. .
Cl5RdtRT AIRCRAFT FSTABLf$bMl’l
BEDFORD.
-: MINISTRY OF AVIATION
AERONAUTICAL RESEARCH COUNCll
CURRENT PAPERS
C.P. No. 852
The Motion of Helicopter Blades at Low Rotor Speeds
in High Winds bY
M. A. P. Willmer
LONDON: HER MAJESTY’S STATIONERY OFFICE
1966
TEN SHILLINGS NET
C.P. 852
September, 1963
TBl3 MYl!ION 03' HELICOPTXR BLADES KC LCV RCTOR SPEXDS IM HIGH KUDS
by
M, A. P. Willmer
A theory fcr predicting the behavicur cf a rctor at lcw r,p.m. in a strong
steady wind has been develcped. Results have been cbtained both for helicopters
cn the ground and cn a rolling platfcrm. It has been found that, under steady
ccnditions and with the helicopter cn the ground, serious blade deflecticns are
net likely tc occur. However, with a helicopter on a raised landing deck, it is
pcssible fcr the 'cliff-edge' effect to be cf major importance especially for
helicopters with large rotors relative to the deck width. Finally the blade
response tc a vertical gust has been investigated and it has been shown that
helicopter rotors are very susceptible to them.
Replaces ,r.R,C.25835, RaE -rlul NaoaC W
F
1 IlNTRODUCTION
2 A %LICOPl!ER IN A STEADY WIND
3 A BELICOPTER ABOARD A ROLLING
3.1 Rigid blade flapping
COM!ENTS
ON THE GROUND
SHIP IN A STEADY WIXD <
3.2 Blade moticn at low rotor speed 3.3 The 'cliff-edge' effect
4 THE EFFECT OF A V3XCICAL GUST ON THE ROTOR AT LOW R.P.M.
5 coI~CLusIONs
6 ACICNOWILEDC~
REFERENCES
SYMBOLS>
rlPPENDIX 1
Page
4
5
8
8 9
10
14
20
20
20
22
24- 27
ILLUSTRXCONS - Figs.l-22
DETXH;BLE :,BSTR.XT CXDS ILLUSTIRATIONS
A typical heliccpter on the ground in a wind
Diagram showing the velocities at azimuth angle \ir
Comparison between flapping theories
Blade deflection harmonics for Whirlwind helicopter in 30 kt wind on the ground (no cyclic control)
Fifi.
1
2
3
4
Fore and aft blade deflections for Whirlwind helicopter in a 30 Id wind on the ground (no cyclic control) 5
Blade deflection harmonics for P531 heliccpter in 25 kt wind on the grcund (no cyclic control) 6
Fore and aft deflecticns for P531 heliccpter in 25 kt wind on the ground (no cyclic control) 7
Diagram shcwing helicopter attached to a rolling ship in a sidewind 8
Variation of the flapping angle with p fcr several roll angles (P531 Helicopter) (A~B)
-2-
U;LUSTRAXCMS (COKED) Pig.
Variation of 'p' with a 10
Diagram showing the disturbed airflow throug;h the rotor disc due to the presence of the ship II
Diagram showing the deck and the equivalent two dimnsicnal circular cylinder
Variation of k3Vv/2b2xV with $
12
43
Variation of G(k) and. G*(k) with k 14
The increment to b, due to the presence of the ship
Blade deflection harmonics for the P531 hclic@,cr in a 25 kt wind on a frigate with 15' ml1 (no cyclic controlj
15
16
Total blade deflecticns for the P531 helicopter in a 25 kt wind on a frigate with 15' roll (no cyclic control) 17
Blade deflection harmonics for Whirlwind helicopter aboard a frigate in a side mind of j0 kt (no cyclic control) 18
Blade deflection harmonics for Whirlwind helicqter aboard a frigate in a side wind of 30 kt (A, = -5’) 19
V,ariation of tip deflecticn with azimuth for Y!hirlwind cn frigate 20 in 30 k-t wind (A B B)
The increase in lift due to the 'cliff-edge' effect 21
Tip deflecticn of 'Whirlwind rotcr in j0 kt wind hit by a 5 Pt/sec downward gust (A FB)
- 3 -’
1 II'B'RODUCTION
I’r
.
For the effective operation of helicopters in the Royal Navy, it.is *necessary that they should be able tc operate frcm ships over a wide range of sea
and wind conditicns. This includes the need for the rotcrs to be safely started and stepped in high ninds. However, it is kncwn that in such conditions, even ashcre, large blade deflecticno can cccur -&en the rctcr speed is low. The cause of these deflections is not fully understcod. The complete analytical solution to the prcblcm is net possible at this stage, due to the complexity of' the blade bending and flapping motions, when the blade is in contact with the droop stops during only part of the azimuth rotation. For this reason, this paper concentrates cn certain aspects of the problem. In this way a better under- standing of the causes of large blade deflecticns at low rotor speed can be obtained relatively quickly.
The first aspect to be examined is the problem of a helicopter in a steady wind on the ground. The rotor blades have been assumed to be fixed at the flapping hinge. At very low r.p.m .J when the blade rests on the droop stops, this assumption will be valid. For typical helicopters this would cover a rotor speed range of 0 to about 50 r.p.m. Unfcrtunately, the aerodynamic fcrces on the blade in this range of rotor speeds are extremely difficult to calculate. Conventional rotor theory as given by Ref.1 say, becomes less accurate as the tip speed ratio increases beyond a value of about 0.6, as this theory does not take into account such phenomena as blade stalling or reversed flow. For the purposes of this paper it is necessary to consider tip speed ratios nuch greater th‘an that ylhere the theory of Ref.l begins to.fail. In order to progress.further therefore, the assumption is made that at a sufficiently large ti.? speed ratio, the aerodynamic force on the retreating side of the rotor disc c,an be neglected. A comparison is given between this assumption and a more ccqlicated theory being developed at Naval Air Department at the moment. In this way, the use of a very sophisticated rotor thecry is avoided.
Secondly, the case of a heliccpter aboard a frigate is investigated. The behaviour of the blades whilst stopping and starting during severe rolling motion has been examined. This is primarily the effect of rotor incidence changes and, since rolling an&es are much greater tkan pitching motion, trouble with the blades is expected to be greatest when the relative M.nd blows from either port cr starbcard, The presence of the ship and the landing platform will then also affect the blade m&ion ccnsiderably, for there will be in these cases a type of 'cliff-edge' effect. This paper considers these factors and the calcula- ticns arc mainly devcted to a study of the behaviour of the Whirlwind and Wasp heliccpter rotors.
Thirdly, the effect cf vertical gusts on the blade motion is examined. Again, in crder to calculate reasonably swiftly the aerodynamic forces involved, it is necessary to m&c many severe assumptions. The gust is considered to hit the whole blade instantaneously and to remain during the subsequent motion. From the point of view or' examining the susceptibility of a rotor to sharp changes in vertical velocity, it is expected that the assumpticns used will be satisfactory.
-4-
2 A HGLICC~BR IM A STEADY KJ3?3 ON TH.3 GROUID
Tc begin the invcstigaticn of blade sailing, the case cf a helicopter in a high steady .<;ind was ccnsldered. Big.,1 slangs a typical helicopter in a wind of speed V ft/oec blcwtilg in the longitudinal directian'cf the aircraft. ??hen the rotcr is staticnary, the blades' nil1 bend as shown and the purvcoe cf this . investigaticn is tc see v:hether in slcwing dcwn cr s:;arting up there is sufficient excess bending'for the blades tc strike the tail bccm. In practice the helicopter need not necessarily be facing into wind SC that the deflecticns all the tiay rcund ' the rctcr'disc must be considered. It is to be ex:>ected that with only collective pitch applied, the blades will rise as they come into wind and f'4.1 as they retreat. Alsc as the rctor slows dcvm, it is to be anticipated that the aerodynamic fcrces d " acting on the blades will only be significant c?n the advancing side where the incident velccity is greatest. lit the rotcr speed considered, beluvt 50 r.p*m*, the rctor will probably be resting on the drcop stops for mcst of the cycle SO that the equation of blade bending will have slightly different bcundary conditions from that generally used in helicopter blade bending thccry. For present‘purpcses therefore, the bending equaticn cf Ref.2 is modified slightly. The datum line, with respect to which the .blade deflecticn z is incasurea, is now taken to be a line thrcugh the blade rsot at the hub, see Fig.2. This figure which shcws the situation at a typical azimuth angle ijr is thus b,asically the same as Fig.U, of'
‘Ref.3. Hence the equation of blade bending is
EI ' 2 z - - 2. m $(R2 _ r2) 22
ar4 2
i3r2 .
. - it being assumed that EI i s constant acrcss the spm. I
The boundary conditions arc ( .
a2 . . z = dr =o. at l-- = 0
and
-at r =R
-as the blade is assumed to behave like acantilever beam at the origin. Before equaticn (I) can be solved the aerodynamic ccmponcnt -?!$ has to be estimated. This .f: is difficult to do accurately because ef the magnitude of the tip speed ratios involved in the problem. A typical value-of TV is given by a 30 kt wind, a blade radius of 20 ft and a rotor speed of 3 rads;/sed, SC that TV h 0.85. s
-5-
On the advancing side of the disc the simple theory of Ref.1 gives the approximate equation
&IL ar= 4 a pc ‘-(Cl
i r+Vsin$
f-7 y 7Yo - (P -V Po cos $)(nr + V sin $>
- i(nr + v sin $) 3
(3)
fcr a rotor with no applied cyclic pitch control. On the retreating side, hcw- ever, this equation is no longer valid because of reversed flew and blade stalling. Although a scphisticated aerodynamic theory can be developed to
dl; calculate the z coqonent, the time and labour involved is not warranted by the
accuracy required at the present stage. Fcrtunately scme reasonable approxima- tions can be limde which keep the estimaticn of e fairly straightfonvard.
The i term of equaticn (3) represents the damping cf the moticn and as damping will always be present it was decided to assume that it is constant round the disc. The value of the constant was found from the vertical flight condition. Also, as the largest aerodynamic force will occur on the advancing side, it was decided to assume that the remaining terms of equation (3) were zero on the retreating side of the disc. Thus Ewas taken to be given by
&L dr = W) -$apcnri Oc)
where
I?($) = & apt I
(nr f V sin $)2 O. - (p -V Po ccs Jl)(Qr + V sin $) I
O<$<n
(5) = 0 7L<$<27F .
It should be noticed tot that the induced vclccity terms are neglected but, as p is large, this assumpticn is again expected tc bc reasonable.
A compariscn between the abcvc theory and some unpublished work on an % advanced rotor thecry by Branwell and 'iii r lde at the R.A.E. is given in Pig.3.
Nsc included are some results from the well known Squire thcoryj, The comparison is between the thecretical predictions cf the blade flapping motion
'; of stiff blades T:,ith the shaft held at a constant angle to the flow at high tip speed ratios. It can be seen that, above a p of 0.5, the accuracy cf the theory of Squire diminishes until, at a p of 0.9, it is very poor* The new simple theory, cn the other hand, give s reasonable agreement for the first harmonic flapping angle a,, whilst all the coning angle predictions are of the same 'order
of accuracy. The new theory can be seen to give, therefore, a reasonable first order amroximation for the large tiR speed rai;ic cases.
Fourier analysis gives F(Q), the aercdynanic loading, in the form
03
I?($) = Be(r) + '-T ),
(n,(r) cos n $ + A,(r) sin n jr) (6) s
n=l
where details of the analysis me t'c be found in Appendix .I. Only the first few terms in'the series +ll be considered so that, from equations (I), (4) and (6), the equation cf blade bending becomes
z
(7) IV
= Be(r) + .c
(B,(r) cos n $ + An(r) sin n $) + mr R2Bc-- mg
I-4
Following Ref.2, giving
a non4irnensional spanwise coordinate x can be used, thus
a42 -- ax4
where x = r/R
= bob) + ' (b,(x) cos n $ + a,(x) sin n $) + 2K*Rx PO
n=l
!I r-d& cl2
z 2EI
J = z a pc iI2 R5 El
w
(9) *
0
with boundary conditions
-7-
x = 0 dZ
Z =dx
= 0
. W) 1 a22 d3Z
x = L=-=
ax2 ax3 0 .
To solve equation (83, z was assumed to be of the form
Z = z,(xj + >, -? (z,,~k, cos n $ + zns(x) sin n $)
n=l
(11)
and the solutions obtained using a digital computer.
As examples of typic,al helicopters, the- ~FJhirlwind and P531 (Wasp) were chosen, The behaviour cf the Whirlwind rotor, which has a blade tip deflection of about 34 ft when stationary, was examined in a 30 kt wind for rotor speeds of lb and 2 rads/sec. Fig.4 gives the results fcr zo, z,C and z,~. Fig.5 shows the total blade deflecticns at the fcre and aft pcsiticns. These results were calculated assuming_that there was no twist on the blades and that the value of tie was that at the $ spanwise pcsition. In the case of the 4 rads/sec rotor
speed both the values fcr the new thecry and the conventional theory are shown, see Fig..!+.. Here the value of p is 0.49 and it can be seen that the agreement between the theories is quite good. Pigs.6 and 7 show the corresponding results for the P531.
From Fig.5 it can be seen that the static deflection is not exceeded in either case so it would appear that in a steady wind there is no danger of the tailboom being stru,ck by the rotor blade as it accelerates or slows down. It must be reacmbered that tho above calculaticns assume zero cyclic variations cf pitch and do not take into account any cyclic stick movements which the pilot could make to tilt the rotor disc.
3 A HELICOPTER ABOARD A ROLLITirG SHIP m A STEADY BIlND
3.1 Rigid blade flapping
The rotor speed will be high when a helicopter lands on a ship. Although the pilot will have taken advantage cf a quiescent period in order tc land, it is quite possible that in a rcugh sea'thc ship will roll violently shortly afterwards. In such cases the effect of a large roll will be either a large increase of flow up or doivn through the rotor depending on whether the roll is into or out of the relative wind. Eecause the rotor speed is high, rigid blade flapping can be considered. Y'ig.8 shows a helicopter with no applied cyclic control in a relative wind of V ft/sec blowing at right angles to the axis of roll. Defining the flapping angle P by
-8-
g = a; - a; co.5 $ - bi sin JI 02)
we have from Ref.4
03)
f
and
1
b, = - 4wc
30 .3P2> . 05)
From Ref. 5 it has been shown that a better approximation for bi is given by
i’ -i b, = 1
1 +&.L
Fig.9 shows the results fcr the 331 helicopter where the blade flapping is calculated at azimuth stations 0, 90, 180 and 270 degrees for rolls of 5, 10 and 15 degrees. Again the value of o. was taken to be that at the 2 spanwise
pcsition. It can be seen that, as p increases, the range of pcssible flapping o?cns up considerably. P53l is high, abcut 340 speed cf 34 l&s or mere during the startin?; and will be e-x--crienced.
Under ncrmal operating ccnditions the rotor speed of the r.p.m., and thus a p of 0.1 is not exceeded until a wind is cnccuntered. Thus trouble is only likely to occur stopping cf' the rotor when the higher tip speed ratios
If the incident wind bloT,-rs at an angle a to the roll axis, the effect is given approximately by using a modified angle cf' roll q', where 'p' is'given by
9’ = cpsina . (17)
The variation of 'p' with a is given in F&IO. %
3.2 Blade motion at low rotor speed s 4 It was shown in Fig.9 that, for some angles of roll and at sufficiently
large tip speed ratios,sRvere flapping will be obtained. When the flapping motion becomes excessive, it nil1 be resisted by stops at the hub. There will be periods
-9-
' thercforc, in the startingl.or stopping of a rotor, under adverse conditions of wind and roll, when the blades will hit against these stops and bend as canti- lever beams. Such a mcticn is extremely difficult to analyse for it will be a combination cf flapping and transient blade bending. When the rotor speed is
, decreased further, the cantilever beam moticn will predominate and thus the blade motion will be given approximately by theequations of section 2. These equations
8 can be readily used by suitable choice of the velocity through the rotor. Examples of the theory, however, will be left until after the 'cliff-edge' effect has been examined.
3.3 The 'cliff-edge! effect
When a helicopter lands on a ship, the rotor can no longer be assumed to lie in an undisturbed airflow moving with a given velocity. The streamlines
,near the ship, when the wind is from the side, are expected to be like those shown diagrammatically in Fig.11. They will rise on the upstream side and fall on the downstream side. A rotor in the airflow above the ship will experience this up and down flow and it is this phenomenon which is sometimes called the 'cliff-edge' effect.
Tb estimate hew these changes cf rotor flow affect the rotor behaviour, it is assumed that the ship can be replaced by a two dimensicnal cylinder of equivalent radius, see Fig.12. This equivalent radius is defined by
; a2 2 = S +d2 . 08)
. If h is the height cf tho rotor disc above the deck then the vertical and hcrizontal velocities at P are given by6,
where
% and
F
% a2 = v- rf2
sin 20
2
33 =Vl-" ( - rf2
cos 20 >
rt2 = (h + s)~ + q2 (21)
.e = arc tm --+- . . 1 J 9
(19)
(20)
(22)
Thus, when the blade is at azimuth angle $, the vertical velocity at the spanwise position r is obtained by putting
- IO -
Defining
equation (19) becomes,
q = -rcbs$ . *
l- h+s
-- = R
(23)
x = r/R (25) ’
b = VR w 3
vv v=
- 2b2 kxccs $
[k2 + x2 cosl- $1 -5 ' (27)
For simplicity, it is assumed that the cylinder's pressure does not affect the horizontal velocity so that
VH = v , w3)
The accuracy of this approximation is expected to be at least as good as that of the assumption that the ship can be replaced by a theoretical two dimensional
9
cylinder.
The expression for the vertical velocity given by equation (27) can be .s siznpiified further by assuming that the demcninator is always calculated with $ = 0. Fig. 13 shows the comparison for x/R = 2. Thus following Ref.1, the effect on the first harmonic of rotor flapping can be silo\~ to be an increase in the b; component of
(29)
where
g(k) = ,!c ‘bog(l + $) - , ; k2j l (30)
The function g(k) is plctted in %
Pig. 14 and the increase cf b' in Fig.1 5 for two values of b2.
1 The value of k chosen is typical of a P531 onboard an Ashanti
Class frigate. ^-
In order to calculate the effect of the 'cliff-edge' contribution on the blade bending at high‘tJ.'s, ari additional term to,the 2 of 'equation (3) must be
included. This additional term, which is due to the extra lift, is given by
. -ll-
A& 2
dr = - & pc a (nr + V sin l/f) 2b kV x 'OS ' .
[k2 + x2]
By making the further assumption that x = I in the denominator,6 this extra term results in a slight modification to the function defined by equation (A.2). Also if the cyclic control movements are included so that the blade pitch is given by _ .
6' = 4 - B, sin JI 2 A, cos $ C 0 (32)
the function of (A.2) becomes I .
60 = a0 x2 - x [B, p + h] + $ 4. p2
g;s = 2~x40-h~-B,x2-$p
(33)
(34)
2 2
gC = pxP,+$p 2A,-&p2A,-~2A,-2b ‘i’;
f.1 +kl (35)
hC = B, @X-&l2 7Yo
hs = +p2 PO-'A, p x-p 2 b2 k x [I + k212 '
(37)
(31 The effect of the assumption, that x = I in the denominator of equation
g(k , on the first harmonic of rigid blade flapping can be shown to change the of equation (30) to
g'(k) = & k (1 + k2)2 l
(38)
This function is ccmpared with g(k) in Fig.l/+; for values of k above 1.5 it is a very reasonable approximation.
Results for a P531 onbcard a frigate of the Ashanti Class with 15' of roll in a 25 kt wind are given in Figs.16 and 17. The fcrmer gives the harmonics of the deflecticns and the latter the total deflections at $ = 0, 90, 180 and 270 degrees. It can be seen that the maximum deflection is about I ft which can be reduced by the appropriate cyclic ccntrcl movements. Thus, for operations from frigates, it can be seen that the P531 is reasonably satisfactory from the point of view of stopping and starting the rotor under difficult conditions.
- 12 -
Gn the other hand, the Whirlwind can be shown to be much less suitable because the larger blades bend very much more. 5'ig.-l8 shows the harmcnics of blade bending for a Y?hirlwind abourd a frigate Cth nc cyclic control in a 30 kt wind. The crder of the magnitude cf the deflections is now considerably increased. The effect of applying cyclic ccntrol is given in Fig.19 where A, is taken.to be -5 degrees. In the case cf the 4 rads/sec rotor speed both results frcm conventional aerodynamic thecry and the new simple thecry are snown. In Pig.20 the variaticn of the total tip deflecticn with azimuth is given. Prom these figures it can be seen that the static deflecticn is well exceeded as the rotor speed decreases even when large cyclic contrcl movements are applied. Although in calculating the aeredynanic fcrces Inany severe assumptions have been made, it is thought that the results cbtained are cf sufficient accuracy tc indicate the im?crtance cf the'cliff-edge' effects.
The 'cliff-edge' effect will alsc influence the total lift cn the rotor whilst the helicc@er is hcvering ever the deck befcre landing. If the r&or head is at a ;?cint (jr', h + s) instead cf (0, h + sj, see I'ig.12, the vertical velccity at the pcint (x, JI) ccrres?cnding to equaticn (27) is
V 2 b2 k(y - x cos $)-- . . ik2 + (y - x- cc9 $)2]2
where
y = yf/R .
The increase cf lift per blade is given apprcximately by
Al? = $-p ac Q2 R3
where av is the increase in the induced velocity. From equation (41) it can be shown that for Q blades the increase in rotor thrust with y is approximately
(39)
s
(40) 2
0
L ac p n2R3Q LIT = .7p, E
. . . (42) '
- 13 -
This increase fcr a P531 helicopter landing on a frigate is plotted as a percentage of the all up weight in Fiy.21. This increase was noticed during recent P531 trials abcard H.X.S. Undaunted. Theoretically there shculd be a ' correspcnding downward rctcr fcrce at positicns dcwnwind cf the centre of the deck, but the effect experienced wa s not nearly so proncunced. This can be explained by the fact that the strea&ines tend to relnain hcrizontal after passing over the deck due tc the wake dcwnwind.of the ship. This phenomenon,
f is shcwn diagrainmatically in Fig.1 of Ref.7.
4 T;% EFFECT CF A VXDICAL CUST OPT TKii RCi'CR AT LC?J R.P.M. ri;
It was shcwn in section 2 that no serious blade deflecticns are expected to cccur when a WhirlGnd starts cr stops its rctor in a steady high'wind. As several accidents have been rellorted in the vast due to abnormal rotor behaviour, it would seem as though the cause-must lie in the field of the blade response to gusts. The analysis of such effects, particularly at low r*p*m., is extremely difficult and many assumptions have had tc be made in order to pregress.
Only vertical gusts have been considered for these are exsec-ted to produce the greatest effects and the assumpticn is made that at a given time the whole blade is affected by the gust which continues until after the peak response has been obtained. This time fcr a Vnirlwind rctcr will be less than a second in most cases. Alzo it is assumed that the rctating blade behaves like a staticnary cantilever beam xi'ih increased stiffness so that the equaticn fcr the blade deflection is given by
(lc3)
wiiere EL' is the mQdified stiffness. Par ccnvenience in the estimation of the
aerodynamic term $$ on the retreating side of the disc, the azimuth angle is taken from the upwind position SC that
* = x+5. $4)
On the part of the blade where the stall has not been reached, 2 is given by
ccl 2 ar = spaci: a (45)
where a is the angle of incidence and w is the velecit/ of the air with respect to the blade neglecting the spanwise ccmpcncnt. Thus
- ll+ -
w2 = (nr 7 v sin c)2 + y.l + v L G (up, I- $) ccs 5 -,ij; ’ e. (46)
. - 1 . ; .
and .:
zz ijo +
bG 9..v cos c (g - PO)-- i . . a
_ Qr - V sin g (47) r
where the blade is iknersed in an upl?rard gust of IJ~ ft/sec. For the stalled part of the blade;it is assumed that
. *
The value of r where blade
(as-. s go) (fir
It is to be ncted that 'the
aI 2 ar= &P acw a .
S
stalling comnences is
- v sin g) = ‘GfV
obtained from
sign.of as depends upon whether the blade is stalled
from above cr belcw and, in cases of ambiguity, the most outbcard stalling point was used.
(48)
the equaticn
.
E+8/ (49)
In crder tc cbtain a quick solution to equation_(l+j), the respcnse of the blade tc the gust is considered in segncnts, each of h set duraticn. These
segments are chosen sufficiently small that in them sip <,‘ ccs E, k and $ can
be regarded as ccnstsnt in the aerodynamic force term. In order to provide continuity the terminal deflections and velccities of cne segment become the starting value s for the next, Also it has been assumed that in each segment, the blade deflecticn and velocity vary linearly with the radius so that, in the (i + 1)th segment,
i az i+l = t(R,i) r/R I > z
i+l = $ B(R,i) (50)
and thus the velocitym and the incidence a are given by x
‘.4. . = (Rr -v sin L+,)2 +
. * 2 co9 E; i+l - $.k{R,i)
. . . (51)
3
- 15 -’
&lG + v cos gi+, ( v - PO
> - $ i(R,-j)
CL i+l = Bo+ Sk - v Sill gi+, (52)
where Ei+.l takes the meCan value of the se,gment. Thus it can be seen thatWi+, and a
i+l are now purely fractions of r.
? The damping of the motion i s calculated as if' the rotor was hovering.
Bjr neglecting blade deflection effects and taking the value at the -2 spanwige z positicn, equation (43) becomes
c
.
31, aLcz a22 aL‘ 0
I - a.2 -+m-+a,% = s
ar4 at2 +mrQ2po-mg
where
‘; 1
= gpacQR .
(53)
(54)
0 I aL The dr term is new a function of r only for each small segment and it contains
all the aerodynamic terms minus the damping. Thus it is possible to write the deflection in the form
z = z,(r) + +Qd
where
, EI' a4zt a2Zt - aet -+m-+a ark at2 i-SF= 0
& .I . 31' 0 dL -. =
ar4 0 ar +mrR2Fo-mg .
(55)
(56)
Using the non-dimonsicnal radius x, the soluticn of equation (56) which reprcscnts the time variable compcncnt can be written in the form
z,(x,t) = A]* -cod-l P,x c i -bkt
. oosh p k
cos(wkt + cp,) e
k a.0 (58)
r = xR
- 16 -
see Refs.8 and 9. The natural freqtiency .of the kth mode, ok, is given by
where the undamped natural frequency wnk and .the damping factor are obtained from
w2 $ EI’
nk = R4 m
21, Wnk r, z,/m = 2; .
(59)
(60)
Prom Ref.8 the values of the first three P, are given as O.GOOX, 1.49~ and 2.50% However, for present purposes, z&x,-t) is given by
only the fundamental mode will be used so that
z;(x,t) = G(x) cos(wt + 'p) 0 43 (62)
a4 = -cos11 px - co3 px
i
sinh Px - sin cash P + cos P L -zh (3 + sin p (63)
where p, w and c have the appropriate values.
The modified stiffness, EX', is determined from the results quoted in Ref. 10 where a relationship between the natural frequency of the rotating blade in any one mode and the natural frequency of the staticnary'blade in that mode is given. Thus
2 'k = + E !a2
where 'k = natural frequency cf the rotating blade in the bth mode
w Ok = natural frequency of the staticnary blade in that mode
R = angular velocity of the blade
& = factor depending on the blade and root condition.
As the stiffness of the blade, EI, and the natural frequency of the stationary blade are connected by
- 17 -
64)
& = !2E R4 m
(65)
s
it can be seen that the modified stiffness, EI', can be calculated from equations (59-61, 6!+ and 65) using E = 1.2 from Ref. 10.
As the motion is considered in segments, each of h set duration, the deflecticn of the blade tip in the ith segment can be obtained frcm equations (55) and (62) giving
z(R,i,t) = zo(R,i) + A(i) g(1) cos(wt + cp (i)) .-=dt (66)
and hence
i(R,i,t) = -A Ci) G(l) [Is cos(wt + p ) + w sin (ut + ‘p W )] St . (67)
The constants ACi' and 9(i) arc obtained from the initial ccnditions at the start of the ith segment. Thus .
z(R,i+l,t) = zo(R,i+l) + e^Gt'r[z(R,i) - zo(R,i)] 1
cos wt
w
-I- [z(R,i) - $(zo(R,i) - z(R,i)) siz ot 3
where t is calculated from the start of each segment and where z(R,i) are the calculated deflections at the end cf the ith segment.
The azimuth position of the blade is given by either
qf = x + E. + n[(i - I) l7 + t] (69)
or c
E
g = i+l E, i i;(i - 4) R (70)
- 18 -
and where the blade is completely engulfed by the gust whilst at the azimuth position (x + go). To complete the analysis it is necessary to determine that
part of the deflection which is independent of t. From equation (57) we have for the ith segment
EI' a4zo -= dr4 1
.2j fat p +lrrn2 P,-Ing
l ** (71)
when the blade is not stalled and
EI' a4zo 2 . x 1 -5 ) ar4
SP ac a + e\ r + eci) 0
+ mr cl2 p, - mg (72)
when it is. The value of &(i), &ii) and "ii) are'given'by _ 0
,(i) .- -, = 3 2
0 -t- v co9 gi
i --PO z(R,i-1)
13 (73)
- -2R v sin Ei - G 0 (74)
,(i> 2 (75)
,(i) -1 = pG+ v cos Ei + i z(R,i-1) o
I 1 (76)
The integration of equations (71) and (72) is straight forward and the constants of tne integration are derived by the necessity for continuity in the deflections and the first three derivations at the point of stall together with the end conditions,
The calculations were made using E z 0,02 set, this interval was found to be sufficiently small for prcsznt purposes. Pig.22 shows the effect of a ,5 ft/sec downward gust on a Whirlwind rotor in a 30 kt wind. The helicopter was
- 19 -
taken to be on the ground and several initial values of E were chosen. It can be seen that the rotor is extremely susceptible to the gust at the lower rotor speed.
,
Unfortunately it is extremely difficult to find information on the order . of magnitude and the frequency-of gusts at grcund level. Such information will be closely connected with the landscape of the surrounding countryside as well as with obstacles which disturb the airstream.
A similar problem is encountered in the case of a helicopter aboard ship. Here the superstructure of the ship, when the relative airflow is in the appropriate direction, will cause a disturbed airflow. It is difficult, at this
t stage, to see how successful the assumption that the blade is completely and ' instantaneously swallowed by the gust will be in practice. The above work does show, however, that the rotor,\ particularly at 10;: r.p.m*, is very sensitive to sharp changes in the vertical velocity.
5 CONCLUSIONS
(1) A theory for predicting the behaviour of a rotor in a strong steady wind whilst starting or stopping has been developed. Results have been cbtained both for helicopters on the ground and on rolling ships at sea,
(2) It has been found that, under steady conditions and with the helicopter on the ground, no serious blade deflections are likely to occur.
c
(3) Por a helicopter with a large rotor, it has been found that the 'cliff- edge' effect can cause significant blade deflections. However, it has been shown that the operation of the P531 from ships should be quite satisfactory from this point cf view.
(4) The effect of vertical gusts has also been investigated theoretically and, although several severe assumptions have had to be made, it has been shown that a rotor of the Whirlwind type is extremely susceptible to such gusts.
The writer wishes to thank Dr. E. Nil& of' N.A.D. for his assistance in the preparation of the computer programnes involved in this study.
g& Author 1 Stewart, Vi.
Title, etc Higher harmonics of flapping on the heli- copter rotor. kR.C.15041, C.P.121
3
- 20 -
RIFEREXCES (COFTD)
I\To.
2 .
3
Hufton, P.A. et al 1
Squire, H. B.
4 Gessow, A. Aerodynamics of the helicopter. Nyers, G.C. The iMacmillan C~mj+ly, Mew York. 1952.
5 Stewart, W.
6 Glauert, H.
National Physical Laboratory
Karma+ von T, Biot, N.
Neumark, S.
IO Jones, J.P.
Author Title, etc
/
General investigaticn into the characteris- tics of the C.30 Autogiro. -4,R.C. R & :i 1859. ldarch 1939.
The flight of a helicopter. &.R.C. R R: i4 1730.
Helicopter ccntrol to trim in forward flight. L.R.C. R 8: 1i 2733.. March 1950,
The elements cf 'aerofoil and airscrew theory. The Cambridge University Tress. 1926,
I The industrifi applicaticn of aerodynamic 1 techniques. CI Notes on Applied Science No.2. 1952.
Xhthematical methcds in Engineering. McGraw Hill 2001~ Co. Inc. New York and London. 1940.
Ccncept or" complex stiffness applied to problem of oscillations viith viscous and hysteretic damping. 4 L.R.C. R Q ?I 3269. Sept.' 1957.
The influence of the wakc\'on the flutter and vibmticn of rotcr blades. ;,eronaut. Q. Vol.IX, Part 3, 1'958 p.p0258-286.
- 21 -
SYlvE3OLS
a
4' Bl
C
d
EI
EI'
In
Q
R
r
r 9
S
AT
t *
u
IJ.G
V
V 0
v
X
Y’
lift curve slope
coefficients of feathering
blade chord
semi-width of platform
flapwise stiffness
modified flapwise stiffness
acceleration due tc gravity
hei&t cl' rctor above platform
interval of time
lift on the blade
mass
nuinber of blades
rotor radius
spanwise coordinates
position along the blade where stalling occurs
height oP' platform above sea level
increase in tctal thrust due to 'cliff-edge' effect
time
velocity through rotor
gust velocity
induced velocity
me& induced velocity/tip speed
-kn.d velocity
non-dimensional sganwise coordinate
sidewards displacement cf helicopter from the centre of the platform
- 22 -
2
z 0
%
a s
OO
P
Y x
P
*0
u
SYI\BOLS (cox!rD>
blade deflection
blade deflection which is independent of t
blade deflection which is a functicn of t
stalling angle
angle subtended by the blade at the root
flapping angle
Lock's inertia No.
coefficient of flow thrcugh the disc (pcinting upwards)
tip speed ratio
m&e of roll
air density
angular velocity of rotor
blade azimuth position measured frcm the.dcwnwind position in direction of motion
collective pitch of blade
solkiity
- 23 -
Frcm equations (5), F(q) becomes
Nljl) = $- apt f12?i2 ‘(2 iYo-h x + 6 p2 go) + px f3,
1
cos $+(2 px 40-hp) sin $ F
sin 2Q k
- 8 y2 o. co3 21) + 3 p2 p, 3
?.. (A.l)
for 0 C $ i x and zero between x < $ < 2x
or writing,
64 = x 279 0 0
-hx +&p2 7Yo
go = PX PO
hs = &- p2 p 0
;
,
1
(Ad
equaticn (A.-l) becomes
F(Q) = $ apt Q2R2 igo + g S
Sin $ + 6, cos $ + h sin 24f + h cos 2$] S C
. . . (A.3)
for 0 x * i 7~. r
In order to obtain a harmonic series for the whole range of Q, it can be seen that it is necessary to obtain the Fourier series for each of the terms of
I‘ equation (A. 3). Now function I'($), of period 2x, can be written as
m = co+?, (ens sin n $ + Cnn cos n @) Y - 114
where
0
2x 1
s = 1 ns x J
I?($) sin n JI d$ . \
0 I
I
Appendix 1
(L4)
(A-5)
Thus for a functicn which is one between 0 and 'x ~md eerc between x arid 2X we
have,
+ [I - (-1)"3sin n $ (a.6) u n=l
$igkParly when z..
we have c.3 7
-f&f) = $ + 4 sin if - :) +L- cl + (-Q~-J cos n Ij L---l n=2 n - 1
(A.7)
- 25 -
Appendix 1
becomes
1 f&) = 4 co9 q + a> zn [I + (-I)"] 2A.n n 4f
L-I~ -1 (Ad
' n=2
and
f,,(s> = sin2Jr Oc$<n
= 0 7t<$<2n
becomes
00 f,,bk> = $g co9 Q + ; sill 2.Q - ;
c n22- 4 [I - (-l)n] co9 n $ (A.91
n=3
and
f,,(Q) = cos w 0 < J, < 7c
= 0 nt$c 2X
becomes
co
f&) = - 2 sin $ + 3 cos 2$ +: 3x c n2n_4
[I - (-I)"] sin n Q.(L,-lO)
Thus the compoents b,(x) and an(x) of equation (8) are given by equations (y),
(A.3) and (A.6) - (A.lO) i.e.
- 26 -
Appendix -I
b,(x) =: JR
.- q. ho b2(x) = JR '- s + 2
1 3
a,(x) = JP, 3
/ i
and in general for n 3 3
4 an(x) = JR
1 go n'
$;?; [I-(-I)"3 + y L J '5 [1+(-l)"] + $ -&- (I-
n24+
(A.11) '
)n)j
- 27 -
ROTATING BLADES Al LOW PITCH
------ STATIONARY BLADE 3
’ FIG.1. A TYPICAL HELICOPTER ON THE GROUND IN A WIND.
SHAFT Ax\S
PLANE PERPENDICULAR TO THE SHAFT AXIS.
- F
FIG. 2. DIAGRAM SHOWING THE VELOCITIES AT AZIMUTH ANGLE 9.
1
/
-
4 EW, $33 iif
I
I I -jc ,
4
I
-b-
s
d
I I
a- J
0.6 0.8
. 0 4 ipADS/ SEC X 2 RADS/SEC.
LIFT ON ADVANCING
A 4 RADS/SEC. 5lDE ONLY CONVENTIONAL AERODYNAMICS
FIG. 4. BLADE DEFLECTION HARMONICS FOR WHIRLWIND HELICOPTER IN 30 KT. WIND ON THE GROUND
(NO CYCLIC CONTROL)
o-4 o-6 .
ch = 4 RADS I
SEC.
0*2 0*4 0’6 0*0
CA= 2RAD/SEC.
KEY 0 y/=0 LIFT ON ADVANCING
x p=leo” SIDE ONLY.
A Cy=O
>
CONVENTIONAL
v yl=d AERODYNAMICS.
STATIC TIP OEFLECTION =3-: FT.
FIG. 5. FORE AND AFT BLADE DEFLECTIONS FOR WHIRLWIND HELICOPTER IN A 30KT. WIND ON THE GROUND L
(NO CYCLIC CONTROL)
-2.0
3, (FT)
-1.0
a-x-2 X-)b-x
0 n- -T ” 0.2 0.4 0.6 0.6
KEY :- 0 4 RADS/SEC.
x 2 RADS/SEC.
6) 0 0.2 0.4 0.6 0.8 I.0
1 - -x-x )I-‘. 0-
0.2 0.4 06 0.8 !.O
FIG.6. BLADE DEFLECTIONS HARMONICS FOR P531 HELICOPTER IN 25KT WIND ON THE GROUND.
(NO CYCLIC CONTROL>
-2-C
b
0
t I*C I
(4 RAOS/SE$
KEY a l/co” x q z 180”
FIG. 7 FORE & AFT BLADE ,DEFLE&ONS FOR P531 HELICOPTER IN 25 KT. WIND ON THE GROUND (NO CYCLIC CONTROL)
OlRECTlOlrl OF RELATIVE WIN0
FIG. 8. DIAGRAM SHOWING HELICOPTER ATTACHED TO A ROLLING SHIP IN A SIDE WIND.
FIG. 9. (a) VARIATION OF THE FLAPPING ANGLE WITH )J FOR
SEVERAL ROLL ANGLES (Ps31 HELICOPTER) ,
I5
DEGREE OF
FLAPPIN
KEY
A = aI0 t b’, _ m = do- b!
7L /
L \ ,
__. _.- .
+15O --
+ IO"
t so
+ls" t IO0 t 5O
l-5 - 5O
- o---- - IO
- ISO -
. .-----
FIG. 9.(b) VARIATION OF THE FLAPPING ANGLE WITH SEVERAL ROLL ANGLES (P. 531 HELICOPTER)
fi FOR
P
FIG. II, DIAGRAM SHOWING THE DISTURBED AIRFLOW THROUGH THE ROTOR DISC DUE TO THE PRESENCE OF THE SHIP
-------__ I,////, -d-
0
2 FIG. 12 DIAGRAM SHOWING THE DECK AND THE EQUIVALENT TWO DIMENSIONAL CIRCULAR CYLINDER
0 I
>’ 3 rc) “rr rc3 *<y
.
E
t
KEY.
FIG. 14. VARIATION OF g(k)AND g’(k)WITH k.
b*=3.47
( 1 Ai=2 b*= 2-37
FlG.IS.THE INCREMENT TO b, DUE TO THE PRESENCE OF THE SHIP.
a, -I
0 FT.
0
+I
3 JS
0
w
+I
-I
-x- 3 If0
I
J
FIG. 16. BLADE DEFLECTION HARMONlCS FOR THE
P531 HELICOPTER IN A 25KT. WIND ON A FRIGATE WITH s 15’ ROLL (NO CYCLIC CONTROL)
0 0.2 0.4 0.6 I- 0
--I
-k($ =30°> 63
0
+I
-I
80)
0
I I I I I
KEY :-84 RAOSISEC. t$-Ve
x4 RAM/SEC (#We h 2. RADS/SEC $-ve
0 2 RADSISEC. c.f~ + ve
* FIG. I7 TOTAL BLADE DEFLECTIONS FOR P53l IN A 25KT WIND ON A FRIGATE WITH.
(NO CYCLIC CONTROL.)
HELICOPTER ISo ROLL.
3p
- 3.0
-2.0
-I*0
0
---
.----
0.2 “O
-3.0
-2.0 ~-
3$) I
-
---_
c”
u 0.6 , . o-8 I*0
LIFT ON ADVANCINQ
/
- A 4 CONVENTIONAL AEROOYNAMICS
FlGl8.BLADE DEFLECTION HARMONICS FOR WHIRLWIND HELICOPTER ABOARD A FRIGATE INASIDE WIND OF 30KTS-NO CYCLIC COEITROL.
‘b 2
+ I-0
0 FT: S
0
- I-0
-2.0
5 I / *
‘
FIG..18 (CONT.)
t2.0
-1.0
FlG.19. BLADE DEFLECTION HARMONICS FOR WHIRLWIND HELICOPTER ABOARD A FRIGATE IN A SIDE WIND OF 30 KTS’-A,= -59
+2*c
I I
FIG. 19 (contd.)
6
I
0 NO CYCLIC CONTROL LI FT ON AOVANCl NG
A A,: -5” StOC ONLY.
El NO CYCLIC CONTROL CONVENTIONAL
V A, = -5O I AERODYNAMICS.
TIP STATIC DEFLECTION I
.
-6
FIG 20(a) VARIATION OF TIP DEFLECTION WITH AZIMUTH FOR WHIRLWIND ON FRIGATE
IN 30 KT WIND (4 rads/sd m (,B r+ cz, (* ii).
-4
-6
-e
.
60 90
FIG. 20(i)
0
I I I 1 KLY
0 NO CYCLIC CONTROL
>
LIFT ON ADVANCING
A A,=-5* SIDE ONLY TIP STATIC DEFLECTIONe3+ FEET
,
bb0 2
VARIATION OF YIP DEFLECTION WITH AZIMUTH FOR WHIRLWIND ..__ _- -_. ON FRIGATE IN 30KT. WIND. (2 rads.1.sec.3
al
0 - .
9 0
9 b
tu 6
0
,’ ‘t- 0, .J
200 220 240 260 280 300 320 340 360 20 40 I I I I
I&O I I I I -T+
20 40 60 gc I 0 I 3 I 60 180 200 220 s
\
\
FIG. 22(a) TIP DEFLECTION OF WHIRLWIND ROTOR IN 30KT. WIND HIT BY A Sftlsec. DOWNWARD GUST (R=&ads.(sec.)
180
-6
-IO
FIG. 22(b) TIP DEFLECTION OF WHIRLWIND ROTOR IN 30KT. WIND HIT BY 5 ft./sac. DOWNWARD GUST (n=2rads./sec.)
A&C. C.P. 8%
THE 1KfTiON OF HELICOPTER BLADE AT IJJW ROTOR SPEEDS IN HIGH WlNDS
A.R.C. C.P. 852 .,
I THE MOTION OF XLICOPTW BLADES AT J.OW ROTOR SPEEDS IN HIGH ilINDS
I Willmer, M. A. P. :;eptember 1963. Willmer, M. A. P. September 1963.
A theory for predicting the behaviour of a rotor at low r.p.m. in a strong steady wind has been developed. Results have been obtained both for helicopters on the ground and on a rolling platform. It has been found that, under steaQ conditions and with the helicopter on the ground, serious blade deflections are not like4 to occur. However, with a helicopter on a raised landing deck, it is possible for the rcliff-edge~ effect to be of amjor importance especial4 for helicopters with large rotors relative to the deck width. Final4 the blade response to a vertical gust has been investi- gated and it has been shown that helicopter rotors are very susceptible to them.
A theory for predicting the behaviour of a rotor at low r.p.m. in a strong steady wind has been developed. Results have been obtained both for :helicopters on the ground and on a rolling platform. It has been found that, under steady conditions and with the helicopter on the ground, serious blade deflections are not likely to occur. However, with a helicopter on a raised landing deck, it is possible for the ~clifr-edget effect to be of s&r importance especial4 for helicopters with Urge rotors relative to the deck width. Final4 tlm blade response to a vertical gust has beew investi- gated and it has been shown that helicopter rotors are very susceptible to them.
I A.R.C. C.P. 852
THE MOTION OF HELICOPTER BLADEX AT LGW ROTOR SPEEDS IN HIGH WINDS
I Willmer, M. A. P. Sepixmber 1963.
A theory for predicting the behaviour of a rotor at low r.p.m. in a strong steady wind has been developed. Results have been obtained both for helicopters on the ground and on a rolling platform. It has been found that, under steady conditions and with the helicopter on the ground, serious blade deflections are not likely to occur. However, with a helicopter on a raised landing deck, it is possible for the ‘cliff-edgef effect to be of major importance especial4 for helicopters with large rotors relative to the deck width. Finally the blade response to a vertical gust has been investi- gated and it has been shown that helicopter rotors are very susceptible to
I them.
b ,
C.P. No. 852
@ Crown Copyright 1966
Published by HER MAJESTY’S STATIONERY OFFICE
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C.P. No. 852
S.O. CODE No. 23-9016-52
