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Non-dimensional analysis of the elastic beam having periodic linear spring mass resonators

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Abstract

Band gaps appear in the frequency spectra of an elastic beam having periodic spring-mass attachment, acronymed as metabeam. Two widely used beam theories, i.e. Euler–Bernoulli and Timoshenko, are non-dimensionalized in order to obtain the dimensionless periodicity parameters. In this paper, the location and width of band-gaps are investigated due to the variation of these periodicity parameters of a metabeam. Implementing transfer matrix method in conjunction with Bloch–Floquet theorem, a unified methodology has been adopted to estimate the propagation and attenuation bands in a metabeam. This paper provides a guideline towards the designing of wide-band metabeam by tunning the non-dimensional parameters.

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References

  1. Liu Y, Yu D, Li L, Zhao H, Wen J, Wen X (2007) Design guidelines for flexural wave attenuation of slender beams with local resonators. Phys Lett A 362(5–6):344–347

    ADS  Google Scholar 

  2. Liu L, Hussein MI (2011) Wave motion in periodic flexural beams and characterization of the transition between Bragg scattering and local resonance. J Appl Mech 79(1):011003–011017

    Article  Google Scholar 

  3. Mead DM (1996) Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995. J Sound Vib 190(3):495–524

    Article  ADS  Google Scholar 

  4. Yu D, Liu Y, Wang G, Zhao H, Qiu J (2006) Flexural vibration band gaps in Timoshenko beams with locally resonant structures. J Appl Phys 100(12):124901

    Article  ADS  Google Scholar 

  5. Wang MY, Wang X (2013) Frequency band structure of locally resonant periodic flexural beams suspended with force–moment resonators. J Phys D Appl Phys 46(25):255502

    Article  ADS  Google Scholar 

  6. Wang T, Sheng MP, Qin QH (2016) Multi-flexural band gaps in an Euler–Bernoulli beam with lateral local resonators. Phys Lett A 380(4):525–529

    Article  ADS  Google Scholar 

  7. Zhang S, Wu JH, Hu Z (2013) Low-frequency locally resonant band-gaps in phononic crystal plates with periodic spiral resonators. J Appl Phys 113(16):163511

    Article  ADS  Google Scholar 

  8. Khajehtourian R, Hussein MI (2014) Dispersion characteristics of a nonlinear elastic metamaterial. AIP Adv 4(12):124308

    Article  ADS  Google Scholar 

  9. Hu G, Tang L, Das R, Gao S, Liu H (2017) Acoustic metamaterials with coupled local resonators for broadband vibration suppression. AIP Adv 7(2):025211

    Article  ADS  Google Scholar 

  10. Nouh MA, Aldraihem OJ, Baz A (2015) Periodic metamaterial plates with smart tunable local resonators. J Intell Mater Syst Struct 27(13):1829–1845

    Article  Google Scholar 

  11. Tang L, Cheng L (2017) Ultrawide band gaps in beams with double-leaf acoustic black hole indentations. J Acoust Soc Am 142(5):2802–2807

    Article  ADS  Google Scholar 

  12. Bacquet CL, Al Ba’ba’a H, Frazier MJ, Nouh M, Hussein MI (2018) Chapter two-metadamping dissipation emergence in elastic metamaterials. Adv Appl Mech 51:115–164

    Article  Google Scholar 

  13. Kushwaha MS, Halevi P, Dobrzynski L, Djafari-Rouhani B (1993) Acoustic band structure of periodic elastic composites. Phys Rev Lett 71(13):2022–2025

    Article  ADS  Google Scholar 

  14. Bo Y, Yong C, Min J, Shuai T, Miao H, Minglin T (2017) The interaction of resonance and bragg scattering effects for the locally resonant phononic crystal with alternating elastic and fluid matrices. Arch Acoust 42(4):725–733

    Article  Google Scholar 

  15. Sheng P, Zhang XX, Liu Z, Chan CT (2003) Locally resonant sonic materials. Physica B 338(1):201–205

    Article  ADS  Google Scholar 

  16. Banerjee A, Das R, Calius EP (2019) Waves in structured mediums or metamaterials: a review. Arch Comput Methods Eng 26(4):1029–1058

    Article  MathSciNet  Google Scholar 

  17. Banerjee A, Das R, Calius EP (2017) Frequency graded 1D metamaterials: a study on the attenuation bands. J Appl Phys 122(7):075101

    Article  ADS  Google Scholar 

  18. Muhammad Lim CW (2019) Elastic waves propagation in thin plate metamaterials and evidence of low frequency pseudo and local resonance bandgaps. Phys Lett A 383(23):2789–2796

    Article  ADS  Google Scholar 

  19. Xiang HJ, Shi ZF, Wang SJ, Mo YL (2012) Vibration attenuation and frequency band gaps in layered periodic foundation: theory and experiment. In: 15th world conference on earthquake engineering

  20. Ahmed RU, Adiba A, Banerjee S (2015) Energy scavenging from acousto-elastic metamaterial using local resonance phenomenon. Active Passive Smart Struct Integr Syst 9431:943106–943110

    Google Scholar 

  21. Hu G, Tang L, Banerjee A, Das R (2016) Metastructure with piezoelectric element for simultaneous vibration suppression and energy harvesting. J Vib Acoust 139(1):11

    Google Scholar 

  22. Li Y, Baker E, Reissman T, Sun C, Liu WK (2017) Design of mechanical metamaterials for simultaneous vibration isolation and energy harvesting. Appl Phys Lett 111(25):251903

    Article  ADS  Google Scholar 

  23. Hu G, Tang L, Das R (2018) Internally coupled metamaterial beam for simultaneous vibration suppression and low frequency energy harvesting. J Appl Phys 123(5):055107

    Article  ADS  Google Scholar 

  24. Chen JS, Sharma B, Sun CT (2011) Dynamic behaviour of sandwich structure containing spring–mass resonators. Compos Struct 93(8):2120–2125

    Article  Google Scholar 

  25. Sharma B, Sun CT (2015) Impact load mitigation in sandwich beams using local resonators. J Sandw Struct Mater 18(1):50–64

    Article  Google Scholar 

  26. Sharma B, Sun CT (2016) Local resonance and Bragg bandgaps in sandwich beams containing periodically inserted resonators. J Sound Vib 364:133–146

    Article  ADS  Google Scholar 

  27. Wang X, Wang MY (2016) An analysis of flexural wave band gaps of locally resonant beams with continuum beam resonators. Meccanica 51(1):171–178

    Article  MathSciNet  Google Scholar 

  28. Zhou J, Wang K, Xu D, Ouyang H (2017) Local resonator with high-static-low-dynamic stiffness for lowering band gaps of flexural wave in beams. J Appl Phys 121(4):044902

    Article  ADS  Google Scholar 

  29. Elishakoff I, Kaplunov J, Nolde E (2015) Celebrating the century of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl Mech Rev 67:060802

    Article  ADS  Google Scholar 

  30. Rayleigh JWS (1877–1878) The theory of sound. Macmillan, London

  31. Bresse JAC (1859) Cours de Mecanique Appliquee. Mallet-Bachelier, Paris

    MATH  Google Scholar 

  32. Timoshenko SP (1920) On the differential equation for the flexural vibrations of prismatical rods. Glas Hrvat Prirodosl Drus Zagreb 32(2):55–57

    Google Scholar 

  33. Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos Mag 41(245):744–746

    Article  Google Scholar 

  34. Timoshenko PS (1921) On the additional deflection due to shearing. Glas Hrvat Prirodosl Drus Zagreb 33(Part 1, Nr. 1):50–52

    Google Scholar 

  35. Hussein MI, Leamy MJ, Ruzzene M (2014) Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl Mech Rev 66(4):040802–040802

    Article  ADS  Google Scholar 

  36. Mei C, Mace BR (2005) Wave reflection and transmission in Timoshenko beams and wave analysis of Timoshenko beam structures. Trans ASME 127:382–394

    Google Scholar 

  37. Majkut Leszek (2009) Free and forced vibrations of Timoshenko beams described by single difference equation. J Theor Appl Mech 47(1):193–210

    Google Scholar 

  38. Brillouin Leon (1946) Wave propagation in periodic structures: electric filters and crystal lattices. McGraw-Hill Book Company, New York

    MATH  Google Scholar 

  39. Wu JS, Chang BH (2013) Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method. Eur J Mech A Solids 38:20–37

    Article  MathSciNet  Google Scholar 

  40. Cowper G (1966) The shear coefficient in Timoshenko’s beam theory. ASME, New York

    Book  Google Scholar 

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Acknowledgements

Authors would like to acknowledge the research grant of Inspire Faculty Award of Department of Science and Technology, Ministry of Science and Technology, Government of India, Grant DST/INSPIRE/04/2018/000052 for supporting the research.

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  1. Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Arnab Banerjee

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  1. Arnab Banerjee
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Correspondence to Arnab Banerjee.

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Banerjee, A. Non-dimensional analysis of the elastic beam having periodic linear spring mass resonators. Meccanica 55, 1181–1191 (2020). https://doi.org/10.1007/s11012-020-01151-z

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  • DOI: https://doi.org/10.1007/s11012-020-01151-z

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