1. Introduction
The traction motor, gearbox and wheelset are the key components of the power wheelset of the electric multiple unit’s (EMU) bogie. As the core unit of power conversion and transmission, the reliability of the gearbox directly determines the safety of the EMU. The gearbox of EMU has a high power-to-weight ratio, complex structure and compact installation space, so splash lubrication is usually used. The power loss of the gearbox during operation is inevitable, and a significant portion of this loss is ultimately transformed into heat and dissipated into the environment. According to its relationship with the load, the power loss can be divided into load-dependent power loss and load-independent power loss; and the churning loss accounts for approximately 30% of the load-independent power loss, which seriously hampers the transmission efficiency [
1]. Accurate prediction of churning loss is an extremely challenging aspect of the gearbox design process, because it is heavily affected by the lubricating-oil distribution in the gearbox, which is relatively unknown. The churning power loss in gearboxes can be obtained by experiments and empirical formulas [
2,
3,
4,
5]. While these methods can provide some reference in the initial estimation stage of the gearbox design process, their continued use in the specific structural design and parameter selection can result in design inaccuracies. Therefore, it is still an urgent and challenging task to investigate the churning loss and lubrication characteristics of EMU gearboxes under splash lubrication.
Throughout the past several decades, with the enhancement of computing power, the computational fluid dynamics (CFD) method has gradually become a robust and effective tool for analyzing the lubrication characteristics of gearboxes. Some scholars have employed CFD methods to investigate churning power loss and lubricating-oil distribution in gearboxes. Based on the finite volume method (FVM), Liu et al. [
6,
7] established a three-dimensional CFD simulation model of a single-stage gearbox considering oil–air two-phase flow, and studied the effects of rotation speed, oil fill level and kinematic viscosity on oil distribution and churning power loss. The oil distribution and churning resistance torque obtained by the CFD method are consistent with the oil distribution records obtained by the high-speed camera and the values obtained by the torque instrument. According to their research conclusions, the speed is the most significant influencing factor for churning resistance torque. Chen and Matsumoto [
8] constructed a rotatable single-stage transmission gearbox test bench to investigate the effects of the relative position of the gear on the oil surface profile and churning loss. Additionally, various shapes of fillers are inserted into the gearbox to simulate the influence of the actual shape of the box on the churning loss. The results show that the filler with a specific structure can significantly reduce the churning loss, consistent with the conclusion reached by Hildebrand et al. [
9]. And Hu et al. [
10] developed a CFD simulation model of a gearbox considering dynamic motion and applied sinusoidal motion to the gearbox by using the method of a non-inertial coordinate system. By comparison with the experimental results of Chen and Matsumoto [
8], the correctness and applicability of the established model were verified. Further analysis was conducted on the impacts of the amplitude and frequency of the dynamic motion of the gearbox on the churning power loss. Regarding the FZG test bench, Mastrone and Concli [
11] substituted for the lubricating oil with grease and used the Herschel–Bulkley model to describe the non-Newtonian fluid characteristics of the grease. Liu et al. [
12] established a CFD model, including gears and bearings, and extended the FVM from a single-stage gearbox to a planetary gearbox with a complex structure. However, to minimize workload, the dynamic motion of the bearing was ignored, and the bearings were set as static. References [
6,
7,
8,
9,
10,
11,
12] have shown the great potential of FVM in predicting the distribution of lubricating oil and the churning loss in the gearbox. Nevertheless, when applying this method, a special dynamic mesh strategy is needed to generate the mesh of the gear meshing area. Liu et al. [
12], Cho et al. [
13], Burberi et al. [
14] and Mastrone et al. [
15] employed overlapping grids, the immersed boundary method, a local grid reconstruction and a global grid reconstruction method to mesh the gear meshing area, so that the gear meshing could be simulated more realistically. Furthermore, Mastrone and Concli [
16] proposed a new mesh processing strategy, that is, a global mesh reconstruction of mesh clustering. By combination with the global mesh reconstruction method, a two-stage industrial gearbox CFD model is established. Theoretically, the FVM method can be extended from the single-stage transmission to the multi-stage transmission gearbox [
16,
17]. However, since FVM is a grid-based Euler technique, each element needs to follow the conservative Euler equation in the process of application, and the entire computational domain needs to be meshed. In order to ensure the integrity of the whole calculation domain, it is usually necessary to artificially reduce gear or expand the center distance. It is precisely because of the application of these measures that the tooth surface shape or assembly relationship of the gear is altered, influencing the power loss, transmission efficiency and lubrication characteristics obtained through simulation.
Numerical methods based on particles can effectively compensate for the aforementioned drawbacks. Particle-based numerical methods typically employ a Lagrangian coordinate system. Compared with the grid-based Euler method, particle-based numerical methods do not need to mesh the whole computational domain, and only the fluid domain needs to be filled with fluid particles with specific physical properties. It is precisely due to this method that gear scaling and various mesh division techniques have become outdated. Smoothed particle hydrodynamics (SPH) and the moving particle semi-implicit (MPS) method are two commonly used meshless particle methods. Ji et al. [
18] used a continuum surface force to model the interfacial surface force between multiphase fluids and used color function to describe different phases. A multiphase fluid numerical model based on SPH was established, and the aeration behavior in the gearbox was analyzed for the first time. Compared with the particle-image velocimetry results, SPH can capture well the local details of the flow field. Legrady et al. [
19] demonstrated a SPH-based simulation model of gearbox flow field suitable for bevel gear transmission, and conducted extensive tests on different rotation speeds, rotation directions and liquid level filling heights. When compared with the experimental results, the numerical flow field obtained by the numerical model considering the surface tension is more consistent with the experimental results. In addition, they studied the churning loss with different gear-reduction ratios. If the gear pair is reduced to 98% of the original size during the simulation process, the average difference in churning loss can reach 30%. Liu et al. [
20] used the CFD model based on SPH to simulate the oil distribution and churning loss of the FZG test gearbox. The SPH method is very robust and accurate in predicting oil distribution, but the error level between the churning loss obtained by SPH and that of the test results can reach 82%. To achieve more accurate quantitative results, optimization and improvement of the SPH are still needed [
21,
22]. Liu et al. [
23] conducted a series of extensive numerical simulations using the MPS method with reference to the gearbox model established in reference [
20].
Based on these findings, combined with the experimental results provided in references [
6,
20], it can be pointed out that both the MPS and SPH methods can be used to predict the oil distribution, but the churning loss obtained by the MPS method is more accurate than that determined by SPH. Deng et al. [
24] and Xie et al. [
25] established high-fidelity numerical simulation models for gearboxes used in different types of rail vehicles and carried out simulation calculations with different speeds and lubricating-oil volume levels and viscosities, providing a precedent for the engineering application of the MPS method. The simulation model they established retained the complex gearbox structure, and the modeling process was very simple. However, this is difficult to achieve in the FVM. Guo et al. [
26] further analyzed the influences of gear parameters on the churning loss. According to the experimental results in this paper, the power losses of the numerical simulations of gear pairs are in good agreement with those of the experiment. Deng et al. applied the MPS method to study the lubrication performance of the roller enveloping worm reducers [
27]. Wei et al. conducted a study on the churning loss and lubricating-oil distribution of a hydraulic pump used in engineering vehicles, based on the MPS method. In addition, they also studied the flow pattern of lubricating oil around the rotating disk under low temperature conditions, dividing the flow pattern of lubricating oil around the disk into coated, immersed and reverse oil films [
28,
29]. The above literature indicates that the MPS method has a significant advantage in determining the churning loss of the gear transmission system.
The gearbox of an EMU usually adopts a single-stage gear transmission, which can be divided into parallel shaft transmissions and cross shaft transmissions according to the arrangement of the transmission shafts. Compared with the parallel shaft transmission, the bevel gear transmission system driven by the universal shaft is beneficial for reducing the unsprung mass of the bogie and improving the dynamic performance of the EMU. However, because of the shift angle of the bevel gear transmission, the oil splashing in the gearbox becomes more complicated, compared to the parallel transmission. Hu et al. [
30] took the helicopter intermediate gearbox as their research object, established an oil–air two-phase flow model that included an oil guide device, casing and spiral bevel gear pair, and discussed in detail the various operating factors occurring during the helicopter’s work, such as gear rotation speed, oil properties and aircraft inclination angle, as well as their impacts on the churning loss of the gearbox. Jiang et al. [
31] used the same CFD model as Hu et al. to analyze the influence of the oil guide device on the lubrication effect of the gearbox. They found that the structural aperture of the oil guide device and the radius of the oil duct have important influences on the oil supply of the bearing, but have minor effects on the churning loss and the lubrication of the gear meshing zone. Lu et al. [
32] established a CFD model of oil–air two-phase flow for the intermediate gearbox used in helicopters. According to the oil flow characteristics, the lubricating oil stirred by the gear is divided into free flow, jet flow and splash flow. In addition, they also introduced a multi-reference frame to describe the rotations of gears and bearings in the established CFD model. And they built a heat-flow coupling model and discussed the convective heat transfer coefficient and temperature characteristics of the gearbox [
33,
34]. Bevel gear transmission is not only widely used in helicopter transmission systems, but also very common in truck-axle transmission systems. Peng et al. [
35] overcame the difficulties of generating meshes in the meshing zone of bevel gear transmission systems by applying the tooth-surface translation method. And an experimental platform matching the simulation model was designed to verify the accuracy of the numerical method. They also applied the proposed modeling method to the drivetrain system and used the multi-objective optimization and response surface method to optimize the system [
36,
37]. Although the authors of the above literature have carried out relevant research on the lubrication characteristics of the splash-lubricated bevel gear transmission system, the numerical model used for research is quite different from the actual model. Compared with the gears in the engineering model, the gears in References [
30,
31,
32,
33,
34,
35,
36,
37] are all reduced, and therefore cannot reflect the real tooth surface shapes and meshing relationships. In addition, in order to reduce the pre-processing of the model and the number of grids used for numerical calculation, a box with a relatively simple shape is necessary. The application of these measures, however, would greatly weaken the guiding role of numerical simulation in engineering practice.
The shape of the casing of the EMU is very important to the analysis of the lubrication characteristics of the gearbox. In order to put forward reasonable measures for subsequent optimization and improve the working performance of the gearbox, when the lubrication mechanism of the gearbox is numerically simulated, retaining the actual structural characteristics of the casing can improve the guiding role of numerical simulation. In this paper, the gearbox of a spiral bevel gear transmission for a body suspension motor EMU is taken as the research object, and a corresponding high-fidelity CFD model is established for the first time. The MPS method is used to analyze the influences of different input shaft speeds, initial lubricating-oil volumes and lubricating-oil temperatures on the distribution of lubricating-oil particles and the churning loss, and the lubrication mechanism of the gearbox is revealed. The organization of this paper is as follows: In
Section 2, we introduce MPS and the calculation method of churning loss used in this paper. In
Section 3, the spiral bevel gearbox studied in this paper and the processes of establishing the numerical model are presented. In
Section 4, the applicability and accuracy of the MPS method are verified. In
Section 5, the influences of different working parameters of the gearbox on lubrication characteristics are discussed. Finally, in
Section 6, the main conclusions of this paper are given.
4. Experimental Verification
To verify the applicability and accuracy of the MPS method in analyzing the lubrication characteristics of the bevel gearbox, a simulation model of a helical bevel gearbox was established, with gear parameters as shown in
Table 5. The physical parameters of the lubricating oil were set at 40 °C, the density was 850 kg/m
3, and the dynamic viscosity was 8.627 × 10
−2 kg/(m·s).
The gear rotation speed was set to 1000 rpm, and the gear immersion depth was defined as 25 mm. The numerical simulation obtains the lubricating-oil distribution under these conditions, which is then compared with the experimental data provided in reference [
31], as shown in
Figure 5. The numerical simulation results obtained by the MPS method are basically consistent with the oil distribution obtained by the experiment. The lubricating oil accumulates at the lower right of the front transparent glass plate, and the splash behavior of the lubricating oil stirred by the other gear is also in good agreement with the experimental results. Setting the same initial oil immersion depth and gear speed as in reference [
30], the churning loss under different working conditions is numerically simulated, as shown in
Figure 6.
The numerical results and experimental results also show that at the same oil immersion depth, with the increase of gear rotation speed, the churning loss increases gradually, and the higher the speed is, the more obvious is the increasing trend. Under the same rotation speed, with the increase of oil immersion depth, the churning loss also increases. Moreover, the error between the churning loss obtained by the numerical simulation of the MPS method and that obtained in the experiment is within 5%, which meets the engineering application requirements. In the simulation process, the temperature of the lubricating oil is assumed to be constant, and there is no consideration of the viscosity–temperature characteristics of the lubricating oil, which may cause some errors between the experimental value and the simulation’s value. By comparing the oil distribution and churning loss in the gearbox obtained by the MPS method and the experiment, it can be seen that the MPS method can be beneficially applied to analyze the lubrication characteristics of the bevel gear gearbox.