Lubrication Reliability and Oil Churning Loss of Differential Gear Trains in a Mechanical-Hydraulic Coupling Mechanism

▪ MPS models are developed for differential gear trains. ▪ The influence of different factors on the lubrication reliability is analyzed. ▪ The height that may reduce the lubrication reliability is obtained. ▪ The feasibility of the MPS method is validated by experiments. A differential gear train (DGT) is a crucial component of a mechanical-hydraulic coupling mechanism. The transmission process generates oil churning losses, which significantly impact the overall transmission efficiency. Due to the complexity of DGTs and the unpredictability of lubrication reliability, traditional analysis of churning characteristics is inadequate. In this study, the moving particle semi-implicit (MPS) method is employed to analyze the effects of steady-state rotation speeds, dynamic rotation speeds, and oil filling heights on the oil churning characteristics of DGTs. The accuracy of the MPS method in predicting churning loss is illustrated by the Mean Absolute Percentage Error (MAPE) of 6.4% obtained experimentally. It is concluded that: increasing the oil filling height improves lubrication reliability by 20.9%, but results in greater power loss. The lubrication reliability and power loss of DGTs with different output forms are mutually advantageous under different influencing factors. This paper helps to improve the lubrication reliability of DGTs.


Introduction
Improving the transmission efficiency of hydraulic-mechanical parallel-coupled transmission systems can significantly reduce fuel consumption, promoting energy savings and emissions reductions, and addressing global warming concerns.DGTs are a key component of these systems because they offer high efficiency and strong differential drive ability, among other benefits.However, these benefits are accompanied by challenges related to complex structure and intense movement, which can impact operational reliability and shorten the working life of component parts [29].To address these challenges, lubrication and cooling strategies such as splash lubrication and oil spray lubrication are employed.Nevertheless, such methods bring oil churning losses and affect transmission efficiency [11,24].Oil churning power loss is a particularly significant factor, accounting for up to 30% of the total loss [2,5,10,23], which severely affects the efficiency of gear churning power losses of gears have been carried out by international researchers.Kahraman et al. [17] developed a special testing device for planetary gear trains and found that the main sources of churning losses are the viscous friction and centrifugal force inside the planetary bearings.Hammami et al. [12,13] obtained the average friction coefficient of a meshing gear by testing the oil churning torque losses of a spur gear on an FZG test bed and established a global power loss model considering load-dependent and load-independent losses.Zhu et al. [28] conducted splash lubrication tests on single face gear, and established a physical model for calculating the loadindependent loss of face gear considering the dynamic oil filling height.
Computational fluid dynamics (CFD) methods have gradually become a major tool in analyzing gear churning loss with the progress of science and technology.Among them, mesh-based finite volume analysis methods are widely used.
Concli et al. [1,3,4] found that the gear oil compression power losses decrease when the temperature increases and increase when the rotation speed increases.For planet gears, when the temperature is low, the power losses decrease with the temperature at a high rate.Liu et al. [20,21,22] built an FZG gear test bed and established a 3D simulation model considering two-phase flow.By comparing the simulation results with the test results, it was found that they have good consistency.
Ouyang et al. [25] investigated the jet lubrication of orthogonal face gear pair and found that a reasonable nozzle layout can improve the lubrication reliability.Hu et al. [14,15] proposed a model to estimate the oil churning power loss in a helicopter spiral bevel gear transmission gearbox and analyzed how different factors affect the lubrication reliability due to oil churning.Zeng et al. [27] analyzed the effect of tooth width and rotation speed on the lubrication reliability of gear pairs with different meshing modes and found that the difference in churning loss mainly came from the viscous force of large gears.
In recent years, some scholars have used meshless particle methods to study the oil churning characteristics of gears.Ji et al. [16] adopted the meshless smooth particle hydrodynamics (SPH) method to analyze the lubricant fluid field inside a gearbox, and compared it with the experimental results to verify the feasibility of the SPH method.Based on the SPH method, Keller et al. [18] studied the diffusion phenomenon and torque when the oil flow with different inclination angles impinges on a spur gear, and compared with the VOF simulation, it was found that the SPH method can save a lot of time.Deng et al. [7,8]

DGT structure
To analyze the oil churning losses of DGTs more comprehensively, a differential gearbox with a planet carrier output shaft (DGPCOS) and a differential gearbox with a ring gear output shaft (DGRGOS) in [6] is taken as the research objects respectively.The mechanical-hydraulic coupling mechanism, i. e. the differential gearbox studied in this paper, mainly consists of two input shafts, a fixed axis gear pair, a sun gear, bearings, planet gears, a planet carrier, a ring gear, and a casing, as shown in Fig. 1.
The mechanical-hydraulic coupling drive system divides engine power output into two channels using a transfer gearbox.

Operating condition
The MPS method is a meshless numerical analysis method for where u is the fluid velocity vector, t is the time, P is the pressure, ρ is the density, ν is the kinematic viscosity coefficient, and g is the gravitational acceleration vector.
In the MPS method, when particles are in motion, they interact with each other through the kernel function.To improve the numerical stability and computation speed, each mathematical model uses the kernel function as the weight function to discretize the control equations [19].Only when the distance between particles is within the effective interaction radius will they interact with each other, as shown in Fig. 2. The kernel function model is as follows: where re is the effective radius of action of the particle and rp is the distance between the particles.
where φ is the calculated particle physical parameters; d is the number of spatial dimensions, 2 for 2D and 3 for 3D; n0 is the particle number density constant of the initial state; i, j are the particle designators; rj, ri are the coordinate vectors of the particles; λ is the correction factor, which is calculated by the following formula: Each gear is subjected to the pressure and viscous resistance of the surrounding lubricant during rotation to form a resisting moment: where T is the resistance moment vector; r is the distance vector; FP is the pressure vector; and Fν is the viscous force vector.
Churning power loss is the product of gear churning torque loss and rotation angular velocity： where PLoss is the churning power loss; TLoss is the churning torque loss; and n is the gear rotation speed.

Simulation model
L-ckc150 lubricant is selected, its density is 812.5 kg/m 3 and kinematic viscosity is 1.5e−4 m 2 /s when the temperature is 40 ℃.Considering the computer performance, the particle diameter is set to 2.5 mm, the structures of the gearboxes are set as Polygon, the pressure is set to implicit iteration model, the viscosity is set to implicit β=1 model, and the surface tension is set to Potential.To improve convergence, model simplification is performed to enhance the computational power as shown in

Analysis of fluid field features and oil churning losses of the DGPCOS
The choice of steady-state rotation speed is based on [9], which does not include analyses of dynamic rotation speed operating conditions.Given the critical role of dynamic speed in gearbox lubrication reliability, this paper explores this aspect.The "−"

Fluid field features and churning losses at steady-state rotation speeds
The amount of oil in contact with the DGT is often indicative of lubrication performance, as shown in Figs. 5 and 6.Since a planet gear meshes with both the sun gear and the ring gear, the percentage of the average number of particles it contacts compared to the number of particles at full immersion can reflect the degree of lubrication reliability of the DGT.
Therefore, the reliability is calculated using the planet gear initially located at the bottom as an example as shown in Fig. 7.

Fluid field features and churning losses at dynamic rotation speeds
The velocity fluid field of the DGPCOS under dynamic rotation speed is compared in Fig. 9

Analysis of fluid field features and oil churning losses of the DGRGOS
Comparing the data in Table 1 and Table 2, it is evident that the ring speed of DGRGOS exceeds the planet carrier speed of DGPCOS, indicating that DGRGOS holds an advantage in terms of dynamic output speed.Since the dimensions of the DGRGOS are different from the DGPCOS, the oil fill heights in this section are slightly different from the previous section, but submerge the gear teeth in the same locations.

Fluid field features and churning losses at dynamic rotation speeds
The velocity fluid field of the DGRGOS under dynamic rotation speed is compared in Fig. 17  In Fig. 18, the rotation speed of the DGT is higher compared to Fig. 17, resulting in a stronger impact on the oil within the casing.Consequently, the oil splashes in the form of dispersed droplets.The more high-speed oil that is splashed, the more area that is lubricated.When the oil filling height is -96 mm, the droplets at the top of the casing are small and dispersed, in contrast to Fig. 17 Fig.19 shows that the churning torque losses and power losses of each gear increase with the increase of oil filling height.
The sun gear experiences small churning torque losses due to minimal oil contact.In the first second, the churning power losses of the sun gear decrease as its rotation speed decreases.
However, after this period, its churning power losses gradually

Experimental results
Experiments were conducted on DGTs under different operating conditions to validate the accuracy of the MPS method in predicting churning loss.A test bench (Fig. 21) was built using a motor, transfer gearbox, speed-torque sensors, differential gearbox (replaceable), and dynamometer [6].Working conditions 1-6 (Table 1) and 11-16 ( oil churning and friction losses from bearings, gears, and couplings.To determine churning power losses, power losses were separately measured with and without lubrication oil, and the difference was obtained as per the following equation. =   −   (10) where   is the experimental value of churning power loss,   is the measured value of churning power loss with lubrication oil, and   is the measured value of churning power loss without lubrication oil. dynamometer.
Equation (9) shows that the torque loss is proportional to the power loss at the same rotation speed, so only the power loss is verified.In conclusion, the study yields the following findings: (1) Under steady-state rotation speed conditions, increasing rotation speed escalates the centrifugal force on the lubricant, resulting in higher splashing oil speeds that tend to churn the oil to the top of the casing.Meanwhile, heightening oil filling levels augments the quantity of oil in contact with the DGT during rotations, improving lubrication reliability (20.9%) and reducing wear, but wear is not quantified as it is a long-term result.
(2) The higher the rotation speed and oil filling height, the greater the impact of the oil on the gears when they rotate, resulting in higher churning torque losses and power losses.The same rule applies under dynamic speed conditions.For the DGRGOS, the churning power losses of the planet gears show a gradual increase with time in the two different speed ranges, but the increase is rapid and exponential when the dynamic rotation speed range is large.
(3) The lubrication reliability of DGRGOS can be increased by up to 9.3% compared to DGPCOS when the gears are immersed in the same position at steady-state speed, but this advantage diminishes as the oil filling height decreases.
(4) The rotation speed significantly affects the churning loss, and the dynamic output speed of DGPCOS is lower than that of DGRGOS for the same dynamic input speeds.Therefore, the DGPCOS has an advantage in churning power loss, and this advantage decreases when the oil filling height is reduced.
However, higher dynamic output speeds up to 1400~2385.92rpm can be achieved by the DGRGOS.
This work provides a feasible approach to analyzing the lubrication reliability and the churning losses of DGTs but does not consider the effects of lubricant properties and temperature on the fluid field features and churning losses of DGTs.
Therefore, this is the focus of our further work to improve the comprehensive analysis of churning losses in DGTs.
used the MPS method to quantitatively analyze the effects of various factors on the oil distribution in rollerenveloped worm gear transmission and observed it through experiments.The outcomes indicated a good agreement between the MPS method and the experimental results.Most existing research has primarily examined churning losses in a single gear or a fixed axis meshing gear pair, with limited attention given to a DGT with ring gear rotation.Consequently, systematic analysis of the dynamic flow field features and churning loss of DGTs has been challenging, a problem addressed in this paper.The structure and motion forms of DGTs are complex, and other analysis methods (such as SPH, Boundary Element Method, Finite Volume Method, Discrete Element Method) are cumbersome in pre-processing, difficult to solve, and inefficient in addressing lubrication reliability.In contrast, the MPS method offers simplicity in computation and high accuracy.Therefore, this study adopts the MPS method to explore the impact of rotational speed and initial oil injection height on the lubrication reliability and churning loss of DGTs with different output forms.Moreover, the study investigates the effect of different oil filling heights on churning characteristics under dynamic speed conditions and discusses any observed differences.Finally, experiments validate the efficacy of the MPS method.
One channel transmits power to input shaft 1, while the other Eksploatacja i Niezawodność -Maintenance and Reliability Vol. 26, No. 2, 2024 channel transmits power to input shaft 2 through a hydraulic transmission unit.In the DGPCOS, input shaft 1 drives the ring gear with mechanical power flow, while input shaft 2 drives the sun gear with hydraulic power flow.The two power flows are then converged by a DGT and outputted by the planet carrier shaft.For the DGRGOS, input shaft 1 drives the planet carrier with mechanical power flow, input shaft 2 drives the sun gear with hydraulic power flow.The two power flows are then converged by the DGT and outputted by the ring gear shaft.
solving incompressible fluids.It can effectively simplify the simulation processing steps, shorten the simulation time and it is easy to analyze complex kinematic boundary problems or multi-physics field problems.The basic principle of the MPS method is to replace the fluid in the computational domain with a particle swarm, to give different flow information to each fluid particle, and to solve and discretize the basic flow equations according to each interaction model based on Lagrange equations [19,26].The MPS method takes the continuity equation (law of conservation of mass) and the Navier-Stokes equation (law of conservation of momentum) as the basic governing equations, as shown below:

Fig. 2 .
Fig. 2. Schematic diagram of particle effective radius.The particle number density represents the combination of the function values of a particle and its neighboring particles within the range of action of the kernel function, which is a fixed value in an arrangement of particles that satisfies the incompressibility condition.The gradient model and the Laplace model are based on the kernel function discrete control equations.The expressions are as follows:  = ∑ (|  −   | ≠

Fig. 3 ,Fig. 4 .
Fig.3, and the Courant limiter is used to prevent the computational divergence.The simulation time is set to 1 s for steady-state rotation speed and 2 s for dynamic rotation speed and the torque output interval is set to 0.01 s.As shown in Fig.4, the oil filling height in this paper refers to the distance between the lubricant level and the center of the sun gear.

Fig. 8 Fig. 8 .
Fig.8shows the variation of total churning power loss with

Fig. 9 .Fig. 10 .
Fig. 9. Flow field distribution of the DGPCOS at different moments when the dynamic rotation speed of the sun gear is 800 ~ −800 rpm: oil filling height at (a) −90 mm and (b) −112 mm.

Fig. 11 Fig. 11 .Fig. 12 .
Fig. 11 shows the variation of churning torque losses and power losses at different oil filling heights with a sun gear rotation speed of 800 ~ −800 rpm.The churning torque losses of the ring gear fluctuate due to the varying exposure to oil.Additionally, dynamic changes in rotation speed notably impact

Figs. 13 Fig. 13 .Fig. 14 .Fig. 15 .
Figs.13 and 14 show the velocity flow field of the DGRGOS at steady-state speeds.A higher initial oil filling height results in more oil being stirred up and better contact between gears, leading to improved lubrication reliability, as shown in Fig.15.When the oil filling height is −40 mm, the DGT stirs up a significant amount of oil due to viscous force, resulting in more oil distribution in the DGT after dynamic equilibrium.This, along with planet gears' revolution, increases oil flow speed.The high-speed oil is more likely to splash onto the gear surfaces, allowing more oil to enter the gear mesh areas, providing a high lubrication reliability.When the oil filling

Fig. 16 (Fig. 16 .
Fig. 16(a) shows an increase in total churning power loss for the DGT with a ring gear output shaft as oil filling height and rotation speed increase.Figs.16(b) and 16(c) further demonstrate that churning power losses of the planet gears and ring gear rise with increasing rotation speed and oil filling height.Conversely, the churning power losses of the sun gear remain minimal even at high rotation speeds due to significant oil expulsion, resulting in minimal contact with the lubricating oil and negligible churning power losses.
at 0.5-second intervals.When the oil filling height is −96 mm, significant lubricant is stirred by orbiting planet gears.At 0.5 seconds, more oil has entered gear meshing areas, and by the first second, more lubricant has reached the casing's inside top, so the lubrication reliability is high.Subsequently, as the gear ring accelerates, the splashed droplets establish consistent contact with all areas, ensuring reliable lubrication.However, when the oil filling height is -132 mm, an insufficient amount of oil reaches the sun gear and planet gears, resulting in low lubrication reliability and high risk.

Fig. 17 .Fig. 18 .
Fig. 17.Flow field distribution of the DGRGOS at different moments when the dynamic rotation speed of the sun gear is 800 ~ −800 rpm: oil filling height at (a) −96 mm and (b) −132 mm.

Fig. 22 .
Fig. 22.Comparison between experimental and MPS simulation results in terms of churning power losses of (a) the DGPCOS and (b) the DGRGOS

Table 1 .
The three different oiling heights are submerging the teeth of the sun gear, the center of the bottom planet gear, and the teeth of the bottom planet gear, respectively.As lubricant viscosity is temperature-dependent, considering temperature would introduce an additional variable, making it challenging to compare the simulation with the experiment when temperature changes.Therefore, the impact of temperature on lubrication reliability is not considered in this study.
Eksploatacja i Niezawodność -Maintenance and Reliability Vol. 26, No. 2, 2024 only indicates the difference in the rotation direction, as shown in

Table 1 .
Working conditions of the DGPCOS.

Table 2 .
Working conditions of the DGRGOS.

Table 2
) were selected for comparative tests while keeping other conditions consistent to minimize error.Power losses measured during the tests included Eksploatacja i Niezawodność -Maintenance and Reliability Vol. 26, No. 2, 2024