﻿ 一类Filippov碰摩转子系统的非光滑分析 The Non-Smooth Analysis of Filippov Rub-Impact Rotor System

Vol.04 No.02(2015), Article ID:15349,11 pages
10.12677/AAM.2015.42025

The Non-Smooth Analysis of Filippov Rub-Impact Rotor System

Jieqiong Xu

College of Mathematics and Information Science, Guangxi University, Nanning Guangxi

Email: clh4@163.com

Received: May 7th, 2015; accepted: May 22nd, 2015; published: May 29th, 2015

ABSTRACT

A three-degree-of-freedom lateral-torsional coupled Filippov differential system for a Jeffcott rotor supported rigidly is established. Comparing the lateral-torsional coupled rub-impact rotor and the lateral rub-impact rotor through bifurcation diagrams of amplitude and the phase difference, time trajectories, phase portraits, Poincaré maps, time-history diagram, the phase characteristic and the non-smooth dynamic behavior of the Filippov rub-impact system are analyzed numerically. It is shown that the rub-impact response has the definite phase characteristic; the two models have a similar bifurcation process in their bifurcation figures; the torsional vibration is obvious, and the stick-slip phenomena will happen in this system in a certain parameter.

Keywords:Rub-Impact Rotor, Filippov System, Phase Difference, Non-Smooth Analysis, Stick-Slip Phenomenon

Email: clh4@163.com

1. 引言

2. 物理模型及运动方程

(1)

Figure 1. Schematic diagram of the rub-impact rotor

Figure 2. Schematic diagram of the rub and impacts force

(2)

(3)

(4)

, , , , , , ,

, , , ,

, ,.

3. 数值仿真与分析

3.1. Filippov碰摩转子系统的碰摩响应相位特征

Figure 3. The bifurcation diagram of amplitude

Figure 4. The bifurcation diagram of the phase difference

Figure 5. The bifurcation diagram of torsion angle

Figure 6. The bifurcation diagram of torsion velocity

3.2. 摩擦系数对Filippov碰摩转子系统动力学响应的影响

(5)

(6)

(7)

Figure 7. The parameter plane of the rotor speed and friction coefficient of Hopf bifurcation

,其它系统参数不变。分析方程(3)和(4)的弯振情况。根据上述参数得到关于和激励与响应相位差的分岔图，如图8和图9所示。从图8和图9可以看出，俩曲线总体趋势基本一致。随着转速的增大，当时，转子与外环接触，转子作全局碰摩运动。当时，全局碰摩运动发生Hopf失稳，转子开始作局部碰摩运动，这与上面的分析结果一致。转子作局部碰摩运动时，响应相位差的平均值小于。当时，相位差开始在内变化，这说明转子开始作反向局部碰摩运动。当时，转子振幅突然跳跃到很大值，相位差仍然在内变化，转子发生反向涡动失稳。

3.3. 扭振特性与弯振特性的比较

3.3.1. 摩擦系数较小的情况

3.3.2. 摩擦系数较大的情况

1) 当时，根据系统(3)和(4)作转子弯振的轨迹图、转子扭振的相图、转子扭振的Poincaré截面图，如图12~14所示。从图12可以看出，在时，转子的弯振为准周期运动。而转子的扭振为混沌运动，如图13和图14所示，Poincaré图14表现为无序点集。

2) 当时，根据系统(3)和(4)作转子弯振的轨迹图、弯振的Poincaré截面图、转子扭振的相图，如图15~18所示。转子的弯振和扭振都为混沌运动，图16和图18都为无序点集。

3) 当时，根据系统(3)和(4)作转子弯振的轨迹图、转子扭振的相图、转子扭振的Poincaré截面图以及接触点的相对速度时间历程图，如图19、图20所示。从图19可以看出系统运动为反向涡动失稳，振幅很大。转子的扭振仍然为混沌运动，扭转角比较大，如图20所示。在时，系统出现stick-slip现象，如图21所示。

Figure 8. The bifurcation diagram of amplitude

Figure 9. The bifurcation diagram of the phase difference

Figure 10. The bifurcation diagram of torsion angle

Figure 11. The bifurcation diagram of torsion velocity

Figure 12. The trajectories of lateral vibration

Figure 13. The phase portraits of torsional vibration

Figure 14. The Poincaré-map of torsional vibration

Figure 15. The trajectories of lateral vibration

Figure 16. The Poincaré-map of lateral vibration

Figure 17. The phase portraits of torsional vibration

Figure 18. The Poincaré-map of torsional vibration

Figure 19. The trajectories of lateral vibration

Figure 20. The phase portraits of torsional vibration

Figure 21. Time-history diagram of relative velocity

4. 结论

The Non-Smooth Analysis of Filippov Rub-Impact Rotor System. 应用数学进展,02,197-208. doi: 10.12677/AAM.2015.42025

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