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Low-frequency band gaps in chains with attached
non-linear oscillators
B.S. Lazarov, J.S. Jensen
To cite this version:
B.S. Lazarov, J.S. Jensen. Low-frequency band gaps in chains with attached non-linear os-
cillators. International Journal of Non-Linear Mechanics, Elsevier, 2007, 42 (10), pp.1186.
�10.1016/j.ijnonlinmec.2007.09.007�. �hal-00501761�
www.elsevier.com/locate/nlm
Author’s Accepted Manuscript
Low-frequency band gaps in chains with attached
non-linear oscillators
B.S. Lazarov, J.S. Jensen
PII: S0020-7462(07)00188-6
DOI: doi:10.1016/j.ijnonlinmec.2007.09.007
Reference: NLM 1403
To appear in: International Journal of Non-
Linear Mechanics
Received date: 17 July 2007
Revised date: 31 August 2007
Accepted date: 18 September 2007
Cite this article as: B.S. Lazarov and J.S. Jensen, Low-frequency band gaps in chains with
attached non-linear oscillators, International Journal of Non-Linear Mechanics (2007),
doi:10.1016/j.ijnonlinmec.2007.09.007
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Accepted manuscript
Low-frequency band gaps in chains with
attached non-linear oscillators
B. S. Lazarov
∗
, J. S. Jensen
Department of Mechanical Engineering, Solid Mechanics, Nils Koppels Alle,
Building 404, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
Abstract
The aim of this article is to investigate the wave pr opagation in one-dimensional
chains with attached non-linear local oscillators by using analytical and numerical
models. The focus is on the influence of non-linearities on the filtering properties of
the chain in the low f requency range. Periodic systems with alternating properties
exhibit interesting dynamic characteristics that enable them to act as filters. Waves
can propagate along them within specific bands of frequencies called pass b ands, and
attenuate within bands of frequencies called stop bands or band gaps. S top bands
in structures with periodic or random inclusions are located mainly in the high
frequency range, as the wave length has to be comparable with the distance between
the alternating parts. Band gaps may also exist in structures w ith locally attached
oscillators. In the linear case the gap is located around the resonant frequency of the
oscillators, and thus a stop ban d can be created in the lower frequency range. In the
case with non-linear oscillators the results show that the position of the band gap
can be shifted, an d the shift depends on the amplitude and the degree of non-linear
behaviour .
Key words: Non-linear wave propagation, Local resonators, Low frequency band
gaps
Preprint submitted to Elsevier 31 August 2007
Manuscript
Accepted manuscript
PACS:
1 Introduction
The filtering properties of periodic structures with a lternating characteris-
tics have been investigated by many authors by using analytical, numerical
and experimental methods. Such systems possess interesting filtering proper-
ties. Waves can propagate unattenuated along these structures within specific
bands of frequencies called propaga tion or pass bands, and attenuate within
bands of frequencies called stop bands, attenuation zones or band gaps. Band
gaps in linear systems can also be created by introducing random inclusions
or geometric disturbances. External excitation in such structures results in
localised response surrounding the external input. The majority of the texts
consider linear systems [2,6,7]. Localisation phenomena can also be observed
in perfectly periodic non-linear structures [8–10,14], where spring-mass chains
are studied and the non-linear behaviour is intro duced either in the spring
between the two neighbour masses, or by adding non-linear springs between
the gro und and the masses. The applications of these filtering phenomena are
mainly in the high frequency range, as the distance between the inclusions has
to be comparable with the wave length.
In the beginning o f the twentieth century Frahm discovered the vibration
absorber as a very efficient way to reduce the vibration amplitude of machinery
∗
Corresponding author.
Email addresses: bsl@mek.dtu.dk (B. S. Lazarov), jsj@mek.dtu.dk ( J. S.
Jensen).
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Accepted manuscript
and structures by adding a spring with a small mass to the main o scillatory
body [1]. The additional spring-mass system is tuned to be in resonance with
the applied load. When t he natural frequency of the attached absorber is equal
to the excitation frequency, the main structure does not oscillate at all, as the
attached absorber provides force equal and with opposite sign to the applied
one. The idea can be exploited f urther by att aching multiple absorbers on a
wave guide. Waves are attenuated in a frequency band lo cated around the
resonant frequency of the local oscillators, and thus stop bands can be created
in the lower frequency range, which is often more important in mechanical
applications. The effect has been studied exp erimentally and analytically in
[11–13].
If the attached oscillators are non-linear, the response displays a dependency
between the amplitude and the frequency. Very little is known about the
filtering properties of the systems in this case. Periodic spring-mass system
with attached non-linear pendulums are investigated in [15]. The attached
pendulums are considered to be stiffer than the main chain, and they do not
introduce band gaps in the lower frequency range. The aim of this paper is to
investigate the behaviour of one-dimensional infinite spring-mass chain with
locally attached oscillators with linear or non-linear behaviour. The oscillators
are considered to be relatively soft compared to the main chain, and to create
band gaps in the lower frequency range. The non-linearities in the attached
oscillators are considered to be cubic. First the mechanical system together
with the equations o f motion is presented. The appearance of band gaps is
shown in the linear case, a nd in the case with non-linear a t tached oscillators
the method of harmonic bala nce is utilised to obtain a system of equations
for the wave amplitude, as well as an approximate expression for the wave
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