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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.
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Keywords:
- bio-inspired structure /
- low-frequency vibration isolation /
- nonlinear vibration /
- arthropods
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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.
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.
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.
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; (c) equivalent stiffness.
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.
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.
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).
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.
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).
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.
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).
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)
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).
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.
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.
表 1 结构参数
Table 1. Structure Parameters.
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 
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