## 2 Model, Loads, and Boundary Conditions of Gear System in Drum Shearer
### Gear System Working Loads and Boundary Conditions  
The load condition of the ranging arm gear system is determined by the load state of the drum during cutting. The load state of the drum is complex, which mainly includes the resistance force and torque generated by the drum cutting the coal seam. In order to obtain the accurate load conditions of the gears, it is necessary to conduct a force analysis on the drum. When the drum cuts the coal seam, the factors that affect the force on each pick and the resultant force on the drum can be mainly divided into three parts. The first part is the design parameters of the drum, which mainly include the geometric parameters of the drum, the arrangement of the picks, the working width of the picks, the influence coefficient of the picks angle, etc. ; the second part is the coal seam parameters, which mainly includes the impedance of the coal seam and the brittleness coefficient of the coal rock, etc. ; the third part is the speed parameters, including the walking speed of the shearer and the rotation speed of the drum. When the drum cuts the coal seam, a single pick will be affected by cutting resistance, feeding resistance and the lateral force exerted on it by the coal seam, as shown in Figure 2. According to the theory of cutting and breaking coal [30], during the cutting process, the lateral force, cutting resistance, and feeding resistance on a single pick can be expressed as follows:  
\[X_{0}=Z_{0}\bigg{(}\frac{C_{1}}{C_{2}+h}+C_{3}\bigg{)}\frac{h}{t} \tag{1}\]  
\[Y_{0}=k_{q}\cdot Z_{0} \tag{2}\]  
\[Z_{0}=10A_{z}\cdot h\cdot t\cdot k_{y}\cdot k_{m}\cdot k_{a}\cdot k_{f}\cdot k _{p}\frac{0.3+0.35b_{p}}{(b_{p}+Bh^{0.5})k_{q}}\cdot\frac{1}{\cos\beta} \tag{3}\]  
The forces on the drum at a certain position are shown in Figure 3. The forces on all the picks can be converted to three directions: A, B, and C. Direction A is the traction resistance of the drum, Direction B is the lateral force of the drum, and Direction C is the vertical resistance of the drum. It can be concluded that the expression of loads on the drum can be defined as follows:  
\[F_{\text{A}}=-\sum\
olimits_{i=1}^{n_{j}}(Y_{i}\sin\varphi_{i}\cos\beta_{i}+Z_ {i}\cos\varphi_{i}) \tag{4}\]  
\[F_{\text{B}}=\sum\
olimits_{i=1}^{n_{j}}X_{i} \tag{5}\]  
\[F_{\text{C}}=\sum\
olimits_{i=1}^{n_{j}}(Z_{i}\sin\varphi_{i}-Y_{i}\cos\varphi _{i}\cos\beta_{i}) \tag{6}\]  
\[M_{\text{R}}=\frac{D_{\text{C}}}{2}\sum\
olimits_{i=1}^{n_{j}}Z_{i} \tag{7}\]  
In these equations, \(M_{\text{R}}\) denotes the cutting resistance torque of the drum, in Nm; \(\varphi_{i}\) is the position angle of the \(i\)th pick, in rad; \(X_{i}\) denotes the lateral force of the \(i\)th pick at position \(\varphi_{i}\), in newtons; \(Y_{i}\) denotes the feeding resistance of the \(i\)-th pick at the position \(\varphi_{i}\); \(Z_{i}\) denotes the cutting resistance of the \(i\)th pick at the \(\varphi_{i}\) position; \(D_{\text{C}}\) is the cutting diameter of the drum, in meters; \(n_{j}\) is the number of picks that instantaneously participate in cutting. Other various parameters and influence coefficients are not the focus of this article, and their detailed meanings can be found in Ref. [30]. Since the parameters of various coal seams and the cutting thickness are different, the load of the shearer changes randomly when working. Putting the parameters of this type of drum shearer when working in a typical coal seam into Equations (1)-(7), it can be calculated that \(F_{\text{A}}=-37.393\) kN, \(F_{\text{B}}=-13.532\) kN, \(F_{\text{C}}=42.28\) kN, \(M_{\text{R}}=81.85\) kNm. According to the gear parameters, a ranging arm model is established in the transmission system software Rmax R20, and the traction resistance, lateral force, vertical resistance and cutting resistance torque of the drum under different working conditions are all applied to the square head of drum. The boundary conditions of the ranging arm include constraints on two hinges.