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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

10

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Freq

(rad/sec)

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|G

pda|

Freq(rad/sec)

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

102

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(rad/sec)

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|Gpda

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Freq(rad/sec)

Transferfunctionfromr

uddertoflightvariables

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argGda

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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