﻿ 船用超低频准零刚度隔振器设计及性能研究
 舰船科学技术  2022, Vol. 44 Issue (23): 37-42    DOI: 10.3404/j.issn.1672-7649.2022.23.008 PDF

Design and performance research of marine ultra-low frequency quasi-zero stiffness vibration isolator
YAN Sen-sen, CHEN Li-bin, ZHAO Chuan, LI Gang, ZHAO Rui-ying
The 707 Research Institute of CSSC, Tianjin 300131, China
Abstract: A kind of quasi-zero stiffness vibration isolator suitable for ship precision equipment is designed by combining silicon-fluorine rubber disc spring and electromagnetic spring in parallel. Firstly, the equipment mechanical model of the isolator was established. The current value was calculated when the stiffness was zero, basing on Maxwell’s electromagnetic principle, and the amplitude-frequency characteristic relationship of the system was deduced. Then the rubber material formula modeling method and magnetic 2D numerical calculation method were verified. On this base, the mechanical properties of the disc rubber spring were calculated, and the influence law of electromagnetic spring spacing, moving core material, coil current on electromagnetic negative stiffness force is obtained. According to quasi-zero stiffness theory, a quasi-zero stiffness vibration isolator is designed by combining electromagnetic spring and disc rubber spring. The research shows that the isolator has no amplification in low frequency vibration, and the vibration reduction effect is about 40dB higher than that of the traditional vibration isolator.
Key words: rubber disc spring     vibration unamplified     isolator     electromagnetic spring     quasi zero stiffness
0 引　言

1 准零刚度隔振器机理分析 1.1 准零刚度隔振器力学模型

 图 1 准零刚度隔振器示意图 Fig. 1 Diagram of quasi-zero stiffness vibration isolator

 ${f}_{m} =\frac{{\mu }_{0} · {N}^{2} · A·{i}^{2}}{2} · \left[\frac{1}{{\left(0.5 \times \left(L-h\right) + x \right)}^{2}} - \frac{1}{{\left(0.5 \times \left(L + h\right) - x\right)}^{2}}\right]\text{。}$ (1)

 $f = kx + \frac{{\mu }_{0} · {N}^{2} · A · {i}^{2}}{2} · \left[\frac{1}{{\left(0.5 \times \left(L - h\right) + x\right)}^{2}} - \frac{1}{{\left(0.5 \times \left(L + h\right) - x\right)}^{2}}\right]\text{。}$ (2)

 $\mathop f\limits^ \wedge = \mathop x\limits^ \wedge + \dfrac{{{{\left( {\mathop i\limits^ \wedge } \right)}^2}}}{2}\left[ {\dfrac{1}{{{{\left( {0.5 \times \left( {\alpha - 1} \right) + \mathop x\limits^ \wedge } \right)}^2}}} - \dfrac{1}{{{{\left( {0.5 \times \left( {\alpha + 1} \right) - \mathop x\limits^ \wedge } \right)}^2}}}} \right] \text{，}$ (3)

 $\mathop K\limits^ \wedge = 1 - \frac{{{{\left( {\mathop i\limits^ \wedge } \right)}^2}}}{4}\left[ {{{\left( {0.5 \times \left( {\alpha - 1} \right) + \mathop x\limits^ \wedge } \right)}^{ - \frac{3}{2}}} + {{\left( {0.5\left( {\alpha + 1} \right) \times - \mathop x\limits^ \wedge } \right)}^{ - \frac{3}{2}}}} \right] \text{，}$ (4)

$\mathop K\limits^ \wedge = 0$ $\mathop x\limits^ \wedge = 0.5$ ，可得在平衡位置，系统达到准零刚度效果时的无量纲电流大小为：

 ${\mathop I\limits^ \wedge _{qzs}} = {\left( {\dfrac{{{\alpha ^3}}}{2}} \right)^{0.25}} \text{。}$ (5)

 图 2 平衡位置处无量纲电流与力和刚度的关系示意图 Fig. 2 Diagram of the relation of current to force and stiffness at equilibrium position

 图 3 三阶泰勒展开与精确解对比关系示意图 Fig. 3 Diagram of the relation of third order Taylor expansion compared with exact result
 $\mathop f\limits^ \wedge = - 7.6\mathop y\limits^ \wedge - 47.8{(\mathop y\limits^ \wedge )^3} \text{。}$ (6)

1.2 准零刚度系统幅频特性

 图 4 准零刚度系统力学模型示意图 Fig. 4 Schematic diagram of mechanical model of quasi-zero stiffness system
 ${\left( {\mathop y\limits^ \wedge } \right)^{\prime \prime }} + \xi (\mathop y\limits^ \wedge )' + 7.6\mathop y\limits^ \wedge + 47.8{\left( {\mathop y\limits^ \wedge } \right)^3} = \mathop F\limits^ \wedge \cos (\varOmega \tau )。$ (7)

 $\mathop y\limits^ \wedge = a\cos (\varOmega \tau ) + b\sin (\varOmega \tau )\text{，}$ (8)

 ${\left( {\mathop y\limits^ \wedge } \right)^\prime } = - a\varOmega \sin (\varOmega \tau ) + b\varOmega \cos (\varOmega \tau ) \text{，}$ (9)

 $\mathop y\limits^ \wedge = a(\tau )\cos (\varOmega \tau ) + b(\tau )\sin (\varOmega \tau )\text{，}$ (10)

 ${\left( {\mathop y\limits^ \wedge } \right)^\prime } = - a(\tau )\varOmega \sin (\varOmega \tau ) + b(\tau )\varOmega \cos (\varOmega \tau ) \text{，}$ (11)

 \begin{aligned}{\left( {\mathop y\limits^ \wedge } \right)^\prime } = &a'(\tau )\cos (\varOmega \tau ) - \varOmega a(\tau )\sin (\varOmega \tau ) + \\ &b'(\tau )\sin (\varOmega \tau ) + \varOmega a(\tau )\cos (\varOmega \tau )\text{，}\end{aligned} (12)

 $a'(\tau )\cos (\varOmega \tau ) + b'(\tau )\sin (\varOmega \tau ) = 0\text{，}$ (13)

 \begin{aligned}{\left( {\mathop y\limits^ \wedge } \right)^{\prime \prime }} = & - \varOmega a'(\tau )\sin (\varOmega \tau ) - {\varOmega ^2}a(\tau )\cos (\varOmega \tau ) +\\ &\varOmega b'(\tau )\cos (\varOmega \tau ) - {\varOmega ^2}b(\tau )\sin (\varOmega \tau )\text{。}\end{aligned} (14)

 $\begin{array}{l}\mathrm{cos}(\varOmega \tau )(\xi {a}^{\prime }(\tau )-{\varOmega }^{2}a(\tau )+7.6a(\tau )+\varOmega {b}^{\prime }(\tau )+\xi \varOmega b(\tau )-\stackrel{\wedge }{F})+\\ \mathrm{sin}(\varOmega \tau )·(\xi {b}^{\prime }(\tau )-\varOmega {a}^{\prime }(\tau )- \xi \varOmega a(\tau )-{\varOmega }^{2}b(\tau )+7.6b(\tau ))+\\ 47.8{a}^{3}(\tau ){\mathrm{cos}}^{3}(\varOmega \tau )+143.4{a}^{2}(\tau )b(\tau )\mathrm{sin}(\varOmega \tau ){\mathrm{cos}}^{2}(\varOmega \tau )+ \\ 143.4a(\tau ){b}^{2}(\tau ){\mathrm{sin}}^{2}(\varOmega \tau )\mathrm{cos}(\varOmega \tau )+47.8{b}^{3}(\tau ){\mathrm{sin}}^{3}(\varOmega \tau )=0。\end{array}$ (15)

 \left\{ {\begin{aligned} a'(\tau ) = & - 143.4{a^2}(\tau )b(\tau )/(8\varOmega ) + 0.5\xi a(\tau ) +\\ &143.4{b^3}(\tau )/(8\varOmega ) - 0.5\varOmega b(\tau ) + 3.8b(\tau )/\varOmega\text{，} \\ b'(\tau ) =& - 143.4a(\tau ){b^2}(\tau )/(8\varOmega ) + 0.5\varOmega a(\tau ) - 3.8a(\tau )/\varOmega -\\ &143.4{a^3}(\tau )/(8\varOmega ) - 0.5\xi b(\tau ) + 0.5{{\mathop F\limits^ \wedge \mathord{\left/ {\vphantom {{\mathop F\limits^ \wedge } \varOmega }} \right. } \varOmega }} \text{。} \end{aligned}} \right. (16)

 $\left\{\begin{array}{llll}{r}^{\prime }=(-r\xi \varOmega +\stackrel{\wedge }{F}\mathrm{sin}(\theta ))/(2\varOmega ）\text{，}\\ {\theta }^{\prime }=(-143.4{r}^{3}-30.4r+4r{\varOmega }^{2}+4\stackrel{\wedge }{F}\mathrm{cos}(\theta ))/8r\varOmega \text{。}\end{array}\right.$ (17)

 $\dfrac{{{r^2}{{(143.4{r^2} - 4{\varOmega ^2} + 30.4)}^2}}}{{16{{(\mathop F\limits^ \wedge )}^2}}} + \dfrac{{{\xi ^2}{r^2}{\varOmega ^2}}}{{{{(\mathop F\limits^ \wedge )}^2}}} = 1\text{。}$ (18)
2 碟型橡胶弹簧力学性能计算 2.1 橡胶公式建模方法验证

 $\begin{split}{E}_{t}=&{K}_{G}\times {K}_{P}\times {K}_{M}\times {E}_{0}=\dfrac{{A}_{总}}{{A}_{自由}}\times \\ &\left(\dfrac{A+\Delta A}{A}\times \dfrac{h+\Delta h}{h}\right)\times \left({e}^{\dfrac{Hs}{100}}-1\right)\times {e}^{(0.02Hs-0.4)}\text{。}\end{split}$ (19)

 图 5 橡胶垫尺寸示意图 Fig. 5 Rubber pad dimensions

2.2 碟型橡胶弹簧力-位移关系

 图 6 橡胶碟型弹簧尺寸和力学特性结果图 Fig. 6 The dimensions and mechanical properties of rubber disc spring

3 负刚度电磁力计算和准零刚度的实现 3.1 电磁力计算方法验证

 图 7 电磁力计算方法验证结果 Fig. 7 The verification result diagram of electromagnetic force calculation method

3.2 隔振器参数对电磁负刚度力影响

 图 8 不同参数对电磁负刚度力影响结果 Fig. 8 The effect of different parameters on electromagnetic negative stiffness force

3.3 准零刚度的实现与减振效果计算

 图 9 准零刚度的实现与减振效果图 Fig. 9 Realization of quasi-zero stiffness and vibration reduction effect diagram

4 结　语

1）在行程为±4 mm时，碟型弹簧、电磁弹簧产生的刚度力均为近似线性，依据该优势，可直接进行数值叠加，准零刚度更易实现，因此在设计时应让两者均向线性靠近；

2）电磁铁间距对电磁负刚度力影响较大，在保证动铁芯不与磁铁磕碰前提下，减小磁铁间距可提高负刚度力的大小；

3）动芯材料为永磁材料时，可提高负刚度力，但整体力学特性呈现明显的非线性，不利于准零刚度的实现，因此动芯应规避此类材料；

4）本文准零刚度隔振器可达到全频带无振动放大效果，且在大于3 Hz时，相对于传统橡胶减振器，可使减振效果提高约40 dB，适用于船用精密设备的减振设计。

 [1] CREMER L, HECKL M, UNGAR E E. Structure-borne Sound [M]. Second Edition. Berlin: Springer-Verlag, 1988. [2] 王汀, 于沛, 李晶. 振动条件下平台惯导系统误差抑制技术研究[J]. 振动与冲击, 2019, 38(15): 6. WANG Ting, YU Pei, LI Jing. Study on error suppression technology of platform Inertial Navigation system under vibration condition[J]. Shock and Vibration, 2019, 38(15): 6. [3] 党建军, 罗建军, 万彦辉. 挠性捷联惯组振动环境下适应性及导航精度分析[J]. 弹箭与制导学报, 2010, 30(1): 4. DANG Jianjun, LUO Jianjun, WAN Yanhui. Adaptability and navigation accuracy analysis of flexible strapdown inertial navigation group under vibration environment[J]. Journal of Arrows and Guidance, 2010, 30(1): 4. [4] 赵钟磊. 磁存储设备的振动冲击电磁主动控制技术的研究[D]. 西安: 西安电子科技大学, 2002. [5] 李东海, 赵寿根, 何玉金, 等. 含有时滞控制的准零刚度隔振器的隔振性能研究[J]. 西北工业大学学报, 2018, 36(6): 8. LI Donghai, ZHAO Shougen, HE Yujin, et al. Research on vibration isolation performance of quasi-zero stiffness vibration isolator with time delay control [J]. Journal of Northwestern Polytechnical University, 2018, 36(6): 8. [6] 刘琪, 李占龙, 王建梅, 等. 准零刚度低频隔振技术的研究进展[J]. 机械强度, 2021. LIU Qi, LI Zhanlong, WANG Jianmei. Research progress of quasi-zero stiffness low frequency vibration isolation technology [J]. Journal of Mechanical Strength, 2021. [7] CARRELLA A, BRENNAN M J, WATERS T P, et al. On the design of a high-static-low-dynamic stiffness isolator using linear mechanical springs and magnets[R]. In special Issue: Euromech Colloquium 483, Geometrically Non-linear Vibrations of Structures, 2008. [8] XU D, YU Q, ZHOU J, et al. Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic[J]. Journal of Sound and Vibration, 2019, 332(14): 3377-3389. [9] HUANG X, LIU X, SUN J, et al. Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: A theoretical and experimental study[J]. Journal of Sound & Vibration, 2020, 333(4): 1132-1148. [10] 苏攀, 吴杰长, 刘树勇, 等. 准零刚度振动系统反馈控制动态特性及试验研究[J]. 船舶力学, 2020, 24(4): 11. SU Pan, WU Jiechang, LIU Shuyong. Dynamic characteristics and experimental study of feedback control for quasi-zero stiffness vibration system[J]. Journal of Ship Mechanics, 2020, 24(4): 11.