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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