Logo image
Back
FRACTAL RELAXATION SYSTEMS. Part II: Distribution of Relaxation Times
Journal article   Peer reviewed

FRACTAL RELAXATION SYSTEMS. Part II: Distribution of Relaxation Times

YUAN-YING Tsao and BANU Onaral
International journal of general systems, v 19(2), pp 133-153
01 Sep 1991
DOI: https://doi.org/10.1080/03081079108935168

Abstract

distribution of relaxation times Fractal relaxation systems fractional power attenuation global singularity gradational fractal region system I/f-type attenuation local singularity multiple fractal region system self-similarity singularity structure
A system is recognized as fractal if its magnitude frequency spectrum attenuates following a power function dependence on the frequency and exhibits a fractional slope over a broad band of frequencies. If the transient response of the fractal system consists solely of real exponentials, the system is classified as a fractal relaxation system. The relaxation elements of the system reach steady state at different rates, i.e. different relaxation times or time constants. The system can therefore can be characterized by a distribution of the relaxation times. The distribution of relaxation times completely defines the system, thus is equivalent to the system function in the relaxation domain. This is the second part of a series of articles, each focusing on a particular aspect of fractal relaxation systems. In the first part, the singularity structure model of the fractal relaxation systems has been proposed. In this part, the concept of distribution of relaxation times of fractal relaxation systems is introduced to explore the dynamics of such systems in the linear range. The distribution of the relaxation times can be transformed from the frequency response of the system; however, the transformation between the two domains often presents mathematical difficulties. In this article, we propose the discrete distribution of relaxation times as a powerful analysis method for fractal systems. The distribution is obtained directly from the singularity structure and is not plagued with the difficulties common to the classical transformation. Fractal relaxation systems are classified as single fractal region, multiple fractal region and gradational fractal region systems according to the variation of the fractional slope of the frequency spectrum. In this article, we study different types of fractal relaxation systems in both the frequency and relaxation lime domains to reveal the relationship which exists between the distribution of relaxation times and the singularity structure of systems. The discrete distribution of relaxation times is invariant with respect to the sampling interval of the distribution spectrum due to the self-similar property of fractals. The sampling interval in the relaxation time domain is synonymous with the structure scale of the fractal system function and is determined according to the view scale chosen by the observer. The concepts of structure scale and view scale will be introduced in the third part of this series, where we will discuss the self-similarity property of the fractal relaxation systems.

Metrics

5 Record Views
2 citations in Scopus

Details

Logo image