# Research on Mathematical Modeling of Critical Impact Force and Rollover Velocity of Coach Tripped Rollover Based on Numerical Analysis Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling of the Critical Rollover Impact Force

- (1)
- The vehicle as a whole is regarded as a rigid body;
- (2)
- The center of mass of the vehicle is in the vertical plane of the center of the axle;
- (3)
- The vertical reaction point on the inner and outer wheels is at the midpoint of the vertical twin tires;
- (4)
- The differences in the parameters of each axis are ignored.

#### 2.1. Mathematical Modeling of Critical Rollover Impact Force between a Vehicle and a Road Obstacle

^{2}), N represents the ground supporting force (N), and ${F}_{Tf}$ represents the impact force (N).

_{z}

_{1}, V

_{y}

_{1}, and z

_{1}into Formula (12), we obtain

- (1)
- Relationship between the impact force ${k}_{G}$ and collision duration $\mathsf{\Delta}t$ under different values of V
_{y0}.

_{y0}, the value of ${k}_{G}$ in the same period of time also increases quite rapidly. This also means that, when the lateral speed is high, the vehicle will collide with the obstacle in a short time with a force several times its own weight, resulting in vehicle damage and possibly even death.

- (2)
- Relationship between the impact force ${k}_{G}$ and collision duration $\mathsf{\Delta}t$ under different H
_{g}(V_{y0}= 15 km/h).

- (1)
- When the collision duration is short ($\mathsf{\Delta}t$ < 0.01 s), the critical rollover impact force is very large and F
_{Tf}rapidly increases to its maximum value; - (2)
- The vehicle rollover is not only related to the impact force but also to the duration of the collision effect; even if the impact force is relatively small, if the collision effect lasts long enough, a second collision may occur or cause rollover;
- (3)
- Theoretically, as long as the impact force–action time relationship is above the curve, it can lead to rollover.

#### 2.2. Mathematical Modeling of Critical Rollover Impact force of Two Vehicles

_{T}can be broken down into components in two directions: F

_{Ty}parallel to the ground and F

_{Tz}perpendicular to the ground. The ground friction force is ${F}_{Tf}$. Then, the equation of motion of the collision process is as follows:

_{1}, V

_{z}

_{1}, V

_{y}

_{1}, and z

_{1}values of the vehicle are, respectively,

_{Gy}(k

_{Gz}= 0), k

_{Gz}(k

_{Gy}= 0), and Δt are considered separately.

- (1)
- The relationship between k
_{Gz}(k_{Gy}= 0) and the collision duration Δt.

_{y}

_{0}differ, they have little impact on k

_{Gz}, which is consistent with reality: when the value of the lateral velocity v

_{y}

_{0}changes, it does not strongly affect the impact force in the Z-direction.

- (2)
- The relationship between k
_{Gy}(k_{Gz}= 0) and the collision duration Δt.

_{y}

_{0}differed, this difference had little effect on k

_{Gy}. This is also consistent with reality: although the value of the lateral velocity v

_{y}

_{0}changes, it does not strongly affect the impact force in the Y-direction. A change in the lateral velocity v

_{y}

_{0}will only affect the collision duration.

- (3)
- The relationships between K
_{Gy}and K_{Gz}and the collision duration Δt under different Hp (v_{y}_{0}= 15).

_{Gy}has a strong influence on Hp and changes quite clearly with an increase in Hp, while K

_{Gz}barely changes. The K

_{G}–Δt relationship in the critical rollover force model of two vehicles colliding is similar to that for a vehicle colliding with a road obstacle; however, the k

_{G}of the vehicle and road obstacle collision model is only related to the vehicle mass while, here, k

_{Gyz}is not only related to the vehicle mass but also to the wheelbase of the vehicle and the height of the collision point. It is obtained by combining relevant factors such as the impact force, collision point, and vehicle structure. As the rollover collision model for two vehicles is similar to that for vehicles and road obstacles, if only the magnitude of the impact force is considered and the influence of its direction and the position of the collision point is not considered, the two types of collision can be unified to simplify the analysis.

## 3. Mathematical Modeling of the Critical Rollover Velocity

#### 3.1. Mathematical Modeling of Critical Rollover Speed of a 90° Rollover

_{y}

_{0}at the beginning of the rollover collision, and the vehicle rotates around point A after the collision. The initial angular velocity is ${\omega}_{1}$. From Figure 21, using the law of conservation of momentum,

_{xx}is difficult to determine; however, it can be approximated using the following formula:

_{1}is given, then the CVS can be directly calculated. If the vehicle is regarded as a rectangle with an evenly distributed mass, we have

#### 3.2. Mathematical Modeling of Critical Rollover Speed of a 180° Rollover

_{2}and the initial angular velocity of rotation is ${\omega}_{2}$, as shown in Figure 22, the following is obtained using the law of conservation of momentum:

_{y}, and V

_{z}can be obtained using Equations (36), (37), and (40), respectively. The velocity at this moment is taken as the initial condition, and the velocity components at the moment ($t+\mathsf{\Delta}t$) before the vehicle falls back to and collides with the ground are calculated using ${\omega}_{p1}$, V

_{py}

_{1}, V

_{pz}

_{1}, and the angle of rotation ${\theta}_{p}$. Assuming that the velocity components of the vehicle after a collision with the ground are ${\omega}_{p2}$, V

_{py}

_{2}, and V

_{pz}

_{2}, the following equations are obtained from the conservation of momentum.

_{2}for a 180° rollover can be obtained.

_{2}is large enough at the beginning, there will be movement immediately after the collision, when ${a}_{z}>-g$, and even during the “top-up” motion off the ground, and the rolling angle will be more than 180°. When the vehicle falls back and collides with the ground, the situation ${a}_{z}>-g$ will occur again. This situation can be analyzed according to the previous process until the kinetic energy after a collision with the ground is insufficient to lift the vehicle to a critical point. In fact, due to the loss of energy during the collision, it is impossible to have a phenomenon such as multiple bounces.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (V

_{y0}= 10 km/h).

**Figure 3.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (V

_{y0}= 20 km/h).

**Figure 4.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (V

_{y0}= 30 km/h).

**Figure 5.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (V

_{y0}= 40 km/h).

**Figure 6.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (Hg = 0.8664 m).

**Figure 7.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (Hg = 1.083 m).

**Figure 8.**Relationship between the impact force and collision duration (${k}_{G}$ vs. $\mathsf{\Delta}t$) (Hg = 1.2996 m).

**Figure 10.**Relationship between the impact force and collision duration (k

_{Gz}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 10 km/h).

**Figure 11.**Relationship between the impact force and collision duration (k

_{Gz}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 20 km/h).

**Figure 12.**Relationship between the impact force and collision duration (k

_{Gz}vs. v) (v

_{y}

_{0}= 30 km/h).

**Figure 13.**Relationship between the impact force and collision duration (k

_{Gz}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 40 km/h).

**Figure 14.**Relationship between the impact force and collision duration (k

_{Gy}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 10 km/h).

**Figure 15.**Relationship between the impact force and collision duration (k

_{Gy}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 20 km/h).

**Figure 16.**Relationship between the impact force and collision duration (k

_{Gy}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 30 km/h).

**Figure 17.**Relationship between the impact force and collision duration (k

_{Gy}vs. $\mathsf{\Delta}t$) (v

_{y}

_{0}= 40 km/h).

**Figure 21.**Critical rollover speed model for a 90° rollover [43].

m/kg | W_{T}/m | H_{g}/m | I_{xx}/kg·m^{2} | g/m·s^{−2} |
---|---|---|---|---|

10,138 | 1.9951 | 1.083 | 11,529 | 9.787 |

Speed Type | CSV1 (km/h) | CSV2 (km/h) | V2 (km/h) |
---|---|---|---|

Calculated value | 70.0311 | 73.8491 | 100.368 |

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**MDPI and ACS Style**

Wu, X.; Wang, Z.; Chen, S.
Research on Mathematical Modeling of Critical Impact Force and Rollover Velocity of Coach Tripped Rollover Based on Numerical Analysis Method. *Symmetry* **2024**, *16*, 543.
https://doi.org/10.3390/sym16050543

**AMA Style**

Wu X, Wang Z, Chen S.
Research on Mathematical Modeling of Critical Impact Force and Rollover Velocity of Coach Tripped Rollover Based on Numerical Analysis Method. *Symmetry*. 2024; 16(5):543.
https://doi.org/10.3390/sym16050543

**Chicago/Turabian Style**

Wu, Xinye, Zhiwei Wang, and Shenghui Chen.
2024. "Research on Mathematical Modeling of Critical Impact Force and Rollover Velocity of Coach Tripped Rollover Based on Numerical Analysis Method" *Symmetry* 16, no. 5: 543.
https://doi.org/10.3390/sym16050543