# Vibro-Acoustic Performance of a Fluid-Loaded Periodic Locally Resonant Plate

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Structure Model and Basic Governing Equations

#### 2.2. Band-Gap Formulations of an Infinite LR Plate

#### 2.3. Vibration and Radiation Formulations of a Finite LR Plate

## 3. Results and Discussion

#### 3.1. Band-Gap Properties

^{3}, approximately twice that of the base plate’s density. As shown above, the introduction of water fluid decreases the Bragg frequency from 1973.2 Hz to 1386.7 Hz, and this decrease is also caused by the attached mass. By comparing Equation (44) to (47), a general relation of Bragg frequency between the plate with water fluid and without fluid can be given as

#### 3.2. Vibration and Sound Radiation Performance

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Gao, N.S.; Zhang, Z.C.; Deng, J.; Guo, X.Y.; Cheng, B.Z.; Hou, H. Acoustic metamaterials for noise reduction: A review. Adv. Mater. Technol.
**2022**, 7, 2100698. [Google Scholar] [CrossRef] - Liao, G.X.; Luan, C.C.; Wang, Z.W.; Liu, J.P.; Yao, X.H.; Fu, J.Z. Acoustic metamaterials: A review of theories, structures, fabrication approaches, and applications. Adv. Mater. Technol.
**2021**, 6, 2000787. [Google Scholar] [CrossRef] - Liu, J.Y.; Guo, H.B.; Wang, T. A review of acoustic metamaterials and phononic crystals. Crystals
**2020**, 10, 305. [Google Scholar] [CrossRef] - Brillouin, L. Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, 1st ed.; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1946. [Google Scholar]
- Kushwaha, M.S.; Halevi, P.; Dobrzynski, L.; Djafarirouhani, B. Acoustic band-structure of periodic elastic composites. Phys. Rev. Lett.
**1993**, 71, 2022–2025. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.Y.; Zhang, X.X.; Mao, Y.W.; Zhu, Y.Y.; Yang, Z.Y.; Chan, C.T.; Sheng, P. Locally resonant sonic materials. Science
**2000**, 289, 1734–1736. [Google Scholar] [CrossRef] - Wen, X.; Wen, J.; Yu, D.; Wang, G.; Liu, Y.; Han, X. Phononic Crystals, 1st ed.; National Defence Industry Press: Beijing, China, 2009. [Google Scholar]
- Wen, Q.H.; Zuo, S.G.; Wei, H. Locally resonant elastic wave band gaps in flexural vibration of multi-oscillators beam. Acta Phys. Sin.
**2012**, 61, 034301. [Google Scholar] - Wu, J.; Bai, X.C.; Xiao, Y.; Geng, M.X.; Yu, D.L.; Wen, J.H. Low frequency band gaps and vibration reduction properties of a multi-frequency locally resonant phononic plate. Acta Phys. Sin.
**2016**, 65, 064602. [Google Scholar] - Xiao, Y.; Wen, J.H.; Wen, X.S. Broadband locally resonant beams containing multiple periodic arrays of attached resonators. Phys. Lett. A
**2012**, 376, 1384–1390. [Google Scholar] [CrossRef] - Romero-Garcia, V.; Krynkin, A.; Garcia-Raffi, L.M.; Umnova, O.; Sanchez-Perez, J.V. Multi-resonant scatterers in sonic crystals: Locally multi-resonant acoustic metamaterial. J. Sound Vib.
**2013**, 332, 184–198. [Google Scholar] [CrossRef] - Xiao, Y.; Wen, J.H.; Wen, X.S. Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators. J. Sound Vib.
**2012**, 331, 5408–5423. [Google Scholar] [CrossRef] - Ma, J.G.; Sheng, M.P.; Guo, Z.W.; Qin, Q. Dynamic analysis of periodic vibration suppressors with multiple secondary oscillators. J. Sound Vib.
**2018**, 424, 94–111. [Google Scholar] [CrossRef] - Qin, Q.; Sheng, M.P.; Guo, Z.W. Low-frequency vibration and radiation performance of a locally resonant plate attached with periodic multiple resonators. Appl. Sci.
**2020**, 10, 2843. [Google Scholar] [CrossRef] - Guo, Z.; Guo, H.; Wang, T. Vibro-acoustic performance of acoustic metamaterial plate with periodic lateral local resonator. Acta Phys. Sin.
**2021**, 70, 214301. [Google Scholar] [CrossRef] - Deymierv, P.A. Acoustic Metamaterials and Phononic Crystals; Springer Science & Business Media: New York, NY, USA, 2013. [Google Scholar]
- Spadoni, A.; Daraio, C.; Hurst, W.; Brown, M. Nonlinear phononic crystals based on chains of disks alternating with toroidal structures. Appl. Phys. Lett.
**2011**, 98, 161901. [Google Scholar] [CrossRef] - Romeo, F.; Ruzzene, M. Wave Propagation in Linear and Nonlinear Periodic Media: Analysis and Applications; Springer Science & Business Media: New York, NY, USA, 2013. [Google Scholar]
- Fang, X.; Wen, J.H.; Benisty, H.; Yu, D.L. Ultrabroad acoustical limiting in nonlinear metamaterials due to adaptive-broadening band-gap effect. Phys. Rev. B
**2020**, 101, 104304. [Google Scholar] [CrossRef] - Sheng, P.; Fang, X.; Wen, J.H.; Yu, D.L. Vibration properties and optimized design of a nonlinear acoustic metamaterial beam. J. Sound Vib.
**2021**, 492, 115739. [Google Scholar] [CrossRef] - Duhamel, D.; Mace, B.R.; Brennan, M.J. Finite element analysis of the vibrations of waveguides and periodic structures. J. Sound Vib.
**2006**, 294, 205–220. [Google Scholar] [CrossRef] - Mead, D.J. The forced vibration of one-dimensional multi-coupled periodic structures: An application to finite element analysis. J. Sound Vib.
**2009**, 319, 282–304. [Google Scholar] [CrossRef] - Manktelow, K.; Narisetti, R.K.; Leamy, M.J.; Ruzzene, M. Finite-element based perturbation analysis of wave propagation in nonlinear periodic structures. Mech. Syst. Signal Process.
**2013**, 39, 32–46. [Google Scholar] [CrossRef] - Qiu, C.Y.; Liu, Z.Y.; Mei, J.; Ke, M.Z. The layer multiple-scattering method for calculating transmission coefficients of 2D phononic crystals. Solid State Commun.
**2005**, 134, 765–770. [Google Scholar] [CrossRef] - Wang, G.; Wen, J.H.; Wen, X.S. Quasi-one-dimensional phononic crystals studied using the improved lumped-mass method: Application to locally resonant beams with flexural wave band gap. Phys. Rev. B
**2005**, 71, 104302. [Google Scholar] [CrossRef] - Cao, Y.J.; Hou, Z.L.; Liu, Y.Y. Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals. Solid State Commun.
**2004**, 132, 539–543. [Google Scholar] [CrossRef] - Li, C.; Chen, Z.; Jiao, Y. Vibration and bandgap behavior of sandwich pyramid lattice core plate with resonant rings. Materials
**2023**, 16, 2730. [Google Scholar] [CrossRef] - Yu, D.L.; Liu, Y.Z.; Qiu, J.; Zhao, H.G.; Liu, Z.M. Experimental and theoretical research on the vibrational gaps in two-dimensional three-component composite thin plates. Chin. Phys. Lett.
**2005**, 22, 1958–1960. [Google Scholar] - Zhao, H.G.; Wen, J.H.; Yu, D.L.; Wen, X.S. Low-frequency acoustic absorption of localized resonances: Experiment and theory. J. Appl. Phys.
**2010**, 107, 023519. [Google Scholar] [CrossRef] - Meng, H.; Wen, J.H.; Zhao, H.G.; Wen, X.S. Optimization of locally resonant acoustic metamaterials on underwater sound absorption characteristics. J. Sound Vib.
**2012**, 331, 4406–4416. [Google Scholar] [CrossRef] - Zhang, S.W.; Wu, J.H.; Hu, Z.P. Low-frequency locally resonant band-gaps in phononic crystal plates with periodic spiral resonators. J. Appl. Phys.
**2013**, 113, 163511. [Google Scholar] [CrossRef] - Nouh, M.; Aldraihem, O.; Baz, A. Wave propagation in metamaterial plates with periodic local resonances. J. Sound Vib.
**2015**, 341, 53–73. [Google Scholar] [CrossRef] - Hsu, J.C. Local resonances-induced low-frequency band gaps in two-dimensional phononic crystal slabs with periodic stepped resonators. J. Phys. D Appl. Phys.
**2011**, 44, 055401. [Google Scholar] [CrossRef] - Xiao, Y.; Wen, J.H.; Yu, D.L.; Wen, X.S. Flexural wave propagation in beams with periodically attached vibration absorbers: Band-gap behavior and band formation mechanisms. J. Sound Vib.
**2013**, 332, 867–893. [Google Scholar] [CrossRef] - Wang, Y.F.; Wang, Y.S. Complete bandgaps in two-dimensional phononic crystal slabs with resonators. J. Appl. Phys.
**2013**, 114, 043509. [Google Scholar] [CrossRef] - Casadei, F.; Dozio, L.; Ruzzene, M.; Cunefare, K.A. Periodic shunted arrays for the control of noise radiation in an enclosure. J. Sound Vib.
**2010**, 329, 3632–3646. [Google Scholar] [CrossRef] - Aladwani, A.; Almandeel, A.; Nouh, M. Fluid-structural coupling in metamaterial plates for vibration and noise mitigation in acoustic cavities. Int. J. Mech. Sci.
**2019**, 152, 151–166. [Google Scholar] [CrossRef] - Claeys, C.C.; Sas, P.; Desmet, W. On the acoustic radiation efficiency of local resonance based stop band materials. J. Sound Vib.
**2014**, 333, 3203–3213. [Google Scholar] [CrossRef] - Guo, Z.W.; Pan, J.; Sheng, M.P. Vibro-acoustic performance of a sandwich plate with periodically inserted resonators. Appl. Sci.
**2019**, 9, 3651. [Google Scholar] [CrossRef] - Song, Y.B.; Wen, J.H.; Yu, D.L.; Liu, Y.Z.; Wen, X.S. Reduction of vibration and noise radiation of an underwater vehicle due to propeller forces using periodically layered isolators. J. Sound Vib.
**2014**, 333, 3031–3043. [Google Scholar] [CrossRef] - Jin, G.Y.; Shi, K.K.; Ye, T.G.; Zhou, J.L.; Yin, Y.W. Sound absorption behaviors of metamaterials with periodic multi-resonator and voids in water. Appl. Acoust.
**2020**, 166, 107351. [Google Scholar] [CrossRef] - Zhong, H.B.; Tian, Y.J.; Gao, N.S.; Lu, K.; Wu, J.H. Ultra-thin composite underwater honeycomb-type acoustic metamaterial with broadband sound insulation and high hydrostatic pressure resistance. Compos. Struct.
**2021**, 277, 114603. [Google Scholar] [CrossRef] - Li, W.L. An analytical solution for the self- and mutual radiation resistances of a rectangular plate. J. Sound Vib.
**2001**, 245, 1–16. [Google Scholar] [CrossRef] - Sugino, C.; Xia, Y.; Leadenham, S.; Ruzzene, M.; Erturk, A. A general theory for bandgap estimation in locally resonant metastructures. J. Sound Vib.
**2017**, 406, 104–123. [Google Scholar] [CrossRef] - Xiao, Y.; Wen, J.H.; Wen, X.S. Flexural wave band gaps in locally resonant thin plates with periodically attached spring-mass resonators. J. Phys. D Appl. Phys.
**2012**, 45, 195401. [Google Scholar] [CrossRef] - Fahy, F.; Gardonio, P. Sound and Structural Vibration: Radiation, Transmission and Response, 2nd ed.; Elsevier/Academic Press: Oxford, UK, 2007. [Google Scholar]

**Figure 2.**Schematic of a finite locally resonant plate baffled in an infinite rigid baffle and single-side submerged in the infinite fluid (SSB: simply supported boundary).

**Figure 3.**Dispersion curves of the water-loaded locally resonant plate and corresponding band gap (BG).

**Figure 4.**Dispersion curves of the locally resonant plate (

**a**) with air load and without load, and (

**b**) with water load and without load (w.o.: without, BG: band gap).

**Figure 5.**(

**a**) Dispersion curves of the LR plate with water load and without (w.o.) fluid load; (

**b**) normalized effective density (d.c.: dispersion curve).

**Figure 6.**Normalized effective density with water load as a function of frequency with frequency range (

**a**) from 0 Hz to 1 kHz and (

**b**) from 460 Hz to 560 Hz (BG: band gap; HM: homogeneous; LR: locally resonant).

**Figure 8.**(

**a**) Band-gap frequency and (

**b**) band-gap width of the locally resonant plate with and without (w.o.) water fluid as a function of the resonant frequency.

**Figure 9.**(

**a**) Average velocity level and (

**b**) radiation power level of the water-loaded LR plate (BG: band-gap).

**Figure 10.**Comparison of (

**a**,

**b**) average velocity level and (

**c**,

**d**) radiation power level between water-loaded locally resonant (LR) plate and water-loaded homogeneous (HM) plate (BG: band-gap).

**Figure 11.**Comparison of (

**a**,

**b**) average velocity level and (

**c**,

**d**) radiation power level between locally resonant (LR) plate with fluid and that without fluid (BG: band gap; w.o.: without).

**Figure 12.**Comparison of radiation efficiencies between a water-loaded locally resonant (LR) plate and a water-loaded homogeneous (HM) plate (BG: band gap).

**Figure 13.**Spatial distributions of the normalized displacement (norm. disp.) at 518 Hz in (

**a**) a homogeneous plate and (

**b**) a locally resonant plate.

**Figure 15.**(

**a**) Average velocity level and (

**b**) radiation power level of a water-loaded homogeneous (HM) plate and a water-loaded locally resonant (LR) plate with damped and undamped periodic resonators.

**Figure 16.**(

**a**) Average velocity level and (

**b**) radiation power level of a water-loaded locally resonant plate with various damping parameters of the resonator.

Band-Gap Start Frequency | Band-Gap Cut-off Frequency | Band-Gap Width | |
---|---|---|---|

Present model | 510.1 Hz | 512.8 Hz | 2.7 Hz |

FEM model | 510.0 Hz | 512.8 Hz | 2.8 Hz |

Band-Gap Start Frequency | Band-Gap Cut-off Frequency | Band-Gap Width | |
---|---|---|---|

without fluid load | 510.1 Hz | 561.6 Hz | 51.5 Hz |

with air load | 510.1 Hz | 561.6 Hz | 51.5 Hz |

with water load | 510.1 Hz | 512.8 Hz | 2.7 Hz |

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**MDPI and ACS Style**

Guo, Z.; Sheng, M.; Zeng, H.; Wang, M.; Li, Q.
Vibro-Acoustic Performance of a Fluid-Loaded Periodic Locally Resonant Plate. *Machines* **2023**, *11*, 590.
https://doi.org/10.3390/machines11060590

**AMA Style**

Guo Z, Sheng M, Zeng H, Wang M, Li Q.
Vibro-Acoustic Performance of a Fluid-Loaded Periodic Locally Resonant Plate. *Machines*. 2023; 11(6):590.
https://doi.org/10.3390/machines11060590

**Chicago/Turabian Style**

Guo, Zhiwei, Meiping Sheng, Hao Zeng, Minqing Wang, and Qiaojiao Li.
2023. "Vibro-Acoustic Performance of a Fluid-Loaded Periodic Locally Resonant Plate" *Machines* 11, no. 6: 590.
https://doi.org/10.3390/machines11060590