A novel nonlinear vibration reduction mechanism based on targeted energy transfer (TET) is proposed. Targeted energy transfer is a physical phenomenon that describes a one-way irreversible energy flow from a linear oscillator (LO) to a nonlinearizable (essentially) nonlinear auxiliary substructure, noted as nonlinear energy sink (NES). The optimal targeted energy transfer where NES is set on the optimal state is investigated in this study. Complexification-averaging methodology is used to derive the optimal TET of the undamped system for different initial conditions. It is revealed that the optimal TET is dependent on the energy, indicating that passive control of NES cannot be optimally set for arbitrary initial conditions. In addition, it is found that for the undamped system, the optimal phrase difference between the linear primary oscillator and the nonlinear attachment is π/2. From the perspective of active control, the NES can be taken as an actuator to keep the system vibrating on the optimal TET. An available modification form of the optimal equations is proposed for the impulse excitation with relatively small damping. The comparisons of the effectiveness of the optimal TET is validated by using numerical simulations with the excitations including impulse, harmonic, to input with sufficient bandwidth, and random signal. The design procedure would pave the way for practical implications of TET in active vibration control.
Issue Section:
Research Papers
References
1.
Vakakis
, A. F.
, Gendelman
, O. V.
, Bergman
, L. A.
, McFarland
, D. M.
, Kerschen
, G.
, and Lee
, Y. S.
, 2008
, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems
, Vol. 156, Springer Science and Business Media
, Dordrecht
, The Netherlands
.2.
Gendelman
, O.
, Manevitch
, L. I.
, Vakakis
, A. F.
, and M'Closkey
, R.
, 2001
, “Energy Pumping in Nonlinear Mechanical Oscillators—Part I: Dynamics of the Underlying Hamiltonian Systems
,” ASME J. Appl. Mech.
, 68
(1
), pp. 34
–41
.3.
Vakakis
, A. F.
, and Gendelman
, O.
, 2001
, “Energy Pumping in Nonlinear Mechanical Oscillators—Part II: Resonance Capture
,” ASME J. Appl. Mech.
, 68
(1
), pp. 42
–48
.4.
Lee
, Y. S.
, Kerschen
, G.
, Vakakis
, A. F.
, Panagopoulos
, P.
, Bergman
, L.
, and McFarland
, D. M.
, 2005
, “Complicated Dynamics of a Linear Oscillator With a Light, Essentially Nonlinear Attachment
,” Phys. D: Nonlinear Phenom.
, 204
(1
), pp. 41
–69
.5.
Vaurigaud
, B.
, Savadkoohi
, A. T.
, and Lamarque
, C. H.
, 2011
, “Targeted Energy Transfer With Parallel Nonlinear Energy Sinks—Part I: Design Theory and Numerical Results
,” Nonlinear Dyn.
, 66
(4
), pp. 763
–780
.6.
Savadkoohi
, A. T.
, Vaurigaud
, B.
, Lamarque
, C. H.
, and Pernot
, S.
, 2012
, “Targeted Energy Transfer With Parallel Nonlinear Energy Sinks—Part II: Theory and Experiments
,” Nonlinear Dyn.
, 67
(1
), pp. 37
–46
.7.
Gendelman
, O. V.
, 2008
, “Targeted Energy Transfer in Systems With Nonpolynomial Nonlinearity
,” J. Sound Vib.
, 315
(3
), pp. 732
–745
.8.
Quinn
, D. D.
, Gendelman
, O.
, Kerschen
, G.
, Sapsis
, T. P.
, Bergman
, L. A.
, and Vakakis
, A. F.
, 2008
, “Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated With 1:1 Resonance Captures—Part I
,” J. Sound Vib.
, 311
(3
), pp. 1228
–1248
.9.
Sapsis
, T. P.
, Vakakis
, A. F.
, Gendelman
, O. V.
, Bergman
, L. A.
, Kerschen
, G.
, and Quinn
, D. D.
, 2009
, “Efficiency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated With 1:1 Resonance Captures—Part II: Analytical Study
,” J. Sound Vib.
, 325
(1
), pp. 297
–320
.10.
Jiang
, X.
, McFarland
, D. M.
, Bergman
, L. A.
, and Vakakis
, A. F.
, 2003
, “Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results
,” Nonlinear Dyn.
, 33
(1
), pp. 87
–102
.11.
Gendelman
, O. V.
, Gourdon
, E.
, and Lamarque
, C. H.
, 2006
, “Quasi-Periodic Energy Pumping in Coupled Oscillators Under Periodic Forcing
,” J. Sound Vib.
, 294
(4
), pp. 651
–662
.12.
Gendelman
, O. V.
, and Starosvetsky
, Y.
, 2007
, “Quasi-Periodic Response Regimes of Linear Oscillator Coupled to Nonlinear Energy Sink Under Periodic Forcing
,” ASME J. Appl. Mech.
, 74
(2
), pp. 325
–331
.13.
Gendelman
, O. V.
, Starosvetsky
, Y.
, and Feldman
, M.
, 2008
, “Attractors of Harmonically Forced Linear Oscillator With Attached Nonlinear Energy Sink—I: Description of Response Regimes
,” Nonlinear Dyn.
, 51
(1–2
), pp. 31
–46
.14.
Starosvetsky
, Y.
, and Gendelman
, O. V.
, 2008
, “Attractors of Harmonically Forced Linear Oscillator With Attached Nonlinear Energy Sink—II: Optimization of a Nonlinear Vibration Absorber
,” Nonlinear Dyn.
, 51
(1–2
), pp. 47
–57
.15.
Manevitch
, L. I.
, Gourdon
, E.
, and Lamarque
, C. H.
, 2007
, “Parameters Optimization for Energy Pumping in Strongly Nonhomogeneous 2dof System
,” Chaos Solitons Fractals
, 31
(4
), pp. 900
–911
.16.
Gourc
, E.
, Michon
, G.
, Seguy
, S.
, and Berlioz
, A.
, 2014
, “Experimental Investigation and Design Optimization of Targeted Energy Transfer Under Periodic Forcing
,” ASME J. Vib. Acoust.
, 136
(2
), p. 021021
.17.
AL-Shudeifat
, M. A.
, Vakakis
, A. F.
, and Bergman
, L. A.
, 2015
, “Shock Mitigation by Means of Low- to High-Frequency Nonlinear Targeted Energy Transfers in a Large-Scale Structure
,” ASME J. Comput. Nonlinear Dyn.
, 11
(2
), p. 021006
.18.
Detroux
, T.
, Habib
, G.
, Masset
, L.
, and Kerschen
, G.
, 2015
, “Performance, Robustness and Sensitivity Analysis of the Nonlinear Tuned Vibration Absorber
,” Mech. Syst. Signal Process.
, 60
, pp. 799
–809
.19.
Wang
, J.
, Wierschem
, N.
, Spencer
, B. F.
, and Lu
, X.
, 2015
, “Experimental Study of Track Nonlinear Energy Sinks for Dynamic Response Reduction
,” Eng. Struct.
, 94
, pp. 9
–15
.20.
Xiong
, H.
, Kong
, X.
, Yang
, Z.
, and Liu
, Y.
, 2015
, “Response Regimes of Narrow-Band Stochastic Excited Linear Oscillator Coupled to Nonlinear Energy Sink
,” Chin. J. Aeronaut.
, 28
(2
), pp. 457
–468
.21.
Manevitch
, L. I.
, 1999
, “Complex Representation of Dynamics of Coupled Nonlinear Oscillators
,” Mathematical Models of Nonlinear Excitations, Transfer, Dynamics, and Control in Condensed Systems and Other Media
, Springer
, New York
, pp. 269
–300
.22.
Manevitch
, L. I.
, Musienko
, A. I.
, and Lamarque
, C. H.
, 2007
, “New Analytical Approach to Energy Pumping Problem in Strongly Nonhomogeneous 2dof Systems
,” Meccanica
, 42
(1
), pp. 77
–83
.23.
Vakakis
, A. F.
, 2001
, “Inducing Passive Nonlinear Energy Sinks in Vibrating Systems
,” ASME J. Vib. Acoust.
, 123
(3
), pp. 324
–332
.24.
Vakakis
, A. F.
, McFarland
, D. M.
, Bergman
, L.
, Manevitch
, L. I.
, and Gendelman
, O.
, 2004
, “Isolated Resonance Captures and Resonance Capture Cascades Leading to Single- or Multi-Mode Passive Energy Pumping in Damped Coupled Oscillators
,” ASME J. Vib. Acoust.
, 126
(2
), pp. 235
–244
.25.
AL-Shudeifat
, M. A.
, 2015
, “Asymmetric Magnet-Based Nonlinear Energy Sink
,” ASME J. Comput. Nonlinear Dyn.
, 10
(1
), p. 014502
.26.
Guo
, C.
, AL-Shudeifat
, M. A.
, Vakakis
, A. F.
, Bergman
, L. A.
, McFarland
, D. M.
, and Yan
, J.
, 2015
, “Vibration Reduction in Unbalanced Hollow Rotor Systems With Nonlinear Energy Sinks
,” Nonlinear Dyn.
, 79
(1
), pp. 527
–538
.27.
Jiang
, X.
, McFarland
, D. M.
, Bergman
, L. A.
, and Vakakis
, A. F.
, 2003
, “Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results
,” Nonlinear Dyn.
, 33
(1
), pp. 87
–102
.28.
Habib
, G.
, Detroux
, T.
, Viguié
, R.
, and Kerschen
, G.
, 2015
, “Nonlinear Generalization of Den Hartog's Equal-Peak Method
,” Mech. Syst. Signal Process.
, 52–53
, pp. 17
–28
.29.
Luongo
, A.
, and Zulli
, D.
, 2015
, “Nonlinear Energy Sink to Control Elastic Strings: The Internal Resonance Case
,” Nonlinear Dyn.
, 81
(1
), pp. 1
–11
.30.
Blanchard
, A.
, McFarland
, D. M.
, Bergman
, L. A.
, and Vakakis
, A. F.
, 2015
, “Damping-Induced Interplay Between Vibrations and Waves in a Forced Non-Dispersive Elastic Continuum With Asymmetrically Placed Local Attachments
,” Proc. R. Soc. London, Ser. A
, 471
(2176
), p. 20140402
.31.
Shao
, J.
, Wu
, X.
, and Cochelin
, B.
, 2014
, “Targeted Energy Transfer in Two Degrees-of-Freedom Linear System Coupled by One Nonlinear Absorber
,” 21st International Congress on Sound and Vibration
, ICSV21
, Beijing, China
, July 13–17.32.
Vakakis
, A.
, Hasan
, A. M.
, Starosvetsky
, Y.
, and Manevitch
, L. I.
, 2013
, “Energy Transfers in Coupled Ordered Granular Chains With No Precompression
,” APS Meeting Abstracts
, Vol. 1
, p. 29010
.33.
Vakakis
, A.
, Bergman
, L. A.
, McFarland
, D. M.
, Lee
, Y. S.
, and Kurt
, M.
, 2011
, “Current Efforts Towards a Non-Linear System Identification Methodology of Broad Applicability
,” Proc. Inst. Mech. Eng., Part C
, 225
(11
), pp. 2497
–2515
.34.
Kerschen
, G.
, Gendelman
, O.
, Vakakis
, A. F.
, Bergman
, L. A.
, and McFarland
, D. M.
, 2008
, “Impulsive Periodic and Quasi-Periodic Orbits of Coupled Oscillators With Essential Stiffness Nonlinearity
,” Commun. Nonlinear Sci. Numer. Simul.
, 13
(5
), pp. 959
–978
.35.
Gendelman
, O. V.
, Sapsis
, T.
, Vakakis
, A. F.
, and Bergman
, L. A.
, 2011
, “Enhanced Passive Targeted Energy Transfer in Strongly Nonlinear Mechanical Oscillators
,” J. Sound Vib.
, 330
(1
), pp. 1
–8
.36.
Nucera
, F.
, Vakakis
, A. F.
, McFarland
, D. M.
, Bergman
, L. A.
, and Kerschen
, G.
, 2007
, “Targeted Energy Transfers in Vibro-Impact Oscillators for Seismic Mitigation
,” Nonlinear Dyn.
, 50
(3
), pp. 651
–677
.37.
Hubbard
, S. A.
, McFarland
, D. M.
, Bergman
, L. A.
, and Vakakis
, A. F.
, 2010
, “Targeted Energy Transfer Between a Model Flexible Wing and Nonlinear Energy Sink
,” J. Aircr.
, 47
(6
), pp. 1918
–1931
.38.
McFarland
, D. M.
, Kerschen
, G.
, Kowtko
, J. J.
, Lee
, Y. S.
, Bergman
, L. A.
, and Vakakis
, A. F.
, 2005
, “Experimental Investigation of Targeted Energy Transfers in Strongly and Nonlinearly Coupled Oscillators
,” J. Acoust. Soc. Am.
, 118
(2
), pp. 791
–799
.39.
Vakakis
, A. F.
, AL-Shudeifat
, M. A.
, and Hasan
, M. A.
, 2014
, “Interactions of Propagating Waves in a One-Dimensional Chain of Linear Oscillators With a Strongly Nonlinear Local Attachment
,” Meccanica
, 49
(10
), pp. 2375
–2397
.Copyright © 2017 by ASME
You do not currently have access to this content.
