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Multi-directional vibration isolation performances of a scissor-like structure with nonlinear hybrid spring stiffness

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Abstract

Most existing quasi-zero stiffness (QZS) vibration isolators with excellent vibration isolation performance focus on suppressing single-directional vibration. There are limitations to applications. In this paper, a multidirectional vibration isolation system with the QZS effect is proposed based on the designed geometrical relationship between the torsion spring and the linear spring. The effect of the mechanism’s structural parameters on its static as well as dynamic characteristics is analyzed. The results reveal that (a) consisting only of elastic elements and connecting rods, the compact mechanism has ideal vibration isolation performance in all three directions; (b) the mechanism characteristics can be flexibly adjusted by tuning the structural parameters; (c) when the excitation amplitudes are large, the vibration isolator remains preferable vibration attenuation performance; (d) compared with the traditional QZS structure, the typical X-shaped QZS structure, and the proposed structure I, the designed structure has a larger QZS area and a wider vibration attenuation area in the main vibration isolation direction.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This research is supported by the Natural Science Foundation of Liaoning (2020-MS-092).

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The authors have not disclosed any funding.

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, China

    Jingxuan Wang & Guo Yao

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  1. Jingxuan Wang
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  2. Guo Yao
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Appendix A

Appendix A

$$ f_{xl1} = \frac{{ - l_{1} \left( {2l_{2}^{2} \sin \varphi_{2} - \chi_{1} } \right)\sin \alpha_{1} + 2l_{2}^{3} \cos^{2} \varphi_{2} - 2l_{2}^{3} + \chi_{1} l_{2} \sin \varphi_{2} }}{{l_{1} l_{2} \left( {\chi_{1} \cos \varphi_{2} - \zeta_{1} \sin \varphi_{2} } \right)\sin \alpha_{1} }} $$
(A.1)
$$ f_{xr1} = \frac{{ - l_{1} \left( {2l_{2}^{2} \sin \varphi_{3} - \chi_{2} } \right)\sin \alpha_{2} + 2l_{2}^{3} \cos^{2} \varphi_{3} - 2l_{2}^{3} + \chi_{2} l_{2} \sin \varphi_{3} }}{{l_{1} l_{2} \left( {\chi_{2} \cos \varphi_{3} - \zeta_{2} \sin \varphi_{3} } \right)\sin \alpha_{2} }} $$
(A.2)
$$ f_{xl2} = \frac{{ - 2l_{2}^{2} \sin^{2} \varphi_{2} + \chi_{1} \sin \varphi_{2} }}{{l_{1} \left( {\chi_{1}^{{}} \cos \varphi_{2} - \zeta_{1}^{{}} \sin \varphi_{2} } \right)\sin \alpha_{1} }} $$
(A.3)
$$ f_{xr2} = \frac{{ - 2l_{2}^{2} \sin^{2} \varphi_{3} + \chi_{2} \sin \varphi_{3} }}{{l_{1} \left( {\chi_{2}^{{}} \cos \varphi_{3} - \zeta_{2}^{{}} \sin \varphi_{3} } \right)\sin \alpha_{2} }} $$
(A.4)
$$ f_{xl3} = \frac{{\left( {2\zeta_{1} l_{2}^{2} \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } \cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) - 2\chi_{1} l_{2}^{2} \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } \sin \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) + \chi_{1}^{3} + \chi_{1} \zeta_{1}^{2} } \right)}}{{l_{2} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)^{\frac{3}{2}} \cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right)}} $$
(A.5)
$$ f_{xr3} = \frac{{\left( {2\zeta_{2} l_{2}^{2} \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } \cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right) - 2\chi_{2} l_{2}^{2} \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } \sin \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right) + \chi_{2}^{3} + \chi_{2} \zeta_{2}^{2} } \right)}}{{l_{2} \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)^{\frac{3}{2}} \cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right)}} $$
(A.6)
$$ f_{yl1} = \frac{{\left( {l_{1} l_{2}^{2} 2\sin \alpha_{1} + 2\sin \varphi_{2} l_{2}^{3} + l_{2} \chi_{1} } \right)\cos \varphi_{2} - \zeta_{1} \left( {l_{1} \sin \alpha_{1} + 2l_{2} \sin \varphi_{2} } \right)}}{{l_{1} l_{2} \left( {\chi_{1} \cos \varphi_{2} - \zeta_{1} \sin \varphi_{2} } \right)\sin \alpha_{1} }} $$
(A.7)
$$ f_{yr1} = \frac{{\left( { - 2l_{1} l_{2}^{2} \sin \alpha_{2} - 2\sin \varphi_{3} l_{2}^{3} - l_{2} \chi_{2} } \right)\cos \varphi_{3} - \zeta_{2} \left( {l_{1} \sin \alpha_{2} + 2l_{2} \sin \varphi_{3} } \right)}}{{l_{1} l_{2} \left( {\chi_{2} \cos \varphi_{3} - \zeta_{2} \sin \varphi_{3} } \right)\sin \alpha_{2} }} $$
(A.8)
$$ f_{yl2} = \frac{{\left( {2l_{2}^{2} \sin \varphi_{2} + \chi_{1} } \right)\cos \varphi_{2} - 2\zeta_{1} \sin \varphi_{2} }}{{l_{1} \left( {\chi_{1} \cos \varphi_{2} - \zeta_{1} \sin \varphi_{2} } \right)\sin \alpha_{1} }} $$
(A.9)
$$ f_{yr2} = \frac{{\left( { - 2l_{2}^{2} \sin \varphi_{3} - \chi_{2} } \right)\cos \varphi_{3} + 2\zeta_{2} \sin \varphi_{3} }}{{l_{1} \left( {\chi_{2} \cos \varphi_{3} - \zeta_{2} \sin \varphi_{3} } \right)\sin \alpha_{2} }} $$
(A.10)
$$ f_{yl3} = \frac{{\left( {2\zeta_{1} l_{2}^{2} \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } \sin \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) + 2\chi_{1} l_{2}^{2} \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } \cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) - \zeta_{1}^{3} - \zeta_{1} \chi_{1}^{2} } \right)}}{{l_{2} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)^{\frac{3}{2}} \cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right)}} $$
(A.11)
$$ f_{yr3} = \frac{{\left( { - 2\zeta_{2} l_{2}^{2} \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } \sin \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right) - 2\chi_{2} l_{2}^{2} \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } \cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right) + \zeta_{2}^{3} + \zeta_{2} \chi_{2}^{2} } \right)}}{{l_{2} \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)^{\frac{3}{2}} \cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right)}} $$
(A.12)
$$ \begin{aligned} f_{\varphi l1} = & L\frac{{\left( { - 2l_{2}^{2} \sin \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) + \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } } \right)\left( {\chi_{1} \cos \varphi + \zeta_{1} \sin \varphi } \right)\left( {l_{1} \sin \alpha_{1} + l_{2} \sin \varphi_{2} } \right)}}{{l_{1} l_{2} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)\sin \alpha_{1} \cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right)}} \\ & \quad + \frac{{L\left( { - 2l_{1} l_{2} \left( {\chi_{1} \sin \varphi - \zeta_{1} \cos \varphi } \right)\sin \alpha_{1} - \left( {2\sin \varphi_{2} l_{2}^{2} \chi_{1} + \chi_{1}^{2} + \zeta_{1}^{2} } \right)\sin \varphi + 2\zeta_{1} l_{2}^{2} \cos \varphi \sin \varphi_{2} } \right)}}{{l_{1} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)\sin \alpha_{1} }} \\ \end{aligned} $$
(A.13)
$$ \begin{aligned} f_{\varphi r1} = & L\frac{{\left( { - 2l_{2}^{2} \sin \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right) + \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } } \right)\left( { - \chi_{2} \cos \varphi + \zeta_{2} \sin \varphi } \right)\left( {l_{1} \sin \alpha_{2} + l_{2} \sin \varphi_{3} } \right)}}{{l_{1} l_{2} \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)\sin \alpha_{1} \cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right)}} \\ & \quad + \frac{{L\left( {2l_{1} l_{2} \left( {\chi_{2} \sin \varphi + \zeta_{2} \cos \varphi } \right)\sin \alpha_{2} + \left( {2\chi_{3} l_{2}^{2} \sin \varphi_{3} + \chi_{2}^{2} + \zeta_{2}^{2} } \right)\sin \varphi + 2\zeta_{2} l_{2}^{2} \cos \varphi \sin \varphi_{3} } \right)}}{{l_{1} (\chi_{1}^{2} + \zeta_{1}^{2} )\sin \alpha_{1} }} \\ \end{aligned} $$
(A.14)
$$ \begin{aligned} f_{\varphi l2} = & \frac{{\left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)l_{1} \sin \alpha_{1} - L\left( {2l_{2}^{2} \left( {\zeta_{1} \cos \varphi - \chi_{1} \sin \varphi } \right)\sin \varphi_{2} - \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)\sin \varphi } \right)}}{{l_{1} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)\sin \alpha_{1} }} \\ & \quad - L\sin \varphi_{2} \frac{{ - 2l_{2}^{2} \sin \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) + \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } }}{{l_{1} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)\sin \alpha_{1} \cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right)}}\left( {\chi_{1} \cos \varphi + \zeta_{1} \sin \varphi } \right) \\ \end{aligned} $$
(A.15)
$$ \begin{aligned} f_{\varphi r2} = & \frac{{\left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)l_{1} \sin \alpha_{2} - L\left( {2l_{2}^{2} \left( {\zeta_{2} \cos \varphi + \chi_{2} \sin \varphi } \right)\sin \varphi_{3} + \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)\sin \varphi } \right)}}{{l_{1} \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)\sin \alpha_{2} }} \\ & \quad - L\sin \varphi_{3} \frac{{ - 2\sin \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right)l_{2}^{2} + \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } }}{{l_{1} \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)\sin \alpha_{2} \cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right)}}\left( {\chi_{2} \cos \varphi - \zeta_{2} \sin \varphi } \right) \\ \end{aligned} $$
(A.16)
$$ f_{\varphi l3} = - \frac{{L\left( {2l_{2}^{2} \sin \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right) - \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } } \right)\left( {\zeta_{1} \sin \varphi + \chi_{1} \cos \varphi } \right)}}{{l_{2} \left( {\chi_{1}^{2} + \zeta_{1}^{2} } \right)\cos \left( {\varphi_{2} + \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right)} \right)}} - \frac{{2Ll_{2} \left( {\chi_{1} \sin \varphi - \zeta_{1} \cos \varphi } \right)}}{{\chi_{1}^{2} + \zeta_{1}^{2} }} $$
(A.17)
$$ f_{\varphi r3} = \frac{{L\left( {2l_{2}^{2} \sin \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right) - \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } } \right)\left( {\chi_{2} \cos \varphi - \zeta_{2} \sin \varphi } \right)}}{{l_{2} \left( {\chi_{2}^{2} + \zeta_{2}^{2} } \right)\cos \left( {\varphi_{3} + \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right)} \right)}} - \frac{{2Ll_{2} \left( {\chi_{2} \sin \varphi - \zeta_{2} \cos \varphi } \right)}}{{\chi_{2}^{2} + \zeta_{2}^{2} }} $$
(A.18)
$$ f_{x4} = \Delta s_{l} \frac{{\left( {H + h_{1} \sin \varphi_{4} - h_{2} \cos \varphi_{4} - X - LL\cos \varphi } \right)}}{{4\left( {\Delta s_{l} + l_{0} } \right) \, }} + \Delta s_{r} \frac{{\left( {H - h_{1} \sin \varphi_{4} - h_{2} \cos \varphi_{4} - X - LL\cos \varphi } \right)}}{{4\left( {\Delta s_{r} + l_{0} } \right)}} $$
(A.19)
$$ f_{y4} = \Delta s_{l} \frac{{(h_{1} \cos \varphi_{4} + h_{2} \sin \varphi_{4} - Y + LL\sin \varphi )}}{{4(\Delta s_{l} + l_{0} )}} + \Delta s_{r} \frac{{( - h_{1} \cos \varphi_{4} + h_{2} \sin \varphi_{4} - Y + LL\sin \varphi )}}{{4(\Delta s_{r} + l_{0} )}} $$
(A.20)
$$ \begin{aligned} f_{\varphi 4} = & \Delta s_{l} LL\frac{{\left( {h_{2} \cos \varphi_{4} - h_{1} \sin \varphi_{4} - H + X} \right)\sin \varphi - \cos \varphi \left( {h_{1} \cos \varphi_{4} + h_{2} \sin \varphi_{4} - Y} \right)}}{{4\left( {\Delta s_{l} + l_{0} } \right)}} \, \\ & \quad + \Delta s_{r} LL\frac{{\left( {h_{2} \cos \varphi_{4} + h_{1} \sin \varphi_{4} - H + X} \right)\sin \varphi + \cos \varphi \left( {h_{1} \cos \varphi_{4} - h_{2} \sin \varphi_{4} + Y} \right)}}{{4\left( {\Delta s_{r} + l_{0} } \right)}} \\ \end{aligned} $$
(A.21)

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Wang, J., Yao, G. Multi-directional vibration isolation performances of a scissor-like structure with nonlinear hybrid spring stiffness. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09561-4

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