NASA/TP–2009-215404

Influence of Lift Offset on Rotorcraft Performance

Wayne Johnson

Ames Research Center

Moffett Field, California

November 2009
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NASA/TP–2009-215404

Influence of Lift Offset on Rotorcraft Performance

Wayne Johnson

Ames Research Center

Moffett Field, California

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

November 2009
Available from:

NASA Center for AeroSpace Information National Technical Information Service

7115 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161

(301) 621-0390 (703) 487-4650
TABLE OF CONTENTS

LISTOF FIGURES ..................................................................................................................................................................iv

LISTOF TABLES ....................................................................................................................................................................v

NOMENCLATURE............................................................................................................................................................... vii

ABSTRACT..............................................................................................................................................................................1

INTRODUCTION.....................................................................................................................................................................1

BASELINE COAXIAL CONFIGURATION ...........................................................................................................................2

PERFORMANCEMETRICS ...................................................................................................................................................2

Momentum Theory for Coaxial Rotor in Hover ....................................................................................................................2

Biplane Theory for Twin Rotors in Cruise ............................................................................................................................4

Performance Metrics..............................................................................................................................................................5

PERFORMANCECALCULATION ........................................................................................................................................5

Analysis .................................................................................................................................................................................5

CoaxialCorrelation ...............................................................................................................................................................6

TandemCorrelation ...............................................................................................................................................................7

ROTOR PERFORMANCE OPTIMIZATION ..........................................................................................................................7

COAXIALCONFIGURATION ...............................................................................................................................................7

MODELREQUIREMENTS .....................................................................................................................................................8

TANDEM AND SIDE-BY-SIDE CONFIGURATIONS .........................................................................................................8

WING-ROTOR LIFT SHARE ..................................................................................................................................................9

CONCLUSIONSAND RECOMMENDATIONS ..................................................................................................................10

REFERENCES........................................................................................................................................................................11

iii
LIST OF FIGURES

Figure 1. Illustration of baseline coaxial rotorcraft configuration, utilizing lift-offset rotors

(courtesy G. Nunez, AFDD) . ...................................................................................................................................................17

Figure 2. Momentum-theory model of coaxial rotors in hover, for lower rotor far below upper rotor, showing

velocities and areas in flow field and pressure on rotor disk. Velocities wu and w, are uniform across wake

section; pressure Op$ and velocity v$ at lower-rotor disk are not uniform. .......................................................................... 17

Figure 3. Span loading and corresponding induced power of articulated helicopter rotor. .....................................................18

Figure 4. Ideal induced power in forward flight as a function of twin-rotor vertical spacing. ................................................18

Figure 5. Span loading for ideal induced power in forward flight, twin-rotor vertical spacing z/D = 0.12 . ........................... 18

Figure 6. Ideal induced power in forward flight as a function of twin-rotor lateral spacing. ..................................................18

Figure 7. Hover performance of Harrington rotor#1 . ..............................................................................................................19

Figure 8. Hover performance of Harrington rotor#2 . ..............................................................................................................19

Figure 9. Forward-flight performance of Harrington rotor#1 at CT/σ = 0.089 . ....................................................................... 19

Figure 10. Hover performance of XH-59A . ............................................................................................................................19

Figure 11. Forward-flight performance of XH-59A (without auxiliary propulsion). ..............................................................20

Figure 12. Forward-flight performance of XH-59A with auxiliary propulsion. .....................................................................20

Figure 13. Hover performance of CH-47D: single rotor on whirl stand (top) and tandem rotors in flight (bottom). ............. 21

Figure 14. Forward-flight performance of CH-47D helicopter; Mtip = 0.69 (top), 0.63, and 0.60 (bottom). .......................... 21

Figure 15. Rotor-blade planform and twist, designed for hover and cruise conditions of coaxial configuration. .................. 22

Figure 16. Influence of twist on hover and cruise performance (baseline coaxial configuration). .........................................22

Figure 17. Influence of taper on hover and cruise performance (baseline coaxial configuration). .........................................22

Figure 18. Influence of tip sweep on hover and cruise performance (baseline coaxial configuration). ..................................22

Figure 19. Influence of rotor-shaft angle on cruise performance at 250 knots. .......................................................................22

Figure 20. Influence of lift offset on cruise performance at 250 knots. ..................................................................................22

Figure 21. Cruise performance of coaxial lift-offset rotorcraft . ..............................................................................................23

Figure 22. Span loading as a function of lift offset; V = 250 knots. ........................................................................................ 23

Figure 23. Span loading as a function of speed; offset = 0.25 . ..............................................................................................23

Figure 24. Cruise performance as a function of altitude. ........................................................................................................24

Figure 25. Rotor-blade loading CT/σ as a function of speed and altitude. ............................................................................... 24

Figure 26. Influence of rotor-wake model on cruise performance. .........................................................................................24

Figure 27. Rotor-wake models, V = 250 knots, V/Vtip = 0.70; trailed vorticity of two rotors and wing. ................................. 25

Figure 28. Influence of rotor-wake model on cruise performance. .........................................................................................26

Figure 29. Influence of rotor-drag model on cruise performance. ..........................................................................................26

Figure 30. Hover performance for coaxial configuration, varying vertical separation. ..........................................................26

Figure 31. Cruise performance for coaxial configuration, varying vertical separation. ..........................................................26

Figure 32. Hover performance for coaxial, tandem, and side-by-side configurations; constant rotor radius, varying

rotor separation. Tandem and side-by-side at z/D = 0.06 identical; side-by-side at z/D = 0 only for d/D > 1 . ....................... 27

Figure 33. Cruise performance for coaxial, tandem, and side-by-side configurations; constant rotor radius, varying

rotor separation. .......................................................................................................................................................................27

Figure 34. Hover performance for coaxial, tandem, and side-by-side configurations; constant disk loading, varying

rotor separation. Tandem and side-by-side at z/D = 0.06 identical; side-by-side at z/D = 0 only for d/D > 1 . ...................... 28

iv
LIST OF FIGURES (cont.)

Figure 35. Cruise performance for coaxial, tandem, and side-by-side configurations; constant disk loading, varying

rotor separation. .......................................................................................................................................................................28

Figure 36. Cruise performance for coaxial, tandem, and side-by-side configurations; constant rotor radius. ........................ 29

Figure 37. Cruise performance for coaxial, tandem, and side-by-side configurations; constant disk loading. ....................... 29

Figure 38. Influence of rotor lift share on cruise performance for coaxial configuration, including compound helicopter. .. 30

LIST OF TABLES

Table 1. Parameters of baseline coaxial rotorcraft configuration ............................................................................................12

Table 2. Momentum theory for coaxial rotors in hover ...........................................................................................................12

Table 3. Ideal induced power for coaxial and tandem rotors in forward flight .......................................................................12

Table 4. Coaxial and tandem rotors for correlation .................................................................................................................12

Table 5. Coaxial configuration performance as a function of lift offset ..................................................................................13

Table 6. Performance of coaxial, tandem and side-by-side configurations, for fixed rotor radius ..........................................14

Table 7. Performance of coaxial, tandem, and side-by-side configurations, for fixed disk loading .......................................15

Table 8. Coaxial configuration performance as a function of rotor lift share, including compound helicopter ......................16

v
vi
NOMENCLATURE N number of blades

P aircraft power

A disk area (one rotor)

Pl

lower-rotor power

Ab blade area (all rotors)

Pref reference power, TT /2pAp

 and

Ap projected area of twin rotors, (2–m)l

T 2 /(2pAV) in cruise

Ar reference area; A for coaxial configuration,

and 2A for tandem configuration P. upper-rotor power

c blade chord Pi

induced power

cd mean drag coefficient, 8(CPo/6)1A#)

Po profile power

CT rotor thrust coefficient T l A V2) (p r tip

Pref reference power

CT/6 thrust coefficient divided by solidity, r blade radius

T /(pA b Vtip

2

) R rotor radius

CP power coefficient, P /(pArVtip

3

)

t/c blade thickness-to-chord ratio

CP/6 power coefficient divided by solidity, T rotor thrust

P /(pA b Vtip3)

TI

lower-rotor thrust

CW weight coefficient, W /(pArVtip2)

T. upper-rotor thrust

CW/6 weight coefficient divided by solidity,

W /(pAb Vtip2)

V flight speed

d horizontal separation between rotor hubs Vap rotor tip speed

(lateral or longitudinal) W gross weight

D rotor diameter, 2R W/A disk loading

D/q airframe drag divided by dynamic pressure W/S wing loading

f(μ) factor in profile power accounting for y rotor disk lateral coordinate

increase of rotor-blade mean dynamic

z vertical separation between rotors pressure with advance ratio

FM rotor-hover figure of merit, (T T / 2pAp

)/ P

17 propeller efficiency, TV/P

L/D aircraft effective lift-to-drag ratio, WV/P p

advance ratio, V/Vtip

(based on cruise power) p air density

L/De rotor effective lift-to-drag ratio, TV/(Pi+Po) 6 rotor solidity, Nc/πR

(based on rotor induced and profile power) RGW referred gross weight

m overlap ratio of twin rotors ISA International Standard Atmosphere

Mat advancing tip Mach number SLS Sea Level Standard

AMx/LR lift offset (differential rotor roll moment, as a

fraction of rotor lift times radius)

vii
viii
INFLUENCE OF LIFT OFFSET ON ROTORCRAFT PERFORMANCE

Wayne Johnson

Ames Research Center

ABSTRACT

The influence of lift offset on the performance of several rotorcraft configurations is explored. A lift-offset rotor, or

advancing blade concept, is a hingeless rotor that can attain good efficiency at high speed by operating with more lift on the

advancing side than on the retreating side of the rotor disk. The calculated performance capability of modern-technology

coaxial rotors utilizing a lift offset is examined, including rotor performance optimized for hover and high-speed cruise. The

ideal induced power loss of coaxial rotors in hover and twin rotors in forward flight is presented. The aerodynamic

modeling requirements for performance calculations are evaluated, including wake and drag models for the high-speed

flight condition. The influence of configuration on the performance of rotorcraft with lift-offset rotors is explored,

considering tandem and side-by-side rotorcraft as well as wing-rotor lift share.

INTRODUCTION

By operating a rotor in edgewise flight with lift offset—

more lift on the advancing side than on the retreating

side of the rotor disk—it is possible to attain good

performance at high forward speed. A conventional rotor

with an articulated hub is constrained to operate with

small hub moments. In forward flight, the retreating side

of the disk is not able to generate much lift because of

low dynamic pressure and stall, so for roll-moment

balance the advancing side is not allowed to generate

much lift either. The resulting load distribution over the

rotor disk is far from optimum for either induced or

profile power losses, and the rotor efficiency and lift

capability steadily decrease with forward speed. Even

hingeless and bearingless rotors are generally not

designed for the blades and hubs to carry significant roll

moment, and thus they encounter similar aerodynamic

performance limitations. However, a very stiff hingeless

rotor can be designed that will permit operation with

significant roll moment, say rotor-lift offsets of 20%.

Roll-moment balance of the entire aircraft requires either

twin main rotors or perhaps a wing. The coaxial

helicopter configuration with lift-offset rotors is known

as the Advancing Blade Concept (ABC).

The lift-offset concept was demonstrated for the coaxial

configuration (ABC) by the XH-59A flight

demonstration program of the 1970s (ref. 1). While

confirming the basic viability of the concept, the

aerodynamic performance of the XH-59A was

compromised by the choice of airfoils, planform, and

twist, as well as by high hub drag. In addition, the stiff

hingeless rotors led to a heavy hub design and high

vibration in flight. Recent interest in high-speed, heavylift rotorcraft makes it appropriate to reexamine the

capability of lift-offset rotors, including the impact of

current and advanced technology. The NASA Heavy Lift

Rotorcraft Systems Investigation (ref. 2) considered the

Large Advancing Blade Concept (LABC) as one of the

three configurations designed and analyzed. Sikorsky

Aircraft is exploring the ABC in the context of modern

technology, including the X2 flight demonstrator (ref. 3).

Interest has also been expressed in the possible

application of lift-offset rotors to other twin-rotor

configurations.

This paper has three objectives: First, the calculated

performance capability of modern-technology coaxial

rotors utilizing lift offset is examined, including rotor

performance optimized for hover and high-speed cruise.

Second, the aerodynamic modeling requirements for

performance calculations are established, including wake

and drag models for the high-speed flight condition.

Third, the influence of configuration on the performance

of rotorcraft with lift-offset rotors is explored,

considering tandem and side-by-side configurations as

well as wing-rotor lift share. The aircraft performance

was calculated using the comprehensive analysis

CAMRAD II. As foundation for these results,

performance metrics are discussed, and comparisons are

presented of calculated and measured performance of

coaxial and tandem rotorcraft.
BASELINE COAXIAL CONFIGURATION

The baseline configuration is a coaxial rotorcraft

utilizing lift-offset rotors, summarized in table 1 and

illustrated in figure 1. The aircraft definition is not the

product of a conceptual design analysis, but rather was

developed from basic system parameters. A heavy-lift

transport is considered, so a gross weight of 150,000 lb is

used. One conclusion of the NASA Heavy Lift

Rotorcraft Systems Investigation (ref. 2) was that the liftoffset rotor is best suited for cruise at moderate speeds

and altitude. Thus the design conditions here are takeoff

(hover) at atmospheric conditions of 5k/ISA+20°C, and

cruise at 250 knots and 5k/ISA+20°C. Based on

reference 2 and subsequent studies, it is appropriate to

design to a disk loading W/A = 15 lb/ft2 and cruise blade

loading CT/u = 0.10 (thrust-weighted). Thus the rotor

disk area is 10,000 ft 2 (one rotor), and the rotor diameter

112.8 ft. The vertical separation of the rotors is z/D =

0.06 for the baseline.

At 250 knots, auxiliary propulsion (preferably

propellers) is required. A small wing is used to mount the

propellers, and also unload the rotor in cruise (thereby

reducing the required rotor solidity). The rotor cruise

thrust is T/W = 0.8 for the baseline, with a wing loading

of 120 lb/ft2

. The result is a wing area of 250 ft 2

, and an

aspect ratio of 6 gives the wing span 38.7 ft.

Advanced airfoils are assumed, permitting an advancing

tip Mach number of Mat = 0.90, which is about 5%

greater than the optimum found using airfoils on current

rotorcraft. It is also assumed that for the thick root

sections of this hingeless rotor, airfoils can be designed

with drag and maximum lift similar to current 10–11%

thick rotor airfoils. A blade structural design (which has

not been done for this rotor) will define the required root

thickness, for which the airfoils must be designed. The

design criterion will be the thickness-to-radius ratio, so

the inverse taper of the present blade design implies a

larger thickness-to-chord ratio than for a tapered blade.

From the flight speed of 250 knots and Mat = 0.90, it

follows that cruise tip speed is 600 ft/sec, and the

advance ratio is V/Vtip = 0.70. Then cruise CT/u = 0.10

and T/W = 0.8 gives a solidity of u = 0.0871 for each

rotor, and u = 0.1742 for both rotors (based on the

projected disk area). Four blades per rotor gives a

reasonable blade aspect ratio.

A hover tip speed of 700 ft/sec corresponds to a cruise

rotor speed reduction of 14%, and results in a hover

blade loading of CW/u = 0.092.

The rotor power, rotor drag, and wing drag are calculated

using the comprehensive analysis, including interference

between the rotor and wing. To complete the calculation

of the aircraft performance, a fuselage and hub drag of

D/q = 50.0 ft 2 is used, and a propeller propulsive

efficiency of il = 0.90. The scaled fuselage and hub drag

is D/q/(W/1000)2/3 = 1.77. For comparison, typically

D/q/(W/1000) 2/3 = 1.4 for current turboprop aircraft, and

D/q/(W/1000) 2/3 = 0.85 for low-drag rotor hubs (ref. 2).

Based on the assumptions for the rotor airfoil

characteristics, the propeller propulsive efficiency, and

the level of fuselage and hub drag, the calculated aircraft

power is probably somewhat optimistic.

PERFORMANCE METRICS

Momentum Theory for Coaxial Rotor in Hover

A coaxial rotor has better hover efficiency than the

equivalent single rotor (no separation), primarily because

of the contraction of the upper-rotor wake before it

reaches the lower rotor. Canceling of swirl losses is a

small effect for helicopter rotor loadings. For the

performance of the aircraft in hover, elimination of the

tail-rotor power loss is a substantial benefit of the coaxial

configuration. Tip-vortex visualization on a Ka-32 (ref.

4) shows that the far-wake contraction is 85% for the

upper rotor and 91% for the lower rotor (for vertical

spacing z/D = 0.10), compared to 78% for a single rotor.

The upper-rotor contraction when it reaches the lower

rotor is 85%.

Consider coaxial rotors with area A of each rotor and

total thrust T =Tu +Tl

. Define the reference velocity as

vh2 = T/2ρA (based on area of a single rotor). For coaxial

rotors with zero vertical spacing (i.e., a single rotor with

the same total solidity), the momentum-theory solution

for ideal induced power is P = Tvh. For two separate

isolated rotors, the solution is P = 2(T/2) (T/2)/(2ρA)

= 2–1/2Tvh = 0.7071 Tvh .

Momentum theory is used to determine the induced

power for coaxial rotors with very large vertical

separation. Figure 2 illustrates the flow model at the

lower rotor. Then the lower rotor has no effect on the

upper rotor, and the momentum-theory solution for the

upper rotor is vu2= Tu/2ρA and Pu = Tuvu. The far-wake

velocity of the lower rotor wu = 2vu is uniform over the

cross-section area A/2. This far-wake velocity acts on the

lower rotor.

2
Momentum theory for the lower rotor follows the

derivation of section 3-2 of reference 5, with the addition

of the interference velocity wu = 2vu above the rotor.

Mass, momentum, and energy conservation then become:

m =

f

vtdA =

f

wtdS

Tt =

f

∆ptdA =

f ρw,dS − ρ(2vu

)

2

(A /2)

Pt = f ∆p

t

vdA =

f z

ρw,dS − 2ρ(2vu)

3

(A /2)

where S is the area in the far wake of the lower rotor.

Calculus of variations shows that the solution for

minimum power with constrained thrust is wt uniform

over the wake. Thus

m =

f vt

dA = wtS

2 2 Tt =

f

∆pt dA = ρSwt −2ρAvu

Pt = f ∆ptvtdA = 2 ρSw, −2ρAvu

3

Momentum theory does not give information about the

distribution of the induced velocity vt over the rotor

disk. Bernoulli’s equation can be used to relate the

loading on the rotor disk, ∆pt = dTt /dA, to the far-wake

velocity wt . Bernoulli’s equation is applied from far

above the rotor (where the pressure equals ambient) to

just above the rotor disk, and from just below the rotor

disk to far below (where the pressure again equals

ambient); and for stream lines starting from within and

without the upper-rotor wake (subscripts I and O, for

inboard and outboard, respectively). The result is

1 2 ∆ptO = 2 ρwt

12 2

∆ptI = 2 ρwt − 2ρvu

For an isolated rotor (i.e. without the effect of the upperrotor wake), uniform far-wake velocity wt implies

uniform disk loading ∆pt. For coaxial rotors, the loading

is significantly different in the inboard and outboard

regions, although uniform in each. Roughly the inboard

loading is 1/3 the outboard loading for this optimum

power solution. Let ∆pt = α(Tt

 /A), where Ttis the

lower-rotor thrust; so αI

AI + α

O

AO = A (AI

 and A O are

the inboard and outboard areas at the rotor disk, which

can be determined from mass conservation if vt is

known). Then αI = αO − vu

2

/(Tt /2ρA) . Define the mean

induced velocity v t = f vtdA, and a nonuniform loading

parameter α = f αvt dA /(vtA) ; so the power can be

written Pt = f ∆pt vt dA = α Tt v t

. For an isolated rotor,

the optimum solution is uniform disk loading, hence

α = 1; in general α is the average of the disk loading

weighted by the induced velocity, giving α > 1. With

these definitions, the conservation equations are:

m = vtA = wt

S

Tt = ρAvt wt − 2p-4 vu

2

Pt = 

z 

ρAvt wt

2 −2ρAvu

3

using the mass-flux relation in the momentum and

energy equations. For an isolated rotor ( vu = 0 and

α = 1), wtis easily eliminated and the usual solution for

the mean induced velocity obtained. Define the lowerrotor reference velocity by vr

2 = Tt /2ρA ; and recall that

vu

2 = Tu /2ρA , so vh2 = T /2ρA = vu2 + vr2

 . Eliminating wt

gives the relation

α vr

2

v t

2

+ vu

3

v t= (vu

2 + vr2 ) 2

Write vr = rvu , Tt = r

2Tu = τTu , and v t = svu . Then the

solution of

ατs

2

+ s = (1 + τ)

2

gives the total power P = Tuvu + α Ttv t = (1 + α τs)Tuvu

,

or

P = (1 +τ)

−3/2 (1 +ατs)Tv h

Given τ, the ratio of the lower and upper induced

velocities is

s = -(

1 + 4(1 +τ)

2

ατ −1 1

tar J

Note the thrust and power ratios are then

Tu /T = 1 /(1 + τ) and Pu /P = 1 /(1 + α τs) . Also, since

αI = αO −1 /τ, the inboard and outboard loading ratios

are αI = 1− AO /(Aτ) and αO = 1 + AI

/(Aτ).

The solution for equal thrust of the two rotors follows

from τ = 1; the solution for equal power of the two rotors

follows from α τs = 1. For equal thrust

1

s = ( 1 16 + α − 1 )

2 α

P = 2 −3/2 (1 + αs)Tvh

and for equal power

2 /(ατ) = (1 + τ)

2

P = (1 +τ)

−3/2 2Tvh

Table 2 gives the results for α = 1.00, 1.05, and 1.10 for

both equal-thrust and equal-power cases. Although the

difference between upper- and lower-rotor power or

thrust is substantial, the momentum-theory solution is a

weak function of the thrust ratio Tu/T, so the total power

depends primarily on the nonuniform loading parameter

α . The power is given relative to both no separation and

independent rotors; the coaxial-rotor solution is closer to

the no-separation case. Although it is the equal-loading

case α = 1.00 that is often found in the literature (e.g.,

section 3-5 of ref. 5), the loading on the lower rotor is far

from uniform. The infinite-separation solution is an

upper bound on the hover performance of the coaxial

3
configuration. The hovering coaxial rotor has at most

7–8% less induced power than the case of no vertical

separation.

A simple alternate approach is to consider the area of the

lower rotor that is outside the upper-rotor slipstream as

an extra active area of the rotor system (ref. 6). Thus for

large separation the effective area is 32 A, and P =

T T /(2ρ(3/2)A) = (2/3) 1/2Tvh = 0.8165Tvh. For finite

spacing with contraction ratio x, the effective area is Ae =

(2 –x

2

)A , and P = T T /(2ρAe) = (2–x

2

)

–1/2Tvh=

0.8847Tvh for 85% contraction. Reference 6 finds this

result consistent with measurements that show the figure

of merit for coaxial rotors to be 8–11 % higher than for

single rotors.

Biplane Theory for Twin Rotors in Cruise

In forward flight, the biplane effect reduces the induced

power of twin rotors at moderate speed, compared to the

induced power for no separation. From Munk’s stagger

theorem, this reduction occurs for tandem as well as

coaxial configurations, as long as the vertical separation

is measured in the wake.

The induced drag of a system of wings can be calculated

from the energy in the far wake. For the ideal case, there

is no rollup or distortion of the wake vorticity, so the

wake far downstream is represented by potential jumps

on lines that are projections of the wing geometry. The

induced drag is

Di

=− 

ρU

2

∫ φ^dS

2

where U is the free-stream velocity, φ the velocity

potential, and S the cross-section area of the wake with

normal n (ref. 7). The increment in potential across the

wake surface is related to the wing-bound circulation Γ:

∆φwake = ∫ γdx = Γ/U (integrating over the chord); and

the normal derivative ∂φ/∂n is the induced velocity v at

the wake:

∂φ 

=±0Z =±v = m 

1 dΓ dη

oh 2πU dη y −η

Thus

Di = 

ρ

2 ∫ Γv dy = 

1

2U

∫ lvdy

where t is the wing-section lift at span position y. This

result is not derived with any assumption about the

geometry of the wake surfaces far downstream. Hence

for multiple wings, the induced drag is the sum of the

drag on the m-th wing due to the wake of the n-th wing:

1

=

Di

= ∑ Dmn ∑ ∫

2U

lmvmn dy

The span loading can be written as a series:

L

t = ρU 2

b∑ Aksin(kθ) =

/ 4

∑ (Ak

 / A1) sin(kθ)

where b is the wing span, y = (b /2) sinθ, and the wing

total lift is L = (π/4)ρU

2

b

2A1 (ref. 8). For the induced

drag of a planar wing due to its own wake, the

integration can be performed analytically:

_ L

2

∑

_ Di

 (

k A

,r/2)pU2b2 ( k /A1

)

2

For a single planar wing, the minimum induced drag is

Di = L2 /((π/2)ρU

2

b

2

), obtained with elliptical loading

(just A1). The induced power is Pi= UDi

.

With the idealization of a rotor as an actuator disk

(circular wing), these results can be applied to a system

of rotors. Switching to rotor notation, the wing lift

becomes the rotor thrust T, speed is V, and the span b

equals the rotor diameter 2R. The minimum induced

power for a single rotor, obtained with uniform disk

loading (hence elliptical span loading), is Glauert’s result

Pideal = T

2

/2ρAV. In addition to neglecting the rollup and

distortion of the wake (as for fixed wings), for a rotor the

discretization of the wake with a finite number of blades

is also neglected when the induced power is evaluated

using this far-wake model. Figure 3 shows the span

loading and corresponding induced power (calculated

using these equations) for an articulated helicopter rotor

up to an advance ratio of V/Vtip = 0.4. The span loading

is far from elliptical at high advance ratio.

For twin main rotors, only the case of equal radius of the

two rotors is considered here. The reference power is the

ideal power of a single rotor (area A) with the total thrust

T of both rotors: P,.ef = T

2

/2ρAV. The total induced power

(sum of both rotors) is written Pi = CP,.ef. For no

separation of the rotors, vertical or lateral, C = 1; for

large separation, C = 1/2. From symmetry, the lowest

power is always obtained with equal thrust on the two

rotors.

For the coaxial or tandem configuration, let z/D be the

vertical spacing of the two rotors. The wake spacing far

downstream is assumed to equal z/D, although the

aircraft pitch angle will affect the wake spacing with

tandem rotors. Figure 4 shows power Pi/P,.ef as a

function of vertical separation, calculated using the

equations of this section. The optimum span loading was

found numerically by varying the span loading in terms

of the series in θ (from symmetry, the loading is the same

on the two wings). For comparison, the induced power

obtained assuming elliptical loading on each wing is

shown (this is the optimum solution for zero and infinite

4
spacing). Also shown in figure 4 is Prandtl’s biplane

result for elliptical loading (quoted in ref. 8, in terms of

the interference factor σ = 2C–1). Based on the idea that

the wing influences a volume of air contained in a

cylinder circumscribing the wing tips, Stepniewski (ref.

9) proposed that the tandem rotor interference be

estimated from the overlap area mA of the cylinders

about the two wings. Taking the effective area as Ae =

A (2–m) gives C = 1/(2–m). While a simple result, this

approach does not give as large an effect as does biplane

theory; for example, it gives Pi/P,.ef = 0.8677 at z/D =

0.12. Figure 5 compares the optimum loading and

elliptical loading for z/D = 0.12. Table 3 shows the

calculated values of Pi/P,.ef for the vertical spacings

considered in this paper (z/D = 0.06 and 0.12), as well as

for the XH-59A and Ka-26 coaxial helicopters. The

benefit of the vertical spacing is 8–20% reduction in

induced power (compared to zero spacing), which is a

significant effect at low speed, but is overwhelmed by

the effect of nonelliptical span loading at high speed.

For the side-by-side configuration, let d/D be the lateral

separation of the two rotors; the vertical separation is

zero. Figure 6 shows power Pi/P,.ef as a function of

lateral separation. The optimum span loading was found

numerically for d/D > 1. (from symmetry, the loading is

the same on the two wings). For d/D < 1, the optimum

loading is elliptical for the two rotors combined, hence

C = 1/(1+d/D) ; this loading is not, however, a practical

loading for d/D near 1.0. For comparison, the induced

power obtained assuming elliptical loading on each wing

is shown (this solution is the optimum solution for zero

and infinite spacing). The increased effective span of the

side-by-side configuration significantly reduces the

induced power.

Performance Metrics

The following performance metrics are used in this

paper. In these definitions, W is the aircraft weight; V the

cruise speed; T the total thrust of both rotors; P the total

aircraft power (rotor-shaft power in hover, rotor and

propeller power in cruise); and A is the area of one rotor.

a) Hover figure of merit: FM = (T T /2ρAp )/P , where

Ap = (2–m)A is the projected disk area (m is the overlap

ratio; m = 1 for coaxial and m = 0 for no overlap). Thus

for coaxial rotors the reference power is the ideal

induced power of a single rotor of area A (the limit of no

vertical separation).

b) Rotor effective lift-to-drag ratio: L/De = LV/(Pi+Po).

Here L is the wind-axis total rotor lift. The effective liftto-drag ratio is a measure of rotor efficiency since the

rotor parasite power is excluded.

c) Rotor-induced power: P/P,.ef, reference power

P,.ef =T T /2ρAp in hover (as for figure of merit) and

P,.ef = T2 /(2ρAV) in cruise. The cruise reference is thus

the ideal induced power of a single rotor of area A,

carrying the total rotor thrust T (the limit of no

separation, vertical or longitudinal or lateral). Generally

it is best to use as the reference power the ideal

momentum-theory power of the actual rotor

configuration, i.e., including the effect of vertical or

lateral separation of the two rotors. However, using a

reference power independent of rotor separation for the

present investigation means P/P,.ef provides an absolute

comparison of induced powers.

d) Rotor profile power: mean cd = 8(CPo/σ)/f(μ), where

f(μ) ≅ 1+4.5μ

2+1.61μ3.7 (ref. 5) accounts for the increase

in mean dynamic pressure with advance ratio. For the

cruise design condition, μ = 0.70 gives f = 3.64.

e) Aircraft lift-to-drag ratio: L/D = WV/P. Here only the

rotor-shaft power and auxiliary-propulsion power are

included in the total power P; other losses are not

considered.

PERFORMANCE CALCULATION

Analysis

Rotor performance was calculated using the

comprehensive rotorcraft analysis CAMRAD II (ref. 10).

CAMRAD II is an aeromechanics analysis of rotorcraft

that incorporates a combination of advanced

technologies, including multibody dynamics, nonlinear

finite elements, and rotorcraft aerodynamics. The rotor

structural-dynamics model is based on beam theory, with

exact kinematics for rigid-body and joint motions. The

rotor-aerodynamics model is based on second-order

lifting-line theory (steady two-dimensional airfoil

characteristics plus vortex wake), with unsteady

aerodynamic forces from thin airfoil theory, and

corrections for yawed-flow and swept blades. The

aerodynamic model includes a wake analysis to calculate

the rotor nonuniform-induced velocities, using rigid,

prescribed, or free-wake geometry. The rotor wake is

represented by a vortex lattice, with a small viscous core

for tip vortices and modeling of the wake-rollup process.

5
The trim task finds the equilibrium solution for a steadystate operating condition, and produces the solution for

performance, loads, and vibration. CAMRAD II has

undergone extensive correlation of performance

measurements on helicopters (ref. 10). Correlation for

coaxial and tandem rotors is presented in the following

section.

Rotor performance was calculated using nonuniform

inflow with prescribed wake geometry in high-speed

cruise and free-wake geometry in hover. Rotor/wing

interference was accounted for using a vortex-wake

model for both the rotor and the wing. The hover free

wake included complete interaction between the wakes

of the two rotors. The blade was modeled using 17

aerodynamic panels, with width ranging from 8%R at the

root to 3%R at the tip. Using a combination of Reynolds

number correction and drag increments for the airfoiltable data, the rotor mean drag coefficient of the baseline

design is cd = 0.0090 at the hover design condition. The

forward-flight wake model used a tip-vortex core radius

of 50% chord. The hover wake model used a tip-vortex

core radius of 20% chord initially, growing quadradically

with wake age to 120% chord after one revolution. An

elastic-blade model was used, based on the lift-offset

rotor design of reference 2, with fundamental flap and

lag frequencies of 1.55/rev and 1.50/rev, respectively at

hover rotor speed.

In cruise, the rotors are trimmed such that the net vertical

force of both rotors and the wing equals the aircraft

weight; the rotor-lift offset equals the specified value; the

rotor mean roll moment and both mean and differential

pitch moments are zero; and the wing lift equals the

specified lift share. The lift offset is defined as AMx/LR,

where AMx is the differential rotor roll moment, and L is

the sum of the lift of both rotors. It was established for

the work in reference 2 that trimming to zero hub

moments (except for differential roll moment) gives the

best performance. The variables adjusted to achieve this

trim are rotor mean collective, lateral and longitudinal

cyclic of each rotor, and the wing pitch angle. The trim

calculation is performed for fixed pitch angle of the rotor

shaft, as determined for best performance, while the

wing pitch angle is trimmed to obtain the required wing

lift. The calculation for this trim state gives the rotorshaft power, rotor drag force, and wing drag force.

Adding the fuselage and hub drag gives the required

propeller propulsive force, from which the propeller

shaft power is determined, hence the total aircraft power

(exclusive of losses).

In hover, mean and differential rotor collective are

adjusted such that the net rotor thrust equals the aircraft

weight, and the net torque of the two rotors is equal.

Coaxial Correlation

Table 4 summarizes the coaxial rotors used for

correlation of measured and calculated performance.

Harrington (ref. 11) tested two coaxial rotors in hover,

and Dingeldein (ref. 12) tested the first rotor of

Harrington in the wind tunnel. The two rotors differ

significantly in solidity and planform. Figures 7 and 8

compare calculated and measured hover performance for

rotor#1 and rotor#2, respectively. Figure 9 shows the

forward-flight performance. Both coaxial and single

rotors were tested. Generally the calculation of

performance is good, although better information about

the airfoil characteristics would be useful.

The XH-59A Advancing Blade Concept demonstrator

aircraft was tested in hover (ref. 13), in forward flight as

a helicopter (ref. 14), and in forward flight with auxiliary

propulsion (ref. 14). Figure 10 compares the calculated

hover performance with flight-test results. The

calculation of performance is good, although there is

significant scatter in the test data. The figure of merit is

higher than that of comparable single rotors, illustrating

the beneficial effect of the coaxial configuration on

hover performance. Figure 11 shows the forward-flight

performance of the XH-59A, operated with the rotors

providing all propulsive force as well as lift. The control

phase (CP in figure 11) refers to the mixing of lateral and

longitudinal cyclic to control the rotor. The flight tests

were conducted at referred gross weights of 11,000 and

13,000 lb. The calculation of power and aircraft L/D =

WY/P is good. Figure 12 shows the performance of the

XH-59A with auxiliary propulsion. The flight tests were

conducted at gross weights from 11,900 to 13,300 lb; the

calculations for 11,900 lb (shown in fig. 12) and 13,300

lb are similar. With lift offsets of 0.2 or 0.3, the

calculation of the rotor effective L/De is good. The ratio

of the calculated induced power to optimum momentum

theory power, Pi/Pops, shows the improvement in

efficiency produced by lift offset. Here Pops = 0.8594Pref

has been used (see table 3), so the induced power does

exhibit at low speed the expected reduction associated

with the coaxial configuration. Finally, profile power is

shown in figure 12, in terms of the mean drag coefficient

cd = 8(CPo/o)/f(u). At u = 0.6 the increase in mean

dynamic pressure gives f(u) = 2.86. There is a very

substantial increase in profile power with speed, even

with lift offset. The mean drag coefficient increases by a

factor of 2.2 at u = 0.6, hence the profile power increases

6
by a factor of 6.4. As a result of this profile power

increase, the rotor effective L/De decreases above 160

knots, so correlation with the XH-59A flight-test data

neither demonstrates the potential of the lift-offset rotor

to achieve good high-speed performance nor confirms

the calculation of the rotor-induced power.

Tandem Correlation

Table 4 also describes the tandem rotor used for

correlation of measured and calculated performance.

Flight-test data for the CH-47D helicopter are given in

reference 15. Figure 13 compares the calculated hover

performance with flight-test results, and also with single

rotor data (ref. 16). Figure 14 compares the calculated

performance with forward-flight test results at three

values of the tip speed and several CT levels (CT based

on total disk area 2A , not the projected area). The

calculation of performance is generally good, given the

difficulties obtaining rotor performance from flight-test

measurements of aircraft performance.

ROTOR PERFORMANCE OPTIMIZATION

For the baseline coaxial configuration, rotor planform

(sweep and taper) and twist variations are explored to

optimize the aircraft performance, balancing the

efficiency at the hover and cruise conditions. The twist

and taper distributions have four segments with linear

variation in each, and the breaks at 0.25R, 0.50R, and

0.75R. The twist is expressed in terms of equivalent rootto-tip linear rate. The taper is expressed in terms of

effective tip/root chord ratio. The sweep is defined in

two segments, from 0.75R to 0.9R and from 0.9R to the

tip. Twist exploration covered the range +3 to –24 deg.

Taper exploration covered the range 2.0 to 0.25. Sweep

exploration covered the range 0 to 35 deg.

The design choices for planform and twist, illustrated in

figure 15, are as follows:

a) Twist: –3/–6/–15/–18 deg

b) Taper: 1.333/1.333/1.333/0.333

c) Sweep: 10/25 deg

where the values given run from inboard to outboard.

The cruise operating condition is a shaft angle of 3 deg

(tilted aft), and a lift offset of 0.25.

Figure 16 shows the hover and cruise performance for

variations of the twist about the design choice. The

performance is primarily sensitive to outboard twist,

unless large values are used for inboard twist (not

shown). In figure 16, the twist of the last segment

(“tw4”, 0.75R to 1.00R) has values from –12 to –24 deg;

the lines are for the twist of the third segment (“tw3”,

0.50R to 0.75R) being equal to that of the last segment,

–3 deg more or –6 deg more. A similar variation is

shown for the inboard twist. The design choice shown is

a compromise between cruise and hover.

Figure 17 shows the hover and cruise performance for

variations of the taper about the design choice. The

performance is sensitive primarily to the outboard taper.

In figure 17, the taper of the last segment (“taper4”) has

values of 0.667, 0.5, and 0.333 for values of the taper in

the third segment from 1.5 to 0.667. A similar variation

is shown for the inboard taper, with values in the second

segment of 2.0, 1.5, 1.333, 1.0, and 0.75.Taper of the last

segment is favorable for hover performance, while

inverse taper inboard has a small but favorable effect on

cruise performance.

Figure 18 shows the hover and cruise performance for

variations of the sweep about the design choice. The

sweep of the outboard segment (0.9R to tip) varies from

0 to 35 deg for each value of the inboard sweep

(“sweep1”, 0.75R to 0.9R). Sweep of the tip is favorable

for hover performance, with figure of merit continuing to

increase up to 35-deg sweep. The design choice is a

smaller value (25 deg), based on considerations of

structural loads and inter-rotor clearance. In addition, it

is known that with kinks in the sweep distribution,

lifting-line theory somewhat underpredicts the induced

power.

Figure 19 shows the variation of cruise performance with

shaft pitch angle (positive for aft tilt) at the design

condition of 250 knots. The aircraft performance

improves as the pitch angle increases. At 3-deg shaft tilt,

the rotor-shaft power is small but positive. Figure 20

shows the variation of cruise performance with lift offset.

At the design speed, most of the benefits are obtained at

a lift offset of 0.25.

COAXIAL CONFIGURATION

Figure 21 shows the cruise performance as a function of

flight speed and lift offset for the coaxial configuration

with the rotor planform and twist of figure 15. Table 5

provides details of the performance at the design cruise

speed of 250 knots and at hover. The rotor lift share is

specified as 0.8 at 250 knots, and the wing lift coefficient

is kept constant as speed varies in these calculations.

Above 200 knots, lift offset has a significant effect on

the rotor performance, reducing the induced power and

minimizing the profile power. Note the low value of

7
Pi

/Pref at 100 knots, reflecting the reduction of induced

power due to vertical separation of the rotors. The mean

cd accounts for the basic rise of profile power with

advance ratio, so it is clear that lift offset is able to delay

the effects of stall beyond the design speed.

Consequently, with a lift offset of 0.25, a rotor effective

lift-to-drag ratio of L/De = TY/(Pi+Po) = 10.4 is achieved

at 250 knots, and an aircraft lift-to-drag ratio of L/D =

WY/P = 6.2.

Figure 22 shows the span loading of the upper and lower

rotors (viewed as circular wings) as a function of lift

offset at 250 knots. The loading is far from elliptical,

hence the ratio Pi

/Pref is well above 1.0; lift offset

reduces the induced power by producing a small shift of

the loading to the advancing side. Figure 23 shows the

span loading as a function of speed at lift offset of 0.25,

illustrating the increase in asymmetry of the loading as

speed increases.

Figure 24 shows the cruise performance as a function of

flight speed and altitude. Figure 25 shows the

corresponding blade loading CT/σ. The design condition

is CT/σ = 0.10 at 5k/ISA+20°C. The break in blade

loading vs. speed is where the rotor speed starts to

reduce in order to maintain Mat = 0.90. The wing lift

coefficient is kept constant as the speed changes (but

varies with altitude), hence the blade loading increases at

low speed. The rotor L/De is relatively insensitive to

altitude until stall occurs—at lower speeds as the blade

loading increases. Thus the effect of altitude is to

decrease the speed capability of the lift-offset rotor; or

alternatively, to operate efficiently at higher altitudes, it

is necessary to increase the blade area in order to

maintain the same design blade loading.

vorticity inboard (not shown). An alternative is the

multiple-trailer model, which has a trailed vortex

emanating from the edges of all aerodynamic panels. The

final model introduces consolidation of these multiple

trailers, combining adjacent trailers of the same sign into

a single rolled-up line, located at the centroid of the

original trailers. Figure 26 shows that free-wake

geometry has almost no effect on the performance, as

expected since the advance ratio is so high. The multipletrailer model increases the induced power, reducing the

lift-to-drag ratios. The consolidation model further

reduces the efficiency. The multiple-trailer model

produces a better calculation of blade airloads for some

rotors (ref. 17), but test data for efficient lift-offset rotors

will be needed to establish the best wake model for

performance.

Figure 28 shows the influence of omitting the shed wake

from the induced-velocity calculation. Without the shed

wake, the induced power is significantly underpredicted.

It is the shed wake directly behind the blade that is

important, not the shed vorticity in the far wake.

The influence of the rotor-drag model is shown in figure

29. Without the radial drag term, the profile power is

significantly underpredicted. Note that the factor f(μ) in

cd = 8(CPo/σ)/f(μ) includes the radial drag contribution,

hence without the radial drag in the computation of

power it appears that the mean cd decreases with speed.

The yawed flow correction impacts the airfoil stall and

the effective Mach number at the swept tip, hence

without the correction the profile power is overpredicted.

While the radial drag and yawed flow corrections are

empirical models, they have a significant effect on the

calculated performance.

MODEL REQUIREMENTS

The influence of the rotor-wake model on cruise

performance is shown in figure 26 for the coaxial

configuration with lift offset of 0.25. The wake models

are illustrated in figure 27. Only the far-wake model (for

interactions with following blades) is shown; behind

each blade where the induced velocity is calculated there

is also a full vortex lattice. Note that the wing is

modeled, as well as the two rotors; the induced velocities

from all wakes are calculated at the collocation points on

the blade (and on the wake elements for free-wake

geometry), accounting for rotor-rotor and wing-rotor

interference. The rolled-up wake model has a discrete tip

vortex emanating from each blade tip, with strength

defined by the peak bound-circulation and sheets of

TANDEM AND SIDE-BY-SIDE

CONFIGURATIONS

Next tandem and side-by-side configurations are

considered. The baseline is the coaxial configuration,

with baseline vertical spacing z/D = 0.06. Tandem rotors

with vertical spacing of z/D = 0.06 or 0.12 are examined,

as a function of longitudinal separation d/D = 0 (coaxial)

to 1.0 (no overlap). Side-by-side rotors with vertical

spacing of z/D = 0.06 and 0 are examined as a function

of lateral separation d/D = 0 (coaxial) to 1.25; with zero

spacing, results are presented only for d/D > 1 (no

overlap). With constant rotor radius, the disk loading

reduces as the separation is increased, reaching 7.5 lb/ft 2

(half the coaxial value) at d/D = 1. With constant disk

loading, the radius is reduced at d/D = 1 to a value of

8
2

-1/2 = 0.7071 times the coaxial value, and the rotor

solidity is twice the coaxial value (the blade area is

maintained, with half the disk area). Note that for hover

the tandem and side-by-side configurations with z/D =

0.06 are identical in these calculations. Also, the same

total fuselage and hub drag is used for all configurations

in cruise. The tandem configuration might have a lower

hub drag than the coaxial configuration, but increased

pylon drag for z/D = 0.12. The side-by-side configuration

would have increased drag from whatever structure

supports the two rotors. Such differences in aircraft drag

would influence the comparison of the configurations.

The hover and cruise performance of the coaxial

configuration as a function of vertical spacing is shown

in figures 30 and 31. The expected reduction in induced

power and increase in hover figure of merit as the

spacing increases is observed (fig. 30). The cruise

performance shows less influence of vertical spacing

(fig. 31), since the induced power is dominated by the

lateral asymmetry of the loading.

Figures 32 and 33 and table 6 show the hover and cruise

performance as a function of horizontal separation for

constant rotor radius. The hover figure of merit (fig. 32)

shows an initial decrease with separation, as the coaxial

effect is lost. Because the disk loading decreases, the

hover power decreases substantially as the separation is

increased. The cruise performance (fig. 33) shows little

effect of longitudinal separation of the rotors. Lateral

separation of the rotors increases the effective span of the

lifting system, so the side-by-side configuration has

about a 10% improvement in performance for

separations greater than d/D = 0.5 (assuming fixed

fuselage and hub drag).

Figures 34 and 35 and table 7 show the hover and cruise

performance as a function of horizontal separation for

constant disk loading. The hover figure of merit (fig. 34)

decreases with separation, largely because of the

decrease in blade aspect ratio. In cruise (fig. 35) the

efficiency is degraded with constant disk loading,

because of the reduced span of the lifting system and the

increased rotor solidity. Although the trends in figure 35

are understandable, the calculated performance is erratic

because of difficulties obtaining a converged inflow

solution with very low aspect-ratio, high-solidity blades.

Figures 36 and 37 compare the cruise performance as a

function of speed for four aircraft: coaxial (z/D = 0.06),

tandem with d/D = 0.75 (z/D = 0.06 and 0.12), and sideby-side with d/D = 1.15 (z/D = 0). Longitudinal

separation has little effect on the cruise performance for

constant radius, while the increased effective span of the

side-by-side configuration improves the performance

(assuming fixed fuselage and hub drag). Allowing the

disk loading to decrease as the rotor separation increases

is the best design approach.

WING-ROTOR LIFT SHARE

At 250 knots, auxiliary propulsion is required. A small

wing is used to mount the propellers, and this wing can

also unload the rotor in cruise, thereby reducing the

required rotor solidity. The baseline configuration has a

rotor lift share of T/W = 0.8 (ratio of rotor thrust to gross

weight) at 250 knots, with a wing loading of 120 lb/ft 2

.

For variations in flight speed, the calculations use a

constant wing lift coefficient to define the wing lift.

Figure 38 and table 8 show the influence of rotor lift

share on the cruise performance of the coaxial

configuration. For rotor lift share above 0.8, the wing

size is kept constant (it is still needed to support the

auxiliary propulsion), so the unloaded wing contributes a

small drag. As the rotor lift share increases, it is

necessary to increase the blade solidity in order to

maintain the design cruise blade loading at CT/o = 0.10.

Unloading the rotor too much is not consistent with using

lift offset, so rotor lift shares are considered only down

to T/W = 0.6. As the rotor lift share decreases, the rotor

efficiency L/De at 250 knots decreases, because both

PZ/Pref and mean cd increase. However, the total aircraft

drag decreases as the lift share decreases, and hence the

aircraft lift-to-drag ratio L/D = WV/P increases,

reflecting the efficiency of generating lift by means of a

fixed wing.

Also shown in figure 38 and table 8 is the calculated

performance of a compound helicopter. The parameters

of this design follow from a disk loading of W/A = 15

lb/ft2, wing loading W/S = 100 lb/ft2

, hover CT/o = 0.148,

and wing span equal to rotor diameter (based on results

of refs. 2 and 18). The rotor-blade twist is 0/0/–12/–12;

the blade has uniform linear taper of 0.8 and no tip

sweep. Following reference 18, the rotor lift is 10,000 lb

(T/W = 0.067), resulting in a small positive rotor-shaft

power. The compound helicopter has total aircraft power

comparable to the lift-offset rotor with 0.6 lift share, both

somewhat better than the lift-offset rotor at 0.8 lift share.

In order to compare lift-offset rotors and compound

helicopters in terms of total aircraft metrics, it will be

necessary to consider component weights as well as

performance, including the basic trade between the

weight of a large wing and the weight of a rotor designed

to carry lift offset. Note also that the present comparison

9
has been made at the design environment (5k/ISA+20°C)

of the lift-offset rotor. At higher altitudes the efficiency

of the compound helicopter is expected to be better

(ref. 2).

CONCLUSIONS AND RECOMMENDATIONS

The calculated performance capability of coaxial rotors

utilizing lift offset has been examined, including modern

technology and rotor performance optimized for hover

and high-speed cruise. Lift offset of about ∆Mx/LR =

0.25 is effective in reducing the rotor-induced power and

minimizing the rotor profile power, resulting in a rotor

effective lift-to-drag ratio of L/De = TY/(Pi+Po) = 10.4

and an aircraft lift-to-drag ratio of L/D = WY/P = 6.2 at

the design cruise conditions of 250 knots and

5k/ISA+20°C. Based on the assumptions for the rotor

airfoil characteristics, the propeller propulsive efficiency,

and the level of fuselage and hub drag, this calculated

performance is probably somewhat optimistic.

The aerodynamic modeling requirements for

performance calculations have been evaluated, including

rotor wake and drag models for the high-speed flight

condition. The design cruise condition is at a high

advance ratio, so free-wake geometry is not required.

Using multiple trailers instead of a rolled-up wake model

results in a lower calculated rotor efficiency, and the

wake geometry consolidation model reduces the

efficiency further. The radial drag increases the power

required, while the yawed flow effects on drag and stall

reduce the power required. Confirmation of the adequacy

of these models requires wind tunnel test data.

The influence of configuration on the performance of

rotorcraft with lift-offset rotors has been explored,

considering vertical separation, tandem and side-by-side

configurations, and wing-rotor lift share. Hover

performance is dominated by the variation of disk

loading with twin-rotor separation and rotor radius (disk

loading based on projected total rotor area). The

expected increase in hover figure of merit for the coaxial

configuration is observed in the calculated performance.

Cruise performance is insensitive to longitudinal

separation of the rotors, so the coaxial and tandem

configurations have nearly the same power required and

lift-to-drag ratio L/D = WY/P. Lateral separation of the

rotors increases the effective span of the lifting system,

so the side-by-side configuration has about a 10%

improvement in performance for separations greater than

d/D = 0.5 (assuming fixed fuselage and hub drag). While

comparing configurations for constant rotor radius means

that the disk loading decreases with separation of the two

rotors, the efficiency is degraded with constant disk

loading because of the reduced span of the lifting system

and the increased rotor solidity. As the rotor lift share

T/W decreases, the rotor efficiency L/De at 250 knots

decreases, but the total aircraft drag decreases and the

aircraft lift-to-drag ratio L/D = WY/P increases,

reflecting the efficiency of generating lift by means of a

fixed wing. The compound helicopter has total aircraft

power comparable to the lift-offset rotor. In order to

compare lift-offset rotors and compound helicopters in

terms of total aircraft metrics, it will be necessary to

consider component weights as well as performance.

To further this exploration of lift-offset rotors, airfoils

should be designed for hover and the unique cruise

environment of the blades, and the performance

evaluated utilizing these new airfoils. The calculated

rotor performance and wing-drag results should be

incorporated in a conceptual design analysis, and the

effect of configuration examined in terms of the

complete aircraft, including the effect of separation and

lift share on the weights. Finally, wind tunnel tests of

advanced lift-offset rotors are needed in order to confirm

the calculated performance and continue development of

the analytical models.

10
REFERENCES

1. Ruddell, A.J.; and Macrino, J.A.: Advancing Blade

Concept (ABC) High Speed Development. AHS 36th

Annual Forum, Washington, D.C., May 13–15, 1980.

2. Johnson, W.; Yamauchi, G. K.; and Watts, M. E.:

NASA Heavy Lift Rotorcraft Systems Investigation.

NASA/TP–2005-213467, Dec. 2005.

3. Eadie, W. J.; Sweeney, G.; and McClaren, H. E.: X2

Technology Applied to the Armed Attack

Reconnaissance and Escort Mission. AHS 62th Annual

Forum, Phoenix, Ariz., May 2006.

4. Akimov, A. I.; Butov, V. P.; Bourtsev, B. N.; and

Selemenev, S. V.: Flight Investigation of Coaxial Rotor

Tip Vortex Structure. AHS 50th Annual Forum,

Washington, D.C., May 11–13, 1994, pp. 1431–1449.

5. Johnson, W.: Helicopter Theory. Princeton, New

Jersey: Princeton University Press, 1980.

6. Bourtsev, B. N.; Kvokov, V. N.; Vainstein, I. M.; and

Petrosian, E. A.: Phenomenon of a Coaxial Helicopter

High Figure of Merit at Hover. 23rd European Rotorcraft

Forum, Dresden, Germany, Sept. 16–18, 1997, pp. 86.1–

86.11.

7. Landahl, M.; and Ashley, H.: Aerodynamics of Wings

and Bodies. Reading, Massachusetts: Addison-Wesley

Publishing Company, Inc., 1965.

8. Glauert, H.: The Elements of Aerofoil and Airscrew

Theory. Cambridge: Cambridge University Press, 1947.

9. Stepniewski, W. Z.; and Keys, C. N.: Rotary-Wing

Aerodynamics. New York: Dover Publications, Inc.,

1984.

10.Johnson, W.: Rotorcraft Aeromechanics Applications

of a Comprehensive Analysis. Heli Japan 98: AHS

International Meeting on Advanced Rotorcraft

Technology and Disaster Relief, Gifu, Japan, Apr. 21–

23, 1998.

11. Harrington, R. D.: Full-Scale-Tunnel Investigation of

the Static-Thrust Performance of a Coaxial Helicopter

Rotor. NACA TN 2318, Mar. 1951.

12. Dingeldein, R. C.: Wind-Tunnel Studies of the

Performance of Multirotor Configurations. NACA TN

3236, Aug. 1954.

13. Arents, D. N.: An Assessment of the Hover

Performance of the XH-59A Advancing Blade Concept

Demonstration Helicopter. USAAMRDL TN-25, May

1977.

14. Ruddell, A. J.: Advancing Blade Concept (ABC)

Technology Demonstrator. USAAVRADCOM TR 81-D5, Apr. 1981.

15. Bender, G. L.; Yamakawa, G. M.; Herbst, M. K.;

Sullivan, P. J.; Robbins, R. D.; and Williams, R. A.:

Airworthiness and Flight Characteristics Test (A&FC) of

the CH-47D Helicopter: Final Report. USAAEFA

Project No. 82-07, Feb. 1984.

16. Kocurek, J. D.; Berkowitz, L. F.; and Harris, F. D.:

Hover Performance Methodology at Bell Helicopter

Textron. AHS 36th Annual Forum, Washington, D.C.,

May 13–15, 1980, 48 pp.

17. Yeo, H.; and Johnson, W.: Assessment of

Comprehensive Analysis Calculation of Airloads on

Helicopter Rotors. J. Aircraft, vol. 42, no. 5, Sept.–Oct.

2005, pp. 1218–1228.

18. Yeo, H.; and Johnson, W.: Optimum Design of a

Compound Helicopter. Heli Japan 2006, AHS

International Meeting on Advanced Rotorcraft

Technology and Life Saving Activities, Nagoya, Japan,

Nov. 15–17, 2006.

11
TABLE 1. PARAMETERS OF BASELINE COAXIAL

ROTORCRAFT CONFIGURATION

TABLE 2. MOMENTUM THEORY FOR COAXIAL

ROTORS IN HOVER

Gross weight (lb) 150000

Hover and cruise atmosphere 5k ISA+20°C

Cruise speed (kt) 250

Rotor diameter (ft) 112.8

Disk loading W/A (lb/ft2) 15

Cruise CT/σ 0.100

Maximum Mat 0.90

cruise tip speed (ft/sec) 600

advance ratio, V/Vtip 0.70

Hover tip speed (ft/sec) 700

cruise/hover rotor speed 0.86

hover CW/σ 0.092

Solidity (one rotor) 0.0871

Number blades per rotor 4

chord (75%R, ft) 3.86

Fuselage+hub drag D/q (ft2

) 50.0

(D/q)/( W/ 1000)2/3 1.77

Wing loading (lb/ft2

) 120

area (ft2

) 250

span (ft) 38.7

cruise wing lift coefficient 0.7

Propeller propulsive efficiency 0.9

Vertical spacing, z/D 0.06

power reference Pref

no sep ind. rotors

Tu/T Pu/P Pi/Pref Pi

/Pref

no 0.5 0.5 1.0 1.4142

infinite

separation

α =1.10

equal T 0.5 0.3765 0.9382 1.3282

equal P 0.6024 0.5 0.9352 1.3226

α =1.05

equal T 0.5 0.3832 0.9226 1.3048

equal P 0.5962 0.5 0.9208 1.3022

α =1.00

equal T 0.5 0.3904 0.9056 1.2808

equal P 0.5898 0.5 0.9058 1.2810

Ind. rotors 0.5 0.5 0.7071 1.0

TABLE 3. IDEAL INDUCED POWER FOR COAXIAL

AND TANDEM ROTORS IN FORWARD FLIGHT

elliptical optimum

loading loading

z/D Pi

/Pref Pi/Pref

0.06 0.8779 0.8724

XH-59A 0.0694 0.8650 0.8594

Ka-26 0.09 0.8397 0.8339

0.12 0.8078 0.8023

TABLE 4. COAXIAL AND TANDEM ROTORS FOR CORRELATION

Aircraft XH-59A Harrington #1 Harrington #2 CH-47D

Configuration coaxial coaxial coaxial tandem

Radius (ft) 18 12.5 12.5 30

Number of blades per rotor 3 2 2 3

Solidity (both rotors) 0.127 0.054 0.152 0.0849

Twist (deg) –10 0 0 –12

Taper ratio 0.5 0.305 1 1

Separation, z/D 0.0694 0.0932 0.08

Overlap, 1–d/D 0.35

Airfoil NACA 0026 / t/c = 0.30 to 0.12 t/c = 0.30 to 0.15 VR7 / VR8

63218a / 23012

12
TABLE 5. COAXIAL CONFIGURATION PERFORMANCE AS A FUNCTION OF LIFT OFFSET

lift offset 0.15 0.20 0.25 0.30 hover

flight speed knots 250 250 250 250 0

rotor tip speed ft/sec 600.3 600.3 600.3 600.3 700

V/Vtip 0.7029 0.7029 0.7029 0.7029 0

advancing tip Mat 0.8997 0.8997 0.8997 0.8997 0.6163

shaft pitch angle deg 3 3 3 3

collective deg 13.51 10.66 7.55 4.36 14.14

pedal deg 0 0 0 0 -0.33

lower rotor lat cyclic deg 5.92 5.91 5.91 5.89

lower rotor long cyclic deg -15.95 -12.24 -8.09 -3.80

upper rotor lat cyclic deg 6.01 5.91 5.96 5.92

upper rotor long cyclic deg -15.95 -12.20 -7.95 -3.69

wing angle deg 6.87 6.86 6.69 6.56

lower rotor long flap deg 0.22 0.12 0.04 -0.01

lower rotor lat flap deg 0.70 1.08 1.47 1.90

upper rotor long flap deg 0.19 0.10 0.02 -0.01

upper rotor lat flap deg 0.70 1.08 1.48 1.90

lift offset, AMx/LR 0.151 0.200 0.252 0.301

aircraft lift lb 149492 149972 149681 149939

total rotor lift lb 119585 120086 119792 120065 149874

lower rotor lift lb 61217 60475 60101 59792 68355

upper rotor lift lb 58368 59611 59690 60273 81519

wing lift lb 29907 29886 29889 29874

aircraft drag lb 21311 20746 20010 20116

total rotor drag lb 10806 10212 9560 9728

lower rotor drag lb 5303 5132 4860 4787

upper rotor drag lb 5503 5080 4699 4940

wing drag lb 2000 2030 1945 1884

profile drag lb 362 364 364 364

induced drag lb 1143 1144 1137 1132

interference drag lb 494 522 444 387

fuselage drag lb 8505 8505 8505 8505

total rotor power hp 3777 2239 1470 969 21806

propulsive power hp -8290 -7834 -7334 -7463

ind+int power hp 4214 3553 3339 3312 19459

profile power hp 7853 6520 5466 5120 2348

Po+Pi hp 12067 10073 8804 8432 21806

aux power (TV/η) hp 18166 17685 17057 17148

aircraft power hp 21943 19923 18527 18117 21806

wing lift coefficient 0.7033 0.7028 0.7029 0.7025

wing drag coefficient 0.0470 0.0477 0.0457 0.0443

rotor total CT/6 0.0997 0.1000 0.0997 0.0998 0.0919

mean cd = 8(CPo/σ)/f(μ) 0.01311 0.01088 0.00913 0.00855 0.00905

Pi

/Plef 2.996 2.505 2.366 2.336 1.140

hover figure of merit 0.783

rotor L/De = LV/(Po+Pi) 7.603 9.146 10.438 10.924

aircraft L/D = WV/P 5.244 5.776 6.211 6.352

13
TABLE 6. PERFORMANCE OF COAXIAL, TANDEM AND SIDE-BY-SIDE CONFIGURATIONS, FOR FIXED

ROTOR RADIUS

configuration coaxial tandem tandem side-byside

coaxial tandem tandem side-byside

separation d/D 0 0.75 0.75 1.15

vertical z/D 0.06 0.06 0.12 0

overlap area factor m 1 0.1443 0.1443 0

radius ft 56.42 56.42 56.42 56.42

disk loading lb/ft2

15 8.08 8.08 7.50

solidity (one rotor) 0.0871 0.0871 0.0871 0.0871

flight speed knots 250 250 250 250 0 0 0 0

rotor tip speed ft/sec 600.3 600.3 600.3 600.3 700 700 700 700

V/Vtip 0.7029 0.7029 0.7029 0.7029 0 0 0 0

advancing tip Mat 0.8997 0.8997 0.8997 0.8997 0.6163 0.6163 0.6163 0.6163

shaft pitch angle deg 3 3 3 3

collective deg 7.55 7.85 7.80 5.49 14.14 12.40 12.37 11.89

pedal deg 0 0 0 0 -0.33 -0.30 -0.33 0

lower rotor lat cyclic deg 5.91 5.38 5.52 4.85

lower rotor long cyclic deg -8.09 -8.34 -8.37 -5.45

upper rotor lat cyclic deg 5.96 5.54 5.54 4.82

upper rotor long cyclic deg -7.95 -8.30 -8.21 -5.46

wing angle deg 6.69 6.51 6.42 5.52

lower rotor long flap deg 0.04 -0.01 -0.05 0.02

lower rotor lat flap deg 1.47 1.51 1.51 1.47

upper rotor long flap deg 0.02 0.01 0.01 0.02

upper rotor lat flap deg 1.48 1.45 1.45 1.47

lift offset, AMx/LR 0.252 0.250 0.250 0.250

aircraft lift lb 149681 149881 149809 149715

total rotor lift lb 119792 119986 119928 119747 149905 150131 150128 150135

lower rotor lift lb 60101 55883 55904 59791 68372 75876 76570 75055

upper rotor lift lb 59690 64103 64024 59955 81534 74255 73558 75079

wing lift lb 29889 29895 29881 29968

aircraft drag lb 20010 19906 19932 19097

total rotor drag lb 9560 9537 9619 9131

lower rotor drag lb 4860 4954 4907 4487

upper rotor drag lb 4699 4583 4712 4644

wing drag lb 1945 1864 1808 1461

profile drag lb 364 365 365 372

induced drag lb 1137 1140 1150 1120

interference drag lb 444 359 293 -31

fuselage drag lb 8505 8505 8505 8505

total rotor power hp 1470 1478 1369 753 21812 16628 16617 16644

propulsive power hp -7334 -7317 -7379 -7005

ind+int power hp 3339 3351 3262 2661 19464 14332 14329 14449

profile power hp 5466 5443 5487 5097 2348 2296 2287 2195

Po

+Pi

hp 8804 8794 8748 7758 21812 16628 16617 16644

aux power (TV/η) hp 17057 16968 16991 16279

aircraft power hp 18527 18446 18360 17032 21812 16628 16617 16644

wing lift coefficient 0.7029 0.7030 0.7027 0.7047

wing drag coefficient 0.0457 0.0438 0.0425 0.0344

rotor total CT/6 0.0997 0.0999 0.0999 0.0998 0.0919 0.0920 0.0920 0.0920

mean cd = 8(CPo/σ)/f(μ) 0.00913 0.00909 0.00916 0.00851 0.00905 0.00885 0.00881 0.00845

Pi

/Pref 2.366 2.352 2.490 1.646 1.140 1.141 1.141 1.194

hover figure of merit 0.783 0.755 0.756 0.727

rotor L/De = LV/(Po+Pi)

10.438 10.468 10.517 11.841

aircraft L/D = WV/P 6.211 6.239 6.268 6.757

14
TABLE 7. PERFORMANCE OF COAXIAL, TANDEM, AND SIDE-BY-SIDE CONFIGURATIONS, FOR FIXED

DISK LOADING

configuration coaxial tandem tandem side-byside

coaxial tandem tandem side-byside

separation d/D 0 0.75 0.75 1.15

vertical z/D 0.06 0.06 0.12 0

overlap area factor m 1 0.1443 0.1443 0

radius ft 56.42 41.42 41.42 39.89

disk loading lb/ft2

15 15 15 15

solidity (one rotor) 0.0871 0.1617 0.1617 0.1743

flight speed knots 250 250 250 250 0 0 0 0

rotor tip speed ft/sec 600.3 600.3 600.3 600.3 700 700 700 700

V/Vtip 0.7029 0.7029 0.7029 0.7029 0 0 0 0

advancing tip Mat 0.8997 0.8997 0.8997 0.8997 0.6163 0.6163 0.6163 0.6163

shaft pitch angle deg 3 3 3 3

collective deg 7.55 10.20 9.30 7.09 14.14 15.80 15.81 15.57

pedal deg 0 0 0 0 -0.33 -0.40 -0.47 0

lower rotor lat cyclic deg 5.91 5.86 5.68 5.37

lower rotor long cyclic deg -8.09 -10.24 -9.94 -6.32

upper rotor lat cyclic deg 5.96 6.22 5.72 5.41

upper rotor long cyclic deg -7.95 -10.15 -8.66 -6.24

wing angle deg 6.69 6.58 6.58 5.44

lower rotor long flap deg 0.04 0.00 -0.02 -0.01

lower rotor lat flap deg 1.47 0.92 0.94 0.84

upper rotor long flap deg 0.02 -0.03 0.04 0.03

upper rotor lat flap deg 1.48 0.91 0.90 0.83

lift offset, AMx/LR 0.252 0.252 0.250 0.247

aircraft lift lb 149681 149049 150348 150557

total rotor lift lb 119792 119061 120488 120609 149905 150131 150117 150000

lower rotor lift lb 60101 50974 52087 59286 68372 76141 76803 74997

upper rotor lift lb 59690 68086 68401 61323 81534 73990 73314 75003

wing lift lb 29889 29988 29861 29948

aircraft drag lb 20010 21562 16400 16114

total rotor drag lb 9560 11155 6004 6181

lower rotor drag lb 4860 5639 4150 3855

upper rotor drag lb 4699 5517 1854 2326

wing drag lb 1945 1902 1891 1429

profile drag lb 364 367 365 372

induced drag lb 1137 1144 1168 1122

interference drag lb 444 390 358 -66

fuselage drag lb 8505 8505 8505 8505

total rotor power hp 1470 4314 8688 7623 21812 23793 23839 25316

propulsive power hp -7334 -8558 -4606 -4742

ind+int power hp 3339 6870 7315 6718 19464 21552 21617 23298

profile power hp 5466 6002 5980 5647 2348 2241 2222 2018

Po+Pi

hp 8804 12872 13295 12365 21812 23793 23839 25316

aux power (TV/η) hp 17057 18380 13980 13736

aircraft power hp 18527 22694 22668 21359 21812 23793 23839 25316

wing lift coefficient 0.7029 0.7052 0.7022 0.7042

wing drag coefficient 0.0457 0.0447 0.0445 0.0336

rotor total CT/6 0.0997 0.0990 0.0999 0.0999 0.0919 0.0920 0.0920 0.0919

mean cd = 8(CPo/σ)/f(μ) 0.00913 0.01002 0.00998 0.00943 0.00905 0.00863 0.00856 0.00777

Pi

/Plef 2.366 2.639 2.981 2.048 1.140 1.260 1.264 1.364

hover figure of merit 0.783 0.719 0.717 0.675

rotor L/De = LV/(Po+Pi)

10.438 7.096 6.953 7.483

aircraft L/D = WV/P 6.211 5.071 5.077 5.388

15
TABLE 8. COAXIAL CONFIGURATION PERFORMANCE AS A FUNCTION OF ROTOR

LIFT SHARE, INCLUDING COMPOUND HELICOPTER

configuration Lift offset Lift offset Lift offset Compound

rotor lift share 1.0 0.8 0.6 0.067

radius ft 56.42 56.42 56.42 56.42

solidity (one rotor) 0.1089 0.0871 0.0654 0.0541

wing span ft 38.73 38.73 54.77 112.84

wing chord ft 6.46 6.46 9.13 12.41

hover tip speed ft/sec 700 700 700 700

hover CT/a

0.0735 0.0919 0.1226 0.1480

flight speed knots 250 250 250 250

rotor tip speed ft/sec 600.3 600.3 600.3 600.3

V/Vtip 0.7029 0.7029 0.7029 0.7029

advancing tip Mat 0.8997 0.8997 0.8997 0.8997

shaft pitch angle deg 3 3 3 3

collective deg 7.61 7.47 6.72 -5.43

lower rotor lat cyclic deg 6.67 5.92 5.02 0.53

lower rotor long cyclic deg -7.88 -7.97 -7.12 5.25

upper rotor lat cyclic deg 6.85 5.95 4.87 0.18

upper rotor long cyclic deg -7.79 -7.84 -7.07 5.18

wing angle deg -1.66 6.68 6.51 4.04

lower rotor long flap deg -0.04 0.04 0.08 0.15

lower rotor lat flap deg 1.88 1.48 1.09 -0.35

upper rotor long flap deg -0.05 0.03 0.05 0.12

upper rotor lat flap deg 1.88 1.47 1.07 -0.33

lift offset, AMx/LR 0.253 0.250 0.252 0

aircraft lift lb 150190 150211 149714 149485

total rotor lift lb 149180 120338 89899 9628

lower rotor lift lb 75609 60415 45132 5046

upper rotor lift lb 73572 59923 44767 4582

wing lift lb 1010 29874 59815 139857

aircraft drag lb 20746 19889 19602 17241

total rotor drag lb 11901 9443 7459 3375

lower rotor drag lb 5762 4741 3761 1662

upper rotor drag lb 6139 4702 3699 1713

wing drag lb 340 1941 3638 5360

profile drag lb 333 364 682 1735

induced drag lb 9 1135 2262 2951

interference drag lb -2 442 694 674

fuselage drag lb 8505 8505 8505 8505

total rotor power hp 1676 1555 944 2943

propulsive power hp -9130 -7244 -5723 -2590

ind+int power hp 4304 3373 2418 555

profile power hp 6503 5426 4249 4978

Po+Pi

hp 10806 8799 6667 5533

aux power (TV/ q) hp 17684 16954 16709 14696

aircraft power hp 19360 18509 17653 17639

wing lift coefficient 0.0237 0.7025 0.7033 0.5873

wing drag coefficient 0.0080 0.0457 0.0428 0.0225

rotor total CT/a

0.0992 0.1001 0.0999 0.0131

mean cd = 8(CPo/a)/f(μ)

0.00869 0.00906 0.00946 0.01338

Pi/Plef 1.954 2.353 3.023 60.467

rotor L/De = LV/(Po+Pi

) 10.591 10.492 10.346 1.335

aircraft L/D = WV/P 5.944 6.217 6.519 6.524

16
Figure 1. Illustration of baseline coaxial rotorcraft configuration, utilizing lift-offset rotors (courtesy G. Nunez, AFDD).

Figure 2. Momentum-theory model of coaxial rotors in hover, for lower rotor far below upper rotor, showing velocities and

areas in flow field and pressure on rotor disk. Velocities wu and wg are uniform across wake section; pressure Opt

and velocity v$ at lower-rotor disk are not uniform.

17
r,f r0 5' 10:0 10d51^ 1r0

'lateral distance; yIR'

Figure 5. Span loading for ideal induced power in

forward flight, twin-rotor vertical spacing z/D = 0.12.

0'5'i 0-20 0:30' 0440'

Figure 3. Span loading and corresponding induced power

of articulated helicopter rotor.

- - 

-001 061 14 1r51 ;21461

0' 0' O'Sti1 0' 21 0:3 '014

- - - - idlD^ -

Figure 6. Ideal induced power in forward flight as a

Figure 4. Ideal induced power in forward flight as a

function of twin-rotor lateral spacing.

function of twin-rotor vertical spacing.

^o'^o'^ ! I I I !

18
0'061 W2!1064, Q561 Otl2l

I

. C117C,

Figure 7. Hover performance of Harrington rotor#1

0661 062, ON 0561 Obal CiTd

Figure 8. Hover performance of Harrington rotor#2.

007 0-481 DY-09 0511

iCTf

Figure 10. Hover performance of XH-59A.

(0,66 040 C-126 O.-M

ly, - -

Figure 9. Forward-flight performance of Harrington

rotor#1 at CT/σ = 0.089.

19
v 20 (6_0_' 100' 140' 1,961

[airspeed (khdl )I

Figure 11. Forward-flight performance of XH-59A

(without auxiliary propulsion).

VN^;p

Figure 12. Forward-flight performance of XH-59A with

auxiliary propulsion.

20
30000' ^ ^ I I I

;20001 ^OOOI 160001 i8000^ '100001

ir;otor'^power^i,(hp^jl

Figure 13. Hover performance of CH-47D: single rotor

on whirl stand (top) and tandem rotors in flight (bottom).

LJ.^! ^^Ql ^^yr; L11L 1"Jti

it iC. U! `l!8^1

;speed ii rots,

Figure 14. Forward-flight performance of CH-47D

helicopter; Mtip = 0.69 (top), 0.63, and 0.60 (bottom).

21
^Qti761 ^078i i0''801 ^0'^821

hover;FM

b) inboard taper

Figure 17. Concluded.

ray'

C0''2

(al

Swi 0^,

4,

"g

ill

X0901 i0'^21 ^0 4 ;-r!R !0'61 i0'^'8i 1r•'OI

Figure 15. Rotor-blade planform and twist, designed for

hover and cruise conditions of coaxial configuration.

0'x761 0'77 0"801

hoverrFM

a) outboard twist

^0'^74 ^076^ ^0' 7i8^ (0'^'B01 ^0'^'821

hover,FM

Figure 18. Influence of tip sweep on hover and cruise

performance (baseline coaxial configuration).

0 74 y- 0'78! 0''801 0' 821

Ihov",er,FM

b) inboard twist

Figure 16. Influence of twist on hover and cruise

performance (baseline coaxial configuration).

(0121 4 (6

. itctliangle((deg)1

Figure 19. Influence of rotor-shaft angle on cruise

performance at 250 knots.

0_'74 0,761 076i 16,661 6821

hovi,6 FM

a) outboard taper

Figure 17. Influence of taper on hover and cruise

performance (baseline coaxial configuration).

0'541'5 0'.20 0'25 (0:301

ioffset`

Figure 20. Influence of lift offset on cruise performance

at 250 knots.

22
'100 ;1[ 0' ;200 ,250 300'

iflighit ispe.ed i, kriots }1

Figure 21. Cruise performance of coaxial lift-offset

rotorcraft.

^pci 0'5i 061 IV 14

I' ateralyd'is[arice; ilik,

Figure 22. Span loading as a function of lift offset;

V = 250 knots.

120001 - 2501knots].

20Qlknotsl

15ciknotsl

',15001 ^^i f1G111e - 1c1 -^

,"W 0_'S, idd C^_s 'idol

;lateraWisfance;lV R

Figure 23. Span loading as a function of speed;

offset = 0.25.

23
1451 2451 :2651 30

Figure 25. Rotor-blade loading CT/σ as a function of

speed and altitude.

0.-

'1001 '1[5,61 2061 :2^01 ^jXO

Figure 24. Cruise performance as a function of altitude.

1061 '1501 :2661 :2501 $3 Ml

ff^jNf.[spe^e^O#ImOsy

Figure 26. Influence of rotor-wake model on cruise

performance.

24
a) Rigid-wake geometry; rolled-up model. b) Free-wake geometry; rolled-up model.

c) Rigid-wake geometry; multiple-trailer model. d) Free-wake geometry; multiple-trailer model.

e) Free-wake geometry; multiple-trailer model f) Free-wake geometry; multiple-trailer model with

(only upper rotor shown). consolidation (only upper rotor shown).

Figure 27. Rotor-wake models, V = 250 knots, V/Vtip = 0.70; trailed vorticity of two rotors and wing.

25
(0'0001 ^ I I I !

'1001 ;150 ;2001 ;2661 3001

tiflightls06E d [knotsyl

Figure 28. Influence of rotor-wake model on cruise

performance.

- ---1001 1'501 ;2001 ;2501 $001

;fligh't`ispeed i{knots}1

Figure 29. Influence of rotor-drag model on cruise

performance.

^003^ 0'061 0909 X0;12! 0',^1'S

'z1D

Figure 30. Hover performance for coaxial configuration,

varying vertical separation.

OM! 10!06 10!09 W1'2 ( 1'5'

,Zb

Figure 31. Cruise performance for coaxial configuration,

varying vertical separation.

26
^o^ ^ I I I I !

isep_a[_ation^ ^dID^

Figure 32. Hover performance for coaxial, tandem, and

side-by-side configurations; constant rotor radius,

varying rotor separation. Tandem and side-by-side at

z/D = 0.06 identical; side-by-side at z/D = 0 only for

d/D > 1.

10 001 0 25 IOP 5 10;75 1 OOI 1 25

separ_6tio_"n,[dD

Figure 33. Cruise performance for coaxial, tandem, and

side-by-side configurations; constant rotor radius,

varying rotor separation.

27
(o^ ; I I I I !

0-00 0-26 O 0'-0 041$ 140 146)

i%rp ar6br^, CdO -

Figure 34. Hover performance for coaxial, tandem, and

side-by-side configurations; constant disk loading,

varying rotor separation. Tandem and side-by-side

at z/D = 0.06 identical; side-by-side at z/D = 0 only

for d/D > 1.

10900 10925 &$0 0575 100 ';125

Oc par a tion.,ld1Dl

Figure 35. Cruise performance for coaxial, tandem, and

side-by-side configurations; constant disk loading,

varying rotor separation.

28
200 ;250' 300'

iflight` ispeed i(knots }1

Figure 36. Cruise performance for coaxial, tandem, and

side-by-side configurations; constant rotor radius.

1;00' 1'S0'^ ;200' ;250' 300'

;flightispe_'ed ^(knot`s}1

Figure 37. Cruise performance for coaxial, tandem, and

side-by-side configurations; constant disk loading.

29
Figure 38. Influence of rotor lift share on cruise performance for coaxial configuration, including compound helicopter.

30