# Horizontal Bending Angle Optimization Method for Scraper Conveyor Based on Improved Bat Algorithm
## 2 Basic Theory  
### Analyze the Running Resistance of the Scraper Conveyor  
The running resistance of scraper conveyor is related to the amount of coal carried and the running direction of the shearer. Its running resistance is divided into the running resistance of the straight section and the additional resistance of the S-bending section.  
Straight line running resistance  
Previous calculation of the running resistance did not take into account the actual transport state of scraper conveyor [25; 26]. According to the actual working conditions, the middle slot of the scraper conveyor is divided into two parts innovatively, the heavy-duty section and no-load section. Combining with the cutting technology of the shearer, the calculation method of the operating resistance of the scraper conveyor is shown as follows:  
\[W_{(m,d)}=\left\{\begin{array}{ll}(q\mu_{ss}+q_{0}\mu_{cs})gL_{m}cos\theta \pm(q+q_{0})gL_{m}sin\theta+q_{0}g(\mu_{cs}cos\theta\mp sin\theta)(L-L_{m})&d =1\\ q_{0}g(\mu_{cs}cos\theta\mp sin\theta)L&d=2\end{array}\right. \tag{1}\]  
where \(q\) is the linear density of the heavy-duty part of the scraper conveyor, kg/m; \(q_{0}\) is the load weight per unit length of the scraper chain, kg/m; \(\mu_{ss}\) is the resistance coefficient of the load in the chute; \(\mu_{cs}\) is the scraper chain movement resistance coefficient in the chute; \(L\) is the total length of the scraper conveyor, m; and \(\theta\) is the vertical bending angle of the scraper conveyor, \(\circ\); The length of the heavy-duty section \(L_{1}=x_{0}+v_{s}t_{s}\), and the no-load length \(L_{2}=L-L_{1}\), m; \(v_{s}\) is the speed of the shearer, m/min; \(t_{s}\) is the shearer operating time, s; \(x_{0}\) is the distance from the starting point of the scraper conveyor to the head, m; \(m=1\) when the shearer and scraper conveyor are running in opposite directions. Otherwise, \(m=2\); \(d=1\) indicates the resistance of the upper chain of the middle slot, and \(d=2\) indicates the resistance of the lower chain of the middle slot.  
S-bending section additional resistance  
Due to the angle between the adjacent middle slot of the S-bending area, the squeegee chain can be regarded as a flexible body sliding around the fixed circular ring guide when sliding therein [27; 28; 29; 30]. The additional resistance (\(\Delta F_{m}\)) of the S-bending area includes the additional traction resistance of the chain and the squeezing force of the squeegee striking the middle trough. The expression of \(\Delta F_{m}\) is expressed as follows:  
\[\Delta F_{m}=\sum_{s}^{K}r\cdot F_{sm}+F_{d}\Big{(}e^{(K+1)\mu\alpha^{\prime}} -1\Big{)}+C_{d}\Bigg{(}\frac{e^{\mu\alpha^{\prime}}\Big{(}e^{K\mu\alpha^{ \prime}}-1\Big{)}}{e^{\mu\alpha^{\prime}}-1}-K\Bigg{)}\quad\ (d=1,2;\ m=1,2) \tag{2}\]  
where \(K\) is the total number of horizontal curved chutes; \(r\) is the lateral friction coefficient of the squeegee and the middle slot; \(F_{d}\) is the chain tension in the end of the chute before the horizontal bending section, N; \(\alpha^{\prime}\) is the angle between adjacent middle slots, \({}^{\circ}\); \(\mu\) is the friction coefficient value of the scraper chain and the chute; the resistance of each middle slot is \(C_{1}=l(q+q_{0})(\mu_{ss}cos\theta\pm sin\theta)\), \(C_{2}=q_{0}(\mu_{ss}cos\theta\mp sin\theta)\); \(l\) is the length of a middle slot, m; \(F_{sm}\) is the lateral pressing force [28].