分形维数特性分析及故障诊断分形方法研究
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摘要
分形是现代数学和非线性科学研究中一个非常活跃的分支，它可以理解为局部和整体在某方面存在相似性。在分形理论中，分形维数是一个非常重要的参数，可以定量的描述非线性系统的分形特征，度量信号的空间填充能力，已经被广泛应用于多个领域。在机械故障诊断方面，不同的故障状态下，非线性因素对机械振动信号的影响是不同的，分形维数可以有效的度量机械系统的故障特征、识别机械设备的故障状态。本文以分形理论为基础，主要围绕分形维数特性和分形故障诊断方法两个方面展开深入的研究。
针对分形盒维数对噪声不敏感的现象，研究了盒维数的抗噪性能。在不同噪声强度的影响下，改变信噪比，对分形盒维数的抗噪曲线进行分析。在该曲线中，定义了分形盒维数的抗噪性能边界点，以此为基准，将抗噪曲线划分为两个区域，分别对每个区域内的曲线变化趋势进行分析，揭示了分形盒维数的抗噪特性。
对于单重分形故障诊断，以分形盒维数为故障特征量，讨论了单重分形故障诊断的一般方法，验证了分形盒维数对机械振动信号的定量度量能力和对故障状态的识别能力。多重分形可以在多个测度下对非线性信号进行定量的度量，不仅可以描述信号的整体特征，还能够刻画信号的局部性和不均匀性。多重分形故障诊断以样本序列为基础，采用相关性判断方式实现设备状态识别，扩大了分形故障诊断的应用范围。
在多重分形故障诊断的基础上，结合信号分解方法，将分形故障诊断的特征量扩展为矩阵形式，实现分形特征量从单重分形维数、广义维数到分形矩阵的延伸与发展。同时，对广义维数的相关性判断法进行改进，提出了适用于矩阵式分形特征量的相关系数计算方法。为了使分形矩阵的构建不局限于一种信号分解方法，分别研究了经验模式分解和小波、小波包分解的基本原理，并以此为基础，实现了基于矩阵式特征量的分形故障诊断，使矩阵式分形特征量的构建适应于更多的信号分解方法。为了使分量信号的选择不受信号分解方式的限制，提出了基于相关系数的分量信号选择方法，采用该分量信号选择方法可以有效的提高矩阵式分形特征量的故障识别能力，更好的区分故障状态、判断故障类型。
高频噪声对分形故障诊断的效果产生了很大的影响，为了抑制这一不利因素，分析了随机共振的机理，重点研究了级联双稳随机共振的滤波特性，并将其与广义维数相结合，应用于高频大噪声背景下的机械故障诊断中。级联双稳随机共振可以利用高频噪声增强低频信号的能量，使分形维数具有更强的状态识别能力，提高了故障诊断的有效性。
Fractal, which is defined as “shape made of parts similar to the whole in someway”, is an active branch in the study of modern mathematics and nonlinear science.As an essential parameter in fractal theory, fractal dimension has been widely used inmany areas of science, including the capability of quantitative measurement for fractalcharacteristics of nonlinear systems and the ability of space-filling capacitymeasurement for signals. In mechanical fault diagnosis, the influences of nonlinearfactors for mechanical vibration signals are different under diverse fault status. Fractaldimension describes the fault features of mechanical system and recognizes the faultstatus of mechanical device effectively. Based on the fractal theory, the investigationon the fractal dimension features and fractal fault diagnosis methodology was carriedout.
As the fractal box dimension is not sensitive to noise, the anti-noise property ofbox dimension was studied. According to influences of different noise strength, theanti-noise behavior of fractal box dimension was analyzed by changing signal to noiseratio (SNR). The fractal box dimension varies with SNR and its curve can be dividedinto two parts with a point which is defined as the frontier point between them.Through the tendency of each part of the curve, its anti-noise performance wasunveiled.
Utilizing the fractal box dimension as characteristic parameter, the single fractalfault diagnosis method was discussed. The result showed that the fractal boxdimension had the ability of quantitative measurement for mechanical vibration signaland recognition of the fault status. In several measures, a nonlinear signal can bedescribed entirely and exactly by multi-fractal, especially for unbalanced property andsubsection attribute. Based on sample sequence, the application range of fractal faultdiagnosis is enlarged by multi-fractal using relationship to recognize the device status.
On the basis of multi-fractal fault diagnosis, relating to the signal decomposition,the fractal fault diagnosis characteristic parameter was extended from single fractaldimension, generalized dimension to fractal matrix. Moreover, a new correlationcoefficient calculation method, adapting to fractal characteristic parameter matrix,was presented based on improvement of generalized dimension correlation coefficientmethod. In order to make the fractal matrix construction out of limited to any signaldecomposition methods, wavelet, wavelet packet analysis and empirical mode decomposition were discussed and used in constituting the fractal matrix. In addition,the selection method of composition signals based on correlation coefficient waspresented for diversity of signal decompositions. The experiment showed that thefractal characteristic parameter matrix obtained through this selection method hadmore efficient for mechanical fault recognition and diagnosis.
As is well known, high frequency noise, which is not conducive to fractal faultdiagnosis, need to be restrained by means of filtering method. Therefore, cascadedbistable stochastic resonance with well filtering property, combining with generalizeddimension, was applied to mechanical fault diagnosis in high frequency noisebackground. The power can be transferred from high frequency domain to lowfrequency domain by cascaded bistable stochastic resonance in order to not onlyremove the high frequency noise but also enhance the low frequency signal.Furthermore, for the filtered signals, it is easier to distinguish between different faultstatuses by fractal dimension so as to advance the effect of fractal fault diagnosis.
针对分形盒维数对噪声不敏感的现象，研究了盒维数的抗噪性能。在不同噪声强度的影响下，改变信噪比，对分形盒维数的抗噪曲线进行分析。在该曲线中，定义了分形盒维数的抗噪性能边界点，以此为基准，将抗噪曲线划分为两个区域，分别对每个区域内的曲线变化趋势进行分析，揭示了分形盒维数的抗噪特性。
对于单重分形故障诊断，以分形盒维数为故障特征量，讨论了单重分形故障诊断的一般方法，验证了分形盒维数对机械振动信号的定量度量能力和对故障状态的识别能力。多重分形可以在多个测度下对非线性信号进行定量的度量，不仅可以描述信号的整体特征，还能够刻画信号的局部性和不均匀性。多重分形故障诊断以样本序列为基础，采用相关性判断方式实现设备状态识别，扩大了分形故障诊断的应用范围。
在多重分形故障诊断的基础上，结合信号分解方法，将分形故障诊断的特征量扩展为矩阵形式，实现分形特征量从单重分形维数、广义维数到分形矩阵的延伸与发展。同时，对广义维数的相关性判断法进行改进，提出了适用于矩阵式分形特征量的相关系数计算方法。为了使分形矩阵的构建不局限于一种信号分解方法，分别研究了经验模式分解和小波、小波包分解的基本原理，并以此为基础，实现了基于矩阵式特征量的分形故障诊断，使矩阵式分形特征量的构建适应于更多的信号分解方法。为了使分量信号的选择不受信号分解方式的限制，提出了基于相关系数的分量信号选择方法，采用该分量信号选择方法可以有效的提高矩阵式分形特征量的故障识别能力，更好的区分故障状态、判断故障类型。
高频噪声对分形故障诊断的效果产生了很大的影响，为了抑制这一不利因素，分析了随机共振的机理，重点研究了级联双稳随机共振的滤波特性，并将其与广义维数相结合，应用于高频大噪声背景下的机械故障诊断中。级联双稳随机共振可以利用高频噪声增强低频信号的能量，使分形维数具有更强的状态识别能力，提高了故障诊断的有效性。
Fractal, which is defined as “shape made of parts similar to the whole in someway”, is an active branch in the study of modern mathematics and nonlinear science.As an essential parameter in fractal theory, fractal dimension has been widely used inmany areas of science, including the capability of quantitative measurement for fractalcharacteristics of nonlinear systems and the ability of space-filling capacitymeasurement for signals. In mechanical fault diagnosis, the influences of nonlinearfactors for mechanical vibration signals are different under diverse fault status. Fractaldimension describes the fault features of mechanical system and recognizes the faultstatus of mechanical device effectively. Based on the fractal theory, the investigationon the fractal dimension features and fractal fault diagnosis methodology was carriedout.
As the fractal box dimension is not sensitive to noise, the anti-noise property ofbox dimension was studied. According to influences of different noise strength, theanti-noise behavior of fractal box dimension was analyzed by changing signal to noiseratio (SNR). The fractal box dimension varies with SNR and its curve can be dividedinto two parts with a point which is defined as the frontier point between them.Through the tendency of each part of the curve, its anti-noise performance wasunveiled.
Utilizing the fractal box dimension as characteristic parameter, the single fractalfault diagnosis method was discussed. The result showed that the fractal boxdimension had the ability of quantitative measurement for mechanical vibration signaland recognition of the fault status. In several measures, a nonlinear signal can bedescribed entirely and exactly by multi-fractal, especially for unbalanced property andsubsection attribute. Based on sample sequence, the application range of fractal faultdiagnosis is enlarged by multi-fractal using relationship to recognize the device status.
On the basis of multi-fractal fault diagnosis, relating to the signal decomposition,the fractal fault diagnosis characteristic parameter was extended from single fractaldimension, generalized dimension to fractal matrix. Moreover, a new correlationcoefficient calculation method, adapting to fractal characteristic parameter matrix,was presented based on improvement of generalized dimension correlation coefficientmethod. In order to make the fractal matrix construction out of limited to any signaldecomposition methods, wavelet, wavelet packet analysis and empirical mode decomposition were discussed and used in constituting the fractal matrix. In addition,the selection method of composition signals based on correlation coefficient waspresented for diversity of signal decompositions. The experiment showed that thefractal characteristic parameter matrix obtained through this selection method hadmore efficient for mechanical fault recognition and diagnosis.
As is well known, high frequency noise, which is not conducive to fractal faultdiagnosis, need to be restrained by means of filtering method. Therefore, cascadedbistable stochastic resonance with well filtering property, combining with generalizeddimension, was applied to mechanical fault diagnosis in high frequency noisebackground. The power can be transferred from high frequency domain to lowfrequency domain by cascaded bistable stochastic resonance in order to not onlyremove the high frequency noise but also enhance the low frequency signal.Furthermore, for the filtered signals, it is easier to distinguish between different faultstatuses by fractal dimension so as to advance the effect of fractal fault diagnosis.
引文
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