Influence of Passenger Capacity on Fatigue Life of Gearbox Suspender of the Traction Transmission System in Urban Railway Vehicles

Abstract

Fatigue damage is the most dangerous failure behavior for gearbox suspenders in urban railway vehicles, and passenger capacity is crucial to the dynamic load characteristics of the traction transmission system. Therefore, in this paper, a dynamic model of the motor car is established, and a numerical simulation is carried out under different speeds and curve radii to investigate the effect of passenger capacity on fatigue life. The research results show that passenger capacity is an essential factor affecting the fatigue life of suspenders. As the vehicle runs at an average speed, the fatigue life of the suspender is 1.07 × 10⁶ km when the passenger capacity is 120 people; when there are 240 people, the fatigue life reduction is 60%, while it is 86% at 339 people and 92% at 389 people. The per capita fatigue damage under a straight line is 7.27 × 10⁻¹⁰ at 20 km/h but 1.23 × 10⁻⁸ at 60 km/h. The per capita fatigue damage under a curved line is 7.18 × 10⁻⁹ in the 600 m curve but 9.00 × 10⁻⁹ in the 400 m curve. It can be concluded that the effect of speed is more significant than the curve radius. This research achievement can provide theoretical support for vehicle design and maintenance decisions.

1. Introduction

Rail transit has rapidly developed in China. Taking Beijing as an example, more than 20 metro lines have been developed, with a total mileage of nearly 800 km and a daily passenger traffic volume of more than 10,000,000 persons. The considerable passenger traffic volume will significantly reduce the fatigue life of various parts of urban railway vehicles. The gearbox is an important part of the railway vehicle. The gear transmission unit transfers the motor torque force to the wheel axle, using gear engagement action to carry the train. However, the gearbox suspender used to connect the gearbox to the bogie frame is associated with the vehicle’s safety. Any gearbox suspender failure will threaten the train operation’s safety and have unpredictable consequences.

In recent years, many scholars have researched the dynamics of the traction transmission system. Sun et al. [1,2] obtained the dynamics of the transmission system using a numerical method. The analyzed vibration characteristics showed that the forces acting on the gears become larger when the operation speed of the vehicle increases. Li et al. [3,4] built a gearbox model to predict the gearbox’s state over time and to ensure strength and stiffness. Fomin et al. [5,6] studied the vertical load using mathematic modeling, which helped to evaluate vehicles and ensure safety. Liu et al. [7,8] studied the problem of difficulties in extracting planetary gearbox fault characteristics. The resonant frequency point of noise generation was determined by removing the first six orders of constrained modal vibration and the inherent frequency of the box. Li et al. [9,10,11] studied the effects of different suspension modes and suspension parameters on the vehicle system’s smoothness and safety, as well as the system’s natural frequency. Li et al. [12] analyzed the vibration acceleration characteristics of the input side of the gearbox in the Simulink platform under the operating conditions of a varying harmonic torque amplitude in the traction transmission system. Qi et al. [13] used the panel acoustic contribution analysis and response surface methodology to conduct a structural-borne acoustics analysis and ensure the multi-objective optimization of the gearbox. Hajnayeb et al. [14,15] proposed a new approach to determine the effect of random manufacturing errors on the vibrations measured on the bearings of mating gears, and to compare the vibration characteristics of railway vehicle axlebox bearings with inner/outer race faults. Wang et al. [16,17] proposed a railway-vehicle gear-system model to investigate the nonlinear dynamics of a time-varying gear system and the profile shifts in the mesh stiffness and dynamic characteristics of spur gears. In addition, the gear system under traction/braking conditions was analyzed to improve the gear system’s stability and the running stability and safety of railway vehicles [18,19]. Ye et al. [20] developed a deep learning model, OORNet, which contributes to the maintenance decision-making regarding wheelsets and clarifies their triggering and evolution mechanisms.

However, fatigue studies on vehicle components have also been given attention. Fang et al. [21,22,23] used the rigid–flexible coupling dynamic simulation method and cross-scale method to analyze the vibration fatigue life of the gearbox structure, providing reference comments for design, detection, and maintenance. Zhang et al. [24,25], by simulating the force condition of the gearbox bracket and using the actual measurement data of the rail line, could predict whether the life of the components could meet the requirements for their use, and could repair or replace the relevant components in advance to ensure operational safety. Zhang et al. [26] analyzed the effects of the combined bearing preload and angular misalignment on the fatigue life of ball bearings and a shaft-bearing system. In the work of Han et al. [27], to deal with a newly developed ferrite and pearlite wheel material named D1, an alternative ordinary state-based peridynamic model for fatigue cracking was introduced due to cyclic loading. Lisowski et al. [28] measured stanchion displacements of overloaded timber trucks under both static and dynamic loads to research fatigue. Li et al. [29] presented a framework for the fatigue evaluation of critical steel bridge details using a multi-scale dynamic analysis of the train–track–bridge system and linear elastic fracture mechanics. Tian et al. [30,31] used the cyclic void growth model and continuum damage mechanics model to predict damage and random vibration. Further, simulation verification was carried out, which can provide a reference for the selection of the life prediction model. Li et al. [32,33] studied the effect of the fatigue notch coefficient and surface roughness for fatigue life by testing different materials. Wang et al. [34,35] simplified the counting process of the rainflow counting method, computed the stress transfer coefficients for the fatigue-critical points in each load spectrum. Zhang [36] processed fatigue test data by fitting different functions to provide a basis for the SN curve equation fitting method.

In addition, there are some studies on gearbox suspenders. Zeng et al. [37,38] optimized the bogie gear box suspender structure using finite element analysis and OptiStruct to create a lightweight design and meet comprehensive performance requirements. Meng et al. [39] took a gearbox suspender of EMU as their research object; the mesh accuracy of the finite element model for the suspender was verified according to the ASME V&V, and the suspender’s static strength was analyzed under ultimate load. Qin [40,41] constructed the suspender device model to calculate a device’s static strength and provide a method for strength analysis during design. Wang [42] studied failure reasons and the improvement scheme through physical detection, numerical simulation, and line measurement based on the fracture problem in the hanging device of a domestic, urban rail gear transmission system that is currently in service. Studies on gear suspenders in the last three years have been relatively few in number. Shi [43] analyzed the gear box boom structure under the working conditions of a 100 kN ultimate tensile and compression load, checking whether the 42CrMo material met the strength requirements. Yang [44] optimized the structure of the gearbox suspension device, and the performances of optimized structure under the extreme condition and fatigue condition were compared using simulation analyses according to the FKM standard. Wu [45] analyzed the causes of fatigue failure of rubber joints to improve the service life of rubber joints of a standard subway gearbox boom, and explained the application scope of the three different types of fatigue study methods. Wang [46] optimized the design of a suspender using free-form optimization to reduce the mass of the suspender by 38.3%, as the safety factor of the original structure of the suspender was too large during the development and design of the suspender. Sun [47] took medium carbon steel and medium carbon alloy steel as the materials of the boom and compared their force conditions. The results show that both meet the requirements, but the fatigue performance of medium carbon alloy steel is stronger; therefore, it is recommended to use medium carbon alloy steel. A tabular literature review of the last three years on gearbox suspenders is shown in

Table 1. Tabular literature review of the last year.

Title	Approaches	Key Parameters
Strength Analysis of Gear Box Boom Assembly [43]	SW software was used to establish the gearbox derrick assembly model; the 3-D model structure was simplified, the model parameters were optimized, and the strength of the gearbox boom was checked through FEA analysis software.	Material: 42 CrMo Maximum stress: 129.3 MPa Safety factor: 7.0
Optimization Design of a Gearbox Suspension Device [44]	We compared the strength of the boom before and after optimization and predicted the fatigue life of major metal components. Finally, we designed a new type of gearbox suspension device.	Bolt preload force: 612 kN. Extreme working condition: 0~1800 kN Fatigue condition: 0~900 kN Compound stress fatigue damage reliability: 0.17
Structural Design and Experimental Research on the Rubber Joints of the Gearbox Boom [45]	New rubber joint was designed. By using finite element analysis software and FEA software, the fatigue resistance of the new structure was optimized.	Fatigue loading times: 1 × 107 Fatigue test load: 40 kN
Optimized Design of Bogie Gearbox Suspender [46]	We optimized the mathematical model and determined the objective function, then reduced the mass of the boom using free-shape optimization.	Load: 27.8 kN Maximum stress: 121.9 Mpa Safety factor: 1.27
Analysis of Material Selection of U-Shaped Gearbox Suspender for Bogie of Urban Railway Vehicle [47]	The suspender was modeled in three dimensions using Pro/Engineer software and subjected to finite element analysis, and then the effects of a material heat treatment process and dimensions were discussed. We determined the material selection based on the Goodman curve.	Ultimate vertical acceleration: 70 g Max. traction torque: 1400 Nm Screw size after machining: Φ39

.

The traction transmission system is the most core technical part of the vehicle. The gearbox and suspender form its key components, so the relevant foreign design is almost in a state of secrecy and domestic research is not sufficient to determine its details. Firstly, the suspender research has been mostly static strength research, and it has determined that static strength can generally meet the requirements and has a large safety margin. However, the fatigue strength is likely not to meet the requirements, one cause of suspender fractures is fatigue damage. Secondly, analysis of load conditions often uses empirical load or a variety of different standards, rather than the actual operation of the vehicle data collection or dynamics simulation data. Thirdly, some studies use fatigue software but do not offer details of the parameters, making it difficult for the reader to understand the rationale behind the choices. Finally, no one has made the connection between fatigue damage and the suspender and passenger capacity, and the reality is that urban railway vehicle suspenders that are overloaded for a long time will break. In summary, a detailed study of the effect of passenger capacity on suspender fatigue life is necessary to obtain the trends in the curve between passenger capacity and suspender damage as well as the magnitude of the effect of different speeds and curve radii. Eventually, the calculated service life can be used to strengthen critical locations and optimize the suspender to enhance its performance.

This paper investigates the effect of passenger capacity on the dynamic load and fatigue life of the traction transmission system gearbox suspender in urban railway vehicles at different speeds and curve radii. Section 2 establishes a dynamic model of urban railway vehicles; Section 3 analyzes the effect of passenger capacity on the dynamic characteristic at different speeds and curve radii; Section 4 discusses the effect of damage on the fatigue life of gearbox suspenders; and Section 5 concludes the paper. Based on the main research content of this paper, the technology route is shown in Figure 1.

2. Urban Railway Vehicle Model with a Traction Transmission System

2.1. Dynamic Modeling

The main transmission process of a train transmission system consists of a gear-engaging transmission and a coupling transmission in the gearbox. Reasonable definitions for the gearbox connection, engaging gear relationship, and coupling transmission description are crucial in establishing a correct dynamic model of a train transmission system. In this paper, to accurately reflect the vibration characteristics of the transmission system, the structure, along with its mass and moment of inertia, are used to represent inertia characteristics, and the motion equation is used to express the motion relationship. To fix the gearbox, the reference position of the gearbox was located using a hinge in the SIMPACK software, and its bearing connection with wheel axles and pinion shafts and its connection with rubber pads in the frame were defined by the force element in SIMPACK.

The gearbox is a load-bearing component of the transmission system. The excitation generated while the vehicle runs is directly transmitted to the gearbox, affecting its vibration acceleration. When the gearbox was modeled, an elastic force element described the connection between structures to show the interaction between vehicle parts and the transmission effect of gearbox vibration. For the gearbox, the gearbox was hinged using the No. 7 General Rail Track Joint in SIMPACK, and its relative position in the vehicle system was determined. The excitation input that affects the gearbox vibration is mainly the position at which the gearbox is connected to other structures. The connection between a gearbox and other structures mainly includes the gearbox suspender connection between the gearbox and frame, the bearing connection between the pinion shaft and gearbox, and the bearing connection between the wheelset and gearbox and gearwheel. These connections were defined by force elements No. 5 Spring-Damper parallel Cmp and No. 43 Bushing Cmp in SIMPACK, and the parameters were defined according to the characteristics of the connections. In the gearbox model, the gearbox suspender was simplified as a spring damping force element, which can reflect the loading situation of the suspender and more accurately reflect the transmission characteristics and effects of each gearbox structure. The engagement between the gearwheel and pinion was modeled by force element No. 225 in the SIMPACK software, which can accurately describe the tooth surface modification, tooth backlash, and time-varying characteristics of engaging stiffness in the gear transmission model. The profile parameters and three-dimensional geometric characteristics of the gear pair required by this force element are shown in

Table 2. Parameters of gear.

Modulus/mm	Teeth Number of Pinion	Teeth Number of Gearwheel	Helix Angle/°
5	22	133	20

for certain urban railways in China. As a key structure for transmitting torque from the motor to the gear, coupling connects the motor and gear and plays a role in transmitting and attenuating the vibrations between the motor and the gearbox. Based on the structure and function of the coupling, the coupling model was established by force element No. 43 in the SIMPACK software, and a reasonable torsional stiffness was set to ensure the torque transmission function of the coupling.

Many parts are needed in the modeling process of urban railway vehicles, and the acting points and forces between parts are very complicated. In addition, the multi-rigid-body dynamic system has strong nonlinearity after establishing urban railway vehicles, which is mainly manifested in the wheel–rail relationship and the primary and secondary shock-absorption suspension. The wheel–rail relationship involves the change in wheel–rail creep force and wheel–rail contact point, and this nonlinearity also makes modeling difficult. To simplify the vehicle modeling to approach the actual situation and facilitate subsequent analysis, a dynamic model of an EMU as a multi-rigid-body vehicle was established based on the analysis of certain urban railway vehicles in China. The main parameters of the vehicle are shown in

Table 3. Parameters of urban railway vehicles.

	Numerical Value	Unit
Center distance of bogie		mm
Wheelbase	2200	mm
Transverse span of wheel rolling circle	1493	mm
Wheel rolling circle diameter	840	mm
Distance between backs of wheel flanges	1353	mm
Wheel profile	LM
Vehicle body mass	40.8	t
Height of vehicle centroid from the rail surface	1756	mm
Frame mass	3188	kg
Height of frame centroid from the rail surface	570	mm
Mass of axle box	85.367	kg

.

The vehicle mainly consisted of a car body, two frames, four sets of wheelsets, four gears, and eight axleboxes. The car body and wheelsets were treated as rigid bodies with six degrees of freedom. Then, the primary and secondary suspension systems and wheel–rail contact were improved. The rubber springs of the primary suspension system were connected to the wheelset with the bogie. The air springs in the secondary suspension system were connected to the bogie and the car body. The spring-damper unit describes the restraining effect of the suspension systems on these three parts; that is, they act on related connecting parts in the form of load. The nonlinear characteristics were considered for the wheel–rail contact relationship and damper damping effect. LM treads were used for wheels and standard steel rails of 60 kg/m were used for tracks, with a track gauge of 1435 mm. The ballastless track spectrum was adopted for the track irregularities. Finally, the transmission system established in the previous section was added as a substructure. A total of 49 degrees of freedom were obtained for the motor train. The vehicle dynamic model is shown in Figure 2.

2.2. Topology Relations and Validation of Model

The topological relations diagram of the kinetic model can visually express the motion relationship between the modeled rigid body and the force action relationship. According to the structural parameters of the research object in this paper, the topological relations diagram of the dynamic model is shown in Figure 3.

The equation of motion of the gearbox suspender as an under-car device contains two main vertical degrees of freedom, which can be expressed as [48]:

$$\left.\begin{array}{l}{M}_e{\ddot{Z}}_e={\displaystyle \sum _{i=1}^2{f}_{ei}}\\ I_e{\ddot{\theta }}_e={\displaystyle \sum _{i=1}^2{f}_{ei}}(X_{ei}-\frac{x_{e1}+x_{e2}}{2})\end{array}\right\}.$$

(1)

To verify the accuracy of the dynamic vehicle model, the simulation results were compared with the actual results of the vehicle gearbox vibration acceleration. The actual results come from the gearbox vibration acceleration of a city railway vehicle. This B-type vehicle line is one of the main lines of an urban railway network and it is a busy line with a high passenger flow, with an average daily passenger flow of 400,000 or more. There are 24 stations on the line with a spacing of 1.6 km running at operation speed. In terms of passenger flow, the number of passengers is slightly less than AW2, with 60 kg per capita. The simulation conditions were set to run the model at 40 km/h with 240 passengers, and a sampling frequency of 2000 Hz. A comparison of the gearbox vibration acceleration is shown in Figure 4.

The actual and simulation results of gearbox vertical vibration acceleration are below 20 g, the results of lateral vibration acceleration are below 5 g, and the results of longitudinal vibration acceleration are below 4 g. The simulated results are slightly smaller compared to the actual results, but the differences are kept within a reasonable range, and fluctuate within a certain range. As the actual line conditions are more complex, the gearbox vibration is more influenced by external conditions, and the vibration amplitude, which is slightly larger than the simulation results, is acceptable. Therefore, the vehicle dynamic model established in this paper has a certain degree of accuracy.

2.3. Simulation Conditions

The dynamics of urban railway vehicles with different passenger capacities are analyzed in this paper. The change in vehicle body mass and moment of inertia directly reflect the passenger capacity of different models. The axle load of a Type-B urban railway vehicle is 14 tons, the mass of an empty vehicle is 21,920 kg, and the bogie frame’s mass is 2250 kg.

Suppose the designed passenger capacity AW2 is taken as a criterion for a train. In that case, the average congestion (train load) is equal to the ratio of the actual number of passengers to the fixed number of passengers. As the passenger number of a train is equal to AW2, there are six persons per square meter, and the average crowding degree is equal to one. When the passenger number of a train is equal to AW3, there are 9 persons per square meter, and the average crowding degree is approximately 1.4. When the designed limit of the train is reached, the average crowding degree is approximately 1.6. The full passenger capacity for certain urban railway vehicles in Beijing is 42 persons. The fixed number of passengers is 240 persons, and the overcapacity is 339 persons. To study the effects of passenger capacity on the gearbox suspenders, five groups of data were simulated. Namely, the passenger capacity for empty vehicles was 0, the passenger capacity for a crowding degree of 0.5 was 120 persons (2–3 persons per square meter), the passenger capacity for a fixed number was 240 persons, the passenger capacity for overloaded hours was 339 persons, and the passenger capacity for special rush hours was 389 persons (10–11 persons per square meter). A weight of 60 kg per person was adopted.

Analyzing the running conditions of urban railway vehicles is complicated, and vehicles frequently enter and leave stations to stop or start a trip. The actual running speed of urban railway vehicles does not exceed 80 km/h, and the average running speed is approximately 40 km/h. Therefore, to study the effects of vehicle passenger capacity on the load and service life of gearbox suspenders under different working conditions, different speeds of 20 km/h, 40 km/h, and 60 km/h under straight-line conditions, and different curve radii of 400 m, 500 m, and 600 m under the fixed speed of 40 km/h conditions, were set. The structure of metro line curves was selected according to the super-elevation/transition curve length of several common metro line curves, specified in Section 6.6.2 of the Code for Design of Metro (GB 50157-2013). The metro line structure is specified as follows: the length of the turn-in straight line section is 800 m, the turn-in transition curve is 20 m, the circular curve is 360 m, the exit transition curve is 20 m, and the exit straight line section is 800 m. The super-elevations are 50 mm, 40 mm, and 30 mm, respectively.

Various load conditions were simulated with a sampling frequency of 1000 Hz through the SIMPACK software. In the post-processing software, the variation history of the vertical dynamic load on the gearbox suspenders over time was derived and recorded in the documents for data statistics and analysis.

3. Effect of Passenger Capacity on Dynamic Load for Gearbox Suspender

The running distance between two stations was sampled at a sampling frequency of 1000 Hz via the SIMPACK software, which was set at 2 km. Taking the straight-line conditions of 40 km/h, for example, the load on the gearbox suspender was random, and the varied history of the loads over time for the passenger capacities of 0 people, 120 persons, 240 persons, 339 persons, and 389 persons is shown in Figure 5.

3.1. Vibration Indicators of Interest

Time-domain signals can be classified into dimensional and dimensionless eigenvalues according to the dimension criteria. The dimensional eigenvalues analyzed in this paper include maximum value, minimum value, mean value, standard deviation, and RMS. The dimensionless indicator is a kurtosis factor.

The maximum and minimum values determine the maximum dynamic load in the load history. The peak-to-peak value represented by the difference between them describes the variation range of the load and affects the structural damage.

The maximum value reflects the maximum stress of the gearbox suspender when it is under pressure, which can be expressed as:

$$x_{\max }=\max [x(i)].$$

(2)

The minimum value reflects the maximum stress of the gearbox suspender when it is under pressure, which can be expressed as:

$$x_{\min }=\min [x(i)].$$

(3)

The mean value is the average of signals and characterizes the static load level in the load spectrum, which can be expressed as:

$${\mu }_{\mathrm{x}}(t)=\frac{1}{N}{x}_i(t).$$

(4)

The standard deviation, also called mean square error (MSR), is the arithmetic square root of the mean value of the square value of the difference between each signal value and the mean value, and it can reflect the discrete degree of a signal set, which can be expressed as:

$${\sigma }_{\mathrm{x}}(t)=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left[x_i(t)-{\mu }_x(t)\right]}^2}}.$$

(5)

The root mean square (RMS), also called quadratic mean, is obtained by summing the squares of all values, averaging them, and then squaring them, which can be expressed as:

$$X_{\mathrm{rms}}=\sqrt{\frac{{\displaystyle \sum _{i=1}^N{x}_{\mathrm{i}}{(t)}^2}}{N}}=\sqrt{\frac{x_1{}^2+x_2{}^2+\cdot \cdot \cdot +x_{\mathrm{N}}{}^2}{N}}.$$

(6)

The kurtosis factor is a statistic operator describing the steepness of all the values’ overall distribution patterns. This statistic operator needs to be compared with the normal distribution. When the kurtosis factor is equal to 0, this means that the overall data distribution is as steep as the normal distribution. When the kurtosis factor is greater than 0, the overall data distribution is steeper than the normal distribution and has a sharp peak. When the kurtosis factor is less than 0, the overall data distribution is relatively flat compared to the normal distribution, and has a flat peak. The greater the absolute value of kurtosis, the greater the difference between its distribution patterns’ steepness and normal distribution. The kurtosis factor can be expressed as:

$$k=\frac{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left[x_i(t)-{\mu }_x(t)\right]}^4}}{{\left\{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left[x_i(t)-{\mu }_x(t)\right]}^2}\right\}}^2}-3.$$

(7)

3.2. Straight Line Condition

The vibration indicators of urban railway vehicles at speeds of 20 km/h, 40 km/h, and 60 km/h under straight-line conditions are shown in Figure 6.

Both the maximum and minimum values increased with passenger capacity and vehicle speed. When the vehicle was not loaded, the maximum value was not more than 15 kN, and the minimum value was not less than −20 kN. When the vehicle was under super-normal load, the maximum and minimum values increased by 5 kN–8 kN. When the speed was 60 km/h, the variation rate of maximum and minimum values was the largest with the increase in passenger capacity, which shows that speed can enhance the increase in maximum and minimum values. The difference between the maximum and minimum values is the peak-to-peak value, and the maximum peak-to-peak value was 4.26 kN at 40 km/h, which is 11.7 kN larger than the peak-to-peak value at no load.

The mean value at different speeds had little difference, which fluctuated around −2000 N. The greater the passenger capacity of a vehicle is, the larger the absolute value of the mean value will be.

Both the maximum and minimum values increased with passenger capacity and vehicle speed. When the vehicle was not loaded, the maximum value was no more than 15 kN, and the minimum value was no less than −20 kN. When the vehicle was under a super-normal load, the maximum and minimum values increased by 5 kN–8 kN. When the speed was 60 km/h, the variation rate of maximum and minimum values was the largest with the increase in passenger capacity, which shows that speed can enhance an increase in maximum and minimum values. The difference between the maximum and minimum values is the peak-to-peak value, and the maximum peak-to-peak value was 4.26 kN at 40 km/h, which is 11.7 kN larger than the peak-to-peak value at no load.

The mean value at different speeds showed little difference, fluctuating at around −2000 N. The greater the passenger capacity of a vehicle, the larger the absolute value of the mean.

Both standard deviation and RMS had a similar variation trend. Figures show that the discrete difference at different speed levels was approximately 500 N, which increased with passenger capacity. The variation in RMS was identical to the standard deviation, and the maximum RMS at 40 km/h was 4.41 kN, which is 1.05 kN higher than that at no load. For these two eigenvalues, the difference in speeds only affected the increase, while the difference in passenger capacity not only increased the data but also accelerated the change rate.

K = 3 is defined as a distribution curve with normal kurtosis (zero kurtoses) and k = K-3. When the standard deviation σt is less than the standard deviation in the normal state, the dispersion of the observed values is small, and K increases. The height of the peak in the normal distribution curve was higher than that of the normal distribution curve, so it is called positive kurtosis. Under the straight-line condition, kurtosis factors were all greater than 0, with a maximum value of 1.014 and minimum value of 0.468. Figures show that the K value changed little for the passenger capacity of 0–240 when the speed was low, and the distribution of the suspender load was mostly concentrated near the mean value. When the passenger capacity was greater than the rated value, the K value curve rapidly decreased, and the distribution of the load on the suspender was more biased on both sides.

3.3. Curved Line Condition

The vibration indicators of urban railway vehicles for different curve radii of 600 m, 500 m, and 400 m at a speed of 40 km/h are shown in Figure 7.

The maximum and minimum values increased with the increase in passenger capacity and the decrease in the curve radii. However, the curve radii only resulted in a smaller increment for the maximum and minimum values. When the passenger capacity was 389 persons, the maximum value increased from 19.8 kN under a curve radius ranging from 600 m to 21.2 kN under a curve radius of 400 m, while the minimum value changed from −23.8 kN to −24.9 kN. The peak-to-peak value under the curve radius of 400 m was 32.8 kN, which is 5.89% higher than that under the straight-line condition when the vehicle was not loaded.

Under different curve radii, the mean value showed little difference, ranging between −1.87 kN and −2.5 kN. The larger the vehicle passenger capacity, the larger the absolute value of the mean value. However, when observing the mean value of different curve radii for the same passenger capacity, its value mostly stayed the same.

Both standard deviation and RMS had a similar variation rate. The discrete difference between the different curve radii was approximately 50 N, and the smaller the curve radius, the greater the standard deviation. The variation in the RMS was slightly larger than the standard deviation, and the maximum RMS was 4.58 kN when the curve radius was 400 m, which is 0.29 kN higher than that under the straight-line condition. Standard deviation and RMS were less affected by the difference in curve radius.

The kurtosis factor increased with the decrease in the curve radius and was greater than 0, and the load distribution was similar to that under straight-line conditions at a speed of 40 km/h. The difference in kurtosis factor under different curve radii for a smaller passenger capacity was greater than that for a larger passenger capacity; the greater the passenger capacity, the smaller the difference in the smoothness of the distribution curve of the load on the gearbox suspender.

4. Effect of Passenger Capacity on Fatigue Life of Gearbox Suspender

Although there is a strength standard for power bogie frames (UIC 615-4), there is no relevant standard for suspenders. According to the Code for Design of Metro (GB 50157-2013) and the General technical specifications for metro vehicles (GB/T 7928-2003), the design life of the gearbox suspender should be no fewer than 10 years and the operating mileage should be no less than 1.2 × 10⁶ km. Vehicle overhaul and the repair cycle are shown in

Table 4. Vehicle overhaul and repair cycle.

Repairing Course	Time Interval (Month)	Distance Traveled (104 km)	Inspection Time (Days)
Monthly inspection	1	1	0.5
3-month inspection	3	3	2
Scheduled repair	15	15	7
Un-wheeling repair	60	60	20
Overhaul	120	120	35

. Calculating the fatigue life of a suspender with different passenger capacities can help to determine whether its life meets the design requirements under different working conditions and facilitate subsequent optimization design.

4.1. Fatigue Life Evaluation Approach

In fatigue strength design, the load on spare parts is divided into the load with certain regularity and the load with uncertain amplitude and a frequency that changes over time; the latter is the random load. The wheel-rail force and car body force on the bogie frame of the rail vehicle in operation have no fixed regularity, so the frame bears a continuous random load. To analyze and calculate the fatigue performance of the collected load spectrum, it is necessary to count the cumulative frequency distribution of the load history using the rainflow counting method. The simulated load-time history was processed into a representative typical load spectrum; this is called spectrum arrangement. The rainflow counting method was programmed using MATLAB. A series of cyclically obtained amplitude data were grouped, and the cumulative frequency of continuous load could be represented.

Based on the finite element principle, the more finely the mesh is divided, the more accurate the results will be. On the premise of not destroying the mesh and node positions, the division can be completed with a hexahedral element. Therefore, the suspender model was meshed using the hexahedral element, with a mesh size of 0.06 mm and element type C3D8R: C denoting a body unit, 3D denoting a three-dimensional unit, 8 denoting an eight-node unit, and R denoting a reduced integral, which is an eight-node linear hexahedral element. This element is the least time-consuming hexahedral element in Abaqus, and the computational accuracy of the hexahedron itself is higher than that of the tetrahedral and wedge element, so C3D8R is the most widely used in the practical application of the body unit. Therefore, it is the object of study in this paper. This model has a total of 57,214 elements and 80,614 nodes. The suspender is made of 45# steel, and the material properties are shown in

Table 5. Material property.

Tensile Strength σb MPa	Yield Strength σ0.2 MPa	Elastic Modulus E N/mm2	Poisson’s Ratio μ
690	490		0.3

.

Since the gearbox suspender was fastened to the frame, full constraint was applied to the upper end of the suspender, a coupling point was set at the location of the spherical bearing at the lower end of the suspender, and a load F was applied, with the direction of the load extending along the line between the upper screw and the lower bore, as shown in Figure 8.

To simplify the fatigue analysis software, the slope of the high-cycle fatigue zone of all steels was set at −0.125. Obtaining more accurate S-N curves is necessary to more accurately calculate the fatigue life of gearbox suspenders.

In a logarithmic coordinate system, the S-N curve usually consists of three segmented line segments, two of which are diagonal lines, while the other is a horizontal line. The two inclined line segments represent low-cycle fatigue and high-cycle fatigue, respectively, while the horizontal line represents fatigue limit (i.e., infinite cycle life). There is no exact demarcation line between low-cycle fatigue and high-cycle fatigue. To conveniently distinguish high-cycle fatigue and low-cycle fatigue, 10³ cycles are usually used as the demarcation line. The curve was close to the level at which the number of cycles reaches 10⁷. No fatigue fracture will occur, even if the number of cycles increases again.

To obtain an empirical S-N curve, it is necessary to determine the stress amplitude σ₁₀₀₀ after 10³ cycles and σ₋₁ after 10⁷ material cycles. An axial tensile and compressive load was applied to the gearbox suspender, and 75% of the material’s tensile strength was usually taken as the value of σ₁₀₀₀.

The cycle life cannot undergo experimentation indefinitely; therefore, the stress amplitude of a 10⁷-cycle load is regarded as the fatigue limit for steel. The fatigue limit σ_be will increase linearly with the increase in tensile strength σ_b, that is, σ_be/σ_b = constant. There is a limit to this proportional relationship, which is called the ultimate critical strength. The fatigue limit stops its growth when the material’s tensile strength exceeds the necessary ultimate strength. The ultimate critical strength of steel is 1400 MPa, and σ_be = 0.5 σ_b when σ_b ≤ 1400 MPa. Therefore, the fatigue limit of 45# steel is σ_be = 345 MPa.

After determining the fatigue limit, σ_be, of materials, estimating the fatigue limit σ₋₁ of the gearbox suspender is necessary. The fatigue limit of the actual members is influenced by many factors related to the material and the members’ geometric shape, size, and surface quality. Hence, the fatigue limit determined by smooth specimens cannot represent the actual fatigue limit of members, and other factors should be comprehensively considered. The fatigue notch factor K_f, affected by the member shape, is generally linearly correlated with the theoretical stress concentration factor K_t. To obtain the fatigue notch factor, the gearbox suspender’s theoretical stress concentration factor was determined first. The theoretical stress concentration factor can be expressed as follows:

$$K_t=\frac{{\sigma }_{max}}{{\sigma }_{non}},$$

(8)

where σ_max means the local maximum stress and σ_non means the nominal stress.

The theoretical stress concentration factor is only associated with the geometric shape of the member and is not affected by materials or other factors. As K_t ≤ 3 and the stress ratio = −1, the following relational expression is satisfied:

$$K_f=B_1{K}_t{}^2+B_2{K}_t+B_3,$$

(9)

where, B₁ = 0.086; B₂ = 0.2544; and B₃ = 0.6806.

Small specimens with d = 7 mm–10 mm are generally used to determine the fatigue limit of materials. The larger the cross-sectional dimension of the member, the lower the fatigue limit will be. For this research object, the fatigue limit σ_be of materials and the fatigue limit σ₋₁ of members satisfy the relational expression:

$${\sigma }_{-1}=\frac{{\sigma }_{be}}{K_f}{C}_L\epsilon \beta ,$$

(10)

where ε means the size coefficient, β means the surface quality coefficient, and C_L means a load type factor of 0.75.

The position at which fatigue failure first occurs is not necessarily the position at which one suffers from the maximum stress. Therefore, in this paper, three gearbox suspender positions, A1, A2, and A3, were selected and calculated, as shown in Figure 9. The corresponding position parameters are shown in

Table 6. Parameters of different dangerous nodes of gearbox suspender.

Position\Parameter	Kt	Kf	ε	β	a
A1	1.15	1.09	0.73	0.8	3.89
A2	1.30	1.15	0.73	0.8	4.07
A3	1.25	1.13	0.7	0.8	3.76

, where means the scaling factor of load and stress. Positions A1, A2, and A3 are all points with high stress, and are also the first fatigue fracture positions on the gearbox suspender in reality.

The fatigue limit at different positions calculated from Equation (10) was 138.5 MPa at Position A1, 131.2 MPa at Position A2, and 128.1 MPa at Position A3. Quasi-static analysis was adopted for this finite element simulation, so the stress of all elements is proportional to the load. The load on the gearbox suspender for the corresponding fatigue limit can be obtained by calculating the ratio of fatigue limit to scaling factor at different positions. Following calculations, the load was 35 kN at position A1, 32 kN at position A2, and 34 kN at position A3. Position A2 will reach fatigue with the minimum load amplitude at the three positions. Consequently, the gearbox suspender’s corresponding values of position A2 are adopted as the relevant parameters and coefficients.

After the fatigue limit of the members was obtained, the S-N curve of the high-cycle fatigue zone in the double logarithmic coordinate system was obtained by fitting, and the equation is as follows:

$$\lg S=2.852-0.105\lg N.$$

(11)

The complete S-N curve is shown in Figure 10.

In the calculation of fatigue strength, the safety factor is often used to express strength condition, and its equation is as follows:

$$n_{\sigma }=\frac{{\sigma }_{-1}}{{\sigma }_{\max }},$$

(12)

where σ₋₁ means the fatigue limit of the member and σ_max means the maximum stress under working conditions.

According to Goodman’s equal life transformation, the stress cycle of non-zero average stress is equivalently transformed into the stress cycle of zero average stress according to the principle of equivalent damage. Goodman’s equivalent formula is as follows:

$$\frac{S_a}{S_{eq}}+\frac{S_m}{S_b}=1,$$

(13)

where S_a means the stress amplitude; S_m means the average stress; S_b means the ultimate tensile strength; and S_eq means the equivalent stress.

After the equivalent stress was obtained, the values that were less than the fatigue limit were eliminated, and the values greater than the fatigue limit were subdivided into groups again. Life under different stress levels was obtained based on the fitted S-N curve. According to Miner’s cumulative damage theory, fatigue damage can be expressed by the corresponding cycle ratio n/N, where n is the number of cycles under different stress levels, and N is the fatigue life under the individual action of each stress level. If the number of cycles is represented by T, the damage to the member caused by stress levels at all levels during the whole working period is Tn/N. When the total damage reaches 1(100%), the member will be damaged immediately, as shown in Equation (14):

$$T{\displaystyle \sum _{i=1}^n\frac{n_i}{N_i}}=1,$$

(14)

The calculation method of the fatigue life of the gearbox suspender consists of the reciprocal of the cumulative sum of damage at all levels being multiplied by the number of simulated kilometers; the expression is:

$$S=\frac{L}{{\displaystyle \sum _{i=1}^n\frac{n_i}{N_i}}}.$$

(15)

4.2. Straight-Line Conditions

After the load was converted into stress, the stress amplitude smaller than the fatigue limit of the member was excluded, and the damage under each stress amplitude level was calculated using sections of 10 MPa. The damage D to the gearbox suspender under different passenger capacity conditions was obtained, and the expression is as follows:

$$D={\displaystyle \sum _{i=1}^n\frac{n_i}{N_i}}.$$

(16)

The damage values at 20 km/h, 40 km/h, and 60 km/h under straight-line conditions are shown in Figure 11.

The presence of fast gearing inside the gearbox significantly affects its vibration characteristics, making them different from those of the bogie frame, while the increase in passenger load increases the weight of the vehicle, and the weight is transferred to the bogie frame through the suspension, increasing the impact load between the gearbox and the frame and thus causing more damage to the gearbox suspender. When the passenger capacity was less than the fixed number of passengers, the gearbox suspender was less damaged, specifically by less than 1.06 × 10⁻⁶. This is because the stress amplitude to which the suspender was subjected was not significantly greater than the fatigue limit, and the number of generated cycles was low. When the speed was 20 km/h and the passenger capacity was 0, 120, and 240 persons, the maximum stress amplitude on the gearbox suspender was less than the structure’s fatigue limit. The damage was nearly zero, so the first half of the damage curve under this working condition was a straight line. When the passenger capacity exceeded 240 persons, the slope of the curve increased because the stress amplitude level increased. The damage corresponding to different levels of stress amplitude significantly differed: the larger the amplitude, the more obvious the damage. In addition, the frequency of stresses that occurred slightly above the fatigue limit greatly increased, causing the cumulative damage to rapidly increase. The effect of each speed stage on the suspender is clearly demonstrated by the increased vertical acceleration of the vehicle and the bogie frame, and the maximum damage was 4.89 × 10⁻⁶. The growth rate of damage under different speed levels is shown in

Table 7. The damage growth rate for various passenger capacities under straight-line conditions.

Damage Growth Rate\Level	0–120 Person	120–240 Person	240–339 Person	339–389 Person
vehicle speed of 20 km/h			9.45 × 10−10	3.79 × 10−9
vehicle speed of 40 km/h	1.56 × 10−9	2.36 × 10−9	8.74 × 10−9	1.81 × 10−8
vehicle speed of 60 km/h	2.38 × 10−9	5.61 × 10−9	2.05 × 10−9	3.61 × 10−8

. At the speed of 60 km/h, the suspender showed the largest damage growth for the passenger capacity of 240–339 persons, at 2.03 × 10⁻⁶. For the straight-line condition with a fixed speed, the damage showed the fastest growth rate for a passenger capacity of 339–389 persons, which could reach 3.61 × 10⁻⁸.

The service life of the vehicle suspenders can be calculated via Equation (15), as shown in Figure 12. Since the damage was zero when the vehicle was at 20 km/h and the passenger capacity was less than the number of overloaded passengers, the calculated service life was infinite. The maximum service life S (1 × 10⁴ km), represented by the ordinate Y in the figure, was set to 5 × 10⁷ km. Moreover, the damage was nearly zero when the vehicle was at 40 km/h and not loaded. In order to beautify the images, the life curve at 40 km/h was fitted by the exponential function according to other working conditions, which supplemented the service life under no-load conditions and was set to 3.49 × 10⁷ km. The damage was also small when the passenger capacity was small, and the difference at various speeds was not significant, but was quite significant in the life curve. With an increase in passenger capacity, the change in service life tended to be flat. When the speed was 20 km/h, the minimum service life was 7.07 × 10⁶ km. When the speed was 40 km/h, and the passenger capacity was 339 persons and 389 persons, the service life was 1.49 × 10⁶ km and 8.93 × 10⁵ km, respectively. When the speed was 60 km/h, and the passenger capacity was more than 240 persons, the service life was less than 2 × 10⁶ km, and the service life under the limit conditions was 4.09 × 10⁵ km. The design life of the gearbox suspender was 1.2 × 10⁶ km, and the suspender could meet the requirements at 20 and 40 km/h under overload conditions; however, when the speed was 60 km/h, the service life was reduced by 46% compared with the design life. Under extreme passenger capacity conditions, the service life was satisfied at 20 km/h, and when the speed was 40 km/h and 60 km/h, the service life as reduced by 26% and 66%, respectively, compared to the design life.

4.3. Curved Line Conditions

The damage for the curve radii of 600 m, 500 m, and 400 m at a speed of 40 km/h can be calculated according to the methods in the previous section, as shown in Figure 13.

The damage increased with the increase in passenger capacity and the decrease in curve radii, and the trend of the damage curve was identical to that under straight-line conditions at the speed of 40 km/h. When the curve radii were 500 m and 600 m, the stress on the gearbox suspender was less than 130 MPa under the no-load state, and the damage could be recorded as 0. When the curve radius was 400 m, the damage was 9.35 × 10⁻⁸. Under the same passenger capacity, the damage to the gearbox suspender for different curve radii showed little difference. When the passenger capacity was 389 persons, the difference in damage between curve radii of 400 m and 600 m was the maximum value of 8.05 × 10⁻⁷. According to the vibration characteristics of the model, the bogie frame and gearbox showed the largest vertical vibration acceleration, and the lateral and longitudinal vibration accelerations were smaller, so the vertical component forces generated by them were not large. Accordingly, the change in curve radius was not significant for the increase in accumulated damage. The damage growth rate under different curve radii grades is shown in

Table 8. The damage growth rate for various passenger capacities under curved line conditions.

Damage Growth Rate\Level	0–120 Person	120–240 Person	240–339 Person	339–389 Person
curve radius of 600 m	1.56 × 10−9	3.16 × 10−9	1.05 × 10−8	2.37 × 10−8
curve radius of 500 m	2.34 × 10−9	3.06 × 10−9	1.16 × 10−8	2.61 × 10−8
curve radius of 400 m	2.36 × 10−9	3.05 × 10−9	1.31 × 10−8	3.12 × 10−8

. The smaller the curve radii, the greater the growth rate. When the curve radii are 400 m, the damage had the largest growth rate for the segment with a passenger capacity of 339–389 persons, at 1.56 × 10⁻⁶. In addition, the growth rate was also the highest in this segment, reaching 3.12 × 10⁻⁸.

The service life of the vehicle suspenders under curved line conditions can be calculated via Equation (15), as shown in Figure 14. Since the damage to the vehicle was zero when the curve radii were 500 m and 600 m and the vehicle was not loaded, the calculated service life was infinite. To complete the life curve, some life curves were fitted by exponential functions according to other passenger capacity conditions, which supplemented the service life under no-load conditions, and these were 2.56 × 10⁷ km and 3.21 × 10⁷ km, respectively. With the increase in passenger capacity, the service life of gearbox suspenders under different curve radii became more similar, and the service life of gearbox suspenders for the segment with a passenger capacity of 240–389 persons was shortened by approximately 10 km due to the decrease in curve radii. The minimum life was 5.56 × 10⁵ km when the curve radius was 400 m. The gearbox suspender could meet the design requirement of 1.2 × 10⁶ km when the curve radius was 600 m under overload conditions, but the safety margin is small. When the curve radius was 500 m and 400 m, the service life was reduced by 7% and 18%, respectively, compared to the design life; under extreme passenger capacity conditions, the service life of the three curve radii could not meet the design requirements, which were reduced by 40%, 46%, and 54%, respectively.

5. Conclusions

The load on the gearbox suspenders under different working conditions was obtained by establishing the complete vehicle dynamic model. Different vibration indicators were analyzed through the data statistics. Among these, the extreme values significantly increased with the increase in passenger capacity. The mean value was maintained at around −2 kN, with little change, and both standard deviation and RMS showed a similar variation trend. The kurtosis factor decreased with passenger capacity, but was always greater than 0, which indicates that the load conforms to the normal distribution and the occurrence frequency of intermediate values gradually decreases. The effect of speed variations on vibration indicators was more significant than the radii of the track curve.

The fatigue limit was calculated after establishing the model of the gearbox suspender, and the position at which fatigue damage is most likely to occur was analyzed. The S-N curve of the suspender structure was fitted based on the parameters of this position. The results show that this curve is different from that obtained by the fatigue software system. The method in this paper has a certain guidance significance if the test-bed experiment cannot be carried out.

The relationship between stress spectrum and increase/decrease in service life shows that, for the vehicle gearbox suspenders, the corresponding cycle number N will decrease to half of its original value for each increase in 10 MPa for the stress level when the fatigue limit is exceeded, representing a stress level where the degree of damage caused by one cycle is roughly two times that of the previous level. As the maximum stress amplitude to which the suspender is subjected increases, the number of cycles greater than the fatigue limit and the total damage greatly increase. The impact load between the bogie frame and the gearbox does not significantly change when the passenger load is low, while damage will accumulate rapidly when the vehicle is overloaded. When the vehicle is not loaded, there is less cumulative damage for the gearbox suspenders, and the service life is almost infinite. However, the damage will significantly increase when the passenger capacity is 240 persons–389 persons, and speed increases will promote the damage growth rate. At the speed of 20 km/h, the service life can be decreased to less than 1.07 × 10⁷ km, and the safety factor is high. At a speed greater than 40 km/h, the expected service life is in the range of 4.09 × 10⁵ km–1.07 × 10⁷ km for straight-line conditions and curved line conditions. The service kilometers of gearbox suspenders will be greatly reduced in case of an overload, and will be less than 1.50 × 10⁶ km under all working conditions. Because the cumulative damage is non-linearly related to the amount of passengers, the calculated life is also non-linearly related to the amount of passengers. The overall curve flattens out in the latter part, but since the life already has a low safety factor with a high number of passengers, further overloading will still have a dangerous effect on the suspender.

The urban railways in some big cities have a high passenger flow, which means that vehicles are overloaded for a long time, which easily causes fatigue damage to the suspender. The traction transmission system suspender is closely related to the protection of life and property. The bogie provides the basis for the safe operation of the vehicle, and the traction transmission system is one of the most important parts of the bogie. If the suspender connecting the traction transmission system and the bogie fails, this will lead to damage to other parts of the system, threatening the safety of the train’s operation. The research results of this paper can help in the safe operation of certain domestic railway vehicles that are under overload conditions for a long time. By studying the influence of passenger capacity on the damage and life of the suspender, the strategy of limiting the passenger flow can be effectively formulated so that the vehicles can avoid overload during peak hours. In addition, vehicle maintenance can be arranged to shorten the maintenance interval of the suspender or replace the suspender that is about to cause damage as early as possible to minimize the safety hazards.

There is no unified standard system for the study of suspension devices of urban railway vehicles. The research in this paper can provide a theoretical basis for the design and optimization of the suspenders and contribute to the future production of a suspender with a better performance in line with operations. In addition, the research method used in this paper is also applicable to other equipment, providing guidance to determine whether their fatigue strengths meet the requirements and facilitate further optimization and upgrading.

Due to the limited time and space in this paper, the following areas need further study:

(1)

In this paper, a rigid body and bogie are established and treated with different degrees of equivalence to reduce the calculation. When the necessary conditions are available, the dynamics model can be further improved using the rigid–flexible coupling model and by establishing a complete model to make the simulation results more accurate and closer to reality;

(2)

Due to the complex structure of the transmission system and various load inputs, the impact characteristics of each part of the excitation are not specifically analyzed in this paper. The effect of a single load on the dynamic characteristics of the drive train can be explored in detail in subsequent studies;

(3)

The fatigue life analysis and prediction in this paper are based on the simulation model, the floating value of linear cumulative damage theory is relatively large, and the calculation results have some deviation from the actual working conditions in theory. When the conditions and equipment allow, the fatigue life of a large number of gearbox suspenders employed in this line can be tested for long-term line tracking, so the model can be verified and corrected for engineering practice;

(4)

Future work should be based on the analysis in this paper, aiming to optimize the suspender design and fatigue life to meet requirements.