The work presented here is the first part of a continuing effort to expand existing capabilities in aeroelasticity by developing the methodology which is necessary to utilize unsteady time-domain aerodynamics directly in aeroservoelastic design and analysis. The ultimate objective is to define a fully integrated state-space model of an aeroelastic vehicle's aerodynamics, structure and controls which may be used to efficiently determine the vehicle's aeroservoelastic stability. Here, the current status of developing a state-space model for linear or near-linear time-domain indicial aerodynamic forces is presented

Gilbert, Michael G.

Woods, J. A.

English

NASA Technical Reports Server

N91-1_ _
DIRECT USE OF LINEAR TIME-DOMAIN AERODYNAMICS IN
AEROSERVOELASTIC ANALYSIS: AERODYNAMIC MODEL
J. A. Woods and M. G. Gilbert
Aeroservoelasticity Branch
NASA Langley Research Center
Hampton, Virginia
207
https://ntrs.nasa.gov/search.jsp?R=19910001003 2020-03-19T20:17:53+00:00Z
Theworkpresentedhereis thefirst partof acontinuingeffort to expandingexisting
capabilitiesinaeroelasticitybydevelopingthemethodologywhichis necessaryto utilize
unsteadytime-domainaerodynamicsdirectlyin aeroservoelasticdesignandanalysis.
Theultimateobjectiveof thisstudyis todefineafully integratedstate-spacemodel
of anaeroelasticvehicle'saerodynamics,structureandcontrolswhichmaybeusedto
efficientlydeterminethevehicle'saeroservoelasticstability.
In thispresentation,thecurrentstatusof developingastate-spacemodelfor linear
or near-lineartime-domainindicialaerodynamicforcesis presented.
MOTIVATION:
TO EXPAND EXISTING AEROSERVOELASTIC DESIGN AND
ANALYSIS CAPABILITIES TO INCLUDE TIlE USE OF
UNSTEADY TIME-DOMAIN AERODYNAMICS
LONG-TERM OBJECTIVE:
DEVELOP METHODOLOGY TO UTILIZE LINEAR AND NEAR-
LINEAR TIME-DOMAIN AERODYNAMICS IN THE SUPERSONIC
AND SUBSONIC REGIMES DIRECTLY IN AEROSERVOELASTIC
DESIGN AND ANALYSIS.
IMMEDIATE OBJECTIVE:
DEVELOP A TIME-DOMAIN STATE-SPACE MODEL OF TIME-
DOMAIN AERODYNAMIC INDICIAL FORCES.
208
THE INTEGRATEDAEROELASTICMODEL
To understandtheimportanceof thisresearch,it is necessaryto considerthat
severalcodes[1,2]havebeendevelopedin recentyearswhichcomputetime-domain
unsteadyaerodynamics,however,thetechniquesneededto utilizetheaerodynamicsin
aeroservoelasticdesignhavenotbeenfully developed.
Oneof theonly methodsdevisedto dateto evaluatetheaeroelasticstabilityof
aerospacevehiclesin thetime-domainhasbeenageneralmethodcapableof handlingthe
nonlinearsystem[3]. Thismethodis expensiveasit involvesthecomputationof the
aeroelasticsystemtimeresponsewhich requiressolutionof thenonlinearsmalldisturbance
aerodynamicequations.Further,a frequencydecompositionof theresponseis necessary
to evaluatethestabilityof componentmodes.Theresponsemustberecomputedat several
dynamicpressuresuntil aneutrallystablemodeisencountered.Otheravailablemethods
modeltheaerodynamicsdirectlyin thefrequencydomain.
For linearandnearlinearsystemsin supersonicandsubsonicflow, however,the
vehiclestabilitymaybeevaluatedwithoutcomputingtheaeroelasticsystemforcedresponse
or transformingforcesto thefrequencydomain.This is accomplishedbyrepresentingthe
time-dependentaerodynamicforcesin state-spaceform coupledwith acommonlyused
state-spacerepresentationof thestructure.Stabilityisdeterminedbytheeigenvaluesof the
coupledsystemmatrix.
Thefocusof thispresentationis, again,on theformulationof theaerodynamic
portionof theintegratedmodel.
uo=]l
I_10"- L_
Figure 1. Schematic block diagram indicating integration of the aerodynamic
model with the structural model.
209
FORMULATION OF AERODYNAMIC MODEL
The aerodynamic model is derived as the Laplace transform of a commonly used
frequency domain approximation modified from ref. 4. It is transformed directly into state-
space form.
MODIFIED FREQUENCY DOMAIN APPROXIMATION
Q(s)= A o+A l_s+ ---V TI(S)
i--1 s +13i _
STATE-SPACE REPRESENTATION IN TIME DOMAIN
Y •
Wl
W2
WN
V
o
V
= o
O •*o
0
V
"W 1 -0 B 1
W 2 0 B 2
WN 0 B N
Wl
Q(t)=_[ll-..I]. w2.
.WN.
210
APPROXIMATION METHOD
The approximation method involves a least squares approximation to the actual
aerodynamic force to determine the scalars A o, A 1 and B i. The fit is constrained at t=0 to
fit exactly and at large times to equal the asymptotic value of the generalized force. As in
the frequency-domain rational function type approximations, aerodynamic poles, 13i , are
initially specified.
The aerodynamic forces currently being approximated are the rigid-body forces
acting on a NACA0064 airfoil and are due to Dowell [5].
APPROXIMATING FUNCTION
Q(t) =[ AoTl(t ) + A 1
CONSTRAINTS ON LEAST SQUARES APPROXIMATION
Ao=Q(t**)
( . )vA 1 = Q(O)-i__T_,lBi -Q(to.) _-
211
APPROXIMATION METHOD (CONTI)
A system identification technique frequently used in control system analysis is
applied to regenerate the generalized aerodynamic force. Specifically, impulse and step
responses of the aerodynamic model are generated using a discrete-time state-transition
method. The sum of these responses is the aerodynamic approximation, Q(t)--Q(t), based
on previously determined coefficients and the specified aerodynamic poles.
Due to the discontinuity at t--0 in the impulse input, an assumption is made that at
t=0+, initial conditions are real valued. At t=0-, initial conditions are zero. This
assumption can be shown mathematically.
v  .NsrrioNEQUAVIONS
w(t + 1)= a,w(t)+ ru(t)
o(t) =Ow(t) +Du(t)
WHERE
ASSUMING
• (t) =e [A]T and 1" = j,T e[A],_ B dx
w(o+):eu(o)
212
APPROXIMATIONMETHOD(CONTI)
Improvementsto theaerodynamicapproximationaremadeby updatingthe
aerodynamicpoles,_i, followedby anotherleastsquaresapproximationto recomputethe
coefficients.To updatethepoles,themethodusedbyPetersonandCrawley[7] to
approximateunsteadyaerodynamicsin thefrequencydomainis implementedin thetime
domain. A normsquare-errorcostfunctionis defined. In thiscase,thesquareof the
differencebetweentheactualaerodynamicforceandtheapproximationis used..The
incrementalchangein aerodynamicpolesis solvedfor by invertingtheHessian,
_2j / _i_k ' in a single term Taylor series expansion of _J / _. The incremental change
in 13i is multiplied by a scale factor, or, and added to the current aerodynamic poles. The
scale factor, or, is computed using quadratic interpolation [8] to insure that the cost is
approaching a local extrema.
The new aerodynamic poles are limited. If a given pole is greater than -0.01, it is
set equal to that value until the next parameter update. To prevent a pole from going to -co
and ill-conditioning the system matrix later on, the pole is limited to a value which would
produce no more than a 99.5% decrease in magnitude of the exponential over a given time
step.
The two step procedure of computing system coefficients and updating
aerodynamic states is repeated until the cost function has been minimized.
SQUARE ERROR COST FUNCTION
J(l_) =[ O( t)-Q( t, I_)] T [ Q( t)-Q( t, 1_)]
NEWTON RAPHSON STEP
WHERE,
AERODYNAMIC POLE UPDATE
_2j
H=
I_new = I_o + Ot _l]i
CONSTRAINTS ON AERODYNAMIC POLES
I_<-0.01 AND I_>ln(O.OO5)/At
213
PRELIMINARYRESULTS
TableI briefly describesomeof the progress which has been made up to this time.
Four sets of initially specified aerodynamic poles, associated coefficients and the initial cost
are indicated as well as the minimum cost quantifies. The f'n'st set of poles is a subset of the
poles which were used by Dowell to generate the aerodynamic forces. Dowelrs zero pole
was not included for stability reasons and because the A1 term serves the same purpose of
providing a constant term at t=0. The other sets of poles represent "random" selections
between a small negative number and -1.0, -2.0 and -3.0.
A minimum cost was obtained for each of these sets of poles. The poles close to
those of the generating function produced the lowest cost. Minimum cost increases from
there as the range of initial poles widens. It is noted that finding a minimum isn't always
guaranteed. For some sets of initial poles, the least-squares fit doesn't converge or the
program determines a local maxima instead of a local minima.
One of the immediate observations which can be made from Table I is that the
aerodynamic poles tend to decrease in magnitude as the cost is minimized. The same trend
occurs as other rigid body forces are being approximated. The implication is that the fit
improves at large times and degrades at small times. In terms of reduced frequencies, this
means that the high frequency components of the curve are not being fit well. Thus, a
weighted least-squares fit and a weighted square error function will be considered to
improve the approximation at small times.
Finally, an assumption made in the quadratic interpolation subroutine which
computes parameter step size is that when the square-error cost is computed for the "step-
ahead" coefficients remain constant. From Table I, this appears to be a valid assumption,
as over the entire range of parameter updates, coefficients have remained fairly unchanged.
214
PRELIMINARY RESULTS
LIFT DUETO PLUNGE
6 POLEAPPROXIMATIONS
Minimum Cost ResultsInitial Data
Poles* Coefficients A 9 A1 Cost Poles* Coefficients A 0
-0.1 -0.433 0.999 0.034 0.0170 -0.091 - 1.24
-0.3 -0.51 -0.081 0.805
-0.8 1.646 -0.148 -0.178
-1.2 -2.841 -0.520 -0.116
- 1.75 2.19 - 1.255 0.402
-3.5 -0.574 -2.082 -0.123
A1 Cost
0.999 -0.036 0.0012
-0.2 -3.041 0.999 -0.074 0.0047 -0.185 -3.202
-0.4 43.06 -0.33 42.016
-0.5 -124.97 -0.401 -114.617
-0.6 121.388 -0.470 103.931
-0.8 -51.012 -0.613 -37.816
-1.0 14.162 -0.775 9.256
0.999 -0.056 0.0020
-0.334 -6.036 0.999 -0.343 0.1995 -0.243 -5.681
-0.668 46.844 -0.424 42.426
-1.O -178.225 -0.590 -152.419
-1.334 333.826 -0.747 263.921
-1.668 -299.231 -0.898 -215.887
-2.0 102.678 -1.049 67.253
0.999 -0.0997 0.0166
-0.5 -9.818 0.999 -1.266 0.8607 -0.35 -6.406
-1.0 89.054 -0.683 42.793
-1.5 -365.568 -1.03 -137.988
-2.0 723.748 -1.409 222.239
-2.5 -678.553 -1.837 -174.33
-3.0 241.917 -2.343 53.628
0.999 -0.421 0.2469
b
* Poles indicated are _]i'_ •
Table I. Summary of some aerodynamic poles, coefficients and cost functions.
215
PRELIMINARYRESULTS(CONTI)
Figure2 illustratessomeof the approximations to the rigid-body forces acting on a
NACA0064 airfoil which are currently being obtained. The figure includes a pair of
figures for each of four rigid-body aerodynamic forces. The lower figure in each pair
contains a comparison between the aerodynamic data and an approximation made by using
the initially specified aerodynamic poles. The norm square-error cost is indicated. The
upper figure in each pair indicates the improved approximation after the minimization
technique has been applied. Again, the minimum norm square-error cost is indicated. In
all cases, aerodynamic data has been normalized with the largest absolute magnitude of
force.
As can be seen, the technique does improve the approximation noticeably. In three
of the four cases, the cost has been reduced by about 90%. In the case of "Moment Due to
Pitch", the cost was observed to remain high even after minimization. This emphasizes the
fact that the current method finds only the first extrema in cost. This extrema may be only a
local extrema and not a global one.
216
PRELIMINARY RESULTS
NACA0064 RIGID BODY AERODYNAMIC FORCES
15O=[-.334 -.668 -1.0 -1.334 -1.668 -2.0]
.8
o
°.6
.4
0
.8
u_ .6
.4
0
Lift Due to Plunge
!
15
!
3O
sec
Jmin =0.0166
,o .75
o
LL
' .5
45 60 0
Jo = 0.1995
8
.75
I I I
15 30 45
sec
Moment Due to Plunge
Jmin = 0.1539
LL
.5
60 0
Lift Due to Pitch
S
!
15
!
30
sec
Jmin = 0.0185
I
45
S Jo = 0.1035
I I I
15 30 45
sec
Moment Due to Pitch
Jmin = 5.5157
' ' -1.2
15 30 45 60 0 15 30 45
sec sec
.6
Jo =1.1159
15 30 45 60 "1"20 15 30 45
sec sec
Jo = 5.7312
6O
6O
6O
6O
Figure 2. Approximations to NACA0064 airfoil rigid body forces using initial aerodynamic
poles and aerodynamic poles computed for minimum cost.
217
FURTHER DEVELOPMENT
Other methods will be considered to determine minimum cost. The method
currently used is effective, but needs modification.
In an effort to improve the fit for small times, a weighted least squares fit will be
implemented to determine the coefficients. A weighted square error cost function will also
be considered.
Sometimes the program converges to a local maximum instead of minimum. Thus,
means of forcing the program to converge on a minimum will be implemented.
FURTHER DEVELOPMENT
• IMPROVE PROCEDURE TO IDENTIFY MINIMUM COST
• INVESTIGATE WEIGHTED LEAST SQUARES APPROXIMATION TO
DETERMINE COEFFICIENTS
• INVESTIGATE A WEIGHTED SQUARE ERROR COST FUNCTION
• INVESTIGATE METHODS OF CHANGING THE SEARCH DIRECTION IF A
MAXIMUM IS BEING APPROACHED INSTEAD OF A MINIMUM
218
FURTHERAPPLICATIONS
To furtherevaluatethis technique,aerodynamicdatageneratedfor arealaircraftby
a time-domainaerodynamicodein thesubsonicandsupersonicflight regimeswill be
modeled.Bothrigid-bodyandflexiblemodeswill beconsidered.
Finally, to fulfill thewholepurposeof developingthismodel,methodologywill
needto bedevelopedtointegratetheaerodynamicmodeleffectivelywith astructural
model. Later,controlsystemswill beintegratedinto thescheme.Usingtheintegrated
models,systemstabilitywill beevaluated.
FURTHERAPPLICATIONS
• APPLYTECHNIQUETO FLEXIBLE AND RIGIDBODY GENERALIZED
AERODYNAMIC FORCESACTINGON A REALAIRCRAFT
DEVELOPMETHODOI_£N3YFORINTEGRATINGMODELWITH DISCRETE-
TIME STRUCTURALMODEL AND PERFORMINGSTABILITY ANALYSIS FOR
ARBITRARY MOTION
219
REFERENCES
. Batina, J.T., Seidel, D.A., Bland, S.R., and Bennett, R.M., "Unsteady Transonic
Flow Calculations for Realistic Aircraft Configurations," AIAA Paper No. 87-0850,
AIAA/ASME/ASCE/AHS 28th Structures, Structural Dynamics and Materials
Conference, Monterey, CA, April 6-8, 1987.
2. Blair, M. and Williams, M.H., "A Time-Domain Panel Method for Wings," AIAA
Paper No. 89-1323, AIAA/ASME/ASCE/AHS/ASC 30th Structures, Structural
Dynamics and Materials Conference, Mobile, Alabama, April 3-5, 1989.
3. Cunningham, H. J., Batina, J. T., and Bennett, R. M., "Modern Wing Flutter Analysis
by Computational Fluid Dynamics Methods," ASME Paper No. 87-WA/Aero-9,
ASME Winter Annual Meeting, Boston, Mass., Dec 13-18, 1987.
. Abel, I.A., "An Analytical Technique for Predicting the Characteristics of a Flexible
Wing Equipped With an Active Flutter-Suppression System and Comparison with
Wind-Tunnel Data," NASA TP 1367, Feb. 1979.
5. Dowell, E. H., "A Simple Method of Converting Frequency-Domain Aerodynamics to
the Time-Domain," NASA TM 81844, 1980.
6. Brogan, W.L., Modem Control Th_qry, Quantum Publishers, Inc., New York, 1974.
7. Peterson, L.D. and Crawley, E.F., "Improved Exponential Time Series Approximation
of Unsteady Aerodynamic Operators," JQ0rnal qf Aircraft, Vol. 25, No. 2, Feb. 1988.
8. Fox. R.L., Op_timizatiqn Meth0d_ fgr En_n_ering D_sima, Addison-Wesley Publishing
Co., Reading, Mass., 1971, pp. 51-58.
220


Direct use of linear time-domain aerodynamics in aeroservoelastic analysis: Aerodynamic model

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