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Analysis of the transient behavior in a two dof nonlinear system. (English) Zbl 1225.37118

Summary: We analyze the behavior of a nonlinear system under impulse loadings. The system is composed of a master “linear” degree of freedom (dof) substructure which is attached to a slave “nonlinear” energy sink (NES) for the sake of control. The Melnikov integral is endowed in order to study the possibility of existence of chaos and transversal homoclinic orbits in the system. Then, the complexification method as an alternative to nonlinear normal modes is implemented to reveal the behavior of the system during the energy exchange between two oscillators. The non-smooth time transformation (NSTT) technique is implemented in order to enlighten the system behavior during its extremely nonlinear regime, meanwhile stable and unstable zones of the system during its quasi-linear regime are highlighted.

MSC:

37N35 Dynamical systems in control
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[1] De Silva, C. W., Vibration and shock handbook (2005), CRC Press · Zbl 1189.74001
[2] Vakakis, A. F., Inducing passive nonlinear energy sinks in vibrating systems, ASME J Vib Acoust, 123, 3, 324-332 (2001)
[3] Gendelman, O. V.; Manevitch, L. I.; Vakakis, A. F.; M’Closkey, R., Energy pumping in nonlinear mechanical oscillators. I: dynamics of the underlying hamiltonian systems, ASME J Appl Mech, 68, 1, 34-41 (2001) · Zbl 1110.74452
[4] Vakakis, A. F.; Gendelman, O. V., Energy pumping in nonlinear mechanical oscillators. II: resonance capture, ASME J Appl Mech, 68, 1, 42-48 (2001) · Zbl 1110.74725
[5] Gendelman, O. V., Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment, Nonlinear Dyn, 37, 2, 115-128 (2004) · Zbl 1081.70012
[6] Gendelman, O. V.; Lamarque, C.-H., Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium, Chaos Soliton Fract, 24, 2, 501-509 (2005) · Zbl 1135.70311
[7] Gendelman, O. V.; Gourdon, E.; Lamarque, C.-H., Quasiperiodic energy pumping in coupled oscillators under periodic forcing, J Sound Vib, 294, 4-5, 651-662 (2006)
[8] Manevitch, L. I.; Gourdon, E.; Lamarque, C.-H., Parameters optimization for energy pumping in strongly nonhomogeneous 2 DoF system, Chaos Soliton Fract, 31, 4, 900-911 (2007)
[9] Manevitch, L. I.; Musienko, A. I.; Lamarque, C.-H., New analytical approach to energy pumping problem in strongly non homogeneous 2 DoF Systems, Meccanica, 42, 1, 77-83 (2007) · Zbl 1162.70326
[10] Manevitch, L. I.; Gourdon, E.; Lamarque, C.-H., Towards the design of an optimal energetic sink in a strongly inhomogeneous two-degree-of- freedom system, Trans ASME, 74, 1078-1086 (2007)
[11] Manevitch, L. I., New approach to beating phenomenon in coupled nonlinear oscillatory chains, Archive Appl Mech, 77, 5, 301-312 (2007) · Zbl 1190.74013
[12] Vakakis, A. F.; Gendelman, O. V.; Bergman, L. A.; McFarland, D. M.; Kerschen, G.; Lee, Y. S., Nonlinear trageted energy transfer in mechanical and structural systems I, 375 (2009), Springer
[13] Vakakis, A. F.; Gendelman, O. V.; Bergman, L. A.; McFarland, D. M.; Kerschen, G.; Lee, Y. S., nonlinear trageted energy transfer in mechanical and structural systems II, 655 (2009), Springer
[14] Starosvetsky, Y.; Gendelman, O. V., Vibration absorption in systems with nonlinear energy sink: nonlinear damping, J Sound Vib, 324, 916-939 (2009)
[15] Starosvetsky, Y.; Gendelman, O. V., Interaction of nonlinear energy sink with a two degrees of freedom linear system: internal resonance, J Sound Vib, 329, 1836-1852 (2010) · Zbl 1189.70100
[16] Gendelman, O. V.; Sapsis, T.; Vakakis, A. F.; Bergman, L. A., Enhanced passive targeted energy transfer in strongly nonlinear mechanical oscillators, J Sound Vib (2010)
[17] Schmidt, F.; Lamarque, C.-H., Energy pumping for mechanical systems involving non-smooth Saint-Venant terms, Int J Non-Linear Mech, 45, 9, 866-875 (2010)
[18] Pham, T. T.; Lamarque, C.-H.; Pernot, S., Passive control of one degree of freedom nonlinear quadratic oscillator under combination resonance, Commun Nonlinear Sci Numer Simul (2010)
[19] McFarland, D. M.; Bergman, L.; Vakakis, A. F., Experimental study of non-linear energy pumping occurring at a single fast frequency, Int J Non-Linear Mech, 40, 891-899 (2005) · Zbl 1349.74011
[20] McFarland, D. M.; Kerschen, G.; Kowtko, J. J.; Lee, Y. S.; Bergman, L. A.; Vakakis, A. F., Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators, J Acoust Soc America, 118, 791-799 (2005)
[21] Kerschen, G.; Kowtko, J. J.; McFarland, D. M.; Bergman, L. A.; Vakakis, A. F., Theoretical and experimental study of multimodal targeted energy transfer in a system of coupled oscillators, Nonlinear Dyn, 47, 285-309 (2007) · Zbl 1177.70023
[22] Kerschen, G.; Kowtko, J. J.; McFarland, D. M.; Lee, Y. S.; Bergman, L. A.; Vakakis, A. F., Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity, J Sound Vib, 299, 822-838 (2007)
[23] Gourdon, E.; Alexander, N. A.; Taylor, C. A.; Lamarque, C.-H.; Pernot, S., Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results, J Sound Vib, 300, 522-551 (2007)
[24] Gourdon, E.; Lamarque, C.-H.; Pernot, S., Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment, Nonlinear Dyn, 50, 793-808 (2007) · Zbl 1170.70373
[25] Lee, Y. S.; Kerschen, G.; McFarland, D. M.; Hill, W. J.; Nichkawde, C.; Strganac, T. W., Suppressing aeroelastic instability using broadband passive targeted energy transfers. Part 2: experiments, AIAA J, 45, 2391-2400 (2007)
[26] Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G, Nucera F, Tsakirtzis S, Panagopoulos PN. Passive non-linear targeted energy transfer and its applications to vibration absorption: a review. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 2008, vol. 222. 2008. p. 77-134.; Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G, Nucera F, Tsakirtzis S, Panagopoulos PN. Passive non-linear targeted energy transfer and its applications to vibration absorption: a review. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 2008, vol. 222. 2008. p. 77-134.
[27] Ture Savadkoohi A, Pernot S, Lamarque C-H. Targeted energy Transfer with Several NES in Parallel: Theory and Experiments. Proceedings of the 2009 ASME-International Design Engineering Technical Conference and Computers and Information in Engineering Conference IDETC/CIE 2009, California, USA.; Ture Savadkoohi A, Pernot S, Lamarque C-H. Targeted energy Transfer with Several NES in Parallel: Theory and Experiments. Proceedings of the 2009 ASME-International Design Engineering Technical Conference and Computers and Information in Engineering Conference IDETC/CIE 2009, California, USA. · Zbl 1337.70042
[28] Lee, Y. S.; Vakakis, A. F.; McFarland, D. M.; Bergman, L. A., Non-linear system identification of the dynamics of aeroelastic instability suppression based on targeted energy transfers, Aeronaut J, 114, 61-82 (2010)
[29] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcation of vector fields (2002), Springer-Verlag: Springer-Verlag New York, LLC, USA
[30] Holmes, P.; Marsden, J., A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch Ration Mech Anal, 76, 2, 135-165 (1981) · Zbl 0507.58031
[31] Awrejcewicz, J.; Calvisi, M. L., Mechanical models of Chua’s circite, Int J Bifur Chaos, 12, 4, 671-686 (2002) · Zbl 1051.70545
[32] Kashdan L, Seepersad CC, Haberman M, Wilson PS. Design, Fabrication, and Evaluation of Negative Stiffness Elements, Solid Freeform Fabrication Symposium, Austin, TX, 2009.; Kashdan L, Seepersad CC, Haberman M, Wilson PS. Design, Fabrication, and Evaluation of Negative Stiffness Elements, Solid Freeform Fabrication Symposium, Austin, TX, 2009.
[33] Gourdon, E.; Lamarque, C.-H., Energy pumping with various nonlinear structures: numerical evidences, Nonlinear Dyn, 40, 281-307 (2005) · Zbl 1101.70015
[34] Manevitch, L. I., The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables, Nonlinear Dyn, 25, 95-109 (2001) · Zbl 1005.70023
[35] Nayfeh, A. H.; Mook, D. T., Nonlinear oscillations, 720 (1979), John Wiley and Sons: John Wiley and Sons New York
[36] Manevitch LI, Kovaleva AS, Manevitch EL. Limiting phase trajectories and resonance energy transfer in a system of two coupled oscillators. Mathematical Problems in Engineering 2010; 1-24 [Article ID 760479, doi:10.1155/2010/760479; Manevitch LI, Kovaleva AS, Manevitch EL. Limiting phase trajectories and resonance energy transfer in a system of two coupled oscillators. Mathematical Problems in Engineering 2010; 1-24 [Article ID 760479, doi:10.1155/2010/760479 · Zbl 1191.34050
[37] Pilipchuk, V. N., Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformations, J Sound Vib, 192, 1, 43-64 (1996) · Zbl 1232.70015
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