# Design and Optimization of a Toothed-Plate Single-Roller Crushing Device for Waste Plastic Film
## 3 Design and Theoretical Analysis of the Crushing Component
### The Analysis of the Movement and Force Pattern
Combining Equations (3) and (4), we can derive the theoretical crushing thickness, \(l_{m}\), as follows:  
\[l_{m}=\frac{2\pi}{Z\omega}\sqrt{V_{\text{0}}^{2}+4\mu_{1}V_{\text{0}}{}^{2} \cos^{2}\beta(\tan\gamma+\tan\beta)+2g\mu_{\text{D}}-2\mu_{1}\text{g}L_{w} \cos\gamma} \tag{5}\]  
where \(l_{m}\) is the theoretical crushing thickness, mm; \(Z\) is the number of crushing roller toothed plates; \(\omega\) is the crushing roller rotation speed, r/min; \(\beta\) is the angle between the feeding velocity and the x-axis, \({}^{\circ}\); \(V_{\text{0}}\) is the feeding velocity, m/s; \(\gamma\) is the angle between the hopper side plate and the x-axis, \({}^{\circ}\); \(L_{w}\) is the total length of the feeding hopper side  
Figure 6: Motion analysis of waste plastic film feeding process. plate, m; \(h_{\rm D}\) is the vertical distance between point D and point A, m; and \(\mu_{1}\) is the friction coefficient between the waste plastic film and the feeding hopper side plate. The value of the crushing thickness directly affects the size of the crushed film. From Equation (5), it can be observed that \(V_{0}\), \(Z\), and \(\omega\) can alter the crushing thickness, thereby influencing the size of the crushed film. Therefore, it is necessary to find the optimal feeding velocity, number of toothed plates, and crushing roller rotation speed to achieve the goal of obtaining the optimal size for subsequent cleaning and separation. The mechanical analysis of the crushing process is conducted as illustrated in Figure 7. The crushing device primarily relies on the cooperation between the moving toothed plates and the fixed toothed plates to achieve the compressing, cutting, and tearing of waste plastic film. After the initial crushing in the left-side area of Figure 6, the waste plastic film is taken by the moving toothed plates and undergoes a second round of crushing in the right-side crushing area. The value of the force, \(F_{q1}\), has a great impact on the impurity content of the crushed film. When \(F_{q1}\) is large, for the same amount of time, the impulse of the toothed plates on the waste plastic film is greater, which means a larger increment in the overall momentum, leading to a larger velocity increment for both soil impurities and film. Furthermore, according to the principle of the conservation of momentum, the ratio of the velocity increment for soil impurities and film is constant. A larger velocity increment for soil impurities and film will result in a greater difference in their velocities, thereby achieving separation between soil impurities and film. Since the force principles are the same on both sides, only the mechanical equation for the first crushing process is listed, as shown in Equation (6). \[\left\{ \begin{array}{l}F_{\rm n1}\sin\alpha_{1}+f_{\rm a1}\cos\alpha_{1}=f_{ \rm b1}\cos(\eta_{1}+\omega t)+F_{\rm q1}\sin(\eta_{1}+\omega t)+m_{\rm 1}g\\ F_{\rm n1}\cos\alpha_{1}+f_{\rm b1}\sin(\eta_{1}+\omega t)=f_{\rm a1}\sin\alpha _{1}+F_{\rm q1}\cos(\eta_{1}+\omega t)\\ f_{\rm a1}=\mu_{2}F_{\rm n1}\\ f_{\rm b1}=\mu_{3}F_{\rm q1}\end{array} \right. \tag{6}\]  
where \(F_{\rm n1}\) is the support force of the fixed toothed plate on the materials, N; \(\alpha_{1}\) is the angle between the fixed toothed plate and the side plate, \({}^{\circ}\); \(f_{\rm a1}\) is the frictional force between the materials and the fixed toothed plate, N; \(f_{\rm b1}\) is the frictional force between the materials and the moving toothed plate, N; \(F_{\rm q1}\) is the crushing force of the moving toothed plate on the materials, N; \(\eta_{1}\) is the angle between the moving toothed plate and the side plate, \({}^{\circ}\); \(m_{\rm 1}\) is the mass of materials crushed by a single toothed plate, kg; \(\mu_{2}\) is the friction coefficient between the fixed toothed plate and the materials; and \(\mu_{3}\) is the friction coefficient between the moving toothed plate and the materials. Figure 7: Mechanical analysis of waste plastic film crushing.