The Design Of Control And Stability Analysis Of Serially Connected Timoshenko Beams

This report is one part of the research on the control problem of serially connected elastic system which was done by author during postgraduate student studying. The aim is to study the feedback stabilization of the serially connected Timoshenko system and to discuss the system's properties such as well-posed, asymptotic stability, exponential stability, spectrum determined growth condition 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 Timoshenko elastic system in this report. The distinguishing feature of this report is that the matrix form is used to denote system, by the technique of asymptotic analysis, the distribution of spectrum is given. Then the Riesz basis property of the eigenvectors and generalized eigenvectors of the operator is proved. So the spectrum determined growth condition holds. By using this method, we not only studied the stability of the closed loop system, but also got the properties of the spectrum distribution and the spectrum determined growth condition.Two results of the serially connected Timoshenko beams can be gotten in this report. On one hand, serially connected Timoshenko beams with joint and boundary feedback controls is studied. 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 Riesz basis properties of the closed loop system are proved to be true. Hence the spectrum determined growth condition holds. Furthermore, the closed loop system is exponentially stable for the case of n = 3.On another hand, stabilization problem of n-connected Timoshenko beams is discussed. Supposed that both ends of the beams are clamped. At intermediate nodes, the shearing force and bending moment are continuous, but the displacement and rotational angle of beams are discontinuous. Shearing force and bending moment at intermediate nodes are observed. The compensators are signed to use to obtain the displacements and that the closed loop system is asymptotically stable. By a detail spectral analysis of the system, it is shown that the closed loop system is of Riesz basis property under some conditions. Hence the spectrum determined growth condition holds.Although the study of this report is mainly on serially connected Timoshenko elastic system, this method used in this report can be generalized in the study of other models, such as serially connected Euler-Bernoulli elastic system, network configuration elastic system and so on...