Actuator saturation in actively controlled trailing edge flaps

(1)

ACTUATOR SATURATION IN ACTIVELY CONTROLLED

TRAILING EDGE FLAPS

R.Cribbs

andP.P.Friedmann y

Departmentof AerospaceEngineering

UniversityofMichigan

AnnArbor,Michigan48109-2140

Abstract

Thein uenceofactuatorsaturationonthe

vibration reduction performance of an actively

controlled ap is investigated. An aeroelastic

modelofafourbladedhingelessrotorwithafree

wakeisusedfortheanalyses. Three methodsfor

constraining ap de ections are studied at two

limiting values, two and four degrees, of

maxi-mum apde ection. Resultsindicatethatneither

scalingnorclippingoftheoptimalcontrol ap

de- ection to themaximum ap de ectionprovides

acceptable vibration reduction. A newly

devel-opedcontrolmethod withsaturationconstraints

shows exceptionalreduction of vibrations. This

newcontrolmethodreducesvibrationstosimilar

levelsastheunconstrainedoptimalcontrolwhile

constrainingmaximum apde ectionstothe

lim-itingvalues.

Nomenclature

a Bladeliftcurveslope

c

b

Bladechord

c

cs

Trailingedge apchord

C do CoeÆcientofdrag C W Helicopter coeÆcient of weight c wu

Weighting multiplier forW

u weightingmatrix EI y ;EI z

Blade bending stinesses in

apandlead-lag

FHX, FHY,FHZ 4/rev vibratory hub shears

in longitudinal, lateral and

vertical directions,

respec-tively

J Quadraticperformanceindex

forvibrationcontrol

L

b

Bladelength

L

cs

Trailingedge aplength

PostdoctoralScholar

y

Francois-XavierBagnoudProfessorofAerospace

En-gineering

m Blade mass distribution per

unitlength

MHX,MHY,MHZ 4/rev vibratory hub

mo-mentsinrolling,pitchingand

yawing,respectively

n

b

Numberofblades

R Rotorradius

T Jacobianof the vibration

re-sponse with respect to the

controlinput

~

u Vectorofcontrolinputs

~ u

Vector of optimal control

in-puts

W

u

Matrix of weightson control

amplitudes

W

z

Matrix of weights on

vibra-tionamplitudes

W

u

Matrixofweightsontherate

of change of control

ampli-tudesfrom one control

itera-tionto thenext

x

cs

Location along blade span

aboutwhichtrailingedge ap

iscentered

X

FA ,X

FC

Horizontal oset of fuselage

aerodynamiccenterand

fuse-lage center of gravity from

hub

~

z Vector of vibration

ampli-tudes

Z

FA ,Z

FC

Vertical oset of fuselage

aerodynamiccenterand

fuse-lage center of gravity from

hub

p

Bladepreconeangle

Locknumber

Æ Flapde ection

Æ

limit

Flapde ectionlimitingvalue

Æ

Nc ;Æ

Nc

Cosineandsineamplitudesof

Nth harmonic of ap

de ec-tion,respectively

Æ

opt

Optimal apde ection

Advanceratio

(2)

Bladeazimuthangle

Rotorangularvelocity

Introduction

In recent years, researchers have

inves-tigated actively controlled trailing edge aps

as a means for vibration control in helicopter

rotors 1{3

. Inthis approach, appropriatecontrol

inputs tothe ap modifythe aerodynamicloads

on the blade to reduce the rotor vibratory hub

loads. Earlierresearchhasshownthatthepartial

span,trailingedge ap,showninFig.1,produces

the sameamountof vibrationreductionas

indi-vidual blade control (IBC) that is implemented

by movingthe entireblade bypitch inputs

pro-vided at its root in the rotatingsystem 4

.

How-ever,theactivelycontrolled ap requires almost

an order of magnitude less powerfor its

opera-tion. Furthermore,thepracticalimplementation

oftheactivelycontrolled apdoesnotrequirethe

extensivemodicationstotheswashplatethatare

describedinRef.5.

For practical implementation, the trend

hasbeentouseadaptivematerialssuchas

piezo-electric or magnetostrictive actuator devices to

activelycontrolthe ap. This typeof actuation

hasbeenconsideredbyseveralresearchers

includ-ing Spangler and Hall 6

, Bernhardand Chopra 7

,

Fulton and Ormiston 8

, and Prechtl and Hall 9

.

The force and stroke producing capability of

adaptive materials based actuation is limited.

Therefore, the actual ap de ections that are

achievable with this type of actuation are

ex-pectedtofallshortoftheanglesrequiredfor

max-imumvibrationreduction. Thus, theseactuators

will be unable to produce the control authority

requiredforoptimalvibrationreduction,andthe

actuatorislikelytoencounter saturation.

In this study, the consequencesof

impos-ing limitsonthe apde ection produced by the

actuatorareexamined. Thein uenceofsuch

lim-its, typically denoted as saturation in the

con-trol terminology, on vibration reduction

capa-bility is assessed. Three methods of

constrain-ing apde ectionstolimitingvaluesarestudied.

Two methods for limiting ap de ection

ampli-tudesgivenanoptimal apde ectionhistoryfor

vibrationreductionare: (1)clipping of the

opti-mal ap de ection such that de ections greater

thanaprescribedvaluearesimplysettothe

pre-scribed maximumvalue, and (2)uniformly

scal-ing down the amplitudes of the ap input

har-monics so that the ap amplitudeneverexceeds

the limiting value. A third ap de ection

lim-iting methodology is developed and studied. In

thismethod,thecontrolprocedureismodiedto

control weightingmatrix. Theweightingmatrix

is adjusted iterativelyuntilthe ap de ection is

properlyconstrained.

The issueof ap saturation hasnotbeen

studiedintheliteraturebefore,yetitplaysavery

importantroleinthepracticalimplementationof

vibrationreductionusing theactivelycontrolled

ap(ACF).

Mathematical Model and Method of

Solution

Themathematicalmodelemployedinthis

studyrepresentsarotorwithanumberof exible,

hingelessbladeseachofwhich containsapartial

span trailing edge ap as shown in Fig.1. The

model was developed in an earlier study by de

Terlizzi andFriedmann 10

.

Structural DynamicModel

Eachrotorbladeismodeledbybeam-type

niteelementscapableofrepresentinga

compos-iterotorbladewithaswepttip. Theblade

struc-turaldynamicmodelwasdevelopedbyYuanand

Friedmann 11

. The model has provisions for an

arbitrary cross-sectional shape which is allowed

to vary alongthe span. Themodel accountsfor

transverseshearandoutofplanewarpingandcan

modelanisotropicmaterialbehaviorand

compos-itecouplingeects.

Aerodynamic Model

The aerodynamic model has two main

components,calculationof thespanwise

aerody-namic loads acting on the blade and the

calcu-lation ofthenonuniformin owdistribution over

therotor.

The blade section aerodynamic loads are

determinedusingtwodierentaerodynamic

mod-els. For the quasi-steady loads acting on

the blade/ ap combination, a modication of

Theodorsen's quasi-steady aerodynamic theory

which includes the in uence of thetrailing edge

ap as developed by Millot and Friedmann 1

is

used. To model the unsteady compressible

air-loads acting on the blade/ apcombination, the

aerodynamic theory developed by Myrtle and

Friedmann 3

isemployed.

Thenonuniformin owdistribution is

cal-culatedfromafreewakemodelthathasbeen

ex-tracted from the comprehensiverotorcraft

anal-ysis tool, CAMRAD/JA 12

, and modied to be

compatible with the aeroelastic response

analy-sis employedin this study. Itconsists of awake

(3)

ducedvelocitydistribution. Thefreewake

geom-etry routine was initially developed by Scully 13

and the wake calculation model was developed

by Johnson 14

. The model is based on a vortex

latticeapproximationofthewake.

Trailing Edge Flap

Activelycontrolled aps havebeen

incor-poratedintothemodel inamannerdescribedin

Ref. 10. Controlinputsprovidedtothe ap

con-sist of acombinationof 2/rev, 3/rev,4/revand

5/rev apde ections. These inputsaretypically

obtainedfrom acontrol law 15

basedonthe

min-imizationofaquadraticperformanceindex

com-posedof vibrationandcontrolamplitudes.

Methodof Solution

A modal analysis is implemented using

eight free vibration modes of a rotating beam,

3 ap, 2lead-lag, 2 torsionaland 1 axial mode.

Either a harmonic balance technique or a time

domain solution is used to solve the blade

re-sponse equations depending on whether or not

unsteadyaerodynamicsareemployed. Inthe

har-monic balancetechnique, theblademotionsand

trim equations are converted into a system of

nonlinearalgebraicequationsandarethensolved

simultaneouslyby the IMSL nonlinearalgebraic

solver DNEQNF 16

. In the time domain

aeroe-lastic response solution,the equationsof motion

arenumericallyintegratedusingthegeneral

pur-pose Adams-Bashfortordinarydierential

equa-tionsolverDE/STEP.

Control

ControlAlgorithm

Thecontrolalgorithmusedinthisstudyis

onethatistypicallyusedinHHCandIBCstudies

andisdescribedindetailinRef.15. Itisbasedon

the minimizationof aperformanceindex that is

aquadratic functionof vibrations~z

i

andcontrol

inputs~u

i

. Thequadraticfunction isgivenby:

J=~z T i W z ~ z i +~u T i W u ~ u i +~u T i W u ~u i (1) where ~u i =~u i ~ u i 1

. Theweightingmatrices,

W z , W u and W u

, used in this study are

di-agonaland assignthe relativeimportance of the

variousvibrationcomponentsandcontrolinputs.

W

u

constrainstherateofchangeofthecontrol

from onecontrol iterationtothenext.

Theoptimalcontrolisfoundbysettingthe

gradientoftheperformanceindexJ withrespect

i

@J

@~u

i

=0 (2)

Thesolutionofthis equationresultsin the

opti-malcontrol~u

i

thatminimizesJ. Itisassumedin

this study that the control inputsand vibration

levelsareknown.

To determine the gradient of the

perfor-manceindexwithrespecttothecontrol,itis

nec-essarytoknowthegradientofthevibrationswith

respecttothecontrol. Tothisend,thevibrations

arelinearizedaboutthecurrentcontrolinput~u

i ~z(~u)=~z(~u i )+T i (~u ~u i ) (3) where T= @~z @~u (4)

ThetransfermatrixTis theJacobianof the

vi-brationresponsewithrespecttothecurrent

con-trol input. This Jacobian is calculated

numeri-callyusingthenitedierencemethod.

Substituting this model of the vibration

response into the performance index and

mini-mizing with respect to the control produces the

optimalcontrolforthegivencontrolstep:

~ u i+1 = D 1 i fT T i W z [~z i T i ~ u i ] W u ~ u i g (5) where D i =T T i W z T i +W u +W u (6)

This procedure is started by setting the initial

optimalcontrolinputtozeroandrepetitively

ap-plyingEq. 5untiltheoptimalcontrolinput

con-verges. Theresultingcontrolvectoristheoptimal

controlvectorforthegivenweightingmatrices.

Vibration Measure and Control Input

In this study, the control law is used to

simultaneously reduce the 4/rev components of

thehubloads. Therefore, thevibrationvector~z

contains the sine and cosine components of the

three 4/revhub shears and the three 4/revhub

moments.

The vibration control is implemented

through anactively controlled trailing edge ap

ontheblade. The ap de ectionis composed of

2,3,4and 5/revharmoniccomponentsandcan

beexpressedas: Æ( )= 5 X N=2 [Æ Nc cos(N )+Æ Ns sin(N )] (7)

Itis assumedthat allfourrotorbladesare

(4)

azimuthangle.

Thecontrol input~uused forvibration

re-duction in the control algorithm is the vector

containingthecosineand sineamplitudes of the

N/rev ap de ection harmonics. This vector is

givenby: ~ u=fÆ 2c ;Æ 2s ;Æ 3c ;Æ 3s ;Æ 4c ;Æ 4s ;Æ 5c ;Æ 5s g T (8)

Constraining Flap De ection

Three dierent methods for limiting ap

de ection to accountfor actuatorsaturationare

studied. Two of these methods take the

com-puted unconstrainedoptimal control and obtain

the limited ap de ection history in one step

through either truncation or amplitude scaling.

The rstmethod simplyclips the ap de ection

atthelimitingamplitudeforanyoptimalcontrol

commandthatexceedsthismaximumvalue. The

apde ection isthusdescribedby:

Æ( )= Æ opt ( ); jÆ opt ( )j<Æ limit sgn(Æ opt ( ))Æ limit ; jÆ opt ( )jÆ limit (9)

orifthelimitofminimumde ection diersfrom

thelimitofmaximumde ection,

Æ( )= 8 < : Æ min ; Æ opt ( )Æ min Æ opt ( ); Æ min <Æ opt ( )<Æ max Æ max ; Æ opt ( )Æ max (10)

The second method for limiting ap de ection

uniformly scales down the optimal control ap

de ection. Eachharmoniccomponentofthe

op-timal apde ectionisscaledbyacommonfactor

to limit the maximum ap de ection to the

de-siredamplitude. Forthiscase,the apde ection

isdescribedby: Æ( )= Æ limit max(jÆ opt j) Æ opt ( ) (11) where max(jÆ opt

j)is the maximumamplitudeof

the optimal ap de ection overthe entire range

of bladeazimuthvalues.

In the rst two limiting methods, no

constraints are imposed on the ap de ection

throughtheuseoftheweightingmatrix,W

u ,in

theoptimalcontrol calculation. Thoughthesole

purposeof theweightingmatrix,W

u

, isto

con-strain apde ections, it isnot possible toknow

a priori the proper weighting matrix that

con-strainsthe apde ectionamplitudetobewithin

prescribed limits. A third ap amplitude

limit-ing method was developed where a new control

procedureautomaticallyadjuststheweightingto

Thestructure ofthenewprocedureis compared

tothetwopreviouslimitingmethodsinFigure2.

Inthenewcontrolprocedure,theoptimal

control u

is calculated for a given set of

pa-rametersin the samewayasin the oldmethod.

However, after the optimal control is obtained,

the maximum and minimum ap de ections for

the givencontrol arecalculated. The maximum

and minimum ap de ections are compared to

the prescribed limiting valuesand a test is

per-formedto determinewhether the apde ections

are properly constrained. This test consists of

ensuringthatthemaximum apde ectionisless

than thelimiting maximumvalueand the

mini-mum ap de ection is greater than the limiting

minimumvalue. An additionaltestisperformed

such that the dierence between oneof the ap

de ectionextremes, minimumormaximum,and

itscorrespondinglimitingvaluemustbewithina

user dened degrees. This additionalcriterion

ensuresthatthe ap isnotoverconstrainedand

allowsthefullallottedcontrolauthorityfor

vibra-tion reduction. If the ap de ection is

overcon-strainedorunderconstrained,the weighting

ma-trixisappropriatelymodiedto relaxortighten

the apde ectionconstraint. Thenewweighting

matrixis inputinto the optimalcontrol

calcula-tionroutineand theprocedurerepeatsuntil the

apisproperlyconstrained.

For this new control procedure, a simple

formoftheweightingmatrixisused. The

weight-ing matrix is assumed to be a scalar times the

identity matrix. The weightingmatrix is

there-foredescribedby:

W

u =c

wu

I (12)

All harmonic components of the ap de ection

are equally weighted. The controller

manipu-lates the scalarmultiplier to provide the proper

apconstraints. Ifthe apde ectionis

overcon-strained, thecontrollerreduces the value of c

wu

and a new optimal control is calculated. If the

apde ectionisunderconstrained,thecontroller

increasesthevalueofc

wu

andanewoptimal

con-troliscalculated. Theiterativeprocedurereduces

or increases c

wu

until the optimal control

con-verges to the desired de ection limits within a

prescribedtolerance.

Results

Simulationswereperformedforbothalow

speed condition, = 0:15, and a higher cruise

speed condition, = 0:30, using quasi-steady

aerodynamics. The results are obtained for a

four bladed rotor consisting of straight,

(5)

chordlengthonequarterthatofthebladechord

and a span of 12% of the blade span, centered

aboutthe3/4bladespanlocation.

Low Speed Results

For the = 0:15 ight condition,

vibra-tory hub loads for a number of test cases are

shown in Figures 3and 4. Twovalues of

max-imum apde ectionareinvestigatedfortheir

in- uence onvibrationreductioncapabilitiesof the

activelycontrolled ap. Resultswithamaximum

ap de ection of 2 degrees are shown in Figure

3 while Figure 4 provides results with a

maxi-mum ap de ection of 4 degrees. The

uncon-trolled, or baseline, 4/rev vibratory hub shears

and moments are presented in the gures along

withtheresultsfromfouractivelycontrolled ap

simulations. The unweightedcontrol resultsare

obtained bysetting thecontrol weighting

matri-ces W

u

and W

u

identically to zero matrices.

This producesabest case scenariofor vibration

reductionwithnolimitationoneithercontrol

am-plitudeortherateofchangeofcontrolamplitude

from onecontrol iterationtothenext.

The ap de ection history as a function

of azimuth fortheunweightedoptimalcontrolis

shownin Fig.5. The apde ection isshownfor

the range of azimuth values from 0 to 360

de-grees. Flap de ection is periodic in this study

sothe apde ection isidenticalfor each

succes-sive rotorrevolution. As seenin Fig. 3,the

un-weighted control ap de ection signicantly

re-duces vibratoryhubloads from thebaseline

val-ues. Hubloadsarereducedbyaminimumof24%

fortheyawingmomenttoamaximumof98%for

theverticalshear. Overallhubvibration,dened

as the square root of the sumof the squares of

the six vibration components, is reduced to 8%

of the uncontrolled level. However the ap

de- ection historyindicates that apde ection

am-plitudes of upto19.5degreesarerequired. This

ismuchgreaterthanwhatiscurrentlyachievable

byadaptivematerialsbasedactuationofthe ap.

This apde ection alsoexceedstherangeof

va-lidityoftheaerodynamictheoryused, aswellas

the anglesthat maybe allowedduring the

prac-tical implementationof such devices. Thus, itis

clear that limiting the angle of ap de ections

wouldplayanimportantrolewheneverthe

prac-ticalimplementationoftheACFforvibration

re-ductionisconsidered. Thissituationcouldoccur

when using the newtype of electromagnetic

ac-tuatorsdescribedin Ref.17.

The eects on vibrationlevels of limiting

the ap de ection to a maximum of 2 degrees

througheither theclipping,scalingorautomatic

weighting methods, as described previously, are

`truncated control', vibratory hub loads are

re-ducedby amaximum of49%in yawingmoment

toanincreaseinverticalshearof12%. The

over-allhubvibrationlevelisreducedbyonly3%. The

ap de ection history for this control is shown

in Figure 6. The ap de ection history for the

uniformscalingof theunweighted ap de ection

amplitude to a maximum of 2degrees is shown

in Fig. 7. This control input reduces vibration

levels from as little as 0% in the yawing

mo-ment to asmuch as 15%in the lateral and

ver-tical shears. Overall hubvibrationisreduced to

86%oftheuncontrolledvalue. Clearlyneitherof

these apde ectionlimitationmethodsis

accept-ablereducinghubvibrations. Theoverall

vibra-tionsarefrom10.7to11.0timesgreaterthanthe

unweighted control. The ap de ection history

for theautomaticweightingmethod isshownin

Fig. 8. With the automatic adjustment of the

control weightingmatrix, vibrationsarereduced

byaminimumof8%inyawingmomenttoa

max-imumof 91%in longitudinal shear. Overallhub

vibration is reduced to 40%of the uncontrolled

level. Vibrations with the automatic weighting

methodarelessthanhalf ofthoseobtainedwith

either of theother twode ection limiting

meth-ods. However,itshould benotedthat vibration

reductionwiththeunconstrained apissuperior

(8%of uncontrolled level)tothe performance of

theconstrained ap (40%ofuncontrolled level).

Theeects onvibrationlevelsof limiting

throughthethreelimitingmethodsareshownin

Figure 4. With the unweighted ap de ection

truncated at 4 degrees, shown in Fig. 9,

vibra-toryhubloadsarereducedbyaminimumof8%

inverticalsheartoamaximumof90%inlateral

shear. Overall hub vibration is reduced to 74%

oftheuncontrolledvalue. The apde ection

his-toryfortheuniformscalingoftheunweighted ap

de ection amplitudeto amaximumof4degrees

isshown inFig.10. With this control,vibratory

hub loads are reduced by a minimum of 8% in

pitchingmomenttoamaximumof30%inlateral

of the uncontrolled level. With the automatic

adjustment ofthe weightingmatrixto constrain

maximum ap de ection to 4 degrees,vibratory

hub loads are reduced by a minimum of 2% in

yawingmomenttoamaximumof97%invertical

his-toryforthiscontrolisshowninFig.11.

With a4degreede ectionlimitation, the

clipping and scaling control methods reduce

vi-brations below the levels obtained with a 2

(6)

re-levels. The control with automatic adjustment

of the weighting matrix produces an overall

vi-brationlevelonefourththatofeitherthescaling

orclippingmethodwiththesamemaximum ap

amplitudeof4degrees.

High Speed Results

For the = 0:30 ight condition,

vibra-tory hubloads are shown in Figures 12 and 13.

Asinthelowspeedcase,twovaluesofmaximum

apde ectionareinvestigatedfortheirin uence

onvibrationreductioncapabilitiesoftheactively

controlled ap. Resultswithamaximum ap

de- ectionof2degreesareshowninFigure12while

Figure13showsresultswithamaximum ap

de- ection of 4degrees. Theuncontrolled4/rev

vi-bratory hub shears and moments are plotted in

the gures along with the results from four

ac-tively controlled ap simulations as studied in

the low speed case. The ap de ection history

for the unweighted optimal control is shown in

Fig.14. The apde ectionhistoryindicatesthat

apde ectionamplitudesofupto9.2degreesare

required for vibration reduction. Hub loads are

reduced by a minimum of 84% for the pitching

moment to a maximum of 97% for the vertical

shear. Overall hubvibrationisreducedto9%of

theuncontrolledlevel.

The eects on vibration levels of

limit-ing the ap de ection to 2 degrees are shown

in Fig. 12. The ap de ection history for the

unweightedoptimalcontroltruncated attwo

de-grees is shown in Fig. 15. With the truncated

control, vibratory vertical shear is reduced by

34%whileallothercomponentsaregreaterthan

the uncontrolled values. The overall hub

vibra-tion levelis actually increasedby 3%. The ap

de ectionhistoryfortheunweightedcontrol

uni-formlyscaledtoamaximumof2degreesisshown

in Fig.16. This control input reduces vibration

levels from as little as 21%in lateral shear and

yawing moment to as much as 27% in

longitu-dinal shear. Overall hubvibrationis reducedto

76% of theuncontrolled value. As with thelow

speed case, neither of these ap de ection

lim-itation methods do a good job in reducing hub

vibrations.

The ap de ection history for the

auto-matically adjusted weighting matrix method is

shown in Fig. 17. With this control, vibration

levelsarereducedbyaminimumof64%in

pitch-ingmomenttoamaximumof91%inlongitudinal

shear. Overallhubvibrationisreducedto15%of

the uncontrolled level. This vibrationreduction

issignicantlybetterthaneitheroftheothertwo

de ection limiting methods. The overall

vibra-tionis lessthanafththeleveloftheothertwo

Theeects onvibrationlevelsof limiting

throughthethreelimitingmethodsareshownin

Figure 13. With the unweighted ap de ection

truncated at 4degrees, shown in Fig.18,

vibra-toryhubloadsarereducedbyaminimumof12%

inlateralsheartoamaximumof85%in vertical

his-toryfortheuniformscalingoftheunweighted ap

de ection amplitudeto amaximumof4degrees

isshown inFig.19. With this control,vibratory

hubloads are reducedby aminimum of 38%in

rolling moment to a maximum of 49%in

longi-tudinal shear. Overall hub vibration is reduced

to60%oftheuncontrolledlevel. The ap

de ec-tion history for the control with the automatic

adjustment ofthe weightingmatrixto constrain

maximum apde ectionto4degreesisshownin

Fig. 20. With this control, vibratory hub loads

are reduced by a minimum of 79% in pitching

momenttoamaximumof 97%in verticalshear.

Overall hub vibration is reduced to 10% of the

uncontrolledvalue. Thisreductioninoverallhub

vibration is nearly the same as that of the

un-weightedoptimalcontrolbutrequiresamaximum

apde ectionthatislessthanhalfthatofthe

un-weightedoptimalcontrol. This overallvibration

levelisonesixththatofthevibrationsfromeither

of the other two limited ap amplitude control

methodswiththesamemaximum ap de ection

of4degrees.

It is evident that neither of the two

sim-ple ap de ection limiting methods, clipping or

amplitude scaling, provides the optimal control

apde ectionforagivenmaximumamplitudeof

de ection. Theautomaticadjustmentofthe

con-trolweightingmatrixperformsmuchbetterin

vi-brationreductionthanthetwoother methodsof

de ectionlimiting.

Concluding Remarks

Simulations on arotorwith actively

con-trolled trailingedge aps were conducted to

in-vestigate the in uence onvibration reductionof

actuatorswithlimitedabilitytoproduce ap

de- ection. Thestudy considered afour bladed

ro-tor with hingeless, isotropic blades

incorporat-ingfullycoupled ap-lag-torsionaldynamics.

Si-multaneous reduction of the vibratory hub

mo-ments was achieved through the minimization

of a quadratic performance index consisting of

weighted squares of vibration magnitudes and

control amplitudes. Three means of limiting

the maximum ap de ection were investigated.

(7)

1. Theactivelycontrolled,partialspan,trailing

edge apwithnoconstraintsoncontroleort

providessignicantreductionofthe4/rev

vi-bratory hub loads. For both low and high

speedcases,overall vibrationswere reduced

to less than 10%of the uncontrolled levels.

Thisunweightedcontrolrequired ap

de ec-tionsthatare unachievablewiththecurrent

state of theart in adaptivematerials based

actuation. Atanadvance ratioof=0:15,

ap de ections of upto 20 degreeswere

re-quired. These de ections are much greater

thantheexpectedmaximumde ectionswith

currentadaptivematerialsoflessthan5

de-grees. This also exceeds levels of ap

de- ection that may be allowed during actual

implementationofsuchasystem. Thus,

lim-itingthede ectionangleisacentralissuein

thepracticalimplementationoftheACF.

2. Twosimple methodsfor limitingthe

ampli-tudeof apde ectionwereinvestigated,

clip-pingandamplitudescaling. Neithermethod

proved acceptable for vibration reduction.

Scalingtheoptimalunweightedcontroldown

toalimitingmaximumde ectionreducedall

six components of the vibratory hub loads

for both speedconditions consideredin this

study but the vibration reduction was only

marginal. The clipping method of limiting

ap amplitudeactuallycaused increased

vi-brationlevelsinmanyhubloadcomponents.

Even with a maximum ap de ection of 4

degrees,neithermethod couldreduce

vibra-tionstoone halfoftheuncontrolledvalue.

3. A new control procedure for vibration

re-ductionwithlimited apde ectionauthority

wasdeveloped. In this procedure, the

con-trol weighting matrix is adjusted

automati-callybythecontrolleruntiltheresulting ap

de ection is within prescribedlimits. With

this method, vibrations are signicantly

re-duced at both forward ightspeeds

consid-eredin thestudy. Overallhubvibrationwas

reduced to 10%of the uncontrolled level at

=0:30withamaximum ap de ectionof

4 degrees. This reduction in vibrations is

nearly thesameas that achievedbythe

un-weighted optimal control that required ap

de ectionsof 9.2degrees.

4. Comparingthe apconstraintsof 2degrees

and 4 degrees to the unconstrained ap, it

is evident that, for the particular case

con-sidered,the4degreelimitisalmost optimal

since arelativelyminor lossof vibration

re-ductionperformanceisencountered.

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Table1:Soft-in-planeIsotropicRotorBladeData

RotorData EI y =m 2 R 4 =0:0106 EI z =m 2 R 4 =0:0301 GJ=m 2 R 4 =0:001473 n b =4 a=2 =5:5 p =0:0 =0:07 c b =R=0:055 HelicopterData C W =0:00515 C d0 =0:01 Z FC =R=0:50 Z FA =R=0:25 X FC =R=0:0 X FA =R=0:0 FlapData L cs =0:12L b c cs =c b =4 x cs =0:75L b

Actuator saturation in actively controlled trailing edge flaps

Deformed

Elastic Axis

Undeformed

Elastic Axis

Deformed

Blade

Undeformed

Blade

z

3

y

3

x

3

x

4

Ω

Optimal control

calculation

Flap

properly

constrained

Max. deflection

calculation

Optimal control

calculation

Truncation

or scaling

u

*

u

applied

no

yes

u

applied

Flap

over

-constrained

Reduce W

u

Increase W

u

yes

no

Truncation or scaling

Automatic control weighting

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

FHX

FHY

FHZ

MHX

MHY

MHZ

N

ondi

m.

Vi

br

ator

y H

ub Loads

uncontrolled

unweighted control (max 19.5 deg)

truncated control (+/- 2 deg)

scaled amplitude (max 2 deg)

autowt (max 2 deg)

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03