WORST-CASE FLUTTER MARGINSFROM F/A-18 AIRCRAFT AEROELASTIC DATA
Rick Lind1 Marty Brenner2
NASA - Dryden Flight Research Center
Abstract
An approach for computing worst-case flutter marginshas been formulated in a robust stability framework.Uncertainty operators are included with a linear modelto describe modeling errors and flight variations. Thestructured singular value, pt, computes a stability mar-gin which directly accounts for these uncertainties.This approach introduces a new method of computingflutter margins and an associated new parameter fordescribing these margins. The n margins are robustmargins which indicate worst-case stability estimateswith respect to the defined uncertainty. Worst-caseflutter margins are computed for the F/A-18 SRA us-ing uncertainty sets generated by flight data analysis.The robust margins demonstrate flight conditions forflutter may lie closer to the flight envelope than previ-ously estimated by p-k analysis.
Introduction
Aeroelastic flutter is a potentially destructive insta-bility resulting from an interaction between aerody-namic, inertia! and structural forces [4]. Design of anew aircraft, or even a configuration change of a cur-rent aircraft, requires study of the aeroelastic stabilitybefore a safe flight envelope can be determined. Theaeroelastic community has identified several areas ofresearch that are essential for developing an accurateflutter test program [6]. These areas focus on the dra-matic tune and cost associated with safely expandingthe flight envelope to ensure no aeroelastic instabilitiesaxe encountered.
An important research topic for aeroelasticity engi-neers is the development of more confident flutter orinstability margins. Experimental methods of deter-mining flutter usually consist of approximating modal
PostDoctoral Research Fellow, NASA, Dryden FlightResearch Center, MS 4840 D/RC, Edwards, CA 93523-0273,805.258.3075, lindQxrd.dfrc.nasa.gov, Member AIAA
damping from flight data [11]. These methods are un-reliable due to the often sudden onset of flutter whichmay not be accurately indicated by an approximatedamping value.
Several analytical methods are developed to determinethe conditions for aeroelastic instability. A traditionalmethod, known as the p-k method, utilizes a struc-tural model coupled with equations for the unsteadyaerodynamics [12]. This method is based on a finiteelement model of the aircraft and does not directlyconsider flight data from the physical aircraft. A pa-rameter estimation algorithm is developed that uti-lizes flight data to formulate elements of a state-spacemodel [19]. This method suffers from poor excitationand data measurements that may lead to inaccuratemodal parameters.
A novel approach to computing flutter instabilityboundaries has been developed that utilizes a theo-retical model while directly accounting for variationswith flight data [14]. The aeroelastic stability problemis formulated in a framework suitable for well devel-oped robust stability theory. Flight data is analyzed todescribe a set of uncertainty operators that account forvariations between the theoretical model and the phys-ical aircraft. A robust stability measure known as thestructured singular value, ̂ , is used to compute flutterboundaries that are robust to these variations [2]. Inthis sense, a worst-case flutter boundary is computedthat directly accounts for flight data.
This paper computes robust, or worst-case, fluttermargins for the F/A-18 Systems Research Aircraft,SRA, being flown at NASA Dryden Flight ResearchCenter. The SRA is a two-seat configuration fighterwith production engines. Recent flutter testing wasinitiated due to a structural modification to the leftwing. Internal fittings were replaced with larger andheavier ones to accommodate flight testing advancedaileron concepts. The flight data presented in this pa-per was generated using the new internal fittings butwith a standard aileron. A wingtip excitation systemfor generating aeroelastic flight data is shown in Fig-ure 1.
The flutter results in this paper represent a significantimprovement to accepted flutter results for the F/A-18 SRA computed using the traditional p-k method.Nominal flutter margins computed using the n methodbut ignoring all uncertainty operators are shown tomatch closely with the p-k method flutter margins.This result lends validity to the p, method as an accu-rate indicator of flutter instability. Directly account-ing for modeling uncertainty and flight data variationsin the p, based flutter analysis generates robust fluttermargins which are more conservative than the nominalmargins.
These robust flutter margins are generated with agreat deal more confidence than the nominal fluttermargins. Flight data from the actual aircraft is ana-lyzed to generate realistic uncertainty operators thatensure the family of plant models covers the true air-craft dynamics. Robust stability theory guaranteesthe robust flutter margins are worst-case margins withrespect to the indicated amount of modeling uncer-tainty. This procedure may greatly reduce the timeand cost associated with experimental flight envelopetesting since the instability limits may be more ac-curately and confidently identified. Additionally, theuncertainty levels in the theoretical model may be de-termined using flight data from a safe flight conditionwithout requiring the aircraft to approach a flutter in-stability point.
Robust Stability and ^
Any aeroelastic model is an approximate represen-tation of the aircraft dynamics. Inaccuracies in themodel, such as errors in coefficients and unmodeleddynamics, must be considered in the stability analysisand control synthesis procedures. Uncertainty opera-tors are included in the system model to account forthese inaccuracies in the robust stability framework.
Define x € Rn< as the vector of states, z e Rn" as thevector of uncertainty outputs, e € R"' as the vectorof errors, w 6 Rn" as the vector of uncertainty inputsand d € Rnj as the vector of disturbances. The state-space description of a linear time-invariant plant canbe represented as
AC\ \\
£22
xwd
where A € R"-xn',Bi <E R"'xn-,B2 eRn'xn'1,C'i €Rn"xn',C2 € Rn"xn- , and the E matrices of appro-priate dimensions.
Define P(s) as the Laplace transform of this sys-tem. The system with plant and uncertainty operatorsis represented as a Linear Fractional Transformation(LFT) of plant, P, and uncertainty operator, A, inFigure 2.
D AM w
Figure 2: Robust Stability Framework
The uncertainty operator is allowed to lie within anorm bounded set. This leads to the consideration ofa family of plant models. Weighting matrices are usu-ally included to restrict the uncertainty norm boundto unity.
A = {A : HAI loo < 1}Robust stability considers stability of the system overthe entire range of uncertainty. The issue of ro-bust stability for LFT systems is associated with well-posedness to guarantee that all internal signals are fi-nite and bounded. The small gain theorem is used todefine robust stability for LFT systems [2, 18].
Complex systems can have several types of uncertaintyoperators. Treating these types separately leads tostructured uncertainty. It is well known robustnessmeasured using the small gain theorem can be overlyconservative for systems with structured uncertainty.
Define the structured singular value, p.._ _________ 1___________~ min {a(A) : A e A, det(J - PA) = 0}
/x is an exact measure of robustness for systems withstructured uncertainty. The inverse of p can be inter-preted as a measure of the smallest destabilizing per-turbation. The system is guaranteed to be robustlystable for all uncertainty operators bounded by thesmallest destabilizing value.
Theorem 0.1 Given stable operator P, the system inFigure 2 is well-posed and stable for all A € A with
< I if and only if n(P) < I.
Unfortunately, p, is difficult to compute. Upper andlower bounds for n have been derived which utilizetwo sets of structured scaling matrices [7]. These scal-ing matrices are similar in structure to the uncertaintyblock structure and commute with the uncertainty ele-ments. An upper bound can be written as a linear ma-trix inequality (LMI) by considering a maximum eigen-value value condition utilizing the structured scalingmatrices [2].
A worst-case method of computing flutter margins uti-lizes /^-analysis for evaluating system stability. A lin-ear system is formulated with associated uncertaintyoperators.
Consider the generalized equation of motion for thestructural response of the aircraft [10].
Mr/ + Cfi + KT] + qQ(s)r) = 0For a system with n modes, define M e Rnx" as themass matrix, C € Rnxn as the damping matrix andK € RnXn as the stiffness matrix, q € R is a scalarrepresenting the dynamic pressure and Q(s) € cnxn
is the matrix of unsteady aerodynamic forces.
The unsteady aerodynamic forces are fit to a standardfinite-dimensional state-space system. This form canbe shown to encompass the traditional forms of Rogerand Karpel that include lag terms for the transientaerodynamics [14].
Given the number of generalized states, n, andaerodynamic states, UQ, define AQ € R""""",BQ € Rn«xn, CQ € R"X"Q and DQ € Rnxn
as state-space elements approximating Q(s).
The method should compute a /n value which relatesan unstable flight conditions. This is accomplishedby introducing an uncertainty operator to consider arange of flight conditions. Dynamic pressure is treatedas an unknown quantity for worst-case flutter analysis.
Consider an additive perturbation, 5g € R, on thenominal dynamic pressure, qnom.
Q - Qnom + <%
Two signals, z and w, are introduced into the formu-lation to represent uncertainty input and output. Theuncertainty output is formulated from system states.
z = Mw is related to z by the dynamic pressure perturbation.
W = ffqZ
The state-space aeroelastic model is formulated withthe additional signals to account for the parameteriza-tion of the dynamic pressure uncertainty. Formulatethe plant, P(s), using state vector [77; 7752;] such thatz = P(s)w. Define M = -M~l.
P =
0M(K + qnomDQ)
BQ-MDQ
IMC
00
0qnomMCQ
AQ-MCQ
0 '-I00
The input to P(s) is the uncertainty input, w, and theuncertainty output, z, is the output of P(s). Definingadditional signals for errors and disturbances allowsP(s) to be formulated in the robust stability frame-work of Figure 2 with 5$ as the uncertainty operator.
Additional uncertainty operators are included to ac-count for modeling errors between the theoretical sys-tem and the physical aircraft. They also allow theanalysis to consider a range of aircraft dynamics thatmay change due to variations in parameters such asmass or variations in the aerodynamics such as smalldeflections in the aircraft surfaces.
Errors in elements of the state-space matrices are of-ten represented by parametric uncertainty [3]. Thisuncertainty may be a real scalar parameter to reflectvariation in physical parameters such as mass and dy-namic pressure or real values such as modal frequencyand damping.
Unmodeled dynamics and nonlinearities are often ac-counted for by including a complex uncertainty. Thecomplex operator allows uncertainty to enter simulta-neously in magnitude and phase of the signals. Thisdynamic uncertainty may be a scalar or a matrix re-flecting unstructured uncertainty for a set of signals.
Experimental flight data can be used to generate un-certainty weightings. Transfer functions of the analyt-ical model can be compared with experimental flightdata transfer functions. Different size perturbationsare allowed to affect specific system parameters to thedegree that the resulting transfer functions cover therange of experimental flight data.
Model validation algorithms are used to verify that theamount of uncertainty in the linear model is sufficientto generate the flight data sets. This paper uses an al-gorithm based on /^-analysis of the linear system withfrequency domain flight data [14, 13]. The model vali-dation condition is derived as a standard n calculation.The y. value at each frequency relates the required sizeof perturbations at that frequency. This informationis used to compute frequency varying weightings toscale the uncertainty set. The model validation proce-dure is repeated until a small amount of uncertainty isdefined that still validates the model but reduces theconservatism in the resulting flutter analysis.
Robust flutter margins are computed using /i-analysison the linear system with the uncertainty operators.The flutter margin is found as the smallest destabiliz-ing perturbation for the dynamic pressure uncertainty,Sq, for the linear system with the given amount of mod-eling uncertainty. This margin is the worst-case fluttercondition for the allowed range of aircraft dynamics.
Worst-Case Flutter Parameter F/A-18 Aeroelastic Data
The flutter computation method described in this pa-per uses [j, as the worst-case flutter parameter. Thereare several advantages to using p. as the flutter param-eter, fj. is a much more informative flutter margin ascompared to traditional parameters such as pole loca-tion and modal damping.
The conservatism introduced by considering the worst-case uncertainty perturbation can be interpreted asa measure of sensitivity. Robust p, values which aresignificantly different than the nominal flutter marginsindicate the plant is highly sensitive to modeling errorsand changes in flight condition. A small perturbationto the system can drastically alter the flutter stabilityproperties. Conversely, similarity between the robustand nominal flutter margins indicates the aircraft isnot highly sensitive to small perturbations.
Robustness analysis determines not only the norm ofthe smallest destabilizing perturbation but also the di-rection. This information relates exact perturbationsfor which the system is particularly sensitive, p, canthus indicate the worst-case flutter mechanism whichmay naturally extend to indicate active and passivecontrol strategies for flutter suppression.
Damping is only truly informative at the point of insta-bility since stable damping at a given flight conditiondoes not necessarily indicate an increase in dynamicpressure will be a stable flight condition, p computesthe smallest destabilizing perturbation which indicatesthe nearest flight conditions that will cause a flutterinstability. In this respect, fj. is a stability predictorwhile damping is merely a stability indicator.
These characteristics of p make the worst-case flutteralgorithm especially valuable for flight test programs.Aeroelastic flight data can be measured at a stableflight condition and used to evaluate uncertainty op-erators. The (JL method, unlike damping estimation,does not require the aircraft to approach instability foraccurate prediction, p can be computed to update thestability margins with respect to the new uncertaintylevels. The worst-case stability margin then indicateswhat flight conditions may be safely considered.
Safe and efficient expansion of the flight envelope canbe performed using an on-line implementation of theworst-case stability estimation algorithm. Comput-ing n does not introduce an excessive computationalburden since each F/A-18 flutter margin presented inthis paper was derived in less than 2 minutes usingstandard off-the-shelf hardware and software packages.On-line algorithms are currently being developed todemonstrate this procedure for a flight test [17].
Extensive flight data from the F/A-18 SRA is used togenerate uncertainty descriptions for an analytical air-craft model [16]. Over 30 flights were conducted in twosessions between September 1994 and February 1995and between June 1995 and July 1995 at Dryden FlightResearch Center. Each flight performed maneuvers fordifferent conditions throughout the flight envelope. Atotal of 260 different data sets are generated from var-ious conditions throughout the flight envelope [5].
The aeroelastic flight data is generated using an ex-ternal structural excitation system developed by Dy-namic Engineering Incorporated (DEI). This DEI ex-citer is a modification of an excitation system usedfor F-16 XL flutter research [20]. The system consistsof a wingtip exciter, an avionics box mounted in theinstrumentation bay, and a cockpit controller.
Aerodynamic forces are generated by the wingtip ex-citer. This exciter consists of a small fixed aerody-namic vane forward of a rotating slotted hollow cylin-der. Rotating the cylinder varies the pressure distribu-tion on the vane and results in a wingtip force changingat twice the cylinder rotation frequency. The magni-tude of the resulting force is determined by the amountof opening in the slot. The F/A-18 aircraft with a leftside wingtip exciter is shown hi Figure 1.
The cockpit controller commands a frequency range,duration and magnitude for the wingtip excitation sig-nal. Frequency varying excitation is generated bychanging the cylinder rotation frequency with sinesweeps. Each wingtip exciter is allowed to act in-phase, 0 degrees, or out-of-phase, 180 degrees, witheach other. Ideally, the in-phase data excites the sym-metric modes of the aircraft and the out-of-phase dataexcites the anti-symmetric modes.
Flight data sets are recorded by activating the excitersystem at a given flight condition. The aircraft at-tempts to remain at the flight condition throughoutthe series of sine sweeps desired by the controller. Thesine sweeps were restricted within 3 Hz and 35 Hz.Smaller ranges were sometimes used to concentrate ona specific set of mode responses. Multiple sets of eitherlinear or logarithmic sweeps were used with the sweepfrequency increasing or decreasing.
Aeroelastic flight data generated with the DEI excitersystem is analyzed by generating transfer functionsfrom the excitation force to the sensor measurements.These transfer functions are generated using standardFourier transform algorithms. There are several inher-ent assumptions associated with Fourier analysis thatare violated with the flight data. The assumptions
of time-invariant stationary data composed of sums ofinfinite sinusoids is not met by this transient responsedata. The analysis presented in this paper is basedon Fourier analysis, although current research investi-gates wavelet techniques to analyze the flight data [5].
The excitation force is not directly measured butrather a strain gauge measurement is used to approx-imate this force. The strain gauge records a pointresponse at the exciter vane root. This point responseis considered representative of the distributed excita-tion force load over the entire wing surface. Vane rootstrain is assumed to be directly proportional to thevane airloads due to excitation [5].
Analysis of the recorded flight data indicates the DEIexciters did not operate entirely as expected. Theexciters displayed erratic behavior at higher dynamicpressures due to binding hi both the motor drive mech-anism and rotating cylinders. At low dynamic pres-sures the system operated better but still displayssome phase drift between the left and right cylinderrotations.
Further erratic behavior is demonstrated by compar-ing measurement signals due to excitation sine sweepsof increasing and decreasing frequency. Transfer func-tions from a symmetric excitation to the wingtipaccelerometers clearly show different modes are ex-cited by the direction of the sweep even though theflight conditions are identical and the data sets wererecorded 30 seconds apart of each other [16].
F/A-18 Nominal Model
The generalized equations of motion are used to derivea linear, finite-dimensional state-space model of theaircraft. This model contains 14 symmetric structuralmodes, 14 antisymmetric structural modes and 6 rigidbody dynamic modes. The control surfaces are notactive and no control modes are included in the model.
A finite element model of the SRA is used to computethe modal characteristics of the aircraft. Frequenciesof the dominant modes for flutter are presented hi Ta-ble 1. These modal frequencies are computed for theaircraft with no aerodynamics considered. The pre-dicted flutter results for this aircraft are computedfrom the finite element model using the p-k method. Adetailed explanation of the SRA flutter analysis usingtraditional methods is given in Reference [21].
Values of the unsteady aerodynamic force matrix atdistinct frequencies are computed for the finite ele-ment model using a computer package developed forNASA known as STARS [9]. This code solves the sub-sonic aerodynamic equations using the doublet lattice
method [8]. The supersonic forces are generated us-ing a different approach known as the constant panelmethod [1].
The doublet lattice and constant panel methods areused to compute the frequency varying unsteady aero-dynamic forces for several subsonic, transonic and su-personic Mach numbers. The Mach numbers, M =.8, .9, .95, 1.1, 1.2, 1.4, 1.6, are available. The unsteadyaerodynamic forces are computed as a function of re-duced frequency, k.
The reduced frequency is a function of the true fre-quency, a;, the true velocity, V, and c the mean aero-dynamic chord. Aerodynamic forces generated for 10reduced frequency points between k = .0001 and k = 4are sufficient for flutter margin computation for thisaircraft.
The unsteady aerodynamic forces are fit to a finite-dimensional state-space system. The system identifi-cation algorithm is a frequency domain curve fittingalgorithm based on a least squares minimization. Aseparate system is identified for each column of theunsteady forces transfer function matrix. 4th orderstate-space systems are used for each column of thesymmetric forces and 2nd order state-space systemsare used for each column of the antisymmetric forces.These systems are combined to form a single multiple-input and multiple-output state-space model of theunsteady aerodynamics forces, previously designatedQ(s), with 56 states for the symmetric modes and 28states for the antisymmetric modes.
The analytical aeroelastic model has inputs for sym-metric and antisymmetric excitation forces. It is as-sumed the excitation force will be purely symmetric orantisymmetric. There are 6 sensor measurements gen-erated by accelerometers at the fore and aft of eachwingtip and on each aileron.
Noise and uncertainty operators are introduced to thelinear aeroelastic model to account for variations be-tween the analytical model and the actual aircraft.These operators are developed by physical reasoningof the modeling process and also using the flight datagenerated by the DEI excitation system [16].
Standard analysis of the linear model is used to deter-mine the framework for how uncertainty operators en-ter the system. Two uncertainty operators and a singlenoise input are used to describe the modeling uncer-tainty in the linear aeroelastic model. The magnitudeof each uncertainty operator and the noise level is de-termined both from physical reasoning of the modeland analysis of the flight data.
An uncertainty operator, dmode, is introduced to themodal elements of the state-space F/A-18 model. Thisparametric uncertainty allows variations in both thenatural frequency and damping values for each mode.This uncertainty covers errors in the coefficients of theequations of motion and the corresponding state-spaceelements of the linear model. An example of such anerror arises in considering the mass of the aircraft. Thelinear model uses a single mass value while in realitythe mass varies considerably due to fuel consumption.Mass variations for a simple second order system af-fect the natural frequency, w = ^/fc/m, and may berepresented as parametric modal uncertainty. Thismodal uncertainty allows a worst-case flutter point tobe computed that accounts for parametric changes inthe aircraft such as those due to mass variations.
The second uncertainty operator, Ain, is a multiplica-tive uncertainty on the force input to the linear model.This uncertainty is used to cover nonlinearities andunmodeled dynamics. The linear model contains nodynamics above 40 Hz so the high frequency compo-nent of this operator will reflect this uncertainty. Thisoperator is also used to model the excitation uncer-tainty due to the DEI exciter system. Analysis of theflight data indicates the input excitation signals rarelyhad the desired magnitude and phase characteristicsthat they were designed to achieve. The low frequencycomponent of the input uncertainty reflects the uncer-tainty associated with the excitation system used togenerate the flight data.
A noise signal is included with the sensor measure-ments. Knowledge of the aircraft sensors is used todetermine a level of 10% noise is possible in the mea-sured flight data. An additional noise may be includedon the force input due to the excitation system but itis decided the input multiplicative uncertainty is suf-ficient to describe this noise.
The magnitude of the parametric modal uncertainty,<5modej is determined from flight data analysis. Thelinear model contains 14 modes for the symmetric re-sponse and 14 modes for the antisymmetric responseof the aircraft. Unfortunately, the flight data does notindicate each of these is sufficiently excited to allowanalysis and comparison with the theoretical model.Only the modes given in Table 1 are observed hi thedata. A linear model is formulated from a subset ofthe full model which contains only the experimentallyobserved modes. The modal parameters of this re-duced order model are compared with the flight dataand uncertainty levels are determined.
Scalar uncertainty parameters, 6, are used to affectthe modal parameters. The state matrix of the linearmodel is formulated as a block diagonal matrix with a2x2 block for each mode. The diagonal componentof each block is the real part of the natural frequencyand the off-diagonal elements are the imaginary partssuch that the natural frequency, w;, and the damping,£i, of the ith mode may be determined.
Ai =
Scalar weightings, wr and Wi, are used to affect theamount of uncertainty in each matrix element. Theamount of variation in the matrix elements, and cor-respondingly the amount of variation in the naturalfrequency and damping, are determined by the mag-nitude of these scalar weightings. Define r and t asthe varying elements of the state matrix affected bythe uncertainty 6.
7 = r(l ± wr6)t = i(l ± wt8)
Aeroelastic modes typically show low damping val-ues caused by the real component being quite smallas compared to the imaginary component. Since lin-ear modeling techniques often identify the natural fre-quency better than the damping value, the weightingfor the real component should be larger than that forthe imaginary component.
The weightings are chosen using the observed modalparameters in the flight data. The natural frequenciesshow variations of ±5% from the theoretical modelwhile the uncertainty in the damping needs approxi-mately ±15% to validate all the flight data. The scalarweightings are chosen accordingly.
wr = .15Wi = .05
The flight data is only able to determine uncertaintylevels for the modal paramters of the experimentallyobserved modes. It is assumed the uncertainty lev-els in the unobserved modes should be consistent with
these values. Parametric uncertainty is introduced foreach modal block in the state matrix, affecting ob-served and unobserved modes, with the weighting val-ues given above.
The block diagonal state matrix also contains somereal valued scalar blocks. These scalar blocks appearas approximations to lag terms in the state-space iden-tification of the unsteady aerodynamic forces. Theidentified system with these lag approximations doesnot accurately model the aerodynamic forces at all fre-quencies. Parametric uncertainty affects each of theselag terms with a weighting of wiag = .15 that allows15% variation.
The low frequency magnitude of the input multiplica-tive uncertainty is determined from the flight data.Levels of uncertainty are chosen that validate the flightdata for a given amount of noise and parametric modaluncertainty. The high frequency component of inputuncertainty is determined to reflect the unknown dy-namics at high frequency for the linear model. Thefrequency varying transfer function for weighting theinput uncertainty is given as Win.
e s + 100Win = 5 JT5000
The block diagram for the aeroelastic model with theuncertainty operators is given hi Figure 3.
Figure 3: F/A-18 Uncertainty Block Diagram
Flight data used to validate this uncertainty structurecovers a large range of flight points from the entire setof 260 flight maneuvers throughout the flight envelope.
Using a single uncertainty description over the entireflight envelope may be conservative. It is reasonableto assume the linear models are more accurate at sub-sonic and supersonic than at transonic. Additionally,the flight data from the DEI exciter system should bebetter at subsonic speeds than at supersonic. How-ever, it simplifies the analysis process to consider asingle set of uncertainty operators. This process isequivalent to formulating the worst-case uncertaintylevels at the worst-case flight condition and assumingthat amount of uncertainty is possible for the remain-ing flight conditions.
F/A-18 Flutter Points
Flutter margins are computed for a linear model withthe associated modeling uncertainty structure usingthe /i-analysis method [15]. Linear systems for sym-metric and antisymmetric structural modes are sepa-rated for ease of analysis. These systems can easily becombined and analyzed as a single system; however,eigenvector analysis would be required to distinguishwhich critical flutter modes are symmetric and whichare antisymmetric. Each system contains the samenumber of structural modes, 14, and the uncertaintydescriptions are identical for each linear model.
The system given in Figure 3 contains three uncer-tainty blocks. The parametric uncertainty coveringvariations due to dynamic pressure, <%, is a scalar pa-rameter repeated 14 times, once for each elastic mode.The parametric uncertainty block affecting the modalparameters, <5mode», is a diagonal matrix with dimen-sion equal to the number of states. Separate scalarsalong the diagonal represent uncertainty in each elasticmode, each mode in the aerodynamic force approxima-tion, and each lag term. The uncertainty parametersfor the modes are repeated two times while the pa-rameters for the lag terms are single scalars. Define <5;as the ith uncertainty parameter for the system withnm modes and n/ lag terms. The input multiplica-tive uncertainty block, Ajn, is a scalar for this singleinput plant model since we are analyzing symmetricexcitation separately from antisymmetric excitation.
The parametric uncertainty parameters representchanges in elements of the state-space model. Thevariation of 6^ between ±1 admits dynamic pressuresbetween 0 < ~q < 25rlom. Allowing the modal uncer-tainty parameters, <$!,...,<Jnm to vary between ± 1allows 5% variation in the imaginary part of the nat-ural frequency and 15% hi the real part. This corre-sponds to approximately 5% variation in the naturalfrequency and 15% in the damping value of each mode.These parameters are real quantities. The multiplica-tive input uncertainty contains magnitude and phaseinformation and is treated as a complex linear time-invariant uncertainty.
Nominal flutter boundaries are initially computed byignoring the modal and input uncertainties, p. is com-puted only with respect to the parametric uncertaintyallowing a range of dynamic pressures to be consid-ered. Robust flutter boundaries are computed withrespect to the structured uncertainty set, A, describedabove using the structured singular value. Traditionalflutter boundaries computed using the p-k method arepresented with the nominal and robust flutter bound-aries computed with /n in Table 2
Table 2: Nominal and Robust Flutter PointsThe nominal flutter dynamic pressures computed us-ing the fj, method can be directly compared with thosecomputed using the traditional p-k method [21]. Eachof these flutter solutions are based on an analyticalmodel with no consideration of modeling uncertainty.
The nominal flutter points for the symmetric modesmatch closely with the p-k method throughout theflight envelope. The subsonic and supersonic casesshow an especially good correlation with the p-k flutterpoints. For each of these flight regions, the /i-analysisflutter dynamic pressures are nearly identical, within1%, to the p-k method flutter dynamic pressures. Thetransonic case at M = 1.1, however, shows a slightdifference between the two methods. The p, methodcomputes a flutter point that is greater than the p-kmethod. In each Mach regime; subsonic,supersonic ortransonic, the nominal flutter points are within 5% forthe two methods.
The antisymmetric modes show a similar relationshipbetween the flutter margins computed with the p. andp-k methods. The subsonic and supersonic flutterpoints are within 5% for the two methods, but there isa greater deviation at the transonic condition, p. com-putes a flutter margin at M = 1.1 that is 40% lowerthan the p-k method indicates.
The nominal flutter points for the p. and p-k methodsshow the greatest difference for both the symmetricand antisymmetric modes at the transonic case. Theaerodynamics at M = 1.1 are more difficult to modelaccurately than at either subsonic or supersonic. Nu-merical sensitivity to representations of the unsteadyaerodynamic foces causes differences in the nominalflutter margins.
The robust flutter margins computed using the p.method have lower dynamic pressures than the nomi-nal margin, which indicates the expected conservativenature of the robust computation. These new flutterpoints are worst-case values for the entire range of al-lowed uncertainty. The subsonic and supersonic flutterboundaries are not greatly affected by the uncertaintyset. In each of these cases, the robust flutter point iswithin 10% of the nominal flutter point.
The flutter boundary at the transonic case, M = 1.1,demonstrates significant sensitivity to the modelinguncertainty. The robust flutter dynamic pressures areapproximately 70% of the nominal flutter margins.This is explained by considering the rapid transition ofcritical flutter boundaries near this region. The criti-cal flutter frequencies and the flutter dynamic pressurewidely vary between Mach numbers slightly lower andhigher than transonic. The small amount of modelinguncertainty is enough to cause the worst-case fluttermechanism to shift and the plant assumes character-istics more consistent with a non-transonic regime.
The modal natural frequencies for the critical fluttermodes are presented in Table 3. The frequencies com-puted using the p-k method and the /i-analysis methodare close throughout the flight envelope for both thesymmetric and antisymmetric modes. Frequencies forthe robust flutter solutions are slightly different thanthe nominal flutter frequencies due to the modelinguncertainty which allowed 5% variation hi the modalnatural frequencies.
Mach
.8
.9.951.11.21.41.6
symmetricUp-k8.27.46.812.126.528.128.9
Wnotn
7.67.36.913.227.428.130.1
Wro67.77.36.913.027.428.130.1
antisymmetricWp-Jfc9.09.29.128.626.930.432.8
Wnom9.1
9.19.228.028.931.732.3
Urob9.19.29.228.328.931.832.1
Table 3: Nominal and Robust Flutter Frequencies
Subcritical flutter margins computed with the p. andp-k methods are presented in Table 4. Only nominalsubcritical margins are detected with p. since the ro-bust margins are always worst-case critical margins.
Mach
.9
.951.11.21.41.6
symmetric9D-fc
7450
540089708400
^nominal
6919
500389598843
antisymmetric9D-fc47005300
60508400
^nominal45835093
60017943
Table 4: Nominal and Robust Flutter Points - Subcritical/^-analysis computes subcritical flutter margins within10% of the p-k method for both the symmetric andantisymmetric modes. The p. method is even able todetect the subcritical flutter hump mode occuring forantisymmetric excitation at 0.9 Mach number.
The dynamic pressures at which flutter occurs are con-verted into altitudes, commonly known as matched-point solutions, using standard atmospheric equations.These altitudes are plotted for the symmetric modes inFigure 4 and for the antisymmetric modes in Figure 5.The flight envelope of the F/A-18 is shown on theseplots along with the required 15% flutter boundary formilitary aircraft.
20000
SOLID : nominal flutter using pkCIRCLE : nominal flutter using muDASHED: robot! Hotter using mu
0 0.2 0.4 OS O.S 1 1.2 1.4 1.6 1.8Mach
Figure 4: Nominal and Robust Flutter Points - MatchedPoint Solutions for Symmetric Modes
SOLID: nominal flutter using pk 'CIRCLE : nominal gutter using mui
DASHED: robust flutter using mu
0 0.2 0.4 0.6 0.8 1 1.2 1.4 16 1.8Mach
Figure 5: Nominal and Robust Flutter Points - MatchedPoint Solutions for AntiSymmetric Modes
Figures 4 and 5 use several short solid lines to indicatethe p-k flutter solutions throughout the flight regime.Each of these short solid lines represents the flutterpouits due to a specific mode. Flutter points for thesymmetric modes given in Figure 4 show four solidlines indicating three different critical flutter modesfor the considered range of Mach numbers along witha subcritical flutter mode occuring at supersonic Mach
numbers. The antisymmetric modes show the onset offlutter from three different critical modes along withthree subcritical flutter modes throughout the flightenvelope in Figure 5. The frequencies of the criticalflutter modes can be found in Table 3.
The subsonic flutter altitudes for symmetric and anti-symmetric modes demonstrate a similar characteristic.The nominal flutter boundary shows a significant vari-ation from Mach number M = .8 to M — .95 causedby sensitivity to Mach number for the dynamics associ-ated with the critical flutter mode. The robust flutterboundary indicates the sensitivity of the plant to er-rors and the worst-case perturbation. The higher alti-tude for the nominal flutter boundary at Mach numberM = .81 than for Mach number M = .80 is reflectedin the large conservatism associated with the robustflutter boundary. Similarly, slight variation of Machnumber near M = .95 is not expected to increase thenominal flutter boundary so there is less conservatismassociated with the robust flutter boundary.
An interesting trend is noticeable for the symmetricmode robust flutter points in Figure 4 at the super-sonic Mach numbers. The flutter mechanism resultsfrom the same modes from M = 1.2 to M = 1.6 withsome increase in frequency. Similarly the altitudes ofthe nominal flutter margins show little change for theseMach numbers. The aeroelastic dynamics associatedwith the critical flutter mode are relatively unaffectedby the variation of Mach over this range and conse-quently each flutter boundary has the same sensitivityto modeling errors.
The robust flutter margins for the antisymmetricmodes at supersonic Mach numbers show a slightlydifferent behavior than the symmetric mode fluttermargins. The flutter mechanism is again caused bya single mode from M = 1.2 to M = 1.6 with similarfrequency variation as symmetric. The robust fluttermargins demonstrate a similar sensitivity to modelingerrors at M = 1.2 and M = 1.4 but at M = 1.6 agreater sensitivity is shown. The greater conservatismat M = 1.6 may indicate impending transition in flut-ter mechanism from the subcritical mode at slightlyhigher Mach number.
The dark solid line on Figures 4 and 5 represents therequired boundary for flutter points. All nominal androbust flutter points lie outside this region indicatingthe flight envelope should be safe from flutter instabil-ities. The robust flutter boundaries computed with nindicate there is more danger of encountering flutterthan was previously estimated with the p-k method.In particular, the robust flutter margin for symmetricexcitation at Mach M = 1.1 lies considerably closer tothe boundary than the p-k method indicates.
The n analysis method of computing flutter marginspresents significant analytical advantages due to therobustness of the resulting flutter margin, but it alsohas several computational advantages over the p-kmethod. The /i algorithm requires a single linearaeroelastic plant model at a given Mach number tocompute critical and subcritical flutter margins. Anentire set of flutter margins may be easily generatedusing a standard engineering workstation in a few min-utes using widely available software packages [2].
The p-k method is an iterative procedure that requiresfinding a matched-point solution [21]. The aircraft isanalyzed at a particular Mach number and air density.The airspeed for these conditions resulting in a flutterinstability is computed. This airspeed, however, oftendoes not correspond to the unique airspeed determinedby that Mach number and air density for a standardatmosphere. Various air densities axe used to computeflutter solutions and the corresponding air speeds areplotted. An example of an air speed plot for flutter isgiven in Figure 6.
Figure 6: Antisymmetric P-K Flutter Solutions for MachM=1.4
The vertical lines in Figure 6 represent two antisym-metric modes that may flutter at Mach M=1.4. Thep-k method computes a flutter solution at the airspeedindicated where the modal line crosses the standardatmosphere curve. This flutter solution is difficult tocompute from only a few air density computations.Typically several air densities are used to compute airspeed flutter solutions and a line is extrapolated be-tween the points to determine the matched-point solu-tion at the standard atmosphere crossing point. Thenonlinear behavior shown for mode 1 near the standardatmosphere crossing point indicates an accurate flutterboundary would be extremely hard to predict unless
many solutions are computed near the true matched-poult solution.
The p-k method also may have difficulty predictingthe subcritical flutter margins. The second mode hiFigure 6 may or may not intersect the standard atmo-sphere curve. More computational analysis is requiredto determine the behavior of this mode at higher air-speeds. The /^-analysis method accurately detectsboth the critical and subcritical flutter margins with-out requiring expensive iterations.
Conclusion
Nominal and robust flutter margins are computed forthe F/A-18 SRA aircraft. Nominal flutter margins arecomputed using a /n-analysis method and a traditionalp-k method. The similarity of these flutter marginsdemonstrates the /i-analysis method is a valid tool forcomputing flutter instability points and is computa-tionally advantageous. Extensive flight data is ana-lyzed to develop a set of uncertainty operators for alinear model. Robust flutter margins are computedusing (i. The resulting flutter margins are worst-casevalues with respect to the modeling uncertainty. Thesemargins are accepted with a great deal more confidencethan previous flutter estimates by directly accountingfor modeling uncertainty hi the analysis process. Therobust flutter margins indicate the desired flight en-velope should be safe from aeroelastic flutter instabil-ities; however, the flutter margins may lie noticeablycloser to the flight envelope than previously estimated.
This method replaces damping as a measure of ten-dency to instability from available flight data. Sincestability norms generally behave smoothly at instabil-ity boundaries, this method is recommended for pre-flight predictions and post-flight analysis with a min-imum amount of flight tune. Additionally, the robustflutter stability framework extends naturally to robustflutter control synthesis for aeroelastic control.
Acknowledgments
The authors wish to acknowledge the financial sup-port of the Controls and Dynamics Branch of NASAat the Dryden Flight Research Center. The structuraldynamics group, Larry Freudinger, Leonard Voelkerand Dave Voracek, provided helpful comments andsuggestions throughout this project. Analysis of thefinite element model was assisted by Tim Doyle andRoger Truax. Dr. Lind is supported through the Post-Doctoral FeUowship program of the National ResearchCouncil.
References[I] K. Appa, "Constant Pressure Panel Method forSupersonic Unsteady Airloads Analysis,", AIAA Jour-nal of Aircraft, Vol. 24, No. 10, October 1987, pp. 696-702.[2] G. Balas, J. Doyle, K. Glover, A. Packard andR. Smith, "/*-Analysis and Synthesis Toolbox - UsersGuide,", MUSYN Inc and The Math Works, Min-neapolis, MN, 1991.[3] G. Balas and P. Young, "Control Design for Vari-ations hi Structural Natural Frequencies," AIAA Jour-nal of Guidance, Control and Dynamics, Vol. 18, No.2,March 1995, pp. 325-332.[4] R. Bisplinghoff, A. Holt, and R. Halfman, Aeroe-lasticity, Addison Wesley Publishing, 1955.[5] M. Brenner, R. Lind and D. Voracek, "Excit-ing Flutter Research with an F/A-18 Aircraft," 1997AIAA Structures, Structural Dynamics and MaterialsConference, Orlando FL, April 1997, Paper 97-1023.[6] J. Cooper and T. Noll, "Technical EvaluationReport on the 1995 Specialists Meeting on AdvancedAeroservoelastic Testing and Data Analysis," Pro-ceedings of the S0th AGARD Structures and Materi-als Panel, AGARD-CP-566, Rotterdam, The Nether-lands, May 8-10 1995, pp. T1-T10.[7] M. Fan, A. Tits and J. Doyle, "Robustness in thePresence of Mixed Parametric Uncertainty and Un-modeled Dynamics," IEEE Trans. on Auto. Control,Vol. 36, January 1991, pp. 25-38.[8] J. Giesing, T. Kalman and W. Rodden, W.,Subsonic Unsteady Aerodynamic for General LatticeMethod, Part I - Vol I - Direct Application of theNonplanar Doublet Lattice Method, AFFDL-TF-71-5,November 1971.[9] K. Gupta, STARS - An Integrated GeneralPurpose Finite Element Structural, Aeroelastic, andAeroservoelastic Analysis Computer Program, NASATM-101709,1990.[10] K. Gupta, M. Brenner and L. Voelker, Develop-ment of an Integrated Aeroservoelastic Analysis Pro-gram and Correlation with Test Data, NASA TechnicalPaper TP-3120, May 1991.[II] M. Kehoe, "A Historical Overview of FlightFlutter Testing," Proceedings of the 80th AGARDStructures and Materials Panel, AGARD-CP-566,Rotterdam, The Netherlands, May 8-10 1995, pp. 1:1-1:15.[12] H. Hassig, "An Approximate True Damping So-lution of the Flutter Equation by Determinant It-eration," AIAA Journal of Aircraft, Vol. 8, No. 11,November 1971, pp. 885-889.[13] A. Kumar and G. Balas, "An Approach to ModelValidation in the n Framework," Proceedings of the
1994 American Controls Conference, Baltimore MD,June 1994, pp. 3021-3026.[14] R, Lind and M. Brenner, "Robust Stability Es-timation of Aeroelastic Systems using Flight DerivedUncertainty Models," submitted to AIAA Journal ofGuidance, Control and Dynamics, June 1996.[15] R. Lind and M. Brenner, "Robust Flutter Mar-gins of an F/A-18 Aircraft from Aeroelastic FlightData," submitted to AIAA Journal of Guidance, Con-trol and Dynamics, July 1996.[16] R. Lind and M. Brenner, "Incorporating FlightData into a Robust Aeroelastic Model," submitted toAIAA Journal of Aircraft, September 1996.[17] R. Lind and M. Brener, "A Worst-Case Ap-proach for On-Line Flutter Prediction," to appear in1997 CEAS International Forum on Aeroelasticity andStructural Dynamics, Rome Italy, June 1997.[18] J. Maciejowski, Multivariable Feedback Design,Addison Wesley Publishers , England, 1989.[19] E. Nissim and G. Gilyard, Method of Experimen-tal Determination of Flutter Speed by Parameter Iden-tification, NASA Technical Paper 2923, 1989.[20] L. Vernon, In-flight Investigation of a RotatingCylinder-Based Structural Excitation System for Flut-ter Testing, NASA Technical Memorandum 4512, June1993.[21] L. Voelker, F-18/SRA Flutter Analysis Results,preprint of NASA Technical Memorandum, 1995.
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