Theory of helicopter flight

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1 A09

2 W eb content

3 Giovanni Di Giorgio Theory of helicopter flight Aerodynamics, flight mechanics

4 Aracne editrice Copyright MMXVIII Gioacchino Onorati editore S.r.l. unipersonale via Vittorio Veneto, Canterano (Rome) (06) isbn No part of this book may be reproduced by print, photoprint, microfilm, microfiche, or any other means, without publisher s authorization. I st edition: April 2018

5 To my father Giuseppe and my mother Wilma

6

7 Contents Preface 13 Units 15 Notation 17 Abbreviations 23 Chapter Helicopter configurations The helicopter and the vertical flight Helicopter configurations The rotor and the flight controls Fundamental types of rotor The flight controls and the swashplate mechanism Chapter 2 Rotor aerodynamics, hovering and vertical flight 2.1. Introduction 2.2. Momentum Theory Vertical climb Hovering flight Vertical descent Curves of induced velocity in vertical flight 2.3. Blade Element Theory Rotor thrust and torque, power required Linear twist of rotor blade Non-uniform induced velocity Rotor blade, root and tip losses Figure of merit Procedure for approximate and preliminary calculation of the aerodynamic parameters, blade loads, rotor power required 2.4. The ground effect 2.5. Introduction to Vortex Theory Dynamics of ideal fluid Fundamental relationships applied to the rotor Kutta-Joukowsky s theorem application

8 27 8 Contents Velocities induced by vortices, Biot-Savart s Law Modelling rotor in hover and approach to calculation Interference phenomenon due to blade tip vortex Prescribed wake, Landgrebe s model in hovering flight Chapter 3 Rotor dynamics 3.1. Introduction Fundamental axes and planes The flapping motion of the blade Flapping hinge offset and control moments The rotor in forward flight and the blade flapping The lagging motion of the blade The cyclic feathering Coupling of fundamental motions of the rotor blade Calculation of centrifugal force along the blade 106 Chapter 4 Rotor aerodynamics, forward flight 4.1. Introduction Momentum Theory Blade Element Theory Parameters for determination of blade angle of 113 attack Blade element and local incidence Aerodynamic forces acting on the rotor, closed form equations Calculation of the thrust Rotor coning and flapping coefficients Calculation of the drag Calculation of the torque Reverse flow region Forces and parameters related to tip path plane and to 139 hub plane Equations referred to the tip path plane Equations referred to the hub plane Helicopter in trim and rotor aerodynamics Corrections of results of Blade Element Theory Blade element theory limitations Stall and compressibility phenomena Swept blade tip and local Mach number 155

9 Contents Rotor wake models in forward flight Computational aerodynamics, advanced methodologies, multidisciplinary approach Chapter Chapter Helicopter trim analysis Introduction Systems of axes General equations of motion of helicopter Helicopter trim conditions The general trim analysis The rotor-fuselage system and the torque reaction Simplified development of equilibrium (trim) Trim equations in forward flight The expression for power in forward level flight Approximate and quick estimation of longitudinal equilibrium General trim solution Autorotation Autorotation of a rotor Aerodynamics of autorotation Final phase of an autorotation Limitations in autorotation and Height-Velocity Diagram Final notes Helicopter flight performance Introduction Total power required Standard atmosphere The engine and the power available The operating condition of the main rotor Configuration of free shaft turbine engine Rotor/transmission/engine system Performance of installed engine and power ratings 6.5. Hover performance Power required PMR and Ptr in hovering flight Vertical drag of the helicopter Maximum hover ceiling 6.6. Performance in vertical climb 6.7. Performance in forward level flight Power required PMR and Ptr

10 10 Contents The parasitic drag D f in forward level flight The total power required in level flight Maximum speed in level flight Maximum endurance and maximum range Power increments due to stall and compressibility 6.8. Forward climb and descent performance Power required P MR in forward climb Rates and angles of climb, ceiling altitude Power required P MR in forward descent Autorotative performance Introduction to mission analysis Take-off and landing weight An approach to helicopter mission analysis 238 Chapter 7 Stability and control, introduction to helicopter flight dynamics 7.1. Introduction The single-degree of freedom dynamic system Helicopter static stability and dynamic stability Helicopter static stability Stability following forward speed perturbation Stability following vertical speed or incidence perturbation Stability following yawing perturbation Helicopter dynamic stability Small disturbance theory Stability derivatives Force perturbation expressions and stability 259 derivatives Moment perturbation expressions and stability 260 derivatives Notes on the methodology of small perturbations Dynamic stability in hovering flight Longitudinal dynamic stability in hovering flight Equations of motion, state variable form Stability derivatives calculation, M q and M u in 267 hover Approximate calculation of longitudinal modes 268 in hovering flight for a medium helicopter The characteristic roots on complex plane Lateral-directional dynamic stability in hovering 270

11 Contents 11 flight 7.7. Dynamic stability in forward flight Longitudinal dynamic stability in forward flight Approximate calculation of longitudinal modes 276 in forward flight for a medium helicopter Lateral-directional dynamic stability in forward 278 flight 7.8. Helicopter control Stability, control and flying qualities Longitudinal control in hovering flight; one 283 degree of freedom approach Lateral-directional control in hovering flight; 284 one degree of freedom approach Chapter 8 Manoeuvres in horizontal and in vertical planes 8.1. Introduction Steady turn Notes on turn manoeuvres Gyroscopic moments in turn Power required in steady turn Symmetrical pull-up 290 Chapter 9 Coaxial rotor and tandem rotor helicopter 9.1. Introduction Coaxial rotor helicopter Application of Momentum Theory to the 293 hovering flight General characteristics of the helicopter Helicopter equilibrium about the body Z-axis Tandem rotor helicopters General description and definitions Application of Momentum Theory and of 300 Blade Element Theory to the hovering flight Application of Momentum Theory to the level 303 forward flight Experimental data Condition of longitudinal equilibrium of the 305 helicopter Notes on stability Forward speed disturbance Stick-fixed dynamic stability in hovering flight 309

12 12 Contents Appendix A Definition of non-dimensional coefficients for the rotor 311 Appendix B International Standard Atmosphere, ISA 313 Appendix C Review of Laplace transform 315 Appendix D Orientation of the aircraft 317 Glossary 319 References 325 List of illustrations 331 Index 337

13 Preface This book provides an introduction to helicopters through the fundamental theories and methods of rotor aerodynamics and flight mechanics. The arguments have been structured in order to provide the reader with the physical aspects of problems, the basic mathematical tools involved, the presentation of theories and methods with solved numerical examples or ready to be implemented on the computer. Therefore, the understanding of both the rotary-wing principles of flight and the approximate magnitude of parameters and variables involved is achieved through a clear and step by step practical presentation. After Chapter 1, that treats the main helicopter configurations, Chapters 2, 3 and 4 review basic rotor aerodynamics applied to helicopters. They treat the momentum and blade element theories, with an introduction to the fundamentals of vortex theory and the elements of rotor dynamics. The developed methods are applied in the subsequent chapters to generate data for examples and to support the arguments. Chapters 5, 6, and 8 present the conditions of helicopter trim and manoeuvres and the flight performance prediction and evaluation. Chapter 7 develops the fundamental problems of helicopter stability and control by means of the mathematical tools provided by the modern control theory. Chapter 9 completes the treatment of theory of flight with specific elements for tandem and coaxial rotor helicopter configurations. Therefore, this book may be used as a reference or a complementary textbook for students in aerospace engineering, and the material provides a starting point to prepare a more in depth analysis useful for both practicing engineers and professionals in helicopter technology. This volume is my English translation with the addition of new arguments of my book Teoria del volo dell elicottero in Italian, published in 2007 and 2009 in Italy by Aracne Editrice. During my translation, I included updates that have occurred over the last years. The Italian book has been used by numerous colleagues and professionals from whom I received positive feedback and appreciation. In my professional experience I have verified the complexities of a rotary-wing aircraft since the early approach to the problems of vertical flight. Therefore, writing an introduction to this subject is a challenge. 13

14 14 Preface Moreover, this book takes into account the multidisciplinary approach required by rotorcraft. Finally, I hope that the same enthusiasm, which has accompanied me from the beginning of my eighteen year career in rotarywing, will be transferred to the reader through the pages of this volume. I would like to thank Professor Gian Battista Garito and Ingegner Giovanni Fittipaldi for the significant discussions about the fundamentals of rotorcraft; moreover, since the first edition of the Italian book, they have given me helpful comments and many suggestions. I am very grateful to Dottor Gianluca Grimaldi and to Ingegner Andrea Bianchi of Leonardo Helicopters Division (AgustaWestland, when I started to write the book) in Cascina Costa; they have always appreciated my efforts, providing me useful comments. I would also like to thank Ingegner Massimo Longo of Leonardo Helicopters Division in Cascina Costa; he has allowed me to appreciate special topics in the field of helicopter flight test. I am also very grateful to Professor Carlo de Nicola of University of Naples Federico II for stimulating many constructive discussions, from the aerodynamics to the aircraft pilot s standpoint, and thanks are due to Professor Renato Tognaccini; over the last years, they have invited me to give an interesting series of conferences on helicopter flight performance in Naples. I want to express my sincere gratitude to Professor Francesco Marulo of University of Naples Federico II for the interesting discussions about rotarywing and aerospace engineering. I would like to thank Dottor Enrico Gustapane and all my colleagues of Leonardo Helicopters Division in Frosinone plant. Giovanni Di Giorgio Roma, February 25, 2018

15 Units International System (SI) Units are used in this text, unless otherwise indicated. The following tables support the conversion to the British System, limited to the arguments and purposes of the present book: Primary quantities Quantity Units SI Brit. S. Conversion Mass kg slug 1 slug = kg Length m ft 1 ft = m Time s s - Temperature K R 1 ( R) = [1/(1.8)] ( K) Temp( K) = temp( C) Supplementary units Quantity Units Conversion SI Brit. S. Angle (plane) rad rad - Derived quantities Quantity Units SI Brit. S. Conversion Velocity m/s ft/s 1 ft/s = m/s Angular Velocity rad/s rad/s - Acceleration m/s 2 ft/s 2 1 ft/s 2 = m/s 2 15

16 16 Units Quantity Acceleration of gravity Units SI Brit. S. m/s 2 ft/s 2 Conversion g = m/s 2 = ft/s 2 Air density kg/m 3 slug/ft 3 1 slug/ft 3 = kg/m 3 Force N lb 1 lb = N Pressure Pa (1 Pa = 1 N/m 2 ) lb/ft 2 1 lb/ft 2 = N/m 2 Power W lb ft/s 1 lb ft/s = W = (1 hp = 550 lb ft/s) (1/550) hp Multiples Quantity Velocity Units SI Brit. S. m/min metre per minute ft/min foot per minute Conversion 1 ft/min = m/min Additional Unit Quantity Unit Conversion Angular Velocity Velocity rpm (revolution per minute) kn (international knot) = one nautical mile per hour 1 rpm = (2π/60) rad/s - (one international nautical mile) = 1852 m = ft Angle (plane) (degree) 1 = (π/180) rad

17 Notation Symbol Units (SI) a lift curve slope of blade section rad -1 a coning angle, main rotor rad 0 a 1 coefficient of term (-cosψ) into expression of the flapping angle β, relative to the no-feathering plane; longitudinal flapping coefficient A 2 main rotor disc area A R rad m 2 A lateral cyclic pitch rad 1 2 A tr tail rotor disc area Atr R tr m 2 b number of blades, main rotor - b 1 coefficient of term (-sinψ) into expression of the flapping angle β, relative to the no-feathering plane; lateral flapping coefficient rad b number of blades, tail rotor - tr B tip loss factor - B longitudinal cyclic pitch rad 1 c blade section chord, main rotor m c blade section chord, tail rotor m tr C section drag coefficient - d 17

18 18 Notation C section lift coefficient - l C main rotor power coefficient - P C main rotor torque coefficient - Q C main rotor thrust coefficient - T D parasitic drag of helicopter N f D.L. disc loading N/m 2 f equivalent flat plate drag area m 2 G gravitational acceleration m/s 2 G helicopter centre of gravity; origin of the body-axis system - H density altitude m d H pressure altitude m p I f mass moment of inertia of blade about flapping hinge kg/m 2 k induced power factor, main rotor - k induced power factor, tail rotor - tr k climb efficiency factor - p K G constant into Glauert s second formula of the induced - velocity K effect - K term of 3 l tail rotor moment arm m tr M Mach number - M disturbance term about the Y-axis for aerodynamic moments N m M aerodynamic moment about the flapping hinge N m A

19 Notation 19 M drag divergence Mach number - d M mass of helicopter M W g heli kg heli G / n load factor - O origin of the Earth-axis system - p pressure of air N/m 2 p pressure of air at sea level, ISA conditions N/m 2 0 P main rotor power required W MR P tail rotor power required W tr Q main rotor torque N m r radial distance of blade element from axis of rotation 0 r R m r e effective blade radius m R main rotor radius m R tail rotor radius m tr T main rotor thrust N T temperature of air K T temperature of air at sea level, ISA conditions K 0 T tail rotor thrust N tr v induced velocity at rotor m/s i v induced velocity at rotor in hover m/s ih V true airspeed of helicopter along the flight path; velocity of the free airstream V climb velocity m/s c V descent velocity m/s d m/s

20 20 Notation VT VT R, or main rotor tip speed in hovering flight VTtr VTtr tr Rtr,or tail rotor tip speed in hovering flight m/s x x r R, ratio of blade element radius to the rotor blade radius - X longitudinal axis of the body-axis system - XT axis of the Earth axes system - Y axis of the body axes system - YT axis of the Earth axes system - WG gross weight of the helicopter N Z axis of the body axes system - ZT axis of the Earth axes system - Incidence of blade section (measured from line of zero lift) rad nf incidence with respect to the no-feathering plane rad S incidence with respect to the rotor hub plane rad TPP incidence with respect to the rotor tip path plane rad blade flapping angle, with respect to the no-feathering plane rad S blade flapping angle, with respect to the hub plane rad blade Lock number acr4 I f r climb angle rad inflow angle at blade element rad circulation m/s - m2/s

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