## 2 Materials and Methods
### Key Component Parameter Design and Analysis
#### 2.3.2 Effect of the Spike Teeth Parameters on Material Impacting through the Screener
In this case, the BSF sand particles and the spike teeth began to separate, and the sand particles of the BSF adopted an oblique projectile motion. The spiked teeth were driven by the motor to rotate in a circular motion at a uniform speed in the screener. After a certain period, they caught up with the sand particles of the BSF that were briefly separated. Subsequently, a secondary collision occurred. The time \(t\), at this time, represents the second solution of \(\varphi_{(t)}=\varphi_{t}^{\prime}\) until the time when the BSF sand particles moved out of the collision area where they came into contact with the spike teeth, and the sieving was completed. Figure 9: Collision force analysis between the nail teeth and BSF sand. Note: \(G\) denotes the self-gravity of the BSFL, N; F\({}_{f}\) is the frictional force of the BSFL at the end of the spike teeth, N; F represents the effect of the spike teeth on the BSFL when the spike teeth rotated with a high-speed force, N; N is the support force of the spiked teeth on the BSFL, N; \(w\) is the angular velocity of the cylinder sieve, rad\(\cdot\)s\({}^{-1}\); and \(w_{1}\) is the angular velocity of the spike teeth, rad\(\cdot\)s\({}^{-1}\). The analysis of Figure 9 suggests that the force \(F\) on the BSFL when the spike teeth rotate at high speeds is expressed as follows:  
\[F=mw_{1}^{2}R_{1} \tag{16}\]  
The force of the spikes on the BSFL must be less than 2.94 N [43], which is the limit of the force that the BSFL are able to withstand, measured by the physical property analyzer. In accordance with \(w_{1}=2\pi n\), the rotational speed \(n\) of the spike teeth must fulfill:  
\[n\leq 41.55\;\mathrm{r/min}\]  
The analysis of the movement trajectory of the spike teeth and the BSF sand particles after the collision indicated that the spike teeth may undergo multiple collisions with the BSF sand particles. The rotational speed of the spike teeth affected the time of the collisions between the BSF sand particles and the spike teeth, thus further affecting the separation and sieving of the BSFL sand particles. 3.3 Effect of the Inclination Angle of the Trommel Screener on the Movement Trajectory of the Material  
Figure 10 presents the movement trajectory of the BSF sand material. Its movement was divided into the linear motion in the direction of the rotary axis of the trommel screener and the plane motion perpendicular to the rotary axis of the trommel screener. These two types of motions were affected by the installation angle and the rotation speed of the trommel, respectively. The angle between the rotary axis of the trommel and the horizontal plane was \(\sigma\) to ensure that the BSF sand material in the trommel had a certain fluidity in the axis direction and increase the probability of screen penetration. With a single particle \(P\) of the BSF sand placed at the origin O, a coordinate system was established (Figure 10). When the unit particle \(P\) of the BSF sand material entered the trommel from D\({}_{1}\), the particle immediately adopted a uniform circular motion, i.e., the trajectory line of the D\({}_{1}\)O. The material was lifted to the O point, following the screener surface. The position of the O point was affected by certain parameters (e.g., the static friction coefficient of the BSF sand material, the rotation speed of the trommel, the inclination angle of the trommel, and the inner diameter of the trommel). Next, the BSF sand particles began to leave the screener surface to adopt a parabolic motion, i.e., the trajectory line of the OD\({}_{2}\), and reached the highest point E and then fell back to the point D\({}_{2}\). In such a cycle, the material was discharged from the trommel screener surface under a certain inclination. The installation of the trommel screener was designed with a certain inclination angle. When the BSF sand particles in the trommel screener moved continuously, the motion  
Figure 10: Kinematic analytical sketch of the BSF sand in the rotary screen.