Theses - Daytona Beach Dissertations and Theses

Fall 2007

Study of Contra-rotating Coaxial Rotors in Hover, A Performance Study of Contra-rotating Coaxial Rotors in Hover, A Performance

Model Based on Blade Element Theory Including Swirl Velocity Model Based on Blade Element Theory Including Swirl Velocity

Florent Lucas Embry-Riddle Aeronautical University - Daytona Beach

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STUDY OF COUNTER-ROTATING COAXIAL ROTORS IN HOVER

A PERFORMANCE MODEL BASED ON BLADE ELEMENT THEORY INCLUDING SWIRL VELOCITY

by

Florent Lucas

A thesis submitted to the Aerospace Engineering Department

In Partial Fulfilment of the Requirements for the Degree of Master of Science in Aerospace Engineering

Embry-Riddle Aeronautical University Daytona Beach, Florida

Fall 2007

UMI Number: EP32005

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A STUDY OF COUNTER-ROTATING COAXIAL ROTORS IN HOVER

by

Florent Lucas

This thesis was prepared under the direction of the candidate's thesis committee chair, Professor Charles Eastlake, Department of Aerospace Engineering, and has been approved by the members of his thesis committee. It was submitted to the Aerospace Engineering Department and was accepted in partial fulfilment of the requirements for the degree of Master of Science in Aerospace Engineering.

THESIS COMMITTEE

Professor Charles Eastlake Chair

Dr. Magdy Attia Member

1U Dr. Frederique Drullion Member

Chair,

?j2 -s> ii^/n/^ooj Department Chair, Aerospace Engineering Date

izjfi//) 7 Vice President for Research and Institutional Effectiveness Date

ii

ACKNOWLEDGMENT

I wish to deeply thank the Thesis Chair, Professor Charles Eastlake, for his practical

suggestions and his priceless help all along this study. It was a pleasure to share discussions

as much on a professional point of view as on a personal one. My appreciation is extended to

the thesis committee members, Dr. Magdy Attia and Dr. Frederique Drullion for the support

and help they gave me in the redaction of this document.

I would also like to express my deepest gratitude to my family and my friends for their

constant encouragement and moral support.

iii

ABSTRACT

Author: Florent Lucas

Title: A Study of Contra-rotating Coaxial Rotors in Hover

A Performance Model Based on Blade Element Theory Including Swirl Velocity

Institution: Embry-Riddle Aeronautical University

Degree: Master of Science in Aerospace Engineering

Year: 2007

The purpose of this study is to create a simple model to evaluate the performance of a

counter-rotating coaxial rotor system. As it is based on the blade element theory, it could be

used as a design tool where the engineer can modify all parameters - rotor radius, blade pitch

distribution, tip velocity, rotor spacing, number of blades, blade chord, and airfoil's shape

through its lift coefficient versus angle of attack slope. Different blade pitch distributions are

tried and are assessed based on the power required to hover at the same weight. Also, the

impact of the swirl velocity induced by the upper rotor is included and discussed. In order to

verify the credibility of the model, results are compared to what was obtained from other

researches. The main result is to compare performance of the current AH-64 Apache single

rotor system with the optimal coaxial rotor model.

IV

" When will man cease to crawl in the depths to live in the azure and quiet of the sky? "

Jules Verne, Robur the Conqueror, 1886

TABLE OF CONTENTS

ACKNOWLEDGMENT iii ABSTRACT iv TABLE OF CONTENTS vi List of figures vii List of tables viii 1 Introduction 1

1.1 About coaxial rotor helicopters 1 1.2 A few steps of history 2 1.3 20th century and current state 3 1.4 Literature survey 4 1.5 Scope of this thesis 5

2 Model developed ~..7 2.1 Blade element theory for a single rotor 7 2.2 Description of coaxial rotor environment 11 2.3 Induced velocity at lower rotor including swirl velocity: 14 2.4 Model function 20

3 Results 24 3.1 Comparison to existing literature 24

3.1.1 Contribution of swirl in the wake 24 3.1.2 A coaxial rotor system or two isolated rotors 24 3.1.3 Rotor spacing, pitch increment and thrust ratio 27

3.2 Effect of pitch distribution 29 3.2.1 Untwisted blade 30 3.2.2 Linearly twisted blade 31 3.2.3 Enhanced blade twist 32

3.3 Comparison to a single rotor 34 4 Special case of ducted rotor 41

4.1 Simple coaxial configuration 42 4.2 In skewed-duct coaxial configuration 42 4.3 In straight-duct coaxial configuration 43

5 Conclusion 44 5.1 Concluding remarks 44 5.2 Recommendation for future work 45

References 46 Appendices 47

vi

LIST OF FIGURES

Figure 1: Da Vinci rotor design 2 Figure 2: J. Verne's Albatross & first aluminum counter-rotating coaxial rotor (d'Amecourt) 2 Figure 3: Various coaxial helicopters 4 Figure 4: Description of blade element 7 Figure 5: Aerodynamics around a blade element 8 Figure 6: Example of single rotor spreadsheet 11 Figure 7: Description of vena contracta 12 Figure 8: Definition of parameters around blade 14 Figure 9: Relation between rotor spacing and air streamtube 15 Figure 10: Inputs of the model 20 Figure 11: Coaxial rotor blade element spreadsheet .21 Figure 12: Coaxial spreadsheet results 22 Figure 13: Coaxial and tandem configuration 25 Figure 14: Power required for coaxial or isolated rotors 27 Figure 15: Velocity induced by rotors with untwisted blades 30 Figure 16: Linear twist distribution 31 Figure 17: Induced velocity for rotor with linear twist 31 Figure 18: Enhanced twist 32 Figure 19: Induced velocity for enhanced twist 32 Figure 20: Mathematical approximation of enhanced twist 33 Figure 21: Power required versus rotor radius 34 Figure 22: Power required versus tip speed 35 Figure 23: Design parameters of modified Apache with linearly twisted blades 38 Figure 24: Performance of coaxial system with linear twist 38 Figure 25: Optimum performance of coaxial configuration 39 Figure 26: Power required if lower rotor influences upper one 40 Figure 27: Different design for duct tests 41 Figure 28: Simple coaxial configuration 42 Figure 29: In skewed-duct coaxial configuration 42 Figure 30: Straight-duct coaxial system 43

vii

LIST OF TABLES

Table 1: Importance of swirl in the wake 24 Table 2: Difference between coaxial and isolated rotors 26 Table 3: Pitch difference required to matain torque balance 28 Table 4: Relation between performance and blade pitch distribution 30 Table 5: Increase of power required for various blade twist 33

viii

1 INTRODUCTION

1.1 About coaxial rotor helicopters

The most common configuration of a helicopter is to have a main single rotor to create the lift

and a small tail rotor whose purpose is to balance the torque created by the main rotor.

Consequently, the machine can be directionally stabilized and the pilot has control over the

yaw angle of the aircraft. Having two counter-rotating rotors is another solution to

directionally stabilizing a helicopter and it provides the advantage of not having a tail rotor

since the torque reaction from the two rotors can cancel each other. Indeed, a tail rotor uses

power without providing lift. Moreover, it is the source of many accidents which could

happen because the pilot hits something when hovering close to ground, because someone

walks by the tail without seeing it or also because an enemy shoots the tail itself. The loss of

the tail rotor often translates to a loss of lives.

1

1,2 A few steps of history

s^^ i i -^

This chapter does not have the purpose of offering an exhaustive history of helicopter but

offers a glimpse at important steps. Leonardo Da Vinci is very

often credited for being the conceptual inventor of helicopter.

Although his design never took life and apparently could not

work, he surely had sensed what the future would offer to

mankind. An opportunity to make a human dream come true, a

chance to fly in the skies, hover and go rearward.

. NI 'i . • *>>S , !f?vy."»sW'!li

»,*{«•*>»-».* v*» w? Figure 1: Da Vinci rotor design

In his book "The God Machine", James R. Chiles [1] describes early inventors' work and

intellectuals' contribution. The word 'helicopter' is adapted from the French hehcopiere,

coined by Gustave de Ponton d'Amecourt in 1861. It is linked to the Greek words hehx/helik-

(EXiKctc;) = "spiral" or "turning" and pteron (nxepov) = "wing". D'Amecourt's efforts were

backed up by the famous Jules Verne through, for example, "Robur the conqueror" in which

the main character, captain of the Albatross, states: "With her, I am master of the seventh part

of the world".

Figure 2; J. Verne's Albatross & first aluminum counter-rotating coaxial rotor {d'Amecourt)

2

However, it seems that the first helicopter design that actually

flew was the Launoy-Bienvenu helicopter. This is considering

that the Chinese invention was a toy and actually did not store

the energy required to fly. The Chinese device consisted of

feathers at the end of a stick, which was rapidly spun between

the hands to generate lift and then released into free flight. The

two French inventors sent in the air their device on April 26th 1784, before the "Academie

Royale des sciences". The energy to turn rotor was stored in a bone bent by strings rolled

around the axe of rotation.

1.3 2(fh century and current state

The sky is today dominated by single rotor helicopters. However, certain companies like

Kamov Design bureau have made the choice of coaxial systems which led to the Ka-32 or Ka-

50 (Figure 3), for example. Other, like Sikorsky, have shown their interest in this technology

through the ABS helicopter of the 1960s or today the X2 demonstrator, Figure 3.

In the 50's Gyrodyne (USA) developed the XRON, Figure 3, a one seat coaxial helicopter

that received in 1961 the Grand Prize for the most manoeuvrable helicopter at the

International Paris Air Show at Le Bourget, France.

Also, at a lot smaller scale (for now) Gen Corporation in Japan developed a personal coaxial

helicopter of 70 kg (1551b) with a maximum takeoff weight of 220kg (4851b), an endurance of

30 to 60 min and a maximum speed of 100 km/h (60mph). This machine, Figure 3, is sold for

$35,000. Did the human dream come true?

3

From left to right, top to bottom Gen corporation helicopter, Ka-SO, X2, XRON

Figure 3: Various coaxial helicopters

1.4 Literature survey

Colin P. Coleman [2], wrote in 1997 a paper titled "A Survey of Theoretical and

Experimental Coaxial Rotor Aerodynamic Research" which has been useful in understanding

what performance to expect from counter-rotating rotors. This paper sorts work done on

coaxial rotors by country. Many methods are described for predicting performance of counter-

rotating rotors but to the best of the author's knowledge, the work being presented in this

thesis is the first public study that shows a simple model using blade element theory in

4

combination with momentum theory. The following paragraph describes discussions and

research that helped in doing this thesis.

Effect of swirl velocity in the wake of a rotor has been discussed by Wayne Johnson [3] and

can be used as a tool of comparison for the current model. His formula was used in the model

developed in this thesis. J. Gordon Leishman [4] offers a simple way to evaluate counter-

rotating coaxial rotor performance but it is based on momentum theory only. Many

discussions refer to Dingeldein [5] and Harrington [6] who in the early fifties conducted

experiments in the NASA Langley facilities and obtained practical results to compare to.

However, they do not show any theoretical approach. Coleman refers to M.J. Andrew [8]

about his work on the rotor spacing. Finally, Japanese researchers [9] show an interesting

method to calculate coaxial systems performance but it is again based on momentum theory

and angular momentum.

1.5 Scope of this thesis

The purpose of this study is to create a simple model to evaluate the performance of a coaxial

rotor system. As it is based on the blade element theory, it can be used as a design tool where

the engineer can modify all parameters - rotor radius, blade pitch distribution, tip velocity,

rotor spacing, number of blades, blade chord, and airfoil's shape through its lift coefficient

versus angle of attack derivative. Different blade pitch distributions are tried and are assessed

based on the power required to hover at the same weight. Also, the impact of the swirl

velocity induced by the upper rotor is included and discussed. In order to verify the credibility

of the model, results will be compared to what was obtained from other researches.

The coaxial rotor model is compared to a single rotor configuration and two isolated counter-

rotating rotors. In addition, performance is analysed based upon some design parameters

changes like the rotor diameter and tip speed. Those results serve to improve the

understanding of coaxial counter-rotating rotors' advantages.

5

Many researches have been conducted on coaxial rotor systems but no model was found to be

based on the blade element theory. The idea came from the fact that an Excel spreadsheet

calculating single rotor performance was developed and carefully checked in the AE433 class,

"Introduction to Helicopter Aerodynamics", at Embry-Riddle Aeronautical University in

Daytona Beach. This spreadsheet is only for a single rotor but with coaxial counter-rotating

rotors becoming more popular, the need for an expanded model appeared.

6

2 MODEL DEVELOPED

The performance model developed is based on blade element theory combined with

momentum theory. A model already existed for single rotors that was developed for use in the

Embry-Riddle class AE433, "Introduction to Helicopter Aerodynamics", and was carefully

checked by Professor Charles Eastlake through many years of use in class. The objective is to

add a second rotor to the proven spreadsheet, and then try to adjust the model of the lower

rotor to account for the interaction effect of the flow induced by the upper rotor. In other

words, the idea was to extend this model to a coaxial rotor system, accounting for the axial

but also swirl velocity induced by the upper rotor. The study focuses only on hovering

performance.

2.1 Blade element theory for a single rotor

Figure 4 shows variables used to describe a blade element as it is used in the model.

<-'' \ \ ' \ \ \

' \ \ I ' \ \ * \ \ * / \ \ *

i \ \ * i \ ^ — " N . >

\ / /

) i ay;

y 7

\ / / / R - / / ' \y / / A

Figure 4: Description of blade element

7

The main advantage of the blade element method is that someone designing a rotor can

modify the pitch distribution of the blades in addition to other parameters such as rotor

diameter, tip speed, etc. Modifying the pitch distribution can allow to obtain a constant or

almost constant induced velocity on both rotors, at least on the outer part which is

predominant in the performance. This situation provides better results in term of power

required. A dimensionless parameter r = — is used to describe the location on the blade.

y and R are shown in figure 4. The model considers 10 points equally separated along a

blade from r = 0 to r = \ and the local pitch angle can be defined as a function of r or

modified by hand for each of those locations. However, adding complexity to the design of

the blades increases the cost and time of manufacturing.

Also, the model allows choosing the chord length which is an important parameter of the rotor

solidity.

Figures 5 and 6 should help to understand the process of performance calculation for a single

rotor, along with the following description.

Q.

, ' • ' '

1

y^^.'-'y J^^—""~ / - ' \ ^ ~ ^ ^ ^ v i

&~-" ' iV r , i

Q,y

\ e

Figure 5: Aerodynamics around a blade element

8

As the study is about hover flight, the knowledge of the rotor tip speed or the rotor speed of

rotation, as well, provides the local Mach number. The geometry of the blade being defined at

dCi i

various r locations, the local lift coefficient versus angle of attack derivative, a = ——, can be da

obtained which leads to the calculation of advance angle,^ = tan" fV.^

shown figure 5.

Then the induced velocity can be deduced as well as the angle of attack, a=9-(f>, and

eventually the thrust and torque coefficients. Those are the purpose of the study. The thrust

derivative — - is similar to the lift and is therefore a function of a and a defined above. dr

The torque derivative — - has two components, one due to the parasite drag related to a dr

and the other one due to the induced drag related to the amount of thrust and (/>.

Values of — - and — - for each location are integrated along the blade to produce C7 and dr dr

CQ to evaluate the rotor performance. CT provides the rotor thrust obtained and CQ provides

the torque required to turn the rotor which is translated into horse power (HP). One should

note that part of tip performance is taken out. This is taken into account by using the

traditional Prandtl's tip loss factor due to the vortex at the tip of the blades. Because of this

vortex, a small portion of the outer part does not produce lift and, therefore, also does not

produce induced drag. However, it still produces profile drag. The amount taken out is a

function of the total thrust and the number of blades.

Prandtl's formula: 5 = l--v 7

B is the proportion of blade length that produces lift so the integration is performed from

r = 0 to B instead of r = 0 to 1. N is the number of blades for the rotor.

9

t

00 01 02 03 04 05 0 6 07 08 09 10

Thrust/Powe B

\*~ T/uo bss

(ACT)^,loss

cT T(\b)

CQO

\ M ) I 'no loss

(ACQAIPIW

C«

CQ

HP induced fatal HP Est Hp hover Hover % pow,

Baseline 0 (deg)

150 14 1 132 123 114 105 96 87 78 69 60

r Calculations 0 9674

8 517E-03

7 344E-04

7 782E-03

17651 1 268E-04

5 773E-04

5 714E-05

5 202E-04

6 470E-04

1557 4

1937 0 69 6

Incremental 8 (deg)

2417 2417 2417 2417 2417 2417 2417 2417 2417 2417 2417

6 (rad)

0 3040 0 2883 0 2726 0 2569 0 2412 0 2254 0 2097 0 1940 0 1783 01626 0 1469

M

0 000 0 065 0 130 0 195 0 260 0 325 0 390 0 455 0 520 0 585 0 650

a (per rad)

5 73 5 73 5 73 5 73 5 73 5 73 5 73 5 73 6 00 6 50 7 00

Note Incremental 8 is 2 417 degrees to

produce hover T = W= 176501b

4 (rad)

UNDEF 02173 0 1776 0 1523 0 1337 0 1190 0 1068 0 0963 00884 0 0821 0 0759

v,(ft/s)

0 15 77 25 79 33 16 38 82 43 19 46 52 48 96 5137 53 66 55 13

a (rad)

— 0 0710 0 0949 01046 0 1075 0 1065 0 1029 0 0977 0 0899 0 0805 0 0710

a (deg)

— 4 07 544 5 99 6 16 6 10 5 90 5 60 5 15 461 4 07

c„ —

0O086 0 0119 0 0137 0 0143 0 0141 0 0134 0 0124 00110 0 0097 00086

dCT/df

0 1 888E-04 1010E-03 2 504E-03 4 574E-03 7 079E-03 9 856E-03 ! 273E-02 1 602E-02 1 967E-02 2 306E-02

dCVdr

0 4 015E-07 4 406E-06 1 72OE-05 4 260E-05 8192E-05 1 342E-04 1 969E-04 2 620E-04 3 279E-04 4 014E-04

dCq/dr

0 4 103E-06 3 588E-05 1 144E-04 2 446E-04 4 212E-04 6 3I6E-04 8 586E-04 1 134E-03 1 454E-03 1 751E-03

o

Figure 6: Example of single rotor spreadsheet

2.2 Description of coaxial rotor environment

In a coaxial rotor system case, the air velocity induced by the upper rotor influences the lower

rotor. Therefore, calculation of lower rotor performance must take it in consideration.

The air velocity induced by a rotor can be divided in two parts, an axial velocity and a

rotational component. Axial velocity is labelled V, and rotational velocity also called swirl

velocity is labelled u .

The downwash created by the rotation of the rotor undergoes a contraction as velocity

increases. Indeed, for hovering flight, it is shown by momentum theory that the velocity in the

far wake is twice the induced velocity. Therefore, the air mass flow being constant below the

rotor, the cross section of the downwash decreases. The narrow throat in the downwash

streamtube is also sometimes called "vena contracta".

m- p-V • A = constant

VX=2V,

therefore, A

A rotor

Because of this effect, only part of the lower rotor will be immersed in the flow created by the

upper rotor.

An important assumption is that the "vena contracta" is reached at a distance equal to the

rotor diameter below the upper rotor. The second major assumption is that the variation of the

downstream cross section is linear. Although it is not exact, it is considered to be

representative enough for our model.

11

The following figure shows a model based on these assumptions:

Roton-

streamtube'

vena contracta

R

. - >

D

0.7 R

Figure 7: Description of vena contracta

We want to obtain an equation providing the radius of the streamtube r with respect to the

distance from the upper rotor.

At the rotor, z = 0 and A = Yl • R2

At z = 2 • R, we have

2

R' = J * -V 2

R'*0.7R

A formula to represent the radius r of the streamtube would be as follow.

A/? D R'-R r = z + R = z + R

r = •

2R -0.3-.ft

2-#

2-R

•z + R

r = R\l-0.\5-R

12

The part of the lower rotor exposed to the downwash will be effectively in climb. However, a

typical spacing distance between two coaxial rotors is from 10% to 20% of the rotor radius.

The lower rotor is so close to the upper rotor that the "entire" part of the lower rotor will be

analized as if it was in climb. This is related to Prandtl's tip loss and discussed in the

following paragraph. Furthermore, this climb velocity will be the velocity induced by the

upper rotor times a factor greater than one because the velocity increases as the streamtube

narrows.

The increase in velocity is inversely proportional to the variation in area. The air velocity goes

linearly from Vl to 2 • Vt between the rotor and a distance 2 • R below.

dis tan ce 2-V -V

V = '• '-z + V, 2R

V = V 1 + 0 .5 -R

It is also assumed that the upper rotor is not influenced by the velocity induced by lower rotor.

This means that any extra air flow required by the lower rotor is considered to be pulled in

horizontally through the vertical gasp between the rotors. This assumption will lead to

comments later in the text.

13

2.3 Induced velocity at lower rotor including swirl velocity:

This section describe the calculation of induced velocity at the lower rotor for the part which

is in the wake of the upper rotor and consequently in climb.

We consider a blade element of the lower rotor.

Figure 8: Definition of parameters around blade

$, the advance angle, is small. Therefore, we can consider that

(/> = t a n

Where,

f Vr+V. C ' ' t

Q-y + u2 j

Q- y + u2

y is the radial position along the blade.

Q is the rotor rotational speed.

u2 is the swirl velocity induced by upper rotor, it is a function of y.

Vc is the climb velocity or axial velocity induced by upper rotor.

Vt is the axial velocity induced by lower rotor.

14

U = J{n-y + u2)2+(Vc+V,)2

u = {Q-y + u2)

cos^

As </> is small, cos^ « 1 and U » Q • y + u2

Upper rotor

lower rotor

\

/

l

R

i

7 / / / i

i

i i

i

i

Figure 9: Relation between rotor spacing and air streamtube

Let's call C, the contraction ratio of the streamtube between lower and upper rotors as

described figure 9. What happens at the location "y" for the upper rotor contracts inward as it

flows down and this is used to calculate what happens at the location "£ • y" for the lower

one.

u2 is the swirl velocity at lower rotor. It can be deduced from swirl velocity «, at upper rotor.

Therefore, the u2 calculated from u] at location y of upper rotor is valid at location £ • y of

lower rotor. For that reason we locate our study at location C, • y for the lower rotor and we

want to use the following equation: U « Q • £ • y + u2.

Because of this method, the integration along the blade stops at^ .ft. However, it does not

actually affect the result. The reason is that for a ratio of (spacing between rotors / diameter of

15

rotor) less than or equal to 0.1, the edge of the downwash from upper rotor would hit the

lower rotor within the region of the blade tip that is taken out of calculation because of the

traditional Prandtl tip loss correction.

We want to work with non-dimensional parameters, therefore we will use:

U _Q-g-y + u2_ u2 £ -r + -

QR QR QR

where R is the rotor radius

Using angular momentum conservation, where "p" stands for a percentage of the blades

radius,

urr2 =*V2 2

U]{pR)2=u2 -{p-C-Rf

" i

c W, is defined in Reference 3 and it should be noted that it does not depend on the pitch of

blades.

2Vh-Q.-y ux =Vh

(n-yf+vh2

• 2

Consequently, u2=-^-C, (n-y) +vh

u2 _ Vh 2-Q.-y hn*~ehR~~f'{{n.yf + Vh

2)-Q.-R

u2 _ Vh 2-r Simplifying, using, y = r-R, — = -,- ({a.yf + y2)

Finally,

U A Vh2 2-r

QR C r2(n-R)2+Vf

16

Vh, induced velocity for hovering, is calculated with momentum theory: Vh = T

I upper ]2p-A

A guess has to be made for thrust provided by upper rotor in order to solve for . Then

Q- R

iterations are made until the guess meets the calculated value. In other words, the ratio of

(thrust provided by upper rotor / overall thrust) is assumed in order to be able to calculate Vh.

Then, the sequence of calculation is repeated until hover at equal rotor torque is obtained and

this solution provides the thrust generated by each rotor. Consequently this gives the thrust

ratio. If the assumed ratio and the calculated one do not match the initial guess has to be

changed and the process redone. This can be automated with a code and such was done with a

macro linked to the Excel sheet ("automatic calculation code" in appendices p59).

In order to derive an expression for V, at the blade element of the "climbing" lower rotor, we

sic* can equate two different expressions for the differential thrust coefficient — - of a blade

dr

element.

dCT = -pU2-(c-dy)-C,

p-(nR2){QRJ

Simplifying, dCT -f ~ \

K-R)

f U V

QRJ

dy_

R

Finally, using the previously defined expression of U

Q f t

dCT = — T 2

1 ( c ^ \n-Rj '• V &R

v dr

For N blades, dCr = — • C, 7 2 /.

C-r + V„

U J r2-(Q-R)2+Vh2 j

dr

17

N-c a = is the solidity of the rotor

n-R

Where 6 is the local pitch angle

The advanced ratio, X, can be written as follow:

V +V

Q-R

VyVc =U-sm(/)

sin^ «<j>

QR

Therefore,

<t> = X

£-r + yh

{C 2-r

r2-{tl-R)2 + Vh'

Then,

dCT = a a 9- C-r + yh

z 2-r

\ W r2.(Q-R)2 + V2 I- C-r +

fvy 2-r U J r2.(Q-R)2

+V2 dr

d^iOo+Oj+r-0^

Where 90 is the pitch angle at blade root

0im is the increment of pitch angle or collective

6M is the twist angle for the blade

The other expression for differential thrust coefficient using momentum conservation for an

annular ring of rotor disk is:

4.(y -\-V\V -C-r dCT = —^-~—'-^-rr1 dr (Because we locate our study at C • r)

{Cl-Rf

18

So equating those two expressions for dCT we have,

era

a-a 2

C-r + rv\z

JL 2-r

2

* J

9-' 'V? C-r+

U

r2-{Cl-Rf+Vh

-X'

4-r 1 K r2-(n-rf+Vh

2 £{C) (r>.(n-R)2+Vh

2)

£-r +

Ar

V2

K$J

2-r

r2-(n-Rf + Vh2)

= 4

-X- 1+-fV?

Q KCJ r2-(n-R)2+V2)

Vc + K n-R

=4-

( v \

\n-Rj C-r

A

The alternate expression for the advance ratio is: X = vc+K n-R

Then,

a-a 9-A- Vc + K n-R)

B (Vc + K n-R , n-R)

= o

VC+VY V,

n-R xn-Rj

B

We want to rearrange this into a quadratic equation.

aa •e-A-i^(vc+v,)B-7^(vc-v,+v2)=o 8 8 Q / ? V " " (n-R)

(„ cT-a-n-R \ V2 + V

I l Vc+'

8 •B +—-n-R-{Vc-B-9-A-n-R) = Q

8

Solving for V, we end up with

V,= Vc aa-n-R-Bs

+ - 1 + 1+-2-(9-A-n-R-VcB)

4-Vr.

a-a-n-R + VC-B +

a-a-Q-R-B2

16

Remembering that Vc is the axial component of the velocity induced by the upper rotor in the

plane of the lower one, we now have a formula for the induced velocity at the lower rotor of a

coaxial system including swirl from the upper one. Vc is a function of the rotor radius. The

contraction of the downwash from upper ratio has to be included in the application of this

equation.

19

The following paragraph show how the induced velocity is incorporated into the process of

calculating performance of the system.

2.4 Model function

The calculation of the induced velocity is considered to be in a "climbing" situation a mean to

calculate the advance angle <t>. Following this result, the angle of attack can be calculated

from the pitch angle 9 with the formula a -9 -<j>.

Finally drag, thrust and torque can be calculated as is described in more detail on the

following pages. Figure 10 is the data input sheet, Figure 11 displays the spreadsheet used and

Figure 12 shows the results.

Name of helicopter | This spreadsheet is a model to calculate performance of coaxial rotor considering that only the lower rotor depends on the other swirl velocity induced by the upper rotor is included

Characteristics of helicopter If (lb) 17650 Engine HP 2784

Following data are characteristics of the rotor

Coaxial Rotors Design

Rift) 24

spacing (H R) 0 1

N 4

Cfft) 175

OR (ft/s) 600

upper rolor UPP' 0o (deg)

0iw (deg)

a (per rad)

15

-9

5 73

lower rolor

9o (deg)

0 W (deg)

a (per rad)

15 -9

5 73

Calculated Data

A (ft')

a

n (rad/s)

spacing H (ft)

increase of induced velocity

slipstream contraction

1809 6

0 0928

25

2 4

1 05

0 985

Atmospheric Data

p (slug/ft3)

a (ft/s)

altitude (ft)

TFAC

T(°R)

0 002378

11167

0

1 0

5190

(@ SLS)

0 002378

1117

alternative formula for density with altitude

2 3769e-3»exp(-h/22965)

Figure 10: Inputs of the model

20

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Figure 11: Coaxial rotor blade element spreadsheet

21

Thrust/Powe 8

\^ T/nolosi

(ACT)up!oiji

CT

r (ib)

CQO

l^Qf/no less

( " C g J a p l o s s

CQ>

C Q

///> induced Total HP

• Calculations 0 9699

7.264E-03 4 728E-04

6 791E-03 10520

1.068E-04

4.440E-04 3 034E-05 4.136E-04 5.205E-04

699.0 879.6

Thrust/Power Calculations B

\*~* T/rto loss

(AC r)tfp(oss

cT T (lb)

^"QO

(^Ql/no loss

( " C Qi)tip loss

CQI

C Q

//F induced Total HP

0.9758 4.702E-03 9.929E-0S

4.603E-03 7130

8.466E-05 4.454E-04 9.558E-06 4.358E-04 5.205E-04

736.5 879.6

T total

equal torque? T ratio u/1

mi

17650

YES 0 596 1759.1

Figure 12: Coaxial spreadsheet results

The mechanism is the following.

The length of the blades is divided in 10 sections; r is the dimensionless variable

corresponding to the blade location. The pitch angle 0 is equal to the pitch distribution of the

blades (column 2) plus the increment 9imremmi (column 3) which represents a collective pitch

change. The Mach number M is used to calculate a compressibility correction to a = dCL

da

The corrected value is a = a in

Vi-iw2

An often-used correction from 2D airfoil to 3D wing data is aw = 6.28 AR

AR + 2 but in the

present case it was considered that aJD = 5.73 as a constant, also a commonly used

assumption.

Then the advance angle is given by the formula $ •• a -a 16-r

•r •l + ,/i + 32-

V a-a

Consequently the local induced velocity can be calculated with the relation tan^ = V, r-QR

22

sin <f>« <j> and cos^ « 1 then Vl = r • nR • </>

The local angle of attack is also obtained: a - (3 - 0

Finally differentials of aerodynamic coefficients can be calculated

A widely used polynomial curve-fit is used for Cd of the NACA 0012 airfoil:

Cd =0.0107-(0.151-a) + (1.72-a2)

dr

OCgQ

dr

dCQl

2

= r3-Cd-

dCT • m

'•a

a

~2

• r

dr dr

Calculation of u, A and B were described previously and are demonstrated in Figure 11.

Calculations at the lower rotor are only a little different. The local induced velocity is

calculated first in order to calculate the advance angle. They are both dependent on velocity

induced by upper rotor.

As shown by Figure 12, differentials of coefficients are then integrated along blades using the

trapezoidal rule and recorded into the table accounting for the Prandtl tip loss.

Since the study is about the hover case, the torque or power required for each rotor should be

identical. A cell shows if the condition is met by displaying "YES" or "NO". The "T total"

cell is the sum of thrusts provided by each rotor to make sure that the hover condition of

thrust equals weight is achieved. The Apache helicopter used as an example weighs 17560 lb.

Finally a cell shows the thrust ratio and another one the total power required.

23

3 RESULTS

3.1 Comparison to existing literature

3.1.1 Contribution of swirl in the wake

In Reference 3, Wayne Johnson states that "there is about a 1% increase in the total rotor

power required because of the swirl in the wake".

The current model consider a coaxial system based on the McDonnell Douglas AH-64

Apache, 22 foot rotor radius, spacing of 2.2 feet, a tip speed of 665.5 ft/s and the same linear

pitch distribution, to hover weight of 17650 lbs. This analytical model shows that the

consideration of swirl in the wake requires 1.4% increase of total power as referenced in

Table 1.

Table 1: Importance of swirl in the wake

model without

swirl with swirl

HP required to hover

1953.1 1980.5

increase

1.4%

There is a small degree of simplification in the calculation since the formula, as described

previously, does not take into consideration the blade pitch. However, this result matches

expectations from Reference 3 which describe an increase of about 1%.

3.1.2 A coaxial rotor system or two isolated rotors

Leishman [4] introduces a discussion based on Dingeldein 15] and Harrington [6] results

where he compares performance of a coaxial rotor with two isolated rotors. Leishman

mentions an increase in induced power for the coaxial case at equal torque compared to two

isolated rotors. This is given as 16% found experimentally and 22% theoretically.

24

However, not enough detailed dimensional data was found to verify this in the

aforementioned NACA Technical Notes to reproduce the results exactly with this analytical

model.

The current model shows for R = 24ft an increase of 34.6% in induced power corresponding

to an increase of 21.9% in total power. These numbers vary with parameters such as rotor

RPM and rotor diameter but are moderately higher than Leishman conclusion. We do not

have enough blade geometry details to ascertain why with certainty.

The discrepancy might be explained by several different reasons. The comparison between the

two cases may not be done in the same manner. Indeed, Dingeldein compares a coaxial

system where each rotor has the same solidity but not the same diameter. Moreover, his rotors

are in tandem configuration which implies that there is probably interaction between them. In

the current model all rotors are identical.

n U

ft

COE

h ty ixial

Two isolated

h V

Current Model N = 4 for each rotor

n

n vj

h Coaxial

Tandem

Dingeldein Model

N = 2 for each rotor

Figure 13: Coaxial and tandem configuration

25

Also it was assumed in the current model that the upper rotor is not influenced by the flow

induced by the lower rotor. However, if the upper rotor was considered to be in climb due to

inflow induced by the lower rotor, even at a small rate, its induced power would be reduced.

In another case, Harrington [6] does a study comparable to the current model and states that

"It appears that [...] the profile torque coefficient CQo at CT = 0 of the coaxial arrangement

was twice that measured separately for either of the single rotors". This means that there is

neither an increase nor a decrease in total profile torque by merging the two isolated rotors

into a coaxial configuration. In comparison, the current model shows that the profile torque

coefficient of the coaxial arrangement is twice that calculated for only one single rotor with a

variation of 4% depending on the detailed geometry of the model.

Therefore, it is concluded that the two studies are illustrating general trends that are very

similar.

A comparison was made for different rotor radii between a coaxial configuration and two

isolated ones and the difference in total power required relative to two isolated rotors is

compiled in the table 2. Again rotors are identical; blades are untapered with a linear pitch

distribution. There are four blades per rotor. Solidity is 0.0928 for each rotor.

Table 2: Difference between coaxial and isolated rotors

Increase required in total power for coaxial system R(ft)

20 24 28 32

percentage 26%

21.9% 38%

14.6%

It can be observed that the difference in total power required decreases as the rotor radius

increases. However, when R = 32ft, the increase required (for the coaxial system) in induced

power is 32.7% which shows that it stays nearly constant with the radius variation. The

following graphic, Figure 14, shows the difference of power required for different radius. It is

26

also noticeable that the total power required decreases as the rotor radius increases but it is to

ponder with structural analysis, increase in weight, specification for the room available, etc,

Although it is more efficient to have two isolated rotors, it requires more room than a coaxial

rotor which is therefore a major design choice.

ZSO0-

2000 -

1500 •

i 1000 -

500

0 3 5 10

HP required to hover vs. Rotor radius W=17650ib, H/RsO.1, tip speed-726ft&ec

* „ ^ - ^

38

<£

15 20 25

radius if! ft

« *

30

•

!

—•—forcoaxaal - for 2 single

•

35

Figure 14: Power required for coaxial or isolated rotors

3.1.3 Rotor spacing, pitch increment and thrust ratio

Rotor sparing

In his survey, Coleman [2] refers to an optimization study done by Andrew [7] in England.

This study reveals that "the greatest gains were made up to H/D = 0.05; thereafter, no

practical gains resulted with increasing separation distance".

The current model was tested for H/D = 0.05 and H/D = 0.1 having all other parameters

identical. Results do show that H/D = 0.05 produces better performance than the other.

However, no attempt was made to create a decision making tool on the spacing between

27

rotors though the analysis did confirm that the influence on the upper rotor by the lower rotor

is highly dependent on this spacing. Indeed, Coleman refers to results presented by

Nagashima [8] in those words: "Perhaps surprising is the extent to which the lower rotor

influences the upper rotor performance".

The current analysis was not intended to optimize rotor spacing, so all subsequent calculation

were then done for a ratio of H/D = 0.05

Pitch difference

Coleman, still referring to the research in Japan, writes that "they determined that

Ob* = eupp + l3° 8 a v e t h e b e s t performance for H/D = 0.105 (and 0, = 9U +1.5° for H/D =

0.316), so long as stall was not present". This statement refers to optimum performance but it

should be noted that it does not mean torque-balanced. Thus, some other means of directional

control would be required.

For the current model, each rotor has the blade pitch adjusted until both rotors require the

same torque. It is a requirement in the model that was chosen because counteracting torque is

the main purpose of a coaxial rotor system. It was found that the collective pitch increment

required is a function of different parameters such as tip speed and rotor radius. Results drawn

from calculation indicate that the increment angle varies between 1.15° and 1.3° for linearly

twisted blades. The following table show pitch difference (9,ower -9wptr) required for various

configuration for which the only differing parameters are shown.

Table 3: Pitch difference required to matain torque balance

configuration R=22ft, QR=665 ft/s R=22ft, OR=726 ft/s R=24ft, QR=665 ft/s R=24ft, £1R=726 ft/s

pitch difference (°) 1.31 1.28 1.19 1.164

28

This shows values close to Coleman's reference and strongly suggests that a torque balance

configuration is close to the optimum thrust performance as well.

Thrust ratio

Finally, each rotor provides a share of the overall thrust and all calculation made with this

T model show that the ratio stays very close to — = 0.6 . Again, this would probably

upper tower

change with a model including the influence of the lower rotor on the upper rotor. Those

results can be read in the appendices section.

3.2 Effect of pitch distribution

Pitch distribution of a blade corresponds to an important phase of rotor design. While the

twist of a blade is designed to give more performance or efficiency from the rotor, it also

brings complexity in the manufacturing which implies an increase in the cost.

In order to have an idea of the gain in performance, calculation were conducted for a coaxial

configuration with a radius of 22 ft, a tip speed of 665 ft/s and 4 blades with a chord of 1.75ft,

still similar to the Apache example.

Three different configuration were tested, an untwisted design, a linear twist design with

90 =15° and 0^, - -9° and finally a hand adjusted design with blade pitch at each radial

location modified to produce a constant induced velocity on the outer part of the rotor. This

last design happens to be close to a second order polynomial pitch distribution though there is

no specific reason for the result to be so.

Table 4 shows the difference in performance for each configuration and the results are

discussed later along with Table 5.

29

Table 4: Relation between performance and blade pitch distribution

Configuratiofi Untwisted

Linear Twist Custom Twist

HP required to hover 2068.8 1966.8 1938.7

The following graphs (Figures 35 to 20) show the pitch distribution and the induced velocity

for those three different cases.

3.2.1 Untwisted blade

There is no need to plot an untwisted pitch distribution. However it is important to notice that

the induced velocity, as show by figure 7, is not constant at all.

Figure 15; Velocity induced by rotors with untwisted blades

The effect of swirl velocity induced by upper rotor is observable at the inner part of the lower

rotor. As discussed earlier it does not have significant contribution to the overall performance.

30

3.2,2 Linearly twisted blade

Pitch distribution shown takes into consideration the difference in pitch between upper and

lower rotors required to obtain hover: Thrust = Weight = 176501b.

18

16 •

14

1 2 -

f 10

I J 6

2

t

0 0

-m

•M

—

0 2

»

pitch distribution

» *~

^ ^ ^ L ^ r * ^^^^_^^ ^ ^

0 4 0 8 OS r

| ~^~» pwich d st up -» pitch cifst low |

* ™

— — -

— _

1 Q 1 2

Figure 16: Linear twist distribution

induced velocity versus raolus location

| —«~- Vi upper ~ » ~ Vi lower j

Figure 17: Induced velocity for rotor with linear twist

Figure 17 shows that a linear twist distribution creates a more desirable, because it is closer to

constant, induced velocity distribution along the length of the rotor.

31

3.2.3 Enhanced blade twist

This graphic corresponds to the actual configuration or blades to hover with a radius of 22

feet and a tip speed of 665 feet per seconde.

pitch distribution

-pitch chst up —«—pitch dist low j

Figure 18: Enhanced twist

The pitch at the inner part of the blades is not modified since is has only little contribution to

the overall thrust.

45

40

35 -

30 -

I ?5

5 20 -

15

10 •

5

0 0

/ r 1 /

J*-''

0 2

^

Induced velocity versus radius location

„ _ „ „

„ — • w- — — m «- ~m

0 4 06 08 f

j __#__ v t upp&r •» Vi Sower j

-

_

—m

1 0

I

1 2

Figure 19: Induced velocity for enhanced twist

32

The modified twist allows obtaining a constant induced velocity distribution on the outer 60%

of each rotor. Last figure demonstrate the possibility to approximate this design by a quadratic

equation.

25 ,

20

15 -

!

10 -

s -

0 0

pitch distribution

^ O v . R? = 0 9994

y * 18 512xJ 41 054x * 28 692 ^ ^ ^ J s ^ RJ = 0 3993 ^ " ^5S»~»_

02 04 06 T

0 8

—

1 0 1 2

Figure 20: Mathematical approximation of enhanced twist

Table 5 indicates the increase of power required for each case, relatively to the enhanced

configuration.

Table 5: Increase of power required for various blade twist

Configuration Untwisted

Linear Twist Customed Twist

Increase in power required

6.71% 1.45%

. . .

Table 5 confirms that the blade pitch distribution has an impact on performance.

Nevertheless, this comparison also proves to be useful in a design process. Showing that the

increase in power required is not very large, when going from a custom twist to a linear twist,

could help a decision maker to choose whether it is worth increasing the cost of

manufacturing the blades with respect to the small gain in performance.

33

3.3 Comparison to a single rotor

One should remember that this comparison considers the main rotor only. Therefore, the

power required calculated does not take into consideration the power to drive the tail rotor in

the case of the single rotor. This is generally an additional 10 to 15% HP of the main rotor

horse power.

Another potential precision issue, as stated previously, is that the upper rotor in the coaxial

configuration was considered to be in free air whereas it is probably in climb in the stream

induced by lower rotor.

HP req to hover W=1?6S0 lb, tip speed = 726 Ws, H/R-0.1

35CQ

3000

2300 •

2000 -

1S00

1000 -

500 -

10 20 30 40

Radius (ft)

50 60

- Coaxial N=4 -> Single rotor N-4 ~*~ Coaxial N=2 - * - Single N=8

Figure 21: Power required versus rotor radius

This first graph compares the power required to hover as a function of the rotor radius for four

different types of rotor configuration. Those are a coaxial system with four blades per rotor

34

and another one with two blades per rotor. Blades chord remains the same, as well as pitch

distribution. Also, two single rotors were analysed, one with eight blades and the other with

four. Since other parameters are the same, the solidities are comparable: a total of 8 blades in

on case and 4 blades in the other case.

2500 •

2000-

| 1500 -

1000 -

500

.. .

0 100

Hp required vs tip speed to hover, W=17650lb, H/R*0.1, R=24ft

---

* \ ^ ^ * *-~~~~-~-~^_2^

"~~-~-—m —*'

200 300 400 500 600 700 800

tip speeil in ft/sec

| * single N=4 - " - coax ia l N=4 - * - coax ia l N=2 -H-s ingle N=8 |

- -

«•

900

...

1000

Figure 22: Power required versus tip speed

Those two graphs show that the solidity of a rotor system seems to be a major factor in

performance, as theory would lead us to expect. Indeed, a single rotor provides results quite

close to the coaxial rotors at equivalent solidity. The reason for the difference between each

model may be the the lower rotor influence on the upper rotor, as stated above figure 21.

However, if, in the coaxial system, the upper rotor was considered to be in climb, the induced

torque would be less and the system would consequently be more efficient. This last

35

statement would give results more consistent with paragraph 3.1.2 of the present study and

what Leishman wrote.

From a designer's point of view, Figures 21 and 22 offer interesting value. Solidity of the

rotor is apparently a key for performance trend. Other parameters such as vibration and

weight would come into the equation. Nevertheless, being at the same solidity or not, it

appears that a coaxial rotor configuration provides a better solution for performance.

Analysis

The two figures (21 & 22) show that having a coaxial system with four blade rotors allows us

to lower the power required in hover by lowering the speed of rotation. This action has the

downside effect of requiring a greater pitch angle and therefore a bigger angle of attack. A

local blade element would stall and not offer the calculated performance for large angles,

above approximately 12 degrees.

Fortunately, the optimum velocity found theoretically never requires a large angle of attack

and therefore lies into the feasibility range. For example, a 24 foot radius rotor with a tip

speed of 400 ft/s and 4 blades in a coaxial configuration requires a maximum local angle of

attack of 13.86 degrees at 30% of radius. However this configuration asks for more power

than the similar one with a tip speed of 500 ft/s which implies a maximum angle of attack of

10.17 degrees at 30% again. This angle falls into the linear part of the CL versus AOA curve

of atypical blade airfoil.

If we consider another design parameter, the rotor radius, it is noticeable that a single rotor

provides better performance for only very large radius. It is not likely that such large rotor

would be built for manufacturing and weight reasons. Consequently the coaxial configuration

obtains more credit again.

36

Best performance scenario

For a better understanding of the results, calculation are based on an actual helicopter, the

AH-64 Apache (Appendices) developed by McDonnell Douglas now Boeing.

The Apache currently has a single rotor and available data are that the radius is 24 ft, each

four blades are twisted as follow, 90 = 15° and 0Msl = -9°, chord is 1.75 ft (this is kept for all

models) and tip speed is 726 ft/s.

The single rotor model calculates a power required for the main rotor of the Apache of 1903

horse power (Appendices) in order to hover. This number would be lower if the radius of the

rotor was larger but it can be assumed that other parameters led to this design choice. OrTe of

them could be the room taken by the rotor. Also, the longer the radius, the more the blades

would bend or flap in flight which can affect performance, which is not considered in this

model. On the other hand more blades could have been added to increase solidity as the

graphs show might be beneficial.

Going from the actual Apache design, what coaxial rotor design could improve performance?

Considering two four bladed rotors with a spacing of 10% of the radius or 5% of the diameter

and keeping same parameters for blade chord and CLa, other parameters such as blade pitch

distribution, rotor radius and tip speed can be modified to look for better performance.

As previous figures demonstrate, any decrease in rotor radius will increase power required. A

first analysis shows that a coaxial system, as described above with a linear twist and a 24 ft

radius but a tip speed of 535 ft/s (Figure 23), would require only 1764.6 HP (Figure 24).

37

Name of helicopter | modified Apache This spreadsheet is a model to calculate performance <

lyuumeu npaene j date performance of eoawal rotor considering that only the lower rotor depends on the other

mx\ velocity induced By the upper rotor is included

Characteristic* of helicopt W (lb) 17650 Engine MP

7650 2784

Following data are characteristics of the rotor

Coaxial Rotors Design

RQh

%paung(HR)

N

C (fl>

fill (ft/s)

24

0 I

4

1 75

535

upper rolor

S» (deg!

K Sdegj

Of (per rad )

15

5 73

hmer rolor

0«(deg) #,» (deg! a (pur rod)

IS -9

5 13

Calculated Data

A i f f )

a

a (rad's)

young H (ft)

increase of induced ve

slipstream contraction

ooty

1809 6

0 0928

22 29166667

24

1 05

0 985

P (slug/ft')

a (ft/s)

Atmosphenc Data

0002378

11167

altitude (ft)

TFAC

T(°R)

0

10

5190

(@ SLS)

0 002378

1117

alternative formula for density with altitude

2 3769e-3*exp(-lv22965)

Figure 23: Design parameters of modified Apache with linearly twisted blades

Thrust'Power Calculations B

\*- T/no loss

(ACT) t Jpioss

cT T(\b)

CQO

l*-*Ql/no loss

("C*Qi)t iploss

C(Ji

Co HP induced Total HP

0.9659 9.319E-03 8.149E-04

8.504E-03 10474

1.408E-04

6.607E-04

6.511E-05 5.956E-04

7.364E-04 713.6 882,3

Thrust/Power Calculations B

\ \ l i n o lof»s

\IAK, xJitp|0 ! > s

CT

7" (lb)

(- QO

v^Qi /no loss

( " C Q,)„p loss

cQ! Co HP induced Total HP

0.9725 6.048E-03 2.220E-04

5.826E-03 7176

9.205E-05

6.715E-04

2.712E-05 6.444E-04

7364E-04 772.0 882.3

T total

equal torque? T ratio u/1

17650

YES 0.593 1764.6

Figure 24: Performance of coaxial system with linear twist

While modifying these parameters, the designer should pay attention to the local angle of

attack which increases as rotor radius and tip speed decreases.

From another designer's point of view, the power required can be given less importance than

the rotor radius. If the rotor radius is set to 21.5 ft and the tip speed to 575 ft/s the model gives

a power required of 1946 HP. The maximum local angle of attack would be 6.24° which still

38

leaves some room for the collective pitch. This configuration saves 5 feet or 1.524 meters on

the rotor diameter.

For a radius of 22 ft and a tip speed of 550 ft/s the power required would be 1903 HP like for

the single rotor. This model saves 4 feet on the diameter.

Changing the pitch distribution to 9lvm =-11° (keeping the same pitch angle at blade root)

for this configuration shows that less power would required to hover, 1890 HP, however the

difference is almost negligible.

Using a hand modified blade pitch distribution, a tip speed of 525 ft/s and a rotor radius equal

to 24 ft, the coaxial rotor model calculates a power required of 1725 HP, Figure 25 'and

appendices, with a maximum local angle of attack of 8,68°. This is a significant power

reduction and illustrates that this blade element model has promise as a rotor design tool.

Thrust/Power Calculations B

v*"- T/no ioss

(AC1) l ¥ b s s

cT r ( i b )

CQO

^ Q l / f t O i Q S S

("C/Qi)tiploss

CQ,

Co HP induced Total HP

0.9655

9.528E-03

7.329E-04

8.796E-03

10432

1.461E-04

6.708E-04

5.490E-05

6.159E-04

7.620E-04

697.3 862.7

Thrust/Power Calculations B

l**" T/no foss

(ACT) t i p k j s s

C T

r(ib) CQO

I'-Qtino loss

( " C Q , ) t i p | o s s

CQ,

C Q

HP induced Total HP

0.9720

6.283E-03

1.968E-04

6.086E-03

7218

9.451E-05

6.899E-04

2.235E-05

6.675E-04

7.620E-04

755.7 862.7

T total

equal torque?

T ratio u/I

ffJvSilW!

17650

YES 0.591 1725.4

Figure 25: Optimum performance of coaxial configuration

CT versus CQ graph

It appears that these graphs are often used to illustrate rotor performance. However, this tool

was not used in the present study because of a lack of information to compare to. One such

graph, corresponding to the optimal configuration, can be found in the appendices.

39

Mutual influence between rotors

The following result seems to contradict the assumption that the influence created on the

upper rotor by the lower one would reduce the power required. If the upper rotor is considered

to be fully influenced by the streamtube induced by the lower rotor then the power required to

hover appears to be higher than presented results in general, Figure 26,

Thrust/Power Calculations B

v> Tmo loss

(ACT) t i f>ioss

c. 7-(lb)

C oo

l*-Q>)no loss

("C*Qt}Uptess

CQ,

C Q

HP induced Total HP

0.9682

8 096E-03

6.499E-04

7 446E-03

9000

1 I81E-04

7 377E-04

6 611E-G5

6.716E-04

7 898E-04

782,3 919,9

Thrust/Power Calculations B

l*- 1 /no !obs

(ACT) t ipioi)S

c. nu>) ^ Oo

V^-Qi/notoss

V"CQ,)tlpt<J55

cQl C Q

HP induced Total HP

0.9695

7.455E-03

2.990E-04

7.156E-03

8650

1.072E-04

7 141E-04

3.152E-05

6.826E-04

7.898E-04

795.1 919.9

T total 17650

equal torque? T ratio u/1

YES

0.510 H i 1839 8

Figure 26: Power required if lower rotor influences upper one

This calculation set was run in the same conditions as the optimum configuration except that

the upper rotor was considered "climbing" in the flow induced by the lower rotor. The only

purpose of this exercise was to obtain an approximate value of the effect of rotor

interdependencies. Therefore, it was simply considered that the streamtube of the airflow

induced by a rotor follows the same trend above and below the rotor. In other words it is

governed by the same equation for its radius.

The significance of this discussion is that if the effect of the lower rotor on the upper rotor

would bring the coaxial model values closer to the single rotor ones. This would lead us to

conclude that a coaxial rotors system can be modelled as a single rotor of equivalent solidity.

Nevertheless, it should be remembered that for a single rotor 10% to 15% of power would be

required for the tail rotor. The study of the extent of this effect could be a topic for another

research,

40

4 SPECIAL CASE OF DUCTED ROTOR

This investigation was prompted in part by request to have ERAU assist with the design of a

ducted, coaxial rotor vehicle with a payload-carrying compartment in the center of the rotor.

This project did not ever actually take place but the interesting question of how to analyse the

rotor remained.

In order to have a definite answer three different models, Figure 27, were compared: the

previous coaxial model and two ducted coaxial models with a cabin in the middle. The first

duct is skewed in order to match the shape of a rotor downwash stream tube and the second

duct has a constant inner radius.

^ Cabin \

I i

r i

coaxial Coaxial in straight duct Coaxial in skewed duct

Figure 27: Different design for duct tests

Blade tips would be very close to the outer part of the duct and consequently there is no tip

loss. Moreover, the coefficient C!a is different since the aspect ratio of blades is considered as

infinite. It becomes Cla = 2TT = 6.28/radian

Other characteristics that were used for these calculations are: the chord was set to be one

foot, the outer rotor diameter is 19 ft and the cabin diameter is 5 ft. Furthermore, blades were

assumed to be untwisted. The weight to hover is 30001b. The power available is unknown and

is the result of the investigation.

I 1]

41

4.1 Simple coaxial configuration

Thrust/Power Calculations B

\S* T/no loss

(ACrXiploss

C T

r (lb)

C Q O

l^Qilnoioss

( "CpJupjoss

cQ, C Q

HP induced Total HP

0.9657

9.390E-03

1.085E-03

8 305E-03

1819 1.437E-04

7.031E-04

1.027E-04

6.003E-04

7.441E-04

136.3 168.9

Thrust/Power Calculations B

l*~ T/no loss

(ACT) t i p i o s s

CT

r (ib)

C Q O

l^Qi'm) loss

siiC Qi/iiploss

CQ>

C Q

HP induced Total HP

0.9734

5.645E-03

2.557E-04

5.389E-03

1181

1.198E-Q4

6.602E-04

3.591E-05

6.243E-04

7.441E-04

141,7 168.9

T total

equal torque?

T ratio u/1

3000

YES 0.606 337.8

Figure 28: Simple coaxial configuration

This model is the one discussed previously. It is considering no influence of the lower rotor

on the upper one.

4.2 In skewed-duct coaxial configuration

Thrust/Power Calculations B

l'-" T /no !oss

(&C*j) t lpj0SS

cT T(\b)

C Q O

S*"QI 'no loss

( « C Qi/tiplow

C Q ,

c0 HP induced Total HP

0.9687 7.820E-03

0.000E+00

7.820E-03 1594

1.U6E-04

7 339E-04

0.000E+00

7 339E-04

g 454E-G4

155.1 178.6

Thrust/Power Calculations B

v^- 1 /no loss

( A C r ) t i p j 0 S S

C T

T(\b)

C Q O

V^-Qi/no loss

("CQ,) t l p | Q S S

C Q ,

C Q

HP induced Total HP

0 9697 7.338E-03

0 000E+00

7.338E-03 1405

1.088E-04

7.367E-04

0 000E+00

7.367E-04

8.455E-04 144.0 165.3

I

T total

equal torque0

T ratio u/1

H

3000

YES 0.532 343.9

Figure 29: In skewed-duct coaxial configuration

This configuration assumes a solid duct has the shape of the downwash streamtube of the

upper rotor. Then, the area of the duct dictates the velocity and pressure distribution between

rotors, In this case there is no uncertainty about the influence of the lower rotor on the upper

rotor as in the model described before.

42

4.3 In straight-duct coaxial configuration

Thrust/Power Calculations B

l*~ T/no loss

(ACT)tfp|0Si|

cT 7* (lb) CQO

i*"Q)ino loss

\"CQ,)„P |0SS

^Q,

CQ

HP induced Total HP

0 9743 5.266E-03 4.884E-04

5.266E-03 1500

1.282E-04 4.221 E-04

4.922E-05 4,221 E-04

5.502E-04 142.1 185.2

Thrust/Powe B

V*** T/no loss

\IM~ f/tsploss

C j

T (lb)

CQO

'^Qi'noloss

("CQ,),lptoss

CQ,

CQ

HP induced Total HP

r Calculations 0.9743

5.264E-03 4.885E-04

5.264E-03 1500

1.282E-04

4.220E-04 4.924E-05 4.220E-04

5.502E-04 142.1 185.2

T total 3000

equal torque? T ratio u/1

YES 0.5001

?Gfe«itItH 370,5

Figure 30: Straight-duct coaxial system

Here again the area of the duct determines the velocity and pressure distribution along duct,

As the cross section area is constant, the velocity of the air between the two rotors is constant.

There may be a slight variation because of the boundary layer, like on the walls of a wind

tunnel, but it is neglected in the current model. Therefore, it is not surprising to observe that

each rotor produces the same thrust. The small difference if noticeable is due to the swirl

velocity.

It would be premature to conclude that an in-duct coaxial rotor system has lower performance

than a coaxial system in free air, Indeed, flow around the lip of the inlet duct creates surface

pressure which is below freestream static pressure. Consequently, this creates a pressure force

on the inlet duct which has a component in the same direction as the rotor thrust and because

the rotor is in hover there is no external aerodynamic drag on the duct itself. It is not included

in this analysis and might change the conclusions. A study of the importance of this extra

thrust would be interesting.

43

5 CONCLUSION

5.1 Concluding remarks

A model was built using Excel to evaluate performance of counter-rotating coaxial rotors.

This model is based on blade element theory combined with momentum theory and, therefore,

can be used as a blade design tool. It was demonstrated that untwisted blades, like in the

single rotor case, are less efficient than linearly twisted blades. Hand modified blade pitch

distribution can be even more efficient but choosing to do so might complicate the blade

manufacturing process.

The following conclusions can be made from this study:

• Coaxial rotor system is more efficient than a single rotor if the number of blade

per rotor is the same.

• At identical power, the diameter of the rotor can be reduced for the coaxial

configuration, by 10% in the Apache test case.

• The optimum rotational velocity is less for the coaxial system and this is due to

the increase of solidity.

• At same overall solidity, counter-rotating coaxial rotors probably require less

power than a single rotor, but this trend reverses if blade radius gets larger.

• A rotor spacing of 5% of diameter was found to provide better performance

than a spacing of 10%.

• Swirl velocity has little influence on performance.

It was also noticed that if a coaxial system is in a duct, the power required increases slightly

but one should keep in mind that the inlet lip of the duct itself would provide thrust which is

not accounted for in this analytical model.

44

5.2 Recommendation for future work

Some other research could contribute to improving this model. First, it would be interesting to

have a better description of the flow between rotors and to know the importance of the

influence of the lower rotor on the upper rotor. This might help gain precision in the model.

Also, it would be beneficial to have a better knowledge of the streamtube shape between the

two rotors and to create a model that allows a larger rotor spacing variation. The inner part of

the lower rotor would be influenced by the flow induced by the upper rotor, the outer part of

the lower rotor would be in free air and finally rules would have to be set for the part hit by

vortices. Work has been done already in those areas and an additional reference [9] is added

toward that purpose. However, the purpose of the current model is to stay a simple

preliminary design tool, so these details were intentionally not pursued at this time.

45

REFERENCES

1. James R. Chiles, "The God Machine" "From Boomerangs to Black Hawks - The story of the helicopter", Bantam Books, November 2007

2. Colin P. Coleman, "A Survey of Theoretical and Experimental Coaxial Rotor Aerodynamic Research", NASA Technical Paper 3675, March 1997

3. Wayne Johnson "Helicopter Theory", Dover Publications, Inc, 1994

4. J. Gordon Leishman, "Principles of Helicopter Aerodynamics", Cambridge University Press, second edition, 2006

5. Richard C. Dingeldein, "Wind-Tunnel Studies of the Performance of Multirotor Configurations", NACA Technical Note 3236, 1954

6. Robert D. Harrington, "Full-Scale-Tunnel Investigation of the Static-Thrust Performance of a Coaxial Helicopter Rotor", NACA Technical Note 2318, 1951

7. M.J. Andrew, "Coaxial rotor aerodynamics", Ph. D. Thesis, Southampton University, England, circa 1981

8. T. Nagashima and K. Nakanishi, "Optimum performance and wake geometry of a coaxial rotor in hover", Proceedings of the 7th European Rotorcraft and Powered Lift Forum, Paper No.41, Sept 1981

9. Yihua Cao, "A new method for predicting rotor wake geometries and downwash velocity field", Associate Professor at the Institute of Aircraft Design, Beijing University of Aeronautics and Astronautics, China, MCB University Press, 1999

46

APPENDICES

Here are shown different result obtained during the study:

- Apache single rotor model

- AH-64 Apache

- Coaxial model without swirl (in two rotor diameter and rotational velocity

configurations)

- Coaxial model including swirl (two configurations also)

- Coaxial model with untwisted blades

Optimum Coaxial configuration (with constraint that R less or equal to 24ft)

- Coaxial model with a rotor spacing of 10% of diameter instead of 5%

CT versus CQ graphic for optimum solution

- Code of macro for automatic calculation of pitch increment to meet requirements

to hover at equal torque.

Apache Single rotor model

Helicopter

W (lb) Engine HP N 7?(ft) C(ft) 0o(deg)

0,w (deg) a (per rad) QR (ft/s)

Data

17650 2784

4 24

1.75 15

-9 5.73 726

Calculated Helicopter Data

A (ft2) 1809.6 a 0.0928 0„P(deg) 6 D (rad/s) 30.25

Atmospheric Data (@ SLS)

p (slug/ft3) 0.002378

a (ft/s) 1117

47

r

00 01 02 03 04 05 06 07 08 09 10

Baseline 6(ie%)

150 14 1 132 123 114 10 5 96 87 78 69 60

Thrust/Power Calculations B

CC-rUte* (ACjXjpiosj

CT

nib) CQO

VM)t)no bss

(^CQ,)opta

CQ,

CQ

HP induced Total HP Est Hp hover Haver % pow

0 9674 8 500F-03

7 179E-04

7 782E-03

17650 1 148E-04

5 761 E-04

5 508E-05

5 210E-04

6 358E-04 H59 8

iWMM 1903 6 68 4

Incremental 0 (deg)

1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986

0(rad)

0 2965 0 2807 0 2650 0 2493 0 2336 0 2179 0 2022 0 1865 0 1708 0 1551 0 1394

W 0 000 0 065 0 130 0 195 0 260 0 325 0 390 0 4S5 0 520 0 585 0 650

a (perrad)

5 73 5 74 5 78 5 84 5 93 6 06 6 22 6 43 6 71 7 06 7 54

Note Incremental 0 is 1 9856 degrees to produce hover T = W = 17650 lb

FWbwer conditoiE

4 (rad)

UNDEF 02128 0 1744 0 1499 0 1322 0 1183 0 1070 0 0975 0 0891 00817 0 0750

v {ft s)

0 15 45 25 32 32 65 38 38 42 96 46 62 49 53 5177 5341 54 47

a (rad)

... 00680 0 0907 00994 0 1015 00996 00952 00890 00817 0 0733 0O643

a (deg) . „

3 89 5 20 5 70 5 81 5 71 5 45 5 10 4 68 4 20 3 69

cd ...

0 0084 00111 0 0127 00131 0 0127 00119 0 0109 0 0098 0 0089 0 0081

dCyfdr

0 1811E-04 9 729E-04 2 427E-03 4 472E-03 7 002E-03 9 897E-03 I 303E-02 1 627E-02 1 948E-02 2 252E-02

dC^/ir 0

3 891E-07 4 140E-06 1591E-05 3 887L-05 7 380E-05 1 194E-04 1 734E-04 2 338E-04 3 004E-04 3 762E-04

dr<j/df

0 3 854E-06 3 393E-05 1 091 E-04 2 364E-04 4 143E-04 6 356E-04 8 890E-04 1 160E-03 1 433E-03 1 690E-03

oo

AH-64 Apache

49

Coaxial model without swirl (configuration 1)

Name of helicopter Thu> spreadsheet is a model to calculate performance of coaxial rolor considering that only the lower rotor depends on the other swirl velocity induced by the upper rolor is included

Characteristics of helicopter W (lb) 17650 bngme HI' 2784

following data are characteristics of the rotor

Coaxial Rotors Design

R On 24

spacing ttl K) 0 1

A' 4

C (ft) 1 75

OH (8/s) 726

upper rolor

0u «leg>

fl„ (deg)

a (per rati)

15

-9

5 73

lover rolor

0„(deg) K (deg) a (per rad)

15 -9

5 73

Calculated Data

A (ft')

u

n (rad/s)

spacing H (ft)

inci ease ot induced velocity

slipstream contraction

1809 6

0 0928

30 25

24

I 05

0 985

Atmospheric Data

P (slug/ft')

a (fi's)

0002378

11167

altitude (fi)

TFAC

T(°R)

0

10

5190

( # SLS)

0 002378

1117

alternative formula for density with altitude

2 3769e-3*exp{-h/22%5)

Thrust/Power Calculations B

\S- T/no loss

(ACjXjpioss

C T

7" (lb)

CQO

'^"QI'no loss

("CQittiplosi

C0, CQ

HP induced Total HP

0.9752

4.920E-03

2.653E-04

4.654E-03

10556

8.909E-05

2.504E-04

1.382E-05

2.366E-04

3.257E-04

708.3 975.0

Thrust/Power Calculations B

(C T)IK> loss

(ACT)llploss

C T

7" (lb)

CQO

''"QiJnoloss

(^CQ,),IR |05!J

CQ,

C Q

HP induced Total HP

0.9801

3.167E-03

3.899E-05

3.128E-Q3

7094

8.506E-05

2.437E-04

3.119E-06

2.406E-04

3.257E-04

720.4 975.0

T total

equal torque?

T ratio u/1

17650

YES

0.598 1950.1

50

t o

y

T5

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Coaxial model with swirl (configuration 1)

Maine of helicopter This spreadsheet is a model t swirl velocity induced by the

Characteristics oi helicopter If (lb) 17650 Lngine HP 2784

modified Apache | o calculate performance of eoasual rotor considering thai only the lower rotor depends on the other upper rotor is included

Following data are characteristics of the rotor

Coasia) Rotors Design

R (fl) 24

spacing (H li) 0 1

.V 4

C(ft) 175

OR (ft's) 726

upper rolor

0, !deg) 15

9,»(degi -9

a (per rad) 5 73

kmer mlar

So(deg} IS <?,» (deg) -9 « (per rad ) 5 73

Calculated Data

A(ft!)

cr

O (rad/s)

spacing H (f()

increase of induced velocit)

slipstream contraction

18096

0 0928

30 25

24

I 05

0 985

Atmospheric Data

p {slug/ft') 0 002378

a (ft's) 1116 7

altitude (ft) 0

TFAC 1 0

TCR) 519 0

alternative formula for density wit

2 3769e-3*exp(-lt/2296S)

{& SLS)

0 002378

1117

h altitude

Thrust/Power Calculations B

' *~ T /no loss

(ACy/ap io^

cT Tito)

CQO

(^-Qi'noloss

( t i t . QtJtiptoss

C Q ,

CQ

HP induced Total HP

0 9751 4 964E-03 2 699E-04 4.694E-03

10647 8 912E-05 2.539E-04

1.416E-05 2.397E-04

3.288E-04 717.6 984 4

Thrust/Powe B

\S~ T/noioss

(ar T ) t i p 3 o s s

CT

7" (lb) C Q0

s^-Qi/no loss

(ACQ,) t ] p j 0 s s

^ Q i

CQ

HP induced Total HP

r Calculations 0.9802

3 127E-03 3.889E-05 3 088E-03

7004 8 508E-05 2.469E-04

3 160E-06 2 437E-04 3.28SE-04

729.7 984.4

T total 17650

equal torque? T ratio u/1

YES 0 603

il§ 1968.9

This result shows only a 0.96% difference induced by the swirl velocity.

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53

Coaxial model without swirl (configuration)

Name of helicopter This spreadsheet is a model to calculate performance of coatial rotor considering that only the lower rotor depends on the other swirl velocity induced by the upper rolor is included

Characteristics of helicopter W (lb) 17650 Engine HP 2784

Following data are characteristics of the rotor

Coaxial Rotors Design

Rfft)

spacing (H R)

N

C(fl)

OR (ft/s) ®mt

0 1

4

1 75

JEEli

«,»(deg)

a (per red )

15

-9

5 73

lower rotor

0a (deg)

a (per rad)

15

•9

5 73

Calculated Data

A(ft!)

a

Q (rad/s)

spacing H (ft)

increase of induced ve

slipstream contraction

ocity

1520 5

0 1013

30 13636364

22

105

0.985

Atmospheric Data

P (slug/ft )

a (ft/s)

0 002378

11167

altitude (ft)

TFAC

T<°R)

0

1 0

5190

(@ SLS)

0 002378

1117

alternative formula for density with altitude

2 3769e-3'exp{-h/22965)

Thrust/Power Calculations B

V*-" T/no loss

(AC T ) E j p ! o s 3

cT T(\b)

C Q O

NS-QI'IIOIOSS

("CQi)tiplos»

CQ,

cQ HP induced Total HP

0.9702 7.117E-03 5.119E-04

6.605E-03 10498

1.046E-04 4.386E-04

3.427E-05 4.044E-04

5.089E-04 774.8 975.1

Thrust/Power Calculations B

***" 7 zoo loss

(AC fJttploM

cT 7" (lb) CQO

sS-Qi/noloss

' " C Q,}t|p)oss

(• Qi

CQ HP induced Total HP

0.9760 4.613E-03 1.131E-04

4.500E-03 7152

8.939E-05 4.310E-04

1.148E-0S 4.195E-04 5.089E-G4

803.8 975.1

T total

equal torque? T ratio u/1

17650

YES 0.595 1950.1

54

Coaxial model with swirl (configuration 2)

Nameofhehcopter modified Apache This spreadsheet is a model to calculate performance ol coaxial rotor considering that only the lower rotor depends on the other swirl velocity induced by the upper rotor is included

Charaetenslics of helicopter JHIb) I76S0 Engine HP 2784

Following data are characteristics of the rotor

Coaxial Rotors Design

Rffn fS-ikl spacing (H P.) 0 1

N 4

C(fl) ^ 1 7 5

OR (ft/s) ^ S ^ M N

Ts

, 'ff? r rolor 9 (deg)

0,» (deg)

a (per rad )

15

-9

5 73

tower rotor

«Mdeg> 15

0» (deg) -9 a (per rad) 5 73

Calculated Data

AW)

CT a (rad/s)

spacing H (ft)

increase of induced velocity

slipstream contraction

1520 5

0 1013

29 72727273

22

1 05

0 985

Atmospheric Data

P (slug'ftl

a (ft/s)

0 002378

11167

altitude (ft)

TFAC

TCR)

0

1 0

5190

(0, SLS)

0 002378

1117

alternative formula for density with altitude

2 TO9e-3»e\p(-h/22965)

Thrust/Power Calculations B

l*~ T/no loss

(AC T) t ip|o s s

CT

7" (lb) CQO

s*~Qf/no ioss

("Couple*, cQl

HP induced Total HP

0.9696 7.404E-03 5.485E-04

6.855E-03 10602

1.071 E-04 4 658E-04

3.770E-05 4.281 E-04 5 351E-04

787.2 984.1

Thrust/Power Calculations B

I*- T/no loss

(AC x)up to$4

CT

7" (lb) CQO

H--Q])noIoss

("CQ,)„P|OSS

CQ,

CQ HP induced Total HP

0,9758 4.680E-03 1 224E-04

4.557E-03 7048

8.957E-05 4 585E-04

1.289E-05 4.456E-04 5.351E-04

819.4 984.1

T total

equal torque? T ratio u/1

! % « » $

17650

YES 0 60! 1968.2

55

Coaxial System with untwisted blades

Name of helicopter This spreadsheet is a model to calculate performance ot coaxtal rotor considering that only the lower rotor depends on the other swul velocity induced by the upper rotor is included

Characteristics of helicopter If (lb) hngme HP

17650 27S4

Following data are characteristics of the rotor

Coaxial Rotors Design

R (ft) 22

spacing (H R) 0 I

V 4

C (/I) 1 75

OR (ft/s) 654

upper rolor am e„(deg) 10

0» (deg) 0

a (per rod) 5 73

losier rotor «Mdeg) 10

S» (deg) 0 a lper rad ) ,,,573

Calculated Data

AifF)

CT O (rad/s)

spacing H (fl)

increase of induced ve

slipstream contraction

ociry

1520 5

0 10)3

29 72727273

22

1 05

0 985

Atmospheric Data

P (slug/ft')

a (ft/s)

0 002378

11167

altitude (ft)

TFAC

im

0

1 0

5190

m SLS) 0 002378

1117

alternative formula for density with altitude

2 3769e-3*exp(-h/22965)

Thrust/Power Calculations B

I *- T /no loss

{ACT)tip|0SS

C *|*

7" (lb)

CQO

I'-'Qt'noloss

("CQ,) I 1 P |0 S S

CQ,

C Q

HP induced Total HP

0.9690

7.684E-03

8.118E-04

6.872E-03

10628

1.118E-04

5.205E-04

6.959E-05

4.509E-Q4

5.627E-04

829.2 1034.8

Thrust/Power Calculations B

\^~- T/no loss

{ACT)tip!oss

CT

r (lb) CQO

s^Qi/no loss

( " C Q,)„p |0Ss

CQ,

CQ

HP induced Total HP

0.9757

4.709E-03

1.687E-04

4.541E-03 7022

9.039E-05

4.937E-04

2.138E-05

4.723E-04

5.627E-04

868.6 1034.8

T total

equal torque?

T ratio u/1

ppailp

17650

YES 0.602

2069.7

56

Coaxial optimum solution

Name of helicopter This spreadsheet is a model to calculate performance of coaxtal rotor considering that only the lower rotor depends on the other swirl velocity induced by Ihe upper rolor is included

Characteristics of helicopter tf (lb) / ngim HP

17650 2784

Following data are characteristics of the rotor

Coaxial Rotors Design

«(ft) 24

spacing (HP) 0 1

N 4

C (ft) 1 75

OR (ft s) 525

JMTSL rotor

6„(deg)

e» |deg)

a (per rad)

15

-9

5 73

Itmer rolor

<J„ideg)

8m jdeg)

a (per rad)

15

•9

5 73

Calculated Data

A (ft1)

cr

O (rad/s)

spacing H (ft)

increase of induced velocity

slipstream conti action

1809 6

0 0928

21 875

24

1 0$

0 985

Atmospheric Data

p {slug'ft )

a (S/s)

0 002378

11167

altitude (ft)

TFAC

T (°R)

0

1 0

5190

( 0 SLS)

0 002378

1117

alternative formula for density with altitude

2 3769e-3*expfh/22965)

Thrust/Power Calculations B

l*- T/no loss

(ACT) l i p ! o s 3

C r

T(\b)

CQO

'^"Qi/uoloss

( a C Q,)np|0SS

CQ,

cQ HP induced Total HP

0.9655

9.528E-03

7.329E-04

8 796E-03

10432

1461 E-04

6.708E-04

5.490E-05

6.159E-04

7.620E-04

697.3 862.7

Thrust/Power Calculations B

\S- T/no loss

(ACT) t ip i03S

C r

n»>) CQO

N*~Qi/no ioss

( " C Q , ) , I P | 0 S S

CQ,

CQ

HP induced Total HP

0.9720

6.283E-03

1.968E-04

6 086E-03

7218

9.451E-05

6.899E-04

2 235E-05

6.675E-04

7.620E-04

755.7 862.7

T total

equal torque?

T ratio u/1

^Itii lR

17650

YES 0.591 1725.4

57

Upper Rotor

r

00 0 1

02 03 04 0 5 06 07 0 8 09 10

Lower Rotor

r

0 0 0985

0 197

0 2955

0 394

0 4925

0 591

0 6895

0 788

0 8865

0 985

Baseline

8 (deg)

18 18 18

16 5

14 9

13 10 95

935 78 66 56

Baseline

0 (deg)

19 19 19 19

16 5

139 11 5

9 65

8 15

68 57

incremental

& (deg)

2.7258

2 7258

2 7258

2 7258

2 7258

2 7258

2 7258

2 7258

2 7258

2 7258

2 7258

tntremenlal

8 (deg)

3.8473 3 8473 3 8473 3 8473 3 8473 3 8473 3 8473 3 8473 3 8473 3 8473 3 8473

0 (rad)

0 3617 0 3617 0 3617 0 3356 0 3076 0 2745 0 2387 0 2108 0 1837 0 1628 0 1453

8 (rad)

0 3988 0 3988 0 3988 0 3988 0 3551 0 3097 02679 02356 0 2094 0 1858 0 1666

M

OOOO

0 047

0 094

0 141

6 188

0 235

0 282

0 329

0 376

0 423

0 470

hi

0 0 076

0112

0 152

0 196

0240 0 285 0 330 0 376 0421 0 467

a (perrad)

5 73 5 74 5 76 5 79 5 83 5 90 5 97 6 07 6 18 632 649

a (per rad)

5 73 5 75 5 77 5 80 5 84 5 90 5 98 607 6 18 6 32 6 48

estimated T ratio

vh (ft/s)

c* (rad)

UNDEF 0 2601 0 2186 0 1841 0 1587 0 1371 0 1181 0 1038 0 0911 00S14 0 0735

Mft's) 0 00 13 16 13 60 16 86 17 72 18 08 18 22 18 36 18 75 18 69 18 67

0 591

34 81

v,(Ws)

0 13 66 22 96 29 00 33 34 35 99 37 19 38 13 38 28 38 48 38 61

4 (rad)

UNDEF 0 532 0 365 0 305 0 255 0216 0 185 0 161 0 142 0 127 0115

used for calculation of swirl velocity

a (rad)

._ 0 1016 0 1431 0 1514 0 1489 0 1374 0 1206 0 1070 0 0926 0 0813 0 0718

a (rad) . „

-0 1330 0 0342 0 0938 0 1003 0 0937 0 0833 0 0742 0 0669 0 0589 0 0521

a (deg)

... 5 82 820 868 8 53 7 87 6 91 6 13 5 30 4 66 4 11

a (deg)

... -7 62 196 5 37 5 74 5 37 4 77 4 25 3 83 3 37 2 99

c„ ...

001312 0 02432 0 02727 0 02634 0 02241 0 01752 0 01424 0 01)46 0 00980 0 00872

c„ ._

00612 00075 00117 00128 00116 00101 0 0090 0 0083 0 0078 0 0075

dCr/d!

0 2 706F-04 1 529E-03 3 662E-03 6 451E-Q3 9398E-03 1 204E-02 1 477E-02 1701 £-02 1 934E-02 2 163E-02

dC,/df

0 -3 441 E-04 3 556E-04 2 204E-03 4 22IE-03 6 225E-03 8 073E-03 9 946E-03 1 193E-02 1 357E-02 1 522E-02

dC,ydr

0 6 090E-07 9031F-06 3 4I8E-05 7 827E-05 1 301 E-04 1 756E-04 2 267E-04 2 724E-04 3 315E-04 4 049E-04

dCVdr

0 2 714E-06 2 678E-06 1 397E-05 3 648E-05 6458E-05 9636E-05 I 365E-04 1 885E-04 2 513E-04 3 328E-04

* V d r

0 7 039E-06 6 687E-05 2 023E-04 4 096E-04 6 442E-04 8 528E-04 1 073E-03 I 240E-03 I 417E-03 I 591E-03

d C V #

0

-1802E-05 2 5S3E-05 1 986E-04 4 239E-04 6 624E-04 8 806E-04 1 1O6E-03 1 339E-03 t 527E-03 1 7I6E-03

u(swu-!)(ft/s

0 32 07 20 80 14 67 1123 907 7 60 6 54 5 73 5 10 4 60

A

0 0000 0 2647 0 2871 0 3559 0 4393 0 5288 0 6212 0 7154 0 8107 0 9066 10031

B

3 0928 16392 12073 10975 10560 10362 10253 10186 10143 10113 10092

Us

Coaxial system with a spacing of 10% of diameter

This model is to compare to previous coaxial model with swirl (configuration 2)

Name of helicopter This spreadsheet is a model to calculate performance of coaxal rotor considering thai only the lower rotor depends on the other swirl velocity induced by the upper rotor is included

Characteristics of helicopter W (lb) fnglne HP

17650 2784

Following data are characteristics of the rotor

Coaxial Rotors Design

R(ft)

spacing (H P.)

N

C(fi)

OR (ft/s)

22

0 05

4

1 75

665 5

itppi >er rotor c?,,(deg) 15

Ox* (deg) -9

a (per rad) 5 73

km er rolar

#<,(deg) 15

0,» (deg) -9 a (per rad) S 73

Calculated Data

A(ft!)

cr

O (rad s)

spacing H(fl)

increase of induced ve

slipstream contraction

ociry

IS20 5

0 1013

30 25

1 1

1 025

0 9925

Atmospheric Data

P (slug/ft1)

a (ft/s)

0 002378

11167

altitude (ft)

TFAC

im

0

1 0

5190

(@ SLS)

0 002378

11)7

alternative formula for density with altitude

2 3769e-3*exp(-h/22965)

Thrust/Power Calculations B

(C T)BO loss

(ACT) t ip i0SS

C T

T(lh)

CQO

V^-Qi/no loss

\ " * - Qi/tip loss

Co.

C Q

HP induced Total HP

0.9702

7.121E-03

5.126E-04

6.609E-03

10583

1.045E-04

4.391 E-04

3.434E-05

4.047E-04

5.093E-04

784.2 986.8

Thrust/Power Calculations B

(C T/no loss

(ACT}„ploss

C T

nib) CQO

sMJi /no loss

( " C Q , ) 1 I P | O S S

CQI

C Q

HP induced Total HP

0.9760

4.624E-03

2.109E-04

4.413E-03

7067

9.234E-05

4.384E-04

2.150E-05

4.169E-04

5.093E-04

807.9 986.8

T total 17650

equal torque? T ratio u/1

YES 0.600

59

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60

CT versus Co graph

This graph was actually not used in this study but it was noticed to be a common tool of

perfonnance comparison. It corresponds to the optimum coaxial configuration.

Name of helicopter This spreadsheet is a model to calculate performance of coaxial rotor considering that only the lower rolor depends on the other swirl velocity induced b> ihe upper rolor is included

Characteristics of helicopter

W (lb) 17650 Lngine HP 2784

Following data are characfensttcs of the rotor

Coaxial Rotors Design

R (ft) 24

spacing (H R) 0 1

N 4

C (ft) I 75

OR (ft/s) 525

"PPer

0o (deg)

<?,. (deg)

a (per rad) 5 73

losier rotor

0„(deg) 0,» (deg) a (per rad)

15

-9

5 73

Calculated Data

A(ft!)

o

O (rad/s)

spacing H (ft)

increase of induced velocity

shpsueam contraction

1809 6

0 0928

21 875

24

I 05

G9g5

Atmospheric Data

P (slug'fl')

a (B s)

0 002378

11167 altitude 1ft)

TFAC

T f f i

0

10

5190

(<ff SLS)

0 002378

1117

alternative formula for density with altitude

2 3769e-3*exp{-h/22965)

0 0 1 5 T -

0.010

6

| 0.005

O.000

0.000

CTvsCG

CQ

0,002

61

Automatic calculation code

This code had for purpose to make calculations faster but was not considered as a deliverable.

Consequently, it has limits and no error management.

Public T ratio, Weight, Diff, Inc temp, Inc m m , Inc max, Inc A, Inc B A3 Variant

Private Sub CommandButtonl_Click() calculate_perf

End Sub

Sub calculate_perf() ' procedure to find rotors characteristics Application.HaxChange D.00000001 ' set the precision of goalseek function

'because of coefficients T_ratio =0.6 Range("C6").Value = 0 ' increment of theta is initially 3et to zero Range("C21").Value = 0 ' same for lower rotor Weight = Worksheets("Characteristics").Range("B7").Value ' thrust required to

'hover le equal to Weight of helicopter Diff = Weight Worksheets("Performance").Range("H39").Value 1 difference between required and actual thrust Inc_max Worksheets("Performance").Range("B16").Value 'set the maximum increment that will be tested Inc_min -Inc_max 'the range of incremental theta is limited by Inc_min and Inc_max which may 'cause error if the solution is not in this range Inc_A = Inc_min Inc_B = Inc_max Worksheets("Performance").Activate Range("C6").Value Inc_min 'initialize the pitch increment goaltest ' function to find the proper increment for theta of lower rotor While Abs(Diff) > 0.5 ' while the thrust doe3 not match the requirement

If Diff > 0 Then 'dichotornie method to find the appropriate increment angle Inc_temp = (Inc_A + Inc_B) / 2 Range("C6").Value = Inc_temp goaltest boundar ies

Else Inc_temp = (Inc_A + Inc_B) / 2 Range("C6").Value = Inc_temp goaltest boundaries

End If Wend

End Sub

Sub goaltest() CQ_up = Range("B44").Value Range("E44").GoalSeek Goal:=CQ_up, ChangingCell:=Range("C21") While Abs(Range("HI").Value - Range("H45").Value) > 0.0001

Range("HI").Value = Range("H45").Value Range("E44").GoalSeek Goal:=CQ_up, ChangingCell:=Range("C21")

Wend Diff = Weight - Worksheets("Performance").Range("H39").Value

End Sub

Sub boundaries() If Diff < 0 Then

Inc_B = Inc_temp Else

Inc_A = Inc_temp End If

End Sub

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