Single-phase multi-resonant metabeam for broadband reduction of multi-polarization low-frequency vibration

doi:10.1007/s10483-025-3322-9

摘要/Abstract

Abstract:

Reducing the vibration transmission in beams is of significant interest, as beams are among the most widely used basic structures in numerous practical engineering applications. However, achieving broadband suppression of multi-polarization low-frequency vibration in beams presents a challenge. This study proposes a single-phase multi-resonant metabeam, which consists of a host beam with subwavelength arrays of tunable local resonators. These resonators exhibit adjustable multi-polarization resonant modes that strongly couple with the host beam, enabling simultaneous suppression of multi-type waves over a broad frequency range. The theoretical analysis demonstrates that under the fixed total added-mass ratio ( $γ total = 1.5$ ), the tailored frequency spacing ( $δ = 25$ –50 Hz) and the controlled loss factor ( $η = 0.03$ –0.07) act synergistically to broaden bandgaps through resonant zone overlapping and attenuation peak smoothing. The experimental validation with monolithic three-dimensional (3D)-printed specimens confirms the efficacy of this design in multi-polarization vibration control within a deep-subwavelength bandgap, opening a new avenue for designing multi-polarization vibration suppression structures.

Key words: metastructure, vibration suppression, metabeam, bandgap, multi-polarization

中图分类号:

O421⁺.5

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(12): 2265-2280.

Yongzhe LI, Yongqiang LI, Gaoge LIANG, Quanxing LIU, Yong XIAO. Single-phase multi-resonant metabeam for broadband reduction of multi-polarization low-frequency vibration[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(12): 2265-2280.

图/表 11

$E$ /GA	$ρ$ / $(kg ⋅ m − 3)$	$ν$
2.6	1 120	0.4

Number	$h 1$ /mm	$h 2$ /mm	$θ$ /(°)	$d 1$ /mm	$d 2$ /mm	$f F$ /Hz	$f L$ /Hz
1	13.1	12.3	34.8	2.4	2.4	170	240
2	12.9	11.1	28	2.4	2.4	207	271
3	13.5	10.4	12	2.4	2.4	265	315
4	12.7	8.9	9.6	2.4	2.4	330	367
5	12.2	5.7	8	2.4	2.4	525	437

Configuration	Mass ratio $γ i = m i / (ρ A a)$	Frequency rule (radial/axial)	$γ total$
$δ = 0$ Hz	(0.300, 0.300, 0.300, 0.300, 0.300)	$f F i = f L i = 300$ (all resonators)	1.5
$δ = 25$ Hz	(0.105, 0.165, 0.240, 0.390, 0.600)	$f F i = f L i = 325 + 25 (i − 3)$	1.5
$δ = 50$ Hz	Same as $δ = 25$ Hz	$f F i = f L i = 325 + 50 (i − 3)$	1.5

参考文献 53

[1]	HUSSEIN, M. I., LEAMY, M. J., and RUZZENE, M. Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Applied Mechanics Reviews, 66(4), 040802 (2014)
[2]	XIAO, Y., WEN, J. H., YU, D. L., and WEN, X. S. Flexural wave propagation in beams with periodically attached vibration absorbers: band-gap behavior and band formation mechanisms. Journal of Sound and Vibration, 332(4), 867–893 (2013)
[3]	MA, G. C., FU, C. X., WANG, G. H., DEL, H. P., CHRISTENSEN, J., LAI, Y., and SHENG, P. Polarization bandgaps and fluid-like elasticity in fully solid elastic metamaterials. Nature Communications, 7(1), 13536 (2016)
[4]	FUJITA, K., TOMODA, M., WRIGHT, O. B., and MATSUDA, O. Perfect acoustic bandgap metabeam based on a quadruple-mode resonator array. Applied Physics Letters, 115(8), 081905 (2019)
[5]	OGASAWARA, A., FUJITA, K., TOMODA, M., MATSUDA, O., and WRIGHT, O. B. Wave-canceling acoustic metarod architected with single material building blocks. Applied Physics Letters, 116(24), 241904 (2020)
[6]	TAKEDA, H., MURAKAMI, E., TOMODA, M., MATSUDA, O., FUJITA, K., and WRIGHT, O. B. Tapered rainbow metabeam for wideband multimode acoustic blocking based on quadruple-mode resonators. Applied Physics Letters, 121(13), 131701 (2022)
[7]	KHORSHIDIPACHI, M., DARDEL, M., and COMI, C. Low frequency bandgap enhancement in dual graded metastructure beam with negative capacitance circuits and light-weight mass-spring resonators. Applied Mathematical Modelling, 144, 116090 (2025)
[8]	DING, Y. H., CHEN, Z. Q., LIANG, F., LEE, H. P., YU, H., LIN, S. C., and LUO, J. Flexural vibration control of functionally graded poroelastic pipes via periodic piezoelectric design. Acta Mechanica, 235(5), 3131–3147 (2024)
[9]	LIANG, F., CHEN, Y., KOU, H. J., and QIAN, Y. Hybrid Bragg-locally resonant bandgap behaviors of a new class of motional two-dimensional meta-structure. European Journal of Mechanics-A/Solids, 97, 104832 (2023)
[10]	WANG, Y. H., YANG, J., CHEN, Z. X., LIN, Y., GONG, L. P., ZHANG, S. W., LI, W. H., and SUN, S. S. Investigation of a magnetorheological elastomer metamaterial sandwich beam with tunable graded stiffness for broadband vibration attenuation. Smart Materials and Structures, 32(6), 065022 (2023)
[11]	ZHU, Z. J., XIAO, Y., FAN, S. X., LI, Y. Q., GUO, J. J., CHEN, N., YANG, P., YU, D. L., and WEN, J. H. Auto-adaptive metastructure for active tunable ultra-low frequency vibration suppression. International Journal of Mechanical Sciences, 271, 109131 (2024)
[12]	DE SOUSA, V. C., TAN, D., DE MARQUI, C. JR, and ERTURK, A. Tunable metamaterial beam with shape memory alloy resonators: theory and experiment. Applied Physics Letters, 113(14), 143502 (2018)
[13]	XIAO, Y., WANG, S. X., LI, Y. Q., and WEN, J. H. Closed-form bandgap design formulas for beam-type metastructures. Mechanical Systems and Signal Processing, 159, 107777 (2021)
[14]	DENG, J., GAO, N. S., and CHEN, X. Ultrawide attenuation bands in gradient metabeams with acoustic black hole pillars. Thin-Walled Structures, 184, 110459 (2023)
[15]	LIU, J. N., LI, J. Q., and WU, Y. Bandgap adjustment of a sandwich-like acoustic metamaterial plate with a frequency-displacement feedback control method. Applied Mathematics and Mechanics (English Edition), 45(10), 1807–1820 (2024) https://doi.org/10.1007/s10483-024-3167-8
[16]	BILAL, O. R., FOEHR, A., and DARAIO, C. Enhancement of deep-subwavelength band gaps in flat spiral-based phononic metamaterials using the trampoline phenomena. Journal of Applied Mechanics, 87(7), 071009 (2020)
[17]	WANG, K., ZHOU, J. X., WANG, Q., OUYANG, H. J., and XU, D. L. Low-frequency band gaps in a metamaterial rod by negative-stiffness mechanisms: design and experimental validation. Applied Physics Letters, 114(25), 251902 (2019)
[18]	YANG, F., MA, Z. Y., and GUO, X. M. Bandgap characteristics analysis and graded design of a novel metamaterial for flexural wave suppression. Applied Mathematics and Mechanics (English Edition), 46(1), 1–24 (2025) https://doi.org/10.1007/s10483-025-3204-7
[19]	BILAL, O. R., FOEHR, A., and DARAIO, C. Observation of trampoline phenomena in 3D-printed metamaterial plates. Extreme Mechanics Letters, 15, 103–107 (2017)
[20]	BILAL, O. R. and HUSSEIN, M. I. Trampoline metamaterial: local resonance enhancement by springboards. Applied Physics Letters, 103(11), 111901 (2013)
[21]	ZHAO, T., YANG, Z. C., XU, Y. L., and TIAN, W. Mode localization in metastructure with T-type resonators for broadband vibration suppression. Engineering Structures, 268, 114775 (2022)
[22]	WANG, K., ZHOU, J. X., TAN, D. G., LI, Z. Y., LIN, Q. D., and XU, D. L. A brief review of metamaterials for opening low-frequency band gaps. Applied Mathematics and Mechanics (English Edition), 43(7), 1125–1144 (2022) https://doi.org/10.1007/s10483-022-2870-9
[23]	XIAO, Y., WANG, Y., ZHAO, H., YU, D., and WEN, J. Research progress of acoustic metamaterials for vibration and noise reduction applications. Journal of Mechanical Engineering, 59(19), 277–298 (2023)
[24]	JIAN, Y. P., HU, G. B., TANG, L. H., TANG, W., ABDI, M., and AW, K. C. Analytical and experimental study of a metamaterial beam with grading piezoelectric transducers for vibration attenuation band widening. Engineering Structures, 275, 115091 (2023)
[25]	BILAL, O. R., BALLAGI, D., and DARAIO, C. Architected lattices for simultaneous broadband attenuation of airborne sound and mechanical vibrations in all directions. Physical Review Applied, 10(5), 054060 (2018)
[26]	HU, G. B., AUSTIN, A. C. M., SOROKIN, V., and TANG, L. H. Metamaterial beam with graded local resonators for broadband vibration suppression. Mechanical Systems and Signal Processing, 146, 106982 (2021)
[27]	XIAO, Y. and WEN, J. H. Closed-form formulas for bandgap estimation and design of metastructures undergoing longitudinal or torsional vibration. Journal of Sound and Vibration, 485, 115578 (2020)
[28]	MENG, H., CHRONOPOULOS, D., FABRO, A. T., ELMADIH, W., and MASKERY, I. Rainbow metamaterials for broadband multi-frequency vibration attenuation: numerical analysis and experimental validation. Journal of Sound and Vibration, 465, 115005 (2020)
[29]	YEH, S. L. and HARNE, R. L. Structurally-integrated resonators for broadband panel vibration suppression. Smart Materials and Structures, 29(8), 085010 (2020)
[30]	CELLI, P., YOUSEFZADEH, B., DARAIO, C., and GONELLA, S. Bandgap widening by disorder in rainbow metamaterials. Applied Physics Letters, 114(9), 091903 (2019)
[31]	BADREDDINE ASSOUAR, M., SENESI, M., OUDICH, M., RUZZENE, M., and HOU, Z. L. Broadband plate-type acoustic metamaterial for low-frequency sound attenuation. Applied Physics Letters, 101(17), 173505 (2012)
[32]	MIRANDA, E. J. P. JR, NOBREGA, E. D., FERREIRA, A. H. R., and DOS SANTOS, J. M. C. Flexural wave band gaps in a multi-resonator elastic metamaterial plate using Kirchhoff-Love theory. Mechanical Systems and Signal Processing, 116, 480–504 (2019)
[33]	XIAO, Y., WEN, J. H., and WEN, X. S. Longitudinal wave band gaps in metamaterial-based elastic rods containing multi-degree-of-freedom resonators. New Journal of Physics, 14(3), 033042 (2012)
[34]	XIAO, Y., WEN, J. H., and WEN, X. S. Broadband locally resonant beams containing multiple periodic arrays of attached resonators. Physics Letters A, 376(16), 1384–1390 (2012)
[35]	EL-BORGI, S., FERNANDES, R., RAJENDRAN, P., YAZBECK, R., BOYD, J. G., and LAGOUDAS, D. C. Multiple bandgap formation in a locally resonant linear metamaterial beam: theory and experiments. Journal of Sound and Vibration, 488, 115647 (2020)
[36]	ZHAO, P. C., ZHANG, K., ZHAO, C., and DENG, Z. C. Multi-resonator coupled metamaterials for broadband vibration suppression. Applied Mathematics and Mechanics (English Edition), 42(1), 53–64 (2021) https://doi.org/10.1007/s10483-021-2684-8
[37]	CAI, C. Q., ZHU, C. J., ZHANG, F. Y., SUN, J. J., WANG, K., YAN, B., and ZHOU, J. X. Modeling and analysis of gradient metamaterials for broad fusion bandgaps. Applied Mathematics and Mechanics (English Edition), 45(7), 1155–1170 (2024) https://doi.org/10.1007/s10483-024-3154-6
[38]	KRUSHYNSKA, A. O., MINIACI, M., BOSIA, F., and PUGNO, N. M. Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials. Extreme Mechanics Letters, 12, 30–36 (2017)
[39]	TANIKER, S. and YILMAZ, C. Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges. International Journal of Solids and Structures, 106-107, 129–138 (2017)
[40]	TANIKER, S. and YILMAZ, C. Design, analysis and experimental investigation of three-dimensional structures with inertial amplification induced vibration stop bands. International Journal of Solids and Structures, 72, 88–97 (2015)
[41]	BANERJEE, A., ADHIKARI, S., and HUSSEIN, M. I. Inertial amplification band-gap generation by coupling a levered mass with a locally resonant mass. International Journal of Mechanical Sciences, 207, 106630 (2021)
[42]	FRANDSEN, N. M. M., BILAL, O. R., JENSEN, J. S., and HUSSEIN, M. I. Inertial amplification of continuous structures: large band gaps from small masses. Journal of Applied Physics, 119(12), 124902 (2016)
[43]	FANG, X., WEN, J. H., BONELLO, B., YIN, J. F., and YU, D. L. Ultra-low and ultra-broad-band nonlinear acoustic metamaterials. Nature Communications, 8(1), 1288 (2017)
[44]	ZHANG, X. D., YU, H. Y., HE, Z. C., HUANG, G. L., CHEN, Y. Y., and WANG, G. A metamaterial beam with inverse nonlinearity for broadband micro-vibration attenuation. Mechanical Systems and Signal Processing, 159, 107826 (2021)
[45]	XIA, Y. W., RUZZENE, M., and ERTURK, A. Dramatic bandwidth enhancement in nonlinear metastructures via bistable attachments. Applied Physics Letters, 114(9), 093501 (2019)
[46]	XIA, Y. W., RUZZENE, M., and ERTURK, A. Bistable attachments for wideband nonlinear vibration attenuation in a metamaterial beam. Nonlinear Dynamics, 102(3), 1285–1296 (2020)
[47]	GUO, J. J., LI, Y. Q., XIAO, Y., FAN, Y. L., YU, D. L., and WEN, J. H. Multiscale modeling and design of lattice truss core sandwich metastructures for broadband low-frequency vibration reduction. Composite Structures, 289, 115463 (2022)
[48]	OH, J. H., QI, S. B., KIM, Y. Y., and ASSOUAR, B. Elastic metamaterial insulator for broadband low-frequency flexural vibration shielding. Physical Review Applied, 8(5), 054034 (2017)
[49]	ZHANG, W. X., YANG, D. S., and GUO, X. Y. Low-frequency vibration suppression of meta-beam with softening nonlinearity. Applied Mathematics and Mechanics (English Edition), 46(6), 1011–1028 (2025) https://doi.org/10.1007/s10483-025-3258-9
[50]	ZHAO, L., LU, Z. Q., DING, H., and CHEN, L. Q. A viscoelastic metamaterial beam for integrated vibration isolation and energy harvesting. Applied Mathematics and Mechanics (English Edition), 45(7), 1243–1260 (2024) https://doi.org/10.1007/s10483-024-3159-7
[51]	SUN, X. T., QU, Y. P., WANG, F., and XU, J. Effects of time-delayed vibration absorber on bandwidth of beam for low broadband vibration suppression. Applied Mathematics and Mechanics (English Edition), 44(10), 1629–1650 (2023) https://doi.org/10.1007/s10483-023-3038-6
[52]	HE, Z. X., LIANG, G. G., LIU, Q. X., LI, Y. Q., and XIAO, Y. Thin-walled box metabeam with graded 2DOF resonators for ultra-broadband complete flexural vibration suppression. Thin-Walled Structures, 210, 113023 (2025)
[53]	LI, Y. Q., XIAO, Y., GUO, J. J., ZHU, Z. J., and WEN, J. H. Single-phase metabeam for three-directional broadband vibration suppression. International Journal of Mechanical Sciences, 234, 107683 (2022)