A method and system for calculating the meshing efficiency of a spiral bevel gear

Abstract

This invention discloses a method and system for calculating the meshing efficiency of spiral bevel gears, including obtaining the blank parameters and machining parameters of the spiral bevel gear; calculating equivalent meshing characteristic parameters based on the blank parameters and machining parameters; calculating the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter; numerically calculating the non-Newtonian fluid thermo-elasto-fluid lubrication equation based on the first characteristic parameter using the separation flow method to obtain the oil film shear force; calculating the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force; and calculating the meshing efficiency of the spiral bevel gear based on the friction coefficient, blank parameters, and machining parameters. This method can combine the influence of instantaneous heavy load conditions, real rough surfaces, and lubricating oil rheological properties on gear lubrication performance, thereby improving the accuracy of spiral bevel gear meshing efficiency calculation.

Description

Technical Field

[0001] This invention relates to the technical field of calculating the meshing efficiency of spiral bevel gears, and in particular to a method and system for calculating the meshing efficiency of spiral bevel gears. Background Technology

[0002] Spiral bevel gears, due to their advantages such as high load-bearing capacity, high overlap ratio, and smooth transmission, have been widely used in high-speed, heavy-load applications in aerospace, automotive, and marine industries. However, the meshing power loss of bevel gear pairs affects the efficiency of the transmission system, mainly including power losses from friction, oil churning, and wind resistance, with friction power loss accounting for the majority. Therefore, establishing accurate calculation methods for gear pair friction power loss and meshing efficiency is of significant importance for improving gear transmission performance.

[0003] Due to the complex contact geometry, heavy-load conditions, and tooth surface roughness of spiral bevel gear pairs, the actual lubrication state of the gears is a hybrid lubrication exhibiting characteristics of both full thermo-elasto-fluid lubrication and boundary lubrication. Determining the ratio of oil film load and roughness peak load is quite complex. Contact analysis is the foundation of gear lubrication characteristic analysis. Traditional solutions for contact geometry and motion parameters are based on the theoretical contact trajectory of the gear pair. However, due to the special tooth profile and heavy-load conditions of spiral bevel gears, deformation under load causes the actual contact trajectory to deviate from the theoretical contact trajectory, and changes in the contact point position have a significant impact on gear lubrication performance. The contact analysis and lubrication calculation of spiral bevel gears involve the coupling of multiple disciplines such as differential geometry meshing principles, interface contact mechanics, and hydrodynamic lubrication mechanics, and the computational workload is enormous, posing a significant challenge to establishing a comprehensive and systematic method for calculating the meshing efficiency of spiral bevel gears. Therefore, it is urgent to develop a method for calculating meshing efficiency based on the lubrication characteristics of the actual load-bearing contact position of spiral bevel gears under a hybrid thermo-elasto-fluid lubrication state. Summary of the Invention

[0004] This invention aims to at least solve the technical problems existing in the prior art. To this end, this invention proposes a method and system for calculating the meshing efficiency of spiral bevel gears, which can improve the accuracy of calculating the meshing efficiency of spiral bevel gears by combining the influence of instantaneous heavy load conditions, real rough surfaces, and the rheological properties of lubricating oil on the gear lubrication performance.

[0005] In a first aspect, the present invention provides a method for calculating the meshing efficiency of a spiral bevel gear, comprising the following steps:

[0006] Obtain the blank parameters and machining parameters of the spiral bevel gear;

[0007] Calculate the equivalent meshing characteristic parameters based on the gear blank parameters and the machining parameters;

[0008] Calculate the first characteristic parameter of the load-bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter;

[0009] Based on the first characteristic parameter, the oil film shear force is obtained by numerically calculating the non-Newtonian fluid thermo-elasto-hydrodynamic lubrication equation using the separation flow method.

[0010] The friction coefficient of the spiral bevel gear tooth surface is calculated based on the oil film shear force.

[0011] The meshing efficiency of the spiral bevel gear is calculated based on the friction coefficient, the gear blank parameters, and the machining parameters.

[0012] The control method according to embodiments of the present invention has at least the following beneficial effects:

[0013] This method obtains the blank parameters and machining parameters of the spiral bevel gear; calculates the equivalent meshing characteristic parameters based on the blank parameters and machining parameters; calculates the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter; numerically calculates the non-Newtonian fluid thermo-elasto-fluid lubrication equation based on the first characteristic parameter using the separation flow method to obtain the oil film shear force; calculates the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force; and calculates the meshing efficiency of the spiral bevel gear based on the friction coefficient, blank parameters, and machining parameters. It can combine the influence of instantaneous heavy load conditions, real rough surfaces, and lubricating oil rheological properties on gear lubrication performance, thereby improving the accuracy of spiral bevel gear meshing efficiency calculation.

[0014] According to some embodiments of the present invention, the equivalent meshing characteristic parameters include equivalent meshing force, equivalent meshing torque, and equivalent meshing point position. The calculation of the equivalent meshing characteristic parameters based on the gear blank parameters and the machining parameters includes:

[0015] A finite element model is established based on the gear blank parameters and the machining parameters;

[0016] Based on the finite element model, a loading contact analysis is performed to obtain the equivalent meshing force, the equivalent meshing torque, and the equivalent meshing point position. The formula for calculating the equivalent meshing force obtained from the loading contact analysis based on the finite element model is as follows:

[0017]

[0018] Among them, F mf The equivalent meshing force is N, where N is the number of gear teeth simultaneously engaged at any position, x is the abscissa of the gear, y is the ordinate of the gear, z is the ordinate of the gear, and F is the effective meshing force. mx For the lateral equivalent meshing force, F my For the longitudinal equivalent meshing force, F mz For the vertical equivalent meshing force, F mj，i F is the coordinate equivalent meshing force of the i-th gear tooth pair. mjLet j be the equivalent meshing force, and j be the component coordinate of the gear;

[0019] The formula for calculating the equivalent meshing torque, obtained by performing loading contact analysis based on the finite element model, is as follows:

[0020] n j =F mj / F mf (j=x,y,z)

[0021]

[0022] Where, n j M is the direction vector of the equivalent meshing force. mj For the equivalent meshing torque, M mj,i Let be the equivalent meshing torque of the i-th gear pair;

[0023] The formula for calculating the equivalent engagement point position, obtained by performing loading contact analysis based on the finite element model, is as follows:

[0024]

[0025] Where, x m Let y be the x-coordinate of the equivalent meshing point position. m Z represents the ordinate of the equivalent meshing point position. m Let r be the vertical coordinate of the equivalent engagement point position. m (x m ,y m ,z m M mx M is the equivalent meshing torque. my For the longitudinal equivalent meshing torque, x m，i Let F be the x-coordinate of the equivalent meshing point position of the i-th gear pair. mj，i Let be the coordinate equivalent meshing force of the i-th gear tooth pair.

[0026] According to some embodiments of the present invention, the first characteristic parameter includes the radius of curvature of the major axis of the contact ellipse, the radius of curvature of the minor axis, the relative sliding speed, the entrainment speed, and the rolling ratio, as well as the directional angle between the entrainment speed and the minor axis of the contact ellipse. The calculation of the first characteristic parameter of the bevel gear tooth surface bearing contact point based on the equivalent meshing characteristic parameter includes:

[0027] Based on the equivalent meshing point position and the first and second basic homogeneity of the tooth surface, the curvature is calculated to obtain the principal curvature (k) of the tooth surface of the large and small gears at the load-bearing contact point. g1 ,k g2 ,k p1 ,k p2 ) and the corresponding principal direction (t)g1 ,t g2 ,t p1 ,t p2 );

[0028] Based on the equivalent meshing characteristic parameters, the principal curvature, and the corresponding principal direction, calculate the normal curvature k of the large gear in the x-direction at the bearing contact point of the bevel gear tooth surface. gx The normal curvature k of the small wheel px and the curvature k of the induced method rx The formula for calculating the normal curvature of the large gear, the normal curvature of the small gear, and the induced normal curvature of the bevel gear tooth surface bearing contact point in the x-direction based on the equivalent meshing characteristic parameters, the principal curvature, and the corresponding principal direction is as follows:

[0029] δ=arccos(t g1 ·t p1 / (|t g1 ||t p1 |))

[0030]

[0031] in, Let x and t be arbitrary tangents on the tangent plane of the bearing contact point on the bevel gear tooth surface. g1 The included angle, δ is the angle of the large wheel t g1 Steering and small wheel t p1 The angle between directions, k gx Let k be the normal curvature of the large wheel in the x-direction. px Let k be the normal curvature of the small wheel. rx For induced curvature;

[0032] By taking the maximum and minimum values ​​of the induced curvature, the maximum value of the induced curvature is obtained with respect to the corresponding included angle. The angle between the induced curvature minimum and the angle corresponding to the minimum curvature. in, To contact the minor axis x-axis and t of the ellipse g1 The included angle, To contact the major axis y-axis of the ellipse with t g1 The included angle;

[0033] The lengths of the minor and major semi-axles of the contact ellipse are calculated based on the equivalent meshing characteristic parameters and the principal curvature, wherein the calculation formula for the lengths of the minor and major semi-axles of the contact ellipse based on the equivalent meshing characteristic parameters and the principal curvature is as follows:

[0034]

[0035] Where a is the length of the minor semi-axis of the contact ellipse, b is the length of the minor semi-axis of the contact ellipse, and k is the length of the minor semi-axis of the contact ellipse. aLet k be the coefficient of the elliptic integral function. b Let F be the coefficient of the elliptic integral function. n For the normal load, A is a constant related to the shape of the object, B is a constant related to the shape of the object, E1 is the elastic modulus of the large gear material, E2 is the elastic modulus of the small gear material, μ1 is the Poisson's ratio of the large gear material, and μ2 is the Poisson's ratio of the small gear material.

[0036] The velocity at the contact point between the large and small gears is obtained based on the basic kinematics of gear transmission. The calculation formula for this velocity is as follows:

[0037] v p =ω p ×r p ,v g =ω g ×r g

[0038] Where, ω g Let ω be the angular velocity vector of the large gear. p Let r be the angular velocity vector of the pinion. g Let r be the radius vector of the large gear. p Let v be the radius vector of the pinion. g v is the velocity vector of the large gear. p The velocity vector of the pinion;

[0039] The entrainment speed along the minor and major axes of the contact ellipse is calculated based on the velocity vectors of the large gear and the small gear, wherein the calculation formula for the entrainment speed along the minor and major axes of the contact ellipse based on the velocity vectors of the large gear and the small gear is as follows:

[0040]

[0041]

[0042] in, Let be the tangential component of the large gear velocity vector along the minor axis of the contact ellipse. Let be the tangential component of the pinion velocity vector along the minor axis of the contact ellipse. Let be the component of the velocity vector of the large gear along the tangent plane of the tooth surface. Let be the component of the pinion's velocity vector along the tangent plane of the tooth surface. Let be the component of the entrainment velocity along the minor axis of the contact ellipse. U is the tangential component of the entrainment velocity along the major axis of the contact ellipse. e Let θ be the entrainment velocity vector. eThe angle between the entrainment velocity vector and the minor axis x-axis of the contact ellipse;

[0043] The formulas for calculating the suction speed, the angle between the suction speed vector and the x-axis of the contact ellipse, and the sliding speed and the angle between the sliding speed vector and the x-axis of the contact ellipse are as follows:

[0044]

[0045]

[0046] Among them, U s Let θ be the sliding velocity vector. s The angle between the sliding velocity vector and the minor axis x-axis of the contact ellipse;

[0047] The slip-roll ratio is calculated based on the suction speed and the sliding speed, wherein the formula for calculating the slip-roll ratio based on the suction speed and the sliding speed is:

[0048] AKC=U s / U e

[0049] Wherein, AKC is the slip-roll ratio at the contact point.

[0050] According to some embodiments of the present invention, the step of numerically calculating the oil film shear force based on the first characteristic parameter using the separation flow method to perform the thermo-elasto-fluid lubrication equation for a non-Newtonian fluid includes:

[0051] The Reynolds equation is calculated from the thermo-elasto-hydrodynamic lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter. The calculation formula for the Reynolds equation from the thermo-elasto-hydrodynamic lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows:

[0052]

[0053] Where p is pressure, h is oil film thickness, ρ is lubricating oil density, η is lubricating oil viscosity, x is the coordinate of the direction of motion, y is the coordinate of the perpendicular direction of motion, and φ is the coordinate of the perpendicular direction of motion. x φ is a dimensionless flow factor. y The dimensionless flow factor is two;

[0054] The film thickness equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the film thickness equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows:

[0055]

[0056] Where h0 is the central film thickness before deformation, and R x R is the radius of curvature in the x-direction. y Let Ω be the radius of curvature in the y direction, E′ be the comprehensive elastic modulus, x′ be the additional coordinate corresponding to x, y′ be the additional coordinate corresponding to y, Ω be the solution domain, and S(x,y) be the roughness height value of the tooth surface.

[0057] The viscosity equation for non-Newtonian fluid thermo-elasto-fluid lubrication is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the viscosity equation is as follows:

[0058] s0=β T (T0-138) / (lnη0+9.67)

[0059]

[0060] Where η0 is the ambient viscosity, z0 is the dimensionless viscosity-pressure index, T is the lubricating oil temperature, T0 is the initial lubricating oil temperature, s0 is the dimensionless viscosity-temperature index, and β... T Where η is the viscosity-temperature coefficient, and η is the viscosity.

[0061] The density equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the density equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows:

[0062]

[0063] Where ρ0 is the ambient density of the lubricating oil, and D0 represents the coefficient of thermal expansion;

[0064] The load balance equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the load balance equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows:

[0065]

[0066] Where F is the total load (N), p hFor the pressure that the oil film withstands, p c The pressure borne by the rough peak;

[0067] The temperature field control equation is calculated based on the first characteristic parameter using the separation flow method to calculate the temperature field control equation for the non-Newtonian fluid thermo-elastohydrodynamic lubrication equation. The temperature field control equation includes the oil film energy equation, the solid thermal conductivity equation, and the temperature continuity equation. The calculation formula for the temperature field control equation based on the first characteristic parameter using the separation flow method is as follows:

[0068]

[0069]

[0070]

[0071]

[0072]

[0073] Where η* represents the equivalent viscosity. For the dimensionless oil film comprehensive shear stress, c f k is the specific heat of lubricating oil. f Let be the thermal conductivity coefficient of the lubricating oil, u be the flow velocity of the lubricating oil along the x-direction, v be the flow velocity of the lubricating oil along the y-direction, c1 be the specific heat capacity of gear 1, ρ1 be the density of gear 1, k1 be the thermal conductivity of gear 1, c2 be the specific heat capacity of gear 2, ρ2 be the density of gear 2, k2 be the thermal conductivity of gear 2, u s1 Let u be the flow velocity of the lubricating oil on the surface of gear 1 along the x-direction. s2 v is the flow velocity of the lubricating oil on the surface of gear 2 along the x-direction. s1 v is the flow velocity of the lubricating oil on the surface of gear 1 along the y-direction. s2 The flow rate of the lubricating oil on the surface of gear 2 along the y-direction;

[0074] The oil film shear force is obtained through numerical calculation using dimensionless transformation and the Erying rheological model based on the Reynolds equation, the film thickness equation, the viscosity equation, the density equation, the load balance equation, and the temperature field control equation. The formula for calculating the oil film shear force using the Reynolds equation, the film thickness equation, the viscosity equation, the density equation, the load balance equation, and the temperature field control equation, based on dimensionless transformation and the Erying rheological model, is as follows:

[0075]

[0076]

[0077] Where ξ is the dimensionless coordinate along the film thickness direction. G represents the dimensionless composite shear stress, and G is the shear modulus. To and Related non-Newtonian fluid models, The dimensionless characteristic shear stress of the lubricating oil. Let be the dimensionless shear stress along the x-direction in the center layer of the oil film. τ is the dimensionless shear stress along the y-direction in the center layer of the oil film. e This refers to the oil film shear force.

[0078] According to some embodiments of the present invention, calculating the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force includes:

[0079] The elastoplastic contact state of the micro-protrusion is obtained based on the separation flow method and the non-Newtonian fluid thermo-elastohydrodynamic lubrication equation;

[0080] The elastic stage contact area, elastic-plastic stage contact area, and fully plastic stage contact area are calculated based on the elastoplastic contact state of the micro-protrusion, wherein the calculation formula for calculating the elastic stage contact area, elastic-plastic stage contact area, and fully plastic stage contact area based on the elastoplastic contact state of the micro-protrusion is as follows:

[0081]

[0082]

[0083] Among them, A e (ω) represents the contact area in the elastic stage, A ep (ω) represents the contact area during the elastoplastic stage, A p (ω) represents the contact area in the fully plastic stage, K is the hardness index, v is the Poisson's ratio of the material, H is the hardness of the softer material in the contact surface, E* is the equivalent elastic modulus, ω is the normal deformation of the micro-convexity, ω1 is the critical normal deformation at the initial yield point, ω2 is the critical normal deformation at the stage of fully plastic deformation, α is the first fitting constant, λ is the second fitting constant, K(e) is the first type of complete elliptic integral, and E(e) is the second type of complete elliptic integral.

[0084] The elastic stage contact load, elastic-plastic stage contact load, and fully plastic stage contact load are calculated based on the elastoplastic contact state of the micro-protrusion, wherein the calculation formulas for calculating the elastic stage contact load, elastic-plastic stage contact load, and fully plastic stage contact load based on the elastoplastic contact state of the micro-protrusion are as follows:

[0085]

[0086] Among them, F e (ω) represents the contact load during the elastic stage, F ep (ω) represents the contact load during the elastoplastic stage, F p (ω) represents the contact load during the fully plastic stage;

[0087] The micro-protrusion contact pressure is calculated based on the elasto-plastic contact state, the contact area of ​​the elastic stage, the contact area of ​​the elasto-plastic stage, the contact area of ​​the fully plastic stage, the contact load of the elastic stage, the contact load of the elasto-plastic stage, and the contact load of the fully plastic stage. The micro-protrusion contact pressure includes elastic micro-protrusion contact pressure, elasto-plastic micro-protrusion contact pressure, and fully plastic micro-protrusion contact pressure. The calculation formula for the micro-protrusion contact pressure based on the elasto-plastic contact state, the contact area of ​​the elastic stage, the contact area of ​​the elasto-plastic stage, the contact area of ​​the fully plastic stage, the contact load of the elastic stage, the contact load of the elasto-plastic stage, and the contact load of the fully plastic stage is as follows:

[0088]

[0089] Where, p c,e (ω) represents the contact pressure of the micro-convex body in the elastic stage, p c,ep (ω) represents the contact pressure of the micro-convexity in the elastoplastic stage, p c,p (ω) represents the contact pressure of the micro-protrusions during the fully plastic stage;

[0090] The friction coefficient of the spiral bevel gear tooth surface is calculated based on the oil film shear force and the contact pressure of the micro-protrusion, wherein the calculation formula for the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force and the contact pressure of the micro-protrusion is as follows:

[0091]

[0092] Where μ is the friction coefficient of the spiral bevel gear tooth surface, μ c p is the boundary friction coefficient of the preset rough peak friction. c (ω) represents the contact pressure of the micro-convex body.

[0093] According to some embodiments of the present invention, calculating the meshing efficiency of the spiral bevel gear based on the friction coefficient, the gear blank parameters, and the machining parameters includes:

[0094] The sliding friction force is calculated based on the friction coefficient, wherein the formula for calculating the sliding friction force based on the friction coefficient is:

[0095] F s =μ·F mf

[0096] Among them, F s It is sliding friction;

[0097] The rolling friction force is calculated based on the gear blank parameters and the machining parameters, wherein the calculation formula for the rolling friction force based on the gear blank parameters and the machining parameters is as follows:

[0098]

[0099]

[0100]

[0101]

[0102]

[0103]

[0104] Among them, F ro For rolling friction, F r For rolling friction, Here, G is the heat-affected factor, and G is a dimensionless material parameter. Let α be the dimensionless entrainment velocity, α be the viscosity-compression coefficient, E′ be the comprehensive elastic modulus, and p be the coefficient of performance. h For the normal load borne by the oil film, β T K is the viscosity-temperature coefficient. f U is the thermal conductivity coefficient of lubricating oil. s U is the relative sliding velocity. e The relative entrainment speed;

[0105] The meshing efficiency of the spiral bevel gear is calculated based on the sliding friction and the rolling friction.

[0106] According to some embodiments of the present invention, the formula for calculating the meshing efficiency of the spiral bevel gear based on the sliding friction and the rolling friction is as follows:

[0107]

[0108] Where, η e For the meshing efficiency of spiral bevel gears, T g For the input torque, ω g Input rotational speed.

[0109] A second aspect of the present invention provides a spiral bevel gear meshing efficiency calculation system, the spiral bevel gear meshing efficiency calculation system comprising:

[0110] The data acquisition module is used to acquire the blank parameters and machining parameters of the spiral bevel gear;

[0111] An equivalent meshing characteristic parameter calculation module is used to calculate equivalent meshing characteristic parameters based on the gear blank parameters and the machining parameters;

[0112] The first characteristic parameter calculation module is used to calculate the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter.

[0113] The oil film shear force calculation module is used to numerically calculate the non-Newtonian fluid thermo-elasto-fluid lubrication equation based on the first characteristic parameter using the separation flow method to obtain the oil film shear force.

[0114] The friction coefficient calculation module is used to calculate the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force.

[0115] The meshing efficiency calculation module is used to calculate the meshing efficiency of the spiral bevel gear based on the friction coefficient, the gear blank parameters, and the machining parameters.

[0116] This system acquires the blank parameters and machining parameters of the spiral bevel gear; calculates the equivalent meshing characteristic parameters based on the blank parameters and machining parameters; calculates the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter; performs numerical calculations on the non-Newtonian fluid thermo-elasto-fluid lubrication equation using the separation flow method based on the first characteristic parameter to obtain the oil film shear force; calculates the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force; and calculates the meshing efficiency of the spiral bevel gear based on the friction coefficient, blank parameters, and machining parameters. It can combine the influence of instantaneous heavy load conditions, real rough surfaces, and lubricating oil rheological properties on gear lubrication performance to improve the accuracy of spiral bevel gear meshing efficiency calculation.

[0117] A third aspect of the present invention provides an electronic device for calculating the meshing efficiency of a spiral bevel gear, comprising at least one control processor and a memory for communicatively connecting to the at least one control processor; the memory stores instructions executable by the at least one control processor, the instructions being executed by the at least one control processor to enable the at least one control processor to perform the above-described spiral bevel gear meshing efficiency calculation method.

[0118] In a fourth aspect, the present invention provides a computer-readable storage medium storing computer-executable instructions for causing a computer to perform the above-described method for calculating the meshing efficiency of spiral bevel gears.

[0119] It should be noted that the beneficial effects of the second to fourth aspects of the present invention compared with the prior art are the same as the beneficial effects of the above-described spiral bevel gear meshing efficiency calculation system compared with the prior art, and will not be described in detail here.

[0120] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0121] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0122] Figure 1 This is a flowchart of a method for calculating the meshing efficiency of a spiral bevel gear according to an embodiment of the present invention;

[0123] Figure 2 This is a schematic diagram of a spiral bevel gear loading contact analysis model for a spiral bevel gear meshing efficiency calculation method provided in an embodiment of the present invention;

[0124] Figure 3 This is a schematic diagram of the tooth surface contact pressure distribution of the small gear of a spiral bevel gear pair, provided in an embodiment of the present invention, for a method of calculating the meshing efficiency of a spiral bevel gear.

[0125] Figure 4 This is a schematic diagram of the sliding speed, entrainment speed and sliding-rolling ratio of a spiral bevel gear, which is provided in an embodiment of the present invention for calculating the meshing efficiency of a spiral bevel gear.

[0126] Figure 5 This is a schematic diagram of the oil film shear force along the x-axis at a certain meshing moment of a spiral bevel gear, provided by an embodiment of the present invention, for a method of calculating the meshing efficiency of a spiral bevel gear.

[0127] Figure 6 This is a schematic diagram of the oil film shear force along the y-axis at a certain meshing moment of a spiral bevel gear, provided by an embodiment of the present invention, for a method of calculating the meshing efficiency of a spiral bevel gear.

[0128] Figure 7 This is a schematic diagram of the friction coefficient change during the meshing cycle of a spiral bevel gear, provided by an embodiment of the present invention, for a method of calculating the meshing efficiency of a spiral bevel gear.

[0129] Figure 8 This is a schematic diagram of the meshing efficiency within the meshing cycle of a spiral bevel gear, provided by an embodiment of the present invention, for a method of calculating the meshing efficiency of a spiral bevel gear.

[0130] Figure 9 This is a schematic diagram of a spiral bevel gear meshing efficiency calculation system according to an embodiment of the present invention. Detailed Implementation

[0131] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0132] In the description of this invention, the use of terms such as "first," "second," etc., is for the purpose of distinguishing technical features only and should not be construed as indicating or implying relative importance, or implicitly indicating the number of technical features indicated, or implicitly indicating the order of the technical features indicated.

[0133] In the description of this invention, it should be understood that the orientation descriptions, such as up, down, etc., are based on the orientation or positional relationship shown in the drawings and are only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention.

[0134] In the description of this invention, it should be noted that, unless otherwise explicitly defined, terms such as "setting," "installation," and "connection" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this invention in conjunction with the specific content of the technical solution.

[0135] Spiral bevel gears, due to their advantages such as high load-bearing capacity, high overlap ratio, and smooth transmission, have been widely used in high-speed, heavy-load applications in aerospace, automotive, and marine industries. However, the meshing power loss of bevel gear pairs affects the efficiency of the transmission system, mainly including power losses from friction, oil churning, and wind resistance, with friction power loss accounting for the majority. Therefore, establishing accurate calculation methods for gear pair friction power loss and meshing efficiency is of significant importance for improving gear transmission performance.

[0136] Due to the complex contact geometry, heavy-load conditions, and tooth surface roughness of spiral bevel gear pairs, the actual lubrication state of the gears is a hybrid lubrication exhibiting characteristics of both full thermo-elasto-fluid lubrication and boundary lubrication. Determining the ratio of oil film load and roughness peak load is quite complex. Contact analysis is the foundation of gear lubrication characteristic analysis. Traditional solutions for contact geometry and motion parameters are based on the theoretical contact trajectory of the gear pair. However, due to the special tooth profile and heavy-load conditions of spiral bevel gears, deformation under load causes the actual contact trajectory to deviate from the theoretical contact trajectory, and changes in the contact point position have a significant impact on gear lubrication performance. The contact analysis and lubrication calculation of spiral bevel gears involve the coupling of multiple disciplines such as differential geometry meshing principles, interface contact mechanics, and hydrodynamic lubrication mechanics, and the computational workload is enormous, posing a significant challenge to establishing a comprehensive and systematic method for calculating the meshing efficiency of spiral bevel gears. Therefore, it is urgent to develop a method for calculating meshing efficiency based on the lubrication characteristics of the actual load-bearing contact position of spiral bevel gears under a hybrid thermo-elasto-fluid lubrication state.

[0137] To address the aforementioned technical deficiencies, referring to... Figure 1 This invention provides a method for calculating the meshing efficiency of spiral bevel gears, including:

[0138] Step S101: Obtain the blank parameters and machining parameters of the spiral bevel gear;

[0139] Step S102: Calculate the equivalent meshing characteristic parameters based on the gear blank parameters and machining parameters;

[0140] Step S103: Calculate the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter;

[0141] Step S104: Based on the first characteristic parameter, the non-Newtonian fluid thermo-elasto-fluid lubrication equation is numerically calculated using the separation flow method to obtain the oil film shear force.

[0142] Step S105: Calculate the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force;

[0143] Step S106: Calculate the meshing efficiency of the spiral bevel gear based on the friction coefficient, gear blank parameters, and machining parameters.

[0144] This method obtains the blank parameters and machining parameters of the spiral bevel gear; calculates the equivalent meshing characteristic parameters based on the blank parameters and machining parameters; calculates the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter; numerically calculates the non-Newtonian fluid thermo-elasto-fluid lubrication equation based on the first characteristic parameter using the separation flow method to obtain the oil film shear force; calculates the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force; and calculates the meshing efficiency of the spiral bevel gear based on the friction coefficient, blank parameters, and machining parameters. It can combine the influence of instantaneous heavy load conditions, real rough surfaces, and lubricating oil rheological properties on gear lubrication performance, thereby improving the accuracy of spiral bevel gear meshing efficiency calculation.

[0145] In some embodiments, the equivalent meshing characteristic parameters include equivalent meshing force, equivalent meshing torque, and equivalent meshing point position. The equivalent meshing characteristic parameters are calculated based on the gear blank parameters and machining parameters, including:

[0146] A finite element model is established based on the gear blank parameters and machining parameters;

[0147] Based on the finite element model, a loading contact analysis is performed to obtain the equivalent meshing force, equivalent meshing torque, and equivalent meshing point location. The formula for calculating the equivalent meshing force, obtained from the loading contact analysis using the finite element model, is as follows:

[0148]

[0149] Among them, F mf The equivalent meshing force is N, where N is the number of gear teeth simultaneously engaged at any position, x is the abscissa of the gear, y is the ordinate of the gear, z is the ordinate of the gear, and F is the effective meshing force. mx For the lateral equivalent meshing force, F my For the longitudinal equivalent meshing force, F mz For the vertical equivalent meshing force, F mj，i F is the coordinate equivalent meshing force of the i-th gear tooth pair. mj Let j be the equivalent meshing force, and j be the component coordinate of the gear;

[0150] Based on the finite element model, a loading contact analysis was performed, and the formula for calculating the equivalent meshing torque was obtained as follows:

[0151] n j =F mj / F mf (j=x,y,z)

[0152]

[0153] Where, n j M is the direction vector of the equivalent meshing force. mj For the equivalent meshing torque, M mj,iLet be the equivalent meshing torque of the i-th gear pair;

[0154] Based on the finite element model, a loading contact analysis was performed, and the formula for calculating the equivalent meshing point location was obtained as follows:

[0155]

[0156] Where, x m Let y be the x-coordinate of the equivalent meshing point position. m Z represents the ordinate of the equivalent meshing point position. m Let r be the vertical coordinate of the equivalent meshing point position. m (x m ,y m ,z m M mx M is the equivalent meshing torque. my For the longitudinal equivalent meshing torque, x m，i Let F be the x-coordinate of the equivalent meshing point position of the i-th gear pair. mj，i Let be the coordinate equivalent meshing force of the i-th gear tooth pair.

[0157] Specifically, spiral bevel gears include arc bevel gears and quasi-hyperboloid gears with offset distance. The main blank parameters and machining parameters of arc bevel gears are shown in Table 1 and Table 2. Table 1 is the blank parameter table of spiral bevel gears, and Table 2 is the gear machining parameters.

[0158] Table 1

[0159]

[0160] Table 2

[0161]

[0162] Specifically, refer to Figure 2 and Figure 3 Based on the blank parameters and machining parameters of the spiral bevel gear, a finite element model was established and a loading contact analysis was performed.

[0163] In some embodiments, the first characteristic parameter includes the radius of curvature of the major axis of the contact ellipse, the radius of curvature of the minor axis, the relative sliding speed, the entrainment speed, and the rolling ratio, as well as the directional angle between the entrainment speed and the minor axis of the contact ellipse. The first characteristic parameter of the bevel gear tooth surface bearing contact point is calculated based on the equivalent meshing characteristic parameters, including:

[0164] Based on the equivalent meshing point location and the first and second basic homogeneity of the tooth surface, the curvature is calculated to obtain the principal curvature (k) of the tooth surface of the large and small gears at the load-bearing contact point. g1 ,k g2 ,k p1 ,kp2 ) and the corresponding principal direction (t) g1 ,t g2 ,t p1 ,t p2 );

[0165] Calculate the normal curvature k of the bevel gear tooth surface bearing contact point in the x-direction based on the equivalent meshing characteristic parameters, principal curvature, and corresponding principal direction. gx The normal curvature k of the small wheel px and the curvature k of the induced method rx The formulas for calculating the normal curvature of the large gear, the normal curvature of the small gear, and the induced normal curvature of the bevel gear tooth surface bearing contact point in the x-direction, based on the equivalent meshing characteristic parameters, principal curvature, and corresponding principal directions, are as follows:

[0166] δ=arccos(t g1 ·t p1 / (|t g1 ||t p1 |))

[0167]

[0168] in, Let x and t be arbitrary tangents on the tangent plane of the bearing contact point on the bevel gear tooth surface. g1 The included angle, δ is the angle of the large wheel t g1 Steering and small wheel t p1 The angle between directions, k gx Let k be the normal curvature of the large wheel in the x-direction. px Let k be the normal curvature of the small wheel. rx For induced curvature;

[0169] By taking the maximum and minimum values ​​of the induced curvature, we can obtain the maximum value of the induced curvature with respect to the corresponding included angle. The angle between the induced curvature minimum and the angle corresponding to the minimum curvature. in, To contact the minor axis x-axis and t of the ellipse g1 The included angle, To contact the major axis y-axis of the ellipse with t g1 The included angle;

[0170] The lengths of the minor and major semi-axles of the contact ellipse are calculated based on the equivalent meshing characteristic parameters and principal curvatures. The formulas for calculating the lengths of the minor and major semi-axles of the contact ellipse based on the equivalent meshing characteristic parameters and principal curvatures are as follows:

[0171]

[0172] Where a is the length of the minor semi-axis of the contact ellipse, b is the length of the minor semi-axis of the contact ellipse, and k is the length of the minor semi-axis of the contact ellipse.a Let k be the coefficient of the elliptic integral function. b Let F be the coefficient of the elliptic integral function. n For the normal load, A is a constant related to the shape of the object, B is a constant related to the shape of the object, E1 is the elastic modulus of the large gear material, E2 is the elastic modulus of the small gear material, μ1 is the Poisson's ratio of the large gear material, and μ2 is the Poisson's ratio of the small gear material.

[0173] The velocity at the contact point between the large and small gears is obtained based on the basic kinematics of gear transmission. The formula for calculating this velocity is as follows:

[0174] v p =ω p ×r p ,v g =ω g ×r g

[0175] Where, ω g Let ω be the angular velocity vector of the large gear. p Let r be the angular velocity vector of the pinion. g Let r be the radius vector of the large gear. p Let v be the radius vector of the pinion. g v is the velocity vector of the large gear. p The velocity vector of the pinion;

[0176] The entrainment velocities along the minor and major axes of the contact ellipse are calculated based on the velocity vectors of the large and small gears. The formula for calculating these entrainment velocities along the minor and major axes of the contact ellipse is as follows:

[0177]

[0178]

[0179] in, Let be the tangential component of the large gear velocity vector along the minor axis of the contact ellipse. Let be the tangential component of the pinion velocity vector along the minor axis of the contact ellipse. Let be the component of the velocity vector of the large gear along the tangent plane of the tooth surface. Let be the component of the pinion's velocity vector along the tangent plane of the tooth surface. Let be the component of the entrainment velocity along the minor axis of the contact ellipse. U is the tangential component of the entrainment velocity along the major axis of the contact ellipse. e Let θ be the entrainment velocity vector. eThe angle between the entrainment velocity vector and the minor axis x-axis of the contact ellipse;

[0180] The entrainment velocity, the angle between the entrainment velocity vector and the x-axis of the minor axis of the contact ellipse, and the sliding velocity, as well as the angle between the sliding velocity vector and the x-axis of the minor axis of the contact ellipse, are calculated based on the entrainment velocity along the minor and major axes of the contact ellipse. The formulas for calculating the entrainment velocity, the angle between the entrainment velocity vector and the x-axis of the minor axis of the contact ellipse, and the sliding velocity, as well as the angle between the sliding velocity vector and the x-axis of the minor axis of the contact ellipse, are as follows:

[0181]

[0182]

[0183] Among them, U s Let θ be the sliding velocity vector. s The angle between the sliding velocity vector and the minor axis x-axis of the contact ellipse;

[0184] The roll-slip ratio is calculated based on the suction speed and the sliding speed. The formula for calculating the roll-slip ratio based on the suction speed and the sliding speed is as follows:

[0185] AKC=U s / U e

[0186] Wherein, AKC is the slip-roll ratio at the contact point.

[0187] Specifically, by projecting the equivalent meshing point onto the cross section passing through the gear shaft, the actual bearing contact point can be obtained using the tooth surface equation. Then, by considering the torsional deformation of the gear under load, the relative positions of the two tooth surfaces are adjusted based on the direction vector of the equivalent meshing force to satisfy the contact condition at the bearing contact point, thereby solving for the first characteristic parameter of the bearing contact point.

[0188] Specifically, refer to Figure 4 The trends of sliding speed, suction speed, and roll ratio with the small wheel are as follows: Figure 4 As shown.

[0189] In some embodiments, the oil film shear force is obtained by numerically calculating the non-Newtonian fluid thermo-elasto-fluid lubrication equation using the separation flow method based on the first characteristic parameter, including:

[0190] The Reynolds equation is calculated from the thermo-elasto-fluidic lubrication equation of a non-Newtonian fluid using the separation flow method based on the first characteristic parameter. The calculation formula for the Reynolds equation from the thermo-elasto-fluidic lubrication equation of a non-Newtonian fluid using the separation flow method based on the first characteristic parameter is as follows:

[0191]

[0192] Where p is pressure, h is oil film thickness, ρ is lubricating oil density, η is lubricating oil viscosity, x is the coordinate of the direction of motion, y is the coordinate of the perpendicular direction of motion, and φ is the coordinate of the perpendicular direction of motion. x φ is a dimensionless flow factor. y The dimensionless flow factor is two;

[0193] The film thickness equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter. The formula for calculating the film thickness equation using the separation flow rate method based on the first characteristic parameter is as follows:

[0194]

[0195] Where h0 is the central film thickness before deformation, and R x R is the radius of curvature in the x-direction. y Let Ω be the radius of curvature in the y direction, E′ be the comprehensive elastic modulus, x′ be the additional coordinate corresponding to x, y′ be the additional coordinate corresponding to y, Ω be the solution domain, and S(x,y) be the roughness height value of the tooth surface.

[0196] The viscosity equation for non-Newtonian fluid thermo-elasto-fluid lubrication is calculated using the separation flow rate method based on the first characteristic parameter. The formula for calculating the viscosity equation using the separation flow rate method based on the first characteristic parameter is as follows:

[0197] s0=β T (T0-138) / (lnη0+9.67)

[0198]

[0199] Where η0 is the ambient viscosity, z0 is the dimensionless viscosity-pressure index, T is the lubricating oil temperature, T0 is the initial lubricating oil temperature, s0 is the dimensionless viscosity-temperature index, and β... T Where η is the viscosity-temperature coefficient, and η is the viscosity.

[0200] The density equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter. The formula for calculating the density equation using the separation flow rate method based on the first characteristic parameter is as follows:

[0201]

[0202] Where ρ0 is the ambient density of the lubricating oil, and D0 represents the coefficient of thermal expansion;

[0203] The load balance equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow method based on the first characteristic parameter. The formula for calculating the load balance equation based on the separation flow method for the non-Newtonian fluid thermo-elasto-fluid lubrication equation using the first characteristic parameter is as follows:

[0204]

[0205] Where F is the total load (N), p h For the pressure that the oil film withstands, p c The pressure borne by the rough peak;

[0206] The temperature field governing equations for non-Newtonian fluid thermo-elastohydrodynamic lubrication equations are calculated using the separation flow method based on the first characteristic parameter. These equations include the oil film energy equation, the solid thermal conductivity equation, and the temperature continuity equation. The calculation formula for the temperature field governing equations for non-Newtonian fluid thermo-elastohydrodynamic lubrication equations using the separation flow method based on the first characteristic parameter is as follows:

[0207]

[0208]

[0209]

[0210]

[0211]

[0212] Where η* represents the equivalent viscosity. For the dimensionless oil film comprehensive shear stress, c f k is the specific heat of lubricating oil. f Let be the thermal conductivity coefficient of the lubricating oil, u be the flow velocity of the lubricating oil along the x-direction, v be the flow velocity of the lubricating oil along the y-direction, c1 be the specific heat capacity of gear 1, ρ1 be the density of gear 1, k1 be the thermal conductivity of gear 1, c2 be the specific heat capacity of gear 2, ρ2 be the density of gear 2, k2 be the thermal conductivity of gear 2, u s1 Let u be the flow velocity of the lubricating oil on the surface of gear 1 along the x-direction. s2 v is the flow velocity of the lubricating oil on the surface of gear 2 along the x-direction. s1 v is the flow velocity of the lubricating oil on the surface of gear 1 along the y-direction. s2 The flow rate of the lubricating oil on the surface of gear 2 along the y-direction;

[0213] The oil film shear force is obtained through numerical calculation using dimensionless methods and the Erying rheological model, based on the Reynolds equation, film thickness equation, viscosity equation, density equation, load balance equation, and temperature field control equation. Specifically, the formula for calculating the oil film shear force is as follows:

[0214]

[0215]

[0216] Where ξ is the dimensionless coordinate along the film thickness direction. G represents the dimensionless composite shear stress, and G is the shear modulus. To and Related non-Newtonian fluid models, The dimensionless characteristic shear stress of the lubricating oil. Let be the dimensionless shear stress along the x-direction in the center layer of the oil film. τ is the dimensionless shear stress along the y-direction in the center layer of the oil film. e This refers to the oil film shear force.

[0217] Specifically, refer to Figure 5 and Figure 6 At a certain engagement moment, the oil film shear force along the x and y axes is as follows: Figure 5 and Figure 6 As shown.

[0218] In some embodiments, calculating the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force includes:

[0219] The elastoplastic contact state of the micro-protrusion is obtained based on the separation flow method and the non-Newtonian fluid thermo-elastohydrodynamic lubrication equation;

[0220] The contact areas in the elastic, elasto-plastic, and fully plastic stages are calculated based on the elasto-plastic contact state of the micro-protrusions. The formulas for calculating these contact areas are as follows:

[0221]

[0222]

[0223] Among them, A e (ω) represents the contact area in the elastic stage, A ep (ω) represents the contact area during the elastoplastic stage, A p(ω) represents the contact area in the fully plastic stage, K is the hardness index, v is the Poisson's ratio of the material, H is the hardness of the softer material in the contact surface, E* is the equivalent elastic modulus, ω is the normal deformation of the micro-convexity, ω1 is the critical normal deformation at the initial yield point, ω2 is the critical normal deformation at the stage of fully plastic deformation, α is the first fitting constant, λ is the second fitting constant, K(e) is the first type of complete elliptic integral, and E(e) is the second type of complete elliptic integral.

[0224] The elastic stage contact load, elastic-plastic stage contact load, and fully plastic stage contact load are calculated based on the elastic-plastic contact state of the micro-protrusion. The calculation formulas for these loads are as follows:

[0225]

[0226] Among them, F e (ω) represents the contact load during the elastic stage, F ep (ω) represents the contact load during the elastoplastic stage, F p (ω) represents the contact load during the fully plastic stage;

[0227] The contact pressure of a micro-assurance is calculated based on its elasto-plastic contact state, the contact area in the elastic stage, the contact area in the fully plastic stage, the contact load in the elastic stage, the contact load in the elasto-plastic stage, and the contact load in the fully plastic stage. The contact pressure includes elastic, elasto-plastic, and fully plastic contact pressures. The formula for calculating the contact pressure of a micro-assurance based on these parameters is as follows:

[0228]

[0229] Where, p c,e (ω) represents the contact pressure of the micro-convex body in the elastic stage, p c,ep (ω) represents the contact pressure of the micro-convexity in the elastoplastic stage, p c,p (ω) represents the contact pressure of the micro-protrusions during the fully plastic stage;

[0230] The friction coefficient of the spiral bevel gear tooth surface is calculated based on the oil film shear force and the contact pressure of the micro-protrusion. The formula for calculating the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force and the contact pressure of the micro-protrusion is as follows:

[0231]

[0232] Where μ is the friction coefficient of the spiral bevel gear tooth surface, μ c p is the boundary friction coefficient of the preset rough peak friction. c (ω) represents the contact pressure of the micro-convex body.

[0233] Specifically, considering both the horizontal distance between micro-protrusions and the vertical distance between the contact surface height and the substrate height, the precise micro-protrusion contact pressure is obtained by iteratively correcting the substrate deformation under the influence of the micro-protrusions.

[0234] Specifically, refer to Figure 7 The coefficient of friction during the meshing cycle of a spiral bevel gear pair is as follows: Figure 7 As shown.

[0235] In some embodiments, calculating the meshing efficiency of the spiral bevel gear based on the friction coefficient, gear blank parameters, and machining parameters includes:

[0236] The sliding friction force is calculated based on the coefficient of friction. The formula for calculating the sliding friction force based on the coefficient of friction is as follows:

[0237] F s =μ·F mf

[0238] Among them, F s It is sliding friction;

[0239] The rolling friction force is calculated based on the gear blank parameters and machining parameters. The formula for calculating the rolling friction force based on the gear blank parameters and machining parameters is as follows:

[0240]

[0241]

[0242]

[0243]

[0244]

[0245]

[0246] Among them, F ro For rolling friction, F r For rolling friction, Here, G is the heat-affected factor, and G is a dimensionless material parameter. Let α be the dimensionless entrainment velocity, α be the viscosity-compression coefficient, E′ be the comprehensive elastic modulus, and p be the coefficient of performance. h For the normal load borne by the oil film, β T K is the viscosity-temperature coefficient. fU is the thermal conductivity coefficient of lubricating oil. s U is the relative sliding velocity. e The relative entrainment speed;

[0247] The meshing efficiency of spiral bevel gears is calculated based on sliding friction and rolling friction.

[0248] In some embodiments, the formula for calculating the meshing efficiency of a spiral bevel gear based on sliding friction and rolling friction is as follows:

[0249]

[0250] Where, η e For the meshing efficiency of spiral bevel gears, T g For the input torque, ω g Input rotational speed.

[0251] Specifically, refer to Figure 8 The meshing efficiency of a spiral bevel gear pair during the meshing cycle is as follows: Figure 8 As shown.

[0252] Additionally, refer to Figure 9 An embodiment of the present invention provides a spiral bevel gear meshing efficiency calculation system, including a data acquisition module 1100, an equivalent meshing characteristic parameter calculation module 1200, a first characteristic parameter calculation module 1300, an oil film shear force calculation module 1400, a friction coefficient calculation module 1500, and a meshing efficiency calculation module 1600, wherein:

[0253] The data acquisition module 1100 is used to acquire the blank parameters and machining parameters of the spiral bevel gear;

[0254] The equivalent meshing characteristic parameter calculation module 1200 is used to calculate the equivalent meshing characteristic parameters based on the gear blank parameters and machining parameters;

[0255] The first characteristic parameter calculation module 1300 is used to calculate the first characteristic parameter of the bearing contact point of the bevel gear tooth surface based on the equivalent meshing characteristic parameter.

[0256] The oil film shear force calculation module 1400 is used to numerically calculate the oil film shear force based on the first characteristic parameter and the separation flow method of the non-Newtonian fluid thermo-elasto-hydrodynamic lubrication equation.

[0257] The friction coefficient calculation module 1500 is used to calculate the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force;

[0258] The meshing efficiency calculation module 1600 is used to calculate the meshing efficiency of spiral bevel gears based on the friction coefficient, gear blank parameters, and machining parameters.

[0259] This system acquires the blank parameters and machining parameters of the spiral bevel gear; calculates the equivalent meshing characteristic parameters based on the blank parameters and machining parameters; calculates the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter; performs numerical calculations on the non-Newtonian fluid thermo-elasto-fluid lubrication equation using the separation flow method based on the first characteristic parameter to obtain the oil film shear force; calculates the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force; and calculates the meshing efficiency of the spiral bevel gear based on the friction coefficient, blank parameters, and machining parameters. It can combine the influence of instantaneous heavy load conditions, real rough surfaces, and lubricating oil rheological properties on gear lubrication performance to improve the accuracy of spiral bevel gear meshing efficiency calculation.

[0260] It should be noted that this system embodiment is based on the same inventive concept as the above system embodiment. Therefore, the relevant content of the above method embodiment is also applicable to this system embodiment, and will not be repeated here.

[0261] This application also provides an electronic device for calculating the meshing efficiency of spiral bevel gears, including: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the spiral bevel gear meshing efficiency calculation method as described above.

[0262] The processor and memory can be connected via a bus or other means.

[0263] Memory, as a non-transitory computer-readable storage medium, can be used to store non-transitory software programs and non-transitory computer-executable programs. Furthermore, memory may include high-speed random access memory, and may also include non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some embodiments, memory may optionally include memory remotely located relative to the processor, and these remote memories can be connected to the processor via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0264] The non-transient software program and instructions required to implement the spiral bevel gear meshing efficiency calculation method of the above embodiments are stored in memory. When executed by a processor, the spiral bevel gear meshing efficiency calculation method of the above embodiments is executed, for example, the method described above is executed. Figure 1 The method steps S101 to S106.

[0265] This application also provides a computer-readable storage medium storing computer-executable instructions for performing, as described above, the method for calculating the meshing efficiency of spiral bevel gears.

[0266] The computer-readable storage medium stores computer-executable instructions that are executed by a processor or controller, for example, by a processor in the above-described electronic device embodiment, causing the processor to perform the spiral bevel gear meshing efficiency calculation method in the above-described embodiment, for example, to perform the above-described... Figure 1 The method steps S101 to S106.

[0267] It will be understood by those skilled in the art that all or some of the steps and systems in the methods disclosed above can be implemented as software, firmware, hardware, and suitable combinations thereof. Some or all of the physical components can be implemented as software executed by a processor, such as a central processing unit, digital signal processor, or microprocessor, or as hardware, or as an integrated circuit, such as an application-specific integrated circuit. Such software can be distributed on a computer-readable medium, which can include computer storage media (or non-transitory media) and communication media (or transient media). As is known to those skilled in the art, the term computer storage media includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storing information (such as computer-readable instructions, data structures, program units, or other data). Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technologies, CD-ROM, digital versatile disc (DVD) or other optical disc storage, magnetic cartridges, magnetic tape, disk storage or other magnetic storage devices, or any other medium that can be used to store desired information and is accessible to a computer. Furthermore, as is known to those skilled in the art, communication media typically contain computer-readable instructions, data structures, program units, or other data in modulated data signals such as carrier waves or other transmission mechanisms, and may include any information delivery medium.

[0268] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A method for calculating the meshing efficiency of spiral bevel gears, characterized in that, The method for calculating the meshing efficiency of the spiral bevel gear includes: Obtain the blank parameters and machining parameters of the spiral bevel gear; Calculate the equivalent meshing characteristic parameters based on the gear blank parameters and the machining parameters; Calculate the first characteristic parameter of the load-bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter; Based on the first characteristic parameter, the oil film shear force is obtained by numerically calculating the non-Newtonian fluid thermo-elasto-hydrodynamic lubrication equation using the separation flow method. The friction coefficient of the spiral bevel gear tooth surface is calculated based on the oil film shear force, specifically as follows: The elastoplastic contact state of the micro-protrusion is obtained based on the separation flow method and the non-Newtonian fluid thermo-elastohydrodynamic lubrication equation; Calculate the contact area in the elastic stage, the contact area in the elastic-plastic stage, and the contact area in the fully plastic stage based on the elastoplastic contact state of the micro-protrusion. Calculate the elastic stage contact load, elastic-plastic stage contact load, and fully plastic stage contact load based on the elastoplastic contact state of the micro-protrusion. The micro-protrusion contact pressure is calculated based on the elastoplastic contact state of the micro-protrusion, the contact area of ​​the elastic stage, the contact area of ​​the elastoplastic stage, the contact area of ​​the fully plastic stage, the contact load of the elastic stage, the contact load of the elastoplastic stage, and the contact load of the fully plastic stage, wherein the micro-protrusion contact pressure includes the elastic micro-protrusion contact pressure, the elastoplastic micro-protrusion contact pressure, and the fully plastic micro-protrusion contact pressure; The friction coefficient of the spiral bevel gear tooth surface is calculated based on the oil film shear force and the micro-protrusion contact pressure. The meshing efficiency of the spiral bevel gear is calculated based on the friction coefficient, the gear blank parameters, and the machining parameters.

2. The method for calculating the meshing efficiency of a spiral bevel gear according to claim 1, characterized in that... The equivalent meshing characteristic parameters include equivalent meshing force, equivalent meshing torque, and equivalent meshing point position. The calculation of the equivalent meshing characteristic parameters based on the gear blank parameters and the machining parameters includes: A finite element model is established based on the gear blank parameters and the machining parameters; Based on the finite element model, a loading contact analysis is performed to obtain the equivalent meshing force, the equivalent meshing torque, and the equivalent meshing point position. The formula for calculating the equivalent meshing force obtained from the loading contact analysis based on the finite element model is as follows: in, F mf The equivalent meshing force is given by N, where N is the number of gear teeth simultaneously engaged at any position, x is the abscissa of the gear, y is the ordinate of the gear, and z is the ordinate of the gear. F mx For the lateral equivalent meshing force, F my For longitudinal equivalent meshing force, F mz For vertical equivalent meshing force, F mj,i Let be the coordinate equivalent meshing force of the i-th gear tooth pair. F mj Let j be the equivalent meshing force, and j be the component coordinate of the gear; The formula for calculating the equivalent meshing torque, obtained by performing loading contact analysis based on the finite element model, is as follows: in, n j The direction vector of the equivalent meshing force. M mj For the equivalent meshing torque, M mj,i Let be the equivalent meshing torque of the i-th gear pair; The formula for calculating the equivalent engagement point position, obtained by performing loading contact analysis based on the finite element model, is as follows: in, x m The x-coordinate of the equivalent meshing point position. y m The ordinate represents the position of the equivalent meshing point. z m Let be the vertical coordinate of the equivalent engagement point position, where the equivalent engagement point position is... r m ( x m , y m , z m ) , M mx For the lateral equivalent meshing torque, M my For the longitudinal equivalent meshing torque, x m，i Let x be the x-coordinate of the equivalent meshing point position of the i-th gear tooth pair. F mj，i Let be the coordinate equivalent meshing force of the i-th gear tooth pair.

3. The method for calculating the meshing efficiency of a spiral bevel gear according to claim 2, characterized in that, The first characteristic parameters include the radius of curvature of the major axis of the contact ellipse, the radius of curvature of the minor axis, the relative sliding speed, the entrainment speed, and the rolling ratio, as well as the directional angle between the entrainment speed and the minor axis of the contact ellipse. The calculation of the first characteristic parameters of the bevel gear tooth surface bearing contact point based on the equivalent meshing characteristic parameters includes: Based on the equivalent meshing point position, the first and second basic homogeneity of the tooth surface, the curvature is calculated to obtain the principal curvature of the tooth surface of the large and small gears at the load-bearing contact point. k g1 , k g2 , k p1 , k p2 ) and the corresponding main direction ( t g1 , t g2 , t p1 , t p2 ); The bearing contact point of the bevel gear tooth surface is calculated based on the equivalent meshing characteristic parameters, the principal curvature, and the corresponding principal direction. x normal curvature of the large wheel in the direction k gx The normal curvature of the small wheel k px and induced curvature k rx The calculation of the bearing contact point of the bevel gear tooth surface based on the equivalent meshing characteristic parameters, the principal curvature, and the corresponding principal direction is as follows: x The formulas for calculating the normal curvature of the larger wheel, the normal curvature of the smaller wheel, and the induced normal curvature in the direction are as follows: in, Any tangent line on the tangent plane of the bearing contact point on the bevel gear tooth surface x and t g1 The included angle, δ For large wheels t g1 Steering and small wheels t p1 The angle between directions, k gx for x The curvature of the large wheel in the direction of travel. k px Let be the normal curvature of the small wheel. k rx For induced curvature; By taking the maximum and minimum values ​​of the induced curvature, the maximum value of the induced curvature is obtained with respect to the corresponding included angle. max The angle between the induced curvature minimum and the angle corresponding to the minimum curvature. min ,in, max To contact the minor axis of the ellipse x shaft and t g1 The included angle, min To contact the major axis of the ellipse y shaft and t g1 The included angle; The lengths of the minor and major semi-axles of the contact ellipse are calculated based on the equivalent meshing characteristic parameters and the principal curvature, wherein the calculation formula for the lengths of the minor and major semi-axles of the contact ellipse based on the equivalent meshing characteristic parameters and the principal curvature is as follows: in, a To contact the length of the minor semi-axis of the ellipse, b To contact the length of the minor semi-axis of the ellipse, k a Let be the coefficient of the elliptic integral function. k b Let be the coefficient of the elliptic integral function. F n Let A be the normal load, B be a constant related to the shape of the object, E1 be the elastic modulus of the large gear material, and E2 be the elastic modulus of the small gear material. μ 1 represents the Poisson's ratio of the large gear material. μ 2 represents the Poisson's ratio of the pinion material; The velocity at the contact point between the large and small gears is obtained based on the basic kinematics of gear transmission. The calculation formula for this velocity is as follows: in, ω g Let be the angular velocity vector of the large gear. ω p Let be the angular velocity vector of the pinion. r g Let be the radius vector of the large gear. r p Let be the radius vector of the pinion. v g The velocity vector of the large gear. v p The velocity vector of the pinion; The entrainment speed along the minor and major axes of the contact ellipse is calculated based on the velocity vectors of the large gear and the small gear, wherein the calculation formula for the entrainment speed along the minor and major axes of the contact ellipse based on the velocity vectors of the large gear and the small gear is as follows: in, Let be the tangential component of the large gear velocity vector along the minor axis of the contact ellipse. Let be the tangential component of the pinion velocity vector along the minor axis of the contact ellipse. Let be the component of the velocity vector of the large gear along the tangent plane of the tooth surface. Let be the component of the pinion's velocity vector along the tangent plane of the tooth surface. Let be the component of the entrainment velocity along the minor axis of the contact ellipse. Let be the tangential component of the entrainment velocity along the major axis of the contact ellipse. U e The entrainment velocity vector, θ e The entrainment velocity vector and the minor axis of the contact ellipse x The included angle of the axis; The suction velocity is calculated based on the suction velocities along the minor and major axes of the contact ellipse, and the suction velocity vector is calculated relative to the minor axis of the contact ellipse. x The included angle of the axis, the sliding velocity, and the sliding velocity vector relative to the minor axis of the contact ellipse x The angle between the axes, wherein the suction velocity is calculated based on the suction velocity along the minor and major axes of the contact ellipse, and the suction velocity vector is perpendicular to the minor axis of the contact ellipse. x The included angle of the axis, the sliding velocity, and the sliding velocity vector relative to the minor axis of the contact ellipse x The formula for calculating the included angle of the axes is: in, U s The sliding velocity vector, θ s The sliding velocity vector and the minor axis of the contact ellipse x The included angle of the axis; The slip-roll ratio is calculated based on the suction speed and the sliding speed, wherein the formula for calculating the slip-roll ratio based on the suction speed and the sliding speed is: Wherein, AKC is the slip-roll ratio at the contact point.

4. The method for calculating the meshing efficiency of a spiral bevel gear according to claim 3, characterized in that, The step of numerically calculating the oil film shear force based on the first characteristic parameter using the separation flow method according to the non-Newtonian fluid thermo-elasto-fluid lubrication equation includes: The Reynolds equation is calculated from the thermo-elasto-hydrodynamic lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter. The calculation formula for the Reynolds equation from the thermo-elasto-hydrodynamic lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows: in, p For pressure, h For oil film thickness, ρ For the density of lubricating oil, η The viscosity of the lubricating oil. x Let the coordinates be the direction of motion. y The coordinates are perpendicular to the direction of motion. x The dimensionless flow factor is one. y The dimensionless flow factor is two; The film thickness equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the film thickness equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows: in, h 0 represents the center film thickness before deformation. R x for x The radius of curvature in the direction, R y for y The radius of curvature in the direction, E To measure the overall elastic modulus, x For corresponding x Additional coordinates, y For corresponding y The additional coordinates, Ω is the solution domain. S ( x , y ) represents the roughness height value of the tooth surface; The viscosity equation for non-Newtonian fluid thermo-elasto-fluid lubrication is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the viscosity equation is as follows: in, For environmental viscosity, It is a dimensionless viscosity-pressure index. T For lubricating oil temperature, This refers to the initial temperature of the lubricating oil. It is a dimensionless viscosity-temperature index. β T Viscosity-temperature coefficient, Viscosity; The density equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the density equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows: in, The ambient density of the lubricating oil, D 0 represents the coefficient of thermal expansion; The load balance equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid is calculated using the separation flow rate method based on the first characteristic parameter, wherein the calculation formula for the load balance equation for the thermo-elasto-fluid lubrication equation of a non-Newtonian fluid using the separation flow rate method based on the first characteristic parameter is as follows: in, F The total load (N) p h For the pressure that the oil film withstands, p c The pressure borne by the rough peak; The temperature field control equation is calculated based on the first characteristic parameter using the separation flow method to calculate the temperature field control equation for the non-Newtonian fluid thermo-elastohydrodynamic lubrication equation. The temperature field control equation includes the oil film energy equation, the solid thermal conductivity equation, and the temperature continuity equation. The calculation formula for the temperature field control equation based on the first characteristic parameter using the separation flow method is as follows: in, η * indicates equivalent viscosity. For the dimensionless oil film comprehensive shear stress, The specific heat of the lubricating oil. Let be the thermal conductivity coefficient of the lubricating oil, u be the flow velocity of the lubricating oil along the x-direction, and v be the flow velocity of the lubricating oil along the y-direction. The specific heat capacity of gear 1, The density of gear 1, The thermal conductivity of gear 1 The specific heat capacity of gear 2, The density of gear 2, The thermal conductivity of gear 2 Let be the flow velocity of the lubricating oil on the surface of gear 1 along the x-direction. Let be the flow velocity of the lubricating oil on the surface of gear 2 along the x-direction. Let be the flow velocity of the lubricating oil on the surface of gear 1 along the y-direction. The flow rate of the lubricating oil on the surface of gear 2 along the y-direction; The oil film shear force is obtained through numerical calculation using dimensionless transformation and the Erying rheological model based on the Reynolds equation, the film thickness equation, the viscosity equation, the density equation, the load balance equation, and the temperature field control equation. The formula for calculating the oil film shear force using the Reynolds equation, the film thickness equation, the viscosity equation, the density equation, the load balance equation, and the temperature field control equation, based on dimensionless transformation and the Erying rheological model, is as follows: in, ξ The coordinates are dimensionless film thickness directions. For dimensionless composite shear stress, G Shear modulus To and Related non-Newtonian fluid models, The dimensionless characteristic shear stress of the lubricating oil. For the oil film center layer along x Dimensionless shear stress in the direction, For the oil film center layer along y Dimensionless shear stress in the direction, τ e This refers to the oil film shear force.

5. The method for calculating the meshing efficiency of a spiral bevel gear according to claim 4, characterized in that, The formulas for calculating the contact area in the elastic stage, the contact area in the elastic-plastic stage, and the contact area in the fully plastic stage based on the elastoplastic contact state of the micro-protrusion are as follows: in, A e ( ω () represents the contact area of ​​the elastic stage. A ep ( ω () represents the contact area during the elastoplastic stage. A p ( ω () represents the contact area during the fully plastic stage. K This is the hardness index. v It is the Poisson's ratio of the material. H The hardness of the softer material in the contact surface. E * represents the equivalent elastic modulus. ω This represents the normal deformation of the micro-convex body. ω 1 represents the critical normal deformation at the initial yield point. ω 2 represents the critical normal deformation amount required to enter the fully plastic deformation stage. α Let the fitting constant be one. λ The fitting constant is two. K ( e () is a first-kind elliptic integral. E ( e () is a complete elliptic integral of the second kind; The formulas for calculating the elastic stage contact load, elastic-plastic stage contact load, and fully plastic stage contact load based on the elastoplastic contact state of the micro-protrusion are as follows: in, F e ( ω () represents the contact load during the elastic phase. F ep ( ω () represents the contact load during the elastoplastic stage. F p ( ω () represents the contact load during the fully plastic stage; The formula for calculating the contact pressure of the micro-protrusion based on the elastoplastic contact state, the contact area of ​​the elastic stage, the contact area of ​​the elastoplastic stage, the contact area of ​​the fully plastic stage, the contact load of the elastic stage, the contact load of the elastoplastic stage, and the contact load of the fully plastic stage is as follows: in, p c,e ( ω () represents the contact pressure of the micro-protrusions in the elastic stage. p c,ep ( ω () represents the contact pressure of the micro-protrusions in the elastoplastic stage. p c,p ( ω () represents the contact pressure of the micro-protrusions during the fully plastic stage; The formula for calculating the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force and the micro-protrusion contact pressure is as follows: in, μ Let be the coefficient of friction of the spiral bevel gear tooth surface. μ c The boundary friction coefficient is the preset rough peak friction. p c ( ω ) represents the contact pressure of the micro-protrusion.

6. The method for calculating the meshing efficiency of a spiral bevel gear according to claim 5, characterized in that, The calculation of the meshing efficiency of the spiral bevel gear based on the friction coefficient, the gear blank parameters, and the machining parameters includes: The sliding friction force is calculated based on the friction coefficient, wherein the formula for calculating the sliding friction force based on the friction coefficient is: in, F s It is sliding friction; The rolling friction force is calculated based on the gear blank parameters and the machining parameters, wherein the calculation formula for the rolling friction force based on the gear blank parameters and the machining parameters is as follows: in, F ro For rolling friction, F r For rolling friction, T For the heat-affected factor, G These are dimensionless material parameters. The dimensionless entrainment velocity, α The viscosity-pressure coefficient, To measure the overall elastic modulus, p h The normal load borne by the oil film. β T Viscosity-temperature coefficient, k f The thermal conductivity coefficient of the lubricating oil, U s The relative sliding speed, U e The relative entrainment speed; The meshing efficiency of the spiral bevel gear is calculated based on the sliding friction and the rolling friction.

7. The method for calculating the meshing efficiency of a spiral bevel gear according to claim 6, characterized in that, The formula for calculating the meshing efficiency of the spiral bevel gear based on the sliding friction and the rolling friction is as follows: in, η e For the meshing efficiency of spiral bevel gears, T g For input torque, ω g Input rotational speed.

8. A system for calculating the meshing efficiency of spiral bevel gears, characterized in that, The spiral bevel gear meshing efficiency calculation system includes: The data acquisition module is used to acquire the blank parameters and machining parameters of the spiral bevel gear; An equivalent meshing characteristic parameter calculation module is used to calculate equivalent meshing characteristic parameters based on the gear blank parameters and the machining parameters; The first characteristic parameter calculation module is used to calculate the first characteristic parameter of the bearing contact point on the bevel gear tooth surface based on the equivalent meshing characteristic parameter. The oil film shear force calculation module is used to numerically calculate the non-Newtonian fluid thermo-elasto-fluid lubrication equation based on the first characteristic parameter using the separation flow method to obtain the oil film shear force. The friction coefficient calculation module is used to calculate the friction coefficient of the spiral bevel gear tooth surface based on the oil film shear force, specifically: The elastoplastic contact state of the micro-protrusion is obtained based on the separation flow method and the non-Newtonian fluid thermo-elastohydrodynamic lubrication equation; Calculate the contact area in the elastic stage, the contact area in the elastic-plastic stage, and the contact area in the fully plastic stage based on the elastoplastic contact state of the micro-protrusion. Calculate the elastic stage contact load, elastic-plastic stage contact load, and fully plastic stage contact load based on the elastoplastic contact state of the micro-protrusion. The micro-protrusion contact pressure is calculated based on the elastoplastic contact state of the micro-protrusion, the contact area of ​​the elastic stage, the contact area of ​​the elastoplastic stage, the contact area of ​​the fully plastic stage, the contact load of the elastic stage, the contact load of the elastoplastic stage, and the contact load of the fully plastic stage, wherein the micro-protrusion contact pressure includes the elastic micro-protrusion contact pressure, the elastoplastic micro-protrusion contact pressure, and the fully plastic micro-protrusion contact pressure; The friction coefficient of the spiral bevel gear tooth surface is calculated based on the oil film shear force and the micro-protrusion contact pressure. The meshing efficiency calculation module is used to calculate the meshing efficiency of the spiral bevel gear based on the friction coefficient, the gear blank parameters, and the machining parameters.

9. A device for calculating the meshing efficiency of spiral bevel gears, characterized in that, It includes at least one control processor and a memory for communicatively connecting to the at least one control processor; the memory stores instructions executable by the at least one control processor, which, when executed by the at least one control processor, enable the at least one control processor to perform a method for calculating the meshing efficiency of a spiral bevel gear as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores computer-executable instructions for causing a computer to perform a method for calculating the meshing efficiency of a spiral bevel gear as described in any one of claims 1 to 7.

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