Nonlinear property and dynamic stability analysis of a novel bio-inspired vibration isolation–absorption structure

Abstract

Inspired by Ostrich anti-vibration and shock and vibration absorption properties, a novel bio-inspired vibration isolation–absorption (BIVIA) system is presented to design wideband vibration isolation bandwidth, low-frequency/ultra-low-frequency vibration isolation and high stability using toe-leg-spine coupling structure. Considering the kinematic relationship between skeletons and muscle/tendon, the geometrical relationships and dynamical equations of the BIVIA system are deduced for theoretical analysis and model verification. The influences of different parameters on loading capacity, dynamic stability, quasi-zero stiffness (QZS) zone, vibration isolation–absorption performance and vibration transmissibility are discussed. It discovers that high loading capacity and extended QZS zone are achieved by coupled vibration isolation–absorption structures. Moreover, the desirable and adjustable vibration isolation–absorption performance of the BIVIA structure can be obtained by designing key parameters. The BIVIA structure presents a practical method for bio-inspired vibration isolation and could be used in engineering and manufacturing.

Appendices

Appendix A

$$ \begin{gathered} F_{x1} = 4k_{tx} \left[ {\gamma_{1} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right) - \frac{{\gamma_{1} L_{1} \cos \alpha_{1} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)}}{{\sqrt {L_{1}^{2} - \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} } }}} \right] + 4c_{tx} \frac{{\gamma_{1}^{2} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} }}{{L_{1}^{2} - \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} }} \hfill \\ F_{x2} = 4k_{lx} \left[ { - \left( {\gamma_{2} - \gamma_{3} } \right)\left( {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right) + \frac{{\left( {\gamma_{2} - \gamma_{3} } \right)L_{3} \cos \alpha_{2} \left( {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right)}}{{\sqrt {L_{3}^{2} - \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} } }}} \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 4c_{lx} \frac{{\left( {\gamma_{2} - \gamma_{3} } \right)^{2} \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} }}{{L_{3}^{2} - \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} }} \hfill \\ \end{gathered} $$

(A1)

$$ \begin{gathered} F_{y12} = k_{ty} \left[ {l_{1} \sin \beta_{1} + l_{2} \sin \beta_{2} - \sqrt {l_{1}^{2} - L_{1}^{2} + \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} } - \sqrt {l_{2}^{2} - L_{1}^{2} + \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \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} \times {\kern 1pt} {\kern 1pt} \left[ {\frac{{\gamma_{1} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)}}{{\sqrt {l_{1}^{2} - L_{1}^{2} + \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} } }} + \frac{{\gamma_{1} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)}}{{\sqrt {l_{2}^{2} - L_{1}^{2} + \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \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} + c_{ty} \left[ {\frac{{\gamma_{1} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)}}{{\sqrt {l_{1}^{2} - L_{1}^{2} + \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} } }} + \frac{{\gamma_{1} \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)}}{{\sqrt {l_{2}^{2} - L_{1}^{2} + \left( {L_{1} \sin \alpha_{1} - \gamma_{1} y} \right)^{2} } }}} \right]^{2} \hfill \\ F_{y34} = k_{ly} \left[ {l_{3} \sin \beta_{3} + l_{4} \sin \beta_{4} - \sqrt {l_{3}^{2} - L_{3}^{2} + \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} } - \sqrt {l_{4}^{2} - L_{3}^{2} + \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \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} \times \left[ {\frac{{\left( {\gamma_{2} - \gamma_{3} } \right)\left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]}}{{\sqrt {l_{3}^{2} - L_{3}^{2} + \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} } }} + \frac{{\left( {\gamma_{2} - \gamma_{3} } \right)\left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]}}{{\sqrt {l_{4}^{2} - L_{3}^{2} + \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \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} + c_{ly} \left[ {\frac{{\left( {\gamma_{2} - \gamma_{3} } \right)\left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]}}{{\sqrt {l_{3}^{2} - L_{3}^{2} + \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} } }} + \frac{{\left( {\gamma_{2} - \gamma_{3} } \right)\left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]}}{{\sqrt {l_{4}^{2} - L_{3}^{2} + \left[ {L_{3} \sin \alpha_{2} + \left( {\gamma_{2} - \gamma_{3} } \right)y} \right]^{2} } }}} \right]^{2} \hfill \\ \end{gathered} $$

(A2)

$$ \begin{gathered} F_{n} = k_{l1} \left( {y_{n1} - y_{b} - z} \right) + c_{l1} \left( {\dot{y}_{n1} - \dot{y}_{b} - \dot{z}} \right){\kern 1pt} + k_{l2} \left( {y_{n1} - y_{n2} } \right) + c_{l2} \left( {\dot{y}_{n1} - \dot{y}_{n2} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \cdots \cdots \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + k_{lj} \left( {y_{ni} - y_{nj - 1} } \right) + c_{li} \left( {\dot{y}_{nj} - \dot{y}_{nj - 1} } \right){\kern 1pt} {\kern 1pt} + k_{lj + 1} \left( {y_{ni} - y_{nj + 1} } \right) + c_{li + 1} \left( {\dot{y}_{nj} - \dot{y}_{nj + 1} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \cdots \cdots \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + k_{{l{\text{N}}}} \left( {y_{{n{\text{N}}}} - y_{{n{\text{N}} - 1}} } \right) + c_{{l{\text{N}}}} \left( {\dot{y}_{{n{\text{N}}}} - \dot{y}_{{n{\text{N}} - 1}} } \right){\kern 1pt} {\kern 1pt} + k_{{l{\text{h}}}} \left( {y_{{n{\text{N}}}} - y_{{\text{h}}} } \right) + c_{{l{\text{h}}}} \left( {\dot{y}_{{n{\text{N}}}} - \dot{y}_{{\text{h}}} } \right) \hfill \\ \end{gathered} $$

(A3)

$$ F_{{\text{h}}} = k_{{l{\text{h}}}} \left( {y_{{\text{h}}} - y_{{n{\text{N}}}} } \right) + c_{{l{\text{h}}}} \left( {\dot{y}_{{\text{h}}} - \dot{y}_{{n{\text{N}}}} } \right) $$

(A4)

Appendix B

$$ F_{{{\text{m}}x}} = \sum\limits_{j = 1}^{2} {\left[ {\frac{{M_{j} + m_{j} }}{2}\left( {\frac{{\partial x_{1} }}{\partial y}} \right)^{2} + \frac{{M_{j + 2} + m_{j + 2} }}{2}\left( {\frac{{\partial x_{2} }}{\partial y}} \right)^{2} } \right]} $$

(B1)

$$ F_{{{\text{m}}J}} = 2\sum\limits_{i = 1}^{4} {J_{i} } \left( {\frac{{\partial \varphi_{i} }}{\partial y}} \right)^{2} + 2\sum\limits_{i = 1}^{4} {I_{i} } \left( {\frac{{\partial \psi_{i} }}{\partial y}} \right)^{2} $$

(B2)

$$ F_{{{\text{m}}\gamma }} = \frac{1}{2}\left[ {M_{1} \gamma_{1}^{2} + M_{2} \gamma_{2}^{2} + M_{3} \left( {\gamma_{2} + \gamma_{3} } \right)^{2} + M_{4} \left( {\gamma_{3} + 1} \right)^{2} } \right] $$

(B3)

$$ F_{{{\text{n}}x}} = \sum\limits_{j = 1}^{2} {\left[ {\frac{{M_{j} + m_{j} }}{2}\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial x_{1} }}{\partial y}} \right)^{2} + \frac{{M_{j + 2} + m_{j + 2} }}{2}\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial x_{2} }}{\partial y}} \right)^{2} } \right]} $$

(B4)

$$ F_{{{\text{n}}J}} = 2\sum\limits_{i = 1}^{4} {J_{i} } \frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial \varphi_{i} }}{\partial y}} \right)^{2} + 2\sum\limits_{i = 1}^{4} {I_{i} } \frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial \psi_{i} }}{\partial y}} \right)^{2} $$

(B5)

$$ F_{{{\text{nc}}}} = 4\left[ {c_{{{\text{t}}x}} \left( {\frac{{\partial x_{1} }}{\partial y}} \right)^{2} + c_{lx} \left( {\frac{{\partial x_{2} }}{\partial y}} \right)^{2} } \right] + \left[ {c_{{{\text{t}}y}} \left( {\frac{{\partial \left( {y_{1} + y_{2} } \right)}}{\partial y}} \right)^{2} + c_{ly} \left( {\frac{{\partial \left( {y_{3} + y_{4} } \right)}}{\partial y}} \right)^{2} } \right] $$

(B6)

$$ F_{{{\text{mm}}}} = \frac{{m_{1} }}{2}\left( {\gamma_{1} + \frac{{\partial \dot{y}_{1} }}{{\partial \dot{y}}}} \right)\left( {\gamma_{1} \ddot{y} + \ddot{y}_{1} } \right) + \frac{{m_{2} }}{2}\left( {\gamma_{1} + \frac{{\partial \dot{y}_{2} }}{{\partial \dot{y}}}} \right)\left( {\gamma_{1} \ddot{y} + \ddot{y}_{2} } \right) + \frac{{m_{3} }}{2}\left( {\gamma_{3} + \frac{{\partial \dot{y}_{3} }}{{\partial \dot{y}}}} \right)\left( {\gamma_{3} \ddot{y} + \ddot{y}_{3} } \right) + \frac{{m_{4} }}{2}\left( {\gamma_{3} + \frac{{\partial \dot{y}_{4} }}{{\partial \dot{y}}}} \right)\left( {\gamma_{3} \ddot{y} + \ddot{y}_{4} } \right) $$

(B7)

$$ \begin{gathered} F_{{{\text{nn}}}} = \frac{{m_{1} }}{2}\left[ {\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial \dot{y}_{1} }}{{\partial \dot{y}}}} \right) - \frac{{\partial \dot{y}_{1} }}{{\partial \dot{y}}}} \right]\left( {\gamma_{1} \dot{y} + \dot{y}_{1} } \right) + \frac{{m_{2} }}{2}\left[ {\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial \dot{y}_{2} }}{{\partial \dot{y}}}} \right) - \frac{{\partial \dot{y}_{2} }}{{\partial \dot{y}}}} \right]\left( {\gamma_{1} \dot{y} + \dot{y}_{2} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{m_{3} }}{2}\left[ {\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial \dot{y}_{3} }}{{\partial \dot{y}}}} \right) - \frac{{\partial \dot{y}_{3} }}{{\partial \dot{y}}}} \right]\left( {\gamma_{3} \dot{y} + \dot{y}_{3} } \right) + \frac{{m_{4} }}{2}\left[ {\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{{\partial \dot{y}_{4} }}{{\partial \dot{y}}}} \right) - \frac{{\partial \dot{y}_{4} }}{{\partial \dot{y}}}} \right]\left( {\gamma_{3} \dot{y} + \dot{y}_{4} } \right) \hfill \\ \end{gathered} $$

(B8)

$$ F_{{{\text{d}}x}} = \sum\limits_{j = 1}^{2} {\left[ {\frac{{M_{j} + m_{j} }}{2}\left( {\frac{{\partial x_{1} }}{\partial y}} \right)\left( {\frac{{\partial^{2} x_{1} }}{{\partial y^{2} }}} \right) + \frac{{M_{j + 2} + m_{j + 2} }}{2}\left( {\frac{{\partial x_{2} }}{\partial y}} \right)\left( {\frac{{\partial^{2} x_{2} }}{{\partial y^{2} }}} \right)} \right]} $$

(B9)

$$ F_{{{\text{d}}J}} = 2\sum\limits_{i = 1}^{4} {J_{i} } \left( {\frac{{\partial \varphi_{i} }}{\partial y}} \right)\left( {\frac{{\partial^{2} \varphi_{i} }}{{\partial y^{2} }}} \right) + 2\sum\limits_{i = 1}^{4} {I_{i} } \left( {\frac{{\partial \psi_{i} }}{\partial y}} \right)\left( {\frac{{\partial^{2} \psi_{i} }}{{\partial y^{2} }}} \right) $$

(B10)