小波在信号检测中的应用

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年 月 日

摘 要

小波分析作为最新的时-频分析工具，在信号分析、图像处理、特征提取、故障诊断等各领域得到了广泛的应用。小波变换具有表征信号局部特征的能力和多分辨率的特征,因此，很适于探测信号中的瞬态和奇异现象, 并可展示其成份。

本文在综述小波变换的基本思想与具体性质和原理的基础上，重点介绍了小波在滚动轴承机械故障检测中的应用。滚动轴承机械故障信号分析中基函数的不同将导致对信号的观测角度和观测方法的不同，在小波基函数的选取方面 Fourier变换、短时Fourier变换和小波变换各自的基函数有着的本质区别。

本文通过比较故障诊断中常用的各种小波基函数的性能和特点，研究不同的故障信号特征与各种小波基函数的内在联系。利用连续小波变换方法将滚动轴承振动信号的特征信息转化为能量谱与尺度的关系，进而建立尺度和能量相对应的特征向量，为滚动轴承的快速诊断提供了新方法。本文提出一种应用 Daubechies 小波包多层分解、重构提取滚动轴承各部件的故障特征频率和能量特征，通过小波包多层分解确定滚动轴承机械振动的奇异点的方法, 实现故障的精确诊断。

关键词：小波分析、故障诊断、滚动轴承、多层分解

Abstract

Wavelet analysis as the latest time - frequency analysis tool in signal analysis, image processing, feature extraction, fault diagnosis and other fields has been widely used. Characterization of the signal wavelet transform has the ability of local features and characteristics of multi-resolution, therefore, it is very suitable for detection of transient signals and singular phenomenon, even to display its components.

General speaking the summary of this paper, the basic ideas of wavelet transform

and the specific nature, the most important of this paper is focusing on the wavelet applications of fault detection in the rolling machine. In the mechanical failure of the rolling bearing signal analysis, the different basis functions lead to a difference of signal point of observing views and observing methods, which are the essential differences among wavelet transform Fourier transform, short-time Fourier transform. In this paper, by comparing the performances and characteristics of a variety of common used small-wavelet fonctions in fault diagnosis, I research on the internal relations between different characteristics of the fault signal and wavelet fonctions. Making using of continuous wavelet transform method, this paper changes the characteristics of rolling bearing vibration signal information into the relationship of energy spectrum and measure, coming to the establishment a feature vector corresponding to energy and scale, creats the new method for the rapid diagnosis of rolling bearings. In order to accurately diagnosis of fault type, this paper proposes the application of multi-decomposition of Daubechies wavelet packet, reconfiguration of the extraction of fault characteristic frequency and energy feature in components rolling bearing components, by analysing multi-decomposition of Daubechies wavelet packet, we can clearly see the failure point of mechanical vibration in rolling bearing.

Key words：Wavelet analysis, fault diagnosis, rolling bearing, multi-decomposition

目 录

摘 要 Abstract

第1章 绪 论 ............................................................................................................................. 1

1.1 论文选题背景和意义 ........................................................................................................ 1 1.2 论文研究现状 .................................................................................................................... 1

1.2.1：小波分析现状 ...................................................................................................... 1 1.2.2：机械故障诊断现状 .............................................................................................. 3 1.3 论文研究方法和内容 ........................................................................................................ 6

第2章 小波分析的理论基础 .................................................................................................. 7

2.1 傅立叶分析及其优缺点 .................................................................................................... 7

2.1.1傅立叶变换(Fourier Transform)........................................................................... 7 2.1.2傅立叶变换的优点与缺点 .............................................................................. 7 2.2小波分析 ............................................................................................................................. 9 2.3小波基性能研究 ............................................................................................................... 11 2.4针对故障诊断处理的小波分类 ....................................................................................... 13 2.5小波变换对信号奇异性检测的基本原理 ....................................................................... 14

2.5. 1奇异性的定义 ...................................................................................................... 14 2.5. 2小波变换的卷积表达形式 .................................................................................. 14 2.5. 3小波变换的极值点、过零点与信号奇异性的联系 .......................................... 15 2.6 小波基的选择 .................................................................................................................. 16 2.7 最佳小波基的选取 .......................................................................................................... 17 2.8 Daubechies小波 ............................................................................................................... 18 2.9 小波分解与尺度选择 ...................................................................................................... 19

第3章 滚动轴承的故障及诊断技术 .................................................................................. 20

3.1滚动轴承的结构 ............................................................................................................... 21 3.2滚动轴承失效的基本形式 ............................................................................................... 21 3.3滚动轴承故障的振动诊断 ............................................................................................... 22 3.4 滚动轴承的振动机理及故障特征频率 .......................................................................... 23

3.4.1滚动轴承的振动机理 ........................................................................................... 23 3.4.2滚动轴承各元件单一缺陷的特征频率 ............................................................... 24 3.4.3由滚动轴承构造所引起的振动 ........................................................................... 25 3.4.5滚动轴承的非线性引发的振动 ........................................................................... 25 3.4.6滚动轴承损伤(缺陷〕而引起的振动 .................................................................. 26

第4章 MATLAB对故障奇异信号进行分析 ................................................................... 26

4.1检测第一类间断点 ........................................................................................................... 26 4.2检测第二类间断点 ........................................................................................................... 28 4.3滚动轴承的保持架机械振动信号的故障分析 ............................................................... 30 4.4滚动轴承的外滚道机械振动信号的故障分析 ............................................................... 32 4.5滚动轴承的内滚道机械振动信号的故障分析 ............................................................... 33

第5章 总结与展望 .................................................................................................................. 37

参考文献 ...................................................................................................................................... 38 至 谢 ........................................................................................................................................... 39 附 录 ........................................................................................................................................... 40

浙江理工大学本科毕业设计(论文)

clear;

load freqbrk; whos;

subplot(311); plot(freqbrk);

xlabel('时间');ylabel('幅值'); title('频率突变信号'); f=fft(freqbrk); subplot(313); plot(abs(f));

title('傅里叶变换后的信号示意图');

图4.3程序 load nearbrk; s=nearbrk;

[c,l]=wavedec(s,5,'db2'); subplot(7,1,1); plot(s);

title('使用db2小波分解5层：s=a5+d5+d4+d3+d2+d1'); Ylabel('s');

a5=wrcoef('a',c,l,'db2',5); subplot(7,1,2); plot(a5); Ylabel('a5');

d5=wrcoef('d',c,l,'db2',5); subplot(7,1,3); plot(d5); Ylabel('d5');

d4=wrcoef('d',c,l,'db2',4); subplot(7,1,4); plot(d4); Ylabel('d4');

d3=wrcoef('d',c,l,'db2',3); subplot(7,1,5); plot(d3); Ylabel('d3');

d2=wrcoef('d',c,l,'db2',2); subplot(7,1,6); plot(d2); Ylabel('d2');

d1=wrcoef('d',c,l,'db2',1); subplot(7,1,7); plot(d1);

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小波在信号检测中的应用

Ylabel('d1');

图4.4程序 clear;

load nearbrk; whos;

subplot(311); plot(nearbrk);

xlabel('时间');ylabel('幅值'); title('频率突变信号'); f=fft(nearbrk); subplot(313); plot(abs(f));

title('傅里叶变换后的信号示意图');

图4.5程序 clear; clc; n=1000; d=50 D=80;

t=0:pi/50:4*pi;

s=(1/2)*(n/60)*(1-(50/80)*cos(t)); num = length(s); for i =1 : 10

s(i+100)=s(i+100)+s(i+100)/10; end

subplot(511);plot(t,s); title('原信号');

[d,a]=wavedec(s,3,'db5'); a3=wrcoef('a',d,a,'db5',3); d3=wrcoef('d',d,a,'db5',3); d2=wrcoef('d',d,a,'db5',2); d1=wrcoef('d',d,a,'db5',1);

subplot(512);plot(a3);ylabel('近似信号a3'); title('小波分解后示意图');

subplot(513);plot(d3);ylabel('细节信号d3'); subplot(514);plot(d2);ylabel('细节信号d2'); subplot(515);plot(d1);ylabel('细节信号d1'); xlabel('旋转弧度');

图4.6程序：

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浙江理工大学本科毕业设计(论文)

clear; clc; n=1000; d=50 D=80;

t=0:pi/50:4*pi;

s=(1/2)*(n/60)*(1-(50/80)*cos(t)); num = length(s); for i =1 : 10

s(i+100)=s(i+100)+s(i+100)/10; end

subplot(511);plot(t,s); title('原信号');

whos;

subplot(311); plot(s);

xlabel('弧度');ylabel('频率'); title('频率突变信号'); f=fft(s);

subplot(313); plot(abs(f));

title('傅里叶变换后的信号示意图');

图4.7程序 clear; clc; n=1000; d=50 D=80; z=3

t=0:pi/50:4*pi;

s=(z/2)*(n/60)*(1+(d/D)*cos(t)); num = length(s); for i =1 : 10

s(i+100)=s(i+100)+s(i+100)/20; end

subplot(511);plot(t,s); title('原信号');

[d,a]=wavedec(s,3,'haar'); a3=wrcoef('a',d,a,'haar',3); d3=wrcoef('d',d,a,'haar',3);

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小波在信号检测中的应用

d2=wrcoef('d',d,a,'haar',2); d1=wrcoef('d',d,a,'haar',1);

subplot(512);plot(a3);ylabel('近似信号a3'); title('小波分解后示意图');

subplot(513);plot(d3);ylabel('细节信号d3'); subplot(514);plot(d2);ylabel('细节信号d2'); subplot(515);plot(d1);ylabel('细节信号d1'); xlabel('旋转弧度');

图4.8程序 clear; clc; n=1000; d=50 D=80; z=3

t=0:pi/50:4*pi;

s=(z/2)*(n/60)*(1+(d/D)*cos(t)); num = length(s); for i =1 : 10

s(i+100)=s(i+100)+s(i+100)/20; end

subplot(511);plot(t,s); title('原信号');

[d,a]=wavedec(s,3,'db5'); a3=wrcoef('a',d,a,'db5',3); d3=wrcoef('d',d,a,'db5',3); d2=wrcoef('d',d,a,'db5',2); d1=wrcoef('d',d,a,'db5',1);

subplot(512);plot(a3);ylabel('近似信号a3'); title('小波分解后示意图');

subplot(513);plot(d3);ylabel('细节信号d3'); subplot(514);plot(d2);ylabel('细节信号d2'); subplot(515);plot(d1);ylabel('细节信号d1'); xlabel('旋转弧度');

图4.11至4.14程序，

其中db小波根据图形采用各自的db波 clear; clc; n=1000; d=50 D=80;

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浙江理工大学本科毕业设计(论文)

z=3

t=0:pi/50:20*pi;

s=(z/2)*(n/60)*(1-(d/D)*cos(t)); num = length(s); for i =1 : 10

s(i+100)=s(i+100)+s(i+100)/20; end

subplot(511);plot(t,s); title('原信号');

[d,a]=wavedec(s,3,'db3'); a3=wrcoef('a',d,a,'db3',3); d3=wrcoef('d',d,a,'db3',3); d2=wrcoef('d',d,a,'db3',2); d1=wrcoef('d',d,a,'db3',1);

subplot(512);plot(a3);ylabel('近似信号a3'); title('小波分解后示意图');

subplot(513);plot(d3);ylabel('细节信号d3'); subplot(514);plot(d2);ylabel('细节信号d2'); subplot(515);plot(d1);ylabel('细节信号d1'); xlabel('旋转弧度');

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