Andrey I. Musienko

Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment

Abstract We study the dynamics of a semi-infinite linear chain of particles that is weakly coupled to a strongly nonlinear oscillator at its free end. We analyze families of localized standing waves situated inside the lower or upper attenuation zones of the linear chain, corresponding to energy predominantly confined in the nonlinear oscillator. These families of standing waves are generated due to resonant interactions between the chain and the nonlinear attachment. In addition, we estimate the energy radiated from the nonlinear oscillator back to the chain, when the oscillator is excited under nonresonant conditions by wavepackets with dominant frequencies inside the propagation zone of the chain. We conclude that the system is capable of energy pumping, i.e., of one-way, irreversible energy transfer from the semi-infinite chain to the nonlinear oscillator. Such energy transfer closely depends on the excitation by the external forcing of a localized standing wave of the type studied in this work. A scenario for the realization of energy pumping phenomena in the system under consideration is discussed, and is confirmed by direct numerical simulations of the chain–attachment dynamic interaction.

Comparison of Passive and Active Energy Pumping in Mechanical Nonlinear System

We describe nonlinear dynamics of essentially non-homogeneous system of coupled oscillators using the technique of multi-scale expansions. The resonant mechanism of passive energy transfer from a massive oscillator into a light auxiliary one provides essential reduction of energy of the massive oscillator. Proposed active control algorithm increases noticeably the efficiency of energy transfer in comparison with passive control. The important advantage of this algorithm is that it provides a small consumption of energy. It is expedient to use our algorithm in the certain limits of parameters determined in the paper. This algorithm provides a necessary level of reduction of oscillator amplitude in a wide range of external perturbations.Copyright

Control of mechanical energy pumping in the nonhomogeneous system of coupled oscillators

We describe nonlinear dynamics of essentially non-homogeneous system of coupled oscillators using the technique of multi-scale expansions. The resonant mechanism of passive energy transfer from a massive oscillator into a light auxiliary one provides essential reduction of energy of the massive oscillator. Proposed active control algorithm increases noticeably the efficiency of energy transfer in comparison with passive control. The important advantage of this algorithm is that it provides a small consumption of energy. It is expedient to use our algorithm in the certain limits of parameters determined in the paper. This algorithm provides a necessary level of reduction of oscillator amplitude in a wide range of external perturbations.

Transversal Nonlinear Dynamics of Stretched Chain on Elastic Foundation

We present analytical and numerical study of short wavelength breathers in the system of asymmetric nonlinear oscillators coupled by stretched weightless beam. The study is focused on modulations of the nonlinear normal mode with shortest wavelength. A small parameter is introduced as ratio of distance between the particles in the chain and characteristic wavelength of modulation. It is shown that in main asymptotic approach the problem can be reduced to Nonlinear Schrodinger Equation (NSE). Breathers (envelope solitons) are localized oscillating solutions of this equation. We reveal the stability and high mobility of such solutions in their numerical study based on integrating of exact discrete equations of motion. We study breathers, localized on chain ends, and radiation of breathers from chain ends, induced by variable external force.© 2005 ASME

Motion confinement in a linear chain of coupled oscillators with a strongly nonlinear end attachment

We study the dynamics of a semi-infinite linear chain of particles that is weakly coupled to a strongly nonlinear oscillator at its free end. We analyze families of localized standing waves situated inside the lower or upper attenuation zones of the linear chain, corresponding to energy predominantly confined in the nonlinear oscillator. These families of standing waves are generated due to resonant interactions between the chain and the nonlinear attachment. A scenario for the realization of energy pumping phenomena in the system under consideration is discussed, and is confirmed by direct numerical simulations of the chain-attachment dynamic interaction.Copyright