Effect of the Actuators’ Location on Vibration Suppression Using Minimum Modal Energy Eigenstructure Assignment

A new Eigenstructure Assignment (ESA) method for vibration confinement of flexible structures has been developed. This method is based on finding an output feedback control gain matrix in such a way that the closed-loop eigenvectors are orthogonal to the open-loop ones. Singular Value Decomposition (SVD) is used for finding the matrix that spans the null space of the closed-loop eigenvectors. It is shown that this matrix has a unique property that can be used to regenerate the open-loop system. This method finds a coefficient vector which leads to a zero gain matrix while several coefficient vectors can be found simultaneously which are orthogonal to the open-loop coefficient vector. As a result, the closed-loop eigenvectors are orthogonal to the open-loop ones. It is shown that the modal energy of the closed loop system is reduced. Moreover, the proposed method needs neither to specify the closed-loop eigenvalues nor to define a desired set of eigenvectors. Also it is shown that if the maximum force of the actuators and the consumed energy of the actuators need to be low, actuators have to be relatively close to input. If the amplitude of vibration in isolated area has to be minimized as much as possible, the actuators need to be relatively closer to isolated area. Also the algorithm of the minimum eigenstructure assignment method has been modified to eliminate the effect of the actuators that are located on the nodes of different vibrational modes. INTRODUCTION Eigenstructure Assignment (ESA) was first introduced by Moore [1]. Moore determined the class of all eigenvectors associated with the closed-loop eigenvalues using a state feedback control. In fact, the control of a MIMO system considers both eigenvalue placement and defining the associated eigenvectors form a class of possible closed-loop eigenvectors [2]. To define the null space of the achievable eigenvectors, Cunningham used Singular Value Decomposition (SVD) [3]. It had been shown that the number of actuators is finite, for suppression of structural vibrations, if SVD is used [4]. Eigenvector scaling, a mode localization technique, has been introduced by Shelly et al. Eigenvector scaling changes the elements of eigenvectors in such a way that relative displacement within the area corresponding to those elements decrease [5]. They showed that eigenvector shaping reduces vibration in the isolated area regardless of the disturbance type. Some experimental works have been done using the eigenvector shaping and reported in [2, 6-8]. Eigenvector shaping using SVD has been introduced and used as a solution to the problem of limited pairs of actuators and sensors in [9]. A Moore-Penrose generalized left inverse is used to produce the closest eigenvector in least square sense to the desired ones, since it gives the minimum Euclidean 2-norm error [9, 10]. Tang et al proposed a method that uses the piezoelectric network actuators in an active-passive hybrid vibration confinement system [11, 12]. The energy is confined to the circuit inductors and resistors as passive elements of the systems rather than bulky mechanical parts. Using the Rayleigh principle optimal eigenvectors can be found. Their method minimizes the ratio of the modal energy at the concerned area to the modal energy of the whole structure using an auxiliary eigenvalue problem [13].