Accurate Subcritical Damping Solution of Flutter Equation Using Piecewise Aerodynamic Function

Abstract

An alternative to the p-k method for accurate subcritical damping solution is presented. By combining the concepts used in rational function aerodynamics and using the piecewise aerodynamic interpolation function commonly used for the k method, a piecewise aerodynamic e utter equation is dee ned, which provides for accurate noncritical damping. A solution of the piecewise e utter equation is described, which is noniterative and which can be used to get results at a single velocity. The resulting e utter method provides a solution of the e utter equation with accurate subcritical damping that is efe cient and reliable. is correct. The frequency and damping values before and after the crossing are not reliable. Subcritical frequency and damping results can be compared with measured data to evaluate the accuracy of the analytical models and thus the accuracy of the predicted e utter velocity. Also, an experi- enced e utter analyst may draw conclusions from the frequency and damping vs velocity curves that lead to structural modie cations to extendthe e uttervelocity.Inboth cases,the frequency and damping values should be as accurate as possible. Twomethodsaremostcommonlyused toobtainaccuratesubcrit- ical damping for a e exible aircraft, the p-k method, and transient e utter. The p-k method 2 repeatedly interpolates for the required aerodynamics asit iterates to e nd each eigenvalue.A spline or other function is dee ned to interpolate the aerodynamics to the required reduced frequency, which is determined from the eigenvalue from the previous iteration. The p-k method is more costly than the k method and problems with convergence can occur. Transient e utter uses a rational function approximation (RFA) of the aerodynamic forces. 3 This method tries to dee ne asingle function to represent the aerodynamic forces over a wide range of reduced frequency. This step can be dife cult and time consuming. However, once the RFA is dee ned the e utter solution is straightforward. The interaction of an aeroelastic system with a control system is known as aeroservoelasticity (ASE). The most common approach toASEisto create alinear differentialequation,in state-spaceform, of the aeroelastic dynamics so that it can be easily coupled with a linear model of the control system, also in state-space form. There are three primary methods used to create this model. The e rst and mostcommonmethodisbasedonRFAaerodynamics,asintransient e utter.ForASEthedee nitionofthesysteminputs (controlsurfaces ) and outputs (control sensors ) is additionally required. The second methodisimplementedintheFAMUSS (FlexibleAircraftModeling UsingStateSpace )program, 4 andusesanequivalentsystemmethod tocreatetheASEmodel.This technique uses an eigenvaluesolution ofthee utterequationwithaccuratefrequencyanddampingtodee ne the dynamic matrix of the state-space model. The rest of the ASE model is dee ned by a e t of the transfer function responses. The third method is the P-transform method. 5 It also requires a e utter solution with accurate frequency and damping to dee ne the state- space dynamic matrix and makes use of the e utter eigenvectors to create the rest of the ASE model.