Aeroelastic Damping Model Derived from Discrete Euler Equations M. A. Woodgate * and K. J. Badcock † University of Liverpool, Liverpool, L69 3BX England, United Kingdom DOI: 10.2514/1.23637 The prediction of flutter onset based on aerodynamic modeling using computational fluid dynamics can be made using an augmented system of equations. Computational times similar to those required for computational fluid dynamics steady-state calculations have been reported for wing test cases. However, for such methods to be fully useful, information about damping must be obtainable without reverting to full-order time-domain simulation. This paper presents a method for computing damping based on a reduced-order modeling approach that systematically derives a two-degrees-of-freedom model from the full discrete system of equations. The method is based on a change of variables that employs the critical eigenvector of the aeroelastic system. The ability of this model to predict the damping for a model problem and for two wing test cases is shown. Nomenclature A = Jacobian matrix E = fluid total energy F = nonlinear part of R F, G, H = convective flux vectors f = force acting at the grid points h = grid spacing in the tubular reactor test problem P = fluid pressure Pe h , Pe m = constants in the tubular reactor test problem p = critical eigenvector of A T , p 1  ip 2 q = critical eigenvector of A, q 1  iq 2 q s = constant scaling vector R = residual function S = transformation function between aerodynamic and structural grids t = real time U, V, W = contravariant Cartesian fluid velocity components u, v, w = Cartesian fluid velocity components w = state variables x, y, z = Cartesian coordinates x = grid locations Y = unknown in the tubular reactor test problem y = part of w not in the critical space z = part of w in the critical space  = structural modal coordinate  0 = constant in the tubular reactor test problem ,   = constants in the tubular reactor test problem  = change in grid locations  = unknown in the tubular reactor test problem  = bifurcation parameter  = fluid density  = structural mode shape  0 i = blending function for transfinite interpolation ! = frequency of critical eigenvalue Subscripts A = of augmented system a = aerodynamic s = structural 0 = values at the bifurcation point Superscripts i = inviscid T = transpose Introduction C OMPUTATIONAL aeroelasticity has developed rapidly, with attention focusing on time-marching calculations using CFD, where the response of a system to an initial perturbation is calculated to determine growth or decay, and from this to infer stability. Recent and impressive example calculations have been made for complete aircraft configurations (see [1–3] amongst others). The time-domain method is powerful because of its generality and ease of use. However, basing an investigation of system dynamics in the time domain has one major drawback, namely, the computational cost. This has led to an intensive effort to extract the useful information out of the full CFD model of the aerodynamics to provide a cheaper model that still retains the essential physics of the problem. Examples include proper orthogonal decomposition [4] (which involves the extraction of modes using a limited set of time snapshots of the flow evolution), a Volterra series (which relates the aerodynamic response to some input by a kernel [4]), and system identification (where a linear model is calculated from a limited time evolution of the aerodynamic response to some input). To date, no single method has proved its utility on general aeroelastic problems. Arguably the most advanced reduced-order method is proper orthogonal decomposition (POD). A set of solutions (steady and unsteady) are generated for prescribed conditions and motions. For example, in [5] the POD modes are calculated for the AGARD wing based on forced motions in the frequency range of interest for the structural modes. From this information, a small-order model was developed and the behavior of the aeroelastic eigenspectrum, which could be calculated for the reduced system, was examined. This approach was used to compute limit-cycle behavior for an airfoil [6], and access to the complete eigenspectrum provided good insights into the physical behavior. However, it is clear from the literature (for example [4,5,7]) that the phenomena which are relevant for a particular problem must be present in the training solutions for the modes. Aspects such as the mean shock location and the range of shock motion must be well covered. In this respect, the problems examined in the literature are Presented as Paper 2021 at the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, Rhode Island, 1–4 May 2006; received 4 March 2006; revision received 17 April 2006; accepted for publication 7 May 2006. Copyright © 2006 by K. J. Badcock and M. A.Woodgate. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 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 $10.00 in correspondence with the CCC. * Research Assistant, Computational Fluid Dynamics Laboratory, Flight Sciences and Technology, Department of Engineering. † Professor, Computational Fluid Dynamics Laboratory, Flight Sciences and Technology, Department of Engineering; K.J.Badcock@liverpool.ac.uk. AIAA JOURNAL Vol. 44, No. 11, November 2006 2601