旋转机械在现代工业和智能制造中占据越来越大的地位。对旋转机械的工作状态进行实时监控不仅能够避免灾难事故的发生，而且有望增加明显的经济收益。滚动轴承被誉为工业生产中旋转机械装备的重要关节之一，在不同领域有着广泛的应用，其运行状态的正常与否直接关系到整个机械装备的工作性能。因此，对于新故障诊断方法的探索和挖掘，滚动轴承是一个很好的研究对象。在实际工程中，由局部缺陷引起的轴承振动信号通常具有非线性、非平稳、低信噪比、故障特征不明显等特点，直接采用频谱分析将难以做出有效诊断。另外，一些常规诊断方法如AR模型、谱峭度、时频分析等各自具有一定局限性。因此，探索有效的滚动轴承故障诊断方法在工程实际中是不得不面对的现实问题。 数学形态学是一种非线性非平稳信号分析方法，通过结构元素探针可以实现对非线性信号细节信息的有效匹配和捕捉，在轴承损伤检测领域有着良好的应用前景。本文以滚动轴承为研究对象，在现有数学形态学方法的基础上，对基于数学形态学的轴承故障诊断方法进行了深入研究并作出了改进，旨在提高轴承故障诊断的准确度，进而最大程度避免事故的出现。本文的创新点和主要工作内容如下： (1)在研究数学形态学基本理论和性质的基础上，通过采用闭开-开闭组合形态滤波器与原信号的差值运算，定义了一种新形态学算子——组合形态-hat变换(Combination morphological filter-hat transform，CMFH)。通过研究数学形态学的滤波特性，给出了不同形态学算子的适用场合。在此基础上，针对数学形态学算子主要依赖于经验性选取结构元素参数的问题，提出了一种基于粒子群优化的组合形态-hat变换(PSO-CMFH)。该算法首先通过粒子群优化算法自适应搜索形态学算子的最佳结构元素参数，随后利用最佳结构元素参数下的CMFH变换对故障信号进行分析，进而实现轴承故障信息的提取。通过仿真验证了提出算法在冲击故障特征提取中的有效性。 (2)以CMFH变换为基础，通过引入多尺度结构元素和加权运算，定义了一种多尺度组合形态-hat变换(Multiscale combination morphological filter-hat transform，MCMFH)，能够用于兼顾不同尺度上的冲击特征信息。随后，在MCMFH变换的基础上，针对融合单一特征指标的多尺度形态学分析容易引起部分故障特征信息丢失的问题，提出了一种基于特征选择框架的多尺度形态学分析方法(FS-MMA)。该算法首先提取原始信号的多域特征；然后根据熵权法选取若干个代表性敏感特征；最后运用灰色关联分析确定MCMFH变换的最优结构元素尺度，进而实现轴承故障特征信息的有效提取。通过仿真和实验轴承故障数据分析验证了提出算法的有效性。分析结果表明：与传统的融合单一特征指标的多尺度形态学分析相比，提出算法具备更好的特征提取效果，提升了诊断精度。 (3)以CMFH变换和开闭平均-hat变换为基础，将两者进行乘积运算，定义了一种形态顶帽乘积算子(Morphology hat product operation，MHPO)。在此基础上，借鉴对角切片谱具备特征增强和抑制噪声的优良特性，提出了一种增强尺度形态顶帽乘积滤波(Enhanced scale morphological-hat product filtering，ESMHPF)。该算法首先采用多尺度形态顶帽乘积算子对原始信号进行滤波处理；然后计算各尺度处形态项帽乘积滤波结果的三阶累计量对角切片和对角切片谱，同时结合故障特征比确定最优结构元素尺度；最后根据最优尺度形态对角切片谱，实现了轴承故障特征信息的增强检测，同时提升了传统多尺度形态学滤波的诊断效果。利用仿真和实测轴承故障信号验证了提出方法的有效性。分析结果表明：ESMHPF方法不仅可以有效地提取故障特征信息，而且具备特征增强的功效。 (4)通过将形态学梯度算子引入到形态谱运算中，同时结合信息熵理论，定义了形态梯度谱(Pattern gradient spectrum，PGS)和形态梯度谱熵(Pattern gradient spectrum entropy， PGSE)的概念。在此基础上，借鉴传统多尺度熵的粗粒化序列过程，提出了一种广义多尺度形态梯度谱熵(Generalized multiscale pattern gradient spectrum entropy，GMPGSE)，实现了PGSE在多个尺度上评估时间序列的随机性和动力学突变行为。最后，为了实现轴承故障状态的智能识别和自动分类，将GMPGSE、拉普拉斯分值(LS)和极限学习机(ELM)相结合，提出了一种基于GMPGSE的滚动轴承智能故障诊断方法。通过实例数据分析证明了提出方法的可行性。研究结果表明：与传统多尺度熵相比，GMPGSE具有更高的诊断精度和计算效率，能够更有效地辨识不同的轴承故障状态。 (5)通过搭建实验室的滚动轴承故障模拟实验台对研究方法进行了实用性验证。首先，详细地介绍了滚动轴承故障模拟的整体实验方案。然后，分别采用PSO-CMFH、FS-MMA、 ESMHPF三种特征提取算法对采集的轴承振动数据和实际工程数据进行了分析，验证了研究方法的有效性。另外，从定量和定性两个方面对三种特征提取算法进行了比较和讨论。最后，将GMPGSE应用在滚动轴承智能故障诊断中。通过实验分析结果表明：GMPGSE能够有效地辨识滚动轴承的不同损伤类型，其分类精度高于传统多尺度熵。 关键词：数学形态学；多尺度形态学；特征提取；滚动轴承；故障诊断 *国家自然科学基金项目(51675098)和江苏省研究生科研创新计划项目(KYCX17_0059)资助

 Rotating machinery occupies an increasingly important position in modern industry and intelligent manufacturing. Real-time monitoring of the working state of rotating machinery not only can avoid the occurrence of disasters, but also bring the obvious economic benefits. Rolling element bearing is reputed as one of the major parts of rotating machinery equipment in industrial production, which is widely used in different fields. The running state of rolling element bearing is directly related to the working performance of mechanical equipment. Therefore, rolling element bearing is a good research object for exploration and excavation of the novel fault diagnosis method. In practical engineering, bearing vibration signals caused by local defects are usually characterized by some characteristics (e.g. nonlinearity, non-stationary, low signal to noise ratio and inconspicuous features), which indicates it is difficult to make effective diagnosis through directly the frequency spectrum analysis. In addition, some conventional diagnostic approaches, such as AR model, spectral kurtosis and time-frequency analysis, have their own limitations. Hence, exploring effective fault diagnosis method is an urgent and challenging task in engineering practice. Mathematical morphology (MM) is a non-linear and non-stationary signal analysis method, which can effectively match and capture the details of non-stationary signals by using a probe named the structuring element (SE). Besides, MM has a good application prospect in bearing damage detection. In this paper, rolling element bearing is taken as the research object. According to the existing MM method, MM-based fault detection approach is studied deeply and improved, which is aimed at improving the accuracy of bearing fault detection and avoiding the accident as much as possible. The innovations and main contributions of this paper are as follows: (1)Inspired by the difference value between the original signal and combination morphological filter, a new morphological operator termed as combination morphological filter-hat transform (CMFH) is formulated based on the theories and properties of the existing MM. The applicable occasions of different morphological operators are suggested by investigating the filtering characteristics of MM. On this basis, a method called particle swarm optimization-based CMFH is proposed for the purpose of overcoming the drawbacks of empirical selection of SE parameters in morphological operators. In this method, the optimal SE parameter of morphological operator is firstly determined by PSO algorithm, and then CMFH containing the optimal SE parameter is used to analyze fault data and extract bearing fault features. Simulation results show that the proposed method is effective in extracting impact fault feature. (2)According to the fusion among CMFH, multicale SE and weighted arithmetic, multiscale combination morphological filter-hat transform (MCFHM) is formulated for acquiring fault feature information at different scales. Then, on this basis, feature selection framework-based multiscale morphological analysis method (FS-MMA) is proposed for the purpose of solving the issue of losing local fault information of multiscale morphological analysis (MMA) with a single index. The algorithm first extracts multi-domain features of the raw signal, and then selects several sensitive features via entropy weight method, and finally grey correlation analysis is conducted to determine the optimal SE scale and then fault feature information extraction of bearing can be achieved. The validity of the proposed method is validated by analyzing the simulated and experimental bearing fault data. Results show that FS-MMA has better performance in bearing fault feature extraction and diagnosis accuracy compared with traditional MMA with a single index. (3)According to the product between CMFH and average-hat transform (AVGH), a novel morphological operator named morphology hat product operation (MHPO) is defined. On this basis, based on the excellent characteristics of feature enhancement and noise suppression of diagonal slice spectrum, an enhanced scale morphological-hat product filtering (ESMHPF) is then presented. In this method, multi-scale morphology hat product operation of the original signal is firstly performed. Next, TCDS and DSS of morphological filtering results for each SE scale are calculated, and fault feature ratio is applied to determine the optimal SE scale. Finally, optimal scale morphology diagonal slice spectrum is used for strengthening bearing fault characteristic information and improving the diagnosis performance of traditional multiscale morphological filtering (MMF). The effectiveness of the proposed method is verified by the simulated and measured bearing fault signal. Results indicate that ESMHPF not only can extract fault characteristic information, but also have the performance of feature enhancement. (4)Through the integration of morphological gradient operators, pattern spectrum (PS) and information entropy, the concept of pattern gradient spectrum (PGS) and pattern gradient spectrum entropy (PGSE) is firstly presented. On this basis, according to the coarse graining procedure of traditional multiscale entropy (MSE), generalized multiscale pattern gradient spectrum entropy (GMPGSE) is further proposed for evaluating randomness and dynamic behavior of the time series on different scales. Finally, in order to realize intelligent identification and automatic classification of bearing fault states, an intelligent detection algorithm for rolling bearing based on GMPGSE is developed by combining the GMPGSE, LS and ELM. The feasibility of the proposed method is verified by the analysis of instance data. Results show that GMPGSE has higher diagnostic accuracy and computational efficiency, and can identify different bearing fault states more effectively, compared with MSE. (5)The feasibility of the above-mentioned method is validated by performing a bearing fault simulation experiment. Firstly, the scheme of bearing fault simulation experiment is introduced in detail. Then, bearing vibration data collected and actual engineering data are respectively analyzed by three methods (PSO-CMFH, FS-MMA and ESMHPF) to prove their efficacy. Performance among feature extraction algorithms is compared and discussed by quantitative and qualitative analysis. Finally, GMPGSE is applied to intelligent fault diagnosis of rolling element bearing. Experimental results show that GMPGSE can identify effectively different bearing damage types, and its classification accuracy is higher than that of MSE. Key words: Mathematical morphology; Multi-scale morphology; Feature extraction; Rolling element bearing; Fault diagnosis * Sponsored by National Natural Science Foundation of China (No. 51675098) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_0059).