An adaptive method based on fractional empirical wavelet transform and its application in rotating machinery fault diagnosis

FrFT is a generalized form of the standard Fourier transform. Essentially, it can be interpreted as a counterclockwise rotation transformation from the time axis to the $u$ axis with angle $\alpha$ [29]. In addition, FrFT can be regarded as the transformation of orthogonal expansion of the chirp signal in the Fourier spectrum [30]. The FrFT of signal $f(t)$ is defined as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{F}^{\,p}}f\left( t \right)=\int_{-\infty }^{\infty }{{{K}_{p}}(t,u)\,f(t)}{\rm d}t.\nonumber \end{align} \tag{ 1 }$$

Wherein, $p$ is the fractional order and ${{K}_{p}}(t,u)$ is the kernel of FrFT, whose expression is as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{K}_{p}}(t,u)=\left\{\begin{array}{@{}cccccccccccccccccccc@{}} {{A}_{\alpha }}\exp \left[\,j\pi \left( {{u}^{2}}{\rm cot}\alpha -2ut\,{\rm csc}\,\alpha +{{t}^{2}}{\rm cot}\alpha \right) \right]~ & \alpha \ne n\pi \quad \quad \,\, \nonumber \\ \delta \left( t-u \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad & \alpha =n\pi \quad \quad \,\, \nonumber \\ \delta \left( t+u \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad & \alpha =(2n+1)\pi . \nonumber \end{array} \right.\nonumber \end{align} \tag{ 2 }$$

Wherein, $u$ is the coordinate axis in the fractional Fourier domain, $\alpha =\frac{p\pi }{2}$ is the rotation angle and ${{A}_{\alpha }}$ is an integer shown as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{A}_{\alpha }}=\frac{{\rm exp}(\frac{-j\pi {\rm sgn}({\rm sin}\alpha)}{4}+\frac{j\alpha }{2})}{{{\left| {\rm sin}\alpha \right|}^{1/2}}}.\nonumber \end{align} \tag{ 3 }$$

Based on the definition of the FrFT, the FrFT of $f(t)$, with a random order, can be taken as the expression that maps to the $(u,v)$ plane after the rotation angle of the $(t,\omega)$ plane. From the above description, for a given order p , the kernel ${{K}_{p}}(t,u)$ is actually an LFM signal (or a chirp signal) with the chirp rate $\alpha$, so the FrFT of an LFM signal will be a $\delta$ function in the fractional Fourier domain. In other words, an LFM signal can be converted into an impulse signal and can be very tight in a proper fractional Fourier domain. Therefore, FrFT is an effective signal filtering method for LFM signals. If the proper fractional order p  can be confirmed, the LFM signal can be separated by filtering the fractional Fourier domain. The rotor startup vibration signal can be regarded as a multi-component LFM signal, where the order of each component can be obtained by rotational acceleration. Thus, it is feasible to analyze the rotor startup vibration signal by FrFT.

However, the filter is a repeatable tool based on FrFT with the flowchart shown in figure 1. In each process, the parameters of fractional Fourier filters need to be set manually. Due to its lack of adaptive capability and fast calculation, this method cannot be applied to adaptively analyze the rotor startup vibration signal.

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EMD, proposed by Huang in 1998, has been applied to extract the features of non-linear and non-stationary signals in many fields, especially rotating mechanical vibration signals, because of its good adaptability. When EMD is applied to decompose a signal, an adaptive generalized basis is used, and the basis function is adopted on the basis of the signal itself. With the EMD method from a perspective of characteristic time scales, the modes with the smallest characteristic time scales in the signal are first separated, and then the modal functions with larger characteristic time scales are separated step by step. Finally, the signal is decomposed into the sum of several intrinsic mode functions (IMFs). Hence, EMD can be considered as a set of high-pass filters. However, EMD still has many problems, such as modal aliasing and the endpoint effect, resulting in incorrect decomposition outcomes when decomposing non-stationary signals [31].

EWT is an adaptive method based on EMD and wavelet transform. With EWT, the Fourier spectrum of the signal is divided adaptively and the wavelet filter is established in each interval to decompose the signal into multiple IMFs respectively.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f\left( t \right)=\underset{k=0}{\overset{N}{\mathop \sum }}\,{{f}_{k}}(t).\nonumber \end{align} \tag{ 4 }$$

An IMF is represented as an amplitude-frequency modulated function.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{k}}\left( t \right)={{F}_{k}}(t)\cos ({{\varphi }_{k}}(t)).\nonumber \end{align} \tag{ 5 }$$

In the above formula, ${{F}_{k}}\left( t \right),\varphi _{k}^{\prime}\left( t \right)>0\quad \forall t$, and the main assumption is that ${{F}_{k}}$ and $\varphi _{k}^{\prime}$ vary much slower than ${{\varphi }_{k}}$.

How to divide the Fourier spectrum is crucial because it directly affects the results of EWT. To divide the spectrum into continuous intervals with the number of N, the spectrum is standardized as $[0,\pi ]$. Then, the maximum points in the spectrum are searched, and the minimum value of ${{\omega }_{n}}$ as the boundary between two adjacent maximum points is searched. Therefore, each interval can be represented as ${{\Lambda }_{n}}=\left[ {{\omega }_{n-1}},{{\omega }_{n}} \right],n-1,2,\ldots ,N({{\omega }_{0}}=0,{{\omega }_{n}}=\pi)$, where a transition phase is defined with each ${{\omega }_{n}}$ as the center and ${{T}_{n}}=2{{\tau }_{n}}$ as the width. Finally, a band-pass filter is established on each ${{\Lambda }_{n}}$. Based on the construction of both Littlewood-Paley and Meyer's wavelets, the empirical scaling function and the empirical wavelets are defined by the formulas of (6) and (7), respectively.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \widehat{{{\Phi }_{n}}}\left( \omega \right)=\left\{\begin{array}{@{}llcccccccccccccccccc@{}} 1 & {\rm if}\,\left| \omega \right|\leqslant (1-\gamma){{\omega }_{n}} \nonumber \\ \cos \left[ \frac{\pi }{2}\beta \left( \frac{1}{2\gamma {{\omega }_{n}}}\left( \left| \omega \right|-\left( 1-\gamma \right){{\omega }_{n}} \right) \right) \right],~ & {\rm if}\,(1-\gamma){{\omega }_{n+1}}\leqslant \left| \omega \right|\leqslant (1+\gamma){{\omega }_{n+1}} \nonumber \\ 0 & {\rm otherwise}. \nonumber \end{array} \right.\nonumber \end{align} \tag{ 6 }$$

And

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \widehat{{{\psi }_{n}}}\left( \omega \right)=\left\{\begin{array}{@{}llcccccccccccccccccc@{}} 1 & {\rm if}\,(1+\gamma){{\omega }_{n}}\leqslant \left| \omega \right|\leqslant (1-\gamma){{\omega }_{n+1}} \nonumber \\ \cos \left[ \frac{\pi }{2}\beta \left( \frac{1}{2\gamma {{\omega }_{n+1}}}\left( \left| \omega \right|-\left( 1-\gamma \right){{\omega }_{n+1}} \right) \right) \right],~ & {\rm if}\,(1+\gamma){{\omega }_{n}}\leqslant \left| \omega \right|\leqslant (1-\gamma){{\omega }_{n+1}} \nonumber \\ \sin \left[ \frac{\pi }{2}\beta \left( \frac{1}{2\gamma {{\omega }_{n}}}\left( \left| \omega \right|-\left( 1-\gamma \right){{\omega }_{n}} \right) \right) \right] & {\rm if}\,(1-\gamma){{\omega }_{n}}\leqslant \left| \omega \right|\leqslant (1+\gamma){{\omega }_{n}} \nonumber \\ 0 & {\rm otherwise}. \nonumber \end{array} \right.\nonumber \end{align} \tag{ 7 }$$

Wherein, $\gamma <{{\min }_{n}}[\frac{{{\omega }_{n+1}}-{{\omega }_{n}}}{{{\omega }_{n+1}}+{{\omega }_{n}}}]$, and $\beta \left( x \right)={{x}^{4}}(35-84x\,+$ $70{{x}^{2}}-20{{x}^{3}})$.

In the same way as for the classic wavelet transform, the detailed coefficients are calculated by the inner products with the empirical wavelets:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W_{f}^{\varepsilon }\left( n,t \right)=\left\langle\,f,{{\psi }_{n}} \right\rangle =\int{f\left( \tau \right)\overline{{{\psi }_{n}}(\tau -t)}}{\rm d}\tau .\nonumber \end{align} \tag{ 8 }$$

And the approximation coefficients by the inner product with the scaling function is as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W_{f}^{\varepsilon }\left( 0,t \right)=\left\langle\,f,{{\Phi }_{n}} \right\rangle =\int{f\left( \tau \right)\overline{{{\Phi }_{n}}(\tau -t)}}{\rm d}\tau .\nonumber \end{align} \tag{ 9 }$$

Then the empirical modes are given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{0}}\left( t \right)=W_{f}^{\varepsilon }\left( 0,t \right)*{{\Phi }_{1}}(t)\nonumber \end{align} \tag{ 10 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{k}}\left( t \right)=W_{f}^{\varepsilon }\left( k,t \right)*{{\psi }_{k}}(t).\nonumber \end{align} \tag{ 11 }$$

The flowchart of the filter based on EWT is shown in figure 2. This method can be utilized to set the parameters and to construct filters adaptively in the Fourier spectrum to extract the signal components. Moreover, this method involves considerably less computational analysis. However, when the EWT is utilized to extract the components of the rotor startup vibration signal, it is no longer valid. The detailed reasons for this are analyzed in the next section.

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As a non-stationary signal, the rotor startup vibration signal is composed of different components, and each component can be seen as a LFM signal [32]. So, this signal can be regarded as a multi-component LFM signal:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle s\left( t \right)=\underset{i=1}{\overset{N}{\mathop \sum }}\,{{A}_{i}}(t)\cos [2\pi {{f}_{i}}\left( t \right)t+{{\Phi }_{i}}(t)].\nonumber \end{align} \tag{ 12 }$$

Wherein, ${{A}_{i}}(t)$, ${{f}_{i}}(t)$ and ${{\Phi }_{i}}(t)$ are the amplitude, frequency change rate and phase of each component, respectively. Here, ${{f}_{i}}(t)$ of each order component is proportional, which means $f_i(t)=i \times f_1(t)$. The frequency of each component is shown in figure 3.

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The signal during startup needs to be decomposed to extract effective features. As an adaptive processing method, EWT can extract empirical modes of the signal with the precondition that each mode of the signal is an AM-FM signal with a compact support Fourier spectrum. Gilles mentions that if the input signal is composed of two LFM signals which overlap in both time and frequency domains, EWT will not be able to separate them [22]. In the same way, if the vibrational signal of the rotor startup with a constant acceleration is composed of some LFM signals which overlap in both time and frequency domains, EWT will obviously not be able to separate them adaptively.

To decompose the rotor startup signal adaptively, the signal is projected into a new time–frequency domain where no components overlap with each other. The FrFT of the LFM signal is a compact support in the fractional Fourier domain. Moreover, the rotor startup vibration signal can be regarded as a multi-component LFM signal. If the components do not overlap in the fractional Fourier domain of a certain order, the empirical wavelet filters could be constructed in the fractional Fourier domain and applied to process each segment of the signal. Based on this, a novel method called FrEWT is proposed in this paper. The procedures of this method are as follows:

• (1)  
  Calculating the fractional order p The selection of the proper fractional order p  is essential to an appropriate time–frequency representation in the fractional Fourier domain. The fractional order p  can be obtained by rotational acceleration. In addition, the rotational acceleration can be obtained by using the rotor key-phase signal to calculate the rotational speed. Therefore, the order p  can be calculated by the equation [27]:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p=2{{\tan }^{-1}}({{f}_{m}}\cdot T/{{f}_{{\rm s}}})/\pi .\nonumber \end{align} \tag{ 13 }$$

  Wherein, ${{f}_{m}}$ is the frequency change rate, T is the time width and ${{f}_{{\rm s}}}$ represents the sampling rate.
• (2)  
  Calculating the time–frequency representation of the analyzed signalThe time–frequency representation of the analyzed signal is based on FrFT.
• (3)  
  Adaptive segmentation of the signal fractional Fourier domainThe local maximum in the fractional Fourier domain is found and the boundaries are set in the horizontal coordinate of the minimum value between the adjacent maximum values. Then, the fractional Fourier domain is decomposed into N consecutive intervals.
• (4)  
  Constructing wavelet filters in intervalsUsing the method for constructing Littlewood-Paley and Meyer wavelets, the empirical wavelets are defined as band-pass filters on each interval. Then, a fractional empirical scaling function and fractional empirical wavelets are defined.
• (5)  
  Calculating the detail coefficients and approximation coefficientsIn the same way as for EWT, the detail coefficients are calculated by the inner products with the fractional empirical wavelets:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W_{p}^{\varepsilon }\left( n,u \right)=\int{f\left( t \right){{K}_{p}}\left( u,t \right)\overline{{{\widehat{\varphi }}_{n}}\left( u \right)}}{\rm d}t=(\widehat{f}(u)\overline{{{\widehat{\varphi }}_{n}}(u)})_{p}^{V}.\nonumber \end{align} \tag{ 14 }$$

  And the approximation coefficients by the inner product with the fractional empirical scaling function are as follows:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W_{p}^{\varepsilon }\left( 0,u \right)=\int{f\left( t \right){{K}_{p}}\left( u,t \right)\overline{{{\widehat{\Phi }}_{n}}(u)}}{\rm d}t=(\widehat{f}(u)\overline{{{\widehat{\Phi }}_{n}}(u)})_{p}^{V}.\nonumber \end{align} \tag{ 15 }$$
• (6)  
  Extracting components of the signal in the fractional Fourier domainEach component can be obtained by the following function:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{0}}\left( u \right)=W_{p}^{\varepsilon }\left( 0,u \right)*{{\Phi }_{1}}(u)\nonumber \end{align} \tag{ 16 }$$

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{k}}\left( u \right)=W_{p}^{\varepsilon }\left( k,u \right)*{{\varphi }_{k}}(u).\nonumber \end{align} \tag{ 17 }$$
• (7)  
  Calculating time domain representation of each componentWith the fractional order p  obtained by the first step, the inverse FrFT of each component in the fractional Fourier domain is defined as:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{i}}(t)=\int_{-\infty }^{\infty }{{{K}_{-p}}(u,t){{f}_{i}}(u)}{\rm d}u.\nonumber \end{align} \tag{ 18 }$$

  The flowchart of FrEWT is shown in figure 4.

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From the analysis mentioned above, FrFT with the order p  can be seen as a mapping of the plane $(t,\omega)$ where the signal $f(t)$ is located after a rotation angle of $\alpha =p\pi /2$ to the plane $(u,v)$ and the FrFT of the LFM signal can be regarded as a $\delta$ function, so each component of the rotor startup vibration signal is a $\delta$ function in the fractional Fourier domain. However, other components are still broadband responses in a certain fractional Fourier domain. EWT is not applicable in the analysis of the signal components overlapping with each other in the Fourier spectrum. Therefore, it is necessary to find a suitable fractional order to separate each component in the fractional Fourier domain. In the analysis of the rotor startup vibration signal, the higher order component is generally regarded as random noise. And the operation state of the rotor can be reflected by the first four-order components of the startup vibration signal. In section 3.1, a signal composed of three LFM signals is constructed to simulate the rotor startup signal, whose time–frequency is shown in figure 3, in which the two adjacent components overlap with each other in the Fourier spectrum.

The fractional Fourier domain representation of the signal performed with the order ${{p}_{1}}$ is shown as figure 5, in which the blue axis represents the axis of $({{u}_{1}},{{v}_{1}})$, and the red line represents the three components in the $({{u}_{1}},{{v}_{1}})$ spectrum. It is clearly seen that the first order component is a compact support, but the second and third order components are still broadband responses and overlap with each other in the first order fractional Fourier domain. In this case, these two components cannot be decomposed. The two broadband responses are located on the same side of the first order component, so they are easily aliasing. Therefore, it is necessary to select a suitable fractional Fourier domain where the two broadband responses are located on different sides of a certain order component.

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The fractional Fourier domain representation of the signal performed on the second order component is shown as figure 6, in which the red line still represents the three components in the $({{u}_{2}},{{v}_{2}})$ domain. This time, the three components of the signal are separated from each other, and the two broadband responses are located on different sides of the $\delta$ function in the second order fractional Fourier domain. Therefore, this following method can be used to realize the adaptive extraction of the rotor startup vibration signal. The procedures are as follows:

• (1)  
  Calculating the chirp rate of the rotor startup and generation of fractional order ${{p}_{1}}$.
• (2)  
  In the fractional Fourier domain of order ${{p}_{1}}$, extract the lower order component, first order component and the higher component from the rotor startup signal by FrEWT.
• (3)  
  Calculate the fractional order ${{p}_{3}}$ by referencing the formula [27].

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{p}_{3}}=2{\rm co}{{{\rm t}}^{-1}}(3\times {{f}_{m}}\cdot T/{{f}_{{\rm s}}})/\pi .\nonumber \end{align} \tag{ 19 }$$
• (4)  
  In the fractional Fourier domain of order ${{p}_{3}}$, extracting the second order component, third order component and fourth order component from the higher component 1 by FrEWT.

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A Bently RK4 rotor test bench is used to verify the effectiveness of FrEWT in adaptive extraction of the features of the rotor startup vibration signal by simulating different rotor fault experiments.

As shown in figure 20, the test bench is mainly composed of a rotor system and a vibration test system. The rotor system consists of a rotor, a motor, a pair of bearings and a base. The vibration test system consists of a displacement sensor, key-phase sensor and data acquisition instrument. In addition, the rotating speed controller of the test bench adopts a closed-loop control, which can control the uniform acceleration and deceleration of the rotor. Then, the appropriate speed ratio is selected, and continuous data is collected in the rotor startup process to obtain the vibration signal and key-phase signal of the rotor. In the fault simulation experiment, the speed change interval is 250–4100 rpm (the starting speed is 250 rpm and the running speed is 4100 rpm), the sampling frequency is chosen as 1024 Hz, the sampling time is every 20 s and the data points are 20 480.

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The rotor normal startup vibration signal is analyzed to verify the effect of extraction by FrEWT and EWT, respectively. The time domain waveform and Fourier spectrum of this signal are shown in figure 21. The time waveform shows that the amplitude of the actual vibration signal is more complex than the simulation signal, for the reason that the startup stage could result in critical resonance, thus causing the complex amplitude modulation. The Fourier spectrum proves a complex wideband response and overlapping components, resulting in the error in the extraction of the signal's components.

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FrEWT is used to extract components of the rotor vibration signal during the startup stage. Firstly, the key-phase signal is processed to obtain the speed sequence of the rotor startup signal, as shown in figure 22. Secondly, the fractional order ${{p}_{1}}$ is decided by the speed sequence, and the first adaptive decomposition of the original signal is obtained by FrEWT (${{p}_{1}}$). Thirdly, the fractional order ${{p}_{3}}$ can be obtained with formula (19) and ${{p}_{1}}$. Finally, the first, second, third and fourth order components of the signal are obtained by FrEWT (${{p}_{3}}$), and the corresponding fractional Fourier domain spectrum is shown in figure 23, where each component is marked as 1X–4X.

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It can be seen that each component of the signal is a compact support fractional Fourier domain, and the corresponding fractional frequencies are proportional. Then, the time domain waveform of each component can be obtained with inverse FrFT, and the waveform's changes with speed are given here in figure 24. For comparison, EWT is used in the same rotor normal startup vibration signal. EWT decomposes the signal into four components. The Fourier spectrum and the waveform's changes with speed are shown in figures 25 and 26. Obviously, the extracted components by EWT are incorrect and meaningless. Therefore, FrEWT can be adapted to extract the correct components of the rotor startup vibration signal, whereas EWT cannot.

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In this part, we will analyze the rotor weak fault vibration signals, and discuss the advantage of the extracted features in the startup stage compared to those in the stable condition.

Correspondingly, the four typical rotor weak faults of imbalance, misalignment, rubbing and crack are simulated on the Bently RK4 rotor test bench, and the vibration signal and key-phase signal are collected during the startup stage and stable running conditions, respectively.

Imbalance is one of the most common faults of a rotor, so it is important to identify such faults [33]. Imbalance is mainly caused by the mass eccentricity of the rotor. On the test bench, a light eccentric block (which is 4 mg, much less than the mass of the rotor) is added to the mass plate to simulate the weak imbalance fault, as shown in figure 27. When the rotor is in an imbalanced condition, the amplitude of its vibration signal is mainly based on the first order component, and is higher than that in a normal condition. Correspondingly, the other components of the rotor startup vibration signal are small [34].

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The components of the weak imbalance startup signal are extracted by FrEWT, and the analysis results are shown in figure 28. The waveforms of each component's change with speed are given here in figure 29, which can be obtained by inverse FrFT. And it can be seen that the critical speed of this rotor system is about 2000 rpm. In figure 29, the first order component of the imbalance startup vibration signal is obviously large, and the other components are small. Therefore, the weak imbalance fault of the rotor can be identified from the extracted components.

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Misalignment is a very common fault of a rotor, and it means that the center line of each rotor in the system is not a straight line [35]. On the test bench, one gasket is placed under the same side of the motor to raise the height about 1 mm to simulate the weak misalignment of the rotor, as shown in figure 30. For the rotor misalignment faults, the amplitude of its vibration signal is mainly based on the first order component; meanwhile the second order component is larger at the sub-critical speed (half of the critical speed), and the magnitude of the second order component can directly reflect the degree of misalignment [34].

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The feature components of the weak misalignment startup vibration signal are extracted by FrEWT, and the analysis results are shown in figure 31. The waveforms of each component's change with speed are given here in figure 32, which can be obtained by inverse FrFT. In figure 32, the second order component of the misalignment startup vibration signal at the sub-critical speed (which is about 1000 rpm) is significantly large, even larger than that at the critical speed. Therefore, the weak misalignment fault can be identified.

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Rubbing is one of the common faults of a large rotor, and the vibration is caused by the contact of the rotor and the fixed parts. Rubbing will result in rotor wear, increased power loss and permanent bending of the rotor, so it is important to identify any rubbing faults [36]. On the test bench, the weak rubbing fault of the rotor was simulated by attaching a metal friction bar to the rotor in the middle of the rotating shaft (with a minor contact), as shown in figure 33. When the rotor has a rubbing fault, there are many non-linear factors that will cause the amplitude of the second and third order components to become larger at the sub-critical speed [34].

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The feature components of the weak rubbing startup signal are extracted by FrEWT, and the analysis results are shown in figure 34. The waveforms of each component's change with speed are given here in figure 35, which can be obtained by inverse FrFT. In figure 35, the second and third order components of the rubbing startup vibration signal at the sub-critical speed are both larger than that at the critical speed. Similarly, the weak rubbing fault can also be identified.

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In contrast, the probability of a crack fault is much less than the other faults, but there are many potential factors that can cause this fault. A crack fault is a gradual evolution process that is not easily diagnosed, and its harmfulness is particularly serious [37]. Therefore, the diagnosis of rotor crack faults is an important work that cannot be ignored. On the test bench, a cracked rotor is used to replace the normal rotor. The crack is in the middle of the shaft and has a depth of about a quarter of the rotor diameter to simulate the crack weakness of the rotor, as shown in figure 36. For the crack fault, the stiffness of the rotor is asymmetrical, and causes the amplitude of each component to become larger [34].

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The feature components of the weak crack startup signal are extracted by FrEWT, and the analysis results are shown in figure 37. The waveforms of each component's change with speed are given here in figure 38, which can be obtained by inverse FrFT. In figure 38, each component of the crack startup vibration signal is always large during the rotor startup stage, so the weak crack fault can also be identified.

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Finally, the vibration signal of the rotor in steady state operation is also analyzed to diagnose the weak rotor fault. In actual industry, the steady working speed of many rotating machines is often higher than its critical speed [34]. Therefore, the vibration signals at a stable running speed (about 4100 rpm) of five different conditions have been acquired on the Bently RK4 test bench, and the time domain waveforms of these signals are shown in figure 39.

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Obviously, because of the weak degree of faults, these time domain waveforms are not so different to each other, and the fault type of the rotor cannot be directly identified by the magnitude or variation of such signals. Therefore, the vibration signal at a stable running speed does not contain the features that can reflect the type of fault, thus the analysis of such signals is no longer effective for the weak fault diagnosis.

The analysis on the results shows that the features extracted by EWT are obviously redundant and erroneous, but the FrEWT could accurately extract the features. In addition, the FrEWT is used to analyze the weak fault startup signal, and the extracted components could reflect the fault characterization, while the signals in the stable condition cannot reflect. Therefore, the proposed FrEWT method demonstrates a better performance than EWT, and successfully identifies the weak fault of the rotor.