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A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities
Authors:Gendelman  Oleg  Manevitch  Leonid I  Vakakis  Alexander F  Bergman  Lawrence
Institution:(1) Russian Academy of Sciences, Institute of Chemical Physics, Moscow, Russia;(2) Division of Mechanics, National Technical University of Athens, Athens, Greece;(3) Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, U.S.A.;(4) Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, U.S.A.
Abstract:We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter epsi, we study thedynamics of this system at the limit epsi rarr 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit epsi rarr 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.
Keywords:degenerate bifurcations  nonlinear normal modes  essential stiffness nonlinearity  nonlinear energy sink
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