Computational and Optimizational Metiods for Quadratic Inverse Eigenvalue Problems Arising In Mechanical Vibration and Structural Dynamics
Speaker:Prof. Biswa Datta
IEEE Fellow, Distinguished Research Professor
Department of Mathematics, Northern Illinois University, USA
Date & Time:03 Feb 2010 (Wednesday) 10:30 - 11:30
Venue:J308
Organized by:Department of Mathematics

Abstract

This talk deals with two quadratic inverse eigenvalue problems that arise in mechanical vibration and structural dynamics. The first one, Qudaratic Partial Eigenvalue Assignment Problem (QPEVAP), arises in controlling dangerous vibrations in mechanical structures, such as buildings, bridges, highways, automobiles, air and spacecrafts, and others. QPEVAP concerns with finding two feedback matrices such that a small amount of the eigenvalues of the associated quadratic eigenvalue problem are reassigned to suitably chosen ones while keeping the remaining large number of eigenvalues and eigenvectors unchanged. For robust and economic control design, these feedback matrices must be found in such a way that they have the norms as small as possible and the condition number of the modified quadratic inverse problem is minimized. These considerations give rise to two nonlinear unconstrained iptimization problems, known respectively, as Robust Qudratic Partial Eigenvalue Assignment Problem (RQPEVAP) and Minimum Norm Qudratic Partial Eigenvalue Assignment Problem (MNQPEVAP) The other one, the Finite Element Model Updating Problem (FEMUP) arising in the design and analysis of structural dynamics, refers to updating an analytical finite element model so that a set of measured eigenvalues and eigenvectors from a real-life structure are reproduced and the physical and structural properties of the orginal model are prserved. A properly updated model can be used in confindence for future designs and constructions. Another major application of FEMUP is the damage detections in structures. Solutions of FEMUP also give rise to several constrained nonlinear constrained optimization problems. We will give an overview of the recent developments on computational methods for these difficult nonlinear optimization problems and discuss directions of future research. The talk is interdisciplinary in nature and will be of interests to mathematicians, computational and applied mathematicians, and control and vibration engineers.