Nonstationary Resonant Dynamics of Oscillatory Chains and Nanostructures

This book suggests a new common approach to the study of resonance energy transport based on the recently developed concept of Limiting Phase Trajectories (LPTs), presenting  applications of the approach to significant nonlinear problems from different fields of physics and mechanics. In order to highlight the novelty and perspectives of the developed approach, it places the LPT concept in the context of dynamical phenomena related to the energy transfer problems and applies the theory to numerous problems of practical importance. This approach leads to the conclusion that strongly nonstationary resonance processes in nonlinear oscillator arrays and nanostructures are characterized either by maximum possible energy exchange between the clusters of oscillators (coherence domains) or by maximum energy transfer from an external source of energy to the chain. The trajectories corresponding to these processes are referred to as LPTs. The development and the use of the LPTs concept aremotivated by the fact that non-stationary processes in a broad variety of finite-dimensional physical models are beyond the well-known paradigm of nonlinear normal modes (NNMs), which is fully justified either for stationary processes or for nonstationary non-resonance processes described exactly or approximately by the combinations of the non-resonant normal modes. Thus, the role of LPTs in understanding and analyzing of intense resonance energy transfer is similar to the role of NNMs for the stationary processes. The book is a valuable resource for engineers needing to deal effectively with the problems arising in the fields of mechanical and physical applications, when the natural physical model is quite complicated. At the same time, the mathematical analysis means that it is of interest to researchers working on the theory and numerical investigation of nonlinear oscillations.

Introduction.- Part I: Conservative systems.- Two coupled oscillators.- Two-particle systems under conditions of sonic vacuum.- Emergence and bifurcations of LPTs in the chain of three coupled oscillators.- Quasi-one-dimensional nonlinear lattices.- Part II: Extensions to non-conservative systems.- Duffing oscillators.- Non-conventional synchronization of weakly coupled active oscillators.- Limiting phase trajectories and the emergence of autoresonance in anharmonic oscillators.- Part III: Applications.- Targeted energy transfer.- Nonlinear energy channeling in the 2D, locally resonant, systems.- Nonlinear Targeted Energy Transfer and Macroscopic Analogue of the Quantum Landau-Zener Effect in Coupled Granular Chains.- Forced pendulum.- Classical analog of linear and quasi-linear quantum tunneling.- Locally supported nonlinear lattices.- Nonlinear vibrations of the carbon nanotubes.- Conclusions.