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

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1 INVITED REVIEW 77 Passive non-linear targeted energy transfer and its applications to vibration absorption: a review Y S Lee 1, A F Vakakis 2, L A Bergman 1, D M McFarland 1, G Kerschen 3, F Nucera 4, S Tsakirtzis 2, and P N Panagopoulos 2 1 Department of Aerospace Engineering, University of Illiois at Urbana-Champaign, Urbana, Illinois, USA 2 School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece 3 Aerospace and Mechanical Engineering Department (LTAS), Université de Liège, Liège, Belgium 4 Department 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 DOI: / JMBD118 Abstract: This review paper discusses recent efforts to passively move unwanted energy from a primary 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, fundamental 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 TET phenomenon will be presented. Finally, some current engineering applications of TET will be discussed. 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, at most, as small perturbations of linear behaviour. Essentially non-linear stiffness elements are intentionally 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, is taken to avoid some of the unwanted dynamic effects that such elements may introduce, such as chaotic 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 vibrational energy from disturbances into a main system either passively or actively since the seminal invention of the tuned vibration absorber (TVA) by Frahm [1] (refer to 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 offer the best performance in terms of vibration absorption. However, in addition to issues of cost and energy consumption associated with active control, robustness and stability need to be addressed. Corresponding author: Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. lbergman@uiuc.edu Passive dynamic absorbers represent an interesting 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 effective in the neighbourhood of a single frequency. Roberson [8] showed that broadening the suppression bandwidth is possible by employing a non-linear system for the TVA. Since then, non-linear vibration absorbers have received increased attention in the literature (e.g. continuously and discontinuously non-linear [9, 10]; piecewise linear [11]; centrifugal pendulum [12]; and autoparametric vibration absorbers [13, 14]). Although non-linearities are usually considered to be detrimental, it is possible to take advantage of the richness and complexity of nonlinear dynamics for the design of improved vibration absorbers. Passive transfers of vibrational energy through mode localization have been of particular interest in

2 78 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos solid-state, condensed-matter, and chemical physics. For example, there are vibrational energy transfers at gas solid interfaces [15, 16]; thermally generated localized 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 presenting chemical dissociation [21]. A novel mechanism was also proposed for inducing highly selective yet very efficient energy transfers in certain discrete nonlinear systems where, under a precise condition of non-linear resonance, when a specific amount of energy is injected as a discrete breather at a donor system it can be transferred as a discrete breather to another weakly coupled acceptor system [22 25]. In applications to mechanical systems, localization or confinement of vibrations, which is referred to as normal mode localization, was studied in references [26] to[29] by considering structural irregularity (or disorder) in weakly coupled component systems. An acoustical application of Anderson localization [30] was demonstrated theoretically and experimentally [31]. It was also shown that (non-linear) mode localization can occur in a class of multi-degree-of-freedom (MDOF) non-linear systems even with perfect symmetry and a weakly coupled structure [32 38]. This kind of standing wave localization, based on intrinsic localized modes (discrete breathers) or non-linear normal modes (NNMS) which exist due to discreteness and system non-linearity [39, 40], can be classified as static because it does not involve controlled spatial transfer of energy through the system. It can be realized through appropriate selection of the initial conditions [41]. Internal resonances (IRs) under certain conditions also promote energy transfer between non-linear modes [14, 42 45]. It was explained, both theoretically and experimentally, how a low-amplitude highfrequency excitation can produce a large-amplitude low-frequency response (called energy cascading [46]). However, in these cases, non-linear energy exchanges are caused by non-linear modal interactions, and they do not necessarily involve controlled TETs [41]. It is only recently that passively controlled spatial (hence dynamic ) transfers of vibrational energy in coupled oscillators to a targeted point where the energy eventually localizes were studied [41, 47 50]. This phenomenon is called non-linear energy pumping or TET. This paper summarizes recent efforts towards understanding passive TET. Some preliminaries and literature reviews are presented in section 2; then, theoretical and experimental fundamentals on non-linear TET phenomena are summarized, respectively, in sections 3 and 4. 2 TARGETED ENERGY TRANSFER (OR NON-LINEAR ENERGY PUMPING) 2.1 Preliminaries Non-linear energy pumping (or passive TETs) refers to one-way targeted spatial transfers of energy from a primary subsystem to a non-linear attachment; it is realized through resonance captures and escapes along the intrinsic periodic solution branches [41, 50]. Because of the invariance property of the resonance manifold, the energy transfers become irreversible once the dynamics is captured into resonance. The non-linear device, which is attached to a primary system for passive energy localization into itself, is called a non-linear energy sink (NES). An NES generally requires two elements: an essentially nonlinear (i.e. non-linearizable) stiffness and a (usually, linear viscous) damper. The former enables the NES to resonate with any of the linearized modes of the primary subsystem, whereas the latter dissipates the vibrational energy transferred through resonant modal interactions. The NES can be categorized as grounded versus ungrounded, single-degree-offreedom (SDOF) versus MDOF, and smooth versus non-smooth, depending on its design and use. Figure 1 depicts a schematic of passive and broadband TETs utilizing an ungrounded SDOF NES. A primary structure is given (usually a linear system and, hence, the mass, damping, and stiffness matrices, M, C, K, respectively). The primary structure, which possesses a set of natural frequencies {ω (k) Primary } k=1,...,n where N is the number of DOFs of the primary structure, can suffer various external disturbances such as impact loading, periodic or random excitation, fluid-structure interaction, etc. One seeks to (passively) eliminate such unwanted external disturbances induced in the primary structure by attaching a simple non-linear device such as 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

3 Passive non-linear TET and its applications 79 stiffness term), it can generate a countably infinite number of non-linear resonance conditions (i.e. IRs, mω (k) Primary nω NES where m, n are integers), through which vigorous energy exchanges occur between the two oscillators. In particular, energy localization to the NES is preferred for efficient mitigation of the disturbances in the primary structure. During the non-linear modal interactions, energy is dissipated in the NES damper. As the total energy decreases, self-detuning is possible with the dynamics escaping from one resonance manifold to another. There are at least three different TET mechanisms, which occur through 1:1 and subharmonic resonance captures, and are initiated by non-linear beat phenomena, respectively (see section 3). Although an NES device looks similar to a linear dynamic absorber (or a TVA) in configuration (both are passive and composed of a mass, a spring, and a 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-state regime. Therefore, even if the TVA is initially designed (tuned) to eliminate resonant responses near the natural frequency of a primary system, the mitigating performance may become less effective over time due to aging of the system, temperature or humidity variations and so forth, thus requiring additional adjustment or tuning of parameters (i.e. the robustness can be questioned). On the other hand, the NES is basically a device that interacts with a primary structure over broad frequency bands; indeed, since the NES possesses essential stiffness non-linearity, it may engage in (transient) resonance capture with any mode of the primary system (provided, of course, that a node of the mode is not at the point of attachment of the NES). It follows that an NES can be designed to extract broadband vibration energy from a primary system, engaging in transient resonance with a set of most energetic modes. Thus, the NES is more robust than the TVA [51]. 2.2 Literature review Resonance capture (or capture into resonance), which turns out to be a fundamental mechanism for non-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 to mechanical oscillators can also be found in references [64] to[66]. Recently, resonance capture was applied to suppress unwanted disturbances in practical engineering problems. In this section, efforts to understand passive TET in coupled oscillators are summarized chronologically and are grouped according to system configurations (SDOF, MDOF or continuous primary systems; grounded or ungrounded and SDOF or MDOF NESs) Grounded NES configurations Gendelman and Vakakis [47] first investigated how non-linear localization in coupled oscillators is progressively eliminated by a dissipative force. A strongly non-linear oscillator with symmetry was studied by computing and then matching separate analytical approximations for the early (localized) and late (nonlocalized) responses (see also reference [67] for a linear oscillator coupled to a strongly non-linear attachment with multiple equilibrium states). It was shown that a damped vibrational system can exhibit localization phenomena at least at the early stages of the motion. In later stages of the motion, non-linear effects diminish and a transition from non-linear localized to linearized weakly non-linear oscillations occurs as energy is dissipated. It was noted that, in a system with symmetry, IRs between subsystems exist leading to linearized beat phenomena which eliminate localization in the linearized regime. Applicability of active control to compensate for dissipation effects was addressed, keeping the localized motion preserved in the system as energy decreases (see, for example, reference [68], which suggested a control algorithm for switching mechanical components such as springs and dampers on and off during their work with minimal energy consumption). Inducing passive NESs in vibrating systems was studied in reference [48], where a complexificationaveraging technique was introduced to obtain modulation equations for the slow-flow dynamics. It was shown that, for an impulsively loaded MDOF chain with an NES attached at the end, the response of the NES after some initial transients is motion dominated by a fast frequency identical to the lower bound of the propagation zone of the linear chain, which reduces the study of TET in the chain to a two-dof equivalent problem. This is because, after some initial transients, the semi-infinite chain in essence vibrates in an inphase mode at the lower frequency boundary of the propagation zone of the infinite linear chain. Possible applications of TET to electric power networks [69] were suggested for passive fault arrest in the network, preventing catastrophic failure due to unchecked fault propagation. Similarly, energy transfer to a non-linear localized mode in a highly asymmetric system was investigated [49]. It was shown that excitation of a NNM [70] occurs via the mechanism of subharmonic resonance. The conditions for TET were suggested: (i) a localized resonant mode should be excited; and (ii) the vibrations of a non-linear oscillator should be damped faster than the primary system with the same

4 80 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos damping terms of the same order. The shortcomings of this passive vibration absorber were noted; that is, it is not activated below a critical amplitude, and, moreover, its effectiveness is reduced as the amplitude grows above the critical resonant regime because the non-linear oscillator cannot absorb more than a given amount of energy at a certain frequency. It was observed that cubic stiffness coupling between the primary structure and the NES is much more effective than linear coupling because the main mechanism of energy transfer becomes a non-linear parametric resonance (see also reference [71] for numerical evidence of TET phenomena in various structures). Dynamics of the underlying Hamiltonian system and non-linear resonance phenomena were investigated to understand energy pumping in a two-dof non-linear coupled system with a linearly coupled grounded NES being one of the DOF [41, 50]. Actionangle formulation was utilized as a reduction method at a fixed energy level to obtain a single second-order ordinary differential equation; then, non-smooth temporal transformations (NSTTs [72]) of the reduced equation were performed to compute its periodic solutions. It was shown that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the TET phenomenon, and that this orbit cannot be excited at sufficiently low energies. Hence, a transient bridging orbit satisfying zero initial conditions must be impulsively excited. Furthermore, introducing action-angle transformations, and applying the averaging theorem to get a twofrequency dynamical system, it was shown that the energy pumping phenomenon in the system studied in that work is associated with resonance capture in a neighbourhood of the 1:1 resonance manifold. The degenerate bifurcation structure of a system of coupled oscillators with an NES was studied [73], where two types of bifurcations of periodic solutions were observed: (i) a degenerate bifurcation at high 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 linear coupling stiffness approaches zero, in which case the linear part of the equations of motion possesses a double 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 manifold approach and multiple-scales expansion [74]. It was noted that there is a special asymptotical structure distinct 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 also found that time evolution of the invariant manifold may be accompanied by bifurcations, and passage of the invariant manifold through bifurcations may bring about destruction of the resonance regime and essential gain in the energy dissipation rate. The damping coefficient should be chosen to ensure the possibility of bifurcation of the NNM invariant manifold, because failure to do so will result in a loss of NES ability to dissipate the vibrational energy. Robustness of TET was examined by introducing uncertain parameters (due to aging, imperfection in design, and so on) to the NES [75]. Polynomial chaos expansions were used to obtain information about random displacements, followed by a numerical parametric study based on Monte Carlo simulation. The design of mechanical TET devices was considered in reference [76], where the complexificationaveraging technique and the method of multiplescales were utilized for analysing TET. Also, the issue of designing a linear structure (specifically, a two-dof linear chain) linearly coupled to a grounded NES was studied for enhancing TET [77]. Expressing the actual DOFs connected to the NES as modal coordinates, and assuming 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 the excitation of a damped NNM invariant manifold that is an analytic continuation of a NNM of the underlying undamped (i.e. Hamiltonian) system. A more general linear substructure (an MDOF chain) was considered in reference [78], where a similar modal expression was utilized to obtain the first version of a frequency energy plot (FEP). For the MDOF primary structure coupled to an NES, resonance capture cascades were demonstrated when TET occurs. Single- and multi-mode energy pumping phenomena were investigated in a two-dof primary structure linearly coupled to a grounded NES [79]. Isolated resonance captures leading to single-mode energy pumping occur in neighbourhoods of only one of the linear modes of the primary structure and are dominated by the corresponding linearized eigenfrequencies (which act as fast frequencies of the dynamics). However, multi-mode energy pumping is caused by resonance capture cascades that involve more than one linear mode, and pumping dynamics are partitioned into different frequency regimes with each regime being dominated by a different fast frequency close to an eigenfrequency of the linear system. Such resonance capture cascades can be clearly depicted in appropriate FEPs, which follow the damped transitions close to branches of the underlying Hamiltonian system as energy decreases due to damping dissipation. Dynamic interaction of a semi-infinite linear chain with an NES coupled at the end was investigated [80]. Energy propagation through traveling waves, with predominant frequencies inside the propagation zone exciting families of localized standing waves situated

5 Passive non-linear TET and its applications 81 inside the lower or upper attenuation zones, were analysed. Transient dynamics of a dispersive semiinfinite linear rod weakly connected to a grounded NES was investigated [81]. By means of a Green s function formulation, which reduces the dynamics to an integro-differential equation in the form of an infinite set of ODEs using Neumann expansions, resonant interaction of the NES with incident traveling waves propagating in the pass-band of the rod was examined. Resonance capture phenomena were also investigated where the NES engages in transient 1:1 IR with the in-phase mode of the rod at the bounding frequency of its pass and stop bands, which are similar to resonance capture cascades in finite-chain non-linear attachment configurations Ungrounded NES configurations Apart from reference [49], dynamics of coupled linear and essentially non-linear oscillators with substantially different masses was investigated [82]. Two mechanisms of energy pumping were examined: (i) through 1:1 resonance capture and (ii) non-resonant excitation of high-frequency vibration of the NES. It was noted that an ungrounded NES configuration can be transformed to a grounded one through change of variables, so no further analysis for the former is required. An ungrounded NES configuration with essential (non-linearizable) cubic stiffness non-linearity coupled to a primary structure was investigated more rigorously in references [83] and [84]. Unlike a grounded NES, the ungrounded configuration eliminates the restriction of relatively heavy mass of the non-linear attachment, thus possessing the feature of simplicity. Lee et al. [83] revealed a very complicated bifurcation structure of symmetric and unsymmetric periodic solutions of the underlying undamped system on a FEP by solving using a shooting method, the twopoint non-linear boundary value problem (NLBVP) formulated through suitable NSTTs based on the two eigenfunctions of a vibro-impact (VI) problem. Some important solution branches of 1:1 and subharmonic resonance manifolds, as well as of nonlinear beating, are examined analytically through the complexification-averaging technique in terms of mode localization. Then, the transient dynamics of the lightly damped system was clearly shown on the FEP by superimposing wavelet transforms (WTs) of the relative displacement between the primary structure and the NES. Furthermore, three distinct pumping mechanisms were identified [84]. The first mechanism, fundamental TET, is realized when the dynamics takes place along the in-phase, 1:1 resonance manifold occurring at the frequency domain less than the lower bound of the eigenfrequency of the linear mode. The second, subharmonic TET, is similar to the fundamental mechanism except that it occurs along the subharmonic solution branches on the FEP. Finally, the third is initiated by non-linear beating, leading to stronger TET by exciting a special (or impulsive) periodic orbit. Impulsive periodic orbits, as well as quasi-periodic orbits, were analysed by separately considering low-, moderate-, and high-energy impulsive motions [85]. Analytical approximations of impulsive periodic orbits, which are separated by corresponding uncountable infinities of quasi-periodic impulsive orbits (IOs), were performed. It was shown that the impulsive dynamics of the system is very complex due to its high degeneracy as it undergoes a codimension-3 bifurcation (indeed, the equations of motion for the ungrounded NES configuration can be transformed to those for a grounded NES configuration as in reference [73]). Robustness of TETs in coupled oscillators due to changes of initial conditions was examined in reference [51]. The problem of choosing appropriate initial conditions for achieving efficient TET in a system of coupled oscillators with an ungrounded NES was investigated by adopting a simplified description of the dynamic flow at the initial stage of motion. The analysis is complementary to the invariant manifold approach of reference [74]. Optimization of the (grounded) NES parameters for TET was considered in reference [86], where an experimental verification was performed for a reduced-scale building model with the NES located at the top floor. Similar to references [77] to[79], multi-modal TETs from a two-dof primary structure to an ungrounded NES were studied theoretically [87]. The main backbone curves on the FEP were computed analytically by utilizing the complexification-averaging technique (or numerically, using optimization techniques). Mode localization phenomena were depicted along the three main backbones, and transient dynamics of the lightly damped system was investigated for in-phase and out-of-phase impulsive forcing (not surprisingly, there exist more one-dimensional manifolds of special periodic orbits (SPOs)). Again, WT results were superimposed on the FEP to demonstrate branch transitions as the total energy decreases. Complex dynamics in a two-dof primary structure coupled to an MDOF NES were investigated in reference [88], where strong passive TET capacity (up to as much as 90 per cent of input energy) was identified. Transient resonance captures (TRCs) in finite linear chains, respectively, coupled to a grounded SDOF NES and to an ungrounded MDOF NES were compared in reference [89], where the dynamics governing the chain-nes interaction was reduced to a single, non-linear integro-differential equation that exactly describes the transient dynamics of the NES. Approximations based on Jacobian elliptic functions

6 82 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos [90] yielded an approximate set of two non-linear integro-differential modulation equations for amplitude and phase, and perturbation analysis in a O( ɛ) neighbourhood of a 1:1 resonant manifold were performed. For the MDOF NES, there were no detectable resonance capture cascades, but simultaneous multimodal resonant interactions were found instead, which suggested robust and wide applicability of TET to many engineering problems such as vibration and shock isolation, packaging, seismic mitigation, disturbance isolation of sensitive devices during launch of payloads in space, flutter suppression, and so forth. Similar work can be found in reference [91], where instantaneous frequencies of the primary structure and NES displacements were, respectively, estimated through the Hilbert transform. Broadband energy exchanges between a dissipative elastic rod and a lightweight ungrounded SDOF NES [92, 93], as well as an MDOF NES [94], were investigated rigorously. In particular, simultaneous (but not 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 input energy) level, the MDOF NES in the parameter ranges of its high efficiency exhibits robustness to changes in the amplitude of the applied shock, the coupling stiffness, and the non-linear springs Experimental studies An experimental study of non-linear TET occurring at a single fast frequency in the system considered under impulsive excitation on the primary structure was performed in reference [96]. All the previously predicted analytical aspects were verified through experiments; in particular, an input energy threshold to bring about energy pumping was clearly depicted on the plot of energy dissipation in the NES versus input energy. Kerschen et al. [97] also experimentally showed that non-linear energy pumping caused by 1:1 resonance capture is triggered by the excitation of transient bridging orbits compatible with the NES being initially at rest, a common feature in most practical applications [41]. Some interesting observations were made through a parametric study of the energy exchanges between the primary structure and the (grounded) NES: (i) the non-linear coefficient does not influence the energy pumping (see also the bifurcation analysis [98]); (ii) the linear coupling spring must be weak in order to have an almost complete energy transfer to the NES along the 1:1 resonance manifold; (iii) the stiffness should be chosen high enough to transfer a sufficient amount of energy to the NES during nonlinear beating; and (iv) relatively large mass for the NES should be considered for better energy transfers (see reference [82]). An indirect analytical comparison via coordinate transformation suggested that the ungrounded NES configuration can eliminate the restriction on the large mass requirement of the nonlinear attachments, which was also experimentally demonstrated in reference [99]. Transient resonance captures were experimentally demonstrated [100] by means of EMD [95]. In particular, the EMD is a very useful tool for experimental studies (i.e. system identification [101, 102]) where the system information is not given a priori. The theoretical work for a two-dof primary structure coupled to an unground SDOF NES [87] was experimentally verified by comparison with numerical simulations. Experimental studies demonstrated the usefulness of the FEP for interpreting TET mechanisms; moreover, isolated resonance captures and resonance capture cascades were also observed Applications Application of NESs to shock isolation was first demonstrated in references [103] and [104]. Essentially non-linear stiffness elements were used for robust energy pumping at a sufficiently fast timescale, because fast energy pumping at the early stage is crucial for shock isolation purposes. In particular, adding two symmetrically placed NESs makes it possible to achieve dual mode shock isolation to reduce unwanted disturbances generated at different ends of the primary system. It was noted that, due to their modular form, the NESs can be added locally in an otherwise linear system in order to globally alter the dynamics in a way compatible to the design objectives. Dual mode non-smooth (piecewise linear) NESs were also utilized for the purpose of shock isolation [105]. Furthermore, steady-state TET from an SDOF linear primary structure under sinusoidal excitation to an attached NES was demonstrated theoretically and experimentally [106]. A linear oscillator coupled to an ungrounded NES was considered in references [107] and [108], and was transformed by proper change of variables to a system similar as the one studied in reference [106]. It was shown that the damped dynamics exhibits a quasi-periodic vibration regime rather than a steady-state sinusoidal response, a regime associated with attraction of the dynamical flow to a damped-forced NNM manifold (for a more advanced analysis, refer to references [109] and [110]). Experiments were also performed on an equivalent electric circuit (see also reference [111] for energy pumping under transient forcing). Application of TET for suppressing self-excited instabilities was examined. Suppression of limit cycle oscillations (LCOs) in the van der Pol (VDP) oscillator by means of non-linear TET was studied in reference [112]. The VDP oscillator exhibits dynamics

7 Passive non-linear TET and its applications 83 analogous to non-linear aeroelastic instability. By studying the slow-flow dynamics, extracted through the complexification-averaging technique, and performing numerical continuation of equilibria and limit cycles, bifurcation structures of LCOs and the possibility of robust LCO suppression were parametrically investigated. It was concluded that a steady-state is 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 oscillator, a sequence of superharmonic and subharmonic resonant interactions between the VDP oscillator and the NES must take place. The triggering mechanism of aeroelastic instability in a two-dof (heave and pitch), two-dimensional rigid wing under subsonic quasi-steady aerodynamics was examined in reference [113]. It was found that the LCO-triggering mechanism consists of three different dynamic phenomena: a series of TRCs, escapes from these captures and, finally, entrapment into permanent resonance capture (PRC). An initial excitation of the heave mode by the flow acts as the trigger of the pitch mode through a series of non-linear modal interactions. Moreover, both the initial triggering and full development of LCOs are transient phenomena, so that one can properly design an NES attachment to the wing for their suppression. Based on these observations, an ungrounded SDOF NES was applied to the two-dof rigid wing, and suppression of aeroelastic instability through passive TETs was investigated both theoretically [114, 115] and experimentally [98]. Three distinct suppression mechanisms were identified: (i) recurring suppressed burstouts, (ii) intermediate, and (iii) complete elimination of aeroelastic instability. Those suppression mechanisms were identified with the bifurcation structure of LCOs obtained through a numerical continuation technique. Furthermore, the robustness of the aeroelastic instability suppression was examined. In order to enhance robustness of aeroelastic instability suppression, the MDOF NES first considered in references [89] and [94] was considered instead of the SDOF NES. Bifurcation analysis showed that robustness of instability suppression by means of simultaneous multi-modal resonant interactions due to the MDOF NES can be greatly enhanced, with a much smaller total mass of the MDOF NES. Non-linear modal energy exchanges were studied for various parameter conditions. Seismic mitigation of a reduced two-dof model [86, 111] and of an MDOF model [71], with an NES on the top floor, was studied. Since an NES with smooth stiffness non-linearities is not suited to suppress the peak seismic responses at the critical early regime of the motion, alternative non-smoothvi NESs were considered in references [116] to[118]. Effective seismic mitigation through the use of VI NESs was demonstrated both numerically and experimentally in these works. Other applications of passive TETs include suppression 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 associated with the averaging theorem are reviewed to support the discussion of non-linear TET that follows. Definition 1 (Resonance Manifold [121]) Consider the system in polar form with multi-phase angles r = ɛr(φ, r), φ = Ω(r) (1) where r R p, 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 system depending on frequency decompositions. The set of points in D R p where i (r) = 0, i = 1,..., q is called the resonance manifold. This resonance condition is not sufficient; that is, if each i (r), i = 1,..., q is away from zero, the IR manifold is defined as the set {r R p :< k, (r) 0, k Z q } where the corresponding 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 the resonant manifold continues in such a way that the commensurable frequency relation is approximately preserved. In this situation, not all phase angles are fast (time-like) variables, so classical averaging cannot be performed with regard to these angle variables. As a result, over the time-scale ɛ 1 the exact and averaged 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 time instant t = t 0, with the non-trivial frequency combination σ = k 1 ω 1 + k 2 ω k q ω q, k i Z, i = 1,..., q, vanishing at that time instant (t = t 0 ). Then, sustained resonance is defined to occur when σ 0 persists for times t t 0 = O(1). On the other hand, transient resonance refers to the case when σ makes a single slow passage through zero.

8 84 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Definition 3 (Capture, escape, and pass-through [64]) The possible behaviour of trajectories near the resonance manifold on the time-scale ɛ 1 is described according to the following three cases: (i) capture: solutions are unbounded in backward time. However, captured trajectories remain bounded for forward times of O(ɛ 1 ), i.e. a sustained resonance exists in forward time; (ii) escape: solutions grow unbounded in forward time. However, in backward time, solutions remain bounded for times of O(ɛ 1 ), i.e. a sustained resonance exists in backward time; (iii) pass-through: solutions do not remain in the neighbourhood of the resonance manifold in either forward or backward time. No sustained resonance exists. A mechanism for resonance capture in perturbed two-frequency Hamiltonian systems was studied by Burns and Jones [61] where the most probable mechanisms for resonance capture were shown to involve an interaction between the asymptotic structures of the averaged system and a resonance. It was further shown that, if the system satisfies a less restrictive condition (or Condition N [125]) regarding transversal intersection of the averaged orbits to the resonance manifold, resonance capture can be viewed as an event with low probability, and passage through resonance is the typical behaviour on the time-scale O(ɛ 1 ). Necessary conditions were proved by Kath [56] both for entrainment to sustained resonance and for its continuance (and thus the possible indication of unlocking or escape from the sustained resonance after a finite time) by successive near-identity transformations; a sufficient condition was also derived for continuation of sustained resonance by means of matched asymptotic expansions [57]. On the other hand, transition to escape was studied by Quinn [65] in a coupled Hamiltonian system consisting of two identical oscillators possessing a homoclinic orbit when uncoupled. Focusing on intermediate energy levels at which sustained resonant motion occurs, the existence and behavior of those motions were analysed in equipotential surfaces whose trajectories are shown to remain in the transiently stochastic region for long times and, finally, to escape or leak out of the opening in the equipotential curves and proceeding to infinity. Regarding passage through resonance, one may refer to references [126] to [128]. The phenomenon of passage through resonance is sometimes referred to as nonstationary resonances caused by excitations having time-dependent frequencies and amplitudes [129]. Finally, the following definitions for non-linear resonant interactions between modes are introduced when the multi-frequency components of a system are taken into account. Definition 4 (IR, TRC, and PRC [61]) Consider an unforced n-dof system whose linear natural frequencies are ω k, k = 1,..., n. The author (i) IR as motions for which there exist k i Z, i = 1,..., n, such that k 1 ω 1 + k 2 ω 2 + +k n ω n 0, i.e. some combination of linear natural frequencies satisfy commensurability; (ii) TRC as capture into a resonance manifold which occurs and continues for a certain period of time (e.g. on the time-scale ɛ 1 ) and then finally involves transition to escape. This includes sustained resonance captures involving escape; (iii) PRC as sustained resonance captures that will never escape for increasing time. Both TRC and PRC may occur along the IR manifold and are distinguished by whether or not they involve an escape. Both IR and PRC may show similar steady-state behaviours, which differ from the commensurability condition between linear natural frequencies. Hereafter, a m : n IR refers to a condition on the slow-flow averaged system unless noted otherwise. For more details on the averaging theorem and resonance captures in multi-frequency systems, one can also refer to references [60], [62], [63], [125], [130], and [131]. 2.4 Analytical and numerical tools Perturbation methods There are many perturbation techniques to compute periodic solutions of a non-linear system: the methods of multiple-scales, of averaging, and of harmonic balance [132]. Although each of these methods has its own features, they are fundamentally equivalent to each other. One restriction to their application is the assumption of weak non-linearity; that is, the derived analytical solutions of the non-linear system lie close to those of the corresponding linearized system. The averaging theorem provides validity of the approximation, generally up to the time-scale ɛ 1. Although the harmonic balance method (HBM) can be applied to strongly non-linear systems, it approximates only the steady-state responses. On the other hand, the methods of averaging and of multiple-scales can be applied to the study of transient dynamic behaviour, which is suitable for understanding nonlinear TET phenomena. An application of the averaging method to the resonance capture problem can be found in reference [133] (and see [134] for the HBM). Since the essentially non-linear coupling between a primary system and an ungrounded NES is not necessarily weak, the complexification-averaging technique first introduced by Manevitch [135] will be utilized in the following analysis as an analytical tool for understanding resonance capture phenomena. Use of

9 Passive non-linear TET and its applications 85 complex variables renders relatively easier manipulation of the resulting modulation equations (particularly, in the presence of multi-frequency components). In addition, this method is applicable to strongly nonlinear systems. For some analyses, the multiplescale method is utilized instead of averaging (e.g. [49, 73, 74]) Non-smooth temporal transformations Non-smooth time transformations (NSTTs) can also be utilized to compute periodic solutions of a (strongly) non-linear system [72, ]. Unlike the usual perturbation methods that implement the basis of sine and cosine functions (or elliptic functions in some cases), the NSTTs employ saw-tooth and square wave functions as the basis (see reference [140] for other types of non-smooth basis functions and their applications). Any periodic solutions can be expressed in terms of asymptotic series expansion of these two non-smooth functions; moreover, this technique can be 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 yields NLBVPs, which are solved by means of numerical schemes such as the shooting method [141] 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 Floquet multipliers [142]; or (iii) by studying the topological structure of numerical Poincaré maps [143]. Then, bifurcation diagrams can be constructed with respect to control parameters, or other induced parameters such as the total energy of the system. Bifurcation analysis [144] of periodic solutions in a coupled oscillator is crucial in order to understand transitions that occur in the damped dynamics or to enhance robustness of instability suppression by means of passive TETs. Methods of numerical continuation of equilibria and limit cycles can be utilized. In particular, AUTO [145] and MatCont [146] can easily be employed Time frequency analysis Understanding transient modal interactions during non-linear TET requires an integrated time frequency analysis [ ]. The most popular techniques include the EMD method and the WT. WTs have found applications in non-linear system identification, e.g. characterization of structural non-linearities and 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]. TheWT can be viewed as a basis for functional representation but is at the same time a relevant technique for time frequency analysis. In contrast to the Fast Fourier Transform (FFT), which assumes signal stationarity, the WT involves a windowing technique with variable-sized regions. Small time intervals are considered for high-frequency components, whereas the size of the interval is increased for lower frequency components, thereby giving better time and frequency resolutions than the FFT. The Matlab R codes used for the WT computations in this paper were developed at the Université de Liège (Liège, Belgium) by Dr V. Lenaerts in collaboration with Dr P. Argoul from the Ecole Nationale des Ponts et Chaussées (Paris, France). Two types of mother wavelets ψ M (t) are considered: (a) the Morlet wavelet, which is a Gaussian-windowed complex sinusoid of frequency ω 0, ψ M (t) = e t 2 /2 e jω0t ; (b) the Cauchy wavelet of order n, ψ M (t) =[j/(t + j)] n+1, where j 2 = 1. The frequency ω 0 for the Morlet WT and the order n for the Cauchy WT are user-specified parameters which allow one to tune the frequency and time resolutions of the results. It should be noted that these two mother wavelets provide similar results when applied to the signals considered in this paper. The plots shown represent the amplitude of thewt as a function of frequency (vertical axis) and time (horizontal axis). Heavily shaded areas correspond to regions where the amplitude of the WT is high, whereas lightly shaded regions correspond to low amplitudes. Such plots enable one to deduce the temporal evolutions of the dominant frequency components of the signals analysed. Alternatively, the EMD gained popularity in the area of 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-linear and non-stationary signals with non-zero mean. In an alternative numerical post-processing technique, the EMD through a sifting process yields a collection of intrinsic mode functions (IMFs), which form a complete, nearly orthogonal, local, and adaptive basis. These properties render the EMD applicable to decomposition of non-linear and non-stationary signals. Once EMD is performed, the obtained IMFs are suitable for Hilbert transformation, which yields the instantaneous amplitude and phase of each IMF at any given instant of time. By differentiating the instantaneous phase, one computes the temporal evolution of the instantaneous frequency of each IMF which, when compared with the overall WT of the time series, enables one to judge the relative contribution of each IMF in the time series and, thus, its relative importance in the decomposition of the signal.

10 86 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Note that an IMF may have a significant contribution in certain time intervals of the signal, and be less important in others. Hence, EMD coupled with the Hilbert transform can be a powerful computational tool for studying complicated non-linear resonance interactions leading to complex dynamic phenomena (such as TET) in coupled structures. Recently, a time series forecasting method based on support vector regression machines was proposed, its apparent superiority attributed to the use of neural networks [158]. An effort was made to improve the quality (i.e. orthogonality) of the obtained IMFs by means of an energy difference tracking method [159]. The EMD method can also be applied to problems of fault diagnosis and damage estimation [160, 161]. In this paper, Matlab R codes developed by Rilling et al.[162] are employed to perform the described EMD analysis. 3 DYNAMICS OF TET In order to establish a clear understanding of nonlinear energy pumping mechanisms, a SDOF primary oscillator coupled to an ungrounded SDOF NES [83, 84] is considered in this section. For a SDOF primary structure coupled to a grounded SDOF NES, one can 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 m 1 (the linear oscillator) coupled through an essentially non-linear stiffness to a mass m 2 (the non-linear attachment). The equations of motion of this two-dof system are given by structure of the periodic orbits of the underlying undamped system (with λ 1 = λ 2 0). Indeed, it will be shown that this seemingly simple system possesses a very complicated topological structure of periodic orbits, some of which are responsible for TET phenomena in the impulsively forced, damped system Numerical approach The periodic orbits of the system will be computed numerically utilizing the method of non-smooth transformations first developed by Pilipchuk [163] and then applied to strongly non-linear oscillators by Pilipchuk et al. [72]. This method can be applied to the numerical and analytical study of the periodic orbits (and their bifurcations) of strongly non-linear dynamical systems. To apply the method, the sought periodic solutions are expressed in terms of two non-smooth variables, τ and e,as ( ) ( ( )) t t v(t) = e y 1 τ, α α ( ) ( ( )) t t x(t) = e y 2 τ α α where α = T /4 represents the (yet unknown) quarterperiod. The non-smooth functions τ(u) and e(u) are defined according to the expressions τ(u) = 2 ( π sin 1 sin π ) 2 u, e(u) = τ (u) (4) (3) and are used to replace the independent time variable from the equations of motion; their graphic depiction is given in Fig. 3. m 1 ẍ + k 1 x + c 1 ẋ + c 2 (ẋ v) + k 2 (x v) 3 = 0 m 2 v + c 2 ( v ẋ) + k 2 (v x) 3 = 0 ẍ + ω 2 0 x + λ 1ẋ + λ 2 (ẋ v) + C(x v) 3 = 0 ɛ v + λ 2 ( v ẋ) + C(v x) 3 = 0 (2) where ω 2 0 = k 1/m 1, C = k 2 /m 1, ɛ = m 2 /m 1, λ 1 = c 1 /m 1, and λ 2 = c 2 /m 1. Before analysing non-linear TET phenomena in the damped system, it is first necessary to examine the Fig. 2 The two-dof system with essential stiffness non-linearity Fig. 3 The non-smooth functions τ(u) and e(u)

11 Passive non-linear TET and its applications 87 Setting λ 1 = λ 2 = 0, and substituting equation (3) into equation (2), smoothening conditions [72] are imposed to eliminate singular 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 transformed equations that is multiplied by the non-smooth variable e, the following two-point NLBVP is formulated in terms of the non-smooth variable τ, in the interval 1 τ +1 y 1 = y 3, y 2 = y 4, y 3 = C ɛ α2 ( y 1 y 2 ) 3, y 4 = ω2 0 α2 y 2 Cα 2 ( y 2 y 1 ) 3 (5) with the boundary conditions, y 1 (±1) = y 2 (±1) = 0, where primes denote differentiation with respect to the non-smooth variable τ, and a state formulation is utilized. The boundary conditions above result from the aforementioned smoothing conditions. Hence, the problem of computing the periodic solutions of the undamped system (2) is reduced to solving the NLBVP (5) formulated in terms of the bounded independent variable τ [ 1, 1], with the quarterperiod α playing the role of the non-linear eigenvalue. It is noted that the solutions of the NLBVP can be approximated analytically through regular perturbation series [72]; however, this will not be attempted herein where only numerical solutions will be considered. It is merely mentioned here that equation (5) is amenable to direct analytical study in terms of simple mathematical functions. It is noted that the NLBVP (5) provides the solution only in the normalized half-period 1 t/α 1 1 τ 1. To extend the result over a full normalized period equal to four, one needs to add the component of the solution in the interval 1 t/α 3; to perform this one takes into account the symmetry properties of the non-smooth variables τ and e by adding the antisymmetric image of the solution about the point ( y i, t/α) = (0, 1), as shown in Fig. 4. It follows by construction that the computed periodic solutions satisfy the initial conditions, x( α) = v( α) = 0 and v( α) = y 1 ( 1)/α, ẋ( α) = y 2 ( 1)/α. It is noted at this point that since equation (2) is an autonomous dynamical system these initial conditions can be shifted arbitrarily in time; for example, they can be applied to the initial time t = 0 instead of t = α = T /4. However, in what follows the formulation of the NLBVP (5) will be respected, and the initial conditions at t = T /4 are retained. Considering the general shape of the periodic orbits depicted 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 orbits that satisfy the conditions ( v T ) ( =± v + T ) y ( 1) =±y 1 (+1) ( ẋ T ) ( =±ẋ + T ) y ( 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 Unsymmetric solutions Unm are orbits that do not satisfy the conditions of the symmetric orbits. Orbits U(m + 1)m bifurcate from the symmetric solution S11 at T /4 mπ/2, and exist approximately 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 programmed in Mathematica R (see references [141] and [164] for some details on the shooting method and general characteristics of global solutions). The NLBVP (5) is solved as follows. 1. For a given non-linear eigenvalue α (quarterperiod), the solutions of the NLBVP are computed at different energy levels; it is expected that at every energy level there co-exist multiple non-linear periodic solutions sharing the same minimal period. Periodic orbits that correspond to synchronous motions of the two particles of the system, and

12 88 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos pass through the origin of the configuration plane ( y 1, y 2 ), are termed NNMs [165]. 2. The different families of computed periodic solutions are depicted in three types of plots. In the first two types of plots, initial displacements x( T /4) = v( T /4) = 0 are assumed, and the initial velocities v( T /4) = y 1 ( 1)/α and ẋ( T /4) = y 2 ( 1)/α corresponding to a periodic orbit as functions of the quarter-period α = T /4 or the (conserved) energy of that orbit are depicted h = 1 2 ( [ɛ v 2 T ) ( + ẋ 2 T )] 4 4 = 1 2α 2 [ɛy 2 1 ( 1) + y 2 2 ( 1)] In the third type of plots, the frequencies of the periodic orbits are depicted as functions of their energies h. These plots clarify the bifurcations that connect, generate, or eliminate the different branches (families) of periodic solutions. 3. The stability of the computed periodic orbits was determined numerically by three different methods: application of Floquet theory; construction of two-dimensional Poincaré maps on the isoenergetic manifolds of the two-dof undamped system (2); and direct numerical simulation of the equations of motion using as initial conditions those estimated by the solution of the NLBVP (5). In the following, the numerical results correspond to the two-dof undamped system with parameters ɛ = 0.05, ω 0 = 1, C = 1.0 in the energy range 0 < h < 1. The bifurcation diagrams of the initial velocities and for varying quarter-period are depicted in Fig. 5. Some general and preliminary observations on the computed periodic orbits are made at this point, and the dynamical behaviour of the system on the various branches will be discussed in the next section. Considering the branches Snn, they exist in the quarter-period intervals 0 <α<nπ/2, and their initial conditions satisfy the limiting relationships (Fig. 5) lim{ v( α), ẋ( α) }=, α 0 lim { v( α), ẋ( α) } = 0 α nπ/2 These symmetric branches exist throughout the examined energy domain 0 < h < 1. It is noted that branches Snn are, in essence, identical to the branch S11, since they are identified over the domain of their common minimal period (the Snn branches are branches S11 repeated n times ); similar remarks can be made regarding the branches S(kn)(km)±, k integer, which are identified with Snm±. Focusing in the neighbourhood of branches S11± 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 (S11 ( ), S13 ( ), S15 ( ), S31 ( ), S21 ( ) with in-phase as filled-in, and branches U without symbol) [83] S11 disappears the branches S11+ and U 21 bifurcate out (similar behaviour is exhibited by the branches S31, S21,...). For π/2 α π, a bifurcation from S11+ to S13+ takes place without change of phase; similar bifurcations take place at higher values of α for branches S15+, S17+,... Forα 3π/2, the branches S13+ and S13 coalesce into the branch S11, with similar coalescences into S11 taking place at higher values of α for the pairs of branches S15+, S17+,... The unsymmetric branches U(m + 1)m bifurcate from the symmetric branches S(m + 1)(m + 1) at quarter-periods equal to α = mπ/2. It turns out that certain orbits (termed SPOs ) on these branches are of particular importance concerning the passive and irreversible energy transfer from the linear to the non-linear oscillator. The special orbits satisfy the additional initial condition y 1 ( 1) = v( α) = 0, and correspond to zero crossings of the branches U(m + 1)m in the bifurcation diagram (the upper

13 Passive non-linear TET and its applications 89 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 or unstable) are depicted in Fig. 6. Taking into account the formulation of the NLBVP (5), it follows that the special orbits satisfy initial conditions v( T /4) = v( T /4) = x( T /4) = 0, and ẋ( T /4) = 0, which happen to be identical to the state of the undamped system (2) (being initially at rest) after application of an impulse of magnitude ẋ( T /4) = y 2 ( 1)/α on the linear oscillator. Moreover, comparing the relative magnitudes of the linear and non-linear oscillators for the special orbits of Fig. 6, it is concluded that certain stable special orbits are localized to the non-linear oscillator. This implies that if the system is initially at rest 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 the invariant manifold of that special orbit, and, hence, the motion is rapidly and passively transferred from the linear to the non-linear oscillator. Moreover, this energy transfer is irreversible because of the invariance properties of the stable special orbit, and, as a result, after the energy is transferred, it remains localized and is passively dissipated at the non-linear attachment. Therefore, it is assumed that the impulsive excitation of one of the stable special orbits is one of the triggering mechanisms initiating (direct) passive TET. This conjecture will be proven to be correct by numerical simulations presented in a later section. Similar classes of special orbits can also be realized in a subclass of S-branches. In particular, this

14 90 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 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 overlapping 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 do not pass through the origin of the configuration plane (such as S21, S12,...). The branch S11 is a particular case, where the special orbit is realized only asymptotically as the energy tends to zero, and the motion is localized completely in the linear oscillator. In Fig. 7, the various branches of solutions are presented in a FEP. For clarity, the following convention regarding the placement of the various branches in the frequency domain is adopted: a specific branch of solutions is assigned with a frequency index equal to the ratio of its two indices, e.g. S21± is represented by the frequency index ω = 2/1 = 2, as is U 21; S13± is represented by ω = 1/3, and so forth. This convention holds for every branch except S11±, 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 (but not sufficient) conditions for bifurcation and stability instability exchanges are satisfied when two Floquet multipliers of the corresponding variational problems coincide at +1 or 1 (since periodic orbits of a Hamiltonian two-dof system are considered, two Floquet multipliers of the variational problem are always equal to +1, whereas the other two form a reciprocal pair), and these are indicated at the solution branches of Fig. 7.

15 Passive non-linear TET and its applications 91 Fig. 8 Close-ups of particular branches in the frequency index logarithm of energy plane: (a) S11 ; (b) S11+; (c) S13±; (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 that take place in different frequency energy domains, certain branches are depicted in detail in Fig. 8, together with the corresponding orbits realized in the configuration plane of the system. The horizontal and vertical axes in the plots in the configuration plane are the non-linear (v) and linear oscillator (x) responses, respectively; the aspect ratios in these plots are set so that the tick mark increments on the horizontal and vertical axes are equal in size, enabling one to directly deduce whether the motion is localized in the linear or the non-linear oscillator. The plot for U 43 (Fig. 8(d)) is composed of two very close branches; for the sake of clarity only one of the two branches is presented. The motion is nearly identical on the two branches, so only the oscillations in the configuration plane of one of the two branches are considered. Since a systematic analytical study of the various types of periodic solutions of the system is presented in the next section, the following preliminary remarks are made. 1. The main backbone of the FEP is formed by the branches S11± which represent in- or out-of-phase synchronous vibrations of the two particles possessing one half-wave per half-period. Moreover, the natural frequency of the linear oscillator ω 0 = 1 (which is identified with a frequency index equal to unity, ω = 1) naturally divides the periodic solutions into higher and lower frequency modes. There are two saddle-node-type bifurcations in the higher frequency, out-of-phase branch S11, and the stable solutions become localized to x or v as ω 1 + or ω 1, respectively (see Fig. 8(a)). The lower frequency, in-phase branch S11+ becomes unstable at higher energies, and the stable solutions localize to 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

16 92 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos branches S11±. Considering first the symmetric solutions, the branches S1(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 two branches S1(2k + 1)± is linked through a smooth transition with its neighbouring branches S1 (2k 1)± or S1(2k + 3)±, and exists over a finite interval of energy. The pair S1(2k + 1)± is eliminated through a saddle-node-type bifurcation at a higher energy value (see Fig. 8(c) for branches S13±). The pairs of branches S1(2k)±, k = 1, 2,... bifurcate out of S1(2k + 1)±, and exist over finite energy intervals. All branches S1n± and Sn1±, n Z + seem to connect with S11 through jumps in the FEP, but in actuality no such discontinuities occur if one takes into account that due to the previous frequency convention solutions Spp+ are identified with the solution S11+, S(2p)(1p)± with S21±, etc. 3. Focusing now on the unsymmetrical branches, a family of U(m + 1)m branches bifurcating from branch S11 exists over finite energy levels and are eliminated through saddle-node-type bifurcations with other branches of solutions. Again, the transitions of branches U 21 and U 32 to S11+ seem to involve jumps, but this is only due to the frequency convention adopted, and no actual discontinuities in the dynamics occur. An additional interesting family of unsymmetrical solutions is Um(m + 1), m = 1, 2,... which, due to the previous frequency convention, is depicted for frequency indices ω<1; the shapes of these orbits in the configuration plane are similar to those of U(m + 1)m, m = 1, 2,..., but rotated by π/2. An important class of periodic orbits realized on the unsymmetrical branches (but also in certain of the symmetric branches) is that corresponding to all initial conditions zero, with the exception of the initial velocity of the linear oscillator. These special orbits provide one of the mechanisms for passive TET from the linear oscillator to the non-linear attachment [84]. The previous discussion indicates that the two-dof undamped system possesses complicated structures of symmetric and unsymmetrical periodic orbits. The next section will focus on the analysis of the computed periodic orbits in detail in an effort to better understand the dynamics and localization properties of the system over different frequency energy ranges. Indeed, understanding the periodic dynamics of the undamped system paves the way to explain passive TET phenomena and complicated transitions between different types of motion in the transient dynamics of the damped system Analytical approach The dynamics of the undamped system and all the different branches of solutions can be studied analytically. As representative examples of this analysis, the periodic orbits on a particular branch, namely S11±, are investigated in detail. To study the periodic orbits of equation (2) for 0 < ɛ 1, the complexification-averaging method first introduced by Manevitch [135] is applied, which not only enables the study of the steady-state motions, but also can be applied to analyse the damped, transient dynamics [50]. The S11± branch is composed of synchronous periodic motions where the two particles oscillate with identical frequencies. The analytical study of these solutions is performed by introducing the new complex variables ψ 1 = ẋ + jωx and ψ 2 = v + jωv where j 2 = 1, and expressing the displacements and accelerations of the two particles of the system as (the asterisk denotes complex conjugatation) x = 1 2jω (ψ 1 ψ 1 ), ẍ = ψ 1 jω 2 (ψ 1 + ψ 1 ) v = 1 2jω (ψ 2 ψ 2 ), v = ψ 2 jω 2 (ψ 2 + ψ 2 ) (6) Since nearly monochromatic periodic solutions of the equations of motion are sought and the two particles oscillate with the identical frequencies, the previous complex variables are approximately expressed in terms of fast oscillations of frequency ω, e jωt, modulated by slow (complex) modulations φ i (t) ψ 1 = φ 1 e jωt, ψ 2 = φ 2 e jωt (7) This amounts to a partition of the dynamics into slow- and fast-varying components, and the interesting dynamics is reduced to the slow flow. Note that no 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 λ 1 = λ 2 = 0, and performing averaging over the fast frequency, to a first approximation only terms containing the fast frequency ω are retained φ 1 + jω 1 2 φ 1 j 1 2ω φ 1 + j 3C 8ω ( φ 1 2 φ φ 2 1 φ 2 φ 2 2 φ 1 + φ 2 2 φ φ 1 2 φ 2 2 φ 2 2 φ 1 ) = 0 ɛ φ 2 + j ɛω 2 φ 2 j 3C 8ω ( φ 1 2 φ φ 2 1 φ 2 3φ2 2 φ 1 + φ 2 2 φ φ 1 2 φ 2 2 φ 2 2 φ 1 ) = 0 (8) These complex modulation equations govern the slow evolutions of the complex amplitudes φ i, i = 1, 2 in time.

17 Passive non-linear TET and its applications 93 Introducing the polar representations φ 1 = Ae jα and φ 2 = Be jβ where A, B, α, β R in equation (8), and separately setting the real and imaginary parts of the resulting equations equal to zero, the following real modulation equations that govern the slow evolution of amplitudes and phases of the two responses of the system are obtained as Ȧ + BC 8ω 3 [(3A2 + 3B 2 ) sin(α β) + 3AB sin(2β 2α)] =0 A α + ωa 2 A 2ω 3CA3 6AB2 C 8ω 3 8ω 3 [( 9A 2 3B 2 ) cos(α β) + 3AB cos(2β 2α)] =0 ɛḃ AC 8ω 3 [(3B2 + 3A 2 ) sin(α β) + 3AB sin(2β 2α)] =0 BC 8ω 3 ɛb β + ɛωb 2 3B3 C 6A2 BC AC 8ω 3 8ω 3 8ω 3 [( 9B 2 3A 2 ) cos(α β) + 3AB cos(2β 2α)] =0 (9) The first and third (amplitude modulation) equations are combined, giving AȦ + ɛbḃ = 0 A 2 + ɛb 2 = N 2 where N is a constant of integration. Clearly, the above is an energy conservation relation reflecting the conservation of the total energy of the undamped system (2) during its oscillation. Hence, the modulation equation (9) can be reduced by one, by imposing the above energy conservation algebraic relation. The periodic solutions on the branches S11± are studied by setting the derivatives with respect to time in equation (9) equal to zero; i.e. by imposing stationarity conditions on the modulation equations. The first and third equations are trivially satisfied if α = β, and the second and fourth equations become ωa A ω 3C 4ω (A 3 B)3 = 0, ɛωb + 3C 4ω (A 3 B)3 = 0 (10) These equations can be solved exactly for the amplitudes A and B, leading to the following approximations for the periodic solutions on the branches S11± x(t) X cos ωt = ψ 1 ψ1 = A cos ωt 2jω ω = εω2 4ω 2 ε(ω 2 1) 3 cos ωt ω 2 1 3C((1 + ε)ω 2 1) 3 v(t) V cos ωt = ψ 2 ψ2 = B cos ωt 2jω ω 4ω = 2 ε(ω 2 1) 3 3C((1 + ε)ω 2 1) cos ωt 3 (11) Considering the original non-linear problem (2), note that relations (11) are approximate since a single fast frequency was assumed in the slow fast partitions (7), and only terms containing this fast frequency were retained after performing averaging in the complex equations (8). It is interesting to note that the ratio of the amplitudes of the linear and non-linear oscillators on branches S11± is given by the following simple form X V = ɛω2 (12) ω 2 1 This relation shows that if the mass ɛ of the non-linear oscillator is small (as is assumed), and if the frequency ω is not in the neighbourhood of the eigenfrequency of the linear oscillator ω 0 = 1, the motion is always localized to the non-linear oscillator (in agreement with the numerical results); however, sufficiently close to ω 0 = 1, the oscillation localizes on the linear oscillator (as one would expect intuitively). There is a region in the frequency domain, 1/(1 + ɛ) < ω < 1, where the coefficients X and V are imaginary, indicating that no periodic motion on S11± can occur there; this represents a forbidden zone not only for S11±, but also for any periodic motion of the system. Accordingly, the branch S11+ of inphase oscillations exists for ω< 1/(1 + ɛ), whereas out-of-phase oscillations on S11 exist for ω>1. The approximations of the branches S11± in the frequency energy plane are computed by noting that the conserved energy of the system is equal to h = X 2 (V X )4 + C (13) 2 4 which, taking into account expressions (11), leads to the plot depicted in Fig. 9; this plot corresponds to the parameters used in the numerical study (ɛ = 0.05, C = 1.0). The approximate plots are close to the exact numerical backbones of the FEP of Fig Transient dynamics of the damped system In this section, the transient, unforced dynamics of the weakly damped system is considered, and it will be shown that complicated transitions between modes in this system can be fully understood and interpreted in terms of the periodic orbits of the undamped system. Specifically, the addition of damping induces transitions between different branches of solutions, and

18 94 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos motion temporarily settling on branch S12 before escaping from resonance capture as time increases (and energy decreases), and being involved in TRC with the stable branch S13. The short capture on branch S12 leads to the conjecture that the domain of attraction of 1:2 resonance capture is much smaller than the corresponding domain of attraction of the 1:3 resonance capture, with the latter eventually capturing the transient damped dynamics. Indeed, it should be expected that due to the complicated topology of the periodic orbits of the undamped system, the transitions between branches and the sequence of resonance captures should be sensitive to viscous damping dissipation. Fig. 9 Analytic approximation provided by the complexification averaging method of the backbone branch S11± 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 system will demonstrate that the structure of periodic orbits of the undamped system greatly influences the dynamics of the weakly damped one. When viewed from such a perspective, one can systematically interpret the complex transitions between multi-frequency modes of the transient, weakly damped dynamics by relating them to the different branches of non-linear modes 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 sections (ɛ = 0.05, ω 0 = 1.0, C = 1.0), and small damping coefficients λ 1 = 0, λ 2 = The motion on the stable special orbit of branch U 76 is initiated and there occurs vigorous TET to the nonlinear attachment. In Fig. 10, the responses and related WTs of the system with initial conditions v( T /4) = v( T /4) = x( T /4) = 0 and ẋ( T /4) = are depicted. The general observation is made that in this case there is strong TET to the non-linear attachment (NES), as evidenced by its large amplitude of oscillation compared with that of the (directly excited) linear oscillator. In particular, Fig. 10(d) is a schematic illustrating the transitions taking place in the weakly damped response on the FEP of the undamped system. The simulation verifies that the impulsive excitation of a stable special orbit is one of the triggering mechanisms initiating (direct) passive energy pumping. Energy decrease due to damping dissipation triggers the transitions between different branches of solutions. The numerical simulations of Fig. 10 demonstrate that, following a prolonged motion on U 76 during the early regime of the motion, there occurs a 1:2 TRC with the 3.2 TET mechanisms In this section, the impulsively forced, damped system (2) with the primary DOF denoted by y is considered, and three basic mechanisms for the initiation of non-linear TET are studied. The first mechanism (fundamental TET) is realized when the motion takes place along the backbone curve S11+ of the FEP of Fig. 7, occurring for relatively low frequencies ω<ω 0. The second mechanism (subharmonic TET) resembles the first, and occurs when the motion takes place along a lower frequency branch Snm, n < m Z +. The third mechanism (TET initiated by non-linear beat) which leads to stronger TET involves the excitation of a special orbit with main frequency ω SO greater than the natural frequency of the linear oscillator ω 0.In what follows, each mechanism is discussed separately, and numerical simulations that demonstrate passive and irreversible energy transfer from the linear oscillator to the non-linear attachment are provided in each case. Analytical results are also provided for the fundamental and subharmonic TET Fundamental TET The first mechanism for TET involves excitation of the branch of in-phase synchronous periodic solutions S11+, where the linear oscillator and the non-linear attachment oscillate with identical frequencies in the neighbourhood of the fundamental frequency ω 0. Although TET is considered only in the damped system, in order to gain an understanding of the governing dynamics it is necessary to consider the case of no damping. Figure 8(b) depicts a detailed plot of branch of the undamped system. At higher energies, the in-phase NNMs are spatially extended (involving finite-amplitude oscillations of both the linear oscillator and the non-linear attachment). However, the non-linear mode shapes of solutions on S11+ depend essentially on the level of energy and at low energies they become localized to the attachment. Considering

19 Passive non-linear TET and its applications 95 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 localization is a basic characteristic of the two-dimensional NNM invariant manifold corresponding to S11+; moreover, this localization property is preserved in the weakly damped system, where the motion takes place in a two-dimensional, damped NNM invariant manifold. This means that when the initial conditions of the damped system are such as to excite the damped analogue of S11+, the corresponding mode shape of the oscillation, initially spatially extended, becomes localized to the non-linear attachment with decreasing energy due to damping dissipation. This, in turn, leads to passive, continuous and irreversible transfer of energy from the linear oscillator to the non-linear attachment, which acts as a NES. The underlying dynamical phenomenon governing fundamental TET was proven to be a resonance capture on a 1:1 resonance manifold of the system [50]. Numerical evidence of fundamental TET is given in Fig. 11 for the system with parameters ɛ = 0.05, ω 2 0 = 1, C = 1, and λ 1 = λ 2 = Small damping is considered in order to better highlight the TET phenomenon, and the motion is initiated near the boxed point of Fig. 8(b). Comparing the transient responses shown in Figs 11(a) and (b), it is noted that the response of the primary system decays faster than that of the NES. The percentage of instantaneous energy captured by the NES versus time is depicted in Fig. 11(e), and the assertion that continuous and irreversible transfer of energy from the linear oscillator to the NES takes place is confirmed. This is more evident by computing the percentage of total input energy that is eventually dissipated by the damper of the NES (see Fig. 11(f)), which in this particular simulation amounts to 72 per cent; the energy dissipated at the NES is computed by the relation t E NES (t) = λ 2 [ v(τ) ẏ(τ)] 2 dτ 0 The evolution of the frequency components of the motions of the two oscillators as energy decreases can be studied by numerical WTs of the transient

20 96 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Fig. 11 FundamentalTET. Shown are the transient responses of the (a) linear oscillator and (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 S11+ to S13+ at smaller energy levels using the (g) NES (observe the settlement of the motion at frequency 1/3) and (h) linear oscillator [84]

21 Passive non-linear TET and its applications 97 responses, as depicted in Figs 11(c) and (d). These plots demonstrate that a 1:1 resonance capture is indeed responsible for TET. Below the value of 4 of the logarithm of energy level, the motion of the linear oscillator is too small to be analysed by the particular windows used in the WT; however, a more detailed WT over smaller energy regimes (see Figs 11(g) and (h)) reveals a smooth transition from S11+ to S13+, in accordance with the FEP of Fig. 7. This transition manifests itself by the appearance of two predominant frequency components in the responses (at frequencies 1 and 1/3) as energy decreases. The complexification-averaging method is utilized to perform an analytical study of the resonance capture phenomenon in the fundamental TET mechanism. System (2) is again considered, and the new complex variables are introduced ψ 1 (t) = v(t) + jv(t) ϕ 1 (t) e jt, ψ 2 (t) = ẏ(t) + jy(t) ϕ 2 (t) e jt (14) where φ i (t), i = 1, 2, represent slowly varying complex amplitudes and j 2 = 1. By writing equation (14), a partition of the dynamics into slow and fast components is introduced, and slowly modulated fast oscillations at frequency ω = ω 0 = 1 are sought. As discussed previously, fundamental TET is associated with this type of motion in the neighbourhood of branch S11+ in the FEP of the undamped dynamics. Expressing the system responses in terms of the new complex variables, y = (ψ 2 ψ2 )/(2j), v = (ψ 1 ψ1 )/(2j) (where (*) denotes complex conjugate), substituting into equation (2), and averaging over the fast frequency, a set of approximate, slow modulation equations that govern the evolutions of the complex amplitudes is derived ɛ ϕ 1 = jɛ λ 2 ϕ 1 ɛ λ 2 (ϕ 1 ϕ 2 ) + j 3C 8 ϕ 1 ϕ 2 2 (ϕ 1 ϕ 2 ) ϕ 2 = ɛ λ 2 ϕ 2 + ɛ λ 2 (ϕ 1 ϕ 2 ) + j 3C 8 ϕ 2 ϕ 1 2 (ϕ 2 ϕ 1 ) (15) For the sake of simplicity, assume that λ 1 = λ 2 = λ in equation (2). To derive a set of real modulation equations, the complex amplitudes are expressed in polar form, ϕ i (t) = a i (t)e jβ it, which is substituted into equation (15), and the real and imaginary parts are separately set equal to zero. Then, equation (15) is reduced to an autonomous set of equations that govern the slow evolution of the two amplitudes a 1 (t) and a 2 (t) and the phase difference φ(t) = β 2 (t) β 1 (t) ɛȧ 1 = ɛ λ 2 a 1 + ɛ λ 2 a 2 cos φ + 3C 8 a 2(a a2 2 2a 1a 2 cos φ) sin φ ȧ 2 = ɛ λ 2 a 1 cos φ ɛλa 2 3C 8 a 1(a a2 2 2a 1a 2 cos φ) sin φ ɛa 1 a 2 φ = 1 2 ɛa 1a 2 ɛ λ 2 (ɛa 1 + a 2 ) sin φ 3C 8 (a2 1 + a2 2 2a 1a 2 cos φ) [(1 ɛ)a 1 a 2 + (ɛa 1 a 2 ) cos φ] (16) This reduced dynamical system governs the slow-flow dynamics of fundamental TET. In particular, 1:1 resonance capture (the underlying dynamical mechanism of fundamental TET) is associated with non-time-like behaviour of the phase variable φ or, equivalently, failure of the averaging theorem in the slow flow (16). Indeed, when φ exhibits time-like, non-oscillatory behaviour [166], one can apply the averaging theorem over φ and prove that the amplitudes a 1 and a 2 decay exponentially with time and no significant energy exchanges (TET) can take place. Figure 12(a) depicts 1:1 resonance capture in the slow-flow-phase plane (φ, φ) for system (16) with ɛ = 0.05, λ = 0.01, C = 1, ω 0 = 1 and initial conditions a 1 (0) = 0.01, a 2 (0) = 0.24, φ(0) = 0. The oscillatory behaviour of the phase variable in the neighbourhood of the in-phase limit φ = 0 + indicates 1:1 resonance capture (on branch S11+ of the FEP of Fig. 7), and leads totet from the linear oscillator to the NES as evidenced by the build-up of amplitude a 1 (see Fig. 12(b)). Escape from resonance capture is associated with time-like behaviour of φ and rapid decrease of the amplitudes a 1 and a 2 (as predicted by averaging in equation (16)). A comparison of the analytical approximation (14) (16) and direct numerical simulation for the previous initial conditions confirms the accuracy of the analysis Subharmonic TET Subharmonic TET involves excitation of a lowfrequency S-tongue. As mentioned earlier, lowfrequency tongues are the particular regions of the FEP where the NES engages in m:n (m, n are integers such that m < n) resonance captures with the linear oscillator. 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 tongues are represented by horizontal lines in the FEP, and the response of system (2) on a tongue locally resembles that of a linear system. In addition, at each specific m:n resonance capture, there appear a pair of closely spaced tongues corresponding to in- and out-of-phase oscillations of the two subsystems. Regarding the dynamics of subharmonic TET, a particular pair of lower tongues are focused, say

22 98 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 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] S13± (Fig. 8(c)). At the extremity of a lower pair of tongues, the curve in the configuration plane is strongly localized to the linear oscillator. However, as for the fundamental mechanism for TET, the decrease of energy by viscous dissipation leads to curves in the configuration plane that are increasingly localized to the NES, and non-linear TET to the NES occurs. In this case, the underlying dynamical phenomenon causing TET is resonance capture in the neighbourhood of a m:n resonance manifold of the dynamics. Specifically, for the pair of tongues S13±, a 1:3 resonance capture occurs that leads to subharmonic TET with the linear oscillator vibrating with a frequency three times that of the NES. It is emphasized that due to the stability properties of the tongues S13±, subharmonic TET involves excitation of S13, but not S13+. The transient dynamics when the motion is initiated at the extremity of S13 (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 in the previous section are considered. Until t = 500 s, subharmonic TET takes place. Despite the presence of viscous dissipation, the NES response grows continuously, with simultaneous rapid decrease of the response of the linear oscillator. A substantial amount of energy is transferred to the NES (see Fig. 13(e)), and eventually nearly 70 per cent of the energy is dissipated 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 whole lower tongue S13 from the right to the left. Once escape from resonance capture occurs (around t = s), energy is no longer transferred to the NES. For analytical study of subharmonic TET, TET in the neighbourhood of tongue S13 will be the focus (similar analysis can be applied for other orders of subharmonic resonance captures). Due to the fact that motions in the neighbourhood of S13 possess two main frequency components, at frequencies 1 and 1/3, the responses of system (2) can be expressed as y(t) = y 1 (t) + y 1 3 (t), v(t) = v 1 (t) + v 1 3 (t) (17) where the indices represent the frequency of each term. As in the previous case, new complex variables are introduced ψ 1 (t) = ẏ 1 (t) + jωy 1 (t) ϕ 1 (t) e jωt, ψ 3 (t) = ẏ 1 3 (t) + j ω 3 y 1 (t) ϕ 3(t) e j ωt 3 3 ψ 2 (t) = v 1 (t) + jωv 1 (t) ϕ 2 (t) e jωt, ψ 4 (t) = v 1 3 (t) + j ω 3 v 1 (t) ϕ 4(t) e j ωt 3 3 (18) where ϕ i (t), i = 1,..., 4 represent slowly varying modulations of fast oscillations of frequencies 1 or 1/3.

23 Passive non-linear TET and its applications 99 Fig. 13 Subharmonic TET initiated on S13 : 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 = ψ 1 ψ 1 2jω + ψ 3 ψ3 2j(ω/3), v = ψ 2 ψ2 2jω + ψ 4 ψ 4 2j(ω/3) (19) substituting into equation (2), and averaging over each of the two fast frequencies, the slow modulation equations that govern the evolutions of the complex amplitudes are derived as ϕ 1 = j 1 ( ω 1 ) ϕ 1 ɛ λ 2 ω 2 (2ϕ 1 ϕ 2 ) j 9C 8ω 3 [3ϕ3 3 9ϕ2 3 ϕ 4 3ϕ ϕ 3ϕ 2 4 (ϕ 1 ϕ 2 ) ϕ 1 ϕ 2 2 6(ϕ 1 ϕ 2 ) ϕ 3 ϕ 4 2 ]

24 100 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos ϕ 2 = j ω 2 ϕ 2 λ 2 (ϕ 2 ϕ 1 ) + j 9C 8ω 3 [3ϕ3 3 9ϕ 2 3 ϕ 4 3ϕ ϕ 3ϕ 2 4 (ϕ 1 ϕ 2 ) ϕ 1 ϕ 2 2 6(ϕ 1 ϕ 2 ) ϕ 3 ϕ 4 2 ] ϕ 3 = j 1 ( ω ) ϕ 3 ɛ λ ω 2 (2ϕ 3 ϕ 4 ) + j 9C 8ω [ϕ 1(2(ϕ 3 3 ϕ 4 )(ϕ 1 ϕ 2) 3(ϕ 3 ϕ 4 )2 ) + ϕ 2 (2(ϕ 4 ϕ 3 )(ϕ 1 ϕ 2) + 3(ϕ 3 ϕ 4 )2 ) + 9(ϕ 3 ϕ 4 ) ϕ 3 ϕ 4 2 ] ϕ 4 = j ω 6 ϕ 4 λ 2 (ϕ 4 ϕ 3 ) j 9C 8ω 3 [ϕ 1 (2(ϕ 3 ϕ 4 )(ϕ 1 ϕ 2) 3(ϕ 3 ϕ 4 )2 ) + ϕ 2 (2(ϕ 4 ϕ 3 )(ϕ 1 ϕ 2) + 3(ϕ 3 ϕ 4 )2 ) + 9(ϕ 3 ϕ 4 ) ϕ 3 ϕ 4 2 ] (20) where again it was assumed that λ 1 = λ 2 = λ in equation (2). To derive a set of real modulation equations, the complex amplitudes are expressed in polar form ϕ i (t) = a i (t)e jβi(t), and an autonomous set of seven slow-flow modulation equations that govern the amplitudes a i = ϕ i, i = 1,..., 4 and the phase differences φ 12 = β 1 β 2, φ 13 = β 1 3β 3, and φ 14 = β 1 3β 4 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 ȧ 1 = ɛλ 2 (2a 1 a 2 ) + g 1 (a, φ), ɛȧ 2 = ɛλ 2 (a 2 a 1 ) + g 2 (a, φ) ȧ 3 = ɛλ 2 (2a 3 a 4 ) + g 3 (a, φ), ɛȧ 4 = ɛλ 2 (a 4 a 3 ) + g 4 (a, φ) φ 12 = f 12 (a) + g 12 (a, φ; ɛ), φ 13 = f 13 (a) + g 13 (a, φ) φ 14 = f 14 (a) + g 14 (a, φ; ɛ) (21) where the functions g i and g ij are 2π-periodic in terms of the phase angles φ = (φ 12, φ 13, φ 14 ) T, and a = (a 1,..., a 4 ) T. In this case (as for the fundamental TET mechanism), strong energy transfer between the linear and non-linear oscillators can occur only when a subset of phase angles φ kl does not exhibit time-like behaviour; that is, when some phase angles possess oscillatory (non-monotonic) behaviour with respect to time. This can be seen from the structure of the slow flow (21) where, if the phase angles exhibit time-like behaviour and the functions g i are small, averaging over these phase angles can be performed to show that the amplitudes decrease monotonically with time; in that case, no significant energy exchanges between the linear and non-linear components of the system can take place. It follows that subharmonic TET is associated with non-time-like behaviour of (at least) a subset of the slow-phase angles φ kl in equation (21). Figure 14 presents the results of the numerical integration of the slow-flow equations (20) and (21) for the system with parameters ɛ = 0.05, λ = 0.03, C = 1, and ω 0 = 1. The motion is initiated on branch S13 with initial conditions v(0) = y(0) = 0 and v(0) = , and ẏ(0) = (it corresponds exactly to the simulation of Fig. 13). The corresponding initial conditions and the value of the frequency ω of the reduced slow-flow model were computed by minimizing the difference between the analytical and numerical responses of the system in the interval t [0, 100]: ϕ 1 (0) = , ϕ 2 (0) = , ϕ 3 (0) = , ϕ 4 (0) = , and ω = This result indicates that, initially, nearly all energy is stored in the fundamental frequency component of the linear oscillator, with the remainder confined to the subharmonic frequency component of the NES. Figures 14(a) and (b) depict the temporal evolutions of the amplitudes a i, from which it is concluded that subharmonic TET in the system is mainly realized through energy transfer from the (fundamental) component at frequency ω of the linear oscillator, to the (subharmonic) component at frequency ω/3 of the NES (as judged from the build-up of the amplitude a 3 and the diminishing of a 1 ). A smaller amount of energy is transferred 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 a 2 ). These conclusions are supported by the plots of Figs 14(c) to (e), where the temporal evolutions of the phase differences φ 12 = β 1 β 2, φ 13 = β 1 3β 3, and φ 14 = β 1 3β 4 are shown. Absence of strong energy exchange between the fundamental and subharmonic frequency components of the linear oscillator is associated with the time-like behaviour of the phase difference φ 13, whereas TET from the fundamental component of the linear oscillator to the two frequency components of the NES is associated with oscillatory early time behaviour of the phase differences φ 12 and φ 14. Oscillatory responses of φ 12 and φ 14 correspond to 1:1 and 1:3 resonance captures, respectively, between the corresponding frequency components of the linear oscillator and the NES; as time increases, time-like responses of the phase variables are associated with escapes from the corresponding regimes of resonance capture. In addition, it is noted that the oscillations of the angles φ 12 and φ 14 take place in the neighbourhood of π, which confirms that, in this particular example, subharmonic TET is activated by the excitation of

25 Passive non-linear TET and its applications 101 Fig. 14 Subharmonic TET: (a) amplitude modulations; (b) (d) phase modulations [84] response of the NES in the resonance capture region. The analytical model fails, however, during the escape from resonance capture since the ansatz (17) and (18) is not valid in that regime of the motion. Indeed, after escape from resonance capture, the motion approximately evolves along the backbone curve of the FEP; eventually S15 is reached whose motion cannot be described by the ansatz (17) and (18), thereby leading to the failure of the analytical model. Fig. 15 Transient response of NES for 1:3 subharmonic TET; comparison between analytical approximation (dashed line) and direct numerical simulation (solid line) an anti-phase branch of periodic solutions (such as S13 ). The analytical results are in full agreement with the WTs depicted in Figs 5(c) and (d), where the response of the linear oscillator possesses a strong frequency component at the fundamental frequency ω 0 = 1, whereas the NES oscillates mainly at frequency ω 0 /3. The accuracy of the analytical model (20) and (21) in capturing the dynamics of subharmonic TET is confirmed by the plot depicted in Fig. 15 where the analytical 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 TET initiated by non-linear beating The previous two mechanisms cannot be activated with the NES at rest, since in both cases the motion is initialized from a non-localized state of the system. This means that these energy pumping mechanisms cannot be activated directly after the application of an impulsive excitation to the linear oscillator with the NES 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 to confine and passively dissipate unwanted vibrations from linear structures that are forced by impulsive (or broadband) loads. Hence, it is necessary to discuss an alternative, third energy pumping mechanism capable of initiating passive energy transfer with the NES initially at rest. This alternative mechanism is based on the excitation of a special orbit that plays the role of a bridging orbit for activation of either fundamental or subharmonic TET. Excitation of a special orbit results in the transfer of a substantial amount of energy from the initially

26 102 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos excited linear oscillator directly to the NES through a non-linear beat phenomenon. In that context, the special orbit may be regarded as an initial bridging orbit or trigger, which eventually activates fundamental or subharmonic TET once the initial non-linear beat initiates the energy transfer. Indeed, as shown below, the third mechanism for TET represents an efficient initial (triggering) mechanism for rapid transfer of energy from the linear oscillator to the NES at the crucial initial stage of the motion, before activating either one of the (fundamental or subharmonic) main TET mechanisms through a non-linear transition (jump) in the dynamics. To study the dynamics of this triggering mechanism, the following conjecture is formulated: Due to the essential (non-linearizable) non-linearity, the NES is capable of engaging in a m:n resonance capture with the linear oscillator, m and n being a set of integers. Accordingly, in the undamped system, there exists a sequence of special orbits (corresponding to non-zero initial velocity of the linear oscillator and 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 was formulated to compute the periodic orbits of system (2) with no damping, and the additional restriction for the special orbits was imposed. The numerical results in the frequency energy plane are depicted in Fig. 16 for parameters ɛ = 0.05, ω 0 = 1, and C = 1. Each triangle in the plot represents a special orbit, and a one-dimensional manifold appears to connect the special orbits; a rigorous proof of the existence of this manifold can be found in reference [85]. In addition, it appears that there exist a countable infinity of special orbits, occurring in the neighbourhoods of the countable 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). To distinguish between the two special solutions in such high-frequency branches, they are partitioned into two subclasses: the a-special orbits that exist in the neighbourhood of ω = ω 0 = 1, and the b-special orbits that 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]

27 Passive non-linear TET and its applications 103 As shown below it is the excitation of the stable b-special orbits that activates the third mechanism for TET. By construction, all special orbits have a common feature; namely, they pass with vertical slope through the origin of the configuration plane (v, y). This feature renders them compatible with an impulse applied to the linear oscillator, which corresponds to a nonzero velocity of the linear oscillator and all other initial conditions zero. The curves corresponding to the special orbits in the configuration plane can be either closed or open depending upon the differences between the two indices characterizing the orbits; specifically, odd differences between indices correspond to closed curves in the configuration plane and lie on U -branches, whereas even differences between indices correspond to open curves on S-branches. In addition, higher frequency special orbits (with frequency index ω>ω 0 ) in the upper part of the FEP (i.e. m > n) are localized to the non-linear oscillator; conversely, special orbits in the lower part of the FEP (with frequency index ω<ω 0 ) tend to be localized to the linear oscillator. This last observation is of particular importance since it directly affects the transfer of a significant amount of energy from the linear oscillator to the NES through the mechanism discussed in this section. Indeed, there seems to be a well-defined critical threshold of energy that separates high- from low-frequency special orbits; i.e. those that do or do not localize to the NES, respectively (see Fig. 16). The third mechanism for TET can only be activated for input energies above the critical threshold, since below that the (low-frequency) special orbits are incapable of transferring significant amounts of input energy from the linear oscillator to the NES; in other words, the critical level of energy represents a lower bound below which no significant TET can be initiated through activation of a special orbit. Moreover, combining this result with the topology of the onedimensional manifold of special orbits of Fig. 16, it follows that it is the subclass of stable b-special orbits that is responsible for activating the third TET mechanism, whereas the subclass of unstable a-special orbits does not affect TET. This theoretical insight will be fully validated by the numerical simulations that follow. When the NES engages in a m:n resonance capture with the linear oscillator, a non-linear beat phenomenon takes place. Due to the essential (nonlinearizable) non-linearity of the NES and the lack of any preferential frequency, this non-linear beat phenomenon does not require any a priori tuning of the non-linear attachment, since at the specific frequency energy range of the m:n resonance capture, 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 in coupled oscillators with linearizable non-linear stiffnesses (e.g. spring pendulum systems [129]), where the defined ratios of linearized natural frequencies of the component subsystems dictate the type of IRs that can be realized [14, 167]. As an example, Fig. 17 depicts the exchanges of energy during the non-linear beat phenomenon corresponding to the special orbits of branches U 21 and U 54 for parameters ɛ = 0.05, ω 0 = 1, C = 1, and no damping. As expected, energy is continuously exchanged between the linear oscillator and the NES, so the energy transfer is not irreversible as is required for TET; it can be concluded that excitation of a special orbit can only initiate (trigger) TET, but not cause it in itself. The amount of energy transferred during each cycle of the beat varies with the special orbit considered; for U 21 and U 54, as much as 32 per cent and 86 per cent of energy can be transferred to the NES, respectively. It can be shown that, for increasing integers 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 maximum amount of energy) should increase as the least common multiple of m and n. Note, at this point, that the non-linear beat phenomenon associated with the excitation of the special orbits can be studied analytically using the complexification-averaging method [135]. To demonstrate the analytical procedure, the special orbit on branch U 21 of the system with no damping is analysed 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 oscillator and the non-linear attachment can be approximately expressed as y(t) Y 1 sin ωt + Y 2 sin 2ωt y 1 (t) + y 2 (t)c v(t) V 1 sin ωt + V 2 sin 2ωt v 1 (t) + v 2 (t) where the amplitudes are Y 1 = A ω, V 1 = B ω, Y 2 = D 2ω, V 2 = G 2ω (22) and A, B, D, and G are computed from the stationarity conditions in the slow-flow equations as 4ɛω B =± 4 (Z 2 8Z 1 ) 9CZ1 3Z, 2 32ɛω G =± 4 (2Z 1 Z 2 ) 9CZ2 3Z A = 1 ω 2 0 ɛω2 B, ω2

28 104 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 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 = Z 1 = 4ɛω2 ω 2 0 4ω2 G ɛω2 ω0 2 1, Z ω2 2 = 4ɛω2 ω ω2 Hence, a two-frequency approximation is satisfactory for this family of periodic motions. The frequency ω SO 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 linear oscillator during the non-linear beat phenomenon is estimated to be E linear (t) = [(ω2 0 4ω2 SO ) sin ω SOt 2(ω 2 0 ω2 SO ) sin 2ω SOt] 2 + 9ωSO 2 ω2 0 ] 2 [ (ω 2 0 4ω 2 SO ) cos ω SOt 4(ω 2 0 ω2 SO ) cos 2ω SOt 9ω 4 0 (23) The non-linear coefficient C has no influence on the fraction of total energy transferred to the NES during the non-linear beat; this means that, during the beat, the instantaneous energies of the linear oscillator and the NES are directly proportional to the non-linear coefficient. Moreover, as the mass of the NES tends to zero, the frequency where the special orbit is realized tends to the limit ω SO ω, and, as a result, E linear (t) 1, and the energy transferred to the NES during the beat tends to zero. However, it is noted that this is a result satisfied only asymptotically since, as indicated by the results depicted in Fig. 17, even for very small mass ratios, e.g. ɛ = 0.05, as much as 86 per cent of the total energy can be transferred to the NES during a cycle of the special orbit of branch U 54. Considering now the damped system, it will be shown that following an initial non-linear beat phenomenon, either one of the main (fundamental or subharmonic) TET mechanisms can be activated through a non-linear transition (jump) in the dynamics. It was previously mentioned that the two main TET mechanisms are qualitatively different from the third mechanism, which is based on the excitation of a non-linear beat phenomenon (special orbit). Indeed, damping is a prerequisite for the realization of the two main mechanisms, leading to an irreversible energy transfer from the linear oscillator to the NES, whereas a special orbit is capable of transferring energy without dissipation, though this transfer is not irreversible but periodic. This justifies the earlier assertion that the third mechanism does not represent an independent

29 Passive non-linear TET and its applications 105 Fig. 18 TET by non-linear beat, transition to S11+. 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] mechanism for energy pumping, but rather triggers it, and through a non-linear transition activates either one of the two main mechanisms. This will become apparent in the following numerical simulations. The following simulations concern the transient dynamics of the damped system (2) with parameters ɛ = 0.05, ω 0 = 1, C = 1, λ 1 = λ 2 = , and an impulse of magnitude Y applied to the linear oscillator (corresponding to initial conditions y(0 + ) = v(0 + ) = v(0 + ) = 0, ẏ(0 + ) = Y ). By varying the magnitude of the impulse, the different non-linear transitions which take place in the dynamics and their effects on TET are studied. The responses of the system to the relatively 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

30 106 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Fig. 19 Percentage of input energy eventually dissipated 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 approximately t = 120 s) the (stable) b-special orbit on branch U 32 is excited (since the NES response possesses two main frequency components at 1 and 3/2 rad/s), and a non-linear beat phenomenon takes place. (Note the continuous exchange of energy between the two subsystems, demonstrating reversibility in this initial stage of the motion.) For t > 120 s, the dynamics undergoes a transition (jump) to branch S11+, and fundamental TET to the NES occurs on a prolonged 1:1 resonance capture (see Figs 18(c) and (d)); eventually, 84 per cent of the input energy is dissipated by the damper of the NES (see Fig. 18(f)) Critical energy threshold necessary for initiating TET To demonstrate more clearly the effect of the b-special orbits on TET, Fig. 19 depicts the percentage of input energy eventually dissipated at the NES for varying magnitude of the impulse for the system with parameters ɛ = 0.05, ω 0 = 1, C = 1, and λ 1 = λ 2 = In the same plot, the positions of the special orbits of the undamped system and the critical threshold predicted in Fig. 16 are depicted. It is concluded that strong TET is associated with the excitation of b-special orbits of the branches U(p + 1)p in the neighbourhood above the critical threshold, whereas excitation of a-special orbits below the critical threshold does not lead to rigorous energy pumping. As mentioned previously, in the neighbourhood of the critical threshold, the b-special orbits are strongly localized to the NES, whereas a-orbits are non-localized. The deterioration of TET is also noted from Fig. 19 as the magnitude of the impulse well Fig. 20 Contours of percentage of input energy eventually dissipated at the NES for the case when both oscillators excited by impulses; superimposed 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 highfrequency special orbits are excited; this is a consequence of the fact that well above the critical threshold, 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 ẏ(0) and ɛ v(0) applied to both the linear oscillator and the NES, respectively. The system parameters used were identical to those of the previous simulation of Fig. 19. Superimposed on contours of energy dissipated 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 this case the essential role of the high-frequency special orbits in TET. Indeed, high levels of energy dissipation are encountered in neighbourhoods of contours of high-frequency U -branches, whereas low values are noted in the vicinity of low-frequency branches. These results agree qualitatively with the earlier theoretical and numerical findings, and enable one to assess and establish the robustness of TET when the NES is not initially at rest. The results presented thus far provide a measure of the complicated dynamics encountered in the two-dof system under consideration. It is logical to assume that by increasing the number of DOFs of the system, the dynamics will become even more

31 Passive non-linear TET and its applications 107 complex. That this is indeed the case is revealed by the numerical simulations presented in the next section where resonance capture cascades are reported in MDOF linear systems with essentially non-linear end attachments. By resonance capture cascades, complicated sudden transitions between different branches of solutions (modes), which are accompanied by sudden changes in the frequency content of the system responses, 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 composed of linear primary systems with attached SDOF or MDOF ungrounded NESs is considered. Results on this specific problem can also be found in references [77 80, 89]. Consider, first, the case of the two-dof linear primary system with attached SDOF ungrounded NES ÿ 2 + ω 2 0 y 2 + λ 2 ẏ 2 + d( y 2 y 1 ) = 0 ÿ 1 + ω 2 0 y 1 + λ 1 ẏ 1 + λ 3 (ẏ 1 v) + d( y 1 y 2 ) + C( y 1 v) 3 = 0 ɛ v + λ 3 ( v ẏ 1 ) + C(v y 1 ) 3 = 0 (24) The system parameters are chosen as ω 0 = (rad/s), λ 1 = λ 2 = 0.155, λ 3 = 0.544, d = , ɛ = 1.8, and C = , with linear natural frequencies ω 1 = and ω 2 = (rad/s). Figure 21(a) depicts the relative response 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. The multi-frequency content of the transient response is evident and is quantified in Fig. 21(b), where the instantaneous frequency of the time series is computed by applying the numerical Hilbert transform [95]. As energy decreases because damping dissipation, a series of eight resonance capture 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 evidenced by the fact that of these eight captures only two (labelled IV and VII in Fig. 21(b)) involve the linearized in-phase and out-of-phase modes of the linear oscillator, with the remaining involving essentially non-linear interactions of the NES with different low- and high-frequency non-linear modes of the system. 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 captures indicated). The two natural frequencies are computed as f 1 = ω 1 /2π = 1.86 Hz and f 2 = ω 2 /2π = 7.98 Hz where ω 1 = and ω 2 = (rad/s) [84] mode involved, before escape from resonance capture occurs and the NES transiently resonates with the next mode in the series. In essence, the NES acts as a passive, broadband boundary controller, absorbing, confining, and eliminating vibration energy from the linear oscillator. Similar types of resonance capture cascades were reported in previous works where grounded NESs, weakly coupled to the linear structure, were examined [78]. The capacity of the NES to resonantly interact with linear and non-linear modes in different frequency ranges is due to its essential non-linearity (i.e. the absence of a linear term in the non-linear stiffness characteristic), which precludes any preferential resonant frequency Analysis of a two-dof linear primary system with an SDOF NES The first system which is considered here is a two- DOF linear primary system with an attached SDOF NES (Fig. 22), in which the effect that the increase in DOF of the primary system has on the TET dynamics

32 108 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Fig. 22 Two-DOF primary system coupled to an ungrounded NES is studied. Equations of motion assume the form m 1 ÿ 1 + ɛλ 1 ẏ 1 + k 1 y 1 + k 12 ( y 1 y 2 ) = 0 m 2 ÿ 2 + ɛλ 2 ẏ 2 + ɛλ(ẏ 2 v) + k 2 y 2 + k 12 ( y 2 y 1 ) + C( y 2 v) 3 = 0 ɛ v + ɛλ( v ẏ 2 ) + C(v y 2 ) 3 = 0 (25) where y 1, y 2, v refer to the displacements of the primary system and the NES, respectively. For obvious practical reasons, a lightweight NES is specified by requiring that ɛ 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 damped and forced system. Hence, the analysis focuses on the analytical computation of the FEP of the undamped and unforced system. The complexification-averaging technique is utilized for the analytical approximation of the main backbone curves on the FEP, which correspond to 1:1 resonant oscillations of the primary system and the NES (i.e. the dominant frequencies of these two system are identical). At this point, the complex variables are introduced ɛ ϕ 3 + jɛω 2 ϕ 3 3jC 8ω 3 (ϕ2 3 ϕ 2 ϕ2 2 ϕ 3 ϕ 3 2 ϕ 3 + ϕ 2 2 ϕ ϕ 3 2 ϕ 2 2 ϕ 2 2 ϕ 3 ) = 0 (27) The complex amplitudes ϕ i can be expressed in polar form as ϕ i = a i e jβ i, a i, β i R for i = 1, 2, 3. Then, by imposing stationarity conditions on the slow-flow equations and considering trivial phase differences such that β 1 β 2 = β 1 β 3 = 0, an approximation of the NNMs on the main backbone is obtained y 1 = a 1 sin ωt, y 2 = a 2 sin ωt, v = a 3 sin ωt (28) where the amplitudes a i, i = 1, 2, 3 can be found as a function of frequency ω by solving the algebraic equations resulting from the steady-state conditions of the real-valued slow-flow equations. The main backbone branches can now be constructed by varying the frequency ω and representing a NNM at a point (h, ω) on the FEP where the total energy h = ω 2 /2[m 1 a 1 (ω) 2 + m 2 a 2 (ω) 2 + ɛa 3 (ω) 2 ] is conserved when the system oscillates in a specific mode. Figure 23 depicts the backbone branch, named S111, of the system with parameters m 1 = m 2 = 1, k 1 = k 2 = k 12 = 1, C = 1, and ɛ = NNMs depicted as projections of the three-dimensional configuration space (v, y 1, y 2 ) of the system are superimposed to demonstrate mode localization behaviours with respect to the total energy of the system; the horizontal and vertical axes in these plots are the nonlinear 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 finite frequencies, the essential non-linearity behaves as a rigid link, and the system dynamics is governed by the equations 1 = ẏ 1 + jωy 1, 2 = ẏ 2 + jωy 2, 3 = v + jωv (26) which are then substituted into equation (25). Expressing the complex variables in polar form i = ϕ i e jωt, i = 1, 2, 3 and performing averaging over the fast frequency, the complex-valued slow-flow modulation equations are obtained m 1 ϕ 1 + jϕ 1 2ω (m 1ω 2 k 1 k 12 ) + jϕ 2 2ω k 12 = 0 m 2 ϕ 2 + jϕ 2 2ω (m 2ω 2 k 2 k 12 ) + jϕ 1 2ω k jC 8ω 3 (ϕ2 3 ϕ 2 ϕ2 2 ϕ 3 ϕ 3 2 ϕ 3 + ϕ 2 2 ϕ ϕ 3 2 ϕ 2 2 ϕ 2 2 ϕ 3 ) = 0 m 1 ÿ 1 + k 1 y 1 + k 12 ( y 1 y 2 ) = 0 (m 2 + ɛ)ÿ 2 + k 2 y 2 + k 12 ( y 2 y 1 ) = 0 (29) The natural frequencies of this system are f 1H = and f 2H = rad/s for the above parameters. At low-energy levels, the equivalent stiffness of the essential non-linearity tends to zero, and the system dynamics is that of the primary system, the natural frequencies 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 the branch S (the ± signs indicate whether the initial condition of the corresponding oscillator is positive or negative, respectively). On this branch,

33 Passive non-linear TET and its applications 109 Fig. 23 Analytic approximation of the main backbone branches of the system m 1 = m 2 = 1, k 1 = k 2 = k 12 = 1, C = 1, ɛ = NNMs depicted as projections of the three-dimensional 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 is set 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 fashion, and the motion is more and more localized to the 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, comprises two different branches, namely S111 + and S These branches coalesce at a point S (see the grey dot in Fig. 23), where the NNM is such that the initial condition on the velocity of the oscillator m 2 is zero. On S111 +, the primary system vibrates in an anti-phase fashion, and the motion localizes to the NES as the frequency goes away from f 2H.OnS111 ++, there is an in-phase motion of the primary system, and the motion localizes to the primary system, as the frequency converges to f 1L. Fig. 24 Numerical computation of the FEP (backbone and loci of special orbits) of a two-dof primary coupled to an NES (m 1 = m 2 = 1, k 1 = k 2 = k 12 = 1, C = 1, ɛ = 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 On this branch, the primary system vibrates in an in-phase fashion, and the motion localizes to the NES as the frequency goes away from f 1H. Owing to the energy dependence of the NNMs along S111, interesting and vigorous energy exchanges may occur between the primary system and the NES. In particular, an irreversible channeling of vibrational energy from the primary system to the NES takes place on S111 + and S Because the NES has no preferential resonance frequency, fundamental TET can be realized either for in-phase or anti-phase motion of the primary system, which shows the adaptability of the NES. The SPOs, determined from accommodating specific initial conditions ẏ 1 (0) = 0, ẏ 2 (0) = 0 with all the others zero, can also be computed for the MDOF system. 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 special orbits for a two-dof primary system.

34 110 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 1. The first family consists of in-phase SPOs (++0) located on in-phase tongues; the masses of the primary system move in-phase. The locus of inphase SPOs is a smooth curve on the FEP. When the phase difference between the NES and the primary is trivial, the motion in the configuration space takes the form of a simple curve; in the case of non-trivial phase differences, a Lissajous curve is realized. For the SPO 1, the motion of the two masses of the primary system is almost identical and monochromatic. The NES has two dominant harmonic components, one of which is at the frequency of oscillation of the primary system, the other being three times smaller; a 1:3 IR between the NES and the primary system is realized. The non-linear beating characteristic of such a dynamical phenomenon can be clearly observed. For the SPO 1, the energy exchange is insignificant as the maximum percentage of total energy of the NES never exceeds 0.17 per cent. For the SPOs 2 and 3, the energy transfer is much more vigorous. To obtain a global picture, the maximum percentage of energy transferred to the NES during the nonlinear 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 a substantial amount of energy to the NES. More precisely, the SPOs must lie above the frequency of the in-phase mode of the primary system f 1L. Fig. 25 Maximum percentage of energy transferred to the NES during non-linear beating (dashed (dotted) 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 locus is also a smooth curve on the FEP. By inspecting Fig. 25, one can conclude the existence of a critical energy threshold for enhanced TETs; the SPOs must lie above the frequency of the anti-phase mode of the primary system f 2L. The transient dynamics of the weakly damped system is now examined and is interpreted based on the topological structure of the non-linear modes of the undamped system. Damping parameters are set to λ 1 = λ 2 = 0.1, λ = 0.04, and others are the same as those used in constructing the FEP in Fig. 23. In this section, only the single-mode responses by imposing the in-phase and anti-phase impulsive forcing are considered, and the multi-mode responses (i.e. resonance capture cascades) will be demonstrated later compared with the experimental system. First, the motion initiated on S (i.e. inphase fundamental TET) is examined (Figs 26(a) and (b)). In Fig. 26(c), the WT of v(t) y 2 (t) is superimposed on the FEP to demonstrate transient dynamics along the damped NNM manifold as the total energy decreases due to damping. The dynamical flow is captured in the neighbourhood of a 1:1 resonance manifold, which leads to a prolonged 1:1 resonance capture. Figure 26(d) depicts the trajectories of the phase differences between the NES and the two masses in the primary structure. The phase variables were computed by utilizing the Hilbert transform (HT) of the responses. Non-time-like behaviour of the two phase variables 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 Now the motion initiated on S111 + (i.e. outof-phase fundamental TET) is examined. Figure 27(a) and (b) depicts the time series where fundamental TET is realized in a first stage (t = s) for an antiphase motion of the primary structure. During this regime, the envelope of all displacements decreases monotonically, but the envelope of the NES seems to decrease more slowly than that of the primary structure; TET to the NES is observed (Fig. 27(e)). Around t = 80 s, the displacement y 2 of the second mass m 2 becomes very small, and a transition from anti-phase (S111 + ) to in-phase (S ) motion in the primary structure occurs. When the inflection point on S is reached (where a bifurcation eliminates the stable/unstable pair of NNMs), escape from resonance capture occurs, which results in time-like behaviour of the phase variables in Fig. 27(d). Figures 27(b), (c), and (e) show that this is soon followed by subharmonic TET on an in-phase tongue; there is a capture into 1:3 resonance manifold.

35 Passive non-linear TET and its applications 111 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 examined to verify the existence of a critical energy threshold above which the SPOs can trigger fundamental TET. In Fig. 28, the motion is initiated from inphase SPOs 1 and 2, located below and above the threshold, respectively. The dynamic responses are remarkably different for those two cases. For the SPO 1, the NES cannot extract a sufficient amount of energy from the primary system, and a transition to S is observed. On this branch, the motion localizes to the primary system as the total energy in the system decreases. For the SPO 2, thanks to a non-linear beating phenomenon, the motion is directed towards the basin of attraction of S , and fundamental TET from the in-phase mode is realized.

36 112 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Fig. 27 Fundamental TET for anti-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] Likewise, if the motion is initiated from an antiphase SPO located below the threshold (e.g. SPO 4), there occurs a transition to S , on which the motion localizes to the primary system with a decrease in the total energy. If the anti-phase SPO lying above the threshold (e.g. SPO 6) is excited, the branch

37 Passive non-linear TET and its applications 113 Fig. 28 Motion initiated from in-phase special orbits: (a, b) time series; (c, d) WT superimposed on the frequency energy plot; (e, f) instantaneous percentage of total energy in the NES [87] S111 + is reached, resulting in the realization of fundamental TET from the anti-phase mode Analysis of an SDOF linear primary system with an MDOF NES Application of an MDOF NES is now considered. It is showed that enhanced TET takes place in this case because of the capacity of the essentially non-linear MDOF NES to engage in simultaneous resonance captures with multiple modes of the linear system. Consider the system in Fig. 29, where a two-dof linear primary oscillator is connected through a weak linear stiffness ɛ (which is the small parameter of the problem), 0 <ɛ 1, to a three-dof non-linear attachment with the two essentially non-linear stiffnesses,

38 114 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos u(0) = 0 for a given period T. Numerically this is performed by minimizing the expression min{[u(t ) u(t )] [0 u(0)]} T Then, the total energy h of the underlying Hamiltonian system, when it oscillates with a periodic solution of frequency ω = 2π/T, is expressed as Fig. 29 Primary (linear) system with an MDOF non-linear attachment C 1 and C 2. The equations of motion for this system can be written as ü 1 + ɛλ u 1 + (ω α)u 1 αu 2 = F 1 (t) ü 2 + ɛλ u 2 + (ω α + ɛ)u 2 αu 1 ɛv 1 = F 2 (t) μ v 1 + ɛλ( v 1 v 2 ) + ɛ(v 1 u 2 ) + C 1 (v 1 v 2 ) 3 = 0 μ v 2 + ɛλ(2 v 2 v 1 v 3 ) + C 1 (v 2 v 1 ) 3 + C 2 (v 2 v 3 ) 3 = 0 μ v 3 + ɛλ( v 3 v 2 ) + C 2 (v 3 v 2 ) 3 = 0 (30) In the limit ɛ 0, the system decomposes into two uncoupled oscillators: a two-dof linear primary system with natural frequencies ω 1 = ω α and ω 2 = ω 0 <ω 1, corresponding to out-of-phase and in-phase linear modes, respectively; and a three-dof essentially non-linear oscillator with a rigid-body mode and two flexible NNMs. Unlike the SDOF NES configuration, this MDOF NES exhibits multi-frequency simultaneous TETs from multiple modes of the primary system; this means that multiple non-linear modes of the MDOF NES engage in transient resonance interactions with multiple modes of the linear system. Once again, complex transitions in the damped dynamics can be related to the topological structure of the periodic orbits of the corresponding undamped system. For practical purposes, the system with NES masses of O(ɛ) is considered with parameter values ɛ = 0.2, α = 1.0, C 1 = 4.0, C 2 ɛ 2 C 2 = 0.05, μ ɛ 2 μ = 0.08, and ω 0 = 1, where rescaling was applied to the NES masses μ and the second essentially nonlinear coupling spring C 2. As performed in previous sections, the FEP of the underlying Hamiltonian system was considered first. A numerical method was utilized to construct the FEP of the periodic solutions of the underlying Hamiltonian system [87]. Denoting u(t) = [u 1 (t)u 2 (t)v 1 (t)v 2 (t)v 3 (t)] T, the periodic solutions of the undamped and unforced system (30) can be determined by computing the values of u(0) for which h = 1 2 [ u 1(0) 2 + u 2 (0) 2 + μ v 1 (0) 2 + μ v 2 (0) 2 + μ v 3 (0) 2 ] Considering ɛ asaperturbationparameter, system (30) with the rescaled parameter μ ɛ 2 μ is expected to possess complicated dynamics as ɛ 0, because it is essentially (or strongly) non-linear, high-dimensional, and singular (since the highest derivatives in three of its equations are multiplied by the perturbation parameter squared). In Fig. 30, the periodic orbits are presented in a FEP. Note that it was difficult to capture the lowest frequency branch through the numerical scheme. It was analytically estimated and superimposed to the numerical results [88]. From the FEP of Fig. 30, it is noted that the backbone branches of periodic orbits are defined over wider frequency and energy ranges than for the system of the NES masses of O(1) [88], and no subharmonic tongues exist in this case (at least none was detected in the numerical scheme). Hence, it can be conjectured that a decrease in magnitude of the masses of the NES results in the elimination of the local subharmonic tongues (i.e. of the subharmonic motions at frequencies integrally related to the natural frequencies f 1 = , f 2 = , and f 3 = rad/s of the linear subsystem). For the limit of 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(ɛ 2 ) [88]

39 Passive non-linear TET and its applications 115 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 ü 1 + (ω α)u 1 αu 2 = 0 ü 2 + (ω α + ɛ)u 2 αu 1 ɛv 1 = 0 3μ v 1 + ɛ(v 1 u 2 ) = 0 (31) with limiting natural frequencies ˆf 1 = , ˆf2 = , and ˆf 3 = rad/s. The efficiency of TETs is demoistrated numerically by means of the MDOF NES configuration considered herein, under the out-of-phase impulsive forcing F 1 (t) = F 2 (t) = Y δ(t), with all other initial conditions being zero. Figure 31 depicts the damped responses for the impulsive forcing amplitude Y = 0.1. In this case, both the relative displacements v 1 (t) v 2 (t) and v 2 (t) v 3 (t) between the NES masses follow regular backbone branches. The relative displacement v 1 (t) v 2 (t) has a dominant frequency component that approaches the linearized natural frequency f 2 of the limiting system for the limit of low energy and finite frequency, where the equation for the NES part becomes μ v 1 + ɛ(v 1 u 2 ) = 0 with decreasing energy. In contrast, v 2 (t) v 3 (t) has two strong harmonic components that approach the linearized natural frequencies f 2 and f 3 for decreasing energy, indicating transfer of energy simultaneously from two modes of the linear limiting system for limit of low energy and finite frequency. Moreover, the same regular backbone branches are tracked by the response throughout the motion and strong energy transfer occurs right from the early stage of the response, which explains the strong eventual energy transfer to the NES ( 90 per cent) that occurs for this low-impulse excitation. By increasing the impulsive forcing to Y = 1.0 [88], the overall energy transfer from the linear to nonlinear subsystem decreases significantly with delay, and the steady-state energy dissipation by the NES is

40 116 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 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 per cent. This occurs because the motion is mainly localized to the directly excited linear subsystems by the strong initial out-of-phase resonance capture, with a small portion of energy spreading out to the NES. Further increasing the impulse magnitude to Y = 1.5 enables the system to escape from the strong initial out-of-phase resonance capture, leading to resumed strong TETs (Fig. 32). The NES relative responses possess multiple strong frequency components, indicating that strong TET occurs at multiple frequencies. The steady-state energy dissipation by the NES reaches nearly 90 per cent of the input energy Analysis of a linear continuous system with SDOF and MDOF attached NESs A separate series of papers examined TET in continuous 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 are capable of passively absorbing and locally dissipating significant portions of the vibration energy of the impulsively forced rod. In Fig. 34, a representative WTs of the damped responses of these two systems superimposed to the FEPs of the underlying Hamiltonian (undamped and unforced) systems are provided. Comparing the action of the SDOF and MDOF NESs, noted it is that the SDOF NES is capable of engaging in resonance capture with only one mode of the linear rod at a time. Hence, in Fig. 34(a), a resonance capture cascade where the SDOF NES engages with a series of modes sequentially (i.e. it escapes from a resonance capture with one mode before it can engage in similar resonance capture with another one) is noted. In the case of the MDOF NES (see Figs 34(b) to (d)), this does not hold, as the NES engages in broadband resonance interactions with multiple modes of the rod; that is, different

41 Passive non-linear TET and its applications 117 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]

42 118 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos non-linear modes of the MDOF NES engage in separate resonance captures with different linear modes of the rod (this is revealed by the broadband character of the non-linear modal interactions between the rod and the NES in this case). Hence, similar to previous applications with discrete coupled oscillators, it is concluded that a MDOF NES is more versatile and effective compared with the SDOF NES, as it can extract vibration energy simultaneously from a set of modes of the linear system. For a more detailed analysis and discussion of these results, the reader is referred to references [92] to[94]. 3.4 Non-smooth VI NES A separate series of papers considered NESs with nonsmooth stiffness characteristics. An NES with piecewise 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 to realize in practice [ ]. This section is concerned with NESs undergoing VIs (hereafter, vibro-impact NESs can be termed as VI NESs). As shown in the aforementioned references, this type of non-smooth NES possesses fast reaction time; i.e. a VI NES is capable of passive TET at a fast time-scale, which makes this type of device ideal in applications where the NES needs to be activated very early in the motion (within the initial one or two cycles of vibration). The simplest primary system VI NES configuration, namely an SDOF linear oscillator coupled to a VI NES (Fig. 35) is considered. 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 studying the damped motion on the FEP of the underlying VI conservative system (i.e. the identical system configuration, but with purely elastic impacts and no viscous damping elements). The premise is that, for sufficiently small damping, the damped non-linear dynamics are perturbations of the dynamics of the underlying conservative system, so that damped nonlinear transitions take place near branches of periodic or 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 phenomena 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 system should be identified. The equations of motion in non-dimensional form between impacts can be written as ü 1 + (1 + σ)u 1 + u 2 = 0, μü 2 + σ(u 2 u 1 ) = 0 (32) where μ = m 2 /m 1, σ = k 2 /k 1 are the mass and stiffness ratios; the rescaling of time, τ = k 1 /m 1 t, is imposed, and the derivative with respect to the new non-dimensional time is denoted by the overdot. Impact occurs whenever the absolute value of the relative displacements satisfies u 2 u 1 =e, where e denotes the clearance; if u 2 u 1 < e, then no impact occurs and the system oscillates simply in a linear combination of the two linear modes of system (32). Setting the coefficient of restitution to 1 (i.e. assuming perfectly elastic impacts), and applying momentum conservation, the velocities of the two masses just before and after impacts can be related; that is v 1 = (1 μ)v 1 + 2μv 2, v μ = (μ 1)v 2 + 2v μ (33) where v i = du i /dτ and the prime denotes the quantity just after impact. The periodic solutions of the VI conservative system were computed numerically and represented in a FEP. This plot was constructed by depicting each VI periodic orbit as a single point with the coordinates determined in the following way: consider the eigenfrequency of the uncoupled linear oscillator as reference frequency, f 0 = ; the frequency coordinate of the FEP is equal to (p/q)f 0, where the rational number 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 periodic orbit considered. The parameters of the system adopted for the FEP computation are μ = σ = 0.1 and e = 0.1, and the resulting FEP is depicted in Fig. 36. The complicated topology of the branches of periodic orbits depicted in the FEP reflects the well-known complexity of the dynamics of this seemingly simple 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 distinguish between the different families of VI periodic motions and study their dependence on energy and frequency. To this end, each VI periodic orbit depicted in the FEP is given the notation Lijkl±. The capital letter L is assigned either letters S or U, referring to symmetric or unsymmetric periodic motions, respectively. Symmetric periodic motions satisfy the conditions u k (τ) =±u k (τ + T /2), τ R, k = 1, 2, where T is the

43 Passive non-linear TET and its applications 119 Fig. 36 Frequency energy plot for the system (32); dashed lines indicate the two linearized eigenfrequencies, and bullets, the maximum energy levels at which oscillations take place without VIs [116] period of the motion, whereas unsymmetric periodic motions do not satisfy the conditions of the symmetric motions. Regarding the four numerical indices {ijkl}, index i refers to the number of left VIs occurring during the first half-period; j to the number of right impacts occurring during the first half-period; k to the number of left impacts occurring during the second half-period; and l to the number of right impacts occurring during the second half-period of a periodic motion. The (+) sign corresponds to in-phase VI periodic motions where, for zero initial displacements, the initial velocities of the two particles have the same sign at the beginning of both the first and second half-periods of the periodic motion; otherwise, the VI periodic motion is deemed to be anti-phase and the ( ) sign is used. It can be shown that S-VI periodic orbits correspond to synchronous motions of the two oscillators, and thus are represented by curves in the configuration plane of the system, (u 1, u 2 ); i.e. these periodic motions are characterized as NNMs. On the contrary, U-VI periodic orbits correspond to asynchronous motions of the two oscillators, and are represented by Lissajous curves in the configuration plane of the system. Considering the FEP of Fig. 36, the two bullets indicate the maximum energy thresholds below which oscillations occur without VIs, and the dynamics of the two-dof system is exactly linear. The first (in-phase) and second (out-of-phase) modes of the linear system (corresponding to the two-dof system with no rigid stops and clearance, e.g. e = ) exist below the energy thresholds for VIs, namely, E 1 = for the in-phase mode and E 2 = for the out-of-phase one. Clearly, when the system oscillates below these maximum energy thresholds, the relative displacement between the two particles of the system satisfies u 1 u 2 < 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 S1001 which bifurcates from the out-of-phase linearized mode, and the branch of in-phase VI NNMs S1001+ which bifurcates from the in-phase linearized mode. The two branches S1001± will be referred to as backbone (global) branches of the FEP; they consist ofvi periodic motions during which the NES vibrates either in-phase or out-of-phase with the linear oscillator with identical dominant frequencies. Moreover, both backbone branches exhibit a single VI per half-period are defined over extended frequency and energy ranges, and correspond to motions that are mainly localized to the VI attachment (except in the neighbourhoods of the two linearized eigenfrequencies of the system with e =,atf 1 = and f 2 = 0.186). Both backbone branches satisfy the condition of 1:1 IR between the linear oscillator and the VI NES, with the oscillations of both subsystems 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 FEP lies on subharmonic tongues (local branches); these are multi-frequency periodic motions, with frequencies being rational multiples of one of the linearized eigenfrequencies of the system. Each tongue is defined over a finite energy range and is composed of a pair of branches of in- and out-of-phase subharmonic solutions. At a critical energy level, the two branches of the pair coalesce in a bifurcation that signifies

44 120 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos Fig. 37 Representative VI impulse orbits: U3223 (upper) and U2222 (lower) [116] the end of that particular tongue and the elimination of the corresponding subharmonic motions for higher energy values. Clearly, there exists a countable infinity of such tongues emanating from the backbone branches, with each tongue corresponding to symmetric or unsymmetric VI subharmonic motions with different patterns of VIs during each cycle of the oscillation. 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 velocity of the linear oscillator. In essence, a VI IO is the response of the system initially at rest due to a single impulse applied to the linear oscillator at time τ = 0 +. Apart from the clear similarity of a VI IO to the Green s function defined for the corresponding linear system, the importance of studying this class of orbits stems from their essential role in passive TET from the linear oscillator to the non-linear attachment. Indeed, for impulsively excited linear systems with NESs having smooth non-linearities, IOs (which are, in essence, non-linear beats) play the role of bridging orbits that occur in the initial phase of TET, and channel a significant portion of the induced impulsive energy from the linear system to the NES at a relatively fast time-scale; this represents the most efficient scenario for passive TET. Although the aforementioned results refer to damped IOs, the dynamics of the underlying conservative system determines, in large part, the dynamics of the damped system as well, provided that the damping is sufficiently small. It follows that the IOs of the conservative system govern the initial phase of TET from the linear oscillator to the NES. As shown in reference [85], impulsive periodic and quasi-periodic orbits form a manifold in the FEP that contains a countable infinity of periodic IOs and an uncountable infinity of quasi-periodic IOs. For the VI system under consideration, the manifold of VI IOs was numerically computed and is depicted in the FEP of Fig. 36; in general, the manifold appears as a smooth curve, with the exception of a number of outliers. Representative VI IOs are depicted in Fig. 37. In general, the IOs become increasingly localized 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 (u 1, u 2 )). Moreover, there is no critical energy threshold for the appearance of VI IOs since there are no low-energy VI motions (the system is linear for lowenergy levels), and the dominant frequency of a VI IO depends on the clearance, e. For the system under consideration, the VI IOs start with a dominant frequency of (or a period of 6.58). Apart from the compact representation of VI periodic motions, the FEP is again a valuable tool for understanding the non-linear resonant interactions that govern energy transactions (such as TET) during damped transitions in the weakly dissipative system. This is because, for sufficiently weak dissipation (due to inelastic VIs or viscous damping), the damped dynamics are expected to be perturbations

45 Passive non-linear TET and its applications 121 it can be concluded that the most efficient energy dissipation by the VI NES occurs during the initial TRC on the subharmonic tongue S This result demonstrates that TRC is a basic dynamical mechanism governing effective passive TET, for example, from a seismically excited primary structure to an attached VI NES. It follows that by studying VI transitions in the FEP and relating them to rates of energy dissipation by VI NESs, one should be able to identify the most effective damped transitions from a TET point of view. The complicated series of VI transitions depicted in Fig. 38 demonstrates the potential of the two-dof system for exhibiting complex dynamics, and the utility of the FEP as a tool for representing and understanding complex transient multi-frequency transitions. 4 EXPERIMENTAL VERIFICATIONS Fig. 38 Damped VI transitions initiated on the tongue U 8778 : (a) WT superimposed on the FEP; (b) instantaneous energy plot [116] In this section, the experimental work that validates the previous theoretical results on passive TET will be reviewed. For a general synopsis regarding the experimental study of TETs, refer to the literature review in section Experiments with SDOF primary systems of solutions of the underlying conservative system. To show this, the dynamics of the system of Fig. 35 for the case of inelastic impacts is computed and analysed the resulting transient responses by numerical WTs. Then the resulting WT spectra are superimposed to the FEP 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, corresponding to VI motion initiated on the VI IO U 8778, with a coefficient of restitution, Three regimes of the damped VI transition can be distinguished. In the initial phase of the motion, the oscillations stay in the neighbourhood of the subharmonic tongue S1221+ until approximately τ = 500 and logarithm of energy equal to There is efficient energy dissipation in this initial phase of the motion, as evidenced by the energy plot of Fig. 38(b). In the second regime, the dynamics makes a transition to branch U 0110 until the logarithm of energy becomes equal to 2.5; in this regime of the damped transition non-symmetric oscillations take place. An additional transition to the manifold of VI IOs (e.g. IOs U 2112+, U 1111+, S1221+) occurs, before the VI dynamics makes a final transition to the backbone branch S0110+ for logarithm of the energy close to 2.7. By studying the instantaneous energy of the system during the aforementioned transitions (see Fig. 38(b)), Figure 39(a) depicts an experimental fixture built to examine the energy transfers in the two-dof system (Fig. 39(b) for its mathematical modelling) described by Mÿ + ɛλ 1 ẏ + ɛλ 2 (ẏ v) + C( y v) 3 + ky = 0, ɛ v + ɛλ 2 ( v ẏ) + 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 restoring force surface method (Fig. 40; [168]): M = kg, ɛ = kg, k = 1143 N/m, ɛλ 1 = Ns/m, ɛλ 2 = 0.4 Ns/m, C = N/m 2.8, and α = 2.8, where α denotes the power of the essential non-linearity. Figure 39(e) is a schematic showing how cubic (essential) non-linearity is achieved through geometric non-linearity. Assuming zero initial tension along the wire, a static force F with respect to a transverse displacement x can be expressed as ( x ) [ ] 1 F = kl 1 L 1 + (x/l) 2 (35) where k = EA/L represents the axial stiffness constant of the wire, L is the half-length of the span, E Young s modulus, and A the cross-sectional area of the wire.

46 122 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 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 ( x ) 3 ( ) x 5 F = EA + O L L 5 (36) from which the coefficient for the essentially nonlinear term can be estimated as C = EA/L 3. Note that a non-integer power (close to three) is obtained via system identification [169]. Two series of physical experiments were conducted in 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 different mechanisms for TETs. Additional tests were performed to investigate whether TETs can take place with increased levels of damping.

47 Passive non-linear TET and its applications 123 Fig. 40 Measured restoring force represented as a function of time (left) and relative displacement v y (right)[99] 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 force levels are depicted in Fig. 41. At 21 N of forcing, the acceleration and displacement of the NES are higher than those of the primary system, which indicates that the NES participates in the system dynamics to a large extent. The percentage of instantaneous total energy plot illustrates that vigorous energy exchanges take place between the two oscillators. However, it can also be observed that the channeling 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 down to 1.5 per cent immediately thereafter. Hence, in this case, energy quickly flows back and forth between the two oscillators, which is characteristic of a non-linear beating phenomenon. Another indication for this is that the envelope of the NES response undergoes large modulations. At the 55 N level, the non-linear beating still dominates the early regime of the motion. A less vigorous but faster energy exchange is now observed as 63 per cent of the total energy is transferred to the NES after 0.12 s. These quantities also hold for the intermediate force levels [99]. It should be noted that these observations are in close agreement with the analytical and numerical studies [83, 84]; indeed, in this case, the special orbits are such that they transfer smaller amounts of energy to the NES, but in a faster fashion when the force level is increased. A qualitative means of assessing the energy dissipation by the NES is to compare the response of the primary system in the following two cases: (a) when the NES is attached to the primary system (the present case denoted by NES displacements at the bottom of Fig. 41); (b) when the NES is disconnected, but its dashpot is installed between the primary system and ground (a SDOF linear oscillator with added damping denoted by ground dashpot displacement in Fig. 41). Case (b) was not realized in the laboratory, but the system response was computed using numerical simulation. The two bottom figures in Fig. 41 compare the corresponding displacements of the linear oscillator in the aforementioned two different system configurations. It can be observed that the NES performs much better than the grounded dashpot for the 21 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 significant portion of the total energy to the NES, it should be a more useful mechanism for energy dissipation Case of high damping Several force levels ranging from 31 to 75 N were considered, and the results for 31 N are presented herein. The damping coefficient was identified to be 1.48 Ns/m, which means that damping can no longer be considered to be O(ɛ). The increase in damping is also reflected in the measured restoring force in Fig. 42. The system responses are almost entirely damped out after five to six periods. The NES acceleration and displacement are still higher than the corresponding responses of the primary system, meaning that TETs may also occur in the presence of higher damping. The percentage of instantaneous total energy in the NES never reaches close to 100 per cent as in the lower damping case. However, one may conjecture that this is due to the increased damping value; as soon as energy is transferred to the NES, it is almost immediately dissipated by the dashpot.

48 124 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, andpnpanagopoulos 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 displacement; the third, percentage of instantaneous total energy in the NES; and the fourth, displacement of the primary structure (NES versus grounded dashpot) [99]

49 Passive non-linear TET and its applications 125 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] Frequency energy plot analysis Utilizing the FEP on which the WT of the relative displacement between the primary structure and the NES is superimposed, the dynamics of the system for highlevel forcing with low damping, and for low-level forcing 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 NES engages in a 1:1 resonance capture with the linear oscillator at a frequency approximately equal to the natural frequency of the uncoupled linear oscillator. 4.2 Experiments with MDOF primary systems In order to support the theoretical findings in section 3.3, physical experiments were carried out

50 126 Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos Fig. 43 Fig. 44 Superposition of the WT of the relative displacement across the non-linearity and the FEP: (a) 55 N, low damping; (b) 31 N, high damping [99] Experimental setup for a two-dof linear primary structure coupled to an SDOF NES [169] using the fixture depicted in Fig. 44, which corresponds 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 surface method: m1 = kg, m2 = kg, = kg, k1 = 420 N/m, k2 = 0 N/m, k12 = 427 N/m, C = N/m3, λ1 = Ns/m, λ2 = Ns/m, λ12 = Ns/m, λ = Ns/m. The mass ratio /(m1 + m2 ) is equal to 8.7 per cent. From these parameters, the natural frequencies of the uncoupled linear subsystem are found to be 1.95 and 6.25 Hz, respectively. The damping coefficients range over a certain interval, because damping estimation is a difficult problem in this setup due to the presence of several ball joints and bearings, and due to the air track. It was found that damping was rather sensitive to the force level, which is why intervals rather than fixed values are given. In addition, at low amplitude friction appeared 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 and of durations of approximately 0.01 s. Four cases of increasing input energy were considered: case I, J; II, J; III, J; and IV, J. The superposition of the WT of the relative displacement across the non-linear spring on the FEP is shown in Fig. 45. Starting with the case I, the lowest energy, S is excited from the beginning of the motion. 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-ofphase mode. For case II, S is again excited, but harmonic components are present. By slightly increasing the imparted energy (case III), the threshold for interaction with the out-of-phase mode is exceeded. As a result, S111 + is excited, and shortly after a jump to S takes place. In case IV, the transitions are similar to those of case III. Further results for case IV, which bear strong resemblance 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 the instantaneous total energy, and the participation of the out-of-phase mode in the system response is drastically reduced. Around t = 2 s, a sudden transition takes place, and the NES starts extracting energy from the in-phase mode. The comparison of Figs 47(c) and (e) with Figs 47(d) and (f ) shows that the predictions of the model identified are in very close agreement with the experimental measurements in the interval 0 4 s. Specifically, the sequential interaction of the

51 Passive non-linear TET and its applications 127 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 m 1 (40 N, 0.01 s): (a) (b) displacements; (c) FEP with the superimposed WT of the relative displacement between m 2 and the NES; (d) instantaneous percentage of total energy in the NES [87]