二元机翼系统的极限环颤振与混沌运动
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摘要
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.
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.
引文
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[12] ABDELKEFI A,RUI V,NAYFEH A H,et al.An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system[J].Nonlinear Dynamics,2013,71(1-2):159-173.DOI:10.1007/s11071-012-0648-z.
[13] CAI Ming,LIU Weifei,LIU Jike.Bifurcation and chaos of airfoil with multiple strong nonlinearities[J].Applied Mathematics and Mechanics,2013,34(5):627-636.DOI:10.1007/s10483-013-1696-x.
[14] CHEN Yanmao,LIU Jike.On the limit cycles of aeroelastic systems with quadratic nonlinearities [J].Structural Engineering and Mechanics,2008,30(30):67-76.DOI:10.12989/sem.2008.30.1.067.
[15] 马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社,2001:244-248.
[16] 张琪昌.分叉与混沌理论及应用[M].天津:天津大学出版社,2005:58-61.
[17] LIU Yirong,LI Jibin,HUANG Wentao.Planar dynamical systems:selected classical problems[M].Beijing:Science Press,2014:69-78.
[18] 蔡燧林,钱祥征.常微分方程定性理论引论[M].北京:高等教育出版社,1994:18-29.
[19] 张锦炎.常微分方程几何理论与分支问题[M].北京:北京大学出版社,1981:207-208.
[2] 陈衍茂,刘济科,孟光.二元机翼非线性颤振系统的若干分析方法[J].振动与冲击,2011,30(3):129-134.
[3] 刘济科,张宪民,赵令诚,等.不可压气流中二元机翼颤振的分岔点研究[J].固体力学学报,1999,20(4):315-319.
[4] 郝淑英,刘海英,张琪昌.立方非线性机翼非零平衡点极限环颤振的研究[J].天津理工大学学报,2007,23(2):1-4.
[5] 郑国勇.不可压缩流中机翼颤振稳定性研究[J].力学季刊,2010,31(2):207-212.
[6] LEE B H K,LIU L,CHUNG K W.Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces[J].Journal of Sound and Vibration,2005,281(3):699-717.DOI:10.1016/j.jsv.2004.01.034.
[7] 刘菲.非线性气动弹性系统的分叉分析[D].成都:西南交通大学,2007.
[8] 郑国勇,杨翊仁.超音速流中结构非线性二元机翼的复杂响应研究[J].振动与冲击,2007,26(12):96-100.
[9] ZHENG Guoyong,YANG Yiren.Chaotic motions and limit cycle flutter of two-dimensional wing in supersonic flow[J].Acta Mechanica Solida Sinica,2008,21(5):441-448.DOI:10.1007/s10338-008-0853-y.
[10] COLLER B D,CHRAMA P A.Structural nonlinearities and the nature of the classic flutter instability[J].Journal of Sound and Vibration,2004,277(4):711-739.DOI:10.1016/j.jsv.2003.09.017.
[11] SHAHRZAD P,MAHZOON M.Limit cycle flutter of airfoils in steady and unsteady flows[J].Journal of Sound and Vibration,2002,256(2):213-225.DOI:10.1006/jsvi.2001.4113.
[12] ABDELKEFI A,RUI V,NAYFEH A H,et al.An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system[J].Nonlinear Dynamics,2013,71(1-2):159-173.DOI:10.1007/s11071-012-0648-z.
[13] CAI Ming,LIU Weifei,LIU Jike.Bifurcation and chaos of airfoil with multiple strong nonlinearities[J].Applied Mathematics and Mechanics,2013,34(5):627-636.DOI:10.1007/s10483-013-1696-x.
[14] CHEN Yanmao,LIU Jike.On the limit cycles of aeroelastic systems with quadratic nonlinearities [J].Structural Engineering and Mechanics,2008,30(30):67-76.DOI:10.12989/sem.2008.30.1.067.
[15] 马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社,2001:244-248.
[16] 张琪昌.分叉与混沌理论及应用[M].天津:天津大学出版社,2005:58-61.
[17] LIU Yirong,LI Jibin,HUANG Wentao.Planar dynamical systems:selected classical problems[M].Beijing:Science Press,2014:69-78.
[18] 蔡燧林,钱祥征.常微分方程定性理论引论[M].北京:高等教育出版社,1994:18-29.
[19] 张锦炎.常微分方程几何理论与分支问题[M].北京:北京大学出版社,1981:207-208.