Harmonic Balance Approach for an Airfoil with a Freeplay Control Surface

The nonlinear aeroelastic response of a two-dimensional airfoil, including a control surface with freeplay placed in an incompressible flow, is studied. The model equations are formulated as a set of first-order ordinary differential equations. First, the dynamic response is investigated by a time integration method, and the time integration results are used for the verification of the harmonic balance results. The interesting hysteresis phenomenon and the effect of initial conditions of the subcritical bifurcation are presented. A higher-order harmonic balance method is then derived to investigate the high harmonics of the airfoil motions. The harmonic balance prediction is verified by comparison to the results from a numerical time marching integration and also by comparison to results from a previous experiment. Nomenclature a = nondimensional distance of the elastic axis from the midchord, with respect to the semichord B = damping submatrix b = semichord Ch = stiffness (per unit span) of wing in deflection C(k) = generalized Theodorsen function Cα, Cβ = torsional stiffness (per unit span) of wing around a and of aileron around c c = nondimensional distance of the control surface (aileron) hinge line from the midchord, with respect to the semichord