## 2 Actual Non-Uniform Bulk Material Distribution Measurement Methodology
### Non-Uniform Bulk Material Distribution Model  
The non-uniform bulk material distribution affects the dynamic performance of the system. Belt tear and breaking accidents can occur when the belt encounters unsteady working conditions, such as starting and braking, since the dynamic characteristics of the belt under a non-uniform bulk material distribution are more complex. Moreover, the type of material, the conveying process requirements, the equipment layout and other factors often cause non-uniform bulk material distribution in actual operation, as shown in Figure 3, it makes the dynamic characteristics of the belt even more complex.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Description** & **Value** & **Parameter Description** & **Value** \\ \hline Facility longitudinal length, mm & 3500 & Belt length, mm & 7000 \\ Facility horizontal width, mm & 760 & Belt width, mm & 500 \\ Facility height, mm & 400 & Belt thickness, mm & 11 \\ Facility height range, mm & 400–1338 (adjustable) & Drive motor & Shanghai shenli yvt2-90l-4 \\ Idler roll set spacing, mm & 700 & Reduction ratio & 1:29 \\ Inclination angle, \({}^{\circ}\) & \(\delta\leq 8^{\circ}\) & Large and small gear ratio & 24:16 \\ Conveyor Belt & Rubber canvas ordinary & Frequency converter & ABB-ACS550-01-05A4-4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the experimental facility.  
Figure 3: The non-uniform bulk material distribution in actual operation.  
Figure 2: Experimental facility for measuring bulk material distribution. (**a**) Sand cross-sections measurement in real-time; (**b**) the fitting of iron powder cross-section in real-time.  
To describe the non-uniform bulk material distribution on a conveyor belt, we discretize the bulk material on the conveyor belt into a series of micro-units according to the travel distance \(l\) in unit time \(T\). Assumed \(T\) contains \(T\) seconds, \(t\) is the \(t\)-th unit of time between time (\(t\)-1)\(T\) and time \(tT\), \(k\) is the number of material flow cross-section areas per unit time. Accordingly, the non-uniform bulk material distribution \(\mathbf{q_{G}(t)}=\{q(t_{1}),\,q(t_{2}),\,\ldots\,\,\ldots\,\,q(t_{k})\}\), which denotes the collection of \(k\) sampling bulk material loads in unit time \(T\) between time (\(t\)-1)\(T\) and time \(tT\), can be represented as Figure 4. Due to the scanning frequency which the laser scanner can capture frames of cross section per second is \(f_{sped}\), \(k\) is obtained by \(k=Tf_{sped,}\). Then \(t_{k}\) means the \(k\) frame of \(t\)-th unit of time. Compared with the \(tT\) (\(t\) is from 1-\(t\)), the \(t_{k}\) (\(t\) is from 1-\(t\), \(k\) is from 1-\(k\)) can much better reflect the original time series. Based on the model of cross-sectional area of the bulk material proposed in [49], the bulk material distribution model can be presented.  
In Figure 4, the belt moves in the negative direction of the \(x\)-axis. \(\mathbf{s(t)}=\{s(t_{1}),\,s(t_{2})\,\ldots\,\,s(t_{k})\}\) is the collection of cross-sectional areas of material flow calculated in unit time \(T\) between time (\(t\)-1)\(T\) and time \(tT\). \(\mathbf{v(t)}=\{\,v(t_{1}),\,v(t_{2})\,\ldots\,\,v(t_{k})\}\) is the collection of belt speed in unit time \(T\) between time (\(t\)-1)\(T\) and time \(tT\). Therefore, the bulk material distribution \(\mathbf{q_{G}(t)}\) (kg/s) in unit time \(T\) between time (\(t\)-1)\(T\) and time \(tT\) could be represented as  
\[\mathbf{q_{G}(t)}=\left\{\frac{\rho}{f_{sped}-1}s(t_{1})v(t_{1})\,,\,\,\frac{\rho }{f_{sped}-1}s(t_{2})v(t_{2})\,,\ldots\cdot\,,\,\,\frac{\rho}{f_{sped}-1}s(t_{k })v(t_{k})\right\} \tag{1}\]  
where \(\rho\) is the bulk density, \(\{v(t_{1}),\,v(t_{2})\,\ldots\,\,v(t_{k})\}\)is the successive instantaneous speed of the belt in unit time \(T\) measured by the speed monitor, (m/s), \(\{s(t_{1}),\,s(t_{2})\,\ldots\,\,s(t_{k})\}\) is the successive cross-sectional area of material flow in the \(i\) frame in unit time \(T\) (m\({}^{2}\)).