The vibration isolation proprieties of an X-shaped structure with enhanced high-static and low-dynamic stiffness via torsional magnetic negative stiffness mechanism

Abstract

Bionic X-shaped structure is widely studied for low-frequency vibration isolation. In this paper, an X-shaped structure with torsional magnet negative stiffness (TMNS) mechanisms and torsion springs is proposed to enhance the high-static and low-dynamic stiffness characteristics. The TMNS contains multiple pairs of inner and outer tile permanent magnets, and can generate periodic torsional negative stiffness. With the compensation of torsion springs, the QZS point can be realized flexibly within the whole stroke, and one or multiple QZS regions can be obtained. Considering the nonlinear inertia caused by TMNS mass, the system motion equation is established and the displacement transmissibility is solved by harmonic balance method (HBM). The results show that the nonlinear inertia causes the system to experience anti-resonance phenomenon, meanwhile, the resonance frequency slightly shifts to the left. Compared to the typical QZS isolator or the classic X-shaped structure with vertical spring, the proposed model possesses higher equivalent static stiffness, and can achieve better low-frequency vibration isolation performance. An experimental prototype is fabricated, and its static nonlinear property as well as absolute displacement transmissibility are tested. The validity of proposed model is verified by comparing the experimental results with theoretical values.

Appendices

Appendix A. Theoretical model of magnetic torque.

For the two tile magnets shown in Fig. 

Fig. 21

The configuration schematic of two tile magnets

21, the azimuthal field projected at a point ${\varvec{P}}\left( {{\varvec{r}}_{0} , \psi_{0} , z_{0} } \right)$ in the outer magnet onto an observation point ${\varvec{N}}\left( {r, \psi , z} \right)$ in the inner magnet can be expressed as [67]

$$ \begin{array}{*{20}c} {H_{\varphi } \left( {r,\varphi ,z} \right) = \frac{{B_{r} }}{{4\pi u_{0} }}\int\limits_{{r_{3} }}^{{r_{4} }} {\int\limits_{{\psi_{3} }}^{{\psi_{4} }} {\int\limits_{{z_{3} }}^{{z_{4} }} {\frac{{\left( {\overrightarrow {PN} \cdot \overrightarrow {{u_{\varphi } }} } \right)}}{{|\overrightarrow {PN} |^{3} }}} } } dr_{0} d\psi_{0} dz_{0} } \\ \end{array} $$

(A1)

with

$$ \begin{array}{*{20}c} {\overrightarrow {PM} \cdot \overrightarrow {{u_{\theta } }} = - r_{0} \sin \left( {\varphi - \psi_{0} } \right)} \\ \end{array} $$

(A2)

$$ \begin{array}{*{20}c} {|\overrightarrow {PN} |^{3} = \left[ {r^{2} + r_{0}^{2} - 2rr_{0} \cos \left( {\psi - \psi_{0} } \right) + \left( {z - z_{0} } \right)^{2} } \right]^{\frac{3}{2}} } \\ \end{array} $$

(A3)

where $B_{r}$ is the residual magnetic flux density of permanent magnets, $\mu_{0}$ is the permeability of vacuum, $r_{1} ,r_{2}$ is the inner and outer radius of inner PM, $r_{3} ,r_{4}$ is the inner and outer radius of outer PM, $z_{1} ,z_{2}$ is the lower and upper axial abscissa of inner PM, $z_{3} ,z_{4}$ is the lower and upper axial abscissa of outer PM, $\psi_{1} ,\psi_{2}$ is the angular abscissa of inner PM, $\psi_{3} ,\psi_{4}$ is the angular abscissa of outer PM.

Substituting Eqs. (A2) (A3) into Eq. (A1) and integrating with respect to $r_{0}$, the expression of the azimuthal field is given by

$$ \begin{array}{*{20}c} {H_{\varphi } \left( {r,\varphi ,z} \right) = \frac{{B_{r} }}{{4\pi u_{0} }}\sum\limits_{i = 3}^{4} {( - 1)^{i + 1} \int\limits_{{\psi_{3} }}^{{\psi_{4} }} {\int\limits_{{z_{3} }}^{{z_{4} }} {\frac{{ - r_{i} \sin \left( {\psi - \psi_{0} } \right)}}{{\left[ {r^{2} + r_{i}^{2} - 2rr_{i} \cos \left( {\psi - \psi_{0} } \right) + \left( {z - z_{0} } \right)^{2} } \right]^{\frac{3}{2}} }}} } } d\psi_{0} dz_{0} } \\ \end{array} $$

(A4)

The moment between any two magnets can be derived as

$$ \begin{array}{*{20}c} {T_{pq} \left( \alpha \right) = B_{r} r_{2}^{2} \int\limits_{{z_{1} }}^{{z_{2} }} {\int\limits_{{\psi_{1} }}^{{\psi_{2} }} {H_{\psi } \left( {r_{2} ,\psi ,z} \right)d\psi dz - B_{r} r_{1}^{2} } } \int\limits_{{z_{1} }}^{{z_{2} }} {\int\limits_{{\psi_{1} }}^{{\psi_{2} }} {H_{\varphi } \left( {r_{1} ,\psi ,z} \right)d\psi dz} } } \\ \end{array} $$

(A5)

A simplified expression for the moment can be obtained by integrating with respect to $\psi$ and $z$ as

$$ \begin{array}{*{20}c} {T_{pq} \left( \alpha \right) = \frac{{B_{r}^{2} }}{{4\pi u_{0} }}\left[ {\mathop \sum \limits_{i,k,m = 1}^{2} \mathop \sum \limits_{j,l,n = 3}^{4} ( - 1)^{i + j + k + l + m + n} r_{i} r_{j} M_{1} + \mathop \sum \limits_{i,k,m = 1}^{2} \mathop \sum \limits_{j,l = 3}^{4} ( - 1)^{i + j + k + l + m} r_{i} r_{j} M_{2} } \right]} \\ \end{array} $$

(A6)

with

$$ M_{1} = - \sqrt {\left( {r_{i} - r_{j} } \right)^{2} + \left( {z_{k} - z_{l} } \right)^{2} } \times F^{*} \left[ {\frac{{\psi_{m} - \psi_{n} }}{2}, - \frac{{4r_{i} r_{j} }}{{\left( {r_{i} - r_{j} } \right)^{2} + \left( {z_{k} - z_{l} } \right)^{2} }}} \right] $$

$$ M_{2} = \left( {z_{l} - z_{k} } \right)\mathop \smallint \limits_{{\psi_{3} }}^{{\psi_{4} }} {\text{log}}\left[ {\sqrt {r_{i}^{2} + r_{j}^{2} + \left( {z_{l} - z_{k} } \right)^{2} - 2r_{i} r_{j} {\text{cos}}\left( {\psi_{m} - \psi } \right)} - \left( {z_{l} - z_{k} } \right)} \right]d\psi $$

Here, $F^{*} \left[ {{\Phi },m} \right]$ is the elliptic integrals of the second kind,

$$ F^{*} \left[ {\phi ,m} \right] = \mathop \smallint \limits_{0}^{\phi } \sqrt {1 - m\sin^{2} \psi } d\psi $$

Derivation of Eq. (A6) with respect to the rotation angle $\alpha$, an expression for the torsional stiffness can be deduced as

$$ \begin{array}{*{20}c} {K_{pq} \left( \alpha \right) = \frac{{B_{r}^{2} }}{{4\pi u_{0} }}\left[ {\mathop \sum \limits_{i,k,m = 1}^{2} \mathop \sum \limits_{j,l,n = 3}^{4} ( - 1)^{i + j + k + l + m + n} r_{i} r_{j} K_{1} + \mathop \sum \limits_{i,k,m = 1}^{2} \mathop \sum \limits_{j,l = 3}^{4} ( - 1)^{i + j + k + l + m} r_{i} r_{j} K_{2} } \right]} \\ \end{array} $$

(A7)

with

$$ K_{1} = - \sqrt {\left( {r_{i} - r_{j} } \right)^{2} + \left( {z_{k} - z_{l} } \right)^{2} } \times \frac{1}{2}\sqrt {1 + \frac{{4r_{i} r_{j} }}{{\left( {r_{i} - r_{j} } \right)^{2} + \left( {z_{k} - z_{l} } \right)^{2} }}\sin^{2} \left( {\frac{{\psi_{m} - \psi_{n} }}{2}} \right)} $$

$$ K_{2} = \left( {z_{l} - z_{k} } \right)\mathop \sum \limits_{s = 3}^{4} ( - 1)^{s} {\text{log}}\left[ {\sqrt {r_{i}^{2} + r_{j}^{2} + \left( {z_{l} - z_{k} } \right)^{2} - 2r_{i} r_{j} {\text{cos}}\left( {\psi_{m} - \psi_{s} } \right)} - \left( {z_{l} - z_{k} } \right)} \right] $$

For the proposed TMNS with $p$ pairs of inner and outer magnets, its torque and torsional stiffness can be expressed as

$$ \begin{array}{*{20}c} {T_{m} \left( \alpha \right) = p\mathop \sum \limits_{q = 1}^{p} T_{pq} } \\ \end{array} $$

(A8)

$$ \begin{array}{*{20}c} {K_{m} \left( \alpha \right) = p\mathop \sum \limits_{q = 1}^{p} K_{pq} } \\ \end{array} $$

(A9)

For the convenience of analysis, a constant torque $T_{m0}$ is defined as

$$ \begin{array}{*{20}c} {T_{m0} = \frac{{B_{r}^{2} }}{{4\pi \mu_{0} }}R_{i}^{3} ,} \\ \end{array} $$

(A10)

where $R_{i}$ is the shaft radius of proposed system shown in Fig. 1, then the dimensionless torque and torsional stiffness generated by TMNS can be obtained as

$$ \begin{array}{*{20}c} {\hat{T}_{m} \left( \alpha \right) = \frac{{T_{m} \left( \alpha \right)}}{{T_{m0} }}, \hat{K}_{m} \left( \alpha \right) = \frac{{K_{m} \left( \alpha \right)}}{{T_{m0} }}.} \\ \end{array} $$

(A11)

Appendix B

$$ \begin{array}{*{20}c} {F = \frac{dV}{{dy}} = 2T_{m} \left( \alpha \right)\frac{d\alpha }{{dy}} + 2k_{t} \alpha \frac{d\alpha }{{dy}} = \frac{{2\left( {T_{m} \left( \alpha \right) + k_{t} \alpha } \right)}}{{nL\sqrt {1 - \left( {\sin \theta_{0} - \frac{y}{2nL}} \right)^{2} } }}} \\ \end{array} $$

(B1)

$$ \begin{array}{*{20}c} {K = \frac{dF}{{dy}} = 2\frac{{K_{m} \left( \alpha \right) + k_{t} }}{{n^{2} L^{2} \left[ {1 - \left( {\sin \theta_{0} - \frac{y}{2nL}} \right)^{2} } \right]}} - 2\frac{{\left( {T_{m} \left( \alpha \right) + k_{t} \alpha } \right)\left( {\sin \theta_{0} - \frac{y}{2nL}} \right)}}{{2n^{2} L^{2} \left[ {1 - \left( {\sin \theta_{0} - \frac{y}{2nL}} \right)^{2} } \right]^{\frac{3}{2}} }}} \\ \end{array} $$

(B2)

Appendix C

$$ \begin{array}{*{20}c} {f_{1} = L^{2} \left( {\frac{d\varphi }{{dy_{r} }}} \right)^{2} = \frac{1}{{4n^{2} \left[ {1 - \left( {\sin \beta_{0} - \frac{{\hat{y}_{r} }}{2n}} \right)^{2} } \right]}}} \\ \end{array} $$

(C1)

$$ \begin{array}{*{20}c} {f_{2} = L^{3} \frac{d\varphi }{{dy_{r} }}\frac{{d^{2} \varphi }}{{dy_{r}^{2} }} = - \frac{{\left( {\sin \theta_{0} - \frac{{\hat{y}_{r} }}{2n}} \right)}}{{8n^{3} \left[ {1 - \left( {\sin \theta_{0} - \frac{{\hat{y}_{r} }}{2n}} \right)^{2} } \right]^{2} }}} \\ \end{array} $$

(C2)

$$ \begin{array}{*{20}c} {f_{3} = \frac{{2\left( {\hat{T}_{m} \left( \alpha \right) + \hat{k}_{t} \alpha } \right)}}{{n\sqrt {1 - \left( {\sin \theta_{0} - \frac{{\hat{y}_{r} }}{2n}} \right)^{2} } }} - \frac{L}{{T_{m0} }}\left( {M + \frac{1}{n}m_{T} } \right)g} \\ \end{array} $$

(C3)

$$ \begin{array}{*{20}c} {\tilde{f}_{1} = \eta_{0} + \eta_{1} \hat{y}_{r} + \eta_{2} \hat{y}_{r}^{2} + \eta_{3} \hat{y}_{r}^{3} } \\ \end{array} $$

(C4)

$$ \begin{array}{*{20}c} {\tilde{f}_{2} = \chi_{0} + \chi_{1} \hat{y}_{r} + \chi_{2} \hat{y}_{r}^{2} + \chi_{3} \hat{y}_{r}^{3} } \\ \end{array} $$

(C5)

$$ \begin{array}{*{20}c} {\tilde{f}_{3} = \kappa_{0} + \kappa_{1} \hat{y}_{r} + \kappa_{2} \hat{y}_{r}^{2} + \kappa_{3} \hat{y}_{r}^{3} } \\ \end{array} $$

(C6)

Appendix D

$$ \begin{array}{*{20}c} { - \left[ {1 + \varepsilon \left( {\frac{{3a^{2} \eta_{2} }}{4} + \eta_{0} } \right)} \right] + \varepsilon \left( {\frac{{a^{4} \chi_{3} }}{8} + \frac{{a^{2} \chi_{1} }}{4}} \right) + \frac{1}{{{\Omega }^{2} }}\left( {\frac{{3a^{2} \kappa_{3} }}{4} + \kappa_{1} } \right) = \frac{{\hat{z}_{0} }}{a}\cos \gamma } \\ \end{array} $$

(D1)

$$ \begin{array}{*{20}c} { - \frac{2}{{\Omega }}\left[ {\xi_{1} + n_{x} \xi_{2} \left( {\frac{{a^{2} \eta_{2} }}{4} + \eta_{0} } \right)} \right] = \frac{{\hat{z}_{0} }}{a}\sin \gamma } \\ \end{array} $$

(D2)