本文選題：不連續(xù)動(dòng)力系統(tǒng)　切入點(diǎn)：周期運(yùn)動(dòng)　出處：《山東師范大學(xué)》2016年博士論文

【摘要】：機(jī)械工程中普遍存在的碰撞和摩擦是兩個(gè)或多個(gè)物體之間能量轉(zhuǎn)換的基本形式.對(duì)實(shí)際問題中出現(xiàn)的碰撞和摩擦進(jìn)行抽象、建模,及動(dòng)力學(xué)行為的研究有助于更好地理解物體之間產(chǎn)生碰撞和摩擦的機(jī)理,進(jìn)一步為實(shí)際操作中控制或利用碰撞及摩擦提供參考.研究碰撞和摩擦的結(jié)果很多,但主要認(rèn)為其發(fā)生在靜態(tài)域和靜態(tài)邊界上,或用連續(xù)動(dòng)力系統(tǒng)思想進(jìn)行研究,因此對(duì)于某些實(shí)際問題,如機(jī)械工程中大量存在的連接問題,研究仍不夠充分.連接的物體之間可能存在間隙和相對(duì)運(yùn)動(dòng),因而連接問題中經(jīng)常會(huì)出現(xiàn)碰撞和摩擦.碰撞和摩擦使得物體的運(yùn)動(dòng)具有較強(qiáng)的不連續(xù)性,所建模型是不連續(xù)動(dòng)力系統(tǒng).近年來一個(gè)研究不連續(xù)動(dòng)力系統(tǒng)的新理論初步形成,該理論將碰撞和摩擦視為發(fā)生在動(dòng)態(tài)域和動(dòng)態(tài)邊界上,從新視角研究其模型,從而可以對(duì)模型的動(dòng)力學(xué)行為進(jìn)行更好地分析.本文利用這一新理論——不連續(xù)動(dòng)力系統(tǒng)理論對(duì)連接模型進(jìn)行研究,主要給出斜碰振子和帶摩擦的水平碰撞振子的周期流的有關(guān)研究結(jié)果.論文的主要內(nèi)容如下：1.利用不連續(xù)動(dòng)力系統(tǒng)的離散映射理論研究斜碰振子只發(fā)生碰撞的周期運(yùn)動(dòng).根據(jù)該振子的實(shí)際運(yùn)動(dòng)情況定義轉(zhuǎn)換平面及轉(zhuǎn)換平面間的四個(gè)基本映射,并給出基本映射的控制方程.利用基本映射給出只發(fā)生碰撞的周期運(yùn)動(dòng)的五種運(yùn)動(dòng)模式,并進(jìn)一步對(duì)具體周期運(yùn)動(dòng)——小球碰矩形斜槽上壁、下壁各一次的周期-1運(yùn)動(dòng)、小球碰矩形斜槽下壁一次的周期-1運(yùn)動(dòng)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運(yùn)動(dòng)——進(jìn)行研究.利用映射的控制方程得到這些周期運(yùn)動(dòng)出現(xiàn)時(shí)參數(shù)滿足的關(guān)系式,并對(duì)這些周期運(yùn)動(dòng)的發(fā)生進(jìn)行數(shù)值模擬.由此還得到結(jié)論：小球碰矩形斜槽上壁、下壁各一次的對(duì)稱周期-1運(yùn)動(dòng)在底座的任意N個(gè)周期內(nèi)都不存在,并給出其解析證明,文獻(xiàn)[78]中有N=1時(shí)的結(jié)果但沒有給出證明,因此本文的結(jié)論更具有一般性,且揭示了斜碰振子與水平碰撞振子在動(dòng)力學(xué)行為上的不同本質(zhì).最后利用映射的雅可比矩陣及特征值給出周期運(yùn)動(dòng)的穩(wěn)定性和分叉的理論分析結(jié)果,并利用數(shù)值模擬加以驗(yàn)證.2.利用不連續(xù)動(dòng)力系統(tǒng)的流轉(zhuǎn)換理論和映射動(dòng)力學(xué)對(duì)斜碰振子的一般周期流進(jìn)行研究.根據(jù)該振子中碰撞的發(fā)生情況將相空間表示成若干子域及其不連續(xù)邊界之并.利用不連續(xù)邊界上定義的G函數(shù)給出粘合運(yùn)動(dòng)發(fā)生、消失及在各邊界上擦邊流出現(xiàn)的充要條件及其解析證明,并用數(shù)值模擬加以驗(yàn)證.從粘合運(yùn)動(dòng)發(fā)生時(shí)相角的范圍可以看出該振子在斜槽兩壁上發(fā)生粘合運(yùn)動(dòng)的幾率不同,而水平碰撞振子在間隙兩壁上發(fā)生粘合運(yùn)動(dòng)的幾率相同,從而揭示斜碰振子與水平碰撞振子在動(dòng)力學(xué)行為上的又一本質(zhì)區(qū)別.在此基礎(chǔ)上,定義不連續(xù)邊界上具有或不具有粘合運(yùn)動(dòng)的基本映射,利用基本映射定義該振子一般周期流的映射結(jié)構(gòu),進(jìn)一步利用映射結(jié)構(gòu)的雅可比矩陣及其特征值得到周期流的穩(wěn)定性和分叉的研究結(jié)果.3.利用不連續(xù)動(dòng)力系統(tǒng)理論對(duì)帶摩擦的水平碰撞振子的周期流進(jìn)行研究.根據(jù)振子中物體的運(yùn)動(dòng)情況將相空間分割成若干子域及其邊界,其中根據(jù)邊界的性質(zhì)不同將其分為速度邊界和位移邊界.在每一個(gè)子域中定義一個(gè)連續(xù)動(dòng)力系統(tǒng),相鄰子域的動(dòng)力系統(tǒng)具有不同的性質(zhì),從而將該振子抽象成不連續(xù)動(dòng)力系統(tǒng).利用不連續(xù)動(dòng)力系統(tǒng)的流轉(zhuǎn)換理論研究相鄰兩子系統(tǒng)在邊界上流的轉(zhuǎn)換情況,從而對(duì)該振子的動(dòng)力學(xué)行為進(jìn)行解析預(yù)測(cè),主要給出兩類粘合運(yùn)動(dòng)發(fā)生、消失以及速度邊界上擦邊流發(fā)生的充要條件,并得到位移邊界上擦邊流發(fā)生的初步結(jié)果.理論分析和數(shù)值模擬均顯示出摩擦對(duì)水平碰撞振子的動(dòng)力學(xué)行為有很大影響：受摩擦力影響,位移邊界上的第二類粘合運(yùn)動(dòng)在第二階段轉(zhuǎn)化為速度邊界上的第一類粘合運(yùn)動(dòng),從而兩類粘合運(yùn)動(dòng)具有相同的消失條件；位移邊界上擦邊流的發(fā)生依賴于速度邊界上流的穿越條件是否滿足.這些結(jié)果與不受摩擦影響的水平碰撞振子的動(dòng)力學(xué)行為有本質(zhì)上的不同.最后利用不連續(xù)動(dòng)力系統(tǒng)的映射動(dòng)力學(xué)給出周期流的一般映射結(jié)構(gòu)及其穩(wěn)定性和分叉的有關(guān)結(jié)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.

【學(xué)位授予單位】：山東師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O313

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本文編號(hào)：1705805