# 2D Dynamic Directional Amplification (DDA) in Phononic Metamaterials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Bloch’s Theorem

#### 2.2. Simple 2D Monoatomic Lattice

#### 2.3. Overview of the Dynamic Directional Amplification (DDA) Mechanism

#### 2.4. 2D Monoatomic Lattice with Dynamic Directional Amplifiers (DDA)

#### 2.5. 2D Phononic Lattice with Dynamic Directional Amplifier (DDA)

#### 2.5.1. Wave Dispersion Analysis

#### 2.5.2. Structural Dynamics of the Finite Lattice

**FRF**) of the metamaterial is defined as:

## 3. Numerical Results

#### 3.1. Dynamic Amplification Induced Bandgaps

#### 3.2. Effect of Number of Unit Cells

#### 3.3. Dynamically Induced Metadamping

#### 3.4. Effects of Response Point

## 4. Conceptual Design

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

**M**,

**C**,

**K**matrices used in the dispersion equation of the lattice without the DDA–Equation (17) and, including the DDA—Equation (18). Starting with the lattice without the amplification mechanism, the dispersion relationship can be calculated from:

## Appendix C

## References

- Brillouin, L. Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices; Dover phoenix editions; Dover Publications: New York, NY, USA, 1946; ISBN 0486600343. [Google Scholar]
- Mead, D.M. Wave Propagation in Continuous Periodic Structures: Research Contribution from Southampton, 1964–1995. J. Sound Vib.
**1996**, 190, 495–524. [Google Scholar] [CrossRef] - Hussein, M.I.; Leamy, M.J.; Ruzzene, M. Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Appl. Mech. Rev.
**2014**, 66. [Google Scholar] [CrossRef] - Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.Y.; Yang, Z.; Chan, C.T.; Sheng, P. Locally resonant sonic materials. Science
**2000**, 289, 1734–1736. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; Guo, H.; Wang, T. A Review of Acoustic Metamaterials and Phononic Crystals. Crystals
**2020**, 10, 305. [Google Scholar] [CrossRef] - Pennec, Y.; Djafari-Rouhani, B.; Larabi, H.; Vasseur, J.O.; Hladky-Hennion, A.C. Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate. Phys. Rev. B
**2008**, 78, 104105. [Google Scholar] [CrossRef] - Wu, T.-T.; Huang, Z.-G.; Tsai, T.-C.; Wu, T.-C. Evidence of complete band gap and resonances in a plate with periodic stubbed surface. Appl. Phys. Lett.
**2008**, 93, 111902. [Google Scholar] [CrossRef] - Li, J.; Chan, C.T. Double-negative acoustic metamaterial. Phys. Rev. E
**2004**, 70, 55602. [Google Scholar] [CrossRef] [Green Version] - Ding, Y.; Liu, Z.; Qiu, C.; Shi, J. Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density. Phys. Rev. Lett.
**2007**, 99, 93904. [Google Scholar] [CrossRef] [PubMed] - Mei, J.; Liu, Z.; Wen, W.; Sheng, P. Effective mass density of fluid-solid composites. Phys. Rev. Lett.
**2006**, 96, 24301. [Google Scholar] [CrossRef] [Green Version] - Hussein, M.I.; Frazier, M.J. Band structure of phononic crystals with general damping. J. Appl. Phys.
**2010**, 108, 93506. [Google Scholar] [CrossRef] - Kushwaha, M.S.; Halevi, P.; Dobrzynski, L.; Djafari-Rouhani, B. Acoustic band structure of periodic elastic composites. Phys. Rev. Lett.
**1993**, 71, 2022–2025. [Google Scholar] [CrossRef] [PubMed] - Zhou, J.; Wang, K.; Xu, D.; Ouyang, H. Local resonator with high-static-low-dynamic stiffness for lowering band gaps of flexural wave in beams. J. Appl. Phys.
**2017**, 121, 44902. [Google Scholar] [CrossRef] [Green Version] - Sigalas, M.M.; Economou, E.N. Elastic and acoustic wave band structure. J. Sound Vib.
**1992**, 158, 377–382. [Google Scholar] [CrossRef] - Sprik, R.; Wegdam, G.H. Acoustic band gaps in composites of solids and viscous liquids. Solid State Commun.
**1998**, 106, 77–81. [Google Scholar] [CrossRef] - Suzuki, T.; Yu, P.K.L. Complex elastic wave band structures in three-dimensional periodic elastic media. J. Mech. Phys. Solids
**1998**, 46, 115–138. [Google Scholar] [CrossRef] - Antoniadis, I.A.; Chronopoulos, D.; Spitas, V.; Koulocheris, D. Hyper-damping properties of a stable linear oscillator with a negative stiffness element. J. Sound Vib.
**2015**, 346, 37–52. [Google Scholar] [CrossRef] - Antoniadis, I.A.; Kanarachos, S.A.; Gryllias, K.; Sapountzakis, I.E. KDamping: A stiffness based vibration absorption concept. J. Vib. Control
**2016**, 24, 588–606. [Google Scholar] [CrossRef] [Green Version] - Paradeisiotis, A.; Kalderon, M.; Antoniadis, I.; Fouriki, L. Acoustic Performance Evaluation of a panel utilizing negative stifffness mounting for low frequency noise control. In Proceedings of the EURODYN 2020, EASD Procedia, Athens, Greece, 23–26 November 2020; pp. 4093–4110. [Google Scholar]
- Antoniadis, I.A.; Paradeisiotis, A. Acoustic Meta-Materials Incorporating the KDamper Concept for Low Frequency Acoustic Isolation. Acta Acust. United Acust.
**2018**, 104, 636–646. [Google Scholar] [CrossRef] - Chronopoulos, D.; Antoniadis, I.; Ampatzidis, T. Enhanced acoustic insulation properties of composite metamaterials having embedded negative stiffness inclusions. Extrem. Mech. Lett.
**2017**, 12, 48–54. [Google Scholar] [CrossRef] - Antoniadis, I.; Paradeisiotis, A. A periodic acoustic meta-material concept incorporating negative stiffness elements for low-frequency acoustic insulation/absorption. In Proceedings of the ISMA 2018-International Conference on Noise and Vibration Engineering and USD 2018-International Conference on Uncertainty in Structural Dynamics, Leuven, Belgium, 17−19 September 2018; pp. 1179–1193. [Google Scholar]
- Yuksel, O.; Yilmaz, C. Shape optimization of phononic band gap structures incorporating inertial amplification mechanisms. J. Sound Vib.
**2015**, 355, 232–245. [Google Scholar] [CrossRef] - Yilmaz, C.; Hulbert, G.M. Theory of phononic gaps induced by inertial amplification in finite structures. Phys. Lett. Sect. A Gen. At. Solid State Phys.
**2010**, 374, 3576–3584. [Google Scholar] [CrossRef] - Acar, G.; Yilmaz, C. Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures. J. Sound Vib.
**2013**, 332, 6389–6404. [Google Scholar] [CrossRef] - Taniker, S.; Yilmaz, C. Generating ultrawide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges. Int. J. Solids Struct.
**2017**, 106, 129–138. [Google Scholar] [CrossRef] - Kulkarni, P.P.; Manimala, J.M. Longitudinal elastic wave propagation characteristics of inertant acoustic metamaterials. J. Appl. Phys.
**2016**, 119, 245101. [Google Scholar] [CrossRef] - Yilmaz, C.; Hulbert, G.M.; Kikuchi, N. Phononic band gaps induced by inertial amplification in periodic media. Phys. Rev. B
**2007**, 76, 54309. [Google Scholar] [CrossRef] - Frandsen, N.M.M.; Bilal, O.R.; Jensen, J.S.; Hussein, M.I. Inertial amplification of continuous structures: Large band gaps from small masses. J. Appl. Phys.
**2016**, 119, 124902. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Li, S. Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects. Phys. Lett. A
**2018**, 382, 241–247. [Google Scholar] [CrossRef] - Foehr, A.; Bilal, O.R.; Huber, S.D.; Daraio, C. Spiral-Based Phononic Plates: From Wave Beaming to Topological Insulators. Phys. Rev. Lett.
**2018**, 120, 205501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morvaridi, M.; Carta, G.; Brun, M. Platonic crystal with low-frequency locally-resonant spiral structures: Wave trapping, transmission amplification, shielding and edge waves. J. Mech. Phys. Solids
**2018**, 121, 496–516. [Google Scholar] [CrossRef] [Green Version] - Antoniadis, I.A.; Georgoutsos, V.; Paradeisiotis, A. Fully enclosed multi-axis inertial reaction mechanisms for wave energy conversion. J. Ocean Eng. Sci.
**2017**, 2, 5–17. [Google Scholar] [CrossRef] - Bergamini, A.; Miniaci, M.; Delpero, T.; Tallarico, D.; Van Damme, B.; Hannema, G.; Leibacher, I.; Zemp, A. Tacticity in chiral phononic crystals. Nat. Commun.
**2019**, 10, 4525. [Google Scholar] [CrossRef] - Krushynska, A.O.; Amendola, A.; Bosia, F.; Daraio, C.; Pugno, N.M.; Fraternali, F. Accordion-like metamaterials with tunable ultra-wide low-frequency band gaps. New J. Phys.
**2018**, 20, 73051. [Google Scholar] [CrossRef] - Orta, A.; Yilmaz, C. Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism. J. Sound Vib.
**2018**, 439. [Google Scholar] [CrossRef] - Fernandez-Corbaton, I.; Rockstuhl, C.; Ziemke, P.; Gumbsch, P.; Albiez, A.; Schwaiger, R.; Frenzel, T.; Kadic, M.; Wegener, M. New Twists of 3D Chiral Metamaterials. Adv. Mater.
**2019**, 31, 1807742. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Delpero, T.; Hannema, G.; Van Damme, B.; Schoenwald, S.; Zemp, A.; Bergamini, A. Inertia Amplification in Phononic Crystals for Low Frequency Band Gaps. 8 ECCOMAS SMART
**2017**, 2017, 1657–1668. [Google Scholar] - Wanjura, C.C.; Brunelli, M.; Nunnenkamp, A. Topological framework for directional amplification in driven-dissipative cavity arrays. Nat. Commun.
**2020**, 11, 3149. [Google Scholar] [CrossRef] - de Lépinay, L.; Damskägg, E.; Ockeloen-Korppi, C.F.; Sillanpää, M.A. Realization of Directional Amplification in a Microwave Optomechanical Device. Phys. Rev. Appl.
**2019**, 11, 34027. [Google Scholar] [CrossRef] [Green Version] - Peano, V.; Houde, M.; Brendel, C.; Marquardt, F.; Clerk, A.A. Topological phase transitions and chiral inelastic transport induced by the squeezing of light. Nat. Commun.
**2016**, 7, 10779. [Google Scholar] [CrossRef] [PubMed] - Malz, D.; Tóth, L.D.; Bernier, N.R.; Feofanov, A.K.; Kippenberg, T.J.; Nunnenkamp, A. Quantum-Limited Directional Amplifiers with Optomechanics. Phys. Rev. Lett.
**2018**, 120, 23601. [Google Scholar] [CrossRef] [Green Version] - Hussein, M.I.; Frazier, M.J. Metadamping: An emergent phenomenon in dissipative metamaterials. J. Sound Vib.
**2013**, 332, 4767–4774. [Google Scholar] [CrossRef] [Green Version] - Bloch, F. Über die Quantenmechanik der Elektronen in Kristallgittern. Zeitschrift Für Phys.
**1929**, 52, 555–600. [Google Scholar] [CrossRef] - Craig, R.R.; Kurdila, A.J. Fundamentals of Structural Dynamics; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Brillouin, L. Science and Information Theory, 1st ed.; Academic Press, Inc.: New York, NY, USA, 1956. [Google Scholar]
- Hussein, M.I. Theory of damped Bloch waves in elastic media. Phys. Rev. B
**2009**, 80, 212301. [Google Scholar] [CrossRef]

**Figure 2.**Dispersion surface of the 2D spring–mass lattice with spring stiffness, ${k}_{x}{,k}_{y}$.

**Figure 3.**Dynamic directional amplification (DDA) mechanism, where the motion v of mass m is kinematically constrained to the motion u.

**Figure 5.**Dispersion surface of the 2D monoatomic lattice of periodic kinematically constrained DoFs with spring stiffness, ${k}_{x}{=k}_{y}$. (

**a**) $\varphi {(}^{\xb0})=15$, (

**b**)$\varphi {(}^{\xb0})=30$, (

**c**)$\varphi {(}^{\xb0})=45$ (

**d**) $\varphi {(}^{\xb0})=60$, (

**e**) $\varphi {(}^{\xb0})=75$ and (

**f**) dispersion curves.

**Figure 7.**Structure with ${M}_{x}{\times M}_{y}$ unit cells with a periodic loading acting at the right boundary and simple supports at the left corners (

**a**) without DDA (

**b**) with DDA.

**Figure 8.**(

**a**) Dispersion curves and (

**b**) frequency response of the 2D phononic lattice without the DDA along the Γ-X.

**Figure 9.**Dispersion contours of 2D phononic lattice for ${m}_{L}=1.0$, ${m}_{D}=1.1$, ${f}_{x}=100$ and ${f}_{y}=50$ for the (

**a**) lattice without DDA, (

**b**) the lattice with DDA and $\varphi {(}^{\xb0})=15$, (

**c**) the lattice with DDA and $\varphi {(}^{\xb0})=45$, (

**d**) the lattice with DDA and $\varphi {(}^{\xb0})=75$. (

**e**) Normalized bandgap width as a function of the amplifier’s angle.

**Figure 10.**Frequency response (FRF) plot in point A (according to Figure 7a of the ${M}_{x}{\times M}_{y}$ finite lattice, without damping and amplifier angle $\varphi {(}^{\xb0})=75$ for (

**a**) $x$-direction (

**b**) $y$-direction.

**Figure 11.**(

**a**) Frequency band structure and (

**b**) damping ratio $(\zeta )$ band structure. (

**c**) Frequency response function (FRF) plots of the $8\times 8$ finite lattice (response in point A).

**Figure 12.**Frequency response (${FRF}_{x}$) of the $8\times 8$ finite lattice, for ${\zeta}_{0x}=0.02$, ${\zeta}_{0y}=0.05$, and amplifier’s angle $\varphi {(}^{\xb0})=75$ at points A–C.

**Figure 13.**Conceptual design of the proposed metastructure, (

**a**) 3D view (

**b**) detail of the dynamic directional amplifier (DDA).

${\mathit{m}}_{\mathit{L}}(\mathbf{kg})$ | ${\mathit{m}}_{\mathit{D}}(\mathbf{kg})$ | ${\mathit{f}}_{0\mathit{x}}(\mathbf{Hz})$ | ${\mathit{f}}_{0\mathit{y}}(\mathbf{Hz})$ | ${\mathbf{\zeta}}_{\mathit{x}}$ | ${\mathbf{\zeta}}_{\mathit{y}}$ |
---|---|---|---|---|---|

1.0 | 1.1 | 100 | 50 | 0.02 | 0.05 |

Case | ${\mathit{f}}_{\mathit{u}}(\mathbf{Hz})$ | ${\mathit{f}}_{\mathit{l}}(\mathbf{Hz})$ | ${\mathit{f}}_{\mathit{av}}(\mathbf{Hz})$ | ${\mathit{b}}_{\mathit{w}}$ |
---|---|---|---|---|

Without DDA | 141.4 | 134.8 | 138.1 | 0.05 |

$\varphi {(}^{\xb0})=15$ | 141.4 | 131.9 | 136.6 | 0.07 |

$\varphi {(}^{\xb0})=45$ | 141.4 | 113.2 | 127.3 | 0.22 |

$\varphi {(}^{\xb0})=75$ | 141.4 | 99.2 | 120.3 | 0.35 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kalderon, M.; Paradeisiotis, A.; Antoniadis, I.
2D Dynamic Directional Amplification (DDA) in Phononic Metamaterials. *Materials* **2021**, *14*, 2302.
https://doi.org/10.3390/ma14092302

**AMA Style**

Kalderon M, Paradeisiotis A, Antoniadis I.
2D Dynamic Directional Amplification (DDA) in Phononic Metamaterials. *Materials*. 2021; 14(9):2302.
https://doi.org/10.3390/ma14092302

**Chicago/Turabian Style**

Kalderon, Moris, Andreas Paradeisiotis, and Ioannis Antoniadis.
2021. "2D Dynamic Directional Amplification (DDA) in Phononic Metamaterials" *Materials* 14, no. 9: 2302.
https://doi.org/10.3390/ma14092302