Theses - Daytona Beach Dissertations and Theses
Fall 2007
Study of Contra-rotating Coaxial Rotors in Hover, A Performance Model Based on Blade Element Theory Including Swirl Velocity
Florent Lucas Embry-Riddle Aeronautical University - Daytona Beach
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Scholarly Commons Citation Lucas, Florent, "Study of Contra-rotating Coaxial Rotors in Hover, A Performance Model Based on Blade Element Theory Including Swirl Velocity" (2007). Theses - Daytona Beach. 127. https://commons.erau.edu/db-theses/127
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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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ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 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
?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
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 s^^ii-^ 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 . NI 'i . • *>>S f?vy."»sW'!li mankind. An opportunity to make a human dream come true, a
chance to fly in the skies, hover and go rearward.
»,*{«•*>»-».* 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
\ -\y / / // / ' / R
/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
\e y^^.'-'y J^^—""~ /-'\^~^^^ vi iV &~-" ' , ir Q,y ,'•''
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 fV.^ obtained which leads to the calculation of advance angle,^ = tan" 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 C and dr dr 7
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 Baseline Incremental t 0 (deg) 8 (deg) 6 (rad) a (per rad) a (deg) M 4 (rad) v,(ft/s) a (rad) c„ dCT/df dCVdr dCq/dr 00 150 2417 0 3040 0 000 5 73 UNDEF 0 — — — 0 0 0 01 14 1 2417 0 2883 0 065 5 73 02173 15 77 0 0710 4 07 0O086 1 888E-04 4 015E-07 4 103E-06 02 132 2417 0 2726 0 130 5 73 0 1776 25 79 0 0949 544 0 0119 1010E-03 4 406E-06 3 588E-05 03 123 2417 0 2569 0 195 5 73 0 1523 33 16 01046 5 99 0 0137 2 504E-03 1 72OE-05 1 144E-04 04 114 2417 0 2412 0 260 5 73 0 1337 38 82 0 1075 6 16 0 0143 4 574E-03 4 260E-05 2 446E-04 05 105 2417 0 2254 0 325 5 73 0 1190 43 19 0 1065 6 10 0 0141 7 079E-03 8192E-05 4 212E-04 06 96 2417 0 2097 0 390 5 73 0 1068 46 52 0 1029 5 90 0 0134 9 856E-03 1 342E-04 6 3I6E-04 07 87 2417 0 1940 0 455 5 73 0 0963 48 96 0 0977 5 60 0 0124 ! 273E-02 1 969E-04 8 586E-04 08 78 2417 0 1783 0 520 6 00 00884 5137 0 0899 5 15 00110 1 602E-02 2 620E-04 1 134E-03 09 69 2417 01626 0 585 6 50 0 0821 53 66 0 0805 461 0 0097 1 967E-02 3 279E-04 1 454E-03 10 60 2417 0 1469 0 650 7 00 0 0759 55 13 0 0710 4 07 00086 2 306E-02 4 014E-04 1 751E-03
Thrust/Powe r Calculations B 0 9674 Note Incremental 8 is 2 417 degrees to \*~ T/uo bss 8 517E-03 produce hover T = W= 176501b
(ACT)^,loss 7 344E-04 7 782E-03 cT T(\b) 17651 CQO 1 268E-04
\M)I 'no loss 5 773E-04
(ACQAIPIW 5 714E-05 C« 5 202E-04 CQ 6 470E-04 HP induced 1557 4 fatal HP Est Hp hover 1937 0 Hover % pow, 69 6
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:
R Roton- . ->
streamtube' D
vena contracta 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/? R'-R r = z + DR = z + R 2R 2-R -0.3-.ft r = • •z + R 2-# 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
f Vr+V. (/> = tan C ' ' t Q-y + u2 j
Q- y + u2
Where,
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 2 2 U = J{n-y + u2) +(Vc+V,) u = {Q-y + u2) cos^
As is small, cos^ « 1 and U » Q • y + u2
Upper rotor R \ i
i 7 / / lower rotor / / i i
i i i
l
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,
2 2 urr =*V2 2 U]{pR) =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 2 (n-yf+vh
• 2
Consequently, u2=-^- C, (n-y) +vh
u2 _ Vh 2-Q.-y hn 2 *~ehR~~f'{{n.yf + Vh )-Q.-R
u2 _ Vh 2-r Simplifying, using, y = r-R, — = -,- ({a.yf + y2)
Finally,
2 U A Vh 2-r 2 2 QR C r (n-R) +Vf
16 T Vh, induced velocity for hovering, is calculated with momentum theory: Vh = 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.
-pU2-(c-dy)-C,
dCT = p-(nR2){QRJ
f ~ \ f U V dy_ Simplifying, dCT - K-R) QRJ R
U Finally, using the previously defined expression of Qft
1 ( c ^ v dCT = — dr T 2 \n-Rj '• V &R
V„ For N blades, dC = — • C, dr r C-r + 2 2 2 7 2 /. UJ r -(Q-R) +Vh j
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^ «
QR Therefore,
Then,
a a 9- yh 2-r fvy 2-r dCT = C-r + 2 2 2 I- C-r + 2 2 2 dr \zW r .(Q-R) + V UJ r .(Q-R) +V
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 1 dCT = —^-~—'-^-rr dr (Because we locate our study at C • r) {Cl-Rf
18 So equating those two expressions for dCT we have,
r z 2 2-r 2-r Vc + K ( v \ era C-r + v\ -X' £-r + V = 4 C-r JL 2 2 2 2 r -{Cl-Rf V r -(n-Rf + Vh ) n-R \n-Rj + h* J K$J
f ' 'V? 4-r 1 Ar V? VC+VY V, a-a 9- C-r+ K -X- 1+- =4- 2 2 2 2 2 2 2 n-R xn-Rj 2 U r -(n-rf+Vh £{C) (r>.(n-R) +Vh ) QKCJ r -(n-R) +V )
A B
The alternate expression for the advance ratio is: X = vc+K n-R
Then, a-a 9-A- Vc + K B (Vc + K = o n-R) n-R , n-R)
We want to rearrange this into a quadratic equation.
A v 2 aa •e- -i^( c+v,)B-7^(vc-v,+v )=o 8 8Q/?V" " (n-R)
2 („ cT-a-n-R \ V + V Vc+' •B +—-n-R-{Vc-B-9-A-n-R) = Q I l 8 8
Solving for V, we end up with
s Vc aa-n-R-B 2-(9-A-n-R-VcB) V,= -1+ 1+- + 2 4-Vr. a-a-Q-R-B + VC-B + a-a-n-R 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
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 Calculated Data Atmospheric Data (@ SLS) Rift) 24 A (ft') 1809 6 p (slug/ft3) 0 002378 0 002378 spacing (H R) 0 1 a 0 0928 a (ft/s) 11167 1117 N 4 n (rad/s) 25 altitude (ft) 0 Cfft) 175 spacing H (ft) 24 TFAC 1 0 OR (ft/s) 600 increase of induced velocity 1 05 T(°R) 5190
U slipstream contraction 0 985 upperPP' rolor 0o (deg) 15 alternative formula for density with altitude 0iw (deg) -9 2 3769e-3»exp(-h/22965) a (per rad) 5 73 lower rolor 9o (deg) 15
0W (deg) -9 a (per rad) 5 73
Figure 10: Inputs of the model
20 SO HO CO CO CM O co so OO » t OO r-~ CQ Ov ro SO rN CO co so r* 3 CO CO CO ecno o o o o
o **- OO *>1 W) m wo W-i CO o SO CM S CO CN ir> CM 8 en so "*3" rM cNl o CO CO < O CM CM rn •**• w^ sO r~- OO V< co O co (O C£> CO O CO CO CO CO
S CM OO NO C3 co -3- CO WO CO "5 OO s o m CT*OO sO W0 v-i -BJ. «* rO ..S3
W") •-a- -3- •B9- CO so wo "«** •nf >*3" •**• "•a- -**< •X? m uo ** •z? "^ -sr co SO C9 CO CO CO CO CO CO CO CO co co CO Y *¥ CO CO U5 U«3? Uj UJ LU UJ UJ UJ UJ £ ih U3 £ 1 ui ti ti 1-6 U3 CM •«a- sO in NO CM •cj- J--. O CO CM w^ m
CO -•3- r- w> CTv w NO CO •«*- r- en CM •rf sO OO CO •o SO NO sO c-i r-~ m OO c-» OO CM r*» CM c~ CM t*; CM OO rO 5 GO eo rN *N rO m •*r v-j ^ CO CO CM CM CO CO -*$• lO CO co co CO CO CO CO CO •**CO CO CO o CO so CO CO CO CO C«#O• CO so
en CO co r*^ o OO CM r~- r- o* ON *> o sO CM m o> CM CM rM
*w- C3 *—. eo: r^ CO ro CO VO NO sO SO SO so NO sO NO so § SP wo SO m Ngj s£s Np vO MO NO NO SO NO sO NO SO W~i »o wo VO wn so wn SO wo wn "*• -*J- •«*• •««• -*• *•*• •«*• lea £ 3 •**• -Kf "T «f ^ •a- -3- -3- CO -*r CO s s ^ -s- -* •**• 1* O so CO CS co O o o co 1* *"•'" * "•*•*_ _ "™* ™- ~~* — — — 5 >*• NO wo ICl OO OO OO OO ON t$» ™ ^o J i OO Os OO CO s> ON OS CM CM SO en oo o r- SO NO 1 * ll •*« S 8 to w~> wo w^ OO **o ,_ 0* CO t^ r^i sO OO CO OO lO ON ON ON OO MD oo V CO CTs. ON OO CO CO C3 CO CO CO CO CO CO CM CO - o CO -3- N© oo ^ CO C5 CO CO CO 1 CO CO CO CO » 3 s Figure 11: Coaxial rotor blade element spreadsheet
21 Thrust/Powe • Calculations Thrust/Power Calculations 8 0 9699 B 0.9758
\^ T/nolosi 7.264E-03 \*~* T/rto loss 4.702E-03
(ACT)up!oiji 4 728E-04 (AC r)tfp(oss 9.929E-0S C T 6 791E-03 cT 4.603E-03 r (ib) 10520 T (lb) 7130 T total 17650
CQO 1.068E-04 ^"QO 8.466E-05
l^Qf/no less 4.440E-04 (^Ql/no loss 4.454E-04
("CgJaploss 3 034E-05 ("C Qi)tip loss 9.558E-06
CQ> 4.136E-04 CQI 4.358E-04 CQ 5.205E-04 CQ 5.205E-04 equal torque? YES ///> induced 699.0 //F induced 736.5 T ratio u/1 0 596 Total HP 879.6 Total HP 879.6 mi 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
dC change. The Mach number M is used to calculate a compressibility correction to a = L da
The corrected value is a = a in Vi-iw2
AR An often-used correction from 2D airfoil to 3D wing data is aw = 6.28 but in the AR + 2 present case it was considered that aJD = 5.73 as a constant, also a commonly used assumption.
•r Then the advance angle is given by the formula $ •• a -a •l + ,/i + 32- 16-r V a-a
Consequently the local induced velocity can be calculated with the relation tan^ = V, r-QR
22 sin
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:
2 Cd =0.0107-(0.151-a) + (1.72-a )
'•a dr 2
OCgQ 3 a = r -Cd- dr ~2
dCQl dCT • m • 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 HP required model to hover increase without swirl 1953.1 with swirl 1980.5 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. h ft h n
COEt y Coaxial ixial n n h vj UTw o isolated V Tandem Current Model Dingeldein Model N = 4 for each rotor 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) percentage 20 26% 24 21.9% 28 38% 32 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.
HP required to hover vs. Rotor radius W=17650ib, H/RsO.1, tip speed-726ft&ec
ZSO0- • ! 2000 - * „ ^-^
38 <£ « * 1500 •
—•—forcoaxaal i - for 2 single
1000 -
500 •
0 3 5 10 15 20 25 30 35 radius if! ft
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
e + l3 ave the best Ob* = upp ° 8 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 pitch difference (°) R=22ft, QR=665 ft/s 1.31 R=22ft, OR=726 ft/s 1.28 R=24ft, QR=665 ft/s 1.19 R=24ft, £1R=726 ft/s 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
HP required Configuratiofi to hover Untwisted 2068.8 Linear Twist 1966.8 Custom Twist 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.
pitch distribution
18
t 16 • -m
•M 14 » » *~ 12- ^^^L^r* f 10 — ^^^^_^^ ^^^ ™ I J * — — - 6 — _
2
00 02 04 08 OS 1 Q 1 2 r
| ~^~» pwich d st up -» pitch cifst low |
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.
Induced velocity versus radius location
„ _ „ „ 45 - I
40
35 - _
30 -
I ?5
5 20 - „—• w- — —m «- ~m —m
15 ^ J*-'' 10 • / r 1 / 5
00 02 0 4 06 08 1 0 1 2 f
j __#__ vt upp&r •» Vi Sower j
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 toapproximat e this design by a quadratic equation.
pitch distribution
25 ,
20
^Ov. R? = 0 9994
15 - ! y * 18 512xJ 41 054x * 28 692 ^^^Js^ RJ = 0 3993 ^"^5S»~»_ 10 - —
s -
00 02 04 06 08 1 0 1 2
T
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 Increase in Configuration power required Untwisted 6.71% Linear Twist 1.45% Customed Twist ...
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 todriv e 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 50 60 Radius (ft)
- Coaxial N=4 -> Single rotorN- 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.
Hp required vs tip speed to hover, W=17650lb, H/R*0.1, R=24ft
2500 • --- - -
«• 2000- .. . *\ ^^* ... *-~~~~-~-~^_2^ ^"~~-~-—m —*'
| 1500 -
1000 -
500
0 100 200 300 400 500 600 700 800 900 1000 tip speeil in ft/sec
| * single N=4 -"-coaxial N=4 -*-coaxial N=2 -H-single N=8 |
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 | lyuumemodifiedu npaenApache j This spreadsheet is a model to calculatdate performance Characteristic* of helicopt W (lb) 176576500 Engine MP 2784 Following data are characteristics of the rotor Coaxial Rotors Design Calculated Data Atmosphenc Data (@ SLS) RQh 24 A iff) 1809 6 P (slug/ft') 0002378 0 002378 %paung(HR) 0 I a 0 0928 a (ft/s) 11167 1117 N 4 a (rad's) 22 29166667 altitude (ft) 0 C (fl> 1 75 young H (ft) 24 TFAC 10 fill (ft/s) 535 increase of induced ve ooty 1 05 T(°R) 5190 upper rolor slipstream contraction 0 985 S» (deg! 15 alternative formula for density with altitude K Sdegj 2 3769e-3*exp(-lv22965) Of (per rad ) 5 73 hmer rolor 0«(deg) IS #,» (deg! -9 a (pur rod) 5 13 Figure 23: Design parameters of modified Apache with linearly twisted blades Thrust'Power Calculations Thrust/Power Calculations B 0.9659 B 0.9725 \*- T/no loss 9.319E-03 \\ lino lof»s 6.048E-03 (ACT)tJpioss 8.149E-04 \IAK, xJitp|0!>s 2.220E-04 C cT 8.504E-03 T 5.826E-03 T(\b) 10474 7" (lb) 7176 T total 17650 CQO 1.408E-04 (- QO 9.205E-05 l*-*Ql/no loss 6.607E-04 v^Qi/no loss 6.715E-04 ("C*Qi)tiploss 6.511E-05 ("C Q,)„p loss 2.712E-05 5.956E-04 6.444E-04 C(Ji cQ! Co 7.364E-04 Co 7364E-04 equal torque? YES HP induced 713.6 HP induced 772.0 T ratio u/1 0.593 Total HP 882,3 Total HP 882.3 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 Thrust/Power Calculations B 0.9655 B 0.9720 v*"- T/no ioss 9.528E-03 l**" T/no foss 6.283E-03 (AC1)l¥bss 7.329E-04 (ACT)tipkjss 1.968E-04 8.796E-03 6.086E-03 cT CT r(ib) 10432 r(ib) 7218 T total 17650 CQO 1.461E-04 CQO 9.451E-05 ^Ql/ftOiQSS 6.708E-04 I'-Qtino loss 6.899E-04 ("C/Qi)tiploss 5.490E-05 ("CQ,)tip|oss 2.235E-05 CQ, 6.159E-04 CQ, 6.675E-04 Co 7.620E-04 CQ 7.620E-04 equal torque? YES HP induced 697.3 HP induced 755.7 T ratio u/I 0.591 Total HP 862.7 Total HP 862.7 ffJvSilW! 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 Thrust/Power Calculations B 0.9682 B 0.9695 v> Tmo loss 8 096E-03 l*- 1 /no !obs 7.455E-03 (ACT)tif>ioss 6.499E-04 (ACT)tipioi)S 2.990E-04 c. 7 446E-03 c. 7.156E-03 7-(lb) 9000 nu>) 8650 T total 17650 C oo 1 I81E-04 ^ Oo 1.072E-04 l*-Q>)no loss 7 377E-04 V^-Qi/notoss 7 141E-04 ("C*Qt}Uptess 6 611E-G5 V"CQ,)tlpt 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 1] 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. 41 4.1 Simple coaxial configuration Thrust/Power Calculations Thrust/Power Calculations B 0.9657 B 0.9734 \S* T/no loss 9.390E-03 l*~ T/no loss 5.645E-03 (ACrXiploss 1.085E-03 (ACT)tipioss 2.557E-04 CT 8 305E-03 CT 5.389E-03 r (lb) 1819 r (ib) 1181 T total 3000 CQO 1.437E-04 CQO 1.198E-Q4 l^Qilnoioss 7.031E-04 l^Qi'm) loss 6.602E-04 ("CpJupjoss 1.027E-04 siiC Qi/iiploss 3.591E-05 6.003E-04 6.243E-04 cQ, CQ> CQ 7.441E-04 CQ 7.441E-04 equal torque? YES HP induced 136.3 HP induced 141,7 T ratio u/1 0.606 Total HP 168.9 Total HP 168.9 337.8 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 Thrust/Power Calculations B 0.9687 B 0 9697 l'-" T /no !oss 7.820E-03 v^- 1 /no loss 7.338E-03 I H (&C*j)tlpj0SS 0.000E+00 (ACr)tipj0SS 0 000E+00 7.820E-03 7.338E-03 cT CT T(\b) 1594 T(\b) 1405 T total 3000 CQO 1.U6E-04 CQO 1.088E-04 S*"QI 'no loss 7 339E-04 V^-Qi/no loss 7.367E-04 («C Qi/tiplow 0.000E+00 ("CQ,)tlp|QSS 0 000E+00 CQ, 7 339E-04 CQ, 7.367E-04 g 454E-G4 8.455E-04 0 c0 CQ equal torque YES HP induced 155.1 HP induced 144.0 T ratio u/1 0.532 Total HP 178.6 Total HP 165.3 343.9 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 Thrust/Powe r Calculations B 0 9743 B 0.9743 l*~ T/no loss 5.266E-03 V*** T/no loss 5.264E-03 (ACT)tfp|0Si| 4.884E-04 \IM~ f/tsploss 4.885E-04 cT 5.266E-03 C j 5.264E-03 7* (lb) 1500 T (lb) 1500 T total 3000 CQO 1.282E-04 CQO 1.282E-04 i*"Q)ino loss 4.221 E-04 '^Qi'noloss 4.220E-04 \"CQ,)„P |0SS 4.922E-05 ("CQ,),lptoss 4.924E-05 ^Q, 4,221 E-04 CQ, 4.220E-04 CQ 5.502E-04 CQ 5.502E-04 equal torque? YES HP induced 142.1 HP induced 142.1 T ratio u/1 0.5001 Total HP 185.2 Total HP 185.2 ?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 Data Calculated Helicopter Data W (lb) 17650 A (ft2) 1809.6 Engine HP 2784 a 0.0928 N 4 0„P(deg) 6 7?(ft) 24 D (rad/s) 30.25 C(ft) 1.75 0o(deg) 15 Atmospheric Data (@ SLS) 3 0,w (deg) -9 p (slug/ft ) 0.002378 a (per rad) 5.73 a (ft/s) 1117 QR (ft/s) 726 47 Baseline Incremental 6(ie%) 0 (deg) 0(rad) a (perrad) 4 (rad) a (rad) a (deg) dr Thrust/Power Calculations B 0 9674 Note Incremental 0 is 1 9856 degrees to CC-rUte* 8 500F-03 produce hover T = W = 17650 lb (ACjXjpiosj 7 179E-04 C 7 782E-03 T FWbwer conditoiE nib) 17650 CQO 1 148E-04 VM)t)no bss 5 761 E-04 (^CQ,)opta 5 508E-05 CQ, 5 210E-04 CQ 6 358E-04 HP induced H59 8 Total HP iWMM Est Hp hover 1903 6 Haver % pow 68 4 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 Calculated Data Atmospheric Data (# SLS) R On 24 A (ft') 1809 6 P (slug/ft') 0002378 0 002378 spacing ttl K) 0 1 u 0 0928 a (fi's) 11167 1117 A' 4 n (rad/s) 30 25 altitude (fi) 0 C (ft) 1 75 spacing H (ft) 24 TFAC 10 OH (8/s) 726 inci ease ot induced velocity I 05 T(°R) 5190 upper rolor slipstream contraction 0 985 0u «leg> 15 alternative formula for density with altitude fl„ (deg) -9 2 3769e-3*exp{-h/22%5) a (per rati) 5 73 lover rolor 0„(deg) 15 K (deg) -9 a (per rad) 5 73 Thrust/Power Calculations Thrust/Power Calculations B 0.9752 B 0.9801 \S- T/no loss 4.920E-03 (C T)IK> loss 3.167E-03 (ACjXjpioss 2.653E-04 (ACT)llploss 3.899E-05 CT 4.654E-03 CT 3.128E-Q3 7" (lb) 10556 7" (lb) 7094 T total 17650 CQO 8.909E-05 CQO 8.506E-05 '^"QI'no loss 2.504E-04 ''"QiJnoloss 2.437E-04 ("CQittiplosi 1.382E-05 (^CQ,),IR|05!J 3.119E-06 2.406E-04 C0, 2.366E-04 CQ, CQ 3.257E-04 CQ 3.257E-04 equal torque? YES HP induced 708.3 HP induced 720.4 T ratio u/1 0.598 Total HP 975.0 Total HP 975.0 1950.1 50 SO in to •g- ^* •» •sf •^r •* t r~ in in •^r "it •* ««• -* N* *t © © © 9 9 o o o o o © O o o o O o o o o to to to W lit to lis US US US US US US to m w w {~» m us o o SO CN r-~ ON r- •* O o r~ o m IN so ^3- ON fciO0 f N — t so o ON o 0"0* o r~ fn cn oo 'It o in sO o CN r~ r-~ so m in in *n -* •* •* it r» so NO »n in in it It ^t it © o o © O o o o o o o o o o o o o o o o di U tQ LO UJ PJ w Ul US li us lis ttJ m to as US US US us o SO u-» "a- O -* "NT cn m m m m m IN (N in •a- *_f cn m m t^N m m m o o o o o o o o o O o o o o o o o o o o $ us p^j Ul PJ UJ US lis US ^ ti US US US US to a: US US vi f™ si w o IN r- CN! o •«t in in "H o so o o r- o so m in CN in i"n» so as sO oo Os m m sn it NO IN cn os r~ oo u*t3 CN !>. SO o *n tN 00 IN o o!> • so oo rr\ sO r-- ON CN CN OS """ so """ en •st NO r~ OS —« ""• m CN OO — IN m in so r-~ t— o r— in so O oo OS OS in 't ^t in NO oo — t o t ^t in -* NO r-* oe r-~ oo oo oo oe (~- r~~ r- r*~ r*~ oo r~ t-. r-~ r-~ o o o o o o o o o O o o o o o o o o o o I o o O i o o o o o o o o o o o o o o o o O o o o o o o o o O o o o o o O o o o o o o o s> CN •<* in OS r~ IN SO r~N IN •* OS rrs oo t- cn NO "O SO so OS C3S r~ o NO Csl r-~ T3 oo in o IN IN CN o so m s^. !* CN m cn en en m•* m rs fN ; o IN IN rsi IN IN » —w ~« O ON en en en fM ON sO 00 t~- 'NT CN o o ON m C-~ N3- OO IN OO 00 m o m so oo o r*^ in OS o 00 so CN oo cn <~•t - sO so sO so SO in •a- m rn Hw 2 M m m cn cn cn CN CN -™- I o o O o O O O o o o i o a o o o•* o o o o o 3 o O O o o o N-* o o o 5S o o o o o o o o o o U-, in Os so o oo OS m rsi in N* U) o oo OS CN oo in m CN — 00 o O so c-~ o 00 so «3 m o in CN o ON OO o ? 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S3 —* O *» o o o o o O o o ° 1o o D J1 Coaxial model with swirl (configuration 1) Maine of helicopter modified Apache | This spreadsheet is a model t o calculate performance of eoasual rotor considering thai only the lower rotor depends on the other swirl velocity induced by the upper rotor is included Characteristics oi helicopter If (lb) 17650 Lngine HP 2784 Following data are characteristics of the rotor Coasia) Rotors Design Calculated Data Atmospheric Data {& SLS) R (fl) 24 A(ft!) 18096 p {slug/ft') 0 002378 0 002378 spacing (H li) 0 1 cr 0 0928 a (ft's) 1116 7 1117 .V 4 O (rad/s) 30 25 altitude (ft) 0 C(ft) 175 spacing H (f() 24 TFAC 1 0 OR (ft's) 726 increase of induced velocit) I 05 TCR) 519 0 upper rolor slipstream contraction 0 985 0, !deg) 15 alternative formula for density wit h altitude 9,»(degi -9 2 3769e-3*exp(-lt/2296S) a (per rad) 5 73 kmer mlar So(deg} IS Thrust/Power Calculations Thrust/Powe r Calculations B 0 9751 B 0.9802 ' *~ T /no loss 4 964E-03 \S~ T/noioss 3 127E-03 (ACy/apio^ 2 699E-04 (arT)tip3oss 3.889E-05 4.694E-03 3 088E-03 cT CT Tito) 10647 7" (lb) 7004 T total 17650 CQO 8 912E-05 C Q0 8 508E-05 (^-Qi'noloss 2.539E-04 s^-Qi/no loss 2.469E-04 (tit. QtJtiptoss 1.416E-05 (ACQ,)t]pj0ss 3 160E-06 CQ, 2.397E-04 ^Qi 2 437E-04 CQ 3.288E-04 CQ 3.28SE-04 equal torque? YES HP induced 717.6 HP induced 729.7 T ratio u/1 0 603 Total HP 984 4 Total HP 984.4 il§ 1968.9 This result shows only a 0.96% difference induced by the swirl velocity. 52 CN o rS *n w^ sO 0 SO ©s (N V© m o O r*> 0 r* S •«^ a* CN Wl <•*•> 0 O O n o r*s o O O o 0 0 O O o m ^J. oo rs a; m rn r- o t£* r» oo m 0 f- o Ov FN o?c 0 0 0 ©'*^ < o r«s* sO r~. 00 c* o o o O o•«* • o«-^ o 0 0 0 0 £ o o o Wl «n T 00 0 "*"*o ^> oo "5 s NO oo in 'sr m rs Wi •st- s© •^1 ^ *© *n •* ^f •-* -9- SO tn •«3f *#• ^ •ST o at O O 0 0 0 0 K. 0 O O O 0 sf 0 O O ;© o o *? di w Sx3 ti U3 11) w aJ MJ tu W W m aj £i} it LU o <*t ©•* Q*. *£> SO r- w00 O O- f-i -:}• (-t^f •* -=f QS, ->r sri r^ m r- s© rsj m <-n 0 0 00 CN ©N O ©s s «9- rn — CS rS 13* rS rN rs! 0 M3 s© s© O 00 ©v *n tS XJ r~ r- <» rs — so rs •"• sO ™" rsj r^ •sf w-i *n vt •^J- Tt m "~ ™* rS -* sn v© vo ^j. Tj" r- so v> ^ ^ ^ ^r tj- •** •^ff vO VO vo >n w-> »n "ST "Sf o o O o O O 0 0 O 0 O 0 O 0 O O O O O O di Uj 1X1 cu rii UJ UJ UJ UJ Ul UJ U4 W a* iii U4 tu di di o oo rn oo o s© CN rs ©mN 0 O m tn •"3- 0 V© ©s •!}" tr — i f*"> 00 *G ©N 0 m W~l t W~l 0 ©s i~4 r-4 m rS — [— -f CN w~» 00 fNi p» W^ fsj -© o r*- *n -0 m r* *n v~i fSj CN •* ™* rsi ©s rsj m — r» "" *n ~ m •**• r~ _- ~* rs m •$• **• m rn r»n m m r^-i (N rsj •tj- Tfr m m <-n m f*i m m V O o © o o o 0 0 0 0 O 0 0 0 0 O 0 0 0 0 -CS W ih ub UJ di UJ lil LU u'i LQ fc. th di W Ui W W £& DJ th di o Os so o 0 OS (N CN ^ O 00 ©v rs ^j- m m rn VI a- r- f") fN 2 to r- **• 00 ©1 ** *n 00 V) so 0 r- rs r~ Sp o sO m ©N m O O ^ 0 ©s VI v© CO •^f •«$. —. *"" s© *** m •o r- ©> "" *-* —*~ © "* rsi m vi so F-- 00 m O O r- •sT rsi 00 vo 0 00 •Sf s© 00 — o ©s O r* i> 00 j r- 00 00 00 r- r- J s© o o 0 O O O 0 0 0 0 0 O 0 0 0 £ \_ o o o o O O O O o o o o o 0 O O O C 0 0 O O O 0 0 © o o O o o o O O O O 0 0 0 O 0 0 O 0 to 1 sa m xj- -3- r- 00 O in 00 "So 0 r^ ©v (^1 0 CO a* VI CN o r~—- sO as av r- •<*• —. r- rSi r~~ T3 "•«*** s© vO 0 rs 0 CO VQ fn a w rs m r^ m r^ w rn rsj rs) ™. m 0 ™ fS rsi rs rs w— ™- © ' ' —. e e Z3 rs r*s r-. oo r- r- m (N v^ rs 00 VI m a •T3 — ™. O -* 0 ©v rs f- oo 00 o r- CN 0 C s© v© 00 "-a- .3- sO «3 sD vC ^o ^ •*t ft m 1 vo rs rn m rn rn rs rs o o o O o o 0 O O 0 O 0 O 0 0 0 O 0 O O J O -© « o o o O o o 0 O O 0 « 0 O 0 0 0 O 0 0 0 U_ t*- 00 O en 00 O sO ©* £XJ s© 0 V© f*s VI 0 f- m »n ^n o •«}- m rs o SO ^e> l»N rsi sO m O 00 ^ Q O rsi 00 s© m ^0 Os 00 NO sn S o rS ™ r- f* 0O ©s CN r- iz<« m *™ O 0 S"N — *f**1 r*~t ("*-[ O (O^ O_ *"0 * 0 —•O O— 0 O 0 o > rs *n m cn m -». H \0 oo 00 V, v© w r- v> •rt oo •© cn rsj 0 rsj t-i-S 5 rs v© xz© v 0 2 r- *3- rs O r- so i 0 m f w 2D o O 0 0 O O *~*1 0 00 CN m *s\ vO r- oo a© ©v """" •%. O O O 0 0 0 O »• -" > O o o o TIT ^ C8 P3 m eo •sJ- m vO rs» m t~ «r m ON in *sf vo rs m ©v rn ©N t-. n- <*r-* r- CO o o rsj t~v O 10 r- r~- r- 00 CN O rs) M- sO 0 --a- a. in M~! m w-^ vri >£> s© ^©. 0 h~ r*. ct. »*-i in ITi in tfl v© N© vo sO r- r- o 53 O « O « o m o V~i O tn Of O f*5 fN •#> so ©s m v© 0 **• O s© m O •« rs O in rsj 00 y~* ©v O VO 00 OO "-3- S o o rs m m •sf Wl s© ^ O O ^ M "*a *w f~ o r*1 NQ OO •<(*• r» O ?n ^ rn 00 m 00 s 0s *=f- 0 sn O sn o •«- a* r^ SO O Os m OS •rf ©v m 00 rn r- rsj so rsi o O NO «r r^ ** O 00 •© T m CN so so rs O ** rs rsi O O r*4 rsj r-s? rsj <33 o o o O o O O O 0 O O ^5 0 O O 0 O 0 0 0 0 O O l ,—s 13 ^^ 00 oo oo OO 00 OO B0 00 00 00 00 s: 00 ss 00 00 oo OO 00 00 so 00 00 00 s; WJ V© v© so s£S *o sO s© v© s© s© O O O O 0 0 0 0 O O CM rsj rsj fN ^r-j r-4 r-i fs* fst m rn Ci m rn rn **t m m rn c3 fN (N rsj f-4 rsj r-J rsj cs s O 0 O 0 0 0 0 O O 0 1 -^ R rl cs O 0 O 0 0 0 0 O O O T T "7 -p -7 -p •7 *7 •7 -p •7 1 <& Sf ^ —s 0 0 0 0 0 0 m r~ O "-> O rS ^c **• 00 — v> sn VO 00 ©s 0 rsi sTt v© ©v O rS m ^j. m o s© 0 rsi C"1 V~! a- 00 N© s© r- Os O — I* 1 « 'g- m rs 0 c* oc [-, r*» vo B in 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 Calculated Data Atmospheric Data (@ SLS) ! Rfft) A(ft ) 1520 5 P (slug/ft ) 0 002378 0 002378 spacing (H R) 0 1 a 0 1013 a (ft/s) 11167 1117 N 4 Q (rad/s) 30 13636364 altitude (ft) 0 C(fl) 1 75 spacing H (ft) 22 TFAC 1 0 OR (ft/s) ®mt increase of induced ve ocity 105 T<°R) 5190 JEEli slipstream contraction 0.985 15 alternative formula for density with altitude «,»(deg) -9 2 3769e-3'exp{-h/22965) a (per red ) 5 73 lower rotor 0a (deg) 15 •9 a (per rad) 5 73 Thrust/Power Calculations Thrust/Power Calculations B 0.9702 B 0.9760 V*-" T/no loss 7.117E-03 ***" 7 zoo loss 4.613E-03 (ACT)Ejp!os3 5.119E-04 (AC fJttploM 1.131E-04 4.500E-03 cT 6.605E-03 cT T(\b) 10498 7" (lb) 7152 T total 17650 CQO 1.046E-04 CQO 8.939E-05 NS-QI'IIOIOSS 4.386E-04 sS-Qi/noloss 4.310E-04 ("CQi)tiplos» 3.427E-05 '"C Q,}t|p)oss 1.148E-0S CQ, 4.044E-04 (• Qi 4.195E-04 5.089E-G4 cQ 5.089E-04 CQ equal torque? YES HP induced 774.8 HP induced 803.8 T ratio u/1 0.595 Total HP 975.1 Total HP 975.1 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 Calculated Data Atmospheric Data (0, SLS) Ts Rffn fS-ikl AW) 1520 5 P (slug'ftl 0 002378 0 002378 spacing (H P.) 0 1 CT 0 1013 a (ft/s) 11167 1117 N 4 a (rad/s) 29 72727273 altitude (ft) 0 C(fl) ^175 spacing H (ft) 22 TFAC 1 0 OR (ft/s) ^S^MN increase of induced velocity 1 05 TCR) 5190 ,'ff?r rolor slipstream contraction 0 985 9 (deg) 15 alternative formula for density with altitude 0,» (deg) -9 2 TO9e-3»e\p(-h/22965) a (per rad ) 5 73 tower rotor «Mdeg> 15 0» (deg) -9 a (per rad) 5 73 Thrust/Power Calculations Thrust/Power Calculations B 0.9696 B 0,9758 l*~ T/no loss 7.404E-03 I*- T/no loss 4.680E-03 (AC T)tip|oss 5.485E-04 (AC x)up to$4 1 224E-04 CT 6.855E-03 CT 4.557E-03 7" (lb) 10602 7" (lb) 7048 T total 17650 CQO 1.071 E-04 CQO 8.957E-05 s*~Qf/no ioss 4 658E-04 H--Q])noIoss 4 585E-04 ("Couple*, 3.770E-05 ("CQ,)„P|OSS 1.289E-05 cQl 4.281 E-04 CQ, 4.456E-04 5 351E-04 CQ 5.351E-04 equal torque? YES HP induced 787.2 HP induced 819.4 T ratio u/1 0 60! Total HP 984.1 Total HP 984.1 !%«»$ 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) 17650 hngme HP 27S4 Following data are characteristics of the rotor Coaxial Rotors Design Calculated Data Atmospheric Data m SLS) R (ft) 22 AifF) 1520 5 P (slug/ft') 0 002378 0 002378 spacing (H R) 0 I CT 0 10)3 a (ft/s) 11167 1117 V 4 O (rad/s) 29 72727273 altitude (ft) 0 C (/I) 1 75 spacing H (fl) 22 TFAC 1 0 OR (ft/s) 654 increase of induced ve ociry 1 05 im 5190 upper rolor slipstream contraction 0 985 e„(deg)am 10 alternative formula for density with altitude 0» (deg) 0 2 3769e-3*exp(-h/22965) a (per rod) 5 73 losier rotor «Mdeg) 10 S» (deg) 0 a lper rad ) ,,,573 Thrust/Power Calculations Thrust/Power Calculations B 0.9690 B 0.9757 I *- T /no loss 7.684E-03 \^~- T/no loss 4.709E-03 {ACT)tip|0SS 8.118E-04 {ACT)tip!oss 1.687E-04 C *|* 6.872E-03 CT 4.541E-03 7" (lb) 10628 r (lb) 7022 T total 17650 CQO 1.118E-04 CQO 9.039E-05 I'-'Qt'noloss 5.205E-04 s^Qi/no loss 4.937E-04 ("CQ,)I1P|0SS 6.959E-05 ("C Q,)„p |0Ss 2.138E-05 CQ, 4.509E-Q4 CQ, 4.723E-04 CQ 5.627E-04 CQ 5.627E-04 equal torque? YES HP induced 829.2 HP induced 868.6 T ratio u/1 0.602 Total HP 1034.8 Total HP 1034.8 ppailp 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) 17650 / ngim HP 2784 Following data are characteristics of the rotor Coaxial Rotors Design Calculated Data Atmospheric Data (0 SLS) «(ft) 24 A (ft1) 1809 6 p {slug'ft ) 0 002378 0 002378 spacing (HP) 0 1 cr 0 0928 a (S/s) 11167 1117 N 4 O (rad/s) 21 875 altitude (ft) 0 C (ft) 1 75 spacing H (ft) 24 TFAC 1 0 OR (ft s) 525 increase of induced velocity 1 0$ T (°R) 5190 JMTSLrotor slipstream conti action 0 985 6„(deg) 15 alternative formula for density with altitude e»|deg) -9 2 3769e-3*expfh/22965) a (per rad) 5 73 Itmer rolor 8m jdeg) •9 a (per rad) 5 73 Thrust/Power Calculations Thrust/Power Calculations B 0.9655 B 0.9720 l*- T/no loss 9.528E-03 \S- T/no loss 6.283E-03 (ACT)lip!os3 7.329E-04 (ACT)tipi03S 1.968E-04 Cr 8 796E-03 Cr 6 086E-03 T(\b) 10432 n»>) 7218 T total 17650 CQO 1461 E-04 CQO 9.451E-05 '^"Qi/uoloss 6.708E-04 N*~Qi/no ioss 6.899E-04 (aC Q,)np|0SS 5.490E-05 ("CQ,),IP|0SS 2 235E-05 CQ, 6.159E-04 CQ, 6.675E-04 cQ 7.620E-04 CQ 7.620E-04 equal torque? YES HP induced 697.3 HP induced 755.7 T ratio u/1 0.591 Total HP 862.7 Total HP 862.7 ^ItiilR 1725.4 57 estimated T ratio 0 591 used for calculation of swirl velocity vh (ft/s) 34 81 Upper Rotor Baseline incremental 8 (deg) & (deg) 0 (rad) a (perrad) c* (rad) a (rad) a (deg) r M v,(Ws) c„ dCr/d! dC,ydr *Vdr u(swu-!)(ft/s A B 00 18 2.7258 0 3617 OOOO 5 73 UNDEF 0 ._ ...... 0 0 0 0 0 0000 3 0928 0 1 18 2 7258 0 3617 0 047 5 74 0 2601 13 66 0 1016 5 82 001312 2 706F-04 6 090E-07 7 039E-06 32 07 0 2647 16392 02 18 2 7258 0 3617 0 094 5 76 0 2186 22 96 0 1431 820 0 02432 1 529E-03 9031F-06 6 687E-05 20 80 0 2871 12073 03 16 5 2 7258 0 3356 0 141 5 79 0 1841 29 00 0 1514 868 0 02727 3 662E-03 3 4I8E-05 2 023E-04 14 67 0 3559 10975 04 14 9 2 7258 0 3076 6 188 5 83 0 1587 33 34 0 1489 8 53 0 02634 6 451E-Q3 7 827E-05 4 096E-04 1123 0 4393 10560 05 13 2 7258 0 2745 0 235 5 90 0 1371 35 99 0 1374 7 87 0 02241 9398E-03 1 301 E-04 6 442E-04 907 0 5288 10362 06 10 95 2 7258 0 2387 0 282 5 97 0 1181 37 19 0 1206 6 91 0 01752 1 204E-02 1 756E-04 8 528E-04 7 60 0 6212 10253 07 935 2 7258 0 2108 0 329 6 07 0 1038 38 13 0 1070 6 13 0 01424 1 477E-02 2 267E-04 1 073E-03 6 54 0 7154 10186 08 78 2 7258 0 1837 0 376 6 18 0 0911 38 28 0 0926 5 30 0 01)46 1701 £-02 2 724E-04 I 240E-03 5 73 0 8107 10143 09 66 2 7258 0 1628 0 423 632 00S14 38 48 0 0813 4 66 0 00980 1 934E-02 3 315E-04 I 417E-03 5 10 0 9066 10113 10 56 2 7258 0 1453 0 470 649 0 0735 38 61 0 0718 4 11 0 00872 2 163E-02 4 049E-04 I 591E-03 4 60 10031 10092 Lower Rotor Baseline tntremenlal r 0 (deg) 8 (deg) 8 (rad) hi a (per rad) Mft's) 4 (rad) a (rad) a (deg) c„ dC,/df dCVdr dCV# 0 19 3.8473 0 3988 0 5 73 0 00 UNDEF .„ ... ._ 0 0 0 0 0985 19 3 8473 0 3988 0 076 5 75 13 16 0 532 -0 1330 -7 62 00612 -3 441 E-04 2 714E-06 -1802E-05 0 197 19 3 8473 0 3988 0112 5 77 13 60 0 365 0 0342 196 00075 3 556E-04 2 678E-06 2 5S3E-05 0 2955 19 3 8473 0 3988 0 152 5 80 16 86 0 305 0 0938 5 37 00117 2 204E-03 1 397E-05 1 986E-04 0 394 16 5 3 8473 0 3551 0 196 5 84 17 72 0 255 0 1003 5 74 00128 4 22IE-03 3 648E-05 4 239E-04 0 4925 139 3 8473 0 3097 0240 5 90 18 08 0216 0 0937 5 37 00116 6 225E-03 6458E-05 6 624E-04 0 591 11 5 3 8473 02679 0 285 5 98 18 22 0 185 0 0833 4 77 00101 8 073E-03 9636E-05 8 806E-04 0 6895 9 65 3 8473 02356 0 330 607 18 36 0 161 0 0742 4 25 0 0090 9 946E-03 I 365E-04 1 1O6E-03 0 788 8 15 3 8473 0 2094 0 376 6 18 18 75 0 142 0 0669 3 83 0 0083 1 193E-02 1 885E-04 1 339E-03 0 8865 68 3 8473 0 1858 0421 6 32 18 69 0 127 0 0589 3 37 0 0078 1 357E-02 2 513E-04 t 527E-03 0 985 57 3 8473 0 1666 0 467 6 48 18 67 0115 0 0521 2 99 0 0075 1 522E-02 3 328E-04 1 7I6E-03 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) 17650 fnglne HP 2784 Following data are characteristics of the rotor Coaxial Rotors Design Calculated Data Atmospheric Data (@ SLS) R(ft) 22 A(ft!) IS20 5 P (slug/ft1) 0 002378 0 002378 spacing (H P.) 0 05 cr 0 1013 a (ft/s) 11167 11)7 N 4 O (rad s) 30 25 altitude (ft) 0 C(fi) 1 75 spacing H(fl) 1 1 TFAC 1 0 OR (ft/s) 665 5 increase of induced ve ociry 1 025 im 5190 itppi>er rotor slipstream contraction 0 9925 c?,,(deg) 15 alternative formula for density with altitude Ox* (deg) -9 2 3769e-3*exp(-h/22965) a (per rad) 5 73 km er rolar #<,(deg) 15 0,» (deg) -9 a (per rad) S 73 Thrust/Power Calculations Thrust/Power Calculations B 0.9702 B 0.9760 (C T)BO loss 7.121E-03 (C T/no loss 4.624E-03 (ACT)tipi0SS 5.126E-04 (ACT}„ploss 2.109E-04 CT 6.609E-03 CT 4.413E-03 T(lh) 10583 nib) 7067 T total 17650 CQO 1.045E-04 CQO 9.234E-05 V^-Qi/no loss 4.391 E-04 sMJi/no loss 4.384E-04 \"*- Qi/tip loss 3.434E-05 ("CQ,)1IP|OSS 2.150E-05 Co. 4.047E-04 CQI 4.169E-04 CQ 5.093E-04 CQ 5.093E-04 equal torque? 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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 Calculated Data Atmospheric Data ( CTvsCG 0015T - 0.010 6 | 0.005 O.000 0.000 0,002 CQ 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 mm, 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 boundaries 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 63