A novel eight-legged vibration isolation platform with dual-pyramid-shape struts

Abstract

An eight-legged vibration isolation platform (ELVIP) with dual-pyramid-shape (DPS) struts is proposed in this paper to achieve six-degree-of-freedom vibration isolation. The DPS strut has higher static stiffness and lower dynamic stiffness than the equivalent linear strut due to geometric nonlinearity, and therefore the ELVIP with DPS struts possesses high-static–low-dynamic-stiffness characteristic which is desirable for widening the frequency range of isolation. The layout of the ELVIP legs is conducive to vibration decoupling and has higher reliability than six-legged platforms. Firstly, the stiffness characteristics of the DPS strut are derived; then the dynamic model of the ELVIP with DPS struts is established and the steady-state responses are obtained analytically; finally, the isolation performance is studied and the effects of damping and excitation amplitudes are investigated. It is shown that the ELVIP with DPS struts, as a passive approach, can achieve good isolation performance in all six directions with a small static deflection.

Appendices

Appendix 1

See Table 4.

Table 4 Terminology table

Appendix 2: Some expressions

The coefficients in Eq. (18):

$$ \eta_{0} = \frac{9}{4}\left[ {1 - \frac{{\sqrt {1 - \varepsilon^{2} } }}{{\sqrt {1 - \left( {\varepsilon - u_{\text{eq}} } \right)^{2} } }}} \right]\left( {\varepsilon - u_{\text{eq}} } \right) + \lambda u_{\text{eq}} $$

(68)

$$ \eta_{1} = \frac{9}{4}\left\{ {\frac{{\sqrt {1 - \varepsilon^{2} } }}{{\left[ {1 - \left( {\varepsilon - u_{\text{eq}} } \right)^{2} } \right]^{{\frac{3}{2}}} }} - 1} \right\} + \lambda $$

(69)

$$ \eta_{2} = - \frac{{27\sqrt {1 - \varepsilon^{2} } \left( {\varepsilon - u_{\text{eq}} } \right)}}{{8\left[ {1 - \left( {\varepsilon - u_{\text{eq}} } \right)^{2} } \right]^{{\frac{5}{2}}} }} $$

(70)

$$ \eta_{3} = \frac{{9\sqrt {1 - \varepsilon^{2} } \left[ {1 + 4\left( {\varepsilon - u_{\text{eq}} } \right)^{2} } \right]}}{{8\left[ {1 - \left( {\varepsilon - u_{\text{eq}} } \right)^{2} } \right]^{{\frac{7}{2}}} }} $$

(71)

The expressions of ${\mathbf{A}}_{i}$ and ${\mathbf{B}}_{i}$:

$$ {\mathbf{A}}_{1} = {\mathbf{A}}_{2} = a\left[ {\begin{array}{*{20}c} 1 \\ 0 \\ 0 \\ \end{array} } \right];\quad {\mathbf{A}}_{3} = {\mathbf{A}}_{4} = a\left[ {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\ \end{array} } \right];\quad {\mathbf{A}}_{5} = {\mathbf{A}}_{6} = a\left[ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} } \right];\quad {\mathbf{A}}_{7} = {\mathbf{A}}_{8} = a\left[ {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} } \right] $$

(72)

$$ {\mathbf{B}}_{8} = {\mathbf{B}}_{1} = a\left[ {\begin{array}{*{20}c} 1 \\ { - 1} \\ { - \tan \phi } \\ \end{array} } \right];\quad {\mathbf{B}}_{2} = {\mathbf{B}}_{3} = a\left[ {\begin{array}{*{20}c} 1 \\ 1 \\ { - \tan \phi } \\ \end{array} } \right];\quad {\mathbf{B}}_{4} = {\mathbf{B}}_{5} = a\left[ {\begin{array}{*{20}c} { - 1} \\ 1 \\ { - \tan \phi } \\ \end{array} } \right];\quad {\mathbf{B}}_{6} = {\mathbf{B}}_{7} = - a\left[ {\begin{array}{*{20}c} 1 \\ 1 \\ {\tan \phi } \\ \end{array} } \right] $$

(73)

where $a$ is the half length of the diagonal line of the platform.

The sub-rotation matrices:

$$ {\mathbf{R}}_{1} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\cos \alpha } & { - \sin \alpha } \\ 0 & {\sin \alpha } & {\cos \alpha } \\ \end{array} } \right] $$

(74)

$$ {\mathbf{R}}_{2} = \left[ {\begin{array}{*{20}c} {\cos \beta } & 0 & {\sin \beta } \\ 0 & 1 & 0 \\ { - \sin \beta } & 0 & {\cos \beta } \\ \end{array} } \right] $$

(75)

$$ {\mathbf{R}}_{3} = \left[ {\begin{array}{*{20}c} {\cos \gamma } & { - \sin \gamma } & 0 \\ {\sin \gamma } & {\cos \gamma } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(76)

The expressions of ${\mathbf{A^{\prime}}}_{i}$:

$$ {\mathbf{A}}_{1}^{\prime } = {\mathbf{A}}_{2}^{\prime } = a\left[ {\begin{array}{*{20}c} {\cos \beta \cos \gamma + {x \mathord{\left/ {\vphantom {x a}} \right. \kern-0pt} a}} \hfill \\ {\cos \alpha \sin \gamma + \sin \alpha \sin \beta \cos \gamma + {y \mathord{\left/ {\vphantom {y a}} \right. \kern-0pt} a}} \hfill \\ {\sin \alpha \sin \gamma - \cos \alpha \sin \beta \cos \gamma + {z \mathord{\left/ {\vphantom {z a}} \right. \kern-0pt} a}} \hfill \\ \end{array} } \right] $$

(77)

$$ {\mathbf{A}}_{3}^{\prime } = {\mathbf{A}}_{4}^{\prime } = a\left[ {\begin{array}{*{20}c} { - \cos \beta \sin \gamma + {x \mathord{\left/ {\vphantom {x a}} \right. \kern-0pt} a}} \hfill \\ {\cos \alpha \cos \gamma - \sin \alpha \sin \beta \sin \gamma + {y \mathord{\left/ {\vphantom {y a}} \right. \kern-0pt} a}} \hfill \\ {\sin \alpha \cos \gamma + \cos \alpha \sin \beta \sin \gamma + {z \mathord{\left/ {\vphantom {z a}} \right. \kern-0pt} a}} \hfill \\ \end{array} } \right] $$

(78)

$$ {\mathbf{A}}_{5}^{\prime } = {\mathbf{A}}_{6}^{\prime } = a\left[ {\begin{array}{*{20}c} {\cos \beta \cos \gamma + {x \mathord{\left/ {\vphantom {x a}} \right. \kern-0pt} a}} \hfill \\ {\cos \alpha \sin \gamma + \sin \alpha \sin \beta \cos \gamma + {y \mathord{\left/ {\vphantom {y a}} \right. \kern-0pt} a}} \hfill \\ {\sin \alpha \sin \gamma - \cos \alpha \sin \beta \cos \gamma + {z \mathord{\left/ {\vphantom {z a}} \right. \kern-0pt} a}} \hfill \\ \end{array} } \right] $$

(79)

$$ {\mathbf{A}}_{7}^{\prime } = {\mathbf{A}}_{8}^{\prime } = a\left[ {\begin{array}{*{20}c} { - \cos \beta \sin \gamma + {x \mathord{\left/ {\vphantom {x a}} \right. \kern-0pt} a}} \hfill \\ {\cos \alpha \cos \gamma - \sin \alpha \sin \beta \sin \gamma + {y \mathord{\left/ {\vphantom {y a}} \right. \kern-0pt} a}} \hfill \\ {\sin \alpha \cos \gamma + \cos \alpha \sin \beta \sin \gamma + {z \mathord{\left/ {\vphantom {z a}} \right. \kern-0pt} a}} \hfill \\ \end{array} } \right] $$

(80)

The expressions of $N_{i} \left( {\mathbf{P}} \right)$:

$$ N_{1} \left( {\mathbf{P}} \right) = - y\cos \phi - z\sin \phi + \beta a\sin \phi - \gamma a\cos \phi $$

(81)

$$ N_{2} \left( {\mathbf{P}} \right) = \;\;y\cos \phi - z\sin \phi + \beta a\sin \phi + \gamma a\cos \phi $$

(82)

$$ N_{3} \left( {\mathbf{P}} \right) = \;\;x\cos \phi - z\sin \phi - \alpha a\sin \phi - \gamma a\cos \phi $$

(83)

$$ N_{4} \left( {\mathbf{P}} \right) = - x\cos \phi - z\sin \phi - \alpha a\sin \phi + \gamma a\cos \phi $$

(84)

$$ N_{5} \left( {\mathbf{P}} \right) = \;\;y\cos \phi - z\sin \phi - \beta a\sin \phi - \gamma a\cos \phi $$

(85)

$$ N_{6} \left( {\mathbf{P}} \right) = - y\cos \phi - z\sin \phi - \beta a\sin \phi + \gamma a\cos \phi $$

(86)

$$ N_{7} \left( {\mathbf{P}} \right) = - x\cos \phi - z\sin \phi + \alpha a\sin \phi - \gamma a\cos \phi $$

(87)

$$ N_{8} \left( {\mathbf{P}} \right) = \;\;x\cos \phi - z\sin \phi + \alpha a\sin \phi + \gamma a\cos \phi $$

(88)

The expressions of $\rho_{0}$, $\rho_{1}$, $\rho_{2}$ and $\rho_{3}$:

$$ \rho_{0} = 2k_{h} L\eta_{0} ;\quad \rho_{1} = k_{h} \eta_{1} ;\quad \rho_{2} = \frac{{k_{h} \eta_{2} }}{2L};\quad \rho_{3} = \frac{{k_{h} \eta_{3} }}{{\left( {2L} \right)^{2} }} $$

(89)

The expressions of inertial matrix, damping matrix, linear stiffness matrix and excitation amplitude vector:

$$ {\mathbf{M}} = {\text{diag}}(m,m,m,I_{x} ,I_{y} ,I_{z} ) $$

(90)

$$ {\mathbf{C}} = 4c \cdot {\text{diag}}(\phi_{c}^{2} , \, \phi_{c}^{2} , \, 2\phi_{s}^{2} , \, \phi_{s}^{2} a^{2} , \, \phi_{s}^{2} a^{2} , \, 2\phi_{c}^{2} a^{2} ) $$

(91)

$$ {\mathbf{K}} = 4\rho_{1} \cdot {\text{diag}}(\phi_{c}^{2} , \, \phi_{c}^{2} , \, 2\phi_{s}^{2} , \, \phi_{s}^{2} a^{2} , \, \phi_{s}^{2} a^{2} , \, 2\phi_{c}^{2} a^{2} ) $$

(92)

$$ {\mathbf{F}}_{E0} = \left[ {F_{x0} ,F_{y0} ,F_{z0} ,M_{\alpha 0} ,M_{\beta 0} ,M_{\gamma 0} } \right]^{\text{T}} . $$

(93)