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Tunable nonlinear band gaps in a sandwich-like meta-plate

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Abstract

This paper aims to explore the working mechanism of sandwich-like meta-plates by periodically attaching nonlinear mass-beam-spring resonators for low-frequency wave attenuation. The nonlinear MBS resonator consists of a mass, a cantilever beam, and a spring to provide negative stiffness in the transverse vibration of the resonator; this stiffness is tunable by changing the parameters of the spring. Considering the nonlinear stiffness of the resonator, the energy method is applied to obtain the dispersion relation of the sandwich-like meta-plate, and the band-gap bounds related to the amplitude of the resonator are derived by dispersion analysis. For a finite-sized sandwich-like meta-plate with a fully free boundary condition subjected to external excitations, its dynamic equation is established by the Galerkin method. The frequency response analysis of the meta-plate is carried out by numerical simulation, with the band-gap range obtained in good agreement with that of the theoretical one. Results show that the band-gap range of the present meta-plate is tunable by designing the structural parameters of the MBS resonator. Furthermore, by analyzing the vibration suppression of the finite-sized meta-plate, it is observed that the nonlinearity of the resonators can widen the wave attenuation range of the meta-plate.

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References

  1. Zhao, H.G., Liu, Y.Z., Wang, G., Wen, J.H., Yu, D.L., Han, X.Y., Wen, X.S.: Resonance modes and gap formation in a two-dimensional solid phononic crystal. Phys. Rev. B 72, 12301 (2005)

    Article  Google Scholar 

  2. Fang, N., Xi, D.J., Xu, J.Y., Ambati, M., Srituravanich, W., Sun, C., Zhang, X.: Ultrasonic metamaterials with negative modulus. Nat. Mater. 5, 452–456 (2006)

    Article  Google Scholar 

  3. Deymier, P.A.: Acoustic Metamaterials and Phononic Crystals. Springer-Verlag, Berlin (2013)

    Book  Google Scholar 

  4. D’Alessandro, L., Belloni, E., Ardito, R., Corigliano, A., Braghin, F.: Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal. Appl. Phys. Lett. 109, 221907 (2016)

    Article  Google Scholar 

  5. Hussein, M.I., Leamy, M.J., Ruzzene, M.: Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl. Mech. Rev. 66, 040802 (2014)

    Article  Google Scholar 

  6. Zhang, S., Park, Y.S., Li, J., Lu, X., Zhang, W., Zhang, X.: Negative refractive index in chiral metamaterials. Phys. Rev. Lett. 102, 023901 (2009)

    Article  Google Scholar 

  7. Casadei, F., Dozio, L., Ruzzene, M., Cunefare, K.A.: Periodic shunted arrays for the control of noise radiation in an enclosure. J. Sound Vibr. 329, 3632–3646 (2010)

    Article  Google Scholar 

  8. Chen, H., Chan, C.T.: Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett. 91, 183518 (2007)

    Article  Google Scholar 

  9. Cai, W., Chettiar, U.K., Kildishev, A.V., Shalaev, V.M.: Optical cloaking with metamaterials. Nat. Photon. 1, 224–227 (2007)

    Article  Google Scholar 

  10. Zhu, R., Huang, G.L., Huang, H.H., Sun, C.T.: Experimental and numerical study of guided wave propagation in a thin metamaterial plate. Phys. Lett. A 375, 2863–2867 (2011)

    Article  Google Scholar 

  11. Madeo, A., Neff, P., Ghiba, I.-D., Rosi, G.: Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model. J. Mech. Phys. Solids 95, 441–479 (2016)

    Article  MathSciNet  Google Scholar 

  12. Sigalas, M.M., Economou, E.N.: Elastic and acoustic wave band structure. J. Sound Vibr. 158, 377–382 (1992)

    Article  Google Scholar 

  13. Ma, J., Hou, Z., Assouar, B.M.: Opening a large full phononic band gap in thin elastic plate with resonant units. J. Appl. Phys. 115, 093508 (2014)

    Article  Google Scholar 

  14. Benchabane, S., Khelif, A., Rauch, J.-Y., Robert, L., Laude, V.: Evidence for complete surface wave band gap in a piezoelectric phononic crystal. Phys. Rev. E 73, 065601 (2006)

    Article  Google Scholar 

  15. Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289, 1734–1736 (2000)

    Article  Google Scholar 

  16. Sun, H., Du, R., Pai, R.F.: Theory of metamaterial beams for broadband vibration absorption. J. Intel. Mat. Syst. Str. 21, 1085–1101 (2010)

    Article  Google Scholar 

  17. Oudich, M., Senesi, M., Assouar, M.B., Ruzenne, M., Sun, J.H., Vincent, B., Hou, Z., Wu, T.T.: Experimental evidence of locally resonant sonic band gap in two-dimensional phononic stubbed plates. Phys. Rev. B 84, 667–673 (2011)

    Article  Google Scholar 

  18. Ma, G., Sheng, P.: Acoustic metamaterials: From local resonances to broad horizons. Sci. Adv. 2, e150159 (2016)

    Article  Google Scholar 

  19. Peng, H., Frank, P.P.: Acoustic metamaterial plates for elastic wave absorption and structural vibration suppression. Int. J. Mech. Sci. 89, 350–361 (2014)

    Article  Google Scholar 

  20. Al Ba"Ba", A.H.B., Attarzadeh, M.A., Nouh, M.: Experimental evaluation of structural intensity in two-dimensional plate-type locally resonant elastic metamaterials. J. Appl. Mech.-Trans. ASME 85, 041005 (2018)

  21. Xiao, Y., Wen, J., Wen, X.: flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators. J. Phys. D Appl. Phys. 45, 195401 (2012)

    Article  Google Scholar 

  22. Miranda, E.J.P., Nobrega, E.D., Ferreira, A.H.R., Dos Santos, J.M.C.: Flexural wave band gaps in a multi-resonator elastic metamaterial plate using Kirchhoff-Love theory. Mech. Syst. Signal Proc. 116, 480–504 (2019)

    Article  Google Scholar 

  23. Lazarov, B.S., Jensen, J.S.: Low-frequency band gaps in chains with attached non-linear oscillators. Int. J. Non-Linear Mech. 42, 1186–1193 (2007)

    Article  Google Scholar 

  24. Wang, K., Zhou, J., Xu, D., Ouyang, H.: Lower band gaps of longitudinal wave in a one-dimensional periodic rod by exploiting geometrical nonlinearity. Mech. Syst. Signal Proc. 124, 664–678 (2019)

    Article  Google Scholar 

  25. Zhou, X., Wang, L.: Opening complete band gaps in two dimensional locally resonant phononic crystals. J. Phys. Chem. Solids 116, 174–179 (2018)

    Article  Google Scholar 

  26. Wang, Q., Li, J., Zhang, Y., Xue, Y., Li, F.: Bandgap properties in metamaterial sandwich plate with periodically embedded plate-type resonators. Mech. Syst. Signal Proc. 151, 107375 (2021)

    Article  Google Scholar 

  27. Qian, D., Shi, Z.: Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring–mass resonators. Phys. Lett. A 380, 3319–3325 (2016)

    Article  Google Scholar 

  28. Raghavan, L., Phani, A.S.: Local resonance bandgaps in periodic media: theory and experiment. J. Acoust. Soc. Am. 134, 1950–1959 (2013)

    Article  Google Scholar 

  29. Jung, J., Kim, H.-G., Goo, S., Chang, K.-J., Wang, S.: Realisation of a locally resonant metamaterial on the automobile panel structure to reduce noise radiation. Mech. Syst. Signal Proc. 122, 206–231 (2019)

    Article  Google Scholar 

  30. Bilal, O.R., Hussein, M.I.: Trampoline metamaterial: local resonance enhancement by springboards. Appl. Phys. Lett. 103, 2022–2025 (2013)

    Article  Google Scholar 

  31. Xiao, Y., Wen, J., Huang, L., Wen, X.: Analysis and experimental realization of locally resonant phononic plates carrying a periodic array of beam-like resonators. J. Phys. D Appl. Phys. 47, 045307 (2013)

    Article  Google Scholar 

  32. Huang, T.Y., Shen, C., Jing, Y.: Membrane- and plate-type acoustic metamaterials. J. Acoust. Soc. Am. 139, 3240–3250 (2016)

    Article  Google Scholar 

  33. Nayfeh, A.H., Mook, D.: Nonlinear oscillations. Clarendon, Oxford (1981)

    MATH  Google Scholar 

  34. Moon, F.C.: Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers. Wiley, Hoboken (1992)

    Book  Google Scholar 

  35. Zhou, J., Dou, L., Wang, K., Xu, D., Ouyang, H.: A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams. Nonlinear Dyn. 96, 647–665 (2019)

    Article  MATH  Google Scholar 

  36. Wu, Z., Liu, W., Li, F., Zhang, C.: Band-gap property of a novel elastic metamaterial beam with X-shaped local resonators. Mech. Syst. Signal Proc. 134, 106357 (2019)

    Article  Google Scholar 

  37. Chen, Z., Zhou, W., Lim, C.W.: Active control for acoustic wave propagation in nonlinear diatomic acoustic metamaterials. Int. J. Non-Linear Mech. 125, 103535 (2020)

    Article  Google Scholar 

  38. Li, Z.-N., Wang, Y.-Z., Wang, Y.-S.: Three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial. Int. J. Non-Linear Mech. 125, 103531 (2020)

    Article  Google Scholar 

  39. Casalotti, A., El-Borgi, S., Lacarbonara, W.: Metamaterial beam with embedded nonlinear vibration absorbers. Int. J. Non-Linear Mech. 98, 32–42 (2018)

    Article  Google Scholar 

  40. Zhang, H., Cheng, X., Yan, D., Zhang, Y., Fang, D.: A nonlinear mechanics model of soft network metamaterials with unusual swelling behavior and tunable phononic band gaps. Compos. Sci. Technol. 183, 107822 (2019)

    Article  Google Scholar 

  41. Khajehtourian, R., Hussein, M.I.: Dispersion characteristics of a nonlinear elastic metamaterial. AIP Adv. 4, 124308 (2014)

    Article  Google Scholar 

  42. Hussein, M.I., Khajehtourian, R.: Nonlinear Bloch waves and balance between hardening and softening dispersion. Proc. Math. Phys. Eng. Sci. 474, 20180173 (2018)

    MathSciNet  MATH  Google Scholar 

  43. 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 Proc. 39, 32–46 (2013)

    Article  Google Scholar 

  44. Kundu, T., Packo, P., Staszewski, W.J., Uhl, T., Leamy, M.J.: Perturbation approach to dispersion curves calculation for nonlinear Lamb waves. Proc. of SPIE 9438, 94381V (2015)

    Article  Google Scholar 

  45. Packo, P., Uhl, T., Staszewski, W.J., Leamy, M.J.: Amplitude-dependent Lamb wave dispersion in nonlinear plates. J. Acoust. Soc. Am. 140, 1319 (2016)

    Article  Google Scholar 

  46. Silva, P.B., Leamy, M.J., Geers, M.G.D., Kouznetsova, V.G.: Emergent subharmonic band gaps in nonlinear locally resonant metamaterials induced by autoparametric resonance. Phys. Rev. E 99, 063003 (2019)

    Article  Google Scholar 

  47. Fronk, M.D., Leamy, M.J.: Internally resonant wave energy exchange in weakly nonlinear lattices and metamaterials. Phys. Rev. E 100, 032213 (2019a)

    Article  Google Scholar 

  48. Fronk, M.D., Leamy, M.J.: Isolated frequencies at which nonlinear materials behave linearly. Phys. Rev. E 100, 051002 (2019b)

    Article  Google Scholar 

  49. Narisetti, R.K., Leamy, M.J., Ruzzene, M.: A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures. J. Vib. Acoust. 132, 031001 (2010)

    Article  Google Scholar 

  50. Wang, K., Zhou, J., Cai, C., Xu, D., Ouyang, H.: Mathematical modeling and analysis of a meta-plate for very low-frequency band gap. Appl. Math. Model. 73, 581–597 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  51. Fang, X., Wen, J., Yu, D., Yin, J.: Bridging-coupling band gaps in nonlinear acoustic metamaterials. Phys. Rev. Appl. 10, 054049 (2018)

    Article  Google Scholar 

  52. Campana, M.A., Ouisse, M., Sadoulet-Reboul, E., Ruzzene, M., Neild, S., Scarpa, F.: Impact of non-linear resonators in periodic structures using a perturbation approach. Mech. Syst. Signal Proc. 135, 106408 (2020)

    Article  Google Scholar 

  53. Emerson, T.A., Manimala, J.M.: Passive-adaptive mechanical wave manipulation using nonlinear metamaterial plates. Acta Mech. 231, 4665–4681 (2020)

    Article  Google Scholar 

  54. Xia, Y., Ruzzene, M., Erturk, A.: Bistable attachments for wideband nonlinear vibration attenuation in a metamaterial beam. Nonlinear Dyn. 102, 1285–1296 (2020)

    Article  Google Scholar 

  55. Li, J., Fan, X., Li, F.: Numerical and experimental study of a sandwich-like metamaterial plate for vibration suppression. Compos. Struct. 238, 111969 (2020)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12072084, 11902093 and 11761131006), the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (No. 3072020GIP0206) and Fundamental Research Funds for the Central Universities.

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  1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, 150001, China

    Yu Xue, Jinqiang Li, Yu Wang & Fengming Li

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  1. Yu Xue
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Xue, Y., Li, J., Wang, Y. et al. Tunable nonlinear band gaps in a sandwich-like meta-plate. Nonlinear Dyn 106, 2841–2857 (2021). https://doi.org/10.1007/s11071-021-06961-8

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