循环平稳和解调频技术在故障诊断中的研究和应用
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摘要
旋转机械周期性运行方式使得所产生的振动信号具有一定的循环平稳性,因此利用循环平稳分析方法能够提取出平稳假设下所不能得到的隐藏的故障特征信息。利用齿轮箱中齿轮和轴承故障时振动信号的循环平稳特性的差异,建立齿轮箱的齿轮和轴承综合故障诊断方法是本文研究工作的重点。为了利用循环平稳分析方法,对描述循环平稳特征的循环统计量进行详细地研究是非常必要的。
由于循环平稳信号的非平稳特性,循环平稳统计量的估计值与信号的时间起点密切相关,也就是说在计算循环自相关函数时,选择不同的时延方式将会得到不同的计算结果,同时也会影响谱相关函数的频移方式,这种状况的出现导致了循环平稳信息的多样性。因此本文推导了不同形式谱相关函数间的对应关系,以便从不同的谱相关函数中提取出唯一的循环平稳信息。
时域的离散化必然会导致循环自相关函数在循环频率域的周期化,由于均方带限信号的时延二次变换在时间变量上的带宽是原信号的两倍,因此对于刚好满足Nyquist采样定理的离散信号,其循环自相关函数在循环频率域将不可避免地发生混迭。然而周期化后的谱相关函数则表现为分布在频率和循环频率谱相关平面上的独立菱形区域,不会发生混迭。基于以上分析,提出了对谱相关函数的菱形支撑区进行逆Fourier变换的循环自相关函数的无混迭算法,解决了直接利用离散信号所引起的混迭问题。本文在证明了循环周期图是一无偏非一致估计的基础上,进一步证明了时域平均和频域平滑对循环周期图的影响。从而发现,由于估计样本的有限性,对于含加性平稳噪声的周期信号(一阶循环平稳),谱相关函数在循环频率域并不能完全抑制平稳噪声对周期成分的影响。功率谱和谱相关函数在处理这类信号时将会得到几乎相近的信噪比(谱图中周期成分与背景噪声能量的比值),利用谱相关函数并不会比功率谱更加有效。
通过分析齿轮箱中齿轮和轴承的运行机理,发现齿轮故障所引起的齿轮箱振动信号具有一阶循环平稳特性,而轴承故障所引起的齿轮箱振动信号却具有二阶循环平稳特性。从循环平稳与纯循环平稳理论的研究中发现,为了有效地抽取信号的循环平稳特性,必须剔除所有低阶循环平稳以排除干扰。将这一思路用在齿轮箱故障诊断中,提出了齿轮箱的齿轮和轴承综合故障诊断方法。利用自适应谱线增强器(或同步平均)将齿轮箱振动信号中的一阶循环平稳成分提取出来,将其能量的改变作为齿轮箱中齿轮故障的判断依据,对自适应谱线增强器的残余信号再利用循环自适应滤波器,抽取出其中的二阶循环平稳分量,则作为齿轮箱中轴承故障的判据,从而达到有效地确定齿轮箱中齿轮和轴承故障状态的目的。
本文在详细分析了决定调频信号的边频带分布的第一类Bessel函数基本性质的基础上,对调频调幅现象共存时信号调制边带的非对称分布进行了理论上的解释,确定了调频调幅在各种混合模式下信号调制边带对称分布的条件。并找到了平方解调方法不能对调频信号解调的根本原因——调频信号的所有间隔相等的两边频的复幅值乘积之和等于零。为了克服常规的解调方法不能处理调频信号的这一缺陷,必须打破调频信号边带的特殊性,为此本文提出基于时延二次非线性的相关域和循环相关域解调频分析方法,以实现对调频信号的解调。
最后利用了齿轮箱振动信号对本文所提出的方法进行了验证,振动信号来自对韶关宏大齿轮箱厂生产的SG135-2型变速器相应的实验,实验故障类型包括了变速器二档齿面剥落,五档断齿,输出轴轴承外圈故障和内圈故障。
Due to the periodicity of runing mode of rotating machinery, the vibration signals obtained from these machines are cyclostationary. Therefore, using cyclostationary methods can effectively extract fault’s features from these vibration signals, but it will be very difficult in the hypothesis of stationarity. The main aim of this paper is to make use of the difference of the cyclostationarity in gearbox’s vibration signals when gears and bearings have faults, and establish a gearbox’s integrated diagnosis method based on cyclostationarity. To this goal, the study of the cyclic statistic is an important step before using it.
Because of the nonstationarity of cyclostationary signals, the estimators of the cyclic statistic depend on the data origination. That means different time-lag pattern of the Cyclic Autocorrelation Function (CAF) will arrive at different results and different frequency-shift mode of the Spectral Correlation Function (SCF). This will lead to some confusion when extracting the cyclostationarity by different estimator expressions for a same signal. Therefore, the relationship of SCF between different expressions is established. Through this method, the exclusive cyclostationarity can be obtained from the results of different expressions.
The discretization in time domain will result in periodic extension in the frequency and cyclic frequency domain. Because the time-lag quadric transformation of meansquare bandlimited signals has double bandwidth in the time parameter, under the Nyquist sample rate, its CAF will appear aliasing in the cyclic frequency inevitability. But for its SCF, it distributes in the frequency and cyclic frequency plane independently. Based on this discover, an alias-free CAF based on inverse Fourier transformation of SCF in the diamond shape support area is brought forward.
The quality of estimator of SCF is deduced and a conclusion is made that cyclic periodogram is unbiase and inconsistent. Further more the influence of time average and frequency smooth are deduced too. Because of the limitation of data length, for periodic signals with adding stationary noise (cyclostationary at the first order), SCF do not totally suppress the effect of stationary noise to periodic components. Either PSD or SCF, the SNR of periodic components to background noise is subequal. Therefore, SCF seems more suitable for cyclostationary signals at the second order than at the first order.
Through study the vibration signals of gearboxs, we find out that the vibration signals induced by a fault gear is cyclostaionary at the first order, and the signals induced by a fault bearing is cylcostationary at the second order. From the knowledge of pure cyclostationarity and cyclostationarity, we know that cyclostationarity at all lower order must be subtracted to define the accurate order of cyclostationarity in the signals. Therefore a new gearbox integrate fault diagnosis method based on cyclostationarity is set up. Using an adaptive line enhancer and a cyclic adaptive filter, the gearbox vibration signal is separated into three parts, i.e., a first-order cyclostationary part and a second-order cyclostationary part and a stationary part. The firs part is used to judge the fault state of gears, and the second one is for bearings. Therefore, the fault whether of gears or bearings in the gearbox can be determined based on them.
In order to explain the feature of amplitude and phase of FM signals, Bessel function of the first kind is studied, because it is a determinant factor to sideband of FM signals. Based on this, the sideband distribution of signals with AM and FM mixed together is analyzed, and the condition of symmetric sideband distribution is determined in spite of the mix mode of AM and MF is addition or multiplication. Square transform isn’t suitable for FM signals, because of the special feature of Bessel function of the first kind. To overcome this limitation, two kinds of method were established. They are based on asymmetric filtering and time-lag quadratic nonlinearity transformation, respectively. Through these methods, FM signals are transformed to signals which can be demodulated by normal demodulation methods, so FM demodulation is achieve.
At last, all methods in this study are tested by the vibration signals obtained from gearbox SG135-2, which was made by HongDa Gearbox Company in ShaoGuan. In these tests, four fault modes are used, including second gear spalling, fifth gear broken, bearings in the output shaft with a outer race fault and a inner race fault.
由于循环平稳信号的非平稳特性,循环平稳统计量的估计值与信号的时间起点密切相关,也就是说在计算循环自相关函数时,选择不同的时延方式将会得到不同的计算结果,同时也会影响谱相关函数的频移方式,这种状况的出现导致了循环平稳信息的多样性。因此本文推导了不同形式谱相关函数间的对应关系,以便从不同的谱相关函数中提取出唯一的循环平稳信息。
时域的离散化必然会导致循环自相关函数在循环频率域的周期化,由于均方带限信号的时延二次变换在时间变量上的带宽是原信号的两倍,因此对于刚好满足Nyquist采样定理的离散信号,其循环自相关函数在循环频率域将不可避免地发生混迭。然而周期化后的谱相关函数则表现为分布在频率和循环频率谱相关平面上的独立菱形区域,不会发生混迭。基于以上分析,提出了对谱相关函数的菱形支撑区进行逆Fourier变换的循环自相关函数的无混迭算法,解决了直接利用离散信号所引起的混迭问题。本文在证明了循环周期图是一无偏非一致估计的基础上,进一步证明了时域平均和频域平滑对循环周期图的影响。从而发现,由于估计样本的有限性,对于含加性平稳噪声的周期信号(一阶循环平稳),谱相关函数在循环频率域并不能完全抑制平稳噪声对周期成分的影响。功率谱和谱相关函数在处理这类信号时将会得到几乎相近的信噪比(谱图中周期成分与背景噪声能量的比值),利用谱相关函数并不会比功率谱更加有效。
通过分析齿轮箱中齿轮和轴承的运行机理,发现齿轮故障所引起的齿轮箱振动信号具有一阶循环平稳特性,而轴承故障所引起的齿轮箱振动信号却具有二阶循环平稳特性。从循环平稳与纯循环平稳理论的研究中发现,为了有效地抽取信号的循环平稳特性,必须剔除所有低阶循环平稳以排除干扰。将这一思路用在齿轮箱故障诊断中,提出了齿轮箱的齿轮和轴承综合故障诊断方法。利用自适应谱线增强器(或同步平均)将齿轮箱振动信号中的一阶循环平稳成分提取出来,将其能量的改变作为齿轮箱中齿轮故障的判断依据,对自适应谱线增强器的残余信号再利用循环自适应滤波器,抽取出其中的二阶循环平稳分量,则作为齿轮箱中轴承故障的判据,从而达到有效地确定齿轮箱中齿轮和轴承故障状态的目的。
本文在详细分析了决定调频信号的边频带分布的第一类Bessel函数基本性质的基础上,对调频调幅现象共存时信号调制边带的非对称分布进行了理论上的解释,确定了调频调幅在各种混合模式下信号调制边带对称分布的条件。并找到了平方解调方法不能对调频信号解调的根本原因——调频信号的所有间隔相等的两边频的复幅值乘积之和等于零。为了克服常规的解调方法不能处理调频信号的这一缺陷,必须打破调频信号边带的特殊性,为此本文提出基于时延二次非线性的相关域和循环相关域解调频分析方法,以实现对调频信号的解调。
最后利用了齿轮箱振动信号对本文所提出的方法进行了验证,振动信号来自对韶关宏大齿轮箱厂生产的SG135-2型变速器相应的实验,实验故障类型包括了变速器二档齿面剥落,五档断齿,输出轴轴承外圈故障和内圈故障。
Due to the periodicity of runing mode of rotating machinery, the vibration signals obtained from these machines are cyclostationary. Therefore, using cyclostationary methods can effectively extract fault’s features from these vibration signals, but it will be very difficult in the hypothesis of stationarity. The main aim of this paper is to make use of the difference of the cyclostationarity in gearbox’s vibration signals when gears and bearings have faults, and establish a gearbox’s integrated diagnosis method based on cyclostationarity. To this goal, the study of the cyclic statistic is an important step before using it.
Because of the nonstationarity of cyclostationary signals, the estimators of the cyclic statistic depend on the data origination. That means different time-lag pattern of the Cyclic Autocorrelation Function (CAF) will arrive at different results and different frequency-shift mode of the Spectral Correlation Function (SCF). This will lead to some confusion when extracting the cyclostationarity by different estimator expressions for a same signal. Therefore, the relationship of SCF between different expressions is established. Through this method, the exclusive cyclostationarity can be obtained from the results of different expressions.
The discretization in time domain will result in periodic extension in the frequency and cyclic frequency domain. Because the time-lag quadric transformation of meansquare bandlimited signals has double bandwidth in the time parameter, under the Nyquist sample rate, its CAF will appear aliasing in the cyclic frequency inevitability. But for its SCF, it distributes in the frequency and cyclic frequency plane independently. Based on this discover, an alias-free CAF based on inverse Fourier transformation of SCF in the diamond shape support area is brought forward.
The quality of estimator of SCF is deduced and a conclusion is made that cyclic periodogram is unbiase and inconsistent. Further more the influence of time average and frequency smooth are deduced too. Because of the limitation of data length, for periodic signals with adding stationary noise (cyclostationary at the first order), SCF do not totally suppress the effect of stationary noise to periodic components. Either PSD or SCF, the SNR of periodic components to background noise is subequal. Therefore, SCF seems more suitable for cyclostationary signals at the second order than at the first order.
Through study the vibration signals of gearboxs, we find out that the vibration signals induced by a fault gear is cyclostaionary at the first order, and the signals induced by a fault bearing is cylcostationary at the second order. From the knowledge of pure cyclostationarity and cyclostationarity, we know that cyclostationarity at all lower order must be subtracted to define the accurate order of cyclostationarity in the signals. Therefore a new gearbox integrate fault diagnosis method based on cyclostationarity is set up. Using an adaptive line enhancer and a cyclic adaptive filter, the gearbox vibration signal is separated into three parts, i.e., a first-order cyclostationary part and a second-order cyclostationary part and a stationary part. The firs part is used to judge the fault state of gears, and the second one is for bearings. Therefore, the fault whether of gears or bearings in the gearbox can be determined based on them.
In order to explain the feature of amplitude and phase of FM signals, Bessel function of the first kind is studied, because it is a determinant factor to sideband of FM signals. Based on this, the sideband distribution of signals with AM and FM mixed together is analyzed, and the condition of symmetric sideband distribution is determined in spite of the mix mode of AM and MF is addition or multiplication. Square transform isn’t suitable for FM signals, because of the special feature of Bessel function of the first kind. To overcome this limitation, two kinds of method were established. They are based on asymmetric filtering and time-lag quadratic nonlinearity transformation, respectively. Through these methods, FM signals are transformed to signals which can be demodulated by normal demodulation methods, so FM demodulation is achieve.
At last, all methods in this study are tested by the vibration signals obtained from gearbox SG135-2, which was made by HongDa Gearbox Company in ShaoGuan. In these tests, four fault modes are used, including second gear spalling, fifth gear broken, bearings in the output shaft with a outer race fault and a inner race fault.
引文
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