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Palm petiole inspired nonlinear anti-vibration ring with deformable crescent-shaped cross-section

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Abstract

This paper presents a novel nonlinear anti-vibration ring with deformable crescent-shaped cross-sections (NAVR-DCCS) inspired by the petiole of palm leaf. The proposed NAVR-DCCS exhibits markedly enhanced nonlinear quasi-zero stiffness through deformable cross-sections, which endow it with advantageous vibration isolation attributes. A comprehensive investigation of the structural nonlinearities and dynamic behaviors of the NAVR-DCCS is undertaken, with emphasis on the principle of cross-sectional deformation and its nonlinear stiffness properties. This study explores the influence of pertinent parameters on the nonlinear characteristics and displacement transmissibility. Tensile-compression testing and transmissibility measurements are conducted to verify theoretical calculations, and the experimental results are found to be in congruity with theoretical predictions. The beneficial nonlinear characteristics of the NAVR-DCCS hold promise for providing a passive vibration isolation methodology, representing a potentially innovative solution with broad-reaching applicability and utility across diverse research domains.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

\(A\) :

Amplitude of maximum acceleration (m/s2)

\(b\) :

Zeroth harmonic term (mm)

\(c\) :

Damping coefficient (Ns/m)

deg:

Degrees (−)

\(E\) :

Young’s modulus (MPa)

Eq:

Equivalent (−)

\(f_{k}\) :

Geometrically nonlinear elastic force (N)

Fig.:

Figure (−)

FDM:

Fused deposition modeling (−)

\(g\) :

Acceleration of gravity (m/s2)

\(H\) :

Thickness of the crescent-shaped cross section (mm)

Hz:

Hertz (−)

HB:

Harmonic balance (−)

HBM:

Harmonic balance method (−)

HSLDS:

High-static-low-dynamic-stiffness (−)

\(H(\omega )\) :

The transmissibility from the base to the platform (−)

\(k_{1}\) :

Stiffness coefficient of ring (N/m)

\(k_{2}\) :

Stiffness coefficient of ring (N/m3)

\(l\) :

The distance between the centers of the inner and outer arcs of a crescent-shaped cross section (mm)

\(L\) :

Lagrangian (−)

\(m\) :

Payload mass (kg)

mm:

Millimeter (−)

\(N\) :

Newton (−)

NAVR-DCCS:

Nonlinear anti-vibration ring with a deformable crescent-shaped cross-section (−)

\(O\) :

Center of the circle (−)

\(O_{1}\) :

The center of the outer arc of the crescent-shaped cross section (−)

\(O_{2}\) :

The center of the inner arc of the crescent-shaped cross section (−)

\(P_{zz} (\omega )\) :

The power spectral density of the excitation signal (g2/Hz)

\(P_{zy} (\omega )\) :

The cross-spectral density of the excitation-response signal (g2/Hz)

QZS:

Quasi-zero stiffness (−)

\(r_{1}\) :

The radius of the outer arc of the crescent-shaped cross section (mm)

\(r_{2}\) :

The radius of the inner arc of the crescent-shaped cross section (mm)

\(R\) :

Radius of the ring (mm)

\(s\) :

Second (−)

SODF:

Single-degree-of-freedom (−)

\(t\) :

Time (s)

\(T\) :

Kinetic energy (J)

\(T_{d}\) :

Displacement transmissibility (−)

TPU:

Thermoplastic polyurethane (−)

\(V\) :

Potential energy (J)

\(W\) :

Width of the crescent-shaped cross section (mm)

\(x\) :

Displacement of payload mass relative to base (mm)

\(x_{0}\) :

Amplitude (mm)

\(y\) :

Absolute displacements of the payload mass (mm)

\(\ddot{y}\) :

Excitation acceleration amplitude applied at the payload mass (m/s2)

\(z\) :

Absolute displacements of the base (mm)

\(z_{0}\) :

Amplitude (mm)

\(\theta\) :

The half of the circular angle of the sectoral section (deg)

\(\omega\) :

Shaker driving frequency (rad/s)

\(\phi\) :

Phase response (rad)

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Funding

This work was supported in part by National Natural Science Foundation of China (52105099), and the Funding by Science and Technology Projects in Guangzhou (202201010668).

Author information

Authors and Affiliations

  1. College of Engineering, South China Agricultural University, Guangzhou, 510642, Guangdong, People’s Republic of China

    Xiao Feng, Jian Feng, Ertai An, Hailin Wang, Shuanglong Wu & Long Qi

  2. Guangdong Polytechnic of Science and Trade, Guangzhou, 510640, Guangdong, People’s Republic of China

    Hailin Wang

Authors
  1. Xiao Feng
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  2. Jian Feng
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  3. Ertai An
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  4. Hailin Wang
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  5. Shuanglong Wu
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  6. Long Qi
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Correspondence to Shuanglong Wu or Long Qi.

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Appendices

Appendix A

See Tables 9 and 10.

Table 9 Parameters of the NAVR-DCCS
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Table 10 Parameters of a ring with rectangular cross-section
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Appendix B

See Fig. 25 and Table 11.

Fig. 25
figure 25

a Crescent-shaped cross-section with different angles \(W\). b Crescent-shaped cross-section with different angles \(H\). c Crescent-shaped cross-section with different angles \(\theta\)

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Table 11 Variations of area and moment of inertia of the cross-section with different cross-section parameters
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Appendix C

$$ \frac{c}{m\omega } = \frac{{z_{0} \sin (\phi )}}{{x_{0} }} $$
(C.1)
$$ \frac{{\frac{3}{4}x_{0}^{2} k_{2} - m\omega^{2} + k_{1} + 3b^{2} k_{2} }}{{m\omega^{2} }} = \frac{{z_{0} \cos (\phi )}}{{x_{0} }} $$
(C.2)
$$ mg + bk_{1} + b^{3} k_{2} + \frac{{3bk_{2} x_{0}^{2} }}{2} = 0 $$
(C.3)

Appendix D

See Fig. 26.

Fig. 26
figure 26

a Effect of structural parameter \(W\) (Stress contour plot in MPa). b Effect of structural parameter \(H\) (Stress contour plot in MPa). c Effect of structural parameter \(\theta\) (Stress contour plot in MPa). d Effect of structural parameter \(R\) (Stress contour plot in MPa)

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Appendix E

See Table 12.

Table 12 Stiffness and 95% confidence interval of the NAVR-DCCS with different parameters
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Appendix F

See Table 13.

Table 13 Parameters of the experimental prototype
Full size table

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Feng, X., Feng, J., An, E. et al. Palm petiole inspired nonlinear anti-vibration ring with deformable crescent-shaped cross-section. Nonlinear Dyn 112, 6919–6945 (2024). https://doi.org/10.1007/s11071-024-09440-y

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