## 1 Introduction
Based on the coherent composite spectrum and principal component analysis, [PERSON] presented an automated data fusion-based gear fault classification framework with ANN for rotating machines [16]. With structural health monitoring techniques and model-based numerical approaches, [PERSON] established an ANN model to predict the machine state of a gear [17]. [PERSON] presented a classification framework with ANN for an induction motor bearing fault, and wavelet scattering transform-based features were used as the input data [18]. [PERSON] applied an ANN and K-nearest neighbors to tackle the fault detection and identify the severity level of spiral bevel gears under different operating conditions [19]. Based on the feature extraction of the direct torque control and Hilbert transform, [PERSON] proposed a novel ANN model for induction motor bearing fault diagnosis [20]. In this paper, an ANN was selected as the training model, which was combined with the feature extraction method of the variational mode decomposition (VMD) to perform the fault diagnosis of rolling bearings. VMD, proposed by [PERSON] in 2014, is an adaptive and completely nonrecursive mode variation signal processing method [21]. VMD overcomes the problems of mode component aliasing in EMD method [22] and decomposes the signal into several signals of different frequency scales. VMD can determine the number of mode decompositions of the signal based on the actual situation and then adaptively match the best central frequency and finite bandwidth of each mode in the search and solution process, finally realizing the effective separation and frequency domain division of the intrinsic mode components in the signal. In recent years, feature extraction based on VMD has been applied more frequently in the field of fault diagnosis. [PERSON] proposed a novel VMD based on power information for the fault diagnosis of rolling bearings [23]. [PERSON] presented a novel VMD based on power spectral density for fault diagnosis for rolling bearing [24]. [PERSON] applied VMD and CNN for the fault diagnosis of a power grid [25]. [PERSON] established a new VMD method for fault diagnosis using the combination of sparse principal component and VMD [26]. For the variable speed of rolling bearings, [PERSON] used VMD to obtain the weak fault features [27]. This paper realized the fault diagnosis of rolling bearings with high accuracy by designing the data preprocessing method, feature extraction and ANN structure appropriately. In terms of the preprocessing and feature extraction of a bearing's vibration data, this paper obtained the optimal data sampling quantity and sampling period by studying the influences on them for prediction accuracy and computational time. The VMD method and discrete Fourier transform were applied to process the preprocessed data to obtain the power spectral density of the intrinsic mode functions of the bearing vibration data. Meanwhile, the normalized energy of bearing vibration data was obtained based on Shannon entropy theory. In terms of the ANN's design, this paper took the normalized energy as the input for the ANN to train the model. This paper studied the influence of the training function, activation function, hidden layer number and node number in each hidden layer on the prediction accuracy and computation time of the model to determine the optimal ANN structure. The procedures and results of the study has certain guiding significance for bearing fault diagnosis based on ANN. The rest of the paper is organized as follows: Section 2 presents the preprocessing of raw vibration signal, feature extraction based on VMD and construction of an ANN for fault diagnosis. Section 3 illustrates the performance of the fault diagnosis with ANN based on VMD. Section 4 provides the conclusion of this paper.