Document
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x \ TECHNICAL MEMORANDUMS NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS +’ .,,, ,..+”’ “ .
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No. J 90?
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DYNAMIC STABILITY 0)? A HELICOPTER WITH E IX(3ED ROTOR BX K. Hohenemser ‘L ,, Ingenieur-Archiv, VOL. 9, December ,-., -.....
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Washington September 1939 :, ‘..)” !.
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.. , NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 310, 907 ., .
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DYNAMIC STAB~L”ITY 0)? A HELICOPTER WITH HING.ED ROTOR BLADES* By K, Hohe&emser .
SIJMitARY The present report is a study of the dynamic stability of a helicopter with hinged rotor blades under hovering con- ditions. While in this case perfect stability can in gener- al not be obtained it is, however, possible by means of cer- tain design features to prolong the period of the spontane- ous oscillations of the helicopter and reduco their amplifi- cation, and so approximately assure neutral oquilihrium. In contrast to rotors with blades mounted rigidly to the rotor shaft tho gyroscopic effects of, two rotors become additive, even if they rotate in opposite direction, and produce dur- ing a rotation of the helicopter a linear damping which is of the greatest importailce for the stability and control characteristics of the helicopter. The possibility of con- trolled stability of a helicopter fitted with hinged blades is proved by the successful flights of various helicopters particularly of the universally known I’ocke/ helicopter, Flr61 l For fixed-wing aircraft the solution of the problem of dynamic stability has been known for some time. For helicopters with blades,fitted rigidly to the rotor shaft the case of hovering has been” covered by H. G. Ktissner.
Having recourse (reference 1). to the test data of Flachsbart and Krt!ber (reference 2) oil propellers in yaw he proved that dyilamic instability is unavoidable on a hcltcopter without tail and with rigidly attached blades.
.~ynamic stability can be achieved, by placing a suitably dimensioned empennage in the- sli,pstroam, but with’ counter- it is still necessary to Xotating rotors, for instance, *fib~r die dynamische Stabilit&t des Hukschraubers mit an- gclcnkten J?lligoln. Ingenieur-Archiv; December 1938, pp.
419-4.28. - ~ .- .— .,?- 2 N. A. C.A. Technical Momorimdurn No’. 907 .. .
have svai’lable a certain residual spiral of the rotors.
In the following, the corresponding case ‘of helicopter Thare with hlados hinged to the rotor shaft is treated.
is no tail, since on such a hclico”pter the “olade loading is USU@lY so low that a. slipstream alreatLY effective with moderate tail surfaces does not result.
Air Loads The helicopter has two conxial, oppositely rotating, identical rotors placed sc close together that the effect of 31ade clearance can be disregarded in tl~e prediction of the air loads. Owink to the counter-rotating arrangement ~, novencnt of the helicopter in the loilgitudinal direction loads transverse to the direction of motion, produces no cir and lateral stability can be studied so that longitudinal on the fixed wiag aircraft. Restricting the soparat~ly as the ‘square of the speed relative study to snail motions, Since the thrust to the speed itself can bc ignorod.
in yaw is proportional to the square of chaace on a blade 3) the thrust remains unchanged the flow velocity (reference hence no vertical notions occur.
in first approximation, Thus the first longitudi:~n.1 notion in question consists merely of a “horizontal motion and a rotation about the transverse axis.
cx.act notion equations for the helicopter To set UT the with hin~ed blades would result iil a very complicated sys- tem, as each universally hinged joint even with utter dis- involves two addition- regard to its flexibility in bending The followintq procedure avoids al de[~recs of freedom.
the air forces ,~d mass forces on the these difficulties: blades are determined on the assumption that the rotor-tip preserves its position relative to the body during plane The actually occurring and the notions of the aircraft.
of incliilation of the rot’or-tip plane conputed changes thus shall be sr,lallin ratio to relative to the body axis Following this the changes in inclifintion of the aircraft.
the inclination of the rotor-tip plane rclntivc to the fuse- defined on the ~ssu~ption of uniforn distribu- la~c axis is tion of r,ir forces’ and n.ass forces over the blades, That is, ,the aircraft notions shall be so slow as to approxinate- 1;” present a succession of stead-y stages for the helicopter This premise prevents an increase in the number of blades.
inasnuch as ..degre~s of freedom through the hin~ed blades, now the forces .ol~the rotors are definitely dependent on Illustrative exa~ples the state of notion of the aircraft.
— , ---- .-, , N. A, C.A. Technical Memorandum No. 907 cited elsewhere in the report actually prove the assump- tions approximately correct. - ... .,. — . . . . .. . ..— . ... . .. . . . .. . . . .
Tile air loads are conputed by means of the simplified Glauert equations which are applicable to propellers with when the-direction of rectangular untwisted blades in yaw, flow ferns a small angle with the plane of the swept disk and when the rolling and longitudinal nonent’s of the blades are equalized by a periodic blade feathering. .For trape- z’oiclalblades it is practical to assune a substitute rotor with rectangular bl.adcs whose chord is equal to the chord of t!le trapezoidal blades at 0.7 radius. In the sane way it is proper for twisted blades to assume a substitute ro- tor wit,~,untwisted blades whose angle of attack is equal to the nag’le of attack of the actual blades at 0.7 radius.
The nest accurate theoretical values at present for the air loads on a rotor with hinged blades in yaw are obtainable with the fornulas worked out by G. Sissingh (reference 4) in a study of for trapezoidal and for twisted. blades. But , the dynanic stability linearized equations must be cnployed for the air loads. The illustrative exanples conputcd with the conpleto practica- the exact fornula’s confirn, however, bility of the approxine,tions enployed here.
The rolling and longitudinal “noncnt b?.lance of a blade noving parallel to the pl~ne of the swept disk re- quires a periodic change in blade angle of attack, where of the advancing blade decreases, the angle of attack $ that of the receding blade increases conformably to is the nean angle of attack referred to zero Here 00 lift curve, with the azinuth angle ~ being measured in peripheral direction fron the rear position of the blade.
The cmplitude 01 of the periodic proportion of the blade-setting angle in relation to the coefficient of ad- vance h (ratio of forward speed to tip speed) follows approximately at (2) If the ‘rotor blades are hinged to the rotor shzift so that the %iades do not change angle of attack during flapping of the pl~:le of *I1O swept notions a backward inclination . .
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* N. A. C.A. Technical Menoraridun ITo. 907 results. For in that case the coa - disk for an an~le fll dition (1) relative to the new plane of the swept disk is In reality the hinged %lades do satisfied (reference 5).
not remain in one plane during rotation, hut approximately !Men the rolling and longitudinal no~,ent describe n, cone.
the baCkWar?L a lateral inclinat- hal..mce requires other than ion of the rotor-tip plane toward the right or left, do- 13ut having assuned pc:l?.in~ on the c!.ircction of rotation.
a pair of iicntical counterrotating rotors, the effects of cmce,l, hei~c~ nay be disrc~ardcd these l(nicral i:.clinations in the followin~.
ta]~~n .so that t]le an:qle If prc.por desi{;n ‘measures are of ~:+ttac];: dec~.eases as ~. l-OtOr riSeS an’?l”in~re~.scs ~S it -.~ su8:;este~- ty Br6&uet (Gcrr~an Patent l~o. 567,584, silnks, class 623, 1933) the,bac.kward inclination of the rotor-tii> plfi,IIc is rcduccd. So, in order to ir~cl”u’dc this case also, bac]cward inclination of beint; a we ass’lmc a co 0“1, :q) factor involved thro@l tb.e rotor desicn.
the force perpendicular Si:lcc is a S!.1.311 L1.ilr_JIC$ he built up fron a share Scp?g to the rotor shaft can Of the hackwor? izl,clinm%ion cf the pl~l’e’of tile rotor ~isk s ksn,hs’ reprcsentii:c tk.c thrust component .aP.d2. share plo,l~e . Here S is the total. thrust of loth iil the sar.1 c where the thr~lst cocfficiont (ks’ = S/i?_!ij~.2, rotors, ks and u, till speed), T is swc:?t-disk arcc., p, {air dcasity, of the thrust corlponcnt in the plane is coefficient ksn of the rotor disk, computable froc~ k sn/~ O = ~ cw/4, is solidity, i.e. , r,atio of blade area to swcpt- Where Cr the ~.::~.n –blade ?-rn~:coefficient. The Cw, ? isk c.re.~. ,r.nd COil~:Jar3tiVClYnarrow linits Valu-c 1:s/2 0 varies within lie assune ks/2 0= for ~>r~.ctically desi(~nccl helicopters.
l<~n (3) — = 0.044 A k s to the rotor shaft with consider- The force perpendicular atio:l to equation (2) then mounts to (4) Sn = S(Q ‘$1+ 0.044X) = ‘A(2$o,Q+ 00044) .
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.. ...!. : ,,” N.A. C .A. Technical, .,Men~r,andwn&&/i’%~7.}. ,.” ,, The hin~cs joininfl the bla’tiesto the rotor shaft aro .—.usually..placeil a certain distance a away fron the shaft haCi5Wiir.d ““iilCl”iii~tiO-ri”’’”@’ ‘-$’-”””o’”f rotor- center. Owirig to the tip pl,ane toward the plane perpendicular to thelshaft, a hcnding noncnt of na~nitude PT al $1 a Cos w, per blade ~f the coroto,ting reference systcn results (PF = contrifu- sn.1 force of a blade). The lon~itudinal fionont per blade on the aircraft is pF 9 $1 a c0s2 ~. The proportions cf this nonent oscillatin~ at twice tho si>ccd of rotation cancel, if each rotor has at least three blades. if with two I)laclesa counteroscillatin~ aircraft notion, not of il’ItCl?CSt hCr(?$ is I)roduced.
The stationary proportions of the lo:l~itudinal r.lonents of ,all blades arc additive, ultinntely yielding fron o ill with respect to the trans- verse nxis throu[jh the centroid the tail-heavy noncat ~s+pF$zJ =.
Ml =Cpdl 2 t?. hp(ss + P~; z) (5)
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Here s is clista.nce of aircraft e.g. fron the plane of the rotor disk and z, the total nunlcr of hlacles of both .,..
rotors.
With allowance for equation “(3) tb.e noncnt cf the thrust co~~Jone]l~ in th~ i>l,?.nc of the rotor disk with re- spect to the transverse axis throu~?~ the C.G. is ksn ~2 =ss— = 9.044 Ss A (6) ‘cs A third proportion of the longitudinal noncnt’ ori&i- nates in the nomcnts about the lon~itudinal axes of the blades. The nonent coefficient referred to tile acro- “crJt dynanic center of the blade is indepeuclcnt of the blade whence the lift is assuz~ed to apply at mi&’1c of attack, the acrod~-nrmic center (about quartor Chord),?
The countcr- force for hin~ed, blades is applied in t~c centroidal axis of the blade. Then the share of the r.lonetit clei>ende~t’cn the blade an:~le of attack is equal to tic “j..,roduct of blade thrust and distance of the centroidal &zis frdp’-the aero- Lynanic ccn%cr of the profile, In the Fourier analysis of the 31ade thrust (blades beiil~ hinged} ““%116 torn varying harmonically with the rotation is alnbst’zero. But this very tcrr: is the only ~ile in the,clevelopnent which, re- ferred to the aircraft, has a stationary proportion. In N.. A. C,. A. Technical Menoraudun No. 907 ,, . . .
the effect of the backwarcl position of the conscoucil Cei .
~]l~de on t]~e lont;itudi~~l nonent center of gra~ity of the of the airpl.me can tie ignorecl. !Che share of the %lade torc!ue not clepcndent on the an[~le of attack is for rota,t- iac systcn R blade chord,and x, dis- where R is rotor radius, t, The loilgitudinal tance of a point from the rotor center.
is obtained herefrom by multiplica- nonent of the airplane tion with sin W. The’ stationary quotas of all hades are additive again, the oscillating shares are disregarded, the longitudinal nonent thus ultimately affordin,~ R z ~~3 = -“ ‘“ U2 AX2 :dx=-~pt2cnRu2 ~ ~Pt%
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o (7) I?or positive i.e., for a lift profile the blade no- en 9 ~ent and hence the longitudinal no:~ent 143 itself is hence the ninus sign.
~ose-hea.v~, The air loads oil fuselage and other parts of the air- pla:~c, b~ii~g proportional to the square of the speed, dis- appear for the prcsuncdly snail, notions in. relation to the (4) to (7)), which are pro- forces and nonents (equations portional to the speed itself.
There renains the effect of the rotatory notion of about the trmsvcrse axis on tho air loads~ the airplane The variable angle of attack of the plane of the rotor disk relative to the wind direction does ~lot change the result achieved so far, provided the notions are ke@ snall$ and hence snail, angles of attack of the plane of the ro- But the rotation speed of the airplane about the tor.
transverse axis in the sense of a nose-heavy nogent ca,uses ~ lift incre,asc in the adPailcinG llade and a lift decrease The result is an equalizing flap- on tho rcturain~ blade.
piilg notion corresponding to a. right or loft lateral incli- of the rotor tip, depending on the ilatiOll of the plane The effects of these lateral in- direction of rotation.
clinations cancel on account of the counter rotation of As a result of this the rotatory notion the two. rotors.
IT.A.C.A. Technical ’Menorandun No. 907 about the transverse axis produces no lateral air loads on .thc hcli-co.p,te r..
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The Equations of Motion for the Longitudinal Motion Let v represent the speed of the rotor center aild the an~le of rotation of the motion about the trans- a, verse axis. Fron the equilibrium of the forces perpendic- ular to the rotor shaft and the equilibrium of the longi- tudinal monents about the center of gravity (signs as in fig. 1) follow the equations .
(;,-+
+czs12a+. cLs:l&+ Vsnv=o (8) (9) &J+ a Ma + & Ma + v Mv = O l where + is horizontal acceleration, a, an~ular veloc- ity, Z, ail~ular accclerr.tion , G, gross weif:ht, ~, (j f accclcratioil ~;ravit-y, J, nonont of inertia of airplane loil~itudi~lal nonent and a’oout the tr~.ilsvcrse axis, M, s the force nornal to the rotor shaft. For abbrevia- t%l, we write a Sn —= s n a?
aa The other partial clerivatives of Sn yrith respect to & and v and of M with respect to a, &, and V are in- With S = G and A= v/u, dicated 3Y si~ns accordingly.
equation (4) gives “ (lo) s : (2 750 co + 0.044) nv= while equations (5) to (7) afford 280C9 Ru (11) Mv=— Gs+P&z + 0.044 : s -;P+2& ( ) u NOW * outside of the air loads, the force of gravity and J., ..nc r~ass forces of the rotors must he considered. The ,’ .,,,, , ,,, , , ,,,, ,,,, ,,, , .,,!.,,,. !.!!! . I I ! , , ,, . . . . . . . . . . . . . . . . . . . . . . . . . .,. , . . ..-.. —. . . . . . . . . . . . . . . . . . . . ..— 1111 m lT. A. C.A. !lechniccil Memoranclun I$o. 907 Qravity Conponent is G u and its longitudinal nonent is zero. Hence s -G (12) na= .
# The rotation about th~ trmsverse axis is acconpnnicd 7~~ G~-roscopic aOnCP.tS which pe~ blade referred to blade r~ot have the na<ylitude 2 JF af~sia$,(w is rotation speed, inertia nonent of a blade referred to %lade JF , root). In a rotation in the nose-heavy sense the gyro- scopic nomcnts raise the advanci~lg blade recardlcss of the direction of rotor rotation. The rotor-tip piano is suh- a backward inclination tho anount of which can he jzcli to dcterniilcd after equating the lift nonent clue to angle of rnttcickChallLe 4 sin W to the ~~roscopic nonent ~L c1 is the lift {;radieilt. The i~~tcgration is effected ~, whence equations (4), (5), and 16 JT cpG (1.5] c~,~p R4t “+1 16 JT M; = “M!,— = @ Gs+PF~z , (16) ?J~.
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a crwp R4t a !The ~yrcscopic forces produce a rotation danpind nonent on the rotor with hingccl tlades. If the rotors rctate in the opposite direction the gyroscopic effects are additivo rathsr than ncutralizin[;. This fact is of utnost inpor- tancc for tho fli{;ht characteristics of a rotatins-wing f ~aircraft with hinGed blades.
I The rotatory acceleration & aho-~t tho transverse axis creates inertia nonents which, for each blacle referrocl to This noncnt induces blade root, anount to JF & COS.$S .,, 1, \ ,,,, . ,-.~ ‘ .
.>+’ \ ,, ,0 (“ /
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N.A.C.A. Technical Menoranclun No. 907 FI’DAU right or left lateral incliila.tions.of the plane of the .
roto-r tip, ‘dependin~;oi l-the direction. ot ,r~tation.
!l?he effects ofthese inclinations cancel out because of the opposite rotation of the rotors. Thus the nonent of in- ertia J of the aircraft about the transverse axis is taken without the blades.
‘lhen the solution of equations (8) and (,9) by nea.ns of v? VT v = Voe and a= a. e (T deilOtiil{; the tine) gives two homogeneous linear equa- tions for ancl a. which depend for their existence ‘o on the disappearance of the determinant of the coeffi- cients. !Then, allowance for equations (12) and (13) gives for v : /‘.
G ~Jv3+ M’g+ Mvs —-1-S.v J tin+(SnvM& -Mv Sn#U+GMv=O (17)
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) a~ ~ “ !l!hc condition for the ai3pcaranCe of decaying notions or oscillations only is a positive value of Routh!s discrin- incant: G G M“ —+- Mvs -+ Snv J -; J G14V>0 (18) (Snv Ma - M.T Sn&) .
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) .-J ~ ‘In aclclition, all coefficients in equation (17) nust be The factor V positive. in equation (17) is, with allow- ance for equations (10), (11), (15), and (16): 0.35 JF G w Py a z 11.4 p t2 c; R U2 1- (19) P~ a cJw2pR5t ( ) all other factors are also positive according Up to Mv to their physical significance, whence as sole further bonclition for the stability according to equations (11) and (19) the existeilce of the inequality 2&oq a G S + PF5 Z + 0.044 5 s - ; p t2 cn R u>O (2o) ( ) u u and 10 N.A.C.A. Technical- Memorandum No. 907 Pya- 11.4 pt2cn Ru2>0 Illustrative Examples Assune a helicopter with the, following data: Gross weight, G = 900 kg - Rotor radius, R = 6 m “- = 120 n/s Tip speed, u Bla.cleail~lc of attack, 00 = 12 P* = 1880 kg Centrifu&al force of each blade, 4 (2 motors of two blades each) Nunber of ‘blades, z = Blade chord, t = 0.28. Q Disttince of hinge, a = 0.2 n . . ~,. ,_ Jr = 20 kg/n/s2 Inertia norient of blade, Inertia r.lomcnt of airplane about its lateral axis, “ J= 150 kg/n/sz - of plane of rotor tip from airplane cd., s = 1.2 n Distance C; = 5.6 .
Slope of lift lines for blade profile, Monent coefficient of blade profile referred to aerodynamic center, cn = O Design factor, CP = 1.0 (Co:lstai:t blades angle of attack during rise and sinking of l)lo.des) helicopter with the.engine These data correspond to a 3qua- of ail autogiro Of type C 30.
power slid dimeilsions tions (lC), (11) , (15), and (16) give in kg/n/see: ~3.45, ‘ M.r = 6.80, Sn& = 56.5, M; ~ 115 s nv whOilCe equatioil (17) reads 90?
N. A. C.A. Technical Memorandum No. 11 13800 v= +15v +6120=0 + (10550 + 750 + 520) v2 (21) m.-.
“-”-”’Inl’6rtioti o-f t’he c’o~f”fici’entsin (18)”:’ discloses the latter to he far fron satisfied, as the positive tern contains oilly about 1/4’70 of the negative term. Thb evaluation of equation (21) gives aside fron o, negative real root (rapid- ly decayiilg aperiodic rnoti”on) two conjugated comple:c roots u = o.16L 0.60 i, which are ideiltical with an amplified OscillatiOil of T = 2Tr/O.60 = 10.4 $’ period and an ampli- eO.16 T.
fication factor of In other words, the ampli- ~cl.16 10.4 tude increases during one oscillation = 5.2 fold. Iil view of the long oscillation period, it should be possible to keep the self-induced oscillation of such a helicopter within narrow linits iiz spite of the ~reat anpl ific(ati.on.
~2 In the factor of in equation (17) the first term predominates, as a comparison with equation (21) discloses.
Thus the stability condition, equation (18), can, with al- lowaace for equation (19), be written in the fcm 11.4 ptacnmz 0.35JTPTaZ 14& -JMV>O (22) v 1- ( ) cl[l~zpR5t, P~ c1 0, Additional sir.lplifications cm be effected ty assuming G 0.044–s-~ptacn Ru=O (23) u because then the last two terms on the right-hand side of equ~tio~ (23) equation (11) CailCelo For our example, Lives Cm = 0.056, i.e., a noment coefficient as is re.5.d- ily obtainable for e. lift profile. With allowance for equations (11) and (16) , equation (22) becones
2“8L:FR4 l-(’ “ 1“4u:cnRu ’24)
from which the pr.ccautionai-y measures r-ecessary to obtain, stability can be rcacl immediately. It involves, first of all, an izicreasc in the mass of the rotor blades because then which ci~ters the stability condition quadratical- ‘1? ‘ ly, as weli as the centrifugal force becones higher.
‘F To get a startin~ yoint for the effect of the blade weight, 12 I?. A. C..A.’Technical Mo,norandun No. ’907 we assume 3.5 tines the value’ of the blade noment of” iner- of the centrifugal force in the tia and 3 tines the value with Jp = 70 kg\n/s2 ,and PF = 5600 kg foregoing example.
it affords Snv = 3.45$ NIV ~ 12*OS Sna = ~00~ Ma ,= 740~ and equation (17) reads 13800 V3 -I- (68000 -I- 1300 + 520)~2 + 160v + 10800 = O (25) lII1e stability equ~ti,on (18.) is still not conPlied With, al- a mere 1/13 of the thou@ the positive tern now an’oun”ts to negative tern. The evaluation of equation (25) yields asi?.e fron a negative real root the conjugate conplex roots v = 0.02 & 0.40 i, equivalent to an anplified oscillation eO.02T a~.plificatiOn fact or.
with T = 15.7s period aiid Durins one oscillation the anplitude” increases eo.o~ 15*7 In other words, the ar.lplification is now = 1.3 tir.les.
substantially less and the oscillation period longer.
i~aturally, equation (24) must he satisfied if stabil- it is not to be sunnarily concluded ity is to prevail 3ut fron the noncompliance with (24) that an increase on the left-hand side of equation (24) will, in every case, pro- duce a lower amplification.
T~~e , for instance, the design factor CP = O. In that case “stability cannot be obtained according to (24).
l!ith otherwise identical data as in the first illustrative exCanple we now have Snv = 0.33~ Mv = 0.40~ Sn& = a?
.
}J& = 0, and equation (21) becomes (26) 13800 U3 -I- (44 + 50) U2 + 360 = O other. than a negative real root the conjugate It conta”ins The oscillation period COiIplCX roots ‘V = 0s14 t 0.25 i.
substantially longer than in the first ex- is T“= 25 s, the amplification factor eoelq ~ has l)econe anple , while Ey decreasing NV through raising the nonent coef- less.
in equation (11) the oscillation period, can ficieilt cm be further-increased and the mplification factor lowered.
Accordingly, there are several entirely clifferent ways of oscillations of,a helicopter renderin~ the spontaneous harmless.
the”original assumptions Next , it can be proved that re~ardiilg the blade notion hold approximately, true.’ To be~”iilwith the speed of the airplane in relation to the The oscilla- speed of rotation of the blades is very low.
tion -periods of the airplane range between 10 aad 25, seconds —...— Hence as ‘against Oile third of a. second for tllc” blades.
,~s.sumpti, on of, a succession of sta- the..crror .fO.11.cwingthe >.
tionary stages. fcm the roto,rs cannot bo very great.
in the: pre”d..ict’ion of’ ~.he‘forces and moments “’lloroojer , the prenise stipulated ‘co fistant position of the rOt Or-tip To. check this premise pl,ane with respect to the fuselage@ wc computod for the first.,example the ‘ratio of speed anPli- tude V. to anplitude of airplane rotation shout the by writing the solution tralzsverse axis a.
v“ocW.’16T v= sin 0.60T a. c 0-16T sin (0.60 T + @) a= It yields @ = 70° and into equations (8) and. (9).
vO/ao The anplitude of = 15. Suppose ‘v. = 1; then a. = 3.8°.
which leaves the amplitude rot~tion is around tLo=0.60 ao, y 115 of the nonents fron the gyroscopic forces o.t & Ma = = T}li.s corrcsp,onds to an inclination of 0.14° of 406 n/kt;* Th O the plane of tile rotor tip accord.in~ to equation (5).
mplitude of the moments duo to the horizontal speed is T MT = 6.8 In/l<Lgwhich, when disre”garding.tho s~~all thrust iil tho pl:~.ilc of the rotor disk according to equa- component tion (6), is identical with a 0.21° inclination of rotor- Thus we have a 3.8° tip plaile according to equ:ltion (5).
and 0.21° inclination airplane inclination ag,3iilSt0.14° chail~os Of the plane of rotor tip toward the fusela.fle axis, and this closely approaches the a,ssunption for the prcdic- ti~il of the air loads and noments.
Lateral Stability The study so far has dealt with longitudinal stability.
Siilcc oill~ the air loads on the rotors have been considered the equations remc.in applicallc if V e. 11 d a. are inter- pr”cted as lat.cral horizontal speed and as angle of rotation !l?hen .J must be included as al)out the longitudil:,nl axis.
iilcrtia ~or.lent of the airplane about the longitudinal aXis.
With ce:ltrally arranged lifting propellers~ the inertia no- anounts to about one third ment about the longitudinal axis that about the transverse axis”, so that lateral stability is easier to obtain according to (24) than longitudin~l stabil- ity. Placing. two rotors side, by.side: and,.the rotor .ccntcrs farther apart, our assumption of snail angles between flow velocity and rotor plane no longer holds true for rotation distance of the about the lon:;itudin-al axis., Suppose tho n No. 907 i?. A. C.”A. Technical Men or‘andu at ion to the distance so Sreat in rel rot or centers 2 r is of tho rotor, that fron the plane s of the airplane C.G.
s the air strikes f ongitudinal axi rotation shout the 1 in For snail coeffi-”. of around ~90°.
rotors at an angle the direction, we have A in thrust cie nts of rotor advance ks ‘t3 —= A ‘ks
4/- 0
d.Lmlp ing follows at danping coefficient the air –lo ad The r ‘pFu 2r2 as —= — r2 F u (27 ) P al u 1 -1- 2& th e validity of the previ- oth er renaining forces, For all whence it them nerely eriv ed relations is assuu.ed Ously c1 M; i ~ equation (17) 3Y the require s th e rcplacenent of . .
1. fra:-:equation (27).
sun of 14; fron equation (16) m?- L; inst ce, helicopter wi th the following ‘Take, for .0, P. a data: G 900 kg Gross ~ot~r r~c_Lit.s, ~ = ~ ~ 120 ra/s Tip speed, u = Blade setting, 00 = 12° PF = 1250 kg #de, Centrifu~al force per bla rs side by side of 3 blades each) Blades, z = 6 (two roto t = 0.25 n Blade chord, a = 0.2 n Hinge distancej kg/n/s2 3.5 Blade nouent of inertia,
‘r
inertia about 1 ongitud ,itialaxis , e n ,onen .-k of Airplan 500 kg/ r.1/s2 J= C.g.
rotor disk fr on “of airplane, an c e of plane ‘of Dist = 1.2 n s , .=.
—.
.
* N. A. C.A. Technical Memorandum No. 907 15 .
Lift gradient, c: = 5.6 i,..–.–. ., Solidity ratio, O-=-”””O. 0-6. ‘ Thrust coefficient; k~ = 0.01 Clearance between rotor ceuters, 2 r = 9 n .
Cn=o and CP = 1, as before.
Those dLZI,~2. corres”poncl to a helicopter of the power and approximate dimensions gf the helico]?tcr type l?W 61.
Equatio:ls (10), (11), (15), (16), and (27) give (in kG/n/s): = s 3.43, MT = 6.80, S:l~ = 38, M& = 7’7, ML& = 950 11 v Then eq-i?.tion (17) reads 46000 Us + (Q5000 + 750 + 1730)~2 + (3560 - 260)V + 6120 = O Inscription of the coeffieieilts in the inoquhtio”n (18) ilis- Clascs tha t th.,~stabilit:: condition is exactly satisfied, since the positive tern is 7 j30rCeIlt greater than, ,the ncc- ative. There is therefore no difficv:lty in so clesi~ning SUCh ‘a helicopter as to assure dynamic stability in latOral motions.
Control.
The question of control is closely allied with that of st,a”~ility, since ne=nsures to influence the stability char,nctcristics of an a.irplano usually also nffect its cor.- trol ch~aracteristics. Oil a helicopter with hingccl blades the ii~cr~ia nor,ent of tho airi)lo.nc d J and the danping nonollt a Ma uust be overtone throu@ the external con- trol noaent 14St, whereby Ma is given through equation (16) or equation (27). Thus the angle of rotation CL of the airplane under the control nonent nust be obtained Uil(lcrthe sano assuraptions ,3s before fron equation (28) , ,M l In the first ~xani~.le conyuted., we found ~ = 115; i.e., induce a steady rota- a coiltrol nonent of 115 m/kg would tion at an angular velocity of & = 1 (5’7.30/s). Hence the dan:~iil~ a{;ai.ilst rotations about the longitudinal and with hin~;ed blacles is quite transverse axis in an autogiro — N. A. C.A. Technical Menorandun No. 907 .
The solution of substantial even at zero flying speed.
the- differential equation (28) for abruptly applied con- trol nouent M~~ fron neutral position reads (MA = 115, J = 150) For the data of the first exanplc 10 m/kg gives after 1 second an a, control r.loncnt MSt = angle of rotation of the airplane of 1.5°, after 2 se~onds, Raising the blade T:iassto about three tines its 4.9~.
(M& = 740, J = 150) gives value as in the latter exacple an o,nsle of rotation of 0.8° after 1 second and of 1.6° Thus inproving the stability hy raising after 2 seconds.
blades involves at the sane tine the nass of the rotating inertia in the Coiltrol action.
the dan~er. of excessive By vanishinc design fe,ctor Q and hcncc danping M&, it J = 150 and = 10 n/kg an angle a~ain afforcls with Mst after 1 second and of 7.6° after 2 of rotation of 1.9° q = O, the amplification factor Of course, for seconds.
of the spontaneous airplaile oscillations can, as was shown~ le considerably reduced but not without dan~er of abnornal Under what condi- control sensitivity of the helicopter.
tions the flight characteristics of a helicopter, taken is impossible to decide as a whole, are nest agreeable, It requires systematic flight tests under theoretically.
the different possible conditions.
Translation by J. Vo,nier, National Advisory Conmittee for Aeronautics.
,- .- .—— N. A. C.A. Technical Memorandum ITo. 907 17 REITUU!lHC13S ,— .— 1. Ktlssilcr, .H. G.: Pro310ne des Hubschraubers.
Jahrhuch 1937 dtsch. Luf tf .-Yorschg. I S. 247.
w. Flachshart , 0. and Krti%cr, G.: Experimental Invosti- ~atioil of Aircraft Propellers Exposed to Oblique Air Currents. T. M. 1~0. 562, N.A.C!.A,, 1930.
3. Hoheilcr:scr$ K. : Perfornailce of RotatinG-?Jin~ Aircraft.
T. M. Ho.
871, lT.ii.C.A. , 1938.
4.. sj.~si]lgh, G.: Neitrat: zur Aerodynar,ilc der Drehfli3gcl- flu~zeu:;c. Luftfo,l~rtforscll.~n~,vol. 15, no. 6, JUnC 6, 1938, ~?i~. 2!30-302.
.
5. Lot!<, c. H. H.: Ikrther Dcvelopncnt of Autcgyro Theory.
P~.rts I aild II.
R.& M. Ho. i127, 3i-itish A.R.C., 192’7.
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I
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3T.A.C.A. Technical Memorandum No. 907 ,,., s a \ v .~ -T \ 1
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