Aeroservoelastic Analysis for Transonic Missile Based on Computational Fluid Dynamics

A EROELASTICITY is a multidisciplinary field of study dealing with the interaction of inertia, structural, and aerodynamic forces. Flutter is a typical aeroelastic problem that can cause an unstable vibration. With the improvement of electrical technique, many kinds of flight control systems are widely used in new aircraft. The new problem is joined with aerodynamics, elasticity, and control systems, and designers should check the stability of the problem of aeroservoelasticity. In the transonic flow region, the shock position is very sensitive to structural vibration, and there is an obvious delay between structural motion and the aerodynamic force. A dip often appears on the flutter boundary in the transonic region, and it often leads to a bottleneck problem in the flight envelope. Some simplemodels are still used [1] to calculate the aerodynamic loads of missiles in the aircraft design institute (e.g., slender-body theory for the body and strip theory for the wings). Most of those classical methods are difficult to use when considering the interference between the wing and the body. Lifting surface methods based on linear theory have also been used for unsteady aerodynamic computation and flutter analysis [2], such as the doublet lattice aerodynamic model in NASTRAN [3]. This kind of method needs to compute thegeneralized aerodynamic force for a rangeof frequencies for each structural mode. Nevertheless, this method is confined to planforms of very thin sections at small angles of attack, whereby neither thickness effect nor angles of attack can be accounted for; it is also incapable in transonic flow. For the transonic flutter problems, it is important to simulate the nonlinear unsteady aerodynamics with the presence of shock movements. With the progress in CPU speeds, computational fluid dynamics (CFD)-based time-integration techniques have been used in aircraft design [4,5]. Coupling structural equations with an Euler/ Navier–Stokes-based unsteady CFD algorithm, the structural aeroelastic responses can be predicted in the time domain. Thosemethods are suitable for solving the nonlinear transonic flutter problems, because they make the fewest assumptions about the characteristics of the flows. However, the challenge of these kinds of methods is their ineffective use in the preliminary aeroelastic design stage. Recently,much researchwascarried out in reducedordermodeling (ROM) for unsteady aerodynamics. It is novel, in that it captures (to some extent) the nonlinear flow characteristics, and it is more computationally efficient than full-CFD simulation. It is very suitable for predicting the flutter boundary in transonic flow. There are two kinds of methods to construct the reduced-order aerodynamic models. One is the proper orthogonal decomposition (POD) technique [6–10], and the other is aerodynamic modeling based on the system identification technique [11–22]. POD is a method that is used extensively at several research organizations for the development of ROMs. A thorough review of POD research activities can be found in the paper by Lucia et al. [6]. Romanowski [7] is perhaps the first to introduce the POD technique to construct the reduced-order aerodynamic model for flutter analysis. In addition, a review of the issues involved in the development of ROMs for aeroelastic problems is provided by Dowell and Hall [9]. A topic of recent interest is the potential development of parameter adaptation of the reduced order models at different Mach numbers; two interpolation methods for adapting the POD basis vectors to varying Mach numbers are presented in [10]. A simplified aerodynamic model that captures the dominant dynamics of the flow can also be constructed by the system identification technique. Silva [11] is among thefirst to introduce nonlinear Volterra theory to unsteady aerodynamics. Based on unit impulse responses, the first-order (linear) kernel and the second-order (nonlinear) kernel are numerically identified for a single-input–singleoutput system. Silva and Bartels [12], Marzocca et al. [13], and Raveh [14] have also applied the Volterra kernel identification technique into aeroelastic systems. The Volterra kernels are approximated in terms of orthonormal piecewise-polynomial multiwavelets. A least square (LS) problem is solved for the multiwavelet coefficients that represent the kernels. As for the secondand higherorder kernels, a very large number of coefficients are required for accuracy. Raveh [15] found that when the nonlinear system is assumed to be a second-order system, the convolved response (based on theVolterra ROM)was extremely sensitive to the amplitude of the impulse inputs used for kernel identification. Only when the step amplitude was closer to that of the direct excitation signal was the prediction accuracy improved. Assuming the system to be a second order did not improve the ROM when compared with a first-order ROM. Cowan et al. [16] used a kind of input–output difference model [autoregressivewith exogenous input model (ARX)] to represent the relationship between generalized aerodynamic force coefficients and structuralmodal coordinates. Amultiple-step input signal is used to prescribe the motion of the modal, and then CFD solutions are carried out to provide a complete data set (input/output) for training. Once the model is defined, it is used in place of the CFD codes in the coupled structural equations to predict the structural responses. Raveh [17] used two types of input signals for modal coordinates: one is the random time series and the other is a filtered random series with Gaussian distribution for system identification of unsteady aerodynamics. Three types of modeling between the generalized aerodynamic force coefficients (outputs) and modal coordinates (inputs) are constructed: a frequency domain model, a discrete-timedomain model, and a discrete-time-domain state-space model. The efficiency of this kind of ROM-basedmethod is improved up to 1 2 Presented as Paper 6482 at the AIAA Modeling and Simulation Technologies Conference and Exhibit, Hilton Head, SC, 20–23 August 2007; received 3 May 2009; revision received 29 August 2009; accepted for publication 9 September 2009. Copyright © 2009 by the American Institute of Aeronautics andAstronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0021-8669/09 and $10.00 in correspondence with the CCC. ∗Associate Professor, National Key Laboratory of Aerodynamic Design and Research, College of Aeronautics; zww12345@sina.com. Member AIAA. (Corresponding author). Professor, National Key Laboratory of Aerodynamic Design and Research, College of Aeronautics; yezy@nwpu.edu.cn. Ph.D. Student, National Key Laboratory of Aerodynamic Design and Research, College of Aeronautics; zhch_a@163.com. JOURNAL OF AIRCRAFT Vol. 46, No. 6, November–December 2009