NASA-CR-166609 19850007406

A Reproduced Copy ~ ..

Reproduced for NASA

by the

NASA Scientific and Technical Information Facility

111111111111111111111111111111111111111111111 NF02167

FFNo 672 Aug 65

.... " "35 . ; ! ~;' ,~.

J:/'.HG!.EY RESEARCH CENTER WBRARY, ,NASA

~,~~TO~ Y1RGllill\

..

\

(.~.

NASA CONTRACTOR REPORT 166609

(UASA:-CR-166blJ'J) STUtY TO ELvnaATE GUCUllD N85-1 :715 nESO~AHCE USING ACTIVE CChlfClS (Hughus Helicopters, eulv~r City, edlif.) 123 p He Ad6/Hf A01 eseL Ole Unclas

G3/0 5 13titi6

Study to Eliminate Ground Resonance Using Ar.tive Controls

F. K. Straub

CONTRACT Nf\S2- 11261

October 1981.

N~P\\

•

.~ -- --'--"~-''''''«'';''i.'~-< - •• -,'.~ ....

Alfs~ 157/;;;;;9

r r

" I.

~

NASA CONTRACTOR REPORT 166609

Study to Eliminate Ground Resonance Using Active Controls

F. K. Straub Hughes He11copters, Inc. Cul ver City, CA

Preoared for , I

Ames Research Center under Con~ract NAS2-11261

Nff\SJ\ I

, N3t:onai Acror~:-lut'cs and , SP<Jcc Admln'l,trrtt,on

Ames Research Center Iv10lfett Field, Caillornl<J 94035

-- -""-~------

.

....

•

~...-----.-

TABLE OF CONTENTS

p"efar

::mmT Irroduc~ion

2) Background

3) Ata 1 Y ti co 1 ;10de 1 4) Corr~l~r--n wlth G~ound R~son~nce Data

I - .... .;.

5) Con+-rol L-" uev;:lQOr:len~ I' .~'-

6) RrSUlts

sra •• F.,., •• k Studl ••

Erfect •• f Rator Con.lguratlon

Toar R". an"

7> rndU.iOn.

RefernC",

FlgUr'

Tables

APPel.I' A,

APpeL I, n,

List of Sym!lols

Equations of Mo~ion

. - ------------- -'-'.~- -- ~ .. - ---_._-0'"

1

2

7

15

..,., c.""

.,­<,_I

31

41

47

50

.. .. -

· :

....

:::

I I

- i

PREFACE

This r(! ort I.:as prepared blJ H'Jghes Helicql~e I. Inc .• under

t·!AS.t. Con c~ r·~~':?2-11261 fUFl.jed bq the Nation Aeronautics

and Spa e Ad~inis~ra~ion. The Hughes Helie lters' proJect

eng ineer was D.·. F. ,. 1" •• Straub. Tech ical program

diredi was provid~d by Dr. W. Warmbro: of NASA Ames

Research Cen~er (Mof~ett Field. CaliFornia).

PAGE 1

---------- - -

I L

SUMMARY

H~licopter arau~d resonan~a conditions typically require

. I.L'·' ... d . I d' d Incerpordr.10~ ~~ augmente~ b!d e lnp ane amplng an

tailorinl ~r rator and ~uselage rrequ~ncie~. The present

study i~Lesti1ates th~ eff~ctiveness of active control blade

featherih g i~ i~creasing rotor/body dampin~ and possibly

eliminating grcu~d resonance instabilities.

A. a.a1,1"., n ••• 1 •• p •••••••• , •• t ••• lapp'.' a.d lead-lag

degrees or ~ reed c:n and body pitch. rolL longitudinal dnd

Th e motIon later.;]l Tot::~ i.= developed

llnearized ~~~ ~ransrormed into consta"t coefFicient form.

Blade felth:r:ng ~ppears as a forcing term in each of the

degrees l, ~r~~d=m. A thorough correlation uith experimental

n ... 1 h'fg.i •••••••••• t •••••• du.t .. t ••• 11, ••• th. mod.!

:~:.::. 1:::~::· ::~:~am ....... ing •• inp, •••• t.d a .. t .. . variable 'e~~~~c~ through a conventional swashplate.

influence

. h . I ·.;J:lg. tlng

=~ vdrious feedback states. ;eedback gain. and

b ? i;;J e en the c y c I icc 0 n t r 0 I s (r~edback phase) is

studied thrQ~1h stability and response analyses. Resul+:s

shew that bl~de cyclic inpla~e motion. roll rate and

a~celeraJio~ reedback can add considerable damping

s·.Jste.~ d Iii eliri!1ate ground resonanCe instabilities.

roll

t!1 the

PAGE 2

ORIGINAL Pi\'::'~ • .:. OF POOR QU,c\L1TY

..

- - - -- - . -----~-- ---- - -........ . __ , __ ,_~ __ " ____ ._,,, __ .,,-_Y"'- #t.-- -

~ ..

.-

The feedback Dh~~a is seen ~o be a powerful parameter. If

chosen prooerly it ~a,imizes augment~tion of the regressing

lag wod~ in'?ren~ damping. For roll acceleration feedback

the ;eldb~:k p~3se has co"siderable effect on the roll mode

I frequency. Thi~ could be u3ed for active control of

I frequency pl~ce~ent and uould indirectly improve system

stabili1ty.

Rotor config~ra~ion para~e~ers, namely blade root hinge

offse~, fl~pping stiffness, and precone have considerable

influence 0" ~~~ control effectiveness. R~sults show that

3ctive con~r~l is particularly powerful for hingeless and

PAGE 3

r---------.-

- . ,

.. -."

f

f

I I

INTRODUCTION

Aero:Tlec han i -::-3!. rotor/fusel.age instability, specifically

I ground reso~a~C~ instabilit~, can occur ~cr articulated,

hingelessJ and bearing less rotors. Typically a range of

various Ja4!Oad config~rations are encoun~ered for a

'drtieUld{ •• iie.,tor. This makes it Vo"" di;iieult to avoid

groynd reso~ance ~y tailorin~ the body and inplane rotor mode

f~e~uencijs to avoid coalescence fo~ all operating

conditionJ Thus, the des i:J!'Ier in many cases has to resort

to m~chanJc~l le.d-Iag dampers for articulated rotor systems.

T ~ is" '? a nl s i !": C 'i' e a sed cos I; , C IJ m pie x i f; y , tTl·? i n I; en a n c e , 'II e i 9 h t

and hlJb idr :3'1' Hingeless rotor systems h.av~ not seen

extensive ';ses in the helicopter industry, in part. because

of poor i~~erer.t

I

aeroel as l:i c stability charac tel' i st iC'.;.

Consequentl4 a r.~ans to increase aeromechanical stab~iity in

a reliable cann~r could significantly improve the utilization

of all rotor hub design con~igurations.

T~e purpos~ cf the present· study is to evaluate the potential

I U3e 0; actIve bl~de pitch control to in~rease rotor/body

I . syste~ danplng. Such an ap~licatilJn could possibly eliminate

the need ~or nechanicallead-Iag damoers to augment rotor

s'.Jste~ danpln(]. 1 -

coupling be~~een

11 C · I. .' we as 0110115

Stabilizin~ effects can arise both from the

blade. fla~ and body de~rees of freedom, as

coupling b~l;ween blade flap and lag motion.

I

PAGE 4

.....

. ,

r I

."

The concept oT I

active con~rol blade feathoring has been

5UCCQSSTully denon5trated ror vibration reduction and the

tachnology is now available ;or advanced app 1 icat ions.

Showing analytically the ie~sibility of using active control I

to eliminata orOJnd re~onance would represent a further step I .

to·.iJards an a·h-anced, fu ll~ integrated, nultimode control

sy sten.

Section 2 starts with 3 literature revieu of the two

ingredients aT the present study: helicopter aeromechanical

stabilit~ j'd a~~lication oT active control blade feathering. I

This is follo~ed by a 5u~~~r~ of the obJectives of the I

present work Next, in S=ction 3 the assumptions on which I

the mathematical model oT the rotor/fuselage is based are

described and an outline oT the derivation oT the equations

of motion a~q solution proc~dure is given. The comple~~

e~uations or r.o~ion are lis~ed in the Appendi,. To validate

the governlng equations of notion a correlation study was

performed. In Section 4 results of the present analysis are

co~pared Ulth e,perimental datd and existing analyses.

The i~plemencation of the active control s~stem and various I

control methods are described in Section 5. All numerical

results, d~scribed in Section 6, are based on state variable I

fee~bac~ co~trol. Those active control simulations are I

intended to shOW the effect of various feedback variables on I

s~stem stability and provide a syste~atic approach in

choosing t e ieedback para~e~ers. The cffec~ of key rotor

PAGE 5

....

;-, !

i 1 I

Ii

pa~ameter is also investigated as is the rotor response

behavior. The report close~ with a summary of the major

Findtngs d c~nclusions.

PAGE 6

~ l - ----_. ____________ _

," .,"

L~ _., l "-'"

B At; ~GR OUND

Possibly ~ha n~st funda~cntal instabili~y ~ssociated with

ro~o~crlTt is ground resonance. This description strictly is

inccrreJ~ 5i"ce ~he phenonenon is in fact no~ a resonance but I

a true ins~a~ili~y. I

A more appropriate name is mechanical

instability whic;, has seen n'Jre use in recen~ t.}ears. This is

an ao0100ria.. d •• criotion becau5. tho oh.no~onon can occur

in a vacuum.

G~ound resO~ance, as a machanical instability in artlculated

is ~911 understoo1. The classical ~orks of Coleman

and Feingold. Reference 1, and Deutsch, Reference 2, I

identifi~d tho rotorcraft parameters and their relationships

in d~rijin~ ~~is ~echanical instability. Their works showed

that tJe ~hanc~enon is fundamentally si~ple but that the

relatioJshiPS required for stability among the paramete~s are

I ver~ co~plex Th~y also showed th~t mechanical instabilit~

is PosJibla onl~ ~hen ~he natural frequency of the rotor

b~ade lJ99i'~ (o~ inplane) notions is less than the speed of

rotatioJ of ~he rotor.

In si~pletar~s, ~echanical instability can occur if: 1) The

lag ~re~uenc~ of the rotor blades is less ~han the rotor

sileetj C)Of+; inplane), 2) The lag frequency ninus the rotor

speed, . J e., ~he regr'essive lag mode frequency, approaches or I

coale5c ~ith the frequency of an airrra~e mode. and 3)

Certain relationships among the blade lag damping anj

PAGE 7

-'" - -. ...

,

i I. I J,

i .' .' ! ~

I ! t. I I

(. !.

I'

II ~ ~

f-I·

airfraMe modal dal1ping, and the effectivE! rotor mass and

airframe rno~al ~~5ses are sa~i5fied.

Coleman'S eq~a~i~ns have been ~sed extensively to define

stability blu~~~ries for articulated rotors and determine the

blade and fi,e1 system damping values required to prevent

ground resonance. For exar.ple see Reference 3.

Hingeless 1otor ~qstems of the soft inplane type have

another aspect to the mec~anical stability problem.

inherently lou blade structural damping and lack

added

Their

of

machanical blade dampers can result in severe mechanical

instabilit~ prc~lems on the ~round (a~ an eJception, th~ Lyn1

rotor has ~ec"allical dampers), Be<3ringless rotors, in many

. I L I . b' . cases, lnc~rp~ra~e e astomerlc 3nub ~rs WhlCh can be deslgned I

to add d<3'Clng to the blade ~tructural damping for inplane

blade notibn. I~ a1dition, the eidstic flapping of hingeless I .

or bearingles~ rotors and the resulting l~rge hub moments

lead to ~re~t~r aeroelas~ic couplir.g between the rotor and

airframe bbth vn the ground and in flight. Thus a hingeless

rotor can clper-ience ~oth ground and air resonance, more

a~propriately de~cribed with the common te~~ aeromechanical

. . I' I lnstabl lty.

Evidence 0; the significance of the encountered problems are

I the numerous p~blica~ions d~aling with the aeromechanical

.stabili4;y! of -3c4;ual hingelass and bearingless rotor designs,

R • I .... . eter-ences ';-lv.

PAGE 8

-.-.------------r'-... -~---.-.'l..'-. 7-..-~:;' ...

..

r ~~ile hinleleS9 rotor designs are attractive through their

I

nechanical siwplicitV' anal~~ical modeling is complicated due I

to 9t;rong a~rojyn"3i'!'1ic, in'~rtial, and structu'ral coupling as

as ir,herentllJ nonlinear blade d eformat ions.

Considerable res~arch efiort has been directed towards a

better ~nd~T'S~anding h

i. aerC:1\ec an.c~l

.I

oi the hingeless rotorcT'clft

stability problem and investigation of design

paraiTleters that' '.IOU 1 d . incr ease b lad e lead-lag damp i ng. I

A

s ;1Ia 1 1 sam p 1 eo;: t his '.11 0 r L w hie his use din the pre sen t

. I . t" . R.I! 11 17 study, 1 S c 1. e '.I 1 n ere'" en c e s -. analytical.

Reference 11. an1 experinental work, References 12 and 13,

have been par;~rr.ed, Recen~ly, increased er::phasis has been

pl~ce~ on correlation betueen analytical and experimental

I~ Reierences 14-" .'), thr(te differ'an4; anallJscs of

various sophis~ication are currelated with the exp9rimental

do' ••••• ~ •• d 'n R.f ••• nc., 12 ond l~ Po. th'. ..1.".01,

sinpl~ hi~g~les9 retor/body configuratl~n, agreement is

generally goed. Furthermore, it seems that some or the

discrepancias can be removed

aerOdyn~m~C nod~ling~ as ,~~un in

blJ improvements in the

Referenc e 15. However,

Reference .-~ I indicates that when atter!l;lting to model more

realistic hl~?eless blad~ designs or bearingless rotors, ..

exis~ing an31yses lack su~~icient accurac~ and consistency in

'hei. ..1om.Chon,c.1 ••• bil.t, p.ed'ct'on.. Thu.. i. i.

evident that c~"siderabl~ c~re research is needed to develop

be~ter anJlytical m~dels and validat~ them against test data,

I

A nodern ro~orcraft mu~t operate in a severe dynamic and

PAGE 9

• __ ._. ____ .-_____ ~ ... "'_..._ ___ ~-t

I

..

t I ..

f •

i

I-I·

·1 l

I ~

, . ,

aero·jlJnar.'I1c environment. This includes atmospheric

turbulencl. 'higi,cr haT.lIonic ~lade air loading and bending and

I loads,

I

shaf~ sta 11 and pot~ntial blade

in'ltablliti~s, irr.pu!sa loading due to bladeifu-:;elage

interT~re~ca, a~d advancinJ blade Mach number effects.

1 · ... 1 . t· .. 1 app lcar.lon OT ac lve con.ro blade feathering mdkes it

fea5ibl~ to alleviato 50.:'10 or malJbe alloT these effects

wh11~ improvin1 ro~orcraft vibration and handling

ch~ract~r~'i~i:s 3nd ~hus '!xpdnding the ro~orcraft flight

envelop~. So~e of the different approaches and possible us~s

for active control syste~s for rotors are described in

I References lS-20.

Most of the past studios in activ~ control for rotorcraft

have dealt '.Iith gust alle .... ia~ion, e. g., References 21. 22, "r

S IJ C C e 5 5 T u 1 ;lig~t of hig~~r harmonic active control for

It alsu contains an eJtensive reviaw of

previous work in this area. I

In these Studl~S, Doth open loop

and closed !~QP adaptive ani gdin schedul~d 'I '.' .' controls

b~~n used t~ ~ini~ize vibratl"ns. Resear:::h in this area 15 I. .

s till ong 0 1"9' e. g., Rerer ence'i 24-26. the data

in Reference 23 clearly sho~s that advances in

on!loard co,~pul;,=rs, sensor and actuator design, and modern

I control the~ry have made vibration reduction through active

con t r 0 1 a Ire:l 1 i t'J.

I

I

I

Applicati~n oi dcti~~ control to improve rotor stability has

I-AGE 10

I I I

',: \ \

,_ ...

received little attention in the past. Of particular

• L \ L • • R .t'. 27' h' h t h 1n~eres~ t~eretore lS ererence 1n w 1C e

aeromeJhanical ~tability o~ hingeless rotor helicopters and

t~. a'~li' •••• n .0 O •• dba,' '.nt •• 1 t. aug.on' ••• t •• da.ping

was studied The baseline rotor/fuselage configuration

\ rese!r.bles an 5-58 helicopte~. However, the ro~or is modelled

having .\ O.u •. hi ng Ole ... ·· .Iad.. with a Hap and I.ad-Iag

frequency of 1.15 and 0.70 cycles per revolution,

l respeci;lvely, af; a rotor tip speed of 650 ft/sec. Blade \

~tructu\al da~ping is assuneJ to be one-half percent of

critical l~~d-Iag damping. This configuration exhibits a

slight Jeromechanical instability on the ground, for thr~st \

to weight rdti·~s greater than 0.6, ·3TH! in hover. The

unstable oode is do~inan.lY a fuselage roll code. The use OT

active control ~as studied by implementing fuselage roil

Dosition a~d roll rate feedback into g set of s~ashplate

actuators i, order to generate longitudinal

c~clic Jlade pif;ch comm~njs. Feedback of ;uselage pitching

motion u~s not pursued since the unstable ~ode I

and lateral

hds a . I'

relatively 5·::0311 pitch cOr.lponent. Numerical results o·F

Reference 27 are presented bl plotting dar.lping values of the

critical as obtained from eigenanalyses. The

corresponding fr~quencies aT"e not presented. These results

show tha~ feedback of roll position and roll rate can

stabilize the ur,stable roll node both on the ground and in

hover. In R e-rerence 28 blade lag rate feedback through I

individuai-bl~d~-control was used to augment the lead-lag I

mode d I. ar.lp, 1 n? Ot isolated rotor blades. Two blade mounted

PAGE 11

... -

.. '

.0 •• 0

!J:~' .. ' ~ .. :::. . -... ~ . "1 '

accelerometerlsare u~ed to s~nse blade lead-lag acceleration.

This signal is then integr~ted and rate inrormation is fed

back to a, ac~uator that controls th~ pitch angle of an

individ'Jal bli::de. This centro I s'lstem is applied to an

ar t i C'J lated model rotor blade. Data from a simplified

anal,~tical h~\'er model sl;ow that feedback control can

increase I la!} damping ratio, while the Hap damping ratio the

is slightly re~uced. E'perimental results confirm that

feedtlack i"cr'E!dses dampin~ of the lag mode both in hover

I (83/4 = 8 deg) and even more so in forward flight (~~ 0.27).

Reference 29 rorrulated the e~uations or motion for air

rasonance ~r hingeless rater halicopters with active control

cf the COlla1tive, longitudinal and lateral cyclic blade

pitch inputs Here, the b l~.je motions are repres'E!nted by the

fundar.ental elastic mode shapes. No numerical results are

presented. I~ Reference 30 fe'E!dback control was used to·

im~rove the ~itc~-flap stabilit~ and response or single rotor

blades in h~ver. The control system is assumed to have four

independent l~tuators, pErr.i~ting independent control of th'E!

"I d 1 ~ h d' It· d I . +. . Th t 1 T ap an P(oC' lSP acenan s an ve OCl,les. e con ro

s·.Jster. para7,e,:et';i .:;re dete.nined using classical and modern

control thc lC1"Y te~hniques.Numerical results are presented

I .

for a nine-ioot :jia~eter mo~el rotor sllstem.

I the fluttel speed could ~e

rad/s'E!c through application of feedback control. I

For this rotor

raised ~rom 67 rad/sec to 150

At the same

time the dy~amic response oi the rotor at subcritical rotor I •

speeds uas o~ered.

PAGE 12

. 1 -..

.'

--~:::'-a-:Jviaw cd tho literature it is clear that most or the

previous JpPli~ations oT active control to roto~s have dealt

with respJnsQ pro~lems, suc~ as gust alleviation or vibration

reduction. W~il~ by no means simple problpms, measurement of

the ,uantiti~s to be controlled, namely gust or vibration

response, i s ,~ell understood. Thus, control systems for

these problems are ~enerally designed to be adaptive. In

addition, TOT vibration cnntrol the fre,uency of the

oscillatory control inputs is fixed at multiples or the rotor

speed. Thus, for preliminary i nv'es t i gat ions, the

acceleratio"s OT the pilot's seat could be us~d in an open

loop type oi control to mininize vibrations in steady flight;

Reference 23.

To directly e~dlua+;e the performance of an active control

s~ste~ or design an adaptive control systen Tor 5tabilit~

is a considerably more dlTricult problem.

D:~namic s':ability measuraments', even off-l ine, are

particularly pTobl~matic for helicopt~r aeromechanical

stability wher~ rotating and nonrotating ~ysterns are directly

rreedom are involved, and the coupled, ~an~ d~grees of

process lnd neasurement noise levels can be ver~ high;

I Referenco 31 Potential

identifiJation techni,ues

use of real-time parameter

to deteT'min~ rotor damping

I parameter/s cr conplete system dynamic stability has been

studied; References 32 and 33. This issue will require I

additional ~or~ in the futUre if the concept of active I

control to incre~~e rotorcrait aeromechanical stability is to

PAGE 13

. :-.

be fully iii zed.

The purp ~ Q~ th~ present stud~ is to eV3!uate in depth the

potential =r usin~ active blade pitch control to increase

rotor/bod s or; ;. em d a :'Tl pin ~ . The detailed obJectives can be

summarize a~ iollcws:

gate the influence of state variable feedback on

system d3nping. Inclu~e bod~ acceleration and rotor

state ':eedbac k systens which have not been considered

before In addition, evaluate changes in the system

frequ CleS and response.

, .: . .

: 1 2) Study effects ~i feedback parameters; that is

feedba ~ gain and weightihg between the time-dependent

cyclic Con ':r·o Is. Det~rmine a systemdtic approach to . ~. ...

choose parar.'leters for optimal stability

augl'1en at'!.on.

3) Invest gJte the use oi control scheduling with rotor

speed o en~~re stability during rotor ~un up.

4) Assess t~e influence 0; rotor design parameters on the

perfor nce of feQdbac~ control.

FAGE 14

! i

i i 1 i

.,

: .~ .. .• _s •• • .:

ANALVI rCAL MODEL

Analytical prediction of coupled rotor/bod~ aeromechanical

stability liS a difficult task because

aerodynamic, inertial, and structural

nonlineariJies inherent in nodern rotors.

of the strong

coupling and

The modification

of system danping through small changes in blade pitch

settings requires a model that includes all the ingredients

of al'1 aerjelas~ic stabilit~ analysis. Hcw<?ver, care must be

taken to mlke th~ model si~?le enough to allow efficient

simulation of various active control concepts in order to

de~ons+;rate t~e ~easibility ~f this approach. At the same I

time, sufficient def:ail sno'Jld be included to systematically

1 study the effect of rotor/body parametric changes on the

1 control laws.

t1ATH :';ODEL

A brief d~scription or the mathematical model developed for

t,i •• t.d~ .n' the 'Y" •••••••• t •••• 0d.ll.d 'ollo.~ Th.

~ath ~odel is sir-ilar to the models used in ReT~rences 11 and

16. The helicopter body (s represented as a rigid fuselage

having pitch al'1d roll rotations (9y.9x) about the center of I

mass and longitudinal a~d lateral translations (Rx,Ry) of the

I center of mass; see Figure 1. The fuselage physical

propertiel requi~ed for modelling are its mass,

I roll inerti~s, and effect;ve !anding gear stiffnesses and

d . .1 L '-' d LIt· amp1ng 1n ro~a,lon an ~rans a 10n.

pitch and

The hub is rotor

PAGE 15

..

--.....

located

blades

damper

.1

a:;;5\a;c: --h above the <usolage cass conter_ The

are ~s~.u"ed to be rigid and rotate against spring and

res~rjin~s about coincident flap and lead-lag hinges

offse~ ~ro~ ~ha axis oT rotation. see Fig ure 2. The

oi"ientation o~ the hin~~s can be dif~erent from the

aerodynamic pitc~ angla, thu~ allowing modeling of variable I

structural fla~-lag coupli~g and pitch input inboard or I

outboard oT the hinges. Blade precone is included. This

I parameter ~as deemed to be important in this study since it

directly cont~ib~tes to the Coriolis forces which augment

d I.

b 1 a d ~ 1 a g3 r:! P 1 n g.

BASIC ASSUMPT!C~3

The major as~u~ptions on which this study is based are listed

below."

1. The fus~lage or rotor support is a rigid body with

lateral. longitudinal, pitch and roll degrees ~T

Treedo~. Vertical motion and yaw rotation are not

inc lud?d (r- igure 1).

2. The u~perturbed rotor shaft is vertlcal (direction of I

gravity) The fuselage center of mass is located on the

rotor lha~t but ofTset below the hub center by a

distancb h.

"3. The ro.~cr operates in a hovering state with low disk

FAGE 16

:1

·~ ... ~- '. ':' .. ,,_ 1(" .. _ .. _ ..... ~_~.- .....

4.

5.

6.

7.

8.

9.

10.

loading The rotor speed is con~tant.

The roter consists of three or more rigid blades.

The blad~ root hinge is off~et from the rotor shaft.

The fe1athering axis is preconed. j

Each Jlade has rigid ~ody flap and lead-lag degrees of

;reedJm. These motions are restrained by a set of

. I d' d sprlngs an V1SCOUS ampers.

The blade cross-secticn reference points coincide with

Lh 1! I ..' . -0; e Te'3t:'1erlng aX1S. Built-in twist is zero.

The induced inflow is uniform along the blade.

The aerodyn~mic force~ are based on two dimen~ional

ItUdSiis!:ead'J theory. A1'parent mass. conp'ressibilit'J and

stall are neglected.

The pitCh control input is composed CT two parts: th'?

tine .i~d6pendent coll.~tive pitch. identical for all

blades. 03nd the time- .... arying "active" pitch.

11. For th~ results presented here the bl~de active control

is aJplied through a conventional swashplate.

PAGE 17

, ... "

;+)7.·

/

'-, ORDERING SCHEME

In deriving tM~ ~~uations 0; motion For this ~odel a large

nu~b~r of STdll terms appear. Many of thes~ can be neglected

s~ste~atically by ap~ropriately using an ordering scheme.

This is basld ~n the magnitude of blade slopes. which are

typ ically L, the range oT 0.1 < e -:. 0.2. The va:'ious

para~~ters In th~ equations are assigned orders of magnitude.

FIJselag~ mOlilJnS are a!'sumeli to b~ of order O( € IS). The

I

activ~ control blade pitch angle is assumed to be of order

c: ,..$" I O( w ). based on experience with the HHC ac';uator control

inputs. Relepa~ce 23. In applying the ord!ring scheme it is

t~en assume~ th~t terms of order O(E2 ) a~e negligible in

comparison with unity. In addition. all ter,"':s that contain

p~o~ucts of th~ fuselage d~grees of freedom ar~ neglected.

EGI)ATIO:~S OF :10TIOtJ

T~e system equatlcn~ of notion are derivad using the

Firs~ ':he blade dis~rlbuted inertia and

aero:1ynamic 1'J-3·js. using ~uasi-steady .:!erodynamics are

derived. The r~ flap an·:1. N lag blade aqlJal::ions (N~3) are

obt~ined by Inte~rating the distributed blade loads over the

length of tne blade and eniorcing moment a~uilibrium at the

root hinge. For a detailed ~escription of this procedure see

Referonce 34. The four fuselage equations are derived From

d . 1'1 . b '. t f-h t f . 1 d' ·h yncJitllC equl 1 rlU1l a • Ie cen er 0 ma'is. lnc u Ing .. e I

rater loads a!; the hub. ruselage inertia and gravity loads.

I and fuselag~ callstraint loads dfJe to landing gear springs and

damp ers.

·PAGE 18

/

~.

I. l

i ~ . ,. I I i

I , I

! . r l.

.L ~.

.! .

I i , , I , .

t,

.~

The

, '1' .. '<-

resulting governing system of 4+(2*N) equations is

c: C! IJ pIe d , non! in'.!.:or and has periodic c:oefficiants.

The active c:o"trol pitc:h input appears as aerodynamic: forcing I

expression l~ all equation5. The values in the blade lag and

~USelagJ tran~la~ion equatiQns are one artier of magnitude

smaller than in the flap equations and in the fuselage pitch

seems and roll eq~~tions. I .

two Dri~~ry mechanisms exist to stabilize ground

I resonance.

I

that

From these equations l~ therefore

Fi rst, the fuselage pitch and roll motion can be

controlled through the pitch and roll moments ari~ing from

flapPinJ. Tha ragnitude OT each is directl~ related the

1 I. • b ade 100~ hInge offset and rlap spring st19fnes5.

second nechanisn is lead-la~ damping augmentation through

C . l' I l' . '-h arlO 1S C~~D Ing Wl~

I ~reS2nce of either steady coning deflection ~r pr~~o~~.

b l-3d e Hap rna+': ion. This requires

As discussed in References 15 and 28. unsi;eadl.J ..,-.-

effects (d~nanic inflow) can at times have a considerable

effect on the b l~d e flap motion. Since flapping plays -30

I Important T'Ji.:! in stabilizing ground resonance perhap<; the

1 .1 aT the present s l:ud '.) would be changed to cone USlons $oma

degree. In particular Tor high flap stiffness rotors

unsteady aerod~namies should be included in a more refined

model.

FAGE 19

~A..J)".l'.-' .:;. · "'- ~- -, - -"'" <.'. ~ - - ...

; ..

'''} . ~. l ., :. (-

!. ~ r. : I I f •

. i

•

- I·

I ,

l ' I I

I' ;-

~.

· I ~

50LUTIO~J MET~C:D

The nonline!r eq~ations of nation can be solved in

I . the tine d :) ~a 1 n. HO:.Jever, fo-r- parametric stabilitl,l st'ldies

a frequ~ncy dQ~ain solution is much mo.e d~sirilble. The

e:tlJd t ions 3-r-e th~refcre lin~ariled to allcu an eigenanalysis

capability.' The. steady:-:;ta.t:::!. nonlinear eq'Jilibrium position

is obtained assuning that the fuselage degrees of freedom and

the active bl~de pitch are l::!ro. In the case of hover, the

blade equilibrium position is independent oi time and can be

obtained iter~+;ively using the Newton-Raphson technique. The

linearized plJrturbation eqt!a':;.:.ns art then :.Jritten as

= 0

The linear, periodic coeFiicient perturbatl~n eqlJations can

I be solved U51"1 Floq,uet theory; Reference 35. . I. " .

In the present

stu d :J ' t r '? e ql~ at ion s . are con vel' ted in'; 0 a con s tan t

coefficienf s~;t~m IJsing the multiblade or FourIer coordinate

transror'-::"rlcn; :1ererence :6. This is possible under the

assu~ptions tha~ all blades are identical and that the active I -------- --- -----P_i:_~_h __ -=-~p(= __ is generated through a conventional _s_w~shplate

~ith threel "active" actuat:lrs in the fixed s'lstem. :.Jith the I

rotor be~ 9 in a hov~r condition only the first cyclic blade J 1

motion3 .1 I d 1n i- ap -:In leari-Iag couple '.!Ii th the fuselage

\.lot ions. Th~ collective and reactionless blade e~uations are

P4GE 20

- .. ,

.'

'.

, ! •

. '. - --- - -

not ne~ded_ The problen ~ize is thus reduced to 2 Flap, 2

lead-lag ln~ 4 Tuselage equations.

(:1(qO)J;~'" ('::{qo)Jq + (iJ·{qo)Jq + L~(qoj)~ :: 0 - - -qT = -

Stability OT ~h~ ground resonance problem in the fixed system

is then lvaluated by transiorming the equa~ions into First

.. • I d f' . 1 l' Th . ft or~er Torr.'! an per or~l1ng an el!}enVa ue ana YS1S. 1S ,"orm

of the 9 rni"!} equations is also used to co~pute the tim~

history and frequency response of the system .

. (A]X (BJu x = + , - - -

xT -= (qT, qTJ ~ - -

PAGE 21

...

\

i:

r l l I t I r' l . I

I

J •

i i r : . r r ~ .

CCRRELATION WITH GROUND RESONANCE DATA

to villidelte the governing equations of motion, a

correlation s~ud~ was periarned for some of the

confi9uratl~~~ reported in References 37,

~otor/bod'J

12 and 13. No I

active controls ilre utilized for these results,

An EH C Ii! 11 e:1 t and exten~iv~ body of e1perimental and

analytical d~ta regarding +:ha state of the art in aeroelastic

stability iln~lysis was presented at the ITR Met hod 0 1 t' 9 IJ

Assessment t,jor~shop held ell; lJASA Ames (May 1983) ; Reference

17.

in

~rom th? co"relation studies presente~. it is clear that

certail cas~s considerable differences exist between

analytical predictions and ~xperimental results. HOl.llever,

tl'\e sinple

, '" ' t j In..-e::·.lga _on

node 1 the presf''1t ana IIJ tic a I used "'01'

to

crossovers and d~~ping trends adequately For the rotor/body

It is not the purpo~~ of the clJrr~nt

,study to l ,";'1" i o',e up on +;h~ state of the ~rt of aeroel~stic

stability preoictions.

Results of ~ cl~ssical ~roun1 r~sonance mooal were used for

initial correl~';lon. The mo·jel chose~ ~rDm Reference 37

:consisted Jf bl~~e lead-Idg and

I lateral d,grees of freedo~ but

longitudlnal and

no aerocU:1dt!lics. All blade

dar..pers are' werking. i. e., the rotor is isotropic. The

para:T"~ters 0': this r.:o.:iel. h~rein termed cO:1figuration A, are

listed in ~le 1. As seen in Figure 3, the results of the

PAGE 22

"

. -.: . ':. -.

. -. ;

f t

prosent analqsis <solid s~mbo15) show v~ry good agreement

with those of P~rerence 37.

The experimental data in References 12 and 13 used here was

obtained to i~vestigate rotor/body stabllity of a hingeless

model rotor. The gimbal b~d~ support has pitch and roll .1 .

degrees I of . it-eedo.,,· .. The blade root attachm~nt consists of

orthogonal ilap and lag ~le,ure5 at _ radial station O. 10SR,

~ithout kin~~atjc couplings. Collective pitch is introduced

outboard o~ ~he flexures. In Reference 12 tantulum rods are

used in5tQ~d aT blades. The parameters

herein t1rmad configuration 8, are listed in

Tor this madill.

Tab Ie 2. For

t~e tuo B conTiguration~, note that the pitch mode is

e'lsentially locked out, tor case Bl, while For case 132 both I - .

pitch and roll ~otions are present. The model blade used in 1 _

Reference 1-., has a cambered airfoil with zero lift at -1. 5 - 1

de~ree angle of attacl<. T~us, even at zero collective pitch I

the rotor develo~s a s~all a~ount of thrust. Two cases From 1

Reference 13 are considered' configuration 1 (as identified I

in ReT~r~nc~ 13) with a soft flap flexure of about one-fourth I

the s4;iffness o~ the lag fll]7lJre, '1ee Table 3, and id::mtified

in this ltud4 as configuration C; and configuration 4 with

eq,ual (~ai;Ched) flaJ: and lag flexure stifinesses, see Table I

3, and iden~lTiej in this study as configur~tion D. System I

paraneters in Reference 13 were chosen so that the systems

. 1 I I experlen~e a s ight aer~nechanica instability at the

coalesc ce o~ the regressin~ lag and the bodq roll mode.

FAGE 23

~ .. l'~ • r·

. ,- '-

~ \

J ~

I' r I . , I

I i ~ !.

: .

i

r •

.... .. -

in References 12 and 13 were E,peri~~nJal results reported

. I I d A .. S t d HH I ' I'" ud!!d l'n tn~ ITR 11e'..no v1og'~ S5eSSr'len. u ",. 5

analyticaJ r~sults obtained under the ITR contract hav~ be~n repcrted ~n Reierences 14 ~"d 17. To corralate with thQse

results Jh~ oresent model includes blade l~ad-Iag as well 35 I .

flap and fu;eia;e pitch and .011 degree-, :H.! ireedom.

Figures 4 a,d 5 sho~ correlation with experi~ental data from

Reference 12 wher~ tantalu~ ~ods were used lnstead of blades.

This essen'tally represents an "ih

I Parameters i:r this r.odel,configuration

vacuum" condition.

3, are listed 1n

Ta~ le 2. 5~stem modal ire~u~ncles pred1cted by the current

dal::a and ";he E-727 anallp; is, Reference 14. both for the body

roll 0 n 1 y cor; T i g ur at i ') n 13 1. Fig IJ r e 4, as .;J ell as for t h ~ bod .~

roll and pi I;cn coniig;,:ratioll 82. Figure 5 . ~ ... I . not sho~n s:nc~ the experiments only sinulat~d a vacyu~.

i. e., a e rid 11 r. a r. i cdr a 9 r 0 r c e s are :; til 1 p .. e ,; en': , 1.11 her e a s the

I . present an~!~sl:; I

cannot sl~ulate this condition. Predict~d

. 11 • 1 I damping eV21~ ror tota vacuum are ower than the measur~d

F1Y~~=S t tnrough 11 show co'relation of ~h~ present dn~l~sis

with eJpa i~E~tal data fro~ Reference 13 (co~figuration 1)

and !;lith the correspondirg :;::-927 analyses ~ro:~ R~ference 14.

Pdracr.eter fer t;~i.s r.:odelr c'Jnriguration C. dre listed in

Ta~le Fr ~Q.'Jenc i as and lead-lag damolng l~ve1s for the

PAGE ~A·

(

;;Jf~:~ ,' ... ', .... 'I

, , ,

: i

,I •• . , ,

I flat PirCh c~.a. Figures 6 and 7, show very good agreement,

Correspondi~g roll and pitch damping values, Figures 8 and 9.

I 1 . ai"e gen~ra!u

I .

higher than axperimental data but in the s~me

range as E-9~7 p~edictions.

Lead-lag dar.plng fer nine degrees or collQctive pitch as a I

functioh

I the present analysis

. '

OT~oto'r sp'eed" is shOIJJn in Figure 10. Agreement of

(s a 1 id symbols) experimental

damping values is very good up to 650 rpm. This includes the

crossover aT the regressing lag mode with the body pitch I

mode, lor higher rotor speeds, at crossover uith the body

roll mode. only general trends in damping are captured.

. J . h .

This

1S cer~a1nl~ a s orteomlng bue it 1~ Fel~ that a bett~r

kne l"ledg1e aild.··"r ~ ~ adJustrnent o~ the body roll frequency and

damping would improve re~ults ~onside'501y. Furth«:?rmore.

trends as a Tunc~10il of collactive pitch angle qU1te well for I

the i"egress~~g lag mode.

F1gures 12 an1 13 sho~ correlation wit~ erperimentdl and

ana l'~ tic a 1 r~~~lts from Refej"~nce 13 \coniigura~ion

case) . Parameter~ TOT" thi~ model·

configuration D. are listed in Table 3. .: 'J .. reI a t i on 0 f t " ~

present ana1. IJsls with tha experiment 1S degraded when

co~parEd to t~at 2chi~ved in the sort ;l~p flexure case.

However. I anal'Jtir:al results

same range as ~hGse sho~n in I

of the prp.sent study are in the

Rei=erence 13. Results from

Rei=ei'ence 15 s:~ow that d'1namic ini=low yields much better

PAGE 25

.\

" .

..,. ~"~- .... -.... cQrrela~ion or ~his matched stiffness rotor.

I~ is concl~ ad ~ha~ the pre~ent model and ~ornputer code are

sufficiant14 valld to inv~stigate the eifects oi active

controls on otor/body aerornechanical stability.

FAGE 26

. \"

.~

CONTROL LAW DEVELOPMENT c :-:~.:. ~·r ~.~ .. ,:~ ~.~ ~.-: ~.)

O~ fC~~\ (~_:. '~.I' I'

Develop~ent ~i control l~us ~nd their ~valuation for this I

stud~ will be r~je with the obJ~ctive to incre~se rotor/bQdy . I

system dampi"~ levels and ~ventu~lly eliminate the need f~~

I blade dampers. Constraints ~n state and control vari~ble; - ~.------------ -- ...-.---- - ----"-. -----"_.,

~~-(Jbserved to ~void ~d~er_sel_y.~-.fiet:tinJ:l overall

systen pe~formance. The sel~ction trade-offs include active

c:ntrol s~ste:: ~C"'Plexity, r'i!liability, st~bility, and syst~m interiace re~uirements.

The basic m::chanisrn for influencing

~rovided t~r:u~~ aerodyna~ic, Corioli;. and

couplin9 ~i~n ~lade flapping and feathering inputs. ~~~

elastic flap-Ia~ coupling would also elas~ic blad-:s,

I . .0 l..~

I

::-us~lage art'! dynami c S coupled

flap.ling throi.ig:, aerodynamic and g'yroscoplC forces.

In impl~menti~g th~ active c~ntrol terms It is assumed thdt

cont;-ols ~re applied through a conveni;ional sl1!ashpldte. i ~. I

I . control mOtlQnS are generated by actuators in the fi(~d

I

s~stem. The active pitch in~ut to the kith blade can then be

expressed '::5

c9Ak - eAC.(If) c.o~<f'k. + 9As (tp} .fln'i'k I

I

where th control inputs eAc and eA~ are yet to be determln~d

function of ~he nondimensional time parameter f .

PAGE 27

...• "\ ..

.. --_ ..

./ ./

, , ' .... ''''. ....~ ~.

I , ·1

F~om ~he above i~ is clear that the aeromecnanical stability I

he!ico~~er is a nulti-input/multi-output control of a

prob lem. r~ the pre~en~ study three different control I 1------

me~hods w~re implemented in the computer code: ___ • _ •• _ 0" _______ ! state

I variable fe2doac~ control, output feedbac~. ,=on~rol. and open _------ ,~~ ____ --------. .. , ____ _ r~._. ..-~-~--..

I loop contr:)l

I i-igvre 14a shows the block diagrams for state

'eedback an;r ou"ut ".d~~cb. each combined uith open loop

control. T:H~ system eq,uations are

. x = ...

Y = ...,

CAlx + LB j'J uT= ... -CCh I .... i:~ra for ....

s4;a~~ faedback:

output r .. "'Ck'

open loop control:

I: 0Ac. I eA,- J

u = [K]x - ,.,.

u = (K]y ...

.J = v - -. .

v =v <'\I)

Note, that uith th~ output vEctar y being a function of the

lead-lag disol3c~m~nt and velocity of th~ first blad~, the

eel !l1atrix CC'1i;ains periodi.c. coefficients. This implies that

either a ti~e histcry solution or Floqu~t th~ory must be

I In tn~ 13~ter case it mlght be more ~esirable to solve

I used.

the periodic I

rat~r/body eq,uations in the first place rather

than transf~rming the blade degrees of freedom and e~~ations I

using Fou':'·.,,. ('. ·:oordinates. In the presen~ studl). time

history in~egration (Hammin~(s Predictor-Corrector method) is ___ __ _ I .

I

FAGE 28

. r

..

.- e.-

, ,

. "

"

I

I

! f. t

I t· r

0- "'_::_

os.: tJ treat output feedbac~ a~ well as open loop control.

T~h -1-. -L---' --.-- •

e motlvar.l0n ~o~ ~ha presen~

I, assumptlons ar~ as fcllc~s.

I

approach and some additional

State feedback is obviously a -------natural chOice. HaTe it is assumed that all the states are

- -- ~--

kn~,wn._l Ho·..:ever, only on'! state at a

I feedbaclt. Comb~~ed feedback of two or more

is used for

state variables

was ndt considered, The a~ove choice of the output vector y

is intJnded to h~lp clarir',J the issue whether -ror the case 0; I lead-lag fead~cc~ knowledge of the complete rotor state. I

i. e., ~c, ~s is necessary ·::r whether it is sufficient to take

I measurer.:ents on one blad~ only. Uith r~spect to lead-lag

d . I 'd t'" L k • d f bl d I amplng 1 e, lTlca~lcn. n'~ia ~e 0 one a e s response was

SUfficilent 'Rererence33i. :Jpen loop control is included!

here Jtnce it provides ~hc capability to perform frequenc~ ! I rasponse analqsei, Ho~~v~ft" op~n loop control is not used to

-------.1--__ - -__ .. _ ~~h __ - - - - -

augl'1~".tl~tabiiit:J._ No at:t':.'~pt was m~,de at ;his stage to use

l'1ultiva~iable optimal con~rol techniques to maximize the

d~~ping augnentatipn since ~~ining a basic understanding of

the problem ~~~ thought ~o b3 m~re important. For the same·

reasons and th~ ~forem~ntioned probl2t;)s with

stabilir y maasurements.,

(combining identification

rotorcraft adaptive control

control) was not considered.

Returning now to

detail,! defin~: the implencntation of control inputs in more

I

II

state feedbaci<:

PAGE 29

Oi\.l~:;-;:-',L ~ .. ~ .... OF. POC;;: QJ..c.U:'l

.\:

'. ,

I­

I i

I

output TeedbdC~:

open loop control: u - =[

. ..: .....

~ 'os 4> - (.os ( WA 'f + 4>,) ] K sin cp- 'OS (wA'f+ CP.S>

In the .above e"!,r~ssions K is the r:ontrol gain. The angle ¢. ! -._---._-- ---.. - - -

herein termed ~Teedback phase" defines the rel~tive weighting

betl;le-en the/ tiroe dependent C'1Cli;--:-O~~~~~~. - In other words. "'"----- -- --' . --'-- . - .

¢ definesl the azimuthal position where the gain that

individual l~d~s experience has its maximum value. This I

point is 90 degrees frc~ the axis of no feathering about

which the s~3shplate oscillates; see Figure 14b. ThE'!

Q.uantl+;:J Go" i; one of the 5y.tem de9re~s of freedom. dnd ~

is a T'Uncti!on of the lead-l~l displacement and velocity of

tne first bl~ce. Th~ open loop control frequency is wA and

phd sin 9 0 f ".: '" :: c" 5 in e .a n d sin e i n p IJ tis den ° t e .j b IJ ¢, ,] n d 1> oS

respectiye~y. It should be ~ointed out t~a~

state feedb~c~ is lntroduc~d into the second I

thus ~ is Jr~~or+;ional to Got r.ather than Xi.

for simplicity

order equation;;.

State feedback

can then ~e tho~ght of as an addition to the system

s':iffness. d3:lpi"g. and/or (;lass matrix. ;.:;1' n = 0. 2

res p e c t i v e 1'1 __ .. _. I

PAGE 30

---

--- I

I I . -~ <

, . q

RE"SULTS

All the active control simulations in this study were

I

performed 10r th~ rotor/body configuration ~. This is a sOTt

inplane hingeless rotor 5upp~rted on a gimbal uith pitch

1

and

roll degrees aT freedom I

The baseline s~stem parameters are

listed in Table' 3. I

Wheri iri~~stigating the 2Tfects of rotor

bl"lde root hinge offset. precone, and flap configurati~~ the

stiffness Jara varied from their nominal values. Parameters

for these cases, inasmuch as they differ fron configuration

C, are sho~n in Tables 5. 6, and 7 respectively. Nominal

I roter speed Tor configur~tion C is 720 rp~. All cases are

I

run ~ith flat pitch, however, this rotor has a cam~ered

airfoil lhiCh gives a. small positive thrust at zero

cOllectiveJ Th~ modal Tre~uencies and danping for the

I baseline C3se without feedback are shewn in Figures 6-9,

I

Recall, that the regressi~g lag mode experiences an

instabilit~ at the freque~cy crossover with the body roll I

mode, wh~c~ occurs at 765 !~n

All the results ~~esented were obtainad using state variable -----1-·· .

feed!lack control. the effect of individual feedback _----_ .. _1_

state vari~bles on system stability is e_~lored by varying

feed!lack Igain and phase ~qstema+;ically. Th.!'>e studies are

porTormed lat the po;nt aT n;n;mum stabiUt" ;, e., at the

coalescence ro~or speed aT 765 rpm. Plats of system damping

and fre~u cy versus feedback phase are used to select

PAGE 31

, , ,

. "

.- -..

-'

i-' I

-r'

I

I

I

/' /'

,-y- ... ,-' ",~. - ...... ~. •.. . .• ' .. -. -I" -. candida~e feedhack states and define feedback phase angles

for madoum Idan'ing augmentation. N.... the.. candidate

feedback s~atas are investig~ted in more dep~h by considering

a range Or rotor speeds to simulate rotor run up. Results

shew the s~n~itivity of th~ system dynamic behavior with

respect to changes in ~eadback gain and phase. Following

this, the erTec~ or rotor c~nfiguration on active control

damping au glentation is s~udied~ To this end the blade root

hinge oHse '.1 procone. and Hap stiHness. ..,hich are ke,

parai!loters in terins of centrol effectiveness, are varied to

cove. a rangJ aT values ~ypical for articulated, I

hingeless,

and bearingless rotors. Lastly, the rotor/body response

behavior is Jcnsidered. This provides a q~antitative measure

of the activl blade feathering amplitudes required to achieve

adequate stlbility m~rgin~, It also giVes a better

unde.standi~J of the rotor/body mode shapes.

PAGE 32

,/ /'

. ....:..

/

/

. '."-~-

/

~"-~I"--' ..

~: !..:

\ ..

f _ !

t •

i·

, I

. r ! r

I ,

I

~ .

:"- . ~ :. . . ··1' ... '" .. STATE FEEDB~CK StUDIES

For state feedback the activ~ blade feathering angle is, for

variou5 la .. I.505. 5" proportional to individual 5t.t,

variabl~s, ~, ~. e~. ~y .,0< ,/3s. and their time derivatives.

Figures 15 through 28 sho~ the effect of feedback on system

damping. i. 0 • ,.oa.1 part OT the eigenvalue, and frequencies,

i. e., ir:1agin:ir'~ part of the eigenvalue. Gain ~alues of K=l,

2. and 3 and a complete range of feedback phase angles.

C<:'¢<:'360 .• are considered. ~lso shown are the damping and

freqUenc~ OT the baselin~ system without active controls. I

i. e .• K=O. T~e rotor speed in these figures is 765 rpm which

I corresponds to coalescence or the bOdu roll mode

. I and

regressing laad-lag mcde frequencies; Figure 6.

rIots ;:Jedb~t~ of the following states was found to be most

From these

I • ••• •• suitable for sta:Jility augnantation: ~. '1's' t:s' $;t and ex' see FiglUreS 16. 18. 20. 2? and 23. Individual results are

d · Id · •. 1 lscusse 1n de~a1 below.

Figure 15 Sho~s the influence of cosine c~clic 12~d-13g

• L' I • ~ pos1"'10nT"21Odback <..,c) on system dynamlcs. The baseline

I . (K=O) lead-laJ re9re5s1ng roje is unstabl~ for this operating

d . ~. I can 1 ... 10n. D~pending on the feedback phasa, variations in

I feedbac~ g3i~ can incr03sa damping and stabilize this mode

I (250<:'¢<:'30 deg) or decrease damping and further dest~bilize it

I

(30<¢<:250 deg). The opposil;e- behavior is observed for the I

i • progress~ng le~d-Iag mode which is stable for K=O. It's

PAGE 33

... T" .l' :.:

/ I

~ ..

I

.1 I .. 1/

I damping i5 decreased for feo1back phase bet~een 180 and 360

inCTeas~d for feedback phase be~ween 0 and 180

degrees. T~is ~akes the pro~re~sing lead-lag mode the least

da~ped mode for feedbac~ ~ha~e between 2~0 and 360 degrees

and, depend ir.g on ~hc ga in value, can result in system

instabilit'l. Therefore there exists onl~ a small range of

wher~ ~he rotor/body

s~stem could be stabilized through active control. Feedback

of t;"c is

Similar

Flgures

thereio;oc

fin dl i r.!1 s

17 aJd 19.

not considered to be a suitable choice. .. . can be ~ade for t:c and ~s feedback, see

Figure 16 ShO~S ~he influe~ce 0* cosine c~c~ic lead-lag rate

(~L I

·:m systen dinamics. dapending on the

feedbac~ p h.3"'~. ~hc da~ping ~f the regressing and progressing

I :7I·:J·jes can be lead-Ia:1 or :je·:reased from the

I baseline valu..!s. This tire, however. damping for both modes

is increased over approxima~ely the same range of feedback

phase valu.?s. As a result the system can be stabilized for I

Feedback p~3~e betwee~ O· ~~d 110 degrees. The rna x imum

1 .I~. t' ~l ncrease 1~ ~~rplng occurs a approxlma~e 'l

feedback PhJsa and is direc~l.y proportional to I •

Feejback of ~c

I candidate f~r stability augnentation.

I be ~ade fer ieedback of ~S

I ¢=60 degrees;

60

the feedback

ga in. is thus consider~j to be a suitable

Similar findings can .. at ~=240 and feedback of .l;"s at

see Figures 18 and the 20. Continuing •

discussion of Figure 16. it is se~n that /;c feedback control

changes the roll and regressing lag mode frequencies only to

PAGE 34

/

. i

. ....

I I

/ /

I I

f

I

t I ~

" . I

t

i , ~

a li~ited exte~d. Furthernore, at feedback phase angles of

1

approxi~ately 60 and 240 dagrees these modal frequencies

reMain unch~ng~1 for all values of feedb~ck gain. This

clearly shows ~ha~ the improved system stability at ¢=60

degrees is a direct result of increasing the regressing lag

node i~herent d~n~ing and n~t due to a change in coalescence

I rotor speed. I~gpection of the roll mode and regressing lag

1

mode danping indicates that the source of the increased lag

daMping is a reduction in roll mode damping. However, ~he

roll ~ode is well da~ped in th~ baseline system and this

I exchange of da.~ing is therefore benefiCial for overall

system ~tabillty.

The eTflect of rull attitude. rate, and accaleration feedback

is sho:.::n in Figures 21, 2=', and 23. Again, damping of th!!

regresJing and progressinJ lead-lag mode is incr~ased or

decreaJed deoending on the feedback phase. In addition, roll I

attitude feedback (Figure 21) can lead to considerable 1

instability uf the roll mode and regressIng flap mode at

·1 certain values of feedbac~ ~hase. ThlS behavior was also

I observed for roll rate an~ roll acceleration Feedback for'

g~ins greater ~nan those shown in Figures 22 and 23.

R e ~ urn i n g t 0 Fig 'JT e 21, r 0 I 1 a t tit u d e fee d b a c k c 0 U 1 d b e use d 1

to s~abilize the·system for Teedback phase between 45 and 120 1

degrees. Howev~r,' the fr~quenc~ plot shows that in this I

range the roll ~ode frequenc~ is raised considerably. Any I

gains in syste~ damping would thus largely be due to a shift

of the coalescence rotor speEd rather than an increase in·

PAGE 35

i

I I

/

.'

, " . ,

.- '.~-. ,

regressing la~ mode inhQ~~nt damping. FeQdback of roll

attitudt" is th.!rl;!f'ore not TUI·th'!'r considered. Feedback of

roll rate rigure 22) at a i~edback phase oetueen 90 and 120

degr~es add da~ping to the regressing lag mode while keeping

t!le regres lag and roll mcd~ frequencies almost

unchanged. Ho~::ver, the Teedback gain uould have to be

incr~asod provide adequate system stability margins.

Si~ilar ob ervations can oe made for roll acceleration

feedback <F 23) at a ieedback phase betueen 240 and 270

degrees. ys. both roll rate and acceleration seem to be

suitable fe dbac~ states and will be studied in more depth.

Feedbacl< of oi':ch atti':ude. Figure 24. is seen to have very

littl~ effe t on da~ping OT the regressing lead-l~g mod~. At

th~ saNe t n~. da~ping of the pitch mode and regressirg flap

~~de can be lo~ered to a point of considerable instability.

Resul':s iro Ditch rate and acceleration feedback. Figures 25

and 26. s o~ no chang~ in regressing lag mode damping and.

fo~ larger ains. can be e1p~cted to exhlbit si~ilar pitch

mode insta i1ities as for ieedback of pitch attitude. Pi tch '.

feedoac~ is t~er~fore not considered a sUl':able choic~ for

.elimina':ing regressi:1g lag/roll node instability

consi.dered ~'!'e.

The influen e 0; flap feedback states on system damping is

shoUln in 19ures 27 and 28. While leading to large changes

.in' dar.'lping f th~ regressing and progressing flap modes. the

damping of th~ regressing lag mode is not i~proved and the.

PAGE 36

I I

I , t-

i

.. :.~

I flap 5~ato va~iable5 are not considered ror rotor/bodll

. I

da~ping aug~~nt3~ion.

• .. . . . Ba~elj on the above results ieedb,:Aek of ~c, t;'s. t;"s. EIx. and ex

I uas fur~h~r ~valuated by

1

considering rotor rpm sweeps and

varying tho gain K. while ~eeping the feedback phase at

constant [value; se~ Fig ures 29 through 33. I ..

The value of ~

U~5 chosen as discussed previously. The obJective was to

select a value of ¢ that would increase damping fer the

re~ressing and advancing la~ mode but leave the frequencies I

of the regressing lag ~ode and roll ~odp. unchanged. In

selocting the gains K. an attempt was made to obtain

apprcxi~ately the 5a~e range of regressinJ lag mode damping

1 • I.. k I· h' 11 va ues Tor ail Tlve feedbac states. t lS seen t at 1n a I

five cases the syst~m ca~ be stabilizej at all previousl~

. t· 1 I Th crl lea rot;cr speeds. illtho:.Jgh to il varYlng degree. is

\;Jill be furth~r quantified through response solutions.

Fp'9d~ae~ of the sel~cted le~d-lag states. Figures 29. 30, and I

31, adds cO:'lsiderable damp.ing to the regiessing lag mode 1

a~ove 700 rp~ and's~abiiiles the system. At; the same time

I the frequencies and in porticular the coalescence rotor speed

I

of the regr~ssin~ lag/roll. r.ode are changed v~ry little. . I· However, at t~e crossover oT the regressing lag mode with the

body Pi~hh Mode (600 rp~i these ~eedb~ck controls could

dcstabilile t~~ system. depending on the value of reedback I

gain. Feedb·3ck of roll ra+;e and roll acceleration, Figures I

32 and 33r

~lso ~ug~ent the damping of the regressing lag

mode abovQ 700 rpm and could be used to staoilize the system.

PAGE 37

. !

. ~ ...

r • r

t

I· I

'1 I !

. ~

r·

I t

Roll feedback has no eTT~ct on the regr@55ing lag mode

dampin~ at coalescence ui~h the pitch node. This is

consistent with ~he previou~ observation that pitch feedback

is not ~ui~~~le ~o

mode instaJ il if.;'~. feedbac~ of the

oli~inate th9 coupled regressing lag/roll

It i$ further interesting to note that

body roll rate and 1n particular roll

acceleratio!"l lead to consid~rable shifts in the frequency of

I

thp roll mode and therefcre change the co~lscence rotor

speej. Thl ~~abilit~ gains se~n in Figures 32 and 33 are

thus attribltable to a co~bination of increased inherent

I damping and rre'1uency shi;:I;5. Whether such a change in roll

fre1uency il desirable mus~ ~e decided on a case to c~se basi'l.

The sensi~l~i~y of the system dynamic beh~vior with respect

to the Teed~3C~ phase is eJplored in Figures 34 through ~7 I 00

for fee d b a cleo;: ))s ' rs ' . e., dnd e., re~pectively. In edch

I I;hree phc:l,.e angles n~ar f;h~ optimum ·.'al'J(! '..!Jere chosen,

I ~hile the gal~ u~s kept at a particular v~lue representing

I

approlinately 3inilar control effort in ter~s or active blade

Pi~Ch' ang11 a~plitud~s. Th~5e values were determined from

I' .. response st~d,es to be K=Q.3,3.0, 9.0, and 27.0 ror ~S' ~$'

and eJ Te~d bac k, resj:ectively. Not~ that for clarity

only the re~r.ssing lag mode damping curves are shown in I

Figures 34 ~hr~ugh 37. Other symbols shew ~h~ damping or the

piogressing 11'3g no'.ie and oth'lr system modes. Again, feedback

of ~~ and iff; Tor the gain .... alues 'lhown in Figures 34 and 35 I

k~eps the s s!en frequencies ~nchanged. DaMping results show

PAGE 38

' ..

----

that feedback phase can be used to maximize the regras5in1

I . lag mode damp1ng at each rotor ~peed. This indicates that

phase ~chedule uith rpm could be used.

a

Feedback of the roll I

rate and rOll acceleration. Figures 36 and ~7. leads to roll

mode irlquen;:\j changes. Hu~ever, the system is stable at the I

ne~ coalesc~nce rotor speed uhich means that inherent damping .I

has been a'dded tCi' the regressing lag mode. Furthermore,

w~ile tJe Te~dback phase has little effect on system damping

it is seen to be a powerful parameter for changing the roll

mode frequency.

The results oetained so far are summarized in Figures 38 and

39. TJese fig~res show root locus plots for the candidate

I lead-lag and roll f~edback state variables. In each case the

regreSSilng lal] :':''Jde is the least damped mode and thus governs

I

.system srabillty. It is s~~n that feedback of the state

variables ~c' ~S' ~S' e~. ~nd ex can be used to eliminate

the inpl~ne!rOII instability of the baseline s\jstem. The

feedba~kl. gain.'" can ~e . . incl·~ased to obtain a specified level

of regre~sing lal mode damping at the coalescence rotor speed

(Figures 38a, band 39a,b). The feedback phase p can be used

to maximize the regressing lag mode damping augmentation at I

other rotor s~eeds (lead-Ia~ feedback sho~n in Figures 38c

and d) or change the roll mode frequency which indirectly

changes the r~~ressing I

lag n~de damping (roll ieedback shown

. F' I 9 1n 19ures 3 c and d >. 'I

These results also show that a

different choice of feedba~~ state variables and control I p araina f; er s would be needed to eliminate an

PAGE 39

.\:

'.

.. '

." e."

~ .I.!".. -_ _ ,,_ '0-. '~ •. . .- -. ... '. - '. - - . - .

inplane/pitc instability. Quantitative resul~s are given in

Table 4. F .. . lfe' ~S' ~.s' and eJ( feedback 03bout 1 percent of

critical da is introduc~d for the regressing lag mode at

a m.3Jimum ac b 1 a d e pit c han g 1 e, e A I!:\QX ' of one third .. de~r~e per d ~ree of cyclic lead-lag angle. For ex about 1.5

percent of critical damping is introduced with the same

control

da;!\ping

(Figure 37 shows that the larger change in

ix feedback is due to changes in the roll mode

frequency.) Th:.: control angles shown in Table 4 are quite

s:r.all in icular when considering the low frequency of the

contiol HOl;Jever, it will be important to engage the

control before the lead-lag motion can build up to

lai'ge ar.;pli'::

The resul+.:s ;hawn so iar ~re very promisi~g. They indicate

t~at se~eral ~ays exis+.: to augment rotor/boay stability. The

i~por':;ant a ~ec':; of control mechanizatlon can thu~ be

approached ~ th considerable flexibility.

PAGE 40

~I

/

"

.;

.'

EFFECTS OF RCTOR CONFIGURATION

of control Very i~portan~ rotor par3~eters in te.ffiS

effectiteness are the blade root hinge offset, precone, and

flap SP~ing stiiiness. I

The';;e parameters were var:ed from

their baseli~e values (configuration C, Table 3) to cover a I

rang~ of values representative of articulated, hingeless, and I

bearing less rotor~ At the same time the blade root spring

s~iffnelses, lead-lag da~~ing and body roll stiffness ~ere changed so that the modified rotor/body systems would closely

appro'i~ate the b~seline s~stem at the coalescence rotor I

speed in terr.s of roll ire~uency and regresSlny lag mode

I •• equency a~d ~~mping. Use 0' these e~uivalent dynamic

s~ste~s is int~nded to permit direct c~mparison of the

stability results obtained with feedback

for thele 'ilJste!.:s. inas::1u~h as they I -

contr'Jl. Parameters

ar~ ~iTferent from

configuration C. are listed in Tables 5, 6. and 7.

Figure 40 sh~us the fr~que~cy and damping For the equivalent

dy~amic systems uhen the blade root hinge 'Jffset is varied

fron 10 ~o 5 and 2 percent. The regressing 1.11 and body roll

frequenc1J re~ai" unchanged at th e coalescence rotor speed of

765 rpm! Al so the damping C'Jrves match varlJ closely between

720 a 11 6 8S0 rpm The ei'fects of precG:;e and flap spring

stif.nels are i n "'f? s t i gat e d for the lowest value of hinge I

offset I (e = . 02;1). Precone I

has negligible effects on system

fl. .equenc 1 es. Fi~IJre 41 therefol'!! only shows the damping for

the e ivale~t dynamic systems when precone is changed from

FAGE 41

/

/ ! ,

---

f

---

I"

;:;,t ... _, ,'. ,- --- -. o to 2 and 4 degrees_ It is seen that da~ping values are

~~ll ~atched throughout the unstable region. Figure 42 shows

the systeml fr~quencies and damping when the flap stiffness is

reduced tl zer~. Frequencies are well matched with the

exception l~ the regressing flap mode. This mode changes its

character ~ro~ baing regressing in the fixed system (rotatin~ I frequency greater than one) to being progressing (rotating

'1 fi"equency Ismaller than one) as the flap sti~fness is reduced.

As a re,ult d~m~ing values match only at the coalescence

rotor speed.

Investigati," ~i ~ctive control is limited to ieedback of the

I -sine cyclic Ifad-lag positia~ (~s) and roll acceleration (ax) I

state variables. For th~se two feedback !tates a brief 3tudy . I. ~as conduct~d to ~Qt~rmin~ approximately the optimal feedback

I

phase angles and appro~riate ~eedback gain !evels. Tables 5, I

6, and 7 li,t th~se feedback parameters dnd ~he resulting

1 .

s~stem damp~n9 values. Results for the various root hinge I

offsets (Taol~

b 'l k e e p i'n J l; ;, :; I = 0.29 degrees

.. I

91. feedbac!()

5) and precone angles (Table 6) are obtained

a·: t i'" e !11 ada' tea the i" in g an g 1 :; s con s tan t (e Atftcu

T~r t;"s feedback. approxim03cei.y 0.4 degrees for

It is seen that the syst~m is stabilized for

both Tor ~s and

feed~ac~. L~ci"eases in hinge offset increase the damping

levels eve,

SimilarlY! I I

increases

the

in

ilapping frequency is reduced.

precone angle increase the damping

levels. When reduclng the ~lap spring stiffness to zero I

(T~ble 7) l~rger active blade feathering angles (~2 degrees)

FAGE 42

••

-. ,- ..

"

"

.' ;

1..

I

are reQ.uired to obtain stabilit'J margins of approximately 0.5 I .

percent cri~i~al da~ping. I

It should be pointed out, however,

that typical ~rti~ulated rotors have hinge· offsets larger

than the CO~~igurations in Table 7.

The above result";, '.IJhile being of a limiteo nature, show that I

the root hinge offset, flap spring stiffness, and precone

have conlidarable influence on the control effectiveness. I

This had to be anticipated due to the action of hub moments

C .11' l' IL h t· and orlO 1<; c·Ju;J lng. .. can beconcl'Jded tat ac lve

control rotor/body damping augmentation will be

particular14 p~ucriul for hingeless and baaringless rotors

which t~~i=allY have a large virtual hinge offset and flap I

spring stii;ness and in ~any ca";es also precona. Controlling

the a~r~lech~"ical stability of typical a~ticulated rotors

will be a ",:"."re diificult 4;~sk. For thes~ rotors it might be

helpful to ~s~ coll~ctive bl~de pitch to introduce steady

blade cO~ina deTlection. T~is should have ;i~ilar beneiicial I .

ef fee ts on = cr; t rol effec t i \'ella s s as precone.

FAGE 43

/

.> ,

. ;

I

ROTOR RESPONSE

I f

Response results are intended to be of a qualitative nature.

to give a oa~ter understan~ing of the rotor/body mode shapes

or to give a~ indication of the required control input

magn i +;ud es.

Free Response rr OiTi a set of initial conditions. forced

I response. a~d frequency response results are presented for

I configurati~n C. The fre.:! response results are computed

using an a~DrOPriate eig~~~'-e~t'~;f-;-~~l' the stability analysis. '- I ________ _

normalizedl to a ...,a~i;r.u.n lead-lag amplitude of one degree~s

:-initial-i cc,n·ji'aon. Frequenc'j response and forced response I -------------

are conpul4;:~d 1#:4 sim!Jlating a one degree blade pl'=ch stick '----------- ,---~-

stir. eith1er in the adv-;~ing or regressing directlon. I

For

I freQ.'H!nc'J Irac;:')':.nse the nondinensional exclcation freQ.uenc'J iUA

is varied tr~t:I 0.1 to 0.7, Forced response is computed by

starting Jith the system at rest (zero initial condition) and

I an excitatiJ~ fr.:!quency wA = ,336. corresponding +;0 the

or the regressing lag mode at coalescence. The

rotor speed is 7,~5 rpm in all cases.

Figure 43 sho~~ the response of the baseline system with no

feedbac!< I c,'Jl'ltrols applied. The progreSSing lag mode (high

';:;-eQ.u.:!nc1l) 15 seen to be st03l:le, The regressing lag mode

(low frcluency) is slightly unstable. with critical damping

I of T\. = -0.5a percent, It's :'1odal components consist largely

of the CiYCl:'C lead-lag mohons (t:'c.~s). the body roll degree

PAGE 44

/ ,f , , f I

.I I

I

/ /

.'

/ i

" !' I /

I

I I ..

. i ,

/' . i ./

ITE~\- '.. '-:-1 I

of rroedor.1'(9x). an·j lateral cyclic flapping (/1s>' Thore is I

very little Ditch and longitudinal flap motion. The inherent

stability or the rotor/body system with sine cyclic lead-lag

feedbac~ co~trol at K=l and I = 240 degrees is illustrated in

Figure 44J The time histdry response or the regressing lag

mode ~hvLs tha~ with ~eedback this previously unstable mode I

(Figure 43b) is. stabi.l.ize~. and both cyclic' lead-lag degrees I

of freedom. t:'c and ~s. red'Jce significantly in amplitude in

I . only ~en rotor revolutl0ns. It is also seen that feedback

control iincroas<!s the p:)T'~icipation or the flap and body I

pitch and roll n~tion5 in the regrossing lag mode. This I

could be tne source of the increased damping of tnis mode.

The a~PIJtude of active bl~d" feathering in Figure 44 is 0.9

degrees initially and redures to less than 0.5 degrees over

ten rotor ravolutions.

Figures 45 and 46 show the response of coniiguration C to

~ . I a,~vanclng and regressing stick stir excil:ation at the

. I regresslng lag r.'\~·je frequenc1J.

. !. No feedbaCk controls are

After ten rotor apPlied'l T~e s'}stem is initially at rest.

revolutions the excitation is stopped and the rotor/body

system il al1ou~j to move.ireel~. Tnis simulates a procedure

t~?iCalll us~d in helicopte~ ground resonance testing. Il; is I

seen that sti~\ stir in either direction excites the unstable

regressilg l~g modo and results in gr~wlng lead-lag motion

amplitud~S after the excitation is stopped. Note however. I

that th1e r,=gressing stick stir (Figure 46) leads to much I

larger amplitudes than the advancing stick stir (Figure 45).

PAGE 45

• 0.-

• ,

. ,. I

,i "

,/

/ ..

'. ,

!

- -,-- I . f. / "Tr-r· ..... ~ ," ...... - .:.

Examination~ 0; the regressi~g lag mode eigenvector shows that

it's flapPl'iQ contributio~s are sequenced in a regressing

direction. iThis mode is th~r~fore most readily excited with

a regressing stick stir.

'. Frequency ~e~po~se andlysis is used to compare the effect of

increased blade lead-lag dan;.:ing versus the application of I

feedba::!- control. Figure 47 shows the influence o-F I I

increasing th ~ 1 ead-lag dai"ip ing -From 11-; = O. 52 percent to 2

percent and 8 percent critical. No feedback controls are

l

~J ; i

increasing

Fiqt!re 49 shows the infl'Jence of ~s feedback with

:,9 03in values, K = 0.3; 1. 0, 3.0, and ¢ = 240

D.lr.lDin:1 is held :It its nominal value Or 1't~ = O. 52

applied.

degr~es.

percen+;. In beth cases only the rrequency response of the

1 " . cosine c'.Jclic l~cd-lag motion is shown. Comparing both

~~~ni+;ude 3~d phase plots qualitiativ~l~ indicates that

-Feedbac~ co~trol and additional I

blade d~~ping have very

sieilar errects in terms of system dynamics. This is an

additional indication that active control can be used to

augment 1'0 or/body damping and reduce or even eliminate the

need ror Ie ~-l~g dampers.

PAGE 46

.' I

I .. /

/.

/' /

/

I

./

" .I

",,.--'

/.

/

/

.... _ ..

..

'. ~ ,I

i

/ r", ' 1:1

':: . .: .

/

,/ /

CONCLUSIONS

The, present study in:.ficates that active control blade

feathering ~h~ough a conventional swashplate is a viable

neans to increase rotor/body damping levels and to eliminate

gro~nd resonance instabilities. The choice of control I

par~meters depends on the rotor/body configuration und~r

coniideratiQ~ and must take aspects of control mechanization

into' ar.:count. Based on the stability and response results

presented here fo~ state variable feedback control the

foliouing co~clusions can be drawn .

Roll rate 3n·j acceleration and blade inplc:1no notion (~C' ~S'

~s) feedback co~trol can a1d considerable damping to the

It eli~inates the regreSSing l~g/rQlI mode ground

res~nance i~s~ability of the hingeless model rotor under i

consideration. The feedbac~phase ¢' i. e., ueighting between

t!1e; cyclic can'=rols, is seen to be a po~erful parameter.

Dep~nding o~ ~he value of ~ the system can be completely

stabilized or further des~abilized.

,

of feedback p~as2, dam~ing or the regresslng lag mode can be

~aximized ~ithout adversely affecting the damping of other

s 'J s t ec. co des The feedbac~ gain K can then be adJusted to I

I

ohtain a s~aciri~d level of regreSSing lag mode damping at,

the:coalescence rotor speed I ..

about 1 percent and for ex I

For ~C:' t,'s' t:s' and ex feedbac k

about 1. 5 percent of critical

damping is lr.troduced far the regressing lag mode. Tllis

PAGE 47

/ I

I I I

I~ ."

I , I . !

, / .I .

/ ;

" I

/

/

i'

i I •

.'~ .. ~,

I

• l' f

I ~

~ .

/.,

, f·, '/ "

'r /

damping augman~a~ion is o~tained with an active blade

feathering: am,Htude of 1/3 degree per degree of blade cyclic

lead-lag an·;le. ,

Inspaction: of the systQC eigenvalues indicates th~t the I

increased l~g danping might ~e due to a reduction in damping

of other: sys~er." :,node'l, notably roll or Flapping. However,

these mode~ are well damped and this exchange of damping is

therefore: benaiicial for ovar~ll system stability. From the ,

regressing' lag !:Hlde response it is also seen that feedback I I

control g~nerally increases participation of longitudinal I , d . flapping and bo ~ pltch motions in this mode. This could be

an additio~al source of the increased regressing lag mode

damp ing.

Rotor rp,'" show that uith the above f~~Qbac~ controls,

a r; th~ coalescence rotor speed, the slJ'ltem is i

stabilizedl throlJghout the range of pre"'lo lJsly unstable

ope r atlng co"ditions. For lead-lag feej bac k <tfc, ~$' fs j

i t~e system, rreq'Jencic-s . a.Tld' in particular the coalescence

rota,.. spe:ed re ... .aln prac tically unchanged. Improvements in

'l~stem stability are a direct result of increasing th~ I

regressingl

l-'l~ :llode damping. Furthermo1'e, scheduling the I

feedbac~ ph3S~ uith rotor speed can be used to maximize the .

ddmping For roll feedbac~ (ex. ell) the

feedback phase has a r.onside~able effect on the roll mode I

f'ieqIJency. I I

Besides inc'ieasing the regressing lag mode

danping, ,

roll ac~eleration c-an be -reedback in p-3rticular

PAGE 48

-,' / • oJ ~

.'

,I

/

.....

"

, .. l~r/.·· ",

designed t~ shi~t the cC3loscence rotor speed. This would

indirectly, i.':'1'Jr'.lve sljstem 5'.ability through active control of

Last ly, it is seen that a d i ffer.,nt

.. . C;,01.CO state variables andccntrol parameters

~oul1 b~ neces~ary to eli~inate an inplane/pitch instabilit~.

For the pr~senf; configuration the active centrols should be

applied o~l~ at rotor speeds above th~ crOSSOV9r of the

regressing! lag "odo with th~ body pitch mode',

Increasing! the root hinge o;~set, flap spring stiffness, and ,

• i precone lm~royOS the control effectiveness considerably. It

can be con~l~ded that active control for rotor/body damping i

augm~ntation ~ill be particularly powerful Tor hingeless and

bearingless roters Controlling the aeromec~anlcal stability

of articulat~d rotors will ~a a more difficult task.

FAGE 49

. ,

.:

/

/

.(

P-I'

I, '\ . ' .. , ~' :

I I •

, ! I I

l r [ r ,

RE~ERENCES

1. i

Co Leman. R P .• "Theory of Self-Excited Mechanical

03cillatio~s 0; ~in~ed Rotor Blades". NACA ARR No. 36. ..July

29, 1943. : 'Subs~~uently reissued under authorship of R. P.

Co 1 eman and A. it Feingold as NACA TR 1351, 1958.

I

2. De u t s c h, :1. L. , . "Grcu.,d Vibrations of Hel icopteT's, "

i Journal oftn e A~ronaut iea 1 Sciences, May 1946.

"

3. Gabel. R. and V. Cap u r so, " E II: c'i e t 11 e c han i cal Ins tab i lit II

Ccundaries i a; Determined from the Coleman Eq,lJation," ..JAHS,

Vo 1. 7, No',. 1. January 196~, p p. 17-21.

4. Dor.n am, " R. E.. S. V. Gartjin-:lle, 13. Sach s.

"Ground and Ai; R~soncinc~ Gha;acteristic:; of a 50Ft Inplane

14. No. .., October 1969,

pp. 33-41.

5. Lytt:Jyn; R. T., W. Miao. and W. Woit3Ch, "Airborne and

Q;ound Reso~H;1c'! 0; Hingele:;s Rotors," Fr<?print No. 414.

'26th AHS Fo~u,-, l'!-=lshlngton, D. C,. Jl1ne 1970.

6. Burkan, .J. E. I

Aero~las+;ic I S';a~ility

I

and l.J. • .... I3oun~aries

. Hingless-Ro~or i1odltl. II Preprint No. I

Wash ing +;on, in C , Mal.J 1972.

FAGE 50

i1i.,Jc • "Exploration of

a Soft-in-Plane

610, 28th AHS Forum,

, (

..

"

-:. -

7. 11iao, &.I. L. and H. B. Huber, "Rotor Aeraelastic

Stability . Ccupl~d with Helicopter Body Motion," Paper No.

14, NASA SP-352. 1974, p~. 137-146.

8. Wh He, : R. P. and W. E Nettles, "Exanination of the I

Air Resonance St~bility Chardctaristics of a O~aringless Hain

Roto~," Preprint No·.,. 78, .. 34-22, 34th AH5 ForlJt'l\, Washington,

D.C., May 1975.

9. Sta ley, ..J. A., R. Gabel, and H. I. MacDonald, "Full

Scale Cround and Air Resopance Testing of the Army- 300ing I

Vertol Bear inqless 11ain Rotor," Preprint No. 79-23. 35th AHS

Forum, ~ashi~9ton, D.C .• Ma~ 1979.

10, IJarnbrodt, W •• ..J. r~cCloud, M. and ..J .

'~F'Jll-Scale Wind-T:Jnnel Test of the Aeroelasttc

S'.;abilir.'l Or a 3.?-lringless I':ain Rotor, II Preprint No. 81-21,

37th A~S Foru~1 i~ew Ci-Ieans, May 1981.

11. Qr1'1iston, R. A. , 'Aerom~chanical Stability of Sort

Inplane Hinqel~~': Rctcr Helicopters," Pa.,;;r r.ll .

. European Ro+;crcraft Forum, . France, 1977. I

12. !t G. I ~A" Experimental Investigation of

Hingeless! H~lico~ter Rotor Cady Stability in Hover," NASA TM

78489, ..June 1778.

FAGE 51

. (

.'" ...

"

II

"

I 1-

; , I •

(

13. Bous,n-1n, " ~ .. G., "An Experimental Inve'l4;igation of the

ETTcc4;s of Aeroelastic Couplings on Aeromechanical Stability

cf ~ Hing~l~ss Rotor Helicopter. II Preprint No. 80-25. 36th

AHS Forun. Washington. D.C .• May 1980.

14. 8. and J. A. Jo:'ns4;on. "Integrated

Technology Rotor Methodology Assessment." Hughe'l Helicopters.

Inc .• Repor!;. r~o\"ember 1981.

1'5. " .". , "Inrlu~nce of Unsteady Aerodynamics on

Hingeless RO~Qr ~round Rescn~nc~I" J. Air,=rart. 'Jol. 19. I

No. 8. August; 1962. pp. 6!:8-673.

,

16. Fried,:ann. P. P.. and C. Venkates<in. "Comparison Ot

Experimental (ou~led Helico~4;er Rotor/Body S4;ability Results

with a Si."":ote Analy4;ical ModJ1L" Pro'=l?edings of the ITR I

Mef:hodology' Asse-ssm:!nt l':or~shop.NASA Ames Research Cvnter.

June 1983.

,

17. . ITR '~"',e?;hc:iolo9y'Asse''jsment Workshop. Proc~<Jdings. NASA

Am<JS Resear~~ Ce~ter. June 1983.

18. II. ret z , i1 , : R e ! a x a!; ion 0 t Rot 0 r Lim ita': ion s by Fe'.? db a c k

Con t i" 0 L" P :r.: 0 r i ~ t No. I

o C., 11ay 1'-1""'7.

19. I

" , . o. ,

craft," Ver'tic<I, Vol. I

I

77.33-36, 33rd AHS Forum, Washington.

ed., ~Active Control Sys4;ems For Rotor-

4, :':0 1. 1980.

FAGE 52

,

/

-... " ....

, "

20. Har.l, . N. D. , "Helicoptar Individual-Blade-Control and

its Applical;ions,:I Proc. 39th AHS Foru~, St. Louis, May

1983, pp. i 61.'3-623.

21. Jchn~o". U., "Optimal Control Alleviation of Tilting

Proprotor! Gust Response,:' J. Aircraft. Vol. 14 .• No. 3. i

March 1977. pp. 301-308.

22. Saito. S. ~Application of an AdaptIve Blade Control

Algorithm' til a Gust Alle .... i:ltion System," Paper No. 64. 9th

European Rotorcr-aft Forum. Italy. September 1983.

I

23. !Jood. E. R., R. w. Powers, J. H. Cline, and C. E.

"O~ De~eloping and Flight Testing a Higher Harmonic

Control S~ste~, h Pro~. 39th AHS Forum, St. Louis, Missouri.

May 1983,! p p. 592-612.

24. 1101 us is, .J. A • C. E. Hammond, and .J. H. C 1 ioe, lOA

Unified Approach to the Op~imal Design o~ Adaptive and Gain

Schedul~d~ Cont~olleis ~a' Achieve Minimum Helicopter Rotor

Vibration'." Pree. 37th AH5 For'Jm. New Orlel>ns, Malj 1981, pp.

188-203.

25. 11ol'us is. • J. A. , P . f100kerJee, and Y. Bar-Sha 10m,

"Evaluation OT the Effect of Vibration Nonlinearity on

Ccnvergen~e Behavior OT Adaptive Higher HarmonlC I

Controlle~s." I·!A:-A CR-166424, Janudry 1983.

PAGE 53

. , !

." .. "

,/

.---------

-

.' .......... -,

26. Davis, 11. II., "Refinement and Evaluation of Helicopter

Real-Tir:1e I 5eli-Adaptive Active Vibration Contro 11 er

Algc':'ith~s," r·!ovember 1983, 4;;0 be published as NASA CR.

27. Young', M. !

I., D. J. Bailey, and M. S. Hirschbein,

"Open and Closed Loop Stability of Hingeless Rotor Helicopter I

Air and Ground ~esonance, :. Paper No. 20. NASA SP-352. 1974.

p p. 205-218.

28. Har.l, N. D., 8. L. Beha 1, and R. M. McKillip, itA

Si~ple Systam for Helicopter Individual-Blade-Control and its

Applica~ion to Lag Da~ping Augmentation Paper No. I

10.2, 8th

Euro;:ean RO~(lrcraft Forum, F .... ance, 1982.

,

29. Levin; J , :'Formulation of the Helicopter Air Resonance

Probler.t in hover with Ac";ive Controls." ~1. S. The~is.

Unlversity 'cf '::ali,Pornia, Los Angeles, 1981

30. Peebles. J H, ItOpti~Ql Control of d Helicopter Rotor

in Hover. .... • j. S . Thesis, George Washing4;;on University.

Wash ing4;;on. D. C .• November 19.77.

31. Johnsol"l. \J .• "A Discussion of Stability I

Measurer:1ent T~chnioi.ues. It NAS~ TN X-73. 081. tJo,.,ember 1975.

32. r~o 1 U5 i. s. , ,

I

J. A .• PRotorcraft nla~e Modal Damping

Ijenficatiol"l FrOll Random Responses Using a Recursive Maximum

Likelihood Aloorith:'n." NAS,; CR-3600. Septemoer 1982. , -

PAGE 54

'- ~ ..

..... .r.

33. , f'lolus is, ..J. A. and ..... Dar-Shalom, "Identification and

Stochastic Control of H~licopter Dvn~mic Modes," NASA

CR-166425,January 1983.

34. i \'en!<af;2san, C., and P. P. Friedmann, "Aeroelastic

Ef'~cts in Multirotor V~~icl~5 with Application to Hybrid

Heavy Lift Syste~ Part I: Formulation of Equations of

Motion," Sub~itted to NASA for publication as a Contractor

Report, Dec~~~er 1982.

35. I, Friedmann, :'., C. E. Hammond, and T. Woo, "Efficient I

Num~rical Treatnent of Periodic Systems with Application to

Stab i1 i ty Prob 1 eo'S," J~JME:, Val. 11, 1977, p P 1117-1136.

36. ',.Johnson. t:., Helicopter Theory, Princeton Universit~ I

Press, 1980.

37. Hanl':'lo nd, c. E., "An Application o~ Floquet Theory to

Predicf;ion of I';echanical Inc;tability," JAH5, '.'01. 19, No. ,

4, October 1974, pp. 14-23.

38. 3rog an, W. L., i10 d ern Con tro 1 Th eory, Quantum

Publishers Inc., New York, 1974.

.. , PAGE :}5

, . •

/

i!. ,

'R 1,~~

y

I

U ....:

FIGURE l.

FIGURE 2.

I

. "

.Ii Fuselage

/---

Model

G >'R.~ G.-.

/~ ~Gk , \".

:' /

Rotor Blade Model

PP.GE 56

r . /

I /

," ..

--

. ~.:.. /.

,. / /

/ /

'/

-

MODAL DAMPING raals~

Z8

,

24 . I

MOOAL 16 FREQUENCY. :

, -

r Qd/ sec 12 t.:-r-....... .D:::-:::-

8 ,

I

,4

:0 i·

100

/ /

/ I

/

o /

/ /

, ZOO

i

1

REFERENCE 37:FIG.4 o PRESENT ANALYSIS

300 ROTOR SP'EED. RPM

o

400

I , ! , I ,

1 2 3 4 5 6 7 , I O~----~----~------~----~----~~----~----~

ROTOR SPEED. Hz

FIGURE 3. Modal Damping and Frequencies Versus Rotor Speed for Classical Ground Resonance Model. Configuration A.

PAGE 57

,,---

. ,.

] .. u Z ... :l •

~ :: .. c o o :a

1

0 100 ~

FIGU E 4.

JOO 6CO !>OIl 000

ROTOR SI'£fO en. ,,"",

Modal Frequencies Versus Body Roll Motion Only, at Flat Pitch.

PAGE 58

100 BOO

Rotor Speed,

/ .. ,/ ~ ---~,'

900 1000

Configuration S 1

. -~-

,

/

I I

/ ~

/

/

. ;

,Ii

OF PC':::~ (_" ......

, ~,. :'" r-.-f-- - L.:.1J/ .. I. . i. /11 ,. I I ' I ~I I I I I I I I I 1_ --r-r-I'-r--.T' r-l' 0 A 0 V EXPERlh'£HT, REF. 12 .' . ~;+-",.: /1---1- '--l-,-,,..,-lI:·-l. - ,wALYSI5.REF.I' -- _.Lf-cir':~

" I I' y;-, 0 PRESENT ANALYSIS I /y' .

I'--l~,-+--+-+-r- '-+-1-. ·..J,··t--!h -I"'· _1. - ,.- - - <-. . I ~-.~- ~ • '=-I-~ .. :,.1\ -, i/ I I; I! I I :k~; . . -

o 100

FIGURE 5.

-Modal Frequencies Versus Roll and Pitch Motion, at Flat Pitch.

PAGE 59

Rotor Speed, Body Configuration B 2

,' ..

---

.---<...

, . I • I

/' ~ .

! /.

" -,. ,

I '

:- .

. ., ,

-.--.

.'

'FIGURE 6.

/. / Ii

I .

. / II

ROTOR Sl'UO I n R",I

Modal Frequencies Versus Rotor Spee~ Configuration C at Flat Pitch.

Pl\GE 60

. .

._------'-----_._ .. _-( ,

,/ " I

j:

/ /:

,/

r I

. '

;

. , , .

:.

i • , .

_-'1"

1

/ l , ;i i

;: ~.

I:

, /

,0

i

'u ::: ! 'J ! 0 .. c

i

--... " z i c

//

OF. F0~ .. \ __ ... -..•

"~I- -- -j-. i +f- - -i ~ .~. --; -~.: .. ~- 0 LlOO EXPERIt.'Brr,rG.13 -+1:·-., WALYSIS. REF. U iJ-'

-~ -I- -i- ~- -l· 1- -- .;. 0 PRESENT ANALYSIS ~ -+-~, ,,' • ! .. i_I....L. I l:.!.:l_Ll-.. Pi .-:-

: 1-1-- -l-: -+-,'T:, f-r-rt'"lh T -, " I ,. - i:-I:!-It:l~t-.:t.

H

-20

•' ;r~_1-+-H--t-t-,.l-, - I-i -f-I-~- LL f- . i I' .. -f-;- - ~-:- ~T t, {l . f-. .,.. ';:TT I.J, '~",)'I',-~' _ _ I. 1 r--~ - . , I I. .J . _ '1 ~j '., }s;.:::-;Y",-\),,;,-

, I - , i\ A 1/' LLJ. " 1 -I-I-~- - - -~t -1--7 - - I· - - - .. 1-·-, . [,;,t ~:rl,(~rYT-: t

! 1-1- i- -1-+-1-+--+-

,0

FIGURE 7.

1 I , -

.... 100

"OT()IO ~flO In" ... ,

Lead-lag Regressing Mode Damping Versus Rotor Speed, Configuration C at Flat Pitch.

: • I . ~ - .. --

'000

I ' l' i . I t ···f - ~ t , , O~OO EXPERIHENT,REF.13 : ' ,;... , ,

-+ ~- r~ ~r -.~-:. -V -i - - AHALYSIS, REF. 1" -I .. ~ roo -r-~

+ 0 PRESENT ANALYSIS , .:,... ! " - - -+- -'- T .. +?-tl--.t- /-. ;\

, r "1"1' ,- -Fu--~ -I' ;- .. .." -~ -I'" \.

I L,+ I , I

-,/,- - \~ +- i -j - -!- - 1- -ll 8> w-p .-!-~ ~~jt .J._ ~LFF ,- .

. 1- ~ -I'Q -+ ~-~ ~ P+--( -r - . :.- -f-. -I~'t' -

r--. -- .+- -~ _. ..L - 1"- :.a - - ~ - 1-:- l-f- -.. ...

Q .10 .. - -J -iI

··1· , -r-

-k- T I I

~- j-

-~ I , , -I G . ~. -J.. ;

1-;-8 a

... - -I' - .. ,

-+- f- -i- -I-I- _. - -r-~ .. f-- f- I-- H-"J - -

1--o

o

-l-I-.... .

I-I-- ' .- _L.. ,

FIGyRE 8.

. ;

~t - -1" f-,

i

L ... ' -t ~T - .-~ ~ . f-l-- .-f- .:.~.- , 1-,'" I i.' ..

1-+ I-..L -~ - \-. , '1 10::. ,.~f'"- ~.;~. ·-t--l-I- -l- i- ~-

-r-F "T "

!-- -~- , _ I .+ f I' T' -I- - 1-- -; h ., . , I - lOll '000.

Body Pitch Mode Damping Versus Rotor Speed, Configuration C at nat Pitch.

PAGE 61

, /

; + r'r"~: ' ,':' ~...:.-. -.. --.. ~ ....... -I ' j'

\'

I , -

l

. . L i, !

,/~ :

f' t I

':: .. ~ . ., z § 4 Q .. 4 Q

i

/

:FIGURE 9. ,

/ ) /:

0 1;'-"""" t'·, -,\.~ .. 4... . ...

OF POI~-- :,:: " .... it \,.~, ,_,. ,

"O'OR "'(0 IU "" ..

Body Roll Mode DJmping Versus Rotor . Speed; Configuration C at Flat Pitch.

c j :--i I :a ' 1:01';\( j' : , I

1 __ I - ,~-"-l-;...,t,,- .!.-~,,-~, -'-"~ ,-~ -4, ,'" ',; /' ~ , ! : ,.: " , '!

1-+-:"t-'J,.-t-;::"*,,I-i=±=V-!'~'9'~ -L" / 7-1\.-i-?a--; +. ~.:.-r--, +-'+-+--+--;"-4-1,.~

/' / ,'----

, . , I

, /

./ ; ,

I~ ,(.

--'U w .. 2

" z 0: :I • Q .. 4 Q 0 :I

00

0'

., 0

I-':'f~o.::"':!-''-i'"'....;-+-+-i '-j-+-HI_'+-H-t-+-+,-+~~I ' '\ 1 Ii - i \ I 1 l i : i 'I'::' ,i. !" T ' I I" ': l\}j,: A I n i 6 1';/ ~i'

" '. I." i I !:+ '~L-"~ ! ( '/ i L

: : " !' I -'-+---'f--,.-t--c,c--t-I+-! -+--'"" I! -r i"7 I , 1

!-' -j' 1:-:+-,' f-,I +'".+ 1.:-, H,-+_II-' .+-+--i~ ~_ +_ J, ,--t++++-f- -'~ -W. -a-- /' I ~'

:.1.:'" 'Q' i,A 0-' A I " ~:+~ i-~-.~-- \H-,~J: : /-,',f'-::.r'f-+- L.4 V EXP£R1~lTlR£F.13, I :,! I !' t :'-i:, " - AHAlYSIStREF.U 1 -! II; T I

" 0 ~ A/W.YSIS I I·

~~l;: ;::,0:: ,:, i:' ". : j -r-l :: -~-f~Y-H: I

iFIGURE 10. ,

"OTCA Sl'U 0 Ul· ,. ... ,

Lead-lag Regressing Mode Damping Versus Rotor Speed with Nine Degrees Collective Blade Pitch, Configuration C

PAGE 62

'; I

, (

" ",J

, )

/'

I

;

i -a ..... ..... _ ".

I

I I

·UI"

·u-

OR:~:~Jt.l r.",·~: OF. POOi~ QJi~:":l'"

-1

06000 EXPERI~ENT.REF. 13

.. -

. a. 0

·H·

-. a ~ :i

l-

,.

'" I'

., ,. 0

.: .. -'~, 0 ,;

:r-

,. ..

.~N".LYSIS. REF. 13 .z.~ t- O

I Q PRESENT ANALYSIS 0 0 0 - i ...

.~ "'I 0 ':I

.15: ~ 1" 0 I LEAD-LAG o· ...

~ BODY ROLL REGRESSI~G .I.~I" -i I

;60_

.650 9 rpt:l 0:;) . s,. ...

~ G 0 I II') I

~1

~ ·l • I I :: ~ f~'"

... ::'0 "...

·u - ~

I !

~~ ·10 -.. .

0 0 . a 0 <) 0

.1.S -

-. '" ~ ·1.0 - ., ; v LEAD-LAG 9 .;

8 0 £ REGRESSI~G -1.5 - Q

• 10- .. BODY PITCH

i 650 rpt:l I . s- . I I ,

-I l • • '0 .. ·2 l

f~ ... f~ ...

FIGURE 11. /l.oda1 PHch

Damping Ang 1 e.

as a Function of 81 ade Configuration C.

PAGE 63

/' •

--

'a

... J.

0 ., ...

'a

...

I r-,

8 ! :~

6

5 ..... J: 4 . , .

'- 3 "" fb ~:' 3 I!" I;

" t.: . " r 2 e t, I'

• 1

I

I L-

a

: .\ '! . i

~

.1 , ,

OR:C:;·::~"I.- ; . OF PCO:, (~

LEAD·LAG REGRESSING (~R)

O~D<> EXPERIMENT,REF.13 ANALYSIS. REF. 13

() PRESENT ANALYSIS

0 0 Q C 0000 .~ .. G 8 8 0 o ; BODY ~'OLL

14» LAP REGRESSING

(iJRh ~ ~

0 o u 8uoOOOao~ ~

IlOO tl 0

~ §5lg 9 0 8 g g 8 BODY PITCH (8) 0

200 400 600 800 1000

11, rpm

FIGURE 12. Modal Frequencies Versus Rotor Speed, for Hatched Stiffness Rotor, Configuration 0 at Flat Pitch.

PJlGE 64

I

/

/

"

..

, .

-1.0 [ LEAD.LAG o ~' REGRESSING IrR'

.~ • 5 .l-~i.....L-or~ 0:; .r' ~. ~~ II a ' ... 0 ~

·1 .' ...... ~ . s~~---------------------------o ~co 400 COO nco 100e

n.tI""

a 2CO 400 600 800 100'1 II. ,pm

-f

. ,

.. , ,

OF. P~S;; (,.; -' .....

•• • • • •

;:0 '00 ICO '000.'

FIGURE 13. Modal Damping Versus Rotor Speed for Matched Sti&fness Rotor, Configuration D. at Flat Pitcn.

. " ,

~ , . , . I t

r

, ,

0 :7'''' •. .. .

State \'anable feedback system.

I " U

~ --------____ ~' K ~! ______________________ ~

, Output feedback \ystem,

FIGURE 14a. State Variable Feedbac~ and Output Feedback Control System.

PAGE 66

(

'. ,

\ ". \

:..i

. '

.~ ...

. .,.,. . \

SWASH PLATE_--"\.

--, ---~~------~

..- ...

tm.G;.'~;··(. ; "-'. OF PCC;~ (~ ... _.'.;

AXIS OF NO FEATHERING

SA: <V J.< sinp "t'=qOo

) a:1: 1'=p

FIGURE 14b. control Implementl.tion Through Swashplate.

PAGE 67

I ,..

-.'

-

... ' ."

.. .,/ ".' '-....

" ..

. ~----, /

/

-,.. ,..

o o

'" ..

/ \ \

UNsr':9LE: .;:: :-0 :Cl.. 0

.:2:": leI: :0

3

~ 2/ g ,,~~/3 N / oj.~0~0~--~6~0-.0~0~~-I~~-O-.-~-Q----lrBO~~.0-0----2'4-0-.-O-Q----3roO--.0-o--~350.00

FEtUBRC~ PHRSE. OEG

I

o o

en.OO

FIGURE I 15.

r

50.00

C[ hIe 'os. r[[08ACA.

,.

C.tll,. .. , .1.)

I

• I~LL,Q008 ltlO.QO 240.08 300.0Ll Ftt RLK PHASE. EG

36u.00

Modal Dampinq and Frequencies Phase with Cosine Cyclic Lag Confiquration C.

Versus Feedback Feedback.

~ .... .. , . . -.,' , .. ' ~ , ,,-' f\ ~\_.. i :,. ' •

OF PO';;! (...:.,_ ....

PAGE 68

-.-~-.

.1

-...

,.

~.

OR:2~ , :' ~ __ 01: FO",:,; , .

.; -

... ,.-,,' I .'

0 0 ..,; I

a -0 u w-en l ...... 0 C:: a ~o

0

t..:) UNSj;:;SLE:

Z _0

Q..~ ::L_ a: 0

o

~-~------~-------,------_\~-------,~'~'----~--~--O 00 60. 00 IZO.I1tl ltJO.OO 2~0.08 3~JO.llU

l j

a <0

. I'E£UBRLK PHRSE. EG 160.00

:i ._---------------

f ft ce"c., tAl ..... .!. )

~1 .J. I i

;.: . ~:Ll. I

...... lr-j(:J ~-==3:=~! .-#-.3:1 ~~ ~i~ q: t----; :t:=-=::1C- -:-;~~ .:.= L.4G- R :J

~'" -':>IU~.J k-O,%,.z.,3 I lL._~._lC:_I_ ~. ~W~1r __ II --,~

c::i

'fLAP-it o ~ • " ,. ,. ,. " o+-______ ,-______ ,-______ ~~~--~~~--~~ 0.. 60.00 I "q., 110 ltJO .. 00 ~~O, 0

0° _ 300.00 300. 00

u.OO l't:tOBRU\ PHRSt:. EG

FIGURE 16. Modal Damping and Frequencies Versus Feedback Phase with Cosine Cyclic lag Rate Feedback. Configuration C.

PAGE 69

..

." .":-

.. . I

\

.... .,..-.

. \ . \ \\

i

" FIGURE

a a N~--------------------------~r-------------~-r~

I /- .........

a -0 U· w-(f)1 ,

'-;1/ "\~ /1 I.A~_P \~

l I

5i;:SLE o era ~a

o~~~~~~~~~~ .0 / UNSTHBr.t

/ /1 " """'0- tI.- 1. I I

2. C[l1>( I>(CfL. 'HOSII( •• CAIH'I.2.1

~+-__ ~ __ ~ ______ ~~;' __ ~/~r-ft'_._'1_6_S __ ~ ______ -r ______ _

0.00 60.00 1,,0.])0 ItlO.l)J) 240.00 300.00 360.00 FEtDBACK PHASE. DEG

:r---~i

J: I :: I ~jLLIL,.J6-R) ;'<·0,1,2.,3

~~ t--.. -----l.I-----,.-----•• -----i • ..----.• .-----... .-----j ~QI I (') I

~ I 1> 1 rc H J :< = ~, 1, Z, 1 u..~t_~-_---..... ---_----<l....-----------...... ----_t

o

0' o

~ . .,. . ,. TLAP-i<

• • " ~_c_----_r_-----__r-----__,----,.- I -.- ,-- -

en.OO 60.UO 1,-o . .\20 I tHl. 110 240.00 300.00 360.00

17.

FEtOBAC~ PHRSE. DEG

Modal Damping and Frequencies Phase with Cosine Cyclic Lag Feedback, Configuration C.

Versus Feedback Acceleration

OF POOR QU~";Lln

PAGE 70

--.

, .~ ..

"

o cD

o o

f'

([lAS 'O~. '[(OBACO. &AI ..... 1.)

"fL.A~-R f' ~

-----, I

~------~------~------~------~------~------~ 93.00

FIGURE 18.

60.00 I'-O • .QO leO.nO 240.0.0 300.00 t-EtOBRLK PHASE. UEG

360.00

Modal Damping and Frequencies Versus Feedback Phase with Sine Cye'ic Lag Fp.(:dback. Configuration C.

PAGE 71

.... -

---

t.:) z _0

a.. ° ::l:"": a: C)

.Si;:SLE

II \ ., .II

LAG- R, j(.~ ,,/

\ ........ - Z I \ /

\. 3 +------------r--------_r--------_,--------,-,~---~~--------l

120.00 180.00 240.00 300.00 360.00 FEEDBACK PHRSE. OEG

C(IAS A~'[ Hl08AC', C~IN.',Z,l

~, ... '6~

~---------r------_r--.----_,.-______ ", " I I __ --f 120,...00 ltJO.OO 240.00 300.00 360.00 fctDBALK PHASE. DEG

FIGURE: 19. Modal Damping and Frequencies Versus Feedback Phase with Sine Cyclic Lag Rate Feedback, Configuration C.

PAGE 72

... I·

, .

'.

,...-.t'"."

,

I (.j

z _0

a... 0

:1:": a: o

10 ID

/

O~----------------------------------------------~

I • ,0 I ~1 ~ 1<:0 LA6-R.k:~ I C~~~~~,J~~~·~"---~~-~~1ft~~'==~Il~~=~.~----.--~"~";~qQ~~=1. i:Jo ' ROLl... he 3 ::l o W CI:II'I PITC.J.I, k-O,l,2,3

~-~----'-----~----------------------------~----4 o

o • o

91.00

• .. , 60.00

• "FL AP_ ~

• • • • 120.QQ. 180.00 ~O.OO JOO.OO fEtUBRCK PHRSt. OEG

360.00

FIGURE 20. Modal Da~~ing and Frequencies Versus Feedback Phase with Sine Cyclic Lag Acceleration Feedback. Configuration C.

PAGE 73

I

/ sr;:9LE

.. -..

/

i \

L:) 'Z _0

, a... 0

~..: a: o

o o

,// 1-""-: I "

UNSj;;SL:':

N+-~ __ ---r ______ ~ ____ ~~ ______ -L~ ______ ~ __ ~ __ ~

FIGURE

0.00 60.00 12Q.,.oO 180.00 240'08 300.00 360.00 FttDBACK PHRSE. EG

o ID

oT-------------------------------------------------~

~l· 91.00

21.

ilOLL

'<-1

.,. ... 165

::.---~~

• r

• • i - ,', .~ 1 i ~- ..

60.00

Modal Phase

121J . .Q£L IRO.OO ,>till. 00 :IJIi.1l0 lbO.')O fEtUBRCK PHHSE. OEG

Damping a~d Frequencies Versus Feedback with Roll Feedback, Configuration C.

PAGE 74

, .'

/7 / /

I . ..-. ~ ..... -,' I

/ /

. \ ' . .; ~

FIGURE

o O+-__ ~~ __ ~ ____ ~ __________________ ~ ______ ~ N I

o o N+-______ ,-______ ,-______ ,-______ ,-~~--~~~--j 0.00 60.00 120 . .00 180.00 240.08 300.00 360.00

FEtDBRCK PHPSE. EG

a: z ..... >-0 Un z· W O

:::l o w a:", u.._

o

o o

-0<

lr •

93.00

22.

~Ol.l I\~I( r([oe~~ •. r;.1l;" .. , .1.'

"P".I~~

i20LI..,:.<oJ

.--r • ~- 1 !I .

it~ LA&, k·~1./2,3

--~

'P\TC"', K.".l/z.~

• • ';'LAll-R I I I f I I • I I • r

-60.00 12q.,.QO 180.00 240.1)0 31)0.00 Ft:.tDBRlK PHRSE. DEG

.lfjO.OI)

;~oda 1 Phase

Damping and Frequencies Versus Feedback with Roll Rate Feedback, Configuration

PAGE 75

C.

\.

.:l/ , i I

"/ /

/

• ..

:' " ., " !.

: ~ • :-4 .. ,

'I •

I I I

" -,"

k....... -, .

o o~ ________ ~ ______________ ~ ____ ~ ________ -,

'" I

o o

~~-------r-------r-------'--------~~~--~~~~--~ 0.00 60.00 120 !)O 180.00 240.00 300.00 ViO.I)O

F~t08RlK PHRSE. OEG

o <C!

o~--------------,----------------------------__

FIGURE; 23. Modal Damping and Frequencies Phase with Roll Acceleration Configuration C.

PAGE 76

Versus Feedback Feedback,

"

/ ,

I

/' .....

!.

f ~

t , i t" .

~ r ~

0" ..

"

I

FIGURE

/

/

o o N~~~--~------~----r--~~-------------------'

o o

~+---~-~------~------~------~------~------~ 0.00 60.00 12!L.QQ. 160.0..0 -~0.08 300.00 360.00

o ID

Ft:.tUBRlK PHfbt. EG

o'-----------------------------------------------~

a: ~ L.A&- R/ ;::;~t.L ~. 0 1 ~ 3 , I 14.

~~~~--~----o-----~----~----~----~----~----~ Z· W O ::> o l.L.J a: lL..~

24. Modal Phase

Damping and wi th Pi tch

360.00.

Frequencies Versus Feedback Feedback, Configuration C.

PAGE 77

.. r' --:J?'J' ' ~ ,.:. .. ... ~.

o O,y-__________________ ~~------------------------~ 1~~_pIT~~ , : I \"- I(.t.=~--~:>; ~g '" ~'2 // U· " _~ ~7 , ./ :-... 3 -'f""" Ocr: .A. "" .. ~ ~~ I U[ 5:'~BLE , ott :.===-;====:;====~;:===~;.====~====-:;====-::;====:,t-~T---

UNST"BLE, ~ LAQ-R z _0

0... 0

l:"; cr:

° o o

rllC" ~AI( r[[OCAC., GRI'hl.l.)

~+I-------.--------r-------~------~~~--~~~---:; 0.00 60.00 120.Q!L t'80.'JO 240'03 lilo.oo 31;0.00

1 0 oD

W O

::> o w

FEtUBR K PHnSE. lEG

l L.-l6- R 1:;,:1.:'" , ;<zO, 1,2.,1

"PITCH

~~1 u:'"

'U-:t~' ~~ o . o

91.01)

&I &I G I :r I • C'

-~J 'HAl'- R f f ,. ,.

\0 ! , I i

I~')'e~ l~lO.OO 2\10,00

FIGURE 25.

60.110 lllu.OO J60.0 11 E' BnCK PHflSE, DEG

Modal Damping and Frequencies Versus Feedback Phase with Pitch Rate Feedbuck, Configuration

Oii:;C:.: .. :.

c.

OE pooa QiJi-~L1T'(

PAGE 78

l

\~ ~

.~ /'

~

-:.

FIGURE I

o o N ,

o -0 u . lLJ­(/"II

....... a 0:

0 ~o

l.:J Z

o

_0.

0. 0

~..: 0: a

o o N 0.00

0 <D

0

In C3'

0 ex: z ....... ... UO ,., • ~o :J a lLJ ex: u..';!!

0

0 • 0

93.00

26.

PIT"GH.I k. j,2,3

l..A6-~ t.C. "

--- 3 ~ ~T~R' :: ~ ..... -.o!.A£:i- R ) '(-0, 1,Z,.3 ~ uNSr;<BL'

.'

" fCK ACC[l. '[[CIIACI. CAl 11.1. 1. J

lIP". 785

60.00 12.Q.,~Q 180.QO ~0.(j8 300.00 Fttu8RCK PHRSt. EG

360.00

L~- R/ROLL ~ K-O, 1,Z, 3

• CI .. GI a • • III I

P,r':=H,I«-O,1.,2,3

"FL.A'P- ~ • • • • • • • ~ .. -.. 60.00 I2.ILDQ_ 180.QO <-40.0.0 3'10.00

fttu8RCK PHRSt. UEG 360.00

Modal Damping and Phase with Pitch Configuration C.

Frequencies Acceleration

Versus Feedback Feedback.

PAGE 79

I

/

.,.

...... ---

/ ,.

;;"":\

"

..

00 -,q ;',.1 ... ,', "I. , () :. (: "

.:g Cl ,.,., ::0 o

C> n N

<> ... ~, ) lJ •

-~~:-

\

-

"'1-r

, '. LA6-P

.-~. ;.

DElRe ros. fllOCIlCI(

..... CI no Iro

- I-. .---_oj' ,- ~- --

.!. -f\""'- - -1-- -- - -- - -t-

D 'f.-- ., .. -~ 1_1 ,. -I-

11 II I A'-,·R ~J -...-?" -'i j(. j ~

•• 0 \ n U :l.: . IT .

7-o ~ •

•• J ji.J J ... J 2 lIAP'~ ~

c> It,"·p / I ~ I I \ ,_ .; ~: ~ , \, \ _, ,_ 0.011

.. (J

(J ,-, r, • J' h. U) •

"".UU Ir'll,1JQ /11(1.1111 "~lJ.IIlJ JUU.UIJ l t1J1I1II.K I'IIW)(. III (~

a) I~ Feedbdck c

rIT'!f \

BUAC Acn I. fl t (lOIlC/(

;~ I LA&-? ((11t).. It • • -. ()

to ... ~ - 0-- - .--_ .. -- . .----.----~j n

n(l 1 .: (f

n

r, (I

N

O.UII

lL,rp .,.1, 3

I 1,11.111'

L -I~ - J,:

l<e"",1,1

, , "T"' , """, "l It·II.1I11 ,1111.1111 .'411.1111 JIIII.IIII

11111111 J< l'I!II',j. III f; (. ) ;; . f.ct:U h.lI; ~ c

" , .~~ ..... ,/ , . / . .. =-~. '-.~/ -­

,.,. .. -- .+

.. .. ''''~--.L~~'' . ~_'I , - -~-~ '1"--4, , 1.-

t' .. , , _h.'· tI,

• I

i r T" I'lfe:: t/ , .

r LAoS· P

r - \ CIt lIIe lUll I r ([Olllltl(

:::,'. j - I ~- .. -- • ~ t'': ~-.. ·L J ': . I ....... ,

~< I '1/ ' 'A:-" --11"---:\ .• ,J '1 J IC·JI1

I S\

~d 11 .. "'--, , -,":'~P II, U.llil 1011.1111 If·'Il.jJtI l'llJ.IJIJ ;lljo.no lllll.nO

11111111 ,K "IIWil. till; 31.0.

FI GURE 27.

11,11.

. b) ~c Feedbdck

Modal Damping Versus Feedback PhdS~ with Cosine Cyclic Flap feedback. Configuration C •

~ ," --

" ;'

"

-

-0 )"". G') rr1

00 .-

u

" '" o

u -c. l' •

~:-; ..... CI an an

... Z .... (.

Q_ c~ :r •. , a n

co n N

--~ .,-:/' / , --'. '1

.r

.,

1 ~ lLAP.P! lLAP~R ~ ~ \

____ 1-11/-\ .... ~' 2__ .. ', __ L '_\

-L h I T~_ I LA6- .. +-~' .+- _. -\ 1- --11---+-

.. .. _. / ~ ...... \I .~L . <:::1 . tF~ ~/(;O -.~~-o-. t--

ILAP-R II k. t 2 II

U'.IIII • bU.lIU

.r .... ~ T, , ~ ~\ I 1,(/,(lU 11111,1111 ':~(J.r:1I lllll.lll) .11,11 t 1 1111111 K l'III1~I. III I; a) I3 s Feedback ..

" • • 6£IR~ R((fl. rffD8RCK

0' ." ,. '

."~ ; u,

~ I II c,

iT,' : -1,111'0 -,. k'Je,?,,6_-.

- -.- -II-- -.-- _= =t=. 1 ,,_

(,

I') 7 •. c. o C! :I .. (J

L,

" n

'" U'.III)

I< 0 i

I (,1).111)

-r;:~l I LA«-' '

"A~ ur , , 1(".1111 /"11.1111 "'~II.II11

If 111111 K 1'1111',1. 1111.

k-y' 11111.11/1

e) ;;s FeedudcK

/1 -.., , , 1 I

I

... " ,

J ,. ,

11,11.

<> It

'~ I " ..,

<.., • h'''· VI'

/

II • •

erlRS nRlf r[fDORCK

-4- LAG-P ...r- -If- _ __ - -+- -t-.-, II a" .. () ~,(;~-"'~;"" ..... . ;' ~.;;;;JF-' 6 -~-~~ ~I 2.: LAtS-1t

n , I /<. t

\ TLAi>

I , r, ,: ~'~" ·-'l

/'_.

'" U'.II11 ,

(,11.1111 IllI.1I11 1"11.1111 "~II.IJII 31111.1111 360. fir IIIII1K ('IIWil. 111(;

FIGURE 28.

o'

r /./ 0

b) Bs Feedback

Modal Damping Versus Feedback Phase with Sine Cyclic Flap Feedback, Configuration C.

i , 0', .0 \

'\ . ' " \

00 ."TI;u "'0 C,) 0 ·-· -. () ~:: ~f-

c -'.' c. ,,' ~f" r

t,' -" ."' ..

./. '/

I

! U

......... ": '. ;

:.(

;'. , . ,

,. ;

.'

/ I, , I

d 'I

1 I f

J

! .

"::L '. t : 1 -'7

'~

(. , , .

/--

/"

. - '

. ',

r'

.. J.

I /'. i

"'-"

o o

.. - .....

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

o -0 u ._ w- • tn l .... o a: o

_ ....

.•.. - I •.

~o c+-----~~----------~~----~~~==~~----~--#--------

t.:)

:z _0

, a..~ , L_

a: o

o <:)

~+_-------~1--------~1--------~1--------~'---------1 50.00 fin.OO 70.00 80.00 90.GO 100.00

o <0

ROTOR SPEED. RPM "la'

c~-----------------------------------------,

IJ"I

a: 'z

...... ! >- i w~·

~c1 :::J .

, 0 UJ a: u..~

C

--- 3,4~l:LOJ~

- - - ... ·3 I ",o6s"

I :a=: == ::&:= ___ ----TL..JP-1<.

~t.oo- 6[1.007b.00 8b.0~ 90.00 100.00

FIGURE 29.

ROTOR SPEED. RPM wlO'

Effect .of Cosine Cyclic Lag Rate Feedback Gain on Nodal Damping and Frequencies, Plotted Versus Rotor Speed, Configuration C .

,.' \.,It •• \'

PAGE 82

":"- '.

". \ ..... , \ . \ ) \ \

... ~\

'..

\

, "-

.

·1 \. ,f

. "

, .

0 0

'" I

0 -0 u w-(J"ll "-

~_ ,I

~~ ., ... -r ... ___ ..... _..r

3~ _ ----z .... - j

1_----

I

a 0: 0 '::0

0 t:)

z .... 0 3 a..~ ;I: ... 0: Q CETR' ros. H[CBACK

0 CRIN'I.2.1 r~AS[.2cOO[C

0

N+--------.---------r--------r-------~------~ 50.00 60.00 70.00 60.00 90. nO 100.00

ROTOR SPEED. RPM .. \0 1

o U)

o T------------------------- -------

o a::: z "­>-0 u .... z· W O

:=l a w a::: III lL._

a

--- 3Al EL.I\.J "

r--: i I ~

~p--

L ..... •

o o - • - &_--- 1'lA?·R

~O.OO 60.00 70.00 60.00 ~O. 0.) 100 .•• 0 ROTOR SPEED. RPM .. 10 1

I FIGURE 30. Effect of Sine Cyclic Llg Feedback. Gain on r~o da 1 Damping and Frequencies, Plotted Versus Rotor Speed. Configuration

PAGE 83

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

..

..

C.

. ... '

\ "

,.­--'

\ - - ~' ....

, I

.. / "

'I ....

'\' \ . , .

."--

", \

"

.. \

-

.-~ --- -

0 0

N • t I t

0 -0 u

'WI :1 V1

" I a I U: o ~O ST"BLE

0 UNSi;:;BLE

: L')

z I

I

i

_0 (Lo

:r:"': a: a

o o

LA6-'R C[lRS ACC[l. f([Ce"CK

LA6-P - - -N+-------~--------~------_r------_,,_----__4

a: z

50.00 60.00 10.00 80.00 90.00 100.00 ROTOR SPEED. RPM "10'

o <D . ---------_._- ,--, _ .. ----.

°1_J4SEL/~t: ~ I ---- K~27J />. ~:;. o

,/ LA;;·R

~g R,'I..L -'" ~o ::l o

'W , a: I lI.... '!!

FIGURE 31.

: - r : r T t_-~,i _____ -'--.-, _r_L_,q_Pj 60.00 70.00 60.00 '30.00 100.00

ROTOR SP,EED. RPM "10'

Effect of Sine Cyclic Lag Acceleration Feecback Gain on Modal Damping and Frequencies, Plotted Versus Rotor Speed, Configuration C.

PAGE 84

- ----- -~-~- --- --.... - ._ ........ -,~ .-. ~ . ~ *' .r .. - -=:- ~~. _.... _ '\ ."

, i

/ j''' .. i ,-

.I

./

, I. \

/

".

\.

'.

I",

o o ~'------I-----I~!-S~~N.~.~-+---------'

0

° tV 50.00

° CD

,;

l/1 -:r o

a::: z "­~o u.., Z· W O ::J o W a::: lL.~

o

o o

LAa-Q o 1,(. t 6 3 + X

q

o

ROLL RnTE FEEOSnCK

C1IH-I,3,9: PHRSE-1200EG

60.00 70.00 80.00 90.01) M 1 O' RO TOR SPEED. RPM

~ ~ l

.,..f ~~ ~!:==::::::::_:!7;====~=::

// .. .­

./

:II/P-~._-II-D_._.-t.t--t._-... -_. :,iCH

100.00

~O.OO : 60.00 70.00 ROTOR SFEED.

80.00 RPM

go.co HIO'

100.00

FIGURE 32. Effect Modal Versus

of Roll Rate Feedback Gain on Damping and Frequencies, Plotted Rotor Speed, Configuration C.

PAGE 85

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

...

" ,

'/ ,

/

\ '

"

'. ,

/ /'

o o

LA6-S

I

IIOll Run. f(tOOACII

GAIH'J.9.l1 /,,,A5t,"1QUrC

N'!-______ ~------.-------,_------~~~--~ 50.00 EO.OO 70.00 80.00 90.00 100.00

o 1£1

ROTOR SPEED. RPM 14) o·

.:::i -- ..... ,-------_ ....

o a: 2 " ~

LJ~ 2· W O =:J a w a: u..~ ~

o

'H.AP-R :===~ __ ----",-z-; of 0------__________________ _

o!-______ ,-______ .-______ ,_ __ ----~~~--~ ~O.OO 60.00 70.00 ao.oo 90.00 100.00 ~ RP .. ) O·

FIGURE 33.

ROTOR SPEED. M

Effect of Roll Acceleration Feedback on Modal Damping and Frequencies, Versus Rotor Speed. Confi guration

PJI.GE 86

Gain Plotted

C.

..

1 •

J ! / I

! I

!

__ 1

/

;/

, I

t' /

0 C

N • J:

0 -0 U· W-(/") . ...... 0 0: 0 OCO S:-:::BLE - .

0

.L:l LAG-~ uNsr;:;BLE

b ·1.il.~ Z _0

a..~ 6 T._ 0: T 0

X

0 0

N 50.00

0 10

0

'" :7

0 OC Z ...... >-0 u .... z· W o :J a w oc'" U-_

0

0 0

C:Sc.oo

FIGURE 34.

Z<ilo·

2;:0" c[ rAS res. '[[CSACK

1-<·0 CAlli •• 3 rHA5[.21S.2~O.l70 O[C

60.00 70.00 80.00 90.00 100.00 ROTOR SPEED. RPM Ie 10'

':'4 ... -Q

~..)!..~

PITCH

• • • . • • • .. TL4i2-~

60.00 70.00 80.00 90.00 100.00 ROTOR SPEED. RPM IdO'

Effect of Sine Cyclic Lag Feedback on Modal Damping and Frequencies. Versus Rotor Speed, Configuration

PAGE 87

Phase Plotted

C.

/

. .'

" .

-..

.I

o o ~~--------t----a--~g~W-u~--------------------'

• o

-0 U· w­tn l

.... ' o 0: 0 i- ~Z~ ~O ~--= O+-------------------~~~~~--~~~~------+-*--------'-:"--

I..:l Z _0

n.. 0

x:": 0: o

,0 ,0

LA6-~ o ?q~.

Co 70· + '10·

x. 1<-0

~ UNST;1BLE

C[IAS IlCC[L. fHoeACK

N+-------~--------r_------_r------_,~----__4 SO.CO 60.00 70.00 60.00 90.00 100.00

o ,ID

ROTOR SPEED. RPM MIO I

10~-----------------------------------------~

'" :r ,0

a: z .......

""'u w,., Z WO :J a w a: u...~

i 0

o o

~f' LA6-R.

-----~.-_ ... a .... _ ..... __ 6___ ..... ~ .. LAP- ~

~O.OO 60.00 71.1.00 BO.OO 100.00

FIGURE 35.

ROTOR SPEED. RPM

Effect Of Sine Cy"cl ic Lag Acceleration Feedback Phase on Modal Damping and Frequencies, Plotted Versus Rotor Speed, Configuration C.

PAGE 88

- -- --- .-..•.. --.---.. ---------.-~-.~

....

, .

.,.

---

0 0

N I

0 -0 U· w-(fll

"-Cl 0: 0 a: o - .

0

L:)

Z _0

a..~ :2:_ a: Cl

0 0

N 50.00

0 I!)

0

in '3" '.

0

a: z ..... >-0 u,.., z· we:> ::l 0 W a: lL..~

0

e:> a

C\;;O.OO

FIGURE 36.

I II

z ~ ; II II IJ z z • • •

$ $ , J t if 2 • • ! • • l( l(

z z z z z -~ I "" -- SHlBLE

~ -e UNS HlBLE

·LA6-R C ¢. "1(,. ~ 100· ROLL RRTE fEEDBRCK

+ 110·

" \.(cO GAIN·91 PHASE'90.100.1100EG

60.00 70.00 BO.OO 90.00 100.00 ROTOR SPEED. RPM M 1 0 1

L.A6· R

.... .;;~- -~. I/J • ,.. :;, <>. -•• • u

". l~.

~. 1:0-

. ¢_IIO·

+l..AP-~

~.::t

i i 60.1)1) 70.00 8(1.00 90.00 100.00

ROTOR SPEED. RPM 0410 1

Effect of Roll Rate Feedback Phase on Modal Damping and Frequencies, Plotted Versus Rotor Speed, Configuration C.

PAGE 89

..

..

." ":'

.I -{ /"

i I

.--'

I / .

., " ,

._~_ ... r·

! I

,

i

0 0

N I

Ii I

i I • U

I

0 ~O

LJ W-V'll ..... Cl a:o !::o

~ Z '7 7' 7' 7' '7 ~ i. ~ ,::::::;t

0

LA6-R ~ t:)

Z 0 _0

CL o A :x:_ a: + Cl

)(

0 0

'" 50.00

0 w 0

In ;:r

0 a: Z ..... >-0 LJ,.., Z W O

::::l 0 W

a: In lL._

0

o o

'i(oc..L. 27/,# 2.r~ ~ ~~.

k -0

?-?GO· z.~S· .'

i!.~. ~Oll ACCCl. '£CceACN K." 'RIHa21

60.00 70.00 ROTOR SPEED.

'HA$[.260.255.210 Dec

80.00 RPM

~o.oo If 1 0 1

L. ':6- Q

_.2-;10 0

'11-0

------------- k.D

STABLE

~ UNSTABL

100.00

~+0-.-0-0----6TO-.-OO-----]TO-.-OO-----8'O-.O-O----~9~O-.O~0~--~IO'O.OO ROTOR SPEED. RPM wi 0 1

E

FIGURE 37. Effect of Roll Accel eration Feedback Phase on Modal Damping and Frequencies, Plotted Versus Rotor Speed, Configuration C.

PAGE 90

.. ~

. r

i .

/

" .

-,

~.

" " ., ~~ ;'

~.

-,

e ,

r---~· ............. .

u

. I .. • .. , w

.. !I l! ... .. I!: c ~

" ! .. ; S

o ...,

c:

------~------------------------------------~

10 L~ ,­, ' , I' I'

!~~ .. ·U

10~ , ::

g=

,-' -f-_-':~""". --"'-~+_----"""---i 0'

:..: IX~ -~ 100: u I " p..1~ ;; I ~ • ~ .. O..J

~; ~ l'~ ~; ,

ig u

• .... .. I

--I

i •

j7 I ,

~' I I '0

I i~ C9'~'O---S-h~~-C---c-'-'-a------s-:-'-C------OO--'O'

'~N/.l:;N~:1C3b,j I

, ....

.. • I ~ fi ..

~

" l .. :. ~

..., .. :

0 ~

:~ .. ,-1 I

'0-~ J'tU .. .... i 0 • ., , ..... i -, c:=

'C-

O

=:.. " .0_ , . c: ,

g .-.e

or>

... ..... " .- -

... c:

--------------------------------' ..

. ! I . : ~ .. .; ... I : ; Ii w 3

-. ..:

-::: ;-

-------~~~~;-----------------~ ,. .. u a. ~ ::.c 0 i C I:. w C>

w 'l"" .....

VI' 'v \ D \

PAGE 91

.....

\ \

-­.'

.-,0

.' :0 '0'

c

/

....

/ I

.'~

.J

/ , /

..

.. . "

I·

i.

..: u <: :::l 0 .... .... L.o.

)(

,-

v

o ... ---------------------Ol ~ § 0

=0 . \l

'tI- X

'" i

..J 0 <l + "

. I . i I

.. ~ ! ~ :r ! . .-i

o 1 0 .. ..: , '",­"","\U j..~ , Ct".., I .....

19j ct:

;~ ;;:' \

-15= ~ -----~.::..--::------..... ----,; ::::;

~ L.o.

r f .:.. .It :>: 8 8 $' 0

;;;-I, • _ )II(

~c:=. o....J

.-

'" .~

. -----------~--------------------

~ -Mer 0

.,: ,

::s: c::: 0 <l ~ )I.

.. I .

I :r ! .

i i

. i

'" :0

.­, . I I .0 I .~

.a:

.:.. ....... . ' ...

c:r': J .•••• - ..

OF FG8.~ (. ~ ... _ ..

• • • Q( ~~ ~ () '" I "''''' .... -~

, .... .:s: '-I 0<0 .. "

Q( "'IT .. () . '" ~ . :s:

~ 0 <l + It

PAGE 92

.... ._< i I

, ,

", .~

~

II l! ... 2i ... · . . -I ... ~ . ~ , =-.. ,J'l __

· Q~ I ::: · • .....

I 5 -c-o-c

• --.' .=:

--.-

-- -

~ .. . l!

I .. :c

- ! . ..:

:; - ,; r .. ~ .. f ~

.-

...

'.'

Col

'" ,., c: '":I

,., c: ,., ~ ... v .3 '":I :.0 :.0 ... §

~ .J'I :.0 ..J

~~ U .., v ;:.,. c=:: c ~ Col "'.."

:;)

CoI:i: ..-::'>

.::::; .., .... -- =-0 C;:.

0: -

"" ... .." 0 CJ

:.. ..- =' V <:.I ColO: .... - :::

Lo.I 0

C'\ M

W

:::> ~

L.o.

t.I

~ "' ~ :;) ... c :..

\... -~ ~.

~ <.. C

... C

:::::

/ I

I

I

/ I

"

'.30 e:: ~ ....... >-u ~ w :::> .1. 0" w e:: L...

0

,-.1.1

- J

u - . ..!.;-..... V')

-I ~ ...... 0 < e::

~ 0 ~

c... ~ .1 0

2

o e -.10

6 ; &.cr o e· ~2.

LAG-R

~7?--------------------~ PITCH

100

----~

~\ /~

~I ~".IO '0

~ 6 ~ •• OS'

0 e • .OJ.

;JO

ROTOR SP£~D RPM

FIGURE', 40. ~lodal Frequencies clnd 2amping for Equivalent

Dynamic Syste:.:s of Varying Hinge Offset.

PME 93

-, .. ", "'"

I

" , i

/

.... :

." ...

\

,\ .'.,

/

1\

\ ' \ , " ", .

\ , ,

\ ' \'

, ,

'i \ ;\ ,

? • "

FIGURE ... 1.

-

-."

- l

- . -

J l

'6,

iC~ ~co

ROTOR SPEED RPM

MJdal Damping for Equivalent Dynamic

Systems of Varying Precone.

(e=.02R) ,

PAGE 94

.'

~ ..-..;.....-..

.0::, z: ....... >-u z: L.J ::l 0-L.J 0:: t...

u L.J V) ....... Cl c:: 0::

c.,:, z: ....... c.. ~ Cl

FIGURE 42.

0 ~ ... 3af II", LAG-R

~ ",1 • (Q.4 Wit'!

.4S 0 ~tS· ()

. 'ROLL . 30

.IS" iJlTCH

;:"LAP-'R

600 100 800 100

~ -.4

I - 3 I

/ I - 2

( -., ~

;J

.1 O. 1<4: 3 e.i II..,

6 /.(~. I c,.,+ '/"",

.2. 0 ~s· 0

'61)0 fOO iOO '?co

ROTOR SPEED RPi~

Modal FreQuencies and Damping for Equivalent

Dynamic Systems of Varying Flap Sti ffness.

(e=.02R)

PAGE 95 .

.... 1t:'tto' ,- III

, .'

, ... -

-

I I

I ! i i

I r t r

!

o o N~-----------------------------------

ORIGl~ri.L F.Q~·: :~~, OF. POOR QUALITI

o o

o o

rROG. LAG ROOr I. C. 0

a) Progressing Lag Mode

o C[lAC

& BE lAC

+ C[lAS

X 8E1RS

o ROLL

.. rllCH

N+-______ ~ __ ----.-----_,------_.------~ '0.00 2.00 4.00 6.00 6.00 !O.OO

o o

ROTOR REVOLUTIONS

IIfG. lllG ROIlr I.C.

r (\ ('\/\ ,t\, ;

1;\\ ;' /\ //. I I \ I 1\ \ I :

\ /~ \ /I'){'t:

o CEIAC

A SE lAC

+ ([ lAS

.J~Yj~ o ROll

• rllCH

Regress ing Lag 140de

I I J 2.00 4.00 S.OO s.co 10.00

ROTOR REVOLUTIONS

FIGURE 43. System Response Using Progressing and Regressing Lag Eigenvector from

. Stability Analysis as Initial Condition, Configuration C. (No Feedback Applied)

OJ PAGE 96 f

I -'e** "a f;E t'jg :t.e"'LtM1:e"'II~,~<~:re;ra:~B-~~:t,er'.

~

" "

! '

!.

r * . r

t

, '.

I,

I :

cn:~:i~;::~ t ~~.'. _ "~_ :'~ OF. POOH c;U/.Li ;'/

o o~ __ ~ __________________________________ ~

'"

o o

o o

CEIAS '05. FEEOBACK

~(C. LAC. "OOE I.C. ","" FEEOBAC., o CE lAC

& BHRC

+ CE lAS

" BE IRS

N+--------r-------.-------.------~~----~ '0.00 2.00 4.00 6.00 6'.00 10.00

FIGURE 44.

ROTOR REVOLUTIONS

System Response with Si~e Cyclic Lag Feedback, Using Regressing Lag Eigenvector from Staoility Analysis as Initial Condition, Configuration C.

PAGE 97

'enw!:=c

. ,

.,

.. f.'~\' , ......... - ... ;- ""f···r"·!",;,,"·

l !E - - -

is

~ I il 'l ,. ... :" I ~' \. . ,oj

:t :4;

~ ~

. ~ "1 I., ( .

.c'

i " j I,

l 1

'i' . ' .; :~~ f?

~: A

" > G) rr1

<.D w

o o N.,'------·-------- -----

t:l W o

ROW ~IICK 5111111, 'DfG, • ll6NII

It''' I.C. a a I 1Ir"·J6~

o

fallCID lIurOIlSf

l1J~~~.. ~ Ul Z o n..o (1')0

W· a:-' ,

a o .

.. _ ... _-_. __ . __ ._- ----.---

, /If! /lurOIlS[

';r-f I I I I . 0.00 2.00 q.OO S.OO 6.00 1~,OO 12'.OO--1.q':OOU- IS', 00 16~ 00

FIGURE 45.

ROTOR flEVOLUTIONS AOTOR REVOLUTIONS

System Response to Advancing Stick Stir at Regressing Lag Mode Frequency, Zero Initial Conditions, Configuration C. (Ex~itation Stopped after Ten Rotor Revolutions).

..

.. ~ 0- ·-':'~.i-~

" C[l1K

• I£lAC

+ C(lAS

x 8[1IU

• /lOll

.. rliCH

10.00

'~

'1 ;-';

11 .~ ~;,

,~ ,

! " :l t

l:f

"j

1'1 ~". l .S

"

"'0 > G"l ITI

~ ~

_.-.... _- -. ----------- ... _" __ .t.-.~... _ •. ___ -~:=~_'___ ... __ .:~:......~_~

a C>

~·r-----· --,

l.:)

UJ o

o o

o o

"lC. "ICII SII"", ID(e, ."6N1I

UIIO I.C.

"'"·1&5

ll.JciY~1..r-' Ul Z o CL a Ulo UJ • u::-

I

o o .

rOIl((D "HrCMS(.. ... "11( IIrsroNsl

7'1 I I I I _I -_. __ ~----____. 0.00 2.00 4.00 6.00 8.00 la.OO 12.00 111. on 16.00

FIGURE 46.

nOTOR nrVOlUI IONS nOT-UR UrVOLlJT IUNS

System Response to Regressing Stick Stir at Frequency, Zero Initial Condition, Configuration Stopped after Ten Rotor Revolutions}.

Regressing Lag Mode C. (Excitation

o ((I~

A allAC

• ((lAS

)( 81 lAS

o IIOll

• rUCH

1.0.00

00 ~n ;OJ

"tl ~~ 0'''' o ,; ;c fa .0 ",; c ::.: .... ' , " .. ' s: t'; -, .'. ~~ ,"

, I

Ii

! ·r

• J

i

0 0

0 (Y)

0 0

c CD 0

0 0

lLJO 07 ::l I-..... za t::J~ CI:a ~rn

I

a a C III

-0-

6

-+-

1. 'I) =. 5".2. ~ = 2~

= e~ '0. 10 0.20 0.40 0.60 0.80 1.00

o o

EXC. FAEOUENCr/NA VAAY BLADE DRHPINCI .5.2.B7o

NO FEEDBRCK

~~ _________ RE~C~.~S_T~I~CK~S~T~I~A~Rr·_I~D~E~C ________________ -,

a c o C1I

c lLJ If)

CI:c :co (L' a

C1I ,

FIGURE

o o o CD

'0.10

47.

-0-'1~=.S2% t::. = 2 ;/'0

-+- :: 1%

0.20 0.40 0.60 0.80 1.00 EXC. FAEOUENCr/NR

Damping on Frequency

'.

Effect of Blade lead-lag Response of Cosine Cyclic Applied, Configuration C.

lag Motion, No Feedback

. .' ..

PAGE 100

FIGURE

CD Cl

o o

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

o o o

o~------------------~~1r----------------~ o

A o ~ -+-o ~+---------~-r------.---~r-r-r--r--r-~'-' '0010 0.20 0.110 0.60 0.801.00

EXC. FREQUENcr/NR

CETRS POSe FB: K-.3.1.31 PHRSE a2400EG

g REG. STICK STIRA, 1:E~

.. g~~----------------~r-----------------~

o en

o . il.J'

U1 C:c ::co a..' o

en ,

o o o

t:. K .. I.

-+-K=3.

~+-__________ -r ______ ~4-~ ____ r-~~~~~ '0.10 0.20 0.110 0.60 0'.801.00

48.

EXC. FREQUENcr/NR

Effect of Sine Cyclic Lag Frequency Response of Cosine Configuration C.

PAGE ·101

- ..

Feedback Cyclic

Gain Lag

, ... -

on /'.otion.

~'C;:urr;;~-1M~:·!~~;;J~~!f!~~7'~~~~·t·':Z$N· -::{··f·: 1:,tt·t;:::·;o:·,::··"#··~")r.'·:'!:a!e~f:-:rv6~~~~,;,,~.;:.w~:-:~!-"~~~·~;;4 I

, {.:

t>

f

· ' ") ..

!

l: I: L !

i

I

tl p

II if It Ii ~

TABLE 1: Rotor/Body Properties for Configuration A

Number of blades

Hi nge offset, ft

Blade mass, slugs

Blade first mass moment, slug - ft 2 Blade second mass moment, slug - ft

Lag spring, ft lb

Lag damper, ft - lb'- sec

Fusel~ge mass long, slugs

Fus~lage mass lat., slugs

Longitudinal Stiffness, lb/ft

lateral Sti ffness, lb/ft

longitudinal damping, lb-sec/ft

lateral da~ping, lb-sec/ft

Rotor radius, ft

Chord, ft

~o~inal rotor speed, rpm

Precone, deg

Height of rvtor above body mass, ft

PAGE 102

4

1.0

6.5

65.0

800.0

0.0

3000.0

550.0

225.0

85000.0

85000.0

3500.0

1750.0

24.0

1.75

300

o

o

(.3048 m)

(94.9 Kg)

(289.1 Kg - m)

(1084.7 Kg _ m2)

(0.0 N-m)

(4067.5 N-m-s)

(8026.6 Kg)

(3283.6 Kg)

(1240481.8 N/m)

(1240481.8 /lIm)

(51078.7 11-5/1:1)

(25539.3tl-s/m)

(7.32 m)

( .53,")

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

.c., ..... ·f ." • • ,.' .. ! .... ~"'H'·~~~o?;"·w··~"'~"10 4'~;';'~ ~::~;~;;~:~~;;.;Z:i %:::" N:;ZSZcn ... r"£'~~;;::..?-'i! ... ;"';i;,7:~~~,,'ii

i .!

TABLE 2: Rotor/Body Properties, Confiquration B

number of blades

Radius, cm

Chord, cm

Nominal rotor speed, rpm

Hinge Offset, cm

Precone, deg

Bl~de airfoil

Lift cuNe slope

Profile drag coefficient

Lock. number

Sol idity ratio

Blade mass, k.g

Blade first mass moment. K~ cm

Blade second mass moment, Kg cm2

Nonrotating flap frequency, Hz

Nonrotating lead-lag frequency, Hz

Damping in lead-lag. '; critical

Height of rotor hub above gimbal,

Fuselage mass in pitch, Kg

Fuselage mas~ in roll. Kg

Fuselage inertia in pitch,Kg cm2

Fusel age inertia in roll, Kg cm2

Pitch frequency, Hz

Ro 11 frequency \ Hz

Damping in roll. ~critical

Damping in pitch, ~ critical

cm

PAGE 103

3

38.01

1.26

10DO

8.51

a Circular

.0

1.0

.0182 (0.0)

0.03179

.699

9.275

177

3.01

6.39

.135

24.1

19.27

19.27

5110

1870

Bl:27 .~; B2:

3.6

3.0

3.0

.. -.

2.39

I .,

; !

·f . , ,

- , - .

{

· i

TABLE 3: Rotor/Body Properties, Configuration C and D

Number of blades 3

, Radius. em 81.1

Chord. em 4.19

!lominal rotor speed. rpm

Hinge Offset. cm

720

8.51

O. PrecQne., deg.

Blade airfoil IlACA 23012

Lift eurve slope

Profile drag coefficient

Lock. number

Sol idity ratio

Blade nass. Kg

Blade first mas~ moment. Kg cm

Blade second mass moment, Kg cm2

::onrotating flap frequency, Hz

!lonrotating lead-lag frequency, Hz

Damping in lead-lag, : critical

Height of rotor hub above gimbal, cm

Fuselage mass in pitch, Kg

Fuselage mass in roll. Kg

Fuselage inertia in·pitch,Kg cm2

2 Fuselage inertia in roll, Kg cn

Pitch frequency. Hz

Roll frequency, Hz

Camping in pitch, ~ critical

Damping in roll, :: critical

2"1

0.OD79

7.73

0.0494

.209

3.887

173

C: 3.13

6.70

0.52

24.1

22.60

19.06

6330

1830

1. 59. 2"

3.9. 4"

3.20

0.929

"Body Frequencies used in study of active control •

PAGE lC4

- ..... - •• - .+~

.. .r~ -'':~'';.. .... ' .=s.

D: 6.63

6.73

0.53

..

- - --- ---------_._--_ ... ------_ .. -,

I I ~ .l

1

i i

~ :

, I ;

I

FEEDBACK STATE

I

I

BASELINE

· t;c

t;s

· . r

"s ,

I · (;) x I · . 3 x

TABLE 4. I

K ¢ a 11 deg rad/sec Of Cc .0

0 - .145 -.58

1 60 -.164 .65

.3 240 -.137 .54

3 60 -.178 .71

9 90 -.149 I .59 i I , I

27 I 270 -.284 ,

1.13 j

I i

Summary of State Feedback Results for Configuration C.

(n = 765 rpm)

PAGE 1eS

(;) Amax deg

-.32

.29

.34

I .33 I

I .39 I

.J

' ..

...

[(b,_. .-----~ --.. -.~~---.~~~ .. ------ .-'- ... - .. ,. .;.~ -. ...... _ . ~ _.. \ r ~} .. q;!:, 1!{m"Zi'Z:<j~i;;;~';_~""';~;'!£1f.~-,;;.;r-~~f~>"'P:;Z;~ ;;:'~:"j.;\~.\~';;:;.:;i$u.;~: 1!; '::::~"~t"·~.::;U;]\iD. ,!ii \,~;zj

/ !

"I~(~\

'·'·i',·' . .

"

"'1 . ,.,-

1';'1 I '" . ~,;.;.

,it ~. J

'J ,c:~ ,11

}i! :11 ;~ ,.'-.t :.,~

<t ::~ :1 ;1

,1 ;1 : ,:j.J' , : I

:. ~~'l i! ,.': S ~:

!'~···l·~ \::; r ~. .' r" ~ l::

'.1 " '~I :;;, ·1

;1 '~~~ ,11 ' n~ '····'l~ .'

.~~ /1 .a

-0 > ~ rrI

.... (J

~

111nlje Offset 'l R

10

5

2

10

5

2

(J

rad/sec

- .137

-.097

-.077

-.284

-.158

- .119

TAGLE 5:

;--:·····-~f"l .. "Il~~·rF~~Ol'~~1 ,

~T () Amax

II I _ deg ____ ~. __ I-- ---I

t.cc

---------"T--reedback I K f State deg

.54

.38

.31

.29

.29

.29

{'s

l,S

~s

.3

.3

.3

240

240

240

........ """',- ',.

-----c lag k lag k flap

N-m-sec II-m N·m

---.0076 30.66 6.69

.0049 41.8 26.0

.0036 47.8 38.8

-----~-------~-------+-------+--------~

1.13 .39 ()

.63 .45 0

.47 .37 ()

x

x

x

27

27

27

270

270

270

.0076

.0049

30.66

41.8

6.69

26.0

1.0036

147

•8 .! 38.8 __ I

Effect of lIin~e Offset on Feedback Results for Configuration C •• [quilla1ent

Dynamic Systems. (II = 765 rpm)

+ Without active control (K=O) system darnpinq is I}= -.58% critical for all three

values of hinge offset.

, .

\.

, ·':I·"~i~\' . ;"} t.-' t', ~~,

:1 '\

,~ ".'.r­.~(~

"~~ i) i:~ ~:~i ,~~~ t ~ ... ;;;~ "if

"~'11 .~.

:' ~,~

"\

:~}J d ';-')

rl I

r',. I'~: '\:

{~ ,i_1i'l~ . ~~~ .

.. ,;, ;~n '~ (:1 ~.i. ' '~:~ ,~~

"'C ~ ~ fT1 -n ""-l

.. \,.

. . lit .. :.'s:':";,... ... : .. .1.. c./.t .• ".,<t ....... \I,.fJ',l;h!'lrri"~\t'" -).4> ¥..Ii G. ;'~~k .',)IIJ&&"l6if W4R£&J4 ._.

I t ! I Clag Precone n ,kl k CJ o '

Feedback I K ' ag flap deg rad/sec %cc

Amax '"

, N-rn-sec 'N-rn ! N-rn deg, S ta te I deg

! I I

I 240 I i

0 -.077 .31 .29, t;s

I .3 .0036 ",47.8 38.8

.29 ' 240 I j

2 - .116 .46 " t;s .3 .0063 ,117 .8 38.8

.1 I 4 - .157 i .62 .29 r.s .3 240 I .0089 47.8 38.8 I

i I

I

I ;

.. I , .

0 -.119 ' .47 .37 0 27 270 I .0036 47.8 38.8 I x , , , .

2 -.288 '1.14 .43 .. f 27 270 I .0063 47.8 38.8 Ox

I I I

4 -.507 r 01 .50 .. 27 270 I .0089 47.8 38.8 0

I x

TABLE 6: Effect of Prp.cone on Feedback Results for Configuration C ••

Equ i va 1 en t [ynarni c Sys terns.

(0 = 765 rpm. e = .02R)

t Without actIve control (K=O) system dampin~' is

n = ~.58% c.rj'tiql for ~l:l three values of, precone.

,

.'-. . -'. I I

. I

\ ... //"

. -""- -~.,-.--.. (+J_ .. --y" ..... _ .......... '-. . ... ,. ... ~- .. -:" ..... --.'~--- .... -.. ---~- '-:-~T~'-"'" .. ·,....,-..O:T'''t~ ...... .:rrj~~f-';-r--''': , .... -.,

I !

.. ~- .. 'r .•.• ,"1 ..... - ............ - .. __ -.-

:g 0:;") (T1

...... o C?

Flap Sti ffness

N-m

3!3.8

19.4

o

38.8

19.4

o

o

rad/ sec

-.077

-.181

Xc c

.31

.71

-.1321 .52

I -.119; .47

-.232! .91 I

-.334 i .36 I t

11+ °l\max deg

.29

1.00

1.83

.37

1.17

2.41

I Feedback State

('s

t:s

t:s

o x

o x

Ox

K

.3

1.

2.

27

27

27

- ~l----I

~, c , k ' k 1 a gil a giro 11

deg N-m-sec N-m : N-m I I I

240 .0036 '47.8 115.6

240 .0110 48.1 132.0

225

270

270

225

.0595

.0036

.0110

.0595

47.5

47.8

48.1

47.5

138.0

115.6

132.0

138.0

TABLE 7: Effect of Flap Stiffness on Feedbdck Results for Configuration C.,

Equivalent Dynamic Systems.

(n =765 rpm, e=.02R)

+ Without active control (K=C) system damping is

n = -.58% critical for all three values of flap stiffness.

-..... ,- . ....-r= r. ." _/.. •

, ..

/"

"

.i , . i

'} t

,'," .. "), ......... -' . ~ .. \:. .

a.

CAJ

eEl

c:

cao Clr

Coy

Ca" C(;)'I

c~

rej

e ,.

T

Cf]

SSL' 551=

h

I" ' !)C

Ix)'

Iy

I<

K

1<13

I<~

Kx

APPENDIX A: LIST OF SVMI30LS

Lift Cur~e Slope

State S~ace System Matrit

State Space Control Input Matrix

Blade Chord

Profile Drag Coe~~icient

Fuselage Longitudinal Damping

Fuselage Lateral Damping

Fuselage Roll Damplng

Fuselage Pitch Damping

Glade Lead-Lag Damping

Dampir.g 1'latrl/. St.a~~ Sp;;ce Output Ma'tru

Blade Root H\n~~ Jffset

Forcing Vector

Control Input Matrix

I3lade Structural Lag and Flap Dampl~ri~

Offset of Rotor Hub from Fuselage C, G,

Blade Second Ma~s Moment or Inartia

Fuselage Roll Inertia " ,

Fuselage Product o~ Inertia

Fuselage Pitch Inertld

Blade Index, k=l. N

Fpedback Gain Constant

Blade Flapping Spring

Blade Lead-Lag Sprlng

Fuselage Longitudlnal Sti~Fness

PAf,E 109

i

1 1

1

I"\y

eM]

N

NR

f3k

A:,/3s f3p

,

t £

t,'k I

:!fc: )~ s

: " Fuselage Lateral Sti~~ness

Fuselage Roll stifFness

Fuselage Pitch Stiffness

Stiffness Matril. Mat"ll of Feedback Ca1ns

Length or I3lade. from Root Hinge to Tip

I3lade Mass

Fuselage Longitudinal Mass

Fuselage Lateral Mass

Mass Matri·x

Number of I3lades

Nominal Rotor Speed

Vector of Generaliz~d Coordt~ates

Rotor Radius

Fuselage Longttudlnal Motton

Fu~elage Lateral Mot!on

Slade Flrs; Mass Moment o~ [nt~rla

Vector of Cont~ol Inputs

Open Loop Forcing Signal

Vector of State Spac~ Va~iable~

Vector of Output Measurements

. ,

Flapping Motion oP the k'th ~l~d~

Rotor Cosine. Stne Cyclic Fl:3p Deo.]rees o~ I-="re"edo • .,

Precone

La c k "Num'l er

Order of Magnitude

Lead-L3~ Motion of k'th Blade

Rotor COSine. Sine Cyclic Lead-Lag Degrees of Freedom

PAGE 110

.... _~.~. <.- .... 4>-r· • ."., ....

/

.. ~ ._.-._- ~-.. -- .. -- ·._--_.-.-

"t

"t~

aD eAk

~'/~AS

I G'

: ¢t:) ¢s . l'

"fk

',e..>

Modal D3mping CoefFicient. % Critical

Blade Lead-Lag Damping. ~ Critical

Rotor Collective Pitch Angle

~ctiv. Control 31ade Feathering Angle

Active Control F~athering Inputs to Nonrotating

Swash Plate

Maxir.um Active Control Dlad~ Fe~th~ring Angle

Per Degree of Lead-Lag Motion

Dlade Aerodynamic Pltch An!!le

Orientation of Blade Root Gprings at Flat Pitch

Fuselage Roll Motion

Fusela~e Pitch Motion

Inflow Ratio

Real Part of Eiqenvalue. 1. e .• Modal Damping •.

rad/seci Rotor ::5olidlty

Fee d b a c k Ph a ~ e • 1. e.. l~ e 1 9 :' ': 1 n!l De t 1.11 e en C y c 1 i c

Controls

Phase of Open LOOD C,.!:.llC C.:lntl·ol Inputs

NondtmenSlonal rlme Paramet~r. Rotor AZ1~uth

AZlmuth Angl~ ot k'th 3~ade

Imaglnary Part .:If Etg~~valu~, i p.. Mod,ll

Frequency. ract/sac

Open Loop Forcing Frequency

Rotor Speed

Nominal Rotor Speed

Nondimensional Quantity

Steady-state Equtlibrium Value

d.( )/d.'t'

PAGE III

.,

'.

(

.•. f .

APPENDIX B: EGUATIONS OF MOTION

The nonlinear ~teady-state blade equilibrium aquations and

the linearized periodic coefficient perturbation equations

for. the b lade and fuselage degrees of freedom are given in

this. App~n~ix ...

STEADY-STATE EGUILIBRIUM

L~~:

~ ( k~ SIJ#,zGJ.r .. ~UU29.J -#0 ..i..Z e S. + J2.2r ~~ 5Jo~,a]

+~. (~- k,4) t,j" &.1 C~ 9.:

+ "JoS: r3...L

&0 14-4-} (

-=FLap: , ....

;;0 [ ( kt? -~ ).!'ihGJJ C01B.! -+- ..JL~ r 4 ~p 1 -I- ;10 r k,4 C~' 61- + k~ .rt"J,,/' &.1 of JL 2. (e s;, + .lb) .. SL l r 4 t9a /Sp ]

/ ~ s/ r ! ,L~,40~ of (GJoz 1.s- -~o4) ( ~z;4'/) ]

. , I

+ .1/ r (- &0.4 +2;. ).0) +.ii. <;1, (e fo -t Yc,) :: 0

O~!Gl:'! ':<L ~ ._ .• r.~ .'::

OF. POOil QUi\LiC~ PAGE 112

- '. -.... -- --... ~ - . ..,. .... "

..

DVNAMIC EGUATIONS

C~I"··"·' ,. . "J"".\.: ~. \ I ... .:. .... - •.

OF. POC.~i ~' ... ; ... : ,;

DO D . - .1 c;,..; ] .12.l. -To ~ + ~ ( ..fi (~!J.'" Ct; eo:s

2 61 ) ~ J2. 2r (610 i A~ ~ ~2-4) +)J<! _.Ji.l. 2J;, ~-Y:O)·.ii Z~.:;.., 9J CC.:.9J <4- .. }i.l.r(9ol'l-2~lA )1

,.? [k .. 2/"1 k- Z",", -, - (" -1. r 1"1 ( - I • ./' - ) , + ':K ~ .rIM Os -I ~ c.ot D'S ~ J2. e ..;)6 +...J?.. (.70 t~ ('/.",~wO/-ts/fo J

./-I< /r k~-~).r'hd.:C7!9r-l Jil.reo(4-L.))~ j ~ ji 7. Ct7: 'tk [ eDJ{' (- h I;. - Vl/-,-+,/.fa) 1: ) - e';}. J,.,..( ~4

dO _ •• (J

-RJ(' S6~.,l R;J:s oJ. 6)~ 2r(2z.~ -GJo4J] - 2.. r" - '( ~O - - -.). ..ry, :llA J'k L & ... h .... Q~ - 9~ ( j., (6 - I, yJ",-+A ;) . .. .

. PAGE 1]3

f • lAo .!r_ ..;.. ~'" ,

;1

~_""_"---....r--"_ ...... _ ... ____ ,_._.~_ a. ____ _

l.! ~ , fi'; ; I

, !

H r: i: " . , . , . : I : t, I' : ~ ~. I I . I l

i

I I

!

I : I· I r. !

f. 1 ". f· , I r

, f

- I' ,

t ~ l.

1=La, p:

- 00

-S2..2 It, /1k

t; I ~ i(' '. ~ . :. :. ':-'."." ~ -, :" ;

rJ:: r:t.;~.Id ~ .... ~ ...

+ ~J< {.n. C~.riJ1c9.r~{;} oJ..n: 76 2 (/J,c¥a) t-JL2 r(f3 A -26&t1~)]

-+- )1< [ .J2.. (~~'F ~ C ~ .dh2.&.r) ... .n, 2. r L't f -to z;'J( ( ( v. ~ -~ ) Slh ~ ~ G.r ;. Ji1. r (.L'I0/l ~D) - &c (22; -L;JI;:) 1 ~ ;:!k[ ~ .::;ot

Z6J.r + k~ J"~2~ + ...n..' (e ~ .. .4) + -fL

2 r ( 2,; f;; ., 2 ~o 4 ~.y./tJ ) - 6'" .l~ A) 1 .J. .Ji.7..c~ 1fk. ( ex- Ie ~ 4 e~ (-J, S6 0,o~(I) - (~t;. .J.~))

"0 _

+ ~k' (- So f/1p71,,))

(I

+ 9'd-( 2 Tb ~ • 2 r (-4- -h ~ (/-p+;io))) o a-L

+ R)r (- 2ri, (/fp +/lo )) .. k'J- 2 r( -804" ~ A)]"

~.Q2 .!'Ih 'Yk { ~x { j:; f6 (/fpy/(})'" e S6+Ib ) + g~ I6 ~ , .

~

+ 12 'd (-~ 0,.0+,40 ))

+ &)f' {-ZT6 t;: .t2r(Zs+h~~0~+I,)}J ·;"'2

~ 6;'J- ( 2 ( z.s: + I ~) .+2. r ( h 1'2. eo +2l( So - i ~ ).))

I

-2. - L2. - - ] ~ Rx zrr..t'l.6)o- z).. J-I<'d-zrLz. (~p"'4o~,. ~ '"

: ~ ~k (-.n.. 2. r L~ ) = 0 .. '.

PAGE 114

,/ .

,.

.. ,. ~ .

C~~;C:.". :. :. " 0': Fr ,,-,·-. c: -. """" \"... -: ., ... -' ..

.;r ,

P K:: --rx]i.'- ~ f ~ l' k r zZ -10 I;: 0~ 1: (~ rfD )

I ---_ .. ------- .---'--~'.-.. ,

I

. - -+< ( 2 i'(. + r ( -2 &0 -iz. fA, ~ »))

";/1< r ( 2;, 0 y.fo ) + t; &0 ~~ )

, + l;'( (- ~ ~ +i(Boi A -I- c:,o ~ )) ~~I( (-:J:0/4)-r'(a';;-~A ))]

-2. 2 . { •• -~ -1)<" J2. - Z. r (;Vt 'U-x z;, i'6 N k /' J(

o -l_

-I ~K ( -2 Sf, So + r (6'0 f ~ + 2 L;.. C:: )) o - -2

+ /-K (-2 ~ (/fp 0-,) +- r( ~~ -2 fA)) .' •• , •• w •

~ ~ ( - S6 • r fJo e;, (~,o Y-~)) J,.~ '< r 9~ 2, So

... ,

f .. :-· ~ 2 V-" ~2 f -no R)( - (Rbh +16 f/~Y-o)) s; -_. ~ G)(' r ( 9~ 1; -1, A) - 9'J. r {h~.fo }PJ

4- G'u r (90 L:;. -.z, ).,) ] + -.rx 5L'- ~ f Ghk r[!"~7k (t,A )-c~;VI(V/Ci1o)l~]

PAGE 115

. - -. "; • ~ . .1-. ..-c::- .......... _. '\. ... -. / , /

,

~ .

J I

i

OR!Gi~!;~.t: r .~.:-~ OF POOR Q~~:.:"i ~ •

Pa = ~ J?.t~ f ~)Pk[-~ So

+ ~ ( 2. S. S. - r ( 9r, r2

A of. 2 1;. C:fI})

-z. ~)k ( 2f;1p~4.)~ rr( 2 ~ A - &0 ~))

+ ~K (S6 - r a~ (,fi"'ll,J) ?-1<t' (- r c90Z ~ ) ] 2. 2 r 'f - If -

~ ..)"~.;1. N f .I/~ rl< ~ ~ ~ -I~I< ~ (~:4) D·

-+- ?;( 2S: -r2 &oLz (/~;4.J) . --

-I- ~K r ( U ~~J -'7. ~ 9,,4 z;: )

~ ~~ (- £;: ~ + r ( 90 ZJ A ~ ~" ~))

+ /11( ( -~ (hy.f.J+ r (L,A -&,,1;)) J 4 2 '1 ~ Ji.

2 [ - Hb R} + ( h b h ~ ~ Y1~+4c)} dJ: . _. . ---

.J. c9)r r ,fJ y£,:40) -I- GJ 'a-r (eo LJ - --" ;\ )

+ 9x r (~ A-a.e; ) } -I- ~'d-3i 2 ~ [ Gkk r ! - C~}PI{ (.t; A) - .:-/;' lfl( fA.,~}2; J

PAGE 116

I

!

!'

:. ! , , I

'; "

, , o~:.::~ :.,.,:. -:--.

Rot!: ,

.i?.' If, .. .iL2 L:; /~ 6b +..e c~ ~ + ~ .... 6?. -ax c ~

"'~ .. ji' ~ f ~7¥k [ Ii; (~(~y!.).J h~) yJ;(-~o.f;,) -z

+ ~ < -h S:2';-: .J r[2&,.;-; j~ +h ;4 ~,;.; ~'" .It,4 J) J. /-1-. (-2J., ~ (/14#0) -I r[-4 ~ -If4,Y-.)~Z" .l(./;~-21l)]) ~ ~ (-16 ~y-. )-h~ .tr[ ~7s -4;\ .. lf~6?t/1py.f.)])

+ /-K( - 50 f:;, • r[ G&izA .. ~,/-:r ~h &,,;; ff.,]? ]

, 2 (... - - ~ - - - I) J. "i~ ... _'i N f .. All., 11< ~ f-h .f6 ~ ) "/-I«-J;,. -e.f~ -h~ Vtp-Y!,J;

"' ~ (-2~ V1~~)- h 21;. -fr[ 26J;iQ-{;' +261,4 fZ-oh(/:p;4.)Jj)

of ~: r(-,t: -eJ; -'h./: VJ,A~())-h-&o;;'t:: > ~ ~ ( .: 1: ;: -r [ -4U y1.) -?: &rJ24-/; J-j,-( &OJ,~ + C:" ~:J] '; ~;11( (- e.~ -:r;. ~h J;'V1t>Y-f.)" r[ -4" ~ -24 .9.. (/1,.:.~ )./~~A,

Jh' (~c90 -~ .A)]) ? ... 2 -ra,...it 2. f e: [ -2 h (/t,..¢oJ-c' - h 2~ -1 (alh;, ~2eJ;. • .i;jj:"

.. I •• - -. + B,;; ~ e ~S:+K~ ( S" (4p yf,,) +h /1. J

~ GJr (-e ~~: - r[ L-:r +1,( 4.{2.z)(~,.Yi.)+e.?z ] > ~ c9d (- j (eZ~ ~2e.{ +,z) -I- r[ ~ ('<;-4)-h(~(i;.J~ )-.vi,"+J))t

+ RL r(fl)..-.2t 6lo )+l_ r;;,v-",,~o) .. &~ (-e{~)] ~ 2 _ ~

+ 0e;" .. .Ii? ;;. ;. 6kk r [C.tnYk (t:;/; +hL,-),)+

T~rk(ls-+eL~ "4~.# .. )2;)J

PAGE 117

. ," .. ':.

Pitch:.

0d ~ -re~.£4 I~ f CP:YIr! ii h.J;. ~ ~;; (e{ .. ~ ~l.r: (/-,.,;4))

• ~(2L{f;"d,,).~2h~ ~ r [~lA-2a4-2~/e-/h(Ap~/j.)]). ·)kr{ -4 .z/; oIi~01'0.) ~h;;'&o~)

~ t;;(- r..{ l;';, oJ r[ 4 0y!1I ) ~ G1. ~(4- -24) .hfal,).. ~tI~;]) .. +;!k ( e~ -I .. - t,--~ V-.cy.,)" r[ 4~ T~ 2414:40) -4 6Z~"

+ t,-rz' A -&0.4 )]) ]

+ vGc_7.l. ~ f .[,;" 'Yk ( ¢. (I4ytpy.f.)" h.{ ) ~j; f-~~ ) -I- ifJ«-24(~~r[ 26?,4~-h(6'of\-2 ;:".1:)1)

· ~ )k < -2C,~ Y-... r.:.)~r[ ~:/GtI(/.p~)-~) -i", fGoZ:- 2 f.AJj) , ~ ~(-'{~".:,,)-i~~r[&tl4--.e;A.ht9,i; f/-.:/..j])

~/!K (- I" if:, -fr[ do1;A" ':<> /;. ,,-.;; &0 ~]> 1 I '4 t·· I'" •• - ~ -l-, -t 2. ";-B~.Ji.l Bit (..: J 'e.ft. :::;, ) ~ ~ [ -2 h v.pv.. ).l6 -I, no

I z- - - - ...!! - --- i. (e he .2 eI;, ~~) _i ... ,~~ (-J;. 1/ ... ;4,) -t, 170 )

. '.

.0 _ ••

• ..l. ' .•

.. + BI('(t?l~·~22~~'.J~ .r[h(&,J;- ~ )')+(.L¥-4)~.h(~Z-<,1)J\ · \ l. /

· • Bo( -!:i; e~ ~r[-;;'-4-(,z f~)U~.)-eL~J)

PAGE 113

\ \ I

. J

-. \ .

'-

1\.· I !

..

2-

L,e ~L./- .~

.I'Z. .,Ll ,:.C ezol-T L2 ~1

LJ :: e~-' .. 2e -r • T

..n.... ..n./-f2.. o

1-1¢. • hb /t4-t~ I<. .s: ,., C~ = C~/( fI,."J2. o Kl )

- 12.-\.<.-! .. l<~ /( '-, .;'Z.~ R ~ )

c, = c"" / (r:",.fZ." )

ir;. ~ C()~ / (:r ... ' .. r,o) " c;. C

I --r ... :::. "2 N lMo R / H ...

Oi'~:G:': ,::"~L ;; .... 4

OF POC~\ (~:..J;.: . .i,'{

'p ... a . % R r= 11./.(

h& h/R

k-;. - K'J I ( H"d .J2.; ) ke;,-:' (1-<6';,-- NHh ,/' )/( .;: .. :2.II~')."

- '2) KC;~" (KS .. - )/. n:; 'hj I ( .J. '" ..... ~o . '" c (j ...

..J'"d ~ ~ }.Jko o R/H-a-

--rg .. = 1 }.J /A.-" R J II)' c

R~· R~/R

pft.GE 119

\

End of Document