energy transfers in a system of two coupled oscillators ... · energy transfers in a system of two...

Nonlinear Dynamics (2005) 42: 283–303 c© Springer 2005

Energy Transfers in a System of Two Coupled Oscillatorswith Essential Nonlinearity: 1:1 Resonance Manifoldand Transient Bridging Orbits

GAETAN KERSCHEN1,5,∗,∗∗, ALEXANDER F. VAKAKIS2, YOUNG S. LEE3,D. MICHAEL MCFARLAND4, JEFFREY J. KOWTKO4, and LAWRENCE A. BERGMAN4

1Department of Materials, Mechanical and Aerospace Engineering, University of Liege, Belgium; 2Division of Mechanics,National Technical University of Athens; Department of Mechanical and Industrial Engineering (adjunct), Department ofAerospace Engineering (adjunct), University of Illinois at Urbana-Champaign, USA; 3Department of Mechanical andIndustrial Engineering, University of Illinois at Urbana-Champaign, USA; 4Department of Aerospace Engineering, Universityof Illinois at Urbana-Champaign, USA; ∗Author for correspondence (e-mail: [email protected], fax: +32-4-3664856)

(Received: 7 September 2004; accepted: 15 March 2005)

Abstract. The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linearoscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchangesare encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly interms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbitsof the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almostcompletely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energypumping is triggered by the excitation of transient bridging orbits compatible with the nonlinear attachment being initially at rest,a common feature in most practical applications. A parametric study of the energy exchanges is also performed to understandthe influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting thetheoretical developments are discussed.

Key words: energy transfer, essential nonlinearity, nonlinear beat phenomenon, nonlinear energy pumping, resonance capture

1. Introduction

The tuned absorber is an effective device for vibration mitigation in many mechanical systems, in-cluding bridges and buildings. However, this passive energy sink is effective over only a narrow bandof frequencies and is incapable of robustly absorbing multi-frequency transient disturbances. In orderto overcome this limitation, the nonlinear energy pumping phenomenon is investigated in this study.It corresponds to the one-way channeling of vibrational energy from a primary system to a passivenonlinear energy sink (NES) where it localizes and diminishes in time due to dissipation.

In recent works [1–3], it was observed that energy pumping can occur in a system composed of alinear oscillator weakly coupled to an essentially nonlinear (nonlinearizable) oscillator. In a systemwith viscous dissipation, a great portion of the energy initially imparted to the primary system canbe transferred to the NES. It was demonstrated that a transient resonance capture on a 1:1 resonancemanifold [4, 5] of the system is at the origin of an irreversible and almost complete energy transferbetween the primary system and the NES. The domain of attraction of the manifold is relatively large but

∗∗This study was carried out while the author was a postdoctoral fellow at the National Technical University of Athens and atthe University of Illinois at Urbana-Champaign.

284 G. Kerschen et al.

the manifold itself is not compatible with the NES being at rest at time t = 0 when an impulse is appliedto the primary system only. It was therefore assumed that transient bridging orbits fully compatiblewith the initial conditions must exist in order to bring the motion into the domain of attraction of the1:1 resonance manifold.

Energy transfers between weakly coupled nonlinear oscillators (or discrete breathers) having fastoscillations at the same frequency have also been studied by Aubry and co-authors [6, 7] and have beenreferred to as targeted energy transfers (TET). The theory has recently been extended for resonancesat higher order, termed Fermi resonances [8]. In these studies, the primary system and the NES aredescribed by Hamiltonians (i.e., there is no energy dissipation) and are termed donor and acceptor,respectively. It is shown that complete (or almost complete) and irreversible energy transfers from thedonor to the acceptor may occur. The transfer is very selective as the two oscillators must be well tuned,and the donor must have a specific amount of energy (the acceptor is initially at rest). Other studiesdealing with energy transfers include those on nonlinear modes in internal resonance in references[9, 10]; these latter exchanges, however, do not necessarily involve one-way, irreversible channeling ofvibrational energy.

The purpose of this study is to highlight and explain the vigorous energy transfers that may take placein a linear oscillator weakly coupled to an essentially nonlinear attachment in the presence of viscousdissipation. Although energy pumping is encountered during the transient dynamics of the dampedsystem, it is shown that the dynamics can be interpreted in terms of the periodic orbits of the underlyingHamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the systemis constructed. The evolution of this frequency-energy plot with respect to the ratio between the NESmass and the mass of the primary system is studied; the influence of the nonlinearity and of the weakcoupling stiffness is also discussed. Finally, the results of experimental measurements are shown thatsupport the theoretical developments.

2. Computation of a Frequency-Energy Plot

The system considered herein is composed of a linear oscillator, termed the primary system, coupledto an essentially nonlinear (nonlinearizable) oscillator, termed the NES, and is depicted in Figure 1.

M y + ελ1 y + ελ(y − v) + ε(y − v) + ky = 0(1)

mv + ελ2v + ελ(v − y) + ε(v − y) + Cv3 = 0

Variables y and v refer to the displacement of the primary system and of the NES, respectively.Weak coupling and damping is assured by requiring that ε � 1. All other variables are treated asO(1) quantities; provided the input energy is high enough, a strongly nonlinear system is thereforeinvestigated.

Figure 1. Linear oscillator weakly coupled to an essentially nonlinear oscillator.

Energy Transfers in a System of Two Coupled Oscillators 285

The computation of the periodic orbits of the undamped system

M y + ε(y − v) + ky = 0(2)

mv + ε(v − y) + Cv3 = 0

is performed using the method of nonsmooth temporal transformations (NSTTs). This method replacesthe temporal variable t by two nonsmooth variables

τ = (2/π ) arcsin [sin (π t/2)], e(t) = τ (t), e2(t) = 1 (3)

and transforms the Equations (1) into a set of two nonlinear boundary value problems (NBVPs) to besolved in a closed interval −1 ≤ τ ≤ 1. The method is not further detailed here, but the interestedreader can refer to [11, 12] and references therein.

The frequency-energy plot is presented in Figure 2 for the system parameters M = m = k =C = 1, ε = 0.1. It gathers all the periodic orbits that have been computed using the method of NSTTs.Periodic orbits that correspond to synchronous motion of the two oscillators are termed nonlinear normalmodes (NNMs) [13]. The frequency-energy plot is composed of several branches, each branch beinga collection of periodic solutions with the same characteristics. The backbone of the frequency-energyplot is formed by the S11− and S11+ branches. The other branches (e.g., S21, U43, S13) are referred toas tongues; each tongue is composed of two very close branches (e.g., S13− and S13+) that bifurcateout and emanate from the backbone branch. The following notations are adopted:• Letters S and U refer to symmetric and unsymmetric solutions of the NBVPs, respectively.• The two numbers indicate how fast the NES is vibrating with respect to the linear oscillator. For

example, the NES engages in a 1:1 resonance capture with the primary system all along S11+ andS11− and is vibrating three times slower than the linear oscillator along S13.

• The + and − signs indicate whether the two oscillators are in phase or out of phase, respectively.It is emphasized that, 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 beingarbitrary integers. However, for very low energy levels, the NES response is dominated by the weakcoupling spring; the ratio i/j must therefore be greater than the ratio of the two frequencies of theunderlying linear system, i/j > f2/ f1, where

f1, f2 =√

k + ε

2M+ ε

2m±

√−4kmMε + (−km − mε − Mε)2

2mM(4)

For M = m = k = C = 1, ε = 0.1, the ratio i/j must be greater than 0.3/1.0535 = 0.285. Thanks tothe essential nonlinearity, 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 andthe NES. As an example, initial conditions y(0) = v(0) = v(0) = 0 and y(0) = 5.59 correspond toa periodic orbit on the U43 branch, and the displacements, together with the instantaneous percentageof energy carried by the NES, are depicted in Figure 3. Energy flows back and forth between the twooscillators, which is the characteristic of a nonlinear beating phenomenon. The energy transfer is notcomplete, and only 20% of the energy can be carried by the NES. Similar nonlinear beating phenomenacan be observed on each tongue in the frequency-energy plot. It is important to note that no a priorituning of the NES is necessary, which is markedly different from other systems exhibiting nonlinearbeating phenomena (see, e.g., reference [9] or spring-pendulum systems).


Figure 2. Frequency-energy plot of the periodic orbits (M = m = k = C = 1, ε = 0.1); transient bridging orbits, termedspecial orbits, are denoted by black dots, and are connected by a dotted line; letters S and U refer to symmetric and unsymmetricsolutions of the nonlinear boundary value problem, respectively; we assign to a specific branch of solutions a frequency indexequal to the ratio of its indices, e.g., S21 is represented by the frequency index ω = 2/1 = 2, as is U21 (this convention ruleholds for every branch except S11±, which, however, are particular branches, forming the basic backbone of the plot); f1 and f2

are the natural frequencies of the underlying linear system; f3 is the natural frequency of the linear subsystem that governs thedynamics at high energies (the nonlinearity behaves as a massless rigid link).

A third frequency, f3, is defined as the characteristic frequency of the oscillators on the S11+ branchfor infinite values of the energy. The nonlinear spring can be considered as infinitely stiff, and the systembehaves as a single degree of freedom with a linear spring of constant stiffness k+ε and a mass equal to M

f3 =√

k + ε

M(5)

For M = m = k = C = 1, ε = 0.1, f3 = 1.0488 rad/s and is smaller than f1 = 1.0535 rad/s.Accordingly, there exist two forbidden zones along the frequency axis where the motion cannot take place

f �[0, f2] and f �[ f3, f1] (6)


Figure 3. Periodic orbit on U43 branch.

3. Basic Mechanisms for Nonlinear Energy Pumping

3.1. RESONANCE MANIFOLD

Recent studies [1–3, 14] have shown that complex energy transfers between the primary system and theNES may occur in this seemingly simple system. At this point, it is noted that these energy transferscan also take place between an NES and a primary system with multiple degrees of freedom [15] or aninfinity of degree of freedom [16].

Even though energy pumping can only occur in the damped system, we conjecture that damping doesnot change radically the underlying mechanisms and that the transient dynamics of the weakly dampedoscillators can be interpreted in terms of the frequency-energy plot computed in Section 2.

A close-up of the S11+ branch is presented in Figure 4(a), where some representative NNMs arealso superposed. The convention adopted in this paper is that the horizontal and vertical axes in theconfiguration space plots depict the displacement of the NES and of the primary system, respectively.Furthermore, the aspect ratio is set so that tick mark increments on the horizontal and vertical axes areequal in size, enabling one to directly deduce whether the motion is localized in the linear or in thenonlinear oscillator. The figure demonstrates the frequency-amplitude relationship in nonlinear systems,as the shape of the NNMs and their frequency strongly depend on the energy. When the energy tends


Figure 4. Close-up of S11+ branch. (a) Mass ratio = 1; (b) Mass ratio = 0.05.

to infinity, the frequency of the oscillators tends to the natural frequency of the linear oscillator; as aconsequence, the motion is localized in the linear oscillator. Conversely, as the energy is decreased,the motion moves away from this region, and the motion becomes localized in the NES. Viscousdissipation facilitates transfer of energy from the linear oscillator to the NES. The key feature of thisenergy transfer is its irreversibility as the motion is more and more localized in the NES when the energyis dissipated. It is also almost complete as the shape of the NNM is close to a horizontal line for low energylevels.

In summary, motion along the S11+ branch represents the basic mechanism for energy pumping, andthe underlying dynamical phenomenon is a 1:1 resonance capture. The two oscillators vibrate with thesame fast frequency, but this frequency varies in time with the amount of energy transferred. Energy isirreversibly and almost completely transferred from the primary system to the NES.


It should be noted that:• The NNMs on the S11− branch are such that energy is transferred from the NES to the primary

system when the energy is decreased by viscous dissipation. This branch is thus not interesting whenthe purpose is to reduce the response of a primary structure.

• In principle, nonlinear energy pumping can also be realized on the tongues (e.g., on S13). But itappears that the domain of attraction of these tongues is much smaller than that of the S11+ branch,and this is not further discussed herein.

3.2. TRANSIENT BRIDGING ORBITS

In practical applications, an impulse is applied to the primary system while the NES is at rest. Thiscondition is only realized asymptotically on the S11+ and S11− branches; i.e., for infinite and zerovalues of the energy, respectively. The shape of the NNMs on these branches is thus generally notcompatible with the NES being initially at rest. As discussed in the introductory section, transientbridging orbits fully compatible with the initial conditions must exist in order to bring the motion intothe domain of attraction of the 1:1 resonance manifold. These transient bridging orbits are referred toas special orbits from now on.

All the periodic orbits previously computed using the method of NSTTs are inspected in order tohighlight those presenting a vertical line passing through the origin of the configuration space. Suchan orbit will be able to play the role of special orbit as it corresponds to y(0) = v(0) = v(0) = 0 andy(0) = Y and as it enables the transfer of some energy to the NES once t > 0.

In fact, the example used to illustrate the nonlinear beating phenomenon in the previous section (cf.Figure 3) had initial conditions corresponding exactly to a special orbit. This demonstrates that theunderlying mechanism of a special orbit is a nonlinear beating phenomenon.

It also turns out that a special orbit exists on each tongue provided that the periodic orbits on thistongue pass through the origin of the configuration space. Due to the existence of a countable infinity oftongues in the frequency-energy plot, it is reasonable to assume the existence of a countable infinity ofspecial orbits. This would represent an attractive feature as for almost any level of energy considered, aspecial orbit would exist and could be excited. To verify this conjecture, a constraint which imposes azero initial velocity to the NES is considered when solving the set of two NBVPs provided by the methodof NSTTs. Each computed special orbit is represented using a black dot in Figure 2. It is emphasizedthat special orbits realizing high-order internal resonances (e.g., 9 : 5, 4 : 7 or 7 : 6) have been calculated.The locus of points corresponding to the special orbits is represented by a dotted line and is of practicalimportance as it enables one to predict where the motion is initiated in the frequency-energy plot for agiven value of the input energy. The dotted line only represents a convenient way to localize the specialorbits; there is no formal proof of its existence. Besides, this line should not be continuous because theNES can only enter in a i : j internal resonance with the linear oscillator, with the restriction that i andj are integers.

Some representative special orbits are illustrated in Figure 5, where the instantaneous percentageof energy carried by the NES during the nonlinear beating is also shown. There are two families ofspecial orbits, namely those living on S tongues and those living on U tongues. The latter take the formof closed, Lissajous curves in the configuration space, and the former take the form of open curves.Ideally, the special orbits should exhibit the following two properties:• They should be stretched out along the horizontal axis in the configuration space as much as possible.

From this standpoint, the special orbit of U43 is more appealing than that of S31.


Figure 5. Special orbits.


• The NES should vibrate faster than the linear oscillator as this will increase its kinetic energy. Fromthis standpoint, the special orbits living in the upper part of the frequency-energy plot are moreinteresting.

This would allow a special orbit to transfer a greater amount of energy to the NES during one cycleof the beating phenomenon. It appears that the special orbits living in the neighborhood of the specialorbit of U43 and below represent the best compromise in terms of energy exchange.

3.3. NUMERICAL SIMULATIONS

Numerical simulations with M = m = k = C = 1, ε = 0.1, λ = λ1 = λ2 = 0.2 are now performedin order to validate the previous findings. The motion is first initiated from the special orbit of U43 andthe results are displayed in Figure 6. The plot in the middle, showing the instantaneous percentage ofenergy carried by the NES, highlights that the initial nonlinear beating phenomenon (0–90 s) effectivelytriggers nonlinear energy pumping (90–150 s) which, in turn, is responsible for an irreversible andalmost complete energy transfer to the NES.

The plot at the bottom is the superposition of the frequency-energy plot and the instantaneous fre-quency of the NES displacement computed using a wavelet transform. Shaded areas correspond to re-gions where the amplitude of the wavelet transform is high whereas lightly shaded regions correspondto low amplitudes. This plot is a schematic representation as it superposes damped and undampedresponses and is used for descriptive purposes only. It indicates that:• The system is strongly nonlinear as the instantaneous frequency of the NES varies significantly with

the total energy present in the system.• There are strong and sustained harmonic components during the nonlinear beating phenomenon.

Once these harmonic components disappear, the NES engages in a 1:1 resonance capture with thelinear oscillator.

• The predominant component of the instantaneous frequency of the NES follows the backbone branchfor most of the signal. This validates our conjecture that the dynamics of the weakly damped systemcan be interpreted in terms of the periodic orbits of the underlying Hamiltonian system.The motion is now initiated from the special orbits of the S13 branch (y(0) = 0.5742) and the U21

branch (y(0) = 32.46) and from an arbitrarily chosen point below the special orbit of the U43 branchbut above f1 (y(0) = 3). Figure 7 shows the corresponding displacement signals. Although the specialorbit of S13 transfers 20% of the total energy to the NES, a transition to the S11+ branch cannot occuras its NNMs are strongly localized in the NES in this particular region of the frequency-energy plot;nonlinear energy pumping cannot be activated. On the other hand, the special orbit of U21 transfers lessenergy to the NES but appears for much higher energies for which the NNMs on S11+ are localizedin the linear oscillator. This special orbit is able to bring the motion into the domain of attraction ofthe 1:1 resonance manifold, but the evolution toward the region where the shape of the NNMs quicklyvaries (i.e., when the energy is between 10−1 and 100) is slow; nonlinear energy pumping is activatedbut takes time. The optimal scenario occurs for intermediate energy levels when the motion is initiatedfrom a special orbit below U43. As illustrated in the two bottom plots of Figure 7, the NES amplitudegrows rapidly with time and after a few cycles exceeds the amplitude of the primary system.

From these numerical simulations, it can be concluded that (a) nonlinear energy pumping in coupledoscillators can occur only above a certain threshold of the input energy, and (b) there is an optimalvalue of the input energy at which energy is quickly transferred to the NES, where it localizes anddiminishes in time due to damping. We note that this is in agreement with what was observed in aprevious theoretical analysis [2] and in experimental measurements [17].


Figure 6. Motion initiated from the special orbit of U43 branch.

4. Parametric Study of the Energy Exchanges Between the Primary Systemand the Nonlinear Energy Sink

The purpose of the previous section was to highlight the basic mechanisms responsible for energytransfers between the primary system and the NES. However, no attempt was made to maximize theseenergy exchanges. In the present section, analytic calculations are performed in order to understand the


Figure 7. Motion initiated from different special orbits in the frequency-energy plot.

effects of the NES parameters on performance. In addition, the frequency-energy plot is computed forsmaller values of the ratio between the NES mass and the mass of the primary system.

4.1. 1:1 RESONANCE MANIFOLD

The motion on S11+ for infinite values of the energy is always completely localized to the primarysystem independent of the NES parameters m, C and ε. We investigate now how the motion on S11+for very low energy levels is influenced by these parameters. For such energy levels, the nonlinear


Table 1. Influence of the NES mass and the coupling spring on thecompleteness of the energy transfer on the 1:1 resonance manifold.

NES mass m Coupling spring ε Ratio V/Y

1 0.1 10.10

0.6 0.1 9.51

0.15 0.1 5.54

0.05 0.1 1.84

1 0.05 20.05

1 0.1 10.10

1 0.3 3.61

1 0.5 2.41

stiffness can be neglected, and the dynamics is mainly governed by the underlying linear system. Amodal analysis is thus carried out in order to compute the mode shape [V ; Y ] corresponding to thelowest natural frequency, yielding

[V ; Y ] =[

1;εM − εm − km +

√−4kmMε + (−km − mε − Mε)2

2εM

](7)

The ratio V/Y gives an indication of the completeness of the energy transfer when the motion is capturedinto the domain of attraction of S11+ and remains on this resonance manifold, the aim being to maximizeit. For example, in Figure 6, a prolonged 1:1 resonance capture takes place; the ratio V/Y is equal to10.10, which implies that eventually the NES carries almost 100% of the total energy in the system. Theinfluence of the parameters m and ε is summarized in Table 1. The most complete energy transfers areobserved for the lowest coupling stiffness and the heaviest nonlinear attachment, a feature which maylimit the utility of this NES in practical applications where the total mass of the structure is an importantdesign criterion. As further evidence of the influence of the NES mass, a close-up of the S11+ branchfor a mass ratio of 0.05 is presented in Figure 4(b).

4.2. SPECIAL ORBITS

An analytic study of special orbits is performed using the complexification averaging technique [18].Our attention is focused on the special orbit of the U21 branch, on which both the NES and the primarysystem carry two harmonic components, one at frequency ω and the other at frequency 2ω. The complexvariables

�1 = y1 + jωy1, �3 = y2 + 2 jωy2(8)

�2 = v1 + jωv1, �4 = v2 + 2 jωv2

are introduced, where y(t) = y1(t) + y2(t) and v(t) = v1(t) + v2(t) have been decomposed into theirtwo harmonic components. Hence,

y = �1 − �∗1

2 jω+ (�3 − �∗

3 )

4 jω(9)


x = �2 − �∗2

2 jω+ (�4 − �∗

4 )

4 jω(10)

y = �1 + �3 − jω

2(�1 + �∗

1 ) − 2 jω(�3 + �∗3 ) (11)

x = �2 + �4 − jω

2(�2 + �∗

2 ) − 2 jω(�4 + �∗4 ) (12)

All these expressions are substituted into the equations of motion of the undamped system (2). Then,the dynamics is partitioned into slow- and fast-varying components and reduced to the slow flow byaveraging over the fast frequencies ω and 2ω. The calculations are not detailed herein, as a similaranalysis has already been carried out in previous publications (see, e.g., references [2, 19]). Imposingstationarity conditions on the equations on the slow flow leads to an approximation of the periodic orbitson the U21 branch

y(t) = A sin ωt + B sin 2ωt and v(t) = D sin ωt + E sin 2ωt (13)

where

D = ± 2

3√

C

√7mω2 − ε + 2ε2/Z2 − ε2/Z1 (14)

E = ± 2

3√

C

√−2mω2 − ε − ε2/Z2 + 2ε2/Z1 (15)

A = εD

Z1B = εE

Z2(16)

Z1 = k − Mω2 + ε Z2 = k − 4Mω2 + ε (17)

Two solutions exist as indicated by the presence of the ±sign, a confirmation that a tongue is composedof two branches.

An analytic solution for the frequency of the special orbit on U21 can be computed by imposing zeroinitial displacement and velocity to the NES; i.e., D = −2E . The expression is complicated and is notgiven here. The energy stored in the primary system during the nonlinear beating phenomenon is

Eprim = ky(t)2

2+ M

y(t)2

2

= k(εD)2

2

(sin ωt

Z1− sin 2ωt

2Z2

)2

+ M(εDω)2

2

(cos ωt

Z1− cos 2ωt

Z2

)2

(18)

For a special orbit y(0) = v(0) = v(0) = 0 and y(0) �= 0, which means that at t = 0 the entirety of thetotal energy is stored in the linear oscillator

Etot = Eprim(t = 0) = M(εDω)2

2

(1

Z1− 1

Z2

)2

(19)

After some rearrangement and normalization by the total energy, the percentage of energy carried bythe primary system, Eprim, % = 100 Eprim/Etot, is

Eprim, % = 100k

4Mw2 (2Z2 sin ωt − Z1 sin 2ωt)2 + (Z2 cos ωt − Z1 cos 2ωt)2

(Z2 − Z1)2(20)


Table 2. Influence of the NES mass and the couplingspring on the maximum percentage of energy trans-ferred to the NES during the nonlinear beating.

NES mass Coupling Maximum energym spring ε transferred to the NES (%)

1 0.1 8.94

0.6 0.1 8.85

0.15 0.1 8.17

0.05 0.1 8.03

1 0.05 4.72

1 0.1 8.94

1 0.3 22.28

1 0.5 31.85

where ω is the frequency of the special orbit. Equation (20) enables us to compute the maximumenergy carried by the NES during the nonlinear beating for various values of the parameters, asENES,% = 100 − Eprim,%. The results are listed in Table 2 and illustrate that the NES mass has al-most no influence on the amount of energy transferred to the NES. The same conclusion can be reachedfor the nonlinear coefficient as it is absent from equation (20). On the other hand, increasing the stiffnessof the coupling spring seems to be beneficial. However, the increase should be limited as the most com-plete energy transfers on the 1:1 resonance manifold are observed for the lowest values of the couplingstiffness.

4.3. FREQUENCY-ENERGY PLOT FOR DIFFERENT MASS RATIOS

The frequency-energy plot is now computed for three mass ratios, namely 0.6, 0.15 and 0.05, usingthe method of NSSTs. From Figure 8, we observe that the shape of the backbone branch is verysensitive to this parameter. In particular, for the mass ratio equal to 0.05, (a) the two inflexion pointson the S11– branch – corresponding to saddle node bifurcations – disappear; (b) the S11+ branchis almost horizontal, meaning that the nonlinear phenomena are less enhanced in this system and (c)the two forbidden zones along the frequency axis where the motion cannot take place increase, andas a consequence, the number of lower tongues (i.e., tongues emanating from S11+) significantlydecreases.

5. Experimental Study

To support the previous theoretical findings, experimental measurements have been carried out using thefixture depicted in Figure 9. This fixture realizes the system described by equations (1) and comprisestwo cars made of aluminum angle stock which are supported on a straight air track. The primary systemof mass M is grounded by means of a linear spring k, and the NES of mass m is connected to theprimary system by means of a weak coupling stiffness ε. An essential cubic nonlinearity C is realizedby a thin wire with no pretension. To dissipate the energy, viscoelastic tape is added to the couplingspring, realizing the damping constant ελ. Some energy dissipation is also provided between the carsand the air track, but the resulting damping constants ελ1 and ελ2 are smaller than ελ. A long-stroke


Figure 8. Frequency-energy plots for different mass ratios. (a) 0.6; (b) 0.15; (c) 0.05.

electrodynamic shaker is used to excite the primary system; i.e, the left car in the upper picture inFigure 9.

Two excitation levels are considered herein, corresponding to peak amplitudes equal to 13N and 18N,respectively. The results obtained are presented in Figures 10 and 11 and are qualitatively similar. Inboth cases, a special orbit is excited at time t = 0 resulting in a nonlinear beating phenomenon. Thiscan clearly be distinguished in the plot showing the instantaneous percentage of energy carried by theNES. The nonlinear beating is capable of transferring energy from the initially excited primary system


Figure 9. Upper picture: Experimental fixture for nonlinear energy pumping; lower picture: close-up of the NES.

to the NES; we note that the amount of energy transferred is greater for the lower excitation level (51%vs. 35%), a feature in agreement with the theoretical findings of Sections 3.2 and 3.3.

After a few cycles, the motion is captured by the domain of attraction of the 1:1 resonance mani-fold. The envelope of both displacement signals decreases monotonically (i.e., no modulation can beobserved), but that of the NES decreases much more slowly than that of the primary system. This isthe sign that an energy transfer from the primary system to the NES has taken place. The instantaneouspercentage of energy carried by the NES further illustrates that this transfer is irreversible, at least until


Figure 10. Experimental results, 13N excitation level. (a) and (b) Measured displacement signals; (c) instantaneous energy carriedby the NES; (d) Energy dissipated at the NES; (e) Wavelet transform of the NES displacement superposed to the frequency-energyplot.

escape from resonance capture occurs. This latter regime is observed when the actual motion is nolonger compatible with the motion on the resonance manifold.

The superposition of the wavelet transform of the NES displacement and of the frequency-energyplot is a very useful tool for the interpretation of the dynamics. First, it indicates that the system isstrongly nonlinear since the instantaneous frequency of the NES displacement varies with the energy


Figure 11. Experimental results, 18N excitation level. (a) and (b) Measured displacement signals; (c) instantaneous energy carriedby the NES; (d) Energy dissipated at the NES; (e) Wavelet transform of the NES displacement superposed to the frequency-energyplot.

level. It also shows in a very clear fashion that the motion is captured by the 1:1 resonance manifold.More importantly, it validates our conjecture that the dynamics can be mainly interpreted in terms of theperiodic orbits of the underlying Hamiltonian system although these energy exchanges are encounteredduring the transient dynamics of the damped system.

Finally, it should be mentioned that, eventually, 87.6% and 72.4% of the total input energy is absorbedin the NES for the 13N and 18N force levels, respectively.


6. Concluding Remarks

The purpose of this paper is to highlight and explain the vigorous energy transfers that may take placein a linear oscillator weakly coupled to an essentially nonlinear attachment. Even though the transientdynamics of the damped system have been considered, it has been demonstrated that a frequency-energy plot gathering the periodic orbits of the underlying Hamiltonian system is a powerful tool forunderstanding these energy exchanges.

Based upon this frequency-energy plot, two mechanisms responsible for energy dissipation havebeen highlighted, namely the excitation of a transient bridging orbit (resulting in a nonlinear beatingphenomenon) and the capture into a 1:1 resonance manifold (resulting in an irreversible energy flowfrom the primary system to the NES) with the former mechanism triggering the latter. All these findingshave been verified experimentally.

A parametric study of the energy exchanges between the primary system and the NES has also beenperformed. Interestingly enough, it was proven that the nonlinear coefficient does not influence theseexchanges of energy. This study has also shown that the stiffness of the coupling spring must be weakin order to have an almost complete energy transfer to the NES along the 1:1 resonance manifold.Regarding the nonlinear beating phenomenon, the stiffness should be chosen high enough to transfer asufficient amount of energy to the NES during the beating. These are contradictory requirements whichmight render the NES design challenging. The parametric study has also pointed out that relativelyheavy nonlinear attachments are to be considered which may represent a limitation when the structuralweight is an important issue.

It is therefore relevant to seek another configuration of the NES that overcomes this drawback. Forexample, let us consider an ungrounded NES with a light mass connected to a primary system by meansof an essential nonlinearity. The equations of motion of the undamped system are

M y + ky + C(y − v)3 = 0(21)

εv + C(v − y)3 = 0

A light mass is assured by requiring that ε � 1. All other variables are treated as O(1) quantities. Forconvenience, we assume that M = k = 1

y + y + C(y − v)3 = 0(22)

εv + C(v − y)3 = 0

The linear change of variables y = αz + βw, v = δz + γw is now applied. The values of α, β, δ andγ are chosen so that the resulting equations of motion resemble equations (2) as closely as possible.Doing so yields

y = ε(z − w), v = εz + w (23)

and equations (22) become

ε(1 + ε)z + ε(z − w) = 0(24)

(1 + ε)w + ε(w − z) + C(1 + ε)4

εw3 = 0


Figure 12. Displacement signals for a light and ungrounded nonlinear attachment.

For the sake of clarity, variables z and w are rewritten as y and v, respectively

ε(1 + ε)y + ε(y − v) = 0(25)

(1 + ε)v + ε(v − y) + C(1 + ε)4

εv3 = 0

These equations bear a strong resemblance to Equations (2). However, it is emphasized that the primarysystem has no grounded linear spring which means that there is no complete equivalence between thesystems described by Equations (2) and (21).

Nevertheless, the comparison of Equations (22) and (25) shows that an NES characterized by a smallmass ratio ε connected to a primary system by means of an essential nonlinearity of O(1) correspondsto an NES characterized by a great mass ratio 1/ε connected to a primary system by means of a weakcoupling spring ε and connected to ground by means of a stiff essential nonlinearity approximatelyequal to 1/ε. Because the system described by Equations (25) bears a strong resemblance to the systeminvestigated in the present paper and because it has a large mass ratio 1/ε and weak coupling spring ε,it will demonstrate good energy pumping performance, as will the system described by Equations (22).

We therefore expect the system described by Equations (22), or more generally by Equations (21), tobe a light and ungrounded NES capable of eliminating undesired broadband disturbances. A numericalsimulation with M = k = C = 1 and ε = 0.05 is carried out in order to verify this prediction. Theconstants of the grounded and coupling dashpots are set equal to 0.005. The displacement signals shownin Figure 12 show that a significant amount of energy is indeed quickly transferred from the primarysystem to the NES. This new NES will be the subject of further research.

Acknowledgements

This work was funded in part by AFOSR Contracts F49620-01-1-0208 and 00-AF-B/V-0813. The authorG. Kerschen 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 madehis visit to the University of Illinois possible is also gratefully acknowledged. The authors would liketo acknowledge Panagiotis Panagopoulos for his contributions to the problem.


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