Active quasi-zero stiffness vibration isolator for time-varying load conditions

doi:10.1007/s10483-026-3429-7

Abstract

Abstract:

To resolve the deteriorated vibration isolation performance of existing passive quasi-zero stiffness (QZS) vibration isolators caused by time-varying loads and stiffness mismatch, this paper proposes a novel load-adaptive QZS vibration isolator (LAQVI) based on radial basis function (RBF) neural network adaptive sliding mode control (SMC). The isolator adopts a composite structure coupling linear springs and electromagnetic units, which delivers adjustable QZS characteristics based on a nonlinear stiffness compensation mechanism. This paper analyzes the influence of various key parameters on the QZS characteristics. Dynamic equation analysis clarifies the relationship between load mismatch and displacement transmissibility, and theoretical studies confirm that excitation current control effectively enhances isolation performance. In addition, a sliding mode controller based on RBF neural network adaptation is designed, leveraging the RBF neural network’s learning and adaptive capabilities for rapid load variation estimation. Simulation results show that the adaptive control of excitation current can effectively compensate for the deviation of the isolation platform and enhance vibration isolation performance. Finally, static experiments verify the QZS characteristics of the isolator, and dynamic load experiments demonstrate the controller’s adaptive adjustment ability for load changes. Vibration test results show that the LAQVI exhibits reliable load adaptability and effective low-frequency vibration isolation performance under time-varying load conditions. The proposed LAQVI provides a theoretical basis for the application of QZS vibration isolators in fields such as intelligent equipment.

Key words: vibration isolation, quasi-zero stiffness (QZS), active vibration control, nonlinear stiffness compensation, sliding mode control (SMC), radial basis function (RBF) neural network

2010 MSC Number:

O322

Tianci JIANG, Guangdong SUI, Wentao WU, Xiaobiao SHAN, M. ELSAMANTY. Active quasi-zero stiffness vibration isolator for time-varying load conditions. Applied Mathematics and Mechanics (English Edition), 2026, 47(9): 2043-2062.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3429-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I9/2043

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[1]	ZHOU, C. Y., SUI, G. D., SHAN, X. B., and WU, W. T. A spatially mechanical-decoupled three-DOF quasi-zero-stiffness isolator integrating flexible curved beams. International Journal of Mechanical Sciences, 312, 111160 (2026)
[2]	CHEN, Y., JIN, Y. H., LIN, R. Z., JIANG, Y. F., MEI, X. T., HOU, L., WANG, Y. L., YONG, N. T., and GUO, A. X. Harmonic balance-automatic differentiation method: a practical nonlinear dynamics solver. International Journal of Mechanical Sciences, 312, 111192 (2026)
[3]	HAN, H. S., SOROKIN, V., TANG, L. H., and CAO, D. Q. Lightweight origami isolators with deployable mechanism and quasi-zero-stiffness property. Aerospace Science and Technology, 121, 107319 (2022)
[4]	ZHANG, C., HE, J. S., ZHOU, G. Q., WANG, K., XU, D. L., and ZHOU, J. X. Compliant quasi-zero-stiffness isolator for low-frequency torsional vibration isolation. Mechanism and Machine Theory, 181, 105213 (2023)
[5]	LIN, Y., WEN, G. L., LIU, C. X., HE, J. F., and LIU, J. A magnetorheological elastomer-based hybrid vibration isolation system with semi-active control and quasi-zero stiffness performance. International Journal of Non-Linear Mechanics, 174, 105063 (2025)
[6]	LIU, Y. F., YAN, B., and CHENG, L. Dynamical analysis and design theory for active bistable vibration isolators considering delay effect. Journal of Sound and Vibration, 573, 118196 (2024)
[7]	JIN, G. X., WANG, Z. H., and YANG, T. Z. Cascaded quasi-zero stiffness nonlinear low-frequency vibration isolator inspired by human spine. Applied Mathematics and Mechanics (English Edition), 43(6), 813–824 (2022) https://doi.org/10.1007/s10483-022-2852-5
[8]	LI, L. B., YANG, F., LIU, S. F., GUO, Z. M., HAN, D., XIA, Y., WANG, L. H., and FAN, H. L. Design of quasi-zero-stiffness metamaterials with ultra-wideband vibration isolation performance. International Journal of Mechanical Sciences, 300, 110440 (2025)
[9]	TIAN, Y. S., CAO, D. Q., CHEN, C., and ZHANG, X. Y. Vibration isolation performance of a rectangular panel with high-static-low-dynamic stiffness supports. Applied Mathematical Modelling, 119, 218–238 (2023)
[10]	YAN, G., LU, J. J., QI, W. H., LIU, F. R., YAN, H., ZHAO, L. C., WU, Z. Y., and ZHANG, W. M. Linear and nonlinear stiffness compensation for low-frequency vibration isolation: a comparative study. Nonlinear Dynamics, 112(8), 5955–5973 (2024)
[11]	TIAN, R. L., WANG, M. H., ZHANG, Y. S., JING, X. J., and ZHANG, X. L. A concave X-shaped structure supported by variable pitch springs for low-frequency vibration isolation. Mechanical Systems and Signal Processing, 218, 111587 (2024)
[12]	CHEN, P., WANG, B., ZHOU, D. S., WU, X. B., and DAI, K. S. Performance evaluation of a nonlinear energy sink with quasi-zero stiffness property for vertical vibration control. Engineering Structures, 282, 115801 (2023)
[13]	PALOMARES, E., NIETO, A. J., MORALES, A. L., CHICHARRO, J. M., and PINTADO, P. Numerical and experimental analysis of a vibration isolator equipped with a negative stiffness system. Journal of Sound and Vibration, 414, 31–42 (2018)
[14]	LIU, C. R. and YU, K. P. Accurate modeling and analysis of a typical nonlinear vibration isolator with quasi-zero stiffness. Nonlinear Dynamics, 100(3), 2141–2165 (2020)
[15]	SAEED, N. A., HOU, L., YI, H. M., SHUKUR, A. A., ALAMRY, S. M., and EL-SHOURBAGY, S. M. On a broadband vibration isolator with tunable stiffness: from quasi-zero-stiffness to zero-stiffness behavior. Applied Mathematics and Mechanics (English Edition), 47(2), 255–282 (2026) https://doi.org/10.1007/s10483-026-3351-9
[16]	ZHAO, F., JI, J. C., CAO, S. Q., ZHENG, J. Y., and LUO, Q. T. A constant quasi-zero stiffness isolator with tension springs to isolate vibrations with ultralow frequency. International Journal of Non-Linear Mechanics, 175, 105129 (2025)
[17]	LIU, C. R. and YU, K. P. Superharmonic resonance of the quasi-zero-stiffness vibration isolator and its effect on the isolation performance. Nonlinear Dynamics, 100(1), 95–117 (2020)
[18]	CHEN, T. T., WANG, K., WANG, Q., WANG, Y. D., CHENG, L., and ZHOU, J. X. A Chinese lantern-inspired low frequency energy harvester with synchronized multi-branched conversion elements. Engineering Structures, 334, 120266 (2025)
[19]	ZHANG, X. L., LU, X. H., LI, C. C., TIAN, R. L., CHEN, L. Q., and WANG, M. H. Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation. Engineering Structures, 312, 118262 (2024)
[20]	ZHOU, C. Y., SUI, G. D., CHEN, Y. F., and SHAN, X. B. A nonlinear low frequency quasi zero stiffness vibration isolator using double-arc flexible beams. International Journal of Mechanical Sciences, 276, 109378 (2024)
[21]	MAO, X. Y., YIN, M. M., DING, H., GENG, X. F., SHEN, Y. J., and CHEN, L. Q. Modeling, analysis, and simulation of X-shape quasi-zero-stiffness-roller vibration isolators. Applied Mathematics and Mechanics (English Edition), 43(7), 1027–1044 (2022) https://doi.org/10.1007/s10483-022-2871-6
[22]	CHAI, Z. Y., ZHANG, Z., XU, K. F., SONG, X. Y., ZHANG, Y. W., and CHEN, L. Q. An innovative nonlinear bionic X-shaped vibration isolator enhanced by quasi-zero stiffness characteristics: theory and experimental investigation. Applied Mathematics and Mechanics (English Edition), 46(8), 1475–1492 (2025) https://doi.org/10.1007/s10483-025-3277-8
[23]	SUI, G. D., HOU, S., ZHANG, X. F., SHAN, X. B., HOU, C. W., SONG, H. N., HOU, W. J., and LI, J. M. A bio-inspired spider-like structure isolator for low-frequency vibration. Applied Mathematics and Mechanics (English Edition), 44(8), 1263–1286 (2023) https://doi.org/10.1007/s10483-023-3020-9
[24]	CHEN, T. T., DING, Y., XU, Z. Y., WANG, K., ZHOU, J. X., and CHANG, Y. P. Bidirectional vibration energy harvesting and sensing system with biomimetic petal architecture. Renewable Energy, 256, 124614 (2026)
[25]	YAN, G., QI, W. H., LU, J. J., LIU, F. R., YAN, H., ZHAO, L. C., WU, Z. Y., and ZHANG, W. M. Bio-inspired multi-joint-collaborative vibration isolation. Journal of Sound and Vibration, 568, 118089 (2024)
[26]	LIU, J., ZHANG, S. D., HE, J. F., WEN, G. L., XIE, Y. M., and WIERCIGROCH, M. Origami-inspired acoustic metamaterials: real-time tunable bandgap and sound attenuation. International Journal of Mechanical Sciences, 314, 111391 (2026)
[27]	WANG, Q., ZHOU, J. X., WANG, K., XU, D. L., and WEN, G. L. Design and experimental study of a compact quasi-zero-stiffness isolator using wave springs. Science China Technological Sciences, 64(10), 2255–2271 (2021)
[28]	WU, W. T., SHAN, X. B., SUN, C. H., SUI, G. D., JIANG, T. C., and ZHANG, H. A disordered composite metamaterial beam optimized using NSGA-II: a structure with an adjustable hybrid bandgap. European Journal of Mechanics- A/Solids, 114, 105731 (2025)
[29]	SHU, Y. Q., WANG, K., CHEN, T. T., PAN, H. B., DENG, Y. P., YIN, H. F., and ZHOU, J. X. A quasi-zero-stiffness metastructure for concurrent low-frequency vibration attenuation and energy harvesting. Thin-Walled Structures, 214, 113371 (2025)
[30]	ZHANG, X. L., ZHU, W. Y., HAO, H. N., TIAN, R. L., and XIAO, Q. K. Bi-material thermally responsive metamaterials with tunable curvature and enhanced stiffness: structural design and experimental validation. Materials Today Communications, 48, 113299 (2025)
[31]	ZHAO, F., JI, J. C., YE, K., and LUO, Q. T. Increase of quasi-zero stiffness region using two pairs of oblique springs. Mechanical Systems and Signal Processing, 144, 106975 (2020)
[32]	ZENG, R., WEN, G. L., ZHOU, J. X., and ZHAO, G. Limb-inspired bionic quasi-zero stiffness vibration isolator. Acta Mechanica Sinica, 37(7), 1152–1167 (2021)
[33]	YAN, G., LU, J. J., QI, W. H., ZHAO, T. Y., YAN, H., ZHAO, L. C., WU, Z. Y., and ZHANG, W. M. Tetrahedron structure with nonlinear stiffness and inertia modulation for enhanced low frequency vibration isolation. Journal of Sound and Vibration, 564, 117897 (2023)
[34]	WU, J. L., ZENG, L. Z., HAN, B., ZHOU, Y. F., LUO, X., LI, X. Q., CHEN, X. D., and JIANG, W. Analysis and design of a novel arrayed magnetic spring with high negative stiffness for low-frequency vibration isolation. International Journal of Mechanical Sciences, 216, 106980 (2022)
[35]	WU, W. T., SUI, G. D., XIE, X. Y., SHAN, X. B., SUN, C. H., JIANG, T. C., and WANG, J. H. Nonlinear aperiodic metamaterials with tunable electromagnetic stiffness. Chaos, Solitons & Fractals, 202, 117433 (2026)
[36]	LIN, Y., OU, H. F., HE, J. F., WEN, G. L., and LIU, J. A bio-inspired metamaterial for attenuating vibration with a compression-torsion coupling effect. Mechanics of Advanced Materials and Structures, 33(1), 2569848 (2025)
[37]	ZHAO, T. Y., YAN, G., QI, W. H., LU, J. J., and ZHANG, W. M. Magnetically modulated tetrahedral structure for low frequency vibration isolation with adjustable load capacity. International Journal of Mechanical Sciences, 251, 108335 (2023)
[38]	SUI, G. D., ZHOU, C. Y., WU, W. T., DOKO, V. K., XIONG, J., and SHAN, X. B. Dual-function of isolation and energy harvesting via a spatial cam-induced multi-region quasi-zero-stiffness mechanism. Mechanical Systems and Signal Processing, 257, 114534 (2026)
[39]	LIAN, X. Y., LIU, B., DENG, H. X., and GONG, X. L. A vibration isolator with a controllable quasi-zero stiffness region based on nonlinear force design. Applied Mathematics and Mechanics (English Edition), 45(8), 1279–1294 (2024) https://doi.org/10.1007/s10483-024-3137-8
[40]	QI, W. H., YAN, G., LU, J. J., LIU, F. R., ZHAO, T. Y., YAN, H., and ZHANG, W. M. Local gravity control method for solving load-mismatch issue in isolators. International Journal of Mechanical Sciences, 265, 108891 (2024)
[41]	LU, J. J., YAN, G., QI, W. H., YAN, H., SHI, J. W., CHEN, A., and ZHANG, W. M. Load-adaptive quasi-zero stiffness vibration isolation via dual electromagnetic stiffness regulation. Journal of Sound and Vibration, 567, 118059 (2023)
[42]	ZHAO, F., JI, J. C., CAO, S. Q., YE, K., and LUO, Q. T. QZS isolators with multi-pairs of oblique bars for isolating ultralow frequency vibrations. Nonlinear Dynamics, 112(3), 1815–1842 (2024)
[43]	HUANG, X. C., LIU, X. T., and HUA, H. X. Effects of stiffness and load imperfection on the isolation performance of a high-static-low-dynamic-stiffness non-linear isolator under base displacement excitation. International Journal of Non-Linear Mechanics, 65, 32–43 (2014)
[44]	LAN, C. C., YANG, S. A., and WU, Y. S. Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads. Journal of Sound and Vibration, 333(20), 4843–4858 (2014)
[45]	JIANG, Y. L., SONG, C. S., DING, C. M., and XU, B. H. Design of magnetic-air hybrid quasi-zero stiffness vibration isolation system. Journal of Sound and Vibration, 477, 115346 (2020)
[46]	HUA, X. D., WANG, S. L., ZHANG, J. C., JIAO, G. Y., and WANG, K. Miura-origami inspired quasi-zero stiffness low-frequency vibration isolator. International Journal of Mechanical Sciences, 295, 110283 (2025)
[47]	CHEN, T. F., ZHENG, Y. X., SONG, L. H., GAO, X. M., and WANG, G. L. Study on a quasi-zero-stiffness isolator for variable mass load. Applied Mathematical Modelling, 123, 447–463 (2023)
[48]	ZHENG, Y. W., SHANGGUAN, W. B., YIN, Z. H., and LIU, X. A. Design and modeling of a quasi-zero stiffness isolator for different loads. Mechanical Systems and Signal Processing, 188, 110017 (2023)
[49]	YANG, S. J., SUN, X. T., QIAN, J. W., XU, J., and LI, K. X. Design, optimization, and realization of a magnetic multi-layer quasi-zero-stiffness isolation platform supporting different loads. Materials, 18(7), 1676 (2025)
[50]	SUN, K., TANG, J., YANG, Y. K., JIANG, B. L., LI, Y. H., and CAO, D. Q. Active control of quasi-zero-stiffness vibration isolator with variable load. International Journal of Structural Stability and Dynamics, 24(21), 2450243 (2024)
[51]	LIU, C. X., HEMMATIAN, M., SEDAGHATI, R., and WEN, G. L. Development and control of magnetorheological elastomer-based semi-active seat suspension isolator using adaptive neural network. Frontiers in Materials, 7, 171 (2020)
[52]	XU, K. F., NIU, M. Q., ZHANG, Y. W., and CHEN, L. Q. An active high-static-low-dynamic-stiffness vibration isolator with adjustable buckling beams: theory and experiment. Applied Mathematics and Mechanics (English Edition), 45(3), 425–440 (2024) https://doi.org/10.1007/s10483-024-3087-6
[53]	XU, S. F., SUN, Z. B., WANG, Y. X., WU, J. F., and SHI, Y. A semi-active adaptive pneumatic vibration isolator based on frequency response analysis and nonlinear control. Journal of Vibration and Control, 31(15-16), 3398–3410 (2025)
[54]	SU, L. C., ZHANG, C., ZHANG, X. Y., and ZHOU, M. L. Observer-based sliding mode predictive control for piezoelectric-driven active-passive vibration isolator with variable-mass payloads. IEEE Transactions on Industrial Electronics, 72(11), 11854–11864 (2025)