A coupled rotor aeroelastic analysis utilizing non-linear

(1)

FOURTEENTH EUROPEAN ROTORCRAFT FORUM

Paper No. 21

A COUPLED ROTOR AEROELASTIC ANALYSIS

UTILIZING NON-LINEAR AERODYNAMICS AND

REFINED WAKE MODELLING

Michael S. Torok and Inderjit Chopra

Center for Rotorcraft Education and Research

Department of Aerospace Engineering

University of Maryland

College Park, MD 20742, U.S.A.

20-23 September, 1988

MILANO, ITALY

ASSOCIAZIONE INDUSTRIE AEROSPAZIALI

ASSOCIAZIONE ITALIANA DI AERONAUTICA ED

(2)

A Coupled Rotor Aeroelastic Analysis Utilizing

Non-Linear Aerodynamics And Refined Wake Modelling

1

Michael S. Torok2

and Inderjit Chopra" Center for Rotorcraft Education and Research

Department of Aerospace Engineering University of Maryland

College Park, MD 20742, U.S.A.

1 ABSTRACT

The effect-s of improved aerodynamic modelling on rotor blade S<'ciiou aud root. loads, and blade response. are investiga.ted. A non-linear aerodynamic model basNI ou au iudicialnwt hod. is incorporaletl int.o a coupled rotor aerodastic aualysis. The aerodynamic analysis consists of t.ltref' phases: a litJear at.t.aclwd flo11' solnt ion. a separat-<.•d flow solut.iou, and a dynamic stall solution. A free wake mod<·! is a.lso iurludcd int.o the analysis. Blade rl'sponses alHiloadings are caknlat.ed using a linil<, ekment fonnnlat.ion in space and t.ime. A modified Newt-on iterat.ive method is ttsed to cakulate blade response and trim cont-rols as one coupkd solution. fhesults show tha.l. at high speed !light. conditions, non-linear effects dolJiiuat.e blade :wdiuu forn,s. am\ siguificant.ly affecL blade root. loads. These elf<'cis are amplified al higher l-hrusl comlit.ions. C'on1pressibilit.y effects considera.bly inflll('tlCe the extent. of sepa.rai<'d flow on the rotor disk. Inclusion of the free wake analysis had only a limited effect. on blade response ami blade loads for a high speed flight condition.

2 INTRODUCTION

The modelling of aerodynamics for rot.orcrafl applications hao uud<"rgoll(' iu-creased scrutiny in recent years due to the necessity to bett.er predict. thC' dynamic characteristics of rotorcraft. The past few years has seen t.he enl<'rgence of a mul-titude of highly complex CFD codes aim<'d at arcnrate predict iouH of rol.ary-wing aC'rodynamics [1.2]. The requirement. of an aerodynamic modPI t.o lw computa-tionally "reasonable", in order to be implemented into a comprehC'nsin' aProdast.ic

'l'resent.etl ot. t.he 14th EuropeAn Rotorcraft. Forum, Milano, Italy. September 20-23, IU88 2_{Hotmcraft. Fellow} _·

3_Professor

(3)

<llla.lysis. ho\rPVf'l'. precludes nse of ~uch rodc·s. Technological rulYa!lc'PH in ro1npnLing power cuut.inuall.\· expandt-i thi!:' ··n·a,':lvucdJle" ra11ge. 1Hil sig11ific.:ant iuqn'O\'<'llli'Jds il\'1'

st.ill reqnirecl. Tints, analysts are fac~d with t.ll(' problem of J,a!ancing tecbnolol(ind COlltplexit.y. acnH·acy, aJHI COJlljJttt.at.ioual feasibilit.y.

In thE' past. fiyp years, significant gains have been made in the nwt.hodologics of acrod.I"JJill!lic modelling. SPn'ri11 approaches haw· been uudPrt.aken. One approach is t.o capt.ure the globalunst.eady dfects on a rotor by use of a dynamic iullow uwdcl

(:l].

This method is limit.ed to low frequency elfeds, and thus cannot predict. dc-t.ailcd flow phenOIHCJla. This mdhod WM ut.ilizcd in a dyuamic amdysis of a rotor in forward llight.

by

Panda. aml Chopra [4].

In

order t.o lwt.t.er ch·liue the aerody-namic contplexit.ies. seY<"ral a.pproaches coucent.ra.tcd ou a blade-element analysis. S<.•1·eral theor·ies. coustndued by the a.ssumpt.ion of harmonic mot.ion. e.g. Lo<c'WY t.lwory. and Greenlwrg tbPOQ', w<:rf> gem•ralized lo arbitrary motions in Hef<"reuce~ [!i,6]. Hec<:'nt.ly. a fini!;e state Lim€'-doma.in a<~rodynamic model was included in an aeroelastic stabililr ami response a.nalysis [7]. One limitation prest'ul.s itself iu t.lJ!'st' approariH"s, mlly im·ompr<:'ssible, a.t.t.a.ched !low is rousiclerc-d. On<' olh€'r approach inroln•d a synthesizalion of exp<'rimcntal data [8]. This nwlho<l eJJcompassPd i\11)'

and all piJ.,·sical ft•atures of a "real'" /low. holl'<'l'f>r, n<'eded <"Xt><•nsin· prPrPquisile

••x-p<>rinwnt.al t.est.ing. a.ud a large uumber of eutpirica.l codficieuts t.o be iucorporat.<·d iuto a. dyuatuic aua.lysis. Additional met-hodologie-s are discussed in rd't•renee

[!l].

The approach tak<:'n in this in,·est.igalion is t.he use of an uusteady a<'rody-lli\lttic modc·l that. ovP.rcoutes t.he short.comings of othe-r models, yet. remains of thE' sa.nw orc!Pr iu contput.at.ional cost.. The imJHOV<'11l<'nl;s invoke a.<"cural.e prPdict ions of at.t.ach<c'<l llow. as well as separated and dynamically stall<>d !low. consi<kralion uf cotnpr<?ssil,ility dfect.s, aml nec<l litnit.ed prerequisite information of airfoil

char-ad.<'ri~tics. A mode'! post.ulaled ami refined by Bt•ddoes, [10]. a11<l Leishman aud 1:3c~ddoPs. [11 ]. utilizing an imlicia.l response 111E'thod has bee-n va!ida.t.r.d over a wide r<\.llgc of condit-ions. The imlicial response method invoh·es the ca.kulat.ion of the lift and pikhing monwnt response to a step change of airfoil angle of att.ack. and then 'utns t.lw ront.ribution' of past arbitrary changes using Duhanwl's principlE' of

sttp<'r-position. This 11l<"thod. thus ca.pl.ttrPs thE> elf<"ct.s of t.he anglc•-of-att.ack tinw history. Cmlq)ressiF>ilit.y dfect.s «re inherent in the modeL The compld.e model re<juirrs only static airfoil da.t.a, ami limited unsteady dat.a. The parameters derinO>d front the uust.Pady data., however. are not SE'nsitive to airfoil shape. and t.lnts uet>d not. be recakulate<l for all airfoils. The linear, a.Uachellllow phase of this model has been utilized in a d)'ttamic analysis inwstigat.ing high<"r harmonic control aduat.ion power I"P<JIIin•nt<·nt'

[1:!].

This aerodynamic refinement. showed a slight. impro\·cment. ov<'r quasi-st.ea.dy st.rip theory. in predid.ing 3P vert.ical hub shear on a 1/6 dynamically snll<"d 13oeing Cll--17D rotor modeL A modiiled V<'rsion of the complet.f' non-litH·ar modd was also incorporated into a dynamic ana.lysis.

[l:J].

Although this analysis was rPstrict.<•d to a !lap-only bending condition. signilkaut non-linear dferts ll"<"re apparent at high spt'ed, high t.hrust. comlitiuns.

(4)

The hlade-Pkmrnl nwt.Jwdologi<'s disntssrd t.o !.his point ac..-otlltl for only sited wakP dFc,cls. To rotnpletf' t.lw Ol'<•rnll ec·rodntmnir 1110<kl. I rnilc•d 1nd;c· <'if<'<" I' ntust. J,p considrr<'d. Th<" rot11biun.t.ioll of sh<·d ami trailc·d vort.icily rn·aiP a wak<' sl.rudun• lha.t. signinf'aully affE'ciB a rotor blades' "'"·odynamir <'ul·ironnH'til. Enrl.1· ana.lys<o's sill!plified t.he wake model to a uniform inflow distribnl ion. A first ord•·r iutpro1·emeul. l.o t.ltis model is assuJJliug a liueM dist.rilJttl.ioll of iuJio\1'. suclt "' t.il<' DreE's wake mode··!

[H].

Rerent.lil.era.ture has entpbasized the ll<"Pd for fmt.hc.•r rP!iu<"-nwnt.s t.o improve corrPlat.ions 1-o a.n arc<'pt.able kvel. Current. l.<'cbnology prol·id<·s sPveral models, from classical skewed helices t.o fr<"e wake calculations. and ronq>M-isons have IJf'E'll made on t.hE'ir respect.ive ability t.o predict. rotor inflow ov<'r n l'illll\<'

of !light. conditions. Two rencn\. papers have nmdE' a cotupr<"hensiv<' st.ud)· on t.lw validity of cnrrPnt wakE' mod<'ls. HPfereuce [l!J] examines five models; Sr111ly's frf'<' wake in CAMHAD. UTHC:'s free wake, geueralized wake, and classical skewed helix wake. and Beddoes model. Surprisingly, all showed ouly limited success. Disn<"p-aucies, in a.Jl the models, appeared in umlerpredict.iug the amount of rotor upwaslt on t.he disk at higher speeds, and in being aiJle to predid the location of maximum dowmva,,Jt. Hcference [lG] inwst.igat.ed two mod<•ls, (Scully and Sadkr). at a low spePd condition. Hesults showed a superiorit-y of Scully's modPI. alt.ltonglt it showc·d lirnit.alions in pn~dicting the fonnation of t.ite tip I'Ortices iu the rolliug-np walw n·-gion. One other rc-i'Preuce,

[li],

investigated sevt·ral wake opt ious withiu (',\J\1 fi,\1), At. hip;h SJJPl'Ll, dilfpn•uces bet. ween a prescribed wake and I he Scully fr•·•~ wal«• W<'r<' insignilica.nt .. Bot.h of these opt-ions greatly intproved correlation o,·,·r a nniforrn inllow dislrihut.iotJ.

The' prcseu\. analysis utilizE's the free wa.ke modd from C'AJ\IHAD [l8]. Al-though it ha.s shown discn•pa.ncies iu prf'cliding f'Xj>erinwnt.al dat.a. il is antong (.he current, st.a.t.P-of-the-ar(. models available. lucorporat.ion of t.Jtis model int.o om dynamic a.ualysis has prO\·idPd us several wake options. In conjuuction wil h tlw blade-element. model discuss<'d ea.rlin, t.he pr<?sent a.eroela.-tic analysis includes a higJtly sophisticated, yet comput.atioually feasible aerodynamic pr<"didiou ca.paiJil-it.y. The st.ructmal model is of comprahle sophistica.t.ion, tlms creatiug a.n overall llJodel with a.dvanc<'d prediction capabilit.ies. The present. study will im·<'st.igak tlw influ<'tH'e of Rewra.l key paratnd-Prs on blade loadings a.nd responsPs, and <kt.<'J'tllitw the signilica.IICE' of the reiinl'd a.rrod_,·na.mirs. Fut.ure efforts shoulll !Pad t.o a. cor-rela.t.iou study. t.o ult.ima.t.ely <l<--t.Pnnine t.he validit.y of t.hese refinc•meut.s, a.- well "·' a.11 invest.iga.t.ion int.o blade sta.bilit.y cha.racterist.ics and incorporat.ion of still het.t.c·r wake models as Lh<"y becomE' a.Yailable.

(5)

3 ROTOR ANALYSIS

The present. im·est.igat.ion utilizes an analysis which soh·es t.lw roupkd.

noll-linear periodic equations of blade motion and six trim equilibrium equat.ions. The analysis is based on a finite element approximation in both space and time. Earlt blade is assumed to be aJl elastic beam undergoing fla.p bending. lag bending. elas-tic twist. and axial deftP.ction. The blades arP. discretized into a. number of lwam

<·'klllent.s, eadt having fifteen degrcP.s of freedom. To reduce compnt.a.!.ional t.in1e. a

la.rg<' number of finite elem<'nt. ecptations ar<' tra.nsform<?d to t.he modal spacP a., a f<,\\' normal mode equations (6-8 modes are normally used). ThP. periodic. non-litwar. response solution is determined by using a temporal finite element. discretization. Fonrt.h or<lf'r Lagrangian shape funrt.ions arP ut-ilized within each time elt·mPnL Tlw finite element in time formulation is basP.d on Hamilton's weak principle. RPsult.-ing normal mode equat.ions are ult.imat.ely solv<:>d as a set of non-linE>ar algebraic E'<pta.t.ions. The solut.ion is obtained utilizing an it.era.tive, modified Newt.on mdhod

[cl.l !I].

Concurrent with the solut.ion of t.he blade E>qua.tions of motion, is the solution of t.lw no-hide I. rim E-quations. Tht> vehicle is t.rimm<"d on all axes for both forcE' and momPnt. equilibrium (6 eqttat.ions). Th<' modilied NE>wt.ou solution method allowo the two solutions t.o );p coupled to produce a u11ique solut.iou that sat.isfies bot b

blade and vehicle eqttilibrium equat.ions. The force summation method is tts<"d to dE't.ermine hub forces as a. sum of inertial and aerodynamic loadings

[20].

3.1 AERODYNAMIC MODEL

The aerodynamic modelling consists of two primary phases; a bladt>-PlemPnt. scht>me, ami a global wake scheme. The blade-element aerodynamic model is fonnu-la.t.ed to allow t>it.her quasi-steady strip theory, or unsteady two-dimensional aerody-namics. The unsteady aerodynamics consist of three parts, a linear att-ached flow solution, a separated flow solution, and a dynamic stall solution, based on the work of Lt>isluna.n a.nd Bedcloes

[11].

3.1.1 Attached Flow

The attached flow aerodynamics are calculat-ed using an indicia] respous<?

(6)

method, b~s<'d on the principle of sup<:'rposition (21]. Tlw indiri~l rPsponse schc•nw clet.Prmines the aerodynamic lo~dings dtw to ~ step change iu the airfoil downwash at t.he three-quarter chord position. This indicia.! response is then convolut<·d ow·r time using the Duhamel iutegral. The iudic-ial rPsponse is id<"alizecl iulo rout rii>111 ions from both circulatory. and non-circulat-ory loadings. The circulatory lift. contpotwnl is given by,

(I)

where WT is t./re d10rdwise velocity compon<'nt, up is t.he clowmrash at thr<'e-qnart.cr

chord, and

cia

is the 2-dimensional CO!ll)Jressible lift curve slope. Circulatory lift. is definc>d in "s-t.ime", which is the distance t.he airfoil travels in S('llli-chords. This is t.mnsformed into "azimuthal t-ime'', i.e. a fun ct. ion of azimuthal angle.

1/•.

Tlw circulat-ory indicia! function,

<Pc

is expressed a.<;,

(2)

This fnnct.ion is similar t.o the classical Kussner function [22], and repres<'nt.s (]]('lift deficiency clue to shed vortices in the near wake.

The impulsive or non-circulatory lift is similarly expressed a,;.

with the impulsive indicia! function as

(-!)

where

r:

is a Mach dependent time constant [21]. This function is based on a. time history of t.he nou-circulat.ory loading due t.o pr!"ssnre wa\'e propagat.ion, with an iuil.ial value computed from piston theory. The total lift. is (.he smu of these two components.

L(s, M)

=

Lc(s, !II)+ L1(s, !II) ('i)

The implementation of equa.t.ions (1) and (3) is ca.rried out. by expresRing these equa.t.ions in a discrete time form. Such a. form is compa.t.ible with tlH' finit.<' elenwnt discretization solution. The lift. equation becomes.

with,

(7)

wh<"l'<, X, and } ;, ~re lirt. <knci<?ucy funrt.ions which r<>present. t.he d<'fici<'ll<'Y i11 do11·nwash due t.o unst.cady effect.s.

D!,t)

is a time history impulsi1·e ~ugk of a \.tack funct.ion, and

D!,

3₁_{is a. tim<? history impulsive pitch rate funrt.ion.}

_To

_and

_T.

1 arr

fll~ch dq><>tHirnt time constants. C'ompl<?te details can be found in [10,1 1.:!1].

The unst.Pauy pit.ching mom<?nt about the quarter chord is. similarly t.o lift.. a sum of circulatory and non-circula.t.ory loadings. Circulatory contributions de-rive from t.hP offset of the circulat.ory lift. from the quart.Pr chord (a·.c), aud thP induced pitching moment effect. associated with the pitch rate induced camber. Non-circnlat.ory contributions an'> impnlsi,·e mome11ts due to angl<' of a.Uack dtanges and pit.ch rate. This yields,

where, qE is an effective pitch ra.t.<?,

!3

is the Pra]l(ltl-Gla.uert. correction factor. C,\;~

and C',\]~ are impulsive moment contributions due to angle of a.t.ta.ck and pit.ch ral.r.

respect.in.-ly, and l'ou aml1~.u a.re flla.ch depend<?nt time cons(.ants

(21].

The unsteady drag can be derived from this lift. force component .. and t.lw

airfoil chord wise force

(2:3].

The chordwise force arises from the COlll)>Ollenl of lift iu (.he dwrd wise direction. scaled by a leadiug edge suction eHiciency factor ( IJr· ).

('l) Due t.o the combination of dPraying non-circula.t.ory loading, and the build up of circulatory loading, the important result. of a. non-zero pressure drag under unsteady forcing condit.ions is oht.aiued. Pressure drag tends to its static value in t.he limit. of the unsteadiness u<?cayiug to zero. The total drag is the smn of this pressure drag and the airfoil viscous drag.

Co = [CN sino -

C'T

coso]

+

Co,

(10)

The,~e components are transformed into a discrete time form with appropriat-e lime history functious and time coustaut.s. These three compoll<'nts, lift, drag. ami pitch-ing moment. are incorporated into the analysis scheme and determiue the unsl<>ady attached flow solution.

3.1.2 Separated Flow

The second phase of the unst.ea.dy blade-e!Pm<?u(. a<?rodynamic model involves corrections t.o t.l1e attached flow solution due io flm,· separation. A method. hasf'd

(8)

011 l(irrltl10ff l.lteory, has lw<'n impknwnted [10]. hirchhoff ilwory n·lai<'s i.lw

""""'"!

force coPI!kicut. iu tenus of the trailing edge ·'<·'Jlilrid ion point.

f.

(II)

Thus. if i.l1<:' S<'IHI.ral.ion point. is known. it. is <:'asy t.o dC't.<:'rmin<' t.I1P norn1al forrr. Accordingly, if the norma.! force allll angle of attack are known from experinwnl, the sr~pa.ra.t.ion point lwha.l"ior ca.tl be implied. This, howev<"r, is a sl.al.il' nwtll. 1111d

must. l,e tnodifi,•d lo n•preseut. unsteady conditions. Two physical plwiJOilJPilil hal'<' h('en id('ntified thot affect separation point behavior

[10].

The first is a lag assorial<·d with t.he lea.diHg <"dge pressure rPsponse, beyond the normal force lag assoria.t<'d will, uust.carly forcing conditions. The second is the dfects of the tmst.ead)· houHd<,r)· la.y<'r. Thus, t.hP met.hodology is as follows; caknla.t.e a. pressm<' laggrrl. "f'ffpclil·<'·· angle of at.tack, and use this value in t.he static airfoil separat.ion point. rdation to obt.aiu a lagged separation point. The final step is to apply a boundary lay<'r lag t.o obtain the unsl.eady separation point. This va.ltt<> is inserted iut.o the Kirchhoff relot.iou, equation ( 11 ), t.o obtain the unsteady s~parat.ed llow normol force.

The calculat.ion of a. separated flow elton! force, and thus a. drag forre. equa-tion (10), is deduced from Kirchhoff theory. i.e.,

( 12)

Further details are given in reference

[11].

The final part. of the S<'paral.f't!llow solution involvE's d<"t.enninal.iou of pilrl!. ing moment characteristics due t.o flow .separation. Kirchhoff theory ha.s no general expr<'ssion for pitching motneut b<'h;n·ior. Thus, an empirical CUJTC fil must. be nl

i-lized to represent the varia-Lion of airfoil center of pressure, as a. funct.ion of scpa.rat.iun point,

wluo>re. 1\0,[\.t, and 1\2 , Me empirical parameters, obtained frotn st.at.ic dat.a. This

result, rearranged. gives the separat.ed pitching monHo>nt behavior. These t.hree part.s sum t.o yield t.he separated flow solution.

3.1.3 Dynamic Stall

The third and final phase of t.hE' llllst.f'a.dy blade-el<'mPni. of'rodyna.mir '"'"'''' ts a. semi-empirica.I method for dynamic stall. Several physico! ph<'nomena 1111tsl

(9)

I"' n1u<klkd t.o accurat.cly rPprescnl. the aerodyuau1ir loadings on t.he airfoil: tlw llln!-(llitud<: of t.he vortex iuduced lift. the cuuditious upu11 whirl1 if '"l"lrafcs J'ro111 t.he leading edge. and t.he time hist.ory of !.he vort.ex as il- t.ra.n•rsf's t-he chord ami dissipa.f.f's iut.o the wake. lnvest.iga.t.ion into Pxperiment.al data.

[11]

shows t.hal. an accept.a.bl<' model rdat.es Uw cxcrss vorl-PX lift (.o l-hP dill'crcnce hrt.weeu t.he a.t.tachf'd flow lift. and t.he sepa.ra.t.cd How lift. cout.ribution. This excess lift is allowed (.o decay expouetlf.ially, while simult.aneously being incremented a.t each st.<"p. Numerically, t.his call be represeut.ed as,

( 11 )

CNv _n = F(CN,· _u-1.(\·,,(\·., _-1 ) (l!i)

when~. C'1• is t.rrnwd t.he excess lift. duP to Lite vorl-ex, aml CN,. is t.he correspoudiug

iuen·nwul.f\.J value of ]if(. due l.o t.he eut.ire t-ime history elfecf; of !.he dynamic st-<t-11. The rrii.Pria. upon which the vorl-ex will s!"pa.rat.e from the lt•adiug edge• is a fuurt-iou of t.Iu~ l<•adiug edge pressme gradit>ul. all(] l-11e airfoil characf.<'l'ist.ics. A lPadiug <'ri)!;<'

pressure parnmel.er is calculated based Oil t.he pressure lag behind t.he a.irfoiluon11al

fore(',

c;,.

For ('\'E'ry airfoil, a limit valuE' of this parameter Cil-11 be delt>l'lllilw<L C',~'

Tht' onsd- of vort-ex srparat.ion a.t. the leading edge is indicated when

c;,

exre('ds

( ·,~. Tlw fiual dfed. tltat Heeds to be moddkd is t-he rate of vorlC'X tmvf'l anoss t-he rhord. This rat.<" has beE'n shown exp<•rinwnt.ally to bP approximat-ely half of I he freE' sf:ream velocity

[11.].

This rate is also 1\lach dE'peudent. As tlw vort('X r<"adtC'S l.be airfoil tra-iling <"clge aml passes inl.o the wake. its elfed on the a.irloads is qnirkly reduced. l\lonitOl'ing of the magnitude of t.lte vortex induced lift, al)d its dural-iou 011 l-he airfoil chord. yields a.n accurate representation of the airfoil lift cha.raclerist.ics under dynamically stalled couditions.

As a direct resulf. of l-his dynamic sl:all prort>ss, !.he pit.ching lllOlllenl. r!Ja.r-acterisl.irs of the airfoil a.re also changed. This variation is effectively modeliPd by l'<'Jll'f'SC'lll-irJg t.]w variaf.ion of t.hP airfoil ccni-N of pressurc, aft. of 1-he cpmrt.Pr-dwrd.

CP1• =

~

[1

+sin { rr

(

1 ~'-~

-0.5)})

(l G)

whrr<". n· is the 11011-dimf.>usioual t-ime after l.]w shedt)ing of the vortex. and

T\'1,

is t.lte total t.imt> for the vort.ex to traverse the chord. The iucremeul.al pitching mome11t is theu simply t.he vort-ex induced lift times the offset of t-he ceult'r of pressur<' from t.he quarter-chord posit-ion, i.e.,

( 1 j)

A nllnlh<'r of modifica.l.ions t.o t.lw modP! a.re C"ondud.<"d t.o a.C"rotml for "int.Pr-act.ional" dl't>cts amoug t.he indi,·idual elf.>meut.s of the model. ThesE' modifications

(10)

arc principally carried out through modifying just two time constants. OIH' Pech as-sociated wit.h (.railing edge separation, and vortex lift. These changes includ .. tlw effects of separation suppression due to moderate pit.ch ra.tes. and tlw acceleration of trailing edge separa.t.ion clue (.o either vortex sepa.ra.tion at t.]w leading Pdg<' or a change in pitch direction during vor(.f'X travel down tlw ilit·foil chord. B<•;dlachttH'Id is also modelled by monitoring the leading edge pressure.

c;,,

aucl adjusting t.he rat<> of reattachment depending on its value. (reattachment is initiat.ed wlwn

c;,

b<'cotnPs less t.han C,~), as well as the location of any vortices in (.he flow. Appropri11i.P logic for these phcnom<'na is incorporated int.o the algorithm

[11].

These modifica.l.ions. in conjunct.ion wit.h the three phase blade-elemE'nl. unsteady aerodynatnir ntod..J. result in a complete, non-linear aerodynamic model. ](. is important. to not<~ that. this solution needs only minimal prerequisite information about t.lw airfoil being modell<'d, is an efficient ami accural.<' procedure, and can be fonnula.t.ed in a fonn compatible with a temporal finite element solution.

3.2 IMPLEMENTATION INTO ROTOR ANALYSIS

As nwnt.ioned earli<'r. the formulation of the aerod.1·namic model is compat-ible wiLh the finite element. in time solution tedmicpw. The discret.(' (.im<' algo· riLhm "steps" through each temporal Ga.ussian integration point around the az-imuLh. Time-hist.ory effects are maintained by sa.ving previous sLep information. The temporal finite elt>ment. solution requires the response solution to be pel'iudir. Consequently, the aerodynamics must also attain periodic time histories. Satisfac-tion of periodicity, as well as trim condit.ions, determine the fina.l cou1·erged soluSatisfac-tion.

3.3 ROTOR WAKE ANALYSIS

The wake modPl refinements included in this rot.or analysis include a lin<"ar (Drees) inflow model

[1-±],

and a. free wake (Scully, .Johnson) modd

[2-!.2!i].

Tllf' geomel.ry of the free wake model is divided into three regions; near wake. rolling-up wake, and far wake. The near wake consists of a series of radial panels, each with linear circulation distributions. The rolling-up wake consists of an inboard litwa.r circulation distribution panel, and a tip panel \hat represents the rolling up of I lw

Lip vortex. The far wake is simply one panel of lim·ar cirnt!at.ion distribul ion. and a concentrated (.ip vortex, of strength proportion<\! to the maximum cirnt!at.ion 1·aht<" on the rotor blade

[18].

See Figme 1. The inboard vorticity is far less signifiranl than t.hC' tip vortices due t.o a small circulation gradient. inboard on t.lw blade. Thus. a more approximate model can be used. Conversely, the t.ip vo1tex requires a t·Pfitwd

(11)

modt··L i.... a. free wak<" model, t.o yield arcura.t.e results. The modPI usf'rl in !It is analysis prescribes the inboard wa.ke with large rore radii ,·ortires, and allows I!J<•

tip Yorl.f'X g<·.·onwtry to vary. Vortex filam<'nts wil h linear vorl iril.y disl rilntf.iolls 11rP used. The extent of each region of the wake can be prescribed

[18].

Set' Fignr<' :!.

••• ••• rw PARWAIC.l ROLLING UP WAKE fNWrlll NEARWAX£ r•~ TIP Figure 1. Figure 2.

The pr<>srnt a.nalysis mncent.rat.es on t.he high sp<><>d flight. regime. At. such a condition, t.he problems of hla.de-vort.ex inl.<>rartions are limit.<"d, and are comltlUtdy found on [.he "sides'' of the rotor disk. Several motlifirations are ut.iliz<>d at. l.ht•sp local.ions. First .. the azimttt.hal step size is r<O>duced. ami secondly, si11ne lirting line t.heory ca.n11ot represent. the effects of chord wise variations in iuflow, a.lil'l iug smface correction is used. One fina.lmodifica.t.ion was driwu by experiment.al results.showiug a decrea.~ed iufluence of the vortex after such an interaction. A met. hod of l<'f.t.ing the vortex core "burst'' or increase in radius after an interaction effectively models this phenomt'na., t.hough the actual mechanics of the pro(ess are not fully lliHlerstood

[24].

Sin'e sht>d wakE' effects art> repre"entecl in t.he bla.cle-element. modelling, they are not. accounted for in t.he wake model. For more details Oll the wake model. S<'t>

[18,24,2!j].

4 RESULTS

The baseline rotor configuration is a four hladt>d hingdt>ss ro1.or '"it h fuudR-mrnt.al rot.a.t.ing fr<"qttenciPs of flap, 1.13. lag. 0.70. and torsion, 4Ai prr/rP\'. Tit<' flight. coudition is a high advance rat.io, 11 = 0.:}.5, with a. hover tip 1\lach numl"'r of 0. 7. The t.h rust coeflicient o\·er solidity. C'-r / 0' = 0.07. UaseliHe wake modt•l

a.t.t.rihut.t>s indttde a. tip vort.t>x core size of .O:JR, ami four radial inflow cakulat.ion

(12)

points, with t.lw oul<'rlllOsl point. lond<'cl al

X

= 0.!)2/7. Tlw rolor bi<Hi<> airl'oil is a N

:\C'A

0012 sect. ion. The rotor ],Jade a.spect ratio is, rndins/chord = l X. Arldi-l.ional rotor information is ginen in Ta.ble J. A parametric study is undertaken to invesl.igaf.e the i11fluence of lincat· unsteady aerodynamics, non-lin<'ilr JJnsff'IHi.l· IHTtJ-dynamics, t.hrust.. lvlach munlwr, tip vortex siz<", and radial inllow point. locat.iou. Comparisons are made between tht• Drees wakC' model, and t.he free wak<" modrl.

NOTE: In the figures, for blade response and root loads, the lint'a.t· solution is l.he attached How solution. and the non-linear solution is the coH!plPte mod<>!. including separat.ion and dynamic stall. All results are using Drt'PS wah·. PXCPpl where noted. The blade section results compare the linear unsl.eady solution. tht> non-linear solution, wit.hout dynamic stall, i.e. just. separation effect.s. and f.h<' complete model, including dynamic stall. In the figmes involving sPpa.rat.ion poinl. a. value of 1, means no SPparaf.iolt, i.P. the separation poiut is at Llw trailing edg<'. !llore and more separa.tion exists as the value decreases toward zero. at the leading edge of the airfoil.

4.1 Baseline Case - Drees Wake Model

The baseline results are a comparison among. quasi-skady, linear (a(.\.adl<'d) unsteady, and non-linear (separation and dynamic sl.all) unsteady aerodyuamics. Figures 3,.1 and 5 show the blade t.ip response for flap, lag, ami torsion resp<~ct.i\·ely. Flap motion is primarily 2/rev, while lag !J,nd torsion are 1/rev motions. Non-litwar effects result in a. phase shift in f!a.p response, and increa.•ed higher harmonic conf.E"nl., especially in the torsional ret.reating side response. The corresponding blade root. verUca.l shea.r and flap bending mommt, Figures 6,7, show amplified phase effects.

a.~ well as an increase in vertical shear oscillatory amplitudes ou the rt>treating side. Examiniug the aerodynamic normal force ami pikhiug moment. at. t.!te tip. Figuws 8,9 respecl.ivdy, depict the flow being dominated by separatiou effpds. with only limited dynamic st.<tll. The sPpa.ration is most. eYidPnt on \.he advancing sidP. nPar the tip, while iuboard, the flow remaius attached, Figurt> 10.

4.2 Baseline Case - Free Wake Model

Inrlt1sion of the fl-ee wake anaJrsis at. this flight. condition yie-lds only smilll changes in the results. Figures 11,12 show t.he variation in blade t.ip respouse magni-tudes. llla.d!" section aerodynamic forces are only sligh\.ly changc•d, see Figures J:l.IJ. a.o; compared to Figures 8,9 rt>spect.ively. Similarly, blade root. loads also show onl)·

(13)

small rhnn,eps. Figm<'s l!i.Hl. It shmdcl h0 noted t.hat rrRJtlt.s arP also obt.aitwd for a prPscribed wake gronwt.ry. based 011 Landgrr•lw's tnodd for ntaxinnun circulation

(:W].

Discrcpanri<'s lwl.lw•PIJ t lu• prescribed aucl freP wakf' nwdPis <lr<' illsig~tilicaltf. coucmring with Hdereuc<>

(IT].

Llms prescribed wake rewlts a.re not shown.

4.3 Compressibility Effects

To d<.'icrmine compressibilit.y effects. (.he flight condition is changed to a Lip i\larh nntubcr of O.!i5. A free wake analysis is used for these results. Hesults show that. the separation effects. which dontina.t.ed t.he flow at the tip regio11 iu t.he baseline case, a.re significantly reduced, Figures 17,18 as compared t.o Figurrs

]:3.1~. At: the reducf'd (.ip i\lach !llllllber, blarle response shows a reduction i11 lag n·spollsc· tll<lgllif.tt<le, Figure l!J. a.nd torsional oscillations. Fignt'<'

:W.

Flap r<'Hf><>IIS<' is primarily nuall'c>cl.t·•d, and Lllf'r<'l'or<> nut. pn•se11t.ed. Blade loads also <>xhil,it. redun·rl oscillatory amplit.udes, Figures :H.22.

4.4 Tip Vortex Core Size and Inflow Point Location

Tip vort.ex core size. a,JI(l t.lw lo,a.t.ion of point.s at which t.he fre<' wake• inflow

i~ rakulaterl, are variP.d to det.ermine their scnsiti,·ity on rotor loadings. First t.h(' four iuflow ca.kulation point locations were varied, the outermost point. \'arying froltl

0.88R :::; X :::; 0.\JGI!.

For each case, the tip vortex core size wa~ varied from.

O.OU275R :::; 1'com; :::; O.O:lR. None of these casE's showed any sig11ificant ell'ed.s. If

the ouf.<'rmosf. inllow point. is moved t.o the Lip~ (X= 0.9\JH), nea.r the vort.ex core. howevE'r, results are signifka.nt.ly altered. The following results a.re for a. ntodified (.ip vortex core sizE' of O.U0275R, ( !i% chord). with the outermost inflow ca kula! ion point. a.t. X = U.99R. Figures 23,24 present section norma.! force and pitching moment l'<'SIIIt.s. rcspectivdy. These fignres show increasf.'d separation on the ret.rcaf.i11g side. plus a small region of dyna111ic stall at. ~·

=

2GO d<:'g. as rompa.rrd t.o th(' lm-'dine ca><(', Fignres I:l.J~. Bla.de flap response magnitude is denea_qed. Figme 2!i. and (.orsiona.l response ma.gnit.nde is increased wit.h decreased oscillat.ions. Figme 2(i. Blade roof. loadings are also alt.erecl, part.icularly an increased peak-to-peak \'Pt'f.kal shc•a.r. Figures 21,28. Th('rel'ore, jw.lgellt<'Ht. is required in the seiPd.ion of l.hP tip vurt.ex core size. and i11flow point locations, in order to a.t.Lain accurate result~.

(14)

4.5 Increased Rotor Thrust

In order t.o magnify t.he non-lin<'ar aNodyn<unic effects. t.he baseliue flight. condi t.iou io changf"d t.o a higlwr th rnsl coellicicut. (from cy /a = .01 to q-f a = .l 0). As exped-t'cl. t.he overallnmgnit.udes in response and loadings are incr<'ased. 1\lore

int.rrr~t.ing. though, is t.he amplificat.inn of J,ot h separation allll dynamic st.all f'lfrcls. At t.lw Lip, t.here is larg<' separat.ion dfect.s around the entire azilllnth, a~

"1'1""''"

l.o jnst. t.he advaucing sick in t.he basE'iine ca~e. Figm<> 2\l. as WlllJHll'<'d t.o Fignrc• 8.

Vorkx lift. cout.ribut.iuns to t.he Lot.al blade normal force arr. increased. In t.hc.• t.ip section pitchiug UlOnHO>nt plot., Figure 30, t.hree regions of dynamic stall are apparl'nt., t.wo lesser st.a.ll regions, "¢• = 215 dc•g, aml"¢• = 310 deg. aml a la.rgP st.all rc>giou <tl

•i•

= 2GO dcg. Inboard ou l.he blade, the dynamic sta.llregious arc a.t. dilfc•n•nt. azimnl.lwl locations,

1/•

= 220 dcg, aml ;'• = 290 deg. Figur<'s 3].:32. Figure :J:l shows I hal tlw separated flow region is spreading inboard. Blade responses. s<>e Figm<>s :3~.:35 and

:JG. show a. corresponding innPase in noll-linear <'lfect.s. including phase shifts and inn<'ased high<'r lwrmonic cont.f"nt .. as W<'ll as an increased lag respouse lllagnitudP. Ulade root. loads also show substant.ial nuu-linear efl'eds, Figures :jj,:J8.

5 DISCUSSION

The inclusion of linear aml non-linPar unsteady aerodynami~s has yieldc•d a. spectrum of interest-ing r<'snlts. For all !light conditions invest.igated, the differences between quasi-st.ea.dy and linear nnsteacly aerodynamic models a.re small. Small phase shift.s and response a.tl.euuations are appa.rent. Non-linear effects are of mol"<' signifkanc<'. The baseline. high speed, low t.hrust, flight condit.ion yields rc•snlt.s dotuinatPd hy flow separation effect-s. These effects arc> coucenlrat.cd at. t.hf' hlnd<> tip. aml are grealt-r Oil the advandng side• of the rot.or disk. The NACA 0012 airfoil.

at the high tip Mach munbers associat.f'd with this flight. conditioll. (as high as Ill

= 0.~),1!) Oil t.lw advancing side), stalls a.t. very low angles of at.t.ack result.iug iu larg"

amounts of separation. Reducing t.he hover tip Mach numlwr from 0.7 t.o O.!i5.

equivalent t.o reduciug Lip speed from 770 ft./sec t.o 600 ft./sec. limit.s the maximum Mach lltllllher t.o ]\[ = 0.7·1 on t.he a<h·auciug sicle. This all hut. diminat.Ps now sepa.ra.l.iun everywhere ou the disk.

lnrrca'<iug tht> thrust. lPVf'l. from ryfa = .07 to .10. gr<'all.1· a111plifi<'s tlw effects of the nou-linear a.Prodynamics. At. the blade clement. k•l'f'l, two t>ll't•cl.s art> appart•nt. One, the region of !low separation has sprc>ad t.o all azimuthal locations of the rot.or disk, as well as spreading iuboard from lhe tip. Secondly. regions of

(15)

lnrg<' d.l'llillllir stall Pxist o11 t II<' n·lrt>aling side of t.he disk. TIH~se rcgiotJH apt><'<ll'

al. dill'l'rl'lll. nhitnll!.hal sLa.l.ion:;, for diJr('rt'lll raditd lurat-iutJs. Thc•sc' f.wo df<'ds

nHJI-bin<' t.o cause illrl'Pased higbc.·r harmonic content. ami phase shills of t.he bind<· root loadings.

The free wake modd impkHWIIt.ed ront.aiHs ma11y para.nwters t.bat gol'<.'l'll wakP. geolllet.r'' and dist.m·t.iotl. To dd.erlllill<' acr~pt.able parauwt.c.•rs, a corrdal.ioll is carried out. with results jn Itefcrenre

[2i].

All va.lttE'S arc t.h<'u !wid constant. t.hroughou(; t.his im·cst.igat.ion, except. (.ip vorte-x core size. and !.he radial location of inllow caknlat.ioll poi11t.s. The ba.<dill<? conflgttrat.ion ut.ilizes Lhf' valn<>s of t.lwse lll'o para.tnd.Prs, det.rrmiti<,d frmn HefPrE'tH·,, [~i]. a11d the correlat.io11 study. HPS1dls sholl' (.!taL [or t.l1c basPlin<" llight. co11dit.iott. inclusion of (.Jw fre<' wakP analysis ,·idds o11l,l" sill all changf's in bladf' l'f'SJH>IIse suggc-slit1g 1-ha.t. (.he disk inflow has l>f'<"U slight h· rcdist.ribut.<?d. Locat.ion of the iBllow cakulat.ioJJ poiuLs at. Lhe blade lip, uear t.lw

vort.ex cote, howe1·er, showed signilica.nL el[ed.s. Large chaugf.'s iu blade response tnagnit.ud<"s sugg<>sl a "hang<' in the nt<'an ittllow value ovPr t.he disk. This also leads (.o difTer<'nt. peak-to-peak shear loads. To c.kt.<"rminf> the significance of f.Jtpse changes. a rorrelat.ion study with flight. t<.>st. dat.a is required.

An important. fa.ct.or which <:hanges as a result. of I hese pa.ra.nwt.ric va.riat.ious is lit<: ;,ehidc trim routrols. These values are gi~'<~ll in Table 2. S!"V<>ral iuf.prest.ing r<,· suits arC' uot.icecl. Non-linear a.c•rotlynalllics required a mud1 highn colleciil'<: <:011t rol iHput. ol'<.'!' liMar or quasi-st.eady arrodynalllirs for all casf's. The reducf>d tip /lla.rh mnnber case showed liU.le change in controls. The baseline free wake case !'<.''Jllired reduced lHagnit.udes of collective a.nd cydic cont. rub, yd. the modiGed t.ip vortex and inflow point case I"C'Jllir<'d illtTf'Hsed magnitudes of colkct.iw and loHgit.udinal cydir coHt.rol, ami a dccrea.'f•d lateral cyclic cont.rol. The inrreascd thrust. case, as would be expE'cf.E'd, requiretlla.rgcr control input.s.

6 CONCLUSIONS

Tl1e following roudusions a.re drawn from this invcstigat.ion:

1. A non-linE'ar aerodynan1ic modd has been succE'ssfully incorporat<,d int.o a roupblrot.or aeroelast.ic analysis.

2. lncorporat.ion of new wake model options, induding a. f!·'~E' wak<~ llloti<·l. has a.lso beE>n comple(.t•d successfully.

3. At. high speE•d conditions. non-li1war aerodynamic ef[eds tlon1inat.P

(16)

hoard blade sect.ion fore.;~. aucl significantly aff<'rt. hlad<" n•spons<" ancl hind<· root lo<Hls. Th<"s<" pff,.,·fs arf' amplified at increast?cl thrt!RI !Pv<'k

-!. Compressibility C'fferls significantly affect. t.he extent of sc•pand<'d flow on ill<" rot or bladt• .

. 5. For t.he baseline configmation, prescribed and fret? wake mock·ls gi"'" sim-ilar results to the Drees wake model.

6. Free wake moddling is insensitiv<" to tip vort<;>x core size when inflow cakula.t.ion points a.re locat.ed away from t.he blade tip. Varying tip 1·orlf'X cor<" siz,.. with an inflow point located at. the blade tip, however, subst.anlia.lly modifies t.lw mean inlluw on the rotor clisk.

7. Trim controls are aff<"d<"d by these parametric variations and must lw considered as a. factor in these results.

Acknowledgements

The authors acknowl<:<lg<" Dr. .J.G. LeisiJJnan and tllr. 1\hanh Ngu,·en. of the Uni~"<~rsit.y of Maryland. for tlu:ir assist.ance throughout. this effort. Tl1is research work was supported by the Army Research Office, Contract No. DAAL-o:J-88-( '-0002; Tedmica.l Monitor, Dr. Robert Singleton.

References

(1]

Cara.donna.F.X .. Tung,C .. "A Tieview of C'mrent. Finit . ., Diff,r<"llc<" Hot.or Flow Methods'', Presented a.t. the 42nd Annual Forum of the American Helicopter Society, vVashiugton D.C., Jun<" 1986.

(t]

1\IrCrosky,W ..

J ..

"Some Rot.orcraft Applications of Comput.ntioHa.l Fluid

D_,._

namics", Presented at t.he 2nd Iuterna.t.ional Conference on Hot.onTaft. Basic Research, Universit.y of Maryland. :Feb. 1988.

(:3]

Gaonkar.G.Il., a.nd Pet.ers,D.A .. "Effect.iwness of Curr<'nf Dynalllic Inflow Models in Ilo\'er and Forward Flight" .. Journal of t.he American lldieopt.er Societ.y. 31. No. 2. Aprj] 1986.

('1]

Panda.,B .. C'hopra.l.. "Flap-Lag-Torsion St.abilit.y in Forward Flight" .. Jom-na.l of the American Helicopter Socit?ty, :30. No.

·1. Oct ..

I US!i.

(17)

[0] DinYa\·Pri nnd Fri<>rlmann. "Applicn1ion of Tinw-Domnin llnsi.Pady A<·ro•h·-namics ~o Hotary- Wing Aeroelast.ici~y", l'resE'n~ed at. the Strurt.ures. Sl rue-\. ural U)·natnics. and :tvlal<>rials C'onfN<>nre, Orlando. Florida. April I'IX0.

[G] FriPdmann. "Arbitrary 1\lolion \Tnstf'ady APro<lynamics and its Applindiotl to Hot.ary- vVing AE'roelasticit.y", Journal of Fluids and S~ruc\.urPs.

1.

.Jatt. 19S7.

[7] Friedmann a.ud Robinson, "lnftHPnce of Time-Domain \TnsiE'ady Aerody-namics 011 Conpl<'rl Flap-Lag- Torsional Aerodastic Stability and lk~ponse

of Hot.or Blades", Prescnt.ed at tlte 2ml lnternat.ional C'ollfrrPnCE' 011 Hot

or-craft. 13asic Research, University of !Vlarylaml, Feb. HISS.

[S] C:a.n~;\\·ani. "Synt.h<'siz<'<l Airfoil Dat.a 1\'let.hod fot' Prediction of Dynamic Stall and Unsteady Airloads". Vertica S, l!JS4.

[!I]

Fri<'dmann, "R<'cent. Trends in Rotary-\Ving Aero<'last.icit.y". Vf't'lica 11,

HIS7.

[10] l3eddoes,T.S., "Represent.at.ion of Airfoil13eha.vior", Vertica 7. 1!183. [11 J Leislt man,.). C: ., l3eddoes.T .S., A C:<"nE'ralizf'd 1\Iodd for llnstPady

APrody-namic BehaYior and DyAPrody-namic Stall using t.hE' lndicial l'l'iethod", l'resent.cd at. thE' !2nd Annual Forum of the AmericanlldicoptE'r Societ:y. Washington D.C., .Juue 198G.

[12] Nguyen ,I\., Chopra.,!., "Actuator Power Requirem<'nts for Higher Harmonic Control (HHC) Systems'', Present.ed at tlw 2nd International Conference on Hotorcraft Basic Research, University of Maryland, Feb. 19SS.

[1.3] Elliott.,A.S., Leishman,J.G., Chopra,!., "Rotorcraft Aeromechanical Analy-sis Using a Non-Linear Unsteady Aerodynamic Model'', Presented at the 44th Annual Forum of the American Helicopter SociE'ty, Washington D.C .. . June HJSS.

[14] Dr<'es,.J.l\L "A Theory of Airflow Through Rotors and it.s Application to Some HelicoptE'r ProblE'ms", Journal of t.he IlelicoptE'r Associa~ion of Great. Britain. Vol. 3, No. 2, July-September 19.J.9.

[l!J] IIoad.D .. Althoff,S., Elliott,.}.. "Ro~or Inflow Variability with A(h·ance Ha-tio", l'resentE'd at the 44th Annual Forum of thE' American Helicopter So-ciety, Washington D.C., June 19S8.

' '

[16] Chades,B .. IIassan,A., "A Correlation Study of Rotor Inflow in Forward Flight", Preseuted at t.he 44th Annual Forum of the AmE'rican Iklicopt.er Societ.y, Washingt.on D.C., .June 1988.

(18)

[ L I] Yamauchi .U.l\., lldfernan.H. .• C:anberl..l\1.. "C'orrPlat.ion or SA:JJ !J /~ 1 1<-li-copi.Pr Flight. Test llala with a ( ~ompn·lwusin, Hol.orcrafl l\ludd", .)ourual of tht• AnwricaH Heliropkr Socid)",

:n.

No. 2. April 1!):38.

[1 S] .Jolmson, W ., "A C:omprC'!lf.'nsive Analytical Modd of Rolorcrafl Acrody·-Hamics am! Dynamics, l'a.rt. 1: Analysis Devdupmenl.'', NASA Tl\1 811:32, .June 1980.

[L\J]

Lim .. )., Chopra.!.. "D<~sign Sensitivity Analysis for an AProelasl.ic Optinli7.a-Liun of a l!Plicopt<,r Blad<.•". !'resented at. the AlA A Dynamics Stwcialists Conference. MouV•rrey. California, April1!JS7.

[20] Lim,J., Chopra,!., "Arroela.~tic Opt.imizatiou of a lldicopt.er Hol.ur". l'n·-SPnt.ed at. l-he 4-Ith Annual Forum of the Americau Helicopter Soci<'t.y, \Vash-ingl.ou [).C., .June 1988.

[21] L<,ish man ..

J.

C:., "Va I ida tion of A pproxi IIJ<tl-e Indicia.! Acrody llillHir

Fuur-t.ions", .Journal of Aircraft, Nuv. 19t:S.

[22] Bisp I i nghoff, H. L., Ash!.,y.ll.. 11 alfmau.H. L., Ac ro£ last ici I y. Add isuu- \Vt·sl<·y Pnblish<'rs, 1!)!)!).

[:2:3]

Leishman.J.C:., "An Analyt.icall\1odd for llnsteady Drag" .. Joumal of Air-era[(.,

J

nly HISS.

[21] Scully,l\l.P., "Comput.at.ion of Tlelicopt.cr Holm· Wake C:romet.ry aud its lu-lluent·e on Rot.oi· Ilarmouic Airloads", Massadmset.l.s lustitute of Technol-ogy. ASRL TR 118-1, March 1975.

[25] .Johnson.vV., Helicopter Theory. Princeton Uniwrsil.y Press. 1980.

[26] La.ndgr<'he,A .

.J..

"The Wake Geometry of a Hoveriug Helicop1-Pl' Rotor aud its Inaucnce on Rotor Performance", Journal of the American Hdicopl<·'r Society. Vol. 17, No. 1, Jauua.ry 1971.

[27] .lohnson.W., "Comparison of Cakulat.ed and l\ka~mctl H<'licoptrr Hotor La.1-<>raJ. Flapping Angl<~s" .. Journal of t.he American llelicopll'r Soci<'i.l'. :!G.

(19)

EI.Jmufl1

_If'

= 0.01080 El,fmufl1 _R·t = 0.02680 GJfmufl2_R4 = 0.00615

Table

1. !.·.</

R

= 0.02900

!.·..,,JR

=

o.oJ:l2o

J.,.,,2/R

= O.O:l-170 • Q",;

_<Ps

Bo

Bct,<eline Quosi- Sfteuly .128 -.0:)7 .167

DrasWctke Unslwcly( I i 11) .1~~ .042 .167

Mnp = O.iO l'nslwrly(non- lin) .128 -.O<IG .1 TO

C'T/CI = .07

Bctseline Qtwsi- S'twcly .1 :)0 -.0:!4 .I !iG

Free Wake Unstuuly(liu) .121 -.0:14 . I r,.J .llnp = O.iO Unstwdy(uou- lin) . t:ll -.041 .l !iU CT/CI = .Oi

Rt duet 1/.1/np

Pru If",, h U11slt'ltdy(11u11 - I ill) .124 -.O:J(; .!!>ll

:llnp = 0.!\!j

C'T/a = .Oi

]I!CI'f ClSt:d('T Quasi-Sif!tdy .OSi -.016 .I iS

v,.,.,,,,,."h

_{_l'}_{n8lt'IIIIJ!(I/II)} _.08:1 _.021 _.I_T!J

;\/TIP = ll.iO l• nstt~uly(uuu /iII) .O!ii .U:J6 . :! I fi

Crfa = .10

MudifitdCure

F1·et Waf.,, _{l'nslfiHiy(nun -liu)} _{.1 :!0} _{-.U!il . l}_\J_I

M1'tP = O.iO

C'Tfa = .Oi

Table 2. Rotor Coupled Trim Controls

21 - 18

Rotor Blade

Unifonn

Structural Properties

'

•

e,c

e,s

Bo(.,,l .04ti -.1

o:.

. !l I :1 .041 .I Ill .014 .042 -.101 .!l 14 .U44 -.U!JI . !l I :! .041 -.1 {)~ .Ill:! .041 -.U!J:l .u1:1 .l):j)) _-.IJ!)(i _.Ill:! .O!i!J

-.

,.,-

_,

. !l I

r,

. O!i I .J:!!I .IJ I !i

.IJ.~!i . l!",;, .IJ~K

(20)

TIP FLAP DEFLECTION

r

--~~~~·~··~~.~~~2~0~JT,~c,~/~•-·~·~·7~----. .• \2 - Ul<S!£AO'/ (liOH-UNEAA) - UNST£.<1lY (UNEAA) • t - QI.MSI-STEI.D'f a.

'

L

...

;f

.02 0_{o~~.~~--~go~-,~~~~1~ao~~.~2s~~2~7o~~.~.s~~.~} AZIIolUTH {OEG) FIGURE 3.

TIP LAG DEFLECTION

~.---~=~~·~··~s~.=~~·~o.~7·~"~·~·~·~·~·'~---, - UNSTE,.>l)Y (NCN-UNEAA) a

"'

'-.02 !,

>

-.04 - UNSTE:.UIY (UN£AR) - OllASl-S'!lW>Y -.060 ~ go llS \80 225 270 3\S 360 AZIMUTH (OEG) FIGURE 4.

TIP TORSION DEFLECTION 0 J" • •• U, Wr. s 0.7, Cr / fl • .07 - UN5ruoY (NON-UNQJ\) - UNSTE:.UIY (UH£AR) -.002 --OLWll-~ -.004 ~ ::.: Q. -.ooa -.Q\ 0

·~

90 ~~ 180 22S 270 315 360 AZIMUTH (OEG) FIGURE 5. .07

•••

'Q: .05 N Q .04 E ' . 0 3 H

...

• 02 .0\ 00

BLADE ROOT VERTICAL SHEAR

P • •

.u.

'*nr

=

0.1, c, 1 o • .01 - · - UNSTtAOY {NON-UNtAR) - UNSl'ti>.OY (UH£1J'j - QlJ;,$1-Sl't..oY

.~···~

4b iO \3.5 H:JO 225 270 31.$ :560 AZIMUTH (DEG) FIGURE 6.

BLADE ROOT FLAP BENDING MOMENT

~

"'

N Q E ...

,..

:::li or---~~~·=·~·s~·~·~~~·~··~'·~~~~-·~·~·~07~----, - - UNST'£ADY· (HOU-UN£JIIR) - · UNS1'ti>.01 (UiltAR) -QUASt-~ -.ao:z ·

..

'

\

...

..

--.004

_..

..

~~ -.006 -.COB O 45 90 13S 160 22S 210 31> ,l(IO AZIMU'TH (OEG)

FIGURE 7.

BLADE SECTION NORMAL FORCE

l JJ. • .». w,

=

.1. c, I a • .01

....

- C. N-L (CCMPLm: ~C0£L) % - C. N-L ('11/C OY!WollC STALL) \!!Ul - C . UNE'AA< ~

...

"'

2 0

..,

~ Mi

"'

0 ·~ -'

..

:::E

"'

.s 0 :z: _fW)W.._{STA"'ilON X • .i9R} 00

·~

90 1JS 160 22~ 270 315 360 AZIMUTH (O£G) FIGURE 8.

(21)

.2 1% !!:! .15 0 ;:: .1

...

....

0 0 .05

!i

... 0

~

-.05 ~ .... 1 :;: u -.15

ii:

...

SLADE SECTION PITCHING MOMENT

II • ..J$. w, ilt .7

.

Ct I • • 07

- C. N-L (COMPL£!£ MOOtL)

- C. N-L l;i//0 OY>IAMIC STAI..L) - C . UNCAR'

-J

·v

_'

AAOIAL STAMM X • .t9R 4-!l $10 1l5 160 225 210 315 360 AZIMUTH (DCC) FIGURE 9.

ROTOR BLADE SEPARATION POINT

~ c .3.5, ..,. • • 1, c,. 1 t1 • • o1 - SEPARATION POINT, X •.79R - SEPAA!o.TION POlNT, X • ,99R

8

_ll. z 0 ;:: .75

...

--. ,S·-···--··-··-·-··---·-1

...

~· U:. •

"'

...

ll.

...

"'

..J

e

""

:;( .5 AIR( OIL. CHORD .25 00

!--.':,---:gto:---:,:;35;;--,;-!a:::o-:::22~5--:::27:;;o:--:z:::,.;;-.

-:,:-!J'

AZIMUTH (DEG} FIGURE 10. .08 .02 0_{o~---:•':•----:•':o---;1:;35;;--,~~:::o-~235~~v:;;o:--:::,tts:--::z•o} AZIMUTH {O£C) FIGURE 11. 21 - 20

TIP TORSION DEFLECTION

or----~~~·~·z=•~-~~~=~O~-'~·~c~,~~~·~·~.o~'~----~

- - UNSTEADY (N-\.) - f"RLE "'ME

- - UNSTE,A.DY (N-L) - Ofi!t:S WAKE:

-.00<

__

...

·

·.

_·.

%: "·... /

if

...

-

····~... /

....

.

...

~··· -.006

···~~.--·~

-.ooa

-.at o'---."':,--g"'o--,-'z""s -"',a'""o--22'-5---2 ... 70,...--J"'t""s -,..Jso

AZIMUTH {OEG)

FIGURE 12.

~I

"I

f'R££ WAKE RADIAL STA110N X • .99R 00~--~--~--~--'----'----'----'--~ 45 90 135 HiO 22!t 270 315 350 .2 1%

"'

.15

u

;:: .1

...

..,

0 u .OS

!i

_w ₀ :::E

i

-.05

g

-.1 ;;; ~ -.15 ll. AZIMUTH (OEG)

FIGURE

13.

BLADE SECTION PITCHING MOMENT J.'•l5w • 7 C / a • 0 7

"'

..

'

- - - C. N-L (COUPU:Tf MODtL) ' - · C. H-L (W/0 OY-IC STAI..L) - Cw UNfAA.

-

...

,

..

'J

v'

"-N\££ WN(£ RADW.. STATION X "" .99R -4$ 90 135 180 225 2:70 31$ 360 AZIMUTH (DEG)

FIGURE 14.

(22)

..

"'

"

c

E

~~o • .Js, w,. = 0.1, c, 1 a = .a1

.07 r----'::::-:=:...=.7-":..:::':':-':W,...::.,=-.:;~---,

- - UNST£ADY {N-L) - FREE WAKE - - UNSTOOY (N-1..) - DREES WAKE

. 0 6 1 - - - ' - - " - - - : . . J .. .05

"'

" _{c .a.} E ' . 0 3 ~ .02 .01 0_{o~~.~5-~9~0-~1~35~~1~8~0-2~~=-~27~0~}₇_31~5~~35Q AZIMUTH (DEG)

FIGURE 15.

~ • • 35, Wt. = 0.7. c,. I II • • 07

Or--~~~~~~~~~~~--~

- UNSTWlY (N-L) - FREE WAKE

- - UNSTOOY (N-L) - ORE£5 WAKE

-.002

..

-.004

...

···

'

~

::!! -.006

-.ooe

₀

'--.'-

₅

--o"'-o--1-':35::--1-'e"o-~22'-5-727::o~-3.1.15--3-'6o

·AZIMUTH (DEG)

FIGURE 16.

BLADE SECTION NORMAL FORCE

I' • ..35. ""-' • • 55, c, I II • .07 3r---~~~~-~~~~~---, - - C,. N-L {C01.4PL.ETE MODEL)

!z

- -

C,. N-L (W/0 OYIWIJC STAU.) ~ 2.5 - - Ctt UN EAR • ~ r--~---~ !::; 2 0 u ~ 1.5 n: ~

....

...

_:::1 IY. .s 0

"

RADIA.L. STATION X • .iiR fREE WAKE 0_{o~~.~5-~.~0~~1~35~-1~B}₇_{0-2~2~5-~27::0~}₇_3~15~~360

AZIMUTH (DEG)

FIGURE 17.

BLADE SECTION PITCHING MOMENT .2 ~ • .35, w,. • .ss c,. 1 " • .o7

...

. :z: - - C., N-L (COMPL£TE MODEL) - - C,. N-L (W/0 OYNAMIC STAU.)

....

.15 13 t: .1

...

....

0 u .05

...

15

0 ::!! ~ -.05 §g -.1 :r ~ -.15 0:: - - C111 UNtAR. ~---... FR"t£ WAKE RADIAL. STATION X • .99R ' <l-5 90 135 1fl0 225 270 315 ... 6J AZIMUTH (DEG)

FIGURE 18.

TIP LAG DEFLECTION _ 02 ~ = .Js, c, 1 (1 = .01 - - UNSTWIY (N-L) - """ • 0.551 - - UNSl'tADY (N-L) - M,. • 0.70 0 .::. -.02

~--···-····

.}

21 - 21 -.04

FREE WAKE AWJ..YSlS

-.06 !:---:':--::--:-::--:-::--:-'::--:-'::--:"-:-_/

0 <45 90 135 180 225 270 J15 JSO AZIMUTH (DEG)

FIGURE 19.

TIP TORSION DEFLECTIOI'i

~ = .3$, Cr I II • .07 o.---~~~~~~~~---~ - - UNS'TtA.DY (N-L.) - Mtwo • 0.55 - - UNS1£ADY (H-l) - M,p • 0. 70 -.002 -.006

...

..-··•· ...

. ..

··

-.008

FREE WAKE ANAL-YSIS

-.01 .___..,__...,.._-::-_-'-_ _.. _ __,__ ... _~

0 <45 80 lJ5 180 225 270 J15 J60 AZIMUTH (DEG)

(23)

BLADE ROOT VERTICAL SHEAR .IJ7 p. • .Js, c, I 11 • ~07 - - UHS!WlY (N-L) - ... • 0.55 .06 - - l!NSTEAOY. (N-1.) - Mrp • 0.70

""

.

~

'C

.04

e

... OJ

.::

.02 .01

f'I!Et WIJ<[ AW.I.YS!S 0

0 i5 90 1JS 180 225 270 315 360

AZIMUTH (DEG)

FIGURE 21.

BLADE ROOT F'LAP BENDING MOMENT

..

-.Otl!! ~

"'

'C

e

-.00<1

..

·~

...

~ :::s -.006 -.ooa o'--.'-.--a:':o--,1,.:3.,.s--:,sc-o-""••"'•,--::•C::7o~""J"'ts=--=310" AZIMUTH (DEG)

FIGURE

22.

BLADE ·SECTION NORMAL FORCE

3

r-~~~P-,*~~~ ... ~-,·--·~~~~~~~·-·~.0~7----,

- - C,. N-L (CCMPl.£1£ IIOOEL)

!z:

-

C,. N-L (W/0 ClNAIIIC STAI.l.) ~ 2.5 - C. UNEAA•

~ ~---~

~

2 MOillfiED TIP WI!ID< COR£ SIZE _"'a_INflDW_POII{f_""""tlO!l

u 0_{o~--.~.--~~~o-:--,::!JS~~,s~o--~22~s--""v~o~-=l~ls~""3~6o} AZIMUTH (DtG)

FIGURE 23.

...

% !:! u ;:;:

...

₀

..,

.2 ,15 • 1 .OS -.2

BLADE SECTION PITCHING MOMENT

11. • .u. w,. ~ .1. ct I (1 *' .o1

- - C. N-L {COMPL.£Tt wODtL) - C,. N-L (W/0 O""""IC STAI.l.)

- Cv UN£AR~

loiOO!Fl£.0 TIP VORTtX COR£ SlZ£

AND INF'LOW P<:»NT I.OCATION

= A

v

'\'

..

\ /

"\.,

\;.~

\':·

_.; ffi£E WAKE RAOW.. STAnOH X • .99R o 45 go us 1 eo 225 2:10 .us 360 AZIMUTH (DEG)

FIGURE

24.

TIP FLAP DEFLECTION

p. • .35. ~ • 0.7. Cr J " c .07

• 12 r----...:...-....c:..:.,;;:.__;c.~;,:.,;...;...;.;._ _ _ _ ..., - , UNST'EioDY (ti-L) - W0t»f1EP nP CORE/INF\..OW PT

- UNSTWIY (H-L) - SASEUHE ,,~----~~~~~~~---4 .08 .02 F'Rrt WAKE ANJ..l.YSI$ 00~--~--~--~--~--.,.c---~--~~..., _4S _SiQ _1Jfl ₁₈₀ ₂₂₅ ₂₇₀ _3\5 _.l60 AZIMUTH (DEG)

FIGURE 25.

or---~~~~·~~~·~·=""'~·~·~.7~.~~~~~a~&~.o~1~--~, - - UNSlVDY {H.-L) - MODIFlE.D 11P CORE/INFI..OW PT - - - UHS'l'EAO'r (N-1.) - SASEUN£ -.002

-.-_!

iE

-.006 ... \ •••.•• -.008 21 - 22

····,

..

___

...

·

f'Rtt WAiti£ Af<W . .'TSIS

-.o•

o!---c•'=s--:t':o--:,735:--,c'I!0-:--22.,.:,.>--7_{27::0,--:J'",,:-~J,J•o}

AZIMUTH (DEC)

FIGURE 26.

(24)

BUDE ROOT VERTICAL SHEAR

~----~~~·~·,~·~-~~~~·~·~-7~-~~~~~·~·~·=·7~~~~

.07 ....

- - - uliS'TEJ\.OY (N-L) - WOCiflED TIP CORE/1Nf1.0W PT

'---~U~N=~~~Y~(~N-~l~)--~~==W~N~(---~ .06 r N .QO

"'

c

.c.+ ~

"'

N c E ... ~ ~ .02

·· ...

_

.01

0_{o~--.~.--~.~o---,~,.~~,.~0--~22~.~~2~70~~,~1~.--,~~} AZIMUTH (DEG)

FIGURE 27.

SUOE ROOT FUP BENDING MO!.!ENT

---~~=·~.J~·~·=~~·~··~7,~c~,~~~·~·~.a~7~~~~

Or

- - - UHS'frACY (N-1.) - WOClfl£0 i1P CORE/INft.OW PT

- - - UNS1'IAOY (N-L) - BASWN£ ' -.002 _\ ...

.#··

...

..

····

···-·

-.004 -.006

FIGURE 28.

BUOE SECTION NORMAL FORCE

Jr---~~~·~~~·~·~~~·~.7~,~~~/T•~·~·='"~----~ >- - C,. N-l (COWPcm:

"00£1.)

:Z: - - - C,. H-l (W/0 lmW'iC ST..u) ~ 2.5 --- C,. UNEAA ~ r---~----~---~

_...

... 2 Q <.>

~

fWJW. STATIO!< X • .OOA ·.~~.~.---.~0~~1~~~~18~0~~22=.~~2=7.~~,~15~~,~ AZI1.4UTH (DEG) .2

....

:z:

""

.15

i::i ;:;: .\

...

Q <.>

.05

....

3

0 :::! ~ -.O!i

"'

~ -.1

~

-.15 a..

BUOE SECTION PITCHING I.IO!.!ENT

JJ> • 35 ~ •

..

7 C, J a a: 10 - - - C.,. N-L (CCMPI..GE l.tODE!..) - - - C,. N-1. (W/0 O~IC SiALJ..) --- c. UNEAR•

=

[7

-

v'o.~~\

\.

/

\·--~'=-'./

RACW.. STAnCH X • .S9R -.2 0

\(

4~ 90 135 150 225 270 315 JSO AZIMUTH (DEG)

FIGURE 30.

BUOE SECTION NORMAL FORCE

14 • .J5, w,.. • .7, C, I a • ,10 Jr-~~~~~~~~~~~----; - C.. H-1. (CCWPL£TE WIOOE!..) ~ - C,. N-1.. {W/0 OT'N.A.WIC STALL.) ~ 2.5 --- C,. UNEAR.o

g

r---~---~ tl 2 Q <.> ~ 1.5

""

0

....

-:t

"'

0: .5 Q

.

.---:::·:\

....

··· ..

'--:z: _{RACIAL STAnCH X • .79R} ·~~--~~~~~~~~--~~-0 45 QQ 135 180 225 270 315 360 AZI1.4UTH (DEG)

FIGURE 31.

BLADE SECTION PITCHING MOMENT

.2 ~ • l5 w.r. •

..

7 Cr I a • 10

....

:z:

...

_i::i

.15

;:;:

_...

_.1

C,. H-1. (COWPL£Tt ~OOE.L.) - - - C. H-1. (W/0 OT'N.A.WIC ST.IJJ.) - - - C., UNEAR'

...

Q <.> .05 ~ ... 0 ::1 ~ -.0:.

··-··

-v----'

_'\-

..

. ~ -.1

=

\···~ •

..

\;

~ -.15

a.. RACLAL STAncH X • .7SIR

45 liiO 135 180 :Z:Z5 270 315 360 AZIMUTH (Df:G)

(25)

ROTOR BLADE SEPARATION POINT

~' • ~ ..,_. • . 7. e, J , • . to

- SEPARAllOH PO\NT, X •.78R - SEPAAAnoH POINT, X • .iiR

~-

...

'·

''·,

..

\

01

·.·· .08 80 135 1eo m 210 315 Joo AZIMUTH (DEG)

FIGURE 33.

TIP FLAP DEFLECTION

... . 06

;

.04 .02 0 o

~ ~ ·~ ·~

m

M$

~

AZII.IUTH (DEG)

FIGURE 34.

TIP LAG DEFLECTION

r

----~~~-~~=-~~~~-~·~·'+·~~~~~·~-~-·~·---, .02 - . UNSTIADY (NOH-UNEAA) - UNSTIADY (UNEAA) - CUASI-STE>m 0 -.OI 0 4S iO 135 110 225 270 31$ .360 AZIMUTH (DEG)

FIGURE 35.

_{21 - 24}

·r=~~~~~-~·3?··=~~~-~·~-'~·~'·~1~·~-~-~··~--~ _- _{UNS'l"EADY (HON-UNE:AA)} - UNS'TtACY (t:JNEAA) -.CC2 ~---:__OUAS __ I_-Sll;ID...;._Y ____ ~ -.010 .07

•••

"

_""

.0$

"

c .04 E

',OJ

.::

.02 .01 0 0 4$ 80 13$ 180 22$ 270 315 360 AZII.IUTH (D!:G)

FIGURE 36.

...

_~"_{• .35,}_{w, •}_0.1._Cr_I_~_•_.1o - UNSltADY (NON-UNEAA) - UNSltAOY (UNEAA) - QUASI-SitADY