FLIGHT TEST EVALUATION OF FLUTTER
PREDICTION METHODS
Rick Lind1
University of FloridaMarty Brenner2
NASA Dryden Flight Research Center
Abstract
Several methods have been formulated to predict the on-set of utter during ight testing. These methods havebeen demonstrated using data from simulations; how-ever, a rigorous evaluation that includes data from ighttesting must be performed. This paper evaluates theability of several methods to predict the onset of ut-ter by analyzing data from ight tests of the Aerostruc-tures Test Wing. The evaluated methods include data-based approaches that use damping extrapolation, anenvelope function, the Zimmerman-Weissenburger ut-ter margin, and a discrete-time autoregressive moving-average model. Also, a model-based approach that usesthe �-method utterometer is evaluated. The data-basedmethods are demonstrated to predict inaccurate utterspeeds using data from low-speed test points but con-verge to the accurate solution as airspeed is increased.Conversely, the utterometer is demonstrated as im-mediately conservative using data from low-speed testpoints but the predictions remain conservative and donot converge to the true utter speed as the envelope isexpanded.
Introduction
The ight test community routinely spends considerabletime and money for envelope expansion of aircraft sys-tems. This cost could be greatly reduced if there was amethod to safely and accurately predict the speed asso-ciated with the onset of aeroservoelastic instabilities or,more generally, utter.
Several methods have been developed with the goal ofpredicting utter speeds and improving ight testing.These methods include approaches based on extrapo-lating damping trends [1], an envelope function [2], theZimmerman-Weissenburger utter margin [3], the ut-terometer [4], and a discrete-time ARMA model [5].
1Assistant Professor, Department of Aerospace Engineer-ing, 231 Aerospace Building, Gainesville FL 32611-6250,[email protected] .edu, Member AIAA
2Research Engineer, MS 4840D, Edwards CA 93523,[email protected], Member AIAA
Copyright c 2002 by the authors. Published by the American In-stitute of Aeronautics and Astronautics, Inc. with permission.
These methods have all been shown to be theoreticallyvalid and demonstrated on simple test cases; however,only limited evidence exists as to their accuracy in pre-dicting utter during a real ight test.
The ability to predict the onset of utter must be care-fully evaluated before an approach can reliably be usedfor envelope expansion. In particular, a rigorous assess-ment of the prediction approach must be performed withrespect to aspects of ight testing that may di�er fromtheoretical assumptions. The intention is to note thestrengths and weaknesses of each method. A procedurefor ight testing could then be developed that takes ad-vantage of the strengths and avoids the weaknesses. Inthis way, the envelope could be expanded quickly to savecosts while ensuring a high level of safety.
This paper presents an evaluation of these 5 methods forpredicting utter speeds. The evaluation is especiallyvaluable because it is based on results from a ight testprogram. This ight test performed an envelope expan-sion of the Aerostructures Test Wing (ATW). The ighttest actually used an F-15 as a host carrier for this smallwing experiment as shown in Fig. 1. The ATW is nota complete aircraft; however, it was a complicated andrealistic structure that was similar to an aircraft wing.Furthermore, the ight test was able to expand the en-velope to a test point at which utter was encountered.Thus, the true utter speed is known exactly and can beused to evaluate the predicted utter speeds.
Figure 1: Flight test of the aerostructures test wing
AIAA-2002-1649
1
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado
AIAA 2002-1649
Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Flight data from the ATW is analyzed with respect to aconstant-Mach or varying-Mach type of envelope expan-sion. The constant-Mach analysis considers data fromtest points at the same Mach number but di�erent al-titudes. Alternatively, the varying-Mach analysis con-siders data from test points at the same altitude butwith di�erent Mach numbers. The constant-Mach anal-ysis is consistent with the theoretical assumptions be-hind the prediction algorithms so it is certainly a validtest of the predictive capabilities of each method. Thevarying-Mach analysis violates assumptions for several ofthe methods but, because this type of envelope expan-sion may be used in practice, it is a valuable exercise tonote the predictive capabilities across a limited range ofMach conditions.
The ability to predict the utter speed for the ATW isobviously not intended to be a completely accurate as-sessment of each approach. The ATW has a realisticnature but ight safety concerns required sacri�ces thatmade this wing di�er from a true aircraft wing. Also, the utter speeds of the �nite element model were very sen-sitive to relatively small changes in mass distribution sothe wing may have some ill-posedness or scaling issues.These concerns are only small drawbacks and are over-shadowed by the bene�t obtained by considering such astructure in a real ight environment.
The evaluation in this paper is essentially an extensionto a previous study [6]. That previous study evaluatedthe ability of these methods to predict utter speeds fora simulated constant-Mach ight test. This paper per-forms a similar type of evaluation but uses ight datameasured during an actual ight test and extends theevaluation to consider varying-Mach analysis.
Flutter Prediction Methodologies
Damping Extrapolation
The most commonly used method of predicting the onsetof utter is to extrapolate trends of modal damping [1].This method can be considered as a data-based methodbecause it relies entirely on analysis of ight data with noconsideration of theoretical models of the speci�c systembeing tested. The data used by this method for predic-tion are values of modal damping ratios.
This approach is actually straightforward to conceptu-ally understand. Simply stated, the damping of at least1 mode becomes zero at the onset of utter. The ut-ter prediction method consists of noting the variation inmodal dampings with airspeed and extrapolating thosevariations to an airspeed at which damping should be-come zero. This resulting airspeed is considered the pre-dicted utter speed.
The principle behind this method is quite sound; how-ever, there are often some diÆculties in practice. Onearea of diÆculty is the extraction of modal dampings.Aeroelastic ight data often has low signal-to-noise ra-tio so sophisticated techniques like parameter estimationor modal �ltering may be required [7]. Another area ofdiÆculty is the extrapolation method. Damping can bea highly nonlinear function of airspeed so the extrapola-tion must be carefully performed to ensure it accuratelyaccounts for any high-order nonlinearity.
Envelope Function
Flutter speeds can be predicted using a method based onan envelope function [2]. This method, like the dampingextrapolation approach, is a data-based approach thatpredicts the onset of utter based entirely on analysis of ight data. The data used by this method is simply thetime-domain measurements from sensors in response toan impulse excitation.
The fundamental nature of this method is somewhatsimilar to the method based on damping extrapolation;however, this method does not directly use estimates ofmodal damping. Instead, this method notes that theenvelope bounding an impulse response gets bigger asdamping decreases. Thus, the size and shape of the re-sponse envelope can be used to indicate a loss of dampingand, consequently, the onset of utter.
An envelope function that bounds an impulse responsecan be computed in several ways. The current formula-tion considers an approach based on the Hilbert trans-form. A signal, y(t), is related to its Hilbert transform,yH(t), as being similar in magnitude but di�ering inphase by 90 deg. An envelope function that bounds theimpulse response is easy to compute by using the phasedi�erence between y(t) and yH(t).
env(t) =py(t)2 + yH(t)2
This envelope will clearly increase in size as the dataindicates impulse responses of a system with decreasingmodal damping. Unfortunately, the amplitude of thisenvelope can be also a�ected by the size and shape of theimpulse given to the system. Thus, the time centroid isneeded as a further indication of the stability of a system.This centroid, t, is computed with respect to a maximumlength of time window, tmax, within which the data lies.
t =
R tmax0
env(t) t dtR tmax0
env(t)dt
A shape parameter is used for the actual prediction of utter. This parameter, S, is simply the inverse of thetime centroid such that S = 1=t. This shape parameteris then assumed to be a polynomial function of airspeed.
S = So + S1V + S2V2
2
The prediction of the utter speed is accomplished bynoting that S = 2=tmax when the system has criticaldamping at the onset of utter. The utter speed isthus predicted by noting the value of the polynomial atwhich this condition is satis�ed.
Zimmerman-Weissenburger Margin
Another method to predict the onset of utter has beendeveloped that uses the concept of a utter margin [3].This method is also a data-based method in the sensethat it only uses information obtained directly from the ight data. In this case, the utter margin makes usesof information about the poles of the transfer functionobtained from the data.
The utter margin, as originally formulated, is an indi-cator of distance to utter in terms of dynamic pressure.The development of this method is based on the equa-tions of motion for a classic aeroelastic system with bend-ing and torsion modes. The method was formulated fora 2-mode utter mechanism but has since been extendedto consider 1-mode [8] or 3-mode [9] instability.
The essence of the method is to consider the characteris-tic polynomial that describes the continuous-time aeroe-lastic system. The stability of this system can be evalu-ated by applying the Routh stability criterion. Assumethat the system is indeed a 2-mode system with 2 sets ofdistinct poles given by �1;2 and �3;4. De�ne parametersto represent the real and imaginary parts of these polessuch that �1;2 = �1 + |w1 and �3;4 = �2 + |w2.
The utter margin, FM , is formulated by applying theRouth stability criterion to the 2-mode system. Thiscriterion results in a parameter that must be positive isthe corresponding system is stable. The parameter isthus written in terms of the system poles.
F =
�(!2
2 � !21)
2+�22 � �21
2
�2
+ 4�1�2
(!2
2 + !21)
2+ 2
��2 + �1
2
�2!
�
�2 � �1�2 + �1
w22 � w2
1
2+ 2
��2 + �1
2
�2!2
The utter margin is obviously zero if either �1 = 0 or�2 = 0. This parameter is thus indicative of the stabilityof a system; however, that does not necessarily make itvaluable for predicting the onset of utter. The natureof a utter margin arises by noting, subject to some as-sumptions, that the parameter F varies with dynamicpressure. Some studies have noted that this variationmay be considered linear [10]; however, this paper willuse the theoretical formulation which assumes quadraticvariation.
F = f0 + f1q + f2q2
The dynamic pressure associated with utter is predictedby computing F from data taken at test points with dif-ferent values of dynamic pressure. The roots of this equa-tion for F give the dynamic pressure at which the onsetof utter is predicted to occur.
Flutterometer
The utterometer is another tool that predicts utterspeeds [4]. This tool di�ers dramatically from the otherapproaches considered in this paper. The main di�er-ence arises because this tool is a model-based approach.Basically, the utterometer uses both ight data and the-oretical models to predict the onset of utter. The ightdata under consideration is frequency-domain transferfunctions from sensors to an excitation. The model to beanalyzed is the corresponding theoretical transfer func-tion.
The formulation for this approach is based on �-methodanalysis [11]. This type of analysis computes a stabil-ity measure that is robust with respect to an uncer-tainty description. The utter speed is thus computedas the largest increase in airspeed for which the theoret-ical model remains robustly stable with respect to theuncertainty.
The utterometer operates by computing a robust utterspeed at every test point. The initial step is to computean uncertainty description for the model at that ightcondition. This step is performed by noting di�erencesbetween the theoretical and measured transfer functions.Uncertainty is introduced into the model as variationssuch that the resulting range of theoretical transfer func-tions bounds the measured transfer function. The nextstep is to compute the robust utter speed. This step isperformed by a straightforward application of �-methodanalysis on the theoretical model that contains the un-certainty variations. In this way, the utterometer pre-dicts a realistic utter speed that is more bene�cial thantheoretical predictions because the robust speed directlyaccounts for ight data.
A mathematical description of the utterometer is be-yond the scope of this paper. Instead, information maybe obtained from the literature [4].
Discrete-Time ARMA Modeling
A utter prediction method has been developed that con-siders stability of discrete-time aeroelastic systems [5].This method is a data-based approach; however, thetype of data used by this method is di�erent from thedata used by the previous methods. The discrete-timeapproach relies on time-domain measurements from thesystem in response to random excitation. This type ofdata is usually provided by sensor measurements thatrecord the response to atmospheric turbulence [12].
3
The analysis of turbulence data presents simultaneouslyan advantage and disadvantage in comparison to othermethods. It is convenient to allow utter testing thatdoes not need to command consistent and broadbandexcitation; however, it is often diÆcult for turbulenceto generate response levels in which all modes are suÆ-ciently observed.
This method requires data measured in response to tur-bulence because of assumptions about the aeroelasticsystem. Speci�cally, the system is assumed to be rep-resented accurately by an autoregressive moving average(ARMA) model. This type of model uses autoregressivemeasurements and a moving average of white noise todescribe the dynamics. The coeÆcients associated withthe autoregressive measurements are associated with thestability characteristics. De�ne the characteristics poly-nomial, G(z), as a function of the discrete-time variablez. This polynomial can be expressed using standard co-eÆcients, �i, or poles, zi.
G(z) = �4z4 + �3z
3 + �2z2 + �1z + �0
= �4(z � z1)(z � z�1)(z � z2)(z � z�2)
This form for the characteristic polynomial assumes thatthe dynamics are described by 2 modes. There are 4poles in the dynamics but they are restricted to be com-plex conjugate pairs.
The stability of the system is readily computed by ap-plying the Jury determinant method. This method guar-antees stability of a discrete-time system if certain con-ditions are satis�ed. The conditions can be written interms of the poles. There are 6 conditions to be satis�edfor this 4th order system; however, the condition de�nedas F�(3) is of particular interest.
F�(3) = �34�j1� z1z2j
2� �
j1� z1z�
2 j2�
��1� jz1j
2� �1� jz2j
2�
Stability of a discrete-time system is ensured if all poleshave magnitudes less than unity. This result implies astable system will always have F�(3) > 0. Further-more, the value of F�(3) goes to zero as the systemapproaches instability. Thus, F�(3) has been used asa stability predictor whose trends toward zero indicatethe onset of utter [13]. Unfortunately, F�(3) was notedto have some potentially adverse behavior with dynamicpressure; therefore, the behavior of a similar parameter,F�(1), was considered.
F�(1) = �4�1� jz1j
2jz2j2�
It is noted that the behavior of F�(3) is somewhat im-proved by associating F�(1). This forms the basis forFz as the discrete-time ARMA utter margin.
Fz =F�(3)F�(1)2
The utter margin is used to predict the onset of ut-ter by expressing Fz as a function of ight condition.Speci�cally, a standard approach is to express Fz as aquadratic function of dynamic pressure.
Fz = f0 + f1q + f2q2
The dynamic pressure associated with utter is predictedby computing Fz at several di�erent ight conditions;computing coeÆcients for Fz = f0+f1q+f2q2 that notethe dependency on dynamic pressure; and �nding thedynamic pressure at which F (z) becomes zero.
The utter speed produced by this method has somesimilarities to the previous predictions. Most notably,the utter speed predicted by this method is mathemat-ically similar to the speed predicted by the Zimmerman-Weissenburger approach if certain assumptions are en-forced.
Flight Test Evaluation
Aerostructures Test Wing
The Aerostructures Test Wing (ATW) was developed atNASA Dryden Flight Research Center. This testbed wasspeci�cally designed for testing methods to predict theonset of utter. The ATW was essentially a wing andboom assembly as shown in Fig. 2.
Figure 2: Aerostructures test wing
The wing was formulated based on a NACA-65A004 air-foil shape. The wing had a span of 18.0 in with rootchord length of 13.2 in and tip chord length of 8.7 in.The boom was a 1 in diameter hollow tube of length21.5 in. The total weight of the ATW was 2.66 lb.
This assembly was own by using an F-15 aircraft andassociated ight test �xture. The ATW mounted hori-zontally to the �xture and the resulting system attachedto the undercarriage of the F-15 fuselage as shown inFig. 3. Previous testing indicated that the air ow is rel-atively smooth around the system so the F-15 fuselageand wings are assumed to have minimal interference withthe ATW.
4
Figure 3: Mounting of the aerostructures test wing
The ATW was meant to be a realistic testbed that repre-sents complexity of an aircraft component; however, theconstruction of the testbed was limited by safety con-cerns. These potentially con icting issues were addressedby designing the ATW with a rib and spar constructionthat uses lightweight materials with no metal. Speci�-cally, the skin and spar were constructed from �berglasscloth, the boom was constructed from carbon �ber com-posite, the wing core was constructed from rigid foam,and components were attached by epoxy. Also, pow-dered tungsten was included in the endcaps of the boomfor mass balancing. The system was designed to utterat a subsonic condition within the ight envelope of thehost F-15.
A measurement and excitation system was incorporatedinto the wing. The measurement system consisted of 18strain gages placed throughout the airfoil structure and3 accelerometers placed at fore, aft and mid locationsin the boom. The excitation system was 6 patches ofpiezoelectric material, 3 patches mounted on the uppersurface that are out of phase with 3 patches on the lowersurface, that acted as a single distributed actuator. Sinu-soidal sweeps of energy from 5 to 35 Hz were commandedto these patches.
Ground vibration tests were conducted to determine thestructural dynamics of the wing. The main modes ofthe system and their natural frequencies are presentedin Table 1. Tests were conducted for the wing on a teststand and also attached to the ight test �xture to ensurethat these modal properties are not a�ected for the ighttesting.
Mode Frequency (Hz)1st Bending 14.051st Torsion 22.382nd Bending 78.54
Table 1: Measured Modes of the ATW
Envelope Expansion
The ight test program followed standard procedures for
envelope expansion. The system was stabilized at a testpoint, response to turbulence excitation was recorded for30 seconds, response to a sine sweep excitation throughthe piezoelectric patches was recorded for 60 seconds,then the system was accelerated to the next test point.
The ATW, in its �nal con�guration, was own on 4 ighttests during April 2001. These ights included 21 testpoints with Mach numbers between 0.50 to 0.83 and al-titudes between 10,000 and 20,000 ft. The ATW experi-enced a destructive utter incident on the �nal ight atconditions of Mach 0.83 and 10,000 ft which correspondsto a speed of approximately 460 knots of equivalent air-speed (KEAS).
Modal parameters were computed at each test point.The response to sinusoidal excitation was used forthese computations. Transfer functions were computedfrom the commanded excitation to the accelerometer re-sponses. A standard frequency-domain method of sys-tem identi�cation was used to formulate a system modelwhose dynamics are similar to the measured transferfunctions. The modal parameters of that model werethen extracted and used as representative of the ATWparameters.
The modal dampings that were extracted at each testpoint are given in Fig. 4. The utter instability a�ectingthe bending mode is clearly evident in the data trends.Furthermore, the damping data indicates that the ATWexperiences a classical type of utter such that one modeis becoming less stable while the other mode is becomingmore stable.
250 300 350 400 450
−0.2
−0.1
0
Airspeed (KEAS)
Str
uctu
ral D
ampi
ng
bending modetorsion mode
Figure 4: Measured modal dampings
The modal frequencies for the ATW are given in Fig. 5.This data seems to contradict the notion that the ATW isexperiencing a classical bending-torsion utter. Notably,the natural frequencies do not appear to be coalescing, asexpected for classical utter, until a possible coalescenceat the airspeed very close to the onset of utter.
5
250 300 350 400 45014
16
18
20
22
24
Airspeed (KEAS)
Mod
al F
requ
ency
(H
z)
bending modetorsion mode
Figure 5: Measured modal frequencies
Data Quality
Each of the methods discussed in this paper analyzes ight data to predict the onset of utter. The quality ofthe ight data is thus of obvious importance in evaluatingthe predictive nature of the approaches. Data quality isa diÆcult measure to describe; however, the quality ofthe ATW data, for purposes of utter prediction, couldbe judged by the ability to observe modes.
Figure 6 presents an example of data quality as deter-mined by modal observability. The transfer functionsfrom the accelerometers to the excitation command arenoticeably di�erent between a test point at Mach 0.60and 20,000 ft and a test point at Mach 0.65 and 20,000 ft.In particular, the response of the bending mode near16 Hz is signi�cantly less at the higher speed.
15 20 25 3010
−2
10−1
100
Frequency (Hz)
Mag
nitu
de
Mach 0.60Mach 0.65
Figure 6: Transfer Functions at 20,000 ft
The low response levels in the data can have signi�-cant e�ects on utter prediction. Essentially, severalapproaches for utter prediction rely on accurate val-
ues of the modal characteristics of the system. The lowresponse levels shown in Fig. 6 make it diÆcult to accu-rately identify any modal characteristics for the bendingmode. Thus, the approaches may predict utter speedsbased on inaccurate or incomplete information.
The comparison in Figure 6 describes only 2 test points;however, similar variations were noted at many testpoints. The bending mode was not excited or observedconsistently at several test points. The problem was es-pecially noted between Mach 0.50 and Mach 0.70 ightconditions. At these low Mach numbers, there were atmost 2 test points, out of the 3 that were own, withgood quality data.
This issue of data quality is evident in the damping plotsof Figure 4. That plot only contains 15 estimates ofdamping even though the ight test consisted of 21 testpoints. The 15 estimates correspond to test points atwhich the bending mode could be suÆciently excited andobserved. These test points include 2 points at Mach0.50, 2 at Mach 0.55, 1 at Mach 0.60, 1 at Mach 0.65, 3and Mach 0.70, 2 at Mach 0.75, 3 at Mach 0.80, and 1at the �nal Mach 0.82 test point.
Prediction Algorithms
Predictions of the onset of utter were performed usingthe 5 methods discussed in this paper. Several typesof data were available for analysis. The �rst type ofdata was simply the time-domain responses of the ac-celerometers. The frequency-domain responses, com-puted by standard Fourier transforms, were also avail-able. Additionally, a 4-state system model was computedby standard system identi�cation approaches applied tothe frequency-domain data between 12Hz and 30Hz [14].Finally, the theoretical state-space model of the ATWwas also available.
The method to predict utter speed by damping extrap-olation was the most straightforward to implement. Thedata analyzed was the modal dampings associated withthe system model identi�ed from the frequency-domainaccelerometer responses. In this case, those dampingswere assumed to be a 2nd-order function of airspeed.The utter speed was predicted as the largest root ofthe 2nd-order polynomial associated with the dampingfunctions.
The method using the envelope function was somewhatmore complex to implement than the damping methodbut did not present any serious diÆculties. The mainissue for implementation of this method was to get atime-domain Hilbert transform of impulse-response dataeven though the ATW only measured data in responseto sinusoidal excitation. The desired data was generatedin several steps by computing the Fourier transform ofthe time-domain sinusoidal data; rearranging the com-
6
ponents to compute the frequency-domain Hilbert trans-form of that data; and taking the inverse Fourier trans-form to get the time-domain Hilbert transform of an im-pulse response. A simple numerical integration was thenused to compute the envelope function and the shapeparameter as 2nd-order functions of airspeed. The valueat which this polynomial matched the utter conditionwas used as the predicted utter speed.
The utter margin associated with the Zimmerman-Weissenburger approach was computed with no diÆ-culty. This method used the real and imaginary partsof the poles of the identi�ed system model. The uttermargin was computed from these values and modeled asa 2nd-order function of dynamic pressure. The largestroot of that polynomial was converted, using match-point ight conditions, to a value of airspeed that wasused as the predicted utter speed.
The utterometer was implemented as the only model-based approach [15]. This tool analyzed transfer func-tions computed from both the frequency-domain re-sponses and the theoretical state-space models. The un-certainty levels used by the utterometer were initiallyzero but were updated at every test point to re ect er-rors noted by the data. The uncertainty descriptioncomputed in this way related the potential errors in themodel associated with the Mach number of the ightcondition. A prediction of the worst-case utter speedwas computed by a robust aeroelastic stability analysisusing the �-method approach for the model with respectto that ight-derived uncertainty description.
The implementation of the method to predict utterusing a discrete-time ARMA approach was the mostdiÆcult. This method used the time-domain measure-ments that were obtained in response to turbulence ex-citation. The data was extremely noisy so it was pro-cessed through a low-pass �lter with cuto� frequency at160 Hz. An ARMA model was then identi�ed to matchthe data using a Gauss-Newton algorithm [16]. The de-velopment of the method assumed a 2-mode system with4 poles; however, the data analysis had to consider asystem with 6 poles. Sometimes the ARMA model wasidenti�ed as having 2 sets of complex conjugate polesand 2 purely real poles. The real poles were ignored andthe method proceeded using the sets of complex conju-gate poles. Other times the ARMA model was identi-�ed as having 3 sets of complex conjugate poles. In thiscase, the 2 sets of poles that were closest in frequencyto the measured aeroelastic modal frequencies were usedfor utter prediction. Either way, the utter parameterwas computed using the desired sets of complex conju-gate poles from the ARMA model. This parameter wasmodeled as a 2nd-order function of airspeed whose rootsindicated the predicted onset speed of utter.
Constant-Mach Predictions
Implementation
A common method for envelope expansion is to operatethe aircraft at a series of test points along lines of con-stant Mach. Such testing essentially involves changingthe altitude to a desired condition and then changing theairspeed to attain the desired Mach condition. The datafrom the ATW was processed to predict utter speedsfor this type of constant-Mach envelope expansion.
The data used for utter prediction was restricted toconsider individual Mach numbers. Each approach con-sidered the ight data from the test points at 3 di�erentaltitudes for a particular Mach number. In this way, the utter speed for Mach 0.70 was based purely on analysisof data from Mach 0.70 ight conditions.
The approaches to predict utter were straightforward toimplement. The data-based approaches simply used theroots of various functions of airspeed measured in ft/s todetermine the utter speed. The model-based approachsimply computed a robust stability analysis of the modelat that Mach number with respect to an uncertainty de-scription determined by the data at that Mach number.
The constant-Mach envelope expansion should be a validtest of the predictive capabilities of each approach. Eachmethod was developed to consider constant Mach condi-tions so this type of envelope expansion does not violateany assumptions associated with the theoretical basis ofthe predictions.
Predictions of Flutter Speeds
The utter speeds predicted for constant-Mach enve-lope expansion are given in Table 2. Each row repre-sents the predicted speeds for a particular Mach num-ber whereas each column gives the speed predicted bya certain method. Also, the values of Vtrue are givento represent the true utter speeds. The ight test en-countered utter only near the Mach 0.8 condition so theremaining speeds are based on model characteristics andassumptions as to the behavior of the ATW.
Mach Vdamp Vfm Venv Vmu Varma Vtrue0.55 664 - 732 652 591 6600.60 657 - 696 697 715 7050.65 - - - 741 2867 7490.70 780 - - 782 751 7910.75 - 843 837 821 806 8320.80 882 870 884 852 886 871
Table 2: Predicted utter speeds in ft/s at each Mach num-ber
7
An important feature of Table 2 is that predicted utterspeeds are not given at certain Mach numbers for someof the methods. This feature does not actually signifythat the method was unable to predict a utter speed;rather, it signi�es that the method predicted the speedas an imaginary or negative number. The speed in thesesituations is predicted by extrapolating a function whosecoeÆcients are determined by curve �ts of data. Thefailure to predict a valid speed is indicative of data valueswith properties that di�ered greatly from the theoreticalexpectations so the curve �t was unable to produce areasonable function.
Another feature of Table 2 is the di�erences in speedspredicted by di�erent methods. In particular, the pre-dictions vary considerably for ight at low Mach num-bers but converge to similar answers as Mach numberincreases. For instance, the predictions at Mach 0.55vary from 591 ft/s to 732 ft/s whereas the predictions atMach 0.80 vary only from 852 ft/s to 886 ft/s.
These two features are actually somewhat related in thatthey are caused by same issue. Namely, the ight datagenerated at lowMach numbers was of lower quality thanthe data generated at high Mach numbers. The conceptof modal observability is actually quite appropriate as ameasure of data quality for this application. The resultof the poor quality was such that there were at most 2high-quality data values at low Mach conditions availablefor a curve �t that required 3 values. Consequently, theresulting predictions show a fair amount of error.
The predicted utter speeds from Table 2 are conve-niently displayed as utter altitudes in Fig. 7. Thesealtitudes correspond to the ATW operating at a Machnumber and its associated utter speed for a standardatmosphere. The test points are not displayed on thispicture but they are easily noted as conditions of 10,000,15,000 and 20,000 ft at each Mach number.
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85−50
−40
−30
−20
−10
0
10
20
Mach
Alti
tude
(kf
t)
dampingzimmermanenvelopemuturbulence
Figure 7: Predicted utter speeds during constant-Machenvelope expansion
The information in Fig. 7 is somewhat easier to dissem-inate than the corresponding information in Table 2. Inparticular, it is straightforward to note the accuracy ofthe di�erent prediction approaches.
Fig. 7 clearly demonstrates the adverse sensitivity topoor data quality for the data-based approaches. Thedamping approach computed reasonable predictions atMach 0.55, 0.70 and 0.80 conditions. The methods basedon the utter margin and the envelope function were ableto produce reasonable predictions only at Mach 0.75 andMach 0.80 which are near the onset of utter for the ighttest. The approach based on ARMA modeling predictedreasonable speeds at Mach 0.60 and Mach 0.80.
Fig. 7 also indicates the reduced sensitivity to data qual-ity for the �-method approach. Essentially, the conser-vativeness of the predictions remains relatively constantthroughout the envelope expansion. The reason for thisconstancy is that the �-method approach does not usea curve �t based on observed properties like damping;therefore, this method does not require 3 high-qualitydata values. The utterometer approach is able to ob-serve modeling errors from a single good data set and usethis information to update the predicted utter speed.Test points at which bending is not observed obviouslydo not relate any information about that mode so the�-method approach simply uses the default level of un-certainty as described by any previous data sets at thatMach number.
Varying-Mach Predictions
Implementation
Flight data was also analyzed with respect to a varying-Mach envelope expansion. Alternatively, this ight testcan be considered as a type of constant-altitude enve-lope expansion. The objective is to use ight data fromall test points, regardless of Mach number, to predictthe speed at which utter will occur for the ATW at analtitude of 10,000 ft.
The approaches used to predict utter speeds were for-mulated to be valid for constant-Mach testing; therefore,the algorithms had to be modi�ed for this varying-Machtesting. Essentially, the methods attempted to roughlyaccount for compressibility by considering utter speedin terms of equivalent airspeed. Speci�cally, the predic-tions were given as knots of equivalent airspeed (KEAS).
The data-based prediction methods were straightforwardto modify. These methods consider extrapolating func-tions of ight condition so the functions were simplyaltered to re ect characteristics of equivalent airspeed.The functions for the damping method and the envelope
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method were directly computed as dependent on equiv-alent airspeed so the roots of this function related the utter speed in KEAS. The functions for the utter mar-gin method and the ARMA method were still computedas dependent on dynamic pressure; however, the utterspeeds were determined as the equivalent airspeed forthe associated roots of those functions using a standardatmosphere.
The model-based prediction method using the mu-method approach and the utterometer was also mod-i�ed to re ect the objectives associated with varying-Mach envelope expansion. The modi�cations consistedof considering a set of plant models to determine the low-est Mach number at which a model is not robustly stablenear 10,000 ft. The assumption behind this approach isthat the uncertainty description identi�ed from data ata certain Mach number is a reasonable approximationto the uncertainty at all Mach numbers. For instance,as the envelope expansion considers a test point at Mach0.65 then the model at Mach 0.8 is analyzed with respectto uncertainty developed using Mach 0.65 data.
These modi�cations obviously introduce some error intothe predictions. The data-based approaches are clearlynot guaranteed to be accurate with respect to airspeedas determined in KEAS. Alternatively, the model-basedapproach is questionable because of the assumption thatthe uncertainty levels from one Mach condition are rea-sonable at any Mach condition. Despite these drawbacks,the predictions are still anticipated to be useful. Themain reason for this expectation is that the envelope ex-pansion will only consider subsonic ight. Mach e�ectsmay be noticeable around Mach 0.8 but equivalent air-speed should be an acceptable measure of ight condi-tion for the majority of test points. Obviously this claimwould be highly suspect for larger ight envelopes thatinclude transonic and supersonic ight so the current re-sults are considered valid only for this limited ight en-velope.
Predictions of Flutter Speed
Predictions of the speed associated with utter werecomputed at every test point. These predictions werebased on data from the current and any previous testpoints. The utter speeds, expressed in KEAS, predictedfrom each test point, also expressed in KEAS, are givenin Fig. 8.
The predictions in Fig. 8 demonstrate a general trendthat shows the methods were able to predict reasonablyaccurate utter speeds; however, it is best for evaluationpurposes to consider sets of predictions. Speci�cally, con-sider two sets that result from separating the predictionsmade at test points from less than 350 KEAS and testpoints with greater than 350 KEAS. Furthermore, con-
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Figure 8: Predicted utter speeds during varying-Mach en-velope expansion
sider the model-based utterometer method separatelyfrom the data-based methods based on damping extrap-olation, envelope function, Zimmerman-Weissenburgermargin, and ARMA modeling.
The predictions made using data from low-speed testpoints of less than 350 KEAS show an interesting be-havior. The data-based methods show very poor resultsat these low-speed test points. The predictions are veryscattered in magnitude and do not show any clear trendthat could be safely trusted. Furthermore, there are sev-eral conditions at which these methods did not predict avalid speed as evidenced by the lack of markers in Fig. 8.
The predictions made using data that included high-speed test points show a behavior that is very dif-ferent behavior from the predictions at low-speed testpoints. Namely, the data-based methods all predictspeeds that consistently converge to the correct answerof 460 KEAS. The methods based on damping extrapola-tion and Zimmerman-Weissenburger margins make par-ticularly good predictions. The methods based on enve-lope function and ARMA modeling show more variationsin the predictions but they clearly converge near the cor-rect answer.
Conversely, the predictions from the utterometer do notshow the same variation with test point airspeed. Thereis a small decrease in predicted speed after the �rst testpoint because the tool needs this initial data to updateits uncertainty model. The predicted speed shows novariation using data from any other test point.
The predictions in Fig. 8 can be summarized. The data-based methods produce poor predictions using low-speeddata but produce reasonable predictions that convergeon the correct answer as the envelope is expanded to in-clude high-speed test points. The utterometer producesa reasonable worst-case prediction of utter speed im-
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mediately and remains conservative throughout the en-velope expansion.
These results are explained by considering the data thateach method analyzes. The data-based methods, directlyor indirectly, use characteristics of the data associatedwith the modal dampings shown in Fig. 4. These damp-ings do not indicate any trend towards instability untilnear 350 KEAS. Thus, any polynomial based on this datawill not predict utter until the function accounts fordata at test points greater 350 KEAS. The utterometerdoes not rely heavily on these damping values; rather,it combines both data and theoretical models. The ut-terometer was able to identify an uncertainty descriptionfor the model by analyzing data at the �rst test point.The data from other test points did not indicate anymore errors so the utter margin does not show depen-dence on test point.
The analysis of Fig. 8 indicates the nature of the pre-diction methods. Namely, the data-based methods at-tempt to compute the exact speed associated with theonset of utter whereas the utterometer attempts tocompute a conservative prediction of the worst-case ut-ter speed. It is expected that the data-based methodsshould be highly accurate at test points that are closeto utter and also expected that this implementation ofthe utterometer should not reduce conservatism despiteanalyzing data from high-speed test points.
Also, the analysis indicates the sensitivity of the meth-ods. The data-based methods are formulated using op-timality criterion so they are highly susceptible to errorsin the data. Conversely, the utterometer is formulatedusing a robust criterion so it is much less sensitive tovariations in data.
Conclusion
This paper has evaluated several methods to predict theonset of utter. In particular, traditional data-based ap-proaches that only analyze ight data are compared witha recent model-based approach that analyzes ight datain conjunction with a theoretical model. Flight data fromthe Aerostructures Test Wing was used for this evalua-tion. At low speeds, the data-based approaches wereunable to consistently predict the onset of utter withany accuracy whereas the model-based utterometer wasable to predict a reasonable estimate of the utter speed.At high speeds, the data-based approaches convergedto an accurate prediction whereas the model-based ut-terometer predicted a utter speed that was conserva-tive. One reason for these results is that the ight datawas of limited quality at low-speed test points and thedata-based approaches were particularly suspectible tothis problem. Another reason for these results is that
this implementation of the utterometer did not updatethe baseline model so it was unable to take advantage ofinformation from high-speed test points and converge tothe true utter speed.
The predicted speeds demonstrate a method for envelopeexpansion. The ight test should be initiated using the utterometer at the low-speed test points to get an ini-tial conservative estimate of the utter speed. The testwould proceed using the utterometer estimates until thetest points approach the predicted speed. The envelopeexpansion at high-speed should rely more heavily on thedata-driven methods to �nalize an accurate prediction ofthe exact speed at which utter will be encountered. Ofcourse, the envelope expansion must still proceed withextreme caution but possibly the combination of theseapproaches will allow for a more eÆcient ight test pro-gram.
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