Applied Mathematics and Mechanics >
Synergistic design of ultra-wide low-frequency continuous bandgap metastructure for audible noise attenuation
Received date: 2025-10-17
Revised date: 2026-01-12
Online published: 2026-03-02
Supported by
Project supported by the National Natural Science Foundation of China (No. 12572020) and the Key Project of Natural Science Foundation of Hebei Province of China (No. A2023210064)
Copyright
Local resonant acoustic metamaterials have broad applications in sound insulation, yet their single-configuration designs often exhibit limited and discontinuous bandgap widths, hindering full-frequency noise attenuation across the human auditory range. This study presents a double-phase fidget-spinner-shaped acoustic metamaterial (DFAM), specifically designed to achieve an ultra-broad, low-frequency continuous bandgap by means of synergistic structural optimization, enabling effective and robust control of audible noise. Based on Bloch’s theorem and the finite element method, the dispersion relation of the DFAM structure is calculated and verified by the transmission loss curves. The propagation characteristics of sound waves within the structure are further analyzed for noise frequencies that fall within the passband. The influence of the geometric and physical parameters on the bandgap is investigated, and the corresponding transmission loss in the propagation direction is further calculated. A hybrid collaborative design strategy, leveraging multi-parameter optimization and bandgap complementarity, is developed to construct a metastructure with continuous bandgap coverage from 20 Hz to 1 000 Hz. The resulting metastructure demonstrates exceptional broadband noise attenuation, achieving a total bandgap width of 876.3 Hz (87.63% of the target range) with the transmission loss up to -762.78 dB in a three-periodic arrangement. The simulation and experimental results for the transmission loss of the DFAM metastructure show strong agreement in the low-frequency range. This work provides a novel framework for designing ultra-wide low-frequency continuous bandgap metastructures, offering significant potential for noise mitigation in complex environments.
Dongxu GUO , Xiaolong ZHANG , Ruilan TIAN , Xiangyang LI , Minghao WANG . Synergistic design of ultra-wide low-frequency continuous bandgap metastructure for audible noise attenuation[J]. Applied Mathematics and Mechanics, 2026 , 47(3) : 473 -496 . DOI: 10.1007/s10483-026-3361-7
| [1] | LU, J. H., DING, S. Q., NI, Y. Q., and LI, S. Bio-inspired acoustic metamaterials for traffic noise control: bridging the gap with machine learning. Communications Engineering, 4, 136 (2025) |
| [2] | COMANDINI, G., OUISSE, M., TING, V. P., and SCARPA, F. Architected acoustic metamaterials: an integrated design perspective. Applied Physics Reviews, 12, 011340 (2025) |
| [3] | ZHANG, J., YAO, D., PENG, W., WANG, R. Q., LI, J., and GUO, S. Y. Optimal design of lightweight acoustic metamaterials for low-frequency noise and vibration control of high-speed train composite floor. Applied Acoustics, 199, 109041 (2022) |
| [4] | WEI, Z. L., SUN, X. F., YANG, F., KE, Z. T., LU, T., ZHANG, P., and SHEN, C. Carriage interior noise-based inspection for rail corrugation on high-speed railway track. Applied Acoustics, 196, 108881 (2022) |
| [5] | LEE, H. M. and LEE, H. P. Low-frequency cabin noise of rapid transit trains. Noise Mapping, 2024, 11, 20220181 (2024) |
| [6] | GARG, S., LIM, K. M., and LEE, H. P. Recording and analyzing carriage noise of various high-speed rail systems using smartphones. Acoustics Australia, 48(1), 121–130 (2020) |
| [7] | LIAO, G. X., LUAN, C. C., WANG, Z. W., LIU, J. P., YAO, X. H., and FU, J. Z. Acoustic metamaterials: a review of theories, structures, fabrication approaches, and applications. Advanced Materials Technologies, 6(5), 2000787 (2021) |
| [8] | CAO, D. X., WANG, L. M., WANG, J. R., GUO, X. Y., and LI, H. T. Design and sound absorption analysis of labyrinthine acoustic metamaterials based on fractal theory. International Journal of Solids and Structures, 306, 113121 (2025) |
| [9] | RUAN, Y. D., LIANG, X., HUA, X. Y., ZHANG, C., XIA, H., and LI, C. Isolating low-frequency vibration from power systems on a ship using spiral phononic crystals. Ocean Engineering, 225, 108804 (2021) |
| [10] | GAO, H. Q., YAN, Q., LIU, X. S., ZHANG, Y., SUN Y. T., DING, Q., WANG, L., XU, J. X., and YAN, H. Low-frequency bandgaps of the lightweight single-phase acoustic metamaterials with locally resonant archimedean spirals. Materials, 15(1), 373 (2022) |
| [11] | WANG, X. X. and FU, T. A novel arc-type auxetic cellular doubly-curved shells with negative Poisson’s ratio for broadband low-frequency sound insulation. European Journal of Mechanics-A/Solids, 106, 105326 (2024) |
| [12] | CAI, T., HUANG, S., GUO, H., YUAN, T., SUN, P., and LIU, N. N. Sound insulation performance of membrane-type acoustic metamaterial based on defect state structure. Physica Scripta, 99(2), 025967 (2024) |
| [13] | LIU, Z. Y., ZHANG, X. X., MAO, Y. W., ZHU, Y. Y., YANG, Z. Y., CHAN, C. T., and SHENG, P. Locally resonant sonic materials. Science, 289, 1734–1736 (2000) |
| [14] | ZHANG, X. P., LI, Y., WANG, Y. G., and LUO, Y. J. Ultra-wide low-frequency bandgap design of acoustic metamaterial via multi-material topology optimization. Composite Structures, 306, 116584 (2023) |
| [15] | ZHANG, S. L., LOU, J., FAN, H., and DU, J. K. A nonlinear acoustic metamaterial beam with tunable flexural wave band gaps. Engineering Structures, 276, 115379 (2023) |
| [16] | MEIER, T., KORAKIS, V., BLANKENSHIP, B. W., LU, H. T., KYRIAKOU, E., PAPAMAKARIOS, S., VANGELATOS, Z., YILDIZDAG, M. E., ZYLA, G., XIA, X. X., ZHENG, X. Y., RHO, Y., FARSARI, M., and GRIGOROPOULOS, C. P. Scalable phononic metamaterials: tunable bandgap design and multi-scale experimental validation. Materials & Design, 252, 113778 (2025) |
| [17] | LI, S. J., HAN, S. H., ZHENG, H. K., HAN, Q., and LI, C. L. Hierarchical design and vibration suppression of the hexachiral hybrid acoustic metamaterial. Applied Acoustics, 224, 110145 (2024) |
| [18] | SUN, P., GUO, H., JIN, F. H., ZHANG, Z. D., LIU, N. N., YUAN, T., MA, L. F., and WANG, Y. S. Mechanics and extreme low-frequency band gaps of auxetic hexachiral acoustic metamaterial with internal resonant unit. Applied Acoustics, 200, 109046 (2022) |
| [19] | YANG, H. Y., CHENG, S. L., LI, X. F., YAN, Q., WANG, B., XIN, Y. J., SUN, Y. T., DING, Q., YAN, H., LI, Y. J., and ZHAO, Q. X. Study on bandgap and vibration attenuation mechanism of novel chiral lattices. Physica B: Condensed Matter, 651, 414596 (2023) |
| [20] | HOU, J. H., LI, D., ZHANG, Z. J., RUAN, H. F., and LIU, H. C. Bandgap enhancement of two-dimensional lattice metamaterial via re-entrant hierarchy. Smart Materials and Structures, 31(9), 095012 (2022) |
| [21] | LI, C. L., SUN, Y., HAN, Q., and JIANG, T. J. Bandgap tunability and impact mitigation enhancement of hybrid graded origami-inspired metamaterials with multiple resonators. Thin-Walled Structures, 202, 112137 (2024) |
| [22] | RUAN, H. F. and LI, D. Band gap characteristics of bionic acoustic metamaterials based on spider web. Engineering Structures, 308, 118003 (2024) |
| [23] | KUMAR, N. and PAL, S. Low frequency and wide band gap metamaterial with divergent shaped star units: numerical and experimental investigations. Applied Physics Letters, 115(25), 254101 (2019) |
| [24] | HOSSEINKHANI, A., PANAHI, E., KHANSANAMI, M. F., and YOUNESIAN, D. A modified star-shaped phononic crystal for the vibration wave filtration in plates: design and experiment. Archive of Applied Mechanics, 93(8), 3153–3169(2023) |
| [25] | DONG, X. J., WANG, S., WANG, A. S., WANG, L., ZHANG, Z. Z., TIE, Y. H., LIN, Q. Y., and SUN, Y. T. Low-frequency bandgap and vibration suppression mechanism of a novel square hierarchical honeycomb metamaterial. Applied Mathematics and Mechanics (English Edition), 45(10), 1841–1856 (2024) https://doi.org/10.1007/s10483-024-3168-7 |
| [26] | TAJSHAM, A., YOUNESIAN, D., GOODINI, J., and HOSSEINKHANI, A. A new polyhedral sonic crystal for broadband sound barriers: optimization and experimental study. Applied Acoustics, 218, 109881 (2024) |
| [27] | KUDELA, P., IJJEH, A., RADZIENSKI, M., MINIACI, M., PUGNO, N., and OSTACHOWICZ, W. Deep learning aided topology optimization of phononic crystals. Mechanical Systems and Signal Processing, 200, 110636 (2023) |
| [28] | ZHANG, K., GUO, Y. Y., LIU, X. B., HONG, F., HOU, X. H., and DENG, Z. C. Deep learning-based inverse design of lattice metamaterials for tuning bandgap. Extreme Mechanics Letters, 69, 102165 (2024) |
| [29] | JIANG, W. F., ZHU, Y. Y., YIN, G. F., LU, H. H., XIE, L. F., and YIN, M. Dispersion relation prediction and structure inverse design of elastic metamaterials via deep learning. Materials Today Physics, 22, 100616 (2022) |
| [30] | JIA, Z. Y., BAO, Y. H., LUO, Y. J., WANG, D. Z., ZHANG, X. P., and KANG, Z. Maximizing acoustic band gap in phononic crystals via topology optimization. International Journal of Mechanical Sciences, 270, 109107 (2024) |
| [31] | WANG, H., CHENG, S. L., WANG, C., SUN, Y. T., and XIN, Y. J. Tunable band gaps and double-negative properties of innovative acoustic metamaterials. Applied Physics A, 127(7), 495 (2021) |
| [32] | ZHAO, R., ZHENG, J., GUO, J., SHI, Y. B., FENG, H. Z., TANG, J., and LIU, J. Suppression of low-frequency ultrasound broadband vibration using star-shaped single-phase metamaterials. Defence Technology, 34, 217–224 (2024) |
| [33] | LIU, M. H., LI, X. Y., ZOU, X. Y., and CHENG, J. C. Broadband acoustic insulation via gradient impedance boundary waveguide. Applied Physics Letters, 120, 123501 (2022) |
| [34] | 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 |
| [35] | 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 |
| [36] | TANG, Y. F., REN, S. W., MENG, H., XIN, F. X., HUANG, L. X., CHEN, T. N., ZHANG, C. Z., and LU, T. J. Hybrid acoustic metamaterial as super absorber for broadband low-frequency sound. Scientific Reports, 7(1), 43340 (2017) |
| [37] | GUO, D. X., ZHANG, X. L., TIAN, R. L., and CHEN, L. Q. Low-frequency bandgap broadening of single-phase shuriken-like acoustic metastructure through coupling design for sound insulation. Applied Acoustics, 236, 110725 (2025) |
| [38] | BARAVELLI, E. and RUZZENE, M. Internally resonating lattices for bandgap generation and low-frequency vibration control. Journal of Sound and Vibration, 332(25), 6562–6579 (2013) |
| [39] | CHENG, S. L., LI, X. F., WANG, Y. L., WANG, B., SUN, Y. T., YAN, Q., DING, Q., and XIN, Y. J. Multi-frequency band gap and active frequency modulation of snowflake-like convex horn ligament structure. European Journal of Mechanics-A/Solids, 97, 104843 (2023) |
| [40] | XIN, Y. J., CAI, P. C., LI, P., QUN, Y., SUN, Y. T., QIAN, D., CHENG, S. L., and ZHAO, Q. X. Comprehensive analysis of band gap of phononic crystal structure and objective optimization based on genetic algorithm. Physica B: Condensed Matter, 667, 415157 (2023) |
| [41] | YONG, J. W., LI, W. T., HU, X. J., WAN, Z. S., DONG, Y. Y., and FENG, N. L. Co-design of mechanical and vibration properties of a star polygon-coupled honeycomb metamaterial. Applied Sciences, 14(3), 1028 (2024) |
| [42] | HAGHPANAH, B., EBRAHIMI, H., MOUSANEZHAD, D., HOPKINS, J., and VAZIRI, A. Programmable elastic metamaterials. Advanced Engineering Materials, 18(4), 643–649 (2015) |
| [43] | LAI, Y., WU, Y., SHENG, P., and ZHANG, Z. Q. Hybrid elastic solids. Nature Materials, 10(8), 620–624 (2011) |
| [44] | LIU, Z. Y., CHAN, C. T., and SHENG, P. Analytic model of phononic crystals with local resonances. Physical Review B, 71, 014103 (2005) |
| [45] | WU, Y., LAI, Y., and ZHANG, Z. Q. Effective medium theory for elastic metamaterials in two dimensions. Physical Review B, 76, 205313 (2007) |
| [46] | LI, Y. L., WANG, Y., and YAO, S. Multipolar resonance and bandgap formation mechanism of star-shaped lattice structure. International Journal of Mechanical Sciences, 193, 106163 (2021) |
| [47] | CHEN, M., XU, W. S., LIU, Y., YAN, K., JIANG, H., and WANG, Y. R. Band gap and double-negative properties of a star-structured sonic metamaterial. Applied Acoustics, 139, 235–242 (2018) |
| [48] | CHENG, S. L., LI, X. F., YAN, Q., WANG, B., SUN, Y. T., XIN, Y. J., DING, Q., YAN, H., and LI, Y. J. Low frequency band gap and wave propagation mechanism of resonant hammer circular structure. Mechanics of Advanced Materials and Structures, 31(20), 4818–4838 (2023) |
| [49] | ZHANG, J., YAO, D., PENG, W., WANG, R. Q., LI, J., and GUO, S. Y. Optimal design of lightweight acoustic metamaterials for low-frequency noise and vibration control of high-speed train composite floor. Applied Acoustics, 199, 109041 (2022) |
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