# Dynamic Coal Flow-Based Energy Consumption Optimization of Scraper Conveyor
## 3 Mechanics and Energy Consumption Modeling of the Scraper
### Force Analysis and Mechanical Model of the Scraper Conveyor
#### 3.1.1 Resistance in Straight Sections  
In the straight sections, the primary resistance comes from the frictional forces acting on both the load-bearing and non-load-bearing sides, generated by the interaction between \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **K-S Test (\(p\)-Value)** & **Spearman (R)** & **MAD** & **Q25\%** & **Q50\%** & **Q75\%** \\ \hline Total Coal Quantity & 0.2176 & 0.9899 & 48.7993 & 27.229 & 42.601 & 45.229 \\ Spatial Distribution & 0.8253 & 0.9620 & 0.2652 & 0.007 & 0.192 & 0.321 \\ Temporal Variation & 0.4503 & 0.9662 & 2.5104 & 0.326 & 3.161 & 0.445 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical comparison between model and simulation results. Figure 14: Load classification of the scraper conveyor. the scraper chain and the base plate. Regardless of the load condition, frictional resistance opposes the chain's movement. (1) Resistance on the load-bearing and non-load-bearing chains  
The scraper chain experiences resistance mainly due to friction with the base plate, which differs between the load-bearing and non-load-bearing sides [36]. As shown in Figure 15, during operation in straight sections, the scraper chain must overcome two key forces: the gravitational component along the inclined direction and frictional resistance. Both forces act against the chain's movement, except when the chain moves downward, where gravity assists motion while frictional resistance remains opposing. The total resistance in straight sections can be expressed as follows:  
\[T_{12}=q_{1}Lg(f\cos a\pm\sin a) \tag{37}\]  
where \(q_{1}\) represents the mass per unit length of the scraper chain (kg/m), \(L\) is the length of the straight section (m), \(\alpha\) is the conveyor angle (positive when moving upward, negative otherwise), and \(f\) is the friction coefficient between the scraper chain and the chute bottom. (2) Resistance from bulk material on the load-carrying side  
On the load-carrying side, material exerts both vertical and lateral pressure on the chute. The lateral pressure increases resistance by enhancing friction with the sidewalls. The lateral pressure \(N\) is given by:  
\[N=rh^{2}\lambda g \tag{38}\]  
The scraper conveyor must overcome gravitational and frictional resistance from normal and lateral pressures, as expressed in:  
\[T_{3}=L[gq_{2}(f_{1}cosa+sina)+f_{1}rh^{2}\lambda](\alpha<25^{\circ}) \tag{39}\]  
where \(r\) is the weight per unit volume of the material, \(h\) is the height of the coal-retaining plate, \(\lambda\) is the material's pressure coefficient on the sidewalls, and \(f_{1}\) is the friction coefficient between the material and the chute. In the empty section, the scraper chain moves along the bottom plate, with the main resistance coming from the friction between the chain and the plate. Therefore, the resistance in the empty section can be expressed as the total frictional resistance of the chain. \[T_{1}=q_{1}Lg(f\cos a-\sin a) \tag{40}\]  
In the loaded section, the scraper chain must overcome both its own frictional resistance and the additional resistance from moving the material. Thus, the total resistance is the sum of the chain's friction and the material's resistance:  
\[T_{4}=q_{1}Lg(fcosa+sina)+L[gq_{2}(f_{1}cosa+sina)+f_{1}rh^{2}\lambda](\alpha< 25^{\circ}) \tag{41}\]  
Figure 15: Scraper conveyor layout diagram. (3) Bending Resistance  
(1) Operating Condition Bending Resistance  
1 Horizontal bending resistance  
The scraper conveyor chute can bend both vertically and horizontally in its structure. However, in practical use, except for cases where the machine body moves, measures are typically taken to avoid horizontal curves. Figure 16 shows a schematic of the horizontal bending in the moving section of the machine body. In a flexible scraper conveyor, the scraper chain follows a zigzag path in the bending section. When the friction coefficient between the chain and bottom plate is much higher than the equivalent friction, the resistance follows [PERSON]'s formula.