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Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system

Abstract

To improve the vibration isolation performance and bandwidth, loading capacity and supporting stability of passive vibration isolation system by utilizing nonlinearity, a bio-inspired embedded X-shaped vibration isolation (BIE-XVI) structure is proposed considering muscle/tendon contractile functions, joint rotational friction and connecting rod mass simultaneously. Furthermore, the dynamic model with pure linear elements and geometric relationship are established and the nonlinear variation properties are investigated. The effects of the key parameters of the BIE-XVI structure on frequency response characteristics and vibration isolation range are analyzed thoroughly by incremental harmonic balance method in various working conditions. From the parametric investigations, it can be found that the sensitivities of the nonlinear resonance properties are markedly different with respect to the different structure parameters. For longer rod length, larger assembly angle and higher stiffnesses, the hardening nonlinearity is weakened, but the resonance peak does not necessarily decrease. Besides, the softening nonlinearity and hardening nonlinearity can be interconverted with changing isolated mass and excitation amplitude. The BIE-XVI structure can widen the isolation frequency range and reduce the resonance peak to improve the vibration isolation properties by adjusting/designing the structural parameters, which could realize quasi-zero-stiffness property for vibration isolation.

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Acknowledgements

The project is supported by Fundamental Research Funds for the Central Universities (No. N2103008) and Natural Science Foundation of China (No. 51805075).

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Authors and Affiliations

Authors

Contributions

SZ helped in conceptualization, formal analysis, writing—original draft, writing—review and editing. YL contributed to formal analysis, visualization. ZJ developed the software. ZR investigated the study.

Corresponding author

Correspondence to Shihua Zhou.

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Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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All data generated or analyzed during this study are included in this published article [and its supplementary information files].

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Appendix A

Appendix A

The Taylor series expansions of fj(y), gj(y) fc(y), fk(y) and hm(y) are presented as follows

$$ \begin{gathered} F_{1} \left( y \right){ = }\left[ {\frac{{n_{{1}} J_{{1}} }}{{4l_{1}^{2} {\text{cos}}^{2} \theta_{1} }} + \frac{{n_{{2}} J_{{2}} {\text{tan}}^{2} \theta_{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}}} \right] - \left[ {\frac{{n_{{1}} J_{{1}} {\text{tan}}\theta_{1} }}{{4l_{1}^{3} {\text{cos}}^{3} \theta_{1} }} + \frac{{n_{{2}} J_{{2}} {\text{sin}}\theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{4} \theta } \right)}}{{l_{1} {\text{cos}}^{4} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }}} \right]y \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \left[ {\frac{{n_{{1}} J_{{1}} \left( {{\text{1 + 3sin}}^{2} \theta_{1} } \right)}}{{16l_{1}^{4} {\text{cos}}^{6} \theta_{1} }} + \frac{{n_{{2}} J_{{2}} }}{{l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} }}\left( {\frac{{{\text{1 + 3sin}}^{2} \theta_{1} }}{{{4}l_{1}^{2} {\text{cos}}^{6} \theta_{1} }} - \frac{{{\text{sin}}^{2} \theta_{1} }}{{{\text{cos}}^{4} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}} - \frac{{{\text{sin}}^{2} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} - 3l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{4{\text{cos}}^{2} \theta_{1} }}} \right)} \right]y^{2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\kern 1pt} \left[ { - \frac{{n_{{1}} J_{{1}} {\text{sin}}\theta_{1} \left( {{\text{1 + sin}}^{2} \theta_{1} } \right)}}{{8l_{1}^{5} {\text{cos}}^{8} \theta_{1} }} + n_{{2}} J_{{2}} \left( \begin{gathered} \frac{{{\text{sin}}\theta_{1} \left( {l_{2}^{2} - l_{1}^{2} - 3l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{{4}l_{1} {\text{cos}}^{4} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} }} + \frac{{{\text{sin}}\theta_{1} \left( {{\text{1 + sin}}^{2} \theta_{1} } \right)}}{{2l_{1}^{3} {\text{cos}}^{8} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}} \hfill \\ + \frac{{{\text{sin}}\theta_{1} \left( {{\text{1 + 3sin}}^{2} \theta_{1} } \right)}}{{4l_{1} {\text{cos}}^{6} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }} - \frac{{l_{1} {\text{sin}}^{3} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} - l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{{\text{2cos}}^{2} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{4} }} \hfill \\ \end{gathered} \right)} \right]y^{3} \hfill \\ \end{gathered} $$
(29)
$$ F_{2} \left( y \right) = \frac{{\left( {n_{1} m_{1} + n_{2} m_{2} } \right)}}{16}\left[ {{\text{tan}}^{2} \theta_{1} - \frac{{{\text{sin}}\theta_{1} }}{{l_{1} {\text{cos}}^{4} \theta_{1} }}y + \frac{{1 + 3{\text{sin}}^{2} \theta_{1} }}{{2l_{1}^{2} {\text{cos}}^{6} \theta_{1} }}y^{2} - \frac{{{\text{sin}}\theta_{1} \left( {1 + 3{\text{sin}}^{2} \theta_{1} } \right)}}{{2l_{1}^{3} {\text{cos}}^{8} \theta_{1} }}y^{3} } \right] $$
(30)
$$ \begin{gathered} F_{3} \left( y \right) = \frac{{m_{2} }}{8}\left[ {\left( {\frac{{l_{1} {\text{sin}}\theta_{1} }}{{\sqrt {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } }} + 2} \right)^{2} + \frac{{l_{1}^{2} {\text{sin}}^{2} \theta_{1} }}{{l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} }}} \right] + \frac{{m_{2} \left( {l_{1}^{2} - l_{2}^{2} } \right)}}{4}\left[ {\frac{{l_{1} {\text{sin}}\theta_{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }} + \frac{1}{{\sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} } }}} \right]y \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{m_{2} \left( {l_{1}^{2} - l_{2}^{2} } \right)}}{16}\left[ {\frac{{l_{1}^{2} - l_{2}^{2} + 3l_{1}^{2} {\text{sin}}^{2} \theta_{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} }} - \frac{{3l_{1}^{2} {\text{sin}}^{2} \theta_{1} }}{{\sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{5} } }}} \right]y^{2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{m_{2} \left( {l_{1}^{2} - l_{2}^{2} } \right)}}{8}\left[ {\frac{{l_{1} {\text{sin}}\theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{4} }} - \frac{{l_{2}^{2} - l_{1}^{2} - 4l_{1}^{2} {\text{sin}}^{2} \theta_{1} }}{{4\sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{7} } }}} \right]y^{3} \hfill \\ \end{gathered} $$
(31)
$$ \begin{gathered} G_{1} \left( y \right) = \frac{1}{8}\left[ {\frac{{n_{1} J_{1} l_{1} {\text{sin}}\theta _{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} }} + \frac{{n_{2} J_{2} {\text{sin}}\theta _{1} \left( {l_{1}^{2} {\text{sin}}^{4} \theta _{1} - l_{1}^{2} + l_{2}^{2} } \right)}}{{l_{1} {\text{cos}}^{4} \theta _{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} }}} \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{y}{8}\left[ \begin{gathered} \frac{{n_{1} J_{1} \left( {l_{1}^{2} - l_{2}^{2} + 3l_{1}^{2} {\text{sin}}^{2} \theta _{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{3} }} - \frac{{2n_{2} J_{2} {\text{tan}}^{4} \theta _{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} }} \hfill \\ - \frac{{n_{2} J_{2} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} + 7l_{1}^{2} {\text{sin}}^{2} \theta _{1} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} - 3l_{2}^{2} {\text{sin}}^{2} \theta _{1} } \right)\left( {l_{1}^{2} {\text{sin}}^{4} \theta _{1} - l_{1}^{2} + l_{2}^{2} } \right)}}{{2l_{1}^{2} {\text{cos}}^{6} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)^{3} }} \hfill \\ \end{gathered} \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{y^{2} }}{8}\left[ \begin{gathered} \frac{{3n_{1} J_{1} l_{1} {\text{sin}}\theta _{1} \left( {l_{1}^{2} - l_{2}^{2} + l_{1}^{2} {\text{sin}}^{2} \theta _{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{4} }} \hfill \\ + \frac{{n_{2} J_{2} {\text{sin}}^{3} \theta _{1} \left( {11l_{1}^{2} {\text{sin}}^{2} \theta _{1} {\text{cos}}^{2} \theta _{1} + 5l_{1}^{2} {\text{cos}}^{2} \theta _{1} - 5l_{2}^{2} - 3l_{1}^{2} {\text{sin}}^{2} \theta _{1} } \right)}}{{2l_{1} {\text{cos}}^{6} \theta _{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{3} }} \hfill \\ - \frac{{n_{2} J_{2} \left( {{\text{cos}}^{2} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)\left( {2l_{1}^{2} - l_{2}^{2} - 6l_{1}^{2} {\text{sin}}^{2} \theta _{1} } \right) + 6{\text{sin}}^{2} \theta _{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} } \right)\left( {l_{1}^{2} {\text{sin}}^{4} \theta _{1} - l_{1}^{2} + l_{2}^{2} } \right)}}{{2l_{1}^{3} {\text{cos}}^{8} \theta _{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{4} }} \hfill \\ \end{gathered} \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{y^{3} }}{8}\left[ \begin{gathered} \frac{{n_{1} J_{1} \left( {l_{2}^{2} - l_{1}^{2} - 5l_{1}^{2} {\text{sin}}^{2} \theta _{1} } \right)}}{{4\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{4} }} - \frac{{n_{1} J_{1} l_{1}^{2} {\text{sin}}^{2} \theta _{1} \left( {3l_{2}^{2} - 3l_{1}^{2} - 5l_{1}^{2} {\text{sin}}^{2} \theta _{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{5} }} \hfill \\ \frac{{n_{2} J_{2} \left( {6{\text{sin}}^{2} \theta _{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} - {\text{cos}}^{2} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)\left( {6l_{1}^{2} {\text{sin}}^{2} \theta _{1} - 2l_{1}^{2} + l_{2}^{2} } \right)} \right)}}{{4l_{1}^{6} {\text{cos}}^{8} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)^{4} }} \hfill \\ - {\text{sin}}^{2} \theta _{1} \left( {l_{1}^{2} {\text{sin}}^{4} \theta _{1} - l_{1}^{2} + l_{2}^{2} } \right)\left( \begin{gathered} \frac{{2l_{1}^{3} {\text{cos}}^{2} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right) + \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)\left( {6l_{1}^{2} {\text{sin}}^{2} \theta _{1} - 2l_{1}^{2} + l_{2}^{2} } \right)}}{{2l_{1} {\text{cos}}^{4} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)^{2} }} \hfill \\ \frac{{\left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)\left[ {4{\text{sin}}^{2} \theta _{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} - 2{\text{cos}}^{2} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)\left( {6l_{1}^{2} {\text{sin}}^{2} \theta _{1} - 2l_{1}^{2} + l_{2}^{2} } \right)} \right]}}{{l_{1}^{4} {\text{cos}}^{{10}} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)^{5} }} \hfill \\ \end{gathered} \right) \hfill \\ + \frac{{{\text{sin}}^{2} \theta \left( { - 2l_{1}^{2} - l_{2}^{2} {\text{sin}}^{2} \theta _{1} + 3l_{1}^{2} {\text{cos}}^{2} \theta _{1} + 5l_{1}^{2} {\text{sin}}^{2} \theta _{1} {\text{cos}}^{2} \theta _{1} } \right)}}{{4l_{1}^{2} {\text{cos}}^{8} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)^{3} }} \hfill \\ - \frac{{{\text{sin}}^{4} \theta \left[ {{\text{cos}}^{2} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)\left( {10l_{1}^{2} {\text{sin}}^{2} \theta _{1} - 6l_{1}^{2} + 3l_{2}^{2} } \right) - 6{\text{sin}}^{2} \theta _{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta _{1} } \right)^{2} } \right]}}{{l_{1}^{2} {\text{cos}}^{8} \theta _{1} \left( {l_{1}^{2} {\text{cos}}^{2} \theta _{1} - l_{2}^{2} } \right)^{4} }} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$
(32)
$$ G_{2} \left( y \right) = \frac{{l_{1}^{2} \left( {n_{1} m_{1} + n_{2} m_{2} } \right)}}{32}\left[ \begin{gathered} - \frac{{l_{1} {\text{sin}}\theta_{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }} + \frac{{\left( {l_{2}^{2} - l_{1}^{2} - 3l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} }}y + \frac{{3l_{1} {\text{sin}}\theta_{1} \left( {l_{2}^{2} - l_{1}^{2} - l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{4} }}y^{2} \hfill \\ - \left( {\frac{{l_{2}^{2} - l_{1}^{2} - 5l_{1}^{2} {\text{sin}}^{2} \theta_{1} }}{{4\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{4} }} - \frac{{l_{1}^{2} {\text{sin}}^{2} \theta_{1} \left( {3l_{2}^{2} - 3l_{1}^{2} - 5l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{5} }}} \right)y^{3} \hfill \\ \end{gathered} \right] $$
(33)
$$ G_{3} \left( y \right) = \frac{{m_{2} \left( {l_{1}^{2} - l_{2}^{2} } \right)}}{4}\left[ \begin{gathered} \frac{{\sqrt {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } + l_{1} {\text{sin}}\theta_{1} }}{{\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }} + \frac{{l_{1}^{2} - l_{2}^{2} + 3l_{1}^{2} {\text{sin}}^{2} \theta_{1} + 3l_{1} {\text{sin}}\theta_{1} \sqrt {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } }}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} }}y \hfill \\ - \left( {\frac{{3\left( {l_{2}^{2} - l_{1}^{2} - 4l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{8\sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{7} } }} + \frac{{3l_{1} {\text{sin}}\theta_{1} \left( {l_{1}^{2} - l_{2}^{2} - l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{4} }}} \right)y^{2} \hfill \\ - \left( {\frac{{5l_{1} {\text{sin}}\theta_{1} \left( {3l_{2}^{2} - 3l_{1}^{2} - 4l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{16\sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{9} } }} - \frac{{l_{2}^{2} - l_{1}^{2} - 5l_{1}^{2} {\text{sin}}^{2} \theta_{1} }}{{4\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{4} }} + \frac{{l_{1}^{2} {\text{sin}}^{2} \theta_{1} \left( {3l_{2}^{2} - 3l_{1}^{2} - 5l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{2\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} }}} \right)y^{3} \hfill \\ \end{gathered} \right] $$
(34)
$$ F_{{\text{c}}} \left( y \right) = \left[ {c_{{l_{y} }} + c_{{l_{x} }} {\text{tan}}\theta_{1} - \frac{{c_{{l_{x} }} {\text{sin}}\theta_{1} }}{{l_{1} {\text{cos}}^{4} \theta_{1} }}y + \frac{{c_{{l_{x} }} \left( {1 + 3{\text{sin}}^{2} \theta_{1} } \right)\left( {1 + {\text{tan}}^{2} \theta_{1} } \right)}}{{4l_{1}^{2} {\text{cos}}^{4} \theta_{1} }}y^{2} - \frac{{c_{{l_{x} }} {\text{sin}}\theta_{1} \left( {1 + {\text{sin}}^{2} \theta_{1} } \right)}}{{2l_{1}^{3} {\text{cos}}^{8} \theta_{1} }}y^{3} } \right]\dot{y} $$
(35)
$$ F_{{\text{k}}} \left( y \right) = \left( {k_{{l_{y} }} + k_{{l_{x} }} {\text{tan}}^{2} \theta_{1} } \right)y - \frac{{3k_{{l_{x} }} {\text{sin}}\theta_{1} }}{{4l_{1} {\text{cos}}^{4} \theta_{1} }}y^{2} + \frac{{k_{{l_{x} }} \left[ {\left( {1 + 2{\text{sin}}^{2} \theta_{1} } \right) + {\text{tan}}^{2} \theta_{1} \left( {3 + 2{\text{sin}}^{2} \theta_{1} } \right)} \right]}}{{8l_{1}^{2} {\text{cos}}^{4} \theta_{1} }}y^{3} $$
(36)
$$ H_{1} \left( y \right) = n_{1} \mu F_{N1} \left[ {\frac{1}{{2l_{1} {\text{cos}}\theta_{1} }} - \frac{{{\text{sin}}\theta_{1} }}{{4l_{1}^{2} {\text{cos}}^{3} \theta_{1} }}y + \frac{{\left( {{\text{1 + 3tan}}^{2} \theta_{1} } \right)}}{{16l_{1}^{3} {\text{cos}}^{3} \theta_{1} }}y^{2} - \frac{{{\text{sin}}\theta_{1} \left( {{\text{3 + 5tan}}^{2} \theta_{1} } \right)}}{{32l_{1}^{4} {\text{cos}}^{5} \theta_{1} }}y^{3} } \right] $$
(37)
$$ \begin{gathered} {\kern 1pt} H_{2} \left( y \right) = - \frac{{n_{2} \mu F_{N2} {\text{tan}}\theta_{1} }}{{2\sqrt {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } }} + \frac{{n_{2} \mu F_{N2} \left( {l_{2}^{2} - l_{1}^{2} - l_{1}^{2} {\text{sin}}^{2} \theta_{1} + l_{1}^{2} {\text{tan}}^{2} \theta_{1} } \right)}}{{4l_{1} {\text{cos}}\theta_{1} \sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} } }}y \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \frac{{n_{2} \mu F_{N2} {\text{sin}}\theta_{1} }}{{16l_{1}^{2} {\text{cos}}^{2} \theta_{1} \sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} } }}\left[ {3l_{2}^{2} - 6l_{1}^{2} + 10l_{1}^{2} {\text{sin}}^{2} \theta_{1} + \frac{{3{\text{sin}}^{2} \theta \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }}{{{\text{cos}}^{2} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}}} \right]y^{2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \frac{{n_{2} \mu F_{N2} }}{2}\left[ \begin{gathered} - \frac{{\left( {l_{2}^{2} - 2l_{1}^{2} + 6l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right) - \frac{{3{\text{sin}}^{2} \theta_{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }}{{{\text{cos}}^{2} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}}}}{{16l_{1}^{3} {\text{cos}}^{3} \theta_{1} \sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} } }} \hfill \\ + \frac{{{\text{sin}}^{2} \theta }}{{4l_{1} {\text{cos}}^{3} \theta_{1} \sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{3} } }} + \frac{{{\text{sin}}^{2} \theta_{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)\left( {l_{2}^{2} - 2l_{1}^{2} + 3l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right)}}{{12l_{1}^{3} {\text{cos}}^{4} \theta_{1} \sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{5} } }} \hfill \\ - \frac{{5{\text{sin}}^{2} \theta_{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)\left( {\left( {l_{2}^{2} - 2l_{1}^{2} + 6l_{1}^{2} {\text{sin}}^{2} \theta_{1} } \right) - \frac{{3{\text{sin}}^{2} \theta_{1} \left( {l_{2}^{2} - 2l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{2} }}{{{\text{cos}}^{2} \theta_{1} \left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)}}} \right)}}{{48l_{1}^{3} {\text{cos}}^{5} \theta_{1} \sqrt {\left( {l_{2}^{2} - l_{1}^{2} {\text{cos}}^{2} \theta_{1} } \right)^{5} } }} \hfill \\ \end{gathered} \right]y^{3} \hfill \\ \end{gathered} $$
(38)

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Zhou, S., Liu, Y., Jiang, Z. et al. Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system. Nonlinear Dyn (2022). https://doi.org/10.1007/s11071-022-07610-4

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  • DOI: https://doi.org/10.1007/s11071-022-07610-4

Keywords

  • BIE-XVI structure
  • Nonlinearity
  • Frequency response characteristics
  • IHBM
  • Transmissibility