二元机翼系统的极限环颤振与混沌运动

Limit Cycle Flutter and Chaostic Motion of Two-Dimensional Airfoil System

运用微分方程定性理论和分支理论对不可压缩流中具有二次非线性俯仰刚度的二元机翼系统在非零平衡点发生极限环颤振和混沌运动进行探讨。首先应用中心流形理论将四维系统进行降维,用高维Hopf分支定理确定系统发生Hopf分叉的分叉点;然后通过计算系统焦点量的值来判别分叉点的稳定性和类别,并用分支问题的Liapunov第二方法给出了系统发生Hopf分叉的类型;最后采用四阶Runge-Kutta法对理论分析进行数值模拟,发现两者结果是一致的,通过数值分析法,得到了系统通向混沌的道路,以及在混沌区域存在周期为5的周期运动。结果表明:系统的分叉点为一阶稳定细焦点且发生超临界Hopf分叉,产生稳定极限环;系统通向混沌的道路为倍周期分叉。

Limit cycle flutter and the motion of chaos of two-dimensional airfoil with quadratic nonlinear pitching stiffness in incompressible flow on nonzero equilibrium points are investigated.The center manifold theory is used to reduce a four-dimensional system to a two-dimensional system,and the bifurcation points of the system are determined by bifurcation theory.The type and stability of bifurcation points are determined by computing focal values of system.The type of Hopf bifurcation is determined by the second Lyapunov method of bifurcation problem.The theoretical analysis presented here provides a good agreement with numerical simulations obtained by using a fourth-order Runge-Kutta method.Furthermore,the way leads to chaos in the airfoil system is found and there exits large field of the period-five motion.The results indicate that the bifurcation point is a stable weak focus,when the supercritical Hopf occurs,there exists a stable limit cycle.The motion of chaos occurs due to period-doubling bifurcation.