# Influence of Fastener Failure on Dynamic Performance of Subway Vehicle
## 3 Numerical Analysis
### Parameter Analysis  
In order to analyze the influence of fastener failure on vehicle dynamic performance more systematically, the dynamic response of the vehicle at different vehicle speeds (including 80 km/h, 100 km/h, 120 km/h, and 140 km/h) and different cases (including perfect fastener, one fastener failure, two fastener failures, three fastener failures, four fastener failures, and five fastener failures) are calculated, and the maximum responses are extracted. Figure 8 depicts the maximum body vertical acceleration at various speeds and in various situations. It can be seen that the maximum acceleration of the vehicle body varies between 0.6 m/s\({}^{2}\) and 0.8 m/s\({}^{2}\). The failure of a fastener in various situations has little effect on the maximum acceleration of the vehicle body, whereas the vehicle speed has a minor effect on the acceleration of the vehicle body. Figure 8: Car body acceleration at different speeds and under different cases. Figure 7: Time–history curve of car body vertical acceleration. Figure 9 illustrates the maximum Nadal index for all wheelsets at various speeds and conditions. As can be seen, there is no obvious relationship between the number of fastener failures and the change in vehicle speed when it comes to the change in the Nadal index. The Nadal index of vehicles under various cases is generally within the acceptable range (less than 0.8). This is primarily due to the fact that when fastener failures occur, the rail vertical stiffness changes more, and the wheel-rail lateral force primarily comes from lateral component of normal and creep force. When the normal force changes, the vertical and horizontal component also change, and they increased almost in the same proportion. According to the calculation method of the coefficient of derailment, it is the ratio of horizontal and vertical force. Furthermore, the lateral component of normal force accounts for a significant portion of the lateral force on the wheel-rail. As a result, when a fastener fails, the vertical force on the wheel-rail changes, but the lateral force on the wheel-rail changes in nearly the same proportion, resulting in little change in the Nadal index. Figure 10 demonstrates the maximum wheel unloading ratio of all wheel sets at various speeds and cases. It can be seen that the number of fastener failures has a great influence on the wheel unloading ratio, and the degree of influence increases with the rise in vehicle speed. When the vehicle is running at a low speed (the speed is 80 km/h), the wheel unloading ratio of the vehicle is only about 0.2 without fastener damage. With the increase in the number of fastener failure, the wheel unloading ratio gradually increases. The wheel unloading ratio is close to the allowable value of 0.6 in the case of four fasteners failure. The wheel unloading ratio is close to the running safety limit value of 0.8 when five fasteners are damaged. Figure 10: The wheel unloading ratio at different speeds and under different cases. Figure 9: The Nadal index under different cases. When the vehicle is running at a high speed (design speed 140 km/h), the wheel unloading ratio of the vehicle reaches about 0.55 with perfect fastener. When the number of failure fastener is one or two, the wheel unloading ratio changes little. The wheel unloading ratio increases dramatically when three fasteners are damaged, exceeding the allowable running safety value of 0.6. The wheel unloading ratio exceeds the running safety limit value of 0.8 when four fasteners are damaged. When five fasteners are damaged, the wheel unloading ratio reaches 1.0, indicating that the vehicle wheel set has experienced wheel-rail separation, and even slight disturbances can cause the vehicle to derail. When one, two, or three fasteners fail, the vehicle is safe to drive at a lower speed than the design speed; when four fasteners fail, the vehicle's allowable running speed is 130 km/h; and when five fasteners fail, the vehicle's allowable running speed is 80 km/h.