FINITE ELEMENT MULTIBODY MODELING
OF ROTORCRAFT SYSTEMS
Carlo L.Bottasso
DipartimentodiIngegneria Aerospaziale,PolitecnicodiMilano, Milano, Italy
OlivierA. Bauchau,
GeorgiaInstitute of Technology,School of Aerospace Engineering,Atlanta,GA, USA
Abstract
This paperdescribesan ongoinge ortin thearea
ofmultibody niteelementdynamicsforthe
mod-elingofrotorcraftsystems. Thekeyaspectsofthe
simulationprocedurearediscussedandselected
ro-torcraftapplicationsarepresented.
Introduction
Multibodydynamicsanalysis was originally
devel-opedasatoolformodeling mechanismswith
sim-ple tree-like topologies composed of rigid bodies,
but has considerably evolved to the point where
itcanhandle nonlinear exiblesystemswith
arbi-trary topologies. It is now widely used as a
fun-damental design tool in many areas of
mechani-cal engineering. In the automotive industry, for
instance,multibodydynamics analysisisroutinely
usedforoptimizingvehicleridequalities,acomplex
multidisciplinarytaskthat involvesthesimulation
ofmany di erentsub-components. Modern
multi-body codes can dealwith complex mechanismsof
arbitrarytopologyincludingsensors,actuatorsand
controls, are interfaced with CAD solid modeling
programs that allow to directly import the
prob-lemgeometry,andhavesophisticatedgraphics,
an-imationand post-processingfeatures. Thesuccess
of multibody dynamics analysis tools stems from
their exibility: a given mechanism can be
mod-eled by an idealization process that identi es the
mechanismcomponentsfromwithinalargelibrary
ofelementsimplementedinthecode. Eachelement
provides abasic functional building block, for
ex-amplearigidor exiblemember,ahinge,amotor,
etc. Assembling the various elements, it is then
possibletoconstructamathematicaldescriptionof
themechanismwiththerequiredlevelofaccuracy.
Despite its generalityand exibility, multibody
dynamicsanalysishasnotyetgainedacceptancein
the rotorcraft industry. Historically, the classical
approachtorotordynamicshasbeentouseamodal
reduction approach, aspioneered by Houbolt and
Brooks [1]. Typicalmodels were limited to a
sin-glearticulatedbladeconnectedtoaninertialpoint,
and the control linkages were ignored. The
equa-tionsofmotionwerespeci callywrittenforablade
in a rotating system, and ordering schemes were
usedtodecreasethenumberofnonlinearterms. In
time, more detailed models of the rotor were
de-veloped to improve accuracyand accountfor
var-ious design complexities such as gimbal mounts,
swash-plates, orbearingless root retentionbeams,
amongmanyothers. Therelevantequationsof
mo-tionwerederivedforthespeci ccon gurationsat
hand. Infact,thevariouscodesdevelopedin-house
byrotorcraftmanufacturersaregearedtowardsthe
modelingofthespeci ccon gurationtheyproduce.
This approach severely limits the generality and
exibilityoftheresultingcodes. Inrecentyears,a
numberofnewrotorcraftcon gurationshavebeen
proposed: bearingless rotors with redundant load
paths,tiltrotors,variablediametertiltrotors,and
quadrotors,tonamejustafew. Developinganew
simulation tool for each novel con guration is a
daunting task, and software validation is an even
morediÆcultissue. Furthermore,therequirement
forevermoreaccuratepredictionscallsfor
increas-ingly detailed and comprehensivemodels. For
in-stance, modelingthe interaction of therotorwith
beconsideredin ordertocapturespeci c
phenom-enaorinstabilities.
Clearly,amoregeneraland exibleparadigmfor
modeling rotorcraft systems is needed. It seems
that manyof theconceptsofmultibody dynamics
analysis would be readily applicableto the
rotor-craft dynamics analysis,since a rotorcraft system
can be viewed as a complex exible mechanism.
Inparticular,theabilitytomodelnovel
con gura-tionsofarbitrarytopologythroughtheassemblyof
basiccomponentschosenfromanextensivelibrary
of elements is highly desirable. In fact, this
ap-proachisattheheartofthe niteelementmethod
which hasenjoyed, for this veryreason, an
explo-sive growth in the last few decades. This
anal-ysis concept leads to new comprehensive
simula-tion softwaretools that are modular and
expand-able. Modularityimpliesthatallthebasicbuilding
blocks canbe validated independently, easing the
morechallengingtaskofvalidatingcomplete
simu-lation procedures. Because they are applicableto
con gurationswitharbitrarytopologies, including
those not yet foreseen, such simulation tools will
enjoyalonger life span, acriticalrequirementfor
anycomplexsoftwaretool.
This paperdescribesamultibody dynamics
ap-proach to the modeling of rotorcraft system and
reviews the key aspects of the simulation
proce-dure. Theproposed approachprovidesthelevelof
generalityand exibilityrequiredtosolvecomplex
problems.
Element Library
The element library involves structural elements:
rigid bodies, composite capable beamsand shells,
andjointmodels. Althoughalargenumberofjoint
con gurations are possible, most applications can
be treated using the well known lowerpair joints
presentedhere. Moreadvancedjoints,suchas
slid-ing joints and backlash elements are brie y
de-scribed.
Beam, Shell and Rigid Body Models
Rigidbodyandbeammodelsaretheheartof
rotor-craftmultibodymodels. Shellmodelsarealso
use-ful fordealingwith composite ex-beamsin
bear-all characterizedby the presenceof linearand
ro-tational elds. Intheproposedformulation,all
el-ementsare referredto asingleinertialframe, and
hence,arbitrarilylargedisplacementsand nite
ro-tationsmustbetreatedexactly.
Rigid bodies can be used for modeling
compo-nents whose exibility can be neglected orfor
in-troducing localized masses. For example, in
cer-tainapplications, the exibility oftheswash-plate
maybenegligibleandhence,arigidbody
represen-tation of this componentis acceptable; the model
consistsoftworigidbodies,representingthe
rotat-ingandthenon-rotatingcomponents,respectively,
properlyconnectedtoeachotherandtotherestof
thecontrollinkages.
Beams are typically used for modeling rotor
blades, but can also be useful for representing
transmissionsshafts, pitch links,or wingsof atilt
rotoraircraft. Inviewoftheincreasinguseof
com-positematerials inrotorcraft, theabilityto model
componentsmade of laminatedcomposite
materi-alsis ofgreat importance. Speci cally, it mustbe
possibleto represent shearing deformatione ects,
the o set of the center of mass and of the shear
center from the beam reference line, and all the
elasticcouplingsthatcanarisefromtheuseof
tai-lored composite materials. Most multibody codes
areunabletodealwithsuch structureswitha
suf- cient level of accuracy. An eÆcient approach to
thisproblem isbasedonatwostepprocedure. At
rst,thesectionalpropertiesofthebeamare
com-puted basedonalinear,two-dimensional nite
el-ement analysis of the beam cross-section. These
properties are used to de ne the physical
charac-teristics of the beams involved in the multibody
system. Next, thedynamic response of the
multi-body systemiscomputed using anonlinear, nite
elementprocedure. Ref.[2]givesdetailsand
exam-plesofapplicationofthis process.
Joint Models
A distinguishing feature of multibody systems is
thepresenceofanumberofjointsthatimpose
con-straintsontherelativemotionofthevariousbodies
of the system. Most joints used for practical
ap-plicationscan bemodeledintermsofthesocalled
lowerpairs:therevolute,prismatic,screw,
Figure1: Thesixlowerpairs.
Articulatedrotorsandtheirkinematiclinkagesare
easily modeled with the help of lowerpair joints.
Forexample,aconventionalbladearticulationcan
be modeled with the help of three revolute joints
representing pitch, lag and ap hinges. Another
exampleisprovidedbythepitch-link,whichis
con-nected to the pitch-horn by means of a spherical
joint, andto theupperswash-platebyauniversal
jointtoeliminaterotationaboutitsownaxis.
The explicit de nition of the relative
displace-ments and rotations in a joint as additional
un-known variables represents animportant detailof
theimplementation. Firstofall,itallowsthe
intro-duction ofgenericspring and/ordamperelements
inthejoints,asusuallyrequiredforthemodelingof
realisticcon gurations. Second,the timehistories
ofjointrelativemotionscanbedrivenaccordingto
suitablyspeci edtimefunctions. Forexample,ina
helicopterrotor,collectiveandcyclicpitchsettings
canbeobtainedbyprescribingthetime historyof
therelativerotationatthecorrespondingjoints.
In the classical formulation of prismatic joints
forrigidbodies,kinematicconstraintsareenforced
betweenthekinematicvariablesofthetwobodies.
These constraints express the conditions for
rela-tivetranslationofthetwobodiesalongabody xed
axis,andimplytherelativeslidingofthetwobodies
whichremainin constantcontactwitheachother.
However,thesekinematicconstraintsnolonger
im-ply relative sliding with contact when one of the
bodiesis exible. Toremedythis situation,a
slid-ing joint [3] wasproposed that involveskinematic
constraints at the instantaneous point of contact
between the sliding bodies. This more
sophisti-catedtypeofconstraintisrequiredfortheaccurate
sider, for instance, the sliding of the swash-plate
ontherotorshaft,ortheslidingjointsinvolvedin
theretraction mechanismofthe variablediameter
tilt rotor [4], as discussed in the applications
sec-tion.
Backlashbehaviorcanbeaddedtothemodeling
ofrevolutejoints,asdescribedinref.[5]. Thejoint
isgenerallyfreetorotate,butwhentherelative
ro-tation reachesa preset value, aunilateral contact
condition is activated corresponding to the
back-lash \stop". The associated contact force is
com-puted accordingto asuitablecontactforce model.
Thiselementcanbeusedtomodelthebladedroop
stops,asshownlateron.
Aerodynamic Models
A descriptionof thevarious aerodynamic solution
procedures used for the modeling of rotorcraft is
beyond the scope of this paper. Simpli ed
mod-els based on lifting line theory and vortex wake
models, or sophisticated computational uid
dy-namics codes can be used for this purpose. At
eachtime stepof thesimulation,theaerodynamic
loadsactingonthebladesandwingsmustbe
com-putedbasedonthepresentcon gurationofthe
sys-tem, and are then used to evaluate the dynamic
response.
Robust Integration of
Multi-body Dynamics Equations
Fromthedescriptiongivensofar,itisclearthatthe
equations governing nonlinear exible multibody
systemspresentveryspeci c features. First, they
are highly nonlinear. There are several possible
sources of nonlinearities: large displacements and
nite rotations (geometric nonlinearities), or
non-linearconstitutivelawsforthedeformable
compo-nentsofthe system(materialnonlinearities).
Sec-ond, when constraints are modeled via the
La-grangemultipliertechnique,theresultingequations
presentadualdi erential/algebraic(DAE)nature.
Third,theexactsolutionoftheequationsofmotion
implies the preservation of a number of dynamic
invariants, such as energy and momenta. Fourth,
whentheelasticbodiesofthesystemaremodeled
quencymodes areintroducedin thesystem. Note
thatthesehighfrequencymodesareartifactsofthe
discretization process,andbearnophysical
mean-ing. In large systems, numerical round-o errors
are suÆcient to provide signi cant excitation of
thesemodes,hinderingtheconvergenceprocessfor
the solutionof thenonlinear equations ofmotion.
Furthermore,the nonlinearitiesof thesystem
pro-vide amechanismto transferenergyfrom thelow
to thehighfrequency modes. Hence,thepresence
of high frequency numerical dissipation is an
in-dispensable feature of robust time integratorsfor
multibodysystems.
Allthese featuresofmultibody systemsmustbe
carefullyconsideredandspeci callytakeninto
con-siderationwhen developingrobustsimulation
pro-cedures that are applicableto a widespectrumof
applications. In particular, problems related to
the modeling of helicopters put stringent
require-ments on the accuracy and robustness of
integra-tion schemes. Indeed, rotorsare characterizedby
highly nonlinear dynamics, large numbers of
con-straints,especiallywhentheentirecontrollinkages
aremodeled,highly exiblemembers,largenumber
of degreesoffreedom, and widelydi erentspatial
and temporal scales. On this last issue, consider,
for instance, the dramatic di erence between the
axial and ap-wise bending sti nesses ofatypical
rotorblade.
The classical approach to the numerical
simu-lation of exible multibody systems is generally
based on the use of o -the-shelf, general purpose
DAE solvers. DAE integratorsare speci cally
de-signed for e ectively dealing with the dual
di er-ential/algebraic nature of the equations, but are
otherwiseunawareofthespeci cfeaturesand
char-acteristicsoftheequationsbeingsolved. Although
appealing because of its generality, this approach
implies that the special features that were just
pointed outwill be approximatedin various
man-ners.
While this standard procedure performs
ade-quatelyforanumberofsimulations,alternate
pro-cedures havebeendeveloped[6,7]. Insteadof
ap-plyingasuitableintegratortotheequations
model-ingthedynamicsofmultibodysystems,algorithms
aredesigned tosatisfyanumberofprecise
require-ments. These design requirements are carefully
choseninordertoconveytothenumericalmethod
solved. In particular, the following requirements
willbesatis edbytheproposed approach:
nonlin-ear unconditional stability of thescheme, a
rigor-oustreatment ofall nonlinearities,theexact
satis-faction of the constraints,andthepresenceofhigh
frequency numericaldissipation. Theproofof
non-linearunconditionalstabilitystemsfromtwo
physi-calcharacteristicsofmultibodysystemsthatwillbe
re ectedinthenumericalscheme:thepreservation
ofthetotalmechanicalenergy,andthevanishingof
the work performed by constraintforces.
Numer-ical dissipationis obtainedby letting the solution
drift from the constantenergy manifold in a
con-trolledmannerinsuchawaythatateachtimestep,
energy can be dissipated but not created.
Algo-rithms meeting the abovedesign requirements are
describedin refs.[8,9,10,11,12,13, 14, 15,6,7].
Solution Procedures
Once a multibody representation of a rotorcraft
systemhasbeende ned, severaltypesof analyses
canbeperformedonthemodel. Themainfeatures
ofthestatic,dynamic, stability,andtrim analyses
arebrie ydiscussedin thefollowingsections.
Static Analysis
Thestaticanalysissolvesthestaticequationsofthe
problem, i.e. the equations resultingfrom setting
all time derivatives equal to zero. The deformed
con gurationofthesystemundertheappliedstatic
loads is then computed. The static loads are of
the followingtype: prescribed static loads,steady
aerodynamic loads, and the inertial loads
associ-ated with prescribed rigid body motions. In that
sense,hovercanbeviewedasastaticanalysis.
Oncethestaticsolutionhasbeenfound,the
dy-namic behavior of small amplitude perturbations
about this equilibrium con guration can be
stud-ied: this is done by rst linearizing the dynamic
equations of motion,then extracting the
eigenval-uesandeigenvectorsoftheresultinglinearsystem.
Duetothepresenceofgyroscopice ects,the
eigen-pairs are, in general, complex. For typical rotor
blades, the real part of the eigenvalues is
negligi-ble, whereasfortransmissionshafts, thisreal part
stabil-usefulforprovidingtheinitialconditionstoa
sub-sequentdynamicanalysis.
Dynamic Analysis
The dynamic analysis solves the nonlinear
equa-tions of motion for the complete multibody
sys-tem. Theinitial conditionare takento beat rest,
orthosecorrespondingto apreviouslydetermined
staticordynamicequilibrium con guration.
Complex multibody systems often involve
rapidlyvaryingresponses. Insuchevent,theuseof
aconstanttimestepiscomputationallyineÆcient,
andcrucialphenomenacouldbeoverlookeddueto
insuÆcient time resolution. Automated time step
size adaptivity is therefore an important part of
the dynamic analysis solution procedure. All the
resultspresentedinthisworkmakeuseoftheerror
estimatorofref.[13].
Stability Analysis
Animportantaspectoftheaeroelasticresponseof
rotorcraft systems is the potentialpresence of
in-stabilitieswhichcanoccurbothonthegroundand
intheair. Typically,Floquettheoryisusedforthis
purposebecausethesystempresentsperiodic
coef- cients. ApplicationofFloquettheorytorotorcraft
problem has been limited to systems with a
rela-tivelysmallnumberofdegreesoffreedom. Indeed,
asthenumberofdegreesof freedomincreases,the
computational burdenassociated with the
evalua-tion of the transition matrixbecomes
overwhelm-ing. A novel approach has been proposed, the
implicit Floquet analysis [16], which evaluatesthe
dominanteigenvaluesofthetransitionmatrixusing
theArnoldialgorithm,withouttheexplicit
compu-tationofthismatrix. Thismethodisfarmore
com-putationally eÆcient than the classical approach
and is ideally suited for systemsinvolvinga large
numberofdegreesoffreedom. TheimplicitFloquet
analysis can be viewed as a post-processing step:
all that is required is to predict the response of
thesystemto anumberofgiveninitial conditions.
Hence, it canbeimplemented using the proposed
multibodydynamicsformulation.
The problem of rotorcraft trim involves both the
search fora periodic solution to the nonlinear
ro-torequationsandthedetermination ofthecorrect
controlsettingsthatsatisfysomedesired ight
con-ditions. The determination of control settings is
animportantaspectofrotorcraftanalysisasthese
settingsareknowntodeeplya ecttheentire
solu-tionaswellasstabilityboundaries. Theauto-pilot
anddiscreteauto-pilotmethods[17]arewellsuited
forthesolutionofthetrimcon gurationwhenthe
problem has been formulated using the proposed
nite elementbasedmultibody dynamicsanalysis.
Theauto-pilotmethodmodi esthecontrolssothat
the system convergesto a trimmed con guration.
Additionaldi erentialequationsareintroducedfor
computing the requiredcontrol settings. The
dis-creteauto-pilotapproachmodi esthecontrol
set-tingsateachrevolutiononly.
Applications
Thefollowingapplicationsarepresentedinthis
sec-tion: the conversion from hover to forward ight
mode for a variable diameter tilt-rotor and the
aeroelasticanalysisoftheshipboardengage
opera-tionsofaH-46helicopter.
Modeling a Variable Diameter
Tilt-Rotor
Theexampledealswiththemodelingofavariable
diameter tilt-rotor (VDTR) aircraft. Tilt-rotors
are machines ideallysuited to accomplishvertical
take-o andlandingmissionscharacterizedbyhigh
speed and long range. They operate either as a
helicopter or as a propeller driven aircraft. The
transitionfromonemodeofoperationtotheother
is achievedbytiltingtheenginenacelles. VDTR's
further re nethetilt-rotorconceptbyintroducing
variable span blades to obtain optimum
aerody-namicperformanceinbothhoverandcruise
con g-urations. A generaldescription of current VDTR
technologyis givenin ref. [4], and g. 2
schemati-cally showstheproposeddesign.
Fig.3presentsSikorskytelescopingbladedesign.
Fig. 4 depicts a schematic view of the multibody
sin-Figure 2: VDTR design schematic. Top
g-ure:cruisecon guration;bottom gure:hover
con- guration.
Figure 3: TheSikorskytelescopingbladedesign.
Figure 4: Con gurationof theVDTR. Forclarity,
asinglebladeonlyisshown.
and a sliding screw joint connect the swash-plate
andtheshaft. Themotionoftheswash-platealong
theshaftcontrolsthebladepitchthroughthepitch
linkages. Prescribingtherelativetranslationofthe
slidingjoint,i.e. thetranslationoftheswash-plate
withrespecttotheshaft,controlsthepitchsetting,
e ectivelytransferring thepilot'scommand in the
stationarysystemtothebladein therotating
sys-tem. Thepresenceofascrewjointforcesthe
swash-platetorotatewiththeshaftwhileslidingalongit.
Thisisusuallyaccomplishedinarealsystemwitha
scissors-like mechanism that connects swash-plate
and shaft. This level of detail in the model,
al-though possible using beams and/or rigid bodies
andrevolutejoints,wasnotconsideredtobe
neces-saryfor thepresentanalysis. Asliding screwjoint
models the nut-jackscrew assembly. The motion
of thenuts alongthejackscrewallowsto vary the
blade span in a continuous manner. By
prescrib-ing the relative translationat the joint, the blade
can then be deployed or retracted accordingto a
suitablefunctionofthenacelletilt. Finally,sliding
screw jointsare used to model the sliding contact
between the torque tube and the outboard blade.
Note that a sliding screwjointmust beused here
as the pilot's input is transferred from the linear
motionoftheswash-platetotwistingofthetorque
tubesthroughthepitch links,and nallyto
twist-ingoftheoutboardblade. Appropriatespringsand
dampersareprovided atthegimbal,while springs
arepresentatthe apandlagrevolutejointsin
or-dertocorrectlyrepresentthecharacteristicsofthe
system.
Since actualdata forthis con gurationwasnot
available,themodelusedforthisexamplehas
tele-scoping blades asin g. 3, but thestructural and
aerodynamic characteristics are those of the
XV-15 aircraft [18, 19]. Fig. 5 gives the variation of
the thrust coeÆcient C
T
in hover as function of
thepowercoeÆcientC
P
;goodcorrelationwiththe
experimental dataisobserved.
The VDTR rotor is initially in the hover
con- guration, with the nacelles tilted upwards and
the blades fully deployed. The rotor angular
ve-locity is 20 rad/sec. The shaft rotational speed
andbladepitchsettingarekeptconstantwhilethe
nacelle is tilted forward to reach the cruise
con- guration. At the same time, the blades are
0
0.5
1
1.5
2
2.5
x 10
−3
−5
0
5
10
15
20
x 10
−3
Cp
Ct
Figure5: ThrustcoeÆcientC
T
versuspower
coeÆ-cientC
P
forvaryingcollectiveangle,fortheVDTR
modelwithXV-15 data.
a)
d)
c)
b)
Figure6:SnapshotsoftheVDTRmultibodymodel
duringtheconversionprocess.
thefuselage, and to optimizeaerodynamic
perfor-mance. Themaneuveriscompleted in 20sec,
cor-responding to about 64 revolutions of the rotor.
The time history of the relative prescribed
rota-tion at the wing-nacelle revolute joint is given as
'=0:25(1 sin(2(t=40+0:25)),whilethe
pre-scribed displacement at the nut-jackscrew sliding
jointis linearin time. The retractedrotor
diame-ter for cruise mode is 66%of that in hover. This
simulationwasconducted in avacuum, i.e.
with-outaerodynamicsforces actingontheblades.
Fig. 6 gives a three dimensional view of the
VDTR multibody model at four di erenttime
in-stantsthroughout themaneuver. This viewis
de-0
2
4
6
8
10
12
14
16
18
20
−1.5
−1
−0.5
0
0.5
1
1.5
TIME [sec]
PITCH ROTATION [deg]
Figure 7: Timehistoryoftherelativerotationsat
thepitch hinge.
ceptivelysimple. In fact,thetilting ofthenacelle
involvesacomplextiltingmotionofthegimbalwith
respecttotheshaft. Inturn, apping,laggingand
pitchingmotionsofthebladesare excited. As the
nacelle beginsits motion,gimbalrotationsare
ex-cited and sharply increase during the rst half of
theconversionprocess. Then,thedamperspresent
intheuniversaljointprogressivelydecreasethe
am-plitude ofthis motion. Fig.7showsthetime
his-tory of the blade pitch. This pitching is entirely
due tothegimbaltilting,sincetheswash-plate
lo-cationalongtheshaftwas xed,whichwouldimply
aconstantvalueof pitchforarigidsystem.
Fig.8showsthetime historyoftheforceat the
jackscrew-nutslidingjointduringtheblade
retrac-tion. Note that the jackscrew carries the entire
centrifugal force. Indeed,the bladeisfree toslide
withrespectto thetorquetube,andhence,no
ax-ial load is transmitted to this member. As a
re-sult, the variable span blade is subjected to
com-pressionduringoperation,aradicaldeparturefrom
classicaldesignsinwhichbladesoperateintension.
As expected, g. 8 shows that the axial load in
thejackscrewdecreasesastherotordiameteris
re-duced. Thehighfrequencyoscillatingcomponents
of the signal are once again due to the apping,
lagging and tilting motions of blades and gimbal
0
2
4
6
8
10
12
14
16
18
20
1
1.5
2
2.5
3
3.5
x 10
5
TIME [sec]
TENSION IN THE JACKSCREW
−
NUT SLIDING JOINT [N]
Figure8:Timehistoryoftheforceatjackscrew-nut
slidingjointduring bladeretraction.
AeroelasticAnalysisof Shipboard
En-gage Operations
When operating in high wind conditions or from
a ship-based platform, rotorcraft blades spinning
at low velocity during engage and disengage
op-erations can ap excessively. During these large
appingmotions,thebladeshitthedroopand ap
stops. The droop stop is a mechanism that
sup-ports the blade weightat rest and at low speeds.
Excessiveupwardmotionofthebladeisrestrained
by a second stop, called the ap stop. Impacts
withthedroopand apstopscancausesigni cant
bendingof theblades, to thepointofstrikingthe
fuselage.
TheH-46helicopterwasmodeledhere. First,the
model was validated based on the available data.
Next, thetransient response of thesystem during
engageoperationswassimulated. Completedetails
on this problem can be found in ref. [20]. In this
e ort, the aerodynamic model was based on
un-steady,two-dimensionalthinairfoiltheory[21],and
the dynamic in ow formulation developed by
Pe-ters[22].
H-46ModelValidation
TheH-46isathree-bladedtandemhelicopter. The
structural and aerodynamic properties of the
ro-torcanbefoundinref.[23]andreferencestherein.
Fig.9depicts the multibodymodel ofthe control
linkagesthatwasusedforthisstudy. Therotating
Flap, lag, and
pitch hinges
Blade
Hub
Pitchlink
Pitchhorn
Scissors
Shaft
Rigid body
Beam
Revolute joint
Spherical joint
Universal joint
Ground clamp
Swash-plate:
Rotating
Non-rotating
Prismatic joint
Figure9: Multibodymodelof therotor.
and non-rotating components of the swash-plate
aremodeledwithrigidbodies,connectedbya
rev-olutejoint. Thelowerswash-plateisconnectedtoa
thirdrigidbodythroughauniversaljoint. Driving
the relativerotations of the universal joint allows
the swash-plate to tilt in order to achieve the
re-quiredvaluesoflongitudinalandlateralcyclic
con-trols. Thecollectivesettingisachievedby
prescrib-ing the motion of this rigid body along the shaft
by means of a prismatic joint. The upper
swash-plate is thenconnectedto therotorshaft through
a scissors-like mechanism, and controls the blade
pitching motionsthroughpitch-links. Each
pitch-link is represented by beam elements, in order to
modelthecontrolsystem exibility. Itisconnected
tothecorrespondingpitch-hornthroughaspherical
joint andto theupperswash-plate througha
uni-versaljointtopreventpitch-linkrotationsaboutits
ownaxis. Finally,theshaftismodeledusingbeam
elements. The location of the pitch-horn is taken
from actual H-46 drawings, while the dimensions
andtopologyoftheothercontrollinkagesarebased
onreasonableestimates. Fig.10givesagraphical
representation ofthe control linkages, asobtained
throughthevisualization module. Only oneblade
isshown,forclarity.
Duringtheengagesimulation,thecontrolinputs
were set to the following values, termed standard
control inputs: collective
0
=3deg.,longitudinal
cyclic
s
=2:5 deg.,lateralcyclic
c
=0:0693deg.
Thesevaluesofthecontrolswereobtainedwiththe
proper actuations of the universal and prismatic
Hub
Blade
Upper and lower
swash-plates
Scissor
Pitchhorn
Pitchlink
Shaft
Figure 10: Graphical representation of the
multi-body model of the control linkages. One single
bladeshownforclarity.
In this work, only the aft rotorsystem is
mod-eled. Thebladesweremeshedwith5cubic
geomet-rically exact nite elements, while the droop and
ap stops were modeled using the revolute joint
with backlashdescribed previously. Thestops are
of the conditional type, activated by centrifugal
forces acting on counterweights. The droop and
apstopangles,onceengagedatrotorspeedbelow
50%ofthe nominalvalue
0
=27:61rad/sec,are
0:54and1:5deg,respectively.
Experimental data available for this rotor
con- guration include static tip de ections under the
bladeweightandrotatingnaturalfrequencies. This
datawasusedforapartial validationofthe
struc-turaland inertialcharacteristicsof themodel. As
expected, static tip de ections are in good
agree-ment with Boeingaverage test data, within a2%
margin. Fig.11 showsa fan plotof the rst
ap-torsionfrequenciesfor therotorconsideredin this
example, where quantities are nondimensionalized
with respect to
0
. These modes are in
satisfac-tory agreement with the experimental data, and
withthosepresentedin ref.[23].
Transient Analysis of Rotor Engage
Opera-tions
Next,acompleterotorengagementwassimulated.
Auniformgustprovidesadownwardvelocityacross
the rotordisk, in addition to alateral wind
com-0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
1
2
3
4
5
6
7
8
9
10
NONDIMENSIONAL ROTATIONAL SPEED
NONDIMENSIONAL FREQUENCY
4 th Flap Mode
1 st Torsion Mode
3 rd Flap Mode
2 nd Flap Mode
1 st Flap Mode
Figure 11: H-46 fan plot. Present solution: solid
line; ref. [23]: dashedline; experimental values:
symbols.
ponent. Theverticalwind velocitycomponentwas
10:35 kn, while the lateral one was38:64 kn,
ap-proaching from the starboard side of the aircraft.
Thesituationistypicalofahelicopteroperatingin
high wind conditions on a ship ight-deck. The
run-up rotor speed pro le developed in ref. [24]
from experimental data wasused in the analysis.
The simulationwasconducted by rstperforming
a staticanalysis, wherethe controls were brought
to theirnominal valuesandgravitywasappliedto
the structure. Then, a dynamic simulation was
restarted from the converged results of the static
analysis.
Fig.12showsathreedimensionalviewofthe
ro-tormultibodymodelatthreedi erenttimeinstants
throughout the engage operation. Large apping
motionsofthebladesinducedbythegustblowing
ontherotordiskareclearlynoticeableeveninthis
qualitativepicture. Fig. 13 givesthe out-of-plane
bladetipde ection,positiveup,foracomplete
run-up. During the rotorengage operation, the
max-imum tip de ections are achieved during the rst
6 sec of the simulation. Then, asthe rotorgains
speed, the de ections decrease under the e ect of
theinertialforcesactingontheblade. Hereandin
the following gures, the thick broken line shown
in thelower partof the plot givesthe time
inter-vals when the revolute joint stops are in contact.
Becauseofthelargedownwardgustblowingonthe
rotordisk, onlythedroopstopis impactedbythe
Hub
T=0.46 sec.
T=4.24 sec.
T=6.24 sec.
Figure12: Predictedcon gurationoftherotor
sys-temduringanengageoperationinauniformgust.
0
2
4
6
8
10
12
14
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
TIME [sec]
BLADE TIP DISPLACEMENT [m]
Figure 13: Out-of-plane blade tip response for a
rotorengageoperation. Thethickbrokenline
indi-catestheextentoftheblade-stopcontactevents.
0
2
4
6
8
10
12
14
−4
−3
−2
−1
0
1
2
TIME [sec]
FLAP ROTATION [deg]
Figure 14: Flaphinge rotation fora rotorengage
operationinauniformgust.
Fig. 14 gives the time history of the ap hinge
rotations. Multiple droopstopimpactstakeplace
atthelowestrotorspeeds,causingsigni cantblade
de ections andtransfersfromkineticto strain
en-ergy. Furthermore,theintensityoftheuniform
ver-ticalgustcomponentontherotordiskcauseslarge
negativetip de ections evenfrom thevery
begin-ningoftheanalysis,whenthebladeangular
veloc-ityandresultingsti eninge ectarestillsmall.
Af-terabout10secthroughthesimulation,thedroop
stopisretractedandthebladetiptimehistory
ex-hibits a smoother behavior. In order to simulate
theconditionalnature oftheparticulardroopstop
mechanismusedbythishelicopter,thestop
retrac-tionwasmodeledbychangingthebacklashangles
ofthe aprevolutejointatthe rsttimeinstantof
separationbetweenthe bladeand itsstops passed
theactivationrotorspeed(50%of
0 ).
The results are in reasonable agreement with
the simulations of refs. [23]. In particular, the
maximum negativetip de ections, that determine
whether the blade will strike the fuselage or not,
areverysimilar,aswellastheresultsatthehigher
speeds. Discrepanciesatthelowerspeedsmightbe
duetothedi erentaerodynamicmodelsemployed.
Therepeatedcontactswiththedroopstopscause
largebending oftheblades. Bladede ections can
becomeexcessive,to thepointofstrikingthe
fuse-lage. Forlessseverecaseswheresuchstrikingdoes
not occur, signi cant over-loading of the control
linkagescouldstilltakeplace. Themultibody
for-mulationusedinthisworkreadilyallowsthe
mod-eling of allcontrol linkages, and the evaluation of
the transient stress they are subjected to during
rotor engage. In viewof the multiple violent
im-pacts and subsequent large blade de ections
ob-served,theloadsexperiencedbythevarious
compo-nentsofthesystemduring anengageoperationin
highwindscouldbesigni cantlylargerthanduring
nominal ightconditions.
Pitch-linkloads were computed during the
run-up sequence discussed earlier. Furthermore, the
sameengageoperationwassimulatedforthecaseof
vanishingwind velocity,in orderto provide
\nom-inal" conditions for comparison. For the case of
vanishing wind velocity, allother analysis
param-eters were identical to those used in the previous
simulations.
Fig. 15 shows the axial forces at the pitch-link
2
3
4
5
6
7
8
9
10
−3500
−3000
−2500
−2000
−1500
−1000
−500
0
500
1000
1500
2000
TIME [sec]
PITCHLINK AXIAL FORCE [N]
Figure 15: Mid-pointaxialforces in thepitch-link
forarotorengageoperation. Uniformgustvelocity:
solidline;nogustvelocity: dashedline.
secof therun-upsequence, forwhichthemost
vi-olent blade tip oscillations where observed in the
previousanalysis. Thesolidlinecorrespondstothe
uniform gust velocity case, while the dashed line
givesthe\nominal",vanishing windvelocity case.
Thethickbrokenlinesinthelowerandupperparts
of theplotindicate the contacteventswith droop
and apstops. Thepitch-linkloadsarefargreater
thanthoseobservedatfullrotorspeed,duetothe
largeblade appingmotionsandrepeatedimpacts
withthestops. Thevanishinggustvelocity
analy-sispredictsbladeimpactswithbothdroopand ap
stops. However,the uniform gust velocitycase is
far moresevere due to thelarge blade de ections
andresultingcompressiveloadsin thepitch-links.
Conclusions
Thispaperhasdescribedamultibodydynamics
ap-proachtothemodelingofrotorcraftsystems. This
approachallowsthemodelingofcomplex
con gura-tionsofarbitrarytopologythroughtheassemblyof
basiccomponentschosenfromanextensivelibrary
ofelementsthatincludesrigidanddeformable
bod-iesaswellas jointelements.
Akeyelementoftheformulationisthe
develop-mentof robustandeÆcienttimeintegration
algo-rithms for dealing with the largescale, nonlinear,
di erential/algebraic equations resulting from the
proposed formulation. Static, dynamic, stability,
andtrimanalysescanbeperformedonthemodels.
Furthermore, eÆcient post-processing and
visual-izationtoolsareavailabletoobtainphysicalinsight
into the dynamic response of the systemthat can
beobscured bythe massiveamountsof data
gen-eratedbymultibodysimulations.
Multibodyformulationsarenowwellestablished
andcandealwithcomplexrotorcraftcon gurations
ofarbitrarytopology. Thisnewapproachto
rotor-craft dynamicanalysisseemstobeverypromising
sinceitenjoysallthecharacteristicsthatmadethe
nite element method the most widely used and
trustedsimulationtoolinmanydi erent
engineer-ing disciplines and areas. This new paradigm for
rotorcraft analysis is expected to gain popularity
and become an industry standard in the years to
come.
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