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

TLDR

The study examines weakly coupled linear–nonlinear oscillator systems, noting that energy pumping cannot occur in purely linear dynamics and that a 1:1 stable subharmonic orbit of the Hamiltonian system is conjectured to drive the phenomenon. The paper aims to provide numerical evidence of irreversible energy pumping from linear to nonlinear components in coupled oscillators above a critical energy and to analyze the underlying Hamiltonian dynamics to understand this phenomenon. The authors reduce the equations of motion to an isoenergetic manifold and compute subharmonic orbits via nonsmooth coordinate transformations that yield nonlinear boundary‑value problems. Numerical evidence shows irreversible energy pumping from linear to nonlinear components occurs only above a critical energy, with the responsible 1:1 stable subharmonic orbit not excitable at low energies, and Part II demonstrates that transient resonance capture on a 1:1 resonance manifold drives the phenomenon.

Abstract

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.