路图傅里叶变换及其在滚动轴承故障诊断中的应用

图信号处理(Graph signal processing, GSP)是由谱图理论发展起来的新研究领域。图傅里叶变换(Graph Fourier transformation, GFT)是图信号关于图拉普拉斯矩阵特征函数的展开，也是GSP的基础。对路图的GFT进行分析，发现GFT得到的特征值谱与经典的傅里叶变换(Fourier transformation, FT)频谱有一一对应关系，同时，特征值谱的幅值与特征矢量也有对应关系。将GFT引入滚动轴承故障诊断，提出基于GFT特征提取和K-均值聚类的滚动轴承故障诊断方法。该方法先用GFT将滚动轴承的路图信号变换到特征值谱域；再计算特征值谱的统计量作为故障特征；最后运用K-均值聚类分类器识别滚动轴承的故障类型。对实际轴承振动信号的分析结果表明，基于GFT和K-均值聚类的故障诊断方法能准确有效地识别滚动轴承故障。

Path Graph Fourier Transformation and Its Applications to Rolling Bearing Fault Diagnosis

The graph signal processing (GSP) is a new research field, which is derived from the spectral graph techniques. The foundation of GSP is the graph Fourier transformation (GFT), which is the expansion of a graph signal in terms of the eigenfunctions of graph Laplacian matrix. The GFT on path graph is analyzed. It is found that the eigenvalue spectra obtained by GFT and the frequency spectra obtained by the classical Fourier transformation (FT) have a one to one correlation. Meanwhile, the amplitude of an eigenvalue is correlated with the amplitude of the corresponding eigenvector. The GFT is introduced into the fault diagnosis of rolling bearings and a fault diagnosis method based on the GFT and the K-means clustering is proposed. The path graph signal of the vibration signal of a rolling bearing is transformed by GFT into the eigenvalue spectrum domain. The statistical quantities of eigenvalues are calculated for fault feature extraction. The K-means clustering classifier is used to identify the work condition and fault patterns of the roller bearing. The analysis results of the practical vibration signals of rolling bearings demonstrate that the diagnosis approach based on the GFT and the K-means clustering can be used to identify the fault patterns of roller bearings accurately and effectively.