Worst-Case Flutter Margins from F/A-18 Aircraft Aeroelastic Data

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WORST-CASE FLUTTER MARGINS

FROM F/A-18 AIRCRAFT AEROELASTIC DATA

Rick Lind I Marry Brenner = NASA - Dryden Flight Research Center Abstract damping from flight data [11]. These methods are un- reliable due to the often sudden onset of flutter which may not be accurately indicated by an approximate An approach for computing worst-case flutter margins damping value.

has been formulated in a robust stability framework.

Uncertainty operators are included with a linear model Severalanalytical methods are developed to determine the conditionsforaeroelastic instability. A traditional to describe modeling errors and flight variations. The method, known as the p-k method, utilizes a struc- structured singular value,/_, computes a stability mar- tural model coupled with equations for the unsteady gin which directly accounts for these uncertainties.

This approach introduces a new method of computing aerodynamics [12]. This method is based on a finite element model of the aircraftand does not directly fluttermargins and an associated new parameter for consider flight data from the physical aircraft. A pa- describing these margins. The p margins are robust rameter estimation algorithm is developed that uti- margins which indicate worst-case stability estimates lizes flight data to formulate elements of a state-space with respect to the defined uncertainty. Worst-case flutter margins are computed for the F/A-18 SRA us- model [19].This method suffers from poor excitation and data measurements that may lead to inaccurate ing uncertainty sets generated by flight data analysis.

The robust margins demonstrate flight conditions for modal parameters.

flutter may lie closer to the flight envelope than previ- A novel approach to computing flutter instability ously estimated by p-k analysis.

boundaries has been developed that utilizesa theo- retical model while directly accounting for variations Introduction with flight data [14].The aeroelastic stability problem is formulated in a framework suitable for well devel- oped robust stability theory. Flight data isanalyzed to Aeroelastic flutter is a potentially destructive insta- describe#a setofuncertainty operators that account for bility resulting from an interaction between aerody- variations between the theoreticalmodel and the phys- namic, inertial and structural forces [4]. Design of a icalaircraft. A robust stabilitymeasure known as the new aircraft, or even a configuration change of a cur- structured singularvalue,/_, isused to compute flutter rent aircraft, requires study of the aeroelastic stability boundaries that are robust to these variations [2]. In before a safe flight envelope can be determined. The thissense, a worst-case flutter boundary is computed aeroelastic community has identified several areas of that directlyaccounts for flight data.

research that are essential for developing an accurate flutter test program [6]. These areas focus on the dra- This paper computes robust, or worst-case, flutter matic time and cost associated with safely expanding margins for the FIA-18 Systems Research Aircraft, the flight envelope to ensure no aeroelastic instabilities SRA, being flown at NASA Dryden Flight Research " • are encountered.

Center. The SILk is a two-seat configuration fighter with production engines. Recent fluttertesting was An important research topic for aeroelasticity engi- initiateddue to a structural modification to the left neers is the development of more confident flutter or wing. Internalfittings were replaced with larger and instability margins. Experimental methods of deter- heavier ones to accommodate flighttesting advanced mining flutter usually consist of approximating modal aileron concepts. The flight data presented in thispa- per was generated using the new internal fittings but t NRC Po4tDoctoral Research Fellow, NASA, Dryden Flight with a standard aileron.A wingtip excitationsystem Research Center, MS 4840 D/RC, Edwards, CA 93523-0273, 805.258.3075, IlndOxrd.dfrc.natm-gov, Member AIAA for generating aeroelastic flight data is shown in Fig- aReaearc.h Engineer, gon_Oxrd.dfrc.n_a.gov ure I.

°AIAA Structures, Structural Dynamics and Materials Con- ference, A[AA-97-1266, Ori&ndo FL, April 1997 where .4 E R ..... , B_ E R"" xn', _'2 E R _" "'_d, Cl E R,_. xn., C2 E R a" " '_' , 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 operators is represented as a Linear Fractional Transformation Figure 1- F/A-18 Wing with DEI Exciter (LFT) of plant, P, and uncertainty operator, A, in Figure 2.

The flutter results in this paper represent a significant improvement to accepted flutter results for the F/A- 18 SRA computed using the traditional p-k method.

Nominal flutter margins computed using the # method but ignoring all uncertainty operators are shown to match closely with the p-k method flutter margins.

Figure 2: Robust Stability Framework This result lends validity to the p method as an accu- rate indicator of flutter instability. Directly account- The uncertainty operator is allowed to lie within a ing for modeling uncertainty and flight data variations norm bounded set. This leads to the consideration of in the _ based flutter analysis generates robust flutter a family of plant models. Weighting matrices are usu- margins which are more conservative than the nominal ally included to restrict the uncertainty norm bound margins.

to unity.

A = {a: IJalloo _< 1} These robust flutter margins are generated with a Robust stability considersstabilityof the system over great deal more confidence than the nominal flutter the entire range of uncertainty. The issue of.ro- margins. Flight data from the actual alrcraR is ana- bust stability for LFT systems is associated with wen- lyzed to generate realistic uncertainty operators that pc_edness to guarantee that all internal signals are FL ensure the family of plant models covers the true air- nite and bounded. The small gain theorem is uses1 to craft dynamics. Robust stabilitytheory guarantees define robust stability for LFT systems [2, 18]: the robust flutter margins are worst-casemargins with respect to the indicated amount of modeling uncer- Complex systems can have several types of uncertainty tainty. This procedare may greatly reduce the time operators. Treating these types separately leads to and cost associated with experimental flight envelope structured uncertainty. It is well known robustness testing since the instability limits may be more ac- measured using the small gain theorem can be ovei'ly curately and confidently identi_ed. Additionally, the conservative for systems with structured uncertainty.

uncertainty levels in the theoretical model may be de- termined using flight data from a safe flight condition Define the structured singular value,/_.

without requiring the aircraft to approach a flutter in- stability point.

9(P) = min{_(a) : AEA, det(I-PA)=0} Robust Stability" and p /_ is an exact measure of robustness for systems with structured uncertainty.The inverse of/_ can be inter- Any aeroelastic model is an approximate represen-- preted as a measure of the smallest destabilizing per- tation of the aircraft dynamics. Inaccuracies in the turbation. The system is guaranteed to be robustly model, such as errors in coe_cients and unmodeled stable for all uncertainty operators bounded by the dynamics, must be considered in the stability analysis smallest destabilizing value.

and control synthesis procedures. Uncertainty opera- Theorem 0.1 Giuen stable operator P, the system in . .

tots are included in the system model to account for Figure 2 ia well-posed and stable/or all A E A _#ith these inaccuraciesin the robust stability fzamework.

Ilalloo < z ff o, y il < 1.

De,he z E R"" as the vector of states, z E R "° as the Unfortunately, /J is di_icult to compute. Upper and vector of uncertainty outputs, e E R '_" as the vec_r lower bounds for /J have been derived which utilize of errors, w E R '_ as the vector of uncertaintyinputs two setsof structuredscaling matrices [7].These scal- and d E R "d as the vector of disturbances.The state ing matrices axe similarin structure to the uncertainty space descriptionof a linear time-invariantplant can block structure and commute with the uncertainty ele- be represented as ments. An upper bound can be written as a linear ma- trEx LnequalJty (LMI) by considering a maximum eigen- = = CL EtL El2 w value value condition utilizing the structured scaling matrice_ [2].

e C2 E21 E_.2 d Worst-Case Flutter Method The input to P(s) is the uncertainty input, w, and the uncertainty output, z, is the output of P(s). Defining additional signals for errors and disturbances allows A worst-case method of computing flutter margins uti- P(s) to be formulated in the robust stability frame- lizes #-anaiysis for evaluating system stability. A lln- work of Figure 2 with _ as the uncertainty operator.

ear system is formulated with associated uncertainty operators.

Additional uncertainty operators are included to ac- count for modeling errors between the theoretical sys- Consider the generalized equation of motion for the tem and the physical aircraft. They also allow the structuralresponse of the aircraft[10].

analysis to consider a range of aircraft dynamics that

M# + + K,7 + "#Q(s),7 = 0

may change due to variations in parameters such as mass or variations in the aerodynamics such as small For a system with n modes, define M E R "x" as the deflections in the aircraft surfaces.

mass matrix, C E R "x" as the damping matrix and K E R n x. as the stiffness matrix. _ E R is a scalar Errors in elements of the state-space matrices are of- representing the dynamic pressure and Q(s) E C "x- ten represented by parametric uncertainty [3]. This is the matrix of unsteady aerodynamic forces.

uncertainty may be a real scalar parameter to reflect variation in physical parameters such as mass and dy- The unsteady aerodynamic forces are fit to a standard namic pressure or real values such as modal frequency finite-dimensional state-space system. This form can and damping.

be shown to encompass the traditional forms of Roger and Karpel that include lag terms for the transient Unmodeled dynamics and nonlinearities are often ac- aerodynamics [14].

counted for by including a complex uncertainty. The complex operator allows uncertainty to enter simulta- • [AQ BQ ]=Dq+CQ(M_AQ)-IBq

Q(s) cQ Dq neously in magnitude and phase of the signals. This

dynamic uncertainty may be a scalar or a matrix re- Given the number of generalized states, n, and flecting unstructured uncertainty for a set of signals.

aerodynamic states, nQ, define A_ E R '_x'_, BQ E R '_xn, CO E R '*x_° and Dq E R '_x'_ Experimental flight data can be used to generate un- as state-space elements approximating Q(s).

certainty weightings. Transfer functions of the analyt- ical model can be compared with experimental flight The method should compute a p value which relates data transfer functions. Different size perturbations an unstable flight conditions. This is accomplished are allowed to affect specific system parameters to the by introducing an uncertainty operator to consider a degree that the resulting transfer functions cover the range of flight conditions. Dynamic pressure is treated range of experimental flight data.

as an unknown quantity for worst-case flutter analysis.

Model validation algorithms are used to veri_y that the Consider an additive perturbation, 5_e E R, on the amount of uncertainty in the linear model is sufficient nominal dynamic pressure, q_om- to generate the flight data sets. This paper uses an al-

=

gorithm based on p-analysis of the linear system with frequency domain flight data [14, 13]. The model vali- Two signals, z and w, are introduced intothe formu- dation condition is derived as a standard p calculation.

lationto represent uncertainty input and output. The The # value at each frequency relates the required size uncertainty output is formulated from system states.

of perturbations at that frequency. This information z = M-tDQrl + M-XCQz is used to compute frequency varying weightings to scale the uncertainty set. The model validation proce- w is related to z by the dynamic pressure perturbation.

dure is repeated until a small amount of uncertainty is defined that still validates the model but reduces the t0 _qZ conservatism in the resulting flutter .analysis.

The state-space aeroelastic model is formulated with the additional signals to account for the parameteriza- Robust flutter margins are computed using p-analysis tion of the dynamic pressure uncertainty. Formulate on the linear system with the uncertainty operators.

the plant, P($), using state vector It/;/};z] such that The flutter margin is found as the smallest destabiliz- z = P(s)w. Define M = -M -t.

ing perturbation for the dynamic pressure uncertainty, 0 I 0 0 "1 6_$, for the linear system with the given amount of mod- eling uncertainty.This margin isthe worst-case flutter ...

condition for the allowed range of aircraftdynamics.

Bq 0 0

J

-l_lDQ 0 -MCQ 0 Worst-Case Flutter Parameter F/A-18 Aeroelastic Data The flutter computation method described in this pa- Extensive flight data from the F/A-18 SRA is used to per uses # as the worst-case flutter parameter. There generate uncertainty descriptions for an analytical air- are several advantages to using p as the flutter param- craft model {16]. Over 30 flights were conducted in two eter. p is a much more informative flutter margin as sessions between September 1994 and February 1995 compared to traditional parameters such as pole loca- and between June 1995 and July 1995 at Dryden Flight tion and modal damping.

Research Center. Each flight performed maneuvers for different conditions throughout the flight envelope. A The conservatism introduced by consideringthe worst- total of 260 different data sets are generated from var- case uncertainty perturbation can be interpreted as ious conditions throughout the flight envelope [5].

a measure of sensitivity. Robust p values which are significantly different than the nominal flutter margins The aeroelastic flight data is generated using an ex- indicatethe plant ishighly sensitive to modeling errors ternal structural excitation system developed by Dy- and changes in flight condition. A small perturbation namic Engineering Incorporated (DEI). This DEI ex- to the system can drastically alterthe flutter stability citer is a modification of an excitation system used properties. Conversely, similarity between the robust for F-16 XL flutter research [20]. The system consists and nominal flutter margins indicates the aircraft is of a wing-tip exciter, an avionics box mounted in the not highly sensitiveto small perturbations.

instrumentation bay, and a cockpit controller.

Robustness analysisdetermines not only the norm of Aerodynamic forces are generated by the wingtip ex- the smallestdestabil_.ing perturbation but also the di- citer. This exciter consists of a small fixed aerody- rection.This information relates exact perturbations namic vane forward of a rotating slottedhollow cylin- for which the system is particularlysensitive. /J can der. Rotating the cylinder varies the pressuredistribu- thus indicate the worst-case flutter mechanism which tion on the vane and resultsin a wing-tip forcechanging may naturally extend to indicate active and passive at twice the cylinder rotation frequency. The magni- control strategies for flutter suppression.

tude of the resulting force isdetermined by the am¢_unt of opening in the slot.The F/A-18 aircraftwith a left Damping is only truly informative at the point of insta- side wingtip exciter is shown in Figure 1. * bility since stable damping at a given flight condition does not necessarily indicate an increase in dynamic The cockpit controller commands a frequency range, pressure will be a stable flight condition. # computes duration and magnitude for the wing'tip excitation sig- the smallest destabilizing perturbation which indicates nal. Frequency varying excitation is generated by the nearest flight conditions that will cause a flutter changing the cylinder rotation frequency with sine instability. In this respect, /_ is a stability predictor sweeps. Each wingtip exciter is allowed to act in- while damping is merely a stability indicator.

phase, 0 degrees, or out-of-phase, 180.;degrees, with each other. Ideally, the in-phase data excites the sym- These characteristics of/_ make the worst-case flutter metric modes of the aircraft and the out-of-phase data algorithm especially valuable for flight test programs.

excites the anti-symmetric modes.

Aeroelastic flight data can be measured at a stable flight condition and used to evaluate uncertainty op- Flight data sets are recorded by activating the exciter erators. The /_ method, unlike damping estimation, system at a given flight condition. The aircraft at- does not require the aircraft to approach instability for tempts to remain at the flight condition throughout accurate prediction, p can be computed to update the the series of sine sweeps desired by the controller. The stability margins with respect to the new uncertainty sine sweeps were restricted within 3 Hz and 35 Hr.

levels. The worst-case stability margin then indicates Smaller ranges were sometimes used to concentrate on • what flight conditions may be safely considered.

a specific set of mode responses, lVfultiple sets of either linear or logarithmic sweeps were used with the sweep Safe and e/fide.at expansion of the flight envelope can frequency increasing or decreasing.

be performed using an on-line implementation of the worst-case stability estimation algorithm. Comput- Aeroelastic flight data generated with the DEI exciter ing p does not introduce an excessive computational system is analyzed by generating transfer functions burden since each F/A-18 flutter margin presented in from the excitation force to the sensor measurements.

this paper was derived in less than 2 minutes using These transfer functions are generated using standard standard off-the-shelf hardware and software packages.

Fourier transform algorithms. There axe several inher- On-line algorithms are currently being developed to ent assumptions associated with Fourier analysis that demonstrate this procedure for a flight test [17].

are violated with the flight data. The assumptions of time-invaxiant stationary data composed of sums of Mode Symmetric AntiSymmetric infinite sinusoidsisnot met by thistransient response 5.59 8.84 Wing 1°t Bending data. The analysis presented in this paper is based 9.30 8.15 Fuselage 1 °' Bendin G on Fourier analysis, although current research investi- 13.98 14.85 Wing 1 't Torsion gates wavelet techniques to analyze the flight data [5].

16.95 16.79 Wing 2 _a Bending 17.22 Wing Outer Torsion The excitation force is not directly measured but 19.81 18.62 Fuselage 2 "d Bending rather a straingauge measurement isused to approx- 24.19 Fuselage Torsion imate this force. The strain gauge records a point 29.88 29.93 Win G 2 _d Torsion response at the exciter vane root. This point response is considered representativeof the distributed excita- Table 1: Modal Frequencies tion force load over the entire wing surface.Vane root stra£n is assumed to be directlyproportional to the vane airloadsdue to excitation [5 I.

method [8]. The supersonic forces are generated us- ing a different approach known as the constant panel Analysis of the recorded flight data indicatesthe DEI method [1].

excitersdid not operate entirely as expected. The exciters displayed erratic behavior at higher dynamic The doublet lattice and constant panel methods are pressuresdue to binding in both the motor drivemech- used to compute the frequency varying unsteady aero- anism and rotating cylinders. At low dynamic pres- dynamic forcesforseveral subsonic,transonic and su- sures the system operated better but still displays persoaic Mach numbers. The Mach numbers, M -- some phase driftbetween the leftand right cylinder .8,.9, .95,1.1,1.2, 1.4,1.6,are available.The unsteady rotations.

aerodynamic forces are computed as a function of re- duced frequency, k.

Further erratic behavior is demonstrated by compar- ing measurement signals due to excitation sine sweeps k = wh-'V of increasing and decreasing frequency. Transfer func- tions from a symmetric excitation to the wingtip The reduced frequency is a function of the t_e_fre - accelerometers clearly show different modes are ex- quency, w, the true velocity, V, and _ the mean a_ro- cited by the direction of the sweep even though the dynamic chord. Aerodynamic forces generated for 10 flight conditions are identical and the data sets were reduced frequency points between k = .0001 and k = 4 recorded 30 seconds apart of each other [16].

are sufficient for flutter margin computation for t_is aircraft.

F/A-18 Nominal Model The unsteady aerodynamic forces are fit to a finite- dimensional state-space system. The system identifi- The generalized equations of motion are used to derive cation algorithm is a frequency domain curve fitting a linear, finite-dimensional state-space model of the algorithm based on a least squares minimization. A aircraft. This model contains 14 symmetric structural separate system is identified for each column of the modes, 14 antisymmetric structuralmodes and 6 rigid unsteady forces transfer function matrix. 4 th order body dynamic modes. The control surfaces are not state-space systems are used for each column of the activeand no controlmodes are included in the model symmetric forces and 2 '_ order state-space systems are used for each column of the antisymmetric forces.

A finite element model of the SRA is used to compute These systems are combined to form a singlemultiple- the modal characteristics of the aircraft. Frequencies input and multiple-output state-space model of the of the dominant modes for flutter are presented in Ta- unsteady aerodynamics forces,previously designated • ble 1. These modal frequencies are computed for the Q(s), with 56 states for the symmetric modes and 28 aircraft with no aerodynamics considered. The pre- states for the antisymmetric modes.

dicted flutter results for this aircraft are computed from the finite element model using the p-k method. A The analyticalaeroelasticmodel has inputs for sym- detailed explanation of the SILk flutter analysis using metric and antisymmetric excitation forces. It is as- traditional methods is given in Reference [21].

sumed the excitation force will be purely symmetric or antisymmetric. There are 6 sensor measurements gen- Values of the unsteady aerodynamic force matrix at erated by accelerometers at the fore and aft of each distinctfrequencies axe computed for the finite ele- wingtip and on each aileron.

ment model using a computer package developed for NASA known as STARS [9]. This code solves the sub- sonic aerodynamic equations using the doublet lattice The magnitude of the parametric modal uncertainty, F/A-18 Uncertainty Description 5,node, is determined from flight data analysis. The linear model contains 14 modes for the symmetric re- Noise and uncertainty operators are introduced to the sponse and 14 modes for the antisymmetric response linearaeroelastic model to account forvariationsbe- of the aircraft. Unfortunately, the flight data does not tween the analytical model and the actual aircraft.

indicate each of these is sufficiently excited to allow These operators are developed by physical reasoning analysis and comparison with the theoretical model.

of the modeling process and also using the flight data Only the modes given in Table 1 are observed in the generated by the DEI excitation system [16].

data. A linear model is formulated from a subset of the full model which contains only the experimentally Standard analysis of the linear model isused to deter- observed modes. The modal parameters of this re- mine the framework for how uncertainty operators en- duced order model are compared with the flight data ter the system. Two uncertainty operatorsand a single and uncertainty levels are determined.

noise input are used to describe the modeling uncer- tainty in the linearaeroeiasticmodel. The magnitude Scalar uncertainty parameters, 6, are used to affect of each uncertainty operator and the noise level is de- the modal parameters. The state matrix of the linear termined both from physical reasoning of the model model is formulated as a block diagonal matrix with a and analysisof the flight data.

2 x 2 block for each mode. The diagonal component of each block is the real part of the natural frequency An uncertainty operator, J,,,ode, is introduced to the and the off-diagonal elements are the imaginary parts modal elements of the state-spaceF/A-18 model. This such that the natural frequency, tvi, and the damping, parametric uncertainty allows variations in both the (i, of the i th mode may be determined.

natural frequency and damping values for each mode.

This uncertainty covers errors in the coefficients of the Ai= -i r ¢'i =-rill equations of motion and the corresponding state-space elements of the linear model. An example of such an Scalar weightings, wr and wl, are used to affect the error arises in considering the mass of the aircraft. The amount of uncertainty in each matrix element. :The linear model uses a single mass value while in reality amount of variation in the matrix elements, and cor- the mass varies considerably due to fuel consumption.

respondingly the amount of variation in the natt_ral Mass variations for a simple second order system af- frequency and damping, are determined by the mag- fect the natural frequency, ¢v = _, and may be nitude of these scalar weightings. Define _ and _ as represented as parametric modal uncertainty. This the varying elements of the state matrix affected _y modal uncertainty allows a worst-case flutter point to the uncertainty 6.

be computed that accounts for parametric changes in the aircraft such as those due to mass variations.

= r(l -4- w_6) : = i(I -I- w,6) The second uncertainty operator, Ai,,, is a multiplica- Aeroelastic modes typically show low damping val- tiveuncertainty on the force input to the linear model.

ues caused by the real component being quite small This uncertainty is used to cover nonlinearities and as compared to the imaginary component. Since lin- unmodeled dynamics. The linear model contains no ear modeling techniques often identify the natural fre- dynamics above 40 Hz so the high frequency compo- quency better than the damping value, the weighting nent of thisoperator will reflect thisuncertainty. This for the real component should be larger than that for operator is also used to model the excitation uncer- the imaginary component.

taintydue to the DEI exciter system. Analysis of the flight data indicates the input excitationsignals rarely The weightings are chosen using the observed modal had the desired magnitude and phase characteristics parameters in the flight data. The natural frequencies • that they were designed to achieve. The low frequency show variations of -b5% from the theoretical model component of the input uncertainty reflects the uncer- while the uncertainty in the damping needs approxi- tainty associated with the excitation system used to mately 4-15% to validate all the flight data. The scalar generate the flight data.

weightings are chosen accordingly.

A noise signal is included with the sensor measure- wr : .15 ments. Knowledge of the aircraftsensors is used to w_ = .05 determine a levelof 10% noise is possiblein the mea- The flight data is only able to determine uncertainty sured flight data. An additional noisemay be included levels for the modal paramters of the experimentally on the force input due to the excitation system but it observed modes. It is assumed the uncertainty lev- isdecided the input multiplicativeuncertainty is suf- els in the unobserved modes should be consistent with ficient to describe thisnoise.

these values.Parametric uncertainty is introduced for F/A-18 Flutter Points each modal block in the state matrix, affecting ob- served and unobserved modes, with the weighting val- Flutter margins are computed for a linearmodel with ues given above.

the associated modeling uncertainty structure using the p-analysis method [15]. Linear systems for sym- The block diagonal state matrix also contains some metric and antisymmetric structural modes are sepa- realvalued scalar blocks. These scalar blocks appear rated forease of analysis.These systems can easily be as approximations to lag terms in the state-spaceiden- combined and analyzed as a single system; however, tification of the unsteady aerodynamic forces. The eigenvector analysis would be required to distinguish identified system with these lag approximations does which critical flutter modes are symmetric and which not accurately model the aerodynamic forcesat allfre- are antisymmetric. Each system contains the same quencies. Parametric uncertainty affects each of these number of structuralmodes, 14, and the uncertainty lag terms with a weighting of w_Gg = .15 that allows descriptionsare identicalfor each linear model.

15% variation.

The system given in Figure 3 contains three uncer- The low frequency magnitude of the input multiplica- tainty blocks. The parametric uncertainty covering tive uncertainty is determined from the flightdata.

variationsdue to dynamic pressure, 5-_q, isa scalarpa- Levels of uncertainty axe chosen that validatethe flight rameter repeated 14 times, once foreach elastic mode.

data for a given amount of noise and parametric modal The parametric uncertainty block affecting the modal uncertainty. The high frequency component of input parameters, 6mo_,, is a diagonal matrix with dimen- uncertainty is determined to reflect the unknown dy- sion equal to the number of states. Separate scalars namics at high frequency for the linear model. The along the diagonal represent uncertainty in each elastic frequency varying transfer function for weighting the mode, each mode in the aerodynamic force approxima- input uncertainty is given as Wi,.

tion, and each lag term. The uncertainty paramdters s+lO0 W_. = 5 for the modes are repeated two times while the pa- s + 5000 rameters for the lag terms are single scalars. Define 5_ The block diagram for the aeroelastic model with the as the i th uncertainty parameter for the systejn trith uncertainty operators isgiven in Figure 3.

n,n modes and n_ lag terms. The input multiplica- tire uncertainty block, A_, is a scalar for this single input plant model since we are analyzing symmetric excitation separately from antisymmetric excitation.

Q The parametric uncertainty parameters represent e .d changes in elements of the state-space model. The variation of _ between +1 admits dynamic pressures between 0 _< _ _< 2"_om. Allowing the modal uncer- tainty parameters, J1,.-.,5-. to vary between 4- 1 allows 5% variation in the imaginary part of the nat- ural frequency and 15% in the real part. This corre- Figure 3:F/A-18 Uncertainty Block Diagram sponds to approximately 5% variation in the natural frequency and 15% in the damping value of each mode.

Flight data used to validatethisuncertainty structure These parameters are real quantities. The multiplica- covers a large range of flight points from the entireset tive input uncertainty contains magnitude and phase of 260 flight maneuvers throughout the flight envelope.

information and is treated as a complex linear time- invariant uncertainty.

Using a singleuncertainty descriptionover the entire Nominal flutter boundaries are initially computed by flight envelope may be conservative. It is reasonable to assume the linear models are more accurate at sub- ignoring the modal and input uncertainties, p is com- sonic and supersonic _han at transonic. Additionally, puted only with respect to the parametric uncertainty the flight data from the DEI excitersystem should be allowing a range of dynamic pressures to be consid- better at subsonic speeds than at supersonic. How- ered. Robust flutter boundaries are computed with ever, it simplifies the analysis process to consider a respectto the structured uncertainty set,A, described single set of uncertainty operators. This process is above using the structured singular value. Traditional equivalent to formulating the worst-case uncertainty flutter boundaries computed using the p-k method axe levels at the worst-case flight condition and assuming presented with the nominal and robust flutter bound- that amount of uncertainty is possible for the remain- aries computed with p in Table 2 ing flight conditions.

The flutterboundary at the transonic case, M = I.I, Mach symmetric antisymmetric demonstrates significant sensitivityto the modeling uncertainty.The robust flutter dynamic pressures are •8 3360 3168 2909 4600 4593 3648 approximately 70% of the nominal fluttermargins.

•9 2700 2706 2575 3150 3057 2944 This is explained by considering the rapid transition of •95 2430 2388 2329 2600 2751 2572 critical flutter boundaries near this region. The criti- I.I 5400 5676 4120 5500 3265 2827 cal flutter frequenciesand the flutter dynamic pressure 1.2 2469 2454 2327 2850 2893 2653 widely vary between Mach numbers slightlylower and 1.4 3528 3432 3034 4600 4439 4191 higher than transonic.The smadl amount of modeling 1.6 4470 4487 3996 5700 5870 4536 uncertainty is enough to cause the worst-case flutter mechanism to shift and the plant assumes character- Table 2: Nominal and Robust Flutter Points kstics more consistent with a non-transonic regime.

The nominal flutter dynamic pressures computed us- The modal natural frequencies for the critical flutter ing the ]J method can be directly compared with those modes are presented in Table 3. The frequenciescom- computed using the traditional p-k method [21]. Each of these flutter solutions axe based on an analytical puted using the p-k method and the/J-analysis method model with no consideration of modeling uncertainty. are close throughout the flightenvelope for both the symmetric and antisymmetric modes. Frequencies for The nominal flutterpoints for the symmetric modes the robust flutter solutions are slightlydifferent than match closely with the p-k method throughout the the nominal flutterfrequencies due to the modeling flightenvelope. The subsonic and supersonic cases uncertainty which allowed 5% variation in the modal show an especially good correlation with the p-k flutter natural frequencies.

points. For each of these flight regions, the/z-analysis flutterdynamic pressures are nearly identical, within Mach symmetric antisymmetric I%, to the p-k method flutter dynamic pressures.The _JF-k OJno_n OJrob Olp--k Cdnorn _Jrob transonic case at M" = 1.1, however, shows a slight • 8 8.2 7.6 7.7 9.0 9.1 9.1 differencebetween the two methods. The # method •9 7.4 7.3 7.3 9.2 9.1 9.2 • computes a flutterpoint that is greaterthan the p-k .95 6.8 6.9 6.9 9.1 9.2 9.2 method. In each Mach regime; subsonic,supersonic or 1.1 12.1 13.2 13.0 28.6 28.0 28.3 transonic,the nominal flutter points axe within 5% for 1.2 26.5 27.4 27.4 26.9 28.9 28.9 the two methods.

1.4 28.1 28.1 28.1 30.4 31.7 31.8 1.6 28.9 30.1 30.1 32.8 32.3 32:.1 The antisymmetric modes show a similar relationship between the flutter margins computed with the # and Table 3: Nominal and Robust Flutter Frequencies p-k methods. The subsonic and supersonic flutter points are within 5% for the two methods, but there is a greater deviatidn at the transonic condition. /z com- Subcritical flutter margins computed with the/_ and putes a flutter margin at M = 1.1 that is 40% lower p-k methods are presented in Table 4. Only nominal than the p-k method indicates.

subcritical margins are detected with /_ since the ro- bust margins are always worst-case critical margins.

The nominal flutter points for the/J and p-k methods show the greatest differencefor both the symmetric Mach symmetric antisymmetric and antisymmetric modes at the transonic case. The aerodynamics at Ai r -- 1.1 are more difficult to model 4700 4583 .9 accurately than at either subsonic or supersonic. Nu- 5300 5093 merical sensitivity to representationsof the unsteady 7450 6919 .95 aerodynamic foces causes differences in the nominal 6050 6001 1.1 flutter margins.

1.2 5400 5003 8400 7943 1.4 8970 8959 The rob.ust flutter margins computed using the # 1.6 8400 8843 method have lower dynamic pressures than the nomb nal margin, which indicates the expected conservative Table 4: Nominal and Robust Flutter Points - Subcritical nature of the robust computation. These new flutter p-analysis computes subcritical flutter margins within points are worst-case values for the entLrerange of al- 10% of the p-k method for both the symmetric and lowed uncertainty.The subsonic and supersonic flutter antisymmetric modes. The/_ method is even able to boundaries axe not greatly affected by the uncertainty detect the subcritical flutter hump mode occuring for set. In each of these cases, the robust flutter point is antisymmetric excitation at 0.9 Mach number.

within 10% of the nominal flutter point.

numbers. The antisymmetric modes show the onset of Matched-Point Flutter Mar_|ns flutter from three different critical modes along with three subcritical flutter modes throughout the flight The dynamic pressures at which flutter occurs are con- envelope in Figure 5. The frequencies of the critical vetted into altitudes, commonly known as matched- flutter modes can be found in Table 3.

point solutions, using standard atmospheric equations.

These altitudes are plotted for the symmetric modes in The subsonic flutter altitudes for symmetric and anti- Figure 4 and forthe antisymmetric modes in Figure 5.

symmetric modes demonstrate a similar characteristic.

The flight envelope of the F/A-18 is shown on these The nominal flutter boundary shows a significant vari- plots along with the required 15% flutter boundary for ation from Mach number M = .8 to M = .95 caused military aircraft.

by sensitivity to Mach number for the dynamics associ- ated with the critical flutter mode. The robust flutter boundary 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 number M = .81 than for Mach number M = .80 is reflected in the large conservatism associated with the robust flutter boundary. Similaxly, slight variation of Mach number near M - .95 is not expected to increase the nominal flutter boundary so there is less conservatism associated with the robust flutter boundary.

An interesting trend is noticeable for the symmetric mode robust flutterpoints in Figure 4 at the super- -4O000 sonic Mach numbers. The fluttermechanism results from the same modes from M -- 1.2 to M -- 1.6 with , | i I | I | !

02 0,4 OAI 0.8 _ 12 14 1_ 18 some increasein frequency. Similarly the altitudes of Mad_ the nominal flutter margins show little change for t_ese Figure 4: Nominal and Robust Flutter Points - Matched Mach numbers. The aeroelastic dynamics associated Point Solutions for Symmetric Modes with the critical flutter mode are relatively unaffected by the variationof Mach over this range and conse- Io_O0 quently each flutter boundary has the same sensitivity .r to modeling errors.

40oo0 The robust flutter margins for the antisymmetric m2des at supersonic Mach numbers show a slightly differentbehavior than the symmetric mode flutter margins. The fluttermechanism is again caused by a single mode from M = 1.2 to M = 1.6 with similar

i

frequency variationas symmetric. The robust flutter -20000 margins demonstrate a similar sensitivity to modeling errors at M = 1.2 and M -- 1.4 but at M = 1.6 a greatersensitivity isshown. The greater conservatism .-4OO0O at M -- 1.6 may indicate impending transitionin flut- ter mechanism from the subcritical mode at slightly | | * J000O 0:_ 0.4 0'.0 0.8' , 1 _P 114 1" 1 jl higher Mach number.

The dark solid lineon Figures 4 and 5 represents the F|gure 5: Nominal and Robust Flutter Points - Matched Point Solutions for AntiSymmetric Modes required boundary for flutter points. All nominal and robust flutter points lieoutside this region indicating Figures 4 and 5 use several short solid linesto indicate the flight envelope should be safefrom flutter instabil- the p-k flutter solutions throughout the flight regime. ities. The robust flutter boundaries computed with I_ Ea_ of these short solid lines represents the flutter indicatethere is more danger of encountering flutter points due to a specific mode. Flutter points for the than was previously estimated with the p-k method.

symmetric modes given in Figure 4 show four solid In particular, the robust fluttermargin for symmetric excitationat Mach M -- 1.2 lies considerably closer to lines indicating three different critical flutter modes for the considered range of Mach numbers along with the boundary than the p-k method indicates.

a subcritical flutter mode occuring at supersonic Mach Computational Anal_,sis many solutions are computed near the true matched- point solution.

The p analysis method of computing flutter margins The p-k method also may have difficulty predicting presents significant analyticaladvantages due to the the subcritical flutter margins. The second mode in robustness of the resulting flutter margin, but it also Figure 6 may or may not intersectthe standard atmo- has several computational advantages over the p-k sphere curve. More computational analysisisrequired method. The # algorithm requires a single linear to determine the behavior of this mode at higher air- aeroelastic plant model at a given Mach number to speeds. The p-analysis method accurately detects compute critical and subcritical flutter margins. An both the critical and subcriticalflutter margins with- entire set of flutter margins may be easilygenerated out requiring expensive iterations.

using a standard engineering workstation in a few min- utes using widely available software packages [2].

Conclusion The p-k method is an iterative procedure that requires finding a matched-point solution [21]. The aircraftis analyzed at a particular Mach number and air density.

Nominal and robust flutter margins are computed for The airspeed for these conditions resulting in a flutter the F/A-18 SRA aircraft. Nominal flutter margins are instability iscomputed. This airspeed, however, often computed using a p-analysis method and a traditional does not correspond to the unique airspeeddetermined p-k method. The similarity of these flutter margins by that Math number and air density for a standard demonstrates the p-analysis method is a valid tool for atmosphere. Various airdensitiesare used to compute computing flutter instability points and is computa- flutter solutions and the corresponding air speeds are tionally advantageous. Extensive flight data is ana- plotted. An example of an air speed plot for flutter is lyzed to develop a set of uncertainty operators for a given in Figure 6.

linear model. Robust fluttermargins are compdted using p. The resulting flutter margins are worst-case 30000 , , , • .....

values with respect to the modeling uncertainty. These margins are accepted with a great deal more conflde_nce ,t t than previous flutter estimates by directly accounting for modeling uncertainty in the analysis process. '_he robust flutter margins indicate the desired flight en- 10_OO! '" velope should be safe from aeroelastic flutter instabil- ities; however, the flutter margins may lie noticeably closer to the flight envelope than previously estimated.

i° This method replaces damping as a measure of ten- dency to instability from available flight data. Since stability norms generally behave smoothly at instabil-

"-

ity boundaries, this method is recommended for pre-

. . L .... flight predictions and post-flight analysis with a min-

CO0 '1000 !100 121:0 13_0 1400 1500 11_0 17'100 imum amount of flight time. Additionally, the robust flutter stability framework extends naturally to robust Figure 6: AntiSymmetric P-KFlutter Solutions forMac.h flutter control synthesis for aeroelastic control.

M=1.4 The verticallines in Figure 6 represent two antisym- Acknowledgments metric modes that may flutterat Mach M--I.4. The p-k method computes a flutter solution at the airspeed indicated where the modal line crosses the standard The authors wish to acknowledge the /_nancialsup- atmosphere curve. This flutter solution is diliicult to port of the Controls and Dynamics Branch of NASA compute from only a few air density computations.

at the Dryden Flight Research Center. The structural Typically several airdensitiesare used to compute air dynamics group, Larry Freudinger, Leonard Voelker speed fluttersolutions and a lineis extrapolated be- and Dave Voracek, provided helpful comments and tween the points to determine the matched-point solu- suggestions throughout this project. Analysis of the tion at the standard atmosphere crossing point. The finite element model was assistedby Tim Doyle and nonlinearbehavior shown formode I near the standard Roger Truax. Dr. Lind issupported through the Post- Doctoral Fellowship program of the National Research atmosphere crossing point indicatesan accurate flutter Council.

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