dynamic aeroelastic response of adaptable airfoils using neural networks

8/7/2019 Dynamic Aeroelastic Response of Adaptable Airfoils Using Neural Networks

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The inuence of each panel on every other panel on theairfoil is calculated. For unsteady calculations, the in-uence of the wake on the airfoil is very important. Atime-marching scheme is used and at each time step, onedoublet panel is shed into the wake from the airfoil trail-ing edge. The combined inuence of all the wake panelson the airfoil panels is calculated for every time step.

The fundamental derivation for panel methods comesfrom the governing Laplacian equation for incompressiblepotential ow, namely,

(2)

where is the velocity potential. Equation 2 is valid insteady and unsteady incompressible ow. Using Greenstheorem and Eq. (2), it can be proved that , any surface inincompressible potential ow can be replaced by a sum-mation of doublets and sources. Typically in thin airfoiltheory, a doublet represents the lifting component and asource represents the airfoil thickness. To make the anal-ysis simpler, we are considering only doublet representa-tions over the airfoil. According to Ref. 1, the absenceof source panels is not a disadvantage for lifting airfoils.The use of smart materials in wings allows the active con-trol of the deections of the wing structure. These adap-tive structures are preliminary models of biological struc-tures. One of the properties of biological structures isthat they can possess time varying stiffness while execut-ing movements . By using adaptive material in aircraftwings, it is possible to introduce this property of timevarying stiffness. It is also possible to utilize the con-cept of a variable stiffness spar to change the torsionalstiffness of the wing. The time-varying stiffness is usedherein to control airfoil oscillations. It is also advanta-geous to have a variable stiffness wing in the case of mor-phing wings as this allows the wing to hold the load whilestill being exible enough to allow morphing of the airfoilfor different ight regimes.

Variable coefcient differential equations present manydifculties in obtaining analytical or efcient numericalsolutions. Since time varying coefcients are now presentin the aeroelastic equation of dynamic equilibrium, an ef-cient numerical scheme to solve the structural dynamicproblem is needed. A very accurate and computationallyefcient scheme of Matrix Exponential time marching as

described in Ref. 5 is used. This scheme allows the use of time steps of the order of . This is especially usefulin training neural networks since the time step being large,the data set used in the training can be reduced. By train-ing neural networks using the method Matrix Exponentialtime marching and the doublet panel method, a 2-D aeroe-lastic analysis is performed wherein the aeroelastic modelpossesses pitch and plunge degrees of freedom.

Steady and Unsteady Aerodynamics

Before computing unsteady aerodynamics, steady statecomputations are performed usingdistributed doublets overthe surface of the airfoil with an innite wake. Figure1 describes the distribution of pressure over a symmetricairfoil at an angle of attack of . This matches the the-oretical solution everywhere over the airfoil except at thetrailing edge. This discrepancy near the trailing edge isdue to numerical problems of satisfying the Kutta condi-tion exactly at the trailing edge . The Kutta condition issatised numerically by forcing the vorticity at the trailingedge due to the doublet panels to be zero. Theoretically,the velocity of ow should be zero at the trailing edge, aslong as the trailing edge is not a cusp. This is difcult torealize numerically without manipulating the trailing edgepanel. However, the trailing edge coefcient of pressureis still nite and close to unity.

Even though the trailing edge creates some numeri-cal problems, the code is able to replicate the steady statepressure. Hence we now try to model unsteady aerody-namics of a symmetric airfoil as the next step. The theo-retical solution to the unsteady aerodynamic lift and pitch-ing moment on an airfoil in inviscid ow is calculated us-ing convolution integrals of the Wagner function . Physi-cally, the Wagner function is a measure of the circulatorylift on the airfoil due to a unit step change in angle of at-tack. It is given as

(3)

Here is the reduced frequency, is the non-dimensionaltime, is termed the Theodorsen function which is aratio of Hankel functions of the second kind. Reference6 provides more details of the Theodorsen and Wagnerfunctions. The Wagner function has a value of 0.5 justafter the impulse has been imparted and then grows to asteady state value of 1 for innite time. We test the dou-blet panel code for the response of a symmetric airfoil toan impulsive angle of attack. Figure 2 shows the growthof the lift with time as predicted by the present code. Itcan be seen that the solution approaches the Wagner solu-tion. However, the initial impulsive start that is obtainedusing the theoretical result based on the Wagner functionis not attainable by the doublet panel code. This is be-cause the doublet panel code is a time domain solution.The Wagner function is derived in frequency domain andhence it is able to represent the impulsive jump in the liftat initial time exactly. A time domain code on the otherhand takes a nite time, no matter how small, to approx-imate this jump in lift. However, it should be noted thatboth solutions approach the same steady state solution.

2American Institute of Aeronautics and Astronautics


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The response shown in Fig 2 is similar to that obtainedby Basu for a symmetric Von-Mises airfoil. Since in thetime domain, an impulse is provided over a time step, theresponse of the system as shown in Fig. 2 will change de-pending on the time step size. Furthermore, theoretically,the angular velocity of the airfoil is innite at zero time.A time domain result will result in a nite velocity andnite inertial effects at the rst time step. The time stepused in the time stepping scheme determines the vortexshedding frequency into the wake. Hence it is required toselect an optimum time step that captures the effect of thesudden change in the angle of attack on the aerodynam-ics and the subsequent growth of circulation. Reference8 provides the details for the discrepancies involved be-tween the Wagner solution and the solution for differenttypes of airfoil subjected to an impulsive pitch rate.

This doublet panel method can now represent the un-steadyaerodynamicsover a pitching airfoil. Figure 3 showsthe unsteady lift over a symmetric airfoil with a variableamplitude pitching motion. The theoretical solution forthe problem in Fig. 3 is obtained using a convolution of the Wagner function with the downwash velocity. How-ever, it was seen previously (Fig. 2), that the time do-main solution does not provide the same response to animpulsive change in angle of attack as computed fromthe theoretical Wagner solution. Hence the convolutionof the Wagner impulse response with sinusoidal functionsdoes show some differences with the time domain so-lution obtained from the doublet panel code for a sinu-soidally pitching airfoil as can be seen from the compari-son made in Fig. 3, but the two results are still similar.

Neural Network Representations

Figure 4 shows the essential constituents of a neural net-work. The concept of neural networks is an attempt atan articial simulation of the functioning of the humanbrain . Accordingly, an articial neural network like theone shown in Fig. 4 is a grouping of articial neuronsin different layers. These neurons are mathematical op-erators. The input to the neuron is multiplied by a quan-tity known as the weight. Any additional input to thisweighted neuron output is termed as a bias. Design of aneural network consists of arranging these neurons in

different layers, deciding on their connectivity, deningthe inputs and outputs of the system and determining thestrength of each connection in between the neurons. Theoutput of each layer of neurons is fed to the next layer of neurons through a selected function. The strength of theneuron connections is based on training a neural network on a series of data sets. Once a neural network has beentrained using sufcient data, then the network is capable

of providing the desired output given a new set of inputdata.

One can designate a subset of the nodes of a neuralnetwork as input nodes and another subset as outputnodes, whose states represent the models results. Onecan then t the input-output mapping into the networksconnection strengths such that when any particular set of input states is applied, the dynamics of the network gen-erates the desired output states at the output nodes. Themain constituents of the neural network are the weightswhich multiply each input and the functional operationperformed with each such weighted input. For exampleif the functional operation, which happens to be thetransfer function of the rst layer, is a linear operator, thenthe output of the rst layer is a weighted sum of the inputs,i.e., the output of the rst layer is

On the other hand, if the operator happened to be anon-linear function such as a Sigmoidal function, then theoutput of the rst layer could be

Thus depending on the layer transfer function, a broadfamily of system representations can be made. Further-more, as seen in Fig. 4, each layer output is connected toanother layer for more functional operations. Thus the un-derlying mathematical representation of a neural network can become quite complicated.

Neural network representations of dynamical systemsapproximate the response of the system and automaticallypossess the impulse response for that dynamical system .During the neural network training, the input data is mappedto the desired output target. The training starts with aninitial assignment of the neuron weights, biases and layertransfer function functions. This initial processing of theinput data through the network produces an output whichmost likely is not the desired target. Hence an error isnow calculated between the neural network output and thedesired target. A gradient based scheme such as the conjugate gradient method is now implemented to minimizethis error of the neural network output. This means thatthe derivatives of the error in the network with respect tothe neuron weights are determined for each iteration inthe training process. Training of the neural network is avery important task and care should be taken such that theneural network is able to reproduce the target data with-out over-tting. This means that the error margin of theneural network away from the target data should not besignicantly greater than near the target data.

In this work, only deterministic neural networks areconsidered. Here we consider both feed-forward and re-current neural networks.

1. Feed Forward Networks : In this type of neural



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network architecture, the ow of data is always uni-directional, that is, the input of one layer of theneural networks is obtained only from the previ-ous layer. However, during the training process,the neural networks require back propagation tech-niques, that is, the weights of a layer are adjusteddepending on the output of the next layer. Feed for-ward neural networks are generally used for input-output mapping where the mapping is not time de-pendent. However, dynamical systems can also bemodeled using feed forward networks if appropri-ate time domain training routines are used to repre-sent the system.

2. Recurrent Neural networks : Recurrent neural net-works have the output of the network fed back toprevious layers to form a new input of the network.Hence recurrent neural networks are able to rep-resent time delays and transient response. Theseneural networks can represent dynamical systems.Some of the popular recurrent neural networks arethe Hopeld, Elman, Jordan networks. The namesare based on the developers of the respective net-works. A stable attractor of a dynamical system iseither a trajectory or a xed point such that any ini-tial condition in the vicinity of this attractor will al-ways converge towards it in forward time. Hopeld

networks are capable of converging to a stable at-tractor of the dynamical systemgiven an initial con-ditions set. For obtaining the transient responseof the system, Elman or Jordan networks are used.These networks have theoutput of the previous timestep as an additional input . Jordan and Elmanneural networks differ only by the layer from whichthe feedback occurs. In Jordan type networks theoutput of one time state is fed-back to the inputlayer for the next time state. In Elman neural net-work, the output is fed-back from the hidden layerto the input layer.

Aeroelastic Applications

Aeroelastic calculations involve uid-structure coupling.This means that every displacement of the structure underuid dynamic loads, as predicted by a structural solver,will in-turn alter the uid dynamic response and vice-versa. This coupling between the aerodynamics and struc-tural behavior can lead to very large computational costsfor an aeroelastic analysis that requires use of compu-tational uid dynamics/computational structural mechan-ics. Hence efcient alternative methods that can providerapid aeroelastic response solutionsneed to be implemented.

Most applications of neural networks in aeroelastic-ity have been in system parameter estimation for aircraftcontrol or in utter speed prediction and utter controlfrom experimental data . Neural network controllershave been designed for non-linear vibration control of he-licopter rotor blades using a 2-D aeroelastic model. Arecent work in the application of neural networks to staticnon-linear aeroelastic problems in morphing wings de-scribes the procedure of aeroelastic optimization of anadaptable bump on the surface of an airfoil. Here we uti-lize trained articial neural networks to provide a rapidsolver to simulate the dynamic response of a time-variant2-D aeroelastic system.

Systemdynamics can be represented by breaking downthe time domain of interest into small time intervals andtraining a neural network to represent the response of thesystem for each of these time intervals. Thus the dynami-cal system is modeled by many neural networks with con-stant weights, each neural network representing one timeinterval in the dynamical system. An equivalent model-ing is to have a single neural network with time variantweights, that is the neural network is itself a time vari-ant system. As an example consider a system with timevarying stiffness, modeled as

(4)

Figure 5 compares the response from a 3 layer El-man recurrent network with the numerical solution ob-tained using a Runge-Kutta time marching scheme. Herethe neural network uses time intervals of 0.04 seconds foreach training sequence. The comparison is agreeable ex-cept for noise that is initially present and also there aresome differences in the maximum amplitude. Presence of noise is typical of feedback neural networks and is one of the disadvantages of such neural networks. Hence in or-der to present a better match between the results as givenby a neural network and those by the time varying dy-namical system, it is necessary to use smaller time stepsthan the 0.04 second step used for this example. However,use of smaller time steps increases the size of the data setrequired for the network training. Large quantities of in-put data do not allow for accurate ANN training. Henceinstead of using reduced time steps to make the neuralnetwork representation of the time varying system more

accurate, it is more reasonable to study accurate numer-ical solvers for linear time varying systems that can uselarge time steps. The ANNS can then be trained by usingthese efcient numerical solvers.

The system shown in Fig. 5 is subjected to a har-monic loading. If this loading is now replaced by a non-conservative time varying load, then this becomes a onedegree of freedom aeroelastic system. Initially we shall



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consider the system is time invariant, that is the stiffnessis taken to be a constant. The governing equation is then

(5)

where is the airfoil pitching angle, is the un-steady pitching moment. The airfoil is given an initial

pitch displacement and this sets off the unsteady aero-dynamic code which provides the instantaneous pitchingmoment. A structural dynamics code will now be devel-oped which will be coupled to the panel code in order tocalculate the instantaneous response of the airfoil for ev-ery change in the unsteady pitching moment.

This aeroelastic response calculation is again repeatedfor a time variant dynamic system, that is, the rotary stiff-ness is a function of time. This variable stiffness is use-ful in control of airfoil oscillations. The model of thetime varying stiffness is exponentially varying such as inRef. 18. where the use of adaptive materials in struc-tures to allow for active control of structural deection is

highlighted. Recent advances in variable stiffness wings

also allow for such exponentially saturating stiffness vari-ation. Accordingly the time varying torsional stiffness isassumed as

(6)

where

is the initial torsional stiffness and

isthe maximum torsional stiffness. The rate of variation iscontrolled by the two parameters

and

.Training of recurrent neural networks based on these

numerical results for dynamic aeroelastic problems re-quires that the time step be small. Otherwise, presenceof noise could be a problem. Hence the use of recurrentneural networks trained by such time stepping schemeswould be computationally expensive and would requirelarge computer memory. Feed-forward neural networkscan also be used for dynamic aeroelastic problems, pro-vided a suitable training algorithm is developed. We in-vestigate rst a stable time integration scheme for lineartime varying systems using large time steps. The utiliza-tion of such a schemefor the training of feed-forward neu-ral networks to represent the dynamic aeroelastic systemis further investigated.

Response of Linear Time VaryingDynamical Systems

Linear Time Varying (LTV) systems can be analyzed bythe method of Matrix Exponential time marching . Thoughan analytical solution is described in Ref. 5 for single de-gree of freedom systems, this work can be extended tomulti-degree of freedom systems numerically. To do so,

we rst look at the 2-D aeroelastic system modeled by thefollowing equations.

(7)

where is the mass, is the mass moment of inertia,

is the pitch angle, is the rst moment of inertia, isthe plunge displacement,

and

are the stiffness inplunge and pitch respectively. A schematic representationof this aeroelastic system is shown in Fig. 6. This systempossesses a time varying pitch stiffness. In state spaceform, this system can be re-written as

(8)

Here

and

Using the method of matrix exponential time march-ing, the solution to this system for a time interval is writ-ten as

(9)

This system can be solved as

(10)Here the time interval,

, chosen for each solution inthe time marching scheme is xed with respect to mini-mizing the errors due to the non-commutative matrix mul-tiplication, i.e.

(11)

where is the error tolerance level.Equation 10 can be written as

(12)

where

and

.This means the response at time

is linked to the re-sponse at time

by the state transition matrix at time

, namely

and the forcing integral vector

.



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This type of representation was shown to be extremelyaccurate for the analysis of time varying systems in Ref.5. Herein, Eq. (4) concerning the periodically time vary-ing system is solved using the method given by Eq. (12)to evaluate the accuracy of this matrix exponential time-marching method. The response of the system representedby Eq. (4) is shown in Fig. 7 which clearly shows an ex-cellent match between the theoretical result and the Ma-trix Exponential time marching approach given by Eq.(12). The time step of 0.1 second was used for solving Eq.(4) by the methodology described by Eq. (12). Hence thistechnique of Matrix Exponential time marching is ableto use large time steps and still obtain very accurate re-sults. This is very advantageous in training neural net-works since the data set required to represent the dynam-ical system for the time domain of interest can be madeto be smaller in comparison with the recurrent neural net-works such as the one used in calculating the response of Eq. (4) shown in Fig. 5.

In the present neuralnetwork representation, three neu-ral networks will be trained to represent the aeroelasticsystem governed by Eq. (7). The rst ANN shall repre-sent the state transition matrix, which depends on thesystem parameters and the time interval used for its def-inition. Another ANN is trained to represent the aerody-namic loads during the concerned time interval of interest.The third ANN represents the forcing integral, forthe time interval,

to

. These three neural networkswhen coupled to each other can be trained to represent afeed-forward array for dynamic aeroelastic analysis withtime varying torsional stiffness.

Neural Network Training

The training of the neural network is the most importantprocess in the representation of any system using neuralnetworks. Some of the gradient based methods for train-ing of ANNs are the method of steepest descent, conjugate gradient method or Gauss-Newton methods. Thesteepest descent and conjugate gradient methods are rstorder methods that require the computation of the Jaco-bian of the estimated error in the output with respect tosome system parameters. These methods can be writtenin the following form

(13)

Here, the output at iteration is dependent on a learn-ing rate

and the Jacobian matrix

constitutes the rstderivatives of the error in the neural network output withrespect to the neuron weights. The error of the neuralnetwork is dened as the

norm of the difference be-tween the neural network output and the desired target.

The learning rate is a constant for the steepest descentmethod. Conjugate gradient methods choose the direc-tion of the gradient based on certain conditions. Two pop-ular Conjugate gradient schemes are the Fletcher-Reevesand the Polak-Ribiere method. More information on theseschemes can be found in Ref. 19.

A second order gradient based scheme is the Gauss-Newton method, a popular version of which is theLevenberg-Marquardt method. Here the rst and secondderivatives of the error in the neural network output needsto be estimated. Thus the Jacobian and Hessian matri-ces needs to be evaluated. The computation of the matrixof the second derivatives of the error with respect to theneuron weights constitutes the Hessian matrix. It is clearthat the Gauss-Newton second order methods are com-putationally expensive and need a signicant amount of system memory. However they are very reliable numeri-cal schemes and have faster convergence than the conju-gate gradient/steepest descent methods. The Levenberg-Marquardt method can be written mathematically as

(14)

where is the Hessian matrix, is a constant diagonalmultiplier and is the error in the output. The Levenberg-Marquardt algorithm provides for a diagonally dominantmatrix by means of the added constant . The neural net-works trained in the aeroelastic analysis that are describedherein are basedon the Levenberg-Marquardt method. TheMatlab software package has a versatile neural network toolbox and in this paper, the training of neural networksis done through the use of this toolbox.

For dynamical systems, the input-output relationshipmust capture the state of the system. Haykin denes astate of a system as A set of quantities that summarizesall the information about the past behavior of the systemthat is needed to uniquely describe its future behavior, ex-cept for the purely external effects arising from the ap-plied output. We dene the following input-output rela-tionship for the three neural networks that were describedin the previous section.

1. ANN(1) Input -

, Output -

2. ANN(2) Input -

, Output -



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3. ANN(3) Input -

Output -

Aeroelastic Computations

The doublet panel code is linked to the numerical struc-tural dynamic code based on the method of matrix expo-nential time marching. The numerical integration in Eq.(8) is performed using Simpsons rule. Initially a one di-mensional system governed by Eq. (5) is solved, that iswith only the pitching motion of the airfoil. The stiff-ness is constant at 5 units. Figure 8 compares the resultof the present numerical code consisting of the coupledpanel method and the structural dynamics with the theo-

retical result computed using a convolution of the pitchrate with the Wagner function. As can be seen the matchis not exact, but near similar. This is because the theo-retical solution does not take into account the thicknessof the airfoil which plays an important role. References7 and 8 explain the role of panel methods in capturingthe thickness effects and note similar differences betweentheoretical results and panel methods.

The plunge displacement of the airfoil only changesthe downwash velocity over the airfoil and hence this al-lows for modication of the above panel code to includethe effect of the velocity of plunge. The normal boundaryconditions over the doublet panels will now reect this

change and hence the doublet potential will account forboth the pitch and plunge displacements.

We now utilize the concept of neural networks to rep-resent this aeroelastic system in the manner described inthe last section. Having done so for both a time variantsystem and a time invariant system, we will be in a po-sition to design a neural controller that can provide forutter suppression in an active manner.

Neural Network Modeling forFlutter Suppression

As mentioned earlier, the Levenberg-Marquardt routine inthe Matlab toolbox for neural networks is used to train thethree neural networks titled ANN1, ANN2, ANN3 to rep-resent the dynamic aeroelastic system. The networks aredesigned with respect to both a xed system stiffness anda varying torsional stiffness. The variation in the torsionalstiffness is according to Eq. (6). The xed stiffness neural

network system will represent the corresponding aeroe-lastic system exactly within the tolerance bounds since thetraining routine uses the same system parameters. How-ever, for a varying stiffness system, it is not possible totrain the neural network system for all possible stiffnessvariations. Thus testing the neural networks for differentstiffness variations is necessary. Accordingly during thetraining process, Eq. (6) uses certain constants,

and

and during testing, the constants will be changed. Thistests the delity of the system to different system parame-ters. If the testing is successful, then it also establishes anadaptive aeroelastic controller wherein, stability in oscil-lations can be obtained using a variable stiffness system.

Having trained, the network with a set of stiffnessparameters at various free stream velocities, we test theANN at a different system stiffness. Here we consider

,

,

,

for the testing of theANNs. Note that the parameters

and

are differentfrom those used for training the ANN. Figure 9 displaysthe result of the testing. As we can see, the aeroelasticsystem as represented by this array of neural networksdoes show some differences from the numerical aeroelas-tic system, but these errors do reduce with time. More-over, the overall behavior of the neural network systemfollows the numerical solver quite accurately. Hence weconclude that the neural network system represented bythree networks, ANN1, ANN2, ANN3 as described ear-lier can be used to describe the aeroelastic system givenby Eq. 7.

This is now an adaptive representation of an aeroe-lastic system with time varying stiffness. Each time thestiffness variation rate or magnitude changes, the numer-

ical solver need not be used. On the other hand, one canuse the neural network system. This allows for the designof a utter suppression toolbox. Consider Fig. 10 whichshows the three designed neural networks linked togetherto describe the output of the aeroelastic system. A feed-back loop is also shown which evaluates the stability of the system. The stability is based on the response of thesystem during one period of oscillation. In the state spaceplot, if the

, the system is unsta-ble. Here, the period of oscillation is and the stability iscalculated at a time step . Since this is a linear sys-tem, this approach is valid. In general, the procedure todetect utter is to calculate the work done over a period of

oscillation by the external non-conservative forces. Hencethe system is at the utter boundary if

(15)

where is the non-conservative aerodynamic loadsand is the vector containing the system velocities. In



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Fig. 11, the initial dynamic aeroelastic response is cal-culated using the constant stiffness structural neural net-works, ANN1 and ANN3. Once the system is tested tobe unstable, (

), the variable stiffness adap-tive neural networks are switched on for the structural dy-namics so as to bring the system again within the stabil-ity boundary. To achieve this, the neural network arraythat was developed and tested in Fig. 11 is used. Differ-ent stiffness variations are used. The reduced frequencyof the aeroelastic system is set to 1.1 which is just overthe utter reduced frequency for this system. This canbe seen in Fig. 11.a. The neural network controller de-tects the system instability after the rst period and thenswitches over to the variable stiffness mode. The stiff-ness is now increased till stability is achieved. This isshown in Fig. 11.b which depicts a stable oscillatory pat-tern for the time history of the airfoils pitching angle.For Fig. 11.b, the stiffness variation that was used was

. This was found to be suf-cient to restore stability within one period of oscillationof the aeroelastic system.

Once the input-output data that is required for train-ing the neural networks has been collected, the training of the neural networks using the MATLAB software on anSGI origin machine takes around minutes of computa-tional time using one processor. The computational timerequired for the neural network simulation is extremelysmall. For example the 5 seconds simulation shown inFig. 11(a) was obtained in a few a seconds on an SGIOrigin machine using 1 processor while running Matlab.Whereas the numerical aeroelastic routine coupling thestructural dynamics solverwith the panel method for aero-

dynamics, takes almost 4 hours for the same response cal-culation. Since the neural network array allows such arapid simulation, the controller can directly calculate theresponse to detect instability and prevent it using the ANNbased on the variable stiffness approach.

Conclusions

A neural network based aeroelastic response solver wasdeveloped for a two dimensional dynamic aeroelastic sys-tem wherein the torsional stiffness was assumed to betime-varying. Three neural networks were trained to

achieve this goal. The trained neural networks describedthe free vibration response, the unsteady aerodynamic loadsand the forced response of the system. Hence when thesethree neural networks were integrated, the true dynamicaeroelastic response is obtained. The neural networks weretrained from a doublet panel method to describe the un-steady aerodynamic loads over an airfoil and from a ma-trix exponential time marching structural dynamics solver

to evaluate the dynamic response of the system to theseaerodynamic loads. The trained neural networks couldfunction at a varying torsional stiffness within a speciedrange of stiffness variation. This array of neural networkswas then utilized to detect utter in the aeroelastic systemand suppress utter by using the concept of a time varyingtorsional stiffness.

Further research work in using articial neural net-works for representing dynamic aeroelasticity can be pur-sued to investigate more sophisticated problems of a non-linear nature such as limit cycle oscillations.

Acknowledgments

The authors gratefully acknowledge the support of AFOSRgrant F49620-99-1-0294, monitored by Major B. Sanders,Ph.D of AFRL and Dr. E. Garcia of DARPA.

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[17] Natarajan, A., Kapania, R.K. and Inman, D.J.,Aeroelastic Analysis of Adaptable Bumps Usedas Drag Rudders, paper # 2002-0707, 40 AIAAAerospace Sciences Meeting and Exhibit, Reno,Nevada, Jan 14-17, 2002.

[18] Youn, I. and Hao, A., Semi Active Suspensionswith Adaptive Capability, Journal of Sound and Vi-bration Vol. 180, No.3, 1995, pp 475-492.

[19] Burden, R.L., and Faires, D.J., Numerical Analysis ,International Thompson Publishing Company, Sev-enth Edition, 1999.

0 0.2 0.4 0.6 0.8 1

1

0.5

0

0.5

1

1.5

2

Airfoil Span

C p

TheoryPresent Method

Figure 1: Steady state pressure distribution over a sym-metric airfoil as predicted by the Doublet Panel Code andcompared with the theoretical result

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Lift coefficient Step input to pitch angle

L i f t C o e

f f i c i e n

t " C l "

NonDimensional Time Ut/b

Doublet Panel SolutionWagner Solution

Steady State Lift = 0.43

Figure 2: Response of the Doublet Panel Code to an Im-pulsive Change in Angle of Attack



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Figure 3: Unsteady lift over a pitching airfoil with pitchangle as computed by the DoubletPanel Code and compared with the Wagner Solution

Figure 4: Schematic Diagram Depicting Two Layers of aNeural Network, with Input Weights

, Layer TransferFunctions, and , Neuron Biases, and

0 0.5 1 1.5 2 2.5 3 3.5 4

1

0.5

0

0.5

1

1.5

Time (Seconds)

x ( t )

Neural Network ResponseNumerical Response

Figure 5: Response of the system

Figure 6: Two degree of Freedom Airfoil in a Free Stream



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Figure 7: Comparison of the Response calculated usingthe Matrix Exponential Time Marching Method using atime step of 0.1 second with the Theoretical Response

Figure 8: Solution to the equation usingthe theoretical result for a at plate and the panel method

0 1 2 3 4 5 6 73

2

1

0

1

2

3

Testing of Neural Network

Time (Seconds)

A n g l e ( D e g r e e s )

Neural NetworkNumerical Result

Figure 9: Neural network testing with variable torsionalstiffness

Figure 10: The array of neural networks depicting boththe constant stiffness aeroelastic system and the variablestiffness aeroelastic system



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0 1 2 3 4 5 63

2

1

0

1

2

3

Time (Seconds)

P i t c

h A n g

l e ( D e g r e e s

)

a) Unstable Pitching Oscillations

0 1 2 3 4 5 6 73

2

1

0

1

2

3

Time (Seconds)

P i t c

h A n g

l e ( D e g r e e s

)

b) Flutter control by varying stiffness method

Figure 11: Control of Flutter at the utter boundary usingvariable torsional stiffness


dynamic aeroelastic response of adaptable airfoils using neural networks

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