## 2 Fault Diagnosis Method of Scraper Conveyor Gear
### Mathematical Model of Fault Current  
The scraper conveyor used a three-phase motor as the power source. The output speed of the motor was decelerated by the reducer and then the power was transmitted to the sprocket, which drove the scraper chain to transport the coal flow. The scraper conveyor transport system and motor model are shown in Figure 4. When the gear failure occurred in the reducer, the electromagnetic torque and current would be affected by the gear failure. Finally, the current will contain the gear failure feature information. The influence mechanism of gear failure on current will be analyzed below. Figure 4: Scraper conveyor transport system and motor model. The meanings of the symbols and serial numbers in the figure are as follows: 1—sprocket; 2—blind axis; 3—scraper chain; 4—motor; 5—hydraulic coupling; and 6—reducer; A, B, C—axis of stator winding; a, b, c—axis of rotor winding; I\({}_{A}\), I\({}_{B}\), I\({}_{C}\)—three—phase current; \(\theta\)—space angular displacement. Figure 3: The fault diagnosis method. BE: bispectrum entropy. The electromagnetic torque equation of the scraper conveyor motor can be expressed as  
\[T_{e}=J\frac{d\omega_{m}}{dt}+T_{L}+B\omega_{m}, \tag{1}\]  
where \(T_{e}\) represents the electromagnetic torque of the motor, \(T_{L}\) is the friction torque of the transmission system, \(B\) is the load torque of the scraper conveyor, \(J\) is the inertia of the system, and \(\omega_{m}\) represents the angular speed of the motor. When the gear failure occurs in the reducer, the motor speed and electromagnetic torque will fluctuate [25; 26]. In this case, the electromagnetic torque, load torque and rotational speed can be decomposed into a constant and cosine component, respectively, expressed as  
\[\left\{ \begin{array}{l}T_{e}=T_{e0}+\sum\limits_{i}T_{ei}\cos(2\pi f_{i}t+ \varphi_{ei})\\ T_{L}=T_{L0}+\sum\limits_{i}T_{Li}\cos(2\pi f_{i}t+\varphi_{Li})\\ \omega_{m}=\omega_{m0}+\sum\limits_{i}\omega_{mi}\cos(2\pi f_{i}t+\varphi_{mi} )\end{array} \right. \tag{2}\]  
where \(T_{e0}\), \(T_{L0}\) and \(\omega_{m0}\) represent the mean values of the electromagnetic torque, load torque and rotational speed, respectively; \(f_{i}\) represents the frequency of the electromagnetic torque, load torque and rotational speed; \(\varphi_{ei}\), \(\varphi_{Li}\) and \(\varphi_{mi}\) represent the phases of the electromagnetic torque, load torque and rotational speed, respectively. In addition, the equation of the electromagnetic torque can also be expressed as  
\[T_{e}=k_{i}i_{q}, \tag{3}\]  
where \(k_{t}\) is the electromagnetic torque coefficient. Combining formulas (2)-(4) obtains:  
\[i_{q}=i_{q0}+\sum\limits_{i}i_{qi}\cos(2\pi f_{i}t+\varphi_{qi}), \tag{4}\]  
where \(i_{q0}\) represents the mean value of the q axis current, and \(\varphi_{qi}\) represents the phase of the q axis current. According to the principle of amplitude conservation, formula (5) can be obtained through inverse matrix transformation:  
\[\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array} \right]=\left[ \begin{array}{cc}\cos(\theta)&-\sin(\theta)\\ \cos(\theta-\frac{2}{3}\pi)&-\sin(\theta-\frac{2}{3}\pi)\\ \cos(\theta+\frac{2}{3}\pi)&-\sin(\theta+\frac{2}{3}\pi)\end{array} \right] \left[ \begin{array}{c}i_{d}\\ i_{q}\end{array} \right], \tag{5}\]  
The current in the ABC three-phase coordinate can be converted to a dq coordinate by formula (5), as shown in formula (6) [27]:  
\[\left\{ \begin{array}{l}i_{a}=-i_{q}\sin(\theta)\\ i_{b}=-i_{q}\sin(\theta-120^{\circ})\\ i_{c}=-i_{q}\sin(\theta+120^{\circ})\end{array}