Appendix B Diamond DA42 Aircraft Data

2.2. | IL MARCHIO, IL LOGOTIPO: LE DECLINAZ IONI

SCUOLADI INGEGNERIA INDUSTRIALEEDELL’INFORMAZIONE

M.Sc. Thesis

FLIGHT MECHANICS MODELINGAND CONTROL STUDY OF A THREE-SURFACE AIRCRAFT

Relatore: Prof. Carlo E. D. RIBOLDI Co-relatore: Prof. Stefano CACCIOLA

Tesi di Laurea Magistrale di: Andrea Bavetta Matr. 863302

Politecnico di Milano Anno Accademico 2017-2018 Ringraziamenti

Questa tesi non è frutto del mio solo lavoro e impegno, ma è il risultato di una collab- orazione che mi ha sempre spronato a continuare la ricerca, soprattutto nei momenti di sconforto, quando i risultati ottenuti non erano quelli aspettati o sperati. Vorrei ringraziare sentitamente il mio relatore Prof. Riboldi e il mio co-relatore Prof. Cacciola per avermi seguito in questi mesi mostrando sempre interesse e disponibilità, in particolar modo per avermi fatto lavorare in un ambiente in cui mi sono sentito a mio agio. Ringrazio i miei gen- itori per essere stati sempre dalla mia parte anche nei momenti diﬃcili, mostrando sempre pazienza e accettando senza indurre pressioni il mio modo di lavorare. Devo moltissimo a mio padre che ringrazio per avermi trasmesso ﬁn da bambino la passione per il volo; a lui devo il coronamento del mio sogno di diventare pilota, avendomi avvicinato agli aeroplani sin da piccolo. Ringrazio mia madre per essere stata sempre comprensiva e per non aver mai mostrato apprensione nei periodi in cui mi sono messo in gioco, soprattutto durante gli ap- pelli d’esame. Ringrazio mio nonno che mi ha permesso di essere la realizzazione di un suo sogno, dato che non ha avuto la possibilità di intraprendere la carriera di pilota, dimostrando la stessa felicità e soddisfazione come se i miei risultati fossero stati suoi. Ringrazio i miei familiari, mio fratello Carlo, mia sorella Irene e i miei zii per avere sempre avuto stima del mio lavoro e del mio studio, cosa che ho apprezzato molto. Devo tanto ai miei amici, compagni di corso e non, conosciuti al primo anno in campus, con cui ho condiviso mesi di studio e momenti indimenticabili; agli amici conosciuti in università con cui ogni giorno ci si è ritrovati per studiare, pranzare o bere una birra. Se anche solo di questi tasselli fosse mancato, sicuramente non sarei riuscito a com- pletare questo mio percorso.

i ii Abstract

Aircraft design in the last decades, especially for what concerns the airline transport, has generally been based on the revision, modernization or partial re-design of well-consolidated products to reduce research, development and production costs. This design philosophy has led to some airliner families whose members feature some common characteristics or components as for example the wing, the system disposition or the passenger seat layout. This applies in general for conventional two-surface aircraft, the most common solution in civil aviation. The idea to employ a different configuration, as for example a canard arrangement, has always been a challenge; this because, even if canard configuration can lead to many advantages in terms of efficiency and fuel consumption, some issues occur as well; the solution of such issues cancels completely the above-mentioned benefits. The problems associated with the canard configuration will be shown, explaining why such lifting surface arrangement can be found mainly in little private airplanes or in model aircraft. This research thesis takes inspiration from the work done by Piaggio Aerospace in the 80s with the P180 Avanti. The company has found a configuration which could exploit the advantages offered by the canard without incurring into the relative problems. The resulting three-surface airplane has demonstrated to compete with business jets of the same class in terms of cruising speed and fuel consumption, even if featuring turboprop engines. The potential of the three-surface configuration may be enhanced by considering the forward wing to feature an elevator (movable surface); in this case the new degree of freedom may be used to search convenient trim solutions. The work starts with a model of the flight mechanics for a three-surface aircraft and moves on with a study on the advantages that an optimal trim solution may offer. The application of those studies on the configuration revision of an existing aircraft is investigated, specifying what are the conditions for which a change of configurations is convenient. Even- tually, a brief method on how to set up a preliminarily ab-initio design of a three-surface aircraft, from a point of view of pure mechanics, will be shown. The main purpose of this work is to show that, under certain conditions, the three- surface configuration offers some advantages in terms of better performance compared to two-surface solutions (tailed or canard), for a given set of performance requirements. That being said, this work may represent the starting point for the definition of three-surface aircraft design guidelines.

Keywords: Aircraft design; three-surface conﬁguration; ﬂight mechanics; trim solutions.

iii iv Sommario

Negli ultimi decenni, specialmente in ambito di trasporto di linea, il progetto di velivoli si è basato principalmente sulla revisione, aggiornamento o riprogettazione parziale di design consolidati, puntando alla riduzioni di costi di ricerca, sviluppo e produzione. Questa filosofia ha portato ad alcune particolari famiglie di velivoli, i quali membri presentano caratteristiche o componenti simili, come l’ala, la disposizione degli impianti e sistemi o dei sedili in cabina. Questo è in genere valido per velivoli a configurazione convenzionale a due superfici, la soluzione più comune in ambito civile. L’idea di utilizzare una configurazione diversa, come ad esempio una soluzione canard, è stata sempre una questione dibat- tuta. La configurazione canard, pur promettendo numerevoli vantaggi tra cui una maggiore efficienza e risparmio di combustibile, comporta l’apparizione di alcuni problemi la cui soluzione annulla tutti i vantaggi aspettati. In questo lavoro i problemi associati all’adozione di una configurazione canard verranno brevemente esposti, spiegando per quale ragione una tale disposizione di impennaggi orizzontali viene trovata principalmente in velivoli di pic- coli dimensioni o aeromodelli. Questa tesi di ricerca prende spunto dal lavoro della Piaggio Aerospace negli anni ’80 con il modello P180 Avanti. L’azienda è riuscita a trovare una configurazione che riuscisse a sfruttare i vantaggi offerti dal canard senza incorrere nei problemi relativi a questa soluzione. Il modello a tre superfici che ne è risultato ha dimostrato essere un ottimo concorrente di velivoli a getto della stessa classe in termini di velocità di crociera e consumo carburante, pur montando propulsori turboelica. Il potenziale della configurazione a tre superfici può essere ulteriormente valorizzato considerando l’aletta canard come im- pennaggio dotato di equilibratore (superficie mobile); in questo modo il nuovo grado di libertà introdotto può essere sfruttato per ricercare soluzioni di equilibrio convenienti. Il lavoro inizia con la definizione di un modello della meccanica del volo di un velivolo generico a tre superfici e prosegue con lo studio dei vantaggi che soluzioni di equilibrio ottime possono offrire. In seguito, i risultati del modello della meccanica del volo di un tre superfici e dello studio di equilibri ottimi verranno applicati alla revisione di configu- razioni di velivoli già esistenti, specificando quali sono le condizioni per cui un cambio di configurazione risulta essere conveniente. Infine, verrà brevemente esposto un metodo per impostare il progetto preliminare ab-initio della configurazione di un velivolo a tre superfici da un punto di vista puramente meccanico. Lo scopo di questo lavoro è di mostrare che, sotto certe condizioni, la configurazione a tre superfici presenta dei vantaggi rispetto a una configurazione a due superfici (convenzionale o canard), a parità di requisiti di progetto. Appurato questo, si propone un punto di partenza per possibili linee guida per la progettazione di velivoli a tre superfici attraverso il modello statico e l’algoritmo di trimmaggio ottimo.

Parole Chiave: Progetto di velivoli; conﬁgurazione a tre superﬁci; meccanica del volo; soluzioni di equilibrio.

v vi Contents

List of Figures ix

List of Tables xi

List of Symbols xiii

List of Abbreviations xxi

1 Introduction 1 1.1 Motivation ...... 1 1.2 Canard Configuration ...... 1 1.3 Overcome the Canard Issues ...... 2 1.4 Goals and Work Outline ...... 3 2 Three-Surface Model 5 2.1 Lift and Pitching Moment of the Whole Aircraft ...... 6 2.2 Lift of the Three Surfaces ...... 7 2.3 Pitching Moment of the Three Surfaces ...... 9 2.4 Stick-Fixed Neutral Point ...... 14 2.5 Maneuvering Flight of a Three-Surface Aircraft ...... 14 2.5.1 Angle of Attack Induced by Pitch Rate ...... 15 2.5.2 Change of the Lift Coefficient in Maneuvering Flight ...... 15 2.5.3 Pitching Moment Variation in Maneuvering Flight ...... 17 2.6 Stick-Free Static Stability ...... 17 2.7 Borri Formulation ...... 17 2.7.1 Two-Surface Aircraft ...... 17 2.7.2 Three-Surface Aircraft - Independent Elevator Deflections . . . . . 20 2.7.3 Three-Surface Aircraft - Dependent Elevator Deflections ...... 21 3 Trimmed Solutions of a Three-Surface Aircraft 23 3.1 Optimal Trim Solutions ...... 23 3.1.1 Variables ...... 24 3.1.2 Constraints ...... 24 3.1.3 Objective Function ...... 25 3.2 Aircraft Drag Polar ...... 25 3.2.1 Assumptions ...... 25 3.2.2 Drag Coefficient of the Three Surfaces ...... 27

vii 3.3 Drag Coefficient Minimization ...... 30 3.4 Validation of the Results ...... 33 3.5 Application of Optimal Trim Solutions ...... 34 3.6 Reformulation of the Trim Problem ...... 37 3.7 Stick-Free Static Stability ...... 38 3.7.1 Hinge Moment Coefficient ...... 38 3.7.2 Stick-Free Lift and Moment Coefficients ...... 39 3.7.3 Stick-Free Neutral Point ...... 40 3.7.4 Borri Formulation - Dependent Elevator Deflections ...... 42 4 Revision of an Existing Aircraft Configuration 45 4.1 Comparison Criteria of Aircraft Configurations ...... 45 4.1.1 Longitudinal Static Stability ...... 46 4.1.2 Longitudinal Control ...... 46 4.2 Application to an Existing Design ...... 46 4.2.1 Application to Diamond DA42 Twin Star ...... 49 4.2.2 Application to Modified Sagitta-Juliett ...... 57 4.2.3 Considerations on the Results ...... 64 4.2.4 Configuration Revision of the DA42 ...... 67 4.3 Preliminary Design ...... 68 5 Conclusions 75

Bibliography 76

A Modiﬁed Juliett Aircraft Data 79

B Diamond DA42 Aircraft Data 83

C Minimization of Drag Coeﬃcient Through SVD 87 C.1 Application of SVD on Trim Equations ...... 87 C.2 Minimization of the Drag Coeﬃcient ...... 88 D Two-Surface Aircraft Trim Solution 91

viii List of Figures

2.1 Three-surface configuration load scheme ...... 5 2.2 Elevator deflection notation ...... 5 2.3 Aerodynamic angles ...... 8 2.4 Three-surface configuration ...... 15 2.5 Angle of attack induced by pitch rate ...... 15 2.6 Aerodynamic force system according to Borri formulation ...... 19

3.1 Trim solution comparison ...... 33 3.2 Trim drag polar comparison ...... 33 3.3 Trim coeﬃcients ...... 34 3.4 Trim variables VS Mach and PA ...... 36 3.5 Structure of the control system ...... 37

4.1 Aircraft configuration parting ...... 49 4.2 Diamond DA42 - Three-view ...... 50 4.3 Diamond DA42 - Horizontal empennage surface area ...... 51 4.4 Diamond DA42 - Longitudinal wing location ...... 51 4.5 Diamond DA42 - Constant quantities ...... 52 4.6 Diamond DA42 - Neutral point and center of gravity shift ...... 53 4.7 Diamond DA42 trim solution - Angle of attack ...... 54 4.8 Diamond DA42 trim solution - Tail elevator deflection ...... 54 4.9 Diamond DA42 trim solution - Forward wing elevator deflection ...... 55 4.10 Diamond DA42 trim solution - Drag polars ...... 55 4.11 Diamond DA42 - Maximum CL∕CD ratio ...... 56 3∕2 4.12 Diamond DA42 - Maximum CL ∕CD ratio ...... 56 4.13 Juliett and Mod-Juliett top view ...... 57 4.14 Mod-Juliett - Horizontal empennage surface area ...... 58 4.15 Mod-Juliett - Longitudinal wing location ...... 59 4.16 Mod-Juliett - Constant quantities ...... 59 4.17 Mod-Juliett - Neutral point and center of gravity shift ...... 60 4.18 Mod-Juliett trim solution - Angle of attack ...... 61 4.19 Mod-Juliett trim solution - Tail elevator deflection ...... 61 4.20 Mod-Juliett trim solution - Forward wing elevator deflection ...... 62 4.21 Mod-Juliett trim solution - Drag polars ...... 63 C ∕C 4.22 Mod-Juliett - Maximum √L D ratio ...... 63 4.23 Mod-Juliett - Maximum CL∕CD ratio ...... 64 4.24 Examples of correlation between engine and wing location ...... 65

ix 4.25 Horizontal tail and forward wing lift coefficient ...... 66 4.26 DA42 - Comparison between old and new configuration ...... 67 4.27 Case 1 - Load configurations ...... 71 4.28 Case 1 - Angle of attack ...... 72 4.29 Case 2 - Load configurations ...... 72 4.30 Case 2 - Angle of attack ...... 73

x List of Tables

3.1 Computational time comparison ...... 34

4.1 Three-surface aircraft data ...... 70 4.2 Lifting surface location ...... 70

A.1 Airfoil data ...... 79 A.2 Surface geometric data ...... 79

B.1 Airfoil data ...... 83 B.2 Surface geometric data ...... 84

xi xii List of Symbols

w A Aircraft zero-lift drag polar coefficient CD Wing drag coefficient Ac Cw Forward wing zero-lift drag polar coef- D0 Zero-lift wing drag coefficient ficient C Dmin Optimal (minimum) drag coefficient t A Horizontal tail zero-lift drag polar coef- t C Tail elevator hinge moment coefficient ficient H t w CH Tail elevator hinge moment coefficient A Wing zero-lift drag polar coefficient ∕ derivative with respect to the angle of B Aircraft drag polar coefficient related to attack 2 Ct H∕ t Tail elevator hinge moment coeffi- c B Forward wing drag polar coefficient re- cient derivative with respect to the an- 2 lated to gle of attack Bt Ct Horizontal tail drag polar coefficient re- H∕ Tail elevator hinge moment coeffi- 2 e lated to cient derivative with respect to eleva- w 2 tor deflection B Wing drag polar coefficient related to Ct H tab Tail elevator hinge moment coeffi- C Aircraft drag polar coefficient related to ∕e 2 e cient derivative with respect to trim tab deflection Cc Forward wing drag polar coefficient re- t 2 CH Hinge moment coefficient at zero angle lated to c 0 of attack, elevator deflection and trim t C Horizontal tail drag polar coefficient re- tab deflection 2 lated to e CL Lift coefficient C d0 Drag coefficient ∗ CL Trim lift coefficient CD Drag coefficient ¨ CL Stick-free lift coefficient c CD Forward wing drag coefficient c CL Forward wing lift coefficient c CD Zero-lift forward wing drag coefficient t 0 CL Horizontal tail lift coefficient Ct w D Tail drag coefficient CL Wing lift coefficient t C Cl D0 Zero-lift tail drag coefficient ∕ 2D liftcurve slope

xiii LIST OF SYMBOLS LIST OF SYMBOLS

C C L∕q Lift coefficient derivative with respect L0 Lift coefficient at zero AoA and zero to the pitch rate control deflection for reduced trim C problem L∕ ̂q Lift coefficient derivative with respect to the pitch rate Cc L0 Forward wing lift coefficient at zero C AoA L∕x Lift coefficient derivative vector t C CL Horizontal tail lift coefficient at zero L∕ Liftcurve slope 0 AoA ¨ C Stick-free liftcurve slope L∕ C  Pitching moment coefficient c C Forward wing liftcurve slope L∕ C ∕ Pitching moment coefficient deriva- Cc tive with respect to the angle of attack L∕ c Forward wing liftcurve slope (referred to forward wing angle of attack) C AC Pitching moment coefficient around Ct the aerodynamic center L∕ Horizontal tail liftcurve slope t C C ACc Pitching moment coefficient around L t Horizontal tail liftcurve slope (re- ∕ the forward wing aerodynamic center ferred to horizontal tail angle of attack) C ACt Pitching moment coefficient around Cw the horizontal tail aerodynamic center L∕ Wing liftcurve slope

w C w Pitching moment coefficient around C Wing liftcurve slope (referred to AC L∕ w wing aerodynamic center wing angle of attack) C C Pitching moment coefficient around C ∕c L∕ Lift coefficient derivative with respect c the control point derivative with re- to forward wing elevator deflection spect to the forward wing elevator de- CL ∕e Lift coefficient derivative with respect flection to horizontal tail elevator deflection C C Pitching moment coefficient around £ ∕e CL ∕e Lift coefficient derivative with respect the control point derivative with re- to horizontal tail elevator deflection spect to the horizontal tail elevator de- for reduced trim problem flection c CL Forward wing lift coefficient deriva- C ∕c CG Pitching moment coefficient around tive with respect to forward wing ele- the center of gravity vator deflection C¨ t CG Stick-free pitching moment coeffi- CL Horizontal tail lift coefficient deriva- ∕e cient around the center of gravity tive with respect to tail elevator deflec- C tion CG∕q Pitching moment coefficient around C the center of gravity derivative with Ln=1 Level flight lift coefficient respect to the pitch rate C L0 Lift coefficient at zero AoA C CG∕ ̂q Pitching moment coefficient around C̃ L0 Lift coefficient at zero AoA and zero the center of gravity derivative with control deflection respect to the pitch rate

xiv LIST OF SYMBOLS LIST OF SYMBOLS

C C CG∕ Pitching moment coefficient around P Pitching moment coefficient around a the center of gravity derivative with point p respect to the angle of attack Cc P Forward wing pitching moment coef- ¨ C Stick-free pitching moment coefficient around a point p CG∕ Ct ficient around the center of gravity P Horizontal tail pitching moment coef- derivative with respect to the angle of ficient around a point p attack Cw P Wing pitching moment coefficient C CG∕ Pitching moment coefficient around around a point p the center of gravity derivative with C Pitching moment coefficient around respect to a generic deflection P∕ a generic point derivative with respect C CG Pitching moment coefficient to the angle of attack  ∕c around the center of gravity deriva- C P Pitching moment coefficient around tive with respect to the forward wing ∕ elevator deflection a generic point derivative with respect to a generic deflection C CG Pitching moment coefficient  ∕e C P Pitching moment coefficient around around the center of gravity deriva- ∕c a generic point derivative with respect tive with respect to the horizontal tail to the forward wing deflection elevator deflection C P Pitching moment coefficient around C¦ ∕e CG Pitching moment coefficient ∕e a generic point derivative with respect around the center of gravity deriva- to the tail elevator deflection tive with respect to the horizontal tail C elevator deflection for reduced trim 0 Zero-lift Pitching moment coefficient problem D Aircraft drag polar coefficient related to 2 C c CG0 Zero-angle of attack / zero-elevator c deflection pitching moment coeffi- D Forward wing drag polar coefficient re- cient around the center of gravity lated to c ̃ C Pitching moment coefficient at zero t CG0 D Horizontal tail drag polar coefficient re- AoA and zero control deflection lated to e C¦ D CG0 Zero-angle of attack / zero-elevator can Forward wing drag deflection pitching moment coeffi- Dtail Horizontal tail drag cient around the center of gravity for reduced trim problem Dw Wing drag C E N∕ Pitching moment coefficient around Aircraft drag polar coefficient related to the stick-fixed neutral point derivative e with respect to the angle of attack c E Forward wing drag polar coefficient re- C lated to N¨ Pitching moment coefficient around ∕ t the stick-free neutral point derivative E Horizontal tail drag polar coefficient re- with respect to the angle of attack lated to

xv LIST OF SYMBOLS LIST OF SYMBOLS

Ew t Wing drag polar coefficient related to AC Pitching moment around the horizontal tail aerodynamic center  Aircraft Endurance w AC Pitching moment around the wing F Aircraft drag polar coefficient related to aerodynamic center c C Pitching moment around the control c F Forward wing drag polar coefficient re- point lated to c C  ∕c Pitching moment around the control t F Horizontal tail drag polar coefficient re- point derivative with respect to the for- lated to e ward wing deflection

G C Aircraft drag polar coefficient related to  ∕e Pitching moment around the control point derivative with respect to the el- H Aircraft drag polar coefficient related to evator deflection e C  1∕e Pitching moment around the control I Aircraft drag polar coefficient related to c point 1 derivative with respect to the horizontal tail elevator deflection J Drag coefficient minimization cost func- C tion  2∕c Pitching moment around the control point 2 derivative with respect to the L Lift forward wing elevator deflection Latt Attitude lift CG Pitching moment around the center of c gravity L Forward wing lift c  Forward wing pitching moment Lctrl CG Control lift around the center of gravity Lt t Horizontal-tail lift CG Hprizontal tail pitching moment w around the center of gravity L Wing lift w  Wing pitching moment around the L CG 0 Lift for zero angle of attack and zero el- center of gravity evator deflection CG Pitching moment around the center L ∕ c ∕ Lift derivative with respect to the angle of gravity derivative with respect to of attack the forward wing elevator deflection

L∕ CG c Lift derivative with respect to the for-  ∕e Pitching moment around the center ward wing deﬂection of gravity derivative with respect to L the tail elevator deﬂection ∕e Lift derivative with respect to the ele-

vator deﬂection N∕ Pitching moment around a the neutral point derivative with respect to the M Aircraft mass angle of attack M TO Aircraft maximum takeoﬀ mass P Pitching moment around a point P c c AC Pitching moment around the forward P Forward wing pitching moment around wing aerodynamic center a point P

xvi LIST OF SYMBOLS LIST OF SYMBOLS

t ̄ ¨t P Horizontal tail pitching moment V Horizontal tail volume around a point P VEAS Equivalent airspeed w P Wing pitching moment around a point W Aircraft weight P V Wing-body relative wind speed P∕ Pitching moment around a point P c derivative with respect to the angle of V Relative wind speed for the forward attack wing t P V  ∕c Pitching moment around a point P Relative wind speed for the horizontal derivative with respect to the forward tail wing elevator deflection K Matrix of the drag coefficient terms B, C, P D E F  ∕e Pitching moment around a point P , and derivative with respect to the tail el- T VN Null space of x evator deflection 2 X Aerodynamic coefficient matrix P0 Pitching moment around a point P for zero angle of attack and zero elevator b Wingspan deflection c Speed of sound Q Pitching moment around a point Q ̄c Mean aerodynamic chord M 0 Flight Mach number c ̄c Forward wing mean aerodynamic chord N Neutral point t ̄c Horizontal tail mean aerodynamic chord ¨ N Stick-free neutral point cP Brake Specific Fuel Consumption P Generic point P on the x-axis c d Distance between the forward wing cen- Q Generic point Q on the x-axis ter of gravity and aerodynamic center t  Range d Distance between the horizontal tail center of gravity and aerodynamic center Re Reynold’s number w d Distance between the wing center of S Wing area gravity and aerodynamic center Sc c Forward wing surface area e Forward wing Oswald factor St t Horizontal tail surface area e Horizontal tail Oswald factor V w True Airspeed e Wing Oswald factor ̄ V Tail volume g Gravitational acceleration ̄ c V Forward wing volume ℎ Altitude ̄ t c V Horizontal tail volume i Forward wing incidence angle ̄ tot t V Total tail volume i Horizontal tail incidence angle ̄ ¨c w V Forward wing tail volume i Wing incidence angle

xvii LIST OF SYMBOLS LIST OF SYMBOLS

t w i1 i − i xC Control point longitudinal location i itc − iw x 2 C1 Control point 1 longitudinal location k x Parabolic drag polar coefficient C2 Control point 2 longitudinal location c k Forward wing parabolic drag polar coef- xCG Longitudinal position of the center of ficient gravity t k Horizontal tail parabolic drag polar coef- xCGc Longitudinal position of the forward ficient wing center of gravity w k Wing-body parabolic drag polar coeffi- xCGt Longitudinal position of the horizontal cient tail center of gravity

x w lc Forward wing interfocal distance CG Longitudinal position of the wing cen- ¨ ter of gravity lc Forward wing arm xMAC MAC leading edge longitudinal loca- l t Horizontal tail interfocal distance tion l¨ t Tail arm xN Longitudinal position of the neutral point lv Vertical tail interfocal distance xN¨ Stick-free longitudinal position of the mc Forward wing mass neutral point mt Horizontal tail mass xP Longitudinal position of a generic point mw Wing mass P p Static pressure xQ Longitudinal position of a generic point Q ̄q Pitch rate xb x axis in the body frame ̂q Non-dimensional pitch rate m Vector which contains G, H and I qd Dynamic pressure vT xT c E Essential space of 2 qd Forward wing dynamic pressure vT xT t N1 First component of null space of 2 qd Horizontal tail dynamic pressure T T v Second component of null space of x x Longitudinal coordinate N2 2 T T c w x xAC Longitudinal position of the forward E Essential space of 1 wing aerodynamic center wT xT N1 First component of null space of 1 xACt Longitudinal position of the horizontal wT xT tail aerodynamic center N2 Second component of null space of 1 xACw Longitudinal position of the wing xT X 1 First row of aerodynamic center xT X 2 Second row of xACw Longitudinal position of the wing aerodynamic center y0 Trim problem constant term vector

xviii LIST OF SYMBOLS LIST OF SYMBOLS y Trim coefficient vector copt Optimal forward wing elevator deflection z Vector which containes the coefficients 1 and 2 cZL Zero-lift optimal forward wing elevator deflection zopt Optimal vector z e Horizontal tail elevator deflection Δmc Forward wing mass variation eopt Optimal horizontal tail elevator deflec- Δmt Horizontal tail mass variation tion ΔC L Lift coefficient variation eZL Zero-lift optimal horizontal tail eleva- c tor deflection ΔCL Forward wing lift coefficient variation c Zero lift and moment forward wing ele- ΔCt 0 L Horizontal tail lift coefficient variation vator deflection ΔC tab CG Horizontal tail lift variation e Trim tab deflection ΔL Lift variation e0 Zero lift and moment horizontal tail el- c evator deflection ΔL Forward wing lift variation t Borri stability index ΔL Horizontal tail lift variation "D Downwash angle ΔV Airspeed variation "D Downwash angle derivative with re- c ∕ ΔV Forward wing airspeed variation spect to angle of attack ΔV t " Horizontal tail airspeed variation D∕ w Downwash angle derivative with re- c spect to wing angle of attack Δ Forward wing angle of attack variation "D Zero-angle of attack (wing-body) Δ t 0 Horizontal tail angle of attack variation downwash angle Λ " Sweep angle D0 Zero-angle of attack downwash angle

1 Trim variables matrix "U Upwash angle " Angle of attack U∕ Upwash angle derivative with respect c to angle of attack Forward wing angle of attack "U w Upwash angle derivative with respect t ∕ Horizontal tail angle of attack to wing angle of attack w " Wing angle of attack U0 Zero-angle of attack (wing-body) downwash angle opt Optimal angle of attack " U0 Zero-angle of attack upwash angle ZL Zero-lift optimal angle of attack c Wing-forward wing dynamic pressure ra-

0 Zero-lift and moment angle of attack tio t Air adiabatic index Wing-tail dynamic pressure ratio

c Forward wing elevator deﬂection p Propeller Eﬃciency Ratio

xix LIST OF SYMBOLS LIST OF SYMBOLS

c Wing-forward wing mean aerodynamic N Non-dimensional location of the neutral chord ratio point t Wing-tail mean aerodynamic chord ratio N¨ Stick-free non-dimensional location of Non-dimensional mass of the aircraft the neutral point ACc Non-dimensional location of the for- Air density ward wing aerodynamic center 0 Standard air density ACt Non-dimensional location of the hori- c zontal tail aerodynamic center Wing-forward wing surface area ratio t ACw Non-dimensional location of the wing Wing-tail surface area ratio aerodynamic center First column of matrix 1 C Non-dimensional location of the control point Trim variable vector C1 Non-dimensional location of the control opt Optimal trim variable vector point 1 0 Constant component of trim variable C2 Non-dimensional location of the control vector point 2 Lagrange multipliers vector CG Non-dimensional location of the center of gravity Deﬂection vector

xx List of Abbreviations

AC Aerodynamic Center MALE Medium-Altitude Long-Endurance c AC Forward wing Aerodynamic Center MSL Mean Sea Level t AC Horizontal tail Aerodynamic Center PA Pressure Altitude w AC Wing Aerodynamic Center SM Static Margin

AFM Aircraft Flight Manual SVD Singular-Value Decomposition

AR Aspect Ratio S&C Stability and Control

BSFC Brake Speciﬁc Fuel Consumption TAS True Airspeed

CFD Computational Fluid Dynamics TSFC Thrust Speciﬁc Fuel Consumption

CG Center of Gravity UAV Unmanned Aerial Vehicle c EAS Equivalent Airspeed ZLL Forward wing Zero-Lift Line t FCS Flight Control System ZLL Horizontal tail Zero-Lift Line w MAC Mean Aerodynamic Chord ZLL Wing Zero-Lift Line

xxi Chapter 1

Introduction

1.1 Motivation

In the courses of flight mechanics and in literature, the canard configuration is addressed as a potential efficient design solution; however, no much time is generally spent on studying in deep the advantages and disadvantages of such lifting surface arrangement. Course pro- grams are often focused on the conventional two-surface configuration, for which a flight mechanics model is built. It is interesting to analyze more in depth the topic which concerns the applicability of the canard configuration and to highlight what advantages are offered and what drawbacks show up1. From a historical perspective, Piaggio Aerospace has found a way to exploit as much as possible the benefits offered by a canard solution without incurring into the issues caused by the canard arrangement itself. A thorough assessment of the validity of such intuition is the main topic of this work. Moreover, the three-surface configuration puts forward some clues which may be exploited; above all, the fact that the aircraft may feature two elevators. This fact leads to an additional degree of freedom for what concerns the trim problem. A way to exploit such features will be investigated.

1.2 Canard Conﬁguration

A canard configuration is a lifting surface arrangement for which the horizontal empennage is placed forward of the main wing of a fixed-wing aircraft [10]. The term "canard" may be used to describe the aircraft itself, the wing configuration or the forward wing (also known as foreplane). In a conventionally tailed two-surface aircraft, the pitching moment balance is achieved by means of a negative lift generated by the ensemble of stabilizer and elevator. This force system causes the wing lift to balance more than the only weight, inducing an increase of induced drag. An aircraft whose surfaces produce a positive lift enjoys another important benefit: a smaller wing surface area and thus, for a given installed thrust (or power), a higher cruising airspeed. The wing surface area is often set by the landing distance requirement and therefore, in cruise condition the aircraft generates an additional parasite drag because of the unnecessary surface area.

1It worths noticing that the ﬁrst successful heavier-than-air powered aircraft, the Wright Flyer, featured a canard conﬁguration. [23]

1 1.3. OVERCOME THE CANARD ISSUES CHAPTER 1. INTRODUCTION

The first drawback which has to be underlined is the impossibility to freely size the forward wing. This constraint is caused by the fact that longitudinal static stability decreases as long as the forward wing surface area increases. However, the maximum size of such lifting surface dictated by this stability constraint may be not enough to ensure a pitching moment balance in every flight condition, especially in landing configuration. The employment of high-lift devices can therefore be compromised, as they would further increase the pitch-down moment. A possible method which allows to increase the maximum foreplane surface area is to set the wing aft of the airplane.2 Such an arrangement results in another setback. In a very aft wing airplane, as long as the fuel in burned, the center of gravity shifts forward, since the fuel is often stored in the wing tanks. This makes the aircraft to continu- ously change its stability and controllability characteristics around the pitch axis; indeed, in an aft-wing configuration, the static margin typically gets larger and the airplane becomes statically stiffer. This brief explanation about the drawbacks of canard configuration leads to a notewor- thy conclusion: in general, a canard solution cannot meet simultaneously the requirements of stability and controllability throughout the whole center of gravity travel and for all the high-lift devices settings. Therefore, such airplane configuration is used mainly on models whose center of gravity does not present a shift throughout the flight and which do not require high-lift devices, as for example electric RC airplanes. A modern solution to such a problem, which has developed in military environment, is the employment of stability augmentation system. However, in military aircraft, the foreplane is designed to be a close- coupled canard; therefore, the design requirements may not be pointed to a reduction of induced drag or increase of cruising speed but, instead, to maneuvering flight performance enhancement. This makes the military scenario rather different from the one of interest in this work, where the inherent stability typically required for civil aircraft is considered a hard constraint. The stability augmentation system may be however a possible solution to ensure longitudinal static stability in a pure canard airplane, even if the aircraft does not feature inherent static stability.

1.3 Overcome the Canard Issues

By adding a horizontal tail to a canard airplane, it is possible to increase the maximum foreplane surface area which was limited due to longitudinal static stability requirements. In the resulting three-surface configuration, setting up a surface sizing and positioning problem is not a trivial issue, as the wing, canard and tail surfaces are potentially all unknowns. This will be analyzed in the following, but it is worth noting now that adding a tail to the back does not require to locate the wing aft, since the aft shift of the neutral point is obtained by moving back the tail only. Note that no conditions which may have requested the tail lift such that it was negative or positive have been imposed, since the neutral point longitudinal locations depends upon the horizontal empennage surface area and position only. This method, especially when all the three surfaces generate positive lift, ensures the wing to balance less than the weight, leading to the advantages shown in Section 1.2. However, there is the need to ensure that the added tail does not cancel the benefits we are striving for. This can be achieved only by computing the trim drag polars, from which the aircraft performance can be obtained. To do so, it is first necessary to build a flight mechanics

2This solution is the most common in canard aircraft. A signiﬁcant example is the Beechcraft Starship. [19]

2 CHAPTER 1. INTRODUCTION 1.4. GOALS AND WORK OUTLINE model for a three-surface aircraft. Once such mathematical model is available, it is possible to perform an analysis which regards convenient feasible trim solutions of a two-elevator airplane. The computation of the trim drag polar represents the ﬁnal step of such process.

1.4 Goals and Work Outline

The main objective of this work is to show that, under certain conditions, a three-surface arrangement which features two elevators can lead to better performance compared to two- surface solutions (tailed or canard) for what concerns range and endurance performance. Moreover, some tools which are required to perform such analyses are presented; in particular a three-surface model and a criterion to ﬁnd a trim solution in stationary ﬂight, from which trimmed drag polars can be obtained.

The work outline is shown below:

Three-Surface Model Model from which the flight mechanics characteristics of a three- surface aircraft are obtained. This includes a stick-fixed (longitudinal) static stability assessment in both straight and maneuvering flight and a revision of the Borri formulation for a three-surface arrangement.

Optimal Trim Assessment Algorithm with which a trim solution which maximizes a certain performance index, for a two-elevator aircraft, is found. Such analysis is followed by a completion of the three-surface model: stick-free static stability assessment and a revision of the Borri formulation which considers the trim condition.

Revision of Existing Design Analysis which shows that the three-surface solution may be beneficial in some cases. Such study has been performed by employing the above- mentioned models and methods. A final brief analysis based on the empennage sizing of a three-surface airplane is shown. This analysis highlights the fact that some convenient load configurations (lift on the three lifting surfaces) in trimmed flight can be obtained. From the lift which has to be generated from a surface, the relative surface area may be estimated.

3 1.4. GOALS AND WORK OUTLINE CHAPTER 1. INTRODUCTION

4 Chapter 2

Three-Surface Model

The aim of this Chapter is to present a flight mechanics model for a three-surface aircraft. Such model represents the starting point for the analyses which will be shown in the next Chapters. Once the geometry of the aircraft, in terms of lifting surface arrangement, is set, the model allows to compute the main characteristics of the aircraft such as aerodynamic coefficients as well as stability and control features. The following scheme represents a simplified model which features the flight mechanics behavior of a three-surface aircraft. The aircraft is represented by the wing, a horizontal tail and a forward wing as shown in Figure 2.1.

x Lt Lw Lc

t w c

AC CG AC AC ◔◔ ACt ACw ACc W l¨ ¨ w ¨ lt lc

lt lc

Figure 2.1: Three-surface conﬁguration load scheme

Figure 2.2 shows the elevator deﬂection notation for the forward wing and the horizontal tail. The deﬂections are positive in the direction indicated by the arrow.

훿푒 , 훿푐

Figure 2.2: Elevator deﬂection notation

5 2.1. LIFT AND PITCHING MOMENT CHAPTER 2. THREE-SURFACE MODEL

2.1 Lift and Pitching Moment of the Whole Aircraft

The aerodynamic loads which act on the aircraft are given by: T L = Lw + Lt + Lc w t c (2.1) P = P + P + P where the superscript w refers to the wing, t to the horizontal tail and c to the forward wing (c meaning canard). The body system, which is represented by the fuselage, is in general coupled to the wing so as to consider the whole as a wing-body system. In this case, the fuselage is not taken into account since the focus is on the lifting surfaces only. Moreover, the wing and the fuselage are referred to two diﬀerent angles of attack, as will be shown hereafter. However, when needed, the fuselage contribution to the lift and to the pitching moment may be included. Even though the three contributions are simply summed, the interference between the lifting surfaces has been considered to a certain extent, as well as between the lifting surfaces and the fuselage. For the lift contributions, the following constitutive equations apply:

w w ⎧L = qdSCL ⎪Lt = qt StCt ⎨ d L (2.2) ⎪Lc = qc ScCc . ⎩ d L t c Note that, in general, qd and qd diﬀer from qd (considered as the nominal dynamic w t c pressure) and CL , CL and CL are functions of , e,c, Re and M. e and c refer to the deﬂection of the elevator of the tail and forward wing respectively. Similarly, for P we can assume:

w w = qdS ̄cC ⎧P P ⎪ t = qt St ̄ctCt ⎨P d P (2.3) ⎪ c = qc Sc ̄ccCc ⎩P d P w t c C C C P e c Re where P , P and P depend upon the reference point as well as on , , , t c and M. The mean aerodynamic chord of the lifting surfaces are represented by ̄c, ̄c and ̄c . Consequently, L and P can be written as:

w t t t c c c L = qdSCL + qdS CL + qdS CL = H I qt t qc c w d S t d S c = qdS CL + CL + CL = (2.4) qd S qd S w t t t c c c = qdS CL + CL + CL ,

w t t t t c c c c P = qdS ̄cC + q S ̄c C + q S ̄c C =  P d P d P H I qt t t qc c c w d S ̄c t d S ̄c c = qdS ̄c C + C + C = P P P qd S ̄c qd S ̄c (2.5)

w t t t t c c c c = qdS C + C + C . P P P

6 CHAPTER 2. THREE-SURFACE MODEL 2.2. LIFT OF THE THREE SURFACES

t t t c c c , , , , and are deﬁned as follows:

t q St ̄ct t ∶= d , t ∶= , t ∶= , qd S ̄c c q Sc ̄cc c ∶= d , c ∶= , c ∶= . qd S ̄c

From Equations (2.4) and (2.5) the lift and moment coeﬃcient of the aircraft may be obtained:

T w t t t c c c CL = CL + CL + CL C = Cw + tttCt + cccCc . (2.6) P P P P

2.2 Lift of the Three Surfaces

If we consider that the elevator deflection affects the lift coefficient of its own horizontal empennage only, the lift coefficient of the three surfaces may be expressed as follows:

⎧Cw = Cw w ⎪ L L∕ w ⎪ t t t t CL = CL + CL e ⎨ ∕ t ∕e (2.7) ⎪Cc = Cc c + Cc L L c L c ⎪ ∕ ∕c ⎩ w where indicates the wing angle of attack (in this case it is convenient to define it as the t c angle between the wing zero-lift line and the relative wind vector); and indicate the horizontal tail and the forward wing angle of attack respectively. Note that e and c are not supposed to have any effect on the wing-body lift coefficient. The angle of attack is considered to be the angle between the relative wind vector V and the x-axis of the body reference frame projected in the xz-plane (body), xb. From t c Figure 2.3 the relation between , and can be obtained, considering also the incidence angle of the three lifting surfaces and the downwash/upwash characteristics of the wing. w t c Incidence angles are indicated with i , i and i . Downwash/upwash angles are indicated with "D and "U and both are approximated as linear functions of the wing angle of attack: T " = " w + " D D∕ w D0 " = " w + " (2.8) U U∕ w U0 ⎧ t = w 1 − " − " + it − iw ⎪ D∕ w D0 (2.9) ⎨ c = w 1 + " + " + ic − iw. ⎪ U∕ w U0 ⎩ For the sake of clarity, some terms may be condensed in one single quantity: T i = −" + it − iw 1 D0 i = " + ic − iw. (2.10) 2 U0

7 2.2. LIFT OF THE THREE SURFACES CHAPTER 2. THREE-SURFACE MODEL

ZLLc

ZLLw c ic w iw

xb it ZLLt t

t "D V " U V

Figure 2.3: Aerodynamic angles

1 − " 1 + " Since the values of D∕ w and U∕ w are approximately 1, it may be as- t c sumed that all the derivatives with respect to and are equivalent to the derivatives with respect to the “main” angle of attack (i.e. that of the fuselage). By referring to Figure 2.3, the wing angle of attack may be expressed as:

w w = + i . (2.11) w Therefore, the derivatives with respect to are equivalent to the derivatives with respect to 1. The lift coeﬃcient CL may now be expanded:

w t t t c c c CL = CL + CL + CL = w w t t t t t c c c c c = CL + CL + CL e + CL + CL c = ∕ ∕ ∕e ∕ ∕c w w t t $ t w t % = CL + CL 1 − "D + i1 + CL e + ∕ ∕ ∕ ∕e c c $ c w c % + CL 1 + "U + i2 + CL c = ∕ ∕ ∕c t c (2.12) ⎡ CL CL ⎤ w w t t ∕ c c ∕ = CL ⎢1 + w 1 − "D + w 1 + "U ⎥ + ∕ ⎢ C ∕ C ∕ ⎥ ⎣ L∕ L∕ ⎦ t t t c c c t t t c c c + CL e + CL c + CL i1 + CL i2 = ∕e ∕c ∕ ∕ = Cw w (1 + F ) + CT L∕ L∕x where: Ct Cc t L∕ c L∕ 1 C = C = By using Equation (2.9) the liftcurve slopes with respect to are L∕ t 1−" and L∕ c 1+" D∕ U∕

8 CHAPTER 2. THREE-SURFACE MODEL 2.3. THREE SURFACE MOMENT

Ct Cc L L F = tt ∕ 1 − " + cc ∕ 1 + " . Cw D∕ Cw U∕ (2.13) L∕ L∕ C L∕x is a vector: T t t t c c c t t t c c c C = CL CL CL CL L∕x ∕e ∕c ∕ ∕ (2.14) and is: T = e c i1 i2 . (2.15) Before coming to an expression for the lift coefficient derivatives with respect to the angle of attack and to the elevator deflections, the lift coefficient shown in Equation (2.12) may be further expanded, by applying Equation (2.11): w w w w w CL = C + i (1 + F )+C = C (1 + F ) +C (1 + F ) i +C . L∕ L∕x L∕ L∕ L∕x (2.16)

By considering the lift coeﬃcient as linear function in the variables , e and c, CL can be written as: CL = CL + CL e + CL c + CL . ∕ ∕e ∕c 0 (2.17) C Therefore, from Equation (2.16) the derivatives and L0 can be determined:

w CL = C (1 + F ) ∕ L∕ (2.18a) t t t CL = CL ∕e ∕e (2.18b) c c c CL = CL ∕c ∕c (2.18c) w w t t t c c c CL = C (1 + F )i + C i + C i = 0 L∕ L∕ 1 L∕ 2 = Cw (1 + F ) − ttCt − ccCc iw + ttCt it + ccCc ic+ L∕ L∕ L∕ L∕ L∕ (2.18d) t t t c c c − C "D + C "U . L∕ 0 L∕ 0

2.3 Pitching Moment of the Three Surfaces

Once the constitutive equations for the lift of the three surfaces have been obtained, it is possible to perform an analysis of the pitching moments which act on the aircraft. Such analysis will lead to the definition of the static stability characteristics of the aircraft (i.e. definition of the longitudinal location of the neutral point). Moreover, since the constitutive equations for the pitching moment around the center of gravity will be obtained, the definition of the trim problem will show up automatically. From Figure 2.1, an equilibrium equation can be written for the moments with respect to the center of gravity CG (given the hypotheses of L∕D ≫ 1 and ≪ 1):

w t c w ¨ t ¨ c ¨ CG = AC + AC + AC + L lw − L lt + L lc (2.19)

¨ ¨ ¨ w t l x w x l x x t l x c x where w is deﬁned as AC − CG, t as CG− AC and c as AC − CG whereas AC, AC c and AC are the moments of the lift distributions of the single surfaces around their own

9 2.3. THREE SURFACE MOMENT CHAPTER 2. THREE-SURFACE MODEL

¨ ¨ aerodynamic center. The deﬁnition of lt and lc has been set so as to ensure those quantities to be positive.

Note that the interfocal distances may be introduced: lt is deﬁned as the distance between the horizontal tail and the wing aerodynamic centers (xACw − xACt ) and lc is deﬁned as the distance between the forward wind and the wing aerodynamic centers (xACc − xACw ). The interfocal distances may be linked to the previous arms as follows:

T ¨ ¨ lt = lt + lw ¨ ¨ (2.20) lc = lc − lw. Then, Equation (2.19) may be expanded as:

w t c w ¨ t ¨ c ¨ CG = AC + AC + AC + L lw − L lt − lw + L lc + lw . (2.21)

It may be useful to write Equation (2.21) in a non-dimensional form by dividing each member by qd, S and ̄c:

¨ ¨ ¨ l lt − l lc + l C = C +tttC +cccC +Cw w −ttCt w +ccCc w . CG ACw ACt ACc L ̄c L ̄c L ̄c (2.22) C By condensing the ﬁrst three terms into one term, AC , and noticing that some terms may be gathered together to give the tail volume, Equation (2.22) can be written as follows:

¨ lw C = C − tV̄ tCt + cV̄ cCc + Cw + ttCt + ccCc CG AC L L L L L ̄c (2.23) ̄ t ̄ c where V and V represent the horizontal tail volume and forward wing tail volume respectively: Stl Scl V̄ t = t and V̄ c = c . S ̄c S ̄c (2.24) w t t t c c c The term CL + CL + CL is simply the lift coeﬃcient of the whole aircraft, as CL + CL e + CL c + CL shown in Equation (2.6), and therefore it can be written as ∕ ∕e ∕c 0 . t c Similarly, CL and CL can be expanded as shown in Equation (2.7).

l¨ C = C + C w − tV̄ tCt + cV̄ cCc = CG AC L ̄c L L l¨ w t ̄ t t t t = C + CL + CL e + CL c + CL − V CL + CL e + AC ∕ ∕e ∕c 0 ̄c ∕ ∕e c ̄ c c c c + V CL + CL c = ∕ ∕c l¨ w t ̄ t t = C + CL + CL e + CL c + CL − V CL e+ AC ∕ ∕e ∕c 0 ̄c ∕e c ̄ c c t ̄ t t t c ̄ c c c + V CL c − V CL + V CL . ∕c ∕ ∕ (2.25)

10 CHAPTER 2. THREE-SURFACE MODEL 2.3. THREE SURFACE MOMENT

The last two terms may be expanded by exploiting Equation (2.9):

−tV̄ tCt t + cV̄ cCc c = L∕ L∕ t ̄ t t w c ̄ c c w = − V C 1 − "D + i + V C 1 + "U + i = L∕ ∕ 1 L∕ ∕ 2 t ̄ t t w c ̄ c c w t ̄ t t c ̄ c c = − V C 1 − "D + V C 1 + "U − V C i + V C i = L∕ ∕ L∕ ∕ L∕ 1 L∕ 2 w t ̄ t t c ̄ c c t ̄ t t c ̄ c c = − V C 1 − "D + V C 1 + "U − V C i + V C i . L∕ ∕ L∕ ∕ L∕ 1 L∕ 2 (2.26)

w By substitution of from Equation (2.11) and by expanding i1 and i2:

w t ̄ t t c ̄ c c t ̄ t t c ̄ c c − V C 1 − "D + V C 1 + "U − V C i + V C i = L∕ ∕ L∕ ∕ L∕ 1 L∕ 2 w t ̄ t t c ̄ c c t ̄ t t = + i − V C 1 − "D + V C 1 + "U − V C i + L∕ ∕ L∕ ∕ L∕ 1 + cV̄ cCc i = L∕ 2 t ̄ t t c ̄ c c = − V C 1 − "D + V C 1 + "U + L∕ ∕ L∕ ∕ w t ̄ t t c ̄ c c + i − V C 1 − "D + V C 1 + "U + L∕ ∕ L∕ ∕ t ̄ t t t w c ̄ c c c w − V C i − i − "D + V C i − i + "U = L∕ 0 L∕ 0 t ̄ t t c ̄ c c = − V C 1 − "D + V C 1 + "U + L∕ ∕ L∕ ∕ w t t t c c c t t t¨¨ c c c + i − V̄ C 1¡ − " + V̄ C 1¡ + " + V̄ ¨C − V̄ C + L D∕ L U∕ L L ∕ ∕ ¨¨ ∕ ∕ t ̄ t t t c ̄ c c c t ̄ t t c c c − V C i + V C i + V C "D + C "U = L∕ L∕ L∕ 0 L∕ 0 t ̄ t t c ̄ c c = − V C 1 − "D + V C 1 + "U + L∕ ∕ L∕ ∕ w t ̄ t t c ̄ c c t ̄ t t t c ̄ c c c + i V C "D + V C "U − V C i + V C i + L∕ ∕ L∕ ∕ L∕ L∕ t ̄ t t c c c + V C "D + C "U . L∕ 0 L∕ 0 (2.27)

Eventually, this result is joined to Equation (2.25) and the coeﬃcients of the terms , e and c are grouped together:

11 2.3. THREE SURFACE MOMENT CHAPTER 2. THREE-SURFACE MODEL

l¨ w t ̄ t t c ̄ c t C = C + CL + CL e + CL c + CL − V CL e + V CL c+ CG AC ∕ ∕e ∕c 0 ̄c ∕e ∕e t ̄ t t c ̄ c c + − V C 1 − "D + V C 1 + "U + L∕ ∕ L∕ ∕ w t ̄ t t c ̄ c c t ̄ t t t c ̄ c c c + i V C "D + V C "U − V C i + V C i + L∕ ∕ L∕ ∕ L∕ L∕ t ̄ t t c c c + V C "D + C "U = L∕ 0 L∕ 0 4 l¨ 5 w t ̄ t t c ̄ c c = CL − V C 1 − "D + V C 1 + "U + ∕ ̄c L∕ ∕ L∕ ∕ 4 l¨ 5 4 l¨ 5 l¨ w t ̄ t t w c ̄ c w + e CL − V CL + c CL + V CL + C + CL + ∕e ̄c ∕e ∕c ̄c ∕c AC 0 ̄c w t ̄ t t c ̄ c c t ̄ t t t c ̄ c c c + i V C "D + V C "U − V C i + V C i + L∕ ∕ L∕ ∕ L∕ L∕ t ̄ t t c c c + V C "D + C "U . L∕ 0 L∕ 0 (2.28)

C Since CG may be written as:

C = C + C + C + C CG CG CG e CG c CG (2.29)   ∕  ∕e  ∕c  0

expressions for the derivatives can be extracted from Equation (2.28):

l¨ w t ̄ t t c ̄ c c C = CL − V CL 1 − "D + V CL 1 + "U (2.30a) CG∕ ∕ ̄c ∕ ∕ ∕ ∕ l¨ C = C w − tV̄ tCt CG L∕ L (2.30b) ∕e e ̄c ∕e l¨ C = C w + cV̄ cCc CG L∕ L (2.30c) ∕c c ̄c ∕c l¨ w w t ̄ t t c ̄ c c C = C + CL + i V CL "D + V CL "U + (2.30d) CG0 AC 0 ̄c ∕ ∕ ∕ ∕ t ̄ t t t c ̄ c c c t ̄ t t c c c − V C i + V C i + V C "D + C "U . L∕ L∕ L∕ 0 L∕ 0 (2.30e)

12 CHAPTER 2. THREE-SURFACE MODEL 2.3. THREE SURFACE MOMENT

C C C C Using Equations (2.18), the terms CG , CG , CG and CG may be expanded:  ∕  ∕e  ∕c  0

l¨ w w t ̄ t t c ̄ c c C = CL (1 + F ) − V CL 1 − "D + V CL 1 + "U (2.31a) CG∕ ∕ ̄c ∕ ∕ ∕ ∕ l¨ 0 l¨ 1 C = ttCt w − tV̄ tCt = tCt t w − V̄ t = CG L L L ∕e ∕e ̄c ∕e ∕e ̄c H I 0Stl¨ t 1 St l l¨ t t t w S lt t t t − t S lt = CL − = CL − = (2.31b) ∕e S ̄c S ̄c ∕e S ̄c S ̄c Stl¨ t t t t ̄ ¨t t = − CL = − V CL ∕e S ̄c ∕e l¨ 0 l¨ 1 C = ccCc w + cV̄ cCc = cCc c w + V̄ c = CG L L L ∕c ∕c ̄c ∕c ∕c ̄c H I 0Scl¨ c 1 Sc l¨ l c c c w S lc c c c − c S lc = CL + = CL + = (2.31c) ∕c S ̄c S ̄c ∕c S ̄c S ̄c Scl¨ c c c c ̄ ¨c c = CL = V CL ∕c S ̄c ∕c $ C = C + Cw (1 + F ) − ttCt − ccCc iw + ttCt it + ccCc ic+ CG0 AC L∕ L∕ L∕ L∕ L∕ % l¨ t t t c c c w w t ̄ t t c ̄ c c − C "D + C "U + i V C "D + V C "U + L∕ 0 L∕ 0 ̄c L∕ ∕ L∕ ∕ t ̄ t t t c ̄ t t c t ̄ t t c c c − V C i + V C i + V C "D + C "U = L∕ L∕ L∕ 0 L∕ 0 4 l¨ 5 4 l¨ 5 = C + it ttCt w − tV̄ tCt + ic ccCc w + cV̄ cCc + AC L∕ ̄c L∕ L∕ ̄c L∕ < l¨ = w w t t t c c c w t ̄ t t c ̄ c c + i C (1 + F ) − C − C + V C "D + V C "U + L∕ L∕ L∕ ̄c L∕ ∕ L∕ ∕ l¨ l¨ t t t w c c c w t ̄ t t c c c − C "D + C "U + V C "D + C "U L∕ 0 ̄c L∕ 0 ̄c L∕ 0 L∕ 0 (2.31d) where: Stl¨ Scl¨ V̄ ¨t = t and V̄ ¨c = c . S ̄c S ̄c

C l¨ V̄ t V̄ c The term CG0 may be simpliﬁed by noticing the summations where w, and ¨ appear. lw = xACw − xCG and the tail volumes contain lt and lc or, in other words, xACw − xACt and xACc − xACw respectively. The summation allows to cancel some terms and the ¨ ¨ remaining quantities result in lt and lc. C CG0 is therefore re-written as:

13 2.4. STICK-FIXED NEUTRAL POINT CHAPTER 2. THREE-SURFACE MODEL

l¨ l¨ t t t t t c c c c w $ w t t t C = C − CL i + CL + i CL (1 + F ) − CL + (2.32) CG0 AC ∕ ̄c ∕ ̄c ∕ ∕ l¨ = c c c w t ̄ t t c ̄ c c − C + V C "D + V C "U + L∕ ̄c L∕ ∕ L∕ ∕ (2.33) l¨ l¨ t t t t c c c c + C "D + C "U . L∕ 0 ̄c L∕ 0 ̄c (2.34) C C Note that CG and CG are negative and positive respectively. Indeed, a positive  ∕e  ∕c deflection of the tail elevator increases the horizontal tail lift coefficient which results in a pitching-down moment; a positive deflection of the forward wing elevator increases the lift coefficient of the forward wing which results in a pitching-up moment.

2.4 Stick-Fixed Neutral Point

C It is known that when the static margin is null the term CG∕ goes to zero as well. From ¨ Equation (2.31a) by moving the reference point to N (neutral point) and by expanding lw into xACw − xN the location of N may be computed: −tV̄ tCt 1 − " + cV̄ cCc 1 + " x x w L D∕ L U∕ N = AC + ∕ ∕ . ̄c ̄c Cw (1 + F ) (2.35) L∕ The term xN∕̄c is the non-dimensional location of the stick-ﬁxed neutral point and can x w be written as N , as well as AC ∕̄c which can be written as ACw . C N − CG CG∕ may be now expressed as function of the static margin by substitution of the term xACw∕̄c of Equation (2.35) in Equation (2.31a):

⎛ t ̄ t t c ̄ c c ⎞ − V CL 1 − "D + V CL 1 + "U w ⎜xN ∕ ∕ ∕ ∕ ⎟ C = CL (1 + F ) − w + CG∕ ∕ ⎜ ̄c C (1 + F ) ⎟ ⎜ L∕ ⎟ ⎝ ⎠ x w CG t ̄ t t c ̄ c c − C (1 + F ) − V C 1 − "D + V C 1 + "U = L∕ ̄c L∕ ∕ L∕ ∕ w xN xCG = C (1 + F ) − = CL N − CG L∕ ̄c ̄c ∕ (2.36) which shows to be the same expression as for the classic two-surface model. C = 0 From Equation (2.36) the fact that N∕ is immediately noticeable.

2.5 Maneuvering Flight of a Three-Surface Aircraft

In this Section a model of a three-surface aircraft in maneuvering flight around the pitch axis is presented. The goal is to give an estimation of the effects of pitch rate on the pitching moment coefficient and an estimation of the derivative of the pitching moment coefficient and lift coefficient with respect to the pitch rate. Such derivatives are often known as “pitch damping derivatives”.

14 CHAPTER 2. THREE-SURFACE MODEL 2.5. MANEUVERING FLIGHT

2.5.1 Angle of Attack Induced by Pitch Rate

̄q ◔◔ t c AC CG AC

¨ ¨ lt lc

Figure 2.4: Three-surface conﬁguration

Figure 2.4 shows the reference scheme used in this formulation. Figure 2.5 shows the effect of the pitch rate on the angle of attack of the horizontal tail and the forward wing. For the sake of simplicity, contrary to what has been said in Section 2.1, the relative wind met by the horizontal tail and the forward wing in non-maneuvering flight is considered to be the same, defined as V . When in pull-up maneuver, the pitch rate induces a ΔV on the above-mentioned surfaces which causes them to generate a lift variation ΔL which influences the pitching moment coefficient. The wing is supposed not see an appreciable change of the angle of attack, since its location is in general in close proximity to the center of gravity.

2.5.2 Change of the Lift Coeﬃcient in Maneuvering Flight

t For what concerns the horizontal tail, the pitch rate will induce a Δ given by: ¨ ΔV t ̄ql Δ t ≈ = t V V (2.37) ¨ where ̄q represents the pitch rate about the center of gravity and lt represents the tail arm, as shown in Figure 2.4. For a positive (pitch-up) pitch rate we expect the tail to increase its angle of attack, and therefore to increase the lift: ̄ql¨ ΔCt = Ct Δ t = Ct t . L L∕ L∕ V (2.38) c c For the forward wing, on the contrary, the Δ is expected to be negative, and so ΔCL:

ZLLt ZLLc t c c V t c V Δ Vt Δ V Vt Vc (a) Horizontal tail (b) Forward wing

Figure 2.5: Angle of attack induced by pitch rate

15 2.5. MANEUVERING FLIGHT CHAPTER 2. THREE-SURFACE MODEL

¨ ΔV c ̄ql Δ c ≈ = − c V V (2.39) ̄ql¨ ΔCc = Cc Δ c = −Cc c . L L∕ L∕ V (2.40) At this point, from Equations (2.6), (2.38) and (2.40), the variation of the lift coeﬃcient of the whole aircraft may be computed. ̄ql¨ ̄ql¨ t t t c c c t t t t c c c c ΔCL = ΔC + ΔC = C − C . L L L∕ V L∕ V (2.41) It is convenient to express ̄q in a non-dimensional form: ̄q ̂q = ̄c. 2V (2.42) From the vertical force equilibrium in level ﬂight:

2W∕S V 2 = C (2.43) Ln=1 and since ̄q may be written as: n − 1 ̄q = g V (2.44) the quantity ̂q can be deﬁned as: (n − 1) g ̄cC (n − 1) C n − 1 (n − 1) g ̄c Ln=1 Ln=1 ̂q = g ̄c = = = = 2 W W 2V W 2 ⋅ 2 ∕S 2 ⋅ 2 ( ∕g) ∕ (S ̄c) 2 (2 ∕S) ∕ CL n=1 (2.45) (n − 1) C Ln=1 (n − 1) = = CL 22M∕S ̄c 2 n=1 where:

• W∕S represents the wing loading of the aircraft C • Ln=1 represents the trim lift coefficient in level flight • n is the vertical load factor • g is the earth gravity constant • is defined as 2M∕S ̄c and represents the non-dimensional mass of the aircraft.

The change ΔCL may be now expressed as follows:

ql¨ ql¨ l¨ l¨ t t t t c c c c t t t 2V t c c c 2V c ΔCL = C − C = C ̂q − C ̂q = L∕ V L∕ V L∕ ̄c V L∕ ̄c V Stl¨ Scl¨ (2.46) = 2tCt t ̂q − 2cCc c ̂q = 2tCt V̄ ¨t − 2cCc .V̄ ¨c ̂q L∕ S ̄c L∕ S ̄c L∕ L∕ C So, the ﬁrst damping derivative L∕ ̂q is obtained: t t ̄ ¨t c c ̄ ¨c CL = 2 C V − 2 C V . ∕ ̂q L∕ L∕ (2.47)

16 CHAPTER 2. THREE-SURFACE MODEL 2.6. STICK-FREE STATIC STABILITY

2.5.3 Pitching Moment Variation in Maneuvering Flight

For what concerns the pitching moment around the center of gravity, the extra lift generated by the surfaces has an eﬀect on the equilibrium around the pitch axis. The extra lift produced by the main wing is considered negligible since, as already mentioned, the wing is supposed not to change appreciably the angle of attack.

ΔLtl¨ ΔLcl¨ qt StΔCt l¨ qc ScΔCc l¨ ΔC = − t + c = − d L t + d L c = CG qdS ̄c qdS ̄c qdS ̄c qdS ̄c = −tV̄ ¨tΔCt + cV̄ ¨cΔCc = −tCt V̄ ¨tΔ t + cCc V̄ ¨cΔ c = L L L∕ L∕ ql¨ ql¨ 0 l¨ l¨ 1 = −tCt V̄ ¨t t − cCc V̄ ¨c c = −tCt V̄ ¨t t − cCc V̄ ¨c c q. L∕ V L∕ V L∕ V L∕ V (2.48) By introducing ̂q:

0 ¨ ¨ 1 l l 2V 2 ΔC = −tCt V̄ ¨t t − cCc V̄ ¨c c ̂q = − tCt V̄ ¨tl¨ + cCc V̄ ¨cl¨ ̂q CG L∕ V L∕ V ̄c ̄c L∕ t L∕ c (2.49) the second damping derivative may be now computed: l¨ l¨ 0l¨ 12 0l¨ 12 t t ̄ ¨t t c c ̄ ¨c c t t t t c c c c C = −2 CL V − 2 CL V = −2 CL − 2 CL . CG∕ ̂q ∕ ̄c ∕ ̄c ∕ ̄c ∕ ̄c (2.50)

2.6 Stick-Free Static Stability

This topic is discussed in Section 3.7, since it exploits some of the results obtained in Chap- ter 3.

2.7 Borri Formulation

The formulation introduced by Borri at the beginning of the ’90s represents an alternative way to describe the study of aircraft trim and static stability characteristics by maintaining a simplified framework. [7] [9] [8] Since the Borri formulation has been developed for two-surface aircraft only, there is the need to expand this study to a three-surface configuration with two different elevator deflections. This Section shows how the Borri approach is applied on a two-elevator configuration for the stick-fixed case. As starting point, a brief description of such formulation for a two-surface aircraft is given in order to develop the three-surface case on the basis of the original one.

2.7.1 Two-Surface Aircraft

The first step is to set an aerodynamic model for level flight. In steady symmetric level flight conditions, under the hypotheses that the angle of attack and the elevator deflection

17 2.7. BORRI FORMULATION CHAPTER 2. THREE-SURFACE MODEL

e are small, we may consider the linear constitutive equations for the lift L and the pitching moment P around a generic point P : T L = L + L + L ∕ ∕e e 0 (2.51) P = P + P e + P .   ∕  ∕e  0 It is convenient to rearrange those equations in homogeneous form by introducing two terms 0 and e0 which cancel simultaneously the lift and the pitching moment or, in other words, yield an aerodynamic force distribution equivalent to a null system of forces:

⎧L = L − + L − ⎪ ∕ 0 ∕e e e0 ⎨ (2.52) P = P − 0 + P e − e . ⎪  ∕  ∕e 0 ⎩ P It must be noted that 0 and e0 are independent from the point as well as on both airspeed and altitude, as long as the ﬂight is subsonic. Under the hypotheses of L∕D ≫ 1 and ≪ 1, the pitching moment around a generic point P may be expressed using the rule of transport between the above-mentioned P and another generic point Q: Q = P + xP − xQ L. (2.53)

Characteristic Points

The characteristic points known as Neutral Point and Control Point may be introduced. We deﬁne the neutral point N and the control point C as the points for which the following conditions are met:

N = 0, C = 0.  ∕  ∕e (2.54)

The ﬁrst statement of Equation (2.54) is exploited by computing the derivative of Equa- tion (2.53) with respect to and by considering N instead of Q: = + x − x L = 0. N∕ P∕ P N ∕ (2.55) The location of the neutral point with respect to a generic point P is given by:

P∕ xN − xP = . (2.56) L∕ In non-dimensional form, Equation (2.56) becomes the following: C P∕ − = . N P C (2.57) L∕ Similarly, the control point location is found as stated in the following: C P ∕e C − P = . (2.58) CL ∕e

18 CHAPTER 2. THREE-SURFACE MODEL 2.7. BORRI FORMULATION

Latt

CG C

N Lctrl

Figure 2.6: Aerodynamic force system according to Borri formulation [5]

N C CL C CL The location of and is found by expressing the terms ∕ , P∕ , ∕e and C P as stated in the model in Equations (2.18a), (2.18b), (2.31a) and (2.31b) by consid-  ∕e ering a generic point P instead of the center of gravity.

w t t t C (1 + F ) w − − V̄ C 1 − " L AC P L D∕ = + ∕ ∕ N P Cw (1 + F ) (2.59a) L∕ t t t − P − ACt CL ∕e C = P + = ACt . (2.59b) t t t CL ∕e

The dimesional form of Equations (2.57) and (2.58) are substituted in the moment equilibrium Equation in (2.52):

⎧L = L − + L − ⎪ ∕ 0 ∕e e e0 (2.60) ⎨ = L − x − x + L − x − x . ⎪P ∕ 0 N P ∕e e e0 C P ⎩ L − L − Latt Lctrl Latt The terms ∕ 0 and ∕e e e0 are called respectively and . ctrl is called attitude lift and depends only upon and L is called control lift and depends only upon e. Equation (2.60) is thus expressed as follows: T L = Latt + Lctrl att ctrl (2.61) P = L xN − xP + L xC − xP .

The system of aerodynamic forces has been expressed as two components of the lift applied on two material points (their location on the x-axis is ﬁxed). the weight is obviously applied on the center of gravity. Figure 2.6 shows such force system.

19 2.7. BORRI FORMULATION CHAPTER 2. THREE-SURFACE MODEL

Trim Condition

If the trim condition is imposed, Equation (2.61) is expressed as follows: T Latt + Lctrl = W att ctrl (2.62) L xN − xCG + L xC − xCG = 0.

By solving the system, the trim attitude lift and control lift are obtained:

⎧Latt = xCG−xC W ⎪ xN −xC ⎨Lctrl = − xCG−xN W. (2.63) ⎪ xN −xC ⎩ The index is now introduced:

xCG − xN ∶= . (2.64) xN − xC The numerator is the static margin (dimensional form) while the denominator may be called as aerodynamic length of the aircraft. The index allows to clearly represent how the lift is divided up in the two contributions: T Latt = (1 + ) W Lctrl = −W . (2.65)

For a conventional tailed statically stable two-surface aircraft, the control point lies behind the neutral point and therefore > 0. So the control lift will be negative and the attitude lift will have to be greater than the weight. The necessary increment of the attitude lift is smaller the smaller is , and therefore the smaller is the static margin compared to the aerodynamic length. On the contrary, for a statically stable canard aircraft the control point lies ahead of the neutral point and therefore < 0. In this way the control lift is positive and the attitude lift is smaller than the weight.

2.7.2 Three-Surface Aircraft - Independent Elevator Deﬂections

For a three-surface aircraft, the aerodynamic forces will depend also upon the forward wing deflection angle. Therefore, the relative term has to be added to (2.51): T L = L + L + L + L ∕ ∕e e ∕c c 0 (2.66) P = P + P e + P c + P .   ∕  ∕e  ∕c  0 Unlike the two-surface case, the equations cannot be expressed in homogeneous form, since the angles 0, e0 and c0 are not uniquely definable. However, the neutral point and control point can be defined as in (2.54). In particular, two control points may be introduced:

N = 0, C = 0, C = 0.  ∕  1∕e  2∕c (2.67)

The derivative with respect to of Equation (2.53) gives the same result as for the two- C CL surface case, Equation (2.57). However, the terms P∕ and ∕ must take into account

20 CHAPTER 2. THREE-SURFACE MODEL 2.7. BORRI FORMULATION the term related to the forward wing. The result is analogous to the one obtained in Section 2.4:

t t t c c c − w − t C 1 − " + c − w C 1 + " AC AC L D∕ AC AC L U∕ = + ∕ ∕ . N P Cw (1 + F ) L∕ (2.68) The control point locations are obtained by taking the derivative of Equation (2.53) with respect to e and c (separately) and by considering the control points C1 and C2 instead of Q: T = + x − x L = 0 C1∕ P∕ P C1 ∕e e e (2.69) C = P + xP − xC L = 0.  2∕c  ∕c 2 ∕ c The two control points non-dimensional locations are obtained as follows: C P ∕e ⎧ C − P = 1 CL ⎪ ∕e C ⎨ P (2.70) ∕c ⎪ C − P = . 2 CL ⎩ ∕c By expanding the derivatives C1 and C2 are obtained:

t t t ⎧ − (P −ACt )CL ∕e ⎪ = + = t C1 P t t t AC CL ⎪ ∕e c c c ⎨ + (ACc −P )CL (2.71) ∕c ⎪ = + = c . C2 P c c Cc AC ⎪ L∕ ⎩ c As for the two-surface case, the control points coincide with the relative lifting surface aerodynamic center. Even if a deﬁnition of the two control points has been given, no further statements may be made in order to link those points to a new formulation of the aerodynamic force system as for the two-surface case, Equation (2.61). This because there is no a unique way to express such equation in homogeneous form.

2.7.3 Three-Surface Aircraft - Dependent Elevator Deﬂections

This topic is discussed in Section 3.7, since it exploits some of the results obtained in Chap- ter 3.

21 2.7. BORRI FORMULATION CHAPTER 2. THREE-SURFACE MODEL

22 Chapter 3

Trimmed Solutions of a Three-Surface Aircraft

A peculiarity of a three-surface configuration is that the pitch moment equilibrium around the center of gravity may be obtained by the combined action of two elevator deflections. If such deflections are independent, the trim problem would accept an infinite number of solutions. In this scenario, are there solutions which are better than others? It may depend from the merit function which is considered. For example, for a given altitude and equivalent airspeed, it is reasonable to conjecture there are some pairs (e, c) for which the total drag generated by the aircraft is larger compared to the one generated by other pairs in equilibrium condition. The angle of attack is obviously part of the solution which satisfies the trim problem besides the two elevator deflections. Thus, it is possible to solve the trim problem by imposing the minimization of the drag induced by the trimmed aircraft configuration.

3.1 Optimal Trim Solutions

The following formulation assumes the geometry of the aircraft is set. As stated in the introduction of this chapter, the main aim is to find a solution in terms of 3 variables ( , e, c) which maximizes/minimizes a certain aircraft performance index. As far as conventional two-surface aircraft are concerned, the trim problem features a unique solution since the problem is represented by two linear equations, i.e. static equilibrium in the direction of lift and around the pitch axis, and two unknowns, i.e. and e. In the case of a three-surface aircraft, another relation is required in order to complete the problem; this relation is the above-mentioned optimization of the performance index. The mathematical problem has been set up to be a constrained minimization of an objective function. The constraints are represented by the two trim equations. The problem has been modeled as a mathematical problem formulated in a rigorous way. The peculiarity of the following formulation is that the solution may be found analytically; therefore, no numerical optimization method is required. A major saving on computational time is thus one of the crucial features of this approach. Furthermore, the existence of a closed form solution for the optimization allows to make the solution a part of the mathematical model, thus easing further developments and speculations. The reference condition of the aircraft is a trimmed symmetric flight at constant equivalent airspeed and weight, or in other words, that of a constant trimmed lift coefficient.

23 3.1. OPTIMAL TRIM SOLUTIONS CHAPTER 3. TRIMMED SOLUTIONS

3.1.1 Variables

The variables to be set as a result of the optimization process are the following:

Angle of attack of the aircraft

e Tail elevator deﬂection

c Forward wing elevator deﬂection

3.1.2 Constraints

As previously stated, the reference condition of the aircraft is symmetric flight at constant Equivalent Airspeed (EAS). The horizontal flight (angle of climb equal to 0) is not a required constraint. Thus, the aircraft may change altitude as long as the EAS is kept constant (the True Airspeed (TAS) is therefore supposed to vary). The trimmed flight condition is translated in two linear equality constraints:

⎧ ∗ C = CL + CL e + CL c + CL ⎪ L ∕ ∕e ∕c 0 ⎨C∗ = C + C + C + C (3.1) CG CG e CG c CG ⎪ CG ∕ ∕e ∕c 0 ⎩ where the superscript * identiﬁes the trim condition. Since the trim pitching moment coeﬃcient around the center of gravity is zero, the constraints may be written as follows:

⎧ ∗ C = CL + CL e + CL c + CL ⎪ L ∕ ∕e ∕c 0 . ⎨0 = C + C + C + C (3.2) CG CG e CG c CG ⎪ ∕ ∕e ∕c 0 ⎩

The trim lift coeﬃcient is related to the equivalent airspeed as shown in Equation (3.3)

2 W∕S 2 W∕S C∗ = = L V 2 2 (3.3) 0VEAS where W represents the aircraft weight, S the reference wing surface area, and 0 the local air density and Mean Sea Level (MSL) air density respectively, V the true airspeed and VEAS the equivalent airspeed. The formulation, as already mentioned, considers constant aircraft weight. To take into account the weight variation due to fuel consumption, the analysis can be performed multiple times by considering decreasing weight in order to cover all the aircraft weight range. As an output transformation, the variation with the TAS (and/or Mach number) or altitude may be computed. By performing the optimization for a set of assigned values of the lift coefficients span- ning the usual operative spectrum, a complete trimmed drag polar is obtained. The coefficients in the two equations may be found by means of the three-surface model of Chapter 2 and they are supposed to be fixed, since aircraft geometry is set.

24 CHAPTER 3. TRIMMED SOLUTIONS 3.2. AIRCRAFT DRAG POLAR

3.1.3 Objective Function

The choice of the objective function depends upon the performance index which has to be maximized/minimized. In the following cases, the hypotheses of constant available thrust and available power (with the airspeed) for jet and propeller-driven aircraft respectively are made. Moreover, the reference flight condition is assumed to be at constant angle of attack. The Thrust Specific Fuel Consumption (TSFC) and Brake Specific Fuel Consumption (BSFC) in the two cases respectively are unvarying throughout the flight as well. Some reasonable objective functions are represented by the following:

• CD Drag coeﬃcient of the aircraft

Condition for which the maximum endurance for a • CL∕CD jet aircraft and maximum range for a propeller-driven aircraft are obtained. √ Condition for which the maximum range for a jet is • CL∕CD obtained.

3∕2 Condition for which the maximum endurance for a • C ∕CD L propeller-driven aircraft is obtained.

However, it is worth noting that the trim lift coefficient is set; therefore, since the above- mentioned merit functions are combination of CL and CD only, the minimization of the drag coefficient will give the same result in terms of optimization parameters and value of the merit function. This means that if one of the objective functions is selected, the resulting ∗ trimmed drag polar will be the set of points (CD, CL) for which all the above-mentioned performance indices are maximized (or minimized, in the case of CD). This statement has been verified by using all te above-mentioned objective functions, obtaining the same result in all cases. Be that as it may, the overall best performance index of the whole drag polar will be obtained for a specific angle of attack, or in other words, a specific trim lift coefficient. In order to perform the constrained minimization, it is required that the objective function is expressed as function of the optimization parameters. This process is explained in detail in Section 3.2.

3.2 Aircraft Drag Polar

3.2.1 Assumptions

An analytical expression in terms of , e and c of the aircraft drag polar is required. The assumptions under which the formulation has been performed are the following:

Parabolic Drag Polar The drag coefficient of the aircraft lifting elements is supposed to be a quadratic function of their lift coefficient. This assumption does not imply the aircraft drag polar to be a parabolic function of the whole aircraft lift coefficient as well. The drag polar of the lifting surfaces may be expressed as follows:

25 3.2. AIRCRAFT DRAG POLAR CHAPTER 3. TRIMMED SOLUTIONS

Cw = Cw + kwCw2 ⎧ D D0 L ⎪Ct = Ct + ktCt 2 ⎨ D D0 L (3.4) ⎪Cc = Cc + kcCc 2. ⎩ D D0 L

Aircraft Drag Coefficient The whole aircraft drag coefficient is expressed as combination of the drag coefficient of the elements of the aircraft, as stated in Equation (3.7). Such expression derives from the sum of the drag generated by the three lifting surfaces:

D = Dw + Dtail + Dcan. (3.5)

The three contributes are expanded:

w ⎧Dw = qdSCD ⎪D = qt StCt ⎨ tail d D (3.6) ⎪D = qc ScCc . ⎩ can d D

By dividing both members of Equation (3.5) by qdS, the following expression is obtained:

w t t t c c c CD = CD + CD + CD. (3.7)

Angle of Attack As shown in Figure 2.3 of Section 2.2 the angle of attack of the aircraft elements can be expressed as function of the main angle of attack :

⎧ w = + iw ⎪ t t ⎨ = + i − "D (3.8) ⎪ c = + ic + " ⎩ U

where "D and "U depend upon the wing angle of attack:

T " = " w + " D D∕ w D0 " = " w + " . (3.9) U U∕ w U0

" = " w = + iw Since D∕ w D∕ and , Equation (3.9) may be expressed as:

T " = " + " iw + " = " + " D D∕ D∕ D0 D∕ D0 " = " + " iw + " = " + " (3.10) U U∕ U∕ U0 U∕ U0 " = " iw + " " = " iw + " where D0 D∕ D0 and U0 U∕ U0 .

Lift Coefficient The lift coefficient of the aircraft elements are expressed as function of the relative angle of attack and elevator deflection (if present) as shown in Equation (2.7):

26 CHAPTER 3. TRIMMED SOLUTIONS 3.2. AIRCRAFT DRAG POLAR

⎧Cw = Cw w ⎪ L L∕ ⎪ t t t t CL = CL + CL e ⎨ ∕ ∕e (3.11) ⎪ c c c c CL = CL + CL c. ⎪ ∕ ∕c ⎩

3.2.2 Drag Coeﬃcient of the Three Surfaces

The drag coeﬃcient of the three aircraft elements of Equation are expanded in order to obtain functions of the variables , e and c:

Wing-Body

Cw = Cw + kwCw2 = D D0 L 2 = Cw + kw Cw w = D0 L∕ 2 2 = Cw + kwCw + iw = D0 L∕ (3.12) = Cw + kwCw 2 2 + 2 iw + iw2 = D0 L∕ = Cw + kwCw 2iw2 + 2kwCw 2iw + kwCw 2 2 = D0 L∕ L∕ L∕ = Aw + Bw 2 + Ew

where:

⎧Aw = Cw + kwCw 2iw2 ⎪ D0 L∕ ⎪Bw = kwCw 2 ⎨ L∕ (3.13) ⎪Ew = 2kwCw 2iw. ⎪ L∕ ⎩

27 3.2. AIRCRAFT DRAG POLAR CHAPTER 3. TRIMMED SOLUTIONS

Horizontal Tail

Ct = Ct + ktCt 2 = D D0 L 2 t t t t t = CD + k CL + CL e = 0 ∕ ∕e 2 t t t t t = CD + k CL + i − "D + CL e 0 ∕ ∕e 2 t t t t t = CD + k CL + i − "D − "D + CL e = 0 ∕ ∕ 0 ∕e 2 t t t t t t t t = CD + k CL + CL i − CL "D − CL "D + CL e = 0 ∕ ∕ ∕ ∕ ∕ 0 ∕e 2 2 t t t 2 2 t 2 2 t 2 t = CD + k CL 1 − "D + CL e + CL i − "D + 0 ∕ ∕ ∕e ∕ 0 t t t 2 t + 2CL CL 1 − "D e + 2CL 1 − "D i − "D + ∕ ∕e ∕ ∕ ∕ 0 t t t + 2CL CL i − "D e = ∕ ∕e 0 2 4 25 t t t 2 t t t 2 2 = C + k C i − "D + k C 1 − "D + D0 L∕ 0 L∕ ∕

t t 2 t t t 2 2 + 2k CL 1 − "D i − "D + k CL e + ∕ ∕ 0 ∕e t t t t t t t + 2k CL CL i − "D e + 2k CL CL 1 − "D e = ∕ ∕e 0 ∕ ∕e ∕ t t 2 t 2 t t t = A + B + C e + D e + E + F e (3.14)

where:

2 ⎧ t t t t 2 t A = C + k C i − "D ⎪ D0 L∕ 0 2 ⎪ t t t 2 B = k C 1 − "D ⎪ L∕ ∕ ⎪Ct ktCt 2 ⎪ = L ∕e ⎨ t t t t (3.15) D = 2k C C 1 − "D ⎪ L∕ L∕ ∕ ⎪ e Et = 2ktCt 2 1 − " it − " ⎪ L D D ∕ ∕ 0 ⎪ t t t t t ⎪F = 2k CL CL i − "D . ⎩ ∕ ∕e 0

28 CHAPTER 3. TRIMMED SOLUTIONS 3.2. AIRCRAFT DRAG POLAR

Forward Wing

Cc = Cc + kcCt 2 = D D0 L 2 c c c c c = CD + k CL + CL c = 0 ∕ ∕c 2 c c c c c = CD + k CL + i + "U + CL c 0 ∕ ∕c 2 c c c c c = CD + k CL + i + "U + "U + CL c = 0 ∕ ∕ 0 ∕c 2 c c c c c c c c = CD + k CL + CL i + CL "U + CL "U + CL c = 0 ∕ ∕ ∕ ∕ ∕ 0 ∕c 2 2 c c c 2 2 c 2 2 c 2 c = CD + k CL 1 + "U + CL c + CL i + "U + 0 ∕ ∕ ∕c ∕ 0 c c c 2 c + 2CL CL 1 + "U c + 2CL 1 + "U i + "U + ∕ ∕c ∕ ∕ ∕ 0 c c c + 2CL CL i + "U c = ∕ ∕c 0 2 4 25 c c c 2 c c c 2 2 = C + k C i + "U + k C 1 + "U + D0 L∕ 0 L∕ ∕

c c 2 c c c 2 2 + 2k CL 1 + "U i + "U + k CL c + ∕ ∕ 0 ∕c c c c c c c c + 2k CL CL i + "U c + 2k CL CL 1 + "U c = ∕ ∕c 0 ∕ ∕c ∕ c c 2 c 2 c c c = A + B + C c + D c + E + F c (3.16)

where:

2 ⎧ c c c c 2 c A = C + k C i + "U ⎪ D0 L∕ 0 2 ⎪ c c c 2 B = k C 1 + "U ⎪ L∕ ∕ ⎪Cc kcCc 2 ⎪ = L ∕c ⎨ c c c c (3.17) D = 2k C C 1 + "U ⎪ L∕ L∕ ∕ ⎪ c Ec = 2kcCc 2 1 + " ic + " ⎪ L U U ∕ ∕ 0 ⎪ c c c c c ⎪F = 2k CL CL i + "U . ⎩ ∕ ∕c 0

Aircraft The whole aircraft drag coeﬃcient, or in other words, the untrimmed drag polar may be now computed by using Equation (3.7):

29 3.3. DRAG COEFFICIENT MINIMIZATION CHAPTER 3. TRIMMED SOLUTIONS

w t t t c c c CD = CD + CD + CD = w w 2 w t t t t 2 t 2 t t t = A + B + E + A + B + C e + D e + E + F e + c c c c 2 c 2 c c c + A + B + C c + D c + E + F c = w t t t c c c w t t t c c c 2 t t t 2 = A + A + A + B + B + B + C e + c c c 2 t t t c c c w t t t c c c + ( C ) c + D e + ( D ) c + E + E + E + t t t c c c + F e + ( F ) c = 2 2 2 = A + B + Ce + Dc + E e + F c + G + He + Ic (3.18)

where:

⎧A = Aw + ttAt + ccAc ⎪ w t t t c c ⎪B = B + B + B ⎪C = ttCt ⎪ ⎪D = ccCc ⎪ t t t ⎨E = D (3.19) ⎪F = ccDc ⎪ ⎪G = Ew + ttEt + ccEc ⎪H = ttF t ⎪ ⎪I = ccF c. ⎩

The term A represents the zero-angle of attack and zero-elevator deflection drag coefficient. It is evident that the formulation takes into account the linear dependency of the untrimmed drag polar upon the angle of attack. Therefore, it represents a more accurate model compared to the most commonly used in the literature, which assumes the drag coefficient to be proportional to only the square of the aircraft lift coefficient.

3.3 Drag Coeﬃcient Minimization

Given the drag coeﬃcient of the aircraft as function of the three variables , e e c of Equation (3.18) and given the constraints of equilibrium in the vertical direction and around the pitch axis, a minimization process can be performed. The trim problem of Equation (3.2) may be rearranged in matrix form since the equations are linear: L M L M 4 ∗ 5 C CL CL CL ⎡ ⎤ CL L = ∕ ∕e ∕c ⎢ ⎥ + 0 . 0 C C C e C (3.20) CG CG CG ⎢ ⎥ CG ∕ ∕e ∕c ⎣c⎦ 0 Equation (3.20) may be expressed in a more compact way:

y = X + y0. (3.21)

30 CHAPTER 3. TRIMMED SOLUTIONS 3.3. DRAG COEFFICIENT MINIMIZATION

The drag coeﬃcient of Equation (3.18) is expressed in such a way so as to introduce the vector :

⎡ ⎤ ⎡ B E∕2 F∕2⎤ ⎡ ⎤ E CD = A + GHI ⎢e⎥ + e c ⎢ ∕2 C 0 ⎥ ⎢e⎥ = ⎢ ⎥ ⎢F ⎥ ⎢ ⎥ (3.22) ⎣c⎦ ⎣ ∕2 0 D ⎦ ⎣c⎦ = A + mT + T K

⎡G⎤ ⎡ B E∕2 F∕2⎤ where m = ⎢H⎥ and K = ⎢E∕2 C 0 ⎥. ⎢ ⎥ ⎢F ⎥ ⎣ I ⎦ ⎣ ∕2 0 D ⎦

Note that K is a positive deﬁnite matrix. The minimization of CD is performed by introducing a cost function J which takes into account the constraints of Equation (3.21) by means of the Lagrange multipliers:

T T T J = A + m + K + X + y0 − y (3.23) 4 5 1 where represents the vector of the Lagrange multipliers . 2

The partial derivatives with respect to and are computed: T )J = m + 2 K + XT = 0 ) )J = X + y − y = 0. (3.24) ) 0 The ﬁrst equation allows to write : 1 = − K−1 m + XT 2 (3.25) which is substituted in the second equation: 1 − XK−1 m + XT + y − y = 0. 2 0 (3.26) Vector is computed from Equation (3.26):

−1 T −1 −1 = XK X 2 y0 − y − XK m (3.27) which is substituted into Equation (3.25) to obtain :

1 $ −1 % = − K−1 m + XT XK−1XT 2 y − y − XK−1m = 2 0 1 −1 1 −1 = − K−1m − K−1XT XK−1XT y − y + K−1XT XK−1XT XK−1m. 2 0 2 (3.28)

Since the constant terms (the trim lift coeﬃcient and trim moment coeﬃcient) are represented by y, it is useful to divide the vector into two contributions, one of which is

31 3.3. DRAG COEFFICIENT MINIMIZATION CHAPTER 3. TRIMMED SOLUTIONS associated with y. Given that of Equation (3.28) is the optimal trim solution and not a generic vector which contains the trim variables, it will be appointed as opt.

opt = 0 + 1y (3.29) where the vector 0 and the matrix 1 are deﬁned as follows:

1 −1 1 −1 = − K−1m − K−1XT XK−1XT y + K−1XT XK−1XT XK−1m 0 2 0 2 (3.30) −1 T −1 T −1 1 = K X XK X .

3×2 Note that 1 ∈ ℝ and the second column is multiplied by 0, which is the second element of y:

⎡○ ×⎤ 4 ∗ 5 CL opt = 0 + ⎢○ ×⎥ . (3.31) ⎢ ⎥ 0 ⎣○ ×⎦

The symbol × represents the elements which are multiplied by 0. Let be the vector 3 ∈ ℝ represented by the circles. In this case opt may be expressed directly as function of ∗ the trim lift coeﬃcient CL:

∗ opt = 0 + CL. (3.32)

Equation (3.32) is expanded so as to highlight the individual trim variables:

⎡ opt⎤ ⎡ ZL⎤ ⎡ 1⎤ = = + C∗ . opt ⎢ eopt ⎥ ⎢ eZL ⎥ ⎢ 2⎥ L (3.33) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ copt ⎦ ⎣ cZL ⎦ ⎣ 3⎦

It is clear that between the trim variables and the trim lift coeﬃcient a linear dependency exists and, moreover, the trim variables are linked one another by a linear relation:

∗ ∗ ⎧ opt = ZL + 1CL = opt CL ⎪ = + C∗ = C∗ ⎨ eopt eZL 2 L eopt L (3.34) ⎪ = + C∗ = C∗ . ⎩ copt cZL 3 L copt L

The minimum value of the drag coeﬃcient may be now computed from Equation (3.22):

C = A + mT + T K . Dmin opt opt opt (3.35) From the untrimmed drag polar CD , e, c , by replacing the trim variables as function of trim lift coeﬃcient as stated in Equation (3.34), the trimmed drag polar may be ∗ ∗ 2 computed. It is worth noting the drag polar is a function of both CL and CL and not only ∗ 2 of CL , thus adding to the generality of the proposed solution.

32 CHAPTER 3. TRIMMED SOLUTIONS 3.4. VALIDATION OF THE RESULTS

3.4 Validation of the Results

To prove the correctness of the optimization method of the previous Section, a computation of the optimal drag polar has been performed on the three-surface aircraft whose data is shown in Appendix A. The optimal analytical trim solution has been compared with a numerical solution obtained from a gradient-based algorithm, MATLAB® [16]. A rough grid search method has been applied in order to be sure the objective function was sufficiently regular to find its minimum by means of a gradient-based algorithm. ∗ The trim lift coefficient CL range has been set between 0 to 1. Figure 3.1 and 3.2 show the results of the comparison.

-5

-10

-15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 3.1: Trim solution comparison

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Figure 3.2: Trim drag polar comparison

33 3.5. APPLICATION OF OPTIMAL TRIM CHAPTER 3. TRIMMED SOLUTIONS

Table 3.1: Computational time comparison

Algorithm Time [s] Analytical 0.017 Numerical 6.001

A major advantage of the analytical method is the eﬃciency of the computation. The method shows to be well faster than the numerical algorithm. Table 3.1 shows the computational time comparison between analytical and numerical method. The gradient algorithm constraint tolerance has been set to the MATLAB® default settings while the function eval- uation tolerance option has been modiﬁed since the objective function (CD) order of mag- −3 −4 nitude is 10 -10 . Moreover, the numerical solution shows some irregularities due to non-convergence errors. This phenomenon is obviously avoided in the analytical approach.

−8 • Function tolerance: 10

−6 • Constraint tolerance: 10 (default)

Figure 3.3 shows the aircraft is actually trimmed at the desired lift coeﬃcients. The lift and moment coeﬃcients have been computed from the constitutive Equations (2.17) and (2.29).

1.2

0.8

0.6

0.4

0.2

-0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 3.3: Trim coeﬃcients

3.5 Application of Optimal Trim Solutions

This Section shows how the optimal trim solutions obtained in Section 3.4 and shown in Figure 3.1 may be exploited to set up a control system so as to maintain the aircraft airspeed and altitude.

34 CHAPTER 3. TRIMMED SOLUTIONS 3.5. APPLICATION OF OPTIMAL TRIM

∗ Since the trim variables , e and c are functions of the trim lift coeﬃcient CL, it is evident that those quantities may be expressed as function of the EAS or as function of the TAS (or Mach number) and altitude. By considering ≪ 1, The EAS and the lift coeﬃcient are linked by the following relation:

W∕S C∗ = 2 . L 2 (3.36) 0VEAS The EAS can be linked to the Mach number and the Pressure Altitude (PA), ℎ in symbol, in the following way1:

⎧ t VEAS = V ⎪ 0 ⎨ V V t (3.37) M = = √ = V ⎪ c p∕ p ⎩ where ℎ appears in the quantities (air density) and p (air static pressure); c represents the speed of sound and the air adiabatic index. Equation (3.37) allows to write the EAS as function of the Mach number and the PA (by means of the static pressure): v p(ℎ) VEAS = M. (3.38) 0

The trim variables may now be expressed as function of the Mach number and the PA. Figure 3.4 shows the trim solutions displayed in Figure 3.1 as function of M and ℎ. The maximum Mach number has been considered to be 0.6. The tail elevator deflection range has been set at [-15;15] degrees. Since the critical angle of attack is expected to be greater than 12 degrees (the original Juliett’s is 15 degrees), the airplane features a stall by controllability limitation. However, the nonlinear aerodynamics in the vicinity of the stall is not of interest since the focus is simply on showing an example of lookup table which may be exploited by a Flight Control System (FCS) in stationary flight to trim the aircraft at prescribed optimal configuration. Moreover, the Mod-Juliett aircraft is just an adaptation of another design (Juliett) which was needed to show an example of the application of optimal trim solution; therefore, no particular attention has been paid to features which concern high angle of attack condition or stall type (e.g aerodynamic, control limit). Since the pilot control stick is assumed to act on the tail elevator only, the forward wing must be deflected automatically as function of the tail elevator deflection and the flight condition. This job is done by the flight control system. The logic with which this process is made is rather simple. Figure 3.4 is used as a lookup table according to which the deflections are set. The air data system provides the pressure altitude and the Mach number, the actuators will deflect the control surfaces according to PA and M. The angle of attack will come by itself. The FCS is not designed to find the minimum drag condition. The optimal has been already solved and the FCS is asked to only hold PA and M by acting on the control variables through a proper control law; implicitly, it is controlling the aircraft in the longitudinal plane whilst minimizing the drag coefficient. The process is shown graphically

1The air is considered to be an ideal gas.

35 3.5. APPLICATION OF OPTIMAL TRIM CHAPTER 3. TRIMMED SOLUTIONS

0 0.1 0.2 0.3 0.4 0.5 0.6 MSL 5 3000 ft 0 6000 ft -5 9000 ft 12000 ft -10 15000 ft -15 0.1 0.2 0.3 0.4 0.5 0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 3.4: Trim variables VS Mach and PA

36 CHAPTER 3. TRIMMED SOLUTIONS 3.6. REFORMULATION OF TRIM PROBLEM

M - ΔM Δδe + δe + + AIRCRAFT δ h + Δh FCS Δδc + c - + Lookup Table

δe

δc

Control System

M h SET BY THE PILOT

Figure 3.5: Structure of the control system in Figure 3.5. The design of a suitable control law is not part of this work; however, the structure of Figure 3.5 may be a starting point for future thesis work.

3.6 Reformulation of the Trim Problem

According to what has been obtained so far, the forward wing elevator deﬂection can be expressed as linear function of the horizontal tail elevator deﬂection:

c = k1e + k2. (3.39)

The trim equations in (3.2) are written so as to contain two unknowns by replacing c with the expression (3.39):

⎧ ∗ C = CL + CL e + k1CL e + k2CL + CL ⎪ L ∕ ∕e ∕c ∕c 0 ⎨0 = C + C + k C + k C + C . (3.40) CG CG e 1 CG e 2 CG CG ⎪ ∕ ∕e ∕c ∕c 0 ⎩

C£ C¦ By gathering the terms related to e into L∕ and CG and by grouping the last two e  ∕e ¦ CL C terms of each equation into 0 and CG0 , the following trim equations are obtained:

37 3.7. STICK-FREE STATIC STABILITY CHAPTER 3. TRIMMED SOLUTIONS

⎧ ∗ £ C = CL + CL e + CL ⎪ L ∕ ∕e 0 ⎨0 = C + C¦ + C¦ (3.41) CG CG e CG ⎪ ∕ ∕e 0 ⎩ C£ = C + k C C¦ = C + k C C = C + k C where L∕ L∕ 1 L∕ , CG CG 1 CG , L0 L0 2 L∕ and e e c  ∕e  ∕e  ∕c c C¦ = C + k C CG CG 2 CG .  0  0  ∕c

Thus, the trim problem has become a linear system of two equations with two unknowns, which has unique solution. For what concerns the mathematical model, the three-surface conﬁguration can be now considered as a two-surface conﬁguration since no terms related to c are present.

3.7 Stick-Free Static Stability

This Section deals with the formulation of stick-free static stability assessment for a three- surface aircraft. If reversible control are installed, even if servo-assisted, the pilot would feel a stick force proportional to the elevator hinge moment. However, since two control surfaces are installed, there is the need to establish a logic with which the pilot control action is transferred to the two elevators and vice versa. In this Section the stick-free static stability analysis is performed assuming the pilot control to be transferred to the tail elevator. The forward wing elevator moves according to the tail elevator deﬂection. This approach is based on the results of the optimal trim solution analysis of Section 3.3, for which the tail and forward wing elevator deﬂections feature a mutual linear relation.

3.7.1 Hinge Moment Coeﬃcient

The stick-free neutral point location estimation needs more information compared to the stick-ﬁxed case. Indeed, an analysis of the control surface hinge moment must be conducted. t For what concerns the tail elevator, the hinge moment coeﬃcient CH may be written as shown in Equation (3.42):

t t t t t tab t C = C + C e + C + C H H t H∕ H tab e H0 (3.42) ∕ e ∕e Ct Ct Ct where H t , H∕ and H tab represent the moment coefficient derivatives with respect to ∕ e ∕e tab Ct the tail angle of attack, the elevator deflection and the trim tab deflection e whereas H0 t tab represents the hinge moment coefficient when , e and e are null. t It would be useful to refer to the aircraft angle of attack; therefore, we may write as function of as stated in Equation (2.9):

t = w 1 − " − " + it − iw = D∕ w D0 = + iw 1 − " − " + it − iw = D∕ D0 (3.43) = 1 − " − iw" − " + it. D∕ D∕ D0

38 CHAPTER 3. TRIMMED SOLUTIONS 3.7. STICK-FREE STATIC STABILITY

Ct t The term H∕ t is obtained by knowing the relation between and as shown in Equa- tion (3.43):

t CH Ct = ∕ . H t (3.44) ∕ 1 − " D∕

Equation (3.42) may now be written as:

t CH t ∕ w t t t tab t C = 1 − "D − i "D − "D + i + C e + C + C = H ∕ ∕ 0 H∕ H tab e H0 1 − " e ∕e D∕ t t t tab t t w t = C + C e + C + C + C −i "D − "D + i = H∕ H∕ H tab e H0 H t ∕ 0 e ∕e ∕ t t t tab ¡t = C + C e + C + C H∕ H∕ H tab e H e ∕e 0 (3.45) ¡t t t w t C = C + C −i "D − "D + i where H0 H0 H∕ t ∕ 0 . C¡t In order not to make the formulation too clumsy, by abuse of notation the term H0 will Ct be hereafter noted as H0 .

3.7.2 Stick-Free Lift and Moment Coeﬃcients

Since a stick-free flight condition has been considered, it would be useful to compute the trim ¨ lift and moment coefficient from which some characteristics as the C (the apostrophe CG∕ stands for stick-free condition) may be obtained. t Referring to Equation (3.45), the hinge moment coefficient CH is null since we are interested in a stick-free condition. The trim condition will be given by the three parameters tab , e and e which simultaneously cancel the hinge moment and satisfy the simplified trim equations: T L = W (3.46) CG = 0.

Since we are interested in a stick-free condition with respect to the tail elevator deﬂection we may use e from Equation (3.45) as a dependent variable: 0 1 1 t t tab t e = − CH + CH e + CH . t ∕ ∕tab 0 (3.47) CH e ∕e

Given Equations (2.17) and (2.29), e may be substituted. Equations (3.48) and (3.49) ¨ ¨ CL C C C show the result of this substitution. and CG are now called L and CG to highlight the fact that a stick-free condition is considered:

39 3.7. STICK-FREE STATIC STABILITY CHAPTER 3. TRIMMED SOLUTIONS

CL 0 1 ¨ ∕e t t tab t CL = CL − CH + CH e + CH + CL c + CL ∕ t ∕ ∕tab 0 ∕c 0 (3.48) CH e ∕e C CG 0 1 ¨ ∕e t t tab t C = C − CH + CH e + CH + C c + C . CG CG∕ t ∕ ∕tab 0 CG∕ CG0 CH e c ∕e (3.49) It is possible to extract from Equations (3.48) and (3.49) the derivatives with respect to : C Ct L∕ H C¨ = C − e ∕ L∕ L∕ t (3.50) CH ∕e C Ct CG H∕ C¨ = C − ∕ e . CG t (3.51) CG∕  ∕ CH ∕e

3.7.3 Stick-Free Neutral Point

As for the stick-ﬁxed case, the neutral point location is obtained by putting to zero the derivative with respect to of the moment coeﬃcient around the neutral point itself. Equation (3.51) may now be expanded by exploiting Equations (2.31a) and (2.31b) to highlight the center of gravity location xCG which will be replaced by xN¨ as reference point:

t t t t ¨ C V̄ ¨ C l L∕ H∕ C¨ = Cw (1 + F ) w −tV̄ tCt 1 − " +cV̄ cCc 1 + " + e . L L D∕ L U∕ t CG∕ ∕ ̄c ∕ ∕ CH ∕e (3.52) ¨ ̄ ¨t Stl¨ ¨ Remembering that lw may be written as xACw − xCG and V as t∕S ̄c where lt can be ¨ expressed as xCG − xACt , the term C may be put to 0 by considering the stick-ﬁxed CG∕ ¨ neutral point N instead of the center of gravity:

x w − x ¨ C = Cw (1 + F ) AC N − tV̄ tCt 1 − " + N¨ L L D∕ ∕ ∕ ̄c ∕ tStCt Ct L H (3.53) c ̄ c c ∕e ∕ + V C 1 + "U + xN¨ − x t = 0. L∕ ∕ t AC S ̄cCH ∕e

The non-dimensional location of the stick-free neutral point N¨ may be now obtained: ttCt Ct ⎡ L H w ∕e ∕ t ̄ t t N¨ = ⎢C (1 + F ) ACw − ACt − V C 1 − "D L∕ t L∕ ∕ ⎢ CH ⎣ ∕e −1 (3.54) ttCt Ct ⎡ L H ⎤ c ̄ c c w ∕e ∕ + V C 1 + "U ⋅ ⎢C (1 + F ) − ⎥ . L∕ ∕ L∕ t ⎢ CH ⎥ ⎣ ∕e ⎦

40 CHAPTER 3. TRIMMED SOLUTIONS 3.7. STICK-FREE STATIC STABILITY

The non-dimensional stick-free static margin is computed as CG − N¨ and it is associated with a stick-free statically stable aircraft when its value is positive. ¨ As for the stick-ﬁxed case, we may think to express C as function of the static CG∕ ¨ w C C (1 + F ) ACw margin and L∕ . To do so, we can isolate the term L∕ from Equation (3.54) (in the stick ﬁxed case only the term ACw was isolated):

t t t t t t t t ⎡ CL CH ⎤ CL CH w w ∕e ∕ ∕e ∕ C (1 + F ) ACw = N¨ ⎢C (1 + F ) − ⎥ + ACt + L∕ L∕ t t ⎢ CH ⎥ CH ⎣ ∕e ⎦ ∕e t ̄ t t c ̄ c c + V C 1 − "D − V C 1 + "U . L∕ ∕ L∕ ∕ (3.55)

l¨ w = xACw −xCG This term is now substituted in Equation (3.52) by expanding the quantity ̄c ̄c :

t t t t t t t t ⎡ CL CH ⎤ CL CH ¨ w ∕e ∕ ∕e ∕ C = N ⎢CL (1 + F ) − t ⎥ + ACt t + CG∕ ∕ ⎢ CH ⎥ CH ⎣ ∕e ⎦ ∕e t ̄ t t c ̄ c c w + V CL 1 − "D − V CL 1 + "U − CL (1 + F ) CG+ ∕ ∕ ∕ ∕ ∕ ttCt Ct L H t ̄ t t c ̄ c c ∕e ∕ ̄ ¨t − V C 1 − "D + V C 1 + "U + V = L∕ ∕ L∕ ∕ t CH ∕e t t t t t t t t ⎡ CL CH ⎤ CL CH w ∕e ∕ ∕e ∕ = N¨ ⎢C (1 + F ) − ⎥ + ACt + L∕ t t ⎢ CH ⎥ CH ⎣ ∕e ⎦ ∕e t t t t CL CH l¨ w ∕e ∕ t − C (1 + F ) CG + = L∕ t ̄c CH ∕e t t t t t t t ¨t ¨ (3.56) ⎡ CL CH ⎤ CL ¨CH w ∕e ∕ ¨¨∕e ∕ = N¨ ⎢CL (1 + F ) − ⎥ + ACt ¨ + ∕ Ct ¨ Ct ⎢ H ⎥ ¨ H ⎣ ∕e ⎦ ¨ ∕e t t t t CL CH w ∕e ∕ ¨¨ − C (1 + F ) + − ACt = L∕ CG t CG ¨ CH ∕e t t t t ⎡ CL CH ⎤ w ∕e ∕ = ⎢C (1 + F ) − ⎥ N¨ − = L∕ t CG ⎢ CH ⎥ ⎣ ∕e ⎦ t ⎛ CL C ⎞ ∕e H∕ = ⎜C − ⎟ ¨ − = L∕ t N CG ⎜ CH ⎟ ⎝ ∕e ⎠ ¨ = C ¨ − . L∕ N CG

It is evident that the expression obtained in Equation (3.56) is analogous to the one of Equation (2.36) for the stick-ﬁxed condition.

41 3.7. STICK-FREE STATIC STABILITY CHAPTER 3. TRIMMED SOLUTIONS

3.7.4 Borri Formulation - Dependent Elevator Deﬂections

By analogy with the approach which concerned the study of stick-free static stability of Section 3.7, the results of the optimal trim solution analysis of Section 3.3 may be exploited in order to link the deflections of the two distinct elevators. As stated in Equation (3.41), the trim problem of such a three-surface aircraft is reduced to a square 2 × 2 linear system. Therefore, the three-surface aircraft is considered, from a mathematical point of view, as a two-surface aircraft. For this reason a single control point may be defined, contrary to what said in the previous Section. In this particular case, the approach shown in Section 2.7.1 is valid. No peculiar considerations are needed with regard to the neutral point, since the optimal trim solution has no effect on the derivatives CL C with respect to : ∕ and CG∕ ; therefore, the neutral point location may be obtained as stated in Equation (2.68).

Control Point

Since the forward wing elevator deﬂection is linked to the horizontal tail elevator deﬂection as follows:

c = k1e + k2 (3.57) the trim problem loses c as variable and the derivative of the lift coefficient and moment e CL C coefficient with respect to is re-defined, as shown in Section 3.6. 0 and CG0 are re-defined as well.

£ CL = CL + k CL ∕e ∕e 1 ∕c (3.58a) C¦ = C + k C CG CG 1 CG (3.58b)  ∕e  ∕e  ∕c

CL = CL + k CL 0 0 2 ∕c (3.58c) C¦ = C + k C . CG CG 2 CG (3.58d)  0  0  ∕c

The control point non-dimensional location is obtained through Equation (2.58). The moment coeﬃcient must be considered around the center of gravity, since the trim condition has been imposed.

C¦ C + k C CG CG 1 CG ∕e ∕e ∕c C − CG = = . (3.59) £ CL + k CL CL ∕e 1 ∕c ∕e

By expanding the terms in Equation (3.59), C is obtained:

42 CHAPTER 3. TRIMMED SOLUTIONS 3.7. STICK-FREE STATIC STABILITY

−tV̄ ¨tCt + k cV̄ ¨cCc L 1 L = + ∕ e ∕ c = C CG t t t c c c (3.60) CL + k1 CL ∕e ∕c t t t c c c − − ACt C + k ACc − C CG L 1 CG L = + ∕ e ∕ c = CG t t t c c c (3.61) CL + k1 CL ∕e ∕c t t t c c c CL k1 CL ∕e ∕c = t + c . t t t c c c AC t t t c c c AC (3.62) CL + k1 CL CL + k1 CL ∕e ∕c ∕e ∕c

It may be interesting to know whether the control point lies ahead or behind the horizontal tail aerodynamic center, since for a two-surface aircraft such point coincides with the control point location. The inequation we are interested in is the following:

t t t c c c CL k1 CL ∕e ∕c t + c < t . t t t c c c AC t t t c c c AC AC (3.63) CL + k1 CL CL + k1 CL ∕e ∕c ∕e ∕c

By re-writing Equation (3.63) the following statement is obtained: c c c k1 CL ∕c c − t < 0. t t t c c c AC AC (3.64) CL + k1 CL ∕e ∕c

ó c c c ó ó c c c ó Since ACc − ACt > 0, k1 < 0 and in general ó CL ó > ók1 CL ó (since ó ∕c ó ó ∕c ó ó ó ó ó k1 value is most of the times −0.3 ÷ −0.1), the fractional term is negative. This means that the inequality is veriﬁed and therefore the control point lies behind the horizontal tail aerodynamic center. Intuitively, this means that the three-surface aircraft may be thought as a two-surface aircraft with an "equivalent" horizontal tail located at the control point, which has larger arm than the horizontal tail itself placed at its original aerodynamic center position. Such larger arm represents the eﬀect of the additional controllability provided by the forward wing.

43 3.7. STICK-FREE STATIC STABILITY CHAPTER 3. TRIMMED SOLUTIONS

44 Chapter 4

Revision of an Existing Aircraft Conﬁguration

Chapter 2 and 3 discuss in detail a model for the flight mechanics of a three-surface aircraft and the peculiar features of the relative trim solutions while Chapter 4.3 shows a possible way to set up a preliminary empennage sizing process. This Chapter shows some applications of the three-surface model and of the optimal trim solution in order to compare the characteristics of aircraft which present different configurations. The aim of this study is to verify if the original configuration of existing aircraft may be modified so as to improve some performance indices, by considering a three-surface solution as well. In the presented case, the term configuration stands for the arrangement and size of the horizontal empennage(s) and the longitudinal location of the wing. The title of this Chapter conveys the fact that the complete renovation of the aircraft component design (as for example new wing design) is not of interest. Instead, the focus is on a revision of the empennage size and location with respect to the wing. This kind of configuration renovation must be performed in such a way so as not to modify the capability of the aircraft in terms of performance requirements (e.g. takeoff and landing distance, service ceiling). Therefore, some crucial parameters as the wing loading and the thrust-to-weight ratio have not been altered. Such configuration revision shall be performed in such a way so as not to require any significant modification of the passenger seat and of the aircraft system arrangement. This constraint considers that such process has to be cost-efficient for the aircraft manufacturer.

4.1 Comparison Criteria of Aircraft Conﬁgurations

It is clear that no comparison of aircraft configurations is meaningful, unless some equiv- alence between the different cases is made. It is the same as comparing two different jet engines which belong to a different aircraft class: the comparison is possible if a quantity which allows the engines to be compared is made, as for example the Thrust Specific Fuel Consumption (TSFC). In this work, there is the need to set some constraints which allow to make a fair comparison between different aircraft configurations, in particular between two-surface "traditional", three-surface and two-surface canard configurations. For what concerns the flight mechanics, two of the major aspects the aircraft designers deal with are the Stability and Control (S&C) characteristics. Therefore, it is expectable that at least one attribute of both stability and control will be considered. In the present case, the

45 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT aircraft S&C features which will be investigated are related to the longitudinal plane only. Since longitudinal dynamic stability is in general linked to ﬂying and handling qualities and not to performance, only static stability has been considered; in particular, stick-ﬁxed static stability.

4.1.1 Longitudinal Static Stability

The longitudinal static stability characteristics are related to some aerodynamic and inertial C features of the aircraft which are translated in coefficients, as for example CG∕ . However, the most convenient quantity to be considered is the static margin. If a fixed value of the static margin for different aircraft is imposed, the same level of static stability is ensured for every different aircraft, no matter what configuration those aircraft feature.

4.1.2 Longitudinal Control

Generally speaking, there is no unequivocal quantity which represents longitudinal control level of an aircraft. Stabilizer and elevator size obviously determine the controllability of an aircraft, so as the distance of the control surface from the center of gravity. However, in aircraft design, when facing the control surface sizing (as recommended by authors like Roskam [20] and Gudmundsson [13]) a peculiar quantity is taken as reference to measure the controllability "level", the tail volume. Note that the tail volume plays a signiﬁcant role in longitudinal static stability as well, as shown in Equation (2.35); however, if the ratio between the elevator and stabilizer surface area is ﬁxed, the tail volume becomes a direct controllability level index.

4.2 Application to an Existing Design

This Section shows the method with which, from an existing design, different aircraft configurations (from conventional tailed to canard and the intermediate three-surface) are generated in order to compare performance characteristics. The first step of the method consists in imposing a forward wing surface area to an existing configuration. The aim is to compute the new wing longitudinal location and the new horizontal tail surface area for which the static margin and the total tail volume are the same as for the original aircraft. It is crucial to underline the fact that the distance between the horizontal tail and the forward wing is supposed to be fixed since it is related to the fuselage length (which is not altered so as not to modify the aircraft system and passenger seat disposition, as stated at the beginning of the Chapter). Once the forward wing area is imposed, the constraints of constant static margin and tail volume are considered in order to close a mathematical problem yielding. It is expected to obtain, as solution of the problem, the arrangement and the surface area of the three lifting surfaces. The mathematical problem is a system of two nonlinear equations in two t unknowns: the wing longitudinal location xACw and the horizontal tail surface area S . For the sake of clarity, the location of a lifting surface is intended as the longitudinal location of the relative aerodynamic center. xACw has been considered instead of xACw since in this case the fuselage and wing are considered as separate entities.

46 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

Constant Static Margin

The ﬁrst nonlinear equation is obtained by imposing a constant static margin. The static margin depends upon both the locations of the neutral point and the center of gravity. As far as the neutral point is concerned, Equation (2.35) shows the factors which aﬀect its longitudinal location: −tV̄ tCt 1 − " + cV̄ cCc 1 + " x x w L D∕ L U∕ N = AC + ∕ ∕ . ̄c ̄c Cw (1 + F ) (4.1) L∕ t The unknowns xACw and S are embedded in the following terms (the forward wing c surface area S is not a variable since it is imposed):

t c V̄ t = S lt V̄ c = S lc • S ̄c and S ̄c

• lt = xACw − xACt

• lc = xACc − xACw t c t CL c CL F = t S ∕ 1 − " + c S ∕ 1 + " • S Cw D∕ S Cw U∕ . L∕ L∕

The term related to the center of gravity depends upon the location and dimension of the lifting surfaces. By taking as a reference the center of gravity of the original aircraft, its shift is computed by considering the wing location and the empennage masses. Since the position of the empennages is ﬁxed, the shift of their respective centers of gravity, whenever their surface area is varied, is negligible. The "new" center of gravity position, due to the wing shift and due to variation of horizontal tail and forward wing masses, is computed as follows: MxCG − mw ¤xCGw − xCGw + ΔmtxCGt + ΔmcxCGc ¡xCG = (4.2) M + Δmt + Δmc where ¡xCG and xCG indicate the new and the old (i.e. before adding the forward plane) center of gravity longitudinal positions respectively, M the aircraft old mass, mw the wing mass, Δmt and Δmc the mass variation of the horizontal tail and forward wing respectively and xCGt and xCGc their center of gravity locations. t In Equation (4.2) no unknowns (i.e. xACw and S ) are apparently recognizable. How- ever, some hypotheses may be introduced in order to relate the quantities in Equation (4.2) to the unknowns of the problem. First of all, the empennage masses mt and mc depend upon their surface area. By exploiting the equation suggested by Roskam [21] the above-mentioned masses may be written as functions of the surface area:

0.2 3.81SempVD Wemp = KℎSemp √ (4.3) 1000 cos Λ where Wemp represents the horizontal empennage weight in lbf , Semp the empennage surface 2 area in ft , VD the dive speed in kn and Kℎ a coeﬃcient whose value is 1 for ﬁxed incidence stabilizers and 1.1 for variable incidence stabilizers.

47 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

Moreover, the distance between the center of gravity and the aerodynamic center of the lifting surfaces is needed, since the locations of the surfaces are taken with respect to the aerodynamic centers:

i xCGi = xACi + d (4.4) i where d indicates the distance between the center of gravity and the aerodynamic center of the i-th lifting surface. What has been done so far allows to write the ﬁrst nonlinear constraining equation in t the unknowns xACw and S :

N − CG = SM. (4.5)

Constant Total Tail Volume

The second constraining equation is obtained by imposing constant total tail volume: Stl Scl V̄ t + V̄ c = t + c = V̄ tot S ̄c S ̄c (4.6) ̄ tot where the unknown xACw is contained in lt and lc and V is a constant.

Resulting Nonlinear Problem

The nonlinear system is therefore the following: T N − CG = SM V̄ t + V̄ c = V̄ tot. (4.7) fsolve The algorithm employed to solve such system is the MATLAB® function. The solution obtained by solving the system is used to compute the characteristics of the aircraft in terms of aerodynamic coefficients as stated in the the three-surface model of Chapter 2, which is valid also for two-surface aircraft. By applying this process imposing different forward wing surface areas, a set of "equivalent" aircraft are obtained from the original one. This aircraft set includes all possible configurations from conventional two-surface to canard passing through all the intermediate three-surface. Once the above-mentioned configurations are obtained, their relative trim drag polars may be computed. For what concerns the conventional two-surface and the canard configurations the trim solutions are unique, while for the intermediate three-surface configurations the trim solutions are computed by employing the optimization method shown in Chapter 3. Once the geometry of all possible aircraft configurations has been obtained, the trim drag polars can be found. For the two-surface configurations (conventional and canard), the trim solutions have been computed by solving the trim problem as shown in Appendix D. The three-surface aircraft trim solutions have been obtained by employing the optimization method explained in Chapter 3. In order to obtain results for the three-surface cases, between the conventional and the canard, seven intermediate configurations have been considered. This choice represents a compromise between reliable results and reasonable computational

48 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

S c

2S 3S-1 3S-2 3S-3 3S-4 3S-5 3S-6 3S-7 2S Tailed Canard

Figure 4.1: Aircraft conﬁguration parting time. Therefore, a total of nine cases has been investigated. The above-mentioned division has been performed on the basis of the vector of the imposed forward wing surface area, as shown in Figure 4.1. 3S refers to a three-surface conﬁguration.

4.2.1 Application to Diamond DA42 Twin Star

The Aircraft

The Diamond DA42 Twin Star is a four seat, twin engine, propeller-driven airplane developed manufactured by Austrian company Diamond Aircraft Industries. It was Diamond’s first twin engine design, as well as the first new European twin-engine aircraft in its category to be developed in over 25 years. By 2012, the DA42 has become a key revenue generator for the company, having gained popularity with government and military operators in addition to the civil market that had suffered as a result of the Great Recession. Government cus- tomers have typically employed the type in the aerial surveillance role, which contributed towards the development of the Aeronautics Defense Dominator, a Medium-Altitude Long- Endurance (MALE) Unmanned Aerial Vehicle (UAV), which derived from the DA42. The data of the aircraft is shown in appendix B. A three-view of the aircraft is shown in Figure 4.2.

Generation of Diﬀerent Conﬁgurations

The computation which have been performed at this level follow the steps shown in Section 4.2. By fixing a set of forward wing surface areas, a set of different aircraft configurations are obtained. The lift coefficient has been set from 0.1 to 1 so as to consider an angle of attack range embedded in the linear section of the lift curve. The forward wing has been set from 0 (which corresponds to the original DA42 configuration) to a value for which the 2 horizontal tail surface area resulted 0 (canard aircraft), in this case 2.78 m . Figure 4.3 shows the horizontal tail surface area as function of the forward wing surface area. The intermediate solutions represent the three-surface configurations. Figure 4.4 shows the longitudinal position of the wing aerodynamic center as function of the imposed forward wing surface area. As expected, the wing shifts to the aft as long as the forward wing area is increased. In this way, the additional de-stabilizing effect of the forward wing is compensated. This is possible since the sensitivity of the neutral point shift on the wing position is much higher than the sensitivity of the center of gravity shift; therefore, if the wing is moved aft, the neutral point shift is larger than the one of the center of gravity, resulting in an increase of the static margin.

49 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

Figure 4.2: Diamond DA42 - Three-view [12]

50 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

2.5

1.5

0.5

0 0 0.5 1 1.5 2 2.5 3

Figure 4.3: Diamond DA42 - Horizontal empennage surface area

5.5

4.5

3.5

3 0 0.5 1 1.5 2 2.5 3

Figure 4.4: Diamond DA42 - Longitudinal wing location

t Once the solution vectors of S and xACw have been obtained, for each pair of the so- t lution S , xACw , or in other words, for each new aircraft conﬁguration, the remaining quantities are also obtained as a result: w • xCGw = xACw + d

• lt = xACw − xACt

51 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

0.2 0.8

0.1 0.6 0 0.4 -0.1

0.2 -0.2

-0.3 0 0 1 2 0 1 2

(a) Static margin (b) Total tail volume

Figure 4.5: Diamond DA42 - Constant quantities

• lc = xACc − xACt

t V̄ t = S lt • S ̄c

c V̄ c = S lc • S ̄c .

The new location of the aircraft center of gravity is computed from Equation (4.2) used in the non-linear problem: MxCG − mw ¤xCGw − xCGw + ΔmtxCGt + ΔmcxCGc ¡xCG = . (4.8) M + Δmt + Δmc

The new neutral point location is computed by using Equation (2.35): −tV̄ tCt 1 − " + cV̄ cCc 1 + " x x w L D∕ L U∕ N = AC + ∕ ∕ . ̄c ̄c Cw (1 + F ) (4.9) L∕

It is thus possible to plot some quantities as function of the forward wing surface area, since it plays the role of the independent variable. Figure 4.5 shows that the static margin and the total tail volume have been kept constant throughout the steps of the computation. The neutral point and center of gravity longitudinal locations are shown in Figure 4.6. The two curves show the same behavior (the vertical distance is kept constant) since the constraint of ﬁxed static margin has been imposed.

52 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

4.6

4.5

4.4

4.3

4.2

4.1

3.9 0 0.5 1 1.5 2 2.5 3

Figure 4.6: Diamond DA42 - Neutral point and center of gravity shift

Comparison of Performance

Once the geometry of all possible aircraft configurations have been obtained, the trim drag polars can be found. For the two-surface configurations (conventional and canard) the trim solutions have been computed by solving the trim problem as shown in Appendix D while for the three-surface aircraft trim solutions have been obtained by employing the optimization method explained in Chapter 3. The division of the possible intermediate three-surface configurations has been set as stated in Section 4.2. Figures 4.7, 4.8 and 4.9 show the trim solutions in terms of angle of attack and elevator deflections as functions of the imposed trim lift coefficient. It is noticeable that, for every configuration, the forward wing elevator deflection decreases as long as the trim lift coefficient is increased. This behavior, in the case of the canard solution, is not accepted by the civil certification authorities (e.g. FAA (FAR), EASA(CS)). For three-surface cases there is no issue since the maneuver can be performed by means of the tail elevator, whose behavior meets the certification requirements. The foreplane elevator may thus be adjusted so as to minimize the drag coefficient.

Once the trim solution have been computed, the trim drag polars may be found by using Equation (3.18) for each given trim lift coefficient, where the coefficients from A to I are known by fixing the geometry of the aircraft, as shown in Section 3.2.2:

2 2 2 CD = A + B + Ce + Dc + E e + F c + G + He + Ic. (4.10)

53 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.7: Diamond DA42 trim solution - Angle of attack

-5

-10

-15

-20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.8: Diamond DA42 trim solution - Tail elevator deﬂection

54 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

-5

-10

-15

-20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.9: Diamond DA42 trim solution - Forward wing elevator deﬂection

The trim drag polars are shown in Figure 4.10.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07

Figure 4.10: Diamond DA42 trim solution - Drag polars

Once the drag polars have been obtained, the required performance indices can be computed. Since the aircraft is propeller-driven, the significant performance indices which have 3∕2 to be considered are the CL∕CD ratio and CL ∕CD ratio, since their maximum value represent the condition of maximum range and maximum endurance respectively1. Given that we are interested in the maximum value of the above-mentioned performance indices, it is useful to show those maxima as function of the forward wing surface area. Thus, the comparison between the different configurations, from tailed to canard, is imme- diate. Figures 4.11 and 4.12 show such plots. A curve fitting has been performed so as to estimate the performance index for every intermediate configuration. A third order fit has

1This statement is valid if the available power is considered to be constant with the airspeed.

55 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT demonstrated to be the best in terms of square of the errors. Considerations about those results are shown in Section 4.2.3, in order to compare them with the second example.

15.8

15.6

15.4

15.2

14.8

14.6 0 0.5 1 1.5 2 2.5 3

Figure 4.11: Diamond DA42 - Maximum CL∕CD ratio

15.8

15.6

15.4

15.2

14.8

14.6 0 0.5 1 1.5 2 2.5 3

3∕2 Figure 4.12: Diamond DA42 - Maximum CL ∕CD ratio

56 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

4.2.2 Application to Modiﬁed Sagitta-Juliett

The Aircraft

The Sagitta-Juliett is an aircraft whose design has been developed (at preliminary level) in the course of "Aircraft Design" by a team composed of the undersigned and four colleagues [3]. The aircraft is a three-surface 6-seat business jet thought to be able to perform a no stop US coast-to-coast trip. The aircraft which has been used to conduct the hereafter presented study is an adaptation of Juliett. In particular, some factors as the lifting surface areas and locations have not been modified, while some features which had no relevance from the point of view of this thesis have been discarded. This choice allows not to make those factors affect the results which must depend only upon the flight mechanics characteristics of the aircraft, and not upon elements which have been developed to satisfy particular requirements (e.g. a proper sweep angle and the employment of supercritical airfoils to allow higher cruising speed. In our case those requirements are irrelevant). The aircraft data is shown in Appendix A. Note that the aircraft is the same as the one used for the validation of the trim optimization algorithm, Chapter 3. Figure 4.13 shows the original Juliett model (on the left) and its schematization Mod-Juliett with rectangular planform (on the right).

Figure 4.13: Juliett and Mod-Juliett top view

Generation of Diﬀerent Conﬁgurations

As for the case of the Diamond DA42 Twin Star, the computations performed at this level follow the steps shown in Section 4.2 for which a set of different aircraft configurations are generated by fixing a set of forward wing surface areas. The lift coefficient has been set from 0.1 to 1, since the formulation is valid for linear aerodynamics and it is expected, given the airfoil and the characteristics of the lifting surfaces, not to incur into non-linearities of the lift curve. The forward wing surface area has been set from 0 (conventional two-surface aircraft) to a value for which the solution if the non-linear system gives as output null horizontal tail surface area (canard aircraft). Figure 4.14 shows the horizontal tail surface area as function of the forward wing surface area. The intermediate solutions represent the three-surface configurations.

57 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

3.5

2.5

1.5

0.5

0 0 0.5 1 1.5 2 2.5 3

Figure 4.14: Mod-Juliett - Horizontal empennage surface area

Figure 4.15 shows the longitudinal position of the wing aerodynamic center as function of the imposed forward wing surface area. For the same reason as for the DA42, the wing shifts aft as long as the forward wing surface area is increased. The intersection of the curves in Figure 4.15 represents the design conﬁguration of the modiﬁed Juliett aircraft.

t Once the solution vectors of S and xACw have been obtained, for each pair of the so- t lution S , xACw , or in other words, for each new aircraft conﬁguration, the remaining ̄ ¨t ̄ ¨c quantities xCGw , lt, lc, V and V are computed as shown in Section 4.2.1.

It is thus possible to plot some quantities as function of the forward wing surface area, since it plays the role of the independent variable.

58 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

5.7

5.6

5.5

5.4

5.3

5.2

5.1

5 0 0.5 1 1.5 2 2.5 3

Figure 4.15: Mod-Juliett - Longitudinal wing location

Figure 4.16 shows that the static margin and the total tail volume have been kept constant throughout the steps of the computation.

0.2 0.27

0.265 0.15 0.26

0.1 0.255

0.25 0.05 0.245

0 0.24 0 1 2 0 1 2

(a) Static margin (b) Total tail volume

Figure 4.16: Mod-Juliett - Constant quantities

The neutral point and center of gravity longitudinal locations are shown in Figure 4.17. The two curves show the same behavior since the constraint of constant static margin has been imposed.

59 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

5.65

5.6

5.55

5.5

5.45

5.4

5.35

5.3 0 0.5 1 1.5 2 2.5 3

Figure 4.17: Mod-Juliett - Neutral point and center of gravity shift

Comparison of Performance

Once the geometry of all possible aircraft configurations have been obtained, the trim drag polars may be found. As for the Diamond DA42, for the two-surface configurations (conventional and canard) the trim solutions have been computed by solving the trim problem as shown in Appendix D and for the three-surface aircraft trim solutions have been obtained by employing the optimization method explained in Chapter 3. The division of the possible intermediate three-surface configurations has been set as for the case of the DA42. Fig- ures 4.18, 4.19 and 4.20 show the trim solutions in terms of angle of attack and elevator deflections as functions of the imposed trim lift coefficient.

60 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.18: Mod-Juliett trim solution - Angle of attack

-5

-10

-15

-20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.19: Mod-Juliett trim solution - Tail elevator deﬂection

61 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.20: Mod-Juliett trim solution - Forward wing elevator deﬂection

A particular behavior of the forward wing for the configurations three-surface 6-7 and the canard is noticeable. The increase of the angle of attack makes the upwash deflec- c tion angle to be greater. This causes the forward wing angle of attack to increase and consequently to increase the forward wing lift. For the specified cases, this force results to be more than what needed to balance the moments around the pitch axis. Therefore, the relative elevator deflection decreases if the trim lift coefficient (and so, the angle of attack) is increased. Once the trim solution have been computed, the trim drag polars may be found by using Equation (3.18) for each fixed trim lift coefficient, where the coefficients from A to I are known by fixing the geometry of the aircraft, as shown in Section 3.2.2:

2 2 2 CD = A + B + Ce + Dc + E e + F c + G + He + Ic. (4.11)

The drag polars are shown in Figure 4.21. Once the drag polars have been obtained, the required performance indices can be computed. Since the aircraft features jet√ engines, the performance indices which have to be considered are the CL∕CD ratio and CL∕CD ratio, since their maximum value represent the condition of maximum endurance and maximum range respectively2. Given that we are interested in the maximum value of the above-mentioned performance indices, it is useful to show those maxima as function of the forward wing surface area. Thus, the comparison between the different configurations, from tailed to canard, is imme- diate. Figures 4.22 and 4.23 show such plots. No curve fitting is needed since there is no stationary point to be identified.

2This statement is valid if the thrust is considered to be constant with the airspeed.

62 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.02 0.03 0.04 0.05 0.06 0.07

Figure 4.21: Mod-Juliett trim solution - Drag polars

17.2

16.8

16.6

16.4

16.2 0 0.5 1 1.5 2 2.5 3

Figure 4.22: Mod-Juliett - Maximum CL∕CD ratio

63 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

24.4

24.2

23.8

23.6

23.4 0 0.5 1 1.5 2 2.5 3

√ Figure 4.23: Mod-Juliett - Maximum CL∕CD ratio

4.2.3 Considerations on the Results

It is evident that the two cases present a substantial difference. For the Mod-Juliett model, the configuration which leads to best performance indices is the canard, while for the Dia- mond DA42 a three-surface solution results to be the more convenient. Such discrepancy is justified by the difference of original lifting surface arrangement of the two airplanes, as will be explained in the next paragraph.

Eﬀect of the Center of Gravity Location

The Diamond DA42 features two engines which are located on the wings, in the leading edge zone. In general the engine(s) position affects significantly the center of gravity, given the relatively large mass with respect to the total airplane, especially in utility aircraft. It is evident that airplanes with the engine mounted in the nose or on the wing leading edge, as the DA42, feature a forward-located wing, in the sense that the wing lies between the nose and the first half of the fuselage. Aircraft with aft engines (e.g. McDonnell Douglas MD-80, as well as conventional business jets) feature an aft wing, due to the fact that the airplane must meet the longitudinal static stability requirements. Figure 4.24 shows such circumstance. The last two airplanes feature tail engines and this makes the wing to be located in the aft section of the fuselage. Of course, the wing location may be affected by other factors which do not deal with flight mechanics issues (e.g. structural constraints or peculiar configuration requirements). The analysis presented in this Chapter allows to highlight a crucial matter which has been mentioned in Chapter 4.3. It is not a case that the canard solution is the best con- C ∕C figuration√ for the Mod-Juliett case only (for what concerns both maximum L D and CL∕CD). Indeed, as already explained, when generating all the possible configurations, the wing shifts aft as long as the given foreplane surface area is increased3. This situation is

3This behavior is due to the constraints which have been imposed: constant static margin and total tail volume.

64 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

(a) Beechcraft Kingair B200 [4] (b) Diamond DA42 Twin Star [11]

Figure 4.24: Examples of correlation between engine and wing location

65 4.2. APPLICATION CHAPTER 4. REVISION OF AN AIRCRAFT

0.7

0.6

0.5

0.4

0.3

0.2

0.1

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.25: Horizontal tail and forward wing lift coefficient beneficial for a canard configuration since the aft center of gravity makes the forward wing ¨ to have a larger arm lc. Therefore, a smaller surface area is needed for a given tail volume; thus, the parasitic drag is reduced. For what concerns the DA42, a pure canard configuration is not convenient since the forward center of gravity results in a smaller arm. As Figure 4.3 shows, the horizontal empennage surface area of the canard is larger than the one of the tail in the conventional configuration. This means that the benefit obtained in terms of reduction of induced drag are compensated and overcome by the drawback of the increase of parasitic drag. The best three-surface configuration presents a total empennage surface area which is larger than the one of the single empennage in the two two-surface solutions. This means that the drag coefficient reduction must be found in the induced component. By taking as reference the DA42 configuration 3S-4 (for which the performance indices 3∕2 CL∕CD and CL ∕CD are simultaneously maximized) and by analyzing the condition in which the horizontal tail and the forward wing work, it has been noticed that both generate t a positive lift (for almost all flight conditions). Figure 4.25 shows the lift coefficient CL and c ∗ CL generated by the two surfaces, for each given trim lift coefficient CL. On the contrary, the three-surface configurations generated from Mod-Juliett feature a negative lift from the horizontal tail. Indeed, the best solution results to be the only one with all the lifting surfaces whose lift is positive, the canard. The canard solution, for what concerns the Diamond DA42 case, has to be excluded in the first place since the foreplane elevator deflection decreases as long as the trim lift coefficient increases, as explained in Section 4.2.1. Note that the above-mentioned results are valid under the hypotheses on which the three-surface flight mechanics model is based. Moreover, since the study takes into account the flight mechanics only, no other constraints are involved. The Mod-Juliett case leads to a canard solution, but as explained in Chapter 1, the pure canard configuration has some drawbacks, including the static margin to travel a wide distance during the flight. This causes the aircraft to vary its stability and control characteristics in a significant way. For the DA42 case the change of configuration would not imply a complete revolution of the component arrangement; therefore, such revision shows to be applicable. However, a revision of avionic system arrangement (generally placed in the nose) may be needed. Since the foreplane is placed in the vicinity of the propeller blades, a further study about

66 CHAPTER 4. REVISION OF AN AIRCRAFT 4.2. APPLICATION

Figure 4.26: DA42 - Comparison between old and new conﬁguration the aerodynamic interaction between the two elements is required.

4.2.4 Conﬁguration Revision of the DA42

In this brief Section, the best configuration revision resulted from the computations of Sec- tion 4.2.1 (the three-surface configuration called 3S-4) is applied. A top view which represents such modification is shown in Figure 4.26. It has been taken into account that the forward wing gross area is significantly larger than the actual area. Therefore, the latter has c been chosen to represent S . The new performance of the aircraft in terms of range and endurance are computed, highlighting the enhancement with respect to the original aircraft.

Range and Endurance Performance

Before computing the performance improvement in terms of range and endurance, some 3∕2 considerations must be made. Since the performance indices CL∕CD and CL ∕CD depend upon the angle of attack, the cruise is considered to be performed at constant . More- over, the cruising altitude is considered not to vary as well, so as the Brake Speciﬁc Fuel Consumption (BSFC). The long range power condition is set at 50% while the maximum endurance power condition is set at 35%. [11] Given those hypotheses, the equation which gives the range of a propeller-driven aircraft, whose available power is considered to be constant with the airspeed, is the following4: 0 1 p CL 1  = ln (4.12) cP CD 1 −

t 4 = 2 V dt It is generally known as Breguet Equation for propeller-driven aircraft. It derives from:  ∫t1 .

67 4.3. PRELIMINARY DESIGN CHAPTER 4. REVISION OF AN AIRCRAFT

where p represents the propeller eﬃciency, cP the BSFC, and is the fraction of burned fuel with respect to the total aircraft weight. The collected data is the following:

• p = 0.8 (estimated by knowing the maximum range performance)

c = 0.51 lb • P hp h ([18]).

Note that, in order to use the Breguet formula, the BSFC must be expressed as fuel weight (not mass) consumption rate per unit of shaft power. To compute the burned fuel quantity is needed. is computed as:

WF = = 0.14 (4.13) W1 where W1 is the maximum takeoﬀ weight (16186 N [12]) and WF the weight of burned fuel. The tank capacity is 289 l [12] and the fuel which is employed is the Jet-A1 (which kg W has a density of 807 m3 [15]). F results to be 2276 N. Except CL∕CD, all the terms do not change if the new conﬁguration is considered (the weight variation is neglected). Therefore, the range increase may be computed as follows: 0 1 p CL 1 Δ = Δ ln . (4.14) cP CD 1 −

From Figure 4.11 it may be noticed that the CL∕CD ratio is increased from 14.47 to 15.65. The maximum range increases from 1165 nm to 1259 nm (8% increase). For what concerns the endurance performance, the Breguet formula is the following:

3∕2 v H I 2p CL S 1  = √ − 1 . (4.15) cP CD 2W1 1 −

As for the maximum√ range case, the endurance is a linear function of the performance index, in this case CL∕CD. Figure 4.12 shows that the index is increased from 14.82 to 15.62. The resulting maximum endurance is 12.4 hours (12 hours and 24 minutes), while the original endurance was 11.75 hours (11 hours and 45 minutes). This increase represents a 5.5% improvement.

4.3 Considerations on Preliminary Three-Surface Conﬁguration Design

In this Section a possible way to build a preliminary analysis of three-surface arrangement design is shown. This brief study does not deal with complete aircraft design but, instead, with the problem which appears when the horizontal empennage size and location has to be set. It is clear that other criteria may be valid as well; however, this approach shows to be versatile since it considers the pure mechanics only, or in other words, the usual aerodynamic and weight forces and related moments. This analysis proves the fact explained in Section 4.2.3, for which a relatively forward longitudinal position of the wing leads to trim solution with a positive lift generated by all the three surfaces.

68 CHAPTER 4. REVISION OF AN AIRCRAFT 4.3. PRELIMINARY DESIGN

The starting point of this analysis is the equilibrium condition of the aircraft intended as vertical force and pitching moment balance: T Lw + Lt + Lc = W w t c . (4.16) CG + CG + CG = 0 The pitching moments may be expressed as function of the lift of the relative surface as shown in Equation (4.17) (note that the moments are computed assuming a positive sign if pitching up). The reference model is shown in Figure 2.1.

w w Lwl¨ ⎧CG = AC + w ⎪ t = t − Ltl¨ ⎨CG AC t (4.17) ⎪ c = c + Lcl¨ . ⎩CG AC c The equilibrium equations in (4.16) may be written as: T Lw + Lt + Lc = W w t c Lwl¨ Ltl¨ Lcl¨ . (4.18) AC + AC + AC + w − t + c = 0 w t c By condensing the terms AC + AC + AC into AC, the equilibrium equations may be expressed in matrix form:

w 4 5 ⎡L ⎤ 4 5 1 1 1 t W ¨ ¨ ¨ ⎢ L ⎥ = . (4.19) lw −lt lc ⎢ c ⎥ −AC ⎣L ⎦ The ﬁrst matrix may be called M, the vector of the three surface lift l and the known term b. Equations (4.19) are thus written as follows:

Ml = b. (4.20) The Singular Value Decomposition (shown in Appendix C) may be applied to M. In this way the equilibrium problem shown in (4.20) is written as:

T UV l = b (4.21) T where U is a 2 × 2 matrix, a 2 × 3 matrix and V a 3 × 3 matrix. U and V are orthogonal matrices. The matrix V may be split into an essential space Vess (3 × 2 matrix) and into a null space vnull (3 × 1 vector): V = V v = v v v . ess null ess1 ess2 null (4.22) The null space does not aﬀect the problem since its components are multiplied by the null column of . Equation (4.21), since is diagonal with null third column, may be expressed as follows: 4 5 s1 0 T U V l = b (4.23) 0 s2

69 4.3. PRELIMINARY DESIGN CHAPTER 4. REVISION OF AN AIRCRAFT

where s1 and s2 are the terms of which are diﬀerent from 0. The term may be introduced as:

T = V l (4.24) hence:

l = V = Vessess + vnullnull (4.25) 4 5 ess where = (ess is a 2 × 1 vector and null is a scalar). null Thus, ess can be computed as: 4 5−1 s1 0 T ess = U b. (4.26) 0 s2

Note that if the value of null changes, the solution changes as well, even if it is still a feasible solution. In practice, by varying null it is possible to ﬁnd the possible combinations of lift of the three surfaces which satisfy the equilibrium equations in (4.18).

An application of this approach has been performed in order to show how the longitudinal position of the wing aﬀects the choice of the empennage size. A realistic three-surface utility aircraft geometry has been chosen in order to highlight what diﬀerences appear when the wing is shifted longitudinally. The goal of this analysis is to show that, in the environment of ab-initio aircraft design, it is possible to arrange the surfaces so as to obtain positive lift from all three components The data of the aircraft is shown in Table 4.1:

Table 4.1: Three-surface aircraft data

Wing Horizontal Tail Canard 2 S [m ] 10 1.5 1 AR 10 5 7 e (Oswald factor) 0.7 0.7 0.7 C deg−1 L∕ [ ] 0.1 0.1 0.1 C AC -0.025 -0.025 -0.025

The longitudinal position of the lifting surfaces has been referred to the center of gravity. The aerodynamic center of each surface is supposed to be coincident with the relative center of gravity. Two diﬀerent cases of wing location have been analyzed. Table 4.2 shows the lifting surface locations in both cases:

Table 4.2: Lifting surface location

xAC [m] Wing Horizontal Tail Canard CASE 1 0.3 -4 2 CASE 2 -1 -4 2

70 CHAPTER 4. REVISION OF AN AIRCRAFT 4.3. PRELIMINARY DESIGN

It worths noticing that the position of the center of gravity is considered to be unchanged in the two cases.

The analysis consists in ﬁnding all the feasible solutions of the problem in (4.18) in terms of lift of the three surfaces by varying the term null, which has the dimension of a force. The latter has been set from -7000 to 3000 N. Given the lift of the surfaces, the relative angle of attack may be computed. The empennages are considered to be fully movable surfaces (stabilators); therefore, the angle of attack of the single empennage is the composition of the aircraft angle of attack, the deﬂection angle and the wing downwash/upwash angle. A reference airspeed of 100 kn (51.4 m/s) has been chosen since the aircraft is a small utility model.

Loads on the Lifting Surfaces - Case 1

The ﬁrst case considers the forward location of the wing. The possible load conﬁgurations are shown in Figure 4.27. Figure 4.28 shows the angle of attack at which the surfaces operate.

8000

7000

6000

5000

4000

3000

2000

1000

-1000

-2000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000

Figure 4.27: Case 1 - Load conﬁgurations

71 4.3. PRELIMINARY DESIGN CHAPTER 4. REVISION OF AN AIRCRAFT

-5

-10 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000

Figure 4.28: Case 1 - Angle of attack

It is evident that for some feasible conﬁgurations (i.e. null approximately between −4000 and −2000 N) the three surfaces generate a positive lift. For null < −4000 N the horizontal tail works by means of negative lift whilst for null > −2000 N the loads on the surfaces are not realistic.

Loads on the Lifting Surfaces - Case 2

The ﬁrst case considers the aft location of the wing. The possible load conﬁgurations are shown in Figure 4.29. Figure 4.30 shows the angle of attack at which the surfaces operate.

8000

7000

6000

5000

4000

3000

2000

1000

-1000

-2000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000

Figure 4.29: Case 2 - Load conﬁgurations

72 CHAPTER 4. REVISION OF AN AIRCRAFT 4.3. PRELIMINARY DESIGN

-5

-10 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000

Figure 4.30: Case 2 - Angle of attack

Considerations

It worths noticing that, for the same values of null of case 1, the horizontal tail of case 2 generates downforce, or in other words, negative lift. This result is in accordance with what has been explained in Section 4.2.3: a forward located wing is beneﬁcial to obtain three surfaces which generate positive lift. To obtain three positive lifts in case 2, a value of null greater than −2000 must be chosen. However, Figure 4.29 shows that the load conﬁguration is not acceptable, since the forward wing is asked to generate half of the wing’s lift. This analysis may be useful to set a preliminary empennage sizing since the surface area of the three lifting surfaces can be estimated from the lift required by the equilibrium condition as shown in Figures 4.27 and 4.29.

73 4.3. PRELIMINARY DESIGN CHAPTER 4. REVISION OF AN AIRCRAFT

74 Chapter 5

Conclusions

As stated in the introduction, the main goal of this thesis is to show that the three-surface configuration is not a solution that can just be rejected outright simply out but, on the contrary, an arrangement which may offer some advantages in terms of trim drag polar, or in other words, in terms of range and endurance performance. The key point of the work is represented by the three-surface model and by the optimal trim solution study, which exploits the additional trim degree of freedom offered by the foreplane elevator. The three-surface model may be considered as an extension of the two-surface model taught in flight mechanics courses. Indeed, the latter can be obtained from the first by simply neglecting the terms related to the forward wing. The study performed on the application of a three-surface configuration to existing aircraft of Chapter 4 are used as proof to show that this non-conventional solution is actually applicable. This justifies the flight mechanics model and the optimal trim algorithm to be developed. Such analysis has shown that, for what concerns the revision of existing designs, the wing longitudinal location plays a crucial role. Indeed, the fact that an additional lifting surface (foreplane) may be beneficial depends upon its relative distance from the wing aerodynamic center and from the aircraft center of gravity. The preliminary analysis of three-surface arrangement design of Chapter 4.3 which, on the contrary, does not start by referring to existing designs, may be the starting point for future thesis work, whose main topic may be the consolidation of three-surface aircraft design methods. I would once more like to underline that the numerical results of this work are all products of models which are based upon some hypotheses; therefore, all the comments and conclusions must be always related to those assumptions. I think that, in addition to the development of guidelines for three-surface aircraft, some deepening about the flight dynamics of such kind of lifting surface arrangement is required. Indeed, it would be crucial to study the effect of the introduction of a forward wing on the dynamic behavior of an airplane since such studies may lead to some constraints on empennage sizing which could not be identified when considering the flight mechanics only. Some civil certification requirements may represent an active constraint which must be taken into account in the design process. Some other constraints may result from other aspects such as the structure sizing or the aerodynamics (e.g. aerodynamic interaction between the foreplane and other components).

75 CHAPTER 5. CONCLUSIONS

76 Bibliography

[1] Ira Abbott, Albert Doenhoﬀ, and Louis Stivers. Report No. 824, Summary of Airfoil Data. Technical Report. National Advisory Committee for Aeronautics, 1945. http://airfoiltools.com [2] Airfoil Tools and Applications. 2018. URL: . [3] Andrea Bavetta et al. Aircraft Design - Sagitta. Technical Report. Dipartimento di Scienze e Tecnologie Aerospaziali- Politecnico di Milano, 2017. https : / / picturelights . club / [4] Beechcraft Kingair B200 Image. 2018. URL: galleries/three-view-aircraft-drawings.html . http://www.b737.org.uk/737ng.htm#737- [5] Boeing 737-800 Images. 1999. URL: 800 . http://flyjetz.mobi/midsize_ [6] Bombardier Learjet 45 XR Image. 2018. URL: files/learjet45XR03.jpg . [7] Marco Borri and Lorenzo Trainelli. “A Simple Framework for the Study of Airplane Equilibrium and Stability”. In: AIAA paper no. 2003-5620, AIAA Atmospheric Flight Mechanics Conference and Hexhibit, Austin, USA (2003). [8] Marco Borri and Lorenzo Trainelli. “A Simple Framework for the Study of Airplane Equilibrium and Stability Revisited”. In: International Conference on Aeronautical Science and Air Transportation (ICASAT), Tripoli, Libia (2007). [9] Marco Borri and Lorenzo Trainelli. “Airplane Equilibrium and Stability: a Simpliﬁed Teaching Approach”. In: XVII AIDAA National Conference, Roma, Italia (2003). [10] Crane and Dale. Dictionary of Aeronautical Terms. Aviation Supplies and Academics, 1997. https://www.diamondaircraft.com/ [11] Diamond DA42 Twin Star. 2018. URL: aircraft/da42 . [12] Diamond DA42 Twin Star - Airplane Flight Manual - Rev 7. 2012. [13] Snorri Gudmundsson. General Aviation Aircraft Design: Applied Methods and Pro- cedures. Butterworth-Heinemann, 2013. [14] B Gunston et al. IHS Jane’s All the World’s Aircraft 2015-2016: Development & Production. 2015. https://www.exxonmobil.com/en/aviation/products- [15] Jet Fuel. 2018. URL: and-services/products/jet-a-jet-a-1 . https://www.mathworks.com/ [16] MathWorks. MATLAB - MathWorks. 2017. URL: products/matlab.html .

77 BIBLIOGRAPHY BIBLIOGRAPHY

http://2.bp.blogspot.com/- [17] McDonnell Douglas MD-80 Image. 2018. URL: Y45SnwbWVVA/Tob1aqdd6JI/AAAAAAAAARo/u6RxNUtojL4/s1600/MD80_Super98_ phase_1.jpg . [18] Luca Piancastelli, Leonardo Frizziero, and Giampiero Donnici. Common Rail Diesel - Automotive to Aerial Vehicle Conversions: An Update (Part II). Technical Manual. ARPN Journal of Engineering and Applied Sciences, 2015. [19] Jan Roskam. Airplane Design - Volume II. DARcorporation, 1985. [20] Jan Roskam. Airplane Design - Volume III. DARcorporation, 1985. [21] Jan Roskam. Airplane Design - Volume V. DARcorporation, 1985. https://en.wikipedia.org/wiki/ [22] Singular-Value Decomposition. 2018. URL: Singular-value_decomposition . [23] Walter Guido Vincenti. What Engineers Know and How They Know It. Johns Hopkins University Press, 1990.

78 Appendix A

Modiﬁed Juliett Aircraft Data

In this Section the data of the aircraft which has been taken as reference for the computations of Chapter 3, Section 3.4 and Chapter 4, Section 4.2.2 is shown. The aircraft features 3 lifting surfaces with rectangular planform. The aircraft derives from a 6-seat forward-swept wing aircraft of the Sagitta family, developed in the course "Aircraft Design". [3] The reference x-axis origin is set on the horizontal tail aerodynamic center, so the distance data is referred with respect to that point. Since the lifting surface feature a rectangular planform, their center of gravity is supposed to be at half chord, while the aerodynamic center is supposed to be at quarter of the chord.

Airfoil Data

The airfoil data has been taken from [1]. The main wing features a NACA 651-212 while the horizontal tail and the forward wing feature a NACA 1410. The main airfoil characteristics are shown in Table A.1.

Table A.1: Airfoil data

C C C Airfoil l∕ d0 AC

NACA 651-212 0.10 0.0040 -0.030 NACA 1410 0.11 0.0055 -0.020

Surface Geometric Data

Table A.2 shows the geometric data of the lifting surfaces.

Table A.2: Surface geometric data

Lifting Surface Surface Area [m2] Span [m] Mean Aerodynamic Chord [m] Aspect Ratio Wing 32.8 16.20 2.02 8 Horizontal Tail 2.2 2.97 0.74 4 Forward Wing 0.9 2.51 0.36 7

79 APPENDIX A. MODIFIED JULIETT AIRCRAFT DATA

Aircraft Geometric Data

The position of the surfaces are referred as the distance between their aerodynamic center and the horizontal tail aerodynamic center. According to the symbols introduced in Chapter 2, the data is the following:

• Interfocal distance: lt = 5.45 m lc = 5.58 m

• Surface position (aerodynamic center): xACw = 5.45 m xACt = 0 m xACc = 11.03 m

• Surface position (center of gravity): xCGw = 4.95 m xCGt = −0.19 m xCGc = 10.94 m

• Incidence angles: w i = 1 deg t i = −2 deg c i = 3 deg

• Center of gravity position: xCG = 5.58 m

• Maximum takeoﬀ mass: MTO = 5000 kg

Aerodynamic Data

The aerodynamic data of the single lifting surfaces has been computed by means of the formulas suggested in [13], Chapter 9, Section 5.3:

Cw = 0.08 -1 • L∕ deg

Ct = 0.07 -1 • L∕ deg

Cc = 0.08 -1 • L∕ deg

t -1 • CL = 0.04 deg ∕e

c -1 • CL = 0.04 deg ∕c Cw = 0.0202 • D0 Ct = 0.0069 • D0

80 APPENDIX A. MODIFIED JULIETT AIRCRAFT DATA

Cc = 0.0069 • D0

According to the formulas obtained from the three-surface model of Chapter 2, the aerodynamic coeﬃcients of the whole aircraft have been estimated:

C = 0.085 -1 • L∕ deg

CL = 0.003 -1 • ∕e deg

CL = 0.001 -1 • ∕c deg C = 0.071 • L0

C = −0.011 -1 • CG∕ deg

C = −0.0074 -1 • CG deg  ∕e

C = 0.0030 -1 • CG deg  ∕c C = −0.0054 • CG0

Other parameters have been estimated from [13], Chapter 9, Section 5.14:

• Oswald factor: ew = 0.8265 et = 0.7500 ec = 0.7500

• Parabolic drag polar coeﬃcient: kw = 0.048 kt = 0.106 kc = 0.061

" = 0.0075 • Downwash derivative: D∕ " = 0.0075 " • Upwash derivative: U∕ (assumed as the same as D∕ )

• Tail volume: V̄ t = 0.181 V̄ c = 0.076

• Neutral point: xN = 5.37 m

• Static margin: SM = 0.097 MAC

The drag coeﬃcient parameters have been computed as stated Section in 3.2.2:

• A = 0.0193

81 APPENDIX A. MODIFIED JULIETT AIRCRAFT DATA

−4 • B = 3.375 ⋅ 10 deg-2 −5 • C = 1.139 ⋅ 10 deg-2 −6 • D = 2.662 ⋅ 10 deg-2 −5 • E = 3.842 ⋅ 10 deg-2 −5 • F = 1.113 ⋅ 10 deg-2 −4 • G = 4.791 ⋅ 10 deg-1 −5 • H = −7.770 ⋅ 10 deg-1 −5 • I = 1.113 ⋅ 10 deg-1

82 Appendix B

Diamond DA42 Aircraft Data

In this Section the data of the Diamond DA42 Twin Star is shown. The aircraft is twin engine propeller-driven and features a conventional tailed (T-tail) conﬁguration. The airframe is full composite. The data has been taken from [14], the Aircraft Flight Manual (AFM) [12] and [11]. As for the previous aircraft data, the reference x-axis origin is set on the horizontal tail aerodynamic center, so the distance data is referred with respect to that point.

Airfoil Data

The airfoil characteristics have been obtained from the online database [2]. The wing features a WORTMANN FX 63-137. Since no data has been found about the horizontal tail airfoil, it has been supposed to be the same as the wing. The airfoil characteristics are shown in Table B.1.

Surface Geometric Data

Table B.2 shows the geometric data of the lifting surfaces.

Aircraft Geometric Data

The position of the surfaces are referred as the distance between their aerodynamic center and the horizontal tail aerodynamic center. Even if a two-surface aircraft is involved, the location of a hypothetical forward wing is included since in Chapter 4 a study which involves a change of conﬁguration is performed. According to the symbols introduced in Chapter 2, the data is the following:

• Interfocal distance: lt = 5.04 m

Table B.1: Airfoil data

C C C Airfoil l∕ d0 AC WORTMANN FX 63-137 0.10 0.0050 -0.020

83 APPENDIX B. DIAMOND DA42 AIRCRAFT DATA

Table B.2: Surface geometric data

Lifting Surface Surface Area [m2] Span [m] Mean Aerodynamic Chord [m] Aspect Ratio Wing 16.29 13.42 1.27 11.06 Horizontal Tail 2.43 3.29 0.66 3.7

lc = 2.34 m

• Surface position (aerodynamic center): xACw = 5.04 m xACt = 0 m xACc = 7.34 m

• Surface position (center of gravity): xCGw = 4.68 m xCGt = −0.16 m xCGc = 7.34 m

• Incidence angles: w i = 0 deg t i = −1.1 deg

• Center of gravity position: xCG = 4.72 m

• Maximum takeoﬀ mass: MTO = 1650 kg

Aerodynamic Data

The aerodynamic data of the single lifting surfaces has been computed by means of the formulas suggested in [13], Chapter 9:. For what concerns the drag coefficients, some error may be present since the zero-lift drag coefficient may be predicted accurately by CFD simulations or experimental activity. Nevertheless, the results in terms of comparison of different configurations do not loose validity, considering the fact that all the compared configurations will feature the same drag coefficient; this makes the error to be systematic and so, not significant.

Cw = 0.08 -1 • L∕ deg

Ct = 0.07 -1 • L∕ deg

Cc = 0.08 -1 • L∕ deg

t -1 • CL = 0.04 deg ∕e

c -1 • CL = 0.05 deg ∕c

84 APPENDIX B. DIAMOND DA42 AIRCRAFT DATA

Cw = 0.031 • D0 Ct = 0.010 • D0 Cc = 0.010 • D0

According to the formulas obtained from the three-surface model of Chapter 2, the aerodynamic coeﬃcients of the whole aircraft have been estimated:

C = 0.093 -1 • L∕ deg

CL = 0.006 -1 • ∕e deg C = 0.073 • L0

C = −0.017 -1 • CG∕ deg

C = −0.024 -1 • CG deg  ∕e C = 0.025 • CG0

Other parameters have been estimated from [13], Chapter 9, Section 5.14:

• Oswald factor: ew = 0.8265 et = 0.7500 ec = 0.8500

• Parabolic drag polar coeﬃcient: kw = 0.035 kt = 0.115 kc = 0.054

" = 0.0075 • Downwash derivative: D∕ " = 0.0075 " • Upwash derivative: U∕ (assumed as the same as D∕ )

• Tail volume: V̄ t = 0.592

• Neutral point: xN = 4.28 m

• Static margin: SM = 0.010 MAC (supposed)

The drag coeﬃcient parameters have been computed as stated Section in 3.2.2:

• A = 0.0319 −4 • B = 2.313 ⋅ 10 deg-2

85 APPENDIX B. DIAMOND DA42 AIRCRAFT DATA

−5 • C = 3.760 ⋅ 10 deg-2

• D = 0 −5 • E = 1.120 ⋅ 10 deg-2

• F = 0 −4 • G = −1.847 ⋅ 10 deg-1 −5 • H = −1.241 ⋅ 10 deg-1

• I = 0

86 Appendix C

Minimization of Drag Coeﬃcient Through SVD

C.1 Application of SVD on Trim Equations

In linear algebra, the Singular-Value Decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semideﬁnite m × n matrix. [22] T The SVD factorizes the m × n matrix in the product of three matrices UV where U is m × m, is a diagonal matrix m × n and V is n × n (for complex matrices the conjugate † T transpose V must be considered instead of the only transpose V ). This decomposition may be exploited in order to perform the computation of constrained minimization of the drag coeﬃcient, where the constraints are represented by the trim equations. Trim equations in (3.20) may be arranged in the following way:

L ∗ M L M C − C CL CL CL ⎡ ⎤ L L0 = ∕ ∕e ∕c −C C C C ⎢ e⎥ (C.1) CG CG CG CG ⎢ ⎥ 0 ∕ ∕e ∕c ⎣c⎦ X xT xT The matrix rows may be called 1 and 2 respectively. The vector of known terms will expressed as function of the above-mentioned vectors: L M L M C∗ − C xT L L0 = 1 −C xT (C.2) CG0 2

SVD xT 1 × 3 The is applied to 2 , which may be considered as a matrix:

vT ⎡ E ⎤ T xT = u 0 0 ⎢v ⎥ 2 2 2 ⎢ N1 ⎥ (C.3) ⎢vT ⎥ ⎣ N2 ⎦ vT essential space xT vT vT The vector E represents the of 2 while the vectors N1 and N2 rep- null space xT resent the of 2 .

87 C.2. MINIMIZATION OF THE DRAG COEFFICIENT APPENDIX C. SVD

xT If a linear combination of the vectors of the null space of 2 is taken, it is possible C to ﬁnd combinations of the variables , e and c which keep the CG constant. This is because the null space is multiplied by 0:

const = v + v C 1 N1 2 N2 (C.4) CG The variation of the pitching moment coeﬃcient around the center of gravity is therefore null:

ΔC = xT = 0 CG 2 (C.5) SVD xT The is then applied to 1 : wT ⎡ E ⎤ T xT = u 0 0 ⎢w ⎥ 1 1 1 ⎢ N1 ⎥ (C.6) ⎢wT ⎥ ⎣ N2 ⎦ C C For every trimmed flight CG must be 0; however, L may vary in such a way so as to allow flight at different equivalent airspeeds. C C The combinations of , e and c which keep the CG constant and make the L vary xT xT may be computed by projecting the essential space of 1 onto the null space of 2 : w = wT v v + wT v v + wT v v E E N1 N1 E N2 N2 E E E (C.7)

The ﬁrst two terms in Equation (C.7) represent the components of wE in the null space xT w C of 2 or, in other words, the components of E which alter the value of L without acting C residual on CG . The third term in Equation (C.7) is called and represents the component w xT C of E in the essential space of 2 . The residual is the term for which the value of CG changes if CL is altered. It is trivial that the trimmed ﬂight at all reasonable equivalent airspeeds is possible when the residual is zero. Equation (C.4) may be written in matrix form: 4 5 1 const = v + v = v v = V z C 1 N1 2 N2 N1 N2 N (C.8) CG 2

const C C represents the trim variables which will keep CG constant and will vary the CG  CL value in proportion to the relevance of the above-mentioned residual:

∗ T T T C − C = x const = x v + v = x V z L L0 C 1 N1 2 N2 N (C.9) 1 CG 1 1

C.2 Minimization of the Drag Coeﬃcient

Equation (3.18) shows the drag coeﬃcient CD as function of the trim variables:

2 2 2 CD = A + B + Ce + Dc + E e + F c + G + He + Ic (C.10) The matrix form of Equation (C.10) is given by Equation (3.22) and it is reported below:

88 APPENDIX C. SVD C.2. MINIMIZATION OF THE DRAG COEFFICIENT

⎡ ⎤ ⎡ B E∕2 F∕2⎤ ⎡ ⎤ E CD = A + GHI ⎢e⎥ + e c ⎢ ∕2 C 0 ⎥ ⎢e⎥ = ⎢ ⎥ ⎢F ⎥ ⎢ ⎥ (C.11) ⎣c⎦ ⎣ ∕2 0 D ⎦ ⎣c⎦ = A + mT + T K C By considering the trim solution which keeps the CG constant, the drag coeﬃcient is expressed as follows:

T T C A m const K const D = + C + Cconst C (C.12) CG CG CG By considering Equation (C.8), Equation (C.12) may be written as follows:

T T CD = A + m VNz + VNz K VNz (C.13) Since A is a constant, it is not considered in the minimization:

T T min m VNz + VNz K VNz (C.14) 1,2 T where z = 1 2 . The trim lift coeﬃcient is considered by Equation (C.9). This constraint is taken into account in the objective function J by means of a Lagrange multiplier :

T J = min mT V z + V z K V z + xT V z − C∗ − C N N N 1 N L L0 (C.15) 1,2

The derivatives with respect to z and are then computed:

T ⎧ )J T = mT V + 2V T KV z + xT V = 0 ⎪ )z N N N 1 N ⎨ )J (C.16) = xT V z − C∗ − C = 0 ⎪ ) 1 N L L0 ⎩ From the ﬁrst equation of (C.16), z is explicited:

1 −1 T 1 −1 T z = − V T KV xT V − V T KV mT V 2 N N 1 N 2 N N N 1 −1 1 −1 (C.17) = − V T KV V T x − V T KV V T m 2 N N N 1 2 N N N Equation (C.17) is substituted in the second equation of (C.16) in order to ﬁnd :

1 −1 1 −1 xT V − V T KV V T x − V T KV V T m − C∗ − C = 0 1 N 2 N N N 1 2 N N N L L0 (C.18) Hence,

1 −1 1 −1 − xT V V T KV V T x − xT V V T KV V T m − C∗ − C = 0 2 1 N N N N 1 2 1 N N N N L L0 (C.19)

89 C.2. MINIMIZATION OF THE DRAG COEFFICIENT APPENDIX C. SVD

−1 −1 1 xT V V T KV V T x 1 xT V V T KV V T m Note that the quantities 2 1 N N N N 1 and 2 1 N N N N are scalar. For the sake of clarity, those scalar quantities are named a and b respectively. Equation (C.19) takes a clearer form: − a − b − C∗ − C = 0 L L0 (C.20)

Hence, b + C∗ − C L L0 = − a (C.21) It worths noticing that a is not null since it derives from the projection of the essential xT xT space of 1 onto the null space of 2 . The optimal vector zopt may now be computed:

b + C∗ − C L L0 −1 1 −1 z = V T KV V T x − xT V V T KV V T m = opt 29a N N N 1 2 1 N N N N −1 b − C V T KV V T x L0 −1 1 −1 N N N 1 = V T KV V T x − V T KV V T m + C∗ = 2a N N N 1 2 N N N 2a L ∗ = zopt CL (C.22) ∗ which is function of the trim coeﬃcient CL. The optimal trim solution is computed from Equation (C.8):

opt = VNzopt (C.23)

It is evident that, since the components of zopt are linear functions of the trim lift coef- C∗ C∗ ﬁcient L, the optimal trim variables opt, eopt and copt are linear functions of L as well. Therefore, the optimal trim variables may be expressed as done in Equation (3.34):

∗ ∗ ⎧ opt = ZL + 1CL = opt CL ⎪ = + C∗ = C∗ ⎨ eopt eZL 2 L eopt L (C.24) ⎪ = + C∗ = C∗ ⎩ copt cZL 3 L copt L and the minimum value of the drag coeﬃcient is computed as done in Equation (3.35):

C = A + mT + T K Dmin opt opt opt (C.25)

90 Appendix D

Two-Surface Aircraft Trim Solution

In this Appendix, the solution of the trim problem for a two-surface aircraft is shown. The trim problem is represented by a system of two linear equations in two unknowns as shown in Equation (D.1). The solution is valid for both conventional and canard two-surface aircraft. The elevator deﬂection is noted as e for both cases.

⎧ ∗ C = CL + CL e + CL ⎪ L ∕ ∕e 0 ⎨0 = C + C + C (D.1) CG CG e CG ⎪ ∕ ∕e 0 ⎩ In matrix form, the trim equations are written as follows:

L M 4 5 L ∗ M CL CL C − C ∕ ∕e = L L0 C C −C (D.2) CG CG e CG  ∕  ∕e  0 The solutions are computed by solving the system:

C C −C C C ⎧ L∕ CG CG L0 CG = e 0 ∕e + ∕e C∗ ⎪ C C −C C C C −C C L L∕ CG L∕ CG L∕ CG L∕ CG ⎪ ∕e e ∕ ∕e e ∕ C C −C C C ⎨ CG L0 L∕ CG CG (D.3) = ∕ 0 − ∕ C∗ ⎪ e C C −C C C C −C C L L∕ CG L∕ CG L∕ CG L∕ CG ⎪ ∕e e ∕ ∕e e ∕ ⎩ ∗ The solutions and e are linear functions of the trim lift coeﬃcient CL.