Effects of structural nonlinearity in the bifurcation analysis of transonic airfoil flutter

Abstract

Hopf-bifurcation analysis is used to determine the flutter boundaries of a pitch and plunge airfoil (PAPA) with structural nonlinearity at transonic Mach number conditions. The nonlinear pitch and plunge airfoil model is a coupling of the Euler equations and a twodegree-of-freedom structural model composed of a linear spring and a bilinear torsional spring. The bilinear torsional spring is in a two-parameter family of nonlinear structural models that includes freeplay. The Euler equations are discretized using an upwind total variation diminishing scheme of Harten and Yee. Equilibrium solutions of the pitch and plunge airfoil model are computed using the fully implicit Newton's method; dynamic solutions are computed explicitly. The Hopfbifurcation point, which models the flutter condition, is computed using a modified form of the Griewank and Reddien algorithm. Paths of Hopf points are computed as functions of the parameters in the structural model to determine their effects on limit-cycle amplitude and the flutter boundary location. The flutter boundary is validated by time-integration of the pitch and plunge airfoil model. Also, flutter boundaries as a function of Mach number are presented for three different nonlinear structural models.