A NONLINEAR INDICIAL PREDICTION TOOLFOR UNSTEADY AERODYNAMIC MODELING
Patrick H. Reisenthel and Matthew T. Bettencourt1‡ *
Nielsen Engineering & Research, Inc.Mountain View, CA
James H. Myatt and Deborah S. Grismer§ ¶
Air Force Research Laboratory / Aeronautical Sciences DivisionWright-Patterson AFB, OH
ABSTRACT
The present paper describes a new tool kit which canbe used to model the time-dependent response ofnonlinear systems. The Indicial Prediction System(IPS) applies nonlinear indicial theory to solvecomplex unsteady problems, such as those associatedwith nonlinear aerodynamic phenomena duringmaneuver of aircraft. The functionality of this systemand its capabilities are described through numerousexamples. An important demonstration of the methodis its application to the prediction of the aerodynamicloads on a 65-degree delta wing undergoing forcedbody-axis rolling motions at high angles of attack.
NOMENCLATURE
Symbols and abbreviations
b Wing spanc Wing chordC Body-axis rolling moment coefficient,l
nondimensionalized with respect to qSbCS Critical StateCSR Critical-state ResponseDEP Dependent variableDOF Degree of Freedomf Aerodynamic load (generic)f Frequencyf Indicial response of f with respect to --
Indicial response of fDeficiency function ( )
H Heaviside step functionIE Indicial ExtractionIP Indicial PredictionIPS Indicial Prediction SystemIR Indicial Responsek Reduced frequency (k ≡ 3b/2U )∞
n Number of retained harmonicsharm
NIR Nonlinear indicial responseN Number of nodal extraction roll angles-
p Roll rateq Dynamic pressureQS Quasistaticrms Root mean squareS Wing areasgn Signt TimeT Period of oscillationU Freestream velocity∞� Angle of attack Dirac delta functionC /- Indicial response of rolling moment withl
respect to roll angle�f Build-up of generic aerodynamic load, f�f Critical-state response of fCS
� Boundary condition (generic)- Roll angle% Support sting angle) Time constant; alternatively, auxiliary time
integration variable in integrals) Time at which critical state is crossedc
3 Angular frequency
(1)
Parameter denoting dependence on prior combines features of flexibility in modeling, goodmotion history execution speed, and high fidelity representation.
(t) Basis function
Subscripts 2. OBJECTIVE AND APPROACHc CriticalCS Critical StateDR Deficiency Responsedyn DynamicQS Quasistatic∞ Time-asymptotic value (except for U )∞
Superscripts
CS Critical Statedyn Dynamic componentv Vortical" Derivative with respect to time"" Second derivative with respect to time~ Indicial or deficiency function
1. INTRODUCTION
In recent years, it has been possible to integrate theflight-dynamics equations fairly efficiently usinglinearized aerodynamics which are occasionallysupplemented with ad hoc methods (i.e.,semiempirical simulations or wind tunnel data) toinclude nonlinear unsteady aerodynamic effects.However, with the expanded flight envelopes beingconsidered for future maneuvering aircraft, it hasbecome increasingly important to be able to modeland predict nonlinear, unsteady aerodynamics. Thisincludes the prediction of the aerodynamic responsein the presence of flow separation, shock movement,and vortex bursting at high angles of attack and/orhigh angular rates.
Future fighter aircraft will be required to performcontrolled maneuvers well beyond traditional aircraftlimits, for example, pitch up and flight at high anglesof attack, rapid point-to-shoot, and other close-incombat maneuvers. These advanced maneuversdemand the use of aerodynamic methods capable ofpredicting characteristics of the nonlinear post-stallregime for multiaxis motions at extremely high rates.At present, the only methods of this scope are Navier-Stokes methods. However, their use in flightsimulations remains impractical at this time.
One method which has the potential to circumventsome of the present difficulties is the application ofnonlinear indicial theory [1,2]. An example of thelatter (the Indicial Prediction System described herein)
The objective of this paper is to provide an overalldescription of the Indicial Prediction System, with anemphasis on its use as a tool box for theaerodynamicist and control system designer. Thispaper is organized as follows. First, a brieftheoretical background is given. This is followed byan overview of the system, including the basicalgorithms and examples illustrating some of thecurrent capabilities. Code validation results are thenpresented. Finally, the complete system is applied toa subset of the U.S. Air Force Research Laboratory(formerly USAF/WL) and Canadian Institute forAerospace Research (IAR) 65−degree delta wingdatabase.
3. INDICIAL THEORY
The indicial approach is based on the concept that acharacteristic flow variable , which describes thestate of the flow, can be linearized with respect to itsboundary condition (or forcing function), �(t), if thevariation of is a smooth function of �(t). Thisallows the representation of in a Taylor seriesabout some value � = � ; thus0
If the response depends only on the elapsedtime from the perturbation �� (a linear time-invariantresponse) then it may be shown [3] that the formalsolution for is
where .
Hence, if the forcing function (i.e., the boundarycondition �) is known and if (the indicial response)is known from some computation or experiment, thenEq. (1) gives the value of for any schedule ofthe boundary condition �(t) without the need tocompute from first principles.
Modeling
Options
IR/CSR Conditioner
Partition
Definition
Extraction
Output
Path
Definition
sSpace
Definition
IR/CSR
Database
Prediction
(2)
(3)
Fig. 1. The IE and IP Programs Share CommonModule Components.
The basic idea behind the use of nonlinear indicialresponse theory [1,2] is that the linear formalism,Eq. (1), can be retained in the form of a generalizedsuperposition integral, provided that the nonlinearindicial response is now taken to be a functional
( ;t,)), where denotes the dependence onthe entire prior motion history. Furthermore, thenonlinear indicial theoretical formulation allows forthe presence of aerodynamic bifurcations by splittingthe integral, for example:
where the nonlinear indicial function isdefined as the following Fréchet derivative:
and �f (t;�() )) is the so-called jump responseCSc
associated with crossing the bifurcation at time ) .c
A critical state is defined [4] as a transition from oneequilibrium flow state to another and is oftenassociated with a discontinuity in the staticaerodynamic loads and/or their derivatives [5]. Theassociated transient response, �f (t ;�() )), isCS
c
referred to either as the critical-state response (CSR)or the jump response. the a priori independent variables and to specify
4. INDICIAL PREDICTION SYSTEM that pitch-plane-only maneuvers are being considered.
The Indicial Prediction System is essentially the union freedom) of the independent variables as time-of two codes: an indicial prediction (IP) code and an dependent. The motion source specification allowsindicial extraction (IE) code. The indicial prediction the program to identify the source of the motion ascode uses a database of indicial and critical-state being a file (tabular form) or a shared objectresponses to predict the output of the system to more subroutine. or less arbitrary inputs, while the indicial extractioncode is responsible for the creation of this database The modeling options module allows the user to from empirical data. Both modules are based on the experiment with various parameterizations of theconcept of a nodal, parameterized representation of indicial and critical-state responses. In some cases,the nonlinear indicial response (References 6 and 7). this may even include the possibility that the desired
The overall structure of the IPS is given in Figure 1.This figure depicts the main components of thesystem, many of which are shared between the IE andIP modules. We will now briefly describe thefunctionality of each of the components.
The purpose of the space definition module is todefine all possible dependent and independentvariables for a given configuration or subcase of agiven configuration. A configuration file is parsed forthese variables, along with static (descriptive)parameters such as flow conditions and geometry.The configuration file also contains IR/CSR databaseinformation as well as textual information such as titleand comment fields.
The path/maneuver definition module has twoprimary purposes: specification of inputs and, in theextraction case, specification of the “training” dataused to identify the IR/CSR database. The inputsspecification serves to optionally deactivate some of
bounds on the retained variables. For instance, in asix degree-of-freedom simulation, one may specify
This identifies a subset (the active degrees of
parameterizations are not strictly consistent with the
Fig. 2. Schematic Depicting the RelationshipBetween Parameters, Active Degrees ofFreedom, Active Dependent Variables and thea priori Dependent and Independent Variablesof the System.
Fig. 3. Two-Dimensional Illustration of IR/CSRSpace Partition.
nominal parameterization expressed in the IR/CSR described either in tabular form or subroutine form.database (see below). "Modeling" in this context is If its form is not specified, then the deficiencydefined as a series of decisions made by the user response is taken to be zero, meaning that thewhich include the choice of the dependent variables response is quasistatic.to be predicted (DEP ), the choice of which activei
degrees of freedom (DOF ) affect each of these The purpose of the IR/CSR conditioner is twofold.j
dependent variables individually, and, for each of the Its first function is to filter the contents of theretained (DEP , DOF ) combinations, how to calculate IR/CSR database in order to establish a short list ofi j
the contributions due to the indicial (IR ) and critical- IR/CSR nodes "likely" to participate in theij
state responses (CSR ). In particular, each IR or interpolation process. Its second function, in the IPij ij
CSR may be treated/overridden as quasistatic rather mode, is to resolve parameterization conflicts, thusij
than dynamic, and each IR /CSR may be maintaining the user's ability to execute the programij ij
parameterized in different ways (see Figure 2). even in cases where the contents of the IR/CSR
The IR/CSR database consists of a number of filescontaining information relative to the known indicialand critical-state responses. Each IR or CSR filerepresents one or more nodal responses. Each filecontains a header, followed by a data description.The data description consists of a list of dependentvariables, one independent variable, and a list ofparameters pertaining to this particular node. Forexample, the file might describe the nonlinear indicialresponse of C and C (two dependent variables) withl m
respect to - (the independent variable), at - = 15°(first parameter), d-/dt = -0.01 (second parameter),% = 30° (third parameter) and sgn(d -/dt ) > 0 (fourth2 2
parameter). The responses per se are specified in twoparts. The first item is the time-asymptotic orquasistatic value of the response (i.e., theaerodynamic derivative for each of the dependentvariables). The second item is a description of thedeficiency responses for each dependent variable (thedeficiency response is the indicial response minus itstime-asymptotic value). A deficiency response can be
database are not strictly consistent with the desiredparameterizations being experimented with. This alsoallows the handling of heterogeneous databases. Forexample, it is conceivable that the IR/CSR databasemay have been created initially with certainparameterizations in mind, but then modified throughthe addition of extra responses with different (eitherfewer or more) parameterizations. In this case, theIR/CSR conditioner permits the program to functionwithout having to regenerate the entire database.
At the end of the IR/CSR conditioner tasks, the usefulportion of the database has been ingested andhomogenized. One task remains to complete themodeling decisions: the definition of partitions of theIR space and the assignment of critical states. Thetasks carried out in the modeling options moduledefine the representation of the IR/CSR parameterspace and its dimensionality. The PartitionDefinition Module creates partitions of the parameterspace. The purpose of partitioning is to carve out
Fig. 4. Classification of Parameter Space Partitionsas Critical States.
sections of the parameter space within which like-nodes are considered for functional interpolation (seeFigure 3). Partition transitions are then classifiedaccording to whether or not they are to be modeled ascritical states (Figure 4).
The Prediction Module and the Extraction Moduleare the core computational modules of the IndicialPrediction System. The task of extracting nonlinearindicial and critical-state responses from experimentaldata is a challenging one, and a separate paper(Reference 8) will be devoted to this topic. Theprediction module is responsible for carrying out thenonlinear indicial theoretical prediction. Thus, it isresponsible for integrating (in a generalized sense) theindicial and critical state contributions. The basicalgorithm in the prediction module is as follows. Foreach dependent variable DEP , the predicted responsei
is the sum of the effects of each participating activedegree of freedom. Each of these effects areseparated into so-called regular contributions (due tothe indicial responses), and critical state contributions.Each contribution is further subdivided into quasistaticand dynamic components. The integration is done intwo steps. The first step is the integration of theregular contributions. The update of each dependentvariable is symbolically denoted
where DR stands for the deficiency response(DR ≡ IR−IR ). The second step is to sum up allQS
relevant critical-state response contributions. Again,the update of each computed dependent variable issymbolically denoted
where CSR designates the deficiency (i.e., dynamic)DR
portion of the critical-state response. Both the regularcontributions and critical-state contributions are storedat each time step for each valid DEP /DOFi j
combination.
5. SAMPLE CASES
In this section, we present the results of four exampleruns illustrating the capabilities of the IP code. Thefirst example exercises the code in linear indicialtheoretical mode. The second example illustrates theeffect of interpolation accuracy for the case of anonlinear quasistatic prediction involving two degreesof freedom. The third example illustrates the effectof partitioning without any critical states. The lastexample models hysteresis with two critical states.
Linear Indicial Response
A single input, single output, linear model wasconstructed using a single indicial response node.The indicial response is of simple exponential form
, so that the response f(t) to an input is known analytically. Figure 5 shows
the result of the IP code, run with the periodic option.The results (DEP (t)) are plotted as a function of the1
damping ratio )/T, where T is the period of theexcitation. According to theory, the amplitude andphase shift of the output are those given in thefollowing table:
Ratio )/T Amplitude Phase
0 4 0°
0.125 3.146 -38.1°
0.25 2.149 -57.5°
0.5 1.214 -72.3°
1 0.629 -81.0°
The results shown in Figure 5 are exact to numericalaccuracy.
Nonlinear Quasistatic Response
In this example, there are two degrees of freedom,denoted � and -. However, we assume that thedependent variable denoted DEP is a function of �1
only. The indicial response of DEP with respect to1
� is assumed to be parameterized by both � and -,
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-4
-2
0
2
4
Time
DEP1τ/T = 0
τ/T = 0.125
τ/T = 0.25
τ/T = 0.5
τ/T = 1
0 10 20 30 40 50 60 70 800
10
20
30
40
50
α
φ
-50
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 70 80
’exact.xg’’1d.xg’’2d.xg’
Fig. 5. Linear Indicial Prediction: Effect ofDamping Ratio.
Fig. 6. Example #2: Nodal Responses andManeuver Trajectory in � - - ParameterSpace.
Fig. 7. Quasistatic Quadratic Prediction of DEP vs.1
DOF = �: Comparison Between 2-D and 1-DInterpolation Schemes.
however. The maneuver consists simultaneously of a required to achieve the same node density as in thehyperbolic tangent-shaped ramp in � and an one-dimensional case. This illustrates the fact thatimpulsive constant roll rate for -. Its trajectory (in reduced parameterization should be used whenever� - - parameter space) is shown in Figure 6, along possible.with the location of the six nodal indicial responsesused (three at - = 0 and three at - = 50). In thisexercise, the deficiency responses are all assumed tobe zero (quasistatic prediction).
In general, it is difficult to make comparisons againstexact solutions in the nonlinear case (i.e., the casewhere the indicial response varies along the dependent variable, denoted DEP . The indicialtrajectory). In some particular cases, however, it is response of DEP with respect to DOF is assumed topossible to obtain such solutions analytically. Oneexample considered here is when the value of theindicial response is proportional to �. In this case,Eq. (2) amounts simply to an integral of the product
. With a zero initial condition, the result mustthen be proportional to � , regardless of the details of2
the path. This is indeed verified, and is illustrated by
the results of Figure 7. Additionally, Figure 7compares the relative accuracies of two interpolationschemes. The first one (curve labeled ‘2d.xg’)corresponds to the full two-dimensional interpolationin � - - parameter space using Shepard quadraticinterpolation. The second scheme (‘1d.xg’) is theresult of projecting all nodes on the � axis (apermissible modeling option, since the indicialresponses are, by construction, not a function of -).In the latter case, the one-dimensional interpolation iscarried out using a cubic spline. The result of theone-dimensional interpolation is seen to be slightlymore accurate than the full two-dimensionalparameterization/interpolation. Since bothinterpolation schemes are capable of representingquadratic behavior exactly, the difference in accuracyis due to the node sparseness factor, i.e., the fact that,in two dimensions, on the order of 6 nodes are2
IR Space Partitioning
We now consider a case where the indicial responsespace is partitioned. The hypothetical system has asingle degree of freedom DOF = � and a single1
1
1 1
be parameterized by and sgn( ), and there are nocritical states. The location of the available nodalindicial responses is indicated in Figure 8. It isassumed that the IR space has two partitions: onecorresponding to � < 35, and the other correspondingto � > 35. For simplicity of interpretation, all nodesin a given partition are chosen to be identical to each
-20 0 20 40 60 80
-15
-10
-5
0
5
10
15
α
α.
-20
0
20
40
60
80
100
120
140
160
0 20 40 60 80
DE
P
DOF
’two_part.xg’’single_part.xg’’const_DR.xg’
’var_DR.xg’
Fig. 8. Example #3: Nodal Responses andManeuver Trajectory in ( �, d�/dt) ParameterSpace. (Dashed vertical line delimits spacepartitions).
Fig. 9. Nonlinear Indicial Prediction: Effect ofParameter Space Partitioning.
other. For � < 35, they are given by , the partition. In other words, ) = ) . The resultingwhile for � > 35, they are given by . prediction, for )3 ≈ 0.785, is shown in Figure 9In each case, the excitation �(t) is assumed to be (curve labelled ‘const_DR.xg’). If, on the other hand,sinusoidal: �(t) = 40 + 40 sin(3t). ) is reduced by a factor of approximately 30, so that
Let us first consider the “fast response” limit () , example, in the case where ) = ) , the T/) ratio1
) → 0, or ). In this case, one would exceeds eight. It can be argued, therefore, that most2
expect DEP (�) to collapse onto a single curve made (> 85%) unsteady effects caused by the previous1
of two linear segments, characterized by partition are “forgotten” by the time the trajectoryd(DEP )/d� = 1 for � < 35, and d(DEP )/d� = 2 for reaches the end lobes, since these are reached1 1
� > 35. This corresponds to the curve labelled approximately two time constants after crossing the‘two_part.xg’ in Figure 9. By contrast, note that, if partition. For )3 ≈ 0.785, theory predicts that thethe partition is removed so that all nodes are amplitude of the output is approximately 80% of itsconsidered together in the interpolation process, thereis a gradual blending of the IRs, as demonstrated bythe curve ‘single_part.xg.’
Consider now the case of an unsteady prediction,where the deficiency responses (once normalized bythe quasistatic IR value) are identical on both sides of
1 2
2
)3 ≈ 0.025 for the right-hand side partition (� > 35),then the result labelled ‘var_DR.xg’ is obtained. Inboth cases, the loops are traveled in the counter-clockwise direction. Note, in particular, that the‘var_DG.xg’ prediction does not immediately join thequasistatic curve, since there are memory effectsassociated with the trajectory having previouslyvisited the left-hand side partition, where strongdynamic effects were present.
As previously mentioned, it is difficult, in theunsteady nonlinear case, to compare the predictionsagainst analytical solutions. However, it is possibleto infer analytical characteristics by reasoning on theinvidual lobes of the dynamic hysteresis curves. For
1 2
quasistatic counterpart. Theory also predicts that theangle of the ellipse will tilt by -23% in the left-handside partition, while for the right-hand side partition,the angle tilts down by less than 5%. The resultingnonlinear ‘folding’ is indeed observed in the figure.The thickness of the ellipses is predicted to bearound 27%. Actual thicknesses (as inferred frommeasurements made at the half major axis location)are around 24%.
Critical State Hysteresis
In this example, we reproduce with the IP code oneof the results published in Reference 9. Anartificially constructed nonlinear system was designedto mimic the rolling moment coefficient response ofthe 65° delta wing undergoing forced roll oscillations.In Reference 9 we considered small amplitudeoscillations in the range of − 4° ≤ - ≤ 8°, for a
support sting angle of 30° and at a freestream Machnumber of three-tenths. Static data taken at fine rollincrements [5,10] suggest the existence of criticalstate transitions at - = 5.20° and - = 4.67° forincreasing and decreasing -, respectively.
The real data were idealized using a nonlinear indicialmodel (described below), and it is this idealized
-5 0 5 10-15
-10
-5
0
5
10
15
φ
f TOTAL
QS
DYN
-5 0 5 10-15
-10
-5
0
5
10
15
ω = 2.0
Fig. 10. Indicial Theoretical Prediction Components fora Nonlinear Dynamic Case with StaticHysteresis. (The curve labeled ` = 2.0'corresponds to a previous prediction made inReference 9 (Fig. 18) using a different code).
model which we consider here. The model is a single For example, the quasistatic and dynamic componentsinput, single output, model. The indicial responses of the prediction are indicated in Figure 10. Noteare parameterized by - (the degree of freedom) only, that the dynamic component includes both the IR andwith nodal indicial responses defined at - = − 4°, CSR deficiency response contributions. A more
−1.3°, 1.6°, 4.6°, 5.3°, and 8.6°. In addition, two detailed analysis reveals that, at the relatively highjump responses are defined at the crossing of critical frequency of the excitation, the net build-up due tostates, i.e., at - = 5.2° for d-/dt > 0, and at jump responses is relatively small, due to aCS
- = 4.7° for d-/dt < 0. The various time constants cancellation effect between consecutive critical states.CS
were chosen so as to qualitatively reproduce some of Both prediction methods assume a piecewise linearthe actual hysteresis loops recorded in this roll angle interpolation between the indicial responses. Thererange [9]. The indicial and critical-state responses are, however, differences in the quadrature methodcontained in the database are as follows: used for time integration. - Type Expression
-4 IR 2.5 − 3.5 exp(−t/1.2)
-1.3 IR -0.5 − 0.5 exp(−t/0.4)
1.6 IR -0.5 − 0.5 exp(−t/0.4)
4.6 IR 1.3529 − 2.3529 exp(−t/0.4)
5.3 IR 1.6 − 2.3833 exp(−t/0.4) − 0.2167 exp(−t/0.6)
8.6 IR 1.6 − 2.6 exp(−t/0.6)
4.7 CSR -2.5 + 12.5 exp(−t) −10 exp(−t/0.76)
5.2 CSR 2.5 − 12.5 exp(−t) +10 exp(−t/0.76)
In Reference 9, the result of extracted indicial andcritical-state responses was used to predict thehysteresis loops associated with various novelmaneuvers. The one considered here is- = 2 + 8 sin(3t), with 3 = 2.0. This corresponds to
the prediction labeled ‘k = 0.0267' in Fig. 18 ofReference 9. This previous prediction is shown in
Figure 10, where it is compared to the predictionmade using the IP code (curve labeled ‘TOTAL’). Inaddition, the output of the code allows theexamination of the various prediction components.
6. ALGORITHM CAPABILITIES
The core computational engines of the indicialprediction module consist of quadrature andinterpolation operations. For a given set of modelingdecisions (parameterizations) and a given databasedensity, the quadrature and interpolation operationsdirectly affect the accuracy of the prediction. It isimportant, therefore, to discuss the various optionsavailable. Each engine is implemented in sharedobject form and, thus, is external to the program,allowing IPS’s versatility to continually improve, asshared objects are added on.
At present, the quadrature shared objects include thefollowing: midpoint rule integration, adaptivetrapezoidal integration, adaptive Simpson integration,a high-order adaptive integration method, and a Diracintegration method by midpoint rule. The latterallows the IPS to be used when the inputs arediscontinuous in time (thus generating Dirac deltas inthe integrand of the convolution integrals), such aswith square wave or step inputs.
The interpolation shared objects are categorizedaccording to dimensionality and whether theinterpolation method is restricted to ordered data ona lattice or functions on scattered nodes. The presentinterpolation capabilities include, in one dimension,piecewise linear interpolation as well as a variety ofunivariate splines (natural, FMM, shape-preserving,improved Akima, and Nielson). For bivariateinterpolation, five different interpolators are provided:four scattered data interpolators, and one sorted nodeinterpolator. The bivariate scattered data interpolatorsinclude: modified Shepard quadratic, SrfPack linear,SrfPack nonlinear, and Akima bivariate cubic. Thesorted data algorithm is a bivariate bicubicinterpolator. A trivariate scattered data interpolatorwill be made available for three-dimensional
0 10 20 30 40 50-0.5
0.0
0.5
1.0
1.5
2.0
∆Cl
α (°)
TOTAL
IRTOT
CSRTOT
0 10 20 30 40 50-0.5
0.0
0.5
1.0
1.5
2.0
csr
ir
total
IP (1997)
mcvJ (1995)
CONSTANT PITCH RATE MANEUVER (q = 0.04)
Fig. 11. Comparison of Indicial Predictions Between‘New’ (IP) Code and ‘Old’ (mcvJ) Code for theNeural Network Simulation of a 60° Pitch UpManeuver (Rectangular Wing) at ConstantRate.
interpolation in the near future. In addition, an ‘TOTAL’), as well as the individual contributions duearbitrary-dimensional scattered data interpolation to the regular (convolution integral) contributionsbased on reciprocal multiquadrics has also been (‘IRTOT’) and the jump (critical-state) responseimplemented. contributions, labeled ‘CSRTOT’.
Together, these capabilities provide the IndicialPrediction System with a wide range of modelingoptions.
7. CODE VALIDATION
The validation of the nonlinear indicial concept haspreviously been discussed in References 6 and 7.Further validation examples are provided here forcompleteness.
The first example concerns the application of the IPprogram to the case of an artificial neural networktrained on wind tunnel data of a pitching wingundergoing dynamic stall [11]. It was previouslyshown [7] that this nonlinear system includes at leastone critical state, which is associated with crossingthe static stall angle. The trained neural network isused here as a nonlinear plant, taken to representaccurately the aerodynamic behavior (five sectionalforce coefficients) associated with a rectangular wingpitching from 0 to 60 degrees.
The nonlinear indicial modeling of this system ischaracterized by the following: (i) a critical statebetween 16 and 17 degrees angle of attack(aerodynamic bifurcation associated with static stall),(ii) nine indicial responses in the region prior toencountering the critical state (� < 16 deg.), and(iii) twenty-nine indicial responses in the post critical-state region. The nodal responses are scattered in atwo-dimensional space characterized by instantaneousangle of attack and pitch rate. In addition, since thecritical-state response changes with pitch rate, it isrepresented using three nodal responses.
Figure 11 compares the prediction made with the newcode (labeled ‘IP (1997)’) to that made using aslightly different treatment of the critical-stateencounter (labeled ‘mcvJ (1995),’ from Ref. 7). Thus,we do not expect perfect agreement between the twomethods. Nevertheless, the scattered two-dimensionalinterpolation method and quadrature method werematched, and the resulting comparisons are shown inFigure 11. The simulated wing motion in this figurecorresponds to a nominally constant pitch ratemaneuver. The figure depicts the total sectional liftcoefficient build-up (thick solid line, labeled
The second validation example is the application ofthe method to the prediction of the rolling moment,C , of a 65−degree sweep delta wing. For thel
dynamic cases discussed here, the delta wing bodyaxis is held at a 30 degree angle to the freestream,and the Mach number is approximately three-tenths.The wing undergoes forced rolling motions -(t), andthe measured aerodynamic force coefficient timehistories are recorded. These data are part of acomprehensive database collected under a jointprogram involving the U.S. Air Force ResearchLaboratory (formerly USAF/WL) and the CanadianInstitute for Aerospace Research (IAR).
The physics of the flowfield and the aerodynamicforces generated by the forced roll oscillations on the65-degree delta wing have been the topic of numerouspapers over the years [4,5,10,12-14]. In particular,the critical states of the rolling moment curve arewell-documented, and have been identified throughdiscontinuities of the static C vs - curve, Figure 12.l
Myatt [12] used parameter identification techniques todetermine the indicial and critical-state responses ofthe rolling moment with respect to roll angle. Thework of Reference 12 provides analyticalapproximations for the critical-state responses at- = −11°, −8.25°, −4°, 5°, 8.5° and 11.3°. Inbetween critical states, the indicial responses assumethe following form:
-3 -2 -1 0 1 2 3-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
-3 -2 -1 0 1 2 3-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
Roll Angle [deg.]
Rol
ling
Mom
ent
ANALYTICAL MODEL
IP PROGRAM
QUASI-STATIC RESPONSE
6 7 8 9 100.010
0.015
0.020
0.025
6 7 8 9 100.010
0.015
0.020
0.025
Roll Angle [deg.]
Rol
ling
Mom
ent
ANALYTICAL MODEL
IP PROGRAM
QUASI-STATIC RESPONSE
QS (analytical model)
Fig. 12. Time-Averaged Static Rolling MomentCoefficient for both Increasing andDecreasing - (from Ref. 10).
Fig. 13. Predicted Rolling Moment Response for- = 0 ± 3 deg., k = 0.02.
Fig. 14. Predicted Rolling Moment Response for- = 8 ± 2 deg., k = 0.14.
where represents the slope of the model, resulting in a computationally efficientquasistatic rolling moment curve with respect to -, differential (rather than integral) form for theand is the vortical component of equations.
. The quasistatic curve is known(fitted) from experiment, and is inferred from Some typical validation results are given in
after calculating the potential flow component Figures 13 through 16, indicating good agreementusing QUADPAN [15]. The nonlinearity in Myatt’s between the two methods. In the figures, the labelmodel comes from the variation in the static slopes ‘analytical model’ refers to Myatt’s prediction.and from the existence of critical states. In betweentwo critical states, the parameters � , � and 0 1
remain constant.
Myatt’s NIR model thus provides a uniqueopportunity to validate the IP code on a problem ofinterest, by using different methods to carry out theprediction calculations. The modeling in Myatt’smethod and in the IP code is identical. By this wemean that, for validation purposes, the IR and CSRnodes used in the IP code are those of Myatt.Furthermore, the parameter space is partitioned in thesame way as in Myatt’s representation (i.e., a total oftwelve critical-state responses: six for positive rollrate, and six for negative roll rate).
The primary difference between the two models isthat, whereas the Myatt model has a complete andcontinuous “knowledge” of the indicial responseseverywhere, the IP program is based on parametricinterpolation of approximately 30 (nodal) indicial
responses, which are known only at discrete values ofthe parameter space). Other differences include thefact that, in IP, the “stationary” limit cycle behavior(in the case of periodic maneuvers) is computed
starting from some assumed equilibrium point.Finally, Myatt’s implementation is based on anequivalent state-space form of the aerodynamic
0 5 10 15-0.010
0.000
0.010
0.020
0.030
0 5 10 15-0.010
0.000
0.010
0.020
0.030
Roll Angle [deg.]
Rol
ling
Mom
ent
ANALYTICAL MODEL
IP PROGRAM
QUASI-STATIC RESPONSE
QS (analytical model)
-10 -5 0 5 10-0.030
-0.020
-0.010
0.000
0.010
0.020
0.030
-10 -5 0 5 10-0.030
-0.020
-0.010
0.000
0.010
0.020
0.030
Roll Angle [deg.]
Rol
ling
Mom
ent
ANALYTICAL MODEL
IP PROGRAM
QUASI-STATIC RESPONSE
QS (analytical model)
0 10 200.0005
0.0010
0.0015
0.0020
0.0025
|err|2
nharm
nharm → ∞ (SMS)
nharm → ∞ (Myatt)
Stochastics [2D]
Myatt
-3 -2 -1 0 1 2 3
-0.01
0.00
0.01
Cl
φ [deg.]
-3 -2 -1 0 1 2 3
-0.01
0.00
0.01
Cl
φ [deg.]-3 -2 -1 0 1 2 3
-0.01
0.00
0.01
Cl
φ [deg.]
Fig. 15. Predicted Rolling Moment Response for- = 8 ± 8 deg., k = 0.14.
Fig. 16 Predicted Rolling Moment Response for- = 0 ± 12 deg., k = 0.08.
Fig. 17. Effect of Noise on Accuracy of Extraction.(Insets depict a typical data set for a givennoise level, characterized by the number ofretained harmonics).
8. RESPONSE KERNEL EXTRACTION / RESULTS
An important capability of the Indicial PredictionSystem is its ability to extract nonlinear indicial andcritical-state responses from empirical data. Such acapability is necessary because it is, in general,difficult to obtain the indicial responses of a systemdirectly. The description of the nonlinear indicial andcritical state extraction scheme will be the topic of aseparate paper [8]. Sample extraction results arepresented here, since it is only together that theprediction (IP) and extraction (IE) modules make theIPS a true data-based prediction method. From afunctional point of view, the method is similar to aneural network: sample maneuvers (input/ouputtransfer functions) can be supplied as “training” data.
These training data are used to extract the responsekernel of indicial and critical-state responses. Thisdatabase kernel, in turn, is used to predict thesystem’s response to arbitrary inputs.
To illustrate how the system works, we will, again,consider the forced rolling motions of the 65-degreedelta wing. This time, however, the indicialresponses are extracted from the data, rather thanusing Myatt’s analytical representation. (The presentresults pertain to a small range of roll angles(−4.05° ≤ - ≤ 5°) without any critical-statetransitions).
First, two essential properties of the extraction methodare illustrated. The first one is robustness withrespect to noise. The second is the convergenceproperty.
Figure 17 depicts the results of a series of errormetrics tests in which the amount of noise in thetraining data is varied in a controlled manner, throughthe number n of retained harmonics. The extractedharm
nodes are then used to predict the very maneuversthey were “trained” on. The resulting total norm−2(rms) error is shown in Figure 17 as a functionof n . For reference, the error associated with theharm
prediction made using Myatt’s IR nodes is indicatedby the dashed line and circles. The insets representtypical data for various levels of filtering. The errorassociated with the extracted nodes is indicated by the
-4 -2 0 2 4-0.010
-0.005
0.000
0.005
0.010
0.015
-4 -2 0 2 4-0.010
-0.005
0.000
0.005
0.010
0.015
-4 -2 0 2 4-0.010
-0.005
0.000
0.005
0.010
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φ [deg.]
Cl dyn
DATA
PREDICTION
PARTIAL PREDICTION
-4 -2 0 2 4-0.010
-0.005
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0.005
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φ [deg.]
Cl dyn
DATA
PREDICTION
f = 7.7 Hz
MYATT PREDICTION
freq.
ampl
.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-4
-2
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2
4
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DATA
SMS-2D, Nφ=6
MYATT, Nφ=10
t [s]
φ [deg.] Cl
Cl
φ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-4
-2
0
2
4
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-4
-2
0
2
4
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
Fig. 18. Nonlinear Indicial Prediction of f = 7.7 Hz,- = ±3° Data Based on Extraction Using(a) Full Data Set (Solid Line), and (b) OneThird of the Data (Short Dashes). (Insetindicates the predicted motion parameterswith respect to those of the partial trainingdata set; Myatt prediction indicated forreference).
Fig. 19. Prediction of - = -4° to +4°, 60°/s RampManeuver. The IE/IP (label: ‘SMS’)Prediction is Based on Harmonic Data Only.
solid line and square symbols. In each case, the The convergence property will be elaborated upon infilled-in grey symbols represent the error when using Reference 8. The sample results of Figures 18the raw data (denoted n → ∞).harm
The error corresponding to the extraction (squaresymbols) is both lower than the error associated withMyatt’s prediction, and remains flat or slightlydecreasing until n > 12. Only after at least 12harm
harmonics are retained in the data does the errorincrease. The flat portion of the curve illustrates thefact that the IR nodes can successfully be extractedfor varying levels of noise. Thus, the method appearsto be robust with respect to noise.
The convergence property is defined as follows. Todemonstrate that the IR extraction is not merely datafitting but does indeed have predictive value, we mustbe able to extract the nodal responses with sufficientaccuracy from a partial data set. Furthermore, theindicial responses must approach those obtained usingthe full data set as the decimation is reduced. Thebasic idea is to attempt to extract indicial responsesusing only a portion of the data available and tosubsequently verify the predictive potential of themethod on maneuvers which were not part of thetraining data set. This methodology is similar to thatused with artificial neural networks and other data-based prediction methods.
and 19 are provided as illustrations. Figure 18compares full- and partial-data set predictions of thedynamic rolling moment response for a harmonicmotion not included in the training. Three predictionsare shown: the first (solid black line) is the IPprediction based on the nodes extracted using all 15harmonic maneuvers available; the second (dashedline) is the prediction when only five maneuvers areincluded in the training data set; the third predictionis the Myatt prediction, which is included forreference. The apparent shift between the data andthe various nonlinear indicial predictions has beenobserved previously, and the exact cause of thisdiscrepancy is not known. (Some of the possibilitiesinclude optically encoded roll angle measurementerror and/or the presence of nonlinear rate effects).The important point of Figure 18 is that the partialprediction, which is based on five maneuvers, isalmost identical to the prediction based on extractionusing the full data set.
Since the use of harmonic motion data for theextraction is known to represent a “worst casescenario” [8], a good test of the method is, first, toextract the nodes using harmonic data, and then toattempt to predict the ramp-and-hold data using theextracted nodes. Figure 19 shows the nonlinearindicial prediction of the rolling moment for a rampmotion from - = −4° to - = +4°, with a maximumroll rate of approximately 60 deg./s. The result of theextraction method (long dashes, labeled ‘SMS’) iscompared in this plot to the reference prediction ofMyatt (short dashes). The SMS extraction-basedprediction uses cubic spline interpolation. A
comparison with the same prediction using linear [4] Jenkins, J. E., Myatt, J. H., and Hanff, E. S.:interpolation (not shown) indicates that the IE/IP “Body-Axis Rolling Motion Critical States of aprediction is accurate, provided that one uses thehigher-order interpolation scheme on the staticcomponent. This is so, because of the relativelysmall number of IR nodes in this region [8]. Withcubic interpolation, the IE/IP prediction is seen to besimilar to Myatt’s. This result is significant, becausethe nodes were extracted on the basis of harmonicdata only.
While predictions using only a partial data set aretypically less accurate, the results of Figures 18and 19 confirm the predictive ability of the method.
9. CONCLUDING REMARKS / SUMMARY
Nonlinear indicial response theory addresses the needfor high-fidelity prediction of nonlinear unsteadyaerodynamic characteristics. The present paperprovides an overview of the Indicial PredictionSystem as a tool kit for the aerodynamicist.Synthetically constructed examples are used toillustrate the modeling capabilities of the system, andcode validation examples are provided. The completesystem (nonlinear response kernel extraction fromexperimental data, followed by indicial prediction fornovel maneuvers) is demonstrated for the rollingmoment of a 65−degree sweep delta wing in rollingmotion.
ACKNOWLEDGMENT
The support of this work by the Air Force ResearchLaboratory Flight Control Division under Phase IISBIR Contract F33615-96-C-3613 is gratefullyacknowledged.
REFERENCES
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[3] Nixon, D.: “Alternative Methods for ModelingUnsteady Transonic Flows,” UnsteadyTransonic Aerodynamics, Vol. 120 of Progressin Astronautics and Aeronautics, Ed. by D.Nixon, AIAA, 1989.
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[10] Grismer, D. S. and Jenkins, J. E.: Critical-StateTransients for a Rolling 65° Delta Wing, AIAAPaper No. 96-2432, June 1996.
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[13] Jenkins, J. E. and Myatt, J. H.: ModelingNonlinear Aerodynamic Loads for AircraftStability and Control Analysis, AGARDReport 789, pp. 13/1-13/10, February 1993.
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