Extension of the g-Method Flutter Solution to Aeroservoelastic Stability Analysis

The g-method frequency-domain flutter solution technique is extended to include control-system effects for closed-loop aeroservoelastic stability analysis. The extension is based on expressing the aeroelastic equations of motion in a Laplacedomain state-space form where the aerodynamic coefficient matrices retain their transcendental dependency on the Laplace variable. The augmentation of these equations with controlsystem state-space models is straight forward. First-order approximations of the system matrix are used for formulating an iterative eigenvalue problem that is solved by a predictor-corrector frequency sweep technique that ensures process robustness. The numerical application is of rollcontrol design for a generic fighter aircraft with several wing control surfaces. The results demonstrate the accuracy and convenient applicability of the extended method. 1 Professor, Associate fellow AIAA; moti_karpel@technion.ac.il 2 Senior Researcher, Member AIAA; boris@aemoti2.technion.ac.il 3 Vice President, 7430E. Stetson Dr., Suite 205; pc@zonatech.com Introduction Linear flutter analysis of flight vehicles is commonly based on the stability boundaries of the frequencydomain aeroelastic equation of motion in modal coordinates. While the structural mass, damping and stiffness coefficient matrices are constant in this equation, the aerodynamic influence coefficient (AIC) matrix is a transcendental function of the frequency of oscillations, calculated for required non-dimensional frequency values by numerical procedures such as the doublet-lattice method and the harmonic gradient method. Consequently, frequency-domain flutter solvers are based on search algorithms that iterate between the system eigenvalues and the AIC matrix. A widely adopted method for flutter solution is the p-k method that was first introduced by Irwin and Guyett and has been generalized and modified to include a determinant-based search process by Hassig. Rodden et. al. added a damping dependent aerodynamic term that improved the search process. Rodden introduced the p-k method to MSC/NASTRAN with a lining-up procedure that matched the frequency values with the imaginary parts of the resulting eigenvalues. Chen extended the p-k concepts in the damping-perturbation gmethod that includes a first-order damping term that is rigorously derived from the Laplace-domain aerodynamics. While the p-k solver serches for one aeroelastic root per each modal coordinate taken into account, the g-solver search algorithm is capable of handling extra roots due to unsteady aerodynamic lag. Aeroservoelasticity (ASE) deals with the interaction of aeroelastic and control systems. The application of modern control design techniques requires the aeroservoelastic equations of motion to be cast in a first-order, time-domain (state-space) form. This representation requires the aerodynamic matrices to be approximated by rational functions in the complex Laplace domain. The resulting statespace equations can be easily augmented by standard control system models to provide ASE constant coefficient equations for which stability analysis is based on standard eigenvalue extraction routines, as utilized in the ZAERO code. The problem is that the aerodynamic approximation is sometimes of questionable accuracy. Hence, it is desired to be able to perform closed-loop flutter analysis in the frequency domain, with the original tabulated AIC matrices. The purpose of the this paper is to discuss the application of frequency-domain techniques to the ASE problem, explain why existing procedures often 1 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere 7-10 April 2003, Norfolk, Virginia AIAA 2003-1512 Copyright © 2003 by M. Karpel, B.Moulin, and P. C. Chen. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. fail to provide a complete analysis, and to present modifications and extensions of the g method that result in a robust and reliable closed-loop flutter and control margins procedure. ASE Equations of Motion The Laplace transform of the open-loop aeroelastic equation of motion in modal coordinates, excited by control surface deflections, is [ ] [ ] [ ] [ ] ( ) )} ( { ) ( s s s s Q q K B M hh hh hh hh ξ + + + [ ] [ ] ( ){ } ) ( ) ( s 2 s s Q q M hc hc δ + − = (1) where {ξ} and {δ} are the vectors of generalized structural displacements and control surface deflection conmmands, [Mhh], [Bhh], [Khh] and [Qhh] are the generalized mass, damping, stiffness and AIC matrices, [Mhc] and [Qhc] are control-coupling mass and aerodynamic matrices, s is the Laplace variable and q is the dynamic pressure. The AIC matrices and can be calculated by unsteady aerodynamic codes at various tabulated reduced-frequency k values along the imaginary axis of the non-dimensional Laplace variable [ ] ) (s Qhh [ ) (s Qhc ] / p sb V g ik ≡ = + (2) where b is a reference semichord, V is the air velocity and k=ωb/V where ω is the vibration frequency. The ASE loop can be closed by relating the input commands to the generalized displacements by { } [ ]{ } ξ δ ) (s Tch = (3) where is a matrix of control transfer functions. The substitution of Eq. (3) in Eq. (1) yields [ ) (s Tch ] [ ] [ ] [ ] [ ] ( ) 0 )} ( { ) ( s s = + + + s s Q q K B M hh hh hh hh ξ (4) where [ ] [ ] [ ] [ ] ( )[ ] ch hc hc hh hh T s Q q M s Q s Q ) ( s ) ( ) ( 2 + + = (5) Previous applications of frequency-domain schemes to closed-loop ASE systems, such as in Refs. 6 and 11, simply applied the search algorithm with [ ] ) (ik Qhh of Eq. (5) (with s replaced by ikV/b) instead of [ . The problem is that ] ) (ik Qhh [ ] ) (ik Qhh may yield many roots coming from those of the control system. Consequently, the search algorithm, that is looking only for nh roots, might lose important aeroelastic roots. The g-method provides a convenient solution to this problem. State-space equations with transcendental aero matrices The ASE module in the ZAERO code is based on modeling the ASE problem in a constant-coefficient state-space form based on rational approximation of the AIC matrices. The control part of the modeling process transforms control components of the most general architecture into a single state-space equation that is augmented to the aeroelastic system. To facilitate the application of the g-method flutter procedure to control-augmented systems, the statespace ASE equations are expressed in this section in the Laplace domain, with the aerodynamic coefficient matrices kept in their transcendental form, before rational approximations are made. These equations can be used for s-domain stability and response analyses where the aerodynamic coefficients are interpolated from a database of force coefficient matrices calculated at tabulated reducedfrequency values. Such analyses may be useful when the rational aerodynamic approximations are of questionable accuracy, or when the results are to be compared with those obtained by other numerical solutions with interpolated aerodynamics. The open-loop ASE equations of motion (1) for the uncoupled ASE and control systems can be expressed as: