## 4 Results and Discussion
### Flight Characteristics of Falling Debris
#### 4.1.2 Falling Debris Rotation  
During the movement of falling debris, changes in the rotation angle can alter the windward surface, thereby influencing its flight trajectory in the train slipstream. To address this, Figure 11 compares the rotation angles of falling debris under single train and trains crossing conditions at speeds of 250-350 km/h.  
Figure 9: Maximum displacement under single-train conditions.  
As depicted in Figure 11, under varying operating conditions, the rotation of falling debris predominantly occurs around the \(X\) and \(Z\) axes, with minimal rotation around the \(Y\) axis. At a train speed of 300 km/h, the maximum rotation angles of falling debris around the \(X\) and \(Z\) axes reach 783\({}^{\circ}\) and 535\({}^{\circ}\), respectively, under single-train conditions. Conversely, under trains-crossing conditions, the maximum rotation angles around the \(X\) and \(Z\) axes are 685\({}^{\circ}\) and 254\({}^{\circ}\), respectively. However, the maximum rotation angle around the \(Y\) axis under these conditions is only about 80\({}^{\circ}\).  
Figure 11: Rotation angle of falling debris: (**a1–a3**) single train; (**b1–b3**) trains crossing.  
Figure 10: Flight velocity of falling debris: (**a1,a2**) single train; (**b1,b2**) trains crossing.  
The rotation angle of falling debris can be roughly divided into two stages. The first stage exhibits small rotation angles with strong regularity. In the second stage, rotation angles increase with worsening regularity. For instance, under trains-crossing conditions, falling debris rotates positively around the \(X\) axis in the first stage, with the rotation angle increasing over time. Subsequently, falling debris primarily rotates around the negative direction of the \(Z\) axis, with rotation angles at different speeds not exceeding 200\({}^{\circ}\). However, in the second stage, the rotation angle around the \(X\) axis ceases to increase, leading to irregular rotation. It is noteworthy that under the influence of the train slipstream, the correlation between the rotation angle of falling debris and the train speed is weak.  
Figure 12 further compares the rotational angular velocity of falling debris under the conditions of a single train and trains crossing at speeds ranging from 250 to 350 km/h. As depicted in Figure 12, the rotational angular velocity of falling debris exhibits two distinct stages: a relatively stable first stage followed by a period of violent fluctuations. Notably, under the scenario of trains crossing, the rotational angular velocity of falling debris experiences more pronounced variations. During the initial stage, \(\omega_{x}\) and \(\omega_{z}\) remain relatively stable near zero, while \(\omega_{y}\) exhibits a slight increase over time. Subsequently, in the second stage, the rotational angular velocity undergoes sharp oscillations, alternating between positive and negative extremes. Specifically, \(\omega_{y}\) and \(\omega_{z}\) sharply ascend to positive peaks before sharply declining to negative peaks, while \(\omega_{x}\) initially decreases before subsequently increasing.