PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Non-linear Metamaterial Structures: Array of Particle Dampers
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*University of Sheffield, United Kingdom

ABSTRACT
Passive dampers to treat the excessive structural vibration has long been researched and used in industry.
Multiple particles placed in a container can be used to dissipate excessive vibration through the inelastic
collisions between particles and cavity of the damper has been shown. However, their application is usuallylimited to treating certain modes of structural vibration and their performance is highly dependent on location.
Another limitation of the particle dampers is nonlinear character caused by discontinuity and randomness ofthe collisions and velocity of the particles. In this paper, it is proposed to modify a particle damper into a
metamaterial type structure in order to expand the applicability range. Metamaterials are known to exhibit
subwavelength performance offering superior vibro-acoustic properties over a wide range of frequencies.
To maintain metamaterial properties, the casing of the particle damper is designed to resonate near selected
modal frequencies. The Bloch-Floquet Theory is applied in studying the singly periodic arrangement of theresonating damper shells with and without particles. Finally, the nonlinear effects observed in themetamaterial structure made of particle dampers are modelled numerically to predict their vibro-acoustic
effects in finite structures. The theoretical, numerical predictions are compared with the experimental results.
Keywords: Metamaterial, Nonlinear Damping, Passive Damping, Structural Vibration, Nonlinearity

1. INTRODUCTIONMechanical oscillations in the systems causes undesired vibrational problems which needs to be solved with
damping options [1]. Passive damping methods, which are able to be used in the certain frequency ranges,
do not require external energy to be effective on the structural vibration. Examples of the passive dampers
and their applications are explained in [2], [3], [4], [5]. Their problem is to be sensitive to the temperatures,
or wear with usage, or leakage as depending on the application type. Efforts were put on finding a better
solution to the structural vibration. Impact damper is one type of passive dampers known as acceleration
damper [3] which was invented after the evaluation of the weaknesses and the problems on some of thepassive dampers. It is made of one structure consisting a cavity which is either inside the primary (main)
structure or externally attached to the main structure. Single particle is placed in the cavity, moves there andcollides with the walls. Energy dissipation mechanism comes from its colloidal behaviour with the cavity
wall under dynamic load [4]. In order to increase the effects of impact damper, the damper is modelled withmore than one cavity to place single particle inside; it is called as ‘Multi-unit impact damper’. Impact damper
was found less sensitive to the system properties such as temperature level and excitation conditions which1sgdemiryurek1@sheffield.ac.uk
a.krynkin@sheffield.ac.uk
3
j.a.rongong@sheffield.ac.uk
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are the advantages of its usage; however, they were found noisy. In order to overcome this problem, the idea
was to place more than one particle in the same cavity which have the equal mass of the single particle. The
new arrangement was called as ‘Particle damper, PD’. Concerning to increase the performance, PD isdesigned to consist several cavities for the multiple particles, known as ‘Multi-unit particle damper’. Figure
1.1 shows these varieties of the particle dampers [1]. PD, which is capable to work under rough conditions,is improved to be effectively used in the wider range of frequencies and needed less maintenance [6].

Figure 1.1: (a) Impact Damper, (b) Multi-unit Impact Damper, (c) Particle Damper and (d) Multi-unit
Particle Damper
Damping properties of the particle damper were defined as a combination of friction, elastic and plastic
collision losses with momentum changes. The particles move in the cavity, and their movement depends on
the level of dynamic excitation. There are two types of collisions for this case: particle/particle collisions andparticle/cavity wall collisions. The dissipation of energy and momentum changes through the vibratory
structure are accomplished by the collisions, frictional effects, deformations and loss mechanisms [7]. In
order to define working parameters and to find a better damper characteristics experimental methods werefollowed, mainly [8]. Experimental studies showed that PD has limited applications to treat the number ofmodes in the structural vibration. Additional to this limitation, discontinuity and randomness of the particle
collisions cause nonlinearity on the particle dampers. This undesired situation was related to the frictional
effects of collisions in the loss mechanism. Furthermore, PD performance varies with the size of the particle,
volume of the cavity, packing ratio, direction of the excitation, material properties of the particles, etc.
Although, PD is effective reducing the structural vibration, highly nonlinear characteristics are undesired.
Therefore, efforts are on finding more effective PD design criteria with several approaches.
Metamaterial phenomena are another effective dispersive mechanism and another topic of this research as anapproach for PD studies. Metamaterial is designed as subwavelength structure and can perform to control thewave bigger than the characterised structural size [9]. This engineered structure is modelled to show local
resonance effects on the main structure under vibratory process [10]. Performance of metamaterial is related
to its periodic arrangement. Periodically arranged structures increase the number of freedoms on the hoststructure; therefore, unique dynamic effects on filtering the wave propagation are observed [11]. Effective
working conditions are classified using band terms: stop band and pass band. Wave propagation is blockedin any direction for the certain wave numbers which is called stop band. Outside of the stop band is pass band
where the wave propagates in the media [12]. Proper calculations and applications may help to widen thestop band properties. Periodically arranged structures refer to infinite structure in the numerical studies.
Infinite structures in the modelling would give better insight to structural behaviour under various working
conditions. However, computational cost of this work is inconvenient. There are several periodic conditions
ready to apply on the structures; Bloch - Floquet Theorem is one of the convenient methods. Bloch – Floquet

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Theory provides infinite structure modelling on a finite structure using unit (elementary) cell identification.
Applying the boundary condition around the unit cell maintains the finite element solver to recognise the unit
cell as an infinite structure. Therefore, potential effects of metamaterials, periodically arranged structures andavailability of bandgaps are questionable with infinite structural modelling.
This paper proposes the widened application range of PDs adding metamaterial properties to them. Particle
behaviour under various excitation conditions were studied experimentally. The nonlinear effects of PD wereobserved. DEM (Discrete Element Method) was used to perform PD numerical study for a single frequency
to understand the damping properties of particles. This was followed by the design of metamaterial made ofperiodic arrangement of PD. The casing of the damper was designed to resonate at one of the beam naturalfrequencies. Vibro-acoustic interactions were studied with finite element model. Bloch - Floquet Theory wasapplied to the modelled unit cell with singly periodic conditions.

2. METHODOLOGY
2.1

Experimental setup

In order to achieve the aimed application of PD and metamaterial, experimental and numerical studies werefollowed. Damper cases were printed using 3D printing conventional additive manufacturing method withrecycled plastic material. Single sized stainless-steel particles (diameter of 1/16 inches) were used in theexperimental studies to fill the volume of the damper casing. A fixed-fixed end conditioned structural steel
beam was used for the experimental studies. SigLab Measurement Structure was used in the experimental
studies: VNA (Virtual Network Analyser) and VSS (Virtual Swept-Sine) are the post-processing methods
used to identify the system characteristics and running the experiments for the given sinusoidal excitation
over a range of frequencies, respectively. Beam-shaker test (Figure 2.1) procedure was followed, andproperties and effects of PD were studied. 60-150 Hz frequency range was selected to run the experiments;
since this range contains the first bending mode of the beam.

Figure 2.1: Experimental rig with PD
2.2
2.2.1.

Numerical modelling
DEM

Physics of densely packed solid particles and their dynamics were modelled with commercial software
EDEM V19.1. The software works with the explicit process within defined time steps (iterations) whilesolving the contact physics of the related particles. Modelled structure has the same volume as theexperimental structure. Filling fraction was converted into the number of particles for the modelling ofsimulation (Figure 2.2). Hertz-Mindlin contact theory was applied to the particle-particle and particle-wall

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interactions. Rotational effects of the particles were modelled with the standard rolling friction method; since
the normal and tangential forces acting on the particles and cavity were aimed to model. Aiming theconsistency with the experimental studies, the excitation frequency of the simulations was held at 105 Hz.

Figure 2.2: Simulation model of DEM
2.2.2. FEM
Commercial software COMSOL Multiphysics V5.0 was used for modelling metamaterial.

Figure 2.3: Periodic condition imposed in x- direction (a); Irreducible Brillouin Zone (b); unit cell with aninclusion (damper) (c)To analyse periodic arrangement of PD casings and the effect of their resonance, different unit cellarrangements were modelled in COMSOL Multiphysics: (i) unit cell without attached damper case (onlysteel beam) and (ii) unit cell containing damper casing. The periodic boundary condition was applied eitheron single side or two sides of the elementary cell (Figure 2.3 (a)). Unit cell was modelled as a square with
100 mm length. The results were analysed in 1st irreducible Brillouin Zone (the smallest symmetrical lattice[13]) in form of dispersion relationship between wave vector and frequency.

3. RESULTS
3.1

Experimental Results

The first few bending modes of the beam discussed in previous section were observed experimentally ataround 105 Hz, 325 Hz and 330 Hz and 580 Hz. The following arrangement of particle dampers mounted on
the beam were tested experimentally: (i) single damper placed either at the centre or away from the centre,(ii) three dampers with various separation distance (L, in the graph).

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Figure 3.1: FRF of several cases with 65% filled dampers
Figure 3.1 shows the comparison between bare beam, single damper and three dampers with various lattice
constant (shown in the Figure legend). The results were obtained at 0.1 V amplitude while the dampers werefilled 65% of their volume with particles. As the number of dampers placed on the beam was increased, theFRF magnitude level was reduced. Natural frequency was also shifted to lower frequency. The observed shift
was relatively small resulted in maximum 10% difference (observed for 3 dampers separated by 50 mmdistance). It is noted that the mass of the dampers was much smaller compared to the beam. Hence, thefrequency shift is expected to be small. It is also noted that due to the relatively small separation distance inthe case of multiple PDs any periodic effect is expected to be observed at higher frequencies.
Amplitude dependency of the first natural frequency is shown in the Figure 3.2 (a) and (b). Both Figures
clearly demonstrate the highest response level measured between 0.01 V and 0.03 V. For both arrangements
the amplitude levels between 0.04 V and 0.08 V were the lowest response region. However, it is clear thatthree-PD arrangement gives much higher damping effect. It is also noted that nonlinearity phenomenon
observed for a single damper are supported by previous studies [13], [14].

Figure 3.2: Amplitude dependency of single damper placed at the centre of the beam (a); Amplitudedependency of three dampers (L=50 mm) with 65% filling ratio (b)

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3.2

Numerical Results

EDEM V19.1 was used to model and to solve densely packed solid particles to observe the particle behaviourin the simulation. Figure 3.3 demonstrates the relationship between the kinetic energy (associated withdissipated energy) and total energy. Each collision consists of impact and friction effects; therefore, thedissipated total energy is a combination of them. The total duration of numerical test was five seconds which
has four seconds of collision and one second of placing the particles into the cavity and their resting at theend of the first second. Energy was dissipated through the collisions and contact mechanisms in the last four
seconds of time. This information can be used to estimate losses and nonlinear phenomenon so that PD canbe modelled accurately in FEM.

Figure 3.3: Energy dissipation of PD from DEM analysis
Bloch - Floquet Theory based infinite structure analysis were conducted to express the acoustic dispersion
relations with single periodicity results. The results are shown in Figure 3.4. According to the Figure 3.4 (b)damper casing changes the dispersion relationship adding more low frequency modes bandgaps. More than
one bands start from k=0 in the graphs; in an unbounded elastic medium there should be two modes startingfrom k=0. For the case of thin beam one additional modes appear for k=0 and there are more than three modes
for k=2 [14]. The absence of the band in the dispersion diagram is called as bandgap, if it is the case for all
the wave numbers then it is called as complete bandgap.

Figure 3.4: Dispersion relationship plotted for 1D lattice in form of solid beam (a) and periodic array ofdamper casing added to the lattice (b)4883

4. CONCLUSION
Nonlinear passive damping method was shown with its limitations including nonlinearity and effective
working properties on the SDOF system running shaker tests. Numerical methods were followed to validate
the studies related to both PD and metamaterial. Periodic arrangement using unit cell with the inclusions
were illustrated here. It was shown numerically that these arrays support low frequency bandgaps. This study
supports the idea that PD can be converted into metamaterial. DEM and FEM comparative studies arecurrently in progress.

REFERENCES[1]

R.A. Ibrahim, “Impact Dampers,” in Vibro-Impact Dynamics: Model., Map. & Appl., LNACM 43., Prof.
Dr.-Ing. Friedrich Pfeiffer and P. D.-I. P. Wriggers, Eds. Detroit, Michigan: Springer-Verlag Berlin
Heidelberg 2009, 2009, pp. 217–232.

[2]

Z. Lu, Z. Wang, S. F. Masri, and X. Lu, “Particle impact dampers: Past, present, and future,” Struct. Control
Heal. Monit., no. May 2017, p. 25, 2017.

[3]

I. Zaman et al., “Study of Passive Vibration Absorbers Attached on Beam Structure,” Appl. Mech. Mater.,vol. 660, no. November, pp. 511–515, 2014.

[4]

G. Takács and B. Rohal’-Ilkiv, Model predictive vibration control: Efficient constrained MPC vibration
control for lightly damped mechanical structures. 2012.

[5]

S. S. Simonian, “Particle beam damper,” no. July, pp. 149–160, 2015.

[6]

M. R. Duncan, C. R. Wassgren, and C. M. Krousgrill, “The damping performance of a single particle
impact damper,” J. Sound Vib., vol. 286, no. 1–2, pp. 123–144, 2005.

[7]

S. E. Olson, “An analytical particle damping model,” J. Sound Vib., vol. 264, no. 5, pp. 1155–1166, 2003.

[8]

S. E. Olson and M. L. Drake, “Development of Analytical Methods for Particle Damping,” Mountain
View,CA, 1999.

[9]

P. Sheng, “A Step Towards a Seismic Cloak,” Physics (College. Park. Md)., vol. 7, p. 34, 2014.

[10]

P. F. Pai, H. Peng, and S. Jiang, “Acoustic metamaterial beams based on multi-frequency vibration
absorbers,” Int. J. Mech. Sci., vol. 79, pp. 195–205, 2014.

[11]

M. Nouh, O. Aldraihem, and A. Baz, “Wave propagation in metamaterial plates with periodic localresonances,” J. Sound Vib., vol. 341, pp. 53–73, 2015.

[12]

A. O. Krushynska, M. Miniaci, F. Bosia, and N. M. Pugno, “Coupling local resonance with Bragg bandgaps in single-phase mechanical metamaterials,” Extrem. Mech. Lett., vol. 12, pp. 30–36, 2017.

[13]

F. Maurin, C. Claeys, E. Deckers, and W. Desmet, “Probability that a band-gap extremum is located on theirreducible Brillouin-zone contour for the 17 different plane crystallographic lattices,” Int. J. Solids Struct.,vol. 135, pp. 26–36, 2018.

[14]

M. Badreddine Assouar, J. H. Sun, F. S. Lin, and J. C. Hsu, “Hybrid phononic crystal plates for lowering
and widening acoustic band gaps,” Ultrasonics, vol. 54, no. 8, pp. 2159–2164, 2014.

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