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16.333: Lecture# 7ApproximateLongitudinalDynamicsModels
Acouplemorestabilityderivatives Given mode shapes found identify simpler models that capture the main re-
sponses
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Fall2004 16.33361MoreStabilityDerivatives
Recallfrom62thatthederivativestabilityderivativetermsZw and
Mw endedupontheLHSasmodificationstothenormalmassandinertiatermsThese are the apparentmass effects someof the surroundingdisplacedairisentrainedandmoveswiththeaircraft
AccelerationderivativesquantifythiseffectSignificantforblimps,lesssoforaircraft.
Maineffect: rateofchangeofthenormalvelocityw causesatransientinthedownwashfromthewingthatcreatesachangeintheangleofattackofthetailsometimelaterDownwashLageffect
IfaircraftflyingatU0,willtakeapproximatelyt=lt/U0 toreachthetail.Instantaneousdownwashatthetail(t) isduetothewingattimett.
(t)=
(tt)
Taylorseriesexpansion(tt)(t)t
Notethat(t) =t. Change inthetailAOAcanbecom-putedas
d d lt(t)= t= =t
d d U0
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Fall2004 16.33362 For the tail,we have that the lift increment due to the change in
downwashisd
ltCLt =CLtt =CLtd U0
Thechangeinliftforceisthen1
Lt = (U02)tStCLt2IntermsoftheZ-forcecoefficient
Lt
St
St
d ltCZ = 1 = CLt = CLt U02S S S d U 02 c/(2U0)tonondimensionalizetime,sotheappropriatestabil-Weuse
itycoefficientformis(noteuseCz tobegeneral,butwearelookingatCz frombefore):
CZ 2U0 CZ= =CZ 0( c/2U0) c 02U0St lt d
= c S U0CLtd
d= 2VHCLtd
ThepitchingmomentduetotheliftincrementisMcg = ltLt
1(U20)tStCLt1CMcg = lt2 U02Sc2
d lt=
VHCLt =
VHCLtd U0
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Fall2004 16.33363
SothatCM 2U0 CM
=
=CM 0( c/2U0) c 0d lt 2U0
= VHCLtd U0 cdlt
= 2VHCLtdclt
CZ c Similarly,pitchingmotionoftheaircraftchangestheAOAofthetail.
Nosepitchupatrateq,increasesapparentdownwardsvelocityoftailbyqlt,changingtheAOAby
qltt =
U0whichchangestheliftatthetail(andthemomentaboutthecg).
Followingsameanalysisasabove: LiftincrementLt = CLtqltU0
12(U20)tSt
CZ = Lt12(U20)S =
StSCLt
qltU0
CZq CZ(qc/2U0) 0 =
2U0c
CZq 0 =
2U0c
ltU0
StSCLt
= 2VHCLtCanalsoshowthat
ltCMq = CZq c
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Fall2004 16.33364ApproximateAircraftDynamicModels
It isoftengoodtodevelopsimplermodelsofthe fullsetofaircraftdynamics.
Provides insightson the roleof theaerodynamicparametersonthefrequencyanddampingofthetwomodes.
Usefulforthecontroldesignworkaswell
Basicapproachistorecognizethatthemodeshaveveryseparatesetsofstatesthatparticipateintheresponse.
ShortPeriodprimarilyandw inthesamephase.Theuandq responseisverysmall.
Phugoidprimarilyandu,and lagsbyabout90.Thewandq responseisverysmall.
Fullequationsfrombefore: Xu
u m
w Zuw
[Mu+Zu]
=
mZ
q
Iyy0
XwmZw
mZw[Mw+Zw]
Iyy0
0Zq+mU0mZw
[Mq+(Zq+mU0)]Iyy1
gcos0 u Xcmgsin0mZ w Zc
mgsin0w
q + McIyy0 0
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Fall2004 16.33365 FortheShortPeriodapproximation,
1.Since u
0 in thismode, then u
0 and can eliminate theX-forceequation. Zq+mU0 mgsin0
mZZw Zcw wmZw mZw[Mq+(Zq+mU0)]
w = +[Mw+Zw] Mcmgsin0Iyyq qIyy Iyy 00 1 02.TypicallyfindthatZw
mandZq
mU0.Checkfor747:
Zw =1909m=2.8866105Zq =4.5105mU0 =6.8107
Mw Mw=
mZw m Zw
U0 gsin0 Zcw wm Mw+ZwMw Mq+(mU0)Mwm = +Mcmgsin0MwIyyq m qIyy Iyy m 00 1 0
3.Set0 =0andremovefromthemodel(itcanbederivedfromq)
Withtheseapproximations,thelongitudinaldynamicsreducetoxsp =Aspxsp+Bspe
wheree istheelevatorinput,andw Zw/m U0xsp = q , Asp = I1(Mw +MwZw/m) I1(Mq +MwU0)yy yy
Ze/mBsp = I1(Me +MwZe/m)yy
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Fall2004 16.33366 Characteristic equation for this system: s2 +2spsps+2 = 0,sp
wherethefullapproximationgives:Zw Mq Mw
2spsp = + + U0m Iyy Iyy
2 ZwMq U0Mw=sp mIyy Iyy
Givenapproximatemagnitudeofthederivativesforatypicalaircraft,candevelopacoarseapproximate:
2spsp MqIyy sp Mq2 1U0MwIyy
2 U0Mw sp sp U0MwIyyIyyNumericalvaluesfor747
Frequency Dampingrad/sec
Fullmodel 0.962 0.387FullApproximate 0.963 0.385CoarseApproximate 0.906 0.187
Bothapproximationsgivethefrequencywell,butfullapproximationgivesamuchbetterdampingestimate
Approximations showed that shortperiodmode frequency is deter-minedbyMw measureoftheaerodynamicstiffnessinpitch.Sign of Mw negative if cg sufficient far forward changes sign(mode goes unstable)when cg at the stick fixed neutral point.FollowsfromdiscussionofCM (see211)
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Fall2004 16.33367 ForthePhugoidapproximation,startagainwith:
Xu Xw 0 gcos0m m mgsin0 Xcu u
Zq+mU0Zu Zw
Z
cMc
wq wqmZw mZw mZw mZw[Mq+(Zq+mU0)] += [Mu+Zu] [Mw+Zw] mgsin0IyyIyy Iyy Iyy 00 0 1 0
1.Changestowandqareverysmallcomparedtou,sowecanSetw 0andq 0Set0 =0
Xu Xw 0 gm m Xcu u
Zq+mU0Zu Zw
00
ZcMc
00 wqmZw mZw mZw[Mq+(Zq+mU0)] += [Mu+Zu] [Mw+Zw]
Iyy Iyy Iyy 00 0 1 02.Usewhat is leftof the Z-equation to show thatwith theseap-proximations(elevatorinputs) ZeZuZq+mU0Zw
mZw mZw mZwmZw
w
u e=
[Mw+Zw] [Mq+(Zq+mU0)] [Me+Ze][Mu+Zu]qIyyIyy Iyy Iyy
3.Use(Zw msoMw )and(ZqmU0)sothat:mZw mU0 w
Mw +Zw Mw [Mq +U0Mw] qm
Zu Ze= Mu+Zu Mw u Mw m Me +Ze m e
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Fall2004 16.333684.Solvetoshowthat
wq =
mU0Mu
ZuMq
ZwMqmU0MwZuMwZwMu
u+mU0Me
ZeMq
ZwMqmU0MwZeMwZwMe
eZwMqmU0Mw ZwMqmU0Mw
5.Substituteintothereducedequationstogetfullapproximation:mU M Z M 0 u u qZwMqmU0Mw
Xu Xw+ gu
m m u=
ZuMwZwMu 0ZwMqmU0Mw Xe Xw mU0MeZeMq+m m ZwMqmU0Mw
e+
ZeMwZwMeZwMq
mU0Mw
6.Stillabitcomplicated.Typicallygetthat(1.4:4)|MuZw| |MwZu|(1:0.13)|MwU0m| |MqZw|
MuXw/Mw|
Xu small|
7.With theseapproximations, the longitudinaldynamics reduce tothecoarseapproximation
xph =Aphxph+Bphewheree istheelevatorinput.
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Fall2004 16.33369And
u
xph = Aph = Xu
g
m
Zu0
mU0
Bph =
MeMwXe Xwm
Zw+
Ze Mw Me
mU0
8.Which
gives
2phph = Xu/m
2 gZu=ph mU0
Numericalvaluesfor747Frequency Dampingrad/sec
Fullmodel 0.0673 0.0489FullApproximate 0.0670 0.0419CoarseApproximate 0.0611 0.0561
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Fall2004 16.333610 Furtherinsights: recallthat
U0 Z U0 L= + 2CL0)QS u 0 QS u 0 (CLu
M2=
1M2CL0 2CL0 2CL0so
Z UoS 2mgZu = (2CL0) = u 0 2 U0
Then
mg2ph = gZu =
mU02mU0
g= 2
U0whichisexactlywhatLanchestersapproximationgave 2 gU0Notethat
X UoSXu = (2CD0) = UoSCD0u 0 2
and2mg= U2SCL0o
soXu XuU0ph =
2mph = 22mg1 U2SCD0o=
2 U2SCL0o1 CD0=
2 CL0sothedampingratiooftheapproximatephugoidmodeisinverselyproportionaltothelifttodragratio.
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Fall2004 16.333611
102
10
1
10
0
10
0
10
1
10
2
10
3
10
4
|Gude
|
F
req(rad/sec)
Transferfunction
frome
levatortoflightvariables
102
10
1
10
0
10
1
10
0
10
1
10
2
10
3
10
4
|Gde
|
Freq
(rad/sec)
102
10
1
10
0
250
200
150
100
500
50
argGude
F
req
(rad/sec)
102
10
1
10
0
350
300
250
200
150
100
500
argGde
Freq
(rad/sec)
FreqComparisonfromelevator(PhugoidModel)B747atM=0.8.BlueFullmodel,BlackFullapproximatemodel,MagentaCoarseapproximatemodel
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Fall2004 16.333612
102
10
1
10
0
10
1
10
0
10
1
10
2
|G
de
|
F
req(rad/sec)
Transferfunction
frome
levatortoflightvariables
102
10
1
10
0
10
1
10
0
10
1
10
2
|Gde
|
Freq
(rad/sec)
102
10
1
10
0
050
100
150
200
250
300
argGude
F
req
(rad/sec)
102
10
1
10
0
350
300
250
200
150
100
500
argGde
Freq
(rad/sec)
FreqComparisonfromelevator(ShortPeriodModel)B747atM=0.8.BlueFullmodel,MagentaApprox-imatemodel
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Fall2004 16.333613Summary
Approximatelongitudinalmodelsarefairlyaccurate
Indicate that theaircraft responsesaremainlydeterminedby thesestabilityderivatives:
Property StabilityderivativeDampingoftheshortperiod MqFrequencyoftheshortperiod MwDampingofthePhugoid XuFrequencyofthePhugoid Zu
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Fall2004 16.333614 Givenachange in,expectchanges inuaswell. Thesewillboth
impact the lift and drag of the aircraft, requiring thatwe re-trimthrottlesettingtomaintainwhateveraspectsoftheflightconditionmighthavechanged(otherthantheoneswewantedtochange).Wehave:
L Lu L u=D Du D
ButtomaintainL=W,wantL=0,sou= Lu L
GivingD= LDu+D Lu2CL0CD = CL D =QSCDeAR
L =QSCLQS
Du = (2CD0) (416)U0QS
Lu = (2CL0) (417)U0CL 2CD0D = QS +CD 2CL0/U0 U0
QS 2C2L0=CL0
CD0 +eAR CL(T0+T)(D0+D) Dtan = = L0+L L0
CD0 2CL0 CL=CL0
eAR CL0
For 747 (Reid 165 andNelson 416), AR = 7.14, so eAR 18,CL0 = 0.654 CD0 = 0.043, CL = 5.5, for a =
0.0185rad
(67) = 0.0006rad. This is the opposite sign to the linearsimulationresults,buttheyarebothverysmallnumbers.