THE POST-HOPF-BIFURCATION RESPONSE OF AN AIRFOIL IN INCOMPRESSIBLE 2-DIMENSIONAL FLOW

A bifurcation analysis of a two-dimensional airfoil with a structural nonlinearity in the pitch direction and subject to incompressible flow is presented. The nonlinearity is an analytical third-order rational curve fitted to a structural freeplay. The aeroelastic equations-of-mo tion are reformulated into a system of eight first-order ordinary diff erential equations. An eigenvalue analysis of the linearized equations is used to give the linear flutter speed The nonlinear equations of m otion are either integrated numerically using a fourth-order Runge-Kut ta method or analyzed using the AUTO software package. Fixed points of the system are found analytically and regions of limit cycle oscillat ions are detected for velocities well below the divergent flutter boun dary. Bifurcation diagrams showing both stable and unstable periodic s olutions are calculated, and the types of bifurcations are assessed by evaluating the Floquet multipliers. In cases where the structural pre load is small, regions of chaotic motion are obtained, as demonstrated by bifurcation diagrams, power spectral densities, phase-plane plots and Poincare sections of the airfoil motion; the existence of chaos is also confirmed via calculation of the Lyapunov exponents. The general behaviour of the system is explained by the effectiveness of the free play part of the nonlinearity in a complete cycle of oscillation. Resu lts obtained using this reformulated set of equations and the analytic al nonlinearity are in good agreement with previously obtained finite difference results for a freeplay nonlinearity.