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Aircraft Design: A Conceptual Approach Daniel P. Raymer President, Conceptual Research Corporation Sylmar, California

EDUCATION SERIES J. S. Przemieniecki Series Editor-in-Chief Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio

Published by American Institute of Aeronautics and Astronautics, Inc. 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

DEDICATION

This book is dedicated to all who taught me, especially Lester Hendrix, Richard Hibma, Louis Hecq, Harry Scott, Richard Child, George Owl, Robert Maier, Ed McGachan, Doug Robinson, Steve White, Harvey Hoge, Michael Robinson, George Palmer, Henry Yang, Robert Swaim, C. T. Sun, Dave Schmidt, Bruce Reese, William Heiser, and Gordon Raymer (test pilot, aeronautical engineer and my father). Thanks also to Rockwell North American Aircraft Operations for permission to use various illustrations. All other artwork is original, in the public domain, or copyrighted by AlAA. American Institute of Aeronautics and Astronautics, Inc., Washington, DC Library of Congress Cataloging-in-Publication Data

Raymer, Daniel P. Aircraft design:a conceptual approach/Daniel P. Raymer. p. cm.-(AIAA education series) Bibliography: p. Includes index. l. Airplanes-Design and construction. 1. American Institute of Aeronautics and Astronautics. II. Title. III. Series. TL67l.2.R29 1989 629.134' 1-dc20 89-14912 CIP ISBN 0-930403-51-7

Second Edition, Second Printing Copyright © 1992 by Daniel P. Raymer. Printed in the United States of America. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a data base or retrieval system, without prior written permission of the publisher.

DISCLAIMER: The Author and the AIAA do not guarantee the accuracy of the information provided in this book, and it should not be referenced as an authoritative source for aircraft design data or methods.

Texts Published in the AIAA Education Series Re-Entry Vehicle Dynamics Frank J. Regan, 1984 Aerothermodynamics of Gas Turbine and Rocket Propulsion Gordon C. Oates, 1984 Aerothermodynamics of Aircraft Engine Components Gordon C. Oates, Editor, 1985 Fundamentals of Aircraft Combat Survivability Analysis and Design Robert E. Ball, 1985 Intake Aerodynamics J. Seddon and E. L. Goldsmith, 1985 Composite Materials for Aircraft Structures Brian C. Hoskins and Alan A. Baker, Editors, 1986 Gasdynamics: Theory and Applications George Emanuel, 1986 Aircraft Engine Design Jack D. Mattingly, William Heiser, and Daniel H. Daley, 1987 An Introduction to the Mathematics and Methods of Astrodynamics Richard H. Battin, 1987 Radar Electronic Warfare August Golden Jr., 1988 Advanced Classical Thermodynamics George Emanuel, 1988 Aerothermodynamics of Gas Turbine and Rocket Propulsion, Revised and Enlarged Gordon C. Oates, 1988 Re-Entry Aerodynamics Wilbur L. Hankey, 1988 Mechanical Reliability: Theory, Models and Applications B. S. Dhillon, 1988 Aircraft Landing Gear Design: Principles and Practices Norman S. Currey, 1988 Gust Loads on Aircraft: Concepts and Applications Frederic M. Hoblit, 1988 Aircraft Design: A Conceptual Approach Daniel P. Raymer, 1989 Boundary Layers A. D. Young, 1989 Aircraft Propulsion Systems Technology and Design Gordon C. Oates, Editor, 1989

Basic Helicopter Aerodynamics J. Seddon, 1990 Introduction to Mathematical Methods in Defense Analyses J. S. Przemieniecki, 1990 Space Vehicle Design Michael D. Griffin and James R. French, 1991 Inlets for Supersonic Missiles John J. Mahoney, 1991 Defense Analyses Software J. S. Przemieniecki, 1991 Critical Technologies for National Defense Air Force Institute of Technology, 1991 Orbital Mechanics Vladimir A. Chobotov, 1991 Nonlinear Analysis of Shell Structures Anthony N. Palazotto and Scott T. Dennis, 1992 Optimization of Observation and Control Processes Veniamin V. Malyshev, Mihkail N. Krasilshikov, and Valeri I. Karlov, 1992 Aircraft Design: A Conceptual Approach Second Edition Daniel P. Raymer, 1992 Published by American Institute of Aeronautics and Astronautics, Inc., Washington, DC

FOREWORD As one of its major objectives, the AIAA Education Series is creating a comprehensive library of the established practices in aerospace design. Aircraft Design: A Conceptual Approach by Daniel P. Raymer provides an authoritative exposition of aircraft conceptual design. The great demand for the first edition of this new authoritative text on aircraft design has prompted the author to update and enlarge the text content into a second edition. In particular, Chapters 8 (Special Considerations in Configuration Layout), 13 (Propulsion), 17 (Performance and Flight Mechanics), and 21 (Conceptual Design Examples) have been extensively enlarged to cover some of the latest developments. The author's extensive experience with several aircraft companies supports the broad cross section of different views and approaches discussed in this comprehensive volume. This textbook offers aircraft designers, design managers, and design instructors an industry perspective on the new aircraft concept development process, which basically consists of two major activities: design layout and design analysis. The whole process is described in a very comprehensive manner, tailored to serve as a college design textbook. However, only an elementary knowledge of mathematics is required to make full use of the text, for the book focuses on industry design practice rather than theoretical definitions. A simplified but complete set of first-order analytical methods is presented. The text covers every phase of conceptual design: configuration layout, payload considerations, aerodynamics, propulsion, structure and loads, weights, stability and control, handling qualities, performance, cost analysis, tradeoff analysis, and many other topics. This latest text in the AIAA Education Series offers students, teachers, and practicing designers a unique source of information on current design practice in the U.S. aircraft industry-its science and art. To write a textbook on aircraft design is indeed a formidable task. Raymer has succeeded in creating a balanced text in which all the necessary topics needed to understand the design process are clearly described. For many years Aircraft Design: A Conceptual Approach will be a valuable textbook for all who struggle with the fundamentals and intricacies of aircraft design. J. S. PRZEMIENIECKI Editor-in-Chief

AIAA Education Series

TABLE OF

CONTENTS Author's Note ....................................................... xiii Chapter 1. 1.1

1.2

Design-A Separate Discipline What Is Design? ............................................ . Introduction to the Book .................................... .

Chapter 2. Overview of the Design Process 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 2.2 Phases of Aircraft Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 2.3 Aircraft Conceptual Design Process ............................ 7 Chapter 3. Sizing from a Conceptual Sketch 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2 Takeoff-Weight Buildup ..................................... 3.3 Empty-Weight Estimation ................................... 3.4 Fuel-Fraction Estimation .................................... 3.5 Takeoff-Weight Calculation .................................. 3.6 Design Example: ASW Aircraft ..............................

II II 12 14 23 24

Chapter 4. Airfoil and Geometry Selection 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2 Airfoil Selection ............................................ 4.3 Wing Geometry ............................................ 4.4 Biplane Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.5 Tail Geometry and Arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

33 33 47 65 67

Chapter 5. Thrust-to-Weight Ratio and Wing Loading 5.1 Introduction ............................................... 5.2 Thrust-to-Weight Ratio .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.3 Wing Loading .............................................. 5.4 Selection of Thrust-to-Weight and Wing Loading ...............

77 78 84 99

Chapter 6. Initial Sizing 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2 Rubber-Engine Sizing ...................................... 6.3 Fixed-Engine Sizing,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.4 Geometry Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.5 Control-Surface Sizing .....................................

101 102 108 109 113

Chapter 7. Configuration Layout and Loft 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 117 7.2 End Products of Configuration Layout .. . . . . . . . . . . . . . . . . . . . .. 117 7.3 Conic Lofting ............................................. 123 7.4 Conic Fuselage Development ................................ 129 Flat-Wrap Fuselage Lofting ................................. 135 7.5 7.6 Circle-to-Square Adapter ................................... 136 7.7 Fuselage Loft Verification .................................. 137 7.8 Wing/Tail Layout and Loft ................................. 139 7.9 Aircraft Layout Procedures ................................. 149 7 .10 Wetted Area Determination ................................. ISO 7.11 Volume Determination ..................................... 152

Chapter 8. Special Considerations in Configuration Layout 8.1 Introduction ............................................. . 155 8.2 Aerodynamic Considerations ............................... . 155 8.3 Structural Considerations .................................. . 158 8.4 Radar Detectability ....................................... . 165 8.5 In f rared Detecta bili ty ..................................... . 170 8.6 Visual Detectability ....................................... . 171 8.7 Aural Signature .......................................... . 171 8.8 Vulnerability Considerations ............................... . 172 8.9 Crashworthiness Considerations ............................ . 174 8.10 Producibility Considerations ............................... . 175 8.11 Maintainability Considerations ............................. . 179

Chapter 14. Structures and Loads 14.1 Introduction .............................................. 14.2 Loads Categories .......................................... 14.3 Air Loads ................................................ 14.4 Inertial Loads ............................................. 14.5 Power-Plant Loads ........................................ 14.6 Landing-Gear Loads ....................................... 14.7 Structures Fundamentals ................................... , 14.8 Material Selection ......................................... 14.9 Material Properties ........................................ 14.10 Structural-Analysis Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14.11 Finite-Element Structural Analysis ...........................

333 334 335 347 348 348 349 354 357 369 389

Chapter 9. Crew Station, Passengers, and Payload 9.1 Introduction ............................................. . 181 9.2 Crew Station ............................................. . 181 9.3 Passenger Compartment ................................... . 185 9.4 Cargo Provisions ......................................... . 186 9.5 Weapons Carriage ........................................ . 188 Gun Installation . .......................................... 191 9.6

Chapter 15. Weights Introduction ............................................. , 15.1 15.2 Approximate Group Weights Method ........................ 15.3 Statistical Group Weights Method ........................... 15.4 Additional Considerations in Weights Estimation ..............

395 399 399 407

Chapter 16. Stability, Control, and Handling Qualities 16.1 Introduction .............................................. 16.2 Coordinate Systems and Definitions. . . . . . . . . . . . . . . . . . . . . . . . .. 16.3 Longitudinal Static Stability and Control ..................... 16.4 Lateral-Directional Static Stability and Control ................ 16.5 Stick-Free Stability ......................................... 16.6 Effects of Flexibility ....................................... 16.7 Dynamic Stability ................................ , ......... 16.8 Quasi-Steady State ........................................ , 16.9 Inertia Coupling ......................................... " 16.10 Handling Qualities ........................................ ,

411 413 414 433 441 442 443 446 448 449

Chapter 17. Performance and Flight Mechanics 17.1 Introduction and Equations of Motion ....................... 17.2 Steady Level Flight ........................................ 17.3 Steady Climbing and Descending Flight .................... '" 17.4 Level Turning Flight ....................................... 17.5 Gliding Flight ............................................. 17.6 Energy-Maneuverability Methods ............................ 17.7 Operating Envelope ........................................ 17.8 Takeoff Analysis .......................................... 17.9 Landing Analysis .......... '" ............................. 17.10 Other Fighter Performance Measures of Merit .................

455 457 463 467 471 475 483 486 489 491

Chapter 18. Cost Analysis 18.1 Introduction .............................................. 18.2 Elements of Life-Cycle Cost ................................ 18.3 Cost-Estimating Methods ................................... 18.4 RDT&E and Production Costs .............................. 18.5 Operations and Maintenance Costs ........................... 18.6 Cost Measures of Merit (Military) ........................... 18.7 Airline Economics .........................................

501 503 505 506 510 514 514

Chapter 10. Propulsion and Fuel System Integration Introduction ............................................. . 10.1 10.2 Propulsion Selection ...................................... . 10.3 Jet-Engine Integration ..................................... . 10.4 Propeller-Engine Integration ............................... . 10.5 Fuel System .............................................. .

193 193 196 220 226

Chapter 11. Landing Gear and Subsystems 11.1 Introduction .. 11.2 Landing Gear A~;~~~~~~~~~ . : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.3 Tire Sizing ............................................... . 11.4 Shock Absorbers ......................................... . 11.5 Castoring-Wheel Geometry ................................ . 11.6 Gear-Retraction Geometry ................................. . 11. 7 Seaplanes ................................................ . 11.8 Subsystems .............................................. .

229 229 233 239 246 247 250 252

Chapter 12. Aerodynamics Introduction 12.1 12.2 Aerodynamic F~~~~~ 12.3 Aerodynamic Coefficients ................................. . 12.4 Lift 12.5 Para~i~~ '(Z'e~~~Liit) 'D~~~ 12.6 Drag Due to Lift (Induced Drag) ........................... . 12.7 Aerodynamic Codes and Computational Fluid Dynamics (CFD) ..

257 258 262 263 280 297 305

Chapter 13. Propulsion 13.1 Introduction ............................................. . 13.2 let-Engine Thrust Considerations ........ '" ................ . 13.3 Turbojet Installed Thrust .................................. . 13.4 Thrust-Drag Bookkeeping ................................. . 13.5 Installed-Thrust Methodology .............................. . 13.6 Piston-Engine Performance ................................ . 13.7 Turboprop Performance ................................... .

313 315 317 317 318 325 331

. :: : :: : : :: : :: : : : :: : :: : :: : :: : :: : :: : :: :: ::

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Chapter 19. Sizing and Trade Studies Introduction ............................................. . 19.1 19.2 Detailed Sizing Methods ................................... . 19.3 Improved Conceptual Sizing Methods ....................... . 19.4 Sizing Matrix and Carpet Plots ............................. . Trade Studies ............................................. 19.5

519 519 520 525 532

Chapter 20. VTOL Aircraft Design 20.1 Introduction ............................................. . 20.2 VTOL Terminology ....................................... . 20.3 Fundamental Problems of VTOL Design .................... . 20.4 VTOL Jet-Propulsion Options .............................. . 20.5 Vectoring-Nozzle Types ................................... . 20.6 Suckdown and Fountain Lift 20.7 Recirculation and Hot-Gas In~~;ti~~'::::::::::::::::::::::::: 20.8 VTOL Footprint .......................................... . 20.9 VTOL Control 20.10 VTOL ProPulsi~~ C~~~i~le~~ti~·~s· ........................... . ............................ 20.11 Weight Effects of VTOL .................................. . 20.12 Sizing Effects of VTOL ................................... .

537 538 538 541 547 551 552 553 554 555 556 557

Chapter 21. Conceptual Design Examples Introduction ............................................. . 559 21.1 Single-Seat Aerobatic 21.2 559 21.3 Lightweight Supercruis~ 'Fi~h~~; . : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 603 Appendix A.l A.2 A.3 A.4 A.5

A Conversion Tables . Standard Atmosphe~~ ~~ci ·Sh~~k T~ble~' : : : : : : : : : : : : : : : : : : : : : : Airfoil Data ............................................. . Typical Engine Performance Curves Design Requirements and Specificati~~~ . : : : : : : : : : : : : : : : : : : : : : :

658 660 687 717 731

References .......................................................... 735 Subject Index ....................................................... 739

AUTHOR'S NOTE There are two equally important aspects of aircraft design: design layout and design analysis. These very different activities attract different types of people. Some people love playing with numbers and computers, while others can't stop doodling on every piece of paper within reach. This book was written to fill a perceived need for a textbook in which both aircraft analysis and design layout are covered equally, and the interactions between these two aspects of design are explored in a manner consistent with industry practice. This book is not intended to be definitive on the subject of aircraft analysis. The analysis techniques presented are simplified to permit the student to experience the whole design process in a single course, including the key concepts of trade studies and aircraft optimization. No textbook can contain the methods actually used in industry, which tend to be proprietary and highly computerized. When the student goes into an industry or government design job, the more sophisticated methods of his or her chosen specialty will be better understood in the broader context of the whole of design as presented here. One key area in which this book differs from prior aircraft design books is in the chapters on aircraft configuration layout. The actual development of the aircraft design drawing is not a trivial task of drafting based upon the analysis results, but rather is a key element of the overall design process and ultimately determines the performance, weight, and cost of the aircraft. The ability to visualize and draw a new aircraft that has a streamlined aerodynamic shape, an efficient internal layout, yet satisfies an incredible number of real-world constraints and design specifications is a rare talent that takes years to cultivate. While to some extent good designers are "born, not made," a number of concepts and techniques in aircraft configuration layout can be taught, and are covered here. Writing this book has been an educating and humbling experience. It is my sincere wish that it help aspiring aircraft designers to "learn the ropes" more quickly. This second edition of AIRCRAFT DESIGN: A Conceptual Approach offers several new subjects, including production methods, post-stall maneuver, an update on VSTOL, and a brief introduction to engine cycle analysis. Also, typographical and technical errors from the first edition are corrected. A key difference in the second edition is Chapter 21, the Conceptual Design Examples. These are reworked to better serve as examples for the chapters of the book. The second example illustrates the use of RDS, a PC-based design, sizing and performance program now available from AIAA. RDS uses the methods in this book, and permits rapid design, analysis, and trade studies. AIAA and the author would like to thank the many people who have offered constructive suggestions for this second edition, as well as the more than 7000 students and working engineers who made the first edition an AIAA best seller. xiii

1 DESIGN-A SEPARATE DISCIPLINE 1.1 WHAT IS DESIGN? Aircraft design is a separate discipline of aeronautical engineeringdifferent from the analytical disciplines such as aerodynamics, structures, controls, and propulsion. An aircraft designer needs to be well versed in these and many other specialties, but will actually spend little time performing such analysis in all but the smallest companies. Instead, the designer's time is spent doing something called "design," creating the geometric description of a thing to be built. To the uninitiated, "design" looks a lot like "drafting" (or in the modern world, "computer-aided drafting"). The designer's product is a drawing, and the designer spends the day hunched over a drafting table or computer terminal. However, the designer's real work is mostly mental. If the designer is talented, there is a lot more than meets the eye on the drawing. A good aircraft design seems to miraculously glide through subsequent evaluations by specialists without major changes being required. Somehow, the landing gear fits, the fuel tanks are near the center of gravity, the structural members are simple and lightweight, the overall arrangement provides good aerodynamics, the engines install in a simple and clean fashion, and a host of similar detail seems to fall into place. This is no accident, but rather the product of a lot of knowledge and hard work by the designer. This book was written primarily to provide the basic tools and concepts required to produce good designs which will survive detailed analysis with minimal changes. Other key players participate in the design process. Design is not just the actual layout, but also the analytical processes used to determine what should be designed and how the design should be modified to better meet the requirements. In a small company, this may be done by the same individuals who do the layout design. In the larger companies, aircraft analysis is done by the sizing and performance specialists with the assistance of experts in aerodynamics, weights, propulsion, stability, and other technical specialties. In this book, the design layout techniques are discussed primarily in Chapters 4-11, while the analysis and optimization methods are presented in Chapters 12-19. Display model of an Advanced Supercruise Fighter Concept (Ref. 13). Pboto courtesy of Rockwell International Nortb American Aircraft Operations.

1.2 INTRODUCTION TO THE BOOK This book describes the process used to develop a credible aircraft conceptual design from a given set of requirements. As a part of the AIAA

4

2.2

AIRCRAFT DESIGN

OVERVIEW OF THE DESIGN PROCESS

PHASES OF AIRCRAFT DESIGN

Conceptual Design

Aircraft design can be broken into three major phases, as depicted in Fig. 2.2. Conceptual design is the primary focus of this book. It is in conceptual design that the basic questions of configuration arrangement size and weight, and performance are answered. ' The first question is, "Can an affordable aircraft be built that meets the requirements?" If not, the customer may wish to relax the requirements. Conceptual design is a very fluid process. New ideas and problems emerge as a design is investigated in ever-increasing detail. Each time the latest design is analyzed and sized, it must be redrawn to reflect the new g~OSS weight, fuel weight, wing size, engine size, and other changes. Early wmd-tunnel tests often reveal problems requiring some changes to the configuration. The steps of conceptual design are described later in more detail. Preliminary Design

Preliminary design can be said to begin when the major changes are over. The big questions such as whether to use a canard or an aft tail have been resolved. The configuration arrangement can be expected to remain about as shown on current drawings, although minor revisions may occur. At some P?int.Iate in preliminary design, even minor changes are stopped when a declSlon IS made to freeze the configuration.

REQUI REP1ENTS MILL IT WORk ?

CONCEPTUAL

NHAT I)(ES IT LOOK LIKE? NHAT REQUIRE~ENTS DRIVE THE DESIGN ?

DESIGN

NHAT TRADE-OFFS SHOlLD BE CONSIDERED?

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PRELIMINARY DESIGN

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( .. YOU-BET-YOUR-COI1PANY .• )

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DETAIL

DESIGN THE TOOLING AND FABRICATION PROCESS

DESIGN

TEST MAJOR ITEMS -

STRUCTURE,

LANDING GEAR,

[ FINALIZE WEIGHT AND PERFOR~NCE ESTI~TES FABRICATION

Fig. 2.2

Three phases of aircraft design.

5

During preliminary design the specialists in areas such as structures, landing gear, and control systems will design and analyze their portion of the aircraft. Testing is initiated in areas such as aerodynamics, propulsion, structures, and stability and control. A mockup may be constructed at this point. A key activity during preliminary design is "lofting." Lofting is the mathematical modeling of the outside skin of the aircraft with sufficient accuracy to insure proper fit between its different parts, even if they are designed by different designers and possibly fabricated in different locations. Lofting originated in shipyards and was originally done with long flexible rulers called "splines." This work was done in a loft over the shipyard; hence the name. The ultimate objective during preliminary design is to ready the company for the detail design stage, also called full-scale development. Thus, the end of preliminary design usually involves a full-scale development proposal. In today's environment, this can result in a situation jokingly referred to as "you-bet-your-company." The possible loss on an overrun contract or from lack of sales can exceed the net worth of the company! Preliminary design must establish confidence that the airplane can be built on time and at the estimated cost.

Detail Design

Assuming a favorable decision for entering full-scale development, the detail design phase begins in which the actual pieces to be fabricated are designed. For example, during conceptual and preliminary design the wing box will be designed and analyzed as a whole. During detail design, that whole will be broken down into individual ribs, spars, and skins, each of which must be separately designed and analyzed. Another important part of detail design is called production design. Specialists determine how the airplane will be fabricated, starting with the smallest and simplest subassemblies and building up to the final assembly process. Production designers frequently wish to modify the design for ease of manufacture; that can have a major impact on performance or weight. Compromises are inevitable, but the design must still meet the original requirements. It is interesting to note that in the Soviet Union, the production design is done by a completely different design bureau than the conceptual and preliminary design, resulting in superior producibility at some expense in performance and weight. During detail design, the testing effort intensifies. Actual structure of the aircraft is fabricated and tested. Control laws for the flight control system are tested on an "iron-bird" simulator, a detailed working model of the actuators and flight control surfaces. Flight simulators are developed and flown by both company and customer test-pilots. Detail design ends with fabrication of the aircraft. Frequently the fabrication begins on part of the aircraft before the entire detail-design effort is completed. Hopefully, changes to already-fabricated pieces can be avoided.

6

AIRCRAFT DESIGN

OVERVIEW OF THE DESIGN PROCESS

7

The further along a design progresses, the more people are involved. In fact, most of the engineers who go to work for a major aerospace company will work in preliminary or detail design.

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comes from the tendency of the rapid pressure rise across the shock to thicken or even separate the boundary layer. .. . A "supercritical" airfoil is one des~gned to. mll:l1~IZe t~ese effects. Mod~ ern computational methods allow desIgn of aIrfOIls In w~Ich the upp~r-su~ face shock is minimized or even eliminated by spreadlI~g the lIft In. t e chord wise direction, thus reducing the upper surface velocIty for a reqUIred total lift. This increases the Critical Mach Number. Design Lift Coefficient

For early conceptual design work, the designer must frequently rel~ UPho~ existing airfoils. From existing airfoils, the. o~e should be selecte t a .. . comes closest to having the desired charactenstlcs. The first consideration in initial airfoil ~election i.s th~ "deSIgn lIft c01~;; . t" This is the lift coefficient at WhICh the aIrfOIl has .the best ~~~~~n in Fig. 4.9 as the point on the airfo~l dragyolar that IS tangent to a .. . line from the origin and closest to the vertIcal axIS).. . flI· ht a well-designed airfoil operating at ItS desIgn lIft coeffII b k· f· r d The n su somc g cient has a drag coefficient that is little more th~n s I~- !IC Ion rag. h aircraft should be designed so that it flies the desI~n mI~s~on at or near t e design lift coefficient to maximize the aerodynamIc effI~Ienf~· ff. t As a first approximation, it can be assumed that the WIng 1 t coe IClen, C equals the airfoil lift coefficient, Ceo In level flight the lift m~s\~qua~ the w~ight, so the required design lift coefficient can be found as 0 ows. W = L = qSCL == qSCe

(4.4) (4.5)

(+)

Fig. 4.7

Laminar airfoil.

Dynamic pressure (q) is a function of velocity ~nd ~ltitude. !3~ assuming the . desIgn lIft.coeffIcIent can be . Ioad·Ing (W/S) as described later' . a WIng calculated for the velocity and altitude of the desIgn. mIssIOn.. . Note that the actual wing loading will decrease dunng the mISS!On as fuel is burned. Thus, to stay at the design lift coeffic~ent, the ~ynamIc press~re must be steadily reduced during the mission by eIther sloWIng down, WhICh

42

AIRCRAFT DESIGN

AIRFOIL AND GEOMETRY SELECTION LEADING EDGE

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A lower-surface upsweep of about 25 deg can be tolerated for a rear-loading transport aircraft provided that the fuselage lower corners are fairly sharp. This causes a vortex-flow pattern that reduces the drag penalty. In general, aft-fuselage upsweep should be minimized as much as possible, especially for high-speed aircraft. The importance of well-designed wing fillets has already been discussed. Fillets are especially important for low-wing, high-speed aircraft such as jet transports. These aircraft will frequently have a modified wing-root airfoil to further minimize fuselage interference and shock-induced drag increases. This modification takes the form of an uncambered or even negatively-cambered airf~il. set. at .a high positive angle of incidence. Design of such a wing modificatIOn IS beyond the scope of this book, but for layout purposes can b~ approximated by examining the wing of an existing, similar speed-class aircraft. "Base area" is any unfaired, rearward-facing blunt surface. Base area causes extremely high drag due to the low pressure experienced by the rearward-facing surface (see Chapter 12). However, a base area between or very near to the jet exhausts may be "filled-in" by the pressure field of the exhaust, partiall alleviating the drag penalty. The T-38 has such a base area between its nozzles. A base area fill-in effect is difficult to predict. The aerodynamic interaction between different components should be visualized in designing the aircraft. For example, a canard should not be located such that its wake might enter the engine inlets at any possible angle of attack. Wake ingestion can stall or even destroy a jet engine.

Fig. 8.2

Sears-Haack volume distribution.

If an aircraft's forebody has sharp lower corners, a separated vortex can be expected at high angle of attack. This could also be ingested by the inlets, with bad results. Also, such a vortex could unpredictably affect the wing or tail surfaces. For supersonic aircraft, the greatest aerodynamic impact upon the configuration layout results from the desire to minimize supersonic wave drag, a pressure drag due to the formation of shocks. This is ~nalytically related to the longitudinal change in the aircraft's total cross-sectIOnal area. In fact, wave drag is calculated using the second derivative (i.e., curvature) of the volume-distribution plot as shown in Fig. 7.36. Thus, a "good" volume distribution from a wave-drag viewpoint has the required total internal volume distributed longitudinally in a fashion t.hat minimizes curvature in the volume-distribution plot. Several mathematical solutions to this problem have been found for simple bodies-of-revolution, with the "Sears-Haack" body (Fig. 8.2-see Ref. 16) having the lowest wave drag. If an aircraft could be designed with a volume plot shaped like the Sears-Haack volume distribution it would have the minimum wave drag at Mach 1.0 for a given length and total internal volume. (What happens at higher Mach numbers is discussed in Chapter 12, but for initial layout purposes the minimization of wave drag at Mach 1.0 is a suitable goal in most cases.) However, it is usually impossible to exactly or even approximately match the Sears-Haack shape for a real aircraft. Fortunately, major drag reductions can be obtained simply by smoothing the volume distribution shape. As shown in Fig. 8.3, the main contributors to the cross-sectional area are the wing and the fuselage. A typical fuselage with a trapezoidal wing will have an irregularly-shaped volume distribution with the maximum crosssectional area located near the center of the wing. By "squeezing" the

158

AIRCRAFT DESIGN

SPECIAL CONSIDERATIONS

"SUPERSONIC AREA RULE"

'M-I.O)

CROSs-SECTI ON CROSS-SECTION

AREA

S/"IOOTHER AREA PROGRESS I ON

AREA

LOWER MAXIMUM CROsS-SECTION

FUSELASE

Fig. 8.3

Design for low wave drag.

fuselage at that point, the volume-distribution shape can be smoothed and the maximum cross-sectional area reduced. This design technique, developed by R. Whitcomb of the NACA (Ref. 17), is referred to as "area-ruling" or "coke-bottling" and can reduce the wave drag by as much as 50070. Note that the volume removed at the center of the fuselage must be provided elsewhere, either by lengthening the fuselage or by increasing its cross-sectional area in other places. While area-ruling was developed for minimization of supersonic drag, there is reason to believe that even low-speed aircraft can benefit from it to some extent. The airflow over the wing tends to separate toward the trailing edge. If an aircraft is designed such that the fuselage is increasing in cross-sectional area towards the wing trailing edge, this may "push" air onto the wing, thus reducing the tendency to separate. The Wittman Tailwind, which is remarkably efficient, uses this approach.

8.3 STRUCTURAL CONSIDERATIONS In most larger companies, the configuration designer is not ultimately responsible for the structural arrangement of the aircraft. That is the responsibility of the structural design group. However, a good configuration designer will consider the structural impacts of the general arrangement of the aircraft, and will in fact have at least an initial idea as to a workable structural arrangement. The primary concern in the development of a good structural arrangement is the provision of efficient "load paths"-the structural elements by which opposing forces are connected. The primary forces to be resolved are the lift of the wing and the opposing weight of the major parts of the aircraft, such as the engines and payload. the size and weight of the structural members will be minimized by locating these opposing forces near to each other.

159

Carried to the extreme, this leads to the Flying Wing concept. In a flying wing the lift and weight forces can be located at virtually the same place. In the ideal case, the weight of the aircraft would be distributed along the span of the wing exactly as the lift is distributed (Fig. 8.4). This is referred to as "spanloading," and eliminates the need for a heavy wing structure to carry the weight of the fuselage to the opposing lift force exerted by the wing. The structure can then be sized by lesser requirements such as the landing-gear loads. While ideal span-loading is rarely possible, the span-loading concept can be applied to more-conventional aircraft by spreading some of the heavy items such as engines out along the wing. This will yield noticeable weight savings, but must be balanced against the possible drag increase. If the opposing lift and weight forces cannot be located at the same place, then some structural path will be required to carry the load. The weight of structural members can be reduced by providing the shortest, straightest load path possible. Figure 8.5 illustrates a structural arrangement for a small fighter. The major fuselage loads are carried to the wing by "longerons," which are typically 1- or H-shaped extrusions running fore and aft and attached to the

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Fig. 8.4

Spanloading for weight reduction.

160

AIRCRAFT DESIGN

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skin. Longerons are heavy, and their weight should be minimized by designing the aircraft so that they are as straight as possible. For example, the lower longerons in Fig. 8.5 are high enough that they pass over the wing-carrythrough box. Had the longerons been placed lower, they would have required a kink to pass over the box. On the other hand, the purpose of the longeron is to prevent fuselage bending. This implies that the lightest longeron structure occurs when the upper and lower longerons are as far apart vertically as possible. In Fig. 8.6 the longerons are farther apart, but this requires a kink to pass over the box. Only a trade study can ultimately determine which approach is lighter for any particular aircraft. In some designs similar to Fig. 8.5 the lower longerons are placed near the bottom of the aircraft. A kink over the wing box is avoided by passing the longeron under or through the wing box. This minimizes weight but complicates both fabrication and repair of the aircraft. For aircraft such as transports, which have fewer cutouts and concentrated loads than a fighter, the fuselage will be constructed with a large number of "stringers" which are distributed around the circumference of the fuselage (Fig. 8.7). Weight is minimized when the stringers are all straight and uninterrupted. Another major structural element used to carry fuselage bending loads is the "keelson." This is like the keel on a boat, and is a large beam placed at the bottom of the fuselage as shown in Fig. 8.7. A keelson is frequently used to carry the fuselage bending loads through the portion of the lower fuselage which is cut up by the wheel wells. As the wing provides the lift force, load-path distances can be reduced by locating the heavy weight items as near to the wing as possible. Similarly, weight can be reduced by locating structural cutouts away from the wing. Required structural cutouts include the cockpit area and a variety of doors (passenger, weapons bay, landing gear, engine access, etc.). An especially poor arrangement (seen on some older fighter aircraft) has the main landing gear retracting into the wing-box area, which requires a large cutout where the loads are the greatest. When possible, structural cutouts should be avoided altogether. For example, a jet engine that is buried in the fuselage requires a cutout for the inlet, a cutout for the exhaust, and in most cases another cutout for removal

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Kinked lower longeron.

162

AIRCRAFT DESIGN

WING BOX CARRYTHROUGH

BENDING BEAM

Fig. 8.7

Structural concepts for fuselage loads. Fig. 8.8

of the engine. The re~ulting weight penalty compared to a podded engine mus.t be bala~ced against the reduced drag of a buried engine installation. FIgure 8.5 Illustrates another important concept in structural arrangement. Large conce~trated loads such as the wing and landing gear attachments must be car ned by a strong, heavy structural member such as a major fuselage bulkhead. The number of such heavy bulkheads can be minimized by arranging the aircraft so that the bulkheads each carry a number of concentrated loads, rather than requiring a separate bulkhead for each concentrated load. I~ Fig. 8:5 the two bulkheads in the aft fuselage carry the loads for the e~gines, t~Il~, a~d arresting hook. Had the tails and engine been located Wlt~out thIS III mind, the structural designer would have had to provide four or fIve heavy bulkheads rather than the two shown. Th7lift force on the wing produces a tremendous bending moment where ~he Wlt;Ig attaches to the fuselage. The means by which this bending moment IS carned ac~oss the f~selage is a key parameter in the structural arrangemen~, and WIll great~y influence both the structural weight and the aerodynamIC drag of the aIrcraft. Figure 8.8 illustrates the four major types of wing carry through structure. The "box ~a~rythr.ough" is virtually standard for high-speed transports and general-aViatIOn aIrcraft. The box carrythrough simply continues the wing box t.hrough the fuselage. The fuselage itself is not subjected to any of the bending moment of the wing, which minimizes fuselage weight. However, the box carrythrough occupies a su bstantial amount of fuselage volume, and tends to add cross-sectional area at the worst possible place for

163

SPECIAL CONSIDERATIONS

RING FRAMES

STRUT-BRACED

Wing carry through structure.

wave drag, as discussed above. Also, the box carry through interferes with the longeron load-paths~ The "ring-frame" approach relies upon large, heavy bulkheads to carry the bending moment through the fuselage. The wing panels are attached to fittings on the side of these fuselage bulkheads. While this approach is usually heavier from a structural viewpoint, the resulting drag reduction at high speeds has led to the use of this approach for most modern fighters. The "bending beam" carrythrough can be viewed as a compromise between these two approaches. Like the ring-frame approach, the wing panels are attached to the side of the fuselage to carry the lift forces. However, the bending moment is carried through the fuselage by one or several beams that connect the two wing panels. This approach has less of a fuselage volume increase than does the box-carrythrough approach. Many light aircraft and slower transport aircraft use an external strut to carry the bending moments. While this approach is probably the lightest of all, it obviously has a substantial drag penalty at higher speeds. Aircraft wings usually have the front spar at about 20-30070 of the chord back from the leading edge. The rear spar is usually at about the 60-75% chord location. Additional spars may be located between the front and rear spars forming a "multispar" structure. Multispar structure is typical for large or high-speed aircraft. If the wing skin over the spars is an integral part of the wing structure, a "wing box" is formed which in most cases provides the minimum weight.

AIRCRAFT DESIGN

164

SPECIAL CONSIDERATIONS

CARRY THROUGH BOX OR RING FRAMES

WING ATTACHMENTS

...+-- "KICK SPAR"

MAIN SPARS

Fig. 8.9

Typical wing box structure.

Aircraft with the landing gear in the wing will usually have the gear located aft of the wing box, with a single trailing-edge spar behind the gear to carry the flap loads, as shown in Fig. 8.9. Ribs carry the loads from the control surfaces, store stations, and landing gear to the spars and skins. A multispar wing box will have comparatively few ribs, located only where major loads occur. Another form of wing structure, the "multirib" or "stringer panel" box, has only two spars, plus a large number of spanwise stringers attached to the wing skins. Numerous ribs are used to maintain the shape of the box under bending. Variable sweep and folding capability add considerably to the wing structural weight. On the other hand, use of a delta wing will reduce the structural weight. These are further discussed in Chapter 15. First-order structural sizing will be discussed in Chapter 14. For initial layout purposes the designer must guess at the amount of clearance required for structure around the internal components. A good designer with a "calibrated eyeball" can prevent a lot of lost effort, for the aircraft may require substantial redesign if later structural analysis determines that more room is required for the structural members. A large airliner will typically require about 4 in. of clearance from the inner wall of the passenger compartment to the outer skin ("moldline"). The structure of a conventional fighter fuselage will typically require about 2 in. of offset from the moldline for internal components. For a small general aviation aircraft, 1 in. clearance or less may be acceptable. The type of internal component will affect the required clearance. A jet engine contained within an aluminum or composite fuselage will require

165

perhaps an additional inch of clearance to allow for a heat shield. The heat shield may be constructed of titanium, steel, or a heat-proof matting. On the other hand, an "integral" fuel tank in which the existing structure is simply sealed and filled with fuel will require no clearance other than the thickness of the skin. There is no easy formula for the estimation of structural clearance. The designer must use judgement acquired through experience. The best way to gain this judgement other than actual design experience is by looking at existing designs.

8.4 RADAR DETECT ABILITY Ever since the dawn of military aviation attempts have been made to reduce the detectability of aircraft. During World War I, the only "sensor" in use was the human eyeball. Camouflage paint in mottled patterns was used on both sides to reduce the chance of detection. Radar (acronym for Radio Detection And Ranging), the primary sensor used against aircraft today, consists of a transmitter antenna that broadcasts a directed beam of electromagnetic radio waves and a receiver antenna which picks up the faint radio waves that bounce off objects "illuminated" by the radio beam. Usually the transmitter and receiver antenn~s ar~ collocated ("monostatic radar"), although some systems have them III dIfferent locations ("bistatic radar"). Detectability to radar has been a concern since radar was first used in World War II. "Chaff" was the first radar "stealth" technology. Chaff, also called "window," consists of bits of metal foil or metallized fibers dropped by an aircraft to create many radar echos that hide its actual echo return. Chaff is still useful against less-sophisticated radars. Chaff obscures the actual location of the aircraft, but does not allow the aircraft to pass unnoticed. To avoid detection, the aircraft must return such a low amount of the transmitted radio beam that the receiver antenna cannot distinguish between it and the background radio static. The extent to WUlch an object returns electromagnetIc energy is the object's "Radar Cross Section" (RCS). RCS is usually measured in square meters or in decibel square meters, with "zero dBsm" equal to ten to the zero power, or one square meter. "Twenty dBsm" equals ten t.o the ~econd power, or 100 square meters. Because radar signal strengt~ IS an Illverse function of the fourth power of the distance to the target, It takes a very substantial reduction in RCS to obtain a meaningful operational benefit. Actually, the RCS of an aircraft is not a single number. The RCS is different for each "look-angle" (i.e., direction from the threat radar). Also the RCS varies depending upon the frequency and polarization of the thre~t radar (see Ref. 21). The following comments relate to typical threat radars seen by military aircraft. There are many electromagnetic phenomena that contribute to the RCS of an aircraft. These require different design approaches for RCS reduction, and can produce conflicting design requirements. Fi~ure 8.1~ illustrates the major RCS contributors for a typical, untreated fIghter aIrcraft.

166

AIRCRAFT DESIGN

167

SPECIAL CONSIDERATIONS

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HIGHRCS COCKPIT CAVITY

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Fig. 8.10

Major ReS contributors.

One of the largest contributions to airframe ReS occurs any time a relatively flat surface of the aircraft is perpendicular to the incoming radar beam. Imagine shining a flashlight at a shiny aircraft in a dark hanger. Any spots where the beam is reflected directly back at you will have an enormous ReS contribution. Typically this "specular return" occurs on the flat sides of the aircraft fuselage and along an upright vertical tail (when the radar is abeam the aircraft). To prevent these ReS "spikes," the designer may slope the fuselage sides, angle the vertical tails, and so on, so that there are no flat surfaces presented towards the radar (Fig. 8.11). Note that this ReS reduction approach assumes that the designer knows where the threat radar will be located relative to the aircraft. This information is usually provided by the operations-analysis department as a design driver. Also, this assumes a monostatic radar. Another area of the aircraft which can present a perpendicular bounce for the radar is the round leading edge of the wing and tail surfaces. If the aircraft is primarily designed for low detectability by a nose-on threat radar, the wings and tails can be highly swept to reduce their contribution to ReS. Note that this and many other approaches to reducing the ReS will produce a penalty in aerodynamic efficiency. It is also important to avoid any "corner reflectors," i.e., intersecting surfaces that form approximately a right angle, as shown in Fig. 8.10 at the . wing-fuselage junction. Another contributor to airframe ReS occurs due to the electromagnetic currents that build up on the skin when illuminated by a radar. These currents flow across the skin until they hit a discontinuity such as at a sharp

LOWRCS

Fig. 8.11

Flat side ReS rednction.

trailing edge, a wing tip, a control surface, or a crack around a removable panel or door. At a discontinuity, the currents "scatter," or radiate electromagnetic energy, some of which is transmitted back to the radar (Fig. 8.12). This effect is much lower in intensity than the specular return, but is still sufficient for detection. The effect is strongest when the discontinuity is straight and perpendicular to the radar beam. Thus, the discontinuities such as at the wing and tail trailing edges can be swept to minimize the detectability from the front. Carried to the extreme, this leads to diamond- or sawtooth-shaped edges on every door, access plate, and other discontinuity on the aircraft, as seen on the B-2 and F-117.

EDGE SCATTERING

Fig. 8.12

Surface current scatterings.

168

AIRCRAFT DESIGN

First-generation stealth designs such as the Lockheed F-117 and the never-constructed North American Rockwell "Surprise Fighter" relied upon faceted shaping in which the aircraft shape is constructed of interlocking flat triangles and trapezoids. This has advantages in ease of construction and signature analysis, but offers a large number of sharp edges to create diffraction returns, and so is no longer in favor (Ref. 92). Current stealth design begins by configuring the aircraft such that all "big" returns, such as from perpendicular bounces, are "aimed" in just a few directions. For example, if the leading edges of the wings and tails are all straight and set at the same angle, there would be a huge radar return from that angle direction, but little return from other directions. This would presumably offer a small probability that the aircraft and threat radar would be mutually oriented in exactly the angle of high return, and the aircraft would be undetectable from all other angles. It is also common practice to "aim" the wing leading-edge return in the same direction as the edge diffraction return from the trailing edge. This is done either by using identical sweep angles at the leading and trailing edges (thus, a wing with no taper, as on the B-2), or by aligning the left wing leading edge at the same sweep angle as the right wing trailing edge (and vice versa). This creates a diamond wing as on the F-23 and an early Mikoyan research aircraft. Once all wing and tail returns are "aimed" in the same direction, the returns from doors, access panels, and other discontinuities can be "aimed" in the same direction by alignment of their edges. This is clearly seen on the B-2 where virtually every feature on the aircraft, including weapons, bay do;rs, gear doors, inlets, nozzles, and access panels, is constructed using only lines which are parallel to a wing leading edge. This design approach leads to an aircraft planform composed entirely of straight, highly swept lines, much like the first-generation stealth designs. However, the desire to eliminate the edge diffractions caused by the facets of first-generation stealth now produces designs in which cross-sectional shapes are smooth, not sharp-edged. The steep angles on the fuselage sides as shown in Fig. 8.11 are employed to prevent broadside perpendicular bounce returns, but these angled sides flow smoothly over the top and bottom of the fuselage. Such shaping can be seen on the B-2, F-22, and F-23. RCS can also be reduced simply by eliminating parts of the aircraft. A horizontal tail that isn't there cannot contribute to the radar return! Modern computerized flight controls combined with the use of vectored-thrust engines can solve many of the difficulties of the tailless configuration. Similarly, RCS can be reduced if the nacelles can be eliminated through the use of buried engines, or better yet, by eliminating the entire fuselage through the use of the flying-wing concept. This approach is used in the Northrop B-2. In addition to reshaping the aircraft, detectability can be reduced through the use of skin materials that absorb radar energy. Such materials, called "radar absorbing materials" (RAM), are typically composites such as fiberglass embedded with carbon or ferrite particles. These particles are heated by the radar electromagnetic waves, thus absorbin some of the ener . This will re .. I

SPECIAL CONSIDERATIONS

Fig. 8.13

169

Delectability reduction approaches.

return due to perpendicular bounce, and can also reduce the surface currents and thus reduce the RCS due to scattering at sharp edges. As there are many types of RAM and similar treatments, no quick estimate for the weight impact of their use can be provided here. However, one can probably assume that such use will reduce or eliminate any weight savings otherwise assumed for the use of composite materials. For most existing aircraft, the airframe is not the largest contributor to RCS, especially nose-on. A conventional radome, covering the aircraft's own radar, is transparent to radar for obvious reasons. Therefore, it is also transparent to the threat radar, allowing the threat radar's beam to bounce off the forward bulkhead and electronic equipment within the radome. Even worse, the aircraft's own radar antenna, when illuminated by a threat radar, can produce a radar magnification effect much like a cat's eye. These effects can be reduced with a "bandpass" radome, which is transparent to only one radar frequency (that of the aircraft's radar). 6t~her huge contributors to the RCS for a conventIOnal aircraft are the inlet and exhaust cavities. Radar energy gets into these cavities, bounces off the engine parts, and sprays back out the cavity towards the threat radar. Also, these cavities represent additional surface discontinuities. The best solution for reducing these RCS contributions is to hide them from the expected threat locations. For example, inlets can be hidden from ground-based radars by locating them on top of the aircraft (Fig. 8.13). Exhausts can be hidden through the use of two-dimensional nozzles. Cockpits provide a radar return for a similar reason. The radar energy enters the cockpit, bounces around off the equipment inside, and then reradiates back outside. One solution for this is to thinly coat the canopy with some conductive metal such as gold, causing the canopy to reflect the radar energy away.

171

AIRCRAFT DESIGN

SPECIAL CONSIDERATIONS

Finally, the aircraft's weapons can have a major impact on RCS. Missiles and bombs have fins that form natural corner reflectors. The carriage and release mechanisms have numerous corner reflectors, cavities, and surface discontinuities. Gun ports present yet another kind of cavity. The only real solution for these problems is to put all the weapons inside, behind closed doors. However, the weight, volume, and complexity penalties of this approach must be carefully considered. Electronic countermeasures (ECM)-devices to trick the threat radarusually consist of some sort of radar receiver that picks up the threat radar emissions, and some sort of transmit antenna to send a deceiving signal back to the threat radar. The many techniques for tricking radar (and ECM) go beyond the scope of this book. However, designers should be aware that there is a tradeoff between the aircraft's RCS level and the required amount of ECM.

IR missiles can sometimes be tricked by throwing out a flare which burns to produce approximately the same IR frequencies as the aircraft. Ho,",:ever, modern IR seekers are getting better at identifying which hot source IS the actual aircraft. IR fundamentals are more thoroughly discussed in Ref. 18.

170

8.5 INFRARED DETECTABILITY Infrared (lR) detectability also concerns the aircraft designer. Many short-range air-to-air and ground-to-air missiles rely upon IR seekers. Modern IR sensors are sensitive enough to detect not only the radiation emitted by the engine exhaust and hot parts, but also that emitted by the whole aircraft skin due to aerodynamic heating at transonic and supersonic speeds. Also, sensors can detect the solar IR radiation that reflects off the skin and cockpit transparencies (windows). Of several approaches for reduction of IR detectability, one of the most potent reduces engine exhaust temperatures through the use of a highbypass-ratio engine. This reduces both exhaust and hot-part temperatures. However, depending upon such an engine for IR reduction may result in selecting one that is less than optimal for aircraft sizing, which increases aircraft weight and cost. Emissions from the exposed engine hot-parts (primarily the inside of the nozzle) can be reduced by cooling them with air bled off the engine compressor. This will also increase fuel consumption slightly. Another approach hides the nozzles from the expected location of the threat IR sensor. For example, the H-tails of the A-lO hide the nozzles from some angles. Unfortunately, the worst-case threat location is from the rear, and it is difficult to shield the nozzles from that direction! Plume emissions are reduced by quickly mixing the exhaust with the outside air. As mentioned, a high-bypass engine is the best way of accomplishing this. Mixing can also be enhanced by the use of a wide, thin nozzle rather than a circular one. Another technique is to angle the exhaust upward or downward relative to the freestream. This will have an obvious thrust penalty, however. Sun glint in the IR frequencies can be somewhat reduced by the use of special paints that have low IR reflectivity. Cockpit transparencies (which can't be painted!) can be shaped with all flat sides to prevent continuous tracking by an IR sensor. Emissions due to aerodynamic heat are best controlled by slowing the aircraft down.

8.6 VISUAL DETECT ABILITY The human eyeball is still a potent aircraft-detection sensor. On ~ clear day, an aircraft or its contrail may be spotted visually ?efore detectIOn by the on-board radar of a typical fighter. Also, fighter aIrcraft usually have radar only in front, which leaves the eyeball as the primary detector for spotting threat aircraft which are abeam or above.. . Visual detection depends upon the size of aircraft and ItS color and mtensity contrast with the background. In simulated combat, pilots of the small F-5 can frequently spot the much-larger F-15s well before the F-5s are seen. However, aircraft size is determined by the mission requirements and cannot be arbitrarily reduced. . . Background contrast is reduced primarily with camouflage pamts, usmg colors and surface textures that cause the aircraft to reflect light at an intensity and color equal to that of the background. T~is r:quires a~s.ump­ tions as to the appropriate background as well as the IIghtmg condItIons. Frequently aircraft will have a lighter paint on the bottom, because t.he background for look-up angles is the sky. Current camouflage paInt schemes are dirty blue-grey for sky backgrounds and dull, mottled greygreens and grey-browns for ground backgrounds. . . Different parts of the aircraft can contrast agaInst each ?ther, ~hICh increases detectability. To counter this, paint colors can be vaned to lIghten the dark areas such as where one part of the aircraft casts a shadow on another. Also,' small lights can sometimes be used to fill in a shadow spot. Canopy glint is also a problem for visual detection. The use of flat transparencies can be applied as previously discussed, but will tend to detract from the pilot's outside viewing. . At night, aircraft are visually detected mostly by engIne and exhaust .glow and by glint off the transparencies. These can be reduced by techmques previously discussed for IR and glint suppre.ssion.. . There are also psychological aspects to VIsual detectIOn. If the aIrcraft does not look like an aircraft, the human mind may ignore it. The irregular mottled patterns used for camouflage paints exploit this tendency. In air-to-air combat seconds are precious. If a pilot is confused as to the opponent's orientatio~, the opponent may obtain ~avor~ble positioning. To this end, some aircraft have even had fake canopIes ?amted on the u?de~­ side. Forward-swept and oblique wings may also prOVIde momentary dlsonentation. 8.7 AURAL SIGNATURE Aural signature (noise) is important for civili~n ~s well. as military aircraft. Commercial airports frequently have antInOISe ordmances that restrict some aircraft. Aircraft noise is largely caused by airflow shear layers, primarily due to the engine exhaust.

172

AIRCRAFT DESIGN

A small-diameter, high-velocity jet exhaust produces the greatest noise, while a large-diameter propeller with a low tip-speed produces the least noise. A turbofan falls somewhere in between. Blade shaping and internal duct shaping can somewhat reduce noise. Piston exhaust stacks are also a source of noise. This noise can be controlled with mufflers, and by aiming the exhaust stacks away from the ground and possibly over the wings. . Within the aircraft, noise is primarily caused by the engmes. Well-designed engine mounts, mufflers, and insulation materials can be us~d to reduce the noise. Internal noise will be created if the exhaust from a piston engine impinges upon any part of the aircraft, especially the. cabin. . Wing-mounted propellers can have a tremendous effect on mternal nOlse. All propellers should have a minimum clearance to the fuselage of about 1 ft, and should preferably have a minimum clearance of about one-half of the propeller radius. However, the greater the propeller clearance, the larger the vertical tail must be to counter the engine-out yaw. Jet engines mounted on the aft fuselage (DC-9, B727, etc.) should be located as far away from the fuselage as structurally permitted to reduce cabin noise.

Sample calculation Presented Area Pilot (a) Computer (b) Fuel (c) Engine (d)

5 4 80 50

fll fll fll fll

p. given hit 1.0 0.5 0.3 0.4

Total vulnerable area

8.8 VULNERABILITY CONSIDERATIONS Vulnerability concerns the ability of the aircraft to sustain battle damage, continue flying, and return to base. An aircraft can be "killed" in many ways. A single bullet through a non-redundant elevator actuator is as bad as a big missile up the tailpipe! "Vulnerable area" is a key concept. This refers to the product of the projected area (square feet or meters) of the aircraft components, times the probability that each component will, if struck, cause the aircraft to be lost. Vulnerable area is different for each threat direction. Typical components with a high aircraft kill probability (near 1.0) are the crew compartment, engine (if single-engined), fuel tanks (unless self-sealing), and weapons. Figure 8.14 shows a typical vulnerable area calculation. When assessing the vulnerability of an aircraft, the first step is to determine the ways in which it can be "killed." Referred to as a "failure modes and effects analysis (FMEA)," this step will typically be performed during the later stages of conceptual design. The FMEA considers both the ways in which battle damage can affect individual aircraft components, and the ways in which damage to each component will affect the other components. During initial configuration layout, the designer should strive to avoid certain features known to cause vulnerability problems. Fire is the greatest danger to a battle-damaged aircraft. Not only is the fuel highly flammable, but so is the hydraulic fluid. Also, combat aircraft carry gun ammo, bombs, and missiles. An aircraft may survive a burst of cannon shells only to explode from a fire in the ammo box. If at all possible, fuel should not be located over or around the engines and inlet ducts. While tanks can be made self-sealing to a small puncture, a large hole will allow fuel to ignite on the hot engine. The pylon-mounted engines on the A-lO insure that leaking fuel cannot ignite on the engines.

173

SPECIAL CONSIDERATIONS

Fig. 8.14

Vulnerable area 5 2 24 20

fll fll fll fll

51 fll

Vulnerable area calculation.

Similarily, hydraulic lines and reservoirs should be located away from the engines. Firewalls should be used to prevent the spread of flames beyond a bu.rning engine bay. Engine bays, fuel bays, and weapon bays should have a flresuppression system. When an engine is struck, turbine and compressor blades can. fly. off at high speeds. Avoid placing critical components such as hydrauhc hnes or weapons anyplace where they could be damaged by a~ exploding eng~ne. Also, a twin-engine aircraft should have enough separatIOn between e~gmes to prevent damage to the good engine. If twin engines are together m the fuselage, a combined firewall and containment shield s~lOuld separate them. This requires at least 1 foot of clearance between engmes. . Propeller blades can fly off either from battle damage or dunng a wheelsup landing. Critical components, especially the crew and passenger. compartments, shouldn't be placed within a 5-deg arc of the propeller disk. Avoid placing guns, bombs, or fuel near the crew compartment. Fuel should not be placed in the fuselage of a passenger plane. . Redundancy of critical components can be used to allow the survival of the aircraft when a critical component is hit. Typical components that could be redundant include the hydraulic system, electrical system, flight control system, and fuel system. Note that while redundancy im~roves the survivability and reliability, it worsens the maintenance reqUirements because there are more components to fail. For more information on vulnerability, Ref. 18 is again suggested.

174

AIRCRAFT DESIGN

SPECIAL CONSIDERATIONS

175

Some form of protection should be provided in the not-unlikely event that the aircraft flips over during a crash. This is lacking in several small homebuilt designs. 8.10 SCARFED FIREWALL PREVENTS SCOOPING

FIREWALL SCOOPING INCREASES CRASH LOADS

Fig. 8.15

NO FLOOR STRUTS

FLOOR STRUTS PUSH FLOOR UPWARDS

Crash worthiness design.

8.9 CRASHWORTHINESS CONSIDERATIONS Airplanes crash. Careful design can reduce the probability of injury in a moderate crash. Several suggestions have been mentioned above, including positioning the propellers so that the blades will not strike anyone if they fly off during a crash. Also mentioned was the desire to avoid placing fuel tanks in the fuselage of a passenger airplane (although fuel in the wing box carrythrough structure is usually acceptable). Figure 8.15 shows several other design suggestions which were learned the hard way. A normal, vertical firewall in a propeller aircraft has a sharp lower corner which tends to dig into the ground, stopping the aircraft dangerously fast. Sloping the lower part of the firewall back as shown will prevent digging in. therefore reducing the deceleration. For a large passenger aIrcraft, the floor should not be supported by braces from the lower part of the fuselage. As shown, these braces may push upward through the floor in the event of a crash. Common sense will avoid many crashworthiness problems. For example, things will break loose and fly forward during a crash. Therefore, don't put heavy items behind and/or above people. This sounds obvious, but there are some aircraft with the engine in a pod above and behind the cockpit. There are also some military jets with large fuel tanks directly behind the cockpit, offering the opportunity to be bathed in jet fuel during a crash. However, the pilot would probably try to eject rather than ride out a crash bad enough to rupture the fuel tanks. One s.hould also consider secondary damage. For example, landing gear and engme nacelles will frequently be ripped away during a crash. If possible, they should be located so that they do not rip open fuel tanks in the process.

PRODUCIBILITY CONSIDERATIONS It is often said that aircraft are bought "by the pound." While it is true that aircraft cost is most directly related to weight, there is also a strong cost impact due to the materials selected, the fabrication processes and tooling required (forging, stamping, molding, etc.), and the assembly manhours. The configuration designer does not usually determine the materials used or exactly how the aircraft will be fabricated. However, the ease of producing the aircraft can be greatly facilitated by the overall design layout. The greatest impact the configuration designer has upon producibility is the extent to which flat-wrap structure is incorporated. This has a major impact upon the tooling costs and fabrication manhours, as discussed in the last chapter. Part commonality can also reduce production costs. If possible, the left and right main landing gear should be identical (left-right common). It may be desirable to use uncambered horizontal tails to allow left-right commonality even if a slight aerodynamic penalty results. In some cases the wing airfoil can be slightly reshaped to allow left-right common ailerons. Forgings are the most expensive type of structure in common usage, and are also usually the longest-lead-time items for production tooling. Forgings may be required whenever a high load passes through a small area. Forgings are used for landing-gear' struts, wing-sweep pivots, and all-moving tail pivots (trunnions). The designer should avoid, if possible, such highlyloaded structure. Installation of internal components and routing of hydraulic lines, electrical wiring, and cooling ducts comprises another major production cost due to the large amount of manual labor required. To ease installation of components and routing, avoid the tight internal packaging so desirable for reduced wetted area and wave drag. When evaluating proposed designs, government design boards will compare the overall aircraft density (weight divided by volume) with historical data for similar aircraft to insure packaging realism. Routing can be simplified through provision of a clearly defined "routing tunnel." This can be internal or, as shown in Fig. 8.16, an external and nonstructural fairing that typically runs along the spine of the aircraft. However, if all routing is concentrated in one area the aircraft vulnerability will be drastically worsened. Routing can be reduced by careful placement of the internal components. For example, the avionics and the crew station will both require cooling air ("environmental control"). If the avionics, crew station, and environmental control system (ECS) can be located near to each other, the routing distances will be minimized. Sometimes clever design can reduce routing. The Rutan Defiant, a "pushpull" twin-engined design, uses completely separate electrical systems for the front and rear engines, including separate batteries. This requires an extra battery, but a trade study determined that the extra battery weighs less

AIRCRAFT DESIGN

176

than the otherwise-required electrical cable, and eliminates the front-to-rear routing requirement. Another factor for producibility concerns manufacturing breaks. Aircraft are buil~ in subassemblies. Typically, a large aircraft will be built up from a COCkPit, an aft-fuselage, and a number of mid-fuselage subassemblies. A small aircraft may be built from only two or three subassemblies. It is important that the designer consider where the subassembly breaks will occur, and avoid placing components across the convenient break locations. Figure 8.17 shows a typical fighter with a fuselage production break located just aft of the cockpit. This is very common because the cockpit pressure vessel should not be broken for fabrication. In the upper design, the nose wheel well is divided by the production break, which prevents fully assembling the nose-wheel linkages before the two subassemblies are connected. The lower illustration shows a better arrangement. Design for producibility requires experience that no book can provide. A g?od understan~ing of structural design and fabrication and the basic principles of operatIOn for the major subsystems provides the background for ?eveloping producible designs. The following material provides a brief mtroduction to aircraft fabrication. While there have been tremendous advances in aircraft production in recent years, much of the modern factory would be recognizable to a T?anufacturing engineer from the Wright Brothers' days. Aircraft production, then and now, involves the application of the mechanical arts of machining, forming, finishing, joining, assembly, and testing. Machining involves the removal of a carefully-controlled amount of material from a part, typically by the application of a cutting tool via relative motion between the part and the tool. The cutting tool is generally based upon the inclined wedge, and acts to peel away a thin shaving of the part (a drill bit can be seen as a set of inclined wedges positioned radially around an axis). The relative motion between tool and part can be rotational, as with the drill, lathe, and mill, or it can be translational, as with the broach and planer.

c Fig. 8.16

External routing tunnel.

SPECIAL CONSIDERATIONS

Fig. 8.17

177

Production breaks.

Forming refers to the numerous ways in which materials, especially' metals, are changed in shape other than by machining. Forming includes casting, forging, extruding, stamping, punching, bending, and drawing. In casting, the metal is brought up to its melting temperature then poured into a mold. Forging involves forcing nonmolten metal into a mold through pressure or impact. Extrusion is the process of forcing metal to flow out a hole with the desired cross-sectional area, creating shaped bar stock. Stamping and punching are used to cut out shapes and holes in sheet metal. Bending is self-explanatory, and drawing is the process of forcing sheet metal into a form creating cup-like geometries. Finishing encompasses a number of processes applied to formed andlor machined parts. Some finish processes include further material removal to create a smoother surface, such as deburring, lapping, and finish grinding. Other finish processes, such as painting, anodization, and plating, involve application of a surface coating. Composite fabrication is sufficiently unlike metal fabrication that it deserves special mention. In thermoset composite production, a liquid or pliable semisolid plastic material undergoes a chemical change into a new solid material, usually accompanied by the application of heat and/o; pressure. For aircraft applications the plastic "matrix" material is reinforced by a fiber, typically of graphite material. Thermoset composite manufacture is unique in that the material itself is produced at the same time and place as the part. A second class of composites, the thermoplastics, involves a plastic matrix which is heated in a mold until it deforms readily, assuming the shape of the mold. Composite fabrication is further described in Chapter 14. Joining is simply the attachment of parts together, by processes including brazing, soldering, welding, bonding, riveting, and bolting. All these processes historically have a high manual-labor content, and all are being automated to various extents in modern factories. For example, modern car factories have long lines of robotic spot-welders attaching body panels.

178

AIRCRAFT DESIGN

~uto~atic riveting machines, applicable for simple geometries such as nvets m a row down a wing spar, can be found in the modern aircraft factory. . Assembly is the process of combining parts and subassemblies into the fmal pr.oduct. A.sse~b~y us.ually involves joining operations such as riveting or boltmg, but IS dIstmgUIshed from joining by the greater level of completeness .of t~e S~b~ss~~b~i~s. ~?r example, when you attach a wing skin to the wmg nbs It IS JOInIng, but when you attach the wing to the fuselage, it is "assembly." . Testin.g is a key part of the manufacturing process. In traditional factones, testmg was generally done by random selection of finished product and was .frequently of a destructive nature. While helping to keep average quahty up, such random destructive testing did not insure that any given part was acceptable because the only parts known by testing to be acceptable were destroyed in the process! Today's factories are tending toward nondestructive testing techniques such as m.agnaflux, ultrasonic, and nuclear magnetic resonance, and are also applymg ~dvanced statistical techniques to better select samples to test and to determme the corrective action required. CAD/CAM, or Computer-Aided Design/Computer-Aided Manufactu~e, is a g~neric .term for the many different ways in which computers are ?emg used m deSIgn and manufacture. Typically CAD/CAM refers specifIcal.ly to the use of computers for two- and three-dimensional component deSign, an? the use of. the resulting CAD data base as the input for the programmmg of n~mencally-controlled machinery and robots (as described below). The benefits of CAD/CAM are well-established and include improved design quality, reduction in design time and/or increase in the numbe.r of design !terations possible, earlier discovery of errors, integration of .d~sIgn, analYSIS, and manufacturing engineering, and facilitation of trammg. Automation refers to almost any use of computerized equipment during man~facture. However, the generic term "automation" is most frequently apphed to tasks ~uch as riveting, parts retrieval, and process control (such as autoclave cychng), whereas the more specific terms "numerical control" and "robotics" are used as described below. . Num:rical c~ntrol (NC) programming refers to the creation of digital m~tructIOns WhIC.h command a computer-controlled machine tool such as a mIll o.r lathe. ThIs. area is probably one of the highest leverage in terms of reduc~ng cost. and Improving quality. While machine tools themselves have expenenced httle fundamental change in this century (this author knows of a co~p~ny making h.igh-tech wind turbines on a lOO-year-old lathe!), the apphcatIOn of numencal control replacing the skilled but bored machinist has had a tremen~o~s effect on productivity and quality. The most sophistIcated subset of automation is robotics in which a c.ompute~-control1ed machine performs tasks involving highly ~omplex mo~I~ns WhIC~. previously might have been performed by a human. Note that It I~ the ~b~hty !o physically manipulate objects in response to programming W.hICh dIstm~UIshes the robot from other forms of automation or mechanIsm. RobotICS examples include part pickup and positioning, painting,

SPECIAL CONSIDERATIONS

179

composite-ply laydown, material handling, simple assembly, and welding, and are usually limited to "semi-skilled" jobs, at least to date. A key robotics technology for composites is in the labor-intensive tape lay-up process. Programmable robot arms with tape dispenser end effectors are widely used to place the prepreg. Also, autoclave cycle control is widely automated. Rapid prototyping of parts without tools is being performed using a new technique known as Stereolithography (SLA), which can produce plastic prototype parts in a day or less. SLA works by mathematically slicing CAD designs into thin cross sections, which are traced one at a time by an ultraviolet laser beam on a vat of photosensitive chemicals that solidify as they are irradiated. After each layer is completed, an "elevator" holding the part moves down slightly and the next layer is solidified on top of it. While to date only certain types of relatively fragile plastics may be used by SLA devices, the plastic prototypes can then be used to create molds for strong epoxy or aluminum parts.

8.11 MAINTAINABILITY CONSIDERATIONS Maintainability means simply the ease with which the aircraft can be fixed. "Reliability and Maintainability" (R&M) are frequently bundled together and measured in "Maintenance Manhours Per Flighthour" (MMH/ FH). MMH/FH's range from less than one for a small private aircraft to well over a hundred for a sophisticated supersonic bomber or interceptor. Reliability is usually out of the hands of the conceptual designer. Reliability depends largely upon the detail design of the avionics, engines, and other subsystems. The configuration designer can only negatively impact reliability by placing delicate components, such as avionics, too near to vibration and heat sources such as the engines. Anybody who has attempted to repair a car will already know what the major driver is for maintainability. Getting at the internal components frequently takes longer than fixing them! Accessibility depends upon the packaging density, number and location of doors, and number of components that must be removed to get at the broken component. Packaging density has already been discussed. The number and location of doors on modern fighters have greatly improved over prior-generation designs. Frequently the ratio between the total area of the access doors and the total wetted area of the aircraft's fuselage is used as a measure of merit, with modern fighters approaching a value of one-half. A structural weight penalty must be paid for such access. This leads to the temptation to use "structural doors" that carry skin loads via heavy hinges and latches. These are always more difficult to open than nonload-bearing doors because the airframe's deflection from its own weight will bind the latches and hinges. In extreme cases, the aircraft must be supported on jacks or a cradle to open these structural doors. As a general rule, the best access should be provided to the components that break the most often or require the most routine maintenance. Engine access doors should definitely be provided that allow most of the engine to be exposed. Also, large doors should be provided for the avionics compart-

180

AIRCRAFT DESIGN

me~t, hydraulic p~mps, actuators, electrical generators, environmental contro system, aUXIlIary power unit, and gun bay. The worst .feature an airc~aft can have for maintainability is a re uire;ent for major structural dIsassembly to access or remove a compo~ent or example, the V /S~OL A V -SB Harrier requires that the entire win b~ remtovfedhbelfore removIng the engine. Several aircraft require removal ~f a par. o. t e ongeron to remove the wing. th SImilarly, the designer should avoid placing internal components such . t·Ion s att one must t b be removed to get to another . In the F -4 Phantom , an eJec ea mus e removed to get to the radio (a high-break-rate item) It is not unco,~md O? for ~he ej~ction seat to be damaged during this proce~s "Oneeep eSlgn wIll aVOid such problems. .

d

9 CREW STATION, PASSENGERS, AND PAYLOAD 9.1 INTRODUCTION At the conceptual design level it is not necessary to go into the details of crew-station design, such as the actual design and location of controls and instruments, or the details of passenger and payload provisions. However, the basic geometry of the crew station and payload/passenger compartment must be considered so that the subsequent detailed cockpit design and payload integration efforts will not require revision of the overall aircraft. This chapter presents dimensions and "rule-of-thumb" design guidance for conceptual layout of aircraft crew stations, passenger compartments, payload compartments, and weapons installations. Information for more detailed design efforts is contained in the various civilian and military specifications and in subsystem vendors' design data packages. 9.2 CREW STATION The crew station will affect the conceptual design primarily in the vision requirements. Requirements for unobstructed outside vision for the pilot can determine both the location of the cockpit and the fuselage shape in the vicinity of the cockpit. For example, the pilot must be able to see the runway while on final approach, so the nose of the aircraft must slope away from the pilot's eye at some specified angle. While this may produce greater drag than a morestreamlined nose, the need for safety overrides drag considerations. Similarly, the need for over-side vision may prevent locating the cockpit directly above the wing. When laying out an aircraft's cockpit, it is first necessary to decide what range of pilot sizes to accommodate. For most military aircraft, the design requirements include accommodation of the 5th to the 95th percentile of male pilots, (i.e., a pilot height range of 65.2-73.1 in.). Due to the expense of designing aircraft that will accommodate smaller or larger pilots, the services exclude such people from pilot training. Women are only now entering the military flying profession in substantial numbers, and a standard percentile range for the accommodation of female pilots had not yet been established as this was written. Future military aircraft might require the accommodation of approximately the 20th percentile female and larger. This may affect the detailed layout of cockpit controls and displays, but should have little impact upon conceptual cockpit layout.

182

AIRCRAFT DESIGN

CREW STATION, PASSENGERS, AND PAYLOAD

General-aviation cockpits are designed to whatever range of pilot sizes the marketing department feels is needed for customer appeal, but typically are comfortable only for those under about 72 in. Commercial-airliner cockpits are designed to accommodate pilot sizes similar to those of military aircraft. Figure 9.1 shows a typical pilot figure useful for conceptual design layout. This 95th percentile pilot, based upon dimensions from Ref. 22, includes allowances for boots and a helmet. A cockpit designed for this size of pilot will usually provide sufficient cockpit space for adjustable seats and controls to accommodate down to the 5th percentile of pilots. Designers sometimes copy such a figure onto cardboard in a standard design scale such as twenty-to-one, cut out the pieces, and connect them with pins to produce a movable manikin. This is placed on the drawing, positioned as desired, and traced onto the layout. A computer-aided aircraft design system can incorporate a built-in pilot manikin (see Ref. 14). Dimensions for a typical cockpit sized to fit the 95th-percentile pilot are shown in Fig. 9.2. The two key reference points for cockpit layout are shown. The seat reference point, where the seat pan meets the back, is the reference for the floor height and the legroom requirement. The pilot's eye point is used for defining the overnose angle, transparency grazing angle, and pilot's head clearance (lO-in. radius). This cockpit layout uses a typical 13-deg seat back angle, but seat back angles of 30 deg are in use (F-16), and angles of up to 70 deg have been considered for advanced fighter studies. This entails a substantial penalty in

10 in.



SHOULDER WIDTH -26 in. ALLOW 30 in. FOR CLEARANCE

TYPICAL SEAT·TYPE PARACHUTE

Fig. 9.1

Average 95th percentile pilot.

183

outside vision for the pilot, but can improve his ability to withstand high-g turns and also can reduce drag because of a reduction in the cockpit height. When designing a reclined-seat cockpit, rotate both the seat and the pilot's eye point about the seat reference point, and then use the new position of the pilot's eye to check overnose vision. Overnose vision is critical for safety especially during landing, and is also important for air-to-air combat. Military specifications typically require 17-deg overnose vision for transports and bombers, and 11-15 deg for fighter and attack aircraft. Military trainer aircraft in which the instructor pilot sits behind the student require 5-deg vision from the back seat over the top of the front seat. Various military specifications and design handbooks provide detailed requirements for the layout of the cockpit of fighters, transports, bombers, and other military aircraft. General-aviation aircraft land in a fairly level attitude, and so have overnose vision angles of only about 5-10 deg. Many of the older designs have such a small overnose vision angle that the pilot loses sight of the runway from the time of flare until the aircraft is on the ground and the nose is lowered. Civilian transports frequently have a much greater overnose vision angle, such as the Lockheed L-1011 with an overnose vision angle of 21 deg. Civilian overnose vision angles must be calculated for each aircraft based upon the ability of the pilot to see and react to the approach lights at decision height (100 ft) during minimum weather conditions (1200-ft runway visual range). The higher the approach speed, the greater the overnose vision angle must be. Reference 23 details a graphical technique for determining the required overnose angle, but it can only be applied after the initial aircraft layout is complete and the exact location of the pilot's eye and the main landing gear is known. For initial layout, Eq. (9.1) is a close approximation, based upon the aircraft angle of attack during approach and the approach speed. CXovernose

== cxapproach + 0.07 Vapproach

(9.1)

where Vapproach is in knots. Figure 9.2 shows an over-the-side vision requirement of 40 deg, measured from the pilot's eye location on centerline. This is typical for fighters and attack aircraft. For bombers and transports, it is desirable that the pilot be able to look down at a 35-deg angle without head movement, and at a 70-deg angle when the pilot's head is pressed against the cockpit glass. This would also be reasonable for general-aviation aircraft, but many generalaviation aircraft have a low wing blocking the downward view. The vision angle looking upward is also important. Transport and bomber aircraft should have unobstructed vision forwards and upwards to at least 20 deg above the horizon. Fighters should have completely unobstructed vision above and all the way to the tail of the aircraft. Any canopy structure should be no more that 2 in. wide to avoid blocking vision.

184

CREW STATION, PASSENGERS, AND PAYLOAD

AIRCRAFT DESIGN

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The transparency grazing angle shown in Fig. 9.2 is the smallest angle between the pilot's line of vision and the cockpit windscreen. If this angle becomes too small, the transparency of the glass or plexiglass will become substantially reduced, and under adverse lighting conditions the pilot may only see a reflection of the top of the instrument panel instead of whatever is in front of the aircraft! For this reason, a minimum grazing angle of 30 deg is recommended . The cockpit of a transport aircraft must contain anywhere from two to four crew members as well as provisions for radios, instruments, and stowage of map cases and overnight bags. Reference 23 suggests an overall length of about 150 in. for a four-crewmember cockpit, 130 in. for three crewmembers, and 100 in. for a two-crewmember cockpit. The cockpit dimensions shown in Fig. 9.2 will provide enough room for most military ejection seats. An ejection seat is required for safe escape when flying at a speed which gives a dynamic pressure above about 230 psf (equal to 260 knots at sea level). At speeds approaching Mach I at sea level (dynamic pressure above 1200), even an ejection seat is unsafe and an encapsulated seat or separable crew capsule must be used. These are heavy and complex. A separable crew capsule is seen on the FB-lI1 and the prototype B-1 A. The latter, including seats for four crew members, instruments, and some avionics, weighed about 9000 lb.

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185

9.3 PASSENGER COMPARTMENT The actual cabin arrangement for a commercial aircraft is determined more by marketing than by regulations. Figure 9.3 defines the dimensions of interest. "Pitch" of the seats is defined as the distance from the back of one seat to the back of the next. Pitch includes fore and aft seat length as well as leg room. "Headroom" is the height from the floor to the roof over the seats. For many smaller aircraft the sidewall of the fuselage cuts off a portion of the outer seat's headroom, as shown. In such a case it is important to assure that the outer passenger has a lO-in. clearance radius about the eye position. Table 9.1 provides typical dimensions and data for passenger compartments with first-class, economy, or high-density seating. This information (based upon Refs. 23, 24, and others) can be used to layout a cabin floor plan. There should be no more than three seats accessed from one aisle, so an aircraft with more than six seats abreast will require two aisles. Also, doors and entry aisles are required for approximately every 10-20 rows of seats. These usually include closet space, and occupy 40-60 in. of cabin length each. Passengers can be assumed to weigh an average of 180 lb (dressed and with carry-on bags), and to bring about 40-60 lb of checked luggage. A current trend towards more carry-on luggage and less checked luggage has been overflowing the current aircrafts' capacity for overhead stowage of bags. The cabin cross section and cargo bay dimensions (see below) are used to determine the internal diameter of the fuselage. The fuselage external di-

186

AIRCRAFT DESIGN Table 9.1

CREW STATION, PASSENGERS, AND PAYLOAD

Typical passenger compartment data

Seat pitch (in.) Seat width (in.) Headroom (in.) Aisle width (in.) Aisle height (in.) Passengers per cabin staff (international-domestic) Passengers per lavatory

Economy

38-40 20-28 >65 20-28 >76 16-20

34-36 17-22 >65 18-20 >76 31-36

30-32 16-18

10-20

40-60

40-60

5-8

1-2

0-1

(40" x40") Galley volume per passenger (ft 3 /pass)

r-------/--,----" I I

High density / small aircraft

First class

~ 12 >60 :0:;50

HEADROOM

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SEAT

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I

---/---/-aio

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I Fig. 9.3

Commercial passenger allowances.

am.eter is then dete~mined by estimating the required structural thickness. ThIs r~nges f~om 1m. for a small business or utility transport to about 4 in. for a Jumbo Jet. 9.4

158 CUBIC FEET

Fig. 9.4

I AISLE WIDTH

LD·3 CONTAINER

78 CUBIC FEET

44.4

I I I

727·200 C CONTAINER

26.4

,--------,

AISLE HEIGHT

187

CARGO PROVISIONS

. Cargo must ~e. ~arried in a secure fashion to prevent shifting while in flIght. Large~ clVlhan transports use standard cargo containers that are p~e-Ioaded w!th cargo and luggage and then placed into the belly of the ancra.ft. DUrIng conceptual design it is best to attempt to use an existing c~ntamer rather than requiring purchase of a large inventory of new contamers.

Cargo containers.

Two of the more widely used cargo containers are shown in Fig. 9.4. Of the smaller transports, the Boeing 727 is the most widely used, and the 727 container shown is available at virtually every commercial airport. The LD-3 container is used by all of the widebody transports. The B-747 carries 30 LD-3's plus 1000 fe of bulk cargo volume (non-containered). The L-lOll carries 16 LD-3's plus 700 ft3 of bulk cargo volume, and the DC-lO and Airbus each carry 14 LD-3's (plus 805 and 565 fe, respectively, of bulk cargo volume). To accommodate these containers, the belly cargo compartments require doors measuring appro.ximately 70 in. on a side. As was discussed in the section on wing vertical placement, low-wing transports usually have two belly cargo compartments, one forward of the wing box and one aft. The cargo volume per passenger of a civilian transport ranges from about 8.6-15.6 ft3 per passenger (Ref. 24). The smaller number represents a small short-haul jet (DC-9). The larger number represents a transcontinental jet (B-747). The DC-lO, L-1011, Airbus, and B-767 all have about II fe per passenger. Note that these volumes provide room for paid cargo as well as passenger luggage. Smaller transports don't use cargo containers, but instead rely upon hand-loading of the cargo compartment. For such aircraft a cargo provision of 6-8 ft3 per passenger is reasonable. Military transports use flat pallets to pre-load cargo. Cargo is placed upon these pallets, tied down, and covered with a tarp. The most common pallet measures 88 by 108 in. Military transports must have their cargo compartment floor approximately 4-5 ft off the ground to allow direct loading and unloading of cargo from a truck bed at air bases without cargo-handling facilities. However, the military does use some commercial aircraft for cargo transport and has pallet loaders capable of raising to a floor height of 13 ft at the major Military Airlift Command bases. The cross section of the cargo compartment is extremely important for a military transport aircraft. The C-5, largest of the U.S. military transports,

188

189

AIRCRAFT DESIGN

CREW STATION, PASSENGERS, AND PAYLOAD

is sized to carry so-called "outsized" cargo, which includes M-60 tanks, helicopters, and large trucks. The C-5 cargo bay is 19 ft wide, 13 Y2 ft high, and 121 ft long. The C-130 is used for troop and supply delivery to the front lines and cannot carry outsized cargo. Its cargo bay measures 10' 3" wide, 9'2" high, and 41' 5" long.

Ejection-launch is used mainly for larger missiles. The missile is attached to the aircraft through hooks which are capable of quick-release, powered by an explosive charge. This explosive charge also powers .two pistons ~hat shove the missile away from the aircraft at an extremely hIgh acceleratlOn. The missile motor is lit after it clears the aircraft by some specified distance. Bombs can also be ejected, or can simply be released and allowed to fall free of the aircraft. There are four options for weapons carriage. Each has pros and cons, depending upon the application. External carriage is the lightest and simplest, and offers the most flexibility for carrying. alter~ate weapon s.t?res. While most fighter aircraft are designed to an aIr-to-aIr role, the abIhty to perform an additional air-to-ground role is ofte~ imposed. To avoi.d p~nal­ izing the aircraft's performance when "clean" (l.e., set up for dogfightmg), most fighter aircraft have "hardpoints" under the wing and fuselage to which weapon pylons can be attached, as shown in Fig. 9.6. The.se are used to carry additional external weapons, and are removed for maXImum dogfighting performance. Most fighter aircraft can also carry external fuel tanks on the weapons pylons. These can be dropped when entering a dogfight. but are not dropped during long overwater ferry flights. Standard external fuel tanks include 150 and 600 gal sizes.

9.5 WEAPONS CARRIAGE Carriage of weapons is the purpose of most military aircraft. Traditional weapons include guns, bombs, and missiles. Lasers and other exotic technologies may someday become feasible as airborne weapons but will not be discussed here. ' T~e weapons are a substantial portion of the aircraft's total weight. This reqUlres that the weapons be located near the aircraft's center of gravity. Otherwise the aircraft would pitch up or down when the weapons are released . . Missiles diff~r ~rom bombs primarily in that missiles are powered. Today, vIrtually all mIssIles are also guided in some fashion. Most bombs are "dumb," or unguided, and are placed upon a target by some bombsight mechanism or computer which releases them at the proper position and velocity so that they free-fall to the desired target. However, "smartbombs," which have some guidance mechanism, are also in use. Missiles are launched from the aircraft in one of two ways. Most of the smaller missiles such as the AIM-9 are rail-launched. A rail-launcher is mounted to the aircraft, usually at the wingtip or on a pylon under the wing. Attached to the missile are several mounting lugs, which slide onto the rail as shown on Fig. 9.5. For launch, the missile motor powers the missile down the rail and free of the aircraft.

RAIL

EJECTOR PYLON OR WINGTIP

EXTERNAL

C\

[email protected]:~

EXPLOSIVE CHARGE PISTON

/J~\

f

"-

RAIL LUGS

C\

J Fig. 9.5

RELEASE MECHANISM

SEMI·SUBMERGED

C\

INTERNAL

Missile carriage/launch. CONFORMAL

Fig. 9.6

Weapon carriage options.

190

AIRCRAFT DESIGN

CREW STATION, PASSENGERS, AND PAYLOAD

Externally-carried weapons have extremely high drag. At near-sonic speeds, a load of external bombs can have more drag than the entire rest of the aircraft. Supersonic flight is virtually impossible with pylon-mounted external weapons, due to drag and buffeting. (Wing tip-mounted missiles are small, and have fairly low drag.) To avoid these problems, semisubmerged or conformally-carried weapons may be used. Conformal weapons mount flush to the bottom of the wing or fuselage. Semisubmerged weapons are half-submerged in an indentation on the aircraft. This is seen on the F-4 for air-to-air missiles. Semisu~~~rged carriage offers a substantial reduction in drag, but reduces fleXIbIlIty for carrying different weapons. Also, the indentations produce ~ .structu~al weigh~ penalty on the airplane. Conformal carriage doesn t intrude into the aIrcraft structure, but has slightly higher drag than the semi submerged carriage. The lowest-drag option for weapons carriage is internal. An internal weapons bay has been a standard feature of bombers for over fifty years, but has been seen on only a few fighters and fighter-bombers, such as the ~-106 and FB-l11. This is partly due to the weight penalty imposed by an mt~rnal weapons bay and its required doors, but is also due to the prevalent d~sl~e to maximize dogfighting performance at the expense of alternate mISSiOn I?er!ormance. However, only an internal weapons bay can completely elIminate the weapons' contribution to radar cross section so the internal weapons bay may become common for fighters as well as b~mbers. During concep~ual layout, there are several aspects of weapons carriage that must be consIdered once the type of carriage is selected. Foremost is the need to remember the l~a~ing crew. They will be handling large, heavy, and extremely dangerous m.lsslles and bombs. They may be working at night, in a snowstorm, on a rolling carrier deck, and under attack. Missiles must be physically attached to the mounting hooks or slid down the rail then secured by a locking mechanism. Electrical connections must be m~de to the guidanc~ mechanism, ~nd the safety wire must be removed from the fusing ~echamsm. For a~ ejector-type launcher, the explosive charge must be inser~ed. All of thIS cannot be done if the designer, to reduce drag, has provIded only a few inches of clearance around the missile. The loading crew absolutely must have sufficient room in which to work. . Clearance around the missiles and bombs is also important for safety. To insure that the weapons never strike the ground, the designer should provide

~i::--~·L I

.,

.a....,

" •

, I I ,

JiIl

10 deg

Fig. 9.7

Weapon release clearance.

I

I I

, \

.....

,

I

10 deg

191

j------ll Fig.9.8

Rotary weapons bay.

at least a 3-in. clearance to the ground in all aircraft attitudes. This includes the worst-case bad landing in which one tire and shock-strut are completely flat, the aircraft is at its maximum tail-down attitude (usually 15 deg or more) and the aircraft is in a 5-deg roll. The minimum clearance should be doubled if the airplane is to operate from rough runways. If weapons are mounted near each other, there should be a clearance on the order of 3 in. between them. There should also be a foot or more clearance between weapons and a propeller disk. The path taken by missiles or bombs when launched must be considered. For rail-launched missiles, there should be at least a lO-deg cone of clearance between any part of the aircraft and the launch direction of the missile. Also, the designer must consider the effects of the missile exhaust blast on the aircraft's structure. For an ejector-launched or free-fall released weapon, there should be a fall line clearance of 10 deg off the vertical down from any part of the missile to any part of the aircraft or other weapons as shown in Fig. 9.7. A special type of internal weapons carriage is the rotary weapons bay, as shown in Fig. 9.8. This allows launching all of the weapons through a single, smaller door. At supersonic speeds it can be difficult or impossible to launch weapons out of a bay due to buffeting and airloads which tend to push the weapon back into the bay. A single smaller door reduces these tendencies. Also, the rotary launcher simplifies installation of multiple weapons into a single bay. In fact, it is possible to design a rotary launcher that can be pre-loaded with weapons and loaded full into the aircraft.

9.6 GUN INSTALLATION The gun has been the primary weapon of the air-to-air fighter since the first World War I scout pilot took a shot at an opposing scout pilot with a handgun. For a time during the 1950's it was felt that the then-new air-toair missiles would replace the gun, and in fact several fighters such as the F-4 and F-104 were originally designed without guns. History proved that missiles cannot be solely relied upon, and all new fighters are being designed with guns. The standard U.S. air-to-air gun today is the M61Al "Vulcan" six-barrel gatling gun, shown in Fig. 9.9. This is used in the F-15, F-16, F-18, and others. Note the ammunition container. This must be located near the aft end of the gun. Rounds of ammo are fed out of the container ("drum") through feed chutes and into the gun. Ammo is loaded into the drum by attaching an ammo loading cart to the feed chute shown. The door to this loading chute must be accessible from the ground.

192

AIRCRAFT DESIGN

TOP

---------I .=~

144 ...- - - - 74 in. ------~~.I

Fig. 9.9

M61 "VULCAN" gun.

An air-to-air gun such as the M6lAI can produce a recoil force on the order of two tons. A large anti-tank gun such as the GAU-8 used in the A-IO can produce ~ ~ecoil force five times greater. To avoid a sudden yawing motIOn from fmng, guns should be located as near as possible to the centerline of the aircraft. On the A-IO, the nose landing gear is offset to one side to allow the gun to be exactly on the centerline. This extreme is not necessary for the smaller air-to-air guns. When a gun is fired, it produces a bright flash and a large cloud of smoke. The gun muzzle should be located so that these do not obscure the pilot's vision. Also, being very noisy, a gun should be located away from the cockpit. The cloud of smoke produced by a gun can easily stall a jet engine if sucked into the inlet. This should also be considered when locating a gun.

10 PROPULSION AND FUEL SYSTEM INTEGRATION 10.1 INTRODUCTION This section treats the integration and layout of the propulsion system into the overall vehicle design, not the calculation of installed propulsion performance. Propulsion analysis methods are covered in Chapter 13. To develop the propulsion system layout it is necessary to know the actual dimensions and installation requirements of the engine as well as its supporting equipment such as inlet ducts, nozzles, or propellers. Also, the fuel system including the fuel tanks must be defined. 10.2 PROPULSION SELECTION Figure 10.1 illustrates the major options for aircraft propulsion. All aircraft engines operate by compressing outside air, mixing it with fuel, burning the mixture, and extracting energy from the resulting high-pressure hot gases. In a piston-prop, these steps are done intermittently in the cylinders via the reciprocating pistons. In a turbine engine, these steps are done continuously, but in three distinct parts of the engine. The piston-prop was the first form of aircraft propulsion. By the dawn of the jet era, a 5500-hp piston-prop engine was in development. Today pistonprops are mainly limited to light airplanes and some agricultural aircraft. Piston-prop engines have two advantages. They are cheap, and they have the lowest fuel consumption. However, they are heavy and produce a lot of noise and vibration. Also, the propeller loses efficiency as the velocity increases. The turbine engine consists of a "compressor," a "burner," and a "turbine." These separately perform the three functions of the reciprocating piston in a piston engine. The compressor takes the air delivered by the inlet system and compresses it to many times atmospheric pressure. This compressed air passes to the burner, where fuel is injected and mixed with the air and the resulting mixture ignited. The hot gases could be immediately expelled out the rear to provide thrust, but are first passed through a turbine to extract enough mechanical power to drive the compressor. It is interesting to note that one early jet engine used a separate piston engine to drive the compressor. There are two types of compressors. The centrifugal compressor relies upon centrifugal force to "fling" the air into an increasingly narrow chan-

194

AIRCRAFT DESIGN

PROPULSION AND FUEL SYSTEM INTEGRATION

BURNER COMPRESSOR

... PISTON-PROP

INCREASING SFC

1

\7~~

CENTRIFUGAL TURBOJET

195

BURNER COMPRESSOR

1



(TYPICAL APPLICATIONS)

TURBINE

~~

--

/"'"

.........

AXIAL-FLOW TURBOJET

---

:>

ROCKET

? SCRAMJET

:..;;;>'"

RAMJET

AFTERBURNING TURBOJET

AFTERBURNING LOW-BYPASS-RATIO TURBO FAN

/ ' DRY LOW-BYPASS-RATIO TURBOFAN BYPASS AIR

TURBOJET OR TURBOFAN

I

TURBOFAN

TURBO-PROP

--&.-------+-- a

t

ZERO-LIFT ANGLE

o

~~:: Fig_ 12.4

.5

1.0

1.5

2.0

2.5

MACH NUMBER

Wing lift curve.

Fig. 12.5

Lift curve slope vs Mach number.

3.0

266

AIRCRAFT DESIGN

2~A 2+

A 2{32 ( 4+-1+ 2

AERODYNAMICS (Sexposed)

tan2A

maxI)

Sref

(F) (12.6)

(12.12)

(3=JM2-1

(12.13)

when

where

M> l/cosALE

(12.7) (12.8)

Amax I is the sweep of the wing at the chord location where the airfoil is thickest. If the airfoil lift-curve slope as a function of Mach number is not known the airfoil efficiency 'Y/ can be approximated as about 0.95. (In several text: books this term is dropped by assuming that 'Y/ equals 1.0 at all Mach numbers.) "Sexposed" is the exposed wing planform, i.e., the wing reference area less the part of the wing covered by the fuselage. "F" is the fuselage lift factor [Eq. (12.9)] that accounts for the fact that the fuselage of diameter "d" creates some lift due to the "spill-over" of lift from the wing. F

=

1.07(1

+ d/b)2

(12.9)

The wing aspect ratio "A " is the geometric aspect ratio of the complete reference planform. The effective aspect ratio will be increased by wing endplates or winglets. Endplate: where h

CLcx = 4/{3

where

{32

'Y/

267

=

Aeffective =

A (1

+ 1.9 h/b)

(12.10)

endplate height. Winglet:

Aeffective

== 1.2A

(12.11)

Thes~ effective aspect ratios should also be used in the induced drag c.alculatlOns below. Note that Eq. (12.11) for winglets is a crude approximation based upon limited data for wings of moderate aspect ratio. The ~ctual increase in effective aspect ratio due to the use of winglets is a functlOn of velocity and lift coefficient and depends upon the selected ai~foils and the relative location, geometry, and twist of the wing and wmglet. Typically, a wing with higher aspect ratio will obtain less improvement by the use of winglets.

Supersonic Lift-Curve Slope

For a wing in purely supersonic flow, the lift-curve slope is ideally defined by Eq. (12.12), as shown in Fig. 12.5. A wing is considered to be in purely supersonic flow when the leading edge is "supersonic," i.e., when the Mach cone angle is greater than the leading-edge sweep [see Eq. (12.14)].

(12.14)

The actual lift-curve slope of a wing in supersonic flight is difficult to predict without use of a sophisticated computer program. The charts in Fig. 12.6 are probably the best approximate method available. They were defined in Ref. 37 and have been used in a number of textbooks. These charts actually estimate the slope of the "normal force" coefficient (Cn ), i.e., the lift curve slope in a direction perpendicular to the surface of the wing. For low angles of attack, this is approximately equal to the liftcurve slope. To use these charts, the wing aspect ratio, taper ratio, and leading-edge sweep are employed. The six charts each represent data for wings of a different taper ratio. If a chart for the actual taper ratio of a wing is not provided, interpolation must be used. The term {3 [Eq. (12.13)] divided by the tangent of the leading-edge sweep is calculated and found on the horizontal axis of the chart. If this ratio is greater than 1.0, it is inverted and the right side of the chart must be used. Then the appropriate line is selected by calculating the wing aspect ratio times the tangent of the leading-edge sweep, and the vertical-axis value is read. To obtain the approximate slope of the lift curve, this value is then divided by the tangent· of the leading-edge sweep, if on the left side of the chart, or by {3 if on the right side of the chart. As this value is referenced to the exposed planform of the wing, it must be multiplied by (Sexposed/Sref) as in Eq. (12.6). Also, the value must be multiplied by F from Eq. (12.9) to account for the fuselage lift effect. Note that these charts give best results only for trapezoidal wings without kinks or strakes. For highly nontrapezoidal planforms, Ref. 37 contains additional estimation procedures. However, these charts are rarely used in industry where computerized "panel methods" are available. These are discussed later. Transonic Lift-Curve Slope

In the transonic regime (roughly Mach 0.85-1.2 for a swept wing) there are no good initial-estimation methods for slope of the lift curve. It is suggested that the subsonic and supersonic values be plotted vs Mach number, and that a smooth curve be faired between the subsonic and supersonic values similar to the curves shown in Fig. 12.5. Nonlinear Lift Effects

For a wing of very high sweep or very low aspect ratio (under two or three), the air escaping around the swept leading edge or wing tip will form

268

I_+-+I~'~~4-+I-+-+_(d~)~4-+-+ wi =

(a)

~ ~~ i"'---

~

f-

X-O

'j.,

(}

i>-...,~h' \'-.. "- :---. h

f-f-~i'

1:--

Ol\i.~ t--- 1"-

TAN ALE (c N ) theory Q

01:

ATAN ALE_

~

"-f-r

cD,

t--- t--- f-'

.1

--I---"'

o

.2

.4

.8

1.0

.8

.6

I

/

-_

A TAN ALE

3

/

(per rad)

./

o

.2

.6 .4 TAN ALE

-----1L-

a)

I

i.- f.--

~5

o

/

V

1/3

~ (CN",) theory

CNo)t(.Ory

V

t:::::

X

71'....,J'....

/

f-+~ TAN A C ) l!No: theory~

~

17"

-t

(per fad)

269

AERODYNAMICS

AIRCRAFT DESIGN

I

o

o

.2

.4

.6

.8

TAN ALE

d)

--~-

TAN ALE

1.0

.8

.6

.4

.2

--~-

,~

6~

X = 1/5

f4

k:

, "-

ATAN ALE

f-§_

--

"

TAN ALE (c

f-- r--:t:---

,"-"-

~Ns~~::-.. "- ....... l 1-";:::: b 1. ..t"i7:lf --=::: ::::::: be--- I-e--V ~ ~~ T ~o t."fil..- I-- V-

k f-

"-

TAN A (C N",

NeJ Iheory

l

o

~""I theiry

. 2?f o

.2

.4

.6

.8

1.0

-K .8

(per fad )

--j

3

/

1

(e)

X = 1/4 c='"

~

f.-- -'7 /i ~

c......

--"---

A TAN i'---....I"-. i'-.. 1'- t----.- -~-

-.:

f::::::, UNsW-r---... r---- f-.... 4 EPTrE:r--- t-- ~3- f-: ::::cF-

W-?f'.

-$NtTE

TAN ALE C ) No: theory

t- t-- r:-- y

1-

(per fad)

TAN A (CN,)

I

a

o

.2

ALE

.4

.6

1

t--- ~ r/

~ f-

,

~

o

a theory

~

....-

..... y

V

o

IE.

r-- '--t- :;:...-

T

I o

.2

.4

.6

1

.8

1.0

.8

--

:/

f.--

V

~

,/

CNo} theory (per fad)

......

.6

,

.4

.2

o

o

TAN ALE

(F)

X= I

-

A TAN ALE

V t7 2- l- V

J

/

1/

// /

--~-

1

6

/

~ (CN",) th~r.-I

o

~?

I-....... I---"' V

%t

c::::

SONIC

__ 3---

--- -- __2......

t--_..

rTAN A (C N )

..6. S

e)

1

17

r;::

,

1

.2

"-

t---

l.L

~E.\'11.'1

(per fad)

~

1

:::::

(CNQ ) theory 2

/

A TAN ALE

...... TAN ALE (CN",) theory

I

(J .

X - 1/2

...- /'

.6 .4 TAN ALE

b)

c)

-

./""

th~rry p!/ .... ~ (C N )."

(per fad)

i

(e)

(b)

(3 (CNeJ theory

3

II

~ b)'h~Ory V I I V . L-t--1 .~5

(per rad)

iotheory i-=

.8

1.0

.8

.4 .6 TAN ALE

.2

o

0+-~~-+-L~~~-+-L4-~~-+~~~

o

--~-

Fig. 12.6 Wing supersonic normal-foree-curve slope. (Ref. 37)

o

f)

FiR. 12.6

.2

.4

.6

.8

1.0

.8

.6

.4

.2

TAN ALE --~-

cont'd.) WinR supersonic normal-foree-curve slope. Ref. 37)

270

a strong vortex that creates additional lift at a given angle of attack. This additional lift varies approximately by the square of the angle of attack. This nonlinear increase in the slope of the lift curve is difficult to estimate and can conservatively be ignored during early conceptual design. (However, the increase in maximum lift due to vortex formation is very important. It will be discussed in the next section.) Maximum Lift (Clean) The maximum lift coefficient of the wing will usually determine the wing area. This in turn will have a great influence upon the cruise drag. This strongly affects the aircraft takeoff weight to perform the design mission. Thus, the maximum lift coefficient is critical in determining the aircraft weight; yet the estimation of maximum lift is probably the least reliable of all of the calculations used in aircraft conceptual design. Even refined windtunnel tests cannot predict maximum lift with great accuracy. Frequently an aircraft must be modified during flight test to achieve the estimated maximum lift. For high-aspect-ratio wings with moderate sweep and a large airfoil leading edge radius, the maximum lift depends mostly upon the airfoil characteristics. The maximum lift coefficient of the "clean" wing (i.e., without the use of flaps and other high-lift devices) will usually be about 90070 of the airfoil's maximum lift as determined from the 2-D airfoil data at a similar Reynolds number. Sweeping the wing reduces the maximum lift, which can be found by multiplying the unswept maximum lift value by the cosine of the quarterchord sweep [Eq. (12.15)]. This equation is reasonably valid for most subsonic aircraft of moderate sweep. C Lmax = 0.9Clmax

cOSAO. 25c

271

AERODYNAMICS

AIRCRAFT DESIGN

I*'-+----}

C

.06C

.0015 C

Fig. 12.7

Airfoil leading edge sharpness parameter.

Note: Untwisted, constant-airfoil-section wings 1.6

M:::::O.2

6.y

1.4

C c

.2

1.2

Lmax

'max

(12.15)

If a wing has a low aspect ratio or has substantial sweep and a relatively sharp leading edge, the maximum lift will be increased due to the formation of leading-edge vortices. This vortex formation is strongly affected by the shape of the upper surface of the leading edge. Leading-edge shape could be defined by the airfoil nose radius. However, the nose radius alone doesn't take into account the effect of airfoil camber on the shape of the upper surface of the airfoil leading edge. Instead, an arbitrary "leading-edge sharpness parameter" has been defined as the vertical separation between the points on the upper surface, which are 0.15% and 6% of the airfoil chord back from the leading edge (Fig. 12.7). The leading-edge sharpness parameter (or ".:ly") as a function of thickness ratio for various airfoils is provided in Table 12.1. The leading-edge sharpness parameter has been used in Ref. 37 to develop methods for the construction of the lift curve up to the stall, for low- or high-aspect-ratio wings. For high-aspect-ratio wings, Eq. (12.16) is used along with Figs. 12.8 and 12.9. The first term of Eq. (12.16) represents the maximum lift at Mach 0.2, and the second term represents the correction to a higher Mach number.

.1Y (PERCENT OF CHORD)

-

1.0

....- V-

~ F-"

:::::

/"

~

.lJ.

-

i'"""

-

.8

V V

r-;;:::::: :::::;-..

-

".0

-

....

..........:

.6 .4

» ~ ~

~~ ~~

o

10

20

30

50

40

60

ALE (deg) Fig. 12.8

Subsonic maximum lift of high-aspect-ratio wings. (Ref. 37)

Table 12.1 Airfoil type NACA NACA NACA NACA

4 digit 5 digit 64 series 65 series

Biconvex

A Y for common airfoils AY

26 26 21.3 19.3 11.8

tic

tic tic tic tic

272

A different set of charts is used for a low-aspect-ratio wing, where vortex flow dominates the aerodynamics. For use of these charts, low aspect ratio is defined by Eq. (12.18), which uses the parameter C 1 from Fig. 12.11. Maximum lift of a low-aspect-ratio wing is defined by Eq. (12.19) using Figs. 12.12 and 12.13. The angle of attack at maximum lift is defined by Eq. (12.20) using Figs. 12.14 and 12.15.

Au=O

MACH NUMBER, M .4 .6

MACH NUMBER, M .2 .4 .6

O~--~--~~~2-+

'\ 2.25

-.2

L\C L

273

AERODYNAMICS

AIRCRAFT DESIGN

L\y -.2 -I--+'>"~~-d---\

A:5 (C

Low Aspect Ratio if:

1

(12.18)

+ 1)3(COSALE)

max

-.4 2.5

-.4 -I--+--+----*~--\

12

3

I l+t ~ ... ~Jc Lmax

1/

-.6 4

10 -I---

CL

4.5

V

-.8 0

ALE =60 0

MACH NUMBER, M .6 .2 .4 0

MACH NUMBER, M

'\ 2

L\CL

&xcL

,c,~Etm 1. ~

max 6

V /

25

-.2

4

4.5

4 I----+---+--+-~ 4.5

2

-.4 -'-_.L----'_--'-_ _

0

Fig. 12.9 Mach-number correction for subsonic maximum lift of high-aspectratio wings. (Ref. 37)

High Aspect Ratio:

CLrnax = Cernax

(~rnax) + tJ.CLrnax Irnax

(12.16)

where Clmax is the airfoil maximum lift coefficient at M = 0.2. The angle of attack for maximum lift is defined in Eq. (12.17) with the help of Fig. 12.10. Note that the first and second terms represent the angle of attack if the lift curve slope were linear all the way up to stall. The second term may be approximated by the airfoil zero-lift angle, which is negative for a cambered airfoil. If the wing is twisted, the zero-lift angle is approximately the zero lift angle at the mean chord location. The third term in Eq. (12.17) is a correction for the nonlinear effects of vortex flow.

--

(12.17)

V

V

V I--"

L..,...-- ~

o

/

/

V

p

. . .V ~ V

/

7l ...V p

/

-

4

-A ~

I--- I---

I---

:

~

30

10

I 60

50

40

(deg) Fig. 12.10 Angle-of-attack increment for snbsonic maximnm lift of high-aspectratio wings. (Ref. 37) 1.5

c,':£1 Hil o

.2

.4

.6

TAPER RATIO, A

High Aspect Ratio:

V

V

(deg)

max

-.2 +--+--f-~_--=! 3

/

J

V

M 5. 06

~

J

/

~ 0.2

~y

V !S 1.2

J

8 ALE =40

>---

Fig. 12.11

.8

1.0

/"

1.0

.5

o

/

o

7 .2

-

r--

.4

.6

-

.8

1.0

TAPER RATIO, A

Taper-ratio correction factors for low-aspect-ratio wings. (Ref. 37)

274

AIRCRAFT DESIGN

,

1.4

275

'

~--!:\I

1.6 .--,

AERODYNAMICS

~:

50

~'~I-+~--~~~;--~:-+I~I~I__~

/~~i

iii)! 1.2 ~t/;[I/,tl:V7');-l]I-it---..:::~~~L~~'+-+-I-+I-+-+--l----+ :-+-I----J..--li-L i -li-Li_iL.;' I,

40

l

( C)

~ ~>1.0 r!,-

Lmax base

V/

1.0

,~~

1,:

~

I~~

'

,

I

I

1 I I~

1

,

..............

I

ot '

L\Yf--~JpPkR ILI~ITI ~LOW-ASPECT-RATIO

"

,

I

I

-ff--i'--t-'-+---+--+----I--f--+--~'

'I r

I

I

T

~!

e

,

3.6

4.0

o

Fig. 12.12

.4

.8

1.6

1.2

(C 1

2.0

+ I) ~ COS

2.4

2.8

I 3.2

.I

3.6

4.0

4.4

ALE

Angle of attack for subsonic maximum lift of low-aspect-ratio wings.

20

.4 M

./

.2

---

o

-.2

o

~

/

Fig. 12.13

~

s-y- V ~ f,

~ :=---V

L-

~

"-

-....

/

~

~

'"

2

4

6

+ I)

8

10

12

14

A TAN ALE

Maximum-lift increment for low-aspect-ratio wings. (Ref. 37)

Low Aspect Ratio: aCL

max

A COS ALE [I 10

L\oc Lmax

(deg)

CL max -- (CL max )base +.:lCL max

(12.19)

+ .:lacLmax

(12.20)

= (ac

) Lmax base

. At trans~ni~ and supersonic speeds, the maximum lift a wing can achieve usually hmited by structural considerations rather than aerodynamics. Unless the aircraft is flying at a very high altitude the available maximum lift at Mach 1 is usually enough to break the win~s off! As a conservative assumption, it can be assumed that the maximum lift available at Mach 0.6 will remain constant at higher Mach numbers. Actually, the maximu~ lift will usually increase through the transonic regime, and then fall agam at supersonic speeds.

~

+ (2)..)']

~

~ ~ t--. o ~ t-:- pi~l,../ ~p-

-10

(C 2

IS

~ o

Fig. 12.14 (Ref. 37)

Maximum subsonic lift of low-aspect-ratio wings. (Ref. 37)

BOJDERLiNE ASPECT RATlO-

~

4.4

--

~

10

BORDERLlNE~

3.2

I

~

I I V I' I I .4~-L-+~~~~1 ~-'--~r-~~~-+~VL-~L-+-__~~~ o .4 .8 1.2 I~ ~o ~4 ~8 (C 1 + 1) ~ COS ALE

UPPER LIMIT OF t/LOW-ASPECT-RATIO j/RANGE

r-- t--- +--

LOW ASPECT RATIO

RA TIL f?' ASPECT :---r---i,

,'/ T

,T,

1

.......

'.5

.8~i1=+=~t=ttjjjjljll~~I:::::::fJ~~~/3l~-~~AN~G~E[~ I~ I~I' /Low ASPECT ~T10 T :- I , 1 .6

r--...

M

p

V- ~

C9::: V' ~ V

l.--:::::: ~ p

....., i:::::::::=== ~ r-- f--- ~

-~

o

2

4

6

8

IO

12

14

(C 2 + I) A TAN ALE

Fig. 12.15 Angle-of-attack increment for subsonic maximum lift of low-aspect-ratio wings. (Ref. 37)

Maximum Lift with High-Lift Devices There is always a basic incompatibility in aircraft wing design. For cruise efficiency a wing should have little camber and should operate at a high wing-loading. For takeoff and landing a wing should have lots of lift, which means a lot of camber and a low wing-loading. In the history of aviation almost every imaginable device for varying the wing camber and wing area has been attempted, including a. wing wi~h a telescoping outer panel, a fabric membrane which unfurls behmd the wmg, a device which pivots out from the fuselage forming an extended flap, and even something called a "mutable" wing having variable span, camber, and sweep (Ref. 38). Figure 12.16 illustrates the commonly used high lift flaps. The plain flap is simply a hinged portion of the airfoil, typically with a flap chord "C/'

[

c

r-c'~ -; ~.

C SPLIT FLAP

-L SLOTTED FLAP

l: ~ FLAP~

crc___~-~ LEADING EDGE SLOT

~

L

DOUBLE SLOTTED FLAP

J~- L Fig. 12.16

>~a.

~

-=c-~ TRIPLE SLOTTED FLAP

Flap types.

LEADING EDGE FLAP

"'L (j'

~

b:

~

SLOTTED LEADING EDGE FLAP (SLAT)

!

KRUGER FLAP

~

WING IN TOP VIEW

~

of 30070 of the airfoil chord. The plain flap increases lift by increasing camber. For a typical airfoil, the maximum lift occurs with a flap deflection angle of about 40-45 deg. Note that ailerons and other control surfaces are a form of plain flap. The split flap is like the plain flap except that only the bottom surface of the airfoil is hinged. This produces virtually the same increase in lift as the plain flap. However, the split flap produces more drag and much less change in pitching moment, which may be useful in some designs. Split flaps are rarely used now but were common during World War II. The slotted flap is a plain flap with a slot between the wing and the flap. This permits high-pressure air from beneath the wing to exit over the top of the flap, which tends to reduce separation. This increases lift and reduces drag. The "Fowler-type" flap is like a slotted flap, but mechanized to slide rearward as it is deflected. This increases the wing area as well as the camber. Fowler flaps can be mechanized by a simple hinge located below the wing, or by some form of track arrangement contained within it. To further improve the airflow over the Fowler flap, double- and even triple-slotted flaps are used on some airliners. These increase lift but at a considerable increase in cost and complexity. Aft flaps do not increase the angle of stall. In fact, they tend to reduce the stall angle by increasing the pressure drop over the top of the airfoil, which promotes flow separation. To increase the stall angle, some form of leading-edge device is required, as shown in Fig. 12.17. The leading-edge slot is simply a hole which permits high-pressure air from under the wing to blow over the top of the wing, delaying separation and stall. Usually such a slot is fixed, but may have closing doors to reduce drag at high speeds.

WING STRAKE OR LEADING EDGE EXTENSION (LEX)

17=--~--

SLOTTED FOWLER

PLAIN FLAP

277

AERODYNAMICS

AIRCRAFT DESIGN

276

VORTEX

Fig. 12.17

Leading edge devices.

A leading-edge flap is a hinged portion of the leading edge that droops down to increase camber. This has the effect of increasing the curvature on the upper surface. The increase has been shown to be a major f~ctor in determining maximum lift. Leading-edge flaps are usually us~d for Impr~v­ ing the transonic maneuvering performance of high-speed fIghters, WhICh need a thin wing for supersonic flight. A slotted leading-edge flap ("slat") provides increased camber, a slot, and an increase in wing area. Slats are the most widely used leading-edge device for both low-speed and transonic maneuvering. At tra~sonic spe~d~, slats are also useful for reducing the buffetting tendency whIch may lImIt the usable lift. At Mach 0.9 the use of slats improved the usable lift of the F-4 by over 50%. . The Kruger flap is used mostly by large airliners. It works as an ~Ir da~, forcing air up and over the top of the wing. Kruger flaps are lIghter III weight than slats but produce higher drag at !he ,~ower ang!es .of. attack. The wing strake, or "Leading Edge ExtensIOn (LEX), IS sI~Illlar to the dorsal fin used on vertical tails. Like dorsal fins, the LEX at hIgh angle of attack produces a vortex that delays separation and stall. Unfortut;Iatel y , a LEX tends to promote pitch-up tendencies and so must be used wIth care. Figure 12.18 illustrates the effects these high-lift devices ha~e upo~ the lift curve of the wing. The nonextending flaps such as the plaIll, splIt, ?r slotted flaps act as an increase in camber, which moves the angle of zero-lIft

278

AIRCRAFT DESIGN

/"

SLOTTED FLAP

CLEAN

a

a

NON EXTENDING FLAPS

LEADING EDGE SLOT

CLEAN

--~"---------------a LEADING EDGE FLAP OR SLAT ~--~~-------------------a EXTENDING FLAPS

in the leading edge acts as a reduction in the effective angle of attack as measured from the leading edge to the trailing edge. Note that a leading edge slat, which increases wing area, also increases the slope of the lift curve much as does a Fowler flap. Leading-edge devices alone do little to improve lift for takeoff and landing, because they are effective only at fairly high angles of attack. However, they are very useful when used in combination with trailing-edge flaps because they prevent premature airflow separation caused by the flaps. The wing strake, or LEX, delays the stall at high angles of attack (over 20 deg). Also, the area of the LEX provides additional lift, thus increasing the slope of the lift curve. However, the LEX does little to increase lift at the angles of attack seen during takeoff and landing. The LEX does not delay the premature stall associated with trailing-edge flaps. There are many complex methods for estimating the effects of high-lift devices, some of which are detailed in Ref. 37. For initial design, Eqs. (12.21) and (12.22) provide a reasonable estimate of the increase in maximum lift and the change in the zero-lift angle for flaps and leading-edge devices when deployed at the optimum angle for high lift during landing. .:lCe values should be obtained from test data for the selected airfoil, or may b~axapproximated from Table 12.2. For takeoff flap settings, lift increments of about 60-80070 of these values should be used. The change in zerolift angle for flaps in the 2-D case is approximately -15 deg at the landing setting, and -10 deg at the takeoff setting.

Snapped)

.:lCL max = .:lCemax ( - S . ref

Fig. 12.18

Effects of high lift devices.

~-------------------a WING STRAKE (LEX)

to th.e left and increases the maximum lift. The slope of the lift curve remaInS unc~anged, and the angle of stall is somewhat reduced. An extendIn~ flap such as the Fowler type acts much like the other flaps as far as zero lift angle and stall angle. However, the wing area is increased as the flap deflects, so the wing generates more lift at any given angle of att~ck compared to the nonextending flap. SInce th~ lift coefficient is .referenced to the original wing area, not the ~x~ended WIng area, th~ effective slope of the lift curve for an extending flap IS Incr~~sed b~ approxImately the ratio of the total extended wing area to the ongInal WIng area. Double- and triple-slotted flaps act much like single-slotted Fowler flaps but the maximum lift is increased. ' A leading-edge slot acts only to delay stall. A leading-edge flap or slat delays the ~tall, but. also has the effect of reducing the lift at a given angle of attack (I.e., the lift curve moves to the right). This is because the droop

279

AERODYNAMICS

.:laOL = (.:laOdairfoil (

Table 12.2

cosAH.L.

Snapped) ----s;:;cosA

H . L.

Approximate lift contributions of high-lift devices

High-lift device

.1.C1max

Flaps Plain and split Slotted Fowler Double slotted Triple slotted

0.9 1.3 1.3 c'lc 1.6 c'lc 1.9c'/c

Leading edge devices Fixed slot Leading edge flap Kruger flap Slat

0.2 0.3 0.3 0.4 c' Ie

(12.21)

(12.22)

In Eqs. (12.21) and (12.22), "H.L." refers to the hinge line of the highlift surface. "Snapped" is defined in Fig. 12.19. The lift increment for a leading-edge extension may be crudely estimated as 0.4 at high angles of attack. Other methods for increasing the lift coefficient involve active flow control using either suction or blowing. Suction uses mechanical air pumps to suck the thickening boundary layer off the wing before it causes separation. This increases the stall angle of attack, and therefore increases maximum lift in a manner similar to leading-edge flaps. Blowing uses compressor bleed air or compressed air provided by a mechanical air pump to prevent flow separation and increase the freestreamflow turning. Typically, the compressed air is exited through rearwardfacing slots over the flaps or leading-edge flaps. 12.5

281

AERODYNAMICS

AIRCRAFT DESIGN

280

snAPPED

LEADING EDGE DEVICES

PARASITE (ZERO-LIFT) DRAG

Equivalent Skin-Friction Method Two methods for the estimation of the parasite drag ("CDO ") are presented below. The first is based upon the fact that a well-designed aircraft in subsonic cruise will have parasite drag that is mostly skin-friction drag plus a small separation pressure drag. The latter is a fairly consistent percentage of the skin-friction drag for different classes of aircraft. This leads to the concept of an "equivalent skin friction coefficient" (Cle ), which includes both skin-friction and separation drag. Cle is multiplied by the aircraft's wetted area to obtain an initial estimate of parasite drag. This estimate [Eq. (12.23) and Table 12.3J is suitable for initial subsonic analysis and for checking the results of the more detailed method described in the next section. C

- C Swet Ie Sref

DO -

Table 12.3

c -c DO -

Ie

(12.23)

"Flapped" wing area.

component BuildUp Method .' Th component buildup method estimates the subsonlc parasl~e dr.ag. of

each ~omponent of the aircraft using ~,~~:I;!~~o~~t(~~~~~:tl~~~~~~~:S dhrag coeffici~~~g(~~~ ~~~i:c~~~~e~:~~~ion. Then the interference effects on t e pressure d t' ated as a factor "Q" and the total component the component rag are es 1m FF d Q d is determined as the product of the wetted area, CJ> ' an . . r~~ote that the interference factor Q should not be confused with dynamIc

pres~ure

Equivalent skin friction coefficients

Swet Sref

Bomber and civil transport Military cargo (high upsweep fuselage) Air Force fighter Navy fighter Clean supersonic cruise aircraft Light aircraft - single engine Light aircraft - twin engine Prop seaplane Jet seaplane

Fig. 12.19

0.0030 0.0035 0.0035 0.0040 0.0025 0.0055 0.0045 0.0065 0.0040

q.) d (C ) for special features of an aircraft such as MIscellaneous rags Dmisc db are unretracted landing gear, an upswept aft fuselage, an ase ~re~ flhaps, . t d and added to the total, along with estimated contn?utlO~s t en estlma e C ) S b nic parasite-drag bUlldup IS for lea~ages and protubehrancte~ ( Db:~ript ~,;?, indicates that those values shown in Eq. (12.24), were e su are different for each component.

(12.24)

For supersonic flight, the skin-friction contribution is simpl~ the flatplate skin friction coefficient times the wet!ed area. ~ll supersonlc pressure drag contributions (except base drag) are included in. th~ w~ve-drag term, which is determined from the total aircraft volume dlstnbutlOn.

For transonic flight, a graphical interpolation between subsonic and supersonic values is used. Supersonic and transonic drag calculations are discussed later. Flat-Plate Skin Friction Coefficient

The flat-plate skin friction coefficient Cf depends upon the Reynolds number, Mach number, and skin roughness. The most important factor affecting skin-friction drag is the extent to which the aircraft has laminar flow over its surfaces. At a local Reynolds number of one million, a surface with turbulent flow will have a friction drag coefficient as much as three times the drag coefficient of a surface with laminar flow. Laminar flow may be maintained if the local Reynolds number is below roughly half a million, and only if the skin is very smooth (molded composite or polished aluminum without rivets). Most current aircraft have turbulent flow over virtually the entire wetted surface, although some laminar flow may be seen towards the front of the wings and tails. A typical current aircraft may have laminar flow over perhaps 10-20070 of the wings and tails, and virtually no laminar flow over the fuselage. A carefully designed modern composite aircraft such as the Piaggio GPI80 can have laminar flow over as much as 50% of the wings and tails, and about 20-35% of the fuselage. For the portion of the aircraft that has laminar flow, the flat-plate skin friction coefficient is expressed by Eq. (12.25). Note that laminar flow is unlikely at transonic or supersonic speeds, unless great attention is paid to shaping and surface smoothness. Laminar:

Cf

=

1.328/.JR

44.62(£/k)1.

053

M1.l

(12.29)

6

Once laminar and turbulent flat-plate skin friction coefficients hav 7been calculated, an "average" coefficient can be calculated as the weIg~ted average of the two. This requires estimation o~ the percentage of lamInar flow which can be attained. This estimation is a Judgment call base~ on past experience as discussed above, and one must revie~ the ~urrent lIterature to determine how much laminar flow can be attaIned WIth current state of the art. Component Form Factors

Form factors for subsonic-drag estimation are prese.nted in Eqs. (12.30-12.32). These are considered valid up to the drag-dIverg~nt Mach number. In Eq. (12.30), the term "(X/C)m" is the chordwi~e lo.catlO~ o~ the airfoil maximum thickness point. For most low-speed aIrfOIls, thIS IS at about 0.3 of the chord. For high-speed airfoils this is at ~bout 0.5 of the chord. Am refers to the sweep of the maximum-thickness lIne. Wing, Tail, Strut, and pylon: FF

~ [I + (!:). m+ 100m'] [1.34M' "(cosA. )'"]

(12.30)

Fuselage and Smooth Canopy: FF

(12.26)

The "£" in Eq. (12.26) is the characteristic length. For a fuselage, £is the total length. For a wing or tail, £ is the mean aerodynamic chord length. For turbulent flow, which in most cases covers the whole aircraft, the flat-plate skin friction coefficient is determined by Eq. (12.27). Note that the second term in the denominator, the Mach number correction, goes to 1.0 for low-subsonic flight. C _

=

f)

(12.31)

60 ( 1+ P + 400

0.455 (log lOR f58 (1 + 0.144M2)o.65

FF

where

= I

(12.33)

£ - d - .J(4hr) Amax

A tail surface with a hinged rudder or elevator will have a form factor about 10% higher than predicted by Eq. (12.30) due to the extra drag of the gap between the tail surface and its control surface.

(12.27) Table 12.4

than indicated by this equation. This is accounted for by the use of a "cut-off Reynolds number," which is determined from Eq. (12.28) or (12.29) using the characteristic length £ (feet) and a skin-roughness value "k" based upon Table 12.4. The lower of the actual Reynolds number and the cut-off Reynolds number should be used in Eq. (12.27). R eutoff = 38.21(£/k)J.053

(12.32)

+ (0.35/f)

f-~ -

If the surface is relatively rough, the friction coefficient will be higher

Subsonic:

Reutoff =

Nacelle and Smooth External Store: R =pV£/JL

f -

Transonic or Supersonic:

(12.25)

where Reynolds number is:

Turbulent:

283

AERODYNAMICS

AIRCRAFT DESIGN

282

(12.28)

Skin roughness value (k)

k (ft) Surface Camouflage paint on aluminum Smooth paint Production sheet metal Polished sheet metal Smooth molded composite

3.33 x 10 5 2.08 X 10- 5 1.33 X 10- 5 0.50xlO- 5 0.17xlO- 5

284

285

AERODYNAMICS

AIRCRAFT DESIGN

300 GALLON TANK ON WING

D/q 2.5

300 GALLON TANK ON FUSELAGE

2.0

150 GALLON TANK ON WING 150 GALLON TANK ON FUSELAGE

SINGLE WEDGE

I/r"S;::cELLE . . I

I

I~ Fig. 12.20

Inlet boundary layer diverter.

:t f

I

~'

Equation (12.31) is mainly used for estimation of the fuselage form factor, but can also be used for a blister or fairing such as a pod used for landing-gear stowage. For a fuselage with a steep aft-fuselage closure angle in front of a pusher propeller, the separation drag will be lower than predicted using this formfactor equation. A square-sided fuselage has a form factor about 40070 higher than the value estimated with Eq. (12.31) due to additional separation caused by the corners. This can be somewhat reduced by rounding the corners. A flyingboat hull has a form factor about 50% higher, and a float has a form factor about three times the estimated value. Equation (12.31) will predict the form factor for a smooth, one-piece fighter canopy such as seen on the F-16. For a typical two-piece canopy with a fixed but streamlined windscreen (i.e., F-15), the form factor calculated with Eq. (12.31) should be increased by about 40%. A canopy with a flatsided windscreen has a form factor about three times the value estimated with Eq. (12.31). The external boundary-layer diverter for an inlet mounted on the fuselage can have a large drag contribution. Equations (12.34) and (12.35) estimate the form factors to use for a double-wedge and single-wedge diverter, where the Reynolds number is determined using f and the wetted area is defined as shown in Fig. 12.20. Remember to double the drag if there are two inlets. Double Wedge:

1.5

FF = 1 + (d/f)

(12.34)

FF = 1 + (2dl£)

(12.35)

o .4

.5

Component Interference Factors

Parasite drag is increased due to the mutual interference between components. For a nacelle or external store mounted directly on the fuselage or wing, the interference factor Q is about 1.5. If the nacelle or store is mounted less than about one diameter away, the Q factor is about 1.3. If it

.8

.9

1.0

MACH NUMBER

Fig. 12.21

External stores drag-fuel tanks.

is mounted much beyond one diameter, the Q factor approaches 1.0. Wing tip-mounted missiles have a Q factor of ~bout 1.25.. . For a high-wing, a mid-wing, or a well-fIlletted low wmg, the mterfere~ce will be negligible so the Q factor will be about 1.0. An unfilletted low wmg can have a Q factor from about 1.1-1.4. . The fuselage has a negligible interference factor (Q = 1.0) m .most cases. Also, Q = 1.0 for a boundary-layer diverter. For tail surfa~es, mterfer~nce ranges from about three percent (Q = 1.03) for a clean v,-tall to about eight percent for an H-tail. For a conventional tail, four to five percent may be assumed (Ref. 8). . . Component parasite drags can now be determl.ned usmg Eq. (12.24) and the skin-friction coefficients, form factors, and mterference factors. Miscellaneous Drags

Single Wedge:

.7

.6

The drag of miscellaneous items can be determine~ separately using a variety of empirical graphs and equations, and then addmg the results to the . ' parasite drags determined above. While the drag of smooth external stores can be estimated usm? Eq. (12.31), the majority of external stores are in fact not very smooth. Figures 12.21 and 12.22 provide drag estimates for external fuel tanks and weapons, presented as drag divided by dynamic pressure (D-over-q or D/q).

286

AIRCRAFT DESIGN

AERODYNAMICS

6-500 Ib BOMB CLUSTER (NOT INCLUDING RACK DRAG)

D/q _ft2

6-250 Ib BOMB CLUSTER (NOT INCLUDING RACK DRAG)

2.5

287

D/q _ft2

2.5

2.0 2.0

1.5 MULTIPLE BOMB CLUSTER RACK

1.5 2000 Ib BOMB ON FUSELAGE

1.0

2000 Ib BOMB ON WING

1.0

.5

AIM-9 MISSILE ANI PYLON

o

.5

FUSELAGE STORES PYLON .4

.5

.6

.7

.8

.9

J.U

1.1

MACH NUMBER

Fig. 12.22

i.........===-======....----~~--

1.2

Bomb and missile drag.

o .5

.6

.7

.8 .9 MACH NUMBER

Fig. 12.23

Dlq has units of square feet, and so is sometimes called the "drag area." Dlq divided by the wing reference area yields the miscellaneous parasite drag coefficient. Note that pylon and bomb-rack drag as estimated using Fig. 12.23 must be added to the store drag. Most transport and cargo aircraft have a pronounced upsweep to the aft fuselage (Fig. 12.24). This increases the drag beyond the value calculated using Eq. (12.31). This extra drag is a complicated function of the fuselage cross-sectional shape and the aircraft angle of attack, but can be approximated using Eq. (12.36) where "u" is the upsweep angle (radians) of the fuselage centerline and Amax is the maximum cross-sectional area of the fuselage.

(12.36) The landing-gear drag is best estimated by comparison to test data for a similar gear arrangement. Such data for a variety of aircraft is available in Refs. 7, 8, 28, and others. If such data is not available, the gear drag can be estimated as the summation of the drags of the wheels, struts, and other gear components using the data in Table 12.5 (largely from Ref. 8). These values times the frontal area of the indicated component yield Dlq values, which must be divided by the wing reference area to obtain parasitedrag coefficients. To account for mutual interference it is suggested that the sum of the gear component drags be multiplied by 1.2. Also, the total gear drag should be increased by about 71170 for a retractable landing gear in which the gear wells are left open when the gear is down.

WING STORES PYLON

1.1

1.0

Pylon and bomb rack drag.

(u IS IN RADIANS)

Ng. 12.24 Table 12.5

Fuselage upsweep.

Landing gear component drags D/q

Frontal area (Ft 2) Regular wheel and tire Second wheel and tire in tandem Streamlined wheel and tire Wheel and tire with fairing Streamline strut (1/6< t/c< 1/3) Round strut or wire Flat spring gear leg Fork, bogey, irregular fitting

0.25 0.15 0.18 0.13 0.05 0.30 1.40 1.0-1.4

AERODYNAM ICS

AIRCRAFT DESIGN

288

Note that landing-gear drag is actually a function of lift. The more lift the aircraft wing is producing, the greater the velocity of the airflow over the top of the wing and, conversely, the lesser the airflow velocity underneath the wing where the gear is located. Hence, at higher lift coefficients the gear drag is reduced. This can be ignored for initial analysis. Strut, wire, and fitting data in Table 12.4 may also be used to estimate the extra drag for a braced wing or biplane. The optimal thickness ratio considering both aerodynamic and structural efficiency is about 0.19 for a strut in tension and about 0.23 for a strut in compression. Flaps affect both the parasite and induced drag. The induced effect is due to the change in the lift distribution, but is relatively small and can be ignored for initial analysis. The flap contribution to parasite drag is caused by the separated flow above the flap, and can be estimated using Eq. (12.37) for most types of flap. Note that this is referenced to wing area. Typically the flap deflection is about 60-70 deg for landing and about 20-40 deg for takeoff. Light aircraft usually take off with no flaps. aC DO flap

=

0 0023 flap span 0 • wing span flap

(12.37)

where Oflap is in deg. Note that this is a very rough estimate! Many aircraft have some form of speed brake. Typically these are plates which extend from the fuselage or wing. Fuselage-mounted speed brakes have a Dlq of about 1.0 times the speed-brake frontal area, while wingmounted speed brakes have a Dlq of about 1.6 times their frontal area if mounted at about the 60% of chord location. Speed brakes mounted on top of the wing will also disturb the airflow and spoil the lift, and so are called "spoilers." These further reduce landing distance by transferring more of the aircraft's weight to the landing-gear which increases the braking action. Base area produces a drag according to Eqs. (12.38) and (12.39) (Ref. 40). "Abase" includes any aft-facing flat surfaces as well as the projected aft-facing area for any portions of the aft fuselage that experience highly-separated airflow. Roughly speaking, this should be expected any place where the aft fuselage angle to the freestream exceeds about 20 deg. As previously mentioned, a pusher propeller may prevent aft-fuselage separation despite an aft fuselage angle of 30 deg or more. Subsonic: Supersonic:

(D /q)base = [0.139 + 0.419(M -0. 161)2]Abase

(12.38)

(D / q)base = [0.064 + 0.042(M - 3.84)2]A base

(12.39)

Fighter-type canopies have been discussed above. For transport and lightaircraft windshields that smoothly fair into the fuselage, an additional Dlq of about 0.07 times the windshield frontal area is suggested. A sharp-edged, poorly-faired windshield has an additional Dlq of about 0.15 times its frontal area.

289

An open cockpit has a Dlq of about 0.50 times the windshield frontal area. For an aircraft with an unenclosed cockpit, such as a hang-glider or ultralight, a seated person has a Dlq of about 6 ft2. This reduces to a Dlq of 1.2 ft2 in the prone position. An arresting hook for carrier operation adds a DIq of about 0.15 ft2. The smaller emergency arresting hook for Air Force aircraft adds a Dlq of about 0.10 ft2. Machine-gun ports add a Dlq of about 0.02 ft2 per gun. A cannon port such as for the M61 adds a Dlq of about 0.2 ft2. Leakage and Protuberance Drag Leaks and protuberances add drag that is difficult to predict by any method. Leakage drag is due to the tendency of an aircraft to "inhale" through holes and gaps in high-pressure zones, and "exhale" int~ the 10,",:pressure zones. The momentum loss of the air "inhaled" ~~ntnbu~es dIrectly to drag, and the air "exhaled" tends to produce addItIonal aIrflow separation. . Protuberances include antennas, lights, and such manufactunng defects as protruding rivets and rough or misaligned skin panels. Typically these drag increments are estimated as a percent of the total parasIte drag. For a normal production aircraft, leaks and protuberance drags can be estimated as about 2-5070 of the parasite drag for jet transports or bombers, 5-10% for propeller aircraft, and 10-15% f~r current-~esign ~ighters (5-10% for new-design fighters). If special care IS taken dunng desIgn and manufacturing, these drag increments can be reduced to near zero but at a considerable expense. An aircraft with variable-sweep wings will have an additional protuberance drag of about 3% due to the gaps and steps of the wing pivot area. Stopped-propeller and Windmilling Engine Drags The specifications for civilian and military aircraft require takeoff and climb capabilities following an engine failure. Not only does this reduce the available thrust, but the drag of the stopped propeller or windmilling engine must be considered. Data on the drag of a stopped or windmilling propeller are normally obtained from the manufacturer. For a jet engine, detailed knowledge of the characteristics of the engine, inlet, and nozzle are required to estimate the drag from a stopped or windmilling engine. In the absence of such data, the following rough approximations can be used. . . For a stopped propeller, Ref. 8 indicates that the subsomc drag coeffIcient will be about 0.1 based upon the total blade area if the propeller is feathered (turned so that the blades align with the airflo~~. If ~he propeller has fixed pitch and cannot be feathered, the drag coeffIcIent IS abou~ 0.8. To determine the total blade area it is necessary to know or to estImate the propeller "solidity" (a), the ratio between the total blade area and th.e propeller disk area. This can be shown to equal the number of blades dIvided by the blade aspect ratio and 1C'. For a typical blade aspect ratio of 8, the solidity will be 0.04 times the number of blades. A small piston-prop engine will generally use a two-

II,

290 AIRCRAFT DESIGN AERODYNAM ICS

291

bladed propeller. A fast piston-prop or a small turboprop will use a threebladed propeller, while a large turboprop may use a four-bladed propeller. Drag of a feathered propeller can be roughly estimated by Eq. (12.40). For an un feathered, stopped propeller, the 0.1 term is replaced by 0.8. (D I q )feathered prop = 0.1 aApropelJer disk (12.40)

For jet engines, Ref. 9 indicates that the subsonic drag coefficient of a windmilIing turbojet engine will be about 0.3, referenced to the flow area at the engine's front face. Thus, the drag of a windmilling turbojet will be approximately: (D / q )windmilling jet = O. 3Aengine

FLIGHT DIRECTION

CROSS·SECTION AREA·

front face

(12.41)

Supersonic Parasite Drag

The supersonic parasite drag is calculated in a similar fashion to the subsonic drag, with two exceptions. First, the supersonic skin-friction drag does not include adjustments for form factors or interference effects (i.e., FF = Q = 1.0). Second, a new term, wave drag, is added. This aCCOunts for thedefined pressure due to shock formation. Supersonic parasite-drag buildup is in drag Eq. (12.42).

• ROJECTED FORWARD ONTO A PLANE PERPENDICULAR TO THE FLIGHT DIRECTION P

(12.42)

The supersonic turbulent skin friction coefficient was previously presented in Eq. (12.27), using the cutoff Reynolds number from Eq. (12.29). Miscellaneous drag calculations for supersonic flight have been presented above, Where appropriate. Many of the items that produce miscellaneous drag will not appear on a supersonic aircraft (floats, open cockpits, etc. I). The drag due to leaks and protuberances in supersonic flight follows aboutonly. the same percentages presented above, applied to the skin-friction drag The wave drag in supersonic flight will often be greater than all the other drag put together. Wave drag is pressure drag due to shocks, and is a direct result of the way in which the aircraft's volume is distributed. An ideal volume distribution is produced by the Sears-Haack body (Ref. 16), which was shown in Fig. 8.2. A Sears-Haack body, as defined by Eq. (12.43), has a wave drag as in Eq. (12.45). This is the minimum possible wave drag for any closed-end body of the same length and total volume. _r = rmax

[1 _(~)2]O.75

where r = the cross-section radius f = the longitudinal dimension

£/2

(12.43)

FUSELAGE STATION

FUSELAGE STATION

Mach-plane cut vo Iu me distribution (two roll angles).

Fig. 12.25

and

-£/2~x~£/2

(Dlq)wave

97r =""2

(12.44)

(A max)

2

(12.45)

-£-

w~~~ ~~~ar area-rule .theo~y says ~hat !e~ ~~;~ of a body of revolution · aft at Mach 1.0 is I?en.hca~ to t e w other words, the actual crossaI~cr me volume-dIstnbutIOn plot. In. no effect on wave drag wuh ::~ ~~ape at a given long;tud;nallocat:~~n~~ the beginning of drag rise, is roughly 0.08 slower in Mach number than MDD and is labeled E. T? co~plete the transonic-drag-rise curve from these points, draw a straIght lille through points Band C, extending almost to the horizontal axis. Then, draw a curve from Mer through MDD which fairs smoothly into the straight line as shown. If a smooth curve cannot be drawn, the Mer point (E) should be moved until an approximately circular arc can be drawn. Finally, draw a smooth curve connecting B to A. This crude technique may be used even for subsonic transport aircraft. The supersonic wave drag (point B) is determined from Eq. (12.46) although the aircraft will never fly at this speed. When calculating the Sears-

DRAG RISE

LEAKS & PROTUBERANCES MISCELLANEOUS

FORM & INTERFERENCE

SKIN FRICTION

DRAG

oL-------------+---~~~--------~ MACH NUMBER MDD 1.0 1.2 o I

Fig. 12.30

Complete parasite drag vs Mach number.

Haack Dlq for Eq. (12.46), remember to subtr~ct from th~ aircraft length the portions of the aircraft where the cross-sectiOnal area IS constant. Complete Parasite-Drag Buildup Figure 12.30 illustrates the complete buildup o~ parasite drag vs. Mach number for subsonic, transonic, and supersomc flight. The s~bsomc drag consists of the skin-friction drag including form factor and illterferenc~, plus miscellaneous drag and leak and protuberance drag. The. supersomc drag includes the flat-plate supersonic skin-friction drag, miscellaneous drag, leak and protuberance drag, and wave drag. . In the transonic regime, the skin friction-drag is estimated simply ~y drawing a straight line between the skin-friction. dra~ ~t MDD (which illcludes form factor and interference) and the sklll-~nct~on drag at Ml.2 (which does not). This does not reflect any reductiOn ill drag, merely a change in bookkeeping. The pressure drags repre~ented by the form and interference terms at subsonic speeds are included ill the wave-drag term at supersonic speeds. . . In Fig. 12.31, the actual parasite drag and drag rIse IS shown for a number

of aircraft.

o

.OO2--+--4

+-___________________________________

~~~~-----~

o

1.0

Fig. 12.29 Transonic drag rise estimation.

_____------.

E

MDD

1.20

MACH NUMBER

12.6 DRAG DUE TO LIFT (INDUCED DRAG) The induced-drag coefficient at moderate angles .of at~ack is proportional to the square of the lift coefficient with a proportiOnalIty factor called the "drag-due-to-lift factor," or "K" [see Eq. (12.4)]. .. . Two methods of estimating K will be presented. The fIrSt IS the claSSIcal method based upon e, the Oswald span efficiency factor. Methods are pre-

299

AERODYNAMICS

AIRCRAFT DESIGN

298

INTERFERENCE FACTOR (11)

F86 CDO

1.0

0.05 .90

F4 0.04 FI05

.80

RA5C

0.03

.70 FI06 .60

0.02

.50

B70 0.01

.40

0 0.5

1.5

1.0

2.5

2.0 MACH NUMBER

Fig. 12.31

Parasite drag and drag rise.

.20

sented for subsonic monoplanes and biplanes along with an empirical equation for supersonic speeds. The second method for the estimation of K is based upon the concept of leading-edge suction and provides, for high-speed designs, a better estimate of K, one that includes the effects of the change in viscous separation as lift coefficient is changed. This method also reflects the choice of the wing design lift coefficient on the drag due to lift at different lift coefficients.

Oswald Span Efficiency Method According to classical wing theory, the induced-drag coefficient of a 3-D wing with an elliptical lift distribution equals the square of the lift coefficient divided by the product of aspect ratio and 7r. However, few wings actually have an elliptical lift distribution. Also, this doesn't take into account the wing separation drag. The extra drag due to the non elliptical lift distribution and the flow separation can be accounted for using e, the "Oswald span efficiency factor." This effectively reduces the aspect ratio, producing the following equation for K. K=_I_ 7rAe

.30

(12.48)

The Oswald efficiency factor is typically between 0.7 and 0.85. Numerous estimation methods for e have been developed over the years, such as those by Glauert and Weissinger. These tend to produce results higher than the e values of real aircraft. More realistic estimation equations based upon actual aircraft (Ref. 45) are presented below.

.10 .0

o

.20

.10

Fig. 12.32

.30

.40

(

GAP ) AVERAGE SPAN

Prandtl's biplane interference factor. (Ref. 12)

Straight-Wing Aircraft: e = 1. 78(1- 0.045A 0.68) - 0.64

(12.49)

Swept-Wing Aircraft: e = 4.61(1- 0.045A 0.68)(cosALE)015 - 3.1

(12.50)

(ALE> 30 deg) If the wing has end-plates or winglets, the effective aspect ratio from Eq. (12 10) or (12.11) should be used in Eq. (12.48). . Drag-due-to-lift for a biplane was first analytic~lly determIned by Max Munk in 1922, based upon the calculation of an eqUivalent monoplane span providing the same wing area and the same drag. . . i Prandtl developed an interference factor (a, sho~~ In FIg. 12.32) that s d· E (1251) to determine a biplane span effiCIency factor (Ref. 12). ~~e ~~la~~ as~ect ratio is the square of the longer span divided by the total area of both wings. Biplane:

(12.51)

300

AIRCRAFT DESIGN

AERODYNAMICS

301

where shorter span/longer span lift on shorter wing/lift on longer wing (approximately = area of shorter wing/area of longer wing) JL = r =

At supersonic speeds, the drag-due-to-lift factor (K) increases substantially. In terms of Oswald efficiency factor, e is reduced to approximately 0.3-0.5 at Mach 1.2. Equation (12.52) provides a quick estimate of K at supersonic speeds (Ref. 6), although the leading-edge suction method presented later is preferable. . S S upersomc peeds:K =

2

A (M - 1) cos A 4A -J M2 - 1 _ 2 LE

(12.52)

Leading-Edge-Suction Method

Drag at angle of attack is strongly affected by viscous separation. At high lift-coefficients the drag polar breaks away from the parabolic shape represented by a fixed value of K in Eq. (12.4). The e method ignores this variation of K with lift coefficient. For a wing with a large leading-edge radius this is acceptable, but for most supersonic aircraft it gives a poor approximation. A semi-empirical method for estimation of K allows for the variation of K with lift coefficient and Mach number; it is based upon the concept of "leading-edge suction." Figure 12.33 illustrates the concept. The thick airfoil on the left is at an angle of attack below that at which substantial separation occurs. The flow streamlines curve rapidly to follow the leadingedge radius over the top of the wing. This rapid curvature creates a pressure drop on the upper part of the leading edge. The reduced pressure exerts a suction force on the leadingedge in a forward direction. This "leading-edge suction" force S is shown at the bottom of the figure in a direction perpendicular to the normal force N.

If there is no viscous separation or induced downwash, the leading-edge suction force exactly balances the rearward component of the norm~l force a d the airfoil experiences zero drag. This is the ideal 2-D case descnbed by t'" d'nAlembert's Paradox, and is called " 100070 lea.d'mg-e dge suc.IOn. A 3-D wing is considered to have 100070 leadmg-edge sectIOn when the Oswald efficiency factor (e) exactly equals 1.0. When e equals 1.0, t~e induced-drag constant K exactly equals the inverse of the aspect ratIO times 11'. • f '1 E On the right side of Fig. 12.33 is a zero-thickness flat pla~e aIr 01. ve~ without the leading-edge separation, which will almost certa~nly occur, thIS airfoil must have higher drag because there is no forward-facmg area for the leading-edge pressure forces to act ag~inst. All p~essure forces for a zerothickness flat plate must act in a directIOn perpendI~ular to. the plate, shown as N. There is zero leading-edge suction, and the hft and mduced drag are simply N times the cosine or sine of the angle of attack [Eqs. (12.53) and (12.54)]. L = Ncosa (12.53)

= N sina = L tan a

D;

(12.54)

or (12.55) but (assuming

a

is small), CD; =

Kcl == aCL

(12.56)

K

ACTUAL K VALUES LIE IN THIS REGION

vo E:::>

LEADING EDGE SUCnON

!

PRESSURE DISTRIBUTION

N

v.~~

NO LEADING EDGE SUCTION

1 KUHl = -

KUHl

... A

"EYEBALLED"

L-----------~--------~----------M RESOLUTION OF FORCES

Fig. 12.33

Leading edge suction definition.

Fig. 12.34

0070 and 100% K vs Mach number.

302

so, (12.57) Thus, in the worst case of zero leading-edge suction, the drag-due-to-lift factor K is simply the inverse of the slope of the lift curve (in radians), as previously determined. All real wings operate somewhere between 100 and 0070 leading-edge suction. The percent of leading-edge suction a wing attains is called S (not to be confused with the force S in Fig. 12.33). During subsonic cruise, a wing with moderate sweep and a large leadingedge radius will have S equal to about 0.85-0.95 (85-95% leading-edge suction). The wing of a supersonic fighter in a high-g turn may have an S approaching zero. The method below for calculating K for high-speed aircraft is based upon an empirical estimate of the actual percent of leading-edge suction attainable by a wing, which is then applied to the calculated K values for 100 and 0% leading-edge suction. The actual K is calculated as a weighted average of the 100 and 0% K, as in Eq. (12.58).

design lift coefficient of 0.5 may have an S value less than 0.3 at a lift . . .. I coefficient of 1.0. Proper calculation of S for an actual wm~ IS c?mplex. An emI?lflca approach may be used during conceptual desIgn. FIgure 12.35 provId~s a first-order estimate of the percent of leading-edge suction for a ~ypI~al supersonic aircraft's wing, given the actual lift coefficient and th~ desIgn hft coefficient (this determines which curve to use). Note that thIS chart ~s­ sumes a well-designed wing, and at some later date the aerodynamIcs department must optimize the twist and camber to attain these values. From Fig. 12.35 the leading-edge suction at various lift coefficients can b.e estimated. This allows adding curves to Fig. 12.34 that represent the estImated K value for different lift coefficients as a function of Mach number, as in Fig. 12.36. These are then used for total drag estimation via Eq. (12.4). For the sake of comparison, Eqs. (12.59) and (12.60) relate S to e and t:.N (used in several other textbooks). (12.59)

t:.N =

K = SK lO()

+ (1 - S)Ko

303

AERODYNAMICS

AIRCRAFT DESIGN

(12.58)

S(_1 __ ?rA

(12.60)

1)

CLcx

TYPICAL DESIGN GOAL VALUES FOR SUPERSONIC AIRCRAFT

The 0% K value is the inverse of the slope of the lift curve, as determined before. The 100% K value in subsonic flight is the inverse of the aspect ratio times ?r. In transonic flight, the shock formation interferes with leading-edge suction. This increases the K value. When the leading-edge becomes supersonic, the suction goes to zero so the K value equals the 0% K value. This occurs at the speed at which the Mach angle (arcsine 1/M) equals the leading-edge sweep. Above that speed the wing has zero leading-edge suction so the K value is always the inverse of the slope of the lift curve. For initial analysis, the supersonic behavior of the 100% K line may be approximated by a smooth curve, as shown in Fig. 12.34. This shows the typical behavior of the 100 and 0% K values vs Mach number. The only unknown remaining is the value of S, the percent of leadingedge suction actually attained by the wing at the flight condition in question. S depends largely upon the leading-edge radius, and is also affected by the sweep and other geometric parameters. S is also a strong function of the wing design lift coefficient and the actual lift coefficient. For any wing, the value of S is at a maximum when the wing is operating at the design lift coefficient. For most wings, S equals approximately 0.9 when operating at the design lift coefficient. For a subsonic wing with large leading-edge radius and moderate sweep, the value of S will change very little with lift coefficient until the wing is near the stall angle of attack. For this reason, the induced drag for this type of wing can reasonably be estimated using the e method. For the thin, swept wings typical in supersonic aircraft, the value of Scan change substantially with lift coefficient. A wing with an S of 0.9 at its

LEAUlNG EDGE SUCTiON FACTOR S 1.0

DESIGN C L

.9

.8

.8 .7

.6

.6

.5 .5 .4 .4

.3

.3

.2

.1

.1

0

o u

.1

.2

Fig. 12.35

.3

.4

.5

.6

.7

.8

Leading edge suction vs CL"

.9

1.0

C.

304

AIRCRAFT DESIGN

AERODYNAMICS

305

.4

12.7 AERODYNAMIC CODES AND COMPUTATIONAL FLUID DYNAMICS (CFD) Industry Practice for Aerodynamic Estimation .3

K

.2

.1

o o

.5

1.0

1.5

2.0

MACH NUMBER

Fig. 12.36

Sample results-K vs Mach and C

• L

Ground Effect

When a wi~g is near the ground, say less than half the span away the g dlra. due to lIft (~) c~n be substantially reduced. This is theoreticall; expaIned as a reductIOn In the induced down wash angle but ca b . 1· d as a trappin f " h· f· , n e Vlsua Ize ~ 0 . a cus Ion 0 aIr" under the wing. This effect is accounted for by multIplYIng K by the factor calculated in Eq. (12.61) (Ref. 70). Keffective

33(h / b )1.5

K

1 + 33(h/b)1.5

(12.61)

where h is wing height above ground. Trim Drag

The dr~g val~e.s used. for performance calculations should include the trim drag: ThIs addItIonal I~duced drag is caused by the horizontal tail force reqUIred to. balance (tn~) the aircraft so that the total pitching moment about th~ aIrcraft c:g. wIll ~e zero for any given flight condition. The t.~l usua~ly tnms the aIrcraft with a download that must be countered ?y addItIOnal hft from the wing. This produces an increase in the wing Indu~ed drag th.at must also be included in the trim drag. !nm calculatI?n is discussed in Chapter 16. The trim drag is determined U~Ing. the above Induced-drag methods once the tail lift force required for tnm IS known.

The aerodynamic methods presented above do not reflect current industry practice. Aircraft companies rely upon linearized computer codes such as the Harris wave-drag code, the Sommer and Short skin-friction code, and one of several panel codes such as USSAERO for induced effects. Newer panel codes such as PANAIR and QUADPAN are used to estimate the induced effects and the wave drag simultaneously and with better accuracy than the older codes. These linearized computer codes can provide correct results only when the airflow around the aircraft is steady, unseparated, and does not contain any strong vortices. This is typically true only during cruising flight. Lift and drag at high angles of attack are estimated empirically using correlations to flight-test and wind-tunnel data for similar configurations. The same is true for transonic lift and drag, where some of the very terms which are thrown away to linearize the equations are the longitudinal velocity-variation terms that produce the transonic shocks. Linearized wave-drag codes tend to over estimate the wave drag from Mach 1.0 to about Mach 1.2, and incorrectly predict zero drag rise below Mach 1.0. Empirical data is therefore used for the transonic regime. Despite these problems, the standard industry practice of combining linearized computer codes with empirical data and corrections will produce good results in most cases. Actual flight-measured values of lift and drag are usually within about 2-10070 of the estimates. Also, the estimates are the most accurate for the cruise portions of the flight where the most fuel is burned. However, the fact that we can estimate a given design's lift and drag with reasonable accuracy does not guarantee that these methods will produce the best of all possible designs. Aerodynamic design has had to rely upon a trial-and-error process of design, analyze, test, and redesign. Wind-tunnel testing offers a powerful tool for aircraft development. Unfortunately, the costs associated with detailed wind-tunnel testing tend to preclude an exhaustive evaluation of all possible designs. At a cost of several hundred thousand dollars per model, one is not likely to try something different just to see if it is better than the baseline design. Instead, the windtunnel is largely used to verify that a given design is workable. It is sometimes difficult to identify the source of a problem during a wind-tunnel investigation because the wind tunnel "solves" all the flow equations simultaneously (i.e., viscous effects, vortex flow, induced effects, etc.). An unacceptable wiggle in the pitching-moment curve may be due to one of a number of causes, and the wind tunnel may not tell you which cause to fix! Another problem with wind-tunnel testing is the Reynolds-number effect. Most wind tunnels cannot test at anything close to full-scale Reynolds numbers, resulting in substantial errors. Even worse, the optimal solution at a lower Reynolds number may not be the optimum at full-scale Reynolds

306

AIRCRAFT DESIGN

n!lmbers. 'Yho wou!d propose a full-scale test on an airfoil or complete aIrcraft desIgn that IS known to be less-than-optimal in the wind tunnel? CFD Definitions

I~ is for these reasons that Computational Fluid Dynamics (CFD) has rapIdly become a key part of the aircraft design process. CFD is a catch-all phrase. for a. number of n7w computational techniques for aerodynamic analysIs. It dIffers from pnor aerodynamic codes by solving for the complete properties of the flowfield around the aircraft, rather than only on the surface of the aircraft. . CFD ~ode~ are based upon the Navier-Stokes (NS) equations, which were ~Irst denve~ In 1822. The NS equations completely describe the aerodynamICS of ~ flUId (excePt. for chemical-reaction effects at high temperatures). NS Inc~udes equatIOns based upon the existence of flow continuity, the co~serv~tlOn of momentum, and the conservation of energy. These are denved In many textbooks on theoretical and computational aerodynamics and will not be repeated here. ' The NS equations seem straightforward enough but cannot be analytically &olved for any useful flow conditions. The author of Ref. 80 describes them as :'some of the nastiest differential equations in theoretical physics." The hIstory of theoretical aerodynamics to date can largely be described ~s ~he ~uest for s~lvable simplifications of the NS equations. The classical lIftIng-lIne theory IS one such simplification, as are the linearized wave-drag and panel codes, the Euler Codes, and the various NS codes. There is a compete hierarchy of aerodynamic codes depending upon how many flow phenomena are neglected from the full NS equations. No current c~d~s att~mpt to act~ally solve the true, full NS equations, due to the dIffIculty In mathematIcally analyzing turbulence. Turbulence occurs at the ~o~ecular level, which would probably require gridding the flow field with bIllIons of molecule-sized grids. . T~e current s~-called "Navier-Stokes Codes" actually use a simplificatIOn In the handlIng of turbulence, which is the most difficult flow phenomena to analyze mathematically. Turbulence is handled with some type of separate statistically-calibrated model apart from the NS solution. The most sophisticated codes to date, the "Large Eddy Simulation" codes, use a statistically-based turbulence model for small-scale turbulence effects. ~arge Eddy codes are capable of directly analyzing the larger turbulent eddIes. The Large Eddy Simulation is beyond the capabilities of current c~~puters for a complex aircraft configuration, but has been used for simplIfIed geometries. The current state of the art for complex aircraft configurations the "Reynolds-Averaged Navier-Stokes," has both large and small eddies'(turbulence) modeled statistically. Reynolds-Averaged codes can handle most of the co~plex flow phenomena that elude linearized codes, including vortex formatIOn, separation, transonic effects, and unsteady effects. Reynolds-Avera~ed codes are being used on the National Aerospace Plane (NASP) project to solve particular design problems where no other methods can give correct results. Unfortunately, Reynolds-Averaged codes are extremely expensive to set up and run. One recent example took 20 hours

AERODYNAMICS

307

on a Cray XMP-22 to yield results at one Mach number, altitude, angle of attack, and angle of sideslip. Because of the expense these codes are not yet useful for routine design work. The NS simplification emerging as the workhorse for design analysis, the "Parabolized Navier-Stokes" (PNS), drops the viscous terms in the streamwise direction, which ignores streamwise separation effects. However, with a good turbulence model the PNS codes give correct and illuminating results for most design problems. If all viscosity effects are ignored and the flow is assumed to be steady, the Euler equations are derived from the NS equations. Euler codes are much cheaper to run than even PNS codes, and are widely used at this time. The Euler codes can handle vortex formation, and with the addition of a separate boundary-layer code, can also realistically estimate viscous and separation effects. The "Potential Flow" equations are further simplified from the Euler equations by dropping the rotational terms. This prevents the analysis of vortex flow, which is important at high angles of attack but is of lesser importance during cruise conditions. Potential Flow codes can handle transonic shock formation and are very useful for transonic design compared to the linearized methods. The Potential Flow codes are not usually considered to be true "CFD," but are probably the most widely-used aerodynamic codes that treat the entire flowfield rather than just the surface conditions. The "Linearized" aerodynamic codes are based upon a further simplification to the Potential Flow equations by neglecting the higher-order terms. It is assumed that, since they involve small quantities multiplied by other small quantities, they must be very small and therefore negligible. At transonic speeds, however, these terms are not so small! The Linearized Potential Flow equations are the basis of the standard industry methods described at the beginning of this section. These include the Harris Wave Drag and the USSAERO and similar panel methods. With further simplifications, such classical methods as the lifting-line theory are derived. To recap, only the Large Eddy, Reynolds-Averaged, and PNS codes are considered to be true "Navier-Stokes" codes. However, the Euler, Potential Flow, and Linearized aerodynamic codes are in fact successive simplifications of the NS equations. The choice of code for a given design problem depends upon the nature of the problem and the available budget (and not always in that order!). Applications of CFD

CFD does not replace the wind tunnel. In fact, it really doesn't even reduce the number of wind-tunnel test hours. CFD does permit you to design a better airplane by a truer understanding of the flow field around it. Not only do the CFD codes determine the entire flow field around the aircraft, but also, unlike the wind tunnel, the flow field determination is done at the full-scale Reynolds number. A perfect example of the use of CFD can be found at every major commercial airport in the country. The installation of the fuel-efficient CFM-56

309

AERODYNAMICS

AIRCRAFT DESIGN

308

engine on the Boeing 737 would not have been possible without the use of CFD, as described in Ref. 80. The original Boeing 737 used the P&W JT8D, a low-bypass-ratio engine that was mounted in a wing-conformal nacelle. The nacelle barely cleared the ground, providing a minimum-weight landing gear. When Boeing decided to develop an updated version of the 737, the CFM-56 engine was the logical choice as a modern fuel-efficient engine of the required thrust class. However, it has a diameter some 20070 greater than the old engine. Furthermore, the CFM-56 exits its fan air up front like most modern turbofans. For this reason, a wing-conformal nacelle was not possible. In a prior chapter, the cited rough rule of thumb for podded jet nacelles said that the inlet should be about two inlet diameters forward of the wing and about one inlet diameter below it. A more-refined empirical method of locating a turbofan engine indicated that the geometry shown in Fig. 12.37a was the closest acceptable nacelle spacing. Clearly this posed a ground clearance problem! The empirical rules for nacelle spacing were based upon years of trial and error in the wind tunnel. Closer spacings were found to increase cruise drag, although the wind-tunnel investigations had not clearly determined just exactly what this "interference" drag consisted of. Various suspects included increased skin friction due to supervelocity, increased separation, shock effects, and a change in the wing's spanwise lift distribution resulting in an increase in the induced drag. Through the use of a nonlinear potential flow panel program (CFD state of the art in the 1970's), Boeing was able to determine that it was in fact the induced drag effect that was creating the "interference drag." This important piece of information had not been determined in 20 years of windtunnel testing! With this information, Boeing was then able to contour a closely-spaced podded nacelle to prevent any change in the wing's span wise lift distribution. This was possible with CFD because the entire flowfield is solved, allowing the designers to study the streamlines and pressure fields resulting from various design changes. The designers sought to minimize the impact of the nacelle on the streamlines of the bare wing.

nt~~~l~ed~~;x~~~~~:e~ls~~~s~pac­ ing to tfhe wi~~ th:~dn:~~~~~e~~s~'o~t~~e~C~~~igh angles of attack re~resent Figure 12.37b shows the result, namely, a

The orma lOn . 1 to the designers of fighter aIrcraft. another ar 7a of substantIa concerndifferent lift, drag, and pitching-moThese vortIces pr.oduce com pletel~. h would be predicted using the linear ment characterIstIcs than t h ose w IC methods. t ·1 the CFD solution of a typical vortex-flow problem d 11 d TEAM (Three-Dimensional Euler AeroReference 81 de a\ s ~sing a.~~:~e~~) E~i:~~~ 1~.~~ ~reprinted with permission) shows t~e c~se ~~~~~etween th~ calculated and the measured pres~ure~a~v~~ t!~n ~; t~~ delta configuration used in the study. The vortex ~eglOn r d plots diagonal pressure contours in both ca~c~;~dt~~ul~e:~~l;ze the flowfield Figure 12.39 illustrates the P?wer 0 Arrows are used to depict around the aircraft rather tha~ Just at t.he surf~~e. . raft The length of the the flow field at four lo.ngitudm~ l~catI~~~~~o: :~r~city ·in the plane of the arrows sh?ws the relatIve mflagn~tu 1e! ~y seen beginning inboard toward the cross sectIOn. The vortex ow IS c e r Cp

-0.20 -0·40 -0·60 -0·60

TEAM COMPUTATIONS (49 x 145 x 33 C-Hl

- 1 .00 - 1 .20 - 1 .40 - 1 .6G - 1 . Be

-2.00 "RULE.OF·THUMB" NACELLE INSTALLATION a)

CFM 56

~M 56

b) CFD·DESIGN NACELLE INSTALLATION

_c~~~~~ Lr---==s;::j~::::::::::-" \ Fig. 12.37 88-2043).

-2.20

MEASUREMENTS

__________ ~

ORIGINAL JT8D NACELLE

CFD example: Boeing 737 nacelle (after R. Bengelink, AIAA Paper

C

y~ - 2 .2~~=---

40-

~~---=--::--

p

_ -1.8-1.4 -1.0 -.80 -.60

.

- 2.40 2.60 2.61J -3.01J

--

-.20

-.10

- 3·20

contours; · 12 38 Correlation of compnted and measured surface pressure .F 19. . 03 -20 (Ref 81) 750/62" donble-Delta wing body; M~ = . ; a. . 0

II

310

AIRCRAFT DESIGN

AERODYNAMICS

311

f~ont of the wing, and growing and moving outward toward the rear of the aIrcraft. CFD Issues and Challenges

y". e have come a long ways since 1879 when the annual proceedings of the BntIsh Royal Aeronau~ical Society could say, "Mathematics up to the present day has been qUlt~ useless to us in regard to flying" (quoted in Ref. 80). However, there are stIll many problems associated with the use of CFO TEAM COMPUTATIONS

v~

(MAGNITUDEI

Fig. 12.40

Fig. 12.39 Correlation of computed and measured cross-plate velocity fields; 75°/62" double-Delta wing body; M oo =0.3; a=20°. (Ref. 81)

Flowfield grid ding. (Ref. 82)

for routinely solving aircraft design problems. Two problems are especially important: the influence of the turbulence model and the requirements for flow gridding. The use of separate turbulence models for NS codes has been discussed. The results of the various NS codes are very sensitive to the turbulence model used, especially when separated flow is present. CFO codes tend to produce reasonable-looking flowfields and pressures, but sometimes the integration of the calculated pressures yields lifts, drags, and moments which do not match experience. Reproduction of experimental data sometimes requires extensive "calibration" (i.e., fudging!) of the turbulence model. For this reason, CFO results are always somewhat suspect until the code has been checked against experimental data for a similar configuration. The need to grid the entire flow field around the aircraft presents another big problem for CFO users. "Gridding" refers to the breaking up of the space around the aircraft into numerous small blocks, or "cells," usually of roughly hexahedral shape. CFO methods calculate the flow properties

AIRCRAFT DESIGN

312

within each cell, using various convergence schemes to equate the flow properties along the boundaries connecting the cells. While gridding the space around a simple cylinder or a lone wing can be easily automated, the gridding of the flowfield around a full aircraft must currently be done manually and can literally take months. Figure 12.40 (reprinted with permission from Ref. 82) illustrates the complexity of the flowfield gridding. Note, for example, where the canopy meets the fuselage and where the cells must fan out in the empty region between wing and canard. Gridding is especially important because the CFD results are highly sensitive to the shaping of the cells. You can actually get different answers for the same aircraft using two different gridding schemes. According to the author of Ref. 82, "this sensitivity is more pronounced than that due to the type of mathematical model being used, e.g., NS vs Euler equations." To address the gridding problem, researchers are investigating artificial intelligence (AI) approaches to gridding. Another approach is the computationally-adaptive gridding in which the grid ding scheme is automatically adjusted based upon the CFD results.

-

~

3f'o~'

A l.J,.z n~

b

In.s''

C.

:.. ~.~ .fl ~

c• '-

r

lIZ.)

0"..= Y'tO'"

~;.....

"" ",

"rs0 'ff

_

..,.'

2~~1~'

.J

Jr"

",,"J,-

I

3

~u

.s. -

,1.-18-"11

DES,{,./"fR..D""I£1. ~ RMMfft

,

~~A;;~~7f-

'

G5 Z

High strength, not weldable, common in high-speed air,craft

High-temperature, high strength-to-weight, subject to corrosion

Table 14.3 (continued) Typical metal properties (room temperature) Material Titanium Titanium· Ti-6AI-4V -Ti-13V-11Cr-3AI

Density Ib/in. 3

0.160 0.174

Temp limit of

750 6001000

F tu

103 psi

160 170

F ty

Fey

10 3 psi

103 psi

145 160

154 162

Fsu

10 3 psi

100 105

E 10 6 psi

16.0 15.5

G

Comments

10 6 psi

6.2

Most-used titanium, induding B-70 SR-71 titanium

U> --I :IJ

C

()

--I

C

:IJ

m

U>

» Z

High temperature nickel alloys Inconel X- 7 50

0.300

Rene 41

0.298

HastelloyB

0.334

10001500 12001800 1400

155

100

100

101

31.0

11.0

X-15

168

127

135

107

31.6

12.1

X-20, very difficult to form

100

45

30.8

Engine parts

0

r 0

»

0 U>

CAl

0)

-...J

AIRCRAFT DESIGN

STRUCTURES AND LOADS

ously undesirable, but one that is overstrong can cause excessive deflection on adjoining areas, which can lead to fracture. Proper repair of an important composite part requires running a computer program to insure that the repaired part will match the original design specifications. The properties of a composite material are not simply the algebraic sum of the properties of the individual ply layers. Although a simple summation provides a rough approximation of the total material properties, actual material properties must be calculated using tensor calculus equations, such as are outlined in Ref. 58. Furthermore, extensive coupon testing is required to determine design allowables for the selected materials and ply orientation. Introductions to composites are provided in Refs. 59 and 83.

that actual material properties for use in detail design should be obtained from the producer or from a specification document such as Ref. 61. For example, Ref. 61 contains 68 pages of design data on 2024 aluminum alone, covering many different forms, heat treatments, tempering, gauges, etc. The values for 2024 in Table 14.3 are merely typical, suitable for rough estimates and student design projects.

368

Sandwich Construction

While not properly classed a "material," sandwich construction has special characteristics and is very important to aircraft design. A structural sandwich is composed of two "face sheets" bonded to and separated by a "core" (Fig. 14.29). The face sheets can be of any material, but are typically aluminum, fiberglass-epoxy, or graphite-epoxy. The core is usually an aluminum or phenolic honeycomb material for commercial and military aircraft, but various types of rigid foam are used as the core in some cases. Many homebuilt aircraft today are constructed of foam-core sandwich with fiberglass composite skins. In a sandwich, the face sheets carry most of the tension and compression loads due to bending. The core carries most of the shear loads as well as the compression loads perpendicular to the skin. As with composites, joints and fittings are a problem with sandwich construction. Analysis of sandwich construction is discussed in Ref. 60.

369

14.10 STRUCTURAL-ANALYSIS FUNDAMENTALS The following sections will introduce the key equations for structural analysis of aircraft components. Derivations will not be presented as they are available in many references, such as 54, 55, and 60. Properties of Sections A number of geometric properties of cross sections are repeatedly used in structural calculations. Three of the most important-centroid, moment of inertia and radius of gyration-are discussed below. Note that the cross sectio~s of interest in tension and compression calculations are perpendicular to the stress, while in shear calculations they are in the plane of the shearing stress.

x

= Ex;

dA;

(14.21)

= !:y;

dA;

(14.22)

A

e

y

A

e

Tables 14.3, 14.4, and 14.5 provide typical material properties for various metals, composites, and woods. Note that these are typical values only, and

The "centroid" of a cross section is the geometric center, or the point at which a flat cutout of the cross-section shape would balance. The coordinates of the centroid (Xc> Ye ) of an arbitrary shape (Fig. 14.30) are found from Eqs. (14.21) and (14.22). A symmetrical cross section always has its centroid on the axis of symmetry and if a cross section is symmetric in two directions, the centroid is at the intersection of the two axes of symmetry.

Table 14.4

Table 14.4 (contd.)

Material-Property Tables

Typical composite material properues (room temperature) Fiber orientation

Material High-strength Graphitel epoxy High-modulus Graphitel epoxy Boron/epoxy Graphite/polyimide S-Fiberglass/ epoxy E-Fiberglassl epoxy Aramid/epoxy

(±~5 (

±~5 0 0 0 0 0

Fiber Temp. 070 Density limit F(U(L) of volume Ib/in 3 10 3 psi 60 60 60 60 50

0.056 0.056 0.056 0.058 0.073

350 350 350 350 350

45 60

0.074 0.Q71 .052

350 350 350

L - Longitudinal direction; T = transverse direction; Fisu stress (ultimate); t = tension; c = compression.

180.0 23.2 110.0 16.9 195 204 219 105 200 =

Ftu(T)

Feu (L)

Fcu(T)

Fsu (LT)

10 3 psi

10 3 psi

10 3 psi

103 psi

8.0 23.2 4.0 16.9 10.4 4.85 7.4 10.2 4.3

180.0 23.9 100 18 353 111 73.9 69 40

30.0 23.9 20 18 40 18.5 22.4 33 20

12 65.5 9.0 43.2 15.3 8.5

interlaminate shear

Typical composite material properties (room temperature)

Fisu

(T)

Et(L)

Et(T)

EC(L)

Ec(T)

G(LT)

in/in

10 6 psi

10 6 psi

106 psi

106 psi

10 6 psi

0.0087 0.022 0.0046 0.012 0.0065

0.0048 0.022 0.0025 0.012 0.004 0.0036 0.019 0.006

1.70 2.34 1.70 2.38 2.7 1.35 2.70 1.82 0.8

21.00 2.34 25.00 2.38 30 17.4 6.80 4.43 11

1.70 2.34 1.70 2.38 2.7 1.4 2.5 1.8 0.8

0.65 5.52 0.65 6.46 0.70 0.84

0.025 0.018

21.00 2.34 25.00 2.38 30 20 7.70 4.23 11

flu

(L)

10 3 psi in/in 13

10 13

flu

11 7.9 9

0.51 0.3

CAl

~

Table 14.5

Ash Birch African mahogony Douglas fir Western pine Spruce

Parallel to grain

Density Ib/in 3

10 3 psi

10 3 psi

10 3 psi

0.024 0.026

14.8 15.5

8.9 9.5

7.0 7.3

5.3 5.5

0.019 0.020 0.016 0.016

10.8 11.5 9.3 9.4

7.9 8.0 6.0 6.2

5.7 7.0 5.3 5.0

4.3 5.6 4.2 4.0

Flu

FlY

Table 14.6

. Illustrations

~& -

H

y

Area

BH

Feu

BI2

Perpendicular to grain

Parallel to grain

Feu

Fs

Fey

10 3

10 3

psi

10 3 psi

10 6 psi

2.3 1.6

1.4 1.3

1.46 1.78

1.4 1.3 0.8 0.8

1.0 0.8 0.6 0.7

1.28 1.70 1.31 1.30

psi

J:j

()

:IJ

» II -I 0

m

en G5 Z

Rad. of gyration

Mom. of inertia

l' HI2

» E

Properties of simple sections

Centroid

X-

X

Wood properties (ANC-5)

Iy

Ix

BH3 -12

HB3 -12

Py

Px

H

B

"""Tt2

"""J'i'"2

I BI Ib I

]Bh I

B

BH-bh

I

J

HB3 -hb 3 ) 12(BH- bh)

-I :IJ

C

()

-I

m

7rR2

-y

R

R

7rR4

7rR4

4

4

RI2

RI2

» Z 0

r

0

» 0

7r(R2 - r 2)

~ B

BH3 -bh 3 12(BH-bh)

:IJ

_X

I

12

J

en

t

-

HI2

HB3_hb 3

c

~-t:

H

BI2

,

BH3 -bh 3 12

en

x

BH 2

R

0

R

HI3

7r(R 4 -r

4)

7r(R4 - r 4 )

4

4

BH3

B3H

36

48

~R2

+r2)

~R2

2

H ~

+r2

en

2

B

"724 CAl

-...J .....

372

AIRCRAFT DESIGN

A "centroidal axis" is any axis that passes through the centroid. An axis of symmetry is always a centroidal axis. Centroids for simple shapes are provided in Table 14.6. The centroid of a complex shape built up from simple shapes can be determined using Eqs. (14.21) and (14.22) using the centroids and areas of the simple shapes. The moment of inertia I is a difficult-to-define parameter that appears in bending and buckling equations. Moment of inertia can be viewed as the cross section's resistance to rotation about some axis, assuming that the cross-sectional shape has unit mass. Moment of inertia is the sum of the elemental areas times the square of the distance to the selected axis [Eqs. (14.23) and (14.24)), and has units of length to the fourth power. The polar moment of inertia (J or Ip) is the moment of inertia about an axis perpendicular to the cross section [Eq. (14.25)]; J is important in torsion calculations.

ARBITRARY BODY

Y

(14.27) The radius of gyration p is the distance from the centroidal axis to a point at which the same moment of inertia would be obtained if all of the crosssectional area were concentrated at that point. By Eq. (14.23), the moment of inertia is the total cross-sectional area times p squared; so p is obtained as follows: p=.JIIA

(14.28)

The main use of p is in column-buckling analysis. Also, the p values in Table 14.6 can be used to approximate I for the given shapes. Other cross-sectional properties such as the product of inertia and the principal axes will not be used in this overview of structures. See Refs. 60,

Y

---f",*~--

Yc

Xc .L-----X

X

I Y

ex] ..... £y

:

J~CE~ROID Xc

••

(14.25)

(14.26)

CENTROID

/'

Xc

(14.24)

Structural calculations usually require the moments of inertia about centroidal axes. Table 14.6 provides moments of inertia for simple shapes about their own centroidal axis. For a complex built-up shape, the combined centroid must be determined, then Eqs. (14.26) and (14.27) can be used to transfer the moments of inertia of the simple shapes to the combined centroidal axes. The "f" terms are the x and y distances from the simple shapes' centroidal axes to the new axes (see Fig. 14.30, bottom). Once the simple shapes' moments of inertia are transferred to the combined centroidal axes, the moments of inertia are added to determine the combined moment of inertia (Ix and Iy). The new J is determined from the new Ix and Iy using Eq. (14.25):

BISYMMETRIC BODY

-f:~i \ CENTROID -fYc XI

(14.23) Iy = Ex? dA i

373

STRUCTURES AND LOADS

Fig. 14.30

COMBINED SHAPES X

Section property definitions.

54, or other structures textbooks for more information about section properties. Tension

Tension, the easiest stress to analyze, is simply the applied load divided by the cross-sectional area [Eq. (14.14), repeated below as Eq. (14.29)]. The shape of the cross section is unimportant in most cases. The appropriate cross section is the smallest area in the loaded part. For example, if the part has rivet or bolt holes the smallest cross-sectional area will probably be where the holes are located, because the areas of the holes are not included for tensional calculations. Usually the relevant cross section is perpendicular to the load. If a line of holes forms a natural "zipper" at an angle off the perpendicular, the part may fail there if the cross-sectional area along the "zipper" line is less than the smallest perpendicular cross section. a=PIA

(14.29)

Remember that the stress level at the limit load should be equal to or less t.han the yield stress or, for composite materials, the stress level corresponding to a strain of two-thirds of the ultimate strain.

374

AIRCRAFT DESIGN

STRUCTURES AND LOADS

Compression

The compression stress is also given by Eq. (14.29) (load divided by area). For the determination of the limit stress, this equation can only be applied to parts that are very short compared to cross-sectional dimensions (such as fittings) or to parts which are laterally constrained (such as spar caps and sandwich face sheets). Long unconstrained members in compression, called "columns" or "struts," are discussed below. For short or laterally constrained parts in compression, the ultimate compressive strength is usually assumed to equal the tensile value. For ductile metals this is a conservative assumption as they never actually fail, but merely "squish" out and support the load by the increased area. Rivet and bolt holes are included in the cross-sectional area calculation for compression because the rivets or bolts can carry compressive loads. Columns in compression usually fail at a load well below that given by applying the ultimate stress to Eq. (14.29). Columns in compression fail either by "primary buckling" or by "local buckling." An important parameter is the column's "slenderness ratio": the column's effective length Le divided by the cross-sectional radius of gyration [Eq. (14.30)]. The effective length of a column is determined by the end connections (pinned, fixed, or free) as shown in Fig. 14.31. Slenderness Ratio:

(14.30)

When you push down on an upright yardstick, the middle part bends outward in a direction perpendicular to the load. This bending action produces internal stresses much greater than the direct compression stress due to the applied load, and is called "primary column buckling." If the bend-

PIN \

,,

,,,

FREE -

I

I ,,

I

I I

,

FIXED

PIN \

, \ I

\,

\

,,

\

I

I

, ,,

I

FIXED

FIXED

LE

=. 5 L PERFECTLY RIGID

LE "'.71 L L E ",.82L

Fig. 14.31

WELDED ENDS RIVETED OR BOLTED

Column effective length.

375

ing action after buckling involves stresses below the proportional limit, the column is said to experience "elastic buckling." The highest compression load that will not cause this elastic column buckling-the so-called "Euler load," or critical load Pc-will be determined from the Euler column equation [Eq. (14.31)]. The resulting compressive stress is found from Eq. (14.32). Note in Eq. (14.31) that the total load a column can carry without buckling does not depend upon either the cross-sectional area or the ultimate compressive stress of the material! Only the column's effective length, its cross-sectional moment of inertia, and the material's modulus of elasticity affect the buckling load if the column is long. Pc

2

EI Le

7r

= -2-

(14.31)

(14.32)

The buckling stresses of Eq. (14.32) are failure stresses and do not have any margin of safety. For design purposes the limit loads should be reduced, usually to two-thirds of these values. A column with an open or highly irregular cross section may fail at a lower load due to cross sectional twisting or deformation. Methods for analysis of such members can be found in Refs. 60 and 83. Equation (14.31) implies that, as column length is reduced to zero, the Euler load goes to infinity. However, the compression stresses experienced due to bending in a buckled column are much greater than the applied load would directly produce. At some point as column length is reduced the internal compressive stresses produced at the onset of buckling will exceed the proportional limit and the column will no longer be experiencing elastic buckling. This has the effect of reducing the buckling load compared to the Euler load. The "critical slenderness ratio" defines the shortest length at which elastic buckling occurs. At a lower slenderness ratio, the stresses at buckling exceed the proportional limit. The column experiences "inelastic buckling" so the Euler equation cannot be used as shown. The critical slenderness ratio depends upon the material used. It is about 77 for 2024 aluminum, 51 for 7075 aluminum, 91.5 for 4130 steel, and 59-76 for alloy steel depending upon heat treatment. Most columns used in aircraft are below these critical slenderness values, so the elastic Euler equation cannot usually be used in aircraft column analysis. The buckling load for inelastic buckling can be determined by Eq. (14.32), with one modification. The modulus of elasticity must be replaced by the tangent modulus, described previously. As the tangent modulus is a function of the stress, iteration is required to find the buckling load for a particular column. However, handbook graphs such as Fig. 14.32 are usua:lly used for design (see Refs. 60 and 61).

AIRCRAFT DESIGN

376

STRUCTURES AND LOADS

160 Fe

are clamped, but with some flexibility to rotate about the side axes. A K value between the clamped and simply-supported values should be used in such a case.

ALLOY STEEL

140

F.. = 180,000 psi F," = 150,000 psi F," = 125,000 psi

(1,000 psi) 120

377

HEAT TREATMENTS

J

Fbuckling

100

= KE(t/b)2

(14.34)

Truss Analysis

A truss is a structural arrangement in which the structural members (struts) carry only compression or tension loads ("columns" and "ties"). In the ideal truss, the struts are weightless and connected by frictionless pins. No loads are applied except at the pins, and no moments are applied anywhere. These ideal assumptions guarantee that the struts carry only compression or tension. The strut loads calculated with these ideal assumptions are called "primary truss loads." Additional loads such as those caused by the attachment of an aircraft component to the middle of a strut must be calculated separately and added to the primary load during analysis of each individual strut. The impact of rigid welded connections in a typical aircraft applica-

80 60

40 20 0 0

20

Fig. 14.32

40

60

80

100

120

140

Column buckling loads (round tubing).

12

As discussed at the beginning of this section, a very short "column" experiences pure compression without any danger of primary column buckling. This is sometimes called "block compression." The compression yield value is used as the limit load, providing a cutoff value for the buckling load of a short column with either a solid cross section or with relatively thick walls (structural tubing). A column can usually be considered in block compression if the slenderness ratio is less than about 12. When you step on an upright soda can, it fails in a form of local buckling called "crippling," in which the walls of the cross section collapse without warning, and the load-carrying ability drops to virtually zero. This is typical for short columns with very thin walls. Methods for estimation of thin-wall crippling are found in Ref. 60. A rough estimate for the crippling stress of a thin-wall cylindrical tube is shown in Eq. (14.33), where t is the wall thickness and R is the radius. Fcrippling

== 0.3(EtIR)

(14.33)

A flat sheet or panel under compression fails by buckling in a manner similar to a column. The buckling load [Eq. (14.34)] depends upon the length (a) in the load direction, the width (b), the thickness, and the manner in which the sides are constrained. Clamped sides cannot rotate about their axis, and provide the greatest strength. Simply-supported sides are equivalent to a pinned end on a column, and can rotate about their axis but cannot bend perpendicularly. A free side can rotate and bend perpendicularly and provides the least strength. Figure 14.33 provides the buckling coefficient K for Eq. (14.34) based upon panel length to width ratio and end constraints. Most aircraft panels

1 I I I

11

1\

F

\ \

10

9

ccr

,

1\

K

X

3

I I I I

\

~

CLAMPED SIDES AND ENDS

I

IA I

11

ASYMPTOTIC TO 6.35 _

1\

' CLAMPED SIDES, SIMPLY SUPPORTED ENDS

\

"

ENDS CLAMPED, SIDES SIMPLY SUPPORTED

I--

'" --,

I I

f'l

!IASYMPTOTIC TO 3.6

~~~I

17 J"-....

SIMPLY SUPPORTED SIDES AND ENDS I

TTTI r I 1 I 1 1

2

ONE SIDE CLAMPED, ONE SIDE FREE, ~ ;:'. ....... ENDS SIMPLY SUPPORTED ...J. ,,__

1

o

.KE(-!Y b

~~..c-'-;/ I I" tr' 1'-~-r't-+-1'-+- t- 'I 1 I

,

1\ \

ccr -

I 1 1 I 1\

I'

1\ \

4

F

._~

f7' 1\

6

5

SId.

\

1;

7

"" t

End.,~ End

Fc cr

1\

8

~ a-,--;,'@ ~

I

I

I

I

i

I

~IDE FREE, ONE SIDE AND ENDS SIMPL~ UPPORTED 0.385

o

Fig. 14.33

T I I r

I

2

3

alb

I

I

I 1 1 4

Panel buckling coefficient. (NACA TN3781)

378

tion is considered only in the definition of effective length in the columnbuckling equation (see Fig. 14.31). Truss structure was used extensively in welded steel-tube fuselages. Today the truss structure is largely used in piston-engine motor mounts, the ribs of large aircraft, and landing gear. Figure i4.34 shows a typical truss structure, a light aircraft motor mount. For illustration purposes this will be analyzed as if it were a two-dimensional truss with only the three struts shown. Analysis of three-dimensional "space structures" will be discussed later. The bottom of Fig. 14.34 shows an equivalent truss that includes the lines of force to the c.g. of the engine, and the vertical resisting forces due to the rigid attachment of the fuselage and engine to the truss. This equivalent truss can be solved by several methods. The most general truss solution, the "method of joints," relies upon the fact that at each joint of the truss, the sums of the vertical and horizontal forces must each total zero. To obtain a solution from the two equations (vertical and horizontal), the solution must begin at and always proceed to a joint with only two unknown struts. The method usually begins at a free joint with an applied external load, in this case at the engine load. Figure 14.35 shows the forces at the joints. All the forces are shown as radiating outward from the joints so that a positive force is a tension and a negative force is a compression. When summing forces at a joint, the positive or negative force is added to the sum if it is up (when summing vertical forces) or to the right (when summing horizontal forces), and subtracted if down or to the left. Confusion about the appropriate sign is the most common error in truss analysis (the author did joint three wrong the first time!).

FUSELAGE

nWENGINE = 4,000 Ib

2

50

4 1

I

130 1 I I

51 4,0001b

Fig. 14.34

379

STRUCTURES AND LOADS

AIRCRAFT DESIGN

5

Typical truss struclun:.

JOINT 2

2

EFH =O=F e -FA CO.;; 27 EF y = 0 = - F D - F A SIN 27 Fe =3919 (T) F D = -2000 (C)

JOINT 3

FD

EFH =O=F E COS 22 +FFCOS n-FB COS 27 EFy =O=F D +FB SIN 27 + FE SIN 22-FF SIN 11 F E =5775(T) FF = - 9463 (C)

JOINT 1 EFH =O=F ACOS 27+FB COS 27 EFy =O=F A SIN 27-FB SIN 27-4000 FA =4400 (T) FB = -4400 (C)

Fig. 14.35

Method of joints.

Joint one is at the engine's c.g .. The unknown forces Fa and Fb must react the engine load of 4000 lb. Solving the equations shown yields Fa of 4400 lb (tension) and Fb of - 4400 lb (compression). Selection of the next joint to analyze depends upon the number of unknown struts. At joint three, there are three unknown struts at this time, so we select joint two. Solving the equations yields Fe of 3919 lb (tension). Fd is found to be - 2000 lb, a compression load on the engine due to the motor mount. If this load is in excess of what the engine can withstand, a vertical motor-mount strut should be welded between joints two and three. At joint three there are now only two unknown strut loads. Solving the equations yields Fe of 5775 lb (tension) and Ffof -9463 lb (compression). In some cases a quicker method can be employed to determine the forces in selected struts without having to solve the whole truss as in the method of joints. This quicker method is actually two methods, the "method of moments" for the upper and lower struts and the "method of shears" for the inner struts. The top illustration of Fig. 14.36 shows the use of the method of moments to solve the force in the top strut of the motor mount. The whole structure is replaced by two rigid bodies connected by a pin, with rotation about the pin prevented by the unknown force in the strut under analysis. The moments about the pin are readily summed and solved for the unknown strut force, which is found to be 3919 lb. A similar technique is shown in the middle illustration for the lowe~ strut, which has a load of 9463 lbs. Note that this technique, where applIcable, allows direct solution for the desired unknown forces.

AIRCRAFT DESIGN

380

STRUCTURES AND LOADS

EM = 0 = -19.6 (4000) + 20 Fe Fe =3919.2

4,000 +---69.6 30

EM=O= -69.6 (4000)-30 FF COSH FF = -9463

EFH = 0=3919.2+F E COS22 + (-9463) COS 11 FE =5775 EFy=O= -4000+F E SIN 22-(-9463) SIN H FE =5775 4,000

Fig. 14.36 Method of moments/method of shears.

The lower illustration of Fig. 14.36 shows the use of the method of shears to solve for the inner strut. This method involves severing the structure along a plane which cuts only three members, the upper and lower strut and the inner strut under analysis. The severed part of the structure is analyzed as a free body, summing either the vertical and horizontal forces, which must total zero. Note that by calculating the unknown strut force both ways (vertical and horizontal summation), a check of your result can be made. This example gives a result of 5775 lb. These methods are only applicable if the truss structure is "statically determinate." In general, a truss is statically determinate if every strut can be cut by some plane that cuts only two other struts. This insures that there is always a joint with only two unknown struts, permitting solution by the method of joints. For "indeterminate" trusses, more complicated methods based upon deflection analysis must be used (see Refs. 54 and 60). Once the loads in each member of the truss are known, the struts can be analyzed using the equations presented above for tension or compression. Use the appropriate effective length for welded, riveted, or bolted columns from Fig. 14.31. To provide an extra margin of safety, it is customary to assume that welded steel-tube motor mounts act as though the ends were pinned (Le = L). The 3-D trusses, or "space structures," are solved similarly to the 2-D truss. Square cross section 3-D trusses, such as a typical welded-tube fuselage, can sometimes be solved separately in side view and top view as 2-D

381

structures. The resulting strut loads are then summed for the various members. This is permitted provided that the combined loads on all struts are within the elastic range. For more complicated 3-D trusses, the method of joints can be applied using three equations and three unknown strut loads. This involves simultaneous solution of equations, e.g., with a simple computer iteration program. In some cases the moments about some selected point can be used to obtain the solution with less effort. Space structures are discussed in detail in Ref. 54. Beam Shear and Bending A common problem in aircraft design is the estimation of the shear and bending stresses in the wing spars or fuselage. This is a two-step process. First, the shear and bending moment distributions must be determined; then the resulting stresses must be found. Figure 14.37 shows a simple beam with a distributed vertical load. The beam is shown cut to depict internal forces. The right side of the beam being a free body, the sum of the vertical forces and the sum of the moments must equal zero. If the severed part of the beam is to remain in vertical equilibrium, the externally applied vertical forces must be opposed by a vertical shear force within the cross section of the material, as shown. Thus, for any span station the shear force is simply the sum of the vertical loads outboard of that station, or the integral of a distributed load.

SHEAR

SUPPORT SHEAR REACfION

MOMENT

SUPPORT MOMENT REACTION

MOMENT REACTION DUE TO SPANWISE COMPRESSION AND TENSION

Fig. 14.37 Shear and moment in beams.

AIRCRAFT DESIGN

382

r

Ib/in.

t

STRUCTURES AND LOADS

r WING WEIGHT

NACELLE

t (t1 t t t t t,-,t ! t tt tt • + , .... ,

Ib

t

'1' t

~

ACTUAL LOADS

EQUIVALENT CONCENTRATED LOADS

SHEAR

.t::9j

r~

9:LJ""

Fig. 14.38

383

However, it is easier to graphically integrate by starting at the tip and working inward, adding to the total the area under the shear distribution at that station. Referring back to Fig. 14.37, the bending moment at a cross-sectional cut is opposed by a combination of tension and compression forces in the spanwise direction. For a positive bending moment such as shown, the internal forces produce compression on the upper part of the beam and tension on the lower part. The vertical location in the beam at which there is no spanwise force due to bending is called the "neutral axis," and is at the centroid of the cross-sectional shape. As long as the stresses remain within the elastic limit, the stresses vary linearly with vertical distance from the neutral axis regardless of the crosssectional shape. These compression or tension stresses are found from Eq. (14.35) (for derivation, see Ref. 55), where M is the bending moment at the spanwise location and z is the vertical distance from the neutral axis. The maximum stresses due to bending are at the upper and lower surfaces. (14.35)

BENDING MOMENT

Wing loads, shear, and bending moment.

The moments produced by the vertical loads must be balanced by a moment at the cut cross section. This moment is equal to the summation of the discrete loads times their distance from the cut station, or the integral of a distributed load with respect to the distance from the cut. Figure 14.38 shows the typical loads on a wing. This shows the critical case of a rolling pullup, with the additional lift load of full aileron deflection. The lift and wing-weight loads are distributed, while the nacelle weight is concentrated. Remember that wing and nacelle weights are multiplied by the aircraft load factor to determine the load on the wing. The easiest way to calculate the shear and moment distribution along a wing is to replace the distributed loads (lift and wing weight) by concentrated loads. The lift distribution can be determined with Schrenk's Approximation, described above. The wing weight will be determined in the next chapter, and can be assumed to be distributed proportional to the chord length. Figure 14.39 shows the trapezoidal approximation for a distributed load, giving the total equivalent force and the spanwise location of that force. About ten to twenty span wise stations will provide an accurate enough approximation for initial design purposes. Once the distributed loads are replaced by concentrated loads, determination of the shear and bending moment distributions is easy. The shear at each span station is the sum of the vertical loads outboard of that station. The shear is found by starting at the wing tip and working inward, adding the load at each station to the total of the outboard stations. The bending moment can be found for each span station by multiplying the load at each outboard station times its distance from the span station.

The vertical shear stresses within a beam are not evenly distributed from top to bottom of the cross section, so the maximum shear stress within the material can not be calculated simply as the total shear divided by the crosssectional area. Referring back to Fig. 14.20, it should be remembered that the vertical shear stresses on an element are balanced by and equal to the horizontal shear stresses. One cannot exist without the other. Therefore, the vertical shear distribution must be related to the horizontal shears in the beam. Figure 14.40 shows a beam in bending, with the vertical distribution of compression and tension stresses. The total horizontal force on any element is the horizontal stress at the element's vertical location times the elemental

b

a

r

x ----a

S(a+b)

F=-2

Fig. 14.39

x-s[

2a+b ] 3a+3b

Trapezoidal approximation for distributed loads.

AIRCRAFT DESIGN

384

STRUCTURES AND LOADS

area. If this beam is split lengthwise as shown, the upper section has only leftward forces, so a shear force must be exerted along the cut. This shear force must be the sum of the horizontal stresses times the elemental areas above the cut. This reaches a maximum at the neutral axis. At the upper and lower surfaces, this shear force is zero. The bottom of Fig. 14.40 shows the resulting vertical distribution of shear forces, expressed as magnitude toward the right. (Don't be confused by this presentation; the shear forces are exerted in a vertical direction, but we show the magnitude to the right to illustrate the distribution of magnitude from top to bottom.) T=

V bl Y

lhl2 z dA

(14.36)

zl

Equation (14.36) describes this mathematically, where the integral term represents the area above the cut located at Z = Zl. Note that the distribution of shear stresses depends upon the shape of the cross section. For a beam of rectangular cross section, the maximum shearing stress (at the neutral axis) is 1.5 times the averaged shearing stress (total shear divided by cross-sectional area). For a solid circular cross section, the maximum shearing stress is 1.33 times the averaged value. Figure 14.41 shows a typical aircraft wing spar consisting of thick "spar caps" separated by a thin "shear web." The cross-sectional area of the

385

shear web is insignificant compared to the area of the spar caps, so the caps absorb virtually all of the bending force (stress times area). The shear stress depends upon the cross-sectional area above the point of interest, and is therefore essentially constant within the thin shear web, as shown to the right. In aircraft wing spar analysis it is common to assume that the caps absorb all of the bending stresses and that the web (extended to the full depth of the spar) absorbs all of the shear. This is shown at the bottom of Fig. 14.41. It is also assumed that the shear is constant within the web and therefore the maximum shear stress equals the average shear stress (shear divided by web area). The shear web will fail in buckling long before the material maximum shear stress is reached. Equation (14.37) defines the critical buckling shear stress for a shear web. The value of K is obtained from Fig. 14.42. Fshear buckle

=

(14.37)

KE (t / b )2

Braced-Wing Analysis

A wing braced with a strut will have the bending moments greatly reduced compared to a fully cantilevered wing. However, the analysis is more complex because of the spanwise compression loads exerted upon the wing by the strut. This can increase the bending moment by as much as a third compared to an analysis that ignores this compression effect.

BENDING STRESSES

BENDING STRESSES

SHEAR STRESS MAGNITUDE SPAR APPROXIMATIONS

z

f(Y x

SHEAR STRESS DISTRIBUTION

~ ~ I

I

I

I

I

I

BENDING

Fig. 14.40

Relationship between shear and bending.

Fig. 14.41

SHEAR

Typical aircraft spar in bending and shear.

AIRCRAFT DESIGN

384

STRUCTURES AND LOADS

area. If this beam is split lengthwise as shown, the upper section has only leftward forces, so a shear force must be exerted along the cut. This shear force must be the sum of the horizontal stresses times the elemental areas above the cut. This reaches a maximum at the neutral axis. At the upper and lower surfaces, this shear force is zero. The bottom of Fig. 14.40 shows the resulting vertical distribution of shear forces, expressed as magnitude toward the right. (Don't be confused by this presentation; the shear forces are exerted in a vertical direction, but we show the magnitude to the right to illustrate the distribution of magnitude from top to bottom.) T=

V bl Y

lhl2 z dA

(14.36)

zl

Equation (14.36) describes this mathematically, where the integral term represents the area above the cut located at Z = Zl. Note that the distribution of shear stresses depends upon the shape of the cross section. For a beam of rectangular cross section, the maximum shearing stress (at the neutral axis) is 1.5 times the averaged shearing stress (total shear divided by cross-sectional area). For a solid circular cross section, the maximum shearing stress is 1.33 times the averaged value. Figure 14.41 shows a typical aircraft wing spar consisting of thick "spar caps" separated by a thin "shear web." The cross-sectional area of the

385

shear web is insignificant compared to the area of the spar caps, so the caps absorb virtually all of the bending force (stress times area). The shear stress depends upon the cross-sectional area above the point of interest, and is therefore essentially constant within the thin shear web, as shown to the right. In aircraft wing spar analysis it is common to assume that the caps absorb all of the bending stresses and that the web (extended to the full depth of the spar) absorbs all of the shear. This is shown at the bottom of Fig. 14.41. It is also assumed that the shear is constant within the web and therefore the maximum shear stress equals the average shear stress (shear divided by web area). The shear web will fail in buckling long before the material maximum shear stress is reached. Equation (14.37) defines the critical buckling shear stress for a shear web. The value of K is obtained from Fig. 14.42. Fshear buckle

=

(14.37)

KE (t / b )2

Braced-Wing Analysis

A wing braced with a strut will have the bending moments greatly reduced compared to a fully cantilevered wing. However, the analysis is more complex because of the spanwise compression loads exerted upon the wing by the strut. This can increase the bending moment by as much as a third compared to an analysis that ignores this compression effect.

BENDING STRESSES

BENDING STRESSES

SHEAR STRESS MAGNITUDE SPAR APPROXIMATIONS

z

f(Y x

SHEAR STRESS DISTRIBUTION

~ ~ I

I

I

I

I

I

BENDING

Fig. 14.40

Relationship between shear and bending.

Fig. 14.41

SHEAR

Typical aircraft spar in bending and shear.

387

STRUCTURES AND LOADS

AIRCRAFT DESIGN

386

(14.39)

_ DJ + wi Mmax - cos(x/j) 16

tan(~m) =

4

12

V

llf7_lb t-

j = .JEIIP

(14.41)

D2 - DJ cos(Llj) sin(L/j)

(14.42)

h

"-

K

--

8

\ 6

(14.40)

where

Clamped Edges

\

10

D2 - DJ cos(LIj) DJ sin (Llj)

~

f

4 ~ ~~~_~_

F scr

r--

CJ =

= KE(flb)Z

I

(14.43)

·2

C2 = DJ = MJ - WJ

, Simply Supported Edges

(14.44)

·2 D2 = M2 - wJ

""- t-.. 4

. d T 1444 shows a solid circular shaft in torsion. The apphe torqu\ Igure a ·twistin deformation 4> that depends upon the length of ~ e P~O~tuc;.~ shown at ~he right of the figure, the torque is resiste~ by sheanng :tr~s~es that increase linearly with distance from the center-If the stresses

Torsion

F

2

2

Fig. 14.42

4

alb

6

8

I d remain within the elastic limit. . The s~ear stressehs duefto torfSIt~ne :~:~a(l~u=a~). at a maXImum at t e sur ace 0

10

·th Eq (14 45) and are

~~e ang~lar den'ection in

Shear web buckling. (NACA TN3781) LIFT DISTRIBUTION

Figure 14.43 shows a typical braced wing. The compression load P is the horizontal component of the force on the strut (S). The vertical component of S is found from summing the moments about the pin at the wing root, using the equivalent concentrated lift loads as discussed earlier. The shear loads of the braced wing are analyzed as before, taking into account the large concentrated vertical load of the strut. The bending moment must be analyzed with special equations provided below. The portion of the wing outboard of the strut is analyzed as before, and the bending moment at the strut location is determined (M2 ). The root bending moment (M 1) is usually zero unless the hinge point is above or below the neutral axis, causing a bending moment due to the compression load P. The lift distribution on the portion of the wing inboard of the strut must be approximated by a uniform load distribution (w). This is usually a reasonable approximation inboard of the strut. The following equations describe bending-moment distribution, maximum bending moment, and spanwise location of the maximum bending moment (Ref. 60): M(x)

=

C J sin(x/j) + C z cos(x/j) + wj2

(14.38)

llltLb SH=PX

w

L

Fig. 14.43 Braced wing analysis.

389

AIRCRAFT DESIGN

STRUCTURES AND LOADS

radians is determined from Eq. (14.46). These equations also apply to circular tubing under torsion, using the appropriate value of Ip as provided previously.

b is its width. These equations may also be applied to members bent up from flat sheet metal by "unwrapping" the member to find the total effective width.

388

T

(14.45)

Trllp

=

T =

(14.46)

4> = TLIGlp

TL( s ) 4> = G 4A 2t

(14.50)

Analysis of the torsional stresses in a complex shape such as a multi celled wing box goes beyond the scope of this book. See Ref. 60 for a discussion of such analysis. 14.11 FINITE-ELEMENT STRUCTURAL ANAL VSIS The structural-analysis methods described above, along with extensive handbooks and nomograms, have been used for many years for aircraft structural design. Today these methods are a dying art. Instead, virtually all major structural analysis is now performed using finite-element computer programs. Even today's homebuilders have access to finite-element programs using personal computers that are as powerful as the mainframe computers of the 1960's. The Finite Element Method (FEM) is based upon the concept of breaking the structure of the aircraft into numerous small "elements," much like the gridding of the air-mass for CFD. Equations describing the structural behavior of these finite elements are prepared using various approximations of the end-constraints and deflection shapes for the element.

(14.47)

TI2At

(14.49)

TL 4> = (3bt3G

For a noncircular member under torsion, the analysis is generally much more complex. Several special cases can be readily solved. A thin-walled, closed cross-sectional member with constant wall thickness t, total crosssectional area A, and cross-sectional perimeter s has shear stress and angular deflection as defined by Eqs. (14.47) and (14.48). T=

T abt 2

(14.48)

~olid rectangular members may be analyzed with Eqs. (14.49) and (14.50) USIng the values from Table 14.7, where t is the thickness of the member and

, I

TRIANGULAR PLATE

I I I

RECTANGULAR PLATE

BAR OR BEAM

......,

I

I

INTERNAL SHEAR STRESSES

I I I Fig. 14.44

Table 14.7 It

SOLID TETRAHEDRON

Solid circular shaft in torsion.

Torsion constants

1.00

1.50

1.75

2.00

2.50

3.00

4

6

8

10

0.208

0.231

0.239

0.246

0.258

0.267

0.282

0.299

0.307

0.313

0.3:

0.141

0.196

0.214

0.229

0.249

0.263

0.281

0.299

0.307

0.313

0.3:

L_--?,... .....

(Xl

SOLID RING

. Fig. 14.45

Typical finite elements.

AIRCRAFT DESIGN

380

STRUCTURES AND LOADS

EM = 0 = -19.6 (4000) + 20 Fe Fe =3919.2

4,000 +---69.6 30

EM=O= -69.6 (4000)-30 FF COSH FF = -9463

EFH = 0=3919.2+F E COS22 + (-9463) COS 11 FE =5775 EFy=O= -4000+F E SIN 22-(-9463) SIN H FE =5775 4,000

Fig. 14.36 Method of moments/method of shears.

The lower illustration of Fig. 14.36 shows the use of the method of shears to solve for the inner strut. This method involves severing the structure along a plane which cuts only three members, the upper and lower strut and the inner strut under analysis. The severed part of the structure is analyzed as a free body, summing either the vertical and horizontal forces, which must total zero. Note that by calculating the unknown strut force both ways (vertical and horizontal summation), a check of your result can be made. This example gives a result of 5775 lb. These methods are only applicable if the truss structure is "statically determinate." In general, a truss is statically determinate if every strut can be cut by some plane that cuts only two other struts. This insures that there is always a joint with only two unknown struts, permitting solution by the method of joints. For "indeterminate" trusses, more complicated methods based upon deflection analysis must be used (see Refs. 54 and 60). Once the loads in each member of the truss are known, the struts can be analyzed using the equations presented above for tension or compression. Use the appropriate effective length for welded, riveted, or bolted columns from Fig. 14.31. To provide an extra margin of safety, it is customary to assume that welded steel-tube motor mounts act as though the ends were pinned (Le = L). The 3-D trusses, or "space structures," are solved similarly to the 2-D truss. Square cross section 3-D trusses, such as a typical welded-tube fuselage, can sometimes be solved separately in side view and top view as 2-D

381

structures. The resulting strut loads are then summed for the various members. This is permitted provided that the combined loads on all struts are within the elastic range. For more complicated 3-D trusses, the method of joints can be applied using three equations and three unknown strut loads. This involves simultaneous solution of equations, e.g., with a simple computer iteration program. In some cases the moments about some selected point can be used to obtain the solution with less effort. Space structures are discussed in detail in Ref. 54. Beam Shear and Bending A common problem in aircraft design is the estimation of the shear and bending stresses in the wing spars or fuselage. This is a two-step process. First, the shear and bending moment distributions must be determined; then the resulting stresses must be found. Figure 14.37 shows a simple beam with a distributed vertical load. The beam is shown cut to depict internal forces. The right side of the beam being a free body, the sum of the vertical forces and the sum of the moments must equal zero. If the severed part of the beam is to remain in vertical equilibrium, the externally applied vertical forces must be opposed by a vertical shear force within the cross section of the material, as shown. Thus, for any span station the shear force is simply the sum of the vertical loads outboard of that station, or the integral of a distributed load.

SHEAR

SUPPORT SHEAR REACfION

MOMENT

SUPPORT MOMENT REACTION

MOMENT REACTION DUE TO SPANWISE COMPRESSION AND TENSION

Fig. 14.37 Shear and moment in beams.

AIRCRAFT DESIGN

STRUCTURES AND LOADS

The element equations are then linked together using matrix algebra so that the entire structure's response to a given external loading condition can be determined. The huge size of the matrices used for FEM analysis requires computers for solution of all but the most trivial cases. Figure 14.45 illustrates the more commonly used finite elements. The aircraft structure must be modeled as a connected collection of one or more of these finite-element shapes. Selection of which element type to use is a matter of engineering judgment. Unfortunately, the selection of the element type can influence the results. Also, the selection of the size of the elements requires experience. As a general rule, the size of the elements should be reduced anywhere that the stress is expected to vary greatly. An example of this would be in the vicinity of a corner. Figure 14.46 shows an FEM example in which the major structural members of a prop fan research aircraft are modeled using the rectangular-plate finite element. As is the case for CFO gridding, the modeling of a complex structure for FEM analysis can be very time-consuming. Oeta~led .derivations of the equations for the various finite-element types shown III FIg. 14.45 are beyond the scope of this book (see Refs. 84 and 85).

A simple example, the one-dimensional (1-0) bar, will be developed to illustrate the principles involved. Figure 14.47 depicts a simple 1-0 bar element with end-loadings PI and P 2 , and end-deflections UI and U2. For a static structural analysis, PI must equal the negative of P 2 , although this is not true in a dynamic analysis. The cross-sectional area of the bar is shown as A. Note that while this example is a 1-0 case, the deflected position is depicted slightly offset for clarity. The strain E is defined earlier in this chapter as the change in length divided by the original length L, as shown in Eq. (14.51). The stress a is defined as the load divided by the cross-sectional area, and Young's Modulus E is defined as the stress divided by the strain. This results in Eq. (14.52).

390

391

(14.51) (14.52)

E = alE = (PIA)/[(ul - u2)IL]

or (14.53) Applying a load PI yields Eq. (14.54). Similarly, applying a load P 2 results in Eq. (14.55). The change in signs of the deflections in Eq. (14.55) is due to the assumed directions of the two loads as drawn in the figure. (14.54) (14.55)

AFT FUSELAGE

Equations (14.54) and (14.55) can be combined into matrix form as shown in Eqs. (14.56) and (14.57). The k matrix is called the "stiffness matrix" because it relates the amount of deflection to the applied loads. The values within the k matrix are called "stiffness coefficients." The U matrix containing the deflection terms is called the "displacement vector." The P matrix is the "force vector." (Letters other than P and U are

L

"-( c---- ~, NACELLE

~--

------

----I-)--..- - -,.{-'- -'. .}--~~ . -=F __ :.~/

I

.,

U2

Fig. 14.46

Typical finite element model. (Courtesy Lockheed)

Fig. 14.47

Simple I-D bar element.

P2

392

AIRCRAFT DESIGN

STRUCTURES AND LOADS

393

frequently used for these terms, but for some reason k is almost always used for the stiffness matrix.) (14.60) (14.56)

(P) = [k] (u)

(14.57)

The values E, A, and L are known, so the stiffness matrix is known. By inverting the stiffness matrix, the deflections can be found for any loading condition. This simple example could easily be solved by classical structure techniques. The power of FEM is in the assemblage of numerous finite elements. Figure 14.48 shows a two-element assemblage using the I-D bar element developed above. Two bars of different length and cross-sectional area are connected. The point where two (or more) finite elements are connected is called a "node" and is distinguished by the fact that at a node, the displacements of the connected finite elements are the same. Thus, U2 represents both the displacement of the right end of the first element and the displacement of the left end of the second element. From Eq. (14.56), the matrix equations for the left- and right-side elements can be written as Eqs. (14.58) and (14.59).

This completes the FEM development for this example. The re~aini~g work is strictly computation based upon the actual values of the vanables m a given design problem. For example, Fig. 14.49 shows a two-bar structure in which the right side attaches to a wall, loads are as shown, and the dimensional and material values are as indicated. This produces the follow-

~:J l(~2\Xxl~67) (~~45xXl~??

ing[,

P3

=

0

(-9.2 X

106)

0 (_9.2 X 106 (9.2 X 106)

)l [~~J

(14.61)

U3

The 3 x 3 stiffness matrix in Eq. (14.61) can be inverted to find the deflections for any loading. This would first require determining the unknown wall-reaction load P 3 • • Alternatively, we can simplify the FEM ma~ri~ solution b~ notmg that the deflection at the wall U3 is zero, so we can ehmmate the th~rd row and ~he third column from the matrix. This produces Eq. (14.62) with a 2 x 2 stiffness matrix. (14.62)

(14.58)

(14.63) (14.59)

10 .0931 = lUll

Now the matrices can be assembled by merging the element matrices. This is shown in Eq. (14.60). Note that the "overlapping" terms at the node result from the nodal condition of identical deflection (U2 in this case). These overlapping terms are added in forming the assembled matrix. 12 in.

r

"-( e. {.. . --:_ _~~_ ~(j-. L

J

H

-I·

L2

1-1

--.....1....---- 14 in. - - - - -..... A2 =12 in. 2

-,

~

(14.64)

l 0.077J l U2J

P2 -

.......

P3

H

U2 ALUMINUM: E=IO.7xI0 6 psi

Fig. 14.48

I-D bar FEM assembly.

Fig. 14.49 FEM example.

394

AIRCRAFT DESIGN

In .Eq .. (14.63) we have found the inverse of the reduced k matrix. By SubstIt~tmg ~he actual values of the loadings P we determine the deflections as prOVided m Eq. (14.64). We can then use the deflections of the nodes to solve for the stram and stress, as follows: fJ

=

(0.093 - 0.077)/12

fZ

=

(0.077 - 0)/14

=

al =

14,267 psi

az

58,850 psi

=

=

0.0013

0.0055

15 WEIGHTS

(14.65) (14.66) (14.67) (14.68)

This I-D exam?le .does not illustrate the complications caused by 3-D ~eo~etry. For th.IS SImple example the deflections at the nodes produce IdentIcal c~ange~ m the length of the bars. Were the bars connected at some angle, the IdentIc~1 n~dal.deflections would produce different changes in bar lengths. Matnx dIrectIon-cosine terms must be used to keep track of these 3-D effects. Most finite-el~ment analyses use surface elements rather than simple bar eleme~ts. The tnangle element shown in Fig. 14.45 is typical, and allows a complIcated structure to be broken into numerous connected elements. These ele~ents are. assumed to be connected at the nodes (corners) where the deflectIons are Identical. Equations are. prepar~d in matrix form describing how each element r~spond~ to loadmg~ at I~S nO.des. The element stiffness matrices are combmed usmg a~propnate dIrectIOn cosine terms to account for 3-D geometry and ~he combIned matrix is inverted to solve for the deflections for a give~ loadmg. Fo.r dynamic analysis, mass and damping terms are developed using matnx me~hods. These greatly increase the number of inputs required for the analYSIS. Fortunately, working structural engineers do not need to develop their own FEM program every time they wish to analyze a structure. There are numerous "canned" FEM programs available, ranging from simple personal-~omputer ones to million-line programs. The Industry-standard FEM program is the "NASTRAN (NAsa STR _ tural ANalysis)" program, developed years ago for NASA and conti~~­ ously enh~nced both by NASA and various private companies. NASTRAN h:ndles VIrtually everything, but requires substantial experience to insure t at the ~esults are meaningful. However, for complex structural analysis some vanant of NASTRAN will probably be in use for many years to come:

15.1 INTRODUCTION The estimation of the weight of a conceptual aircraft is a critical part of the design process. The weights engineer interfaces with all other engineering groups, and serves as the "referee" during the design evolution. Weights analysis per se does not form part of the aerospace engineering curriculum at most universities. It requires a broad background in aerospace structures, mechanical engineering, statistics, and other engineering disciplines. There are many levels of weights analysis. Previous chapters have presented crude statistical techniques for estimating the empty weight for a given takeoff weight. These techniques estimate the empty weight directly and are only suitable for "first-pass" analysis. More sophisticated weights methods estimate the weight of the various components of the aircraft and then sum for the total empty weight. In this chapter, two levels of component weights analysis will be presented. The first is a crude component buildup based upon plan form areas, wetted areas, and percents of gross weight. This technique is useful for initial balance calculations and can be used to check the results of the more detailed statistical methods. The second uses detailed statistical equations for the various components. This technique is sufficiently detailed to provide a credible estimate of the weights of the major component groups. Those weights are usually reported in groupings as defined by MIL-STD-1374, or some similar groupings defined by company practice. MIL-STD-1374 goes into exhaustive detail (taxi lights, for example!), but at the conceptual level the weights are reported via a "Summary Group Weight Statement." A typical summary format appears as Table 15.1, where the empty weight groups are further classified into three major groupings (structure, propulsion, and equipment). The structures group consists of the load-carrying components of the aircraft. Note that it includes the inlet (air-induction-system) weight, as well as the nacelle (engine-section) weight including motor mounts and firewall provisions-despite their obvious relationship to the engine. The propulsion group contains only the engine-related equipment such as starters, exhaust, etc. The as-installed engine includes the propeller, if any. Armament is broken down into fixed items, which are in the equipment groups, and expendable items, which are in the useful load. Sometimes a judgement call is required. For example, a gun may be considered to be . fixed equipment, or it may be viewed as readily removable and unimportant to flight and therefore a part of the useful load.

395

WEIGHTS

AIRCRAFT DESIGN

396

The takeoff gross weight-the sum of the empty weight and the useful load-reflects the weight at takeoff for the normal design mission. The flight design gross weight represents the aircraft weight at which the structure will withstand the design load factors. Usually this is the same as the takeoff weight, but some aircraft are designed assuming that maximum loads will not be reached until the aircraft has taken off and climbed to altitude, burning off some fuel in the process. "DCPR" stands for "Defense Contractors Planning Report." The DCPR weight is important for cost estimation, and can be viewed as the Table 15.1

Group weight format

Group

Group

STRUCTURES GROUP

EQUIPMENT GROUP

Wing Tail-horizontal!canard vertical ventral Body Alighting gear-main auxiliary arresting gear catapult gear Nacelle/engine section Air induction system

PROPULSION GROUP Engine-as installed Accessory gearbox and drive Exhaust system Cooling provisions Engine controls Starting system Fuel system/tanks

Flight controls APU Instruments Hydraulic Pneumatic Electrical Avionics Armament Furnishings Air conditioning/ECS Anti-icing Photographic Load and handling TOTAL WEIGHT EMPTY

weight of the parts of the aircraft that the manufacturer makes, as opposed to buys and installs. DCPR weight equals the empty weight less the weights of the wheels, brakes, tires, engines, starters, cooling fluids, fuel bladders, instruments, batteries, electrical power supplies/converters, avionics, armament, fire-control systems, air conditioning, and auxiliary power unit. DCPR weight is also referred to as "AMPR" weight (Aeronautical Manufacturers Planning Report). In a Group Weight Statement, the distance to the weight datum (arbitrary reference point) is included, and the resulting moment is calculated. These are summed and divided by the total weight to determine the actual centerof-gravity (c.g.) location. The c.g. varies during flight as fuel is burned off and weapons expended. To determine if the c.g. remains within the limits established by an aircraft stability and control analysis, a "c.g.-envelope" plot is prepared (Fig. 15.1). The c.g. must remain within the specified limits as fuel is burned, and whether or not the weapons are expended. It is permissible to "sequence" the fuel tanks, selecting to burn fuel from different tanks at different times to keep the c.g. within limits. However, an automated fuel-management system must be used, and that imposes additional cost and complexity. Note that the allowable limits on the C.g. vary with Mach number. At supersonic speeds the aerodynamic center moves rearward, so the forwardc.g. limit may have to move rearward to allow longitudinal trim at supersonic speeds. However, the aft-c.g. limit is often established by the size of the vertical tail, which loses effectiveness at supersonic speeds. This prevents moving the aft limit rearwards at supersonic speeds, forcing a very narrow band of allowable limits.

GROSS WEIGHT

wo-+_ _ _ _

TAKEOFF ~

USEFUL LOAD GROUP Crew Fuel-usable -trapped Oil Passengers Cargo/baggage Guns Ammunition Pylons and racks Expendable weapons Flares/chaff TAKEOFF GROSS WEIGHT Flight design gross weight Landing design gross weight DCPR weight

397

C.G. LOCATION, OJo M.A.C. FROM DATUM

Fig. 15.1

C.G. envelope diagram.

398

WEIGHTS

AIRCRAFT DESIGN

15.2 APPROXIMATE GROUP WEIGHTS METHOD Early in design it is desirable to do a rough C.g. estimate. Otherwise, substantial rework may be required after the c.g. is properly estimated. A rough c.g. estimate can be done with a crude statistical approach as provided in Table IS.2. The wing and tail weights are determined from historical values for the weight per square foot of exposed plan form area. The fuselage is similarly based upon its wetted area. The landing gear is estimated as a fraction of the takeoff gross weight. The installed engine weight is a multiple of the uninstalled engine weight. Finally, a catch-all weight for the remaining items of the empty weight is estimated as a fraction of the takeoff gross weight. This technique also applies the approximate locations of the component c.g. as given in Table IS.2. The resulting c.g. estimate can then be compared to the desired c.g. location with respect to the wing aerodynamic center. Also, these approximate component weights can be used as a check of the more detailed statistical equations provided below.

I~

o V) I

~

g

'" '" '"

....

~

.~

~

"3

::E

'"(;E '"(;E '"E '" 'a 'a ~

~

.3~S ~ "0~ ~

0. 0. 0.

"B

"0

"'"'

~O

"0

~

"0 0 ri- ~~ f-<

:>\

~

0-

VJ~ VJV VJV

....

.0



.!:::i 0

S

~

~ til

... .~...0 ~:r: > 01)

~

.-

0

~ 0

~

~ 0 0

'00 f-< ~

V;~~"",:r;;

C

01) ' - '

C

NNN-~

t;i

.c _"'"' ~.o 'ii)

399

til

0 0

01)

til

'0

'"

;::l ~

-...".0

. .

0 ~

'00 C

01)

0

01)

-0

~

:aC til

..J

~



'" ~ c. 0

!

..= ~ '"

S 0

15.3 STATISTICAL GROUP WEIGHTS METHOD A more refined estimate of the group weights applies statistical equations based upon sophisticated regression analysis. Development of these equations represents a major effort, and each company develops its own equations. To acquire a statistical database for these equations, weights engineers must obtain group-weight statements and detailed aircraft drawings for as many current aircraft as possible. This sometimes requires weights engineers to trade group-weight statements much like baseball cards ("I'll trade you a T-4S for an F-16 and a C-SB"!) The equations presented below typify those used in conceptual design by the major airframe companies, and cover fighter/attack, transport, and general-aviation aircraft. They have been taken from Refs. 62-64 and other sources. Definitions of the terms follow the equations. It should be understood that there are no "right" answers in weights estimation until the first aircraft flies. However, these equations should provide a reasonable estimate of the group weights. Other, similar weights equations may be found in Refs. 10, 11, and 23. It's a good idea to calculate the weight of each component using several different equations and then select an average, reasonable result. Reference 11 tabulates group-weight statements for a number of aircraft. These can also be used to help select a reasonable weight estimate for the components by comparing the component weights as a fraction of the empty weight for a similar aircraft. Table IS. 3 tabulates various miscellaneous weights. When the component weights are estimated using these or similar methods, they are tabulated in a format similar to that of Table IS.1 and are summed to determine the empty weight. Since the payload and crew weights are known, the fuel weight must be adjusted to yield the as-drawn takeoff weight that is the sum of the empty, payload, crew, and fuel weights. If the empty weight is higher than expected, there may be insufficient fuel to

WEIGHTS

AIRCRAFT DESIGN

400

Table 15.3

complete the design mission. This must be corrected by resizing and optimizing the aircraft as described in Chapter 19, not by simply increasing fuel weight for the as-drawn aircraft (which would invalidate the component weight predictions that were based on the as-drawn takeoff weight).

Miscellaneous weights (approximate)

Fighter/Attack Weights

Missiles Harpoon (AGM-84 A) Phoenix (AIM-54 A) Sparrow (AIM-7) Sidewinder (AIM-9) Pylon and launcher

1200 Ib 1000 Ib 500 Ib 200 Ib .12 W missile

M61 Gun Gun 940 rds ammunition

401

x (1 + }..)0.05 (cosArl.O S~s~

Whorizontaltail

=

3.316 ( 1+

250lb 550lb

B:

F

)-2.0

(15.1)

(w,1000

N)O.260 Z

806

Sgi

(15.2)

Seats Flight deck Passenger Troop

601b 321b 11 Ib

w.fuselage -- 0. 499 K dwf »1.dg35NO. 25 L 0.5D O.849 WO. 685

Instruments Altimeter, airspeed, accelerometer, rate of climb, clock, compass, turn & bank, Mach, tachometer, manifold pressure, etc. Gyro horizon, directional gyro Heads-up display

Z

1-2 Ib each 4-6 Ib each 401b

W main landing =

KctJ(tpg( WtN, )0.25 L~·973

(15.4)

(15.5)

gear

W nose landing -- (W1IJ)0.290Lo.5No.525 J1"'/ n nw

(15.6)

gear

Lavatories Long range aircraft Short range aircraft Business/executive aircraft

(15.3)

1.11 N~;,~ 0.31 N~a~~ 3.90 N~a~~

Arresting gear Air Force-type Navy-type

O.795""'O.579N 0 013Nen W engine =. 1 Z

(15.7)

mounts

Wfirewall

= 1. 13 Sfw

Wengi.ne section

=

0.01 n1n7l7NenNz

(15.8) (15.9)

Catapult gear Navy carrier-based

Wair induction system

= 13.29KvgL~·643K~·182N:n498(Ls/Ldr{)·373 De

(15.10)

Folding Wing Navy carrier based

.06 Wwing

where Kd and Ls are from Fig. 15.2. Wtailpipe

Wengine cooling

= 3.5DeLtpNen

(15.11)

= 4.55DeLs~en

(15.12)

402

AIRCRAFT DESIGN

o

KO = 1.0

D

Wfurnishings = 217. 6Nc

SPLIT DUCT

=2.2

KO

+ 200 N c )/IOOO]o.735

(15.23)

and anti-ice

(15.24)

Whandling = 3.2 X 10-4 Wdg gear

=2.75

[email protected] KO = 1.68

(15.22)

Wairconditioning = 201.6 [(Wuav

Ko = 1.31

OD KO

403

WEIGHTS

INLET FRONT FACE

ENGINE FRONT FACE

Cargo/Transport Weights

KO = 3.43

~ ~

(15.25) Fig. 15.2

Inlet duct geometry.

K O =2.6

fir

n

horizontal tail

00379Kuht (I + F wIB h )-0.25 w:O.639NO.I0S0.75L-I.O =. dg z ht t

X K~·704(CoSAhtrl.°A2·166 (1

Woilcooling = 37.S2Ni,,023

(15.13)

+ SeISht)O.1

875 . = 0 0026(1 + H t IHv )O.225wJ·556No.536L -o.5S0.5KO. w:vertical' dg z vt t

tail

Wengine = 1O.5NiriOO8Le~222

(15.26)

Z

(15.14)

controls

Wstarter = 0.025 Tj.760~ri72 (pneumatic)

(15.27)

(15.15) (15.2S)

Wfuelsystem = 7.45 v?47(1 and tanks

+ V;)-o.095 V;

(I

+

Vp)m.066~.052 V; en t

(T.

SFC)O.249 1000

ril.888NO.25L 0.4" ,0.321 ",-0.5 0..1 Wmainlanding = O.0I06Kmp W{ I m lYmw lYmss stall Tr

(15.29)

gear

(15.16) Wflight

= 36.2SMO.OO3Sc~·489~.484~.127

controls

45 - 0032K w:O.646NO.2L 0.5NO. Wnos e landing . np I I n nw

(15.30)

gear

(15.17) Wnacelle = O.6724KngN2/0N~,294N~119 W~~611N~/84S~224

Winstruments = 8.0 + 36.37 Ne~676N?237

+ 26.4(1 + N c;)1.356

WhydraUlics = 37 .23 KvshN~·664

Welectrical = 172.2KmeRkO.152No.iOL va c a0.10" IVril.091 gen Wavionics = 2.117 w:1a~33

group

(15.31)

(includes air induction)

(15.IS) Wengine = 5.0Nen (15.19)

+ O.SOLec

(15.32)

controls

(15.20)

N w: )0.541 Wstarter . = 49.19 ( {OOOen (pneumatic)

(15.21)

Wfuel

system

= 2.405

v?,606(1

+ v;1v;rl.°(1 + VpIV;)Nto. 5

(15.33)

(15.34)

404

AIRCRAFT DESIGN

. w:flight

controls

WEIGHTS

= 145 . 9N9·554 (1 + N m INf )-I.°So.20(1 f cs y X 10-6)0.07

(15.35)

WAPU = 2.2 WAPU installed uninstalled

W

(15.36)

tic)

. = 0.073 1 + 0.2~ (N w: )0.376 q O.122S0.873 (100 - - - -0.49 vertIcal H z dg vt cosA vt tail v X

(15.37)

H)

(

405

A )0.357 -2}"ei039 ( -COS Av [

S1.086(N W fuselage =0052 • 'f z w:dg )0.177L-o.051(LID)-0.072qo.241+ f

(15.48)

w:press

(15.49)

(15.38) Wmainlanding = 0.095 (N,Wdo.768(Lm/12)0.409 (15.39) - 0. 125 (N"W)0.566(L n 112)°·845 Wnose landing -

(15.51)

Winstalled engine = 2.575 »1ri922Nen (total)

(15.52)

gear

Wavionics = 1. 73 w,?a~83

(15.40)

Wfurnishings = 0.0577/tj·1 »1. 393 S;·75

(15.41) _

w:.aIr

conditioning

Wfuel system - 2.49 v?

= 62 • 36N?·25(V. p pr l1000)O.604TrTll.1O Wuav

(15.42)

w:flight -controls

Wanti-ice = 0.002 Wdg

(15.45)

=O.036S0.758W:O.OO35(~)0.6 2 fw

cos A

q

tiC)· -0.3(N w:dg) 0.49

0.006},,0.04(100 cosA

z

(15.46) = 0 016(N

~~;,zontal'

X

Z

(15.53) (15.54)

w:dg )0.414qo.168so.896(100 tic) -0.12 ht cosA

A _ )0.043 },,-o.02 -_ ( cos2Aht h

Whydraulics = 0.00 1 Wdg

(15.55)

Welectrical = 12.57 (Wfuel system + Wavionics)0.51

(15.56)

Wavionics = 2.117 w,?a~33

(15.57)

TrTll.17 M O.oS w:air conditioning =. 0 265 TrTll.52No.68 W dg p Wavionics

(15.58)

Wfurnishings = 0.0582 W dg - 65

(15.59)

Weights Equations Terminology = aspect ratio Bh = horizontal tail span, ft Bw = wing span, ft D = fuselage structural depth, ft De = engine diameter, ft Fw = fuselage width at horizontal tail intersection, ft HI = horizontal tail height above fuselage, ft HIIHv = 0.0 for conventional tail; 1.0 for "T" tail Hv = vertical tail height above fuselage, ft Iy = yawing moment of inertia, Ib-ft2 (see Chap. 16) A

W.

0 .053 L 1.536Bo.371 (Nz w:dg X 10-4)°·80 w

and anti-ice

General-Aviation Weights

w

0.363 1 NO.242NO.157 1 + ViIV; I en ) (

(15.44)

Wmilitarycargo = 2.4 X (cargo floor area, ft 2) handling system

g

.726

(15.43)

Whandling = 3.0 X 10-4 Wdg gear

Wwin

(15.50)

gear

fir 7291Ro.782Lo.346NO.10 n electrical =. kva a gen

(15.47)

Kmp Kng Knp Kp K, K rht K tp K tpg K tr K uht Kvg Kvs Kvsh

Kws

Ky Kz L

La Ld Lec

Lf Lm Ln Ls Lsh L, L,p M

Nc Nci N en Nf N gen N,

Nu Nm

N mss N mw Nnw

AIRCRAFT DESIGN = 2.25 for cross-beam (F-ll1) gear; = 1.0 otherwise = duct constant (see Fig. 15.2) = 1.0 if no cargo door; = 1.06 if one side cargo door; = 1.12 if two side cargo doors; = 1.12 if aft clamshell door; = 1.25 if two side cargo doors and aft clamshell door = 0.768 for delta wing; = 1.0 otherwise = 0.774 for delta wing aircraft; = 1.0 otherwise = 1.12 if fuselage-mounted main landing gear; = 1.0 otherwise = 1.45 if mission completion required after failure; = 1.0 otherwise = 1.126 for kneeling gear; = 1.0 otherwise = 1.017 for pylon-mounted nacelle; = 1.0 otherwise = 1.15 for kneeling gear; = 1.0 otherwise = 1.4 for engine with propeller or 1.0 otherwise = 1.133 if reciprocating engine; = 1.0 otherwise = 1.047 for rolling tail; = 1.0 otherwise = 0.793 if turboprop; = 1. 0 otherwise = 0.826 for tripod (A-7) gear; = 1.0 otherwise = 1.18 for jet with thrust reverser or 1.0 otherwise = 1.143 for unit (all-moving) horizontal tail; = 1.0 otherwise = 1.62 for variable geometry; = 1.0 otherwise = 1.19 for variable sweep wing; = 1.0 otherwise = 1.425 if variable sweep wing; = 1.0 otherwise = 0.75[1 + 2A)/(l + A)] (Bw tanA/L) = aircraft pitching radius of gyration, ft ( 0.3Lt) = aircraft yawing radius of gyration, ft ( L,) = fuselage structural length, ft (excludes radome, tail cap) = electrical routing distance, generators to avionics to cockpit, ft = duct length, ft = length from engine front to cockpit-total if multiengine, ft = total fuselage length = length of main landing gear, in. = nose gear length, in. = single duct length (see Fig. 15.2) = length of engine shroud, ft = tail length; wing quarter-MAC to tail quarter-MAC, ft = length of tailpipe, ft = Mach number = number of crew = 1.0 if single pilot; = 1.2 if pilot plus backseater; = 2.0 pilot and copas senger = number of engines = number of functions performed by controls (typically 4-7) = number of generators (typically = N en ) = ultimate landing load factor; = N gear x 1.5 = nacelle length, ft = number of mechanical functions (typically 0-2) = number of main gear shock struts = number of main wheels = number of nose wheels

= =

WEIGHTS Np Ns Nt Nu Nw Nz q R kva Scs Scsw

Se Sf Sfw

Sh'

Sn S, Sv,

Sw SFC T Te Vi Vp Vp ,

Vt W

We Wdg Wee

Wuav

A

407

number of personnel onboard (crew and passengers) number of flight control systems = number of fuel tanks = number of hydraulic utility functions (typically 5-15) = nacelle width, ft = ultimate load factor; = 1.5 x limit load factor = dynamic pressure at cruise, Ib/ft2 = system electrical rating, kv . A (typically 40-60 for transports, 110-160 for fighters & bombers) = total area of control surfaces, ft2 2 = control surface area (wing-mounted), ft = elevator area, ft = fuselage wetted area, ft2 = firewall surface area, ft2 = horizontal tail area = nacelle wetted area, ft2 = rudder area, ft2 = vertical tail area, ft2 = trapezoidal wing area, ft2 = engine specific fuel consumption-maximum thrust = total engme thrust, lb = thrust per engine, lb = integral tanks volume, gal = self-sealing "protected" tanks volume, gal = volume of pressurized section, fe = total fuel volume, gal = fuselage structural width, ft = maximum cargo weight, lb = design gross weight, lb = weight of engine and contents, lb (per nacelle), =2.331 W2n~~e KpKtr = engine weight, eacli, lb = weight of fuel in wing, lb = landing design gross weight, lb = weight penalty due to pressurization, = 11.9 + (VprPdelta)o.271, where P delta = cabin pressure differential, psi (typically 8 psi) = uninstalled avionics weight, lb (typically = 800-1400 lb) = wing sweep at 25070 MAC =

=

15.4 ADDITIONAL CONSIDERATIONS IN WEIGHTS ESTIMATION These statistical equations are based upon a database of existing aircraft. They work well for a "normal" aircraft similar to the various aircraft in the database. However, use of a novel configuration (canard pusher) or an advanced technology (composite structure) will result in a poor weights estimate when using these or similar equations. To allow for this, weights .engineers adjust the statistical-equation results using "fudge factors" (defined as the variable constant that you multiply your answer by to get the right answer!)

WEIGHTS

AIRCRAFT DESIGN

408

Fudge factors for composite-structure, wood or steel-tube fuselages, braced wings, and flying-boat hulls are provided in Table 15.4. These should be viewed as rough approximations only and subject to heated debate. For example, there are those who claim that a properly-designed steeltube fuselage can be lighter than an aluminum fuselage. One final consideration in aircraft-weights estimation is the weight growth that most aircraft experience in the first few years of production. This growth in empty weight is due to several factors, such as increased avionics capabilities, structural fixes (such as replacing an aluminum fitting with steel to prevent cracking), and additional weapons pylons. Figure 15.3 shows the empty-weight growth of a number of aircraft. In the past, a weight growth of 5070 in the first year was common. Today's better design techniques and analytical methods have reduced that to less than 2% in the first year. Still, some allowance for weight growth should be made in the conceptual-design weight estimation.

"1. EMPTY WEIGHT GROWTH

YEARS AFTER FIRST PROTOTYPE

Fig. 15.3 Table 15.4 Category Advanced composites

Braced wing Wood fuselage Steel tube fuselage Flying boat hull

409

Aircraft weight growth.

Weights estimation "fudge factors" Weight group

Wing Tails Fuselage/nacelle { Landing gear Air induction system Wing Fuselage Fuselage Fuselage

Fudge factor (multiplier) 0.85 0.83 0.90 0.95 0.85 0.82 1.60 1.80 1.25

Fudge factors are also required to estimate the weight of a class of aircraft for which no statistical equations are available. For example, there have been too few Mach 3 aircraft to develop a good statistical database. Weights for a new Mach 3 design can be estimated by selecting the closest available equations (probably the fighter/attack equations) and determining a "fudge factor" for each type of component. This is done using data for an existing aircraft similar to the new one (such as the XB-70 for a Mach 3 design) and calculating its component weights using the selected statistical equations. Fudge factors are then determined by dividing the actual component weights for that aircraft by the calculated component weights. To estimate the component weights for the new design, these fudge factors are multiplied by the component weights as calculated using the selected statistical equations.

16 STABILITY, CONTROL, AND HANDLING QUALITIES 16.1 INTRODUCTION During early conceptual design, the requirements for good stability, control, and handling qualities are addressed through the use of tail volume coefficients and through location of the aircraft center of gravity (c.g.) at some percent of the wing mean aerodynamic chord (MAC), as discussed in Chapter 6. In larger aircraft companies, the aircraft is then analyzed by the controls experts, probably using a six-degree-of-freedom (6-DOF) aircraftdynamics computer program to determine the required c.g. location and the sizes of the tails and control surfaces. An understanding of the important stability and control design parameters can be attained through study of simpler methods, which are also suitable for use by homebuilders and designers at smaller companies. This chapter introduces the key concepts and equations for stability, control, and handling qualities evaluation. These are based upon classical controls methods, many of which were developed by NACA in the period from 1925-1945. For derivations and additional detail on these methods, see Refs. 7, 37, 65, 66, and especially 67 and 4. The basic concept of stability is simply that a stable aircraft, when disturbed, tends to return by itself to its original state (pitch, yaw, roll, velocity, etc.). "Static stability" is present if the forces created by the disturbed state (such as a pitching moment due to an increased angle of attack) push in the correct direction to return the aircraft to its original state. If these restoring forces are too strong the aircraft will overshoot the original state and will oscillate with greater and greater amplitude until it goes completely out of control. Although static stability is present, the aircraft does not have "dynamic stability." Dynamic stability is present if the dynamic motions of the aircraft will eventually return the aircraft to its original state. The manner in which the aircraft returns to its original state depends upon the restoring forces, mass distribution, and "damping forces." Damping forces slow the restoring rates. For example, a pendulum swinging in air is lightly damped and will oscillate back and forth for many minutes. The same pendulum immersed in water is highly damped and will slowly return to vertical with little or no oscillation. Figure 16.1 illustrates these concepts for an aircraft disturbed in pitch. In Fig. 16.1a, the aircraft has perfectly neutral stability and simply remains at 411

AIRCRAFT DESIGN

STABILITY, CONTROL, AND HANDLING QUALITIES

whatever pitch angle the disturbance produces. While some aerobatic aircraft are nearly neutral in stability, few pilots would care to fly such an aircraft on a long trip in gusty conditions. Illustration Fig. 16.1 b shows static instability. The forces produced by the greater pitch angle actually cause the pitch angle to further increase. Pitchup is an example of this. In Fig. 16.1c, the aircraft shows static stability with very high damping. The aircraft slowly returns to the original pitch angle without any overshoot. Illustration Fig. 16.1 d shows a more typical aircraft response; the aircraft returns to its original state, but experiences some converging oscillation. This is acceptable behavior provided the time to converge is short. In Fig. 16.1 e, the restoring forces are in the right direction so the aircraft is statically stable. However, the restoring forces are high and the damping forces are relatively low, so the aircraft overshoots the original pitch angle by a negative amount greater than the pitch angle produced by the disturbance. Restoring forces then push the nose back up, overshooting by an even greater amount. The pitch oscillations continue to increase in amplitude until the aircraft "diverges" into an uncontrolled flight mode such as a spin. Note that dynamic instability is not always unacceptable provided that it occurs slowly. Most aircraft have at least one dynamic-instability mode, the spiral divergence. This divergence mode is so slow that the pilot has plenty of time to make the minor roll correction required to prevent it. In fact, pilots are generally unaware of the existence of the spiral-divergence mode because the minor corrections required are no greater than the roll corrections required for gusts. Dynamic-stability analysis is complex and requires computer programs for any degree of accuracy. Most of the stability-analysis methods presented in this chapter evaluate static stability. For conventional aircraft configura-

tions, satisfaction of static-stability requirements will probably give acceptable dynamic stability in most flight modes. Rule-of-thumb methods are presented for stall departure and spin recovery, the dynamic-stability areas of greatest concern.

412

(0) PERFECTLY NEUTRAL

f=:a

~

(c) STABLE, HIGHLY

TOME

(b) STATICALLY UNSTABLE a

~~

Y

WIND AXI!'

ZB

(e) STATICALLY STABLE, DYNAMICALLY UNSTABLE

(d) STABLE, LIGHTLY DAMPED

DAMPED a

16.2 COORDINATE SYSTEMS AND DEFINITIONS Figure 16.2 defines the two axis systems commonly used in aircraft analysis. The "body-axis system" is rigidly fixed to the aircraft, with the X axis aligned with the fuselage and the Z axis upward. The origin is at an arbitrary location, usually the nose. The body-axis system is more "natural" for most people, but suffers from the variation of the direction of lift and drag with angle of attack. (Remember that lift, by definition, is perpendicular to the wind direction.) The "wind axis" system solves this problem by orienting the X-axis into the relative wind regardless of the aircraft's angle of attack a or sideslip {3. The aircraft is not fixed to the axis system, so the axis projections of the various lengths (such as the distance from the wing MAC to the tail) will vary for different angles of attack or sideslip. This variation in moment arms is usually ignored in stability analysis since the angles are usually small. The "stability" axis system, commonly used in stability and control analysis, is a compromise between these two. The X-axis is aligned at the aircraft angle of attack, as in the wind axis system, but is not offset to the yaw angle. Directions of X, Y, and Z are as in the wind axis system. Note that the rolling moment is called L. This is easily confused with lift. Also, the yawing moment is called N, which is the same letter used for the normal-force coefficient. The aerodynamics crowd must have used up all the good letters by the time the stability folks developed their equations! Wing and tail incidence angles are denoted by i, which is relative to the body-fixed reference axis. The aircraft angle of attack a is also with respect to this reference axis, so the wing angle of attack is the aircraft angle of attack plus the wing angle of incidence. Tail angle of attack is the aircraft angle of attack plus the tail angle of incidence, minus the downwash angle (f) which is discussed later. In this BODY AXIS

TIM'

413

J'

"'0

"'0

-II----_TIME

~~~----TIME-;~~~-~

...

"DIVERGENCE"

Fig. 16.1

Static and dynamic stability. X

Fig. 16.2 Aircraft coordinate system.

414

STABILITY, CONTROL, AND HANDLING QUALITIES

AIRCRAFT DESIGN

chapter, angles of attack are measured from the zero-lift angle, which was discussed in Chapter 12. Nondimensional coefficients for lift and drag have been previously defined by dividing by dynamic pressure and wing area. For stability calculations, the moments about the three axes (M, N, and L) must also be expressed as nondimensional coefficients. Since the moments include a length (the moment arm) they must be divided by a quantity with dimension of length as well as by the dynamic pressure and wing area. This length quantity is the wing MAC chord for pitching moment and the wing span for yawing and rolling moments, as shown in Eqs. (16.1-16.3). Positive moment is nose up or to the right. =M/qSC

(16.1)

Cn =

N/qSb

(16.2)

Cf=

L/qSb

(16.3)

Cm

415

L

() z

R~Nc~~l

Fp

AFT INLET: a p = a + ip -

I

f

Xp

Stability analysis is largely concerned with the response to changes in angular orientation, so the derivatives of these coefficients with respect to angle of attack and sideslip are critical. Subscripts are used to indicate the derivative. For example, Cn~ is the yawing moment derivative with respect to sideslip, a very important parameter in lateral stability. Similarly, subscripts are used to indicate the response to control deflections, indicated by o. Thus, Cm, indicates the pitching-moment response to an elevator deflection. e Unless otherwise indicated, all sweep angles in this chapter are quarterchord sweeps, and all chord lengths C are the wing MAC. Also, all angles are in radians unless otherwise mentioned. Angle terms that are not estimated in radians must be converted to radians before use in stability equations.

16.3

LONGITUDINAL STATIC STABILITY AND CONTROL

Pitching-Moment Equation and Trim Most aircraft being symmetrical about the centerline, moderate changes in angle of attack will have little or no influence upon the yaw or roll. This permits the stability and control analysis to be divided into longitudinal (pitch only) and lateral-directional (roll and yaw) analysis. Figure 16.3 shows the major contributors to aircraft pitching moment about the c.g., including the wing, tail, fuselage, and engine contributions. The wing pitching-moment contribution includes the lift through the wing aerodynamic center and the wing moment about the aerodynamic center. Remember that the aerodynamic center is defined as the point about which pitching moment is constant with respect to angle of attack. This constant moment about the aerodynamic center is zero only if the wing is uncambered and untwisted. Also, the aerodynamic center is typically at 25070 of the MAC in subsonic flight.

Fig. 16.3

Longitudinal moments.

Another wing moment term is the change in pitching moment due to flap deflection. Flap deflection also influences the wing lift, adding to that term. Flap deflection has a large effect upon downwash at the tail, as discussed later. Drag of the wing and tail produces some pitching moment,. but the.se values are negligibly small. Also, the pitching moment of the tall about Its aerodynamic center is small and can be ignored. On the other hand, the long moment arm of the tail times its lift produces a very large moment that is used to trim and contr.o! the aircra~t. ~hil~ this figure shows tail lift upward, under many condltlOns the tall hft wIll be downward to counteract the wing pitching moment. A canard aircraft has a "negative" tail moment-arm that should be applied in the equations that follow. If an aircraft is tailless, the wing flap must be used for trim and control. Due to the short moment arm of such a control, the trim drags will be substantially higher for off-design c.g. locations. The fuselage and nacelles produce pitching moments that are difficult to estimate without wind-tunnel data. These moments are influenced by the upwash and downwash produced by the wing. . The engine produces three contributions to pitching moment. The O~Vl­ ous term is the thrust times its vertical distance from the c.g. Less obvlOus is the vertical force Fp produced at the propeller disk or inlet fro~t ~ace due to the turning of the freestream airflow. Also, the propwash .or Jet-mdu.ced flow field will influence the effective angle of attack of the tall and possIbly the wing. Equation (16.4) expresses the sum of these moments about th~ c.g. The effect of elevator deflection is included in the tail lift term. EquatlOn (16.5)

AIRCRAFT DESIGN

STABILITY, CONTROL, AND HANDLING QUALITIES

expresses the moments in coefficient form by dividing all terms by (qSw c ) and expressing the tail lift in coefficient form. Note that, to facilitate understanding, these equations are defined in the body-axis coordinate system rather than the stability-axis system.

Due to downwash effects, the tail angle of attack does not vary directly with aircraft angle of attack. A derivative term accounts for the effects of wing and propeller downwash, as described later. A similar derivative is provided for the propeller or inlet normal-force term Fp.

416

Cm", = CL",(Xcg - Xacw)

Sh

OCih -

417

-

+ Cm"'fUS - 'Y/h Sw CL"'h a; (Xach - Xeg)

(16.4) This produces a term representing the ratio between the dynamic pressure at the tail and the freestream dynamic pressure, which is defined in Eq. (16.6) as 'Y/h. This ranges from about 0.85-0.95, with 0.90 as the typical value. To simplify the equations, all lengths can be expressed as a fraction of the Wing mean chord c. These fractional lengths are denoted by a bar. Thus, Xeg represents Xeg/ c. This leads to Eq. (16.7). C

meg

= C

(Xeg-Xacw) + C + C (, + C _ qhSh C (Xach - Xeg) L C mw mWof f mfus qSw Lh C

(16.5) (16.6)

T-

F

Fp ", ~ (Xeg -X) +S!:I P q w vCi

(16.8)

Equation (16.8) seems to offer no mechanism for stabilizing a tailless aircraft. In fact, the tailless aircraft must be stabilized in the first term by providing that the wing aerodynamic center is behind the c.g., making the first term negative. The magnitude of the pitching-moment derivative [Eq. (16.8)] changes with c.g. location. For any aircraft there is a c.g. location that provides no change in pitching moment as angle of attack is varied. This '.'~irpla.ne aerodynamic center," or neutral point X np represents neutral stablhty (Fig. 16.1a) and is the most-aft c.g. location before the aircraft becomes unstable. Equation (16.9) solves Eq. (16.8) for the neutral point (Cm • = 0). Equation (16.10) then expresses the pitching moment derivative in terms of the distance in percent MAC from the neutral point to the c.g. This percent distance, called the "static margin," is the term in parenthesis in Eq. (16.10).

--

-8 ZI +-sP (Xeg-Xp) q w q w

(16.7)

For a static "trim" condition, the total pitching moment must equal zero. For static trim, the main flight conditions of concern are during the takeoff and landing with flaps and landing gear down and during flight at high transonic speeds. Trim for the high-g pullup is actually a dynamic problem (discussed later). Usually the most forward c.g. position is critical for trim. Aft-c.g. position is most critical for stability, as discussed below. Equation (16.7) can be set to zero and solved for trim by varying some parameter, typically tail area, tail lift coefficient (i.e., tail incidence or elevator deflection), or sometimes c.g. position. The wing drag and tail trim drag can then be evaluated. Methods for the first-order evaluation of the terms of Eq. (16.7) are presented later. Static Pitch Stability

For static stability to be present, any change in angle of attack must generate moments which oppose the change. In other words, the derivative of pitching moment with respect to angle of attack [Eq. (16.8)] must be negative. Note that the wing pitching moment and thrust terms have dropped out as they are essentially constant with respect to angle of attack.

(16.9)

(16.10) If the c.g. is ahead of the neutral point (positive static margin), the pitching-moment derivative is negative so the aircraft is .s!able (this .is yet another confusing terminology!) At the most-aft c.g. position, a tYPical transport aircraft has a positive static margin of 5-10070. Current fighters typically have positive static margins of about 5%, b~t new fighters such as the F -16 are being designed with "relax:d static sta~II­ ity (RSS)" in which a negative static margin (zero to -15%) IS coupled w.lth a computerized flight control system that deflects the elevator to provide artificial stability. This reduces trim drag substantially. It is common to neglect the inlet or propeller force term Fp in Eq. (16.9) to determine "power-off" stability. This removes any strong dependence of X np on velocity (q) in the subsonic flight regime. Power effects are then accounted for using a static margin allowance based upon test data for a

418

AIRCRAFT DESIGN

STABILITY, CONTROL AND HANDLING QUALITIES

or-,

1.2

I(a) A

-1.6

,

e

B-747 e B-727 x ii.C,

TRANSPORT

.8

i

I

jA TAN ALE

I

6~

c

P

.6

~4

_

.4

-.8

,

4-+-

I":

I

o

,

.......... , I

,

I

---I.. SUPERSONIC TAN ALE

a) A=O

FIGHTER-STABLE

eF4

--~--

1.4

__'-

T

1.2



1.0

Typical pitching moment derivative values.

c

~L A TAN ALE

MACH NUMBER

T .....-v

--

p¥f--t -

!

II

6

"-I

If

V

~

:r=;::::::

ATAN'\LE

«.j

I--

5 4 I--

31--

II

1-

3

+-

l-- I-- _+- .- ....

UN SWEPT T.E.

,i

lz.

ll~

ZERO LIFT ANGLE OF ATTACK

-.8

DISTANCE FORWARD OF ROOT QUARTER-CHORD POINT IN ROOT CHORDS

Fig. 16.11

Upwash estimation (snbsonic only). (Ref. 37)

425

Fig. 16.12 Downwash estimation. (after Ref. 4)

426

AIRCRAFT DESIGN STABILITY, CONTROL, AND HANDLING QUALITIES

427

as express~d i~ Eqs. (16.19) and (16.20). Equation (16.20) is the tail angle of attack den~atJve from Eq. (16.8), called {3 in many texts, which is easily c~nfused wIth yaw angle. The downwash derivative is with respect to the Wlllg a~gle of attack, so the tail angle of attack can now be determined a shown ill Eq. (16.21). s

oa u = oa

Upwash:

+ iw)

(16.19)

oa

oah = 1 _ Of oa Oa

Downwash:

ah = (a

1 + Ofu

(1 -::) +

(16.20)

(ih - iw)

(16.21)

A canard will o~vi~usly experience no downwash from the wing, but its own downwash wIll mflu~nce. the wing. The estimation of the effect of canard downwash on the wIng IS very difficult because the downwash varies across the canard .span and because the canard tip vortices actually create an upwash on the wmg outboard of the canard. The effect of ~anard downwash on the wing may be crudely approximated by ~ssumlng that the canard down wash as calculated with these methods umformly affects the wing inboard of the canard tips. This reduces the angle of attack at the wing root.

10

20

40

30

Fig. 16.14

50

Position of

v..

60

"70 FUSELAGE LENGTH

root chord.

Fuselage and Nacelle Pitching-Moment The pitching-moment contributions of the fuselage and nacelles can be approximated by Eq. (16.22) from NACA TR 711. The Uj-is the maximum width of the fuselage or nacelle and L f is the length. Figure 16.14 provides the empirical pitching moment factor Kf .

C

m"fuselage

=

KW2L f f 1 per deg cS w '

(16.22)

(.1f)Al br/ (b/2)1 .1CL

15

10

5

~

-.2

__ ____+-__ ____ __ ____ ~

-.1

~

0

Fig. 16.13

.1

~

.2

~

.3

hh IS HORIZONTAL TAIL HEIGHT ~~A:B:O~V.EWING

.4

.5

Downwash increment due to flaps.

(bh/~)

Thrust Effects The remaining terms in Eq. (16.7) are thrust effects upon pitching moment. Thrust has three effects, namely the direct moment of the thrust, the propeller or inlet normal force due to turning of the air, and the influence of the propwash or jet-induced flows upon the tail, wing, and aft fuselage. The direct moment of the thrust is simply the thrust times the moment arm about the c.g., as defined in Eq. (16.7). If the thrust axis passes through or near the c.g. this term may be ignored. The normal force due to the turning of the air at an inlet front face Fp can be calculated from momentum considerations. This normal force equals the massflow into the inlet times the change in vertical velocity. Since the angles are small, the change in vertical velocity is approximately the turning angle (ap-see Fig. 16.3) times the aircraft velocity [Eq. (16.23)]. The engine mass-flow can be approximated by assuming a capture-area ratio of one [Eq. (16.24)] if installed engine mass-flow data is unavailable. Note that mass flow is in slugs per second, which equals pounds per second divided by 32.2.

(16.23) in

== P VAinlet. slugs/s

(16.24) (16.25)

Note in Eq. (16.7) that a propeller mounted aft of the c.g. is stabilizing. This is one of the advantages of the pusher-propeller configuration. The propwash affects the down wash seen by the ~orizontal .tail and reduces the tail's effectiveness. Equation (16.27) estimates this propeller downwash effect as a derivative that is added to the wing downwash derivative. The constant terms come from Fig. 16.17.

The derivative of the normal force with respect to angle of attack is the mass flow times the velocity [Eq. (16.25)]. The derivative of a p with respect to angle of attack [see Eq. (16.9)] is the upwash derivative Eq. (16.19) if the inlet is ahead of the wing, and the downwash derivative Eq. (16.20) if the inlet is behind the wing. For an inlet mounted under the wing, the wing turns the flow before it reaches the inlet front face so the normal force is approximately zero. For a propeller-powered aircraft, a normal force contribution to pitching moment is also produced by the momentum change caused by the turning of the airstream. Unlike the jet inlet, the actual turning angle is not apparent because the propeller does not fully turn the airflow to align with the propeller axis. Equation (16.26) is an empirical method for estimation of the propeller normal force based upon charts in Ref. 68; NB is the number of blades per propeller, Ap is the area of one propeller disk. The derivative term is the normal force exerted by one blade when the propeller is operating at zero thrust, found in Fig. 16.15 as a function of advance ratio. The function f(T) adjusts for non-zero thrust and is found in Fig. 16.16.

2.00

1.75

E=: 1.50

;:;'

1.25

1.00

.75 -.5

F.

pC/.

= qNB A P OCNblade oa f(T)

.5

0

(16.26) · 16 . 16 F Ig.

.125

429

STABILITY, CONTROL, AND HANDLING QUALITIES

AIRCRAFT DESIGN

428

1.52

1.0 T pV 2 D2

2.5

.0

Propeller normal force factor. (after Ref. 68)

(AT THRUST = 0) .5

.100 PER BLADE) ( PER RADiAN

.4

.075 rIJ

o==

.3

after NACA WR L-25

13

...-<

.050

.2 .025 .1

ADVANCE o +---+---+---t------1~----tRATIO 2 o 5 3 4 REFERENCED TO PROPELLER DISK AREA

Fig. 16.15

J=~

o 4---4-----+----~--t---~--~ _ .05

0

.50

1.00

1.50

2.00

2.50

nD

Propeller normal force coefficient. (after Ref. 68)

}ig. 16.17

Propeller downwash factors. (after Ref. 68)

T

pv 2D2

AIRCRAFT DESIGN

430

~= K oa

I

+

K N OCNblade(Oap ) 2 B oa oa

STABILITY, CONTROL, AND HANDLING QUALITIES

(16.27) .... -<

If largely in the propwash, the tail will experience an increased dynamic

"'0

pressure, as shown in Eq. (16.28). The tail dynamic pressure ratio 'Y/h for zero thrust is approximately 0.9. If the tail is only partly in the propwash the right-side term in the parenthesis should be reduced proportionally. This term can also be applied to estimate increase in dynamic pressure at the wing, which may especially affect the pitching moment due to flap deflection.

u"' ....

~I

=

0

0

M

1

~

M N

,.; ,.;

(16.28) rIJ

~

l-

z

ie...

0

=-

~

~

:s

I-

Q.

'2

= ~

'a

I'!

IS obtaIned from q. (16.58), and the resulting pItch rate IS obtamed from Eq. (16.59).

Table 16.2 Class I II III

IVA IVB IVC

STABILITY, CONTROL, AND HANDLING QUALITIES

AIRCRAFT DESIGN

448

MIL-F-8785 B roll requirements

Aircraft type

Required roll

Light utility, observation, primary trainer Medium bomber, cargo, transport, ASW, recce. Heavy bomber, cargo, transport Fighter-attack, interceptor Air-to-air dog fighter Fighter with air-to-ground stores

n

=

l/cos¢

60° 45° 30° 90° 90° [ 360° 90 0

in 1.3 s in 1.4 s in 1.5 s in 1.3 s in 1.0 s in 2.8 s inl.7s (16.58) (16.59)

Steady Roll The steady roll is found by setting Eq. (16.56) to zero. Equation (16.37) C, indicates that the only rolling-moment term that remains when the sideslip equals zero is the roll due to aileron deflection. This leads to Eq. (16.60), which is solved for roll rate (radians) as a function of aileron deflection in Eq. (16.61). (16.60) (16.61) For many years the roll-rate requirement was based upon the wing helix angle pb 12 V. NACA flight tests (NACA 715) determined that most pilots consider an aircraft to have a good roll rate if the wing helix angle is at least equal to 0.07 (0.09 for fighters). Military specifications (MIL-F-8785B or Mil Std 1797) require that the aircraft reach a certain roll angle in a given number of seconds, as noted in Table 16.2. These assume that the aircraft is in level flight upon initiation of the roll, so the rotational acceleration should be accounted for. However, aircraft generally reach maximum roll rate quickly; the quasi-steadystate roll rate therefore may be used initially to estimate the time to roll. 16.9 INERTIA COUPLING The F-IOO prototype, the first fighter capable of level supersonic flight, featured a thin swept wing and long heavy fuselage compared to previous fighters. During flight testing, a series of high-speed rolls suddenly diverged in angle of attack and sideslip, much to the surprise of all concerned. Detailed analysis and simulation discovered the cause to be "inertia coupling." Figure 16.26 shows a typical fighter in roll. The mass of the forebody and aft-fuselage are concentrated like a barbell for illustrative purposes.

449

Like all objects, the fighter tends to roll about it~ princiI?al (~ongitu.dinal) . H ver if the fighter rolled 90 deg about Its longltudmal axIS, the aXIS. owe , . 1 f h The n Ie of attack would be exchanged wIth the ang eo yaw, as sown. . a g ffect of the vertical tail would oppose this increase in yaw angle wIth C n~ e roll. . d . Th In addition the aileron rolling moments are about the wm aXl~.. e aircraft thus ~ctuallY rolls around an axis somewhere between the pnnclpal axis and the wind axis. 1 h' The masses of the forebody and aft-fuselage are above and be ow t .1S actual roll axis. Centrifugal force tends to pull the~ a~ay from t~e roll axl~, creating a nose-up pitching moment. The combmatlOn of ~he l~c~ease m yaw angle with roll and the nose-up pitching moment due to mertta IS called inertia coupling. d db Inertia coupling becomes a problem only when the .moment~ pro uce y the inertia forces are stronger than the aerodynaml~ resto~lllg mome~ts. This is most likely to happen at high altitudes ~lower aIr densIty) and at hIgh Mach numbers where the tail loses lift effectIveness. .' The solution to inertia coupling in the F-IOO was a larger .vertic~l tall. This remains the typical solution. For this reason the vertical-tall are~ should not be reduced below the statistical tail-volume-method result until a more detailed analysis is available. 16.10

HANDLING QUALITIES

Cooper-Harper Scale Aircraft handling qualities are a subjective assessmen.t of the way the plane feels to the pilot. Few modern pilots fully apprecIate the great adCENTRIFUGAL FORCE

WIND AXIS

v·~ ---------~~~~

ACTUAL ROLL AXIS

{3

v~-90° ROLL ABOUT BODY AXIS

- - - WIND AXIS ' ........ PRINCIPAL AXIS Fig. 16.26 Inertia coupling.

STABILITY, CONTROL, AND HANDLING QUALITIES

AIRCRAFT DESIGN

450

451

vances in handling qualities made since the dawn of aviation. Early fighters such as the Fokker Eindecker had handling qualities which were so poor that the pilots felt that without constant attention, the aircraft would' 'turn itself inside out or literally swap ends" (movie stunt pilot Frank Tallman, quoted from Ref. 71). A number of "goodness" criteria such as the wing helix angle have already been discussed. It is important that the aircraft have a nearly linear response to control inputs and that the control forces be appropriate for the type of aircraft. The control forces required due to flap deflection or power application should be small and predictable. These handling qualities criteria are generally considered later in the design cycle. Figure 16.27 illustrates the Cooper-Harper Handling Qualities Rating Scale, which is used by test pilots to categorize design deficiencies (Ref. 72). Handling qualities are discussed in detail in Ref. 69.

"u

c: '" 0c: E o~ ~

'"

"0." c:

Departure Criteria

0.

!!! E '" 0 ". 0

c: c:

'"

'5

'" (3 "" 'c:" "U

.0

U ii

:0 c:

0",

"=u"OIV

0=

:;'" " U t~ !!1." c: c;~ " "0~ .-!,!" -g~ 0 u

.0

c:.:

i~

"

0-

"

,.,~

~

:::;{l ~g

ill

'(3""

"u'c:"

'(3"" c: §"

" "§ a;

a;

a;

"0

"0

"0

0

0

0

0

:::;

:::;

(3

c:

iii

:::;

c:



iii

:::;

"

§

iii

a;

"0

.

One of the most important aspects of handling qualities is the behavior of the aircraft at high angles of attack. As the angle of attack increases, a "good" airplane experiences mild buffetting to warn the pilot, retains control about all axes, and stalls straight ahead with immediate recovery and no tendency to enter a spin. If a spin is forced, the "good" airplane can be immediately recovered. A "bad" airplane loses control in one or more axis as angle of attack increases. A typical bad characteristic is the loss of aileron roll control and an increase in aileron adverse yaw. When the aircraft is near the stall angle of attack, any minor yaw resulting from aileron deflection may slow down one wing enough to stall it. With only one wing generating lift, the "bad" airplane will suddenly depart into a spin or other uncontrolled flight mode. Design features for good departure and spin characteristics have been discussed in earlier chapters. There have been many criteria proposed for good departure characteristics. Several aerodynamic coefficients are important to departure ~hara~teristics, especially Cn~' Cn'a' Cl~' and Cloa· These are combmed m the lateral control departure parameter (LCDP), sometimes called the lateral control spin parameter or the aileron-alone divergence parameter [Eq. (16.62)]. The LCDP focuses upon the relationship between adverse yaw and directional stability. Equation (16.63) shows another departure parameter, Cn"I-'dynamlc., which includes the effects of the mass moments of inertia. Both of these parameters should be positive for good departure resistance. A typical goal is to have Cn"I-'dynamIC. greater than 0.004. (16.62)

(16.63)

453

AIRCRAFT DESIGN

STABILITY, CONTROL, AND HANDLING QUALITIES

Figure 16.28 shows a crossplot of the LCDP and en" . with increase in "dYl''V'''' angle of attack. In Ref. 73 the boundaries for acceptame departure resistance were determined from high-g simulator tests using experienced pilots. The earlier Weissman criteria is also shown. Note the departure-parameter crossplot for the F-S. This aircraft is widely considered to be one of the best fighters at high angle of attack. Both departure parameters are increasing with angle of attack. On the other hand, the F-4 has poor departure characteristics. Its departure-parameter crossplot starts in the acceptable zone, but crosses into the unacceptable zone as angle of attack increases. The HiMat fighter shows that even an advanced supersonic canard configuration can have good departure characteristics. The HiMat has highlycambered outboard wing leading edges and has large twin tails with a substantial portion below the wing. Unfortunately, the stability derivatives used to calculate these departure parameters become very nonlinear near the stall. First-order estimation techniques used in conceptual design may not give usable results for departure estimation. However, the configuration designer can expect to be instructed to "fix it" when the first wind-tunnel data is available! There are a few design rules which can be applied during early configuration layout. The fuselage forebody shape has a huge effect upon the stability characteristics at high angles of attack. An elliptical nose cross section that has width greater than height is desirable.

Wing-tip stalling should be prevented by the use of wing twist, fences, notches, or movable leading-edge devices. It is also desirable for departure prevention to have a substantial ventral-tail surface.

452

Spin Recovery After stall, a spin will develop in severely abused. Figure 16.29 shows spin. The fuselage and wing masses trifugal forces acting on the fuselage ing the wing stall.

a "bad" airplane or a good airplane the forces acting in a fully developed are repres.ented by barbells. ~he centend to raIse the nose, further Increas-

RUDDER DEFLECTION AND TAIL DAMPING

MORE~~-.....

LIFT

/'

LCDP

/'

.008

/'

./

./

/'

NO DEPARTURES

.006

SPIN AXIS

•'ig. Ib.29 .004

Forces in spin .

r

.002

o +---~q---------~~----------------------------. CN~DYN POOR ROLL CONTROL (WEISSMAN CRITERIA)

-.004.-__~~____+-____+-____~~__~____~____~____~ -.002

u

.002

.004

Fig. 16.28

.006

.008

.010

Departure susceptibility.

.012

.014

Fig ••6.30

Geometry for spin recovery estimation.

454

AIRCRAFT DESIGN

RUDDER ALONE RECOVERY

24

___ RUDDER AND ELEVATOR

20

17 PERFORMANCE AND FLIGHT MECHANICS

16

12

8

4

--- -

o -240

-200

-160

BODY HEAVY

-120

-::::..--~ -80

-40

SPIN RECOVERY CRITERION

Fig. 16.31

0

40

80

[IX - I y ] b 2W /2

120

160 (xl0- 4 )

WING HEAVY

Spin recovery criteria.

The spin is primarily driven by the difference in lift between the outer, faster wing and the inner, slower wing, which is more fully stalled. The spin is opposed by damping forces, primarily from portions of the aft fuselage and vertical tail underneath the horizontal tail (SF-see Fig. 16.30). For recovery, the rudder is deflected against the spin. However, only the part of the rudder not blanketed by the stalled air from the horizontal tail will aid the recovery (SRI and SR 2). Figure 16.31 presents an empirical estimation of the required tail damping and rudder area for spin recovery for straight-winged aircraft (Ref. 74). This determines the minimum allowable tail-damping power factor (TDPF), defined in Eq. (16.64) where TDR is the tail damping ratio [Eq. (16.65)] and URVC is the unshielded rudder volume coefficient [Eq. (16.66)]. The airplane relative density parameter (p.) is defined in Eq. (16.67). TDPF

=

(TDR)(URVC)

(16.64) (16.65)

URVC

+ SR2 L2 Sw(bl2)

= SR]Ll

WIS p. = pgb

(16.66)

(16.67)

17.1 INTRODUCTION AND EQUATIONS OF MOTION The last chapter discussed stability and control, which largely concern the rotational motions of the aircraft in pitch, yaw, and roll. This chapter introduces flight mechanics, the study of aircraft translational motions. The geometry for flight mechanics is shown in Fig. 17.1. The climb angle), is the angle between horizontal and the wind (stability) X-axis (Xs). The "climb gradient" (0), the tangent of the climb angle, represents the vertical velocity divided by the horizontal velocity. Summing forces in the XS and Zs directions yields Eqs. (17.1) and (17.2). The resulting accelerations on the aircraft in the XS and Zs directions are determined as these force summations divided by the aircraft mass (Wig): 'f.Fx = T cos(a

+ (h) - D - W sin),

(17.1)

EFz = T sin(a

+ (h) + L - W cos)'

(17.2)

W=

-CT

(17.3) (17.4)

T = 550 bhp l1plV

(17.5)

Equation (17.3) defines the time rate of change in aircraft weight as the specific fuel consumption (C) times the thrust. For a piston-propeller engine, Eq. (17.4) determines the equivalent C based upon the piston-engine definition of C bhp (see Chapter 5), and Eq. (17.5) determines the thrust of the propeller. These simple equations are the basis of the most detailed sizing and performance programs used by the major airframe companies. The angle of attack and thrust level are varied to give the required total lift (including load factor) and the required longitudinal acceleration depending upon what maneuver the aircraft is to perform (level cruise, climb, accelerate, turn, etc.). Angle of attack and lift are restricted by the maximum lift available. The thrust level is restricted to the available thrust, as obtained from a table of installed engine thrust vs altitude and velocity (or Mach number).

455

456

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

457

17.2

STEADY LEVEL FLIGHT If the aircraft is flying in unaccelerated level flight, then), equals zero and the sum of the forces must equal zero. This leads to ~qs. (17.8) a!1d (17.9), the most simple versions of the translational equatIons .of motIOn. '!'hey state simply that in level flight, thrus~ equals. ~rag and hft equals. weIg~t. These are expressed using aerodynamIc coeffIcIents for the analysIs WhICh follows.

L

HORIZONTAL

T vv=v SINy v H =V COSy

D L

Vv vH

o

G=TANy=--

Fig. 17.1

=

Geometry for performance calculation.

~" T

V=

'Bom

"AXIS

What makes the sizing and performance programs complicated is not the actual calculation of the aircraft response to the forces at a given angle of attack and thrust level. The complications arise in determining what the angle of attack and thrust level should be to perform some maneuver. For example, the rate of climb varies with velocity. What combination of velocity and thrust setting will allow an airliner to climb to cruise altitude with the least fuel consumption over the total mission? This chapter will address such performance issues. For most aircraft the thrust axis has little incidence with respect to the wind axis under most flight conditions. This is by design, and permits simplifying Eqs. (17.1) and (17.2) to the forms shown in Eqs. (17.6) and (17.7).

r.Px

=

r.Pz

T - D - W sin),

(17.6)

L - W cos)'

(17.7)

=

(A word of caution: Be especially careful with units in the performance calculations. Apply each equation to the units of the data you are using to be sure that all units cancel leaving you with the units of the desired answer. Be wary of equations involving horsepower. Anytime the constant "550" appears in an equation, the other units must be converted to feet, pounds, and seconds (One bhp = 550 ft-Ib/s). Another "gotcha" is the specific fuel consumption C, which is usually given in units of hours-I. This must be divided by 3600 to yield seconds-I.)

qS(CDO + KCl)

(17.8)

qSCL

(17.9)

Jp~L (~)

(17.10)

=

= W =

From Eq. (17.9), the velocity in level flight ~an be expressed as a function of wing loading, lift coefficient, and air densIty. [Eq. (17 ..10)]. These equations imply that the actual T/W In level flIght must be the inverse of the LID at that flight condition [Eq. (17.11)]. The T IW and LIJ? in level flight can be expressed in terms of the wing loading and dynamIc pressure by substituting Eq. (17.9) into Eq. (17.8), as follows:

~ = L~D = (~:;) + (~) ~

(17.11)

Minimum Thrust Required for Level Flight From Eq. (17.11) it follows that the co~dition for mini~um thrust a~ a given weight is also the condition for ma?umum. LID. To fInd .the .veloc~ty at which thrust is minimum and LID IS maxlI~u~, the ~envatIve wIth respect to velocity of Eq. (17.11) is set to zero. Th~s IS shown I~ Eq. (17 .I2~, and solved in Eq. (17.13) for the velocity at WhICh the reqUIred thrust IS minimum and the LID is at a maximum.

o(TIW) _pVCDo _ W ~=O oV - WIS S YzpV 3

Vmin thrust = or drag

J2 Ws JCK

(17.13)

~

(17.14)

CLmin thrust = or drag

(17.12)

p

DO

DO

K

Substituting this velocity into Eq. (17.9) yi7lds t~e lift. coeff}~~e?tn[~~ minimum drag in level flight [Eq. (17.14)]. ThIS optImal hft coe ICIe

458

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

only dependent upon the aerodynamic parameters. At any given weight, the aircraft can be flown at the optimal lift coefficient for minimum drag by varying velocity or air density (altitude). If the lift coefficient for minimum drag is substituted back into the totaldrag Eq. (17.8), the induced-drag term will equal the zero-lift drag term. The total drag at the lift coefficient for minimum drag will then be exactly twice the zero-lift drag [Eq. (17.15)].

(17.15)

Minimum Power Required for Level Flight The conditions for minimum thrust and minimum power required are not the same. Power is force times velocity, which in steady level flight equals the drag times the velocity as shown in Eq. (17.16). Substituting the lift coefficient in level flight from Eq. (17.9) yields Eq. (17.17). P

= Y2p V 3S(CDo

= DV = qS(CDO + KCf) V P -_ v"/2P V3SCDO

KW2

+ Y2pVS

+ KCf)

(17.16) (17.17)

The velocity for flight on minimum power is obtained by setting the derivative of Eq. (17.17) to zero, as shown in Eqs. (17.18) and (17.19). Substituting this into Eq. (17.9) yields the lift coefficient for minimum power, Eq. (17.20). Substituting this into Eq. (17.8) gives the drag at minimum power required [Eq. (17.21)]:

oP

3 2 oV = 2" pV SCDO

Kw 2

Y2pV 2S = 0

-

(17.18)

(17.19)

CLmin power

D min

power

=

J

= qS(CDO

3CD O K

(17.20)

+ 3CDO )

(17.21)

Note that the velocity for minimum power required is approximately 0.76 times the velocity for minimum thrust [Eq. (17.13)]. The aircraft is flying at a lift coefficient for minimum power, which is about 73% higher than the lift coefficient for minimum drag [Eq. (17.14)].

459

The induced drag at the lift coefficient for mini~um power is .exactly three times the zero-lift drag, so the total drag i~ four time~ t?e zero-lIft drag [Eq. (17.21)]. This drag coefficient is twice as hIgh as at mInImUm drag [Eq. (17.15)]. .. . . fl . Remember that at the minimum-power conditIOn the aIrcraft IS yIng at a slower speed (reduced dynamic pressure) than at the minimum-drag condition. The actual drag increase will thus be less than the. factor. of two indicated by the drag coefficients. The actual drag .increase IS 2.0 times the ratio of dynamic pressures (0.76 2), or only 15.5OJo hIgh.er than the tot~l drag at minimum-drag conditions. Thus, the LID wh~n flYIng at t~e velocIty for minimum power required is 111.155, or 0.866 times the maxImum LID. Graphical Analysis for Thrust and Power Required The analytical optimizations in the last two sections depend upon the assumptions that the zero-lift drag is constant with velocit~, that the dr.ag due to lift follows the parabolic approximation, and that K IS constant WIth velocity. As seen in Chapter 12, these assu~ptio~s are ~ot ~ery ~ood other than for an aircraft with a high-aspect-ratIO WIng WhICh IS flYIng at low Mach numbers. . To determine the actual thrust (or horsepower) required for level flIght, the aerodynamic results are plotted vs velocity or Mach number and compared to the engine data. . . . For piston-powered aircraft, horsepower IS vIrtually constant WIth velo~­ ity. The only horsepower variation with velocity is due to ram p~essu~e In the intake manifold. For jet aircraft, equivalent horsepower vanes wIdely with velocity but thrust is roughly constant with veloci~y. . It is therefore common practice to graph the propulsIve reqUIrements of an aircraft vs velocity (or Mach number), using thrust fo~ jet. aircraft and using horsepower for propeller aircraft. These are shown In FIg. 17 ..2. Th.e horsepower required is found by multiplying the drag by the veloc~ty (dIvided by 550 to make the units come out as horse~ower). !he eqUI~alent thrust for the propeller aircraft is also shown for IllustratIOn, but IS not commonly plotted. . . The velocities for minimum thrust and mInImUm power are sho~n: Note that the minimum-power-required velocity is about ~6.6070 of the mInIm~m­ thrust-required velocity, as predicted in the last sectIOn. Also, the. s~penor­ ity of the jet engine for high-speed flight should be clear from thIS Illustration. . h The excess thrust at full throttle is determined simply by subtractIng t e thrust required from the thrust available. This excess can be used to accelerate or climb, as discussed later. . Such a plot of thrust or horsepower vs velocity is different at each altItude. Range . . The range of an aircraft is its velocity multiplied by the amount of tI~e it can remain in the air. Time in the air equals the amount of fuel car ned

460

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

THRUST OR DRAG·lb

\

5000

DRAG (THRUST REQUIRED) JET THRUST AVAILABLE

\

- -.........----------:;;:1"'-

4000 3000 2000 1000

THRUST A VAILABLE 2000 Hp PISTON.PROP

-

.--

0

VELOCITY ·flls

HORSEPOWER

MAX

MIN MIN POWER DRAG REQUIRED

8000

MAX SPEED PISTON.PROP

POWER REQUIRED .... JET THRUST POWER AVAILABLE

6000

4000

PISTON·PROP HORSEPOWER --=-~q---:::::?,~------ AVAILABLE

2000

-

__ -

---

o o

100

: - ._ _ _ -

.... -

300

400

Fig. 17.2

500

600

700

Thrust and power.

divided by the rate at which the fuel is burned. This in turn is the required thrust multiplied by the specific fuel consumption. .unfortunately, the simple equation implied by the last paragraph is complIcated by the fact that the aircraft weight drops as fuel is burned. This c~anges the drag, which then changes the thrust required. Net result: the aIrcraft goes farther but the calculation is more difficult! H~wever, th~ "ins.tantane?us range" derivative can be calculated using the. sImple. relatIOnshIP. ~escnb~d above, which is expressed in Eq. (17.22). !hIS descnbes the addItIonal dIstance the aircraft will travel with the next Incremental amount of fuel burned. This can also be expressed in terms of the LID and w~ight, a~ shown. Instantaneous range is a commonly-used measure of ment and IS usually discussed in units of nautical miles per pound of fueL dR _~ _ V _ V(LID) d W - - CT - - CD - - CW

R

=

fw/ V(LID) dW = Y~ f.t(Wi)

JWi -CW

These assumptions require that the aircraft hold lift coefficient constant. To hold the lift coefficient constant as the aircraft becomes lighter requires reducing the dynamic pressure. Since velocity is also being held constant, the only way to reduce dynamic pressure is to reduce air density by climbing. This results in a flight path known as the "cruise-climb," which maximizes range. The cruise-climb is not normally permitted for transport aircraft because of the desire by air-traffic control to keep all aircraft at a constant altitude and airspeed. It is possible to develop a rather messy range equation under these assumptions. However, the Breguet range equation can be applied with little loss of accuracy by breaking the cruise legs into several shorter mission-segments, using the appropriate LID values as aircraft weight drops. On a long flight, air traffic control may permit a "stairstep" flight path in which the aircraft climbs to a more optimal altitude several times during the cruise as fuel is burned off.

-BEST PROP RA!'.(,t.

VELOCITY·fl/s 200

461

CD

Wf

(17.22)

(17.23)

~ntegr~ting the ,i~stantaneous range with respect to the change in aircraft weIght YIelds the Breguet range equation" [Eq. (17.23)]. This integration ~ssumes that the velocity, specific fuel consumption, and LID are approxImately constant.

Range Optimization-Jet The Breguet range equation can be applied equally well to jets or propeller aircraft, with the use of Eq. (17.4) to determine an equivalent thrust specific fuel consumption for the propeller aircraft. However, the conditions for maximum range differ for jets and props because of the effect of velocity on thrust for the propeller. The terms in the Breguet range equation that do not involve the weight change [Le., (V /C)(L / D)] are known as the "range parameter" and are a measure of the cruising performance. For subsonic jet aircraft the specific fuel consumption is essentially independent of velocity and the range parameter can be expanded as shown in Eq. (17.24). Setting the derivative of Eq. (17.24) with respect to velocity equal to zero yields Eq. (17.25), the velocity for best range for a jet. The resulting lift coefficient and drag are given in Eqs. (17.26) and (17.27).

(17.24)

Vbest = range

W~K ~ -

(17.25)

§DO -

(17.26)

CD 0) DO + -3-

(17.27)

-

CL best =

pS

3K

range

D best -_ qS ( C range

CDo

. Note that the drag coefficient for best range for a jet is 1.33 times the zero-lift drag. This is a lower drag coefficient than the drag coefficient for

482

483

PERFORMANCE AND FLIGHT MECHANICS

AIRCRAFT DESIGN

dh.

This method minimizes time to climb with no constraint on ending velocity. To climb to a given altitude with a specified ending velocity, the optimal trajectory is flown until the aircraft reaches the energy-height curve of the desired ending condition. Then that energy-height curve is followed to the ending altitude and velocity, by either climbing or diving.

LINE OF CONSTANT fS = dWf ALTITUDE 103 ft

SUPERSONIC OBJECTIVE: MACH 2.0 AT 45,000 It

50

TANGENT TO

t

l-2

(17.90)

== (Ps )average

.,."""*"-+_ ENERGY fS AND CONSTANT HEIGHT

40

CURVES

The actual time to climb is determined by numerically integrating along the optimal trajectory using Eq. (17.89). The time to change energy height is approximately expressed in Eq. (17.90) as the change in energy height divided by the average P s during the change. As always, accuracy is improved with smaller integration steps. Note that the time to follow lines of constant energy-height is usually negligible for a first-order analysis.

30

MINIMUM FUEL CLIMB \'--\--\--r-T" PROFILE

20

10

Minimum Fuel-to-Climb Trajectory The energy equations can be modified to determine the climb trajectory that minimizes fuel consumption. The "fuel specific energy" (is) is defined as the change in specific energy per change in fuel weight. This is shown in Eq. (17.91) to equal the P s divided by the fuel flow, which is the thrust times the specific fuel consumption. Like P s, the is values can be calculated and plotted vs Mach number for each altitude and then cross-plotted as contour lines on a Mach number vs altitude chart, as shown in Fig. 17.15. is

=

dh e = dhe/dt d Uj- d Uj-/dt

Ps CT

(17.91)

(17.92)

In Eq. (17.92), Eq. (17.91) is rearranged and integrated to yield the change in fuel weight for a change in energy height (he). Note that this is minimized when is is maximized for each energy height. This implies that the minimum-fuel-to-climb trajectory passes through those points for which is contours are exactly tangent to the he contours. This is shown in Fig. 17.15, which greatly resembles the chart used to determine the minimumtime-to-climb trajectory. (17.93)

The fuel consumed during the climb is determined by numerically integrating along the minimum-fuel trajectory, using Eq. (17.93) as an approximation.

12

o

o

.2

.4

.6

.8

1.0

1.2 1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

MACH NUMBER

. 17 .15 Minimum fuel to climb. Fig.

Energy Method for Mission-Segment Weight Fraction. . Equation (17.94) is an expression of the mission-segmen~ weight. fraction for any flight maneuver involving an in~rea~e in energy height. ThiS c:~h~~ used for climbs or accelerations or combInations of the two: Re~emb~ ht t the mission-segment weight fraction expresses the tot~l alrcra t. ~elg t: the end of the mission segment divided by the total ~r~raft we~g t at . e beginning of the mission segment. This is used for slZlng as dlscusse d In earlier chapters.

1

Wi+ I [ - Cl1he Wi =exp V(I-DIT)

(17.94)

Unfortunately a maneuver involving a reduction in energy heig:t c~nnot create fuel as w;uld be implied by putting a negative value for t e c ange in he into Eq. (17.94)! 17 7 OPERATING ENVELOPE b' . . " "fl' ht envelope" maps the com 1The aircraft "operatIng envelope or Ig . ed to nations of altitude an~ velocity ~hat the I air~~a!~s~~~ ~::t~e~~~:t~iction . withstand. The "level-flight operatIng enve o?e that the aircraft be capable of steady level flight.

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

The rate of climb, or vertical velocity, is the velocity times the sine of the climb angle [Eq. (17.37)].

Therefore, the aircraft gains more altitude for a given horizontal distance; important for clearing mountains! The most accurate way to determine best rate and angle of climb is to plot the rate of climb vs velocity, using Eq. (17.37) and the actual thrust and drag data as shown in Fig. 17.3. The best rate of climb is obviously the peak of the curve. The best angle of climb is the point of tangency to a line from the origin. The angle of climb is the arctangent of the vertical velocity divided by the horizontal velocity at that point.

464

T

=

D + W sin'}'

(17.34)

L = W cos'}'

(17.35)

_ ._I(T-D)_ .-I(T cos'}')_ .-I(T 1) sm W - LID = SIn W - LID

----w- -

'}' - sm

v"

=

. V SIn'}'

=

1)

----w-

V (T-D) == V (T W - LID

(17.36)

(17.37)

The velocity for steady climbing flight can now be derived from Eq. (17.35), as shown in Eq. (17.38). The thrust-to-weight ratio is no longer the inverse of the lift-to-drag ratio as was the case for level flight. Solving Eq. (17.36) for TIWyields Eq. (17.39), the thrust-to-weight ratio required for a steady climb at angle '}'. (17.38) T _ cos'}' . _ I . W - LID + SIn'}' = LID + SIn'}'

(17.39)

Graphical Method for Best Angle and Rate of Climb Two climb conditions especially concern the aircraft designer: the "best rate of climb," which provides the maximum vertical velocity (Vv ), and the "best angle of climb," which provides a slightly lower vertical velocity but at a reduced horizontal speed, so that the angle of climb is maximized. RATE OF CLIMB-Vy BEST RATE OF CLIMB BEST ANGLE OF CLIMB

~

/

//

/

/~

/

/____________________________

~/--

Fig. 17.3

.VH~V

Graphical method for best climb.

465

Best Angle and Rate of Climb-Jet Analytical optimization of velocity for best angle and rate of climb can be messy. Graphical analysis is more reliable, but doesn't give an analytical feeling for the key variables. For a jet aircraft, the thrust is essentially constant with velocity so Eq. (17.36) can be directly maximized for the conditions for best climb angle. Since the TIW term is constant with velocity, the velocity for best LID should be selected to maximize climb angle. This velocity was determined in Eq. (17.13). To determine the velocity for best rate of climb of a jet aircraft, Eq. (17.37) must be maximized. Equation (17.40) is obtained from Eq. (17.37) by expanding the drag term and assuming that'}' is small enough that lift approximately equals weight: (17.40)

(17.41)

In Eq. (17.41), the derivative of the vertical velocity with respect to aircraft velocity is set to zero and solved for velocity for best climb. Note that if the thrust is zero this equation collapses to the equation for the velocity for minimum power required [Eq. (17.19)), which serves as a lower boundary on the solution. The effect of nonzero thrust is a significant increase in the velocity for best climb rate with increasing thrust. The velocity for best climb rate including the effects of thrust may be on the order of twice the velocity for minimum power. Velocities of 300-500 knots are not uncommon for the best climb speed for a jet. The XB-70 has a best climb speed of 583 knots! This climb optimization will only determine the velocity for the best rate of climb at some altitude. It will not tell you what the complete climb profile should be to minimize time to a given altitude. For many supersonic aircraft, minimizing total time to climb requires leveling off or even diving as the aircraft accelerates through transonic speeds to minimize the time spent at these high-drag conditions. In a later section, the "specific excess power"

467

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

method will be presented as a means for determining the climb profile that minimizes total time to climb.

ending altitudes of the climb being analyzed, but need not be exactly the same altitudes.

466

(17.46)

Best Angle and Rate of Climb-Prop Equation (17.42) expresses the climb angle of a propeller aircraft as obtained by substituting Eq. (17.5) into Eq. (17.36). This equation ca~ be expanded and the derivative taken with respect to velocity:

DJ

. -J [550bh P 'YIp --l' -- sm VW W

(17.42)

Ho~ever, the equatlO~ tend to

theoretical optimal velocities obtained with the resulting be too low (sometimes lower than the stall speed) for the parabolIc drag approximation to be valid, because of the separation drag at ~igh.angles of atta~k .. A!S?, the thrust no longer follows Eq. (17.5) which ImplIes that thrust IS mfmIte at zero airspeed . .If thrust and drag data are available at low speeds, the graphical method wIll produce good results. Most propeller aircraft have a best angle-of-climb speed about 85-90070 of the best rate-of-climb speed. This can be used for an initial estimate. Best. rate of climb for a propeller aircraft is obtained by substituting Eq. (1:.5) mto Eq. (17.37). This yields Eq. (17.43); simply the power available mmus the power required, divided by aircraft weight. Therefore the best ~ate of climb occurs at the velocity for minimum power required, as defined m Eq. (17.19):

v.v --

V· _ 550bhp 'YIp DV sm'Y W - W

(17.43)

Time to Climb and Fuel to Climb The t~me to cli?Ib to a given altitude is the change in altitude divided by the vertical velOCIty (rate of climb), as shown in Eq. (17.44) for an incremental altitude change. Fuel burned is the product of the thrust, specific fuel consumption, and time to climb [Eq. (17.45)].

(17.44)

(17.47)

If the climb is broken into short segments (less than 5000 ft in altitude gain), the fuel burned will be an insignificant portion of the total aircraft weight and can be ignored in the time integration. Subst~tuting E~. (17.46) into Eq. (17.44) and integrating yields Eq. (17.48), the tIme to chmb from altitude i to altitude i + 1. Oddly enough, the change in altitude has dropped out ~f the equati.on! However the change in altitude is implicit in the change m rate of chmb (V ) due ~o change in altitude. The fuel burned will then be described by Eq. v (17.49). 1

tt+J - ti = -

a

.:l W fueJ = ( -

f.t

(V v.-

The air density, aircraft weight, drag, thrust, specific fuel consumption, and best climb velocity all change during the climb. A good approximation over small changes ~n altitude is that the rate of climb at a given weight and constant-thrust settmg and constant velocity will reduce linearly with the altitude. This is shown in Eq. (17.46), where the linear constant a is determined from the rates of climb at any two altitudes hi and h2 [Eq. (17.47)]. These two altitudes used to determine a should be near the beginning and

(17.48)

Vi+1

CT)average (ti + I -

ti)

(17.49)

If desired, the accuracy of Eq. (17.48) can be improved upon by iterati?n. The rate of climb at the end of the climb segment can be recalculated usmg the reduced aircraft weight obtained by subtracting the fuel burned [Eq. (17.49)] from the original weight. This revised rate of climb can then be applied back into Eq. (17.48).

17.4 LEVEL TURNING FLIGHT In level turning flight, the lift of the wing is canted so that the horizontal component of the lift exerts the centripet~l force required .to turn. The total lift on the wing is n times the aircraft weIght W, where n. IS the load factor. Since the vertical component of lift must be W, the hOrIzontal component of lift must be W times the square root of (n 2 - 1). The geometry of a level turn is shown in Fig. 17.4 . .

W~g~

1/;= (W/g)V =

(17.45)

v; )

(17.50)

V

Turn rate (d1/;/dt) equals the radial acceleration divid~d by the velocity, as shown in Eq. (17.50). Turn rate is usually expressed m degrees pe~ s~c­ ond. Equation (17.50) yields radians per second, which must be multIplIed by 57.3 to get degrees per second. Instantaneous Turn-Rate _ If the aircraft is allowed to slow down during the turn ("inst~ntaneo~s turn"), the load factor n will be limited only by the maximum hft coeffI-

468

AIRCRAFT DESIGN PERFORMANCE AND FLIGHT MECHANICS

469

to-weight and lift-to-drag ratios [Eq. (17.51»), assuming that the thrust axis is approximately aligned with the flight direction. To solve for the sustained load factor in terms of the basic aerodynamic coefficients, the drag is expanded using (CL = nWlqS) and set equal to the thrust. This leads to Eq. (17.52), which defines the maximum available sustained load factor for a given flight condition. Note that the drag-due-to-lift factor (K) is a function of lift coefficient, as described in Chapter 12. Since n is also a function of lift coefficient, iteration is required to solve Eq. (17.52). n

=

-J

q K(WIS)

n -

L= nW

Lv=w

LH~w~~J Level turn geometry.

. . The intersection of the stall limit and h "corner speed" h' h' h . t e structural hmlt defines the , w lC IS t e velocIty for m' . rate. For a typical fighter Corner s d' aXlmum instantaneous turncal turning dogfight, opp~nents wir.e~ ~s about 300-:-350 knots. In a classiquickly as possible. y 0 get to theIr Own corner speed as Sustained Turn-Rate In a "sustained tur " t h ' . n, e alrcr~ft IS not permitted to slow down or lose altitude durin th and the lift m~st :qt:~~ioI:d af!~:~~ine~ turn ~he t~rust must equal the drag load factor for sustained turn ca b n Imes t de weIght. Thus the maximum n e expresse as the product of the thrust-

qCDo) WIS

(17.52)

~

(17.53)

L=nW=qS..J K

TURN RATE

.J, (deg/sec)

,\

30

cient or structural strength of th . f . structural limits expressed as tu~na~:~~a t. Flru~e 17 ·5 show~ thes~ stall and craft. vs ve OClty f or a tYPIcal fIghter air-

(TW -

Equation (17.51) implies that the sustained-turn load factor can be optimized by flying at the lift coefficient for maximum LID, which was determined in Eq. (17.14). Using this lift coefficient and setting lift equal to n times W leads to Eq. (17.53). This can be readily solved for either velocity or wing loading to obtain the maximum sustained-turn load factor.

(SAMPLE DATA) (SOME GIVEN ALTITUDE)

Fig. 17.4

(17.51)

(TIW)(LID)

25

\

20

CORNER SPEED

,t,

, " ""

TYPICAL FIGHTER STRUCTURAL LIMIT n '" 7.33

"-

SUSTAINED TURN RA TE ENvELOPE

\

---

\

\

15 10

STALL LIMIT

5

C LMAX

--

----

0 0

100

200

300

400

6

"'-

4

CI

...,

> ~

.., 0

2 500

V-knOls

Fig. 17.5

10 t'" 0 8 >

Turn rate and corner speed.

600

700

"=

PERFORMANCE AND FLIGHT MECHANICS

AIRCRAFT DESIGN

470

Figure 17.5 showed the "sustained turn-rate envelope." This is derived using Eq. (17.50) to determine the turn-rates provided by the sustained load factors available at the various flight conditions. Turn Rate with Vectored Thrust Vectored thrust offers improved turn performance for future fighters, and is already used in the VSTOL Harrier fighter to maximize turn-rate. The direction the thrust should be vectored depends upon whether instantaneous or sustained turn-rate is to be maximized. In a level turn with vectored thrust, the load factor times the weight must equal the lift plus the contribution of the vectored thrust, as shown in Eq. (17.54). The maximum load factor (and turn rate) is obtained by taking the derivative with respect to vector angle and setting it to zero [Eq. (17.55»). This yields Eq. (17.56), which states simply that the thrust vector for maximum instantaneous turn-rate should be perpendicular to the flight direction. nW=L

+ Tsin(a + 4>T)

(17.54)

) (T)

(17.55)

0 (L T.

on = 04>T W + W sm(a + 4>T) = W cos(a + 4>T) = 0 04>T 4>T

= 90 deg - a

(17.56)

Since none of the thrust is propelling the aircraft forward, it will slow down very rapidly! British pilots in combat have used the 90-deg vectoring of the Harrier to generate a high turn-rate while decelerating, causing pursuing pilots to overshoot. In a sustained turn with vectored thrust, the drag equals the thrust times the cosine of the total thrust angle, so the load factor n is expressed as in Eq. (17.57). Setting the derivative with respect to thrust-vector angle equal to zero [Eq. (17.58») yields Eq. (17.59). n

=

:;T =

(T COS(~+ 4>T»)(;)

(17.57)

~ sin(a + 4>T) (t) = 0

(17.58)

4>T =

-a

17.5

GLIDING FLIGHT

Straight Gliding Flight . Gliding flight is similar to climbing flight wIth the thrust set to z.ero. Equations (17.34) and (17.35) become Eqs. (17.60) and (17.61). The dIrection of the gliding angle'}' is assumed to be reversed from that used for

climb.

L

D

D = W sin,},

(17.60)

L = W cos'}'

(17.61)

W cos'}' = _ _ W sin,},

Equation (17.59) implies that the thrust vector for maximum sustained turn rate should be aligned with the flight direction. If the aircraft is at a positive angle of attack, the thrust should be vectored upward (relative to the fuselage axis) to align it with the freestream! However, this calculation ignores the jet flap effect which may produce a drag reduction with slight downward deflection if the nozzles are located near the wing trailing edge.

== _

(17.62)

The lift-to-drag ratio is the inverse of the tangent of the glide.angle [Eq. (17.62)]. In sailplane terminology, t?e "glide ratio': is the ratI~:~~~~~~ l horizontal distance travelled and ~ltItude I?Sht , a~~ IS e~uao/~O will travel drag ratio. A high-performance sailplane Wit a g 1 e r~ 10 over seven statute miles for every thousand feet of altItud~ lost. . (Cultural note: In sailplane terminology, a "sail~lane" IS an expensre, high-performance unpowered aircraft. A "glider" IS a crude, low-per ormance unpowered aircraft!) . . ld b To maximize range from a given altitude, the gh~e ratIO shou e m~imized This requires flying at the velocity for maxImum as /ound I~ 1'7 13 repeated below as Eq. (17.63). The lift coe~flclent or ma,xI~~~ LiD)is repeated as Eq. (17.64). The result~ng maxImum LID (ghde ratio) is determined from Eq. (17.15), as shown III Eq. (17.65).

L/P

(17.63)

CLmaxLiD =

(17.59)

471

(L) D

1

max

= 2

~

(17.64)

~ -2....jC;;

(17.65)

....jK

~CDrJ(

_1

. d b h " . k rate'" The time a glider may remain in the air i~ det~rmme '! t e SI~ h 'r~ the vertical velocity Vv , which is negative m thIS case. Smk .rate IS t e a~ craft velocity times the sine of the glide angle, as expressed III Eq. (17.6 ). (17.66)

472

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

. D CD SIn-y = - COS-y = - COS-y

L

(17.67)

W 2 cos 3-yC}j _

W

2

pcl

S

p(Cl/c}j)

S

. Equation

CL

=

Vminsink =

I2w

~ pS

J

K 3CDO

(17.68)

.. t· ,as s o~n In Eq. (17.68). For typical, small glide angles the COSIne erm may be Ignored. The lift coefficient for minimum sink rate is solved for by ... h term . I· C . . maxlmlZlng t e InVO vlng L and C~. !hlS IS shown in Eq. (17.69), with the result in Eq. (17.70). Note that thIS IS also the lift coefficient for minimum p re~~ired, s~ the velocity can be expressed as in Eq (17 71) Th LIc;er mInImum SInk speed is given by Eq. (17.72). ... e at

(17.69)

DO CL min sink -- J3C --;r-

(17.72)

(17.70)

The velocity for minimum sink rate is 760/0 of the velocity for best glide ratio. Sailplane pilots fly at minimum sink speed when they are in "lift" (i.e., in an airmass moving upwards). When the lift "dies," they accelerate to the velocity for best glide ratio to cover the most ground while looking for the next lift. An instrument called a "variometer" tells the sailplane pilots when they are in lift. Figure 17.6 shows a graphical representation of sink rate for a sailplane. This is known as a "speed-polar," or "hodograph," and can be used to graphically determine the velocities for minimum sink rate and best glide ratio.

Turning Gliding Flight When sailplane pilots find lift, they turn in a small circle to stay within the lifting airmass. Due to the additional wing lift required to turn, the sailplane will experience higher drag and a greater sink rate. Equation (17.61) must be modified to account for the bank angle e/> [Eq. (17.73)]. L cose/> = W cos-y == W

(17.73)

VELOCITY -ft/s

10

20

30

40

50

60

o ~,~,~~__~__~~__~~__~____~__~7~0__~8~0____:~~~1:OO

2

(17.71)

~17 .66) co~tains both sine and cosine terms. In Eq. (17.67) the

~~e ~:7t~~:hde a~gle IS. expressed in cosine terms to allow substitution into

o

473

... , ... ... ... ... ... , ...

MINIMUM SINK RATE / _, _.-'_ _

/

BEST GLIDE RATIO (HIGHEST LID)

Turn-rate is equal to the centripetal acceleration divided by the velocity, and is also equal to the velocity divided by the turn radius [Eq. (17.74)] . This allows the centripetal acceleration to be expressed as the velocity squared divided by the turn radius [Eq. (17.75)]. In Eq. (17.76), the turning force due to the lateral component of wing lift is equal to the aircraft mass times the centripetal acceleration.

3

......

4

5

.... ,

6

7 I

8

SINK SPEED

"

--

....

.... ,

,,

Fig. 17.6

-

,,

(17.74) (17.75)

gR

....

(17.76)



,

Sailplane sink rate.

= alV = VIR

WV2 L sine/>=-- = W~

"' "' , \

fils

....

~

BANK ANGLE

", ,

V2 R - -- -

V2

----,~=

- gtane/> - g~

(17.77)

Equation (17.76) can be solved for turn radius as expressed in terms of either bank angle or load factor [Eq. (17.77)]. The vertical velocity (sink rate) can be determined by substituting CL cose/> for CL in Eq. (17.68). This yields Eq. (17.78), which is simply the previous

474

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

resul~ divided by the cosine of e/>, raised to the 3/2 power The radius of the turn IS found by substituting Eq. (17.73) into Eq (1777)' as sho . E (17.79): . ., wn In q.

These effects are shown in Eq. (17.80): V = Vcg [ 1 +

W

2

S

p(ClICb)

(17.78) Mnner

R =

= Vcg [ 1 -

f 1 cose/>

(17.79)

. O~e ~nique problem for a slow-flying sailplane in a turn is the variation In ve oelty across the long span of the wing. The wing on the inside of the turn ?I ay stall due to the lower velocity. This is shown in Fig 17 7 Th velocIty across the span varies linearly with distance from the 'axis' ;f the turn. Also, the bank angle shortens the wing span when seen from above~

v

VOUTER

--- ---

~ -:. :: : - : ~ .=-..----------------.. Ir"--_-_-_-_-_-_-__-1_ TURN AXIS

TURN

(17.81)

In Eq. (17.81), the velocity at the inner wing tip is shown as a function of wing span, turn radius, and bank angle. In normal flight this velocity difference is easily corrected with a little aileron to increase the lift coefficient on the inner wing. However, when flying near the stall at even a moderate bank angle, this can reduce the velocity of the inner wing tip enough to create a one-wing stall, which leads to a spin.

17.6

ENERGY-MANEUVERABILITY METHODS

Energy Equations Fighter pilots have always known that management of energy is critical to survival and success. In World War I the experienced pilots always tried to enter a dogfight from above. They could then exchange the potential energy of altitude for the kinetic energy of speed or turn rate. let-fighter dogfight maneuvers largely rely upon the exchange of potential and kinetic energy to attain a positional advantage. For example, the "High Speed Yo-Yo" maneuver is used when overtaking a slower aircraft in a hard turn. The attacker pulls up, trading kinetic energy for potential energy and slowing to allow a higher turn rate. After turning, the attacker rolls partially inverted and pulls down astern of the opponent, now exchanging potential energy back for speed. Fighter pilots understand that potential and kinetic energy can be exchanged, and that the sum of the aircraft energy must be managed to attain success. This intuitive measure of goodness can be analytically developed and applied to aircraft design. E

AXIS

=

Wh +

4(~)V2

(17.82)

V2

(17.83)

h = E = h + 1.e W 2g RINNER

ROUTER

PSusoo =

Fig. 17.7

1

2~ cose/>

(17.80)

2W

pSC~ sine/>

Since the e/> term. i~ Eq. (17.78) does not vary with velocity, the rior results for the velOCItIes for best glide ratio and minim . k P applied. urn SIn rate can be

VINNER

475

Turn radios effect on wingtip velocity.

dh e

dt

=

dh VdV dt + g dt

(17.84)

At any point in time, the total energy of an aircraft (the "energy state") is the sum of the potential and kinetic energy, as shown in Eq. (17.82). Dividing by aircraft weight gives the "specific energy" [Eq. (17.83)).

476

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

Specific energy has units of distance (feet), and is also called the "energy height" (he) because it equals the aircraft altitude if the velocity is zero. Power is the time rate of energy usage, so the "specific power" (Ps)used can be defined as the time rate at which the aircraft is gaining altitude or velocity [Eq. (17.84)]. Since specific energy has units of distance (feet), specific power has units of distance per time (feet per second). This power being used by the aircraft to gain height or velocity has to come from somewhere. In the discussions of power required vs power available, it was pointed out that the excess power could be used to climb or accelerate. This excess power is the excess thrust (T - D) times the velocity [Eq. (17.85)]. The "specific excess power" (Ps ) is the excess power divided by the weight, and equals the specific power used, as shown in Eq. (17.86). P

=

V(T -D)

P _ V(T - D) _ dh s -

W

- dt

(17.85) V dV

+g dt

(17.86)

(17.87) Drag, and therefore P s , is a function of the aircraft load factor. The higher the load factor, the greater the drag, and thus the less excess power available. Equation (17.86) can be expanded in terms of the load factor and the aerodynamic coefficients as shown in Eq. (17.87). Note that T/Wand W/S are at the given flight condition, not the takeoff values! Specific excess power P s has units of feet per second, just like rate of climb. In fact, Eq. (17.86) is identical to the rate-of-climb equation if the longitudinal acceleration (dV/dt) is zero. The P s at a load factor of one is actually the rate of climb that would be available if the pilot chose to use all of the excess power for climbing at constant velocity. When P s equals zero, the drag of the aircraft exactly equals the thrust so there is no excess power. This does not necessarily mean that the aircraft isn't climbing or accelerating. However, if the sum of the energy usage equals zero, then the aircraft must be flying level, or climbing and decelerating, or descending and accelerating. Equations (17.86) and (17.87) assume that the thrust axis is approximately aligned with the flight direction. If this is not the case, the thrust term should be multiplied by the cosine of (ex + 4>t). Ps Plots

P s values are calculated and plotted against Mach n~mber as shown i.n Fig. 17.8 for a number of altitudes. Computers are especially handy for thIS "number crunching." From the P s charts at the various altitudes (Fig. 17.8), several additional charts can be prepared by cross-plotting. The level turn-rate can be determined for the various .load factors ~t .a given altitude and Mach number, ~nd plotted vs ~s (FIg. 17.9). ThIS IS compared to the data for a threat aIrcraft at that altItude an~ Mach number. With an equivalent P s at a higher turn-rate, the new f!ghter would always be able to turn inside the oppon~nt wit~ou! ~osing relatIve energy. A turn-rate advantage of 2 deg/s is consIdered sIgmfIcant. In Fig. 17.10, P s = 0 contours are plotted for different load factor.s on a Mach number vs altitude chart. This is a major tool for the evaluatIOn of new fighters, and permits comparisons between two aircraft f?r all Ma~h numbers and altitudes on one chart. To win a protracted dogfIght, an aIrcraft should have P s = 0 contours that envelop those ?f an opponent. In Fig. 17.11, contour lines of constant P s at a gIven load. factor are plotted onto a Mach number vs altitude chart. A separate char~ IS prep~red for each load factor. The chart for load factor equal~ one IS ~~peclally important because it provid~s the rate ?f c1im~ and t?e aIrcraft ceIhng, and because it is used to determme an optImal chmb traJectory.

(TYPICAL VALUES)

P s - ft/sec

ALTITUDE = 30,000 ft

500 400

LOAD FACTOR

300

n

200

1 100

3

5

0

7

-100 - 200

For any given altitude, P s can be calculated using Eq. (17.87) for varying

- 300

In.stalled thrust data are available. Design specifications for a new fighter wIll have a large number of "must meet or exceed" P s points , such as "P s = 0 at n = 5 at Mach 0.9 at 30,000 ft."

-400

~ach numbers and load factors once the aerodynamic coefficients and

477

0

.2

.4

.6

Fig. 17.8

.8

l~O

1.2 1.4

1.6 1.8

2.0

P s vs Mach number and load factor.

MACH 2.2 NUMBER

478

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

P s -ft/sec

ALTITUDE 1,000 ft

600 ALTITUDE = 30,000 ft MACH =0.9

400

(TYPICAL)

479

P s VALUES, n = 5 -400

50

200

,,

0 -200

40

,

ADVANCED DOGFIGHTER

\

-400

\ -600

30

, \

20

\

-800

THREAT \ AIRCRAFT \

-1000

10

\ \ \

-1200

\

\

-1400 0

5

10

o IS

20

.2

.4

.6

25

ALTITUDE 1000 ft

Fig. 17.11

Turn rate vs P s .

(TYPICAL)

1.0

1.2

1.4

,1.6

1.8

2.0

MACH NUMBER

TURN RATE ~-deg/sec Fig. 17.9

.8

Ps contours, constant load factor.

Minimum Tlme-to-Climb Trajectory Figure 17.12 is a plot of energy height vs Mach number and altitude. This is merely a graphical representation of Eq. (17.83), and has nothing to do with the particulars of anyone aircraft. An F-16 or a Boeing 747 would have an energy height of 42,447 ft if flying at Mach 0.9 at 30,000 ft.

n=1

50

40

dt

=

dh e Ps

(17.88)

30

(17.89) 20

10

.4

.6

1.0

.8

1.2

1.4

MACH NUMBER

Fig. 17.10

P s =0 contours.

1.6

1.8

2.0

Equation (17.84) can be rearranged into Eq. (17.88) which expresses the incremental time to change energy height (he) as the change in energy height divided by the P s at that flight condition. This is then integrated in Eq. (17.89) for the time to change energy height. Equation (17.89) shows that the time to change energy height is minimized if the P s is maximized at each energy height. This occurs at those points on the Mach number vs altitude plot of I-g P s (Fig. 17.11) where the P s curve is exactly tangent to an energy-height curve (Fig. 17.12). In Fig. 17.13, the I-g P s curves for a typical current-technology-highthrust fighter are superimposed on the he curves of Fig. 17.12. The trajectory for minimum time to climb is shown as passing through the dots repre-

(10 3 ft)

"r"'-_

70

60

80

90

100

(P s CONTOURS FOR n = 1)

3 ALTITUDE 10 fl

ENERGY HEIGHT: he = h + 2~ V2 ALTITUDE 10 3 ft 50

481

PERFORMANCE AND FLIGHT MECHANICS

AIRCRAFT DESIGN

480

120 140

OBJECTIVE: MACH 2.0 AT 45,000 fl.

50

160

\

40

30

40 TANGENT TO P s AND CONSTANT ENERGY HEIGHT CURVES

30

MINIMUM TIME TO CLIMB PROFILE

20

20

CONSTANT ENERGY HEIGHT CURVES

10

o

10

o

.4

.2

.6

.8

1.0

1.2

1.4 ,1.6

1.8

2.0

2.2 2.4

2.6

2.8,

MACH NUMBER

Fig. 17.13

o

.2

.4

.6,.8

1.0

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

2.8

MACH NUMBER Fig. 17.12

M inimum time-to-c1imb trajectory, high-thrust fighter.

Lines of constant energy height.

senting the points where the P s curves are tangent to he curves. For such a fighter, the minimum time to climb is obtained by staying low and accelerating to transonic speeds, then pitching up into a steep climb at approximately constant indicated airspeed (i.e., dynamic pressure), as shown by the optimal trajectory. Figure 17.14 shows the log P s curves for a typical 1960's era jet fighter. These fighters had significantly less thrust, and suffered a "thrust pinch" at transonic speeds-in which the thrust minus drag would reduce to almost zero. This causes the P s contours to form "bubbles." The minimum-time-to-climb trajectory requires jumping from one bubble to the other. This is done by diving or climbing along lines of constant energy height tangent to P s lines of the same numerical value for both bubbles, as shown in Fig. 17.14. Note that Fig. 17.14 requires diving through Mach 1.0 to minimize time to climb for this aircraft. This was common in earlier jets, and makes .sense intuitively. Since thrust minus drag is nearly zero at transonic speeds, acceleration will be slow and the aircraft will spend a lot of time in transonic acceleration. Diving reduces this time. The altitude lost is easily regained at higher speeds where the drag is less.

(Ps CONTOURS FOR n

ALTITUDE 103 ft

= 1) OBJECTIVE: MACH 2.0 AT 45,000 fl.

50

TANGENT TO Ps AND CONST ANT ENERGY HEIGHT CURVES

40

30

MINIMUM TIME

'-I....-Hrl- TO CLIMB PROFILE 20

10

o .2 .4

.6

.8

1.0

1.2

1.4 1.6

1.8 2.0

2.2

VI

MACH NUMBER

Fig. 17.14

Minimum time-to-climb, low thrust fighter (circa 1960).

482

483

PERFORMANCE AND FLIGHT MECHANICS

AIRCRAFT DESIGN

. This m.ethod mi~imizes ~ime to climb with no constraint on ending velocity. To chmb to a given altitude with a specified ending velocity, the optimal trajectory is flown until the aircraft reaches the energy-height curve of the desired ending condition. Then that energy-height curve is followed to the ending altitude and velocity, by either climbing or diving. (17.90) The actual time to climb is determined by numerically integrating along the optimal trajectory using Eq. (17.89). The time to change energy height is. ~pproximately expressed in Eq. (17.90) as the change in energy height divided by the average P s during the change. As always, accuracy is improved with smaller integration steps. N?t.e that the ~ime to follow lines of constant energy-height is usually neghgible for a first-order analysis.

dh. LINE OF CONSTANT fS = dWf ALTITUDE 103 fl

SUPERSONIC OBJECTIVE: MACH 2.0 AT 45,000 fl

50

TANGENT TO fs AND CONSTANT ENERGY HEIGHT CURVES

40

30 MINIMUM FUEL CLIMB

l'-..\-4--+-+- PROFILE 20

10

Minimum Fuel-to-Climb Trajectory The energy equations can be modified to determine the climb trajectory that minimizes fuel consumption. The "fuel specific energy" (is) is defined as the change in specific energy per change in fuel weight. This is shown in Eq. (17.91) to equal the P s divided by the fuel flow, which is the thrust times the specific fuel consumption. Like PH the is values can be calculated and plotted vs Mach number for each altitude and then cross-plotted as contour lines on a Mach number vs altitude chart, as shown in Fig. 17.15.

is

= dhe = dhe/dt

dUj-

dUj-/dt

Ps CT

(17.91)

(17.92) In Eq. (17.92), Eq. (17.91) is rearranged and integrated to yield the change in fuel weight for a change in energy height (he). Note that this is minimized when is is maximized for each energy height. This implies that the minimum-fuel-to-climb trajectory passes through those points for which is contours are exactly tangent to the he contours. This is shown in Fig. 17.15, which greatly resembles the chart used to determine the minimumtime-to-climb trajectory.

w.h-2 =-

I1he

(I")

(17.93)

Vs average

The fuel consumed during the climb is determined by numerically integrating along the minimum-fuel trajectory, using Eq. (17.93) as an approximation.

100

o

.2

.4

.6.8

1.0

1.2 1.4

1.6

1.8

2.0

2.2

2.4

12

2.6

2.8

MACH NUMBER

Fig.17.15

Minimum fnel to climb.

Energy Method for Mission-Segment Weight Fraction Equation (17.94) is an expression of the mission-segment weight fraction for any flight maneuver involving an increase in energy height. This can be used for climbs or accelerations or combinations of the two. Remember that the mission-segment weight fraction expresses the total aircraft weight at the end of the mission segment divided by the total aircraft weight at the beginning of the mission segment. This is used for sizing as discussed in earlier chapters.

1

Wi+ I [ - Cl1he Wi =exp V(1-D/T)

(17.94)

Unfortunately, a maneuver involving a reduction in energy height cannot create fuel as would be implied by putting a negative value for the change in he into Eq. (17.94)! 17.7 OPERATING ENVELOPE The aircraft "operating envelope" or "flight envelope" maps th~ combinations of altitude and velocity that the aircraft has been deSigned to withstand. The "level-flight operating envelope" has the further restriction that the aircraft be capable of steady level flight.

485

PERFORMANCE AND FLIGHT MECHANICS

AIRCRAFT DESIGN

484

mainin limits shown in Fig. 17.16 are structu~al. ~he external-

flo:h~ r~amic p;essure q as defined in E.q .. (17.95) .has.a dIrect I~pact u~on

ALTITUDE.10 J fl 60

y t I loads A maximum q limIt IS specIfIed In the desIgn reqUlre. I . T . I nd used by the structural designers for s~ress ana YSIS. ypica ~;~:~r ~ircraft have a q limit of 1800-2200 psf. ThIS corresponds to transonic speeds at sea level.

t h e struc ura

ABSOLUTE CEILING

------.... - -Ps-=l00 - . . . . . . . . . .,

,'-::....

PILOT EJECTION ALTITUDE LIMIT

, ,,'/' ~

'L--

SERVICE CEILING'"

, ,~---------~ ,

50

\

/'

\

\

-

I



,---- .........

;..

.~..... .;;;; ...... ~. ~ , ~(;fvV .... ,"~~

"',, \

\

30 .

~.

\

~ ~

\ ....

""

'

PTO=Pstatic[l

P~~~:~~~a~s~~~~ ~~~~i~ot~~ ti:~e!i:~~~~r~t;~~:~~~ ~~a:b~~~

I

... ,,, ....

~

......

10

.... '1;'

~

$

~ ;..

, , ,

,

"'. I.,

'"

II

'l~ .,'

0 0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

MACH NuMBER

Fig. 17.16

(17.96)

2 +0.2M f·5

9Jc;,;

,~ I~

,I

'I;' ~

.;;;;.

~,

20

,

(17.95)

q = Y2poo V';' = 0.7 Pstatic M2

•I

Operating envelope.

The operating envelope for a typical fighter is shown in Fig. 17.16. Fighter operating envelopes are the most complicated and contain all the elements of the operating envelopes of other classes of aircraft. The level-flight operating envelope is determined from the P s = 0 and stall limit lines. The P s = 0 limit is usually shown for both maximum thrust and for military (nonafterburning) thrust. Since the P s = 0 and stall lines vary with aircraft weight, some assumption about aircraft weight must be made. Typically the operating envelope is calculated at takeoff weight, cruise weight, or combat weight. The "absolute ceiling" is determined by the highest altitude at which P s =0. Some small rate of climb capability (Le., P s ) is required at the "service ceiling." FAR's require 100 fpm for propeller aircraft and 500 fpm for jets. Military specifications require 100 fpm at the service ceiling. For some jet aircraft, the limitation on usable ceiling is the pilot. The odds of surviving an ejection above 50,000 ft are rather small without an astronaut-type pressure suit or some type of capsule. This limits the usable ceiling as shown. Another limitation to the level flight envelope of many jet aircraft is the low-q engine operating limit. At low velocities and high altitudes there may not be enough air available to restart the engine in the event of a flameout. It may also be impossible to operate or light the afterburner. These limits are provided by the engine manufacturer.

The airload freestream press . f f ) Th total pressure of the oncommg MhO 4 0 5 at the engme ront- ace . e . t Eq (17 96) using the static atmosphenc pressure a . a~ . - '. d f . ., h T bl A 2 aIr IS determIne rom that altitude from the ~~ndt~: d~~~~~~a~r~he ~u~sid~ ~otal pressure times The total pressure WI In . d' Cha ter 13 Equation the inle~-duct pres.sure re~o~ry, ::th~~s~~~s~uc~nand s~lved f;r the .static (17.96) IS used agam for t e oWen ine front-face. This is the maxImum pressure at the Mac~ n~~?e~hat ::l~t d~ct and may easily be three times the . 17 16 the inlet-duct pressure wall pressure e~erte WI In e h t'd dynamIC pressure. As s own In Ig. . , . . ~~~: ;oes not follow the same slope as the dynamic-pressure hmIt.

P'

"YCLIMB

~/ '"

R

START V=O

l? 6

c

BEGIN TO TAKEOFF ROTATE V=VTO

I

~ -------~

S~I.. SR~

~~~"

~_"'\. I

......,

_- " hTR.

Fig. 17.17

hOBSTACLE

tIt

STR-----I~C CLIMB

ROTATE TRANSITION TO TOTAL GROUND ROLL CLIMB TOTAL TAKEOFF DISTANCE G

1\1

VCL,,"'''' 1 _"YC. _LIMB

4 • •

Takeoff analysis.

\

PERFORMANCE AND FLIGHT MECHANICS

AIRCRAFT DESIGN

486

The remaining operating envelope limit is the temperature limit du t skin aerod.ynamic heating: This depends upon the selected structural also A desI.gn chart for skm temperature vs Mach number and altitude was presented III Chapter 14.

(17.98)

ma:er~

17.8 TAKEOFF ANALYSIS An ~mpirical.chart for determining takeoff distance has been presented. Later In the desIgn process, a more detailed analysis breaks the takeoff into segments for more accurate analysis. Fi?ure 17.17 illustrates the segments of the takeoff analysis. The ground roll I.ncludes two parts, the level ground-roll and the ground roll during rotatIOn to the .angle of .attack for liftoff. After rotation, the aircraft follows an approxImately CIrcular arc ("transition") until it reaches the climb angle. Ground Roll

=

~[T-D -JL(W-L)] =g[ (~-JL)

+ 2:IS (-CDo - Kcl + JL Cd V2 ] Table 17.1

(17.97)

Ground rolling resistance

wtypical values Surface Dry concrete/asphalt Wet concrete/asphalt Icy concrete/asphalt Hard turf Firm dirt Soft turf Wet grass

The ground-roll distance is determined by integrating velocity divided by acceleration, as shown in Eq. (17.98). Note the mathematical trick that simplifies the integration by integrating with respect to V2 instead of V. The takeoff velocity must be no less than 1.1 times the stall speed, which is found by setting maximum lift at stall speed equal to weight and solving for stall speed. The maximum lift coefficient is with the flaps in the takeoff position. Remember that landing gear geometry may limit maximum angle of attack (and hence lift coefficient) for takeoff and landing. Equation (17.98) is integrated for ground-roll distance from J'initial to Vfinal in Eq. (17.99), where the terms KT and KA are defined in Eqs. (17.100) and (17.101). KT contains the thrust terms and KA contains the aerodynamic terms. (17.99)

I?uring. t~e ground roll, the forces on the aircraft are the thrust, drag, and rolhn.g .fnctlO~ of the wheels, this last being expressed as a rolling friction coe~fIcIent. JL tImes the weight on the wheels (W-L). A typical JL value for rolhng reSIstance on a hard runway is 0.03. Values for various runway surfaces are presented in Table 17.1. The resulti~g acceleration of the aircraft, as expressed by Eq. (17.97), can b~ expand.ed In terms of the aerodynamic coefficients. This requires evaluatIng the hft an~ drag of the aircr~~t in grou!ld effect and with landing gear down. ~nd 0aps In the takeoff pOSItIOn, as dIscussed in Chapter 12. The lift coeffIcIen.t IS based on the wing angle of attack on the ground (measured to the zero hft-angle), and is typically less than 0.1 unless large takeoff flaps are deployed.

a

487

Rolling (brakes off)

Brakes on

0.03-0.05 0.05 0.02 0.05 0.04 0.07 0.08

0.3-0.5 0.15-0.3 0.06-0.10 0.4 0.3 0.2 0.2

(17.100) (17.101) Equation (17.99) integrates ground roll from any initial vel?city to a?y final velocity. For takeoff, the initial velocity is zero and the fmal velOCIty is V . Since the thrust actually varies somewhat during the ground roll, an TO averaged thrust value must be used. Since we integrate with respect to velocity squared, the averaged thrust to use is the thrust at about 70070 (1/squareroot 2) of VTO · For greater accuracy the ground roll can be broken into smaller segments and integrated using the averaged thrust for each segment in Eq. (17.99). The averaged thrust is the thrust at 70% of the velocity increase for that segment. Also, K may be reduced due to ground effect (Chapter 12) .. The time to rotate to liftoff attitude depends mostly upon the pIlot. Maximum elevator deflection is rarely employed. A typical assumption for large aircraft is that rotation takes three seconds. The acceleration is assumed to be negligible over that short time interval, so the rotation groundroll distance SR is approximated by three times VTO· For small aircraft the rotational time is on the order of 1 s, and SR = VTO · Transition During the transition, the aircraft accelerates from takeoff spe~~ (1 ..1 VstaU) to climb speed (1.2 VstaU). The average velocity during trans~tIon IS therefore about 1.15 Vstau. The average lift coefficient during transitIon can be assumed to be about 90% of the maximum lift coefficient with takeoff flaps. The average vertical acceleration in terms of load factor can then be

488

AIRCRAFT DESIGN

PERFORMANCE AND FLIGHT MECHANICS

489

found from Eq. (17.102):

(17.102) V2 n = 1.0 + RTR g = 1. 2

ViR

R =

g(n - 1)

=

ViR _

either brake to a halt or continue the takeoff in the same total distance. If the engine fails before decision speed, the pilot can easily brake to a halt. If the engine fails after decision speed, the pilot must continue the takeoff. An empirical methoq for balanced field-length estimation was presented in Chapter 5. A more detailed equation, as developed in Ref. 23, takes this

(17.103) form: BFL

2

0.2g = 0.205 VStalJ

=

0.863 ( 1 + 2.3G

(17.104)

W

LID

(17.107)

If the obstacle height is cleared befor h .. then Eq. (17.108) is used to d t . eht e end. ~f the. tranSItIon segment, e ermIne t e transition distance. ST = VR2 - (R - hTR)2

(17.108)

Climb

Finally, the horizontal distance trav 11 d d . . obstacle height is found from E 17 1~ e unng ~he chmb to clear the is 50 ft for military and small .

TIW=0.9

l2-

Wo = 44,000 Ib p. = 140 fps

Wo p.

STO = 670 ft a =56s Require: P s

= 39,000 Ib = -230 fps

~

Wo = 36,000 Ib p. = -320 fps

G5 z

~

STO = 1070 ft a = 51 s

STO = 810 ft = 53 s a

~

0 at (MO.9, 30k ft, Sg's) 500 ft a :S SO s from MO.9 to Ml.5

STO:S

Fig. 19.1

Sizing matrix.

TAKEOFF DISTANCE

Ps at M.9, 30,000 fl. 5 g'.

TAKEOFF WEIGHT P s (100 fps)

W. (1000 Ib)

S (100 fI) I

60t so~I--~_

TlW=1.1

.

4Ofr---r-

6

10

J

4

8

1

2

6

0

4

:r

-2

30 I .

-

~

U>

N

I

z » Z

G> 10

6

4

SO:t.

~

5 TlW= 1.0

6

401--.1----- --- ee

4

8

2

6

0

4

0 --I

:IJ

» 0

4

m

2

-2 I •

30 '

6

0 9

T/W=.9

;L i-------

30

I

SO

60

W/S

:y 9

70

;l -2

t, SO

lOt

8~

8

U> --I C 0

m

U>

6

4

~ 60

W/S

... 9

70

2 (I

SO

60

W/S

70 01

Fig. 19.2

Sizing matrix cross plots.

!\)

"'..J

528

AIRCRAFT DESIGN

SIZING AND TRADE STUDIES

529

Sizing Matrix Plot

Optimization of TIW and WIS requires crossplotting the sizing-matrix data, as shown in Fig. 19.2. For each value of thrust-to-weight ratio, the sized takeoff gross weight, P s , and takeoff distance are plotted vs wing loading. The data points from the sizing matrix in Fig. 19.1 are shown as numbered black dots. (The acceleration data points were plotted in a similar fashion, but not shown.) From the takeoff-weight graphs in Fig. 19.2, the wing loadings corresponding to regularly spaced arbitrary gross weights are determined. For this example, gross weights at 5,OOO-lb increments were selected. For these arbitrary weight increments, the corresponding WIS values are shown as circles on Fig. 19.2. The WIS and TIW values for the arbitrary gross-weight increments are transferred to a TIW- W IS graph as shown in Fig. 19.3. Smooth curves are drawn connecting the various points that have the same gross weight to produce lines of constant-size takeoff gross weight (Fig. 19.3). From these curves one can readily determine the sized takeoff weight for variations of the aircraft with any combination of TIW and WIS. Next, the WIS values that exactly meet the various performance requirements are obtained from the performance plots for different TIW values (right side of Fig. 19.2). These values are again shown as circles. These combinations of W IS and T IW that exactly meet a performance requirement are transferred to the TIW-WIS graph and connected by W O=55K

50K

TlW

1.05

40K

0.95

0.9 45

50

55

Fig. 19.3

60

65

Sizing matrix plot (continued).

w/s

70

T/W

1.05

+---,c.---t-"""'7'---+-

1.00 +--oooooEJ.:-~!'II:::""--~~:""----+-~~--+--;~--::::0f8'-------1

0.95

0.9 45

50

55

Fig. 19.4

60

65

W/S

70

75

Sizing matrix plot (concluded).

smooth curves, as shown in Fig. 19.4. Shading is used to indicate which side of these "constraint lines" the desired answer must avoid. The desired solution is the lightest aircraft that meets all performance requirements. The optimum combination of TIWand WIS is found by inspection, as shown in Fig. 19.4, and usually will be located where two constraint lines cross. This is a simple example with only three performance constraints. In a real optimization, a dozen or more constraint lines may be plotted. While it is not necessary to include every performance requirement in the sizing matrix plot, all those which the baseline aircraft does not handily exceed should be included. This example showed only a 3 x 3 sizing matrix. For better accuracy, 5 x 5 and larger sizing matrices are used at the major aircraft companies.

45K

1.10

1.00

1.10

75

Carpet Plot Another presentation format for the sizing matrix, the so-called "carpet plot," is based upon superimposing the takeoff weight plots from Fig. 19.2. In Fig. 19.5a, the upper-left illustration from Fig. 19.2 is repeated showing a plot of sized takeoff gross weight Wo vs W IS for a T IW of 1.1. The points labeled 1, 2, and 3-data points from the matrix (Fig. 19.1)represent wing loadings of 50, 60, and 70. The next illustration of Fig. 19.5 superimposes the next Wo vs WIS plot from Fig. 19.2. This plot represents a TIWof 1.0. The data points labeled 4, 5, and 6 again represent wing loadings of 50, 60, and 70.

Wo (1000 LB)

Wo (1000 LB)

TlW=1.1

60

3

50

SHIFTED SCALE FOR NEXTTlW

60

~

....

50

~~

TlW=l.l

+-------~------~-50

60

W/S,

:

70

~----+----+60 50 70

W/S

for 1, 2, 3

I

. .----,...----1...

50 Wo (I,OOOLB)

60

W IS for 4, 5, 6

70

.... II CI'I

......

=>

30

=!

~

TlW = 1.0

40

40

II

3

6

30

531

SIZING AND TRADE STUDIES

AIRCRAFT DESIGN

530

=>

~

l"-

II

II

~ ......

rLJ

...... ~

~

~

W/S=50

60 50

40

30

Fig. 19.5

Carpet plot format. (same results!)

To avoid clutter, the horizontal axis has been shifted to the left some arbitrary distance. This shifting of the axis is crucial to the development of the carpet-plot format. In the lower illustration of Fig. 19.5, the third curve of Wo vs W/S has been added, again shifting the horizontal axis the same increment. The points labeled 7, 8, and 9 again represent wing loadings of 50, 60, and 70. Now these regularly spaced wing-loading points on the three curves can be connected, as shown. The resulting curves are said to resemble a carpet; hence the name. The horizontal axis can be removed from the carpet plot because one can now read wing loadings by interpolating between the curves. In Fig. 19.6, the wing loadings that exactly meet the takeoff, P" and acceleration requirements (from Fig. 19.2) have been plotted onto the carpet plot and connected with constraint lines. The optimal aircraft is found by inspection as the lowest point on the carpet plot that meets all constraints. This usually occurs at the intersection of two constraint curves. The carpet plot and the sizing-matrix crossplot format give the same answer. Some people prefer the carpet-plot format because the "good" direction for minimum weight is obvious (down). Others prefer the sizingmatrix crossplot format because it is easier to read the optimal thrust-toweight ratio and wing loading once they are found. Note that both formats are commonly referred to as "carpet plots."

I

532

AIRCRAFT DESIGN

It is also possible to create sizing plots in which the measure of merit is cost rather than weight. The plotting procedure is the same except that cost values are used rather than weight values in the development of the sizing plot. However, for most aircraft types the minimization of weight will also minimize cost for a given design concept. Sizing-Matrix Data Approximations

A massive amount of work would be required to analyze fully the impact of variations in T IW and W IS On the aerodynamic, propulsion, and weight data required to develop a carpet plot. A variation in T IW affects the thrust and fuel flow, but also affects the wetted area and wave drag due to the change in nacelle size. A change in W/S affects the wetted area and wave drag. Additionally, changing W IS affects the drag-due-to-lift (K) factor because the fuselage covers up more or less of the wing span. Note that, while the total parasite drag usually increases as the wing size increases, the drag coefficient may drop because it is referenced to the wing area! At the major aircraft companies, sophisticated modules for analyzing the effects of the parametric variations of T IW and W IS are incorporated into the sizing programs. For initial studies and student designs, this analysis can be approximated by ratioing the baseline analysis for the affected parts of the airplane. The change in zero-lift drag can be assumed to be proportional to the change in wetted area due to the wing-area and nacelle-size variations. Wing wetted area varies approximately directly with wing area. Nacelle wetted area varies roughly with the variation in thrust. For a supersonic aircraft the wave drag should be recalculated. The wing cross-sectional area varies directly with a change in wing area. This is used to determine the new total cross-sectional area that is used to approximate the wave drag. The variation in K due to relative fuselage size, being small, may be ignored for initial studies. If the wing area is changed, however, then the aircraft will fly at different lift coefficients. The statistical equations in Chapter 15 show that the wing and tail component weights vary approximately by the 0.7 power of the change in wing area. The engine itself varies in weight by the 1.1 power of a change in thrust. Installed propulsion performance can be assumed to ratio directly with the thrust. These and similar, reasonable approximations can be used to estimate the revisions to aerodynamic, weight, and propulsion data for sizing analysis and carpet plotting. 19.5

TRADE STUDIES

Trade studies produce the answers to design questions beginning with "What if. .. 1" Proper selection and execution of the trade studies is as important in aircraft design as a good configuration layout or a correct

SIZING AND TRADE STUDIES

533

535

AIRCRAFT DESIGN

SIZING AND TRADE STUDIES

sizing analysis. Only through the trade studies will the true optimum aircraft emerge. The "granddaddy" of all trade studies is the TIW-WIS carpet plot. This is such an integral part of aircraft analysis that it is not usually even thought of as a trade study. A TIW- W/S carpet plot in good measure determines the minimum-weight aircraft that meets all performance requirements. Table 19.1 shows a number of the trade studies commonly conducted in aircraft design. These are loosely organized into design trades, requirements trades, and growth sensitivities. Design trades reduce the weight and cost of the aircraft to meet a given set of mission and performance requirements. These include wing-geometry and propulsion variations as well as configuration arrangement trades. Requirements trades determine the sensitivity of the aircraft to changes in the design requirements. If one requirement forces a large increase in weight or cost, the customer may relax it. Growth-sensitivity trade studies determine how much the aircraft weight will be impacted if various parameters such as drag or specific fuel consumption should increase. These are typically presented in a single graph, with percent change of the various parameters on the horizontal axis and percent change in takeoff weight on the vertical axis. Be aware of an important consideration in all of these trade studies: the realism factor. There is an unfortunate tendency to minimize redesign effort, especially for yet another boring trade study! If asked to study the impact of carrying two more internal missiles, the designer may find a way to "stuff them in" without changing the external lines of the aircraft. This might completely invalidate the results of the trade study. If there was sufficient room in the baseline to fit two more missiles internally, then the baseline was poorly designed. If the baseline was already "tight," then the revised layout must be a fake! The best way to avoid such problems is to insist that all redesigned layouts used for trade studies be checked to maintain the same internal density as the baseline, calculated as takeoff weight divided by internal volume. The trade studies shown in Table 19.1 must be calculated using a complete TIW-WIS carpet plot for each data point. For example, to determine the optimal aspect ratio the designer might parametrically vary the baseline aspect ratio up and down 20070. For each aspect ratio, a T IW- W IS carpet plot would be used to determine the minimum-weight airplane. These minimum weights would then be plotted vs aspect ratio to find the best aspect ratio. The workload for trade studies can rapidly exceed manual capabilities. To optimize aspect ratio as described above requires a minimum of 3 X 3 X 3 (27) data points. Each data point requires full analysis for aerodynamics, propulsion, and weights, followed by a sizing iteration. To truly optimize an aircraft, a large number of the parameters from Table 19.1 should be considered simultaneously. However, the hundreds or thousands of data points required to do this would exceed even computer capabilities. There is currently great interest in developing optimization procedures that permit such multi variable optimization in a design environment. Two

techniques show promise, "Latin Squares" and "Decomposition," but go beyond the scope of this book. It has been assumed here that the measure of merit for trade studies will always be takeoff gross weight. Cost, though, will be the final selection measure in a design competition. Using minimum weight as the measure ~f merit is usually a good approximation to minimum cost because the acqUisition cost is so strongly driven by the weight. However, life-cycle cost is driven largely by fuel cost, which .may not be minimized by the minimum-weight airplane. LCC can be estimated and plotted on the sizing matrix, and the best aircraft can then be selected as the lowest LCC point.

534

20 VTOL AIRCRAFT DESIGN 20.1 INTRODUCTION This chapter introduces the essential concepts and technologies of vertical takeoff and landing (VTOL) aircraft design. Although similar in many respects to conventional aircraft design, VTOL presents some key differences and pitfalls to avoid. This chapter emphasizes the differences that affect VTOL vehicle layout and sizing analysis. The operational benefits of an ability to take off and land vertically are self-evident. Conventional aircraft must operate from a relatively small number of airports or airbases with long paved runways. For commercial transportation, the airport is rarely where you actually wish to go, and is usually crowded, causing delays in the air and on the ground. The military air base is highly vulnerable to attack, and during a wartime situation the time expended cruising to and from the in-the-rear airbase increases the required aircraft range and also increases the amount of time it takes for the aircraft to respond to a call for support. The first type of VTOL heavier-than-air aircraft was the helicopter, which was conceived by Leonardo daVinci but not regularly used until shortly after World War II. The helicopter rapidly proved its worth for rescue operations and short range point-to-point transportation, but its inherent speed and range limitations restricted its application. For propeller-powered aircraft, the tilt-rotor concept as tested in the Bell XV-IS seems to offer the best compromise between helicopter-like vertical flight and efficient wing-borne cruise. The tilt-rotor concept is the basis of the V-22 Osprey now under development. Helicopters and tilt-rotors go beyond the scope of this book, but are discussed in Ref. 68. For jet VTOL aircraft, a clear "best" solution for vertical lift has yet"to emerge. Instead, there are a wide variety of alternative vertical-lift concepts, some tested and some not, available for incorporation into a new design. Selection of a "best" concept depends upon the intended mission and operational environment as well as the assumptions made as to the technical details of the selected lift concept. To date there have only been a few operational jet VTOL designs-the British Harrier and the Russian YAK-36. These are both subsonic aircraft. While at least one supersonic VTOL design has flown (The Mach 2 Mirage lII-V back in 1966), there has yet to be an operational supersonic VTOL aircraft. 537

538

AIRCRAFT DESIGN

This is largely due to the increased internal volume required for the vertical-lift apparatus and vertical-flight fuel. Also, most concepts for vertical lift tend to increase the aircraft's cross-sectional area near the aircraft's center of gravity (c.g.), and that increases the supersonic wave drag. Finally, the state of the art in engine thrust-to-weight ratio has imposed an excessive weight penalty on VTOL designs. It has simply not been possible up to now to provide both vertical flight and supersonic forward flight in an operational aircraft of any usable range. However, the overall level of aircraft/engine technology and VTOLspecific technology is advancing so rapidly that this author expects the next generation of new military jets to include at least one supersonic VTOL concept. 20.2

VTOL AIRCRAFT DESIGN

a)

FORWARD FLIGHT

539

b) MAGIC FINGER VERTICAL FLIGHT

VTOL TERMINOLOGY

VTOL refers t~ a capability for Vertical TakeOff and Landing, as opposed to ConventlOnal TakeOff and Landing (CTOL). An aircraft which has the flexibility to perform either vertical or short takeoffs and lan.d.ings is s~id to have Vertical or Short TakeOff and Landing (VSTOL) cap~bihty. An ~ircraft which has insufficient lift for vertical flight at takeoff weight but which can land vertically at landing weight is called a Short TakeOff and Vertical Land (STOVL). . The "tail-sitter". or Vertical Attitude TakeOff and Landing (V ATOL) aircraft cannot use its vertical lift capability to shorten a conventional takeoff or landing roll. In contrast, a Horizontal Attitude TakeOff and Land (HATOL) concept can usually deflect part of its thrust downward while in forward flight enabling it to perform a Short TakeOff and Landing (STOL).

20.3 FUNDAMENTAL PROBLEMS OF VTOL DESIGN A number of unique problems characterize the design and operation of jet VTOL aircraft. T.wo fundamental problems stand out because they tend to have the greatest impact upon the selection of a VTOL propulsion concept and upon the design and sizing of the aircraft: balance and thrust matching. Modern supersonic jet fighters have a TIWexceeding 1.0, so it would seem fairly easy to point the jet exhaust downward and attain vertical flight. Unfortunately, this is complicated by the balance problem. NoIany subsonic jets and virtually all supersonic jets are designed with the engme at the rear, the cockpit and avionics at the nose, and the payload and fuel near the center of the aircraft. This traditional layout places the expendables on the c.g., co-locates the parts of the aircraft requiring cooling (cre.w and avionics), and keeps the avionics away from the hot and vibrating engme. Fig.ure 20.1a illustrates this traditional (and usually optimal) layout. If the aircraft's thrust exceeds its weight, vertical flight could be obtained simply by deflecting the thrust downward, as shown in Fig. 20.1 b. However, a "magic finger" must hold up the nose in order to balance the

c) THRUST LOCATION MOVED

Fig. 20.1

d) BALANCED THRUST

The balance problem.

vertical thrust force at the tail. This balance problem is possibly the single most important driver of the design of the VTOL jet fighter. There are really only two conceptual approaches to solving the balance problem. Either the thrust can somehow be moved to the c.g. (Fig. 20.1c), or an additional thrust force can be located near the nose (Fig. 20.1d). Both of these approaches will tend to compromise the aircraft away from the traditional and usually optimal layout. For cruise-dominated VTOL aircraft such as transports, a more severe problem involves thrust matching. If the thrust required for vertical flight is provided by the same engines used for cruise, the engines will be far too large for efficient cruise. As an example, imagine designing a VTOL transport using four of the TF-39 engines used in the C-5. These produce about 40,000 lb of thrust at sea-level static conditions, or 160,000 lb altogether. If the aircraft is to have a typical 30070 thrust surplus for vertical flight (TIW = 1.3), then the aircraft can weigh no more than 123,077 lb at takeoff. Note that this is far less than the C-5 at 764,000 lb! Assuming a typical cruise LID of 18 yields a required TIW during cruise of about 1118, or 0.056. If the aircraft weight at the beginning of cruise is about 95% of the takeoff weight, then the total thrust required during cruise is only 6,496 lb (123,077 X 0.95 X 0.056). This is only 1624 lb of thrust per engine, which is about 18% of the available thrust for that engine at a typical cruise altitude of 35,000 ft. It is doubtful that the engine would even run at that Iowa thrust setting. At 35,000 ft and Mach 0.9, the best SFC for this engine would be about 0.73 at a thrust of 9,000 lb per engine. The SFC at the 50% throttle setting is about 1.2-64% worse than the SFC at the higher thrust setting. If the engine would run at only 18% of its available thrust, its SFC would be even worse than the 1.2 value. Aircraft range is directly proportional to SFC. The mismatch between thrust for vertical flight and thrust for cruise will produce a tremendous fuel

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consumption and range penalty for a cruise-dominated design that uses only the vectored thrust of its cruise engines for vertical flight. For this reason many conceptual VTOL transport designs incorporate separate "lift engines" used during vertical flight. If three of the TF-39 engines in the example above could be turned off during cruise (without a drag penalty), the remaining engine could be operated at a 72% thrust setting where it gets an SFC of about 0.8. This is a big improvement over all engines being used for both lift and cruise. However, the use of separate lift engines introduces additional problems, as discussed later. There are numerous other problems associated with VTOL aircraft design including transition, control, suckdown, hot gas ingestion, FOD, inlet flow matching, and ground erosion. These are discussed below following a brief discussion of the various VTOL jet propulsion options which are currently available to the designer.

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VTOL AIRCRAFT DESIGN

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20.4 VTOL JET-PROPULSION OPTIONS The major options for jet VTOL propulsion systems are notionally derived in Fig. 20.2. Broadly speaking, jet VTOL concepts can be divided into those that utilize fairly conventional engines and those that use engines modified so that the fan and core air are split, with the fan air ducted and exhausted from some place separate from the core air. The conventional-engine VTOL concepts that do not use additional lift engines for vertical flight must have a net takeoff TIWin excess of 1.0. If the jet exhaust is not diverted to some other location for vertical flight, the aircraft must either be a tail sitter (V ATOL), or have the engine exhaust at the aircraft c.g. and capable of vectoring downward for vertical flight. This

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Conventional engine, no lift engine, no flow diversion.

542

AIRCRAFT DESIGN

VTOL AIRCRAFT DESIGN

can be accomplished by using a vectoring nozzle or nacelles which tilt (Fig. 20.3). ~he X-14 .research air~raft had vectoring nozzles at the c.g., with the engl':les .out III front. This is probably not a good arrangement for most applIcatIOns b.ecause the cockpit winds up in back, for balance, and thus doe~ not proVide acceptable visibility for the pilot. Also, in forward flight the Jet exhaust scrubs alongside the fuselage, causing thermal and acoustic problems. An alternative approach is to place the nozzles at the center of gravity but put the engine in the rear fuselage as on a regular aircraft, but installed backwards! This "Reverse Installation Vectored Engine Thrust" (RIVET) concept offers design simplicity, reduced weight, ease of transition, and inherent vectoring in forward flight (VIFF). However, inlet duct losses of 5 percent or more will be caused by the 180-deg bend required to supply air to a backwards engine. Sizing studies (Ref. 93) indicate that despite these duct losses, RIVET offers a viable option for supersonic V/STOL. Tilt nacelles, although heavy, may be the best compromise for some applications. Grumman has been pursuing a tilt-nacelle concept for Naval applications for a number of years. Some VTOL concepts provide a means of diverting the exhaust flow to gain vertical lift. This is generally done by a retracting blocker device in the engine that shuts off the flow through the rearward-facing nozzle. The flow is then diverted forward through internal ducting (Fig. 20.4).

The diverted flow can be exhausted directly downwards, or it can be "augmented" by either a tip-driven fan or an ejector. Both of these can actually increase the thrust obtained from the diverted flow by using the energy of the exhaust flow to accelerate a larger mass of air. This augments thrust by increasing the propulsive efficiency. The tip-driven fan (a ducted fan) is turned by turbine blades at its tip. The diverted engine exhaust is passed over the turbine blades to spin the fan. The Ryan XV-SA used such fans and attained an "augmentation ratio" of almost three (the lifting thrust attained with the tip-driven fans was almost three times the thrust produced by the jet engines during normal forward flight). The ejector makes use of the viscosity of the air. Any exhaust jet will "drag" along adjacent air molecules, accelerating the free air in its vicinity. The ejector consists of a short duct with an exhaust stream blowing down it. Additional air is pulled by viscosity into the duct, accelerated, and ejected through a nozzle. This produces thrust greater than the thrust due to the jet exhaust alone. While ejectors promise theoretical augmentation ratios of 3 or more, a more realistic value ranges from about 1.3 to perhaps 2.2. The Rockwell XFV-12A featured ejectors along the entire span of the wing and canard. It was expected to produce a high value of augmentation ratio, the actual value achieved was only about 1.5, and it never flew. Both ejectors and tip-driven fans are heavy and tend to chop up the aircraft structure. Also, the internal ducting is bulky and poses a thermal problem. However, ejectors and tip-driven fans tend to reduce the thrustmatching problem since the engines do not have to be sized to lift the aircraft by jet thrust alone. The resulting improvement in cruise fuel consumption may offset the weight of the ejector or tip-driven fan. One of the simplest ways of providing VTOL capability adds lift engines to an essentially conventional aircraft (Fig. 20.5). This brute-force approach was used in the Mirage III-V. Obviously, the separate lift engines add considerable weight and volume to the design, but the forward-flight engine can be sized for efficient cruise, thus solving the thrust-matching problem. Since the lift engines are designed for a single operating condition, they can be highly optimized for that condition. Existing lift engines have engine TIWon the order of IS, compared to about 6-8 for a typical forward-flight engine. Future lift engines are expected to have engine TIW of 25 or more. A more subtle approach than the use of separate engines for lift and cruise is to size the forward-flight engine for efficient cruise, but also provide a means of vectoring its thrust downward for vertical flight. The vertical-thrust shortfall is made up by the addition of lift engines. This is known as a "lift plus lift/cruise (L + LIC)" approach. Since the forward-flight engine is providing some vertical thrust, the thrust required from the lift engines is reduced. The forward-flight thrust can be vectored by a vectoring nozzle as in the YAK-36 or by tilt nacelles. A major problem with the L + LIC approach is the transition from vertical to forward flight. During the transition period, the lift/cruise engine thrust is being vectored rearward, decreasing the vertical component of

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Fig. 20.4

Conventional engine, no lift engines, flow diversion used.

543

544

VTOL AIRCRAFT DESIGN

AIRCRAFT DESIGN

a) SEPARATE LIFT ENGINES

b) L + LIC (VECTORED)

c

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Fig. 20.5

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thrust.. Since !he lift! cru~se engine is at the back of the aircraft, additional thrust l~ requ~red to aVOid a nose-up pitching moment. If a l~ft engme s~ould fail during vertical flight or transition, the aircraft would illstantly pItch nose-down. It is rumored that the YAK-36 h as an '" au t omatlc ejection seat to save the pilot in this event. The + LIC approach.is especially poor for providing vectoring in for~ard fll~ht ~VIFF). The pIlot must start up the lift engines before the selection of ill-flIght vectoring. A~other problem is that the aircraft operators would rather not have to prOVl?e tools, spare parts, and trained mechanics for two types of engine in one aIrcraft.

or supplemental turbine, and spins the lift fan to provide vertical thrust. This avoids the need to develop a complete new lift engine, although the fan, driveshaft, gearbox, and turbine must be developed. Also, SDLF has a cooler front exhaust since the forward lift exhaust is not combusted. SDLF does give up the return-to-base capability of the L + L IC. A number of VTOL propulsion concepts are based upon a "split-flow" modification to the turbofan engine. The airflow from the fan is split away from the core airflow and used in some fashion to address the balance andlor thrust-matching problems. One such approach exhausts the fan air separately and provides a means for vectoring it downward for vertical flight (Fig. 20.6a). The AV-8 Harrier uses the high-bypass Pegasus engine in which the fan air and core air are each separately vectored through "elbow" nozzles (described later). This permits nearly instantaneous vectoring of thrust with no mode changes (such as starting a lift engine or diverting air into an ejector). This approach also simplifies transition and enhances maneuverability. On the negative side, the Pegasus-type engine suffers the thrust-matching problem since the engine thrust must provide all of the required lifting force. Also, the engine must straddle the aircraft c.g. This increases the aircraft's cross-sectional area right at the wing location, and thus increases supersonic wave drag (the Harrier is subsonic). It is possible to augment the thrust of such an engine by essentially providing an "afterburner" for the fan and core airflows in so-called plenum-chamber burning (PCB). There is considerable debate about the desirability of such high exhaust temperatures for VTOL operation. Another means of provIOmg after burning to the Pegasus-type split flow engine is to duct the fan and core exhausts together during forward flight, and provide a conventional afterburner which is only used in forward flight

l:

b) TANDEM FAN

a) VECTORED THRUST

~ne possible benefit fo~ the L + LIC concept is the ability to use the lift

en~ille to return to base ill the event that the cruise engine fails. This re-

qUlres so.m~ aft vec~~ring ability for the lift engine, which is desirable anyway to aId ill tranSItion. ~elated to the L + L I C concept is the "shaft-driven lift fan" (SDLF). ThIS offers many of the benefits of L + L I C but without some of the problems . In the SI?~F concept, a driveshaft runs from the engine to a ~eparate lift fan I?oSitlOne.d where the lift engine on the L + L IC concept IS located. The dnveshaft IS powered by the main engine through a modified

545

CORE

c) HYBRID FAN VECTORED THRUST

Fig. 20.6

Split-flow engines (vectored fan air),

548

AIRCRAFT DESIGN

VTOL AIRCRAFT DESIGN

The vectoring flaps can also be external to the nozzle as a part of the wing flap system. This approach was used on the XC-IS transport prototype. Although this was not a VTOL aircraft, its wing flap system was able to turn the engine flow more than 60 deg for STOL landings. This, combined with a landing gear that permits a 30-deg nose-up position, would provide the required 90 deg of total thrust vectoring for vertical flight. The bucket vectoring mechanism (Fig. 20Sb) is similar to the commonlyused clamshell thrust reverser. The great advantage of this concept is that the flow-turning forces are all carried through the hingeline; thus the actuator can be fairly small. Also, the bucket can be designed with a smooth turning surface to raise the turning efficiency. A bucket vectoring nozzle can be designed to have a thrust loss of only about 2-3070 when vectored 90 deg. Figure 20.Sc shows an "axisymmetric" vectoring system. The tailpipe is broken along slanted lines into three pieces, as shown. The three pieces are c?nnected with circular rotating-ring bearings so that the middle (shaded) piece can be rotated about its longitudinal axis while the other parts remain unrotated. This causes the middle and end pieces of the tailpipe to vector downward as shown. The rotating-ring bearings must be circular in shape, so the tailpipe must have a circular cross section along the slanted lines shown. For this to occur the perpendicular cross-sectional shape of the tailpipe must be an ellipse, so

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Fig. 20.8

Vectoring nozzles.

549

this nozzle system is not truly axisymmetric, despite its name. This type of vectoring nozzle has roughly a 3-5% thrust loss when vectored 90 deg. The ventral nozzle (Fig. 20.Sd) is simply a hole in the bottom of the tailpipe leading to a downward-facing nozzle. The flow out the regular nozzle is blocked off with some type of door. To reduce hot-gas ingestion and damage to the runway, an afterburner is not usually used for vertical lift. A ventral nozzle can therefore be placed upstream (forward) of the afterburner. This moves the vertical thrust substantially forward compared to a vectoring nozzle at the end of the entire engine. That helps the balance problem. The ventral nozzle has a thrust loss on the order of 3-6% when vectored 90 deg. The "elbow" nozzle is used on the Pegasus engine in the highly successful A V-S Harrier. In the elbow nozzle the flow is turned 90 deg outboard (see top view in Fig. 20.Se). A circular ring bearing connects to the mov~ble ~art of the nozzle which turns the flow 90 deg back to the freestream dlrectlOn. To vector the flow downward, the ring bearing is rotated 90 deg, as shown. The elbow nozzle is simple and lightweight, and requires a minimum of actuator force for vectoring. However, the flow is always being turned through a total of ISO deg, even in forward flight. Because of this the engine is always suffering a thrust loss of approximately 6-S%. All the other types of vectoring nozzle only impose a thrust loss during vertical flight. To reduce this thrust loss, the elbow nozzle can be designed using turns of less than 90 deg by canting the ring bearings downward and rearward. This can reduce the total turning angle to about 110 deg, which reduces the thrust loss to about 4-6%. However, the nozzles will then yaw inward during transition from horizontal to vertical thrust. This reduces the total usable thrust during transition and also increases the exhaust impingement upon the fuselage. . Another alternative is to provide elbow nozzles that are only used dunng vertical flight. A blocker door like that used for the ventral nozzle can divert the airflow from a conventional nozzle to the elbow nozzles. Like the ventral nozzle, the "part-time" elbow nozzles can be located forward Of. the afterburner for balance. The use of a conventional nozzle in forward flight saves fuel during cruise. This can more than compensate for the extra nozzle weight. .. There is another important factor in the selection of nozzle type. ThiS IS the effect of the vectoring mechanism on the resultant thrust magnitude during transition. For the elbow-type nozzle as used in the Harrier, the vector angle has virtually no effect upon the thrust magnitude. For the vectoring flap, bucket and rotating segment-type nozzles the only effect is the previouslymentio~ed loss of thrust due to turning. This thrust loss is zero at zero degrees of deflection and gradually approaches the maximum value as the thrust deflection approaches 90 deg. For the ventral nozzle, another factor causes a further reduction in thrust during transition. To transition from forward to vertical flight, t.he flow out the rear nozzle is gradually blocked off and the ventral nozzle IS gradually opened.

550

AIRCRAFT DESIGN

VTOL AIRCRAFT DESIGN

The net thrust direction during transition is the vector sum of the thrusts produced by the aft and ventral nozzles. For example, if the exhaust massflow out the aft nozzle and the ventral nozzle are equal, then each nozzle has a thrust approximately equal to half the total engine thrust. The direction of the net resultant thrust is therefore 45 deg downward. The magnitude of the net resultant thrust is found by vector addition to be 0.707 times the total engine thrust (square root of 0.5 2 plus 0.5 2). Thus the magnitude of the thrust is reduced by almost 30070 when the net resultant thrust is vectored 45 deg during transition! Figure 20.9a shows this result. The net resultant thrust for the ventral nozzle drops to about 70% of the total thrust as the thrust is vectored through 45 deg. It returns to full net thrust as the thrust vectoring is continued to 90 deg. By comparison, the elbow-type nozzle has the same thrust at all vector angles. The reduction in the net resultant thrust of the ventral nozzle can be substantially lessened if the aft and ventral nozzles have vectoring capability. T~e middle line on Fig. 20.9a shows the net resultant thrust for 20-deg vectonng. This vectoring ventral nozzle has full net thrust within 20 deg of the zeroand 90-deg vector angles. The worst case 45-deg vector angle, improves to a loss of only 10% of the thrust. Figure 20.9a ignores the thrust losses due to turning. This reduces the relative advantage of the elbow nozzle. In Fig. 20.9b, the curves of Fig. 20.9a are repeated but with an assumed thrust loss of 3% for each 90-deg turn in the exhaust flow.

In forward flight, the ventral nozzle has a higher net thrust than the elbow concept due to the two 90-deg turns in the latter. This is also true in vertical flight where the ventral nozzle concept has only one 90-deg turn. At the 45-deg vector angle, the elbow nozzle has a substantially better net resultant thrust if the ventral nozzle has no vectoring. If the ventral nozzle has a 20-deg vectoring ability, then the difference is negligible.

20.6 SUCKDOWN AND FOUNTAIN LIFT The VTOL aircraft in hover is not in stagnant air. The jet exhaust that supports the aircraft also accelerates the airmass around it. This entrainment is due to viscosity and is strongest near the exhaust plume, producing a downward flow field about the aircraft (Fig. 20.lOa). This downward flow field pushes down on the aircraft with a "vertical drag" force equivalent to a loss of typically 2-6% of the lift thrust. The magnitude of this vertical drag force depends largely upon the relative locations of the exhaust nozzles and the wing. If the nozzles are right under the wing, the entrained airflow will exert a large downward force. Unfortunately, the nozzles and the wing are both near the c.g. for most VTOL concepts. The use of a tandem wing, forward-swept wing, or joined wing may reduce the entrained download by separating the wing away from nozzles, which are located at the c.g.

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EFFECTS-FOUNTAIN LIFT

Fig. 20.9

551

Resultant thrust. Fig. 20.10

Suckdown and fonntain lift.

552

AIRCRAFT DESIGN

Fi~ure 20. lOb shows the effect of the ground On the entrained flowfield. The Jet exhaust strikes the ground and spreads outward. This increases the mixi~g between the jet exhaust and the adjacent air, which increases the ~ntraInment effect. The entrained download (or "suckdown") therefore mcreases as the ground is approached. A single-jet VTOL concept can experience a 30070 reduction in effective lift due to suckdown. Furthermore, the suckdown increases as the ground is approached-a very undesirable handling quality! Figure 20.lOc shows a VTOL concept with widely separated multiple nozzles near the ground. The jet exhausts strike the ground and spread outward. The exh.austs meet in the middle. Since there is nowhere else to go, they merge and nse upward, forming a "fountain" under the aircraft. This fountain pushes upward on the aircraft with a magnitude that will often cancel the suckdown force. The strength of the fountain lift depends upon the exact arra~gement of the nozzles and the shape of the fuselage. Lower-fuselage shapmg that makes it more difficult for the fountain to flow around the fuselage will increase the fountain effect. For example, square lower corners are better than round ones. Fountain lift increases as the ground is approached. This desirable handling quality counters the undesirable effect of suckdown. The fou.ntain lift can be increased even more by the use of Lift Improvement DevIces (LIDS), (called Cushion Augmentation Devices in Britain). Th~se are longitudinal strakes located along the lower fuselage corners WhICh capture the fountain (Fig. 20.lOd). LIDS added to the A V-SB increased the net vertical lift over 6%. Note that multiple nozzles near to each other may not produce a fountain effect. because t~e exhaust plumes may merge into a single jet, producing a flowfleld more hke that shown in Fig. 20.IOa.

20.7 RECIRCULATION AND HOT-GAS INGESTION A VTOL aircraft hovering near the ground tends to "drink its own bath~at:r:" The hot ~xh~ust gases find their way back into the inlet, causing a sIgmfIcant reductIOn In thrust. Also, this "recirculated" air can include dirt and other erosion particles that can damage or destroy the engine. In some cases the dirt kicked up by a hovering VTOL aircraft can completely obscure the pilot's vision. Figure 20.11 shows the three contributors to exhaust recirculation: bouyancy, fountain, and relative wind. Bouyancy refers to the natural tendancy of hot gases to rise. The jet exhaust mixes with the ambient air and slows down as it moves farther away from the airplane. Eventually it has slowed enough that the outward momentum becomes negligible and the bouyancy effect takes over. The now-warm air rises up around the aircraft and can eventually be drawn back into the inlet. The bouyancy effect takes time. It takes about 30 seconds in hover for the air around .the Ha~rier to heat up by 5°C. This 5°C increase in air temperature entenng the Inlet reduces the engine thrust by about 4%.

VTOL AIRCRAFT DESIGN

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If the nozzle arrangement produces a fountain, the recirculation will be greatly increased. This causes additional hot-gas ingestion (HOI) in addition to the bouyancy effect. Unlike the bouyancy effect, the fountain effect takes little time to increase the temperature of the air entering the inlet. The Harrier experiences a 10°C temperature rise due to the fountain effect. This reduces thrust by about S%. The third contributor to recirculation, relative wind, can be due to atmospheric wind or to aircraft forward velocity. Essentially, the relative wind pushes back on the spreading exhaust gases, forcing them up. At some combination of relative wind and exhaust velocity the hot gases will wind up back in the inlet. Hot-gas ingestion is typically limited to speeds below about 50 knots. If the nozzles can rapidly vector from full-aft to a downward angle, a rolling takeoff can be used to minimize HOI problems. The pilot starts the takeoff with the nozzles fully-aft and quickly accelerates to about 50 knots. Then the nozzles are vectored downward and the aircraft takes off.

20.8 VTOL FOOTPRINT The "footprint" of a VTOL aircraft refers to the effect of the exhaust upon the ground. This is largely determined by the dynamic pressure and temperature of the exhaust flow as it strikes and flows along the ground. Even a helicopter cannot operate from a very loose surface such as fine sand or dust. The exhaust of a turbojet VTOL aircraft can be of such high

554

AIRCRAFT DESIGN

pressure and temperature that it can erode a concrete landing pad if the aircraft is hovered in one spot for too long. No exact method exists to determine the acceptable exhaust pressures and temperatures for VTOL operation off of a given surface. Roughly speaking, a turbojet exhaust is marginal for operation off concrete and is too hot and high-pressure for asphalt. The front-fan exhaust of a split-flow turbofan is generally acceptable for concrete, asphalt, and dense sod. However, the core-flow exhaust of the turbofan may be too hot and high-pressure for asphalt and sod. Ejectors and tip-driven fans substantially reduce the exhaust temperature and pressure, allowing operation from regular sod and perhaps hardpacked soil. In general, the nozzles should be as far above the ground as possible. The ground temperatures due to a turbofan will be reduced by about 30070 if the nozzles are five nozzle diameters above the ground. This suggests that a pair of side-mounted elbow nozzles are preferable to a single ventral nozzle because they are higher off the ground and have less diameter for the same total airflow. Although Refs. 88 and 89 give some data on footprint, this author does not know any published report that details the suitability of VTOL operation from a variety of surfaces as a function of exhaust pressure, temperature, and nozzle geometry. Such a test report would be a useful data source for the design of jet VTOL aircraft.

20.9

VTOL CONTROL

The VTOL aircraft in hover and transition must be controlled by some form of thrust modulation. Most VTOL concepts use a reaction control system (RCS), in which high-pressure air is ducted to the wing tips and the nose and/or tail. This high-pressure air can be expelled through valve-controlled nozzles to produce yaw, pitch, and roll control moments. The high-pressure air for the RCS is usually bled off the engine compressor, causing a reduction in thrust. The Harrier loses roughly 10% of its lift thrust due to RCS bleed air. Bleed-air RC~ systems can be light in weight. For the Harrier, the whole s~st7~ only weIghs about 200 lb. However, the RCS ducting occupies a slgmflcant volume in the aircraft. Also, RCS ducting is hot and cannot be placed too near the avionics. If a VTOL concept has three or more lift nozzles placed well away from the C.g., modulation of the lift thrusts can be used for control in vertical flight. For example, if the thrust from the forward nozzle is reduced the nose will pitch down. Vectoring the left-side nozzles forward and the rightside nozzles rearward will cause the nose to yaw to the left. I~ additio~ to three-axis control (roll, pitch, and yaw), a VTOL needs ~er.tIcal-veloclty control ("heave" control). Thi~) is done by varying the hftmg thrust. For an aircraft with fixed nozzle-exit area (such as the Harrier), the lifting thrust is varied by engine throttle setting.

VTOL AIRCRAFT DESIGN

555

An engine with variable nozzle-exit area can change its lifting thrust more rapidly by changing exit area, leaving the throttle setting unchanged. Provision of acceptable heave control generally adds about 5% to the required hover T/W. A multiengine aircraft should remain under control following the loss of an engine. This common requirement is far more difficult for a VTOL aircraft to meet than for a conventional aircraft. For example, if a VTOL aircraft requires two engines to hover, a third engine of the same thrust would be required to assure hover ability after loss of an engine. Not only that but the engines must be arranged so that their combined thrust passes thro~gh the c.g. with all engines running and with anyone engine failed. The more engines a VTOL concept has, the smaller the impact of adding one extra engine for engine-out hover. However, the increased pilot workload for operating multiple engines and the additional maintenance makes this option less attractive. Also, the more engines, the greater the . . probability of an engine failure. Another technique studied for engine-out control mvolves cross-shaftmg the engine fans so that the fans of all the engines can be driven from the cores of the other engines. This minimizes the asymmetric thrust loss from the failure of one engine core. However, the weight impact of the crossshafting mechanisms must be considered. " . Some multi-engine VTOL concepts have been desIgned wIth several Jet engines operating together through some form of augmentation devices. For example, the Ryan XV-5A had two jet engines that were diverted to three tip-driven fans. Either engine could drive all three fans. 20.10 VTOL PROPULSION CONSIDERATIONS Thrust matching has already been discussed as one of the key problems facing VTOL designers. Inlet matching presents a similar problem. For efficient jet-engine operation at zero airspeed, the inlet should look much like a bellmouth as seen on jet-engine test stands. The inlet should have a large inlet area and generous inlet-lip radii. These features cause unacceptable drag levels during high-speed flight. As a compromise, inlets can be sized for cruise operations and provide? with auxiliary doors for VTOL operation. For reasonable low-speed efficiency these auxiliary doors must be very large compared with typical auxiliary doors as seen on a CTOL aircraft. . Another propulsion consideration is the amount of vertical thrust required for vertical flight. As a minimum, the net TIW for vertical flight must obviously exceed 1.0. For acceptable response in heave (vertical acceleration), the net T /W should equal or exceed 1.05. The net thrust available for vertical lift will be reduced by suckdown, hot-gas ingestion, and RCS bleed. Because of these factors: the required T IW for vertical flight will greatly exceed the 1.05 value reqUlred merely to hold the airplane up and provide heave control. . For most types of VTOL aircraft, the overall installed TIW for vertIcal flight ranges between about 1.2 and 1.5, with 1:3 being a typ~cal value. Figure 20.12 shows a typical breakdown of contnbutors to reqUlred TIW.

556

AIRCRAFT DESIGN

VTOL AIRCRAFT DESIGN T/W INCREMENTS

TOTAL REQUIRED T/W =

T/W

1.26-1==========f,

.10

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.08

HGI EFFECT

.03

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1.0 (TINSTALLED

Fig. 20.12

20.11

=

W)

Typical hover T/W breakdown.

WEIGHT EFFECTS OF VTOL

It is difficult to assess statistically the impact of VTOL on aircraft weights us!ng design data from existing aircraft. VTOL designs are so strongly dnven by weight considerations that the designers will push much harder to reduce weight than in a normal CTOL design. .For example, the Harrier was designed so that it requires removing the WIng to remove the engine. This would be considered a fatal design flaw in a CTOL aircraft ?ut is tolerated in the Harrier because of the weight savings compared to the Immense doors that would otherwise be required to remove the engine. Be~ause of such design compromises, the Harrier has an empty-weight f~a~tlOn Wei Wo o~ only 0.48 whereas a statistical approach based upon sImIlar CTOL desIgns would indicate that the Harrier should have an empty-weight fraction of about 0.55. By way of reference, the A-4M, which performs a similar mission, has an empty-weight fraction of 0.56. A VTOL aircraft designed to the same ground rules as a CTOL aircraft will always be heavier in two areas, propulsion and control systems.

557

The propulsion system will be heavier due to the compromises described above for solving the balance andlor thrust-matching problems. The various VTOL propulsion concepts all incorporate some additional features such as vectoring nozzles, extra internal ducting, tilt nacelles, or lift engines. These add weight. Reference 87 compares CTOL and VTOL versions of a carrier-based utility aircraft (similar to the S-3). The CTOL version's propulsion-system weighs 8070 of the takeoff weight. The VTOL version's tilt-nacelle propulsion system weighs 20% of the takeoff weight. Data from Refs. 87-89 indicate that a typical supersonic CTOL fighter design may have a propulsion-system weight about 16-18% of the takeoff weight. An equivalent VTOL design would have a propulsion-system weight about 18-22% of the takeoff weight. The far-greater propulsion-system weight for the cruise-dominated utility aircraft reflects the fact that the fighter concept already requires large engines for supersonic flight. Control-system weights are increased about 50% for most VTOL designs. This is due to the ducting, nozzles, and valves of the typical RCS. However, the total control-system weight is only a small fraction of the takeoff weight (2% for a typical CTOL design) so the impact is slight. For most VTOL designs the landing gear will weigh the same as for a CTOL design. Carrier-based aircraft may show reduced landing-gear weight with VTOL. The landing gear of carrier-based aircraft are substantially heavier than the landing gear of other aircraft because of the extremely high sink-rates during landing and because of the catapult and arresting-hook loads. These can increase the landing-gear weight from about 4% to about 6% of the aircraft takeoff gross weight. A VTOL aircraft designed for carrier operation need not incorporate the heavier landing gear. This represents a weight savings compared to the CTOL carrier-based aircraft. As mentioned, it is difficult to provide an estimate for the total impact of VTOL on WelWO based upon statistics. However, data in Refs. 87-89 indicate that a fighter aircraft will experience an increase in Wei Wo of roughly 4% if designed to the same groundrules as an equivalent CTOL aircraft. Similarly, a transport/utility aircraft will have an increase in WelWO of about 7%. These estimates should be considered to be extremely crude . 20.12 SIZING EFFECTS OF VTOL The final sized weight of a VTOL aircraft will be increased by the emptyweight effects described above. Also, a thrust mismatch between vertical flight and cruise may force the engine to be operated well away from the optimal thrust setting for cruise efficiency. This increases fuel consumption, which increases sized aircraft weight. These factors will clearly increase the sized aircraft takeoff-weight if a VTOL aircraft is flown over the same mission as an equivalent CTOL. In some cases, though, the mission requirements can be reduced for the VTOL aircraft with no real loss in operational effectiveness.

558

AIRCRAFT DESIGN

For example, a Close-Air Support (CAS) aircraft like the A-1O should be

21 CONCEPTUAL DESIGN EXAMPLES

ba~:d as near as. possi.ble t.o the ground troops being supported. VTOL a~lh~y may permIt basmg hterally at the forward lines, thus reducing the

mlSSlOn range requirements while providing better operational effectiveness. The VTOL aircraft can also "loiter" on the ground unlike the CTOL aircraft, which may have a requirement for a one-hou; loiter on a CAS mission. VTOL greatly simplifies instrument landings. Helicopters can "feel their way around" in foggy conditions that ground all CTOL aircraft. A VTOL capability should therefore reduce landing reserves for loiter or diversion to alternate airports. . On the other hand, the fuel burned by a vertical landing can be substantial whereas a CTOL aircaft uses virtually no fuel in landing. A~other f~vorable effect of a VTOL capability comes in the optimization o.f wIng loadmg. For many aircraft the wing loading will be determined by eIthe.r the takeoff or landing requirements. A VTOL capability removes this co~slderation, possibly permitting a smaller wing, which in turn reduces weIght and fuel usage. Taken altogether, these factors indicate that the VTOL aircraft will usually be heavier than an equivalent CTOL design. Based upon the data in the references, an increase in sized takeoff weight of about 10-20070 can be expe~ted for a fighter design. A transport/utility design will typically size to a weIght roughly 30-60% greater than the CTOL design. As before, these are very crude trends, not estimates for any particular design.

21.1

Introduction

This final chapter offers two design examples that illustrate the concepts and methods presented in this book. The design examples are a single-seat aerobatic homebuilt airplane and a lightweight supercruise dog fighter designed to replace the F-16 as the "low" end of a future "high-low" mix of advanced fighters. These examples illustrate the steps and thought processes used in conceptual design, covering the extremes from propeller-powered, fixed-size-engine design to "rubber-engine" supersonic design. The differences and surprising similarities between these extremes of design are shown. Design requirements for these aircraft were assumed based upon data for similar aircraft. These design requirements were then treated as if they were mandated by some customer and used as the starting point for the design effort. The examples are designed and analyzed using the methods presented in the book. Every effort has been made by the author to develop credible, realistic designs and to analyze and optimize them properly. However, no claim is made that these are optimal designs or even good designs or that all calculations are correct! Furthermore, the examples are incomplete in that only the more important analysis areas are presented due to space limitations. Were the author to grade himself in a college design course, these examples would rate at most a "B." The" A" students would conduct far more analysis (structures, roll rate, c.g. envelopes, etc.) and would ultimately redraw the as-optimized aircraft to insure that the analysis assumptions were realistic. 21.2

Single-Seat Aerobatic

This design represents an aircraft that the author hopes to build and fly some day. It would provide fun weekend aerobatic flying for the occasional pilot and would offer better performance than the Great Lakes Biplane but without the touchy handling qualities of the Pitts Special. The design is fairly classical in layout but is based around the more recent techniques for quick fabrication using moldless foam-fiberglass sandwich construction. This permits rapid "garage" fabrication of a one-of-a-kind aircraft. Also, the selected engine is already set up for aerobatic flight, which should minimize the installation effort. One interesting result of the sizing and optimization presented below is that the low-wing loading required for good aerobatic capability has strongly biased the aspect ratio optimization, leading to a lower-than-expected optimal aspect ratio. In fact, without the addition of a maneuver-related performance constraint, the "optimal" aspect ratio approaches zero! (Special note to homebuilders concerning the first example-DON'T BUILD IT!!! This is a first-pass conceptual design only. It would take at least a man-year of design and analysis effort by an experienced designer before this concept could be built and safely flown.)

559

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1 1 .995

1 1 .996

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------A Conceptual Approachl----591

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Conceptual Approach----592

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- - - - - A Conceptual Approach:----593

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21.3 Lightweight Supercruise Fighter The U.S. Air Force currently operates the F-15 and F-16 as a "high-low" mix of dogfighters. The F-15 has greater range, avionics, and weaponry but is too costly to fill the entire inventory requirement. The F-16, with less capability and cost, rounds out the total required dogfighter inventory and also serves in an air-to-ground role. The U.S. Air Force is developing the Advanced Tactical Fighter (A TF) as a replacement for the F-15. The next fighter after ATF is then likely to be a replacement for the F-16, which is almost as old as the F-15. This new fighter would be the "low" end of a "high-low" mix with A TF. This design example presents such an F-16 follow-on design. Design requirements are based upon assumed improvements to published F-16 capabilities, with the addition of a required capability for sustained supersonic cruise ("supercruise") on dry power. Also, relatively short takeoff and landing requirements are imposed. The selected design incorporates one unproven technology, the variable dihedral vertical tail. This patented concept purports to control the rearward shift in aerodynamic center as the aircraft accelerates to supersonic flight by converting from a "V" tail subsonically to upright vertical tails supersonically. This should reduce trim drag and enhance maneuverability. Such a technology study is very common in early conceptual design. As will be seen, the impact of such a technology on aerodynamics, weights, propulsion, etc., is estimated as best as possible, and the aircraft is sized and optimized. The resulting aircraft is then compared to a baseline design that doesn't incorporate the new technology to determine if the new idea should be pursued . (Homebuilders: don't build this one either!)

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AIRCRAFT DESIGN------........

DESIGN ANALYSIS From the design layout, the aircraft was analyzed using the methods presented in thi s book. The author-prepared computer code 'RDS' was used, which automates the number-crunching of these methods. RDS is available from AIAA along with this textbook, and the RDS program disk includes the complete DR-3 sample design files as described below. Lift and Drag Drag analysis was based upon fully turbulent flow over camouflage paint. Areas and geometric information for wings, tails, fuselage, canopy, and boundary-layer diverter were determined from the drawing and input as shown on the next page. Fuselage and canopy equivalent diameters were determined from maximum cross-section areas. D/Q for the missile was input based upon the AIM-9 data in fig. 12.22, and a cannon port D/Q of 0.2 was input as a constant value from zero to Mach 2. Leakage and protuberance drag of six percent was assumed. For wave drag analysis, the maximum total cross section area was estimated at 20.9 square feet, less 3.83 for capture area, or a net of 17.07 square feet. As th i s i s intended to be a supersonic-cruise aircraft and was designed with low wave drag in mind, an Ewd of 2.0 was assumed. Maximum lift was estimated by adjusting the airfoil maximum 1 ift for the effects of the assumed automatic leading- and trailing-edge maneuver flaps. For a 54-series airfoil, Cl-max is about 0.82, and from table 12.1, delta-y is about 1.28. using table 12.2 the lift adjustment for trailing-edge plain flaps is about 0.9, and for leading-edge flaps, about 0.3. With hinge line angles of 10 and 39 degrees, equation 12.21 gives a delta CL-max of about 0.82, so as a first approximation, the wing was analyzed using an adjusted airfoil Cl-max of 1.54. For landing, a historical CL-max value of 1.8 was used. The following pages include a sample of the parasite drag calculation fOr one altitude and velocity, the total parasite drag as a function of speed and altitude, the parameters determined for calculation of drag-due-to-lift factor ('K'), a plot of K versus speed and lift coefficient, and drag polars and LID ratios far the DR-3.

AIRCRAFT TYPE: SUPERSONIC. THIN WING

KEY AERODYNAMIC DATA Max V or KI Max Altitude %Laminar k/W5 (ft I %Leak&Protub Amax-aircrft ;ength-eff Ewd CL-cruise WING

2,000 50000,000 0.000 3,3JO 6,000 17,070 45,200 2. 000 0,210

I Componts Sref-wing 3exp-wing A true A effecLive Lambda=Ct/Cr Sweep-LE tic average De It a Y Q I interfer I CL-design CLrnx-airfoil Drag Fudge

1.000 294,000 215,000 3,500 3,500 0,250 38,000 0,060 1,280 1,000 0,400 1.640 1.000

HORIZONTAL TAIL I Cornponts S-tail Sexp-tail A true A effective Lambda=Ct/Cr Sweep-LE tic average De!La Y Q (interfer I Drag Fudge

1.000 92,000 92,000 4,000 4, 000 0,340 30,000 0,060 1,280 1,000 1.000

FilSELAGE I Componts Swet length diam-effctiv Q (interfer) Upsweep-deg Abase Drag Fudge

------ A Conceptual Approac 620

I NPl.JTS

AER,ODYNAMIC

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........._ - - A Conceptual Approac 621

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c;:.,

0 .,~sou ,= 0.800U

'I

fj

.8rm

!:l.]SHtl 1 . FlFHRI

1.100

1.600

IJ or Mit

[,

11.1500

(""'I

1. t.~!Jl:lO

DWlI, POUIIH .,~t m.T

LIFT COEFFICIENT (CL)

1.200

1 .000

JIIU()[I

+

- 0.551313 1.·m013

: DR3

.. --:~J.•--fl

. 0 3 0 rn o.~.::c Cl II> eT::l I C::Clo.o.~1I> W

.... ::l (,C

0 0

::::J ()

CD

0)-0 CAl _ 01

C

n.>

» -0 -0 -c

0

n.>

()

::s ............ ::s

C')

(flO :E

o~·

»

c+ 3

:!" .......

~

~,

......

eT ::r ~ 11>, II> ~eT C"t:!"

~II> ~

1I>::l o.U

c

eTeT ::r ~ ~ (fl

(fl ~ eT 3 ~ U ::leT 0. cn::l "T1 rn n-

o'g) ...... ::scn

~ 0 (fl ~.:E' 0 - .... ::l 1I>,::l eT en ...... ' g) :( I\):!" :;::leT'OiiE,,:.oll> ~ro::r .... ClUIl> ~eT'''''II>::rIl>::l II> c::OCl ,to

a.

en'

o.eT

g:. ~ ,

to ~

::l ~

'< (fl (fl

C')

...J.

1ll~1I>::l ;ti::l II> X ~ eT ~.

~

Q,

~ - 0. 3· ~ 0. ::l eT~cnC::~ ~~

::l "T13 II>~. U ~o.n ~::leTU ::l' ~ 0. ~ 0 ~

o· a ~

C"t g)

cn~

£,~~~9-.

"T1N-i,::lll>o.X n~::rO"o.rn< II> II> c:: ~ l> eT ::l, U , ::l • ::ro. ::lOIl>2~ " ............ :( g) I O~::l::lll>rnO'~ C::tO(fltO'o to eT ::leT ::r:E~eTlI>~lI>:E

Cl

gll>::::~~~g.;;:

~

eT;ti~~~1I>5::r

Cl c::

::r~. 0

~

eT 0 ::l (fl

-

~. eT Cl ~ ~ II> eT ~ 0 ::l II> ::l ::l ~ ~. ~to U ::lOo'

»

II

'"aCS)

D:l)-

,....

II

~

..... c: t::I

U'1

~

:z '",.........

""-

~

.'"..., ,....

\'","

~

.....

"

c

o

II

..,

..

::r

""' ""'

""' '' ""'' ""' ""' ""''' ""''

""'U'1 ""'""'

~I

""'""' ""'

Z

.....

::t:

""'""' ""'

""'~

..,"-

\

\\\\

~

;:

'"

\

::r ~

x

:rc:

\ ,\

\ \ \ \~ \ \

\

\

\

c

o

..

::r

U'1

""' ""'

N

""'""'

\

.i

t

G)

~

,~

J,

m

:z '".....,.... ,....

"~\

o

(J)

.'"...,

\ \ ~ \ \I .' -

» JJ o JJ »

::r

~

\

..,

i:CJ

'".....c: '"

"-i

~

-\

'"

\~ \~

il

-

..'" "

I

~

....

~ ~ . \ \ \ \

II

-J.

+\, \, '\

::t:

, ' I'

j

I,

:z '"..... '".....E'5

\

""" """

""'

""'""' ""'

II

'"'"

\

U'1

N

U'1

G)

U'1N

~ ~

\

.\\ \

o

....

""'

m

""' '" ! ~I-------BBB. D = SBBBB.

'-"----A Conceptual Approac 640

2.BB

or Mil

..........- - - A Conceptual Approac 641

AIRCRAFT DESIGN

AIRCRAFT DESIGN

Refined Sizing and Performance The RDS computer program sizing and performance of the was then used to calculate refined DR-3 using the methods of the A detailed mission mOdel book. was developed, using several assumptions. Takeoff was assumed to consist of five minutes at Military (dry) POwer. Cruise was performed at MO.9 at ft. , determined from the range optimization plot 45,000 to be approximately best cruise speed and altitude (see below) . Best lOiter speed at sea level was determined to be 176 kts. This was found using the range optimization plot, which that the velocity for best range at sea level was about indicated Mach .35. Comparing equations 17.13 and 17.25 it can be seen that velocity for best loiter is the about one over the fourth root of three, or D.76 times the velocity for best range. This yields Mach .266, or 176 kts. Mission sizing, as detailed below, resulted in a sized takeoff gross weight of 17301.5 lbs. , versus our as-drawn weight of 1648D lbs. Performance calculations were done at a combat weight times takeoff weight, which is the weight at the end of of .87 (beginning cruise of combat) . FOllowing are the inputs and results of the refined sizing and performance calculations, including takeoff, landing, Ps, turn rate, and acceleration. Also included are plots of envelope, specific range, rate of cl imb, Ps, and turn rate. flight

><

~

I I

=

. ~

...

:(1')

~

:0: I I

g

:

I I I I I •• I I I

: rJJ.: ~:

: : ~: : ~: :Z: : H:

I I

:Z: : 0: : H:

: rJJ.: : rJJ.:

642

u

§

::E=

"..

~



»

0-

=-.0

:-~

r-J",

~

tj 70

W ....'"

~Z ~"

0-

-c

0

n.> 0

TY

...

~'"

~

»

"-I 0

(J)

,,3-

=--

,,~

0

JJ 0 JJ

m ~~ G) 1;0\'" ""1 I

~N

-0 -0

»-

x:i 0

l

Z

APPENDIX A

A.I A.2-I A.2-2 A.2-3 A.3 A.4 A.4-I A.4-2 A.4-3

A.S A.S-I

A.S-2 A.S-3 A.S-4 A.S-S

Conversion Tables Standard Atmosphere Compressible Flow Tables Shock Charts Airfoil Data Typical Engine Performances Curves Afterburning Turbofan High-Bypass Turbofan Turboprop Design Requirements and Specifications FAR Applicability Takeoff Specifications Landing Specifications FAR Climb Requirements Special Carrier Suitability Requirements

657

658

AIRCRAFT DESIGN A.I (cont'd) A.I

Multiply British Thermal Unit/ Minute (BTU/min)

Centimeter (cm)

Cubic Foot (ft 3) Foot (ft)

Foot-Pound/Second (ft-Ib/s)

Foot/Second (ft/s)

Gallon (U.S.) (gal)

Gram (g)

Horsepower (hp)

Imperial Gallon

Inch (in.)

Conversion Factors By

Multiply To Obtain

106

3.969 x 1.297 x 10 2.357 x 10- 2 2.987 x 10-2

calories/second foot -pounds/ second horsepower kilogram meters/second

3.281 x 3.938 x 1.000 x 1.000 x

feet inches kilometers meters

10-2 10- 1 10-5 10-2

28.317 7.481

liters gallons

3.048 x 1.200 x 3.048 x 3.048 x 1.894 x

10 10 10-4 10- 1 10-4

centimeters inches kilometers meters miles

7.713 x 3.239 x 1.818 x 1.383 x

10-2 10- 1 10-3 10-1

BTU/min calories/ second horsepower kilogram meters/second

1.097 5.921 x 10- 1 3.048 x 10- 1 6.818 x 10- 1

kilometers/hour knots meters/ second mileS/hour

1.3368 x 10- 1 3.78542 3.785 x 10-3 231 128

cubic feet liters cubic meters cubic inches fluid ounces

3.528 x 10-2 2.205 x 10-3

Ounces pounds

659

Conversion Factors By

To Obtain 103

Kilogram (kg)

1.000 x 6.854 x 10-2 2.205 9.807

grams slugs pounds newtons

Kilogram-Meter/Second (kg-m/s)

3.347 x 10 7.233

BTU/min foot -pounds/ second

Kilometer/Hour (km/h)

9.113xlO- 1 5.396 x 10-1 6.214 x 10- 1

feet/second knots miles/hour

Knot

1.689 1.151 1.852

feet/second miles/hour kilometers/hour

Liter (1)

3.532 x 10-2 2.6417 X W- I 1.000 x 10-3 33.8142

cubic feet gallons cubic meters fluid ounces

Meter (m)

1.000 3.281 3.937 1.000 6.214

x

IOZ

x 10 x 10-3 x 10-4

centimeters feet inches kilometers miles

Meter/Second (m/s)

3.281 3.600 1.943 2.237

feet/second kilometers/hour knots miles/hour

Mile/Hour (mph)

1.467 1.609 0.8684 0.4470

feet/second kilometers/hour knots meters/ second

Nautical Mile (nmi)

6.076 x 103 1.852 x 103 1.15078

feet meters mile

Pound (lb)

4.536 x 16

10Z

grams ounces

4.242 x 10 550 7.604 x 10 745.7 2.774 x IOZ 1.201 4.546

BTU/min foot-pounds/second kilogram meters/second watt cubic inches gallons (U.S.) liters

Slug

1.459 x 104 1.459 x 10 32.174

grams kilograms Lb mass

2.540 8.333 x 10-2 2.540 x 10-2

centimeters feet meters

Statute Mile (mi)

5.280 x 103 1.609 1.760 x 10 3 0.868976

feet kilometers yards nautical mile

(continued)

A.2-1

660

AIRCRAFT DESIGN A.2-1

h (ftlIQ3)

0 I 2 3 4 5

T CR)

p (psf)

P

Jl

(sllft3)

(sllft/s)

.15723 -3 .1611 .1650 .1691 .1732 .1776

41 42 43 44 45

390.0 390.0 390.0 390.0 390.0

3743 -I 3572 3405 3246 3095

.5598 -3 .5337 .5087 .4849 .4623

.2969 -6 .2969 .2969 .2969 .2969

968.1 968.1 968.1 965.1 968.1

.5304 -3 .5564 .5837 .6123 .6423

46 47 48 49 50

390.0 390.0 390.0 390.0 390.0

2950 -I 2812 2681 2556 2436

.4407 -3 .4201 .4005 .3818 .3639

.2969 -6 .2969 .2969 .2969 .2969

968.1 968.1 968.1 968.1 968.1

.6738 -3 .7068 .7415 .7778 .8159

h (ftl I 03)

Characteristics of the standard atmosphere

T CR)

p (psf)

p

(sllft3)

a

Jl

(sl/ftls)

v (ft2/s)

(ftls)

661

Characteristics of the standard atmosphere

a (ft/s)

v (ft 2/s)

518.69 515.1 511.6 508.0 504.4 500.9

2116.2 2041 1963 1897 1828 1761

.23769 -2 .37373 -6 1116.4 .2308 .3717 1I12.6 .2241 .3697 1108.7 .2175 .3677 1104.9 .2111 .3657 1I01.0 .2043 .3637 1097.1

6 7 8 9 10

497.3 493.7 490.2 486.6 483.1

1696 1633 1572 1513 1456

.1987 -2 .1927 .1869 .1811 .1756

.3616 -6 .3596 .3576 .3555 .3534

1093.2 1089.3 1085.3 1081.4 1077.4

.1820 -3 .1866 .1914 .1963 .2013

II 12 13 14 15

479.5 475.9 472.4 468.8 465.2

51 54 56 58 60

390.0 390.0 390.0 390.0 390.0

2214 -I 2012 1829 1662 1510

.3307 -3 .3006 .2732 .2482 .2256

.2969 -6 .2969 .2969 .2969 .2969

968.1 968.1 968.1 968.1 968.1

.8978 -3 .9879 .1087 -2 .1196 .1316

1400 1346 1294 1244 1195

.1701 -2 .1648 .1596 .1546 .1496

.3514 -6 .3493 .3472 .3451 .3430

1073.4 1069.4 1065.4 1061.4 1057.4

.2066 -3 .2120 .2175 .2233 .2293

16 17 18 19 20

461.7 458.1 454.6 451.0 447.4

62 64 66 68 70

390.0 390.0 390.0 390.0 390.0

1373 -I 1243 1134 1031 9367 -2

.2051 -3 .1864 .1694 .1540 .1399

.2969 -6 .2969 .2969 .2969 .2969

968.1 968.1 968.1 968.1 968.1

.1448 -2 .1593 .1753 .1929 .2122

1148 1102 1058 1015 9733 -I

.1448 -2 .1401 .1355 . 13 II .1267

.3409 -6 .3388 .3367 .3346 .3325

1053.3 1049.2 1045.1 1041.0 1036.9

.2354 -3 .2418 .2484 .2553 .2623

21 22 23 24 25

443.9 440.3 436.8 433.2 429.6

72 74 76 78 80

390.0 390.0 390.0 390.0 390.0

8514 -2 7739 7035 6394 5813

.1272 -3 .1156 .1051 .9552 -4 .6683

.2969 -6 .2969 .2969 .2969 .2969

968.1 968.1 968.1 968.1 968.1

.2335 -2 .2568 .2826 .3108 .3420

9333 -I 8946 8572 8212 7863

.1225 -2 .1184 .1144 .1104 .1066

.3303 -6 .3282 .3260 .3238 .3217

1032.8 1028.6 1024.5 1020.3 1016.1

.2697 -3 .2772 .2851 .2932 .3017

(b)

26 27 28 29 30

426.1 422.5 419.0 415.4 411.9

85 90 95 100

390.0 394.3 402.5 410.6 418.8

5193 -2 4533 3629 2888 2309

.7764 -4 .6771 .5253 .4097 .3211

.2969 -6 .2997 .3048 .3099 .3150

968.1 973.4 983.5 993.4 1003.2

.3824 -2 .4426 .5803 .7565 .9809

7527 -I 7203 6890 6588 6297

.1029 -2 .9931 -3 .9580 .9239 .8907

.3195 -6 .3173 .3151 .3129 .3107

1011.9 1007.7 1003.4 999.1 994.8

.3104 -3 .3195 .3289 .3387 .3488

31 32 33 34 35

408.3 404.8 401.2 397.6 394.1

1I0 120 130 140 150

435.1 451.4 467.6 483.9 500.1

1495 -2 9837 -3 6574 4455 3060

.2001 -4 .1270 .8190 -5 .5364 .3564

.3250 -6 .3348 .3444 .3539 .3632

1022.5 1041.5 1060.1 1078.3 1096.3

.1624 -I .2637 .4206 .6598 .1019 +0

6016 -I 5746 5485 5235 4993

.8584 -3 .8270 .7966 .7670 .7382

.3085 -6 .3063 .3040 .3018 .2995

990.5 986.2 981.9 977.5 973.1

.3594 -3 .3703 .3817 .3935 .4058

(c)

160 170

.2973 -6 .2969 .2969 .2969 .2969 .2969

968.7 968.1 968.1 968.1 968.1 968.1

.4185 -3 .4205 .4379 .4594 .4820 .5056

508.8 508.8 508.8 508.8 499.0 473.0 457.0

2515 -3 2125 1479 1218 1027 7047 -4 4754

.2880 -5 .2433 .1693 .1395 .1200 .8589 -6 .6061

.3682 -6 .3682 .3682 .3682 .3626 .3505 .3381

1105.7 1105.7 1105.7 1I05.7 1095.0 1071.7 1047.9

.1278 +0 .1513 .2175 .2640 .3023 .4081 .5580

36 (a)

37 38 39 40

390.5 390.0 390.0 390.0 390.0 390.0

4761 -I 4727 4539 4326 4124 3931

.7103 -3 .7061 .6780 .6463 .6161 .5873

(continued)

(d)

180 190 200 Symbols-

h T

= geo. altitude = temperature

p = p. =

density viscosity

a

=

sound speed

v = kinematic viscosity

p = pressure

Single digit preceded by plus or minus sign indicates power of 10 (i.e., .23769 -2

Altitudes of Temperature Profile Discontinuity(a) 36,152 ft (b) 82,346 ft (c) 155,348 ft (d) 175,344 ft Data from "US Extension of the ICAO Standard Atmosphere," 1958.

=

.0023769)

662

AIRCRAFT DESIGN

A.2-2

663

APPENDIX A

Compressible Flow Tables (NACA TR-1l35j1953) A.2-2

NOTATIONS

Compressible Flow Tables SUBSONIC FLOW

Al or M, local Mach number or 1\fach numbcr upstrcam of a normal shock wave

1!.. !'atio of static presslIl'C to to tal p,'cssme P,

",(=7/5

1'-

AI

P,

T

!!...

'I

PI

p

p,

ratio of static density to total [lensity

o

1.0000 .9999 .9997 .9994 .9989

1.0000 1.0000 .9998 .9996 .9992

1.0000 1.0000 .9999 .9998 .9997

1.0000 1.0000 .9998 .9995 .9992

.09

. ~983 .9975 .9966 .9955 .9944

.9988 .9982 .9976 .9968 .9960

.9995 . ~993 .9990 .9987 .9984

.10 .11 .12 .13 .14

.9930 .9916 .9900 .9883 .9864

.9950 .9940 .9928 .9916 .9903

.15 .16 .17 .18 .19

.9844 .9823 .9800 .9776 .9751

.20 .21 .22 .23 .24

.1747 .2514 .3418 .4460 .5638

II. 5914 9.6659 8.2915 7.2616 6.4613

.05476 .06570 .07664 .08758 .09851

.9980 .9Y76 .9971 .9966 .9961

.9950 .9939 .9928 .9915 .9902

.9979 -2 .1169 -] .1353 -]

5.8218 5.2992 4.8643 4.4969 4.1824

.10944 .12035 .13126 .14217 .15306

.9888 .9873 .9857 .9840 .9822

.9955 .9949 .9943 .9936 .9928

.9887 .9871 .9854 .9837 .9818

.2217 -\ .2464 -I

3.9103 3.6727 3.4635 3.2779 3. 1123

.16395 .17482 .18569 .19654 .20739

.9725 .9697 .9668 .9638 .9607

.9803 .9783 .9762 .9740 .9718

.9921 .9913 .9904 .9895 .9886

.9798 .9777 .9755 .9732 .9708

.3276 -\ .3569 -I .3874 -I

'I

2.9635 2.8293 2.7076 2.5968 2.4956

.21822 .22904 .21984 .25063 .26141

.25 .26 .27 .28 .29

.9575 .9541 .9506 .9470 .9433

.9694 .9670 .9645 .9619 .9592

.9877 .9867 .9856 .9846 .9835

.9682 .9656 .9629 .9600 .9570

.4189 -1 .4515 -I .4851 -\ .5197 -I .5553 -I

I

2.4027 2.3173 2.2385 2.1656 2.0979

.27217 .28291 .29364 .30435 .31504

.30 .31 .32 .33 .34

.9395 .9355 .9315 .9274 .9231

.9564 .9535 .9506 .9476 .9445

.9823 .9811 .9799 .9787 .9774

.9539 .9507 .9474 .9440 ,9404

.5919 -I .6293 -\ .6677 -\ .7069 -] .7470 -I

2.0351 I. 9765 1. 9219 I. 8707 I. 8229

.32572 .33637 .34701 .35762 .36822

.35 .36 .37 .38 .39

.9188 .9143 .9098 .9052 .9004

.9413 .9380 .9347 .9313 .9278

.9761 .9747 .9733 .9719 .9705

.9367 .9330 .9290 .9250 .9208

.7879 .8295 .8719 .9149 .9587

I. 7780

I. 7358 I. 6961 I. 6587

1.6234

.37879 .38935 .39988 .41039 .42087

.40 .41 .42 .43 .44

.8956 .8907 .8857 .8807 .8755

.9243 .9207 .9170 .9132 .9094

.9690 .9675 .9659 .9643 .9627

.9165 ,9121 ,9075 .9028 .8980

.1003 .1048 .1094 ,1140 .1187

I. 5901 I. 5587 1.5289 1.5007 I. 4740

.43133 .44177 .45218 .46257 .47293

.45 .46 .47 .48 .49

.8703 .8650 .8596 .8541 .8486

.9055 .9016 .8976 .8935 .8894

.9611 .9594 .9577 .9560 .9542

.8930 .8879 .8827 .8773

.1234 .1281 .1329 .1378 .1426

I. 4487 1.4246 I. 4018 I. 3801 I. 3595

.48326 .49357 .5C385 .51410 .52433

{3

-/Jw-l

.05 .06 .07

A*

v v

ratio of dynamic pressure,

~ p V 2 , to total pressure

ratio of local 'cross-sectional area of an isentropic stream tube to cross-sectional area at the point where M=l ratio of local speed to speed of sound at the point where M=l Prandtl-::\leyer angle (angle through which a SUpel'SOlliC stream IS tUl'lled to expand from Jl = 1 to M>l), deg Mach angle, sin- 1 ~, deg 1\1 aelJ nil mlip!, downstream of a normal sIwek wa v,, static pressure ratio aeross a normal shock wave

P2

static density ratio across a normal shock waVe static temperature ratio across a normal shock wave total pressure ratio across a normal shock wave ratio of static pressure upstre!tm of a normal shock wave to total pressure downstream

.01095 .02191 .03286 .04381

.9987 .9982 .9975 .9968 .9959

ratio of static tempm'atlll'c to 1.olal temperat.llre

p, A

o

57.8738 28.9421 19.300,> 14.4815

T,

!L

o .7000 -. .2799 -, .6296 -, .1119 -2

.01 .02 .03 .04

T

A

A.

.01>

i

.8717

.6951 .&3\19

-2 -2 -2 -2 -2 -2 -2

.1550 -\ .1760 -\ .19&3

I

-1

:~: ~:

-\ -\ -\ -\ -]

I

!

665 AIRCRAFT DESIGN

664

A.2-2

Compressible Flow Tables

'" 8888g; OoOOO'l

----

SlJBSONIC FLOW

2t-;:3~~

"1=7/5

:=;8000 ""';-i-i.....i-i

:_. "\f_I--*~II, __;_'__ I I

,

0.50 ••>1 .52 .53 .54

I

o. 8852

_._P_q'__

f3 O. 8660 .8602 .8542 .8480 .8417

O. 1475 . 1525 .1574 .1624 .1674

1. 3398 1. 3212 1. 3034 1. 2865 1. 2703

0.53452 .54469 .55483 .56493 .57501

.86.34 .8589 .8.>44 . 8498 .8451

. 9430 .9410 .9390 . 9370 .9349

. 8352 .8285 .8216 . 8146 .8074

. 1724 . 1774 .1825 . 1875 .1925

1. 2550 1. 2403 1. 226.3 1. 2130 1. 2003

. 58506 .59507 .60505 . 61501 .62492

.840.> .8357 .8310 .8262 .8213

.9328 .9307 .9286 .9265 .9243

.8000 .7924 . 7846 . 7766 .7684

.1976 .2026 .2076 .2127 .2177

1.1882.6.3481 1.1767.644611 1. 1657 .65448 1. 1552 .66427 1. 1452 .67402

.7528 .7465 .7401 .7338 .7274

.8164 .8115 .8066 .8016 .7966

.9221 .9199 .9176 .9153 .9131

.7599 .7513 .7424 .7332 .7238

.2227 .2276 .2326 .2375 .2424

1. 1356

1. 1265 1.1179 1. 1097 1. 1018

.7209 .7145 .7080 .7016 .6951

.7916 .7865 .7814 .7763 .7712

.9107 .9084 .9061 .9037 .9013

.7141 · 7042 · 6940 .6834 · 6726

.2473 .2521 . 2569 .2617 . 2664

1.0944.73179 1. 0873 .74129 1. 0806 . 75076 1. 0742 .76019 1. 0681 . 76958

.7660 .7609 .7557 .7505 .7452

.8989 .8964 .8940 .8915 .8890

.6614 .6499

.2711 .2758

:~~~~ :~m

1.0624.77894 1. 0570 . 78825

.79

.6886 .6821 .6756 .6691 .6625

.6131

.2894

I:: gm

:~~~~~

.80 .81 .82 . 83 .04

.6560 .6495 .6430 .6365 .11.300

.7400 .7347 .7295 .7242 .7189

.8865 .8840 .8815 .8789 .876.3

.6000 .5864 .5724 .5578 .5426

.2939 · 2983 · 3027 · 3069 .3112

1. 0382

1. 0342 1. 0305 1. 0270 1. 0237

.82514 .83426 . 84335 . 85239 .86140

I

.55 .56 .57 .58 .59

.8142 .8082 .8022 .7962 .7901

:i:I .62

I I

.53 .64

!

.7840 .7778 .7716 .7654 .7.>91

:

~~

1

I !

.67 .68 .69 .70 .71

.72 .73 .74 .75

.76 .77 .78

I

I

: I,

, I I

1

1

1. 0425

.....

0t-C"':)O'll(;)N ___ _

OlOO_N

,....;.....;,....;....:.....;

.7136 .7083 .7030 .6977 .6924

.8737 .8711 .8685 .86.>9 .86.32

.5268 .5103 .4931 .4750 .4560

.3153 .3195 .3235 .3275 .3314

1.0207 1.0179 1. 0153 1. 0129 1.0108

.87037 .87929 .88818 .89703 .90583

.90 .91 .92 .9:l .94

.5913 .5849 .5785 .5721 .5658

.6870 .6817 .6764 .6711 .6658

.8606 .8579 .8552 .8525 .8498

.4359 .4146 .3919 .3676 .3412

.3352 .3390 .3427 .3464 .3500

1. 0089

1. 0071 1. 0056 1. 0043 1. 0031

.91460 .92332 .93201 .94065 .94925

.95 .96 .97 .98 .99

.5595 .5532 .5469 .5407 .5345

.6604 .6551 .6498 .6445 .6.392

.8471 .8444 .8416 .8389 .8361

.3122 .2800 .2431 .1990 .1411

.3534 .3569 .3602 . 36.35 .3667

1. 0022 1. 0014 1. 0008 1. 0003 1. 0001

. 9.5781 .96633 .97481 . 98325 .99165

1. 00

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1. 0000

1. 00000

I

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,....;,....;,....;,....;,....;

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.6235 .6170 .6106 .6041 .5977

· 3698

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.68374 .69342 .70307 . 71268 .72225

.85 .86 .87 .88 .89

I I

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O. 9524 .9506 .9487 .9468 .9449

1

c:~~o.c::: ......, ....... -'"""--

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O. 8430 .8374 .8317 .8259 .8201

f_,_

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~~~~~ ~ r_.. r- r_..ao

M~MMM

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----- :::~~ ~

o

V

V V

v

V

...

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l.r

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i.-'

v

0.

V

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v

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28

V

V

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3.2

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135

g

U

r--.

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125

70

k'

t-

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nL..

I-

1"-

I-'"

.....

4

y

8

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16 De/lectlon ongle, Il, delJre~s

20

24

Variation of shock-wave angle with flow-deflection angle for various upstream Mach numbers.

28

Perfect gas,

32 "/=

%.

675

APPENDIX A

A.2-3

Shock Charts

EQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOW

90

so

70

M =2.2

2.6

2.4

2.S

3.0 3.2 3.4 3.

4.0

+'38

4.55

-

/20

8 , 6

10

00

60

I 0

v

~

'" c

.. a

:>

a

3: I

~40 o

.c (()

\

30

,",,' .o .. M,

f;£

~ """',

-

~

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e

Slreamline~,

/

8

.-

20

10

o

26

30

34

38 42 46 Delleclion angle, 8, degrees Concluded

50

54

58

675

APPENDIX A

A.2-3

Shock Charts

EQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOW 90

~~ I-

8 o~

~

t---

r-.. h. l::>.

I'-

70

,-20

8 , M -2.2

2.8

2.6

2.4

3.0

7

3.2 3.4 3.E

'\4.0

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4.5 5

u.- •

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6

IX)

f..

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60

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1.8

0

a. ~ ..c

0

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o

:: .90 ::>

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1.6

~

» ""C

2. ~1 1. 25 3. d 2.5 R.OI 5.0 0.0f) 7..'" f). 83 10 7.\)7 IS 8.70 20 9.17:!5 9.38 30

-2. O~ I -2.80, -3.84 I =r -4.47 -4.90 §: -:'.42 -5. !iii 0 QO -!i.70 ;-

\1.2.')

-5.2!i -4. (il -:l. \10 -3.0fi -2.15 - I. 17 -.68 (-.16)

40

70 80 90 \Iii 100

8.57.'10 7.50 flO 6.10 70 4.41 80 2. 4.') 90 l. 34 95 (.16) 100

100

100

no

L. E. radills: 0.89

L. E. radius: 0.40 - - - - - -

Perfect ga~J

NACA 0009

NACA 0006

,0

I-~

...>

-.'). (i2

""C

m Z 0

X

»

0

L. E. radius: 2.48 Slope of radius through L. E.: 0.10 0) (Xl

""..J

0) (X) (Xl

NACA 23015

NACA 4415

[Stations and ordinatrs giv(,JI in Jlf'rcf'Tlt of airfoil chord]

[Stations and ordinatrs given in percent of airfoil chord] -------

Lowpr surface

Upper surface

3. 07 I 4.171 .'i. 74 6. 91 7.84 9.27 10.25 1IJ. \)2 11.2,'> 11. 2.'i ID . .'i:1 g. :10 7. (53 ,,>. 5.'5 :J.08

1~

1., 20 2.'i

:m

40 ,'>0 flO 70 HO 90 95 100 100

I

I. ()7

C.16)

1. 25 2.5 ~. 0 I • •'>

- 1 79 1-2.48

-?

I -

10

II

l5 2D 2,'> 30 40 .'ill

60 70 SO 90 95 100 100

~7

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Lower surface ---------.---

0

I

I I ,

l. 2.'5

3. :14 4.44

2. 5 5 0 7 5 10 15 20 25 :10 40 50 50 70 80

~.89

6.90 7.64 8. 52 8.92 9.08 9.0.'i 8. 59 7.74 6.61 5. 25 3. 73 2.04 1. 12 C.16)

no

95 100 100

0 1. 25 2. 5 5.0 7.5 10 15 20 25 30 40 50 60 70 80 90 95 100 100

0 -1. 54 -2.25 -3.04 -3.61 -4.09 -4.84 -5.41 -5.78 -5.96 -5.92 -5.50 -4.81 -3.91 -2.83 -1. 59 -.90 (-.W) 0

> .... > :;.

...

~ 0QO S-

» :IJ

()

:IJ

»

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-I 0

m

en

G5

z

L. E. radins: 2.48 Slope of radius through L. E.: 0.305

NACA 65(21 6)-41 5 a c -O,5

NACA 64-006

[Stations and ordinates given in percent of airfoil chord]

Station

-- -.----

-------- ------ ---

L. E. radius: 2.48 Slope of radius through L. E.: 0,20

Upper surface

Ordinate Statio'} I Ordinate

Station

s,:~on. OCd"""1 "':"'' -'1 ~',:"n"", 1. 25 2 . .'i ~. D ,.5

Upper surface

----------

Lower surface Ordinate

Ordinate Station

[Stations and ordinates given in percent of airfoil chord] Upper surface Station

LowE'r surface

Ordinate Station

Ordinate

--0 .50 .75 1. 25 2,5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0 .494 .596 .754 1. 024 1. 405 1. 692 1. 928 2.298 2.572 2.772 2,907 2.981 2.995 2.919 2.775 2.575 2.331 2.050 1. 740 1. 412 1.072 .737 .423 .157 0

0

0

-.494 -.596 -.754 -1. 024 -1. 405 -1. 692 -1. 928 -2.298 -2,572 -2.772 -2.907 -2,981 -2.995 -2.919 -2,775 -2.575 -2.331 -2.050 -1. 740 -1. 412 -1. 072 -.737 -.423 -.157 0

.50 .75 1. 25 2,5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 I

L, E. radius: 0,256

I

0 .244 .469 .930 2.121 4. 564 7.044 9. 540 14.56l 19. 608 24.669 29.742 34.825 30.916 45.019 50. 153 55.263 60.305 65.308 70,281 75. 237 80.180 85.117 gO. Ofi2 95.020 100.000

0 1. 236

1. 498 1. 947 2.837 4.175 5. 208 6.073 7. 465 8.518 9.315 9.900 10.279 1O.4fi7 10.438 10. 131 9.512 8. 645 7.575 6.373 5.152 3.890 2.639 1. 533 , fiOG 0

0 .756 1. 031 1. 570 2.879 5.436 7.956 10.460 15.439 20.392 25.331 30.258 35. 175 40.084 44.981 49,847 54. 737 59.695 64.692 69.719 74.763 79.820 84.883 89.938 94.980 100.000

0 -.960 -1. 110 -1. 359 -1. 801 -2.411 -2.832 -3.169 -3.673 -4.022 -4.257 -4,428 -4.507 -4. 52::! -4.446 -4.251 -3.940 -3.521 -2.995 -2.409 -1. 848 -I. 278 -.723 -.305 -.030 0

~

~

....

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0

= 0

QO

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» "'C "'C

m

Z

0

X

»

--

L. E. radiu~: 1.4Y8 Slope of radius through L. E,: 0,233

0) (Xl



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rovJh'\e~5

d~flected

60"

Standard rou9hness

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I I I I a.. c . pOSition

R -r--I"rr-'~~ vic 030'/0' 0.24/:1-0.0/4 060 .246 -.0/3

-8

deg

.,.

I I

I\) (II

a

.B

-4 .4 Section 11ft coeffiCient, c,

/.6

12

Aerodynamic characteristics or tbe N ACA :MIll airfoil section, 24·1ncb chord.

-TtlIII\

:1.IS f-H

.

32

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e9 z.

,

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