Introduction

Lots of faults about rotating machinery are related to rolling bearings which have a great influence on the working state of the machine [1, 2]. Therefore, study on rolling bearing fault diagnosis is of great significance.

The research on fault diagnosis of rolling bearing began in 1960s. In 1974, Boeing invented the technology of resonance demodulation. Modern analytical methods of fault diagnosis are mainly based on feature extraction in time domain and frequency domain. The time domain analysis of rolling bearing fault needs complex parameter comparison, and the stability and sensitivity are not good. Frequency domain analysis method solves these problems to a certain extent. However, it is not enough to study the complex non-stationary signals separately from the time domain or frequency domain. It is necessary to analyze the signals both in time and frequency domain. Meanwhile mechanical fault signals are non-stationary signals, it is not comprehensive that the characteristics of fault signals are described only in time domain or frequency domain [3].

Scholars hope to analyze the characteristics of fault signals in time and frequency domain simultaneously [4]. Short-time Fourier transform (STFT) is the first one to be used. This method can obtain better local characteristics in time-frequency domain. However, STFT is limited by the selection of window functions and is difficult to analyze various situations [5]. In order to get rid of this limitation, researchers put forward Wavelet technology. Experiments prove that Wavelet method has strong resolving power to the fault signal. Wavelet method is widely used in bearing fault diagnosis because of its multi-scale decomposition [6, 7]. As a priori algorithm, Wavelet analysis needs to set parameters in advance, and the improper selection of parameters will have a bad impact on the results [8]. Wigner-Ville method is used in the field of mechanical fault diagnosis due to its high time-frequency resolution and good aggregation. However, the biggest disadvantage of this method is the existence of cross terms [9]. In recent years, some new signal processing methods, such as empirical mode decomposition and auto-encoder, are also used in fault diagnosis [10,11,12]. Intelligent fault diagnosis of rolling bearing has also been developed based on hierarchical convolutional network, learning machine, and genetic algorithm [13,14,15,16].

Fractional Fourier transform is a signal processing technology developed recently. Its remarkable advantage over Fourier transform is that it can provide more choices for time-frequency analysis and obtain some improvements in performance. Almeida laid the foundation for time-domain analysis in fractional domain [17, 18]. Power spectrum, which describes the average power of each frequency component in a random process, is one of the important representations of random signals [19, 20]. Referring to fractional time-frequency rotation characteristics, the power spectrum distribution is directly calculated in fractional domain. Furthermore, the energy intensity of each frequency component in fractional domain is achieved by integrating the time-frequency spectrum along the time axis, which is applied to the fault diagnosis. The proposed algorithm can separate noise signals from the fault signals in the fractional order domain, which provides a good condition for further extracting their time domain and frequency domain characteristics.

Method Principle

Fractional Fourier Transform

The fractional Fourier transform of signal x(t) is defined as following [17, 18].

$$ {M}_{\alpha }(u)={FRFT}_{\alpha}\left[x(t)\right]={\int}_{-\infty}^{+\infty }x(t){N}_{\alpha}\left(t,u\right) dt $$
(1)

Here,

$$ {N}_{\alpha}\left(t,u\right)=\left\{\begin{array}{c}\sqrt{\frac{1-j\cot \alpha }{2\pi }}\exp \left(j\frac{u^2+{t}^2}{2}\cot \alpha -j\frac{ut}{\sin \alpha}\right)\alpha \ne \mathrm{n}\pi \\ {}\delta \left(u-t\right)\alpha =2\mathrm{n}\pi \\ {}\delta \left(u+t\right)\alpha =\left(2\mathrm{n}+1\right)\ \pi \end{array}\right. $$
(2)

α is the transform angle andα < |α| < π. Transform order isp = 2α/π. When p = 1, FRFT is Fourier transform.

Considering the relationship between the one-dimensional power spectrum and the spectrum distribution, the 2D distribution of power spectrum in the fractional domain with angle of α is given [21].

$$ PS{D}_{\left(u-v,\alpha \right)}\left(u,\upsilon \right)=\frac{1}{2T}{\left|\int {M}_{\alpha}\left({u}^{\prime}\right)H\left(u\prime -u\right){e}^{-j2\pi u\prime \upsilon } du\prime \right|}^2 $$
(3)

In the Formula, Mα(u) is the result of fractional Fourier transform at α angle, which is applied for a signal with a length T. H(u) is a fractional domain window function. In order to have a higher time-frequency resolution,H(u) is chosen as Gauss window function generally. u' is a variable in the u ‐ υ domain. According to 20th reference, the fractional domain power spectrum of α angle is defined as

$$ PS{D}_{\alpha }(u)=\underset{T\to \infty }{\lim}\frac{1}{2T}{\left|{M}_{\alpha }(u)\right|}^2 $$
(4)

In the formula, Mα(u) is the result of fractional Fourier transform of x(τ) at α angle based on Eqs. (1) and Eq. (2), where x(τ) is a segment of signal intercepted from x(t), with a length T.

Time Frequency Distribution of Power Spectrum in Fractional Domain

In order to have a clear physical meaning, the power spectrum is rotated from the u ‐ υ plane of the fractional field to the t ‐ f plane. Then the direct expression of the time-frequency distribution in the fractional domain is obtained [22].

$$ {\displaystyle \begin{array}{c} PS{D}_{\left(t-f,a\right)}\left(t,f\right)={R}_a\left\{ TFPS{D}_a\left(u,v\right)\right\}\\ {}={R}_a\left\{\frac{1}{2T}{\left|{\int}_{-\infty}^{+\infty }{\int}_{-\infty}^{+\infty }x\left({t}^{\prime}\right){N}_a\Big(t\prime, u\prime \Big) dt\prime \times H\left(u\prime -u\right){e}^{-j2 pu\prime u} du\prime \right|}^2\right\}\\ {}={R}_a\left\{\frac{1}{2T}{\left|{\int}_{-\infty}^{+\infty }{\int}_{-\infty}^{+\infty }x\left({t}^{\prime}\right){e}^{jp\left({u}^2-{v}^2\right) sinacosa}\times h\left(t\prime - ucosa+ vsina\right){e}^{-j2 pt\prime \left( usina+ vsina\right)} dt\prime \right|}^2\right\}\\ {}=\frac{1}{2T}{\left|{\int}_{-\infty}^{+\infty }x\left({t}^{\prime}\right)h\left(t\prime -t\right){e}^{-j2 pt\prime f} dt\prime \right|}^2\end{array}} $$
(5)
$$ \left(\begin{array}{l}t\\ {}f\end{array}\right)={R}_{-\alpha}\left(\begin{array}{l}u\\ {}v\end{array}\right)=\left(\begin{array}{l}\cos \alpha \kern1em \hbox{-} \sin \alpha \\ {}\sin \alpha \kern1.25em \cos \alpha \end{array}\right)\left(\begin{array}{l}u\\ {}v\end{array}\right) $$
(6)

In the form, t'is a variable in t ‐ f domain. Rα represents rotation and the Formula (5) is the rotation equation.

And h(t) is determined by the fractional Fourier transform of H(u).

$$ h(t)={FRFT}_P\left[H(u)\right] $$
(7)

Formula (5) shows that the power spectrum distribution in the fractional domain is identical with the two-dimensional spectrum in form. The key lies in the selection of Gauss window function in fractional domain. According to the conclusions given in document [23, 24], when

$$ H(u)={\mathrm{e}}^{-\frac{\pi {u}^2{B}_{x_p}}{T_{x_p}}}, $$
(8)

the aggregation of the power spectrum in the fractional domain will be optimal.\( {T}_{x_p} \)and\( {B}_{x_p} \)are the time-wide and frequency-wide of signal x(t) in the p order fractional domain. In the vast literature, there are many ways to determine the best order p. Considering the characteristics of the fault signal, the maximum kurtosis coefficient method is adopted as following [25,26,27].

$$ K\left\{{M}_{\alpha }(u)\right\}=\frac{E\left[{\left(\left|{M}_{\alpha }(u)\right|-\left|{\overline{M}}_{\alpha }(u)\right|\right)}^4\right]}{E^2\left[{\left(\left|{M}_{\alpha }(u)\right|-\left|{\overline{M}}_{\alpha }(u)\right|\right)}^2\right]} $$
(9)
$$ p=\arg \underset{0\le p<2}{\max }K\left\{M{}_{\alpha}\;(u)\right\} $$
(10)

In the formula, \( {\overline{M}}_{\alpha }(u) \)is the mean value ofMα(u)and E is mathematical expectation.

Quantitative Evaluation Index

Entropy is a tool to measure the uncertainty of information. For signals with strong randomness, the greater the uncertainty, the greater the entropy. For signals with strong certainty, the entropy is smaller. Applied to the time-frequency analysis, the more concentrated the energy distribution is, the smaller the uncertainty is and the smaller the entropy is. The more dispersed the energy is, the greater the uncertainty is and the larger the entropy is. The entropy value can be used to judge the degree of time-frequency energy convergence, and then judge the result of time-frequency analysis.

Renyi entropy, as evaluation index, is used to evaluate the advantages of the proposed algorithm and other algorithms. Renyi generalized the concept of entropy in 1973, and put forward the Renyi entropy.

$$ {R}_{\alpha }(p)=\frac{1}{1-\alpha }{\log}_2\frac{\sum \limits_i{p}_i^{\alpha }}{\sum \limits_i{p}_i} $$
(11)

Where, α is the order of entropy, when α → 1, Renyi entropy became Shannon entropy. pi is the probability density (i = 1, 2,…).

Simulation Analysis

In order to verify the effectiveness of the presented algorithm, a set of sinusoidal frequency modulation signals are selected for testing. Its time domain waveform and instantaneous frequency distribution are shown in Fig. 1. According to the theory deduced above, the optimal order p is searched by Eqs. (9) and Eq. (10). Then the optimal window function H(u) in the fractional domain is obtained from Eq. (8). And the u ‐ υ distribution of the power spectrum in the fractional domain can be achieved by Eq. (3). Finally, the t ‐ fdistribution of the power spectrum in the fractional domain can be obtained based on Eq. (5).

Fig. 1
figure 1

Time domain waveform of sinusoidal frequency modulation signal (a) and it Instantaneous frequency distribution (b)

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For comparison, the results from STFT method and the improved Wigner-Ville method are given. Figure 2 shows the time-frequency spectrum and the power spectrum of the simulated signal from STFT, the improved Wigner-Ville and the proposed method respectively. Here, the smoothing pseudo Wigner-Ville distribution is used to effectively solve the problem of cross-interference in Wigner-Ville distribution. From Fig. 2(a), it can be seen that the two-dimensional spectrum of STFT exhibits the instantaneous frequency characteristics of the simulated signal with sinusoidal variation. However, the resolution of the two-dimensional spectrum is low, and the bright band which represents the frequency change is very wide. In Fig. 2(b), the obtained marginal spectrum (power spectrum) is also smooth and lacks peak information. In the improved Wigner-Ville spectrum shown in Fig. 2(c), the clarity and resolution are improved. Its marginal spectrum in Fig. 2(c) is more detailed than that shown in Fig. 2(b), but its practicability is still low. The time-frequency spectrum and power spectrum of the fractional domain obtained from the presented method are shown in Fig. 2(e) and (f). In the 0.05–0.1 Hz frequency band, the amplitude variation of each frequency component is shown in Fig. 2(f) clearly while the variation is smoothed in Fig. 2(b) and (d).

Fig. 2
figure 2

Time-frequency spectrum and power spectrum of simulated signal: (a), (b) is from STFT; (c), (d) is from the smoothing pseudo Wigner-Ville method and (e), (f) is from the presented method

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Renyi entropy of time-frequency spectrum from three methods are compared in Table 1. Compared with the STFT method, the entropy value has been reduced from 9.9979 to 7.9664 using the proposed method. And it is smaller than that of the smoothing pseudo Wigner-Ville method. The time-frequency aggregation of the instantaneous spectrum from FRFT is greatly enhanced. It can be seen that the accuracy of instantaneous energy distribution and marginal spectrum has been greatly improved. Resolution is significantly higher than that of Fig. 2(a) and (c). The bright band with sinusoidal shape is more concentrated, the time frequency clustering is greatly improved and the brightness is uniform. The time-frequency distribution of energy coincides well with the instantaneous frequency distribution shown in Fig. 1. The marginal spectrum is no longer smooth, and the peak information is displayed, which is conducive to further identification.

Table 1 Comparison of Renyi entropy of time-frequency spectrum from three methods
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Measured Data Processing

To further illustrate the effectiveness of the proposed method, it is applied to fault diagnosis of rolling bearings. The experimental data come from the electrical engineering laboratory of Case Western Reserve University, USA. The tested bearing is 6205-2RS JEM SKF. The parameters of some main components are described below: Rolling element number is 9, the contact angle is 0, bearing speed is 1750 r/min. Inner diameter is 25.0012 mm, outer diameter is 51.9989 mm, thickness is 14.5906 mm, pitch diameter is 39.0398 mm, rolling body diameter is 7.9412 mm. In outer ring, damage diameter is 0.1778 mm which is made by the electric spark machining. The sampling frequency of vibration signal is 12 kHz. The selected data contains fault signals from the inner, outer and rolling rings and its time series are shown in Fig. 3. Based on the theory, the fault frequency of bearing can be calculated as following [2]:

Fig. 3
figure 3

Measured rolling bearing fault signal: (a) inner ring, outer ring (b) and the roller(c)

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Fault frequency on inner race:

$$ {f}_i=\frac{z}{2}\left(1+\frac{d}{D}\cos \alpha \right){f}_r $$
(12)

Fault frequency on outer race:

$$ {f}_o=\frac{z}{2}\left(1-\frac{d}{D}\cos \alpha \right){f}_r $$
(13)

Fault frequency on rolling ball:

$$ {f}_b=\frac{D}{d}\left(1-{\left(\frac{d}{D}\right)}^2{\cos}^2\alpha \right){f}_r $$
(14)

Here, fr is the bearing speed, z is the number of balls, d is the rolling diameter, D is the bearing pitch diameter, α is rolling contact angle. So based on the theoretical formula, fault frequency can be determined as: fi ≈ 158 Hz, fo ≈ 104 Hz, fb ≈ 137 Hz.

Firstly, the vibration signal of rolling bearing with outer ring fault is considered. The smoothing pseudo Wigner-Ville spectrum and its corresponding marginal spectrum of outer ring fault signal are shown in Fig. 4(a) and (b). In time-frequency spectrum, there is an obvious bright band near 104 Hz, but it is discontinuous and its divergence is large. In the marginal spectrum, there is an obvious spectral peak at the characteristic frequency point 104 Hz, and the amplitude is very large. This is because the fault position acts as an excitation source when the outer ring of the rolling bearing fails, and the excitation produces an impact every rotation. The vibration signal of the impact is expressed by the modulation of the amplitude modulation signal to the carrier signal of the bearing natural frequency, where the modulation frequency is the fault characteristic frequency. Therefore, the bearing fault lists in outer ring, which can be obtained from the spectrum line at the characteristic frequency fo = 104 Hz. However, through careful observation it can be shown that other components of the fault diagnosis interference are greater and other harmonic components can’t be shown in the smoothing pseudo Wigner-Ville spectrum and its corresponding marginal spectrum.

Fig. 4
figure 4

Time-frequency spectrum and power spectrum of outer race fault signal: (a), (b) based on the smoothing pseudo Wigner-Ville method; (c), (d) based on the proposed method

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The time-frequency spectrum and the corresponding marginal spectrum in the fractional domain are shown in Fig. 4(c) and d. In the time-frequency spectrum, not only the 104 Hz energy band is brighter and more concentrated, but also the second harmonic can be seen. In the marginal spectrum, the characteristic frequency of 104 Hz, its harmonics 208 Hz and 312 Hz all can be marked clearly, which is more helpful to the diagnosis of bearing fault. Obviously, the proposed method can get more detailed time-frequency spectrum and higher time-frequency resolution. The Renyi entropy of time-frequency spectrum of the measured data are shown in Table 2. Compared with the smoothing pseudo Wigner-Ville method, the entropy value has been reduced from 8.998 to 7.8468 by using the proposed method. The energy distribution is more concentrated. The time-frequency accumulation spectrum of the fractional energy band has the characteristics of focusing. The fractional energy band can highlight the target component to the greatest extent, and restrain other components and noise. This feature is especially suitable for extracting weak fault characteristics of vibration signals.

Table 2 Renyi entropy of time-frequency spectrum of the measured data
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Furthermore, the rolling bearing vibration signals from inner ring and rolling body are analyzed. Similarly, the results are compared with the smoothing pseudo Wigner-Ville method. According to the physical information of rolling bearing, we can calculate the fault frequency of ring as: fi = 158 Hz and fb = 137 Hz. Figures 5 and 6 show the processed results. It can be seen that the conclusions obtained from the analysis are consistent with these shown in Fig. 4. Similarly, these Renyi entropy values listed in Table 2 reflect the same conclusion that the proposed algorithm has higher time-frequency resolution than smoothing pseudo Wigner-Ville method. In the fractional domain time-frequency spectrum and the corresponding marginal spectrum, the characteristic frequency and its harmonics can be marked more clearly. It is more conducive to the diagnosis of bearing failure. The presented method is better than the smoothing pseudo Wigner-Ville method in fault diagnosis.

Fig. 5
figure 5

Time-frequency spectrum and power spectrum of inner race fault signal: (a), (b) based on the smoothing pseudo Wigner-Ville method; (c), (d) based on the proposed method

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Fig. 6
figure 6

Time-frequency spectrum and power spectrum of rolling body fault signal: (a), (b) based on the smoothing pseudo Wigner-Ville method; (c), (d) based on the proposed method

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Conclusion

In this paper, a new fault diagnosis method is proposed based on instantaneous power spectrum estimation in fractional domain. Here, the maximum kurtosis coefficient method is used to estimate the spectrum of optimal fractional domain. And the power spectrum distribution in fractional domain is transformed into time-frequency distribution by using fractional domain time-frequency analysis technique. This method enables intuitive physical meaning in fractional domain of the power spectrum distribution. At the same time, the resolution of two-dimensional power spectrum is improved. Especially in weak signal analysis, there is a good aggregation for weak fault signals. Experimental results of simulation signal and real data show that the proposed algorithm has higher time-frequency resolution than the smoothing pseudo Wigner-Ville method, and can obtain more accurate characteristic frequency identification for fault diagnosis.