Blade with swept-back tip for the rotary wings of an aircraft

The present invention relates to a blade with a swept-back tip for a rotary wings of an aircraft, a blade being formed of successive elemental cross sections. The blade is divided into different regions, with the chord length L and the offset y'f of the center of pressure from the pitch-change axis and the relative thickness of each elemental section are optimized for high-speed flight. The first region has a linearly increasing chord length, the second region has the maximum and constant chord length, the third region has a linearly decreasing chord length and the fourth region has a decreasing chord length according to a parabolic function. The offset of the aerodynamic center is positive and increases in proportional to the distance in the first region, is at its maximum and constant in the second region, and decreases linearly and become negative in the third region.

The present invention relates to a blade for the rotary wings of an 
aircraft, with a swept-back tip, optimized for high-speed flight. 
It is known that both in hovering flight and in forward flight, the 
performance of a rotary-wing aircraft rotor, especially a helicopter 
rotor, is limited by the following phenomena: 
the shockwaves which develop on the suction face and pressure face of the 
advancing blades during high-speed flight; 
the stalling that results from the detachment of the boundary layer on the 
suction face of the retreating blades when there is a demand for lift in 
translational flight; 
the interaction of the vortex generated by the previous blade with the 
following blade, which leads to a substantial dissipation of energy in two 
forms: induced power and drag on the profiles, in hovering flight. 
In addition to being responsible for drops in performance, the shocks and 
the blade-vortex interaction are also responsible for acoustic problems in 
the form of pulsating noises caused by shock delocalization (high-speed 
flight) or by pulsating changes in lift when the marginal vortex directly 
strikes the blade (descent). 
It has been found that the performance of a blade for the rotary wings of 
an aircraft depends, to a large extent, on parameters associated with the 
construction of the blade, such as: 
a) the radial distribution of blade area; 
b) the back sweep of the blade tip; 
c) the change in relative thickness of the profiles; 
d) the distribution of the twist of the profiles; 
e) the blade tip droop. 
The influence that these parameters have on the performance of a 
rotary-wing blade is set out in detail below. 
a) Radial distribution of blade area 
For the rotor of a rotary-wing aircraft whose elemental sections or 
profiles all work with the same coefficient of lift Cz, the linear lift 
varies as the local chord length L(r) and as the square of the local 
speed, which is directly proportional to the radius (radial position) r of 
the section. This means that the total lift of the blade varies in 
proportion with the mean chord L defined by a square-law weighting of the 
radius r: 
##EQU1## 
in which RO represents the radius r at the start of the blade section at 
the root end of the blade, and R the total radius of the blade. 
It is common practice for the performance of different shapes of blades to 
be compared on the basis of this mean chord L. 
Compared with a conventional blade of rectangular shape, calculations have 
shown and experience has confirmed that reducing the chord at the outboard 
end of the blade (tapered shape) improves performance, particularly at 
high speed. This improvement in performance is achieved by the reduction 
in drag of the profiles due to the reduction in chord at the tip. Shocks 
in this region are exerted on a smaller area, while the central part of 
the blade which is not subjected to the shocks gives most of the lift with 
a maximum aerodynamic efficiency: the lift/drag ratio here is at a 
maximum. 
The increase in chord along the rest of the length of the blade, needed to 
keep the mean chord constant, is however considerable because of the 
r.sup.2 weighting. This then makes the rotor considerably heavier. 
Nonetheless, tapering the blade toward the tip is a means commonly used to 
improve blade performance, generally combined with the blade tip being 
swept back, as illustrated in patents FR-2 617 118 and FR-2 473 983. 
Another source, patent FR-2 311 713, proposes a very different construction 
which, among other particular features, consists in broadening the chord 
sharply beyond a section situated at approximately 87% of the total radius 
of the blade R. This arrangement contributes to the appearance of intense 
and stable vortices which push back the stall limit, particularly when the 
blade is retreating. However, this concept, which concentrates the blade 
area toward its tip, amounts to reducing the effective part of the rotor 
to a peripheral ring. This makes the induced flow less uniform and the 
induced power therefore increases, something which is a particular problem 
for taking off. 
The sharp variations in chord and the associated drawbacks are avoided with 
the blade design that forms the subject of patent FR-2 689 852 in the name 
of the Applicant Company. In this design, the region with longest chord is 
still close to the tip, again with the purpose of increasing the lift 
capabilities. However, this design is not ideal for high-speed flight with 
medium lift because, in these conditions, the sections with a long chord, 
close to the tip, suffer from increased drag because of compressibility 
phenomena, while the region with the greatest aerodynamic efficiency 
(lift/drag ratio) is further from the tip. Too small a proportion of the 
area of the blade is making use of this central region of the blade where 
the aerodynamic efficiency is at its maximum. 
b) Sweeping back the blade tip 
In addition, in order to push back the threshold at which shockwaves appear 
and to limit their intensity, it is advantageous for the blade tip to be 
curved backward. The sweep angle .LAMBDA. defined by the line of centers 
of pressure (in about the front quarter of the chord) and the pitch-change 
axis reduces the effective Mach number and thus sweeping back the blade 
tip constitutes an effective means of reducing the unfavorable 
consequences of the compressibility of air, especially the appearance of 
shockwaves. 
A swept-back blade tip of this kind is illustrated in particular in patents 
FR-2 311 713, FR-2 473 983 and FR-2 617 118, and is actually used on some 
helicopters. 
However, the size of the sweep angle and the amount of the blade span that 
the tip in question can occupy are in fact limited by the torsional forces 
which result from the aerodynamic lift and the center of gravity being 
offset rearward. 
Patent FR-2 311 713 proposes to offset part of the blade forward so as to 
balance the rearward offset of its tip. This makes it possible to extend 
the swept-back region to a larger part of the blade span. However, the 
join between the forwardly-offset leading edge portion and the inboard 
part of the blade is abrupt and the vortex that this join generates causes 
premature stalling of the profiles closest to the rotor hub. At high 
speed, the concave shape of the leading edge at the join focuses and 
locally strengthens the shockwaves with the risk of reducing or even 
nullifying the beneficial effects of the swept-back tip. 
Patent FR-2 397 328 also recommends that the leading edge be offset 
forward, but this is for a different reason. Here, it is not a question of 
reducing the torsional forces, but of causing elastic deformation of the 
blade in a controlled way which is deemed to be favorable. 
However, the effect of the sweep angle may become unfavorable at high speed 
if this angle exceeds a value of about 45.degree.. If the angle is too 
large, the air flows practically parallel to the leading edge when the 
blade is retreating and in the rear sector of the rotor disk during 
high-speed translational flight: this temporarily reduces the aerodynamic 
lift of the tip and increases its drag, hence resulting in an overall drop 
in efficiency. The swept-back blade that is the subject of patent FR-2 689 
852 may in particular suffer from this problem. 
c) Change in relative thickness of profiles 
The relative thickness of a blade cross section is defined as being the 
ratio of the absolute thickness to the chord length of the profile that 
constitutes the contour of this section, i.e. e/L. 
Conventional blades, which have a constant chord length over most of the 
length of their span, generally employ, at the blade root end, profiles 
that have a relative thickness of between 10 and 14%, which gives them 
sufficient torsional rigidity but does not give them good enough 
performance at high speed. 
In patent FR-2 689 852 for example, reducing the chord near the blade root 
is likely to improve the high-speed performance because the aerodynamic 
efficiency is not good at the blade root on account of the very great 
variations in aerodynamic incidence and the attack from the trailing edge 
suffered by profiles inside the region known as the "reverse-flow circle". 
However, reducing the chord dangerously reduces the rigidity of the blade, 
and the torsional deformations, exacerbated also by offsetting the tip 
from the pitch-change axis, assume an amplitude which is so great that 
performance diminishes or alternatively is not improved as much as it 
might be desired. 
d) Distribution of twist of profiles 
In addition to the radial distribution of chord length and the sweeping 
back of the blade tips, the distribution of the twist of the elemental 
sections of a blade also plays a part in improving the performance of this 
blade. The blade twist consists in varying the angle at which the profiles 
are set along the span of the blade, in such a way that the outboard end 
or tip of the blade attacks the air at a relatively low angle of 
incidence, and the blade root attacks the air at a higher angle of 
incidence. This makes it possible to distribute the lift more uniformly 
over the entire rotor area and thus the absorbed power in hovering flight. 
The twist is therefore characterized by the difference in angular setting 
between the two ends of the blade. However, it is known that a high degree 
of twisting may cause the outboard end of the blade to produce negative 
(downward) lift when the blade is advancing when the rotary-wing aircraft 
is traveling at high speed. Performance is thus degraded and, what is 
more, vibration increases considerably. 
The choice of twist is therefore a compromise between on the one hand, 
hovering flight and low speeds which require a high degree of twisting 
and, on the other hand, forward flight for which a more modest amount of 
twisting is desirable. 
For simplicity, the radial distribution of the twist is often linear, which 
means that the total twist is all that is required to define the angular 
setting of all the elemental sections. 
Nonetheless, to improve performance, patent FR-2 636 593 proposes a 
non-linear twist which consists in giving the outboard end of the blade, 
for example between 85 and 100% of the total radius of the blade R, an 
extra amount of twist. This has the effect of reducing the strength of the 
marginal vortex, or even of canceling it, for a given amount of lift, so 
that the low-speed performance is improved and the noises of blade/vortex 
interactions during descent are attenuated. However, this arrangement does 
not make it possible to push back the limits of stalling, and the power 
savings decrease at high speed. 
Patent FR-2 689 852 proposes a radical increase in the amount of twist in a 
central region lying approximately between 45 and 80% of the span. A 
modification of this kind, which is aimed at improving the lift 
capability, would not be optimal for a rotor whose objective was to 
increase the efficiency at high speed. 
e) Blade tip droop angle 
Conventionally, blades are constructed in such a way that the center of 
pressure of the profiles, generally defined as being the point at 
mid-thickness in the front quarter of the chord, remains more or less on 
the pitch-change axis, and does so along the entire length of the span. 
Furthermore, sweeping back the blade tip is generally achieved by shifting 
the center of pressure in the plane defined, on the one hand, by the 
pitch-change axis and, on the other hand, by the direction of the chord of 
the profile in the tip region. 
Described in patent FR-2 617 118 is an improvement to this conventional 
construction, which consists in making the line of centers of pressure lie 
in a plane that passes through the pitch-change axis, but at an angle to 
the chord, so that the centers of pressure of the profiles at the blade 
tip are appreciably lower than the inboard part of the blade. The marginal 
vortex generated by the outboard end of one blade is thus further from the 
following blade, the result of this being that it reduces the strength of 
the interaction, particularly in hovering flight. This leads to a clear 
reduction in the power absorbed by the rotor, especially in hovering and 
low-speed flight. 
This approach is denoted by the term "mise en diedre [drooping]" of the 
blade tip even though, in patent FR-2 617 118, the center of pressure is 
shifted progressively (in a parabolic way) rather than there being a 
simple "break" as also in patent FR-2 689 852. 
It can thus be seen that none of the above documents describes a blade 
structure which is entirely free of drawbacks, and in this context the 
present invention relates to a blade for rotary wings, with a swept-back 
tip, the geometry of which is designed to be optimal for guaranteeing the 
best performance, especially for high-speed flight. 
To this end, the blade with a swept-back tip for the rotary wings of an 
aircraft, which blade is intended to form part of a rotor whose hub is 
connected to said blade, which can be rotated about the axis of said hub, 
said blade having a leading edge and a trailing edge, and being formed of 
successive elemental cross sections identified by the distance r 
separating each of them from the axis of rotation of the said hub and each 
having a defined chord profile and a center of pressure whose offset from 
the pitch-change axis, orthogonal to each of said sections, determines the 
sweep of said blade, is noteworthy, according to the invention, in that, 
subdividing said blade along its longitudinal length into four regions, 
namely a first region extending from the inboard edge of the blade to a 
first section situated at approxiamtely 66% of the distance along the 
total length of the blade, measured from the axis of rotation of the hub, 
a second region extending from the first section to a second section 
situated at approximately 85% of the distance along the total length of 
the blade, a third region extending from the second section to a third 
section situated between 93% and 97% of the distance along the total 
length of the blade, and a fourth region extending from the third section 
to the free outboard edge of the blade, the chord length L increases in a 
more or less linear manner in said first region, is at its maximum and 
constant in said second region, decreases linearly in said third region, 
and decreases according to a parabolic function in said fourth region, 
satisfying the conditions of continuity of the rate of chord variation at 
the boundary where this region meets the third region, and the offset Y'f 
of the center of pressure from the pitch-change axis in said first region 
is positive and increases in proportion to the distance r, is at its 
maximum and constant in said second region, decreases linearly and becomes 
negative in said third region, so that the sweep angle .LAMBDA. keeps a 
constant value approximately equal to 25.degree. backward, and decreases 
according to a parabolic function in said fourth region, satisfying the 
condition of continuity of the sweep angle .LAMBDA. at the boundary where 
this regions meets the third region until it reaches its lowest negative 
value at the tip. 
Thus the geometry, as defined, of the blade makes it possible to guarantee 
the best performance for a rotary-wing aircraft, especially a helicopter, 
in which lift is provided by a rotary wing consisting of such blades, 
flying at a high cruising speed of between 315 and 350 km/h for example, 
with a modest amount of lift exhibiting a mean lift coefficient for the 
profiles Czm of between 0.3 and 0.5. 
More specifically, the fact that according to the invention the place along 
the span of the blade where the chord is at its maximum is shifted close 
to the middle of the span makes it possible to obtain a higher speed 
capability, as does the resulting longer length of the region in which the 
chord decreases toward the tip. This new arrangement also makes it 
possible to shorten the region in which the chord is reduced toward the 
blade root, which leads to structural stiffening which limits the 
undesirable torsional deformation. 
Furthermore, remembering that the center of pressure is defined here as 
being the point on each section situated in the front quarter between the 
leading edge and the trailing edge, the offset Y'f is the distance, in the 
direction of the chord, between the pitch-change axis and the center of 
pressure, taken as being positive when the shift in section takes place 
toward the leading edge. The sweep angle .LAMBDA. is defined as being the 
angle between the tangent to the curve joining the centers of pressure of 
the sections and the pitch-change axis, viewed from above. The sweep is 
directed backward at the blade tip. The angle .LAMBDA. can be deduced 
directly from the law governing changes in Y'f: 
EQU .LAMBDA. (r)=arctan (dY'f/dr). 
The beneficial effect of the sweep angle, reducing the strength of the 
shocks, is obtained to the full for values of sweep angle .LAMBDA. of 
between 30.degree. and 45.degree., the maximum value of 45.degree. being 
reached only at the very tip of the blade. The invention makes it possible 
to limit the angle .LAMBDA. to a maximum value of 45.degree. at the tip 
thanks to the very gradual decrease in chord, combined with a trailing 
edge which is straight along the whole of the part of the blade that lies 
between the broadest chord and the tip. 
Another beneficial consequence of this new arrangement consists in the fact 
that the rearward offset of the center of pressure at the tip with respect 
to the pitch-change axis is reduced by comparison, in particular, with the 
case of patent FR-2 689 852, and this limits torsional deformation of the 
blade. 
The law governing the change in chord length may advantageously lie 
between: 
a lower limit ABCDE such that the coordinates of the points A, B, C, D and 
E are as follows: 
______________________________________ 
r/R L/L 
______________________________________ 
A 0 0.55 
B 0.66 1.1 
C 0.85 1.1 
D 0.93 0.73 
E 1 0.25 
______________________________________ 
the lines joining these points to form the limit ABCDE being such that: 
______________________________________ 
x L/L 
______________________________________ 
AB (r-Ra)/(Rb-Ra) 0.55 + 0.55x 
BC (r-Rb)/(Rc-Rb) 1.1 
CD (r-Rc)/(Rd-Rc) 1.1 - 0.37x 
DE (r-Rd)/(Re-Rd) 0.73 - 0.32375x - 0.15625x.sup.2 
______________________________________ 
with Ra, Rb, Rc, Rd and Re representing the respective positions of A, B, 
C, D and E along the blade, and 
an upper limit FGHIJ, such that the coordinates of the points F, G, H, I 
and J are as follows: 
______________________________________ 
r/R L/L 
______________________________________ 
F 0 0.65 
G 0.66 1.2 
H 0.85 1.2 
I 0.97 0.83 
J 1 0.45 
______________________________________ 
the lines joining these points to form the limit FGHIJ being such that: 
______________________________________ 
x L/L 
______________________________________ 
FG (r-Rf)/(Rg-Rf) 0.65 + 0.55x 
GH (r-Rg)/(Rh-Rg) 1.2 
HI (r-Rh)/(Ri-Rh) 1.2 - 0.37x 
IJ (r-Ri)/(Rj-Ri) 0.83 - 0.0925x - 0.2875x.sup.2 
______________________________________ 
with Rf, Rg, Rh, Ri and Rj representing the respective positions of F, G, 
H, I and J along the blade. 
Between the lower limit ABCDE and the upper limit FGHIJ, a preferred curve 
is formed by the points P, Q, R, S and T whose coordinates are as follows: 
______________________________________ 
r/R L/L 
______________________________________ 
P 0.20223 0.7784 
Q 0.66 1.168864 
R 0.85 1.168864 
S 0.95 0.779222 
T 1 0.351538 
______________________________________ 
the lines joining these points to form the curve PQRST being such that: 
______________________________________ 
x L/L 
______________________________________ 
PQ (r-Rp)/(Rq-Rp) 
0.778400 + 0.390464x 
QR (r-Rq)/(Rr-Rq) 
1.168864 
RS (r-Rr)/(Rs-Rr) 
1.168864 - 0.389642x 
ST (r-Rs)/(Rt-Rs) 
0.779222 - 0.194821x - 0.232863x.sup.2 
______________________________________ 
with Rp, Rq, Rr, Rs and Rt representing the respective positions of P, Q, 
R, S and T along the blade. 
Likewise, the law governing the change in offset of the center of pressure 
may preferably lie between: 
a lower limit A'B'C'D'E' such that the coordinates of the points A', B', 
C', D' and E' are as follows: 
______________________________________ 
r/R Y'f/R 
______________________________________ 
A' 0 0 
B' 0.66 +0.005 
C' 0.85 +0.005 
D' 0.93 -0.053 
E' 1 -0.105 
______________________________________ 
the lines joining these points to form the limit A'B'C'D'E' being such 
that: 
______________________________________ 
x Y'f/R 
______________________________________ 
A'B' (r-Ra')/(Rb'-Ra') 
+0.005x 
B'C' (r-Rb')/(Rc'-Rb') 
+0.005 
C'D' (r-Rc')/(Rd'-Rc') 
+0.005 - 0.058x 
D'E' (r-Rd')/(Re'-Rd') 
-0.053 - 0.05075x - 0.00125x.sup.2 
______________________________________ 
with Ra', Rb', Rc', Rd' and Re' representing the respective positions of 
A', B', C', D' and E' along the blade, and 
an upper limit F'G'H'I'J', such that the coordinates of the points F', G', 
H', I' and J' are as follows: 
______________________________________ 
r/R Y'f/R 
______________________________________ 
F' 0 0 
G' 0.66 +0.030 
H' 0.85 +0.030 
I' 0.97 -0.013 
J' 1 -0.050 
______________________________________ 
the lines joining these points to form the limit F'G'H'I'J' being such 
that: 
______________________________________ 
x Y'f/R 
______________________________________ 
F'G' (r-Rf')/(Rg'-Rf') 
+0.03x 
G'H' (r-Rg')/(Rh'-Rg') 
+0.03 
H'I' (r-Rh')/(Ri'-Rh') 
+0.03 - 0.043x 
I'J' (r-Ri')/(Rj'-Ri') 
-0.013 - 0.01075x - 0.02625x.sup.2 
______________________________________ 
with Rf', Rg', Rh', Ri' and Rj' representing the respective positions of 
F', G', H', I' and J' along the blade. 
Between the lower limit A'B'C'D'E' and the upper limit F'G'H'I'J', a 
preferred curve is formed by the points P', Q', R', S' and T' whose 
coordinates are as follows: 
______________________________________ 
r/R Y'f/R 
______________________________________ 
P' 0.20223 +0.003396 
Q' 0.66 +0.011083 
R' 0.85 +0.011083 
S' 0.95 -0.035547 
T' 1 -0.072113 
______________________________________ 
the lines joining these points to form the curve P'Q'R'S'T' being such 
that: 
______________________________________ 
x Y'f/R 
______________________________________ 
P'Q' (r-Rp')/(Rq'-Rp') 
+0.003396 +0.007687x 
Q'R' (r-Rq')/(Rr'-Rq') 
+0.011083 
R'S' (r-Rr')/(Rs'-Rr') 
+0.011083 - 0.046630x 
S'T' (r-Rs')/(Rt'-Rs') 
-0.035547 - 0.023315x - 0.013251x.sup.2 
______________________________________ 
with Rp', Rq', Rr', Rs' and Rt' representing the respective positions of 
P', Q', R', S' and T' along the blade. 
Advantageously, the parameters of the laws governing variations along said 
longitudinal length and for each of said regions, on the one hand in the 
chord length L of each section with respect to the mean chord L and on the 
other hand in the offset Y'f of the center of pressure of each section 
from the pitch-change axis, ensure that the overall center of pressure of 
said blade is positioned more or less on said pitch-change axis. 
The law governing the change in vertical shift of the centers of pressure 
Zv/R may, moreover, be such that the centers of pressure of the profiles 
at the blade tip are appreciably lower than the inboard part of the blade. 
The marginal vortex generated by the outboard end of one blade is thus 
further from the following blade, the result of this being that it reduces 
the strength of the interaction, particularly in hovering flight. This 
leads to a clear reduction in the power absorbed by the rotor, especially 
in hovering and low-speed flight. 
Advantageously, this law governing the change in the vertical shift of the 
center of pressure will lie between: 
a lower limit A"B"C"D"E" such that the coordinates of the points A", B", 
C", D" and E" are as follows: 
______________________________________ 
r/R Zv/R 
______________________________________ 
A' 0 -0.001 
B' 0.66 -0.001 
C' 0.85 -0.001 
D' 0.93 -0.001 
E' 1 -0.015 
______________________________________ 
the lines joining these points to form the limit A"B"C"D"E" being such 
that: 
______________________________________ 
x Zv/R 
______________________________________ 
A"B" (r-Ra")/(Rb"-Ra") 
-0.001 
B"C" (r-Rb")/(Rc"-Rb") 
-0.001 
C"D" (r-Rc")/(Rd"-Rc") 
-0.001 
D"E" (r-Rd")/(Re"-Rd") 
-0.001 - 0.014x.sup.2 
______________________________________ 
with Ra", Rb", Rc", Rd" and Re" representing the respective positions of 
A", B", C", D" and E" along the blade, and 
an upper limit F"G"H"I"J", such that the coordinates of the points F", G", 
H", I" and J" are as follows: 
______________________________________ 
r/R Zv/R 
______________________________________ 
F" 0 +0.001 
G" 0.66 +0.001 
H" 0.85 +0.001 
I" 0.97 +0.001 
J" 1 -0.005 
______________________________________ 
the lines joining these points to form the limit F"G"H"I"J" being such 
that: 
______________________________________ 
x Zv/R 
______________________________________ 
F"G" (r-Rf")/(Rg"-Rf") 
+0.001 
G"H" (r-Rg")/(Rh"-Rg") 
+0.001 
H"I" (r-Rh")/(Ri"-Rh") 
+0.001 
I"J" (r-Ri")/(Rj"-Ri") 
+0.001 - 0.006x.sup.2 
______________________________________ 
with Rf", Rg", Rh", Ri" and Rj" representing the respective positions of 
F", G", H", I" and J" along the blade. 
Between the lower limit A"B"C"D"E" and the upper limit F"G"H"I"J", a 
preferred curve is formed by the points P", Q", R", S" and T" whose 
coordinates are as follows: 
______________________________________ 
r/R Zv/R 
______________________________________ 
P" 0.20223 0 
Q" 0.66 0 
R" 0.85 0 
S" 0.95 0 
T" 1 -0.009050 
______________________________________ 
the lines joining these points to form the curve P"Q"R"S"T" being such 
that: 
______________________________________ 
x Zv/R 
______________________________________ 
P"Q" (r-Rp")/(Rq"-Rp") 0 
Q"R" (r-Rq")/(Rr"-Rq") 0 
R"S" (r-Rr")/(Rs"-Rr") 0 
S"T" (r-Rs")/(Rt"-Rs") -0.00905x.sup.2 
______________________________________ 
with Rp", Rq", Rr", Rs" and Rt" representing the respective positions of 
P", Q", R", S" and T" along the blade. 
The law governing the change in relative thickness of the sections e/L may, 
in addition, lie between: 
a lower limit UVW, such that the coordinates of the points U, V and W are 
as follows: 
______________________________________ 
r/R e/L 
______________________________________ 
U 0 0.14 
V 0.4 0.14 
W 1 0.06 
______________________________________ 
the lines joining these points to form the limit UVW being such that: 
______________________________________ 
x e/L 
______________________________________ 
UV (r-Ru)/(Rv-Ru) 0.14 
VW (r-Rv)/(Rw-Rv) 0.14 - 0.08x 
______________________________________ 
with Ru, Rv and Rw representing the respective positions of U, V and W 
along the blade, and 
an upper limit XYZ, such that the coordinates of the points X, Y and Z are 
as follows: 
______________________________________ 
r/R e/L 
______________________________________ 
X 0 0.16 
Y 0.4 0.16 
Z 1 0.08 
______________________________________ 
the lines joining these points to form the limit XYZ being such that: 
______________________________________ 
x e/L 
______________________________________ 
XY (r-Rx)/(Ry-Rx) 0.16 
YZ (r-Ry)/(Rz-Ry) 0.16 - 0.08x 
______________________________________ 
with Rx, Ry and Rz representing the respective positions of X, Y and Z 
along the blade. 
Between the lower limit UVW and the upper limit XYZ, a preferred curve is 
formed by the points K, L and M whose coordinates are as follows: 
______________________________________ 
r/R e/L 
______________________________________ 
K 0.20223 0.15 
L 0.4 0.15 
M 1 0.07 
______________________________________ 
the lines joining these points to form the limit KLM being such that: 
______________________________________ 
x e/L 
______________________________________ 
KL (r-Rk)/(Rl-Rk) 0.15 
LM (r-Rl)/(Rm-Rk) 0.15 - 0.08x 
______________________________________ 
with Rk, Rl and Rm representing the respective positions of K, L and M 
along the blade.

The blade 1 with swept-back tip in accordance with the present invention 
and shown in FIG. 1 forms part of a rotor whose hub 2 is illustrated 
purely diagrammatically and whose other blades have not been depicted. The 
blade 1 is connected to the hub 2 by blade articulation and retaining 
members 3, particularly a pitch-change articulation to change the pitch of 
the blade about an axis 4 known as the pitch-change axis, as is 
conventional. 
Furthermore, the blade 1, which has a leading edge 5 and a trailing edge 6, 
is formed of successive elemental cross sections, one 7 of which is 
depicted in FIG. 1. Each elemental section 7 is identified by the distance 
r separating said section from the axis of rotation 2A of the hub, and has 
a defined chord profile L and a center of pressure (the point at which the 
variations in aerodynamic lift forces are applied) whose "curve" along the 
longitudinal span of the blade is depicted by 8 in FIG. 2. The offset 
between the center of pressure and the pitch-change axis 4 orthogonal to 
said successive sections 7 determines the sweep of the blade, as seen more 
clearly in FIG. 2. 
The geometric construction that allows the area of a blade 1 according to 
the invention to be defined precisely will be described below. 
The construction reference frame is chosen as being an orthonormal 
trihedron whose origin 0 is the center of the rotor. 
The axis OX is the pitch-change axis 4, which means that the first 
coordinate coincides with the radius r measured from the center of 
rotation 0. The second axis OY, orthogonal to the axis OX, constitutes the 
reference direction for the angular setting, and points, arbitrarily, 
toward the leading edge 5. The third axis OZ is orthogonal to the plane 
defined by the axes OX and OY and points, arbitrarily, upward (toward the 
suction face of the profiles). The trihedron is the correct way up if the 
rotor turns counterclockwise. It should, however, be clearly understood 
that all that follows is still valid for a rotor that turns clockwise. 
The plane OX, OY will be called the construction plane or the reference 
plane. The plane OX, OY will be chosen to coincide with the zero lift 
plane of the blade. The blade area (blade envelope) is generated by a 
collection of elemental planar sections 7 which are all parallel to one 
another and parallel to the plane OX, OZ and orthogonal to the 
pitch-change axis OX. 
Each elemental section can be identified by its radius r (distance of the 
section from the axis OY) lying between R0 (start of the actual blade 
part) and R (outboard end or tip of the blade). 
The parameters that define the contour of any elemental cross section 7 of 
the blade 1 are generally known, especially from patent FR-2 689 852. 
The blade that is the subject of the invention is subdivided into four 
regions that allow it to be described, independently of the twist and 
relative thickness of the sections which require special breakdowns 
specified below. These four regions are: 
a first region, which extends from the section R0 that corresponds to the 
start of the actual blade part, as far as the section R1 situated at 
approximately 66% of the total radius R; 
a second region, which extends from the section R1 as far as the section R2 
situated at approximately 85% of the total radius R; 
a third region, which extends from the section R2 as far as the section R3 
situated between 93% and 97% of the total radius R; and 
a fourth region, which extends from the section R3 as far as the tip of the 
blade (radius R). 
According to the invention, in these various regions, the chord length L 
increases generally linearly in said first region, is at its maximum and 
constant in said second region, decreases linearly in said third region, 
and decreases according to a parabolic function in said fourth region, 
meeting the requirement for continuity in the rate of variation of the 
chord at the boundary where this region meets the third region, and the 
offset Y'f of the center of pressure from the pitch-change axis in said 
first region is positive and increases in proportion to the distance r, is 
at its maximum and constant in said second region, decreases linearly and 
becomes negative in said third region, so that the sweep angle .LAMBDA. 
keeps a constant value approximately equal to 25.degree. backward, and 
decreases according to a parabolic function in said fourth region, meeting 
the criterion of continuity of the sweep angle .LAMBDA. at the boundary 
where this region meets the third region and until it reaches its lowest 
negative value at the tip. 
As can be seen in FIG. 3, the law governing the change in chord length may 
advantageously lie between: 
a lower limit ABCDE such that the coordinates of the points A, B, C, D and 
E are as follows: 
______________________________________ 
r/R L/L 
______________________________________ 
A 0 0.55 
B 0.66 1.1 
C 0.85 1.1 
D 0.93 0.73 
E 1 0.25 
______________________________________ 
the lines joining these points to form the limit ABCDE being such that: 
______________________________________ 
x L/L 
______________________________________ 
AB (r-Ra)/(Rb-Ra) 0.55 + 0.55x 
BC (r-Rb)/(Rc-Rb) 1.1 
CD (r-Rc)/(Rd-Rc) 1.1 - 0.37x 
DE (r-Rd)/(Re-Rd) 0.73 - 0.32375x - 0.15625x.sup.2 
______________________________________ 
with Ra, Rb, Rc, Rd and Re representing the respective positions of A, B, 
C, D and E along the blade, and 
an upper limit FGHIJ, such that the coordinates of the points F, G, H, I 
and J are as follows: 
______________________________________ 
r/R L/L 
______________________________________ 
F 0 0.65 
G 0.66 1.2 
H 0.85 1.2 
I 0.97 0.83 
J 1 0.45 
______________________________________ 
the lines joining these points to form the limit FGHIJ being such that: 
______________________________________ 
x L/L 
______________________________________ 
FG (r-Rf)/(Rg-Rf) 0.65 + 0.55x 
GH (r-Rg)/(Rh-Rg) 1.2 
HI (r-Rh)/(Ri-Rh) 1.2 - 0.37x 
IJ (r-Ri)/(Rj-Ri) 0.83 - 0.0925x - 0.2875x.sup.2 
______________________________________ 
with Rf, Rg, Rh, Ri and Rj representing the respective positions of F, G, 
H, I and J along the blade. 
Between the lower limit ABCDE and the upper limit FGHIJ, a preferred curve, 
as shown in FIG. 3, is formed by the points P, Q, R, S and T whose 
coordinates are as follows: 
______________________________________ 
r/R L/L 
______________________________________ 
P 0.20223 0.7784 
Q 0.66 1.168864 
R 0.85 1.168864 
S 0.95 0.779222 
T 1 0.351538 
______________________________________ 
the lines joining these points to form the curve PQRST being such that: 
______________________________________ 
x L/L 
______________________________________ 
PQ (r-Rp)/(Rq-Rp) 
0.778400 + 0.390464x 
QR (r-Rq)/(Rr-Rq) 
1.168864 
RS (r-Rr)/(Rs-Rr) 
1.168864 - 0.389642x 
ST (r-Rs)/(Rt-Rs) 
0.779222 - 0.194821x - 0.232863x.sup.2 
______________________________________ 
with Rp, Rq, Rr, Rs and Rt representing the respective positions of P, Q, 
R, S and T along the blade. 
Likewise, as can be seen in FIG. 4, the law governing the change in offset 
of the center of pressure may preferably lie between: 
a lower limit A'B'C'D'E' such that the coordinates of the points A', B', 
C', D' and E' are as follows: 
______________________________________ 
r/R Y'f/R 
______________________________________ 
A' 0 0 
B' 0.66 +0.005 
C' 0.85 +0.005 
D' 0.93 -0.053 
E' 1 -0.105 
______________________________________ 
the lines joining these points to form the limit A'B'C'D'E' being such 
that: 
______________________________________ 
x Y'f/R 
______________________________________ 
A'B' (r-Ra')/(Rb'-Ra') 
+0.005 
B'C' (r-Rb')/(Rc'-Rb') 
+0.005 
C'D' (r-Rc')/(Rd'-Rc') 
+0.005 - 0.058x 
D'E' (r-Rd')/(Re'-Rd') 
-0.053 - 0.05075x - 0.00125x.sup.2 
______________________________________ 
with Ra', Rb', Rc', Rd' and Re' representing the respective positions of 
A', B', C', D' and E' along the blade, and 
an upper limit F'G'H'I'J', such that the coordinates of the points F', G', 
H', I' and J' are as follows: 
______________________________________ 
r/R Y'f/R 
______________________________________ 
F' 0 0 
G' 0.66 +0.030 
H' 0.85 +0.030 
I' 0.97 -0.013 
J' 1 -0.050 
______________________________________ 
the lines joining these points to form the limit F'G'H'I'J' being such 
that: 
______________________________________ 
x Y'f/R 
______________________________________ 
F'G' (r-Rf')/(Rg'-Rf') 
+0.03x 
G'H' (r-Rg')/(Rh'-Rg') 
+0.03 
H'I' (r-Rh')/(Ri'-Rh') 
+0.03 - 0.043x 
I'J' (r-Ri')/(Rj'-Ri') 
-0.013 - 0.01075x - 0.02625x.sup.2 
______________________________________ 
with Rf', Rg', Rh', Ri' and Rj' representing the respective positions of 
F', G', H', I' and J' along the blade. 
Between the lower limit A'B'C'D'E' and the upper limit F'G'H'I'J', a 
preferred curve, as shown in FIG. 4, is formed by the points P', Q', R', 
S' and T' whose coordinates are as follows: 
______________________________________ 
r/R Y'f/R 
______________________________________ 
P' 0.20223 +0.003396 
Q' 0.66 +0.011083 
R' 0.85 +0.011083 
S' 0.95 -0.035547 
T' 1 -0.072113 
______________________________________ 
the lines joining these points to form the curve P'Q'R'S'T' being such 
that: 
______________________________________ 
x Y'f/R 
______________________________________ 
P'Q' (r-Rp')/(Rq'-Rp') 
+0.003396 + 0.007687x 
Q'R' (r-Rq')/(Rr'-Rq') 
+0.011083 
R'S' (r-Rr')/(Rs'-Rr') 
+0.011083 - 0.046630x 
S'T' (r-Rs')/(Rt'-Rs') 
-0.035547 - 0.023315x - 0.013251x.sup.2 
______________________________________ 
with Rp', Rq', Rr', Rs' and Rt' representing the respective positions of 
P', Q', R', S' and T' along the blade. 
As already stated, it is also advantageous that the parameters of the laws 
governing variations along said longitudinal length and for each of said 
regions, on the one hand in the chord length L of each section with 
respect to the mean chord, L and on the other hand in the offset Y'f of 
the center of pressure of each section from the pitch-change axis, ensure 
that the overall center of pressure of said blade is positioned more or 
less on said pitch-change axis. 
Furthermore, as can be seen in FIG. 5, the law governing the change in the 
vertical shift of the center of pressure Zv/R may, moreover, lie between: 
a lower limit A"B"C"D"E" such that the coordinates of the points A", B", 
C", D" and E" are as follows: 
______________________________________ 
r/R Zv/R 
______________________________________ 
A" 0 -0.001 
B" 0.66 -0.001 
C" 0.85 -0.001 
D" 0.93 -0.001 
E" 1 -0.015 
______________________________________ 
the lines joining these points to form the limit A"B"C"D"E" being such 
that: 
______________________________________ 
x Zv/R 
______________________________________ 
A"B" (r-Ra")/(Rb"-Ra") 
-0.001 
B"C" (r-Rb")/(Rc"-Rb") 
-0.001 
C"D" (r-Rc")/(Rd"-Rc") 
-0.001 
D"E" (r-Rd")/(Re"-Rd") 
-0.001 - 0.014x.sup.2 
______________________________________ 
with Ra", Rb", Rc", Rd" and Re" representing the respective positions of 
A", B", C", D" and E" along the blade, and 
an upper limit F"G"H"I"J", such that the coordinates of the points F", G", 
H", I" and J" are as follows: 
______________________________________ 
r/R Zv/R 
______________________________________ 
F" 0 +0.001 
G" 0.66 +0.001 
H" 0.85 +0.001 
I" 0.97 +0.001 
J" 1 -0.005 
______________________________________ 
the lines joining these points to form the limit F"G"H"I"J" being such 
that: 
______________________________________ 
x Zv/R 
______________________________________ 
F"G" (r-Rf")/(Rg"-Rf") 
+0.001 
G"H" (r-Rg")/(Rh"-Rg") 
+0.001 
H"I" (r-Rh")/(Ri"-Rh") 
+0.001 
I"J" (r-Ri")/(Rj"-Ri") 
+0.001 - 0.006x.sup.2 
______________________________________ 
with Rf", Rg", Rh", Ri" and Rj" representing the respective positions of 
F", G", H", I" and J" along the blade. 
Between the lower limit A"B"C"D"E" and the upper limit F"G"H"I"J", a 
preferred curve, as shown in FIG. 5, is formed by the points P", Q", R", 
S" and T" whose coordinates are as follows: 
______________________________________ 
r/R Zv/R 
______________________________________ 
P" 0.20223 0 
Q" 0.66 0 
R" 0.85 0 
S" 0.95 0 
T" 1 -0.009050 
______________________________________ 
the lines joining these points to form the curve P"Q"R"S"T" being such 
that: 
______________________________________ 
x Zv/R 
______________________________________ 
P"Q" (r-Rp")/(Rq"-Rp") 0 
Q"R" (r-Rq")/(Rr"-Rq") 0 
R"S" (r-Rr")/(Rs"-Rr") 0 
S"T" (r-Rs")/(Rt"-Rs") -0.00905x.sup.2 
______________________________________ 
with Rp", Rq", Rr", Rs" and Rt" representing the respective positions of 
P", Q", R", S" and T" along the blade. 
Furthermore, the recommended blade shape may be given a linear aerodynamic 
twist, with a total amplitude of more or less between -8.degree. and 
-14.degree. between the center of the rotor and the tip of the blade. 
Consistent with common practice, the twist is said to be negative when the 
leading edge of the outboard sections is lower than that of the sections 
located closer to the center. To obtain the geometric angular setting of 
each section, measured with respect to the reference chord, the zero-lift 
incidence of the profile in question needs to be added to the aerodynamic 
twist, as explained in detail in document FR-A-2 689 852. 
With the reduction in chord at the root as recommended by the present 
invention, it proves necessary to increase the relative thickness to more 
than 14% in order to maintain a torsional stiffness that is at least 
equivalent to that of conventional blades. Nonetheless, 16% should not be 
exceeded, this being in order to avoid any degradation to the lift 
capability, and in order not to increase the aerodynamic drag. The 
relative thickness needs to be kept between 14 and 16% in the region of 
the blade where the chord is still small. However, beyond the section 
situated at 40% of the span, the chord becomes large and the relative 
thickness can begin to decrease in order to delay the detrimental effects 
of the compressibility of air. 
At the tip end, the intensity of the shockwaves need to be minimized when 
the blade is advancing during high-speed flight. The shape in accordance 
with the present invention uses profiles in which the relative thickness 
does not exceed 8% at this end. However, this thickness must not be 
reduced to less than 6%, this being to maintain sufficient lift capability 
which is still needed when the tip is retreating. 
Between the 40% of span section and the tip, the relative thickness will 
decrease more or less linearly. Breaking the blade down into two regions 
alone is sufficient to describe this change. 
As can be seen in FIG. 6, the law governing the change in relative 
thickness of the sections e/L may thus lie between: 
a lower limit UVW, such that the coordinates of the points U, V and W are 
as follows: 
______________________________________ 
r/R e/L 
______________________________________ 
U 0 0.14 
V 0.4 0.14 
W 1 0.06 
______________________________________ 
the lines joining these points to form the limit UVW being such that: 
______________________________________ 
x e/L 
______________________________________ 
UV (r-Ru)/(Rv-Ru) 0.14 
VW (r-Rv)/(Rw-Rv) 0.14 - 0.08x 
______________________________________ 
with Ru, Rv and Rw representing the respective positions of U, V and W 
along the blade, and 
an upper limit XYZ, such that the coordinates of the points X, Y and Z are 
as follows: 
______________________________________ 
r/R e/L 
______________________________________ 
X 0 0.16 
Y 0.4 0.16 
Z 1 0.08 
______________________________________ 
the lines joining these points to form the limit XYZ being such that: 
______________________________________ 
x e/L 
______________________________________ 
XY (r-Rx)/(Ry-Rx) 0.16 
YZ (r-Ry)/(Rz-Ry) 0.16 - 0.08x 
______________________________________ 
with Rx, Ry and Rz representing the respective positions of X, Y and Z 
along the blade. 
Between the lower limit UVW and the upper limit XYZ, a preferred curve, as 
shown in FIG. 6, is formed by the points K, L and M whose coordinates are 
as follows: 
______________________________________ 
r/R e/L 
______________________________________ 
K 0.20223 0.15 
L 0.4 0.15 
M 1 0.07 
______________________________________ 
the lines joining these points to form the limit KLM being such that: 
______________________________________ 
x e/L 
______________________________________ 
KL (r-Rk)/(Rl-Rk) 0.15 
LM (r-Rl)/(Rm-Rk) 0.15 - 0.08x 
______________________________________ 
with Rk, Rl and Rm representing the respective positions of K, L and M 
along the blade.