In this paper we consider two coupled oscillators which exhibit energy transfer from master ‘linear’ degrees of freedom submitted to resonant external excitation with two different harmonics to a slave ‘nonlinear’ energy sink (NES). By endowing the idea of the relative mode, splitting harmonics and iterative techniques, we prove that control of the two one-to-one resonances of the system is possible simultaneously; moreover, the iterative technique gives us a better outlook over effects of the NES parameters on the overall behavior of the system. Then, analytical procedure for multi-resonance capture are explained and commented on. Finally, the obtained analytical results are compared with numerical evidence.
De SilvaCW (2005) Vibration and Shock Handbook. Boca Raton, FL: CRC Press.
2.
JiangXMcFarlandMBergmanLAVakakisAF (2003) Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dynamics33: 7–102.
3.
GendelmanOV (2004) Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dynamics37: 115–128.
4.
GendelmanOVGourdonELamarqueC-H (2006) Quasiperiodic energy pumping in coupled oscillators under periodic forcing. Journal of Sound and Vibration294: 651–662.
5.
GendelmanOVLamarqueC-H (2005) Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium. Chaos, Solitons and Fractals24: 501–509.
6.
GendelmanOVManevitchLIVakakisAFM'CloskeyR (2001) Energy pumping in nonlinear mechanical oscillators I: dynamics of the underlying Hamiltonian systems. ASME Journal of Applied Mechanics68: 34–41.
7.
GendelmanOVStarosvetskyY (2007) Quasiperiodic response regimes of linear oscillator coupled to nonlinear energy sink under periodic forcing. Journal of Applied Mechanics74: 325–331.
8.
GourdonE (2006) Contrôle passif de vibrations par pompage énergétique. PhD Thesis, Ecole doctorale MEGA, Génie mécaniqueEcole Centrale de Lyon et ENTPE, 2006–2031.
9.
GourdonEAlexanderNATaylorCALamarqueC-HPernotS (2007a) Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. Journal of Sound and Vibration300: 522–551.
10.
GourdonELamarqueC-HPernotS (2007b) Contribution to Efficiency of irreversible passive energy pumping with a strong nonlinear attachment. Nonlinear Dynamics50: 793–808.
11.
ManevitchLI (1999) Complex representation of dynamics of coupled oscillators. In: Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed SystemsNew York: Kluwer Academic/Plenum Publishers, 269–300.
12.
ManevitchLIGourdonELamarqueC-H (2007a) Parameters optimization for energy pumping in strongly nonhomogeneous 2 DoF system. Chaos, Solitons and Fractals31: 900–911.
13.
ManevitchLIGourdonELamarqueC-H (2007b) Towards the design of an optimal energetic sink in a strongly inhomogeneous two-degree-of-freedom system. Transactions of the ASME74: 1078–1086.
14.
ManevitchLIManevitchEL (2009) Limiting phase trajectories (LPT) and resonance in a strognly asymmetric 2DOF system. In: Proceedings of the 10th Conference on Dynamical Systems, Theory and Applications, Lódź, Poland, 7–10 December 2009.
15.
ManevitchLIMusienkoAILamarqueC-H (2007c) New analytical approach to energy pumping problem in strongly non homogeneous 2 DoF systems. Meccanica42: 77–83.
16.
McFarlandDBergmanLVakakisAF (2005) Experimental study of non-linear energy pumping occurring at a single fast frequency. International Journal of Non-Linear Mechanics40: 891–899.
17.
MusienkoAILamarqueC-HManevitchLI (2006) Design of mechanical energy pumping devices. Journal of Vibration and Control12: 355–371.
18.
PhamTTLamarqueC-HPernotS (2010a) Passive control of one degree of freedom nonlinear quadratic oscillator under combination resonance. Communication in Nonlinear Science and Numerical Simulation, doi: 10.1016/j.cnsns.2010.04.050.
19.
PhamTTLamarqueC-HTure SavadkoohiAPernotS (2010b) Multiple resonance capture of a compound system based on the definition of relative modes under harmonic excitation. In: IMPACT 2010, Dynamic of Systems, Materials and StructuresDjerba, Tunisia, 22–24 March.
20.
StarosvetskyYGendelmanOV (2008a) Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning. Journal of Sound and Vibration315: 746–765.
21.
StarosvetskyYGendelmanOV (2008b) Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry. Physica D237: 1719–1733.
22.
Ture SavadkoohiAPernotSLamarqueC-H (2009) Targeted energy transfer with several NES in parallel: theory and experiments In: Proceedings of the 2009 ASME-International Design Engineering Technical Conference and Computers and Information in Engineering Conference IDETC/CIE 2009California. DETC2009-86260, California, USA.
23.
VakakisAF (2001) Inducing passive nonlinear energy sinks in vibrating systems. ASME Journal of Vibration and Acoustics123: 324–332.
24.
VakakisAF (2003) Shock isolation through the use of nonlinear energy sinks. Journal of Vibration and Control12: 355–371.
25.
VakakisAFGendelmanOV (2001) Energy pumping in nonlinear mechanical oscillators II: resonance capture. ASME Journal of Applied Mechanics68: 42–48.
26.
VakakisAFGendelmanOVBergmanLAMcFarlandDMKerschenGSup LeeY (2009a) Nonlinear Trageted Energy Transfer in Mechanical and Structural Systems I, Berlin: Springer.
27.
VakakisAFGendelmanOVBergmanLAMcFarlandDMKerschenGSup LeeY (2009b) Nonlinear Trageted Energy Transfer in Mechanical and Structural Systems II, Berlin: Springer.
