aiaa99_4192

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AIAA-99-4192

Integrated Flight Mechanics and Aeroelastic Aircraft Modeling

using Object-Oriented Modeling Techniques

Gertjan Looye

German Aerospace Center

DLR-Oberpfaffenhofen

Institute for Robotics and System Dynamics

D-82234 Wessling, Germany

E-mail: [email protected]

AbstractIn this paper a software implementationis proposed of integrated flight mechanics / aeroelas-tic aircraft models, using object-oriented modeling

techniques. Model development involves integrat-ing a rigid aircraft simulation model with aeroelas-tic flutter analysis models. The main application isprimary flight control law design, in which most ofthe criteria are of flight mechanical nature. For thisreason, the rigid aircraft model is taken as the basis,while the fidelity of the aeroelastic part may dependon the accuracy required.Object-oriented modeling allows physical objectsand phenomena to be implemented one-to-one intosoftware objects, since interconnections can be de-fined freely (e.g. according to physical interactionslike energy flow). This feature facilitates integra-

tion of model components from different engineeringdisciplines. A model compiler generates simulationcode by symbolic manipulation of the model equa-tions. The code can be exported to several (simula-tion) languages. This allows the same model to beused in different engineering environments. We il-lustrate this feature with a flutter analysis example.

1. Introduction

This paper proposes a software implementationof integrated flight mechanics / aeroelastic aircraft

Research Engineer

Copyright c1999 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

models, using object-oriented modeling techniques.The main model applications are simulation andflight control law design.

For slender airframe designs, like large civil trans-port aircraft, structural flexibility may considerablyinfluence the flight dynamics. For this reason, aeroe-lastic effects have to be accounted for in the aircraftsimulation model. Traditionally, flight control de-sign models only contain rigid body dynamics andthe design criteria are mostly of flight mechanical na-ture (e.g. handling qualities criteria). Therefore, weprefer a so-called bottom-up approach in the modelintegration. This means that a full nonlinear rigidbody simulation model is taken as the basis, and ex-tended with an aeroelastic part of scalable fidelity(e.g. number of modes, detail of unsteady aerody-

namics taken into account, order reduced), derivedfrom detailed flutter models.

In the first place, we need the equations of motionfor flexible aircraft as a spine for interconnectingthe different model components. These equationsare derived in refs 12, 20, 4.

Given the equations of motion, we can start think-ing how to interconnect models that actually drivethese equations, like the aerodynamic model andthrust models. In this paper we will concentrateon implementation and interconnection of these sub-models, rather than the modeling itself.

The total aircraft model consists of many sub-models from different engineering disciplines. This is

1American Institute of Aeronautics and Astronautics

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an important reason why we use the object-orientedmodeling language Dymola (DYnamic MOdelingLAnguage1) 5. Moormann 14, 13 successfully im-plemented a rigid aircraft library using this lan-guage. This paper describes the further developmentto flexible aircraft. The Dymola language allows

one-to-one implementation of physical objects andphenomena into software ob jects, because of free-dom in defining the object interconnections. Theseinterconnections are not limited to signal flows (likein most control-oriented graphical simulation tools)but represent physical system interactions, like en-ergy flows, or kinematic constraints. The objectsare defined and interconnected using a graphical in-terface. Model component libraries can be devel-oped to allow easy re-use in other models 14, 13.These libraries may contain different objects for asame sub-model with different levels of detail. Theseobjects can be easily exchanged within the aircraft

model. The modeling language is equation based,i.e. the model components can simply be written inthe form of equations, without the need of bringingunknowns to the left-hand side.

A model developed in Dymola is further processedusing the DYnamic MOdeling LAboratory (sameacronym: Dymola). The model compiler collectsand sorts the equations from the objects and us-ing symbolic algorithms generates simulation codein Ordinary Differential Equation (ODE) (or Differ-ential Algebraic Equation (DAE)) form. As this isnormally done by hand, this feature saves a lot of

error-prone work. Finally, the simulation code canbe exported as C, Fortran, or ACSL code, or directlyto environments such as MATLAB/Simulink 11 andMatrixx/SystemBuild

7. This gives great flexibilityin application of the model.

This paper is structured as follows. In Section 2 wereview the equations of motion for flexible aircraft.In section 3 we briefly discuss aerodynamic model in-tegration. Next we describe the implementation ofan example model in Dymola. In section 5 we dis-cuss code generation. In section 6 we present flutteranalysis as an example application of the model. Wewill end with conclusions in section 7.

1to be updated to the Modelica standard in near future 6

2. Equations of motion

Flight mechanics models are based on the non-linear Newton-Euler equations of motion for a rigidbody 3. These equations describe the motion of arigid aircraft with six-degrees of freedom, driven by

thrust, aerodynamic and gravity forces. In the refer-ences 12, 20, 4 the equations of motion are extendedto flexible aircraft. Ref. 4 presents a detailed deriva-tion and shows how structural and rigid dynamicscan be integrated in a mechanically correct way. Thereader is referred to these references for a detaileddiscussion. We will only mention the most impor-tant assumptions made and cite the final equationsfrom 20.

The three basic assumptions we make are (1) a ref-erence system fixed relative to the earth surface is aninertial system; (2) structural deformations are suf-ficiently small so that linear elastic theory applies;

(3) a set of vibration modes with eigen frequencies isavailable from e.g. finite element analysis. Figure 1

i

i

i

ii

i

i

x g

yg

z g

inertial

R 0

x

yzb

b

b

p

R

p = s + d

p

qr

dm

d = deformation:

u

v

w

Figure 1: Flexible Aircraft

shows a free flying aircraft considered as a large setof lumped-masses dmi (assumption 4), of which oneis shown in the figure. The axes xg, yg, zg form theinertial reference frame. The axes xb, yb, zb formthe body axes that move with the airframe. The in-

ertial position of the origin is R0. The angular ratesof the body axes are = [p, q, r]T, while the Eu-


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ler angles are E = [ , , ]T (not depicted). Thecomponents of the inertial velocity vector along thebody axes are V = [u , v , w]T. The position of themass element dmi is pi, consisting of its undeformedlocation si and its deformed translation di. The lo-cal deformation is written as a linear combination of

mode shape deformation vectors ij , where j refersto the jth elastic mode:

di =

Nej=1

ijj (1)

Ne is the number of modes taken into account andj is the generalized displacement of mode j. In

refs20, 4 the equations of motion are derived usingLagrangian mechanics. The most important issue isthe choice of an appropriate body reference system.In all references, so-called mean axes are considered

as the best choice, since this results in minimal in-ertial coupling between rigid and elastic dynamics.The orientation of mean axes is such that deforma-tion induced translational and rotational momentumare always zero. In reality, these constraints are hardto satisfy, but with the assumption (3) that the de-formations are small, and that deformation and de-formation rates are co-linear, (assumption 5) it issufficient when so-called practical mean axes con-straints 20 are satisfied. If the mode shapes havebeen obtained from free-free finite element analysis,then a reference system with the axes in the direc-tion of the rigid modes, and with the origin always in

the center of gravity, satisfies these practical meanaxes constraints.

Finally, we make the assumption that the inertiatensor is constant, although it will vary slightly dueto deformation of the airframe (assumption 6). Theresulting equations of motion are as follows:* Kinematic relation R0 and V:

R0 = RTbg(E) V (2)

where Rbg is the rotation matrix from the inertial tothe body mean axes.* Kinematic relation E and :

E = Rb(E) (3)

where Rb is the transformation matrix from bodyangular rates into Euler angular rates.* Rigid body force equation:

m[V+ VRbg(E) [0, 0, g]T] = FA + FT (4)

where FA and FT are aerodynamic and thrust forcevectors respectively, and g is the gravity accelera-tion.* Rigid body moment equation:

I + I = MA + MT (5)

where MA and MT are aerodynamic and thrust mo-ment vectors respectively.* Elastic degrees of freedom: (j = 1...Ne)

Mj(j + 2jj j + 2

jj) = Qj (6)

where Mj is the generalized mass matrix, j the

structural damping, j the eigenfrequency, and Qjthe generalized aerodynamic force for mode j respec-tively.

The only coupling between rigid and flexible mo-tions is via the aerodynamic forces and moments. In-ertial coupling disappears through the assumptionsmade (5,6). Dropping these assumptions, the equa-tions become considerably more complicated 4.

For flexible aircraft, the kinematics of local pointson the airframe with respect to the mean axes aremore complicated than for the rigid case. We need toknow these local kinematics at the positions wherewe want to interconnect for example engine and sen-sor models. As we see from fig. 1, the kinematics ofa local point are described by the movements of themean axes (R0), the undeformed location w.r.t. themean axes (si), and the local airframe deformation(di, from eq. 1).

For our implementation (section 4) we will makeuse of multi-body principles. At the airframe loca-tions where we intend to attach e.g. engine or sen-sor models, we define a local reference system which,contrary to the mean axes, is physically attached tothe airframe. In undeformed position the orientationof the local and the mean axes are the same. Along

the local body axes, we compute inertial kinematicsof the local point (position, orientation, (angular)


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velocity, (angular) acceleration), and forces and mo-ments. For example, the relation between absoluteand relative velocities of the local and mean axes aregiven by:

di = Rblvil V+ pi (7)

where the velocity of point i relative to the meanaxes is:

di =

Nej=1

ij j

vil is the inertial velocity of point i along the localaxes (l), Rbl is the time dependent rotation matrixfrom the local into the mean axes. This matrix de-pends on the local angular displacement vector i.We will not cite the equations for acceleration here,but it will be clear that differentiating eq. 7 leadsto a complicated expression 15.

An engine or sensor model now, is defined in itsown reference system. Connecting a model to theairframe means that its local axis system and thelocal axis system defined on the airframe merge, i.e.kinematics of both axis systems are constraint tobe equal. A vertical acceleration sensor for exam-ple, is modeled such that it outputs the accelera-tion along its vertical z-axis. Once interconnectedwith the airframe model, this acceleration is equalto the inertial acceleration in z-direction of the lo-cally defined reference system on the airframe. Aswe will show in section 4, we can directly implementthis interconnection procedure, without need for re-

ordering equations or variables to sort out unknownvariables; Dymola will do this automatically whengenerating simulation code.

3. Aerodynamic model

Although not exercised for the model examplepresented in this paper (since data was readilyavailable), a major integration effort in developingan integrated flight mechanics/aeroelastic aircraftmodel is required in development of the aerodynamicmodel. Therefore we will briefly discuss some keyaspects of the aerodynamic modeling.

Both for flight mechanic and aeroelastic modelsagreed-upon aerodynamic modeling methods exist.

In rigid models so-called stability and control deriva-tives are used, usually implemented in polynomial ortabular form. These derivatives are obtained fromfor example handbook methods, CFD computations,wind tunnel experiments, and flight testing. If nec-essary, flexibility of the airframe is accounted for us-

ing so-called rigid/flex static correction ratios thatscale the derivatives.

The computation of aeroelastic forces and mo-ments is highly complicated. A trade-off is necessarybetween accuracy and computational costs. For thisreason the Doublet Lattice method1 is a popularmethod in industry to compute aeroelastic modelsin sub-sonic flow regimes for flutter analysis. Forour proposed model integration, we use those fluttermodels to develop the aeroelastic part of the aero-dynamic model.

Integration of rigid and the aeroelastic aero-dynamic models is for example discussed in refs2, 22, 9. The model consists of four parts:

QRQ

=

RR(xa, xa,...) + RF(, , ...)FR(xa, xa,...) + FF(, , ...)

(8)where QR = [FA, MA]

T are forces and momentsdriving rigid dynamics, xa denotes rigid body aero-dynamic states, such as airspeed VA, angle of attack and sideslip angle . is the vector with gener-alized displacements of the elastic modes. Of coursethe model depends on other variables as well, suchas control inputs. RR (Rigid-Rigid) and RF (Rigid-Flex) describe the forces and moments driving therigid part of the equations of motion, induced byrigid and flexible aircraft states respectively. FR(Flex-Rigid) and FF (Flex-Flex) describe the gener-alized forces driving the flexible equations of motion,induced by rigid and flexible aircraft states respec-tively.

The RR part is the same as in the rigid aircraftmodel, whereas the three other components are to beobtained from flutter models. It is generally knownthat the computation of generalized forces inducedby rigid modes in flutter models is poor, but up tonow we did not look into other modeling methods.

Flutter models are obtained in the frequency do-main and therefore need to be transformed into the


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time domain. Several approaches have been devel-oped that are based on rational function approxi-mations in the Laplace domain16, 8. Ref. 17 com-pares several methods and concludes that Karpelsmethod8 gives the most accurate results with thelowest number of additional aerodynamic lag states.

Rigid aerodynamic models implemented in look-up tables are usually valid over (a part of) the flightenvelope, whereas flutter models only have very localvalidity. In practice, these models are computed overa grid of flight conditions and aircraft configurations.It is very difficult to parametrize these models asa function of such variables. Therefore, the mostobvious way to increase validity of the aeroelasticparts in the aircraft model is interpolation betweena set of locally derived aeroelastic models.

Finally, both the rigid and the aeroelastic partsof the aerodynamic model contain static aeroelasticinformation. In the rigid model this is representedby the rigid-flex ratios. In order that the trim statevector does not depend on the number of modes thatis included, the refs. 22, 9 use a procedure developedby Adams at NASA Langley, in which the static in-fluence is subtracted from the RF part of the aero-dynamic model. Static aeroelastic deformation isrepresented by the rigid-flex ratios only.

4. Model implementation

Figure 2 shows the basic implementation of an ex-ample flexible aircraft model in the Dymola graph-ical interface. The aircraft model has been takenfrom ref. 21. It has one asymmetrical mode andfour symmetrical modes (see also ref.20). The quasi-steady aerodynamic model was derived using strip-theory.

The central object flexbodycontains the equationsof motion as discussed in section 2, and since allother objects are connected to it, it has indeed thefunction of a spine for the model. In addition twoengine models are visible (the aircraft has four en-gines; each model actually represents a pair of en-gines on each wing), as well the aerodynamic, grav-

ity, atmospheric, wind and systems (actuators foraerodynamic control surfaces) sub-models. In each

Figure 2: Model implementation (top level)

object a local reference system is defined in whichthe sub-model is described.

All objects have connection points to other ob-jects. These connection points will be referred to ascuts. Connections between cuts represent physicalrelations between sub-models, in this case kinematicconstraints and energy flows. In these cuts the fol-lowing variables are defined:Forces and moments F

l, M

l(force and moment

vector along local reference system of the sub-model), Q (generalized aeroelastic forces).Kinematic quantities Rgl (rotation matrix fromlocal into inertial axes), R0 (inertial position vec-tor of local axes origin), V, (translational androtational inertial velocities of the local axis system,described in local axes), a, (translational and ro-tational inertial accelerations of the local axes), , , (mode shape generalized coordinates and deriva-tives). Also environmental quantities are defined,but we will not discuss these. When connecting twocuts, the kinematic quantities in both cuts are set

equal, so that the objects are forced to move exactlythe same way. The forces and moments on the other


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hand, are summed to zero (can be interpreted as alocal equilibrium condition).

Flexbodyobject

The flexbody object has two cuts: the variables inthe lower cut are in earth fixed axes, in the upper one

in body mean axes. Wind, atmosphere and gravityare modeled in earth fixed axes, and are thereforeinterconnected via the lower cut. The other ob jectsmove with the aircraft and drive the equations ofmotion via the mean axes. These are therefore in-terconnected via the upper cut. The kinematic re-lations between earth fixed and mean axes are partof the equations of motion, see section 2.

Figure 3 zooms in on the flexbodyobject (by right-clicking on it with the mouse in the graphical inter-face). The lower right block (eqm) contains the rigid

Figure 3: Equations of motion (EQM) object

equations of motion, entered in exactly the sameform as in eqs. 2, 3, 4, 5. Again, the variables inthe lower cut are in earth axes (inertial), whereasthose in the upper cut are in body axes (6 DOF).The object to the left (mass) defines and computesmass dependent variables (empty aircraft weight,fuel weight, inertia tensor). The top-right object(Flexible Modes) contains the flexible equations ofmotion (eq. 6) in mean axes. By interconnectingthe eqmand Flexible Modes objects, the mean axes

merge with the body axes in the first object, i.e. themean axes become a 6 DOF reference system. The

variables in the upper cut of the second object arealso in mean axes, but herein also generalized modeshape coordinates and derivatives (, , ) are de-fined, as well as generalized aeroelastic forces (Q).

All objects are of generic nature, and can be usedin any aircraft model. When double-clicking on an

object, a parameter window shows up that allowsthe user to enter aircraft-specific parameters. Thewindow for the Flexible Modes object is shown infigure 4.

The number of longitudinal and lateral modes canbe adapted via nmodeslon and nmodeslat respec-tively, as a means of scaling the fidelity of the elas-tic part of the model. The eigenfrequencies can beadapted as well (Wlon1, Wlon2, etc.). The param-eter doResid specifies what to do with the modes tobe removed. If doResid is true, j and j in equa-tion 6 are set to zero so that remains:

Mj(2

j j) = Qj

which will be solved by the model compiler. In theother case, j , j and j are set to zero. The modelcompiler then automatically removes the equationin the simulation code, thus truncating mode j.

Figure 4: Parameter dialog window Flexible Modes

Engine model object

Figure 5 shows the contents of an Engine Model

object. The Thrust block contains a highly simpli-fied engine model that only scales a throttle input


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into a thrust force (without dynamics). This thrustforce is simply defined as T = [Tx, 0, 0], where Tx isthe thrust along the local x-axis. As discussed in theprevious section, a local axis system (attached to theairframe) is defined at the engine location. This isdone via four transformation objects, TranslateFix,

TranslateFlex, RotateFlex and RotateFix. These re-late the mean axes with the local axis system on theairframe via a fixed translation (from CoG to enginein undeformed position), a time dependent transla-tion and rotation due to local airframe deformation,and a fixed rotation (engines are usually installedwith a small angular offset) respectively. By con-necting the engine model with the transformationobjects, directional thrust variations with respect tothe mean axes due to flexibility are automatically ac-counted for. The transformation objects are generic;they are used within the sensorsobject as well. Lo-cal airframe information (e.g. position, mode shape

vectors) is specified via a parameter window.

Figure 5: Engine implementation

Sensors object

In figure 2 two sensor objects are visible, CoGandCockpit. Both are based on the generic object sen-sors, depicted in figure 6. On the right we see five(simple) sensor models that are modeled in their ownaxis systems. Like in the object Engine model, eachlocal axis system is related to the mean axes via fourtransformations. The transformation objects are ex-

actly the same as for the engines, only local positionand mode shape information for the cockpit has been

entered via the parameter window.The transformation objects are based on a same

parent object. This parent has the two cuts andimplements general kinematic relations between thecut variables (e.g. eq. 7). The four transformationobjects inherit these features and implement the

actual kind of transformation (e.g. flexible transla-tion).

Figure 6: Sensor implementation

Aerodynamics object

The Aerodynamics object contains the aerody-namic model in the form of equation 8. The RR part

is implemented in the form of stability and controlderivatives that are interpolated from look-up ta-bles. The RF, FR, and FF parts are implemented asconstant matrices, valid for a single operating con-dition and aircraft configuration. The aerodynamicmodel is defined in mean axes.

5. Model processing

The model building process in Dymola is summa-rized in figure 7. At the top is the actual model im-plementation, as described in the previous section.

A model is composed in the graphical user inter-face (GUI) from objects that are stored in libraries.


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Automatic

modelbuildingmathematical

parameter-explicit

nonlinear / linear

models in:

* LFT-standard form

* state-space form

simulation code for

different simulationenvironments:

* Matlab/Simulink

* MatrixX/Systembuild

(UCB)

(mfile / cmex)

* c, fortran (DSblock)

numeric

simulation

code

symbolic

analysis

code

Model

parameters Model

ObjectLibraries

mu flutteranalysis

Modeling(in GUI)

.....

HTML

documentationmodel-

Application

example:

simulation in

Simulink Worst-case parameters

Verification

Figure 7: Model building process

Parameters can be adjusted to aircraft-specific val-ues. An HTML-code generation facility is avail-able for model documentation, see for an exampleref. 13. Once finished, the model is loaded in the Dy-mola main program. The symbolic equation handlercollects all equations from the model objects and

solves them according to the inputs and outputs ofthe aircraft model (Automatic mathematical modelbuildingfigure 7). Equations that are not necessaryin the simulation code are automatically removed.Symbolic simulation code is generated in nonlinearstate space form. This code can be exported in sev-eral languages, like c, Fortran, but also for specificenvironments like MATLAB/Simulink (m-code, orSimstruct c-code) and Matrixx/SystemBuild (UserCode Block). Another possibility is to generate sym-bolic analysis code in state-space from, which can beused to symbolically generate a model in the form ofa Linear Fractional Transformation using symbolic

mathematical software, see ref. 19, 18.Dymola offers a built-in trimming/simulation and

visualization facility for direct model evaluation.However, for application such as control law de-sign, the export facility to for example MAT-LAB/Simulink is extremely useful. We used Sim-struct code to simulate the aircraft in this environ-ment. With this code also an m-file is generated inwhich parameter, state, input, and outputs namesand initial values are defined. This information canfor example be used to automatically generate aGUI in MATLAB for trimming and linearizing the

model, see figure 8. Within this GUI, trim valuescan be entered and specified as fixed or free inthe trim computation. For this trim computation amex-function based on MINPACK has been imple-mented.

Finally, the aircraft model can be included into asimulink diagram with a block diagram of the controlsystem to perform closed loop simulations.

6. Application example

As an application example, we use the model for

flutter analysis in the -framework10

. The aircraftdynamics are augmented by a pitch rate feedback via


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Figure 8: Screen shot of MATLAB trim GUI

the elevator. The pitch rate is sensed near the cock-pit. Using the software described in ref. 19, whichcombines available symbolic and numeric computa-tional software tools, we automatically generate amodel in the form of a Linear Fractional Transfor-mation (LFT) from the parametric Dymola model.

The LFT, in which varying parameters are separatedfrom the model and interconnected in a feedbackpath, is the standard form for -analysis. The LFTof the example aircraft model contains airspeed,structural eigenvalues and modal damping as vary-ing parameters. With -analysis a worst-case flutterspeed is computed in face of the uncertain structuralparameters. The worst-case parameter combinationis substituted into the Dymola generated simulinkmodel, to verify the occurrence of flutter in a simu-lation. It turns out that the third symmetrical modegoes unstable, see figures 9, 10.Since both the LFT model and simulation model

have the same Dymola model as origin, this illus-trates the advantage of single-source modeling ca-pability (figure 7).

0 5 10 15 20 25 30 35 40 45 502

0

2

4

6

8

10x 10

4

time [s]

q_cp [rad/s]

Figure 9: Pitch rate in cockpit: verification of flutterfor worst model parameters

7. Conclusions

In this paper we have proposed a software imple-mentation of integrated flight mechanics / aeroelas-tic aircraft models exploiting the features of object

oriented modeling. Sub-models from different engi-neering disciplines can be implemented one-to-one


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0 5 10 15 20 25 30 35 40 45 500.0538

0.0536

0.0534

0.0532

0.053

0.0528

0.0526

0.0524

0.0522

0.052

time [s]

eta_3 []

Figure 10: Gen. displacement of third symm. mode

into software objects, since we are able so specifythe interconnections according to the way the ob-

jects physically interact.

The equations of motion form the core of themodel implementation. The rigid simulation modelis used as a basis, while the fidelity of the aeroelas-tic part may depend on the accuracy required by theapplication.

The implementation of an elastic aircraft modelof limited complexity was demonstrated. Except forthe aerodynamic model, other model componentscan be easily re-used in other aircraft models byadapting values in the object parameter window. Atthe moment, the fidelity of the aeroelastic model canonly be adjusted via the number of modes. For morecomplex models this, for example, could also be donevia the order of the rational function approximationof the unsteady aerodynamics, by implementing sev-eral aerodynamic models that are easily exchangablein the integrated aircraft model.

The package used for implementing the model, Dy-mola, is able to automatically generate simulationcode for several engineering environments. The pos-sibility to generate symbolic code facilitates para-metric uncertainty modeling, since model param-eters can be treated explicitly in e.g. a symbolicmathematical package. A great advantage is thatthe symbolic and the simulation model are gener-

ated from the same source. An example was givenby generating an LFT model from the Dymola code

to perform flutter analysis.So far, the implementation is conceptual, since the

complexity of the implemented aircraft model is low,and the model was readily at hand. Future work willinvolve actual modeling work and implementation ofa more complex aircraft model.

Acknowledgment

The author wishes to thank Dieter Moormannfrom the same department for his support in all Dy-mola implementation work.

References

[1] E. Albano and W.P. Rodden. A Doublet-Lattice Method for Calculating Lifting Dis-turbances of Oscillating Surfaces in SubsonicFlows. AIAA Journal, 7(2):279285, 1969. (Er-rata in AIAA Journal 7(11), November 1969,p.2192).

[2] P. Douglas Arbuckle, Carey S. Buttrill, andThomas A. Zeiler. Simulation Model Build-ing Procedure for Dynamic Systems Integra-tion. Journal of Guidance Control and Dynam-ics, 12:894900, 1989.

[3] Rudolf Brockhaus. Flugregelung. Springer-Verlag, Berlin Heidelberg, 1994.

[4] C.S. Buttrill, T.A. Zeiler, and P.D. Arbuckle.Nonlinear Simulation of a Flexible Aircraft inManeuvering Flight. In Proceedings of theAIAA Flight Simulation Technologies Confer-ence, Monterey, CA, August 1987. AIAA-87-2501.

[5] H. Elmqvist. Object-oriented modelingand automatic formula manipulation in dy-mola. In SIMS 93, Scandinavian Sim-ulation Society, Kongberg, Norway, June1993. available from http://www.dynasim.se/publications.html.

[6] Modelica Design Group. Modelica: Lan-

guage design for multi-domain modeling.http://www.modelica.org.


7/29/2019 aiaa99_4192

11/11

[7] Integrated Systems Inc. SystemBuild UsersGuide, version 6, 1998.

[8] M. Karpel and S.T. Hoadley. PhysicallyWeighted Minimum State Unsteady Aero-dynamic Approximation. Technical Report

NASA-TP-3025, NASA, 1990.

[9] E. Levretsky. High Speed Civil Transport(HCST) Flight Simulation and Analysis Soft-ware Development. In Proceedings of the 36thAerospace Sciences Meeting and Exhibit, Reno,Nevada, Jan 1998. AIAA-98-0173.

[10] Rick Lind and Martin J. Brenner. Ro-bust Flutter Margin Analysis that IncorporatesFlight Data. Technical Report TP-1998-206543,NASA, Dryden Flight Research Center, Ed-wards CA, 1998.

[11] The Math Works Inc. Simulink Users Guide,1998.

[12] R.D. Milne. Dynamics of the Deformable Air-plane (Part I and II). Technical Report R & MNumber 3345, and AD-A953155, Her MajestysStationary Office, London, England, 1964.

[13] Dieter Moormann. GARTEUR ResearchCivil Aircraft Model (RCAM): AircraftPhysical Modeling Leading to AutomaticCode Generation. Technical report, DLR-Oberpfaffenhofen, Institute for Robotics and

System Dynamics, 1997. Hypertext document:http://www.op.dlr.de/DD-DR-ER/research/

flight/rcammod/node2.html.

[14] D. Moormnann, P.J. Mosterman, and G. Looye.Object-oriented computational model build-ing of aircraft flight dynamics and systems.Aerospace Science and Technology, 3:115126,1999. To appear early 1999.

[15] Martin Otter. Objektorientierte Modellierungmechatronischer Systeme am Beispiel Geregel-ter Roboter. PhD thesis, Fakultat fur Maschi-

nenbau der Ruhr-Universitat Bochum, Novem-ber 1993.

[16] K. Roger. Airplane Math Modeling Methodsfor Active Control Design. AGARD, (CP-228),August 1977.

[17] J. Schuler. Flugregelung und aktiveSchwingungsdampfung fur flexible Gro-

raumflugzeuge Modellbildung und Simulation.PhD thesis, Institut fur Flugmechanik undFlugregelung, Universitat Stuttgart, 1997.

[18] A. Varga and G. Looye. Symbolic and numeri-cal software tools for lft-based low order uncer-tainty modeling. In Proc. CACSD99 Sympo-sium, Cohala Coast, Hawaii, 1999.

[19] A. Varga, G. Looye, D. Moormann, andG. Grubel. Automated Generation of LFT-based Parametric Uncertainty De-scriptions from Generic Aircraft Models.Mathematical and Computer Modelling of

Dynamical Systems, 4(4):249274, 1998.See also GARTEUR report TP-088-36 at

http://www.nlr.nl/public/hosted-sites/

garteur/sum36.html.

[20] Martin R. Waszak and Dave K. Schmidt. FlightDynamics of Aeroelastic Vehicles. Journal ofAircraft, 25(6):563571, June 1988.

[21] M.R. Waszak and D.K. Schmidt. A Simula-tion Study of the Flight Dynamics of ElasticAircraft: Volume 1 Experiment, Results andAnalysis. Technical Report NASA CP-4102,

NASA, December 1987.

[22] B.A. Winther, P.J. Goggin, and J.R. Dyk-man. Reduced Order Dynamic Aeroe-lastic Model Development and Integrationwith Nonlinear Simulation. In Proceedingsof the AIAA/ASME/ASCE/AHS/ASC Struc-tures, Structural Dynamics and Materials Con-

ference, pages 1694 1704, Los Angeles, April1998.


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