A Partitioning Method For Active Vibration Control Of Structures

In recent years, the technique of active vibration control has been developed rapidly, and many active control methods of structure vibrations have been used in practical engineering. Results obtained both in laboratory demonstrations and in actual measurements for practical engineering show that vibration suppression by active control is emerging as a powerful technique to improve the performance of structures against earthquakes, wind and other dynamic excitations. A great deal of research is currently in progress on developing methods for the simultaneous (integrated) design of the structures and control systems, whereas the control systems are the important issues for design of the intelligent structures, especially finding new and powerful control law is quite challenging. The design of current control law is mainly based on the optimal control method, which involves solution of the ARE (algebraic Riccati equation). The ARE has higher dimensions in general, however, its solution is therefore very difficult. In this paper, we present the concept of the partitioning-optimal control, which can reduce the dimension of the ARE effectively and makes it easily to be solved. For the sake of reducing the mutual influences among the blocks, we partition the dynamic system in the modal space. 1．System description and basic equations Using a finite element model of a control-augmented structure, the equations of motion for undamped system can be written as (1)where and are the mass and stiffness matrices of the system, respectively; is a matrix that places the control actuators at nodal degrees of freedom; are the vectors of nodal displacements and actuator forces, respectively. For free vibration of the undamped system, we have the following generalized eigenvalue problem (2)where ,,,.Let us perform a transform of variables from physical coordinates to modal coordinates according to (3)where is the vector of modal amplitudes. Use of Eq. (1)-(3) leads to (4)Let , then we get (5)2．Partitioning analysis and the optimal control method Assume that Eq. (4) in the modal space is split into the blocks, the block has the frequencies and modes, and . For block, we have frequencies: , (6)vibration modes: , (7)modal coordinates: (8)then we can get the block equation (9)Defining the state space variables as, (10)we can write Eq. (9) as (11)where , .Use of the optimal control method yields the vector of the block input forces: . (12) Here, the matrix = satisfies the following ARE: (13)where: weighting matrix (): weighting matrix ()Once the sub-input control force is obtained, the actual input forces are as follows: (14)where the vector of modal forces is composed of .Numerical results for some example problems are given to illustrate the use and effectiveness of the presented method.