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Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems
In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account f...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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MDPI
2019
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7515025/ https://www.ncbi.nlm.nih.gov/pubmed/33267250 http://dx.doi.org/10.3390/e21050536 |
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author | Sotoudeh, Zahra |
author_facet | Sotoudeh, Zahra |
author_sort | Sotoudeh, Zahra |
collection | PubMed |
description | In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account for weak nonlinearities. Nonlinearities are modeled as nonlinear springs. To calculate the Khinchin’s entropy, one needs to obtain an analytical expression of the system’s phase volume. We use a perturbation method to do so, and verify the results against the numerical calculation of the phase volume. It is shown that such an approach is valid for weakly nonlinear systems. In an extension of the author’s previous work for linear systems, a mixing entropy is defined for these two oscillators. The mixing entropy is the result of the generation of entropy when two systems are combined to create a complex system. It is illustrated that mixing entropy is always non-negative. The mixing entropy provides insight into the energy behavior of each system. The limitation of statistical energy analysis motivates this study. Using the thermodynamic relationship of temperature and entropy, and Khinchin’s entropy, one can define a vibrational temperature. Vibrational temperature can be used to derive the power flow proportionality, which is the backbone of the statistical energy analysis. Although this paper is motivated by statistical energy analysis application, it is not devoted to the statistical energy analysis of nonlinear systems. |
format | Online Article Text |
id | pubmed-7515025 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2019 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
spelling | pubmed-75150252020-11-09 Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems Sotoudeh, Zahra Entropy (Basel) Article In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account for weak nonlinearities. Nonlinearities are modeled as nonlinear springs. To calculate the Khinchin’s entropy, one needs to obtain an analytical expression of the system’s phase volume. We use a perturbation method to do so, and verify the results against the numerical calculation of the phase volume. It is shown that such an approach is valid for weakly nonlinear systems. In an extension of the author’s previous work for linear systems, a mixing entropy is defined for these two oscillators. The mixing entropy is the result of the generation of entropy when two systems are combined to create a complex system. It is illustrated that mixing entropy is always non-negative. The mixing entropy provides insight into the energy behavior of each system. The limitation of statistical energy analysis motivates this study. Using the thermodynamic relationship of temperature and entropy, and Khinchin’s entropy, one can define a vibrational temperature. Vibrational temperature can be used to derive the power flow proportionality, which is the backbone of the statistical energy analysis. Although this paper is motivated by statistical energy analysis application, it is not devoted to the statistical energy analysis of nonlinear systems. MDPI 2019-05-26 /pmc/articles/PMC7515025/ /pubmed/33267250 http://dx.doi.org/10.3390/e21050536 Text en © 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Sotoudeh, Zahra Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems |
title | Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems |
title_full | Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems |
title_fullStr | Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems |
title_full_unstemmed | Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems |
title_short | Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems |
title_sort | entropy and mixing entropy for weakly nonlinear mechanical vibrating systems |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7515025/ https://www.ncbi.nlm.nih.gov/pubmed/33267250 http://dx.doi.org/10.3390/e21050536 |
work_keys_str_mv | AT sotoudehzahra entropyandmixingentropyforweaklynonlinearmechanicalvibratingsystems |