Transonic Flutter Analysis Using Euler Equation and Reduced Order Modeling Technique

A new method identifies coupled fluid-structure system with a reduced set of state variables is presented. Assuming that the structural model is known a priori either from an analysis or a test and using linear transformations between structural and aeroelastic states, it is possible to deduce aerodynamic information from sampled time histories of the aeroelastic system. More specifically given a finite set of structural modes the method extracts generalized aerodynamic force matrix corresponding to these mode shapes. Once the aerodynamic forces are known, an aeroelastic reduced-order model can be constructed in discrete-time, state-space format by coupling the structural model and the aerodynamic system. The resulting reduced-order model is suitable for constant Mach, varying density analysis. Keywords—ROM (Reduced-Order Model), Aeroelasticity, AGARD 445.6 wing, NOMENCLATURE A B C Structural system matrices Aa Ba Ca Da Aerodynamic system matrices At Bt Ct Aeroelastic system matrices Ata Aerodynamic sub-matrix defied in (27) Cta Aerodynamic output sub-matrix defined in (32) Cv Aeroelastic output matrix for aerodynamic measurements Cw Aeroelastic output matrix for structural measurements F Forcing input matrix for structure G Generalized damping matrix K Generalized stiffness matrix M Generalized mass matrix Mach Mach number M Number of time steps N Number of structural modes or displacement measurements na Number of aerodynamic measurements P (2N x 1) generalized coordinates vector Qij Generalized aerodynamic force coefficients q Dynamic pressure ( 2 2 1 V ρ ≡ ) qref Reference dynamic pressure at which aeroelastic responses are sampled R Dimension of structural states vector x D. H. Kim is with Gyeongsang National University (GNU), Jinju City, Gyungnam, Republic of Korea (phone: 82+55-755-2083; fax: 82+55-755-2083; e-mail: [email protected]). Y. H. Kim is with Gyeongsang National University (GNU), Jinju City, Gyungnam, Republic of Korea (e-mail: [email protected]). T. Kim is with the Gyeongsang National University (GNU), Jinju City, Gyungnam, Republic of Korea (e-mail: txkim @ comcast.net). Re Reynolds number Rt Dimension of aeroelastic states vector Y η T Transformation matrix from x to Yx ζ T Transformation matrix from Y to x t Real time u control inputs vector V U ∑ Singular value decomposition matrices V Free stream air speed Vref Reference air speed at which aeroelastic responses are sampled v (na x 1) aerodynamic measurements vector w (2N x 1) structural measurements vector ( ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≡ z z & ) x (R x 1) structural states vector y Aerodynamic states vector Y (Rt x 1) aeroelastic states vector Yx (Rt x 1) structural sub-states vector Yy (Rt x 1) aerodynamic sub-states vector z (N x 1) displacement measurements vector Φ Structural sensor matrix ρ Air density Computational Fluid Dynamic (CFD) programs for aeroelastic simulations and analyses of military and civil aircraft. However, although the use of CFD is quite broad for static aerodynamic and aeroelastic calculations nowadays, it is limited in the field of unsteady aeroelasticity due to enormous size of computer memory and unreasonably long CPU time associated with long time periods required to observe transient responses and a large number structural modes. While a military airplane model may need 20-50 modes, a commercial aircraft model typically includes as many as 200 modes to describe the motion of the structure with enough accuracy. Thus, much research has been conducted on model reduction of the coupled fluid-structure systems including the eigen analysis [1]-[2], the Proper Orthogonal Decomposition (POD) or Karhunen-Loeve (KL) method [3]-[6], system identification methods [7]-[10]. However, in all of the methods the aerodynamics is treated separately from the structure making them difficult and inconvenient for structural engineers to apply. Recently, Kim [11], [12] developed a novel system identification and model reduction technique, also known as “Aerodynamics is Aeroelasticity minus Structure” (AAEMS), Transonic Flutter Analysis Using Euler Equation and Reduced Order Modeling Technique IN the past much effort has been made to utilize advanced NTRODUCTION I. I D. H. Kim, Y. H. Kim, and T. Kim World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:5, No:9, 2011 1506 International Scholarly and Scientific Research & Innovation 5(9) 2011 scholar.waset.org/1999.7/8981 In te rn at io na l S ci en ce I nd ex , P hy si ca l a nd M at he m at ic al S ci en ce s V ol :5 , N o: 9, 2 01 1 w as et .o rg /P ub lic at io n/ 89 81 for linear time-invariant, coupled fluid-structure systems. Unlike the previous methods, it works directly on time history data of the coupled aeroelastic system. Assuming that structural properties are known a priori, and using linear transformations between the structural and aeroelastic states, it extracts and models the underlying aerodynamic system with a finite number of state variables. Using two types of CSD/CFD models and simulations, Kim showed that the method is able to produce aerodynamic and aeroelastic ROMs with high accuracy without requiring a long CPU time normally associated with a large number of structural modes. In this paper, to demonstrate further the efficiency and accuracy of the new model reduction method, we will examine AGARD 445.6 wing modeled by FLUENT CFD and NASTRAN FEM programs. See Fig. 1 and Table I for the finite element model and its specifications. The wing motion is described by four natural modes and their natural frequencies are listed in Table II. Aeroelastic responses of the coupled CSD/CFD system, i.e., the displacements and velocities of the four structural coordinates will be recorded in time. In addition, unsteady pressures will be calculated at various points on the wing during the numerical simulation. These aerodynamic samples are necessary to generate aerodynamic ROM that is valid for all dynamic pressure values. All the responses will be obtained for a fixed Mach, at a low sub-critical dynamic pressure value. Once the aerodynamic ROM is obtained, an aeroelastic ROM can be constructed by coupling the aerodynamic ROM with the structural model. Using the ROM one can predict flutter by making Vg plot as a function of the dynamic pressure. It is also possible to use the model for other aeroelastic analyses such as dynamic flight loads and active control design. See Ref. [11-12] for examples of Vg plots obtained by the AAEMS and the aeroelastic ROM. Since the AGARD wing does not have any control surface, it is necessary to make up an artificial input for the purpose of the system identification. For example, any (8x1) arbitrary vector array with zeros in the top four and non-zeros in the bottom four entries multiplied by an impulse or a random time function will fulfill the requirement. More conveniently, however, an initial condition in the velocity components of the four structural coordinates can be imposed and the corresponding aeroelastic responses can be obtained. It is expected that the accuracy of the aeroelastic ROM will largely depend on the number and locations of the aerodynamic pressures. Kim [12] showed previously that even without the pressure data the AAEMS will accurately predict aeroelastic behavior in the neighborhood of the reference dynamic pressure. It was also shown that in order to improve the accuracy and extend its range away from the reference point it is necessary to add a sufficient number of the aerodynamic measurements. What is not known is optimum locations of the pressure points that will lead to an optimal aeroelastic ROM and this will be the main focus of the proposed research. Thus, different combinations of pressure values at different locations will be tested and the results will be reported in the final paper. Finally, advantages of using the new model reduction method over traditional methods will be discussed. More specifically, it is expected that the CPU time required for sampling the aeroelastic responses and creating the ROM will be significantly reduced. This is because in the new method there is no need to execute the mode-by-mode excitation for the calculation of the generalized aerodynamic force (GAF) matrix. In the case of the AGARD wing modeled by four natural modes, a single set of time samples due to a single initial condition will be sufficient to generate accurate ROM and therefore the saving in the computing time will be nearly a factor of four. An equally important advantage is that for structural engineers it will make the process of generating aerodynamic ROM handy and convenient because it bypasses the necessity to deal with the CFD directly. II. A. Basic Assumption We will assume that time histories of airplane structural and aerodynamic responses due to certain inputs, e.g., control surfaces, are available at both zero and nonzero air speeds. The structural responses here are displacements and velocities at various positions on the airplane, whereas the aerodynamic responses could be pressure measurements (in the case of tests, specially), or in the case of numerical simulations any of the independent aerodynamic variables such as vorticities, potentials in the flow field. The following assumptions are also made. 1. Structure, aerodynamics, and aeroelasticity are all dynamically linear, i.e., have small perturbed oscillations. 2. The airplane is flying along a CMVD curve. 3. Sufficient numbers of structural and aeroelastic measurements are available. 4. Background noise in the data is minimal or has been subdued by standard signal processing. 5. The system is controllable and observable. Fig.2.1 represent the process to develop ROM system using AAEMS (Aerodynamics is Aeroelasticity minus Structure) method. There are two way to predict flutter boundary (see Fig 2.2). One is CMVD (Constant Mach, Varying Density) and the other is CDVM (Constant Density, Varying Mach). CMVD is the method that predicts flutter boundary according to various flight altitudes with constant Mach and CDVM is the method according to various Mach with constant flight altitude. CDVM method is used in this study. B. Structural and Aerodynamic measurement First, at M time steps t M t t t ∆ − ∆ ∆ = ) 1 ( , , 2 , , 0 L we take airplane responses on the ground and in the air: 0 V = − @ ] [ 1 2 1 0 M w w w w L (1) ref V V = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − @ 1 2 1 0 1 2 1 0