A novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears

Abstract

This invention proposes a novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears. First, based on load-bearing meshing contact analysis, the comprehensive curvature, entrainment velocity, sliding velocity, and load distribution of the contact points throughout the entire meshing cycle are extracted. Second, a transient thermo-elasto-fluidic lubrication numerical model is established to couple elastic deformation, energy equations, and nonlinear properties of the lubricating oil. Then, an algorithm is used to solve for the oil film pressure field, temperature field, and film thickness distribution. This invention can objectively quantify the lubrication performance of the novel modified tooth surface under high-speed, heavy-load conditions, focusing on how to scientifically and accurately evaluate the lubrication benefits brought by the modification. It provides reproducible and quantifiable technical support for the optimized design and engineering verification of high-performance spiral bevel gears, and offers a certain theoretical reference for evaluating the lubrication performance of complex spiral bevel gear tooth surfaces.

Description

Technical Field

[0001] This invention relates to the field of gear lubrication analysis technology, and in particular to a novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears. Background Technology

[0002] Spiral bevel gears, due to their high load-bearing capacity and smooth transmission characteristics, are widely used in key transmission systems of high-end equipment such as aero engines, helicopter main reduction gears, and heavy vehicle drive axles. As equipment develops towards higher speeds, heavier loads, and lighter weights, the working conditions of gear pairs are becoming increasingly demanding: Hertzian pressures in the contact area can reach several GPa, frictional temperature rises dramatically, and under extreme conditions, lubricating oil temperatures can even exceed 300°C. High temperatures cause a significant decrease in lubricating oil viscosity, while high pressure compresses the oil film thickness. The coupling effect of these two factors easily disrupts the elastohydrodynamic lubrication state, leading to surface failures such as galling and micropitting, seriously threatening system reliability.

[0003] To improve meshing performance, advanced tooth surface modification techniques such as high overlap ratio design and high-order transmission error modification (such as fourth-order and seventh-order parabolic profiles) have been widely adopted in recent years. These modifications can theoretically improve lubrication conditions by controlling the contact path, dispersing loads, and suppressing dynamic excitation.

[0004] However, the complex modified geometry also presents new challenges to the accurate evaluation of lubrication performance. Traditional lubrication analysis is mostly based on ideal smooth surfaces or simplified line contact models, which makes it difficult to reflect the non-uniform pressure distribution, local velocity gradients, and transient thermal effects of the actual contact spots after modification.

[0005] Meanwhile, existing evaluation methods often rely on empirical formulas or static assumptions, which cannot effectively couple the time-varying geometry, kinematics and load characteristics of the modified tooth surface during the entire meshing cycle, resulting in large prediction deviations for key indicators such as minimum oil film thickness and peak temperature.

[0006] Therefore, there is an urgent need to develop a refined and quantitative lubrication performance evaluation method that can deeply integrate the precise geometric model of the modified tooth surface with the transient thermo-elasto-fluidic lubrication mechanism, so as to support the design verification and optimization iteration of high-performance spiral bevel gears. Summary of the Invention

[0007] This invention proposes a novel method for evaluating the lubrication performance of modified tooth surfaces in spiral bevel gears, enabling objective quantification of the lubrication performance of these modified tooth surfaces under high-speed, heavy-load conditions. Compared to solutions that only focus on the modification design itself, this invention focuses on how to scientifically and accurately evaluate the lubrication benefits brought about by the modification. It provides reproducible and quantifiable technical support for the optimized design and engineering verification of high-performance spiral bevel gears, and offers a theoretical reference for evaluating the lubrication performance of complex spiral bevel gear tooth surfaces. It deeply integrates the precise geometric model of the modified tooth surface with the transient thermo-elasto-fluidic lubrication mechanism to achieve a refined and quantitative lubrication performance evaluation method, which can support the design verification and optimization iteration of high-performance spiral bevel gears.

[0008] The present invention adopts the following technical solution.

[0009] A novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears is proposed. First, based on load-bearing meshing contact analysis, the comprehensive curvature, entrainment speed, sliding speed, and load distribution of the contact points within the entire meshing cycle are accurately extracted. Second, a transient thermo-elastohydrodynamic lubrication numerical model is established to couple elastic deformation, energy equations, and nonlinear properties of lubricating oil. Then, an efficient numerical algorithm is used to solve for the oil film pressure field, temperature field, and film thickness distribution.

[0010] The method includes the following steps;

[0011] Step 1: Coordinated design of contact ratio and transmission error of spiral bevel gears;

[0012] Step 2: Establish a novel spiral bevel gear meshing simulation model;

[0013] Step 3: Contact and motion analysis of the meshing point on the tooth surface;

[0014] Step 4: Establish a thermo-elasto-fluidic lubrication simulation model for the new spiral bevel gear;

[0015] Step 5: Evaluation of the lubrication performance of the modified spiral bevel gear.

[0016] Step 1 specifically involves:

[0017] In terms of overlap ratio design, the overlap ratio Cr of spiral bevel gears is defined as the ratio of the angle traversed by a single tooth of the pinion from engagement to disengagement to the angle traversed by the pinion in one meshing cycle:

[0018]

[0019] In the formula, T c T is the angle traversed by a single tooth of the pinion from engagement to disengagement. m T is the angle traversed by the small wheel in one meshing cycle, obtained by the following formula. m =2π / Z1, where Z1 is the number of teeth on the pinion.

[0020] Only need to add T c This can increase the overlap ratio, which manifests on the tooth surface as a reduction in the angle between the contact path and the root cone, such as... Figure 1 As shown. Therefore, controlling the contact path to tilt towards the root cone can increase T. c .

[0021] In terms of transmission error design, a seventh-order concave transmission error curve is used as the preset transmission error curve, such as... Figure 2 The entire meshing process is controlled by five points: the entry point A1 (T1, δ1), the first peak point A2 (T2, δ2), the minimum point A4 (T4, δ4), the second peak point A4 (T4, δ4), and the disengagement point A5 (T5, δ5). The first peak point A2 and the second peak point A4 are respectively the pass coefficients. and To determine the relative relationship between them and the engagement angle of the intermediate transition point A3;

[0022] The expression for the seventh-order transmission error curve is given by the following formula:

[0023]

[0024] In this equation, a1~a7 are the polynomial coefficients, and X and Y are represented by the following formula:

[0025]

[0026] Differentiating the above seventh-order polynomial, we obtain the first derivative as:

[0027]

[0028] Based on the quantitative relationship of five reference points on the high-order transmission error curve, constraint conditions and constraint equations are established, and the coefficients of the seventh-order polynomial are determined by solving the constraint equations.

[0029] Step 2 specifically involves: simulating a hypothetical gear using a rocking table mechanism. During the simulation, the cutting surface of the cutter head is considered the tooth surface of the hypothetical gear, and the cutting process is the meshing process between the hypothetical gear and the gear being machined. Figure 3 .

[0030] Based on differential geometry, gear meshing principle, and homogeneous coordinate transformation, mathematical models of the working surfaces (concave surface of the small gear and convex surface of the large gear) of the tool tooth profile are established respectively.

[0031] The following equation describes the tooth profile modeling method for small wheel cutting tools.

[0032]

[0033] in , For the tool surface, R p For the blade tip radius and Tooth profile angle;

[0034] , Let be the angular velocity of the rocking platform and the small wheel. and , and The relationship is as follows:

[0035]

[0036] in, For rolling ratio, The coefficients are from the 2nd to the 7th order;

[0037]

[0038] Among them, K 1d1 ,K d1c1 ,K c1b1 ,K b1a1 and K a1p1 Let Q be the homogeneous transformation matrix. 1d1 Q d1c1 Q c1b1 Q b1a1 and Q a1p1 Representing K 1d1 ,K d1c1 ,K c1b1 ,K b1a1 and K a1p1 The top-left 3 × 3 submatrix;

[0039]

[0040]

[0041]

[0042]

[0043]

[0044] The method used for modeling the tooth surface of the large wheel is the same as or similar to that used for modeling the tooth surface of the small wheel, so it will not be described again here.

[0045] Step 3 specifically involves the following: In the thermo-elasto-fluidic lubrication analysis of spiral bevel gears, the load at the contact point, contact curvature, sliding speed, and entrainment speed are essential. The analysis process is as follows: Figure 4The load and contact curvature were obtained using a load-bearing meshing simulation software package, while the sliding velocity and entrainment velocity required kinematic analysis. The bending characteristics of spiral bevel gears are complex, and the relative motion relationship between the large and small gears is highly intricate. For the basic motion transformation at the loading meshing point M, the surface velocity is:

[0046]

[0047]

[0048] Where P represents the pinion, G represents the gear, and v t v represents the tangential velocity at the point of contact. n v represents the normal velocity at the point of contact. t×n R represents the velocity at the contact point that is simultaneously perpendicular to the tangential and normal directions; Z represents the vector from the gear's rotation center to the contact point M; W represents the direction of the gear's rotation axis; and ω represents the rotational speed.

[0049] in, , .

[0050] Sliding speed U s and entrainment speed U e It was concluded that:

[0051]

[0052]

[0053] in, The angle between the sliding velocity and the minor axis of the contact ellipse. The angle between the entrainment velocity and the minor axis of the contact ellipse;

[0054] Finally, the slide-roll ratio (SRR) at the contact point can be defined as SRR=U s / U e .

[0055] Step 4 specifically involves: the pressure distribution in the lubrication contact area of ​​the spiral bevel gear is controlled by the Reynolds equation, establishing a quasi-steady-state Reynolds equation:

[0056]

[0057] Where p represents pressure and h represents oil film thickness. This represents the current viscosity of the lubricant.

[0058] The viscosity-temperature calculation formula is:

[0059]

[0060] in, T represents the initial viscosity, T0 represents the initial temperature, and T represents the current temperature. and Represented as

[0061]

[0062]

[0063] α is the viscosity-pressure coefficient, and β is the viscosity-temperature coefficient.

[0064] Then, considering the macroscopic contact geometry and surface elastic deformation, the oil film thickness equation is:

[0065]

[0066] Among them, R x and R y These are the radii of curvature along the minor and major axes of the Hertzian contact ellipse, respectively.

[0067] V e via Boussinesq The integral calculation yielded the following:

[0068]

[0069] ; v1 and v2 are the elastic moduli of the large and small gears, respectively, and the Poisson's ratios of the large and small gears, respectively.

[0070] The method for calculating the density of lubricating oil is as follows:

[0071]

[0072] D is the coefficient of thermal expansion, which is usually taken as -0.00065. This is the initial density of the lubricant;

[0073] To accurately calculate the temperature distribution of the lubricating oil, it is necessary to solve the energy equation. Ignoring the pressure difference in the z-direction and heat transfer in the x and y directions, the energy equation is:

[0074]

[0075] In the formula c p and k f These represent the specific heat capacity and thermal conductivity of lubricating oil, respectively. Indicates equivalent viscosity;

[0076] Boundary conditions for temperature solution:

[0077]

[0078] Where T0 is the initial temperature; and c1 and c2, K1 and K2, u1 and u2 are the density, specific heat, heat transfer coefficient and shear rate of the upper and lower surface materials, respectively.

[0079] Step 5 specifically involves the following: The thermo-elastohydrodynamic lubrication equation is typically a nonlinear differential equation. During the solution process, the Reynolds equation is discretized using the finite difference method, and a semi-system method is employed to ensure the convergence and numerical stability of the solution under heavy load conditions. The coefficient matrix of the discrete Reynolds equation established using this method is related not only to the pressure flow term on the left-hand side of the equation but also to the shear flow term on the right-hand side. This ensures that even when the pressure flow term approaches zero under heavy load, the coefficient matrix maintains diagonal dominance. Then, the Gauss-Seidel row-by-row iterative method is used to solve the discrete Reynolds equation, and a multigrid method is employed to improve the convergence speed. Finally, a multigrid integration method is used to improve the solution speed of the elastic deformation equation.

[0080] In step 2, the spiral bevel gear is a gear machined on a spiral bevel gear milling machine, wherein the large gear is machined using the double-sided method and the small gear is machined using the modification method.

[0081] The novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears was validated using a pair of spiral bevel gears. During the validation process, the gear pair was designed with high overlap ratio and high-order transmission error in synergy. The mathematical model of the gear was established using the double-sided method and the modification method. Then, an accurate thermo-elastohydrodynamic lubrication model was established, and the Reynolds equation was solved using MATLAB programming to analyze the lubrication performance of the spiral bevel gear pair.

[0082] The verification process includes the following steps;

[0083] The study takes a pair of ordinary spiral bevel gears as an example, and the basic parameters of the gear blank are shown in Table 1.

[0084] Table 1. Geometric parameters of wheel blank

[0085]

[0086] Step 1: Collaborative design of conventional ordinary spiral bevel gears with high overlap ratio and high-order transmission error;

[0087] After the collaborative design of high overlap ratio and high-order transmission error, the gear pair machining parameters are shown in Table 2.

[0088] Table 2. Gear Pair Machining Parameters

[0089]

[0090] Based on the machining parameters in Table 2, calculate the transmission error and contact path; for example... Figure 5 6, 7; After calculation, the overlap ratio after shaping reaches 2.59, and the transmission error curve also meets the requirements. Therefore, the above processing parameters realize the design results of overlap ratio and transmission error.

[0091] Step 2: Determine the parameters at the meshing point of the gear pair; for ease of operation, rely on the existing MATLAB program for load meshing contact analysis to analyze the modified gear pair, and modify the original program to calculate the various input parameters required for thermo-elasto-fluidic lubrication analysis; set the input speed to 500 r / min and the torque to 150 N·m.

[0092] Based on contact analysis, the contact points during meshing were obtained. and The combined radius of curvature in the direction, such as Figure 8 As shown; through kinematic analysis, the entrainment velocity and sliding velocity at the contact point were calculated, and the results are as follows. Figure 9 As shown; in addition, using existing load-bearing meshing contact analysis programs, the load distribution at the contact points is obtained, as shown. Figure 10 As shown.

[0093] Step 3: Analyze the lubrication performance of the gear pair; based on the established thermo-elasto-fluidic lubrication model, use MATLAB programming to solve the mathematical model, and use the Gauss-Seidel line-by-line iterative method to calculate the maximum oil film pressure, maximum oil film thickness and minimum oil film thickness at the meshing point of the spiral bevel gear.

[0094] Table 3 shows the gear material properties and lubricant material properties at 40 °C (313 K) required for solving the thermo-elasto-fluidic lubrication model. These data can be used to solve the Reynolds equation for evaluating the thermo-elasto-fluidic performance of the spiral bevel gear. The computational domain of the Reynolds equation is −2.5 ≤ X ≤ 1.5, −2 ≤ Y ≤ 2, with 65 × 65 nodes. The solution of the temperature field governing equation is consistent with the computational domain of the Reynolds equation in the X and Y directions. Five equivalent nodes are set in the Z direction, and the computational domain of the energy equation is 0 ≤ Z ≤ 1.

[0095] Table 3. Gear and Lubricating Oil Material Parameters

[0096]

[0097] Figure 11 This refers to the maximum oil film pressure at each engagement point within a single engagement cycle. Figure 12 This represents the maximum temperature rise of the oil film. Figure 13 This represents the minimum oil film thickness.

[0098] like Figure 11 As shown in Figures 12 and 13, the thermo-elastohydrodynamic lubrication performance evaluation method developed in this study successfully captured the typical lubrication response characteristics of modified tooth surfaces during meshing: the oil film pressure distribution exhibits an "M"-shaped trend, highly consistent with the load distribution; the highest oil film temperature shows a double peak on both sides of the node, at 64.5°C and 71.8°C, respectively; the minimum oil film thickness is 0.175 μm, appearing near the node. These results not only conform to physical laws but also fully verify the effectiveness and reliability of this evaluation method in accurately characterizing the lubrication behavior of complex modified tooth surfaces.

[0099] Compared with the prior art, the present invention has the following beneficial effects:

[0100] This invention provides a method for evaluating the lubrication performance of a novel modified tooth surface for spiral bevel gears. First, based on load-bearing meshing contact analysis, the comprehensive curvature, entrainment velocity, sliding velocity, and load distribution of the contact points throughout the entire meshing cycle are accurately extracted. Second, a transient thermo-elasto-fluidic lubrication numerical model is established, coupling elastic deformation, energy equations, and nonlinear properties of the lubricating oil. An efficient numerical algorithm is used to solve for the oil film pressure field, temperature field, and film thickness distribution. This method can objectively quantify the lubrication performance of the novel modified tooth surface under high-speed, heavy-load conditions. Compared to focusing solely on the modification design itself, this invention focuses on how to scientifically and accurately evaluate the lubrication benefits brought by the modification, providing reproducible and quantifiable technical support for the optimized design and engineering verification of high-performance spiral bevel gears, and offering a certain theoretical reference for evaluating the lubrication performance of complex spiral bevel gear tooth surfaces. Attached Figure Description

[0101] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0102] Appendix Figure 1 This is a schematic diagram illustrating the variation of the contact ratio of the spiral bevel gear in a preferred embodiment of the present invention;

[0103] Appendix Figure 2 This is a schematic diagram of the high-order transmission error design of the spiral bevel gear according to a preferred embodiment of the present invention;

[0104] Appendix Figure 3 This is a schematic diagram of coordinate transformation for machining the tooth surface of an arc-shaped bevel gear according to a preferred embodiment of the present invention;

[0105] Appendix Figure 4 This is a schematic diagram of the contact analysis model of the spiral bevel gear according to a preferred embodiment of the present invention;

[0106] Appendix Figure 5 This is a schematic diagram of the pinion meshing contact trajectory after the spiral bevel gear has been modified according to a preferred embodiment of the present invention.

[0107] Appendix Figure 6This is a schematic diagram of the meshing contact trajectory of the large gear after the spiral bevel gear has been modified according to a preferred embodiment of the present invention.

[0108] Appendix Figure 7 This is a schematic diagram illustrating the transmission error of the spiral bevel gear after modification according to a preferred embodiment of the present invention.

[0109] Appendix Figure 8 This is a schematic diagram of the combined radius of curvature in the x and y directions of the spiral bevel gear after modification according to a preferred embodiment of the present invention;

[0110] Appendix Figure 9 This is a schematic diagram of the sliding speed and entrainment speed at the meshing point of the spiral bevel gear after modification, according to a preferred embodiment of the present invention.

[0111] Appendix Figure 10 This is a schematic diagram of the contact load after the spiral bevel gear has been modified according to a preferred embodiment of the present invention.

[0112] Appendix Figure 11 This is a schematic diagram of the maximum oil film pressure at the meshing point of the spiral bevel gear after modification, according to a preferred embodiment of the present invention.

[0113] Appendix Figure 12 A schematic diagram showing the maximum oil film temperature at the meshing point of the spiral bevel gear after modification according to a preferred embodiment of the present invention;

[0114] Appendix Figure 13 This is a schematic diagram of the minimum oil film thickness at the meshing point of the spiral bevel gear after modification, according to a preferred embodiment of the present invention. Detailed Implementation

[0115] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0116] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0117] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations according to this application; as used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise; furthermore, it should be understood that when the terms “comprising” and / or “including” are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0118] As shown in the figure, a novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears is proposed. First, based on load-bearing meshing contact analysis, the comprehensive curvature, entrainment speed, sliding speed, and load distribution of the contact points within the entire meshing cycle are accurately extracted. Second, a transient thermo-elastohydrodynamic lubrication numerical model is established to couple elastic deformation, energy equations, and nonlinear properties of lubricating oil. Then, an efficient numerical algorithm is used to solve for the oil film pressure field, temperature field, and film thickness distribution.

[0119] The method includes the following steps;

[0120] Step 1: Coordinated design of contact ratio and transmission error of spiral bevel gears;

[0121] Step 2: Establish a novel spiral bevel gear meshing simulation model;

[0122] Step 3: Contact and motion analysis of the meshing point on the tooth surface;

[0123] Step 4: Establish a thermo-elasto-fluidic lubrication simulation model for the new spiral bevel gear;

[0124] Step 5: Evaluation of the lubrication performance of the modified spiral bevel gear.

[0125] Step 1 specifically involves:

[0126] In terms of overlap ratio design, the overlap ratio Cr of spiral bevel gears is defined as the ratio of the angle traversed by a single tooth of the pinion from engagement to disengagement to the angle traversed by the pinion in one meshing cycle:

[0127]

[0128] In the formula, T c T is the angle traversed by a single tooth of the pinion from engagement to disengagement. m T is the angle traversed by the small wheel in one meshing cycle, obtained by the following formula. m =2π / Z1, where Z1 is the number of teeth on the pinion.

[0129] Only need to add T c This can increase the overlap ratio, which manifests on the tooth surface as a reduction in the angle between the contact path and the root cone, such as... Figure 1 As shown. Therefore, controlling the contact path to tilt towards the root cone can increase T. c .

[0130] In terms of transmission error design, a seventh-order concave transmission error curve is used as the preset transmission error curve, such as... Figure 2The entire meshing process is controlled by five points: the entry point A1 (T1, δ1), the first peak point A2 (T2, δ2), the minimum point A4 (T4, δ4), the second peak point A4 (T4, δ4), and the disengagement point A5 (T5, δ5). The first peak point A2 and the second peak point A4 are respectively the pass coefficients. and To determine the relative relationship between them and the engagement angle of the intermediate transition point A3;

[0131] The expression for the seventh-order transmission error curve is given by the following formula:

[0132]

[0133] In this equation, a1~a7 are the polynomial coefficients, and X and Y are represented by the following formula:

[0134]

[0135] Differentiating the above seventh-order polynomial, we obtain the first derivative as:

[0136]

[0137] Based on the quantitative relationship of five reference points on the high-order transmission error curve, constraint conditions and constraint equations are established, and the coefficients of the seventh-order polynomial are determined by solving the constraint equations.

[0138] Step 2 specifically involves: simulating a hypothetical gear using a rocking table mechanism. During the simulation, the cutting surface of the cutter head is considered the tooth surface of the hypothetical gear, and the cutting process is the meshing process between the hypothetical gear and the gear being machined. Figure 3 .

[0139] Based on differential geometry, gear meshing principle, and homogeneous coordinate transformation, mathematical models of the working surfaces (concave surface of the small gear and convex surface of the large gear) of the tool tooth profile are established respectively.

[0140] The following equation describes the tooth profile modeling method for small wheel cutting tools.

[0141]

[0142] in , For the tool surface, R p For the blade tip radius and Tooth profile angle;

[0143] , Let be the angular velocity of the rocking platform and the small wheel. and , and The relationship is as follows:

[0144]

[0145] in, For rolling ratio, The coefficients are from the 2nd to the 7th order;

[0146]

[0147] Among them, K 1d1 ,K d1c1 ,K c1b1 ,K b1a1 and K a1p1 Let Q be the homogeneous transformation matrix. 1d1 Q d1c1 Q c1b1 Q b1a1 and Q a1p1 Representing K 1d1 ,K d1c1 ,K c1b1 ,K b1a1 and K a1p1 The top-left 3 × 3 submatrix;

[0148]

[0149]

[0150]

[0151]

[0152]

[0153] The method used for modeling the tooth surface of the large wheel is the same as or similar to that used for modeling the tooth surface of the small wheel, so it will not be described again here.

[0154] Step 3 specifically involves the following: In the thermo-elasto-fluidic lubrication analysis of spiral bevel gears, the load at the contact point, contact curvature, sliding speed, and entrainment speed are essential. The analysis process is as follows: Figure 4 The load and contact curvature were obtained using a load-bearing meshing simulation software package, while the sliding velocity and entrainment velocity required kinematic analysis. The bending characteristics of spiral bevel gears are complex, and the relative motion relationship between the large and small gears is highly intricate. For the basic motion transformation at the loading meshing point M, the surface velocity is:

[0155]

[0156]

[0157] Where P represents the pinion, G represents the gear, and v t v represents the tangential velocity at the point of contact. n v represents the normal velocity at the point of contact. t×n R represents the velocity at the contact point that is simultaneously perpendicular to the tangential and normal directions; Z represents the vector from the gear's rotation center to the contact point M; W represents the direction of the gear's rotation axis; and ω represents the rotational speed.

[0158] in, , .

[0159] Sliding speed U s and entrainment speed U e It was concluded that:

[0160]

[0161]

[0162] in, The angle between the sliding velocity and the minor axis of the contact ellipse. The angle between the entrainment velocity and the minor axis of the contact ellipse;

[0163] Finally, the slide-roll ratio (SRR) at the contact point can be defined as SRR=U s / U e .

[0164] Step 4 specifically involves: the pressure distribution in the lubrication contact area of ​​the spiral bevel gear is controlled by the Reynolds equation, establishing a quasi-steady-state Reynolds equation:

[0165]

[0166] Where p represents pressure and h represents oil film thickness. This represents the current viscosity of the lubricant.

[0167] The viscosity-temperature calculation formula is:

[0168]

[0169] in, T represents the initial viscosity, T0 represents the initial temperature, and T represents the current temperature. and Represented as

[0170]

[0171]

[0172] α is the viscosity-pressure coefficient, and β is the viscosity-temperature coefficient.

[0173] Then, considering the macroscopic contact geometry and surface elastic deformation, the oil film thickness equation is:

[0174]

[0175] Among them, R x and R y These are the radii of curvature along the minor and major axes of the Hertzian contact ellipse, respectively.

[0176] V e via Boussinesq The integral calculation yielded the following:

[0177]

[0178] ; v1 and v2 are the elastic moduli of the large and small gears, respectively, and the Poisson's ratios of the large and small gears, respectively.

[0179] The method for calculating the density of lubricating oil is as follows:

[0180]

[0181] D is the coefficient of thermal expansion, which is usually taken as -0.00065. This is the initial density of the lubricant;

[0182] To accurately calculate the temperature distribution of the lubricating oil, it is necessary to solve the energy equation. Ignoring the pressure difference in the z-direction and heat transfer in the x and y directions, the energy equation is:

[0183]

[0184] In the formula c p and k f These represent the specific heat capacity and thermal conductivity of lubricating oil, respectively. Indicates equivalent viscosity;

[0185] Boundary conditions for temperature solution:

[0186]

[0187] Where T0 is the initial temperature; and c1 and c2, K1 and K2, u1 and u2 are the density, specific heat, heat transfer coefficient and shear rate of the upper and lower surface materials, respectively.

[0188] Step 5 specifically involves the following: The thermo-elastohydrodynamic lubrication equation is typically a nonlinear differential equation. During the solution process, the Reynolds equation is discretized using the finite difference method, and a semi-system method is employed to ensure the convergence and numerical stability of the solution under heavy load conditions. The coefficient matrix of the discrete Reynolds equation established using this method is related not only to the pressure flow term on the left-hand side of the equation but also to the shear flow term on the right-hand side. This ensures that even when the pressure flow term approaches zero under heavy load, the coefficient matrix maintains diagonal dominance. Then, the Gauss-Seidel row-by-row iterative method is used to solve the discrete Reynolds equation, and a multigrid method is employed to improve the convergence speed. Finally, a multigrid integration method is used to improve the solution speed of the elastic deformation equation.

[0189] In step 2, the spiral bevel gear is a gear machined on a spiral bevel gear milling machine, wherein the large gear is machined using the double-sided method and the small gear is machined using the modification method.

[0190] The novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears was validated using a pair of spiral bevel gears. During the validation process, the gear pair was designed with high overlap ratio and high-order transmission error in synergy. The mathematical model of the gear was established using the double-sided method and the modification method. Then, an accurate thermo-elastohydrodynamic lubrication model was established, and the Reynolds equation was solved using MATLAB programming to analyze the lubrication performance of the spiral bevel gear pair.

[0191] The verification process includes the following steps;

[0192] The study takes a pair of ordinary spiral bevel gears as an example, and the basic parameters of the gear blank are shown in Table 1.

[0193] Table 1. Geometric parameters of wheel blank

[0194]

[0195] Step 1: Collaborative design of conventional ordinary spiral bevel gears with high overlap ratio and high-order transmission error;

[0196] After the collaborative design of high overlap ratio and high-order transmission error, the gear pair machining parameters are shown in Table 2.

[0197] Table 2. Gear Pair Machining Parameters

[0198]

[0199] Based on the machining parameters in Table 2, calculate the transmission error and contact path; for example... Figure 5 6, 7; After calculation, the overlap ratio after shaping reaches 2.59, and the transmission error curve also meets the requirements. Therefore, the above processing parameters realize the design results of overlap ratio and transmission error.

[0200] Step 2: Determine the parameters at the meshing point of the gear pair; for ease of operation, rely on the existing MATLAB program for load meshing contact analysis to analyze the modified gear pair, and modify the original program to calculate the various input parameters required for thermo-elasto-fluidic lubrication analysis; set the input speed to 500 r / min and the torque to 150 N·m.

[0201] Based on contact analysis, the contact points during meshing were obtained. and The combined radius of curvature in the direction, such as Figure 8 As shown; through kinematic analysis, the entrainment velocity and sliding velocity at the contact point were calculated, and the results are as follows. Figure 9 As shown; in addition, using existing load-bearing meshing contact analysis programs, the load distribution at the contact points is obtained, as shown. Figure 10 As shown.

[0202] Step 3: Analyze the lubrication performance of the gear pair; based on the established thermo-elasto-fluidic lubrication model, use MATLAB programming to solve the mathematical model, and use the Gauss-Seidel line-by-line iterative method to calculate the maximum oil film pressure, maximum oil film thickness and minimum oil film thickness at the meshing point of the spiral bevel gear.

[0203] Table 3 shows the gear material properties and lubricant material properties at 40 °C (313 K) required for solving the thermo-elasto-fluidic lubrication model. These data can be used to solve the Reynolds equation for evaluating the thermo-elasto-fluidic performance of the spiral bevel gear. The computational domain of the Reynolds equation is −2.5 ≤ X ≤ 1.5, −2 ≤ Y ≤ 2, with 65 × 65 nodes. The solution of the temperature field governing equation is consistent with the computational domain of the Reynolds equation in the X and Y directions. Five equivalent nodes are set in the Z direction, and the computational domain of the energy equation is 0 ≤ Z ≤ 1.

[0204] Table 3. Gear and Lubricating Oil Material Parameters

[0205]

[0206] Figure 11 This refers to the maximum oil film pressure at each engagement point within a single engagement cycle. Figure 12 This represents the maximum temperature rise of the oil film. Figure 13 This represents the minimum oil film thickness.

[0207] like Figure 11As shown in Figures 12 and 13, the thermo-elastohydrodynamic lubrication performance evaluation method developed in this study successfully captured the typical lubrication response characteristics of modified tooth surfaces during meshing: the oil film pressure distribution exhibits an "M"-shaped trend, highly consistent with the load distribution; the highest oil film temperature shows a double peak on both sides of the node, at 64.5°C and 71.8°C, respectively; the minimum oil film thickness is 0.175 μm, appearing near the node. These results not only conform to physical laws but also fully verify the effectiveness and reliability of this evaluation method in accurately characterizing the lubrication behavior of complex modified tooth surfaces.

[0208] Unlike traditional methods that rely solely on experience or simplified models, this approach integrates high-precision contact analysis with transient thermo-elasto-hydrodynamic simulation. It systematically integrates the influence of key design parameters such as overlap ratio and transmission error on lubrication conditions, achieving quantitative prediction of oil film pressure, temperature rise, and thickness across the entire process and multiple fields. Therefore, this evaluation framework can not only be used to verify the lubrication benefits of specific modification schemes but also serve as a general tool, providing reproducible and quantifiable technical support for the scientific evaluation and optimization design of spiral bevel gear lubrication performance. It possesses significant methodological importance and engineering reference value.

[0209] The design advantages of this invention are demonstrated in the above example.

Claims

1. A method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear, characterized in that: First, based on the load-bearing meshing contact analysis, the comprehensive curvature, entrainment speed, sliding speed and load distribution of the contact points are extracted throughout the entire meshing cycle; Secondly, a transient thermo-elastohydrodynamic lubrication numerical model is established to couple elastic deformation, energy equations and nonlinear properties of lubricating oil, and then the oil film pressure field, temperature field and film thickness distribution are solved by algorithm.

2. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 1, characterized in that: The method includes the following steps; Step 1: Coordinated design of contact ratio and transmission error of spiral bevel gears; Step 2: Build a simulation model of arc-tooth bevel gear meshing; Step 3: Contact and motion analysis of the meshing point on the tooth surface; Step 4: Establish a thermo-elasto-fluidic lubrication simulation model for spiral bevel gears; Step 5: Evaluation of the lubrication performance of the modified spiral bevel gear.

3. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 2, characterized in that: Step 1 specifically involves: In terms of overlap ratio design, the overlap ratio Cr of spiral bevel gears is defined as the ratio of the angle traversed by a single tooth of the pinion from engagement to disengagement to the angle traversed by the pinion in one meshing cycle: In the formula, T c T is the angle traversed by a single tooth of the pinion from engagement to disengagement. m T is the angle traversed by the small wheel in one meshing cycle, obtained by the following formula. m =2π / Z1, where Z1 is the number of teeth on the pinion. Increase T c To increase the contact ratio, the angle between the contact path and the root cone on the tooth surface decreases. Controlling the contact path to tilt towards the root cone increases T. c ; In terms of transmission error design, a seventh-order concave transmission error curve is used as the preset transmission error curve. The meshing process is controlled by five points: the entry meshing point A1 (T1, δ1), the first peak point A2 (T2, δ2), the minimum point A4 (T4, δ4), the second peak point A4 (T4, δ4), and the disengagement point A5 (T5, δ5). The first peak point A2 and the second peak point A4 are respectively the pass coefficients. and To determine the relative relationship between them and the engagement angle of the intermediate transition point A3; The expression for the seventh-order transmission error curve is given by the following formula: In this equation, a1~a7 are the polynomial coefficients, and X and Y are represented by the following formula: Differentiating the above seventh-order polynomial, we obtain the first derivative as: Based on the quantitative relationship of the five reference points on the transmission error curve, constraints and constraint equations are established, and the coefficients of the seventh-order polynomial are determined by solving the constraint equations.

4. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 3, characterized in that: Step 2 specifically involves: simulating a hypothetical gear using a rocking table mechanism. During the simulation, the cutting surface of the cutter head is considered the tooth surface of the hypothetical gear, and the cutting process is the meshing process between the hypothetical gear and the gear being machined. Based on differential geometry, gear meshing principle, and homogeneous coordinate transformation, mathematical models of the working surfaces of the small and large gears are established from the tooth profiles of the cutting tools. The following equation describes the tooth profile modeling method for small wheel cutting tools. in , For the tool surface, R p For the blade tip radius and Tooth profile angle; , Let be the angular velocity of the rocking platform and the small wheel. and , and The relationship is as follows: in, For rolling ratio, The coefficients are from the 2nd to the 7th order; Among them, K 1d1 ,K d1c1 ,K c1b1 ,K b1a1 and K a1p1 Let Q be the homogeneous transformation matrix. 1d1 Q d1c1 Q c1b1 Q b1a1 and Q a1p1 Representing K 1d1 ,K d1c1 ,K c1b1 ,K b1a1 and K a1p1 The top-left 3 × 3 submatrix; 。 5. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 4, characterized in that: Step 3 specifically involves the following: In the thermo-elasto-fluidic lubrication analysis of spiral bevel gears, the analysis process for the load, contact curvature, sliding speed, and entrainment speed at the contact point is as follows: The load and contact curvature are obtained using a load-bearing meshing simulation software package, while the sliding speed and entrainment speed are subjected to kinematic analysis; for the basic motion transformation at the loaded meshing point M, its surface velocity is: Where P represents the pinion, G represents the gear, and v t v represents the tangential velocity at the point of contact. n v represents the normal velocity at the point of contact. t×n R represents the velocity at the contact point that is simultaneously perpendicular to the tangential and normal directions; Z represents the vector from the gear's rotation center to the contact point M; W represents the direction of the gear's rotation axis; and ω represents the rotational speed. in, , . Sliding speed U s and entrainment speed U e It was concluded that: in, The angle between the sliding velocity and the minor axis of the contact ellipse. The angle between the entrainment velocity and the minor axis of the contact ellipse; Finally, the roll ratio at the contact point is defined as SRR=U s / U e .

6. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 5, characterized in that: Step 4 specifically involves: the pressure distribution in the lubrication contact area of ​​the spiral bevel gear is controlled by the Reynolds equation, establishing a quasi-steady-state Reynolds equation: Where p represents pressure and h represents oil film thickness. This represents the current viscosity of the lubricant. The viscosity-temperature calculation formula is: in, T represents the initial viscosity, T0 represents the initial temperature, and T represents the current temperature. and Represented as α is the viscosity-pressure coefficient, and β is the viscosity-temperature coefficient. Then, considering the macroscopic contact geometry and surface elastic deformation, the oil film thickness equation is: Among them, R x and R y These are the radii of curvature along the minor and major axes of the Hertzian contact ellipse, respectively. V e via Boussinesq The integral calculation yielded the following: ; v1 and v2 are the elastic moduli of the large and small gears, respectively, and the Poisson's ratios of the large and small gears, respectively. The method for calculating the density of lubricating oil is as follows: D is the coefficient of thermal expansion. This is the initial density of the lubricant; To calculate the temperature distribution of the lubricating oil, solve the energy equation; neglecting the pressure difference in the z-direction and heat transfer in the x and y directions, the energy equation is: In the formula c p and k f These represent the specific heat capacity and thermal conductivity of lubricating oil, respectively. Indicates equivalent viscosity; Boundary conditions for temperature solution: Where T0 is the initial temperature; and c1 and c2, K1 and K2, u1 and u2 are the density, specific heat, heat transfer coefficient and shear rate of the upper and lower surface materials, respectively.

7. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 6, characterized in that: Step 5 specifically involves: discretizing the Reynolds equation using the finite difference method when solving the thermo-elastohydrodynamic lubrication equation, and employing a semi-system method to ensure the convergence and numerical stability of the solution under heavy load conditions. The coefficient matrix of the established discrete Reynolds equation is related to the pressure flow term and the shear flow term of the equation. Then, the Gauss-Seidel row-by-row iterative method is used to solve the discrete Reynolds equation, and the multigrid method is used to improve the convergence speed, while the multigrid integration method is used to improve the solution speed.

8. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 2, characterized in that: In step 2, the spiral bevel gear is a gear machined on a spiral bevel gear milling machine, wherein the large gear is machined using the double-sided method and the small gear is machined using the modification method.

9. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 7, characterized in that: The novel method for evaluating the lubrication performance of modified tooth surfaces of spiral bevel gears was validated using a pair of spiral bevel gears. During the validation process, the gear pair was designed with high overlap ratio and high-order transmission error in synergy. The mathematical model of the gear was established using the double-sided method and the modification method. Then, an accurate thermo-elastohydrodynamic lubrication model was established, and the Reynolds equation was solved using MATLAB programming to analyze the lubrication performance of the spiral bevel gear pair.

10. The method for evaluating the lubrication performance of a novel modified tooth surface of an arc-shaped bevel gear according to claim 9, characterized in that: The verification process includes the following steps; Step 1: Design spiral bevel gears based on high overlap ratio and high-order transmission error coordination; Based on the processing parameters, the transmission error and contact path are calculated; Step 2: Determine the parameters at the meshing point of the gear pair; using the MATLAB program for load meshing contact analysis, analyze the modified gear pair, and modify the original program to calculate the various input parameters required for thermo-elasto-fluidic lubrication analysis; Based on contact analysis, the contact points during meshing are obtained. and The combined radius of curvature in the direction is used to determine the load distribution at the contact point using existing load-bearing meshing contact analysis programs. Step 3: Analyze the lubrication performance of the gear pair; based on the established thermo-elasto-fluidic lubrication model, use MATLAB programming to solve the mathematical model, and use the Gauss-Seidel line-by-line iterative method to calculate the maximum oil film pressure, maximum oil film thickness and minimum oil film thickness at the meshing point of the spiral bevel gear.

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