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NASA Technical Memorandum 1 01 574

AEROSERVOELASTIC MODELING ANDAPPROXIMATIONS OF THE UNSTEADYAERODYNAMICSAPPLICATIONS USING MINIMUM-STATE

Sherwood H. Tiffany and Mordechay Karpel

APRIL 1989(NASA-TH-10 57U) A EBOSEBVOEL U T C 80 DEL 11sG N89-24308A ND A P P L I C B T I O I S USING HINIHOH-STATEA P P R O X I M A T I O N S OF THE UNSTEADY AERODY N A M ICs(IhSA, Langley Research C e n t e r ) 1 3 p Unclas

C S C L O l C 63/05 0212638

NASANational Aeronautics andSpace AdministrationLangley Research CenterHampton,Virginia 23665-5225

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AEROSERVOELASTIC MODELING AND APPLICATIONS USING MINIMUM-STATEAPPROXIMATIONS OFTHE UNSTEADY AERODYNAMICS

Sherwood H. Tiffany?Aeroservoelasticity BranchStructural Dynamics DivisionNASA Langley Research CenterHampton, Virginia 23665-5225

Mordwhay KarpelSenior Research AssociateDepartment of Aeronautical EngineeringTechnion - Israel Institute of TechnologyHaifa, IsraelAbstract

Various control analysis, design, and simulationtechniques foraeroelastic applications require the equationsof motion to be cast in a linear time-invariant state-spaceform. Unsteady aerodynamics forces have to beapproximated as rational functions of the Laplace variablein order to put them in this framework. For theMinimum-State method, the number of augmenting statesrepresenting the unsteady aerodynamics is a function onlyof the number of denominator roots in the rationalapproximation. Results are shown of applying variousapproximation enhancements (including optimization,frequency dependent weighting of the tabular data, andconstraint selection) with the Minimum-State formulationto the Active Flexible Wing wind-tunnel model. Theresults demonstrate that good models can be developedwhich have an order of magnitude fewer augmentingaerodynamic equations than more traditional approaches.This reduction facilitates the design of lower order controlsystems, analysis of control system performance, and nearreal-time simulation of aeroservoelastic phenomena.

The equations of motion of a flexible aircraftcontain unsteady generalized aerodynamic force termswhich are transcendental functions. The availability ofefficient linear systems algorithms used in aeroservoclasticanalysis and design has providcd strong motivation toapproximate the unsteady aerodynamic forces as rationalfunctions of the Laplace variable, refs. 1-5. Such rationalfunction approximations (RFA's) allow t h eaeroscrvoelastic equations of motion to be cast in a lineartime invariant (LTI) state-space form, albeit with increasedsize of the state vector due to the RFA's. This increase inthe number of states due to the RFA's is referred to as theaerodynamic dimension. There is always a trade offbetween how well the RFA's approximate t h eaerodynamic forces and the desire to kecp the aerodynamicdimension small. The RFA formulations in the literature(e.g., refs. 1 - 12) have varying capabilities to perform

such a trade-off.Currently there are three basic formulations usedin approximating unsteady generalized aerodynamic forcesfor arbitrary motion using rational functions:

1. Least-squares (LS) - references 1 and 52. Modified matrix-Pade' @IMP) - reference2,3, 6, and 73. Minimum-State(MS)- 8 and 12Extensions to these approaches were developed (refs. 9 -11) which included the capability to enforce selectedequality constraints on the RFA's and of optimizing thedenominator coefficients in the rational functions usingnonlinear programming techniques. For th e Minimum-State method, h e number of augmenting states required torepresent the unsteady aerodynamics is a function only ofthe number of denominator roots in the rationalapproximation: there are no multiple aerodynamic rootsas there are for the Least-Squares formulation. Methods todetermine critical frequency ranges b

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develop valid low order mathematical models for thecontrol system design, evaluation and simulation tasksassociated with the cooperative effort.The following pages present a review of thetechniques involved. Results are also shown whichdemonstrate that good models can be developed whichintroduce an order of magnitude fewer acrodynamic stateequations than more traditional approaches.

In order to express the equations of motion infirst-order form, the unsteady generalized aerodynamicforcc (gar) coefficients ( Q i j ) must bc dcfined in thecomplcx Laplace s-domain, or nondimcnsionalized Laplacep-domain, p=(c / 2u) s ; here u is the freestream velocity andc is a rcfcrence length (typically the mean acrodynamicchord). Since the gaf coefficicnls are computcd atspecified valucs of rcduced frequcncics kn,. it is ncccssaryto gcneratc complcx p-domain functions by approximatingthe tabular values, Qi j ( i kn) , of the gars as closely aspossible, and then employing the concept of analyticcontinuation in a region near the imaginary axis. Figure1 depicts the approximating process for a single elementQij of Q. wherek reduced frequency, ( c l 2u)oQ;j(ik,,) reduced-frequency domain tabular data& ( i k )dij(ik, , )

(identified by the open circles)approximating curve, ~ c ( p )forp = ik (corresponding to the solid line)points along the approximating curve atreduced frequencies, k,,, corresponding to thetabular data (solid dot$)is the approximation error bctwccn twocorresponding points (dcnoted by an arrowbetween the points).

cij(ikn)

1....1'REAL p v 1 Q,Figure 1 - Approximating aerodynamic tabular data for asingle element of Q , employing consuaints

The approximating function Qv ( p ) is determinedin such a way as to minimize some Icast-squarc combina-

4

4Lion of the errors, &ij(ikn)= lQv(ik,,) - Qij(ikn) I, betweenthe approximating function at p = ik n and the tabularvalue Q i j ( i k n ) . It is possible that certain equalityconstraints or weighted constraints might be desirable toimpose on some of the tabular values (such as the steadystate (k= 0) point, shown in the shaded box).Review of Rational Function ApDroximationsGcnenlized Aerodvnamic Forces

The most common form of the approximatingfunctions used currently for each generalized forcecoefficient, Qij , s a rational function of th e nondimen-sional Laplace variable p . The normal form chosen isonc in which the numerator polynomial is order 2 morethan the dcnominator. This gives rise to the followingpartial fraction form for each element, Qij ,where n~ isthe number of partial fractions and corresponds to the orderof thc overall dcnominator polynomial:

which can be rewritten in the Laplace domain as

n

Because tabular data are determined for specifiedvalucs of reduced frequency, k, , , the QQ are actuallydefined only for these values of the nondimensionalizedLaplace variable p = ik,. It is desirable to use the aboverational function form for the approximating functions inorder to convert the transcendental equations of motioninto linear, time-invariant, state-space form. The fact thatthe numcrator is only order 2 more than the denominatorimplies thc number of resulting equations associated withthc aerodynamics is a function of the ordcr of thedenominator polynomials. The Ao, AI , and A2 terms canbe included with the stiffness, damping, and mass terms,without any additional equations.

2

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~ Whcn the matrix equations, which combine allthe Q ij 's , are formulated, the character of the partialfractions are expressed in various ways (ref. 11). Thefirst, A2 P ) = A 0 + A 1 p e A 2 9 QL ,where

is referred to simply as the "Least-Squares'' (LS)formulation in which t h e number and v3Iues ofdenominator coefficients, b ~ ,re fixed for all Qij . Thesecond,

nP (3)L iwhere A' L i = Q d A ( Q + 2 ) } j p+brj

for each column j , is a Modified Matrix-Pade' (MMP)formulation, which allows the number of denominatorcoefficients and their values to vary between columns.They are only fixed per column. The Minimum-State(MS) formulation,A6 ( p ) = Ag + A1 P+ A2 + QL ,where

fixes the denominator coefficients as in the least squares,but additionally. the partial fraction numerator coefficientsare determined as a coupled product of a premultiplyingmatrix, D, and a postmultiplying matrix, E. The diagonalmatrix of aerodynamic roots is denoted by R ; Le..

The resulting number of aerodynamic statesintroduced into the first order equations as a result of theMS formulation is N. It is usually a little larger than nL,but generally much smaller than either the aerodynamicdimension for the least-squares method, nCnL, or that ofthe modified matrix-Pade',

AThe character of eachfor the diffcrent formulations is summarized in table 1.

and the aerodynamic dimensions

Constraints and Lae Coefficient OptimizationExtended versions of th e LS, MMP, and MSmethods to fitting functions to aerodynamic tabular dataare included in the ISAC program (ref. 14). Theextensions include capabilities to select equalityconstraints and to perform a nonlinear optimization of thedenominator coefficients (refs. 9-11). The equalityconstraints allow for more realistic modeling of theaerodynamics and for improved fits at critical points (suchas flutter). Nonlinear optimization of the denominatorcoefficients allows improvement in the approximationswithout increasing their number by determining a betterset of coefficients than might be chosen apriori.

Table 1 RFA Matrix Formulations 'A Aerodynamic Dimensionerodynamic Method - -~_ _. &?&r of%_- - _ _ _ ~Last-Squares ACommon denominator coefficients in each Q,

n PQ = I ( A ( Q + 2 ) ) i /

Modified Matrix Pade' Different number of and values for denominatorcoefficients for each column ,Qjn.

AMinimum-State Common denominator coefficients in each

N

3

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inimum-State Eauations o f M o t mThe Minimum-State matrix equation for theunsteady aerodynamics is given by equation 4. Thercsulting own-loop statc-space equations of motion are:

. .

0r o 0 }I ,

The 6 represents structural modes and the 6 representscontrol modes. The submatrices are defined as follows:

2= M - q ( 5 ) [ A 2 ] (massmauix)

(stiffness matrix)= K - $ A 0 3K g = - q t A 0 I gG = G - q L d I1 (damping matrix)

C 4g = - q x { A- nondimensional gust velocities4 - dynamic pressure

D , E, and R were defined in equations (4)and (5). Thesubscripts refer to column subsets involved; crefers tovibration modes, 6 efers to control modes, and g to gustmodes.P-ical Weighting

In ISAC (ref. 14) the inclusion of constraints isoptional and those selected are imposed on a pe r columnbasis. A recently developed program, MIST (MinimumState), which has bcen intcrfaccd with the ISAC program,provides the capability to include weights,

on each &ij(ik,) to weight the importance of the fits at

each tabular point Qg(ikn). Each "ijn is a measure ofimporlance which allows the fits to be improved at somepoints (at the possible expense of others) based uponphysical properties without actually enforcing equality atthe specified points (ref. 12) The method developedtherein determines the measure of importance based uponpartial derivatives of selected open-loop parameters withrespect to the weighted term at a specified design point.For the vibration modes the weight of the fi t at each valueof k is determined by the effective influence on the systemdeterminant; for control modes, by the effect on Nyquistgains: for gust modes, by the effect on the response tocontinuous gusts: and for hinge moment terms, by thehinge moment response to control surface or gustexcitations. The physical weights are k-dependent, butthey are independent of normalization of the generalixdaerodynamic force coefficients. The wCUt n equation (7)provides a means to specify the minimum of themaximum weighted magnitudes

Q.'. I nmax {lQi (ikn) kijn 1of each Qi j . It can be shown that

wcur I Q:. 1 .Hence, a value of w C U f= 0 allows the fu l l influence ofphysical weighting by setting the minimum = 0; whereasw C U f= 1 forces the most sensitive point for each Qij tobe normalizeci or have a weightcd magnitude of 1.

Y

The current MIST formulation requires theimposition of three equality constraints on each element& i j . These equality constraints can be selected fromvarious options, but mus t be the same for all $6. Forthe results shown herein, this requirement is not severe.

Data normalization is the basic measure of errorused in ref. 9-11 which enforces fits equally (except forequality constraints) over a selected frequency range oftabular values in order to best meet the criterion foranalytic continuation. However, since the concept ofanalytic continuation used to extend frequency domainfunctions, descibed by a finite set of tabular aerodynamicdata, is approximate at best, physical weighting withoptimization provides another tool for extending thesefunctions in some reasonable fashion. Furthermore,varying the min imum of the maximum weightedmagnitude, by varying the value of wcui between 0 and 1,providcs a mechanism which allows the approximalionsto incorporate some combination of physical weightingand simple data normalization.

A F W -ace Modeling'The AFW project requires various first-ordermodels for different purposes such as low-order models forcontrol law design and near real-time simulation an dhigher-order models for control system evaluation in batchsimulation.

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For both types of simulation it was desircd togcncrate a reliable mathematical model in which thc massmatrix was constant, i.e. had no aerodynamic terms, inordcr to avoid repetitive matrix inversion. This impliedthat the A2 coefficients in the rational approximations hadto be zero for all rigid and elastic modes. To meet the loworder requirements of design models and near real-timesimulation, MS models were generated using 2denominator coefficients. This resultcd in anantisymmetric aeroservoelastic model with a total of 38states. These states consisted of 22 rigid and elastic modedisplacement and rate states, 12 actuator states, 2 guststates, and 2 aerodynamic states.Discussion of Resu &

The following results are for the antisymmetricAFW model at Mach .9. There are 11 vibration modcs(the roll mode is included in this group), 4 control modesand one gust. In each case, the aerodynamicapproximations are constrained to match at k=O , thcimaginary parts are are matched at k=0.005, and thc A2 =0. Furthermore, there is no constmint on the minimumthat the maximum weighted magnitudes of QG must be(i.e.. W,-ur = 0.0). corresponding to fu l l inclusion ofphysical weighting.

Figure 2 is a set of three dimensional plots (fromtwo different views) of the physically weightedmagnitudes of each Qi, ( ikn) These weights weredetermined for a value of dynamic pressure (q ) ncar flutter.On each plot, Qi,, i=l to 11)are shown in a group foreach j. For the vibration modes, j increments from 1 to11,corresponding to the generalized coordinates. For thecontrol modes, j increments from 12 to15, correspondingto the leading edge inboard (LEI) , railing edge inboard(TEI), leading edge outboard (LEO),nd th e trailing edgeoutboard (TEO) control surfaces. There is one gust mode,corresponding to j = Z 6 . The peaks correspond to thetabular values of reduced frequency of highest influence onsystem characteristics. For example, for the roll mode(column 1) the system determinant is most sensitive to

changes in the generalized aerodynamic force coefficientnear the third k value, k=0.01, in the first row; is., thesystem is most sensitive to errors in Q l l ( k 3 ) .Some of the rows have nearly zero weightedmagnitudes for all values of k , which indicates that thesystem has minimal sensitivity to variation in theseelements. The view on the left indicates that for thevibration modes, the diagonal terms, Qii ,are the most

sensitive. Referring to Table 2 for th e values of invacuum natural frequencies (w),he view on the rightindicates that the area of sensitivity is at k values near kvof each mode. This can be seen more easily in figure 3,which shows the weighted magnitudes of the mostsensitive vibrational mode elements, Qii , t each tabularvalue of k. According to this plot, the system is mostsensitive to errors in Q66 at reduced frequencies between0.6 and 0.8. This plot also indicates that the generalizedforces probably should have been calculated for additionalvalues of reduced frequency between 0.6 and 1.5, as thep e a k s are not clearly defined for some of the modes in thisregion. As indicated by figure 2, the same denominatorcoefficients based on sensitivity for modes 1 through 5should work for the control modes and gust in for thismodel. They have the same sensitivity range as the first 5modes; Le., they are not very sensitive to fits past k=0.6.In order to obtain a reliable model for all the modesselected however, it appears that the range of tabularvalues (0.04~2.0)was appropriate for this configuration.

The determination of an optimal set ofdenominator coefficients for a Least-Squares fi t is a fairlyquick process (reference 11). For the Minimum-Stateapproximation, however, nonlinear optimization of thedenominator coefficients requires a three-fold, time-consuming iteration process. In lieu of using nonlinearoptimization, figure 3 was used to help select denominatorcoefficients for several cases. The selections, although notoptimal, produced good fits, and results are presented intable 3 and figure 4 in order to show the reader the type ofresults which can be obtained with various selections ofdenominator coefficients.Table 2 Frequency parameters ( w y , k y ) at q = 0 and (wf ik f ) at flutter

Most sensitiveranpe of k tWV k V ?fl k fMode (radlsec) (radlsec)

9.34 0.034 [O.O.O. 12.39 0.0092 44.13 0.159 45.37 0.164 [ O .1,0.2]3 49.37 0.178 53.93 0.195 near 0.2*4 82.23 0.297 73.05 0.263 [0.2,0.3]5 101.53 0.366 101.64 0.367 near 0.46 173.04 0.625 200.11 0.722 [0.6,0.8]7 240.65 0.869 2 17.93 0.787 near 0.88 248.93 0.899 248.79 0.898 [0.8,1O]9 258.77 0.934 252.54 0.91 1 [0.9,1 O]10 313.74 1.132 313.95 1.133 [1.0,1.2]11 326.88 1.180 348.58 1.258 near 1.2

*Flutter mode?range of k for each mode in which system determinant is most sensitive to error

5

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1 2 3 4 5 6 YVibration .ModeColumns(mws L - 11)

(a) Vibnrion modes and rigid body roil

LEI m LEO nCantmi Columns( m w s 1 - 11)

3.0

/Gust CJlumn 2 .i m w s I - 11')

Convoi

(c) Gust nlode

6ORIGINAL PAGE ISOE POOR QUALtTY

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4-++-+

*++4+4

0.0 0.5 1 .o 1.5 2.0REDUCED FREQUENCY (k)

Figure 3.- Weighted magnitudes of most sensitive aerodynamic elementsTable 3 shows the choices made Tor the order ofapproximation equal to 1.2. and 4 (equals lhe number ofdenominator coefficients). Also listed are the method forselection and the % errors in flutterq and frequency. TJeword 'unconstrained refers to k values for which the Q qare not constrained; i.e., those other than k=O andk=0.005. The words 'estimated most sensitive Qq ' refcrsto a case in which the estimated peaks (as identified infigure 3) are used to determine the most sensitive Q q .The k 's corresponding to the Q iwith 'estimated peaks'

are also estimated.

Figure 4 shows the open loop root loci of abaseline configuration and the best overall of the two andfour-aerodynamic state cases as indicated in table 3. Thebaseline used the tabular, frequency domain generalizedaerodynamic forces and was generated using a p-kdeterminant iteration process employing interpolatedvalues (ref. 15). The other two were based on linear timeinvariant state-space methods. The Hassig form ofimplementation of the p-k method (ref. 16) becomes lessaccurate as the approximation Q ( p ) = Q(O+ik) degrades;hence the baseline result for the roll mode in figure 4should not be regardedas accurate.

Yalues of Selected&twoximation Denom inator coefficients Method o Selection oemr %errorOrderof1 0.8 (unconstrained) k of most sensitive Qij 0.06 0.270.722*+ kf of most sensitive Q i j 0.13 0.260.625 k , of most sensitive Q i j 0.24 0.250.625, 0.8690.297, 0.625 (unconstrained) k , 'sof most sensitive Q i s , for different j 0.32 0.12

1 o Mid-rangek -0.07 0.292 0.2, 0.8* (unconstrained) k's of most sensitiveQq's -0.07 0.32-0.87 0.38v 'sof most sensitive Qij's

0.625, 1.180 k , 'sof estimated most sensitive Qq's -0.78 0.370.4, 1.5 Mid-range k 's of least sensitive QQ's -0.55 0.364 0.2, 0.6, 0.8, 1.0 (unconstrained) k's of most sensitiveQQ's 0.64 0.000.2, 0.8, 1.0, 1.5* 1OO 0.05

0.2, 0.73, 0.8, 1.22 k' s o f stimated most sensitive (unconstrained) Qy's 0.59 0.010.009, 0.625, 0.869, 1.180 k , 's of estimated most sensitive 0;;'s 0.09 0.12

(unconstrained) k 's of most sensitiveQGs. for different j-9 ~0.98 0.05.159, 0.366, 0.899, 1.132 k,'s or least sensitive Q i j ' s

*Best frequency and damping properties at flutter for all modes for comesDonding order of auuroximation- a .- . .?Very little difference overall between this and other three cases7

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0 300

* - < 200

optimization of the denominator coefficients must be Cr: 0.4employed. In this case the criterion for goodness of fi t isthe total physically weighted error. Table 4 shows thecomparison of total physically weighted error (wcuI= O) ,as well as the total normalized error of refs. 9-11, and %error in flutter q and frequency for four optimized,physically wcightcd MS approximations. Figure 5 showsthe dccrcasing total physically wcightcd error criterionwi th incrcnsing ordcr. As would he cxpcctcd sincc thc

3i .3!0.24 0.dt- 0.0s

- aseline-Frequency domain aerodynamics(no rational approximations)a - - _ S (4-states) Physically Weighted0 - - - - MS (2-statcs)Physically Weighted

,--I I II I-80 -60 -40 -20 0 20

Real (rad/sec)(0 < q c 400 sf)%difference in flutter q < 1 O (=3.25 psf)%difference in flutter frcqumcy c 4 (=.3 radsec)

Figure 4 .- Stability analysis root loci for the AFW antisymmetric modes at Mach 0.9.4Table 4 Comparison o f optimized physically weighted Minimum-State approximations

renominator Coefficients Method of Selection Total Error Total Error %error %err0Odx Values of Selected Physically Weightcd Normalized qf Y1 1.225 Nonlinear Optimization 0.32 17.75 -0.18 0.312 0.438 , 0.582 Nonlinear Optimization 0.14 12.98 -0.71 0.353 0.431 . 0.917. 1.314 Nonlinear Omimization 0.07 19.22 0.58 -0.010.70 0.09

8

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l l l l l l l l l l l

1 2 3 4 5 6 7 8 9 1 0 1 1MODE

40

! 08 10$ 0

0 06c1B

g.10

-20 1 2 3 4 5 6 7 8 9 1 0 1 1MODEFigure 6.- Comparison of open loop flutter root characteristics for optimized physically weighted approximations

Imag(rad/sec)

i 00- aseline-Frequency domain aerodynamics__ -S (2-states) Optimized Physically Weighted

0 - - - - MS ( I -state) Optimized Physically Weighted

(no rational approximations) k lutter e)I I I I I I-80 -60 -40 -20 0 20Real (radhec) - -(0 c q

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approximation is clearly invalid that there is no consistentimprovement in damping ratio as compared to the baselinewith increasing order of approximation. Results (notshown) of Minimum-State approximations with the sameequality constraints shown herein which reduce normalizederror have higher 96 errors in flutter q and frequency and aremuch more sensitive to optimization of lag coefficients.Physical weighting tends to improve open-loop fluttercharacteristics and reduces the sensitvity to choices ofaerodynamic roots. Corresponding control and gustresponse analyses, as well as closed loop flutter analyseswould have to be presented in order to demonstrate thegoodness of the physical weighting criteria for the controland gust modes.

By combining various capabilities, namely theMinimum-State method, selectable constraints, optimalselection of denominator coefficients, as well as thedetermination and use of critical frequency ranges forapproximating the tabular generalized aerodynamic forceswith rational functions based upon physical properties, itis possible to obtain good, low order state-spacemathematical models for design and simulation ofaeroservoelastic systems. A state-space model with onlytwo aerodynamic states predicted a flutter dynamic pressurewith less than 0.8 percent error and showed goodagreement in the root loci of all elastic modes as comparedto conventional stability analysis. The significance ofthese results is that good models can be developed havingan order of magnitude fewer augmenting aerodynamicequations than more traditional approaches. This reductionfacilitates the design of lower order control systems,analysis of control system performance, and near real-timesimulation of aeroservoclastic phenomena.

1 .

2.

3.

4.

5.

ReferencesSevart, Francis. D.: Development of Active FlutterSuppression Wind Tunnel Testing Technology.AFFDL TR-74-126.Roger. Kenneth. L.: Aimlane Math Modeling

6.

7.

8.

9.

TP-1367, 1979.Dunn, Henry. J.: An Analytical Technique fo rApproximating U nsteady Aerodynamics in the l im eDomain, NASA TP-1738, 1980.Dunn, Henry. J.: An Assessment of UnsteadyAerodynamic Approximations for Time DomainAnalysis. Proceedings of the AeroservoelasticitySpecialist Meeting, AFWAL-TR-84-3105, Vol. 1, U.S. Air Force, October 1984, pp. 98-1 15.Karpel, Mordechay. : Design fo r Active and PassiveFlutter Suppression and Gust Alleviation, NASA CR-3482, 1981.Tiffany, Sherwood. H.; and Adams, William M., Jr.:Fitting Aerodynamic Forces in the Laplace Domain:An Application of a Nonlinear Nongradient Techniqueto Multilevel Constrained Optimization, NASATM-86317, 1984.

10. Tiffany, Sherwood. H.; and Adams, William. M., Jr.:Nonlinear Programming Extensions to RationalFunction Approximations of Unsteady Aerodynamics.AIAA Paper 87-0854-CP presented at theAIAA/ASME/ASCE/AHS 28th Structures, StructuralDynamics,and Materials Conference, Monterey,California, April 1987.

11 . Tiffany, Shcrwood. H.; and Adams, William. M., Jr.:Nonlinear Programming Extensions to RationalFunction Approximation of Uns teady Aerodynam icForces. NASA TP-2776, 1988.12. Karpel, Mordcchay: Time-Dom ain AeroservoelasticModeling Using Weighted Unsteady AerodynamicForces. Submitted for publication in J. Guidance,Control, and Dynamics, Log No . G2460, March 1988.13. Perry, Boyd 111; Dunn, Henry J.; and Sanford, MaynardC.: Control Law P arameterization or an AeroelasticWind-Tunnel Model Equipped with an Active RollControl Svstem and ComDarison with Experiment .MGhods for Active Coniol Design. Structural Aspectsof Active Controls, AGARD CP-228, August 1977,

pp. 4-1 - 4-11.Vepa, Ranjan.: Finite State Modeling of AeroelasticSystems, NASA CR 2770, 1977.Edwards, John. W.: Unsteady Aerodynamic Modelingand Aeroelastic Conlrol.. SUDAAR-504 (NASA GrantNGL-05-020-007), Dept. of Aeronautics andAstronautics, Stanford University, February 1977.(Available as NASA CR-148019.)Abel, Irving.: An Analytical Technique f o r Predictingthe Characteristics of a Flexible W ing Equipped with anActive Flutter-Suppression System and Comparison

NASA TM-100593, 1988.14. Peele, Ellwood. L.; and Adams, William. M., Jr.: ADigital Program or Calculating the Interaction BetweenFlexible Structures, Unsteady Aerodyna mics, and ActiveControls. NASA TM80040, January 1979.15. Adams, William M., Jr.; Tiffany, Sherwood H.;Newsom, Jerry R.; and Peele, Ellwood L.:STABCAR :A Program fo r Finding CharacteristicRoots of Systemv Ilaving Transcendental StabilityMatrices. NASA TP2165, 1984.16. Hassig, Hermann J.: An Approximate True DampingSolution of the Flutter Equation b y DeterminantIteration. J. Aircraft, Vol. 8, No. 11, Nov. 1971, pp.with Wind-Tunnel Data, NASA 885-889.

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Report Documentation Page1 Report No 2 Government Accession No

NASA TM-101574 1

19 Secu rity Classif (of this report)Unclassified

Aeroservoelastic Mod eling and ApplicationsUsing M inimum-State Approximations of theUnsteady Aerodynamics

2u ---iirifv Classif (o fUnclassified

Sherwood H. TiffanyMordechay Karpel- - __ _ _. _.__-_ _ - - __9 Performirig Organization Ndtne and Address

NASA Langley Research CenterHampton, VA 23665-5225--12. Sponsoring Agency Name and Address

National Aerona utics and Space Adm inistrationWashington, DC 20546-0001.- ~15. Supplementary Notes

3. Recipients Catalog No .

5 Report DateApril 1989

-_ ~ _ _ _ __ - --6 Performing Organization Code

.____8. Performing Organization Report No.

10. Work Unit No.505-63-2 1-04

11 . Contract or Grant No

13. Type of Report and Period CoveredTechnical Mem orandum

14 . Sponsoring bgency Code

Presented at the AIAA 30th Structures, Structural Dynamics and M aterials Conferenc e in Mobile,Alabama, April 3-5, 1989.

16. Abstract

Various control analysis, design, and simulation techniques for aeroelastic applications require theequations of motion to be cast in a linear time-invariant state-space form. Unsteady aerodynamics forceshave to be approximated as rational functions of the Laplace variable in order to put them in thisframework. For the Minimum-State Method, the number of augm enting states representing the unsteadyaerodynamics is a function only of th e number of denominator roots in the rational approximation.Results are shown of applying various approximation enhancements (including optimization, frequencydepende nt weighting of the tabu lar data, and constraint selection) with the M inimum -State formulation tothe Active Flexible Wing wind-tunn el model. The results demonstrate that good mode ls can be developedwhich have an order of magnitude fewer augmenting aerodynamic equations than more traditionalapproaches. This reduction facilitates th e design of lower order control systems, analysis of controlsystem perform ance, and near real-time simulation of aeroservoelastic phenom ena.

i. Distribution StatementUnclassified - UnlimitedSubject Category - 05

page)

I IN A S A FORM 1626 OC T 86

21 No. of pages 22 Price,1