Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow

In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.