Identification of a Nonlinear Aeroelastic Aircraft Wing Model

Transcription

1 Identifiation of a Nonlinear Aeroelasti Airraft Wing Model Christopher M. Rihards * University of Louisville, Louisville, Kentuky 9 Martin J. Brenner NASA Dryden Flight Researh Center, Edwards, California 95 and Raendra Singh The Ohio State University, Columbus, Ohio The reverse path spetral method, a frequeny domain based nonlinear system identifiation tehnique, is onsidered for identifying nonlinear aeroelasti models. The method is based upon a multi-input/multi-output spetral onditioning proess that mutually unorrelates the linear and nonlinear omponents of the system response. Conditioned frequeny response estimates derived from this proess enable estimation of the system s underlying linear frequeny response matrix without ontaminating effets from the nonlinearities. Analyti funtions for desribing the system s nonlinearities are also identified one the linear frequeny response matrix has been estimated. Sine these analyti funtions must be hosen prior to the identifiation, onditioned oherene funtions, also derived from the spetral onditioning proess, are used to indiate the presene (or absene) of selet nonlinear omponents in the response data, hene overoming the a priori assumption of this and many other parametri-based nonlinear system identifiation tehniques. To illustrate the performane of the identifiation method, numerial data is first simulated from a nonlinear analytial aeroelasti system. Results show that the method is apable of identifying an aurate nonlinear aeroelasti model even when the underlying nonlinearities are unknown. The usefulness of the onditioned oherene funtions for seleting analyti funtions for desribing the nonlinearities is also illustrated. The method is then applied to test data olleted from the Ative Aeroelasti Wing (AAW) airraft, an experimental vehile used to investigate ative flexibility ontrol of airraft wings for improved maneuverability, weight redution and extended range. I. Introdution Auray of analytial/experimental hybrid based flutter predition tools depends on the underlying aeroelasti models that are identified from experimental flight data. Developing an aurate aeroelasti model is often ompliated by nonlinear dynami behavior observed from the experimental data,. This behavior an be the result of various strutural and flow soures, inluding pieewise linear stiffness resulting from strutural disontinuities, nonlinear stiffness arising from large wing displaements and aerodynami flows with large shok motions and flow separation,5. Therefore, nonlinear system identifiation tehniques are neessary in order to take into aount the presene of these types of nonlinearities and others 6. There have been a number of approahes used for identifying nonlinear systems in the reent years,,7,8. For instane, wavelets have shown to be suessful for desribing polynomial type nonlinearities of an aeroelasti model * Assistant Professor, Department of Mehanial Engineering. Member AIAA. Aerospae Engineer, Aerostrutures Branh, MS 8D/RS. Member AIAA. Donald D. Glower Chair. Department of Mehanial Engineering.

2 as well as predit the limit-yle osillation of the same system using time-frequeny maps generated from the wavelet analysis 7. Wavelets have also been used in onuntion with singular-value deomposition, transformedsingular-value deomposition and wavelet based modal parameter estimation for extrating modal parameters from non-stationary data 8. Data onsidered were simulated sinusoidal signals, data from a test-bed nonlinear aeroelasti system, aeroelasti flight data from a drone flutter experiment and data from an F8 flight test. The methods, one using the singular-value deomposition and the other using transformed-singular-value deomposition, provided promising results for all for data ases onsidered. In addition, the Volterra series has been employed to identify nonlinear aeroelasti systems where the first order Volterra kernel, whih is the linear omponent of the system response, is used to improve flutter predition auray by unertainty estimation. In this study, the Reverse Path Spetral Method (RPSM), a frequeny domain based nonlinear system identifiation tehnique, is onsidered 9,. The RPSM utilizes onditioned oherene funtions to indiate the presene of seleted nonlinear omponents in the response data and to indiate the overall auray of the identifiation model. The method is first evaluated using numerially simulated data from an analytial aeroelasti model and is then applied to test data olleted from the Ative Aeroelasti Wing (AAW) airraft. II. Bakground Conventional frequeny response estimation methods suh as the H and H methods often yield frequeny response funtions that are ontaminated by the presene of nonlinearities and hene make it diffiult or impossible to extrat underlying linear system properties from measured multi-input/multi-output (MIMO) experimental data. To overome this diffiulty, a spetral approah for identifying multi-degree-of-freedom nonlinear systems, known as the Reverse Path Spetral Method (RPSM), has been developed 9. The method alulates a onditioned frequeny response matrix H(ω) without ontaminating effets from the nonlinearities. In addition, funtions for desribing the system s nonlinearities are also identified. Derivation of the method begins with the set of oupled frequeny domain nonlinear equations of motion for desribing an N input, M output vibratory system with n nonlinearities: B( ω) X( ω) + n = a Y ( ω) = F( ω) () where ω is frequeny, F(ω) is a vetor ontaining the spetra of the externally applied exitations, X(ω) is a vetor of the system response spetra and B(ω) is the system s linear dynami stiffness matrix. The spetra Y (ω) are Fourier transforms of nonlinear funtions y (x(t)) that are inluded to desribe the nonlinearities and the vetors a ontain oeffiients of these terms. The goal of the RPSM is to identify the underlying linear frequeny response matrix H(ω) (H(ω) = B(ω) - ) of the system without influenes from the nonlinearities, and identify the oeffiients of the nonlinear funtions ontained in the vetors a. To ahieve this goal, a reverse path spetral model as shown in Figure is derived by re-arranging equation () with F(ω) as the output and X(ω) and Y (ω) as the inputs to the model. Using spetral onditioning tehniques, an equivalent onditioned reverse path model is determined and the reverse path model of Figure an be redrawn as a model with unorrelated inputs as shown in Figure, where the inputs Y (-:-) and X (-:n) are mutually unorrelated (note, the dependene on ω has been dropped for the sake of brevity). The subsript in parentheses (-:-) indiates that that spetrum is unorrelated with spetra Y through spetra Y -. Notie that the path between X (-:n) and F (-:n) remains unaltered from the onditioning proess. This path is the linear dynami stiffness matrix B and its input and output vetors are unorrelated with all of the spetra of the nonlinear funtions (the subsript (-:n) indiates that the respetive vetor is unorrelated with all of the nonlinearities Y through Y n ). Therefore, the underlying linear system an be identified without orruption from the nonlinearities. Sine H is desired for modal parameter estimation, the linear path is re-reversed as illustrated in Figure (). Now, any of the onventional frequeny response estimation methods an be modified to estimate H. For instane, the onditioned H estimate is: T H = G XF ( : n) G XX ( : n) ()

3 where alulations for the onditioned power-spetral density (PSD) matries G XX(-:n) and G XF(-:n) are derived in Ref. 9. One H has been estimated, the oeffiient vetors a an also be estimated (see Ref. 9 for development and solution). In addition, oherene funtions have been formulated for the onditioned model of Figure : γ G F ( : ) G X i F ( : n) : ) =, n; γ X n i N G G i F ( : ) =, (a,b) G G F ( =, ( : ) FF X i X i ( : n) FF Equation is the oherene funtion for eah of the nonlinear paths of the unorrelated input model (i.e., the oherene between the inputs Y (:-) and the output F where F is one of the externally applied exitations). Equation is the oherene funtion for eah element of X (-:n) and F. These oherene funtions yield values from zero to one indiating the orrelation between eah input and output of the onditioned model. From these oherene funtions the overall auray of the model is assessed by alulating the umulative oherene for the system: γ Mi ( ω) = γ ( ω) + γ ( ω), i =, N; γ ( ω) = γ ( ω) (a,b) X i F ( : n) YF YF n = F ( : ) Sine all of the inputs of the onditioned model are unorrelated, the umulative oherene, whih is the sum of all of the input/output oherenes, also maintains values between zero and one with values lose to unity at frequenies that the model is aurate and values lose to zero at frequenies that the model is inaurate. Y L F F Y (-) L F F (-) Y (-:) L F F (-:) X( ω) B( ω) F (-:) Y ( ω) a Y (-: -) L if F (-: i-) Y ( ω) a Σ F( ω) Y n(-: n-) Y n ( ω) a n L nf F (-: n-) X (-: n) B F (-: n) F (-:n) H () Figure. Reverse path system model. orrelated input form. unorrelated input form. () forward path of underlying linear system. III. X (-:n) Analytial Study In this setion an analytial aeroelasti model is developed to evaluate the performane of the RPSM. The model adopted for this evaluation is the two-degree-of-freedom pith-plunge airfoil illustrated in Figure, where the translational and rotational strutural stiffnesses of the wing are desribed by nonlinear funtions g(x) and g α (x),

4 respetfully. For the oordinate system hosen (i.e., translational degree-of-freedom measured at the enter-ofgravity), the elastially oupled, inertially unoupled equations of motion are, M && x + ( C + UL) x& + ( K + U H) x + w m x =, b =, M =, C = βk, α a I k ke ρ dc L K =, H = ke kα + ke dα ( ρ dc L L = dα ( o ) [( o n = ( )( a y ( x) = bf ; o o ) ) + π 8 o a, ) dcl dα ] (5) where m and I are the mass and mass moment of inertia about the enter of gravity of the wing, β is the proportionality onstant for this proportionally damped system, and k and k α are the oeffiients of the linear terms of g(w) and g α (α), respetfully. The derivative dcl/dα is assumed to be onstant equal to π, U is the airspeed veloity and ρ is the density of air. The external fore f a is from aileron exitation as desribed later. The nonlinear funtions y (x) are made up of the nonlinear terms of g(w) and g α (α). For a detailed desription of how to develop these equations see Ref. 9. These physial parameters as well as the geometri parameters depited in Figure have been hosen to yield the at rest (U = ) modal parameters listed in Table, where f r is the undamped natural frequeny, ζ r is the damping ratio and φ r is the mode shape of the r th mode. For this evaluation study, U was hosen to be U f / where U f is the flutter speed. U g α (α) α w f a g(w) o e a Figure. Analytial model. r f r (Hz) ζ r φ r.. [.5 ] T.9.5 [ -. ] T Table. Modal parameters of the analytial aeroelasti model for U =. The experimental omponent of the RPSM system identifiation proess involves applying measurable external exitations, in this ase the fore f a generated by aileron exitation, and measuring the resulting system response, i.e., the translation w and rotation α for this example. Sine the purpose of this study is to evaluate the performane of the RPSM for use on experimental data olleted on the AAW, the same type of exitation used during AAW experimental flights is used in this performane evaluation. The exitation applied to the AAW during flight onsisted of Shroeder sweep 5,6 ommands given to the ailerons while simultaneous measurements of aileron displaement and aeleration response of the airraft wings were reorded (among many other measurements). To emulate a similar experiment for this analytial system, the AAW s right aileron displaement measurement multiplied by a gain is used here as the exitation f a.

5 First onsider an airraft wing with linear strutural torsional stiffness (g α (α) = k α α) and asymmetri hardening translational stiffness given by g(w) = kw + a w + a w (6) where the oeffiients of the nonlinear terms are a = MN/m and a = MN/m. Also, initially assume that the types of the nonlinearities that make up the translational stiffness are known. Therefore, for equation () and the reverse path identifiation model depited in Figure, [ aˆ ] T, [ aˆ ] T y x ) = w, y ( x) = w, a = a (7a-d) ( = where â and â are estimates of a and a, respetfully. The goal of the RPSM is to ) estimate the linear frequeny response funtions (matrix H) of the wing so that the underlying modal parameters of the wing an be determined by modal parameter estimation tehniques 7 and ) yield aurate estimates â and â of the atual values for a and a. After arrying out the neessary spetral onditioning, equation () determines the H frequeny response estimates. The estimates are shown in Figure along with the atual underlying linear frequeny response funtions derived from equation (5) with a =, [,n] as well as estimates by the onventional H frequeny response estimation tehnique. By omparing these results, it is apparent that modal parameter estimation would yield erroneous modal parameters from the onventional H estimated FRFs. However, sine the H estimate losely mathes the atual underlying linear frequeny response funtions, aurate modal parameter estimation of the H estimates would result. - - H wf (m/n) - -5 H αf (rad/n) Figure. Magnitude of frequeny response funtions (FRFs). FRF of wing translation to aileron exitation. FRF of wing rotation to aileron exitation., atual FRF;, onventional estimate;, onditioned estimate. Figure is a graph of the onventional oherene funtion, whih provides an indiation of the auray of the H estimate, as well as the umulative oherene funtion (equation ), whih indiates the auray the H estimate. As illustrated, the umulative oherene is muh loser to unity than the onventional unonditioned oherene, indiating that the multi-input/output reverse path model with the proper nonlinear funtions hosen as inputs is more aurate than a linear model whose inputs and outputs are the original unonditioned exitation and nonlinear response. Consequently, under real-world onditions in whih the atual underlying frequeny response funtions are unknown, the umulative oherene is a useful tool for evaluating the auray of the reverse path model. Estimates â and â of the oeffiients of the nonlinearities are plotted in Figure 5(a, b). Although the atual values a and a are onstants and not a funtion of frequeny, their estimates â and â are frequeny dependent sine all of the terms in the equation for determining these estimates are frequeny dependent 9. Therefore, aurate estimates of onstant oeffiients should not vary muh with frequeny, whih is the ase illustrated in the figures.

6 γ M Figure. Coherene funtions for estimates of H wf., onventional oherene for H estimate;, umulative oherene for H estimate..5 5 a (MN/m ).5 a (MN/m ) Figure 5. Coeffiients of nonlinear stiffness terms. quadrati term. ubi term., estimate ;, atual. The assumption that the types of system nonlinearities that exist is known prior to the identifiation is unrealisti for most pratial systems. Therefore, to extend this evaluation, onsider now a nonlinear translational stiffness of the form, g(w) = kw + a sgn(w) w β (8) where β is an unknown non-integer power. For simulation of the system response, a value of β =. is hosen. However, from a system identifiation standpoint, it is unlikely for this type of nonlinearity to be hosen a priori for desribing the wing s nonlinear translational stiffness. Instead, it is ommon pratie to onsider an approximation to the unknown nonlinearity suh as an m th order trunated power series, m g + ( w ) = kw a w (9) + = Using this equation to model the nonlinear translational stiffness, for equation () and the reverse path model depited in Figure : 6

7 [ aˆ ] T x a (a-d) + y ( ) = w, = Considering different values of m, spetral onditioning is arried out and equation () is employed to determine the H frequeny response estimates. Results are shown in Figure 6 along with the atual FRFs. To illustrate the strength of the nonlinearity and its detrimental effets on onventional frequeny response estimation, an H estimate is shown in Figure H αf (rad/n) -5 H αf (rad/n) () H αf (rad/n) Figure 6. Magnitude of H α f FRFs. m =. m =. () m =., atual FRF;, onventional estimate;, onditioned estimate. For m = a signifiant improvement of the frequeny response estimation by the H method over the onventional H method has been made. However, the first mode of the H estimate is shifted slightly higher in frequeny than the first mode of the atual underlying linear frequeny response. This is likely due to the fat that the trunated series for m = is an inomplete representation of the hardening stiffness nonlinearity given by equation (8) and therefore some remaining nonlinear effets are present in the estimation. Additional terms, m = and m =, do however slightly improve the estimate. Figure 7 are plots of the oeffiients of the terms for m =. Notie from Figure 7 that the estimated oeffiient for the quadrati term flutuates about zero whereas the estimated oeffiient for the ubi term flutuates about 5 MN/m. Consequently, the quadrati term does not play a signifiant role in desribing the nonlinearity. This is onsistent with the fat that the atual nonlinearity is symmetri and therefore even order terms of the series add little to no value to the approximation. As a further verifiation, onsider the additional approximations, m =, and m =,. In other words, onsider two additional models, one onsisting only of a quadrati term and the other onsisting only of a ubi term. Cumulative oherene funtions for these two models and the m = model are shown in Figure 8 (note, in 7

8 order to learly illustrate the differenes between the oherene funtions, they are plotted on a sale from.8 to ). As the figure illustrates, the oherene for the m =, model is similar to the m = model, indiating that the quadrati term in the m = model adds little value to the estimation a (MN/m ) a (MN/m ) Figure 7. Coeffiients of nonlinear stiffness terms. quadrati term. ubi term γ M Figure 8. Cumulative oherene funtions for H estimates of H α f., m = model;, m = model;, m =, model. From this analysis, one might onlude that even ordered terms are unneessary for the system identifiation. Therefore, onsider now the odd-order-term-only power series approximation, m + g ( w) = kw a w () + = A sample H estimate is shown in Figure 9 for m = 6 where it is apparent that this higher order approximation has improved the estimation of the first mode of vibration. The exessive noise that appears on the amplitude of the first mode an be suppressed with additional data averaging. 8

9 - H αf (rad/n) Figure 9. Magnitude of FRF of wing rotation to aileron exitation., atual FRF;, onventional estimate;, onditioned estimate using thirteenth order odd-term-only power series approximation. IV. Appliation to AAW Flight Test Data In this setion, data olleted from the Ative Aeroelasti Wing airraft is used to identify an approximate underlying linear dynami model of this airraft by minimizing influenes from aeroelasti nonlinearities. In partiular, data olleted during the following flight onditions are utilized: Mah =.5, altitude = 5, ft. Aileron displaement was used for exitation of the airraft, and therefore is used as the output of the reverse path model. Likewise, aelerations were measured at forward and aft loations at the wing tip; therefore, these aelerations are used as the input X of the reverse path model of Figure. Sine the RPSM is a frequeny domain based system identifiation tehnique, seleting radix- sized data sets enables the omputationally effiieny of the fast Fourier transform for spetral onditioning alulations. Consequently, for this analysis, an automated approah was taken for identifying the most aurate reverse path model from a large number of different ombinations of nonlinear terms. A bath file was oded that performed the neessary alulations onsidering different nonlinear models. To assess the auray of eah model, the areas under the umulative oherene funtions were alulated and a plot of these areas was then used to identify the most aurate models. Of these models, the most aurate was one that inluded the following nonlinearities, + + a x, b ( x x ) (a,b) = = where x is the aft wing tip displaement and x is the forward wing tip displaement. Note that sine aeleration was originally measured, double integration was neessary to obtain wing tip displaements. Also, only displaement dependent nonlinearities are onsidered. Figure ontains sample results from the most aurate model. The onditioned H estimated frequeny response between the aft aelerometer signal (displaement after the double integration) and aileron displaement reveals peaks with larger magnitudes ompared with the frequeny response from the onventional H estimate. In addition, the oherene for these two estimates indiates a more aurate H estimate in the 5 to 9 Hz frequeny range. Keeping in mind that the H estimate does not ompensate for the presene of nonlinearities, then it is possible that the redued magnitude in the H estimate is due to hardening spring strutural nonlinearities of the airraft wing (although fluid-type nonlinearities may also ontribute as suggested below). In the evaluation study of the previous setion, the oeffiients of the atual nonlinearities were independent of frequeny. However, this is not a neessary ondition. In fat, nonlinearities that exhibit memory are frequeny dependent and have been seen in various physial phenomenon suh as wake vorties 8. Sine the RPSM makes no assumption about the frequeny dependene of the nonlinear oeffiients, nonlinearities that exhibit memory are automatially identified. Figure () Illustrates the most dominant oeffiient estimated where frequeny dependene is apparent. This is the oeffiient of the fifth order relative displaement term of equation. The 9

10 oeffiient s dominant frequenies oinide with the frequeny response peak frequenies. Also, this oeffiient is always positive over this frequeny range indiating the presene of a hardening nonlinearity as previously suggested. Note, the magnitude of this oeffiient is in the order of 8 (E is prefix exa). This is not unusual sine the relative response x -x is on the order of - m and when taken to the fifth power is on the order of - m 5. Therefore, although b has a relatively large value the nonlinear term b (x -x ) 5 is only on the order of N. The frequeny dependene of the estimated oeffiient indiates that this nonlinearity might be the result of an aeroelasti nonlinearity with memory..9.8 H f (m/rad) - - γ M () 5 b (EN/m 5 ) Figure. Frequeny responses, oherenes and oeffiient estimate for AAW flight data. Magnitude of FRFs of wing tip aft aelerometer response to aileron displaement. Coherene funtions for wing tip aft aelerometer response to aileron displaement., onventional estimate;, onditioned estimate using model of equation (). () estimate of oeffiient of fifth order relative displaement term of equation. V. Conlusions The Reverse Path Spetral Method has been evaluated using a nonlinear two-degree-of-freedom aeroelasti model. The apability of identifying the nonlinear aeroelasti system even when the exat form of the nonlinearity is unknown was illustrated. Likewise, the usefulness of the umulative oherene funtion (as derived from the onditioned reverse path model) as a utility for indiating the auray of the reverse path model was shown. When applied to Ative Aeroelasti Wing flight data, the RPSM identified a model that yielded better oherene than the onventional H estimate of the same data. The underlying linear frequeny response funtions estimated by the onditioned H method showed peaks with higher amplitudes than the peaks of the onventional H method. Consequently, it is likely that a hardening nonlinearity is present resulting in a suppression of the peak amplitudes. The dominant oeffiient of the identified nonlinear terms was found to be frequeny dependent, indiating an aeroelasti nonlinearity with memory.

11 The wings of the AAW are instrumented with over 5 response sensors, providing a large amount of data to be utilized for nonlinear system identifiation. In addition, many other experiments have been onduted under different flight onditions. Consequently, further work in the appliation of the RPSM for identifying a nonlinear aeroelasti model of the AAW will inlude the use of additional flight response data not onsidered here. Also, beause of the reent suesses of wavelets for desribing nonlinear aeroelasti systems, future modifiation to the RPSM will inlude wavelet funtions as inputs to the reverse path model. VI. Aknowledgement This work was supported by the NASA Kentuky EPSCoR and the NASA Faulty Fellowship programs. VII. Referenes. R. Lind and M. Brenner, Robust Aeroservoelasti Stability Analysis: Flight Test Appliations, Series: Advanes in Industrial Control, Springer-Verlag, R. J. Prazenia, R. Lind and A. Kurdila AIAA Journal of Guidane, Control, and Dynamis 6, - 9. Unertainty Estimation from Volterra Kernels for Robust Flutter Analysis.. R. Lind, R. J. Prazenia, and M. J. Brenner Proeedings of the AIAA Strutures, Strutural Dynamis, and Materials Conferene, Norfolk, VA. AIAA--6. Estimating Nonlinearity using Volterra Kernels in Feedbak with Linear Models.. E. H. Dowell and D. Tang AIAA Journal, Nonlinear aeroelastiity and unsteady aerodynamis. 5. E. Dowell, J. Edwards and T. Strgana AIAA Journal of Airraft, Nonlinear aeroelastiity. 6. B. H. K. Lee, S. J. Prie and Y. S. Wong 999 Progress in Aerospae Sienes 5, 5-. Nonlinear aeroelasti analysis of airfoils: bifuration and haos. 7. R. Lind, K. Snyder and M. Brenner Mehanial Systems and Signal Proessing 5, Wavelet analysis to haraterise non-linearities and predit limit yles of an aeroelasti system. 8. M. J. Brenner Mehanial Systems and Signal Proessing 7, Non-stationary dynamis data analysis with wavelet-svd filtering. 9. C. M. Rihards and R. Singh 998 Journal of Sound and Vibration, Identifiation of multidegree-of-freedom nonlinear systems under random exitations by the reverse path spetral method.. C. M. Rihards and R. Singh 999 Journal of Sound and Vibration, -5. Feasibility of identifying non-linear vibratory systems onsisting of unknown polynomial forms.. E. W. Pendleton, D. Bessette, P. B. Field, G. D. Miller and K. E. Griffin AIAA Journal of Airraft 7, Ative aeroelasti wing flight researh program: tehnial program and model analytial development.. L. D. Mithell 98 Amerian Soiety of Mehanial Engineers, Journal of Mehanial Design, Improved methods for the Fast Fourier Transform (FFT) alulation of the frequeny response funtion.. H. R. Busby, C. Nopporn and R. Singh 986 Journal of Sound and Vibration 8, 5-7. Experimental modal analysis of non-linear systems: a feasibility study.. J. S. Bendat and A. G. Piersol 986 Random Data. Analysis and Measurement Proedures. New York. Wiley-Intersiene, seond edition. 5. K. R. Godfrey, A. H. Tan and H. A. Barker Proeedings of the th IFAC Symposium on System Identifiation. A survey of readily aessible perturbation signals. 6. Rik Pintelon and Johan Shoukens System Identifiation: A Frequeny Domain Approah. IEEE Press. 7. R. J. Allemang and D. L. Brown 998 Journal of Sound and Vibration, -. A unified matrix polynomial approah to modal identifiation. 8. J. S. Bendat, R. N. Coppolino and P. A. Palo 995 International Journal of Non-Linear Mehanis, Identifiation of physial parameters with memory in non-linear systems.