0
Research Papers

Entropy for Strongly Coupled Oscillators

[+] Author and Article Information
Dante A. Tufano

M.A.N.E. Department,
Rensselaer Polytechnic Institute,
Troy, NY 12180
e-mail: tufand@rpi.edu

Zahra Sotoudeh

Department of Aerospace Engineering,
California State Polytechnic Institute,
Pomona, CA 91768
e-mail: zahra.vpi@gmail.com

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 22, 2017; final manuscript received May 24, 2017; published online August 17, 2017. Assoc. Editor: Nicole Kessissoglou.

J. Vib. Acoust 140(1), 011003 (Aug 17, 2017) (8 pages) Paper No: VIB-17-1170; doi: 10.1115/1.4037142 History: Received April 22, 2017; Revised May 24, 2017

This paper examines an approach for determining the entropy of coupled oscillators that does not rely on the assumption of weak coupling. The results of this approach are compared to the results for a weakly coupled system. It is shown that the results from each methodology agree in the case of weak coupling, and that a correction term is required for moderate to strong coupling. The correction term is shown to be related to the mixed energy term from the coupling spring as well as the geometry and stiffness of the system. Numerical simulations are performed for a symmetric system of identical coupled oscillators and an asymmetric system of nonidentical oscillators to demonstrate these findings.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Topics: Entropy
Your Session has timed out. Please sign back in to continue.

References

Lyon, R. H. , and Maidanik, G. , 1962, “ Power Flow Between Linearly Coupled Oscillators,” J. Acoust. Soc. Am., 34(5), pp. 623–639. [CrossRef]
Lyon, R. H. , 1995, Theory and Application of Statistical Energy Analysis, MIT Press, Cambridge, MA.
Tufano, D. , and Sotoudeh, Z. , 2016, “ Overview of Coupling Loss Factors for Damped and Undamped Simple Oscillators,” J. Sound Vib., 372, pp. 223–238. [CrossRef]
Carcaterra, A. , 2002, “ An Entropy Formulation for the Analysis of Energy Flow Between Mechanical Resonators,” Mech. Syst. Signal Process., 16(5), pp. 905–920. [CrossRef]
Tufano, D. A. , and Sotoudeh, Z. , 2015, “ Introducing Entropy for the Statistical Energy Analysis of an Artificially Damped Oscillator,” ASME Paper No. IMECE2015-50591.
Tufano, D. , and Sotoudeh, Z. , 2016, “ Exploring Entropy for Continuous Systems,” AIAA Paper No. 2016-2174.
Carcaterra, A. , 2014, “ Thermodynamic Temperature in Linear and Nonlinear Hamiltonian Systems,” Int. J. Eng. Sci., 80, pp. 189–208. [CrossRef]
Le Bot, A. , 2009, “ Entropy in Statistical Energy Analysis,” J. Acoust. Soc. Am., 125(3), pp. 1473–1478. [CrossRef] [PubMed]
Le Bot, A. , 2011, “ Statistical Energy Analysis and the Second Principle of Thermodynamics,” IUTAM Symposium on the Vibration Analysis of Structures With Uncertainties, St. Petersburg, Russia, July 5–9, pp. 129–139. https://doi.org/10.1007/978-94-007-0289-9_10
Newland, D. , 1968, “ Power Flow Between a Class of Coupled Oscillators,” J. Acoust. Soc. Am., 43(3), pp. 553–559. [CrossRef]
Langley, R. S. , and Cotoni, V. , 2004, “ Response Variance Prediction in the Statistical Energy Analysis of Built-up Systems,” J. Acoust. Soc. Am., 115(2), pp. 706–718. [CrossRef] [PubMed]
Lafont, T. , Totaro, N. , and Le Bot, A. , 2014, “ Review of Statistical Energy Analysis Hypotheses in Vibroacoustics,” Proc. R. Soc. A: Math., Phys. Eng. Sci., 470(2162), p. 20130515. [CrossRef]
Davies, H. G. , 1973, “ Random Vibration of Distributed Systems Strongly Coupled at Discrete Points,” J. Acoust. Soc. Am., 54(2), pp. 507–515. [CrossRef]
Woodhouse, J. , 1981, “ An Approach to the Theoretical Background of Statistical Energy Analysis Applied to Structural Vibration,” J. Acoust. Soc. Am., 69(6), pp. 1695–1709. [CrossRef]
Chandiramani, K. , 1978, “ Some Simple Models Describing the Transition From Weak to Strong Coupling in Statistical Energy Analysis,” J. Acoust. Soc. Am., 63(4), pp. 1081–1083. [CrossRef]
Hodges, C. , and Woodhouse, J. , 1986, “ Theories of Noise and Vibration Transmission in Complex Structures,” Rep. Prog. Phys., 49(2), p. 107. [CrossRef]
Mace, B. , 2003, “ Statistical Energy Analysis, Energy Distribution Models and System Modes,” J. Sound Vib., 264(2), pp. 391–409. [CrossRef]
Mace, B. , and Ji, L. , 2007, “ The Statistical Energy Analysis of Coupled Sets of Oscillators,” Proc. R. Soc. A: Math., Phys. Eng. Sci., 463(2081), pp. 1359–1377. [CrossRef]
Legault, J. , Woodhouse, J. , and Langley, R. , 2014, “ Statistical Energy Analysis of Inhomogeneous Systems With Slowly Varying Properties,” J. Sound Vib., 333(26), pp. 7216–7232. [CrossRef]
Tufano, D. , and Sotoudeh, Z. , 2017, “ Entropy for Nonconservative Vibrating Systems,” AIAA Paper No. 2017-0409.
Tufano, D. , and Sotoudeh, Z. , 2017, “ Entropy for Nonlinear Oscillators,” AIAA Paper No. 2017-0410.
Le Bot, A. , Carcaterra, A. , and Mazuyer, D. , 2010, “ Statistical Vibroacoustics and Entropy Concept,” Entropy, 12(12), pp. 2418–2435. [CrossRef]
Tufano, D. , and Sotoudeh, Z. , 2017, “ Exploring the Entropy Concept for Coupled Oscillators,” Int. J. Eng. Sci., 112, pp. 18–31. [CrossRef]
Tufano, D. , 2017, “ Entropy for Mechanically Vibrating Sytems,” Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, NY.
Khinchin, A. I. , 1949, Mathematical Foundations of Statistical Mechanics, Courier Corporation, North Chelmsford, MA.
Hannah, J. , 1996, “ A Geometric Approach to Determinants,” Am. Math. Mon., 103(5), pp. 401–409. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

System composed of two coupled oscillators with spring stiffnesses k1, k2 and masses m1, m2, coupled by a spring of stiffness kc

Grahic Jump Location
Fig. 2

Correction term, h(E,Ê), plotted over time for the normalized coupling stiffnesses ε=0.01,ε=0.1,ε=0.5, and ε=1.0

Grahic Jump Location
Fig. 3

The exact system entropy, H(d), compared to the composite method estimate, H(c), and the correction term, h, for ε=0.1

Grahic Jump Location
Fig. 4

The exact system entropy, H(d), compared to the composite method estimate, H(c), and the correction term, h, for ε = 1

Grahic Jump Location
Fig. 5

hE plotted over time for ε=0.01,ε=0.1,ε=0.5, and ε=1.0

Grahic Jump Location
Fig. 6

hs plotted against ε

Grahic Jump Location
Fig. 7

System composed of two nonidentical-coupled oscillators with spring stiffnesses k1≠0,k2=0, and masses m1, m2, coupled by a spring of stiffness kc

Grahic Jump Location
Fig. 8

Correction term, h(E,Ê) plotted over time for the normalized coupling stiffnesses ε=0.01,ε=0.1,ε=0.5, and ε=1.0

Grahic Jump Location
Fig. 9

The exact system entropy, H(d), compared to the composite method estimate, H(c), and the correction term, h, for ε=0.01

Grahic Jump Location
Fig. 10

The exact system entropy, H(d), compared to the composite method estimate, H(c), and the correction term, h, for ε=0.1

Grahic Jump Location
Fig. 11

hE, the component of the correction term relating to the energies E and Ê, over time for ε=0.01,ε=0.1,ε=0.5, and ε=1.0

Grahic Jump Location
Fig. 12

hs, the component of the correction term relating to the system parameters, plotted against ε

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In