## 2 Basic Theory
### Order Tracking  
The order refers to the number of vibrations in each rotation of the shaft. Thus, the order is irrelevant to the rotating speed. The definition of order is shown in Equation (16):  
\[O=\frac{60f}{n} \tag{16}\]  
where \(O\) represents the order, \(n\) (r/min) denotes the rotating speed of the shaft, and \(f\) (Hz) represents the vibration frequency. The key point of order analysis is even angle sampling, while the key point of even angle sampling is to acquire sampling time. After the sampling time is known, stationary signals in the angle domain will be obtained by interpolating and fitting the original signal [18]. As shown in Equations (4)-(6), the FCFs of intershaft bearings are proportional to \(|f_{0}-f_{i}|\). Therefore, an order tracking method in the speed difference domain is required for feature extraction of the intershaft bearing's faults. This order tracking method is different from traditional order tracking methods because traditional ones are based on one speed which can be measured directly. To extract the fault features of intershaft bearing, vibration data needs to be order-tracked according to the difference of the two rotational speeds. The difference and its key phasor pulse cannot be directly measured. Therefore, the even angle sampling time must be selected on the basis of the trend of the rotational speed difference. The details of the procedures are as follows. 1. Resampling of original vibration: A polyphasic finite impulse response filter is adopted to upsample the frequency. As a result, frequency aliasing in even angle sampling can be avoided [27]. Resampling, which is to raise the sampling frequency, is obtained through interpolation on the original signal. As shown in Figure 3, \(v(n)\) can be obtained by raising the sampling frequency of the original signal \(x(n)\) to its \(L\) times; in other words, \(L-1\) zeros are inserted between the adjacent elements of \(x(n)\). Then, the output \(y(n)\) with interpolation is obtained by designing low-pass filter \(h(n)\) to the process \(v(n)\). \[v(n)=\left\{ \begin{array}{cc}x(n/L)&n=0,\pm L,\pm L,\cdots\\ 0&\text{others}\end{array} \right.\] (17) \[ \begin{array}{c}x(n)\\ \end{array} \] (18) \[ \begin{array}{c}x(n)\\ \end{array} \] (19) \[ \begin{array}{c}x(n)\\ \end{array} \] (20) \[ \begin{array}{c}x(n)\\ \end{array} \] (21)
2. Acquiring the even angle sequence: The corresponding rotation angle at each sampling point is obtained according to the trend of the rotational speed difference. Then, the minimum and maximum angles are determined to obtain the even angle sequence on the basis of the principle that the sampling number per revolution is twice the maximum order to be analyzed (sampling when the shaft passes the even angle point) [28]. Assuming that the original sampling interval is \(\Delta t\) and the rotational speed difference of the \(k\)th sampling point \(i\) is \(v_{i}\), then the computation equation for the corresponding rotation angle is  
\[\theta_{i}=2\pi\Delta t\times\frac{v_{i}}{60} \tag{22}\]  
The sampling time is \(t_{i}=k\Delta t\) and the shaft rotation angle at the moment is  
\[\theta_{t_{i}}=\sum_{i=1}^{k}\theta_{i} \tag{23}\]  
Therefore, the sampling point of the even angle during \(t_{1}-t_{i}\) is as follows:  
\[N_{i}=2\times MaxOrder\times\frac{(\theta_{t_{i}}-\theta_{t_{1}})}{2\pi} \tag{24}\]  
where \(MaxOrder\) is the maximum order to be analyzed. According to the sampling theorem of even angle sampling, the minimum sampling number per revolution is \(2\times MaxOrder\), the sampling frequency in the angle domain. According to the Shannon sampling theorem, \(MaxOrder\) can be written as  
\[2\times MaxOrder\times\frac{n_{\text{max}}}{60}\leq f_{s} \tag{25}\]  
Figure 3: Flowchart of interpolation filtering. where \(n_{\max}\) denotes the maximum speed difference within the sampling work, and \(f_{s}\) is the sampling frequency of the original signal.