Visitar URL original
Realization of load adaptivity with magnet array-based stair-step quasi-zero stiffness vibration isolators | Science China Technological Sciences | Springer Nature Link Skip to main content
Springer Nature Link
Log in

Realization of load adaptivity with magnet array-based stair-step quasi-zero stiffness vibration isolators

  • Research Paper
  • Published:
View saved research
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

Conventional quasi-zero stiffness (QZS) isolators are typically engineered for a specific mass or payload, and their low-frequency isolation capability deteriorates notably when the load deviates from the designated value. To address this drawback, this work proposes a magnet array-based stair-step QZS vibration isolator (MAS-QZS-VI) to achieve discrete load adaptability. The magnetic force and nonlinear magnetic stiffness of the rectangular magnet array are derived using the magnetic charge model. A parametric analysis of geometric variables is conducted to inform the design of the restoring force profile and determine the number of QZS regions. The governing equation of MAS-QZS-VI is established, and the corresponding transmissibility under different loads is derived. Numerical simulations and experimental tests are conducted to assess the load adaptability and low-frequency isolation performance of the proposed isolator. The number of QZS regions depends on the maximum magnetic negative stiffness. The designed MAS-QZS-VI can maintain considerable low-frequency vibration isolation performance when the mass is increased from 1.79 to 2.68 kg (∼49.7%). This work provides a feasible approach for realizing discrete load adaptability in stair-step QZS vibration isolator with magnet array configurations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+
from $39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

References

  1. Liu H, Mao L, Wang T, et al. Theoretical and experimental study on a dual-stage variable stiffness friction damper for satellite flywheel. Mech Syst Signal Process, 2023, 185: 109761

    Article  Google Scholar 

  2. Wan M, Shen C J, Qu X, et al. Chatter stability of the robotic milling process involving the influences of low frequency vibrations in three directions. Mech Syst Signal Process, 2025, 224: 112014

    Article  Google Scholar 

  3. Wang M, Hu Y, Sun Y, et al. An adjustable low-frequency vibration isolation stewart platform based on electromagnetic negative stiffness. Int J Mech Sci, 2020, 181: 105714

    Article  Google Scholar 

  4. Yu Y, Naganathan N G, Dukkipati R V. A literature review of automotive vehicle engine mounting systems. Mech Mach Theory, 2001, 36: 123–142

    Article  Google Scholar 

  5. Liu C, Jing X, Daley S, et al. Recent advances in micro-vibration isolation. Mech Syst Signal Process, 2015, 56–57: 55–80

    Article  Google Scholar 

  6. Ma H, Xu P, Fan W, et al. Bionic Issus-inspired gear-driven structure with adjustable stiffness and rotational inertia properties for low-frequency vibration isolation. Mech Syst Signal Process, 2025, 239: 113331

    Article  Google Scholar 

  7. Yan G, Zou H X, Wang S, et al. Large stroke quasi-zero stiffness vibration isolator using three-link mechanism. J Sound Vib, 2020, 478: 115344

    Article  Google Scholar 

  8. Chai Y, Jing X, Chao X. X-shaped mechanism based enhanced tunable QZS property for passive vibration isolation. Int J Mech Sci, 2022, 218: 107077

    Article  Google Scholar 

  9. Liu C, Wang Y, Zhang W, et al. Nonlinear dynamics of a magnetic vibration isolator with higher-order stable quasi-zero-stiffness. Mech Syst Signal Process, 2024, 218: 111584

    Article  Google Scholar 

  10. Zhou J, Xu D, Bishop S. A torsion quasi-zero stiffness vibration isolator. J Sound Vib, 2015, 338: 121–133

    Article  Google Scholar 

  11. Carrella A, Brennan M J, Waters T P. Optimization of a quasi-zero-stiffness isolator. J Mech Sci Technol, 2007, 21: 946–949

    Article  Google Scholar 

  12. Carrella A, Brennan M J, Kovacic I, et al. On the force transmissibility of a vibration isolator with quasi-zero-stiffness. J Sound Vib, 2009, 322: 707–717

    Article  Google Scholar 

  13. Carrella A, Brennan M J, Waters T P. Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J Sound Vib, 2007, 301: 678–689

    Article  Google Scholar 

  14. Zhang Y, Cao Q, Huang W. Bursting oscillations in an isolation system with quasi-zero stiffness. Mech Syst Signal Process, 2021, 161: 107916

    Article  Google Scholar 

  15. Liu C, Yu K. Accurate modeling and analysis of a typical nonlinear vibration isolator with quasi-zero stiffness. Nonlinear Dyn, 2020, 100: 2141–2165

    Article  Google Scholar 

  16. Zhou J, Wang X, Xu D, et al. Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam-roller-spring mechanisms. J Sound Vib, 2015, 346: 53–69

    Article  Google Scholar 

  17. Mao X, Yin M, Ding H, et al. Modeling, analysis, and simulation of X-shape quasi-zero-stiffness-roller vibration isolators. Appl Math Mech-Engl Ed, 2022, 43: 1027–1044

    Article  MathSciNet  Google Scholar 

  18. Li M, Cheng W, Xie R. A quasi-zero-stiffness vibration isolator using a cam mechanism with user-defined profile. Int J Mech Sci, 2021, 189: 105938

    Article  Google Scholar 

  19. Zuo S, Wang D, Zhang Y, et al. An innovative design of parabolic cam-roller quasi-zero-stiffness isolators for ultralow frequency vibration isolation. Nonlinear Dyn, 2024, 112: 18717–18744

    Article  Google Scholar 

  20. Ling P, Miao L, Zhang W, et al. Cockroach-inspired structure for low-frequency vibration isolation. Mech Syst Signal Process, 2022, 171: 108955

    Article  Google Scholar 

  21. Ling P, Miao L, Ye B, et al. Ultra-low frequency vibration isolation of a novel click-beetle-inspired structure with large quasi-zero stiffness region. J Sound Vib, 2023, 558: 117756

    Article  Google Scholar 

  22. Duan X, Ye J, Huang D, et al. Investigation of dynamical behaviors of bionic paw-inspired vibration isolators by theoretical and numerical analysis. Int J Dynam Control, 2025, 13: 399

    Article  Google Scholar 

  23. Yan G, Wang S, Zou H X, et al. Bio-inspired polygonal skeleton structure for vibration isolation: Design, modelling, and experiment. Sci China Tech Sci, 2020, 63: 2617–2630

    Article  Google Scholar 

  24. Yan B, Wang S, Ling P, et al. Seahorse-exoskeleton-inspired structure with linear-to-torsion transition property for low-frequency vibration isolation. Fundam Res, 2025, 5: 1505–1523

    Article  Google Scholar 

  25. Feng X, Jing X. Human body inspired vibration isolation: Beneficial nonlinear stiffness, nonlinear damping & nonlinear inertia. Mech Syst Signal Process, 2019, 117: 786–812

    Article  Google Scholar 

  26. Xu D, Yu Q, Zhou J, et al. Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. J Sound Vib, 2013, 332: 3377–3389

    Article  Google Scholar 

  27. Wang Q, Zhou J X, Wang K, et al. Design and experimental study of a compact quasi-zero-stiffness isolator using wave springs. Sci China Tech Sci, 2021, 64: 2255–2271

    Article  Google Scholar 

  28. Yuan S J, Wu Y J, Song D P, et al. High-static-low-dynamic stiffness isolator based on an electromagnetic negative stiffness spring with long linear stroke. Sci China Tech Sci, 2024, 67: 740–752

    Article  Google Scholar 

  29. Guo M, Tang L, Wang H, et al. A comparative study on transient vibration suppression of magnetic nonlinear vibration absorbers with different arrangements. Nonlinear Dyn, 2023, 111: 16729–16776

    Article  Google Scholar 

  30. Wu J L, Che J X, Chen X D, et al. Design of a combined magnetic negative stiffness mechanism with high linearity in a wide working region. Sci China Tech Sci, 2022, 65: 2127–2142

    Article  Google Scholar 

  31. Ma Z, Zhou R, Yang Q, et al. A semi-active electromagnetic quasi-zero-stiffness vibration isolator. Int J Mech Sci, 2023, 252: 108357

    Article  Google Scholar 

  32. Virgin L N, Davis R B. Vibration isolation using buckled struts. J Sound Vib, 2003, 260: 965–973

    Article  Google Scholar 

  33. Liu X, Huang X, Hua H. On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. J Sound Vib, 2013, 332: 3359–3376

    Article  Google Scholar 

  34. Xu J, Yang X, Li W, et al. Design of quasi-zero stiffness joint actuator and research on vibration isolation performance. J Sound Vib, 2020, 479: 115367

    Article  Google Scholar 

  35. Li M, Li Y, Liu X, et al. A quasi-zero-stiffness vibration isolator using bi-stable hybrid symmetric laminate. Compos Struct, 2022, 299: 116047

    Article  Google Scholar 

  36. Zhang C, He J, Zhou G, et al. Compliant quasi-zero-stiffness isolator for low-frequency torsional vibration isolation. Mech Mach Theory, 2023, 181: 105213

    Article  Google Scholar 

  37. Liu J, Wang Y, Yang S, et al. Customized quasi-zero-stiffness metamaterials for ultra-low frequency broadband vibration isolation. Int J Mech Sci, 2024, 269: 108958

    Article  Google Scholar 

  38. Liu W, Yan S, Meng Z, et al. Programmable quasi-zero-stiffness metamaterials. Engineering, 2024, 47: 160–167

    Article  Google Scholar 

  39. Liu S, Peng G, Jin K. Design and characteristics of a novel QZS vibration isolation system with origami-inspired corrector. Nonlinear Dyn, 2021, 106: 255–277

    Article  Google Scholar 

  40. Liu S, Peng G, Jin K. Towards accurate modeling of the Tachi-Miura origami in vibration isolation platform with geometric nonlinear stiffness and damping. Appl Math Model, 2022, 103: 674–695

    Article  MathSciNet  Google Scholar 

  41. Song X, Ding J, Sun Y, et al. A hybrid active control approach for the attenuation of low-frequency vibrations and multi-narrowband disturbances. Mech Syst Signal Process, 2026, 249: 114096

    Article  Google Scholar 

  42. Liu C, Zhao R, Yu K, et al. In-plane quasi-zero-stiffness vibration isolator using magnetic interaction and cables: Theoretical and experimental study. Appl Math Model, 2021, 96: 497–522

    Article  MathSciNet  Google Scholar 

  43. Ji J C, Luo Q, Ye K. Vibration control based metamaterials and origami structures: A state-of-the-art review. Mech Syst Signal Process, 2021, 161: 107945

    Article  Google Scholar 

  44. Chen T, Zheng Y, Song L, et al. Study on a quasi-zero-stiffness isolator for variable mass load. Appl Math Model, 2023, 123: 447–463

    Article  MathSciNet  Google Scholar 

  45. Demir M U, Yilmaz C. Realization of a wideband three-axis horizontal vibration isolator with adjustable stiffness and damping. J Sound Vib, 2025, 600: 118876

    Article  Google Scholar 

  46. Mo X C, Li K X, Fan Z W, et al. Design and analysis of a quasi-zero stiffness vibration isolation system with load adaptability. Results Eng, 2025, 28: 107778

    Article  Google Scholar 

  47. Ba H, Niu M Q, Chen L Q. An adjustable stiffness vibration isolator implemented by a semicircular ring. Mech Syst Signal Process, 2025, 222: 111797

    Article  Google Scholar 

  48. Yu N, Tian F, Ma H, et al. Tunable quasi-zero stiffness vibration isolators with electromagnetic damping for low-frequency vibration isolation enhancement. Mech Syst Signal Process, 2025, 237: 113082

    Article  Google Scholar 

  49. Yan G, Lu J J, Qi W H, et al. Tetrahedron structure with nonlinear stiffness and inertia modulation for enhanced low frequency vibration isolation. J Sound Vib, 2023, 564: 117897

    Article  Google Scholar 

  50. Chen R, Li X, Tian J, et al. On the displacement transferability of variable stiffness multi-directional low frequency vibration isolation joint. Appl Math Model, 2022, 112: 690–707

    Article  MathSciNet  Google Scholar 

  51. Zhao T Y, Yan G, Qi W H, et al. Magnetically modulated tetrahedral structure for low frequency vibration isolation with adjustable load capacity. Int J Mech Sci, 2023, 251: 108335

    Article  Google Scholar 

  52. Lan C C, Yang S A, Wu Y S. Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads. J Sound Vib, 2014, 333: 4843–4858

    Article  Google Scholar 

  53. Yan G, Qi W H, Shi J W, et al. Bionic paw-inspired structure for vibration isolation with novel nonlinear compensation mechanism. J Sound Vib, 2022, 525: 116799

    Article  Google Scholar 

  54. Yan G, Wu Z Y, Wei X S, et al. Nonlinear compensation method for quasi-zero stiffness vibration isolation. J Sound Vib, 2022, 523: 116743

    Article  Google Scholar 

  55. Ye K, Ji J C, Brown T. Design of a quasi-zero stiffness isolation system for supporting different loads. J Sound Vib, 2020, 471: 115198

    Article  Google Scholar 

  56. Zhou J, Zhou J, Pan H, et al. Multi-layer quasi-zero-stiffness meta-structure for high-efficiency vibration isolation at low frequency. Appl Math Mech-Engl Ed, 2024, 45: 1189–1208

    Article  MathSciNet  Google Scholar 

  57. Zheng Y, Shangguan W B, Yin Z, et al. Design and modeling of a quasi-zero stiffness isolator for different loads. Mech Syst Signal Process, 2023, 188: 110017

    Article  Google Scholar 

  58. Liu L, Chai Y, Guo Z, et al. A novel isolation system with enhanced QZS properties for supporting multiple loads. Aerosp Sci Technol, 2023, 143: 108719

    Article  Google Scholar 

  59. Meng Q, Hou L, Lin R, et al. On a quasi-zero stiffness vibration isolator with multiple zero stiffness points for mass load deviation. Appl Math Model, 2025, 145: 116112

    Article  Google Scholar 

  60. Zeng C, Liu L, Hu Y, et al. Stair-stepping mechanical metamaterials with programmable load plateaus. Adv Funct Mater, 2024, 34: 2408887

    Article  Google Scholar 

  61. Yang S, Sun X, Qian J, et al. Design, optimization, and realization of a magnetic multi-layer quasi-zero-stiffness isolation platform supporting different loads. Materials, 2025, 18: 1676

    Article  Google Scholar 

  62. Zhang Y, Zhu G, Cao Q. Isolation performances and optimization of triple quasi-zero stiffness isolators. Sci China-Phys Mech Astron, 2024, 67: 274511

    Article  Google Scholar 

  63. An Y, Hu J, Ge M, et al. Design and analysis of gradient quasi-zero stiffness metamaterial isolator with continuous stepped plateau for anti-perturbations. Nonlinear Dyn, 2025, 113: 163–187

    Article  Google Scholar 

  64. Ou H, Zeng J, Lin M, et al. Stair-stepping metamaterial and the derived isolator for all-DOF vibrations with multi-payloads. Eng Struct, 2025, 330: 119890

    Article  Google Scholar 

  65. Wang J, Yao G. A multiple layer enhanced quasi-zero stiffness vibration isolator with magnetically modulated nonlinear stiffness. Aerosp Sci Technol, 2025, 161: 110113

    Article  Google Scholar 

  66. Wu P, Li S, Li Z, et al. Modular QZS metamaterials with enhanced load adaptability for low-frequency vibration isolation. Smart Mater Struct, 2025, 34: 055017

    Article  Google Scholar 

  67. Liu X, Chen S, Tan X, et al. A mechanical metamaterial with programmable arbitrary quasi-zero-stiffness regions. Compos Part A-Appl Sci Manuf, 2025, 198: 109076

    Article  Google Scholar 

  68. Lu J J, Yan G, Qi W H, et al. Integrated vibration isolation and actuation via dual nonlinear stiffness regulation. Int J Mech Sci, 2023, 263: 108760

    Article  Google Scholar 

  69. Lu J J, Yan G, Qi W H, et al. Load-adaptive quasi-zero stiffness vibration isolation via dual electromagnetic stiffness regulation. J Sound Vib, 2023, 567: 118059

    Article  Google Scholar 

  70. Qi W H, Yan G, Lu J J, et al. Local gravity control method for solving load-mismatch issue in isolators. Int J Mech Sci, 2024, 265: 108891

    Article  Google Scholar 

  71. Gao X, Teng H D. Dynamics and isolation properties for a pneumatic near-zero frequency vibration isolator with nonlinear stiffness and damping. Nonlinear Dyn, 2020, 102: 2205–2227

    Article  Google Scholar 

  72. Cheng Z, Ma R, Bi K, et al. An in-situ adjustable arrayed magnetic negative stiffness device: Analytical modelling, experimental testing, and parametric study. Eng Struct, 2025, 328: 119676

    Article  Google Scholar 

  73. Yuan J, Zhou Y, Zhang Y, et al. A large-stroke and adjustable-load magnetic quasi-zero stiffness isolator. Mech Syst Signal Process, 2025, 239: 113326

    Article  Google Scholar 

  74. Liu W, Wu J, Zheng J, et al. Nonlinear dynamic characteristics of quasi-zero stiffness cruciform maglev isolators. J Vib Eng Technol, 2025, 13: 470

    Article  Google Scholar 

  75. Wu J, Zeng L, Han B, et al. Analysis and design of a novel arrayed magnetic spring with high negative stiffness for low-frequency vibration isolation. Int J Mech Sci, 2022, 216: 106980

    Article  Google Scholar 

  76. Wu M, Wu J, Che J, et al. Analysis and experiment of a novel compact magnetic spring with high linear negative stiffness. Mech Syst Signal Process, 2023, 198: 110387

    Article  Google Scholar 

  77. Allag H, Yonnet J P. 3-D analytical calculation of the torque and force exerted between two cuboidal magnets. IEEE Trans Magn, 2009, 45: 3969–3972

    Article  Google Scholar 

  78. Yu N, Fei X, Sun H, et al. Rhombus-type magnetic-levitation structure for low-frequency vibration isolation. Mech Syst Signal Process, 2025, 225: 112289

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52422504, U2541231, U23A20615) and Zhejiang Natural Science Foundation (Grant Nos. LZ26E050004, LQN25A020002).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Fanghao Tian, Ning Yu, Qinchuan Li & Bo Yan

  2. Zhejiang Key Laboratory of Reliability Technology for Mechanical & Electrical Product, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Bo Yan

Authors
  1. Fanghao Tian
    View author publications

    Search author on:PubMed Google Scholar

  2. Ning Yu
    View author publications

    Search author on:PubMed Google Scholar

  3. Qinchuan Li
    View author publications

    Search author on:PubMed Google Scholar

  4. Bo Yan
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Bo Yan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, F., Yu, N., Li, Q. et al. Realization of load adaptivity with magnet array-based stair-step quasi-zero stiffness vibration isolators. Sci. China Technol. Sci. 69, 2120302 (2026). https://doi.org/10.1007/s11431-026-3434-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Version of record:

  • DOI: https://doi.org/10.1007/s11431-026-3434-x

Keywords