Nonlinear Dynamics of a System of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (nonlinear normal modes NNMs), as well as, asynchronous periodic motions (elliptic orbits EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive nonlinear energy pumping phenomena from the linear to the nonlinear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions. INTRODUCTION We consider the dynamics of a two degree-of-freedom (DOF) system of weakly coupled oscillators with cubic stiffness nonlinearities. In the limit of zero coupling the system decomposes into two single-DOF subsystems: A linear oscillator with normalized natural frequency equal to unity, and a nonlinear oscillator possessing a nonlinearizable cubic stiffness. We are interested in studying the dynamics of the weakly coupled system. Previous works (for example (Nayfeh and Mook,1985),(Rand and Armbruster,1987)) analyzed the dynamics of systems with internal, external and combination resonances, by partitioning the dynamics into ’slow’ and ’fast’ components and reducing the analysis to a small set of modulation equations governing the slow-flow, i.e., the evolution of the ’slow’ dynamics of the system. Generally, internal resonances introduce interesting bifurcations to the free and forced dynamics, and lead to essentially nonlinear dynamical phenomena that have no counterparts in linear theory. In recent works, a comprehensive classification of the possible internal resonances in discrete nonlinear oscillators was performed by (Luongo et al.,2002a),(Luongo et al.,2002b). In this work we focus in the 1:1 internal resonance between the linear and nonlinear oscillators and apply asymptotic techniques to study the free dynamics when no damping exists. Depending on the system parameters, stable and 1 Copyright c © 2003 by ASME unstable synchronous periodic solutions (nonlinear normal modes NNMs) or asynchronous periodic motions are detected, along with homoclinic loops in the ’slow’ flow dynamics. Numerical simulations confirm the analytical predictions. When damping is introduced, certain of these homoclinic loops can be transformed to domains of attraction for resonance capture (Quinn,1997),(Quinn,2002). In turn, resonance capture leads to passive energy pumping (Vakakis and Gendelman,2001),(Vakakis,2001) from the linear to the nonlinear oscillator. Following (Quinn,2002) we provide a direct link between resonance capture and passive nonlinear energy pumping in the damped system of coupled oscillators. We utilize analytical and numerical techniques to analyze these interesting dynamical phenomena. STATEMENT OF THE PROBLEM We are interested in the dynamics of a system of two oscillators, one of which is strictly nonlinear with cubic nonlinearity. The oscillators are assumed to be coupled by small nonlinear (cubic) terms. If we neglect damping, the problem is defined by the following equations: dx dt2 + x = − ∂V ∂x (1) dy dt2 + y = − ∂V ∂y (2) where << 1 and where V is given by: V = a40x + a31xy + a22xy + a31xy (3) We study this system by first using the method of averaging to obtain a slow flow valid to O( ), and then analyzing the slow flow. AVERAGING In order to perturb off of the = 0 system, we need to solve the equation: