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Passive non-linear targeted energy transfer and itsapplications to vibration absorption: a review

YSLee1, A F Vakakis2, L A Bergman1∗, D M McFarland1, G Kerschen3, F Nucera 4, S Tsakirtzis2, and P N Panagopoulos2

1Department of Aerospace Engineering, University of Illiois at Urbana-Champaign, Urbana, Illinois, USA 2School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece3 Aerospace and Mechanical Engineering Department (LTAS), Université de Liège, Liège, Belgium4Department of Mechanics and Materials, Mediterranean University, Reggio Calabria, Italy

The manuscript was received on 26 July 2007 and was accepted after revision for publication on 3 March 2008.

DOI: 10.1243/14644193JMBD118

Abstract: This review paper discusses recent efforts to passively move unwanted energy from aprimary structure to a local essentially non-linear attachment (termed a non-linear energy sink)by utilizing targeted energy transfer (TET) (or non-linear energy pumping). First, fundamen-

tal theoretical aspects of TET will be discussed, including the essentially non-linear governing dynamical mechanisms for TET. Then, results of experimental studies that validate the TETphenomenon will be presented. Finally, some current engineering applications of TET will bediscussed. The concept of TET may be regarded as contrary to current common engineering practise, which generally views non-linearities in engineering systems as either unwanted or, atmost, as small perturbations of linear behaviour. Essentially non-linear stiffness elements areintentionally introduced in the design that give rise to new dynamical phenomena that are very beneficial to the design objectives and have no counterparts in linear theory. Care, of course, istaken to avoid some of the unwanted dynamic effects that such elements may introduce, such aschaotic responses or other responses that are contrary to the design objectives.

Keywords: passive non-linear targeted energy transfer, vibration absorbtion

1 INTRODUCTION

Many studies have been made to suppress vibrationalenergy from disturbances into a main system eitherpassively or actively since the seminal invention of thetuned vibration absorber (TVA) by Frahm [1] (referto references [2] and [3] for a historical review of passive/active TVAs and structural control methods,respectively). With advances in electro-mechanical

devices, active control schemes are more likely to offerthe best performance in terms of vibrationabsorption.However, in addition to issues of cost and energy con-sumption associated with active control, robustnessand stability need to be addressed.

∗Corresponding author: Department of Aerospace Engineering,

University of Illinois at Urbana-Champaign, Urbana, IL 61801,

USA. email: [email protected]

Passive dynamic absorbers represent an interest-ing alternative. The classical TVA from Frahm [1]has been extensively studied in the literature [4–7].It is a simple and efficient device but is only effec-tive in the neighbourhood of a single frequency.Roberson [8] showed that broadening the suppres-sion bandwidth is possible by employing a non-linearsystem for the TVA. Since then, non-linear vibra-tion absorbers have received increased attention in

the literature (e.g. continuously and discontinuously non-linear [9, 10]; piecewise linear [11]; centrifu-gal pendulum [12]; and autoparametric vibrationabsorbers [13, 14]). Although non-linearities are usu-ally considered to be detrimental, it is possible totake advantage of the richness and complexity of non-linear dynamics for the design of improved vibrationabsorbers.

Passive transfers of vibrational energy throughmode localization have been of particular interest in

JMBD118 © IMechE 2008 Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics



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solid-state, condensed-matter, and chemical physics.For example, there are vibrational energy transfersat gas–solid interfaces [15, 16]; thermally generatedlocalized modes and their delocalization in a strongly anharmonic solid lattice such as quantum crystals[17–19]; linear and non-linear exchanges of energy between different components in coupled Klein–Gordon equations [20]; and targeted energy transfer(TET) between a rotor and a Morse oscillator present-

ing chemical dissociation [21]. A novel mechanism was also proposed for inducing highly selective yetvery efficient energy transfers in certain discrete non-linear systems where, under a precise condition of non-linear resonance, when a specific amount of energy is injected as a discrete breather at a donorsystem it can be transferred as a discrete breather toanother weakly coupled acceptor system [22–25].

In applications to mechanical systems, localizationor confinement of vibrations, which is referred to asnormal mode localization, was studied in references[26] to [29] by considering structural irregularity (ordisorder) in weakly coupled component systems. An

acoustical application of Anderson localization [30] was demonstrated theoretically and experimentally [31].Itwasalsoshownthat(non-linear)modelocaliza-tion can occur in a class of multi-degree-of-freedom(MDOF) non-linear systems even with perfect sym-metry and a weakly coupled structure [32–38]. Thiskind of standing wave localization, based on intrin-sic localized modes (discrete breathers) or non-linearnormalmodes(NNMS) which exist dueto discretenessand system non-linearity [39, 40], can be classifiedas ‘static’ because it does not involve controlled spa-tial transfer of energy through the system. It can be

realized through appropriate selection of the initialconditions [41].Internal resonances (IRs) under certain conditions

also promote energy transfer between non-linearmodes [14, 42–45]. It was explained, both theoreti-cally and experimentally, how a low-amplitude high-frequency excitation can produce a large-amplitudelow-frequency response (called energy cascading [46]). However, in these cases, non-linear energy exchanges are caused by non-linear modal interac-tions, and they do not necessarily involve controlledTETs [41].

It is only recently that passively controlled spa-

tial (hence ‘dynamic’) transfers of vibrational energy in coupled oscillators to a targeted point where theenergy eventually localizes were studied [41, 47–50].This phenomenon is called non-linear energy pump-ing or TET. This paper summarizes recent effortstowards understanding passive TET. Some preliminar-ies and literature reviews are presented in section 2;then, theoretical and experimental fundamentals onnon-linear TET phenomena are summarized, respec-tively, in sections 3 and 4.

2 TARGETED ENERGY TRANSFER (ORNON-LINEAR ENERGY PUMPING)

2.1 Preliminaries

Non-linear energy pumping (or passive TETs) refersto one-way targeted spatial transfers of energy froma primary subsystem to a non-linear attachment; itis realized through resonance captures and escapesalong the intrinsic periodic solution branches [41, 50].

Because of the invariance property of the resonancemanifold, the energy transfers become irreversibleonce the dynamics is captured into resonance.

The non-linear device, which is attached to a pri-mary system for passive energy localization into itself,is called a non-linear energy sink (NES). An NESgenerally requires two elements: an essentially non-linear (i.e. non-linearizable) stiffness and a (usually,linear viscous) damper. The former enables the NESto resonate with any of the linearized modes of theprimary subsystem, whereas the latter dissipatesthe vibrational energy transferred through resonant

modal interactions. The NES can be categorizedas grounded versus ungrounded, single-degree-of-freedom (SDOF) versus MDOF, and smooth versusnon-smooth, depending on its design and use.

Figure 1 depicts a schematic of passive and broad-band TETs utilizing an ungrounded SDOF NES. A primary structure is given (usually a linear systemand, hence, the mass, damping, and stiffness matrices,M, C, K , respectively). The primary structure, whichpossesses a set of natural frequencies {ω

(k )Primary }k =1,..., N

where N is the number of DOFs of the primary struc-ture, can suffer various external disturbances suchas impact loading, periodic or random excitation,fluid-structure interaction, etc.

One seeks to (passively) eliminate such unwantedexternal disturbances induced in the primary struc-ture by attaching a simple non-linear device suchas an NES. Because an NES does not possess any preferential resonance frequency (i.e. it has no linear

Fig. 1 Schematic of passive and broadband TETs

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stiffness term), it can generate a countably infinitenumber of non-linear resonance conditions (i.e. IRs,mω

(k )Primary ≈ nωNES where m, n are integers), through

which vigorous energy exchanges occur between thetwo oscillators. In particular, energy localization to theNES is preferred for efficient mitigation of the distur-bances in the primary structure. Duringthe non-linearmodal interactions, energy is dissipated in the NESdamper. As the total energy decreases, self-detuning

is possible with the dynamics escaping from one res-onance manifold to another. There are at least threedifferent TET mechanisms, which occur through 1:1and subharmonic resonance captures, and are initi-ated by non-linear beat phenomena, respectively (seesection 3).

Although an NES device looks similar to a lineardynamic absorber (or a TVA) in configuration (bothare passive and composed of a mass, a spring, anda damper), they are totally different in nature. A TVA operates effectively in a narrow band of frequencies,and its effect is most prominent in the steady-stateregime. Therefore, even if the TVA is initially designed

(tuned) to eliminate resonant responses near the nat-ural frequency of a primary system, the mitigating performance may become less effective over time dueto aging of the system, temperature or humidity vari-ations and so forth, thus requiring additional adjust-ment or tuning of parameters (i.e. the robustness canbe questioned). On the other hand, theNES is basically a device that interacts with a primary structure overbroad frequency bands; indeed, since the NES pos-sesses essential stiffness non-linearity, it may engagein (transient) resonance capture with any mode of theprimary system (provided, of course, that a node of

the mode is not at the point of attachment of theNES). It follows that an NES can be designed to extractbroadband vibration energy from a primary system,engaging in transient resonance with a set of ‘mostenergetic’ modes. Thus, the NES is more robust thanthe TVA [51].

2.2 Literature review

Resonance capture (or capture into resonance), whichturns out to be a fundamental mechanism fornon-linear TET, has been studied in various fields

(e.g. physics [52–54]; aerospace engineering [55–59])and originated as a consequence of the averaging theorem [60–63]. Applications of resonance capture tomechanical oscillators can also be found in references[64] to [66].

Recently, resonance capture was applied to sup-press unwanted disturbances in practical engineer-ing problems. In this section, efforts to understandpassive TET in coupled oscillators are summarizedchronologically and are grouped according to system

configurations (SDOF, MDOF or continuous primary systems; grounded or ungrounded and SDOF orMDOF NESs).

2.2.1 Grounded NES configurations

Gendelman and Vakakis [47] first investigated how non-linear localization in coupled oscillators is pro-gressively eliminated by a dissipative force. A strongly non-linear oscillator with symmetry was studied by computing and then matching separate analyticalapproximations for the early (localized) and late (non-localized)responses(see alsoreference [67]foralinearoscillator coupled to a strongly non-linear attachment

with multiple equilibrium states). It was shown thata damped vibrational system can exhibit localizationphenomenaatleastattheearlystagesofthemotion.Inlater stages of the motion, non-linear effects diminishand a transitionfrom non-linear localizedto linearized

weakly non-linear oscillations occurs as energy is dis-sipated. It was noted that, in a system with symmetry,IRs between subsystems exist leading to linearized

beat phenomena which eliminate localization in thelinearized regime. Applicability of active control tocompensate for dissipation effects was addressed,keeping the localized motion preserved in the systemas energy decreases (see, for example, reference [68],

which suggested a control algorithm for switching mechanical components such as springs and damperson and off during their work with minimal energy consumption).

Inducing passive NESs in vibrating systems wasstudied in reference [48], where a complexification-averaging technique was introduced to obtain mod-

ulation equations for the slow-flow dynamics. It wasshown that, for an impulsively loaded MDOF chain with an NES attached at the end, the response of theNES after some initial transients is motion dominatedby a fast frequency identical to the lower bound of thepropagation zone of the linear chain, which reducesthe study of TET in the chain to a two-DOF equivalentproblem. This is because, after some initial transients,the semi-infinite chain in essence vibrates in an in-phase mode at the lower frequency boundary of thepropagation zone of the infinite linear chain. Possi-ble applications of TET to electric power networks [69]

were suggested for passive fault arrest in the network,

preventing catastrophic failure due to unchecked faultpropagation.

Similarly, energy transfer to a non-linear localizedmode in a highly asymmetric system was investi-gated [49]. It was shown that excitation of a NNM[70] occurs via the mechanism of subharmonic res-onance. The conditions for TET were suggested: (i) alocalized resonant mode should be excited; and (ii)the vibrations of a non-linear oscillator should bedamped faster than the primary system with the same




, , , , , , ,

damping terms of the same order. The shortcomingsof this passive vibration absorber were noted; thatis, it is not activated below a critical amplitude, and,moreover, its effectiveness is reduced as the ampli-tude grows above the critical resonant regime becausethe non-linear oscillator cannot absorb more than agiven amount of energy at a certain frequency. It wasobserved thatcubic stiffness coupling between the pri-mary structure and the NES is much more effective

than linear coupling because the main mechanism of energy transfer becomes a non-linear parametric res-onance (see also reference [71] for numerical evidenceof TET phenomena in various structures).

Dynamics of the underlying Hamiltonian systemand non-linear resonance phenomena were investi-gated to understand energy pumping in a two-DOFnon-linear coupled system with a linearly coupledgrounded NES being one of the DOF [41, 50]. Action-angle formulation was utilized as a reduction methodat a fixed energy level to obtain a single second-orderordinary differential equation;then,non-smooth tem-poral transformations (NSTTs [72]) of the reduced

equation were performed to compute its periodicsolutions. It was shown that a 1:1 stable subhar-monic orbit of the underlying Hamiltonian system ismainly responsible for theTET phenomenon, and thatthis orbit cannot be excited at sufficiently low ener-gies. Hence, a transient bridging orbit satisfying zeroinitial conditions must be impulsively excited. Fur-thermore, introducing action-angle transformations,and applying the averaging theorem to get a two-frequency dynamical system, it was shown that theenergy pumping phenomenon in the system studiedin that work is associated with resonance capture in a

neighbourhood of the 1:1 resonance manifold.The degenerate bifurcation structure of a systemof coupled oscillators with an NES was studied [73],

where two types of bifurcations of periodic solu-tions were observed: (i) a degenerate bifurcation athigh energy (i.e. bifurcation from infinity); and (ii)non-degenerate bifurcation near the exact 1:1 IR. It

was noted that the degeneracy occurs when the lin-ear coupling stiffness approaches zero, in which casethe linear part of the equations of motion possesses adouble zero and a conjugate pair of purely imaginary eigenvalues (i.e. a codimension-3 bifurcation occurs).

Bifurcation of damped NNMs for 1:1 resonance

was studied by combining the invariant manifoldapproach and multiple-scales expansion [74]. It wasnoted that there is a special asymptotical structuredistinct between three time-scales: (i) fast vibration;(ii) evolution of the system towards the NNM; and (iii)time evolution of the invariant manifold. It was alsofound that time evolution of the invariant manifoldmay be accompanied by bifurcations, and passage of the invariant manifold through bifurcations may bring

about destruction of the resonance regime and essen-tial gain in the energy dissipation rate. The damping coefficient should be chosen to ensure the possibility of bifurcation of the NNM invariant manifold, becausefailure to do so will result in a loss of NES ability todissipate the vibrational energy.

Robustness of TET was examined by introducing uncertain parameters (due to aging, imperfectionin design, and so on) to the NES [75]. Polynomial

chaos expansions were used to obtain informationaboutrandom displacements, followed by a numericalparametric study based on Monte Carlo simulation.

The design of mechanical TET devices was consid-ered in reference [76], where the complexification-averaging technique and the method of multiple-scales were utilized for analysing TET. Also, the issueof designing a linear structure (specifically, a two-DOFlinear chain) linearly coupled to a grounded NES wasstudied for enhancing TET [77]. Expressing the actualDOFs connected to the NES as modal coordinates, andassuming no IRs between uncoupled linear modes,the physical aspects of non-linear TET were studied. It

was revealed that damping is a prerequisite for energy pumping because non-linear TET is caused by theexcitation of a damped NNM invariant manifold thatis an analytic continuation of a NNM of the underlying undamped (i.e. Hamiltonian) system. A more generallinear substructure (an MDOF chain) was consideredin reference [78], where a similar modal expression

was utilized to obtain the first version of a frequency–energy plot (FEP). For the MDOF primary structurecoupled to an NES, resonance capture cascades weredemonstrated when TET occurs.

Single- and multi-mode energy pumping phenom-

ena were investigated in a two-DOF primary structurelinearly coupled to a grounded NES [79]. Isolated reso-nance captures leading to single-mode energy pump-ing occur in neighbourhoods of only one of the linearmodes of the primary structure and are dominated by the corresponding linearized eigenfrequencies (whichact as fast frequencies of the dynamics). However,multi-mode energy pumping is caused by resonancecapture cascades that involve more than one linearmode, and pumping dynamics are partitioned intodifferent frequency regimes with each regime being dominated by a different fast frequency close to aneigenfrequency of the linear system. Such resonance

capture cascades can be clearly depicted in appropri-ate FEPs, which follow the damped transitions closeto branches of the underlying Hamiltonian system asenergy decreases due to damping dissipation.

Dynamic interaction of a semi-infinite linear chain with an NES coupled at the end was investigated [80].Energypropagation through traveling waves, with pre-dominant frequencies inside the propagation zoneexciting families of localized standing waves situated




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inside the lower or upper attenuation zones, wereanalysed. Transient dynamics of a dispersive semi-infinite linear rod weakly connected to a groundedNES was investigated [81]. By means of a Green’s func-tion formulation, which reduces the dynamics to anintegro-differential equation in the form of an infi-nite set of ODEs using Neumann expansions, resonantinteraction of the NES with incident traveling wavespropagating in thepass-band of therod wasexamined.

Resonancecapture phenomena werealso investigated where the NES engages in transient 1:1 IR with thein-phase mode of the rod at the bounding frequency of its pass and stop bands, which are similar to res-onance capture cascades in finite-chain non-linearattachment configurations.

2.2.2 Ungrounded NES configurations

Apart from reference [49], dynamics of coupled lin-ear and essentially non-linear oscillators with sub-stantially different masses was investigated [82]. Twomechanisms of energy pumping were examined: (i)

through 1:1 resonance capture and (ii) non-resonantexcitation of high-frequency vibration of the NES. It

was noted that an ungrounded NES configuration canbe transformed to a grounded one through changeof variables, so no further analysis for the former isrequired.

An ungrounded NES configuration with essential(non-linearizable) cubic stiffness non-linearity cou-pled to a primary structure was investigated more rig-orously in references [83] and [84]. Unlike a groundedNES, the ungrounded configuration eliminates therestriction of relatively heavy mass of the non-linear

attachment, thus possessing the feature of simplicity.Lee et al. [83] revealed a very complicated bifurca-tion structure of symmetric and unsymmetricperiodicsolutions of the underlying undamped system on aFEP by solving using a shooting method, the two-point non-linear boundary value problem (NLBVP)formulated through suitable NSTTs based on thetwo eigenfunctions of a vibro-impact (VI) problem.Some important solution branches of 1:1 and sub-harmonic resonance manifolds, as well as of non-linear beating, are examined analytically throughthe complexification-averaging technique in terms of mode localization. Then, thetransient dynamics of the

lightly damped system was clearly shown on the FEPby superimposing wavelet transforms (WTs) of the rel-ative displacement between the primary structureandthe NES.

Furthermore, three distinct pumping mechanisms were identified [84]. The first mechanism, fundamen-tal TET, is realized when the dynamics takes placealong the in-phase, 1:1 resonance manifold occurring at the frequency domain less than the lower boundof the eigenfrequency of the linear mode. The second,

subharmonicTET, is similar to the fundamental mech-anism except that it occurs along the subharmonicsolution branches on the FEP. Finally, the third is initi-ated by non-linear beating, leading to stronger TET by exciting a special (or impulsive) periodic orbit.

Impulsive periodic orbits, as well as quasi-periodicorbits, were analysed by separately considering low-,moderate-, and high-energy impulsive motions [85].

Analytical approximations of impulsive periodic

orbits, which are separated by corresponding un-countable infinities of quasi-periodic impulsive orbits(IOs), were performed. It was shown that the impul-sive dynamics of the system is very complex due toits high degeneracy as it undergoes a codimension-3bifurcation (indeed, the equations of motion for theungrounded NES configuration can be transformedto those for a grounded NES configuration as inreference [73]).

Robustness of TETs in coupled oscillators due tochanges of initial conditions was examined in ref-erence [51]. The problem of choosing appropriateinitial conditions for achieving efficient TET in a sys-

tem of coupled oscillators with an ungrounded NES was investigated by adopting a simplified descriptionof the dynamic flow at the initial stage of motion.The analysis is complementary to the invariant man-ifold approach of reference [74]. Optimization of the(grounded) NES parameters forTET was considered inreference [86], where an experimental verification wasperformed fora reduced-scale building model with theNES located at the top floor.

Similar to references [77] to [79], multi-modal TETsfrom a two-DOF primary structure to an ungroundedNES were studied theoretically [87]. The main back-

bone curves on the FEP were computed analytically by utilizing the complexification-averaging technique (ornumerically, using optimization techniques). Modelocalization phenomenaweredepicted along the threemain backbones, and transient dynamics of the lightly damped system was investigated for in-phase andout-of-phase impulsive forcing (not surprisingly, thereexist more one-dimensional manifolds of special peri-odic orbits (SPOs)). Again, WT results were superim-posed on the FEP to demonstrate branch transitionsas the total energy decreases. Complex dynamics in atwo-DOF primary structure coupled to an MDOF NES

were investigated in reference [88], where strong pas-

siveTETcapacity(uptoasmuchas90percentofinputenergy) was identified.

Transient resonance captures (TRCs) in finite lin-ear chains, respectively, coupled to a grounded SDOFNES and to an ungrounded MDOF NES were com-pared in reference [89], where the dynamics gov-erning the chain-NES interaction was reduced to asingle, non-linear integro-differential equation thatexactly describes the transient dynamics of the NES.

Approximations based on Jacobian elliptic functions




, , , , , , ,

[90] yielded an approximate set of two non-linearintegro-differential modulation equations for ampli-tude and phase, and perturbation analysis in a O(

√ )

neighbourhood of a 1:1 resonant manifold were per-formed. For the MDOF NES, there were no detectableresonance capture cascades, but simultaneous multi-modal resonant interactions were found instead,

which suggested robust and wide applicability of TETto many engineering problems such as vibration and

shock isolation, packaging, seismic mitigation, distur-bance isolation of sensitive devices during launch of payloads in space, flutter suppression, and so forth.Similar work can be found in reference [91], whereinstantaneous frequencies of the primary structureand NES displacements were, respectively, estimatedthrough the Hilbert transform.

Broadband energy exchanges between a dissipativeelastic rod and a lightweight ungrounded SDOF NES[92, 93], as well as an MDOF NES [94], were inves-tigated rigorously. In particular, simultaneous (butnot necessarily sequential) TRCs with the MDOF NES

were demonstrated on a FEP utilizing empirical mode

decomposition (EMD [95]). Contrary to an SDOF NES, which is sensitive to the external shock (or inputenergy) level, the MDOF NES in the parameter rangesof its high efficiency exhibits robustness to changesin the amplitude of the applied shock, the coupling stiffness, and the non-linear springs.

2.2.3 Experimental studies

An experimental study of non-linear TET occurring ata single fast frequency in the system considered under

impulsive excitation on the primary structure was per-formed in reference [96]. All the previously predictedanalytical aspects were verified through experiments;in particular, an input energy threshold to bring aboutenergy pumping was clearly depicted on the plot of energy dissipation in the NES versus input energy.

Kerschen et al. [97] also experimentally showed thatnon-linear energy pumping caused by 1:1 resonancecapture is triggered bythe excitation of transientbridg-ing orbits compatible with the NES being initially atrest, a common feature in most practical applica-tions [41]. Some interesting observations were madethrough a parametric study of the energy exchanges

between the primary structure and the (grounded)NES: (i) the non-linear coefficient does not influencethe energy pumping (see also the bifurcation analy-sis [98]); (ii) the linear coupling spring must be weak in order to have an almost complete energy transferto the NES along the 1:1 resonance manifold; (iii) thestiffness should be chosen high enough to transfer asufficient amount of energy to the NES during non-linear beating; and (iv) relatively large mass for theNES should be considered for better energy transfers

(see reference [82]). An indirect analytical compar-ison via coordinate transformation suggested thatthe ungrounded NES configuration can eliminate therestriction on the large mass requirement of the non-linear attachments, which was also experimentally demonstrated in reference [99].

Transient resonance captures were experimentally demonstrated [100] by means of EMD [95]. In par-ticular, the EMD is a very useful tool for experimental

studies (i.e.system identification [101, 102]) where thesystem information is not given a priori .The theoretical work for a two-DOF primary struc-

ture coupled to an unground SDOF NES [87] wasexperimentally verified by comparison with numer-ical simulations. Experimental studies demonstratedthe usefulness of the FEP for interpreting TET mech-anisms; moreover, isolated resonance captures andresonance capture cascades were also observed.

2.2.4 Applications

Application of NESs to shock isolation was first

demonstrated in references [103] and [104]. Essen-tially non-linear stiffness elements were used forrobust energy pumping at a sufficiently fast time-scale, because fast energy pumping at the early stageis crucial for shock isolation purposes. In particular,adding two symmetrically placed NESs makes it pos-sible to achieve dual mode shock isolation to reduceunwanted disturbances generated at different ends of the primary system. It was noted that, due to theirmodular form, the NESs can be added locally in anotherwise linear system in order to globally alter thedynamics in a waycompatible to thedesignobjectives.

Dual mode non-smooth (piecewise linear) NESs werealso utilized for the purpose of shock isolation [105].Furthermore, steady-state TET from an SDOF lin-

ear primary structure under sinusoidal excitation toan attached NES was demonstrated theoretically andexperimentally [106]. A linear oscillator coupled to anungrounded NES was considered in references [107]and [108], and was transformed by proper change of variables to a system similar as the one studied inreference [106]. It was shown that the damped dynam-ics exhibits a quasi-periodic vibration regime ratherthan a steady-state sinusoidalresponse, a regime asso-ciated with attraction of the dynamical flow to a

damped-forced NNM manifold (for a more advancedanalysis, refer to references [109] and [110]). Experi-ments were also performed on an equivalent electriccircuit (see also reference [111] for energy pumping under transient forcing).

Application of TET for suppressing self-excitedinstabilities was examined. Suppression of limit cycleoscillations (LCOs) in the van der Pol (VDP) oscil-lator by means of non-linear TET was studied inreference [112]. The VDP oscillator exhibits dynamics




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analogous to non-linear aeroelastic instability. By studying the slow-flow dynamics, extracted throughthe complexification-averaging technique, and per-forming numerical continuation of equilibria andlimit cycles, bifurcation structures of LCOs and thepossibility of robust LCO suppression were parametri-cally investigated. It was concluded that a steady-stateis reached through a series of TRCs that can be clearly represented in a FEP. In particular, it was demonstrated

that, in order to suppress instability in the VDP oscil-lator, a sequence of superharmonic and subharmonicresonant interactions between the VDP oscillator andthe NES must take place.

The triggering mechanism of aeroelastic instabil-ity in a two-DOF (heave and pitch), two-dimensionalrigid wing under subsonic quasi-steady aerodynam-ics was examined in reference [113]. It was found thatthe LCO-triggering mechanism consists of three dif-ferent dynamic phenomena: a series of TRCs, escapesfrom these captures and, finally, entrapment into per-manent resonance capture (PRC). An initial excitationof the heave mode by the flow acts as the trigger of

the pitch mode through a series of non-linear modalinteractions. Moreover, both the initial triggering andfull development of LCOs are transient phenomena,so that one can properly design an NES attachment tothe wing for their suppression.

Based on these observations, an ungrounded SDOFNES was applied to the two-DOF rigid wing, andsuppression of aeroelastic instability through passiveTETs was investigatedboth theoretically [114, 115]andexperimentally [98]. Three distinct suppression mech-anisms were identified: (i) recurring suppressed burst-outs, (ii) intermediate, and (iii) complete elimination

of aeroelastic instability. Those suppression mecha-nisms were identified with the bifurcation structureof LCOs obtained through a numerical continuationtechnique. Furthermore, the robustness of the aeroe-lastic instability suppression was examined. In orderto enhance robustness of aeroelastic instability sup-pression, the MDOF NES first considered in references[89] and [94] was considered instead of the SDOFNES. Bifurcation analysis showed that robustness of instability suppression by means of simultaneousmulti-modal resonant interactions due to the MDOFNES can be greatly enhanced, with a much smallertotal mass of the MDOF NES. Non-linear modal

energy exchanges were studied for various parameterconditions.

Seismic mitigation of a reduced two-DOF model[86, 111] and of an MDOF model [71], with an NES onthe top floor, was studied. Since an NES with smoothstiffness non-linearities is not suited to suppress thepeak seismic responses at the critical early regime of the motion, alternativenon-smoothVI NESs werecon-sidered in references [116] to [118]. Effective seismic

mitigation through the use of VI NESs was demon-strated both numerically and experimentally in these

works.Other applications of passive TETs include suppres-

sion of stick-slip self-excited vibrations in a drill-string problem [119], and acoustic energy pumping [120].

2.3 Useful definitions

In this section, concepts of resonance captures asso-ciated with the averaging theorem are reviewed tosupport the discussion of non-linear TET that follows.

Definition 1 (Resonance Manifold [121])

Consider the system in polar form with multi-phaseangles

r = R (φ, r ), φ = Ω(r ) (1)

where r ∈ Rp, φ ∈ T q (generally, q p), (r ) =(1(r ), . . . , q (r )), and the dimension of r may be

greater than that of the original dynamical systemdepending on frequency decompositions. The set of points in D ⊂ Rp where i (r ) = 0, i = 1, . . . , q is calledthe resonance manifold. This resonance conditionis not sufficient; that is, if each i (r ), i = 1, . . . , q isaway from zero, the IR manifold is defined as the set{r ∈ Rp :< k , (r ) 0, k ∈ Zq } where the correspond-ing Fourier coefficients from R(φ, r ) are not identically zero.

Assume that the averaged system of equation (1)intersects transversely the resonant manifold. Then,capture into resonance may occur for some phase

relations satisfying the condition that an orbit of the dynamical system reaching the neighbourhood of theresonantmanifold continuesin such a waythat thecommensurable frequency relation is approximately preserved. In this situation, not all phase angles arefast (time-like) variables, so classical averaging cannotbe performed with regard to these angle variables. Asa result, over the time-scale −1 the exact and aver-aged solutions for equation (1) diverge up to O(1)

[60, 122, 123].

Definition 2 (Sustained and transient resonances

[124])Suppose that (internal) resonance occurs at a timeinstant t = t 0, with the non-trivial frequency combina-tion σ = k 1ω1 + k 2ω2 + . . . + k q ωq , k i ∈ Z, i = 1, . . . , q ,vanishing at that time instant (t = t 0). Then, sustainedresonance is defined to occur when σ ≈ 0 persists fortimes t − t 0 = O(1). On the other hand, transient res-onance refers to the case when σ makes a single slow passage through zero.




, , , , , , ,

Definition 3 (Capture, escape, and pass-through [64])

The possible behaviour of trajectories near the res-onance manifold on the time-scale −1 is describedaccording to the following three cases: (i) capture:solutions are unbounded in backward time. How-ever, captured trajectories remain bounded for for-

ward times of O(−1), i.e. a sustained resonanceexists in forward time; (ii) escape: solutions grow unbounded in forward time. However, in backward

time, solutions remain bounded for times of O(−1),i.e. a sustained resonance exists in backward time;(iii) pass-through: solutions do not remain in theneighbourhood of the resonance manifold in eitherforward or backward time. No sustained resonanceexists.

A mechanism for resonance capture in perturbedtwo-frequency Hamiltonian systems was studied by Burns and Jones [61] where the most probable mech-anisms for resonance capture were shown to involvean interaction between the asymptotic structures of the averaged system and a resonance. It was further

shown that, if the system satisfies a less restrictivecondition (or Condition N [125]) regarding transver-sal intersection of the averaged orbits to the resonancemanifold, resonance capture can be viewed as anevent with low probability, and passage through res-onance is the typical behaviour on the time-scaleO(−1).

Necessary conditions were proved by Kath [56]both for entrainment to sustained resonance and forits continuance (and thus the possible indication of unlocking or escape from the sustained resonanceafter a finite time) by successive near-identity trans-

formations; a sufficient condition was also derivedfor continuation of sustained resonance by means of matched asymptotic expansions [57].

On the other hand, transition to escape was stud-ied by Quinn [65] in a coupled Hamiltonian sys-tem consisting of two identical oscillators possessing a homoclinic orbit when uncoupled. Focusing onintermediate energy levels at which sustained reso-nant motion occurs, the existence and behavior of those motions were analysed in equipotential surfaces

whose trajectories are shown to remain in the tran-siently stochastic region for long times and, finally,to escape or leak out of the opening in the equipo-

tential curves and proceeding to infinity. Regarding passage through resonance, one may refer to refer-ences [126] to [128]. The phenomenon of passagethrough resonance is sometimes referred to as non-stationary resonances caused by excitations having time-dependent frequencies and amplitudes [129].

Finally, the following definitions for non-linear res-onant interactions between modes are introduced

when the multi-frequencycomponents of a systemaretaken into account.

Definition 4 (IR, TRC, and PRC [61])

Consider an unforced n-DOF systemwhose linear nat-ural frequencies are ωk , k = 1, . . . , n. The author (i) IRas motions for which there exist k i ∈ Z, i = 1, . . . , n,such that k 1ω1 + k 2ω2 + · · · + k nωn ≈ 0, i.e. some com-bination of linear natural frequencies satisfy com-mensurability; (ii) TRC as capture into a resonancemanifold which occurs and continues for a certainperiod of time (e.g. on the time-scale −1) and then

finally involves transition to escape. This includes sus-tained resonance captures involving escape; (iii) PRCas sustained resonance captures that will never escapefor increasing time.

Both TRC and PRC may occur along the IR man-ifold and are distinguished by whether or not they involve an escape. Both IR and PRC may show sim-ilar steady-state behaviours, which differ from thecommensurability condition between linear naturalfrequencies. Hereafter, a m : n IR refers to a conditionon the slow-flow averaged system unless noted other-

wise. For more details on the averaging theorem andresonance captures in multi-frequency systems, onecan also refer to references [60], [62], [63], [125], [130],and [131].

2.4 Analytical and numerical tools

2.4.1 Perturbation methods

There are many perturbation techniques to computeperiodic solutions of a non-linear system: the meth-ods of multiple-scales, of averaging, and of harmonic

balance [132]. Although each of these methods hasits own features, they are fundamentally equivalent toeach other. One restriction to their application is theassumption of weak non-linearity; that is, the derivedanalytical solutions of the non-linear system lie closeto those of the corresponding linearized system. Theaveraging theorem provides validity of the approxima-tion, generally up to the time-scale −1. Although theharmonic balance method (HBM) can be applied tostrongly non-linear systems, it approximates only thesteady-state responses. On the other hand, the meth-ods of averaging and of multiple-scales can be appliedto the study of transient dynamic behaviour, which

is suitable for understanding nonlinear TET phenom-ena. An application of the averaging method to theresonance capture problem can be found in reference[133] (and see [134] for the HBM).

Since the essentially non-linear coupling between aprimary system and an ungrounded NES is not neces-sarily weak, the complexification-averaging techniquefirst introduced by Manevitch [135] will be utilizedin the following analysis as an analytical tool forunderstanding resonance capture phenomena. Use of




-

complex variables renders relatively easier manipula-tion of the resulting modulation equations (particu-larly, in the presence of multi-frequencycomponents).In addition, this method is applicable to strongly nonlinear systems. For some analyses, the multiple-scale method is utilized instead of averaging (e.g.[49, 73, 74]).

2.4.2 Non-smooth temporal transformations

Non-smooth time transformations (NSTTs) can alsobe utilized to compute periodic solutions of a(strongly) non-linear system [72, 136–139]. Unlike theusual perturbation methods that implement the basisof sine and cosine functions (or elliptic functions insome cases), the NSTTs employ saw-tooth and square

wave functions as the basis (see reference [140] forother types of non-smooth basis functions and theirapplications). Any periodic solutions can be expressedin terms of asymptotic series expansion of these twonon-smooth functions; moreover, this technique canbe applied to solutions of a discontinuous system such

as a VI oscillator. Application of NSTTs to the problem of computing

the periodic solutions of a dynamical system yieldsNLBVPs, which are solved by means of numericalschemes such as the shooting method [141].

2.4.3 Stability evaluation and bifurcation analysis

Once periodic solutions are obtained, their stability can be evaluated: (i) by direct numerical integration of equations of motion; (ii) by computing their Floquetmultipliers [142]; or (iii) by studying the topological

structure of numerical Poincaré maps [143]. Then,bifurcation diagrams can be constructed with respectto control parameters, or other induced parameterssuch as the total energy of the system.

Bifurcation analysis [144] of periodic solutions ina coupled oscillator is crucial in order to understandtransitions that occur in the damped dynamics orto enhance robustness of instability suppression by means of passive TETs. Methods of numerical contin-uation of equilibria and limit cycles can be utilized. Inparticular, AUTO [145] and MatCont [146] can easily be employed.

2.4.4 Time–frequency analysis

Understanding transient modal interactions during non-linear TET requires an integrated time–frequency analysis [147–151]. The most popular techniquesinclude the EMD method and the WT. WTs havefound applications in non-linear system identifica-tion, e.g. characterization of structural non-linearitiesand prediction of LCOs of aeroelastic systems [152];free vibration analysis of non-linear systems [153];

damage size estimation or fault detection in structures[154, 155].

TheWTcanbeviewedasabasisforfunctionalrepre-sentation but is at the same time a relevant techniquefor time–frequency analysis. In contrast to the FastFourier Transform (FFT), which assumes signal sta-tionarity, theWT involves a windowing technique withvariable-sized regions. Small time intervals are con-sidered for high-frequency components, whereas the

size of the interval is increased for lower frequency components, thereby givingbetter time and frequency resolutions than the FFT.

The Matlab

codes used for the WT computationsin this paper were developed at the Université deLiège (Liège, Belgium) by Dr V. Lenaerts in collab-oration with Dr P. Argoul from the Ecole Nationaledes Ponts et Chaussées (Paris, France). Two typesof mother wavelets ψ M (t ) are considered: (a) theMorlet wavelet, which is a Gaussian-windowed com-plex sinusoid of frequency ω0, ψ M (t ) = e−t 2/2e j ω0t ; (b)the Cauchy wavelet of order n, ψ M (t ) = [ j /(t + j )]n+1,

where j 2

= −1. The frequency ω0 for the Morlet WT

and the order n for the Cauchy WT are user-specifiedparameters which allow one to tune the frequency and time resolutions of the results. It should be notedthat these two mother wavelets provide similar results

when applied to the signals considered in this paper.TheplotsshownrepresenttheamplitudeoftheWTasafunctionof frequency (vertical axis)and time(horizon-tal axis). Heavily shaded areas correspond to regions

where the amplitude of the WT is high, whereas lightly shaded regions correspond to low amplitudes. Suchplots enable one to deduce the temporal evolutionsof the dominant frequency components of the signals

analysed. Alternatively, the EMD gained popularity in the areaof signal processing and is also utilized in this work.Originally introduced by Huang et al. [95, 156, 157],it was shown to be applicable to strongly non-linearand non-stationary signals with non-zero mean. Inan alternative numerical post-processing technique,the EMD through a sifting process yields a collec-tion of intrinsic mode functions (IMFs), which forma complete, nearly orthogonal, local, and adaptivebasis. These properties render the EMD applicableto decomposition of non-linear and non-stationary signals.

Once EMD is performed, the obtained IMFs aresuitable for Hilbert transformation, which yields theinstantaneous amplitudeand phase of each IMFat any given instant of time. By differentiating the instanta-neous phase, one computes the temporal evolutionof the instantaneous frequency of each IMF which,

when compared with the overallWT of the time series,enables one to judge the relative contribution of eachIMFin thetime seriesand, thus, itsrelative importancein the decomposition of the signal.




, , , , , , ,

Note that an IMF may have a significant con-tribution in certain time intervals of the signal, andbe less important in others. Hence, EMD coupled withthe Hilbert transformcanbe a powerfulcomputationaltool for studying complicated non-linear resonanceinteractions leading to complex dynamic phenomena(such as TET) in coupled structures. Recently, a timeseries forecasting method based on support vectorregression machines was proposed, its apparent supe-

riority attributed to the use of neural networks [158]. An effort was made to improve the quality (i.e. orthog-onality) of the obtained IMFs by means of an energy difference tracking method [159]. The EMD methodcan also be applied to problems of fault diagnosis anddamage estimation [160, 161]. In this paper, Matlab

codes developed by Rilling et al.[162] are employed toperform the described EMD analysis.

3 DYNAMICS OFTET

In order to establish a clear understanding of non-

linear energy pumping mechanisms, a SDOF pri-mary oscillator coupled to an ungrounded SDOF NES[83, 84] is considered in this section. For a SDOF pri-mary structure coupled to a grounded SDOF NES, onecan refer to references [41], [50], and [97].

3.1 Undamped periodic solutions

The system under consideration is depicted in Fig. 2,and consists of an oscillator of mass m1 (the linearoscillator) coupled through an essentially non-linearstiffness to a mass m2 (the non-linear attachment).

The equations of motion of this two-DOF system aregiven by

m1 ¨ x + k 1 x + c 1 ˙ x + c 2( ˙ x − v ) + k 2( x − v )3 = 0

m2v + c 2(v − ˙ x ) + k 2(v − x )3 = 0

⇒ ¨ x + ω20 x + λ1 ˙ x + λ2( ˙ x − v ) + C ( x − v )3 = 0

v + λ2(v − ˙ x ) + C (v − x )3 = 0

(2)

where ω20 = k 1/m1, C = k 2/m1, = m2/m1, λ1 = c 1/m1,

and λ2 = c 2/m1.

Before analysing non-linear TET phenomena in thedamped system, it is first necessary to examine the

Fig. 2 The two-DOF system with essential stiffness

non-linearity

structure of the periodic orbits of the underlying undamped system (with λ1 = λ2 ≡ 0). Indeed, it willbe shown that this seemingly simple system possessesa very complicated topological structure of periodicorbits, some of which areresponsible forTET phenom-ena in the impulsively forced, damped system.

3.1.1 Numerical approach

The periodic orbits of the system will be computed

numerically utilizing the method of non-smoothtransformations first developed by Pilipchuk [163]and then applied to strongly non-linear oscillators by Pilipchuk et al. [72].This method can be applied to thenumerical and analytical study of the periodic orbits(and their bifurcations) of strongly non-linear dynam-ical systems. To apply the method, the sought periodicsolutions are expressed in terms of two non-smoothvariables, τ and e , as

v (t ) = e

t

α

y 1

τ

t

α

,

x (t ) = e

t

α

y 2

τ

t

α

(3)

where α = T /4 represents the (yet unknown) quarter-period. The non-smooth functions τ (u) and e (u) aredefined according to the expressions

τ (u) = 2

π

sin−1

sinπ

2u

, e (u) = τ (u) (4)

and are used to replace the independent time variable

from the equations of motion; their graphic depictionis given in Fig. 3.

Fig. 3 The non-smooth functions τ (u) and e (u)




-

Setting λ1 = λ2 = 0, and substituting equation (3)into equation (2), smoothening conditions [72] areimposed to eliminatesingular terms from the resulting equations, such as terms proportional to

e ( x ) = τ ( x ) = 2

∞k =−∞

[δ( x + 1 − 4k ) − δ( x − 1 − 4k )]

Setting to zero, the component of the transformedequations that is multiplied by the non-smooth vari-able e , the following two-point NLBVP is formulatedin terms of the non-smooth variable τ , in the inter-val −1 τ +1

y 1 = y 3, y 2 = y 4, y 3 = −C

α2( y 1 − y 2)3,

y 4 = −ω20α2 y 2 − C α2( y 2 − y 1)3

(5)

with the boundary conditions, y 1(±1) = y 2(±1) = 0, where primes denote differentiation with respect tothe non-smooth variable τ , and a state formulation is

utilized. The boundary conditions above result fromthe aforementioned smoothing conditions.

Hence, the problem of computing the periodic solu-tions of the undamped system (2) is reduced to solving the NLBVP (5) formulated in terms of the boundedindependent variable τ ∈ [−1, 1], with the quarter-period α playing the role of the non-linear eigenvalue.It is noted that the solutions of the NLBVP can beapproximated analytically through regular perturba-tion series [72]; however, this will not be attemptedherein where only numerical solutions will be consid-ered. It is merely mentioned here that equation (5) is

amenable to direct analytical study in terms of simplemathematical functions.It is noted that the NLBVP (5) provides the solu-

tion only in the normalized half-period −1 t /α 1⇒ −1 τ 1. To extend the result over a full nor-malized period equal to four, one needs to add thecomponent of the solution in the interval 1 t /α 3;to perform this one takes into account the symme-try properties of the non-smooth variables τ and e by adding the antisymmetric image of the solutionabout the point ( y i , t /α) = (0, 1), as shown in Fig. 4.It follows by construction that the computed peri-odic solutions satisfy the initial conditions, x (

−α)

=v (−α) = 0 and v (−α) = y 1(−1)/α, ˙ x (−α) = y 2(−1)/α.It is noted at this point that since equation (2) is anautonomous dynamical system these initial condi-tions can be shifted arbitrarily in time; for example,they can be applied to the initial time t = 0 insteadof t = −α = −T /4. However, in what follows the for-mulation of the NLBVP (5) will be respected, and theinitial conditions at t = −T /4 are retained.

Considering the general shape of the periodic orbitsdepicted in Fig. 4, the following classification of

Fig. 4 Construction of the periodic solutions

v (t ) = e (t /α) y 1(τ (t /α)), x (t ) = e (t /α) y 2(τ (t /α))

over an entire normalized period −1 t /α 3

from the solutions y i (τ (t /α)), i = 1, 2 of the

NLBVP (5) computed over the half-normalized

period −1 t /α 1

periodic solutions is introduced.

1. Symmetric solutions Snm ± correspond to orbitsthat satisfy the conditions

v

−T

4

= ±v

+T

4

⇒ y 1(−1) = ± y 1(+1)

˙ x

−T

4

= ±˙ x

+T

4

⇒ y 2(−1) = ± y 2(+1)

with n being the number of half-waves in y 1(v ),and m the number of half-waves in y 2( x ) in the

half-period interval −T /4 t +T /4 ⇐⇒ −1 τ +1.

2. Unsymmetric solutions Unm are orbits that donot satisfy the conditions of the symmetric orbits.Orbits U (m + 1)m bifurcate from the symmet-ric solution S 11 − at T /4 ≈ mπ/2, and existapproximately within the intervals mπ/2 < T /4 <

(m + 1)π/2, m = 1,2, . . . .

The numerical solution of the two-point NLBVP(5) is constructed utilizing a shooting method pro-grammed in Mathematica

(see references [141] and

[164] for some details on the shooting method andgeneral characteristics of global solutions).

The NLBVP (5) is solved as follows.

1. For a given non-linear eigenvalue α (quarter-period), the solutions of the NLBVP are computedat different energy levels; it is expected that at every energylevel there co-exist multiple non-linear peri-odic solutions sharing the same minimal period.Periodic orbits that correspond to synchronousmotions of the two particles of the system, and




, , , , , , ,

pass through the origin of the configuration plane( y 1, y 2), are termed NNMs [165].

2. The different families of computed periodic solu-tions are depicted in three types of plots. In the firsttwo types of plots, initial displacements x (−T /4) =v (−T /4) = 0 are assumed, and the initial velocitiesv (−T /4) = y 1(−1)/α and ˙ x (−T /4) = y 2(−1)/α cor-responding to a periodic orbit as functions of thequarter-period α = T /4 or the (conserved) energy

of that orbit are depicted

h = 1

2

v 2

−T

4

+ ˙ x 2

−T

4

= 1

2α2[ y 1

2(−1) + y 2

2(−1)]

In the third type of plots, the frequencies of theperiodic orbits are depicted as functions of theirenergies h. These plots clarify the bifurcationsthat connect, generate, or eliminate the differentbranches (families) of periodic solutions.

3. The stability of the computed periodic orbits was

determined numerically by three different meth-ods: application of Floquet theory; construction of two-dimensional Poincaré maps on the isoener-getic manifolds of the two-DOF undamped system(2); and direct numerical simulation of the equa-tions of motion using as initial conditions thoseestimated by the solution of the NLBVP (5).

In the following, the numerical results correspondto the two-DOF undamped system with parameters = 0.05, ω0 = 1, C = 1.0intheenergyrange0 < h < 1.The bifurcation diagrams of the initial velocities andfor varying quarter-period are depicted in Fig. 5. Somegeneral and preliminary observations on the com-puted periodic orbits are made at this point, and thedynamical behaviour of the system on the variousbranches will be discussed in the next section.

Considering the branches Snn−, they exist in thequarter-period intervals 0 < α < nπ/2, and their ini-tial conditions satisfy the limiting relationships (Fig. 5)

limα→0

{|v (−α)|, | ˙ x (−α)| } = ∞,

limα→nπ/2

{|v (−α)|, | ˙ x (−α)|} = 0

These symmetric branches existthroughout the exam-ined energy domain 0 < h < 1. It is noted thatbranches Snn− are, in essence, identical to the branchS 11−, since they are identified over the domain of their common minimal period (the Snn− branchesare branches S 11− ‘repeated n times’); similar remarkscan be made regarding the branches S (kn)(km)±, k integer, which are identified with Snm±.

Focusing in the neighbourhood of branches S 11±and referring to Fig. 5, at the point α = π/2 where

Fig. 5 Normalized initial velocities of periodic orbits

y i (−1), i = 1, 2 as functions of the quarter-period

α; solid (dashed) lines correspond to positive

(negative) initial velocities (S 11 (

), S 13 (), S 15

(), S 31 (), S 21 (♦) with in-phase as filled-in, and

branches U without symbol) [83]

S 11−disappears the branches S 11+ and U 21 bifurcateout (similar behaviour is exhibited by the branchesS 31, S 21, . . .). For π/2 α π, a bifurcation fromS 11+ to S 13+ takes place without change of phase;similar bifurcations take place at higher values of α forbranches S 15+, S 17+, . . .. For α ≈ 3π/2, the branchesS 13+ and S 13− coalesce into the branch S 11−, withsimilar coalescences into S 11− taking place at highervalues of α for the pairs of branches S 15

+, S 17

+, . . ..

The unsymmetric branches U (m + 1)m bifurcatefrom the symmetric branches S (m + 1)(m + 1)− atquarter-periods equal to α = mπ/2. It turns out thatcertain orbits (termed ‘SPOs’) on these branchesare of particular importance concerning the passiveand irreversible energy transfer from the linear tothe non-linear oscillator. The special orbits satisfy the additional initial condition y 1(−1) = v (−α) = 0,and correspond to zero crossings of the branchesU (m + 1)m in the bifurcation diagram (the upper




-

Fig. 6 Special periodic orbits on the U -branches with initial conditions y 1(−1) = y 2(−1)

= 0, y 1(−1) = 0, y 2(−1) = 0; Unm(a) and Unm(b) denote the unstable and stable SPOs,

respectively ( y 1(τ ) is represented by a solid line and y 2(τ ) by a dashed line; x -axis

represents τ )

plot); some of these special orbits (either stable orunstable)are depicted in Fig. 6.Taking into account theformulation of the NLBVP (5),it follows that thespecialorbits satisfy initial conditions v (−T /4) = v (−T /4) =

x (−T /4) = 0, and ˙ x (−T /4) = 0, which happen to beidentical to the state of the undamped system (2)

(being initially at rest) after application of an impulseof magnitude ˙ x (−T /4) = y 2(−1)/α on the linear oscil-lator.

Moreover, comparing the relative magnitudes of the linear and non-linear oscillators for the specialorbits of Fig. 6, it is concluded that certain stablespecial orbits are localized to the non-linear oscil-lator. This implies that if the system is initially atrest and is forced impulsively, and one of the stable,localized special orbits is excited, a major portion

of the induced energy is channeled directly to theinvariant manifold of that special orbit, and, hence,the motion is rapidly and passively transferred fromthe linear to the non-linear oscillator. Moreover, thisenergy transfer is irreversible because of the invari-ance properties of the stable special orbit, and, as

a result, after the energy is transferred, it remainslocalized and is passively dissipated at the non-linearattachment. Therefore, it is assumed that the impul-sive excitation of one of the stable special orbits isone of the triggering mechanisms initiating (direct)passive TET. This conjecture will be proven to be cor-rect by numerical simulations presented in a latersection.

Similar classes of special orbits can also be real-ized in a subclass of S -branches. In particular, this




, , , , , , ,

Fig. 7 Frequency–energy plot of the periodic orbits; for the sake of clarity no stability is indicated,

special orbits are denoted by filled circles (•; some appear unfilled due to the overlap-

ping symbols) and are connected by dashed-dot lines; other symbols indicate bifurcation

points (stability–instability boundaries): () four Floquet multipliers at +1; (♦) two Floquet

multipliers at +1 and the other two at −1 [83]

type of orbit can be realized on branches S (2k + 1)

(2p + 1)±, k = p, but not on periodic orbits that donot pass through the origin of the configuration plane(such as S 21, S 12, . . .). The branch S 11− is a particular

case, where the special orbit is realized only asymp-totically as the energy tends to zero, and the motion islocalized completely in the linear oscillator.

In Fig. 7, the various branches of solutions are pre-sented in a FEP. For clarity, the following conventionregarding the placement of the various branches inthe frequency domain is adopted: a specific branch of solutions is assigned with a frequency index equal tothe ratio of its two indices, e.g. S 21± is representedby the frequency index ω = 2/1 = 2, as is U 21; S 13±

is represented by ω = 1/3, and so forth. This con-vention holds for every branch except S 11±, which,however, are particular branches. On the energy axis,the (conserved) total energy of the system is depicted

when it oscillates in a specific mode. Necessary (butnot sufficient) conditions for bifurcation and stability–instability exchanges are satisfied when two Floquetmultipliers of the corresponding variational problemscoincide at +1 or −1 (since periodic orbits of a Hamil-tonian two-DOF system are considered, two Floquetmultipliers of the variational problem arealways equalto +1, whereas the other two form a reciprocal pair),and these are indicated at the solution branches of Fig. 7.




-

Fig. 8 Close-ups of particular branches in the frequency index–logarithm of energy plane: (a)

S 11−; (b) S 11+; (c) S 13±; (d) U 43 (double branch). Stability–instability boundaries are

represented as in Fig. 7; some representative periodic orbits are also depicted in insets in

the format x (v ) (configuration plane of the system); SPOs on the U - and S -branches are

indicated by triple asterisks. Arrowed lines indicate the intervals of instability [ 83]

To understand the types of periodic motions thattake place in different frequency–energy domains, cer-tain branches are depicted in detail in Fig. 8, together

with the corresponding orbits realized in the con-figuration plane of the system. The horizontal andvertical axes in the plots in the configuration plane arethe non-linear (v ) and linear oscillator ( x ) responses,respectively; the aspect ratios in these plots are set sothat the tick mark increments on the horizontal andvertical axes are equal in size, enabling one to directly

deduce whether the motion is localized in the linear orthe non-linear oscillator. The plot for U 43 (Fig. 8(d)) iscomposed of two very close branches; for the sake of clarity only one of the two branches is presented. Themotion is nearly identical on the two branches, so only the oscillationsin the configuration plane of one of thetwo branches are considered.

Since a systematic analytical study of the varioustypes of periodic solutions of the system is presentedin the next section, the following preliminary remarks

are made.

1. The main backbone of the FEP is formed by thebranches S 11± which represent in- or out-of-phasesynchronous vibrations of the two particles pos-sessing one half-wave per half-period. Moreover,the natural frequency of the linear oscillator ω0 = 1(which is identified with a frequency index equalto unity, ω = 1) naturally divides the periodic solu-tions into higher and lower frequency modes. There

aretwo saddle-node-type bifurcations in the higherfrequency, out-of-phase branch S 11−, and the sta-ble solutions become localized to x or v as ω → 1+or ω 1, respectively (see Fig. 8(a)). The lower fre-quency, in-phase branch S 11+ becomes unstableat higher energies, and the stable solutions localizeto the non-linear attachment as ω decreases away from ω = 1 (see Fig. 8(b)).

2. There is a sequence of higher and lower frequency periodic solutions bifurcating or emanating from




, , , , , , ,

branches S 11±. Considering first the symmetricsolutions, the branches S 1(2k + 1)±, k = 1,2, . . .

appear in the neighbourhoods of frequencies ω =1/(2k + 1), e.g. at progressively lower frequencies

with increasing k . For fixed k , each of the twobranches S 1(2k + 1)± is linked through a smoothtransition with its neighbouring branches S 1(2k − 1)± or S 1(2k + 3)±, and exists over a finiteinterval of energy. The pair S 1(2k + 1)± is elim-

inated through a saddle-node-type bifurcation ata higher energy value (see Fig. 8(c) for branchesS 13±). The pairs of branches S 1(2k )±, k = 1,2, . . .

bifurcate out of S 1(2k + 1)±, and exist over finiteenergy intervals. All branches S 1n± and Sn1±, n ∈Z

+ seem to connect with S 11− through ‘jumps’ inthe FEP, but in actuality no such discontinuitiesoccur if one takes into account that due to theprevious frequency convention solutions Spp+ areidentified with the solution S 11+, S (2p)(1p)± withS 21±, etc.

3. Focusing now on the unsymmetrical branches, afamily of U (m

+1)m branches bifurcating from

branch S 11− exists over finite energy levels andare eliminated through saddle-node-type bifur-cations with other branches of solutions. Again,the transitions of branches U 21 and U 32 toS 11+ seem to involve ‘jumps’, but this is only due to the frequency convention adopted, andno actual discontinuities in the dynamics occur.

An additional interesting family of unsymmet-rical solutions is Um(m + 1), m = 1,2, . . . which,due to the previous frequency convention, isdepicted for frequency indices ω < 1; the shapesof these orbits in the configuration plane are

similar to those of U (m + 1)m, m = 1,2, . . . , butrotated by π/2. An important class of periodicorbits realized on the unsymmetrical branches(but also in certain of the symmetric branches)is that corresponding to all initial conditionszero, with the exception of the initial veloc-ity of the linear oscillator. These special orbitsprovide one of the mechanisms for passiveTET from the linear oscillator to the non-linearattachment [84].

The previous discussion indicates that the two-DOFundamped system possesses complicated structures

of symmetric and unsymmetrical periodic orbits. Thenext section will focus on the analysis of the com-puted periodic orbits in detail in an effort to betterunderstand the dynamics and localization propertiesof the system over different frequency–energy ranges.Indeed, understanding the periodic dynamics of theundamped system paves the way to explain passiveTET phenomenaand complicated transitions betweendifferent types of motion in the transient dynamics of the damped system.

3.1.2 Analytical approach

The dynamics of the undamped system and all thedifferent branches of solutions can be studied analyt-ically. As representative examples of this analysis, theperiodic orbits on a particular branch, namely S 11±,are investigated in detail.

To study the periodic orbits of equation (2) for 0 <

1, the complexification-averaging method firstintroduced by Manevitch [135] is applied, which not

only enables the study of the steady-state motions, butalso can be applied to analyse the damped, transientdynamics [50].

The S 11± branch is composed of synchronous peri-odic motions where the two particles oscillate withidentical frequencies. The analytical study of thesesolutions is performed by introducing the new com-plex variables ψ1 = ˙ x + j ω x and ψ2 = v + j ωv where

j 2 = −1, and expressing the displacements and accel-erations of the two particles of the system as (theasterisk denotes complex conjugatation)

x =1

2 j ω (ψ1 − ψ∗1 ), ¨ x = ψ1 −j ω

2 (ψ1 + ψ∗1 )

v = 1

2 j ω(ψ2 − ψ∗

2 ), v = ψ2 − j ω

2(ψ2 + ψ∗

2 )

(6)

Since nearly monochromatic periodic solutions of theequations of motion are sought and the two particlesoscillate with the identical frequencies, the previ-ous complex variables are approximately expressedin terms of ‘fast’ oscillations of frequency ω, e j ωt ,modulated by ‘slow’ (complex) modulations φi (t )

ψ1 = φ1e j ωt

, ψ2 = φ2e j ωt

(7)

This amounts to a partition of the dynamics intoslow- and fast-varying components, and the interest-ing dynamics is reduced to the slow flow. Note thatno a priori restrictions are posed on the frequency ω of the fast motion. Substituting equations (6) and(7) into the equations of motion (2) with λ1 = λ2 = 0,and performing averaging over the fast frequency, toa first approximation only terms containing the fastfrequency ω are retained

˙φ1

+ j ω

1

2φ1

− j

1

2ω

φ1

+ j

3C

8ω3

(

−|φ1

|2φ1

+φ2

1 φ∗2

− φ22 φ∗

1 + |φ2|2φ2 + 2|φ1|2φ2 − 2|φ2|2φ1) = 0

φ2 + j ω

2φ2 − j

3C

8ω3(−|φ1|2φ1 + φ2

1 φ∗2 − 3φ2

2 φ∗1

+ |φ2|2φ2 + 2|φ1|2φ2 − 2|φ2|2φ1) = 0

(8)

These complex modulation equations govern the slow evolutions of the complex amplitudes φi , i = 1, 2 intime.




-

Introducing the polar representations φ1 = Ae j α andφ2 = Be j β where A, B, α, β ∈ R in equation (8), andseparately setting the real and imaginary parts of theresulting equations equal to zero, the following realmodulation equations that govern the slow evolutionof amplitudes and phases of the two responses of thesystem are obtained as

˙ A + BC

8ω3[(3 A2 + 3B2) sin(α − β)

+ 3 AB sin(2β − 2α)] = 0

Aα + ω A

2− A

2ω− 3CA3

8ω3− 6 AB2C

8ω3− BC

8ω3

× [(−9 A2 − 3B2) cos(α − β)

+ 3 AB cos(2β − 2α)] = 0

B − AC

8ω3[(3B2 + 3 A2) sin(α − β)

+ 3 AB sin(2β − 2α)] = 0

Bβ + ωB

2− 3B3C

8ω3− 6 A2BC

8ω3− AC

8ω3

× [(

−9B2

−3 A2) cos(α

−β)

+ 3 AB cos(2β − 2α)] = 0

(9)

The first and third (amplitude modulation)equations are combined, giving

A ˙ A + BB = 0 ⇒ A2 + B2 = N 2

where N is a constant of integration. Clearly, theabove is an energy conservation relation reflecting theconservation of the total energy of the undamped sys-tem (2) during its oscillation. Hence, the modulation

equation (9) can be reduced by one, by imposing theabove energy conservation algebraic relation.

The periodic solutions on the branches S 11± arestudied by setting the derivatives with respect to timein equation (9) equal to zero; i.e. by imposing station-arity conditionson the modulation equations. The firstand third equations are trivially satisfied if α = β, andthe second and fourth equations become

ω A − A

ω− 3C

4ω3( A − B)3 = 0,

ωB + 3C

4ω3( A − B)3 = 0

(10)

These equations can be solved exactly for the ampli-tudes A and B, leading to the following approximationsfor the periodic solutions on the branches S 11±

x (t ) ≈ X cos ωt = ψ1 − ψ∗1

2 j ω= A

ωcos ωt

= −εω2

ω2 − 1

4ω2ε(ω2 − 1)3

3C ((1 + ε)ω2 − 1)3cos ωt

v (t ) ≈ V cos ωt = ψ2 − ψ∗2

2 j ω= B

ωcos ωt

=

4ω2ε(ω2 − 1)3

3C ((1 + ε)ω2 − 1)3cos ωt

(11)

Considering the original non-linear problem (2), notethat relations (11) are approximate since a single fast

frequency was assumed in the slow–fast partitions (7),and only terms containing this fast frequency wereretained after performing averaging in the complex equations (8).

It is interesting to note that the ratio of the ampli-tudes of the linear and non-linear oscillators onbranches S 11± is given by the following simple form

X

V = −ω2

ω2 − 1(12)

This relation shows that if the mass of the non-linearoscillator is small (as is assumed), and if the frequency

ω is not in the neighbourhood of the eigenfrequency of the linear oscillator ω0 = 1, the motion is always local-ized to the non-linear oscillator (in agreement withthe numerical results); however, sufficiently close toω0 = 1, the oscillation localizes on the linear oscillator(as one would expect intuitively).

There is a region in the frequency domain,√ 1/(1 + ) < ω < 1, where the coefficients X and V

are imaginary, indicating that no periodic motion onS 11± canoccur there; this represents a forbidden zonenot only for S 11±, but also for any periodic motionof the system. Accordingly, the branch S 11+ of in-phase oscillations exists for ω <

√ 1/(1

+), whereas

out-of-phase oscillations on S 11− exist for ω > 1.The approximations of the branches S 11± in the

frequency–energy plane are computed by noting thatthe conserved energy of the system is equal to

h = X 2

2+ C

(V − X )4

4(13)

which, taking into account expressions (11), leads tothe plot depicted in Fig. 9; this plot corresponds to theparameters used in the numerical study ( = 0.05, C =1.0). The approximate plots are close to the exactnumerical backbones of the FEP of Fig. 7.

3.1.3 Transient dynamics of the damped system

In this section, the transient, unforced dynamics of the weakly damped system is considered, and it will beshown that complicated transitions between modes inthis system can be fully understood and interpreted interms of the periodic orbits of the undamped system.Specifically, the addition of damping induces tran-sitions between different branches of solutions, and




, , , , , , ,

Fig. 9 Analytic approximation provided by the com-

plexification–averaging method of the backbone

branch S 11± in the frequency index–logarithm of

energy plane

thus influences the transfer of energy between the

linear oscillator and the non-linear attachment.The transient responses of the weakly damped sys-

tem will demonstrate that the structure of periodicorbits of the undamped system greatly influences thedynamics of the weakly damped one. When viewedfrom such a perspective, one can systematically inter-pret the complex transitions between multi-frequency modes of the transient, weakly damped dynamics by relating them to the different branches of non-linearmodes in the FEP of Fig. 7. Unless otherwise noted,in the following simulations system (2) is considered

with the same parameters used in the previous sec-

tions ( = 0.05, ω0 = 1.0, C = 1.0), and small damping coefficients λ1 = 0, λ2 = 0.0005.Themotion on thestablespecial orbit of branchU 76

is initiated and there occurs vigorous TET to the non-linearattachment.In Fig. 10, the responses andrelated

WTs of the system with initial conditions v (−T /4) =v (−T /4) = x (−T /4) = 0 and ˙ x (−T /4) = −0.1039 aredepicted. The general observation is made that in thiscase there is strong TET to the non-linear attachment(NES), as evidenced by its large amplitude of oscilla-tion compared with that of the (directly excited) linearoscillator.

In particular, Fig. 10(d) is a schematic illustrating

the transitions taking place in the weakly dampedresponse on the FEP of the undamped system. Thesimulation verifies that the impulsive excitation of astablespecialorbitisoneofthetriggeringmechanismsinitiating (direct) passive energy pumping. Energy decrease due to damping dissipation triggers the tran-sitions between different branches of solutions. Thenumerical simulations of Fig. 10 demonstrate that, fol-lowing a prolonged motion on U 76 during the early regime of the motion, there occurs a 1:2 TRC with the

motion temporarily settling on branch S 12− beforeescaping from resonance capture as time increases(and energy decreases), and being involved in TRC

with the stable branch S 13−. The short capture onbranch S 12− leads to the conjecture that the domainof attraction of 1:2 resonance capture is much smallerthan the corresponding domain of attraction of the 1:3resonance capture, with the latter eventually captur-ing the transient damped dynamics. Indeed, it should

be expected that due to the complicated topology of the periodic orbits of the undamped system, thetransitions between branches and the sequence of resonance captures should be sensitive to viscousdamping dissipation.

3.2 TET mechanisms

In this section, the impulsively forced, damped system(2) with the primary DOF denoted by y is consid-ered, and three basic mechanisms for the initiationof non-linear TET are studied. The first mechanism(fundamental TET) is realized when the motion takes

place along the backbone curve S 11+ of the FEP of Fig. 7, occurring for relatively low frequencies ω < ω0.The second mechanism (subharmonic TET) resem-bles the first, and occurs when the motion takes placealong a lower frequency branch Snm, n < m ∈ Z+.Thethird mechanism (TET initiated by non-linear beat)

which leads to stronger TET involves the excitation of a special orbit with main frequency ωSO greater thanthe natural frequency of the linear oscillator ω0. In

what follows, each mechanism is discussed separately,and numerical simulations that demonstrate passiveand irreversible energy transfer from the linear oscil-

lator to the non-linear attachment are provided ineach case. Analytical results are also provided for thefundamental and subharmonic TET.

3.2.1 Fundamental TET

The first mechanism for TET involves excitation of thebranch of in-phase synchronous periodic solutionsS 11+, where the linear oscillator and the non-linearattachment oscillate with identical frequencies in theneighbourhood of the fundamental frequency ω0.

Although TET is considered only in the damped sys-tem, in order to gain an understanding of the govern-

ing dynamics it is necessary to consider the case of nodamping.

Figure 8(b) depicts a detailed plot of branchof the undamped system. At higher energies, thein-phase NNMs are spatially extended (involving finite-amplitude oscillations of both the linear oscil-lator and the non-linear attachment). However, thenon-linear mode shapes of solutions on S 11+ dependessentially on the level of energy and at low energiesthey become localized to the attachment. Considering




-

Fig. 10 Damped motion initiated on the stable special orbit of branch U 76 with weaker damping:

(a) and (b) transient responses of the linear and non-linear oscillators; (c) their WTs; (d)

WTs superimposed to the undamped FEP [83]

now the motion in-phase space, this low-energy local-ization is a basic characteristic of the two-dimensionalNNM invariant manifold corresponding to S 11+;moreover, this localization property is preserved in the

weakly damped system, where the motion takes placein a two-dimensional, damped NNM invariant man-ifold. This means that when the initial conditions of the damped system are such as to excite the dampedanalogue of S 11+, the corresponding mode shape of the oscillation, initially spatially extended, becomeslocalized to the non-linear attachment with decreas-ing energy due to damping dissipation. This, in turn,leads to passive, continuous and irreversible transferof energy from the linear oscillator to the non-linearattachment, which acts as a NES. The underlying dynamical phenomenon governing fundamental TET

was proven to be a resonance capture on a 1:1resonance manifold of the system [50].

Numerical evidence of fundamental TET is givenin Fig. 11 for the system with parameters = 0.05,ω2

0 = 1, C = 1, and λ1 = λ2 = 0.0015. Small damping

is considered in order to better highlight the TETphenomenon, and the motion is initiated near theboxed point of Fig. 8(b). Comparing the transientresponses shown in Figs 11(a) and (b), it is notedthat the response of the primary system decays fasterthan that of the NES. The percentage of instantaneousenergy captured by the NES versus time is depictedin Fig. 11(e), and the assertion that continuous andirreversible transfer of energy from the linear oscilla-tor to the NES takes place is confirmed. This is moreevident by computing the percentage of total inputenergy that is eventually dissipated by the damper of the NES (see Fig. 11(f)), which in this particular simu-lation amounts to 72 per cent; the energy dissipated atthe NES is computed by the relation

E NES(t ) = λ2

t

0

[v (τ ) − ˙ y (τ )]2dτ

The evolution of the frequency components of themotions of the two oscillators as energy decreasescan be studied by numerical WTs of the transient




, , , , , , ,

Fig. 11 FundamentalTET.Shownarethetransientresponsesofthe(a)linearoscillatorand(b)NES;

WTs of the motion of (c) NES and (d) linear oscillator; (e) percentage of instantaneous total

energy in the NES; (f) percentage of total input energy dissipated by the NES; transition

of the motion from S 11+ to S 13+ at smaller energy levels using the (g) NES (observe the

settlement of the motion at frequency 1/3) and (h) linear oscillator [84]




-

responses,asdepictedinFigs11(c)and(d).Theseplotsdemonstrate that a 1:1 resonance capture is indeedresponsible for TET. Below the value of −4 of the loga-rithm of energylevel,the motionof thelinear oscillatoris too small to be analysed by the particular windowsused in the WT; however, a more detailed WT oversmaller energy regimes (see Figs 11(g) and (h)) revealsa smooth transition from S 11+ to S 13+, in accordance

with theFEP of Fig. 7.This transition manifests itself by

the appearance of two predominant frequency com-ponents in the responses (at frequencies 1 and 1/3) asenergy decreases.

The complexification-averaging method is utilizedto perform an analytical study of the resonance cap-ture phenomenon in the fundamental TET mecha-nism. System (2) is again considered, and the new complex variables are introduced

ψ1(t ) = v (t ) + jv (t ) ≡ ϕ1(t ) e jt ,

ψ2(t ) = ˙ y (t ) + jy (t ) ≡ ϕ2(t ) e jt

(14)

where φi (t ), i = 1, 2, represent slowly varying com-plex amplitudes and j 2 = −1. By writing equation (14),a partition of the dynamics into slow and fast com-ponents is introduced, and slowly modulated fastoscillations at frequency ω = ω0 = 1 are sought. Asdiscussed previously, fundamental TET is associated

with this type of motion in the neighbourhood of branch S 11+ in the FEP of the undamped dynam-ics. Expressing the system responses in terms of thenew complex variables, y = (ψ2 − ψ∗

2 )/(2 j ), v = (ψ1 −ψ∗

1 )/(2 j ) (where (*) denotes complex conjugate), sub-stituting into equation (2), and averaging over the

fast frequency, a set of approximate, slow modulationequations that govern the evolutions of the complex amplitudes is derived

ϕ1 = − j λ

2ϕ1 −

λ

2(ϕ1 − ϕ2) + j

3C

8|ϕ1 − ϕ2|2(ϕ1 − ϕ2)

ϕ2 = −λ

2ϕ2 +

λ

2(ϕ1 − ϕ2) + j

3C

8|ϕ2 − ϕ1|2(ϕ2 − ϕ1)

(15)

For the sake of simplicity, assume that λ1 = λ2 = λ

in equation (2). To derive a set of real modulationequations, the complex amplitudes are expressed in

polar form, ϕi (t ) = ai (t )e j βi t

, which is substituted intoequation (15), and the real and imaginary parts areseparately set equal to zero. Then, equation (15) isreduced to an autonomous set of equations that gov-ern the slow evolution of the two amplitudes a1(t ) anda2(t ) and the phase difference φ(t ) = β2(t ) − β1(t )

a1 = −λ

2a1 +

λ

2a2 cos φ

+ 3C

8a2(a2

1 + a22 − 2a1a2 cos φ) sin φ

a2 = λ

2a1 cos φ − λa2

− 3C

8a1(a2

1 + a22 − 2a1a2 cos φ) sin φ

a1a2φ = −1

2a1a2 −

λ

2(a1 + a2) sin φ

− 3C

8(a2

1 + a22 − 2a1a2 cos φ)

× [(1

−)a1a2

+(a1

−a2) cos φ

] (16)

This reduced dynamical system governs the slow-flow dynamics of fundamental TET. In particular, 1:1 reso-nance capture (the underlying dynamical mechanismof fundamental TET) is associated with non-time-likebehaviour of the phase variable φ or, equivalently, fail-ure of the averaging theorem in the slow flow (16).Indeed, when φ exhibits time-like, non-oscillatory behaviour [166], one can apply the averaging theoremover φ and prove that the amplitudes a1 and a2 decay exponentially with time and no significant energy exchanges (TET) can take place. Figure 12(a) depicts1:1 resonance capture in the slow-flow-phase plane(φ, φ) for system (16) with = 0.05, λ = 0.01, C = 1,ω0 = 1 and initial conditions a1(0) = 0.01, a2(0) =0.24, φ(0) = 0. The oscillatory behaviour of the phasevariable in the neighbourhood of the in-phase limitφ = 0+ indicates 1:1 resonance capture (on branchS 11+oftheFEPofFig.7),andleadstoTETfromthelin-ear oscillator to the NES as evidenced by the build-upof amplitude a1 (see Fig. 12(b)). Escape from reso-nance capture is associated with time-like behaviourof φ and rapid decrease of the amplitudes a1 and a2

(as predicted by averaging in equation (16)). A com-parison of the analytical approximation (14)–(16) anddirect numerical simulation for the previous initialconditions confirms the accuracy of the analysis.

3.2.2 Subharmonic TET

Subharmonic TET involves excitation of a low-frequency S -tongue. As mentioned earlier, low-frequencytongues arethe particular regions of theFEP

where the NES engages in m:n (m, n are integers suchthat m < n) resonance captures with the linear oscil-lator. A feature of the lower tongues is that on them

the frequency of the motion remains approximately constant with varying energy. As a result, the tonguesare represented by horizontal lines in the FEP, and theresponse of system (2) on a tongue locally resemblesthat of a linear system. In addition, at each specificm:n resonance capture, there appear a pair of closely spaced tongues corresponding to in- and out-of-phaseoscillations of the two subsystems.

Regarding the dynamics of subharmonic TET, aparticular pair of lower tongues are focused, say




, , , , , , ,

Fig. 12 Fundamental TET: (a) 1:1 resonance capture in the slow flow; (b) amplitude modulations;

(c) comparison between analytical approximation (dashed line) and direct numerical

simulation (solid line) for v (t ); (d) transient responses of the system [84]

S 13± (Fig. 8(c)). At the extremity of a lower pairof tongues, the curve in the configuration plane isstrongly localized to the linear oscillator. However, asfor the fundamental mechanism for TET, the decreaseof energy by viscous dissipation leads to curves in the

configuration plane that are increasingly localized tothe NES, and non-linear TET to the NES occurs. In thiscase, the underlying dynamical phenomenon causing TET is resonance capture in the neighbourhood of am:n resonance manifold of the dynamics. Specifically,for the pair of tongues S 13±, a 1:3 resonance captureoccurs that leads to subharmonic TET with the linearoscillator vibrating with a frequencythreetimes that of theNES. It is emphasized that due to thestability prop-erties of the tongues S 13±, subharmonic TET involvesexcitation of S 13−, but not S 13+.

The transient dynamics when the motion is initi-ated at the extremity of S 13

−(see the initial condition

denoted by the box on the right part in Fig. 8(d))is displayed in Fig. 13. The same parameters as inthe previous section are considered. Until t = 500s,subharmonic TET takes place. Despite the presenceof viscous dissipation, the NES response grows con-tinuously, with simultaneous rapid decrease of theresponse of the linear oscillator. A substantial amountof energy is transferred to the NES (see Fig. 13(e)), andeventually nearly 70 per cent of the energy is dissi-pated by the NES damper (see Fig. 13(f)). A prolonged

1:3 resonance capture is nicely evidenced by the WT of Figs 13(c) and (d), and the motion follows the wholelower tongue S 13− from the right to the left. Onceescape from resonance capture occurs (around t =620–630 s), energy is no longer transferred to the NES.

For analytical study of subharmonic TET, TET inthe neighbourhood of tongue S 13− will be the focus(similar analysis can be applied forother ordersof sub-harmonic resonance captures). Due to the fact thatmotions in the neighbourhood of S 13− possess twomain frequencycomponents,at frequencies 1 and1/3,the responses of system (2) can be expressed as

y (t ) = y 1(t ) + y 13

(t ), v (t ) = v 1(t ) + v 13

(t ) (17)

where the indices represent the frequency of eachterm. As in the previous case, new complex variablesare introduced

ψ1(t ) = ˙ y 1(t ) + j ω y 1(t ) ≡ ϕ1(t ) e j ωt ,

ψ3(t ) = ˙ y 13

(t ) + j ω

3 y 1

3(t ) ≡ ϕ3(t ) e j ωt

3

ψ2(t ) = v 1(t ) + j ωv 1(t ) ≡ ϕ2(t ) e j ωt ,

ψ4(t ) = v 13

(t ) + j ω

3v 1

3(t ) ≡ ϕ4(t ) e j ωt

3

(18)

where ϕi (t ), i = 1, . . . , 4 represent slowly varying mod-ulations of fast oscillations of frequencies 1 or 1/3.




-

Fig. 13 Subharmonic TET initiated on S 13−: shown are the transient responses of the (a) linear

oscillator and (b) NES; WTs of the motion of (c) the NES and (d) the linear oscillator; (e)

percentage of instantaneous total energy in the NES; (f) percentage of total input energy

dissipated by the NES [84]

Expressing the system responses in terms of the new complex variables

y = ψ1 − ψ∗1

2 j ω+ ψ3 − ψ∗

3

2 j (ω/3), v = ψ2 − ψ∗

2

2 j ω+ ψ4 − ψ∗

4

2 j (ω/3)

(19)

substituting into equation (2), and averaging overeach of the two fast frequencies, the slow modulation

equations that govern the evolutions of the complex amplitudes are derived as

ϕ1 = − j 1

2

ω − 1

ω

ϕ1 −

λ

2(2ϕ1 − ϕ2)

− j 9C

8ω3[3ϕ3

3 − 9ϕ23 ϕ4 − 3ϕ3

4 + 9ϕ3ϕ24

− (ϕ1 − ϕ2)|ϕ1 − ϕ2|2 − 6(ϕ1 − ϕ2)|ϕ3 − ϕ4|2]




, , , , , , ,

ϕ2 = − j ω

2ϕ2 − λ

2(ϕ2 − ϕ1) + j

9C

8ω3[3ϕ3

3

− 9ϕ23 ϕ4 − 3ϕ3

4 + 9ϕ3ϕ24

− (ϕ1 − ϕ2)|ϕ1 − ϕ2|2 − 6(ϕ1 − ϕ2)|ϕ3 − ϕ4|2]

ϕ3 = − j 1

2

ω

3− 3

ω

ϕ3 −

λ

2(2ϕ3 − ϕ4)

+ j 9C

8ω3[ϕ1(2(ϕ3 − ϕ4)(ϕ∗

1 − ϕ2) − 3(ϕ∗3 − ϕ∗

4 )2)

+ ϕ2(2(ϕ4 − ϕ3)(ϕ∗1 − ϕ2) + 3(ϕ∗3 − ϕ∗4 )2

)

+ 9(ϕ3 − ϕ4)|ϕ3 − ϕ4|2]

ϕ4 = − j ω

6ϕ4 − λ

2(ϕ4 − ϕ3) − j

9C

8ω3

× [ϕ1(2(ϕ3 − ϕ4)(ϕ∗1 − ϕ2) − 3(ϕ∗

3 − ϕ∗4 )2)

+ ϕ2(2(ϕ4 − ϕ3)(ϕ∗1 − ϕ2) + 3(ϕ∗

3 − ϕ∗4 )2)

+ 9(ϕ3 − ϕ4)|ϕ3 − ϕ4|2](20)

where again it was assumed that λ1 = λ2 = λ inequation (2). To derive a set of real modulation equa-

tions, the complex amplitudes are expressed in polarform ϕi (t ) = ai (t )e j βi (t ), and an autonomous set of seven slow-flow modulation equations that govern theamplitudes ai = |ϕi |, i = 1, . . . , 4 and the phase differ-ences φ12 = β1 − β2, φ13 = β1 − 3β3, and φ14 = β1 − 3β4

are derived.The equations of the autonomous slow flow will not

be reproduced here, but it suffices to state that they are of the form

a1 = −λ

2(2a1 − a2) + g 1(a ,φ),

˙a2

= −λ

2(a2

−a1)

+ g 2(a ,φ)

a3 = −λ

2(2a3 − a4) + g 3(a ,φ),

a4 = −λ

2(a4 − a3) + g 4(a ,φ)

φ12 = f 12(a ) + g 12(a ,φ; ),

φ13 = f 13(a ) + g 13(a ,φ)

φ14 = f 14(a ) + g 14(a ,φ; )

(21)

where the functions g i and g ij are 2π-periodic interms of the phase angles φ

=(φ12, φ13, φ14)T, and

a = (a1, . . . , a4)T.In this case (as for the fundamental TET mecha-

nism), strong energy transfer between the linear andnon-linear oscillators can occur only when a subset of phase angles φkl does not exhibit time-like behaviour;that is, when some phase angles possess oscillatory (non-monotonic) behaviour with respect to time. Thiscan be seen from the structure of the slow flow (21)

where, if the phase angles exhibit time-like behaviourand the functions g i are small, averaging over these

phase angles canbe performed to show that theampli-tudes decrease monotonically with time; in that case,no significant energy exchanges between the linearand non-linear components of the system can takeplace. It follows that subharmonic TET is associated

with non-time-like behaviour of (at least) a subset of the slow-phase angles φkl in equation (21).

Figure 14 presents the results of the numerical inte-gration of the slow-flow equations (20) and (21) for

the system with parameters = 0.05, λ = 0.03, C = 1,and ω0 = 1. The motion is initiated on branch S 13− with initial conditions v (0) = y (0) = 0 and v (0) =0.01 499, and ˙ y (0) = −0.059 443(it correspondsexactly to the simulation of Fig. 13). The corresponding ini-tial conditions and the value of the frequency ω of the reduced slow-flow model were computed by min-imizing the difference between the analytical andnumerical responses of the system in the intervalt ∈ [0,100]: ϕ1(0) = −0.0577, ϕ2(0) = 0.0016, ϕ3(0) =−0.0017, ϕ4(0) = 0.0134, and ω = 1.0073.

This result indicates that, initially, nearly all energy is stored in the fundamental frequency component of

the linear oscillator, with the remainder confined tothe subharmonic frequency component of the NES.Figures 14(a) and (b) depict the temporal evolutions of the amplitudes ai , from which it is concluded that sub-harmonic TET in the system is mainly realized throughenergy transfer from the (fundamental) component atfrequency ω of the linear oscillator, to the (subhar-monic) component at frequency ω/3 of the NES (as

judged from the build-up of the amplitude a3 and thediminishingof a1).Asmalleramountofenergyistrans-ferred from the fundamental frequency component of the linear oscillator to the corresponding fundamental

component of the NES (as judged by the evolution of the amplitude a2).These conclusions are supported by the plots of

Figs 14(c) to (e), where the temporal evolutions of thephase differences φ12 = β1 − β2, φ13 = β1 − 3β3, andφ14 = β1 − 3β4 are shown. Absence of strong energy exchange between the fundamental and subharmonicfrequency components of the linear oscillator is asso-ciated with the time-like behaviour of the phasedifference φ13, whereas TET from the fundamentalcomponentof thelinear oscillator to thetwo frequency components of the NES is associated with oscillatory early time behaviour of the phase differences φ12 andφ14. Oscillatory responses of φ12 and φ14 correspond to1:1 and 1:3 resonance captures, respectively, betweenthe corresponding frequency components of the lin-ear oscillator and the NES; as time increases, time-likeresponses of the phase variables are associated withescapes from the corresponding regimes of resonancecapture. In addition, it is noted that the oscillations of the angles φ12 and φ14 take place in the neighbourhoodof π, which confirms that, in this particular example,subharmonic TET is activated by the excitation of




-

Fig. 14 Subharmonic TET: (a) amplitude modulations; (b)–(d) phase modulations [84]

Fig. 15 Transient response of NES for 1:3 subharmonic

TET; comparison between analytical approx-

imation (dashed line) and direct numerical

simulation (solid line)

an anti-phase branch of periodic solutions (such asS 13−). The analytical results are in full agreement

with the WTs depicted in Figs 5(c) and (d), where theresponse of the linear oscillator possesses a strong frequency component at the fundamental frequency

ω0 = 1, whereas the NES oscillates mainly at frequency ω0/3.

The accuracy of the analytical model (20) and (21)in capturing the dynamics of subharmonic TET is con-firmed by the plot depicted in Fig. 15 where the analyt-ical response of the NES is found to be in satisfactory agreement with the numerical response obtained by the direct simulation of equation (2). Interestingly,the reduced analytical model is capable of accurately modelling the strongly non-linear, damped, transient

response of the NES in the resonance capture region.The analytical model fails, however, during the escapefrom resonance capture since the ansatz (17) and (18)is not valid in that regime of the motion. Indeed,after escape from resonance capture, the motionapproximately evolves along the backbone curve of the FEP; eventually S 15 is reached whose motion can-not be described by the ansatz (17) and (18), thereby leading to the failure of the analytical model.

3.2.3 TET initiated by non-linear beating

The previous two mechanisms cannot be activated with the NES at rest, since in both cases the motionis initialized from a non-localized state of the system.This means that these energy pumping mechanismscannot be activated directly after the application of animpulsive excitation to the linear oscillator with theNES initially at rest. Such a forcing situation, however,is important from a practical point of view; indeed,this is the situation where local NESs are utilized toconfine and passively dissipate unwanted vibrations

from linear structures that are forced by impulsive (orbroadband) loads.

Hence, it is necessary to discuss an alternative, thirdenergy pumping mechanism capable of initiating pas-sive energy transfer with the NES initially at rest. Thisalternative mechanism is based on the excitation of a special orbit that plays the role of a ‘bridging orbit’for activation of either fundamental or subharmonicTET. Excitation of a special orbit results in the trans-fer of a substantial amount of energy from the initially




, , , , , , ,

excited linear oscillator directly to the NES through anon-linear beat phenomenon. In that context, thespe-cial orbit may be regarded as an initial ‘bridging orbit’or trigger, which eventually activates fundamental orsubharmonic TET once the initial non-linear beat ini-tiates the energy transfer. Indeed, as shown below, thethird mechanism for TET represents an efficient ini-tial (triggering) mechanism for rapid transfer of energy from the linear oscillator to the NES at the crucial ini-

tial stage of the motion, before activating either one of the (fundamental or subharmonic) main TET mech-anisms through a non-linear transition (jump) in thedynamics.

To study the dynamics of this triggering mecha-nism, the following conjecture is formulated: Dueto the essential (non-linearizable) non-linearity, theNES is capable of engaging in a m:n resonance cap-ture with the linear oscillator, m and n being a setof integers. Accordingly, in the undamped system,there exists a sequence of special orbits (correspond-ing to non-zero initial velocity of the linear oscillatorand all other initial conditions zero), aligned along a

one-dimensional smooth manifold in the FEP.

As a first step to test this conjecture, a NLBVP wasformulated to compute the periodic orbits of system(2) with no damping, and the additional restrictionfor the special orbits was imposed. The numericalresults in the frequency–energy plane are depictedin Fig. 16 for parameters = 0.05, ω0 = 1, and C = 1.Each triangle in the plot represents a special orbit, anda one-dimensional manifold appears to connect thespecial orbits; a rigorous proof of the existence of this

manifold can be found in reference [85]. In addition, itappears that there exist a countable infinity of specialorbits, occurring in the neighbourhoods of the count-able infinities of IRs m:n (m, n integers) of the system.It is noted that a subset of high-frequency branches(for ω > 1) possesses two special orbits instead of one(for example, all U (p + 1)p branches with p 3). Todistinguish between the two special solutions in suchhigh-frequency branches, they are partitioned intotwo subclasses: the a-special orbits that exist in theneighbourhood of ω = ω0 = 1,andthe b-special orbitsthat occur away from this neighbourhood (see Fig. 16).It was proven numerically that the a-special orbits are

unstable, whereas the b-special orbits are stable [83].

Fig. 16 Manifold of special orbits (represented by triangles) in the FEP [84]




-

As shown below it is the excitation of the stableb-special orbits that activates the third mechanism forTET.

By construction, all special orbits have a commonfeature; namely, they pass with vertical slope throughthe origin of the configuration plane (v , y ). This fea-ture renders them compatible with an impulse appliedto the linear oscillator, which corresponds to a non-zero velocity of the linear oscillator and all other

initial conditions zero. The curves corresponding tothe special orbits in the configuration plane can beeither closed or open depending upon the differencesbetween the two indices characterizing the orbits;specifically, odd differences between indices corre-spond to closed curves in the configuration plane andlie on U -branches, whereas even differences betweenindices correspond to open curves on S -branches.

In addition, higher frequency special orbits (withfrequency index ω > ω0) in the upper part of the FEP(i.e. m > n) are localized to the non-linear oscilla-tor; conversely, special orbits in the lower part of theFEP (with frequency index ω < ω0) tend to be local-

ized to the linear oscillator. This last observation isof particular importance since it directly affects thetransfer of a significant amount of energy from thelinear oscillator to the NES through the mechanismdiscussed in this section. Indeed, there seems to bea well-defined critical threshold of energy that sepa-rateshigh- fromlow-frequency special orbits; i.e.thosethat do or do not localize to the NES, respectively (seeFig. 16).

The third mechanism for TET can only be acti-vated for input energies above the critical threshold,since below that the (low-frequency) special orbits are

incapable of transferring significant amounts of inputenergy from the linear oscillator to the NES; in other words, the critical level of energy represents a lowerbound below which no significant TET can be initi-ated through activation of a special orbit. Moreover,combining this result with the topology of the one-dimensional manifold of special orbits of Fig. 16, itfollows that it is the subclass of stable b-special orbitsthat is responsible for activating the third TET mecha-nism, whereas thesubclassof unstable a-special orbitsdoes not affectTET. This theoretical insight will be fully validated by the numerical simulations that follow.

When the NES engages in a m:n resonance cap-

ture with the linear oscillator, a non-linear beatphenomenon takes place. Due to the essential (non-linearizable) non-linearity of the NES and the lack of any preferential frequency, this non-linear beatphenomenon does not require any a priori tuning of the non-linear attachment, since at the specificfrequency–energy range of the m:n resonance cap-ture, the non-linear attachment adjusts its amplitude(tunes itself) to fulfil the necessary conditions of IR. This represents a significant departure from the

‘classical’ non-linear beat phenomenon observed incoupled oscillators with linearizable non-linear stiff-nesses (e.g. spring–pendulum systems [129]), wherethe defined ratios of linearized natural frequencies of the component subsystems dictate the type of IRs thatcan be realized [14, 167].

As an example, Fig. 17 depicts the exchanges of energy during the non-linear beat phenomenon cor-responding to the special orbits of branches U 21

and U 54 for parameters = 0.05, ω0 = 1, C = 1, andno damping. As expected, energy is continuously exchanged between the linear oscillator and the NES,so the energy transfer is not irreversible as is requiredfor TET; it can be concluded that excitation of a specialorbit can only initiate (trigger) TET, but not cause it initself. The amount of energy transferred during eachcycle of the beat varies with the special orbit consid-ered; for U 21 and U 54, as much as 32 per cent and86 per cent of energy can be transferred to the NES,respectively. It can be shown that, for increasing inte-gers m and n with corresponding ratios m/n → 1+,the maximum energy transferred during a cycle of the

special orbit tends to 100 per cent. At the same time,however, the resulting period of the cycle of the beat(and, hence, of the time needed to transfer the max-imum amount of energy) should increase as the leastcommon multiple of m and n.

Note, at this point, that the non-linear beat phe-nomenon associated with the excitation of the spe-cial orbits can be studied analytically using thecomplexification-averaging method [135]. To demon-strate the analytical procedure, the special orbit onbranch U 21 of the system with no damping is anal-

ysed in detail. In the previous section, the periodic

motions on this entire branch were studied, and it was shown that the responses of the linear oscillatorand the non-linear attachment can be approximately expressed as

y (t ) ≈ Y 1 sin ωt + Y 2 sin2ωt ≡ y 1(t ) + y 2(t )c

v (t ) ≈ V 1 sin ωt + V 2 sin 2ωt ≡ v 1(t ) + v 2(t )

(22)

where the amplitudes are

Y 1 =

A

ω, V

1 =B

ω, Y

2 =D

2ω, V

2 =G

2ω

and A, B, D , and G are computed from the stationarity conditions in the slow-flow equations as

B = ±

4ω4( Z 2 − 8 Z 1)

9CZ 31 Z 2,

G = ±

32ω4(2 Z 1 − Z 2)

9CZ 32 Z 1⇒ A = ω2

ω20 − ω2

B,




, , , , , , ,

Fig. 17 Exchanges of energy during non-linear beat phenomena corresponding to special orbits

on (a), (b) U 21, and (c) and (d) U 54

D = 4ω2

ω20 − 4ω2

G

Z 1 = ω2

ω20 − ω2

− 1, Z 2 = 4ω2

ω20 − 4ω2

− 1

Hence, a two-frequency approximation is satisfactory for this family of periodic motions. The frequency ωSO

at which the special orbit appears is computed by imposing the initial conditions y (0) = v (0) = v (0) = 0,

which leads to the relation

B = −2G (special orbit)

The instantaneous fraction of total energy in the lin-ear oscillator during the non-linear beat phenomenonis estimated to be

E linear(t )

= [(ω20 − 4ω2

SO) sin ωSOt − 2(ω20 − ω2

SO) sin2ωSOt ]2

9ω2SOω2

0

+

(ω2

0 − 4ω2SO) cos ωSOt

−4(ω20 − ω2

SO) cos2ωSOt

2

9ω40

(23)

The non-linear coefficient C has no influence on thefraction of total energy transferred to the NES dur-ing the non-linear beat; this means that, during the

beat, the instantaneous energies of the linear oscil-lator and the NES are directly proportional to thenon-linear coefficient. Moreover, as the mass of theNES tends to zero, the frequency where the specialorbit is realized tends to the limit ωSO → ω, and, as a

result, E linear(t ) → 1, and the energy transferred to theNES during the beat tends to zero. However, it is notedthat this is a result satisfied only asymptotically since,as indicated by the results depicted in Fig. 17, evenfor very small mass ratios, e.g. = 0.05, as much as86 per cent of the total energy can be transferred to theNES during a cycle of the special orbit of branch U 54.

Considering now the damped system, it will beshown that following an initial non-linear beat phe-nomenon, eitherone of themain (fundamentalor sub-harmonic) TET mechanisms can be activated througha non-linear transition (jump) in the dynamics. It

was previously mentioned that the two main TET

mechanisms are qualitatively different from the thirdmechanism, which is based on the excitation of anon-linear beat phenomenon (special orbit). Indeed,damping is a prerequisite for the realization of the twomain mechanisms, leading to an irreversible energy transfer from the linear oscillator to the NES, whereasa special orbit is capable of transferring energy with-out dissipation, though this transfer is not irreversiblebut periodic.This justifies the earlier assertion that thethird mechanism does not represent an independent




-

Fig. 18 TET by non-linear beat, transition to S 11+. Shown are the transient responses of the (a)

linear oscillator and (b) NES;WTs of the motion of (c) the NES and (d) the linear oscillator;

(e) percentage of instantaneous total energyin theNES; (f) percentage of total input energy

dissipated by the NES [84]

mechanism for energy pumping, but rather triggersit, and through a non-linear transition activates eitherone of the two main mechanisms. This will becomeapparent in the following numerical simulations.

The following simulations concern the transientdynamics of the damped system (2) with parame-ters = 0.05, ω0 = 1, C = 1, λ1 = λ2 = 0.0015, and animpulse of magnitude Y applied to the linear oscillator

(corresponding to initial conditions y (0+) = v (0+) =v (0+) = 0, ˙ y (0+) = Y ). By varying the magnitude of the impulse, the different non-linear transitions whichtake place in the dynamics and their effects on TETare studied. The responses of the system to the rela-tively strong impulse Y = 0.25 are depicted in Fig. 18.Inspection of the WTs of the responses (see Figs 18(c)and (d)), and of the portion of total instantaneous




, , , , , , ,

Fig. 19 Percentage of input energy eventually dissi-

pated at the NES for varying magnitude of the

impulse (the positions of certain special orbits

are indicated) [84]

energy captured by the NES (see Fig. 18(e)), reveals

that at the initial stage of the motion (until approxi-mately t = 120 s) the (stable) b-special orbit on branchU 32 is excited (since the NES response possesses twomain frequency components at 1 and 3/2 rad/s), anda non-linear beat phenomenon takes place. (Notethe continuous exchange of energy between the twosubsystems, demonstrating reversibility in this ini-tial stage of the motion.) For t > 120 s, the dynamicsundergoes a transition (jump) to branch S 11+, andfundamental TET to the NES occurs on a prolonged1:1 resonance capture (see Figs 18(c) and (d)); eventu-ally, 84 per cent of the input energy is dissipated by the

damper of the NES (see Fig. 18(f)).

3.2.4 Critical energy threshold necessary for initiating TET

To demonstrate more clearly the effect of the b-specialorbits on TET, Fig. 19 depicts the percentage of inputenergy eventually dissipated at the NES for varying magnitude of the impulse for the system with parame-ters = 0.05, ω0 = 1, C = 1, and λ1 = λ2 = 0.01. In thesame plot, the positions of the special orbits of theundamped system and the critical threshold predictedin Fig. 16 are depicted.

It is concluded that strong TET is associated withthe excitation of b-special orbits of the branchesU (p + 1)p in the neighbourhood above the criticalthreshold,whereas excitation of a-special orbitsbelow the critical threshold does not lead to rigorous energy pumping. As mentioned previously, in the neighbour-hood of the critical threshold, the b-special orbits arestrongly localized to the NES, whereas a-orbits arenon-localized. The deterioration of TET is also notedfrom Fig. 19 as the magnitude of the impulse well

Fig. 20 Contours of percentage of input energy even-

tually dissipated at the NES for the case when

both oscillators excited by impulses; superim-

posed are contours of high- and low-frequency

branches of the undamped system (solid line:in-phase, dashed line: out-of-phase branches);

special orbits in high- and low-frequency

branches are denoted by circles and triangles,

respectively [84]

above the critical threshold increases, where high-frequency special orbits are excited; this is a conse-quenceofthefactthatwellabovethecriticalthreshold,the special orbits are weakly localized to the NES.

Extending the previous result, Fig. 20 depicts the

contours of energy eventually dissipated at the NES,but for the case of two impulses of magnitudes ˙ y (0)

and v (0) applied to both the linear oscillator andthe NES, respectively. The system parameters used

were identical to those of the previous simulation of Fig. 19. Superimposed on contours of energy dissi-pated at the NES are certain high- and low-frequency U - and S -branches of the undamped system together

with their special orbits, in order to confirm for thiscase the essential role of the high-frequency specialorbits in TET. Indeed, high levels of energy dissipa-tion are encountered in neighbourhoods of contoursof high-frequency U -branches, whereas low values are

noted in the vicinity of low-frequency branches. Theseresults agree qualitatively with the earlier theoreticaland numerical findings, and enable one to assess andestablish the robustness of TET when the NES is notinitially at rest.

The results presented thus far provide a measureof the complicated dynamics encountered in thetwo-DOF system under consideration. It is logical toassume that by increasing the number of DOFs of the system, the dynamics will become even more




-

complex. That this is indeed the case is revealed by thenumerical simulations presented in the next section

where resonance capture cascades are reported inMDOF linear systems with essentially non-linear endattachments. By resonance capture cascades, compli-cated sudden transitions between different branchesof solutions (modes), which are accompanied by sud-den changes in the frequency content of the systemresponses, are denoted. As shown in previous works

[78], such multi-frequency transitions can drastically enhance TET from the linear system to the essentially non-linear attachment.

3.3 MDOF and continuous oscillators

To gain additional insight into the dynamics of TET,the case of combinations of MDOF systems com-posed of linear primary systems with attached SDOFor MDOF ungrounded NESs is considered. Resultson this specific problem can also be found in ref-erences [77–80, 89]. Consider, first, the case of thetwo-DOF linear primary system with attached SDOFungrounded NES

¨ y 2 + ω20 y 2 + λ2 ˙ y 2 + d ( y 2 − y 1) = 0

¨ y 1 + ω20 y 1 + λ1 ˙ y 1 + λ3( ˙ y 1 − v ) + d ( y 1 − y 2)

+ C ( y 1 − v )3 = 0

v + λ3(v − ˙ y 1) + C (v − y 1)3 = 0

(24)

Thesystem parameters are chosenas ω0 = 136.9 (rad/s),λ1 = λ2 = 0.155, λ3 = 0.544, d = 1.2 × 103, = 1.8, and

C = 1.63 × 10

7

, with linear natural frequencies ω1 =11.68 and ω2 = 50.14 (rad/s).Figure 21(a)depicts the relativeresponse v (t ) − y 1(t )

of the system for initial displacements y 1(0) = 0.01, y 2(0) = v (0) = −0.01, and zero initial velocities. Themulti-frequency content of the transient response isevident and is quantified in Fig. 21(b), where theinstantaneous frequency of the time series is com-puted by applying the numerical Hilbert transform[95].

As energy decreases because damping dissipation, aseries of eight resonancecapture cascades is observed;i.e. of transient resonances of the NES with a number

of non-linear modes of the system. The complexity of the non-linear dynamics of the system is evi-denced by the fact that of these eight captures only two (labelled IV and VII in Fig. 21(b)) involve thelinearized in-phase and out-of-phase modes of thelinear oscillator, with the remaining involving essen-tially non-linear interactions of the NES with differentlow- and high-frequency non-linear modes of the sys-tem. On the average, during these resonance captures,the NES passively absorbs energy from the non-linear

Fig. 21 Resonance capture cascades in the two-DOF

system with non-linear end attachment: (a)

relative transient response v (t ) − y 1(t ); (b)

instantaneous frequency (resonance cap-

tures indicated). The two natural frequencies

are computed as f 1 = ω1/2π = 1.86Hz and

f 2 = ω2/2π = 7.98 Hz where ω1 = 11.68 and

ω2 = 50.14 (rad/s) [84]

mode involved, before escape from resonance captureoccurs and the NES transiently resonates with the nextmode in the series.

In essence, the NES acts as a passive, broadbandboundary controller, absorbing, confining, and elimi-nating vibration energy from the linear oscillator. Sim-ilar types of resonance capture cascades were reportedin previous works where grounded NESs, weakly cou-pled to the linear structure, were examined [78]. Thecapacity of the NES to resonantly interact with linearand non-linear modes in different frequency rangesis due to its essential non-linearity (i.e. the absenceof a linear term in the non-linear stiffness charac-

teristic), which precludes any preferential resonantfrequency.

3.3.1 Analysis of a two-DOF linear primary systemwith an SDOF NES

The first system which is considered here is a two-DOF linear primary system with an attached SDOFNES (Fig. 22), in which the effect that the increase inDOF of the primary system has on the TET dynamics




, , , , , , ,

Fig. 22 Two-DOF primary system coupled to anungrounded NES

is studied. Equations of motion assume the form

m1 ¨ y 1 + λ1 ˙ y 1 + k 1 y 1 + k 12( y 1 − y 2) = 0

m2 ¨ y 2 + λ2 ˙ y 2 + λ( ˙ y 2 − v ) + k 2 y 2 + k 12( y 2 − y 1)

+ C ( y 2 − v )3 = 0

v + λ(v − ˙ y 2) + C (v − y 2)3 = 0

(25)

where y 1, y 2, v refer to thedisplacements of theprimary systemand the NES, respectively. Forobvious practicalreasons, a lightweight NES is specified by requiring that 1; in this way, weak damping is also assured.

All other variables are treated as O(1) quantities. As shown in the previous section, understanding

the topological structure of the FEP of the underlying Hamiltonian system is a prerequisite for interpreting (even complex) damped transitions in the dampedand forced system. Hence, the analysis focuses on theanalytical computation of the FEP of the undampedand unforced system. The complexification-averaging technique is utilized for the analytical approximation

of the main backbone curves on the FEP, which cor-respond to 1:1 resonant oscillations of the primary system and the NES (i.e. the dominant frequenciesof these two system are identical). At this point, thecomplex variables are introduced

1 = ˙ y 1 + j ω y 1, 2 = ˙ y 2 + j ω y 2, 3 = v + j ωv

(26)

which arethen substituted intoequation (25). Express-ing the complex variables in polar form i = ϕi e

j ωt ,i = 1, 2, 3 and performing averaging over the fast fre-

quency, the complex-valued slow-flow modulationequations are obtained

m1ϕ1 + j ϕ1

2ω(m1ω2 − k 1 − k 12) + j ϕ2

2ωk 12 = 0

m2ϕ2 + j ϕ2

2ω(m2ω2 − k 2 − k 12) + j ϕ1

2ωk 12

+ 3 jC

8ω3(ϕ2

3 ϕ∗2 − ϕ2

2 ϕ∗3 − |ϕ3|2ϕ3 + |ϕ2|2ϕ2

+ 2|ϕ3|2ϕ2 − 2|ϕ2|2ϕ3) = 0

ϕ3 + j ω

2ϕ3

− 3 jC

8ω3(ϕ2

3 ϕ∗2 − ϕ2

2 ϕ∗3 − |ϕ3|2ϕ3 + |ϕ2|2ϕ2

+ 2|ϕ3|2ϕ2 − 2|ϕ2|2ϕ3) = 0

(27)

The complex amplitudes ϕi can be expressed in polarform as ϕi

=ai e

j βi , ai , βi

∈R for i

=1, 2, 3. Then, by

imposing stationarity conditions on the slow-flow equations and considering trivial phase differencessuch that β1 − β2 = β1 − β3 = 0, an approximation of the NNMs on the main backbone is obtained

y 1 = a1 sin ωt , y 2 = a2 sin ωt , v = a3 sin ωt (28)

where the amplitudes ai , i = 1, 2, 3 can be found as afunction of frequency ω by solving the algebraic equa-tions resulting from the steady-state conditions of thereal-valued slow-flow equations.

The main backbone branches can now be con-

structed by varying the frequency ω and represent-ing a NNM at a point (h, ω) on the FEP where thetotal energy h = ω2/2[m1a1(ω)2 + m2a2(ω)2 + a3(ω)2]is conserved when the system oscillates in a spe-cific mode. Figure 23 depicts the backbone branch,named S 111, of the system with parameters m1 =m2 = 1, k 1 = k 2 = k 12 = 1, C = 1, and = 0.05. NNMsdepicted as projections of the three-dimensional con-figuration space (v , y 1, y 2) of the system are superim-posed to demonstrate mode localization behaviours

with respect to the total energy of the system; thehorizontal and vertical axes in these plots are the non-linear and primary system responses, respectively.

Four characteristic frequencies, f 1L, f 2L, f 1H , and f 2H ,are defined in this plot. At high-energy levels and finitefrequencies, the essential non-linearity behaves as arigid link, and the system dynamics is governed by theequations

m1 ¨ y 1 + k 1 y 1 + k 12( y 1 − y 2) = 0

(m2 + ) ¨ y 2 + k 2 y 2 + k 12( y 2 − y 1) = 0

(29)

The natural frequencies of this system are f 1H =0.9876 and f 2H = 1.7116 rad/s for the above parame-

ters. At low-energy levels, the equivalent stiffness of the essential non-linearity tends to zero, and the sys-temdynamicsisthatoftheprimarysystem,thenaturalfrequencies of which are f 1L = 1 and f 2L =

√ 3 rad/s.

From Fig. 23, it is observed that the two frequencies f 1L and f 2L divide the FEP into three distinct regions.

1. The first region, for which ω f 2L, comprises thebranch S 111 + −+ (the ± signs indicate whetherthe initial condition of the corresponding oscillatoris positive or negative,respectively).On this branch,




-

Fig. 23 Analytic approximation of the main back-

bone branches of the system m1 = m2 = 1,

k 1 = k 2 = k 12 = 1, C = 1, = 0.05. NNMs

depicted as projections of the three-dimen-

sional configuration space (v , y 1, y 2) of the

system are superimposed; the horizontal and

vertical axes in these plots are the non-linear

and primary system responses, respectively

(top plot: (v , y 1); bottom plot: (v , y 2); see legend

in the bottom right corner). The aspect ratio isset so that increments on the horizontal and

vertical axes are equal in size, enabling one to

directly deduce whether the motion is localized

to the primary system or to the non-linear

oscillator [87]

the primary system vibrates in an anti-phase fash-ion, and the motion is more and more localized tothe primary system or to the NES as the frequency approaches f 2L or ∞, respectively.

2. The second region, for which f 1L ω f 2H , com-prises two different branches, namely S 111

+ −−and S 111 + +−. These branches coalesce at a pointS 111 + 0− (see the grey dot in Fig. 23), where theNNM is such that the initial condition on the veloc-ity of the oscillator m2 is zero. On S 111 + −−, theprimary system vibrates in an anti-phase fashion,and the motion localizes to the NES as the fre-quency goes away from f 2H . On S 111 + +−, thereis an in-phase motion of the primary system, andthe motion localizes to the primary system, as thefrequency converges to f 1L.

Fig. 24 Numerical computation of the FEP (back-

bone and loci of special orbits) of a two-DOF

primary coupled to an NES (m1 = m2 = 1,

k 1 = k 2 = k 12 = 1, C = 1, = 0.05); black dots

and squares denote anti-phase and in-phase

special orbits, respectively [87]

3. The third region, for which ω f 1H , comprises the

branch S 111 + ++. On this branch, the primary system vibrates in an in-phase fashion, and themotion localizes to the NES as the frequency goesaway from f 1H .

Owing to the energy dependence of the NNMs along S 111, interesting and vigorous energy exchanges may occur between the primary system and the NES. Inparticular, an irreversible channeling of vibrationalenergy from the primary system to the NES takesplace on S 111 + −− and S 111 + ++. Because theNES has no preferential resonance frequency, fun-damental TET can be realized either for in-phase or

anti-phasemotion of the primary system, whichshowsthe adaptability of the NES.

The SPOs, determined from accommodating spe-cific initial conditions ˙ y 1(0) = 0, ˙ y 2(0) = 0 with all theothers zero, can also be computed for the MDOF sys-tem. The role of special orbits is to transfer as quickly as possible a significant portion of the induced energy to the NES, initially at rest, which should trigger TET.

Figure 24 depicts two different families of specialorbits for a two-DOF primary system.




, , , , , , ,

1. The first family consists of in-phase SPOs (++0)located on in-phase tongues; the masses of theprimary system move in-phase. The locus of in-phase SPOs is a smooth curve on the FEP. Whenthe phase difference between the NES and the pri-mary is trivial, the motion in the configurationspace takes the form of a simple curve; in the caseof non-trivial phase differences, a Lissajous curveis realized. For the SPO 1, the motion of the two

masses of the primary system is almost identicaland monochromatic. The NES has two dominantharmonic components, one of which is at the fre-quency of oscillation of the primary system, theother being three times smaller; a 1:3 IR betweenthe NES and the primary system is realized. Thenon-linear beating characteristic of such a dynam-ical phenomenon can be clearly observed. For theSPO 1, the energy exchange is insignificant as themaximum percentage of total energy of the NESnever exceeds 0.17 per cent. For the SPOs 2 and3, the energy transfer is much more vigorous. Toobtain a global picture, the maximum percentage

of energy transferred to the NES during the non-linear beating is superposed on the FEP in Fig. 25.This clearly depicts that there exists a critical energy threshold above which the SPOs can transfer asubstantial amount of energy to the NES. More pre-cisely, the SPOs must lie above the frequency of thein-phase mode of the primary system f 1L.

Fig. 25 Maximum percentage of energy transferred to

theNES during non-linearbeating (dashed (dot-

ted) line: in-phase (anti-phase) special orbits).

The backbone of the FEP (solid line) and the

loci of the special orbits are also superimposed

(square – in-phase; circle – anti-phase [87])

2. The second family consists of anti-phase SPOs(+ − 0) located on anti-phase tongues. Their locusis also a smooth curve on the FEP. By inspecting Fig. 25, one can conclude the existence of a criticalenergy threshold for enhanced TETs; the SPOs mustlie above the frequency of the anti-phase mode of the primary system f 2L.

The transient dynamics of the weakly damped sys-tem is now examined and is interpreted based onthe topological structure of the non-linear modes of the undamped system. Damping parameters are setto λ1 = λ2 = 0.1, λ = 0.04, and others are the same asthose used in constructing the FEP in Fig. 23. In thissection, only the single-mode responses by impos-ing the in-phase and anti-phase impulsive forcing areconsidered, and the multi-mode responses (i.e. res-onance capture cascades) will be demonstrated latercompared with the experimental system.

First, the motion initiated on S 111 + ++ (i.e. in-phase fundamental TET) is examined (Figs 26(a) and

(b)). In Fig. 26(c), the WT of v (t ) − y 2(t ) is superim-posed on the FEP to demonstrate transient dynamicsalong the damped NNM manifold as the total energy decreases due to damping. The dynamical flow is cap-tured in the neighbourhood of a 1:1 resonance mani-fold, which leads to a prolonged 1:1resonance capture.Figure 26(d) depicts the trajectories of the phase dif-ferences between the NES and the two masses inthe primary structure. The phase variables were com-puted by utilizing the Hilbert transform (HT) of theresponses. Non-time-like behaviour of the two phasevariables is observed, as the evidence for resonance

capture. Figure 26(e) confirms that fundamental TET,i.e. an irreversible energy transfer from the primary structure to the NES, takes place along S 111 + ++.

Now the motion initiated on S 111 + −− (i.e. out-of-phase fundamental TET) is examined. Figure 27(a)and (b) depicts thetime series where fundamentalTETis realized in a first stage (t = 0 − 100 s) for an anti-phase motion of the primary structure. During thisregime, the envelope of all displacements decreasesmonotonically, but the envelope of the NES seems todecrease more slowly than that of the primary struc-ture; TET to the NES is observed (Fig. 27(e)). Aroundt

=80 s, the displacement y 2 of the second mass m2

becomes very small, and a transition from anti-phase(S 111 + −−) to in-phase (S 111 + +−) motion in theprimary structure occurs. When the inflection pointon S 111 + +− is reached (where a bifurcation elimi-nates the stable/unstable pair of NNMs), escape fromresonance capture occurs, which results in time-likebehaviour of the phase variables in Fig. 27(d). Fig-ures 27(b), (c), and (e) show that this is soon followedby subharmonic TET on an in-phase tongue; there is acapture into 1:3 resonance manifold.




-

Fig. 26 Fundamental TET for in-phase motion of the primary system: (a) time series; (b) close-up

of the time series (square: y 1(t ); circle: y 2(t ); reversed triangle: v (t )); (c) WT superimposed

on the frequency-energy plot; (d) trajectories of the phase modulation; (e) instantaneous

percentage of total energy in the NES [87]

A motion initiated from special orbits is examinedto verify the existence of a critical energy thresh-old above which the SPOs can trigger fundamentalTET. In Fig. 28, the motion is initiated from in-phase SPOs 1 and 2, located below and above thethreshold, respectively. The dynamic responses areremarkably different for those two cases. For theSPO 1, the NES cannot extract a sufficient amount

of energy from the primary system, and a transi-tion to S 111 + +− is observed. On this branch, themotion localizes to the primary system as the totalenergy in the system decreases. For the SPO 2, thanksto a non-linear beating phenomenon, the motion isdirected towards the basin of attraction of S 111 + ++,and fundamental TET from the in-phase mode isrealized.




, , , , , , ,

Fig. 27 FundamentalTET foranti-phase motionof theprimary system:(a) time series; (b)close-up

of the time series (square: y 1(t ); circle: y 2(t ); reversed triangle: v (t )); (c) WT superimposed

on the frequency–energy plot; (d) trajectories of the phase modulation; (e) instantaneous

percentage of total energy in the NES [87]

Likewise, if the motion is initiated from an anti-phase SPO located below the threshold (e.g. SPO 4),there occurs a transition to S 111 + −+, on which

the motion localizes to the primary system with adecreasein the total energy. If the anti-phase SPO lying above the threshold (e.g. SPO 6) is excited, the branch




-

Fig. 28 Motion initiated from in-phase special orbits: (a, b) time series; (c, d) WT superim-

posed on the frequency energy plot; (e, f) instantaneous percentage of total energy in the

NES [87]

S 111 + −− is reached, resulting in the realization of fundamental TET from the anti-phase mode.

3.3.2 Analysis of an SDOF linear primary systemwith an MDOF NES

Application of an MDOF NES is now considered. It isshowed that enhanced TET takes place in this case

because of the capacity of the essentially non-linearMDOF NES to engage in simultaneous resonancecaptures with multiple modes of the linear system.Consider the system in Fig. 29, where a two-DOF linearprimary oscillator is connected through a weak linearstiffness (which is the small parameter of the prob-lem), 0 < 1, to a three-DOF non-linear attach-ment with the two essentially non-linear stiffnesses,




, , , , , , ,

Fig. 29 Primary (linear) system with an MDOF

non-linear attachment

C 1 and C 2. The equations of motion for this system canbe written as

u1 + λu1 + (ω20 + α)u1 − αu2 = F 1(t )

u2 + λu2 + (ω20 + α + )u2 − αu1 − v 1 = F 2(t )

μv 1 + λ(v 1 − v 2) + (v 1 − u2) + C 1(v 1 − v 2)3 = 0

μ

¨v 2

+λ(2

˙v 2

− ˙v 1

− ˙v 3)

+C 1(v 2

−v 1)3

+ C 2(v 2 − v 3)3 = 0

μv 3 + λ(v 3 − v 2) + C 2(v 3 − v 2)3 = 0

(30)

In the limit → 0, the system decomposes into twouncoupled oscillators: a two-DOF linear primary sys-tem with natural frequencies ω1 =

ω2

0 + 2α and ω2 =ω0 < ω1, corresponding to out-of-phase and in-phaselinear modes, respectively; and a three-DOF essen-tially non-linear oscillator with a rigid-body mode andtwo flexible NNMs.

Unlike the SDOF NES configuration, this MDOFNES exhibits multi-frequencysimultaneous TETs frommultiple modes of the primary system; this meansthat multiple non-linear modes of the MDOF NESengage in transient resonance interactions with mul-tiple modes of the linear system. Once again, complex transitions in the damped dynamics can be related tothe topological structure of the periodic orbits of thecorresponding undamped system.

For practical purposes, the system with NES massesof O() is considered with parameter values = 0.2,α = 1.0, C 1 = 4.0, C 2 → 2C 2 = 0.05, μ → 2μ = 0.08,and ω0

=1, where rescaling was applied to the NES

masses μ and the second essentially nonlinear cou-pling spring C 2.

As performed in previous sections, the FEP of the underlying Hamiltonian system was consideredfirst. A numerical method was utilized to con-struct the FEP of the periodic solutions of theunderlying Hamiltonian system [87]. Denoting u(t ) =[u1(t )u2(t )v 1(t )v 2(t )v 3(t )]T, the periodic solutions of the undamped and unforced system (30) can be deter-mined by computing the values of u(0) for which

u(0) = 0 for a given period T . Numerically this isperformed by minimizing the expression

minT

{[u(T )u(T )] − [0u(0)]}

Then, the total energy h of the underlying Hamiltoniansystem, when it oscillates with a periodic solution of frequency ω = 2π/T , is expressed as

h = 12[u1(0)2 + u2(0)2 + μv 1(0)2 + μv 2(0)2 + μv 3(0)2]

Considering as a perturbationparameter,system (30) with the rescaled parameter μ → 2μ is expected topossess complicated dynamics as → 0, because it isessentially (or strongly) non-linear, high-dimensional,and singular (since the highest derivatives in threeof its equations are multiplied by the perturbationparameter squared).

In Fig. 30, the periodic orbits are presented in aFEP. Note that it was difficult to capture the lowest

frequency branch through the numerical scheme. It was analytically estimated and superimposed to thenumerical results [88]. From the FEP of Fig. 30, it isnoted that the backbone branches of periodic orbitsare defined over wider frequency and energy rangesthan for the system of the NES masses of O(1) [88],and no subharmonic tongues exist in this case (at leastnone was detected in the numerical scheme). Hence,it can be conjectured that a decrease in magnitudeof the masses of the NES results in the eliminationof the local subharmonic tongues (i.e. of the sub-harmonic motions at frequencies integrally related tothe natural frequencies f 1

=1.8529, f 2

=1.5259, and

f 3 = 0.9685 rad/s of the linear subsystem). For the limitof high energy and finite frequency, the underlying Hamiltonian system (30) reaches the linear limiting

Fig. 30 Frequency–energy plot of the periodic orbits for

the MDOF system with the NES masses of O(2)

[88]




-

Fig. 31 Damped responses for out-of-phase impulses Y = 0.1: (a) Cauchy WTs superimposed on

the FEP; (b) partition of instantaneous energy of the system [ 88]

system

u1 + (ω20 + α)u1 − αu2 = 0

u2 + (ω20 + α + )u2 − αu1 − v 1 = 0

3μv 1 + (v 1 − u2) = 0

(31)

with limiting natural frequencies ˆ f 1 = 1.7734, ˆ f 2 =1.1120, and ˆ f 3 = 0.7960 rad/s.

The efficiency of TETs is demoistrated numerically

by means of the MDOF NES configuration consid-ered herein, under the out-of-phase impulsive forcing F 1(t ) = −F 2(t ) = Y δ(t ),withallotherinitialconditionsbeing zero.

Figure 31 depicts the damped responses for theimpulsive forcing amplitude Y = 0.1. Inthis case, boththe relative displacements v 1(t ) − v 2(t ) and v 2(t ) −v 3(t ) between the NESmasses follow regular backbonebranches.The relative displacement v 1(t ) − v 2(t ) has adominant frequency component that approaches the

linearized natural frequency f 2 of the limiting systemfor the limit of low energy and finite frequency, wherethe equation for the NES part becomes

μv 1 + (v 1 − u2) = 0

with decreasing energy. In contrast, v 2(t ) − v 3(t ) hastwo strong harmonic components that approach thelinearized natural frequencies f 2 and f 3 for decreas-ing energy, indicating transfer of energy simultane-ously from two modes of the linear limiting systemfor limit of low energy and finite frequency. More-

over, the same regular backbone branches are trackedby the response throughout the motion and strong energy transfer occurs right from the early stage of theresponse, which explains the strong eventual energy transfer to the NES (≈ 90 per cent) that occurs for thislow-impulse excitation.

By increasing the impulsive forcing to Y = 1.0 [88],the overall energy transfer from the linear to non-linear subsystem decreases significantly with delay,and the steady-state energy dissipation by the NES is




, , , , , , ,

Fig. 32 Damped responses for out-of-phase impulses Y

=1.5: (a) Cauchy WTs superimposed on

the FEP; (b) partition of instantaneous energy of the system [ 88]

only about 50 percent.Thisoccurs because themotionis mainly localized to the directly excited linear sub-systems by the strong initial out-of-phase resonancecapture, with a small portion of energy spreading outto the NES.

Further increasing the impulse magnitude to Y =1.5 enables the system to escape from the strong initialout-of-phase resonance capture, leading to resumedstrong TETs (Fig. 32). The NES relative responsespossess multiple strong frequency components, indi-

cating that strong TET occurs at multiple frequencies.The steady-stateenergydissipation by the NES reachesnearly 90 per cent of the input energy.

3.3.3 Analysis of a linear continuous system withSDOF and MDOF attached NESs

A separate series of papers examined TET in con-tinuous systems with attached NESs. For example,Fig. 33 depicts linear (dispersive) elastic rods coupled

to SDOF and MDOF NESs [92–94]. In these works,it was shown that appropriately designed NESs arecapable of passively absorbing and locally dissipat-ing significant portions of the vibration energy of theimpulsively forced rod.

In Fig. 34, a representative WTs of the dampedresponses of these two systems superimposed to theFEPs of the underlying Hamiltonian (undamped andunforced) systems are provided.Comparing the actionoftheSDOFandMDOFNESs,noteditisthattheSDOF

NES is capable of engaging in resonance capture withonly one mode of the linear rod at a time. Hence,in Fig. 34(a), a resonance capture cascade where theSDOF NES engages with a series of modes sequen-tially (i.e. it escapes from a resonance capture withone mode before it can engage in similar resonancecapture with another one) is noted. In the case of theMDOF NES (see Figs 34(b) to (d)), this does not hold,as the NES engages in broadband resonance interac-tions with multiple modes of the rod; that is, different




-

Fig. 33 Linear elastic rod coupled to (a) an ungrounded SDOF NES; (b) an MDOF NES

Fig. 34 Wavelet spectra of the relative responses between the rod end and (a) an SDOF NES, (b–d)

an MDOF NES, superimposed to the corresponding FEPs of each system [94]




, , , , , , ,

non-linear modes of the MDOF NES engage in sepa-rate resonance captures with different linear modes of the rod (this is revealed by the broadband characterof the non-linear modal interactions between the rodand the NES in this case). Hence, similar to previousapplications withdiscrete coupled oscillators, it is con-cludedthataMDOFNESismoreversatileandeffectivecompared with the SDOF NES, as it can extract vibra-tion energy simultaneously from a set of modes of the

linear system. For a more detailed analysis and discus-sion of these results,the reader is referredto references[92] to [94].

3.4 Non-smoothVI NES

A separate series of papers considered NESs with non-smooth stiffness characteristics. An NES with piece-

wise linear springs was first utilized for the purpose of shock isolation in reference [105] (see also reference[71]); this piecewise linear stiffness is relatively easy torealize in practice [116–118].

This section is concerned with NESs undergoing VIs(hereafter, vibro-impact NESs can be termed as VINESs). As shown in the aforementioned references,this type of ‘non-smooth’ NES possesses fast reac-tion time; i.e. a VI NES is capable of passive TET ata fast time-scale, which makes this type of deviceideal in applications where the NES needs to be acti-vated very early in the motion (within the initial oneor two cycles of vibration). The simplest primary sys-tem – VI NES configuration, namely an SDOF linearoscillator coupled to a VI NES (Fig. 35) is consid-ered. It will be demonstrated that a clear depiction of the damped non-linear transitions that govern energy

transactions in this system can be gained by study-ing the damped motion on the FEP of the underlying

VI conservative system (i.e. the identical system con-figuration, but with purely elastic impacts and noviscous damping elements). The premise is that, forsufficiently small damping, the damped non-lineardynamics are perturbations of the dynamics of theunderlying conservative system, so that damped non-linear transitions take place near branches of periodicor quasi-periodic motions of the undamped system.Hence, by studying the structure of periodic orbits of the conservative system, the behaviour of the damped

dynamics should be understood as well, and phe-nomena such as TRCs and jumps between different

Fig. 35 An SDOF linear oscillator connected to a VI NES

branches of solutions that govern TET in the VI systemshould be identified.

The equations of motion in non-dimensional formbetween impacts can be written as

u1 + (1 + σ )u1 + u2 = 0, μu2 + σ (u2 − u1) = 0

(32)

where μ = m2/m1, σ = k 2/k 1 are the mass and stiff-ness ratios; the rescaling of time, τ =

k 1/m1t , isimposed, and the derivative with respect to the new non-dimensional time is denoted by the overdot.

Impact occurs whenever the absolute value of therelative displacements satisfies |u2 − u1| = e , where e denotes the clearance; if |u2 − u1| < e , then no impactoccurs and the system oscillates simply in a linearcombination of the two linear modes of system (32).

Setting the coefficient of restitution to 1 (i.e. assum-ing perfectly elastic impacts), and applying momen-tum conservation, the velocities of the two masses justbefore and after impacts can be related; that is

v 1 = (1 − μ)v 1 + 2μv 2

1 + μ, v 2 = (μ − 1)v 2 + 2v 1

1 + μ(33)

where v i = dui /dτ and the prime denotes the quantity just after impact.

The periodic solutions of the VI conservative sys-tem were computed numerically and represented ina FEP. This plot was constructed by depicting each

VI periodic orbit as a single point with the coordi-nates determined in the following way: consider theeigenfrequency of the uncoupled linear oscillator as

reference frequency, f 0 = 0.1515; the frequency coor-dinateof the FEP isequal to (p/q ) f 0, where the rationalnumber p/q is the ratio of the basic frequency of the linear oscillator to the basic frequency of the NES.The energy coordinate is the (conserved) total energy of the system when it oscillates in the specific peri-odic orbit considered. The parameters of the systemadopted for the FEP computation are μ = σ = 0.1 ande = 0.1, and the resulting FEP is depicted in Fig. 36.The complicated topology of the branches of peri-odic orbits depicted in the FEP reflects the well-knowncomplexity of the dynamics of this seemingly sim-ple non-linear dynamical system. It is exactly because

of the complexity of VI motions that it is necessary to establish a careful notation in order to distinguishbetween the different families of VI periodic motionsand study their dependence on energy and frequency.

To this end, each VI periodic orbit depicted in theFEP is given the notation Lijkl ±. The capital letterL is assigned either letters S or U , referring to sym-metric or unsymmetricperiodic motions, respectively.Symmetric periodic motions satisfy the conditionsuk (τ ) = ±uk (τ + T /2), ∀τ ∈ R, k = 1, 2, where T is the




-

Fig. 36 Frequency–energy plot for the system (32); dashed lines indicate the two linearized eigen-

frequencies, and bullets, the maximum energy levels at which oscillations take place

without VIs [116]

period of the motion, whereas unsymmetric periodicmotions do not satisfy the conditions of the sym-metric motions. Regarding the four numerical indices{ijkl }, index i refers to the number of left VIs occur-ring during the first half-period; j to the number of right impacts occurring during the first half-period; k to thenumber of left impacts occurring during the sec-ond half-period; and l to the number of right impactsoccurring during the second half-period of a periodicmotion.

The (+) sign corresponds to in-phase VI periodicmotions where, for zero initial displacements, theinitial velocities of the two particles have the same

sign at the beginning of both the first and secondhalf-periods of the periodic motion; otherwise, the

VI periodic motion is deemed to be anti-phase andthe (−) sign is used. It can be shown that S-VI peri-odic orbits correspond to synchronous motions of thetwo oscillators, and thus are represented by curvesin the configuration plane of the system, (u1, u2); i.e.these periodic motions are characterized as NNMs.On the contrary, U-VI periodic orbits correspond toasynchronous motions of the two oscillators, and arerepresented by Lissajous curves in the configurationplane of the system.

Considering the FEP of Fig. 36, the two bullets indi-cate the maximum energy thresholds below whichoscillations occur without VIs, and the dynamics of thetwo-DOF system is exactly linear. The first (in-phase)and second (out-of-phase) modes of the linear sys-tem (corresponding to the two-DOF system with norigid stops and clearance, e.g. e = ∞) exist below theenergy thresholds for VIs, namely, E 1 = 0.001 185 12for the in-phase mode and E 2 = 0.000 865 078 for theout-of-phase one. Clearly, when the system oscillates

below these maximum energy thresholds, the relativedisplacement between the two particles of the systemsatisfies |u1 − u2| < e .

As the energy is increased above the threshold VIs,giving rise to two main branches of periodic VI NNMs:the branch of out-of-phase VI NNMs S 1001− whichbifurcates from the out-of-phase linearized mode,and the branch of in-phase VI NNMs S 1001+ whichbifurcates from the in-phase linearized mode. Thetwo branches S 1001± will be referred to as backbone(global) branches of theFEP; they consist ofVI periodicmotions duringwhich the NES vibrates either in-phaseor out-of-phase with the linear oscillatorwith identical

dominant frequencies.Moreover, both backbone branches exhibit a single

VI per half-period are definedover extended frequency and energy ranges, and correspond to motions thatare mainly localized to the VI attachment (exceptin the neighbourhoods of the two linearized eigen-frequencies of the system with e = ∞, at f 1 = 0.136and f 2 = 0.186). Both backbone branches satisfy thecondition of 1:1 IR between the linear oscillator andthe VI NES, with the oscillations of both subsys-tems possessing the same dominant frequency, as

well as weaker harmonics at integer multiples of the

dominant frequency. A different class of VI periodic solutions of the FEPlies on subharmonic tongues (local branches); theseare multi-frequency periodic motions, with frequen-cies being rational multiples of one of the linearizedeigenfrequencies of thesystem. Each tongueis definedover a finite energy range and is composed of a pairof branches of in- and out-of-phase subharmonicsolutions. At a critical energy level, the two branchesof the pair coalesce in a bifurcation that signifies




, , , , , , ,

Fig. 37 Representative VI impulse orbits: U3223 (upper) and U2222 (lower) [116]

the end of that particular tongue and the elimina-tion of the corresponding subharmonic motions forhigher energy values. Clearly, there exists a countableinfinity of such tongues emanating from the back-bone branches, with each tongue corresponding tosymmetric or unsymmetric VI subharmonic motions

with different patterns of VIs during each cycle of theoscillation.

Finally, there exists a third class of VI motions in

the FEP, which are denoted as VI impulsive orbits(VI IOs). These are periodic solutions corresponding to zero initial conditions, except for the initial veloc-ity of the linear oscillator. In essence, a VI IO is theresponse of the system initially at rest due to a singleimpulse applied to the linear oscillator at time τ = 0+.

Apart from the clear similarity of a VI IO to the Green’sfunction defined for the corresponding linear system,the importance of studying this class of orbits stemsfrom their essential role in passive TET from the linearoscillator to the non-linear attachment.

Indeed, for impulsively excited linear systems withNESs having smooth non-linearities, IOs (which are,

in essence, non-linear beats) play the role of bridg-ing orbits that occur in the initial phase of TET, andchannel a significant portion of the induced impulsiveenergy from the linear system to the NES at a relatively fast time-scale; this represents the most efficient sce-nario for passive TET. Although the aforementionedresultsrefertodampedIOs,thedynamicsoftheunder-lying conservative system determines, in large part,the dynamics of the damped system as well, providedthat the damping is sufficiently small. It follows that

the IOs of the conservative system govern the initialphase of TET from the linear oscillator to the NES.

As shown in reference [85], impulsive periodic andquasi-periodic orbits form a manifold in the FEP thatcontains a countable infinity of periodic IOs and anuncountable infinity of quasi-periodic IOs.

For theVI system under consideration, the manifoldofVI IOs was numerically computed and is depicted inthe FEP of Fig. 36; in general, the manifold appears

as a smooth curve, with the exception of a num-ber of outliers. Representative VI IOs are depicted inFig. 37. In general, the IOs become increasingly local-ized to the VI NES as their energy decreases, a result

which is in agreement with previous results for NESs with smooth essential non-linearities [85]. As energy increases, the VI IOs tend to the in-phase mode (i.e.a straight line of slope π /4 in the configuration plane(u1, u2)). Moreover, there is no critical energy thresh-old for the appearance of VI IOs since there are nolow-energy VI motions (the system is linear for low-energy levels), and the dominant frequency of a VI IOdepends on the clearance, e .Forthesystemundercon-

sideration, the VI IOs start with a dominant frequency of 0.152 (or a period of 6.58).

Apart from the compact representation of VI peri-odic motions, the FEP is again a valuable tool forunderstanding the non-linear resonant interactionsthat govern energy transactions (such as TET) dur-ing damped transitions in the weakly dissipativesystem. This is because, for sufficiently weak dissi-pation (due to inelastic VIs or viscous damping), thedamped dynamics are expected to be perturbations




-

Fig. 38 Damped VI transitions initiated on the tongue

U 8778−: (a) WT superimposed on the FEP; (b)

instantaneous energy plot [116]

of solutions of the underlying conservative system. Toshow this, the dynamics of the system of Fig. 35 for thecase of inelastic impacts is computed and analysedthe resulting transient responses by numerical WTs.Then theresulting WT spectra aresuperimposed to theFEP in order to study the resulting damped transitions

and related them to the dynamics of the underlying conservative system. A damped transition is depicted in Fig. 38, corre-

sponding to VI motion initiated on the VI IO U 8778−, with a coefficient of restitution, 0.995. Three regimesof the damped VI transition can be distinguished.

In the initial phase of the motion, the oscillationsstay in the neighbourhood of the subharmonic tongueS 1221+ until approximately τ = 500 and logarithm of energy equal to −2.15. There is efficient energy dissi-pation in this initial phase of the motion, as evidencedby the energy plot of Fig. 38(b).

In the second regime, the dynamics makes a tran-

sition to branch U 0110− until the logarithm of energy becomes equal to −2.5; in this regime of thedamped transition non-symmetric oscillations takeplace. An additional transition to the manifold of VIIOs(e.g.IOs U 2112+, U 1111+, S 1221+) occurs, beforethe VI dynamics makes a final transition to the back-bone branch S 0110+ for logarithm of the energy closeto −2.7.

By studying the instantaneous energy of the systemduring the aforementioned transitions (see Fig. 38(b)),

it can be concluded that the most efficient energy dissipation by the VI NES occurs during the initialTRC on the subharmonic tongue S 1221+. This resultdemonstrates that TRC is a basic dynamical mech-anism governing effective passive TET, for example,from a seismically excited primary structure to anattached VI NES. It follows that by studying VI tran-sitions in the FEP and relating them to rates of energy dissipation by VI NESs, one should be able to iden-

tify the most effective damped transitions from a TETpoint of view. The complicated series of VI transi-tions depicted in Fig. 38 demonstrates the poten-tial of the two-DOF system for exhibiting complex dynamics, and the utility of the FEP as a tool forrepresenting and understanding complex transientmulti-frequency transitions.

4 EXPERIMENTAL VERIFICATIONS

In this section, the experimental work that validatesthe previous theoretical results on passive TET will be

reviewed. For a general synopsis regarding the exper-imental study of TETs, refer to the literature review insection 2.2.3.

4.1 Experiments with SDOF primary systems

Figure 39(a) depicts an experimental fixture builtto examine the energy transfers in the two-DOFsystem (Fig. 39(b) for its mathematical modelling)described by

M ¨ y + λ1 ˙ y + λ2( ˙ y − v ) + C ( y − v )3 + ky = 0,

v + λ2(v − ˙ y ) + C (v − y )3 = 0

(34)

A schematic of the system is provided in Fig. 39(c),detailing major components. The system parameters

were identified using modal analysis and the restor-ing force surface method(Fig. 40;[168]): M = 1.266 kg, = 0.140 kg, k = 1143 N/m, λ1 = 0.155 Ns/m, λ2 =0.4 Ns/m, C = 0.185 × 107 N/m2.8, and α = 2.8, whereα denotes the power of the essential non-linearity.

Figure 39(e) is a schematic showing how cubic(essential) non-linearity is achieved through geomet-

ric non-linearity. Assuming zero initial tension along the wire, a static force F with respect to a transversedisplacement x can be expressed as

F = kL x

L

1 − 1

1 + ( x /L)2

(35)

where k = EA/L represents the axial stiffness constantof the wire, L is the half-length of the span, E Young’smodulus, and A the cross-sectional area of the wire.




, , , , , , ,

Fig. 39 Experimental setup for an SDOF linear primary structure coupled to an SDOF NES:

(a)–(c) general configuration and schematics; (d) experimental force pulse (21 N);

(e) realization of the essential cubic non-linearity through a system with geometric

non-linearity [96, 99, 169]

Taylor-series expansion of the bracketed term about x = 0 assuming x /L 1 gives

F = EA x

L

3

+O

x 5

L5

(36)

from which the coefficient for the essentially non-linear term can be estimated as C = EA/L3. Note that

a non-integer power (close to three) is obtained viasystem identification [169].

Two series of physical experiments were conductedin which the primary system was impulsively loaded.In the first series of tests, the damping in the NES

was kept relatively low in order to highlight the dif-ferent mechanisms for TETs. Additional tests wereperformed to investigate whether TETs can take place

with increased levels of damping.




-

Fig. 40 Measured restoring force represented as a function of time (left) and relative displacement

v − y (right)[99]

4.1.1 Case of low damping

In the low-damping case, several force levels ranging from 21 to 55 N were considered, but for conciseness,only the results for the lowest and the highest forcelevels are depicted in Fig. 41.

At 21 N of forcing, the acceleration and displace-ment of the NES are higher than those of the primary system, which indicates that the NES participates inthe system dynamics to a large extent. The percentageof instantaneous total energy plot illustrates that vig-orous energy exchanges take place between the twooscillators. However, it can also be observed that thechanneling of energy to the NES is not irreversible.

After 0.23 s, as much as 88 per cent of the total energy is present in the NES, but this number drops downto 1.5 per cent immediately thereafter. Hence, in thiscase, energy quickly flows back and forth between thetwo oscillators, which is characteristic of a non-linearbeating phenomenon. Another indication for this isthat the envelope of the NES response undergoes largemodulations.

At the 55 N level, the non-linear beating still dom-inates the early regime of the motion. A less vigor-ous but faster energy exchange is now observed as63 per cent of the total energy is transferred to the NESafter 0.12 s.These quantities also hold for the interme-

diate force levels [99]. It should be noted that theseobservations are in close agreement with the analyti-cal and numerical studies [83, 84]; indeed, in this case,the special orbits are such that they transfer smalleramounts of energy to the NES, but in a faster fashion

when the force level is increased. A qualitative means of assessing the energy dissi-

pation by the NES is to compare the response of theprimary system in the following two cases: (a) whenthe NES is attached to the primary system (the present

case – denoted by ‘NES’ displacements at the bottomof Fig. 41); (b) when the NES is disconnected, but itsdashpot is installed between the primary system andground (a SDOF linear oscillator with added damp-ing – denoted by ‘ground dashpot’ displacement inFig. 41). Case (b) was not realized in the laboratory, butthe system response was computed using numericalsimulation. The two bottom figures in Fig. 41 com-pare the corresponding displacements of the linearoscillator in the aforementioned two different systemconfigurations. It can be observed that the NES per-forms much better than the grounded dashpot for the21 N level, but this is less obvious for the 55 N level.This might mean that, when the non-linear beating

phenomenon is capable of transferring a significantportion of the total energy to the NES, it should be amore useful mechanism for energy dissipation.

4.1.2 Case of high damping

Several force levels ranging from 31 to 75 N wereconsidered, and the results for 31 N are presentedherein. The damping coefficient was identified to be1.48 Ns/m, which means that damping can no longerbe considered to be O(). The increase in damping isalso reflectedin themeasured restoring force in Fig. 42.

The system responses are almost entirely damped

out after five to six periods. The NES acceleration anddisplacement are still higher than the corresponding responses of the primary system, meaning that TETsmay also occur in the presence of higher damping.The percentage of instantaneous total energy in theNES never reaches close to 100 per cent as in thelower damping case. However, one may conjecturethat this is due to the increased damping value; assoon as energy is transferred to the NES, it is almostimmediately dissipated by the dashpot.




, , , , , , ,

Fig. 41 Experimental results for low damping (left column: 21 N; right column: 55 N; note differing

durations). The first row depicts measured acceleration; the second, measured displace-

ment; the third, percentage of instantaneous total energy in the NES; and the fourth,

displacement of the primary structure (NES versus grounded dashpot) [99]




-

Fig. 42 Experimental results for high damping (31 N). From the top left, measured accelerations,

measured displacements, percentage of instantaneous total energy in the NES, measured

and simulated energy dissipated by the NES, displacement of the primary system (NES

versus grounded dashpot), and restoring force [99]

4.1.3 Frequency–energy plot analysis

Utilizing the FEP on which the WT of the relative dis-placement between the primary structure and theNESis superimposed, the dynamics of the system for high-level forcing with low damping, and for low-level forc-ing with high damping, can be investigated (Fig. 43).There are strong harmonic components developing during the non-linear beating phenomenon. Once

these harmonic components disappear, the NESengages in a 1:1 resonance capture with the linearoscillator at a frequency approximately equal to thenatural frequency of the uncoupled linear oscillator.

4.2 Experiments with MDOF primary systems

In order to support the theoretical findings insection 3.3, physical experiments were carried out




, , , , , , ,

Fig. 43 Superposition of theWT of the relative displacement across the non-linearity and the FEP:

(a) 55 N, low damping; (b) 31N, high damping [99]

Fig. 44 Experimental setup for a two-DOF linear pri-

mary structure coupled to an SDOF NES [169]

using the fixture depicted in Fig. 44, which cor-responds to the schematic depicted in Fig. 22.It realizes the system described by equation 25,and the system parameters are identified using modal analysis and the restoring force surfacemethod: m1 = 0.6285 kg, m2 = 1.213 kg, = 0.161 kg,k 1 = 420 N/m, k 2 = 0 N/m, k 12 = 427 N/m, C = 4.97 ×106 N/m3, λ1

=0.05

−0.1 Ns/m, λ2

=0.5

−0.9 Ns/m,

λ12 = 0.2 − 0.5 Ns/m, λ = 0.3 − 0.35 Ns/m.The mass ratio /(m1 + m2) is equal to 8.7 per cent.

From these parameters, the natural frequencies of theuncoupled linear subsystem are found to be 1.95 and6.25 Hz, respectively. The damping coefficients rangeover a certain interval, because damping estimation isa difficult problem in this setup due to the presenceof several ball joints and bearings, and due to the airtrack.It wasfoundthatdamping wasrather sensitivetothe force level, which is why intervals rather than fixed

values are given. In addition, at low amplitude frictionappeared to play an important role in the dynamics of

the system.In these experimental verifications, the mass m1

was loaded by impulses of different amplitudes andof durations of approximately 0.01 s. Four cases of increasing input energy were considered: case I,0.0103 J; II, 0.0258J; III, 0.0296 J; and IV, 0.0615J. Thesuperposition of the WT of the relative displacementacross the non-linear spring on the FEP is shown inFig. 45.

Starting with the case I, the lowest energy,S 111 + ++ is excited from the beginning of themotion. This means that the input energy is already

above the threshold for TET from the in-phase mode,but below the threshold for resonance with the out-of-phase mode. For case II, S 111 + ++ is again excited,but harmonic components are present. By slightly increasing the imparted energy (case III), the thresh-old for interaction with the out-of-phase mode isexceeded.As a result, S 111 + −− is excited, andshortly after a jump to S 111 + ++ takes place. In case IV, thetransitions are similar to those of case III.

Further results for case IV, which bear strong resem-blance to those in Fig. 46, are displayed in Fig. 47.During the first few cycles, the NES clearly resonates

with the out-of-phase mode. As a result, after 2 s,

the NES can capture as much as 87 per cent of theinstantaneous total energy, and the participation of the out-of-phase mode in the system response is dras-tically reduced. Around t = 2 s, a sudden transitiontakes place, and the NES starts extracting energy fromthe in-phase mode. The comparison of Figs 47(c) and(e) with Figs 47(d) and (f) shows that the predictionsof the model identified are in very close agreement

with the experimental measurements in the interval0–4 s. Specifically, the sequential interaction of the




-

Fig. 45 Frequency–energy plot for the experimental fixture for a peak duration around 0.01 s:

(a)–(d) Cases I–IV [87]

Fig. 46 Response following direct impulsive forcing of mass m1 (40 N, 0.01 s): (a)–(b) displace-

ments; (c) FEP with the superimposed WT of the relative displacement between m2 and

the NES; (d) instantaneous percentage of total energy in the NES [87]




, , , , , , ,

Fig. 47 Experimental results (case IV): (a)–(c) measured displacements; (d) predicted NES dis-

placement; (e)–(f) measured and predicted instantaneous percentage of total energy in

the NES [87]

NES with both modes is accurately reproduced by thenumerical model. Discrepancies occur after t = 4 s,probably due to unmodelled friction in the bearings;this explains why the TET predicted by the numericalmodel between 4 and 8 s was not reproduced with theexperimental fixture.

During this experiment, no attempt was made tomaximize energy dissipation in the NES. The purpose

was rather to examine the energy transfers in this sys-tem, to highlight the underlying dynamic phenomena,

and to demonstrate that the NES is capable of resonat-ing with virtually any given mode of a structure.

5 CONCLUDING REMARKS

Fundamental aspects of passive TET in systems of coupled oscillators with essentially non-linear attach-ments were reviewed in this work. The concepts,methods, and results presented in this review article




-

can be applied to diverse engineering fields. To justgive an indication of the powerful applications thatpassive TET can find some recent applications of TETand NES to some practical engineering problems.

Inaseriesofpapers[98, 114, 115]theabilityofSDOFand MDOF NESs to robustly eliminate aeroelasticinstabilities occurring in in-flow wings is demon-strated both theoretically and experimentally. Thisis a consequence of a series of transient or sus-

tained resonance captures between the essentially non-linear NESs and aeroelastic (pitch and heave)modes, which act to suppress the triggering mech-anism that yields to LCOs and assure instability-freedynamics. The designs proposed in these papers holdpromise for using strongly non-linear local elementsto achieve passive vibration reduction in situations

where this is not possible by weakly non-linear orlinear methods.

Moreover, in an additional series of papers[116–118], NESs with smoothand/or non-smooth (VI)characteristics are employed in frame structures tomitigate the damaging effects of strong seismic exci-

tations. In particular, the author demonstrated, boththeoretically and experimentally, that NESs with non-smooth stiffness characteristics can provide passivereduction of the seismic response during the criti-cal initial cycles (i.e. immediately after application of the earthquake excitation), where the motion is atits highest energetic state. This is due to fast-scaleTET from the structure to the non-smooth NES. Theuse of VI NESs in seismic mitigation designs has theadded advantages of ‘spreading’ seismic energy tohigher structural modes, which leads to amplitudereduction and to more efficient dissipation of seismic

energy.The results, methods, and applications reviewed inthis paper hopefully demonstrate the potential ben-efits to be gained through intentional introductionof non-linearities in certain engineering applications.Though this runs counter to the prevailing view thatnon-linearities in structural design should be avoided

when possible; but here it is shown that, for certainapplications, the intentional use of (even strong) non-linearities can yield beneficial results that cannot beobtained otherwise by weakly non-linear or lineardesigns.

ACKNOWLEDGEMENTS

This work was supported in part by the US Air ForceOffice of Scientific Research through Grants Num-ber FA9550–04–1–0073 and F49620-01-1-0208. GaëtanKerschen is supported by a grant from the Bel-gian National Science Foundation, which is gratefully acknowledged.

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