## 1 Introduction  
Rotating machinery has been widely used in transmission machinery, and its running state directly affects the entire machine, including reliability and stability. Thus, the fault diagnosis of the rotating machinery is increasingly having attention paid. Over the past several decades, new fault detection methods [1; 2; 3; 4; 5; 6; 7] and new criteria [8; 9; 10] were proposed, which greatly enriched the field of fault diagnosis. He et al. [11] proposed a data mining-based diagnostic system using acoustic emission (AE) signals to detect faults in full ceramic bearings. They further developed a series of methods to identify faults in plastic bearings [12], gearbox tooth cut faults [13], planetary gearbox faults [14], etc. For the fault detection of the shaft, various researchers have reported feasible methods. Mohammed et al. [15] investigated a novel route using artificial neural networks (ANN) and power spectral density (PSD) to detect the cracks in a rotating shaft. As is well known, intelligent methods play an important role in detecting faults in mechanical components, such as genetic algorithms (GAs), neural networks (NNs), support vector machine (SVM), etc., and those approaches have been applied to detect faults using experimental samples of all kinds of faults [7]. However, for the lack of suitable training samples that represent every possible fault in all kinds of practical mechanical systems, the above intelligent techniques have not been agreeable applied to detect faults in mechanical systems. In order to understand the fault effects in depth, many researchers proposed numerical simulation for fault diagnosis. Torkaman et al. [16] adopted the three-dimensional time-step finite element method (3D-TSFEM) to simulate the fault of a switched reluctance motor (SRM), and the new diagnosis index associated with power losses was obtained in their results. To detect faults online in nonlinear continuous systems, Bregon et al. [17] used the simulation and state observer models to obtain the final fault diagnosis results. Gong et al. [18] investigated rotor dynamic stability analysis on the hydraulic turbine by numerical simulation and also demonstrated how each factor affects the dynamic character of the labyrinth system. Xiang et al. [19; 20] carried out two kinds of wavelet-based numerical simulation models to calculate the dynamic responses of a shaft. Tannous et al. [21] performed rotor-stator contact simulations using 3D numerical modeling. Baccarini et al. [22] proposed a dynamic numerical model to simulate mechanical faults in induction machines, and the simulation results confirmed the validity of the model. For numerical simulation, on the one hand, less time and equipment are used to get a large amount of experimental data, especially for those that are very difficult to conduct in a real machine for carrying out the experimental investigation. On the other hand, this allows the investigator to determine the fault samples for all types of faults under the complex running conditions. In order to perform the intelligent methods, the selection of feature vectors is one of the key problems. Recently, time domain or time frequency domain feature indexes were adopted to create feature vectors. Chen et al. [23] proposed an approach that has six indexes (e.g., mean value, root mean square, standard deviation, skewness, kurtosis and shape indicator) in the time domain and two indexes (e.g., mean frequency and standard deviation frequency) in the frequency domain to construct the feature sets to train the SVM. Liu et al. [24] used a hybrid time frequency analysis method to get the feature information of gear faults. Wavelet packet transform (WPT) is a useful tool in handling non-stationary signals to detect faults in mechanical systems [25]. As a signal processing tool, WPT decomposes the original signal into multi-layers, and the same frequency bandwidth is offered in each layer. Xiang et al.