Abstract
Based on the symplectic and higher-order WKB theory, a semi-analytical method is developed for the forced vibration of a built-up system comprising rectangular thin plate terminated by multiple acoustic black hole beams. The analytical waves are used to describe the vibration of the plate and ABH beam components. The dynamic flexibility matrix and dynamic stiffness matrix are derived based on the analytical wave expressions for the plate and ABH beam component, respectively. By enforcing the displacement continuity and equilibrium of force at the connection interface, the dynamic coupling between the plate component and the ABH beam component is established. The proposed method can obtain the system balance equation by assembling the component matrix just like the traditional finite element method and has a much less degree of freedom. Numerical examples compare results from the proposed method with those from the finite element method. The comparison illustrates that the proposed method gives good predictions for the forced response of the built-up system considered here. The present approach is of high accuracy and can be used to provide benchmark solutions for other prediction methods.
Similar content being viewed by others
References
Kiran K, Poojary UR, Gangadharan KV (2022) Developing the viscoelastic model and model-based fuzzy controller for the MRE isolator for the wide frequency range vibration isolation. J Braz Soc Mech Sci 44:275. https://doi.org/10.1007/s40430-022-03575-y
Ribeiro LP, de Lima AMG, Silva VAC (2020) Robust project of resonant shunt circuit for passive vibration control of composite structures. J Braz Soc Mech Sci 42:342. https://doi.org/10.1007/s40430-020-02396-1
Pelat A, Gautier F, Conlon SC, Semperlotti F (2020) The acoustic black hole: a review of theory and applications. J Sound Vib 476:115316. https://doi.org/10.1016/j.jsv.2020.115316
Mironov MA (1988) Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval. Sov Phys Acoust. 34:318–319
Krylov VV, Tilman FJBS (2004) Acoustic ‘black holes’ for flexural waves as effective vibration dampers. J Sound Vib 274:605–619. https://doi.org/10.1016/j.jsv.2003.05.010
Deng J, Gao NS, Chen X (2023) Ultrawide attenuation bands in gradient metabeams with acoustic black hole pillars. Thin Wall Struct 184:110459. https://doi.org/10.1016/j.tws.2022.110459
Wan ZW, Zhu X, Li TY, Nie R (2022) Low-frequency multimode vibration suppression of an acoustic black hole beam by shunt damping. ASME J. Vib. Acoust. 144:021012. https://doi.org/10.1115/1.4053590
Deng J, Guasch O, Maxit L, Gao NS (2023) Sound radiation and non-negative intensity of a metaplate consisting of an acoustic black hole plus local resonators. Comput Struct 304:116423. https://doi.org/10.1016/j.compstruct.2022.116423
Gao WL, Qin ZY, Chu FL (2022) Broadband vibration suppression of rainbow metamaterials with acoustic black hole. Int J Mech Sci 228:107485. https://doi.org/10.1016/j.ijmecsci.2022.107485
Fu QD, Wu JW, Yu CY, Du XF, Zhang N, Zhang JR (2022) Parametric studies and optimal design of the exponents collocation of a segmented acoustic black hole beam. Appl Acoust 200:109086. https://doi.org/10.1016/j.apacoust.2022.109086
Tang LL, Gao NS, Xu JL, Chen K, Cheng L (2021) A light-weight periodic plate with embedded acoustic black holes and bandgaps for broadband sound radiation reduction. J Acoust Soc Am 150:3532–3543. https://doi.org/10.1121/10.0007067
Ning L, Wang YZ, Wang YS (2020) Active control of a black hole or concentrator for flexural waves in an elastic metamaterial plate. Mech Mater 142:103300. https://doi.org/10.1016/j.mechmat.2019.103300
Lyu XF, Sheng H, He MX, Ding Q, Tang LH, Yang TZ (2023) Satellite Vibration Isolation Using Periodic Acoustic Black Hole Structures With Ultrawide Bandgap. ASME J. Vib. Acoust. 145:014501. https://doi.org/10.1115/1.4054978
Souza MR, Fabro AT, Lenzi A (2021) Broadband vibration attenuation from a one-dimensional acoustic black hole resonator for plate-on-plate structures. J Braz Soc Mech Sci 43:1–16. https://doi.org/10.1007/s40430-021-03162-7
Ji HL, Wang N, Zhang C, Wang XD, Cheng L, Qiu JH (2021) A vibration absorber based on two-dimensional acoustic black holes. J Sound Vib 500:116024. https://doi.org/10.1016/j.jsv.2021.116024
Zhou T, Cheng L (2021) Planar Swirl-shaped acoustic black hole absorbers for multi-directional vibration suppression. J Sound Vib 516:116500. https://doi.org/10.1016/j.jsv.2021.116500
Karlos A, Hook K, Cheer J (2022) Enhanced absorption with multiple quadratically tapered elastic wedges of different lengths terminating a uniform beam. J Sound Vib 531:116981. https://doi.org/10.1016/j.jsv.2022.116981
Hook K, Cheer J, Karlos A (2022) An experimental investigation into a dual taper acoustic black hole termination. JASA Express Lett. 2:095601. https://doi.org/10.1121/10.0013899
Sheng H, He MX, Ding Q (2023) Vibration suppression by mistuning acoustic black hole dynamic vibration absorbers. J Sound Vib 542:117370. https://doi.org/10.1016/j.jsv.2022.117370
Zhao CY, Zheng JY, Sang T, Wang LC, Yi Q, Wang P (2021) Computational analysis of phononic crystal vibration isolators via FEM coupled with the acoustic black hole effect to attenuate railway-induced vibration. Constr Build Mater 283:122802. https://doi.org/10.1016/j.conbuildmat.2021.122802
Kim SY, Lee D (2020) Numerical simulation of characteristics of wave propagation and reflection coefficient in a helix-acoustic black hole. J Vib Control 28:615–625. https://doi.org/10.1177/1077546320980570
Huang W, Ji HL, Qiu JH, Cheng L (2018) Analysis of ray trajectories of flexural waves propagating over generalized acoustic black hole indentations. J Sound Vib 417:216–226. https://doi.org/10.1016/j.jsv.2017.12.012
Karlos A, Elliott SJ, Cheer J (2019) Higher-order WKB analysis of reflection from tapered elastic wedges. J Sound Vib 449:368–388. https://doi.org/10.1016/j.jsv.2019.02.041
Sheng H, He MX, Lyu XF, Ding Q (2021) Ultra-low frequency broadband gap optimization of 1D periodic structure with dual power-law acoustic black holes. J Intel Mat Syst Str 33:532–546. https://doi.org/10.1177/1045389X211018841
Conlon SC, Fahnline JB, Semperlotti F (2015) Numerical analysis of the vibroacoustic properties of plates with embedded grids of acoustic black holes. J Acoust Soc Am 137:447–457. https://doi.org/10.1121/1.4904501
Tang LL, Cheng L, Ji HL, Qiu JH (2016) Characterization of acoustic black hole effect using a one-dimensional fully-coupled and wavelet-decomposed semi-analytical model. J Sound Vib 374:172–184. https://doi.org/10.1016/j.jsv.2016.03.031
Leng J, Romero-García V, Pelat A, Picób R, Grobya JP, Gautier F (2020) Interpretation of the acoustic black hole effect based on the concept of critical coupling. J Sound Vib 471:115199. https://doi.org/10.1016/j.jsv.2020.115199
Georgiev VB, Cuenca J, Gautier F, Simon L, Krylov VV (2011) Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect. J Sound Vib 330:2497–2508. https://doi.org/10.1016/j.jsv.2010.12.001
Hachemi M, Guenanou A, Chebou R, Bachari K (2023) Mechanical behaviors of variable stiffness composite laminated sandwich plates using layer-wise model. J Braz Soc Mech Sci 45:77. https://doi.org/10.1007/s40430-022-03949-2
He DZ, Shi DY, Wang QS, Ma CL (2021) Free vibration characteristics and wave propagation analysis in nonlocal functionally graded cylindrical nanoshell using wave-based method. J Braz Soc Mech Sci 43:292. https://doi.org/10.1007/s40430-021-03008-2
Yao LQ, Ji CJ, Shen JP, Li C (2020) Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams. J Braz Soc Mech Sci 42:137. https://doi.org/10.1007/s40430-020-2206-9
Habibi M, Mohammadgholiha M, Safarpour H (2019) Wave propagation characteristics of the electrically GNP-reinforced nanocomposite cylindrical shell. J Braz Soc Mech Sci 41:221. https://doi.org/10.1007/s40430-019-1715-x
Ma YB, Zhang YH, Kennedy D (2016) Energy flow analysis of mid-frequency vibration of coupled plate structures with a hybrid analytical wave and finite element model. Comput Struct 175:1–14. https://doi.org/10.1016/j.compstruc.2016.06.007
Ma YB, Zhang YH, Kennedy D (2015) A symplectic analytical wave based method for the wave propagation and steady state forced vibration of rectangular thin plates. J Sound Vib 339:196–214. https://doi.org/10.1016/j.jsv.2014.11.029
Qiao YF, Hou GL, Chen A (2021) Symplectic approach for plane elasticity problems of two-dimensional octagonal quasicrystals. Appl Math Comput 400:126043. https://doi.org/10.1016/j.amc.2021.126043
Lim CW, Xu XS (2011) Symplectic elasticity: theory and applications. Appl Mech Rev 63:050802. https://doi.org/10.1115/1.4003700
Pan CG, Sun XB, Zhang YH (2020) Vibro-acoustic analysis of submerged ring-stiffened cylindrical shells based on a symplectic wave-based method. Thin Wall Struct 150:106698. https://doi.org/10.1016/j.tws.2020.106698
Bai E, Chen A (2013) A symplectic eigenfunction expansion approach for free vibration solutions of rectangular Kirchhoff plates. J Vib Control 19:1208–1215. https://doi.org/10.1177/1077546312448503
Zhou ZH, Ni YW, Zhu SB, Tong ZZ, Sun JB, Xu XS (2019) An accurate and straightforward approach to thermo-electro-mechanical vibration of piezoelectric fiber-reinforced composite cylindrical shells. Compos Struct 207:292–303. https://doi.org/10.1016/j.compstruct.2018.08.076
Ma YB, Deng ZC (2022) A semi-analytical method for the dispersion analysis of orthotropic composite plates with periodically attached acoustic black hole resonators. Appl Math Model 110:562–582. https://doi.org/10.1016/j.apm.2022.06.013
Gao RX, Sun XB, Liao HT, Li Y, Fang DN (2021) Symplectic wave-based method for free and steady state forced vibration analysis of thin orthotropic circular cylindrical shells with arbitrary boundary conditions. J Sound Vib 491:115756. https://doi.org/10.1016/j.jsv.2020.115756
Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant numbers: 12072280 and 12072266).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The manuscript has not been submitted elsewhere or published before. I declare that there is no conflict of interest in the publication of this article and that there is no conflict of interest with any other author or institution for the publication of this article.
Additional information
Technical Editor: Samuel da Silva.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ma, Y., Fan, J. & Deng, Z. Semi-analytical solutions for the forced vibration of plate structures terminated by multiple acoustic black hole beams. J Braz. Soc. Mech. Sci. Eng. 45, 423 (2023). https://doi.org/10.1007/s40430-023-04291-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40430-023-04291-x