# Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions
## 1 Introduction  
Rolling bearings are among the most commonly used support elements in rotating machinery. They are prone to faults under harsh working conditions. When a fault occurs on the inner or outer race of a bearing, a series of impulses is generated in the vibration signal as the bearing defect interacts with another surface, and the impacts excite high-frequency resonances where the signal-to-noise ratio (SNR) is higher than the other frequency regions in the bearing system, thereby inducing a modulating phenomenon [1; 2]. However, many other sources of bearing vibration such as the waviness of rolling elements etc. always result in the emergence of side bands around the principal bearing frequencies, which are more pronounced at higher frequencies [3]. Therefore, accuratelydetermining the high-frequency resonance band where the impulse occurs is key to successfully detecting bearing faults. In earlier research, the resonance band is often determined by experimental tests, which are time-consuming [4]. As a statistical index, kurtosis is sensitive to the peaks caused by abnormal vibrations, and it is usually used as a direct measure of the transient impulses of the signal. Nevertheless, kurtosis is easily affected by noise. To overcome this limitation, Frequency Domain Kurtosis (FDK) was proposed by [PERSON] [5] to complement the flaw of classical power spectral density (PSD), i.e., not being sensitive to the statistical nature of the signal. Inspired by this proposal, Spectral Kurtosis (SK) was presented by [PERSON] based on the Wold-Cramer theorem for non-stationary feature extraction in [6]. The basic idea of this approach is that the kurtosis at each frequency line of a signal is calculated to discover the presence of transients, and to indicate in which frequency bands these occur. For the convenience of industrial applications, [PERSON] further proposed the Fast Kurtogram (FK) using short-time Fourier transform (STFT) combined with 1/3 binary tree algorithms to split frequency bands to reduce computing time, as described in [7]. The fault impulses are extracted after a raw signal is processed by a band-filter for which the center frequency and bandwidth are optimized by FK. Since then, many studies have been conducted to enhance these theories [8; 9; 10; 11]. Considering the problem whereby the parameters of the band-filter cannot be determined adaptively, [PERSON] et al. [12] combined genetic algorithms and FK to optimize the parameters. To extract transient impulsive signals under a low SNR condition, [PERSON] et al. [13] proposed a time-frequency analysis method which combines the merits of ensemble local mean decomposition and FK to detect bearing faults. In [14], [PERSON] proposed replacing STFT with wavelet package transform (WPT) to improve the kurtogram (WPTK). Recently, [PERSON] proposed an enhanced kurtogram, in which the kurtosis values are calculated based on the envelope power spectrum of WPT nodes at different depths [15]. It is worth mentioning that each kurtosis value of the filtered signal is calculated without source identification in these methods, which is sometimes incorrect, especially when the vibration signal contains random knocks which usually have higher amplitudes, as well as kurtosis values which are far larger than those of real faults [4; 16]. This effect means that the optimal frequency band corresponding to the maximum kurtosis is the resonance band containing random knocks, while the real fault signature is missing. To solve this problem, [PERSON] et al. [17] proposed a higher resolution kurtosis index, the Protrugram, which is obtained by calculating the kurtosis of the envelope spectrum amplitudes of a narrow band filtered signal along the frequency axis. However, the optimal filter bandwidth depends on a certain knowledge of the sought fault. In [18], [PERSON] took advantage of the periodicity of the faults and proposed Correlated Kurtosis (CK) to detect cyclic transients.