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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\
..
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(.~.
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
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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
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