Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances
Abstract
The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing nonproportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slowflow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes andmore »
 Authors:

 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1559516
 Report Number(s):
 SAND20199484J
Journal ID: ISSN 0924090X; 678450; TRN: US2000356
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Nonlinear Dynamics
 Additional Journal Information:
 Journal Name: Nonlinear Dynamics; Journal ID: ISSN 0924090X
 Publisher:
 Springer
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING
Citation Formats
Mathis, Allen T., and Quinn, D. Dane. Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances. United States: N. p., 2019.
Web. doi:10.1007/s1107101905198w.
Mathis, Allen T., & Quinn, D. Dane. Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances. United States. doi:10.1007/s1107101905198w.
Mathis, Allen T., and Quinn, D. Dane. Fri .
"Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances". United States. doi:10.1007/s1107101905198w. https://www.osti.gov/servlets/purl/1559516.
@article{osti_1559516,
title = {Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances},
author = {Mathis, Allen T. and Quinn, D. Dane},
abstractNote = {The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing nonproportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slowflow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes and is proportional to the energy in the lowest mode. The results of the analysis tie together interpretations used by both experimentalists and theoreticians to study such systems and provide a more concrete way to interpret these phenomena.},
doi = {10.1007/s1107101905198w},
journal = {Nonlinear Dynamics},
number = ,
volume = ,
place = {United States},
year = {2019},
month = {8}
}
Web of Science
Works referenced in this record:
Computation of real normal modes from complex eigenvectors
journal, January 2008
 Fuellekrug, Ulrich
 Mechanical Systems and Signal Processing, Vol. 22, Issue 1
The Relationship Between the real and Imaginary Parts of Complex Modes
journal, April 1998
 Garvey, S. D.; Penny, J. E. T.; Friswell, M. I.
 Journal of Sound and Vibration, Vol. 212, Issue 1
A Note on Nonproportional Damping
journal, November 2009
 Udwadia, Firdaus E.
 Journal of Engineering Mechanics, Vol. 135, Issue 11
Nonlinear ModeCoupling in Nanomechanical Systems
journal, March 2013
 Matheny, M. H.; Villanueva, L. G.; Karabalin, R. B.
 Nano Letters, Vol. 13, Issue 4
On the coupling of nonlinear normal modes
journal, June 2006
 Pak, C. H.
 International Journal of NonLinear Mechanics, Vol. 41, Issue 5
Bifurcation of coupledmode responses by modal coupling in cubic nonlinear systems
journal, December 2015
 Pak, C. H.; Lee, Young S.
 Quarterly of Applied Mathematics, Vol. 74, Issue 1
Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays
journal, April 2003
 Lifshitz, Ron; Cross, M. C.
 Physical Review B, Vol. 67, Issue 13
Sustained highfrequency energy harvesting through a strongly nonlinear electromechanical system under single and repeated impulsive excitations
journal, July 2014
 Remick, Kevin; Joo, Han Kyul; McFarland, D. Michael
 Journal of Sound and Vibration, Vol. 333, Issue 14
Proportional damping approximation for structures with added viscoelastic dampers
journal, March 2006
 Bilbao, A.; Avilés, R.; Agirrebeitia, J.
 Finite Elements in Analysis and Design, Vol. 42, Issue 6
Nonlinear Coupling of Linearly Uncoupled Resonators
journal, January 2019
 Menotti, M.; Morrison, B.; Tan, K.
 Physical Review Letters, Vol. 122, Issue 1
Nonlinear dynamics and chaos in two coupled nanomechanical resonators
journal, April 2009
 Karabalin, R. B.; Cross, M. C.; Roukes, M. L.
 Physical Review B, Vol. 79, Issue 16
Nonlinear thermoacoustic mode synchronization in annular combustors
journal, January 2019
 Moeck, Jonas P.; Durox, Daniel; Schuller, Thierry
 Proceedings of the Combustion Institute, Vol. 37, Issue 4
Nonlinear modelbased estimation of quadratic and cubic damping mechanisms governing the dynamics of a chaotic spherical pendulum
journal, March 2011
 Gottlieb, O.; Habib, G.
 Journal of Vibration and Control, Vol. 18, Issue 4
Identification of NonProportional Modal Damping Matrix and real Normal Modes
journal, November 2002
 Kasai, T.; Link, M.
 Mechanical Systems and Signal Processing, Vol. 16, Issue 6
Energydependent path of dissipation in nanomechanical resonators
journal, May 2017
 Güttinger, Johannes; Noury, Adrien; Weber, Peter
 Nature Nanotechnology, Vol. 12, Issue 7
The influence of an internal resonance on nonlinear structural vibrations under subharmonic resonance conditions
journal, October 1985
 Mook, D. T.; Plaut, R. H.; HaQuang, N.
 Journal of Sound and Vibration, Vol. 102, Issue 4
Nonlinear dynamics of two harmonic oscillators coupled by Rayleigh type selfexciting force
journal, December 2012
 Chatterjee, S.; Dey, Somnath
 Nonlinear Dynamics, Vol. 72, Issue 12
Nonlinear Oscillations
journal, September 1962
 Minorsky, Nicholas; Teichmann, T.
 Physics Today, Vol. 15, Issue 9