## 3 Results  
### Theoretical Digging Resistance  
#### 3.1.1 Active Pressure \(F_{B}\)  
From the analysis of the force in the process of excavating material using the grab, it can be understood that the force can be further simplified on the basis of the characteristics of forces W, R and \(F_{B}\). In Figure 11, the roles of unit forces for \(dW\), \(q\) and \(dF_{B}\) are presented, where \(q-dq\) and \(q\) represent the contact forces between the material layers. The weight \(dW\) of the material is determined by the area of the material unit \(H_{1}\,I_{1}\,H_{2}\,I_{2}\) body. \[dh=\frac{\cos(\beta+\mu)}{\cos\mu}\,dz \tag{13}\]  
\[\mu(z)=\frac{\pi}{2}-\tan(\frac{d}{d\eta}\,H(\eta)|_{\eta=z}) \tag{14}\]  
where \(dh\) is the height of the trapezoid of \(H_{1}\,I_{1}\,H_{2}\,I_{2}\) (m), \(\mu\) is the angle between the tangent and the vertical direction of the surface point (rad), \(\beta\) is the angle between the failure plane and the horizontal plane (rad), \(\eta\) is the grab digging depth (m), and \(H(\eta)\) is a function of the grab geometry parameters (m). The following section analyzes the calculation of the pressure of the grab for any shape of material, as shown in Figure 12.  
The trapezoidal area of the material unit is calculated according to the following formulas:  
\[\frac{H_{1}\,I_{1}}{\sin(0.5\pi-(\mu+\alpha))}=\frac{OH_{1}}{\sin(\beta+\alpha )},OH_{1}=\frac{z}{\cos\mu}\,H_{1}\,I_{1}=\frac{\cos(\alpha+\mu)}{\sin(\beta+ \alpha)\cos\mu}\,z \tag{15}\]  
Figure 11: Simplified force characteristics of active pressure. Figure 12: Mathematical model of material unit body. \[H_{\rm{2}}\,l_{\rm{2}}=\frac{\cos(\alpha+\mu)\cos(\beta+\mu)}{\sin(\beta+\alpha) \cos\mu}\,{\rm{2}}-\frac{\sin(\beta+\mu)}{\cos\mu}\,{\rm{2}}{\rm{-}}\frac{\cos (\beta+\mu)}{\cos\mu\tan(\beta+\alpha)}\,{\rm{d}}{\rm{z}} \tag{16}\]  
\[S_{H_{\rm{1}}\,h_{\rm{2}}\,l_{\rm{2}}}=\frac{1}{2}(H_{\rm{1}}\,l_{\rm{1}}+H_{ \rm{2}}\,l_{\rm{2}})\,{\rm{d}}{\rm{h}}=\frac{\cos(\alpha+\mu)\cos(\beta+\mu)}{ \sin(\beta+\alpha)\cos^{2}\mu}\,{\rm{z}}{\rm{d}}{\rm{z}} \tag{17}\]  
The weight of the material unit body is:  
\[dW=\rho bS_{H_{\rm{1}}\,l_{\rm{1}}\,h_{\rm{2}}\,l_{\rm{2}}}=\frac{\cos(\alpha+ \mu)\cos(\beta+\mu)}{\sin(\beta+\alpha)\cos^{2}\mu}\,{\rm{z}}{\rm{p}}{\rm{b}}{ \rm{d}}{\rm{z}} \tag{18}\]  
where \(dW\) is the material weight (N). By analyzing the force balance of the trapezoidal material unit body, in the X and Y directions, the balance formula is obtained:  
\[\left\{ \begin{array}{l}\sum F_{x}=dF_{\rm{2}}\cos(\gamma-\mu)+((q-dq)\,H_{ \rm{2}}\,l_{\rm{2}}-qH_{\rm{1}}\,l_{\rm{1}})\sin(\beta-\varepsilon)=0\\ \sum F_{y}=dF_{\rm{2}}\sin(\gamma-\mu)-((q-dq)\,H_{\rm{2}}\,l_{\rm{2}}-qH_{\rm {1}}\,l_{\rm{1}})\cos(\beta-\varepsilon)-dW=0\end{array} \right. \tag{19}\]  
where \(\gamma\) is the friction angle between grab and material (rad). Solving the above equation, the formula of active pressure \(dF_{\rm{2}}\) is obtained as follows:  
\[dF_{\rm{2}}=\frac{dW}{\sin(\gamma-\mu)-\cos(\gamma-\mu)\cot(\beta-\varepsilon)} \tag{20}\]  
Further analysis of the force can be carried out as follows:  
\[dF_{\rm{2}}=\frac{\cos(\alpha+\mu)\cos(\beta+\mu)}{(\sin(\gamma-\mu)-\cos( \gamma-\mu)\cot(\beta-\varepsilon))\sin(\beta+\alpha)\cos^{2}\mu}\,{\rm{z}}{ \rm{p}}{\rm{d}}{\rm{z}} \tag{21}\]  
By dividing the left and right ends of the above equation by \({\rm{d}}{\rm{z}}\), we obtain:  
\[P_{\rm{2}}:=\frac{dF_{\rm{2}}}{{\rm{d}}{\rm{z}}}=\,f_{\rm{2}}(\beta)z{\rm{p}} \tag{22}\]  
where \(f_{\rm{2}}(\beta)\). is satisfied thus:  
\[f_{\rm{2}}(\beta)=\frac{\cos(\alpha+\mu)\cos(\beta+\mu)}{(\sin(\gamma-\mu)- \cos(\gamma-\mu)\cot(\beta-\varepsilon))\sin(\beta+\alpha)\cos^{2}\mu} \tag{23}\]  
It is now necessary to find the extreme value of \(f_{\rm{2}}(\beta)\). to determine the sliding surface, and so Formula (23) can be further transformed into:  
\[