HELICOPTER FLIGHT DYNAMICS

The Theory and Applicationof

Flying Qualitiesand

Simulation Modelling

Second Edition

Gareth D. PadfieldBSc, PhD, C Eng, FRAeS

HELICOPTER FLIGHT DYNAMICS

HELICOPTER FLIGHT DYNAMICS

The Theory and Applicationof

Flying Qualitiesand

Simulation Modelling

Second Edition

Gareth D. PadfieldBSc, PhD, C Eng, FRAeS

C© 1996, 2007 by G.D. Padfield

Blackwell Publishing editorial offices:Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK

Tel: +44 (0)1865 776868Blackwell Publishing Asia Pty Ltd, 550 Swanston Street, Carlton, Victoria 3053, Australia

Tel: +61 (0)3 8359 1011

ISBN: 978-14051-1817-0

Published in North America by American Institute of Aeronautics and Astronautics, Inc.370 L’ Enfant Promenade, SW, Washington DC 20024-2518

The right of the Author to be identified as the Author of this Work has been asserted inaccordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying,recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act1988, without the prior permission of the publisher.

First published 1996Second edition published 2007

Library of Congress Cataloging-in-Publication Data:A catalogue record for this title is available from the Library of Congress

British Library Cataloguing-in-Publication Data:Padfield, G. D.Helicopter flight dynamics : the theory and application of flying qualities and simulationmodelling/Gareth D. Padfield. – 2nd ed.

p. cm.Includes bibliographical references and index.ISBN-13: 978-1-4051-1817-0 (hardback : alk. paper)ISBN-10: 1-4051-1817-2 (hardback : alk. paper)1. Helicopters–Aerodynamics. 2. Helicopters–Handling characteristics. I. Title.

TL716.P23 2007629.132′5252–dc22

2007004737

Set in 9.5/12 pt Timesby Techbooks, New Delhi, IndiaPrinted and bound in Singaporeby Markono Print Media Pte. Ltd

The publisher’s policy is to use permanent paper from mills that operate a sustainable forestrypolicy, and which has been manufactured from pulp processed using acid-free and elementarychlorine-free practices. Furthermore, the publisher ensures that the text paper and cover boardused have met acceptable environmental accreditation standards.

For further information on Blackwell Publishing, visit our website:www.blackwellpublishing.com

To my familyJoey, Jude and George

Contents

Preface to first edition xiiiPreface to second edition xviiCopyright acknowledgements xxiNotation xxiiiList of abbreviations xxxiii

Chapter 1 Introduction1.1 Simulation modelling 11.2 Flying qualities 31.3 Missing topics 41.4 Simple guide to the book 5

Chapter 2 Helicopter flight dynamics – an introductory tour2.1 Introduction 92.2 Four reference points 10

2.2.1 The mission and piloting tasks 112.2.2 The operational environment 142.2.3 The vehicle configuration, dynamics and flight envelope 15

Rotor controls 15Two distinct flight regimes 17Rotor stall boundaries 20

2.2.4 The pilot and pilot–vehicle interface 222.2.5 Resume of the four reference points 24

2.3 Modelling helicopter flight dynamics 25The problem domain 25Multiple interacting subsystems 26Trim, stability and response 28The flapping rotor in vacuo 30The flapping rotor in air – aerodynamic damping 33Flapping derivatives 36The fundamental 90◦ phase shift 36Hub moments and rotor/fuselage coupling 38Linearization in general 41Stability and control resume 42The static stability derivative Mw 43Rotor thrust, inflow, Zw and vertical gust response in hover 46Gust response in forward flight 48Vector-differential form of equations of motion 50

viii Contents

Validation 52Inverse simulation 57Modelling review 58

2.4 Flying qualities 59Pilot opinion 60Quantifying quality objectively 61Frequency and amplitude – exposing the natural dimensions 62Stability – early surprises compared with aeroplanes 63Pilot-in-the-loop control; attacking a manoeuvre 66Bandwidth – a parameter for all seasons? 67Flying a mission task element 70The cliff edge and carefree handling 71Agility factor 72Pilot’s workload 73Inceptors and displays 75Operational benefits of flying qualities 75Flying qualities review 77

2.5 Design for flying qualities; stability and control augmentation 78Impurity of primary response 79Strong cross-couplings 79Response degradation at flight envelope limits 80Poor stability 80The rotor as a control filter 81Artificial stability 81

2.6 Chapter review 84

Chapter 3 Modelling helicopter flight dynamics: building a simulation model3.1 Introduction and scope 873.2 The formulation of helicopter forces and moments in level 1 modelling 91

3.2.1 Main rotor 93Blade flapping dynamics – introduction 93The centre-spring equivalent rotor 96Multi-blade coordinates 102Rotor forces and moments 108

Rotor torque 114Rotor inflow 115

Momentum theory for axial flight 116Momentum theory in forward flight 119

Local-differential momentum theory and dynamic inflow 125Rotor flapping–further considerations of the centre-spring

approximation 128Rotor in-plane motion – lead–lag 135Rotor blade pitch 138Ground effect on inflow and induced power 139

3.2.2 The tail rotor 1423.2.3 Fuselage and empennage 146

The fuselage aerodynamic forces and moments 146The empennage aerodynamic forces and moments 149

Contents ix

3.2.4 Powerplant and rotor governor 1523.2.5 Flight control system 154

Pitch and roll control 154Yaw control 158Heave control 158

3.3 Integrated equations of motion of the helicopter 1593.4 Beyond level 1 modelling 162

3.4.1 Rotor aerodynamics and dynamics 163Rotor aerodynamics 163

Modelling section lift, drag and pitching moment 164Modelling local incidence 167

Rotor dynamics 1683.4.2 Interactional aerodynamics 171

Appendix 3A Frames of reference and coordinate transformations 1753A.1 The inertial motion of the aircraft 1753A.2 The orientation problem – angular coordinates of the aircraft 1803A.3 Components of gravitational acceleration along the aircraft axes 1813A.4 The rotor system – kinematics of a blade element 1823A.5 Rotor reference planes – hub, tip path and no-feathering 184

Chapter 4 Modelling helicopter flight dynamics: trim and stability analysis4.1 Introduction and scope 1874.2 Trim analysis 192

4.2.1 The general trim problem 1944.2.2 Longitudinal partial trim 1964.2.3 Lateral/directional partial trim 2014.2.4 Rotorspeed/torque partial trim 2034.2.5 Balance of forces and moments 2044.2.6 Control angles to support the forces and moments 204

4.3 Stability analysis 2084.3.1 Linearization 2094.3.2 The derivatives 214

The translational velocity derivatives 215The angular velocity derivatives 224The control derivatives 231The effects of non-uniform rotor inflow on damping and control

derivatives 234Some reflections on derivatives 235

4.3.3 The natural modes of motion 236The longitudinal modes 241The lateral/directional modes 247Comparison with flight 250

Appendix 4A The analysis of linear dynamic systems (with special reference to6 DoF helicopter flight) 252

Appendix 4B The three case helicopters: Lynx, Bo105 and Puma 2614B.1 Aircraft configuration parameters 261

The DRA (RAE) research Lynx, ZD559 261The DLR research Bo105, S123 261

x Contents

The DRA (RAE) research Puma, SA330 263Fuselage aerodynamic characteristics 264Empennage aerodynamic characteristics 268

4B.2 Stability and control derivatives 2694B.3 Tables of stability and control derivatives and system eigenvalues 277

Appendix 4C The trim orientation problem 293

Chapter 5 Modelling helicopter flight dynamics: stability under constraintand response analysis

5.1 Introduction and scope 2975.2 Stability under constraint 298

5.2.1 Attitude constraint 2995.2.2 Flight-path constraint 306

Longitudinal motion 306Lateral motion 310

5.3 Analysis of response to controls 3155.3.1 General 3155.3.2 Heave response to collective control inputs 317

Response to collective in hover 317Response to collective in forward flight 323

5.3.3 Pitch and roll response to cyclic pitch control inputs 325Response to step inputs in hover – general features 325Effects of rotor dynamics 327Step responses in hover – effect of key rotor parameters 327Response variations with forward speed 330Stability versus agility – contribution of the horizontal tailplane 331Comparison with flight 332

5.3.4 Yaw/roll response to pedal control inputs 3385.4 Response to atmospheric disturbances 344

Modelling atmospheric disturbances 346Modelling helicopter response 348Ride qualities 350

Chapter 6 Flying qualities: objective assessment and criteria development6.1 General introduction to flying qualities 3556.2 Introduction and scope: the objective measurement of quality 3606.3 Roll axis response criteria 364

6.3.1 Task margin and manoeuvre quickness 3646.3.2 Moderate to large amplitude/low to moderate frequency: quickness

and control power 3716.3.3 Small amplitude/moderate to high frequency: bandwidth 378

Early efforts in the time domain 378Bandwidth 381Phase delay 386Bandwidth/phase delay boundaries 387Civil applications 389The measurement of bandwidth 391

Contents xi

Estimating ωbw and τp 397Control sensitivity 399

6.3.4 Small amplitude/low to moderate frequency: dynamic stability 4016.3.5 Trim and quasi-static stability 402

6.4 Pitch axis response criteria 4046.4.1 Moderate to large amplitude/low to moderate frequency: quickness

and control power 4046.4.2 Small amplitude/moderate to high frequency: bandwidth 4086.4.3 Small amplitude/low to moderate frequency: dynamic stability 4106.4.4 Trim and quasi-static stability 413

6.5 Heave axis response criteria 4176.5.1 Criteria for hover and low speed flight 4206.5.2 Criteria for torque and rotorspeed during vertical axis manoeuvres 4246.5.3 Heave response criteria in forward flight 4246.5.4 Heave response characteristics in steep descent 427

6.6 Yaw axis response criteria 4296.6.1 Moderate to large amplitude/low to moderate frequency: quickness

and control power 4306.6.2 Small amplitude/moderate to high frequency: bandwidth 4336.6.3 Small amplitude/low to moderate frequency: dynamic stability 4336.6.4 Trim and quasi-static stability 436

6.7 Cross-coupling criteria 4376.7.1 Pitch-to-roll and roll-to-pitch couplings 4376.7.2 Collective to pitch coupling 4406.7.3 Collective to yaw coupling 4406.7.4 Sideslip to pitch and roll coupling 440

6.8 Multi-axis response criteria and novel-response types 4426.8.1 Multi-axis response criteria 4426.8.2 Novel response types 444

6.9 Objective criteria revisited 447

Chapter 7 Flying qualities: subjective assessment and other topics7.1 Introduction and scope 4557.2 The subjective assessment of flying quality 456

7.2.1 Pilot handling qualities ratings – HQRs 4577.2.2 Conducting a handling qualities experiment 464

Designing a mission task element 464Evaluating roll axis handling characteristics 466

7.3 Special flying qualities 4787.3.1 Agility 478

Agility as a military attribute 478The agility factor 481Relating agility to handling qualities parameters 484

7.3.2 The integration of controls and displays for flight in degraded visualenvironments 487Flight in DVE 487Pilotage functions 488Flying in DVE 489

xii Contents

The usable cue environment 490UCE augmentation with overlaid symbology 496

7.3.3 Carefree flying qualities 5007.4 Pilot’s controllers 5087.5 The contribution of flying qualities to operational effectiveness and the safety

of flight 511

Chapter 8 Flying qualities: forms of degradation8.1 Introduction and scope 5178.2 Flight in degraded visual environments 519

8.2.1 Recapping the usable cue environment 5208.2.2 Visual perception in flight control – optical flow and motion parallax 5238.2.3 Time to contact; optical tau, τ 5328.2.4 τ control in the deceleration-to-stop manoeuvre 5368.2.5 Tau-coupling – a paradigm for safety in action 5388.2.6 Terrain-following flight in degraded visibility 545

τ on the rising curve 5488.3 Handling qualities degradation through flight system failures 559

8.3.1 Methodology for quantifying flying qualities following flight functionfailures 562

8.3.2 Loss of control function 564Tail rotor failures 564

8.3.3 Malfunction of control – hard-over failures 5688.3.4 Degradation of control function – actuator rate limiting 574

8.4 Encounters with atmospheric disturbances 5768.4.1 Helicopter response to aircraft vortex wakes 578

The wake vortex 578Hazard severity criteria 579Analysis of encounters – attitude response 587Analysis of encounters – vertical response 588

8.4.2 Severity of transient response 5938.5 Chapter Review 597Appendix 8A HELIFLIGHT and FLIGHTLAB at the University of Liverpool 599

FLIGHTLAB 601Immersive cockpit environment 602

References 608Index 633

Preface to first edition

In this preface, I want to communicate three things. First, I would like to share withthe reader my motivation for taking on this project. Second, I want to try to identifymy intended audience and, third, I want to record some special acknowledgements tocolleagues who have helped me.

When I decided to pursue a career as an aeronautical engineer, my motivationstemmed from an aesthetic delight in flight and things that flew, combined with anuncanny interest in tackling, and sometimes solving, difficult technical problems. Bothheld a mystery for me and together, unbeknown to me at the time, helped me to‘escape’ the Welsh mining community in which I had been sculptured, on to the roadsof learning and earning. Long before that, in the late 1940s, when I was taking myfirst gasps of Welsh air, the Royal Aircraft Establishment (RAE) had been conductingthe first research flight trials to understand helicopter stability and control. It shouldbe remembered that at that time, practical helicopters had been around for less thana decade. From reading the technical reports and talking with engineers who workedin those days, I have an image of an exciting and productive era, with test and theorycontinuously wrestling to provide first-time answers to the many puzzles of helicopterflight dynamics.

Although there have been quiet periods since then, the RAE sustained its heli-copter research programme through the 1950s, 1960s and 1970s and by the time Itook charge of the activities at Bedford in the mid-1980s, it had established itself atthe leading edge of research into rotor aerodynamics and helicopter flight dynamics.My own helicopter journey began in the Research Department at Westland Helicoptersin the early 1970s. At that time, Westland were engaged with the flight testing of theprototype Lynx, a helicopter full of innovation for a 1960s design. This was also anexciting era, when the foundations of my understanding of helicopter flight dynamicswere laid down. Working with a small and enthusiastic group of research engineers,the mysteries began to unfold, but at times it felt as if the more I learned, the less Iunderstood. I do not want to use the word enthusiastic lightly in this context; a greatnumber of helicopter engineers that I have known have a degree of enthusiasm thatgoes way beyond the call of duty, so to speak, and I do believe that this is a spe-cial characteristic of people in this relatively small community. While it is inevitablethat our endeavours are fuelled by the needs of others – the ubiquitous customer, forexample – enthusiasm for the helicopter and all of the attendant technologies is apowerful and dynamic force. In writing this book I have tried to share some of myenthusiasm and knowledge of helicopter flight dynamics with as large an audienceas possible, and that was probably sufficient personal motivation to undertake thetask. This motivation is augmented by a feeling that my own experience in theoryand test has given me insight into, and a somewhat unique way of looking at, the

xiv Preface to first edition

subject of flight dynamics that I hope will appeal to the reader in search of under-standing.

There are, however, more pragmatic reasons for writing this book. While fixed-wing flight dynamics, stability and control have been covered from a number of per-spectives in more than a dozen treatise over the years, there has never been a helicoptertextbook dedicated to the subject; so there is, at least, a perceived gap in the availableliterature, and, perhaps more importantly, the time is ripe to fill that gap. The last 10–20years has seen a significant amount of research in flight simulation and flying qualitiesfor helicopters, much of which has appeared in the open literature but is scattered inscores of individual references. This book attempts to capture the essence of this workfrom the author’s perspective, as a practitioner involved in the DRA (RAE) researchin national and international programmes. It has been a busy and productive period,indeed it is still continuing, and I hope that this book conveys the impression of a livingand mature subject, to which many contributions are yet to be made.

The book is written mainly for practising flight dynamics engineers. In someorganizations, such a person may be described as a flying qualities engineer, a flightsimulation engineer or even a flight controls engineer, but my personal view is that thesetitles reflect subdisciplines within the larger field of flight dynamics. Key activities of theflight dynamics engineer are simulation modelling, flying qualities and flight control.Simulation brings the engineer into a special and intimate relationship with the systemhe or she is modelling and the helicopter is a classic example. The present era appears tobe characterized by fast-disappearing computational constraints on our ability to modeland simulate the complex aeroelastic interactions involved in helicopter flight. Keepingstep with these advances, the flight dynamics engineer must, at the same time, preservean understanding of the link between cause and effect. After all, the very objectives ofmodelling and simulation are to gain an understanding of the effects of various designfeatures and insight into the sensitivity of flight behaviour to changes in configurationand flight condition. In the modelling task, the flight dynamics engineer will need toaddress all the underlying assumptions, and test them against experimental data, in away that provides as complete a calibration as possible. The flight dynamics engineerwill also have a good understanding of flying qualities and the piloting task, and heor she will appreciate the importance of the external and internal influences on thesequalities and the need for mission-oriented criteria. Good flying qualities underpinsafe flight, and this book attempts to make the essence of the theoretical developmentsand test database, assembled over the period from the early 1980s through to thepresent time, accessible to practising engineers. Flight testing is an important part offlight dynamics, supporting both simulation validation and the development of flyingqualities criteria. In this book I have attempted to provide the tools for building andanalysing simulation models of helicopter flight, and to present an up-to-date treatmentof flying qualities criteria and flight test techniques.

While this is primarily a specialist’s book, it is also written for those with empathyfor the broader vision, within which flight dynamics plays its part. It is hoped that thebook, or parts of the book, will appeal to test pilots and flight test engineers andoffer something useful to engineers without aeronautical backgrounds, or those whohave specialized in the aerodynamic or controls disciplines and wish to gain a broaderperspective of the functionality of the total aircraft. In writing Chapters 2, 6 and 7, Ihave tried to avoid a dependence on ‘difficult’ mathematics. Chapters 3, 4 and 5, onthe other hand, require a reasonable grasp of analytical and vectorial mechanics as

Preface to first edition xv

would, for example, be taught in the more extensive engineering courses at first andhigher degree levels. With regard to education programmes, I have had in mind thatdifferent parts of the book could well form the subject of one or two term courses atgraduate or even advanced undergraduate level. I would strongly recommend Chapter 2to all who have embarked on a learning programme with this book. Taught well, I havealways believed that flight dynamics is inspirational and, hence, a motivating subjectat university level, dealing with whole aircraft and the way they fly, and, at the sametime, the integration of the parts that make the whole. I have personally gained muchfrom the subject and perhaps this book also serves as an attempt to return my ownpersonal understandings into the well of knowledge.

In the sense that this book is an offering, it also reflects the great deal of gratitudeI feel towards many colleagues over the years, who have helped to make the businessenjoyable, challenging and stimulating for me. I have been fortunate to be part of severalendeavours, both nationally and internationally, that have achieved significant progress,compared with the sometimes more limited progress possible by individuals workingon their own. International collaboration has always held a special interest for me and Iam grateful to AGARD, Garteur, TTCP and other, less formal, ties with European andNorth American agencies, for providing the auspices for collaboration. Once again,this book is full of the fruits of these activities. I genuinely believe that helicopters ofthe future will perform better, be safer and be easier to fly because of the efforts of thevarious research groups working together in the field of flight dynamics, feeding theresults into the acquisition processes in the form of the requirements specifications,and into the manufacturing process, through improved tools and technologies.

In the preparation of this book several colleagues have given me specific supportwhich I would like to acknowledge. For assistance in the generation and presentation ofkey results I would like to acknowledge the Rotorcraft Group at DRA Bedford. But mygratitude to the Bedford team goes far beyond the specific support activities, and I resistidentifying individual contributions for that reason. As a team we have pushed forwardin many directions over the last 10 years, sometimes at the exciting but lonely leadingedge, at other times filling in the gaps left by others pushing forward with greater paceand urgency. I want to record that this book very much reflects these team efforts, asindicated by the many cited references. I was anxious to have the book reviewed in acritical light before signing it off for publication, and my thanks go to colleagues andfriends Ronald Milne, Ronald DuVal, Alan Simpson, Ian Simons and David Key forbeing kind enough to read individual chapters and for providing me with importantcritical reviews. A special thanks to Roy Bradley for reviewing the book in its entiretyand for offering many valuable ideas which have been implemented to make the bookbetter.

I first had the serious idea of writing this book about 4 years ago. I was familiarwith the Blackwell Science series and I liked their productions, so I approached themfirst. From the beginning, my publisher at Blackwell’s, Julia Burden, was helpful andencouraging. Later, during the preparation, the support from Julia and her team wassustained and all negotiations have been both positive and constructive; I would like toexpress my gratitude for this important contribution. I would like also to acknowledgethe vital support of my employer, the Defence Research Agency, for allowing me touse material from my research activities at RAE and DRA over the past 18 years. Myparticular thanks to my boss, Peter England, Manager, Flight Dynamics and SimulationDepartment at DRA Bedford, who has been continually supportive with a positive

xvi Preface to first edition

attitude that has freed me from any feelings of conflict of interest. Acknowledgementsfor DRA material used and figures or quotes from other sources are included elsewherein this book. The figures in this book were produced by two artists, those in Chapter 2by Peter Wells and the rest by Mark Straker. Both worked from often very rough draftsand have, I believe, done an excellent job – thank you both.

All these people have helped me along the road in a variety of different ways,as I have tried to indicate, but I am fully accountable for what is written in this book.I am responsible for the variations in style and ‘colour’, inevitable and perhaps evendesirable in a book of this scope and size. There have been moments when I have beenguided by some kind of inspiration and others where I have had to be more concernedwith making sure the mathematics was correct. I have done my best in this secondarea and apologise in advance for the inevitable errors that will have crept in. My finalthanks go to you, the reader, for at least starting the journey through this work. I hopethat you enjoy the learning and I wish you good fortune with the application of yourown ideas, some of which may germinate as a result of reading this book. It might helpto know that this book will continue to be my guide to flight dynamics and I will belooking for ways in which the presentation can be improved.

Gareth D. Padfield

Sharnbrook, England

Preface to second edition

In the preface to the first edition of my book I talked about flight dynamics as a ‘livingand mature subject, to which many contributions are yet to be made’; I believe thisstatement is still true and every new generation of engineers has something new to addto the store of knowledge. During the 10 years since its publication, the disciplines offlight dynamics and handling/flying qualities engineering have matured into a systemsapproach to the design and development of those functions and technologies requiredto support the piloting task. At the same time, as pilot-centred operational attributes,flying qualities are recognised as the product of a continual tension between perfor-mance and safety. These two descriptions and the interplay between them highlightthe importance of the subject to continuing helicopter development. The most obviouscontributors to flying qualities are the air vehicle dynamics – the stability and controlcharacteristics – and these aspects were treated in some depth in the first edition. Fly-ing qualities are much more, however, and this has also been emphasized. They are aproduct of the four elements: the aircraft, the pilot, the task and the environment, and itis this broader, holistic view of the subject which is both a technical discipline and anoperational attribute, which emphasizes the importance to flight safety and operationaleffectiveness. I have tried to draw out this emphasis in the new material presented inChapter 8, Degraded Flying Qualities, which constitutes the bulk of the new contentin this second edition.

During the preparation of the first edition, ADS-33C was being used extensivelyin a range of military aircraft programmes. The handling qualities (HQs) criteria repre-sented key performance drivers for the RAH-66 Comanche, and although this aircraftprogramme would eventually be cancelled, Industry and the surrounding helicopter‘community’ would learn about the technology required to deliver Level 1 HQs acrossa range of operational requirements. The last decade has seen ADS-33 applied toaircraft such as NH-90 and the UK’s attack helicopter, and also to new operationsincluding maritime rotorcraft and helicopters carrying external loads, and used as adesign guide for civil tilt rotor aircraft. It is now common at annual European andAmerican Helicopter Fora to hear presentations on new applications of ADS-33 or ex-tensions to its theoretical basis. The Standard has also been refined over this period andcurrently exists in the ADS-33E-PRF (performance) version, emphasizing its status asa performance requirement. A brief resume of developments is added to Chapter 6.

Significant advances have also been made on the modelling and simulation front,and it is very satisfying to see the considerable pace at which the modelling of complexhelicopter aerodynamics is moving. It surely will not be very long before the results ofaccurate physical flow modelling will be fully embodied into efficient, whole aircraftdesign codes and real-time simulation. A combination of high-quality computer toolsfor comprehensive synthesis and analysis and robust design criteria pave the way for

xviii Preface to second edition

massive reductions in timescales and costs for design, development and certification.The modelling and simulation material in Chapters 3, 4 and 5 is largely unchanged inthis second edition. This is simply a result of the author needing to put limits on whatis achievable within the timescale available.

In August 1999, I left government ‘service’ to join The University of Liverpoolwith a mandate to lead the aerospace activity, both on the research and the learningand teaching (L&T) axes. I was confident that my 30 years of experience would enableme to transition fairly naturally into academia on the research axis. I had very littleexperience on the L&T side however, but have developed undergraduate modules inrotorcraft flight, aircraft performance and flight handling qualities. I confirm the oldadage – to learn something properly, you need to teach it – and it has been very satisfyingto ‘plough’ some of my experience back into the formative ‘soil’ of future careers.

As with the first edition, while this work is a consolidation of my knowledge andunderstanding, much has been drawn from the efforts and results of others, and notonly is acknowledging this fact appropriate but it also feels satisfying to record thesethanks, particularly to the very special and highly motivated group of individuals inthe Flight Science and Technology Research Group at the University of Liverpool.This group has formed and grown organically, as any university research group might,over the period since 2000 and, hopefully, will continue to develop capabilities andcontribute to the universal pool of knowledge and understanding. Those, in academe,who have had the pleasure and privilege to ‘lead’ a group of young post-graduatestudents and post-doctoral researchers will perhaps understand the sense in which Iderive satisfaction from witnessing the development of independent researchers, andadding my mite to the process. Thanks to Ben Lawrence and Binoy Manimala who havebecome experts in FLIGHTLAB and other computational flight dynamics analysesand helped me in numerous ways, but particularly related to investigating the effectsof trailing wake vortices on helicopters. Neil Cameron derived the results presentedin Chapter 8 on the effects of control system failures on the handing qualities of tiltrotor aircraft. Gary Clark worked closely with me to produce the results in Chapter8, relating to terrain following flight in degraded visibility. Immeasurable gratitude toMark White, the simulation laboratory manager in FS&T, who has worked with me onmost of the research projects initiated over the last 5 years. The support of AdvancedRotorcraft Technology, particularly Ronald Du Val and Chengian Ho, with variousFLIGHTLAB issues and the development of the HELIFLIGHT simulator has beenhuge and is gratefully acknowledged.

Those involved in flight dynamics and handling qualities research will understandthe significant contribution that test pilots make to the subject, and at Liverpool we havebeen very fortunate indeed to have the sustained and consistently excellent supportfrom a number of ex-military test pilots, and this is the place to acknowledge theircontribution to my developing knowledge captured in this book. Sincere thanks toAndy Berryman, Nigel Talbot, Martin Mayer and Steve Cheyne; they should hopefullyknow how important I consider their contributions to be.

Thanks to Roger Hoh and colleagues at Hoh Aeronautics, whose continuouscommitment to handling qualities excellence has been inspirational to me. Roger hasalso made contributions to the research activities in FS&T particularly related to thedevelopment of handling criteria in degraded conditions and the attendant design ofdisplays for flight in degraded visual environments. The whole subject of visual per-ception in flight control has been illuminated to me through close collaboration withDavid Lee, Professor of Perception in Action at The University of Edinburgh. David’s

Preface to second edition xix

contributions to my understanding of the role of optical flow and optical tau in thecontrol of motion has been significant and is gratefully acknowledged.

Over the last 10 years I have received paper and electronic communications fromcolleagues and readers of the first edition worldwide who have been complementary andhave politely identified various errors or misprints, which have been corrected. Thesecommunications have been rather too numerous to identify and mention individuallyhere but it is hoped that a collective thanks will be appreciated.

Mark Straker produced the figures in the form they appear in this book to hisusual very high standard; thanks again Mark for your creative support.

Finally, grateful thanks to Julia Burden at Blackwell Publishing who has beenunrelenting in her encouragement, dare I say persistence, with me to produce materialfor this second edition. Any Head of a fairly large academic department (at LiverpoolI am currently Head of Engineering with 900 students and 250 staff) will know whata challenging and rather absorbing business it can be, especially when one takes it onto direct and increase the pace of change. So, I was reluctant to commit to this secondedition until I felt that I had sufficient new research completed to ‘justify’ a new edition;the reader will now find a consolidation of much of that new work in the new Chap-ter 8. Only the authors who have worked under the pressures of a tight schedule, whilstat the same time having a busy day job, will know how and where I found the time.

So this book is offered to both a new and old readership, who might also findsome light-hearted relief in a ‘refreshed’ version of my poem, or sky-song as I call it,Helicopter Blues, which can also be sung in a 12-bar blues arrangement (normally inEmaj but in Am if you’re feeling cool)

I got the helicopter bluesThey’re going round in my headI got the helicopter bluesThey’re still going round in my headbrother please tell me what to do about these helicopter blues

My engine she’s failingGotta reduce my torqueMy engine she keeps failingGotta pull back on my powerseems like I’m autorotating from all these helicopter blues

My tail rotor ain’t workingAin’t got no place to goMy tail rotor she ain’t workingAin’t got no place to turnThese helicopter blues brotherThey’re driving me insane

My humms are a hummingFeel all fatigued, used and abusedMy humms are hummingI’m worn out from all this aerofoil toilIf I don’t get some maintenancesister I’ve had it with all these helicopter blues

My gearbox is whiningMust need more lubrication

xx Preface to second edition

I said I can’t stand this whiningplease ease my pain with boiling oilIf I don’t get that stuff right nowI’m gonna lock up with those helicopter blues

Dark blue or lightThe blues got a strong hold on meIt really don’t matter which it isThe blues got no respect for meWell, if only I could change to greenMaybe I could shake off these helicopter blues

I’ve designed a new helicopterIt’ll be free of the bluesI’ve used special techniques and powerful computersI’m sure I know what I’m doingnow I gotta find someone to help me chase away these helicopter blues

I went to see BoeingSaid I got this new blues-free designI went up to see Boeing, told them my story and it sounded fineBut they said why blue’s our favourite colourBesides which, you’re European

So I took my design to EurocopterI should have thought of them firstIf I’d only gone to EurocopterI wouldn’t be standing here dying of thirstThey said ‘ces la vie mon frere’ you can’t make a sans bleu helicoptre

I went to see SikorskyI thought – They’ll fix the bluesThey sent for Nick LapposTo fix the helicopter bluesNick said don’t be such a baby Gareth(besides, I don’t work here anymore)Just enjoy those helicopter blues

I’ll go see Ray ProutyPeople say, Ray – he ain’t got no bluesPlease help me Ray – how much more aerodynamics do I need – I’ll clean your shoesRay said, wake up and smell the coffee fellaLearn how to hide those helicopter blues

I’ve learned to live with them nowI’m talking about the helicopter bluesEven got to enjoy themThose sweet, soothing helicopter bluesI’m as weary as hell but please don’t take away my helicopter blues

Gareth D. Padfield

Caldy, England

The cover photograph is reproduced with permission from AgustaWestland.

Copyright acknowledgements

The following people and organizations are gratefully acknowledged for granting per-mission for the use of copyright material.

The UK MoD and Defence Research Agency for Figs 2.31, 2.43, 2.44, 2.50, 3.15,3.28, 3.29, 3.35, 3.37, 3.38, 5.7–5.9, 5.28–5.31, 5.34, 6.7, 6.8, 6.9, 6.10, 6.18, 6.19,6.35, 6.36, 6.38, 6.39, 6.47–6.52, 6.59, 7.10–7.24, 7.38, 7.44, 7.45 and 7.46.* The USArmy for Figs 6.15, 6.17, 6.20, 6.25, 6.30, 6.33, 6.40–6.45, 6.56, 6.61, 6.64, 6.65, 6.70and 7.28 and Table 7.4. The American Helicopter Society (AHS) for Figs 3.16 and7.5 (with the US Army). Bob Heffley for Figs 6.6 and 6.11. Cambridge UniversityPress for the quote from Duncan’s book at the beginning of Chapter 3. ChengjianHe and the AHS for Fig. 5.27. Chris Blanken, the US Army and the AHS for Figs7.29 and 7.30. Courtland Bivens, the AHS and the US Army for Fig. 6.63. David Keyand the Royal Aeronautical Society for Figs 6.3 and 6.31. David Key for the quote atthe beginning of Chapter 7. DLR Braunschweig for Figs 6.21, 6.23 (with RAeSoc),6.32, 6.37, 6.58 (with the AHS), 6.68 (with the US Army) and 7.4 (with AGARD).Eurocopter Deutschland for Figs 6.46 and 6.66. Ian Cheeseman and MoD for Figs 3.28and 3.29. Jeff Schroeder and the AHS for Figs 7.32–7.36. Jeremy Howitt and the DRAfor Figs 7.39, 7.40 and 7.41. Knute Hanson and the Royal Aeronautical Society forFig. 6.69. Lt Cdr Sandy Ellin and the DRA for Figs 2.7, 3.44 and 3.45. Mark Tischlerand AGARD for Figs 5.25, 5.26, 6.34 and 6.57. McDonnell Douglas Helicopters,AGARD and the US Army for Fig. 6.71. NASA for Figs 4.12 and 6.2. Institute forAerospace Research, Ottawa, for Figs 6.54 and 7.7 (with the AHS). Pat Curtiss forFigs 3.46, 3.47 and 5.4. Roger Hoh for Figs 6.24, 6.26 (with the AHS), 6.29 (with theRAeSoc) and 7.27 (with the AHS). Sikorsky Aircraft, the US Army and the AHS forFig. 6.72. Stewart Houston and the DRA for Figs 5.10–5.13. Tom Beddoes for Fig.3.42. Jan Drees for Fig. 2.8. AGARD for selected text from References 6.72 and 7.25.Westland Helicopters for granting permission to use configuration data and flight testdata for the Lynx helicopter. Eurocopter Deutschland for granting permission to useconfiguration data and flight test data for the Bo105 helicopter. Eurocopter France forgranting permission to use configuration data and flight test data for the SA330 Pumahelicopter.

In this second edition, once again the author has drawn from the vast store ofknowledge and understanding gained and documented by others and the followingpeople and organizations are gratefully acknowledged for the use of copyright material.

Philippe Rollet and Eurocopter for the use of Table 8.9. John Perrone at theUniversity of Waikato for Figs 8.4, 8.6 and 8.11. James Cutting at Cornell Universityand MIT Press for Figs 8.7, 8.8 and the basis of Fig 8.10. NASA for Fig. 8.14. David Leefor Figs 8.18 and 8.19. The US Army Aviation Engineering Directorate for the use ofTable 6.6 and Figs 6.74, 6.75 and 6.77 and general reference to ADS33. AgustaWestland

xxii Copyright acknowledgements

Helicopters for the use of the photographs of the EH101 at the start of Chapter 8 andalso on the book cover. Roger Hoh and the American Helicopter Society for Fig. 8.2.The American Helicopter Society for a variety of the author’s own figures publishedin Ref 8.31, 8.33 and 8.55. The Institution of Mechanical Engineers for Fig. 8.45 fromthe author’s own paper. The Royal Aeronautical Society for the use of the author’s ownfigures from Ref 8.53. J. Weakly and the American Helicopter Society for Fig. 8.43.Franklin Harris for Fig. 8.62.

*C© British Crown Copyright 1995/DRA; reproduced with the permission of the Controller ofHer Britannic Majesty’s Stationery Office.

Notation

a0 main rotor blade lift curve slope (1/rad)ag constant acceleration of the τ guidea0T tail rotor blade lift curve slope (1/rad)an−1, an−2, . . . coefficients of characteristic (eigenvalue) equationap acceleration of P relative to fixed earth (components

ax , ay , az) (m/s2, ft/s2)ap/g acceleration vector of P relative to G (m/s2, ft/s2)axb, ayb, azb acceleration components of a blade element in rotating blade axes

system (m/s2, ft/s2)azpk peak normal acceleration (m/s2, ft/s2)c rotor blade chord (m, ft)c constant τ motiond(ψ, rb) local drag force per unit span acting on blade element (N/m, lbf/ft)eR flap hinge offset (m, ft)eζ R lag hinge offset (m, ft)f(t) forcing function vectorfβ (ψ), fλ(ψ) coefficients in blade flapping equationfy(rb), fz(rb) in-plane and out-of-plane aerodynamic loads on rotor blade at

radial station rb

g acceleration due to gravity (m/s2, ft/s2)g1c0, g1c1 lateral cyclic stick–blade angle gearing constantsg1s0, g1s1 longitudinal cyclic stick–blade angle gearing constantsgcc0, gcc1 collective lever–lateral cyclic blade angle gearing constantsgcT 0 pedal/collective lever–tail rotor control run gearing constantgθ , gφ nonlinear trim functionsgsc0, gsc1 collective lever–longitudinal cyclic blade angle gearing constantsgT 0, gT 1 pedal–tail rotor collective blade angle gearing constantgT tail rotor gearingh height above ground (m(ft))he eye-heighth, h height (m, ft), height rate (m/s, ft/s)hfn height of fin centre of pressure above fuselage reference point

along negative z-axis (m, ft)h R height of main rotor hub above fuselage reference point (m, ft)hT height of tail rotor hub above fuselage reference point (m, ft)i, j, k unit vectors along x-, y- and z-axesk τ coupling constantk1, k2, k3 inertia coupling parameters

xxiv Notation

k1s , k1c feedforward gains (rad/unit stick movement)k3 = tan (rad / m2) tail rotor delta 3 anglekφ , kp feedback gains in roll axis control system (rad/rad, rad/(rad/s))kg feedback gain in collective – normal acceleration loop (rad /m2)kλ f main rotor downwash factor at fuselagekλfn main rotor downwash factor at finkλT main rotor downwash factor at tail rotorkλtp main rotor downwash factor at tailplanek0, kq feedback gains in pitch axis control system (rad/rad,

rad/(rad s))kθ i , kφi trim damping factors(ψ, r) lift per unit span (N/m, Ibf/ft)lf fuselage reference length (m, ft)lf n distance of fin centre of pressure aft of fuselage reference point

along negative x-axis (m, ft)lT distance of tail rotor hub aft of fuselage reference point (m, ft)ltp distance of tailplane centre of pressure aft of fuselage reference

point (m, ft)m(r) blade mass distributionmam apparent mass of air displaced by rotor in vertical motionn, nzpk load factor (g)p, q, r angular velocity components of helicopter about fuselage x-, y-

and z-axes (rad/s)ppk /φ attitude quickness parameter (1/s)pss , ps steady state roll rate (rad/s)r , rb (−) blade radial distance (with overbar – normalized by radius R)

(m, ft)r, rc radial distance from vortex core and vortex core radiusrp/g position vector of P relative to G (components x, y, z) (m, ft)s Laplace transform variables rotor solidity = Nbc/πR

sT tail rotor solidityt time (s)t normalized time (t/T )tr time in a manoeuvre when the reversal occurs (s)tw heave time constant (−1/Zw ) (s)tw tw normalized by T

t1 manoeuvre time (s)tr10,50,90 time constants – time to 10%, 50%, 90% of steady-state

response (s)u(t) control vectoru, v, w translational velocity components of helicopter along fuselage

x-, y- and z-axes (δw ≡ w, etc.) (m/s, ft/s)vi induced velocity at disc (m/s, ft/s)vihover induced velocity at disc in hover (m/s, ft/s)vi∞ induced velocity in the far field below rotor (m/s, ft/s)vj eigenvectors of AT

vg , vp velocity vector of G, P relative to fixed Earth

Notation xxv

vp/g velocity vector of P relative to G (components u p/g, v p/g , w p/g)vg velocity of motion guide (m/s, ft/s)vg0 initial velocity of motion guide (m/s, ft/s)w velocity along aircraft z-axis (ms, fts)wss steady-state velocity along aircraft z-axis (m/s, ft/s)w(r, t) blade out-of-plane bending displacement (m, ft)w0 vertical velocity (m/s, ft/s)wg(t) gust velocity component along z-axis (m/s, ft/s)wgm maximum value of velocity in ramp gust (m/s, ft/s)wi eigenvectors of A

wλ w − kλf Rλ0 total downwash over fuselage (m/s, ft/s)wss steady-state normal velocity (m/s, ft/s)wss steady state velocity along aircraft z axis (m/s, ft/s)x(t) state vectorx, xcmd position and position command in pilot/vehicle systemx, z distance along x- and z-directionsx, x distance (normalized distance) to go in manoeuvre (m, ft)x ′, x ′′ normalized velocity and acceleration in menoeuvrex, y, z mutually orthogonal directions of fuselage axes – x forward, y to

starboard, z down; centred at the helicopter’s centre of massx0 initial condition vector x(0)xcg centre of gravity (centre of mass) location forward of fuselage reference

point (m, ft)xe equilibrium value of state vectorxe distance in eye-height/sxe velocity in eye-heights/sxg0 initial displacement of motion guide (m(ft))xg distance to go in motion guide (m(ft))xm distance to go in manoeuvre (m(ft))xr edge rate (1/s)xf , xr , xp, xc elemental state vectors ( f – fuselage, r – rotor, p – powerplant,

c – control)zg distance of ground below rotor (m, ft)A, B system and control matricesAff , Afr, etc. system matrices; ff – fuselage subsystem, fr – rotor to fuselage

couplingA11, A12 . . . submatrices in partitioned form of A

Ab blade area (m2, ft2)Ad rotor disc area (m2, ft2)A f agility factor – ratio of ideal to actual manoeuvre timeAx , Ay x- and y-axes acceleration components of aircraft relative to Earth

(m/s2, ft/s2)Bff , Bfr, etc. control matrices; ff fuselage subsystem, fr rotor to fuselage coupling

C ′1 = 1

1 + a0s/16λ0lift deficiency factor

C ′2 = a0s

16λ0

C1(ψ) time–dependent damping matrix in individual blade flapping equations

xxvi Notation

Cif normalized fuselage force and moment coefficients, i = x, y, z, l, m, n

CLa aerodynamic flap moment coefficient about roll axisCLmax maximum aerofoil lift coefficientCM (ψ) time-dependent damping matrix in multi-blade flapping equationsCM0(ψ) constant damping matrix in multi-blade flapping equationsCMa aerodynamic flap moment about pitch axisCnfa, Cnfb fuselage aerodynamic yawing moment coefficientsCQ main rotor torque coefficientCQi, CQp induced and profile torque coefficientsCQT tail rotor torque coefficientCT rotor thrust coefficientCTT tail rotor thrust coefficientCW weight coefficientCx , Cy , Cz main rotor force coefficientsCyf η normalized sideforce on finCζ lag dampingCztp normalized tailplane forceD aircraft drag (N, lbf)D(s) denominator of closed-loop transfer functionDI(ψ) time-dependent stiffness matrix in individual blade flapping equationsDM (ψ) time-dependent stiffness matrix in multi-blade flapping equationsDM0(ψ) constant stiffness matrix in multi-blade flapping equationsE(r )1(r ) distributed blade stiffnessF (1) out-of-plane rotor blade forceF (2) in-plane rotor blade forceF(r, t) distributed aerodynamic load normal to blade surfaceF(x, u, t) nonlinear vector function of aircraft motionF (1)

0 main rotor force componentF (1)

0 one-per-rev cosine component of F (1)

F (1)1s one-per-rev sine component of F (1)

F (1)2c two-per-rev cosine component of F (1)

F (1)2s two-per-rev sine component of F (1)

F (2)1c one-per-rev cosine component of F (2)

F (2)1s one-per-rev sine component of F (2)

Fg vector of external forces acting at centre of mass (components X,Y, Z)FT tail rotor-fin blockage factorFvi , Fw , etc. flap derivatives in heave/coning/inflow rotor modelGe(s), He(s) engine/rotorspeed governor transfer functionGη1cp(ω) cross-spectral density function between lateral cyclic and roll rateHη1cp(ω) frequency response function between lateral cyclic and roll rateHI (ψ) time-dependent forcing function matrix in individual blade

flapping equationsHM (ψ) time-dependent forcing function matrix in multi-blade

flapping equationsHM0(ψ) forcing function matrix in multi-blade flapping equationsIβ flap moment of inertia (kg m2, slug ft2)In moment of inertia of nth bending mode (kg m2, slug ft2)

Notation xxvii

IR moment of inertia of rotor and transmission system (kg m2; slug ft2)Ivi, Iw , etc. inflow derivatives in heave/coning/inflow rotor modelIxx, Iyy, Izz moments of inertia of the helicopter about the x-, y- and z-axes

(kg m2; slug ft2)Ixz product of inertia of the helicopter about the x- and z-axes

(kg m2; slug ft2)K3 rotorspeed droop factorKβ centre-spring rotor stiffness (N m/rad, ft Ib/rad)Kp, Kx pilot and display scaling gainsL , M, N external aerodynamic moments about the x-, y- and z-axes (N m, ft lb)Lβ transformation matrix from multi-blade to individual blade

coordinatesL f , M f , N f fuselage aerodynamic moments about centre of gravity (N m, ft Ib)Lfn, Nfn fin aerodynamic moments about centre of gravity (N m, ft Ib)Lθ0 , MθI s control derivatives normalized by moments of inertia (1/s2)LT , NT , MT tail rotor moments about centre of gravity (N m, ft Ib)Lv , Mq , etc. moment derivatives normalized by moments of inertia (e.g., ∂L/∂v)

(rad/(m s), rad/(ft s), 1/s)Lw turbulence scale for vertical velocity component (m, ft)M, Md Mach number, drag divergence Mach numberMa mass of helicopter (kg, Ib)Mβ first moment of mass of rotor blade (kg m; slug ft)Mg vector of external moments acting at centre of mass

(components L , M, N)M (r )

h (0, t) rotor hub moment (N m, ft Ib)Mh , Lh main rotor hub pitch and roll moments (N m, ft Ib)MR , L R main rotor pitch and roll moments (N m, ft Ib)Mtp tail plane pitching moment (N m, ft Ib)Nb number of blades on main rotorNH yawing moment due to rotor about rotor hub (N m, ft Ib)Nreffective effective yaw damping in Dutch roll motion (1/s)Pe, Qe, Re trim angular velocities in fuselage axes system (rad/s)Pi rotor induced power (kW, HP)Pn (t) blade generalized coordinate for out-of-plane bendingPR main rotor power (kW, HP)PT tail rotor power (kW, HP)Px , Py position of aircraft from hover box (m, ft)Qacc accessories torque (N m, ft Ib)Qe, Qeng engine torque (N m, ft Ib)Qemax maximum continuous engine torque (N m, ft Ib)Q R main rotor torque (N m, ft Ib)QT tail rotor torque (N m, ft Ib)Qw quickness for aircraft vertical gust response (1/s)R rotor radius (m, ft)R(s) numerator of closed-loop transfer functionRT tail rotor radius (m, ft)

Sβ Stiffness numberλ2β−1

γ /8

xxviii Notation

Sfn fin area (m2, ft2)Sn (r) blade mode shape for out-of-plane bendingSp, Ss fuselage plan and side areas (m2, ft2)Stp tail plane area (m2, ft2)Sz (0, t) shear force at rotor hub (N, Ibf)T main rotor thrust (N, Ibf)T manoeuvre duration (s)Theq time constant in heave axis first-order equivalent system (s)Tige rotor thrust in-ground effect (N, Ibf)Toge rotor thrust out-of-ground effect (N, Ibf)Tx distance between edges on surface (m(ft))TT tail rotor thrust (N, Ibf)Ue, Ve, We trim velocities in fuselage axes system (m/s, ft/s, knot)UP , UT normal and in-plane rotor velocities (m/s, ft/s)V, Vx aircraft forward velocity (m/s, ft/s)Vc rotor climb velocity (m/s, ft/s)Vc tangential velocity at the edge of the vortex core (m/s, ft/s)Vd rotor descent velocity (m/s, ft/s)V f total velocity incident on fuselage (m/s, ft/s)Vfe total velocity in trim (m/s, ft/s, knot)Vfn total velocity incident on fin (m/s, ft/s)V (r )

h (0, t) rotor hub shear force (N, Ibf)Vres resultant velocity at rotor disc (m/s, ft/s)Vtp total velocity incident on tailplane (m/s, ft/s)VT (r ) tangential velocity in vortex as a function of distance from core r

(m/s, ft/s)

Vx , Vy velocity components of aircraft relative to EarthW eigenvector matrix associated with A

X , Y , Z external aerodynamic forces acting along the x-, y- andz-axes (N, Ibf)

X f , Y f , Z f components of X , Y , Z from fuselage (N, Ibf)Xhw, Yhw rotor forces in hub/wind axis system (N, Ibf)X R , XT components of X from main and tail rotors (N, Ibf)Xtp, Xfn components of X from empennage (tp – horizontal tailplane,

fn – vertical fin) (N, Ibf)

Xu , X p, etc. X force derivatives normalized by aircraft mass (1/s, m/(s rad), etc.)Y(t) principal matrix solution of dynamic equations of motion

in vector formYfn aerodynamic sideforce acting on fin (N, Ibf)Yp, Ya(s) transfer function of pilot and aircraftYT component of Y force from tail rotor (N, Ibf)Yv , Yr , etc. Y force derivatives normalized by aircraft mass (1/s, m/(s rad), etc.)Zw heave damping derivative (1/s)Zθ0 heave control sensitivity derivative (m/(s2 rad), ft/(s2 rad))Ztp component of Z force from tailplane (N, Ibf)Zw , Zq , etc. Z force derivatives normalized by aircraft mass (1/s, m/(s rad), etc.)α (ψ , r, t) total incidence at local blade station (rad)