This paper further expands this approach by introducing complete derivation of entropy using the perturbation method for Duffing oscillators and Henon–Heiles oscillators. However, the accuracy of this approach has yet to be investigated numerically for nonlinear systems. This is expected to provide insight into restrictive assumptions of SEA and possibly expand the applicability of it.Ĭarcaterra has introduced Khinchin’s entropy for weakly nonlinear systems with (1) dead-zone nonlinearity and (2) polynomial anharmonic potentials. Using the entropy concept for vibrating structures may result in an alternative way of deriving the power flow law for nonlinear systems. Entropy approaches do not assume linearity, and consequently can be applied to nonlinear systems. The concept of entropy has been introduced for mechanical oscillators in order to improve SEA. Although in this paper we do not study the dynamic behavior of a nonlinear system, we are developing the mathematical requirement for expanding SEA for studying nonlinear systems. The response of the structure in the frequency domain is divided into frequency bands, and the energy flow between the frequency bands is calculated. suggests defining subsystems based on modal analysis instead of physical subsystems. studies the energy cascade phenomenon due to large deformations in coupled vibrating plates. It shows that although the steady-state response of the structure strongly depends on the structural nonlinearities, the basic power proportionality law of SEA is preserved. studies the coupling between a linear acoustic fluid and a nonlinear structure. It has been accepted that SEA is used for linear systems. To the best of our knowledge, there are not many papers devoted to nonlinear SEA. Efforts to examine SEA for nonlinear systems have been limited in scope. The classical SEA approach is valid under a number of limiting assumptions. In SEA, energy is assumed to flow from systems with higher average modal energies to systems with lower average modal energies. This principle is used to obtain the energy balance equations of the systems. These methods are based on the first law of thermodynamics, i.e., the principle of conservation of energy. In classical SEA, a power flow proportionality law is used to relate the ensemble average energies of individual structural or acoustic subsystems to the ensemble average energy flow rate between subsystems. SEA is a computationally inexpensive reduced order model that serves as an alternative to numerical approaches such as finite element analysis. Statistical energy analysis (SEA) is a method of predicting energy transmission in systems that are undergoing high-frequency vibration. Although this paper is motivated by statistical energy analysis application, it is not devoted to the statistical energy analysis of nonlinear systems. Vibrational temperature can be used to derive the power flow proportionality, which is the backbone of the statistical energy analysis. Using the thermodynamic relationship of temperature and entropy, and Khinchin’s entropy, one can define a vibrational temperature. The limitation of statistical energy analysis motivates this study. The mixing entropy provides insight into the energy behavior of each system. It is illustrated that mixing entropy is always non-negative. The mixing entropy is the result of the generation of entropy when two systems are combined to create a complex system. In an extension of the author’s previous work for linear systems, a mixing entropy is defined for these two oscillators. It is shown that such an approach is valid for weakly nonlinear systems. We use a perturbation method to do so, and verify the results against the numerical calculation of the phase volume. To calculate the Khinchin’s entropy, one needs to obtain an analytical expression of the system’s phase volume. Nonlinearities are modeled as nonlinear springs. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account for weak nonlinearities. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. We initially introduced the idea of entropy in Chapter 7 by saying that if a quantity of heat dQ is added to a system at temperature T, the entropy increases by dS = dQ/T.In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators.

Entropy of mixing