scholarly journals Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 536 ◽  
Author(s):  
Zahra Sotoudeh
Keyword(s):  
Nonlinear Systems ◽  
Linear Systems ◽  
Power Flow ◽  
Energy Analysis ◽  
Vibrational Temperature ◽  
Statistical Energy Analysis ◽  
Phase Volume ◽  
Weakly Nonlinear ◽  
Mixing Entropy ◽  
Energy Behavior

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.

Power Flow and Energy Sharing between an Oscillator and a Continuous Receiver Structure

2011 ◽  
Vol 189-193 ◽  
pp. 1914-1917
Author(s):  
Lin Ji
Keyword(s):  
High Frequency ◽  
Power Flow ◽  
Energy Analysis ◽  
Statistical Energy Analysis ◽  
Coupled Subsystems ◽  
Modal Density ◽  
Vibration Problems ◽  
Energy Sharing ◽  
High Modal Density ◽  
Vibration Modeling

A key assumption of conventional Statistical Energy Analysis (SEA) theory is that, for two coupled subsystems, the transmitted power from one to another is proportional to the energy differences between the mode pairs of the two subsystems. Previous research has shown that such an assumption remains valid if each individual subsystem is of high modal density. This thus limits the successful applications of SEA theory mostly to the regime of high frequency vibration modeling. This paper argues that, under certain coupling conditions, conventional SEA can be extended to solve the mid-frequency vibration problems where systems may consist of both mode-dense and mode-spare subsystems, e.g. ribbed-plates.

Mechanical Power Flow Between Stiffened Plates and Viscoelastic Discrete Systems

1986 ◽  
Vol 108 (2) ◽  
pp. 155-164 ◽  
Author(s):  
E. Goldfracht ◽  
G. Rosenhouse
Keyword(s):  
Power Flow ◽  
Energy Analysis ◽  
Power Transmission ◽  
Discrete Systems ◽  
Vibration Energy ◽  
Stiffened Plates ◽  
Statistical Energy Analysis ◽  
Frequency Range ◽  
Lower Frequency Range ◽  
Fundamental Theorems

In this paper we primarily discuss a theory of power transmission and vibration energy distribution of dynamically loaded structures. The loads are random and the system comprises linked elements, which consist of machine-supported stiffened plates. Fundamentally, the theory is deterministic, but in addition it uses some features of the SEA. In fact, the analysis is intended to verify fundamental theorems of the Statistical Energy Analysis in the lower frequency range.

Statistical Energy Analysis for the Time-Integrated Transient Response of Vibrating Systems

1990 ◽  
Vol 112 (2) ◽  
pp. 206-213 ◽  
Author(s):  
M. L. Lai ◽  
A. Soom
Keyword(s):  
Steady State ◽  
Energy Balance ◽  
Transient Response ◽  
Power Flow ◽  
Energy Analysis ◽  
Statistical Energy Analysis ◽  
Balance Equations ◽  
Integrated Response ◽  
Energy Relations ◽  
Loss Factors

For more than twenty years, statistical energy analysis (SEA) has been used for the analysis of steady-state response distributions in complex coupled structures and sound-structure systems. However, the steady-state SEA formalism is not directly applicable to the analysis of transient vibrations. In this paper, energy relations, analogous to steady-state SEA power flow relations, are derived for the time-integrated transient response of each oscillator. These energy flow relations can be combined using statistical concepts, to obtain a set of energy balance equations for N coupled multimodal subsystems. It is shown that the time-integrated response of each subsystem can be described in terms of transient input energies and conventional SEA parameters, i.e., modal densities, loss factors and coupling loss factors. By solving the energy balance equations, the time-integrated response of each subsystem can be obtained. The results of experiments, conducted on a coupled structure consisting of two welded plates, are presented to illustrate the applicability of these relations.

Introducing Entropy for the Statistical Energy Analysis of an Artificially Damped Oscillator

2015 ◽  
Author(s):  
Dante A. Tufano ◽  
Zahra Sotoudeh
Keyword(s):  
Energy Analysis ◽  
Lagrangian Method ◽  
Statistical Energy Analysis ◽  
Artificial Damping ◽  
Energy Behavior ◽  
Resonator System ◽  
The Individual ◽  
Entropy Formulation ◽  
Damped Oscillator

The purpose of this paper is to introduce the concept of entropy for a main resonator attached to a “fuzzy structure”. This structure is described explicitly using the Lagrangian method, and is treated as a layer of discrete resonators. A generic entropy formulation is then developed for the layer of resonators, which is used to determine the individual oscillator entropies. The combined entropy of the linear resonator system is then determined and compared numerically to the sum of the individual oscillator entropies. The entropy behavior of the system is then related to the energy behavior of the system and explained in regards to the the “artificial damping” of the main resonator.

Prediction of insertion loss of lagging in rectangular duct using statistical energy analysis

2019 ◽  
Vol 67 (6) ◽  
pp. 438-446
Author(s):  
M. Yoganandh ◽  
Jade Nagaraja ◽  
B. Venkatesham
Keyword(s):  
Insertion Loss ◽  
Power Flow ◽  
Energy Analysis ◽  
Transmission Loss ◽  
Flow Analysis ◽  
Computational Cost ◽  
Statistical Energy Analysis ◽  
Rectangular Duct ◽  
Modal Density ◽  
Element Method

In this article, statistical energy analysis (SEA) is used to predict insertion loss from a lagged rectangular HVAC duct. For a lagged duct, all duct walls are treated from outside with acoustic material. Although deterministic methods like the finite element method (FEM), boundary element method (BEM), and wave based methods can predict the breakout noise, these methods have limitations in handling systems with high modal density due to higher computational cost. In this study, a rectangular duct is divided into six subsystems, which are four duct walls (each wall considered as a subsystem), internal air cavity and external airspace. Power flow analysis is performed on all subsystems to calculate transverse transmission loss of an unlined duct and insertion loss for a lagged duct. Predicted transverse transmission loss values are validated with ASHRAE data and Insertion loss values with literature. The results obtained are in good agreement.

scholarly journals On the Solution of a Class of Nonlinear Systems Governed by an -Matrix

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Woula Themistoclakis ◽  
Antonia Vecchio
Keyword(s):  
Fixed Point ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Global Convergence ◽  
Linear Systems ◽  
Real Function ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Weakly Nonlinear ◽  
Unknown Vector

We consider a weakly nonlinear system of the form , where is a real function of the unknown vector , and is an -matrix. We propose to solve it by means of a sequence of linear systems defined by the iteration procedure , . The global convergence is proved by considering a related fixed-point problem.

scholarly journals The Poincare method for a strongly nonlinear Duffing oscillator

2003 ◽  
Vol 25 (1) ◽  
pp. 19-25 ◽  
Author(s):  
Nguyen Van Dinh
Keyword(s):  
Nonlinear Systems ◽  
Linear Systems ◽  
Duffing Oscillator ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Non Linear ◽  
Non Linear Systems

It is well-known that the classical Poincare method is limited to weakly nonlinear systems and for extending the range of validity of this method to strongly non-linear systems, various modifications have been developed.

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