Stability and Hopf bifurcation of a two-dimensional supersonic airfoil with a time-delayed feedback control surface

The time-delayed feedback control for a supersonic airfoil results in interesting aeroelastic behaviors. The effect of time delay on the aeroelastic dynamics of a two-dimensional supersonic airfoil with a feedback control surface is investigated. Specifically, the case of a 3-dof system is considered in detail, where the structural nonlinearity is introduced in the mathematical model. The stability analysis is conducted for the linearized system. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo the stability switches with the variation of the time delay. The nonlinear aeroelastic system undergoes a sequence of Hopf bifurcations if the time delay passes the critical values. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation and stability of Hopf-bifurcating periodic solutions are determined. Numerical simulations are performed to illustrate the obtained results.