【作者】 甄亚欣；

【导师】 张池平；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2008， 硕士

【摘要】 本文研究了基于切比雪夫正交多项式理论和非线性Galerkin法的高维非线性动力学系统降维技术及非线性油膜力、密封力作用下转子-轴承系统的非线性动力学问题。复杂动力系统往往呈现出高维、非线性、强耦合的特点,这类系统的动力学特性即使采用数值方法研究也存在着很多困难。非线性非自治系统的动力学行为模式往往由低维流形上的动力学所控制,因而降维简化就成为从理论上揭示其动力学现象的必要步骤和前提。本文利用平移后的切比雪夫正交多项式理论求得一个线性项随时间周期变化的非线性非自治系统的Liapunov-Floquet变换矩阵,将其变换为线性项时不变的系统,再利用非线性Galerkin法对此系统进行降维。通过对降维前后系统的时间历程图和相轨迹图的对比,说明降维成功。针对单圆盘Jeffcott转子-轴承系统,建立其在非线性油膜力和密封力作用下的动力学方程,其中油膜力采用了Capone在1991年提出的短轴承假设下的非线性油膜力模型,该模型具有较好的精度和收敛性,密封力采用了Muszynska非线性模型。本文利用龙格-库塔法求解上述非线性转子-轴承动力学系统,得到了系统的周期响应和非线性动力学特性,利用最大值法绘制了系统随转子转速和圆盘质量偏心变化的分岔图。在所研究的转速变化范围内,通过绘制Poincare截面图、轨迹图、相图等,发现随着转子转速和圆盘质量偏心的变化,系统发生周期运动、倍周期运动以及准周期运动,进而进入混沌状态,而后又变为二周期运动,最后以周期运动结束。更多还原

【Abstract】 Based on the Chebyshev orthogonal polynomials theory and the nonlinear Galerkin method, the order reduction of high order nonlinear dynamics and the dynamical characteristics of rotor-bearing systems with nonlinear oil film force and sealing force are studid in this paper.Complex dynamical systems always present high dimension, nonlinear and strong coupling characteristics. For these systems, there are many difficulties even if we use the numerical methods to study their dynamical characteristics. The nonlinear non-autonomous systems’dynamical behavior patterns are often controlled by the low-dimensional manifolds on the dynamics. Thus the order reduction becomes the necessary steps and precondition to reveal their dynamical phenomenon theoretically. In this paper the shifted Chebyshev orthogonal polynomials are used to obtain the Liapunov-Floquet transition matrix of a nonlinear non-autonomous system whose linear part is periodic, and to transfer the system to a new one with time invariant linear part. Then, we use the nonlinear Galerkin method to make this system’s dimension lower. Through comparing the time trace and the phase diagram of the reduced model with the original model, we can find the order reduction is successful.The dynamical equations of a single-disc Jeffcott rotor-bearing system with nonlinear oil film force and sealing force are established, where the nonlinear oil film force model put forward by Capone in 1991 under the short-bearing assumption is used to make the model has better accuracy and convergence. About the nonlinear sealing force, the Muszynska model is used. The Runge-Kutta method is used to study the nonlinear dynamical equations and the periodic response and nonlinear dynamical characteristics of the system are obtained. The bifurcation map with respect to the rotating speed and disc mass eccentricity is drawn by using the maximum method. Among the rotating speed range, by drawing the diagrams of Poincare section, trajectory map and phase trait, we can learn that with the changes of rotating speed and the mass eccentricity, the system has periodic, periodic doubling and quasi-period motion, then becomes chaos, next be changed to double period motion, and periodic motion at last.更多还原