﻿ 基于PID反馈的两级超低频隔振系统理论分析及仿真
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 大地测量与地球动力学  2020, Vol. 40 Issue (9): 976-980  DOI: 10.14075/j.jgg.2020.09.019

### 引用本文

HU Yuanwang, ZOU Tong, ZHANG Li, et al. Analysis and Simulation of Two-Stage Ultra-Low Frequency Vibration Isolation System Based on PID Feedback[J]. Journal of Geodesy and Geodynamics, 2020, 40(9): 976-980.

### Foundation support

Scientific Research Fund of Institute of Seismology and Institute of Crustal Dynamics, CEA, No.IS201826283; The Spark Program of Earthquake Technology of CEA, No.XH18029Y.

### Corresponding author

ZOU Tong, PhD, senior engineer, majors in geophysical instruments, E-mail:tong.zou@eqhb.gov.cn.

### 第一作者简介

HU Yuanwang, associate researcher, majors in geophysical instruments, E-mail:Hu19881003@163.com.

### 文章历史

1. 中国地震局地震研究所地震大地测量重点实验室，武汉市洪山侧路40号，430071;
2. 中国地震局地壳应力研究所武汉科技创新基地，武汉市洪山侧路40号，430071

1 两级隔振系统运动模型

 图 1 两级隔振系统示意图 Fig. 1 Diagram of two-stage vibration isolation system

 $\left\{ {\begin{array}{*{20}{l}} {{m_1}{{\ddot x}_1} = {k_1}\left( {{x_3} - {x_1}} \right) + {\beta _1}\left( {{{\dot x}_3} - {{\dot x}_1}} \right) + {k_2}\left( {{x_2} - {x_1}} \right) + {\beta _2}\left( {{{\dot x}_2} - {{\dot x}_1}} \right)}\\ {{m_2}{{\ddot x}_2} = {k_2}\left( {{x_1} - {x_2}} \right) + {\beta _2}\left( {{{\dot x}_1} - {{\dot x}_2}} \right)} \end{array}} \right.$ (1)

 $\frac{{X}_{2}\left(s\right)}{{X}_{3}\left(s\right)}=\frac{\left({\beta }_{1}{s}+{k}_{1}\right)\left({\beta }_{2}{s}+{k}_{2}\right)}{\left({m}_{1}{s}^{2}+{\beta }_{1}{s}+{k}_{1}\right)\left({m}_{2}{s}^{2}+{\beta }_{2}{s}+{k}_{2}\right)+{m}_{1}{s}^{2}\left({\beta }_{2}{s}+{k}_{2}\right)}$ (2)

 $\left\{ {\begin{array}{*{20}{l}} {{m_1}{{\ddot x}_1} = {k_1}\left( {{x_3} - {x_1}} \right) + {\beta _1}\left( {{{\dot x}_3} - {{\dot x}_1}} \right) + {k_2}\left( {{x_2} - {x_1}} \right) + {\beta _2}\left( {{{\dot x}_2} - {{\dot x}_1}} \right) + f\left( t \right)\left( {{x_2} - {x_1}} \right)}\\ {{m_2}{{\ddot x}_2} = {k_2}\left( {{x_1} - {x_2}} \right) + {\beta _2}\left( {{{\dot x}_1} - {{\dot x}_2}} \right)} \end{array}} \right.$ (3)

 $\frac{{{X_2}\left( s \right)}}{{{X_3}\left( s \right)}} = \frac{{\left( {{\beta _1}s + {k_1}} \right)\left( {{\beta _2}s + {k_2}} \right)}}{{\left( {{m_1}{s^2} + {\beta _1}s + {k_1}} \right)\left( {{m_2}{s^2} + {\beta _2}s + {k_2}} \right) + {m_1}{s^2}\left( {{\beta _2}s + {k_2} + f\left( s \right)} \right)}}$ (4)

σ1=β1/m1σ2=β2/m2σ12=β2/m1ω12=k1/m1ω22=k2/m2ω122=k2/m1，代入式(4)并展开得：

 $\frac{{{X_2}\left( s \right)}}{{{X_3}\left( s \right)}} = \frac{{{\sigma _1}{\sigma _2} + {\sigma _1}{\omega _2}^2 + {\sigma _2}{\omega _1}^2 + {\omega _1}^2{\omega _2}^2}}{{{s^4} + \left( {{\sigma _1} + {\sigma _2} + {\sigma _{12}}} \right){s^3} + \left[ {f\left( s \right)/{m_1} + {\omega _1}^2 + {\omega _2}^2 + {\omega _{12}}^2 + {\sigma _1}{\sigma _2}} \right]{s^2} + \left( {{\sigma _1}{\omega _2}^2 + {\sigma _2}{\omega _1}^2} \right)s + {\omega _1}^2{\omega _2}^2}}$ (5)
2 闭环传递函数分析

 $f(s) = {k_p} + {k_d}s$ (6)

 $\frac{{{X_{\rm{2}}}(s)}}{{{X_{\rm{3}}}(s)}} = \frac{{{\sigma _1}{\sigma _{\rm{2}}} + {\sigma _1}{\omega _{\rm{2}}}^2 + {\sigma _2}{\omega _{\rm{1}}}^2 + {\omega _{\rm{1}}}^2{\omega _{\rm{2}}}^2}}{{{s^4} + ({\sigma _1} + {\sigma _2} + {\sigma _{12}} + {\sigma _d}){s^3} + ({\omega _p}^2 + {\omega _{\rm{1}}}^2 + {\omega _{\rm{2}}}^2 + {\omega _{{\rm{12}}}}^2 + {\sigma _1}{\sigma _{\rm{2}}}){s^2} + ({\sigma _1}{\omega _{\rm{2}}}^2 + {\sigma _2}{\omega _{\rm{1}}}^2)s + {\omega _{\rm{1}}}^2{\omega _{\rm{2}}}^2}}$ (7)

 $\frac{{{X_2}(s)}}{{{X_3}(s)}} \approx \frac{{({\sigma _1}s + {\omega _1}^2)({\sigma _2}s + {\omega _2}^2)}}{{({s^2} + {\sigma _d}s + {\omega _p}^2)({s^2} + {C_1}s + {\omega _1}^2{\omega _2}^2/{\omega _p}^2)}}$ (8)

 ${C_1} \approx \frac{{{\sigma _1}{\omega _2}^2 + {\sigma _2}{\omega _1}^2}}{{{\omega _p}^2}} - {\sigma _d}\frac{{{\omega _1}^2{\omega _2}^2}}{{{\omega _p}^4}}$ (9)

 $\left| {{H_{{\rm{j}}w}}} \right| = \left| {\frac{{{X_2}({\rm{j}}w)}}{{{X_3}({\rm{j}}w)}}} \right| \approx \frac{{{\omega _1}^2{\omega _2}^2/{\omega _p}^2}}{{\sqrt {{{({\omega _1}^2{\omega _2}^2/{\omega _p}^2 - {\omega ^2})}^2} + {C_1}^2{\omega ^2}} }}$ (10)

 ${\omega _0} = {\omega _1}{\omega _2}/{\omega _p} = \sqrt {\frac{{{k_1}{k_2}}}{{{k_p}{m_2}}}}$ (11)

 $f(s) = {k_p} + {k_d}s + {k_i}/s$ (12)

 $\frac{{{X_{\rm{2}}}(s)}}{{{X_{\rm{3}}}(s)}} = \frac{{{\sigma _1}{\sigma _{\rm{2}}} + {\sigma _1}{\omega _{\rm{2}}}^2 + {\sigma _2}{\omega _{\rm{1}}}^2 + {\omega _{\rm{1}}}^2{\omega _{\rm{2}}}^2}}{{{s^4} + ({\sigma _1} + {\sigma _2} + {\sigma _{12}} + {\sigma _d}){s^3} + ({\omega _p}^2 + {\omega _{\rm{1}}}^2 + {\omega _{\rm{2}}}^2 + {\omega _{{\rm{12}}}}^2 + {\sigma _1}{\sigma _{\rm{2}}}){s^2} + ({\sigma _1}{\omega _{\rm{2}}}^2 + {\sigma _2}{\omega _{\rm{1}}}^2 + {\sigma _i})s + {\omega _{\rm{1}}}^2{\omega _{\rm{2}}}^2}}$ (13)

 $\frac{{{X_2}(s)}}{{{X_3}(s)}} = \frac{{({\sigma _1}s + {\omega _1}^2)({\sigma _2}s + {\omega _2}^2)}}{{({s^2} + {\sigma _d}s + {\omega _p}^2)({s^2} + {C_2}s + {\omega _1}^2{\omega _2}^2/{\omega _p}^2)}}$ (14)

 ${C_2} \approx \frac{{{\sigma _1}{\omega _2}^2 + {\sigma _2}{\omega _1}^2 + {\sigma _i}}}{{{\omega _p}^2}} - {\sigma _d}\frac{{{\omega _1}^2{\omega _2}^2}}{{{\omega _p}^4}}$ (15)

3 两级隔振系统仿真 3.1 隔振特性仿真

 图 2 闭环两级隔振系统Bode图 Fig. 2 The Bode diagram of closed-loop two-stage vibration isolation system
3.2 各反馈参数对系统影响的仿真验证

 图 3 反馈比例系数kp对系统的影响 Fig. 3 The influence of the feedback scale factor kp on the system

 图 4 反馈积分系数ki对系统的影响 Fig. 4 The influence of the feedback integral coefficient ki on the system

 图 5 反馈微分系数kd对系统的影响(kd取值较小) Fig. 5 The influence of the feedback differential coefficient kd on the system (kd is small)

 图 6 反馈微分系数kd对系统的影响(kd取值较大) Fig. 6 The influence of the feedback differential coefficient kd on the system (kd is large)

 图 7 反馈微分系数kd过大时系统发散 Fig. 7 The system diverges when the feedback differential coefficient kd is too large

4 结语

1) 应用于绝对重力仪中的两级超低频隔振系统，采用PID反馈代替PD反馈，可更方便地调节系统等效阻尼系数，有效避免等效阻尼系数为负值，从而有利于在调试过程中实现系统的稳定。

2) 两级隔振系统采用PID反馈方式时，等效固有周期主要受比例系数kp影响，kp越大，等效固有周期越长，隔振效果越好。选取合适的kp后，积分系数ki和微分系数kd共同影响整个系统的阻尼，且ki占主导地位，决定系统的稳定性。

 [1] Ju L W, Blair D G. Low Resonant Frequency Cantilever Spring Vibration Isolator for Gravitational Wave Detectors[J]. Review of Science Instruments, 1994, 65(11): 3 482-3 488 DOI:10.1063/1.1145218 (0) [2] Lee R R. SuperSpring: A New Type of Low Frequency Vibration Isolator[D]. Boulder: University of Corolado, 1983 (0) [3] Niebauer T M, Sasagawa G S, Faller J E, et al. A New Generation of Absolute Gravimeters[J]. Metrologia, 2005, 32(3): 159-180 (0) [4] 张兵, 滕云田, 邢丽莉, 等. 激光干涉绝对重力仪参考棱镜隔振系统仿真[J]. 地球物理学报, 2017, 60(11): 4 221-4 230 (Zhang Bing, Teng Yuntian, Xing Lili, et al. The Simulation of Reference Corner Cube Vibration Isolation System of Laser Interference Absolute Gravimeter[J]. Chinese Journal of Geophysics, 2017, 60(11): 4 221-4 230) (0) [5] Hensley J M, Peters A, Chu S. Active Low Frequency Vertical Vibration Isolation[J]. Review of Science Instruments, 1999, 70(6): 2 735-2 741 DOI:10.1063/1.1149838 (0) [6] Fteier C. Measurement of Local Gravity Using Atom Interferometer[D]. Berlin: Humboldt University, 2010 (0) [7] Zhou M K, Hu Z K, Duan X C, et al. Performance of a Cold-Atom Gravimeter with an Active Vibration Isolator[J]. Physical Review A, 2012, 86(4): 43 630 DOI:10.1103/PhysRevA.86.043630 (0) [8] Tang B, Zhou L, Xiong Z Y, et al. A Programmable Broadband Low Frequency Active Vibration Isolation System for Atom Interferometry[J]. Review of Science Instruments, 2014, 85(9): 93 109 DOI:10.1063/1.4895911 (0) [9] 张华兴, 田蔚, 丁国龙. 绝对重力仪中几种隔振系统的对比测试[J]. 大地测量与地球动力学, 2017, 37(7): 767-770 (Zhang Huaxing, Tian Wei, Ding Guolong. Comparison Experiments of Several Kinds of Vibration Isolation Systems in Absolute Gravimeter[J]. Journal of Geodesy and Geodynamics, 2017, 37(7): 767-770) (0) [10] 李刚, 胡华, 伍康, 等. 基于零长弹簧的超低频垂直隔振系统研究[J]. 振动与冲击, 2015, 34(7): 33-37 (Li Gang, Hu Hua, Wu Kang, et al. Ultra-Low Frequency Vertical Vibration Isolation System Based on Zero-Length Spring[J]. Journal of Vibration and Shock, 2015, 34(7): 33-37) (0) [11] Li G, Hu H, Wu K, et al. Ultra-Low Frequency Vertical Vibration Isolator Based on LaCoste Spring Linkage[J]. Review of Scientific Instruments, 2014, 85(10): 104 502 DOI:10.1063/1.4897488 (0) [12] 王观, 胡华, 伍康, 等. 基于两级摆杆结构的超低频垂直隔振系统[J]. 物理学报, 2016, 65(20): 44-50 (Wang Guan, Hu Hua, Wu Kang, et al. Ultra-Low-Frequency Vertical Vibration Isolator Based on a Two-Stage Beam Structure[J]. Acta Physica Sinica, 2016, 65(20): 44-50) (0) [13] Wang G, Wu K, Hu H, et al. Ultra-Low-Frequency Vertical Vibration Isolator Based on a Two-Stage Beam Structure for Absolute Gravimetry[J]. Review of Scientific Instruments, 2016, 87(10): 105 101 DOI:10.1063/1.4963676 (0) [14] Peterson J.Observations and Modeling of Seismic Background Noise[R]. Albuquerque: Department of Interior Geological Survey, 1993 (0)
Analysis and Simulation of Two-Stage Ultra-Low Frequency Vibration Isolation System Based on PID Feedback
HU Yuanwang1,2     ZOU Tong1,2     ZHANG Li1,2     MA Wugang1,2
1. Key Laboratory of Earthquake Geodesy, Institute of Seismology, CEA, 40 Hongshance Road, Wuhan 430071, China;
2. Wuhan Base of Institute of Crustal Dynamics, CEA, 40 Hongshance Road, Wuhan 430071, China
Abstract: In this paper, we establish the motion model of the two-stage vibration isolation system, then derive and analyze the closed-loop transfer function of the system. The analysis results show that when PID feedback is adopted, the equivalent damping coefficient of the system can be adjusted more conveniently than the PD feedback used in the current ultra-long spring(superspring). The equivalent natural period of the system is mainly affected by the feedback proportional coefficient kp; the larger the kp is, the longer the equivalent natural period, and the system will have better vibration isolation performance. After kp is selected, the integral coefficient ki together with the differential coefficient kd will affect the damping of the system, and ki is dominant, which determines the stability of the system. The accuracy of the closed-loop transfer function analysis results are verified by simulation, which show that the vibration isolation effect of the system can reach -70 dB at 1 Hz.
Key words: ultra-low frequency; two-stage vibration isolation; PID feedback; absolute gravimeter