A Symmetric Inverse Vibration Problem for Nonproportional Underdamped Systems

[+] Author and Article Information
L. Starek

Slovak Technical University of Bratislava, 812 31 Bratislava, Slovakia

D. J. Inman

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219

J. Appl. Mech 64(3), 601-605 (Sep 01, 1997) (5 pages) doi:10.1115/1.2788935 History: Received October 14, 1995; Revised October 01, 1996; Online October 25, 2007


This paper considers a symmetric inverse vibration problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices and nonproportional damping. The inverse problem of interest here is that of determining real symmetric, coefficient matrices assumed to represent the mass normalized velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. The approach presented here gives an alternative solution to a symmetric inverse vibration problem presented by Starek and Inman (1992) and extends these results to include noncommuting (or commuting) coefficient matrices which preserve eigenvalues, eigenvectors, and definiteness. Furthermore, if the eigenvalues are all complex conjugate pairs (underdamped case) with negative real parts, the inverse procedure described here results in symmetric positive definite coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of the system is underdamped. The result provides an algorithm for determining a nonproportional (or proportional) damped system which will have symmetric coefficient matrices and the specified spectral and modal data.

Copyright © 1997 by The American Society of Mechanical Engineers
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