Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part I

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Abstract

We study targeted energy transfers and nonlinear transitions in the damped dynamics of a two degree-of-freedom system of coupled oscillators (a linear oscillator with a lightweight, essentially nonlinear, ungrounded attachment), caused by 1:1 resonance captures of the dynamics. Part I of this work deals with the underlying structure of the Hamiltonian dynamics of the system, and demonstrates that, for sufficiently small values of viscous damping, the damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the corresponding Hamiltonian system. Focusing exclusively on 1:1 resonance captures in the system, it is shown that the topology of these damped transitions affect drastically the efficiency of passive energy transfer from the linear system to the nonlinear attachment. Then, a detailed computational study of the different types of nonlinear transitions that occur in the weakly damped system is presented, together with an analytical treatment of the nonlinear stability of certain families of periodic solutions of the underlying Hamiltonian system that strongly influence the said transitions. As a result of these studies, conditions on the system and forcing parameters that lead to effective or even optimal energy transfer from the linear system to the nonlinear attachment are determined. In Part II of this work, direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed, in order to analytically model the dynamics in the region of optimal energy transfer, and to determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator.

Introduction

The aim of this work is to study the complex transitions in the dynamics and targeted energy transfers associated with the attraction of the transient dynamics of a two degrees- of- freedom (dof) strongly nonlinear system into 1:1 resonance capture [1], [2]. In particular, the transient dynamics of a linear damped oscillator (labeled as primary system) with an essentially nonlinear, damped, lightweight attachment are investigated. The complexity of the problem dictates an extensive series of numerical simulations, together with the development of methodologies capable of analytically modeling both qualitatively and quantitatively the transient, strongly nonlinear transitions.

As reported in earlier works [3], [4] the two degrees of freedom system under consideration possesses surprisingly complex dynamics. Moreover, at certain ranges of parameters and initial conditions passive targeted energy transfer passive targeted energy transfer is possible, whereby vibration energy initially localized in the linear oscillator (LO) gets passively transferred to the lightweight attachment in a one-way irreversible fashion. Interesting features of passive targeted energy transfer in this system are the ability of the attachment (labeled nonlinear energy sink—nonlinear energy sink) to irreversibly absorb broadband vibration energy, and, in cases of multi-degree-of-freedom primary structures, to engage in a sequential resonance captures with a set of modes of the primary structure, passively absorbing energy from each before engaging the next [5]. In essence, the nonlinear energy sink acts as a passive, adaptive and broadband boundary controller. As explained in the mentioned references, what enables the local nonlinear energy sink attachment to affect the global dynamics of the primary structure is its lack of a preferential resonance frequency (since it possesses an essentially nonlinear—nonlinearizable—stiffness nonlinearity), giving rise to degenerate, high-codimension bifurcations of the underlying Hamiltonian dynamics. In recent studies passive targeted energy transfer has been experimentally shown to exist and to be robust to small variations of parameters and initial conditions [6].

Previous works examined targeted energy transfer in systems of coupled nonlinear oscillators through energy exchanges between donor and acceptor discrete breathers due to nonlinear resonance [7], [8], [9]. In Ref. [10] resonant interactions between monochromatic electromagnetic waves and charged particles were studied, leading to chaotization of particles and transport in phase space. In Ref. [11] the processes governing energy exchange between coupled Klein–Gordon oscillators were analyzed; the same weakly coupled system was studied in Ref. [12], and it was shown that, under appropriate tuning, total energy transfer can be achieved for coupling above a critical threshold. In related work, localization of modes in a periodic chain with a local nonlinear disorder was analyzed [13]; transfer of energy between widely spaced modes in harmonically forced beams was analytically and experimentally studied [14]; and a nonlinear dynamic absorber designed for a nonlinear primary system was analyzed [15].

In Ref. [4] it was shown that there are (at least) three distinct mechanisms for passive targeted energy transfer in the two-degree-of-freedom system under consideration herein: namely, fundamental passive targeted energy transfer, subharmonic passive targeted energy transfer, and passive targeted energy transfer through nonlinear beats followed by 1:1 resonance capture. The latter mechanism is the most efficient for passive targeted energy transfer, as it permits a significant percentage of the vibration energy of the primary system to be transferred and then locally dissipated by the nonlinear energy sink. It is the principal aim of the current work to study in detail the complex nonlinear transitions associated with this passive targeted energy transfer mechanism and to determine the conditions for strong (or even optimal) passive targeted energy transfer in the two-degree-of-freedom system.

The first section of this work deals with the underlying structure of the Hamiltonian dynamics of the system and demonstrates that, for sufficiently small values of viscous damping, the damped transitions are strongly influenced by the underlying periodic and quasiperiodic orbits of the corresponding Hamiltonian system. Then, a detailed computational study of the different types of nonlinear transitions that occur in the weakly damped system is presented, together with an analytical treatment of the nonlinear stability of certain families of periodic solutions of the underlying Hamiltonian system that strongly influence the said transitions. In a companion paper, the second part of this work will be reported, focusing on direct analytical treatment of the governing strongly nonlinear equations of motions, theoretical modeling, and interpretation of the computational results.

Consider the following two degrees- of- freedom nonlinear system of coupled oscillators:y¨1+2ɛζ1y˙1+2ɛζ2(y˙1-y˙2)+y1+43ɛα(y1-y2)3=0,ɛy¨2+2ɛζ2(y˙2-y˙1)+43ɛα(y2-y1)3=0,viewed as a LO, described by y1, coupled to a nonlinear energy sink whose response is measured by y2, as illustrated in Fig. 1. This system, with ζ1=0, has also been the subject of the recent study by Manevitch, Musienko and Lamarqu [16].

In the absence of damping the total energy E(t) in the system is conserved and given as E(t)=E1(t)+E2(t), whereE1(t)=y˙12(t)2+y12(t)2,E2(t)=ɛy˙22(t)2+ɛα3(y1(t)-y2(t))4.The quantity E1(t) can be interpreted as the energy in the LO, while E2(t) then represents the energy associated with the attachment. The energy in each component can be scaled by the total initial energy in the system, so that hi(t)Ei(t)/E(0). Finally, the instantaneous fraction of the total energy in each component is identified as h^i(t)Ei(t)/E(t).

With direct impulsive forcing of the primary system, i.e., y˙1(0)=Y0, y1(0)=y2(0)=0, y˙2(0)=0, the possibility of passively transferring a significant fraction of the imparted energy to the nonlinear energy sink, where it is coincidentally localized and dissipated, is examined. As an illustrative example, the response of Eqs. (1a) is computed with α=1, ɛ=0.05, ζ1=0.00, ζ2=0.10 for an intermediate level of initial energy, Y=0.75. As seen in Fig. 2, the response of the system can be characterized by three different regimes of motion. In the first interval, seen for t>10 in this simulation, a fraction of the energy, initially localized to the LO, is rapidly transferred to the nonlinear energy sink. This is followed by a slow decay of the total energy in the system characterized by large amplitude response of both the nonlinear energy sink as well as the LO, exhibited here for 10<t<55. In addition, in this interval the energy is slowly localized to the nonlinear energy sink as the energy decays. Finally, in the final regime of the motion for t>55 the energy is again transferred back to the LO.

The response of the system is strongly dependent on the initial energy in the system. Fig. 3 depicts the system response at low- (Y=0.375), moderate- (Y=0.50) and high-energy (Y=1.00) levels, together with the corresponding scaled energy in the system. Clearly, there are qualitative changes in the response of the system as the initial energy level increases. In the low-energy regime, no interesting energy transfer occurs from the LO to the nonlinear energy sink, and the response remains localized to the LO. As discussed in Refs. [3], [4], significant targeted energy transfer takes place only above a well-defined critical threshold of input energy. Efficient passive targeted energy transfer to the nonlinear energy sink is triggered through the initial energy transfer noted above for Y=0.75 in the moderate-energy regime (Figs. 3c, d). Gendelman, Kerschen, Vakakis, Bergman and McFarland [17] discuss the existence of impulsive periodic orbits in the underlying Hamiltonian system that play the role of bridging orbits and channel a significant portion of the energy initially localized in the LO to the nonlinear energy sink at a relatively fast time scale. As the initial energy is further increased (Figs. 3e, f), the initial triggering of the passive targeted energy transfer still occurs, but its effect on the overall energy dissipation in the system is diminished. As illustrated in Fig. 3, the nonlinear energy sink is most effective for an intermediate level of energy.

Indeed, these different regimes in the response are clearly seen in the instantaneous fraction of the energy in each component, h^i(t), shown in Fig. 4. For the low-energy regime the energy remains localized in the LO and passive targeted energy transfer does not occur. As the initial impulse is increased, seen in Fig. 4b, a significant fraction of the energy is transferred to the nonlinear energy sink, which is then able to effectively dissipate the response. Finally, for large initial energies, passive targeted energy transfer occurs but the initial triggering transfers a smaller fraction of the energy. By the time the energy transfer to the nonlinear energy sink is significant the total energy has decayed to a relatively small value, so that the effect of this efficient transfer on the overall evolution of the system is less significant. Therefore the most effective passive targeted energy transfer occurs when the initial triggering of the nonlinear energy sink is accompanied by a significant transfer of energy to the nonlinear energy sink.

The efficiency of the passive targeted energy transfer can be evaluated as the initial impulse varies through the maximum energy transferred to the nonlinear energy sink, h2,max. As illustrated in Fig. 5, as the initial energy in the system increases, the maximum energy transferred to the nonlinear energy sink undergoes a sharp jump. In this figure this quantity is shown for several values of the damping ratio ζ2. In particular, we note that the observed jump in the transferred energy even occurs in the absence of damping, shown as the solid line. Therefore, the properties of the irreversible energy transfer and dissipation are strongly dependent on the dynamics of the underlying Hamiltonian system in the absence of damping.

In Refs. [3], [4], the transient dynamics of the weakly damped system was studied and interpreted in terms of a frequency-energy plane by comparing the response of the damped system to that of the underlying Hamiltonian system. Such a representation is given in Fig. 6 for the moderate-energy regime of Fig. 3. In each panel the solid curve represents the frequency of the periodic solutions in the underlying Hamiltonian system, while the shading denotes the relative amplitude of the dominant harmonic components of the damped motions, as computed through the Morlet wavelet transform. The plot clearly shows the correspondence of the response of the damped system with that of the system in the absence of damping. In addition, this frequency-energy plot clearly illustrates three regimes of motion—the initial transient, followed by the 1:1 resonant motion as the initial energy decays, so that the dominant frequency of the nonlinear energy sink and LO are identical. Finally as the energy decreases near 10-3, this resonant state is lost, and the frequency of the LO once again approaches unity, while that of the nonlinear energy sink remains low.

Section snippets

Slow flow

Analysis of the above system proceeds through the development of a reduced model via the method of averaging. Thus, to this system the following transformations are applied:y1(t)=a1cos(t+θ1(t)),y2(t)=a2cos(t+θ2(t)),y˙1(t)=-a1sin(t+θ1(t)),y˙2(t)=-a2sin(t+θ2(t)).If a 1:1 resonance exists between y1 and y2 the difference between the phase variables, θ1-θ2φ, is, on average, stationary. In contrast, non-resonant dynamics imply that φ is time-like. The resulting equations are not in the correct form

Undamped system (ζ1=ζ20)

As described above, in the absence of damping, that is, with ζ1=ζ20, r is stationary and the slow flow equations reduce tor˙=0,ψ˙=-αr22ɛ[(1+ɛ)-(1-ɛ)sinψ-2ɛcosψcosφ]sinφ,φ˙=12-αr24ɛ[(1+ɛ)-(1-ɛ)sinψ-2ɛcosψcosφ](1-ɛ)-2ɛsinψcosφcosψ.This resulting dynamical system on the sphere, described by (ψ,φ), is integrable, with the second integral of motion, Eq. (5b), represented asr283+sinψ+αr24ɛ((1+ɛ)-(1-ɛ)sinψ-2ɛcosψcosφ)2=h.Therefore, for the undamped system, trajectories in the space (ψ,φ) lie on level

Damped system (ζ10, ζ20)

The response of the undamped system and the analysis presented above can be used as a framework for understanding the response of the system in the presence of damping. Although with ζ1 and/or ζ2 non-zero the trajectories are no longer constrained to integral curves (r=constant, h=constant) in phase space, the timescale associated with the response along integral curves is much smaller than that associated with the evolution of r and h, which is O(ɛ). Therefore, for O(1) durations in time, the

Concluding remarks

The findings of this work confirm that, for sufficiently weak damping, nonlinear damped transitions are essentially influenced by the underlying Hamiltonian dynamics. It follows that significant understanding and interpretation of (even complicated and multi-frequency) damped nonlinear transitions in the dynamics can be gained by performing perturbation studies that use as generating functions the Hamiltonian periodic or quasi-periodic orbits. An additional finding in this work is the central

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant no. CMS–0201347 (DDQ). The author GK is supported by a grant from the Belgian National Science Foundation (FNRS), which is gratefully acknowledged.

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