Leonid Manevitch

In this paper we study the effect of nonstationary energy localization in a nonlinear conservative resonant system of two weakly coupled oscillators. This effect is alternative to the well-known stationary energy localization associated with the existence of localized normal modes and resulting from a local topological transformation of the phase portraits of the system. In this work we show that nonstationary energy localization results from a global transformation of the phase portrait. A key to solving the problem is the introduction of the concept of limiting phase trajectories (LPTs) corresponding to maximum possible energy exchange between the oscillators. We present two scenarios of nonstationary energy localization under the condition of 1:1 resonance. It is demonstrated that the conditions of nonstationary localization determine the conditions of efficient targeted energy transfer in a generating dynamical system. A possible extension to multi-particle systems is briefly discussed.

We present analytical and numerical studies of nonstationary resonance processes in a system with four degrees of freedom. The system under consideration can be considered as one of the simplest geometrically nonlinear discrete models of an elastic beam supported by nonlinear elastic grounding support. Two symmetrically distributed discrete masses reflect the inertial properties of the beam, two angular springs simulate its bending stiffness. The longitudinal springs, as is usual in systems of oscillators, reflect the tensile stiffness and two transversal springs simulate the reaction of grounding support. Dealing with lowenergy dynamics, we singled out the equations of transversal motion corresponding to the approximation of two coupled oscillators with nonlocal nonlinearity in elastic forces. We have analyzed this model using the concept of limiting phase trajectories (LPT). LPT’s concept was recently developed to study the nonstationary resonance dynamics. An analytical description of intensive interparticle energy exchange was obtained in terms of nonsmooth functions, which is consistent with numerical results. We have identified two dynamic transitions the first of which corresponds to the instability of out-of-phase normal mode and the second one is a transition from the intense energy exchange to the energy localization on the initially excited oscillator. Special attention was paid to the influence of bending stiffness on the conditions that ensure the implementation of each of the dynamic transitions.