Investigation of Relationship Between Initial Setting of Leveling Valves and Air Spring Pressure of a Railway Vehicle When Assuming the Centrifugal Force Action

Abstract

The wheel load balance of a railway vehicle could vary according to whether it is before or after the curve passage. The variation is influenced by the leveling valve (LV) that has a dead zone of air supply and exhaust. The purpose of this research is to clarify the mechanism of the wheel load variation due to the dead zone of the LVs. In this paper, we assume the situation where the lateral force is loaded and unloaded on the car body. The simulation and experimental results show that the diagonal difference in LV rod lengths and the maximum lateral force are related to the air spring pressure variation.

1 Introduction

It is known that the wheel load balance of a railway vehicle which has leveling valves (LV) with a dead zone could vary according to whether it is before or after the curve passage because of the function of the leveling valves [1]. The variation in the wheel load is caused by the air spring pressure variation. It is important to make clear the phenomenon from the viewpoint of traveling safety. The phenomenon could also occur in a stationary test, and its clarification is also important in terms of the development of a method for conducting stationary tests of vehicle performance affecting the wheel load. Although several researchers proposed simulation models with due consideration for the characteristics of the LV [2,3,4], however, the principal objective was to construct a model, and the mechanisms of the phenomenon have not been discussed.

The result of the stationary test is decided by 2 elements: one is the vehicle characteristics, which include adjustable initial settings, and another is the external force. In this paper, we focus on the difference in the LV rod length as an adjustable initial setting. As shown in Fig. 1, the initial setting of LV lever angles could vary within the dead zone of the LV since the length of the LV rods are adjusted manually. We assume the situation where the lateral force is loaded and unloaded on the car body, which is assumed to be the external force in a stationary test. The lateral force emulates the centrifugal force in a curve. We investigated the relation between the difference in LV rod lengths at each position and the pressure variation through simulations and experiments under the situation.

Fig. 1.

Overview of the simulation model: the LV rod lengths at each position are variable within the LV dead zone.

2 Lateral Force Loading Simulation

2.1 Simulation Model

The overview of the simulation model is shown in Fig. 1. A secondary suspension model is assumed in the simulation: the bogie frame is fixed on the inertial reference frame. The car body is modeled as a 6-DOF body and it is affected by the force from the air springs and the lateral stopper rubbers. The principal specifications of the car body model are shown in Table 1. The time series lateral force pattern, which is shown in Fig. 2 is applied to the car body model. The maximum lateral force of the pattern is 20 kN, which is the lateral force acting to the vehicle when the vehicle stops on a circular curve of which the cant is 105 mm. Note that the situation is very simplified for the sake of simplification of the problem. The car body is influenced by not only the centrifugal force but also truck displacement, vibration and so on in an actual running situation of a railway vehicle. Those problems are left to future work.

Fig. 2.

Lateral force pattern in the simulation

Fig. 3.

Configuration of the pneumatic system model

The overview of the pneumatic system model is shown in Fig. 3. The model is constructed by the air spring, the laminated rubber, the sub air tank, the leveling valve and the main tank. The air spring is assumed as a cylinder of which the cross section is equal to the pressurized area of a real air spring. The isothermal process is assumed in deriving the spring characteristics. The vertical air spring force F is calculated by solving the following equation:

$$\begin{aligned} \frac{A_0 P_0 h_{01}}{F + A_0 P_0} + h_{02} - \frac{F + c\dot{h}_2}{k} = H. \end{aligned}$$

(1)

The significances of the symbols in the equation are shown in Table 2 and Fig. 3. The “Variable” in the table means the variable that is calculated recursively in the process of numerical integration of the dynamic system. The air flow between the air spring and the other component of the pneumatic system (e.g. atmosphere) is modeled as a nozzle flow. The mass flow rate is described as follows:

$$\begin{aligned} \dot{m} = \left\{ \begin{array}{ll} s\alpha \sqrt{ \displaystyle \frac{2\gamma }{\gamma - 1}P_\mathrm {1}\rho _\mathrm {1} \left( \frac{P_\mathrm {2}}{P_\mathrm {1}} \right) ^\frac{2}{\gamma }\left\{ 1 - \frac{P_\mathrm {2}}{P_\mathrm {1}} \right\} ^\frac{\gamma -1}{\gamma }} &{} (P_\mathrm {2}/P_\mathrm {1} \ge p_\mathrm {cr}) \\ s\alpha \sqrt{ \displaystyle \frac{2\gamma }{\gamma - 1}P_\mathrm {1}\rho _\mathrm {1} \left( p_\mathrm {cr} \right) ^\frac{2}{\gamma }\left\{ 1 - p_\mathrm {cr} \right\} ^\frac{\gamma -1}{\gamma }} &{} (P_\mathrm {2}/P_\mathrm {1} < p_\mathrm {cr}) \\ \end{array} \right. , \end{aligned}$$

(2)

Table 1. Principal specifications of the car body model

Table 2. Principal parameters of the air spring and LV model

where $P_1$ and $P_2$ are the valve-inlet and outlet pressure respectively, $\rho _1$ is the valve-inlet air density, $\kappa$ (=1.4) is the polytropic index, $\alpha$ is the effective area of the cross section of the air tube, s ($= 1$, 0 or $-1$) is the index to switch the valve states: inlet, closed or exhaust, and $p_\mathrm {cr}$ is the critical pressure ratio which is calculated as follows: $p_\mathrm {cr} = \left\{ 2/(\kappa + 1) \right\} ^\frac{\kappa }{\kappa -1}$. In the case of the calculation of the mass flow rate between the main tank and the air spring, for example, $P_1$ is the constant main tank pressure, $P_2$ is the variable air spring pressure. The index s is decided based on the total air spring height H and the LV rod length $L_\mathrm {s}$ to simulate the LV behavior. In the simulation, the 3 s of dead time of starting inlet or exhaust are also considered. The dead zone of LV is $\pm 1.91 \mathrm {[deg]}$ (corresponding to $\pm 5 \mathrm {[mm]}$ at the LV lever tip). The air flow between the air spring and the sub tank is also calculated by using Eq. (2). According to the mass conservation law, the no-load height of the air spring, $h_{01}$ and the sub tank pressure, $P_\mathrm {sub}$ are calculated by integrating the following differential equations:

$$\begin{aligned} \dot{h}_{01} = \frac{R_\mathrm {air} T_0}{A_0 P_0} (\dot{m}_\mathrm {LV} + \dot{m}_\mathrm {sub}), \ \dot{P}_\mathrm {sub} = -\frac{R_\mathrm {air} T_0}{V_\mathrm {sub}} \dot{m}_\mathrm {sub}, \end{aligned}$$

(3)

where $\dot{m}_\mathrm {LV}$ is the mass flow rate of air that flows in the LV and $\dot{m}_\mathrm {sub}$ is the mass flow rate of air that flows between the air spring and the sub tank.

The shear restoring force of the air spring, $F_{xy}$ and the restoring force of the lateral stopper rubber, $F_\mathrm {stp}$ are modeled as follows respectively:

$$\begin{aligned} F_{xy} = -\frac{k_0 q}{\sqrt{1 + aq^2}}, \ F_\mathrm {stp} = -a_1 \tan \left( \frac{q}{a_2} \right) , \end{aligned}$$

(4)

where q is a variable that describes the deformation displacement, $k_0$, a, $a_1$ and $a_2$ are the parameters that characterize the restoring force models. A gap of 10 mm between the center-pin of the car body and the lateral stopper is considered separately. The concrete values of each parameter are the following: $k_0 = 1.5 \times 10^5$ [N/m], $a = 1000$ [$\mathrm {m}^{-2}$], $a_1 = 4000$ [N] and $a_2 = 0.0137$ [m].

2.2 Relation Between the Amount of Pressure Difference Variation and Initial Setting of LV Rod Lengths in Simulation

We set 11 conditions of the LV rod lengths as shown in Table 3 to investigate relations between the air spring pressure and the lateral force. In Table 3, “Nominal” means such LV rod length as gives rise to a state where the LV lever is placed at the center of the dead zone. “±4.9 mm” means the conditions of longer or shorter LV rod length than that of nominal condition. Note that the initial LV lever angle is inside of the dead zone of the LV in all of the conditions.

Table 3. Settings of LV rod length for each simulation

The relations between the air spring pressure difference and the lateral force applied in the several cases are shown in Fig. 4. The vertical axis indicates the pressure difference between the left and right side air springs of the each bogie and the horizontal axis indicates the lateral force. The simulation results are classified into 2 groups qualitatively: one is the state where there is a large hysteresis in the relation between the pressure difference and the lateral force (no. 2 and 8) and another is the state where there is an almost linear relation between them (no. 1, 6, 7 and 10). The equable initial displacement of the 4 LV rods and the left/right difference in initial LV rod lengths do not cause the large hysteresis. On the other hand, the case where one LV rod is longer than the other LV rods (no. 2) and the case where the LV rods at the positions 1 and 4 are longer than the other (no. 8) cause a large hysteresis. We introduce two difference indicators of LV rod length, the left/right difference $e_\mathrm {LR}$ and the diagonal difference $e_\mathrm {diag}$ to summarise the results quantitatively. The difference indicators are defined as follows:

$$\begin{aligned} e_\mathrm {LR}= & {} \frac{1}{2} (\theta _\mathrm {LV1} + \theta _\mathrm {LV3}) - \frac{1}{2} (\theta _\mathrm {LV2} + \theta _\mathrm {LV4}) \end{aligned}$$

(5)

$$\begin{aligned} e_\mathrm {diag}= & {} \frac{1}{2} (\theta _\mathrm {LV1} + \theta _\mathrm {LV4}) - \frac{1}{2} (\theta _\mathrm {LV2} + \theta _\mathrm {LV3}) \end{aligned}$$

(6)

where $\theta _1$, $\theta _2$, $\theta _3$ and $\theta _4$ are the LV rod lengths that are converted into the angle of LV levers. We investigated the relation between the difference indicators and the amount of the pressure difference variation, which is illustrated in Fig. 4-no. 2, in all the 11 cases of the simulation as shown in Fig. 5. The plots show that the corelation between the LV diagonal difference and the amount of the pressure difference variation is large, while the corelation between the LV left/right difference and the amount of the pressure difference variation is not large.

Fig. 4.

Relations between the lateral force applied and the pressure difference between the left and the right of the bogie in several cases of the simulation

Fig. 5.

Relation between the amount of pressure difference variation and LV left/right or diagonal difference in the simulation

3 Lateral Force Loading Experiment

3.1 Overview of the Experimental Device

An overview of the lateral force loading device that is used for the verification experiment of the simulation results is shown in Fig. 6. The device consists of 2 pillars, 2 winches and 2 wires. The lateral force is applied to the car body by applying tension manually. The lateral force applied is measured by using load cells that are installed between the wire and the car body. The laser displacement sensors are also installed on the car body to measure the LV lever angle as shown in Fig. 6-right.

Fig. 6.

Overview of the lateral force loading test device and the measurement setup for LV lever angles

Fig. 7.

Relation between the amount of pressure difference variation and LV left/right or diagonal difference in the experiment

3.2 Relation Between the Amount of Pressure Difference Variation and Diagonal Difference in LV Rod Lengths in the Experiment

The relations between the difference indicators and the amount of the pressure difference variation in 44 cases of the experiment are shown in Fig. 7. The corelation between the LV diagonal difference and the amount of the pressure difference variation is large, similarly to the result of the simulation. The corelation factors between the diagonal difference and the amount of the pressure difference variation are $-0.998$ for the 1st bogie and 0.997 for the 2nd bogie.

3.3 Relation Between the Amount of Pressure Difference Variation and Maximum Lateral Force in the Experiment

We also investigated the relation between the maximum lateral force and the amount of the pressure difference variation when a certain level of the LV diagonal difference is given. In this experiment, we set the LV condition as follows: the LV rods at the position 2 and the position 3 are longer than that at the other positions. The relation between the maximum lateral force and the amount of the pressure difference variations are shown in Fig. 8. The vertical axis of the left side figure shows the amount of the pressure difference variation and that of the right side figure shows the value of the pressure variation divided by the LV diagonal difference. The variance of the amount of the pressure difference variation becomes small by dividing it by the LV diagonal difference. In the region of which the maximum lateral force is smaller than 20 kN, the relation between the amount of pressure difference variation and the max. lateral force is almost linear. When the max. lateral force is equal to or larger than 20 kN, the amount of the pressure difference variation is saturated. This means that the amount of pressure difference variation is bounded and its maximum value can be determined.

Fig. 8.

Relation between the amount of pressure difference variation and maximum lateral force applied to the car body

4 Discussions

As shown in simulation and experimental results, the sign of the amount of the pressure difference variation in the 1st bogie and the 2nd bogie is different: in the simulation no. 2 and 8, for example, the pressure difference in the 1st bogie is positive while the pressure difference in the 2nd bogie is negative. This means that the rolling moment due to the difference in the air spring forces in the 1st bogie and that in the 2nd bogie is canceled out, and the pressure variation occurred as the variation of the inner force. In other words, the diagonal difference in the LV rod length would cause the diagonal 2-point support of the air springs, which causes the wheel load imbalance.

5 Conclusions

We investigated the mechanism of the pressure difference variation before and after the curve passage of a railway vehicle with a focus on the difference in the LV rod lengths at each position. The simulation and experiment results show that the diagonal difference in the LV rod lengths and the maximum lateral force that is applied to the car body are strongly related to the amount of the pressure difference variation. In this investigation, the influence of the twist in transition curves is not considered. That is future work from the viewpoint of elucidation of quasi-static phenomena. It is also future work to consider the influence of dynamic behaviors, e.g. vibration of the car body.