aircraft control lecture 4

7/29/2019 aircraft control Lecture 4

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2/20

Fall2004 16.33341AircraftDynamics

Notecandevelopgoodapproximationofkeyaircraftmotion(Phugoid)using

simple

balance

between

kinetic

and

potential

energies.

Consideranaircraft insteady, levelflightwithspeedU0 andheight

h0. ThemotionisperturbedslightlysothatU0 U =U0+u (1)h0 h=h0+h (2)

Assume that E = 1mU2 +mgh is constant before and after the

2perturbation. Itthenfollowsthatu ghU0

FromNewtonslawsweknowthat,intheverticaldirectionmh=L W

1whereweightW =mgandliftL=2SCLU2 (S isthewingarea).

Wecanthenderivetheequationsofmotionoftheaircraft: 1

mh=

L

W

=

SCL(U

2 U0

2

)

(3)21= SCL((U0+u)2 U02)1SCL(2uU0)(4)22

ghU0 =(SCLg)h (5) SCL

U0 Sinceh= hand for theoriginalequilibriumflightconditionL=1W =2(SCL)U2 =mg,wegetthat0 2

SCLg g=2

m U0Combinetheseresulttoobtain:

h+2h=0 , g2U0

These equations describe an oscillation (called the phugoid oscilla-tion)ofthealtitudeoftheaircraftaboutitnominalvalue.Onlyapproximatenaturalfrequency(Lanchester),butvalueclose.


3/20

k)

Fall2004 16.33342 Thebasicdynamicsare:

I I

F =mvc and T =H1

B vc TransportThm.F =vc + BIm

B

BI T =H + H

Basicassumptionsare:

1.Earthisaninertialreferenceframe2.A/Cisarigidbody3.BodyframeBfixedtotheaircraft(i,j,

BI Instantaneousmappingofvc and intothebodyframe:BI

=Pi+Qj+Rk vc =Ui+Vj+Wk P U BIB =Q (vc)B = VR W

By symmetry, we can show that Ixy = Iyz = 0, but value of Ixzdepends on specific frame selected. Instantaneous mapping of theangularmomentum

H=Hxi+Hyj+HzkintotheBodyFramegivenby

Hx Ixx 0 Ixz PHB =Hy= 0 Iyy 0 Q

Hz Ixz 0 Izz R


4/20

1

Fall2004 16.33343 Theoverallequationsofmotionarethen:

1 B

F = vc +BI

vcm X U 0 R Q U

mY = V + 0 P V RZ W Q P 0 W

U + QWRV = V + RUPWW+ PV QU

B

BIT = H + H

L IxxP + IxzR 0 R Q Ixx 0 Ixz PM = IyyQ + 0 P 0 Iyy 0 QR

P 0 Ixz 0 Izz RN IzzR+ IxzP Q IxxP + IxzR +QR(IzzIyy) + PQIxz

= IyyQ +PR(IxxIzz) + (R2P2)Ixz

IzzR+ IxzP +PQ(IyyIxx) QRIxz Clearly theseequations arevery nonlinearand complicated, and we

havenotevensaidwhereFandTcomefrom. = Needtolinearize!!Assumethattheaircraft isflying inanequilibriumconditionand

wewilllinearizetheequationsaboutthisnominalflightcondition.


5/20

Fall2004 16.33344Axes

ButfirstweneedtobealittlemorespecificaboutwhichBodyFrame

wearegoinguse. Severalstandards:1.BodyAxes - Xalignedwith fuselagenose. Zperpendicular to

Xinplaneofsymmetry(down). YperpendiculartoXZplane,totheright.

2.WindAxes- Xalignedwithvc. ZperpendiculartoX(pointeddown).

Y

perpendicular

to

XZ

plane,

off

to

the

right.

3.StabilityAxes- Xalignedwithprojectionofvc intothefuselage

planeofsymmetry. ZperpendiculartoX(pointeddown). Ysame.

RELATIVEWIND

( )

( )

( )

B ODY

Z-AXIS

B ODY

Y -AXIS

X-AXIS

WIND

X-AXIS

S T AB ILITY

X-AXIS

B ODY

Advantagestoeach,buttypicallyusethestabilityaxes.Indifferentflightequilibriumconditions,theaxeswillbeoriented

differentlywithrespecttotheA/Cprincipalaxesneedtotrans-form(rotate)theprincipalinertiacomponentsbetweentheframes.Whenvehicleundergoesmotionwith respect to theequilibrium,StabilityAxesremainfixedtoairplaneas ifpaintedon.


6/20

Fall2004 16.33345 Canlinearizeaboutvarioussteadystateconditionsofflight.Forsteadystateflightconditionsmusthave

F Faero+ Fthrust = 0 and T = 0 = Fgravity+ 3Soforequilibriumcondition,forcesbalanceontheaircraft

L= W andT = D AlsoassumethatP = Q= R= U = V = W = 0

Imposeadditionalconstraintsthatdependonflightcondition:3Steadywings-levelflight = = = = 0

Key Point: While nominal forces and moments balance to zero,motion about the equilibrium condition results in perturbations totheforces/moments.RecallfrombasicflightdynamicsthatliftLf = CL0 where:03CL =liftcurveslopefunctionoftheequilibriumcondition30 =nominalangleofattack(anglethatwingmeetsairflow)But,asthevehiclemovesabouttheequilibriumcondition,would

expectthattheangleofattackwillchange= 0+

ThustheliftforceswillalsobeperturbedLf = CL(0+ ) = Lf + Lf0

Canextendthisideatoalldynamicvariablesandhowtheyinfluence

allaerodynamicforcesandmoments


7/20

Fall2004 16.33346GravityForces

GravityactsthroughtheCoMinverticaldirection(inertialframe+Z)Assumethatwehaveanon-zeropitchangle0NeedtomapthisforceintothebodyframeUsetheEulerangletransformation(215)

0 sinFg =T1()T2()T3()

0

=mg

sincos

Bmg coscos

Forsymmetricsteadystateflightequilibrium,wewilltypicallyassumethat0,0 =0,so sin0

Fg =mg 0 Bcos0

UseEuleranglestospecifyvehiclerotationswithrespecttotheEarthframe

= QcosRsin = P+Qsintan+Rcostan = (Qsin+Rcos)sec

Notethatif0,then Q Recall: Roll,Pitch,andHeading.


8/20

Fall2004 16.33347Linearization

Definethetrimangularratesandvelocities P Uo

BIo = Q (vc)o = 0 B BR 0

whichareassociatedwiththeflightcondition. In fact, thesedefinethetypeofequilibriummotionthatwelinearizeabout.Note:W0 = 0 sinceweareusingthestabilityaxes,andV0 = 0 becauseweareassumingsymmetricflight

ProceedwithlinearizationofthedynamicsforvariousflightconditionsNominal Perturbed PerturbedVelocity Velocity Acceleration

Velocities U0, U = U0+ u U = uW0 = 0, W = w W = w

V0 = 0, V = v

V = vAngular P0 = 0,Rates Q0 = 0,

R0 = 0,P = p P = pQ= q

Q =q

R= r R = rAngles 0, = 0+ =0 = 0, = =0 = 0, = =


9/20

Fall2004 16.33348

W C.G.

0VT

VT0= U0

XE

X0

X

q

Z

Z0ZE

or

0

0 0

Horizontal

U = U + u

Down

Figure1:PerturbedAxes.Theequilibriumconditionwasthattheaircraftwasangledupby0withvelocityVT0 = U0.Thevehiclesmotionhasbeenperturbed(X0 X)sothatnow = 0+ andthevelocityisVT = VT0.NotethatVT isnolongeralignedwiththeX-axis,resultinginanon-zerouandw.Theangle iscalledtheflightpathangle,anditprovidesameasureoftheangleofthevelocityvectortotheinertialhorizontalaxis.


10/20

Fall2004 16.33349 Linearization forsymmetricflight

U = U0+ u,V0 = W0 = 0,P0 = Q0 = R0 = 0.Notethattheforcesandmomentsarealsoperturbed.

1[X0+ X] = U + QWRV u + qwrvu

m 1

[Y0+ Y] = V + RUPWm

v + r(U0+ u)

pw v + rU0

1 [Z0+ Z] = W+ PV QU w +pvq(U0+ u)m

w qU0 X u 1

1Y = v+ rU0 2m

Z w qU0 3

Attitudemotion:

L IxxP + IxzR +QR(IzzIyy) + PQIxz

M = IyyQ +PR(IxxIzz) + (R2P2)IxzN IzzR+ IxzP +PQ(IyyIxx) QRIxz

L Ixxp + Ixzr 4M = Iyyq 5N Izzr+ Ixzp 6


11/20

Fall2004 16.333410 Keyaerodynamicparametersarealsoperturbed:

TotalVelocity2 2)1/2 U0+uVT = ((U0+u)2+v +w

PerturbedSideslipangle = sin1(v/VT)v/U0

PerturbedAngleofAttackx = tan1(w/U)w/U0

Tounderstandtheseequationsindetail,andtheresultingimpactonthevehicledynamics,wemustinvestigatethetermsX ...N.Wemustalsoaddresstheleft-handside(F,T)Netforcesandmomentsmustbezeroinequilibriumcondition.AerodynamicandGravityforcesareafunctionofequilibriumcon-

ditionANDtheperturbationsaboutthisequilibrium. Predictthechangestotheaerodynamicforcesandmomentsusinga

firstorderexpansioninthekeyflightparametersX X X X Xg

X = U+ W+ W+ +...+ +XcU W W X X X X Xg

= u+ w+ w+ +...+ +XcU W W X calledstabilityderivativeevaluatedateq.condition.U Givesdimensionalform;non-dimensionalformavailableintables. Clearly approximation since ignores lags in the aerodynamics forces(assumesthatforcesonlyfunctionofinstantaneousvalues)


12/20

Fall2004 16.333411StabilityDerivatives

First proposed by Bryan (1911) has proven to be a very effec-

tivewaytoanalyzetheaircraftflightmechanicswellsupportedbynumerousflighttestcomparisons.

Theforcesandtorquesactingontheaircraftareverycomplexnonlin-earfunctionsoftheflightequilibriumconditionandtheperturbationsfromequilibrium.Linearizedexpansioncaninvolvemanytermsu, u,...,w, w,...u, w, Typicallyonlyretainafewtermstocapturethedominanteffects.

Dominantbehaviormosteasilydiscussedintermsofthe:Symmetricvariables: U,W,Q&forces/torques: X,Z,andMAsymmetricvariables: V,P,R&forces/torques: Y,L,andN

ObservationfortrulysymmetricflightY,L,andN willbeexactlyzeroforanyvalueofU,W,QDerivativesofasymmetricforces/torqueswithrespecttothesym-

metricmotionvariablesarezero.

Further(convenient)assumptions:

1.Derivativesofsymmetricforces/torqueswithrespecttotheasym-metricmotionvariablesaresmallandcanbeneglected.

2.Wecanneglectderivativeswithrespecttothederivativesofthemotion variables, but keep Z/w and Mw M/w (aero-dynamic lag involved informingnewpressuredistributiononthewinginresponsetotheperturbedangleofattack)

3.X/q isnegligiblysmall.


13/20

Fall2004 16.333412()/() X Y Z L M N

uvwpqr

00

00

000

000

000

000

000

Notethatwemustalsofindtheperturbationgravityandthrustforcesandmoments

Xg

Zg=mgcos0

0 =mgsin00

Aerodynamicsummary:1A X= XU 0u+ XW 0wXu,x w/U02A Y v/U0,p,r3A Zu,x w/U0,x w/U0,q4A Lv/U0,p,r5A M u,x w/U0,x w/U0,q6A N v/U0,p,r


14/20

Fall2004 16.333413 Resultisthat,withtheseforce,torqueapproximations,

equations1,3,5decouplefrom2,4,6

1,3,5arethelongitudinaldynamicsinu,w,andq

X muZ

=

m(wqU0)

M I

yyq

XgX X + Xcu+ w+U W 0 0 0

ZgZ Z Z Zw+ + Zcu+ w+ q+ WU W Q 0 0 00 0M M M w + M q+ Mcu+ w+U 0 W 0 W 0 Q 0

2,4,6arethelateraldynamicsinv,p,andr

Y m(v+ rU0)

L = Ixxp + IxzrN Izzr + Ixzp

Y Y Y v+0p+ r+ +YcV 0 P R 0

L v+ L0p+ L r+ LcV 0 P R 0

N N Nv+0p+ r+ NcV 0 P R 0


15/20

Fall2004 16.333414BasicStabilityDerivativeDerivation

ConsiderchangesinthedragforcewithforwardspeedU

D = 1/2VT2SCD2 2V2 = (u0+ u)2+ v + wT

VT2 VT2= 2(u0+ u) = 2u0u u 0

VT2 VT2Note: = 0 and = 0v 0 w 0

Atreferencecondition: D VT2SCD=Du u 0 u 2 0 S

2 CD V

T2

= u0 + CD02 u 0 u 0S 2 CD= + 2u0CD0u02 u 0

Note D isthestabilityderivative,whichisdimensional.u

Define nondimensional stability coefficient CDu as derivative ofCD withrespecttoanondimensionalvelocityu/u0

D CDand CD0 (CD)0CD = 1VT2S CDu u/u02 0

So( 0 correspondstothevariableatitsequilibriumcondition.

)


16/20

Fall2004 16.333415Nondimensionalize:

D Su0 CD=

u0 + 2CD0u 0 2 u 0

QS CD= + 2CD0u0 u/u0 0

u0 D= (CDu + 2CD0)QS u 0

Sogivenstabilitycoefficient,cancomputethedragforceincrement. NotethatMachnumberhasasignificanteffectonthedrag:

CDu = CDu/u0 0 =u0a

CDua 0 = M

CDM

where CDM canbeestimatedfromempiricalresults/tables.

Aerodynamic Principles

= Constant

CD

0

Mcr0 1 2

= 0.1CDM

Mach number, M

Thrustforces CTu = CTu/u0 0

Tu 0 = CTu

1u0QS

Foraglider,CTu = 0Forajet,CTu 0Forapropplane,CTu = CD0


17/20

Fall2004 16.333416 Liftforcessimilartodrag

L= 1/2VT

2SCL

L Su0 CL

= + 2CL0 u 0 2 u0 u 0 QS CL

= + 2CL0u0 u/u0 0 u0 L

= (CLu + 2CL0)QS u 0

where CL0 is the lift coefficient for the eq. condition and CLu =MCL

M asbefore. Fromaerodynamictheory,wehavethatCL MCL|M=0CL = = CL

1 M2 M 1 M2M2CLu = 1 M2CL0

Derivatives: Nowconsiderwhathappenswithchangesintheangleofattack. Takederivativesandevaluateatthereferencecondition:Lift: CL

C2L 2CL0Drag: CD = CDmin +eARCD =eARCL


18/20

Fall2004 16.333417 CombineintoX,Z ForcesAtequilibrium,forcesbalance.Usestabilityaxes,so0 =0Include the effect in the force balance of a change in on the

forcerotationssothatwecanseetheperturbations.Assume perturbation is small, so rotations are by cos 1,

sinX = T

D+L

Z = (L+D)C

L

Cx

Cz

z

x

V

Note: = T at t = 0

CD

(t)

(i.e., Static Trim)

So,nowconsiderthederivativesoftheseforces:X T D L

= +L+

TThrustvariationwithverysmall 0 0

Applyatthereferencecondition(=0),i.e. CX Nondimensionalizeandapplyreferencecondition:

CX =

CD +CL02CL0= CL0 CLeAR

CX=

0


19/20

Fall2004 16.333418AndfortheZ direction

Z D L= D

GivingCZ = CD0 CL

RecallthatCM wasalreadyfoundduringthestaticanalysis Canrepeatthisprocessfortheotherderivativeswithrespecttothe

forwardspeed. Forwardspeed:

X T D L= +

u u u uSothat

u0 X u0 T u0 D=

QS u 0 QS u 0 QS u 0 CXu CTu (CDu + 2CD0)

SimilarlyfortheZ direction:Z L D

= u u u

Sothat u0 Z u0 L=

QS u 0 QS u 0(CLu + 2CL0)CZu

M2=

1 M2CL0

2CL0

Manymorederivativestoconsider!


20/20

Fall2004 16.333419Summary

PickedaspecificBodyFrame(stabilityaxes) from the listofalter-

nativesChoice simplifies some of the linearization, but the inertias now

changedependingontheequilibriumflightcondition.

Since the nonlinear behavior is too difficult to analyze, we needed

to consider the linearized dynamic behavior around a specific flightconditionEnablesustolinearizeRHSofequationsofmotion.

Forcesand

moments

also

complicated

nonlinear

functions,

so

we

lin-earizedtheLHSaswell

Enablesustowritetheperturbationsoftheforcesandmomentsintermsofthemotionvariables.Engineering insightallowsustoarguethatmanyofthestability

derivatives that couple the longitudinal (symmetric) and lateral(asymmetric)motionsaresmallandcanbeignored.

Approachrequiresthatyouhavethestabilityderivatives.Thesecanbemeasuredorcalculatedfromtheaircraftplanform

andbasicaerodynamicdata.

aircraft control lecture 4

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