INVITED REVIEW
77
Passive non-linear targeted energy transfer and its
applications to vibration absorption: a review
Y S Lee1 , A F Vakakis2 , L A Bergman1∗ , D M McFarland1 , G Kerschen3 , F Nucera4 , S Tsakirtzis2 , and P N Panagopoulos2
1
Department of Aerospace Engineering, University of Illiois at Urbana-Champaign, Urbana, Illinois, USA
2
School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece
3
Aerospace and Mechanical Engineering Department (LTAS), Université de Liège, Liège, Belgium
4
Department of Mechanics and Materials, Mediterranean University, Reggio Calabria, Italy
The manuscript was received on 26 July 2007 and was accepted after revision for publication on 3 March 2008.
DOI: 10.1243/14644193JMBD118
Abstract: This review paper discusses recent efforts to passively move unwanted energy from a
primary structure to a local essentially non-linear attachment (termed a non-linear energy sink)
by utilizing targeted energy transfer (TET) (or non-linear energy pumping). First, fundamental theoretical aspects of TET will be discussed, including the essentially non-linear governing
dynamical mechanisms for TET. Then, results of experimental studies that validate the TET
phenomenon will be presented. Finally, some current engineering applications of TET will be
discussed. The concept of TET may be regarded as contrary to current common engineering
practise, which generally views non-linearities in engineering systems as either unwanted or, at
most, as small perturbations of linear behaviour. Essentially non-linear stiffness elements are
intentionally introduced in the design that give rise to new dynamical phenomena that are very
beneficial to the design objectives and have no counterparts in linear theory. Care, of course, is
taken to avoid some of the unwanted dynamic effects that such elements may introduce, such as
chaotic responses or other responses that are contrary to the design objectives.
Keywords: passive non-linear targeted energy transfer, vibration absorbtion
1
INTRODUCTION
Many studies have been made to suppress vibrational
energy from disturbances into a main system either
passively or actively since the seminal invention of the
tuned vibration absorber (TVA) by Frahm [1] (refer
to references [2] and [3] for a historical review of
passive/active TVAs and structural control methods,
respectively). With advances in electro-mechanical
devices, active control schemes are more likely to offer
the best performance in terms of vibration absorption.
However, in addition to issues of cost and energy consumption associated with active control, robustness
and stability need to be addressed.
∗ Corresponding
author: Department of Aerospace Engineering,
University of Illinois at Urbana-Champaign, Urbana, IL 61801,
USA. email: [email protected]
JMBD118 © IMechE 2008
Passive dynamic absorbers represent an interesting alternative. The classical TVA from Frahm [1]
has been extensively studied in the literature [4–7].
It is a simple and efficient device but is only effective in the neighbourhood of a single frequency.
Roberson [8] showed that broadening the suppression bandwidth is possible by employing a non-linear
system for the TVA. Since then, non-linear vibration absorbers have received increased attention in
the literature (e.g. continuously and discontinuously
non-linear [9, 10]; piecewise linear [11]; centrifugal pendulum [12]; and autoparametric vibration
absorbers [13, 14]). Although non-linearities are usually considered to be detrimental, it is possible to
take advantage of the richness and complexity of nonlinear dynamics for the design of improved vibration
absorbers.
Passive transfers of vibrational energy through
mode localization have been of particular interest in
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
solid-state, condensed-matter, and chemical physics.
For example, there are vibrational energy transfers
at gas–solid interfaces [15, 16]; thermally generated
localized modes and their delocalization in a strongly
anharmonic solid lattice such as quantum crystals
[17–19]; linear and non-linear exchanges of energy
between different components in coupled Klein–
Gordon equations [20]; and targeted energy transfer
(TET) between a rotor and a Morse oscillator presenting chemical dissociation [21]. A novel mechanism
was also proposed for inducing highly selective yet
very efficient energy transfers in certain discrete nonlinear systems where, under a precise condition of
non-linear resonance, when a specific amount of
energy is injected as a discrete breather at a donor
system it can be transferred as a discrete breather to
another weakly coupled acceptor system [22–25].
In applications to mechanical systems, localization
or confinement of vibrations, which is referred to as
normal mode localization, was studied in references
[26] to [29] by considering structural irregularity (or
disorder) in weakly coupled component systems. An
acoustical application of Anderson localization [30]
was demonstrated theoretically and experimentally
[31]. It was also shown that (non-linear) mode localization can occur in a class of multi-degree-of-freedom
(MDOF) non-linear systems even with perfect symmetry and a weakly coupled structure [32–38]. This
kind of standing wave localization, based on intrinsic localized modes (discrete breathers) or non-linear
normal modes (NNMS) which exist due to discreteness
and system non-linearity [39, 40], can be classified
as ‘static’ because it does not involve controlled spatial transfer of energy through the system. It can be
realized through appropriate selection of the initial
conditions [41].
Internal resonances (IRs) under certain conditions
also promote energy transfer between non-linear
modes [14, 42–45]. It was explained, both theoretically and experimentally, how a low-amplitude highfrequency excitation can produce a large-amplitude
low-frequency response (called energy cascading
[46]). However, in these cases, non-linear energy
exchanges are caused by non-linear modal interactions, and they do not necessarily involve controlled
TETs [41].
It is only recently that passively controlled spatial (hence ‘dynamic’) transfers of vibrational energy
in coupled oscillators to a targeted point where the
energy eventually localizes were studied [41, 47–50].
This phenomenon is called non-linear energy pumping or TET. This paper summarizes recent efforts
towards understanding passive TET. Some preliminaries and literature reviews are presented in section 2;
then, theoretical and experimental fundamentals on
non-linear TET phenomena are summarized, respectively, in sections 3 and 4.
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
2 TARGETED ENERGY TRANSFER (OR
NON-LINEAR ENERGY PUMPING)
2.1
Preliminaries
Non-linear energy pumping (or passive TETs) refers
to one-way targeted spatial transfers of energy from
a primary subsystem to a non-linear attachment; it
is realized through resonance captures and escapes
along the intrinsic periodic solution branches [41, 50].
Because of the invariance property of the resonance
manifold, the energy transfers become irreversible
once the dynamics is captured into resonance.
The non-linear device, which is attached to a primary system for passive energy localization into itself,
is called a non-linear energy sink (NES). An NES
generally requires two elements: an essentially nonlinear (i.e. non-linearizable) stiffness and a (usually,
linear viscous) damper. The former enables the NES
to resonate with any of the linearized modes of the
primary subsystem, whereas the latter dissipates
the vibrational energy transferred through resonant
modal interactions. The NES can be categorized
as grounded versus ungrounded, single-degree-offreedom (SDOF) versus MDOF, and smooth versus
non-smooth, depending on its design and use.
Figure 1 depicts a schematic of passive and broadband TETs utilizing an ungrounded SDOF NES. A
primary structure is given (usually a linear system
and, hence, the mass, damping, and stiffness matrices,
M, C, K, respectively). The primary structure, which
(k)
possesses a set of natural frequencies {ωPrimary
}k=1,...,N
where N is the number of DOFs of the primary structure, can suffer various external disturbances such
as impact loading, periodic or random excitation,
fluid-structure interaction, etc.
One seeks to (passively) eliminate such unwanted
external disturbances induced in the primary structure by attaching a simple non-linear device such
as an NES. Because an NES does not possess any
preferential resonance frequency (i.e. it has no linear
Fig. 1
Schematic of passive and broadband TETs
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Passive non-linear TET and its applications
stiffness term), it can generate a countably infinite
number of non-linear resonance conditions (i.e. IRs,
(k)
mωPrimary
≈ nωNES where m, n are integers), through
which vigorous energy exchanges occur between the
two oscillators. In particular, energy localization to the
NES is preferred for efficient mitigation of the disturbances in the primary structure. During the non-linear
modal interactions, energy is dissipated in the NES
damper. As the total energy decreases, self-detuning
is possible with the dynamics escaping from one resonance manifold to another. There are at least three
different TET mechanisms, which occur through 1:1
and subharmonic resonance captures, and are initiated by non-linear beat phenomena, respectively (see
section 3).
Although an NES device looks similar to a linear
dynamic absorber (or a TVA) in configuration (both
are passive and composed of a mass, a spring, and
a damper), they are totally different in nature. A TVA
operates effectively in a narrow band of frequencies,
and its effect is most prominent in the steady-state
regime. Therefore, even if the TVA is initially designed
(tuned) to eliminate resonant responses near the natural frequency of a primary system, the mitigating
performance may become less effective over time due
to aging of the system, temperature or humidity variations and so forth, thus requiring additional adjustment or tuning of parameters (i.e. the robustness can
be questioned). On the other hand, the NES is basically
a device that interacts with a primary structure over
broad frequency bands; indeed, since the NES possesses essential stiffness non-linearity, it may engage
in (transient) resonance capture with any mode of the
primary system (provided, of course, that a node of
the mode is not at the point of attachment of the
NES). It follows that an NES can be designed to extract
broadband vibration energy from a primary system,
engaging in transient resonance with a set of ‘most
energetic’ modes. Thus, the NES is more robust than
the TVA [51].
2.2
Literature review
Resonance capture (or capture into resonance), which
turns out to be a fundamental mechanism for
non-linear TET, has been studied in various fields
(e.g. physics [52–54]; aerospace engineering [55–59])
and originated as a consequence of the averaging
theorem [60–63]. Applications of resonance capture to
mechanical oscillators can also be found in references
[64] to [66].
Recently, resonance capture was applied to suppress unwanted disturbances in practical engineering problems. In this section, efforts to understand
passive TET in coupled oscillators are summarized
chronologically and are grouped according to system
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configurations (SDOF, MDOF or continuous primary
systems; grounded or ungrounded and SDOF or
MDOF NESs).
2.2.1
Grounded NES configurations
Gendelman and Vakakis [47] first investigated how
non-linear localization in coupled oscillators is progressively eliminated by a dissipative force. A strongly
non-linear oscillator with symmetry was studied by
computing and then matching separate analytical
approximations for the early (localized) and late (nonlocalized) responses (see also reference [67] for a linear
oscillator coupled to a strongly non-linear attachment
with multiple equilibrium states). It was shown that
a damped vibrational system can exhibit localization
phenomena at least at the early stages of the motion. In
later stages of the motion, non-linear effects diminish
and a transition from non-linear localized to linearized
weakly non-linear oscillations occurs as energy is dissipated. It was noted that, in a system with symmetry,
IRs between subsystems exist leading to linearized
beat phenomena which eliminate localization in the
linearized regime. Applicability of active control to
compensate for dissipation effects was addressed,
keeping the localized motion preserved in the system
as energy decreases (see, for example, reference [68],
which suggested a control algorithm for switching
mechanical components such as springs and dampers
on and off during their work with minimal energy
consumption).
Inducing passive NESs in vibrating systems was
studied in reference [48], where a complexificationaveraging technique was introduced to obtain modulation equations for the slow-flow dynamics. It was
shown that, for an impulsively loaded MDOF chain
with an NES attached at the end, the response of the
NES after some initial transients is motion dominated
by a fast frequency identical to the lower bound of the
propagation zone of the linear chain, which reduces
the study of TET in the chain to a two-DOF equivalent
problem. This is because, after some initial transients,
the semi-infinite chain in essence vibrates in an inphase mode at the lower frequency boundary of the
propagation zone of the infinite linear chain. Possible applications of TET to electric power networks [69]
were suggested for passive fault arrest in the network,
preventing catastrophic failure due to unchecked fault
propagation.
Similarly, energy transfer to a non-linear localized
mode in a highly asymmetric system was investigated [49]. It was shown that excitation of a NNM
[70] occurs via the mechanism of subharmonic resonance. The conditions for TET were suggested: (i) a
localized resonant mode should be excited; and (ii)
the vibrations of a non-linear oscillator should be
damped faster than the primary system with the same
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
damping terms of the same order. The shortcomings
of this passive vibration absorber were noted; that
is, it is not activated below a critical amplitude, and,
moreover, its effectiveness is reduced as the amplitude grows above the critical resonant regime because
the non-linear oscillator cannot absorb more than a
given amount of energy at a certain frequency. It was
observed that cubic stiffness coupling between the primary structure and the NES is much more effective
than linear coupling because the main mechanism of
energy transfer becomes a non-linear parametric resonance (see also reference [71] for numerical evidence
of TET phenomena in various structures).
Dynamics of the underlying Hamiltonian system
and non-linear resonance phenomena were investigated to understand energy pumping in a two-DOF
non-linear coupled system with a linearly coupled
grounded NES being one of the DOF [41, 50]. Actionangle formulation was utilized as a reduction method
at a fixed energy level to obtain a single second-order
ordinary differential equation; then, non-smooth temporal transformations (NSTTs [72]) of the reduced
equation were performed to compute its periodic
solutions. It was shown that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is
mainly responsible for the TET phenomenon, and that
this orbit cannot be excited at sufficiently low energies. Hence, a transient bridging orbit satisfying zero
initial conditions must be impulsively excited. Furthermore, introducing action-angle transformations,
and applying the averaging theorem to get a twofrequency dynamical system, it was shown that the
energy pumping phenomenon in the system studied
in that work is associated with resonance capture in a
neighbourhood of the 1:1 resonance manifold.
The degenerate bifurcation structure of a system
of coupled oscillators with an NES was studied [73],
where two types of bifurcations of periodic solutions were observed: (i) a degenerate bifurcation at
high energy (i.e. bifurcation from infinity); and (ii)
non-degenerate bifurcation near the exact 1:1 IR. It
was noted that the degeneracy occurs when the linear coupling stiffness approaches zero, in which case
the linear part of the equations of motion possesses a
double zero and a conjugate pair of purely imaginary
eigenvalues (i.e. a codimension-3 bifurcation occurs).
Bifurcation of damped NNMs for 1:1 resonance
was studied by combining the invariant manifold
approach and multiple-scales expansion [74]. It was
noted that there is a special asymptotical structure
distinct between three time-scales: (i) fast vibration;
(ii) evolution of the system towards the NNM; and (iii)
time evolution of the invariant manifold. It was also
found that time evolution of the invariant manifold
may be accompanied by bifurcations, and passage of
the invariant manifold through bifurcations may bring
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
about destruction of the resonance regime and essential gain in the energy dissipation rate. The damping
coefficient should be chosen to ensure the possibility
of bifurcation of the NNM invariant manifold, because
failure to do so will result in a loss of NES ability to
dissipate the vibrational energy.
Robustness of TET was examined by introducing
uncertain parameters (due to aging, imperfection
in design, and so on) to the NES [75]. Polynomial
chaos expansions were used to obtain information
about random displacements, followed by a numerical
parametric study based on Monte Carlo simulation.
The design of mechanical TET devices was considered in reference [76], where the complexificationaveraging technique and the method of multiplescales were utilized for analysing TET. Also, the issue
of designing a linear structure (specifically, a two-DOF
linear chain) linearly coupled to a grounded NES was
studied for enhancing TET [77]. Expressing the actual
DOFs connected to the NES as modal coordinates, and
assuming no IRs between uncoupled linear modes,
the physical aspects of non-linear TET were studied. It
was revealed that damping is a prerequisite for energy
pumping because non-linear TET is caused by the
excitation of a damped NNM invariant manifold that
is an analytic continuation of a NNM of the underlying
undamped (i.e. Hamiltonian) system. A more general
linear substructure (an MDOF chain) was considered
in reference [78], where a similar modal expression
was utilized to obtain the first version of a frequency–
energy plot (FEP). For the MDOF primary structure
coupled to an NES, resonance capture cascades were
demonstrated when TET occurs.
Single- and multi-mode energy pumping phenomena were investigated in a two-DOF primary structure
linearly coupled to a grounded NES [79]. Isolated resonance captures leading to single-mode energy pumping occur in neighbourhoods of only one of the linear
modes of the primary structure and are dominated by
the corresponding linearized eigenfrequencies (which
act as fast frequencies of the dynamics). However,
multi-mode energy pumping is caused by resonance
capture cascades that involve more than one linear
mode, and pumping dynamics are partitioned into
different frequency regimes with each regime being
dominated by a different fast frequency close to an
eigenfrequency of the linear system. Such resonance
capture cascades can be clearly depicted in appropriate FEPs, which follow the damped transitions close
to branches of the underlying Hamiltonian system as
energy decreases due to damping dissipation.
Dynamic interaction of a semi-infinite linear chain
with an NES coupled at the end was investigated [80].
Energy propagation through traveling waves, with predominant frequencies inside the propagation zone
exciting families of localized standing waves situated
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Passive non-linear TET and its applications
inside the lower or upper attenuation zones, were
analysed. Transient dynamics of a dispersive semiinfinite linear rod weakly connected to a grounded
NES was investigated [81]. By means of a Green’s function formulation, which reduces the dynamics to an
integro-differential equation in the form of an infinite set of ODEs using Neumann expansions, resonant
interaction of the NES with incident traveling waves
propagating in the pass-band of the rod was examined.
Resonance capture phenomena were also investigated
where the NES engages in transient 1:1 IR with the
in-phase mode of the rod at the bounding frequency
of its pass and stop bands, which are similar to resonance capture cascades in finite-chain non-linear
attachment configurations.
2.2.2
Ungrounded NES configurations
Apart from reference [49], dynamics of coupled linear and essentially non-linear oscillators with substantially different masses was investigated [82]. Two
mechanisms of energy pumping were examined: (i)
through 1:1 resonance capture and (ii) non-resonant
excitation of high-frequency vibration of the NES. It
was noted that an ungrounded NES configuration can
be transformed to a grounded one through change
of variables, so no further analysis for the former is
required.
An ungrounded NES configuration with essential
(non-linearizable) cubic stiffness non-linearity coupled to a primary structure was investigated more rigorously in references [83] and [84]. Unlike a grounded
NES, the ungrounded configuration eliminates the
restriction of relatively heavy mass of the non-linear
attachment, thus possessing the feature of simplicity.
Lee et al. [83] revealed a very complicated bifurcation structure of symmetric and unsymmetric periodic
solutions of the underlying undamped system on a
FEP by solving using a shooting method, the twopoint non-linear boundary value problem (NLBVP)
formulated through suitable NSTTs based on the
two eigenfunctions of a vibro-impact (VI) problem.
Some important solution branches of 1:1 and subharmonic resonance manifolds, as well as of nonlinear beating, are examined analytically through
the complexification-averaging technique in terms of
mode localization. Then, the transient dynamics of the
lightly damped system was clearly shown on the FEP
by superimposing wavelet transforms (WTs) of the relative displacement between the primary structure and
the NES.
Furthermore, three distinct pumping mechanisms
were identified [84]. The first mechanism, fundamental TET, is realized when the dynamics takes place
along the in-phase, 1:1 resonance manifold occurring
at the frequency domain less than the lower bound
of the eigenfrequency of the linear mode. The second,
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subharmonic TET, is similar to the fundamental mechanism except that it occurs along the subharmonic
solution branches on the FEP. Finally, the third is initiated by non-linear beating, leading to stronger TET by
exciting a special (or impulsive) periodic orbit.
Impulsive periodic orbits, as well as quasi-periodic
orbits, were analysed by separately considering low-,
moderate-, and high-energy impulsive motions [85].
Analytical approximations of impulsive periodic
orbits, which are separated by corresponding uncountable infinities of quasi-periodic impulsive orbits
(IOs), were performed. It was shown that the impulsive dynamics of the system is very complex due to
its high degeneracy as it undergoes a codimension-3
bifurcation (indeed, the equations of motion for the
ungrounded NES configuration can be transformed
to those for a grounded NES configuration as in
reference [73]).
Robustness of TETs in coupled oscillators due to
changes of initial conditions was examined in reference [51]. The problem of choosing appropriate
initial conditions for achieving efficient TET in a system of coupled oscillators with an ungrounded NES
was investigated by adopting a simplified description
of the dynamic flow at the initial stage of motion.
The analysis is complementary to the invariant manifold approach of reference [74]. Optimization of the
(grounded) NES parameters for TET was considered in
reference [86], where an experimental verification was
performed for a reduced-scale building model with the
NES located at the top floor.
Similar to references [77] to [79], multi-modal TETs
from a two-DOF primary structure to an ungrounded
NES were studied theoretically [87]. The main backbone curves on the FEP were computed analytically by
utilizing the complexification-averaging technique (or
numerically, using optimization techniques). Mode
localization phenomena were depicted along the three
main backbones, and transient dynamics of the lightly
damped system was investigated for in-phase and
out-of-phase impulsive forcing (not surprisingly, there
exist more one-dimensional manifolds of special periodic orbits (SPOs)). Again, WT results were superimposed on the FEP to demonstrate branch transitions
as the total energy decreases. Complex dynamics in a
two-DOF primary structure coupled to an MDOF NES
were investigated in reference [88], where strong passive TET capacity (up to as much as 90 per cent of input
energy) was identified.
Transient resonance captures (TRCs) in finite linear chains, respectively, coupled to a grounded SDOF
NES and to an ungrounded MDOF NES were compared in reference [89], where the dynamics governing the chain-NES interaction was reduced to a
single, non-linear integro-differential equation that
exactly describes the transient dynamics of the NES.
Approximations based on Jacobian elliptic functions
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
[90] yielded an approximate set of two non-linear
integro-differential modulation equations for ampli√
tude and phase, and perturbation analysis in a O( )
neighbourhood of a 1:1 resonant manifold were performed. For the MDOF NES, there were no detectable
resonance capture cascades, but simultaneous multimodal resonant interactions were found instead,
which suggested robust and wide applicability of TET
to many engineering problems such as vibration and
shock isolation, packaging, seismic mitigation, disturbance isolation of sensitive devices during launch of
payloads in space, flutter suppression, and so forth.
Similar work can be found in reference [91], where
instantaneous frequencies of the primary structure
and NES displacements were, respectively, estimated
through the Hilbert transform.
Broadband energy exchanges between a dissipative
elastic rod and a lightweight ungrounded SDOF NES
[92, 93], as well as an MDOF NES [94], were investigated rigorously. In particular, simultaneous (but
not necessarily sequential) TRCs with the MDOF NES
were demonstrated on a FEP utilizing empirical mode
decomposition (EMD [95]). Contrary to an SDOF NES,
which is sensitive to the external shock (or input
energy) level, the MDOF NES in the parameter ranges
of its high efficiency exhibits robustness to changes
in the amplitude of the applied shock, the coupling
stiffness, and the non-linear springs.
2.2.3
Experimental studies
An experimental study of non-linear TET occurring at
a single fast frequency in the system considered under
impulsive excitation on the primary structure was performed in reference [96]. All the previously predicted
analytical aspects were verified through experiments;
in particular, an input energy threshold to bring about
energy pumping was clearly depicted on the plot of
energy dissipation in the NES versus input energy.
Kerschen et al. [97] also experimentally showed that
non-linear energy pumping caused by 1:1 resonance
capture is triggered by the excitation of transient bridging orbits compatible with the NES being initially at
rest, a common feature in most practical applications [41]. Some interesting observations were made
through a parametric study of the energy exchanges
between the primary structure and the (grounded)
NES: (i) the non-linear coefficient does not influence
the energy pumping (see also the bifurcation analysis [98]); (ii) the linear coupling spring must be weak
in order to have an almost complete energy transfer
to the NES along the 1:1 resonance manifold; (iii) the
stiffness should be chosen high enough to transfer a
sufficient amount of energy to the NES during nonlinear beating; and (iv) relatively large mass for the
NES should be considered for better energy transfers
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(see reference [82]). An indirect analytical comparison via coordinate transformation suggested that
the ungrounded NES configuration can eliminate the
restriction on the large mass requirement of the nonlinear attachments, which was also experimentally
demonstrated in reference [99].
Transient resonance captures were experimentally
demonstrated [100] by means of EMD [95]. In particular, the EMD is a very useful tool for experimental
studies (i.e. system identification [101, 102]) where the
system information is not given a priori.
The theoretical work for a two-DOF primary structure coupled to an unground SDOF NES [87] was
experimentally verified by comparison with numerical simulations. Experimental studies demonstrated
the usefulness of the FEP for interpreting TET mechanisms; moreover, isolated resonance captures and
resonance capture cascades were also observed.
2.2.4
Applications
Application of NESs to shock isolation was first
demonstrated in references [103] and [104]. Essentially non-linear stiffness elements were used for
robust energy pumping at a sufficiently fast timescale, because fast energy pumping at the early stage
is crucial for shock isolation purposes. In particular,
adding two symmetrically placed NESs makes it possible to achieve dual mode shock isolation to reduce
unwanted disturbances generated at different ends of
the primary system. It was noted that, due to their
modular form, the NESs can be added locally in an
otherwise linear system in order to globally alter the
dynamics in a way compatible to the design objectives.
Dual mode non-smooth (piecewise linear) NESs were
also utilized for the purpose of shock isolation [105].
Furthermore, steady-state TET from an SDOF linear primary structure under sinusoidal excitation to
an attached NES was demonstrated theoretically and
experimentally [106]. A linear oscillator coupled to an
ungrounded NES was considered in references [107]
and [108], and was transformed by proper change of
variables to a system similar as the one studied in
reference [106]. It was shown that the damped dynamics exhibits a quasi-periodic vibration regime rather
than a steady-state sinusoidal response, a regime associated with attraction of the dynamical flow to a
damped-forced NNM manifold (for a more advanced
analysis, refer to references [109] and [110]). Experiments were also performed on an equivalent electric
circuit (see also reference [111] for energy pumping
under transient forcing).
Application of TET for suppressing self-excited
instabilities was examined. Suppression of limit cycle
oscillations (LCOs) in the van der Pol (VDP) oscillator by means of non-linear TET was studied in
reference [112]. The VDP oscillator exhibits dynamics
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Passive non-linear TET and its applications
analogous to non-linear aeroelastic instability. By
studying the slow-flow dynamics, extracted through
the complexification-averaging technique, and performing numerical continuation of equilibria and
limit cycles, bifurcation structures of LCOs and the
possibility of robust LCO suppression were parametrically investigated. It was concluded that a steady-state
is reached through a series of TRCs that can be clearly
represented in a FEP. In particular, it was demonstrated
that, in order to suppress instability in the VDP oscillator, a sequence of superharmonic and subharmonic
resonant interactions between the VDP oscillator and
the NES must take place.
The triggering mechanism of aeroelastic instability in a two-DOF (heave and pitch), two-dimensional
rigid wing under subsonic quasi-steady aerodynamics was examined in reference [113]. It was found that
the LCO-triggering mechanism consists of three different dynamic phenomena: a series of TRCs, escapes
from these captures and, finally, entrapment into permanent resonance capture (PRC). An initial excitation
of the heave mode by the flow acts as the trigger of
the pitch mode through a series of non-linear modal
interactions. Moreover, both the initial triggering and
full development of LCOs are transient phenomena,
so that one can properly design an NES attachment to
the wing for their suppression.
Based on these observations, an ungrounded SDOF
NES was applied to the two-DOF rigid wing, and
suppression of aeroelastic instability through passive
TETs was investigated both theoretically [114, 115] and
experimentally [98]. Three distinct suppression mechanisms were identified: (i) recurring suppressed burstouts, (ii) intermediate, and (iii) complete elimination
of aeroelastic instability. Those suppression mechanisms were identified with the bifurcation structure
of LCOs obtained through a numerical continuation
technique. Furthermore, the robustness of the aeroelastic instability suppression was examined. In order
to enhance robustness of aeroelastic instability suppression, the MDOF NES first considered in references
[89] and [94] was considered instead of the SDOF
NES. Bifurcation analysis showed that robustness of
instability suppression by means of simultaneous
multi-modal resonant interactions due to the MDOF
NES can be greatly enhanced, with a much smaller
total mass of the MDOF NES. Non-linear modal
energy exchanges were studied for various parameter
conditions.
Seismic mitigation of a reduced two-DOF model
[86, 111] and of an MDOF model [71], with an NES on
the top floor, was studied. Since an NES with smooth
stiffness non-linearities is not suited to suppress the
peak seismic responses at the critical early regime of
the motion, alternative non-smoothVI NESs were considered in references [116] to [118]. Effective seismic
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mitigation through the use of VI NESs was demonstrated both numerically and experimentally in these
works.
Other applications of passive TETs include suppression of stick-slip self-excited vibrations in a drill-string
problem [119], and acoustic energy pumping [120].
2.3
Useful definitions
In this section, concepts of resonance captures associated with the averaging theorem are reviewed to
support the discussion of non-linear TET that follows.
Definition 1 (Resonance Manifold [121])
Consider the system in polar form with multi-phase
angles
r = R(φ, r),
φ = Ω(r)
(1)
where r ∈ Rp , φ ∈ T q (generally, q p), (r) =
(1 (r), . . . , q (r)), and the dimension of r may be
greater than that of the original dynamical system
depending on frequency decompositions. The set of
points in D ⊂ Rp where i (r) = 0, i = 1, . . . , q is called
the resonance manifold. This resonance condition
is not sufficient; that is, if each i (r), i = 1, . . . , q is
away from zero, the IR manifold is defined as the set
{r ∈ Rp :< k, (r) 0, k ∈ Zq } where the corresponding Fourier coefficients from R(φ, r) are not identically
zero.
Assume that the averaged system of equation (1)
intersects transversely the resonant manifold. Then,
capture into resonance may occur for some phase
relations satisfying the condition that an orbit of
the dynamical system reaching the neighbourhood of
the resonant manifold continues in such a way that the
commensurable frequency relation is approximately
preserved. In this situation, not all phase angles are
fast (time-like) variables, so classical averaging cannot
be performed with regard to these angle variables. As
a result, over the time-scale −1 the exact and averaged solutions for equation (1) diverge up to O(1)
[60, 122, 123].
Definition 2 (Sustained and transient resonances
[124])
Suppose that (internal) resonance occurs at a time
instant t = t0 , with the non-trivial frequency combination σ = k1 ω1 + k2 ω2 + . . . + kq ωq , ki ∈ Z, i = 1, . . . , q,
vanishing at that time instant (t = t0 ). Then, sustained
resonance is defined to occur when σ ≈ 0 persists for
times t − t0 = O(1). On the other hand, transient resonance refers to the case when σ makes a single slow
passage through zero.
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Definition 3 (Capture, escape, and pass-through [64])
Definition 4 (IR, TRC, and PRC [61])
The possible behaviour of trajectories near the resonance manifold on the time-scale −1 is described
according to the following three cases: (i) capture:
solutions are unbounded in backward time. However, captured trajectories remain bounded for forward times of O( −1 ), i.e. a sustained resonance
exists in forward time; (ii) escape: solutions grow
unbounded in forward time. However, in backward
time, solutions remain bounded for times of O( −1 ),
i.e. a sustained resonance exists in backward time;
(iii) pass-through: solutions do not remain in the
neighbourhood of the resonance manifold in either
forward or backward time. No sustained resonance
exists.
Consider an unforced n-DOF system whose linear natural frequencies are ωk , k = 1, . . . , n. The author (i) IR
as motions for which there exist ki ∈ Z, i = 1, . . . , n,
such that k1 ω1 + k2 ω2 + · · · + kn ωn ≈ 0, i.e. some combination of linear natural frequencies satisfy commensurability; (ii) TRC as capture into a resonance
manifold which occurs and continues for a certain
period of time (e.g. on the time-scale −1 ) and then
finally involves transition to escape. This includes sustained resonance captures involving escape; (iii) PRC
as sustained resonance captures that will never escape
for increasing time.
A mechanism for resonance capture in perturbed
two-frequency Hamiltonian systems was studied by
Burns and Jones [61] where the most probable mechanisms for resonance capture were shown to involve
an interaction between the asymptotic structures of
the averaged system and a resonance. It was further
shown that, if the system satisfies a less restrictive
condition (or Condition N [125]) regarding transversal intersection of the averaged orbits to the resonance
manifold, resonance capture can be viewed as an
event with low probability, and passage through resonance is the typical behaviour on the time-scale
O( −1 ).
Necessary conditions were proved by Kath [56]
both for entrainment to sustained resonance and for
its continuance (and thus the possible indication of
unlocking or escape from the sustained resonance
after a finite time) by successive near-identity transformations; a sufficient condition was also derived
for continuation of sustained resonance by means of
matched asymptotic expansions [57].
On the other hand, transition to escape was studied by Quinn [65] in a coupled Hamiltonian system consisting of two identical oscillators possessing
a homoclinic orbit when uncoupled. Focusing on
intermediate energy levels at which sustained resonant motion occurs, the existence and behavior of
those motions were analysed in equipotential surfaces
whose trajectories are shown to remain in the transiently stochastic region for long times and, finally,
to escape or leak out of the opening in the equipotential curves and proceeding to infinity. Regarding
passage through resonance, one may refer to references [126] to [128]. The phenomenon of passage
through resonance is sometimes referred to as nonstationary resonances caused by excitations having
time-dependent frequencies and amplitudes [129].
Finally, the following definitions for non-linear resonant interactions between modes are introduced
when the multi-frequency components of a system are
taken into account.
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
Both TRC and PRC may occur along the IR manifold and are distinguished by whether or not they
involve an escape. Both IR and PRC may show similar steady-state behaviours, which differ from the
commensurability condition between linear natural
frequencies. Hereafter, a m : n IR refers to a condition
on the slow-flow averaged system unless noted otherwise. For more details on the averaging theorem and
resonance captures in multi-frequency systems, one
can also refer to references [60], [62], [63], [125], [130],
and [131].
2.4
2.4.1
Analytical and numerical tools
Perturbation methods
There are many perturbation techniques to compute
periodic solutions of a non-linear system: the methods of multiple-scales, of averaging, and of harmonic
balance [132]. Although each of these methods has
its own features, they are fundamentally equivalent to
each other. One restriction to their application is the
assumption of weak non-linearity; that is, the derived
analytical solutions of the non-linear system lie close
to those of the corresponding linearized system. The
averaging theorem provides validity of the approximation, generally up to the time-scale −1 . Although the
harmonic balance method (HBM) can be applied to
strongly non-linear systems, it approximates only the
steady-state responses. On the other hand, the methods of averaging and of multiple-scales can be applied
to the study of transient dynamic behaviour, which
is suitable for understanding nonlinear TET phenomena. An application of the averaging method to the
resonance capture problem can be found in reference
[133] (and see [134] for the HBM).
Since the essentially non-linear coupling between a
primary system and an ungrounded NES is not necessarily weak, the complexification-averaging technique
first introduced by Manevitch [135] will be utilized
in the following analysis as an analytical tool for
understanding resonance capture phenomena. Use of
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Passive non-linear TET and its applications
complex variables renders relatively easier manipulation of the resulting modulation equations (particularly, in the presence of multi-frequency components).
In addition, this method is applicable to strongly
nonlinear systems. For some analyses, the multiplescale method is utilized instead of averaging (e.g.
[49, 73, 74]).
2.4.2
Non-smooth temporal transformations
Non-smooth time transformations (NSTTs) can also
be utilized to compute periodic solutions of a
(strongly) non-linear system [72, 136–139]. Unlike the
usual perturbation methods that implement the basis
of sine and cosine functions (or elliptic functions in
some cases), the NSTTs employ saw-tooth and square
wave functions as the basis (see reference [140] for
other types of non-smooth basis functions and their
applications). Any periodic solutions can be expressed
in terms of asymptotic series expansion of these two
non-smooth functions; moreover, this technique can
be applied to solutions of a discontinuous system such
as a VI oscillator.
Application of NSTTs to the problem of computing
the periodic solutions of a dynamical system yields
NLBVPs, which are solved by means of numerical
schemes such as the shooting method [141].
2.4.3
Stability evaluation and bifurcation analysis
Once periodic solutions are obtained, their stability
can be evaluated: (i) by direct numerical integration of
equations of motion; (ii) by computing their Floquet
multipliers [142]; or (iii) by studying the topological
structure of numerical Poincaré maps [143]. Then,
bifurcation diagrams can be constructed with respect
to control parameters, or other induced parameters
such as the total energy of the system.
Bifurcation analysis [144] of periodic solutions in
a coupled oscillator is crucial in order to understand
transitions that occur in the damped dynamics or
to enhance robustness of instability suppression by
means of passive TETs. Methods of numerical continuation of equilibria and limit cycles can be utilized. In
particular, AUTO [145] and MatCont [146] can easily
be employed.
2.4.4
Time–frequency analysis
Understanding transient modal interactions during
non-linear TET requires an integrated time–frequency
analysis [147–151]. The most popular techniques
include the EMD method and the WT. WTs have
found applications in non-linear system identification, e.g. characterization of structural non-linearities
and prediction of LCOs of aeroelastic systems [152];
free vibration analysis of non-linear systems [153];
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damage size estimation or fault detection in structures
[154, 155].
TheWT can be viewed as a basis for functional representation but is at the same time a relevant technique
for time–frequency analysis. In contrast to the Fast
Fourier Transform (FFT), which assumes signal stationarity, the WT involves a windowing technique with
variable-sized regions. Small time intervals are considered for high-frequency components, whereas the
size of the interval is increased for lower frequency
components, thereby giving better time and frequency
resolutions than the FFT.
The Matlab codes used for the WT computations
in this paper were developed at the Université de
Liège (Liège, Belgium) by Dr V. Lenaerts in collaboration with Dr P. Argoul from the Ecole Nationale
des Ponts et Chaussées (Paris, France). Two types
of mother wavelets ψM (t) are considered: (a) the
Morlet wavelet, which is a Gaussian-windowed com2
plex sinusoid of frequency ω0 , ψM (t) = e−t /2 e jω0 t ; (b)
the Cauchy wavelet of order n, ψM (t) = [ j/(t + j)]n+1 ,
where j 2 = −1. The frequency ω0 for the Morlet WT
and the order n for the Cauchy WT are user-specified
parameters which allow one to tune the frequency
and time resolutions of the results. It should be noted
that these two mother wavelets provide similar results
when applied to the signals considered in this paper.
The plots shown represent the amplitude of theWT as a
function of frequency (vertical axis) and time (horizontal axis). Heavily shaded areas correspond to regions
where the amplitude of the WT is high, whereas lightly
shaded regions correspond to low amplitudes. Such
plots enable one to deduce the temporal evolutions
of the dominant frequency components of the signals
analysed.
Alternatively, the EMD gained popularity in the area
of signal processing and is also utilized in this work.
Originally introduced by Huang et al. [95, 156, 157],
it was shown to be applicable to strongly non-linear
and non-stationary signals with non-zero mean. In
an alternative numerical post-processing technique,
the EMD through a sifting process yields a collection of intrinsic mode functions (IMFs), which form
a complete, nearly orthogonal, local, and adaptive
basis. These properties render the EMD applicable
to decomposition of non-linear and non-stationary
signals.
Once EMD is performed, the obtained IMFs are
suitable for Hilbert transformation, which yields the
instantaneous amplitude and phase of each IMF at any
given instant of time. By differentiating the instantaneous phase, one computes the temporal evolution
of the instantaneous frequency of each IMF which,
when compared with the overall WT of the time series,
enables one to judge the relative contribution of each
IMF in the time series and, thus, its relative importance
in the decomposition of the signal.
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Note that an IMF may have a significant contribution in certain time intervals of the signal, and
be less important in others. Hence, EMD coupled with
the Hilbert transform can be a powerful computational
tool for studying complicated non-linear resonance
interactions leading to complex dynamic phenomena
(such as TET) in coupled structures. Recently, a time
series forecasting method based on support vector
regression machines was proposed, its apparent superiority attributed to the use of neural networks [158].
An effort was made to improve the quality (i.e. orthogonality) of the obtained IMFs by means of an energy
difference tracking method [159]. The EMD method
can also be applied to problems of fault diagnosis and
damage estimation [160, 161]. In this paper, Matlab
codes developed by Rilling et al.[162] are employed to
perform the described EMD analysis.
3
DYNAMICS OF TET
In order to establish a clear understanding of nonlinear energy pumping mechanisms, a SDOF primary oscillator coupled to an ungrounded SDOF NES
[83, 84] is considered in this section. For a SDOF primary structure coupled to a grounded SDOF NES, one
can refer to references [41], [50], and [97].
3.1
Undamped periodic solutions
The system under consideration is depicted in Fig. 2,
and consists of an oscillator of mass m1 (the linear
oscillator) coupled through an essentially non-linear
stiffness to a mass m2 (the non-linear attachment).
The equations of motion of this two-DOF system are
given by
structure of the periodic orbits of the underlying
undamped system (with λ1 = λ2 ≡ 0). Indeed, it will
be shown that this seemingly simple system possesses
a very complicated topological structure of periodic
orbits, some of which are responsible for TET phenomena in the impulsively forced, damped system.
3.1.1
Numerical approach
The periodic orbits of the system will be computed
numerically utilizing the method of non-smooth
transformations first developed by Pilipchuk [163]
and then applied to strongly non-linear oscillators by
Pilipchuk et al. [72]. This method can be applied to the
numerical and analytical study of the periodic orbits
(and their bifurcations) of strongly non-linear dynamical systems. To apply the method, the sought periodic
solutions are expressed in terms of two non-smooth
variables, τ and e, as
t
t
v(t) = e
y1 τ
,
α
α
x(t) = e
t
t
y2 τ
α
α
(3)
where α = T /4 represents the (yet unknown) quarterperiod. The non-smooth functions τ (u) and e(u) are
defined according to the expressions
2
π τ (u) = sin−1 sin u , e(u) = τ (u)
(4)
π
2
and are used to replace the independent time variable
from the equations of motion; their graphic depiction
is given in Fig. 3.
m1 ẍ + k1 x + c1 ẋ + c2 (ẋ − v̇) + k2 (x − v)3 = 0
m2 v̈ + c2 (v̇ − ẋ) + k2 (v − x)3 = 0
⇒ ẍ + ω02 x + λ1 ẋ + λ2 (ẋ − v̇) + C(x − v)3 = 0
v̈ + λ2 (v̇ − ẋ) + C(v − x)3 = 0
(2)
where ω02 = k1 /m1 , C = k2 /m1 , = m2 /m1 , λ1 = c1 /m1 ,
and λ2 = c2 /m1 .
Before analysing non-linear TET phenomena in the
damped system, it is first necessary to examine the
Fig. 2 The two-DOF system with essential stiffness
non-linearity
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
Fig. 3 The non-smooth functions τ (u) and e(u)
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Passive non-linear TET and its applications
87
Setting λ1 = λ2 = 0, and substituting equation (3)
into equation (2), smoothening conditions [72] are
imposed to eliminate singular terms from the resulting
equations, such as terms proportional to
e (x) = τ (x) = 2
∞
[δ(x + 1 − 4k) − δ(x − 1 − 4k)]
k=−∞
Setting to zero, the component of the transformed
equations that is multiplied by the non-smooth variable e, the following two-point NLBVP is formulated
in terms of the non-smooth variable τ , in the interval −1 τ +1
C
y3 = − α 2 ( y1 − y2 )3 ,
y4 = −ω02 α 2 y2 − Cα 2 ( y2 − y1 )3
y1 = y3 ,
y2 = y4 ,
Fig. 4
(5)
with the boundary conditions, y1 (±1) = y2 (±1) = 0,
where primes denote differentiation with respect to
the non-smooth variable τ , and a state formulation is
utilized. The boundary conditions above result from
the aforementioned smoothing conditions.
Hence, the problem of computing the periodic solutions of the undamped system (2) is reduced to solving
the NLBVP (5) formulated in terms of the bounded
independent variable τ ∈ [−1, 1], with the quarterperiod α playing the role of the non-linear eigenvalue.
It is noted that the solutions of the NLBVP can be
approximated analytically through regular perturbation series [72]; however, this will not be attempted
herein where only numerical solutions will be considered. It is merely mentioned here that equation (5) is
amenable to direct analytical study in terms of simple
mathematical functions.
It is noted that the NLBVP (5) provides the solution only in the normalized half-period −1 t/α 1
⇒ −1 τ 1. To extend the result over a full normalized period equal to four, one needs to add the
component of the solution in the interval 1 t/α 3;
to perform this one takes into account the symmetry properties of the non-smooth variables τ and e
by adding the antisymmetric image of the solution
about the point ( yi , t/α) = (0, 1), as shown in Fig. 4.
It follows by construction that the computed periodic solutions satisfy the initial conditions, x(−α) =
v(−α) = 0 and v̇(−α) = y1 (−1)/α, ẋ(−α) = y2 (−1)/α.
It is noted at this point that since equation (2) is an
autonomous dynamical system these initial conditions can be shifted arbitrarily in time; for example,
they can be applied to the initial time t = 0 instead
of t = −α = −T /4. However, in what follows the formulation of the NLBVP (5) will be respected, and the
initial conditions at t = −T /4 are retained.
Considering the general shape of the periodic orbits
depicted in Fig. 4, the following classification of
JMBD118 © IMechE 2008
Construction of the periodic solutions
v(t) = e(t/α)y1 (τ (t/α)), x(t) = e(t/α)y2 (τ (t/α))
over an entire normalized period −1 t/α 3
from the solutions yi (τ (t/α)), i = 1, 2 of the
NLBVP (5) computed over the half-normalized
period −1 t/α 1
periodic solutions is introduced.
1. Symmetric solutions Snm ± correspond to orbits
that satisfy the conditions
T
T
v̇ −
= ±v̇ +
⇒ y1 (−1) = ±y1 (+1)
4
4
T
T
ẋ −
= ±ẋ +
⇒ y2 (−1) = ±y2 (+1)
4
4
with n being the number of half-waves in y1 (v),
and m the number of half-waves in y2 (x) in the
half-period interval −T /4 t +T /4 ⇐⇒ −1 τ +1.
2. Unsymmetric solutions Unm are orbits that do
not satisfy the conditions of the symmetric orbits.
Orbits U (m + 1)m bifurcate from the symmetric solution S11 − at T /4 ≈ mπ/2, and exist
approximately within the intervals mπ/2 < T /4 <
(m + 1)π/2, m = 1, 2, . . . .
The numerical solution of the two-point NLBVP
(5) is constructed utilizing a shooting method programmed in Mathematica (see references [141] and
[164] for some details on the shooting method and
general characteristics of global solutions).
The NLBVP (5) is solved as follows.
1. For a given non-linear eigenvalue α (quarterperiod), the solutions of the NLBVP are computed
at different energy levels; it is expected that at every
energy level there co-exist multiple non-linear periodic solutions sharing the same minimal period.
Periodic orbits that correspond to synchronous
motions of the two particles of the system, and
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
pass through the origin of the configuration plane
( y1 , y2 ), are termed NNMs [165].
2. The different families of computed periodic solutions are depicted in three types of plots. In the first
two types of plots, initial displacements x(−T /4) =
v(−T /4) = 0 are assumed, and the initial velocities
v̇(−T /4) = y1 (−1)/α and ẋ(−T /4) = y2 (−1)/α corresponding to a periodic orbit as functions of the
quarter-period α = T /4 or the (conserved) energy
of that orbit are depicted
T
T
1
2
2
v̇ −
+ ẋ −
h=
2
4
4
1
2
2
=
[y (−1) + y2 (−1)]
2α 2 1
In the third type of plots, the frequencies of the
periodic orbits are depicted as functions of their
energies h. These plots clarify the bifurcations
that connect, generate, or eliminate the different
branches (families) of periodic solutions.
3. The stability of the computed periodic orbits was
determined numerically by three different methods: application of Floquet theory; construction of
two-dimensional Poincaré maps on the isoenergetic manifolds of the two-DOF undamped system
(2); and direct numerical simulation of the equations of motion using as initial conditions those
estimated by the solution of the NLBVP (5).
In the following, the numerical results correspond
to the two-DOF undamped system with parameters
= 0.05, ω0 = 1, C = 1.0 in the energy range 0 < h < 1.
The bifurcation diagrams of the initial velocities and
for varying quarter-period are depicted in Fig. 5. Some
general and preliminary observations on the computed periodic orbits are made at this point, and the
dynamical behaviour of the system on the various
branches will be discussed in the next section.
Considering the branches Snn−, they exist in the
quarter-period intervals 0 < α < nπ/2, and their initial conditions satisfy the limiting relationships (Fig. 5)
lim{|v̇(−α)|, |ẋ(−α)|} = ∞,
α→0
lim {|v̇(−α)|, |ẋ(−α)|} = 0
α→nπ/2
These symmetric branches exist throughout the examined energy domain 0 < h < 1. It is noted that
branches Snn− are, in essence, identical to the branch
S11−, since they are identified over the domain of
their common minimal period (the Snn− branches
are branches S11− ‘repeated n times’); similar remarks
can be made regarding the branches S(kn)(km)±, k
integer, which are identified with Snm±.
Focusing in the neighbourhood of branches S11±
and referring to Fig. 5, at the point α = π/2 where
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
Fig. 5
Normalized initial velocities of periodic orbits
yi (−1), i = 1, 2 as functions of the quarter-period
α; solid (dashed) lines correspond to positive
(negative) initial velocities (S11 (), S13 (), S15
(), S31 (), S21 (♦) with in-phase as filled-in, and
branches U without symbol) [83]
S11− disappears the branches S11+ and U 21 bifurcate
out (similar behaviour is exhibited by the branches
S31, S21, . . .). For π/2 α π, a bifurcation from
S11+ to S13+ takes place without change of phase;
similar bifurcations take place at higher values of α for
branches S15+, S17+, . . .. For α ≈ 3π/2, the branches
S13+ and S13− coalesce into the branch S11−, with
similar coalescences into S11− taking place at higher
values of α for the pairs of branches S15+, S17+, . . ..
The unsymmetric branches U (m + 1)m bifurcate
from the symmetric branches S(m + 1)(m + 1)− at
quarter-periods equal to α = mπ/2. It turns out that
certain orbits (termed ‘SPOs’) on these branches
are of particular importance concerning the passive
and irreversible energy transfer from the linear to
the non-linear oscillator. The special orbits satisfy
the additional initial condition y1 (−1) = v̇(−α) = 0,
and correspond to zero crossings of the branches
U (m + 1)m in the bifurcation diagram (the upper
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Passive non-linear TET and its applications
Fig. 6
89
Special periodic orbits on the U -branches with initial conditions y1 (−1) = y2 (−1)
= 0, y1 (−1) = 0, y2 (−1) = 0; Unm(a) and Unm(b) denote the unstable and stable SPOs,
respectively ( y1 (τ ) is represented by a solid line and y2 (τ ) by a dashed line; x-axis
represents τ )
plot); some of these special orbits (either stable or
unstable) are depicted in Fig. 6. Taking into account the
formulation of the NLBVP (5), it follows that the special
orbits satisfy initial conditions v(−T /4) = v̇(−T /4) =
x(−T /4) = 0, and ẋ(−T /4) = 0, which happen to be
identical to the state of the undamped system (2)
(being initially at rest) after application of an impulse
of magnitude ẋ(−T /4) = y2 (−1)/α on the linear oscillator.
Moreover, comparing the relative magnitudes of
the linear and non-linear oscillators for the special
orbits of Fig. 6, it is concluded that certain stable
special orbits are localized to the non-linear oscillator. This implies that if the system is initially at
rest and is forced impulsively, and one of the stable,
localized special orbits is excited, a major portion
JMBD118 © IMechE 2008
of the induced energy is channeled directly to the
invariant manifold of that special orbit, and, hence,
the motion is rapidly and passively transferred from
the linear to the non-linear oscillator. Moreover, this
energy transfer is irreversible because of the invariance properties of the stable special orbit, and, as
a result, after the energy is transferred, it remains
localized and is passively dissipated at the non-linear
attachment. Therefore, it is assumed that the impulsive excitation of one of the stable special orbits is
one of the triggering mechanisms initiating (direct)
passive TET. This conjecture will be proven to be correct by numerical simulations presented in a later
section.
Similar classes of special orbits can also be realized in a subclass of S-branches. In particular, this
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 7
Frequency–energy plot of the periodic orbits; for the sake of clarity no stability is indicated,
special orbits are denoted by filled circles (•; some appear unfilled due to the overlapping symbols) and are connected by dashed-dot lines; other symbols indicate bifurcation
points (stability–instability boundaries): () four Floquet multipliers at +1; (♦) two Floquet
multipliers at +1 and the other two at −1 [83]
type of orbit can be realized on branches S(2k + 1)
(2p + 1)±, k = p, but not on periodic orbits that do
not pass through the origin of the configuration plane
(such as S21, S12, . . .). The branch S11− is a particular
case, where the special orbit is realized only asymptotically as the energy tends to zero, and the motion is
localized completely in the linear oscillator.
In Fig. 7, the various branches of solutions are presented in a FEP. For clarity, the following convention
regarding the placement of the various branches in
the frequency domain is adopted: a specific branch of
solutions is assigned with a frequency index equal to
the ratio of its two indices, e.g. S21± is represented
by the frequency index ω = 2/1 = 2, as is U 21; S13±
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
is represented by ω = 1/3, and so forth. This convention holds for every branch except S11±, which,
however, are particular branches. On the energy axis,
the (conserved) total energy of the system is depicted
when it oscillates in a specific mode. Necessary (but
not sufficient) conditions for bifurcation and stability–
instability exchanges are satisfied when two Floquet
multipliers of the corresponding variational problems
coincide at +1 or −1 (since periodic orbits of a Hamiltonian two-DOF system are considered, two Floquet
multipliers of the variational problem are always equal
to +1, whereas the other two form a reciprocal pair),
and these are indicated at the solution branches of
Fig. 7.
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 8
91
Close-ups of particular branches in the frequency index–logarithm of energy plane: (a)
S11−; (b) S11+; (c) S13±; (d) U 43 (double branch). Stability–instability boundaries are
represented as in Fig. 7; some representative periodic orbits are also depicted in insets in
the format x(v) (configuration plane of the system); SPOs on the U - and S-branches are
indicated by triple asterisks. Arrowed lines indicate the intervals of instability [83]
To understand the types of periodic motions that
take place in different frequency–energy domains, certain branches are depicted in detail in Fig. 8, together
with the corresponding orbits realized in the configuration plane of the system. The horizontal and
vertical axes in the plots in the configuration plane are
the non-linear (v) and linear oscillator (x) responses,
respectively; the aspect ratios in these plots are set so
that the tick mark increments on the horizontal and
vertical axes are equal in size, enabling one to directly
deduce whether the motion is localized in the linear or
the non-linear oscillator. The plot for U 43 (Fig. 8(d)) is
composed of two very close branches; for the sake of
clarity only one of the two branches is presented. The
motion is nearly identical on the two branches, so only
the oscillations in the configuration plane of one of the
two branches are considered.
Since a systematic analytical study of the various
types of periodic solutions of the system is presented
in the next section, the following preliminary remarks
JMBD118 © IMechE 2008
are made.
1. The main backbone of the FEP is formed by the
branches S11± which represent in- or out-of-phase
synchronous vibrations of the two particles possessing one half-wave per half-period. Moreover,
the natural frequency of the linear oscillator ω0 = 1
(which is identified with a frequency index equal
to unity, ω = 1) naturally divides the periodic solutions into higher and lower frequency modes. There
are two saddle-node-type bifurcations in the higher
frequency, out-of-phase branch S11−, and the stable solutions become localized to x or v as ω → 1+
or ω 1, respectively (see Fig. 8(a)). The lower frequency, in-phase branch S11+ becomes unstable
at higher energies, and the stable solutions localize
to the non-linear attachment as ω decreases away
from ω = 1 (see Fig. 8(b)).
2. There is a sequence of higher and lower frequency
periodic solutions bifurcating or emanating from
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
branches S11±. Considering first the symmetric
solutions, the branches S1(2k + 1)±, k = 1, 2, . . .
appear in the neighbourhoods of frequencies ω =
1/(2k + 1), e.g. at progressively lower frequencies
with increasing k. For fixed k, each of the two
branches S1(2k + 1)± is linked through a smooth
transition with its neighbouring branches S1
(2k − 1)± or S1(2k + 3)±, and exists over a finite
interval of energy. The pair S1(2k + 1)± is eliminated through a saddle-node-type bifurcation at
a higher energy value (see Fig. 8(c) for branches
S13±). The pairs of branches S1(2k)±, k = 1, 2, . . .
bifurcate out of S1(2k + 1)±, and exist over finite
energy intervals. All branches S1n± and Sn1±, n ∈
Z+ seem to connect with S11− through ‘jumps’ in
the FEP, but in actuality no such discontinuities
occur if one takes into account that due to the
previous frequency convention solutions Spp+ are
identified with the solution S11+, S(2p)(1p)± with
S21±, etc.
3. Focusing now on the unsymmetrical branches, a
family of U (m + 1)m branches bifurcating from
branch S11− exists over finite energy levels and
are eliminated through saddle-node-type bifurcations with other branches of solutions. Again,
the transitions of branches U 21 and U 32 to
S11+ seem to involve ‘jumps’, but this is only
due to the frequency convention adopted, and
no actual discontinuities in the dynamics occur.
An additional interesting family of unsymmetrical solutions is Um(m + 1), m = 1, 2, . . . which,
due to the previous frequency convention, is
depicted for frequency indices ω < 1; the shapes
of these orbits in the configuration plane are
similar to those of U (m + 1)m, m = 1, 2, . . . , but
rotated by π/2. An important class of periodic
orbits realized on the unsymmetrical branches
(but also in certain of the symmetric branches)
is that corresponding to all initial conditions
zero, with the exception of the initial velocity of the linear oscillator. These special orbits
provide one of the mechanisms for passive
TET from the linear oscillator to the non-linear
attachment [84].
The previous discussion indicates that the two-DOF
undamped system possesses complicated structures
of symmetric and unsymmetrical periodic orbits. The
next section will focus on the analysis of the computed periodic orbits in detail in an effort to better
understand the dynamics and localization properties
of the system over different frequency–energy ranges.
Indeed, understanding the periodic dynamics of the
undamped system paves the way to explain passive
TET phenomena and complicated transitions between
different types of motion in the transient dynamics of
the damped system.
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
3.1.2
Analytical approach
The dynamics of the undamped system and all the
different branches of solutions can be studied analytically. As representative examples of this analysis, the
periodic orbits on a particular branch, namely S11±,
are investigated in detail.
To study the periodic orbits of equation (2) for 0 <
1, the complexification-averaging method first
introduced by Manevitch [135] is applied, which not
only enables the study of the steady-state motions, but
also can be applied to analyse the damped, transient
dynamics [50].
The S11± branch is composed of synchronous periodic motions where the two particles oscillate with
identical frequencies. The analytical study of these
solutions is performed by introducing the new complex variables ψ1 = ẋ + jωx and ψ2 = v̇ + jωv where
j 2 = −1, and expressing the displacements and accelerations of the two particles of the system as (the
asterisk denotes complex conjugatation)
1
(ψ1 − ψ1∗ ),
2jω
1
v=
(ψ2 − ψ2∗ ),
2jω
x=
jω
(ψ1 + ψ1∗ )
2
jω
v̈ = ψ̇2 − (ψ2 + ψ2∗ )
2
ẍ = ψ̇1 −
(6)
Since nearly monochromatic periodic solutions of the
equations of motion are sought and the two particles
oscillate with the identical frequencies, the previous complex variables are approximately expressed
in terms of ‘fast’ oscillations of frequency ω, e jωt ,
modulated by ‘slow’ (complex) modulations φi (t)
ψ1 = φ1 e jωt ,
ψ2 = φ2 e jωt
(7)
This amounts to a partition of the dynamics into
slow- and fast-varying components, and the interesting dynamics is reduced to the slow flow. Note that
no a priori restrictions are posed on the frequency
ω of the fast motion. Substituting equations (6) and
(7) into the equations of motion (2) with λ1 = λ2 = 0,
and performing averaging over the fast frequency, to
a first approximation only terms containing the fast
frequency ω are retained
1
1
3C
φ̇1 + jω φ1 − j φ1 + j 3 (−|φ1 |2 φ1 + φ12 φ2∗
2
2ω
8ω
− φ22 φ1∗ + |φ2 |2 φ2 + 2|φ1 |2 φ2 − 2|φ2 |2 φ1 ) = 0
ω
3C
φ̇2 + j φ2 − j 3 (−|φ1 |2 φ1 + φ12 φ2∗ − 3φ22 φ1∗
2
8ω
+ |φ2 |2 φ2 + 2|φ1 |2 φ2 − 2|φ2 |2 φ1 ) = 0
(8)
These complex modulation equations govern the slow
evolutions of the complex amplitudes φi , i = 1, 2 in
time.
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Passive non-linear TET and its applications
Introducing the polar representations φ1 = Ae jα and
φ2 = Be jβ where A, B, α, β ∈ R in equation (8), and
separately setting the real and imaginary parts of the
resulting equations equal to zero, the following real
modulation equations that govern the slow evolution
of amplitudes and phases of the two responses of the
system are obtained as
modulation)
AȦ + B Ḃ = 0 ⇒ A2 + B 2 = N 2
where N is a constant of integration. Clearly, the
above is an energy conservation relation reflecting the
conservation of the total energy of the undamped system (2) during its oscillation. Hence, the modulation
equation (9) can be reduced by one, by imposing the
above energy conservation algebraic relation.
The periodic solutions on the branches S11± are
studied by setting the derivatives with respect to time
in equation (9) equal to zero; i.e. by imposing stationarity conditions on the modulation equations. The first
and third equations are trivially satisfied if α = β, and
the second and fourth equations become
A
3C
−
(A − B)3 = 0,
ω 4ω3
3C
(A − B)3 = 0
ωB +
4ω3
ωA −
(10)
These equations can be solved exactly for the amplitudes A and B, leading to the following approximations
for the periodic solutions on the branches S11±
−εω2
= 2
ω −1
ψ1 − ψ1∗
A
= cos ωt
2jω
ω
4ω2 ε(ω2 − 1)3
cos ωt
3C((1 + ε)ω2 − 1)3
JMBD118 © IMechE 2008
=
4ω2 ε(ω2 − 1)3
cos ωt
3C((1 + ε)ω2 − 1)3
(11)
X
−ω2
= 2
V
ω −1
(9)
x(t) ≈ X cos ωt =
B
ψ2 − ψ2∗
= cos ωt
2jω
ω
Considering the original non-linear problem (2), note
that relations (11) are approximate since a single fast
frequency was assumed in the slow–fast partitions (7),
and only terms containing this fast frequency were
retained after performing averaging in the complex
equations (8).
It is interesting to note that the ratio of the amplitudes of the linear and non-linear oscillators on
branches S11± is given by the following simple form
BC
[(3A2 + 3B 2 ) sin(α − β)
8ω3
+ 3AB sin(2β − 2α)] = 0
A
3CA3
6AB 2 C
BC
ωA
−
−
−
−
Aα̇ +
3
3
2
2ω
8ω
8ω
8ω3
2
2
× [(−9A − 3B ) cos(α − β)
+ 3AB cos(2β − 2α)] = 0
AC
Ḃ −
[(3B 2 + 3A2 ) sin(α − β)
8ω3
+ 3AB sin(2β − 2α)] = 0
ωB 3B 3 C
6A2 BC
AC
B β̇ +
−
−
−
3
3
2
8ω
8ω
8ω3
× [(−9B 2 − 3A2 ) cos(α − β)
+ 3AB cos(2β − 2α)] = 0
Ȧ +
The first and third (amplitude
equations are combined, giving
v(t) ≈ V cos ωt =
93
(12)
This relation shows that if the mass of the non-linear
oscillator is small (as is assumed), and if the frequency
ω is not in the neighbourhood of the eigenfrequency of
the linear oscillator ω0 = 1, the motion is always localized to the non-linear oscillator (in agreement with
the numerical results); however, sufficiently close to
ω0 = 1, the oscillation localizes on the linear oscillator
(as one would expect intuitively).
√ There is a region in the frequency domain,
1/(1 + ) < ω < 1, where the coefficients X and V
are imaginary, indicating that no periodic motion on
S11± can occur there; this represents a forbidden zone
not only for S11±, but also for any periodic motion
of the system. Accordingly, the √
branch S11+ of inphase oscillations exists for ω < 1/(1 + ), whereas
out-of-phase oscillations on S11− exist for ω > 1.
The approximations of the branches S11± in the
frequency–energy plane are computed by noting that
the conserved energy of the system is equal to
h=
X2
(V − X )4
+C
2
4
(13)
which, taking into account expressions (11), leads to
the plot depicted in Fig. 9; this plot corresponds to the
parameters used in the numerical study ( = 0.05, C =
1.0). The approximate plots are close to the exact
numerical backbones of the FEP of Fig. 7.
3.1.3
Transient dynamics of the damped system
In this section, the transient, unforced dynamics of
the weakly damped system is considered, and it will be
shown that complicated transitions between modes in
this system can be fully understood and interpreted in
terms of the periodic orbits of the undamped system.
Specifically, the addition of damping induces transitions between different branches of solutions, and
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
94
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
motion temporarily settling on branch S12− before
escaping from resonance capture as time increases
(and energy decreases), and being involved in TRC
with the stable branch S13−. The short capture on
branch S12− leads to the conjecture that the domain
of attraction of 1:2 resonance capture is much smaller
than the corresponding domain of attraction of the 1:3
resonance capture, with the latter eventually capturing the transient damped dynamics. Indeed, it should
be expected that due to the complicated topology
of the periodic orbits of the undamped system, the
transitions between branches and the sequence of
resonance captures should be sensitive to viscous
damping dissipation.
Fig. 9
Analytic approximation provided by the complexification–averaging method of the backbone
branch S11± in the frequency index–logarithm of
energy plane
thus influences the transfer of energy between the
linear oscillator and the non-linear attachment.
The transient responses of the weakly damped system will demonstrate that the structure of periodic
orbits of the undamped system greatly influences the
dynamics of the weakly damped one. When viewed
from such a perspective, one can systematically interpret the complex transitions between multi-frequency
modes of the transient, weakly damped dynamics by
relating them to the different branches of non-linear
modes in the FEP of Fig. 7. Unless otherwise noted,
in the following simulations system (2) is considered
with the same parameters used in the previous sections ( = 0.05, ω0 = 1.0, C = 1.0), and small damping
coefficients λ1 = 0, λ2 = 0.0005.
The motion on the stable special orbit of branch U 76
is initiated and there occurs vigorous TET to the nonlinear attachment. In Fig. 10, the responses and related
WTs of the system with initial conditions v(−T /4) =
v̇(−T /4) = x(−T /4) = 0 and ẋ(−T /4) = −0.1039 are
depicted. The general observation is made that in this
case there is strong TET to the non-linear attachment
(NES), as evidenced by its large amplitude of oscillation compared with that of the (directly excited) linear
oscillator.
In particular, Fig. 10(d) is a schematic illustrating
the transitions taking place in the weakly damped
response on the FEP of the undamped system. The
simulation verifies that the impulsive excitation of a
stable special orbit is one of the triggering mechanisms
initiating (direct) passive energy pumping. Energy
decrease due to damping dissipation triggers the transitions between different branches of solutions. The
numerical simulations of Fig. 10 demonstrate that, following a prolonged motion on U 76 during the early
regime of the motion, there occurs a 1:2 TRC with the
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
3.2 TET mechanisms
In this section, the impulsively forced, damped system
(2) with the primary DOF denoted by y is considered, and three basic mechanisms for the initiation
of non-linear TET are studied. The first mechanism
(fundamental TET) is realized when the motion takes
place along the backbone curve S11+ of the FEP of
Fig. 7, occurring for relatively low frequencies ω < ω0 .
The second mechanism (subharmonic TET) resembles the first, and occurs when the motion takes place
along a lower frequency branch Snm, n < m ∈ Z+ . The
third mechanism (TET initiated by non-linear beat)
which leads to stronger TET involves the excitation of
a special orbit with main frequency ωSO greater than
the natural frequency of the linear oscillator ω0 . In
what follows, each mechanism is discussed separately,
and numerical simulations that demonstrate passive
and irreversible energy transfer from the linear oscillator to the non-linear attachment are provided in
each case. Analytical results are also provided for the
fundamental and subharmonic TET.
3.2.1
Fundamental TET
The first mechanism for TET involves excitation of the
branch of in-phase synchronous periodic solutions
S11+, where the linear oscillator and the non-linear
attachment oscillate with identical frequencies in the
neighbourhood of the fundamental frequency ω0 .
Although TET is considered only in the damped system, in order to gain an understanding of the governing dynamics it is necessary to consider the case of no
damping.
Figure 8(b) depicts a detailed plot of branch
of the undamped system. At higher energies, the
in-phase NNMs are spatially extended (involving
finite-amplitude oscillations of both the linear oscillator and the non-linear attachment). However, the
non-linear mode shapes of solutions on S11+ depend
essentially on the level of energy and at low energies
they become localized to the attachment. Considering
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 10
95
Damped motion initiated on the stable special orbit of branch U 76 with weaker damping:
(a) and (b) transient responses of the linear and non-linear oscillators; (c) their WTs; (d)
WTs superimposed to the undamped FEP [83]
now the motion in-phase space, this low-energy localization is a basic characteristic of the two-dimensional
NNM invariant manifold corresponding to S11+;
moreover, this localization property is preserved in the
weakly damped system, where the motion takes place
in a two-dimensional, damped NNM invariant manifold. This means that when the initial conditions of
the damped system are such as to excite the damped
analogue of S11+, the corresponding mode shape of
the oscillation, initially spatially extended, becomes
localized to the non-linear attachment with decreasing energy due to damping dissipation. This, in turn,
leads to passive, continuous and irreversible transfer
of energy from the linear oscillator to the non-linear
attachment, which acts as a NES. The underlying
dynamical phenomenon governing fundamental TET
was proven to be a resonance capture on a 1:1
resonance manifold of the system [50].
Numerical evidence of fundamental TET is given
in Fig. 11 for the system with parameters = 0.05,
ω02 = 1, C = 1, and λ1 = λ2 = 0.0015. Small damping
JMBD118 © IMechE 2008
is considered in order to better highlight the TET
phenomenon, and the motion is initiated near the
boxed point of Fig. 8(b). Comparing the transient
responses shown in Figs 11(a) and (b), it is noted
that the response of the primary system decays faster
than that of the NES. The percentage of instantaneous
energy captured by the NES versus time is depicted
in Fig. 11(e), and the assertion that continuous and
irreversible transfer of energy from the linear oscillator to the NES takes place is confirmed. This is more
evident by computing the percentage of total input
energy that is eventually dissipated by the damper of
the NES (see Fig. 11(f )), which in this particular simulation amounts to 72 per cent; the energy dissipated at
the NES is computed by the relation
t
ENES (t) = λ2 [v̇(τ ) − ẏ(τ )]2 dτ
0
The evolution of the frequency components of the
motions of the two oscillators as energy decreases
can be studied by numerical WTs of the transient
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
96
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 11
Fundamental TET. Shown are the transient responses of the (a) linear oscillator and (b) NES;
WTs of the motion of (c) NES and (d) linear oscillator; (e) percentage of instantaneous total
energy in the NES; (f) percentage of total input energy dissipated by the NES; transition
of the motion from S11+ to S13+ at smaller energy levels using the (g) NES (observe the
settlement of the motion at frequency 1/3) and (h) linear oscillator [84]
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
responses, as depicted in Figs 11(c) and (d). These plots
demonstrate that a 1:1 resonance capture is indeed
responsible for TET. Below the value of −4 of the logarithm of energy level, the motion of the linear oscillator
is too small to be analysed by the particular windows
used in the WT; however, a more detailed WT over
smaller energy regimes (see Figs 11(g) and (h)) reveals
a smooth transition from S11+ to S13+, in accordance
with the FEP of Fig. 7. This transition manifests itself by
the appearance of two predominant frequency components in the responses (at frequencies 1 and 1/3) as
energy decreases.
The complexification-averaging method is utilized
to perform an analytical study of the resonance capture phenomenon in the fundamental TET mechanism. System (2) is again considered, and the new
complex variables are introduced
ψ1 (t) = v̇(t) + jv(t) ≡ ϕ1 (t) e jt ,
ψ2 (t) = ẏ(t) + jy(t) ≡ ϕ2 (t) e jt
(14)
where φi (t), i = 1, 2, represent slowly varying complex amplitudes and j 2 = −1. By writing equation (14),
a partition of the dynamics into slow and fast components is introduced, and slowly modulated fast
oscillations at frequency ω = ω0 = 1 are sought. As
discussed previously, fundamental TET is associated
with this type of motion in the neighbourhood of
branch S11+ in the FEP of the undamped dynamics. Expressing the system responses in terms of the
new complex variables, y = (ψ2 − ψ2∗ )/(2j), v = (ψ1 −
ψ1∗ )/(2j) (where (*) denotes complex conjugate), substituting into equation (2), and averaging over the
fast frequency, a set of approximate, slow modulation
equations that govern the evolutions of the complex
amplitudes is derived
λ
λ
3C
ϕ̇1 = −j ϕ1 − (ϕ1 − ϕ2 ) + j
|ϕ1 − ϕ2 |2 (ϕ1 − ϕ2 )
2
2
8
λ
λ
3C
ϕ̇2 = − ϕ2 + (ϕ1 − ϕ2 ) + j
|ϕ2 − ϕ1 |2 (ϕ2 − ϕ1 )
2
2
8
(15)
For the sake of simplicity, assume that λ1 = λ2 = λ
in equation (2). To derive a set of real modulation
equations, the complex amplitudes are expressed in
polar form, ϕi (t) = ai (t)e jβi t , which is substituted into
equation (15), and the real and imaginary parts are
separately set equal to zero. Then, equation (15) is
reduced to an autonomous set of equations that govern the slow evolution of the two amplitudes a1 (t) and
a2 (t) and the phase difference φ(t) = β2 (t) − β1 (t)
λ
λ
ȧ1 = − a1 + a2 cos φ
2
2
3C
+
a2 (a12 + a22 − 2a1 a2 cos φ) sin φ
8
JMBD118 © IMechE 2008
97
λ
ȧ2 = a1 cos φ − λa2
2
3C
−
a1 (a12 + a22 − 2a1 a2 cos φ) sin φ
8
1
λ
a1 a2 φ̇ = − a1 a2 − (a1 + a2 ) sin φ
2
2
3C 2
−
(a + a22 − 2a1 a2 cos φ)
8 1
× [(1 − )a1 a2 + (a1 − a2 ) cos φ]
(16)
This reduced dynamical system governs the slow-flow
dynamics of fundamental TET. In particular, 1:1 resonance capture (the underlying dynamical mechanism
of fundamental TET) is associated with non-time-like
behaviour of the phase variable φ or, equivalently, failure of the averaging theorem in the slow flow (16).
Indeed, when φ exhibits time-like, non-oscillatory
behaviour [166], one can apply the averaging theorem
over φ and prove that the amplitudes a1 and a2 decay
exponentially with time and no significant energy
exchanges (TET) can take place. Figure 12(a) depicts
1:1 resonance capture in the slow-flow-phase plane
(φ, φ̇) for system (16) with = 0.05, λ = 0.01, C = 1,
ω0 = 1 and initial conditions a1 (0) = 0.01, a2 (0) =
0.24, φ(0) = 0. The oscillatory behaviour of the phase
variable in the neighbourhood of the in-phase limit
φ = 0+ indicates 1:1 resonance capture (on branch
S11+ of the FEP of Fig. 7), and leads to TET from the linear oscillator to the NES as evidenced by the build-up
of amplitude a1 (see Fig. 12(b)). Escape from resonance capture is associated with time-like behaviour
of φ and rapid decrease of the amplitudes a1 and a2
(as predicted by averaging in equation (16)). A comparison of the analytical approximation (14)–(16) and
direct numerical simulation for the previous initial
conditions confirms the accuracy of the analysis.
3.2.2
Subharmonic TET
Subharmonic TET involves excitation of a lowfrequency S-tongue. As mentioned earlier, lowfrequency tongues are the particular regions of the FEP
where the NES engages in m:n (m, n are integers such
that m < n) resonance captures with the linear oscillator. A feature of the lower tongues is that on them
the frequency of the motion remains approximately
constant with varying energy. As a result, the tongues
are represented by horizontal lines in the FEP, and the
response of system (2) on a tongue locally resembles
that of a linear system. In addition, at each specific
m:n resonance capture, there appear a pair of closely
spaced tongues corresponding to in- and out-of-phase
oscillations of the two subsystems.
Regarding the dynamics of subharmonic TET, a
particular pair of lower tongues are focused, say
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
98
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 12
Fundamental TET: (a) 1:1 resonance capture in the slow flow; (b) amplitude modulations;
(c) comparison between analytical approximation (dashed line) and direct numerical
simulation (solid line) for v(t); (d) transient responses of the system [84]
S13± (Fig. 8(c)). At the extremity of a lower pair
of tongues, the curve in the configuration plane is
strongly localized to the linear oscillator. However, as
for the fundamental mechanism for TET, the decrease
of energy by viscous dissipation leads to curves in the
configuration plane that are increasingly localized to
the NES, and non-linear TET to the NES occurs. In this
case, the underlying dynamical phenomenon causing
TET is resonance capture in the neighbourhood of a
m:n resonance manifold of the dynamics. Specifically,
for the pair of tongues S13±, a 1:3 resonance capture
occurs that leads to subharmonic TET with the linear
oscillator vibrating with a frequency three times that of
the NES. It is emphasized that due to the stability properties of the tongues S13±, subharmonic TET involves
excitation of S13−, but not S13+.
The transient dynamics when the motion is initiated at the extremity of S13− (see the initial condition
denoted by the box on the right part in Fig. 8(d))
is displayed in Fig. 13. The same parameters as in
the previous section are considered. Until t = 500 s,
subharmonic TET takes place. Despite the presence
of viscous dissipation, the NES response grows continuously, with simultaneous rapid decrease of the
response of the linear oscillator. A substantial amount
of energy is transferred to the NES (see Fig. 13(e)), and
eventually nearly 70 per cent of the energy is dissipated by the NES damper (see Fig. 13(f )). A prolonged
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
1:3 resonance capture is nicely evidenced by the WT of
Figs 13(c) and (d), and the motion follows the whole
lower tongue S13− from the right to the left. Once
escape from resonance capture occurs (around t =
620–630 s), energy is no longer transferred to the NES.
For analytical study of subharmonic TET, TET in
the neighbourhood of tongue S13− will be the focus
(similar analysis can be applied for other orders of subharmonic resonance captures). Due to the fact that
motions in the neighbourhood of S13− possess two
main frequency components, at frequencies 1 and 1/3,
the responses of system (2) can be expressed as
y(t) = y1 (t) + y 13 (t),
v(t) = v1 (t) + v 13 (t)
(17)
where the indices represent the frequency of each
term. As in the previous case, new complex variables
are introduced
ψ1 (t) = ẏ1 (t) + jωy1 (t) ≡ ϕ1 (t) e jωt ,
ω
ωt
ψ3 (t) = ẏ 13 (t) + j y 13 (t) ≡ ϕ3 (t) e j 3
3
ψ2 (t) = v̇1 (t) + jωv1 (t) ≡ ϕ2 (t) e jωt ,
ω
ωt
ψ4 (t) = v̇ 13 (t) + j v 13 (t) ≡ ϕ4 (t) e j 3
3
(18)
where ϕi (t), i = 1, . . . , 4 represent slowly varying modulations of fast oscillations of frequencies 1 or 1/3.
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 13
Subharmonic TET initiated on S13−: shown are the transient responses of the (a) linear
oscillator and (b) NES; WTs of the motion of (c) the NES and (d) the linear oscillator; (e)
percentage of instantaneous total energy in the NES; (f ) percentage of total input energy
dissipated by the NES [84]
Expressing the system responses in terms of the new
complex variables
y=
99
ψ1 − ψ1∗
ψ3 − ψ3∗
+
,
2jω
2j(ω/3)
v=
ψ2 − ψ2∗
ψ4 − ψ4∗
+
2jω
2j(ω/3)
(19)
substituting into equation (2), and averaging over
each of the two fast frequencies, the slow modulation
JMBD118 © IMechE 2008
equations that govern the evolutions of the complex
amplitudes are derived as
1
1
λ
ω−
ϕ1 − (2ϕ1 − ϕ2 )
2
ω
2
9C
− j 3 [3ϕ33 − 9ϕ32 ϕ4 − 3ϕ43 + 9ϕ3 ϕ42
8ω
− (ϕ1 − ϕ2 )|ϕ1 − ϕ2 |2 − 6(ϕ1 − ϕ2 )|ϕ3 − ϕ4 |2 ]
ϕ̇1 = −j
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
100
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
ω
λ
9C
ϕ̇2 = −j ϕ2 − (ϕ2 − ϕ1 ) + j 3 [3ϕ33
2
2
8ω
− 9ϕ32 ϕ4 − 3ϕ43 + 9ϕ3 ϕ42
− (ϕ1 − ϕ2 )|ϕ1 − ϕ2 |2 − 6(ϕ1 − ϕ2 )|ϕ3 − ϕ4 |2 ]
1 ω
λ
3
ϕ̇3 = −j
−
ϕ3 − (2ϕ3 − ϕ4 )
2 3
ω
2
9C
+ j 3 [ϕ1 (2(ϕ3 − ϕ4 )(ϕ1∗ − ϕ2 ) − 3(ϕ3∗ − ϕ4∗ )2 )
8ω
+ ϕ2 (2(ϕ4 − ϕ3 )(ϕ1∗ − ϕ2 ) + 3(ϕ3∗ − ϕ4∗ )2 )
+ 9(ϕ3 − ϕ4 )|ϕ3 − ϕ4 |2 ]
ω
λ
9C
ϕ̇4 = −j ϕ4 − (ϕ4 − ϕ3 ) − j 3
6
2
8ω
× [ϕ1 (2(ϕ3 − ϕ4 )(ϕ1∗ − ϕ2 ) − 3(ϕ3∗ − ϕ4∗ )2 )
+ ϕ2 (2(ϕ4 − ϕ3 )(ϕ1∗ − ϕ2 ) + 3(ϕ3∗ − ϕ4∗ )2 )
+ 9(ϕ3 − ϕ4 )|ϕ3 − ϕ4 |2 ]
(20)
where again it was assumed that λ1 = λ2 = λ in
equation (2). To derive a set of real modulation equations, the complex amplitudes are expressed in polar
form ϕi (t) = ai (t)e jβi (t) , and an autonomous set of
seven slow-flow modulation equations that govern the
amplitudes ai = |ϕi |, i = 1, . . . , 4 and the phase differences φ12 = β1 − β2 , φ13 = β1 − 3β3 , and φ14 = β1 − 3β4
are derived.
The equations of the autonomous slow flow will not
be reproduced here, but it suffices to state that they
are of the form
λ
(2a1 − a2 ) + g1 (a, φ),
2
λ
ȧ2 = − (a2 − a1 ) + g2 (a, φ)
2
λ
ȧ3 = − (2a3 − a4 ) + g3 (a, φ),
2
λ
ȧ4 = − (a4 − a3 ) + g4 (a, φ)
2
φ̇12 = f12 (a) + g12 (a, φ; ),
ȧ1 = −
φ̇13 = f13 (a) + g13 (a, φ)
φ̇14 = f14 (a) + g14 (a, φ; )
(21)
where the functions gi and gij are 2π-periodic in
terms of the phase angles φ = (φ12 , φ13 , φ14 )T , and
a = (a1 , . . . , a4 )T .
In this case (as for the fundamental TET mechanism), strong energy transfer between the linear and
non-linear oscillators can occur only when a subset of
phase angles φkl does not exhibit time-like behaviour;
that is, when some phase angles possess oscillatory
(non-monotonic) behaviour with respect to time. This
can be seen from the structure of the slow flow (21)
where, if the phase angles exhibit time-like behaviour
and the functions gi are small, averaging over these
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
phase angles can be performed to show that the amplitudes decrease monotonically with time; in that case,
no significant energy exchanges between the linear
and non-linear components of the system can take
place. It follows that subharmonic TET is associated
with non-time-like behaviour of (at least) a subset of
the slow-phase angles φkl in equation (21).
Figure 14 presents the results of the numerical integration of the slow-flow equations (20) and (21) for
the system with parameters = 0.05, λ = 0.03, C = 1,
and ω0 = 1. The motion is initiated on branch S13−
with initial conditions v(0) = y(0) = 0 and v̇(0) =
0.01 499, and ẏ(0) = −0.059 443 (it corresponds exactly
to the simulation of Fig. 13). The corresponding initial conditions and the value of the frequency ω of
the reduced slow-flow model were computed by minimizing the difference between the analytical and
numerical responses of the system in the interval
t ∈ [0, 100]: ϕ1 (0) = −0.0577, ϕ2 (0) = 0.0016, ϕ3 (0) =
−0.0017, ϕ4 (0) = 0.0134, and ω = 1.0073.
This result indicates that, initially, nearly all energy
is stored in the fundamental frequency component of
the linear oscillator, with the remainder confined to
the subharmonic frequency component of the NES.
Figures 14(a) and (b) depict the temporal evolutions of
the amplitudes ai , from which it is concluded that subharmonic TET in the system is mainly realized through
energy transfer from the (fundamental) component at
frequency ω of the linear oscillator, to the (subharmonic) component at frequency ω/3 of the NES (as
judged from the build-up of the amplitude a3 and the
diminishing of a1 ). A smaller amount of energy is transferred from the fundamental frequency component of
the linear oscillator to the corresponding fundamental
component of the NES (as judged by the evolution of
the amplitude a2 ).
These conclusions are supported by the plots of
Figs 14(c) to (e), where the temporal evolutions of the
phase differences φ12 = β1 − β2 , φ13 = β1 − 3β3 , and
φ14 = β1 − 3β4 are shown. Absence of strong energy
exchange between the fundamental and subharmonic
frequency components of the linear oscillator is associated with the time-like behaviour of the phase
difference φ13 , whereas TET from the fundamental
component of the linear oscillator to the two frequency
components of the NES is associated with oscillatory
early time behaviour of the phase differences φ12 and
φ14 . Oscillatory responses of φ12 and φ14 correspond to
1:1 and 1:3 resonance captures, respectively, between
the corresponding frequency components of the linear oscillator and the NES; as time increases, time-like
responses of the phase variables are associated with
escapes from the corresponding regimes of resonance
capture. In addition, it is noted that the oscillations of
the angles φ12 and φ14 take place in the neighbourhood
of π, which confirms that, in this particular example,
subharmonic TET is activated by the excitation of
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 14
101
Subharmonic TET: (a) amplitude modulations; (b)–(d) phase modulations [84]
response of the NES in the resonance capture region.
The analytical model fails, however, during the escape
from resonance capture since the ansatz (17) and (18)
is not valid in that regime of the motion. Indeed,
after escape from resonance capture, the motion
approximately evolves along the backbone curve of
the FEP; eventually S15 is reached whose motion cannot be described by the ansatz (17) and (18), thereby
leading to the failure of the analytical model.
3.2.3
Fig. 15 Transient response of NES for 1:3 subharmonic
TET; comparison between analytical approximation (dashed line) and direct numerical
simulation (solid line)
an anti-phase branch of periodic solutions (such as
S13−). The analytical results are in full agreement
with the WTs depicted in Figs 5(c) and (d), where the
response of the linear oscillator possesses a strong
frequency component at the fundamental frequency
ω0 = 1, whereas the NES oscillates mainly at frequency
ω0 /3.
The accuracy of the analytical model (20) and (21)
in capturing the dynamics of subharmonic TET is confirmed by the plot depicted in Fig. 15 where the analytical response of the NES is found to be in satisfactory
agreement with the numerical response obtained by
the direct simulation of equation (2). Interestingly,
the reduced analytical model is capable of accurately
modelling the strongly non-linear, damped, transient
JMBD118 © IMechE 2008
TET initiated by non-linear beating
The previous two mechanisms cannot be activated
with the NES at rest, since in both cases the motion
is initialized from a non-localized state of the system.
This means that these energy pumping mechanisms
cannot be activated directly after the application of an
impulsive excitation to the linear oscillator with the
NES initially at rest. Such a forcing situation, however,
is important from a practical point of view; indeed,
this is the situation where local NESs are utilized to
confine and passively dissipate unwanted vibrations
from linear structures that are forced by impulsive (or
broadband) loads.
Hence, it is necessary to discuss an alternative, third
energy pumping mechanism capable of initiating passive energy transfer with the NES initially at rest. This
alternative mechanism is based on the excitation of
a special orbit that plays the role of a ‘bridging orbit’
for activation of either fundamental or subharmonic
TET. Excitation of a special orbit results in the transfer of a substantial amount of energy from the initially
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
102
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
excited linear oscillator directly to the NES through a
non-linear beat phenomenon. In that context, the special orbit may be regarded as an initial ‘bridging orbit’
or trigger, which eventually activates fundamental or
subharmonic TET once the initial non-linear beat initiates the energy transfer. Indeed, as shown below, the
third mechanism for TET represents an efficient initial (triggering) mechanism for rapid transfer of energy
from the linear oscillator to the NES at the crucial initial stage of the motion, before activating either one of
the (fundamental or subharmonic) main TET mechanisms through a non-linear transition (jump) in the
dynamics.
To study the dynamics of this triggering mechanism, the following conjecture is formulated: Due
to the essential (non-linearizable) non-linearity, the
NES is capable of engaging in a m:n resonance capture with the linear oscillator, m and n being a set
of integers. Accordingly, in the undamped system,
there exists a sequence of special orbits (corresponding to non-zero initial velocity of the linear oscillator
and all other initial conditions zero), aligned along a
one-dimensional smooth manifold in the FEP.
Fig. 16
As a first step to test this conjecture, a NLBVP was
formulated to compute the periodic orbits of system
(2) with no damping, and the additional restriction
for the special orbits was imposed. The numerical
results in the frequency–energy plane are depicted
in Fig. 16 for parameters = 0.05, ω0 = 1, and C = 1.
Each triangle in the plot represents a special orbit, and
a one-dimensional manifold appears to connect the
special orbits; a rigorous proof of the existence of this
manifold can be found in reference [85]. In addition, it
appears that there exist a countable infinity of special
orbits, occurring in the neighbourhoods of the countable infinities of IRs m:n (m, n integers) of the system.
It is noted that a subset of high-frequency branches
(for ω > 1) possesses two special orbits instead of one
(for example, all U (p + 1)p branches with p 3). To
distinguish between the two special solutions in such
high-frequency branches, they are partitioned into
two subclasses: the a-special orbits that exist in the
neighbourhood of ω = ω0 = 1, and the b-special orbits
that occur away from this neighbourhood (see Fig. 16).
It was proven numerically that the a-special orbits are
unstable, whereas the b-special orbits are stable [83].
Manifold of special orbits (represented by triangles) in the FEP [84]
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
As shown below it is the excitation of the stable
b-special orbits that activates the third mechanism for
TET.
By construction, all special orbits have a common
feature; namely, they pass with vertical slope through
the origin of the configuration plane (v, y). This feature renders them compatible with an impulse applied
to the linear oscillator, which corresponds to a nonzero velocity of the linear oscillator and all other
initial conditions zero. The curves corresponding to
the special orbits in the configuration plane can be
either closed or open depending upon the differences
between the two indices characterizing the orbits;
specifically, odd differences between indices correspond to closed curves in the configuration plane and
lie on U -branches, whereas even differences between
indices correspond to open curves on S-branches.
In addition, higher frequency special orbits (with
frequency index ω > ω0 ) in the upper part of the FEP
(i.e. m > n) are localized to the non-linear oscillator; conversely, special orbits in the lower part of the
FEP (with frequency index ω < ω0 ) tend to be localized to the linear oscillator. This last observation is
of particular importance since it directly affects the
transfer of a significant amount of energy from the
linear oscillator to the NES through the mechanism
discussed in this section. Indeed, there seems to be
a well-defined critical threshold of energy that separates high- from low-frequency special orbits; i.e. those
that do or do not localize to the NES, respectively (see
Fig. 16).
The third mechanism for TET can only be activated for input energies above the critical threshold,
since below that the (low-frequency) special orbits are
incapable of transferring significant amounts of input
energy from the linear oscillator to the NES; in other
words, the critical level of energy represents a lower
bound below which no significant TET can be initiated through activation of a special orbit. Moreover,
combining this result with the topology of the onedimensional manifold of special orbits of Fig. 16, it
follows that it is the subclass of stable b-special orbits
that is responsible for activating the third TET mechanism, whereas the subclass of unstable a-special orbits
does not affect TET. This theoretical insight will be fully
validated by the numerical simulations that follow.
When the NES engages in a m:n resonance capture with the linear oscillator, a non-linear beat
phenomenon takes place. Due to the essential (nonlinearizable) non-linearity of the NES and the lack
of any preferential frequency, this non-linear beat
phenomenon does not require any a priori tuning
of the non-linear attachment, since at the specific
frequency–energy range of the m:n resonance capture, the non-linear attachment adjusts its amplitude
(tunes itself ) to fulfil the necessary conditions of
IR. This represents a significant departure from the
JMBD118 © IMechE 2008
103
‘classical’ non-linear beat phenomenon observed in
coupled oscillators with linearizable non-linear stiffnesses (e.g. spring–pendulum systems [129]), where
the defined ratios of linearized natural frequencies of
the component subsystems dictate the type of IRs that
can be realized [14, 167].
As an example, Fig. 17 depicts the exchanges of
energy during the non-linear beat phenomenon corresponding to the special orbits of branches U 21
and U 54 for parameters = 0.05, ω0 = 1, C = 1, and
no damping. As expected, energy is continuously
exchanged between the linear oscillator and the NES,
so the energy transfer is not irreversible as is required
for TET; it can be concluded that excitation of a special
orbit can only initiate (trigger) TET, but not cause it in
itself. The amount of energy transferred during each
cycle of the beat varies with the special orbit considered; for U 21 and U 54, as much as 32 per cent and
86 per cent of energy can be transferred to the NES,
respectively. It can be shown that, for increasing integers m and n with corresponding ratios m/n → 1+ ,
the maximum energy transferred during a cycle of the
special orbit tends to 100 per cent. At the same time,
however, the resulting period of the cycle of the beat
(and, hence, of the time needed to transfer the maximum amount of energy) should increase as the least
common multiple of m and n.
Note, at this point, that the non-linear beat phenomenon associated with the excitation of the special orbits can be studied analytically using the
complexification-averaging method [135]. To demonstrate the analytical procedure, the special orbit on
branch U 21 of the system with no damping is analysed in detail. In the previous section, the periodic
motions on this entire branch were studied, and it
was shown that the responses of the linear oscillator
and the non-linear attachment can be approximately
expressed as
y(t) ≈ Y1 sin ωt + Y2 sin 2ωt ≡ y1 (t) + y2 (t)c
v(t) ≈ V1 sin ωt + V2 sin 2ωt ≡ v1 (t) + v2 (t)
(22)
where the amplitudes are
Y1 =
A
,
ω
V1 =
B
,
ω
Y2 =
D
,
2ω
V2 =
G
2ω
and A, B, D, and G are computed from the stationarity
conditions in the slow-flow equations as
B=±
G=±
4ω4 (Z2 − 8Z1 )
,
9CZ13 Z2
32ω4 (2Z1 − Z2 )
ω2
⇒A= 2
B,
3
9CZ2 Z1
ω0 − ω2
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
104
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 17
Exchanges of energy during non-linear beat phenomena corresponding to special orbits
on (a), (b) U 21, and (c) and (d) U 54
D=
4ω2
G
ω02 − 4ω2
Z1 =
ω2
− 1,
ω02 − ω2
Z2 =
4ω2
−1
ω02 − 4ω2
Hence, a two-frequency approximation is satisfactory
for this family of periodic motions. The frequency ωSO
at which the special orbit appears is computed by
imposing the initial conditions y(0) = v(0) = v̇(0) = 0,
which leads to the relation
B = −2G
(special orbit)
The instantaneous fraction of total energy in the linear oscillator during the non-linear beat phenomenon
is estimated to be
Elinear (t)
=
2
2
) sin ωSO t − 2(ω02 − ωSO
) sin 2ωSO t]2
[(ω02 − 4ωSO
2
2
9ωSO ω0
2
2
(ω02 − 4ωSO
) cos ωSO t
2
) cos 2ωSO t
−4(ω02 − ωSO
+
(23)
9ω04
The non-linear coefficient C has no influence on the
fraction of total energy transferred to the NES during the non-linear beat; this means that, during the
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
beat, the instantaneous energies of the linear oscillator and the NES are directly proportional to the
non-linear coefficient. Moreover, as the mass of the
NES tends to zero, the frequency where the special
orbit is realized tends to the limit ωSO → ω, and, as a
result, Elinear (t) → 1, and the energy transferred to the
NES during the beat tends to zero. However, it is noted
that this is a result satisfied only asymptotically since,
as indicated by the results depicted in Fig. 17, even
for very small mass ratios, e.g. = 0.05, as much as
86 per cent of the total energy can be transferred to the
NES during a cycle of the special orbit of branch U 54.
Considering now the damped system, it will be
shown that following an initial non-linear beat phenomenon, either one of the main (fundamental or subharmonic) TET mechanisms can be activated through
a non-linear transition (jump) in the dynamics. It
was previously mentioned that the two main TET
mechanisms are qualitatively different from the third
mechanism, which is based on the excitation of a
non-linear beat phenomenon (special orbit). Indeed,
damping is a prerequisite for the realization of the two
main mechanisms, leading to an irreversible energy
transfer from the linear oscillator to the NES, whereas
a special orbit is capable of transferring energy without dissipation, though this transfer is not irreversible
but periodic. This justifies the earlier assertion that the
third mechanism does not represent an independent
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
105
Fig. 18 TET by non-linear beat, transition to S11+. Shown are the transient responses of the (a)
linear oscillator and (b) NES; WTs of the motion of (c) the NES and (d) the linear oscillator;
(e) percentage of instantaneous total energy in the NES; (f ) percentage of total input energy
dissipated by the NES [84]
mechanism for energy pumping, but rather triggers
it, and through a non-linear transition activates either
one of the two main mechanisms. This will become
apparent in the following numerical simulations.
The following simulations concern the transient
dynamics of the damped system (2) with parameters = 0.05, ω0 = 1, C = 1, λ1 = λ2 = 0.0015, and an
impulse of magnitude Y applied to the linear oscillator
JMBD118 © IMechE 2008
(corresponding to initial conditions y(0+ ) = v(0+ ) =
v̇(0+ ) = 0, ẏ(0+ ) = Y ). By varying the magnitude of
the impulse, the different non-linear transitions which
take place in the dynamics and their effects on TET
are studied. The responses of the system to the relatively strong impulse Y = 0.25 are depicted in Fig. 18.
Inspection of the WTs of the responses (see Figs 18(c)
and (d)), and of the portion of total instantaneous
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
106
Fig. 19
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Percentage of input energy eventually dissipated at the NES for varying magnitude of the
impulse (the positions of certain special orbits
are indicated) [84]
energy captured by the NES (see Fig. 18(e)), reveals
that at the initial stage of the motion (until approximately t = 120 s) the (stable) b-special orbit on branch
U 32 is excited (since the NES response possesses two
main frequency components at 1 and 3/2 rad/s), and
a non-linear beat phenomenon takes place. (Note
the continuous exchange of energy between the two
subsystems, demonstrating reversibility in this initial stage of the motion.) For t > 120 s, the dynamics
undergoes a transition (jump) to branch S11+, and
fundamental TET to the NES occurs on a prolonged
1:1 resonance capture (see Figs 18(c) and (d)); eventually, 84 per cent of the input energy is dissipated by the
damper of the NES (see Fig. 18(f )).
3.2.4
Critical energy threshold necessary for
initiating TET
To demonstrate more clearly the effect of the b-special
orbits on TET, Fig. 19 depicts the percentage of input
energy eventually dissipated at the NES for varying
magnitude of the impulse for the system with parameters = 0.05, ω0 = 1, C = 1, and λ1 = λ2 = 0.01. In the
same plot, the positions of the special orbits of the
undamped system and the critical threshold predicted
in Fig. 16 are depicted.
It is concluded that strong TET is associated with
the excitation of b-special orbits of the branches
U (p + 1)p in the neighbourhood above the critical
threshold, whereas excitation of a-special orbits below
the critical threshold does not lead to rigorous energy
pumping. As mentioned previously, in the neighbourhood of the critical threshold, the b-special orbits are
strongly localized to the NES, whereas a-orbits are
non-localized. The deterioration of TET is also noted
from Fig. 19 as the magnitude of the impulse well
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
Fig. 20
Contours of percentage of input energy eventually dissipated at the NES for the case when
both oscillators excited by impulses; superimposed are contours of high- and low-frequency
branches of the undamped system (solid line:
in-phase, dashed line: out-of-phase branches);
special orbits in high- and low-frequency
branches are denoted by circles and triangles,
respectively [84]
above the critical threshold increases, where highfrequency special orbits are excited; this is a consequence of the fact that well above the critical threshold,
the special orbits are weakly localized to the NES.
Extending the previous result, Fig. 20 depicts the
contours of energy eventually dissipated at the NES,
but for the case of two impulses of magnitudes ẏ(0)
and v̇(0) applied to both the linear oscillator and
the NES, respectively. The system parameters used
were identical to those of the previous simulation of
Fig. 19. Superimposed on contours of energy dissipated at the NES are certain high- and low-frequency
U - and S-branches of the undamped system together
with their special orbits, in order to confirm for this
case the essential role of the high-frequency special
orbits in TET. Indeed, high levels of energy dissipation are encountered in neighbourhoods of contours
of high-frequency U -branches, whereas low values are
noted in the vicinity of low-frequency branches. These
results agree qualitatively with the earlier theoretical
and numerical findings, and enable one to assess and
establish the robustness of TET when the NES is not
initially at rest.
The results presented thus far provide a measure
of the complicated dynamics encountered in the
two-DOF system under consideration. It is logical to
assume that by increasing the number of DOFs of
the system, the dynamics will become even more
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
107
complex. That this is indeed the case is revealed by the
numerical simulations presented in the next section
where resonance capture cascades are reported in
MDOF linear systems with essentially non-linear end
attachments. By resonance capture cascades, complicated sudden transitions between different branches
of solutions (modes), which are accompanied by sudden changes in the frequency content of the system
responses, are denoted. As shown in previous works
[78], such multi-frequency transitions can drastically
enhance TET from the linear system to the essentially
non-linear attachment.
3.3
MDOF and continuous oscillators
To gain additional insight into the dynamics of TET,
the case of combinations of MDOF systems composed of linear primary systems with attached SDOF
or MDOF ungrounded NESs is considered. Results
on this specific problem can also be found in references [77–80, 89]. Consider, first, the case of the
two-DOF linear primary system with attached SDOF
ungrounded NES
ÿ2 + ω02 y2 + λ2 ẏ2 + d( y2 − y1 ) = 0
ÿ1 +
ω02 y1
Fig. 21
+ λ1 ẏ1 + λ3 (ẏ1 − v̇) + d( y1 − y2 )
+ C( y1 − v)3 = 0
v̈ + λ3 (v̇ − ẏ1 ) + C(v − y1 )3 = 0
(24)
The system parameters are chosen as ω0 = 136.9 (rad/s),
λ1 = λ2 = 0.155, λ3 = 0.544, d = 1.2 × 103 , = 1.8, and
C = 1.63 × 107 , with linear natural frequencies ω1 =
11.68 and ω2 = 50.14 (rad/s).
Figure 21(a) depicts the relative response v(t) − y1 (t)
of the system for initial displacements y1 (0) = 0.01,
y2 (0) = v(0) = −0.01, and zero initial velocities. The
multi-frequency content of the transient response is
evident and is quantified in Fig. 21(b), where the
instantaneous frequency of the time series is computed by applying the numerical Hilbert transform
[95].
As energy decreases because damping dissipation, a
series of eight resonance capture cascades is observed;
i.e. of transient resonances of the NES with a number
of non-linear modes of the system. The complexity
of the non-linear dynamics of the system is evidenced by the fact that of these eight captures only
two (labelled IV and VII in Fig. 21(b)) involve the
linearized in-phase and out-of-phase modes of the
linear oscillator, with the remaining involving essentially non-linear interactions of the NES with different
low- and high-frequency non-linear modes of the system. On the average, during these resonance captures,
the NES passively absorbs energy from the non-linear
JMBD118 © IMechE 2008
Resonance capture cascades in the two-DOF
system with non-linear end attachment: (a)
relative transient response v(t) − y1 (t); (b)
instantaneous frequency (resonance captures indicated). The two natural frequencies
are computed as f1 = ω1 /2π = 1.86 Hz and
f2 = ω2 /2π = 7.98 Hz where ω1 = 11.68 and
ω2 = 50.14 (rad/s) [84]
mode involved, before escape from resonance capture
occurs and the NES transiently resonates with the next
mode in the series.
In essence, the NES acts as a passive, broadband
boundary controller, absorbing, confining, and eliminating vibration energy from the linear oscillator. Similar types of resonance capture cascades were reported
in previous works where grounded NESs, weakly coupled to the linear structure, were examined [78]. The
capacity of the NES to resonantly interact with linear
and non-linear modes in different frequency ranges
is due to its essential non-linearity (i.e. the absence
of a linear term in the non-linear stiffness characteristic), which precludes any preferential resonant
frequency.
3.3.1
Analysis of a two-DOF linear primary system
with an SDOF NES
The first system which is considered here is a twoDOF linear primary system with an attached SDOF
NES (Fig. 22), in which the effect that the increase in
DOF of the primary system has on the TET dynamics
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
jω
ϕ3
2
3jC 2 ∗
−
(ϕ ϕ − ϕ22 ϕ3∗ − |ϕ3 |2 ϕ3 + |ϕ2 |2 ϕ2
8ω3 3 2
+ 2|ϕ3 |2 ϕ2 − 2|ϕ2 |2 ϕ3 ) = 0
ϕ̇3 +
(27)
Fig. 22 Two-DOF primary system coupled to an
ungrounded NES
is studied. Equations of motion assume the form
m1 ÿ1 + λ1 ẏ1 + k1 y1 + k12 ( y1 − y2 ) = 0
y1 = a1 sin ωt,
m2 ÿ2 + λ2 ẏ2 + λ(ẏ2 − v̇) + k2 y2 + k12 ( y2 − y1 )
+ C( y2 − v)3 = 0
v̈ + λ(v̇ − ẏ2 ) + C(v − y2 ) = 0
3
(25)
where y1 , y2 , v refer to the displacements of the primary
system and the NES, respectively. For obvious practical
reasons, a lightweight NES is specified by requiring
that 1; in this way, weak damping is also assured.
All other variables are treated as O(1) quantities.
As shown in the previous section, understanding
the topological structure of the FEP of the underlying
Hamiltonian system is a prerequisite for interpreting
(even complex) damped transitions in the damped
and forced system. Hence, the analysis focuses on the
analytical computation of the FEP of the undamped
and unforced system. The complexification-averaging
technique is utilized for the analytical approximation
of the main backbone curves on the FEP, which correspond to 1:1 resonant oscillations of the primary
system and the NES (i.e. the dominant frequencies
of these two system are identical). At this point, the
complex variables are introduced
1 = ẏ1 + jωy1 ,
2 = ẏ2 + jωy2 ,
The complex amplitudes ϕi can be expressed in polar
form as ϕi = ai e jβi , ai , βi ∈ R for i = 1, 2, 3. Then, by
imposing stationarity conditions on the slow-flow
equations and considering trivial phase differences
such that β1 − β2 = β1 − β3 = 0, an approximation of
the NNMs on the main backbone is obtained
3 = v̇ + jωv
(26)
y2 = a2 sin ωt,
v = a3 sin ωt
(28)
where the amplitudes ai , i = 1, 2, 3 can be found as a
function of frequency ω by solving the algebraic equations resulting from the steady-state conditions of the
real-valued slow-flow equations.
The main backbone branches can now be constructed by varying the frequency ω and representing a NNM at a point (h, ω) on the FEP where the
total energy h = ω2 /2[m1 a1 (ω)2 + m2 a2 (ω)2 + a3 (ω)2 ]
is conserved when the system oscillates in a specific mode. Figure 23 depicts the backbone branch,
named S111, of the system with parameters m1 =
m2 = 1, k1 = k2 = k12 = 1, C = 1, and = 0.05. NNMs
depicted as projections of the three-dimensional configuration space (v, y1 , y2 ) of the system are superimposed to demonstrate mode localization behaviours
with respect to the total energy of the system; the
horizontal and vertical axes in these plots are the nonlinear and primary system responses, respectively.
Four characteristic frequencies, f1L , f2L , f1H , and f2H ,
are defined in this plot. At high-energy levels and finite
frequencies, the essential non-linearity behaves as a
rigid link, and the system dynamics is governed by the
equations
m1 ÿ1 + k1 y1 + k12 ( y1 − y2 ) = 0
(m2 + )ÿ2 + k2 y2 + k12 ( y2 − y1 ) = 0
(29)
which are then substituted into equation (25). Expressing the complex variables in polar form i = ϕi e jωt ,
i = 1, 2, 3 and performing averaging over the fast frequency, the complex-valued slow-flow modulation
equations are obtained
jϕ1
jϕ2
(m1 ω2 − k1 − k12 ) +
k12 = 0
2ω
2ω
jϕ2
jϕ1
m2 ϕ̇2 +
(m2 ω2 − k2 − k12 ) +
k12
2ω
2ω
3jC 2 ∗
+
(ϕ ϕ − ϕ22 ϕ3∗ − |ϕ3 |2 ϕ3 + |ϕ2 |2 ϕ2
8ω3 3 2
+ 2|ϕ3 |2 ϕ2 − 2|ϕ2 |2 ϕ3 ) = 0
m1 ϕ̇1 +
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
The natural frequencies of this system are f1H =
0.9876 and f2H = 1.7116 rad/s for the above parameters. At low-energy levels, the equivalent stiffness of
the essential non-linearity tends to zero, and the system dynamics is that of the primary system,√the natural
frequencies of which are f1L = 1 and f2L = 3 rad/s.
From Fig. 23, it is observed that the two frequencies
f1L and f2L divide the FEP into three distinct regions.
1. The first region, for which ω f2L , comprises the
branch S111 + −+ (the ± signs indicate whether
the initial condition of the corresponding oscillator
is positive or negative, respectively). On this branch,
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 23
Analytic approximation of the main backbone branches of the system m1 = m2 = 1,
k1 = k2 = k12 = 1, C = 1, = 0.05. NNMs
depicted as projections of the three-dimensional configuration space (v, y1 , y2 ) of the
system are superimposed; the horizontal and
vertical axes in these plots are the non-linear
and primary system responses, respectively
(top plot: (v, y1 ); bottom plot: (v, y2 ); see legend
in the bottom right corner). The aspect ratio is
set so that increments on the horizontal and
vertical axes are equal in size, enabling one to
directly deduce whether the motion is localized
to the primary system or to the non-linear
oscillator [87]
the primary system vibrates in an anti-phase fashion, and the motion is more and more localized to
the primary system or to the NES as the frequency
approaches f2L or ∞, respectively.
2. The second region, for which f1L ω f2H , comprises two different branches, namely S111 + −−
and S111 + +−. These branches coalesce at a point
S111 + 0− (see the grey dot in Fig. 23), where the
NNM is such that the initial condition on the velocity of the oscillator m2 is zero. On S111 + −−, the
primary system vibrates in an anti-phase fashion,
and the motion localizes to the NES as the frequency goes away from f2H . On S111 + +−, there
is an in-phase motion of the primary system, and
the motion localizes to the primary system, as the
frequency converges to f1L .
JMBD118 © IMechE 2008
Fig. 24
109
Numerical computation of the FEP (backbone and loci of special orbits) of a two-DOF
primary coupled to an NES (m1 = m2 = 1,
k1 = k2 = k12 = 1, C = 1, = 0.05); black dots
and squares denote anti-phase and in-phase
special orbits, respectively [87]
3. The third region, for which ω f1H , comprises the
branch S111 + ++. On this branch, the primary
system vibrates in an in-phase fashion, and the
motion localizes to the NES as the frequency goes
away from f1H .
Owing to the energy dependence of the NNMs along
S111, interesting and vigorous energy exchanges may
occur between the primary system and the NES. In
particular, an irreversible channeling of vibrational
energy from the primary system to the NES takes
place on S111 + −− and S111 + ++. Because the
NES has no preferential resonance frequency, fundamental TET can be realized either for in-phase or
anti-phase motion of the primary system, which shows
the adaptability of the NES.
The SPOs, determined from accommodating specific initial conditions ẏ1 (0) = 0, ẏ2 (0) = 0 with all the
others zero, can also be computed for the MDOF system. The role of special orbits is to transfer as quickly
as possible a significant portion of the induced energy
to the NES, initially at rest, which should trigger TET.
Figure 24 depicts two different families of special
orbits for a two-DOF primary system.
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
1. The first family consists of in-phase SPOs (++0)
located on in-phase tongues; the masses of the
primary system move in-phase. The locus of inphase SPOs is a smooth curve on the FEP. When
the phase difference between the NES and the primary is trivial, the motion in the configuration
space takes the form of a simple curve; in the case
of non-trivial phase differences, a Lissajous curve
is realized. For the SPO 1, the motion of the two
masses of the primary system is almost identical
and monochromatic. The NES has two dominant
harmonic components, one of which is at the frequency of oscillation of the primary system, the
other being three times smaller; a 1:3 IR between
the NES and the primary system is realized. The
non-linear beating characteristic of such a dynamical phenomenon can be clearly observed. For the
SPO 1, the energy exchange is insignificant as the
maximum percentage of total energy of the NES
never exceeds 0.17 per cent. For the SPOs 2 and
3, the energy transfer is much more vigorous. To
obtain a global picture, the maximum percentage
of energy transferred to the NES during the nonlinear beating is superposed on the FEP in Fig. 25.
This clearly depicts that there exists a critical energy
threshold above which the SPOs can transfer a
substantial amount of energy to the NES. More precisely, the SPOs must lie above the frequency of the
in-phase mode of the primary system f1L .
Fig. 25
Maximum percentage of energy transferred to
the NES during non-linear beating (dashed (dotted) line: in-phase (anti-phase) special orbits).
The backbone of the FEP (solid line) and the
loci of the special orbits are also superimposed
(square – in-phase; circle – anti-phase [87])
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
2. The second family consists of anti-phase SPOs
(+ − 0) located on anti-phase tongues. Their locus
is also a smooth curve on the FEP. By inspecting
Fig. 25, one can conclude the existence of a critical
energy threshold for enhanced TETs; the SPOs must
lie above the frequency of the anti-phase mode of
the primary system f2L .
The transient dynamics of the weakly damped system is now examined and is interpreted based on
the topological structure of the non-linear modes of
the undamped system. Damping parameters are set
to λ1 = λ2 = 0.1, λ = 0.04, and others are the same as
those used in constructing the FEP in Fig. 23. In this
section, only the single-mode responses by imposing the in-phase and anti-phase impulsive forcing are
considered, and the multi-mode responses (i.e. resonance capture cascades) will be demonstrated later
compared with the experimental system.
First, the motion initiated on S111 + ++ (i.e. inphase fundamental TET) is examined (Figs 26(a) and
(b)). In Fig. 26(c), the WT of v(t) − y2 (t) is superimposed on the FEP to demonstrate transient dynamics
along the damped NNM manifold as the total energy
decreases due to damping. The dynamical flow is captured in the neighbourhood of a 1:1 resonance manifold, which leads to a prolonged 1:1 resonance capture.
Figure 26(d) depicts the trajectories of the phase differences between the NES and the two masses in
the primary structure. The phase variables were computed by utilizing the Hilbert transform (HT) of the
responses. Non-time-like behaviour of the two phase
variables is observed, as the evidence for resonance
capture. Figure 26(e) confirms that fundamental TET,
i.e. an irreversible energy transfer from the primary
structure to the NES, takes place along S111 + ++.
Now the motion initiated on S111 + −− (i.e. outof-phase fundamental TET) is examined. Figure 27(a)
and (b) depicts the time series where fundamental TET
is realized in a first stage (t = 0 − 100 s) for an antiphase motion of the primary structure. During this
regime, the envelope of all displacements decreases
monotonically, but the envelope of the NES seems to
decrease more slowly than that of the primary structure; TET to the NES is observed (Fig. 27(e)). Around
t = 80 s, the displacement y2 of the second mass m2
becomes very small, and a transition from anti-phase
(S111 + −−) to in-phase (S111 + +−) motion in the
primary structure occurs. When the inflection point
on S111 + +− is reached (where a bifurcation eliminates the stable/unstable pair of NNMs), escape from
resonance capture occurs, which results in time-like
behaviour of the phase variables in Fig. 27(d). Figures 27(b), (c), and (e) show that this is soon followed
by subharmonic TET on an in-phase tongue; there is a
capture into 1:3 resonance manifold.
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Passive non-linear TET and its applications
Fig. 26
111
Fundamental TET for in-phase motion of the primary system: (a) time series; (b) close-up
of the time series (square: y1 (t); circle: y2 (t); reversed triangle: v(t)); (c) WT superimposed
on the frequency-energy plot; (d) trajectories of the phase modulation; (e) instantaneous
percentage of total energy in the NES [87]
A motion initiated from special orbits is examined
to verify the existence of a critical energy threshold above which the SPOs can trigger fundamental
TET. In Fig. 28, the motion is initiated from inphase SPOs 1 and 2, located below and above the
threshold, respectively. The dynamic responses are
remarkably different for those two cases. For the
SPO 1, the NES cannot extract a sufficient amount
JMBD118 © IMechE 2008
of energy from the primary system, and a transition to S111 + +− is observed. On this branch, the
motion localizes to the primary system as the total
energy in the system decreases. For the SPO 2, thanks
to a non-linear beating phenomenon, the motion is
directed towards the basin of attraction of S111 + ++,
and fundamental TET from the in-phase mode is
realized.
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 27
Fundamental TET for anti-phase motion of the primary system: (a) time series; (b) close-up
of the time series (square: y1 (t); circle: y2 (t); reversed triangle: v(t)); (c) WT superimposed
on the frequency–energy plot; (d) trajectories of the phase modulation; (e) instantaneous
percentage of total energy in the NES [87]
Likewise, if the motion is initiated from an antiphase SPO located below the threshold (e.g. SPO 4),
there occurs a transition to S111 + −+, on which
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
the motion localizes to the primary system with a
decrease in the total energy. If the anti-phase SPO lying
above the threshold (e.g. SPO 6) is excited, the branch
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 28
Motion initiated from in-phase special orbits: (a, b) time series; (c, d) WT superimposed on the frequency energy plot; (e, f ) instantaneous percentage of total energy in the
NES [87]
S111 + −− is reached, resulting in the realization of
fundamental TET from the anti-phase mode.
3.3.2
113
Analysis of an SDOF linear primary system
with an MDOF NES
Application of an MDOF NES is now considered. It is
showed that enhanced TET takes place in this case
JMBD118 © IMechE 2008
because of the capacity of the essentially non-linear
MDOF NES to engage in simultaneous resonance
captures with multiple modes of the linear system.
Consider the system in Fig. 29, where a two-DOF linear
primary oscillator is connected through a weak linear
stiffness (which is the small parameter of the problem), 0 < 1, to a three-DOF non-linear attachment with the two essentially non-linear stiffnesses,
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
u(0) = 0 for a given period T . Numerically this is
performed by minimizing the expression
min{[u(T )u̇(T )] − [0u̇(0)]}
T
Then, the total energy h of the underlying Hamiltonian
system, when it oscillates with a periodic solution of
frequency ω = 2π/T , is expressed as
Fig. 29
Primary (linear) system
non-linear attachment
with
an
MDOF
C1 and C2 . The equations of motion for this system can
be written as
ü1 + λu̇1 + (ω02 + α)u1 − αu2 = F1 (t)
ü2 + λu̇2 + (ω02 + α + )u2 − αu1 − v1 = F2 (t)
μv̈1 + λ(v̇1 − v̇2 ) + (v1 − u2 ) + C1 (v1 − v2 )3 = 0
μv̈2 + λ(2v̇2 − v̇1 − v̇3 ) + C1 (v2 − v1 )3
+ C2 (v2 − v3 )3 = 0
μv̈3 + λ(v̇3 − v̇2 ) + C2 (v3 − v2 )3 = 0
(30)
In the limit → 0, the system decomposes into two
uncoupled oscillators: a two-DOF linear
primary sys
tem with natural frequencies ω1 = ω02 + 2α and ω2 =
ω0 < ω1 , corresponding to out-of-phase and in-phase
linear modes, respectively; and a three-DOF essentially non-linear oscillator with a rigid-body mode and
two flexible NNMs.
Unlike the SDOF NES configuration, this MDOF
NES exhibits multi-frequency simultaneous TETs from
multiple modes of the primary system; this means
that multiple non-linear modes of the MDOF NES
engage in transient resonance interactions with multiple modes of the linear system. Once again, complex
transitions in the damped dynamics can be related to
the topological structure of the periodic orbits of the
corresponding undamped system.
For practical purposes, the system with NES masses
of O() is considered with parameter values = 0.2,
α = 1.0, C1 = 4.0, C2 → 2 C2 = 0.05, μ → 2 μ = 0.08,
and ω0 = 1, where rescaling was applied to the NES
masses μ and the second essentially nonlinear coupling spring C2 .
As performed in previous sections, the FEP of
the underlying Hamiltonian system was considered
first. A numerical method was utilized to construct the FEP of the periodic solutions of the
underlying Hamiltonian system [87]. Denoting u(t) =
[u1 (t)u2 (t)v1 (t)v2 (t)v3 (t)]T , the periodic solutions of
the undamped and unforced system (30) can be determined by computing the values of u̇(0) for which
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
h=
1
[u̇1 (0)2 + u̇2 (0)2 + μv̇1 (0)2 + μv̇2 (0)2 + μv̇3 (0)2 ]
2
Considering as a perturbation parameter, system (30)
with the rescaled parameter μ → 2 μ is expected to
possess complicated dynamics as → 0, because it is
essentially (or strongly) non-linear, high-dimensional,
and singular (since the highest derivatives in three
of its equations are multiplied by the perturbation
parameter squared).
In Fig. 30, the periodic orbits are presented in a
FEP. Note that it was difficult to capture the lowest
frequency branch through the numerical scheme. It
was analytically estimated and superimposed to the
numerical results [88]. From the FEP of Fig. 30, it is
noted that the backbone branches of periodic orbits
are defined over wider frequency and energy ranges
than for the system of the NES masses of O(1) [88],
and no subharmonic tongues exist in this case (at least
none was detected in the numerical scheme). Hence,
it can be conjectured that a decrease in magnitude
of the masses of the NES results in the elimination
of the local subharmonic tongues (i.e. of the subharmonic motions at frequencies integrally related to
the natural frequencies f1 = 1.8529, f2 = 1.5259, and
f3 = 0.9685 rad/s of the linear subsystem). For the limit
of high energy and finite frequency, the underlying
Hamiltonian system (30) reaches the linear limiting
Fig. 30
Frequency–energy plot of the periodic orbits for
the MDOF system with the NES masses of O( 2 )
[88]
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Passive non-linear TET and its applications
Fig. 31
115
Damped responses for out-of-phase impulses Y = 0.1: (a) Cauchy WTs superimposed on
the FEP; (b) partition of instantaneous energy of the system [88]
system
linearized natural frequency f2 of the limiting system
for the limit of low energy and finite frequency, where
the equation for the NES part becomes
ü1 + (ω02 + α)u1 − αu2 = 0
ü2 + (ω02 + α + )u2 − αu1 − v1 = 0
μv̈1 + (v1 − u2 ) = 0
3μv̈1 + (v1 − u2 ) = 0
(31)
with limiting natural frequencies fˆ1 = 1.7734, fˆ2 =
1.1120, and fˆ3 = 0.7960 rad/s.
The efficiency of TETs is demoistrated numerically
by means of the MDOF NES configuration considered herein, under the out-of-phase impulsive forcing
F1 (t) = −F2 (t) = Y δ(t), with all other initial conditions
being zero.
Figure 31 depicts the damped responses for the
impulsive forcing amplitude Y = 0.1. In this case, both
the relative displacements v1 (t) − v2 (t) and v2 (t) −
v3 (t) between the NES masses follow regular backbone
branches. The relative displacement v1 (t) − v2 (t) has a
dominant frequency component that approaches the
JMBD118 © IMechE 2008
with decreasing energy. In contrast, v2 (t) − v3 (t) has
two strong harmonic components that approach the
linearized natural frequencies f2 and f3 for decreasing energy, indicating transfer of energy simultaneously from two modes of the linear limiting system
for limit of low energy and finite frequency. Moreover, the same regular backbone branches are tracked
by the response throughout the motion and strong
energy transfer occurs right from the early stage of the
response, which explains the strong eventual energy
transfer to the NES (≈ 90 per cent) that occurs for this
low-impulse excitation.
By increasing the impulsive forcing to Y = 1.0 [88],
the overall energy transfer from the linear to nonlinear subsystem decreases significantly with delay,
and the steady-state energy dissipation by the NES is
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
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Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 32
Damped responses for out-of-phase impulses Y = 1.5: (a) Cauchy WTs superimposed on
the FEP; (b) partition of instantaneous energy of the system [88]
only about 50 per cent. This occurs because the motion
is mainly localized to the directly excited linear subsystems by the strong initial out-of-phase resonance
capture, with a small portion of energy spreading out
to the NES.
Further increasing the impulse magnitude to Y =
1.5 enables the system to escape from the strong initial
out-of-phase resonance capture, leading to resumed
strong TETs (Fig. 32). The NES relative responses
possess multiple strong frequency components, indicating that strong TET occurs at multiple frequencies.
The steady-state energy dissipation by the NES reaches
nearly 90 per cent of the input energy.
3.3.3
Analysis of a linear continuous system with
SDOF and MDOF attached NESs
A separate series of papers examined TET in continuous systems with attached NESs. For example,
Fig. 33 depicts linear (dispersive) elastic rods coupled
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
to SDOF and MDOF NESs [92–94]. In these works,
it was shown that appropriately designed NESs are
capable of passively absorbing and locally dissipating significant portions of the vibration energy of the
impulsively forced rod.
In Fig. 34, a representative WTs of the damped
responses of these two systems superimposed to the
FEPs of the underlying Hamiltonian (undamped and
unforced) systems are provided. Comparing the action
of the SDOF and MDOF NESs, noted it is that the SDOF
NES is capable of engaging in resonance capture with
only one mode of the linear rod at a time. Hence,
in Fig. 34(a), a resonance capture cascade where the
SDOF NES engages with a series of modes sequentially (i.e. it escapes from a resonance capture with
one mode before it can engage in similar resonance
capture with another one) is noted. In the case of the
MDOF NES (see Figs 34(b) to (d)), this does not hold,
as the NES engages in broadband resonance interactions with multiple modes of the rod; that is, different
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 33
117
Linear elastic rod coupled to (a) an ungrounded SDOF NES; (b) an MDOF NES
Fig. 34 Wavelet spectra of the relative responses between the rod end and (a) an SDOF NES, (b–d)
an MDOF NES, superimposed to the corresponding FEPs of each system [94]
JMBD118 © IMechE 2008
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
118
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
non-linear modes of the MDOF NES engage in separate resonance captures with different linear modes of
the rod (this is revealed by the broadband character
of the non-linear modal interactions between the rod
and the NES in this case). Hence, similar to previous
applications with discrete coupled oscillators, it is concluded that a MDOF NES is more versatile and effective
compared with the SDOF NES, as it can extract vibration energy simultaneously from a set of modes of the
linear system. For a more detailed analysis and discussion of these results, the reader is referred to references
[92] to [94].
3.4
Non-smooth VI NES
A separate series of papers considered NESs with nonsmooth stiffness characteristics. An NES with piecewise linear springs was first utilized for the purpose of
shock isolation in reference [105] (see also reference
[71]); this piecewise linear stiffness is relatively easy to
realize in practice [116–118].
This section is concerned with NESs undergoing VIs
(hereafter, vibro-impact NESs can be termed as VI
NESs). As shown in the aforementioned references,
this type of ‘non-smooth’ NES possesses fast reaction time; i.e. a VI NES is capable of passive TET at
a fast time-scale, which makes this type of device
ideal in applications where the NES needs to be activated very early in the motion (within the initial one
or two cycles of vibration). The simplest primary system – VI NES configuration, namely an SDOF linear
oscillator coupled to a VI NES (Fig. 35) is considered. It will be demonstrated that a clear depiction of
the damped non-linear transitions that govern energy
transactions in this system can be gained by studying the damped motion on the FEP of the underlying
VI conservative system (i.e. the identical system configuration, but with purely elastic impacts and no
viscous damping elements). The premise is that, for
sufficiently small damping, the damped non-linear
dynamics are perturbations of the dynamics of the
underlying conservative system, so that damped nonlinear transitions take place near branches of periodic
or quasi-periodic motions of the undamped system.
Hence, by studying the structure of periodic orbits of
the conservative system, the behaviour of the damped
dynamics should be understood as well, and phenomena such as TRCs and jumps between different
Fig. 35
An SDOF linear oscillator connected to a VI NES
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
branches of solutions that govern TET in the VI system
should be identified.
The equations of motion in non-dimensional form
between impacts can be written as
ü1 + (1 + σ )u1 + u2 = 0,
μü2 + σ (u2 − u1 ) = 0
(32)
where μ = m2 /m1 , σ = k2 /k1 are the mass
and stiffness ratios; the rescaling of time, τ = k1 /m1 t, is
imposed, and the derivative with respect to the new
non-dimensional time is denoted by the overdot.
Impact occurs whenever the absolute value of the
relative displacements satisfies |u2 − u1 | = e, where e
denotes the clearance; if |u2 − u1 | < e, then no impact
occurs and the system oscillates simply in a linear
combination of the two linear modes of system (32).
Setting the coefficient of restitution to 1 (i.e. assuming perfectly elastic impacts), and applying momentum conservation, the velocities of the two masses just
before and after impacts can be related; that is
v1 =
(1 − μ)v1 + 2μv2
,
1+μ
v2 =
(μ − 1)v2 + 2v1
1+μ
(33)
where vi = dui /dτ and the prime denotes the quantity
just after impact.
The periodic solutions of the VI conservative system were computed numerically and represented in
a FEP. This plot was constructed by depicting each
VI periodic orbit as a single point with the coordinates determined in the following way: consider the
eigenfrequency of the uncoupled linear oscillator as
reference frequency, f0 = 0.1515; the frequency coordinate of the FEP is equal to (p/q)f0 , where the rational
number p/q is the ratio of the basic frequency of
the linear oscillator to the basic frequency of the NES.
The energy coordinate is the (conserved) total energy
of the system when it oscillates in the specific periodic orbit considered. The parameters of the system
adopted for the FEP computation are μ = σ = 0.1 and
e = 0.1, and the resulting FEP is depicted in Fig. 36.
The complicated topology of the branches of periodic orbits depicted in the FEP reflects the well-known
complexity of the dynamics of this seemingly simple non-linear dynamical system. It is exactly because
of the complexity of VI motions that it is necessary
to establish a careful notation in order to distinguish
between the different families of VI periodic motions
and study their dependence on energy and frequency.
To this end, each VI periodic orbit depicted in the
FEP is given the notation Lijkl±. The capital letter
L is assigned either letters S or U , referring to symmetric or unsymmetric periodic motions, respectively.
Symmetric periodic motions satisfy the conditions
uk (τ ) = ±uk (τ + T /2), ∀τ ∈ R, k = 1, 2, where T is the
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Passive non-linear TET and its applications
Fig. 36
119
Frequency–energy plot for the system (32); dashed lines indicate the two linearized eigenfrequencies, and bullets, the maximum energy levels at which oscillations take place
without VIs [116]
period of the motion, whereas unsymmetric periodic
motions do not satisfy the conditions of the symmetric motions. Regarding the four numerical indices
{ijkl}, index i refers to the number of left VIs occurring during the first half-period; j to the number of
right impacts occurring during the first half-period; k
to the number of left impacts occurring during the second half-period; and l to the number of right impacts
occurring during the second half-period of a periodic
motion.
The (+) sign corresponds to in-phase VI periodic
motions where, for zero initial displacements, the
initial velocities of the two particles have the same
sign at the beginning of both the first and second
half-periods of the periodic motion; otherwise, the
VI periodic motion is deemed to be anti-phase and
the (−) sign is used. It can be shown that S-VI periodic orbits correspond to synchronous motions of the
two oscillators, and thus are represented by curves
in the configuration plane of the system, (u1 , u2 ); i.e.
these periodic motions are characterized as NNMs.
On the contrary, U-VI periodic orbits correspond to
asynchronous motions of the two oscillators, and are
represented by Lissajous curves in the configuration
plane of the system.
Considering the FEP of Fig. 36, the two bullets indicate the maximum energy thresholds below which
oscillations occur without VIs, and the dynamics of the
two-DOF system is exactly linear. The first (in-phase)
and second (out-of-phase) modes of the linear system (corresponding to the two-DOF system with no
rigid stops and clearance, e.g. e = ∞) exist below the
energy thresholds for VIs, namely, E1 = 0.001 185 12
for the in-phase mode and E2 = 0.000 865 078 for the
out-of-phase one. Clearly, when the system oscillates
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below these maximum energy thresholds, the relative
displacement between the two particles of the system
satisfies |u1 − u2 | < e.
As the energy is increased above the threshold VIs,
giving rise to two main branches of periodic VI NNMs:
the branch of out-of-phase VI NNMs S1001− which
bifurcates from the out-of-phase linearized mode,
and the branch of in-phase VI NNMs S1001+ which
bifurcates from the in-phase linearized mode. The
two branches S1001± will be referred to as backbone
(global) branches of the FEP; they consist ofVI periodic
motions during which the NES vibrates either in-phase
or out-of-phase with the linear oscillator with identical
dominant frequencies.
Moreover, both backbone branches exhibit a single
VI per half-period are defined over extended frequency
and energy ranges, and correspond to motions that
are mainly localized to the VI attachment (except
in the neighbourhoods of the two linearized eigenfrequencies of the system with e = ∞, at f1 = 0.136
and f2 = 0.186). Both backbone branches satisfy the
condition of 1:1 IR between the linear oscillator and
the VI NES, with the oscillations of both subsystems possessing the same dominant frequency, as
well as weaker harmonics at integer multiples of the
dominant frequency.
A different class of VI periodic solutions of the FEP
lies on subharmonic tongues (local branches); these
are multi-frequency periodic motions, with frequencies being rational multiples of one of the linearized
eigenfrequencies of the system. Each tongue is defined
over a finite energy range and is composed of a pair
of branches of in- and out-of-phase subharmonic
solutions. At a critical energy level, the two branches
of the pair coalesce in a bifurcation that signifies
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120
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 37
Representative VI impulse orbits: U3223 (upper) and U2222 (lower) [116]
the end of that particular tongue and the elimination of the corresponding subharmonic motions for
higher energy values. Clearly, there exists a countable
infinity of such tongues emanating from the backbone branches, with each tongue corresponding to
symmetric or unsymmetric VI subharmonic motions
with different patterns of VIs during each cycle of the
oscillation.
Finally, there exists a third class of VI motions in
the FEP, which are denoted as VI impulsive orbits
(VI IOs). These are periodic solutions corresponding
to zero initial conditions, except for the initial velocity of the linear oscillator. In essence, a VI IO is the
response of the system initially at rest due to a single
impulse applied to the linear oscillator at time τ = 0+ .
Apart from the clear similarity of a VI IO to the Green’s
function defined for the corresponding linear system,
the importance of studying this class of orbits stems
from their essential role in passive TET from the linear
oscillator to the non-linear attachment.
Indeed, for impulsively excited linear systems with
NESs having smooth non-linearities, IOs (which are,
in essence, non-linear beats) play the role of bridging orbits that occur in the initial phase of TET, and
channel a significant portion of the induced impulsive
energy from the linear system to the NES at a relatively
fast time-scale; this represents the most efficient scenario for passive TET. Although the aforementioned
results refer to damped IOs, the dynamics of the underlying conservative system determines, in large part,
the dynamics of the damped system as well, provided
that the damping is sufficiently small. It follows that
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
the IOs of the conservative system govern the initial
phase of TET from the linear oscillator to the NES.
As shown in reference [85], impulsive periodic and
quasi-periodic orbits form a manifold in the FEP that
contains a countable infinity of periodic IOs and an
uncountable infinity of quasi-periodic IOs.
For the VI system under consideration, the manifold
of VI IOs was numerically computed and is depicted in
the FEP of Fig. 36; in general, the manifold appears
as a smooth curve, with the exception of a number of outliers. Representative VI IOs are depicted in
Fig. 37. In general, the IOs become increasingly localized to the VI NES as their energy decreases, a result
which is in agreement with previous results for NESs
with smooth essential non-linearities [85]. As energy
increases, the VI IOs tend to the in-phase mode (i.e.
a straight line of slope π /4 in the configuration plane
(u1 , u2 )). Moreover, there is no critical energy threshold for the appearance of VI IOs since there are no
low-energy VI motions (the system is linear for lowenergy levels), and the dominant frequency of a VI IO
depends on the clearance, e. For the system under consideration, the VI IOs start with a dominant frequency
of 0.152 (or a period of 6.58).
Apart from the compact representation of VI periodic motions, the FEP is again a valuable tool for
understanding the non-linear resonant interactions
that govern energy transactions (such as TET) during damped transitions in the weakly dissipative
system. This is because, for sufficiently weak dissipation (due to inelastic VIs or viscous damping), the
damped dynamics are expected to be perturbations
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Passive non-linear TET and its applications
121
it can be concluded that the most efficient energy
dissipation by the VI NES occurs during the initial
TRC on the subharmonic tongue S1221+. This result
demonstrates that TRC is a basic dynamical mechanism governing effective passive TET, for example,
from a seismically excited primary structure to an
attached VI NES. It follows that by studying VI transitions in the FEP and relating them to rates of energy
dissipation by VI NESs, one should be able to identify the most effective damped transitions from a TET
point of view. The complicated series of VI transitions depicted in Fig. 38 demonstrates the potential of the two-DOF system for exhibiting complex
dynamics, and the utility of the FEP as a tool for
representing and understanding complex transient
multi-frequency transitions.
4
Fig. 38
Damped VI transitions initiated on the tongue
U 8778−: (a) WT superimposed on the FEP; (b)
instantaneous energy plot [116]
EXPERIMENTAL VERIFICATIONS
In this section, the experimental work that validates
the previous theoretical results on passive TET will be
reviewed. For a general synopsis regarding the experimental study of TETs, refer to the literature review in
section 2.2.3.
4.1
of solutions of the underlying conservative system. To
show this, the dynamics of the system of Fig. 35 for the
case of inelastic impacts is computed and analysed
the resulting transient responses by numerical WTs.
Then the resulting WT spectra are superimposed to the
FEP in order to study the resulting damped transitions
and related them to the dynamics of the underlying
conservative system.
A damped transition is depicted in Fig. 38, corresponding to VI motion initiated on the VI IO U 8778−,
with a coefficient of restitution, 0.995. Three regimes
of the damped VI transition can be distinguished.
In the initial phase of the motion, the oscillations
stay in the neighbourhood of the subharmonic tongue
S1221+ until approximately τ = 500 and logarithm of
energy equal to −2.15. There is efficient energy dissipation in this initial phase of the motion, as evidenced
by the energy plot of Fig. 38(b).
In the second regime, the dynamics makes a transition to branch U 0110− until the logarithm of
energy becomes equal to −2.5; in this regime of the
damped transition non-symmetric oscillations take
place. An additional transition to the manifold of VI
IOs (e.g. IOs U 2112+, U 1111+, S1221+) occurs, before
the VI dynamics makes a final transition to the backbone branch S0110+ for logarithm of the energy close
to −2.7.
By studying the instantaneous energy of the system
during the aforementioned transitions (see Fig. 38(b)),
JMBD118 © IMechE 2008
Experiments with SDOF primary systems
Figure 39(a) depicts an experimental fixture built
to examine the energy transfers in the two-DOF
system (Fig. 39(b) for its mathematical modelling)
described by
M ÿ + λ1 ẏ + λ2 (ẏ − v̇) + C( y − v)3 + ky = 0,
v̈ + λ2 (v̇ − ẏ) + C(v − y)3 = 0
(34)
A schematic of the system is provided in Fig. 39(c),
detailing major components. The system parameters
were identified using modal analysis and the restoring force surface method (Fig. 40; [168]): M = 1.266 kg,
= 0.140 kg, k = 1143 N/m, λ1 = 0.155 Ns/m, λ2 =
0.4 Ns/m, C = 0.185 × 107 N/m2.8 , and α = 2.8, where
α denotes the power of the essential non-linearity.
Figure 39(e) is a schematic showing how cubic
(essential) non-linearity is achieved through geometric non-linearity. Assuming zero initial tension along
the wire, a static force F with respect to a transverse
displacement x can be expressed as
F = kL
x L
1− 1
1 + (x/L)2
(35)
where k = EA/L represents the axial stiffness constant
of the wire, L is the half-length of the span, E Young’s
modulus, and A the cross-sectional area of the wire.
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122
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 39
Experimental setup for an SDOF linear primary structure coupled to an SDOF NES:
(a)–(c) general configuration and schematics; (d) experimental force pulse (21 N);
(e) realization of the essential cubic non-linearity through a system with geometric
non-linearity [96, 99, 169]
Taylor-series expansion of the bracketed term about
x = 0 assuming x/L 1 gives
F = EA
x 3
L
x5
+O
L5
(36)
from which the coefficient for the essentially nonlinear term can be estimated as C = EA/L 3 . Note that
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
a non-integer power (close to three) is obtained via
system identification [169].
Two series of physical experiments were conducted
in which the primary system was impulsively loaded.
In the first series of tests, the damping in the NES
was kept relatively low in order to highlight the different mechanisms for TETs. Additional tests were
performed to investigate whether TETs can take place
with increased levels of damping.
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Passive non-linear TET and its applications
Fig. 40
4.1.1
123
Measured restoring force represented as a function of time (left) and relative displacement
v − y (right)[99]
Case of low damping
In the low-damping case, several force levels ranging
from 21 to 55 N were considered, but for conciseness,
only the results for the lowest and the highest force
levels are depicted in Fig. 41.
At 21 N of forcing, the acceleration and displacement of the NES are higher than those of the primary
system, which indicates that the NES participates in
the system dynamics to a large extent. The percentage
of instantaneous total energy plot illustrates that vigorous energy exchanges take place between the two
oscillators. However, it can also be observed that the
channeling of energy to the NES is not irreversible.
After 0.23 s, as much as 88 per cent of the total energy
is present in the NES, but this number drops down
to 1.5 per cent immediately thereafter. Hence, in this
case, energy quickly flows back and forth between the
two oscillators, which is characteristic of a non-linear
beating phenomenon. Another indication for this is
that the envelope of the NES response undergoes large
modulations.
At the 55 N level, the non-linear beating still dominates the early regime of the motion. A less vigorous but faster energy exchange is now observed as
63 per cent of the total energy is transferred to the NES
after 0.12 s. These quantities also hold for the intermediate force levels [99]. It should be noted that these
observations are in close agreement with the analytical and numerical studies [83, 84]; indeed, in this case,
the special orbits are such that they transfer smaller
amounts of energy to the NES, but in a faster fashion
when the force level is increased.
A qualitative means of assessing the energy dissipation by the NES is to compare the response of the
primary system in the following two cases: (a) when
the NES is attached to the primary system (the present
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case – denoted by ‘NES’ displacements at the bottom
of Fig. 41); (b) when the NES is disconnected, but its
dashpot is installed between the primary system and
ground (a SDOF linear oscillator with added damping – denoted by ‘ground dashpot’ displacement in
Fig. 41). Case (b) was not realized in the laboratory, but
the system response was computed using numerical
simulation. The two bottom figures in Fig. 41 compare the corresponding displacements of the linear
oscillator in the aforementioned two different system
configurations. It can be observed that the NES performs much better than the grounded dashpot for the
21 N level, but this is less obvious for the 55 N level.
This might mean that, when the non-linear beating
phenomenon is capable of transferring a significant
portion of the total energy to the NES, it should be a
more useful mechanism for energy dissipation.
4.1.2
Case of high damping
Several force levels ranging from 31 to 75 N were
considered, and the results for 31 N are presented
herein. The damping coefficient was identified to be
1.48 Ns/m, which means that damping can no longer
be considered to be O(). The increase in damping is
also reflected in the measured restoring force in Fig. 42.
The system responses are almost entirely damped
out after five to six periods. The NES acceleration and
displacement are still higher than the corresponding
responses of the primary system, meaning that TETs
may also occur in the presence of higher damping.
The percentage of instantaneous total energy in the
NES never reaches close to 100 per cent as in the
lower damping case. However, one may conjecture
that this is due to the increased damping value; as
soon as energy is transferred to the NES, it is almost
immediately dissipated by the dashpot.
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124
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 41
Experimental results for low damping (left column: 21 N; right column: 55 N; note differing
durations). The first row depicts measured acceleration; the second, measured displacement; the third, percentage of instantaneous total energy in the NES; and the fourth,
displacement of the primary structure (NES versus grounded dashpot) [99]
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
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Passive non-linear TET and its applications
Fig. 42
4.1.3
125
Experimental results for high damping (31 N). From the top left, measured accelerations,
measured displacements, percentage of instantaneous total energy in the NES, measured
and simulated energy dissipated by the NES, displacement of the primary system (NES
versus grounded dashpot), and restoring force [99]
Frequency–energy plot analysis
Utilizing the FEP on which the WT of the relative displacement between the primary structure and the NES
is superimposed, the dynamics of the system for highlevel forcing with low damping, and for low-level forcing with high damping, can be investigated (Fig. 43).
There are strong harmonic components developing
during the non-linear beating phenomenon. Once
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these harmonic components disappear, the NES
engages in a 1:1 resonance capture with the linear
oscillator at a frequency approximately equal to the
natural frequency of the uncoupled linear oscillator.
4.2
Experiments with MDOF primary systems
In order to support the theoretical findings in
section 3.3, physical experiments were carried out
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126
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 43
Fig. 44
Superposition of the WT of the relative displacement across the non-linearity and the FEP:
(a) 55 N, low damping; (b) 31 N, high damping [99]
Experimental setup for a two-DOF linear primary structure coupled to an SDOF NES [169]
using the fixture depicted in Fig. 44, which corresponds to the schematic depicted in Fig. 22.
It realizes the system described by equation 25,
and the system parameters are identified using
modal analysis and the restoring force surface
method: m1 = 0.6285 kg, m2 = 1.213 kg, = 0.161 kg,
k1 = 420 N/m, k2 = 0 N/m, k12 = 427 N/m, C = 4.97 ×
106 N/m3 , λ1 = 0.05 − 0.1 Ns/m, λ2 = 0.5 − 0.9 Ns/m,
λ12 = 0.2 − 0.5 Ns/m, λ = 0.3 − 0.35 Ns/m.
The mass ratio /(m1 + m2 ) is equal to 8.7 per cent.
From these parameters, the natural frequencies of the
uncoupled linear subsystem are found to be 1.95 and
6.25 Hz, respectively. The damping coefficients range
over a certain interval, because damping estimation is
a difficult problem in this setup due to the presence
of several ball joints and bearings, and due to the air
track. It was found that damping was rather sensitive to
the force level, which is why intervals rather than fixed
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
values are given. In addition, at low amplitude friction
appeared to play an important role in the dynamics of
the system.
In these experimental verifications, the mass m1
was loaded by impulses of different amplitudes and
of durations of approximately 0.01 s. Four cases of
increasing input energy were considered: case I,
0.0103 J; II, 0.0258 J; III, 0.0296 J; and IV, 0.0615 J. The
superposition of the WT of the relative displacement
across the non-linear spring on the FEP is shown in
Fig. 45.
Starting with the case I, the lowest energy,
S111 + ++ is excited from the beginning of the
motion. This means that the input energy is already
above the threshold for TET from the in-phase mode,
but below the threshold for resonance with the out-ofphase mode. For case II, S111 + ++ is again excited,
but harmonic components are present. By slightly
increasing the imparted energy (case III), the threshold for interaction with the out-of-phase mode is
exceeded. As a result, S111 + −− is excited, and shortly
after a jump to S111 + ++ takes place. In case IV, the
transitions are similar to those of case III.
Further results for case IV, which bear strong resemblance to those in Fig. 46, are displayed in Fig. 47.
During the first few cycles, the NES clearly resonates
with the out-of-phase mode. As a result, after 2 s,
the NES can capture as much as 87 per cent of the
instantaneous total energy, and the participation of
the out-of-phase mode in the system response is drastically reduced. Around t = 2 s, a sudden transition
takes place, and the NES starts extracting energy from
the in-phase mode. The comparison of Figs 47(c) and
(e) with Figs 47(d) and (f ) shows that the predictions
of the model identified are in very close agreement
with the experimental measurements in the interval
0–4 s. Specifically, the sequential interaction of the
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
Fig. 45
Frequency–energy plot for the experimental fixture for a peak duration around 0.01 s:
(a)–(d) Cases I–IV [87]
Fig. 46
Response following direct impulsive forcing of mass m1 (40 N, 0.01 s): (a)–(b) displacements; (c) FEP with the superimposed WT of the relative displacement between m2 and
the NES; (d) instantaneous percentage of total energy in the NES [87]
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127
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128
Y S Lee, A F Vakakis, L A Bergman, D M McFarland, G Kerschen, F Nucera, S Tsakirtzis, and P N Panagopoulos
Fig. 47
Experimental results (case IV): (a)–(c) measured displacements; (d) predicted NES displacement; (e)–(f) measured and predicted instantaneous percentage of total energy in
the NES [87]
NES with both modes is accurately reproduced by the
numerical model. Discrepancies occur after t = 4 s,
probably due to unmodelled friction in the bearings;
this explains why the TET predicted by the numerical
model between 4 and 8 s was not reproduced with the
experimental fixture.
During this experiment, no attempt was made to
maximize energy dissipation in the NES. The purpose
was rather to examine the energy transfers in this system, to highlight the underlying dynamic phenomena,
Proc. IMechE Vol. 222 Part K: J. Multi-body Dynamics
and to demonstrate that the NES is capable of resonating with virtually any given mode of a structure.
5
CONCLUDING REMARKS
Fundamental aspects of passive TET in systems of
coupled oscillators with essentially non-linear attachments were reviewed in this work. The concepts,
methods, and results presented in this review article
JMBD118 © IMechE 2008
Passive non-linear TET and its applications
can be applied to diverse engineering fields. To just
give an indication of the powerful applications that
passive TET can find some recent applications of TET
and NES to some practical engineering problems.
In a series of papers [98, 114, 115] the ability of SDOF
and MDOF NESs to robustly eliminate aeroelastic
instabilities occurring in in-flow wings is demonstrated both theoretically and experimentally. This
is a consequence of a series of transient or sustained resonance captures between the essentially
non-linear NESs and aeroelastic (pitch and heave)
modes, which act to suppress the triggering mechanism that yields to LCOs and assure instability-free
dynamics. The designs proposed in these papers hold
promise for using strongly non-linear local elements
to achieve passive vibration reduction in situations
where this is not possible by weakly non-linear or
linear methods.
Moreover, in an additional series of papers
[116–118], NESs with smooth and/or non-smooth (VI)
characteristics are employed in frame structures to
mitigate the damaging effects of strong seismic excitations. In particular, the author demonstrated, both
theoretically and experimentally, that NESs with nonsmooth stiffness characteristics can provide passive
reduction of the seismic response during the critical initial cycles (i.e. immediately after application of
the earthquake excitation), where the motion is at
its highest energetic state. This is due to fast-scale
TET from the structure to the non-smooth NES. The
use of VI NESs in seismic mitigation designs has the
added advantages of ‘spreading’ seismic energy to
higher structural modes, which leads to amplitude
reduction and to more efficient dissipation of seismic
energy.
The results, methods, and applications reviewed in
this paper hopefully demonstrate the potential benefits to be gained through intentional introduction
of non-linearities in certain engineering applications.
Though this runs counter to the prevailing view that
non-linearities in structural design should be avoided
when possible; but here it is shown that, for certain
applications, the intentional use of (even strong) nonlinearities can yield beneficial results that cannot be
obtained otherwise by weakly non-linear or linear
designs.
ACKNOWLEDGEMENTS
This work was supported in part by the US Air Force
Office of Scientific Research through Grants Number FA9550–04–1–0073 and F49620-01-1-0208. Gaëtan
Kerschen is supported by a grant from the Belgian National Science Foundation, which is gratefully
acknowledged.
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129
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