非线性颤振系统的分岔与混沌
|
摘要
非线性颤振在大型工程设备中经常出现而且危害性很大。它不仅造成设备及其零部件的疲劳损坏,还会造成灾难性事故。本文重点研究非线性颤振系统的动力学特性,并具体对飞机机翼和机床这两颤振系统的分岔与混沌进行分析。本文的创新性研究成果主要有以下几个方面:
(1)首次用数值仿真法研究了在俯仰方向具有间隙非线性的机翼随机颤振。得到了间隙非线性参数对系统动力学响应的影响。研究了随机系统的边缘和双维两种概率密度函数,还研究了此系统的随机分岔。结果表明:对于低强度和中强度的扰动,在不同的气流速度区,这两种概率密度函数的形状是不同的。然而在高强度扰动的情况下,不管在高气流速度区或者低气流速度区,这两种概率密度函数的形状都是相似的。随机分岔分析表明:在颤振前后速度区都发生了P-分岔,而没有发生D-分岔。此研究深入地剖析了机翼的随机颤振动力学特性。
(2)研究机翼系统在随机扰动下的控制问题,机翼的控制方法是在随机域内而不是在时域内设计的。建立了随机扰动下二元机翼的伊藤微分方程,得到了随机域内伊藤微分方程的一阶和二阶矩。将最优控制理论用于二阶矩方程。数值仿真结果表明:当飞行速度在不稳定区域,最优控制律能使系统很快收敛到零。利用这种方法设计控制系统可以有效地抑制机翼系统的颤振。
(3)对于俯仰方向具有间隙和立方非线性刚度耦合的二自由度气动弹性机翼,采用庞加莱映射法和Floquet理论分析了机翼系统在超音速和高超音速时的极限环颤振和混沌运动。结果表明:Floquet乘子能够很好的预见极限环颤振发生,浮沉方向的相轨迹图比俯仰方向的相轨迹图复杂的多。证实了初值条件对系统的动力学特性有十分重要的影响。发现了存在一条周期轨道贯穿于整个颤振发生的飞行速度区。
(4)机床发生颤振不仅引起较差的被加工件表面质量,而且增加了刀具的磨损破坏,阻碍了生产效率的提高。采用切比雪夫多项式法和Floquet理论相结合来预测铣床运行中的颤振和分岔,得到了颤振稳定性极限图。通过系统的特征值分析,发现此系统发生了倍周期分岔和Hopf分岔。系统由稳定的平衡点通过倍周期分岔收敛到稳定的极限环运动,由Hopf分岔转化为概周期运动。由庞加莱截面方法所得结果也证实了概周期运动的发生。本文的研究可以为切削稳定性分析提供更为精确地计算方法,为先进的五轴并联机床的改进设计提供理论依据。
Nonlinear flutter or chatter often occurs in large engineering equipments. It not only leads to the equipments or their accessory breakage but also results in catastrophic accident. The paper investigates the nonlinear dynamics character of flutter or chatter system and analyses detailed the bifurcation and chaos in two-dimensional airfoil and milling machine separately. The main innovative contributions achieved are as follows:
(1) The two degree-of-freedom (2-DOF) airfoil system with freeplay nonlinearity in pitch is investigated numerically. The effect of parameters of the freeplay nonlinearity on the system responses is obtained. The two probability density function (PDF) including Marginal PDF、Bi-dimension PDF and random bifurcation are all used in investigation of the random system. The results show the two PDFs have different shapes in low level turbulence at pre- and post-flutter speeds, but they keep similar shape in high level turbulence. The random bifurcation analysis indicates that the P-bifurcation can happen at both pre- and post-flutter speeds but the D-bifurcation never occurs.
(2) Investigation on the control of the airfoil system excited by the random turbulence is important and critical for designing of the airfoil. The feedback control strategy of the airfoil system is studied in the stochastic domain instead of in the time domain. The differential equation of the two-degree-of-freedom (2-DOF) airfoil excited by random turbulence is found, and then the first or second order moment equation are derived from the differential equation in the stochastic domain. The optimal control theory applies to the second order moment equation. Especially, the numerically simulation shows that the mean airspeed in unstable regime, the optimal control input feedback on the airfoil system can make the system convergent to zero in short time. The results confirm that the optimal control can suppress random flutter of the airfoil system effectively.
(3) The dynamics character of a two degree-of-freedom aeroelastic airfoil with freeplay and cubic stiffness nonlinearities combined in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincarémapping method and Floquet theory are adopted to analyze the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincarésection also approves previous conclusion.
(4) Chatter in machine leads to poor surface finish, promotes wear of the tool and hamper productivity. The shifted Chebyshev polynomials and Floquet theory are adopted for the prediction chatter stability and bifurcation in milling. The stability lobes diagram is obtained. The stability in milling can well be predicted by the lobes diagram. The muliti-periodic and Hopf bifurcations are detected by the Eigen-values analysis. The result shows that the stability solution of the system transform from the stable equilibrium point to the limit cycle oscillatory after multiple cycle bifurcation, and it transforms to the quasi-periodic oscillation after Hopf bifurcation. The numerical result of the Poincarésection proves that the occurrence of the quasi-periodic oscillation. The paper presents more accurately calculation methods for analysis of milling stability, which offers new methods to solve the critical questions for the design of 5-axis machine.
(1)首次用数值仿真法研究了在俯仰方向具有间隙非线性的机翼随机颤振。得到了间隙非线性参数对系统动力学响应的影响。研究了随机系统的边缘和双维两种概率密度函数,还研究了此系统的随机分岔。结果表明:对于低强度和中强度的扰动,在不同的气流速度区,这两种概率密度函数的形状是不同的。然而在高强度扰动的情况下,不管在高气流速度区或者低气流速度区,这两种概率密度函数的形状都是相似的。随机分岔分析表明:在颤振前后速度区都发生了P-分岔,而没有发生D-分岔。此研究深入地剖析了机翼的随机颤振动力学特性。
(2)研究机翼系统在随机扰动下的控制问题,机翼的控制方法是在随机域内而不是在时域内设计的。建立了随机扰动下二元机翼的伊藤微分方程,得到了随机域内伊藤微分方程的一阶和二阶矩。将最优控制理论用于二阶矩方程。数值仿真结果表明:当飞行速度在不稳定区域,最优控制律能使系统很快收敛到零。利用这种方法设计控制系统可以有效地抑制机翼系统的颤振。
(3)对于俯仰方向具有间隙和立方非线性刚度耦合的二自由度气动弹性机翼,采用庞加莱映射法和Floquet理论分析了机翼系统在超音速和高超音速时的极限环颤振和混沌运动。结果表明:Floquet乘子能够很好的预见极限环颤振发生,浮沉方向的相轨迹图比俯仰方向的相轨迹图复杂的多。证实了初值条件对系统的动力学特性有十分重要的影响。发现了存在一条周期轨道贯穿于整个颤振发生的飞行速度区。
(4)机床发生颤振不仅引起较差的被加工件表面质量,而且增加了刀具的磨损破坏,阻碍了生产效率的提高。采用切比雪夫多项式法和Floquet理论相结合来预测铣床运行中的颤振和分岔,得到了颤振稳定性极限图。通过系统的特征值分析,发现此系统发生了倍周期分岔和Hopf分岔。系统由稳定的平衡点通过倍周期分岔收敛到稳定的极限环运动,由Hopf分岔转化为概周期运动。由庞加莱截面方法所得结果也证实了概周期运动的发生。本文的研究可以为切削稳定性分析提供更为精确地计算方法,为先进的五轴并联机床的改进设计提供理论依据。
Nonlinear flutter or chatter often occurs in large engineering equipments. It not only leads to the equipments or their accessory breakage but also results in catastrophic accident. The paper investigates the nonlinear dynamics character of flutter or chatter system and analyses detailed the bifurcation and chaos in two-dimensional airfoil and milling machine separately. The main innovative contributions achieved are as follows:
(1) The two degree-of-freedom (2-DOF) airfoil system with freeplay nonlinearity in pitch is investigated numerically. The effect of parameters of the freeplay nonlinearity on the system responses is obtained. The two probability density function (PDF) including Marginal PDF、Bi-dimension PDF and random bifurcation are all used in investigation of the random system. The results show the two PDFs have different shapes in low level turbulence at pre- and post-flutter speeds, but they keep similar shape in high level turbulence. The random bifurcation analysis indicates that the P-bifurcation can happen at both pre- and post-flutter speeds but the D-bifurcation never occurs.
(2) Investigation on the control of the airfoil system excited by the random turbulence is important and critical for designing of the airfoil. The feedback control strategy of the airfoil system is studied in the stochastic domain instead of in the time domain. The differential equation of the two-degree-of-freedom (2-DOF) airfoil excited by random turbulence is found, and then the first or second order moment equation are derived from the differential equation in the stochastic domain. The optimal control theory applies to the second order moment equation. Especially, the numerically simulation shows that the mean airspeed in unstable regime, the optimal control input feedback on the airfoil system can make the system convergent to zero in short time. The results confirm that the optimal control can suppress random flutter of the airfoil system effectively.
(3) The dynamics character of a two degree-of-freedom aeroelastic airfoil with freeplay and cubic stiffness nonlinearities combined in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincarémapping method and Floquet theory are adopted to analyze the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincarésection also approves previous conclusion.
(4) Chatter in machine leads to poor surface finish, promotes wear of the tool and hamper productivity. The shifted Chebyshev polynomials and Floquet theory are adopted for the prediction chatter stability and bifurcation in milling. The stability lobes diagram is obtained. The stability in milling can well be predicted by the lobes diagram. The muliti-periodic and Hopf bifurcations are detected by the Eigen-values analysis. The result shows that the stability solution of the system transform from the stable equilibrium point to the limit cycle oscillatory after multiple cycle bifurcation, and it transforms to the quasi-periodic oscillation after Hopf bifurcation. The numerical result of the Poincarésection proves that the occurrence of the quasi-periodic oscillation. The paper presents more accurately calculation methods for analysis of milling stability, which offers new methods to solve the critical questions for the design of 5-axis machine.
引文
[1]陈桂彬,邹丛青,杨超,气动弹性设计基础,北京:北京航航空航天大学出版社, 2004
[2] E. H.道尔, H. C.小柯蒂斯, R. H.斯坎伦等,气动弹性力学现代教程(陈文俊,李传家),北京:宇航出版社, 1991
[3] B. H. K. Lee, S. J. Price, Y. S. Wong, Non-linear aeroelastic analysis of airfoils: birurcation and chaos, Progress in Aerospace Sciences, 1999, 35(3): 205~334
[4] H. Ashley, On the role of shock waves in the“sub-transonic”flutter phenomenon. In: AIAA 79-0765, AIAA/ASME/ASCE/AHS 20th Structures, Structural Dynamics, and Materials Conference, 1979
[5] S. J. Price, B. H. K. Lee, H. Alighanbari, An analysis of the post-instability behaviour of a two-dimensional airfoil with a structural nonlinearity. In: Proceedings of 34th AIAA/ASME/ASCE/AHS/ASC stuctures, Stuctural Dynamics and Materials Conference, 2004, 1452~1460.
[6] S. J. Price, H. Alighanbari, B. H. K. Lee, The aeroelastic response of a two-dimensional airfoil with bilinear and cubic structural nonlinearities, Journal of Fluids and Structures, 1995, 9(2): 175~193
[7] B. H. K. Lee, L. Gong, Y. S. Wong, Analysis and computation of nonlinear dynamic response of a two-degree-of-freedom system and its application in aeroelasticity, Journal of Fluids and Structures, 1997(3), 11: 225~246
[8] B. H. K. Lee, L. Y. Jiang, Y. S. Wong, Flutter of a airfoil with a cubic restoring force, Journal of Fluids and Structures, 1999, 13(1): 75~101
[9] B. H. K. Lee, L. Liu, K. W. Chung, Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces, Journal of Sound and Vibration, 2005, 281(3-5): 699~717
[10] M. D. Conner, D. M. Tang, E. H. Dowell et al, Nonlinear behavior of a typical airfoil section with control surface freeplay: a numerical and experimental study, Journal of Fluids and Structures, 1997, 11(1): 89~109
[11] D. Tang, E. H. Dowell, L. N. Virgin, Limit cycle behavior of an airfoil with a control surface, Journal of Fluids and Structures, 1998, 12(7): 839~858
[12] E. H. Dowell, J. P. Thomas, K. C. Hall, Transonic limit cycle oscillation analysis using reduced order aerodynamic models, Journal of Fluids and Structures, 2004, 19(1): 17~27
[13] L. Liu, Y. S. Wong, B. H. K. Lee, Non-linear aeroelastic analysis using the point transformation method, part 1: freeplay model. Journal of Sound and Vibration, 2002, 253(2): 447~469
[14] L. Liu, Y. S. Wong, B. H. K. Lee, Non-linear aeroelastic analysis using the point transformation method, part 2: hysteresis model, Journal of Sound and Vibration, 2002, 253(2): 471~483
[15] L. Liu, E. H. Dowell, The secondary bifurcation of an aeroelastic airfoil motion effect of high harmonics, Nonlinear Dynamics, 2004, 37(1): 31~49
[16] K. W. Chung, C. L. Chan, B. H. K. Lee, Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 2007, 299(3): 520~539
[17] K. W. Chung, Y. B. He, B. H. K. Lee, Bifurcation analysis of a two-degree-of-freedom aeroelastic system with hysteresis structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 2009, 320(1-2): 163~163
[18] L. Librescu, G. Chiocchia, P. Marzocca, Implications of cubic physical/aeroedynamic non-linearities on the character of the flutter instability boundary, International Journal of Non-Linear Mechanics, 2003, 38(2): 173~199
[19] A. Raghothama, S. Narayanan, Non-Linear dynamics of a two-dimensional airfoil by incremental harmonic balance method, Journal of Sound and Vibration, 1999, 226(3): 493~517
[20] P. A. Chamara, B. D. Coller, A study of double flutter, Journal of Fluids and Structures, 2004, 19(7): 863~879
[21]杨智春,赵令诚,带外挂二元机翼颤振模态的转变,航空学报, 1992, 13(9): 552~554
[22]杨智春,赵令诚,李伶等,偏航侧摆联接对代外挂三角机翼颤振特性的影响,振动工程学报, 1992, 5(2): 168~172
[23]王立军,赵令诚,外挂俯仰联接刚度对机翼颤振的影响,航空学报, 1992, 13(9): 560~563
[24] J. K. Liu, L. C. Zhao, Bifurcation analysis of airfoil in incompressible flow, Journal of Sound and Vibration, 1992, 154(1): 117~124
[25]刘继科,二元机翼颤振分岔点类别的判定,力学与实践, 1998, 20: 38~40
[26]刘继科,张宪民,赵令诚等,不可压缩气流中二元机翼颤振的分岔点研究,固体力学学报, 1999, 20(4): 315~319
[27] M. Cai, J. K. Liu, J. Li, Incremental harmonic balance method for airfoil flutter with multiplr strong nonlinearities, Applied Mathematics and Mechanics, 2006, 27(7): 953~958
[28]刘继科,高磊,机翼非线性颤振的分岔点研究,中山大学学报, 1998, 37(3): 28~31
[29]陈衍茂,刘济科,非线性颤振极限环稳定性判别的复正规性法,航空动力学报, 2007, 22(4): 614~618
[30] Y. M. Chen, J. K. Liu, Homotopy analysis method for limit cycle flutter of airfoils, Applied Mathematics and Computation, 2008, 203(2): 854~863
[31]谷迎松,杨智春,带迟滞非线性环节的二元机翼的气动弹性响应分析,机械科学与技术, 2006, 25(8): 900~904
[32]李道春,向锦武,迟滞非线性二元机翼颤振特性分析,航空学报, 2007, 28(3): 600~604
[33]李道春,向锦武,非线性二元机翼气动弹性近似解析研究,航空学报, 2007, 28(5): 1080~1084
[34]丁千,陈予恕,用胞映射法分析双线性结构刚度的机翼颤振,空气动力学报, 2002, 20(2): 123~132
[35]丁千,王冬立,结构和气动非线性机翼颤振分析,动力学与控制学报, 2004, 2(3): 18~23
[36]丁千,王冬立,用规范形直接法研究立方非线性机翼的颤振,飞行力学, 2005, 23(3): 85~88
[37] Q. Ding, D. L. Wang, The flutter of an airfoil with cubic structural and aerodynamic non-linearities, Aeroespace Science and Technology, 2006, 10(5): 427~434.
[38] Q. Ding, D. L. Wang, Flutter control of a two-dimensional airfoil using wash-out filter technique, Chinese Journal of Aeronautics, 2005, 8(2): 131~137
[39]张琪昌,刘海英,任爱娣,非线性机翼极限环颤振的研究,空气动力学学报, 2004, 22(3): 332~336
[40]张琪昌,刘海英,任爱娣,具有立方非线性机翼颤振的局部分岔,天津大学学报, 2004, 37(11): 970~974
[41]任爱娣,张琪昌,具有不对称间隙的二元机翼颤振研究,工程力学, 2006, 23(9): 25~29
[42]吴志强,张建伟,二元机翼极限环颤振复杂分岔,工程力学, 2008, 25(2): 52~55, 92
[43]吴志强,张建伟,王喆,极限环高阶分岔控制,动力学与控制学报, 2007, 5(1): 23~26
[44]刘菲,杨翊仁,不可压缩流中二元翼运动的分支问题,西南交通大学学报, 2002, 37(S1): 95~97
[45]刘菲,杨翊仁,立方非线性机翼的二重半稳定极限环分叉,西南交通大学学报, 2004, 39(5): 638~640, 669
[46]刘菲,杨翊仁,不可压缩流中二元翼运动的余维二分叉,振动与冲击, 2005, 24(4): 18~19, 23
[47]叶献辉,杨翊仁,刘菲,非对称结构参数对颤振速度的影响,西南交通大学学报, 40(6): 779~782, 792
[48] P. Marzocca, L. Librescu, G. Chiocchia, Aeroelastic response of 2-D airfoil in a compressible flow field and exponse to blast load, Aeroespace Science and Technology, 2002, 6(4): 259~272
[49] G. Schewe, H. Mai, G. Dietz, Nonlinear effects in transonic flutter with emphasis on manifestations of limit cycle oscillations. Journal of Fluids and Structures, 2003, 18(1): 3~22
[50]梁强,叶正寅,杨永年,采用非定常N-S方程的翼型颤振特性分析研究,西北工业大学学报, 2001, 19(3): 341~344
[51]史爱明,杨青,杨永年,非结构运动网格下的三维机翼颤振数值分析,振动与冲击, 2005, 24(6): 27~28, 36
[52]史爱明,王刚,杨永年,颤振数值模拟的时间步长选取方法研究,西北工业大学学报, 2005, 23(2): 208~211
[53]杨青,梁强,杨永年,机翼-机身-尾翼结构的跨音速颤振分析研究,西北工业大学学报, 2005, 23(1): 84~88
[54]梁强,杨永年,樊则文,三维机翼跨音速颤振的分布式并行计算,计算物理, 2004, 21(2): 179~183
[55]杨青,闫锋,杨永年,超临界翼型的跨声速颤振特性研究,西北工业大学学报, 22(6): 782~785
[56]牟让科,杨永年,叶正寅,带有结构刚度非线性的跨音速翼型颤振特性研究,振动工程学报, 2001, 14(3): 319~321
[57]史爱明,杨永年,叶正寅,带结构刚度非线性的超音速弹翼颤振分析方法研究,西北工业大学学报, 2003, 21(4): 481~485
[58]谢飞,叶正寅,不同迎角下翼型颤振实验研究,流体力学实验与测量, 2003, 17: 55~59
[59] L. Dubcová, M. Feistauer, J. Horácek et al, Numerical simulation of airfoil vibrations induced by turbulent flow, Journal of Computational and Applied Mathematics, 2008, 218(1): 34~42
[60] P. Svácek, M. Feistauer, J. Horácek, Numerical simulation of flow induced airfoil vibration with large amplitudes, Journal of Fluids and Structures, 2007, 23(3): 391~411
[61] G. Dietz, G. Schewe, H. Mai, Experiments on heavy/pitch limit-cycle oscillations of a supercritical airfoil close to the transonic dip, Journal of Fluids and Structures, 2004, 19(1): 1~16
[62] G. Dietz, G. Schewe, H. Mai, Amplicationand and amplitude limitation heavy/pitch limit-cycle oscillations close to the transonic dip, Journal of Fluids and Structures, 2006, 22(4): 505~527
[63] S. J. Price, J. P. Keleris, Non-linear dynamics of an airfoil forced to oscillate in dynamic stall, Journal of Sound and Vibration, 1996, 194(2): 265~283
[64] M. H. Akbari, S. J. Price, Simulation of dynamic stall for a NACN 0012 airfoil using a vortex method, Journal of Fluids and Structures, 2003, 17(6): 855~874
[65] S. Sarkar, K. Venkatraman, Influence of pitching angle of incidence on the dynamics stall behavior of a symmetric airfoil, European Journal of Mechanics , B/Fluids, 2008, 27(3): 219~238
[66] S. Sarkar, H. Bijl, Nonlinear aeroelastic behavior of an oscillation airfoil during Stall-induced vibration, Journal of Fluids and Structures, 2008, 24(6): 757~777
[67] S. Sarkar, J. A. S. Witteveen, A. Loeven et al, Effect of uncertainty on the bifurcation behavior of pitching airfoil stall flutter, Journal of Fluids and Structures, 2009(2), 25: 304~320
[68]牟让科,杨永年,飞机抖振问题研究进展,应用力学学报, 2001, 18(S1): 142~150
[69]牟让科,杨永年,叶正寅,超临界翼型的跨音速抖振特性,计算物理, 18(5): 477~480
[70]李劲杰,杨青,肖春生等,边条翼布局流场及其双垂尾抖振特性研究,航空学报, 2006, 27(3): 395~398
[71]李劲杰,杨青,杨永年等,边条翼布局双垂尾抖振表面脉动压力实验研究,计算流体力学, 2006, 20(3): 29~32, 38
[72]李劲杰,杨青,杨永年,边条翼布局双垂尾抖振的数值模拟,空气动力学报, 2007, 25(2): 205~210
[73]李劲杰,杨青,杨永年等,边条翼布局双垂尾抖振特性与机理风洞实验研究,空气动力学报, 2006, 24(4): 397~402
[74]谭光辉,解江,夏巍等,改装苏-80飞机的尾翼抖振研究,飞行力学, 2008, 26(5): 67~74
[75]王巍,杨智春,夏巍等,尾翼气动弹性抖振模型设计与机抖振实验研究,西北工业大学学报, 2006, 24(3): 338~341
[76] W. Geissler, Numerical study of buffet and transonic flutter on the NLR 7301 airfoil, Aeroespace Science and Technology, 2003, 7(7): 540~550
[77] D. Poirel, S.J. Price, Post-instability behavior of a structurally nonlinear airfoil in longitudinal turbulence, Journal of Aircraft, 1997, 34(5): 619~626
[78] D. Poirel, S. J. Price, Structurally nonlinear fluttering airfoil in turbulence flow, AIAA Journal, 2001, 39(10): 1960~1968
[79] D. Poirel, S.J. Price, Random binary (coalescence) flutter of a two-dimensional linear airfoil. Journal of Fluids and Structures, 2003, 18(1): 23~42
[80] D. Poirel, S.J. Price, Response Probability structure of a structural nonlinear fluttering airfoil in turbulent flow, Probabilistic Engineering Mechanics, 2003, 18(2): 185~202.
[81] D. Poirel, S. J. Price, Bifurcation characteristics of a two-dimensional structurally non-linear airfoil in turbulent flow, Nonlinear Dynamics, 2007, 48(4): 423~435
[82] L. E. Christainsen, T. Lehn-Schioler, E. Mosekide et al, Nonlinear characteristics of randomly excited transonic flutter, Mathematics and Computer in Simulation, 2002, 58: 385~405
[83] C. L. Wu, H. M. Zhang, T. Fang, Flutter analysis of an airfoil with bounded parameters in incompressible flow via Gegenbauer polynomial approximation, Aerospace Science and Technology, 2007, 11(7-8), 518~526
[84] S. Pospísil, J. Náprstek, S. Hracov, Stability domains in flow-structure interaction and influence of random noises, Journal of Wind Engineering and Industrial Aerodynamics, 2006, 94(11): 883~893
[85] H. Y. Hu, E. H. Dowell, L. N. Virgin, Stability estimation of high dimensional vibrating systems under state delay feedback control, Journal of Sound and Vibration, 1998, 214(3): 497~511
[86]孙伟,翁建生,胡海岩,带有传动机构的翼段颤振半主动抑制,南京航空航天大学学报, 2004, 36(4): 422~426
[87]孙伟,胡海岩,基于多级磁流变阻尼器的操纵面振动半主动抑制—阻尼器设计与实验建模,振动工程学报, 2005, 18(1): 8~13
[88]孙伟,胡海岩,基于多级磁流变阻尼器的操纵面振动半主动抑制—数值仿真与风洞实验,振动工程学报, 2005, 18(1): 14~18
[89]于明礼,胡海岩,基于超声电机作动器的翼段颤振主动控制,振动工程学报, 2005, 18(4): 418~425
[90]于明礼,文浩,胡海岩等,二维翼段颤振的?控制,航空学报, 2007, 28(2): 340~343
[91]于明礼,文浩,胡海岩,二维翼段颤振的H?控制,航空学报, 2007, 28(2): 340~343
[92] Y. H. Zhao, Stability of a two-dimensional airfoil with time-delayed feedback control, Journal of Fluids and Structures, 2009, 25(1): 1~25
[93]杨发友,顾仲权,飞机机翼颤振自适应抑制研究,南京航空航天大学学报, 2004, 36(6): 708~712
[94] V. M. Rao, A. Behal, P. Marzocca et al, Adaptive aeroelastic vibration suppression of a supersonic airfoil with flap, Aerospace Science and Technology, 2006, 10(4): 309~315
[95] L. Librescu, P. Marzocca, W. A. Silva, Aeroelasticity of 2-D lifting surfaces with time-delayed feedback control, Journal of Fluids and Structures, 2005, 20(2): 197~215.
[96] H. Heo, Y. H. Cho, D. J. Kim, Stochastic control of flexible beam in random flutter, Journal of Sound and Vibration, 2003, 267(2): 335~354
[97] D. Tang, H. P. Gavin, E. H. Dowell, Study of airfoil gust response alleviation using of an electro-magnetic dry friction damper. Part 1: theory, Journal of Sound and Vibration, 2004, 269(3-5): 853~874.
[98]谢长川,吴志刚,杨超,大展弦比柔性机翼的气动弹性分析,北京航空航天大学学报, 2003, 29(12): 1087~1090
[99]谢长川,张欣,陈桂彬,复合材料大展弦比机翼机翼动力学建模与颤振分析,飞机设计, 2004, (2): 6~10
[100] X. N. Liu, J. W. Xiang, Stall flutter analysis of high aspect-ratio composite wing, Chinese Journal of Aeronautics, 2006, 19(1): 36~43
[101] Y. H. Zhao, H. Y. Hu, Structural modeling and aeroelastic analysis of high-aspect-ratio composite wings, Chinese Journal of Aeronautics, 2005, 18(1): 25~30
[102]赵永辉,胡海岩,大展弦比夹芯翼大攻角颤振分析,振动工程学报, 2004, 17(1): 25~30
[103]杨智春,张立新,谭光辉,基于遗传算法的复合材料壁板热颤振优化设计,宇航学报, 2008, 29(4): 1451~1456
[104]夏巍,杨智春,复合材料壁板热颤振的有限元分析,西北工业大学学报, 2005, 23(2), 180~183
[105]杨智春,夏巍,高速飞行器壁板颤振的分析模型和分析方法,应用力学学报, 2006, 23(4), 537~542
[106]夏巍,杨智春,热环境下复合材料壁板的振动特性分析,应用力学学报, 2005, 22(3), 359~363
[107]张伟,高美娟,姚明辉等,压电复合材料层矩形板的高维混沌动力学研究,中国科学G辑物理学力学天文学, 2009, 39(8): 1105~1115
[108]胡国才,向锦武,张晓谷,前飞状态直升机旋翼/机体耦合动稳定性分析模型,航空学报, 2004, 25(5): 451~455
[109]薛海峰,向锦武,张晓谷,前飞状态下旋翼动稳定性及自由度间相互作用研究,航空动力学报, 2004, 19(5): 671~677
[110]薛海峰,向锦武,张晓谷,直升机前飞空中共振稳定性和各自由度相互作用研究,航空学报, 2005, 26(4), 454~457
[111]薛海峰,向锦武,张晓谷等,变距/摆振耦合对直升机空中共振稳定性的影响,中国工程科学, 2007, 9(9): 58~62
[112] G. C. Hu, J. W. Xiang, X. G. Zhang, Analytical model of elastomeric lag damper kinematic coupling and its effect on helicopter air resonance in hover, Chinese Journal of Aeronautics, 2002, 15(1): 27~32
[113]胡国才,向锦武,张晓谷,黏弹减摆器非线性特性对直升机前飞空中共振的影响分析,航空学报, 2005, 26(2): 199~202
[114]戚厚军,低刚度摆线轮缘高速铣削变形与铣削力建模方法: [博士学位论文],天津;天津大学, 2009
[115]吴化勇,高速切削系统动态特性研究: [硕士学位论文],济南;山东大学, 2005
[116] S. A. Tobias, Machine Tool Vibration, London: Blackie and Sons Ltd, 1965
[117] Y. Altintas, E. Budak, Analytical prediction of stability lobes in milling, Annals of the CIRP, 1995, 44(1): 357~362
[118] Y. Altintas, Analytical prediction of three dimensional chatter stability in milling, JSME. International Journal Series C, 2001, 44(3): 717~723.
[119] S. D. Merdol, Y. Altintas, Multi-frequency solution of chatter stability for low immersion milling, Journal of Manufacturing Science and Engineering, 2004, 126(3): 459~466
[120] Y. Altintas, G. Stepan, D. Merdol et al, Chatter stability of milling in frenquency and discrete time domain, CIRP Journal of Manufacturing Science and Technology, 2008, 1(1): 35~44
[121] Y. Altintas, M. Eynian, H. Onozuka, Identification of dynamic cutting force coefficients and chatter stability with process damping, CIRP Annals-Manufacturing Technology, 2008, 57(1): 371~374
[122] G. Morgan, R. Q. Cheng, Y. Altintas, K. Ridgway, An expert troubleshooting system for the milling process, International Journal of Machine Tools &Manufacture, 2007, 47(9): 1417~1425.
[123] S. Turner, D. Merdol, Y. Altintas et al, K. Modelling of the stability of variable helix end milling, International Journal of Machine Tools &Manufacture, 2007, 47(9): 1410~1416
[124]宋清华,艾兴,于水清,高速铣削稳定性与表面加工精度研究,制造技术与机床, 2008, (4): 40~43
[125]唐委校,艾兴,姜华等,高速切削稳定性数据库的研究,制造技术与机床, 2005, (6): 75~78
[126]宋清华,艾兴,万熠,薄壁件变参数铣削系统动态特性的研究,工具技术, 2008, 42(7): 35~37
[127] Q. H. Song, X. Ai, Y. Wan et al, Influence of tool helix angle on stability in high-speed milling process. Transactions of Nanjing University of Aeronautics & Astronautics, 2008, 25(1): 18~25
[128]宋清华,艾兴,万熠,高速加工系统稳定性预测的关键技术研究,制造技术与机床, 2008, (10): 59~63
[129]宋清华,艾兴,万熠等,考虑刀具偏心的变径向切深铣削稳定性研究,振动、测试与诊断, 2008, 28(3): 206~210, 299
[130]宋清华,艾兴,万熠等,铣削系统稳定性判定新方法研究,机械强度, 2008, 30(5): 718~722
[131]宋清华,艾兴,万熠等,小径向切深下进给量对切削稳定性的影响,中国机械工程, 2008, 19(10): 1148~1152
[132] Q. H. Song, X. A, Y. Wan et al, State department regenerative delay model for high-speed milling processes. Journal of Beijing Institute of Technology, 2008, 17(3): 264~269
[133] W. X. Tang, Q. H. Song, S. Q. Yu et al, Prediction of chatter stability in high-speed finishing end milling considering multi-mode dynamics. Journal of Materials Processing Technology, 2009, 209(5): 2585~2591
[134] Z. Li, Q. Liu, Solution and analysis of chatter stability for end milling in the time-domain, Chinese Journal of Aeronautics, 2008, 21(2): 169~178
[135] J. Warmiński, G. Litak, M. P. Cartmell et al, Approximate analytical solutions for primary chatter in the non-linear metal cutting model, Journal of Sound and Vibration, 2003, 259(4): 917~933
[136] B. P. Mann, P. V. Bayly, M. A. Davies et al, Limit cycles, bifurcations, and accuracy of the milling process, Journal of Sound and Vibration, 2004, 277(1-2): 31~38
[137] C. Henninger, P. Eberhard, Improving the computational efficiency and accuracy of the semi-discretization method for periodic delay-differential equations, European Journal of Mechanics A/Solids, 2008, 27(6): 975~985
[138] E. Shamoto, K, Akazawa, Analytical prediction of chatter stability in ball end milling with tool inclination, CIRP Annals-Manufacturing Technology, 2009, 58(1): 351~354
[139] T. Insperger, D. A. W. Barton, G. Stépán, Criticality of Hopf bifurcation in State-dependent delay model of turning processes, International Journal of Non-Linear Mechanics, 2008, 43(2): 140~149
[140] G. Totis, RCPM-A new method for robust chatter prediction in milling, International Journal of Machine Tools & Manufacture, 2009, 49(3-4): 273~284
[141] A. Tang, Z. Liu, Modeling and simulation of three-dimensional stability lobes of milling thin-walled plate. In: Proceedings of the IEEE International Conference on Automationand Logistics Qingdao, 2008: 532~535
[142] U. Bravo, O. Altuzarra, L. N. López de lacalle et al, Stability limits of milling considering the flexibility of th e workpiece and the machine, International Journal of Machine Tools & Manufacture, 2005, 45(15): 1669~1680.
[143] R. G. Landers, A. Galip Ulsoy, Nonlinear feed effect in machining chatter analysis, Journal of Manufacturing Science and Engineering, 2008, 130(1): 011017-1~ 011017-8
[144] J. Gradisek, M. Kalveram, T. Insperger et al, On stability prediction for milling, International Journal of Machine Tools & Manufacture, 2005, 45(1): 769~781
[145] R. P. H. Faassen, N. van de Wouw, J. A. J. Oosterling et al, Prediction of regenerative chatter by modelling and analysis of high-speed milling, International Journal of Machine Tools & Manufacture, 2003, 43(14): 1437~1446
[146] G. Quintana, J. Ciurana, D. Teixidor, A new experimental methodology for identification of stability lobes diagram in milling operations, International Journal of Machine Tools & Manufacture, 2008, 48(15): 1637~1645
[147] T. Insperger, G. Stépán, P. V. Bayly et al, Multiple chatter frequencies in milling processes, Journal of Sound and Vibration, 2003, 262(2): 333~345
[148] M. X. Zhao, B. Balachandran, Dynamics and stability of milling process, International Journal of Solids and Stuctures, 2001, 38(10-13): 2233~2248
[149]朱位秋,随机振动,北京:科学出版社, 1992
[150]龚光鲁,钱敏平,应用随机过程教程,北京:清华大学出版社, 2004
[151]陆启韶,常微分方程的定性方法与分叉,北京:北京航空航天大学出版社, 1989
[152] Y. Xu, G. M. Mahmoud, W. Xu, et al. Suppressing chaos of a complex Duffing’s system using a random phase. Chaos, Solitons and Fractals, 2005, 23(1): 265~273
[153]李爽,徐伟,李瑞红,利用随机相位实现Duffing系统的混沌控制,物理学报, 2006, 55(3): 1049~1054
[154]刘嘉焜,应用随机过程,北京:科学出版社, 2004 207~218
[155]王积伟,陆一心,吴振顺,现代控制理论及工程,北京:高教出版社, 2002
[156] L. K. Abbas, Q. Chen, K. O’Donnell et al, Numerical studies of a non-linear aeroelastic system with plungeing and pitching freeplays in supersonic/hypersonic regimes, Aerospace Science and Technonlgy, 2007, 11(5): 405~418
[157] M. D. Bernardo, C. J. Budd, A. R. Champneys et al, Piecewise-smooth dynamical systems theory and applications, London: Springer-Verlag London Limited, 2008
[158] Q. Y. Wang, Q. S. Lu, H. X. Wang, Transition to complete synchronization via near-synchronization in two coupled chaotic neurons, Chinese Physics, 2005, 14(11): 2189~2195
[159]陆启韶,金俐,具有刚性约束的非线性动力系统的局部映射法,固体力学学报, 2005, 26(2): 132~138
[160]金俐,陆启韶,王琪,非光滑动力系统Floquet特征乘子的计算方法,应用力学学报, 2004, 21(3): 21~26
[161]陈予恕,非线性振动系统的分岔和混沌理论,北京:科学出版社, 1993
[162] T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, New York: Springer-Verlag World Publishing Corp, 1989
[163] Q. Brandon, T. Ueta, D. Fournier-Prunaret et al, Numerical bifurcation analysis framework for autonomous piecewise-smooth dynamical systems, Chaos, Solitons and Fractals, 2009, 42(1): 187~201
[164] C. P. Silva, Member, IEEE, Shil’nikov’s theorem-A tutorial, IEEE Transactions on Circuits and Systems-1: Fundamental Theory and Applications, 1993, 40(10): 675~682
[165] E. A. Butcher, H. T. Ma, E. Bueler, V. Averina, Z. Szabo, Stability of Linear Time-Periodic Delay-Fifferential Equations via Chebyshev Polynomials, International Journal for Numerical Methods in Engineering, 2004, 59(7): 895~922.
[166] S. C. Sinha, D. H. Wu, An Efficient Computational Scheme for the Analysis of Periodic Systems, Journal of Sound and Vibration, 1991, 151(1) : 91~117
[167] E. A. Butcher, S. C. Sinha, A Hybrid Formulation for the Analysis Time Preiodic Linear systems via Chebyshev Polynomials, Journal of Sound and Vibration, 1996, 195(3): 518~527
[168] S. C. Sinha, E. A. Butcher, Symbolic computation of fundmental solution Matrices for Linear Time-Periodic Dynamical Systems, Journal of Sound and Vibration, 1997, 26(1): 61~85
[2] E. H.道尔, H. C.小柯蒂斯, R. H.斯坎伦等,气动弹性力学现代教程(陈文俊,李传家),北京:宇航出版社, 1991
[3] B. H. K. Lee, S. J. Price, Y. S. Wong, Non-linear aeroelastic analysis of airfoils: birurcation and chaos, Progress in Aerospace Sciences, 1999, 35(3): 205~334
[4] H. Ashley, On the role of shock waves in the“sub-transonic”flutter phenomenon. In: AIAA 79-0765, AIAA/ASME/ASCE/AHS 20th Structures, Structural Dynamics, and Materials Conference, 1979
[5] S. J. Price, B. H. K. Lee, H. Alighanbari, An analysis of the post-instability behaviour of a two-dimensional airfoil with a structural nonlinearity. In: Proceedings of 34th AIAA/ASME/ASCE/AHS/ASC stuctures, Stuctural Dynamics and Materials Conference, 2004, 1452~1460.
[6] S. J. Price, H. Alighanbari, B. H. K. Lee, The aeroelastic response of a two-dimensional airfoil with bilinear and cubic structural nonlinearities, Journal of Fluids and Structures, 1995, 9(2): 175~193
[7] B. H. K. Lee, L. Gong, Y. S. Wong, Analysis and computation of nonlinear dynamic response of a two-degree-of-freedom system and its application in aeroelasticity, Journal of Fluids and Structures, 1997(3), 11: 225~246
[8] B. H. K. Lee, L. Y. Jiang, Y. S. Wong, Flutter of a airfoil with a cubic restoring force, Journal of Fluids and Structures, 1999, 13(1): 75~101
[9] B. H. K. Lee, L. Liu, K. W. Chung, Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces, Journal of Sound and Vibration, 2005, 281(3-5): 699~717
[10] M. D. Conner, D. M. Tang, E. H. Dowell et al, Nonlinear behavior of a typical airfoil section with control surface freeplay: a numerical and experimental study, Journal of Fluids and Structures, 1997, 11(1): 89~109
[11] D. Tang, E. H. Dowell, L. N. Virgin, Limit cycle behavior of an airfoil with a control surface, Journal of Fluids and Structures, 1998, 12(7): 839~858
[12] E. H. Dowell, J. P. Thomas, K. C. Hall, Transonic limit cycle oscillation analysis using reduced order aerodynamic models, Journal of Fluids and Structures, 2004, 19(1): 17~27
[13] L. Liu, Y. S. Wong, B. H. K. Lee, Non-linear aeroelastic analysis using the point transformation method, part 1: freeplay model. Journal of Sound and Vibration, 2002, 253(2): 447~469
[14] L. Liu, Y. S. Wong, B. H. K. Lee, Non-linear aeroelastic analysis using the point transformation method, part 2: hysteresis model, Journal of Sound and Vibration, 2002, 253(2): 471~483
[15] L. Liu, E. H. Dowell, The secondary bifurcation of an aeroelastic airfoil motion effect of high harmonics, Nonlinear Dynamics, 2004, 37(1): 31~49
[16] K. W. Chung, C. L. Chan, B. H. K. Lee, Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 2007, 299(3): 520~539
[17] K. W. Chung, Y. B. He, B. H. K. Lee, Bifurcation analysis of a two-degree-of-freedom aeroelastic system with hysteresis structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 2009, 320(1-2): 163~163
[18] L. Librescu, G. Chiocchia, P. Marzocca, Implications of cubic physical/aeroedynamic non-linearities on the character of the flutter instability boundary, International Journal of Non-Linear Mechanics, 2003, 38(2): 173~199
[19] A. Raghothama, S. Narayanan, Non-Linear dynamics of a two-dimensional airfoil by incremental harmonic balance method, Journal of Sound and Vibration, 1999, 226(3): 493~517
[20] P. A. Chamara, B. D. Coller, A study of double flutter, Journal of Fluids and Structures, 2004, 19(7): 863~879
[21]杨智春,赵令诚,带外挂二元机翼颤振模态的转变,航空学报, 1992, 13(9): 552~554
[22]杨智春,赵令诚,李伶等,偏航侧摆联接对代外挂三角机翼颤振特性的影响,振动工程学报, 1992, 5(2): 168~172
[23]王立军,赵令诚,外挂俯仰联接刚度对机翼颤振的影响,航空学报, 1992, 13(9): 560~563
[24] J. K. Liu, L. C. Zhao, Bifurcation analysis of airfoil in incompressible flow, Journal of Sound and Vibration, 1992, 154(1): 117~124
[25]刘继科,二元机翼颤振分岔点类别的判定,力学与实践, 1998, 20: 38~40
[26]刘继科,张宪民,赵令诚等,不可压缩气流中二元机翼颤振的分岔点研究,固体力学学报, 1999, 20(4): 315~319
[27] M. Cai, J. K. Liu, J. Li, Incremental harmonic balance method for airfoil flutter with multiplr strong nonlinearities, Applied Mathematics and Mechanics, 2006, 27(7): 953~958
[28]刘继科,高磊,机翼非线性颤振的分岔点研究,中山大学学报, 1998, 37(3): 28~31
[29]陈衍茂,刘济科,非线性颤振极限环稳定性判别的复正规性法,航空动力学报, 2007, 22(4): 614~618
[30] Y. M. Chen, J. K. Liu, Homotopy analysis method for limit cycle flutter of airfoils, Applied Mathematics and Computation, 2008, 203(2): 854~863
[31]谷迎松,杨智春,带迟滞非线性环节的二元机翼的气动弹性响应分析,机械科学与技术, 2006, 25(8): 900~904
[32]李道春,向锦武,迟滞非线性二元机翼颤振特性分析,航空学报, 2007, 28(3): 600~604
[33]李道春,向锦武,非线性二元机翼气动弹性近似解析研究,航空学报, 2007, 28(5): 1080~1084
[34]丁千,陈予恕,用胞映射法分析双线性结构刚度的机翼颤振,空气动力学报, 2002, 20(2): 123~132
[35]丁千,王冬立,结构和气动非线性机翼颤振分析,动力学与控制学报, 2004, 2(3): 18~23
[36]丁千,王冬立,用规范形直接法研究立方非线性机翼的颤振,飞行力学, 2005, 23(3): 85~88
[37] Q. Ding, D. L. Wang, The flutter of an airfoil with cubic structural and aerodynamic non-linearities, Aeroespace Science and Technology, 2006, 10(5): 427~434.
[38] Q. Ding, D. L. Wang, Flutter control of a two-dimensional airfoil using wash-out filter technique, Chinese Journal of Aeronautics, 2005, 8(2): 131~137
[39]张琪昌,刘海英,任爱娣,非线性机翼极限环颤振的研究,空气动力学学报, 2004, 22(3): 332~336
[40]张琪昌,刘海英,任爱娣,具有立方非线性机翼颤振的局部分岔,天津大学学报, 2004, 37(11): 970~974
[41]任爱娣,张琪昌,具有不对称间隙的二元机翼颤振研究,工程力学, 2006, 23(9): 25~29
[42]吴志强,张建伟,二元机翼极限环颤振复杂分岔,工程力学, 2008, 25(2): 52~55, 92
[43]吴志强,张建伟,王喆,极限环高阶分岔控制,动力学与控制学报, 2007, 5(1): 23~26
[44]刘菲,杨翊仁,不可压缩流中二元翼运动的分支问题,西南交通大学学报, 2002, 37(S1): 95~97
[45]刘菲,杨翊仁,立方非线性机翼的二重半稳定极限环分叉,西南交通大学学报, 2004, 39(5): 638~640, 669
[46]刘菲,杨翊仁,不可压缩流中二元翼运动的余维二分叉,振动与冲击, 2005, 24(4): 18~19, 23
[47]叶献辉,杨翊仁,刘菲,非对称结构参数对颤振速度的影响,西南交通大学学报, 40(6): 779~782, 792
[48] P. Marzocca, L. Librescu, G. Chiocchia, Aeroelastic response of 2-D airfoil in a compressible flow field and exponse to blast load, Aeroespace Science and Technology, 2002, 6(4): 259~272
[49] G. Schewe, H. Mai, G. Dietz, Nonlinear effects in transonic flutter with emphasis on manifestations of limit cycle oscillations. Journal of Fluids and Structures, 2003, 18(1): 3~22
[50]梁强,叶正寅,杨永年,采用非定常N-S方程的翼型颤振特性分析研究,西北工业大学学报, 2001, 19(3): 341~344
[51]史爱明,杨青,杨永年,非结构运动网格下的三维机翼颤振数值分析,振动与冲击, 2005, 24(6): 27~28, 36
[52]史爱明,王刚,杨永年,颤振数值模拟的时间步长选取方法研究,西北工业大学学报, 2005, 23(2): 208~211
[53]杨青,梁强,杨永年,机翼-机身-尾翼结构的跨音速颤振分析研究,西北工业大学学报, 2005, 23(1): 84~88
[54]梁强,杨永年,樊则文,三维机翼跨音速颤振的分布式并行计算,计算物理, 2004, 21(2): 179~183
[55]杨青,闫锋,杨永年,超临界翼型的跨声速颤振特性研究,西北工业大学学报, 22(6): 782~785
[56]牟让科,杨永年,叶正寅,带有结构刚度非线性的跨音速翼型颤振特性研究,振动工程学报, 2001, 14(3): 319~321
[57]史爱明,杨永年,叶正寅,带结构刚度非线性的超音速弹翼颤振分析方法研究,西北工业大学学报, 2003, 21(4): 481~485
[58]谢飞,叶正寅,不同迎角下翼型颤振实验研究,流体力学实验与测量, 2003, 17: 55~59
[59] L. Dubcová, M. Feistauer, J. Horácek et al, Numerical simulation of airfoil vibrations induced by turbulent flow, Journal of Computational and Applied Mathematics, 2008, 218(1): 34~42
[60] P. Svácek, M. Feistauer, J. Horácek, Numerical simulation of flow induced airfoil vibration with large amplitudes, Journal of Fluids and Structures, 2007, 23(3): 391~411
[61] G. Dietz, G. Schewe, H. Mai, Experiments on heavy/pitch limit-cycle oscillations of a supercritical airfoil close to the transonic dip, Journal of Fluids and Structures, 2004, 19(1): 1~16
[62] G. Dietz, G. Schewe, H. Mai, Amplicationand and amplitude limitation heavy/pitch limit-cycle oscillations close to the transonic dip, Journal of Fluids and Structures, 2006, 22(4): 505~527
[63] S. J. Price, J. P. Keleris, Non-linear dynamics of an airfoil forced to oscillate in dynamic stall, Journal of Sound and Vibration, 1996, 194(2): 265~283
[64] M. H. Akbari, S. J. Price, Simulation of dynamic stall for a NACN 0012 airfoil using a vortex method, Journal of Fluids and Structures, 2003, 17(6): 855~874
[65] S. Sarkar, K. Venkatraman, Influence of pitching angle of incidence on the dynamics stall behavior of a symmetric airfoil, European Journal of Mechanics , B/Fluids, 2008, 27(3): 219~238
[66] S. Sarkar, H. Bijl, Nonlinear aeroelastic behavior of an oscillation airfoil during Stall-induced vibration, Journal of Fluids and Structures, 2008, 24(6): 757~777
[67] S. Sarkar, J. A. S. Witteveen, A. Loeven et al, Effect of uncertainty on the bifurcation behavior of pitching airfoil stall flutter, Journal of Fluids and Structures, 2009(2), 25: 304~320
[68]牟让科,杨永年,飞机抖振问题研究进展,应用力学学报, 2001, 18(S1): 142~150
[69]牟让科,杨永年,叶正寅,超临界翼型的跨音速抖振特性,计算物理, 18(5): 477~480
[70]李劲杰,杨青,肖春生等,边条翼布局流场及其双垂尾抖振特性研究,航空学报, 2006, 27(3): 395~398
[71]李劲杰,杨青,杨永年等,边条翼布局双垂尾抖振表面脉动压力实验研究,计算流体力学, 2006, 20(3): 29~32, 38
[72]李劲杰,杨青,杨永年,边条翼布局双垂尾抖振的数值模拟,空气动力学报, 2007, 25(2): 205~210
[73]李劲杰,杨青,杨永年等,边条翼布局双垂尾抖振特性与机理风洞实验研究,空气动力学报, 2006, 24(4): 397~402
[74]谭光辉,解江,夏巍等,改装苏-80飞机的尾翼抖振研究,飞行力学, 2008, 26(5): 67~74
[75]王巍,杨智春,夏巍等,尾翼气动弹性抖振模型设计与机抖振实验研究,西北工业大学学报, 2006, 24(3): 338~341
[76] W. Geissler, Numerical study of buffet and transonic flutter on the NLR 7301 airfoil, Aeroespace Science and Technology, 2003, 7(7): 540~550
[77] D. Poirel, S.J. Price, Post-instability behavior of a structurally nonlinear airfoil in longitudinal turbulence, Journal of Aircraft, 1997, 34(5): 619~626
[78] D. Poirel, S. J. Price, Structurally nonlinear fluttering airfoil in turbulence flow, AIAA Journal, 2001, 39(10): 1960~1968
[79] D. Poirel, S.J. Price, Random binary (coalescence) flutter of a two-dimensional linear airfoil. Journal of Fluids and Structures, 2003, 18(1): 23~42
[80] D. Poirel, S.J. Price, Response Probability structure of a structural nonlinear fluttering airfoil in turbulent flow, Probabilistic Engineering Mechanics, 2003, 18(2): 185~202.
[81] D. Poirel, S. J. Price, Bifurcation characteristics of a two-dimensional structurally non-linear airfoil in turbulent flow, Nonlinear Dynamics, 2007, 48(4): 423~435
[82] L. E. Christainsen, T. Lehn-Schioler, E. Mosekide et al, Nonlinear characteristics of randomly excited transonic flutter, Mathematics and Computer in Simulation, 2002, 58: 385~405
[83] C. L. Wu, H. M. Zhang, T. Fang, Flutter analysis of an airfoil with bounded parameters in incompressible flow via Gegenbauer polynomial approximation, Aerospace Science and Technology, 2007, 11(7-8), 518~526
[84] S. Pospísil, J. Náprstek, S. Hracov, Stability domains in flow-structure interaction and influence of random noises, Journal of Wind Engineering and Industrial Aerodynamics, 2006, 94(11): 883~893
[85] H. Y. Hu, E. H. Dowell, L. N. Virgin, Stability estimation of high dimensional vibrating systems under state delay feedback control, Journal of Sound and Vibration, 1998, 214(3): 497~511
[86]孙伟,翁建生,胡海岩,带有传动机构的翼段颤振半主动抑制,南京航空航天大学学报, 2004, 36(4): 422~426
[87]孙伟,胡海岩,基于多级磁流变阻尼器的操纵面振动半主动抑制—阻尼器设计与实验建模,振动工程学报, 2005, 18(1): 8~13
[88]孙伟,胡海岩,基于多级磁流变阻尼器的操纵面振动半主动抑制—数值仿真与风洞实验,振动工程学报, 2005, 18(1): 14~18
[89]于明礼,胡海岩,基于超声电机作动器的翼段颤振主动控制,振动工程学报, 2005, 18(4): 418~425
[90]于明礼,文浩,胡海岩等,二维翼段颤振的?控制,航空学报, 2007, 28(2): 340~343
[91]于明礼,文浩,胡海岩,二维翼段颤振的H?控制,航空学报, 2007, 28(2): 340~343
[92] Y. H. Zhao, Stability of a two-dimensional airfoil with time-delayed feedback control, Journal of Fluids and Structures, 2009, 25(1): 1~25
[93]杨发友,顾仲权,飞机机翼颤振自适应抑制研究,南京航空航天大学学报, 2004, 36(6): 708~712
[94] V. M. Rao, A. Behal, P. Marzocca et al, Adaptive aeroelastic vibration suppression of a supersonic airfoil with flap, Aerospace Science and Technology, 2006, 10(4): 309~315
[95] L. Librescu, P. Marzocca, W. A. Silva, Aeroelasticity of 2-D lifting surfaces with time-delayed feedback control, Journal of Fluids and Structures, 2005, 20(2): 197~215.
[96] H. Heo, Y. H. Cho, D. J. Kim, Stochastic control of flexible beam in random flutter, Journal of Sound and Vibration, 2003, 267(2): 335~354
[97] D. Tang, H. P. Gavin, E. H. Dowell, Study of airfoil gust response alleviation using of an electro-magnetic dry friction damper. Part 1: theory, Journal of Sound and Vibration, 2004, 269(3-5): 853~874.
[98]谢长川,吴志刚,杨超,大展弦比柔性机翼的气动弹性分析,北京航空航天大学学报, 2003, 29(12): 1087~1090
[99]谢长川,张欣,陈桂彬,复合材料大展弦比机翼机翼动力学建模与颤振分析,飞机设计, 2004, (2): 6~10
[100] X. N. Liu, J. W. Xiang, Stall flutter analysis of high aspect-ratio composite wing, Chinese Journal of Aeronautics, 2006, 19(1): 36~43
[101] Y. H. Zhao, H. Y. Hu, Structural modeling and aeroelastic analysis of high-aspect-ratio composite wings, Chinese Journal of Aeronautics, 2005, 18(1): 25~30
[102]赵永辉,胡海岩,大展弦比夹芯翼大攻角颤振分析,振动工程学报, 2004, 17(1): 25~30
[103]杨智春,张立新,谭光辉,基于遗传算法的复合材料壁板热颤振优化设计,宇航学报, 2008, 29(4): 1451~1456
[104]夏巍,杨智春,复合材料壁板热颤振的有限元分析,西北工业大学学报, 2005, 23(2), 180~183
[105]杨智春,夏巍,高速飞行器壁板颤振的分析模型和分析方法,应用力学学报, 2006, 23(4), 537~542
[106]夏巍,杨智春,热环境下复合材料壁板的振动特性分析,应用力学学报, 2005, 22(3), 359~363
[107]张伟,高美娟,姚明辉等,压电复合材料层矩形板的高维混沌动力学研究,中国科学G辑物理学力学天文学, 2009, 39(8): 1105~1115
[108]胡国才,向锦武,张晓谷,前飞状态直升机旋翼/机体耦合动稳定性分析模型,航空学报, 2004, 25(5): 451~455
[109]薛海峰,向锦武,张晓谷,前飞状态下旋翼动稳定性及自由度间相互作用研究,航空动力学报, 2004, 19(5): 671~677
[110]薛海峰,向锦武,张晓谷,直升机前飞空中共振稳定性和各自由度相互作用研究,航空学报, 2005, 26(4), 454~457
[111]薛海峰,向锦武,张晓谷等,变距/摆振耦合对直升机空中共振稳定性的影响,中国工程科学, 2007, 9(9): 58~62
[112] G. C. Hu, J. W. Xiang, X. G. Zhang, Analytical model of elastomeric lag damper kinematic coupling and its effect on helicopter air resonance in hover, Chinese Journal of Aeronautics, 2002, 15(1): 27~32
[113]胡国才,向锦武,张晓谷,黏弹减摆器非线性特性对直升机前飞空中共振的影响分析,航空学报, 2005, 26(2): 199~202
[114]戚厚军,低刚度摆线轮缘高速铣削变形与铣削力建模方法: [博士学位论文],天津;天津大学, 2009
[115]吴化勇,高速切削系统动态特性研究: [硕士学位论文],济南;山东大学, 2005
[116] S. A. Tobias, Machine Tool Vibration, London: Blackie and Sons Ltd, 1965
[117] Y. Altintas, E. Budak, Analytical prediction of stability lobes in milling, Annals of the CIRP, 1995, 44(1): 357~362
[118] Y. Altintas, Analytical prediction of three dimensional chatter stability in milling, JSME. International Journal Series C, 2001, 44(3): 717~723.
[119] S. D. Merdol, Y. Altintas, Multi-frequency solution of chatter stability for low immersion milling, Journal of Manufacturing Science and Engineering, 2004, 126(3): 459~466
[120] Y. Altintas, G. Stepan, D. Merdol et al, Chatter stability of milling in frenquency and discrete time domain, CIRP Journal of Manufacturing Science and Technology, 2008, 1(1): 35~44
[121] Y. Altintas, M. Eynian, H. Onozuka, Identification of dynamic cutting force coefficients and chatter stability with process damping, CIRP Annals-Manufacturing Technology, 2008, 57(1): 371~374
[122] G. Morgan, R. Q. Cheng, Y. Altintas, K. Ridgway, An expert troubleshooting system for the milling process, International Journal of Machine Tools &Manufacture, 2007, 47(9): 1417~1425.
[123] S. Turner, D. Merdol, Y. Altintas et al, K. Modelling of the stability of variable helix end milling, International Journal of Machine Tools &Manufacture, 2007, 47(9): 1410~1416
[124]宋清华,艾兴,于水清,高速铣削稳定性与表面加工精度研究,制造技术与机床, 2008, (4): 40~43
[125]唐委校,艾兴,姜华等,高速切削稳定性数据库的研究,制造技术与机床, 2005, (6): 75~78
[126]宋清华,艾兴,万熠,薄壁件变参数铣削系统动态特性的研究,工具技术, 2008, 42(7): 35~37
[127] Q. H. Song, X. Ai, Y. Wan et al, Influence of tool helix angle on stability in high-speed milling process. Transactions of Nanjing University of Aeronautics & Astronautics, 2008, 25(1): 18~25
[128]宋清华,艾兴,万熠,高速加工系统稳定性预测的关键技术研究,制造技术与机床, 2008, (10): 59~63
[129]宋清华,艾兴,万熠等,考虑刀具偏心的变径向切深铣削稳定性研究,振动、测试与诊断, 2008, 28(3): 206~210, 299
[130]宋清华,艾兴,万熠等,铣削系统稳定性判定新方法研究,机械强度, 2008, 30(5): 718~722
[131]宋清华,艾兴,万熠等,小径向切深下进给量对切削稳定性的影响,中国机械工程, 2008, 19(10): 1148~1152
[132] Q. H. Song, X. A, Y. Wan et al, State department regenerative delay model for high-speed milling processes. Journal of Beijing Institute of Technology, 2008, 17(3): 264~269
[133] W. X. Tang, Q. H. Song, S. Q. Yu et al, Prediction of chatter stability in high-speed finishing end milling considering multi-mode dynamics. Journal of Materials Processing Technology, 2009, 209(5): 2585~2591
[134] Z. Li, Q. Liu, Solution and analysis of chatter stability for end milling in the time-domain, Chinese Journal of Aeronautics, 2008, 21(2): 169~178
[135] J. Warmiński, G. Litak, M. P. Cartmell et al, Approximate analytical solutions for primary chatter in the non-linear metal cutting model, Journal of Sound and Vibration, 2003, 259(4): 917~933
[136] B. P. Mann, P. V. Bayly, M. A. Davies et al, Limit cycles, bifurcations, and accuracy of the milling process, Journal of Sound and Vibration, 2004, 277(1-2): 31~38
[137] C. Henninger, P. Eberhard, Improving the computational efficiency and accuracy of the semi-discretization method for periodic delay-differential equations, European Journal of Mechanics A/Solids, 2008, 27(6): 975~985
[138] E. Shamoto, K, Akazawa, Analytical prediction of chatter stability in ball end milling with tool inclination, CIRP Annals-Manufacturing Technology, 2009, 58(1): 351~354
[139] T. Insperger, D. A. W. Barton, G. Stépán, Criticality of Hopf bifurcation in State-dependent delay model of turning processes, International Journal of Non-Linear Mechanics, 2008, 43(2): 140~149
[140] G. Totis, RCPM-A new method for robust chatter prediction in milling, International Journal of Machine Tools & Manufacture, 2009, 49(3-4): 273~284
[141] A. Tang, Z. Liu, Modeling and simulation of three-dimensional stability lobes of milling thin-walled plate. In: Proceedings of the IEEE International Conference on Automationand Logistics Qingdao, 2008: 532~535
[142] U. Bravo, O. Altuzarra, L. N. López de lacalle et al, Stability limits of milling considering the flexibility of th e workpiece and the machine, International Journal of Machine Tools & Manufacture, 2005, 45(15): 1669~1680.
[143] R. G. Landers, A. Galip Ulsoy, Nonlinear feed effect in machining chatter analysis, Journal of Manufacturing Science and Engineering, 2008, 130(1): 011017-1~ 011017-8
[144] J. Gradisek, M. Kalveram, T. Insperger et al, On stability prediction for milling, International Journal of Machine Tools & Manufacture, 2005, 45(1): 769~781
[145] R. P. H. Faassen, N. van de Wouw, J. A. J. Oosterling et al, Prediction of regenerative chatter by modelling and analysis of high-speed milling, International Journal of Machine Tools & Manufacture, 2003, 43(14): 1437~1446
[146] G. Quintana, J. Ciurana, D. Teixidor, A new experimental methodology for identification of stability lobes diagram in milling operations, International Journal of Machine Tools & Manufacture, 2008, 48(15): 1637~1645
[147] T. Insperger, G. Stépán, P. V. Bayly et al, Multiple chatter frequencies in milling processes, Journal of Sound and Vibration, 2003, 262(2): 333~345
[148] M. X. Zhao, B. Balachandran, Dynamics and stability of milling process, International Journal of Solids and Stuctures, 2001, 38(10-13): 2233~2248
[149]朱位秋,随机振动,北京:科学出版社, 1992
[150]龚光鲁,钱敏平,应用随机过程教程,北京:清华大学出版社, 2004
[151]陆启韶,常微分方程的定性方法与分叉,北京:北京航空航天大学出版社, 1989
[152] Y. Xu, G. M. Mahmoud, W. Xu, et al. Suppressing chaos of a complex Duffing’s system using a random phase. Chaos, Solitons and Fractals, 2005, 23(1): 265~273
[153]李爽,徐伟,李瑞红,利用随机相位实现Duffing系统的混沌控制,物理学报, 2006, 55(3): 1049~1054
[154]刘嘉焜,应用随机过程,北京:科学出版社, 2004 207~218
[155]王积伟,陆一心,吴振顺,现代控制理论及工程,北京:高教出版社, 2002
[156] L. K. Abbas, Q. Chen, K. O’Donnell et al, Numerical studies of a non-linear aeroelastic system with plungeing and pitching freeplays in supersonic/hypersonic regimes, Aerospace Science and Technonlgy, 2007, 11(5): 405~418
[157] M. D. Bernardo, C. J. Budd, A. R. Champneys et al, Piecewise-smooth dynamical systems theory and applications, London: Springer-Verlag London Limited, 2008
[158] Q. Y. Wang, Q. S. Lu, H. X. Wang, Transition to complete synchronization via near-synchronization in two coupled chaotic neurons, Chinese Physics, 2005, 14(11): 2189~2195
[159]陆启韶,金俐,具有刚性约束的非线性动力系统的局部映射法,固体力学学报, 2005, 26(2): 132~138
[160]金俐,陆启韶,王琪,非光滑动力系统Floquet特征乘子的计算方法,应用力学学报, 2004, 21(3): 21~26
[161]陈予恕,非线性振动系统的分岔和混沌理论,北京:科学出版社, 1993
[162] T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, New York: Springer-Verlag World Publishing Corp, 1989
[163] Q. Brandon, T. Ueta, D. Fournier-Prunaret et al, Numerical bifurcation analysis framework for autonomous piecewise-smooth dynamical systems, Chaos, Solitons and Fractals, 2009, 42(1): 187~201
[164] C. P. Silva, Member, IEEE, Shil’nikov’s theorem-A tutorial, IEEE Transactions on Circuits and Systems-1: Fundamental Theory and Applications, 1993, 40(10): 675~682
[165] E. A. Butcher, H. T. Ma, E. Bueler, V. Averina, Z. Szabo, Stability of Linear Time-Periodic Delay-Fifferential Equations via Chebyshev Polynomials, International Journal for Numerical Methods in Engineering, 2004, 59(7): 895~922.
[166] S. C. Sinha, D. H. Wu, An Efficient Computational Scheme for the Analysis of Periodic Systems, Journal of Sound and Vibration, 1991, 151(1) : 91~117
[167] E. A. Butcher, S. C. Sinha, A Hybrid Formulation for the Analysis Time Preiodic Linear systems via Chebyshev Polynomials, Journal of Sound and Vibration, 1996, 195(3): 518~527
[168] S. C. Sinha, E. A. Butcher, Symbolic computation of fundmental solution Matrices for Linear Time-Periodic Dynamical Systems, Journal of Sound and Vibration, 1997, 26(1): 61~85