Sage Journals: Discover world-class research

Abstract

With the metro vehicle as the subject, a mathematical model of the main chamber, the auxiliary chamber and the orifice of the air spring were established based on the thermodynamic theory. The lateral nonlinear model is obtained based on the analytical geometry. Establish the four-point levelling valve and differential valve models through theoretical analysis. A complete three-dimensional air spring model is established. The correctness of the model is verified by comparing it with the actual data. The impact of orifice diameters and secondary vertical dampers on the dynamic performance of the metro vehicle is investigated. The results show that the secondary vertical damping configuration has a greater effect on vertical stability and acceleration but a minor impact on operational stability. Metro vehicles equipped only with air springs in the smaller diameter orifice do not need to be configured with a vertical damper, resulting in excellent dynamic performance and savings in production costs.

Keywords

Metro vehicles air spring modelling secondary suspension orifice vehicle dynamics

Introduction

Due to the considerable load variation of metro vehicles, the air spring ensures the consistency of vehicle floor height and running stability. It has been widely used in motor train unit (EMU) vehicles.¹ The orifice is located between the additional chamber and the main chamber. When vertical deformation occurs, the pressure difference between the two will be generated, making the air pass through the orifice and produce a specific resistance to dissipate part of the energy to achieve the effect of vibration reduction. The fixed orifice is simple in structure and hardly increases the cost of the air spring. It is generally used in vehicles with low speeds. Using an orifice or oil vertical damper to provide vertical damping depends on the air spring structure and working characteristics. Due to the low speed of subway operation, studying the running performance of air springs and the orifice is necessary.

Air spring suspension has complex nonlinear characteristics. Modelling and specific parameter analyses have always been difficult research points. The modelling of air springs can be classified into four categories: mechanical model, thermodynamic model, analytical model, and finite element model.² The application scope and ideas of each method are different. The mechanical model is mainly expressed by equivalent damping, equivalent stiffness or other elements, and the parameter values are often derived from experiments. Because of its simplicity and rapidity, it is currently most widely used in multibody dynamics. The most widely used are ‘Nishimura linear model’,³ ‘SIMPACK FE83 model’,⁴ ‘Vampire model’,⁵ ‘Berg model’⁶ and the corresponding improved models.^7–9 The thermodynamic model^10–12 uses the laws of mechanics and thermodynamics to describe the physical processes in motion. Various models are discussed, compared, and verified by experiments.^13,14

However, many thermodynamic models only focus on vertical performance and ignore lateral characteristics. In addition, the physical input parameters of the model need to be obtained through experiments. Docquier proposed a relatively complete thermodynamic model in his thesis.¹⁵ The analytical model^16,17 is based on the geometric analysis and the geometric changes produced by the air spring under specific conditions (usually requiring assumptions and simplification). The finite element model^18,19 is the most accurate. The model is accurate and parameterisable, but the simulation cost becomes enormous. Compared with other methods, the current application is less. It is often used to determine the specific physical parameters of thermodynamic models.

Many scholars have researched air springs’ vertical and lateral nonlinear characteristics and orifice damping characteristics. Qi et al.^20,21 established the mathematical models of the main chamber, additional chamber and orifice considering the rubber balloon friction. It analysed the vertical stability of the nonlinear air spring through joint simulation. The transverse restoring force model is established by analysing the balloon’s stress. Li et al.²² studied the effect of key parameters through analytical geometry. Li et al.¹ established the mathematical model based on thermodynamics and hydrodynamics theory and analysed the main influencing parameters. Wu et al.²³ analysed linear and nonlinear and equivalent models’ running stability under straight lines and curves of air springs. Wu et al.²⁴ identified critical parameters and studied their influence on dynamic performance. Chen et al.²⁵ established a nonlinear model and analysed the influence of rubber balloons on air springs. Quaglia et al.²⁶ established a dimensionless mathematical model with an orifice and additional chamber and analysed its vibration characteristics. Alonso²⁷ studied the effect of air spring and orifice size on vehicle comfort. The modelling of air springs is mainly studied from each sub-component without establishing a complete three-dimensional model. Only single vertical or horizontal characteristics are studied. The research object is mainly high-speed EMUs, with less research on the metro.

This paper mainly consists of three aspects. Firstly, the complete air spring nonlinear models were built based on the previous research theory.²⁸ Secondly, the model’s validity is tested by simulating the lateral and vertical characteristics, which agree with the experimental data. Finally, taking the metro vehicle as the research object, the influence of orifice and external vertical damper on vehicle dynamics are studied. The results show that the established complete nonlinear air spring model is feasible. The orifice of metro vehicles can be set to a smaller diameter without an external vertical damper, which can meet vehicle operation requirements and save the cost of design and parts purchase.

Three-dimensional dynamic models of air spring

Figure 1 illustrates the working principle of the air spring pneumatic system. Air enters the main and auxiliary chambers by the levelling valve; the orifice plays a damping role at this time. When the differential pressure of the air spring reaches a specific value, the differential valve will be connected and play a role. Only the air spring fails, and the emergency rubber spring comes into play.

Figure 1.

Air spring pneumatic system.

Main chamber and auxiliary chamber models

During metro operation, the frequency of vehicle vibration is relatively high. When the air spring is stressed, it will affect the volume, pressure, and stiffness of the main chamber and the stiffness of the auxiliary chamber. That is what this Section is all about. It is assumed that there is no heat exchange between the rubber balloon and the outside during the whole vibration process, and the gas meets motion equation under adiabatic state as follows

P_{b} {(\frac{V_{b}}{m_{b}})}^{n} = P_{b 0} {(\frac{V_{b 0}}{m_{b 0}})}^{n}

(1)

Equation (2) is derived from the derivation of time by equation (1), as follows

{\dot{P}}_{b} = n P_{b} \frac{{\dot{m}}_{b}}{m_{b}} - n P_{b} \frac{{\dot{V}}_{b}}{V_{b}}

(2)

where n is the polytropic index (in the range of 1–1.4),

P_{b}

is the absolute pressure of the main chamber at working status, Pa;

V_{b}

is the volume of the main chamber at working status, m³;

m_{b}

is the air mass of the entire pneumatic system, kg;

{\dot{m}}_{b}

is the mass flow rate, kg/s;

P_{b 0}

is the absolute pressure of the main chamber under the original state, Pa;

V_{b 0}

is the volume of the air chamber under the original state, m³;

m_{b 0}

is the total air mass under the original state, kg.

The volume is related to its working height h, lateral displacement $y_{s}$ and longitudinal displacement $x_{s}$ relevant. The main chamber is made of rubber material, producing elastic deformation when subjected to vertical, lateral, and longitudinal load. The volume changes thus changing the gas pressure. The equation is as follows

{\dot{V}}_{b} = \frac{\partial V_{b}}{\partial h} \dot{h} + \frac{\partial V_{b}}{\partial y_{s}} {\dot{y}}_{s} + \frac{\partial V_{b}}{\partial x_{s}} {\dot{x}}_{s} + k_{v} {\dot{P}}_{b}

(3)

where

\partial V_{b} / \partial h

is the height-dependent volume change rate;

\partial V_{b} / \partial y_{s}

represents the rate of volume change related to the lateral displacement;

\partial V_{b} / \partial x_{s}

is the rate of volume change associated with the the longitudinal displacement,

k_{v}

is the pressure-dependent volume change rate; h is the air spring height, m.

The effective area of the main chamber is a function about h, $y_{s}$ and $x_{s}$ , which is expressed as

{\dot{S}}_{e} = \frac{\partial S_{e}}{\partial h} \dot{h} + \frac{\partial S_{e}}{\partial y_{s}} {\dot{y}}_{s} + \frac{\partial S_{e}}{\partial x_{s}} {\dot{x}}_{s} + k_{s} {\dot{P}}_{b}

(4)

where

\partial S_{e} / \partial h

is the effective area change rate related to the height;

\partial S_{e} / \partial y_{s}

is the effective area change rate related to the lateral displacement;

\partial S_{e} / \partial x_{s}

is the effective area change rate related to the longitudinal displacement;

k_{s}

is the effective area change rate related to pressure;

S_{e}

is the effective area, m².

Relative pressure of the main chamber $P_{0}$ , which is expressed as

P_{0} = P_{b} - P_{a}

(5)

where

P_{a}

is the standard atmospheric pressure.

The reaction force of the main chamber $F_{z h i}$ , which is expressed as

F_{z h i} = P_{0} \times S_{e}

(6)

Since the additional chamber is rigid, the volume

V_{t}

is a fixed value. According to equation (1), the absolute pressure of the additional chamber can be obtained

{\dot{P}}_{t} = n P_{t} \frac{{\dot{m}}_{t}}{m_{t}}

(7)

where

P_{t}

is the absolute pressure of the additional chamber, Pa;

{\dot{m}}_{t}

is the total gas mass flow through the differential valve and orifice;

m_{t}

is the total mass of the additional chamber gas, kg.

Orifice model

The orifice is an important damping structure which restricts and hinders the gas flow between the main chamber and the additional chamber. Based on hydrodynamics theory, the gas flow through the orifice is subject to the pressure difference between the main and additional chambers. Due to fluid friction and inertial forces, the effective flow area of the orifice needs to be considered in the calculation. Q represents the gas mass flow through the orifice²⁹

Q = {\begin{cases} β S P_{\max} \sqrt{\frac{2 n}{R T (n - 1)} {(\frac{P_{dn}}{P_{up}})}^{\frac{2}{n}} [1 - {(\frac{P_{dw}}{P_{up}})}^{\frac{n - 1}{n}}]} & \frac{P_{dn}}{P_{up}} > 0.518 \\ β S P_{\max} {(\frac{2}{n + 1})}^{\frac{1}{n - 1}} \sqrt{\frac{2 n}{R T (n + 1)}} & \frac{P_{dn}}{P_{up}} < 0.518 \end{cases}

(8)

where

P_{dn}

and

P_{u p}

is the downstream and upstream pressure at both ends of the orifice, Pa; 0.581 is the critical pressure ratio;

β

is a mathematical function,

β = s g n (P_{2} - P_{1})

; T is the maximum temperature of the chamber, K; S is the effective circulation area,

S = S_{0} η

, Where

η

is the shrinkage coefficient between 0.6 and 0.8 and

S_{0}

represents the theoretical orifice area; R is the universal air constant,

R = 8.31 J / (m o l \cdot K) .

Emergency rubber spring model

When the air spring is broken for other reasons, the top plate directly contacts the emergency rubber spring to prevent rigid contact between the vehicle body and the bogie. Under the regular operation of metro vehicles, emergency rubber springs have little effect. The supporting effect of the emergency rubber spring can be considered only when the air spring fails.

The rubber balloon is connected in series with the emergency rubber spring during regular work, and emergency rubber spring stiffness can be almost ignored. When the air spring fails, the stiffness of the emergency rubber spring can be linearised. The vertical support reaction force of an emergency rubber spring is calculated as follows³⁰

F_{r} = {\begin{cases} 0 h < H_{r} \\ (h - H_{r}) \cdot K_{r} + \dot{h} \cdot C_{r} h > H_{r} \end{cases}

(9)

where

H_{r}

is the gap between the top plate and lower plate, equal to the air spring height, m;

K_{r}

is the equivalent stiffness, N/m;

C_{r}

is the equivalent damping,

N \cdot s / m

A transverse nonlinear model of air spring

Analytical geometry is used to study the lateral behaviour of the air spring. When the lateral force pushes the air spring, the left and right rubber balloons deform, affecting the contact area between the car body and the air spring. The metro vehicle equips the rolling lobe air spring, and rubber balloon friction is also generated during lateral movement.

Figure 2 shows that when a right lateral force pushes the air spring, the centre of the left rubber balloon arc rises from the O_A point to the O_B point. The centre of the right rubber balloon arc drops from the O_C point to the O_D point. The deformation of the rubber balloon can be assumed to be projected into an oval area. The rubber balloon will bend when the transverse force pushes the air spring. Where $Δ h_{c}$ is the variation in vertical displacement when the lateral displacement is $y_{s}$ , m; $R_{h}$ is the effective radius of the air spring, m; $φ$ and $θ$ are the inner and outer corners of the rubber balloon, respectively.

Figure 2.

Transverse deformation.

The lateral nonlinear model contains two parts: one is the aerodynamic force obtained by multiplying the air spring pressure and the effective area $F_{a e r}$ ; The second is the friction of the rubber balloon $F_{f}$ , as follows

{\begin{cases} F_{h} = F_{a e r} + F_{f} \\ F_{a e r} = π R_{h} Δ h_{c} P_{0} \end{cases}

(10)

where

P_{0}

is the relative pressure of the main chamber, Pa; since

Δ h_{c}

is linked to the internal angle

φ

and outer corner

θ

of the rubber balloon, the correlation function can be obtained²¹

\frac{d Δ h_{c}}{d y_{s}} = u (φ, θ) = \frac{(π + φ + θ) \cos φ \cos θ - \sin (φ + θ)}{2 + 2 \cos (φ + θ) + (π + φ + θ) \sin (φ + θ)}

(11)

In order to study the rule of

φ

and

θ

when lateral displacement

y_{s}

changes, further analysis can be obtained

Δ h_{c} = R_{h} \cos θ Δ θ - \sin θ Δ r

(12)

Δ r = \frac{R Δ θ}{π + φ + θ}

(13)

where

Δ r

is the radius change of a single rubber balloon.

It is assumed that the circular arc deformation of the rubber balloon on both sides is the same, and the internal and outer angles change equally. Substitute equation (12) and equation (13) into equation (11) is as follows

{\begin{cases} Δ θ = \frac{(π + φ + θ) u (φ, θ) d y_{s}}{R_{h} (π + φ + θ) \cos θ - R_{h} \sin θ} \\ Δ θ = Δ φ \end{cases}

(14)

Under lateral force, the top plate is in friction contact with the rubber balloon, and the balloon is deformed. A specific friction force and linear elastic force will be generated within the material, and the friction model can express its hysteretic characteristics. General friction models include Maxwell and Coulomb friction models. Due to the low smoothness of the Maxwell model, which is unable to reflect the nonlinear properties of rubber, the Coulomb friction model is adopted in this section, which is expressed as³¹

F_{f} = {\begin{cases} F_{f s} y_{s} = y_{s 0} \\ F_{f s} + \frac{y_{s} - y_{s 0}}{y_{m} (1 - a) + (y_{s} - y_{s 0})} ({F f s {_{s}}_{s 0}}_{f m a x} \\ F_{f s} + \frac{y - y_{s}}{y_{m} (1 + a) + (y_{s} - y_{s 0})} ({F f s {_{s}}_{s 0}}_{f m a x} { \end{cases}

(15)

where

F_{f m a x}

is the maximum friction force generated by an air balloon, N;

y_{m}

is the lateral displacement with frictional force when the friction force is

\frac{1}{2} F_{f m a x}

, m;

F_{f s}

is the original lateral force, N;

y_{s 0}

is the original lateral displacement of the rubber balloon, m;

a

is the ratio of

F_{f s}

F_{f m a x}

Levelling valve model

The four-point levelling configuration refers to a bogie with two sets of levelling valves on the left and right air springs and the auxiliary chambers not connected, that is, with differential pressure installed between the auxiliary chambers. The mathematical model of the four-point levelling configuration of a metro vehicle is expressed as follows

{\begin{cases} c {\dot{z}}_{1} + k z_{1} = k (n_{1} z + s i g n (n_{1} z - z_{1})) u_{v}) \\ w = n_{2} z_{1} \\ n_{1} = \frac{l_{2}}{l_{1}} \\ n_{2} = \frac{l_{4}}{l_{3}} \end{cases}

(16)

where

z

is the vertical displacement of the air spring, m; k is the levelling valve stiffness, N/m; c is levelling valve damping,

N \cdot s / m

; w is the height variation range where the levelling valve does not work, m;

l_{1}

l_{2}

l_{3}

, and

l_{4}

is the levelling valve stem size, m;

z_{1}

is the relative displacement of levelling valve piston, m;

u_{v}

is the precompression of the buffer spring of the levelling valve, m.

According to the theory of fluid mechanics, the flow rate of air spring during charging and discharging is not only related to the diameter of the orifice but also depends on the pressure ratio. It can be calculated by the following equation (17)³⁰

{\begin{cases} Q_{1} = a_{0} c_{1} {\frac{2 g n}{n - 1} [{(\frac{P_{c}}{P_{b}})}^{\frac{2}{n}} - {(\frac{P_{c}}{P_{b}})}^{\frac{n + 1}{2}}]}^{\frac{1}{2}} \frac{P_{c}}{P_{b}} \sqrt{R T_{1}} \\ Q_{2} = a_{0} c_{2} {\frac{2 g n}{n - 1} [{(\frac{P_{a}}{P_{b}})}^{\frac{2}{n}} - {(\frac{P_{a}}{P_{b}})}^{\frac{n + 1}{2}}]}^{\frac{1}{2}} \sqrt{R T_{2}} \end{cases}

(17)

where

a_{0}

is the flow area of the pipeline hole, m²;

c_{1}

is the flow coefficient of the pipeline hole when inflated;

c_{2}

is the flow coefficient of the pipeline hole when exhausted;

P_{c}

is the absolute pressure in the air cylinder, Pa;

T_{1}

T_{2}

, respectively, the maximum temperature on both sides of the flow hole during the charging and discharging, K.

Q_{1}

Q_{2}

are the gas flow of the levelling valve during the charging and discharging, kg/s.

Differential valve model

The differential pressure valve is a safety device that ensures the pressure difference between the air springs on both sides of the bogie does not exceed the specified value. The differential valve will automatically open when the pressure difference is too large. The gas flow to and from the additional chamber ${\dot{m}}_{t}$ is determined by the orifice and the differential pressure valve as follows

{\dot{m}}_{t} = {\dot{m}}_{j} + {\dot{m}}_{y}

(18)

where

{\dot{m}}_{j}

is the gas flow through the orifice, kg/s, and

{\dot{m}}_{y}

is the gas flow through the differential valve, kg/s.

The differential valve model can adopt equation (8) of the orifice for similar calculation. The differential pressure on rail vehicles is mainly in the 0.1 ∼ 0.2 MPa.³²

Co-simulation

The research object is the motor train of a metro vehicle, and some parameters are shown in Table 1. AW0 indicates an empty state, and AW3 indicates an overload state. Simpack creates the vehicle dynamics model, and the air spring model is built with Simulink. Figure 3 shows co-simulation.

Table 1.

Partial parameters of a metro vehicle.

Name	Value	Unit
Maximum operating speed	120	km/h
Body mass	AW0(22400)/AW3(49000)	kg
Distance between bogie centres	15.7	m
Mass between primary and secondary suspension	4015	kg
Wheelset mass	1710	kg
Vertical stiffness of primary suspension system	1.3	kN/mm
Lateral/longitudinal stiffness of primary suspension system	1.1	kN/mm
Secondary air spring transverse span	2050	mm
Secondary vertical damping half of transverse span	1250	mm

Figure 3.

Co-simulation diagram.

In real-time, Simpack transmits the working height and lateral and longitudinal displacement data to the three-dimensional air spring model in Simulink. The vertical, lateral, and longitudinal support forces are calculated in Simulink and then transmitted to Simpack to calculate the real-time movement of metro vehicles, generating new working height data and lateral and longitudinal displacement data. The two exchange data in the real-time cycle until the end of the simulation.

Model verification

Vertical and lateral nonlinearity were tested to verify the model’s correctness. Table 2 shows the stiffness, pressure and volume parameters of air springs under different loads provided by a CRRC company.

Table 2.

Actual data of air spring.

Loading conditions	Load (kN)	Vertical stiffness (kN/m)	Lateral stiffness (when the lateral displacement is 30 mm) (kN/m)	Relative pressure of air spring (kPa)	Air balloon volume (L)
AW0	56	233	128	262	36.5
AW3	122.5	450	168	506	42.5

Firstly, the vertical performance of the air spring is verified. Apply 56 kN, 122.5 kN, and 10 mm amplitude excitation to the air spring, as shown in Figures 4(a) and (b). Then, lateral performance is verified. The load is applied, as shown in Figures 4(c) and (d).

Figure 4.

Variation in load-displacement and stiffness-displacement curves (a) Vertical load-displacement (b) Vertical stiffness-displacement (c) Lateral load-displacement (d) Lateral stiffness-displacement.

Figures 4(a) and (b) show that the variation in load-displacement curves of air springs under 56 kN load and 122.5 kN load are consistent with the vertical stiffness in Table 2. The vertical stiffness under 56 kN load is 233.8 kN/m, and the error is 0.34% compared with the given parameter 233 kN/m. The vertical stiffness under 122.5 kN load is 450.8 kN/m, and the error is 0.17% compared with the given parameter 450 kN/m, which is approximately equivalent to what the real data shows.

Figures 4(c) and (d) shows that the variation in load-displacement curves of air springs under 56 kN load and 122.5 kN load is consistent with the lateral stiffness in Table 2. When the displacement is 30 mm under 56 kN load, the lateral stiffness is 128.4 kN/m, and the error is 0.31% compared with the given parameter of 128 kN/m. When the displacement is 30 mm under 122.5 kN load, the lateral stiffness is 169.4 kN/m, and the error is 0.83% compared with the given parameter 168 kN/m, which is relatively similar to the real data.

Figure 5 shows the rubber friction model, normal air spring model and air spring with rubber friction model. Rubber friction cannot be ignored. When considering rubber friction, the curve of the air spring model with rubber friction moves to the left, the peak value increases, the motion period decreases, and the stiffness of the air spring increases.

Figure 5.

Lateral reaction force variation diagram(a) 56 kN load (b) 122.5 kN load.

The simulated parameters are similar to the real data, reflecting the model’s vertical and lateral nonlinear characteristics. The simulation proves that the model can be used for subsequent research.

Research on the influence of orifice damping

For the bogies of EMUs or metro vehicles, there are generally two different configurations of secondary vertical damping: one is configured with a vertical damper, which is characterised by good performance but high cost, and the design is slightly complex. The second is the configuration orifice, characterised by constant damping, simple design, and cost-saving.

The running speed of metro vehicles is lower than that of EMUs, and the passenger capacity is ample. In order to study the influence of orifice damping on metro vehicles, the nonlinear critical speed, riding stability, and curve passing performance of three different diameter orifice and vertical dampers are compared. Track excitation is the American sixth track spectrum density. (When the vehicle damper is configured, the orifice is considered maximum, ignoring the influence of orifice damping).

Nonlinear critical velocity analysis

The nonlinear critical speed is a significant reference index for vehicle design, and its value should be greater than the design speed. The excitation is applied to just one line segment when the vehicle travels straight. It is defined as a nonlinear critical velocity when the wheel’s transverse motion is neither divergent nor convergent.

Figure 6 shows that the damping configuration has little impact on the critical speed and meets the actual vehicle speed requirements. The critical speed of the AW3 is more significant than the AW0 condition.

Figure 6.

The effect of damping configuration on the critical speed.

Analysis of straight-line running stability

Considering the design speed and operation conditions, metro vehicles’ acceleration and riding quality indexes are calculated, operating in a straight line between 40 km/h and 120 km/h under the AW0 working condition. The results obtained are maximised and, as shown in Figure 7.

Figure 7.

Impact of secondary vertical damping configuration on vehicle running stability (a) Lateral stability (b) Vertical stability(c) Lateral vibration acceleration of car body (d) Vertical vibration acceleration of car body.

Figure 7 shows the vehicle dynamics performance is worse with the increased running speed. The vertical damping configuration dramatically influences the vertical stability and vertical acceleration, but it is not significant in the lateral. With the decrease in orifice diameter, the vehicle stability index and running quality index are better, and the vertical damper performance is the best. However, all the projects meet the requirements of GB/T 5599-2019 《Specification for dynamic performance assessment and testing verification of rolling stock》in China,³³ and the running stability reaches an excellent level. In sum, a 12 mm orifice damping configuration is the best.

Analysis of curving performance

The spring stiffness is unchanged when the air spring is under static load. When the car body is steering and inclined, the air spring will be stretched or compressed, and the spring stiffness will gradually increase. In order to comprehensively investigate the lateral and vertical features of the air suspension, as per the relevant provisions of 《Code for the design of metro》(GB 50157-2013)³⁴ in China, the vehicle speed adjusted above the actual running speed, the curve running line is set as shown in Table 3. Q/P represents the derailment coefficient, △P/P represents the wheel load reduction rate, and H represents wheelset lateral force.

Table 3.

Curve setting.

Radius of circular curve (m)	Superelevation (mm)	Length of transition curve (m)	Length of circular curve (m)	Speed (Km/h)
800	120	60	150	120
450	120	60	150	100
300	120	70	150	80
250	120	70	150	70

Figure 8 shows that under different curve conditions, the configuration of vertical damping has little effect on the derailment coefficient, wheel load reduction rate and wheelset lateral force of metro vehicles, and the operation stability can meet the requirements of GB/T 5599-2019. Considering the overall cost, it is sufficient to configure the orifice as the main vertical damping without needing the vertical damper.

Figure 8.

Influence of secondary vertical damping configuration on vehicle running safety (a) Derailment coefficient (b) Wheel load reduction rate (c) Wheelset lateral force.

Conclusion

By establishing a complete three-dimensional air spring model, the influence of the orifice diameter and the secondary vertical damper on the dynamic performance of the metro vehicle is studied. The research results are as follows:

(1) Through the simulation test, the accuracy of the nonlinear rolling lobe air spring model is verified, and it has excellent vertical and lateral nonlinear characteristics.

(2) When analysing the damping characteristics of air suspension, it has less impact on the critical speed. It is much better than the design requirement, so the critical speed index can be ignored when analysed.

(3) When the vertical damper is used in metro vehicles, its nonlinear critical speed and running stability index are better than the orifice, and the running stability is the same. The smaller the orifice diameter, the better the running stability of the vehicle.

(4) There is no need to configure the secondary vertical damper of the metro vehicle. The air spring with a smaller diameter orifice is better, ensuring great dynamic capacity and reducing the cost, simplifying the design.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51965016).

ORCID iDs

Xu Hu

Yunhua Huang

References

Huang

. Analysis of the dynamic characteristic parameter of air Spring. J Southwest Univ 2003; 38: 276–281.

Mendia-Garcia

Gil-Negrete Laborda

Berg

, et al. A survey on the modelling of air springs-secondary suspension in railway vehicles. Veh Syst Dyn 2022; 60: 835–864.

Oda

Nishimura

. Vibration of air suspension bogies and their design. Bull JSME 1970; 13(55): 43–50.

Presthus

. Derivation of air spring model parameters for train simulation. Sweden: Lulea University of Technology, 2002. Master's Thesis.

Taylor

Eickhoff

Evans

. A review of modelling methods for railway vehicle suspension components. Veh Syst Dyn 1995; 24: 469–496.

Berg

. A three-dimensional air spring model with friction and orifice damping. Veh Syst Dyn 1999; 33: 528–539.

Spiroiu

. Railway vehicle pneumatic rubber suspension modelling and analysis. Mater Plast 2018; 55: 24–27.

Moheyeldein

Abd-El-Tawwab

Abd El-Gwwad

, et al. An analytical study of the performance indices of air spring suspensions over the passive suspension. Beni Suef Univ J Basic Appl Sci 2018; 7: 525–534.

Toyokawa

Shiozaki

Yoshida

, et al. Simulation technology for air springs of railway systems. SEI Tech Rev 2020; 90: 13–16.

10.

Chang

. Dynamic model of an air spring and integration into a vehicle dynamics model. Proc Inst Mech Eng - Part D J Automob Eng 2008; 222: 1813–1825.

11.

Wei

. Simulation on polytropic process of air springs. Eng Comput 2016; 33: 1957–1968.

12.

. A three-dimensional coupled dynamics model of the air spring of a high-speed electric multiple unit train. Proc Inst Mech Eng - Part F J Rail Rapid Transit 2017; 231: 3–18.

13.

Mazzola

Berg

. Secondary suspension of railway vehicles - air spring modelling: performance and critical issues. Proc Inst Mech Eng - Part F J Rail Rapid Transit 2014; 228: 225–241.

14.

Bruni

Vinolas

Berg

, et al. Modelling of suspension components in a rail vehicle dynamics context. Veh Syst Dyn 2011; 49(7): 1021–1072.

15.

Doquier

. Multiphysics modelling of multibody systems application to railway pneumatic suspensions. Belgium: Université Catholique de Louvain, 2010. PhD Thesis.

16.

Guo

Chen

, et al. Theoretical modelling and experimental analysis of the vertical stiffness of a convoluted air spring including the effect of the stiffness of the bellows. Proc Inst Mech Eng - Part D J Automob Eng 2017; 232: 547–561.

17.

Pereira

Morais

. The simulation of an automotive air spring suspension using a pseudo-dynamic procedure. Appl Sci Basel 2018; 8: 1049.

18.

Renno

Strano

Terzo

. Development and validation of an air spring multiphysical model. Eng Lett 2017; 25: 176–182.

19.

Yang

Chen

C-H

Chen

Y-L

, et al. Finite element analysis of an air spring for automobile suspension. J Beijing Univ Chem Technol 2004; 31: 105–109.

20.

Zhuang

Xiaolei

Rongli

, et al. Air spring modelling and vehicle dynamics analysis based on fractional calculus theory. J China Railw Soc 2021; 43: 67–76.

21.

Zhuang

Xiaolei

Rongli

, et al. Dynamic modelling and application of lateral characteristics for high-speed train air spring. J Vib Meas Diagnosis 2021; 41: 904–912.

22.

Liu

, et al. Research on the vertical stiffness of a rolling lobe air spring. Proc Inst Mech Eng - Part F J Rail Rapid Transit 2015; 230: 1172–1183.

23.

Xingwen

Maoru

Haohao

. Influences of air spring models on dynamics performance of railway vehicle. J Traffic Transport Eng 2013; 13: 54–59.

24.

Mingyu

Xuebing

Hang

, et al. Identification of key nonlinear parameters and research on dynamic characteristics of air spring dynamic stiffness model. J Mech Eng 2022; 58: 88–96.

25.

Junjie

Zhihong

Jianghua

, et al. Study on modelling and dynamic characteristics of air spring with throttling damping orifice and auxiliary chamber. J Mech Eng 2017; 53: 166–174.

26.

Quaglia

Sorli

. Air suspension dimensionless analysis and design procedure. Veh Syst Dyn 2003; 35: 443–475.

27.

Alonso

Giménez

Nieto

, et al. Air suspension characterisation and effectiveness of a variable area orifice. Veh Syst Dyn 2010; 48: 271–286.

28.

Xuan

. Research on suspension parameters optimisation of the bogie for 120 km/h standard A-type metro. Master's Thesis. China: Southwest Jiaotong University, 2022.

29.

Jiasheng

. Research on dynamic characteristics of air spring with the auxiliary chamber. China: Nanjing Agricultural University, 2009. PhD Thesis.

30.

Zhuang

. Research of high-speed EMU's air spring dynamics characteristics and its failure modes. China: Southwest Jiaotong University, 2015. PhD Thesis.

31.

Berg

. A nonlinear rubber spring model for rail vehicle dynamics analysis. Veh Syst Dyn 1998; 30: 197–212.

32.

Guijun

. Description of air-tightness of different pressure adjustment valves and the pressure difference test. Rolling Stock 2020; 58: 13–18.

33.

Standardization Administration of China . GB/T5599-2019: specification for dynamic performance assessment and testing verification of rolling stock. Beijing, China: Standardization Administration of China, 2019.

34.

Standardization Administration of China . GB50157-2013: code for design of metro. Beijing, China: Standardization Administration of China, 2013.

Modelling of air springs for metro vehicles and studies on the effect of orifice damping

Abstract

Keywords

Introduction

Three-dimensional dynamic models of air spring

Main chamber and auxiliary chamber models

Orifice model

Emergency rubber spring model

A transverse nonlinear model of air spring

Levelling valve model

Differential valve model

Co-simulation

Model verification

Research on the influence of orifice damping

Nonlinear critical velocity analysis

Analysis of straight-line running stability

Analysis of curving performance

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References