Helicopter Dynamics and Aerodynamics

Citation preview

LIBR

MICHIGAN

·

THE

UNIVERSITY UNIVERSIT

MICHIGA

OF

HELICOPTER DYNAMICS AND AERODYNAMICS

HELICOPTER DYNAMICS AND AERODYNAMICS

by

P. R. PAYNE

THE MACMILLAN COMPANY NEW YORK 1959

Engin . Library

TL

716

· P35

First published in

United States by The Macmillan Company , 1959 the

P. R. PAYNE

, 1959

MADE IN GREAT BRITAIN AT THE PITMAN PRESS, BATH

87-163648 FOREWORD THIS book is an extremely valuable contribution to the advancement of the helicopter industry ; it is also important and timely with reference to other types of vertically rising aircraft . From my personal knowledge of him I know that the author is inspired by a long - standing and devoted enthusiasm for this new frontier of aeronautics . To this conviction he brings the highest scientific competence in several other fields coupled with outstanding success as a practising project engineer in

VTOL

aircraft development .

The increasing complexity of all

branches of science spotlights one of the most enigmatic hurdles in our technology . The problem is to induce that uniquely qualified top echelon group to meet the ever - increasing demands of industrial performance while at the same time making the essential contri-

bution to the literature which only a few of such experienced leaders can make . This text is a fine example of the type of simplification and clarification which may well be applied in other fields to link the most advanced achievements in industry with the vital educational needs of our society . Concise and practical analytical methods are presented in several areas which were previously subject only to costly experimental solutions . Complete mastery and working familiarity with past theory and practice is evident from the fresh approach and the imaginative departure from convention when justified . Many basic concepts are presented from a new viewpoint which cannot fail to prove stimulating . For those interested in the classical aspects of vertical flight in the dawning space age as well as for the engineer seeking more effective practical design references for industry , this book is a significant advance which should be included in every library on the subject . ,

Barr Bldg .

Jr.

Washington , D.C.

K.

6.

Carty, M

Jr.

L. C. McCarty 412 ,

PREFACE IT is customary to preface books on helicopter theory by the statement that current developments are the result of twenty - five years of extension to the work of such distinguished investigators as Glauert , Lock and Wheatley . Whilst this is very true , it does not mean that the results of their work can be applied to the design of contemporary helicopters , which is the impression sometimes given . It may be that the books so far published have been written by authors whose interest in helicopter theory is academic , so that even when they achieve more than uncritical anthologies of published reports , little attempt has been made to correlate theory and practical results . The present work is written from the standpoint of a practising aerodynamicist and project designer , in an effort to produce a book which is more closely related to the requirements of industry . By far the best introduction to helicopter theory is Aerodynamics of the

Helicopter by Gessow and Myers ( Macmillan ) . The present book has been written on the assumption that the reader is familiar with this introduction , and its scope has been limited , in general , to the requirements of the projectdesign engineer . The project designer has to be familiar with the latest reliable theoretical developments in all fields , without becoming enmeshed in the toils of research in any one field , unless it is essential to the success of a

radically new design . It is generally agreed ( although not always practised ) that the theory used in the analysis and optimization of a new design should be known from past experience to give reliable results . For this reason a successful attempt to cater for the needs of the project designer should be

of value

to all helicopter engineers , and it is earnestly hoped that this book will go some way to satisfying this requirement . In a companion volume the actual mechanics of design will be considered , together with the allied subjects of helicopter economics , and those aspects of power - plant design

which affect the project - design stage . Certain aspects of essential aerodynamic theory are in such a state of flux that adequate treatment is very difficult . This is particularly so in the case of dynamic stability , and an attempt has been made to introduce the subject in such a way that the reader will be able to undertake his own

Dynamic stability in hovering is examined in writing detail , but at the time of it was felt that it could not usefully be stability flight , because of the number of uncertainties done for in forward involved . The available work on the subject is fully referenced in Appendix 1 , and the application of existing theoretical treatments will be illustrated in the companion volume . analytical

developments .

Most chapters are prefaced by a list of symbols , which are uniform throughout for the purely aerodynamic theory . For subjects such as blade resonance and " ground resonance , " which have developed separately from vi

Preface

vii

aerodynamic theory , it is inevitable that some duplication of the " aerodynamic " symbols has occurred , and it was considered more convenient to accept the present conventions . An attempt has been made to give appropriate acknowledgments whenever the work of other investigators has been used . Sincere thanks are also due to the editor of Aircraft Engineering for permission to use the work contained in twelve papers by the author which were first published in that journal . To Raoul Hafner , under whom the author's four years of " helicopter apprenticeship " were spent , it is difficult to express adequate thanks for his instruction and example ; or to my wife , without whose constant help this book would never have been written . P. R. PAYNE Clevedon , Somerset 2nd June , 1956

,

CONTENTS CHAP .

PAGE

Foreword by

L. C.

McCarty

V

Preface .

vi

Notation

xii

1. GENERAL AERODYNAMICS

1

Symbols used in Chapter 1 1.1 . Introduction 1.2 . Subsonic aerofoil lift 1.3 . Subsonic aerofoil profile drag 1.4 . Yawed aerofoils 1.5 . Subsonic drag of streamline bodies1.6 . Aerodynamic pitching moment

-

-

-

-

2. INDUCED AERODYNAMICS Symbols used

29

in Chapter 2

-

2.1 . The actuator disc 2.2 . Induced velocity in vertical flight ( average momentum theory ) -2.3 . Induced velocity in forward flight (average momentum theory ) -2.4 . " Effective area " concept 2.5 . Alternative nondimensional induced velocity equations 2.6 . Wake calculations 2.7 . velocity Variation of induced over the disc ( Glauert's hypothesis ) -2.8 . velocity vicinity theory simple Induced in the of a rotor ( vortex ) -2.9 . Vertical drag 2.10 . Vertical drag in hovering flight 2.11 . Vertical drag in vertical flight drag flight Example 2.12 . Vertical in forward 2.13 . of vertical drag calculations 2.14 . Ground effect 2.15 . Strip theory in vertical flight2.16 . Induced tip loss 2.17 . Calculation of optimum twist 2.18 . The application of strip theory to calculate effective disc area e in hovering2.19 . Variation of slipstream rotation from momentum theory 2.20 . Induced velocity variation due to a finite number of tip vortices 2.21 . Twin -rotor hovering interference in 2.22 . Twin - rotor interference in forward flight— 2.23 . Influence of results on tandem design 2.24 . Measurement of rotor thrust and circulation by wake survey 2.25 . Ducted fan theory 2.26 . General remarks on " vertical lift "

-

-

-

--

-

-

-

-

-

--

-

-

-

3. FUNDAMENTALS OF ROTOR DYNAMICS Symbols

used

106

in Chapter 3

3.1 . Rigid

-

-

-

-

rotor 3.2 . Freely flapping rotor with central hinges 3.3 . Flapping rotor analysis 3.4 . Reversed -flow region 3.5 . Effect of aerodynamic compressibility on flapping 3.6 . Elemental angle of attack3.7. Approximate relationships for blade angle of attack and flight envelopes -3.8 . General accelerations on a flapping - blade particle 4. DYNAMICS OF ROTORS

-

149

WITH HINGE CONSTRAINT

used in Chapter 4 General considerations 4.2 . The stiff -hinged rotor 4.2.1 . Blade flapping with respect to the shaft axis 4.2.2 . Blade flapping with respect to the no - feathering orbit 4.2.3 . Angle of attack distribution and retreating blade stall 4.2.4 . Control advance 4.2.5 . Moments in hub and blade root arms 4.2.6 . Calculations for the example rotor 4.3 . The high -offset flapping - pin rotor 4.4 . Reversed flow effects Symbols

-

-

4.1 .

5. FLAPPING

STABILITY

-

-

AND BLADE MOVEMENTS

IN GUSTS

·

181

5.1 . Flapping stability 5.2 . Blade motion in a vertical sharp - edged gust (hovering ) —5.3 . Blade motion in a gust in forward flight

6. PERFORMANCE 6.1 . Simple energy equations

-6.3 .

Derivation

195

-

6.2 . Variation of mean 8 with tip - speed ratio of profile torque and H -force by refined energy method-

ix

X

Contents CHAP .

-

-

6.4 . A simple performance method 6.4.1 . Vertical and low - speed flight6.4.2 . Forward flight 6.5 . Comparison with flight tests and examples of application 6.6 . Presentation of performance curves 6.7 . Engine failure in Jump -start " autogiro 6.10 . hovering 6.8 . Vertical flare - out 6.9 . " Exact " theory of rotor performance 6.10.1 . Physical explanation of induced H -force 6.10.2 . Conditions at a blade element 6.10.3 . Induced tor-

--

-- -

-

-

-

"

-

--

-

PAGE

6.10.4 . Profile torque 6.10.5 . Induced H -force 6.10.6 . Profile drag Ĥ -force 6.10.7 . Performance computor 6.10.8 . Examples of typical

que

calculations 7. STABILITY

AND CONTROL

248

-

-

7.1 . General control considerations 7.2 . Static stability in hovering7.3 . Static stability in forward flight 7.4 . Dynamic stability 7.4.1 . Solution of equations of motion for two degrees of freedom 7.4.2 . Complete solution of equations of motion 7.4.3 . Extraction of the roots of the stability quartic . Hovering stability with two degrees of freedom 7.5.1 . Calculation of derivatives dai / əq and da¿ V in hovering 7.5.2 . Period and damping of hovering oscillation 7.6 . General remarks on stability calculations 7.6.1 . The downwash derivatives ( λ¸K ) | Əµ‚ Ə(λ¡K ) / əλ and Ə(kî ,) dμ 7.7 . Automatic servo control 7.8 . Effect of control system stiffness and damping 7.9 . Control sensitivity ( in pitch or roll ) -7.10 . Yawing stability in forward flight

-

-7.5

-

8. ROTOR VIBRATION Symbols used

in

-

-

/

-

- --

289

.

Chapter 8

--

-

-

8.2 . Vertical vibration of a balanced rotor 8.2.1 . Periodic blade flexing 8.2.2 . Blade stalling 8.3 . In - plane vibration of a balanced rotor 8.3.1 . Ground resonance 8.3.2 . Coriolis forces 8.3.3 . Induced forces 8.3.4 . Profile drag forces 8.4 . Additional causes of vibrationdrag hinges are fitted8.5 . Transmission of blade vibration to the hub when 8.6 . Vertical vibration due to unbalance 8.7 . In - plane vibration due to unbalance ( in hovering ) -8.7.1 . Freely - flapping rotor without drag hinges 8.7.2 . Fully -articulated rotor 8.7.3 . Out - of- balance forces in hovering due to unequal coning angles ( tip - path tracking ) 8.1 . Introduction

--

-

-

9. GROUND RESONANCE Symbols used

in

-

-

AND VIBRATION DUE TO ROTOR

Chapter 9

--

RESONANCE

321

-

9.2 . Fuselage or hub natural frequencies 9.3 . " Ground 9.4 . General remarks on the determination of fuselage frequencies natural 9.5 . Rotor blade oscillation 9.6 . Two bladed rotor resonance 9.7 . Blade snubbers 9.8 . Multi - bladed rotors 9.9 . Drag hinge dampers 9.10 . Coupling between fuselage and rotor oscillations ( three or more blades ) -9.11 . Deutsch equations for critical speed and damping9.12 . Twin -rotor helicopters 9.13 . Ground resonance of two - bladed rotors 9.1 . Introduction resonance " in flight

--

10.

CONTROL

-

-

-

-

LOADS

Symbols used

--

AND VIBRATION

in Chapter

10.1 . Introduction

-

354

10

Tail -rotor

--

loads and vibration 10.3 . Main rotor controls 10.3.1 . Blade torque due to positions of blade axes 10.3.2 . Torque due to aerodynamic pitching moment 10.3.3 . Torque due to propeller moment 10.3.4 . Inertia torque and torsion bearings 10.3.5 . Torque - bar torque 10.3.6 . Total torque of balanced blade 10.3.7 . Additional causes of control system loads 10.4 . Control vibration 10.2 .

-

-

11. BLADE FLUTTER

-- - -

AND ROTOR WEAVING

-

-

371

-

11.1 . Rotor blade flutter 11.2 . Flutter of a flexible blade 11.3 . Flutter tests with a model rotor 11.3.1 . Effect of blade - pitch control stiffness11.3.2 . Effect of tip - speed ratio 11.3.3 . Effect of flutter on blade stresses-11.4 . Rotor weaving 11.4.1 . The analysis of Coleman and Stempin 11.4.2 . Effect of tip jets on weaving 11.5 . Loewy's theory of damping at low blade pitch angles 11.6 . Advanced flutter theory

-

-

xi

Contents CHAP .

PAGE

12. BLADE

FLEXING AND RESONANCE

Symbols

used

in Chapter

--

387

12

-

12.2 . Calculation of blade natural frequencies by Rayleigh energy method 12.3 . Blade torsional flexure 12.3.1 . Increment of Ka due to tension 12.3.2 . Effect of chordwise components of C.F. 12.3.3 . Bifilar effect 12.3.4 . Torque - bar effect 12.3.5 . Effective Southwell co12.1 . Introduction

-

-

efficient of torsion

Appendix

1.

References to Literature

Appendix 2.

Trigonometric Identities

Index

-

409

. PLATES THE HELIVECTOR McCarty

,

.

A

-

one - man helicopter designed

now known as the DeLackner D.H.

4.1 .

THE "HILLER " FLYING PLATFORM

4.2 .

THE BELL XV - 3 CONVERTIPLANE

by Lewis

4 Aerocycle

.

.

9.5 . CANTILEVER - BEAM UNDERCARRIAGE 9.31 . DROP TESTS ON THE PROTOTYPE 10.1 . THE S.N.C.A.S.O.

" DJINN "

SIELEY

431

•

439

FACING PAGE

FIG .

3.7 .

•

C. ·

116

•

152

•

153 327

SHOCK

STRUT

.

352 354

Notation

THROUGHOUT this book the units conform to the " Perry system " in which the pound force may be regarded as the fundamental unit .

Force Mass Length Time

=

pounds (lb) pounds weight

acceleration due to gravity = feet (ft)

= seconds

lb 32.18

=

slugs

(sec )

The derived units are thereforeVelocity Acceleration Momentum Pressure Density

Work Power

-feet /second (ft/sec ) = feet/second /second (ft / sec² ) = (slugs ft/sec ) (lb/ft² )

= (slugs /ft³)

= (ft lb) = (ft lb/sec ) Brake horse power (b.h.p. )

Torque

-

Power 550

= (lb ft)

Since this book covers subjects as diverse as blade flexing , resonance " and induced aerodynamics , it is inevitable that some refer to two or more different quantities . To avoid confusion their is always made clear in the text . In a limited number of cases one

" ground symbols meaning quantity

is denoted by two different symbols , an example being the lag angle for which the symbol ẞ is used in the chapter on Ground Resonance . It is general practice to write ẞ for the blade flapping angle , but in Coleman's classic works on ground resonance it refers to the lag angle . Although it would be easier to pacify the book reviewers by uniform notation , it is thought that the course adopted will be of more value to the practising technicians who will use the book .

of the twelve chapters contain lists of symbols and the remaining broadly four are consistent with the first four chapters . In cases where meaning may their be in doubt , symbols are always defined in the text where they appear . Seven

xii

Notation

xiii

Uniformity of notation and units is often impossible for the helicopter engineer , particularly in project design . Unavoidably , the symbol T must stand for rotor thrust ( lb ) , jet pipe thrust ( lb ) , total temperature (° K ) and a may denote one of a dozen second moments of taper integral as well . intensity mass , or even combustion in CHU/ ft³ hr ( atmosphere )² . In dealing engine with manufacturers one must accept a system of units in which both

I

mass and force are taken as the pound weight .

Thus the only effective symbols problem , and check the resolution is to define units and for each by analysis apparent sults dimensional where the need is .

CHAPTER

1

General Aerodynamics SYMBOLS USED IN CHAPTER 1 ι - length of streamline

a = lift - curve slope dC₁ /da aLS = lift - curve slope at low Mach numbers A - wing aspect ratio cross - sectional areas of a A 0,43,48

L

M

ramjet at the reference planes shown in Fig . 1.17 c = rotor blade chord = effective chord due to yaw Ce (Fig . 1.15 )

1.2 )

ness

U = resultant

Vo0 ==free - stream air velocity α wing angle of attack 8 - mean blade profile drag coefficient

namic centre

(equation 1.2 ) CDO = profile drag coefficient CDi induced drag coefficient C = skin friction drag coefficient D = drag force due to skin

ბი = minimum value of 8 (value for CL = 0) Δε = increment of 8 due to CL 0 = yaw angle defined in Fig .

,

friction

K

-

1.15

kinematic viscosity of air (Table 1.4 ) ρ = mass density of air (Table 1.1-1.3 )

profile drag

V

drag

diameter of streamline body (equation 1.20 ) = induced drag factor ≈ 1 ·1

to

velocity

V = air velocity

CQ = air mass flow coefficient

d

area (projected )

2S for a wetted area wing t = wing or rotor blade thick-

Swet

coefficient

induced Di = =

force

S = wing

C mo = moment coefficient of pitching about the aerody-

Do

lift

body

1.20 )

= Mach number air mass flow (equation Q

-

CL = lift

=

(equation

1.2 1.1

.

INTRODUCTION

IN helicopter theory , dynamics , which deals with the behaviour of systems . of springs and masses , and aerodynamics , which defines the forces on bodies moving in air , are inseparable . Until recently , no similar state of affairs existed for fixed - wing aircraft , other than in the " specialist " subjects , such as flutter , and even today an aerodynamicist is largely concerned with air Since this degree of specialization is impossible for the helitechnician , it follows that he must cover a very much wider field

forces alone . copter

1

2

Helicopter Dynamics and Aerodynamics

than his fixed - wing counterpart , with the inevitable result that much of the intricate detail of fixed - wing theory is lost in relatively crude generalizations which must be made if answers are to be obtained in a reasonable time . This is particularly true of helicopter aerodynamics , in which the whole field of fixed - wing aerodynamics is merely the foundation upon which the helicopter theory is constructed . No attempt will be made to cover conventional aerodynamics in this book , but in this chapter an attempt will be made to re - state and emphasize simple concepts which are essential to the following chapters , and which are not normally dealt with in elementary books on aerodynamics . It is hoped that , for readers who have a reasonable grounding of elementary fixed - wing and helicopter aerodynamics , this book will not require reference to other books .

By far the best introduction to helicopter theory is given by Gessow and Myers in Aerodynamics of the Helicopter , published by the Macmillan Company of New York . Although this book is , in many ways , out of date now

and some of the theory rather crude and even erroneous by present - day standards , it presents the physical picture of the simple helicopter in a manner which is altogether admirable . In the present work , it will be assumed that the reader is familiar with the general concepts which have been dealt with by Gessow and Myers . 1.2 .

AEROFOIL LIFT

SUBSONIC

When dealing with the performance of the blade section of a rotor blade , the lift and drag characteristics for infinite aspect ratio are always used , corrections for induced air flow being made separately . In general , there is little point in introducing small variations in the lift - curve slope with change

of aerofoil section , and it is common practice to assume a constant slope up to the stall . Opinions vary as to the most representative lift - curve slope , but , in the writer's opinion , a value of dC₁ / dx = 5.73 is probably a good compromise ( a being in radians ) . In model rotor work , the value can be much less than this , owing to scale effects , and may fall as low as 3-5 ; the writer has tested model propellors of one foot diameter at between 1,000 and 10,000 r.p.m. , which revealed

lift - curve slopes as low as 2-0 per radian . For interpretation of model rotor tests is often hazardous and the this helicopter , some workers have in fact , adopted the rather extreme position of disregarding model tests entirely . reason ,

For high subsonic Mach numbers , the lift - curve approximately

a = aLs /

/

√ ( 1 — M² )

slope

is increased to ( 1.1 )

where a = dC₁ da aLS

M=

the low speed value of a Mach number .

This approximation

is compared

with

some wind - tunnel test results in

3

General Aerodynamics

Fig . 1.1

, and is seen to be fairly reasonable until a point near the lift divergence Mach number is reached . Lift divergence Mach numbers ( value of M for which dc ) , for the same sections are plotted in Fig . 1.2 , where they are seen to diminish with increasing thickness and lift coefficient . Below the critical Mach number , both lift and drag vary inversely as the

/da0

-

/

Prandtl - Glauert factor √ ( 1 M² ) so that the L D ratio is independent of Mach number until the critical value is reached . As the tip speed of a rotor or propellor passes the critical Mach number , the local thrust / torque ratio falls off fairly steeply . The " factor of merit , " which is the efficiency of static thrust generation , does not fall off , however , until the tip speed is well above the speed of sound , as the shock - wave losses at the tip are offset by an increase in efficiency of the inner sections of the blade , particularly if the blade planform is elliptical or tapered . is normal practice to neglect compressibility entirely for all performance and aerodynamic calculations , except control angles to trim . In the last case , it is evident that compressibility effects will considerably reduce the

It

collective pitch angle necessary to generate a given thrust , and in forward a greater reduction in the pitch of the advancing blade will be required to trim the rotor ; in other words , the swash plate will be tilted further forward ( allowing for 90 ° control advance ) than normal " control angle to trim " calculations would indicate . This effect is dealt with in Section 3.5 .

flight

A reasonable approximation for estimating compressibility effect on collective pitch angles to trim is to divide the low - speed left - curve slope by the Glauert factor (1 M² ) , taking M as the mean value at 0.7 of the blade radius . This approach has been justified by Laitone and Talbot in ref. 2.28 for axial flight , and by the writer (Chapter 3 ) for forward flight . Blade stalling is most important in helicopter aerodynamics because it can impose a limit on the maximum forward speed and cause severe vibration when stalling is experienced on the retreating blade . In this connexion , it is the " dynamic " maximum coefficient which is of interest , since the blade

-

is oscillating sinusoidally ( feathering ) as it travels round the disc , and is in the (CL ) max region for a time of the order of 1/200 sec only . It is often assumed that the dynamic (CL ) max , due to " Wagner " effect , is 50 per cent greater than the static ( CL )max , an assumption borrowed from fixed - wing gust - stressing calculations . On the other hand , some early American tests on a machine with a relatively low - speed rotor have shown that there is little or no dynamic increase , and in the absence of specific data it is probably safest to take the static value .

An approximate analytical method for predicting blade stall will be given in Chapter 3 , and the appropriate value of (CL ) max for this is found by using the equations in reverse to analyse flight tests for a similar rotor . This is probably the most straightforward way of obtaining values of (CL ) max for practical design - office application . A common method in America , copied by some workers in this country , is to define blade stalling as occurring when the tip angle of attack of retreating blade is 12 ° . It is obvious that this does not merit serious

2-( A.183 )

×

0

2 ·

R.N.


π

or

is

of

Ln

TA

+ m 1

0

CL -1

→

m

-2a

+

+ 2

dCr In

Ln + 1

the preceding blade

a )

+

C

1

1 1

to

/(

=

dC

< 1.0

"

.

in

is

to

is

,

.

,

,

the aerodynamic forces coupled with the random Section 2.2 and the lack of vertical damping

in of

sinusoidal variation variations described

.

in

,

,

,

at of

,

as

,

is

is "

C

is

,

.

if

they are elliptically loaded 2.0 There are regions behind conventional rotor blades particularly where high such the local downwash near the tip vortex where the effective aspect ratio instability can arise lower than 2-0 and hence this form substantially different from the value When which the instability occurred each blade will be well away from the intense downwash wake of the preceding one and the system will stabilize unless another unstable system has occurred elsewhere obviously difficult the blade Flight experienced due this unstable region and random vibration the non-

A >

For wings this condition implies

ΠΑ

(

+ a

πA

the

a

dC

dCL

TA

СL

will be

dCLin

)a

The condition for stability met when 2a

the

πA

−2a

/

dC

1+

)/ (

av

(

dCLt Ln

1

dCL dC

-Ln

dCL

-

dCL dc dCi1-1

1

+

dC Un m

-

and

CL with respect

(

Thus the change

In

ΠΑ

of

CLA

at

2aC ΠΑ

2aC

or

wing

Ln

aCL In ΠΑ

av

CLn

will

angle

-

its

2C /TA , whilst self induced downwash wing angle of incidence the lift coefficient

be

vibration may be demonstrated by considering the analogy of a cascade of elliptically - loaded wings , of aspect ratio A. When the vortex sheet of one wing passes close to the following wing it will induce a downwash angle

49

Induced Aerodynamics The relationship between induced velocity and aspect ratio is

-

=

-a

For the critical condition A

x crit

-

in vertical flight .

CL ΠΑ

Therefore Vi

xVT

CL - ΠΑ

CL

*

a

= α

In other words , vortex -ring " roughness " due to this effect will occur if the local induced velocity ratio 2 exceeds xC1 a . It is evident that for low solidity rotors (low disc loading and high blade lift coefficient ) any vortex ring roughness experienced will be due only to the pulsation of the air body

/

(Section 2.2 ) , and that even this may be unimportant because damping of vertical velocity variations will be present , in contrast to the more usual condition of zero damping in this region . In forward flight the speed with which the trailing vortices are carried away from the plane of the rotor ordinarily diminishes , unless the disc incidence is very large , so that the accuracy of current vortex - sheet theories

diminishes for the reasons previously given . The greatest error occurs in the case of ideal auto - rotation , as in Fig . 2.22 for example . Moreover the effect of the number of blades becomes very powerful , an aspect which is discussed more fully in Section 2.20 .

It is evident that for normal design purposes it is likely to be physically impossible to obtain an accurate induced velocity variation over the disc for most problems of forward flight . For rotors whose blades are tapered and twisted to give substantially uniform downwash in hovering it is therefore recommended that Glauert's hypothesis v₁ = v ( 1 +

be used , taking

In

K from

Kx

cos

y)

equation (2.21 ) . may be available to permit the

some cases sufficient information refinement v₁ = v ( 1 +

Kx

cos

y

+ kx sin y )

to be used . For example k may be determined from wind - tunnel measurements of the flap - back angle a₁ . On some unpublished work the writer has found that

k The alternative

μ3 -(at)43

-

extreme

16e

and

to a blade

2k

2μ

Ст

(2.23 )

ta

of constant circulation is

chord untapered blade , for which the circulation increases

a parallel-

radially in an

Helicopter Dynamics and Aerodynamics

50

approximately hovering of

By assuming an induced flow variation in

.

linear manner

-

.

as

k

K

may be taken the same where and for the uniform loading case the absence of more accurate information and vit 200

in

kx2 sin

forward

)y

+

y

,

the tip the induced velocity

Kx2 cos

+

=

( x

vi

vit

v

is

at

where Vit the induced velocity flight becomes

in

it

Vi = XV

x )

1 +

Kx cos

/

1-2 +

=

K

[ v₁ =

3 v₂ (

0.8

DOWNWASH

0.6 CASTLES

AND DE LEEUW

0.4

MOMENTUM

THEORY

0.2

TRAILING EDGE

0.6

OF

THE

DISTRIBUTION ON AXIS OF SYMMETRY FOUR MAJOR THEORIES AT

(

μ = 0 · 4

0.4

)

INDUCED VELOCITY

COMPARISON

0.6

0.2

~ 4 · 0

2.14

.

.

FIG

OLEADING EDGE

UPWASH

MANGLER AND SQUIRE

.

K

.

is

,

to

is

It

is a

)

is

(

.

at

in

.

v

in

,

This equation gives sufficiently good agreement with downwash measureremembering that the absolute ments with untwisted untapered blades generally flight values of are small forward Fig 2.14 concludes this section by comparing induced velocity variations high speed for the four major theories the longitudinal plane of symmetry given by equation evident that the Glauert variation with very approximation good 2.21 the Castles and DeLeeuw variation uniformly loaded and thus suitable for use when the rotor

,

a

of

For loadings characteristic untwisted untapered blades the theory of Mangler and Squire gives much more complicated induced flow distribution

Induced Aerodynamics

51

as Fig . 2.14 shows . Comparing the values along the longitudinal plane of symmetry with those for uniform loading , it is evident that the greatest difference is experienced at the rotor centre and behind , an upwash occurring behind the hub instead of a strong downwash . Mangler's results are confirmed , for an infinite number of blades , by the work of Heyson and Katzoff ( ref. 1.30 ) , who apply the technique of Castles and DeLeeuw to a

radially variable loading . In the lateral plane of symmetry both the uniform and variable loading theories give results which are consistent with their assumptions . In there is a dissimilarity because of the difference in blade advancing loading on the and retreating sides of the disc , and because of the finite number of blades . The use of Glauert's variation in helicopter rotor dynamics leads to practice

( ref. 1.29 )

important modifications in basic results . The lateral flapping angle is considerably increased and in simple theory it can be regarded as an effective increase in the coning angle , as can be seen from the work in Chapter 3 , where

Δαρ ao

-K

i

(2.24 )

μια

where t , and t₁ are taper integrals defined in Chapter 3 ( page 112 ) . The lateral downwash gradient k adds an increment Aa , = kλ , to the flapback angle defining rotor longitudinal flapping , for zero hinge stiffness . In the general theory of rotor moments , forces and hub vibration , the results are

greatly affected by Glauert's variation in the intermediate speed range . In it should always be used in theoretical work in preference to the more usual constant induced velocity assumption .

general

2.8 .

INDUCED

VELOCITY IN THE VICINITY OF A ROTOR

Figs . 2.15 to 2.26 are abstracted from NACA Technical Note No. 2912 ,

AIR RELATIVE VELOCITY

FIG . 2.15A . DEFINITION

OF ORDINATES

)

(

5- A.183

.

J.

,

,

,

its

The normal component of the induced velocity in the vicinity of a lifting rotor application by W. Castles Jr. and H. DeLeeuw and some examples of

FIG

.

.

2.15B

DEFINITION

OF ANGLE OF SKEW OF SLIPSTREAM AND

DISC INCIDENCE

,

.2

O-

-

α ·

°

%

a,

-

/v v

x =

°

x

for

x ); = – /v v

;

"o

induced velocity at any point induced velocity at the centre of the disc momentum value 90 tan for 90 cot sin sin —voi (

Vi

0.06

10:06

0.04 0.02 ο οι -

0.020.04

-0.6.

DIN

O.8. 1-0 ·1 ·

1-2 4

/-0.1-0.21

0.01 -0.02

°

-1.6 -0.06 -0.04

1-8

O

-0-06-0-04

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY v

/v

.

.

-0.02 -0.01

∙ 1 · 9

-2 FIG 2.16

-02-01

0.1

0.2 0.05

· · 6 ·

0 0.8 0.9 1.9 1-1 1.2-1.3 1.4-

NO -0.02

-0.04

0.05

0.4

0.02

-0.1 -0.2

-0.06

0.02

1-004

-1 -1.8-

1-9

-3

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY %

-

14-04

°

/v

2.17

v

.

FIG

.

-2

-0.02 -0.01

2

-0.21

0.4 0.6 O-80.1

22

0.06

1-2 0-02 -0.06 -0-1-0-2 -0.04 1-6

-2

3

2

3

FIG . 2.18 . LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY X = 26-56°

0.04 0.02

O 90 0.1

0.2 0.4 0.6 O-8

-0.02

-0-06-0-1-0-2 -0.04

OXR

2

0.2

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY = 45.00 °

%

.

.

FIG 2.19

53

0.1 0.2

DIN

1.2 -0.02 -0.04-0.06 -0.1 -0.2

OXR

2

1.6

2.20

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY x = 63.43

2

°

.

.

FIG

0.06

1.2 1.4 1.8 1-6

0.020.04 -0.06-01-02.03

0.8

0.0 0.02 0.1 0.4

2

0.06

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY

= 75-97

°

X

2.21

.

FIG

.

DIN

46002

0.02 0.1

54

2-2 -2.01-8 -1· 61-4 -1-21.0 -OO-

-0-2

6

8

.

-0-04-0-06-0-1

+

O

00

0.1

-0.4.

0.2

XIX

0.02

.

.

FIG 2.22

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY °

x = 90-00

2 O- 06

//

01 0.02 2/01/04 0.0 0.2 0.6

0-90-0-

· 1

0-02 -0-04

.8

.

1.2 1.0. O

0.6. O.43

0.06

3

0.02

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY °

% = 104-03

55

35

.

.

FIG 2.23

2

-0.2 1.2. 1.Q O.8 0.6

.

-0.02 -0.04-0-06-01

00

0.1 0.2

°

LINES OF CONSTANT VALUES OF ISOINDUCED VELOCITY RATIO IN LONGITUDINAL PLANE OF SYMMETRY = 116-57

°

0

X =

x

.

FIG 2.24

2

-2

.

14-04

X

26-56

°

-

°

-

X

63-43 116-57° °(

-

104-03

2

)

°

X -

75-97

(

X

°

=

X 45.0

INDUCED VELOCITY DISTRIBUTIONS ALONG

AXIS

X -

2.25

.

FIG

2

-2

°

90

=

X

.

DIN

2.0

57

Induced Aerodynamics

X= 0° X- 45°

x =90°

· -

X- 45 °

X 0°

X=90 ° -1

2

3

R FIG . 2.26 . INDUCED VELOCITY DISTRIBUTIONS ALONG

2.9 .

Y -AXIS

VERTICAL DRAG

At low forward

speeds the slipstream from a helicopter rotor is substantially downwards in direction and will cause a drag force to be generated on any body immersed in it , the drag acting in the direction of the slipstream . In

most performance methods the effect of this vertical drag is ignored but can in fact substantially modify calculated performance , being equivalent to a weight increase of over 10 per cent even on some single rotor designs . The basic parameter is the equivalent flatplate area ( area of body - drag

it

coefficient ) which is immersed

ratio of the rotor - disc area ,

in the slipstream , and this is

expressed

as a

i.e.

ACD πR2

(2.25 )

Obviously this ratio is greater for bluff , poorly faired fuselages and for multi - rotor helicopters where a large fuselage or outriggers are in the downwash . The drag coefficient of the drag - producing body can usually be esti mated from wind - tunnel data already in existence ( Hoerner's Aerodynamic Drag -for instance ) but it is almost always found that the effective - drag coefficient in the downwash is higher than the free - stream value . this

If

is due to the non - uniformity of the slipstream causing a redistribution of pressure then it is possible that for some bodies it might be possible for the drag to be less in the downwash .

2.10 .

VERTICAL DRAG IN HOVERING FLIGHT

The special case of hovering is susceptible to elementary analysis and will first . We have seen ( equation ( 2.2 ) of Section 2.2 ) that the

be considered

58

Helicopter Dynamics and Aerodynamics

thrust of an actuator disc in hovering is equal to the mass of air flowing through the effective disc in unit time , times the acceleration imposed upon it .

Τ= T If a drag - producing

2pv

,²π R²e

( 2.26

body in the slipstream is subject to a drag

▲T

)

the

Co

1.4

1.2 DRAG OF CIRCULAR DISCS

(

AVERAGE VALUES FROM VARIOUS SOURCES

1.0

4.0

0.83.0

5.0

6.0

LOG R.N.

995

Co 1-8

1-6

)

)

J.

.

1.4

I.

( (

DRAG OF RECTANGULAR FLAT PLATES AVERAGE VALUES Ae SCIENCES SEE Vol No.1

1.2

20 25 ASPECT RATIO

15

,

the )

2.27

Av¿ ²πR²e )

+

2p

v¿

(

=

+

T

,

be

an increase Av

in

2.27

thrust must increased by that amount resulting induced velocity

AT

40

35

(

.

FIG

30

in

10

5

1.00

of

in

, )

+

( iv

is

is

,

; .)

+

2 ( v₂

Av

a

in

it .

of

The drag force AT the body the slipstream will be function the mean slipstream velocity past This velocity Av the plane of the disc and the fully developed value some way below the disc The variation

between these two values will be discussed

59

Induced Aerodynamics shortly

, but for present purposes we designate the slipstream velocity at any arbitrary point as n (v , + Av , ) where 1 ·0 ≤ n ≤2.0 . Applying the normal drag equation , the vertical drag is

Τ= T

(2.28 )

ACD1pn² (v ; + Av ; ) ²

where A = area of body in the plane parallel to the tip - path plane which is immersed in the wake CD = appropriate drag coefficient

Τ

Since

and from

( 2.25 )

and ( 2.27 )

Then

of the body

'

(see

Fig . 2.27 ) .

ΔΤ Τ + ΔΤ

=1

T + AT AT AC Dn2 Τ + ΑΤ 4π R²e 1 T + AT = Τ AC Dn² 1

( 2.29 )

4еπ R2

As already stated , it is evident that the basic parameter is ACD /TR² but the velocity ratio n is also of considerable importance . This simple theory indicates that the nearer the drag producer is to the rotor the smaller

will be

n and hence the vertical drag . few test results are available for vertical drag in hovering , notably the work of Fail and Eyre ( ref . 1.25 ) , and Makofski and Menkick ( ref . 1.24 ) . From equation ( 2.29 )

A

vertical drag

AT

total rotor thrust

Τ + ΑΤ

Fail and Eyre give

a value

of

Dn² 4e

-

Dn²

A πR2

0.7 based on experiments

with a

large wing under a model rotor . Makofski and Menkick carried out fullscale experiments on the Langley helicopter test tower , mounting flat panels underneath the rotor and normal to its slipstream . The rotor radius was 18-74 ft , and two panels were used , both of 4-0 ft chord , having a span equal to , and half , the rotor diameter . The panels were positioned between 1 ft and 12 ft below the plane of no - flapping (hub orbit ) and caused no detectable variation in the power required by the rotor at a given thrust and tip speed . It was found that , contrary to the simple momentum theory given above , the vertical drag of a panel was greatest when it was close to the rotor .

The force acting on the panel can be divided into two components ; a steady drag force due to the slipstream velocity , and a pulsation caused by the passage of each blade over the panel . Whereas the steady component does tend to increase with increasing distance below the rotor ( as simple theory indicates ) , the pulsating component diminishes rapidly . Thus , the total effective vertical drag tends to diminish rapidly at first , as the panel is moved away from the rotor , and then remains fairly constant .

Helicopter Dynamics and Aerodynamics

60

A second

deviation from simple theory is caused by the radial variation of the induced flow in a practical rotor disc . For the rotor tested , the thrust grading and , therefore , the induced velocity , increased towards the blade tips , so that the half- span panel had proportionally less drag than would be indicated by the simple A TR2 criterion , whilst the drag of any body inside the radius defined by the blade root - ends was found to be negligible . The agreement with strip theory calculations , assuming fully - developed downwash v₂ ( = 2v ) was found to be quite good however , particularly when the

/

panel distance below the rotor exceeded 30 per cent of the rotor radius . Using the form of equation (2.29 ) the ratio of vertical drag rotor thrust can be modified to allow for radial induced velocity variations by the modification

AT

Τ + ΔΤ

k

A

(

TR2

= semi - span of panel or wing A = plan area of panel or wing

where

R

If

-

rotor radius

k == an experimentally - determined constant

z = distance of panel or wing below rotor , then the results of Makofski's and Menkick's tests are given in the following table . VERTICAL DRAG OF WING IN HOVERING Non - dimensional distance of wing below rotor

Z /R =

k

n°CD

when

l

=

0.05

0.81

2.59

= 0.2 = 0.65

0.66

2.11

0.66

2.11

R

This approach is rather crude ( although of course fairly large errors can be tolerated because of the relative smallness of vertical drag in relation to rotor thrust ) and no allowance is made for such effects as wake contraction . By assuming C₁ CD = 1.2 for the panel these results can be used to give the vertical drag of such items as fuselages and floats , by multiplying the panel vertical drag ( for the same plan area and radial station ) by the ratio of the drag coefficients . The pressure pulsations on the horizontal surface are of a fairly significant magnitude . Taking averaged values between the spanwise stations of 0-3 R

to 0.9 R (random movements of the tip vortex make measurements unreliable outboard of 0-9 R ) the pulse pressures can be expressed as a ratio of the rotor disc loading . Like the steady pressure distribution , the pulse - pressure distribution is of the blade lift grading . At constant disc loading and wing

also a function

position it would be expected to diminish with increasing number of blades

,

Induced Aerodynamics

61

PRESSURE PULSE AMPLITUDE ON HORIZONTAL PANEL UNDER TWO -BLADED ROTOR ( p - rotor disc loading )

A

Non - dimensional distance of wing below rotor

Upper surface pressure less ambient pressure

Upper surface pressure less lower surface

-

3.0 p

2.1 p

p 0.75 p 0.26 p

0.42 p

Z/R

0-05

pressure

1.9 p

0.1

0.2

1.05

1.05

0-3 0.5

p

0.2 p 0.05

0.64

p

0

although the frequency would of course be increased . When more than two blades are used , the figures can be multiplied by 2/b , where b is the number of blades . It is hardly necessary to point out that if a wing mounted underneath a

rotor has a natural frequency

close to the pressure pulse frequency , considerlikely able trouble is to be encountered .

2.11 .

VERTICAL DRAG IN VERTICAL FLIGHT

Reverting to simple momentum theory , it is evident that in vertical flight the slipstream velocity at any point is ( V.c + nv , ) instead of nv , as in hovering . The value of n is difficult to calculate exactly but its value along the centre line perpendicular to the tip - path plane is given in Fig . 2.28 for a uniformlyloaded disc . This graph was constructed from the results in ref. 1.14 for hovering and vertical flight , but subsequently it was found that if n is defined

of the induced velocity at an arbitrary point on the tip - path axis to the mean value at the centre of the rotor , then Fig . 2.28 is quite general for all low - speed flight conditions . At the time of writing this seems rather a

as the ratio

remarkable result , since the only limitations are that the arbitrary must be within the rotor wake and the tip - speed ratio u must not

point exceed

the inflow ratio λ .

The curve in Fig . 2.28 has been used to construct a generalized vertical drag chart ( Fig . 2.29 ) in which the " drag loading " of the equivalent flatplate area is plotted against a non - dimensional rate of climb parameter for points at various distances below the tip - path plane . The basic curve against Vс, was taken from Fig . 2.2 , and the validity of the parameters can be demonstrated as follows . In Section 2.2 it was shown that the induced velocity could be presented as

of v ,

a unique curve for vertical flight by plotting From equation ( 2.28 )

AT = ACD it is evident that

( Vc + nv¿) is √ (p/2p )

v√ (p/ 2p )

against V./√(p/2p ) .

P ( Vε + nv.)²

a unique curve against

(2.30 )

V

√ (p/2p)

for particular

62

Helicopter Dynamics and Aerodynamics

TIP

PATH

PLANE

FLIGHT

SPACE AXIS

VELOCITY

SLIPSTREAM

BOUNDARY

TO

Χ

RELATIVE PATH

PLANE

TIP

SLIPSTREAM

BOUNDARY

OF

< λ IS

THIS VARIATION ACCURATE SO LONG AS AND APPLIES WHEN THE IS IMMERSED IN THE SLIPSTREAM

2.0 BELOW ROTOR

nv₁

²)

2.31

2p

)

Vc

2.30

(

APCD

=

+

AT

Substituting this expression

P /

the induced velocity ratio

SPEEDS

),

VELOCITY AT LOW FORWARD

.n

SLIPSTREAM

1 (

.

2.28

of

.

FIG

1.0 DISTANCE BELOW ROTOR ROTOR RADIUS

(

1.0

ABOVE ROTOR

in

о

values

RELATIVE TO TIP PATH PLANE

TIP PATH PLANE

O

INDUCED AVELOCITY VALUE AT ROTOR CENTR NTRE

2515

VARIATION ALONG

&

=

/

INDUCED VELOCITY

2.0

V

+ Vc +

vsini

Y

"

=

/>

DISC INCIDENCE tan X

SPACE AXES IS ANGLE OF TIP PATH PLANE TO FLIGHT PATH

is

is

in

.

.

in

is

,

to

.

It to

.

.

Fig 2.2 Thus Fig 2.29 constructed directly from the information currently available for vertical drag No experimental evidence vertical flight evident that both pressure pulses and the steady component due the induced slipstream will be smaller whilst there will be an easily calculated increment due the vertical climb velocity At

Induced Aerodynamics

63

)( DRAG PARAMETER

ACT

IDEAL AUTOROTATION

2.0

HOVERING

3.0

T

0.75

VERTICAL

ό

-1.0

O

RATE OF CLIMB

AT

):

); p = in

(

):

ft /

( ft );

C -p (

R /

slipstream ft³

sec mass density of ath plane on T.P. axis

( ft )

);

(

=

Z

equivalent flat plate area

AC

actual ft² V. vertical distance of drag producer below rotor tip Rrotor radius

.

2.12

to

is

)

.

)

to

(

is

so

(

of

likely climb the solution given by equation 2.31 be approach experimental hovering that one fair the values accurate into the curve given by equation 2.31 for vertical climb and descent ,

high rates

);

lugs ft2

/

(s

air

)

√ (

/p

2p protor disc loading

against non dimensional rate of climb -

ACD

(

vertical drag

) ( ( lb ); lb /

=

;

AT

/P

VARIATION OF VERTICAL DRAG WITH VERTICAL RATE OF CLIMB

Ratio of drag loading to rotor disc loading

Vc

3.0

2.0

Vc 2P

/P

2.29

.

.

FIG

1.0

PARAMETER

ACD

-2.0

VERTICAL DRAG IN FORWARD FLIGHT

In the

of

)

(

if

.

2p

is

or

if

is .

)

/p

.

is

or n

,

in

is

.

a

√ (

.

of v / or

-

intermediate flight speed range between hovering and cruising flight equation 2.15 vertical drag can be calculated by using Fig 2.6 Section flight specified 2.5 For condition the value first obtained The induced velocity ratio determined from Fig 2.28 the drag proslipstream Figs ducer the from 2.16-2.24 some all of the body

,

is

. )

nvi

)

2p

.

the tip path plane -

speed component normal

)

2.32

(

+

Vc

/p

√ (

i

sin

to

i

V

sin being the forward

V

[

4 p

AT = ACD

a

If

).

(

of

n

(

.

is

large outside the slipstream the body mean value must be taken Finally AT being the r.m.s. of the local values obtained from an equation 2.31 extension

Helicopter Dynamics and Aerodynamics

64

2.13 .

EXAMPLE OF VERTICAL DRAG CALCULATIONS

In Fig . 2.30 the vertical drag of a single - rotor helicopter is plotted against vertical rate of climb based on momentum theory calculations only . The normal sea - level disc loading for this example is 3 lb/ft2 and it is evident that at 600 ft /min vertical rate of climb the thrust must be increased by 8.6 20

CLIMB

5

DISC LOADING

3

FT SEC

/

)

LB

ATT

(

8

DISC LOADING

10% THRUST

3

/

4 FT2

)

(

-10

IDEAL AUTOROTATION ‫הוח‬ ‫וחד‬ ‫תוח‬

VERTICAL

6

2

% DRAG TO ROTOR

12 %

% 4% RATIO OF VERTICAL

%

RATE OF CLIMB

-

-20

0812

;

p

at

;

used

regarded as an increase

in

:

Note In performance calculations AT

is

altitude an effective disc loading

R² = 0 ·

conditions ACD

/

ICAN

IN VERTICAL FLIGHT

[, 2 ]

AT

is

0-45:

.

2.30

8.1 .

R

/

Effective

Z

.

FIG

A.U.W.

in

of

in

.

.

is

if

so

.

to

performance allow for this vertical drag The same reduction by weight per would be obtained the were increased 8.6 cent that this evidently very serious hovering an increase effect Even 5.3 per cent is obtained

per cent

.

.

-

in

,

of

,

the S.56 type however even higher thrust increases are ,

helicopter

of

a

On

a

is

is

,

a

Since the example helicopter has bluff sharp edged fuselage whose Fig 2.30 drag approximately vertical coefficient 1.0 the increase given greater than for Sycamore type example machine the Bristol for

Induced Aerodynamics

65

to be expected , because of the stub wing in the slipstream . The most severely affected types will be tandems fitted with stub wings and those , like the S.56 and the Fairey Rotodyne , which have bluff fuselages and a fair area of horizontal lifting and control surfaces . In forward flight the affect diminishes rapidly , as shown by Fig . 2.31 , but is still large enough to be included in performance calculations up to

It

is evident that many uncertainties are introduced into

WAKE

ANGLE

ROTOR

WAKE RATIO

X

/

C

OFOUT

W

ANGLE WAKE

°

W

%

FUSELAGE

60

THEORETICAL IMMERSIONVALUE

°

40

IN

ASSUMING WAKE 74

TOTAL

20

°

RATIO OF VERTICAL

WAKE ANGLE

IN DRAG

3

DRAG TO

x

% VERTICAL FUSELAGE

5

THRUST

X

about 40 ft / sec .

EMERGING

FROM WAKE

30 FT SEC SPEED

)

= 3.0 lb ft²

/

disc oading

(

;

0-129

-l

D100 A.U.W.

p

VARIATION OF VERTICAL DRAG WITH FORWARD

2.31

.

.

Fig

/

-

FORWARD SPEED

forward flight but for engineering purposes an approxiinfinitely preferable the only practical alternative of

.

is

mate solution ignoring this effect

to

in

this calculation

.

2.14

GROUND EFFECT is

-

in

)

a

(

.

,

in

,

is

a

to

a

hovering rotor approaches the ground or water the slip stream restrained by the presence of the ground and the induced velocity required produce given thrust induced with resultant decrease reduced power loss coupled with some reduction profile power loss Several

When

results

.

that Zbrozeck's curves give reasonable

,

.

.

a

is

.

,

a

-

is

of

,

methods of evaluating this effect have been published notably ref 1.12 knowledge but in the writer's opinion the present state such that only practical for design office purposes Ref 1.9 semi empirical approach presents such method and experience over the last few years has shown

Helicopter Dynamics and Aerodynamics

66

From performance calculations or flight tests the maximum all - up weight free air will be known for a given power . For the same power output the machine will hover in the ground cushion at a new weight

for hovering in

Wg.c.

W∞ ( T/ T∞ )

(2.33 )

/T

where (T ) is obtained from Fig . 2.32 , provided that the rotor blades are not stalled . It should be noted that this result is independent of vertical drag .

2.0

TO FREE

AIR VALUE

У

тоо

2.5

/-

RATIO OF THRUST

Z

2R

0.4

01

0.6

08 1.0

0.05

02

0.15

01 Стое

007

x

Ст

T

tpVR² in

to

-

W g.c. )

T∞

of

Some performance methods merely reduce the induced power the rotor factor which depends upon its height above ground ground effect The variation forward flight has been investigated by in

of

.

a

by

)

2.34

(

=

T /

∞

(

equivalent

W

-

-

,

to

it at is a

,

required find the power necessary hover given weight free air performance calculations must be made for an equivalent free air weight of

if

Alternatively the ground cushion

bco #R

-

00-7

POWER

.7

height of 0.7R blade element above ground¸ rotor radius 2

Z 2R

INCREASE IN THRUST AT CONSTANT DUE TO GROUND EFFECT

2.32

.

.

FIG

Induced Aerodynamics

67

and Bennett in ref. 1.1 . In their theoretical analysis the rotor source and the ground effect is simulated by a mirror image of the source equidistant to the ground line . The chief value of the analysis is in the determination of the important design variables , and flight Cheeseman is replaced

by a simple

1-15

/2PπR2

Too 0.3 1-1 7/10

10.5

1.05

1.0 1.5

2.0

1.8.5

1.0

2.0

3.0

2.5

VARIATION OF THRUST WITH HEIGHT ABOVE GROUND GIVEN FORWARD SPEED EMPIRICAL CURVES (

1.1

0.35

basic

); ( C )

.

(

Ref

AT

)

2.33

.

.

FIG

A

/R

Z

1.5

CL BASIC -

O 275 O 323 O.363 O- 426.

.

-

1.3

1-2

T

Y /

To

Too

1.1 1.0 2.0

0.8

2.0

1.6

basic

)

(

VARIATION OF THRUST WITH HEIGHT AND BLADE GIVEN SPEED AT A

.

.

FIG 2.34

C

Z R

/

1.2

.

Theoretical curves from ref 1.1

of

to

T

R )

mean blade lift ,

.

in

is

)

(

of

.

,

is

of

Fig 2.34 where coefficient 0.35 The effect CL basic indicated evident that ground effect diminishes with increased blade loading )

6 (

A.183

as it

-

2.33 for

a

Fig

.

given

-

Z /

2рπ R2

in

(

T

is

forward speed ratio

V

to

.

T /

of

-

to

of

single rotor helicopters have been used tests on three types derive empirical curves similar ∞ with the ratio Zbrozeck's The variation rotor height above ground rotor radius and the non dimensional

68

Helicopter Dynamics and Aerodynamics

for the hovering

Τ

and

2 V²πR² T

It

- πR

case .

should be noted that in Fig . 2.32 Cr∞ is defined as

bc0-7

00-7

The application of these results to the power curve of a helicopter is illustrated in Fig . 2.35 which indicates the considerable power saving which results from flight in the ground cushion . It is interesting that at the low height of 60 per cent of the rotor radius the power required increases as the helicopter accelerates away from hovering , before falling off in the normal

NR

= 0.6

NR NR

= 1.0 =

∞

POWER

REQUIRED

manner beyond 10 knots .

FORWARD SPEED 2.35

.

.

FIG

TYPICAL VARIATION OF THE POWER REQUIRED TO MAINTAIN HEIGHT WITH FORWARD SPEED

The work of

.

2.15

.

as

of

,

a

is

of

adequate for prediction perand Bennett ground great deal of work to formance in the cushion but there remains ground effect can be regarded satisfactory be done before our knowledge Cheeseman

STRIP THEORY IN VERTICAL FLIGHT

-

a

,

"

as "

.

is

of

.

a

is to

develop equations for the concepts can be used velocity induced variation blade which not uniformly loaded but arbitrary plan simple equations form the basis has an twist and form These commonly strip theory what known well established method of analysis for propellers Instead of the whole disc we consider only an at

The simple momentum

elementary annulus

a

V.

+

v₁ )

r

at

(

+

it ,

in

.p

its

b -

2v ;

)

2.35

(

v¿x

)

v¿

+

(

= 4рπR²

Vc

,

=

.

b

b

dx

2pπr

)

2.36

(

i

aσR

λ

32x

2¿ )

=

+

dx

(

dCr

λc

,

pVT²RabCo

T /}

=

C₁

a

dr ,

X

dL dL

v₁ )

to

of

its

of

.

Defining

dr ( V¸

of width dr and radial distance from the centre The mass air flowing through the annulus will be equal the product area 2πr the air velocity and mass density Since thrust = mass acceleration we have for bladed rotor

69

Induced Aerodynamics The solution to this quadratic in 2, is

--

λί

A

2.

2+

aσR

+

4

/

relationship for dCr dx can be deduced

considerations

.

dL

= pVxRca (

dx

For a completely arbitrary blade to be

dx

-4

Equating equations in 2 ,

=

+

aσR

c RX2

( 2.36 )

( 2.38 )

)

we assume the local blade pitch angle

= DR

dCr

( 2.37 )

from simple blade element

-

-

-

9s

The inflow angle &

λ

( dCT

32.x

i

λc + λ x

(2.39 )

-

-

(2.40 )

Vşx² — (λc + λ ;) x ]

and ( 2.40 ) and solving the resultant quadratic

с

+

16

2 ( DR

422 (C0 c

aσR

-

9g ) x

+

λe

+

21c

aσR

c

(2.41 )

16

Once λ , has been found from equation ( 2.41 ) equation ( 2.40 ) can be used to obtain the elemental thrust loading . By solving both equations for various stations along the blade curves of thrust grading and induced velocity grading can be obtained . Since no allowance has been made for root and tip losses the curves will have to terminate at an appropriate distance from each end of the blade . Note that the incremental induced power will be

dT ( λ + λ¿ ) VT • practice In the tip loss of a blade influences the performance of all blade elements , the effect becoming significant outboard of 0.7R . This can be equal to

expressed

by introducing

K into

a factor

32x

dCT dx

aσR

equation ( 2.36 )

(λcC +

2₁ ) λ¸K

(2.42 )

K is

a function of the inflow ratio ( 2 , + 2 ; ) , the radial position along the of blades in the rotor . Standard tables were presented by Lock and Yeatman in R and M No. 1674 ( 1935 ) based on the work of Prandtl , Betz and Goldstein , and these values are plotted in Figs . 2.36 and 2.37 for the range of inflow angles likely to be met on a helicopter rotor with two or four blades . The effect of K is to increase the induced velocity near

blade and the number

the tip

for a given thrust grading , 2

,

-

2

+

so that equation ( 2.37 ) becomes (2.2

+

aσR

32xK

(dCT

) dx (

(2.43 )

Helicopter Dynamics and Aerodynamics

70 1.0

0.3

0.9 0.45

0.8 0.05

K

0.6

O.15

RADIAL

0.8 0.2

TIP

0.3

ALONG

0.75

POSITION

CIRCULATION LOSS O O

BLADE

COEFFICIENT

Ò

0.9

INFLOW

ANGLA O.95

0.2

-

0.1 0.2 0.3 O.4 LOCAL INFLOW ANGLE RADIANS -

0.5

VARIATION OF CIRCULATION LOSS COEFFICIENT LOCAL INFLOW ANGLE FOR TWO BLADED ROTOR

K

-

WITH

No. 1674 1935) by Lock and Yeatman

аOR

2.44

)

аσR 16K

(

+20+

22

.

-

becomes

c

C

4K22

) x

ORs

2 ( (

1+

)

(

с

αση

+ 16K

[

)

(

Combining equations 2.42 and 2.40 the general equation for

2,

(

R

Based on

& M

-

-

2.36

.

.

FIG

C .

-

if

,

K

of

-

This equation can be used for all vertical flight strip theory calculations Strictly speaking the values calculated by Lock only apply exactly

)

(

)

.

in

a

,

(

at

in

.

is

it

is

the induced velocity constant along the blade radius actuator disc loading practice that no appreciable error can be detected for any but found Flight tests practical rotor the R.A.E. by Brotherhood 1.10 have shown that strip theory gives excellent agreement with practice and its propeller design use in rather different form has long been standard

Induced Aerodynamics

71

0.3 0.45

0-05

0.6

O.9

0.7

0.75

8

COEFFICIENT

0-15 O.85

POSITION

LOSS

0.2

TIP

-

ANGLE

RADIAL

INFLOW

CIRCULATION O

ALONG

0.7

/T

R BLADE

K

X 0.95

0.3

0.2

K

WITH

M No. 1674 1935) by Lock and Yeatman (

&

Based on

R

-

-

VARIATION OF CIRCULATION LOSS COEFFICIENT LOCAL INFLOW ANGLE FOR FOUR BLADED ROTOR -

2.37

.

.

FIG

0.5

O.4 RADIANS

0.3 ANGLE

-

0.2 INFLOW

-

0.1 LOCAL

of

,

.

in

is

is

8K

Co

(

λi

ORX

In Fig

)

2.45

2¿ )

+

(

CL

λc

(

.

is

to

,

.

.

to

a

to

of

strip theory tapered blade The use calculate the thrust loading Fig 2.38 2.39 and illustrated direct analysis not readily susceptible plotted against radius for the 2.40 In Fig 2.39 the induced velocity maximum collective pitch angles obtainable on this design for the hovering give the C₁ variation using the relationship condition The result used

is

if

),

(

so

.

It ,

=

of

is a

.

a is

plotted using equation 2.40 2.40 the thrust coefficient grading gives integration graphical 0-0975 should mean CT curves of the and unique function pitch be noted that mean Cr = collective that R19.5 the maximum angle attainable with the existing hub geometry

THICKNESS

10

/ROOT

CHORD

RATIO

½x10-

/

CCHORD CO

0.5

RATIO

10 ANGLE

OTWIST

0-2 RADIAL

O-6 ALONG

O.4 STATION

0.8

BLADE

x

1.0

CL

FIG . 2.38 . STRIP THEORY ANALYSIS FOR MAXIMUM COLLECTIVE PITCH IN HOVERING FLIGHT (0 = 19.5 )

CL

FLOW RATIO

LIFT COEFFICIENT

20.05

ELEMENTAL

O

INDUCED

goo

x

-6

FIG

.

2.39

/C

THEORY

0.15

COEFFICIENT

CO 0975

MEAN

-

GRADING

dx

SIMPLE

CORRECTED BY USING TABLES OF LOCK AND YEATMAN

THRUST

O

O.4

0.6 .

FIG

2.40

72

*

1.0

73

Induced Aerodynamics

the elemental lift coefficient cannot exceed the values plotted in Fig . 2.39 . This fact can be used to check that a helicopter rotor does not stall in hovering when operating at its maximum collective - pitch setting , since the critical pitch angle is independent both of altitude and tip speed .

In Fig .

2.41 the variation

altitude for various tip

is plotted against

of maximum rotor thrust

speeds , using the mean CT obtained from

Fig .

2.40 .

Vertical drag effects are included from the work above and the ratio plotted is maximum rotor thrust

vertical drag

nominal all - up weight

- FT

15,000

/ /

SEA LEVEL 0.5

SECT

SEC

FT

FT 800

700

SEC

SEC

FT

/ /

FT 600

ALTITUDE

10,000

5,000

VT900

SPEED

TIP

20,000

1.0 1-5 2.0 MAX HOVERING WEIGHT NOMINAL ALL UP WEIGHT

-

.

2.5

2.41 VARIATION OF HOVERING CEILING WITH HELICOPTER WEIGHT FROM CONSIDERATION OF MAXIMUM COLLECTIVE PITCH Nominal A.U.W. 1.550lb vertical drag ratio 5.3 per cent ;

.

.

FIG

,

,

6 )

.

.

a

is

of

of

of

,

of

-

(

a

it is

to

plot another set of curves based possible normal helicopter power Chapter on the limitations and these curves should always lie under those calculated from the maximum collective pitch angle strip theory which provides the only There are many applications reliably way determining each blade element the behaviour accurate One of the most obvious the design an optimum blade shape for

For

as

)

(

of

,

a

.

.

a

in

Fig 2.42A for typical case neglecting This equation has been solved losses for blade with ideal twist The inclusion the circulation loss ,

tip

2.46

2₁ )

(

(c

) =

8K12 ασκ θα

)

(

as

,

in

(

).

(

-

)

or ,

practice its optimum lift coefficient near CL opt other considerations allow From equation 2.44 we obtain the relationship for hovering at

operating

to

:

.

,

is

it

.

of

specified condition hovering vertical flight Usually this design condition although for future transport helicopters may well be vertical climb We have seen that the ideal blade fulfils two requirements the induced velocity must be uniform over the disc and each blade element must be

Helicopter Dynamics and Aerodynamics

74

factor K would give the blade a rounded tip of the type familiar on conventional subsonic airscrews . It is evident from Fig . 2.42A that both twist and taper are essential if maximum efficiency is to be obtained . It will be shown in Chapter 3 that blade twist is the most important cause of fluctuating blade loads . the blade is constructed from light

If

alloy , or has a form of construction which is liable to fail under fatigue conditions it is obviously important to minimize the fluctuating loads . Thus a case sometimes exists for using a tapered but untwisted blade , although SIMPLE MOMENTUM STRIP THEORY LINEAR

TAPER

t = 0.6 OPTIMUM TIP SHAPE WHEN GOLDSTEIN'S CIRCULATION LOSS IS ALLOWED FOR

CHORD

IDEAL FIG . 2.42A . OPTIMUM CHORD VARIATION FOR IDEAL HOVERING ROTOR

in the writer's opinion it is better to

use a form of construction which has a high resistance to fatigue , such as bonded steel . 2.16 .

INDUCED

TIP

LOSS

The use of the circulation loss factor K is unwieldy in many calculations , particularly when strip theory is not being used . As explained in Section 2.4 it is usual to make the assumption that a small area at the blade tip is not generating thrust . This area is assumed to start at a radius BR , so that the tip loss is ( ) . For the tip - loss factor concept to be useful it must yield substantially the same result , when applied to simple theory with no circulation loss , as is obtained with the detailed strip theory .

1B

In the past , several equations have been suggested for calculating the effective radius factor B, some of the empirical ones bordering on the absurd . The one most generally used is due to Sissingh , and is based upon the circulation - loss calculations of Prandtl . B =1 where

Ст

- √ bCT (

)

( 2.47 )

T

=

V²πR² T

It

will be shown below that while Sissingh's equation gives values of B which are in reasonable agreement with tests on untwisted untapered blades , it yields values of B which are too small for uniformly - loaded blades . The most accurate equation ( at the present state of knowledge ) is , for uniformly loaded blades ,

B

- - 2√2 b 1

(2.48 )

.

75

V .

с

at the disc

VT

number of blades

.

,

,

.

is

is

in

Fig 2.42B for hovering flight against the thrust plotted For comparison Sissingh's equation also plotted and the

This equation coefficient CT

.

=

ratio

*

= inflow

I

b

2

where

+

Induced Aerodynamics

8 =

1-0

-

O- 96

STRIP THEORY CALCULATIONS AFTER GOLDSTEIN LOCK AND YEATMAN

-

√

=

CT D

/

1

EQUATION

B

SISSINGH'S

8 TIP LOSS FACTOR

b

/2/

1-1

O 98

0-92

0-90

0.01

O⚫015

0.02

/

,

½PV ²πR²

-

-

VARIATION OF HOVERING TIP LOSS FACTOR WITH THRUST COEFFICIENT FOR TWO BLADED ROTOR -

2.42B

.

.

FIG

T

THRUST COEFFICIENT C₁

=

0.005

is

)

(

,

.

)

x =

so

2.49

(

.

x

20

0=

[2

,

to

of

is

It

.

,

)2

x

a

reduction in radius

)

2.50

(

+ λ ¿ ) λ i

λc

аσR

for CT 2.51

)

.

K. dx

(

1.0

x

***

2

B2

K. dx

16 B2

x =

these two equations

1.0

=

the circulation loss we substitute

Ст = equating

2₁

aσR

2₂ +

32

(

If for

-

(

)

(

СТ

justified for rotors then possible assume that constant along the radius the blade that is

is

low which

the induced flow ratio from equation 2.42

2,

is

loading

,

a

to

evidently of four strip theory calculations which equation 2.48 good approximation The strip theory calculations are based on the assumption that the disc

results

0

x

=

,

-

,

,

,

Solutions to this equation have been obtained graphically by the writer for two- three- and four bladed rotors for various thrust coefficients

and

Helicopter Dynamics and Aerodynamics

76

four two - blade solutions are plotted in Fig . 2.42B . In determining the value of K for each value of x the simple momentum value of 1 , was used , i.e.

+

λ₁²

The value of

B

λcλï

aσR

-

16B2

=0

Ст

(2.52 )

= 1.0 was taken for calculating 2 ,, and a more rigorous

solution can be obtained by cross - plotting against B. The work done is sufficient to justify the general use of equation ( 2.48 ) however .

the value

which the ratio CL15 CDO

/

shaft - driven rotors (CL ) maximum

which the ratio CL CDO

is a

the value

/

)

at

opt

CL

rotors

(

-

For tip driven

is

.

a

For

at

which enables every blade element to work in the design condition . The value of ( CL ) opt the section changes , and is defined according supplied to the rotor . opt

Optimum twist is the value at its optimum lift coefficient may vary along the blade as to the form in which power is is

OF OPTIMUM TWIST

CALCULATION

is

2.17 .

.

maximum

,

If

= 1.0

max

3

max

)

=

/

√

)

C₁²

)

/

/

+

.

to

(1

( (

for CL CDO (

)

min

for CL15 CDO

opt

opt

( CL )

(

CL

CDO

)

=

CDO then

a

.

opt

)

(

of

/

is

/

is

-

to

due

at a

is

difference

is

the fact that for shaft driven rotors efficiency maximum when thrust power maximum whilst for tip drives torque the ratio thrust critical high values Both definitions CL lead we take Hoerner's relationship

This

.

.

-

.

of

a

is

to

-

.

in

It

follows that the optimum value may not be practical because of retreating blade stall forward flight In Fig 1.12 efficiency curves are plotted for NACA laminar flow aerofoils from the NACA 63 series The optimum section seen be function the design lift coefficient

it

is

)

)

,

.

.

it

in

If ( (

a

of

3 ).

of or (

a

to

is

In selecting the optimum section range sections for given rotor knowledge usual start with the maximum CL basic permitted by retreating blade stall considerations Chapter CL basic below the determining the CL value for maximum efficiency then must be used most efficient section When compressibility effects are not important the section with the highest efficiency would normally be selected for use This method of selection can lead to surprising results and the thinnest

If

,

.

be

it

21


►Z , Z →> X. Except for special applications , only the 8 , angle is of sufficient general interest to be considered in this chapter . When 8 , is positive , an increase

← Y,

ANGLE FIG . 3.9

in

the flapping angle relative to the hub orbit will result in pitch , the reduction being of the form tan ẞ,8 tan 83 .

a reduction

of

If the blade is at an angle

o to the radial position on the lag hinge , then coupling , and we may write for the actual blade angle relative to the hub orbit

this also

causes

(VR ) s = ( VOR ) s

- tan ẞ tan ( ,

-

83 — 50 )

(3.29 )

This equation is only approximate , the exact equations for various combinations of hinge angles being extremely complex . * Fortunately , it is quite sufficient for practical purposes to use the relationship ( VR ) s

= ( VOR ) s

-

—– ẞ ,

tan

83

where 8 , is defined as that angle which most nearly satisfies the true

*

See , for example , the work of Roberts in ref . 2.20 .

(3.30 ) ( VR ) s

Helicopter Dynamics and Aerodynamics

118

This effective d , angle is , in many cases , roughly constant for all conditions of flight . In Fig . 3.10 , for example , it is compared with the geometric 8, angle for the Saunders - Roe Skeeter . It is easy to show that variation

.

151

20

.

&

ANGLE

deg

-

TRUE &

EFFECTIVE

5

angle

,8

ANGLE

Lag angle

LEADING

LAGGING

degl

10

5

-5

0-0.25

0.5

0.25

,

BHP bV .

FIG

3.10

-

)

(

-

NO FEATHERING ORBIT CONTROL ORBIT SWASH PLATE TIP PATH PLANE

3.11

.

.

FIG

REFERENCE ORBITS

FOR FLAPPING

ROTOR

to

.

.

,

9 °

,

=

take

8

evidently sufficient

to

is

,

it

8,

.

For most practical purposes Skeeter

,

.

to

it

is

,

in is

is

in

hovering the lag angle proportional the parameter B.H.P./bVó power input where B.H.P. the brake horse and ref 2.14 shows that the mean lag angle forward flight also proportional this parameter Thus independent plotting against constitutes the best variable for effective

for the

119

Fundamentals of Rotor Dynamics

The flapping angle of B=

αo

-

y

a₁ cos

by the Fourier

a rotor blade is expressed

-

b₁ sin y

a

cos 24

- b₂ sin 2y .....

series ( 3.31 )

The plane with reference to which flapping is measured is denoted by a suffix . C denotes the control orbit , S the shaft , or hub orbit , and no suffix denotes the no - feathering orbit . The feathering is also expressed

v=

VR

- A₁

y

cos

sin y

B₁

-

as a Fourier series

A2 cos

24

- B₂

2

Before commencing the analysis of a flapping rotor , it is necessary to establish a general relationship between flapping and feathering , since the two concepts are , to some extent , interchangeable .

In

Up

/

... .

( 3.32 )

$ UT

of attack is

both cases the blade angle

— d −4

sin 2y 24

FIG .

3.12

= d − Up UT. The in - plane velocity T Ur is the same for both flapping and feathering , but Up differs by the relative velocity due to flapping . a=

dT ,

dL

Up PLANE OF

NO - FEATHERING

UT

-FEATHERING

NO ORBIT

TIP - PATH

PLANE

MECHANICAL OR ORBIT HUB FIG . 3.13 . FLAPPING ROTOR

Up = V¸c + V sin

i+

v,

+

aV

cos

y

(pure

feathering )

(3.33 )

= climb velocity + forward velocity component due to disc incidence + induced velocity + forward velocity component due to coning

UpVV

sin ( i + a₁ ) + v , + BV cos y + (r

— e)

the addition term being the relative velocity due

dẞ (pure flapping ) (3.34 dt

)

to blade flapping : r is the the flapping - pin offset from the hub 4. Note that the coning term a cos y is replaced by BV cos y , and that disc incidence is replaced by (i + a₁).

radius

of the

element

and

e

120

Helicopter Dynamics and Aerodynamics

The other difference between the two systems is the pitch - angle variation with azimuth

RA₁ cos y — B₁ sin y ... for pure feathering .

= θα

for pure flapping

The two equations for angle of attack are , therefore , Pure feathering α = DR - A₁ cos y - B₁ sin y A 2 cos 24 ... ) ( V + iV + a V cos y + v₂ )

-

(

(3.35 )

UT Pure flapping (V +

α = OR

+

V

dy

dẞ dt

• dẞ

dy

dt

UT

cos y

UT (r - e)

UT

Since

-

+ a V cos y + v₂ )

iV

(a ,

cos

sin

b₁

+ a, cos 2y

y

...)

Va₁

dB

Ur

dt

w (a

y

-

sin y

(3.36 )

y +

b₁ cos

2a , sin 2y . . . )

(3.37 )

we may equate the two systems to give

-(A₁

cos

y

+ B₁ sin y

+ A₂2 cos

V (a α1 = + cos 2 UT

ხა

+2

b3

+2

sin 34 +

w(r−

e)

UT

(a

. . .)

b₁ (ջ cos y а2 sin 2y + cos 3y + 24 + 2 2 2

b2

sin 24 +

2y

2

sin

y + a3 cos 2

ხვ sin 4p 4p 2

--

sin ч

24 +

a3 cos 44 2

...) b₁ cos

y

+ 2a2 sin

24-2b , cos 2y ...)

ναι UT Equating coefficients

V

+ -A₁ = UT α2 2

w(r

V ba -B₁ = UT 2

w(r

-A₂2 = -B₂ =

V V (a 2

UT

V (b₁

\2

UT

2

)

e

UT e)

UT + +

by a1

-

w(r e) a3 + 2b2 2 UT 2

etc.

-

w(r -

UT

e)

202

(3.38 )

Fundamentals of Rotor Dynamics

But

It

V

μ

UT

x + μ sin y

I

w (r --- e)

is easy to see that when

-A₁

=

U

µ =

121

x

§

x + μ sin y

0 (or 5

0 and e =

+

=

e

R = 0)

= b₁

B₁ =

a1

--A2

= 2b2

B2

2a2

(3.39 )

=0

e

-An = nbn Bn = nan

Lock has shown that the relations a, = B₁ and b₁ = -A₁ apply for finite μ when e = 0. In other words , flapping and feathering are equivalent , for zero brings offset . This is easily proved by equating the A , and b₁ terms , for instance sin 2y + (x — § ) cos

-A₁ cos y

)2y

+

y bi

= 0

§

b₁

=

when

) cos

sin

(xz

-A₁

-§

2 μ

+ (x

cos

sin 2y

y

(x + μ sin y )

.

is

§

is

There no equivalence between flapping and feathering when the hinge parameter offset finite

if

,

to

(

)

3.40

0 )

=

(

§

B₁s A1

,

bis

+

a₁ =

a18

b₁ =

.

,

,

us

,

to

,

is

of

A

second result the equivalence theorem that we define flapping feathering respect and with an intermediate orbit the hub orbit for example the above relationships will give the flapping relative the nofeathering orbit or vice versa

to

of

)

(

83

bic tan

ae tan

( A10

)

3.41

-

83

= b1e

B₁e

+

=

b₁c

=

b₁

),

-

+

a₁e

= a1c

,

If is

or

(

is ,

a₁ =

-

to

.

is

required for the treatment coupling the relationship which flapping feathering flapping amplitudes between and a₁e and b₁e are the with respect the control swash plate orbit the flapping relative the no feathering plane from equation 3.30

This

Helicopter Dynamics and Aerodynamics

122

Alternatively

,

orbit are known

if

the flapping amplitudes relative to the no - feathering

,

= a₁ + b₁tan

a1c

83

( 1 + tan² 83 ) ---

b₁

bic

tan

a₁

+ tan²

(1

(3.42 ) 83

83 )

In some helicopters , such as the Bristol designs , the 8 , feathering takes place relative to the control orbit ; in others , where the actual hinge line is skewed , it is a function of flapping with respect to the shaft orbit . 3.3 .

FLAPPING ROTOR ANALYSIS

Irrespective of 8, or the position of the shaft orbit , the motion of the blades of a centrally - hinged rotor is uniquely defined with reference to the no - feathering orbit . The equations which govern this will now be developed for a twisted and tapered rotor blade .

As for the feathering rotor , the velocity component in the plane of nofeathering is UT

-

(3.43 )

VT (x + μ sin ч )

y

+ r

)

(

)

3.44

dt

(

zx

y

dp +

cos

,

+ ¿

cos

(

+

flapping terms

).

+

i

dẞ dt

a₁

(

+

µ (

µß

-

λ¿

[

V₁

= 20+

v₁ )

y +

cos

=

VT 20+

ABV

)

=

no flapping terms

(

where

+ iV i√ +

( V¸

Kx )

Up =

a¿V

Note that the plane of reference has been changed from the " no - flapping , ” however , which was the tip - path plane . The vertical velocity component is as for the no - flapping plane , plus the terms added as a result of the change of reference plane .

,

-

,

,

.

-

/

dy

y

dp

µx

µ²

μ20 +

sin

)

(

3.45

((

y

sin W

+

x

2µ

cos

+

Kx² )

+

x²

μ²

+ ¿ ¡

dy

+ (

d

= xλ0

x²

UTUP

+

2

VT

=

µxß

UT2

the

,

of

can be shown that the values U2 and UTUp required for are neglecting second- and higher harmonic terms

lift force equations -

special cases

in

except

,

It

unimportant

,

of

-

,

Only first harmonic flapping will be considered since second harmonic amplitudes are of the order one tenth of the first and are therefore

123

Fundamentals of Rotor Dynamics

+ α₁x²

y

μ² )

in

Integrating by means of the taper and twist integrals defined

)

3.48

(

-

xλo }

λ¸Kx² cos

sin

y

μλομλο

-

θς ds )

-

{

μαρχ

( 3.2a )

T

μπ

( x² +

(

- + {

θα

( 3.47 )

- UTUP

− 0s ) ( x² +

{ (Dx

-

VI

2

2

4

in the thrust equation U₁2

Vs )

y

— } µ²b₁ ) cos

}

-

c

= 4 (VR — dCr dx =4√[(9%

y

μ²α ) sin

-

Substituting these expressions

cos y

+ λ¡Kx²

b₁x²

(3.46 )

}

+ ( µà¸x −

b₁

−

( µλo

{

UTUP = xλ +

-

y

a₁ sin

y

(( 22 x²

-

— b₁ sin

kµ² )

αβ

dy

y

a₁ cos

+

ao

b₁

В=

µ² )

Substituting

equations

]

3.49

(

)

αι

cos

ψ

-

]

—

t₂Kλ¡

μέλο sin

y

+

)

)

ΐ3

μεt

(

) b₁

µ²ƒ₁

{

( tz

-[

μta

+

[2

μεθη

t₂λo

]

2μkg

§µ²k₁

+

)

& R

— (

k3

---

-

+

µ²t₁

{

[ ( t3

=

Cò

+

)

(

)

(

3.13 and 3.14

(

)

3.50

t₂λo

]

—

)

+ {

— ( k3

R

[ ( t3

µ²k₁

(

)

3.50a

t₂λ0

]

)

—

Dr

µ²Ð½

{

+

t₁

— (

)

{

+

--

µ²Ð₁ DR

10T

.

kn

+

since

-

becomes [ ( 3

mean CT

=

,

,

which for linear twist

§µ²Ð₁

) &

-

Cr C₁

=

mean

+

Integrating round the disc and dividing by 2π

is

-

),

)

—

µ²

) α1

y 3.51

)

cos

(

]

t₁λ¿K

μt220 sin ]

-

]

tзλ0

y

— {

) b₁

+

t₁

— (

{

+

µ²ƒ₂

µ²k₂

{

-

k₁ +

— (

R

—

2µkз

— († 4

µtzαo

-

2μtзR

[

-

) &

+

첃½

[

+

CMT Смт

= [ (t

.

(

-

,

As for the feathering rotor analysis the thrust moment coefficient given by equation 3.49 when the taper and twist integral suffixes are increased by one Thus

.

as

,

-

,

to

In addition this aerodynamic moment about the flapping there are moments due to centrifugal and inertia forces and the blade weight moment negligible the last usually being treated

Helicopter Dynamics and Aerodynamics

124

The centrifugal force dm . w²r acting on the element of mass dm in 3.14 can be resolved into a component in the blade and one normal to which exerts a moment about the flapping pin . The elemental moment is

Fig .

it

dMcF

and the integral MCF where

IF =

= Smw2r2ßs R

=

r² = w²IFẞs

Sm .

( 3.52 )

second moment of blade mass about the flapping pin .

w Sm

dCF dCF SIN B

HUB ORBIT

]t FIG . 3.14 . ELEMENTAL MOMENT DUE TO CENTRIFUGAL FORCES

For convenience , this (CM ) CF

IF

γ

pR¹aCo

to coefficient form

is reduced

MCF

=

@21Fẞs

PVR²αCop²R¹aCo T

and is an " inertia number , " or

=· γβ.

" second

(3.53 )

moment of mass

number , " similar to Lock's , but inversely proportional to it . The elemental inertia moment is the force required to accelerate the elemental mass about the flapping pin . Since the velocity of this mass is

rdẞ dt

,

the acceleration is r

de

. 8

dt2

and the elemental moment

dM1 = Smr2

MI = IF and the coefficient

ais cos

(3.54 )

-

CMI

0

(3.55 )

y -- biis, sin y

d dy2

In

dt2

about the flapping to zero , for equilibrium

Смт — Смсг

Therefore

d2B ,8

= γαβ, = γ d²ßs w2 dt2 dy2

Equating moment coefficients

= Bao

dt2

becomes

CMI

Since

d²ß ,

= ais

cos

y

CMCFCMI

+ bis sin yao

Y (3.56 )

other words , the fluctuating inertia and centrifugal forces cancel out ,

125

Fundamentals of Rotor Dynamics

with once - rotor frequency ; i.e. the blade frequency flap equal natural in is to w . This is only true when the flapping pin is centrally located and flapping is not mechanically or elastically and the rotor blade is in resonance

restrained .

Substituting for CMT in the equilibrium Fourier coefficients

-

(†₁ + ¹µ²Ð½ ) √R − (k4 +

2μtзR

2µk¸ - 2μkз μtzαo

§µ²k₂ )

equation , and equating the

— t3λ0 — ya

— (ts — } µ²t )α ---

(t

— μtzλo

=

= 0

1µ³t₂ ) b₁ + t₁λ , K =

+

0

0

These equations , together with equation ( 3.50 ) , yield the following explicit relationships for the collective pitch and flapping amplitudes-

-

ao

-

Сr

(tз +

t₂λo

(3.57 )

1 μ² t₁ )

(ts + { µ³t 2) DR — ( k₁ + ‡ µ²k₂ ) — tзλp

( 3.58 )

γ a1

= μ[ 2t30R2k3-1220

20

]

(3.59 )

( t₁ — —µ³t₂ )

b₁ -

For linear twist

‡ µ²k₁ ) +

+ ( k¸3 +

+ tλ¿K

μtзao

(t +

(3.60 )

1 μ²t 2)

= λc + λ¿ + µ ( i + a₁ )

( 3.61 )

T kn = in

tiến

These equations can be solved simultaneously , or by assuming a₁ = 0 , to determine an initial value for 20 , and then solving by successive approximation ; or by taking 2 as the independent variable . In practice , the last approach is normally the best , particularly if computors are used , since , once charts of the angles have been plotted against 2o , they can be used for any condition of flight , from climb to auto - rotative descent , provided that the initial range selected for 2 , is large enough . A chart of flapping angles plotted

against tip - speed ratio u for various values of 2 , is a complete of rotor behaviour at a given thrust coefficient . three or four

If

statement

are constructed , once the structural design ( and , hence ,

IF ) has

been finalized , for various values of Cr , they will constitute statements of its behaviour , and will remain so throughout its life , irrespective of the type of fuselage to which it is fitted , or the value of 8 , which is used , provided the hinge offset is small enough to be regarded as zero , that is , less than about 0.05 R.

can also be shown , from equation

that the flap - back angle for

of μ )

3.59a

(

a₁ = ²² (2C₁ +

™ )

small values

( 3.59 )

t₂λ

It

Helicopter Dynamics and Aerodynamics

126

to the tip - path plane , i.e. λ = . This relationship eliminates the necessity for successive approxi-

is the inflow ratio with

where

20ua mation ,

If

of course

respect

-

.

the blade weight is not negligible , an additional term x occurs in the numerator of equation ( 3.58 ) . When the flapping amplitudes are known , the thrust grading along the blade can be calculated from equation ( 3.48 ) .

This has

been

grading of the Saunders - Roe

used to calculate the thrust

CONSTANT

CONSTANT TERM

TERM

0.2

0.2

0.1

sin TERM

sin

0-8

6.59 bi ao 3.58 503 ft sec 0-25 0.07 2,100lb A.U.W.

/

;

1

;

;

1-25

;

.

.

1-0

THRUST LOADING OF SKEETER METAL BLADE V@

; b₁ =

;

at a

.

,

to

in

,

is

y

-

,

in

8 °

twist

In

25 .

0 ·

,

so ,

of

,

zero

TERM

tipFigs 3.15 and 3.16 velocity gradient this calculation the induced that the cos term in particular too small but twist seen be very powerful forward flight

so

as

=

of µ

was taken

,

to

.

.

in

10 °

,

is

,

of

.

.

.

.

.

is

the effect Another example the equation applied the ramjet helicopter rotor for which Figs 3.17 and 3.18 are drawn the blade twist being this Fig 3.20 illustrates the effect integration limits on the equations case and Fig 3.21 compares the theory with experimental results Subsequent analysis has shown that nearly all the discrepancies are due neglect of the lateral downwash gradient

of

even

με

2,100 lb

Skeeter metal blade for zero and speed ratio

Total twist

/

;

@ 1

.

= 0 · 25 ; 0 ; 2 = 0 · 07 ;

μ

A.U.W.

-

3.16

;

8

FIG

THRUST LOADING OF SKEETER METAL BLADE Total twist = 6.48 1.34 ao = 3.84 503 ft sec VT .

FIG 3.15

0.4 0.6 RADIAL POSITION R

0.2

O-

R

0.2 0.4 0.6 RADIAL POSITION

cos

CosTERM

8 ; °

TERM

18

16

DEGREES

14°

PITCH COLLECTIVE ANGLE AT ROOT

12

FR

ANGLE

IN

$10°

6 °

DISC INCIDENCE

FLAPPING

AMPLITUDES and

a,

X

ANGLE

b₁

-

TIP INFLOW

CONING ANGLE

40

Disc Loading

3.0

ICAN sea level conditions

0.7

FLIGHT 0.025

-

Nominal A.U.W

100

ANGLES TO TRIM IN FORWARD RAMJET HELICOPTER lb / ) ft ;

CONTROL

80 KNOTS

p =

.

.

3.17

.; (

FIG

60

SPEED

-

20

FORWARD

3

=

06-

777 TIP LOSS ASSUMED

0-08

0.06

Y

180

=

0.6

0.8

1.0

RADIAL POSITION

R

/

-

O 02

=

°

270

° 004 OROOT LOSS

ELEMENTAL

THRUST ON BLADE

T

%

0.12

-0.02 SPEED THRUST LOADING AT MAXIMUM FORWARD RAMJET HELICOPTER 0.19 900 sec Tip peed ratio ICAN sea- level conditions Nominal A.U.W Tip peed Vr μ =

-s

;

ft /

-s

.;

(

)

BLADE ;

3.18

.

.

FIG

0-2

RADIANS

(

COLLECTIVE PITCHANGLE

(

)

DISCINCIDENCE

)

RATIO INFLOW FLAP- BACKY ANGLE (

)

ANGLE ao CONING FLAPY bi LATERAL FLAP 0.3 ANGLE

-

μ TIP SPEEDRATIO

0.2

; 8 %

ft /

=

;

.

;

;

ft

BRISTOL SYCAMORE ROTOR ANGLES AND INCIDENCE = 0-011 5,100lb Alt = 2,000 ICAN VT 638 sec T = 1.04 in hovering

197 A.U.W.

=

3.19

.

.

FIG D100

W

0.2

COLLECTIVE

(

-

=

RADIANS

X = = ·02

X₁ 0.2 x₂O 97 x2 1.0

)

(

INFLOW RATIO

( BRISTOL SYCAMORE CONTROL

EFFECT OF TIP LOSS ASSUMPTION ON ANGLES TO TRIM -

3.20

.

FIG

128

)

(

BACK a₁ FLAP-BACK ANGLE

EXPERIMENT RA.E. T.N. No AERO 2378 12FT DIA ROTOR

6

°

-12

°

-8

EXPERIMENTAL ERROR USE OF x2 -0.97 INSTEAD OF EXACT VALUE AND ERROR DUE TO THEORETICAL IDEALIZATION OF INDUCED FLOW FIELD

,

THEORY INCLUDING REVERSED FLOW EFFECT

, °

-4

,

0.2

TIP

-

SPEED

-

μ

RATIO

THRUST IN FORWARD FLIGHT COMPARISON BETWEEN THEORY AND EXPERIMENT

3.21A

i

momentum value

°

12

%

°

,3 · .0

12 = JR

=

μ

=8°

THEORY

BACK ANGLE

R.A.E.

TN

,a

ROTOR

No

.

AERO

-DEGREES

2378

10

F

&

0.2

-

TIP SPEED

-

μ

IN

0.4

i

x

-

a₁

VARIATION OF FLAP BACK ANGLE WITH COLLECTIVE PITCH SPEED RATIO COMPARISON BETWEEN THEORY AND EXPERIMENT Disc incidence momentum value vi = 1.08 .

TIP

= 0;

AND

-

.

FIG 3.21B

0.3 RATIO

°

= 4

FLAP

-

10

OF UNIFORM IS PROBABLY DUE TO THE ASSUMPTION DISCREPANCY DOWNWASH IN THE THEORETICAL CALCULATIONS THIS IS INDICATED BY THE REDUCTION THE DISCREPANCY ABOVE AND THE FACT THAT IT DECREASES RAPIDLY WITH INCREASING DISC INCIDENCE

1.08

0;

Disc incidence

x

.

.

FIG

Helicopter Dynamics and Aerodynamics

130

3.4 .

-

REVERSED - FLOW REGION

the vicinity of y 270 ° there is a region where the forward speed of the helicopter exceeds the rotational velocity of the in - board blade elements , so that the relative airflow is from the blade trailing edge to the leading edge , resulting in negative lift . This effect is not shown by the normal analysis

In

for low tip - speed ratios , because lift is a function of V² and ( —V ) ² = + V² . The boundaries of the reversed - flow region can be obtained from the equation for in - plane velocity

Ax

-

UT

Equating to zero ,

Vτ (x + μ sin ч )

x=

-μ sin y

----

(3.62 )

This is the equation of a circle , centre x = tu at

= μ.

y = 270 ° and of diameter

One method of allowing for reversed flow when integrating elemental forces is to integrate between X1 x₁ and X2 x₂ for y = 0 to 2π , and to subtract integral from this the between x = x₁ and x = μ sin Y for Y = 0 to 2π. region This assumes that there is no force in the of reversed flow . Although this is satisfactory for in - plane forces , rotor - thrust corrections should be made by subtracting a further term which represents the integrated negative lift forces . For small angles , it is reasonable to assume that the lift - curve slope in unchanged . Integrating the elemental - thrust equation ( 3.48 ) between 0 and x = -μ sin y , we obtain the same expression as equation ( 3.49 ) , but the

-

S

0

for linear taper

c = Co ( 1

-μn

=4

n

and for linear twist

y+

n+1

n

n+

lo

(3.63 )

v )

sin³

•

3

t

-

13 3

)

-

y

y

мя

y

(

sin

(3.64 )

and treating powers

sin

a

-

a1

4

OR

0₂

)

)

sin²

1²

2

((!

24

[

- μ sin y

sin "+1 Y

( 3.49 ) ,

9x +

)

y

sin

y

13 + 2

sin³

( 3

[

−4

txn +17

3 +4 4

=

3

+

AC₂

Substituting for t, and k, in equation greater than µ³ as negligible , -4

xn

μ" +2 * sin " +2 Y -µ”+¹ sin ” +1 y + n+ 1 n+2

4√r

kn =

t *x)

t*x") dx = 4 sin"

- 1 dx

+

S

(2n -1

-

xn

sin²

-μ sin v

Co

2 ²

=4

Rtn

-μsiny c

of μ

]

=4

Rtn

20 λο

taper integrals are now

131

y ) b ,

v

4³

4sin

+

4³

3sin³

(

]

3.65

(

y

cos

cos y

)

+

sin

in

μ sin

)]y

-

y

sin²

04³

3

sin3 nin

4KΚ

μελ

2

Гаона

[

4

(

sin2

Y

*t

1123

μ 120

-

y

Fundamentals of Rotor Dynamics

y

to

(

dCT BELOW

CALCULATED

dx

-μsin

)

-

(

°

= 270

to 2,

only

20

understand physically

,

easy

to

is

The result

dx

CALCULATED VALUE

EFFECT OF REVERSE FLOW ON THRUST GRADING Y

.

.

3.22

CT

VALUE OF

MIRRORIMAGE

usin

d

CORRECT

x

OF

dC

FIG

very simple

= a 0 .

-

x₁

is

of μ

Neglecting powers greater than give seen equation for the thrust of the reversed flow region when

)

3.66

µ³α cos

2

y

—

sin

µ³

*

20

µ³

μέλο

2

AC

µ²

=

5

2-5

1

,

,

Multiplying out and retaining only the first harmonic terms

terms being retained

Limiting the equation

)

3.67

(

) λo ]

2µ²

is ,

that

+

₂

( †

)

−

-

0;

{

+

for

µ²k₁

x1

(

−

is

reverse flow correction

kg

-

VR

)

{

µ²Ð₁

-

The above

+

С₁

[ ( t3

mean

=

,

,

.

is .

-

no

.

0 · 5,

µ >

/x

20

of

in is

it

is

because the inflow angle due the form insufficiently accurate for means that and that reverse flow the flapping equations The equation for mean thrust terms appear modified however and becomes

when the blade as

.

in

0

x

at

is

is

,

that reverse flow effects are generally -

= 0-4

so

.

of

u

in

excess

.

-

speed ratios

neglected

3

x

in

developing considerable thrust and tilted forward rarely possible In this condition flight

positive disc incidence

,

is

-

at a

,

flight when the rotor tip

>

x₁

.

-

a

in

as

,

,

of

,

is

to

continues right hub see saw rotor When 0-1 the correction less and may be easily calculated by substituting for the lower integration neglecting powers of limit above the final result Reverse flow effects are only important the pure helicopter condition of

)

A.183

(

10-

),

is

in

of

if

y

,

is .

is

of

-

x =

.

,

=

2

V )

(

to

.

.

is

to

(

of

In the calculation thrust gradings from equation 3.48 there no necessity modify the equations for reverse flow the following procedure used Due V2 the grading curve will be the form shown Fig 3.22 by the continuous line This corrected by plotting the mirror image of the curve in board of shown by the broken line -u sin which the correct position the curve

Helicopter Dynamics and Aerodynamics

132 3.5 .

EFFECT OF AERODYNAMIC COMPRESSIBILITY ON FLAPPING

In all

the foregoing calculations , the implicit assumption is made that the lift - curve slope a is a constant , and this assumption has always been made in the past in works of which the writer has knowledge . It is evident , however , that since a =

— √ (1 -

aL.s.

M²)

the equations for elemental thrust and thrust moment should be multiplied by the term 1 √ ( 1 — M² ) so that a will be greater on the advancing blade

/

4.0

3.0A +

tn 2-0

x =0·2 x2-1-0

1.0

2

4 S 6 INTEGRAL SUFFIX

3 TAPER

7 n

FIG . 3.23 . VARIATION OF t, WITH N FOR ASSUMED

BLADE

and less on the retreating one . Although this is found to have little effect on the flapping amplitudes ( Figs . 3.28 and 3.29 ) it can considerably increase both thrust and coning angle at given collective pitch setting ( Figs . 3.26 and 3.27 ). Considering the elemental - thrust equation in Section 3.3 for vertical flight only , we have , for compressible flow ,

dCr dx

-√

/

4c Co

(1

- M²x² ) [

4 ( 1 — t * x)

√ (1

x² ( VR — ds ) — 201

·[ x²DR

− M₁₁²x² )

—

x³ÛT

χλο ] — xλ0

(3.68 )

for linear twist and taper , where MT is the tip Mach number . To integrate this , the integral of the form

-

xn 1

Sn

_V (

1 –

MT2r2 )

dx

(3.69)

133

Fundamentals of Rotor Dynamics has

It

to be solved .

can be shown that 1-0

S₁

dx

=

√ (1 -— M²x² )

1

π

М.T

2

sin - 1 MTX1

1

---)¹] 2 M₁₂ MT [(1 − M₁²x‚²

-

1

[S₁ + x₁ ( 1

2MT2

SA =

1

[28₂ + x₁²

3MT2

( 1

) ]

M¸²x²

1 )

-

Sn

-- M²x² -

1 )

n

(

+

¹]

M²

−

(

2 ) (

n

-

=

(1

x₁³

1

S5 $ 5= 4MT2 [ 383

Sn

- M²x² )

) * ]

S3

—

S2

FLOW

1.0 EQUATION

)

7M²

PITCH

ANGLER FOR COMPRESSIBLE FOR INCOMPRESSIBLE FLOW O

·

√1-0

02

1.0

ft2

/

3.0

lb

=

;

p

=

;

0 ·

3.24

X2

)

] 1 · 0,

,

= 0.2

3.70

(

S₂t

* ) λ0

—

S₂

— (

Fig

2

.

for

.

Table 3.1 given use

x₁

ÛÃ

* )

§5t

in

—

,

0.8

the equation for thrust becomes

in

in

)

S

(

— (

ØR

* )

—

S3

(

4 [

=

S₁t

Values of n are given equation 3.70 example

of

0.8 MT 05

1 · 0;

X2

S4

;

0.2

1

0;

dr

Substituting these expressions

C₁

0.6 NUMBER

EFFECT OF COMPRESSIBILITY ON COLLECTIVE PITCH IN HOVERING

= 0;

.

3.24

t

.

FIG

0.4 TIP MACH

is

COLLECTIVE

0.2

and an

Helicopter Dynamics and Aerodynamics

134

TABLE 3.1 COMPRESSIBILITY INTEGRALS (Sn )

Мт

0.0

0.4

0.6

0.7

0.8

0.85

0.9

0.95

S1

0.800 0.480 0.331 0.250 0.200

0.829 0.500 0.350 0.259 0.215

0.875 0.534 0.379 0.285 0.238

0.906 0.564 0.398 0.309 0.249

0.958 0.603 0.434 0.337 0.277

0.995 0.635 0.461 0.362 0.299

1.044 0.675 0.497 0.392 0.328

1-119 0.742 0.556 0.448 0.378

2.8 2.6-

2.42.2

2.01.8

(1+1.85M4 )

VI - M2)

161.4 1.2

M²+40

(I+ 0.2 FREE

FIG . 3.25 . VARIATION

- STREAM

OF

)

M².... 08

0.6 MACH

LIFT -CURVE

NUMBER SLOPE

1.0

M

WITH

MACH

NUMBER

For forward flight , this simple approach is not easy to use analytically , and an alternative approach must be used . We have seen from Section 3.3 , equation ( 3.51 ) that the elemental thrust - moment coefficient CMT is given by an equation of the form dx

= K₁ + K₂ sin y

CMT Смт

= MT /

dCMT whose

If a is the low- speed

lift - curve

by

It is

= (K₁

3 cos

+

y

( 3.71

)

Т PVT²R²αCo

slope , this equation

flight

dCMT dx

- K,

will

become , in forward

-— K,3 sin y ) -√ ( 1 — M²)

K₂2 cos y

shown in Fig . 3.25 that the Prandtl - Glauert factor may be replaced 1 + 1.85 M4

- M² 1

√ (1 —

)

Fundamentals of Rotor Dynamics

135

to a

first order of accuracy . For some aerofoils , this gives even better agreement with test results than the Prandtl - Glauert factor . Since

- VTV sin y

UT

= MTx

M

+ Mo sin y

where Mr and M 0 are the tip and flight Mach numbers respectively the equation for dC dCмT becomes

= K₁

dCMT

dx

+ K2 sin

y

—

K3 sin

.

Thus

,

y

+ 1.85K₁ ( Mx + M sin y )4

+

1.85K₂2( MT

+ Mo sin y ) sin y

1.85K 3(M

+ Mo sin

y)4 cos y

Expanding M4

M¹

= M¸¹Ã¹ +

4M2M²x²

sin² y + Mo¹ sin¹

y

+ 4M MT3x3 sin y + 2M2M2x2 sin² Y

+ 4M3MTx 0 sin³ y Selecting only the first - harmonic terms , we have after some algebraic manipulation

dCмT = [ K₁ {1 + dx

1 · 85 ( M¸²x²

+ 185K +

+

+

} Mo¹

, ( 2MoMr3x3 +

[ K , {1 +

− [ K3 {1

+

, Mr )

3M 0

185 ( Mr * r * +

1 · 85K₁ (4M

+ 3M¸²M¸²x² )}

,

4 $ M 0*

+

]

MMr )

%

0 T Móñ³ + 3M¸³Mä )]

1 · 85 ( M¸²x²

+

} Moª

+

}

sin

y

T ) }] cos { M¸²M¸²x²

y

(3.72 )

In the exact solution , the equations for K₁ , K, and K, are now substituted , multiplied out , and the resulting expression for dCMT integrated with respect to x . The equation thus derived for CMT is used to obtain the flapping equations , as before . For the purposes of this section only the most important terms will be retained , however . For low values of μ

dCMT dx

- K₁

[1

+

+

1 · 85M¹x¹ ]

T K₂[ 1 + 1.85M¹¹

- K3 [ 1 +

]

sin y

T 1 · 85M4x4 ] cos

y

(3.73 )

At μ = 0-1 , this has the effect of increasing all the CMT coefficients by about 30 per cent . The integrated equation for MT is obviously that of equation ( 3.51 ) , plus the same terms multiplied by 1.85M4 , with all the taper integral suffixes increased by 4 , viz-

Cr

136

Helicopter Dynamics and Aerodynamics

CMT = [ (†₁ + { µ²Ð½) & R

[ (t

+ 1.85M

+ [2μgθη + -

1.85M

-[

-

—

-

+ Įµ²k₂)

(k4

+ { µ²6

- -

2μίg

-

[ 2μt

)

-VR −

2µk ,

( kg

---

+

{ µ²k6 ) —

Aut2 ) 01

( 4

—

— tзλ0 ]

-—

-

1.85

Mμta

―

−

(ts8

+

y μέλο ] sin ψ -

{ µ²Ð¸ )b₁

µtλ0 ] sin y

)α , —

(ts — { µ²

μtzao — ( ts + } µ²t₂ ) b₁ + t₁λ¿K ] cos

tλ0 ]

y

+ t¸λ¿K ]

cos

y

(3.74 )

Substituting this expression in the equation for moments about the flapping pin , we have , in the simplest case of a centrally - hinged , freelyflapping rotor , ao

1.85M4

= (αo )Ls +

γ

·[ (ts + §µ²6 ) &R

+

— ( kg

{ µ²k¸ ) —

tλ0 ]

( 3.75 )

where the suffix LS denotes the coning angle in incompressible flow — 2kg — tąλo )

a1

= µ{(2tzVR

b₁

= {µtçao + tλ¿K + {( t

{( t₁ — { µ²t₂ ) 1·

+ { µ²t₂ ) +

+ +

T 85M¹

T ( 2t , VR — 1 · 85Mê

-

2k , —

tgλo )}

1 · 85Mù ( t¸ — { µ²to )} ( µt - α

1 · 85Mù (ts

+ t¸λ¿K ) }

(3.76 )

(3.77)

+ ‡ µ²tc )}

The collective pitch angle is given by OR R

-

Câ + (k3 + } µ²k₁ ) + { (t3

tąλo

+ { µ²t₁ ) +

+

T¹ {( k , 1 · 85M+

T (t , 1 · 85Mù

Finally , the inflow ratio equation is the по = λc 20

+

+

{ µ²k5 )

+

tgλo }

(3.78 )

½µ²t5 )}

same as before ,

+ λ; + µ ( i + a₁ )

(3.79 )

-

These equations should be used for control angles to trim calculations if the rotor has a high tip - speed as in the case of a ramjet helicopter , for example . The Mach - number terms can be easily extended to apply to stiff - hinged or high - offset flapping - hinge rotors , so that the analysis will not be repeated for these cases , nor for the equally simple feathering rotor of Section 3.1 . Examples of the use of these equations are given in Figs . 3.26-3.30 , for which the taper terms of Fig . 3.23 apply . The effects of compressibility on rotor theory has been investigated by Gessow and Crim , subsequent to the above analysis , using a numerical method presented by Gessow in NACA TN 3747. This analysis is valid for

section characteristics only , and does not permit an overall compressibility effects to be obtained , but within these limitations of analysis the of Gessow and Crim confirms the validity of the equations developed in this section .

the

assumed

picture

O-4

MT -0.95

сто

0.3

-

-0608

MT

-

MT

THRUST

COEFFICIENT

MT

04 INCOMPRESSIBLE

FLOW VARIATION

0.02

0.06

0.04 INFLOW

0.12

9T =

0

;

σ =

= 0;

R

;

0.05

μ

Collective Pitch 0.2 Xa = 1.0 ;

;

No reverse flow allowance

=8 ° 0 · ; 3 t

EFFECT OF COMPRESSIBILITY ON THRUST IN FORWARD FLIGHT 1

3.26

.

.

FIG

0.08 λo

RATIO

TIP MACH

-

NUMBER

-06 -04

MT

0.8

MT

0.95

MT

CONING

INCOMPRESSIBLE

FLOW VARIATION

0.04

RATIO

12

λo

9T

στ

;

;

;

137

2.0

0;

μπ

y

0.05

2

ANGLE

1.0 0.3

= 0;

EFFECT OF COMPRESSIBILITY ON CONING 8 °; t

.

.

FIG 3.27 Collective pitch

0.08

0.06 INFLOW

α

0:02

R

ANGLE

ao

MT

21

0.2

FLAP BACK ANGLE

,a 6 INCOMPRESSIBLE FLOW

-

S

2000 9986

VARIATION

0.1

-

FLAP BACK 0.05

ANGLE μ = 0-3

21 ** 0.2

;

σ =

= 0;

r

FLOW VALUE

CONING ANGLE

INCOMPRESSIBLE

ON

O- 14

0.12

;

COMPRESSIBILITY x2 = 1 · 0;

OF

8 °; t * = 0;

EFFECT R

3.28

.

.

FIG

Collective pitch

0.08 0.06 INFLOW RATIO 20

0.04

0.02

////

EQUATION

LATERAL

FLAP

ANGLE O

/b

0.2

NUMBER

1.0

MT

138

x₂

= 1 · 0 b₁

= 0;

r

03

;

μπ

= 0;

=

K

= 02

2 · 0 ;

x1 7

= 0;

8 °; t

EFFECT OF COMPRESSIBILITY ON LATERAL FLAPPING ANGLE

Collective Pitch

R

.

.

FIG 3.29

0.8

0.6

O.4

TIP MACH

139

Fundamentals of Rotor Dynamics

EQUATION

CTLS

,

I-

COEFFICIENT

M²

‫רדר‬

FLOW

VALUE

THRUST

INCOMPRESSIBLE

1.0 MT

= 0 · 3

;

THRUST *

x

r

=

= 0;

x1 = 0 · 2;

20 = 0;

10

μ

NUMBER

EFFECT OF COMPRESSIBILITY ON ROTOR 8 °;

.

3.30

R

.

FIG

Collective Pitch

0.6

0.4 TIP MACH

= 0;

02

,

as a

roughly

0.115t4AM2

which the Mach number of the advancing blade .

exceeds the Mach number for drag divergence

attack

α = DR

ds

the blade pitch angle less the

sin

î

2¿Kx cos )

ub sin 24

(

,

3.80

)

xb₁

+

-

]y

(

μ

x

[

UT

µa

ua cos 24

,

Up —

y +

xa₁ sin

[(

)

20 −-

μa₁

+

,

a

Since the blade elemental angle inflow angle we have for

is

ELEMENTAL ANGLE OF ATTACK of

.

3.6

Up UT

=

1pVT3 RabCo

+

90

° )

y

(

tip

AM = amount by =

where

ΔΡΟ

Po

AC

the profile power coefficient

is

,

average the incremental rise

in

.

It

is

in

Of particular interest this investigation the effect of compressibility on the rotor power requirement can be inferred that statistical

a

of

.

.

a

A

,

-

-

.

in to

is

83

of

is

,

to

of

independent coupling and the Note that the angle attack flapping relative flapping pin the shaft for zero offset being function only of flapping with respect typical angle feathering the no orbit Fig 3.31 for ramjet rotor example The plotted attack distribution

°,

.

y

=

of

a

.

a

is

large area of negative lift primarily function the fact that the blades example of this have no twist The maximum value of occurs rather owing to the combined effects of coning and inducedforward of 270 velocity gradient As the speed increases and the induced velocity gradient

Helicopter Dynamics and Aerodynamics

140

becomes less important , the region

y = 270 °. In Fig .

3.32 the example

5%

of maximum a tends towards the position

of attack distribution

is plotted

for another

.

3

FIG . 3.31 . ANGLE OF ATTACK VARIATION IN FLIGHT Tip -speed ratio μ = 0.15; CT00744 ; R 0-58; 1 6.75; 0 = 0 ; t = 0; a 1.2; 8 = 0; y = 6·4 ; x -7.6 0.616°; Induced velocity gradient K 1.97°; b₁ Cross-hatched circle is region of reversed flow . Zero flapping hinge stiffness and off-set

-

3°

2°

BLADE ROOT CIRCLE

40

NEGATIVELIFT

FIG . 3.32 . ANGLE OF ATTACK DISTRIBUTION IN FLIGHT με 0-19; total blade twist = 10°

ramjet rotor , which has twisted and tapered blades . In this case , the assumption has been made that the induced velocity gradient is zero , and , because of the small coning angle , the maximum a occurs at y = 270 ° , being some distance in - board from the

tip

because

of the blade twist .

141

Fundamentals of Rotor Dynamics

Since maximum a occurs at the blade tip in Fig . 3.31 and at 0-7R in owing to blade twist , it is evident that the method sometimes used correlating retreating blade stall with the tip angle of attack can give for very fallacious results . See , for example , Gessow and Myers , Chapter 10 .

Fig . 3.32 ,

3.7. APPROXIMATE RELATIONSHIPS FOR BLADE ANGLE OF ATTACK AND FLIGHT ENVELOPES

Whilst the calculation of retreating blade stall is essential to even the most simple project - design study , the method of Section 3.6 often involves too great an expenditure of design - office effort . The approximate method given in this section , while less accurate intrinsically , can be used to determine values of (CL ) max experimentally from flight tests , and the values thus derived can be used in the basic equations to predict the blade stalling for new designs with similar rotors . Typical values thus derived are (CL )max = 1.2 for thin (high speed ) rotor blades , and ( CL ) max = 1.5 for blade sections whose thickness is of the order t/c = 12 per cent . speeds

If the mean

blade - lift coefficient is designated CLB , the elemental thrust is

dT .= CLB PUTcR dx

= CLBPV2RCO •

c

Co

(x

c

= CLBPVT²RC

Co

sin y )2 dx

+

yu²

(x² + 1µ² + 2µx sin

cos 2 ) dx

Integrating as before , the total thrust for one blade is

T₂

т PVT²RCo (tз +

= CLв

which , in hovering flight ,

{ µ²t₁

+ 2μt , sin y

-

μ²t, cos 24 )

(3.81 )

becomes

То To = (CLB )hov PVT²RC 0

(3.82 )

tз

On the advancing blade , y = 90 ° , the ratio of CLB to the hovering value (CLB )hov becomes , if T is assumed to be constant , CLB

(CLB )hov

=

t3

For the retreating blade , an allowance must

-

(t33 - 2μt2 CLBPVTRC RCo { (

c Co

(x2

-

·µx + µ²) dx}

* x )

-t 1

(

Co

dx 3.84 )

* x )

µ²

]

(

—

µ²

+

µ³x

x²

-

*

µt

+

-

−

με

§µx¸²

ƒx¸³ )

** )) ( (

µμι

+

1

x141

—

—

x³

*

— t

x²

fr(

8

µx

*

+

= μ

4

12t *

+

[

= μx + µ²) dx € • µ •

= 8

(x².

—

-

Co

$ ¥

S

с

8µ³

8

μ

μ

taper C

For linear

flow

be made for reversed

8

+ μ²t₁ )

(

=

+

То To

(3.83 )

(tз + 2μt₂ + μ²t₁ )

Helicopter Dynamics and Aerodynamics

142

*

)

]

3.85

(

µ²x₁ +122 +

)

4 *t

*1

_

•

*

10 µ

— }

µt

+

20 2º

+

x¸³

(

)

3.86

µ²x₁

]

+

1µx₁²

—

µ³

µ²t₁

20

,

negligible therefore —

tз

t3 /[

2µt₂

µ¤¸²

-

)

(

=

hov

+

CLB

CL

µ¹x₁4 and x₁³ are —

0 · 5,




(1

cos 20

(sin 20+ μ₁

+ μ₁ sin cos 20 —

20 )

µ₁ )

( 9.7 )

A practical application of this principle is shown in Fig . 9.7 . In operation the circular brake lining is held on to the inner

tube by a compression rubber spring when the leg is closing and the kinetic energy of the aircraft is absorbed in friction . There is no resistance to the return stroke , which is achieved by a light return spring when next the machine is airborne . The normal working stroke ( for a 6 - ft /sec landing velocity ) is 5 in . In the event of the aircraft landing at 12 ft / sec , the spring plate is pushed out from the top of the leg , enabling a total stroke of 20 in . to be achieved . The spring plate is self - re - setting during the next take - off or by manual extension of the leg .

Other advantages are— (a ) The undercarriage is made up from two standard steel tubes ; there is no need for fine tolerances associated with pressure seals . (b ) The construction is both simple and robust . Should the need arise a leg could be dismantled and reassembled in the field with the simplest of tools .

Ground Resonance and Vibration Due

to

Rotor Resonance

329

(c) Irrespective of the descent velocity , the undercarriage reaction has always the same value thus giving considerable savings in structure weight . (d ) The leg needs no routine servicing but is provided with replaceable bushes to cope with normal wear at the attachment points . (e ) In the leg depicted in Fig . 9.7 a mechanism is provided for folding the leg when the helicopter is standing on a lorry and minimum overall height is COVER

RETAINING DISC

CONERING LIGHT RETURN SPRING

BETHING RETAINING ROD

END PLUG

DISCHOUSING RETURN SPRING

OUTERTUBE

(FIXEDTO AIRFRAME ) EJECTIONTUBE BEARING RETAINING PLUG BEARING

BEARING FRICTION LINING DETAIL OF SPLIT CONE SPLIT CONE RELEASEHANDLE COMPRESSION RING LIGHT_RETURN SPRING

SPLITRINGCONERING FRICTION LINING SPLIT CONE

SPRING

ENDPLUG BUFFER RETAINING CLIP

RELEASE HANDLE DETAIL OF RELEASE

HANDLE

INNERTUBE (FIXEDTO SKID)

FIG . 9.7 . SIELEY UNDERCARRIAGE LEG (Patent rights held by Auster Aircraft Ltd. )

required in order to pass under road bridges . The release handle is pulled out and rotated through 60 ° , releasing the split cone from contact with the cone ring so that there is no resistance to sliding in either direction . A second common form of undercarriage is the oleo and wheel combination , of which the simplest example (analytically configuration ( Fig . 9.8 ) .

)

is the four -wheeled

The vertical stiffness K , is the combined stiffness of the tyres and oleos , and the lateral stiffness K is almost entirely the lateral stiffness of the tyres . The dynamic vertical spring constant of a tyre is about one or two times the

spring constant determined by static deflexion tests , and the lateral stiffness is generally between 30 per cent and 50 per cent of the vertical value . The longitudinal stiffness is about the same as the vertical , as would be expected ,

Helicopter Dynamics and Aerodynamics

330

but this only applies when the brakes are operative or the wheels located by chocks . With the wheels free to run over the ground the longitudinal stiffness is much less , tending to zero for large - amplitude , low - frequency oscillations .

Tyre - damping coefficients range from 4 per cent to 7 per cent of the critical damping coefficient , but on complete helicopters the effect of the tyre scrubbing " over the ground during an oscillation can raise the effective

"

A

value to as high as 10 per cent of the critical value . Maximuin damping is achieved with channel - tread tyres as used on the tail and nose wheels of fixedwing aircraft to prevent " shimmy . "

FIG . 9.8 . CONVENTIONAL

CG

KL

EQUIVALENT

FOUR - POINT

KL

DYNAMIC

OLEO AND WHEEL

SYSTEM

UNDERCARRIAGE

There is no doubt that the use of pneumatic tyres on a helicopter undercarriage considerably hampers the achievement of a satisfactory solution , unless the configuration is designed to give a natural frequency well below rotor speed . Even in this case they can be detrimental , since the lateral (horizontal ) motion of the fuselage against the lateral tyre stiffness can be of the same order as the rotor speed and there is little chance of introducing extra damping to supplement the small amount natural to the tyre . The spring rate of an oleo leg is variable , since the " spring " is a column of

air trapped in the upper half of the

leg by the piston . For small displacements equivalent spring the linear constant is

K=

1.4W ,

D₂v

-

Ds

(9.8 )

= the weight on the strut Dv = total virtual displacement of the

where Ws

piston that is possible piston plus = total stroke of the the displacement which would be necessary to eliminate the residual air pocket that remains in

D,

the fully compressed condition displacement of the piston from the fully extended position .

= static

In general this relationship is not accurate enough for the amplitudes of oscillation met with in ground resonance , and the rate also varies with the frequency of oscillation . Moreover the axial load in the strut must reach a

Ground Resonance and Vibration Due

Rotor Resonance

to

331

certain value before any movement takes place , because of the friction of the gland and the pre - load on the piston due to the air pressure in the fullyextended position . Thus when the strut is partly compressed under a constant mean load , no periodic change of its length will take place unless the periodic force applied exceeds the static friction and air pre - load force . The strut has

"infinite " stiffness

,

therefore , for small force fluctuations

.

For larger forces the strut will telescope periodically , and the rate will be that of equation (9.8 ) . As the magnitude of the oscillation increases to large values the effective spring rate will drop , with an appropriate reduction in the fuselage natural

frequency

.

When a tyre

is combined with the oleo it is obvious that the picture is further complicated by the relative deflexion of the tyre and the oleo , which will depend on whether the rotor is supporting any portion of the helicopter weight , and if so , what percentage it supports . A more detailed treatment of this is given by Jones and Howarth in ref. 5.6 .

9.3 .

" GROUND RESONANCE

The simplest example of this phenomenon of a long flexible "mast " ( Fig . 9.9 ) .

IN FLIGHT

”

is a rotor mounted on the end

of

HUB MASS FLEXIBLE BEAM

LARGE FUSELAGE MASS

FIG .

It is obvious

9.9

if the

fundamental natural frequency of this system in will result in flight or on the ground , and since the hysteresis damping in the shaft must necessarily be small its destruction is unavoidable . An analogous case is when the entire rotor shaft engine system is isolated from the fuselage , as in the Bell helicopters , and it is this which limits the Bell configuration to

the bending

relatively

that

,

mode is near the rotor frequency , dynamic instability

small rigid rotors without drag hinges

, where

the fundamental

natural frequency of the blade in the drag plane is well above the maximum rotor speed . Another example is provided by the large tandem helicopter . Apart from the obvious example of a flexible rear rotor pylon , which is analogous to the flexible mast just described , the fundamental

torsion or lateral bending

Helicopter Dynamics and Aerodynamics

332

frequencies of the fuselage as a heavy beam may be close to the rotor speed , with similarly destructive results ( Fig . 9.10 ) . In passing it should be noted that conventional resonance can also cause trouble with large helicopters , particularly if any of the natural frequencies are near to be , where b is the number of blades per rotor and w the rotor speed . It is believed that some of the early Piasecki helicopters experienced considerable vibration because the first torsional frequency coincided with the

three times rotor speed . It is interesting to speculate that the fitting of tandem stub wings ( as on the Bristol 173 ) would have considerably reduced this vibration , not for any aerodynamic reason but because the wings would have lowered the torsional natural frequency .

FIG . 9.10 . FUNDAMENTAL FLEXURE OF A TANDEM HELICOPTER FUSELAGE

On a medium - sized helicopter such as the Bristol 173 the lowest values at which resonance occurs are of the following orders— Fundamental Mode Lateral flexure Vertical flexure Fundamental torsion

Second Mode

9.0 c.p.s.

19.0 c.p.s.

8.0 c.p.s.

16.0 c.p.s.

14.0 c.p.s.

These frequencies can usually be calculated on simple assumptions at an early stage in the design of a new project , and the answers are often surprisingly close to the values measured on the prototype for the assumed payload distributions . Such a check should always be made at an early stage , since a fundamental defect of this nature is extremely difficult to cure if it is not discovered until the prototype stage is reached . Some examples of fuselage natural frequency calculations are quoted by Hafner in ref . 5.151 .

9.4 .

GENERAL REMARKS ON THE DETERMINATION OF FUSELAGE NATURAL FREQUENCIES

It is not within the province of a book on rotor dynamics to deal in detail with the calculation of fuselage natural frequencies , which is strictly structural vibration and would require a separate volume to do the subject justice . As introductions to the subject the reader is referred to Wagner's Vibrations Handbook for Helicopters , Howarth and Jones ( ref . 5.6 ) , and Horvay ( ref . 5.20 ) . An excellent general introduction is Manley's Fundamentals of Vibration Study , published by Chapman and Hall , Ltd.

Ground Resonance and Vibration Due

to Rotor

Resonance

333

Reference must be made , however , to a method of defining and measuring the dynamic characteristics of the fuselage for ground resonance calculations . In Coleman's theory the helicopter fuselage is represented by an equivalent hub mass constrained by springs and suitably damped to exhibit the same dynamic characteristics as the fuselage when the motion is measured at the

If

MH is the equivalent hub mass , K the equivalent spring rate , and the natural frequency , we can define the effective hub mass by the relationship hub .

Q,

K

'T

2₁ =

√ (1H)

(9.9)

Since K can be estimated for simple cases , together with the natural frequency N ,, MH = KQ (9.10 ) ΚΩ,,2

It is always desirable , and in most cases essential to measure the fuselage frequencies before ground - running the complete helicopter . This is conveniently done by attaching a mass to the fuselage hub which is of the same order as the total blade weight ( defined as AM) and applying an excitation by whirling an unbalanced mass at the hub . The most simple means of doing this is to mount an out - of - balance weight on the hub which is rotated by the engine , but this has the disadvantage of restricting the tests within the r.p.m. range of the engine . The phase of the unbalance weight and the fuselage oscillation at the hub are both measured , a natural frequency occurring when these are 90 ° out of phase with each other . The hub mass Mн

is found by varying the added mass AM within limits small enough to prevent a change of fuselage mode , and plotting AM against 1 2,2 . Since

/

Q, = 1

2,2 $ 1

When Thus

Ω,

MH

√ (K/ Mµ

+

AM)

H + AM = Mu

-

K

0,

MH

= -AM

is numerically equal to the value of

Q, is measured for

AM

1

at which

2,2 £

0.

If

a series of AM values , this point is easily found by extrathe plot is linear ( Fig . 9.11 ) . The value of the hub damping constant can be obtained by measuring the decay curves of free oscillations if the value is fairly low ; otherwise it must be obtained by measuring the phase shift when the excitation is moved slightly off resonance and the ratio of damping to critical damping obtained from the usual theoretical curves . *

polation

, since

* See Manley , Fundamentals of Vibration Study , p . 34 (London , Chapman and

Hall ) .

334

Helicopter Dynamics and Aerodynamics

AM

-MH

FIG . 9.11 . EXPERIMENTAL

9.5 .

DETERMINATION

OF

EFFECTIVE

HUB

MASS

ROTOR BLADE OSCILLATION

Having considered the fuselage system without reference to the rotor we consider the case of a rotor whose shaft is earthed .

now

Unless the blade is very flexible in the drag plane it is generally sufficient

to represent the rotor blade by two equivalent point

masses connected

by a

HUB

m2

FIG . 9.12 . EQUIVALENT DYNAMIC SYSTEM REPRESENTING A HINGED , RIGID ROTOR BLADE

rigid rod of zero mass . One of these point masses is at the drag hinge and the value of this must be taken into account when working out the fuselage natural frequency , as in the preceding section , since it is definitely a " hub " mass . It should also be simulated when the natural frequencies of an actual fuselage are being measured .

Fig . 9.12 gives a plan view of the equivalent dynamic system for a drag . articulated rotor blade , with drag hinge offset 1. The dimensions of the system are found by equating them to the dynamic characteristics of the actual blade about the drag hingeFor equal

m₁ + m² = Mr

mass

For equal first moment of mass For equal

second moment

of

mass

= MMD 2 mar * = ID mar*

I

Ground Resonance and Vibration Due to Rotor Resonance

From the last two relationships the

r₁ Also

335

radius

effective

ID / MMD = MMD² mD /ID

=

m2

m1 = Mr

( 9.11 )

m2

When the mass system is rotating the centrifugal force on the mass m tends to restore it to the position of zero lag . When the lag angle is finite , as in Fig . 9.13 the restoring force acting on m₂ is the C.F. component CF sin ( ¿ -— 7) . Thus the restoring torque about the drag hinge ( CF ) r sin ( 7) = maw²rr

sin

(

— 7) .

TF

M2

(CF) FIG . 9.13

-

Considering the triangle centre - line , - m , m2 , the exterior angle must be equal to two interior and opposite ones , by a well - known theorem of

Euclid-

=T+

By the

(say ) , so that

=

-

.

T

sine rule , and since the angles are small ,

ф

ι

But

ф

Therefore

=

T

=r

T

T

T

5

T

1+

Fr*

Also so that the

C.F. torque

* m₂w²r²

is 1 *

+1 =

-I

5

= mow²r

/

+ 1 r*.

15

(9.12 )

Thus the effect of C.F. is to give an equivalent torsional stiffness in the drag hinge , of magnitude dQ dě

= moo²rl = Mm Dw²l

=

( CF )

( 9.13 )

336

Helicopter Dynamics and Aerodynamics

All three methods of presenting the solution are of interest but in the

If

the hub ( and drag hinge to the first . to be earthed the system possesses a single degree of freedom . The mass m₂ is free to swing about the drag hinge , constrained by the C.F. stiffness , and the characteristics of the system are those of a torsional system , as depicted in Fig . 9.14 . The inertia of the flywheel is obviously the second moment of mass r²mą present analysis we confine ourselves

offset ) are assumed

and the equivalent shaft stiffness m₂w²rl . The natural frequency of a simple undamped torsion system of this type is

1-0

Ω

FIG . 9.14

/

/ -2

Ω

- =

= /mor +w²l 2

=

=

mar*

IMmD ID

= ID

since

M

D

( 9.14 )

Since is small in relation to on conventional designs , the natural frequency is low compared with the rotor speed , and the motion of the blade under the impulse of fluctuating external forces is substantially the motion that would occur if the drag hinge was at the centre of rotation . 9.6 .

TWO - BLADED ROTOR RESONANCE

As an introduction to the torsional vibration of a complete rotor and hub we consider the case of a simple drag - articulated rotor without inter - blade snubbers (Fig . 9.15 ) . M2

T* m2

INERTIA OF HUB , SHAFT AND APPROPRIATE TRANSMISSION SYSTEM INERTIAS

(a )

dynamic system for two -bladed rotor

Equivalent

J₁ C m2w2r1

0-0 m₂w²rl

2m,1²+IH

m₂r2

(b ) Equivalent torsional system FIG . 9.15

In

J

the analysis it is convenient to use the and C symbols and assume are , from left to right , 01 , 0 , and 03 .

that the angular deflexions

Ground Resonance and Vibration Due to Rotor Resonance

337

Equating the inertia and stiffness torques ,

-C

C(01-02 )

J₁

— 0₂ ) =

dt2

J₂ d202 dt2

=

( 02 — 03 )

d201

C (02-03 ) = √3

( 9.15 )

d203

dt²

--- C ( 01

,

We know that the motion of the flywheels will be simple harmonic motion so that the usual substitutions may be used ,

0

01 = A sin Qt 02 =

B sin Qt

03 = D sin Qt

C

)

(

)

(

(

of

)

(

0

A

and

) )

(

9.21

J3N² D in

C

—

BC

the equation for equili

J22 0

C =

)

9.22

(

] •

+

J }

)

G₂

1

)

J½

)

-

2C

B (

2J1

√ [

-

-

Ω

=

(

J1J 2N²

—

C

Q2

+

-J ,

202

B

(

),

9.17

2

(

D.

=

,

A

=

J₁

the

9.20

J₁N²

Substituting for

brium of the hub flywheel

or

9.19

)

9.18

D= J3

=

)

+

BC

-

A=

since

Q²

(

)

J3D

we have the relationship for the amplitude

first flywheel

and from equation

in

C ( +

( 9.16

(

From equation

J₂B

the shafts

zero-

also

2

,

J₁A

is

Adding these equations we see that the total work done zero and the total inertia work

9.18

)

-J3DQ²

BD

9.17

is

—J₂BQ²

9.16

—

=

—

B )

B

) + C (

C (

= —J₁AN² ΑΩ

D )

A

(

AB

—

—C

) =

The equations of motion then become

Helicopter Dynamics and Aerodynamics

338

This equation section in that

with that for the single blade in the preceding

checks

if J₂

∞, Ω

This is only

one

of the two possible

natural frequencies however , being the one in which the hub is swinging in , is evidently of anti - phase to the blades . The value of the hub inertia great importance , since as

J

J₂2 → 0 , N → ∞ . J2

FIG . 9.16

The relative amplitudes in the hub mode can be obtained from equation ( 9.19 ) . Since

J₁A

(

J¸D )

2 + J½B +

=

B = A

2

(

2J₂A + J½B )

= 0

J₁

J₂

( 9.23 )

of Fig .

so that the swinging form diagram is of the form

9.16 .

The other mode of oscillation is obviously that of the single blade , the node being at the hub . The swing form diagram is of the form shown in Fig .

9.17 . ‫لو‬

J2

FIG . 9.17 Since there is only one node this is the fundamental mode of oscillation the natural frequency being

W

The second mode is equation

C

+ 1

/

2m₂rl

)

9.25

( H

I

m₁l m₂r

is

to

.

is

possible

9.24

well below the rotor speed no trouble with The first overtone can give rise serious transmission

Since the fundamental mode resonance

G²₂

√

+

+

= -JG-

(

ω

as before .

( 9.22 )

=

22₂

Ω

()

)

21 =

(

√( )

*

=

/

Ω1

,

Ground Resonance and Vibration Due to Rotor Resonance and hub vibration however

if Ωρ

339

approaches 1.0 ( due to the very high first-

harmonic force fluctuation on the blade ) and to a lesser extent ,

equals if 222 Ο

2.0 , 3.0 and so on , the severity generally decreasing with increasing harmonic order .

9.7 .

BLADE SNUBBERS

To restrain blades oscillating with respect to each other , but not with respect to the hub , inter - blade " snubbers " are sometimes fitted . In the most usual case these can be represented as an inter - blade spring , as in Fig . 9.18 .

(a)

COMBINED

SNUBBER STIFFNESS

C2

C₁

(CF)

STIFFNESS

(b) FIG . 9.18 (a ) Two - bladed rotor with inter -blade snubbers (b) Equivalent torsional system

The equivalent torsional system is still susceptible to analysis , although such a system is not mechanically posible as drawn , without the use of gears . As would be expected , the snubber stiffness does not affect the second natural frequency , since the blades are swinging together against the hub in this mode . The fundamental frequency is increased by its snubber stiffness and is given by the equation

Ω, =

C₁

+202 )

Ji

(9.26 )

where C₂2 is the total snubber stiffness . We may summarise this system in the swinging form diagrams of Fig . 9.19 .

23-( A.183 )

Helicopter Dynamics and Aerodynamics

340

In practice , snubbers are usually end - loaded rubber blocks , so that their rate varies with loading , tending to become infinite at very high compression loads . This is advantageous from the point of view of ground resonance , since it limits the maximum blade amplitude of oscillation and therefore the degree of energy which can be transferred to the fuselage motion . Unfortunately the amplitude already achieved when the snubber bottoms is often sufficient to break it in a few cycles , during the tension cycle , and the writer has seen several

ground resonance

MODE

accidents in which the phenomenon was divided

WITHOUT SNUBBERS SWINGING FORM FREQUENCY

WITH SNUBBERS SWINGING FORM

FREQUENCY

J2

FUNDAMENTAL

FIRST OVERTONE

FIG . 9.19

stages : a relatively mild oscillation of the aircraft , which rapidly changed , on the failure of a snubber , into a violent and destructive oscillation .

into two

There is , in the writer's opinion , no good reason at all why rigid links should not be substituted for snubbers . The use of such links would effectively prevent ground resonance when the blades are sufficiently stiff in the drag plane .

9.8 .

MULTI - BLADED ROTORS

For one- and two - bladed rotors the value of the torsional analogy has probably been obscure to the reader , since the same results could have been obtained by analysing the basic equivalent dynamic system . For three or more blades such an analysis becomes progressively more difficult , however , particularly if elastic snubbers are used . A three - bladed rotor with snubbers has six modes of oscillation ( since it has six stiffnesses ) , so that the frequency equation is a polynomial in ( 2) , which cannot be solved explicitly . A torsional system can easily be analysed for any number of degrees of freedom , however , by the well known effective inertia method , or the " dynamic stiffness " method (sometimes called " mechanical impedance " ) both of which are described by Manley . The analysis of the more complicated systems representing rotors is given by Bishop in ref. 5.5 . The result of such an analysis for a three - bladed rotor with snubbers is

given in Fig . 9.204 , the complete equivalent dynamical system being that of

Fig .

9.20B .

* Manley , Fundamentals of Vibration Study .

Ground Resonance and Vibration Due

to

Rotor Resonance

341

Since the swinging form diagrams can be easily calculated when the natural frequencies have been found , it is possible to determine which inertias are oscillating and which shafts are carrying most of the torque in the system , thus enabling a simplified equivalent dynamical system to be defined

for

each mode .

Both the second and third modes in Fig . 9.20в are unsymmetrical , so that the mean c.g. of the rotor is displaced from the shaft centre - line and both these modes can cause ground resonance . Another feature of these modes is that they can be excited by the once - rotor air force variation on the blades if

J

BLADE ,

C.F

STIFFNESS HUBJa SNUBBERC₂

FIG . 9.20A . THREE -BLADED ROTOR WITH SNUBBERSEQUIVALENT DYNAMIC SYSTEM

/

In

stiffness mass ratio is such that resonance can occur . example , if the approximate equation gives

N3 = ω

1

(C₁ + 3C2)

J₁

C₁ + 3C2

i.e.

w2

the third mode , for

= 1.0

- Ji

(9.27 )

then the hub will experience a severe once - rotor vibration relative to the rotating axes in forward flight , which will result in a twice - rotor fuselage vibration , despite the fact that the rotor is three - bladed . The fourth and fifth modes are also unsymmetrical and capable of exciting both ground resonance and forward flight vibration if the mass stiffness ratio is wrong . It is evident that unless the drag hinges are critically damped it is advisable to carry out a detailed calculation of the rotor natural frequencies for any new design to ensure that none of them are in resonance with a harmonic of the air forces on the blades .

/

It is also apparent that when a vibration is measured in the fuselage of a harmonic order which is not a multiple of the number of blades , then " rotor resonance " should be suspected . An instance of this occurred recently when a strong fifth - harmonic fuselage vibration was detected on a two - blade British helicopter

.

In

this case the rotor had no drag hinges

, so

that the blades were

/

Ω

(

(

Q

Equivalent torsional system

Mode of deformation

0.343

C₁ 2C₁

30₁ ) ))

2C₂

27

|

+

( 3 ( 1

C₁

2J1

+

J₁ J2

Ji

C₂

)

C₁

+

2C 2C2

( 3 (

3-03 ,

3C

6

*

106-0

2JI , +

141-0 71-0

(

0.463

(

J₂

.

.

Note Value of has great effect on this mode

9.20B

THREE BLADED -

.

FIG

.

Ω

WITH SNUBBERS

-

ANALYSIS

; C,

;

;

'

;

J₁

,

;

;

*

ROTOR

circular frequency relative to rotating axes circular frequency relative to hub inertia space axes blade inertia about drag hinge Ja rotor speed C₁ C.F. stiffness snubber stiffness plus equivalent True hub inertia should considered i.e. hub transmission and engine inertia be

5

51-2 18.8

+

C₁

√ (

3C2

123.0

16.2

201

2C

+

12.0

*

,

C,

47.0

(

0.3

(

45.5 124.5

3

0.252

10.5

2J

2C2

143.8 126.2

4

2

,

2J 8.80

Swinging form diagram

0.057

'

2.0

Approx equation for Ω

.

Frequency rad sec relative to spaceaxes Ω

/ )

relative to rotating axes

Frequency rad sec relative to rotating axes

Mode No. 1

)

(

Ground Resonance and Vibration Due to Rotor Resonance

343

evidently flexing near the sixth harmonic relative to the rotating axes , giving a fifth- and seventh - harmonic fuselage vibration . Such resonant flexing is obviously dangerous from the point of view of fatigue .

9.9 .

DRAG HINGE DAMPERS

of the preceding section assume that the equivalent damping torsional is small compared with the critical damping coefficient . Damping of the blade on the drag hinge is due to three effectsThe calculations

Air Damping The blade torque about the drag hinge , due to air forces is approximately equal to where

K

QD

= Κω

= a constant

w = rotational speed

of the rotor .

When the blade is oscillating about the drag hinge with an instantaneous velocity of Aw , the effective rotational speed is ( w + Aw ) and the torque becomes

QD

= K(w

+ Aw )² =

K ( w²

+ 2wA∞ + Aw² )

(9.28 )

Since Aw² is negligibly small the increment of damping coefficient air forces is 2QD

QD dQD ABB = dAw = 2Κω = 20

(9.29 )

ω

where Qp is the steady torque of one blade , approximately Q torque about the shaft for small drag hinge offsets .

due to

equal to the

b

The critical drag hinge damping coefficient theory ,

where

Ip =

MmD 1

(Ba)erit second moment

= first

= drag

is , from simple vibration

= 200√ (IDMmD¹

of blade

)

(9.30 )

mass about the drag hinge

moment of blade mass about the drag hinge hinge offset

.

In practice the air damping increment is usually found to be between per cent and 8 per cent of the critical damping coefficient . 4 Air damping can be improved by inclining the drag hinge so that the blade

flaps when moving on the hinge . This was first suggested by Bennett (ref. 2.19 ) who reported comparative tests between damping by flapping (for hinge inclinations of 50 ° to 65 °) and friction damping , to show the effectiveness and smoother action of damping by flapping , which is equivalent to viscous damping . the drag hinge is so inclined that a reduction in shaft torque

If

344

Helicopter Dynamics and Aerodynamics

reduces the blade pitch angle by virtue

of the forward movement of the blade on the hinge , the inclination can serve the additional purpose of putting the blades automatically in to the autorotative pitch setting in the event of a power failure . Since only 2 to 4 seconds is available for this reduction after a power failure , before the blades stall , an automatic means of pitch reduction is obviously preferable , except when the power fails whilst the helicopter is hovering near the ground ; it can then be dangerous , since the helicopter can strike the ground before the pilot has fully realised what is happening .

Friction Damping

due to Bearing

Friction

This is not easily calculated , particularly as the equivalent viscous damping coefficient of friction damping varies inversely as the frequency and amplitude of oscillation . Unless values are measured experimentally it is usual to take an empirical figure of between 10 per cent and 15 per cent of the critical damping coefficient for typical installations . Damping by External Damping Mechanism Blade dampers can be either " frictional " where the damping force is to be constant , or made up of level " steps " which come into action with increasing amplitude , as in a step damper ; " viscous , " where the damping force varies with the angular velocity on the hinge ; or " hydraulic , ” when it varies approximately as the square of the velocity . In order to use existing theory the damping coefficient must always be expressed as the " equivalent viscous " value B. For a plain friction damper B varies inversely as the amplitude of oscillation , so that once ground resoassumed

it will rapidly "run away . " The step damper avoids this and some authorities regard it as mechanically superior to the

nance has started

difficulty , theoretically ideal viscous damper . The orifice or " hydraulic " damper has either insufficient damping for small amplitudes or introduces high in - plane blade bending loads during large amplitude oscillations . It is often used with viscous dampers , however , to limit to maximum amplitude , for which it is preferable to a mechanical stop . Symmetrical oscillation of the blades with respect to the hub ( Mode 6 in

Fig .

by changes in engine torque , higher pitch changes harmonic air and collective . To avoid trouble with these modes the total drag hinge damping should be of the order of 50 per cent to 60 per cent of the critical damping coefficient . Since air and friction damping may account for 20 per cent , this implies mechanical dampers which supply between 30 per cent and 40 per cent of the critical damping . An interesting damper was used on the pre - war C.40 . This was mounted centrally on the hub and linked to the blades in such a way that although they could move symmetrically with respect to the hub they could not move with 9.20B , for instance ) can be excited

force ,

respect to each other , so that ground resonance was impossible so long as the fundamental in - plane natural frequency of the blade as an elastic beam was sufficiently above the once - rotor frequency . This damper , the principle of which is illustrated in Fig . 9.21 , still allows the drag hinges to fulfil their

Ground Resonance and Vibration Due

to

Rotor Resonance

345

primary function of relieving the large steady component of the root bending moment due to rotor torque . This is a mechanical simplification of the system

DAMPER

FIG . 9.21

already suggested

as a " first aid " measure for existing designs , where the blades are linked with rigid inextensible " snubbers , " and was originally due to Bennett (ref. 2.19 ) who used it on the C.40 autogiro .

COUPLING BETWEEN FUSELAGE AND ROTOR OSCILLATIONS ( THREE OR MORE BLADES )

9.10 .

When the rotor and

oscillations are close enough for coupling to of rotation of the rotor can increase the energy

fuselage

exist , so that kinetic energy

content of the fuselage oscillation , the possibility of " ground resonance " exists . The condition for increasing fuselage amplitude of oscillation is that the damping of the fuselage motion is insufficient to dissipate the energy fed from the rotor . Coleman's classic analysis of ground resonance (ref. 5.12 ) is still the standard theoretical work on the subject , and an interesting re - statement of his analysis is given by Howarth and Jones in ref. 5.6 . It is not intended to repeat the analysis in this book , since it is readily accessible in either of these two references , and the standard of the mathematics employed is rather above the self - imposed limitation of this book . In brief , Coleman's method involves the use of complex variables as generalized co - ordinates in the Lagrangian equations of motion . With a b - bladed rotor there would be b 2 degrees of Special freedom . linear combinations of the rotor blade deflexion on the drag hinges are used as generalized co - ordinates , resulting in four degrees of freedom to be considered simultaneously . The use of complex variables reduces these four equations to two linear , homogeneous , second order differential equations in two generalized co - ordinates , and their complex conjugates with the time as independent variables . Exponential solutions are assumed and substituted into the differential equations , resulting in algebraic equations . The roots of the characteristic linear homogeneous investigated determinant are to determine the stability or instability of the system . Figs . 9.22 to 9.24 , taken from Coleman's report , show the critical value of speed for the shaft critical and self- excited ( ground resonance ) oscillations of values of the hub stiffness ratio x of 1-0 , co and 0. The use of the charts is illustrated by the numerical example A₁ - = 0 ·07 , A , = 0 · 22 , Ag =

rotor

0-10 ,

KK

S10

function

2, = 16.2 rad /sec . The w₁ /2, + A₂2 plotted against ,

straight line in Fig . 9.22 is the

(w

/ A, ) ² ,

and intersects the shaft

Helicopter Dynamics and Aerodynamics

346

critical curve of A3 = 0 ·1 at ( w /N , ) ² = 0 · 77 , so that the shaft critical rotor

136 r.p.m.

/

=

77

14.2 rad sec

=

0 ·

/

2, √

=

@ ₁

speed is

5.0

O

A3

LIMIT

LOWER

EXCITED

REGION

LIMIT

UPPER

SELF

4.0

-

.

0.1

0.5

ROTOR SPEED FREQUENCY REFERENCE

3.0

1.0

SHAFT CRITICAL

GROUND

1.0

1.2

SPEEDS

1.4

1.6

STABILITY CHART

RESONANCE

( S

PYLON STIFFNESS

-

EQUAL

1.0

)

9.22

.

.

FIG

0.8

W12

=

0.6

0.4

0.2 0.3

)

²

,

N ,

/

w

be (

to

a

in

of

as

∞ )

=

S

(

,

.

.

is

)

A2

)

(

9.31

ΛΛ2

-

1

– A1

1

41

(

Аз

+

-

.

2228

plotted

+

equation from which Fig 9.25

is

-

to

,

a

of

=

=

to

-

of

= 1.6 - 4.85 The limits the self excited region are seen which corresponds w₂ 196 r.p.m. and wg 342 r.p.m. For hub mounting with only one degree freedom tandem helicopter without tyre resilience Fig 9.25 shows the value damping required eliminate self excited vibrations The approximate

,

or

,

is

as

so

is

long reasonably This equation sufficiently accurate the damping distributed between the drag hinges and the fuselage supports but when either the hinge the support damping approaches zero Coleman's report

5.Or

1.0

LIMIT

REGION

LIMIT

LOWER

SELF

UPPER

4.0

EXCITED

)2

(w

=10

EV

0.5

ROTOR SPEED REFERENCE FREQUENCY

3.0

2.0

0.5

= 0 ·1

SHAFT CRITICAL SPEEDS A3

0.5 0.8

ω

1.2

1.0

1.4

1.6

S

(

A320-1 LIMIT UPPER REGION LIMIT EXCITED LOWER SELF

4.0

3.0

FREEDOM

)

RESONANCE STABILITY CHART PYLON = ∞0 IN ONE DIRECTION

GROUND

=

9.23

.

.

FIG

0.6

0.4

+

0.2

0.5

2.0

= 0 ·1

+

Ʌ3

SHAFT CRITICAL SPEEDS

0.5

ω

0.8

1.0

1.2

RESONANCE STABILITY CHART IN ONE DIRECTION

347

= 0 )

GROUND

0.6

S

9.24

.

.

FIG

0.4

(

0.2

2

0.5 1.4

PYLON FREEDOM

Helicopter Dynamics and Aerodynamics

348

If

must be referred to for a more accurate treatment . either is equal to zero , the total effective damping is zero . The damping required for S = 1.0 is approximately the same as that given by the above equation for = ∞..

It should

Soo

be borne in mind that Coleman's theory includes the effect of an

K

, at the drag hinge , which gives stiffness with respect to the elastic restraint hub , but does not allow for elastic restraint between blades , such as is introduced by inter - blade snubbers

.

10.0

7.0 4.0 2.0 0.1

228

A3

1.0

0.7

0.2 0.5 1.0

0.4 2.0 0.2

5.0 10.0

0.1.01

0.02 0.04 1 40.070 VERTICAL HINGE

0.2

0.4

PARAMETER

0.7 1.0

A,

FIG . 9.25 . DAMPING REQUIRED TO ELIMINATE SELF - EXCITED INSTABILITY FOR S = ∞ ( PYLON FREEDOM IN ONE DIRECTION )

of damping required , such as equation ( 9.31 ) and do not enable the effect of air damping due to blade flapping to be , estimated so that the required mechanical damping will be somewhat less than the theoretical value of 22. Moreover , Howarth and Jones have suggested that the Coleman criterion is a little pessimistic and that the unstable Theoretical estimates

Fig .

9.25 ,

range closes less and less rapidly with increase in damping , so that the range is small and the divergence very gentle some time before the damping required for theoretical stability is reached . In practice an oscillation whose amplitude very slowly with time is often quite acceptable because of the short period of time for which the machine operates at the critical condition . Howarth and Jones found in model tests that they required only 14 to 28 per cent of the theoretical damping to close the unstable range for practical purposes .

increases

9.11 .

DEUTSCH EQUATIONS FOR CRITICAL SPEED AND DAMPING

Within the limitations of the above remarks it is interesting to note the results obtained for a three - bladed rotor by Deutsch in ref. 5.17 . For a hub

Ground Resonance and Vibration Due to Rotor Resonance

349

with isotropic spring restraint and mass in the plane of the rotor he expresses the shaft whirl speed as 2

− ^ 2) +

= ( A1

√ [ (A1

+ A2 ) ² + 4 ^ 2^3 ]

(9.32 )

2(A1 + A3 )

If

the hub has only one degree of freedom in the plane of rotation corresponding to infinite stiffness in one direction , the following equation in ,) must be solved to find Р = (w

/

(1

− p² ) ( A₁p² +

A½ ) ( ^ ₁p²

- 4p² — )

A₂

+

—

-

A¸рª ( ^ ₁p² + A₂ ▲ , − 2p² ) =

0

(9.33 )

K

When the elastic restraint of the drag hinge = 0 , which is a very equation , common case A₂ = 0 and the above reduces to )

9.34

(2A1 + A3 ) .

-

)

^ 2

0.187

Qp

Q ,

)²

/

of

. )

according

to

Deutsch

,

,

.

is

)

to

is

(

damping required

for isotropic

is

0-1

9.37

)

>

(

A3

)

λλβ

1 (

,

The critical supports

in

of

the fundamental natural frequency the blade the plane stationary of rotation chord wise sense when the blade easy This frequency determine experimentally (

=

(

A₂

)

9.36

(

A₁ Λι

where

)

(

^ 1

—

2

^

)

( 1

— +

^ ₁

+

9.35

A₁

rotor without drag hinges we can use the approximations

=

case

=

For the

of a

Ω ,

1

W

√ (

is

For both types given by range

is

/

N ,

always less than unity the above equations hub restraint the centre of the self excited instability w

In

2A₁

=

of

all

Ω,

(

2

@1

).

(

,

TWIN ROTOR HELICOPTERS -

.

9.12

).

(

is

of

, is

w

/2

of

where the value for the centre the instability range equation 9.35 only degree When the hub has one freedom the required damping product approximately given by equation 9.37 22ẞ half the value

co -

Twin rotor helicopters

a

as a

.

is

in

,

so

co -

,

of

a

-

or

,

-

are either axial with the rotors mounted on the some distance that there the hubs are separated common axis plane of rotation between the two centre lines single Deutsch has suggested that axial rotor can be treated only being degree rotor with one hub freedom the number of blades taken

Helicopter Dynamics and Aerodynamics

350

as the total number in the two rotors . Thus equations ( 9.32 ) and ( 9.35 ) can be used for whirling and instability respectively , and the required damping product to eliminate instability will be half that specified by equation ( 9.37 ) . When the rotor hubs are separated , yawing modes of the fuselage become significant , and rolling , pitching and yawing motions may combine in various amplitude combinations to give either approximately circular motion of the hubs ( isentropic support case ) or approximately straight line motion , corresponding to only one degree of freedom . In calculating the minimum damping for a twin - rotor helicopter of this type , isentropic hub supports should therefore be assumed . The case of tandem rotors is dealt with in more detail by Howarth and Jones in ref . 5.6 . 9.13 .

OF TWO - BLADED ROTORS

GROUND RESONANCE

The two - bladed rotor constitutes a special case , and the theory developed for multi - bladed rotors is no longer applicable . The main difference is in the shaft critical region , as explained in Section 9.1 . The multi - bladed rotor has a single shaft - critical speed for each hub natural frequency N ,, and no associated unstable range , and the rotor speed at which this occurs is always less than

Q.

The two bladed rotor however has two shaft - critical speeds for each value

of Q,, with a region of instability between these two

speeds . The lower speed is associated with hub oscillation at right angles to the line of the blades , the rotor mode of oscillation being that of Fig . 9.17 in Section 9.6 , and occurs at a rotor speed which is smaller than Q ,, the frequency of the oscillation being equal to w₁ . The upper shaft - critical speed is associated with a hub oscillation parallel to the line of the blades , and occurs at a rotor speed equal to the

natural frequency of the hub Q ,. The theory for two - bladed rotors is given by Coleman in ref . 5.14 , but Figs . 9.26 and 9.30 have been abstracted as being within the scope of this book . Figs . 9.26 , 9.27 and 9.28 are for zero damping (2 = 2B = 0) and indicate the important rotor speeds

In Fig .

.

9.26 the lower shaft - critical speed is plotted as a ratio

/A, for various

/

natural frequency ( w 2 , ) against A1 equation for the shaft critical speed is

Ω

12

+

ω 11

2A3

Ω,

of the hub

/

values of A2 A3 . The

@2

Ω,

-

0

(9.38 )

The first factor is the lower speed , and the solution is plotted in Fig . 9.26 . The second factor defines the upper limit of the first region of instability , and is the second shaft - critical speed , i.e. @2

2,2 £

-

1.0

-

The second range of instability is defined by Fig . 9.27 . An example has been imposed on this graph for the case A 0.05 , A2 A₂ = 0.20 , A30-10 ,

Ground Resonance and Vibration Due to Rotor Resonance

/

and the straight chain line represents ( 1 -

4A3 ) w² Q ,² against

/ ,) for, these values region

(w

351

A₂2 + A₁

.

2

As A increases the of instability rotor speeds and moves to infinity at A =

occurs at progressively higher . For values greater than this

2.8 2.4

2.0 1.6

Δι

A₁

1.2

0.8

√^2/^3= 0

0.4

0.4

2

FIG .

9.26 . SHAFT CRITICAL SPEEDS

0.8

1.0

FOR TWO -BLADE

ROTOR

A₂=0.15

6

0.05 EXAMPLE

5

4 3

0-15

0.05

2

UPPER LIMIT LOWER LIMIT 0.4

0.6

1.2

INSTABILITY

REGION FOR TWO BLADE

ROTOR

ID

4MmD2

3m₂R

(

R³ the

,

Mн H

and ID

9.40

(

MmD =

{ m

where

)

9.39

ID =

,

a

For constant mass rotor condition is

MHID

R²

MH MA

= MmD2 >

,

or

М.mD