碰摩转子系统的非光滑分析

A NON-SMOOTH ANALYSIS TO THE RUBIMPACTING ROTOR SYSTEM

通过建立转子系统碰摩的Ｐｏｉｎｃａｒｅ映射，将对非光滑碰摩系统的研究转化为对Ｐｏｉｎｃａｒｅ映射的分析．文中主要对转子碰摩当中一类特殊的运动形式──单点碰摩下的擦边现象进行了详细研究．从序列的极限理论出发分析了该映射的周期不动点的稳定性及其吸引域：得到了转子系统在接近擦边运动时解随系统参数变化的分岔情形．

This paper mainly deals with the single grazing impact phenomenon of the rub-impactrotor system by using the method of Poincare map analysis. As we know, rub-impacts of rotorsystems are always suddenly occurring and hardly predictable. The recent research indicates thatthe rub-impacts are intimately related to a phenomenon called grazing impact, which is in facta critical rub-impact when the rotor touches the stator with very low radial velocity that nearlyequals 0. of cause, it is quit difficult to set up a theoretical system for the general case of rubimpact, which includes the single impact, the multiple impacts and the wholly whirling.impact.But it is somehow an easily thing to investigate the motions of the rotor system with the singleimpact since the rotor touches the stator only once a period. Now, a certain trajectory (grazingorbit) is chosen and a local map is constructed near the trajectory. Then the local dynamicalbehavior can be obtained by study of the map. Because of the discontinuity of stickness in the rub-impact rotor system, the flow in the phasespace is not smooth and even discontinuous. Those common measures for the smooth dynamicalsystems, such as the method of central manifold and linearizing of the system ill the neighborhoodof the fixed point, become invalid. However, enlightened by the significant work of S. W. Shawand A. B. Nordmark, the method concerned in the non-smooth dynamic system of single degreeof freedom is successfully applied to the rub-impact rotor of two degrees of freedom in this paper.And some special dealments including the theory of limits for sequences and singular analysis areused to determine the stability of the fixed point of the Poincare map, and also the attracted regionis found for the fixed point.A further study gives the interesting result of the grazing bifurcation for the non-smoothdynamical system. It is turned out that the appearance of grazing impact causes the uncertaintyto the motion of the rotor system and changes its stability. It may be the stable periodic motionbefore the grazing impact, but if the grazing impact occurs, the stability of the motion can beeither stable or unstable. Soon after that, chaos will follow it. The essence of this phenomenonis just because of the non-smooth characteristic of the system. That is, the grabing bifurcationcauses the square root singularity in the impact map whiCh makes the Jacobian of the map nosense as rotor touches the stator with zero radial velocity. When the motion of the rotor systemis near the grazing impact, it experiences stable periodic motions, grazing bifurcation to irregularrub-impact motions with the variation of the system parameters.