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

Abstract

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.

Appendices

Appendix A

The elements of $C^{*}$ and $K^{*}$ are given as follows,

$$\begin{aligned} c_{11}&= c_h +4\rho _\infty a_\infty b, c_{12} =4 (1-\bar{{x}}_0 )\rho _\infty a_\infty b^{2},\\ c_{13}&= \left( {2-\bar{{x}}_1 } \right) ^{2}\rho _\infty a_\infty b^{2},\\ c_{21}&= 4 (1-\bar{{x}}_0 )\rho _\infty a_\infty b^{2},\\ c_{22}&= c_\alpha +\left( \frac{16}{3}-8\bar{{x}}_0 +4\bar{{x}}_0^2 \right) \rho _\infty a_\infty b^{3},\\ c_{23}&= \frac{1}{3}(2-\bar{{x}}_1 )^{2}(4-3\bar{{x}}_0 +\bar{{x}}_1 )\rho _\infty a_\infty b^{3},\\ c_{31}&= \left( {2-\bar{{x}}_1 } \right) ^{2}\rho _\infty a_\infty b^{2},\\ c_{32}&= \frac{1}{3}(2-\bar{{x}}_1 )^{2}(4-3\bar{{x}}_0 +\bar{{x}}_1 )\rho _\infty a_\infty b^{3},\\ c_{33}&= c_\beta +\frac{2}{3}(2-\bar{{x}}_1 )^{3}\rho _\infty a_\infty b^{3},\\ k_{11}&= k_h , k_{12} =4\rho _\infty a_\infty bV,\\ k_{13}&= 2 \left( {2-\bar{{x}}_1 } \right) \rho _\infty a_\infty bV, k_{21} =0,\\ k_{22}&= k_\alpha +4(1-\bar{{x}}_0 )\rho _\infty a_\infty b^{2}V,\\ k_{23}&= (2-\bar{{x}}_1 ) (2-2\bar{{x}}_0 +\bar{{x}}_1 )\rho _\infty a_\infty b^{2}V, k_{31} =0,\\ k_{32}&= (2-\bar{{x}}_1 )^{2}\rho _\infty a_\infty b^{2}V,\\ k_{33}&= k_\beta +(2-\bar{{x}}_1 )^{2}\rho _\infty a_\infty b^{2}V \end{aligned}$$

Appendix B

$$\begin{aligned} b_1&= -2p_2 +p_1^2 , b_2 =2p_4 -2p_1 p_3 +p_2^2 -q_0^2,\\ b_3&= -2p_6+2p_1 p_5 -2p_2 p_4 +p_3^2 +2q_0 q_2 -q_1^2,\\ b_4&= 2p_2 p_6 -2p_3 p_5 +p_4^2 -2q_0 q_4 +2q_1 q_3 -q_2^2,\\ b_5&= -2p_4 p_6 +p_5^2 +2q_2 q_4 -q_3^2 , b_6 =p_6^2 -q_4^2 \end{aligned}$$