【作者】 王丽丽；

【导师】 王军民；

【作者基本信息】 北京理工大学 ， 应用数学， 2015， 硕士

【摘要】 随着科学技术的日益进步,机械系统的动力学研究变的越来越复杂.由于操作机械臂、机器人逐渐被实际运用到工程建造中,从而带来一类机械系统主要是刚-柔耦合系统的研究.本文以转盘和梁构成的系统为研究对象,研究这类刚-柔耦合系统的稳定性问题.论文将研究由一个转盘和柔性梁组成的耦合系统,其中,梁的一端在转盘的中心,并且与转盘所在的平面垂直,另一端是自由的,转盘在所在的平面绕着自身轴转动,在盘转动的同时,梁与转盘所在的平面垂直.为了使系统稳定,我们分别对转盘施加扭矩控制,对梁的自由端施加内部控制,从而得到一个闭环系统.论文的主要结果是证明：当转盘的角速度|ω|<(?)μ1时,闭环系统具有指数稳定性,这里的μ1是一个具有正定自伴算子的自由梁系统的第一本征值.论文给出线性子系统的本征值和本征向量的渐进表示,并证明线性子系统的广义本征向量在相应的能量空间生成Riesz基,以及线性子系统的指数稳定性.论文的结构安排如下：第一章我们给出论文的研究背景.第二章我们给出论文研究需要的预备知识,包括线性算子理论,Riesz基及C0-算子半群等.第三章研究由转盘和柔性梁组成的转盘-梁系统的稳定性,证明了当转盘的角速度|ω|<(?)μ1时,闭环系统具有指数稳定性,这里的μ1是一个具有正定自伴算子的自由梁系统的第一本征值,系统是指数稳定的.当转盘的角速度|ω|≥(?)μ1时,系统不稳定.最后一部分,给出了论文的总结,以及一些有待进一步解决的问题.更多还原

【Abstract】 With the development of science and technology, the study of the mechanical en-gineering system is becoming more and more complex. Mechanical arm operation and robots are used in engineering construction gradually, while these mechanical systems are called rigid-flexible coupled systems. This thesis focuses on the coupled systems which consist of a rotating disk and a flexible beam, and deals with the stability of the system.This thesis considers the feedback stabilization problem of a flexible beam, which is clamped at the center of a disk and free at the other end. The disk is assumed to rotate freely around its axis with a nonuniform angular velocity and the motion of the beam is confined to a plane perpendicular to the disk. In order to leave the whole disk-beam system rotating with a desired angular velocity, we propose a torque control exerted on the disk and suggest a distributed control of a flexible beam which lead a closed-loop system. Our main contribution is to show that when the angular velocity is less than the square root of the first eigenvalue of the free fourth-order beam operator, the closed-loop system can be exponentially stabilized. Furthermore, we get the eigenvalues and eigenfunctions of the subsystem, and further prove that the corresponding generalized eigenfunctions form a Riesz basis.The contents of the thesis are as follows:In Chapter 1, we introduce some research background of the thesis.In Chapter 2, we present some basic preliminaries, including linear operator theory, semigroup theory and Riesz basis property, etc.In Chapter 3, we are concerned with the stability of the rotating disk-beam system, and prove that the system can be exponentially stable when the angular velocity is less than the square root of the first eigenvalue of the free fourth-order beam operator.In the last chapter, a summary of this thesis is presented and some possible inter-esting problems are addressed.更多还原