Nonlinear Dynamics (2005) 42: 283–303
c Springer 2005
Energy Transfers in a System of Two Coupled Oscillators
with Essential Nonlinearity: 1:1 Resonance Manifold
and Transient Bridging Orbits
GAETAN KERSCHEN1,5,∗,∗∗ , ALEXANDER F. VAKAKIS2 , YOUNG S. LEE3 ,
D. MICHAEL MCFARLAND4 , JEFFREY J. KOWTKO4 , and LAWRENCE A. BERGMAN4
1 Department
of Materials, Mechanical and Aerospace Engineering, University of Liège, Belgium; 2 Division of Mechanics,
National Technical University of Athens; Department of Mechanical and Industrial Engineering (adjunct), Department of
Aerospace Engineering (adjunct), University of Illinois at Urbana-Champaign, USA; 3 Department of Mechanical and
Industrial Engineering, University of Illinois at Urbana-Champaign, USA; 4 Department of Aerospace Engineering, University
of Illinois at Urbana-Champaign, USA; ∗ Author for correspondence (e-mail: kerschen@uiuc.edu, 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 linear
oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchanges
are encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly in
terms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits
of the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almost
completely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energy
pumping 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 understand
the influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting the
theoretical 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, including bridges and buildings. However, this passive energy sink is effective over only a narrow band
of frequencies and is incapable of robustly absorbing multi-frequency transient disturbances. In order
to 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 passive
nonlinear 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 a
linear oscillator weakly coupled to an essentially nonlinear (nonlinearizable) oscillator. In a system
with viscous dissipation, a great portion of the energy initially imparted to the primary system can
be transferred to the NES. It was demonstrated that a transient resonance capture on a 1:1 resonance
manifold [4, 5] of the system is at the origin of an irreversible and almost complete energy transfer
between 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 at
the 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 applied
to the primary system only. It was therefore assumed that transient bridging orbits fully compatible
with the initial conditions must exist in order to bring the motion into the domain of attraction of the
1:1 resonance manifold.
Energy transfers between weakly coupled nonlinear oscillators (or discrete breathers) having fast
oscillations at the same frequency have also been studied by Aubry and co-authors [6, 7] and have been
referred to as targeted energy transfers (TET). The theory has recently been extended for resonances
at higher order, termed Fermi resonances [8]. In these studies, the primary system and the NES are
described 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 the
donor 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 studies
dealing 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 of
vibrational energy.
The purpose of this study is to highlight and explain the vigorous energy transfers that may take place
in a linear oscillator weakly coupled to an essentially nonlinear attachment in the presence of viscous
dissipation. Although energy pumping is encountered during the transient dynamics of the damped
system, it is shown that the dynamics can be interpreted in terms of the periodic orbits of the underlying
Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the system
is constructed. The evolution of this frequency-energy plot with respect to the ratio between the NES
mass and the mass of the primary system is studied; the influence of the nonlinearity and of the weak
coupling stiffness is also discussed. Finally, the results of experimental measurements are shown that
support the theoretical developments.
2. Computation of a Frequency-Energy Plot
The system considered herein is composed of a linear oscillator, termed the primary system, coupled
to an essentially nonlinear (nonlinearizable) oscillator, termed the NES, and is depicted in Figure 1.
M ÿ + λ1 ẏ + λ( ẏ − v̇) + (y − v) + ky = 0
m v̈ + λ2 v̇ + λ(v̇ − ẏ) + (v − y) + Cv 3 = 0
(1)
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 1. All other variables are treated as
O(1) quantities; provided the input energy is high enough, a strongly nonlinear system is therefore
investigated.
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 ÿ + (y − v) + ky = 0
(2)
m v̈ + (v − y) + Cv 3 = 0
is performed using the method of nonsmooth temporal transformations (NSTTs). This method replaces
the temporal variable t by two nonsmooth variables
τ = (2/π) arcsin [sin (π t/2)],
e(t) = τ̇ (t),
e2 (t) = 1
(3)
and transforms the Equations (1) into a set of two nonlinear boundary value problems (NBVPs) to be
solved in a closed interval −1 ≤ τ ≤ 1. The method is not further detailed here, but the interested
reader 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, = 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 normal
modes (NNMs) [13]. The frequency-energy plot 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 the S11− and S11+ branches. The other branches (e.g., S21, U43, S13) are referred to
as tongues; each tongue is composed of two very close branches (e.g., S13− and S13+) that bifurcate
out 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+ and
S11− 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 being
arbitrary integers. However, for very low energy levels, the NES response is dominated by the weak
coupling spring; the ratio i/j must therefore be greater than the ratio of the two frequencies of the
underlying linear system, i/j > f 2 / f 1 , where
f1, f2 =
k+
+
±
2M
2m
−4km M + (−km − m − M)2
2m M
(4)
For M = m = k = C = 1, = 0.1, the ratio i/j must be greater than 0.3/1.0535 = 0.285. Thanks to
the 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 and
the NES. As an example, initial conditions y(0) = v(0) = v̇(0) = 0 and ẏ(0) = 5.59 correspond to
a periodic orbit on the U43 branch, and the displacements, together with the instantaneous percentage
of energy carried by the NES, are depicted in Figure 3. Energy flows back and forth between the two
oscillators, which is the characteristic of a nonlinear beating phenomenon. The energy transfer is not
complete, and only 20% of the energy can be carried by the NES. Similar nonlinear beating phenomena
can be observed on each tongue in the frequency-energy plot. It is important to note that no a priori
tuning of the NES is necessary, which is markedly different from other systems exhibiting nonlinear
beating phenomena (see, e.g., reference [9] or spring-pendulum systems).
286 G. Kerschen et al.
Figure 2. Frequency-energy plot of the periodic orbits (M = m = k = C = 1, = 0.1); transient bridging orbits, termed
special orbits, are denoted by black dots, and are connected by a dotted line; letters S and U refer to symmetric and unsymmetric
solutions of the nonlinear boundary value problem, respectively; we assign to a specific branch of solutions a frequency index
equal to the ratio of its indices, e.g., S21 is represented by the frequency index ω = 2/1 = 2, as is U 21 (this convention rule
holds for every branch except S11±, which, however, are particular branches, forming the basic backbone of the plot); f 1 and f 2
are the natural frequencies of the underlying linear system; f 3 is the natural frequency of the linear subsystem that governs the
dynamics at high energies (the nonlinearity behaves as a massless rigid link).
A third frequency, f 3 , is defined as the characteristic frequency of the oscillators on the S11+ branch
for infinite values of the energy. The nonlinear spring can be considered as infinitely stiff, and the system
behaves as a single degree of freedom with a linear spring of constant stiffness k+ and a mass equal to M
f3 =
k+
M
(5)
For M = m = k = C = 1, = 0.1, f 3 = 1.0488 rad/s and is smaller than f 1 = 1.0535 rad/s.
Accordingly, there exist two forbidden zones along the frequency axis where the motion cannot take place
f [0, f 2 ]
and
f [ f 3 , f 1 ]
(6)
Energy Transfers in a System of Two Coupled Oscillators 287
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 the
NES may occur in this seemingly simple system. At this point, it is noted that these energy transfers
can also take place between an NES and a primary system with multiple degrees of freedom [15] or an
infinity of degree of freedom [16].
Even though energy pumping can only occur in the damped system, we conjecture that damping does
not change radically the underlying mechanisms and that the transient dynamics of the weakly damped
oscillators 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 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 of the primary system, respectively.
Furthermore, the aspect ratio is set so that 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 in the
nonlinear 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
288 G. Kerschen et al.
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 a
consequence, 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. Viscous
dissipation facilitates transfer of energy from the linear oscillator to the NES. The key feature of this
energy transfer is its irreversibility as the motion is more and more localized in the NES when the energy
is dissipated. It is also almost complete as the shape of the NNM is close to a horizontal line for low energy
levels.
In summary, motion along the S11+ branch represents the basic mechanism for energy pumping, and
the underlying dynamical phenomenon is a 1:1 resonance capture. The two oscillators vibrate with the
same fast frequency, but this frequency varies in time with the amount of energy transferred. Energy is
irreversibly and almost completely transferred from the primary system to the NES.
Energy Transfers in a System of Two Coupled Oscillators 289
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 when
the 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 it
appears 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. This
condition is only realized asymptotically on the S11+ and S11− branches; i.e., for infinite and zero
values of the energy, respectively. The shape of the NNMs on these branches is thus generally not
compatible with the NES being initially at rest. As discussed in the introductory section, transient
bridging orbits fully compatible with the initial conditions must exist in order to bring the motion into
the domain of attraction of the 1:1 resonance manifold. These transient bridging orbits are referred to
as special orbits from now on.
All the periodic orbits previously computed using the method of NSTTs are inspected in order to
highlight those presenting a vertical line passing through the origin of the configuration space. Such
an orbit will be able to play the role of special orbit as it corresponds to y(0) = v(0) = v̇(0) = 0 and
ẏ(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 the
underlying 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 this
tongue pass through the origin of the configuration space. Due to the existence of a countable infinity of
tongues in the frequency-energy plot, it is reasonable to assume the existence of a countable infinity of
special orbits. This would represent an attractive feature as for almost any level of energy considered, a
special orbit would exist and could be excited. To verify this conjecture, a constraint which imposes a
zero initial velocity to the NES is considered when solving the set of two NBVPs provided by the method
of NSTTs. Each computed special orbit is represented using a black dot in Figure 2. It is emphasized
that 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 practical
importance as it enables one to predict where the motion is initiated in the frequency-energy plot for a
given value of the input energy. The dotted line only represents a convenient way to localize the special
orbits; there is no formal proof of its existence. Besides, this line should not be continuous because the
NES can only enter in a i : j internal resonance with the linear oscillator, with the restriction that i and
j are integers.
Some representative special orbits are illustrated in Figure 5, where the instantaneous percentage
of energy carried by the NES during the nonlinear beating is also shown. There are two families of
special orbits, namely those living on S tongues and those living on U tongues. The latter take the form
of 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.
290 G. Kerschen et al.
Figure 5. Special orbits.
Energy Transfers in a System of Two Coupled Oscillators 291
• The NES should vibrate faster than the linear oscillator as this will increase its kinetic energy. From
this standpoint, the special orbits living in the upper part of the frequency-energy plot are more
interesting.
This would allow a special orbit to transfer a greater amount of energy to the NES during one cycle
of the beating phenomenon. It appears that the special orbits living in the neighborhood of the special
orbit 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, = 0.1, λ = λ1 = λ2 = 0.2 are now performed
in order to validate the previous findings. The motion is first initiated from the special orbit of U43 and
the results are displayed in Figure 6. The plot in the middle, showing the instantaneous percentage of
energy carried by the NES, highlights that the initial nonlinear beating phenomenon (0–90 s) effectively
triggers nonlinear energy pumping (90–150 s) which, in turn, is responsible for an irreversible and
almost complete energy transfer to the NES.
The plot at the bottom is the superposition of the frequency-energy plot and the instantaneous frequency of the NES displacement computed using a wavelet transform. Shaded areas correspond to regions where the amplitude of the wavelet transform is high whereas lightly shaded regions correspond
to low amplitudes. This plot is a schematic representation as it superposes damped and undamped
responses 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 the
linear oscillator.
• The predominant component of the instantaneous frequency of the NES follows the backbone branch
for most of the signal. This validates our conjecture that the dynamics of the weakly damped system
can 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 ( ẏ(0) = 0.5742) and the U21
branch ( ẏ(0) = 32.46) and from an arbitrarily chosen point below the special orbit of the U43 branch
but above f 1 ( ẏ(0) = 3). Figure 7 shows the corresponding displacement signals. Although the special
orbit of S13 transfers 20% of the total energy to the NES, a transition to the S11+ branch cannot occur
as 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 less
energy to the NES but appears for much higher energies for which the NNMs on S11+ are localized
in the linear oscillator. This special orbit is able to bring the motion into the domain of attraction of
the 1:1 resonance manifold, but the evolution toward the region where the shape of the NNMs quickly
varies (i.e., when the energy is between 10−1 and 100 ) is slow; nonlinear energy pumping is activated
but takes time. The optimal scenario occurs for intermediate energy levels when the motion is initiated
from a special orbit below U43. As illustrated in the two bottom plots of Figure 7, the NES amplitude
grows 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 coupled
oscillators can occur only above a certain threshold of the input energy, and (b) there is an optimal
value of the input energy at which energy is quickly transferred to the NES, where it localizes and
diminishes in time due to damping. We note that this is in agreement with what was observed in a
previous theoretical analysis [2] and in experimental measurements [17].
292 G. Kerschen et al.
Figure 6. Motion initiated from the special orbit of U43 branch.
4. Parametric Study of the Energy Exchanges Between the Primary System
and the Nonlinear Energy Sink
The purpose of the previous section was to highlight the basic mechanisms responsible for energy
transfers between the primary system and the NES. However, no attempt was made to maximize these
energy exchanges. In the present section, analytic calculations are performed in order to understand the
Energy Transfers in a System of Two Coupled Oscillators 293
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 for
smaller 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 primary
system independent of the NES parameters m, C and . We investigate now how the motion on S11+
for very low energy levels is influenced by these parameters. For such energy levels, the nonlinear
294 G. Kerschen et al.
Table 1. Influence of the NES mass and the coupling spring on the
completeness of the energy transfer on the 1:1 resonance manifold.
NES mass m
Coupling spring Ratio V/Y
1
0.6
0.15
0.05
1
1
1
1
0.1
0.1
0.1
0.1
0.05
0.1
0.3
0.5
10.10
9.51
5.54
1.84
20.05
10.10
3.61
2.41
stiffness can be neglected, and the dynamics is mainly governed by the underlying linear system. A
modal analysis is thus carried out in order to compute the mode shape [V ; Y ] corresponding to the
lowest natural frequency, yielding
[V ; Y ] = 1;
M − m − km +
−4km M + (−km − m − M)2
2 M
(7)
The ratio V/Y gives an indication of the completeness of the energy transfer when the motion is captured
into the domain of attraction of S11+ and remains on this resonance manifold, the aim being to maximize
it. For example, in Figure 6, a prolonged 1:1 resonance capture takes place; the ratio V/Y is equal to
10.10, which implies that eventually the NES carries almost 100% of the total energy in the system. The
influence of the parameters m and is summarized in Table 1. The most complete energy transfers are
observed for the lowest coupling stiffness and the heaviest nonlinear attachment, a feature which may
limit the utility of this NES in practical applications where the total mass of the structure is an important
design criterion. As further evidence of the influence of the NES mass, a close-up of the S11+ branch
for 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 primary
system carry two harmonic components, one at frequency ω and the other at frequency 2ω. The complex
variables
1 = ẏ1 + jωy1 ,
3 = ẏ2 + 2 jωy2
2 = v̇1 + jωv1 ,
4 = v̇2 + 2 jωv2
(8)
are introduced, where y(t) = y1 (t) + y2 (t) and v(t) = v1 (t) + v2 (t) have been decomposed into their
two harmonic components. Hence,
y=
1 − 1∗
(3 − 3∗ )
+
2 jω
4 jω
(9)
Energy Transfers in a System of Two Coupled Oscillators 295
2 − 2∗
(4 − 4∗ )
+
2 jω
4 jω
jω
˙1 +
˙3 −
ÿ = (1 + 1∗ ) − 2 jω(3 + 3∗ )
2
˙2 +
˙ 4 − jω (2 + 2∗ ) − 2 jω(4 + 4∗ )
ẍ = 2
x=
(10)
(11)
(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 by
averaging over the fast frequencies ω and 2ω. The calculations are not detailed herein, as a similar
analysis has already been carried out in previous publications (see, e.g., references [2, 19]). Imposing
stationarity conditions on the equations on the slow flow leads to an approximation of the periodic orbits
on the U21 branch
y(t) = A sin ωt + B sin 2ωt
and v(t) = D sin ωt + E sin 2ωt
(13)
where
2 D= ± √
7mω2 − + 2 2 /Z 2 − 2 /Z 1
3 C
2 E=± √
−2mω2 − − 2 /Z 2 + 2 2 /Z 1
3 C
D
E
A=
B=
Z1
Z2
Z 1 = k − Mω2 + Z 2 = k − 4Mω2 + (14)
(15)
(16)
(17)
Two solutions exist as indicated by the presence of the ±sign, a confirmation that a tongue is composed
of two branches.
An analytic solution for the frequency of the special orbit on U21 can be computed by imposing zero
initial displacement and velocity to the NES; i.e., D = −2E. The expression is complicated and is not
given here. The energy stored in the primary system during the nonlinear beating phenomenon is
ẏ(t)2
y(t)2
+M
2
2
k( D)2 sin ωt
sin 2ωt 2 M( Dω)2 cos ωt
cos 2ωt 2
=
−
+
−
2
Z1
2Z 2
2
Z1
Z2
E prim = k
(18)
For a special orbit y(0) = v(0) = v̇(0) = 0 and ẏ(0) = 0, which means that at t = 0 the entirety of the
total energy is stored in the linear oscillator
E tot
M( Dω)2
= E prim (t = 0) =
2
1
1
−
Z1
Z2
2
(19)
After some rearrangement and normalization by the total energy, the percentage of energy carried by
the primary system, E prim, % = 100 E prim /E tot , is
E prim, % = 100
k
(2Z 2
4Mw 2
sin ωt − Z 1 sin 2ωt)2 + (Z 2 cos ωt − Z 1 cos 2ωt)2
(Z 2 − Z 1 )2
(20)
296 G. Kerschen et al.
Table 2. Influence of the NES mass and the coupling
spring on the maximum percentage of energy transferred to the NES during the nonlinear beating.
NES mass
m
Coupling
spring Maximum energy
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 maximum
energy carried by the NES during the nonlinear beating for various values of the parameters, as
E NES,% = 100 − E prim,% . The results are listed in Table 2 and illustrate that the NES mass has almost no influence on the amount of energy transferred to the NES. The same conclusion can be reached
for the nonlinear coefficient as it is absent from equation (20). On the other hand, increasing the stiffness
of the coupling spring seems to be beneficial. However, the increase should be limited as the most complete energy transfers on the 1:1 resonance manifold are observed for the lowest values of the coupling
stiffness.
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, using
the method of NSSTs. From Figure 8, we observe that the shape of the backbone branch is very
sensitive to this parameter. In particular, for the mass ratio equal to 0.05, (a) the two inflexion points
on the S11– branch – corresponding to saddle node bifurcations – disappear; (b) the S11+ branch
is 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, and
as a consequence, the number of lower tongues (i.e., tongues emanating from S11+) significantly
decreases.
5. Experimental Study
To support the previous theoretical findings, experimental measurements have been carried out using the
fixture depicted in Figure 9. This fixture realizes the system described by equations (1) and comprises
two cars made of aluminum angle stock which are supported on a straight air track. The primary system
of mass M is grounded by means of a linear spring k, and the NES of mass m is connected to the
primary system by means of a weak coupling stiffness . An essential cubic nonlinearity C is realized
by a thin wire with no pretension. To dissipate the energy, viscoelastic tape is added to the coupling
spring, realizing the damping constant λ. Some energy dissipation is also provided between the cars
and the air track, but the resulting damping constants λ1 and λ2 are smaller than λ. A long-stroke
Energy Transfers in a System of Two Coupled Oscillators 297
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 in
Figure 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. In
both cases, a special orbit is excited at time t = 0 resulting in a nonlinear beating phenomenon. This
can clearly be distinguished in the plot showing the instantaneous percentage of energy carried by the
NES. The nonlinear beating is capable of transferring energy from the initially excited primary system
298 G. Kerschen et al.
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 manifold. The envelope of both displacement signals decreases monotonically (i.e., no modulation can be
observed), but that of the NES decreases much more slowly than that of the primary system. This is
the sign that an energy transfer from the primary system to the NES has taken place. The instantaneous
percentage of energy carried by the NES further illustrates that this transfer is irreversible, at least until
Energy Transfers in a System of Two Coupled Oscillators 299
Figure 10. Experimental results, 13N excitation level. (a) and (b) Measured displacement signals; (c) instantaneous energy carried
by the NES; (d) Energy dissipated at the NES; (e) Wavelet transform of the NES displacement superposed to the frequency-energy
plot.
escape from resonance capture occurs. This latter regime is observed when the actual motion is no
longer compatible with the motion on the resonance manifold.
The superposition of the wavelet transform of the NES displacement and of the frequency-energy
plot is a very useful tool for the interpretation of the dynamics. First, it indicates that the system is
strongly nonlinear since the instantaneous frequency of the NES displacement varies with the energy
300 G. Kerschen et al.
Figure 11. Experimental results, 18N excitation level. (a) and (b) Measured displacement signals; (c) instantaneous energy carried
by the NES; (d) Energy dissipated at the NES; (e) Wavelet transform of the NES displacement superposed to the frequency-energy
plot.
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 the
periodic orbits of the underlying Hamiltonian system although these energy exchanges are encountered
during 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 absorbed
in the NES for the 13N and 18N force levels, respectively.
Energy Transfers in a System of Two Coupled Oscillators 301
6. Concluding Remarks
The purpose of this paper is to highlight and explain the vigorous energy transfers that may take place
in a linear oscillator weakly coupled to an essentially nonlinear attachment. Even though the transient
dynamics of the damped system have been considered, it has been demonstrated that a frequencyenergy plot gathering the periodic orbits of the underlying Hamiltonian system is a powerful tool for
understanding these energy exchanges.
Based upon this frequency-energy plot, two mechanisms responsible for energy dissipation have
been highlighted, namely the excitation of a transient bridging orbit (resulting in a nonlinear beating
phenomenon) and the capture into a 1:1 resonance manifold (resulting in an irreversible energy flow
from the primary system to the NES) with the former mechanism triggering the latter. All these findings
have been verified experimentally.
A parametric study of the energy exchanges between the primary system and the NES has also been
performed. Interestingly enough, it was proven that the nonlinear coefficient does not influence these
exchanges of energy. This study has also shown that the stiffness of the coupling spring must be weak
in 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 a
sufficient amount of energy to the NES during the beating. These are contradictory requirements which
might render the NES design challenging. The parametric study has also pointed out that relatively
heavy nonlinear attachments are to be considered which may represent a limitation when the structural
weight is an important issue.
It is therefore relevant to seek another configuration of the NES that overcomes this drawback. For
example, let us consider an ungrounded NES with a light mass connected to a primary system by means
of an essential nonlinearity. The equations of motion of the undamped system are
M ÿ + ky + C(y − v)3 = 0
v̈ + C(v − y)3 = 0
(21)
A light mass is assured by requiring that 1. All other variables are treated as O(1) quantities. For
convenience, we assume that M = k = 1
ÿ + y + C(y − v)3 = 0
v̈ + C(v − y)3 = 0
(22)
The linear change of variables y = αz + βw, v = δz + γ w is now applied. The values of α, β, δ and
γ are chosen so that the resulting equations of motion resemble equations (2) as closely as possible.
Doing so yields
y = (z − w),
v = z + w
(23)
and equations (22) become
(1 + )z̈ + (z − w) = 0
(1 + )4 3
(1 + )ẅ + (w − z) + C
w =0
(24)
302 G. Kerschen et al.
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
(1 + ) ÿ + (y − v) = 0
(1 + )4 3
(1 + )v̈ + (v − y) + C
v =0
(25)
These equations bear a strong resemblance to Equations (2). However, it is emphasized that the primary
system has no grounded linear spring which means that there is no complete equivalence between the
systems described by Equations (2) and (21).
Nevertheless, the comparison of Equations (22) and (25) shows that an NES characterized by a small
mass ratio connected to a primary system by means of an essential nonlinearity of O(1) corresponds
to an NES characterized by a great mass ratio 1/ connected to a primary system by means of a weak
coupling spring and connected to ground by means of a stiff essential nonlinearity approximately
equal to 1/. Because the system described by Equations (25) bears a strong resemblance to the system
investigated in the present paper and because it has a large mass ratio 1/ and weak coupling spring ,
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), to
be a light and ungrounded NES capable of eliminating undesired broadband disturbances. A numerical
simulation with M = k = C = 1 and = 0.05 is carried out in order to verify this prediction. The
constants of the grounded and coupling dashpots are set equal to 0.005. The displacement signals shown
in Figure 12 show that a significant amount of energy is indeed quickly transferred from the primary
system 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 author
G. 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 made
his visit to the University of Illinois possible is also gratefully acknowledged. The authors would like
to acknowledge Panagiotis Panagopoulos for his contributions to the problem.
Energy Transfers in a System of Two Coupled Oscillators 303
References
1. Vakakis, A. F., ‘Inducing passive nonlinear energy sinks in vibrating systems’, Journal of Vibration and Acoustics 123, 2001,
324–332.
2. Gendelman, O., Manevitch, L. I., Vakakis, A. F., and M’Closkey, R., ‘Energy pumping in nonlinear mechanical oscillators:
Part I – Dynamics of the underlying Hamiltonian systems’, Journal of Applied Mechanics 68, 2001, 34–41.
3. Vakakis, A. F. and Gendelman, O., ‘Energy pumping in nonlinear mechanical oscillators: Part II – Resonance capture’,
Journal of Applied Mechanics 68, 2001, 42–48.
4. Arnold, V.I., Dynamical Systems III (Encyclopaedia of Mathematical Sciences), Springer Verlag, Berlin, 1988.
5. Quinn, D., Rand, R., and Bridge, J., ‘The dynamics of resonance capture’, Nonlinear Dynamics 8, 1995, 1–20.
6. Aubry, S., Kopidakis, G., Morgante, A. M., and Tsironis, G. P., ‘Analytic conditions for targeted energy transfer between
nonlinear oscillators or discrete breathers’, Physica B 296, 2001, 222–236.
7. Kopidakis, G., Aubry, S., and Tsironis, G. P., ‘Targeted energy transfer through discrete breathers in nonlinear systems’,
Physical Review Letters 87, 2001, 165501.
8. Maniadis, P. and Aubry, S., ‘Targeted energy transfer by fermi resonance’, Physica D 202, 2005, 200–217.
9. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
10. Nayfeh, S. A. and Nayfeh, A. H., ‘Energy transfer from high- to low-frequency modes in a flexible structure via modulation’,
Journal of Vibration and Acoustics 116, 1994, 203–207.
11. Pilipchuk, V. N., ‘The calculation of strongly nonlinear systems close to vibration-impact systems’, PMM 49, 1985, 572–578.
12. Pilipchuk, V. N., Vakakis, A. F., and Azeez, M. F. A., ‘Study of a class of subharmonic motions using a non-smooth temporal
transformation’, Physica D 100, 1997, 145–164.
13. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. V., Pilipchuk, V. N., and Zevin, A. A., Normal Modes and Localization in
Nonlinear Systems, Wiley, New York, 1996.
14. Panagopoulos, P. N., Vakakis, A. F., and Tsakirtzis, S., ‘Transient resonant interactions of finite linear chains with essentially
nonlinear end attachments leading to passive energy pumping’, International Journal of Solids and Structures 41, 2004,
6505–6528.
15. Vakakis, A. F., McFarland, D. M., Bergman, L. A., Manevitch, L. I., and Gendelman, O., ‘Isolated resonance captures and
resonance capture cascades leading to single- or multi-mode passive energy pumping in damped coupled oscillators’, Journal
of Vibration and Acoustics 126, 2004, 235–244.
16. Vakakis, A. F., Manevitch, L. I., Musienko, A. I., Kerschen, G., and Bergman, L. A., ‘Transient dynamics of a dispersive
elastic wave guide weakly coupled to an essentially nonlinear end attachment’, Wave Motion 41, 2005, 109–132.
17. McFarland, D. M., Bergman, L. A., and Vakakis, A. F., ‘Experimental study of nonlinear energy pumping occurring at a
single fast frequency’, International Journal of Non-Linear Mechanics 40, 2005, 891–899.
18. Manevitch, L.I., ‘Complex representation of dynamics of coupled oscillators’, in Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems, Kluwer Academic/Plenum Publishers, New York, 1999, pp.
269–300.
19. Jiang, X., McFarland, D. M., Bergman, L. A., and Vakakis, A. F., ‘Steady state passive nonlinear energy pumping in coupled
oscillators: Theoretical and experimental results’, Nonlinear Dynamics 33, 2003, 87–102.