COMPARISON OF MODEL INTEGRATION APPROACHES FOR FLEXIBLE AIRCRAFT FLIGHT DYNAMICS MODELLING

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1 COMPARISON OF MODEL INTEGRATION APPROACHES FOR FLEXIBLE AIRCRAFT FLIGHT DYNAMICS MODELLING Christian Reschke and Gertjan Looye DLR German Aerospace Center, Institte of Robotics and Mechatronics D Wessling, Germany Keywords: Model integration, Flight mechanics, Aeroelasticity, Simlation Abstract. Two approaches for the development of aircraft models integrating flight and airframe strctral dynamics are compared. These models are intended for flight loads analysis and flight simlator application. The approaches combine available, agreed on flight mechanics and aeroelastic aircraft models from the respective engineering disciplines. The present work compares two methods, with different nderlying assmptions, regarding accracy, implementability, pre-processing and compting effort, as well as possibility to adapt the model components as a fnction of the flight condition. Nomenclatre A A,1,2 B D E F g J M Q R T be V c m n z q q s δ η φ θ ψ λ Ω system matri approimation matri damping matri, inpt matri otpt matri of lag state space eq. inpt matri of lag state space eq. force vector gravity acceleration vector total aircraft inertia tensor moment vector, mass matri aerodynamic inflence coefficient diagonal matri with lag state poles transformation matri from inertial into body aes (air-)speed mean aerodynamic chord length total aircraft mass vert. load factor in dir. of body z-ais dynamic pressre decopled modal states Laplace variable state vector variation generalised coordinates Eler angle Eler angle Eler angle eigenvale anglar velocity vector Φ eigenvectors of system matri, inpt matri KS approach Ψ inverse of Φ Abbreviations AE aeroelastic Eq eqation FM Flight Dynamics RFA rational fnction approimation Sbscripts E elastic R rigid L lag state control trim, qasi-steady b in body-fied frame e in inertial frame Sperscripts T transpose 1 inverse Other χ time derivative of χ χ absolte vale of χ det determinant 1 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

2 I. Introdction Flight mechanic models (FM) are sally based on the nonlinear si degree of freedom Newton-Eler eqations of motion, a detailed aerodynamics model, actator models, etc. Aeroelastic models (AE) combine strctral dynamics and nsteady aerodynamics in modal form. Aircraft tend to get larger with a lighter and more fleible airframe. Therefore the interaction between rigid and elastic motion becomes increasingly important and reqires integrated models. Varios works describe approaches for integrated flight mechanics and aeroelastic modelling. 1 3, 7, 8, 1 13 This paper is concerned with model integration approaches that combine available indstrial agreed on flight mechanics and aeroelastic models. The major task in model integration of available FM and AE models is the handling of overlaps. Flight mechanic models describe the rigid body motion bt se aerdynamic databases that accont for qasi-steady strctral deformation. On the other hand aeroelastic models se a free-free modal analysis and inclde rigid body modes representing small amplitde rigid body motion. Two approaches are considered here. Both accont for the model overlaps by adapting the AE model while leaving the FM model basically nchanged. The first approach was developed by König and Schler, KS-method, 11 and agments the FM and AE model differential eqations. Hereby a modal transformation to the AE model is applied, so that rigid body state derivatives from the FM differential eqations can be incorporated. The FM differential eqations are left nchanged and describe the mean motion of the fleible aircraft center of gravity. The second approach, the Residalised Model RM-method, 2, 8 combines the FM Newton- Eler and AE modal eqations of motion (fleible degrees of freedom), which is allowed nder assmptions made in. 12 In combining the aerodynamic models, the qasi-steady contribtion of the nsteady aerodynamics model is continosly sbtracted from the fleible to rigid copling, while leaving the FM qasi-fleible aerodynamic model nchanged. This approach basically removes the qasi-fleible rigid degrees of freedom from the AE data. This present work reviews the integral model components followed by a review of the integration techniqes, the KS- and RM-method. Then a comparison regarding assmption, pre-processing and implementation is presented. A nmerical test case is stdied net analyzing accracy and compting effort. II. Integrated Model Components The integration approaches combine components of the flight mechanic (FM) models and aeroelastic (AE) models. The strctre of the two basic elements is described this section. Flight Mechanics Model Flight mechanic models are based on the si degree of freedom Newton Eler eqations of motion for a rigid body. Force and moment eqations are given by: 4 [ ] m Vb + Ω b V b T be g e F et (1a) J Ω b + Ω b JΩ b M et where V b and Ω b are the translational and anglar velocity vectors resolved in aircraft body aes. The matri T be denotes the transformation from the geodetic reference frame into body ais. Eternal force and moment vectors F et, M et contain the aerodynamic forces, thrst forces and other eternal forces. While the the Newton Eler eqations of motion 2 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany. (1b)

3 only contain si rigid degrees of freedom, the aerodynamic forces depend on the aircraft flight state and the qasi static deformation of the airframe 4 (Fig. 1). The aerodynamic coefficients are therefore corrected by fle-factors that depend on the vertical load factor n z. n z qasi-static deflection Figre 1. Aircraft in flight qasi static deformation inflences the total aerodynamic forces; this is taken into accont via fle factors as a fnction of the vertical load factor The correction of the qasi static deformation has to be kept in mind for sbseqent copling of the FM model with the AE model, becase the qasi static deformation is also contained in the AE model. For later se in the copling process the vector ẋ F M will be defined (note that this is not the state vector of the FM model): Aeroelastic Model ẋ T F M [ V T b, Ω T b, V T b, Ω T b ] (2) The aeroelastic aircraft model contains of a strctral finite element model and an nsteady aerodynamic model. The finite element model consists of condensed grid points for which a free-free modal analysis is performed. The strctral degrees of freedom are transformed to modal degrees of freedom sing si rigid body mode shapes (nit translations and rotations in the direction of the aircraft body aes) and a selection of elastic mode shapes. The aeroelastic eqation of motion may then be written in the following generalized form: 1 {[ MRR M EE ] s 2 + [ ] [ ]} [ ] BRR KRR ηr s + B EE K EE η E {[ ] [ ] QRR (s) Q q RE (s) ηr + Q ER (s) Q EE (s) where the rigid η R and elastic η E generalizes coordinates are: η E [ ] } QR (s) η Q E (s) η T R [δ b, δy b, δz b, δϕ b, δϕ yb, δϕ zb ] (4) η T E [η E1,..., η En ] (5) and η is the vector of control inpts. The generalized stiffness, damping and mass matrices are denoted by K, B, M. The matrices of aerodynamic inflence coefficients Q, Q are obtained from the Doblet Lattice theory 5 and initially depend on the mach nmber and the redced freqency. A rational fnction approimation 6, 9 (RFA) is then applied to transform the nsteady aerodynamic forces and control forces from Laplace into time domain. For eample: { ( } c Q EE (s) η E A EE + A 1EE V s + A c 2 2EE V 2 s2 + D EE si V c 1 E) R E E s η E (6) with the aerodynamics lag states LE : (3) ẋ LE V c R E LE + E E η E (7) 3 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

4 The aeroelastic eqation of motion Eq. (3) can then be written in state space form with separated lag states (rigid, elastic and control lag states): 8 η R A RR1 A RR A RE1 A RE A RLR A RLE A RL η R B R1 B R η R I η R η E A ER1 A EE1 A EE A ELR A ELE A EL η E B E1 B E [ ] η η E I η V ẋ LR E R R E + (8) η c R ẋ LE V E E R LR c E LE V ẋ L R c L E The vector η is not inclded in the formlation, since is sally not available from actator models. The state space model can also be written in compact form with a single set of lag states: ẋ R A RR A RE A RL ẋ E A ER A EE A EL ẋ L A LR A LE A LL R E L + B R B E B L [ η η with the state vectors T R [ ηt R, ηt R ], T E [ ηt E, ηt E ] and the vector L, containing all aerodynamic lag states. Note that Eq. (8) and Eq. (9) are alternative formlations of the same aeroelastic system. ] (9) III. Review of Integration Techniqes In this section the KS-approach and the RM-approach are reviewed and the connection with the flight mechanics model is shown. KS-Approach This approach was developed by König and Schler. 11 It ses a modal analysis of the system matri A (Eq. (9)) to decople rigid and elastic states by modal transformations with the complete set of comple system eigenvectors. Fig. 2 shows a schematic distribtion of the comple eigenvales of the aeroelastic system matri A. Rigid and elastic states as well as aerodynamic lag states can be recognized by their freqencies and damping factors. The elastic states aerodynamic lag states Im eig(a) eig(a KS ) Re rigid states Figre 2. Eigenvales of aeroelastic model A Eq. (9) and copled model A KS Eq. (1) system matri decopled rigid states are replaced by the ones obtained from the flight mechanics model 4 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

5 (see also the appendi for a derivation of the KS-approach). The copled model can then be written in the following form: ẋ KS A KS KS + B KS + Φ KS (ẋ F M ẋ F M, ) (1) where the inde KS denotes data related to the copled model and Φ KS is the inpt matri that incorporates the FM-model states. The new system matri A KS has the same elastic and aerodynamic lag states as the original AE model bt does not contain the rigid body motion. All rigid body poles are zero after application of the copling process; Fig. 2 also depicts the poles of the system matri A KS. The connection of the nonlinear flight mechanics model Eq. (1) and the aeroelastic model with the KS-approach Eq. (1) is shown in Fig. 3. It starts with the comptation of rigid states with the qasi fleible flight mechanics model. After removing the initial vales at trim condition ẋ F M, the data from the rigid model is introdced in the eqation of the copled model. The otpt of the copled model ẋ KS contains dynamic increments for rigid and fleible states. It may sbseqently be sed for comptation of loads and accelerations over the airframe as well as sensor signals FM - FM, Nonlinear Flight Mechanics Model (qasi-fleible). FM FM -. FM, KS. Copled Model... KS + KS-Approach - Figre 3. Connection of nonlinear flight mechanics model and aeroelastic model: KSapproach RM-Approach The residalised model approach (RM-approach) was developed by Winther 13 and etended by Looye 8 to be sed with AE models that contain aerodynamic lag states. Varios forms of the RM-approach are presented in. 8 In this work the form with variable dynamic pressre will be reviewed for completeness. First the partitioned state space system Eq. (8) is redced to elastic states and lag states: η E η E ẋ LR ẋ LE ẋ L A EE1 A EE A ELR A ELE A EL I V c R R V E E R c E V R c η E η E LR LE L + B E1 B E E A ER1 [ ] η + η E R δẋ F M (11) where the aeroelastic rigid body states η R are replaced by the corresponding states of the flight mechanics model δẋ F M δẋ F M ẋ F M. The previos eqation is sed to compte elastic and lag states as a fnction of the control inpt and the flight dynamic states. Net the elastic modes are residalized: η E η E LE (12) 5 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

6 With Eq. (12) in Eq. (11) the qasi static elastic contribtion then is: { η E (A EE ) 1 [AELR ] [ ] [ ] } A LR η EL + B E + A ER δẋ F M L η (13) Dynamic load increments are compted in analogy to the nsteady aerodynamic forces (Eq. (6)) as follows: [ ] { } δf c q A RE (η E η E ) + A 1RE δm V η c 2 E + A 2RE V η 2 E + D RE LE (14) where only elastic lag states are contained. Rigid lag states LR wold nintentionally cople into the rigid-elastic term. Therefore the separation of the lag states is important. The incremental dynamic loads are then incorporated into the Newton Eler eqations to complete the copling process: [ ] m Vb + Ω b V b T be g e F et + δf (15a) J Ω b + Ω b JΩ b M et + δm (15b) The RM copling process is depicted in Fig. 4. The comptation of the elastic states is governed by the flight mechanics model. The feedback of the dynamic load increments to the flight mechanics model agments the FM model.... Nonlinear Flight Mechanics Model (qasi-fleible). FM FM -. FM, FM - FM, RM-Approach residalised AE model δf δμ. E E Figre 4. Connection of nonlinear flight mechanics model and aeroelastic model: RMapproach IV. Comparison of the Integration Procedres In this section the KS and RM-approach will be compared regarding nderlying assmptions, model strctre and accracy. Assmptions The assmptions of the KS-approach (Table 1) is valid when rigid and elastic/lag states can be separated by modal decopling. No minimm separation for the freqency of rigid and elastic effects is assmed. The RM-approach (Table 1) is based on static residalisation of the elastic mode shapes. Static residalisation implies a certain separation in freqency of the rigid and elastic motion. This is assred for most passenger aircraft bt shold be kept in mind and reviewed for etremely fleible airframe strctres. 6 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

7 KS-Approach Flight mechanics states are eqivalent to a linear combination of rigid body eigenvectors of the aeroelastic system matri RM-Approach The residalised linear aeroelastic model, containing rigid body dynamics only, is eqivalent to the linearized flight mechanics model inclding qasi static corrections Table 1. Copling Approaches Assmptions Pre-Processing The pre-processing of the KS-approach starts with the eigenvale analysis of the system matri that yields comple eigenvectors and comple eigenvales (poles). Then the poles corresponding to the rigid states mst be identified. This can be done by manal selection or by atomated selection of the low freqency poles. An important check is to make sre, that no aerodynamic lag states have been mistaken for rigid body states. Then the copling matrices are assembled. Nmerical errors can be cased by the model transformation with comple matrices. Still eisting small imaginary parts mst be eliminated. The KSapproach can be applied even when the internal strctre of the aeroelastic state space model is nknown. Contrary to the KS-approach the RM-approach reqires an aeroelastic state space model, where aerodynamic lag states are separated in those affecting rigid body and those affecting elastic dynamics (de to Eq. (14)). If the rational fnction approimation was performed sing the approimation by Rogers 9 the lag states can be separated. In case of an approimation with Karpel s method 6 the lag states have to be repeated for each matri partition. Once the state space model is available with separated lag states the copling matrices can be assembled. The pre-processing effort of the KS-approach is mainly based on the nmerical handling and checking of the copling matrices. It is convenient that all states, other than the rigid ones, can be treated as a single set of states. The RM pre-processing is determined by the separate handling and the partitioning of the aerodynamic lag states. Contrary to the KS method a nmerical check of the assembled copling matrices is not necessary. Implementation The KS-approach is implemented in an eisting flight dynamics environment in form of the differential eqations Eq. (1). The original flight mechanics model i.e. the implementation of the Newton Eler eqations is not effected by the copling process. This is favorable when the original flight dynamics environment is not accessible or changes to the original implementation is overly time consming (e.g. for flight simlator applications). Since there is is no feedback on the flight mechanics eqations, the KS-approach may also be applied in the post-processing; an advantage for load analysis. Frther more a different integrator algorithm can be sed for the FM model and the AE-model. The RM-approach reqires the implementation of an additional state space model to compte the elastic states and lag states Eq. (11). An algebraic eqation is needed to calclate the residalised elastic states Eq. (13). With this information the dynamic load increments can be compted Eq. (14). The original flight mechanics eqation is then agmented by the dynamic load increments. The RM-approach can be applied when altering the original flight mechanics eqation is possible and acceptable. It is assmed that the qasi fleible aerodynamic model (fle factors) is not negatively inflenced by the additional dynamic load increments. In case the original model mst not be affected the fle-rigid copling may be removed. 7 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

8 Dynamic increments are then obtained for the elastic states only. The RM-approach can be implemented in the present form with variable dynamic or in a simple state space form with constant dynamic pressre. The variation of dynamic pressre reqires online matri inversion. See Table 2 for a smmary of the implementation aspects. KS-Approach RM-Approach Form of implementation transformed physical Transformation of AE eigenvale analysis and separate elastic lag states in model modal transformation case of Karpel RFA Integrator different integrator possible same as FM integrator Applicability rigid modes separable from FM model eqivalent to fleible modes and lag states residalized AE model interpolation w.r.t mach nmber, dyn. pressre mach nmber Table 2. Implementation V. Nmerical Eample The copling techniqes described above are now applied in a nmerical eample. The stdy vehicle is a large passenger aircraft for which an indstrial nonlinear flight mechanics model and an aeroelastic model is available. The copling methods are compared in time and freqency domain. In time domain simlation, elevator and aileron stair inpts (Fig. 5) are sed as a test case. The otpt of interest here is the longitdinal and lateral response of flight mechanics and copled rigid states. The response of the rigid states to the respective inpt is shown in Fig. δ elev [ ] δ ail [ ] Figre 5. Elevator and aileron deflection de to inpt of stair command 6. Roll and pitch acceleration ṗ, q of the copled models now contain dynamic increments that reslt from small rigid body motion of the AE model. The KS- and RM-approach lead to nearly identical rigid body accelerations that vary arond the acceleration of the original FM model. Differences between the two approaches can be identified for the Eler angles θ, φ and roll rates p, q. The mean vale of the KS-approach more closely matches the FM-Model. The RM-approach response shows a difference in mean vales de to feedback of incremental dynamic forces to the flight mechanics model. The acceleration at cockpit (Fig. 7) is now analyzed, representing the airframe aeroelastic response. The KS and RM-approach again yields very similar reslts. In regions where the acceleration is dominated by its elastic contribtion the reslts cannot be distingished. With decreasing elastic acceleration slight differences can be noticed de to the differences in the rigid body response. The transfer fnction of a rdder inpt to lateral cockpit acceleration is depicted in Fig. 8 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

9 FM Model KS RM FM Model KS RM θ φ q p dq/dt dp/dt Figre 6. Rigid state response to elevator stair inpt (left colmn) and response to aileron stair inpt (right colmn) a z,pilot FM Model KS RM a y,pilot FM Model KS RM Figre 7. Acceleration response to elevator stair inpt (left colmn) and response to aileron stair inpt (right colmn) 8. The lower freqency range is dominated by the dtch roll mode. The higher freqency range by the elastic mode shapes. The aeroelastic model (AE) does not represent the rigid body motion very well compared to the qasi fleible flight mechanics model (FM). Both copling techniqes can accrately represent higher freqency range of the AE model. In the low freqency range the KS-approach eactly matches the dtch roll mode of the FM model. The RM-approach leads to a slightly increased amplitde. Adaption to Flight Condition Common flight mechanics models are valid for a wide range of flight conditions. Aeroelastic models are only valid for a linearisation point, specified by dynamic pressre and mach nmber. The KS-Method reqires the transformation of the aeroelastic system matri. Therefore copling matrices are only valid at the specific working point. For copled simlation in a wider range of working points a two dimensional interpolation of the copling matrices over mach nmber and dynamic pressre is reqired. Any error that arises from interpolation can adversely affect the zero poles of the copling matri A KS which may reslt in nmerical problems dring simlation. The RM-approach, if sed in the form with variable dynamic pressre, reqires the interpolation of the nsteady aerodynamic 9 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

10 magn(a y ) [ ] magn(a y ) [ ] Dtch Roll Mode Dtch Roll Mode phase(a ) [deg] phase(a ) [deg] y y AE FM qasi fle RM KS AE FM qasi fle RM KS Figre 8. freq. [rad/s] Bode plot of transfer fnction: rdder y-acceleration at cockpit freq. [rad/s] forces Q(M a) only. Therefore interpolation is redced to a one dimensional interpolation over the mach nmber. Compting Effort The RM-approach consists of two algebraic eqations Eq. (13), Eq. (14) and the differential eqation Eq. (11) containing elastic and lag states. The KS-approach consists of the differential eqation Eq. (1) containing elastic and lag states as well as rigid body states. This leads to a similar compting effort for both methods (Table 3). The RM-approach with variable dynamic pressre reqires the greatest compting effort since a matri inversion has to be performed for every time step (or chance in dynamic pressre). However sice the variation of dynamic pressre is sally slow compared to the simlation rate, the matri pdate can be done at slower rate withot noticeable effect on the reslts. Model FM-model KS-approach RM-approach ( q const) RM-approach ( q q(t)) time for 1s simlation.5 s 1.4 s 1.3 s 4.6 s Table 3. Compting effort for test case; depends on FM aerodynamic model, nmber of elastic mode shapes in AE model and nmber of lag states VI. Conclsion In this paper a comparison of two model integration approaches has been presented. Both methods are based on the idea that the flight mechanics model shold be left nchanged as far as possible. The application of the KS-approach is convenient even if the strctre of the aeroelastic model in nknown, e.g. for order redced AE models. Only the rigid body modes have to be recognized. The preprocessing shold always inclde nmerical sanity checks of the 1 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

11 copling matrices. The original flight mechanics model remains nchanged which makes the KS-approach favorable for simple implementation in flight simlators. Different integrators may be applied to the FM and the AE model. The RM-approach offers a variety of implementation forms. It can be sed with or withot feedback of dynamic force increments to the original flight mechanics eqations. Its physical model strctre takes the variation of dynamic pressre into accont. Separate implementation of the eqations of motion and aerodynamic forces is provided; acconting for eqations of motion with inertial copling. The RM-approach also etends to loads analysis, since the concept of qasi fleible deformation also applies for loads calclation. It was shown that reslts of the KS-approach and the RM-approach yield close reslts, so the choice of a method is primarily based on the simlation environment. Appendi Derivation of the KS-Copling The derivation of the König-Schler approach is reviewed from 11 for completeness. The aeroelastic eqations of motion in state space form Eq. (9) can be written as follows: [ ] [ ] [ ] [ ] ẋr ARR A R,EL R BR + (16) ẋ EL A EL,R A EL,EL where T EL [T E, T L ] denotes the vector of combined elastic and lag states. Then the eigenvale problem: AΦ Φλ det A λi (17) is solved. In the above eqation Φ denotes the matri of (comple) eigenvectors and λ denotes the diagonal matri of (comple) eigenvales. The system matri A can now be diagonalized sing a particlar case of a similarity transform of the matri A : EL B EL Φ 1 AΦ λ (18) The state vector is transformed to modal states q as follows: [ ] [ ] [ ] R ΦRR Φ R,EL qr EL Φ EL,R Φ EL,EL q EL (19) For convenience the inverse of the matri of eigenvectors Ψ Φ 1 is introdced: [ ] [ ] 1 ΨRR Ψ R,EL ΦRR Φ R,EL (2) Ψ EL,R Ψ EL,EL Φ EL,R Φ EL,EL The eigenvales λ of the system matri A is partitioned in its rigid λ R and elastic/lag state λ EL eigenvales. Inserting Eq. (19) in Eq. (16) and pre-mltiplication by Ψ (Eq. (2)) we obtain the decopled modal form of the original aeroelastic model: [ ] [ ][ ][ ][ ] [ ][ ] qr ΨRR Ψ R,EL ARR A R,EL ΦRR Φ R,EL qr ΨRR Ψ + R,EL BR q EL Ψ EL,R Ψ EL,EL A EL,R A EL,EL Φ EL,R Φ EL,EL q EL Ψ EL,R Ψ EL,EL B EL [ ] [ ] [ ] [ ] λr qr ΨRR Ψ + R,EL BR (21) λ EL q EL Ψ EL,R Ψ EL,EL B EL The aeroelastic model will now be copled to the flight mechanics model ẋ F M G( F M,,...) Eq. (1). It is assmed that the derivative of the rigid states of the flight mechanic model ẋ F M can be approimated by: ẋ F M Φ RR q R (22) 11 of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.

12 Whith Eq. (22) in Eq. (19) the modal approach can be partitioned as follows: [ ] [ ] [ ] [ ] ẋr ẋ F M Φ R,EL q EL I ΦR,EL ẋ EL Φ EL,R Φ 1 RRẋF M Φ EL,EL q EL Φ EL,R Φ 1 ẋ F M + q EL (23) RR Φ EL,EL The vector q EL is obtained from the elastic part (second row) of Eq. (21). With q EL Ψ EL,R R + Ψ EL,EL EL from Eq. (19) the elastic part of Eq. (21) can be written as: q EL λ EL (Ψ EL,R R + Ψ EL,EL EL ) + (Ψ EL,R B R + Ψ EL,EL B EL ) [ λ EL Ψ EL,R λ EL Ψ EL,EL ] [ T R T EL] T + (ΨEL,R B R + Ψ EL,EL B EL ) (24) The eqation for the copled model is now obtained from Eq. (23) and Eq. (24): [ ẋr ẋ EL ] [ I Φ EL,R Φ 1 RR or in compact form ] [ ΦR,EL λ ẋ F M + EL Ψ EL,R Φ R,EL λ EL Ψ EL,EL Φ EL,EL λ EL Ψ EL,R Φ EL,EL λ EL Ψ EL,EL + ] [ ] R EL [ ΦR,EL (Ψ EL,R B R + Ψ EL,EL B EL ) Φ EL,EL (Ψ EL,R B R + Ψ EL,EL B EL ) ] (25) ẋ KS Φ KS ẋ F M + A KS KS + B KS (26) where KS denotes the states and matrices related to the König-Schler approach. References 1 P. D. Arbckle and C. S. Btrill. Simlation model-bilding procedre for dynamic systems integration. Jornal of Gidance, 12(6):894 9, P.J.Goggin B.A. Winther and J.R.Dykman. Redced order dynamic aeroelastic model development and integration with nonlinear simlation. In Strctral Dynamics and Materials Conference, Los Angeles, April 1998, pages AIAA, April C.S. Btrill, T.A. Zeiler, and P.D. Arbckle. Nonlinear simlation of a fleible aircraft in manevering flight. In AIAA Flight Simlation Technologies Conference, nmber AIAA AIAA, B. Etkin. Dynamics of Flight. John Wiley & Sons, INC, J.P. Giesing, T.P. Kalman, and W.P. Rodden. Sbsonic Steady and Oscillatory Aerodynamics for Mltiple Interferencing Wings and Bodies. Jornal of Aircraft, 9(1):693 72, M. Karpel and E. Strl. Minimm-state nsteady aerodynamic approimations with fleible constraints. Jornal of Aircraft, 33(6): , M.J.Brenner K.K.Gpta and L.S.Voelker. Development of an integrated aeroservoelastic analysis program and correlation with test data. Technical report, G. Looye. Integration of rigid and aeroelastic aircraft models sing the residalised model method. In Abstract sbmitted for presentation at the International Form on Aeroelasticity and Strctral Dynamics, K. L. Roger. Airplane math modeling methods for active control design. In AGARD Strctres and Materials Panel, nmber AGARD/CP-228, pages AGARD, April J. Schler. Flgregelng nd aktive Schwingngsdämpfng für fleible Großramflgzege. PhD thesis, Universität Stttgart, J. Schler and K. König. Integral control of large fleible aircraft. In RTO AVD Specialists Meeting on Strctral Aspects of Fleible Aircraft Control, Ottawa 18-2 October 1999, nmber RTO MP-36. RTO, October M. R. Waszak and D. K. Schmidt. Flight dynamics of aeroelastic vehicles. Jornal of Aircraft, 25(6): , B. A. Winther, P.J. Goggin, and J. R. Dykman. Redced-order dynamic aeroelastic model developement and integration wirh nonlinear simlation. Jornal of Aircraft, 37(5): , of 12 International Form on Aeroelasticity and Strctral Dynamics 25, Mnich, Germany.