A method for characterizing temperature rise of tooth surfaces of a closed differential double helical gear

Abstract

This invention discloses a method for characterizing the temperature rise of the tooth surface of a closed differential herringbone gear, belonging to the field of gear dynamics technology. The method establishes a system dynamic model to solve the dynamic meshing force of the internal and external meshing pairs and determines the contact line length of a single pair of teeth. Based on the contact line length, the load distribution ratio is determined, converting the dynamic meshing force into a normal load on a single pair of teeth. The radius of curvature, entrainment velocity, sliding velocity, and rolling ratio at the meshing point are calculated, and the tooth surface friction coefficient is solved using a finite-length line contact dynamic thermo-elasto-fluidic lubrication model. The friction power loss of the internal and external meshing pairs is calculated based on the friction coefficient, normal load, and sliding velocity. The heat distribution coefficient is determined based on the physical properties of the paired gears, distributing the friction power loss of the internal and external meshing pairs to the sun gear, planet gears, and internal gear ring, respectively, to obtain the thermal power of each gear. Finally, the thermal balance equations for each gear are established to solve for the tooth surface temperature rise. This invention achieves integrated calculation of dynamics, lubrication, and heat transfer, improving the accuracy of tooth surface temperature rise prediction.

Description

Technical Field

[0001] This invention belongs to the field of dynamics technology of herringbone gear transmission systems, and specifically relates to a method for characterizing the temperature rise of the tooth surface of a closed differential herringbone gear. Background Technology

[0002] Herringbone gears are widely used in high-power-density transmission devices such as coaxial reversible helicopter main gearboxes due to their high load-bearing capacity, smooth transmission, and the ability to cancel out axial forces. A closed differential planetary gear transmission system includes an external meshing pair of sun gear and planet gears and an internal meshing pair of planet gears and an internal gear ring. Both types of meshing pairs participate in power splitting and merging simultaneously. During operation, affected by factors such as time-varying meshing stiffness, tooth flank clearance, and transmission errors, the actual meshing force of each meshing pair exhibits periodic dynamic fluctuations rather than a constant load.

[0003] Finite-length line contact thermo-elastohydrodynamic lubrication theory is currently an effective means of solving for oil film pressure, film thickness, temperature, and friction coefficient in the tooth surface contact area. However, when using this theory to characterize the tooth surface temperature rise of a closed differential herringbone gear transmission system, existing methods struggle to reflect the thermal response characteristics under real dynamic conditions. The accuracy of this analysis is highly dependent on the input tooth normal load, contact point curvature radius, and relative velocity. However, current thermo-elastohydrodynamic analyses generally use the average load under static meshing force or rated torque as input, failing to reflect the impact of load fluctuations on the lubrication state during actual meshing. Furthermore, when considering frictional heat calculations, since the planetary gears in the closed differential configuration participate in both internal and external meshing pairs, and the kinematic and contact geometric characteristics of these two types of meshing pairs differ significantly, analysis methods based on a single herringbone gear pair cannot handle the heat attribution and distribution problem under conditions of multiple heat sources. Furthermore, microscopic parameters such as oil film temperature and pressure in the tooth surface contact area are difficult to measure directly, while tooth surface temperature rise is a macroscopic parameter that can be obtained by temperature measurement. There is a lack of quantitative mapping relationship between the two, which means that the above-mentioned microscopic lubrication analysis results cannot be effectively transferred to the macroscopic temperature response level. This is the fundamental reason why existing methods are unable to reflect the true thermal response characteristics of such systems. Summary of the Invention

[0004] To address the aforementioned problems, the purpose of this invention is to provide a method for characterizing the temperature rise of the tooth surface of a closed differential herringbone gear. This method constructs an integrated calculation process that combines dynamics, dynamic thermo-elasticity, frictional heat distribution, and tooth surface heat transfer, thereby achieving accurate quantitative characterization of the temperature rise of the tooth surface of each component of this type of gear.

[0005] The technical solution of this invention is: a method for characterizing the temperature rise of the tooth surface of a closed differential herringbone gear, comprising the following steps: A dynamic model of a closed differential herringbone gear transmission system is established. The dynamic model is solved to obtain the dynamic meshing force of the external meshing pair of the sun gear and planet gear and the dynamic meshing force of the internal meshing pair of the planet gear and the internal gear ring. The contact line length of a single pair of teeth in the external meshing pair and the internal meshing pair is determined based on the end face overlap, axial overlap and total overlap of the herringbone gear.

[0006] Based on the contact line length of a single pair of teeth, the load distribution ratio of a single pair of teeth in the external and internal meshing pairs is determined, and the dynamic meshing force of the corresponding meshing pair is converted into a normal load of a single pair of teeth according to the load distribution ratio.

[0007] Calculate the contact point radius of curvature, entrainment speed, sliding speed, and rolling ratio of the external and internal meshing pairs. Input the normal load of a single pair of gear teeth, as well as the radius of curvature, entrainment speed, sliding speed, and rolling ratio, into a finite-length line contact dynamic thermo-elastohydrodynamic lubrication model. Solve the model to obtain the oil film pressure, oil film thickness, lubricating oil temperature, and friction coefficient of the external and internal meshing pairs.

[0008] Based on the friction coefficient, the normal load on a single pair of gear teeth, and the sliding speed, calculate the friction power loss of the external meshing pair and the internal meshing pair, respectively.

[0009] The heat distribution coefficient is determined based on the thermal conductivity, density, specific heat capacity at constant pressure, and tangential velocity at the meshing point of the two meshing gears. The frictional power loss of the external meshing pair is distributed to the sun gear and planet gear according to the heat distribution coefficient between the sun gear and planet gear. The frictional power loss of the internal meshing pair is distributed to the planet gear and internal gear according to the heat distribution coefficient between the planet gear and the internal gear ring. The heat power obtained by the sun gear, planet gear and internal gear ring are obtained respectively.

[0010] Based on the heat power obtained by each gear, a heat balance equation is established between frictional heat generation and convective heat transfer of each gear. The heat balance equation is then solved to obtain the temperature rise of the tooth surfaces of the sun gear, planet gears, and internal gear ring.

[0011] Furthermore, the heat distribution coefficient is determined according to the following formula: Determine the heat distribution coefficient; where, This represents the thermal conductivity of the i-th meshing gear; This represents the material density of the i-th meshing gear; This represents the specific heat capacity at constant pressure of the i-th meshing gear; This represents the tangential velocity at the meshing point of the i-th meshing gear.

[0012] Furthermore, the thermal power obtained by the sun gear is determined based on the heat distribution coefficient of the sun gear and the frictional power loss of the external meshing pair; the thermal power obtained by a single planet gear is determined based on the heat distribution coefficient of the planet gear, the sum of the frictional power losses of the external meshing pair and the internal meshing pair, and the number of planet gears; the thermal power obtained by the internal gear ring is determined based on the heat distribution coefficient of the internal gear ring and the frictional power loss of the internal meshing pair.

[0013] Furthermore, the load distribution ratio is the ratio of the contact line length of a single pair of gear teeth to the sum of the contact line lengths of all pairs of gear teeth participating in meshing simultaneously; the normal load of a single pair of gear teeth is determined based on the load distribution ratio and the dynamic meshing force of the corresponding meshing pair.

[0014] Furthermore, the coefficient of friction is determined based on the hydrodynamic friction force and the normal load on a single pair of gear teeth; the hydrodynamic friction force is determined based on the lubricating oil viscosity, contact half-width, central film thickness, and average sliding speed along the meshing line direction.

[0015] Furthermore, the frictional power loss of the external meshing pair and the internal meshing pair is the product of the average value of the instantaneous meshing power loss of a single pair of teeth within their respective meshing cycles and the number of planetary gears. The instantaneous meshing power loss of a single pair of teeth is the product of the friction coefficient, the normal load of a single pair of teeth, and the sliding speed.

[0016] Furthermore, the heat balance equation is: ;in, Indicates the mass of the gear. This indicates the specific heat capacity at constant pressure of the gear material. This indicates the average temperature of the tooth surface. This indicates the thermal power obtained by the gear. Indicates the equivalent convective heat transfer coefficient. This represents the equivalent heat transfer area. This indicates the initial temperature of the lubricating oil.

[0017] Furthermore, the tooth surface temperature rise is the difference between the steady-state tooth surface temperature and the initial temperature of the lubricating oil. The steady-state tooth surface temperature is determined based on the initial temperature of the lubricating oil, the average heat power during the meshing cycle, the equivalent convective heat transfer coefficient, and the equivalent heat transfer area.

[0018] Compared with existing technologies, the advantages of this invention are as follows: This invention solves the dynamic meshing force of the internal and external meshing pairs by establishing a system dynamics model, and determines the load distribution ratio based on the contact line length to calculate the normal load of a single pair of gear teeth. This reflects the influence of periodic fluctuations in meshing force caused by factors such as time-varying stiffness and tooth flank clearance on the tooth surface contact state, overcoming the mismatch problem of traditional static load input methods. Furthermore, the normal load of a single pair of gear teeth, along with the radius of curvature of the meshing point, entrainment speed, sliding speed, and sliding-rolling ratio, are input into the dynamic thermo-elasto-hydrodynamic lubrication model to solve for the oil film pressure, oil film thickness, lubricating oil temperature, and friction coefficient in the contact area. Based on this, the frictional power loss of the internal and external meshing pairs is calculated, realizing the quantitative transmission of system dynamic response and tooth surface lubrication state, avoiding the data fragmentation caused by the independent calculation of dynamics, lubrication, and thermal analysis in existing methods. Based on this, the heat distribution coefficient is determined according to the physical properties of the paired gears. The frictional power loss of the external meshing pair is distributed to the sun gear and planet gears, and the internal meshing pair is distributed to the planet gears and internal gear ring. This yields the individual heat power of the three components, solving the problem of unclear heat source attribution and lack of basis for differentiated heat distribution when the planet gears simultaneously bear the frictional heat of the two meshing pairs in a closed differential configuration. Finally, by establishing the heat balance equations for each gear and solving for the tooth surface temperature rise, a complete quantitative characterization link from dynamic meshing behavior and lubrication state to the temperature response of each component is established. This provides a reliable theoretical basis for the thermal design and lubrication optimization of closed differential herringbone gear reducers. Attached Figure Description

[0019] Figure 1 This is a flowchart of the present invention; Figure 2 This is a simplified kinematic diagram of a closed differential herringbone gear transmission system, an application example of the present invention. Figure 3 , Figure 4 This is a diagram showing the contact line length and load distribution results of a single pair of gear teeth in an application example of the present invention; wherein, Figure 3 It is an external meshing. Figure 4 It is an internal meshing; Figure 5 , Figure 6 This is a comparison diagram of the load on the tooth surface of a single pair of gears in an application example of the present invention; wherein, Figure 5 It is an external meshing. Figure 6 It is an internal meshing; Figure 7 This is a graph showing the calculation results of the surface temperature of the sun gear, planet gears, and internal gear ring, as well as the equivalent heat convection coefficient, in an application example of this invention.

[0020] Among them, 1-closed-stage sun gear, 2-closed-stage planetary gear, 3-closed-stage internal gear ring, 4-differential-stage internal gear ring, 5-differential-stage planetary gear, 6-differential-stage sun gear, 7-inner output shaft, and 8-outer output shaft. Detailed Implementation

[0021] The following is combined with Figures 1 to 7 The specific embodiments of the present invention will be described in detail below.

[0022] Example like Figure 1 The method for characterizing the temperature rise of the tooth surface of a closed differential herringbone gear, as shown, includes the following steps: S1. Obtain the dynamic meshing force of the internal and external meshing pairs in the closed differential herringbone gear transmission system. Based on the structural connection relationship of the closed differential herringbone gear transmission system, a dynamic model is established, including the bending, torsional, axial, and oscillating degrees of freedom of the sun gear, planet gears, internal gear ring, and planet carrier. The model is input with gear mass, moment of inertia, support stiffness, support damping, time-varying meshing stiffness, meshing damping, comprehensive transmission error, backlash, input speed, and input torque.

[0023] Solve the system dynamic equations based on the force relationships of each component to obtain the relative displacements of the sun gear-planet gear meshing lines and the planet gear-internal gear ring meshing lines, and then apply the formula... Calculate the dynamic meshing force of the external meshing pair Dynamic meshing force of internal meshing pairs ;in, This represents the dynamic meshing force of the external meshing pair corresponding to the i-th planetary gear; This represents the dynamic meshing force of the internal meshing pair corresponding to the i-th planetary gear; Indicates the time-varying meshing stiffness of the external meshing pair; Indicates the time-varying meshing stiffness of the internal meshing pair; Indicates the meshing damping of the external meshing pair; Indicates the meshing damping of the internal meshing pair; This indicates the relative displacement of the meshing line, i.e., the real-time meshing position at each moment; This indicates the tooth flank clearance.

[0024] S2. Determine the normal load on a single pair of gear teeth. The herringbone gear is equivalent to a pair of helical gears with equal and opposite helix angles. The contact line length of a single pair of teeth from engagement to disengagement is determined based on the end face overlap, axial overlap, and total overlap.

[0025] Assuming the tooth surface load is uniformly distributed along the contact line, based on the length of the contact line between the i-th pair of teeth and the adjacent teeth, the formula is used. Determine the load distribution ratio LSFn(t), where, This represents the contact line length of the i-th pair of gear teeth at time t; and ε represents the contact line length of adjacent gear teeth; T represents the overlap ratio; and T represents the meshing cycle of a single pair of gear teeth.

[0026] And adopt the formula Calculate the normal load on a single pair of gear teeth. For an external meshing pair, the equation is... F in SP / RP Take the dynamic meshing force F of the external meshing pair SP For internal meshing pairs, take the dynamic meshing force F of the internal meshing pair. RP .

[0027] S3. Determine the contact geometry and kinematic parameters. Select the meshing point on the contact line of a single pair of gear teeth, and calculate the radii of curvature of the two tooth surfaces based on the base circle dimensions, meshing positions, and base circle helix angles of the driving and driven gears: The combined radius of curvature is The entrainment speed is The sliding speed is The slip ratio is In the formula, and These are the radii of curvature of the two meshing gears at the contact point; and These represent the angular velocities of the two meshing gears; "±" corresponds to external meshing and internal meshing, respectively.

[0028] S4. Establish a dynamic thermo-elasto-fluidic lubrication model. The normal load of a single pair of gear teeth, the combined radius of curvature, the entrainment speed, the sliding speed, and the sliding-rolling ratio are used as inputs to the finite-length line contact dynamic thermo-elastohydrodynamic lubrication model.

[0029] The dynamic thermo-elastohydrodynamic lubrication model includes the Reynolds equation, film thickness equation, viscosity-pressure-temperature relationship equation, density-pressure-temperature relationship equation, energy equation, and load balance equation. Solving the model yields the oil film pressure, oil film thickness, lubricating oil viscosity, lubricating oil density, and temperature within the contact area.

[0030] A multi-grid W-loop method is used to numerically solve the dynamic thermo-elastohydrodynamic lubrication control equations. First, initial pressure, center film thickness, and temperature field are set. Then, the pressure is solved and elastic deformation is calculated, updating the film thickness. The lubricant viscosity and density are updated based on pressure and temperature. The energy equation is solved and the temperature field is updated. The center film thickness is adjusted based on load balance. Once the pressure, film thickness, temperature, and load errors meet the convergence conditions, the lubrication parameters at the current engagement moment are output.

[0031] S5. Calculate the friction coefficient and meshing power loss. Based on the lubricating oil viscosity, contact half-width, central film thickness, and average sliding velocity along the meshing line output by the dynamic thermo-elastohydrodynamic lubrication model, the following formula is used: Calculate the hydrodynamic friction force and use the formula Calculate the coefficient of friction; where, Indicates the tooth width of a single-sided helical tooth; Indicates the contact half-width; Indicates the viscosity of the lubricating oil; This represents the average sliding velocity along the direction of the meshing line; Indicates the thickness of the central membrane; Indicates the coefficient of friction; Indicates a normal load on a single pair of gear teeth; It represents fluid dynamic friction.

[0032] Substituting the friction coefficient, the normal load on a single pair of gear teeth, and the average sliding speed along the meshing direction into the equation... This yields the instantaneous meshing power loss of a single pair of gear teeth at each meshing moment. This indicates the instantaneous meshing power loss of a single pair of gear teeth; Indicates the coefficient of friction; Indicates a normal load on a single pair of gear teeth; This represents the average sliding speed along the meshing line.

[0033] The average instantaneous meshing power loss of a single pair of gear teeth is the average instantaneous meshing power loss of a single pair of gear teeth over the meshing cycle, obtained by dividing its integral over a complete meshing cycle by the meshing cycle. Integrating the integral over a complete meshing cycle for both external and internal meshing pairs, and considering the number of planetary gears, the formula is used. with formula Calculate the average power loss of the external meshing pair and the average power loss of the internal meshing pair respectively; where, , These represent the normal loads on a single pair of teeth in external and internal meshing pairs, respectively. Indicates the load distribution ratio; Indicates the degree of overlap of the external meshing pair; Indicates the degree of overlap of the internal meshing pair; Indicates the number of planetary gears; This indicates the meshing cycle of a single pair of gear teeth.

[0034] S6. Perform frictional heat power distribution. The average power loss of the internal and external meshing pairs is taken as the frictional heat source of the gear teeth. Based on the thermal conductivity, material density, specific heat capacity at constant pressure, and tangential velocity at the meshing point of the two meshing gears, the formula is used. Determine the heat distribution coefficient; where, This represents the thermal conductivity of the i-th meshing gear; This represents the material density of the i-th meshing gear; This represents the specific heat capacity at constant pressure of the i-th meshing gear; This represents the tangential velocity at the meshing point of the i-th meshing gear.

[0035] The frictional heat of the external meshing pair is distributed between the sun gear and the planet gears, while the frictional heat of the internal meshing pair is distributed between the planet gears and the internal ring gear. The planet gears simultaneously receive frictional heat from both the external and internal meshing pairs. (The following is a formula / method / approach]... The thermal power obtained by the sun gear, a single planet gear, and the internal gear ring was obtained respectively; among them, This represents the heat distribution coefficient corresponding to the sun gear; This represents the heat distribution coefficient corresponding to the planetary gears; This represents the heat distribution coefficient corresponding to the internal gear ring.

[0036] S7. Calculate tooth surface temperature and temperature rise Assuming the average temperature of the tooth surface of a single gear at any given time is T(t), the gear is considered as a rigid body with an approximately uniform temperature distribution on the tooth surface. The gear exchanges heat with the lubricating oil through the oil-immersed end face and the outer cylindrical surface of the rim via convection.

[0037] Adopted Establish the heat balance equation between frictional heat generation and convective heat dissipation. Use equation... Japanese style To calculate Nusselt numbers, Reynolds numbers, and Prandtl numbers, the formula is used. The equivalent convective heat transfer coefficient is calculated using the formula... Calculate the equivalent heat transfer area.

[0038] By integrating the heat balance equation over time, the temperature variations of the sun gear, planet gears, and internal gear ring tooth surfaces over time are obtained. After several meshing cycles, the average heat power of the meshing cycle is used instead of the instantaneous heat power, and dT / dt = 0 is set. The equation is then applied. The steady-state tooth surface temperature is obtained. The difference between the steady-state tooth surface temperature and the initial temperature of the lubricating oil is the steady-state temperature rise of the tooth surface. Where m represents the gear mass; c represents the specific heat capacity at constant pressure of the gear material; This indicates the average temperature of the tooth surface; The initial temperature of the lubricating oil is represented by h; the equivalent convective heat transfer coefficient is represented by A; the equivalent heat transfer area is represented by ω; the gear angular velocity is represented by D; the addendum circle diameter is represented by η; and the viscosity of the lubricating oil is represented by η. ρ represents the thermal diffusivity of the lubricating oil; ρ represents the density of the lubricating oil. This represents the specific heat capacity of the lubricating oil at constant pressure; k represents the thermal conductivity of the lubricating oil. This represents the area of ​​the gear's end face immersed in the oil. This represents the heat transfer area of ​​the outer cylindrical surface of the wheel rim; Indicates the pitch circle diameter; This represents the average thermal power during the meshing cycle.

[0039] Application examples Based on the embodiments, the proposed approach is as follows: Figure 2The temperature rise of the gear tooth surface in the closed differential herringbone gear transmission system shown was characterized.

[0040] like Figure 2 The diagram shown is a simplified kinematic diagram of the closed differential herringbone gear transmission system in this application example. The closed differential herringbone gear transmission system in this application example includes a closed-stage sun gear 1, closed-stage planetary gears 2, a closed-stage internal gear ring 3, a differential-stage internal gear ring 4, a differential-stage planetary gear 5, a differential-stage sun gear 6, an inner output shaft 7, and an outer output shaft 8. The closed-stage sun gear 1 and the differential-stage sun gear 6 are rigidly connected to the input shaft, maintaining the same rotational speed. The closed-stage internal gear ring 3 and the differential-stage internal gear ring 4 are rigidly connected, forming an output end opposite to the input direction. The differential-stage planetary carrier forms an output end with the same input direction. Through this transmission structure, the input power is differentially distributed between the two output branches. A dynamic model of the closed differential herringbone gear transmission system is established to solve for the dynamic meshing forces of the sun gear-planetary gear external meshing pair and the planetary gear-internal gear ring internal meshing pair, and these dynamic meshing forces are used as the load input for dynamic thermo-elasto-fluidic lubrication calculations.

[0041] like Figure 3 , Figure 4 The diagram shows the contact line length and load distribution results for a single pair of gear teeth in this application example. Based on the contact line lengths of a single pair of gear teeth and adjacent gear teeth, the application example calculates the load distribution ratio for a single pair of gear teeth. The contact line lengths of both internal and external meshing pairs gradually increase during gear engagement, remain stable in the middle of engagement, and gradually decrease during disengagement. The overlap ratio of the external meshing pair is less than 2, and during meshing, the load is alternately borne by one or two pairs of teeth, with a maximum load distribution ratio of 1. The overlap ratio of the internal meshing pair is greater than 2 and less than 3, and during meshing, the load is alternately borne by two or three pairs of teeth, with a load distribution ratio of less than 1 for a single pair of teeth. Multiplying the load distribution ratio by the dynamic meshing force of the corresponding internal and external meshing pairs yields the normal load borne by a single pair of teeth at each meshing position.

[0042] like Figure 5 , Figure 6The figure shows a comparison of the tooth surface loads of a single pair of gears in this application example. The application example uses the dynamic meshing force obtained from the system dynamics model and the traditional static meshing force to calculate the normal loads of a single pair of gears in both the external and internal meshing pairs. Within a complete meshing cycle, both the dynamic and static loads change with the tooth contact line length and load distribution ratio, generally increasing gradually during engagement, reaching a maximum value in the middle of engagement, and gradually decreasing during disengagement. Compared to the static load, the dynamic load fluctuates periodically around the static load throughout the meshing process, reflecting the influence of time-varying meshing stiffness, tooth flank clearance, comprehensive transmission error, and system vibration on the actual tooth surface load. The dynamic loads, along with the comprehensive radius of curvature of the contact point, entrainment speed, sliding speed, and sliding-rolling ratio, are input into the dynamic thermo-elasto-fluidic lubrication model to obtain the oil film pressure, oil film thickness, temperature, and friction coefficient of the internal and external meshing pairs, and further calculate the tooth surface friction power loss.

[0043] like Figure 7 The figure shows the calculation results of the surface temperatures and equivalent heat convection coefficients of the sun gear, planet gears, and internal gear ring in this application example. This application example combines the friction coefficients obtained from the dynamic thermo-elastohydrodynamic lubrication model with the dynamic load and sliding speed of a single pair of gear teeth to calculate the frictional power loss of the external and internal meshing pairs, respectively. Frictional heat distribution is then performed based on the thermal conductivity, material density, specific heat capacity at constant pressure, and tangential velocity at the meshing point of the sun gear, planet gears, and internal gear ring. The distributed heat power is then substituted into the gear frictional heat generation and convection heat dissipation balance equation to obtain the equivalent heat convection coefficient and surface temperature of each gear over time. The sun gear and planet gears heat up rapidly in the initial stage of operation, then gradually stabilize; the internal gear ring, due to its larger mass and heat capacity, heats up relatively slowly. Under the initial operating conditions of the application example, the steady-state surface temperatures of the sun gear, planet gears, and internal gear ring are 70.54℃, 54.08℃, and 31.79℃, respectively. By comparing the tooth surface temperature and temperature rise of different gears, the differences in frictional heat generation and thermal response of each component in a closed differential herringbone gear transmission system can be characterized.

[0044] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and do not limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the protection scope of the present invention.

Claims

1. A method for characterizing the temperature rise of the tooth surface of a closed differential herringbone gear, characterized in that, Includes the following steps: A dynamic model of a closed differential herringbone gear transmission system is established, and the dynamic model is solved to obtain the dynamic meshing force of the external meshing pair of the sun gear and planet gear and the dynamic meshing force of the internal meshing pair of the planet gear and the internal gear ring. The contact line length of a single pair of teeth in the external meshing pair and the internal meshing pair is determined based on the end face overlap, axial overlap and total overlap of the herringbone gear. Based on the contact line length of a single pair of teeth, the load distribution ratio of a single pair of teeth in the external meshing pair and the internal meshing pair is determined respectively, and the dynamic meshing force of the corresponding meshing pair is converted into a normal load of a single pair of teeth according to the load distribution ratio. Calculate the contact point radius of curvature, entrainment speed, sliding speed and rolling ratio of the external and internal meshing pairs. Input the normal load of a single pair of gear teeth, as well as the radius of curvature, entrainment speed, sliding speed and rolling ratio, into the finite-length line contact dynamic thermo-elasto-fluid lubrication model, and solve the model to obtain the oil film pressure, oil film thickness, lubricating oil temperature and friction coefficient of the external and internal meshing pairs. Based on the friction coefficient, the normal load on a single pair of gear teeth, and the sliding speed, calculate the friction power loss of the external meshing pair and the internal meshing pair respectively. The heat distribution coefficient is determined based on the thermal conductivity, density, specific heat capacity at constant pressure, and tangential velocity at the meshing point of the two meshing gears. The frictional power loss of the external meshing pair is distributed to the sun gear and planet gear according to the heat distribution coefficient between the sun gear and planet gear. The frictional power loss of the internal meshing pair is distributed to the planet gear and internal gear according to the heat distribution coefficient between the planet gear and the internal gear ring. The heat power obtained by the sun gear, planet gear and internal gear ring are obtained respectively. Based on the heat power obtained by each gear, a heat balance equation is established between frictional heat generation and convective heat transfer of each gear. The heat balance equation is then solved to obtain the temperature rise of the tooth surfaces of the sun gear, planet gears, and internal gear ring.

2. The method according to claim 1, characterized in that, The heat distribution coefficient is determined according to the following formula: Determine the heat distribution coefficient; where, This represents the thermal conductivity of the i-th meshing gear; This represents the material density of the i-th meshing gear; This represents the specific heat capacity at constant pressure of the i-th meshing gear; This represents the tangential velocity at the meshing point of the i-th meshing gear.

3. The method according to claim 1, characterized in that, The heat power obtained by the sun gear is determined based on the heat distribution coefficient of the sun gear and the frictional power loss of the external meshing pair; the heat power obtained by a single planet gear is determined based on the heat distribution coefficient of the planet gear, the sum of the frictional power losses of the external and internal meshing pairs, and the number of planet gears; the heat power obtained by the internal gear ring is determined based on the heat distribution coefficient of the internal gear ring and the frictional power loss of the internal meshing pair.

4. The method according to claim 1, characterized in that, The load distribution ratio is the ratio of the contact line length of a single pair of gear teeth to the sum of the contact line lengths of all pairs of gear teeth participating in meshing simultaneously; the normal load of a single pair of gear teeth is determined based on the load distribution ratio and the dynamic meshing force of the corresponding meshing pair.

5. The method according to claim 1, characterized in that, The coefficient of friction is determined based on the hydrodynamic friction force and the normal load on a single pair of gear teeth; the hydrodynamic friction force is determined based on the lubricating oil viscosity, contact half-width, central film thickness, and average sliding speed along the meshing line direction.

6. The method according to claim 1, characterized in that, The frictional power loss of the external and internal meshing pairs is the product of the average instantaneous meshing power loss of a single pair of teeth within their respective meshing cycles and the number of planetary gears. The instantaneous meshing power loss of a single pair of teeth is the product of the friction coefficient, the normal load of a single pair of teeth, and the sliding speed.

7. The method according to claim 1, characterized in that, The heat balance equation is: ;in, Indicates the mass of the gear. This indicates the specific heat capacity at constant pressure of the gear material. This indicates the average temperature of the tooth surface. This indicates the thermal power obtained by the gear. Indicates the equivalent convective heat transfer coefficient. This represents the equivalent heat transfer area. This indicates the initial temperature of the lubricating oil.

8. The method according to claim 1, characterized in that, The tooth surface temperature rise is the difference between the steady-state tooth surface temperature and the initial temperature of the lubricating oil. The steady-state tooth surface temperature is determined based on the initial temperature of the lubricating oil, the average heat power during the meshing cycle, the equivalent convective heat transfer coefficient, and the equivalent heat transfer area.

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