Di Giorgio Giovanni__Chapter 7__Stability and Control_Introduction to helicopter flight dynamics_Theory of Helicopter Flight

A09 Web content Giovanni Di Giorgio Theory of helicopter flight Aerodynamics, flight mechanics Aracne editrice www.aracneeditrice.it info@aracneeditrice.it Copyright © MMXVIII Gioacchino Onorati editore S.r.l. — unipersonale www.gioacchinoonoratieditore.it info@gioacchinoonoratieditore.it via Vittorio Veneto, 20 00020 Canterano (Rome) (06) 45551463 isbn 978 – 88 – 255 – 1442 – 1 No part of this book may be reproduced by print, photoprint, microfilm, microfiche, or any other means, without publisher’s authorization. Ist edition: April 2018 To my father Giuseppe and my mother Wilma Contents Preface 13 Units 15 Notation 17 Abbreviations 23 Chapter 1 1.1. 1.2. 1.3. Helicopter configurations The helicopter and the vertical flight Helicopter configurations The rotor and the flight controls 1.3.1. Fundamental types of rotor 1.3.2. The flight controls and the swashplate mechanism Chapter 2 Rotor aerodynamics, hovering and vertical flight 2.1. Introduction 2.2. Momentum Theory 2.2.1. Vertical climb 2.2.2. Hovering flight 2.2.3. Vertical descent 2.2.4. Curves of induced velocity in vertical flight 2.3. Blade Element Theory 2.3.1. Rotor thrust and torque, power required 2.3.2. Linear twist of rotor blade 2.3.3. Non-uniform induced velocity 2.3.4. Rotor blade, root and tip losses 2.3.5. Figure of merit 2.3.6. 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 2.5.1. Dynamics of ideal fluid 2.5.2. Fundamental relationships applied to the rotor 2.5.2.1. Kutta-Joukowsky’s theorem application 25 26 29 29 32 39 39 40 43 46 48 49 52 57 58 61 62 63 69 71 72 76 77 7 8 Contents 78 2.5.2.2. Velocities induced by vortices, Biot-Savart’s Law 2.5.3. Modelling rotor in hover and approach to calculation 2.5.4. Interference phenomenon due to blade tip vortex 2.5.5. Prescribed wake, Landgrebe’s model in hovering flight Chapter 3 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. Rotor dynamics 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 Chapter 4 4.1. 4.2. 4.3. Rotor aerodynamics, forward flight Introduction Momentum Theory Blade Element Theory 4.3.1. Parameters for determination of blade angle of attack 4.3.2. Blade element and local incidence 4.3.3. Aerodynamic forces acting on the rotor, closed form equations 4.3.3.1. Calculation of the thrust 4.3.3.2. Rotor coning and flapping coefficients 4.3.3.3. Calculation of the drag 4.3.3.4. Calculation of the torque Reverse flow region Forces and parameters related to tip path plane and to hub plane 4.5.1. Equations referred to the tip path plane 4.5.2. 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 4.9.1. Swept blade tip and local Mach number 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 81 82 83 87 87 90 93 98 99 101 103 106 109 109 113 113 18 27 118 120 123 127 131 135 138 139 139 141 144 148 149 150 155 Contents 4.10. Rotor wake models in forward flight 4.11. Computational aerodynamics, advanced methodologies, multidisciplinary approach Chapter 5 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. Chapter 6 6.1. 6.2. 6.3. 6.4. Helicopter trim analysis Introduction Systems of axes General equations of motion of helicopter Helicopter trim conditions 5.4.1. The general trim analysis The rotor-fuselage system and the torque reaction Simplified development of equilibrium (trim) 5.6.1. Trim equations in forward flight 5.6.2. The expression for power in forward level flight Approximate and quick estimation of longitudinal equilibrium General trim solution Autorotation 5.9.1. Autorotation of a rotor 5.9.1.1. Aerodynamics of autorotation 5.9.1.2. Final phase of an autorotation 5.9.2. Limitations in autorotation and Height-Velocity Diagram 5.9.3. Final notes Helicopter flight performance Introduction Total power required Standard atmosphere The engine and the power available 6.4.1. The operating condition of the main rotor 6.4.2. Configuration of free shaft turbine engine 6.4.3. Rotor/transmission/engine system 6.4.4. Performance of installed engine and power ratings 6.5. Hover performance 6.5.1. Power required PMR and Ptr in hovering flight 6.5.2. Vertical drag of the helicopter 6.5.3. Maximum hover ceiling 6.6. Performance in vertical climb 6.7. Performance in forward level flight 6.7.1. Power required PMR and Ptr 9 156 158 161 162 164 168 169 171 173 173 179 181 185 195 195 195 197 198 200 201 201 202 205 205 206 208 209 212 212 213 214 215 216 216 10 Contents 6.7.1.1. The parasitic drag Df in forward level flight 6.7.2. The total power required in level flight 6.7.2.1. Maximum speed in level flight 6.7.2.2. Maximum endurance and maximum range 6.7.2.3. Power increments due to stall and compressibility 6.8. Forward climb and descent performance 6.8.1. Power required PMR in forward climb 6.8.2. Rates and angles of climb, ceiling altitude 6.8.3. Power required PMR in forward descent 6.9. Autorotative performance 6.10. Introduction to mission analysis 6.10.1. Take-off and landing weight 6.10.2. An approach to helicopter mission analysis Chapter 7 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. Stability and control, introduction to helicopter flight dynamics Introduction The single-degree of freedom dynamic system Helicopter static stability and dynamic stability Helicopter static stability 7.4.1. Stability following forward speed perturbation 7.4.2. Stability following vertical speed or incidence perturbation 7.4.3. Stability following yawing perturbation Helicopter dynamic stability 7.5.1. Small disturbance theory 7.5.2. Stability derivatives 7.5.2.1. Force perturbation expressions and stability derivatives 7.5.2.2. Moment perturbation expressions and stability derivatives 7.5.3. Notes on the methodology of small perturbations Dynamic stability in hovering flight 7.6.1. Longitudinal dynamic stability in hovering flight 7.6.1.1. Equations of motion, state variable form 7.6.1.2. Stability derivatives calculation, Mq and Mu in hover 7.6.1.3. Approximate calculation of longitudinal modes in hovering flight for a medium helicopter 7.6.1.4. The characteristic roots on complex plane 7.6.2. Lateral-directional dynamic stability in hovering 219 221 225 226 228 229 229 230 234 234 237 237 238 241 242 250 251 251 251 252 252 255 257 259 260 261 261 261 263 267 268 269 270 Contents flight 7.7. Dynamic stability in forward flight 7.7.1. Longitudinal dynamic stability in forward flight 7.7.1.1. Approximate calculation of longitudinal modes in forward flight for a medium helicopter 7.7.2. Lateral-directional dynamic stability in forward flight 7.8. Helicopter control 7.8.1. Stability, control and flying qualities 7.8.2. Longitudinal control in hovering flight; one degree of freedom approach 7.8.3. Lateral-directional control in hovering flight; one degree of freedom approach Chapter 8 Manoeuvres in horizontal and in vertical planes 8.1. Introduction 8.2. Steady turn 8.2.1. Notes on turn manoeuvres 8.2.2. Gyroscopic moments in turn 8.2.3. Power required in steady turn 8.3. Symmetrical pull-up Chapter 9 Coaxial rotor and tandem rotor helicopter 9.1. Introduction 9.2. Coaxial rotor helicopter 9.2.1. Application of Momentum Theory to the hovering flight 9.2.2. General characteristics of the helicopter 9.2.3. Helicopter equilibrium about the body Z-axis 9.3. Tandem rotor helicopters 9.3.1. General description and definitions 9.3.2. Application of Momentum Theory and of Blade Element Theory to the hovering flight 9.3.3. Application of Momentum Theory to the level forward flight 9.3.4. Experimental data 9.3.5. Condition of longitudinal equilibrium of the helicopter 9.3.6. Notes on stability 9.3.6.1. Forward speed disturbance 9.3.6.2. Stick-fixed dynamic stability in hovering flight 11 273 273 276 278 282 282 283 284 287 287 289 289 290 290 293 293 293 296 297 298 298 300 303 305 305 308 308 309 12 Contents Appendix A Definition of non-dimensional coefficients for the rotor Appendix B International Standard Atmosphere, ISA Appendix C Review of Laplace transform Appendix D Orientation of the aircraft Glossary References List of illustrations Index 311 313 315 317 319 325 331 337 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 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 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 Conversion SI Brit. S. Mass kg slug 1 slug = 14.5939 kg Length m ft 1 ft = 0.3048 m Time s s °K °R 1 (°R) = [1/(1.8)] (°K) Temperature Temp(°K) = 273.15 + temp(°C) Supplementary units Quantity Angle (plane) Units SI Brit. S. rad rad Conversion - Derived quantities Quantity Units SI Brit. S. Conversion Velocity m/s ft/s 1 ft/s = 0.3048 m/s Angular Velocity rad/s rad/s - Acceleration m/s2 ft/s2 1 ft/s2 = 0.3048 m/s2 15 16 Units Units Quantity Acceleration of gravity Air density Force Pressure Power Conversion SI Brit. S. m/s2 ft/s2 kg/m3 slug/ft3 1 slug/ft3 = 515.379 kg/m3 N Pa lb 1 lb = 4.44822 N lb/ft2 1 lb/ft2 = 47.8803 N/m2 lb·ft/s 1 lb·ft/s = 1.35575 W = (1/550) hp (1 Pa = 1 N/m2) W g = 9.80665 m/s2 = 32.174 ft/s2 (1 hp = 550 lb·ft/s) Multiples Units Quantity Velocity Conversion SI Brit. S. m/min metre per minute ft/min foot per minute 1 ft/min = 0.3048 m/min Additional Unit Quantity Unit Conversion Angular Velocity rpm (revolution per minute) 1 rpm = (2π/60) rad/s Velocity kn (international knot) = one nautical mile per hour (one international nautical mile) = ° (degree) 1° = (π/180) rad Angle (plane) 1852 m = 6076.115 ft Notation Units (SI) Symbol rad-1 a lift curve slope of blade section a0 coning angle, main rotor rad a1 coefficient of term (-cosψ) into expression of the flapping angle β, relative to the no-feathering plane; longitudinal flapping coefficient rad A main rotor disc area A  R2 A1 lateral cyclic pitch Atr tail rotor disc area Atr  Rtr2 b number of blades, main rotor b1 coefficient of term (-sinψ) into expression of the flapping angle β, relative to the no-feathering plane; lateral flapping coefficient btr number of blades, tail rotor - B tip loss factor - B1 longitudinal cyclic pitch c blade section chord, main rotor m ctr blade section chord, tail rotor m Cd section drag coefficient   m2 rad   m2 rad rad - 17 18 Notation Cl section lift coefficient - CP main rotor power coefficient - CQ main rotor torque coefficient - CT main rotor thrust coefficient - Df parasitic drag of helicopter N D.L. disc loading f equivalent flat plate drag area G gravitational acceleration G helicopter centre of gravity; origin of the body-axis system Hd density altitude m Hp pressure altitude m If k mass moment of inertia of blade about flapping hinge induced power factor, main rotor - ktr induced power factor, tail rotor - kp climb efficiency factor - KG constant into Glauert’s second formula of the induced velocity - K term of  3 K     effect ltr tail rotor moment arm M Mach number M disturbance term about the Y-axis for aerodynamic moments N∙m MA aerodynamic moment about the flapping hinge N∙m  N/m2 m2 m/s2  - kg/m2 m - Notation 19 Md drag divergence Mach number M heli mass of helicopter M heli  WG / g  n load factor - O origin of the Earth-axis system - p pressure of air N/m2 p0 pressure of air at sea level, ISA conditions N/m2 PMR main rotor power required W Ptr tail rotor power required W Q main rotor torque r radial distance of blade element from axis of rotation m re effective blade radius m R main rotor radius m Rtr tail rotor radius m T main rotor thrust N T temperature of air °K T0 temperature of air at sea level, ISA conditions °K Ttr tail rotor thrust vi induced velocity at rotor m/s vih induced velocity at rotor in hover m/s V true airspeed of helicopter along the flight path; velocity of the free airstream m/s Vc climb velocity m/s Vd descent velocity m/s 0  r  R  kg N∙m N 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 Chapter 7 Stability and control, introduction to helicopter flight dynamics 7.1. Introduction The properties analyzed in this chapter are concerned with the response of the helicopter after the perturbation of a steady trimmed flight condition, produced by the action of a gust or the action of the pilot through flight controls. In particular, helicopter behavior is expressed in terms of stability and control characteristics, which configure the flight qualities; these topics constitute a significant part of the flight dynamics. This chapter introduces some fundamental problems of helicopter stability and control by means of theories using typical assumptions to simplify the approach. Therefore, as in the basic analysis of fixed-wing aircraft, we assume the following for the disturbed motion of the helicopter: small disturbances and the separation of longitudinal and lateral motions. For the latter case, we saw that its consequences represent major critical issues for the analysis applied to the helicopter (remember the natural mating between the two types of motion due to the modalities of main rotor flapping). For a conventional helicopter configuration with a single main rotor, which we will analyze in this chapter, the tail rotor confers asymmetry to the whole rotorcraft, which requires solving all the equations of motion simultaneously, for a rigorous approach. However, it is general practice to set up basic analysis on the separation of the two types of motions, for the following reasons: considerable problem simplification and interesting obtained results. Therefore, the treatment that follows adopts the assumptions above. 241 242 Theory of helicopter flight Finally, the arguments incorporate methodologies and procedures ready to be implemented on the computer. 7.2. The single-degree of freedom dynamic system Before introducing the helicopter stability, it is very useful to review the properties of the system composed of a mass, a spring and a damper, that can be modelled by a second-order differential equation. This system can be used to understand and to represent many dynamic systems, and it provides results which are needed for the presentation of the arguments that follow. Thus, in the general model (shown in Figure 7.1) a force F(t), that is the forcing function or the applied force, acts on the mass m; in x-direction, there are also a linear force provided by the spring and a damping force, proportional to the mass velocity, provided by the damper. x k F(t) m c x kx F(t) m c(dx/dt) Figure 7.1 Mass/spring/damper dynamic system, single-degree of freedom VII. Stability and control, introduction to helicopter flight dynamics 243 Homogeneous solution or free response Considering that m(dx2/dt) is the inertia force, the following second-order differential equation describes the dynamic system shown in Figure 7.1: m dx dx2  c  kx  F t  dt dt It is an ordinary differential equation with constant coefficients. The solution of the homogeneous equation dx dx2  c  kx  0 dt dt m provides the transient or free response of the system. The solution is found by substituting x = Aeλt into the equation; therefore, we obtain: c  m k  m 2        0 which has the following roots: 2  c   c  k       2m   2m   m  1,2   Therefore, the solution of the homogeneous differential equation is: xt   a1e1t  a2e2t xt    c   c 2  k         t     2m   2m   m     a1e   c   c 2  k         t     2m   2m   m     a2e  and it represents the free response of the damped system, where a1 and a2 are constants and are determined from the initial conditions. This solution depends on the values of m, c and k. In particular, consider that if we have  c  k      2m   m 244 Theory of helicopter flight the solution is xt  xt  2   c   k   c      t   i      2m   m   2m    a1e  c    t  e 2m   A1 cos   2   c   k   c       t  i      2m   m   2m    a2e 2 2  k  c   k  c   t   t  A2 sin        m   2m    m   2m   This solution describes a damped sinusoidal motion, characterized by the following damped natural frequency ω: k  c     m   2m    2 Consider that if we have k  c       m  2m  the solution x(t) describes a critical damped motion; in this case, we have  m2 k    2 km ccr  2   m    where ccr is defined as the critical damping constant, and the ratio ζ  c ccr is defined as the damping ratio. Now, let us write the homogeneous equation for the undamped system (c=0): m dx2  kx  0 dt VII. Stability and control, introduction to helicopter flight dynamics 245 By using the previous procedure, we obtain the following solution:    k k  k xt   C cos    t  D sin    t   E cos    t    m  m   m    which describes a steady sinusoidal motion, characterized by the following undamped natural frequency ωn: k  m n    Finally, Figure 7.2 shows all the solutions as functions of m, c and k. Undamped oscillation; -1<ζ<0 Aperiodic, overdamped; ζ >1 x(t) x(t) 0 t 0 Damped oscillation; 0<ζ<1 t Critically damped; ζ =1 x(t) x(t) 0 t 0 Aperiodic,exponentially growing motion; ζ < -1 Steady oscillation; ζ=0 x(t) x(t) 0 Figure 7.2 t t 0 t Types of free response of the dynamic system with a single-degree of freedom 246 Theory of helicopter flight Finally, using the parameters defined above, the second-order differential equation with constant coefficients that describes the mass/spring/damper dynamic system shown in Figure 7.1 can be written as: dx2 dx 1  2n  n2 x  F t  dt dt m Therefore, the damped natural frequency ω, the damping ratio ζ and undamped natural frequency ωn are determined from the analysis of the free response of the system. In fact, note that the solution of the following characteristic equation 2  2n  n2  0 can be written, in a general form, as 1,2  n  in 1   2 Particular solution corresponding to a sinusoidal applied force Now, let us consider the case where the forcing function F(t)≠0 dx2 dx 1  2n  n2 x  F t  dt dt m and is equal to F(t)/m=F0cosωt. Therefore, the equation of the dynamic system (with a single-degree of freedom) becomes: dx2 dx  2n  n2 x  F0 cost dt dt Before continuing, let us remember that the solution of the second-order differential equation is the sum of the solution of the homogeneous equation, that represents the transient motion, with F(t)=0, and of a particular solution of the complete equation, the steady motion, with F(t)≠0. Hence: x(t) = [x(t)]homogeneous eq + [x(t)]particular solution VII. Stability and control, introduction to helicopter flight dynamics 247 We have that: {[x(t)]particular solution = Xf cos(ωf t + ϕ)}, where the response amplitude Xf and the phase angle ϕ are given by the following expressions: F0 m Xf   1 2 n   2f   4 2  2n f    2 2  f   n    tan 1  , n2 2f 2 These relations define the frequency response of the system. Xf F0/m ζ=0 ζ=0.1 ζ=0.2 ζ=0.3 ζ=0.5 ζ=1.0 1 ω2n ωn 0 ωf ϕ π ζ=0 ζ=0.1 ζ=0.2 ζ=0.3 ζ=0.5 π/2 ζ=1.0 ζ=1.0 ζ=0.5 ζ=0.3 ζ=0.2 ζ=0.1 ζ=0 0 ωn Figure 7.3 Amplitude and phase, frequency response ωf 248 Theory of helicopter flight Transfer function of the mass/spring/damper system Considering that the equation of the system is dx2 dx 1  2n  n2 x  F t   f t  dt dt m then, let us write xt   x1 t  dx  x  x2 t  dt Therefore, we obtain dx1  x1  x2 t  dt dx2  2n x2  n2 x1  f t  dt Choose the initial conditions as x0  0 dx0 0 dt write the Laplace transform of x(t) and of f(t): L x1 t   Y s  and  dx   dx  L  2  2n x2  n2 x1   L  2  +2ζωn L x2 t  +ω2n L x1 t  =U(s)  dt   dt  From the relations above we have: s2Y(s) + 2ζωn sY(s) + ω2n Y(s) = U(s) VII. Stability and control, introduction to helicopter flight dynamics 249 Finally, the transfer function G(s) of the system, that is the ratio of the output and the input, is equal to: Gs   1 Y s   2 U s  s  2n  n2 State-space modeling The following relations x1  x2 t  x2  2n x2  n2 x1  f t  can be written in matrix form as 1   x1  0  x1   0  x     2  2   x   1u(t ) n  2     2  n where u(t)=f(t). We have: x  Ax  Bu with  x  x   1  ,  x2  1   0 A 2 ,  n  2n  0 B  1 x  and x   1  is the state vector.  x2  The system is fully described by the state-space matrices A and B. Now, we know that the free response of the system, where f(t)=0, may be studied by the equations: x  Ax  jt The substitution of x(t )  x j e into equations above gives A   jIx j  0 where I is the identity matrix. 250 Theory of helicopter flight Now, the vector xj is the eigenvector associated with the eigenvalue λj of the matrix A. The solution is the following linear combination: x(t )  2 c j x je t j j 1 (cj is a constant that is fixed by the initial conditions) Control form of a second-order differential equation If the system has mass m=1, then it can be visualized by the diagram in Figure 7.4: F(t) + - x(t) - ʃ x(t) ʃ x(t) ζ k Figure 7.4 Control form of the second-order differential equation 7.3. Helicopter static stability and dynamic stability The stability, in general terms, is defined as the capability to restore an initial trim condition that has been perturbed by a particular cause. Static stability is defined as the initial tendency of the system to return to the trim condition. Then, dynamic stability is defined as the tendency of the system to restore the trim condition as the time goes on. In other words, the static stability studies the initial motion (initial response) of the aircraft after the perturbation. Instead, the dynamic stability is concerned with the evolution of the aircraft motion versus time, in relation with the tendency to return to or to leave the trim condition that has been perturbed. VII. Stability and control, introduction to helicopter flight dynamics 251 It should be noted that an aircraft can be statically stable but dynamically unstable. However, the static stability is a necessary condition but is not a sufficient condition for the dynamic stability. 7.4. Helicopter static stability In the pages that follow we will discuss some fundamental cases related especially to the main rotor properties, because it supplies a relevant contribution to the stability characteristics of the helicopter as a whole. 7.4.1. Stability following forward speed perturbation In the context of the aircraft response immediately following a disturbance, as first case, we treat the response to speed perturbation in the direction of the motion. Supposing to analyze a forward flight condition, for the reason we saw in the previous chapters, an increase in forward speed will involve an increase of the rotor flapping with backward inclination of rotor disc. Therefore, the rotor thrust is characterized by a component in the tail direction that opposes the disturbance: the rotor supplies a contribution to the static stability. The fuselage, instead, can provide a contribution to stability or a contribution to instability, depending on the direction of the generated aerodynamic forces (lift and drag). It is also clear that an additional contribution to stability can be provided by the horizontal stabilizer, depending on its dimensions and position on the entire helicopter. These considerations are valid for both forward and hovering flight, taking into account the fact that as the speed decreases the contribution from the fuselage and from the horizontal tail tends to decrease (until being negligible in estimation at very low flight speed). 7.4.2. Stability following vertical speed or incidence perturbation Assume a steady level flight condition; as a consequence of a vertical gust, the main rotor blades have an increase in incidence and the rotor thrust also increases. The total effect on the advancing and retreating blades (considering also the difference in relative speed) produces backward flapping of the rotor, with the generation of a nose up pitching moment. Indeed, after the inclination of the rotor disc, this moment is due to the thrust: the rotor is statically unstable. It is clear that the rotor instability grows as the forward speed increases. The considerations discussed above for the fuselage 252 Theory of helicopter flight are still valid, but generally its contribution is in terms of instability. The only one contribution to stability is provided by the horizontal stabilizer: this contribution grows as the forward speed increases. Finally, note that the availability of accurate methods for analysis of aeroelastic phenomena of rotor blades and the use of advanced composite materials can allow the designer to obtain appropriate load distributions to contain the unstable effect of the rotor on the response to the incidence perturbation. 7.4.3. Stability following yawing perturbation Assuming an attitude with a yawing angle different from zero, a change in the incidence of the tail rotor is obtained; this variation provides a damping effect, similar to that of a vertical fin (known as ‘fin effect’). Therefore, the vertical fin provides an important contribution to the stability, because it responds generating a lateral force that produces a consequent yawing moment. This moment confers stability. However, different from fixed-wing aircraft, it shall be noted that evaluation regarding the fin (and the tail rotor) of the helicopter shall take into account the effects due to the main rotor wake on the empennage. Increasing the forward speed, we shall consider also the contribution from the fuselage (generally neglected at low flight speed): its action can be of stable or unstable type, depending on the disturbance, the position of the centre of gravity and the airframe geometry. 7.5. Helicopter dynamic stability Procedures and methods for the helicopter stability analysis (from the determination of equations of motion to the application to the flight test issues) have been developed, since early studies, by extending the methodologies applied to the fixed-wing aircraft. Therefore, also the topic that follows is characterized by an initial formal setting clearly common for rotary-wing and fixed-wing aircraft, and by a subsequent stage, where we find the typical problems to be solved for the helicopter. Again, we recall Equations (5.5b) determined in Chapter 5, with the rigid body assumption. Now, for an accurate assessment, each rotor blade has its own degree of freedom, which provides a contribution to the perturbed motion. Consequently, for the helicopter with a single main rotor, over the six equations of motion as for a fixed-wing aircraft (three for the translation and three for the rotation about the reference axes) we shall add, in a basic approach, as minimum other three equations: one for rotor longitudinal VII. Stability and control, introduction to helicopter flight dynamics 253 flapping, one for lateral flapping, and one for conic attitude of the rotating blades. Remember that due to rapid response of the blades versus the whole airframe, the rotor can be assumed as a compact generator of forces and moments, neglecting the motion of the single blade. Then, this is a quasistatic condition of motion, by which, now, the equations are only six (because there are six active degrees of freedom). Obviously, this condition cannot be maintained in those cases where the designer needs to study aeroelastic phenomena or resonance. However, those cases are beyond the purposes of the present book. From the previous notes and assumptions, we recall the set of Equations (5.5b), now with the new number (7.1): WG  (Vx   yVz   zV y )  Fxa  Wx g WG  (V y   xVz   zVx )  Fya  W y g WG  (Vz   xV y   yVx )  Fza  Wz g  x I x   y z ( I y  I z )  ( z   x y ) I xz  M xa (7.1)  y I y   x z ( I z  I x )  ( z2   x2 ) I xz  M ya  z I z   x y ( I x  I y )  ( x   y z ) I xz  M za with the auxiliary Equations (5.5b):      sin tg   cos tg  x y z    cos    sin   y z    sin  sec    cos  sec   y z Equations (7.1), representing the balance of forces and moments acting on the helicopter, constitute the basic model to study the aircraft motion. We remember that the equations have been written with respect to the body axes, with the following assumptions: rigid body, constant mass, existence of the aircraft plane of symmetry. From a general point of view, each Equation (7.1) is related to a degree of freedom, having therefore three degrees of freedom for translation and three degrees of freedom for rotation. Going in depth into dynamic stability analysis, we will discuss the fixed control cases. The classical approach (that is followed in the present book) considers that the action by a gust or the action 254 Theory of helicopter flight to perform a manoeuvre generate an unsteady flight condition, analyzed superimposing a ‘disturbance’ to the initial steady condition of motion. From a mathematical point of view, this approach requires to consider the terms in the set (7.1) equal to the sum of the value in the initial steady trim condition and the value of perturbation. Therefore, we can write: Vy  Vy 0  v Vx  Vx0  u  y  q0  q x  p0  p Vz  Vz 0  w z  r0  r Fxa  Fx0  X Fya  Fy 0  Y M xa  M x0  L M ya  M y 0  M M za  M z 0  N   0  d   0   d   0  d Fza  Fz 0  Z (7.2) where the terms with subscript ‘0’ represent the initial condition of motion and the second term (we take, for example, u) defines the disturbance. By using expressions (7.2), the first equation of the set (7.1) can be written in the following form: WG  [(Vxo  u)  (q0  q)(Vz 0  w)  (r0  r )(Vy 0  v)]  ( Fx0  X )  g  WG sin(0  d ) and, W  [(Vxo  u)  (q0Vz 0  q0 w  qVz 0  qw)  (r0Vy 0  rVy 0  r0v  rv)]  g (7.3)  (Fx0  X )  W sin(0  d ) Now, consider that in the trim condition, before introducing the disturbance, we can write the following expression: WG  (Vx0  q0Vz 0  r0Vy 0 )  Fx0  WG sin 0 g Therefore, the Equation (7.3) can be written as: VII. Stability and control, introduction to helicopter flight dynamics 255 W (u  qVz 0  q0 w  qw  rVy 0  r0 v  rv)  X  W [sin(0  d )  sin 0 ] g Operating in a similar manner on the other five equations and using the expressions (7.2), finally, we obtain the following set of equations: WG (u  qw  rv  Vz 0 q  q0 w  V y 0 r  vr0 )  X  WG [sin( 0   d )  sin 0 ] g WG (v  ur  wp  Vx0 r  r0u  Vz 0 p  p0 w)  Y  WG [cos(0   d ) sin( 0   d )  g  cos0 sin  0 ] WG (w  vp  uq  V y 0 p  vp0  Vx0 q  q0u)  Z  WG [cos(0  d ) cos( 0   d )  g  cos0 cos0 ] p I x  (q0 r  r0 q  qr)( I y  I z )  (r  p0 q  q0 p  pq)I xz  L qI y  ( p0 r  r0 p  pr)( I z  I x )  (2r0 r  2 p0 p  p 2  r 2 )I xz  M rI z  ( p0 q  q0 p  pq)( I x  I y )  ( p  q0 r  r0 q  qr)I xz  N (7.4) The system (7.4) is the set of equations for the perturbed motion in the general form. 7.5.1. Small disturbance theory The perturbed dynamics is based on the resolution of the set of Equations (7.4), once the initial condition of motion has been fixed and the forces and moments acting on the aircraft have been defined for each scalar equation. In order to approach this typology of problem, methods and assumptions shall be obviously defined in accordance with the task to be accomplished. Considering the notes discussed in the previous pages, the small disturbance assumption can be valid and applicable to many problems and constitutes the initial approach for many dynamic analyses due to the simplification of the mathematical models and to the interesting results which can be obtained. From a mathematical standpoint, this assumption allows the disturbance quantities and their derivatives to be considered small, and, consequently, we can neglect their products and squares into Equations (7.4). Moreover, angles are considered so small that the cosine can be considered equal to 1, and the sine and the tangent equal to the value of the angle expressed in radians. Therefore, let us use the following expressions: 256 Theory of helicopter flight sin(0  d )  sin 0  d cos0 cos(0  d )  cos0  d sin 0 Now, Equations (7.4) become: WG (u  Vz 0 q  q0 w  V y 0 r  r0v)  X  WG [d cos0 ] g WG (v  Vx0 r  r0u  Vz 0 p  p0 w)  Y  WG [ d cos 0 cos0   d sin 0 sin  0 ] g WG (w  V y 0 p  p0 v  Vx0 q  q0u)  Z  WG ( d cos0 sin  0   d sin 0 cos 0 ) g p I x  (q0 r  r0 q)( I y  I z )  (r  p0 q  q0 p)I xz  L qI y  ( p0 r  r0 p)( I z  I x )  (2r0 r  2 p0 p) I xz  M rI z  ( p0 q  q0 p)(I x  I y )  ( p  q0 r  r0 q)I xz  N (7.5) Assuming that the initial trim condition is, for formal convenience, a steady level flight condition with constant speed V (Vx0, 0, Vz0), we may write Vy 0  p0  q0  r0  0 , and also 0  0  0 and the set of Equations (7.5) becomes: WG (u  Vz 0 q)  X  WG [ d cos 0 ] g WG (v  Vx0 r  Vz 0 p)  Y  WG [ d cos 0 ] g WG (w  Vx0 q)  Z  WG (d sin 0 ) g p I x  rI xz  L qI y  M (7.6) rI z  p I xz  N Instead, if the initial trim condition is a hovering flight condition, we may write Vx0  Vy 0  Vzo  p0  q0  r0  0 VII. Stability and control, introduction to helicopter flight dynamics 257 and 0  0  0  0 finally, obtaining: WG (u)  X  WG [d ] g WG (v)  Y  WG [ d ] g WG (w )  Z g p I x  rI xz  L (7.7) qI y  M rI z  p I xz  N 7.5.2. Stability derivatives The parameters representing the perturbation from trim values are written using a Taylor series with the first terms only (linear terms, small disturbance assumption). Then, the expression for the force increment X is: X X X X X X X X X u v w p q r  MR  A1  u v w p q r  RP A1 X X  tr B1  tr B1 In the expression above we find also the terms θMR, B1, A1, θtr (that are, in this chapter, variations from trim values) associated respectively with the following control inputs: collective pitch, longitudinal cyclic pitch and lateral cyclic pitch of the main rotor, and collective pitch of the tail rotor. In order to simplify the notation, we adopt the following compact form: X  X uu  X vv  X ww  X p p  X q q  X r r  XMRMR  X A1 A1  X B1 B1   Xtrtr where the generic derivative has been written as ∂X / ∂a = Xa . 258 Theory of helicopter flight Using the same procedure for each term representing an increment, finally we obtain the set of expressions: Fxa  Fx0  X uu  X vv  X ww  X p p  X qq  X r r  X MRMR  X A1 A1  X B B1  X tr 1 tr Fya  Fy 0  Yuu  Yvv  Yww  Yp p  Yqq  Yr r  Y MRMR  YA1 A1  YB1 B1  Y tr tr Fza  Fz 0  Zu u  Zv v  Z w w  Z p p  Z q q  Z r r  Z MR MR  Z A1 A1  Z B1 B1  Ztrtr  Z w w M xa  M x0  Lu u  Lv v  Lw w  Lp p  Lq q  Lr r  L MR  MR  LA1 A1  LB1 B1  Ltrtr M ya  M y 0  M uu  M vv  M w w  M p p  M q q  M r r  MMRMR   M A1 A1  M B1 B1  Mtrtr  M w w M za  M z 0  Nu u  Nv v  N w w  N p p  Nq q  Nr r  N MR  MR  N A1 A1  N B1 B1  Ntrtr As in fixed-wing aircraft analysis, the derivatives in u, v, w, p, q, r contained in the expressions above are called stability derivatives, and the derivatives in θMR, B1, A1, θtr are called control derivatives. The derivatives are expressed in a so called normalized form when those related to the forces are divided by the mass Mheli of the helicopter, and those related to the moments are divided by an appropriate moment of inertia. Note that the third and the fifth expressions contain also the derivatives ∂Z/∂ẇ and ∂M/∂ẇ, related to change of force along the Z-axis and to change of moment about the Y-axis respectively, due to the acceleration ẇ (as in fixed-wing aircraft analysis); in the expressions above, they are the contributions due to an acceleration, remembering that we are considering the disturbances calculated as functions of speed. In particular, their contributions take into account the downwash effect of the main rotor on the horizontal stabilizer. Therefore, ∂Z/∂ẇ and ∂M/∂ẇ are kept when a sophisticated investigation is required to perform the analysis; generally, they are neglected in order to simplify the treatment. The stability derivatives are evaluated in the steady trim conditions; moreover, the derivatives are constant. Generally, in hovering flight each derivative is obtained by determining the contributions due to the main rotor and to the tail rotor. In forward flight, the designer shall consider also the VII. Stability and control, introduction to helicopter flight dynamics 259 contributions due to the fuselage, to the horizontal stabilizer and to the vertical fin. The derivatives may be determined by using various procedures. The analytical or classical methodology (appropriate for an initial analysis) requires to write the equations of forces and moments for the rotors, for the fuselage and empennage (that we wrote in detail in Chapter 5, Section 5.6 and 5.8) and then to apply the derivative operation. Moreover, from the relations obtained in Chapter 5, we know that the rotor force and moment derivatives are directly related to the thrust and flapping derivatives. 7.5.2.1. Force perturbation expressions and stability derivatives Thus, considering the parameters in Figures 5.2b-5.5 and the relations that we wrote in detail in Chapter 5, Section 5.8, the force increments X, Y, Z along the body axes, that we find into relations (7.2), may be expressed as: X = -TTPPΔa1s - a1sΔTTPP - HTPP - ΔXFuselage+Tail empennage Y = TTPPΔb1s + b1sΔTTPP + ΔYFuselage+Tail empennage + ΔTtr Z = - ΔTTPP Therefore, for Xu, Xw, Xq we have: a HTPP X Fus tail _ emp T X   X u  TTPP 1S  a1S TPP  u u u u u a HTPP X Fus tail _ emp X T   X w  TTPP 1S  a1S TPP  w w w w w X a X T H  X q  TTPP 1S  a1S TPP  TPP  Fustail _ emp q q q q q and for Zu, Zw and Zq we have: Z T  Zu   TPP u u Z T  Z w   TPP w w 260 Theory of helicopter flight T Z  Z q   TPP q q 7.5.2.2. Moment perturbation expressions and stability derivatives Considering the parameters in Figures 5.2b-5.5 and the relationships that we wrote for the moments about the body axes in Chapter 5, Section 5.8, we obtain the expressions for the moment increments (or change in moments) L, M, N: L = (LMR Δb1s ) + (TTPP hz Δb1s) + (hz b1s ΔTTPP) + (ΔLFuselage+Tail empennage) + (ztr ΔTtr) M = (MMR Δa1s) + (TTPP hz Δa1s) + (hz a1s ΔTTPP) - (hx ΔTTPP) + (hz ΔHTPP) + (ΔMFuselage+Tail empennage) N = - (TTPP hx Δb1s) - (hx b1s ΔTTPP) + (ΔNFuselage+Tail empennage) + (ltrΔTtr) + (ΔQ) dLMR dMMR b1S , and M MR  a1S (see Chapter 3, Sect.3.4). db1S da1S Therefore, for Lv, Lr we have: where LMR  LFustail _ emp T b T b L  ztr tr  Lv  LMR 1S  hzTTPP 1s  hzb1S TPP  v v v v v v LFustail _ emp T b T b L  ztr tr  Lr  LMR 1S  hzTTPP 1s  hzb1S TPP  r r r r r r For Mw, MB1, we have: M a a T T HTPP M F t _ e  M MR 1S  hzTTPP 1s  hz a1S TPP  hx TPP  hx  w w w w w w w M a a T T HTPP M F t _ e  M MR 1S  hzTTPP 1s  hz a1S TPP  hx TPP  hx  B1 B1 B1 B1 B1 B1 B1 For Nr, Nθtr we have: N Fustail _ emp Q T T b N  ltr tr   Nr  TTPP hx 1S  hxb1S TPP  r r r r r r VII. Stability and control, introduction to helicopter flight dynamics 261 N Fustail _ emp Q T T b N  ltr tr   Ntr  TTPPhx 1S  hxb1S TPP  tr tr tr tr tr tr 7.5.3. Notes on the methodology of small perturbations From the set of Equations (7.4) the reader may appreciate that it is not possible, from a rigorous standpoint, to separate a pure lateral-directional motion. Consequently, the six equations require to be solved simultaneously. The small disturbance assumption has reduced the interaction between the longitudinal motion and the lateral motion. Moreover, considering the groups of Equations (7.6) and (7.7), it is more clear the link (by means of the variables involved) among the first, the third and the fifth equations (they define the longitudinal motion), and among the second, the fourth and the sixth equations (they define the lateral-directional motion). Therefore, in order to simplify the treatment, the two types of motions will be analyzed separately, knowing both the approximations made. In the developments that follow, for the purposes of this chapter (which provides an introduction to stability and control), we will study separately the longitudinal motion and the lateral-directional motion. From a mathematical point of view, the problem requires to determine the equations by introducing the derivatives, and then to perform the stability analysis of the equations obtained. 7.6. Dynamic stability in hovering flight 7.6.1. Longitudinal dynamic stability in hovering flight Considering Equations (7.7), we start with the longitudinal motion by using the expressions with the derivatives; however, we do not take into account those stability derivatives that we have neglected by applying the separation of the lateral-directional motion from the longitudinal motion. Therefore, we obtain: WG (u)  X  WG [d ] g WG (w )  Z g qI y  M 262 Theory of helicopter flight and, finally WG (u)  X uu  X ww  X q q  X MRMR  X B1 B1  WG[d ] g WG (w )  Zuu  Zww  Zw w  Zqq  Z MRMR  Z B1 B1 g (7.8)   Mqq  M MRMR  M B1 B1 qI y  Muu  M ww  M w w Then, we rewrite the right-hand side where we insert only the control terms; therefore, we obtain: WG (u)  WG[d ]  X uu  X ww  X q q   X MRMR  X B1 B1 g W (7.9)  G (w )  Zuu  Zww  Zw w  Zq q  Z MRMR  Z B1 B1 g   qI y  Muu  M ww  M w w  M qq  M MRMR  M B1 B1 Before proceeding with the analysis of Equations (7.9), it is important to remember the meaning of some stability derivatives. This will also help the interpretation of other derivatives. In previous analysis of static stability we saw that an increment u in forward speed along the X-axis produces an increase and a decrease in airspeed at the advancing blade and at the retreating blade, respectively. As final result, the main rotor tilts backwards, with an increase in thrust, in H-force and in longitudinal aerodynamic drag of the fuselage. Then, it is clear that these changes are functions of the forward flight speed of the helicopter (higher the flight speed, higher the changes). Instead, in hover, a small disturbance u does not involve meaningful changes for the force along the Z-axis: then, ∂Z/∂u can be neglected. Otherwise, as forward speed increases, this assumption becomes not acceptable, because at low speeds ∂Z/∂u < 0, and then, at high speeds ∂Z/∂u > 0. Moreover, an increment w in speed along the Z-axis will produce an effect on the forces along the Z-axis, but will not produce a significant effect along the X-axis (especially in hovering flight); then, in this flight condition we can accept that ∂X/∂w = 0. Analogously, we can verify that in hovering is acceptable to consider ∂Z/∂q = 0, ∂Z/∂ẇ = 0 and ∂M/∂ẇ = 0. Using the previous results, the group of Equations (7.9) may be simplified as: VII. Stability and control, introduction to helicopter flight dynamics WG (u)  WG[d ]  X uu  X q q   X MRMR  X B1 B1 g W  G (w )  Zww  Z MRMR  Z B1 B1 g  qI y  Muu  M ww  Mqq  M MRMR  M B1 B1 263  (7.10) This set of equations is composed of linear differential equations with constant coefficients. Now, the objective is to investigate about the typology of stability following a disturbance. From the theory of differential equations, it is known that some tools for the immediate stability verification are available. Remember Routh’s criterion for stability that allows us to proceed without the necessity to solve the equations involved. However, the criterion (non-quantitative type) presents operational limits because it does not allow us to evaluate the entity of stability or instability of the system. 7.6.1.1. Equations of motion, state variable form Let us divide the derivatives by the mass Mheli of the helicopter or by the moment of inertia Iy, as follows: X u  M Xu Z M M ; Z w  w ; M u  u ; M w  w ; M q  q ; M heli M heli Iy Iy Iy (7.11) By using the same procedure we obtain XoMR , X Bo1 , Z0MR , Z B01 , M0MR , M B01 . By using expressions (7.11), the set of linear differential equations with constant coefficients (7.10) can be written in a more compact manner, introducing the state vector x. Then, the set of equations may be written in matrix form as x  Ad x  Bc where x the state vector Ad the stability derivatives matrix B the control matrix c the control vector 264 Theory of helicopter flight and also:  u   X u0 w      0  q  M u0    d   0 X q0 0 M q0 1 0 Z w0 M w0 0  g   u   X oMR    0   w   Z0MR  0   q  M0MR    0  d   0 X Bo1   Z B01   MR    M B01   B1   0  Now, we assume that θMR = 0, B1 = 0. The characteristic polynomial φ(λ) is equal to det(λ I - Ad). Therefore the characteristic equation is obtained expanding the following determinant: φ(λ) = det(λ I - Ad) = 0 where I is the identity matrix, order 4x4. Being  0 I   0  0 0 0  0 0  0 0 0 0  0   then, we have   X u 0  M u0 0 0  X q0 g 0 0   Z w0 0 0 0  Mw   Mq 0 0 1  (7.12a) Expanding the determinant produces the following characteristic equation: A4  B3  C2  D  E  0 where A 1 B   X u0  Z w0  M q0 (7.12b) VII. Stability and control, introduction to helicopter flight dynamics 265 C  Z w0 ( X u0  M q0 ) D  M u0 g E  Z w0 M u0 g The expressions for the terms C and D are obtained considering that [ C  Z w0 ( X u0  M q0 )  X u0 M q0  M u0 X q0 ] [ D  M u0 g  Z w0 ( X u0 M q0  M u0 X q0 ) ] and, for the helicopter with a single main rotor, note that: [ X u0 M q0  M u0 X q0  0 ] Now, in order to solve the characteristic Equation (7.12b), first of all the values of the stability derivatives involved shall be calculated. Then, the characteristic Equation (7.12b) has four roots: λ1, λ2, λ3, λ4, eigenvalues of matrix Ad, and they may be real or complex conjugate. Therefore, the generic root λ has the form λ = η ± iω. The general solution for each dependent variable (for example, we choose u) is of the following type: u  a1e1t  a2e2t  a3e3t  a4e4t where a1, a2, a3, a4 are constant that can be evaluated by the initial conditions. Consequently, for stability verification tasks, if the real roots are negative, then the perturbation is damped; vice versa if the real roots are positive, then the perturbed motion results a divergence. In case of complex roots, if the real part is negative, then the motion is a damped oscillation; vice versa, if the real part is positive, then the motion is a divergent oscillation (Figure 7.5). In hover, the typical case for a single main rotor helicopter is dominated by a couple of real roots and a couple of complex conjugate roots. Real roots are related to very damped motions (heavily damped subsidence) with a pure convergence, while the complex roots configure a dynamically unstable motion, with increasing amplitude oscillations. To understand the unstable response it is necessary to remember, for example, the response of rotor to a forward speed disturbance: 266 Theory of helicopter flight the rotor tilts backwards, producing a nose up attitude of the helicopter. Then, a backward motion is generated and, now, the rotor tilts forwards causing a nose down attitude of the helicopter. We immediately understand that the backward motion is stopped, but a forward motion is starting, so the phenomenon restarts in a manner that is divergent and unstable. In this case, the meaningful stability derivative is ∂M / ∂q, and the designer shall consider in detail this derivative in order to attenuate the unstable motion. subsidence (real root < 0) 0 thalf divergence (real root > 0) 0 time damped oscillation (complex root, real part < 0) 0 time divergent oscillation (complex root, real part > 0) time 0 time tdouble T, period Figure 7.5 Evolutions of perturbed motion Generally, for calculation of the oscillation frequency, the second equation of the set of Equations (7.10) is considered negligible, because the VII. Stability and control, introduction to helicopter flight dynamics 267 described motion does not involve relevant changes in altitude. Consequently, considering the first equation and the third equation, we can write the characteristic equation that provides the oscillation frequency:  7.6.1.2.  gM u0 M q0 The stability derivatives Mq and Mu in hover In hover, the only contribution that cannot be neglected for the calculation of the derivative ∂M / ∂q is produced by the main rotor. Therefore, considering the parameters in Figure 5.2b (Chapter 5), we have: or M q  M q MR  X M Z h  hx   q z q q M q  M q MR  dM MR a1S M Z X   hx   hz  q q q da1S q (for rotor with hinge offset   0 ) (for hingeless rotor) These expressions may be simplified remembering that ∂Z/∂q ≈ 0 in hover. In a similar manner, we have: M u  M u MR  M X   hz u u M u  M u MR  M X dM MR a1S   hz  u u da1S u or (for rotor with hinge offset   considering that, in hovering flight, ∂Z/∂u ≈ 0. To estimate the previous terms, we have to consider that X u   X u MR   HTPP ; u X q  X q MR   HTPP q 0) (for hingeless rotor) 268 7.6.1.3 Theory of helicopter flight Approximate calculation of longitudinal modes in hovering flight for a medium helicopter In order to illustrate the previous theory, we will study the approximate control fixed response in hovering flight at sea level of a typical medium helicopter (DL= 350 N/m2; main rotor: four-bladed rotor, radius R = 6.6 m, chord c = 0.4 m; tail rotor: radius Rtr = 1.0 m; ltr = 8.1 m). Now, let us consider that the applicable stability derivatives assume the values shown in the matrix Ad as follows: 0 0.8500  9.8066  0.0200  0  0.300 0 0  Ad    0.0500 0.065  1.700 0    0 1 0   0 Therefore, the characteristic equation (7.12b) becomes: 4  (2.0200)3  (0.516)2  (0.4903)  (0.1471)  0 The roots of the characteristic equation above, or eigenvalues, are: λ1 = -0.300, λ2 = -1.861, λ3,4 = 0.0707 ± 0.5083i Therefore, the two negative real roots represent two stable responses with a thalf (time to half, or time during which the disturbance quantity will half itself) equal to: t half  0.693 n and: t half  0.693 0.693   2.31seconds n  0.300 t half  (for λ1), 0.693 0.693   0.37 seconds (for λ2) n  1.861 The complex roots λ3 and λ4 (that have a positive real parts) imply an unstable oscillatory mode (divergent oscillation) with the following period T, VII. Stability and control, introduction to helicopter flight dynamics 269 time to double amplitude tdouble, undamped natural frequency ωn, damping ratio ζ, and number of cycles to double amplitude Ndouble: T 2   2  12.4 seconds, 0.5083 n  n2   2  0.513 rad/s, Ndouble  t double  0.693 0.693   9.8 seconds n 0.0707 n n   0.0707  0.138 , 0.513 0.693 1   2  0.7895  2 Im (λ) 0.5083 0.0707 Figure 7.6 7.6.1.4. Re (λ) Roots λ3 and λ4 on complex plane The characteristic roots on complex plane In the preceding example of calculation we obtained the numerical parameters related to each root, real or complex. From a general standpoint, now it is useful to show the relationships among n, ωn and ζ in the complex plane. 270 Theory of helicopter flight Figure 7.7 shows a generic root λ = n ± iω in the left half plane (therefore, n is negative): The left half plane: stable The right half plane: unstable Im (λ) Period T increasing ωn ω = ωn (1 – ζ)0.5 θ Re (λ) n = - ζ ωn Figure 7.7 7.6.2. Relationships among parameters on complex plane Lateral-directional dynamic stability in hovering flight Now, for the lateral-directional motion, we saw that the applicable set of equations is: WG (v)  Y  WG [ d ] g p I x  rI xz  L rI z  p I xz  N By substituting the expressions of perturbations into equations above, we obtain: VII. Stability and control, introduction to helicopter flight dynamics WG (v)  WG [d ]  Yvv  Yp p  Yr r  YA1 A1  Ytrtr g p I x  rI xz  Lvv  Lp p  Lr r  LA1 A1  Ltrtr 271  (7.13) rI z  p I xz  Nvv  N p p  Nr r  N A1 A1  Ntrtr Now, let us divide the stability derivatives of the first equation by the mass Mheli of the helicopter; thus, we obtain: Yp0  Yp M heli Y Y , Yr0  Yr , Yv0  Yv , YA0  A1 , Y0  tr 1 tr M heli M heli M heli M heli (7.14a) Considering the second and the third equations of the set (7.13), in order to write the group of equations in the required matrix form, let us calculate the expression of the term ṙIxz from the third equation; we have: I xz r  I xz2 I I I I I p  xz Nv v  xz N p p  xz Nr r  xz N A1 A1  xz Ntrtr Iz Iz Iz Iz Iz Iz By substituting into the second equation, we obtain: p ( I x  I xz2 I I I I )  xz Nv v  xz N p p  xz Nr r  xz N A1 A1 Iz Iz Iz Iz Iz I  xz Ntrtr  Lv v  Lp p  Lr r  LA1 A1  Ltrtr Iz Multiplying by Iz and after appropriate rearranging of terms, finally, is: p  (I N  I L ) ( I xz Nv  I z Lv ) ( I N  I z Lr ) v  xz p z2 p p  xz r r 2 ( I x I z  I xz ) ( I x I z  I xz ) ( I x I z  I xz2 )  ( I xz N A1  I z LA11 ) (I x I z  I xz2 ) A1  ( I xz Ntr  I z Ltr ) ( I x I z  I xz2 ) tr By applying a similar procedure to the third equation, we can calculate the expression for ṙ. Therefore, in order to write the previous expressions in a suitable form, let us use the following relations: 272 Theory of helicopter flight I z ( L p )  I xz ( N p ) L0p  IxIz  I xz2 N 0p  ; I x ( N p)  I xz ( L p ) I x I z  I xz2 L0r  I z ( L r )  I xz ( N r ) ; I x I z  I xz2 N r0  I x ( N r )  I xz ( Lr ) I x I z  I xz2 L0v  I z ( Lv )  I xz ( Nv ) ; I x I z  I xz2 N v0  I x ( N v)  I xz ( Lv ) I x I z  I xz2 (7.14b) L0A1  L0tr  I z ( L A1 )  I xz ( N A1 ) IxIz  I xz2 I z ( Ltr )  I xz ( Ntr ) I x I z  I xz2 I x ( N A1)  I xz ( LA1 ) ; N 0p  ; N0tr  I x I z  I xz2 I x ( N tr )  I xz ( Ltr ) I x I z  I xz2 As in the longitudinal motion we just treated, let us rewrite the group of Equations (7.13) by expressions (7.14a) and (7.14b); then using the matrix notation, we have:  v   Yv0  p   0    Lv  r    Nv0     d   0   0   d  Yp0 L0p N 0p 1 0 Yr0 L0r Nr0 0 1 g 0 0 0 0 0  v   YA01    0  p   L0A1 0  r    N A01    0 d   0 0  d   0 Y0tr   L0 tr   A1  N0tr     tr  0  0  Expanding the following determinant   Yv0  L0v  N v0 0 0  Yp0   L0p  N 0p 1 0  Yr0  L0r   N r0 0 1 g 0 0  0 0 0 0 0 0  leads to the characteristic equation: A5  B4  C3  D2  E  F  0 VII. Stability and control, introduction to helicopter flight dynamics 273 where A 1 B   N r0  L0p  Yv0 C  L0p N r0  N 0p L0r  Yvo N r0  Yv0 L0p  L0vYp0  N v0Yr0 D  Yvo ( Lor N 0p  L0p N r0 )  L0v (Yp0 N r0  N 0pYr0  g )  N v (Yr0 L0p  Yp0 L0r ) E  L0v N r0 g  L0r N v0 g F 0 In this case, there are five roots; in detail, one is relative to   0 and represents a neutral condition of stability (heading mode); other four roots, typically, are as follows: — two real roots, relative to stable motions (one root produces a rolling damped motion, the other one produces a yawing damped motion); — two complex conjugate roots, that produce dynamically unstable oscillation. The rolling damped mode is characterized by the derivative Lp, while the yaw stable mode by the derivative Nr. The unstable oscillation represents changes in helicopter sideways speed and in bank angle. 7.7. Dynamic Stability in forward flight 7.7.1. Longitudinal dynamic stability in forward flight In this flight condition, the set of equations is: WG (u  Vz 0 q)  X  WG [d cos0 ] g WG (w  Vx0 q)  Z  WG (d sin 0 ) g qI y  M 274 Theory of helicopter flight Introducing the stability derivatives, the equations become: W WG (u)  WG [d cos 0 ]  X uu  X w w  ( X q  G Vz 0 )q   XMR MR  X B1 B1 g g W W  G (w )  WG (d sen0 )  Zuu  Z w w  (Z q  G Vx0 )q  ZMR MR  Z B1 B1 g g   qI y  Muu  M ww  M w w  M q q  MMRMR  M B1 B1 Also in this case, the derivative ∂M/∂ẇ can be neglected to simplify the calculation and, therefore, can be removed from the third equation. Using the expressions (7.11), the above set of equations may be written in matrix notation (using a similar procedure just applied to hovering flight condition):  u   X u0  w   0     Zu  q  M u0    d   0 X w0 Z w0 M w0 0 X q0  Vz 0  g cos 0   u   X oMR X Bo1      Z q0  Vx 0  g sin 0   w   Z0MR Z B01   MR     q  M0 M B0   B1  0 M q0 1      MR  1 0 0 0    d   Now, we study the fixed control response (natural modes of motion) of the helicopter; therefore, we need to expand the following determinant:   X u0  Z u0  M u0 0  X w0  X q0  Vz 0   Z w0  Z q0  Vx0  M w0   M q0 1 0 g cos 0 g sin 0 0 0  that leads finally to the characteristic equation A4  B3  C2  D  E  0 (7.15) where A 1 B  M q0  Z w0  X u0 C  Mu0 X q0  Zw0 Mq0  M w0 Zq0  M w0Vx0  X u0M q0  X u0Zw0  Zu0 X w0 VII. Stability and control, introduction to helicopter flight dynamics 275  Mu0Vz 0 D  Mu0 g cos0  M w0 g sin 0  Mu0 X w0Vx0  Mu0Zw0 X q0  Mu0Zw0Vz 0  Zu0M w0Vz 0  Zu0M w0 X q0  Mu0 X w0 Zq0  X uo Zw0 M q0  X u0M w0 Zq0  X u0M w0Vx0  Zu0 X w0 M q0 E  (Mu0 X w0  X u0M w0 ) g sin 0  (Zu0M w0  Mu0Zw0 ) g cos0 Before analyzing the roots of the characteristic equation, it shall be noted that, generally, the values of the stability derivatives can vary throughout the flight envelope of the helicopter, from hovering to high-speed forward flight. Consequently, the trend of the stability derivatives has an impact on the typology of the roots so that the characteristic equation could have two pairs of complex conjugate roots or four real roots. In the case of two pairs of complex conjugate roots, a response similar to that of fixed-wing aircraft is obtained, with a pair of complex roots that corresponds to an oscillatory motion, with a long period, defined as phogoid mode (Figure 7.8). altitude Period (typically, 20 seconds) time Figure 7.8 Phogoid mode 276 Theory of helicopter flight The long period oscillatory motion, the phugoid mode, is characterized by changes in altitude and speed, with an angle of attack almost constant. In order to understand the motion represented, let us assume that, following a disturbance, the altitude increases: now, during the climb, a decrease in speed and the action produced by the weight take the helicopter to descend. Decreasing the altitude, the speed and the rotor thrust increase so that the oscillation restarts once again, and generally the motion is unstable. For this case, the very relevant stability derivatives are M wo , M q0 , M u0 , Zu0 . 7.7.1.1. Approximate calculation of longitudinal modes in forward level flight for a medium helicopter For example, we will study the approximate control fixed response of a utility helicopter (see also calculation in Section 7.6.1.3) in straight and level flight at V=100 knots, sea level, where we assume that the characteristic equation (7.15) becomes: 4  (3.3400)3  (0.4333)2  (0.2205)  (0.2414)  0 The roots of the characteristic equation above, or eigenvalues, are: λ1 = -0.4266 , λ2 = -3.2195, λ3,4 = 0.1530 ± 0.3903i Therefore, in this case we have two negative real roots which represent two stable responses with a thalf equal to: t half  0.693 n and: t half  0.693 0.693   1.6 seconds n  0.4266 t half  (for λ1), 0.693 0.693   0.2 seconds (for λ2) n  3.2195 From results above, we see that the stable modes are short-period responses. VII. Stability and control, introduction to helicopter flight dynamics 277 The complex roots λ3 and λ4 (that have a positive real parts) imply an unstable mode (the phugoid) with the following period T, time to double amplitude tdouble, undamped natural frequency ωn and damping ratio ζ: T 2  t double  2  16.1 seconds, 0.3903 0.693 0.693   4.5 seconds n 0.1530 n  n2   2  0.419 rad/s,  n n N double    0.1530  0.365 0.419 0.693 1   2  0.2806 2  Example of longitudinal root locus plot as a function of forward flight speed In Sections 7.6.1.3 and 7.7.1.1 we studied the longitudinal natural modes, in hovering flight and at V= 100 knots respectively, of a medium reference helicopter with a hingeless rotor, where we assumed also uncoupled longitudinal and lateral-directional motions. Again, we remember that this a critical assumption for the helicopter, because it is characterized by an asymmetric configuration. The fully coupled equations can show significant different results both in longitudinal and lateral-directional eigenvalues with respect to results provided by the analysis of the uncoupled set, from the hovering to the forward flight. This must be always considered when a rigorous analysis shall be performed. In particular, the following derivatives, which are neglected in the uncoupled analysis, shall be considered: Lu (roll moment due to longitudinal velocity), Lw (roll moment due to vertical velocity), Lq (roll moment due to pitch rate), Mv (pitch moment due to the lateral velocity), Mp (pitch moment due to the roll rate), Nw (yaw moment due to the vertical velocity), and the other control derivatives. Figure 7.9 illustrates the influence of the forward speed, from hovering to forward flight at V=100 knots at sea level, on the longitudinal eigenvalues for the helicopter used for calculation. 278 Theory of helicopter flight ω Roots for hovering flight (rad/s) Roots for level flight at V=100 knots 0.5 Phugoid Heavy subsidence Subsidence 3.0 Figure 7.9 7.7.2. 2.5 2.0 1.5 1.0 0.5 0 0.5 n (1/s) Example of longitudinal root locus as a function of forward speed Lateral-directional dynamic stability in forward flight In a similar manner, the set of equations for lateral-directional flight condition is: WG (v  Vx0 r  Vz 0 p)  Y  WG [ d cos 0 ] g p I x  rI xz  L rI z  p I xz  N VII. Stability and control, introduction to helicopter flight dynamics 279 By substituting the expressions for perturbations, the equations above become: WG (v  Vx0r  Vz 0 p)  WG [d cos 0 ]  Yv v  Yp p  Yr r  YA1 A1  Ytrtr g p I x  rI xz  Lvv  Lp p  Lr r  LA1 A1  Ltrtr rI z  p I xz  Nvv  N p p  Nr r  N A1 A1  Ntrtr and then: WG (v  Vx0r  Vz 0 p)  WG [d cos 0 ]  Yv v  Yp p  Yr r  YA1 A1  Ytrtr g  p I x  rI xz  Lvv  Lp p  Lr r  LA1 A1  Ltrtr   rI z  p I xz  Nvv  N p p  Nr r  N A1 A1  Ntrtr By using expressions (7.14a) and (7.14b) and by means of the procedure applied to the previous cases, we can rewrite the equations above, and the matrix Ad becomes:  Yv0 Yp0  Vz 0  0 L0p  Lv A d   N v0 N 0p  1 0 0 0   Vx 0  Yr0 L0r N r0 0 1 0  0 0  0 0 g cos 0 0 0 0 0 Therefore, we obtain the following determinant:   Yv0  Yp0  Vz 0 Vx0  Yr0  g cos 0   L0p  L0v  L0r 0 0 0 0   Nr  Nv  Np 0  0 1 0 0 0 1 and finally, we have the characteristic equation 0 0 0 0 0 0  280 Theory of helicopter flight A5  B4  C3  D2  E  F  0 where: A 1 B  Yv0  L0p  N r0 C  L0p N r0  N 0p L0r  Yvo N r0  Yv0 L0p  L0vYp0  N v0Yr0  N vVx0  L0vVz 0 D  L0v g cos0  Yv0 L0p Nr0  Yv0 L0r N 0p  Yp0 L0v Nr0  Yr0 L0v N 0p  Yp0 L0r Nv0  Yr0 L0p Nv0  (L0v N 0p  L0p Nv0 )Vx0  (L0v Nr0  L0r Nv0 )Vz 0 E  (L0v Nr0  L0r Nv0 ) g cos 0 F 0 As in the hover, the characteristic equation has one root (λ=0) relative to a neutral condition. Then, generally it presents other two real roots and two complex roots, as follows: — a rolling damped motion (the roll mode) and a spiral motion (the spiral mode) correspond to the two real roots; — a lateral-directional oscillation, LDO (therefore, an oscilla- tion in roll and in yaw, with a pitching), called ‘dutch-roll’, corresponds to the pair of complex roots, similar to the motion defined for the fixed wing aircraft. Figure 7.10 Dutch-roll oscillation VII. Stability and control, introduction to helicopter flight dynamics 281 Example of lateral-directional root locus plot as a function of forward flight speed Figure 7.11 illustrates the typical influence of the forward speed, from the hovering flight to high speed flight, on the lateral-directional eigenvalues for a medium helicopter. Im (λ) Roots for hovering flight Roots for level flight at V=140 knots Roll subsidence - LDO (Dutch-roll) Spiral subsidence 0 Re(λ) + Figure 7.11 Example of lateral-directional root locus as a function of forward speed 282 Theory of helicopter flight 7.8. Helicopter control 7.8.1. Stability, control and flying qualities The study of helicopter control is based on the analysis of the whole aircraft response following the application of one or more control inputs by the ‘human’ pilot or by an automatic system (auto-pilot). Through the flight controls, the pilot shall perform a flight manoeuvre or shall compensate adequately an atmospheric disturbance (as it can be a gust). Then, it is necessary to verify the helicopter response in the entire spectrum of manoeuvre and for each aircraft configuration. Moreover, it shall be noted that stability and manoeuvrability (capability of rapid response to a pilot action) are substantially opposite characteristics, because an aircraft with high stability ‘suffers’ from low manoeuvrability. From this standpoint, required stability and manoeuvrability will vary with the helicopter type and with the flight mission requirements. Generally, the reference specifications for stability and control requirements of V/STOL aircraft define various classes versus weights and manoeuvrability (light, heavy, low/medium and high manoeuvrability). Typically, the specifications define also the flying qualities in terms of defined levels, related to the capability to complete the flight mission. For example, typical levels are described as follows: Level 1 flying qualities are adequate Level 2 flying qualities are adequate, but increment in pilot work load is required Level 3 flying qualities allow the helicopter to fly safely, but intense pilot work load is required VII. Stability and control, introduction to helicopter flight dynamics 7.8.2. 283 Longitudinal control in hovering flight; one degree of freedom approach The application of a control by the pilot requires (in general, both in hovering and in forward flight) an additional action of compensation through other controls in order to perform correctly a manoeuvre (in some cases, also an appropriate mix of controls to minimize the secondary effects is present). For example, a change in collective pitch in hover generates, of course, a change in rotor thrust (the primary effect), but causes also a change in rotor torque that shall be counteracted acting through the pedals in order to maintain the flight direction. After this introduction (to be always present for an entire and advanced analysis), we will obtain very useful information also by simplified models. In particular, in the pages that follow we will analyze some cases of helicopter response by an approach with only one degree of freedom. Therefore, we can use Equations (7.10) with the application of the only one forcing B1 and with the assumption of one degree of freedom about the pitching axis. Then, in order to analyze the helicopter attitude, the third equation of (7.10) can be written as:  qI y  M q q  M B1 B1 Dividing by Iy, using the normalized notation adopted for the relationships (7.11), and going into Laplace domain (Appendix C), we obtain: M B01 q  B1 ( s  M q0 ) (7.16) The expression (7.16) is the transfer function of the pitch rate q due to the longitudinal cyclic pitch B1. Fixing the forcing changes and performing the inverse transformation, finally we have the relationship between q and B1 in time domain. In order to complete the treatment, note that M B1  M dMMR a1S  X   h   B1 da1S B1  B1 MR z Now, let us consider the case where a vertical gust of magnitude wgust is developed; then, consider the second equation of the group of Equations 284 Theory of helicopter flight (7.10), and by introducing the disturbance wgust and taking into account only the one degree of freedom to the vertical movement, we obtain:    WG w   Z w w  wgust  0 g and finally w  Z w0 w  Z w0 wgust As in the previous case, if we know the changes in disturbance, then we can calculate the approximate response of the system by means of the described methods. Note that it is: ZW  7.8.3. TTpp Z  w w Lateral-directional control in hovering flight; one degree of freedom approach If we consider the forcing θtr and only the degree of freedom about the yawing axis, the third equation of set (7.13) can be written as: rI z  Nr r  Ntrtr obtaining N0tr  tr (s  Nr0 ) r (7.16) By using the previous assumption, the relation (7.16) describes the response following the forcing (collective control input for the tail rotor). In particular, if we assume that the forcing changes according to the following rule: VII. Stability and control, introduction to helicopter flight dynamics 0  tr (t )    u (t )  st step t 0 (step function, step change input) t 0 and then tr s    st s finally, in the time domain we obtain: r t   st  285 Ntr Nr Nr t  1  e I z      (for t > 0).