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Low-frequency multi-direction vibration isolation via a new arrangement of the X-shaped linkage mechanism

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Abstract

Most existing quasi-zero stiffness (QZS) isolators with excellent vibration isolation performance in the low-frequency range are designed to attenuate vibration transmission only in one direction, but vibration suppression in multi-direction is more useful and expected in engineering practice. Hence, a novel 3-degree-of-freedom (3-DOF) passive vibration isolation unit with enhanced QZS effect in a large stroke is designed based on the X-shaped mechanism. The 3-DOF vibration isolation unit exhibits beneficial nonlinear stiffness and damping properties, and it can provide excellent ultra-low-frequency vibration isolation performance in three directions simultaneously. Combining two such isolation units can of course lead to more DOF vibration isolation. The effects of several design parameters such as spring stiffness, lengths of the rods, static equilibrium positions, spring connection parameters, damping coefficients and excitation amplitudes on vibration isolation performance are analyzed in detail. Some comparisons of the static characteristics and vibration isolation performance with a spring–mass–damper (SMD) isolator and an existing typical QZS isolator are carried out. The results reveal that (a) the proposed 3-DOF vibration isolation unit can have much enhanced QZS range with larger loading capacity in the vertical and horizontal directions and HSLD stiffness in the rotational direction; (b) when the excitation amplitudes are large, the novel vibration isolation unit exhibits beneficial nonlinear properties in all three directions without jumping and bifurcation phenomena; (c) compared with the typical QZS isolator, the X-shaped mechanism enables the proposed isolation unit to possess excellent vibration isolation performance in three directions simultaneously with guaranteed stable equilibrium; (d) the new 3-DOF isolator includes only 4 bars in the entire mechanism due to the special and totally new arrangement of the X-shaped mechanism without any guiding sliders, leading to more compact designs of multi-DOF vibration isolation systems of high performance, definitely demanded by engineering practices.

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Acknowledgements

This research is supported by Hong Kong Construction Industry Council R&D Fund (EPS_202017) and Innovation and Technology Fund of Hong Kong Innovation and Technology Commission under Grant No. ITP/020/19AP.

Funding

Innovation and Technology Fund, ITP-020-19AP, Xingjian Jing, Construction Industry Council, EPS_202017, Xingjian Jing.

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Authors and Affiliations

  1. Department of Mechanical Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong, China

    Yuyang Chai & Xingjian Jing

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  1. Yuyang Chai
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Yuyang Chai did the modeling, analysis and simulation under supervision of Xingjian Jing who also provided the research idea, organized the research work and refined the paper based on the draft.

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Correspondence to Xingjian Jing.

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Appendices

Appendix A

$$ \begin{aligned} \varphi_{1} & = - \arctan \left( {\frac{{\zeta_{1} }}{{\chi_{1} }}} \right) \\ & \quad - \arcsin \left( {\frac{{L_{1}^{2} - L_{2}^{2} - \chi_{1}^{2} - \zeta_{1}^{2} }}{{2L_{2} \sqrt {\chi_{1}^{2} + \zeta_{1}^{2} } }}} \right), \\ \varphi_{2} & = - \arctan \left( {\frac{{\zeta_{2} }}{{\chi_{2} }}} \right) \\ & \quad - \arcsin \left( {\frac{{L_{1}^{2} - L_{2}^{3} - \chi_{2}^{2} - \zeta_{2}^{2} }}{{2L_{2} \sqrt {\chi_{2}^{2} + \zeta_{2}^{2} } }}} \right), \\ \end{aligned} $$
(A.1)

where

$$ \begin{aligned} \chi_{1} & = z + L_{1} \cos \theta_{1} \sin \psi \\ & \quad - \frac{1}{2}(L_{2} - L_{3} )\cos \theta_{2} \sin \psi \\ & \quad - L_{1} \sin \theta_{1} - L_{2} \sin \theta_{2} , \\ \end{aligned} $$
(A.2)
$$ \begin{aligned} \zeta_{1} & = x + L_{1} \cos \theta_{1} \cos \psi \\ & \quad - \frac{1}{2}(L_{2} - L_{3} )\cos \theta_{2} \cos \psi \\ & \quad - \frac{1}{2}(L_{2} + L_{3} )\cos \theta_{2} , \\ \end{aligned} $$
(A.3)
$$ \begin{aligned} \chi_{2} & = z - L_{1} \cos \theta_{1} \sin \psi \\ & \quad + \frac{1}{2}(L_{2} - L_{3} )\cos \theta_{2} \sin \psi \\ & \quad - L_{1} \sin \theta_{1} - L_{2} \sin \theta_{2} , \\ \end{aligned} $$
(A.4)
$$ \begin{aligned} \zeta_{2} & = - x + L_{1} \cos \theta_{1} \cos \psi \\ & \quad - \frac{1}{2}(L_{2} - L_{3} )\cos \theta_{2} \cos \psi \\ & \quad - \frac{1}{2}(L_{2} + L_{3} )\cos \theta_{2} . \\ \end{aligned} $$
(A.5)
$$ \begin{aligned} f_{3x} & = \frac{1}{{L_{2} \left( {\sin \varphi_{2} \cos \alpha_{1} + \cos \varphi_{2} \sin \alpha_{1} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{2} \beta \cos \alpha_{1} \cos \beta_{5} f_{s5} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \alpha \cos \alpha_{1} \sin \beta_{2} f_{s2} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \alpha \cos \beta_{2} f_{s2} \sin \alpha_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \beta \cos \alpha_{1} f_{s5} \sin \beta_{5} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} \cos \beta_{2} f_{s2} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \cos \alpha_{1} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{1} L_{2} \cos\varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \cos \beta_{1} f_{s1} \sin \alpha_{1} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} \\ & \quad + L_{1} L_{3} \sin\varphi_{2} \cos \alpha_{1} \cos \beta_{2} f_{s2} \\ & \quad + L_{1} L_{3} \cos \varphi_{2} \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{2} L_{4} \cos \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad - L_{2} L_{4} \cos \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} ), \\ \end{aligned} $$
(A.6)
$$ \begin{aligned} f_{3z} & = \frac{1}{{L_{2} \left( {\sin \varphi_{2} \cos \alpha_{1} + \cos \varphi_{2} \sin \alpha_{1} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{2} \alpha \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{1} L_{2} \sin \varphi_{2} \alpha \cos \beta_{2} f_{s2} \sin \alpha_{1} \\ & \quad + f_{s5} \cos \beta_{5} \beta L_{2} \sin \varphi_{2} L_{1} \sin \alpha_{1} \\ & \quad + f_{s5} \sin \beta_{5} \beta L_{2} \cos \varphi_{2} L_{1} \sin \alpha_{1} \\ & \quad - L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \beta_{1} f_{s1} \sin \alpha_{1} \\ & \quad + f_{s2} \cos \beta_{2} \sin \varphi_{2} L_{2} L_{1} \sin \alpha_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} \\ & \quad + f_{s2} \sin \beta_{2} \cos \varphi_{2} L_{2} L_{1} \sin \alpha_{1} \\ & \quad + f_{s2} \cos \beta_{2} \sin \varphi_{2} L_{3} L_{1} \sin \alpha_{1} \\ & \quad + f_{s2} \sin \beta_{2} \cos \varphi_{2} L_{3} L_{1} \sin \alpha_{1} \\ & \quad + L_{2} L_{4} \sin \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad + L_{2} L_{4} \sin \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} ), \\ \end{aligned} $$
(A.7)
$$ \begin{aligned} f_{4x} & = \frac{1}{{L_{2} \left( {\sin \varphi_{1} \cos \alpha_{2} + \cos \varphi_{1} \sin \alpha_{2} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{1} \beta \cos \alpha_{2} \cos \beta_{4} f_{s4} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \alpha \cos \alpha_{2} \sin \beta_{3} f_{s3} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \alpha \cos \beta_{3} f_{s3} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \beta \cos \alpha_{2} f_{s4} \sin \beta_{4} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} \cos \beta_{3} f_{s3} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \alpha_{2} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{1} L_{2} \cos\varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \beta_{1} f_{s1} \sin \alpha_{2} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} \\ & \quad + L_{1} L_{3} \sin\varphi_{1} \cos \alpha_{2} \cos \beta_{3} f_{s3} \\ & \quad + L_{1} L_{3} \cos \varphi_{1} \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{2} L_{4} \cos \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad - L_{2} L_{4} \cos \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} ), \\ \end{aligned} $$
(A.8)
$$ \begin{aligned} f_{4z} & = \frac{1}{{L_{2} \left( {\sin \varphi_{1} \cos \alpha_{2} + \cos \varphi_{1} \sin \alpha_{2} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{1} \alpha \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{1} L_{2} \sin \varphi_{1} \alpha \cos \beta_{3} f_{s3} \sin \alpha_{2} \\ & \quad + f_{s4} \cos \beta_{4} \beta L_{2} \sin \varphi_{1} L_{1} \sin \alpha_{2} \\ & \quad + f_{s4} \sin \beta_{4} \beta L_{2} \cos \varphi_{1} L_{1} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \beta_{1} f_{s1} \sin \alpha_{2} \\ & \quad + f_{s3} \cos \beta_{3} \sin \varphi_{1} L_{2} L_{1} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} \\ & \quad + f_{s3} \sin \beta_{3} \cos \varphi_{1} L_{2} L_{1} \sin \alpha_{2} \\ & \quad + f_{s3} \cos \beta_{3} \sin \varphi_{1} L_{3} L_{1} \sin \alpha_{2} \\ & \quad + f_{s3} \sin \beta_{3} \cos \varphi_{1} L_{3} L_{1} \sin \alpha_{2} \\ & \quad + L_{2} L_{4} \sin \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad + L_{2} L_{4} \sin \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} ), \\ \end{aligned} $$
(A.9)
$$ \begin{aligned} F_{x} & = \frac{1}{{L_{2} \left( {\sin \varphi_{2} \cos \alpha_{1} + \cos \varphi_{2} \sin \alpha_{1} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{2} \beta \cos \alpha_{1} \cos \beta_{5} f_{s5} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \alpha \cos \alpha_{1} \sin \beta_{2} f_{s2} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \alpha \cos \beta_{2} f_{s2} \sin \alpha_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \beta \cos \alpha_{1} f_{s5} \sin \beta_{5} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} \cos \beta_{2} f_{s2} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \cos \alpha_{1} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{1} L_{2} \cos\varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \cos \beta_{1} f_{s1} \sin \alpha_{1} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} \\ & \quad + L_{1} L_{3} \sin\varphi_{2} \cos \alpha_{1} \cos \beta_{2} f_{s2} \\ & \quad + L_{1} L_{3} \cos \varphi_{2} \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{2} L_{4} \cos \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad - L_{2} L_{4} \cos \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} ) \\ & \quad - \frac{1}{{L_{2} \left( {\sin \varphi_{1} \cos \alpha_{2} + \cos \varphi_{1} \sin \alpha_{2} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{1} \beta \cos \alpha_{2} \cos \beta_{4} f_{s4} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \alpha \cos \alpha_{2} \sin \beta_{3} f_{s3} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \alpha \cos \beta_{3} f_{s3} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \beta \cos \alpha_{2} f_{s4} \sin \beta_{4} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} \cos \beta_{3} f_{s3} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \alpha_{2} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{1} L_{2} \cos\varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \beta_{1} f_{s1} \sin \alpha_{2} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} \\ & \quad + L_{1} L_{3} \sin\varphi_{1} \cos \alpha_{2} \cos \beta_{3} f_{s3} \\ & \quad + L_{1} L_{3} \cos \varphi_{1} \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{2} L_{4} \cos \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad - L_{2} L_{4} \cos \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} ) \\ & \quad - f_{s2} \cos \beta_{2} + f_{s3} \cos \beta_{3} + f_{s4} \cos \beta_{4} - f_{s5} \cos \beta_{5} , \\ \end{aligned} $$
(A.10)
$$ \begin{aligned} F_{z} & = \frac{1}{{L_{2} \left( {\sin \varphi_{2} \cos \alpha_{1} + \cos \varphi_{2} \sin \alpha_{1} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{2} \alpha \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{1} L_{2} \sin \varphi_{2} \alpha \cos \beta_{2} f_{s2} \sin \alpha_{1} \\ & \quad + f_{s5} \cos \beta_{5} \beta L_{2} \sin \varphi_{2} L_{1} \sin \alpha_{1} \\ & \quad + f_{s5} \sin \beta_{5} \beta L_{2} \cos \varphi_{2} L_{1} \sin \alpha_{1} \\ & \quad - L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \beta_{1} f_{s1} \sin \alpha_{1} \\ & \quad + f_{s2} \cos \beta_{2} \sin \varphi_{2} L_{2} L_{1} \sin \alpha_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} \\ & \quad + f_{s2} \sin \beta_{2} \cos \varphi_{2} L_{2} L_{1} \sin \alpha_{1} \\ & \quad + f_{s2} \cos \beta_{2} \sin \varphi_{2} L_{3} L_{1} \sin \alpha_{1} \\ & \quad + f_{s2} \sin \beta_{2} \cos \varphi_{2} L_{3} L_{1} \sin \alpha_{1} \\ & \quad + L_{2} L_{4} \sin \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad + L_{2} L_{4} \sin \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} ) \\ & \quad + \frac{1}{{L_{2} \left( {\sin \varphi_{1} \cos \alpha_{2} + \cos \varphi_{1} \sin \alpha_{2} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{1} \alpha \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{1} L_{2} \sin \varphi_{1} \alpha \cos \beta_{3} f_{s3} \sin \alpha_{2} \\ & \quad + f_{s4} \cos \beta_{4} \beta L_{2} \sin \varphi_{1} L_{1} \sin \alpha_{2} \\ & \quad + f_{s4} \sin \beta_{4} \beta L_{2} \cos \varphi_{1} L_{1} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \beta_{1} f_{s1} \sin \alpha_{2} \\ & \quad + f_{s3} \cos \beta_{3} \sin \varphi_{1} L_{2} L_{1} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} \\ & \quad + f_{s3} \sin \beta_{3} \cos \varphi_{1} L_{2} L_{1} \sin \alpha_{2} \\ & \quad + f_{s3} \cos \beta_{3} \sin \varphi_{1} L_{3} L_{1} \sin \alpha_{2} \\ & \quad + f_{s3} \sin \beta_{3} \cos \varphi_{1} L_{3} L_{1} \sin \alpha_{2} \\ & \quad + L_{2} L_{4} \sin \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad + L_{2} L_{4} \sin \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} ) \\ & \quad - f_{s2} \sin \beta_{2} - f_{s3} \sin \beta_{3} - f_{s4} \sin \beta_{4} - f_{s5} \sin \beta_{5} , \\ \end{aligned} $$
(A.11)
$$ \begin{aligned} M_{\psi } & = \frac{1}{2}\sin \psi \\ & \quad (2L_{1} \cos \theta_{1} + L_{3} \cos \theta_{2} - L_{2} \cos \theta_{2} ) \\ & \quad (2f_{s1} \cos \beta_{1} - f_{s2} \cos \beta_{2} - f_{s3} \cos \beta_{3} \\ & \quad + f_{s4} \cos \beta_{4} + f_{s5} \cos \beta_{5} \\ & \quad + \frac{1}{{L_{2} \left( {\sin \varphi_{2} \cos \alpha_{1} + \cos \varphi_{2} \sin \alpha_{1} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{2} \beta \cos \alpha_{1} \cos \beta_{5} f_{s5} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \alpha \cos \alpha_{1} \sin \beta_{2} f_{s2} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \alpha \cos \beta_{2} f_{s2} \sin \alpha_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \beta \cos \alpha_{1} f_{s5} \sin \beta_{5} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} \cos \beta_{2} f_{s2} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \cos \alpha_{1} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{2} \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{1} L_{2} \cos\varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \cos \beta_{1} f_{s1} \sin \alpha_{1} \\ & \quad - L_{1} L_{2} \cos \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} \\ & \quad + L_{1} L_{3} \sin\varphi_{2} \cos \alpha_{1} \cos \beta_{2} f_{s2} \\ & \quad + L_{1} L_{3} \cos \varphi_{2} \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{2} L_{4} \cos \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad - L_{2} L_{4} \cos \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} ) \\ & \quad + \frac{1}{{L_{2} \left( {\sin \varphi_{1} \cos \alpha_{2} + \cos \varphi_{1} \sin \alpha_{2} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{1} \beta \cos \alpha_{2} \cos \beta_{4} f_{s4} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \alpha \cos \alpha_{2} \sin \beta_{3} f_{s3} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \alpha \cos \beta_{3} f_{s3} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \beta \cos \alpha_{2} f_{s4} \sin \beta_{4} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} \cos \beta_{3} f_{s3} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \alpha_{2} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \cos \varphi_{1} \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{1} L_{2} \cos\varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \beta_{1} f_{s1} \sin \alpha_{2} \\ & \quad - L_{1} L_{2} \cos \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} \\ & \quad + L_{1} L_{3} \sin\varphi_{1} \cos \alpha_{2} \cos \beta_{3} f_{s3} \\ & \quad + L_{1} L_{3} \cos \varphi_{1} \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{2} L_{4} \cos \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad - L_{2} L_{4} \cos \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} )) \\ \end{aligned} $$
$$ \begin{aligned} & \quad - \frac{1}{2}\cos \psi (2L_{1} \cos \theta_{1} + L_{3} \cos \theta_{2} - L_{2} \cos \theta_{2} ) \\ & \quad (2f_{s1} \sin \beta_{1} - f_{s2} \sin \beta_{2} + f_{s3} \sin \beta_{3} \\ & \quad - f_{s4} \sin \beta_{4} + f_{s5} \sin \beta_{5} \\ & \quad + \frac{1}{{L_{2} \left( {\sin \varphi_{2} \cos \alpha_{1} + \cos \varphi_{2} \sin \alpha_{1} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{2} \alpha \cos \alpha_{1} f_{s2} \sin \beta_{2} \\ & \quad - L_{1} L_{2} \sin \varphi_{2} \alpha \cos \beta_{2} f_{s2} \sin \alpha_{1} \\ & \quad + f_{s5} \cos \beta_{5} \beta L_{2} \sin \varphi_{2} L_{1} \sin \alpha_{1} \\ & \quad + f_{s5} \sin \beta_{5} \beta L_{2} \cos \varphi_{2} L_{1} \sin \alpha_{1} \\ & \quad - L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \beta_{1} f_{s1} \sin \alpha_{1} \\ & \quad + f_{s2} \cos \beta_{2} \sin \varphi_{2} L_{2} L_{1} \sin \alpha_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} \\ & \quad + f_{s2} \sin \beta_{2} \cos \varphi_{2} L_{2} L_{1} \sin \alpha_{1} \\ & \quad + f_{s2} \cos \beta_{2} \sin \varphi_{2} L_{3} L_{1} \sin \alpha_{1} \\ & \quad + f_{s2} \sin \beta_{2} \cos \varphi_{2} L_{3} L_{1} \sin \alpha_{1} \\ & \quad + L_{2} L_{4} \sin \varphi_{2} \cos \alpha_{1} f_{s4} \sin \beta_{4} \\ & \quad + L_{2} L_{4} \sin \varphi_{2} \cos \beta_{4} f_{s4} \sin \alpha_{1} ) \\ & \quad - \frac{1}{{L_{2} \left( {\sin \varphi_{1} \cos \alpha_{2} + \cos \varphi_{1} \sin \alpha_{2} } \right)L_{1} }} \\ & \quad (L_{1} L_{2} \sin \varphi_{1} \alpha \cos \alpha_{2} f_{s3} \sin \beta_{3} \\ & \quad - L_{1} L_{2} \sin \varphi_{1} \alpha \cos \beta_{3} f_{s3} \sin \alpha_{2} \\ & \quad + f_{s4} \cos \beta_{4} \beta L_{2} \sin \varphi_{1} L_{1} \sin \alpha_{2} \\ & \quad + f_{s4} \sin \beta_{4} \beta L_{2} \cos \varphi_{1} L_{1} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} f_{s1} \sin \beta_{1} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \beta_{1} f_{s1} \sin \alpha_{2} \\ & \quad + f_{s3} \cos \beta_{3} \sin \varphi_{1} L_{2} L_{1} \sin \alpha_{2} \\ & \quad + L_{1} L_{2} \sin \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} \\ & \quad + f_{s3} \sin \beta_{3} \cos \varphi_{1} L_{2} L_{1} \sin \alpha_{2} \\ & \quad + f_{s3} \cos \beta_{3} \sin \varphi_{1} L_{3} L_{1} \sin \alpha_{2} \\ & \quad + f_{s3} \sin \beta_{3} \cos \varphi_{1} L_{3} L_{1} \sin \alpha_{2} \\ & \quad + L_{2} L_{4} \sin \varphi_{1} \cos \alpha_{2} f_{s5} \sin \beta_{5} \\ & \quad + L_{2} L_{4} \sin \varphi_{1} \cos \beta_{5} f_{s5} \sin \alpha_{2} )). \\ \end{aligned} $$
(A.12)

Appendix B

Table 2 Detailed expressions of the parameters y, κ0, κ1, κ2, κ3, κ4, μ1, μ2, μ3 and μ4 in Eq. (42) of the 3-DOF passive vibration isolation unit in the directions of x, z and ψ
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Chai, Y., Jing, X. Low-frequency multi-direction vibration isolation via a new arrangement of the X-shaped linkage mechanism. Nonlinear Dyn 109, 2383–2421 (2022). https://doi.org/10.1007/s11071-022-07452-0

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