Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems

Abstract

This paper discusses a class of unexpected irreversible phenomena that can develop in linear conservative systems and provides a theoretical foundation that explains the underlying principles. Recent studies have shown that energy can be introduced to a linear system with near irreversibility, or energy within a system can migrate to a subsystem nearly irreversibly, even in the absence of dissipation, provided that the system has a particular natural frequency distribution. The present work introduces a general theory that provides a mathematical foundation and a physical explanation for the near irreversibility phenomena observed and reported in previous publications. Inspired by the properties of probability distribution functions, the general formulation developed here is based on particular properties of harmonic series, which form the-common basis of linear dynamic system models. The results demonstrate the existence of a special class of linear nondissipative dynamic systems that exhibit nearly irreversible energy exchange and possess a decaying impulse response. In addition to uncovering a new class of dynamic system properties, the results have far-reaching implications in engineering applications where classical vibration damping or absorption techniques may not be effective. Furthermore, the results also support the notion of nearly irreversible energy transfer in conservative linear systems, which until now has been a concept associated exclusively with nonlinear systerns.(c) 2007 Acoustical Society of America

Vibration absorption using non-dissipative complex attachments with impacts and parametric stiffness

“…For both cases, numerical results show very good energy absorption capability of the attachments introduced in this paper. They may be considered as alternatives to selecting special frequency distributions within the attachment 10 to produce a nearirreversible energy transfer from the master to an attached set of linear oscillators. Of note, the results obtained with the techniques described here are not significantly sensitive to the particular frequency distribution within the attachment.…”

“…7 the intrinsic properties of attachments are identified, which control the speed of energy sharing between a master and the attachment and the time the energy takes to be transferred back to the master. Several studies examined the conditions that, even in the absence of energy dissipation, prevent energy transport back to the master, which lead to the so called near-irreversibility condition [8][9][10] also confirmed by experimental tests. 11 Finally, the problem of an efficient design of a multi-degrees of freedom tuned-mass-damper has been also considered in the context of control theory.…”

“…10 it is shown how a damping effect on the a͒ Author to whom correspondence should be addressed. Electronic mail: a.carcaterra@dma.ing.uniroma1.it master develops due to the attachment only when the uncoupled natural frequency of the master belongs to the interval B described by the natural frequencies of the oscillators within the attachment.…”

From fractal media to continuum mechanics

“…(ii) By looking for physical systems which can be modelled at a microscopic level as first gradient continua and become higher gradient models after the dimensional regularization procedure has been applied [74][75][76].…”

Entropy in sound and vibration: towards a new paradigm

“…Several examples of exchange of entropy in mechanical oscillators or continuous systems have been studied in the literature [11,12]. Some of them even show an apparent irreversible behaviour [13,14].…”