[en] We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency-energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers.
Disciplines :
Physics Mechanical engineering
Author, co-author :
Tsakirtzis, Stylianos; National Technical University of Athens > School of Applied Mathematical and Physical Sciences
Panagopoulos, Panagiotis; National Technical University of Athens > School of Applied Mathematical and Physical Sciences
Kerschen, Gaëtan ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux
Gendelman, Oleg; Technion–Israel Institute of Technology > Faculty of Mechanical Engineering
Vakakis, Alexander F.; National Technical University of Athens > Department of Mechanical and Industrial Engineering
Bergman, Lawrence A.; University of Illinois at Urbana-Champaign > Department of Aerospace Engineering
Language :
English
Title :
Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments
Arnold, V.I. (ed.): Dynamical Systems, vol. III, Encyclopaedia of Mathematical Sciences. Springer Verlag, Berlin, New York (1988)
Quinn, D.: Resonance capture in a three degree of freedom mechanical system. Nonlinear Dyn. 14, 309-333 (1997)
Vakakis, A.F., Gendelman, O.: Energy pumping in nonlinear mechanical oscillators. II: Resonance capture. J. Appl. Mech. 68(1), 42-48 (2001)
Vakakis, A.F., McFarland, D.M., Bergman, L.A., Manevitch, L.I., Gendelman, O.: Isolated resonance captures and resonance capture cascades leading to single- or multi-mode passive energy pumping in damped coupled oscillators. J. Vib. Acoust. 126(2), 235-244 (2004)
Kopidakis, G., Aubry, S., Tsironis, G.P.: Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett. 87(16) (2001), paper 165501-1
Aubry, S., Kopidakis, S., Morgante, A.M., Tsironis, G.P.: Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers. Physica B 296, 222-236 (2001)
Morgante, A.M., Johansson, M., Aubry, S., Kopidakis, G.: Breather-phonon resonances in finite-size lattices: phantom breathers. J. Phys. A 35, 4999-5021 (2002)
Maniadis, P., Kopidakis, G., Aubry, S.: Classical and quantum targeted energy transfer between nonlinear oscillators. Physica D 188, 153-177 (2004)
Vakakis, A.F., Rand, R.H.: Nonlinear dynamics of a system of coupled oscillators with essential stiffness nonlinearities. Int. J. Non-Linear Mech. 39, 1079-1091 (2004)
Manevitch, L.I.: Complex representation of dynamics of coupled oscillators. In: Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems, pp. 269-300. Kluwer Academic/Plenum Publishers, New York (2001)
Gendelman. O.V., Vakakis, A.F., Manevitch, L.I., McCloskey, R.: Energy pumping in nonlinear mechanical oscillators I: Dynamics of the underlying hamiltonian system. J. Appl. Mech. 68(1), 34-41 (2001)
Tsakirtzis, S., Kerschen, G., Panagopoulos, P.N., Vakakis, A.F.: Multi-frequency nonlinear energy transfer from linear oscillators to MDOF essentially nonlinear attachments. J. Sound Vib. 285, 483-490 (2005)
Panagopoulos, P.N., Vakakis, A.F., Tsakirtzis, S.: Transient resonant interactions of linear chains with essentially nonlinear end attachments leading to passive energy pumping. Int. J. Solids Struct. 41(22-23), 6505-6528 (2004)
McFarland, D.M., Bergman, L.A., Vakakis, A.V.: Experimental study of nonlinear energy pumping occurring at a single fast frequency. Int. J. Non-Linear Mech. 40, 891-899 (2005)
Rosenberg, R.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 9, 155-242 (1966)
Lee, Y.S., Kerschen, G., Vakakis, A.F., Panagopoulos, P.N., Bergman, L.A., McFarland, D.M.: Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204, 41-69 (2005)
Keller, H.B.: Numerical solution of two-point boundary value problems. Society of Industrial and Applied Mathematics, Philadelphia (1976)
Storn, R., Price, K.: Differential evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces. J. Glob. Optim. 11, 341-359 (1997)
Keller, H.B.: Numerical Methods in Bifurcation Problems. Springer-Verlag, Berlin (1987)
