A Rolling Bearing Fault Diagnosis Method Based on Variational Mode Decomposition and an Improved Kernel Extreme Learning Machine  
[PERSON]  
1 Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Jiangnan University, 1800 Li Hu Avenue, Wuxi 214122, Jiangsu, China; [EMAIL_ADDRESS] (L.S. ); [EMAIL_ADDRESS] (J.W.) [PERSON]  
1 Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Jiangnan University, 1800 Li Hu Avenue, Wuxi 214122, Jiangsu, China; [EMAIL_ADDRESS] (L.S. ); [EMAIL_ADDRESS] (J.W.) [PERSON]  
1 Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Jiangnan University, 1800 Li Hu Avenue, Wuxi 214122, Jiangsu, China; [EMAIL_ADDRESS] (L.S. ); [EMAIL_ADDRESS] (J.W.) [PERSON]  
2 School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, 15 Beisanhuan East Road, ChaoYang District, Beijing 100029, China  
2 Graduate School of Bioresources, Mie University, 1577 Kurimamachiya-cho, Tsu, Mie 514-8507, Japan; [EMAIL_ADDRESS]  
Footnote 1: email: {hq,h,  
intrinsic mode function (IMF) energy distribution and bearing state for bearing state identification. In reference [5], a support vector machine (SVM) and EMD were combined for diagnosis of a rolling bearing's fault envelop spectrum. [PERSON] et al. [6] combined EMD and a neural network for bearing fault diagnosis. Inspired by EMD, [PERSON] proposed another adaptive signal decomposition method, local mean decomposition (LMD), in 2005 [7]; this method also attracted significant attention from researchers, and numerous LMD-based diagnosis methods were proposed in succession. [PERSON] et al. [8] employed LMD for rolling bearing and gear fault diagnosis. [PERSON] et al. [9] obtained a wind power generator's vibration signal instantaneous frequency via LMD to monitor the wind power generator's state. EMD and LMD have been widely deployed to extract fault features. However, the two methods belong to recursive mode decomposition, which is affected by mode aliasing, end effects, and sampling frequency. When the frequency ratio is the reciprocal of an odd number, the decomposition has significant error [10]. Variational mode decomposition (VMD) is a new adaptive signal processing method proposed by [PERSON] and [PERSON] [11]. VMD assumes that each intrinsic mode function has limited bandwidth and a different central frequency. To ensure the sum of the estimated bandwidths for the intrinsic mode functions is minimal, a variation problem is solved via a transformation. Each intrinsic mode function is demodulated to the corresponding base frequency band. Finally, each intrinsic mode function and its corresponding central frequency are extracted. Once VMD was proposed, it immediately became the focus of fault diagnosis research. [PERSON] et al. [12] analyzed and compared the feature extraction performance of VMD and the empirical wavelet transform, and suggested that VMD was superior for feature extraction. [PERSON] et al. [13] applied VMD to bearing fault diagnosis. [PERSON], et al. [14] decomposed a vibration signal via VMD and employed an immunogenic algorithm-optimized support vector machine for fault identification. Artificial neural networks (ANN) have been widely deployed in fault diagnosis. However, conventional neural networks employ learning algorithms based on gradient descent and have problems such as slow convergence and being trapped in local minima. Extreme learning machine (ELM) is a recently proposed single-hidden-layer forward neural network learning algorithm [15]. In ELM, the input weight of the input layer and threshold of the hidden layer node are selected randomly, and the output weight is calculated via the Moore-Penrose generalized inverse of the hidden layer's output matrix. Compared with conventional neural networks, ELM has advantages such as a fast learning speed and excellent generalization performance. However, the randomly generated hidden layer weight and hidden layer threshold have significant impacts on the ELM algorithm's performance.