Leonid I. Manevitch

Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems

The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) channeling of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

Dynamics of linear discrete systems connected to local, essentially non-linear attachments

The dynamics of a linear periodic substructure, weakly coupled to an essentially non-linear attachment are studied. The essential (non-linearizable) non-linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena, e.g., passive, one-way, irreversible transfer of energy from the substructure to the attachment. As a specific application the dynamics of a finite linear chain of coupled oscillators with a non-linear end attachment is examined. In the absence of damping, it is found that the dynamical effect of the non-linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain. When damping exists energy pumping phenomena are realized in the system. It is shown that energy pumping strongly depends on the topological structure of the non-linear normal modes (NNMs) of the underlying undamped system. This is due to the fact that energy pumping is caused by the excitation of certain damped invariant NNM manifolds that are analytic continuations for weak damping of NNMs of the underlying undamped system. The bifurcations of the NNMs of the undamped system help explain resonance capture cascades in the damped system. This is a series of energy pumping phenomena occurring at different frequencies, with sudden lower frequency transitions between sequential events. The observed multi-frequency energy pumping cascades are particularly interesting from a practical point of view, since they indicate that non-linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure, in compatibility with the design objectives.

Nonlinear dynamics of topological solitons in DNA

Dynamics of topological solitons describing open states in the DNA double helix are studied in the framework of a model that takes into account asymmetry of the helix. It is shown that three types of topological solitons can occur in the DNA double chain. Interaction between the solitons, their interactions with the chain inhomogeneities, and stability of the solitons with respect to thermal oscillations are investigated.

Asymptotical Mechanics of Thin-Walled Structures

1 Asymptotic Approximations.- 2 Regular Perturbations of Parameters.- 3 Singular Perturbation Problems.- 4 Boundary Value Problems of Isotropic Cylindrical Shells.- 5 Boundary Value Problems - Orthotropic Shells.- 6 Composite Boundary Value Problems - Isotropic Shells.- 7 Composite Boundary Value Problems - Orthotropic Shells.- 8 Averaging.- 9 Continualization.- 10 Homogenization.- 11 Intermediate Asymptotics - Dynamical Edge Effect Method.- 12 Localization.- 13 Improvement of Perturbation Series.- 14 Matching of Limiting Asymptotic Expansions.- 15 Complex Variables in Nonlinear Dynamics.- 16 Other Asymptotical Approaches.- Afterword.- References.

Isolated Resonance Captures and Resonance Capture Cascades Leading to Single- or Multi-Mode Passive Energy Pumping in Damped Coupled Oscillators

We examine passive energy pumping in a system of damped coupled oscillators. This is a one-way, passive and irreversible energy flow from a linear main system to a nonlinear attachment that acts, in essence, as a nonlinear energy sink (NES). Energy pumping is caused by 1:1 resonance captures on resonant manifolds of the damped systems. We show that the NES is capable of absorbing significant portions of the energies generated by transient, broadband external excitations. By performing a series of numerical simulations we confirm that the energy dependence of the nonlinear normal modes (NNMs) of the underlying undamped, unforced system determines, in essence, the resonance capture and energy pumping dynamics in the corresponding damped system. We present numerical simulations of single- and multi-mode energy pumping, that involve isolated resonance captures or resonance capture cascades, respectively. In addition, we discuss methodologies for enhancing the nonlinear energy pumping phenomenon by properly selecting the system parameters. The described technique of passively localizing and locally eliminating externally induced energy provides a new paradigm for vibration and shock isolation of mechanical oscillators.

Mechanics of periodically heterogeneous structures

0 Introduction.- 1 Definitions, assumptions and theorems in homogenization problems.- 2 Application of cell functions for the calculation of binary composite elastic moduli.- 3 Asymptotic study of linear vibrations of a stretched beam with concentrated masses and discrete elastic supports.- 4 Reinforced plates.- 5 Problems of elasticity theory for reinforced orthotropic plates.- 6 Reinforced shells.- 7 Corrugated plates.- 8 Other periodic structures.- 9 Perforated plates and shells.- Concluding remarks. Perspectives and open problems.- References.

Perforated plates and shells

We consider the biharmonic equation

Dynamics of an Eccentric Rotational Nonlinear Energy Sink

D{\nabla ^4}W = P(X,Y)