Periyodik yığılı kütleli kirişlerde eğilme dalgalarının yayılımı

Öz

Bu çalışmada, periyodik yığılı kütleli Euler-Bernoulli kirişinde dispersiyon analizi yapılmakta ve periyodikliğin etkileri araştırılmaktadır. Önce sonsuz uzunlukta periyodik yığılı kütleli bir kiriş için yayılma matrisi yöntemi kullanılarak dispersiyon ilişkisi türetilmiş ve kütle oranına bağlı olarak oluşan bantlı frekans spekturumu verilmiştir. Daha sonra sonlu sayıda periyodik yığılı kütle olması durumunda dalga yayılımına olan etki araştırılmış ve iletkenlik fonksiyonu elde edilmiştir. Son olarak bu yığılı kütlelerden oluşan bariyerin yerdeğiştirme mod şekilleri, geçme ve durma bandı frekans değerleri için elde edilmiştir. Sonuçlar, periyodik yığılı kütleler ile yapılacak tasarımların dalga bariyeri olarak kullanılmasının mümkün olduğunu göstermektedir.

Anahtar Kelimeler

• Eğilme dalgası
• Periyodik yapı
• Yığılı kütle
• , Bant boşluğu
• Yayılma matrisi yöntemi

Propagation of flexural waves in beams with periodic lumped mass

Abstract

In this study, dispersion analysis is carried out in the Euler-Bernoulli beam with periodic lumped mass, and periodicity effects are investigated. First, the dispersion relation is derived using the propagator matrix method for an infinitely long periodic beam with lumped mass. The banded frequency spectrum is given depending on the mass ratio. Then, in the case of a finite number of periodic lumped masses, the effect on wave propagation was investigated and the transmission function was obtained. Finally, the displacement mode shapes of the barrier consisting of these lumped masses were obtained for the pass and stop band frequency values. The results show that it is possible to use designs made with periodic lumped masses as wave barriers.

Keywords

• Flexural wave
• Periodic structure
• Lumped mass
• Bandgap
• Propagator matrix method

Kaynakça

1. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Annales Scientifiques de l'École Normale Supérieure, 12, 47–88, 1883. https://doi.org/10.24033/asens.220.
2. L. Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 24, 145–159, 1887. https://doi.org/10.1080/ 14786448708628074.
3. F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Zeitschrift fr Physik, 52, 555–600, 1929. https://doi.org/10.1007/BF01339455.
4. L. Brillouin, Wave propagation in periodic structures: electric filters and crystal lattices, Dover, New York, 1953.
5. D. M. Mead, Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995, Journal of Sound and Vibration, 190, 495–524, 1996. https://doi.org/ 10.1006/jsvi.1996.0076.
6. P. Zhao, K. Zhang, and Z. Deng, Size effects on the band gap of flexural wave propagation in one-dimensional periodic micro-beams, Composite Structures, 271, 114162, 1–8, 2021. https://doi.org/ 10.1016/j.compstruct.2021.114162.
7. R. Chen and T. Wu, Vibration reduction in a periodic truss beam carrying locally resonant oscillators, Journal of Vibration and Control, 22, 270–285, 2016. https://doi.org/10.1177/1077546314528020.
8. S. Sgubini, F. Graziani, and A. Agneni, Elastic waves propagation in bounded periodic structures, Acta Astronautica, 15, 913–917, 1987. https://doi.org/ 10.1016/0094-5765(87)90049-X.

9. A. Ozmutlu, M. Ebrahimian and M. I. Todorovska, Wave propagation in buildings as periodic structures: Timoshenko beam with rigid floor slabs model, Journal of Engineering Mechanics, 144, 04018010, 1–14, 2018. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001436.
10. M. Iqbal, M. M. Jaya, O. S. Bursi, A. Kumar, R. Ceravolo, Flexural band gaps and response attenuation of periodic piping systems enhanced with localized and distributed resonators, Scientific Reports, 10, 10:85, 1–11, 2020. https://doi.org/10.1038/s41598-019-56724-0.
11. L. Liu and M. I. Hussein, Wave motion in periodic flexural beams and characterization of the transition between bragg scattering and local resonance, Journal of Applied Mechanics, 79, 011003, 1–17, 2012. https://doi.org/10.1115/1.4004592.
12. M. I. Hussein, M. J. Leamy and M. Ruzzene, Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook, Applied Mechanics Reviews, 66, 040802, 1–38, 2014. https://doi.org/10.1115/1.4026911.
13. C. Y. Koh, D. A. Jorba and E. L. Thomas, Phononic metamaterials for vibration isolation and focusing of elastic waves, U.S. Patent 8,833,510, 16 Sep. 2014.
14. M. Ruzzene and A. Baz, Attenuation and localization of wave propagation in periodic rods using shape memory inserts, Smart Materials and Structures, 9, 805–816, 2000. https://doi.org/10.1088/0964-1726/9/6/310.
15. L. Banakh, The vibroisolation properties of the lattices containing the lumped inclusions, Vibroengineering Procedia, 29, 237–242, 2019. https://doi.org/ 10.21595/vp.2019.21075.
16. Y. Xiao, J. Wen, D. Yu, X. Wen, Flexural wave propagation in beams with periodically attached vibration absorbers: Band-gap behavior and band formation mechanisms, Journal of Sound and Vibration, 332, 867–893, 2013. https://doi.org/ 10.1016/j.jsv.2012.09.035.
17. Y. Xiao, J. Wen, G. Wang, X. Wen, Theoretical and experimental study of locally resonant and bragg band gaps in flexural beams carrying periodic arrays of beam-like resonators, Journal of Vibration and Acoustics, 135, 041006, 1–17, 2013. https://doi.org/10.1115/1.4024214.
18. Y. K. Lin and T. J. McDaniel, Dynamics of beam-type periodic structures, Journal of Engineering for Industry, 91, 1133–1141, 1969. https://doi.org/ 10.1115/1.3591761.
19. N. Fukuwa and S. Matsushima, Wave dispersion and optimal mass modelling for one-dimensional periodic structures, Earthquake Engineering & Structural Dynamics, 23, 1165–1180, 1994. https://doi.org/ 10.1002/eqe.4290231102.
20. D. Yu, Y. Liu, G. Wang, H. Zhao, J. Qiu, Flexural vibration band gaps in Timoshenko beams with locally resonant structures, Journal of Applied Physics, 100, 124901, 1–5, 2006. https://doi.org/10.1063/1.2400803.
21. D. Yu, J. Wen, H. Shen, Y. Xiao, X. Wen, Propagation of flexural wave in periodic beam on elastic foundations, Physics Letters A, 376, 626–630, 2012. https://doi.org/ 10.1016/j.physleta.2011.11.056.
22. T. Chen, Investigations on flexural wave propagation of a periodic beam using multi-reflection method, Archive of Applied Mechanics, 83, 315–329, 2013. https://doi.org/10.1007/s00419-012-0657-x.
23. Z. Guo, M. Sheng and T. Wang, Flexural wave attenuation in a periodic laminated beam, Journal of Engineering Research, 5, 258–265, 2016.
24. R. Prasad and A. Sarkar, Broadband vibration isolation for rods and beams using periodic structure theory, Journal of Applied Mechanics, 86, 021004, 1–10, 2019. https://doi.org/10.1115/1.4042011.
25. Z. Zhang, T. Li, and Y. Tang, Traveling wave analytical solutions of vibration band gaps of composite periodic beams, Journal of Vibration and Control, 25, 460–472, 2019. https://doi.org/10.1177/1077546318783557.
26. F. Gilbert and G. E. Backus, Propagator matrices in elastic wave and vibration problems, Studia Geophysica et Geodaetica, 10, 271, 1966. https://doi.org/10.1007/BF02587859.
27. S. B. Coskun, M. T. Atay, and B. Ozturk, Transverse vibration analysis of Euler-Bernoulli beams using analytical approximate techniques, in: E. Farzad (Eds.) Advances in Vibration Analysis Research, InTech, 1–25, 2011. https://doi.org/10.5772/15891.
28. B. Ozturk and S. B. Coskun, The Homotopy perturbation method for free vibration analysis of beam on elastic foundation, Structural Engineering and Mechanics, 37, 415–425, 2011. https://doi.org/ 10.12989/sem.2011.37.4.415.
29. B. Ozturk and S. B. Coskun, Analytical Solution for free vibration analysis of beam on elastic foundation with different support conditions, Mathematical Problems in Engineering, 2013, 1–7, 2013. https://doi.org/10.1155/2013/470927.
30. K. Torabi, D. Sharifi, and M. Ghassabi, A semi-analytical solution for free vibration analysis of a step beam with multiple concentrated masses using variational iteration method, International Journal for Computational Methods in Engineering Science and Mechanics, 22, 333–343, 2021. https://doi.org/10.1080/15502287.2021.1882616.
31. Ł. Domagalski, M. Świątek, and J. Jędrysiak, An analytical-numerical approach to vibration analysis of periodic Timoshenko beams, Composite Structures, 211, 490–501, 2019. https://doi.org/ 10.1016/j.compstruct.2018.12.007.
32. S. Y. Lee, H. Y. Ke, and M. J. Kao, Flexural waves in a periodic beam, Journal of Applied Mechanics, 57, 779–783, 1990. https://doi.org/10.1115/1.2897092.
33. S. Y. Lee and H. Y. Ke, Flexural wave propagation in an elastic beam with periodic structure, Journal of Applied Mechanics, 59, S189-S196, 1992. https://doi.org/10.1115/1.2899487.
34. T. Belytschko and W. L. Mindle, Flexural wave propagation behavior of lumped mass approximations, Computers & Structures, 12, 805–812, 1980. https://doi.org/10.1016/0045-7949(80)90017-6.
35. M. J. Leamy, Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices, Journal of Sound and Vibration, 331, 1580–1596, 2012. https://doi.org/10.1016/j.jsv.2011.11.023.
36. H. Lv and Y. Zhang, A wave-based vibration analysis of a finite Timoshenko locally resonant beam suspended with periodic uncoupled force-moment type resonators, Crystals, 10, 1132, 1–16, 2020. https://doi.org/10.3390/cryst10121132.
37. S. H. Kim and M. P. Das, Seismic waveguide of metamaterials, Modern Physics Letters B, 26, 1250105, 1–4, 2012. https://doi.org/ 10.1142/S0217984912501059.
38. A. Colombi, D. Colquitt, P. Roux, S. Guenneau, A seismic metamaterial: The resonant metawedge, Scientific Reports, 6, 27717, 1–6, 2016. https://doi.org/ 10.1038/srep27717.
39. A. Palermo, S. Krödel, A. Marzani et al., Engineered metabarrier as shield from seismic surface waves, Scientific Reports, 6, 39356, 1–10, 2016. https://doi.org/10.1038/srep39356.
40. V. K. Dertimanis, I. A. Antoniadis, and E. N. Chatzi, Feasibility analysis on the attenuation of strong ground motions using finite periodic lattices of mass-in-mass barriers, Journal of Engineering Mechanics, 142, 04016060, 1–10, 2016. https://doi.org/10.1061/ (ASCE)EM.19437889.0001120.
41. P. Persson, K. Persson, and G. Sandberg, Numerical study of reduction in ground vibrations by using barriers, Engineering Structures, 115, 18–27, 2016. https://doi.org/10.1016/j.engstruct.2016.02.025.
42. P.-R. Wagner, V. K. Dertimanis, E. N. Chatzi, J. L. Beck, Robust-to-uncertainties optimal design of seismic metamaterials, Journal of Engineering Mechanics, 144, 04017181, 1–17, 2018. https://doi.org/10.1061/(ASCE)EM.19437889.0001404.
43. F. Sun and L. Xiao, Bandgap Characteristics and seismic applications of inerter-in-lattice metamaterials, Journal of Engineering Mechanics, 145, 04019067, 1–13, 2019. https://doi.org/10.1061/(ASCE) EM.19437889.0001642.
44. K. F. Graff, Wave motion in elastic solids, Dover; London: Constable, New York, 1991.
45. Y. K. Lin and B. K. Donaldson, A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels, Journal of Sound and Vibration, 10, 103–143, 1969. https://doi.org/10.1016/ 0022-460X(69)90132-1.
46. B. R. Mace, Wave reflection and transmission in beams, Journal of Sound and Vibration, 97, 237–246, 1984. https://doi.org/10.1016/0022-460X(84)90320-1.