源于碰撞和摩擦的不连续动力系统的周期流研究

发布时间：2018-04-03 15:32

本文选题：不连续动力系统　切入点：周期运动　出处：《山东师范大学》2016年博士论文

【摘要】：机械工程中普遍存在的碰撞和摩擦是两个或多个物体之间能量转换的基本形式.对实际问题中出现的碰撞和摩擦进行抽象、建模,及动力学行为的研究有助于更好地理解物体之间产生碰撞和摩擦的机理,进一步为实际操作中控制或利用碰撞及摩擦提供参考.研究碰撞和摩擦的结果很多,但主要认为其发生在静态域和静态边界上,或用连续动力系统思想进行研究,因此对于某些实际问题,如机械工程中大量存在的连接问题,研究仍不够充分.连接的物体之间可能存在间隙和相对运动,因而连接问题中经常会出现碰撞和摩擦.碰撞和摩擦使得物体的运动具有较强的不连续性,所建模型是不连续动力系统.近年来一个研究不连续动力系统的新理论初步形成,该理论将碰撞和摩擦视为发生在动态域和动态边界上,从新视角研究其模型,从而可以对模型的动力学行为进行更好地分析.本文利用这一新理论——不连续动力系统理论对连接模型进行研究,主要给出斜碰振子和带摩擦的水平碰撞振子的周期流的有关研究结果.论文的主要内容如下：1.利用不连续动力系统的离散映射理论研究斜碰振子只发生碰撞的周期运动.根据该振子的实际运动情况定义转换平面及转换平面间的四个基本映射,并给出基本映射的控制方程.利用基本映射给出只发生碰撞的周期运动的五种运动模式,并进一步对具体周期运动——小球碰矩形斜槽上壁、下壁各一次的周期-1运动、小球碰矩形斜槽下壁一次的周期-1运动和由倍周期分叉引起的小球碰矩形斜槽下壁k次的周期-k运动——进行研究.利用映射的控制方程得到这些周期运动出现时参数满足的关系式,并对这些周期运动的发生进行数值模拟.由此还得到结论：小球碰矩形斜槽上壁、下壁各一次的对称周期-1运动在底座的任意N个周期内都不存在,并给出其解析证明,文献[78]中有N=1时的结果但没有给出证明,因此本文的结论更具有一般性,且揭示了斜碰振子与水平碰撞振子在动力学行为上的不同本质.最后利用映射的雅可比矩阵及特征值给出周期运动的稳定性和分叉的理论分析结果,并利用数值模拟加以验证.2.利用不连续动力系统的流转换理论和映射动力学对斜碰振子的一般周期流进行研究.根据该振子中碰撞的发生情况将相空间表示成若干子域及其不连续边界之并.利用不连续边界上定义的G函数给出粘合运动发生、消失及在各边界上擦边流出现的充要条件及其解析证明,并用数值模拟加以验证.从粘合运动发生时相角的范围可以看出该振子在斜槽两壁上发生粘合运动的几率不同,而水平碰撞振子在间隙两壁上发生粘合运动的几率相同,从而揭示斜碰振子与水平碰撞振子在动力学行为上的又一本质区别.在此基础上,定义不连续边界上具有或不具有粘合运动的基本映射,利用基本映射定义该振子一般周期流的映射结构,进一步利用映射结构的雅可比矩阵及其特征值得到周期流的稳定性和分叉的研究结果.3.利用不连续动力系统理论对带摩擦的水平碰撞振子的周期流进行研究.根据振子中物体的运动情况将相空间分割成若干子域及其边界,其中根据边界的性质不同将其分为速度边界和位移边界.在每一个子域中定义一个连续动力系统,相邻子域的动力系统具有不同的性质,从而将该振子抽象成不连续动力系统.利用不连续动力系统的流转换理论研究相邻两子系统在边界上流的转换情况,从而对该振子的动力学行为进行解析预测,主要给出两类粘合运动发生、消失以及速度边界上擦边流发生的充要条件,并得到位移边界上擦边流发生的初步结果.理论分析和数值模拟均显示出摩擦对水平碰撞振子的动力学行为有很大影响：受摩擦力影响,位移边界上的第二类粘合运动在第二阶段转化为速度边界上的第一类粘合运动,从而两类粘合运动具有相同的消失条件；位移边界上擦边流的发生依赖于速度边界上流的穿越条件是否满足.这些结果与不受摩擦影响的水平碰撞振子的动力学行为有本质上的不同.最后利用不连续动力系统的映射动力学给出周期流的一般映射结构及其稳定性和分叉的有关结果.
[Abstract]:The collision and friction exists in mechanical engineering is a basic form of energy conversion between two or more objects. Abstraction, collision and friction on the actual problems in the study of dynamic behavior modeling, and contributes to a better understanding of the mechanism of collision and friction between objects, or by collision and further control the friction to provide reference for the actual operation. The research results of a lot of collision and friction, but it mainly occurred in the static fields and boundaries, research or by continuous dynamical systems thinking, so for some practical problems, there are a lot of problems such as connecting in mechanical engineering, the research is still insufficient. There may be a gap and relative motion the connection between the object and the connection problems usually occur in collision and friction collision and friction makes the movement of objects with strong continuity, the model is The initial formation of discontinuous dynamical systems. In recent years a new theory of discontinuous dynamical systems, the theory of collision and friction as occurred in the dynamic domain and dynamic boundary, the research model from a new perspective, which can dynamic behavior on the model of a better analysis. This paper uses the new theory of discontinuous the theory of the dynamic system of connection model was studied. The results of cycle level impact oscillator are oblique oscillator and frictional flow. The main contents of this paper are as follows: 1. using discontinuous dynamical systems theory of discrete mapping oblique oscillator only the collision of the periodic motion. According to the actual situation of the definition of motion the conversion of four basic mapping and conversion between the plane plane resonator, and presents the basic mapping control equation. The periodic motion occurs only by mapping the collision of five Motion mode, and further to the specific periodic motion of small ball touch rectangular chute on the wall, the wall of each cycle of the -1 movement, the ball hit the rectangular chute wall under cycle time -1 movement and the period doubling bifurcation caused by the ball touch Rectangle Flume inferior K cycles of -k -- the movement was studied. The control equation of mapping these periodic motion parameters satisfy the relationship type, and the occurrence of the periodic motion is simulated. We also obtain the conclusion: the ball touch the rectangular chute on the wall, the wall under a symmetric -1 movement does not exist in any N base period, and the analysis proved that N=1 is the result of [78] but no proof is given, so the conclusion of this paper is more general, and reveals the essence of different oblique oscillator and oscillator in horizontal collision dynamics. Then using the mapping The analysis results are given on the stability and bifurcation of periodic motion of the theoretical value of Jacobi matrix and feature shoot, and the numerical simulation is verified using.2. discontinuous dynamical systems theory and dynamic flow conversion mapping for the research of the periodic oblique oscillator oscillator flow. According to the occurrence of phase space representation into several collision subdomain and its discontinuous boundary and. Using discrete G function is defined on the boundary adhesion movement, and disappeared in the border on the edge and prove necessary and sufficient conditions of flow analysis, and verified by numerical simulation. The adhesive movement occurs phase range can be seen in probability of the oscillator adhesion movement in the chute. Two on the wall of different level, the probability of collision oscillator motion in the gap between the two adhesive on the wall of the same, so as to reveal the oblique vibrator and the level of impact oscillator in power Another essential difference between learning behavior. On this basis, the definition of discontinuous boundaries with or without bonding the basic mapping, using the basic mapping mapping defined structure of the oscillator periodic flow, further by using Jacobi matrix mapping structure and characteristics of the current study is not worth the theory of continuous dynamical system of horizontal collision period vibrator with friction using the stability and bifurcation of periodic flow to the research results of.3.. According to the object in motion of the oscillator phase space is divided into several sub domains and boundary, which according to the different nature of the boundary will be divided into the velocity and displacement boundary. The definition of a continuous dynamical system in each sub domain power system, adjacent subdomains with different properties, which will abstract the oscillator into discontinuous dynamical systems. The use of discontinuous dynamical systems flow conversion theory research In the upper boundary of the adjacent two conversion system, dynamic behavior of the oscillator to analyze the prediction, mainly give two kinds of adhesive movement, the necessary and sufficient conditions of flow and velocity boundary edge disappeared, and obtained the preliminary results on the edge displacement boundary flow. Theoretical analysis and numerical simulation show that there are greatly influence the dynamic behavior of impact oscillator friction: under the influence of the friction, displacement on the boundary of second kinds of adhesive movement in the second stage into the bounds on the rate of the first and two kind of adhesive bonding exercise, movement has disappeared under the same conditions; whether the crossing occurred on the edge displacement boundary flow velocity dependent boundary high satisfaction. The dynamics of these results and not influenced by friction level impact oscillators are essentially different. Finally the use of discontinuous dynamical systems The mapping dynamics gives the general mapping structure of the periodic flow and the related results of its stability and bifurcation.

【学位授予单位】：山东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O313

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