Harmonic Balance Approach for an Airfoil with a Freeplay Control Surface

TLDR

The study examines the nonlinear aeroelastic response of a 2‑D airfoil with a freeplay control surface in incompressible flow. The authors formulate the dynamics as first‑order ODEs, use time‑integration to validate a higher‑order harmonic balance method, and compare its predictions with numerical time‑marching and experimental data. The analysis reveals hysteresis and sensitivity to initial conditions in the subcritical bifurcation. The paper defines key non‑dimensional parameters and symbols for the aeroelastic model.

Abstract

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