Stability Analysis Of The N-Beam Euler-Bernoulli Coupled Beam With Dissipative Joints

This dissertation is one part of the research on the control problem of serially connected elastic system . The aim is to study the feedback stabilization of the serially connected Euler-Bernoulli system and to discuss the system's properties such as well-posed, asymptotic stability e.t.c.In the case of single beam, these properties were usually discussed by using multiplier method which is still an important way to study the system stability. Under this case the eigenvalues usually can be calculated and the properties of them is simple.However, for serially connected system, it is difficult to find a multiplier for the system. The eigenvalues of system are very complex and usually can not be obtained. So another way should be found to solve this problem. Frequency domain method is used to study the stability of serially connected Euler-Bernoulli elastic system in this report. The distinguishing feature of this report is that the matrix form is used to denote system,the distribution of spectrum is given. Then the completeness of the eigenvectors and generalized eigenvectors of the operator is proved. By using this method, we not only studied the stability of the closed loop system.The main rusult of this dissertation is that serially connected Euler-Bernoulli beams with joint and boundary feedback controls. Supposed that the left end of whole beam is clamped and the right end is free. At intermediate nodes, the displacement and rotational angle of beams are continuous but the shearing force and bending moment could be discontinuous. The collocated velocity feedback of the beams at intermediate nodes and the right end are used to stabilize the system, then completeness of the closed loop system are proved to be true.Although the study of this dissertation is mainly on serially connected Euler-Bernoulli elastic system, this method used in this dissertation can be generalized in the study of other models, such as serially connected Timoshenko elastic system, network configuration elastic system and so on.