Based on the rigid-flexible coupling dynamics model of rack vehicle, the finite element model of rack vehicle is established, as shown in Fig. 2. The dynamic characteristics of rack vehicle are investigated under various slopes and driving speeds conditions, encompassing the dynamic response of vehicle submodel, track submodel, gear-rack transmission model, and the contact characteristics of gear-rack contact surfaces. By studying the dynamic characteristics of each component and the root bending stress of rack, a reasonable value for slope and driving speed is proposed to ensure the stability and safety of rack vehicle.

## 3.1. Dynamic response of rack vehicle under different slopes

The variation in train running slope has a significant impact on the stability and safety of rack vehicle. This section aims to investigate the effects of slope changes on various components of rack vehicle, including wheel-rail contact force, root bending stress of rack, and dynamic characteristics of each component, within the range of 60‰-300‰.

Figure 3 reveals the vertical contact forces between wheelset and rail under different slopes. In Fig. 3(a), the variation of wheel-rail vertical contact force is illustrates when the slope is 60‰, and the trend of wheel-rail contact force at other slopes is comparable to that of 60‰. It can be seen from the Fig. 3(a) that the vertical contact force of wheel-rail changes periodically during uniform motion. The reason is that during the rotation of wheel, different positions of wheel come into contact with rail, causing the contact force to appear up and down in this cycle. During the time interval of 2.5s-4.5s, the periodic variation in wheel-rail vertical contact force is attributed to the cyclic rotation of wheel. Moreover, due to track random irregularity excitation, weld seams of track, etc. The change trend of wheel-rail vertical contact force exhibits variations at 2.8s and 3.5s.

Figure 3(b) shows the average wheel-rail vertical contact force under different slopes. It can be seen from the Fig. that the wheel-rail vertical contact force gradually decreases as the slope increases. When the slope increases from 60‰ to 300‰, the wheel-rail vertical contact force decreases from 71.28kN to 66.35kN. The wheel-rail contact force is reduced by approximately 6.9%.

Figure 4 shows the vertical vibration acceleration and displacement of vehicle under different slopes. In order to explore the running stability of rack vehicle, the vibration acceleration of vehicle is shown in Fig. 4(a). It can be seen from the Fig. that the vibration acceleration of vehicle increases with the increase of slope. Due to the vibration direction of vehicle is different from the direction of coordinate axis, the vibration acceleration of vehicle can be either positive or negative, which has little impact on the results.

According to the above analysis, the wheel-rail vertical contact force gradually decreases with the increase of slope. It shows that the vertical force exerted by each component on rail gradually decreases. Furthermore, all parameters of the secondary suspension are the same, with the support of the secondary suspension, the vertical displacement of vehicle gradually decreases as slope increases. In Fig. 4(b), the vertical displacement of vehicle under different slopes are shown. When the slope increases from 60‰ to 300‰, the vertical displacement of vehicle decreases by about 20mm.

Variations in wheel-rail vertical contact force not only influence the vertical vibration acceleration and vertical displacement of vehicle, but also exert a significant influence on vertical displacement of rail. When suffered to a significant vertical force, the rail will has a substantial deformation, leading to increased wear and potential interference with the contact between wheel and rail. In Fig. 5, the vertical displace-ment of rail under different slopes are shown. The Fig. illustrates that as the slope increases, the vertical displacement of the rail decreases. Specifically, when the slope increases from 60‰ to 300‰, the maximum vertical displacement of rail decreases from 1.03mm to 0.98mm.

The analysis of the wheel-rail vertical contact force reveals that as slope increases, the vertical contact force between wheel and rail decreases gradually, while the longitudinal force increases gradually. This results in an increase in slope resistance that the train must overcome during the climbing process. Nevertheless, the incorporation of gear-rack system can significantly compensate for the traction deficiency.

Figure 6 shows the root bending stress maps of rack under different slopes, (a)-(e) correspond to five slopes in 60‰-300‰, respectively. As can be seen from the Figure, when the slope increases from 60‰ to 300‰, the stress map of the gear-rack contact surface gradually becomes larger, indicating that the contact force between gear and rack increases gradually. The variation trend of the maximum root bending stress of rack under different slopes is not significant. Therefore, the value of the maximum root bending stress of rack is presented separately in a bar chart format, as shown in Fig. 7. It can be observed from the Figure. that there is an increasing trend in the maximum root bending stress of rack with an increase in slope. When the slope increases from 60‰ to 300‰, the maximum root bending stress of rack rising from 79.73MPa to 84.15Mpa, with a increase rate of about 5.5%.

## 3.2. Dynamic response of rack vehicle under different driving speeds

The following section will investigate the varying characteristics of each component of the train at different driving speeds. The train will be operated at constant speeds of 15km/h, 20km/h, 25km/h, 30km/h, and 35km/h respectively on a slope of 120‰.

Figure 8 illustrates the variations in wheel-rail vertical contact force at various driving speeds. For instance, at 30km/h, the fluctuation in the wheel-rail vertical contact force is depicted in Fig. 8(a). Similar trends can be observed for the wheel-rail contact force at other driving speeds. The Fig. illustrates the periodic variation of the wheel-rail contact force at a driving speed of 30km/h. The factors influencing this change are comparable to those observed at a 60‰ slope in Section3.1.

The average of wheel-rail contact force at different driving speeds is shown in Fig. 8(b). As can be seen from the Fig., when the driving speed of train increases from 15km/h to 35km/h, the change of wheel-rail contact force is not obvious, in which the difference between the minimum value (68.68kN) and the maximum value (69.33kN) is only 0.65kN, which is negligible within the allowed range of error. It can be seen that different driving speeds have little effect on the vertical contact force between wheel and rail.

In Fig. 9(a), the vertical vibration acceleration and vertical displacement of vehicle at different driving speeds is shown. It can be seen that the vertical vibration acceleration of vehicle does not change significantly with the driving speed, and the vibration acceleration fluctuates within the range of approximately ± 0.01 at different driving speeds. This is because the driving speed of rack vehicle belongs to the low-speed driving mode, and the dynamics of vehicle is not sensitive to the change of speed. Furthermore, the vertical displacements of vehicle at different driving speeds are also the same, as shown in Fig. 9(b). It can be seen from the Fig. that the vertical displacement of vehicle is basically maintained at about 26mm, and the amplitude of the up and down fluctuations is not obvious.

The vertical displacement of rail at different driving speeds is illustrated in Fig. 10. Due to the varying locations of the probe points within the finite element, the displacement of rail appears staggered at different driving speeds, but its changing trend remains consistent. It can be observed that the impact of different driving speeds on the vertical displacement of rail is not significant, with a range between 1.03mm and 1.05mm.

Figure 11 illustrates the root bending stress of rack at various driving speeds, ranging from 15km/h to 35km/h denoted as (a)-(e). The diagram reveals that the bending stress does not follow a consistent pattern across different driving speeds. The changing trend of the maximum root bending stress of rack is shown in Fig. 12. When the train operates at various speeds, the rise in gear speed results in an escalation of the meshing frequency between gear and rack, consequently amplifying the intricacy of the nonlinear dynamics within gear-rack system [24]. The contact condition between gear and rack is significantly influenced by the multi-state meshing behavior, subsequently causing fluctuations in the root bending stress of rack, displaying a trend of initially decreasing, then increasing, and finally decreasing.

Based on the above analysis, the meshing frequency of gear-rack system varies with the driving speed, so that the vertical vibration acceleration of gear is changed. In Fig. 13, the root mean square of the vertical vibration acceleration of gear at different driving speeds is shown. It can be seen that the change trend of the vertical vibration acceleration of gear aligns with the maximum bending root stress of rack, providing additional confirmation of the impact of gear-rack meshing frequency on both the maximum root bending stress of rack and gear vibration acceleration. Therefore, it is recommended that the speed of rack vehicle is 25km/h.