Research on Wading Mobility of a Certain Type of Special Vehicle Based on MPS Method

Abstract

In order to study the wading mobility of a certain type of special vehicle, a high-precision dynamic model of the special vehicle is built. The vehicle dynamic model is verified by in-situ acceleration test and traction test. Through the dam-break experiment and wet road traction test, the feasibility and reliability of calculating the water impact pressure, buoyancy and solving the fluid–solid coupling problem based on the moving particle semi-implicit algorithm (MPS) are verified. Then, based on the MPS algorithm, a multi-body dynamics-fluid analysis simulation model is built to simulate and analyze the vehicle mobility under the static water wading condition and the water flow lateral impact condition. The relationship between the maximum speed of the vehicle at different water depths and gears under the static water wading condition the and the relationship between the lateral offset of the vehicle and the flow velocity and the deflection angle of the vehicle entering the water in the lateral impact condition are obtained.

1 Introduction

Military vehicles will encounter various extremely complex road conditions on the battlefield, such as rapid rivers or bridge decks submerged by rivers. Due to the buoyancy of the water and the lateral impact of the water flow, whether the vehicle can pass safely and quickly is of great significance to the safety and combat efficiency of the occupants. Vehicle mobility can measure the ability of the vehicle to maintain rapid passage under various road and terrain conditions (such as water-covered roads) that the vehicle may encounter [1, 2]. Studying the vehicle’s wading mobility can effectively evaluate and optimize the actual mobility performance of military vehicles under wading conditions.

The research methods of vehicle wading maneuverability mainly include test analysis and simulation. The vehicle wading test involves the safety of vehicle occupants and the regulation of water flow velocity, which is difficult to implement and costly. The simulation method can reduce the cost while ensuring the calculation accuracy. Tison et al. [3] established a six-degree-of-freedom model of an amphibious vehicle. Based on the finite element method, the water-exit process of the vehicle was simulated and analyzed. At this time, the finite element method needs to perform mesh reconstruction to ensure the mesh quality [4], and it is difficult to simulate the fluctuation and splash of the water area, so the particle algorithm is proposed and gradually widely used.

Particle method, also known as meshless method, is mainly divided into smooth particle algorithm (SPH) and moving particle semi-implicit algorithm (MPS). Yreux [5] simulated the dam-break experiment and vehicle wading conditions through the SPH algorithm in LS-DYNA. The MPS algorithm was first proposed by Koshizuka et al. [6]. The implicit method was used to solve the pressure equation to ensure the incompressibility of the fluid. At the same time, the semi-implicit solution method was introduced to shorten the calculation time. It has a wide range of applications in simulating water flow. Chixin et al. [7] used the MPS algorithm to visually analyze the splash lubrication characteristics of the gearbox. Jing and Jinbiao [8] improved the MPS algorithm and verified the feasibility of the MPS algorithm in fluid–solid coupling simulation.

Based on the MPS algorithm, this paper tests, numerically simulates and analyzes the mobility of a special vehicle under wading conditions through the joint simulation of multi-body dynamics software and fluid analysis software. The maximum speed under different water depths and the body offset under the lateral impact of water flow are obtained, and the influence of water depth and water flow velocity on the mobility of the vehicle is analyzed.

2 Moving Particle Semi-Implicit Algorithm

The moving particle semi-implicit algorithm (MPS) characterizes the interaction between particles by kernel function, and uses gradient model and Laplace model to simulate the control equation.

2.1 Control Equation

The MPS algorithm mainly calculates and simulates incompressible fluid, and its control equations are continuous equations and momentum conservation equations:

$$\frac{d\rho }{dt}=0$$

(1)

$$\rho \frac{d{\varvec{u}}}{dt}=-\nabla p+\mu {\nabla }^{2}{\varvec{u}}+\rho {\varvec{g}}$$

(2)

where ${\varvec{u}}$ is the velocity of the fluid, $\rho$ is the density of the fluid, $\mu$ is the dynamic viscosity of the fluid, $p$ is the pressure, ${\varvec{g}}$ is the acceleration of gravity, and $t$ is the time.

2.2 Kernel Function

The kernel functions commonly used in the MPS algorithm are:

$$w\left({r}_{p}\right)=\left\{\begin{array}{c}\frac{{r}_{e}}{{r}_{p}}-\mathrm{1,0}<{r}_{p}<{r}_{e}\\ 0,{r}_{p}>{r}_{e}\end{array}\right.$$

(3)

$${r}_{e}=R{D}_{i}$$

(4)

where ${r}_{p}$ is the distance between particles, ${r}_{e}$ is the effective radius of particles, $R$ is the effective radius coefficient, and ${D}_{i}$ is the radius of particles.

2.3 Gradient Model and Laplace Model

The gradient model and the Laplace model in the MPS algorithm can discretize the control equations. The gradient model and the Laplace model are as follows:

$${\left[\nabla \mathrm{\varphi }\right]}_{i}=\frac{d}{{n}_{0}}\sum_{j\ne i}\left[\frac{{\varphi }_{j}-{\varphi }_{i}}{{\left|{r}_{j}-{r}_{i}\right|}^{2}}\left({r}_{j}-{r}_{i}\right)\omega \left(\left|{r}_{j}-{r}_{i}\right|\right)\right]$$

(5)

$${\left[{\nabla }^{2}\mathrm{\varphi }\right]}_{i}=\frac{2d}{{n}^{0}\lambda }\sum_{j\ne i}\left({\varphi }_{j}-{\varphi }_{i}\right)\omega \left(\left|{r}_{j}-{r}_{i}\right|\right)$$

(6)

In the formula, $d$ is the dimension of the solution space, ${n}_{0}$ is the particle number density, ${r}_{i}$ is the coordinate vector of the i th particle, $\mathrm{\varphi }$ is the scalar value of the physical parameter of the particle, $\lambda$ is the correction factor, and its expression is:

$$\lambda =\frac{\sum_{j\ne i}\omega \left(\left|{r}_{j}-{r}_{i}\right|\right){\left|{r}_{j}-{r}_{i}\right|}^{2}}{\sum_{j\ne i}\omega \left(\left|{r}_{j}-{r}_{i}\right|\right)}.$$

(7)

3 Vehicle Multi-Body Dynamics Modeling and Verification

3.1 The Construction of Dynamic Model

In the vehicle dynamics model, the parameters that have the greatest influence on the calculation accuracy of wading conditions include engine performance parameters, body shape characteristics and tire-ground adhesion coefficient. In order to ensure the simulation accuracy, the torque-speed curve on each driving wheel of the simulation model is obtained from the engine characteristic curve of the original vehicle and the overall speed ratio of the transmission system. The body shape is consistent with the original vehicle, and the three-dimensional tire model is exactly the same as the original vehicle (Fig. 1). The tire model file adopts UA tire, and the tire parameters are adjusted by the test data. At the same time, the suspension system is built, and a high-precision multi-degree-of-freedom dynamic vehicle model is established. The vehicle dynamics model is shown in Fig. 1.

Fig. 1

Tire and vehicle model

3.2 Validation of Vehicle Dynamic Model

The reason for the decrease of power and the deflection of the vehicle body during wading and lateral impact is the slip of the tire, so the correct and reliable contact parameters between the tire and the ground in the dynamic model are very important to the accuracy of the simulation results. It is verified by hard road traction test and in-situ starting acceleration test.

According to the test standard, in the hard road traction test (Fig. 2.a), the test vehicle is connected to the load trailer through the traction rod, and the load trailer steps on the brake until the test vehicle is completely skidded. The maximum traction force of the test vehicle on the hard road is obtained by the force sensor on the traction rod. The load trailer brake is equivalent to a fixed end. The test vehicle continues to move forward to tighten the traction rod until the maximum traction force is obtained. Therefore, the load trailer can be equivalent to a fixed end in the simulation model of the traction force test (Fig. 2. b). The vehicle is connected to the fixed end by a spring, and the spring is tightened until the wheel slip rate reaches 100%, and the maximum traction force of the simulation is obtained.

Fig. 2

Hard road traction test and simulation model

The traction curve of hard road is shown in Fig. 3.a. Comparing the maximum traction force of the test and simulation, the test result is 80.78 kN, the simulation result is 82.32 kN, and the calculation error is 1.9%. It shows that the simulation model can well simulate the characteristics of vehicle tire and ground mechanics on hard road.

Fig. 3

Hard road traction curve and in-situ starting acceleration curve

In order to ensure the dynamic performance of the vehicle when it is wading, it is usually driven in first gear. Therefore, the model is compared with the speed change curve of the original vehicle in the first gear (Fig. 3.b). It can be seen that the established dynamic model reaches the maximum speed of 24.5km/h in the first gear at 2.4s, and the acceleration time and the maximum speed can be well fitted with the test curve.

4 Verification of MPS Algorithm

Based on the MPS algorithm, the simulation analysis of the vehicle wading condition needs to ensure the correctness of the MPS algorithm in simulating the water pressure, buoyancy and fluid–solid coupling calculation. The dam impact test and the vehicle wet slippery road traction test are used for verification.

4.1 Dam Break Experiment Based on MPS Algorithm

Under the conditions of wading and lateral impact of water flow, the water force that has the greatest impact on mobility is the impact pressure and buoyancy of water flow. The essence of buoyancy is the pressure difference caused by different water layer heights, and then the longitudinal force acting on the surface of the object is generated. Therefore, It is necessary to verify whether the MPS algorithm is reliable for the simulation of water pressure. This is verified by the dam break experiment done by Kleefsman et al. [9] and Sug et al. [10].

Kleefsman et al. built an experimental platform for dam break, and measured the impact pressure of water flow at different positions of the baffle by sensors. Firstly, a 1:1 dam break simulation model with the original experiment is built, as shown in Fig. 4. The particle spacing of the model is set to 10 mm. In order to accurately capture the pressure peak point, the output data spacing is set to 0.01 s.

Fig. 4

Dam break simulation model

The curves of simulation data and experimental data are shown in Fig. 4b. Comparing the data results, the simulation model reaches the peak pressure point at 0.43 s, and the relative error of the peak pressure is 9.8%. The pressure change trend is basically the same as the experimental data, but due to the size of the model particles, the calculation results fluctuate significantly. In addition, the peak of about 5s has a time delay, which is caused by the echo after the impact of water flow, and has little effect on the wading and water flow impact conditions of the vehicle in the open water.

4.2 Traction Test of Wet Road Based on MPS Algorithm

The vehicle wet road traction test (Fig. 5) can not only verify whether the tire-ground mechanics of the vehicle dynamics model is reliable under the influence of the water layer, but also verify whether the calculation of the fluid–solid coupling problem based on the MPS algorithm is accurate. The simulation model of wet road traction also equivalents the load trailer to a fixed end, and establishes a joint simulation of dynamic software and fluid analysis software.

Fig. 5

Wet road traction test and data curve

The vehicle wet road traction curve is shown in Fig. 5. The maximum traction force obtained by simulation is 39.41 kN, while the maximum traction force measured by the test fluctuates around 38.37 kN, and the calculation error is 2.7%. The existence of the water layer makes the adhesion coefficient of the tire-ground decrease, resulting in the maximum traction force and the time to reach the complete slip of the tire are reduced compared with the hard road conditions.

5 Simulation Analysis of Vehicle Wading Condition

5.1 Simulation Analysis of Vehicle Wading in Static Waters

The simulation of vehicle wading conditions in static waters can calculate and analyze the mobility of this type of special vehicle in different water depths. The mobility index involved is the maximum speed that the vehicle can reach. The water depth is set to 0.5m and 1m. The vehicle first accelerates to the maximum speed in the first or second gear on the hard road, and then drives into the water area, and finally reaches the stable maximum speed of wading.

The components in the vehicle dynamics model that are in contact with the water are exported as interface files and coupled with the fluid analysis software to obtain a simulation model of static water wading as shown in Fig. 6. The horizontal area (not including uphill and downhill) of the water area is 40 m long, 10 m wide. Due to the large volume of the water area, in order to reduce the calculation cost, the particles are appropriately enlarged, and the particle spacing is set to 50 mm.

Fig. 6

Simulation model and simulation process of vehicle wading condition

When the water depth is 1 m, the curves of vehicle speed, resistance and buoyancy are shown in Fig. 7. The simulation results of the first gear are shown in Fig. 7a. Due to the influence of water resistance, the actual maximum speed is 16 km/h. The simulation results of the second gear are shown in Fig. 7b. It can be seen that when the vehicle accelerates on the hard road to the maximum speed (42 km/h) and enters the water, the speed drops rapidly to 10 km/h, reaching the actual maximum speed of the second gear wading.

Fig. 7

Simulation results of wading conditions (water depth 1 m)

When the vehicle wades in the second gear, the longitudinal resistance and buoyancy do not increase gradually as the first gear enters the water, but will produce a large peak at the moment when the vehicle enters the water. This is because the water entry speed of the second gear is 1.7 times that of the first gear. The increase in speed makes the body have a greater impact on the water body. It can be seen that the vehicle speed has a great influence on the longitudinal resistance and buoyancy of the vehicle.

When the water depth is 0.5 m, the speed, longitudinal resistance and buoyancy of the vehicle are shown in Fig. 8. Compared with the simulation results of the water depth of 1 m, it can be seen that the decrease of the water depth makes the longitudinal resistance and buoyancy of the vehicle significantly reduced. In the stable stage of vehicle speed, the buoyancy value is reduced from 35 to 10 kN, while the longitudinal resistance value is reduced from 15 to 8 kN. The maximum speed has increased, when the vehicle to a gear into the waters, the stable maximum speed of 19 km/h; when the vehicle enters the waters with a second gear, the maximum speed that can be achieved is 24 km/h. The maximum speed of the vehicle under different water depths and gears is shown in Table 1.

Fig. 8

Simulation results of wading conditions (water depth 0.5 m)

Table 1 The actual maximum speed of the vehicle static wading

5.2 Simulation Analysis of Water Flow Lateral Impact

When military vehicles pass through the bridge deck submerged by the river, the external force on the body is not only the buoyancy of the water body, but also the lateral impact force caused by the river. In order to ensure the trafficability of the vehicle, it is necessary to ensure that the lateral offset (Δy) of the body under the lateral impact of the water flow is controlled within an appropriate range. The lateral offset refers to the displacement of the body center of mass in the direction perpendicular to the center line of the bridge deck. Its range is determined by the width of the vehicle and the bridge deck. The purpose is to ensure that the vehicle will not be impacted by the water flow and fall off the bridge deck. The wheel track of the comprehensive test model (1.97 m) and the assumed bridge width (8 m), the body offset should not be >6m. Considering the general river width, the length of the water area is set to 30 m.

The dynamic model does not consider the driver’s control of the steering wheel, so only the lateral impact of the water flow when the vehicle is fixed on the steering wheel is simulated. Considering the influence of water flow, the water depth of 0.5 and 1 m are compared and analyzed. The water flow impact model is shown in Fig. 9a, and the particle spacing of the model is maintained at 50 mm. When the water depth is 0.5 m, the lower water flow rate will not have a significant impact on the body, so the flow rate begins to increase from 1.5 m/s. When the water depth is 1 m, the flow velocity begins to increase from 0.5 m/s until the body offset is greater than the set target value, and the limit flow impact velocity is obtained. The simulation process is shown in Fig. 9b.

Fig. 9

Simulation model and simulation process of water flow lateral impact condition

In the actual driving process, when the vehicle encounters the lateral impact of the river, it often changes the direction of the driving in advance, deflects the direction of the vehicle, and maintains a certain angle with the center line of the bridge deck to enter the water area. Here, the lateral impact is simulated with the deflection angles of 0°, 5° and 10° respectively. The relationship between the lateral offset of the vehicle and the flow velocity and angle under different water depths is shown in Fig. 10.

Fig. 10

Simulation results of lateral impact of water flow

According to the simulation results, whether the water depth is 0.5 or 1 m, when the vehicle enters the water in the direction parallel to the center line of the bridge deck (0°), the lateral offset of the vehicle will be improved to varying degrees compared to the deflection of a certain angle into the water at the same flow rate. When the flow rate increases, the difference is more obvious. Taking the impact condition of 1 m water depth as an example, when the flow velocity is 0.5 m/s, the lateral offsets of the three water entry angles are basically the same. When the flow velocity increases to 1 m/s, the lateral offset difference between 0°and 10°reaches 3.55 m, which is much larger than the wheelbase of the vehicle itself. It can be seen that changing the angle of the vehicle into the water has an important influence on improving the trafficability and safety of the vehicle under the impact of higher flow velocity.

In terms of the limit flow velocity to ensure the safe passage of vehicles, when the water depth is 0.5 m, the limit flow velocity of 0° water entry is 1.9 m/s, and when the deflection angle is 10°, the limit flow velocity increases to 2 m/s. When the water depth is 1 m, due to the increase of water flow, the lateral impact force of water flow on the body increases. At this time, when the vehicle enters the water at 0°, the allowable limit flow rate is only 0.7 m/s. After increasing the deflection angle, the trafficability of the vehicle is improved, and the allowable maximum flow rate at 10° is 0.9 m/s.

In terms of the relationship between the lateral offset of the vehicle body and the flow velocity, the simulation results of the lateral offset of the water entering in the 0° direction at a water depth of 1 m are taken as an example. When the flow velocity increases to 1 m/s, the lateral offset suddenly increases from 7.86 to 10.89 m, and the increase is significantly increased. This is because when the flow velocity reaches 1 m/s, the lateral impact force of the water flow is greater than the lateral adhesion of the wheel, and the tire becomes sliding in the lateral direction, resulting in a significant increase in the lateral offset. At this time, in order to improve the mobility of the vehicle driving through the river, it is necessary to improve the adhesion coefficient of the tire, that is, to change the material, pattern and width of the tire until the requirements of wading are met, so as to ensure the trafficability and safety of the vehicle.

6 Conclusion

A multi-body dynamics model of a special vehicle is built, and the vehicle model and MPS algorithm are verified according to the vehicle tests and dam break experiment in this paper. Then, the maximum speed of the vehicle under different water depths and different gears is obtained by joint simulation. In addition, the lateral impact conditions of water flow at different water depths are simulated and analyzed with the deflection angles of 0°, 5° and 10° respectively. The variation law of lateral offset under different flow velocities and deflection angles is obtained.