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Acta Mechanica

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09-02-2023 | Original Paper

A semi-analytical model for dynamic analysis of thin plates with plate-type resonators

Authors: Jian Xue, Li-Qun Chen

Published in: Acta Mechanica | Issue 6/2023

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Abstract

A semi-analytical method is proposed for free and forced vibrations of a thin square binary material configuration plate containing circular plate-type resonators. The resonator embedded in the host plate consists of a rigid mass and a circular plate made of soft material. Based on the classical plate theory, the host plate and the resonator are modelled separately and then coupled by the condition of displacement compatibility. A set of local admissible functions is incorporated into the global admissible functions in the circular plate domain, to describe the non-smoothness and vibration localization of the displacement caused by the material difference between the host plate and the resonator. The resulting global admissible functions promote the capabilities of the Ritz method in predicting accurately the vibration characteristics of the plate with binary material configuration. The Ritz method with the constructed admissible functions is developed to extract frequencies and analytic mode functions of the plate under different boundary conditions. The plate subjected to a transverse harmonic excitation is discretized into a multi-degree-of-freedom system by the Lagrange approach with the analytic mode functions. The effects of boundary conditions, locations of the mass, and material properties on the vibration are explored. It is demonstrated that the mass location in the circular plate only affects the frequencies of the resonator and has almost no effect on the frequencies and the modes of global vibration, and the resonator with the appropriate material parameters reduces significantly the multi-mode plate vibration.

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Askari, E., Jeong, K.H., Amabili, M.: A novel mathematical method to analyze the free vibration of eccentric annular plates. J. Sound Vib. 484, 115513 (2020). https://doi.org/10.1016/j.jsv.2020.115513CrossRef

Chen, Y.X., Li, F.L., Hao, Y.X.: Analysis of vibration and sound insulation characteristics of functionally graded sandwich plates. Compos. Struct. 249, 112515 (2020). https://doi.org/10.1016/j.compstruct.2020.112515CrossRef

Yeh, S.L., Harne, R.L.: Cut-out resonators for tuned vibration suppression of plates. Thin-Walled Struct. 167, 108200 (2021). https://doi.org/10.1016/j.tws.2021.108200CrossRef

Lu, S.F., Li, H.J., Zhang, W., Song, X.J.: Vibration reduction of FG-CNTR piezoelectric laminated composite cantilever plate under aerodynamic load using full-dimensional state observer. Eng. Struct. 255, 113942 (2022). https://doi.org/10.1016/j.engstruct.2022.113942CrossRef

Richiedei, D., Tamellin, I., Trevisani, A.: Beyond the tuned mass damper: a comparative study of passive approaches to vibration absorption through antiresonance assignment. Arch. Comput. Methods Eng. 29, 519–544 (2022). https://doi.org/10.1007/s11831-021-09583-wMathSciNetCrossRef

Yang, F., Sedaghati, R., Esmailzadeh, E.: Vibration suppression of structures using tuned mass damper technology: a state-of-the-art review. JVC J. Vib. Control. 28, 812–836 (2022). https://doi.org/10.1177/1077546320984305MathSciNetCrossRef

Huang, Y., Li, J., Chen, W., Bao, R.: Tunable bandgaps in soft phononic plates with spring-mass-like resonators. Int. J. Mech. Sci. 151, 300–313 (2019). https://doi.org/10.1016/j.ijmecsci.2018.11.029CrossRef

Gao, P., Climente, A., Sánchez-Dehesa, J., Wu, L.: Single-phase metamaterial plates for broadband vibration suppression at low frequencies. J. Sound Vib. 444, 108–126 (2019). https://doi.org/10.1016/j.jsv.2018.12.022CrossRef

Yeh, S.L., Harne, R.L.: Structurally-integrated resonators for broadband panel vibration suppression. Smart Mater. Struct. 29, 085010 (2020). https://doi.org/10.1088/1361-665X/ab9148CrossRef

10.

Ma, F., Wang, C., Liu, C., Wu, J.H.: Structural designs, principles, and applications of thin-walled membrane and plate-type acoustic/elastic metamaterials. J. Appl. Phys. 129, 231103 (2021). https://doi.org/10.1063/5.0042132CrossRef

11.

Liao, G., Luan, C., Wang, Z., Liu, J., Yao, X., Fu, J.: Acoustic metamaterials: a review of theories, structures, fabrication approaches, and applications. Adv. Mater. Technol. 6, 1–29 (2021). https://doi.org/10.1002/admt.202000787CrossRef

12.

Yang, Z., Mei, J., Yang, M., Chan, N.H., Sheng, P.: Membrane-type acoustic metamaterial with negative dynamic mass. Phys. Rev. Lett. 101, 1–4 (2008). https://doi.org/10.1103/PhysRevLett.101.204301CrossRef

13.

Nouh, M., Aldraihem, O., Baz, A.: Wave propagation in metamaterial plates with periodic local resonances. J. Sound Vib. 341, 53–73 (2015). https://doi.org/10.1016/j.jsv.2014.12.030CrossRef

14.

Zhou, G., Wu, J.H., Lu, K., Tian, X., Huang, W., Zhu, K.: Broadband low-frequency membrane-type acoustic metamaterials with multi-state anti-resonances. Appl. Acoust. 159, 107078 (2020). https://doi.org/10.1016/j.apacoust.2019.107078CrossRef

15.

Naify, C.J., Chang, C.M., McKnight, G., Nutt, S.: Transmission loss and dynamic response of membrane-type locally resonant acoustic metamaterials. J. Appl. Phys. 108, 114905 (2010). https://doi.org/10.1063/1.3514082CrossRef

16.

Naify, C.J., Chang, C.M., McKnight, G., Nutt, S.: Transmission loss of membrane-type acoustic metamaterials with coaxial ring masses. J. Appl. Phys. 110, 124903 (2011). https://doi.org/10.1063/1.3665213CrossRef

17.

Naify, C.J., Chang, C.M., McKnight, G., Scheulen, F., Nutt, S.: Membrane-type metamaterials: transmission loss of multi-celled arrays. J. Appl. Phys. 109, 104902 (2011). https://doi.org/10.1063/1.3583656CrossRef

18.

Gao, C., Halim, D., Yi, X.: Study of bandgap property of a bilayer membrane-type metamaterial applied on a thin plate. Int. J. Mech. Sci. 184, 105708 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105708CrossRef

19.

Ma, F., Cai, Y., Wu, J.H.: Ultralight plat-type vibration damper with designable working bandwidth and strong multi-peak suppression performance. J. Phys. D. Appl. Phys. 54, 055303 (2021). https://doi.org/10.1088/1361-6463/abc11aCrossRef

20.

Xiao, W., Zeng, G.W., Cheng, Y.S.: Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs. Appl. Acoust. 69, 255–261 (2008). https://doi.org/10.1016/j.apacoust.2006.09.003CrossRef

21.

Hsu, J.C., Wu, T.T.: Lamb waves in binary locally resonant phononic plates with two-dimensional lattices. Appl. Phys. Lett. 90, 201904 (2007). https://doi.org/10.1063/1.2739369CrossRef

22.

Dal Poggetto, V.F., Serpa, A.L.: Flexural wave band gaps in a ternary periodic metamaterial plate using the plane wave expansion method. J. Sound Vib. 495, 115909 (2021). https://doi.org/10.1016/j.jsv.2020.115909CrossRef

23.

Langfeldt, F., Gleine, W., Von Estorff, O.: Analytical model for low-frequency transmission loss calculation of membranes loaded with arbitrarily shaped masses. J. Sound Vib. 349, 315–329 (2015). https://doi.org/10.1016/j.jsv.2015.03.037CrossRef

24.

Li, J., Shi, Y., Jiang, R., Zhang, Z., Huang, Q.: Acoustic insulation mechanism of membrane-type acoustic metamaterials loaded with arbitrarily shaped mass blocks of variable surface density. Materials (Basel) 15, 1556 (2022). https://doi.org/10.3390/ma15041556CrossRef

25.

Cheng, C.Z., Yao, S.L., Han, Z.L., Recho, N., Niu, Z.R.: Evaluation of the singularity exponents and characteristic angular functions for piezoelectric V-notches under in plane and out of plane conditions. Theor. Appl. Fract. Mech. 76, 50–59 (2015). https://doi.org/10.1016/j.tafmec.2015.01.001CrossRef

26.

Cheng, C.Z., Yao, S.L., Sun, J.L., Niu, Z.R.: Singularity characteristic analysis for a V-notch in angularly heterogeneous moderately thick plate. Int. J. Mech. Sci. 115–116, 215–225 (2016). https://doi.org/10.1016/j.ijmecsci.2016.07.001CrossRef

27.

Moreno-García, P., dos Santos, J.V.A., Lopes, H.: A review and study on ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates. Arch. Comput. Methods Eng. 25, 785–815 (2018). https://doi.org/10.1007/s11831-017-9214-7MathSciNetCrossRefMATH

28.

Kumar, Y.: The Rayleigh–Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: a literature review. JVC J. Vib. Control 24, 1205–1227 (2018). https://doi.org/10.1177/1077546317694724MathSciNetCrossRefMATH

29.

Gozum, M.M., Serhat, G., Basdogan, I.: A semi-analytical model for dynamic analysis of non-uniform plates. Appl. Math. Model. 76, 883–899 (2019). https://doi.org/10.1016/j.apm.2019.07.013MathSciNetCrossRefMATH

30.

Gozum, M.M., Aghakhani, A., Serhat, G., Basdogan, I.: Electroelastic modeling of thin-laminated composite plates with surface-bonded piezo-patches using Rayleigh–Ritz method. J. Intell. Mater. Syst. Struct. 29, 2192–2205 (2018). https://doi.org/10.1177/1045389X18758189CrossRef

31.

Gao, C., Pang, F., Li, H., Jia, D., Tang, Y.: Steady and transient vibration analysis of uniform and stepped annular/circular plates based on FSDT. Acta Mech. 233, 1061–1082 (2022). https://doi.org/10.1007/s00707-022-03157-yMathSciNetCrossRefMATH

32.

Chan, Y.J., Tai, C.Y.: Free vibration of stepped rectangular Mindlin plates with non-Lévy boundary conditions. Int. J. Mech. Sci. 144, 668–678 (2018). https://doi.org/10.1016/j.ijmecsci.2018.05.052CrossRef

33.

Deng, J., Guasch, O., Zheng, L., Song, T., Cao, Y.: Semi-analytical model of an acoustic black hole piezoelectric bimorph cantilever for energy harvesting. J. Sound Vib. 494, 115790 (2021). https://doi.org/10.1016/j.jsv.2020.115790CrossRef

34.

Blesa Gracia, J., Rammerstorfer, F.G.: Increase in buckling loads of plates by introduction of cutouts. Acta Mech. 230, 2873–2889 (2019). https://doi.org/10.1007/s00707-019-02435-6MathSciNetCrossRefMATH

35.

Yang, Y., Cheng, C., Niu, Z., Hu, Z.: Free vibration analysis for V-notched Mindlin plates with free or clamped radial edges. Acta Mech. 233, 2271–2285 (2022). https://doi.org/10.1007/s00707-022-03211-9MathSciNetCrossRefMATH

36.

Xue, J., Wang, Y.: Free vibration analysis of a flat stiffened plate with side crack through the Ritz method. Arch. Appl. Mech. 89, 2089–2102 (2019). https://doi.org/10.1007/s00419-019-01565-6CrossRef

37.

Xue, J., Wang, Y., Chen, L.: Buckling and free vibration of a side-cracked Mindlin plate under axial in-plane load. Arch. Appl. Mech. 90, 1811–1827 (2020). https://doi.org/10.1007/s00419-020-01698-zCrossRef

38.

Milazzo, A.: Free vibrations analysis of cracked variable stiffness composite plates by the eXtended Ritz method. Mech. Adv. Mater. Struct. 0, 1–17 (2022). https://doi.org/10.1080/15376494.2022.2038742CrossRef

39.

Song, Y., Xue, K., Li, Q.: A solution method for free vibration of intact and cracked polygonal thin plates using the Ritz method and Jacobi polynomials. J. Sound Vib. 519, 116578 (2022). https://doi.org/10.1016/j.jsv.2021.116578CrossRef

40.

Huang, C.S., Lee, H.T., Li, P.Y., Chang, M.J.: Three-dimensional free vibration analyses of preloaded cracked plates of functionally graded materials via the mls-ritz method. Materials 14, 7712 (2021). https://doi.org/10.3390/ma14247712CrossRef

41.

Liew, K.M., Hung, K.C., Lim, M.K.: A solution method for analysis of cracked plates under vibration. Eng. Fract. Mech. 48, 393–404 (1994). https://doi.org/10.1016/0013-7944(94)90130-9CrossRef

42.

Kharazi, M., Ovesy, H.R., Taghizadeh, M.: Buckling of the composite laminates containing through-the-width delaminations using different plate theories. Compos. Struct. 92, 1176–1183 (2010). https://doi.org/10.1016/j.compstruct.2009.10.019CrossRef

43.

Cheng, C.Z., Zhou, W., Niu, Z.R., Recho, N.: Stress singularity analysis for orthotropic V-notches in the generalised plane strain state. Fatigue Fract. Eng. Mater. Struct. 38, 881–896 (2015). https://doi.org/10.1111/ffe.12282CrossRef

44.

Zhou, D., Cheung, Y.K., Au, F.T.K., Lo, S.H.: Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. Int. J. Solids Struct. 39, 6339–6353 (2002). https://doi.org/10.1016/S0020-7683(02)00460-2CrossRefMATH

Title: A semi-analytical model for dynamic analysis of thin plates with plate-type resonators
Authors: Jian Xue
Li-Qun Chen
Publication date: 09-02-2023
Publisher: Springer Vienna
Published in: Acta Mechanica / Issue 6/2023
Print ISSN: 0001-5970
Electronic ISSN: 1619-6937
DOI: https://doi.org/10.1007/s00707-023-03496-4

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A semi-analytical model for dynamic analysis of thin plates with plate-type resonators

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