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 non-proportional 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 slow-flow 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 »
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1559516
- Report Number(s):
- SAND2019-9484J
Journal ID: ISSN 0924-090X; 678450; TRN: US2000356
- Grant/Contract Number:
- AC04-94AL85000
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Nonlinear Dynamics
- Additional Journal Information:
- Journal Name: Nonlinear Dynamics; Journal ID: ISSN 0924-090X
- 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/s11071-019-05198-w.
Mathis, Allen T., & Quinn, D. Dane. Transient dynamics, damping, and mode coupling of nonlinear systems with internal resonances. United States. doi:10.1007/s11071-019-05198-w.
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/s11071-019-05198-w. 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 non-proportional 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 slow-flow 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/s11071-019-05198-w},
journal = {Nonlinear Dynamics},
number = ,
volume = ,
place = {United States},
year = {2019},
month = {8}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 1 work
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
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 Mode-Coupling 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 non-linear normal modes
journal, June 2006
- Pak, C. H.
- International Journal of Non-Linear Mechanics, Vol. 41, Issue 5
Bifurcation of coupled-mode 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 high-frequency 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
Non-linear model-based 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 Non-Proportional Modal Damping Matrix and real Normal Modes
journal, November 2002
- Kasai, T.; Link, M.
- Mechanical Systems and Signal Processing, Vol. 16, Issue 6
Energy-dependent 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 non-linear 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 self-exciting force
journal, December 2012
- Chatterjee, S.; Dey, Somnath
- Nonlinear Dynamics, Vol. 72, Issue 1-2
Nonlinear Oscillations
journal, September 1962
- Minorsky, Nicholas; Teichmann, T.
- Physics Today, Vol. 15, Issue 9