Fatigue Life Analysis of Main Reducer Gears for Battery Electric Bus Considering Regenerative Braking

Abstract

The braking mode of the battery electric urban bus (BEUB) is different from the friction braking of the traditional fuel bus due to the introduction of a regenerative braking system. The intervention of electromagnetic braking changes the working condition of the main reducer gears, thus affecting their service lives. Based on the Urban Dynamometer Driving Schedule (UDDS) driving cycle condition, the stress–time history of the main reducer gears is calculated. Combined with the static analysis results and the S-N curve of the material, the fatigue lives of the main reducer gears considering electromagnetic braking and traditional friction braking are analyzed. The reverse torque on the driving axle during electromagnetic braking is taken into account to be closer to the real situation. Results show that, under the electromagnetic braking mode, the bending fatigue lives of the tooth root on the convex and concave surfaces of the pinion are 78.5% and 78.9% of that under the traditional friction braking mode, respectively, while the contact fatigue life of the pinion working surface is 78.2% of that under the friction braking mode, indicating that the introduction of the regenerative braking system into the BEUB will cause a significant reduction in the service life of the main reducer gears. This study provides a high-precision fatigue life calculation method for the BEUB main reducer gears and the accurate prediction of their remaining life.

1. Introduction

New energy vehicles have attracted much attention due to the energy and environmental issues [1]. Most of the countries in the world attach great importance to the development of new energy vehicles that are environmentally friendly and low oil-depending [2]. The BEUB has been widely used for a long time because of its working conditions consistent with the high-efficiency zone of the electric driving system.

Due to the particularity of urban traffic conditions, buses brake frequently. Traditional fuel buses use friction braking, and part of the kinetic energy will be transformed into heat energy and dissipation through friction during braking, resulting in energy waste [3]. According to statistics, under typical urban traffic conditions, buses have 50–60% of the energy consumed in the braking process [4]. Therefore, the regenerative braking system is generally introduced into the battery electric buses, and the energy is recovered through electromagnetic braking to reduce energy consumption and improve endurance mileage [5].

The introduction of the regenerative braking system, however, leads to the increasing of the maintenance rate for the BEUB main reducer, and a significant gap between the service life and the design life occurs. This is because the original bus chassis is generally inherited in major car companies, and the centralized driving mode is widely used, while ignoring the difference in working conditions between battery electric buses and traditional buses; thus, the main reducer applied in the traditional bus would have a plummeted life when applied in the BEUB [6]. It is found that the main difference is that the regenerative system based on electromagnetic braking is introduced into the BEUB. When the regenerative system works, the drive shaft will endure a reverse torque, which changes the stress cycle characteristics of the gear, and then affects their service lives [7]. Therefore, in order to avoid the introduction of the regenerative braking system based on electromagnetic braking, which leads to a sharp decrease in the service life of BEUB main reducer gears, and reduce the waste of maintenance funds and manpower, it is imperative to investigate the influence of electromagnetic braking on the BEUB main reducer gears’ fatigue life.

Plenty of research with remarkable results has been conducted by scholars in the field of gear fatigue life by using finite element analysis, bench tests, etc. John [8] tested the fatigue life S-N curves of four gear steels under low and high cycles, as well as the durability limit, and gave the design curve based on the fatigue test data. S. Glodez [9] and J. Kramberger [10] proposed a calculation model to determine the bending fatigue service life of gear tooth roots. The initial crack life was calculated by the finite element method and strain-life method. Yu et al. obtained the unidirectional gear load spectrum consistent with the real working conditions through the ADAMS software. Then, the finite element method is used to calculate the maximum stress of dangerous parts of the gear tooth roots and predict the bending fatigue life of the gear considering the influence of dynamic loads [11]. Based on the actual driving load spectrum, Zou et al. systematically studied the virtual fatigue test method of transmission combined with vehicle and bench test verification methods [12]. Tadeusz Lagoda used Miner theory and the rain-flow counting method to analyze the influence of different mean stress on uniaxial random loads fatigue tests [13]. Zhen summarized and analyzed the basic fatigue formula, load spectrum processing method, and other issues related to fatigue research in practical engineering [14]. Using the cumulative fatigue damage theory, Huang [15] and Guo [16] et al. calculated the unidirectional contact stress spectrum and obtained the contact fatigue life of a transmission gear under the measured speed condition by using a variety of finite element simulation software.

It can be seen that only the unilateral meshing of the working face (concave surface of the pinion, convex surface of the driven gear) was considered in most of the existing research on gear fatigue life based on the actual load spectrum, and few scholar have studied the load spectrum of the gears with reverse torque. This may work in the fatigue life analysis of the main reducer gears for traditional buses because the braking torque is applied to the driven gear under friction braking, and it is still the case that the pinion concave surface drives the gear convex surface. That is, the transmission between the reducer gears is always unidirectional from driving to braking, and only the working faces contact. However, the braking mode of the BEUB with the regenerative braking system is different from that of the traditional bus. Electromagnetic braking is based on the reversibility of the driving motor. The motor will convert from the driving state to the power generation state when braking [17], and the resistance torque is generated at the same time. The speed of the driving pinion is suddenly reduced, resulting in the nonworking face (convex surface of the driving pinion, concave surface of the driven gear) participating in meshing, and the pinion convex surface is driven by the gear concave surface, which changes the gear tooth contact and bending stress cycle characteristics, and inevitably affects the fatigue life of the main reducer gears.

For this reason, based on the nominal stress method [18], a pair of spiral bevel gears is taken as the research object, and the contact and bending stress load spectrum of the gears under typical braking conditions were calculated based on the UDDS road cycle condition and the vehicle driving equation. When calculating the load spectrum, on the basis of the existing research, the reverse torque during electromagnetic braking is taken into account to be closer to the BEUB real braking condition so that, after the introduction of the regenerative braking system, the calculation results are more in line with the engineering practice compared with previous studies. The results of finite element static analysis are imported into nCode DesignLife software, and the fatigue lives of the corresponding gears are analyzed by applying the aforementioned typical load spectrum. Furthermore, the variation law of the fatigue lives of the gears under different braking modes is systematically analyzed, a method for calculating the fatigue life of the gears in the vehicle transmission system with the regenerative braking system is provided to provide a theoretical basis for the design and manufacture of the main reducer gears for the BEUB, with high performance and long service life, and more scientific research and design of new energy vehicles.

2. Finite Element Modeling and Static Analysis

2.1. Establishment of Gear Solid Model

The basic parameters of the gear pair for a BEUB main reducer are shown in Table 1.

The modeling method of a spiral bevel gear is different due to its complex tooth surface [19]. In this paper, the gear modeling command of the GC toolbox in UG NX is used to realize the parametric modeling of the gears. The pinion is the driving gear and the large gear is the driven gear. The central connection vector selects the axial vector of the driven gear and the vector on the driving pinion perpendicular to the end face of the driven gear, respectively, to complete the meshing assembly of the gear. To improve the calculation efficiency and accuracy, the gear model is simplified to a fewer-tooth model. The simplified gear model is shown in Figure 1. As shown in the Figure 1, different colors are used to distinguish the gears, the green gear is the pinion, the blue gear is the driven gear.

2.2. Finite Element Modeling

The above solid model is imported into ANSYS Workbench to complete the establishment of the finite element model and static analysis.

The properties of materials will affect the calculation results of static analysis [20]. The pair of gear materials in this paper are both 20CrNiMo, and the corresponding US standard number in the nCode material library is SAE Steel Grade 8620, with a density of 7850 kg/m³, an elastic modulus of 180 GPa, and a Poisson’s ratio of 0.3. The gears used in this paper are manufactured by face milling, followed by heat treatment. Gears of this material need to be treated with carbonitriding. Finally, the tooth surface is ground to improve the gear accuracy and reduce the surface roughness; after grinding, the gear accuracy grade is 6.

Due to the complex tooth surface shape, the tetrahedral mesh is adopted [21] and the local mesh of gear root and meshing surface is refined. The meshing results are shown in Figure 2 and Figure 3. The pinion has 135,579 elements and 231,749 nodes, and the driven gear has 163,401 elements and 278,735 nodes.

The gear tooth contact pair is established [22]. The tooth surface participating in meshing of the driving pinion is selected as the contact surface, and the tooth surface participating in meshing of the driven gear is selected as the target surface. After the contact pair has been set up, the friction constraint is applied. The friction type is Frictional, and the friction coefficient is taken as 0.03.

The constraint of boundary condition should reflect the actual working state of gears as much as possible. In the process of transmission, the pinion rotates at a certain speed by the torque transferred from the upper level, which drives the driven gear to rotate. The driven gear is affected by the resistance moment so that the gear pair achieves balance. Therefore, a hinge constraint is applied on the inner surface of the master–slave driving gear, the freedom of rotation around the axis of the driving pinion is released, and other freedoms are constrained. The driven gear is fully constrained. A torque around the axis is applied on the inner surface of the gear, which is 200 N·m.

2.3. Statics Results and Analysis

After the preprocessing setting is completed, the solution can be calculated. The stress distributions of the driving pinion and the driven gear are shown in Figure 4 and Figure 5. It can be seen that the maximum equivalent stress is 1197.7 MPa, which appears in the contact area. Moreover, the contact stress is very large near the meshing contact line and very small in other positions, which is consistent with the actual situation.

3. Fatigue Life Analysis Based on Load Spectrum

3.1. Obtaining Fatigue Load Spectrum

Fatigue load is the repeated load that causes fatigue damage. Most parts are subjected to varying loads rather than constant loads in the working process. Fatigue load spectrum is divided into constant amplitude load spectrum, block spectrum, and random load spectrum [23]. The operating environment of urban buses is mainly in the urban areas of major cities, where traffic conditions are complex and random. The UDDS working condition is the urban road cycle condition formulated by the United States Environmental Protection Agency (the EPA), which simulates the urban road working condition of 12.07 km [16]. Compared with other working conditions, the UDDS working condition contains frequent braking conditions, and the average speed is 31.5 km/h, which is more in line with the actual operation of urban buses, and can be applied to simulate the bidirectional torque of the main reducer gear. Thus, the UDDS cycle is selected as the driving speed in the simulation, as shown in Figure 6.

Urban buses are generally subjected to two kinds of forces during driving, namely vehicle driving resistance and driving force. Vehicle driving resistance includes rolling resistance, air resistance, acceleration resistance, and slope resistance. During the driving process, the driving resistance of the vehicle needs to maintain a balanced relationship with the driving force, which is called the vehicle driving equation [24].

(1)$\frac{T_{tq}{i}_0{i}_g{\eta }_t}{r}=mgfcos\alpha +\frac{C_{D}A{u}^2}{21.15}+mgsin\alpha +\delta m\frac{du}{dt}$

In the formula, T_tq is the output torque of the motor. i₀ is the transmission ratio of the main reducer; i_g is the transmission ratio of the gearbox; η_t is the transmission efficiency; r is the wheel rolling radius; f is the road resistance factor; α is road inclination; C_D is wind resistance coefficient; A is the windward area; u is instantaneous speed; and δ is the rotation mass conversion coefficient.

In this paper, vehicle parameters and motor characteristics are shown in Table 2 and Table 3.

According to the vehicle driving Equation (1), taking the UDDS cycle condition as the input of the driving speed, the corresponding main reducer input torque is calculated, as shown in Figure 7.

It can be seen from Figure 7 that, during electromagnetic braking, the reverse torque generates in the driving pinion shaft. However, the friction braking does not produce reverse torque, so the reverse torque in Figure 7 is converted to positive direction, as shown in Figure 8.

Figure 9 shows the torque–time history when the reverse braking torque is ignored. In this case, the reverse torque during electromagnetic braking is regarded as 0, which is obviously inconsistent with the actual working condition of the BEUB, and the calculated fatigue life is bound to deviate from reality.

In this paper, the fatigue life of gears is calculated by using the results of static analysis, so it is necessary to transform the torque–time history into the stress–time history, that is, the calculated torque load is transformed into the contact stress of the tooth surface and the bending stress of the tooth root, which can be realized by using Equations (2) and (3) [24]:

(2)$\mathrm{contact}\mathrm{stress}:{\sigma }_H=\sqrt{\frac{4K_H{T}_1}{{\varphi }_R{\left(1-0.5{\varphi }_R\right)}^2{d}_1^{3}u}}{Z}_H{Z}_E,$

(3)$\mathrm{bending}\mathrm{stress}:{\sigma }_F=\frac{K_F{T}_1{Y}_{Fa}{Y}_{Sa}}{{\varphi }_R{\left(1-0.5{\varphi }_R\right)}^2{m}^3{z}_1^2\sqrt{u^2+1}},$

where, $K_H$ and $K_F$ are the load coefficients. The load coefficients include the use coefficient $K_A$, the dynamic load coefficient $K_V$, the longitudinal load distribution coefficient $K_{\beta }$, and the load distribution coefficient among gear teeth $K_{\alpha }$. The calculation formula is as follows:

(4)$K=K_A{K}_V{K}_{\beta }{K}_{\alpha },$

${\varphi }_R$ is the tooth width coefficient, ${\varphi }_R=b/d_1$, b is the tooth width;

$d_1$ is the pitch circle diameter of the pinion.;

u is the transmission ratio;

$Z_H$ is the node region coefficient, and $Z_H$ is 2.2;

$Z_E$ is the elasticity impact coefficient, and its value is related to the material; the 20CrNiMo gear can be taken as $189.8\mathrm{MP}{\mathrm{a}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$;

$Y_{Fa}$ is the tooth shape coefficient, which is related to the tooth system, modification coefficient, and tooth number and $Y_{Fa}$ is 1.5;

$Y_{Sa}$ is the stress correction coefficient, and $Y_{Sa}$ is 1.6 [24].

The contact stress load spectrum and bending stress load spectrum of electromagnetic braking condition, and the contact stress load spectrum and bending stress load spectrum of friction braking condition can then be obtained, as shown in the Figure 10, Figure 11, Figure 12 and Figure 13. Regardless of electromagnetic braking or friction braking, the gear contact stress is larger than the bending stress, and, since the reverse torque is taken into account, the stress calculated by the reverse torque also changes direction from tensile stress to compressive stress, or from compressive stress to tensile stress.

3.2. Calculation and Analysis of Fatigue Life

Based on the static analysis results and the stress–time load spectrum data [25], combined with the material properties of 20CrNiMo, the fatigue life of the main reducer gear under electromagnetic braking condition and friction braking condition was calculated, respectively. The specific calculation process is shown in Figure 14.

Four nCode SN TimeSeries modules are established and a channel is created with the statics module; the statics results are imported into each of the four modules to calculate the fatigue lives of the main reducer gears under different stress spectra.

After importing the static results, nCode will automatically generate various modules for solving. The stress–time history obtained above is imported into TimeSeries input module, and the load spectrum and finite element results are added. In the solver, the material type is set to SN R-ratio multi-curve, and the fatigue curve of the material is the S-N curve [26] of the multi-stress ratio shown in Figure 15. SAE Steel Grade 8620 is found to assign it to all entities, and the Goodman curve is used to correct the mean load.

After the solver is set up, the fatigue life of the main reducer gears under electromagnetic braking condition and friction braking condition is calculated, respectively, and the results are shown in Figure 16.

The fatigue life data of the same dangerous node are extracted, as shown in Table 4. For the main reducer gears, during the normal driving process of the bus, the working face is the concave surface of the pinion, the tooth root of the concave side is tensile stress, and the tooth root of the convex surface is compressive stress. In the case of conventional buses, the stress on the tooth root is also the same during braking, whereas, in the case of battery electric buses with electromagnetic braking, the working face will change to the pinion convex surface during braking, and the compressive stress on the tooth root of this side will also change to tensile stress; the tensile stress on the concave surface will change to compressive stress as well. It can be seen that the stress changes in the pinions are more complex, and the calculations with four load spectrums show that the lowest life spans are all found on the pinions, so that the effects of electromagnetic braking can be analyzed by simply calculating the fatigue life of the pinions. Calculation results showed that the bending fatigue life of pinion concave tooth root under electromagnetic braking condition is 78.9% of that under friction braking condition, while the bending fatigue life of pinion convex tooth roots is 78.5% of that under friction braking condition. The tooth contact fatigue life of pinion working surface under electromagnetic braking condition is 78.2% of that under friction braking condition, which is in line with the engineering practice. This shows that electromagnetic braking does have an influence on the main reducer gears fatigue life. In this paper, it is taken into account that, when the electromagnetic braking mode is braking, the electricity would reverse, which would drive the driving shaft to produce reverse torque, and the reverse torque would change the tooth surface meshing condition of the main reducer gear; this change affects the main reducer gears’ fatigue life. In addition, from the data in Table 4, it can be seen that the contact fatigue life of a gear is lower than bending fatigue life; this is because the tooth surface suffers frequent impacts due to the switch between driving and braking, the damage degree of the impact on the tooth surface is greater than the damage to the tooth root, and the contact stress of the tooth surface is greater than the bending stress of the tooth root in normal meshing, so the contact fatigue life is far lower than the bending fatigue life. This is consistent with the fact that tooth contact fatigue often occurs first.

4. Conclusions

Due to the introduction of the regenerative braking system, the service life of the BEUB main reducer gears is reduced; this paper found that, after the introduction of the regenerative braking system, the braking mode of BEUB changes to electromagnetic braking, and the electromagnetic braking would cause the reverse torque of the driving axle during braking, thus changing the meshing condition of the main reducer gears. Therefore, when calculating the fatigue life of the main reducer gears, the factor of reverse torque is especially taken into account and compared with friction braking.

• (1). Based on the UDDS urban cycle condition, the torque load spectrums of electromagnetic braking and friction braking are calculated and converted into bending stress and contact stress spectrums.

• (2). Considering the influence of electromagnetic braking, the tooth root bending fatigue lives of the pinion convex and concave surface are 78.5% and 78.9%, respectively, of those under the traditional friction braking, and the contact fatigue of the concave surface of the pinion is 78.2%.

• (3). The braking energy recovery system of the BEUB has a significant influence on the fatigue life of the main reducer gears. The main reason is that the introduction of reverse braking torque changes the gear working condition, thus leading to the main reducer gear meshing surface changing frequently during the BEUB operation, which reduces the main reducer gears’ service life sharply. Therefore, more attention should be paid to the meshing performance of the nonworking faces when designing and manufacturing the BEUB main reducer gears. At the same time, the change in meshing would also have an impact, which should be taken into account in the design of the main reducer gears, so as to improve the comprehensive performance of the gear more comprehensively.

• (4). The proposed method is more consistent with the actual working conditions of BEUB and is also applicable to the gear transmission of other fuel vehicles fitted with retarders for braking, which is the same as that of the BEUB with the introduction of the regenerative braking system, so the method proposed in this article is equally applicable to the calculation of the fatigue life of the main reducer gears of a vehicle fitted with retarders.

Author Contributions

Conceptualization, J.D. and X.W.; methodology, J.D., X.W. and J.M.; software, J.D. and X.W.; validation, J.D. and J.M.; investigation, X.W.; resources, J.D.; data curation, X.W.; writing—original draft preparation, X.W.; writing—review and editing, J.D.; visualization, X.W. and J.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We acknowledge the support provided to this study by Xi’an University of Technology, Xi’an University of Technology, in the form of time and facilities.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Spiral bevel gear assembly model.

Figure 2. Grid division results.

Figure 3. Grid quality.

Figure 4. Stress distribution nephogram of the driving pinion.

Figure 5. Stress distribution nephogram of the driven gear.

Figure 6. UDDS speed condition diagram.

Figure 7. Electromagnetic braking torque–time history.

Figure 8. Friction braking torque–time history.

Figure 9. Load spectrum ignoring braking torque.

Figure 10. Contact stress load spectrum of electromagnetic braking.

Figure 11. Contact stress load spectrum of friction braking.

Figure 12. Bending stress load spectrum of electromagnetic braking.

Figure 13. Load spectrum of friction braking bending stress.

Figure 14. Fatigue life calculation process based on nominal stress method.

Figure 15. 20CrNiMo S-N curve.

Figure 16. Fatigue life maps of pinion with four load spectrums. (a) With contact stress load spectrum of electro-magnetic braking; (b) With contact stress load spectrum of friction braking; (c) With bending stress load spectrum of electro-magnetic braking; (d) With bending stress load spectrum of friction braking.

Table 1

Gear geometric parameters.

Gear Parameter	Value
tooth numbers	16/27
Module (mm)	4.5
tooth width (mm)	17
pressure angle (°)	20°
shaft angle (°)	90°
mean spiral angle (°)	35°
hand of spiral	LH/RH
pitch angle (°)	30.65°/59.35°

Table 2

Vehicle parameters.

Definition	Value
maximum total mass (kg)	14,000
frontal area (m2)	8
coefficient of rolling resistance	0.016
coefficient of air resistance	0.7
tire size (m)	0.555
full speed (km/h)	100
grad-ability limit (%)	25
mechanical efficiency of power transmission (%)	90

Table 3

Motor characteristics.

Rated Power (kW)	Peak Power (kW)	Peak Torque (N·m)	Rated Speed (rpm)	Peak Speed (rpm)
75	100	280	2600	7500

Table 4

Fatigue life of pinion dangerous nodes.

Braking Method	Stress Type		Cycle Index
electromagnetic braking	bending stress	concave	4.926 × 106
		convex	3.618 × 109
	contact stress		4.558 × 104
friction braking	bending stress	concave	6.236 × 106
		convex	4.609 × 109
	contact stress		5.825 × 104

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