A09

W eb content

Giovanni Di Giorgio

Theory of helicopter flight

Aerodynamics flight mechanics

Aracne editrice

wwwaracneeditriceitinfoaracneeditriceit

Copyright copy MMXVIIIGioacchino Onorati editore Srl mdash unipersonale

wwwgioacchinoonoratieditoreitinfogioacchinoonoratieditoreit

via Vittorio Veneto 2000020 Canterano (Rome)

(06) 45551463

isbn 978 ndash 88 ndash 255 ndash 1442 ndash 1

No part of this book may be reproduced by print photoprint microfilm microfiche or any other means

without publisherrsquos authorization

Ist edition April 2018

To my father Giuseppe and my mother Wilma

7

Contents

Preface 13

Units 15

Notation 17

Abbreviations 23

Chapter 1 Helicopter configurations

11 The helicopter and the vertical flight 25

12 Helicopter configurations

26

13 The rotor and the flight controls

131 Fundamental types of rotor

132 The flight controls and the swashplate mechanism

29

29

32

Chapter 2 Rotor aerodynamics hovering and vertical flight

21 Introduction 39

22 Momentum Theory 39

221 Vertical climb 40

222 Hovering flight 43

223 Vertical descent 46

224 Curves of induced velocity in vertical flight 48

23 Blade Element Theory 49

231 Rotor thrust and torque power required

52

232 Linear twist of rotor blade 57

233 Non-uniform induced velocity 58

234 Rotor blade root and tip losses 61

235 Figure of merit 62

236 Procedure for approximate and preliminary

calculation of the aerodynamic parameters

blade loads rotor power required

63

24 The ground effect 69

25 Introduction to Vortex Theory 71

251 Dynamics of ideal fluid

252 Fundamental relationships applied to the rotor

2521 Kutta-Joukowskyrsquos theorem application

72

76

77

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

W eb content

Giovanni Di Giorgio

Theory of helicopter flight

Aerodynamics flight mechanics

Aracne editrice

wwwaracneeditriceitinfoaracneeditriceit

Copyright copy MMXVIIIGioacchino Onorati editore Srl mdash unipersonale

wwwgioacchinoonoratieditoreitinfogioacchinoonoratieditoreit

via Vittorio Veneto 2000020 Canterano (Rome)

(06) 45551463

isbn 978 ndash 88 ndash 255 ndash 1442 ndash 1

No part of this book may be reproduced by print photoprint microfilm microfiche or any other means

without publisherrsquos authorization

Ist edition April 2018

To my father Giuseppe and my mother Wilma

7

Contents

Preface 13

Units 15

Notation 17

Abbreviations 23

Chapter 1 Helicopter configurations

11 The helicopter and the vertical flight 25

12 Helicopter configurations

26

13 The rotor and the flight controls

131 Fundamental types of rotor

132 The flight controls and the swashplate mechanism

29

29

32

Chapter 2 Rotor aerodynamics hovering and vertical flight

21 Introduction 39

22 Momentum Theory 39

221 Vertical climb 40

222 Hovering flight 43

223 Vertical descent 46

224 Curves of induced velocity in vertical flight 48

23 Blade Element Theory 49

231 Rotor thrust and torque power required

52

232 Linear twist of rotor blade 57

233 Non-uniform induced velocity 58

234 Rotor blade root and tip losses 61

235 Figure of merit 62

236 Procedure for approximate and preliminary

calculation of the aerodynamic parameters

blade loads rotor power required

63

24 The ground effect 69

25 Introduction to Vortex Theory 71

251 Dynamics of ideal fluid

252 Fundamental relationships applied to the rotor

2521 Kutta-Joukowskyrsquos theorem application

72

76

77

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

Giovanni Di Giorgio

Theory of helicopter flight

Aerodynamics flight mechanics

Aracne editrice

wwwaracneeditriceitinfoaracneeditriceit

Copyright copy MMXVIIIGioacchino Onorati editore Srl mdash unipersonale

wwwgioacchinoonoratieditoreitinfogioacchinoonoratieditoreit

via Vittorio Veneto 2000020 Canterano (Rome)

(06) 45551463

isbn 978 ndash 88 ndash 255 ndash 1442 ndash 1

No part of this book may be reproduced by print photoprint microfilm microfiche or any other means

without publisherrsquos authorization

Ist edition April 2018

To my father Giuseppe and my mother Wilma

7

Contents

Preface 13

Units 15

Notation 17

Abbreviations 23

Chapter 1 Helicopter configurations

11 The helicopter and the vertical flight 25

12 Helicopter configurations

26

13 The rotor and the flight controls

131 Fundamental types of rotor

132 The flight controls and the swashplate mechanism

29

29

32

Chapter 2 Rotor aerodynamics hovering and vertical flight

21 Introduction 39

22 Momentum Theory 39

221 Vertical climb 40

222 Hovering flight 43

223 Vertical descent 46

224 Curves of induced velocity in vertical flight 48

23 Blade Element Theory 49

231 Rotor thrust and torque power required

52

232 Linear twist of rotor blade 57

233 Non-uniform induced velocity 58

234 Rotor blade root and tip losses 61

235 Figure of merit 62

236 Procedure for approximate and preliminary

calculation of the aerodynamic parameters

blade loads rotor power required

63

24 The ground effect 69

25 Introduction to Vortex Theory 71

251 Dynamics of ideal fluid

252 Fundamental relationships applied to the rotor

2521 Kutta-Joukowskyrsquos theorem application

72

76

77

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

Aracne editrice

wwwaracneeditriceitinfoaracneeditriceit

Copyright copy MMXVIIIGioacchino Onorati editore Srl mdash unipersonale

wwwgioacchinoonoratieditoreitinfogioacchinoonoratieditoreit

via Vittorio Veneto 2000020 Canterano (Rome)

(06) 45551463

isbn 978 ndash 88 ndash 255 ndash 1442 ndash 1

No part of this book may be reproduced by print photoprint microfilm microfiche or any other means

without publisherrsquos authorization

Ist edition April 2018

To my father Giuseppe and my mother Wilma

7

Contents

Preface 13

Units 15

Notation 17

Abbreviations 23

Chapter 1 Helicopter configurations

11 The helicopter and the vertical flight 25

12 Helicopter configurations

26

13 The rotor and the flight controls

131 Fundamental types of rotor

132 The flight controls and the swashplate mechanism

29

29

32

Chapter 2 Rotor aerodynamics hovering and vertical flight

21 Introduction 39

22 Momentum Theory 39

221 Vertical climb 40

222 Hovering flight 43

223 Vertical descent 46

224 Curves of induced velocity in vertical flight 48

23 Blade Element Theory 49

231 Rotor thrust and torque power required

52

232 Linear twist of rotor blade 57

233 Non-uniform induced velocity 58

234 Rotor blade root and tip losses 61

235 Figure of merit 62

236 Procedure for approximate and preliminary

calculation of the aerodynamic parameters

blade loads rotor power required

63

24 The ground effect 69

25 Introduction to Vortex Theory 71

251 Dynamics of ideal fluid

252 Fundamental relationships applied to the rotor

2521 Kutta-Joukowskyrsquos theorem application

72

76

77

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

To my father Giuseppe and my mother Wilma

7

Contents

Preface 13

Units 15

Notation 17

Abbreviations 23

Chapter 1 Helicopter configurations

11 The helicopter and the vertical flight 25

12 Helicopter configurations

26

13 The rotor and the flight controls

131 Fundamental types of rotor

132 The flight controls and the swashplate mechanism

29

29

32

Chapter 2 Rotor aerodynamics hovering and vertical flight

21 Introduction 39

22 Momentum Theory 39

221 Vertical climb 40

222 Hovering flight 43

223 Vertical descent 46

224 Curves of induced velocity in vertical flight 48

23 Blade Element Theory 49

231 Rotor thrust and torque power required

52

232 Linear twist of rotor blade 57

233 Non-uniform induced velocity 58

234 Rotor blade root and tip losses 61

235 Figure of merit 62

236 Procedure for approximate and preliminary

calculation of the aerodynamic parameters

blade loads rotor power required

63

24 The ground effect 69

25 Introduction to Vortex Theory 71

251 Dynamics of ideal fluid

252 Fundamental relationships applied to the rotor

2521 Kutta-Joukowskyrsquos theorem application

72

76

77

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

7

Contents

Preface 13

Units 15

Notation 17

Abbreviations 23

Chapter 1 Helicopter configurations

11 The helicopter and the vertical flight 25

12 Helicopter configurations

26

13 The rotor and the flight controls

131 Fundamental types of rotor

132 The flight controls and the swashplate mechanism

29

29

32

Chapter 2 Rotor aerodynamics hovering and vertical flight

21 Introduction 39

22 Momentum Theory 39

221 Vertical climb 40

222 Hovering flight 43

223 Vertical descent 46

224 Curves of induced velocity in vertical flight 48

23 Blade Element Theory 49

231 Rotor thrust and torque power required

52

232 Linear twist of rotor blade 57

233 Non-uniform induced velocity 58

234 Rotor blade root and tip losses 61

235 Figure of merit 62

236 Procedure for approximate and preliminary

calculation of the aerodynamic parameters

blade loads rotor power required

63

24 The ground effect 69

25 Introduction to Vortex Theory 71

251 Dynamics of ideal fluid

252 Fundamental relationships applied to the rotor

2521 Kutta-Joukowskyrsquos theorem application

72

76

77

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

8 Contents

2522 Velocities induced by vortices Biot-Savartrsquos Law

253 Modelling rotor in hover and approach to

calculation

254 Interference phenomenon due to blade tip vortex

255 Prescribed wake Landgrebersquos model in hovering flight

78

81

82

83

Chapter 3 Rotor dynamics

31 Introduction 87

32 Fundamental axes and planes 87

33 The flapping motion of the blade 90

34 Flapping hinge offset and control moments

93

35 The rotor in forward flight and the blade flapping

98

36 The lagging motion of the blade 99

37 The cyclic feathering 101

38 Coupling of fundamental motions of the rotor blade 103

39 Calculation of centrifugal force along the blade 106

Chapter 4 Rotor aerodynamics forward flight

41 Introduction 109

42 Momentum Theory 109

43 Blade Element Theory 113

431 Parameters for determination of blade angle of

attack

432 Blade element and local incidence

433 Aerodynamic forces acting on the rotor

closed form equations

4331 Calculation of the thrust

4332 Rotor coning and flapping coefficients

4333 Calculation of the drag

4334 Calculation of the torque

113

118 18

120

123 27

127

131

135

44 Reverse flow region 138

45 Forces and parameters related to tip path plane and to

hub plane

139

451 Equations referred to the tip path plane

452 Equations referred to the hub plane

139

141

46 Helicopter in trim and rotor aerodynamics 144

47 Corrections of results of Blade Element Theory 148

48 Blade element theory limitations 149

49 Stall and compressibility phenomena 150

491 Swept blade tip and local Mach number 155

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

Contents 9

410 Rotor wake models in forward flight 156

411 Computational aerodynamics advanced

methodologies multidisciplinary approach

158

Chapter 5 Helicopter trim analysis

51 Introduction 161

52 Systems of axes 162

53 General equations of motion of helicopter 164

54 Helicopter trim conditions 168

541 The general trim analysis 169

55 The rotor-fuselage system and the torque reaction 171

56 Simplified development of equilibrium (trim) 173

561 Trim equations in forward flight 173

562 The expression for power in forward level flight 179

57 Approximate and quick estimation of longitudinal

equilibrium 181

58 General trim solution 185

59 Autorotation 195

591 Autorotation of a rotor 195

5911 Aerodynamics of autorotation 195

5912 Final phase of an autorotation 197

592 Limitations in autorotation and Height-Velocity

Diagram

198

593 Final notes 200

Chapter 6 Helicopter flight performance

61 Introduction 201

62 Total power required 201

63 Standard atmosphere 202

64 The engine and the power available 205

641 The operating condition of the main rotor 205

642 Configuration of free shaft turbine engine 206

643 Rotortransmissionengine system 208

644 Performance of installed engine and power

ratings

209

65 Hover performance 212

651 Power required PMR and Ptr in hovering flight

212

652 Vertical drag of the helicopter

653 Maximum hover ceiling

213

214

66 Performance in vertical climb 215

67 Performance in forward level flight

671 Power required PMR and Ptr

216

216

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

10 Contents

6711 The parasitic drag Df in forward level flight 219

672 The total power required in level flight

6721 Maximum speed in level flight

6722 Maximum endurance and maximum range

6723 Power increments due to stall and

compressibility

221

225

226

228

68 Forward climb and descent performance 229

681 Power required PMR in forward climb

229

682 Rates and angles of climb ceiling altitude 230

683 Power required PMR in forward descent

234

69 Autorotative performance 234

610 Introduction to mission analysis 237

6101 Take-off and landing weight 237

6102 An approach to helicopter mission analysis 238

Chapter 7 Stability and control introduction to helicopter

flight dynamics

71 Introduction 241

72 The single-degree of freedom dynamic system 242

73 Helicopter static stability and dynamic stability 250

74 Helicopter static stability 251

741 Stability following forward speed perturbation 251

742 Stability following vertical speed or incidence

perturbation 251

743 Stability following yawing perturbation 252

75 Helicopter dynamic stability 252

751 Small disturbance theory 255

752 Stability derivatives 257

7521 Force perturbation expressions and stability 259

derivatives

7522 Moment perturbation expressions and stability 260

derivatives

753 Notes on the methodology of small perturbations 261

76 Dynamic stability in hovering flight 261

761 Longitudinal dynamic stability in hovering flight

7611 Equations of motion state variable form

261

263

7612 Stability derivatives calculation Mq and Mu in

hover 267

7613 Approximate calculation of longitudinal modes

in hovering flight for a medium helicopter 268

7614 The characteristic roots on complex plane 269

762 Lateral-directional dynamic stability in hovering 270

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

Contents 11

flight

77 Dynamic stability in forward flight

273

771 Longitudinal dynamic stability in forward flight 273

7711 Approximate calculation of longitudinal modes

in forward flight for a medium helicopter 276

772 Lateral-directional dynamic stability in forward

flight 278

78 Helicopter control 282

781 Stability control and flying qualities 282

782 Longitudinal control in hovering flight one

degree of freedom approach

283

783 Lateral-directional control in hovering flight

one degree of freedom approach

284

Chapter 8 Manoeuvres in horizontal and in vertical planes

81 Introduction 287

82 Steady turn 287

821 Notes on turn manoeuvres 289

822 Gyroscopic moments in turn 289

823 Power required in steady turn 290

83 Symmetrical pull-up 290

Chapter 9 Coaxial rotor and tandem rotor helicopter

91 Introduction 293

92 Coaxial rotor helicopter 293

921 Application of Momentum Theory to the

hovering flight

293

922 General characteristics of the helicopter 296

923 Helicopter equilibrium about the body Z-axis 297

93 Tandem rotor helicopters 298

931 General description and definitions 298

932 Application of Momentum Theory and of

Blade Element Theory to the hovering flight

300

933 Application of Momentum Theory to the level

forward flight

303

934 Experimental data 305

935 Condition of longitudinal equilibrium of the

helicopter

305

936 Notes on stability 308

9361 Forward speed disturbance 308

9362 Stick-fixed dynamic stability in hovering flight 309

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

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 fun-

damentals 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 configu-

rations

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 argu-

ments of my book Teoria del volo dellrsquoelicottero 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 posi-

tive 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

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

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 rotary-

wing 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 spe-

cial 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 pilotrsquos 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 rotary-

wing 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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

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

Conversion SI Brit S

Mass kg slug 1 slug = 145939 kg

Length m ft 1 ft = 03048 m

Time s s -

Temperature degK degR 1 (degR) = [1(18)] (degK)

Temp(degK) = 27315 + temp(degC)

Supplementary units

Quantity Units

Conversion SI Brit S

Angle (plane) rad rad -

Derived quantities

Quantity Units

Conversion SI Brit S

Velocity ms fts 1 fts = 03048 ms

Angular

Velocity rads rads -

Acceleration ms2 fts2 1 fts2 = 03048 ms2

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

16 Units

Quantity Units

Conversion SI Brit S

Acceleration

of gravity ms2 fts2 g = 980665 ms2 =

32174 fts2

Air density kgm3 slugft3 1 slugft3 = 515379 kgm3

Force N lb 1 lb = 444822 N

Pressure Pa

(1 Pa = 1 Nm2) lbft2 1 lbft2 = 478803 Nm2

Power W lbfts

(1 hp = 550 lbfts)

1 lbfts = 135575 W =

(1550) hp

Multiples

Quantity Units

Conversion SI Brit S

Velocity mmin

metre per minute

ftmin

foot per minute 1 ftmin = 03048 mmin

Additional Unit

Quantity Unit Conversion

Angular

Velocity

rpm

(revolution per minute) 1 rpm = (2π60) rads

Velocity

kn (international knot)

=

one nautical mile per hour

-

(one international nautical mile) =

1852 m = 6076115 ft

Angle

(plane) deg (degree) 1deg = (π180) rad

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

17

Notation

Symbol Units (SI)

a lift curve slope of blade section

rad-1

0a coning angle main rotor

rad

1a 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 2RA

m2

1A lateral cyclic pitch

rad

trA tail rotor disc area 2

trtr RA

m2

b number of blades main rotor

-

1b coefficient of term (-sinψ) into expression of the

flapping angle β relative to the no-feathering plane

lateral flapping coefficient

rad

trb number of blades tail rotor

-

B tip loss factor

-

1B longitudinal cyclic pitch

rad

c blade section chord main rotor

m

trc blade section chord tail rotor

m

dC section drag coefficient

-

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

18 Notation

lC section lift coefficient

-

PC main rotor power coefficient

-

QC main rotor torque coefficient

-

TC main rotor thrust coefficient

-

fD parasitic drag of helicopter

N

LD disc loading

Nm2

f equivalent flat plate drag area

m2

G gravitational acceleration

ms2

G helicopter centre of gravity origin of the body-axis

system

-

dH density altitude

m

pH pressure altitude

m

fI mass moment of inertia of blade about flapping hinge kgm2

k induced power factor main rotor

-

trk induced power factor tail rotor

-

pk climb efficiency factor

-

GK constant into Glauertrsquos second formula of the induced

velocity

-

K term of 3 K effect

-

trl tail rotor moment arm

m

M Mach number

-

M disturbance term about the Y-axis for aerodynamic

moments

N∙m

AM aerodynamic moment about the flapping hinge

N∙m

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

Notation 19

dM drag divergence Mach number

-

heliM

mass of helicopter gWM Gheli kg

n load factor

-

O origin of the Earth-axis system

-

p pressure of air

Nm2

0p pressure of air at sea level ISA conditions

Nm2

MRP main rotor power required

W

trP tail rotor power required

W

Q main rotor torque

N∙m

r radial distance of blade element from axis of rotation

Rr 0

m

re effective blade radius

m

R main rotor radius

m

trR tail rotor radius

m

T main rotor thrust

N

T temperature of air

degK

0T temperature of air at sea level ISA conditions

degK

trT tail rotor thrust

N

iv induced velocity at rotor

ms

ihv induced velocity at rotor in hover

ms

V true airspeed of helicopter along the flight path

velocity of the free airstream

ms

cV climb velocity

ms

dV descent velocity ms

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s

20 Notation

TV RVT or main rotor tip speed in hovering flight

ms

TtrV trtrTtr RV or tail rotor tip speed in hovering flight

ms

x Rrx 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

-

GW 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 fIacR4

-

r climb angle

rad

inflow angle at blade element

rad

circulation

m2s