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.
C.P. No. 877
LONDON: HER MAJESTY’S STATIONERY OFFICE
1966
PRICE 7s 6d NET
U.D.C. No. 533.562.6 : 533.6,013.4-Z
C.P. No.877
April 1965
THE FLAPPIXG BEUAVIOUR OF A HELICOPTEE ROTOR 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 no-
feathering axis angle, 1s almost the same fcrr boti rotors up to about p = 0.75. For a 35 ft/sec gust at a forwLard speed of 200 ft/sec, and typical rotor/fuselage clearance, this represents the limiting tip-speed ratio for either rotor. The better response of the see-saw rotor, however, makes it possible to increase
the limiting tipxxpeed ratio by some form of flapping restraint. This has been
investigated by considering the effects of springs and dampers, and offset and 63-hinges.
Replaces R.A.E. Technical Report 65068 - A.R.C. 27375
06
CCNTENTS
INTRODUCTION THEFREELY FLAPPINGRCYTOR 2.1 Analysis of the fully articulated rotor 2.2 The analysis of the see-saw rotor ANALOGUE COXPUTER RESULTS 3.1 The fully articulated rotor 3.2 The see-saw rotor THE RESTRAINT OF FLAPPING %OTION BY SPRINGS AND DAJY~S 4.1 Analysis of spring and damper restraint 4.2 Computed results RESTRAINT OF FLAPPING MOTION BY K!XAXS OF 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
STOPPING A ROTOR IN FLIGHT CONCLUSIONS
Symbols References Illustrations Detachable abstract cards
Page
3 3
3 7 8 8
9 IO IO
II II II
12
13 13 15 15
15 17 18
Figures l-14
1 INTRODUCTIO~~
Rotorcraft have established themselves as flight vehicles of great utility but with low top speed. This utility 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 of rotor character- istics, such as blade fla>>ing amplitudes and stability, is essential before the improvement can be completely determined,
In order to reduce compressibility losses and noise, it will be necessary to reduce the speed oi" the rotor as the forward speed increases. This means
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 most recent being those of Shutler and Jones' 2 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 should be correct in the more important advancing region*. This would not be
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 3~1 this 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 of the fully articulated rotor
The rotor divides itself into three regions as shown in Ii'ig.1. They are:-
(1) The tradvancingtt region where the airflow over the whole blade is from loading edge to trailiq edge, 0 IC $ 6 n*
* 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 trailing edge over the outboard part of the blade, but from trailing to leading edge over the inboard part. 1‘Jhen /.J<I this re@cn extents from $t = 71 to $ = a. -When L.J > 1 the region is in two parts given by - i < sin $ $ 0, '
(3) The "total reversed" region which exists when p B 1 and the flow is from trailing edge to leading edge over the whole span. In this region sin$< -;.
The aerodynamic moment will be determined in each of these regions subject to the following assumptions. \
(I) The lift slope is constant and has the same value for both advancing and reversed flow.
(2) The effects of stalling and compressibility are ignored. Compres- sibility would be avoided in practice and so the latter assumption is not unreasonable. The assumption of no stalling leads to great simplifications and since many of the cases of interest involve lightly loaded rotors, this
assumption 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
perpendicular to the no-feathering axis has a chordwise component, UT,
given by
uT = Rr + V cos anf.sin $
or
where p = v cos anr
RR
uT = m (x -t p sin $)
The velocity at a blade section , parallel to the no-feathering axis, is
U P
z v sin anf - vi - vg cos cLnf cos Q - r;
0)
= QR (h - pfi cos $ - xb/fl> (2)
where h = V sin anf - vi
a?
58 5
If the collective pitch is eo, and the airflow is frarn leading to trail- ing edge, then the incidence of the element is
-1 4 =eottan g el
uP
T dots
T
Asszing Q = aa, the lift on an element of blade length dr is
(3)
The elementary lift moment is
Substituting equations (1) and (2) into (4) and integrating from x = 0 to x = 1
gives for the advancing region,
where ti AR denotes the blade flapping moment in the advancing region.
When the flow is from trailing edge to leading edge the expression for a in equation (3) changes sign, i.e.
(
up! a = 9
e. + q)
In the l'partial reverse" region, therefore, we use expression (6) between x=Oandxz- p sin $ and expression (3) from x = - p sin $ to x = 1. Thus
ir% is the flapping mamcnt in the partial reverse region,
But equation (7) can be rewritten as
=I M YIQ2 3 -up sin3$ ( d AR pea sin*+ 2h-2p.Pcoslir t O Sin* 1 03
Finally, if % is the flapping moment in the "total reverse" region, we have, since equation (6) applies along the entire blade,
The equation of the flapping motion 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
independent variable $. However, these difficulties are easily overcome in an
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--sciv7 rotor
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 reference blade is from leading to trailing edge but over the other blade it is partially reversed. In this
1 region 0 < sin $i < -. IJ
(2) >?here the airf'lci over the reference blade is from leading to trailing edge but over tile otl.e- r blade it is totally reversed. TiliS OCCUTS
1 only if ~1 > 1 and when - < sin $. CL
(3) 'ir'here tlle airflo;Y over the reference blade is partially reversed
but over the other blade it is from lending to trailing edge. For this case I1 i sin $I < c\.
I-r
(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.
0 The flapping moments for
the complete see-saw rotor I’OL’ the four re:,ions then becume:-
when % < sin $ ,
when
when
MA* -- IQ2
y L3 =540 c + +p200sin2j,+Jj-h--$ p2ao sin * cos $ 3
I --<
P sin $ < 0 ,
*A 3
- = MA1 xl* -+i4sin3~ eosin*-
( *a0 cos q
>
sin $ -c -5,
- i h A4
= - MA 2
03)
04)
3 ANALOGUE Coi\- ?&?JSuLTS
3.1 The fully articulated rotor
The computed response to a disturbance of the fully articulated rotor is given in Fig.&. It can be seen that the flapping motion 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 1 ti 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 a ti, there must be a linear relation- ship between /3 and a,f~
For the freely flapping rotor, the total flapping smplitude, at various
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 nu&er is constant at either 0.35 or 0.95. These assumptions define relationships between anf and p which have also been plotted. From these
curves it can be seen that the geometric limit is not exceeded for a gust of
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;iiitin it is 0.95.
3-2 ThC SC?e-SCLW rotor -
The variation of the flap<&ng angle wZth six&t angle, anfs for the see- saw 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
I-3 is zero. The curves of IFig. should be compared with those of sig.5 for tile fully artioulated rotor. 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, or at some value close to this figure.
It has been assumed in the analysis that the blades are ;?erfectly rigid and that tine coning angle rer;ljins fixed. It is rcaliscd,that in tactics, the coning angle may differ from the esswned CLue,due to blade flcxi'oility, by an amoat which 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, consideration has been given to various means of restricting it. Springs do
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 aft plane. Dampers, on the other hand, actively dissipate energy and so reduce
the amount of flapping. It is recognised that such devices may well cause
stressing problems but such considerations are beyond tne scope of the present investigation.
4.q Analysis of snring, and damper restraint
The spring-damper arrangement considered is shown in Fig.16. As the see- saw rotor appears to have the better stability, only this type of rotor has been considered. Ii&ever, the method given below is applicable to the freely
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 (15)
where ks is the spring constant, lb wt/ft
kd is the damping constant, lb nt/ft/sec
and Rt is th2 distance from the root of the point of attachment.
This equation reduces to
where ks =
ksR12
IQ2
kdRt2 'd = IQ 07)
11
Equation (16) implies that the two final terms should be added to equations
00, 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 con- sider&, 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 nf was 0 ""for~~<lanci~ofo~~~>l. The results of the computation are shown in Figs. 3-12.
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 flapp-
ing to within acceptable limits.
3.1 The flappin c;quation
Let Io be the mcment oi' inertia of the blade about the flapping-hinge when 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
P+P = IiJ
IO cm2 6 3
12
With a b3-hinge, the incidence of a blade element is
a e. % =
-@ tans3+rj;;
and we get for the flapping moment in the advancing region
As in Section 2.1, when the flow is from trailing edge to leading edge the incidence is reversed and
(19)
a (
OO "P
= - - @tan6 + - > 3 UT
The flapping moment for the partially reversed region becomes
( 20)
If& 3 -e %a =- % 12 CC fj - ptmlj
0 3 > p4sin4$ - 2y sin3* (A -I-IP-*)
4 +F’p sin 4J’ 3 (21)
. .
and for the totally reversed region
as before.
The computer program was the same as for the case discussed in 2.1 except for the additional terms in 6 3’
592 Computed results
The rate of change of flapping with incidence for a range of p and b3-
hinge angles is shown in Fig.15 It can 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 01. AiT OFFSET ?X.&?PmG HINGE
6.1 The flapping equation
The equaticn of motion of the blade with an offset flapping hinge is
I fit! + 19 4. KG R2 F = ii1 A ( 23)
where I iu the momen1; GP inertia about tile oi'fset flapping hinge Ibi is the blade mass
eR is the distance of the flapyling hinge from the hub ?R is the distance CC the ct'ntre oi' gravity of the blade from the
flappkg hinge.
For a uniform blade of mass m per unit length,
I = $mR'(l-e) 3 1 I\1 = n,R (I - e)
xx = IR (1 - e) 2
and equation (23) reduces to
( 24)
( 25)
The chordwise velocity iti the s:me as in equation (1) but the velocity peYpendiCUh.T? to the blade is now
14 0t
To simplify the calculation of the aercdynamic flapping moment we can
put e. = 0, since B. merely adds a constant to the flapping angle and there- fore does not affect the flapping stability or the way in which it varies with the shaft incidence.
Thus I Id& = $pao i12R4 s[ h- p fl cos $ - (x - e) pt l(
x+psin$ dx 1 ( 27) -I / e
expanding equation (27) gives‘
I -ac n2d+ IAm = ‘z p (‘-e) 31 l++eh I
c
I +;ye 3 1-e -3 l-e p P cos 4
+’ 1 I 2 l-e
pAsin$-47 _ e p2 PsinJIcos~
-; (1 c -& e) PI - 3 p (3' stn$ 3
. . . (28)
therefore
I I '? l-e
p2PsinI;rcosI[r-A I+ 4( -$e) P' -5 ,I p' sin 9 3
l *e (29)
where y, is Lock's inertia number for the blade when e = rj.
Calculating IdIm in the manner of the previous sections gives
3 IVIm = IVIAR - (cl sin$+ e)
12 c 2x- 2p+Pt (psi-r-he)
1
and we also have, as before
( 30)
(31)
15
It will be seen that equations (29), (30) and (31) are of the same form as equations ($I, (8) and (9) of Section 2.4, except for the modification of the
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, : tnk5n a3 (a,2 + ?J,~):, varies only slightly over a large range of hinge offset.
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 emphasised that i&e mass per unit ler;&h o? the blade has been kept constant
in the above analysis.
7 STCI?i-‘~G X 2OTC2 IH FLIG~
?erformance calculation3 show that a rotor must be off-loaded to achieve
high forward speed wrAch leads one to thin!: about the possibilities of stopping
the rotor in forward flight and even of retracting it. One of the main problems of slowing down or stopsing a rotor in Plip&t is the loss of centrifugal stiff-
ness which helps to restrain flapping. A1tiloug!-i it is possible to reduce the
aercdynamic forces on the blade to zero in still air, they may bcccme quite large when thwt: are any atmospheric disturbances.
It is assumed that the speed at which rotor stopping and retraction will
occur is about 203 ft/sec. B 35 f't/soc gust at this speed will cause an effective change or" disc incidence of IO cie&rccs and from Pigs. 5 and 7 it can be seen that the asz~~ecTi geometric limit of 5' degrees is reached at about p = cl.7 for both fully articulated and unrostraincd see-saw rotors. Fran Figs. 7 and 12 it appears hat a diimpcr having i; d z 1.2 xi.11 just bring the flapsing 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 radius can be reduced to 27 r-pu before exceeding p z 2.0 at 200 ft/scc,
The flapping behaviour of articuiated rotors has been investigated up to high values of tip speed ratio, taking into account the region of reversed flow. The conclusions ere:-
16
(I) The motion of a freely fla?pin g rotor becomes unstable at about
P= 2,25, depending a little on Lock's inertia number. The see-saw rotor is 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 be- comes 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.
(4 The fig-urcs for 'ihe see-s&~ rotor corresponding to (3) above, are )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.
(51 The effect of a b--hinge on a fully iarticulat<d blade 5 hal-Ja the fia;clsino -- L, over the whole range of p, vhcn ths 6
3 -angle
raises ttl;; limiting value oi' p to about 1 for the 35 ft,%x gust
200 ft/sec. At cruising speed the limiting value of 11 is raised
is roughly to 0 is j0 . This
case at from 1 to -i..5,
the latter val;le cGrrespondin& to a forward speed of <about 600 ft/sec.
( 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 and it ap;;ccirs that lit",le is gs:tiled b,,~ using s?ri.ngs as vrsll. The structural conscqucnces of using these devices, hd>;;evcr, have not beon examincd.
a 0 eR
I
kd
ks
r;a
L Iid r R
uP
uT v V.
1
a
OLnf
9
Y
ao
h
P
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
spring constant kdR!2
In
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 component of velocity perpendicular to plane of no-feathering
forward speed of helicopter
induced velocity-
incidence of blade element angle between relative wind and no-feathering axis
flapping angle of blade aacR4- Lock's inertia number 7
collective pitch angle
inflolv ratio, V sin 3Lti - vi
SIR
tip speed ratio, V cos arf 1
m
density of air
blade azimuth angle angular velocity of rotor
18
Ft.lPEmfcEs
No, Author Title, etc.
9 A.G. Shutler The stability of rotor blade flapping motion. J.P. Jones Pi.R,C, R & 9I 3178, May 1358
2 O.J. Lowis The stability of rotor blade flapping motion at high tip speed ratios.
A.R.C. 23, 371, January 1962
3 0.3. Lowis 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 AXIS TO DIRECTION OF FLIGHT
FlG(2 b)POSITION 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
25
20 FLAPPING ANGLE fi deg
IS
I p= 2.0 35fthec GUST
AT V* tOOft/Sec
I 33ft/sec GUST . / p I-5
TYPICAL CEOM&TRfC LIMIT I-- -m -- -- -w-
2 4 6 8 IO &f deg
FIG.5 FLAPPING RESPONSE OF FREELY FLAPPING ROTOR
TO CHANGE IN ANGLE OF NO-FEATHERING AXIS
30
2J
2ci
Q, 1 bl
Is”
la’
5O
\ I 0” \ 0 i
L I I I 1
II 4* 6O 0O QCnf
IO0
FIG.6 FIRST HARMONICS OF FLAPPING RESPONSE
TO CHANGE IN ANGLE OF NO-FEATHERING AXJS
(FREELY FLAPPING ROTOR)
FLAPPING ANGLE
/3
30
25
20
15
ICI
5
0
Qo = 0” -------- Q,S4'
1
35ft/sec GUST AT V= 2yO ft/sec
/ /
CAL GEOMETRIC C MIT m-.--s.-
2 4 R 6
0 IO nf
FIG.7 FLAPPING RESPONSE OF A SEE-SAW ROTOR
TO CHANGE IN ANGLE OF NO-FEATHERING AXIS
6
5
2
I
0 I 2 3 4 5
)I
FIG. 8 VARIATION OF db
-q; WITH )I FOR A SEE-SAW
ROTOR ((1=5*6,d=6)
--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 b, de9 8
6
4
2
0
FIG IO. VARIATION OF FLAPPING COEFFICIENTS WITH SPRING
AND DAMPER (p = I, dnf=8q Q, J 2q Q. = 03
16
14
IO f LAPPING ANGLE 4 Wl ’ de9 6
4
il
0 ------ QI -.-m- bl
0 I / Of’ O-4 O-6 0.8 I*0 1-e - I-4
.’ ./ -,-,--- -----------
es- fi d
-4 - -- ---- -__- -I
FIG. I I VARIATION OF FLAPPING COEFFICIENTS,
WITH SPRING AND DAMPER
( p=M, oc,f *so, Q,=2”, eo=oo)
- - -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 OF BLADE FLAPPING AS A FUNCTION OF OFFSET HINGE POSITION.
&=O-4, h 8 loo, $=O, i&=6)
S&i AND/OR DAMPER
FIG. 16. ARRANGEMENT OF SPRINGS AND DAMPERS.
I L.R.C. C.P. 1;u.8?7 533.6626 I
533.6.013.b.2 Wilde, E., Bramwell, k.R.3. ;umrzrscales, R.
THE PUPPING BEHAVIOUR OF A HEi.ICCf”iER ftOTCR AT HIGH-TIP SPEU RATIOS April I%5
The blade flapping equation has been solved on an analogue computer The blade flapping equation has been solved on an analoglle computer taking into account the reversed flow region but neglecting stall. The taking into account the reversed flow region but neglecting stall. The fully articulated blade becomes unstable at about il = 2.3, whilst a see- fully articulated blade becomoes unstable at about p = 2.3, whilst a see- saw rotor is stable up to p = 5 at least and the trends suggest that it saw rotor is stable up to p = 5 at least and the trends suggest tktt it may be stable for all values of p . Howewr, tlm response to a gust, or imy be stable for all values of p. Howswr, tim response to-a gust, or the equivalent change of no-feathering axis angle, is almost rJla saem for the equivalent ohfmge & n44eetkmr4ng axis angle, k alnmst * 68~~ r0r both rotors up to about p = 0.3. For a 35 rt/sec gust at a roiward both rotors up to about p = 0.x. For a 35 ftlsec gust at a forward speeu of 200 It/ sec. and typical rotor/fuselage clearahce, this repre- speed of 200 ft/sec, and typical rotor/fuselage clearance, this repre- sents the limiting tip-speed ratio for either rotor. The better response sents the limiting tip-speed ratio for either rotor. The better response
1 1 A.R.C. C.P. IJo. 533.662.6 I
533.6.013J+2 Wilde, E., Bramwell, A.R.S. i;n~merscales, R.
I THE FLPPING BRHAVIOUR OF A HELICOP’ISR ROKJR AT HIGH-TIP SPES) RHTIOS kpril 1965 I
A.R.C. C.P. tlor8n
Wilue, 2.. Bramwell, i..ti.$, jusxikerscales, H.
533.662.6: !Ti33.4013.42
THE IUPPIIG XHI.VIO~R Ok. A XLiCOP’lXR I-UNQA AT 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 see- saw 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
of the see-saw rotor, homver, makes it possible to increase the limiting tip-speed ratio by soape form 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, 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, 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 49 High Holborn, London vv.c.1 423 Olxford Street, London w.1 13.~ Casl.le Street, Edmburgh 2
109 St. Mary Street, Cardiff Brazennose ,Street, Manchester 2
50 Fairfax Street, Bristol 1 35 Smallbl ook, Ringway, Birmingham 5
80 Uuchester Street, Belfast 1 or :hr~oug,h any bookseller
S.O. CODE No. 23-9016-77
