## 2 Two-Dimensional Point Collision Dynamics Analysis
### Construction of a Vehicle-To-Road Coupling Model after Collision  
On the superelevation circular curve section, a spatial cartesian coordinate system with the driving direction along the center line of the road as the X-axis, the cross-sectional direction of the road as the Y-axis, and the normal direction of the road as the Z-axis, was established. The vehicle-road coupling dynamic analysis model of the circular curve section after collision is shown in Figure 2. The angle between the road surface and the horizontal plane is the lateral slope angle \(\theta\), the road superelevation is the lateral slope \(i_{h}=\tan\theta\), and the radius of the flat curve is \(R\). Assuming that the instantaneous velocity of the vehicle after the collision is \(V\), the driver takes braking measures, and the vehicle drives along the turning radius of the circular curve section at the collision point until it stops. There is no secondary collision during the period, and the phase after the collision is a uniform deceleration motion. The braking system of the vehicle is not damaged after the collision. The vehicle is in constant contact with the road surface, and there is a vertical force on the wheels. Figure 2: Vehicle mechanical analysis model after a collision with a superelevation circular curve section. Safety Threshold of the Minimum Radius of the Circular Curve under the Critical Condition of Vehicle Rollover after Collision  
The critical condition for vehicle rollover is that the wheels on the inside of the car are supported by zero-forcing on the road surface, and the vehicle is in danger of lateral overturning. The entire vehicle after the collision is regarded as a rigid body, regardless of the influence of the lateral slip angle of the wheel. Analysis of the force on the vehicle in the z-axis and y-axis directions:  
\[F_{z}=mgcos\theta+F_{g}sin\theta \tag{12}\]  
\[F_{y}=F_{g}cos\theta-mgsin\theta \tag{13}\]  
According to the centrifugal force formula when the vehicle is turning:  
\[F_{g}=\frac{mv^{2}}{R} \tag{14}\]  
where \(F_{z}\) represents the vertical force (N) of the inner wheel; \(F_{y}\) indicates the lateral force of the inner wheel (N); \(F_{g}\) represents the centrifugal force (N); \(m\) indicates the mass of the vehicle (kg); \(g\) represents acceleration of gravity based on 9.8 m/s\({}^{2}\); \(\theta\) represents the lateral slope angle (\({}^{\circ}\)); \(v\) represents the instantaneous velocity (m/s) of the vehicle in the direction of the tangential line of the road surface after the collision; and \(R\) represents the radius of the circular curve (m). In the general two-dimensional point collision analysis model, the plane rectangular coordinate system was established in the tangent direction and normal direction at the collision point, so the instantaneous speed of the vehicle along the tangent direction of the road surface centerline after the collision was as follows:  
\[vv=v_{10x}-\frac{1}{m_{1}t}(1+k)v_{r0x} \tag{15}\]  
Then, the centrifugal force of the vehicle after the collision was as follows:  
\[F_{g}=\frac{m\Big{(}v_{10x}-\frac{1}{m_{1}t}(1+k)v_{r0x}\Big{)}^{2}}{R} \tag{16}\]  
The type of road studied was a curve with superelevation, the research object was the vehicle after the collision, and the force distance to the outside wheel O of the vehicle was taken. \[\frac{b}{2}mgcos\theta+hmgsin\theta-hF_{g}cos\theta+\frac{b}{2}F_{g}sin\theta- F_{z}b=0 \tag{17}\]  
where \(v_{10x}\) is the projection (m/s) of the vehicle's centroid velocity in the normal direction of the collision point before the collision; \(v_{r0x}\) is the relative velocity of the two vehicles before the collision in the normal direction of the collision point; \(b\) is the wheelbase of the vehicle (m); and \(h\) is the centroid height of the vehicle (m). In a section of a circular curve with superelevation, the critical condition for the lateral overturning of the vehicle after the collision is zero vertical force of the inner wheel.