The Flapping Behaviour of a Helicopter Rotor at High Tip

C.P. No. 877
LIBRAUV
MINISTRY
OF
AVIATION
AERONAUTICAL RESEARCH COUNClL
CURRENT PAPERS
The Flapping Behaviour of a
Helicopter
Rotor at High
Tip-Speed Ratios
bY
E. Wilde. Ph.D. A. R. S. hamwell, M.Sc. Ph.D.
and
R. Summerscales,&SC.
LONDON:
HER MAJESTY’S STATIONERY
1966
PRICE 7s 6d NET
OFFICE
U.D.C. No. 533.562.6 : 533.6,013.4-Z
C.P. No.877
April
1965
THE FLAPPIXG BEUAVIOUROF A HELICOPTEEROTOR
AT HIGH TIP-SPEED RATIOS
%. Wilde, Ph.D.
A. R. S. Bramwell, id.%. Ph.D.
R. Sumnerscalcs, B.&z.
The blade flapping equation has bden solved on an analogue COmPUter
taking into acoount the reversed flow region but neglecting
stall.
The fully
articulated
blade becomes unstable at about p = 2.3, Whilst a see-saw rotor is
stable up to p = 5 at least and the trends suggest that it may be stable for
all values of G. However, the response to a gust , or the equivalent change of noangle, 1s almost the same fcrr boti rotors up to about p = 0.75.
gust at a forwLard speed of 200 ft/sec,
and typical rotor/fuselage
represents the limiting
tip-speed ratio for either rotor.
The
of the see-saw rotor, however, makes it possible to increase
This has been
tipxxpeed ratio by some form of flapping restraint.
feathering
axis
For a 35 ft/sec
clearance, this
better response
the limiting
investigated
63-hinges.
by considering
Replaces R.A.E.
the effects
Technical
of springs
and dampers, and offset
Report 65068 - A.R.C.
27375
and
06
CCNTENTS
Page
3
3
INTRODUCTION
THEFREELY FLAPPINGRCYTOR
2.1 Analysis of the fully articulated
rotor
2.2 The analysis of the see-saw rotor
ANALOGUECOXPUTERRESULTS
3.1 The fully articulated
rotor
3.2
The see-saw rotor
THE RESTRAINT OF FLAPPING %OTIONBY SPRINGS AND DAJY~S
4.1 Analysis of spring and damper restraint
4.2 Computed results
RESTRAINT OF FLAPPING MOTION BY K!XAXSOF A G3-HINGE
5.1 The flapping equation
5.2 Computed results
FLAPPING RESTRAINT BY iZ.&?JS OF A3 OFFSET FLAPPING HINGE
6.1 The flapping equation
6.2 Computed results
STOPPINGA ROTORIN FLIGHT
CONCLUSIONS
Symbols
References
Illustrations
Detachable abstract cards
3
7
8
8
9
IO
IO
II
II
II
12
13
13
15
15
15
17
18
Figures
l-14
1
INTRODUCTIO~~
utility
Rotorcraft
but with
have established
themselves
low top speed. This utility
as flight
vehicles of great
would be improved if the top
speed could be increased and considerable attention
has already been devoted
to basic performance assessments.
This work has indicated that significant
improvements are possible but more detailed consideration
istics,
such as blade fla>>ing amplitudes and stability,
the improvement can be completely
of rotor characteris essential before
determined,
In order to reduce compressibility
losses and noise, it will be necessary
This means
to reduce the speed oi" the rotor as the forward speed increases.
that the tip speed ratio, 11, will become high and might even exceed unity.
Little
is known about blade flapping stability
at these high values of p as
the f'lapFing quation has periodic coefficients
and is extremely difficult
to
solve analytically.
Several attempts to obtain solutions have been made, the
2
most recent being those of Shutler and Jones' and Lowis . In all these attempts
the aerodynamic flapping motier,t from tiie region of reversed f'luw was not merely
ignored but had to appear with the wrong sign in order that the flapping moment
This would not be
should be correct in the more important advancing region*.
important at low values of p but ;;rould result in serious error at the very
values of p for which the investigation
was rquired.
Kith the aid of an analcLue comi>uter, the reversed flow region has nc3w
been included in an investigation
of blade flapping behaviour and the results
are described 3~1this paper. Both freely flapping and see-saw rotors have
been analysed and the cff~cts
of sizings, daqers and 6 - and offset hinges
5
have been considered.
2.1
Analysis
The rotor
of the fully
divides
itself
articulated
into
rotor
three regions
(1) The tradvancingtt region where the airflow
from loading edge to trailiq
edge, 0 IC $ 6 n*
as shown in Ii'ig.1.
They are:-
over the whole blade is
* Whi_le the work described in this Report was in progress another paper by
titvis3 appeared in which the reversed flow region was approximately
taken
into account.
His results largely confirm some of the findings
of this
Report.
4
(2)
The "partial
reverse" region where the airflow is from leading to
edge over the outboard part of the blade, but from trailing
to
edge over the inboard part. 1‘Jhen/.J<I this re@cn extents from $t = 71 to
-When L.J> 1 the region is in two parts given by - i < sin $ $ 0,
'
trailing
leading
$ = a.
(3) The "total
reversed" region which exists when p B 1 and the flow
from trailing
edge to leading edge over the whole span. In this region
sin$<
-;.
The aerodynamic moment will
the following
assumptions.
(I)
The lift
and reversed flow.
(2)
The
sibility
would
unreasonable.
and since many
assumption
be determined
slope is constant
in each of these regions
\
subject
is
to
and has the same value for both advancing
effects of stalling
and compressibility
are ignored.
Compresbe avoided in practice and so the latter assumption is not
The assumption of no stalling
leads to great simplifications
of the cases of interest
involve lightly
loaded rotors,
this
would appear to be justified.
(3)
The effect
of spanwise flow
is neglected.
(4)
Unsteady aerodynamic effects
are ignored.
.
The system of axes is shown in Fig.2.
The velocity
at a blade section distance r from the root, in a plane
axis has a chordwise component, UT,
perpendicular
to the no-feathering
given by
uT = Rr + V cos anf.sin $
or
uT
0)
= m (x -t p sin $)
v cos anr
where
p
=
RR
The velocity
at a blade section , parallel
U z
P
=
where h =
to the no-feathering
axis,
is
v sin anf - vi - vg cos cLnf cos Q - r;
QR (h - pfi cos
V sin anf - vi
a?
$ - xb/fl>
(2)
5
58
If the collective
pitch is eo, and the airflow
ing edge, then the incidence of the element is
-1
4 =eottan
Asszing
Q = aa, the lift
The elementary
lift
g
el T
to trail-
uP
(3)
T
on an element of blade length
dr is
moment is
Substituting
equations (1) and (2) into
gives for the advancing region,
where
dots
is frarn leading
ti AR denotes the blade flapping
When the flow is from trailing
a in equation (3) changes sign, i.e.
a
=
(4) and integrating
moment in the advancing region.
edge to leading
9
from x = 0 to x = 1
( e.
edge the expression
for
up!
+ q)
In the l'partial
reverse" region, therefore,
we use expression (6) between
x=Oandxzp sin $ and expression (3) from x = - p sin $ to x = 1. Thus
is the flapping mamcnt in the partial
reverse region,
ir%
But equation
(7) can be rewritten
=I MAR-upYIQ2
3 sin3$
(
as
pea
sin*+
2h-2p.Pcoslir
Finally,
if % is the flapping moment in the "total
since equation (6) applies along the entire
blade,
The equation
of the flapping
motion
d
t O Sin*
reverse"
region,
03
1
we have,
is
pII * p = M
IR2
where the dashes denote differentiation
with respect to $ and M stands for
either IfIm, Sk, Mi according to which region the blade is in, as described
above.
Equation (IO) is very difficult
to solve analytically
because some of the
coefficients
are periodic and because the forms of M depend on the values of the
are easily overcome in an
independent variable $. However, these difficulties
analogue computer since the periodic coefficients
can be handled by multiplier
units whilst electronic
relays can be arranged to switch in the correct moment
terms at the appropriate values of \ir. The program used is given in Appendix A.
7
2.2
The analysis
of the see--sciv7rotor
The flapping motions of the blades of a see-saw rotor are no longer
independent, since the blades are structurally
integral but jointly
free to
flap.
The flapping equation may be found by the same methods as before but
this time the rotor Uist is divided into iour' regions as shown in Pig.3.
These regions are:(1)
Where the airflo?r,r over the
trailing
edge but over the other blade
region 0 < sin $i < -.1
IJ
over the
(2) >?here the airf'lci
trailing
edge but over tile otl.e-r blade
only if ~1> 1 and when -1 < sin $.
CL
(3)
'ir'here tlle airflo;Y
but over the other
I1 i sin $I < c\.
I-r
blade it
reference blade is from leading to
it is partially
reversed.
In this
reference blade is from leading
it is totally
reversed.
TiliS
over the reference
is from lending
blade is partially
to trailing
edge.
to
OCCUTS
reversed
For this
case
(4) Where the airflow; over the reference blade is totally
reversed but
over the other blade it is from lending to trailing
edge. Again, this occurs
only if &I > 1 and Aen sin $ < - :.
If the built-in
coning angle is ao, then the flapping
of the rel"erecce
blade will be ao + P and of the other blade a - 0. The flapping moments for
0
the complete see-saw rotor I’OL’ the four re:,ions then becume:-
when
% < sin $ ,
MA*
-ycL3+
=540
+p200sin2j,+Jj-h--$
p2ao
IQ2
when
--<
*A
-xl*
3
I
P
sin $ < 0 ,
-+i4sin3~
eosin*-
= MA1
(
when
sin
-hi
A4
3
ANALOGUECoi\-
3.1
The fully
sin * cos $
3
$
-c
*a0 cos q
>
03)
-5,
= - MA
2
04)
?&?JSuLTS
articulated
rotor
The computed response to a disturbance of the fully
given in Fig.&.
It can be seen that the flapping motion
articulated
rotor is
is stable for values
of p up to about 2.25, (depending slightly
on Lock's inertia number y) and
becomes unstable at higher values of & This is in contrast to the result
given by simple theory, which neglects the reversed flow region, and which
predicts flapping instability
at p = fl2.
The steady blade flapping angles have been computed for a number of
values of 1ti and for values of p of 0.35, 0.5, 0.7, 1.0. 1.5 and 2,O. It may
be noted that since the differential
equation for flapping is linear and the
right hand side of the equation is linear in ati, there must be a linear relationship between /3 and a,f~
the total flapping smplitude,
at various
For the freely flapping rotor,
values of tiF-speed ratio,
is plotted against ati in Fig.5. The coefficients
of the first
harmonics
of flapping,
a, and b, are shown in Fig.6.
In Fig.5 there are shown three curves which give the flapping resulting
from a 35 ft/sec gust, under different
conditions
of flight
the collective
pitch
being assumed to be zero. In one case the gust occurs at a speed of 200 ft/sec
9
which is assumed to be the speed at which rotor retraction
might take place.
It can be seen +&at the flapping exceeds the assumed geoinetric limit of
9 degrees for p greater tila 0.7. ?or the otnar two cases it is assumed that
the blade ti?iJach
define relationships
nu&er is constant at either 0.35 or 0.95. These assumptions
between anf and p which have also been plotted.
From these
is not exceeded for a gust of
curves it can be seen that the geometric limit
35 ft/sec unless u is greater than about 1, being maze or less than 1 according
to the tip Mach number. Along each curve there is also a definite
variation
of
forward speed and it appears that the geometric limit is reached at 465 ft/sec
when t!le tip Xach number is 0.85 and 525 fti'soc y;L;iiitinit is 0.95.
3-2
ThC
SC?e-SCLW
rotor-
The variation
of the flap<&ng angle
wZth six&t angle, anfs for the seesaw rotor is shown in Pig. 7. C~urvcs are given for two values of built-in
coning
angle, 0 kgree and L!+dcgrccs.
It can bc setin that the curves for the
4. degrees coning angle arr: no-l; linear with znf*
This is to be expected as the
coning angle itself
will product fla_oLAng, even when CL is zero. The curves of
I-3
IFig.
artioulated
rotor.
should be compared with those of sig.5 for tile fully
Up to a tip soeed ratio of about C.73 tint: rcsponscs of both rotors are very
sli-nilsr but chewers the fra~lq fiapping rotor bcccmos unstable at about
1-1= 2.3, the stie-saw rotor is stable up to i.r = 5 at lcast.
The slope of the
curve cf flapping with ati, as a function
of p, is shcwn in Fig.8 where it can
be seen that d@/danr becomes rou&ly
linear above about p = I.
For
the
unrestrained
see--saw rotor this is not of great use since the factor limiting
its use is tiio rtisponsc to a 35 ft/sec gust at about p = 0.7 which is similar
to that of' the freely f'lap;kg
rotor.
EIcwevcr, the suppression cf flapping at
higher values of p encblos values of p to be r oachcd of between 1.5 and 2,
dopending on the ocning angle, with ccrrespcnding forward s&peeds of at least
600 ft/sec.
This latter
speed must be acccxnpnied by a very low rotor speed,
of course, in crder that the ti? Liach number should remain at 0.85,
value close to this figure.
It has
and that tine
coning angle
amoat which
or at some
been assumed in the analysis that the blades are ;?erfectly rigid
coning angle rer;ljins fixed.
It is rcaliscd,that
in tactics,
the
may differ
from the esswned CLue,due to blade flcxi'oility,
by an
dqends upon the flagpin~:.
It is thought, however, that unless the
flapping is excessively
large,tlle
range oi‘ coning angles of Fig. 7 will cover
most cases of practicul
interest.
-IO
4
RESTRAII~T OF FLAPPING XOTION BY SPRmGS AND DMiiS
Since the amplitude of flapping motion appears to impose operating limits,
Springs do
consideration
has been given to various means of restricting
it.
not dissipate
energy but merely act as stores and consequently will redistribute
the flapping.
This may be useful, however, in converting backward flapping into
sideways flapping since the main obstructions
to flapping are in the fore and
dissipate
energy and so reduce
aft plane. Dampers, on the other hand, actively
It is recognised that such devices may well cause
the amount of flapping.
stressing problems but such considerations
investigation.
4.q
Analysis
are beyond tne scope of the present
of snring, and damper restraint
The spring-damper arrangement considered is shown in Fig.16.
As the seesaw rotor appears to have the better stability,
only this type of rotor has
the method given below is applicable
to the freely
been considered.
Ii&ever,
flapping
rotor.
The flapping
equation,
with
spring
and damper restraint,
can be written
2ig 4. IQ2 (a0 + 13) - IQ2 (a0 - B)
= Ii, - X2 - 2 ks Rt2 p - 2 kd RL2b
where
lb wt/ft
ks is the spring
constant,
kd is the damping constant,
and
Rt is th2 distance
This equation
(15)
lb nt/ft/sec
from the root
of the point
of attachment.
reduces to
ksR12
where
ks
=
IQ2
'd
=
kdRt2
IQ
07)
11
Equation
00,
(16) implies that the two final terms should be added to equations
WL
(13 and (11~) to include the effects of spring and damper.
1;. 2
Computed results
The effects of spring and dami";:r restraint
were considered separately.
The magnitudes of the spring alid damper coefficients
;Jcre chosen quite
arbitrarily
merely to illustrate
their effects.
Owing to the ability
of the
damper to dissipate
energy, a greater range of damper coefficients
was consider&,
so that kd ranged from 0 to 1.4, while zs ranged between 0 and 0.6.
Collective
pitch and built-in
coning angle -ti~re kept constant at 0' and 2',
respectively,
and tip-speed ratios between 0 and 2.0 were considered.
In each
case, only a single value of anr was taken, selected to give convenient scaling
on the analogue computer, so that a was 0""for~~<lanci~ofo~~~>l.
nf
results
of the computation are shown in Figs. 3-12.
The
The trends expected in tie discussion of Section 4 are confirmed by the
results.
As indicated above, the rangc:s of zd and x8 were not taken to extreme
values but it is not expcctod t1:at tilt: trends
ShOWl
-,vould bc changed if Ed
and Es wore furthor increased.
Rough calculations
for a t;@.csl c ast: Sh.0~~tliat there: may be very large
concentrated loads at the points of attachment of spring or damper possibly
resulting
in unacceptable stresses.
However, it is outside the scope of this
note to discuss the full
structural
implications
of attaching springs and dampers
in the manner shc~rm in Fig.16 or by any other means such as semi-rigid
rotors
which provide the same restraining
moments. The object of the above calculations
is merely to show what moments are necessary to reduce the otherwise free flapping to within
3.1
acceptable
The flappin
limits.
c;quation
Let Io be the mcment oi' inertia
of the blade about the flapping-hinge
bJ = 0 and let I be the moment of inzrtia
for a given value of 6 3 .
Then
I = IO co2 6 3
and the flapping
equation
is
IiJ
P+P
=
IO cm2 6
3
when
12
With a b3-hinge,
the incidence
of a blade element is
%
a = e.
-@ tans3+rj;;
and we get for the flapping
(19)
moment in the advancing region
( 20)
As in Section 2.1, when the flow
incidence is reversed and
a
=
(
The flapping
%a
=- %
moment for
-e
is from trailing
OO
the partially
- @tan6
3
reversed
If&
fj - ptmlj3 p4sin4$
12 CC
>
0
edge to leading
edge the
+ -"P
UT >
region
becomes
- 2y 3 sin3*
(A -I-IP-*)
+F’p 4 sin 4J’
. 3
and for the totally
reversed
(21)
region
as before.
The computer program was the same as for
for the additional
terms in 6
3’
Computed results
592
The rate
hinge angles
of change of flapping
It
is shown in Fig.15
the case discussed
in 2.1 except
with incidence for a range of p and b3can be seen that the effect of the b3-hinge
13
is roughly to reduce the flapping linearly
by about 35-40~ per IO0 of hinge
angle over the whole range of p. Thus, for a JC" hinge, the flapping is
reduced to nearly 3 of the flapping when a 6,- hinge is absent.
Fig.14 shows
the change in flapping with incidence when 6; t 30' and is to be compared
with Fig.4.
It can be seen that the operating limit is raised from about p = 1
to p = 1.5, The rise in dP/aa, is so vapid above p = 1.5 hosvever, that
little
improvement of operating limit can bc expected by increasing the h3
angle further.
c
Fl&i?PEK$ RES'mII'IT DY BWS
6.1
The flapping
equation
The equaticn
of motion
01. AiT OFFSET ?X.&?PmG HINGE
of the blade with
an offset
flapping
hinge is
I fit! + 19 4. KG R2 F = ii1
A
where
I iu the
Ibi is the
eR is the
?R is the
flappkg
For a uniform
momen1;GP inertia
about tile oi'fset
flapping hinge
blade mass
distance of the flapyling hinge from the hub
distance CC the ct'ntre oi' gravity of the blade from the
hinge.
blade of mass m per unit
length,
I
= $mR'(l-e)
I\1
=
n,R
xx = IR
2
and equation
( 23)
(I
-
3
e)
1
( 24)
(1 - e)
(23) reduces to
( 25)
The chordwise velocity
iti the s:me as in equation
peYpendiCUh.T?
to the blade is now
(1) but the velocity
0t
14
To simplify
the calculation
of the aercdynamic
put e. = 0, since B. merely adds a constant
fore does not affect the flapping stability
with the shaft incidence.
Is[
Thus
Id&
=
$pao
expanding
IAm
I
moment we can
to the flapping angle and thereor the way in which it varies
p fl cos $ - (x - e) pt
-I
/
x+psin$
1 dx
l(
e
equation
‘z p-ac
=
h-
i12R4
flapping
( 27)
(27) gives‘
31
n2d+ (‘-e)
c
l++eh
-3
1
+’2
I +;ye
l-e
I
3 1-e
l-e
p P cos
I
_ e p2 PsinJIcos~
pAsin$-47
-;
4
(1 c -& e) PI - 3 p (3' stn$
3
. . . (28)
therefore
I
'?
I
l-e
p2PsinI;rcosI[r-A
4(
I+
-$e)
P' -5
,I p' sin 9
3
l
where
y,
is Lock's
Calculating
IVIm
inertia
number for
12
e)
(29)
the blade when e = rj.
IdIm in the manner of the previous
= IVIAR- (cl sin$+
*e
3
2x-
c
2p+Pt
sections
gives
(psi-r-he)
1
( 30)
and we also have, as before
(31)
15
It will be seen that equations (29), (30) and (31) are of the same form as
of the
equations ($I, (8) and (9) of Section 2.4, except for the modification
coefficients.
&.2
Computed results
No computed results
are given in the figures
as over the whole range
Of
p the effect of hinge offset on flapping amplitude was found to be very smalL
Bs a check on the results frCm1 the computer, the values of a4 and b, were
calculated
for 10~ Ii, .frcm the equations of Section 6.4. The results of this
calculation
are shown in T&.15 where it can be seen that the total flapping,
over a large range of hinge offset.
tnk5n a3 (a,2 + ?J,~):,: varies only slightly
Thus, it appears that ~1 offset hinge has practically
no effect on blade
flapping amplitude,
and if qy'lihkg,
tends to increase it.
It should be
unit ler;&h o? the blade has been kept constant
emphasised that
i&e
mass
per
in the above analysis.
7
STCI?i-‘~G
X 2OTC2 IH FLIG~
to achieve
a rotor must be off-loaded
of stopping
high forward speed wrAch leads one to thin!: about the possibilities
One of the main problems
the rotor in forward flight
and even of retracting
it.
stiffof slowing down or stopsing a rotor in Plip&t is the loss of centrifugal
A1tiloug!-i it is possible to reduce the
ness which helps to restrain
flapping.
?erformance
calculation3
show that
aercdynamic forces on the blade to zero in still
air,
large when thwt: are any atmospheric disturbances.
It
occur is
effective
be seen
p = cl.7
Figs. 7
flapsing
is assumed that the speed at which rotor
they may bcccme quite
stopping
and retraction
will
about 203 ft/sec.
B 35 f't/soc gust at this speed will cause an
change or" disc incidence of IO cie&rccs and from Pigs. 5 and 7 it can
that the asz~~ecTi geometric limit
of 5' degrees is reached at about
for both fully articulated
and unrostraincd
see-saw rotors.
Fran
and 12 it appears hat
a diimpcr having i;
d z 1.2 xi.11 just bring the
at p = 2.(3, and coning angle C?dtigreeo, to mithjn the flapping limit.
This means that t?ie rotational
speed of a rotor having a 35 foot
reduced to 27 r-pu before exceeding p z 2.0 at 200 ft/scc,
The flapping behaviour of articuiated
to high values of tip speed ratio,
taking
flow.
The conclusions ere:-
radius
can be
rotors has been investigated
up
into account the region of reversed
16
(I)
The motion of a freely fla?pin g rotor becomes unstable at about
on Lock's inertia number. The see-saw rotor is
P= 2,25, depending a little
stable up to p = 5 at least and the trend shovm by Fig,8 suggests that it may
be stable for all p.
(2)
The flaTping amplitude in response to a change of disc incidence becomes increasingly
great as the tip speed ratio increases.
A see-saw rotor has
the same tendency but above about p = 1 5t is less than the fully articulated
rotor.
( 3 The response to a 35 ft/sec ,+xst at 200 I"t~'scc (assunlcd speed for
rotor retraction)
would c ause fully articulated
blades to strike the fuselage
of a typical helicopter
confi&uration
at about 11= 0.7. At a cruising speed of
475 ft/szec, corresponding to a ti.:2 XrIsCil number Of 0.9, 3 35 f t/SW2
@St
WOUld
cause the blades to strike: the fuselags at abwt @ = I.
The fig-urcs for 'ihe see-s&~ rotor corresponding to (3) above, are
(4
)l = 0.7 at a forward speed of 200 ft/sec and ii = 1.5 to 2.0, (depending on the
built-in
oGning angle) when the tip i&l&
mmber is 0.9 and the cruising speed
is tim-i abo~~t 630 ft/sec.
of a b--hinge on a fully iarticulat<d
blade is roughly to
5
L, over the whole range of p, vhcn ths 6 -angle
is j0 0 . This
hal-Ja the fia;clsino
-3
raises ttl;; limiting
value oi' p to about 1 for the 35 ft,%x
gust case at
200 ft/sec.
At cruising
speed the limiting
value of 11 is raised from 1 to -i..5,
the latter val;le cGrrespondin& to a forward speed of <about 600 ft/sec.
(51 The effect
( 6)
An offset
hjnge has very little
effect
on the amplitude
of blade
flap++
(7)
Viscous dznpers arc quite eF'i'cctivc in restraining
blade flapping
that lit",le
is gs:tiled b,,~using s?ri.ngs as vrsll.
The structural
and it ap;;ccirs
conscqucnces of using these devices, hd>;;evcr, have not beon examincd.
a
0
eR
I
kd
local lift
slope of blade element
blade chord
distance of flapping hinge from centre of rotation
moment of inertia
of blade about flapping hinge
damping constant
ks
spring
kdR!2
r;a
In
L
Iid
r
R
uP
uT
v
V.
1
a
OLnf
9
Y
ao
constant
lift
on u blade
moment on blade about flapping hinge
distanoe of blade element f'rcm centre of rotation
radius of blade
component of velocity
parallel
to no-feat2icrin.g axis
chordwise
forward
component of velocity
perpendicular
to plane of no-feathering
speed of helicopter
induced velocityincidence of blade element
angle between relative
wind and no-feathering
flapping angle of blade
Lock's inertia
number 7aacR4collective
pitch
angle
V sin 3Lti - vi
h
inflolv
P
tip
ratio,
speed ratio,
density
SIR
V cos arf1
m
of air
blade azimuth angle
angular velocity
of rotor
axis
18
Ft.lPEmfcEs
No,
Author
Title,
etc.
9
A.G. Shutler
J.P. Jones
The stability
of rotor blade flapping
Pi.R,C, R & 9I 3178, May 1358
motion.
2
O.J. Lowis
The stability
of rotor
tip speed ratios.
motion at high
A.R.C.
3
0.3.
Lowis
blade flapping
23, 371, January 1962
The effect of reverse flow on the stability
of rotor
blade flapping motion at high tip speed ratios.
A.R.C. 24, 431, January 1963
FIG.1 DIAGRAM
SHOWING REVERSED FLOW REGION
(4
r1>
blRECTION
OF
FIG?@ RELATION OF NO-FEATHERING
FLIGHT
FlG(2 b)POSITION
AXIS TO DIRECTION OF
OF BLADE IN PLANE PERPENDICULAR
TO NO-FEATHERING
AXIS.
I /
FIG.3
POSITIONS
‘\
//
\
\
OF DESIGNATED REGIONS FOR SEE-SAW ROTOR
(p = l-5)
FLAPPING
ANGLE
fi
TIME
FIG. 4.
BLADE
FLAPPING
STABILITY (a’ = 6)
3c
p=
2.0
25
20
FLAPPING
ANGLE
I
35fthec
GUST
AT V* tOOft/Sec
I
33ft/sec
GUST
.
p
/
I-5
fi deg
IS
I--
2
4
&f
FIG.5
TO
FLAPPING
RESPONSE
CHANGE
IN ANGLE
TYPICAL
CEOM&TRfC
LIMIT
-m
---w-
6
deg
OF FREELY
8
IO
FLAPPING
OF NO-FEATHERING
ROTOR
AXIS
30
2J
2ci
Q, 1 bl
Is”
la’
5O
\
I 0”
\
0
L
I
iII
4*
I
6O
I
1
0O
IO0
QCnf
FIG.6
FIRST
TO CHANGE
HARMONICS
IN ANGLE
(FREELY
OF FLAPPING
OF NO-FEATHERING
FLAPPING
ROTOR)
RESPONSE
AXJS
30
1
Qo
--------
= 0”
Q,S4'
25
35ft/sec
GUST
AT V= 2yO ft/sec
/
/
20
FLAPPING
ANGLE
/3
15
ICI
CAL
GEOMETRICC MIT
m-.--s.-
5
0
2
4
R nf
FIG.7
FLAPPING
TO CHANGE
RESPONSE
IN ANGLE
6
0
OF A SEE-SAW
OF NO-FEATHERING
IO
ROTOR
AXIS
6
5
2
I
0
I
2
4
3
5
)I
FIG. 8
VARIATION
ROTOR
OF
db
-q;
((1=5*6,d=6)
WITH )I FOR A SEE-SAW
--mm--.
Ql
m.-.-.F
bl
4
FIG.9
VARIATION OF FLAPPING COEFFICIENTS WITH SPRING
AND
DAMPER c/l: 0.5, aL,p8y
a0 =2q 8, = 00)
-a----
Q,
-.-o--b,
16
I4
FLAPPING
ANGLE IO
&Q,t
de9
b,
8
6
4
2
0
FIG IO. VARIATION
AND
OF FLAPPING
DAMPER
COEFFICIENTS
(p = I, dnf=8q
Q, J 2q
WITH SPRING
Q. = 03
0
QI
bl
------.-m-
16
14
IO
f LAPPING
ANGLE
4 Wl
’
de9
6
4
il
0
I/
Of’
.’
./ -,-,---4
O-4
O-6
0.8
I*0
-----------
FIG. I I VARIATION
--
-__-
----
OF
FLAPPING
1-e
es-
COEFFICIENTS,
WITH SPRING AND DAMPER
p=M, oc,f *so, Q,=2”, eo=oo)
(
-I
fi d
I-4
---w-m
QI
-.-mw
bl
1
FIG. 12. VAR:lATlON OF’ FLAPPING ’ COEFFICIENTS WIT-H SPRING
AND DAMPER c/u R 2, +=‘Sy
CL= 2O, O. = 07)
IO
FLAPPING
ANGLE
de9
5
O-I
_
0.2
HINGE OFFSET t
FIG. 15 FIRST HARMONICS
FUNCTION OF OFFSET
&=O-4,
h 8 loo,
OF BLADE FLAPPING
HINGE POSITION.
$=O,
i&=6)
AS A
S&i
FIG. 16.
ARRANGEMENT
OF SPRINGS
AND
AND/OR DAMPER
DAMPERS.
1
I
L.R.C.
C.P.
1;u.8?7
533.6626
I
533.6.013.b.2
533.662.6
I
533.6.013J+2
1
Wilde, E., Bramwell, A.R.S.
i;n~merscales, R.
Wilde, E., Bramwell, k.R.3.
;umrzrscales, R.
BEHAVIOUR
OF A HEi.ICCf”iERftOTCRAT
HIGH-TIP SPEU RATIOS
A.R.C. C.P. IJo.
THE PUPPING
April
I%5
The blade flapping equation has been solved on an analogue computer
taking into account the reversed flow region but neglecting stall.
The
fully articulated
blade becomes unstable at about il = 2.3, whilst a seesaw rotor is stable up to p = 5 at least and the trends suggest that it
may be stable for all values of p . Howewr, tlm response to a gust, or
the equivalent change of no-feathering
axis angle, is almost rJla saem for
both rotors up to about p = 0.3.
For a 35 rt/sec gust at a roiward
speeu of 200 It/ sec. and typical rotor/fuselage
clearahce, this represents the limiting tip-speed ratio for either rotor.
The better response
I
THE FLPPING BRHAVIOUR
OF A HELICOP’ISR
ROKJRAT
HIGH-TIP SPES)RHTIOS
kpril
1965
The blade flapping equation has been solved on an analoglle computer
taking into account the reversed flow region but neglecting stall.
The
fully articulated
blade becomoesunstable at about p = 2.3, whilst a seesaw rotor is stable up to p = 5 at least and the trends suggest tktt it
imy be stable for all values of p.
Howswr, tim response to-a gust, or
the equivalent ohfmge & n44eetkmr4ng axis angle, k alnmst *
68~~ r0r
both rotors up to about p = 0.x.
For a 35 ftlsec gust at a forward
speed of 200 ft/sec, and typical rotor/fuselage
clearance, this represents the limiting tip-speed ratio for either rotor.
The better response
A.R.C. C.P. tlor8n
533.662.6:
!Ti33.4013.42
Wilue, 2.. Bramwell, i..ti.$,
jusxikerscales, H.
THE IUPPIIG XHI.VIO~R Ok. A XLiCOP’lXR I-UNQAAT
HIGH-TIP 3PCfU it;.TIOS
April
1%5
The blue I-kpping equation ins been zolvea on an analogue computer
taking into account the reverseL. flow region but neglecting stall.
The
fully urticulnteu
bln..e uecomes uusL;ble at about &l = -.3, wiiilst a seesaw rotor is stable up to p = 5 at least and the trends suggest that it
may be stable for all values of p.
Howewr, the response to a gust, or
the equivalent change of no-feathering axis angle, Is almost the same for
both rotor; up to . bout &L = 0.75.
!‘or L 35 ft/;ec zu:u;t at a iorward
speeu of 200 it/set,
<int: typic.1 rotor/fuselage
clu:r:rice, this repre:xnts the limiting tip-apeeu ratio for either rotor.
(ne better response
I
of the see-saw rotor, honever, RGGS it possible to increase the
limiting tip-speed ratio by some form of flapping
restraint.
This
has been investigated by considering the effects
of springs
and
dampers, and off set and 6 3-binges.
of the see-saw rotor, homver, makes it possible to increase the
limiting tip-speed ratio by soapeform of flapping restraint.
This*
has been investigated by consiuering the efPects of springs anu
dampers,
and offset and 6+nges,
of the see-saw
rotor, hovrever, maps it possible to increase the
limiting tip-speed ratio by sme fow of flapping restraint,
TNs
has been investigated by considering the effects of springs and
dampers, and Offset and 6+nges.
C.P. No. 877
Published by
HER MAJESTY’S
STATIONERY
OFFJCE
To be purchased from
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vv.c.1
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5
S.O. CODE
No.
23-9016-77