﻿ 准零刚度隔振系统自适应控制
 舰船科学技术  2021, Vol. 43 Issue (11): 79-82    DOI: 10.3404/j.issn.1672-7649.2021.11.014 PDF

1. 海军工程大学 动力工程学院，湖北 武汉 430033;
2. 海军工程大学 舰船与海洋学院，湖北 武汉 430033

Adaptive control of quasi zero stiffness vibration isolation system
CHEN Xu-chao1, DIAO Ai-min2, YANG Qing-chao2, CHAI Kai2
1. College of Power Engineering, Naval University of Engineering, Wuhan 430033, China;
2. College of Ships and Oceans, Naval University of Engineering, Wuhan 430033, China
Abstract: Due to the particularity of its own structure, the quasi-zero stiffness vibration isolation system is widely used in low-frequency vibration isolation compared to traditional linear vibration isolation systems when the system parameters are ideally designed. However, because it is a non-linear system, the change of system parameters may change the vibration isolation equipment from a small amplitude to a large amplitude, which is not conducive to the normal operation of the equipment. First, construct a two-degree-of-freedom quasi-zero stiffness vibration isolation system dynamic model and analyze its dynamic characteristics; then, analyze the phase diagram and power spectrum of the system under different parameters, design an adaptive controller and analyze its stability Finally, through numerical simulation, the time history diagram under the condition of changing system parameters is obtained. The results show that the adaptive control algorithm can make the system always run to the ideal small amplitude motion state when the parameters are disturbed.
Key words: quasi zero stiffness     nonlinear system     parameter adaptation
0 引 言

1 两自由度准零刚度隔振系统建模

 $\left\{ \begin{array}{l} {{\dot x}_1} = {y_1}\text{，} \\ {{\dot y}_1} = - {\xi _1}({y_1} - {y_2}) - ({x_1} - {x_2}) + \gamma {({x_1} - {x_2})^2} -\\ \qquad {({x_1} - {x_2})^3} + f\cos \omega t \text{，} \\ {{\dot x}_2} = {y_2} \text{，} \\ {{\dot y}_2} = - w{\xi _2}{y_2} - w{k_2}{x_2} + w{\xi _1}({y_1} - {y_2}) + w({x_1} -\\ \qquad {x_2}) - w\gamma {({x_1} - {x_2})^2} + w{({x_1} - {x_2})^3} \text{。} \end{array} \right.$ (1)

2 动力学特性数值分析

 图 1 ${k_2}$ 为分岔参数时的局部分岔图 Fig. 1 ${k_2}$ is a local bifurcation diagram with bifurcation parameters

 图 2 ${k_2} = 1$ 时系统稳态相图 Fig. 2 Steady state phase diagram for ${k_2} = 1$

 图 3 ${k_2} = 1$ 时系统功率谱图 Fig. 3 Power spectrum of the system at ${k_2} = 1$

 图 4 ${k_2} = 2.1$ 时系统稳态相图 Fig. 4 Steady state phase diagram for ${k_2} = 2.1$

 图 5 ${k_2} = 2.1$ 时系统功率谱图 Fig. 5 Power spectrum of the system at ${k_2} = 2.1$

 图 6 ${k_2} = 2.5$ 时系统稳态相图 Fig. 6 Steady state phase diagram for ${k_2} = 2.5$

 图 7 ${k_2} = 2.5$ 时系统功率谱图 Fig. 7 Power spectrum of the system at ${k_2} = 2.5$
3 参数自适应控制 3.1 控制器的设计

 $\left\{ \begin{array}{l} {{\dot x}_1} = {y_1}\text{，} \\ {{\dot y}_1} = - {\text{0}}{\text{.1}}({y_1} - {y_2}) - ({x_1} - {x_2}) + {\text{2}}{({x_1} - {x_2})^2} -\\ \qquad{({x_1} - {x_2})^3} + {\text{6}}{\text{.8}}\cos {\text{1}}{\text{.6}}t \text{，} \\ {{\dot x}_2} = {y_2} \text{，}\\ {{\dot y}_2} = 0.5( - {\text{0}}{\text{.1}}{y_2} - {k_2}{x_2} + {\text{0}}{\text{.1}}({y_1} - {y_2}) +\\ \qquad({x_1} - {x_2}) - {\text{2}}{({x_1} - {x_2})^2} + {({x_1} - {x_2})^3}) \text{。} \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {{\dot x}_1} = {y_1} \text{，} \\ {{\dot y}_1} = - {\text{0}}{\text{.1}}({y_1} - {y_2}) - ({x_1} - {x_2}) + {\text{2}}{({x_1} - {x_2})^2} -\\ \qquad{({x_1} - {x_2})^3} + {\text{6}}{\text{.8}}\cos {\text{1}}{\text{.6}}t \text{，} \\ {{\dot x}_2} = {y_2} \text{，} \\ {{\dot y}_2} = 0.5( - {\text{0}}{\text{.1}}{y_2} - 0.1{x_2} + {\text{0}}{\text{.1}}({y_1} - {y_2}) + \\ \qquad({x_1} - {x_2}) - {\text{2}}{({x_1} - {x_2})^2} + {({x_1} - {x_2})^3})\text{，} \end{array} \right.$ (3)

 ${\dot k_2} = - \left( {{k_2} - 0.1} \right){\left( {{x_1} - {x_2}} \right)^2} \text{，}$ (4)

 $\left\{ \begin{array}{l} {{\dot x}_1} = {y_1} \text{，}\\ {{\dot y}_1} = - {\text{0}}{\text{.1}}({y_1} - {y_2}) - ({x_1} - {x_2}) + {\text{2}}{({x_1} - {x_2})^2} -\\ \qquad{({x_1} - {x_2})^3} + {\text{6}}{\text{.8}}\cos {\text{1}}{\text{.6}}t \text{，}\\ {{\dot x}_2} = {y_2} \text{，}\\ {{\dot y}_2} = 0.5( - {\text{0}}{\text{.1}}{y_2} - {k_2}{x_2} + {\text{0}}{\text{.1}}({y_1} - {y_2}) +\\ \qquad({x_1} - {x_2}) - {\text{2}}{({x_1} - {x_2})^2} + {({x_1} - {x_2})^3}) \text{，}\\ {{\dot k}_2} = - \left( {{k_2} - 0.1} \right){\left( {{x_1} - {x_2}} \right)^2} \text{。} \end{array} \right.$ (5)

 $V\left( t \right) = \frac{1}{2}{\left( {{k_2} - 0.1} \right)^2}\text{，}$ (6)

 $\begin{split} \dot V\left( t \right) =& \left( {{k_2} - 0.1} \right){{\dot k}_2} = \left( {{k_2} - 0.1} \right)\left( { - \left( {{k_2} - 0.1} \right){{\left( {{x_1} - {x_2}} \right)}^2}} \right) =\\ & - {\left( {{k_2} - 0.1} \right)^2}{\left( {{x_1} - {x_2}} \right)^2} \leqslant 0\text{，} \\[-12pt] \end{split}$ (7)

 $\underset{t\to \infty }{\mathrm{lim}}{\displaystyle \int_{t}^{0}\dot{V}\left(t\right){\rm{d}}t=}V\left(\infty \right)-V\left(0\right)=\text{有限值，}$ (8)

 $\mathop {\lim }\limits_{t \to \infty } \dot V\left( t \right) = 0 \text{，}$ (9)

 $\mathop {\lim }\limits_{t \to \infty } \left( {{k_2} - 0.1} \right) = 0\text{。}$ (10)

3.2 数值仿真

 图 8 初始参数 ${k_2} = 1.5$ 的 $t - {k_2}$ 曲线 Fig. 8 $t - {k_2}$ curve of initial parameter ${k_2} = 1.5$

 图 9 初始参数 ${k_2} = 1.5$ 的 $t - {x_2}$ 曲线 Fig. 9 $t - {x_2}$ curve of initial parameter ${k_2} = 1.5$

 图 10 初始参数 ${k_2} = 2.1$ 的 $t - {k_2}$ 曲线 Fig. 10 $t - {k_2}$ curve of initial parameter ${k_2} = 2.1$

 图 11 初始参数 ${k_2} = 2.1$ 的 $t - {x_2}$ 曲线 Fig. 11 $t - {x_2}$ curve of initial parameter ${k_2} = 2.1$

 图 12 初始参数 ${k_2} = 2.5$ 的 $t - {k_2}$ 曲线 Fig. 12 $t - {k_2}$ curve of initial parameter ${k_2} = 2.5$

 图 13 初始参数 ${k_2} = 2.5$ 的 $t - {x_2}$ 曲线 Fig. 13 $t - {x_2}$ curve of initial parameter ${k_2} = 2.5$
4 结　语

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