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Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 27373, 8 pages

On an elastic dissipation model as continuous approximation for discrete media

1Institute of General Mechanics, RWTH Aachen University, Templergraben 64, Aachen 52062, Germany
2Department of Automatics and Biomechanics, Technical University of Łódź, Stefanowski Street 1/15, Łódź 90-924, Poland
3Prydniprovska State Academy of Civil Engineering and Architecture, Chernishevskogo Street 24 a, Dnepropetrovsk 49005, Ukraine

Received 18 August 2006; Accepted 13 September 2006

Copyright © 2006 I. V. Andrianov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Construction of an accurate continuous model for discrete media is an important topic in various fields of science. We deal with a 1D differential-difference equation governing the behavior of an n-mass oscillator with linear relaxation. It is known that a string-type approximation is justified for low part of frequency spectra of a continuous model, but for free and forced vibrations a solution of discrete and continuous models can be quite different. A difference operator makes analysis difficult due to its nonlocal form. Approximate equations can be obtained by replacing the difference operators via a local derivative operator. Although application of a model with derivative of more than second order improves the continuous model, a higher order of approximated differential equation seriously complicates a solution of continuous problem. It is known that accuracy of the approximation can dramatically increase using Padé approximations. In this paper, one- and two-point Padé approximations suitable for justify choice of structural damping models are used.