Abstract

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.

Highlights

  • Statistical energy analysis (SEA) is a method of predicting energy transmission in systems that are undergoing high-frequency vibration

  • This paper further expands this approach by introducing complete derivation of entropy using the perturbation method for Duffing oscillators and Henon–Heiles oscillators

  • A perturbation method is used to calculate an explicit equation of phase volume

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Summary

Introduction

Statistical energy analysis (SEA) is a method of predicting energy transmission in systems that are undergoing high-frequency vibration. Using the entropy concept for vibrating structures may result in an alternative way of deriving the power flow law for nonlinear systems. This is expected to provide insight into restrictive assumptions of SEA and possibly expand the applicability of it. This paper further expands this approach by introducing complete derivation of entropy using the perturbation method for Duffing oscillators and Henon–Heiles oscillators. The entropy inequality property ensures that the entropy of a system must not decrease as a result of coupling with other systems This is shown by introducing the concept of a mixing entropy for nonlinear systems. The Henon–Heiles system contains a repulsive anharmonic potential, which causes instability if one of the oscillators has sufficient energy to escape from the origin

Motivation
Overview of Khinchin’s Entropy
Linear Systems
Nonlinear Systems
Mixing Entropy
A Duffing Oscillator
The Phase Volume
Khinchin’s Entropy
Coupled Duffing Oscillators
Entropy of the Coupled Duffing System
Mixing Entropy of the Coupled Duffing System
Entropy of Henon–Heiles Oscillators
Single Degree of Freedom Oscillator with Third Order Anharmonic Potential
Henon–Heiles Oscillators
Entropy of the Henon–Heiles Oscillators
Mixing Entropy of the Henon–Heiles Oscillators
Findings
Conclusions
Full Text
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