CONEM 2012 «Engenharia em destaque» Vil Congresso Nacional de Engenharia Mecdnica ‘Sao Luis - Maranhao - Brasil 31 de julho a 03 de agosto ‘wwwabem.org.br/conem2012 AEROFLEX: A TOOLBOX FOR STUDYING THE FLIGHT DYNAMICS OF HIGHLY FLEXIBLE AIRPLANES Flavio Luiz Cardoso Ribeiro, flaviocr @' Pedro Paglione, paglione@ita.brl’ Roberto Gil Annes da Silva, gil@ita.br' Marcelo Santiago de Sousa, marcelo.santiago@ uni edu.br? “Instituto Tecnol6gico de Aeronautica, Praga Marechal Eduardo Gomes, 50, Campus do CTA, Sto José dos Campos -SP. Universidade Federal de lajubs, Av. BPS, 1303, Bairo - Pinbeirinho, Itsjubs- MG Abstract: This work addresses a mathematical formulation to model highly flexible airplanes. A toolbox was developed and can be used to analyze how structural flexibility affects the airplane flight dynamics. A nonlinear beam model was applied to represent the structural dynamics, taking into account large displacements. For aerodynamic calculations, the strip theory was used including three modeling approaches: a quasi-steady, quasi-steady with apparent mass and full unsteady aerodynamics representations. Nonlinear simulations are performed and, through linearization of the equations of motion, dynamic stability is analyzed. Keywords: aervelasticiy, flight dynamics, flexible airplanes, structural dynamics 1. INTRODUCTION Although all airplanes are flexible, rigid body assumption is very usual during the studies of flight dynamics. The effects of flexibility are usually taken into account by the discipline of aeroelasticity. This separation between aeroelas- and rigid body flight dynamics used to be enough to describe these phenomena, but recent progress in aeronautical engineering with the advent of lighter structural materials has led to more flexible airplanes and urged the development of complete flight dynamics models including structural flexibility effects. ‘Waszak and Schmidt (1988) described dynamic equations of motions that include a linearized structural model. This approach can be used to study the influence of small structural deflections inthe rigid body flight dynamics, Silvestre and Paglione (2008), Pogorzelski (2010) and Silva et al. (2010) used this formulation to study the flight dynamics and control of flexible airplanes. Among their hypothesis, these works neglect the inertial coupling between rigid body and flexible modes. ‘The recent development of High-Altitude Long-Endurance (HALE) airplanes increased even more the need for appro priate modeling of highly flexible aicrafts: since they have very high aspect ratio and low structural rigidity, their wings present large structural deflections as it can be seen in Figure 1 Patil (1999) used a nonlinear beam model from Hodges (1990) to describe the flight dynamics of highly flexible air planes. Brown (2003) modified the formulation, rewriting the equations in a strain-based form; he developed a framework for studying of wing warping as a means of achieving aeroelastic goals. Subsequently, Shearer (2006) improved the mo- deling, replacing numerical iterative calculations by closed form expressions. Su (2008) included absolute and relative nodal displacement constraints, allowing the study of Joined-Wing configurations. ‘AeroFlex isa toolbox that intends to implement the formulations of Brown (2003) and Shearer (2006), allowing the study of highly flexible airplanes flight dynamics. Among its main capabilities: ‘© Simulation and stability analysis of classic wing aeroelastic phenomena like: divergence, fluter, aileron reversals ‘© Simulation and stability analysis of nonlinear wing aeroclastic phenomena, due to nonlinear geometry deflections; ‘© Simulation and stability analysis ofa flexible aircraft in free-flight condition, ‘The main goal of this work is to present the implementation of AeroFlex. The equations of motion are presented in Section 2. Section 3 describes how the equations are solved. Section 4 presents the AeroFlex computational environment. Section 5 shows the results of several studies performed using the toolbox with the goal of validating it, These results are ‘compared with test cases presented in the literature.VII Gongresse Nacional ae Engenharia Mectnica, 31 de juthe @ 03 ee Agosto 2012, Seo Luts - Maranhéo Figure 1: NASA Helios - Ref. Noll ef al. (2004). 2, THEORETICAL FORMULATION, ‘The equations of motion can be obtained from the principle of virtual work. The deduction is presented by Brown (2003), Shearer (2006), Su (2008) and Ribeiro (2011). ‘A three-dimensional structural model is decomposed in a bidimensional (cross-sectional) analysis. The results from the cross-section can be used (o analyze a unidimensional beam, Each flexible structural member of the airplane is treated as a beam. These beams are split in several elements, each one can undergo deformations of extension, flexion and torsion. The deformations vector « represents the deformation of each structural element, Linear and rotational speeds aze represented by (. ‘The following equations of motion represent both the rigid body motion and the structural dynamics, [ser aie JE ]*[ece ete ][Es]+E%" s][e]=L ee] ® ms Mrr(© =JEMJne Mra(¢) JM Jus Muv(e) JM Ine Maal) =JieM Jas > Mies Cer(ed) -IM Ine +6 Cra (é,8) JEM Hy + 202M dna + One Carle, ~JaM Ine Caale.é,8) JEM Hn + IRM @) Kee -K [S]-[B}e Loe [ Berl lel oe te) In Eq. 1, My represents the mass matrix; Ci,; represents the damping matrix; Kr represents the structural rigidity matrix. Its possible to see that rigid body states (represented by 8) are inertally coupled with structural states (¢, since the mass matrix is not diagonal Re and Re represents the generalized forces that are applied in the airplane. They are obtained from the aerodynamic, sravitational and propulsive forces applied to each structural node. Strip theory is applied, so that aerodynamic forces and ‘moments are calculed using bidimensional models in each node. The aerodynamic models are presented in Section 2.1 ‘The Jacobian matrices Jy. and J represent the relationship between structural deformations (¢) and nodal displace- ‘ments and rotations. Jn and Jo tepresent the relationship between rigid body degrees of freedom and nodal displace- ‘ments and rotations. The Jacobian matrices are nonlinear functions of ¢, They can be obtained either numerically (through numerical linearization), or through analytical expressions as presented by Shearer (2006). This work uses the latter, In addition, M is the flexible structure mass matrix. It depends only on inertias and masses of the structural elements {it is not dependent on strain, differently from the Mj; matrix). K isthe structural rigidity matrix and Cis the structural damping matrix. In AcroFlex, we usually uses a linear relationship between C’ and K, given by: Ss Cack @ where cis the damping ratio.VII Gongresse Nacional ae Engenharia Mectnica, 31 de juthe @ 03 ee Agosto 2012, Seo Luts - Maranhéo Euler angles are used to describe the airplane attitude (4,0 and ¥, which are used to describe the bank, pitch and heading angles), Stevens and Lewis (2003) show that the time rate derivative of Euler angles are related with angular speeds (P.Q.R) by the following expressions: 4 =Qeosg— Rsing o O=P +tand(Qsing + Roos) o (Qsing + Reos 4) oO 088 ‘Stevens and Lewis (2003) also presents the relationship between speeds in the Body Frame (given by U.V,W) and Inertial Frame (given by H, % and 9): H =U sind — V sinécos — W cosgcos8 ® =U cos cosy + V(sindsin8 cos vi — cos sin wv) + W(cos sind cos + singsiny) o 4 =U cosOsiny + V(sin dsinO sing + cos dcos ey) + W(cos dsinOsin y — sin gos v) (10) 2.1 Aerodynamic Models ‘The aerodynamic model is included through bidimensional forces and moments distributed slong the beams. In this work, strip theory is used: aerodynamic forces and moments are calculated in each node considering an independent two-dimensional aerodynamic model, Once aerodynamic forces (drag and lift) and moment are calculated; forces and moments are transformed from the local aerodynamic frame to body frame. The force vectors are then arranged in a 9N vector (where N is to number of elements). These vectors, of distributed forces and moments, are then applied to Eq, 3. Three different models were used to calculate aerodynamic forces. The first consists in a quasi-steady model, which takes into account only the circulatory part of lift force, neglecting the wake effects. The second model includes the apparent mass terms. Finally the third model is an unsteady aerodynamic model proposed by Peters etal. (2007), which includes states to represent the aerodynamic lag due the wake. 21.1 Quasi-steady ‘The following equations can be used to calculate the aerodynamic lift and moment around the elastic center for a flat plate (Ref, Haddadpour and Firouz-Abadi (2006)): i & L=2npbt? |" 4 905 —a)S +a Le =Inpbl! lr (0. let | ay Meg =(05 + a)h.— TAU ay In these equations, p is the air density; U is the airspeed a isthe local angle of atack; & is the airfoil semichord: and ais the distance between elastic axis and the half chord (normalized by the semichord 6). Itis possible to rewrite these equations asa function of relative speeds written in the zro-lft coordinate system! (13), (4) 2.1.2 Quasi-steady with apparent mass ‘The following equations include the apparent mass effect (Ref, Haddadpour and Firouz-Abadi (2006) 1b =npb? [is baa + 08] + 2npbU [i+ (0.5 — 0)6 + U4] as) Men =0(0.5 + a)L ~ xpd fo oh + (0.125 — 0.5a?)é + UA] 6) Agsn, itis possible to rewrite the equations a a funtion of variables writen in the zer-t coordinate system L =nob*(-24- 96 — da) + 2rphi? [-2 + (30d) $] an y 2 v Men =b(0.5 + a)L ~ xpb* [-0.52 + 6(0.125 — 0.50°)& + 54] (18) onda system are sepeeseted hereby # anVII Gongresse Nacional ae Engenharia Mectnica, 31 de juthe @ 03 ee Agosto 2012, Seo Luts - Maranhéo 2.1.3 Unsteady An unsteady aerodynamic model based on Peters ef al. (1995) is applied. Expressions for lift and drag are presented by Shearer (2006): 7 6 — da) + Onphi? |-F (G 3 ay stg td pt (-$ue— bata fons 2a) a (2 (22) A= B\A+ Eye + By + Ege (23) On above axpressions, Ng is the number of aerodynamic lag states..,; Bi, Ref. Balvedi (2010). » Bs € By are matrices presented in 2.14 Drag Inthe previous modelling approaches, the dag is calculated using a constant seo drag coetcient (Ca) D = hpite, ey xen 2.1.5 Including trailing edge Map deflections ‘Trailing edge deflection is implemented by adding incremental values to the airfoil aerodynamic forces and moments: 25) (26) LS pby?C, on MS = pC su (28) 1.6 and Cm,s can be obtained through experimental data or airfoil analysis softwares. 6, isthe airfoil flap deflection. 2.2 Control inputs ‘Two types of control inputs are used in this modelation: «+ Flap deflections 5, (as presented in the previous section); # Engine throttle =, Propulsion forces are modeled as point forces attached to a structural node.VII Gongresse Nacional ae Engenharia Mectnica, 31 de juthe @ 03 ee Agosto 2012, Seo Luts - Maranhéo 3. SOLVING EQUATIONS, LINEARIZATION AND STABILITY “The elastic equations of motion (Eq. 1), the unsteady aerodynamic equations (Eq, 23) and the kinematies equations (Eqs. 5, 6, 7, 8, 9 and 10) represent all the needed expressions to describe the flight dynamics of the flexible airpla- nes. These are 4.V second order differential equations to describe the structural dynamics, 3.V.V 4 first order differential equations to describe the lag aerodynamic states and 12 first order equations to describe the rigid body motion * Following, the methodologies used to find the equilibrium condition, integrate and linearize the equations of motion are presented 3.1 Calculation of equilibrium The determination of equilibrium concition inthe ease ofthe full sirplane is done by the following iterative procedure 1, Consider «= 0; 2. Calculate the rigid body equilibrium (§ = 0° 3. Calculate the structural equilibrium (@ — 0) 4, Return to item 2 until both conditions are valid (3 = 0 e ¢ = 0), In the case of a straight level fight, for example, 2 specific ight condition is given (altitude, speed and angle of tuajectory). Step 1 finds the engine throttle 7, the elevator angle 6 and the pitch angle 9. Step 2 finds the structural deformations e, Each step is found by using numerical methods. 3.2 Integration of the nonlinear equations of motion ‘We have a large system of differential equations (Eqs. 1, 23, 5, 6,7, 8, 9 and 10). Eq. 1 is a second order system of equations. To solve these equations we could try to convert this system in a first order system, Unfurtunately, this is not usually possible, since aerodynamic force expressions are nonlinear functions of the states’ time rate, So two options are avaiable: ‘* Neglect the derivative terms from aerodynamic expressions and transform the system of equations into a frst order system, This lead to a system of equations that can be integrated using classic explict methods (like Runge-Kutta, methods); ‘Integrate the second order system of equations using an implicit method. ‘The second option, though usually slower, is obviously the most precise, 3.3 Linearization and stability Its possible to linearize the equations of motion with the goal of studying the stability of the flexible airplane. The system of equations can be represented in the following form: $6629.38, 8,88, 6,,,75) =0 29) where fis a nonlinear function which dimension is equal to the total number of system's states, Eis the vector of inematis variable: k=[¢ oun G0) ILis possible to reduce it toa first order system (making X = @) and linearize it around a equilibrium point M +Bl as %] ep me em The linearization is done numerically, By analysing the eigenvalues of M~1A, we can verify if the system is stable. Choosing subsets of matrices M and A. itis possible to decouple the rigid body and structural dynamics. This allows, in a single process of linearization, determine the stability characteristics and autonomous response of the following systems: 2 Where Viste total umber of srectural element; Nis the numberof lg states in each node See that in the equilibrium: Myg@ = Rp. So: 5 = Dis equivaleatto Ry = 0. Where Ry is the sum ofexernal ores, whichis also a function “We calculate eo that Kure = Ry, Remeber that Ry is a fanetion ofVII Congresse Nacional av Engenharia Mectnica, 91 lutho 2 03 6e Agosto 2012 Séo Luls - Marenhae + Rigid body; ‘© Cantilevered wing: ‘¢ Flexible airplane in free flight, To determine the instability speed (flutter, divergence or other). the following procedure is applied: the airplane speed is increased; for each speed, a new equilibrium condition is obtained; the system is linearized; the largest real part of the eigenvalues of M~' Ais taken. Once one of the eigenvalues has a positive real part, the system is unstable. The imaginary part of this eigenvalue gives the frequency associated with the unstable aeroelastic mode. 4. CODE IMPLEMENTATION Aeroflex is intended to perform the following tasks: ‘« Implement the strain-based geometrically nonlinear beam structural dynamics model proposed by Brown (2003) and the improvements proposed by Sheater (2006) ‘¢ Allow the study of airplanes with the following characteristics - Rigid fuselage and flexible members (wings, horizontal and vertical tail) - Rigid concentrated mass/inertia elements attached to the flexible structure's nodes (to represent engines or fuel tanks for example) = Propulsve forces attached to the flexible structure's nodes, + Determine the equilibrium point, considering the structural deformations + Linearize the equations of motion, allowing the dynamic stability study; «Simulate the linear and nonlinear dynamics, using several numerical integration methods The code is intended to be very general, allowing the user to define new configurations easly. The authors chose to write the AeroFlex code using the Matlab @. 4.1 Initializing the airplane data AcroFlex was developed using objectoriented programming. Several classes were defined, in order to initialize and ‘update the various data types that should be handled by the program, The four most important classes of this tool are the following: node, element, engine, airplane. To initialize the airplane modelling, the user needs the following information: masses and inertias per unit of lenght at each structural node; rigidity and damping matrices of each element; lengh of each element and relative orientation beetwen one clement and the next one, These data are obtained from the airplane geometry and from a cross-sectional analysis software. Additionaly, the user needs the aerodynamic data of cach node (zero-lft angle of attack, Ci., number of lag-states, etc). (Once all airplane properties are known, the following procedure should be done by the user: 1. Create node objects. Each object is a structural node and needs to be initialized with the mass and inertia data, ‘These objects also have the information about the aerodynamic model (since the aerodynamic calculations are done in each node); 2. Create vectors of element objects. Bach vector is a flexible member. Each unit of this vector is an element and it is associated with three node objects. These objects includes the rigidity and damping properties, associated with ‘each clement. The user can create how many flexible members are needed to describe the airplane; 3. Create engine objects. Each object of this class is an engine. To initialize this object, the user gives informations about the engine’s position and about the propulsive model. 4, Create one airplane object. This object covers all the airplane data, Their input arguments are: vectors of element ‘objects (members) and engine objects, In addition, the user can start this object with rigid fuselage data, if it exists. Following this procedure, the user will have an object ofthe airplane class, which includes all the structural, aerodyna- ical and propulsive data of the airplane, This object allows the use of methods for calculating equilibrium; linearization; simulation and others. These methods are presented in the next items. ‘Matlab 2010, The Maur, Natick, MAVII Gongresse Nacional ae Engenharia Mectnica, 31 de juthe @ 03 ee Agosto 2012, Seo Luts - Maranhéo -10 Figure 2: Example of airplane modeled in AeroFlex. 4.2. Equilibrium methods One of the airplane class functions is the trimairplane. This function's goal is to calculate the structural and rigid body equilibrium given a straight flight condition (altitude and speed). The deformations vector ¢, the elevator angle, the pitch angle and the throttle are the outputs for this function, The methodology to find equilibrium is described in Section 3.1 AeroFlex uses FSOLVE Matlab function to solve the nonlinear equations. 4.3 Linearization method The function linearizeairplane has the following input arguments: an airplane object and equilibrium conditions around which the linearized matrices should be calculated. The following outputs are presented by this function: A and B matrices of the full linearized system; in addition, matrices Ageroctase © Atody ate the linearized system neglecting the rigid body and flexible degrees of freedom, respectively. Analyzing the eigenvalues of cach of these matrices, itis possible to study the system stability. Linearization is performed numerically as presented in Section 3.3, 4.4 Nonlinear simulation method “The function simulate is also a method of airplane class. It is intended to simulate the nonlinear dynamics. Its input arguments are: the airplane object; initial conditions; function handles to describe the engine throttle and elevator inputs as a function of time: integration method (implicit or explicit) To integrate the equations of motion, the ODEI5i and ODE15s Matlab functions are used (implicit and explicit metho- dologies, respectively), 4.5 Graphical outputs methods The outputs of the simulation routines are vectors ofthe system’s states for each instant of time (strain ¢, linear and angular speeds 3, position and orientation of the body frame & and lag states A). A function called airplanemovie was created allowing the presentation of a video with the airplane deflections along the time from the simulation results. The input arguments for it are: the airplane object; atime vector; a strain vector for each instant of time. Additionally, the function plotairplane3d presents a 3D figure of object airplane. Figure 2 shows an example of this graphical output. 5, RESULTS In order to validate the toolbox, several results were obtained and compared with literature. AcroFlex was used to solve structural problems; aeroelastic problems; and to make comparisons with rigid body flight dynamics. More resultsVII Congresse Nacional ae Engenharia Mecénica, 91 49 juihe 8 02 ee Agoste 2012, ccan be found in Ribeiro (2011). 5.1 Structural Problems A simple cantivelered beam was modeled, as proposed by Ref. Brown (2003). Concentrated forces and moments were applied, as presented in Figure 3, Results show good agreement between this tool and the literature results, as shown in Figure 4. '" = 1m ——__—___| Figure 3: Cantilevered beam with a force applied - Ref. Brown (2003). ‘ Te acomt coe ee oe Figure 4: Deflection of tip as a result of a force. 5.2. Aeroelasticity ‘Two test cases presented in the literature were analyzed, with the goal of finding the utter speed and frequency of cantilevered wings, In order to get these results, itis necessary to find the equilibrium condition for several different speeds, linearize the equations of motion and get the eigenvalues ofthe state matrix. Once at least one of the eigenvalues thas a positive real part, the system is unstable. We can get the frequency of the unstable modes from the imaginary part of this eigenvalue (if it i oscilatory). The first test case results are for the Goland wing (Ref, Goland (1945)) and can be seen in Table 1. After that, we show the results for a highly flexible wing, as proposed by Patil (1999). Due to its highly flexible nature, this wing shows an interesting result: if we study instability around an undeformed shape, very different results than those of a deformed shape are obtained (Fig, 5). Results are shown in Tables 2 and 3. ‘Table 1: Flutter speed and frequency for the Goland Wing. Results Attitude AcroFlex Ref Brown (2003) v f v f (és) (rads) ls) (rad/s) on 451 72 447 7 Bx 1A | 58L oT 374 68.1VII Congresse Nacional av Engenharia Mectnica, 91 lutho 2 03 6e Agosto 2012 Séo Luls - Marenhae Figure 5: Highly Flexible Wing. ‘Table 2: Flutter speed and frequency for the highly flexible ng - undeformed wing, Tero Fe Re PHT (TIDY Speed (an/s) 326 32.2 Frequency (rad/s) 22.6 22.6 ‘Table 3: Flutter speed and frequency for the highly flexible wing - deformed wing. Tero Fle Ref Su BOOBY Speedimis) 23.4 232 5.3 Flight Dynamics In order to check ifthe flight dynamics simulated by AeroFlex agrees with a rigid body classical model (as of Ref. Stevens and Lewis (2003)), we've modeled a flying wing (Figure 6). Results for a doublet input in the elevator are presented in Fig. 7. For the very rigid aisplane (K=1000), the results of AeroFlex and the rigid body dynamics are very Similar. On the other hand, fora highly flexible airplane (KI), we can see the coupling between structural and ight dynamics responses. Figure 6: Flying wing. 6. CONCLUSIONS This work presented the implementation of AeroFlex, a computational tool that allows the study of highly flexible airplane flight dynamics. This tool was used to study several test cases, from static structural problems to fight dynamics of flexible vehicles. Results obtained are very similar from those of the literature. ‘The methodology used is more suitable for studying airplanes with high aspect ration lifting surfaces, since it uses strip theory for aerodynamics and beam theory for structural dynamics. Ir order to study low aspect ratio wings, a threedimensional aerodynamic model would be necessary.VII Gongresse Nacional ae Engenharia Mectnica, 31 de juthe @ 03 ee Agosto 2012, Seo Luts - Maranhéo 1 [oT Tine) Ties Figure 7: Simulated response for a doublet input in the elevator. 7. REFERENCES Balvedi, E.A., 2010. Linear and Nonlinear Aeroelastic Analyses of a Typical Airfoil Section With Control Surface Free- play. Mestrado em engenharia acronéutica mecénica, Instituto Tecnolégico de Aeronsutica, Sao José dos Campos. Brown, EL, 2003. Integrated Strain Actuation In Aircraft With Highly Flexible Composite Wings. Dissertation for a doctoral degree, Massachusetts Institute of Technology. Goland, M., 1945, “The fluter of a uniform cantilever wing”, Journal of Applied Mechanics, Vol. 12, No. 4, pp. A197— A208. Haddadpour, H. and Firouz-Abadi, R.D., 2006. “Evaluation of quasi-steady aerodynamic modeling for fluter prediction in subsonic Mow”. Journal of Thin-Walled Structures, Vol. 44, No. 9, pp. 931-936. Hodges, D., 1990, “A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams”. International journal of solids and structures, Vol. 26, No. 11, pp. 1253-1273, Noll, T., Brown, J., Perez-Davis, M., Ishmael, $., Tiffany, G. and Gaier, M., 2004. Investigation of the helios prototype aircraft mishap. NASA. Disponivel em: . Acesso em: 29.10.2011, Washington,DC. Pati, MJ, 1999. Nonlinear Aeroelastic Analysis, Flight Dynamics, and Control of a Complete Aircraft. Dissertation for a doctoral degree, Georgia Institute of Technology. Peters, D.A., Hsich, M.A. and Torrero, A., 2007. “A state-space airloads theory for flexible airfoils". JOURNAL OF THE AMERICAN HELICOPTER SOCIETY, Vol. 52, No. 4, pp. 329-342. Peters, D., Karunamoorthy, $. and Cao, W., 1995. “Finite state induced flow models. i: Two-dimensional thin airfoil” Journal of Aircraft, Vol. 32, No. 2, pp. 313-322 Pogorzelski, G., 2010. Dindmica de Aeronaves Flexiveis Empregando Teoria das Faixas Nao-Estaciondria. Mestrado em. cngenharia aerondutica e mecénica, Instituto Tecnoligico de Aeronautica, S40 José dos Campos. Ribeito, FL.C,, 2011. Dinamica de voo de aeronaves muito flexiveis. Master of science thesis, Instituto Tecnologico de Aeronautica Shearer, CM., 2006. Coupled Nonlinear Flight Dynamics, Aeroelasticity and Control of Very Flexible Aircraft. Disserta- tion for a doctoral degree, The University of Michigan, Ann Arbor: Silva, A.L., Paglione, P. and Yoneyama, T., 2010. “Conceptual flexible aircraft model for modeling, analysis and control studies”. In Proceedings... Atmospheric Flight Mechanics Conference, ATAA, Reston, VA. Silvestre, F. and Paglione, P., 2008. “Dynamics and contol of a flexible aircraft”. In Proceedings... Atmospheric Flight Mechanics Conference and Exhibit, AIAA, Reston, VA. Stevens, B. and Lewis, F, 2008. Aircraft control and simulation, Wiley-Interscience, Hoboken, New Jersey. Su, W., 2008. Coupled Nonlinear Aeroelasticity and Flight Dynamics of Fully Flexible Aircraft. Dissertation for a doctoral degree, The University of Michigan, Ann Arbor. Waszak, MR. and Schmidt, D.K., 1988. “Flight dynamics of aeroelastic vehicles”. Journal of Aireraft, Vol. 25, No. 6 8. RESPONSIBILITY NOTICE “The authors are the only responsible forthe printed material included in this paper.