Measurement of dispersion curves for locally resonantmetamaterials with damping

L. Van Belle 1,2, C. Claeys 1,2, E. Deckers 1,2, W. Desmet 1,2

1 KU Leuven, Department of Mechanical Engineering,Celestijnenlaan 300, B-3001, Leuven, Belgiume-mail: lucas.vanbelle@kuleuven.be

2 Member of Flanders Make

AbstractRecently, locally resonant metamaterials come to the fore in noise and vibration control engineering, show-ing great potential due to their superior noise and vibration attenuation possibilities. As damping has animportant influence on the performance of these materials, unit cell models for wave propagation in infi-nite periodic structures have been developed in previous work, allowing various damping mechanisms. Theobtained dispersion curves for locally resonant metamaterials with damping were shown to agree well withearlier findings and measurements. In this paper, the numerically obtained dispersion curves for a locally res-onant metamaterial design are compared with experimental dispersion curve measurements using ScanningLaser Doppler Vibrometry, on a metamaterial plate manufactured at a representative scale. Good agreementis obtained between the predicted and measured dispersion curves.

1 Introduction

Over the past decades, increasingly stringent ecologic and economic requirements have given rise to theintroduction of lightweight materials and designs [1]. Because of their increased stiffness over mass ratio,lightweight materials come at the cost of worse noise, vibration and harshness (NVH) behaviour, causingunfavourable noise and vibration suppression performance. This conflicts with customer expectations andthe tightening legal regulations regarding NVH exposure [2], for which current classic countermeasures areheavy and bulky. Consequently, the challenging task of combining lightweight design and favorable NVHperformance requires novel, low mass and compact volume solutions.

Recently, locally resonant metamaterials have come to the fore, showing great potential for noise and vibra-tion control engineering [3–5]. These materials exhibit superior noise and vibration attenuation performance,at least in some targeted and tunable frequency ranges. These frequency ranges are referred to as stop bandsand are frequency zones in which no free wave propagation is possible. It is known that this stop band be-haviour emerges from the addition of local resonances to the host structure on a subwavelength scale, causinga Fano-type interference between the incoming and re-radiated waves by the resonant cells [6, 7].

It has been shown that the presence of damping in locally resonant metamaterials has an important effecton the NVH performance. The addition of damping to the resonant structures leads to a broader frequencyrange of attenuation, at the expense of a reduced peak attenuation performance [7]. A good understandingof damping phenomena is consequently important to accurately predict the vibro-acoustic performance ofresonant metamaterials.

Because of the often, yet not required, periodic nature of these locally resonant metamaterials, the wavepropagation and stop band behaviour can be analyzed using a unit cell (UC) modelling approach. Makinguse of the Bloch-Floquet theorem, assuming infinite periodic structures, the locally resonant metamaterial

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can be represented by a single UC [8, 9]. An eigenvalue problem in frequency and propagation constant isobtained, the solution of which yields dispersion curves. These dispersion curves give information about thewave propagation in the considered infinite periodic structure.

In the classic approach, damping is discarded and free wave propagation is considered, solving the eigenvalueproblem to the frequency for given, purely propagating propagation constants along the Irreducible BrillouinContour (IBC) [7,10]. Stop bands emerge as frequency zones for which no freely propagating wave solutionsare found. In order to more accurately account for the damping effects and better predict the performance oflocally resonant metamaterials, the authors have included damping in a UC modelling technique [11], solvingfor the propagation constant for given frequencies and propagation directions. By increasing the damping inthe locally resonant metamaterial, the wave solutions corresponding to the stop band are observed to becomeless attenuated, while the wave solutions around the stop band are observed to become more attenuated. Thisresult is in good agreement with the damping effect of a broadening frequency zone of attenuation, at theexpense of lower peak attenuation.

In this paper, a locally resonant metamaterial is designed and the UC analysis including damping is used toanalyze the dispersion curves. In order to verify whether this UC modelling procedure represents the wavepropagation behaviour in a finite metamaterial well, a measurement is performed to retrieve experimental dis-persion curves. To this end, Scanning Laser Doppler Vibrometer (SLDV) measurements on a manufacturedmetamaterial plate are combined with the Inhomogeneous Wave Correlation (IWC) method proposed byBerthaut et al. in [12]. This allows dispersion curves to be extracted from displacement field measurements.Numerical and experimental dispersion curves are compared for various propagation directions.

The outline of this paper is as follows. In section 2 the UC analysis is briefly discussed and the experimentalmeasurement of dispersion curves is clarified. Section 3 describes the locally resonant metamaterial designand explains the experimental setup and parameters used. The numerical results of the UC analysis arediscussed in section 4. Next, the experimentally retrieved dispersion curves are presented and compared tothe numerical predictions. Section 5 summarizes the main conclusions of this work.

2 Methodology

In this section, the UC modelling procedure is briefly described, followed by a discussion on the experimentaldispersion curve measurement.

2.1 Unit cell modelling

Locally resonant metamaterials require the presence of resonant cells on a scale smaller than the wavelengthto be affected. Due to their often periodic nature, although not a strict requirement for stop band behaviour,these metamaterials can be modelled through a UC analysis using the Bloch-Floquet theorem, consideringinfinite periodic structures. To this end, a model of a single representative UC is required, in combinationwith Bloch-Floquet periodicity boundary conditions [13]. In this work, 2D planar periodicity is assumed andthe UC is represented by a Finite Element (FE) model, for which the equations of motion for time harmonicmotion are in general given as: (

K + jωC− ω2M)q = F, (1)

with stiffness, mass and viscous damping matrices K, M and C, generalized displacements and forces qand F, angular frequency ω and j =

√−1.

The Bloch-Floquet theorem states that, considering plane wave propagation, the generalized displacementsand forces on the boundaries of a rectangular UC are scaled with a factor ejµx and ejµy when moving fromone cell to another along the lattice vectors dx and dy of UC lengths Lx and Ly respectively. The complexpropagation constants µi along a lattice vector di of the UC are defined as:

µi = k · di, (2)

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with k the wavevector, and the real and imaginary part of µi representing propagation and spatial decayrespectively. Application of the Bloch-Floquet boundary conditions to (1) gives rise to an eigenvalue problemin the frequency and propagation constants [13]:

A(ω, µx, µy)q(red) = 0, (3)

with q(red) a reduced set of generalized displacements and A a general matrix depending on ω, µx and µy.

A first approach to solve this eigenvalue problem is referred to as the inverse approach [7, 10]. For an un-damped UC, assuming free wave propagation, thus real propagation vectors µ = (µx, µy), this eigenvalueproblem reduces to a linear eigenvalue problem which can be solved for real frequencies ω. This results indispersion curves, which indicate at which frequencies different wave types can freely propagate. Classi-cally, the dispersion curves are only solved for propagation vectors along the so-called Irreducible BrillouinContour (IBC). Stop bands emerge as frequency zones for which no free wave propagation is found alongthe IBC.

If damping is introduced, previous inverse solution approach no longer applies and the so-called direct ap-proach can be followed instead [7, 10]. For real frequencies ω, the propagation constants µx and µy becomecomplex, implying spatially decaying waves. The eigenvalue problem (3) can be reformulated as a polyno-mial eigenvalue problem in µ, in case the frequency and propagation direction θ are known and the ratioµy

µx=

Ly

Lxtan(θ) is rational [13]. Damped dispersion curves are now calculated for various propagation

directions, as the information along the IBC alone no longer suffices [11, 14, 15].

2.2 Dispersion curve measurement

As described in previous section, the wave propagation behaviour of the locally resonant metamaterialswith damping can be obtained as damped dispersion curves from a UC analysis. In order to compare thesenumerical predictions to the behaviour of their finite size realizations, measurements can be performed toidentify experimental dispersion curves for various propagation directions. As shown in [16], sampling ofthe undamped dispersion curves for infinite periodic metamaterials well predicts the eigenfrequencies andmodes of their finite size counterparts. Consequently, as no free wave propagation is found in the stop band,no eigenfrequencies for the finite size metamaterial are obtained in that frequency zone.

In this work, damping is considered in the locally resonant metamaterial. Therefore, a method should be usedthat allows retrieving propagation constants for the considered bending waves in damped two dimensionalstructures. In [12] a wavenumber space analysis method is presented for the measurement of the flexuralwavenumber of two dimensional structures, allowing experimental dispersion curves to be retrieved fromharmonic displacement field measurementsw(x, y) of a vibrating planar structure. Using this method, namedthe Inhomogeneous Wave Correlation method, an inhomogeneous wave σkprop,kdec,θ is introduced as:

σkprop,kdec,θ(x, y) = e(jkprop(θ)−kdec(θ))(x cos(θ)+y sin(θ)), (4)

with kprop and kdec the propagative and decaying part of the wavenumber respectively and θ the propagationdirection in the xy-plane. This inhomogeneous wave is a spatially decaying plane wave, corresponding tothe wave solutions arising from the UC analysis with damping. The correlation between this inhomogeneouswave and the measured wave field is calculated as follows:

IWC(kprop, kdec, θ) =|∑N

i=1w(xi, yi)σ∗kprop,kdec,θ

(xi, yi)Si|√∑Ni=1 |w(xi, yi)|2Si

∑Ni=1 |σkprop,kdec,θ(xi, yi)|2Si

, (5)

with ∗ denoting the complex conjugate, N the amount of points measured and Si an estimation of the surfacearound measurement point i. Although measurements of the wave field w can be considered in arbitrarypoints (xi, yi)i∈N, in this work, a regular rectangular grid of Nx and Ny points in x and y directions is used.

By maximization of the IWC(kprop, kdec, θ) at the considered frequencies of interest, the kprop and kdec canbe identified for given propagation directions θ. Experimental dispersion curves are thus retrieved, whichcan be compared to the numerically calculated dispersion curves using UC analysis.

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3 Application

A locally resonant metamaterial is designed and a finite plate metamaterial is manufactured for the experi-mental dispersion curve measurements. This section discusses the geometry and materials used for the designof the locally resonant metamaterial as well as the experimental measurement setup and the parameters usedfor later application of the IWC method.

3.1 Description metamaterial

In this paper, a locally resonant metamaterial is obtained by adding resonant structures to a host structure.In order to establish stop band behaviour, these resonant additions have to be spaced on a scale smallerthan the wavelength to be affected, while their resonant motion should exert a nonzero net force on the hoststructure [7, 17]. Stop band behaviour is desired for the out-of-plane bending motion of the structure, asthis is the relevant motion in the considered NVH context [18]. To this end, the resonators are shaped asa cantilevered beam with an end point mass, as depicted in Figure 1a. The first bending mode of this typeof resonator leads to a nonzero net force in the out-of-plane direction. The geometry of the beam and endpoint mass, together with the resonator material properties, allows the resonance frequency and the relativeadded mass to be tuned. Based on the resonator tuning and host structure properties, the subwavelengthrequirement is consequently met by using proper spacing of the resonators.

(a) Undeformed resonator (b) First bending mode, clamped base

Figure 1: Resonant structure design

For the host structure, a 1 mm thick flat aluminum plate is used. The material properties of the aluminum areretrieved by weighing the structure to obtain the density ρ and applying a model updating procedure [19] withmeasurements of the bare host structure and an FE model to obtain the Young’s modulus E and Poisson’sratio ν. These material parameters are summarized in Table 1.

Because of their shape, the resonant structures require a customizable production process. Various possi-bilities are available, among which 3D-printing, selective laser sintering, laser cutting. . . In this case theresonant structures are created by a laser cutting operation from a polymethyl methacrylate (PMMA) plate.As for the aluminum host structure, the PMMA material properties are obtained by weighing a PMMA plateand applying a model updating procedure with an FE model. The obtained values are also shown in Table 1.

Material Young’s modulus Poisson’s ratio DensityAluminum 63.5 GPa 0.36 2666 kg/m3

PMMA 4.85 GPa 0.31 1188 kg/m3

Table 1: Retrieved material parameters for host structure (aluminum) and resonator (PMMA)

With the known material parameters for the PMMA, the resonator geometry is tuned in order to obtain aresonance frequency of the first bending mode in the frequency range of interest up to 1000 Hz. Further,a relative mass addition of less than 50% is maintained and the resonators should be compact enough tosatisfy the subwavelength requirement. The eventually obtained resonator dimensions are shown in Figure2, leading to a numerically predicted resonance frequency of 616 Hz for an FE model of the clamped baseresonator consisting of 936 HEX8 solid elements, as shown in Figure 1b. In order to comply with the

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subwavelength requirement, the UC dimensions are chosen as 3× 3 cm, leading to a relative mass additionof 40.3% with respect to the bare host structure UC. The resonators are added to the aluminum plate bymeans of adhesive bonding and oriented with the long side of the beam along the x-direction (Figure 3).

Figure 2: Geometry of the resonant structures, dimensions in mm

Figure 3: Unit cell orientation and convention of propagation direction θ in the xy-plane

Furthermore, since damping is important for the performance of the locally resonant metamaterial, a struc-tural damping coefficient is estimated for the PMMA of the resonant structures. By an SLDV measurementof the structural response of a single resonator, excited at the base by a shaker, application of the half powerbandwidth method results in a structural damping coefficient of η = 5% [20]. Similarly, for the aluminum,the same method is applied to the resonance peaks of the host structure plate around the targeted 600−650 Hzstop band frequency range, yielding a structural damping coefficient of η = 0.6%.

For the eventual metamaterial plate, a total of 130 resonators are added to the 1 mm thick aluminum platewith dimensions 32× 42 cm. The 3× 3 cm relative spacing of the resonators corresponds to the adopted UCdimensions, leading to a grid of 13× 10 unit cells centered on the plate, oriented with the x-direction alongthe long side of the plate as indicated in Figure 4a. An edge is left around the plate to allow the structure tobe used with various mounting configurations.

(a) Finite metamaterial plate orientation (b) Close-up unit cell

Figure 4: Manufactured locally resonant metamaterial plate

3.2 Description measurement setup

In order to determine the experimental dispersion curves for the manufactured metamaterial plate describedabove, a Scanning Laser Doppler Vibrometer measurement is performed to obtain the harmonic structuraldisplacement field w(x, y) of the plate subjected to a shaker excitation (see Figure 5).

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Figure 5: SLDV measurement setup with freely suspended metamaterial plate and electrodynamic shaker

The 42× 32 cm metamaterial plate is suspended with two wires along its short side to mimic free boundaryconditions. An electrodynamic shaker is used to provide a harmonic force excitation of the structure. Theexcitation signal is white noise, ranging from 10 Hz to 1000 Hz with a frequency step of 0.3125 Hz. Inorder to excite all the modes of the structure, the shaker is fixed through an impedance head at an off-center location, at (−12.6,−6) cm from the plate center, with x and y conventions according to Figure 5.The out-of-plane structural displacement response w(x, y) is measured with a Polytec PSV-500 ScanningVibrometer as the frequency response function between the input force measured at the impedance head andthe responses for an equidistant grid of Nx = 14 by Ny = 11 points located at each of the UC corners.

4 Results

In this section, the numerically predicted dispersion curves are presented for the locally resonant metamate-rial, resulting from the UC analysis with damping. Next, the experimental dispersion curves are presentedfor the manufactured finite size metamaterial, obtained by application of the IWC to the SLDV measurementresults. These results are eventually compared with the numerical predictions.

4.1 Numerically predicted dispersion curves

The UC analysis procedure for damped periodic structures, as described in section 2.1, is used to predict thedispersion curves with the metamaterial UC introduced in section 3.1. For the FE model of the 3 × 3 cmUC, 936 HEX8 solid elements are used for the PMMA resonator and 930 QUAD4 shell elements are usedfor the aluminum plate. Without explicit modelling of the glue layer, equivalencing of the interfacing nodesis applied to assure the mesh connectivity.

The dispersion curves are calculated for three propagation directions θ = 0◦, 45◦ and 90◦ by solving theeigenvalue problem of equation (3) for the frequency range f = 1−1−1000 Hz to the complex propagationconstants µx and µy. From the resulting propagation constants, the wavenumber k = kprop + jkdec can becalculated for the considered direction of propagation θ, with kprop and kdec respectively the propagative anddecaying part of the wavenumber.

The dispersion diagrams for these three propagation directions are given in Figure 6 and show the propaga-tive part kprop of the wavenumber against the frequency f , with a color scale indicating the attenuation γ

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calculated as the ratio γ = kdec/kprop. In these diagrams shear, compressive and flexural waves are observed,with the latter being the most dispersive type for the structure at hand. For completeness, both the left andright going wave solutions are shown. Because of the convention adopted for the Bloch-Floquet theorem insection 2.1, positive and negative kprop indicate left and right going waves respectively.

(a) θ = 0◦ (b) θ = 45◦ (c) θ = 90◦

Figure 6: Dispersion curves obtained with the damped UC analysis for different propagation directions,colored according the magnitude of the wave attenuation γ, solid horizontal lines indicate the band gapspredicted using the undamped UC analysis

If no damping is considered, a flexural wave stop band is found along the IBC between 596 and 658 Hz.If only the three propagation directions θ = 0◦, 45◦ and 90◦ are considered, directional band gaps arerespectively found between 573 and 658 Hz, 592 and 658 Hz, and 596 and 658 Hz. These are indicated bymeans of solid horizontal lines in Figure 6. Differences in stop band width can be attributed to the shape ofthe resonator, clearly showing stronger interaction with the flexural waves along the θ = 0◦ direction.

From the damped dispersion curves in Figure 6, it is evident that the addition of damping has a considerableinfluence on these band gaps. No freely propagating wave solutions are found, rendering the notion of astop band more vague. A region of strong attenuation is found around the predicted undamped band gapfrequency zones, extending over a broader frequency range. However, the introduction of damping alsocauses the presence of solutions with a propagative part inside the predicted stop band zone. Depending onthe propagation direction, a difference is observed in the attenuation performance for flexural waves.

4.2 Experimantally measured dispersion curves

The measured frequency response functions obtained with SLDV measurements described in section 3.2are used as the input displacement field w(x, y) for IWC equation (5). Corresponding to the propagationdirections considered in the UC analysis, again the three directions θ = 0◦, 45◦ and 90◦ are investigated. Ina first step, the IWC is calculated considering a purely propagating wave, by imposing kdec = 0. Results areshown in Figure 7; the color scale indicates the magnitude of the IWC. Since the out-of-plane displacementfield is measured, only the flexural wavenumber content is retrieved. Similar to the results for the UCanalysis, both left and right going waves are shown.

As in the lower frequency range, the structure is mainly dominated by modal behaviour, structural modesshow up as (f, kprop)-pairs with a high IWC magnitude, sampling on the dispersion curves. Around the

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(a) θ = 0◦ (b) θ = 45◦ (c) θ = 90◦

Figure 7: Calculated IWC for different propagation directions θ, considering no attenuation, colored accord-ing to the magnitude of the IWC

predicted stop band regions, the IWC shows a zone of higher dispersion, resembling the shape of the pre-dicted dispersion curves from Figure 6. Since the dimension of the plate in y-direction is shorter than inx-direction, the effective wavenumber resolution for the θ = 90◦ is lower, due to the underlying relationbetween the IWC and the Discrete Fourier Transform. Nonetheless, the overall shapes of the IWC resultsshow differences between the considered propagation directions similar to the numerical predictions.

(a) θ = 0◦ (b) θ = 45◦ (c) θ = 90◦

Figure 8: IWC optimized with respect to kprop and kdec for different propagation directions θ, consideringtwo maximizers per frequency, colored according to the magnitude of the IWC

To determine dispersion curves for the considered propagation directions θ, the IWC is maximized withrespect to kprop and kdec, for two peak values per frequency. The results of this optimization are depictedin Figure 8. The color scale again indicates the magnitude of the, now optimized, IWC values. In theobtained dispersion diagrams, the modal behaviour leads to more distinct peaks for the lower frequency

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region, with regions of lower correlation in between. At higher frequencies, the shorter wavelength allowsa better correlation to be obtained in between modes as well. As the structure is excited off-center, someasymmetry is noticed between the left and right going solution.

4.3 Comparison numerical and experimental dispersion curves

The measured dispersion curves are compared to the numerically predicted dispersion curves for the flexuralwave type in Figure 9. A good agreement is found between the shapes of the dispersion curves. In the lowfrequency region some more discrepancies are noticed, which can be explained by the free-free boundaryconditions. The shapes of these low frequency modes deviate from the sinusoidal wave shapes that are foundfor infinite periodic structures. With increasing frequency, together with the presence of damping, theseboundary conditions become of less influence.

Around the predicted stop band frequency ranges, the numerical results predict the bounds for the maximaloccurring wavenumber content well. Inside the stop band region, the high attenuation leads to strongly de-caying waves, which can give rise to more difficult wavenumber estimation. Another cause of discrepanciescan be found in the excitation. Where both the UC analysis and IWC assume plane wave propagation, thepoint force excitation leads to a circular wave pattern. Overall, the experimental results agree well with thenumerical predictions for the various propagation directions.

(a) θ = 0◦ (b) θ = 45◦ (c) θ = 90◦

Figure 9: Comparison of the experimentally retrieved dispersion curves from Figure 8, colored according tothe magnitude of the optimized IWC, with the numerically predicted damped dispersion curves for bendingwaves from Figure 6, colored in black, for different propagation directions θ

To further evaluate the experimentally retrieved results, Figure 10 shows the same optimized values for kprop,but now colored according to the attenuation γ. These results can be compared to the damped dispersioncurves from Figure 6. The results with attenuation γ > 1.62, corresponding to the maximum attenuationobserved with the UC analysis, are omitted for the sake of clarity. For the three propagation directions, aclear region of higher attenuation is observed around the high attenuation zones predicted by the dampedUC analysis. Where differences in attenuation performance were numerically predicted, this is also foundexperimentally. While the UC analysis assumes perfect periodicity, some scatter can be present on theperiodicity and the resonant structures. As discussed in [21], apart from damping, this has an additionalwidening effect on the region of attenuation, in spite of lower peak performance. In general, similar to theshape of the dispersion curves, a reasonable agreement with the predicted attenuation range is found.

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(a) θ = 0◦ (b) θ = 45◦ (c) θ = 90◦

Figure 10: Experimentally retrieved dispersion curves from Figure 8, colored according to the magnitude ofthe wave attenuation γ

5 Conclusion

This paper presents the comparison of numerically predicted and experimentally measured dispersion curvesfor locally resonant metamaterials with damping. A locally resonant metamaterial design is analyzed bymeans of a unit cell modelling approach including damping. For the out-of-plane flexural waves a region ofhigh attenuation is predicted in the numerically calculated dispersion curves for different propagation direc-tions, around the stop band obtained with the undamped unit cell analysis. A finite size plate of the designedmetamaterial is manufactured and Scanning Laser Doppler Vibrometer measurements are performed to ob-tain the displacement field under structural excitation and free boundary conditions. Experimental dispersioncurves for different propagation directions are retrieved by application of the Inhomogeneous Wave Corre-lation method to the measured displacement field. The numerically predicted and experimentally measureddispersion curves are in good agreement.

Acknowledgements

The research of L. Van Belle is funded by a grant of the Research Foundation - Flanders (F.W.O.). E.Deckers is a postdoctoral fellow of the Research Foundation - Flanders (F.W.O.). The KU Leuven ResearchFund is gratefully acknowledged for their support through an IOF-Leverage project. This research is partiallysupported by Flanders Make, the Strategic Research Centre for the Manufacturing Industry.

References

[1] A. I. Taub, P. E. Krajewski, A. A. Luo, and J. N. Owens. The evolution of technology for materialsprocessing over the last 50 years: The automotive example. JOM, 59(2):48–57, 2007.

[2] M. Basner, W. Babisch, A. Davis, M. Brink, C. Clark, S. Janssen, and S. Stansfeld. Auditory andnon-auditory effects of noise on health. The Lancet, 383(9925):1325–1332, 2014.

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[3] P. Sheng, X. Zhang, Z. Liu, and C. Chan. Locally resonant sonic materials. Physica B: CondensedMatter, 338(1):201–205, 2003.

[4] X. Zhou, X. Liu, and G. Hu. Elastic metamaterials with local resonances: an overview. Theoreticaland Applied Mechanics Letters, 2(4):041001, 2012.

[5] C. Claeys, E. Deckers, B. Pluymers, and W. Desmet. A lightweight vibro-acoustic metamaterial demon-strator: Numerical and experimental investigation. Mechanical Systems and Signal Processing, 70-71:853–880, Mar 2016.

[6] C. Goffaux, J. Sanchez-Dehesa, A. Yeyati, Ph. Lambin, A. Khelif, J. Vasseur, and B. Djafari-Rouhani.Evidence of Fano-Like Interference Phenomena in Locally Resonant Materials. Physical Review Let-ters, 88(22), May 2002.

[7] C. Claeys, K. Vergote, P. Sas, and W. Desmet. On the potential of tuned resonators to obtain low-frequency vibrational stop bands in periodic panels. Journal of Sound and Vibration, 332(6):1418–1436, Mar 2013.

[8] L. Brillouin. Wave propagation in periodic structures. McGraw-Hill Book Company, 2nd edition,1946.

[9] D. J. Mead. Wave propagation in continuous periodic structures: research contributions fromSouthampton, 1964–1995. Journal of Sound and Vibration, 190(3):495–524, 1996.

[10] M. I. Hussein, M. J. Leamy, and M. Ruzzene. Dynamics of phononic materials and structures: Histor-ical origins, recent progress, and future outlook. Appl. Mech. Rev., 66(4):040802, May 2014.

[11] L. Van Belle, C. Claeys, E. Deckers, and W. Desmet. Study of damping in locally resonant metamateri-als through unit cell modelling. In The 12th International Conference on Mathematical and NumericalAspects of Wave Propagation Book of Abstracts, pages 138–139, Jul 2015.

[12] J. Berthaut, M. N. Ichchou, and L. Jezequel. K-space identification of apparent structural behaviour.Journal of Sound and Vibration, 280(3-5):1125–1131, Feb 2005.

[13] B. R. Mace and E. Manconi. Modelling wave propagation in two-dimensional structures using finiteelement analysis. Journal of Sound and Vibration, 318(4-5):884–902, Dec 2008.

[14] F. Farzbod and M. J. Leamy. Analysis of Bloch’s method in structures with energy dissipation. Journalof Vibration and Acoustics, 133(5):051010, 2011.

[15] E. Andreassen and J. S. Jensen. Analysis of phononic bandgap structures with dissipation. Journal ofVibration and Acoustics, 135(4):041015, Jun 2013.

[16] C. Claeys, P. Sas, and W. Desmet. On the acoustic radiation efficiency of local resonance based stopband materials. Journal of Sound and Vibration, 333(14):3203–3213, Jul 2014.

[17] G. Wang, X. Wen, J. Wen, L. Shao, and Y. Liu. Two-dimensional locally resonant phononic crystalswith binary structures. Physical Review Letters, 93(15), Oct 2004.

[18] L. Cremer, M. Heckl, and B. A. T. Petersson. Structure-borne sound: Structural vibrations and soundradiation at audio frequencies. Springer Science & Business Media, 3rd edition, 2005.

[19] W. Heylen, S. Lemmens, and P. Sas. Modal analysis theory and testing. KU Leuven, PMA division,2014.

[20] M. Carfagni, E. Lenzi, and M. Pierini. The loss factor as a measure of mechanical damping. InProceedings of the 16th International Modal Analysis Conference, 1998, volume 3243, pages 580–584, 1998.

METAMATERIALS 2125

[21] C. Claeys. Design and analysis of resonant metamaterials for acoustic insulation (Ontwerp en analysevan resonante metamaterialen voor geluidsisolatie). PhD thesis, KU Leuven Faculty of EngineeringScience, Arenberg Doctoral School, Apr 2014.

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