【作者】 赵德敏；

【导师】 张琪昌；

【作者基本信息】 天津大学 ， 振动与控制， 2010， 博士

【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。更多还原

【Abstract】 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.更多还原