تحلیل آشوبناک و فلاتر بال دوبعدی در جریان مافوق صوت با استفاده از نظریه آیرودینامیکی پیستون

In this paper, based on the nonlinear piston aerodynamic theory in supersonic flow, the differential equations governing nonlinear vibrations and aeroelastic chaotic motions of a two-dimensional wing with cubic nonlinear stiffness in the pitching direction has been performed. The nonlinear theory of Hopf bifurcation has been used to calculate the wing flutter speed with nonlinear structural and aerodynamic behavior. Then, employing the fourth order Runge-Kutta method via MATLAB, the numerical solution of governing equations of the system to determine the stable limit cycle responses and chaotic motions of the aeroelastic system of the wing has been done. The effect of changes of system parameters on the flutter speed has been investigated. The obtained results are verified with the ones available in the literature for special cases. Results showed that the number and the stability of equilibrium points of the system change when the flow speed increases. Also, besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic behavior in the aeroelastic flutter system in period-doubling bifurcation. It was observed that when the initial conditions are small the chaotic behavior in the system occurs only when the flow speed is higher than the linear divergent speed. In addition, the flow speed region in which the system behaves chaotic are very narrow. Finally, to control vibration of the linear plunge and pitching motions of the wing due to initial disturbances, the LQR controller was employed and dynamic response of open loop of the wing pitching motion under these initial disturbances has been obtained.