The vibration of linear mechanical systems with arbitrary damping is known to pose challenging problems to the analyst, for these systems cannot be analyzed with the techniques pertaining to their undamped counterparts. It is also known that a class of damped systems, called proportionally damped, can be analyzed with the same techniques, which mimic faithfully those of single-degree-of-freedom systems. For this reason, in many instances the system at hand is assumed to be proportionally damped. Nevertheless, this assumption is difficult to justify on physical grounds in many practical applications. What this assumption brings about is a damping matrix that admits a simultaneous diagonalization with the stiffness matrix. Proposed in this paper is a decomposition of the damping matrix of an arbitrarily damped system allowing the extraction of the proportionally damped component, which, moreover, approximates optimally the original damping matrix in the least-square sense. Finally, we show with examples that conclusions drawn from the proportionally damped approximation of an arbitrarily damped system can be dangerously misleading.
The Proportional-Damping Matrix of Arbitrarily Damped Linear Mechanical Systems
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, December 17, 1999; final revision, February 28, 2002. Associate Editor: V. K. Kinra. Discussion on the paper should be addressed to the Editor, Professor Robert M. McMeeking, Department of Mechanical and Environmental Engineering, University of California–Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Angeles, J., and Ostrovskaya, S. (August 16, 2002). "The Proportional-Damping Matrix of Arbitrarily Damped Linear Mechanical Systems ." ASME. J. Appl. Mech. September 2002; 69(5): 649–656. https://doi.org/10.1115/1.1483832
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