀𝑀𝑀𝑀2 and, 𝑦𝑦 6 1+𝜆𝜆1 𝑦𝑦 2 6 1+𝜆𝜆2
= . (5.26) 𝑦𝑦𝑀𝑀𝑀𝑀𝑀𝑀1𝑆𝑆1+�𝑦𝑦𝑀𝑀𝑀𝑀𝑀𝑀1+𝑦𝑦𝑀𝑀𝑀𝑀𝑀𝑀2�𝑆𝑆2 The x-location of the mean aerodynamic𝑦𝑦𝑀𝑀𝑀𝑀𝑀𝑀 chord is the𝑆𝑆 distance from the wing root leading edge to the leading edge of the wing at the spanwise location of the mean aerodynamic chord. For the constant sweep wing:
= , = , (5.27)
and, 𝑥𝑥𝑀𝑀𝑀𝑀𝑀𝑀1 𝑦𝑦𝑀𝑀𝑀𝑀𝑀𝑀1 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬 𝑥𝑥𝑀𝑀𝑀𝑀𝑀𝑀2 𝑦𝑦𝑀𝑀𝑀𝑀𝑀𝑀2 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬
= . (5.28) 𝑥𝑥𝑀𝑀𝑀𝑀𝑀𝑀1𝑆𝑆1+𝑥𝑥𝑀𝑀𝑀𝑀𝑀𝑀2𝑆𝑆2 The sweep at the ¼ and ½ chord locations𝑥𝑥𝑀𝑀𝑀𝑀𝑀𝑀 along the𝑆𝑆 wing are often used for mass properties
estimates. To compute these estimates for any x between 0 and 1 in fraction of chord, the equations:
( ) ( ) = = (5.29) ( ), ( ), 4𝑥𝑥1 1−𝜆𝜆1 4𝑥𝑥2 1−𝜆𝜆1 𝑥𝑥1 𝑥𝑥2 and, 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬 − 𝐴𝐴𝐴𝐴1 1+𝜆𝜆1 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬 − 𝐴𝐴𝐴𝐴2 1+𝜆𝜆1
= (5.30) 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬𝑥𝑥1𝑆𝑆1+𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬𝑥𝑥2𝑆𝑆2 are used. 𝑡𝑡𝑡𝑡𝑡𝑡 𝛬𝛬𝑥𝑥 𝑆𝑆
5.4.1.3 Geometric Description of Wing Parameters
The previous sections described the math required to define the wing using the selected parameterization scheme, but how do these parameters actually affect the wing shape? Figure 5-23 shows the geometric variation due to each parameter. In these plots, the red outline is the original
107 ESAV baseline shape and the black lines are updated planform shapes based on parameter variation. On the left side of the plot, the parameters are reduced in magnitude relative to the baseline and on the right side, the parameters are increased.
Changes to the planform area result in a direct scaling of the wing size. In traditional design, the wing loading or weight per unit area of the wing is a major driving factor of the design and it is used in this work, as described in the next chapter. The aspect ratio relates the span of the wing to the planform area. A higher aspect ratio leads to a longer, thinner wing, while a lower aspect ratio results in a shorter wing. Aerodynamically, a higher aspect ratio wing is more efficient and is generally used for transport and long range aircraft, while a lower aspect ratio produces less drag and lower structural forces required of the fighter class of aircraft. At high speeds, the wing sweep helps to delay the onset of compressibility effects and associated shockwaves. The taper ratio relates to how small the wing tip chord is relative to the root chord for a given configuration. This
ratio can be used to manipulate the lift distribution and ultimately the structural bending moments
created by the lift. The wing break location controls the ratio of inboard section to outboard section
in the lamda wing. Finally, the wing break factor controls how much wing break exists from a
straight, tapered wing to a full lamda wing with outboard taper ratio of 1.
108
Figure 5-23. Geometric changes to the wing for each design parameter; the red outline is the baseline ESAV shape, the left side of the plot shows parameter values lower than baseline and right side shows parameters higher than baseline.
109 5.4.2 Fuselage Parameterization
In order to properly model the fuselage, internal components would ideally be determined
and placed in available space. In ESP and CAPS, any parameter used to generate the fuselage could
be accessible as a design parameter, allowing a user to create a model capable of accounting for
these components. However, the main focus of this work is the aerodynamics and controllability
assessment of the MADO and as such, the parameterization of the fuselage is limited to scaling
about the variation in the wing shape.
In traditional aircraft designs, the magnitudes of the aerodynamic forces on the tails
drastically overshadow that of the fuselage and for early design iterations, the fuselage can be
ignored. However, since the baseline ESAV vehicle is tailless, generating some representative
shape is important for estimating aerodynamic properties, particularly in yaw. For this reason, the
fuselage is organized into three sections – nose, wing, and rear. The x-location of each section in
the nose scales linearly with the distance to the leading edge location of the wing root. Each fuselage
station in the wing section scales with the length of the wing-fuselage chord. The rear of the fuselage maintains the same relative x-distance from the trailing edge of the wing to the end of the fuselage. This is demonstrated in Figure 5-24 for multiple wing placements. Ideally, components such as the cockpit could be maintained at the required size and move rearward into the fuselage as the wing moves forward toward the nose instead of collapsing, like in the case of real-world flying wing configurations such as the B-2. Instead, without independently modeling the cockpit, the parameters must be limited to not allow the wing to move far enough forward to cause geometry issues with the collapsed cockpit representation. As the wing moves rearward, the scaling looks a little more realistic and begins to take the form of something similar to the real-world B-1 aircraft.
Both of these aircraft are pictured below in Figure 5-25. The three cases show that this parameterization is capable—at least conceptually—of representing a wide range of aircraft configurations.
110
Figure 5-24. Scaling of fuselage in the x-direction with wing placement
Figure 5-25. Real examples of extreme forward wing placement (B-2, top) [61] and rear wing placement (B-1, bottom) [62]
111 The second parameterization used for the fuselage is a scaling of the height and width. The rationale for this scaling comes directly from traditional conceptual design in which the engine size scales directly with required thrust. Clearly, as the engine size grows, the fuselage must expand as well. For the same size wing, this reduces the exposed area and available real estate for potential control effectors. As this is a significant aerodynamic and controllability concern, the fuselage width, wf, is directly included as a design parameter in the geometry generation. The fuselage height is then automatically scaled by the same fraction as the width relative to the baseline. Figure 5-26 shows examples of this scaling. Similar to the lengthwise scaling above, there would ideally be lower limits at least due to the width required of the cockpit and as the overall fuselage width scales, there should be no reason for the cockpit itself to scale much beyond the original sizing.
Figure 5-26. Example scaling of fuselage width
5.4.3 Control Effectors
There are a total of 14 control effectors included in the model. They are not individually parameterized, but are linked to specific locations on the wing and fuselage and their geometry changes accordingly. There are eight traditional wing control surfaces consisting of leading and trailing edge flaps on both the inboard and outboard sections of the wing. The chord of these vary from 25% wing chord at the root to 40% of the wing chord at the tip and cover the entire exposed span of the wing. The spoiler/deflector combination consists of an upper surface (spoiler) with the
112 hinge line on the front and a lower surface (deflector) with the hinge line on the rear. They are defined to take the entire available chord between leading and trailing edge flap surfaces from 50 to 95% of the span of the outboard wing section. The Rhino Horn effector(s) are tied to the fuselage shape and are created by drawing a line from the third fuselage station (nominally 60 inches from the nose), 5 inches from the maximum width to the fuselage centerline at an angle of 30 degrees.
The placement of each of these effectors is shown in Figure 5-27.
Figure 5-27. Planform layout of parameterized ESAV model with control effectors
113 Figure 5-28 shows the spoiler and deflector geometries on the baseline ESAV geometry in
ESP. In both cases, ESP makes creating the surfaces simple using a primitive function called
‘spoilerz.’ In this function, the user inputs (x, y) coordinates defining the outline of the
spoiler/deflector shapes above and the function automatically creates them by projecting the outline
into the wing surface and extracting a copy of the top layer of the surface at a specified thickness.
Then, this copy of the surface is rotated about the leading or trailing edge for spoiler or deflector,
respectively.
Figure 5-28. Spoiler and deflector on ESP geometry
The Rhino Horn effector is also created with a simplified ESP primitive function, called
‘popupz.’ In this function, the surface is created by projecting the list of (x, y) coordinates of the outline, onto the surface as well. The result—a 3-dimensional sketch on the surface of the body— is extruded vertically to create the surface of the Rhino Horn. In order to make the effector realistic and applicable in the MADO sense, its height is dictated by the depth of the fuselage at the forward most point of the effector.
Figure 5-29. Rhino Horn control effector on ESP geometry
114 CHAPTER VI
THE MADO PROCESS
Throughout MSTC, research is being conducted to incorporate many disciplines into the aircraft MADO scheme. However, the work presented here focuses primarily on the Controllability
Analysis in the loop. The previous three chapters discussed the major contributions required for implementation and testing of this portion of the MADO framework and the minimum required set of additional data and analyses. The remaining components and data flow for the MADO framework of this project are described in the N2 diagram in Figure 6-1.
The MADO process begins with a problem definition in the form of a desired mission
profile, along with a list of assumptions for static inputs. The mission profile is defined in terms of
flight phase—takeoff, cruise, climb, etc. with assumptions for the wing loading (W/S), thrust-to-
weight ratio (T/W), and engine Specific Fuel Consumption (SFC). Initial planform design
parameters are selected to define the rest of the wing. Conceptual design equations are used to
compute fuel fractions for each segment of the mission profile—e.g. the fraction of fuel burn relative to the gross weight during each mission segment [1]. Relating the gross weight back to wing loading and thrust loading, the size of the wing and engine is determined and a planform layout of the vehicle is drawn. The components are then located on this layout to compute center
of gravity and moments of inertia that are required downstream. The wing area and design
parameters are exported to the CAPS program through a Python interface with Matlab. CAPS then
uses these design parameters to generate the appropriate geometry in ESP which is then
automatically sent to the desired aerodynamic analysis software. After the analysis is performed,
the results are returned to Matlab to be analyzed. The controllability analysis is performed and the
results are incorporated into the evaluation of the optimization cost function. This cost function is
then the subject of the overall optimization algorithm.
115
Figure 6-1. MADO N2 diagram
This chapter discusses the components of the N2 diagram not mentioned in the preceding chapters, including mass properties estimates, the cost function, and optimization methods. It then details the implementation of the MADO framework, along with the decisions and assumptions made in an attempt to reduce the computation time of the incorporated analysis while simultaneously enabling a practical demonstration of the MADO framework. Finally, it describes the incorporation of the MADO into the distributed computing framework, MSTC Engineering
[63].
6.1 Mass Properties
In order to complete the controllability analysis described in Chapter III, there is a minimum set of mass properties that must available. To determine the required lift to trim the vehicle, the weight must be known. The location of the CG is vital to determine the pitching and yawing moments and ultimately determine steady-state trim. After computing the trim requirements, the inertias of the vehicle must be known to analyze any dynamic CPR requirements.
These required values could ideally be determined for any design parameterization set directly from the resulting geometry. Indeed, some work has been performed to include structural sizing in the
116 MADO loop [24]. However, even with this implementation, the structure is only one driver in the overall weight of a vehicle. In order to arrive at a feasible estimate, designers must lay out various additional components, such as avionics, fuel tanks, engines, actuators, etc. and size them accordingly. While this is an eventual goal of the process, it was not within the scope of this project.
Instead, traditional conceptual design techniques from Reference [1] for estimating mass properties
were used. In this process, the ratio of required fuel weight to the vehicle gross weight is first
estimated using mission segment weight fractions such that for a given mission segment, i, the
weight of fuel burned is:
= 1 , (6.1) 𝑊𝑊𝑖𝑖 𝑓𝑓𝑖𝑖 𝑖𝑖−1 and the total fuel required for a given𝑊𝑊 mission� is−: 𝑊𝑊𝑖𝑖−1� 𝑊𝑊
= , (6.2) 𝑛𝑛 for a mission consisting of n segments. 𝑊𝑊𝑓𝑓 ∑1 𝑊𝑊𝑓𝑓𝑖𝑖
In the appendix of Reference [1], Raymer uses an example futuristic F-16 to demonstrate
the use of the mission fuel fractions approach to determine gross takeoff weight. This example
problem uses a mission profile consisting of 13 mission segments; (1) Warmup and Takeoff, (2)
Climb, (3) Cruise, (4) Accelerate, (5) Dash, (6) Combat, (7) Weight Drop, (8) Accelerate, (9) Dash,
(10) Cruise, (11) Descent, Loiter, (12) Descent for Landing, and (13) Land (Figure 6-2). This
mission profile was adopted for this work and modified to more closely match the expected mass
properties of the ESAV baseline configuration.
117
Figure 6-2. Mission Profile for conceptual weight computation
For each of the mission segments above, a conceptual equation is used to compute the fraction of fuel burned relative to the initial weight of the aircraft in that section. For example, to compute the weight fraction during cruise in segment i, the equation:
= (6.3) ( ) 𝑊𝑊𝑖𝑖 −𝑅𝑅𝑅𝑅 is used. In this equation, is the range𝑊𝑊 traveled𝑖𝑖−1 𝑒𝑒𝑒𝑒𝑒𝑒 during�𝑉𝑉 𝐿𝐿⁄ 𝐷𝐷the� cruise segment, is the specific fuel consumption of the engine,𝑅𝑅 is the cruising velocity, and is the lift-𝐶𝐶to-drag ratio in this condition. Ideally, with the aerodynamic𝑉𝑉 tools of Chapter IV, 𝐿𝐿⁄𝐷𝐷 could be directly computed and this fuel fraction could be estimated more accurately. However,𝐿𝐿 ⁄the𝐷𝐷 methods employed here require generating the geometry after estimating the mass properties. As a result, the lift-to-drag ratio is instead estimated based on wing loading by:
1 = 1 . (6.4) 0 + 𝐿𝐿 𝑞𝑞𝐶𝐶𝐷𝐷 𝑊𝑊 𝐷𝐷 𝑊𝑊 𝑆𝑆 𝑆𝑆 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞 In this equation, = 1 2 is the dynamic⁄ pressure at the cruise condition, is the 2 𝐷𝐷0 zero-lift drag coefficient, 𝑞𝑞 is the⁄ aspect𝜌𝜌𝑉𝑉 ratio, and is an assumed efficiency factor. In𝐶𝐶 all of the mission segment weight fraction𝐴𝐴 equations, the only 𝑒𝑒geometry design parameters from the previous chapter that are used are the wing area, aspect ratio, and sweep angle. Since the wing loading in any condition cannot be known without also knowing the weight, is selected to be used as a
𝑊𝑊⁄𝑆𝑆 118 design variable in the MADO and this is in turn used to compute the wing area after estimating the takeoff weight of the vehicle. Similarly, the segment fuel fraction equations call for the thrust loading, and it too is selected as a design variable moving forward.
Multiplying𝑇𝑇⁄𝑊𝑊 all fractions together for all of the mission flight segments gives the fraction
of the empty weight relative to the takeoff gross weight. For the parameterization set of the baseline
ESAV vehicle, the assumed thrust-to-weight ratio is taken from published values of modern fighter
airplanes. The wing loading is selected to approximate expected fuel fractions, resulting in an
estimated wing loading of 77.2 psf., which is within range of expected fighter plane wing loading
values. The output mission segment weight fraction summary for the baseline configuration is
shown in Figure 6-3.
Figure 6-3. Mission weight segment analysis summary for baseline ESAV vehicle
119 Using this fraction for fuel weight, an optimization routine is implemented to estimate the
total weight. In this algorithm, an initial guess is made of the gross weight and then based on this
guess, empirical equations are used to compute the weight of each component of the vehicle. These
equations are based on many previous designs and include several empirical factors in addition to
the generic shape and size of each component. For example, the wing weight equation for a fighter
aircraft is
. . = 0.0103 . . (1 + ) . (cos ) . . . (6.5) 0 5 0 622 0 785 −0 4 0 05 −1 0 0 04 𝑊𝑊𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝐾𝐾𝑑𝑑𝑑𝑑𝐾𝐾𝑣𝑣𝑣𝑣�𝑊𝑊𝑑𝑑𝑑𝑑𝑁𝑁𝑧𝑧� 𝑆𝑆𝑤𝑤 𝐴𝐴 �𝑡𝑡� �𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝜆𝜆 Λ 𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐 In this equation, and are weight multipliers𝑐𝑐 for delta and variable sweep wings,
𝑑𝑑𝑑𝑑 𝑣𝑣𝑣𝑣 respectively; is𝐾𝐾 the design𝐾𝐾 gross weight, or the total gross weight minus 50% fuel; is the
𝑑𝑑𝑑𝑑 𝑧𝑧 maximum load𝑊𝑊 factor the aircraft is designed to withstand; is the ratio of wing maximum𝑁𝑁 thickness to the chord length; is the planform area of control𝑡𝑡⁄𝑐𝑐 surfaces on the wing; and the remaining parameters have been𝑆𝑆𝑐𝑐𝑐𝑐 defined𝑐𝑐 previously.
The component weights, fuel, and payload are summed to give a computed gross weight.
This value is compared to the initial guess and the process is iterated using Matlab’s built in
‘fminsearch’ algorithm to minimize the difference between initial guess and computed gross
weight. Once the optimization converges, the result is a conceptual estimate for the gross weight
and individual component weights of a vehicle that could achieve the selected mission profile.
Using this method, the baseline ESAV configuration gross weight is computed to be 71,042 lb.,
compared to the estimated value of 69,000 lb., described in Chapter III. All of the component
weights for this configuration are printed in Figure 6-4.
120
Figure 6-4. Estimated component weights for baseline ESAV configuration
121 Once gross weight is determined in this manner, the each component is dynamically
located in 3D space relative to the parameterized geometry to compute center of gravity and
moments of inertia. The CG location of the major components—wing, fuselage and tails (if there
were any)—are based on conceptual estimates. For the wing, the CG is placed at 40% chord along
the mean aerodynamic chord. The CG of the fuselage can be between 40 and 50% of the length.
The CG of each control surface is placed at the geometric center. The total estimated actuator
weight is divided by control surface area and placed at the center of the respective control surface
for CG calculations as well. The main landing gear location is determined mathematically such that the location of the landing gear is at a 15 degree angle behind to the CG location to allow for takeoff rotation. However, no consideration is given to the landing rear location in relation to the rear of the fuselage. The remaining component locations were estimated by visual placement and linked to nearby fuselage or wing stations to allow them to move parametrically. Figure 6-5 shows the location of each component in red. A weighted sum is used to compute the overall CG which is shown in a black circle with an ‘X’ through it. With the mass and CG locations of each component tabulated, the moments and products of inertia are computed using point masses and perpendicular distances to the CG.
Figure 6-5. CG locations of baseline ESAV vehicle components
122 In order to fully test the planform generation and mass properties estimates, a Monte Carlo
Analysis is performed on the input design variables. Table 6-1 lists the 8 design variables used, the upper and lower limits allowed, and the reference values used to generate the baseline ESAV planform. Some of the bounds selected for these parameters are taken from conceptual ranges found in design textbooks References [1] and [64] for typical fighter airplanes. The other bounds are taken
from a combination of previous work and design exploration studies to determine at which points
the CAPS geometries begin to commonly fail (e.g. when the wing leading edge moves too far
forward).
Table 6-1. Monte Carlo design variables with upper and lower bounds
Figure 6-6 shows a series of nine randomly generated configurations from within these
boundaries compared to the baseline ESAV configuration in light blue. It is clear here that even
within the seemingly tight bounds, there is a wide variation in the potential for aircraft shapes.
These configurations are ordered by computed gross weight from top to bottom, left to right. For
similar planform shapes, drastic changes in weight can be achieved through scaling due to the wing
loading and ultimately wing area that results. The full Monte Carlo analysis is performed with
100,000 randomly generated parameter sets in the design space. The histogram of the resulting
weights is shown in Figure 6-7. The weight of the baseline ESAV configuration is marked with a
dotted red line. It can be seen that it is on the lower end of the potential weight within the design
space, suggesting (unsurprisingly) that weight was a strong driver in the initial design.
123
Figure 6-6. Example planforms generated, component CG locations, and resulting gross weight for Monte Carlo analysis
Figure 6-7. Histogram of vehicle gross weight for Monte Carlo analysis with 100,000 samples
124 The minimum weight achieved through Monte Carlo analysis is 25,902 lb., while the
maximum is 948,077 lb. and both of these configurations are shown in Figure 6-8. Based solely on
intuition, neither of these configurations need to be analyzed to be excluded from further design
efforts. However, for MADO the computer must make these decisions. Indeed, using only AVL to
estimate aerodynamic properties, the CPR analysis shows that the low-weight configuration on the
left only has about 3.7% of the required yaw control power for high angle of attack maneuvers.
Interestingly, the configuration on the right is able to pass the CPR test. This is due to the extremely long moment arm on the spoilers and ailerons producing large roll and yaw moments. Although far from being in the same weight class, the real-life B-2 uses physics with a control methodology
including a ‘split rudder’ near the wing tips, lending some credence to the analysis results here.
However, this extreme weight is a prime example of the need to model additional subsystems and
disciplines—especially structures—as it would likely not resolve into a realizable configuration.
This is lightly captured in the conceptual weight equations and for the computer to realize this in
the MADO, a weight penalty is included in the cost function, described later.
Figure 6-8. Minimum and maximum weight configurations achieved through Monte Carlo analysis
125 6.2 Multi-fidelity Aerodynamic Approach for MADO
Chapter IV discussed the selection of required aerodynamic analysis tools required for the
CPR analysis and narrowed the tools to a combination of AVL and Cart3D. However, as
mentioned, the full analysis with finite differencing of all control effectors in Cart3D would take at
least a day for a single iteration of the desired MADO. In addition, that analysis was performed
prior to the geometry simplifications that were performed to robustly parameterize the vehicle in
CAPS. As a result, an additional aerodynamic study using CAPS to determine the best use of
computational resources in the MADO framework is discussed below. In this study, A total of 686
analysis runs were performed in Cart3D including parameter sweeps of angle of attack and sideslip
and combinations of control effector deflections. The matrix of analysis runs are shown below in
Table 6-2. Out of all of these runs, only two cases fail to converge and both are with the spoiler and
deflector deployed, at high angle of attack, and high sideslip angles.
6.2.1 Convergence Study
The first part of this study analyzed Cart3D convergence. In Cart3D, the number of
iterations is specified as an input and there is not a clear method to determine convergence while
analysis is running. Instead, there is an option to restart a solution after completion to add iterations
if it requires longer run time to converge. The analysis runs that are discussed in Chapter III are
able to be restarted as many times as required to achieve fully converged state (if possible) for each configuration. However within the MADO process, the individual analysis cannot be allowed to run indefinitely or restarted and cannot even reasonably be monitored. A cutoff has to be determined for the number of iterations that each analysis run is allowed to perform. The plots in
Figure 6-9 show the history of the residuals along with the individual forces and moments for all of these runs (except the two failed cases) on the same plot. It can be seen from these plots that almost all cases that are able to converge, are converged prior to approximately 300 iterations. A
126 buffer is added to this number in the process and each case in the MADO aerodynamic analysis moving forward is allowed to run for 400 iterations.
Table 6-2. Run Matrix of aerodynamic study in CAPS for MADO
Figure 6-9. Convergence of clean configuration data (21 total runs)
127 6.2.2 Superposition Study
A superposition study was also performed using this set of analysis. The CPA analysis of
Chapter III assumes that for any combination of control effector deflection, the amount of control
power produced is the same as what would be produced by each of these deflections individually.
In reality, the airflow around the various control effectors is likely to interact with those around and
downstream of them. In fact, control effectors like the Rhino Horn generate a large portion of their
effectiveness through downstream interactions with the fuselage. The spoilers and deflectors are
likely to interact with other wing control surfaces as well. Many of the analysis runs above are
designed to test this phenomena.
From these results (shown in Figures 6-10 through 6-16) and for the combination of control
effectors employed in this work, the superposition assumption nearly holds. When the spoiler and
deflector are deployed individually and the results added together or when they are deployed at the
same time, they result in similar effectiveness. Combinations of flap deflections along the leading
and trailing edge match fairly well using superposition. However when all surfaces are deflected
simultaneously, not only does the superposition assumption not hold, but in some cases varying
multiple combinations of effectors in fact created the opposite trend of what would be expected if
the individual contributions of those effectors were added together (Figures 6-15 and 6-16). In these
cases, the deflection of the spoiler and deflector have shielded the outboard trailing edge flaps,
reducing their effectiveness (Figure 6-17).
One result of this study is a decision to separate the usefulness of the control effectors in this MADO study. The traditional wing control effectors are limited to use in the pitch and roll axes only and the Rhino Horn and spoiler/deflectors used in the study are analyzed all fully deflected to give the maximum yaw control power including superposition effects relative to each other. Ideally, in the future with more computational power, further combinations will be analyzed within the
MADO loop, but this is extremely prohibitive given the current available resources.
128 0.1 0.02
0.05 0.01
0 0
-0.05 -0.01
-0.1 -0.02 Deflector+Spoiler
SD dCl dCD -0.15 -0.03
-0.2 -0.04
-0.25 -0.05
-0.3 -0.06
-0.35 -0.07 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.2 0.04
0 0.02
-0.2 0
-0.4 -0.02 dCL dCm -0.6 -0.04
-0.8 -0.06
-1 -0.08
-1.2 -0.1 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.06 0.005
0 0.05
-0.005 0.04
-0.01 0.03
-0.015 dCn dCC 0.02 -0.02
0.01 -0.025
0 -0.03
-0.01 -0.035 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-10. Analysis of superposition of Spoiler and Deflector. The red line shows the summation of individual analysis of the spoiler and deflector. The black line shows the analysis performed with both fully deflected.
129 -3 10 0.05 20
0.04 15
0.03
10 0.02 dCl dCD 0.01 5
0
0
-0.01 TEFIL_Down+TEFOL_Down TEF_Down -0.02 -5 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.14 0.04
0.12 0.03
0.1 0.02
0.08 0.01
0.06 0 dCL dCm
0.04 -0.01
0.02 -0.02
0 -0.03
-0.02 -0.04 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
-3 -3 10 10 8 4
6 2
4 0
2 -2
0 dCn dCC -4 -2
-6 -4
-8 -6
-8 -10 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-11. Analysis of superposition of trailing edge flap surfaces deflected down. The red line shows the summation of individual analysis of the trailing edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected.
130 -3 10 0.02 5
TEFIL_Up+TEFOL_Up TEF_Up 0.01 0
0
-5 -0.01 dCl dCD -0.02 -10
-0.03
-15 -0.04
-0.05 -20 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
-0.02 0.035
0.03
-0.04 0.025
0.02 -0.06
0.015
-0.08 0.01 dCL dCm
0.005
-0.1 0
-0.005 -0.12
-0.01
-0.14 -0.015 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
-3 10 0.015 6
0.01 4
0.005 2
0
0 dCn dCC -0.005
-2 -0.01
-4 -0.015
-0.02 -6 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-12. Analysis of superposition of trailing edge flap surfaces deflected up. The red line shows the summation of individual analysis of the trailing edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected.
131 -3 10 0.02 16
LEFIL_Down+LEFOL_Down LEF_Down 14 0.01 12
0 10
8 -0.01
6 dCl dCD -0.02 4
2 -0.03
0
-0.04 -2
-0.05 -4 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.08 0.04
0.06 0.03 0.04
0.02 0.02
0 0.01
-0.02 dCL dCm 0 -0.04
-0.06 -0.01
-0.08 -0.02 -0.1
-0.12 -0.03 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
-3 10 0.015 8
0.01 6
0.005
4 0
2 -0.005 dCn dCC -0.01 0
-0.015 -2
-0.02
-4 -0.025
-0.03 -6 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-13. Analysis of superposition of leading edge flap surfaces deflected down. The red line shows the summation of individual analysis of the leading edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected.
132 -3 10 0.035 6
0.03 4
0.025 2
0.02 0
0.015 -2
0.01 -4 dCl dCD
0.005 -6
0 -8
-0.005 -10
-0.01 LEFIL_Up+LEFOL_Up -12 LEF_Up -0.015 -14 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.1 0.03
0.08 0.02
0.06 0.01
0.04 0
0.02 dCL dCm -0.01 0
-0.02 -0.02
-0.03 -0.04
-0.06 -0.04 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
-3 10 0.025 6
0.02 4
0.015 2
0.01 0
0.005 dCn dCC -2 0
-4 -0.005
-6 -0.01
-0.015 -8 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-14. Analysis of superposition of leading edge flap surfaces deflected up. The red line shows the summation of individual analysis of the leading edge flap surfaces on the left side of the vehicle. The black line shows the analysis performed with both deflected.
133 0.05 0.03
0.04 0.02
0.03
0.01 0.02
0.01 0 dCl dCD
0 -0.01
-0.01 Deflector+Spoiler+RhinoHorn+LEF_Down+TEF_Down AllDown -0.02 -0.02
-0.03 -0.03 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.25 0.1
0.2
0.15 0.05
0.1 dCL dCm 0.05
0 0
-0.05
-0.1 -0.05 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.07 0.01
0.06 0.005
0.05
0 0.04 dCn dCC 0.03 -0.005
0.02
-0.01 0.01
0 -0.015 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-15. Analysis of superposition of all surfaces fully extended and/or deflected down. The red line shows the summation of individual analysis of the surfaces. The black line shows the analysis performed with all deflected.
134 0.04 0.03
0.03 0.02
0.02 0.01
0.01 0
0 -0.01 dCl dCD
-0.01 -0.02
-0.02 -0.03 Deflector+Spoiler+RhinoHorn+LEF_Up+TEF_Up AllUp
-0.03 -0.04
-0.04 -0.05 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.1 0.12
0.1 0.05
0.08 0
0.06
-0.05 dCL dCm 0.04
-0.1 0.02
-0.15 0
-0.2 -0.02 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
0.1 0.01
0.08 0.005
0.06
0
0.04 dCn dCC
-0.005
0.02
-0.01 0
-0.02 -0.015 -10 -5 0 5 10 15 20 -10 -5 0 5 10 15 20 angle of attack, deg angle of attack, deg
Figure 6-16. Analysis of superposition of all surfaces fully extended and/or deflected up. The red line shows the summation of individual analysis of the surfaces. The black line shows the analysis performed with all deflected.
135
Figure 6-17. Low pressure area (blue color) shows rear wing flap surface shielded by spoiler/deflector, reducing effectiveness of the flap
6.2.3 Multi-Fidelity Aerodynamics
As final result of these analysis runs, they are used to implement a multi-fidelity approach to the aerodynamic analyses. A large source of the error in AVL prediction of the pitch and yaw moment coefficients is due to not modeling the fuselage. Since the fuselage is not modeled, the prediction of the neutral point—or location where these moment act upon—is not correct and if the moments are measured about the actual CG, then the distance between the CG the true neutral point is potentially much different than when only the wing is modeled. As part of the final MADO
process, Cart3D analysis is performed first and the results are used to determine the static margin—
136 defined as the distance between the CG and the neutral point in non-dimensional fraction of wing mean aerodynamic chord length. Then, AVL is run once to determine the computed wing-only neutral point location. After this, in the AVL input file, the reference CG location is changed to create the same static margin in the analysis before running the analysis again and collecting the final parameters. The results of this version of the analysis is compared to the wind tunnel data in
Figure 6-18.
-3 CD 10 Cl 1 0.4 WT Cart3D AVL 0.2 0
0 -1 -10 0 10 20 30 -10 0 10 20 30
-3 10 CC Cm 2 0.2
1 0.1
0 0
-1 -0.1 -10 0 10 20 30 -10 0 10 20 30
-3 CL 10 Cn 1 1
0 0
-1 -1 -10 0 10 20 30 -10 0 10 20 30
Figure 6-18. Comparison of Cart3D and AVL analysis to ESAV wind tunnel data after correcting for neutral point
After making this correction, the moments computed in AVL due to the traditional wing control surfaces AVL is able to model are more accurate (Figure 6-19). AVL is unable to capture many trends at higher angles of attack that Cart3D is able to, but the AVL data is deemed acceptable for the MADO fidelity desired, drastically reducing the number of Cart3D runs that are required per iteration for finite differencing. The final analysis consists of four Cart3D runs and 2 AVL runs
(Figure 6-20). Cart3D is run once at zero degrees angle of attack and sideslip to create a baseline.
137 Then it is run at one degree angle of attack to perform finite difference to find the lift and pitching moment curves. It is then run at one degree of sideslip to find the derivatives with respect to .
Finally, it is run with all of the yaw control effectors simultaneously deployed to determine the𝛽𝛽 control power available in the yaw axis. The results of this are used to compute the Neutral Point of the full vehicle to update AVL results. In AVL, a baseline analysis is performed to determine the computed Neutral Point without the fuselage and the CG is then updated accordingly before continuing the analysis. This process requires approximately 45 minutes of run time on the computers used for this work.
dCD dCl 0.02
0
0 -0.02 WT Cart3D -0.04 AVL -0.02 -10 0 10 20 30 -10 0 10 20 30
dCC dCm 0.02 0.02
0 0
-0.02
-0.02 -10 0 10 20 30 -10 0 10 20 30
-3 dCL 10 dCn 5
0
0 -0.05
-0.1 -5 -10 0 10 20 30 -10 0 10 20 30
Figure 6-19. Comparison of Cart3D and AVL analysis to ESAV wind tunnel data for trailing edge outboard flap left after correcting for neutral point
138
Figure 6-20. MADO aerodynamic analysis process
6.3 Cost Function
The goal of this work is to develop an aircraft MADO framework with a focus on
controllability. As discussed in Chapter III, the CPR and CPA methods are implemented because
of their suitability in creating a simple cost function that can be used to measure how close the
vehicle is to being controllable. The basis of the cost function is the ratio of the CPR to CPA,
summed across roll, pitch, and yaw axes and across all of the flight conditions discussed in Chapter
III. A 5% margin is added to the CPR such that when the vehicle has 5% more control power than
needed, the cost function goes to zero. For a given axis, , , the equation for control power cost
is 𝑙𝑙 𝑚𝑚 𝑛𝑛
( , , ) 1.05 ( , , ) 1.05 ( , , ) 𝐶𝐶( ,𝑙𝑙 𝑚𝑚, 𝑛𝑛) 𝑟𝑟𝑟𝑟𝑟𝑟 ( , , ) = . 𝑙𝑙 𝑚𝑚 𝑛𝑛 𝑟𝑟𝑟𝑟𝑟𝑟 𝑙𝑙 𝑚𝑚 𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 𝐶𝐶0𝑙𝑙 𝑚𝑚 𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶( , , ) ≤> 1.05𝐶𝐶( , , ) 𝐽𝐽𝐶𝐶𝐶𝐶 𝑙𝑙 𝑚𝑚 𝑛𝑛 � 𝑟𝑟𝑟𝑟𝑟𝑟 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐶𝐶 𝑙𝑙 𝑚𝑚 𝑛𝑛 𝐶𝐶 𝑙𝑙 𝑚𝑚 𝑛𝑛
Then, the total control power cost is found by summing across all three axes and all of the flight conditions considered:
= , , ( , , ) .
𝐶𝐶𝐶𝐶 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑙𝑙 𝑚𝑚 𝑛𝑛 𝐶𝐶𝐶𝐶 𝑙𝑙 𝑚𝑚 𝑛𝑛 𝐽𝐽 ∑ 139�∑ 𝐽𝐽 � From a controllability perspective, there is no disadvantage to having an over powered control
system and this is reflected in this portion of the cost function. However, the extra control power
would be directly associated with additional weight through the sizing of the control surfaces and
actuators. The first part of this Chapter showed that the computed weight of the vehicle must also
be a driving factor in the design to arrive at a potential intuitively-feasible solution. For this reason,
the total cost function used to demonstrate the MADO framework includes the computed vehicle
weight and is computed as:
= + . (6.6) 2 In this function, the weighting factors𝐽𝐽 𝑊𝑊 𝐶𝐶and𝐶𝐶𝐽𝐽𝐶𝐶 𝐶𝐶 𝑊𝑊 are𝑊𝑊𝑊𝑊 chosen0 give the two portions of the cost function similar magnitudes. The weighting𝑊𝑊𝐶𝐶𝐶𝐶 factors𝑊𝑊𝑊𝑊 are selected such that the cost of each portion is equal to 50 when the CPR exactly matches the CPA and when the vehicle weighs 100,000 lb.
6.4 Distributed Computing Environment and Optimization
Since many combinations of geometric parameters result in non-feasible 3D geometries or solutions, gradient optimization techniques are avoided because the gradient would be undefined when the generation of geometry fails. Instead, a Particle Swarm Optimization (PSO) technique is employed using Matlab’s built in global optimization tools. However, the minimum recommended particles in the swarm is ten times the number of design variables, or in this case 80 particles. The vast majority of the computation time involved in determining the cost function is derived from the
45 minutes of Cart3D analysis time described above. Although significantly less than the expected time described in Chapter IV, this means on a single computer one iteration of the optimization is expected to take 3600 minutes, or about 2.5 days. Based on results achieved with only a low fidelity analysis, a target goal of 30 iterations was set for the optimizer. This should require a minimum of
75 days to run.
Instead, the MADO framework is implemented in MSTC’s distributed computing environment, MSTC Engineering [63]. In particular, a tool currently under development and titled
140 ‘Bond’ is used. When the analysis is started, Bond seeks out computers on the network and is able to place computational jobs wherever available resources exist. Using this framework, the analysis time required is reduced to approximately two weeks of run time. The results of the MADO test cases are described in the next chapter.
141 CHAPTER VII
MADO RESULTS AND DISCUSSION
The MADO framework is tested in multiple phases. In the first phase, parameter sweeps are performed on both the low-fidelity models incorporating AVL and the multi-fidelity models with Cart3D included in the loop. The goal of this work is to capture the types of design sensitivities that are expected of the traditional conceptual design process, while at the same time attempting to explain some of the complex coupling phenomena associated with geometry changes due to the parameterization scheme. In the early phases of design, it is common to explore many design configurations using low-fidelity tools before settling on a small handful of configurations to take forward into the preliminary design phase. In this phase, design sensitivity studies may be performed at a higher fidelity to capture the physical results of small changes relative to the baseline configurations. In the second phase, the MADO analysis is performed. Again, it is first allowed to run only with AVL analysis in the loop and then with both AVL and Cart3D in the loop. This chapter discusses the results of these tests.
7.1 Parameter Sweeps
The MADO framework is used to perform parameter sweeps of cost function evaluation, covering the same range of parameters used in the previous chapter for Monte Carlo Analysis of geometry and mass properties. In each case, all parameters except the one of interest are held constant at the baseline values representing the initial ESAV configuration. In the low-fidelity analysis, each parameter sweep contains 31 total evaluations between the minimum and maximum of each parameter. In the multi-fidelity analysis, 11 evaluations are performed over the same boundaries.
142 During the conceptual design phase, estimates are commonly made of phenomena that are not able to be modeled using physics. For example, the effectiveness of the Rhino Horn and spoiler/deflector control effectors cannot be evaluated using AVL. DATCOM contains empirical estimates for a spoiler/deflector combination, but for the Rhino Horn or other innovative devices, there are no references. In order to test the algorithm in the low-fidelity mode, the moments generated by each control effector in the wind tunnel are divided by the length from CG to geometric center of the effector to give an approximation to the forces generated by the individual control effectors. Since both drag and lift are direct functions of area, it is assumed that for parameter changes to the vehicle, the force due to each control effector scales with the area of the effector, and the moments generated are estimated by multiplying this force by the updated distance to the CG. Although overly simplified, this or a similar estimate must be made to incorporate new technology in a conceptual design phase with low fidelity analysis.
Figures 7-1 through 7-8 show the results of the parameter sweeps using only AVL, shown as dashed lines, and with Cart3D included, shown as solid lines. Each of these figures contains three graphs. On the left side, the total computed cost function and gross weight for each parameter is shown. The two plots on the right of these figures show the ratio of CPA to CPR in pitch and yaw axes, respectively. In all cases, the vehicle was computed to have roll control power well above the required control power and results related to this axis are not presented. When the vehicle is controllable, CPA/CPR is equal to or greater than 1 and is shown in the green highlighted section.
Any values below this result in an uncontrollable vehicle and are shown in the red highlighted section. In order to discover the driving factors in this analysis, the CPA/CPR results are separated by each of the six flight conditions evaluated.
The first parameter that is varied is Thrust-to-Weight ratio (Figure 7-1). This parameter has a direct link to the mass properties and fuselage geometry. For a given vehicle, a higher thrust-to- weight ratio leads to a larger engine, wider fuselage, and overall heavier vehicle. The heavier vehicle in turn leads to a larger wing, all of which lead to larger control surfaces for the Rhino Horn
143 and spoiler/deflector combinations. Since AVL does not model the fuselage affects and additional
instabilities created by making the nose longer, the increased control effectiveness associated with
increasing the size of these effectors reduces the yaw deficiency. In pitch, the increased size and
inertia of the engine and fuselage makes the 9-g pitch up maneuver nearly uncontrollable at the
upper end of the parameter range. Using Cart3D, similar results are achieved in the yaw axis, but
the pitch axis shows less controllability, likely due to the increased fuselage size moving the neutral
point toward the rear.
Figure 7-1. Parameter Sweep – Thrust-to-Weight Ratio. MADO cost function(s) and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
The next parameter explored is the wing loading (Figure 7-2). Varying the wing loading parameter results in an inverse relationship with weight and cost. As the wing loading increases, the size of the wing relative to the rest of the vehicle is smaller, resulting in an overall lighter vehicle. Although the vehicle becomes heavier with a lower wing loading, the increase in wing size in turn leads to an increase in spoiler/deflector size, again reducing the gap between CPA and CPR in yaw. In this case, both the low-fidelity and high-fidelity solutions behave similarly.
144
Figure 7-2. Parameter Sweep – Wing Loading. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
The aspect ratio of the baseline configuration is 4. Aside from being within standard ranges
for a fighter aircraft, an assumption can be drawn from Figure 7-3 that this value was selected as an optimization between lower weight and meeting pitch control objectives in early design phases.
According to the AVL results, for aspect ratios below approximately 3.75, the vehicle is not be able to perform the 9-g pitch up maneuvers desired. However, with the higher fidelity results incorporated, the data suggests the aspect ratio should be increased slightly to a value greater than
4.5. In yaw, however, the parameterization used leads to a longer moment arm on the spoiler and deflector as the aspect ratio is increased, making it more appealing. If only the low-fidelity analysis were used, it would suggest placing the aspect ratio outside the current boundary for the best case yaw controllability. The higher fidelity data however, suggests there is a maximum in the yaw control power ratio at an aspect ratio of approximately 5.5.
Figure 7-3. Parameter Sweep - Aspect Ratio. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
145 Similar to the aspect ratio, most modern fighter aircraft have wings swept near 40 degrees.
In this case, the low-fidelity analysis seems to agree with this value as well and for sweep values
of slightly less than 39 degrees, the vehicle fails to meet maximum g-load requirements in pitch.
Based on the AVL analysis, increasing the wing sweep increases the yaw damping derivative slightly and reduces the yaw control power required, making it an appealing option based on this study. With the higher fidelity analysis influencing the design, the sweep could be increased slightly to 45 degrees for the pitch up maneuver and potentially settle on a local minimum of the cost function around 48 degrees sweep.
Figure 7-4. Parameter Sweep - Sweep. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
The location of the wing leading edge is typically determined using a sensitivity study
similar to this one. It is also the only parameter with a clearly preferred value in the low-fidelity
analysis, with a cost function minima occurring within the analysis boundary. This minima is
created by a combination of pitch and yaw requirements. In pitch, the further rear the wing is placed,
the further rear the CG, reducing aircraft stability and making it easier to control. Moving the wing
too far forward has the opposite affect and at about 300 inches from the nose, the vehicle cannot be
controlled in the 9-g pitch up maneuver. However, moving the wing toward the rear elongates the
fuselage and therefore increases the required yaw control power. Clearly the Rhino Horn effector
does not counteract this despite its increased moment arm. In this case, the higher fidelity analysis
shows similar trends, but with a best wing placement farther toward the rear of the fuselage. Using
146 AVL only, the analysis shows the wing should be placed at about 275 inches from the nose. With
Cart3D, the analysis suggests placing it around 325 inches from the nose. The baseline ESAV
locates the wing at 343 inches.
Figure 7-5. Parameter Sweep – Wing Leading Edge Location. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
In both pitch and yaw, increasing taper ratio reduces the gap between required and available
control power. This seems to be directly related to the decrease in weight that results. However, this is unintuitive because the only weight-related equation where taper ratio appears in the component weight equations is the wing weight equation, where the trend is opposite—increasing
weight with increasing taper ratios. For the parameter sweep boundary, the wing weight is 11,198
lb. at a taper ratio of 0.15 and 10,979 lb. for a taper of 0.25 and the overall vehicle weight decreases
by 2,000 lb. over this same parameter range. This is an unexpected (and unintended) consequence
of the MADO parameterization. Since the fuselage stations are linked to the wing root chord,
changing the taper ratio results in a change to the fuselage length. Lower taper ratios lead to a larger
root chord, and in this case, a longer fuselage. In the mass properties estimate, the longer fuselage
increases the weight, which in turn increases the overall wing size in the sub-optimization loop.
The overall increase in wing size, combined with higher taper ratio creates larger spoilers and
deflectors on the outboard section of the wing, which in turn yields increased yaw control
effectiveness.
147
Figure 7-6. Parameter Sweep - Taper Ratio. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
Decreasing the wing break location increases the size of the outboard section of the wing
and therefore increases the size and effectiveness of the spoilers and deflectors. Based on the low-
fidelity results, there should be no break in the wing. When the high-fidelity results are
incorporated, the wing break would be best located around 34% of the exposed span. This is due to
a tradeoff between the pitch and yaw control effectiveness where placing the break farther inboard leads to larger spoilers and deflectors for yaw, while moving the break outboard leads to larger (and more effective) inboard effectors for pitch.
Figure 7-7. Low-Fidelity Parameter Sweep – Wing Break Location. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
Based on both low and high-fidelity analyses, there is strong evidence to suggest removing the
wing break altogether and using a straight, tapered wing instead of the lamda configuration. It
should be noted however, that there may be other benefits to this configuration that are not modeled
here.
148
Figure 7-8. Low-Fidelity Parameter Sweep – Wing Break Factor. MADO cost function and weight (left), CPA/CPR in pitch axis (middle) and yaw axis (right)
Typically, the sensitivity analyses that are performed are centered on an aircraft design that
is based on empirical data and intuition covering over 100 years of aircraft design experience and is nearly optimal to begin with. Small excursions form this baseline might act as a final step in a design process to move toward an optimal design. In the ESAV configuration, the vehicle was likely designed to typical requirements and parameters initially. Removing the traditional tail surfaces and replacing them with the innovative control surfaces leads to a vehicle that is suboptimal. As a result, if the parameter sweeps above are used as a sensitivity analysis and allowed to inform the design, the resulting configurations created by choosing the minimum cost of each parameter look like Figure 7-9. The low-fidelity result weighs only 26,000 lb., but has a larger control deficiency than the baseline configuration. The multi-fidelity result is nearly controllable, but weighs 118,000 lb. Clearly, the sensitivity analysis alone does not lead to a successful planform.
With MADO, multiple parameters can be varied simultaneously and this is explored in the next section.
149
Figure 7-9. Vehicle configurations resulting from minimum cost of each parameter sweep excursion from ESAV baseline
7.2 Results of Low-fidelity MADO with AVL
Similar to the sensitivity study above, the MADO optimization is tested separately with low-fidelity using only AVL and with multi-fidelity incorporating Cart3D. Figures 7-10 and 7-11 show the first 6 iterations of the low-fidelity Particle Swarm Optimization with 80 particles. On the left side of these plots is a scatter plot for each of the 8 design variables showing the relationship between each design parameter and the computed cost function. In the first iteration, there is a large variation in cost for the particles across the 8 design parameters with the highest cost function value found to be almost 7,000 and the lowest found to be 14.73. On the right side of these plots, the resulting shape of the lowest cost vehicle configuration for each iteration is plotted. The first three iterations show significant variation in the potential planform shapes from extremely long wingspan, high-aspect ratio wings to shorter, conventional shapes.
150
Figure 7-10. Results of low-fidelity MADO with AVL, iterations 1 through 3
In iteration 4, the vehicle begins to take the shape of a flying wing configuration, with the wing leading edge moving far forward. By iteration 6, the particles shown in the scatter plot are beginning to cluster and almost all of them result in cost function values less than 1,000. At this point, the optimal configuration has already almost converged with the lowest cost function evaluation equal to 7.35.
151
Figure 7-11. Results of low-fidelity MADO with AVL, iterations 4 through 6
The optimization is allowed to continue for a total of 100 iterations, reducing the cost
function value to 6.28. In the scatter plot of the final iteration, almost all particles are on the
horizontal axis and have nearly converged onto each other. The final vehicle design is shown in
Figure 7-12. Mathematically, the MADO algorithm performs perfectly. The minimum ratio of control power available to control power required is 104.4% in the high-angle of attack flight condition. However, the weight of this vehicle is computed to be 111,700 lb., well above what
152 would be considered reasonable for an airplane of this class. It may be that better estimates of the
control effector forces through surrogate modeling or other techniques could lead to a more likely
candidate configuration. However, the multi-fidelity MADO is able to directly model these effectors, drastically reducing this potential source of error.
Figure 7-12. Final Result of low-fidelity MADO with AVL
Figure 7-13. Final vehicle configuration result of low-fidelity MADO with AVL
7.3 Results of Multi-fidelity MADO with Cart3D and AVL
The multi-fidelity MADO optimization is allowed to run for a total of 40 iterations. The optimization algorithm performed similarly to the low-fidelity case, with the first several iterations resulting in drastically different planform variations and the final 20 only making small refinements to the shape. The first two iterations are shown in Figure 7-14. In the first iteration, the lowest cost
153 configuration is lighter than the final result of the low-fidelity test case, but still extremely heavy
(102,110 lb.) and uncontrollable. The second iteration yields a controllable solution that only weighs 86,059 lb. From this point forward, the optimizer is able to seek out similarly controllable solutions that are lighter weight. Figure 7-15 shows some of the smaller refinements that are made during iterations 3 through 6 as it selects a near delta-wing configuration.
Figure 7-14. Results of multi-fidelity MADO with AVL and Cart3D, iterations 1 and 2
154
Figure 7-15. Results of multi-fidelity MADO with AVL and Cart3D, iterations 3 through 6
155 After 40 iterations, the particles in the scatter plot are nearly converged, with the general
planform shape changing very little from the 6th iteration to the final (Figure 7-16). In this configuration, the pitch axis control power during the 9-g pitch up maneuver has the lowest
CPA/CPR ratio with 105.1% control available. In the yaw axis, the high angle of attack condition leads to 107.9% yaw control available. The weight of this vehicle is computed to be 54,412 lb., significantly less than the uncontrollable baseline ESAV configuration. Figure 7-17 shows the final configuration of the multi-fidelity MADO process.
Figure 7-16. Final Result of multi-fidelity MADO with AVL and Cart3D
Figure 7-17. Final vehicle configuration result of multi-fidelity MADO with AVL and Cart3D
156 7.4 Discussion of low-fidelity and multi-fidelity optimization results
Figure 7-18 shows the Pareto front of the results of both the low-fidelity and multi-fidelity
MADO results, with the cost function due to the control power plotted on the y-axis and the vehicle gross weight plotted on the x-axis. The vehicle configurations that have at least 5% more control power available than control power required across all flight conditions have a cost due to control power of zero and are highlighted in red. The final optimal solutions for both cases are highlighted as green dots. In both cases, the Pareto front supports the results and shows that in order to reduce the weight any further, the controllability of the aircraft must be sacrificed, meaning that the optimizer is likely successful in finding the minimum of the cost function.
Figure 7-18. Pareto front of low-fidelity (left) and multi-fidelity (right) MADO results
In the low-fidelity test case, there is clearly evidence to justify continuing the current design trends—basing initial concepts on intuition and empirical data. A practiced designer is likely to be able to draw an aircraft with greater likelihood of success than the result presented. This is likely the reason MADO tools have not been incorporated into the design process to date. However, the high-fidelity test case results in a configuration that has a reasonable chance of success in the later stages of design, according to the physics and disciplines that are modeled. In fact, a quick internet search of aircraft with similar planforms yields the X-4 Bantam, which is compared to the MADO
157 planform in Figure 7-19. If the fuselage parameterization allowed the cockpit to move back within the wing region and the rear of the fuselage to move forward, these to planforms would look very similar. Ironically, the X-4 was one of the first aircraft to be designed semi-tailless (with no horizontal tail). It was designed to test flight at transonic speeds, making 81 test flights, some of which were piloted by Chuck Yeager [65].
Figure 7-19. Comparison of X-4 Bantam [66] and multi-fidelity MADO result
158 CHAPTER VIII
CONCLUSIONS AND FUTURE WORK
8.1 Conclusions
Traditionally, aircraft designs are based in large part on empirical and intuitive
relationships related to previous designs. In conceptual design, a limited amount of low-fidelity analyses may be incorporated into the decision making process before limiting the design space to only a few potential configurations that are analyzed in more detail in the preliminary design phase.
This method of design has been shown to have major implications on the life-cycle cost of modern aircraft, with the earliest design decisions based on the least amount of physical data representing the largest changes to overall cost of vehicles. As new technologies are developed, incorporating them into this design process continues to raise the financial risk involved and tends to pressure decision makers to avoid incorporating radical and potentially evolutionary changes into future designs.
It is a growing belief that in order to successfully deviate from this process and truly enable revolutionary designs in the future, methods must be developed to reduce this risk. One clear way of doing so is to incorporate higher levels of analysis fidelity earlier in the design process. In order to do so, however, tools must be developed to automate large portions of the process. Within
MSTC, there is a goal to develop tools that enable tens of configurations to be analyzed with preliminary design levels of fidelity using roughly the same level of human effort as would typically be required to analyze two or three designs.
This dissertation presented the development of one of these tools—the Controllability
Analysis—and the testing of it on a notional tailless fighter configuration that requires the use of innovative control effectors. In previous aircraft MADO and stability and control publications that
were reviewed, either the controllability analysis, the supporting data, or both were of low fidelity.
159 Often, the MADO uses an overly simplified controls analysis that only incorporates static design
requirements such as static margin. Otherwise, the MADO may use low-fidelity geometry and aerodynamic results to demonstrate more complicated controller designs. This dissertation showed some of the practical considerations across multiple disciplines that are involved to truly achieve preliminary design levels of a fidelity in the MADO process with a controllability focus.
In Chapter III, the Control Power Required method is described and derived. This method
is selected due the simplicity of representing the mathematical ratio between the Control Power
Required and Control Power Available of the vehicle in a cost function while at the same time
being capable of analyzing dynamic flight conditions according to military handling qualities
recommendations. While deriving the CPR method, an error was found and corrected relative to
the previous implementation of the method within AFRL. Other methods reviewed might have led
to more realizable controllers, but required sub-optimization routines within the controls discipline.
The result of these are typically binary—successful sub-optimization or not, and therefore lend no easy way to measure how close the vehicle is to being controllable.
Chapter IV details the aerodynamic requirements for the CPR method. In this chapter, a multi-fidelity aerodynamic approach was deemed necessary. Out of five aerodynamic analysis software packages that are evaluated, only Cart3D is found to be capable of modeling small changes to the fuselage or any of the innovative control effectors of interest. However, Cart3D is not capable of computing dynamic derivative required of the controllability analysis. For this reason, the selected approach uses a combination of AVL and Cart3D data.
Due to the results of the aerodynamic analysis selection, parameterized, high-fidelity analysis geometry is required was required for Cart3D analysis. Chapter V introduces the CAPS program and the effort that was performed to represent the MADO geometry in a manner that is parameterized and capable of outputting analysis geometry representative of both required tools.
In this chapter, the parameterization rationale and associated math of the wing and fuselage planforms are also derived and described.
160 Chapter VI discusses the only low-fidelity analysis of this work—mass properties—and
the overall implementation of the MADO. In this chapter, a second aerodynamic study is described
that is used to test and limit the required analysis in Cart3D. In the first part of this study, the
number of iterations required for convergence was determined. After this, a superposition study is
described, analyzing many combinations of control effector deployment to test how the individual
contribution of each control effector applied to the combined effectiveness of multiple
simultaneously deployed effectors. It is shown that in some configurations, the individual
summation of effectiveness not only fails to match, but results in trends of the opposite sign relative to analysis of the same effectors when simultaneously deployed. As a result, a decision was made to separate the control effectors used for the different axes. In the pitch and roll axes, the traditional wing control surfaces are used and in the yaw axis, the innovative control effectors are used. The
Cart3D analysis specific to the yaw control power available is analyzed with all yaw control
effectors deployed simultaneously to minimize the effect of superposition on the results. The final
portion of the aerodynamic study uses initial Cart3D results including fuselage to correct for the
Neutral Point location in AVL, where the vehicle is modeled without a fuselage. This is shown to
increase the accuracy of AVL results and with this correction, it was decided to use AVL results
for all traditional wing control effectors. After this study, the final aerodynamic analysis for the
MADO process is limited to four Cart3D analysis runs and two AVL runs.
This chapter also discusses the simple cost function that is used in this work. As described above, the CPR analysis lends itself well to implementation in the cost function by computing the ratio between CPR and CPA. Since many vehicle shapes can be found to be controllable, the vehicle weight is added to the cost function. The weighting of the two portions of the cost function are selected such that both portions are equal to a value of 50 when the vehicle weight is 100,000 lb. and the CPR exactly equals the CPA. The last part of this chapter briefly discusses the distributed
computing environment, Bond.
161 The testing and results of the implemented MADO framework are presented in Chapter
VII. In this chapter, both low-fidelity results using only AVL and multi-fidelity results incorporating both AVL and Cart3D analyses are presented. In order to better understand and describe the complicated coupling and physics that are being modeled while both planform and control effector shapes change simultaneously, the first results presented are simple parameter sweep excursions from the baseline ESAV vehicle. In low-fidelity analysis, an estimate of the forces generated by the innovative control effectors are used in which they are simply scaled by their area change relative to the data measured in the wind tunnel. This sort of estimate is typical of an attempt to incorporate new technology into conceptual design. In multiple cases, the low- fidelity results have opposite trends in the cost relative to the multi-fidelity results. The vehicle configuration selected from the ‘best’ parameter of each sweep is presented and shown to be intuitively infeasible in either case, leading to a strong argument for the MADO process where multiple parameters can be varied simultaneously.
The MADO process is also demonstrated with both low-fidelity only and multi-fidelity analysis. In both cases, the MADO algorithms perform well from a mathematical standpoint, resulting in vehicles that are computed to be controllable. In the low-fidelity case, however, the resulting configuration is over 100,000 lb. and does not pass an intuitive feasibility study. Results such as these explain why there has to date been a strong reluctance to accepting MADO techniques in aircraft design.
However, this is the first effort known to incorporate preliminary design level of fidelity in both the aerodynamics and controllability analyses simultaneously in an MADO process. The final result of this process is a vehicle configuration that weighs 54,000 lb. and is computed to be controllable. In addition to mathematically passing these tests, an internet search of similar planforms leads to the X-4 Bantam aircraft. If the geometric parameterization set allowed for the cockpit and rear of the fuselage to move in toward the center of the vehicle, these two aircraft would match fairly well. Interestingly, the X-4 was also designed as a semi-tailless aircraft with no
162 horizontal tail and controlled in pitch and roll with four trailing edge wing flaps instead. Although
far from a complete design, the likeness of the MADO result to a real-lift aircraft lends some
credence to the feasibility of the result.
Using the distributed computing framework, Bond, the total computation time for this
process was 21 days, 4 hours, and 16 minutes. In this time frame, a total of 3,200 cost function
evaluations are performed in the Particle Swarm Optimization, each requiring the generation of two
separate high-fidelity meshes (clean and control effector deployed configurations) and four Cart3D
analyses. As mentioned in Chapter VI, one cost function evaluation on a single computer takes 45
minutes to complete, meaning it would take roughly 100 days of run-time to achieve the same
results on a standalone computer. This computational requirement is a major hurdle to
implementation of this framework. Adding additional parameters, additional disciplines, and
additional fidelity will all compound the need to increase computational resources required to
realize a true aircraft MADO design using this process. In the meantime, the tools that are being
developed can immediately be used to reduce the human interaction required to enable preliminary
design levels of fidelity to be incorporated earlier in the design process.
8.2 Future work
The work implemented and presented here is intended to demonstrate a baseline for a
framework capable of autonomously analyzing the controllability of an ESAV-class aircraft with preliminary design levels of fidelity. In each step of the process described, there is significant room for continued improvement. There are in fact several tools currently under development within
MSTC that could be incorporated almost immediately, including better component layouts and sizing, structural analysis, propulsion modeling, higher fidelity aerodynamics (especially when computing dynamic derivatives) and even aeroelastic analysis—coupling the aerodynamics and structures. However, updating each of these tools would only feed higher-fidelity data into the primary focus of this work—the controllability analysis.
163 The CPR method described implements several requirements derived from MIL-STD-
1797B for flying qualities of fighter aircraft. However, this handbook is nearly 800 pages long and
there are many more requirements that can be implemented. For instance, an interesting test case could be to implement the requirements related to an entirely different class of vehicle such as a bomber. If the mission and associated component weights are also updated, would the MADO
framework result in a vehicle that represented a traditional bomber in planform shape?
The next major improvement to the CPR method could be to implement constraints on and/or computation of required actuator rates. Actuators are sized based on both the required rates and hinge moments that must be generated. In Cart3D, the hinge moments can be computed, making this a relatively simple additional task to implement. The data that is generated in the
MADO loop currently should be enough to estimate linear models of the aircraft and from them, the rate requirements could be generated.
In fact, with the linear models accessible, a more detailed control design should be achievable. Some effort was performed to incorporate simple pole placement design in the loop.
However, with closed-form solutions using transformation matrices to convert the state space matrices into modal form, pole switching often occurred in the lateral/directional axis. The poles could potentially be placed using optimal controllers instead, but that leads to a sub-optimization routine that may not converge or return a controllability result that is measurable for implementation in the overall cost function.
164 REFERENCES
[1] D. P. Raymer, Aircraft Design: A Conceptual Approach, Fifth Edition. American Institute of Aeronautics and Astronautics, Inc., 2012.
[2] N. R. Council, Pre-Milestone A and Early-Phase Systems Engineering: A Retrospective Review and Benefits for Future Air Force Systems Acquisition. The National Academies Press, 2008.
[3] E. J. Alyanak and R. M. Kolonay, “Efficient supersonic air vehicle structural modeling for conceptual design,” in 12th AIAA Aviation Technology, Integration and Operations (ATIO) Conference and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2012.
[4] J. A. Bowlus, D. Multhopp, and S. S. Banda, “Challenges and Opportunities in Tailless Aircraft Stability and Control,” in AIAA Guidance, Navigation, and Control Conference, 1997.
[5] B. R. Rich, Skunk Works. Back Bay Books, 1996.
[6] C. Davies, M. Stelmack, S. Zink, A. de la Garza, and P. Flick, “High fidelity MDO process development and application to fighter strike conceptual design,” in 12th AIAA Aviation Technology, Integration and Operations (ATIO) Conference and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2012.
[7] T. W. Simpson and J. R. R. A. Martins, “Multidisciplinary Design Optimization for Complex Engineered Systems: Report From a National Science Foundation Workshop,” Journal of Mechanical Design, vol. 133, no. 101002, pp. 1–10, 2011.
[8] R. Colgren and R. Loschke, “Effective Design of Highly Maneuverable Tailless Aircraft,” Journal of Aircraft, vol. 45, no. 4, pp. 1441–1449, 2008.
[9] J. Simon, W. Blake, and D. Multhopp, “Control Concepts for a Vertical Tailless Fighter,” in AIAA Aircraft Design, Systems, and Operations Meeting, 1993.
[10] S. P. Fears, “Low-Speed Wind-Tunnel Investigation of a Porous Forebody and Nose Strakes for Yaw Control of a Multirole Fighter Aircraft,” Hampton, VA, 1995.
[11] K. M. Dorsett and D. R. Mehl, “Innovative Control Effectors (ICE),” Fort Worth, TX, 1996.
165 [12] E. L. Roetman, S. A. Northcraft, and J. R. Dawdy, “Innovative Control Effectors (ICE),” Seattle, WA, 1996.
[13] A. Filippone, “Inverted Jet Spoilers for Aerodynamic Control,” Journal of Aircraft, vol. 46, no. 4, pp. 1240–1252, 2009.
[14] W. Blake, “Wing Mounted Aircraft Yaw Control Device,” 2002.
[15] D. C. Clark, “Aircraft with Forward Opening Inlay Spoilers for Yaw Control,” 2005.
[16] C. Meckstroth, “An Evaluation of Aerodynamic Analysis Software for use in Aircraft MDO,” in 54th AIAA Aerospace Sciences Meeting, 2016.
[17] K. Ekici and P. S. Beran, “Adjoint Sensitivity Analysis of Low-Speed Flows Using an Efficient Harmonic Balance Technique,” AIAA Journal, vol. 52, no. 6, pp. 1330– 1336, 2014.
[18] M. R. Anderson and W. H. Mason, “An MDO approach to control-configured- vehicle design,” in 6th Symposium on Multidisciplinary Analysis and Optimization, 1996, pp. 734–743.
[19] D. L. Allison, C. C. Morris, J. A. Schetz, R. K. Kapania, C. Sultan, L. T. Watson, J. D. Deaton, and R. V. Grandhi, “A Multidisciplinary Design Optimization Framework for Design Studies of an Efficient Supersonic Air Vehicle,” in 12th AIAA Aviation, Technology, Integration, and Operations Conference, 2012.
[20] S. A. Burton, E. J. Alyanak, and R. M. Kolonay, “Efficient supersonic air vehicle analysis and optimization implementation using SORCER,” in 12th AIAA Aviation Technology, Integration and Operations (ATIO) Conference and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2012.
[21] S. A. Burton, R. M. Kolonay, and M. Sobolewski, “Multidisciplinary Analysis with SORCER using Domain-Specific Objects,” in 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2014.
[22] R. Haimes and J. F. Dannenhoffer, “The Engineering Sketch Pad: A Solid- Modeling, Feature-Based, Web-Enabled System for Building Parametric Geometry,” in 21st AIAA Computational Fluid Dynamics Conference, 2013.
[23] E. J. Alyanak, “Modeling for conceptual design: An aeroelastic approach,” in 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 2012, 2012.
[24] D. E. Bryson, M. P. Rumpfkeil, and R. J. Durscher, “Framework for Multifidelity Aeroelastic Vehicle Design Optimization,” in 18th AIAA/ISSMO Multidisciplinary
166 Analysis and Optimization Conference, 2017.
[25] C. C. Morris, C. Sultan, D. L. Allison, J. A. Schetz, and R. K. Kapania, “Towards flying qualities constraints in the multidisciplinary design optimization of a supersonic tailless aircraft,” in 12th AIAA Aviation Technology, Integration and Operations (ATIO) Conference and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2012.
[26] J. Roskam, Airplane Design Part I: Preliminary Sizing of Airplanes. Lawrence, KA: Design Analysis Research Corporation, 2005.
[27] D. of Defense, Flying Qualities of Piloted Aircraft. Wright-Patterson AFB, 2006.
[28] J. Roskam, Airplane Design Part VII: Determination of Stability, Control and Performance Characteristics: FAR and Military Requirements. Lawrence, KA: Design Analysis Research Corporation, 2006.
[29] D. McRuer and D. Graham, “Eighty years of flight control - Triumphs and pitfalls of the systems approach,” Journal of Guidance and Control, vol. 4, no. 4, pp. 353– 362, 1981.
[30] “Northrop YB-49.” [Online]. Available: https://www.edwards.af.mil/News/Photos/igphoto/2001757305/. [Accessed: 02- Dec-2019].
[31] “Grumman X-29.” [Online]. Available: https://www.dfrc.nasa.gov/Gallery/Photo/X-29/Large/EC90-039-4.jpg. [Accessed: 02-Dec-2019].
[32] F. R. Swortzel and A. F. Barfield, “The CCV Fighter Program - Demonstrating New Control Methods for Tactical Aircraft,” in Aircraft Systems and Technology Meeting, 1976.
[33] E. G. Rynaski and N. C. Weingarten, “Flight Control Principles for Control Configured Vehicles,” 1972.
[34] E. Livne, “Special Issue: Multidisciplinary Design Optimization of Aerospace Systems,” Journal of Aircraft, vol. 36, no. 1, pp. 9–10, 1999.
[35] T. Noll, “Aeroservoelasticity,” 1990.
[36] E. Livne, “Integrated Aeroservoelastic Optimization - Status and Direction,” in 38th Structures, Structural Dynamics, and Materials Conference, 1997.
[37] I. Isnardi, P. Paoletti, S. Fichera, D. Miranda, and G. Innocenti, “Nonlinear Aeroservoelastic Control Design and Validation,” in 2018 AIAA Guidance,
167 Navigation, and Control Conference, 2018.
[38] G. J. Balas, C. Morena, and P. J. Seiler, “Robust Aeroservoelastic Control Utilizing Physics-Based Aerodynamic Sensing,” in AIAA Guidance, Navigation, and Control Conference, 2012.
[39] R. B. Alstrom, G. Ahmadi, E. Bollt, and P. Marzocca, “The Effect of Rate Limited Actuator Dynamics on the Adaptive Control of a Nonlinear Aeroservoelastic (ASE) System,” in 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2011.
[40] R. Brown, K. V. Singh, and R. Kolonay, “Optimal Sizing and Placement of Control Surfaces for Active Aeroservoelastic Control,” in 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2017.
[41] B. K. Stanford, “Optimization of an Aeroservoelastic Wing with Distributed Multiple Control Surfaces,” in 33rd AIAA Applied Aerodynamics Conference, 2015.
[42] S. Haghighat, J. R. R. A. Martins, and H. H. T. Liu, “Aeroservoelastic Design Optimization of a Flexible Wing,” Journal of Aircraft, vol. 49, no. 2, pp. 432–443, 2012.
[43] L. P. Ozoroski, K. A. Geiselhart, S. L. Padula, W. Li, E. D. Olson, R. L. Campbell, E. W. Shields, J. J. Berton, J. S. Gray, S. M. Jones, C. G. Naiman, J. A. Seidel, K. T. Moore, B. A. Naylor, and S. Townsend, “Initial Multidisciplinary Design and Analysis Framework,” 2010.
[44] C. C. Morris, “Flight Dynamic Constraints in Conceptual Aircraft Multidisciplinary Analysis and Design Optimization,” 2013.
[45] J. Mieloszyk and T. Goetzendorf-Grabowski, “Introduction of full flight dynamic stability constraints in aircraft multidisciplinary optimization,” Aerospace Science and Technology, vol. 68, pp. 252–260, 2017.
[46] D. Cosenze and R. Vos, “Handling Qualities Optimization in Aircraft Conceptual Design,” in 17th AIAA Aviation Technology, Integration, and Operations Conference, 2017.
[47] R. E. Perez, H. H. T. Liu, and K. Behdinan, “Multidisciplinary Optimization Framework for Control-Configuration Integration in Aircraft Conceptual Design,” Journal of Aircraft, vol. 43, no. 6, pp. 1937–1948, 2006.
[48] M. Chilali and P. Gahinet, “H∞ design with pole placement constraints: An LMI approach,” IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 358–367, 1996.
168 [49] B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation, 2nd Edition. Hoboken, NJ: John Wiley & Sons, Inc., 2003.
[50] G. H. Bryan, Stability in Aviation. Macmillan, 1911.
[51] M. Drela and H. Youngren, AVL 3.30 User Primer. 2010.
[52] C. R. B. Air Force Research Laboratory, Air Vehicles Unstructured Solver (AVUS) User’s Manual.
[53] J. E. Williams and S. R. Vukelich, The USAF Stability and Control Digital DATCOM, Volume I, Users Manual, no. AFFDL-TR-79–3032, Volume I. St. Louis, MO, 1979.
[54] ZAERO User’s Manual, Version 8.5. Scottsdale, AZ, 2011.
[55] ZEUS User’s Manual, Version 3.2. Scottsdale, AZ, 2012.
[56] M. Aftosmis, “Cart3D.” [Online]. Available: http://people.nas.nasa.gov/aftosmis/cart3d/cart3Dhome.html. [Accessed: 2019].
[57] E. Alyanak, R. Durscher, and R. Haimes, “Multi-fidelity Geometry-centric Multi- disciplinary Analysis for Design,” in AIAA Modeling and Simulation Technologies Conference, 2016.
[58] C. Meckstroth, “Parameterized, Multi-fidelity Aicraft Geometry and Analysis for MDAO Studies using CAPS,” in AIAA Scitech 2019 Forum, 2019.
[59] N. Bhagat, D. L. Allison, and E. Alyanak, “Geometry Driven High Fidelity Stability Derivatives Obtained Using Automated CFD Analysis Process,” in 16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2015.
[60] C. M. Heath, J. W. Slater, and S. K. Rallabhandi, “Inlet Trade Study for a Low- Boom Aircraft Demonstrator,” Journal of Aircraft, vol. 54, no. 4, pp. 1283–1293, 2017.
[61] “That’s the Spirit.” [Online]. Available: https://media.defense.gov/2012/May/17/2000150836/-1/-1/0/120509-F-VV395- 906.JPG. [Accessed: 03-Dec-2019].
[62] “CENTAF airpower summary for Jan. 26.” [Online]. Available: https://www.af.mil/News/Photos/igphoto/2000519160/. [Accessed: 03-Dec-2019].
[63] R. M. Kolonay, “MSTC Engineering - A Distributed and Adaptive Collaborative Design Computational Environment for Military Aerospace Vehicle Development
169 and Technology Assessment,” in AIAA Aviation 2019 Forum, 2019.
[64] J. Roskam, Airplane Design Part II: Preliminary Configuration Design and Integration of the Propulsion System. Lawrence, KA: Design Analysis Research Corporation, 2004.
[65] “NASA Armstrong Fact Sheet: X-4 Bantam.” [Online]. Available: https://www.nasa.gov/centers/armstrong/news/FactSheets/FS-078-DFRC.html. [Accessed: Aug-2019].
[66] “EG-0087-01.” [Online]. Available: https://www.nasa.gov/centers/armstrong/multimedia/graphics/EG-0087-01.html. [Accessed: 03-Dec-2019].
170