Irreversible Passive Energy Transfer in Coupled Oscillators with Essential Nonlinearity

Abstract

Abstract. We study numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused by either fundamental or subharmonic resonance capture, and in some cases is initiated by nonlinear beat phenomena. It is shown that, due to the essential nonlinearity, the end attachment is capable of passively absorbing broadband energy at both high and low frequencies, acting, in essence, as a passive broadband boundary controller. Complicated transitions in the damped dynamics can be interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. Moreover, complex resonance capture cascades are numerically encountered when we increase the number of degrees of freedom of the system. The ungrounded essentially nonlinear end attachment discussed in this work can find application in numerous practical settings, including vibration and shock isolation of structures, seismic isolation, flutter suppression, and packaging. 1. Introduction. We study passive and irreversible energy transfer from a linear oscillator to an essentially nonlinear attachment, which, in essence, acts as a nonlinear energy sink (NES); such energy transfer we refer to as nonlinear energy pumping. In previous works (Vakakis and Gendelman (2001), Vakakis et al. (2003)) grounded and relatively heavy nonlinear attachments were considered, a feature that limits their attractiveness in practical applications. To eliminate these restrictions, an ungrounded and light nonlinear attachment is considered in this work, which, in addition, possesses the feature of modularity. As shown in Lee et al. (2005), even though the system considered has a simple configuration, it possesses a very complicated structure of undamped periodic orbits, which, in turn, give rise to a complicated

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

“…Firstly, it cannot be easily extended to nonconservative systems. However, as discussed in the companion paper, Part I, and in [15,26], the damped dynamics can often be interpreted based on the topological structure of the NNMs of the underlying conservative system. Moreover, due to the lack of knowledge of damping mechanisms, engineering design in industry is often based on the conservative system, and this even for linear vibrating structures.…”

“…The dynamics of such systems have been extensively studied by Vakakis and co-workers [15,26,35,36] using the NNM concept. The motivation for using an essential nonlinearity is that the nonlinear attachment possesses no preferential resonant frequency; it can, therefore, interact with an SDOF structure in a frequency-independent fashion or with any mode of an MDOF structure.…”

Dynamics of a linear beam with an attached local nonlinear energy sink

“…The following simulations were performed for shock excitation (5) The dynamical mechanisms governing passive targeted energy transfer from linear to nonlinear oscillators are discussed in previous works [5,6]. Two of these mechanisms rely on resonance capture of the damped dynamics on either fundamental or subharmonic resonant manifolds in the phase space of the dynamical system; viewed from an alternative perspective, irreversible targeted energy transfer from the linear oscillator to the nonlinear attachment takes place when the dynamics is restricted to a damped nonlinear normal mode invariant manifold, whose mode shape becomes strongly localized to the nonlinear attachment as the energy deceases due to damping dissipation.…”

“…In particular, it was shown that transient resonance captures of the transient dynamics in neighborhoods of resonance manifolds of the phase space of a dynamical system [3,4] may initiate one-way, irreversible targeted energy transfer from a linear (main) subsystem to a local essentially nonlinear attachment, which acts, in essence, as nonlinear energy sink (NES) [5,6]. Moreover, in contract to the classical linear vibration absorber, the NES may passively absorb broadband energy over wide frequency ranges; this due to the essential stiffness nonlinearity of the NES, which enables resonance capture of the NES with any mode of the linear subsystem, irrespective of its frequency.…”

Generalization of Den Hartog’s Equal-Peak Method for nonlinear primary systems

“…For instance, it was shown that an NES, i.e., an essentially nonlinear absorber, can extract energy from virtually any mode of a host structure [8]. The NES can also carry out targeted energy transfer, which is an irreversible channeling of vibrational energy from the host structure to the absorber [9]. This makes nonlinear vibration absorbers suitable candidates for vibration mitigation of nonlinear primary structures.…”