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Research Papers

Vibration Cancellation in a Plate Using Orthogonal Eigenstructure Control

[+] Author and Article Information
M. A. Rastgaar

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139; Center for Vehicle Systems and Safety (CVeSS), Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

M. Ahmadian1

Center for Vehicle Systems and Safety (CVeSS), Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061ahmadian@vt.edu

S. C. Southward

Center for Vehicle Systems and Safety (CVeSS), Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

1

Corresponding author.

J. Appl. Mech 77(6), 061007 (Aug 19, 2010) (7 pages) doi:10.1115/1.4001991 History: Received February 28, 2008; Revised February 04, 2010; Posted June 16, 2010; Published August 19, 2010; Online August 19, 2010

Orthogonal eigenstructure control is a novel control method that can be used for vibration suppression in flexible structures. The method described in this study does not need defining the desired locations of the closed-loop poles or predetermining the closed-loop eigenvectors. The method, which is applicable to linear multi-input multi-output systems, determines an output feedback control gain matrix such that some of the closed-loop eigenvectors are orthogonal to the open-loop eigenvectors. Using this, the open-loop system’s eigenvectors as well as a group of orthogonal vectors are regenerated based on a matrix that spans the null space of the closed-loop eigenvectors. The gain matrix can be generated automatically; therefore, the method is neither a trial and error process nor an optimization of an index function. A finite element model of a plate is used to study the applicability of the method to systems with relatively large degrees of freedom. The example is also used to discuss the effect of operating eigenvalues on the process of orthogonal eigenstructure control. The importance of the operating eigenvalues and the criteria for selecting them for finding the closed-loop system are also investigated. It is shown that choosing the operating eigenvalues from the open-loop eigenvalues that are farthest from the origin results in convergence of the gain matrix for the admissible closed-loop systems. It is shown that the converged control gain matrix has diagonal elements that are two orders of magnitude larger than the off-diagonal elements, which implies a nearly decoupled control.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 4

Open-loop eigenvalues of the first order realization of the plate and four different areas for the operating eigenvalues

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Figure 8

Transverse displacement at nodes of the plate for case 2

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Figure 9

Transverse displacement at nodes of the plate for case 3

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Figure 10

Transverse displacement at nodes of the plate for case 4

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Figure 11

Distribution of eigenvalues in cluster close to origin at case 4

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Figure 12

Distribution of eigenvalues in cluster far from origin at case 4

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Figure 13

Distribution of operating eigenvalues for case 4 that generate acceptable closed-loop systems

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Figure 1

The difference between an achievable and a desirable eigenvector

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Figure 2

Open-loop eigenvectors are the intersections of the open loop and achievable eigenvectors sets

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Figure 3

Elements and nodes of the square plate. The plate is simply supported at all edges.

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Figure 7

Transverse displacement at nodes of the plate for case 1

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Figure 5

Eigenvalues of the open-loop and closed-loop systems for different cases: (a) case 1, (b) case 2, (c) case 3, and (d) case 4

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Figure 6

Actuation forces for different cases: (a) case 1, (b) case 2, (c) case 3, and (d) case 4

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