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16.333Lecture# 8
AircraftLateralDynamicsSpiral,Roll,andDutchRollModes
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Fall2004 16.33371AircraftLateralDynamics
Using a procedure similar to the longitudinal case, we can developtheequationsofmotionforthelateraldynamics
v a,u= rx =Ax+Bu,x=pr
and =rsec0
A=
Yv Yp YrmU0 gcos0m m
(IL
v +I Nv) (Lp +I Np) (Lr +I Nr) 0zx zx I zxIxx xx xx
(I Lv +Nv) (I Lp+Np) (I Lr +Nr) 0zx I zx zxI Izz zz zz
0 1 tan0 0where
I = (IxxIzzI2xx zx)/IzzI = (IxxIzzI2zz zx)/IxxI = Izx/(IxxIzzI2zx zx)
and
(m)1 0 0
0 (Ixx
)1 Izx
Ya Yr
B= La Lrzz)
1 zx (I0 I Na Nr0 0 0
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Fall2004 16.33372LateralStabilityDerivatives
Akeytounderstandingthelateraldynamicsisroll-yawcoupling. Lp rollingmomentduetorollrate:
Rollratepcausesrighttomovewingdown,leftwingtomoveupVerticalvelocitydistributionoverthewingW = py
LeadstoaspanwisechangeintheAOA:r(y) = py/U0Createsliftdistribution(chordwisestrips)
1
Lw(y) = U02Clr(y)cydy2Netresultishigherliftonright,loweronleftRollingmoment:
b/2 b/2L= Lw(y)(y)dy=
2
1U
0
2b/2Cl
py2cydyLp
0
6 4Keypoint: positiveyaw rate positive rollmoment.
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Fall2004 16.33373 Np yawingmomentduetorollrate:
Rolling wing induces a change in spanwise AOA, which changesthespanwiseliftanddrag.
Distributeddragchangecreatesayawingmoment. Expecthigherdragonright(loweronleft)positiveyawmoment
There isbothachange inthe lift(largerondownwardwingbe-causeoftheincreasein)andarotation(leansforwardondown-wardwingbecauseofthelarger). negativeyawmoment
In general hard to know which effect is larger. Nelson suggeststhatforarectangularwing,crudeestimateisthat
1Np U02Sb(CL)
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Fall2004 16.33374NumericalResults
Thecodegivesthenumericalvaluesforallofthestabilityderivatives.Cansolve fortheeigenvaluesofthematrixAtofind themodesofthesystem.
0.03310.9470i0.56330.0073
Stable,butthereisoneveryslowpole.
Thereare3modes,buttheyarealotmorecomplicatedthanthelongitudinalcase.
Slowmode -0.0073 SpiralModeFastreal -0.5633 RollDampingOscillatory 0.03310.9470i DutchRoll
Canlookatnormalizedeigenvectors:Spiral Roll DutchRoll
=w/U0 0.0067 -0.0197 0.3269 -28p=p/(2U0/b) -0.0009 -0.0712 0.1198 92r=r/(2U0/b) 0.0052 0.0040 0.0368 -112
1.0000 1.0000 1.0000 0
Notasenlighteningasthe longitudinalcase.
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Fall2004 16.33375LateralModes
RollDamping- welldamped.Astheplanerolls,thewinggoingdownhasanincreased
(windiseffectivelycomingupmoreatthewing)Oppositeeffectforotherwing.Thereisadifferenceintheliftgeneratedbybothwings
moreonsidegoingdownThedifferentialliftcreatesamomentthattendstorestorethe
equilibrium. RecallthatLp
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Fall2004 16.33376SpiralMode- slow,oftenunstable.From levelflight,consideradisturbancethatcreatesasmallroll
angle>0 This results ina smallside-slipv (vehicle slidesdownhill)
Nowthetailfinhitsontheoncomingairatanincidenceangleextratailliftpositiveyawingmoment
Moment creates positive yaw rate that creates positive roll mo-ment(Lr >0)thatincreasestherollangleandtendstoincreasetheside-slipmakesthingsworse.
If unstable and left unchecked, the aircraft would fly a slowlydivergingpathinroll,yaw,andaltitudeitwouldtendtospiralintotheground!!
Cangetarestoringtorquefromthewingdihedral Wantasmalltailtoreducetheimpactofthespiralmode.
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Fall2004 16.33377DutchRoll- dampedoscillationinyaw,thatcouplesintoroll.
Frequency similar to longitudinal short period mode, not as welldamped(finlesseffectivethanhorizontaltail).
Consideradisturbancefromstraight-levelflightOscillationinyaw(finprovidestheaerodynamicstiffness)Wingsmovingbackand forthduetoyawmotion result inoscil-
latorydifferential lift/drag(wingmovingforwardgeneratesmorelift)Lr >0
Oscillationinrollthatlagsbyapproximately90 ForwardgoingwingislowOscillatingrollsideslipindirectionoflowwing.
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Fall2004 16.33378
Doyouknowtheoriginsonthenameofthemode? DamptheDutchrollmodewithalargetailfin.
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Fall2004 16.33379AircraftActuator Influence
102
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pda|
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Transferfunctionfroma
ilerontoflightva
riables
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Figure1:Aileron impulsetoflightvariables.Responseprimarily in. Transferfunctionsdominatedby lightlydampedDutch-rollmode. Notetherudder isphysicallyquitehigh,so italso influencestheA/Croll. Ailerons influencetheYawbecauseofthedifferentialdrag
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Fall2004 16.333710
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Transferfunctionfromr
uddertoflightvariables
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Figure2:Aileron impulsetoflightvariables.Responseprimarily in.
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Fall2004 16.333711
0 5 10 15 20 25 304
2
0
2x 10
4
rad
Aileron 1 deg Impulse 2sec on then off
0 5 10 15 20 25 304
2
0
2x 10
3
p
rad/sec
a
> 0 ==> right wing up
0 5 10 15 20 25 305
0
5x 10
4
rrad/sec
Initial adverse yaw ==> RY coupling
0 5 10 15 20 25 300.01
0.005
0
rad
time sec
Figure3:Aileron impulsetoflightvariables Ailerona=1deg impulsefor2sec.
Since a >0 then right ailerongoesdown, and rightwinggoesup Reidsnotation,and it is notconsistentwiththepictureon64(fromNelson).
Influenceof the rollmode seen in the responseofp toapplicationand releaseoftheaileron input.
Seeeffectofadverseyawintheyawrateresponsecausedbythedifferentialdragduetoailerondeflection.
Spiralmodehardertosee.Dutchmoderesponse inothervariablesclear(1rad/sec6secperiod).
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Fall2004 16.333712
0 5 10 15 20 25 300.01
0
0.01
0.02
rad
Rudder 1 deg step
0 5 10 15 20 25 300.1
0
0.1
pr
ad/sec
0 5 10 15 20 25 300.1
0
0.1
rrad/sec
0 5 10 15 20 25 302
1
0
1
rad
time sec
Figure4:Ruddersteptoflightvariables Rudderstep input1degstep.
Dutchrollresponseveryclear.Other2modesaremuch lesspronounced. showsavery lightlydampeddecay.pclearlyexcitedaswell.Doesntshow it,butoftenseeevidenceofadverserollinp responsewhere initialp isopposite sign to steady state value. Reason isthat the forces act on the finwhich iswell above the cg and the aircraftrespondsrapidly(initially) inroll.
ultimatelyoscillatesaround2.5