Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture

Abstract

We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging can be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an Oε neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed.

Experimental study of a hybrid electro-acoustic nonlinear membrane absorber

“…The phenomena can be described as a 1:1 resonance capture [2] and, considering harmonic forcing, as response regimes characterized in terms of periodic and…”

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

“…Resonant phenomena occur in a variety of different systems: energy exchange between coupled oscillators, Vakakis and Gendelman, 2001;Popov at al, 2001), mixing in fluids (Neishtadt et al, 1998;Ward and Homsy, 2001), celestial mechanics and astrophysics (Fux, 2001), billiards (Itin et al, 2001), Josephson junctions (Vasiliev et al, 1997) to name a few recent papers. In the present paper we apply the theory of resonant processes in Hamiltonian systems with slow and fast motions in the form, developed in (Neishtadt, 1999) (see also (Neishtadt, 1997) for references on preceding works) to the analysis of the dynamics of charged particles in nonhomogeneous electromagnetic fields in the magnetotail of the earth.…”

Passive control of resonances by nonlinear absorbers

“…It is important to look at the nonlinear normal modes of the system because the TET phenomenon is caused by a 1:1 resonance capture [2]. In order to analyze the two nonlinear normal modes, we remove the force F and all the dampings in the system (1).…”

“…A new passive technique for reducing noise and vibration is to use purely nonlinear absorbers working on the principle of "Targeted Energy Transfer" (TET). The concept of TET was first proposed by Vakakis and Gendelman [1,2], in 2001. We often speak of Nonlinear Energy Sink (NES) for a purely nonlinear absorber.…”