Design and dynamic mechanism of M-shaped low-frequency isolation structure imitating limb configuration of arthropods

LAN Chunbo; JIA Jie; WANG Yang; WANG Shuo; ZHANG Lu

doi:10.7498/aps.74.20241203

Abstract

Arthropods, including spiders and mantises, can maintain their body stability on shaking surfaces, such as spiderwebs or leaves. This impressive stability can be attributed to the specific geometric shape of their limbs, which exhibit an M-shaped structure. Inspired by this geometry, this work proposes an arthropod-limb-inspired M-shaped structure for low-frequency vibration isolation. First, the design method of the M-shaped quasi-zero-stiffness (QZS) structure is presented. A static analysis of potential energy, restoring force, and equivalent stiffness is conducted, showing that the M-shaped structure enables a horizontal linear spring to generate nonlinear stiffness in the vertical direction. More importantly, this nonlinear stiffness effectively compensates for the negative stiffness in large-displacement responses, thereby achieving a wider quasi-zero-stiffness region than the conventional three-spring-based QZS structure. Subsequently, the harmonic balance method is employed to derive approximate analytical solutions for the M-shaped QZS structure, which are well validated through numerical simulation. A comparison between the proposed M-shaped QZS structure and the conventional three-spring-based QZS structure is performed. Results show that the M-shaped QZS structure is advantageous for reducing both the cut-in isolation frequency and the resonance frequency. In particular, under large excitation or small damping conditions, the performance improvement of the M-shaped QZS structure in terms of reducing the resonance frequency and maximum response becomes more pronounced. The underlying mechanism behind this feature is primarily attributed to the expanded QZS region induced by the M-shaped structure. Finally, since the M-shaped structures vary among different arthropods, the effect of the geometry of M-shaped structures on low-frequency vibration performance is investigated. Interestingly, a trade-off between vibration isolation performance and loading mass is observed. As the M-shaped structure becomes flatter and the QZS region expands, the cut-in isolation frequency, resonance frequency/peak, and loading mass all decrease. This occurs because a flatter M-shaped structure leads to a reduction in the equivalent stiffness generated by the horizontal stiffness. Therefore, as the loading mass capacity decreases, the low-frequency vibration isolation performance is enhanced. This novel finding provides a reasonable explanation for why most arthropods possess many pairs of limbs, allowing the loading mass to be distributed while achieving excellent low-frequency vibration isolation.

Keywords:

Authors and contacts

College of Aerospance Enegineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Corresponding author: LAN Chunbo, chunbolan@nuaa.edu.cn

Funds: Project supported by the Natural Science Foundation of China (Grant No. 12002152), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190379), the General Programs of China Postdoctoral Science Foundation (Grant No. 2020M681577), the Funds of National Key Laboratory of Helicopter Aeromechanics (Grant No. 2024-CXPT-GF-JJ-093-04), and the Fundamental Research Funds for the Central Universities (Grant No. NT2024002).

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References

[1]	Jiao X L, Zhang J X, Li W B, Wang Y Y, Ma W L, Zhao Y 2023 Prog. Aerosp. Sci. 138 100898 Google Scholar
[2]	Li L, Wang L, Yuan L, Zheng R, Wu Y P, Sui J, Zhong J 2021 Acta Astronaut. 180 417 Google Scholar
[3]	Dennehy C J, Wolf A A, Swanson D K Spacecraft Line-of-Sight Jitter Management and Mitigation Lessons Learned and Engineering Best Practices https://ntrs.nasa.gov/citations/20210017871 [2025-05-07]
[4]	孟光, 董瑶海, 周徐斌, 申军烽, 刘兴天 2019 中国科学: 物理学力学天文学 49 024508 Google Scholar Meng G, Dong Y H, Zhou X B, Shen J F, Liu X T 2019 Sci. Sin-Phys. Mech. Astron. 49 024508 Google Scholar
[5]	McPherson K, Hrovat K, Kelly E, Keller J 2015 A Researcher’s Guide to Acceleration Environment on the International Space Station (Washington, D. C.: NASA) pp37–41
[6]	刘海平, 张世乘, 门玲鸰, 何振强 2022 物理学报 71 160701 Google Scholar Liu H P, Zhang S C, Men L L, He Z Q 2022 Acta Phys. Sin. 71 160701 Google Scholar
[7]	Luo H T, Fan C H, Li Y X, Liu G M, Yu C S 2023 Eur. J. Mech. A-Solid. 97 104833 Google Scholar
[8]	Molyneux W G 1958 Aircr. Eng. Aerosp. Tec. 30 160 Google Scholar
[9]	Carrella A, Brennan M J, Waters T P 2007 J. Sound Vib. 301 678 Google Scholar
[10]	张月英 2014 硕士学位论文 (湖南: 湖南大学) Zhang Y Y 2014 M. S. Thesis (Hunan: Hunan University
[11]	Kovacic I, Brennan M J, Waters T P 2008 J. Sound Vib. 315 700 Google Scholar
[12]	Zhou J X, Wang X L, Xu D L, Bishop S 2015 J. Sound Vib. 346 53 Google Scholar
[13]	Zhao F, Ji J C, Ye K, Luo Q T 2020 Mech. Syst. Signal Pr. 144 106975 Google Scholar
[14]	Deng T C, Wen G L, Ding H, Lu Z Q, Chen L Q 2020 Mech. Syst. Signal Pr. 145 106967 Google Scholar
[15]	Liu C R, Yu K P, Liao B P, Hu R P 2021 Commun. Nonlinear Sci. Numer. Simulat. 95 105654 Google Scholar
[16]	郝志峰 2016 博士学位论文 (哈尔滨: 哈尔滨工业大学) Hao Z F 2016 Ph. D. Dissertation (Harbin: Harbin Institute of Technology
[17]	阮子悦 2023 硕士学位论文 (石家庄: 石家庄铁道大学) Ran Z R 2023 M. S. Thesis (Shijiazhuang: Shijiazhuang TieDao University
[18]	Yan B, Yu N, Wang Z H, Wu C Y, Wang S, Zhang W M 2022 J. Sound Vib. 527 116865 Google Scholar
[19]	Liu X T, Huang X C, Hua H X 2013 J. Sound Vib. 332 3359 Google Scholar
[20]	Wang K, Zhou J X, Chang Y P, Ouyang H J, Xu D L, Yang Y 2020 Nonlinear Dynam. 101 755 Google Scholar
[21]	Wu W J, Chen X D, Shan Y H 2014 J. Sound Vib. 333 2958 Google Scholar
[22]	安隽翰 2021 硕士学位论文 (南京: 南京航空航天大学) An J H 2021 M. S. Thesis (Nanjing: Nanjing University of Aeronautics and Astronautics
[23]	Yan G, Zou H X, Wang S, Zhao L C, Wu Z Y, Zhang W M 2021 Appl. Mech. Rev. 73 020801 Google Scholar
[24]	Wu Z J, Jing X J, Bian J, Li F M, Allen R 2015 Bioinspir. Biomim. 10 056015 Google Scholar
[25]	Yan G, Wang S, Zou H X, Zhao L C, Gao Q H, Zhang W M 2020 Sci. China Tech. Sci. 63 2617 Google Scholar
[26]	Shi X J, Xu J, Chen T K, Qian C, Tian W J 2023 J. Bionic Eng. 20 2194 Google Scholar
[27]	Zeng R, Wen G L, Zhou J X, Zhao G 2021 Acta Mech. Sinica-PRC 37 1152 Google Scholar
[28]	Yan G, Zou H X, Wang S, Zhao L C, Wu Z Y, Zhang W M 2022 Mech. Syst. Signal Pr. 162 108010 Google Scholar
[29]	Jin G X, Wang Z H, Yang T Z 2022 Appl. Math. Mech. -Engl. Ed. 43 813 Google Scholar
[30]	Ling P, Miao L L, Zhang W M, Wu C Y, Yan B 2022 Mech. Syst. Signal Pr. 171 108955 Google Scholar
[31]	Long S M, Leonard A, Carey A, Jakob E M 2015 J. Arachnol. 43 111 Google Scholar

Cited By

图 1 蜘蛛和螳螂等节肢动物腿部的几何形状与仿生M形结构

Figure 1. Geometries of arthropods’ limbs and the bio-inspired M-shaped structure.

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图 2 仿生M形结构模型图　(a)无负载时的静平衡状态; (b)有负载时的静平衡状态; (c)运动状态

Figure 2. Bio-inspired M-shaped structure: (a) Static equilibrium without mass; (b) static equilibrium with mass; (c) oscillating state.

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图 3 M形准零刚度结构的静态特性　(a)势能; (b)等效恢复力; (c)等效刚度

Figure 3. Static characteristics of MQZS structure: (a) Potential energy; (b) equivalent restoring force; (c) equivalent stiffness.

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图 4 M形准零刚度结构(MQZS)与三弹簧准零刚度结构(QZS)静态特性对比　(a)等效恢复力; (b)等效刚度

Figure 4. Static characteristics comparison between MQZS and QZS: (a) Equivalent restoring force; (b) equivalent stiffness.

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图 5 等效恢复力的解析解与多项式拟合结果对比

Figure 5. Comparison of analytical solution and fitted results of the restoring force.

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图 6 系统频率响应的解析解与数值解对比

Figure 6. Comparison between analytical solution and numerical simulation.

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图 7 M形仿生结构的激励幅值分岔图(f = 2.2 Hz)

Figure 7. Bifurcation diagram of MQZS induced by excitation amplitude (f = 2.2 Hz).

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图 8 不同激励幅值下系统的响应(f = 2.2 Hz)　(a), (d), (g)时域图; (b), (e), (h)频域图; (c), (f), (i)相图

Figure 8. Dynamic responses at different excitations (f = 2.2 Hz): (a), (d), (g) Time-history response; (b), (e), (h) spectrum responses; (c), (f), (i) phase diagrams.

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图 9 激励频率变化时系统分岔图(A = 2.0 m/s²)

Figure 9. Bifurcation diagram as the excitation frequency changes (A = 2.0 m/s²).

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图 10 不同激励频率下系统的响应(A = 2 m/s²)　(a), (d), (g)时域图; (b), (e), (h)频域图; (c), (f), (i)相图

Figure 10. Dynamic responses at different excitation frequencies (A = 2 m/s²): (a), (d), (g) Time-history response; (b), (e), (h) spectrum responses; (c), (f), (i) phase diagrams.

DownLoad: Full-Size Img PowerPoint

图 11 MQZS结构与三弹簧QZS结构传递率对比(A = 1.0 m/s², c = 0.1 N·s/m)

Figure 11. Transmittance comparison between MQZS and QZS (A = 1.0 m/s², c = 0.1 N·s/m).

DownLoad: Full-Size Img PowerPoint

图 12 不同激励幅值下MQZS和传统QZS系统传递率特性对比图(c = 0.1 N·s/m)

Figure 12. Transmittances comparison of MQZS and QZS under different excitations (c = 0.1 N·s/m)

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图 13 不同阻尼下MQZS和传统QZS系统传递率特性对比图(A = 1.0 m/s²)

Figure 13. Transmittances comparison of MQZS and QZS with different damping (A = 1.0 m/s²).

DownLoad: Full-Size Img PowerPoint

图 14 参数s对MQZS等效刚度的影响　(a)等效刚度; (b)准零刚度范围; (c)负载质量

Figure 14. Effect of s on the equivalent stiffness of MQZS: (a) Equivalent stiffness; (b) quasi-zero-stiffness region; (c) mass.

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图 15 参数s对MQZS传递率特性的影响规律

Figure 15. Effect of s on the transmittance of MQZS.

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表 1 M形结构的参数

Table 1. Parameters of M-shaped structure.

MQZS结构参数	取值	三弹簧QZS结构参数	取值
水平间距 s/mm	120	水平间距 s₁/mm	120
杆长 l/mm	80	初始高度 h₁/mm	30
初始高度 h/mm	30	斜弹簧刚度 k₁₁/(N·m^–1)	400
斜弹簧刚度 k₁/(N·m^–1)	400	垂直弹簧刚度 k₁₂/(N·m^–1)	24.62

DownLoad: CSV

[1]	Jiao X L, Zhang J X, Li W B, Wang Y Y, Ma W L, Zhao Y 2023 Prog. Aerosp. Sci. 138 100898 Google Scholar
[2]	Li L, Wang L, Yuan L, Zheng R, Wu Y P, Sui J, Zhong J 2021 Acta Astronaut. 180 417 Google Scholar
[3]	Dennehy C J, Wolf A A, Swanson D K Spacecraft Line-of-Sight Jitter Management and Mitigation Lessons Learned and Engineering Best Practices https://ntrs.nasa.gov/citations/20210017871 [2025-05-07]
[4]	孟光, 董瑶海, 周徐斌, 申军烽, 刘兴天 2019 中国科学: 物理学力学天文学 49 024508 Google Scholar Meng G, Dong Y H, Zhou X B, Shen J F, Liu X T 2019 Sci. Sin-Phys. Mech. Astron. 49 024508 Google Scholar
[5]	McPherson K, Hrovat K, Kelly E, Keller J 2015 A Researcher’s Guide to Acceleration Environment on the International Space Station (Washington, D. C.: NASA) pp37–41
[6]	刘海平, 张世乘, 门玲鸰, 何振强 2022 物理学报 71 160701 Google Scholar Liu H P, Zhang S C, Men L L, He Z Q 2022 Acta Phys. Sin. 71 160701 Google Scholar
[7]	Luo H T, Fan C H, Li Y X, Liu G M, Yu C S 2023 Eur. J. Mech. A-Solid. 97 104833 Google Scholar
[8]	Molyneux W G 1958 Aircr. Eng. Aerosp. Tec. 30 160 Google Scholar
[9]	Carrella A, Brennan M J, Waters T P 2007 J. Sound Vib. 301 678 Google Scholar
[10]	张月英 2014 硕士学位论文 (湖南: 湖南大学) Zhang Y Y 2014 M. S. Thesis (Hunan: Hunan University
[11]	Kovacic I, Brennan M J, Waters T P 2008 J. Sound Vib. 315 700 Google Scholar
[12]	Zhou J X, Wang X L, Xu D L, Bishop S 2015 J. Sound Vib. 346 53 Google Scholar
[13]	Zhao F, Ji J C, Ye K, Luo Q T 2020 Mech. Syst. Signal Pr. 144 106975 Google Scholar
[14]	Deng T C, Wen G L, Ding H, Lu Z Q, Chen L Q 2020 Mech. Syst. Signal Pr. 145 106967 Google Scholar
[15]	Liu C R, Yu K P, Liao B P, Hu R P 2021 Commun. Nonlinear Sci. Numer. Simulat. 95 105654 Google Scholar
[16]	郝志峰 2016 博士学位论文 (哈尔滨: 哈尔滨工业大学) Hao Z F 2016 Ph. D. Dissertation (Harbin: Harbin Institute of Technology
[17]	阮子悦 2023 硕士学位论文 (石家庄: 石家庄铁道大学) Ran Z R 2023 M. S. Thesis (Shijiazhuang: Shijiazhuang TieDao University
[18]	Yan B, Yu N, Wang Z H, Wu C Y, Wang S, Zhang W M 2022 J. Sound Vib. 527 116865 Google Scholar
[19]	Liu X T, Huang X C, Hua H X 2013 J. Sound Vib. 332 3359 Google Scholar
[20]	Wang K, Zhou J X, Chang Y P, Ouyang H J, Xu D L, Yang Y 2020 Nonlinear Dynam. 101 755 Google Scholar
[21]	Wu W J, Chen X D, Shan Y H 2014 J. Sound Vib. 333 2958 Google Scholar
[22]	安隽翰 2021 硕士学位论文 (南京: 南京航空航天大学) An J H 2021 M. S. Thesis (Nanjing: Nanjing University of Aeronautics and Astronautics
[23]	Yan G, Zou H X, Wang S, Zhao L C, Wu Z Y, Zhang W M 2021 Appl. Mech. Rev. 73 020801 Google Scholar
[24]	Wu Z J, Jing X J, Bian J, Li F M, Allen R 2015 Bioinspir. Biomim. 10 056015 Google Scholar
[25]	Yan G, Wang S, Zou H X, Zhao L C, Gao Q H, Zhang W M 2020 Sci. China Tech. Sci. 63 2617 Google Scholar
[26]	Shi X J, Xu J, Chen T K, Qian C, Tian W J 2023 J. Bionic Eng. 20 2194 Google Scholar
[27]	Zeng R, Wen G L, Zhou J X, Zhao G 2021 Acta Mech. Sinica-PRC 37 1152 Google Scholar
[28]	Yan G, Zou H X, Wang S, Zhao L C, Wu Z Y, Zhang W M 2022 Mech. Syst. Signal Pr. 162 108010 Google Scholar
[29]	Jin G X, Wang Z H, Yang T Z 2022 Appl. Math. Mech. -Engl. Ed. 43 813 Google Scholar
[30]	Ling P, Miao L L, Zhang W M, Wu C Y, Yan B 2022 Mech. Syst. Signal Pr. 171 108955 Google Scholar
[31]	Long S M, Leonard A, Carey A, Jakob E M 2015 J. Arachnol. 43 111 Google Scholar

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