Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators

Transcription

1 Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators D. Michael McFarland a Department of Aerospace Engineering, University of Illinois at Urbana Champaign, 306 Talbot Laboratory, 104 S. Wright Street, Urbana, Illinois Gaetan Kerschen b Department of Materials Mechanical and Aerospace Engineering, University of Liège, 1 Chemin des Chevreuils (B52/3), B-4000 Liege, Belgium Jeffrey J. Kowtko, Young S. Lee, and Lawrence A. Bergman c Department of Aerospace Engineering, University of Illinois at Urbana Champaign, 306 Talbot Laboratory, 104 S. Wright Street, Urbana, Illinois Alexander F. Vakakis d Division of Mechanics, National Technical University of Athens, P.O. Box 64042, GR Zografos, Athens, Greece, and Department of Mechanical and Industrial Engineering (adjunct), Department of Aerospace Engineering (adjunct), University of Illinois at Urbana Champaign, 306 Talbot Laboratory, 104 S. Wright Street, Urbana, Illinois Received 6 October 2004; revised 11 May 2005; accepted 11 May 2005 Our focus in this study is on experimental investigation of the transient dynamics of an impulsively loaded linear oscillator coupled to a lightweight nonlinear energy sink. It is shown that this seemingly simple system exhibits complicated dynamics, including nonlinear beating phenomena and resonance captures. It is also demonstrated that, by facilitating targeted energy transfers to the nonlinear energy sink, a significant portion of the total input energy can be absorbed and dissipated in this oscillator Acoustical Society of America. DOI: / PACS number s : r, Ga, At JGM Pages: I. INTRODUCTION Targeted nonlinear energy transfers between coupled oscillators have been recently investigated. In Refs. 1 and 2 irreversible and almost complete energy transfers between a discrete breather in a donor nonlinear system weakly coupled to an acceptor nonlinear system sustaining another discrete breather were reported. The transfer of energy is very selective because the two oscillators must be well tuned, and the donor must have a specific amount of energy, the acceptor being initially at rest. Application examples include donor and acceptor oscillators described by Hamiltonians e.g., discrete nonlinear Schrödinger models and a weakly coupled rotor-morse oscillator system. Targeted nonlinear energy transfers between an impulsively loaded linear oscillator termed the primary system and an essentially nonlinear attachment termed the nonlinear energy sink NES weakly coupled to it were observed numerically 3,4 and experimentally. 5 It was shown that nonlinear energy pumping i.e., an irreversible channeling of vibrational energy to the NES may occur in the presence of viscous dissipation. The concept of essential nonlinearity a Electronic mail: dmmcf@uiuc.edu b Currently, Postdoctoral Fellow at the University of Illinois at Urbana Champaign. Electronic mail: g.kerschen@ulg.ac.be c Electronic mail: lbergman@uiuc.edu d Electronic mail: vakakis@central.ntua.gr; avakakis@uiuc.edu i.e., the absence of a linear term in the stiffnessdisplacement relation is central because it means that the NES has no preferential resonant frequency; it may resonate a priori with and extract energy from any mode of the primary structure, 6 which is an attractive feature for vibration absorption and shock mitigation in the presence of broadband disturbances. However, grounded and relatively heavy nonlinear attachments were considered in these studies, which represents a limitation when the structural weight is an important design criterion. To overcome this drawback, an ungrounded and light attachment strongly nonlinearly coupled to a linear oscillator depicted in Fig. 1 was studied in detail in Refs. 7 and 8. The main conclusions from these studies were as follows. i ii iii This seemingly simple two degrees of freedom system can exhibit very complicated dynamics including, for instance, the existence of a countable infinity of periodic orbits and the ability of the NES to engage in an i: j internal resonance with the linear oscillator, i and j being relatively prime integers. A nonlinear beating phenomenon can be excited with the NES initially at rest that triggers transient resonance capture on a resonant manifold, which, in turn, is responsible for an irreversible and almost complete energy transfer to the NES. A significant percentage of the total input energy can be dissipated in the NES in spite of its modest mass. J. Acoust. Soc. Am , August /2005/118 2 /791/9/$ Acoustical Society of America 791

2 FIG. 1. Linear oscillator coupled to a lightweight NES. It should be noted that the study of the resonance capture phenomenon has received increasing attention in recent years. Interested readers can refer to Refs for further details. Our purpose in this paper is to report an experimental study of the targeted energy transfers that may occur between a linear oscillator and a lightweight NES. The paper is organized as follows: In Sec. II, the dynamics of the system of Fig. 1, together with the basic mechanisms for the energy exchanges, are briefly reviewed. The experimental setup and the parameter identification procedure are described in Sec. III. In Sec. IV we present the experimental results, and the conclusions of the present study are summarized in Sec. V. FIG. 2. Frequency-energy plot M =k=c=1, =0.05 ; unfilled dots represent points of stability exchange. II. REVIEW OF THE DYNAMICS OF THE UNDAMPED SYSTEM The system considered herein, depicted in Fig. 1, is composed of a linear oscillator strongly coupled to a lightweight NES. The equations of motion are Mÿ + 1 ẏ + 2 ẏ v + C y v 3 + ky =0, 1 v + 2 v ẏ + C v y 3 =0. Variables y and v refer to the displacement of the primary system and of the NES, respectively. A small mass of the NES and weak damping are assured by requiring that 1. All other variables are treated as O 1 quantities. In Refs. 7 and 8, it was shown that the structure and bifurcations of the periodic orbits of the undamped and unforced system enable one to understand the energy transfers in the weakly damped and impulsively loaded system. Therefore, all of the computed periodic orbits were gathered in a frequency-energy plot, represented in Fig. 2 for M = k = C =1, =0.05. It is composed of several branches, each branch being a collection of periodic solutions with the same characteristics. The backbone of the frequency-energy plot is formed by two branches, namely S11 and S11+, the latter being continued by S13+, S13, S15, S15+, etc. The other branches e.g., S21, U43, S14 are referred to as tongues; each tongue is composed of two very close branches e.g., S12 and S12+ that emanate from the backbone branch but also coalesce into this branch. The following notations are adopted. i ii iii Letters S and U refer to symmetric and unsymmetric solutions of the nonlinear boundary value problems that were solved in the calculation of the periodic orbits, respectively. The main qualitative difference between the periodic orbits on S and U branches is that they are represented by lines and Lissajous curves in the configuration space, respectively. The two indices indicated for the S and U branches refer to how fast the NES is vibrating relative to the linear oscillator. For example, on branches S11+ and S11 the NES engages in a 1:1 resonance capture with the primary system, whereas the NES is vibrating four times slower than the linear oscillator along S14. The and signs in the notations of the branches indicate whether the two oscillators are in phase or out of phase during the periodic motion, respectively. Due to the essential nonlinearity, the NES has no preferential resonant frequency. As a consequence, it may engage in an i: j internal resonance with the linear oscillator, i and j being arbitrary relatively prime integers. A countable infinity of tongues is thus expected in the frequency-energy plot, each tongue being a realization of a different i: j internal resonance between the primary system and the NES. A close-up of the S11+ branch is presented in Fig. 3, where some representative periodic orbits are also superposed. The convention adopted in this paper is that the horizontal and vertical axes in the configuration space plots depict the displacement of the NES and primary system, 792 J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators

3 FIG. 4. Experimental setup. a General configuration; b experimental impulsive force 21 N. FIG. 3. Close-up of the S11+branch. respectively. Furthermore, 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 in the linear or the nonlinear oscillator, respectively. The plot of Fig. 3 illustrates how the periodic orbits evolve with energy and frequency and how an irreversible and complete energy transfer to the NES is possible. As the energy decreases, the frequency of the motion diverges from the natural frequency of the linear oscillator i.e., 1 rad/s in the present case, and the periodic orbits become more and more localized in the NES. Hence, the mode shape of the free nonlinear periodic motion nonlinear normal mode changes with energy variation; this feature is not encountered in linear normal modes. By decreasing the total energy, viscous dissipation therefore facilitates targeted energy transfer as the motion localizes from the linear to the nonlinear oscillator. The underlying dynamical phenomenon is a transient resonance capture on a 1:1 resonant manifold because the two oscillators vibrate with the same frequency, but this frequency varies in time with the amount of energy transferred. In the absence of damping, irreversible energy transfer cannot occur; the energy flows back and forth between the NES and the primary system, and a nonlinear beating occurs. We also note that the role of damping has been studied carefully by Gendelman 13 through the computation of damped nonlinear normal modes. The resonance manifold cannot be reached immediately after the application of an impulsive excitation to the primary system because the shapes of the periodic orbits on the S11 + branch are not compatible with the NES being initially at rest. A transient bridging orbit, referred to as a special orbit, is then necessary in order to bring the motion into the domain of attraction of this manifold. This issue is not considered further herein, but a detailed discussion is available in Refs. 7 and 8. Specifically, it was demonstrated that the majority of tongues e.g., S31, U12 in the frequency-energy plot carries at least one special orbit that through nonlinear beats triggers targeted energy transfer energy pumping from the linear to the nonlinear oscillator. A final remark is that the NES cannot absorb any frequency with equal effectiveness across the spectrum. As shown in Ref. 8, there seems to be a well-defined critical threshold of energy that separates high- from low-frequency special orbits, i.e., those that localize or not to the NES, respectively. As a result, the transfer of a significant amount of energy from the linear oscillator to the NES is only possible above this threshold. III. EXPERIMENTAL SETUP A. Description of the experimental fixture The experimental fixture built to examine the energy transfers in the two degrees of freedom system described by Eqs. 1 is depicted in Fig. 4 a. A schematic of the system is provided in Fig. 5, detailing major components. The primary system of mass M, grounded by means of a linear leaf spring k, consisted of a car made of aluminum angle stock that was supported on a straight air track, depicted in Fig. 5 a. The NES of mass, highlighted in Fig. 5 b, consisted of a shaft supported by two bearings. A dashpot was connected to one end of the shaft, allowing adjustable viscous damping between the primary system and the NES, while steel plates clamped two steel wires at the other end. The wires were configured with almost no pretension, realizing the essential nonlinearity C. They were connected FIG. 5. a Schematic of the experimental fixture; b mass repartition grey: primary system; black: NES. J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators 793

4 TABLE I. System parameters identified using modal analysis and the restoring force surface method. Parameter to the primary system using another clamp, the position of which could be modified. For this experiment, the wires span was adjusted to 12 in. For further details about the construction of the essential nonlinearity, the interested reader can refer to Ref. 5 A long-stroke shaker provided a controlled and repeatable short impulse to the primary system. A representative input broadband force is given in Fig. 4 b. The response of both oscillators was measured using accelerometers. An estimate of the corresponding velocity and displacement was obtained by integrating the measured acceleration. The resulting signals were then high-pass filtered to remove the spurious components introduced by the integration procedure. B. System identification Value M kg kg k 1143 N/m N s/m N s/m C N/m The goal of system identification is to exploit input and output measurements performed on the structure using vibration sensing devices in order to estimate all the parameters governing the equation of motion 1. It should be noted that, prior to system identification, the primary system and the NES were weighed, and their masses were found to be equal to M =1.266 kg and =0.140 kg, respectively, which implies a mass ratio /M equal to This represents a smaller increase of the total mass of the system compared to previous experimental measurements performed on another NES configuration 5 for which the mass ratio was System identification was carried out in two separate steps. First, the primary system was disconnected from the NES, and modal analysis was performed on the disconnected primary system using the stochastic subspace identification method. 14 The natural frequency and the critical viscous damping ratio were estimated to be 4.78 Hz and 0.2%, respectively. Because the mass of the primary system was known, the stiffness and the damping parameters were easily deduced from this modal analysis; their values are listed in Table I. In the second step, the primary system was clamped, and an impulsive force was applied to the NES using an instrumented hammer. The NES acceleration and the applied force were measured. The restoring force surface method 15 was then used to estimate the nonlinearity C and the damping coefficient 2. In essence, Newton s second law was applied, f NL v,v,y,ẏ = p v, where f NL v,v,y,ẏ was the restoring force and p the external force for simplicity, the temporal dependence is omitted. Equation 2 shows that the time history of the restoring force can be calculated directly from the measurement of the acceleration and the external force and from the knowledge of the mass. This is illustrated for the 21N force level in Fig. 6 a. The representation of the restoring force in terms of the relative displacement v y in Fig. 6 b demonstrates that the linear component of the nonlinear stiffness was negligible; in other words, an essential nonlinearity was realized. The model f NL v,v,y,ẏ = 2 v ẏ + C v y 3 could then be fitted to the measured estimate of the restoring force, and least-squares parameter estimation could be used to obtain the values of coefficients C and 2. For greater flexibility, the functional form of the nonlinear stiffness was relaxed to f NL v,v,y,ẏ = 2 v ẏ + C v y sign v y. The three unknown parameters, namely the nonlinear coefficient, the exponent of the nonlinearity and the dashpot constant, were estimated by following the same procedure as in Ref. 16; i.e., one looks for the minimum of the normalized mean-square error between the measured and predicted restoring forces as a function of the exponent of the nonlinearity. The resulting parameters are listed in Table I. The best results have been obtained using an exponent equal to 2.8 which is not far from the theoretical value of FIG. 6. Measured restoring force represented as a function of a time and b relative displacement v y. 794 J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators

5 FIG. 7. Experimental results for low damping left column: 21 N; right column: 55 N; note differing durations : a, b measured accelerations; c, d measured displacements; e, f percentage of instantaneous total energy in the NES; g, h displacement of the primary system NES versus grounded dashpot. IV. INTERPRETATION OF THE EXPERIMENTAL RESULTS Two series of experimental tests 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 targeted energy transfers. Additional tests were performed to investigate whether the energy transfers to an NES can also take place with increased damping. A. Low-damping case In the low-damping case, several force levels ranging from 21 N to 55 N were considered, but for conciseness, only the results for the lowest and highest force levels are displayed in Fig. 7. At the 21 N level, 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 J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators 795

6 TABLE II. Nonlinear beating phenomenon: energy transferred to the NES and transfer time. Excitation level N Energy transferred to the NES % Transfer time s a large extent. Figure 7 e, showing the percentage of instantaneous total energy carried by the NES, 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% of the total energy is present in the NES, but this number drops down to 1.5% immediately thereafter. Hence, in this case, energy quickly flows back and forth between the two oscillators, which is characteristic of a nonlinear beating phenomenon. Another indication that a nonlinear beating occurs is that the envelope of the NES response undergoes large modulations. At the 55 N level, the nonlinear beating phenomenon still dominates the early regime of the motion. A less vigorous but faster energy exchange is now observed as 63% of the total energy is transferred to the NES after 0.12 s. These quantities also hold for the intermediate force levels listed in Table II. It should be noted that these observations are in close agreement with the analytical and numerical studies reported in Refs. 7 and 8; 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. The main qualitative difference from the case of the lowest force level is that now there exists a second regime of motion. After approximately 2.5 s the motion is captured in the domain of attraction of the 1:1 resonance manifold, as clearly evidenced in Fig. 7 f. This graph also demonstrates the irreversibility of this energy transfer, at least until escape from resonance capture occurs around t = 6.2 s. Another manifestation of the resonance capture is that the envelope of the displacement and acceleration signals decreases almost monotonically in this regime; no modulation is observed. The system is capable of sustaining the resonance capture during a large part of the motion i.e., from t=2.5 s to t =6.2 s. 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, which corresponds to the present results; b when the NES is disconnected, but its dashpot is installed between the primary system and ground; this corresponds to a single degree of freedom linear oscillator with added damping. Case b was not realized in the laboratory, but the system response was computed using numerical simulation. Figures 7 g and 7 h 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 nonlinear beating phenomenon is capable of transferring a significant portion of the total energy to the NES, it might be a more useful mechanism for energy dissipation. B. High-damping case Several force levels ranging from 31 N to 75 N were considered, but only the results for the 31 N level are presented herein. The damping constant was identified to be 1.48 Ns/ m, which means that damping can no longer be considered to be of order. The increase in damping is also reflected in the measured restoring force in Fig. 8 f. The system response shown in Figs. 8 a and 8 b is almost entirely damped out after 5 to 6 periods. The NES acceleration and displacement are still higher than the corresponding responses of the primary system, which means that targeted energy transfers may also occur in the presence of higher damping. The percentage of instantaneous total energy in the NES never reaches values close to 100% as in the previous case, but we 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. A comparison of the performance of the NES and the grounded dashpot is given in Fig. 8 e. A quantitative measure of energy dissipation is available through the computation of the energy dissipated in the NES normalized by the total input energy E diss t = t v 2 d t max p ẏ d. Experimental and simulated estimates of this quantity are depicted in Fig. 8 d. This demonstrates that as much as 96% of the total input energy is dissipated in the NES. There is very good agreement between predictions and measurements, validating the mathematical model developed in Sec. III. C. Further results The wavelet transform WT is a relevant technique for time-frequency analysis. In contrast to the fast Fourier transform FFT that 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 lowerfrequency components. As a result, the WT offers a means of computing the temporal evolution of the frequency components of a vibration signal; it is usually represented in a timefrequency plane. In the present study, the WT is represented in a energyfrequency plane by substituting the instantaneous total energy in the system for time. This enables one to superpose the WT and the frequency-energy plot. In Fig. 9, the backbone curve of the frequency energy plot of the experimental J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators

7 FIG. 8. Experimental results for high damping 31 N : a measured accelerations; b measured displacements; c percentage of instantaneous total energy in the NES; d measured and simulated energy dissipated in the NES; e displacement of the primary system NES versus grounded dashpot ; f restoring force. system, represented by a solid line, is superposed on the WT of the relative displacement v y. Shaded areas correspond to regions where the amplitude of the WT is high, whereas lightly shaded regions correspond to low amplitudes. This plot is a schematic representation because it superposes damped the WT and undamped the frequency-energy plot responses and is used for descriptive purposes only. However, it represents a useful tool for the interpretation of the dynamics. It indicates the following. i ii The dynamics of the system is indeed nonlinear, as the predominant frequency component of the NES varies with energy. There are strong harmonic components developing during the nonlinear beating phenomenon. Once these harmonic components disappear, the NES engages in FIG. 9. Superposition of the wavelet transform of the relative displacement across the nonlinearity and the frequency-energy plot. a 55 N, low damping; b 31 N, high damping. J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators 797

8 FIG. 10. Case of increased nonlinear coefficient: a measured accelerations; b measured displacements; a comparison between predicted and measured accelerations: c primary system; and d NES; e motion in the configuration space; f percentage of instantaneous total energy in the NES. iii a 1:1 resonance capture with the linear oscillator at a frequency approximately equal to the natural frequency of the disconnected linear oscillator. The predominant frequency component of the NES follows the backbone branch for most of the signal. This validates our conjecture that the weakly damped, transient dynamics can be interpreted mainly in terms of the periodic orbits of the underlying Hamiltonian system. Additional measurements were performed using a stiffer nonlinearity by shortening the span of the wire from 12 to 10 in. and by increasing the wire diameter from in. to in., which results in a nonlinear coefficient of N/m 3. An inspection of the accelerations and displacements shown in Figs. 10 a and 10 b reveals that the NES is no longer vibrating symmetrically with respect to its equilibrium position, particularly between t =1 s and t=3 s. Interestingly enough, there is almost a pointwise agreement between the experimental accelerations and those predicted by the identified numerical model in Figs. 10 c and 10 d. A better understanding of this particular regime of motion can be gleaned from a snapshot of the configuration space see Fig. 10 e. Apparently, the motion might be captured in the domain of attraction of a tongue on which the characteristic motion is not symmetric with respect to the origin of the configuration space. It turns out that the motion takes the form of a closed loop, which might mean that a U branch is excited. However, due to the existence of a countable infinity of tongues, and due to the presence of damping, it is difficult to ascertain with certainty what tongue is reached. Finally, Fig. 10 f illustrates that a nonlinear beating occurs during this particular regime, but the energy exchanges are not so vigorous, as approximately 40% is transferred to the NES. V. CONCLUDING REMARKS Our purpose in this study is the experimental investigation of targeted energy transfers to a NES. By facilitating these energy transfers, one can promote energy dissipation of a major portion of externally induced energy in the NES. 798 J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators

9 FIG. 11. Targeted energy transfer for a mass ratio of a Displacements; b percentage of instantaneous total energy in the NES. Two basic mechanisms governing targeted energy transfer have been highlighted, namely the excitation of a transient bridging orbit resulting in a nonlinear beating phenomenon and resonance capture into a 1:1 resonance manifold resulting in irreversible energy flow from the primary system to the NES with the former mechanism triggering the latter. As a result, this seemingly simple system may possess very complicated dynamics. However, it should be noted that satisfactory agreement was obtained between analytical and computational predictions and experimental measurements throughout this study, this, in spite of the transient and strongly nonlinear nonlinearizable nature of the NES dynamics. The mass ratio between the NES and the primary system considered herein is equal to 11%, but it is worth inquiring whether vigorous energy exchanges can still occur for smaller ratios. Figure 11 confirms that this is indeed possible for a mass ratio of 4% we note that the nonlinear coefficient had to be modified because decreasing the NES mass shifts the backbone branch toward lower energies in the frequencyenergy plot, but this effect can be compensated for by decreasing the nonlinear coefficient. Actually, the limiting mass ratio for this particular setup is 2%. Below this threshold, the special orbits are no longer capable of transferring a sufficient amount of energy to the NES, and targeted energy transfer does not take place. Specifically, it was proven in Ref. 8 that the energy transferred to the NES during the beating tends to zero as the NES mass tends to zero. Finally, it should be noted that essentially nonlinear attachments are promising for structures with multiple degrees of freedom. Due to the absence of a preferential resonant frequency, a NES has the potential to resonate with and extract energy from virtually any mode of the structure. This will be investigated in further detail in subsequent studies. ACKNOWLEDGMENTS This work was funded in part by AFOSR Contracts No. F and No. 00-AF-B/V One of the authors GK is supported by a grant from the Belgian National Fund for Scientific Research FNRS which is gratefully acknowledged. The support of the Fulbright and Duesberg Foundations which made GK s visit to the University of Illinois possible is also gratefully acknowledged. 1 S. Aubry, G Kopidakis, A. M. Morgante, and G. P. Tsironis, Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers, Physica B 296, G. Kopidakis, S. Aubry, and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett. 87, O. V. Gendelman, Transition of energy to nonlinear localized mode in highly asymmetric system of nonlinear oscillators, Nonlinear Dyn. 25, A. F. Vakakis, Inducing passive nonlinear energy sinks in vibrating systems, J. Vibr. Acoust. 123, D. M. McFarland, L. A. Bergman, and A. F. Vakakis, Experimental study of nonlinear energy pumping occurring at a single fast frequency, Int. J. Non-Linear Mech. 40, A. F. Vakakis, D. M. McFarland, L. A. Bergman, L. I. Manevitch, and O. Gendelman, Isolated resonance captures and resonance capture cascades leading to single- or multi-mode passive energy pumping in damped coupled oscillators, J. Vibr. Acoust. 126, Y. S. Lee, G. Kerschen, A. F. Vakakis, P. N. Panagopoulos, L. A. Bergman, and D. M. McFarland, Complicated dynamics of a linear oscillator with an essentially nonlinear local attachment, Physica D 204, G. Kerschen, Y. S. Lee, A. F. Vakakis, D. M. McFarland, and L. A. Bergman, Irreversible passive energy transfer in coupled oscillators with essential nonlinearity, SIAM J. Appl. Math.. 9 V. I. Arnold, Dynamical Systems III, Encyclopedia of Mathematical Sciences Springer-Verlag, Berlin, D. Quinn, R. Rand, and J. Bridge, The dynamics of resonance capture, Nonlinear Dyn. 8, R. Haberman, R. Rand, and T. Yuster, Resonant capture and separatrix crossing in dual-spin spacecraft, nonlinear dynamics, Nonlinear Dyn. 18, D. L. Vainchtein, E. V. Rovinsky, L. M. Zelenyi, and A. I. Neishtadt, Resonances and particle stochastization in nonhomogeneous electromagnetic fields, Journal of Nonlinear Science 14, O. V. Gendelman, Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment, Nonlinear Dyn. 37, P. Van Overschee and B. DeMoor, Subspace Identification For linear Systems: Theory, Implementation, Applications Kluwer Academic, Boston, S. F. Masri and T. K. Caughey, A nonparametric identification technique for nonlinear dynamic systems, J. Appl. Mech. 46, G. Kerschen, V. Lenaerts, S. Marchesiello, and A. Fasana, A frequency domain vs. a time domain identification technique for nonlinear parameters applied to wire rope isolators, J. Dyn. Syst., Meas., Control 123, J. Acoust. Soc. Am., Vol. 118, No. 2, August 2005 McFarland et al.: Energy Transfers in Nonlinearly Coupled Oscillators 799