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

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.

Passive linearization of nonlinear resonances

“…Superharmonic excitation and internal resonances were employed to improve stability properties of resonators 32 and time keeping devices 13 . Finally, the inherent frequency-energy dependence of nonlinear oscillations was exploited in broadband vibration absorbers 33 .…”

Passive control of resonances by nonlinear absorbers

“…In order to analyze the two nonlinear normal modes, we remove the force F and all the dampings in the system (1). A simplified analysis of the nonlinear normal modes is done by using a harmonic balance method (HBM) with a single term, as follows u 1 (t) = u 1c cos(ωt) and u a (t) = u ac cos(ωt).…”

“…After defining a dimensionless time τ = tω 1 , where ω 1 = √ k 1 /m 1 , the equations of the system (Fig. 1) are represented in the following dimensionless form:…”

“…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.…”

Targeted energy transfer and modal energy redistribution in automotive drivetrains

“…In the available literature [28][29][30][31][32][33][34], the mechanism of this targeted energy transfer (TET) or "energy pumping" is well documented for structures coupled to strongly nonlinear mechanical oscillators. The effects of NES on the dissipation and redistribution of energy were studied by Quinn et al [22] in a two-DoF linear structure, subjected to impulsive excitation.…”