The Spectral Analysis Of Disk-beam System With Distributed Feedback Control

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.