NASA-CR-177362 

19850022732 

A Reproduced Copy 

OF 

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Reproduced for NASA 

by the 

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NASA Scientific and Technical Information Facility 

~p~o 672 Aug 65 

1111111111111 1111 11111 11111 1111111111 11111111 

NF00675 

LIBRARY COpy 

LANGLEY RESEARCH ,:ENTtR 

LI~RY, NASA 

HA· ... nON. VIRGINIA 
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1 fl....-t M J 2. ~I Au:G:orJl No l Roc'potn', C. .. 101 N,). 

NASA CR- 177362 

.. Tt' .. ....-d :.utIClOI 5. n~ Dott 

EstiFatlcn of Dyna~ic Rotor loads for the Rotor r~av 1985 

Syste'TIs Researcn Aircraft: HethOdalogy G. hrlor ... ~ O_utOOO' C<>c. 

Development and Validation FHF 

7 .... '_"1 8 l'trioml"'Q O<;otI.u_ IItC>Orl t'l) 

R. W. Ou Val and H. Bahrami - HI Work Unol ..... 

g .... ,or .... '" O< ...... ~_ N_ &ftC! Ao<l<"ru Job Order Ii 147130 

Advanced Rotorcraft Technolcgy. Inc. II Co"ltort or ~t "'" 

1804 Stlerl1n Road. SUite 210 tlAS2 -11686 

Hountaln VIew. CA 9~O43 13 T YII' 0' AIOOIt ...., ,..,00<1 Co .... .., 

11 5:.'''.0''''1 ~y N.".. ...., ... """,,. Contractor Report National Aeronautics and Space Administration 14 Soonw"'t ~ c.._ Ames Research Center. Moffett Field. CA 94035 RTOP # 505 42-51 

, s s..""'"""""'y NO'" 

Point of Contact: Technical Monitor. C. W. Acree. M.S. 237-3, 

NASA Ames Research Center, r'~offett Field, CA 9~035 

(415) 694-56~4 ot" F-~ 4('4-566! 

1& ,>t><·,act 

The Rotor Systems Research Aircraft uses load cells to isolate the 

rotor/transmissIon system fro~ the fuselage. A mathematical ~del relating applied rotor loads and Inertial loads of the rotor/transmiSSion sySt~ to 

the load cell response IS required to allow the load cells to be used to 

estimate rotor loads frcm fllght data. In this study, such a model IS derl~ed analytically by applYing a force-and-~o~ent balance to the isolated 

rotor/transmIsSion syste~. The r:lodel IS tested by cOTparl~~ ItS estimated 

values of applied rotor loads WIth ~easured values obtulned from a ground based shake test. DiscrepanCIes in the comparIson are used to Isolate 

sources of unmodeled external loads. ---

Once the structure of the mat!:el'1atI::al r.lOGel has been valIdated by co~parlson with Pxp~rI~ental data, the parameters must be IdentifIed. Sln~e the para~eters muy vary WIth fllght condItIon It IS deSirable to 

IdentIfy the pararnl!ters~dIrectly from the fllght data. A Haxlr.ium Llkelthood ---

Identification al~orithm IS derIved for thIS purpose and tested USlng a 

computer slrr.ulatlon of load cell data. ihe IdentlficatIcn IS found to coP\~r~e Within 10 sa~ples. The rapid ccnvergenc~ facllitates traCKIng ~ 

of ~lme varYIng parameters of-the load cell mJdel in flIght. - -

-

17 Key WOft!> IS~I":I l:y A~..""(,, I 19 O*ttrl~le'n 5tJ1.t-f'nt 

AeronautIcs. load Cell, DynamiC, CalIbratIon. Rotor Systems Research Unc1aSSlfieJ - .~ i Iml ted 

AIrcraft. Para~eter IdentifIcatIon Subject Categ.ur:,.. 05 

10 ~lty 0 .... ' lof' .... , .. I>"<1I 120 s.e.r.ty Ct...." (0' rt>o. pr,;:1 121 N~~' p~ '4:' Fnco' Unclassified UnclaSSIfIed 

-For uft b-t ,,, N.t~01\~ Ttc""~I' In.~""'ltl()t\ !:olrvtc:. SP"'l~,.l.j Vlr~nl' 221C1 
" -..---- -

NASA CONTRACTOR REPORT 177362 

tUASA- ca-117362) ESlIltA110U OF D!UAMIe NOS-310QS 

Boxon LOADS fen THE ECTon sYs!rn~ llESEABCB 

AIReD API: HEIllODOLOGY DEVEICE~ENl AND 

VAl.IDhl'IOU iAdvallced fotorcrdit Techuology, UncJa!! 

Inc.) 50 F lie A04/flP A01 CSCL Ole G.l;OS 214912 

EstimatIon of Dynamic Rotor Loads for the Rotor Syst~s Research Aircraft: Methodology Develo~~ent and Validation 

R. \I. Du· Val 

ft. Bahri!n i 

Cm>JTRACT rJAS2- 11588 

llJW 1985 

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NASA CONTRACTOR REPORT 177362 

Estimation of Dynanic Rotor loads for the Rotor 

Systens Research AI rcraft: r~ethodolcgy 

Develop~ent and Validation 

R. W. Du Val and M. Bahra~l 

Advanced Rotorcraft Technology. Inc. r'~ountain View, Cal tforma 

/lay. 1985 

Prepared for 

Ames Research Center 

unjer Contract NAS2-116SS 

(.1\ f1 ~\ ;<":"::l i'~\ It \\ ~l/J "W::C!.ryl I, f~ 'l.:..~ \.:':"~~.J "'\ 

Na1tcnal Aerof"a:.J:.cs and 

Space Aam.nt$trat·on 

Ames n~c:lrch Ccntt.'1' 

Uc'fett Flelo Cat,'crnIJ 94035 

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TABLE OF CO~TE=,"TS • 

S('ction P.lge 

Jut rudun IOn 

'\orllrnc!:lt II r{' :.! 

Ph<1~(' I DI'\t'iopml'nl and \dhdatlon of an -\nal~ticdl rnodrl of the HOI or 

Tran'rnl~lon Load (rll s~~tl"m 'i 

Backgrollnd 5 

Pre\ IOIl!l ~ ('thod~ 5 

Prnpo,,·d ~(ethod j 

Ohjt·rt 1\" and -\pproach 6 

Holllr Tr,mslIlls,lOn \Iod!'l 6 

Tt'~t ConditIOn ... . I • ~ 

1 

Error \nal~ 'II'! i i 

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Total Estimation Error i t 

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t S~5lernatic Errors In Data Collection 8 • f 

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l nmodelt'd :'tat·c ,\nli O~ namlc Effe-cts 9 I . 

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HanJom Error .. In S .. n~ori 9 t~ · ~ 

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L nmodel • ...! External Load~ 10 ~ 

~ 

IncorrN:t PMamet('r \'alues 10 

Rt.'5Ul:s 10 

Conclll~lon5 For Pha~r I II 

Concluding Prmar!-'s 1 ) 

SPf'Clfl-C C0!lclu~lon .. 1~ 

Recommendations For 1- urthl'r n .. ,,!.' urh I::! 

-------,,-------------------------
Phase II Identification of Rotor :Transmlsslon load Ct'll model parameters dlrt-ctl) 

from flight test data 1-1 

B~ckr,round 

Identification A"d CalibratIOn 

ObJectl\e And ApprooSch ... 

Dl'51gn Of The On-Line Callbr3tlOn Algorathm 

Approach. 

The Luad Cell ~fodel 

A D) namlC" ~fudel of -\pplred to.lds 

D!'~lgn of Tht' Kalman FIItN 

The \Iaxlmurn LIkelIhood ESlIm'ltiOn Algorithm 

Op('ratlonal Approach 

Simulallon Result., 

IdentificatIOn From Ambient \'Ibration .. 

((InclUSions For Phase II 

ConcludlOl; Remarks 

Sp!'clfic Conclusions 

Rt'CllfTlmendatlon5 For Further Research 

Reference':> 

Appt'ltdlX A Load Cell \It)dcl Dem atlUn 

Appendix B The errt'ct of Selisor :"\olse on Loa~ E~tlmatlon 

.\ppendlx C. CalibratIOn \lgortthm 

II 

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21 

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LIST OF TABLES AND FIGVRES 

Tablt' I . ComparISon of Actual EstimatIOn LeH'ls \\Ith PredIcted L!;'\els Bas!;'d on In~truml'nto1tlUn 

:\OIS(' 

FIgure I Loads acting on Isolated rotorltransml~510n 

Flgurf' 2 Load <:1'11 arrangement for rotor ,'transnllSSlOn 5: stem 

FlgUrl'" ~lal;nIlIJUl' of load c('11 transfer functIons for 800 Ib appllt'd lat!;'ral force 

Flgurl';; \I.lI;nll1Idl' of JO( rtldl 10dd tran~fer functions for 800 Ib applll'd laleral force 

Flgllrt' G ~tagnltude of X-d,(IS applll'd force estImation error for appllt·d lateral forces 

FISllrl' j ~!agnJtlldt, of Y-d,(IS appll"u force e<;timallOn error for applIed latt'ral forct':!l. 

Figure 8 . \tagnltllue of :-axls apphpd force estlmatloa t'Tror for apphed lateral forct's 

Figure 9 \tagnltud(' of x-axIs applied mom!'nl (>"tlmatlOfI error for applied lateral forces 

Figure 10 \Iagnlluu of y-a,(l< app/a(·d moment htllnatlon t'rror for applied lateral force:! 

FIgure 11 ~taSnJtude of .:-a\15 dpphed moment (~tlmatlon error for applted lateral forces 

f'lsure I:! Compafl50n of magnitudes of x-axIs foret' estimatIOn error (or applll'd x. y and z axIS 

loads 

Figure 13 C'omp,ul,",on of Illil!;nltllde of y a,\15 force estimatIOn ('nor for applied I y and.: aXIs loads 

Figure' 1-4 COrnpdriSOn of magnitude of :·axls force eSlilnatlO1l errur for applied x 'J and: aXIs Idad" 

Flgur(' I ') COlllpan~on of rni!.!jnltude of .r-axls moment e"tllllatlOn l'fror for applied x. y 'lnd :-axls 

loads. 

Flgurp 16 Comp,Hlson of maJ;nltlJde of y-axIs moment estimatIOn effor for applied :r, Y and: aXIs 

IOdd" 

flgun' I j ComparIson of magnltudl' of .:-axls moment estlmallon error applsed x. y and .:-axls loads 

Flgufe 18 Pha~l' lag of unmodelf'd ro:l moment for applsed lateral loads 

Figure 19 SchematIC o';>l;ram of on-line calIbration 

flgure:!O Identification of \erlle.!1 load cell angle 

Figure ~1 IdentificatIOn of lateralludd cell an~11' 

figure:!::! Comparison of .letual and ~stlmalt'd !I-a,\l~ hub load .. 

Figure ~3 Comparbon of actual and l· .. tlmaIE'd roIlIng momfnl hub loads 

Figure .\ 1 l'nd('formed load (('II gE'O'l1cLr) .... Ito gt'ncf<lllzed reactIOn force 

FigurE' .\:! Load tt'll t:eometr: with 1:l'nerahzed fractIOn force componf'nts 

III 

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INTRODUCTION 

The Rotor S)stems Research Aircraft (RSR.\) has a set of !e\en load cells connecting the main rotor 

transmisSion 10 the ru~elage Their purpose IS to make high accurac) measurements of the net rotor IOolds 

as resohed at tht' rotor hub from flight data 1 •. The use of these load cells to estimate applied rotor 

forces and moments at tl.f' hub reqUires an accurate mathematical expressIOn relating rotor load:!. inertial 

load5 and load cell readW9;" Both the stru~ture and parameter,; of thl~ model must be ~pl"cllitd Pre\ lOllS 

apprt)drhl"" hr-procc~swg r,round te~t data ha\(~ not YIPlded an acceptably accurate relationship for the 

ca~(' of high frt'C)uenc) d) flamlC IO<1ds 

The prc\ IOU<; <tppro,1chl'S ha\l:' utilized empirical technlquf'~ ba"ed on l'xpl'mm ntal data to obtain the 

de~lrt'd model The aprrodch pre~ellted 10 the lir'lt phase ofthl~ stud) I" to d('\clop the monel anal)lIcall) 

and \ alldate It from experHIlent41 data uSing modern lechntqllt~3 of mod!'1 ~trllcture dc-termination dnd 

parameter Identification TillS approach pro\ Ides a le\(~1 of ph~sJcal Inslrht that has be!.'n ml:>slng In 

t ht' purel) e"penmental approaches and tht' resulting anal) IIcal relallUnship pru\ Ide~ d umfied bdStS for -

utllt:nng both static and dynamiC caltbratlon data to \altdate dnd upgrade a common model 

The second phase of thiS stud) deH·lops and demonstrates a technique for Idenllf)lng the parameter 

values of the model dlrectl) from flight datd thl'reb) ehmm4tmg a costl) ground test procedure and 

pott'ntlally pro\ Idmg more accurate re~ults than could bt' obtalOt'd from ground based caltbralloll The 

capablhty to prO\ Ide ~ actl\ e sensor caltbratlOn~ In an operational en\ Ironment has potentIal appltcatlons. 

beyond till' RSR-\ IOdd cell problem 

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NOMENCLATURE 

Symhols 

II Input 

y ,. Olltput 

,\'~ . '\umber uf rotor blidt'i 

n, , Rolor rate (rad "l'(,) 

fi \-!'ctor of u applied hub luad'! 

ii ,Estimate'! nC appllf'd hub loads 

f, \'ector of i IOdd cell measurements 

f \t·ctor Qf 6 Inerllal loads 

e Set of unkno .... n parameter~ 

8. -\-pnorl e~t.matC' of parameters 

e Pust-callbratlon 1' .. t1molt(' of parameters 

Q Sl't of 11 Iliad ("I'll dt'formatJ(} : rtm;l('s 

Po.,t-ldentlficatlOn c~tlrnate of it 

It Lon;IIUdtnal rotor shaft tilt 

S 'fi.tn<;fornlolllon matrt'\( Cor load cells 

R 'fidn5formatlon matrtx for hub loads 

E{} ExpPC,tl'd \alllC' opC'rator 

;; Vector of T1lt'd"Uremcnt nOIse components 

Q. . Co-vanance of measul ement nOIse components 

r1 \'ector of process nOISI' components 

Q'I' Co-\'artance of proce~s nOI~l' components 

I Vector of states to be est.lmatl'd 

--= y .... ...-:.---- ----~--

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X Estimate of state vector 

F D) namlc couplsng matrl'l: for states 

<P • 'fransltlon matrlX for states 

n Index of measurt'ment update tImes 

~t TIme bct .... ('f·n rnell.~urement updates 

Jz P.1rtidl dcrs\au\f' of inertIal loads w r t ~tate.; 

J... Pdrtial derJ\all\e of inertIal loads w r t deformation angle<; 

P. . Co.varsancE' of stat.! ('stlmatton error 

\'. ~Idxllnsm Llkt·lshood ro~t functIon 

I\' \Ied~urement error \\Clghllng matrix 

.V, 'umber of sdmplcs used for !dentlficatlon 

eJ. Error In e,t Imate of inertIal loads 

eH Error In estsmate of applIed hub load~ 

..; . Frequcnn 

.-1.,B,C,D,E,F,G IndIvidual load cell outputs (FIg .\ 1) 

X,Y,Z,L .. \I,.\' Force and moment components (FIg .\:!) 

b POSItion of C G from shaft attach pOInt 

II POSItIon of hub from C G 

c 

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w 

d. 

0.0.1./ 

PosItIOn of C G from aft load cell dttach POlnb 

Lateral offset of fOf\\ard lateral load cell from crnterl'nc 

l.aterilloffs(·t of aft lateral load cell from CI'nt erllne 

Lateral dlHance bet\\l€'n \ erlledl load rell attach pOints 

Longltudsnal dIstance bt:'t\\een \€'rtlc,d load cell attach POInts 

LOdd cell deformatron angles about r.y and;: 3\€'S respectl\c1i 

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., 

r-- ~---- _ .. ------------------ -- .... 

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ii ... Lmear accrirratlon at C G 

I \Iomrm of ml'rtla about C G 

m EfTectJ\e mass of rotor I transmISSion engme system 

.\1 \fas<; matfl" 

p.q.r . RotatIonal rates about x. y. and:: axes 

Total applied engmt' and tall rotor drive t.)rqul'~ 

r... G) roscoplC coupling coelhclents 

Subscript'! 

.-l . .s.ccelerometer 

I.J Row and column indiCt'S rpspectl\t'I\ fur \ectors and matrices 

o Low frrquenc) cornpoOl'nt 

One re\ romp anent 

,\ .\'~ rev compunent 

m \Ieasurt'd data 

I.y.:: Component for x y.:: aXIs 

I InertIal load 

II Hub load 

C 'Total load 

T Load Cell 

a.b c.d.t:./.g Attach pOint of (>;:. .. n load cell 

Superscripts 

Pre-measurement estimate 

Post-measurement estImate 

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PHASE I: DEVELOPMENT AND VALIDATION OF AN ANALYTICAL 

MODEL OF THE ROTOR/TRANS~fISSIO);/LOAD-CELL SYSTE:VI 

Background 

Previous :vrethods 

The mltlal attempt to determme I he load cell respons£' to the applied rotor loads 1m ohed appl> Inij 

static loads at the hub and measuring the resultmg load cell respon .. e A least ~quares regressIOn approach 

\\a'l u~ed to Identlf} d coefficient matn" relatmg the seven load cell:. to tIll' SIX hub loads A SIX clement 

bias vector \\a5. also t'~tlmatf>d It \\as found 2: that the coeffiCient matrix \arled as a function of the 

applied multiple axi!:. load Thl!> indicates either a nonlinear dependenr(' of the load cell respon~e to the 

applied rotor loads or an unmodcled mdependent variable In the regrc::,slon 

The next calibration was a ground bast'd shake test In which a pair of inerllnl shakers were mounted on 

the rolor hub to appl} d) namlc loads at spf'cliled amplitudes and fr('que'lclcs '3, During thiS c",permient. 

the RSRA \\as suspended from the hub, so the static loadtng \\as the same for all tests As a result. the 

nonlinear \'artatlon of the relationship with static loads, as obsen ed :n the statiC test. should not have 

been present In the sha"e test The Force Determmation ~Iethod .J \\as utilized to e~t1mate applied 

rotor loads from a "ariel) of sensors around the aircraft, ThiS method fir"t Identifies the transfer functions 

between the applied rotor load::. dnd sensors at vartous pOints on the aircraft and then Identifies the applied 

rotor load from a least squares fit to the transfer functIOns and measured ~ensor responses The results 

\\ere unacceptable because the Identified transfer functions \arted \\Ith th(' magllltude of the applied load, 

hence they could not be used to estimate the applted load Without some c> ten~m(' calibration procedure 

,\g;am. th(' .. aT/at ion With applied rotor load indicate!> either a nonltnear depcnJence or an unmodeled load 

~ource 

Propos(ld Method 

Smce the sensors utlltzed 10 the Force Determination \I('thod Included numerous accelerometers and 

stram guages mounted on the fuselage, transfer functions of the,e sensors \\LiI be affected by any nonltnear 

d~ namlc .beha\ lor III the fuselag('. ThiS effect complicates the u~\'! of sensors on the fuselage to determme 

applied rotor loads The RSRA was deSIgned to use load cells to Isolate applied loads from different 

sources, such as the mam rotor, tall rotor, engine and wings The proposed approach takes ad\antage 

of tIllS concept by treating the rotor transmiSSIOn as an Isolated system With externally applted loads 

from the load ("ells and the rotor (FIg 1) The appllcd rotor loads are then measured from a forcf' and 

mom('nt balanc(' usmg measured load cell loads (T) dnd inertial loads (I) as demed from transmission 

accleratlOn me!!Surernents In order to utilize thIS approach a model IS reqlllred that relates applH'd 

e,<terna/loads on the rotor.'transrnls~lOn 5) stem to the resuitlng forces and moment" at the center of mas3 

of the S) stem ThiS modd IS dern ed analytlcaJl} from ph} sical pnoclples uSing th£' kno\\ n geometry C'f 

the rotOT! transrnleSlon s} stem Potentially uncertain parameters m tills model Me e\pltcltl~ represented 

to proVide the capability to caltbrate the model 

The ad\antages of thiS approach predommantl) arise from the ph\ sical mSlght obtamed In IlStn3 all 

analytically clem ('d model \'v Ith such a model, sen31t1\ It) ana!) 51" and phYSical Judgement can be !lopd 

to select the most appropriate set of a\ atlablf' paramet£'rs for rallbrallon Consequently fe\\er parameters 

need be calibrated than when no ph)slcal mSlght I~ us£'d In addition. thl' parameter" to hI' rallbrated 

nO\\ Ita\e a phySical mterpretatlOn so the \altdJt) of the ("althratlon r('sults rna, bl' ass('::~ed Til!' model 

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should initially be derl\ed to be as simple as po~slble If It can not adequatel) cxplaln the observed 

experimental behavIOr v.lth a ph~ 51Call) reasonable set of parameter \ alues, It rna) be expanded to Include 

addltlonal effects as reqUired It IS Important .u.at all major effects be Identified and Incorporated Into tht' 

model before caltbratlon of the parameters IS' attempted, or the parameter \alues v.11I compensate for the 

un mode/cd effects as best they can and achle\e ph)slcally unrealistiC \alues In the process 

Objl"ctivp and Appro<\cb 

The obJectlvc of thl~ pha"c of the ~tud} IS to dem'c a ~Imple d) n<lIllIC model of the Isolated rotOf/ transmission S} "tern and tl'<;t Its accurac\ With expenmental data The approach IS to first derl\t' 

a 'Hmple model of the rotor transml!>Slon s)"tem, treating It as a hnt'ar, rigid Isolated body Knov.n or 

assumed \alu('s art' u ... ed for all parameters of the dcrl\ed model The modd IS then applted to test data 

tel drtt'rmlne It'> accur.lC} If It appearc; that calibration can flJrther ImprOH' tlit' accuracy. the appropriate 

parameter q·t \\111 or st'lectt'd and calibrated If the accuracy appears lrmlted b) an unmodeled effect, the 

model Will bt' expanded to e\aluatc potential sources of the unmodeled effect 

Rotor /Transmisssion ~1odel 

The arrangement of the load cells belov. the rotor transml!>Slon system IS shown In Fig 2 A detailed 

deSCription of tille; s~!-tcm, mrludlng all inertia contributions b given In Reference 2 There arc seven load 

ceJls, four of thefTI mounted \l'rtJcall) at the corners of the transmiSSIOn mounting plate, lYoo mounted 

laterall~ along the fore and aft edges of the mounting pldte ,tnd one mounted longitudinally at the for-

\\;ud edge of the mountmg plate -\n mertlal load \ector, j, IS located at the center of gra\lty of the 

rotort transmiSSion system and the applied rotor load Hctor (II) IS located at the rotor hub (Fig. 3) The 

rotor hub IS located at the entl of the roto:- shaft \\ hlch IS tilted fon\ ard at an angle of 2 degrees The 

load cells are connected to the transmiSSion and the fu~elage b\ sphcrlcal bearings 

The proposed approach I!> to estimate the applied hub loads from a force and moment balanc:e of the 

external loads and tPc mertlal loads In order to '1ccomphsh thiS all extern all> applied loads must be 

transformed to the center of mass. \\here the mprtlalload~ act B) treatmg the rotor transmission as an 

Isoldted S} stem, the load cdl force~ are considered a me<lsured, C\I email) appl" d load on the system .\ 

6xj matnx (S) IS derived that transforms the \ector of "'e\cn load cdr IOdds (/) at their attach pomls 

to a ~et of 6 load components at the center of mass A 6\6 matn\ (R) I~ dem ~d that transforms rhe 

\ector of applied rotor loads (1/) at the hub to the center of ma:o~ An mertlal load \ector (i) at the 

c:enter of rna"" IS den\ed from measured accelerations and assumed Inertial parameters. l"smg the derl\ed 

matrices,a force and moment balance at the center of mass re~lllt" m a set of :;1\ Simultaneous equatIOns 

\\ hlch ma) be \\ rllt en 10 matrix form as 

i-sf-Rfi=o (II) 

A detailed dE'~ctJPtlUn of the~e \e"tors and matrlc:es are gl\en 111 -\ppelldl\ \ 

The a~sumptlons used 10 deriVIng the matrices and Eq 1 1 arc that the rotor/ transmiSSion systtm 

IS a rigid body and that there IS no fnctlOn 1'1 th!' load cell beanngs These assumptIOns \\ere made to 

slmphf} the IlIltl.11 approdch Both non-rigid bnd) eff(>('ls and frictIOn In the load cell bearings could be 

added to the model If th:l:t apP('ars to be reqlllred 10 order to match the experimental data 

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The transformation matrices Sand R and the inertial load vector] \\l.'Te all den\ed to expliCitly 

contain all potcntlall;. uncertain parameters of the s) stem so tha( any subset of parameters can be selected 

for calibratIOn, as requIred to Improve the model accuracy The parameters explicit: defined In the model 

art' 

a) All distances and angles requIred to define the resultant moment arm from the center of masS to load 

applicatIOn pOints 

b) All angles and magnitudes reqUired to define IOdd cell load component<; acting on the rotor! tran~­

rnl~slOn system 

c) All mass properties reqUired to dE'lermme inertial load~ from mfM~url'd rates and acceleratIOns 

Test Conditione; 

Ha\'lng derl\ed a model, the next step is to evaluate It With e'<perlmental data Both static and 

dynamiC ground test data are a\aJiable The dynamiC data generatE'd by the shake test were choo;en since 

It would provide a more ngorous test of the model structure than the static test data The dynamiC 

data are not, howe\Cr. sUitable for testing the calibratIOn procedure ThiS IS because the same static load 

condition eXists (or all d:namlc tests and parameter \anatlons are mostl, dependent on variatIOn'! In the 

static loading Once the model structure h~ been \alidated With the dynamiC data the model :an be 

app\ted to the static test data to evaluate the cahbratlOn procedure 

The test data set 5€'lected was a frequency <;weep from 15 to 185Hz In the applied rotor load. ThiS 

frequency range \ .. as cho"'en because It contaln~ the 'vb re\' frt>quenq, and IdentificatIOn of dpplJed rotor 

loads at tillS frequency I" of speCial Interest Transfer function data \\ere generated from the raw test data. 

by a harmOniC anal) zer for four le\eb of applied load In each of tht' three axes Eq 1 1 was then IIsed to 

generate the applied load co;tlmale from the (ransfer fUllctlO1i data Eq 1 1 ..... as processed \\Ith the \aIUC5 

of all of tbe model parawetcrs set to their de~lgn speCificatIOns 

Error Analysis 

Total Estimation Er.ror 

In a controlll'd ground lest enVironment, the actual \alues of the applied rotor loads art> kno\\n 50 

the total erro. HI the estlmat€' IS readily obtained GI\en the vector of measured load cell readings T:, 

and the \('ctor of mertlal loads derl\,ed from accelerometer measurements f M • Eq lIma) be used to 

estlmatl' the applied rotor loads as 

(1 2) 

The total error In the €'<;tlmate IS obtained by "ubt ractlng the I.no\\ n \alueo; oi applied rotor load from 

the estimate of Eq 1 2 to get 

ell = if - if = -R-I J:" -sf ..... - if 

I - - - ell = - R - J ... - ST", - R /l (1 3) 

i 

-" --... --...-~-----------------~ -..... 

• 

__ J ----'" 
./ 

The avaIlable shake test data wa:. m the form of transfer functions -that had been generated from trn. 

ra\\ data b) a harmonic anahze, In order to utilize thIs data \\Ith the proposed modelll \\as nece~5ar: to 

transform the model to the frequenq domain and \\ rltc It In ter;ns of thc transfer functIOns Transforming 

Eq 1 3 to the freq'Jency domam gl\ es. 

(1 4) 

Ifonl) a Y-dXIS rOlor load. H,,(...,), IS applied, Eq 1 .. rna} be> \\rtllen In terms of the transfer [unctIons 

as 

_ ( ) ___ I (I",( ... )) _ c::(i",( ... )) _ ( ) 

tJl..., - R H y

(...,) ~ H y(...,) R fly ..., (1 5) 

\\ here ~::;1 and ~ ::: ~ are \ ectors of transfer functlOn~ of the inertIal and load cellload~ wIth respect to the 

y-axIs rotor load Eq 1 5 \\as used to e\aluate the total error In the estimate usmg the avatlable transfer 

functIOn data for y-axIs e\cltatlon Corresponding (.\r>re~sions \\ere used to e\aluat r ~he e'itlmation error 

for :r and;: :tXIS eXCItatIOns. The elTect of applied moment'> at thE' rotor hub \\dS not e\aluated in thIS 

,umerous potential sources of error are present In thiS s)stem The c\aluatlOn procedure was to 

eXdmln£, the> total error and attempt to categorize It Into the potenllal sources Once Id,mtlfied, tl,e 

source:. would be modeled and mcluded HI the system \to:.t of the> f'fror \\ III probabl) bl' atttnbutable to 

one of fi\e sources 

1) Systematic errors In dqta col/f'CIIOn 

2) l:nmodeled st,lllC dnd d) n<lIllIC E'ffE'cts 

3) Cnmodelf'd e'(tcrlldl loads 

4) Random errors In sen~ors 

5) Incorrect pdrdlllcter \alues 

Systematic Errors iu Data Collection 

One> source of error 15 the use of transfer [unctIOn data Since thiS IS tredted as raw data In thiS ~tud: 

any errors In the IdentIficatIOn of the transfer functlon<; \\ould propagate through the proposed approdrh 

The recorded time domain data should be reprocessed b) t he harmOniC anal)Zer to prOVide on I) FOUrier 

transformed data. not transfer functIons 

Angular accelerometers and rate g) ras \\pre not a\adable on the rotor transmiSSion 5) stem for the 

shake test Since there IS no \\a\ to obtain an~ular acceleration ddta. the approdch tahcn 15 to a~sum{' 

ItS \alues are negligible and ~ee ho\\ good the e ... tlmdte~ ilre \>vllh thiS a'i,;umptlon Some JustificatIon 

for thiS assumption comrs from comparing the response of t\\O linear accelerometerc; that ,1re mounted 

on the transmISSIOn with a t\\O foot \ertlcal dl"pl:!.C"prnt'nt bl'l\\een ti!Pnl The (hfierence III the y-axIs 

components dl\ Ided b) the \ertlcallsplacl'ment should gl\ e the roll a\15 anguldr arceleratlOn The 3\eralje 

8 

,..--------------------------' - -__ .. ~ ~:J:.i:" 

. -"":./ ..,...~~ .. ------,-------~~- --- .-

r 

I 

1 

. \ 

... 
\ 

\ 

\ 

. -, 

,alUl' ofth.s demed roll acct'lcr:ltlon O\rr th .. freqll!'nc} r.w!;!' \\a~ found to be 09 rad H'C ~ec, ~lIppOttlrll; 

the 10" angular acceleration a!'iumptlOn Tht' deuH-d IlnSlIll1T Ilccelt'Tatlon datli "35 'ot UbN! \\Ilh thr 

modrl brcllu~(' the errors If' thr hnt'llr acceleromrter5 are ~uch that the accurac) of the drn\ed .1nl;ul.'U 

accclt'rMlon 1'1 f3 rad, ~ec ,te, Tht' derl\cd \alue-1 lItt' thl'refof~ 10 tht' nOI'!!c leHI The lInS;UlI.r accderallon 

afT"ct!' thr translatlonalrquatlOns since the linear acccierom!"lH IS not mo,unt~d al the center of ma~s and 

",III thert-foft' bl' affectt'd b} Ilngular acceleration, The mom('nt t'quatlons are aff"cted since tht' angular 

InNllal IO'ld~ art' dep .. ndent on the anqular llC'celrr:l.ton« Tilt' f'rror5 a~~ocI3ted \\ Ilh ne~lectJllg a O~ 

rad ~M: !>('c roll anr,ular accelcr.ltlOll drt' I:! Ih~ III the IJ forre' and :!7 ft Ibs In th(' rolllll!; momc'nl ~o lh., 

a!>~umrtlOn appear" JU5tll1('d 

UJUIIod"iru Static And Dyuamic' Efff'cts 

Tht' model derl\t'\l for thl~ stud} \l.0tS ~ept dehberatel~ ~1I!lp!l' to f.\r:llt~te thp 'lnal~5Is It can be 

t'(pandt'd If nC'Ccssar) to acrount for the obst't\('d etror Lnmodrlt'd clrfft .. "'Ith plltentlall) ~Ignlficant 

I.npact on the model snciud(' fnctlon 10 the load cell bearings 1l~,(lblllt) In the rotor' transnll'\SlOn s~stem 

and nonhne:mt} In tht> d~ nllmlt r("'pon~~ to applied loadi The n:lture or lht' error sIgnal should 5I1ggt':.t 

\\III(h of tht'~e' ('!f('cls are prc:~t'nt Friction and de:\dband5 \\111 bl' rhardctef\2t'd by hY'Itt're:,ls In the 

rf"pnn<e TIm dr('ct I~ morl' t!'ddd) ob'll'r\ cd 10 !>tatlc data th:m In d) namlc dat:!. Flexlblhty \\111 sho .... 

III' a" a r("onance at some frrquenc~ and "III rrsult In a phase and amplltlldE' shift bel"'l'en the mput and 

output ~I!i;nals :-onllflea!' d) namlCS \1.111 bt' rcadll) detectf>d b) 11 frt·quency shift bet\'~n the IllPllt and 

output da.ta Static nonllne:lntlt's r<'"-ult In parameter \11t1atlons and art' correct I'd b) calibration r~tht'r 

th.ul b) t"pandulI; t he model 

Random Errors In SC'l1sors 

Butlt 1('.1d cells and .. ccel~r()rneter~ hd\(' nJellSurer:-t'nt 1l01~,' ,dllch prodllct's a lo\\er bound on the 

dccurac: of the ;tpph .. d !oMI t·~tsrtnll·~ It IS pO~"'lbll' to obtam accuraCI(,,, In .e'(j:~'''~.ullS 1I0llt but thIS 

requl~Cs the uee of stall&;l!c:d prvc(,~~lfIg tt'chnl'llle:. such a~ KahHan Flltf'rms and £moothmg ThIS effect 

c:m not be reduct'<:! b~ 1m} modlficatlons to the modd Th(' drt et .If arctl.'romt'tl'r dnd load cell nOI<e 

on the applied rotor load t'stunat(' I';; dem('d In -\ppl'ndl\ Band u<t·d to generatt' tLl' errors gJ\'en In 

T.ll-le J Th:s table comp.!re:; the ('stlmatlon ('rror~ In paeh ,1\:-' for four I('\('b of appli('d load \l.ith the 

predlCtrd ~t'and.lfd den\ allOn of t 1\(' prror ba!>~d on In<;trunwntat I()n nm'" Th"'e nllmbers \l.ere based on 

the 3:>5Ulllptlon of Indrpendent random ('nors far I'.ich ~t'n ... or \\ It h .icrur:lCIt''; of I percf'ut of fuil sc,lt' for 

thl' acceleromet~rs "nd 0 I percent of full scah· for the lo .. d rdb The tOfalll"trulllt"ntntlOn error IS sho\\n 

along \\Ith thl' componellt~ dduf' to load cdl MId arceleromr(rr nOI~(' 

E~tlJnlltl()n errer 

Lua..! :!OOlb ·IOOIb GCOlb 

X ~-t iG 1-" ." 

'\ 10 ,,- _<l 35 

Z 1 :.: I, :':3 

L )~I:! lulo 1--'> , 1-

\\ ,'--, 17 ~-t 

~ Ii 1 ,() 173 

LO.ld cl'li full ~cale :.: :':5,000 Ib , t'rr -: 1 "c 

\C'ct'!eromt"t"r full ~C;:l!e 7 G '" , Hr ..-:; 1 c-;: 

\I~s:= 11(10 Ib 

'fable I 

In"t rUrTlcnl .. tlon nOI~e 

SOOlb Total Load Ct,lIs Ac(.e1 

211 ::09 :!5 308 

913 31:! :\j 305 

:!9 :~ I :! :iO :ms 

:.!Iij -I-tG li9 -lOS 

,Ii 136 H, ·n~ 

:!I-t 1sI 131 51 

9 
- - -~:"-~

1 

I 

UnUloul!h'd External Loads 

The demrd model .... 111 b<.' In error If 3n~ ('xtt'rn:111) apph('d lo~d IS not ancluded If the le\l·ls art' 10\\. 

then thIs can ~e the most dIfficult source of t'rror to Idcntlf~ ThiS IS bI.'C:1u!'c It can take on \ IHualh an~ 

characterIStics and \10111 blend an \Iolth other error sourcrs. Thl' onl\ pO!~ll)lht) for det('ctlng thIS t) pe of 

trror IS If II 15 ~ufficlentl) large th:1l It C:1n not bl.' 101;Icall\ explaIned b~ an) of till' ()th(';, error sou ret's. 

On«(' the pn',!'n('(' of an unmudeled load it; suspectfd. thl' error data m:n IC>~IH In I~olatang Its source, hut 

d thllrlluJ;h l''f.ununatlon of tite l{><it condtlCJns I:, usualh rt'qulfcd to r"~()he thl:' l'ITl.'ct 

Iu('orrl'C't Para llI!'t C'r Va lUI'S 

Once tht' modl·1 ~trurtllrl' ha. .. bct'n \altdat('d to tht' ful1(,>( extent pO,\~lble. the rem:l.Ining error~ shoilid 

he d\Jl' only to l~rrl'Ct \altH'S for the parameters \t thl~ pOint calibration ma) hl' apphed to reduce 

thl~ error !o!ource. If cahbratlon 1'0 attemptt'd before lht' model "trueturr 1'1 adequately delermlO(·d , the 

parameter:. \\111 take on .... hntt'\f'r \all1e5 are requlrt'd to cornpen!<llte for th!" model structure error'! ThiS 

.y,lll rt'!ult In ph~slcally Imrealssttc values for the param('t('r~ and ",uld, 10 fact, be a test for whether lhl' 

model "{Tucture I" accurate 

Tht' comentlonal approach to calibration ha .. b('('n to aprly lea.,t 'quart's nHl1llm:at!on of the enol' 

.... lIb r('~pect to thl' p:ualllf'tC'r<; to be callhrated ~m(1' tht" paramel('rs :ue no\lo Imb\'dd(:d HI a model, thc 

least !'-quareo; rnlf11ml2atlOII of the ~rror must be done ~ubJt'ct to lhe constraint that the model t'quatlons 

arE' s~lt1~fied Thl" IS r",ff'f('d to as a comtramed It·,i.<t ~'l'lare~ approach 5, and the al~onthm IS deri\cd In 

.\ppendlx C for thp COIHramt of Eq 1 1 

"lfIce the a\ IlIlabll! (hla I~ In the frequenq domam. the callbral\;)n must bl.' perfow'ed In tile frio'-

OUCOC) domain Tfll~ I~ cctual1~ an ad\ ant.lSI' .,Inee the frequt:'nc) dom<iln triln.,formatlOn has conc(:ntrated 

tnfCrmallon for the reqUired frequ!'nc)' ranJ:t' IIlto f('\\er data points than required for a tune dom.111l repre-

~entallon Cal:bratton ma) then bt' pl'ffortlled "'lth 1t'55 data to the frequ~nc} domal:!. Th~ lel<:t !>quares 

mlnlmlZ;ltlon m.l: bE' applJed ;0 fr,'qu(,llc~ domain d:lla In the .. arne \\a~ as It I" ~prhcd to tIme dom.un 

da~a '0 The onl) modification IS that the ,!at:: I:; org:lnlzpd \\Illl the fl'al and Imagtn;u) parts stacked 

end to end rather than U~lng thl" data HI comple, form Till:' tn~ure<; that the Idc:111fied paral11l!ter \alllt'S 

.... 111 not be compll" 

Eq 1 3 \oa" apphl'o to tran5fer functIon data for the load c('1I<; an.1 the m::lIn rotor gear box accelerometers to generate the l'stlm'lllllfl error of the deTl\('u model \II the freq\l(,i1c~ dOlllll1l The \alues u~eJ In 

thl' de~lgn speCIfication for thl! I"('oml'tr~ and mi1-<;s prop"rtll':l \\!'re u~i'd fur all paralll(·ters In the model 

Fig's -t and 5 ~ho\\ lllt· magnttude of the transfer functIOn d<1ta for the i IO:1d lells and 3 accel('rnm"lf'r~ 

for an !leo lb y-,l'>;I' (!\r\tatlcfl The 10(11tlon of lhe load n·lb ma\ be s'>t'n b\ fintilllt:; the corr('spondtngl~ 

lab'led loa(! cell In rIg ~ The dcceierometer tran5fcr functions h.l\e heen multiplied b) the ~) stem mass 

10 produCt· Inertial load~ The ~trcng cOllplin!; In the :,~ <.t('10 IS :l.pparent from the largl' \('rtleal load 

cell \aluP$ for a y-<1\l'< "xCitatlon The strong correi:ltwll In the ;:-axls 1<; partlclIll'.rh apparent frolll the 

5udden drop 1:1 :-.1\1';; atCelerallOn ~t the same frequenc\ (IG () Hz) \\herc t\\O \ertlcalload cells (\ and 

B) sllddenl~ a"'~lImE' ('qual and 0rPO'lte \alut's 

Fillun 5 () through 11 shol\ the '51" a,\l!' e,tlmatlOn crrur~ rar !I-axIs e'cltatlon at :!OO, 400, 600 an.1 800 

pound 1('\ "Is to dl'l1\ol1~tratt' the diect of load le\Cl on e..,urn ilion trror. The fllrht \ alll£'; of the applud t h(' 

apphed rotor load at tbe Sb/rev freqllenC\' -n,)uld filII \\ltilln till" nn~t' rlgurr .. 12 through Ii .. ho\\. the 

10 

I 

" 

\ :"- , 

, 
51' aXIs estimation errors for r. y. and: aXIs excitatIOn at the -too pound le\(!1 to dl'mor:!:trate th~ effect 

of excitatIon aXIs on the e'ltlmatlon error The errors plotted 10 (Figures 6 throush 17) .... er .. obt:unrd 

from Eq 1 3 lI!lOg nommal pa~ameter \al\l!,~. Table I sho\\-, the llH'rage !'rror Q\er the- frequenc~ range 

as compared to the accuracy limIt set b~ the Instrumentation nOI~e With the exception of applied rolling 

moment due to Y-I1't15 excitation ,md r-axls force due to x-axIs excitation. the a\l'rage eno:, 5ho .... n 10 

Table I and the frequenq plOIS comparing tht' three axes of exCItation sho .... n 10 Flg's 1 :!-1 j. demonstrate 

that the ngld bod~ modrl of the I~olated rotor transml<;~lon 5~.;tem product'$ applied load .-<>tlmates \\Ith 

('tror le\eh: comparable to the instrumentation nOIse le\el 

The pronounced roll moment error from !I-axIs eXCitatIon (FI~ 1,») I" too sreat to be explaaned b\ 

parameter t'rror~. s"n~or prrors or angular acccleratloll~ Thl' denton"trated lack of signtficant anl.':uldr 

drrl'lf'ratIOn'l .. bout tlw roll axl't Indicate.;; that It is not duE' to a non-rll;ld bodv eff(Oct It can also bE' seen 

thdt there 15 no frt'qul'nc: ~hlft (the hump <H Hi 6 Hz m the error slI;nal mMche .. tht' hump 10 the load 

cell d.1ta at that frequenc)) so the error l!o not a nonltnenr functIon of the modeled \artable5 Tlae only 

rernammg e'<pldnatlOn is that the error 1'1 due to an unmodt'led extend load \\-nlt predominant eff~t m 

tht' roll aXIs .J. potl'nual candIdate for the source of thiS load IS the drt\e tram since It \\-ould affect only 

thE' roll moment \lea~url'm'·nt .. of the shaft torqul'S from the engme and the t:1I1 rotor are rt'qUire-d to 

\f.'rlf) t hiS and could bt' usrd to compensate for thiS effect. 

The unknov. n roll torque ma) be rtconstructt>d from Eq 1 3 to gl\ e furtht'r inSight mto Its source 

FIg 18 IS a plot of the phast' of tnt' unmodded torque rt'latl\e to the applied eXCitation as a function of 

fr .. quenc) for four 1(0\('15 of applIed ~-aXI5 load It can hl' .,~n that the ph~e IS 5ufficll'ntl) close to lSO 

drgret'!> to Indicate that the torquE' IS due to ela~tlc rather than dampmg effects ThiS IS consistent .... Ith 

tbe POSSlblht: of a blndtng m tht' drt\l' train rotatIon 

ThE' pronounced x-axIs force error from .r-a\.b excitation {Fig I:;!} 1,\ also too great to be explained 

b~ pdrameler errors. sen,>or errors or anguJ.~r acceleratIOns and no frequenc! shift IS present bclv.een 

mput :lOd output. so the tffect IS not due to a nonhne:mt! Apln tht· only remaining (',planatIOn IS an 

unmade/cd external hl.1d :\ potential candIdate for th1:> load source IS pOSSible binding of a tr3nslatlOn."11 

coupllnl'; on the drt\l' tram that IS deSIgned to pre\ent tran!<!l"'ISSIOn of X-.lXIS loads from the rotor though 

the drJ\{' tram to th!' fu~c!agc ~Ieasurements of tht' tensIOn 10 the urt\e tr:l.In shdt \\Ill bl' requlf~d to 

confirm thl~ and to proprrl!' account for' tillS effect In the mod"i 

J 

Out of eightet'll estimates (SIX axes of esllmatt'~ for each of three axes of applied loads) nominal ~ 

parameter \alui's producE'd (·"timatlOn errors wtthln thl' instrumentation nOI.,e Ic\d for all but tVoo c:'.Ses 

Roll moment due to y-a)'ls eXCitation and x-axIs fcrce due to r-axls .~xcltatlOn had errors tot) gre:lt to 

be expl:lIm'c! b! parametu uncertainties. so cahb~atlOn IS not needed to IlnprO\e the accuracy In the 

other ~,J'(teo:u ca!>es and .... ould not hdp In thc~e t\\O case:; The source of these unkno\. n loads must be 

dl·termlfll'tl and either bE' m!'asllred or model I'd as a functIOn of th(' mei\~urcd loads before the model cnn 

bE' u!>l'd to e~tlmate roll moments and longltutiludl f{lrCeS 

ConcinsimlS for Pba'le I 

Concluding Rem'lrl.s 

-\n an:l.l) tlcaH) den\ ('d model with nomll1:l1 paramNt'r \ alues ha" heen u~('d to e'ltlmate applied rotor 

loads from measured load cl'lI and accelcroml't('r d.lt:!. ThIS approach has also pro\ Ided n check on the 

con"lstrnq of the llIeasureJ tnput output ~ata TIle prl"~enrc of an IInmodeled roll torque dill' to Y-2.Wi 

apphed loads and an unmodE'led x·force due to apphetl r·a'ls I03d~ h:Hf' bel'n detected Hc('On~tructlOn of 

II 
______ ........... _... ....... ..-.:~ .. "*-" .. i;... ___ .. ..... "'Ito. _ ..... ,.. ... 

thl' unmodeled loads shO\\s that the~ ha\e a pha!c las clost' to 180 degrees from the applied load and are 

therefore prerlomlOantl} f'iastlc 10 nature. Rotational and tran,lattonal blndang In the drtH tr31n appE'3rS 

to be the most Itkel~ sourcl' of these unmodeled loads Shaft torsion and tensIOn measurements" III be 

required to confirm thiS and to Incorporate these effects In the model 

Specific Conduslo1l9 

Tht' follm\llig concluslOno; arc drawn from th.s <;lull} 

-\n anal~ llralh' uenH·d linear. rigid bod~ modd of Ih" Isolated rotor: transmiSSIOn system \\lth 

nominal parameter valu.!~ provides estimates \\Ith errors \\Ithan the level of the s('n~or nOise for all cases 

bUl roll torque dul' to appllt'd !I-a'Wi load and x-forn· due lo applaeu x-a\IS load 

Xon-ngld bod) effect'i or nonlmt'ar beha\ lor Coln not ('xpldtn tll!~ largl' errori' pre~enl 10 th~e t\\oo rn~t'5 

Thl' t'rror must be dul' to an e'(t('rnall) apphtt..!. unmouelt·d load The prrsl'ncl' of such an unmodelt>d load 

cOllld also acrount (or thE' apparent nonitnt'dfltiE.'S d('tee-tt'd b} tnt' pre\lousl} alternptE'd tran .. fl'r function 

ult'nttficallon If thNe are Ilnmodelt>d load sources till' tran!>fer functIOns \\111 a~sunlt· \\ohatevt'r \ldu(s 

art' requlft'd 10 reproduce tht' lo.1d cell reading .. gl\t-n tll(' rneasUfE'd IOdd.. If tht' unmodf'l('d load .. are 

rl<l~tlc. and thert'fore drpendent on tht> applied load~ thell dlnetf'nt trlnsfer functIOn .. \\111 be Identllird 

dt dlffer('nt apphl'd load:. ThiS 1:'< pnclselv tht> .·tf('ct ob~ef\ I'd an lI.lenllf) mg transffr functions from th(' 

!ohake lbt data 

Rt'constructton of the unmouE'led load from the error slhn,1 indicates It IS predommantl~ elastiC 

In nature If ro, II \\as probabl) prr~ent In [ht' ~t.lllc e-ahbr:J.tlon and could account for the apparent 

nonllllearltll~s detcctl'd In the !otatlc calibration -\pph In;;; a rl'grl'~SlOn fit to mput!output data III till' 

pre~encc of an unknO\\n Input \\111 f(><;ult In tht' IdE'ntlfird co('ffiCients tahng on ",helevcr \'a'ur~ are 

r~ulrt'J to eccount for the lIn~,co\\n input. \s III the ~haJ..e It'sl cao:;e, If the unl.nown loads are eldstlc. 

the) \\ III be dependrnt en the applied loads and thr Idpnllfitti coefficlCnts "III \ar)' \\ Ith applied load 

TIllS ene'cr "as ob"ened In Identlfymg a model from the static tE'st data 

CahbratlOn of the parameters IS unnrct>Ssar) <it thiS pomt since the deSign values appear to gl\e 

~atlo;factor) re~ults \\ hcn the external loads are all proper!> mcluded. 

R('commpndation For Further R{'~(>arch 

Th(' follo\\ mg rf'commendatlons are made 

If thf' unmodeled loads are elastIC. rather than uampmg. a$ thf' phas<.· lag appedrs to suggest, then 

the~l' loads \\ere probabl) pre"aflt In the static test and could account for lhe apparrnt nonllneanty In that 

calibration The .tnahtlcal model should be applied to tht' ("\IS1IIIg ';[<ltIC calibratIOn data to determine If 

the same ufllnodelpd 10dds appedr to have been prl'sent 

Th(' upcommg static calibratIOn .,hould be ~uffiClent I) ln~trllrl1l nled 10 detect the suspected '>ollre-r<; 

of unmodded load., from the drh e tram dnd the anah tic,!l model should be usrd to proceSs the data 

\\ Ith sumClentl~ fasl turn-around that ne'" Ii Identified !!fiects can be mvc:.llgatt'd befarl' th<: lests arc 

completeu 

I:! 

·1 

I 

I 
The approach should be applied to static test data to ,,\aluatl." thp calibratIon techmque :\ommal 

paramt'ter \illues have dom,' surprisingly \\e11 10 allo\\ inS accurat" hub load esttmatlon for the spt'clfk 

'<- static load condition present In the d~ namlc data :,tatlc te!St e'penence suggests that thIs will not be 

• true under all statiC load condItIOns Once the model <;tructure IS full~ \ahdat~ It should be apphed to 

statIc t\!.~ data to detewlIne .... hlch parameters to cahbrate and to determine the range of \artatlOn of the 

parameters ~hrough calibration 

13 

--"--- ---~----------
.. 

" --

PHASE II: IDENTIFICATIO~ OF ROTOR/TRA:-;SMISSIO~/LOAD CELL 

MODEL PARA:"1ETERS DIRECTLY FRO~l rLIGHT TEST DATA 

Background 

Overview 

Pha~E' I of thl~ study conct'ntraled on the deH'lopnH'nl of an- anal) IIcal mooel of the rolor. transml~sion load cell S) ~tem and on the \ahdatlon of thaI model from ground le~t dal'1 It \\as determlO(,d 

that under .,ome loadmg cOlldltHlns. unmodeled e'«erodl loads arc present and tht'~e loads must be acCflunt<d for bl'forc tho model can be further Impro\cd Once thl:> I:> accornph~hed. the nett ... tcp \\111 be 

10 ,,,hddte the model pardmeters ov('r a range of operalln~ conditIOns Thf' a\allable data In Phase I 

lended to conhrlll thaI param('(er values obtained from the design spE'clfications .... e~e adequate but thiS 

.,('t of dat a Included onl} one static load condition and the parameters of the model arc more h!..el} to be 

depf'ndent on the high magnitude st aile loads than on the lo\\er magnitude d) namlc loads It IS possible 

that the parameters \\111 \ar)' \\.Ith flight conditIOn and must therefore be lahbrat~d against static load 

If thE' dl'pendenc) 1<; a Simple one. then tillS ma) be readily accomplished b~ ground based calibration. 

If thl' df'pendenc) IS complicated. ho .... e\er. the required ground testing could b(' cost!) and the resulting 

cahbratlOn curH' rna) be Inaccurate o'er ~omt' rE'glOns of thl' operating r!'glOw 

\fadern parameter IdentificatIOn tt'chOlqlJE's provld£, an alternatlH' to laborator) calibration by allowmg direct Identification of thl' paraml'terc; from mput, output data 111 the operatIOnal en\ Ironment ThiS 

has the advantage that there IS no need to c~aractcnze a compllcatl'd functional dept'ndenc} In order to 

determine the p-nameter \alues The paramEters are Identlht'd that pro' Ide the be~t fit of the model to 

the Input OUlput d.1ta. o\rr a ~peclfit'd data rI'cNd If thl' parameters 'at: ~lo\\I) relatne to the r~ngth of 

thl' data re£.ord. till' IdentificatIOn rna, b{' morl' accurdt(' th~n could bl' obtaln{'d from a calibratIOn cune 

of limited order Tnt' high co<;t of gent'r1tlng compllcate,j calibration cun e~ from ground testmg I!. also 

£,hmll1atecl 

Pha<;e II of tlll~ stud) examines the UCil' of mOOt'rn panmeter Identification methods 111 Id('nllfy'ng the 

pdraml'ter!> of thE' RSH.\ rotor trammlsslon load ct'll modd dlr(~ctl~ from a\allable flight cata In order 

to faCIlitate thE' stud). the data ~ct \\a!:> gf'Pl'raled b) a computer slInulallan uSing a model With the same 

structure 8S'lumed by the parameter IdentIfication method but \\lth different parametl'r \alues Since thiS 

approach on I) Idt'ntlfies parameters and doe, not qUt'stlon the model structure It can anI) be used ~\lth 

models \·.ho~p structure has bu'n adequatel} \ahdated from ground lest::. 

Identification and Calibration 

Identification methods can be c1d~slfied IIlto tllr!'e cate&.ones 

I) Equallon Error .\fcthod In thiS method a set of rm a~urf'd Inputs. U I and outputs 'J. nrn a~slJmpd 

related by a known mathematical modrl f(u '1) = 0 and the param('ters of th!' mod!'1 art> Identified 

b) a least squares minimizatIOn of the equation error I( Il, y) i Thl., tlchlllque IS common I) referred 

to as regressIOn and requlr!'s accurate nl':asurem!'nt., of both the mput and output 

11 

, . 
.1 

• 

2) Output Error Method In this method. the parameter~ of the assumed mathematical model are 

Identified b) a least squares minimIZatiOn of the output error. dt'lermlned as the difference bet .... een 

the output of the model of the S) stem. and the outpUT of thl' actual S) stem 8 ThIS approach also 

requIres measurements of both Inputs and outputs. 

3) .\/axlmum LI~tllhoQd \/ethod In thIS method. a Kalman Filter IS constructl'd uSing a d}namlc s}stem 

model and an algebraiC sensor model The paramell'rS of both modl'ls are then Identified by a wel1~hted 

least ~4uares mlnllniZatlOn of the uutput error of the }\alman Filter (InnO\atlon~ sequence) !j. :-'ote 

that If onl) the p<lram('ters of the sensor mudel art' to be Identlfifd the h<llman Filter can bt' IIsl'd 

to estimate the IOput'l to the sen .. or llIodel thl'reforc, onl: the sl'n!:or outputs need be measurl'd 

ThiS feature makt's the \la1(IIlIUm Likelihood dppro<lch partlcularli rell'\ant to ~l'n~or calibration In 

an operational em'lronment 

Calibration mNhods can be categorized based on the parameter Identification technique uSfd The 

follOlHng definitions are proposeJ 

I) Off·lme Calibration. TIllS reffrs to calibration U~lng either the E4uatlon Error ~lethod or the Output 

Error \tethod In a tl'st en\ Ironment \\ lH're both the· :.ensor Input and output are !-nown Once 

the "ensor parameters hav(' been Identified. thl') are used \\lIh the .. en..,or modfl In thl' operational 

cm Ironment to eStimate the sensor Input (c,) stem state,,) from mea,.uremenb of the sensor output 

Since the paramt'lers are not Identified from data obt?lned In the operational environment. thiS 

method IS referred to as "off·llne calibratIOn M Th,,, I~ the mo~t com man I) used approach to sensor 

calibration 

2) On·lme Calrbrallon TIllS method uses the ~la1(lmlJm LI!-ehhood ~lethod to ("'lImate slrllultaneolJsly 

the parameters of tlw sen~or model and the IIlput to the sen"or (<;)stcm states) Onl) the measurf>d 

output of the ~en<;or and a model of lIlt' d) namlcs of the $) stem dtl\ 109 the ,.en"nr are 1lt'l'ded Since no 

knowled!;c of the Input IS rt qUlred, the parameter Identlfiction may be performed usmg data obtained 

in the actual operatIonal en\lronment hence the term .. on·hne calibratIOn :-'ol{' that - on·l:ne-. a~ 

used here. does nol Imply "real·tlme" Caltbrallon b) post·e"penlllent data processing !>till fib the 

defimtlOn of on-hne calibratIOn 

Off·hne calibratIOn IS \·.ell 5ult{'d to situatIOns v. here til(' parameters are constant or SImple fUllctlons 

of the en\ Ironment Cnder these ClrcumstdnCt'S, II IS th!' preferable approach because It prO'v Ides a Simple 

means of appl} Illg the Identlfird parameters to the sen<;or model to estimate the states In an operatIOnal 

environment \\ hen the parameters of the sensor model become complicated functIOns of the S} stem 

"tates. ho\\e·.er. the oIT·hne calibratIOn procedure becomes more Ill\ohed It IS then necessary to relate 

the parameters· to ril"~urable (1Ilanlll1!'~. such as the sen'ior output. throu~h additIOnal mathematical 

models (u~uall) pol) nomlals) the calibration th('n 1000l\es IdentiC) 109 the coeffiCIents of these additIOnal 

model~ 

On·llne calibratIon I~ Ideall) !'ulled to handle SItuatIOns \\ here the "en~or model parameters are 

complicated functlon~ of time or the ,,) stem states SInC!' the parameter~ of the sensor model art' dlrectl\ 

Identified III the actual operational em Ironment, there IS no need to further characterize the parametero; 

as functions of measurable quantities Thus far, on·lme calibration has been used prlllcipalh III support 

of parameter Identification of d: namlc system models ThiS pap!'r \\111 address on·lme calibration as a 

techmque for obtamlng more accurate cahbratlOns of sensor mod!>ll' \\ Ith \anable parameters 

I') 

-, 

'I 

" 

. 

t 
I 

; 

---------------------.-------,--.------- -------------1 

Objective And Approach 

The obJl'ctl\e of this stud} IS to apply the on-hne calibration method to the problem of calibrating 

the RSRA load cells The rationale IS that b) dlrectl} Identlf} 109 \anable parameters of the load Cl'1I 

model In the operating en\lronment. the need to model their functional dependence on applied loads IS 

eliminated and a more accurate estimate of the applied 10dds at thl' hub may be obtained The technique 

\\as e\aluated b}" uSing computer simulations of the load cell response to applied IOdds to generate data 

for the on-line calibratIOn ,dgomhm The ernphasl~ \\ns on calibrating the load cells for 11Igh frequency 

applied loads (Ii Hz ). beCIIUSl' a major obJectne of the RSR.\ program IS 10 usc the load cells to l'\aluat~ 

coupllnb of rotor loads at the S" re\ olutlon blade pa~sage frl'4UenC) (the number of blades, S~ , tune~ 

till: ~haft rotational rate,) The !>Imulallon discussed here IS nl'ce~~ar) to refine the on-hne method befure 

commltlment til flight te.,t .. or othl'r c'perlments. 

Design or The On-Line Cdlibration Algorithm 

Approach 

The dl'sign of the on-line calibration algorithm for the RSR.\ load cells include'S till' follo\\lng steps 

1) Define the load cl'lI model In thb <;lep a mathematical model relating applied rotor loads to load 

cell readings dnd inertial loads for the 150lateJ rotor transmhSlon s}stem IS derJ\ed The model must 

be formulated <;0 that Its \arJatlOn v. II h appli(·d load~ can be represented by a set of Identifiable 

parameters 

~) Define a model of the d} namlcs of the applied rotor 10dds Smc!;' knov.ledge of the rotor load d:.namlcs 

IS used as an alternatl\e to kno\\ledge of th(> actual loads In on-hne calibration. It IS c<;sefltlal that an 

accurate representatIOn of these d) namics be derl\ cd Thl' derJ\ed model must ha\e sufficient de:;rees 

of freedom to allo\\ Ito; tranSient response to be fit to the e ... penmental data 

3) DeSign a Kalman Filter to estimate the applied rotor loads The d) namlc model of the rotor loads 

and the modt'l of the load cell response to rotor loade; are combined In a Kalman Filter to estimate 

the net applJf'd loads at the rotor' hub The accurac~ of the estImate depends on thE' accuracy of the 

parameters In the load cell model 

-I) DeSign a \la'\lmum Likelihood estimator for the parameter" uf the load cell model The \Ia ... lmu/TI 

Likelihood parameter IdentificatIOn algOrithm uses the mea!>urement error benerated by the !{almall 

Filter to estimate th£' parameters of the load cell model Thcr updated parameters arc then used In 

th!;' !{dlman filter to obtam Impro\ed estimates of the applied rOlor loads 

The interaction of the~e four modules .... Ith measurements ohtalned from tht' aircraft is shoy,n In 

Figure 19 The KalfTlan Filter consists of the dyndmlc model and tht' sensor model hnkt'd together ~} 

feedbacl. loops \\Ith Kalman gams. J( The functIOn of the Kalman Filter IS to generate an estimate. ii 

of the system states, ii , uSing. a\ all able measurements i and i. to be deSCribed belo\~ These estlmate:-

are based on assumed values, e. for the paramet :s of th!' models. e The \[a ... imum LII\t'lIhood \lethod 

then us('~ a gradIent algorl:hm on the sequence of Kalrr.an filter measurement errors. ;}. to generate nr\\ 

IG 

i 
~ ... -----~--'..-:;.-- - .... -

estimates, 9 of the parameters of the models B • For purposes of this ~tud), the RSR.\ mea5urements \.\ere 

simulated uSing computer models that \.\ere IOdependent of the models used 10 the calibration algorithm 

The success of the algorithm "'as then determined bi ob~eT\ mg the con\ergence of thE' models 10 the 

calibration algorithm to the models In the RSRA-measurement SimulatIOn 

The Load Cpll :'viodel 

Tht' load cell model used for thl!> stud\' IS tht' model uemt"d In Phac..e I and dt'Scrlbed In detail In 

Appendix A The parameter 5('t to be I~entlhed \\111 be limited to tht' 10dd cell deformatIOn angles for 

purpo~es of till" st ud\. although an) of the geometric or ma"s properties parameters Included In the 

der" allon in -\ppendlx .-\ could be Identified The parameters .... III be )(ienllfied from the error bet\.\een 

the med..,ured merllal loado;. as obtained from the accelerometers and mase; properties, and the estimated 

mertlal loads. a:. obtamed from Eq lIas a function of the parameter set and the applied rotor load_s 

The model. as glH'n b} Eq 1.1, IS therefore solved for the mertlal loads. J , to give an equation of the 

forlll 

f = -S(ci)i - RH - ii. ii = .\'(O.Q,,) (2 I) 

\.\here f IS the \ector of the SIX components of inertial loads actmg at the center of m~s. i I!' a vector 

consisting of the seven rest •• ant load cell loads, and IllS a H'ctor of the SIX components of applied loads 

at the rotor hub. The measurement nOI~e Vl.'ctor, ii . I:. ~dded to characterIze the ~loc~<lStlc nature of 

the inertial lo;:.ds measurement. The nOIse IS assumed to be normall~ distributed \\Jth zero mean and 

pov.er !:>pcctral denSity Q" The S matrix IS a 6 \ i g~omrtrlc mdttl\ that tran!>forms th(' s!'ven load cell 

resultant loads at the load cell attach pOints to a ,et of <;IX component loads at th!' center of mass :O-:ou~ 

that thiS matrix I" a function of the fourteen <lrbltr<1r~ load cell deformatIOn angles o. that 'he Will use 

as the parameter ~et to be Identified The R m..tlrl\ I" a 6 \ 6 geom("t ric matrix that transforms the SIX 

components of applied loads at the hub to Sl\ components of applJed loads at the ce'Jler of mass of the 

rotor/transmission s~stem A detdlled deSCriptIOn of th!'~e \ectors and mattlces I~ gl\en in AppendiX A 

EquatIOn 2 1 \\111 he used as the sensor model, relatmg load cell measurements. i to apphed rotor 

loads, jj \\e ha\e therefore derned a sensor model \.\here the effects of fuselage deformatIOns can be 

represented b: the H't of I·' load cell deformation angles CalibratIOn of thiS model consists of Identifying 

these angles Either off·hne or on·llpe calibratIOn techniques can be used to accomplish thiS, but since 

thl' angl!'s are complicated functions of the applied rotor loads, an off·llne cahbratlOn procedure \\ould 

requm' modehng and calibrating thiS functional dependenq The on·hne calibratIOn procedure allo\.\s 

(lirect Identification of th!'se angles m the operatmg em Ironment In \\ hlch the) will be used to estimate 

the hub IOdds. ('on~e(IUentl!, there IS no need to model their functIOnal dependenq on the applJed loads 

The a~sumptlUns u~ed In the dernatlOn of tIllS model are 

1) The rotor/transmiSSIOn s:-stem IS a rigid bod}. therefore. all changes 111 the load cell g<'ometry are due 

to relatl\e fuselage deformations Th,s assumptIOn allo\\s the net effect of the fusel,1ge deformation 

on the load cell geometr) to be completely charactf rlzed b} a ~el of t\\O orthogonal rotations of thl' 

load cell aXIs about each attdch POlOt The rc·latl\e ~t'ometr:- of the applied iOdds at the hub and 

inertial loads at the center of mass do not change 

}; 

I 

1 I , 

. 

I 

I 

I ' 

lI 

, 

l 
-. - -- \''''''- -----~---- n 

2) Applied loads at the load cell attach pOlOts are directed along the load cell aXIs and no moments 

are supported at the !pherlcal bearings ThIs essentlall} assumes no frictIOn In the bearings and I! 

contrary to experimental e\ Idence A sIgnIficant le\ el of h} stereSls has been obsen ed due to bearing 

fnc l IOn ThIs effect can be Incorporated Into the model at a latl'r time but \\111 be neglectt'd herr for 

purposes of e\ aluatlng the on-line calibration procedure 

A thIrd assumption not mherent In the model but u~ed In thl' on-line calibratIon algOrithm. IS that 

thl' load cell deformatIOn angles \ar) slo\\l} so thl'} ma) be treatrd a!< constant paramete~s mer the dat<l 

"ets u!>ed for IdentificatIOn 

A Dynamic Model of the Applied Rotor Loads 

:\ "e~ element In on-hne calibration IS the u<;e of a d) namlc model of thl' system dTl\'lng the sensors, 

It I" thIS model, m connection wIth the time ,aQ Ing data record that allo ..... s cahbratlon wIth speCIfic 

kno",ledge of the sensor Input, The application addre'lsed here requIres the modeling of the d)namlcs 

uf the applied rotor loads Since thIS IS the Input to the lc.dd cells \lodehng the rotor load d) namlcs 

acro!>s all frequencIes ",ould be a formIdable task. requiring accurate kno\\ ledge of the d) namlc load 

rl'''ponse to pilot Inputs and changing flIght en\'lronment There are frequencIes. ho\\e\er, at .... hlch the 

d) nanllc", of the rotor loads are \·.dl kno\\n. In a rotorcraft. strong \ Ibratnr) loads are present at the one: 

re\olution frequenq and at the .\'~ 'n,\ frequency and Its harmOniCs. wh(>rE.' .v~ IS thl' number of blades 

of the rotor (currE.'ntly .Y~ -= 5 for the RSRA) These VIbrator) lo:td., may be modE.'led U5Hlg second order 

s)stems WIth zero dampmg and resonant frequencIes at the prescribed ~alues By filtering the load cel! 

and accelerometer data to pa>:s onl) these frequencIes, the d)namlc content In the filtered data \l.11! be 

accuratcl~ tepresentE.'d by the "p!'clfied models ThIS d) namlc content IS essential to the Identification of 

I'H'n quaslstatlc parameters SI'1ce there are more parameters than equatIOns and the problem IS therefor!' 

statical!) Indt'lermlnate 

By filtering the data to pass unl} "elected frequencIes WIth \ Ibrator} load!>. the pilot Input IS eliminated 

from the ddta Pilot Input I" normally used In pdrameter Idt'ntlficatl.::ln to Insure that the parameters arE.' 

IdentIfiable b) pro~'ldlng sufficient degrees of freedom In the dynamIC response and b) e\cltlng all mode;;; 

of th(> d) namlc S) ~t('m Since the parameters to be Identified arc all In the algebraIC sensor mo-del then> 

arc no speCIfic modes to E.'''Clt~. but the Input to the load cel!s must ha\e at INst as man) degrees of 

freedom as ther(> ate parameters to be IdentIfied In thIS case there are fourteen un~no",n parameters If 

.... E.' model the VIbratIon at one frequenc) as a second order 5\ stl.'m for each component of hub load, the 

d) namlc response .... 111 cant am 12 Independent components (a sine and a rOSln1.' romponent for each of the 

SIX hub load~) We therefore need to pass t\\O frequencIes WIth \ IbralOr) loads so that the dynamICs .... Ill 

contain 24 degrees of freedom and Insure IdentIfiability for the 14 parametl.'rs l'slng thiS approach makes 

the parameter IdentificatIon procl.'ss mdependent of the pdot Input 'lllo\\mg thE' calibration to proceed In 

parallel \\Ith any chosen flIght maneu\ er 

Slne(' two \ IbratlOn frequencIes must be pas;ed for adequate sensor e\CltatlOn. the t\\O lo\\est frequenc) 

\Ibrauon modes a5~oClated \\lth the rotor WIll be USE'd These are the one re\ and the .vb/rev frequencIes 

The d} namlc models for the one re\ and the .\'~, rev \ IhratlOns arE' !;l\en by the follo\Hng equations 

'1.\ = .\'(O.Q'l'o) (23) 

18 

f 

I 

, 

, 

I I I 

• If 
-----,--

.... hen> n IS the rotational frequenc) of the rotor and '11 and 1): are normall\' dlstnbuted"1'andom \arlables 

.... It h zero mean and pov. er spectral densities Q I'll and Q..,~ respectl\ ely Both H I and J! z are SIX component 

hub load vectors and the total hub load vector present In the filter.ed data IS gl\en b) 

(:! oJ) 

While Equations 2 2 through 2 -1 are not physlcall) based models of the Vibrator: loads, the} pro\ Ide 

a mathematical representation that can be u.-,ed to fit the d: namlc content of the filtered ddta B} 

~peclficaJl~.dIaracterlzmg the frequenc) charactenstlcs. a more acclJrate fit to the data can be obtamed 

for a mude) of gl\t~n order 

EqlJ.itlons :! :! and:! 3 rna) be \\nUen 10 Slate \anable form d" 

o 01 0 [HI HI 1 [0 ,il 1 o / 1 ii, - 0 

-(.\\rW I oj il.. ri .. 

o 

(25) 

DefmlOg a <;tate \ector. r. b) 

(26) 

\\c may ..... nte Equation 2 5 a"i 

i-=. Fi-rf. rf = S(O.Q,,} (2 7) 

The \alut' of i at tlnlE' n i(n} ma\ then bl' computed from i(n - 1) b:-

~! 

i(n) = cf!(.lt).i(n - 1) - I cf!(.ll - :-)1)(r)d:-

IJ 

v.here the time Inter\al from n - 1 to n Is.lt and 

q,(.ll) = ~F~! (:! 9) 

Desip,ll of the Kalm:lll Filter 

lJa\ 109 definl'd a dynamiC model of the rotor IOdd .. dnd a gf'ometflc model of the response of the load 

cells to appllf'd rotor loads and mertlal loads a Kalman Filter to estimate the rotor loads may be den\('d 

The senser model EquatIOn 2 1 Will be used as the measurement equation but the load cell readlO~s.t. 

cann:>t be used a" the measurements ThiS IS because Kalman filter must estimate the measurements from 

Its state estimates and use the error between the estimated and actual measurements to update the state 

estlmate<; TheTt> are se\en load cells but anI) 51" mea- surernent equations 10 Equation::! 1 As a re"ult 

a UOlqUr: set of estimated load cell readlllgs, t. cannot be eenerated from the 51\. state estimates USlIl b the 

SIX me .. "uremcnt equatIOns The solutIOn to thiS problt>m IS to lise the llIertlal lnads because they n,3:-

be umquel) e~tlmated from the SIX estlmateci rotor loads and tht, ,.c\en "no\\n load cell loads uSllIg the 

SIX measurement equations of EquatIOn 2 1 It shollid be noted that the tnertlJ.1 loads cannot actualI) be 

dlrectlv measured but must be computed from a\.1t1ab/e dcc!'k·rorn('te.~ measurements and assum!'d l11a.,,. 

J!) 
--_ ... -,---------- --- - --..----,.-- --- --.-- -- -~ .. ----------- .. -.., ~ ...., --......... ~ 

properties The accuraC) of the assumed mass properties and accelerometer responses could slgnificantl} 

affect the results 

Equations:! I, :!A, and:! 6 mOl} be combined and v.nttt'n In the form 

i= -SIcilY - R iii - ii\ = J(ci.f,i) (2 10) 

The Kalman Filter 5 ma) no\\ be \\ TIt (en uSing Equation 2 10 as the sensor model and Equation 

:! i as the dynamic model Oct\\een measurrment updatf's the ~tate estimates, i, are propagated by the 

deterministic part of Equation:! 8 3!) 

(2 11) 

\\ here the superscnpts - and - refer to estimates before and after the measurement update at the speCified 

lime 

The post-measurement estimates ma) then be obtained from apnoCi estimates at the measurement 

update tlr.'le, n. by 

i"-(n) = i"- (n) - K(n) i",(n) - f(ci. T(n) Y (nJ); (2 12) 

\\here J:" I" a \ector of 6 inertial loads derl\ed from measured accelrratlons :-':ote that the stochastic 

part of the d} namlc model. Equation 28. IS simulated by the measurement error feedback In the Kalman 

Filter. Equation 2 12 The Kalman gains K(n). are gl\en by' 

(2 13) 

\\ here 

J:r. = - R 0 R 0 (2 H) 

(2 l:i) 

and 

P-(n) = ,I - K(n)J" P-(n) (2 16) 

P- and P- arE' the apnOrI and post mea~urement \ an abies of the state estimation error, respectl\Cl:-

I 

I 

I. ' ...... I 

. Smct' J:r IS constant and K IS Independent of tht' Identlfil'd paramNcrs, ci, EquatIOns (2 13) through ! 

(2 16) \\ III rpach a stead) state conditIOn that IS mdependent of the identification process The algonthms J 

can be great!) slmphf:rd b) uSing the resultmg stead) state \alue of the Kalman gams, f{. In EquatIOn 

2 12, thereby eliminating the need to use EquatlOns 2 13 through 2 16 The use of a stead} state Kalman 

filter \\here the gam is mdept·ndent of the parameters to 'H' Identified msures a globally stable \fa"murr 

Likelihood IdenfificatlOn -\Igonthm '9: 

20 
The Maximum Likelihood Estimation Algorithm 

The accuracy \\Ith \\hlch the Kalman Filter can estimate thE' applied hub loads depends on the 

accuracy of the eStimate of the load cell deformatIOn angles Ci In the proposed approach these values are 

updated on-hne uSing \faxlmum Likelihood parameter estimatIOn The \1axlmum Likelihood estimation 

algorithms are derl\ ed bj minimIZIng a weighted sum of the ~quares of the measurement error of the 

Kalman Filter. (thE' mnO\atlons sequen("e) ..... Ith respect to the parameters over a data set of S, samples 

The cost function IS defined a!> 

N. 

V::: 1/2 2~:e'J(tl)TW -ltJ(n) (2 Ii) 

... =1 

\\ here 

(2 18) 

and 

'I. 

IV = lIN. I: eJ(n)eT J(n) (2 19) 

... :1 

Denctmg the apnon estimate of the \ ector Ci b) 6, and the post-measurement ('stlmate by Q, the first 

partial of \. \\Ith respect to the deformation angles IS expanded about Its \alue at 6 to gi\e 

(220) 

Setting the deSired \<tltle dt Q to zero corresponding to an extremum, g!\es 

. =' _ (BZ\'(6))-1 (O\'(Q))T Q a oCi~ oCi (221) 

Takln.g the first partial of" With respect to 0 gl\es 

(') 'l'l) ~ -~ 

oeJ(n 6) _ -J "T- :.-( ')) _ J (oi-(n,6)) Be; - <> ~ a, ,x n 0 % Be; (223) 

The d) fl::lmlC beha\ lOr of tIll> partial denvatlves of the apnon state estimates, i-. With respect to the 

angles, Q IS obtained frum Equations (2 11) and (2 12) as 

(224) 

21 

I 

! 

I 

! 

I • . 
- - -- ..-- -.. -- -" - - - ... -' ---- - - - -- __ 0. _____ ... ______ ~I 

t 

i 

i 

ai-(o.u) = 0 

an 

The partials of the estimated tnertlalloads with respect to the angles J a • are given by 

\\herc 

(J) = _ ~ as·,(6.)t( ) 1 a L.. an I n ,::\ 

The second partial, to first order. IS then 

(:! 25) 

(:! :!6) 

(227) 

(228) 

The parameter estimates art> obtained from Equations 2 21 through 2 28 uSing the measurement error 

sequence, eJ. generated b) the Kalman Filter 

Operational Approach 

The data IS first filtered to pass onl~ the t\\O specIfied frequencIes The Kalman FIlter IS then applIed 

to tllP data and allo .... ed to reach steady state before tile \taxlmum LikelIhood algOrithm beginS operatmg 

on t he measurement errors generated b) the Kalman FIlter The reas~n for \\altlng IS thllt the error In 

t he initial rotor load estimate WIll bIas the estlmate~ of the load cell angles By waiting until the tranSlent 

due to thiS inItial error decays we can ehmmate thiS source of bias WIthout havmg to add the mltial 

conditIOns to our 'Set of parameters to be Identlhed Thl~ dela) IS necessar) only on the first data used for 

Identification If subsequent data sets are contiguous. the Inltldl e'5tlmate of the hub load can be set to the 

final estimate of the hub load on the prevIOus ddta ~et ,lOd the inItial error .... 111 be small The ~tatlstlcs of 

the process and measurement nOIse. Q'l and Q". are chosen to generate stead) state Kalman gains from 

EquatIons 2 13 through 2 16 that provIde!:> a hl!;h filter band .... ldth (short tlme constant) compared to the 

frequencIes of the \ Ibratlons bemg tracked 110, 

The Kalman FIlter deSigned for use III callbratmg the load cells v.lll estimate onl) !hl-' one:re\' and 

.v~/re\' components of the hub loads If tl-c IO\~ frcqucnCj re"pOrlse IS also deSIred, It IS necessary to pa~, 

the unfiltered data through a Kalman Filter that utlhzes the updated parameters An additIonal G degrees 

of freedom dynamiC model must be Included In thiS Kdlman FIlter to track the low frequenc) components 

of the hub loads ThIS additIOnal states caY) be modeled by the \\elner process 

Ho = ijo 

and the total hub loads are then gl\en h) 

ijl) = ,\ (O.Qr)..) (2 Z9) 

1 

I 

I 

j. 

I 

.-

-1 

. ". . 
/ 

(~ 30) 

Slmul:1tiol) R"qults 

Tht' algorithms ~pre IInp/!'menll'd /l$ sho\l.n In FI~lIr(' 19. !lnd 11 computtr sImulatIOn of the ap~!lt'd 

huh load .. and tht' fill'<1SUrf'd 10.1.1 cdl r!'spun~~ \\'1'1 u .... d tu grnl'rZlle SlfTlu!att·d mell.surf'ment data for ti,t' 

HSR.-\ -\gam thf" prohlem of h<1\tng to g~n('rate S('Hn Untqul' 10'1d cell rt'adm;;< IHlIIg tht' spl'clfit'J huh 

IOdds and ltIl'rtlalload .. \\Ith the SI' measurement ('quatton!! \l.as enrounter .. d For purp05''', of simulatIOn. 

lht" pro!1l('nt "'.1'4 ",,,hPd 1)\ .1t.HIO>; a "{"\enth con<tr1ul\t t'qUdtlllll \l.hlrh t'''('nu!\ll~ ll11nlllllZed lhl! ~tr:lln 

(nl'rg\ an thl:' /tnt! nil.. Th.,. approach: le1ue<1 rCII!!Oo.}nablt· ~lfnlJIlltron \ alul'~ of load cell rl'adlflg~ for the 

applll·d huh loads Tltr ph~ ~Iral basiS for thl!> constr111nt equdllOn \\as too ljuNlionable to allow It to 

he ullhzl'd an thl' IIlenuficatlon al~oflthm, ~ It \\:1.:1 on I) \Isrd to generate sllnu/Ilt{od datn \\ith wlllch to 

\dltd,,:e thl!' algurlthm. The deformAtion Angll'S In th~ I03d cell simulation \.'Cre ~et to arbitrary con~tanl 

\alul'~ ant.! a different -f't of \alut.>!l \\ert' u<;ed "'Ith the Kltlm:!n Filter to IOltl.'ll' th!' IdentificatIOn proce-;~ 

ValuE'S of uro d('grl't .. , \\t.>rE' U"'l.J for tlw SimulatIOn and fOllr degreE'S for the Kalman Fllt(,f 

:'100 mertl<ll 100lds \\l'Te sImulated for th(' l~l :SlOet' t!lI~ represents a \\orst case for the Identification 

process .\n} Inplll to the s~stem from the mcrtlallo.\ds \\ould h.l\1' ndJl'd additIOnal dE'grH's of fret'dorn 

to the mput to the load cell'! and therrb} Imprmed the .dentlhablht) of th(' load cell allgl('s :'Iooll' thll.l the 

presencl' of ~uch addItional m<!a".Ufrment IOpUts I!.ould rNuc(' lhe requlrt'd number of degrees of frf.-eUom 

111 the \ Ibr:HlOn model 

.\s e\plalnf:d In d.l' pr(,\IOllS "'t'Ctlon lht- IOltlal \ahleS of the hub load:; and the load cell dl'formatlon 

anslt ... \\('r(' ~('t dlfferentl} 111 tr'e "'lmulatlOn I1H .. dr·1 and III 'ht' raltk{'tlCn algorithm II) e\~/uate the abtht\ 

of the al!:o:nhm to con\t'rge toJ tllt' \alllt'" lJ5Cd 10 tlt~, "'Imu!:ltilln Tran"lt'nlS re<:ultlllg from the Initial 

state error In the I\almnn FllteT t-",tmI3te \\111 bld~ the par"mner t'''Umatl>s In order to a\old thiS proble.TI, 

[tu' filter \\n~ allc\\ed tl) I"e.lch a stead} ... tatl:' condition b .. fore tht' ~la'l:IIIlUm Llkdlhuod .Jgomhm \\~ 

IllItlatt:-d 

The tllllt' constants chosen for the Kalman FIlter \\!'re all at 0 1 ~econd'l 0;0 a dela)r of about 6 5 SP(OIlJ~ 

\\as n~es"arj to In~ur(' that lh., respoll"e h~d reached .1 stead~ st:lte condition T\\o frequencIes, one re\ 

dnd Sl/tt'\, ... ere Included In the slmulatl:'d hub load d;. namlcs dn\lng tilt' load cell sllnu!atlOrl. ThIS \\a5 

to Ins"r~ that the load cell Input had ~ufficlent degrc-es of rrt'edom to pl'Cmlt ldentlfic.ltlOn of the 14 load 

cell <lnl;lt>S It al:w effectl\el: sImulated filtermg all but tho~e frcqu::ncles In re~1 test data A samp1tng rate 

of::G samples, second \\3S used to Insure adequate reconstructIOn of the 1 j II;:, .Y~ re\olutlon frequl.'ncy. 

Fq;t.re ~O I~ a plot IJf tht, trrN In the estimate of the longltlll.hnal dt'formdllon angle of om' of the 

\ ertlcal 10lld cells \ clelil: of --e\ cnt) 5Mnples In ImplC'IT".'nttrlg I h .. IdentificatIon \\as used to aUo\\ tht' 

filter to rcach stead\ "ltd!e :\ "~.:juenc\. d 10 ~amp!t's \\<1:> thl'n u"eJ 10 perform the estImation Due to th!' 

nonhnr:lt n:ll ure of the mea"'\Irt:'r.ent ... qudllon. multlp/" lit-ration" mer ,\ dOlt" " I \er(' rt'qulred to obta.n 

comeq;~nCl' for thl:' param·~t~r t:"_t'INl!('.,;; On e<len Itt'r~\tlOn tile ,,/'\cn l \ ~ample dela\ \\as repcated .\I.d 

the la'$l 10 sample" \\ere used to .,suol'ate the IOdU celi <lllgl(" [\cellent con\f.rgt'nce \\:l.,o; obtallled "lIhln 

four IteratIOn:; mer the ~ame uJ.~a ~e~ FIgufl':?l I!> a Slllll/ar plot for .1 btera/ load cell deformation .',ngle 

FI!;ure::!::! :ompolres thf' actual y-a\IS hub 11).ld \\IIt. t·,tlnlat'" Obt.11:l('d using the ,"Itlal :'lIld final 

parameter estimates The f<,ttrnat.." "ert' obl.tllled from d halplOln Filter \\!th a \\'t'HlN proce~s modd of 

thl' addltwna! 10\\ freqLlene~ stat,:" The I\alman Ftll,'r U't d a !llnlu/att,d Ilnfilterc(J data ~t'l that comalll.·..! 
... --------\.."!-' l 

a 101'0 frequency component In addition (0 the t\\O \Ibr:ltor~ compant-nt! Tht' hub load e~tlmatlon error 

IS significant for tht' Initial parameter estimates but IS con~ldl'rabl~ ImprO\t>d \\ht'n tht' final parameter 

estlmat~ :m' used A similar result IS seen In Fq;ure :!:: for thl' app!Jl'd roll moment at thl' hub The 

compllter 5lnlUlatlon demof'!trates clear!} the potential for pl.'rformmg on-hne calibration of the load cells. 

U51t1g data obtained In the flight. em'lronment to simuitaneousl) cahbratl.' ,the load cell~ and estimate tilt' 

hub lands Figures 12 and 23 demonstrate the sensltl\ lt~ of tht' load ~tlmatlon to ('rrON In the load cell 

angl.:s The load f'stlmatlon error 1'1 slgnlficantl) rrduced \\hen acurl\lO!' f'Stlmatt's of the paramett'rs are 

u~ed 

Identification From Amhil'ut Vibrations 

Thl' demonslratt'd ("apabillt) to Identify paramt'tt"M from amblcnt \ Ibrat ton:. I' 511~nlficant In that it 

allo"" Id!'ntlficatlOn to proceed 10 an opf'rattonal em Ironm('nt "lIhoUl the n('l-d tu supertmpose speclall) 

dl'51gned Input ~Ignals to faclittdte the Identification The mt'thod m3\ be applred to other problems 

bt'}ond that dIscussed here For any such apphcatlon the condittons for Imyl!'mentlng thIS approarh are 

as (0110\1,5. 

f) All parameters to. be Idl'ntlfil'd mu~t be Included In an algebraiC rerre--ent:J.lIon of the system since 

dynamiC model,; "ould rt'qllltt' ('''(cltatton '11 ~peCific frequencies ThIS does not present a problem SlOce 

any d~ nam/c mod"l can be treatt'd :l.\> a ~t-t of algebrau: equatton!': tf all states llnd their denvatl\CS 

art' exphc/tl,)' rneasurt'd or t'stlmalE'd at each pOint In ttme 

:!) -\mblent \ Ibrations must mc/uci(' as mam degrt"~ of freedom as thert" are parameters to be Identlf.ffi 

and titt' d~ namlc model of the \ IbratlOns must al50 tnclude th!.'!>t' d('~rl'es oC frt'!.'dem Thl~ cnn he 

acrompltshEd by modt'llng onl) the ,t'qulred numot'l" of d('grt't'5 of [.!.'eciom and filter:ng the data to 

p:1S~ onh thl' moJ('led fr('que"lctes ' 

Poa-ollal appltcatlOns of tim approach mcludl' tht' t(ac~ing of tum' \ar~ Inl; stabiht~ dematl\es of 

3 flight H'llIcll' during manf'lI\ pr.,. that I'oerr dl'5lgncd to In\ estlgate pcrformance TIlts \\Quld pro\lde j 

dlrpct correlatIon bel\\l'en dp51gn parameters and ob~en l'd handllll!; qu;\IJlle~ .-\ real·llme implementatIon 

of thiS Identification t('chnlqut' "ould al~o facilitate adaplt\(' control ar;Jh(".lllOns. 

Conclu:.ions For Phasl' II 

Condudior, Remad.s 

The purpose of Phase II of tillS ~tud~ .... as to ana!~ tl':::al\) demonstrate the feaslbl/tt; of an on-ltn~ 

calibration prC"Cf'dure Ratr-er than use c"(perlmental data, tIll' ,"\t'~tlgallon "as condllctt'd uSing daq 

g('nerated b) a computer simulatIOn of the load cells to allo\\ J compar'son of tht' Idc>nuiil'd model \\Hh 

the kno\\ n 51mulatton moclel Th!' structure of the model to h" Identified. as Implemented In the Kl1lman 

Filter, \\as the sam!' as that u~ed to senl'rate the simulated load cell data, btlt the state,; and parameter'S of 

the t\\O models \\ere Inltlahzed dlfferentl! Thl' un"-no .... n mudd paramt'ters (load ('t'il deformation anglE'S) 

\\1'(1.' thrn Identified succe~"full~ b) applj inS thl' algorithm to thc ~Imuhted load ("I'll data The rt'suitms 

C51lmat(' of tht' states (applied rotor loads) also con\erged to the \altle' Sl'nerat«i b~ thl' wT'ulatlon, The 

mod!'l structure assumed for t!lI~ 1m c~tli;atloll has not bN:'n \ ahdaled '\ith ("I.penmel,tal data and the 

effect of uSing a model structure In the l\~lmolll Fllter~that do~~ not m.}tch the simulallon model has nct 

been In\,'stlgatcd 

j 

} 

i 

f 

; 

• I 

I 

. I' ~ .-I 
\ 

Sp{'dfit' Conclusions 

The follo\~ inS conclusions can be dra~ n 

I) For the assumed model. thl' dYllamlc rotor-load estimates ~er(' found to be \eQ senSltI\e to static 

IO<1d cell geometrj derorm~tlons The rotor-load estimates \H're greatl} IInpro\ed b} usmg Identified 

geometric parameters 

:!) Cdllhratlon of all model oararn('{ers \\ai succ('~"full) accompllshl'd uSing on I; rnca.,urf>o load cdl ant! 

IfIcrtHll loads data 

:q TIt(' IdentIficatIOn of the model paramctt'r" \\a" dcc()mplt~"l'd uSing onl~ rotor \ Ibratlon" at thl' t\\I) 

lo\\est frequcncH~s (one re\ and ,\ ~ t re\) as the source of exclta- tlOn. ~o 5peclally deslgnffi {'\t<,rnal 

IOputs ~cre required 

-1) Tht' Identification wa .. accomplished \Iollh the inertial load data set to zero ThiS represents 3 ~or5t 

case situation SlOce an} Independent degrees of freedom 10 the inertial loads ",auld prOVide excitation 

that would enhancl' the Identification 

Rp("ommelluation For Furthpr Research 

The fa !I ov,109 slInulatlOn-ba .. ed In\estu;atlon" \10111 be r>ec('Ssar) to full} quallfj the on-hne calibratIon 

procedure for applIcatIOn tu tlIght data 

1) The rapablht\ 10 track tlme-\an In!; defurmatlon angles mu"l b(' le~led II~Ing SImulated load (ell data 

:!) Tht'dfl"CI of ~en~or r\(Jl~t and model ~lructure error on the Identlficdtlon must be detcrmlllcJ 

3) The f'fft'ct .. un pcrfurmanct> uf \aT) IlIg the mudded \ IbratlOn frfqlJ('ncle~. the samplln!; rates and thl' 

Kalman rllter tlme constants ~hollld be In\cslIgated 

.\ practIcal lIIi pl(>n1cn'tatlOn of thiS mcthodolog\ \\111 rl'qu\r!' a \ahdated malhematlC:l1 model of the 

load cell Tespons(' to applied rotor loads Tht' accurac~ of thIS technique IS a!~o dependent on the aecuraclt'~ 

of the assumed mass propertle'> and accelerometer re~pOllses.. All of thIS must be determined fTom .,ultable 

ground t(,SIS before the on-hne cahbratlon approach can be appllcd to flIght dat.! 

REFERE:--;CES 

1) Bur"". J S ,-Rotar .~~'ttrn' RfUllfC;' -{/reraft (R!:oR~J Rotor rorn -Ind \lo'IIEnl \/tCl!'urtllIent 

.':y~ttm-. -\1-\-\ Ftr!ll Fltght Te't Conft'rcllcl' Las \ .. lja .... \",\, \0\ Itl"l 

::!) \erN' C \\ . -Rtsult$ of tilt Flr,t Compl.tt ... ·ta/l(' CallbretlOn of the R .... R t RO/CJr Load \frasure· 

"lfnt '::lI,(.m - "-\5 \ TP·:'!3~;. lO~-t 

3) Wellman. B .• Prdlmlrzary Ri <u/r, of the R~n -t ~hal.{ Tt~/ -. To be published as d \..-\:3.-\ T:-.t 

IO>!:> 

4) Glan ... anll', \ e'rlllan -\ Flanlll'l~ \\ (; ... lId \.ag\ E J. -.'trudural ~y~toll Urll/,firallOl: 

TfC:hllo/(l9!1 \I:rtfieall(lll-, LS\-\\R-\DCO\\ BerMt \u TH-81-D-~8. ~'J\ 1981 

.,--) 
... ' 

5) Br)!'on, A E. Jr and Ho, , C, .ipp/ted Opllmai Control, Blaisdell Publishing Co, Waltham 

~Ia!\~ ,1969 

6) Du Val. R \\ ,~Th~ ['u of Fr~quency ,\Idhods In Rotorcraft System identificatIon", Papl'r ~o 

AIAA-81-2386, AIAA 1st Flight Testing Conference. Las \'ega~, ="e\ , :-':o\' , 1931 

j) Jennnch, R E, St('P\\I~1' Rcgrcmon,Statlsllcal '\Irlhod~ for DI91tai Computus, Chapter 4, WII('~, 

'\('\\ Yorl. 

8) "('hra R 1\," -tn Inn(lIatlo"~ -tpprollrh to \Iaxlmum L,k'/lhcJoJd 1d~"lIfira"on of LUItIlT and Son, 

lmrur Dynamic '''y~trm:i''. Prf'~l'ntt'J at tltt· 'ith lIa\\all Intt'matronal Cunference on S~stem SCience, 19;2 

9) LJung L,".-t l'sympotollc Bthal lor of ti,t Extt nd~d fi IIlman Flltt" as a Paramtltr Es/rmatoT lor 

Llntar Sy~t(ms", IEEE Transactions on Automatic Contro\. \01 AC-:!4. '\0. 1. feb, 19j9 

10) Br)~on .\ E, Jr , and Hall. \V Earl, Jr. ·Optlmal Control and Frlta ~ynthtsls by Elgent'utor 

DtCOlllpoHtlOn·, Report:\o 436. Dept of Aeronautics and -\stronautlcs. Stanford L"nlH'rslty, Stanford 

Co-\ • '\0 , 19;1 

! , 

I 
---

.......... ,.-....---~---.----- - -- ~- --

APPENDIX A: LOAD CELL MODEL DERIVATION 

A diagram of the undeformed IO:ld cell geometry is sho\\n 10 Figure A 1. The load cells are mounted 

to the fusebgE' and to the transmiSSion base by spherical bearings It IS assumed that the transmission 

bac:e does not warp so all changes 10 the load cell geometry are due to deformatIOns 10 the load cells 

themsehes or 10 the fuc;elage mounting pomts With this ll!>5UmptlOn It IS possible to complete!} model 

thl' load cell respon~e uSing a genf'ral three component reprec;cntatlon of the feactlon force at each attach 

POint on thE' transmission ba~e, no knowledge of the fu~ela~l' dt!formatlon IS required 

...... Three plane VIE'\\C; of thE' load crll geometr) \l.lth thE' three component reactIOn force reprl'sentatlon 

art' <;ho~n In Figure A:! Inerllalload!> including gra\lt~ and the dn\E' -haft torque, arl' ac;suml'<i to ht' 

ronrentrall,d at the tran'nll'<;lOn center of gr<l\'lty The rotor luad<; are concl'ntratl'd a~ the hub Tal..lIIg 

the "um of the forces and moments about the cel,ter of gra\ It) j\1\CS 

Xc = X, -.\11 - G: - E: - F: - :1" - B., - C: - Dz = 0 

}'. = Y, -}'JI - Gy - El/ - Fy - '\'1 - By - C" - D~ = 0 

Z, = Z, - ZH - G~ - E~ - F. - .-t. - B~ - C~ - Dz = 0 

L< = L, - LII - }'Hh:: - (A. - D.)tl :! - E.e - (B .. - C.)w 2 - F.I 

-(Bv - C y - Fy - Gy - Ey - .4v - Dl/)b= = 0 

(A 1) 

.\1, = ,\[,- ,\l1J - .\111.:; - ZlIhx - (B. - A. - Ez - G .)(d -l) - (Cr - Dr - F., - Bz - .4% - Er - G:)b:: 

-(Cz .... D. - F.)l = 0 

Sr = .v, - .\'11 - (A~ - Ell - GlI - By)(d - £) - (Cy - Fy - Dy)l 

-(Az - Dr )w/2 - Ezf - (Cr - B.:)u;/2 - Fzt = 0 

LSlIlg the tran5formatlOns. 

Ar = Acos(o .. )sln(O .. ) 

All = -.. ~~tn(oG)COS(Oa) 

-

.-tz = .1cos(o~)cos(O .. ) 

Bz = BCOS(¢b)sln(Ob) 

By = -Bsln(o~) 

Bz = 8co-(0.)co,;(0.) 

C., = Ccos(o.)~rn(9,) 

e" ~ -C31n(O,) 

c. = Ccos(o,)co,,(t\) 

DJ: = Dco,,(O.l)<HI(Oi) 

Dy = - D::ln(oJ) 

D. = Dros(ol}('Os{9 1) 

(A 2) 

. -, 
// 

_ • _____ ........... ~ __ "'~~w _-- • ---~ 

the f'quallon~ becoml' 

and 

Ez = -Ecos(o.)sln(1,.J.) 

£~ = £cos(o.)co,,(,-.) 

Ez = ESln(o.) 

Fz = -Fcos(o,)sln('-',) 

F" = Fcos(o,}cos( .... ,) 

F. = F5/n(O,) 

Gz = Gcns(8 1 )cos(,-'}) 

G~ = GCO.-;(OJ)$/n{'-'J) 

G Z "" -GS/II(O]) 

i- RH - sf = 0 

-rn ztl z 

-m\fa~-r!iq 

-m.a. = -/ zp - QI - r Lq 

-lyq - r \Iq 

-l,r 

(A 3) 

(.-t -t) 

(.45) 

In ("quatlon A:J, oz.ay , and (I, are hnpar accrleratlOn~ In (','>.eh dlrrctlOn mea<;ured at the rotor' 

I ran~ml!-slon !» stern center of gra\'lt~ and P lj and r are rotational <l('ce!eratlO05 Bt'cause of the nonrigId 

t'ngln(> mountings tht' engIne contrlhullOns 10 IIINtial iorcrs arl' not l'qual in ,,/I d,rectIOns \\hen meastlfPd 

at the <;)~tem Cl'nler of gra\lt~ ThiS effl'Ct can hI.' adpquatel~ modeled b) aS~lgnlng dlfferel,t \alues to th(' 

total effectne ma"s of the comblnC'd rotor tr1n~rlll:;Slon engHle S) stern In each d,rf'ctlOn. rr1::. my, and 

.m, There are abo a fe\'. minor ehor termc; not 51\en here that .ue dl~euss/'d full) In Reference 4 IZI l~, 

and / z are moments of inertia. related terms In tll!~ cross-products of rotational f<ltes are neglIble QI I:> 

total apphf'd engine and tad rotor shaft torque GyroscopIc coupling forel'S due to engine and transnmslon 

rotatIOnal moments of momentum are represe'1tPd by the coeffiCients r. \\lth ~ubscrlpts for the appropriate 

3\1<; 

ThE' hub forces (Ii) and 10<ld cell readings (t) In Eq .\ 3 arE' gl\ en by 

if = XI{ }·II. ZH.LII,.\II1. \'/1 

t = ,'1. B.C. D E. F.G 

(A 6) 

(..t ';) 

The geometrIc tran~rormatlons from the applzed loads to the center of mass (R and S) are gnen b)! 

I 0 0 0 0 0 

o 1 0 0 0 0 

o 0 0 0 0 

o hz 0 0 0 R= (.t 8) 

-hz () /z;: 0 1 0 

o 0 0 0 0 

and equation (A 0) (011 the ne'(t page) m "hleh C and S denotp cosmc and sme functions, respectl\e1)! 

I 

i 

1 

7 

l 

. 
.. 

:! 

~ 

-;,. 

..r. .. ~ 

;:.,; 

..... 

' .. 

~ ..... 

1J: 

..... 

~ 

~ 

:r. ~ 

~ 

" 

..... 

-. 

c:::.' 

..r. 

.~~ 

..... .' 

.. c:::. 

:r. 

-0. 

V 

," oC 

e::. 

:r. 

~ '-

c:::.-

:r. 

~ 

::.; 

, 

-' / ~ 

-' 

- / 

--•• ~, < .... -

... -;,. 

:r. .. <:> 

. , 

"" 

.... 

~ 

U 

.... 

~ 

u 

.. J 

\wi 

~ " 

..... 

-<5 

:r. 

~ 

:r. 

... ~ 

:r. 

~ 

..r. 

.. e::. 

:r. 

..... 

"'J. 

:r. 

-<>. 

tJ'l 

c:::.-

~ 

~ V 

c:::.~ 

~ 

..;j. 

~ 

... e::. 

V 

.c 

. 

~ , 

..... 

c:::.-

'-

" 

v. 

. , 

'-

19 

:= ~ _- _:-c....-:=.~-___ ;.. ___ .~. _________ • ____ .. p _ ..... ~_ 

,. 

~ 

:r. 

? 

..... 

-, 

..:: 

..... ....~ 

i)~ 

:r. .... ..... :;>- , '--, 

..:: 

.. -:.. 

" 

~ 

~ "" .. ::-- :r. 

... "'" -c " 

~ 

~ 

~u .., :r.'$ 

.. i ~ 

-c 1'" 

'Q,-

" ~ .... 

-§ :r. 

" .... -C 3 t. 

... e::. 

. , .c ..... -::; -§ :r. 

., ..... -C J" 

c:::.- v. '-',:, :r ...... .. . , ..:: ..... 

3'" 

:r. 

0RiGlr~AL Pi1Cl~ ~5 

OF POOR QUI\Llrt 

.,. -:.. 

" :r. ~~ ... . - ? ..... -.... ..... e::. C" '0 

..... 

-, 

-..:: - "': - .... 

~ ..... ..r. ~ j 

- .... :.... -..., :r. ~-i ~,.; :j. --~ -, "",\wi ..:: ........ 

.:; 

-;; ~ ~ :S:r. .~ ~ 

~ ~ 

~c ..r.c 

U 

.. , ~, ..Q -- "'" .. "': - ~ .. c:::. <:> f."", ~ - - .;s ~$ "'J. , " ..r. " ... '- ..... .... 1'" " '- ..::: - "",,- c:::.0 

~~ - ..r. ...... ::~ , .~ • l 

~ ~ '-' '- ..r. U .. .... ... ..:: ) ... 

~ ' , "" "" .... ... $ ~ c:::. -:;>- c:::. :r. :r. .r. ... ..... .c .... ~ C "'" " .... "'" -, ""::l ..:: ... - -= 

? 

-"'" ~-~ <::: -:s :..; J. :r -..... ...:;. --'-

~ .... '- ..... 

" It'.~ ..:: -= 
----

----- "", 

,/ 

/ 

__ .... _. _r ________ :a ..... ~._, ....... ""_. ___ ... _ .. .JO........ - ___ 

-, -----~ ------- - ~ 

APPENDIX B: THE EFFECT OF SENSOR NOISE Q::'Ij LOAD ESTIMATIO"'N 

The analytical model of the rolor 'transmIssIon system dertved In the text has the form 

I- sf - RH = 0 (8-1) 

Gnen the measured load cell readIngs. f ..... and inertial loads demed from accelerometer measurements im.. the applied mlor loads. II. are estimated from Eg. BIas 

ii = -R-I i ... - sf ... (8 ~) 

Thl' meastJr!'d load cl'lI dnd accelerometer \alues rna) be \\flUen In terms of theIr true values and a 

random measurement nUl,,!, component as follo\\s 

ilA. = S(O.QA.) 

ilT = S(O.QT) 

(83) 

(84) 

\\rtllng the deflH,d inertial load \ector.):'. as the product of an mertta matriX, M, and the accelerometer ml:'d!>urements a .... gl\I''1 

J:' = Jlo",,:= .\10- '\1ilA. = i- .\Iil,\ 

.... here its the actual mertlalload \ector 

SubstItutIng Eq's B 4 and B 5 Into Eq B ~ gIves 

- I - - H = -R- P - ST: - ilJl 

\\ here 

"ubstttuting Eq B I Into Eq B 6 then gl\es 

II = Ii - illl ilH = .\'(0 QH) 

(85) 

(86) 

. (87) 

(88) 

From Cq B S It 15 <;l'en that ilH. as gl\en b) Eq (B i) IS the combInt'd effect of the Instrumentation 

('rrors on the appII!'d rOlor lOdd estlmate,ii ThIs error represl'nts a !o\\er bound on the accuraC) of the 

estImatIon thalls attatnablp \\Ithout applymg statIstical proceSSIng. such as Kalman filtenng orsmoothmg 

The co\artance of IhlS error. QH • may be computed from Eq (B i) usmg the kno\\n co\anancc of the 

Instrumentdtlon error!>. Q ~ .lnd Qr , as foil 0"" 5 

QII = E{ilHil'{;} =E{ -R- I .\!L";l - SiiT, 

- R- I \ft'A - SilT T } 

QIl =R-I.HE{v"ilD.\lT(R-1)T - SE{ilTt;rptTR- IT 

- R-I.U E{il"vf}ST(R-t)' - R- I 5 £{vT~'F}S' R- ' 

(89) 

(810) 

The sensor nOIse compon('nts are assumed mdependent so the term E{V.\IJrT} IS zero Suhstltutlllg the 

\..no\\n co\arlanc(' matrlCf'S of the sensors for the other ('xpectf'd \al..Jc terms gl\es 

(8 11) 

30 

I I ' 

I 

f 

j 

, 

I 

I 

I 

j' 

I 

j 

I 

! 

I 

t 

... " 
· · · 

APPENDIX C: CALmRATION ALGORITlL'\i 

The apphed rotor load estlmatlon error IS gl\en b~ Eq 1 ~ In the text as 

(C 1) 

The coefficient matrices, Rand S contasn geomptrlc parampter~ of tht- model and the demed mertlal 

load., \fctor. I .... Includes inertial parameters of the model A subset. e. of the parameters IS sell'cted for 

cahbrdtlOn dnd the error 1<; treatt'll a:. a function of ti>o,e paramelprs -\ cost function IS \\Tltten sn the 

furm-

\. 

\'(e) = 1 :! ') t-II(6.nfn -lllI(e tI} o-J 

(C :!) 

"=1 

where \\' IS a weighting matn, gl\en b) 

N. 

W = liN, Lell(9,n)ei,(9,n)IT (C 3) 

n=1 

Denotmg the apnorl estimate of the vC"ctor e by e and the postmf'asurement estimate bye, the first 

partial of \' with respect to the parameter 0- IS expanded about ItS \alue at the apnorl parameter estimate 

eta gl\C 

(C 4) 

Settmg the desJr(,d \alue at e to zero, corre<;pondmg to an extremum.gl\es 

(C.5) 

Taking the fir~t partial cf \. with re"pect to e at e gl\e5 

(C 6) 

Th" second partial :0 first urder is then 

J~\ (8) = ~(aeH{? nJ)T\t, _Iael/(~.n) 

J8 z ~ ae as 

(C i) 

Oncl' thl' pardnletf'r set hd<; bt-en chosen, the parll:li aernatl\p of thi' errors. i!1/, \\Ith respect to the 

pararn('ler~ rnu"t he generalt'd anal:. ucal\) from Eq (C 1) The parameter estimates are then obtained 

from Eq" (C 'i) tu (C 7) U<;IIlS the mea<;urt'nH'nt errors ~equl'nce, {II generated by Eq (C 1) 

:n 

u-------------- -

."~ 

I 

. , 

I 

I I • r: .. ! ';. 

.... 1, 

,'"""" 

-~~ 
• 

y~ 

;0-JN 

Vz 

j • 

FIGt.;RES 

Figure 1 Loads acting on I<;olated Rotor 'Transmission 

Figure 2 Load cell arrangement for Rotor 'Transmission S)stem 

, , 

" , 

t 

, 

I, 

I 

t i, 

I , 

t 

, 

~ 

~ 

i • • " 

, 

i 

I 

I 

I 

I 

I~ i:"7

:1

t.:.:
/ 

I 

, 

., 

.. 

) 

I 

; 

I 

I 

, 

I 

ooCll--FWD 

AIRFRAt,1E 

ATTACHMENT 

H 

FIgure 3 RSR.\ rotor bdlance ~\ stem and mounted Hems 

33 

I ~ 

! 

I 

I 

I 

, 

l"r I " .... 
/ 

-.... ~ I 

I 

/ 

I 

: 

I 

r 

.' 

;' 

/ 

3500 

::! 

~. 3000 

« 

o 

..J 2500 ..J 

« 

~ 2000 

\.U 

Z 

u. 1500 

o 

\.U 

§ 1000 

~ ~ 

" , 

-, 

" 

" '. 

" 

' .. ' .. ' .. .... .... 

DIRECTION 

-X 

---Y 

--z 

...... _---.. ---- ----------------

(3 500 

« ...... , :: ...... -------, OL-___ ~~~ __ ~ ___ ~ ____ ~~~~"~C"t~!====me~==~~~~ 

14.5 150 -:155 160 165 170 175 180 18.5 

~REQUENCY,Hz 

Figure -I \lagnttude or load cell transfer functIOn;; for 800 Ib apphed lateral forn' 

::! 2700 

~. 2400 

« 

g 2100 

..J 

Ii! 1390 

u 

c 1500 

« 

o 

..J 1200 u. 

o 900 

\.U 

C 

~ 600 

Z 

~ 300 « 

~ 

o 

145 1S 0 

-- .....-- - - :mr,...lo-"",..,J,_ ............ -: __ ..1 

15516016517.0175 laO 18 S 

FREQUENCY. Hz 

Figure 5 \Iagnltude or inertial load tran~f(>r functlOn~ for 800 lb applied lateral for.~· 

ill 

I 

I 

! 

! 

i 
, 

160 

.c 

~ 140 

X 

~ 120 X 

z 

~ 100 

o 

a: 

c: SO w 

u. 

o 60 w 

o 

:;). t:4'O 

z 

~ 20 

:: 

LATERAL HUB LOAD.lb 

200 

400 

600 

800 

fIgure 6 \tagmtude of x-axIs applied force estimatIOn error for applied lateral forces 

.c 400 LATERAL HUB LOAD, Ib 

c.ri 

X 350 

<X: 

>- 300 z 

c: 

0250 

c: 

c: 

w 200 w 

:l: 

... 

11. 150 

o 

w 

0100 

::> 

... 

z 50 t:) 

<X: 

-200 

---- 400 

--- GeQ 

--800 

I 

~ I 

I 

I 

/ l / ,~ 

I " / /' ,., , 

:: a L-_--' __ --L __ ~J. __ ..:..I.'___...I_ __ .....J... __ ...J 

15 a 15.5 16 a 165 17.0 175 13 0 185 

FREQUENCY, Hz 

FIgure i \ldgmtude of y-axIs applied (orce estlmlltlOn error for applied lateral forct" 

, 

L 

: --...- . 

, 
:9 100 

Cli 90 

X 

-:'\ 

LATER.AL)iUB LOAO,1b 

-200 

c:t 80 N 

Z 

70 

a: 

I \ I . ----- 400 

--- 600 i,' \ iI\\ --- 800 

g /: \\ "v. 60 

a: 

~ 50 

-, ,. v ~ 

II \\ /1 '" 

} :;\\\ .. :I: f / I, \1 I I'" A 

I- 40 u. 

o 

I~ I ' \ \\ _ I'. V ' ... 

I ': \1\ ./,''' ... , 

w J: "\'/' \ 30 

o 

~ 20 

z 

.N, I \ \".J''''''..I .1;.. 

, ... 1 /,' t . ..... I ,~""-"'J \., • • M." '* .,~.." •• ..,;..'" ..... "'.,' '!\,./" ~ l~ ,'\ • I , "I '\ 

CJ 10 

c:t 

• ..l,..~" • " 

~ O~-----~---~~---~-----~------~-----~-----~ 150 155 160 165 170 175 180 185 

FREQUENCY, Hz 

FIgure 8 \1agmtude of z-axIs applied jvrce estimatIOn error for applied lateral furcc~ 

7000 

z 

§ .0 \ 

~ :; 5000 \ 

w • \ wI-

~ § 4000 '. 

O- u.o " w ~ 3000 "- a ::! '... " ..... x......... " 

LATERAL HUB LOAD, Ib 

--200 

----- 400 

--- 600 

-'-800 

t=:<! ..... , "'" 2 x 2000 .......... .:--'.... . .............. 

~ "'--... ~~ .......... --- <! ~-----:--."'- :: 1000 -----::::---.._- ,-------==-:=--

o~--~----~----~--~----~----~--~ 

150 155 160 16.5 170 175 180 185 

FREQUENCY, Hz 

Figure 9 ~1agnltudc of r-dXIS applied momert c<;tlmatJOn error for applied later.d forces 

3G 

; " ! . 

, 1-- , I 

j 

I' I 

I 

. 
1 

i 

I 

I 

I 

I 

J 

/ 

350 

300 

c:: 

g :: 250 

0:-

w .... 

~ ~ 200 ... ~ 

u.O 

o :2 150 wen ax 

~ <t 100 

z> 

C) 

~ 50 

LATERAL HUG LOAD,lb 

200 

400 

600 

BOO 

o~--~--~~~~----~--~----~--~ 

15 a 155 160 165 170 175 18 a 10.5 

FREOUENCY,Hz 

Figure 10 \ragnltud~ of Y-d"is applied moment e~tlmatlOn error for applied lateral forcE'S 

LATERAL HUSlOAD, Ib 

-20:.1 

---- 400 

--COO 

90 

GO 

30~--~----~--~----~--~--__ ~ __ -J 

15 a 155 160 165 170 

FREQUENCY, Hz 

.,-

.11 

175 m a 18.5 
a: 

o 

1200 

1000 

aoo 

c:: 600 c:: 

w 

X 

u. 

400 

200 

INSTRUMENTATION NOISE 

_ z 

---------.- V ------ ... _----- ------------ o ------- ---- - ---- -------,--------... -------., 

1!:i 155 16 lG 5 17 17.5 18 10.5 

FREQUENCY, Hz: 

Figure I:! . Comparison of magnitudes of I-a\IS force estimation crror for appllcd I, 'J anJ ':-,\\l~ 

loads 

I 

• I 

I I , -!>I ... J 

I I 

----~(!;] 

I 
/ 

320 

140 

120 

:: 100 

c:: 

0 

c: 

c: 

w SO 

>- u.. 

60 

40 

20 

0 

15 -

INSTRUMENTATION NOISE 

", I \ , , , I , , , I , I 

I , 

I , 

" I , " , ' .. " , .. , I, \ 

\.... 't I \ ,I, ' I " , , 

v t, '- ", '.. I \." t '-----~\,:-J 

15.5 16 16.5 17 

FREQUEf'~CY, Hz 

17.5 1a 18.5 

Figure 13 • Comparison of magnitudE.' of tlw y-a\I' forrt' t"~tlmatlon error for applied z. y and .: a,\l~ 

loaos. 
! 

, 

.' 

/ 

, 

, 

---

I 

/ 

i 

:e 

c£ 

o 

a: 

c: 

w 

.., u.. 

600 

200 

lOa 

\ 

\ 

\ 

- .. ------------- -~~- .. .:.-- --....... ~-~-~.;; 

INSTRUMENTATlmJ NOISE 

.------.-.-.-.~ 

/\z - // \ 

I \ /.-' 

/ \ /-" 

\. / -'\..._/ \ .............. 

.-

o~~~~------~----~----~~--~~----~----~ 

15 155 10 165 17 17.5 18 18.5 

FREQUErJCV,H~ 

Figure 14 Comparison of magnitude of :-a\I" force estlmatlun error for dPphed x. y and: aXIs lo,td~ 

10 
.-' 

I 

=: 

cr.' 

0 

cr. 

cr. 

UJ 

X 

~ 

3500 

3000 \ 

\ 

\ , 

, 

2500 

2000 

1500 

1000 

500 

, , 

, 

, 

, 

, 

, - , 

" "' ... ''''... y "', ..... ' .. .... _-.... , 

'-.. ..... -.................. .... ...... - .... -................... .... 

--.. .... 

INSTRUMENTATION NOISE 

._-'--' -_. -_. --' --. __ . --.--.-

~~-=~~=:==~~~~~=~~i~D~==~~~~~s-======~~~=--===--~i 16 16.5 17 17.5 ~a 185 

FREQUENCY, Hz 

Figurl' 15 Comparison of magnitude of x-a\IS moment C'stlOlatlOn error for appll('d I. lJ and :·a\IS 

./ 

l 

I' 

I 

j 

t 

. 

~ 
- . 

.' 

: 

I 

.. 

-

~. 

a 

c: 

~ 

700 

I 

INSTRUMENTATION NOISE 

UI 300 >-

~ 

200 

100 . " , ~, ~ 

' .... _- " Y ,-... _------... ----.... ---- o - --:-""L_' ........ ""'.-::;;-_1'-4.-_ ... _"' __ I'--__ -I'--__ --I ___ -I 

15 15.5 16 16.5 17 17.5 18 165 

FREQUENCY, Hz 

Figure 16 Comparison of magnitude of Y-d\lS mOrnl'llt l'~tlmatlOn error applied x. y and .:·a\l~ 

~ 

1

' 

! 

, 

; 

II 

~ 

, 

t 

i 

. 
I 

I 

/ . / , 

I / 

I I 

I 

I 

- . 

, 

/ 

I 

- ! 

f 

/: 

.. J • I 

J / 

/ 

! 

! 

I 

I 

/ 

I ' 

, 

, 

" I 

~"~4"""".-z:. ........... _.-.... ...-....._ ... _. ________ _ 

140 

120 

100 

:9 

.:: SO 

a:," 

0 

a:, 

c: 

w 60 

N .,.. 

...: 

40 

20 

\ 

, 

, 

I 

\ 

I " ' ... , . .' \ 

-, Y 

• \ 

\ 

\ 

' ... .,.., INSTRUMENTATION NOISE . -_. --.... --. -_. --. -_. -_. --. --'-

"'''' .... " r"', ,-..... " , , --,-........ \! .... " '-............. ' .......... ,-.... --....".. "'''-'''~ .. 

'-'" 

15.5 16 16.5 17 17.5 13 18.5 

FREQUENCY, Hz 

FIgure l'j Comparison of magmtude of .:-a"15 moment estunatlon error for applied r. y and.: <1"15 

loads 

~ 
" • 

., 

/10 

I 

! 

/ 

I. 

t 

I 

. 

I . ;. 

. '. 

, 

/ 

j.' 

. 

" 

/ 

--- f I . , .'- / 

189 

15G~ ____ ~ ______ ~ ______ ~ ______ ~ ____ ~ ______ ~ ____ ~ 

15 155 16 16.5 17 17.5 18 18.5 

FREQUerJCY. Hz 

Figure 1 g Pha~e lag unmodeled roll morncllt for applied lateralloadi 

~ 

t 

I I 

! 

I 

! 

I 

I. 

I 

I 

I 

I 

I 

I 

1. . 

! 

, 

1 

t 

~ 

J 

i 

I 
/ 

/ 

./' 

/... I 

I 

/ 

, 

/ 

/ 

" 

/ 

I 

/ 

I 

/ 

; / ,/ 

- --- -- --- ------------------ -- -

RSRA 

T 

I -------------- -- '--I 

I " 1\ J I 

I ~ HUB LOAD H LOAD CELL .!...d Ie 

I DYNAMICS MODEL!~ : 

I I 

I I 

I I 

I I 

~KALMAN __________________ FILTER " 1 I 

MAXIMUM 

LIKELIHOOD 

IDEtHI FICA TlON 

A 

a 

Figure 19 ~chematlc diagram of on-linE' calibratIOn 

.10 

.1', '0 

~ 

.. 

=i .08 // , 

(,J ;' , fa: .OS 

, -I \ -I 

W , 

(,J .04 \. c 

<{ 

"-

"-

0 

-I .02 "- -I 

<{ 

'--

u .00 ---. ~ --::z:: ::.=;g 

c: 

w 

> -02 L I I I I I 

0 1 2 3 4 5 6 

NUMBER OF ITERATIONS ON 10 SAMPLE DATA SET 

Flgure:!O Identification of \ertlc..!.1 load cell an'5le 

I 

I ~' 

\ 

I 

I 
! 

/ 

i 

I 

I 

, 

I 

, 

, 

, 

/ 

I 

I 

/ 

Y 

I 

, 

/ 

I 

I 

/ 

!. ------... - .. 

.I;' ,I I 

r _. I I 

.03 "0 

~ 

.. .07 

"?! 

<t .05 

> .05 \ ..J 

..J \ w .04 

u \ 

0 .03 \ <t \ 0 .02 ..J \ ..J .01 <t \ 

a: 

w .00 \ --- 0-

I-<t - 01 \ ,," ..J v .... -.02

0 1 2 3 4 5 6 

NUMBER OF ITERATIONS ON 10 SAMPLE DATA SET 

Figure ~l IdentificatIOn of lalerallodd cell angle 
i 

, -, 

:-

; 

~-

-~ I 

'/ J 

I 

I 

I ,I 

I 

I 

I 

J 

J 

, 

/ , 

/ 

, 

/ ,- / _ .... ,-

I , : 

~ 

.:: 

c" 

ct 

0 

..J 

to 

::J 

:r: 

lZ 

UI 

:E 

0 

:: 

0 

z 

:::i 

..J 

0 

c:: 

- ACTUAL HUB LOAD 

••• ESTIMATES USING INITIAL 

PARAMETER VALUES 

3000 ---ESrlMATES USING FINAL 

PARAMETER VALUES 

0 

-1000 

-2000 

-3000 

-4000 

-5000 

-6000 

0 .15 ,30 .45 .60 

TIME, !ec 

FIgure :!2 , Companson of actual and c<;limaleJ y-axIs hub loads 

4i 

__ .,.ry¥i7'l,c ........ ~ ............... oi __ ~ __ .... ____ ~ ___ ._ •• _" _______________ ~_--------- - ----

. - --- -.~- ----

I 

r~ 

t 

I 

I 

,~ 

, 

I 

I 

I. 

I 

I.-

.. 

I..; 
~ 

0" 

~ 

o 

..J 

o 

-1000 

~ -3000 

:t: 

(I) 

X 

~ 

). -4000 

-5000 

o 0 

............... t .................... y ,- -.-- ~ -> --"OJ .. ___ .. _. ....._ _ _ 

- ACTUAL HUB LOAD 

o 0 0 ESTIMATES USING INITIAL 

PARAMETER VALUES 

---ESTIMATES USING FINAL 

PARAMETER VALUES 

o 0 0 

o 0 

o • 

00 

-6000~------~------~------~------~ 

o .15 .30 .45 .60 

TIME. ~ec 

FIgure 23 Companson I)f actual and e"tlmatcd roiling moment hub loads. 

48 

-.. --~ 

I 

I 

I 

I 

1 

I 

I 

\ 

1 

I 

I 

f 

I 

I 

I 
.. f· I 

• 

Figure A 1 Un deformed load cell geometer) with generalized fC'actlon force 

l" .i 

\ 
e 

~~C~X~ ________ F_X~~~D_X~~~ 

.. TOP VIEW 

Lh 

-,-r------j'-r-i-I> Y h 

hz 

REAR VIE\aJ 

Dy 

d 

-r--------+-r--+-> Xn 

SIDEVIEW 

Figure A 2 Load cell geometry \\lth generalJsl'd reaction force components 

.~ ~i>~l'l~f :::i'r{;5;f.ic;~;i :~'~3~~~y,~'~' ' 

10 

" 

" f 

I 

• 

f 

I' 

I 

. 

i 

i 

t 

!. 
-----

. , 

'j 

/\ ,/ 

-. 

, \ 

" , ., 

' . 

-, 

.. ",' , 

.1 

- , 

" 

'~'b~~-'~_'~'_:~f~~:J:~~ 
End of Document