Flexural edge waves in a Kirchhoff plate carrying periodic edge resonators and resting on a Winkler foundation. (English) Zbl 1524.74332
Summary: Flexural edge waves in a Kirchhoff plate, carrying periodically spaced spring-mass resonators at its edge and laid on a Winkler foundation, are considered. A spectral element method is applied on a representative unit cell to develop the dynamic stiffness matrix of the structure. In light of structural periodicity, Bloch wave theorem is used to derive a quadratic eigenvalue problem of the wave propagation constants, which are numerically solved for. Frequency bands are analyzed using the attenuation and phase constants. Two types of bandgaps are identified: one is attributed to the existence of attached resonators and the other is due to the fact that these resonators are arranged periodically. It is found that these bandgaps are influenced by the value of the resonator’s natural frequency and the stiffness of the elastic foundation. The analytically realized bandgap structures agree very well with the finite-element obtained dispersion curves by COMSOL.
MSC:
| 74K20 | Plates |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74J15 | Surface waves in solid mechanics |
Keywords:
flexural edge waves; Kirchhoff plate; periodic resonators; Winkler foundation; banded spectrumSoftware:
COMSOLReferences:
| [1] | L. Brillouin, Wave Propagation in Periodic Structures: electric filters and crystal lattices, 1946. http://dx.doi.org/10.1038/158926a0. · Zbl 0063.00607 |
| [2] | Lee, S. Y.; Ke, H. Y.; Kao, M. J., Flexural waves in a periodic beam, J. Appl. Mech. Trans. ASME, 57, 779-783 (1990) |
| [3] | M.A. Hawwa, SH-Wave Interaction in a Harmonically Inhomogeneous Elastic Plate, Trans. ASME. 62 (n.d.) 174-177. |
| [4] | Mcintyre, J. S.; Rasmussen, M. L.; Bert, C. W.; Kline, R. A., Resonance in fiber-reinforced composite materials with sinusoidal stiffness properties, Wave Motion, 30, 97-123 (1999) · Zbl 1074.74559 |
| [5] | Burr, K. P.; Triantafyllou, M. S.; Yue, D. K.P., Asymptotic governing equation for wave propagation along weakly non-uniform Euler-Bernoulli beams, J. Sound Vib., 247 (2001) · Zbl 1237.74109 |
| [6] | Schoengerg, M., Wave propagation in alternating solid and fluid layers, Wave Motion, 6, 303-320 (1984) · Zbl 0538.73060 |
| [7] | Touratier, M., Floquet waves in a body with slender periodic structure, Wave Motion, 8, 485-495 (1986) · Zbl 0594.73071 |
| [8] | A.H. Nayfeh, M.A. Hawwa. Nayfeh, Vibration and wave propagation characteristics of multi-segmented elastic beams. PDF, in: Proc. 16th Int. Conf. Exp. Mech., 1990, pp. 397-401. |
| [9] | Ruzzene, M.; Baz, A., Attenuation and localization of wave propagation in periodic rods using shape memory inserts, Smart Mater. Struct., 9, 805-806 (2000) |
| [10] | Moustaghfir, N.; Daya, E. M.; Braikat, B.; Damil, N.; Potier-Ferry, M., Evaluation of continuous modelings for the modulated vibration modes of long repetitive structures, Int. J. Solids Struct., 44, 7061-7072 (2007) · Zbl 1166.74371 |
| [11] | Delph, T. J.; Herrmann, G.; Kaul, R. K., Harmonic wave propagation in a periodically layered, infinite elastic body: Plane strain, numerical results, Am. Soc. Mech. Eng., 45 (1980) · Zbl 0441.73035 |
| [12] | Golebiewska, A. A., On dispersion of periodically layered composites in plane strain, J. Appl. Mech. Trans. ASME, 47, 206-207 (1980) |
| [13] | Duan, Z. P.; Eischen, J. W.; Herrmann, G., Harmonic wave propagation in nonhomogeneous layered composites, J. Appl. Mech. Trans. ASME, 53, 108-115 (1986) · Zbl 0591.73032 |
| [14] | Braga, A. M.B.; Herrmann, G., Floquet waves in anisotropic periodically layered composites, J. Acoust. Soc. Am., 91, 1211-1227 (1992) |
| [15] | Auld, B. A.; Chimenti, D. E.; Shull, P. J., Shear horizontal wave propagation in periodically layered composites, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 43, 319-325 (1996) |
| [16] | Lee, S. Y.; Ke, H. Y., Flexural wave propagation in an elastic beam with periodic structure, J. Appl. Mech. Trans. ASME, 59, S189-S196 (1992) · Zbl 0825.73131 |
| [17] | Hawwa, M. A., Reflection of flexural waves in geometrically periodic beams, J. Sound Vib., 199, 453-461 (1997) · Zbl 1235.74166 |
| [18] | Trainiti, G.; Rimoli, J. J.; Ruzzene, M., Wave propagation in periodically undulated beams and plates, Int. J. Solids Struct., 75-76, 260-276 (2015) |
| [19] | Li, Y.; Xu, Y. L., Tuning and switching of band gap of the periodically undulated beam by the snap through buckling, AIP Adv., 7 (2017) |
| [20] | Pelat, A.; Gallot, T.; Gautier, F., On the control of the first bragg band gap in periodic continuously corrugated beam for flexural vibration, J. Sound Vib., 446, 249-262 (2019) |
| [21] | Sigalas, M. M.; Economou, E. N., Elastic waves in plates with periodically placed inclusions, J. Appl. Phys., 75, 2845-2850 (1994) |
| [22] | Kinra, V. K.; Day, N. A.; Maslov, K.; Henderson, B. K.; Diderich, G., The transmission of a longitudinal wave through a layer of spherical inclusions with a random or periodic arrangement, J. Mech. Phys. Solids, 46, 153-165 (1998) · Zbl 0939.74514 |
| [23] | Chen, J. J.; Qin, B.; Cheng, J. C., Complete band gaps for lamb waves in cubic thin plates with periodically placed inclusions, Chinese Phys. Lett., 22, 1706-1708 (2005) |
| [24] | Yu, D.; Liu, Y.; Wang, G.; Zhao, H.; Qiu, J., Flexural vibration band gaps in Timoshenko beams with locally resonant structures, J. Appl. Phys., 100 (2006) |
| [25] | Li, Y.; Chen, J.; Han, X.; Huang, K.; Peng, J., Large complete band gap in two-dimensional phononic crystal slabs with elliptic inclusions, Phys. B Condens. Matter., 407, 1191-1195 (2012) |
| [26] | Miranda, E.; Santos, J. M.C., Flexural wave band gaps in AL2O3 /epoxy composite rectangular plate using minklin theroy, (Brazilian Conf. Compos. Mater. (2016)) |
| [27] | Mead, D. J., Free wave propagation in periodically supported, infinite beams, J. Sound Vib., 11, 181-197 (1970) |
| [28] | Sen Gupta, G., Natural flexural waves and the normal modes of periodically-supported beams and plates, J. Sound Vib., 13, 89-101 (1970) |
| [29] | Mace, B. R., Periodically stiffened fluid-loaded plates, I: Response to convected harmonic pressure and free wave propagation, J. Sound Vib., 73, 473-486 (1980) · Zbl 0444.73041 |
| [30] | Mead, D. J.; Yaman, Y., The harmonic response of rectangular sandwich plates with multiple stiffening: A flexural wave analysis, J. Sound Vib., 145, 409-428 (1991) |
| [31] | Mukherjee, S.; Parthan, S., Free wave propagation in rotationally restrained periodic plates, J. Sound Vib., 163, 535-544 (1993) · Zbl 0925.73370 |
| [32] | Rumerman, M. L., Vibration and wave propagation in ribbed plates, J. Acoust. Soc. Am., 57, 370-373 (1975) · Zbl 0294.73057 |
| [33] | Liu, Y.; Yu, D.; Li, L.; Zhao, H.; Wen, J.; Wen, X., Design guidelines for flexural wave attenuation of slender beams with local resonators, Phys. Lett. Sect. A Gen. At. Solid State Phys., 362, 344-347 (2007) |
| [34] | Xiao, Y.; Wen, J.; Wen, X., Broadband locally resonant beams containing multiple periodic arrays of attached resonators, Phys. Lett. Sect. A Gen. At. Solid State Phys., 376, 1384-1390 (2012) |
| [35] | Xiao, Y.; Wen, J.; Wang, G.; Wen, X., Theoretical and experimental study of locally resonant and bragg band gaps in flexural beams carrying periodic arrays of beam-like resonators, J. Vib. Acoust. Trans. ASME, 135 (2013) |
| [36] | Wang, M. Y.; Wang, X., Frequency band structure of locally resonant periodic flexural beams suspended with force-moment resonators, J. Phys. D. Appl. Phys., 46 (2013) |
| [37] | Casadei, F.; Bertoldi, K., Wave propagation in beams with periodic arrays of airfoil-shaped resonating units, J. Sound Vib., 333, 6532-6547 (2014) |
| [38] | Wang, M. Y.; Choy, Y. T.; Wan, C. W.; Zhao, A. S., Wide band-gaps in flexural periodic beams with separated force and moment resonators, J. Vib. Acoust. Trans. ASME., 137, 1-6 (2015) |
| [39] | Wang, X.; Wang, M. Y., An analysis of flexural wave band gaps of locally resonant beams with continuum beam resonators, Meccanica, 51, 171-178 (2016) · Zbl 1382.74078 |
| [40] | Wang, T.; Sheng, M. P.; Qin, Q. H., Multi-flexural band gaps in an Euler-Bernoulli beam with lateral local resonators, Phys. Lett. Sect. A Gen. At. Solid State Phys., 380, 525-529 (2016) |
| [41] | Zhou, J.; Wang, K.; Xu, D.; Ouyang, H., Multi-low-frequency flexural wave attenuation in Euler-Bernoulli beams using local resonators containing negative-stiffness mechanisms, Phys. Lett. Sect. A Gen. At. Solid State Phys., 381, 3141-3148 (2017) |
| [42] | Song, Y.; Feng, L.; Wen, J.; Yu, D.; Wen, X., Analysis and enhancement of flexural wave stop bands in 2D periodic plates, Phys. Lett. Sect. A Gen. At. Solid State Phys., 379, 1449-1456 (2015) |
| [43] | Wang, T.; Sheng, M. P.; Guo, Z. W.; Qin, Q. H., Flexural wave suppression by an acoustic metamaterial plate, Appl. Acoust., 114, 118-124 (2016) |
| [44] | Miranda Jr, E.; Santos, J., Flexural wave band gaps in elastic metamaterial thin plate, (IX Congr. Nac. Eng. Mec. Fortaleza - Ceara (2016)) |
| [45] | 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 Process., 116, 480-504 (2019) |
| [46] | Konenkov, Y. K., A-Rayleigh-type flexural wave, Sov. Phys. Acoust., 6, 122-123 (1960) |
| [47] | Pupyrev, P. D.; Lomonosov, A. M.; Nikodijevic, A.; Mayer, A. P., On the existence of guided acoustic waves at rectangular anisotropic edges, Ultrasonics, 71, 278-287 (2016) |
| [48] | Kaplunov, J.; Prikazchikov, D. A.; Rogerson, G. A., Edge bending wave on a thin elastic plate resting on a Winkler foundation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 472, Article 20160178 pp. (2016) · Zbl 1371.74194 |
| [49] | Nobili, A.; Radi, E.; Lanzoni, L., Flexural edge waves generated by steady-state propagation of a loaded rectilinear crack in an elastically supported thin plate, Proc. R. Soc. Math. Phys. Eng. Sci., 473, 2204, Article 20170265 pp. (2017) · Zbl 1404.74071 |
| [50] | Feng, F.; Shen, Z.; Shen, J., Edge waves in a 3D plate: Two solutions based on plate mode matching, Math. Mech. Solids, 22, 11, 2065-2074 (2017) · Zbl 1395.74042 |
| [51] | Ukrainskii, D. V., On the type of flexural edge wave on a circular plate, Mech. Solids, 53, 5, 501-509 (2018) |
| [52] | Alzaidi, A. S.M.; Kaplunov, J.; Prikazchikova, L., Elastic bending wave on the edge of a semi-infinite plate reinforced by a strip plate, Math. Mech. Solids, 24, 10, 3319-3330 (2019) · Zbl 07273368 |
| [53] | Wilde, M. V.; Golub, M. V.; Eremin, A. A., Experimental observation of theoretically predicted spectrum of edge waves in a thick elastic plate with facets, Ultrasonics., 98, 88-93 (2019) |
| [54] | Hughes, J. M.; Kotousov, A.; Ng, C-T., Generation of higher harmonics with the fundamental edge wave mode, Appl. Phys. Lett., 116, Article 101904 pp. (2020) |
| [55] | Kaplunov, J.; Prikazchikov, D. A.; Rogerson, G. A.; Lashab, M. I., The edge wave on an elastically supported Kirchhoff plate, J. Acoust. Soc. Am., 136, 1487-1490 (2014) |
| [56] | Banerjee, J. R., Dynamic stiffness formulation for structural elements: A general approach, Comput. Struct., 63, 101-103 (1997) · Zbl 0899.73513 |
| [57] | Banerjee, J. R.; Papkov, S. O.; Liu, X.; Kennedy, D., Dynamic stiffness matrix of a rectangular plate for the general case, J. Sound Vib., 342, 177-199 (2015) |
| [58] | Yu, D.; Wen, J.; Shen, H.; Xiao, Y.; Wen, X., Propagation of flexural wave in periodic beam on elastic foundations, Phys. Lett. Sect. A Gen. At. Solid State Phys., 376, 626-630 (2012) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
