## 1 Introduction  
As everyone knows, most mechanical fault signals have the characteristics of being nonlinear and non-stationary; it is difficult to accurately reveal the fault feature frequency applying directly fast Fourier transform (FFT) in data processing [1]. Hence, a host of studies have focused on how to improve the ability of fault information excavation, mainly classified as three types, namely the time domain, frequency domain, and time-frequency domain [2, 3, 4]. Among these approaches, the time-frequency analysis (TFA) tool has gained increasing interest for capturing intrinsic feature information from nonlinear and non-stationary signals due to the fact that it is capable of excavating time-frequency information at the same time. Short-time Fourier transform (STFT), Wigner-Ville distribution (WVD), and wavelet transform (WT) are recognized as several classic representations of algorithms of this class [5]. To our disappointment, the aforesaid algorithms contain some inevitable disadvantages when they are used to analyze complex and irregular signals. For example, it is difficult to acquire a clear frequency trajectory and high resolution by using STFT with a moving window [6]. The inevitable cross term will appear if we apply WVD to process the actual non-stationary compound signal [7]. Although WT is equipped with a rigorous mathematical basis, it lacks self-adaptability due to the wavelet basis which need to be selected in advance [8]. Inspired by foregoing correlation work, several self-adaption TFA tools possessing a recursive property are smoothly proposed and formulated in analyzing the non-stationary compound signal, such as empirical mode decomposition (EMD), local mean decomposition (LMD), intrinsic time-scale decomposition (ITD), local characteristic-scale decomposition (LCD), and intrinsic characteristic-scale decomposition (ICD). Hitherto, many efforts have been made to enlarge the application of these methods in mechanical fault detection and prognostics [9]. Unfortunately, the prevalent self-adaption TFA algorithms still contain some adverse issues. As an example, some drawbacks (e.g., an end point and aliasing effect) still remain to be addressed in EMD and LMD [10]. Besides, to obtain accurate decomposition results, one needs to set a suitable step length for the moving average in LMD [11]. Due to ITD adopting the principle of linear transformation to obtain sub-signals, it is provided with the disadvantages of waveform burr [12]. As the modified tool of ITD, LCD and ICD can alleviate the mode mixing and suppress the end effect problem efficiently, but the physical meaning of the sub-signal obtained still cannot be explained exactly [13]. Hence, it is valuable to develop a novel TFA tool for fault detection. More recently, Dragomiretskiy and Zosso [14] presented an alternative TFA tool termed variational mode decomposition (VMD), which can decompose a complex and irregular compound signal into several sub-signals named intrinsic mode function (IMF). At present, this new tool has appeared in various fields, including speech recognition [15], image processing [16], seismic analysis [17], fault diagnosis, and forecasting [18; 19; 20]. For instance, Upadhyay and Pachori [21; 22] applied VMD to detect the instantaneous voice/non-voice of the speech signal and to achieve speech enhancement. An et al. [23] adopted VMD to detect the fault patterns of hydropower units. Wang et al. [24] utilized VMD to identify the rubbing fault of rotor systems and researched its narrow-band filtering nature. Yang et al. [25] investigated the decomposition performance of VMD through a series of comparison analyses. Yao et al. [26] used VMD to achieve the noise reduction of diesel engines. The results show that more interference noises can be eliminated by his approach. Additionally, Zhang et al. [27] and Liu et al. [28] applied VMD to identify fault types of bearing and obtained favorable diagnosis results.