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VALIDATION OF METHODOLOGY FOR ESTIMATING AIRCRAFT UNSTEADY AERODYNAMIC PARAMETERS FROM DYNAMIC WIND TUNNEL TESTS * Patrick C. Murphy NASA Langley Research Center Hampton, Virginia USA 23681 – 2199 † Vladislav Klein The George Washington University NASA Langley Research Center Hampton, Virginia USA 23681 – 2199 IAR inclined-axis rolls or oscillatory coning Abstract J cost function A basic problem in flight dynamics is the k non-dimensional frequency, k= ω l /V mathematical formulation of the aerodynamic model l characteristic length, l = c /2 for aircraft. This study is part of an ongoing effort at PSD power spectral density NASA Langley to develop a more general formulation of the aerodynamic model for aircraft that includes q pitch rate, rad/sec nonlinear unsteady aerodynamics and to develop s Laplace transform variable appropriate test techniques that facilitate identification SF single frequency of these models. A methodology for modeling and t time, sec testing using wide-band inputs to estimate the unsteady V airspeed, m/sec form of the aircraft aerodynamic model was developed v measurement noise previously but advanced test facilities were not available at that time to allow complete validation of WB wide band the methodology. The new model formulation retained Z vector of output measurements the conventional static and rotary dynamic terms but α angle of attack, rad replaced conventional acceleration terms with more β sideslip angle, rad general indicial functions. In this study advanced η state variable in unsteady model testing techniques were utilized to validate the new methodology for modeling. Results of static, θ unknown parameter vector conventional forced oscillation, wide-band forced σ variance oscillation, oscillatory coning, and ramp tests are τ dummy integration variable presented.
τ non-dimesional time constant Nomenclature φ bank angle, rad Only the main symbols are introduced here; other ω angular frequency, rad/sec symbols are defined in the paper.
superscripts, subscripts A, B, C numerator transfer function coefficients estimate (superscript) b wing span, m A amplitude (subscript) a , b indicial function parameters 1, c Aerodynamic derivatives c mean aerodynamic chord, m C ∂ C , C lift and normal force coefficient L N A ( ) C C ∞ ≡ = A A a a a ∂ Cm pitching moment coefficient qc qc c b α β , , ; , , , , , for A L m orN a or α β = = 2 * 2 2 2 V V V Senior Research Engineer, Associate Fellow AIAA 4 V † Professor Emeritus, Associate Fellow AIAA American Institute of Aeronautics and Astronautics effective validation of the new methodology.
Introduction Previously, only conventional forced-oscillation testing results were available for comparison. A limitation of A basic problem in flight dynamics is the conventional modeling and forced-oscillation testing is mathematical formulation of the aerodynamic model that only in-phase and out-of-phase aerodynamic terms for aircraft. Aerodynamicists have investigated this can be estimated. These terms are frequency and problem and in turn the problem of how to test in wind amplitude dependent and represent combinations of tunnels to obtain model parameters since the early days stability derivatives with acceleration terms.
of flight. Today the problem of predicting aerodynamic Consequently, direct comparison of separate estimates response for arbitrary aircraft motion has not been of rotary damping and unsteady terms could not be completely solved. The conventional formulation is to made. More recently, a facility with an advanced assume the aerodynamic forces and moments can be dynamic test capability became available for validating represented by a differentiable function and therefore the new methodology and for further investigation into expanded in Taylor series with only first order linear nonlinear unsteady aerodynamic modeling.
terms (stability and control derivatives) retained [1].
This formulation has only been effective in certain For this study, validation experiments were made portions of the flight envelope where nonlinear or possible by performing tests using an advanced unsteady effects are either not present or relatively dynamic test rig at Rolling Hills Research Company benign. As flight-maneuvering capability has expanded (RHRC), formerly Eidetics Corporation. This test rig so have limitations of the conventional aerodynamic was developed at RHRC through a Navy Phase II model to predict aircraft responses in flight. In SBIR completed in June 2000 [3]. The new rig was addition, as efforts to develop new mathematical designed to allow forced-oscillation, coning, models have proceeded, the limitations of conventional oscillatory coning, and combined-axis rotation test techniques have also been realized.
experiments in the RHRC water tunnel. With a Phase III SBIR, through NASA Langley Research Center, In reference [2] limitations of the conventional further enhancements to maximize dynamic testing linear aerodynamic model for rigid-body aircraft and accuracy and capability were added, including wide- conventional forced-oscillation testing were noted. A band input testing. As part of the effort under the Phase methodology for modeling and testing to estimate an III SBIR, several advanced experiments have been unsteady form of the model was presented. The new performed to apply the new methodology proposed in formulation retained the conventional static and rotary [2] and to run validating experiments. In addition, flow dynamic terms but replaced conventional acceleration visualization experiments, under static and dynamic terms (derivatives with respect to angle of attack rate conditions, have been performed to further support this or sideslip rate) with more general indicial functions.
effort.
In addition, a frequency domain method for data analysis was presented to estimate all the terms in the In this paper one linear unsteady aerodynamic new aerodynamic model at each test condition. The model is postulated for the experiments. However, approach only required the use of a conventional different model structures are presented since each forced-oscillation test rig, although the test rig was lends itself to a different method of identification. Two required to provide wide-band inputs. This approach independent methods are used to identify the model substantially improved upon conventional one- and to validate the estimation techniques, themselves.
frequency-at-a-time forced-oscillation tests by Besides the first method under evaluation from [2], a reducing the test matrix size and providing second method called 2-Step regression [4] is used for substantially more information content. The comparison. Further validation of the estimated model methodology was demonstrated on a longitudinal is shown by testing the prediction capability of the example using a 10% scale model of the F-16XL model on data not used in the identification process.
aircraft.
For these validation tests, predicted and measured responses are compared for two types of experiments: Ideally, validation of any methodology is (1) ramp-up and hold tests at different rates and (2) accomplished by comparisons with independent small-amplitude forced oscillation tests at different identification methods and experimental tests showing frequencies.
the same results and by accurate prediction of aerodynamic response. In this study an effort is made Four different types of experimental data provide to apply a variety of methods and tests that lead to an final validation of the methodology under evaluation.
American Institute of Aeronautics and Astronautics First, wide-band (WB) forced-oscillation experiments where ) ( t C and ) ( t C are the indicial functions, L L α q are performed to allow application of the methodology.
This results in a general unsteady model that can be ) 0 ( C is the initial value of C , l is the characteristic L L validated against the remaining tests. Second, single- length, and V is the airspeed. Two assumptions were frequency (SF) forced-oscillation data are analyzed adopted to allow simplification of the model used in using harmonic analysis to get conventional in-phase the analysis of measured data: a) the effect of ) ( t q on and out-of-phase coefficients. These coefficients can the lift can be neglected and b) the indicial function be compared with those determined directly from the ) ( t C can be expressed as L α unsteady model. Third, inclined-axis oscillatory coning experiments or inclined-axis rolls (IAR) allow t b − ae C t C ) ( ) ( − ∞ = . (2) L L estimation of frequency dependent acceleration terms α α [5]. Separation of these terms, normally combined in The simplified model, which takes into account conventional models, allows for direct comparison increments with respect to steady state conditions, has with that predicted by the unsteady model. Fourth, the form static data are used to confirm the estimation process is working correctly by comparing, for example, lift l α t q C t C t C ) ( ) ( ) ( ) ( ) ( ∞ + ∞ = L L L curve slope from the model with that calculated α q V directly from the static lift curve.
t (3a) τ t b ) ( − − τ τ α d e a ) ( −
Model Postulation ∫
Results from wind tunnel forced-oscillation tests or in operator form show that the resulting combinations of stability derivatives depend on the frequency of the oscillations, l ( ) ( ) ( ) ( ) ( ) C t C t C q t α = ∞ + ∞ amplitude, and mean angle of attack. This dependency L L L q α V contradicts the basic assumption that stability (3b) a derivatives are time invariant. The effect of frequency D α − D b + on the aerodynamic parameters is related to unsteady 1 aerodynamics as explained in [6]. In this reference the where ) ( ∞ C and ( ) C ∞ are the rates of change L L α q aerodynamic model was expressed in the form of state- space equations. Later, in [7], the same problem was with α and q evaluated in steady flow. By introducing addressed by using indicial functions. In [7] the t indicial function was postulated in models for ( ) b t τ − − ( ) ( ) t e d η α τ τ = ∫ aerodynamic coefficients as a simple exponential function the state-space form of (3) is b t (1 ) a e c − + ( ) t b η η α = − + (4) The unknown aerodynamic parameters in the complete l ( ) ( ) ( ) ( ) ( ) ( ) C t C t C q t a t α η = ∞ + ∞ − L L L aerodynamic model can be estimated in different ways. q α V Two approaches have been established, namely, a 2- Applying the Laplace transform to Eq. (4), the step linear regression [4] and a maximum likelihood transfer function for the lift coefficient is obtained as (ML) method [2]. Both approaches are validated by this study.
( ) C s As Bs C + + L = (5) As an example, lift is considered in the form ( ) s s b α + t where s is the Laplace transform parameter and τ τ α τ d t C C t C ) ( ) ( ) 0 ( ) ( − + = L L L α
∫
(1) A t ( ) A C = ∞ L q l V τ τ τ d q t C ) ( ) ( − + q L
∫
V A ( ) ( ) B C a b C = ∞ − + ∞ (6) 1 L L q α V ( ) C b C = ∞ 1 L α American Institute of Aeronautics and Astronautics The next model structure lends itself to the 2-step C a C τ = − (10) 0 1 L L q α regression approach. For a one degree of freedom oscillatory motion with β =0 and where sin( ) t α α ω = ( ) a C a C τ = + − A 0 1 L L q α (7) cos( ) q t α α ω ω ≡ = A As follows from the harmonic analysis of a linear The steady-state solution of (3) is system, the in-phase and out-of-phase components are obtained from the measurement as ( ) sin( ) cos( ) C t C t C t ω ω = + (8) L L L q α n T c ( )sin C C t t dt ω = ∆ ∫ L L where, as shown in [7], the in-phase and out-of-phase α n T α o A c components of ) ( t C are (11) L n T c ( ) cos C C t t dt ω = ∆ ∫ L L 2 2 q k τ kn T α o A c C C a = − L L 2 2 α α 1 k τ + 2 π (9) where T = and n is the number of cycles of c τ ω C C a = − L L 2 2 q q 1 k τ + 1 oscillation. Knowing the in-phase and out-of-phase components at various frequencies ω ω ω 1, 2, …, n, V A where n > 2, the parameters a and τ can be obtained where , b k τ ω = = . 1 1 1 V A by applying the least-squares principle to (10). In the second step τ is assumed known and the least-squares An important element of the methodology is the 1 estimates of ( , , ) a C C follows from model general structure used for the aerodynamic model. It L L q α has a form that retains conventional static and rotary equation (9).
aerodynamic terms that have traditionally provided The second estimation technique uses maximum substantial engineering information to the flight likelihood in the frequency domain. The model dynamics community. Using lift as a representative example for any of the non-dimensional forces and equation is given by (5) after expressing s as j ω . In this moments, the general form can be written as equation case, the model used in the estimation has the form (4). Similar equations can be written for the other force ω ω iB C A + + − and moment equations. This structure allows easy ) ( ) ( ω α ω C = (12) L ω i b + interpretation of the model parameters by retaining conventional stability and control derivatives for static ) ( v ) ( ) ( j j C j z + = , N j ,..., 2 , 1 = (13) L and dynamic terms. Unsteady terms are obtained by solving a first order differential equation with α - where ( ) C ω and ( ) α ω are the Fourier transforms of L dependent coefficients. This approach offers a ) ( t C and ) ( t α , ) ( v j is the measurement noise L straightforward model for simulation.
assumed to be a Gaussian random complex sequence Model Identification 2 with zero mean and variance σ , N is the number of frequencies at which the transformed input output data In model equations (3), (4), (5), and (9) there are are known, and ω is the angular frequency. The four unknown parameters ( , , , ) a b C C or 1 L L q α maximum likelihood estimator minimizes the negative ( , , , ) a C C τ that can be estimated, in general, logarithm of the likelihood function 1 L L q α from measured time histories of α( t), q(t), and C (t). 2 L ˆ min ln ( ; , ) L Z θ θ σ = − (14)
{ } N
The first model equation, for 2-step linear regression, , θ σ is obtained from (9) by expressing where Z = [z(1), z(2), …, z(N)] is a vector of output N 2 2 measurements and θ = [A, B, C, b 1 k τ 1] is the vector of 1 = − 2 2 2 2 unknown parameters. Because (5) is nonlinear in the 1 1 k k τ τ + + 1 1 parameters, the estimation represents a nonlinear estimation problem. The initial values of parameters then American Institute of Aeronautics and Astronautics rolls, and static runs. Each data type facilitates an for this technique were obtained from a linear independent estimate of parameters that can be regression using the cost function compared against that predicted by the general N 2 unsteady model. From tests based on conventional ( ) ( )( ) ( ) ( ) J C j b i A C iB j θ ω ω ω α = + + − − (15)
1 L j j j ∑
single-frequency forced-oscillation data, in-phase and 1 j = out-of-phase components are obtained using (11).
Oscillatory tests using a simple harmonic input These components can be compared with those are usually repeated at different frequencies. If the data obtained from (12) realizing that are to be used to estimate (four) unsteady model ( ) C ω L parameters then six or more frequencies are U iV = + (16) ( ) α ω recommended for better statistical results. In order to avoid the large number of runs, the use of a wide-band where, input was proposed in [2]. Specifically, the Schroeder sweep [8] was selected with specified amplitude to U C = (17) L α provide a flat power spectrum over a specified frequency range. Transforms of the time histories of V C = (18) L the input and aerodynamic coefficients to the q k frequency domain were accomplished using a Discrete Fourier Transform (DFT) algorithm [9]. This algorithm IAR experiments allow separate estimation of utilizes a zoom transform and allows the transform to unsteady acceleration terms. In this case, the unsteady be performed over the frequency range corresponding term in the more general model (last term in (3)) to the wide-band input. provides estimates that compare directly to those estimated from the IAR experiment. This experiment is Model Validation a modification of the traditional rotary balance testing where the axis of rotation is inclined from the velocity Model validation is accomplished, in this study, vector by an angle λ . During the experiment the by considering three kinds of validation tests. The first rotational velocity, Ω , remains constant, whereas the test uses two independent identification methods, angle of attack and sideslip oscillate. For small applied to the same data. Obtaining the same model inclination λ, the changes in α and β are defined as estimates confirms both the model and the estimation techniques. For this comparison, maximum likelihood ( ) cos t t α α λ = + Ω estimation in the frequency domain, using model (12), (19) ( ) sin t t β λ = Ω is applied to the wide-band data. Then a comparison is made with results from the two-step regression method From the output data the frequency dependent applied to the same data. For the two-step method, parameters C and C (where C is one of the A A α A β harmonic analysis is done first to produce the required in-phase and out-of-phase coefficients as inputs. aerodynamic forces or moments) can be estimated.
With a linear aerodynamic model assumed to be given To ensure the methodology has produced an as adequate model, validation data are required to test the predictive ability of the model. These data are ( ) ( , , , ) C t C α β α β = (20) L L additional measurements not used for identification of and assuming small λ , the response (using lift the model under test. For the second validation test, coefficient as an example) in terms of in-phase and comparisons of measured and predicted responses are out-of-phase harmonic components can be written as used. Test data are created using ramp inputs to drive the model at different maximum angular rates to excite cos sin C C t C t λ ω λ ω = + (21) L L L α β unsteady behavior; inputs are applied at various angles of attack. Additional comparisons, to assess the where predictive capability of the model, are made using sinusoidal SF forced-oscillation data at different C C kC = +
( ) L L L
α α frequencies. β (22) The third set of validation tests use measurements C C kC = −
( ) L L L
β β α from conventional SF forced-oscillation, inclined-axis American Institute of Aeronautics and Astronautics validated using the same water tunnel conditions. With As described in reference [5], by performing + - this relaxed requirement, hydrodynamic flow at the oscillatory coning in both Ω and Ω directions, inlet was improved by adding an inlet fairing to block frequency dependent parameters can be estimated as flow into the inlet. Flow visualization and + − [ ] /(2 ) C C C = − − Ω L L L measurement equipment inside the model did not allow α β β (23) enough space for smooth flow through the body.
+ − [ ] /(2 ) C C C = + − Ω L L L α α β Application of the methodology under test required wide-band forced oscillation experiments.
With unsteady terms estimated from oscillatory coning, Wide-band oscillatory data were created using it is possible to separate steady-flow damping Schroeder sweeps in α as an input. Tests were derivatives from the combined out-of-phase damping conducted at 17 mean values of angle of attack, α coefficients determined in conventional forced- 0, oscillation experiments. using an amplitude α = 5 degrees. The range of α - A mean values was from α = 0 to 75 degrees. Data were The last validation test checks the unsteady model sampled at 10 Hz with a lowpass analog filter at 5 Hz.
estimates of force or moment derivatives with respect Tests were repeated eleven times at each angle of to α or β , C or C respectively. This parameter (first A α A β attack and then an average signal was formed using the term in (3)) can be directly compared to that obtained ensemble data to minimize measurement noise. The from static measurements. Comparable values from the ensemble-averaged data was used for data analysis.
two methods provide further validation that the identification process is working correctly. Validation data were created using the same test rig and instrumentation as that used for the wide-band Another test technique, requiring a test capability data. For this paper, only small amplitude ramps and not available for this study, is an experiment with oscillatory motions are considered. In both cases α = A direct heave or sideslip motion. In oscillatory heave 5 degrees. Ramp inputs were generated for three and sideslip testing different non-dimensional pitch rates, (0.01, 0.02, ( ) sin t t α α α ω = + 0.03), starting at five different α ranging from 30 to 0 A (24) 50 degrees. Sinusoidal, single-frequency, forced- ( ) sin t t β β ω = A oscillation data were created at five different non- Using the model structure in (20) and application of dimensional frequencies, (0.05, 0.10, 0.15, 0.20, 0.25), harmonic analysis (11), estimates in-phase and out-of- and at five mean α , ranging from 20 to 60 degrees.
phase coefficients can be obtained. From these results Inclined-axis oscillatory coning experiments were run C and C can be determined. for λ equal to 5 degrees. These tests were completed A α A β for four non-dimensional rotation rates, Ω b/2V = (0.05, Experiments 0.10, 0.15,0.20), and α up to 35 degrees.
Example time histories of input/output data for Advanced dynamic tests were conducted in the application of the methodology are shown in figure 2.
Rolling Hills Research Company Water Tunnel in Angle of attack, normal force and pitching moment order to apply and validate the methodology presented coefficients, α , C , and C , at α = 30 degrees are in [2]. For these tests a 2.5% scale model of the F- N m 0 presented. These plots show displacements relative to 16XL (figure 1) was mounted on a dynamic test rig through a five-component strain-gauge balance (axial starting values at α . The harmonic content of the angle force was not measured). The dynamic test rig is a of attack is shown in figure 3 as a function of computer-controlled system with a sting-mounted frequency. Figure 3 indicates a flat spectrum for a double C-strut support system [3]. The mounting frequencies up to 0.2 Hz which corresponds to reduced arrangement rotated the model about the reference frequencies up to k = 0.42. For the analysis, time center of gravity location of 0.558 c . The tests were histories of inputs and aerodynamic coefficients were conducted at a dynamic pressure of 0.81 psf resulting transformed to the frequency domain using a DFT in the flow velocity of 11 inches/sec and a Reynolds algorithm over the frequency range of 0.003 to 0.2 Hz.
number of 52x10 based on the mean aerodynamic Example time histories of input/output data for chord. Reynolds number values different from that validation are shown in figures 4 and 5. Figure 4 found in flight or wind tunnels are acceptable for this provides example measurements α , C , and C for N m study since the methodology is both applied and conventional forced oscillation (FO) data. This American Institute of Aeronautics and Astronautics example is for oscillations about α = 30 degrees and at both trends between points and 2 σ bounds. In the top o frequencies of k = 0.25 (f = 0.118 Hz). For these graph for α between 30 and 50 degrees, mean values experiments 30 cycles of data were recorded for each for C are in good agreement between the two N α run. An average over all cycles, forming one cycle, is methods. The differences at higher alphas reflect the shown in the figure. Figure 5 shows φ , α , β , C , and limited unsteady information content in the data. The N C time histories for IAR runs at α = 35 degrees. model structure, used in both approaches, assumes the m o Bank angle, φ , indicates that the model undergoes presence of unsteady dynamics. This problem is rotations in the positive direction followed by the same reflected in the tendency for larger error bounds above steady rotations in the negative direction. In this α = 50 degrees for all the estimated parameters. For the example, rotation rate is 25.9 deg/sec producing C term similar results occurred, except the 2-Step Nq oscillation periods for α and β of approximately 13.9 method produced unrealistic negative values above 50 degrees angle of attack. This is likely the result of seconds and Ω b/2V = 0.2. Similar results were limited unsteady dynamics further aggravated by the obtained for side force, rolling and yawing moments.
poor signal-to-noise ratio. A similar plot is obtained for For IAR tests ensemble averages were formed using 15 the model parameter “a” except the ML method is also repetitions of the experiment.
producing larger error bounds for the higher alphas.
Results and Discussion Mean values from the ML method are consistent with expectations that the parameter “a” will become small For this paper, methodology validation was as unsteady behavior reduces for α > 50 degrees. In limited to longitudinal low-amplitude forced- water tunnel facilities motions have much longer time oscillation data and only test cases for angle of attack constants than wind tunnels. Consequently the transfer between 30 and 70 degrees were considered. Unsteady function parameter, b effects, for this water-tunnel model, primarily occur for 1, for the unsteady model is very small. Mean values agree very well between the two 30< α <50 degrees, as will be shown by considering the methods and the larger error bounds for the 2-Step in-phase and out-of-phase coefficients (figure 10).
method reflect the less desirable input data. Mean Both identification methods used in the first value of b over the range 30< α <50 is approximately validation test require transformation of the data to a 0.168.
frequency domain representation. The method in [2], An indication that an adequate model has been based on Maximum Likelihood (ML), uses input and determined is shown by the ability to predict responses output measurements of the experiments almost using data that was not used for identification. Figures directly except for the DFT applied to the data. The 2- 8 and 9 provide this type of validation data. In Step Regression approach [4] requires estimates of the addition, these figures show inputs α (t) and q(t), as in-phase and out-of-phase coefficients as inputs, thus well as transient behavior of state, η (t), defined in (4).
some data processing is also required before Figure 8 shows a representative example of measured application of the method. In order to minimize data and predicted responses of C to ramp-and-hold inputs processing a simple method was chosen to obtain these N performed at different pitch rates. For the example in coefficients. Forming transfer function (16) directly by figure 8, the unsteady model estimated at α = 42.5 dividing outputs C ( ω ) by inputs α ( ω ) in the frequency 0 N degrees is used to model a ramp from α = 40 to 45 domain, the coefficients could be obtained using (17) degrees. The maximum pitch rate achieved during the and (18). This method produces a relatively noisy ramp was / 2 0.03 qc V = or approximately 5 deg/sec frequency response function; however, sufficient information content and signal-to-noise ratio are (model scale). Figure 9 shows a representative present for the analysis. Because of this approach the example of the unsteady model (at α = 40 degrees) 2-Step Regression method produced relatively larger predicting harmonic response of C during N standard errors, reflecting more on the quality of the conventional forced oscillation. For this comparison, inputs rather than the identification method, itself.
the third cycle of oscillation is compared with the measured data to allow start-up transients to die out.
Figure 6 shows the estimates of the four unsteady For this example the harmonic input has a reduced model parameters obtained by ML and 2-Step methods.
frequency of k = 0.2 and an amplitude of 5 degrees.
Circles indicate ML estimates and solid lines are used to show the trends between points. The 2 σ error The third set of validation tests allow various bounds are also plotted as solid vertical lines. 2-Step parameters to be estimated and compared with the estimates are marked by “x” with dotted lines showing general unsteady model. In-phase and out-of-phase American Institute of Aeronautics and Astronautics measurements of C to simulate realistic tunnel data.
coefficients (Fourier coefficients) defined by (11) can N Parameter true values, θ , for the IAR model (20) and be obtained from conventional, single-frequency (SF), ˆ forced-oscillation data. The same parameters can be estimates, θ , of the acceleration terms are given in the estimated from the general model by using (17-18).
table below.
This comparison is shown in figure 10 for three C C C C reduced frequencies. The general unsteady model N α N β N α N β results are provided over the α range shown, however θ 1.0 -0.3 5.0 18.0 the coefficients from SF forced-oscillation data are only computed at α = (20, 30, 40, 50, 60) degrees and ˆ --- --- 4.98 17.87 θ are shown as triangles. This figure shows the strong frequency dependence or unsteady behavior of the The top half of figure 13 shows mean values for 5 aerodynamics for α greater than 30 degrees and less repeated measurements of static C with 2 σ error N than 50 degrees. The agreement between the two bounds. The second graph shows the corresponding modeling methods is very good within the range of α mean values and 2 σ bounds for C estimated by the N α where unsteady behavior occurs. At α = 30 degrees, general unsteady model and by a simple gradient where little unsteady behavior occurs, only a small method using the static data above. The two methods difference is apparent.
generally agree well and have reasonable error bounds.
Although some difference in mean values occur at α = Oscillatory coning or IAR experiments produced 40 degrees where a steep gradient occurs in C . This is N data shown in figure 5. The advantage of harmonic not surprising since the gradient can vary sharply in IAR experiments is that unsteady acceleration terms this region and this is a region of peak unsteady can be readily extracted from this type of data. The behavior. Some unsteady behavior was observed dynamic rig in this example is capable of during static measurements as well.
approximately 406 degrees of rotation before the wires leading to the balance reach their limits. Consequently, Concluding Remarks the motion begins at a minimum φ = –406 degrees and This study is part of an ongoing effort at NASA continues until a maximum of φ = +406 degrees. At Langley to develop a more general formulation of the this point the rig must stop and reverse direction. It aerodynamic model for aircraft that includes nonlinear appears that the measured responses may not be in unsteady aerodynamics. In this study independent steady harmonic motion since the lower peaks of C N identification methods and a series of different are changing with each cycle. This implies that the dynamic tests were used to show the validity of a initial transient has not decayed sufficiently to reach methodology for modeling and testing to estimate steady harmonic oscillation where each peak would linear unsteady models for aircraft. Independent have approximately the same amplitude. The analysis identification methods applied to the same wide-band defined in (23) assumes steady harmonic data and data produced the same model parameter values therefore is not appropriate for this type of data. A indicating legitimacy of the two approaches. The transient analysis of this data will be part of the next unsteady model successfully predicted transient phase of this study and comparisons can then be made dynamics that occurred in ramp and hold as well as with results using the ML methodology in [2]. Figure harmonic experiments providing validation that the 11shows estimates of ( , ) C k α from the WB N α methodology produces an adequate model. Static experiment using the ML approach. This can be derivatives and Fourier coefficients were well considered as the unsteady equivalent to the predicted by the model further validating the conventional derivative, / ( / 2 ) C c V α ∂ ∂ .
N methodology for modeling unsteady aircraft behavior.
Simulated inclined-axis results demonstrated a To demonstrate the harmonic analysis proposed technique for estimating unsteady acceleration terms.
in [5] for IAR experiments, representative model parameters were chosen and simulated data were References prepared as shown in figure 12. Time histories of α , β , 1. Etkin, B.: Dynamics of Atmospheric Flight, and C are shown for steady harmonic motion in both N John Wiley & Sons, Inc., New York, 1972.
the positive and negative roll directions. Oscillations 2. Murphy, P. C., Klein, V.: Estimation of Aircraft occur at a reduced frequency of k = ω b/2V = 0.05. A Unsteady Aerodynamic Parameters From moderate amount of noise was added to the American Institute of Aeronautics and Astronautics Dynamic Wind Tunnel Testing, AIAA Paper 2001-4016, August 2001.
3. Eidetics Corporation: Final Report; Determination of Nonlinear Dynamic (t) (deg) Aerodynamic Coefficients for Aircraft, SBIR 0 α Phase II Final Report, Naval Air Warfare Center −5 Aircraft Division, TR00-002, June 11, 2000.
0 50 100 150 200 250 300 350 4. Abramov, N. B, Goman, M. G., Greenwell, D.
0.6 I., and Khrabrov, A. N.: Two-Step Linear Regression Method for Identification of High 0.4 Incidence Unsteady Aerodynamic Model. AIAA (t) 0.2 N C Paper 2001-4080, August 2001.
5. Khrabrov, A., Kolinko, K., Miatov, O., −0.2 Vinogradov, J., Zhuk, A.: Using of Oscillatory 0 50 100 150 200 250 300 350 Conning Experimental Rig for Separation of Rotary and Unsteady Aerodynamic 0.15 th Dervivatives, 18 ICIASF, Toulouse, France, 0.1 June 14-17, 1999.
(t) 0.05 m 6. Goman, M. and Khrabrov, A.: State-Space C Representation of Aerodynamic Characteristics at High Angles of Attack. Journal of Aircraft , −0.05 Vol. 31, No. 5, Sept.–Oct. 1994, pp. 1109–1115. 0 50 100 150 200 250 300 350 time (sec) 7. Klein, Vladislav and Noderer, Keith D.: Figure 2. Time histories of angle of attack, lift, and Modeling of Aircraft Unsteady Aerodynamic pitch-moment coefficients for α =30 degrees during Characteristics. Part 2-Parameters Estimated wide-band input exeperiment in water tunnel.
from Wind Tunnel Data. NASA TM 110161, 1995.
8. Schroeder, M.R.: Synthesis of Low-Peak-Factor Signals and Binary Sequences with Low Autocorrelation, IEEE Transactions on Information Theory, January 1970, pp. 85-89.
9. Morelli, Eugene A.: High Accuracy Evaluation of the Finite Fourier Transform Using Sampled Data, NASA TM 110340, June 1997.
−10 −20 Figures PSD Magnitude (dB) −30 −40 −50 −60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 frequency, Hz Figure 3. Harmonic content of transformed α wide- band input for α = 30 degrees in water tunnel.
Figure 1. Three-view drawing of 2.5% F-16XL model.
American Institute of Aeronautics and Astronautics maximum likelihood 2−step aoa (deg) ) 0 1 2 3 4 5 6 7 8 9 ∞ ( α N 0 C N 1.5 C −1 −2 0 1 2 3 4 5 6 7 8 9 0.25 −3 0.2 30 35 40 45 50 55 60 65 70 75 m C 0.15 maximum likelihood 0.1 2−step 0 1 2 3 4 5 6 7 8 9 time (sec) Figure 4. Conventional FO experiment measurements ) at k = 0.25 and α = 30 degrees.
o ∞ ( Nq C −1 (deg) φ −2 −500 0 10 20 30 40 50 60 70 −3 30 35 40 45 50 55 60 65 70 75 (deg) α 0 10 20 30 40 50 60 70 −1 0 a −2 (deg) β −5 −3 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0.25 −4 maximum likelihood m 2−step 0.2 C −5 30 35 40 45 50 55 60 65 70 75 0.15 0 10 20 30 40 50 60 70 4 maximum likelihood 2−step N 1.5 C 0 10 20 30 40 50 60 70 b time (sec) Figure 5. Input and output measurements for IAR −1 experiments at k=0.2 and α = 35 degrees.
o −2 −3 30 35 40 45 50 55 60 65 70 75 (deg) α (deg) Figure 6. Estimated parameters and their 2 σ confidence bounds for Normal force coefficient.
American Institute of Aeronautics and Astronautics 6 15 α (t) (deg) k=0.1 10 k=0.15 q(t) (deg/sec) k=0.2 .
(t)), q(t) α .
0 α N C −2 0 5 10 15 20 25 30 35 40 45 50 0.1 −5 20 25 30 35 40 45 50 55 60 65 70 (t) 0.05 η α (deg) Figure 11. Variation of C with α and k.
N α 0 5 10 15 20 25 30 35 40 45 50 1.8 predicted ) α (deg) Ω 1.7 measured (t) N β (deg) , (+ 1.6 C β , 1.5 α −5 0 5 10 15 20 25 30 35 40 45 50 0 50 100 150 200 250 300 350 time (sec) ) α (deg) Figure 8. Measured and predicted C (t) for ramp input Ω N β (deg) at non-dimensional maximum pitch rate = 0.03. , (− β , α −5 α (t) (deg) 0 50 100 150 200 250 300 350 q(t) (deg/sec) 0.2 (t)), q(t) 0 C (+ Ω ) α N N C (− Ω ) −5 C N 0 2 4 6 8 10 12 0.1 −0.2 0 50 100 150 200 250 300 350 (t) 0 time (sec) η Figure 12. Input and output measurements for IAR −0.1 simulated experiment at k=0.04 and α = 35 degrees.
o 0 2 4 6 8 10 12 predicted (t) measured 1.5 N C 1.5 N 1 C 0 2 4 6 8 10 12 time (sec) Figure 9. Measured and predicted C (t) for harmonic N 0.5 20 25 30 35 40 45 50 55 60 65 input at k = 0.2, and α = 5 deg.
A α α N N C C 1 static data indicial model −5 20 25 30 35 40 45 50 55 60 65 70 20 25 30 35 40 45 50 55 60 65 WB: k=0.1 α (deg) WB: k=0.15 Figure 13. C estimated from wide-band data and WB: k=0.2 N α SF: k=0.10 static data.
SF: k=0.15 Nq SF: k=0.20 C 20 25 30 35 40 45 50 55 60 65 70 α (deg) Figure 10. In-phase and out-of-phase coefficients estimated from wide-band and single harmonic data.
American Institute of Aeronautics and Astronautics