A calculation method and device for face gear meshing stiffness considering lubrication factors at the meshing interface

Abstract

An embodiment of the present application provides a method and device for calculating the meshing stiffness of face gears considering lubrication factors at the meshing interface. The method includes the following steps: obtaining geometric parameters of a plurality of face gear pairs, where the face gear pairs include face gears and spur gears; determining a target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the plurality of face gear pairs; calculating the meshing stiffness of the face gear and the spur gear within the target contact area; calculating the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation; and correcting the meshing stiffness of the face gear pair with the oil film stiffness to obtain the target meshing stiffness of the face gear pair. The embodiment of the present application can improve the calculation accuracy of the meshing stiffness of face gears in the time-varying meshing stiffness.

Description

Technical Field

[0001] This application relates to the technical field of meshing stiffness calculation, and particularly relates to a face gear meshing stiffness calculation method and device considering lubrication factors at the meshing interface. Background Art

[0002] Face gears are usually used in combination with bevel gears or cylindrical gears to transmit power between intersecting or skew axes, and their working performance has an important impact on the vibration and noise characteristics of the entire transmission system. The time-varying meshing stiffness, as the main internal excitation of the gear transmission system, is the main source of system vibration and noise. Therefore, accurately and quickly calculating the time-varying meshing stiffness is the key basis for the dynamic analysis of gear systems.

[0003] Currently, most calculation methods for time-varying meshing stiffness rely on finite element analysis of commercial software, but the calculation accuracy is relatively low. Summary of the Invention

[0004] The main objective of the embodiments of this application is to propose a face gear meshing stiffness calculation method, device, electronic device, and storage medium considering lubrication factors at the meshing interface, aiming to improve the calculation accuracy of the face gear meshing stiffness in time-varying meshing stiffness.

[0005] To achieve the above objective, the first aspect of the embodiments of this application proposes a face gear meshing stiffness calculation method considering lubrication factors at the meshing interface. The method includes:

[0006] Obtain the geometric parameters of multiple face gear pairs, where the face gear pairs include face gears and spur gears;

[0007] Determine the target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the multiple face gear pairs;

[0008] Calculate the meshing stiffness of the face gear and the spur gear within the target contact area;

[0009] Calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation;

[0010] Use the oil film stiffness to correct the meshing stiffness of the face gear pair to obtain the target meshing stiffness of the face gear pair.

[0011] In some embodiments, the determining the target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the multiple face gear pairs includes:

[0012] Conduct contact analysis on the geometric parameters of the multiple face gear pairs to obtain target parameters, where the target parameters include the first contact area;

[0013] Adjust the first contact area according to the gear boundary conditions to obtain the target contact area of the meshing interface, and the target contact area is an elliptical area.

[0014] In some embodiments, the contact analysis of the geometric parameters of multiple face gear pairs to obtain target parameters includes:

[0015] Establish a face gear tooth surface equation and a spur gear tooth surface equation based on the geometric parameters of multiple face gear pairs;

[0016] According to the face gear tooth surface equation and the spur gear tooth surface equation, and based on the meshing principle, obtain the tooth surface contact points and the meshing trajectory;

[0017] Calculate the radius of curvature of the face gear and the spur gear at the tooth surface contact points, and calculate the major and minor semi-axes of the meshing trajectory based on a preset first load distribution ratio and Hertz contact theory to obtain the first contact area.

[0018] In some embodiments, the calculation of the meshing stiffness of the face gear and the spur gear in the target contact area includes:

[0019] Divide the tooth surface of the face gear into multiple sliced tooth surfaces along the tooth width direction within the target contact area;

[0020] Calculate the bending stiffness, the shear stiffness, and the axial compression stiffness of each sliced tooth surface respectively;

[0021] Adjust the boundary of the target contact area through the deformation coordination condition to obtain the second contact area, and calculate the second load distribution ratio according to the second contact area. The deformation coordination condition is the geometric condition that the displacements or strains of each part of the face gear pair satisfy during the deformation process;

[0022] Judge whether the difference between the second load distribution ratio and the first load distribution ratio is greater than or equal to a first preset threshold. If the difference is greater than or equal to the first preset threshold, use the second load distribution ratio as the first load distribution ratio, and go back to the step of calculating the major and minor semi-axes of the meshing trajectory based on the preset first load distribution ratio and Hertz contact theory to obtain the first contact area until the difference between the second load distribution ratio and the first load distribution ratio is less than the first preset threshold, and use the second load distribution ratio obtained in the last iteration as the target load distribution ratio;

[0023] Integrate the bending stiffness of each sliced tooth surface to obtain the total bending stiffness of all sliced tooth surfaces;

[0024] Integrate the shear stiffness of each of the slice tooth surfaces to obtain the total shear stiffness of all the slice tooth surfaces;

[0025] Integrate the axial compression stiffness of each of the slice tooth surfaces to obtain the total axial compression stiffness of all the slice tooth surfaces;

[0026] Perform calculations based on the target load distribution ratio to obtain the Hertz contact stiffness of the face gear pair;

[0027] Perform calculations based on the Hertz contact stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness of the face gear pair to obtain the meshing stiffness of the face gear pair.

[0028] In some embodiments, the performing calculations based on the Hertz contact stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness of the face gear pair to obtain the meshing stiffness of the face gear pair includes:

[0029] Calculate the load of the target contact area based on the target load distribution ratio, and obtain the Hertz contact stiffness of the face gear pair according to the load;

[0030] Calculate, based on the Hertz contact stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness, to obtain a first meshing stiffness of a single tooth meshing, and the face gear pair includes a plurality of the single tooth meshings;

[0031] Calculate the meshing stiffness of the face gear pair according to the deformation coordination relationship for the first meshing stiffness of the single tooth meshing.

[0032] In some embodiments, the using the oil film stiffness to correct the meshing stiffness of the face gear pair to obtain the target meshing stiffness of the face gear pair includes:

[0033] Calculate, based on the oil film stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness, to obtain a second meshing stiffness of the single tooth meshing;

[0034] Calculate the target meshing stiffness of the face gear pair according to the deformation coordination relationship for the second meshing stiffness of the single tooth meshing.

[0035] In some embodiments, the calculating the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation includes:

[0036] Perform calculations according to a first oil film pressure value and a first minimum film thickness to obtain a first oil film thickness;

[0037] Obtain a first lubricating oil viscosity value and a first lubricating oil density value according to the first oil film thickness;

[0038] Based on the first lubricating oil viscosity value and the first lubricating oil density value, a second oil film pressure value is obtained;

[0039] If the difference between the second oil film pressure value and the first oil film pressure value is greater than or equal to a second preset threshold, then based on the second oil film pressure value, a second lubricating oil viscosity value and a second lubricating oil density value are obtained, and the second lubricating oil viscosity value is used as the first lubricating oil viscosity value, and the second lubricating oil density value is used as the first lubricating oil density value, and the step of obtaining the second oil film pressure value according to the first lubricating oil viscosity value and the first lubricating oil density value is executed, and the obtained second oil film pressure value is used as the target oil film pressure value until the difference between the second oil film pressure value and the first oil film pressure value is less than the second preset threshold;

[0040] If the difference between the second oil film pressure value and the first oil film pressure value is less than the second preset threshold, then it is judged whether the target oil film pressure value meets the load balance condition according to the load balance equation;

[0041] If the target oil film pressure value does not meet the load balance condition, then the third oil film pressure value and the second minimum film thickness are calculated by adjusting the rigid normal displacement of the face gear pair, and the third oil film pressure value is used as the first oil film pressure value, and the second minimum film thickness is used as the first minimum film thickness, and the step of calculating the first oil film thickness according to the first oil film pressure value and the first minimum film thickness is executed until the load balance condition is met, and the third oil film pressure value obtained in the last iteration is used as the target oil film pressure value;

[0042] If the target oil film pressure value meets the load balance condition, then the latest first oil film thickness is used as the target oil film thickness;

[0043] The oil film stiffness of the meshing interface is calculated according to the target oil film thickness and the target oil film pressure value.

[0044] To achieve the above object, a second aspect of the embodiments of the present application proposes a face gear meshing stiffness calculation device considering lubrication factors at the meshing interface, and the device includes:

[0045] An acquisition module, configured to acquire geometric parameters of a plurality of face gear pairs, where the face gear pair includes a face gear and a spur gear;

[0046] A contact area calculation module, configured to determine a target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the plurality of face gear pairs;

[0047] A meshing stiffness calculation module, configured to calculate the meshing stiffness between the face gear and the spur gear in the target contact area;

[0048] An oil film stiffness calculation module, configured to calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation;

[0049] A correction module, configured to correct the meshing stiffness of the face gear pair by using the oil film stiffness to obtain the target meshing stiffness of the face gear pair.

[0050] To achieve the above object, a third aspect of the embodiments of the present application provides an electronic device, which includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the method described in the first aspect above is implemented.

[0051] To achieve the above object, a fourth aspect of the embodiments of the present application provides a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, the method described in the first aspect above is implemented.

[0052] The face gear meshing stiffness calculation method, device, electronic device and storage medium considering the lubrication factor of the meshing interface proposed by the present application obtain the geometric parameters of multiple face gear pairs, determine the target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of multiple face gear pairs; calculate the meshing stiffness of the face gear and the spur gear in the target contact area; calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation; correct the meshing stiffness of the face gear pair by using the oil film stiffness to obtain the target meshing stiffness of the face gear pair; solve the meshing stiffness by simulating the lubrication state of the gear meshing process and combining the meshing effect of the gear, considering the influence of the lubrication factor of the meshing interface on the face gear meshing stiffness, thereby improving the calculation accuracy of the face gear meshing stiffness. Description of the Drawings

[0053] Figure 1 is a schematic flowchart of the face gear meshing stiffness calculation method considering the lubrication factor of the meshing interface provided by the embodiments of the present application;

[0054] Figure 2 is a schematic diagram of the face gear shaping principle;

[0055] Figure 3 is a schematic diagram of the coordinate system of the face gear machined by the shaping cutter envelope;

[0056] Figure 4 is a schematic diagram of the tooth surface meshing relationship, where (a) is a schematic diagram of the gear surface and (b) is a schematic diagram of the contact ellipse;

[0057] Figure 5 is a schematic diagram for judging the tooth surface boundary conditions of the face gear pair;

[0058] Figure 6 It is a schematic diagram of the tooth surface section of the face gear;

[0059] Figure 7 It is a schematic diagram of the deformation coordination relationship;

[0060] Figure 8(a) is a comparison diagram of the meshing stiffness of the face gear pair considering the influence of the oil film at the meshing interface and the single-tooth meshing stiffness of the face gear pair without considering the influence of the oil film at the meshing interface;

[0061] Figure 8(b) is a comparison diagram of the meshing stiffness of the face gear pair considering the influence of the oil film at the meshing interface and the comprehensive meshing stiffness of the face gear pair without considering the influence of the oil film at the meshing interface;

[0062] Figure 9 It is a schematic diagram of the oil film stiffness;

[0063] Figure 10 It is a schematic diagram of the structure of the face gear meshing stiffness calculation device considering the lubrication factor at the meshing interface provided by the embodiment of the present application;

[0064] Figure 11 It is a schematic diagram of the hardware structure of the electronic device provided by the embodiment of the present application. Detailed implementation manners

[0065] In order to make the purpose, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application, and are not used to limit the present application.

[0066] It should be noted that although the functional modules are divided in the device schematic diagram and the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in a different order from the module division in the device or the order in the flowchart. The terms "first", "second", etc. in the specification, claims and the above drawings are used to distinguish similar objects, and do not have to be used to describe a specific order or sequence.

[0067] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the technical field to which the present application belongs. The terms used herein are only for the purpose of describing the embodiments of the present application, and are not intended to limit the present application.

[0068] The face gear meshing stiffness calculation method, device, electronic device and storage medium considering the lubrication factor at the meshing interface provided by the embodiments of the present application will be specifically described through the following embodiments. First, the face gear meshing stiffness calculation method considering the lubrication factor at the meshing interface in the embodiments of the present application will be described.

[0069] Figure 1It is an alternative flowchart of the face gear meshing stiffness calculation method considering the lubrication factor of the meshing interface provided by the embodiments of the present application. Figure 1 The method in

[0070] Step S100: Obtain the geometric parameters of multiple face gear pairs, where the face gear pairs include face gears and spur gears.

[0071] In this embodiment, the geometric parameters of the face gear pair can be obtained by input or measurement. The geometric parameters of multiple face gear pairs include but are not limited to the number of teeth of the spur gear, the number of teeth of the face gear, the module at the large end, the tooth width, the pressure angle, the addendum coefficient, the clearance coefficient, the elastic modulus, and the Poisson's ratio. Exemplarily, Table 1 shows the geometric parameters of the face gear pair and a corresponding value-taking method, as follows:

[0072] Table 1

[0073]

[0074] Step S200: Determine the target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of multiple face gear pairs.

[0075] Construct the face gear tooth surface equation and the spur gear tooth surface equation according to the obtained geometric parameters of the face gear pair; calculate the tooth surface contact points and the meshing trajectory of the face gear tooth surface and the spur gear tooth surface based on the meshing principle to determine the contact area; adjust the boundary of the contact area according to the deformation coordination condition to obtain the target contact area.

[0076] Step S300: Calculate the meshing stiffness of the face gear and the spur gear in the target contact area.

[0077] In the target contact area, divide the tooth surface of the face gear into multiple slice tooth surfaces along the tooth width direction. Each slice tooth surface can be regarded as an independent unit for calculating the local meshing stiffness; for each slice tooth surface, calculate its bending stiffness, shear stiffness, and axial compression stiffness respectively; integrate the bending stiffness, shear stiffness, and axial compression stiffness of the slice tooth surface to obtain the total bending stiffness, total shear stiffness, and total axial compression stiffness of all slice tooth surfaces; calculate the Hertz contact stiffness based on the target load distribution ratio, and calculate according to the Hertz contact stiffness, total bending stiffness, total shear stiffness, and total axial compression stiffness to obtain the meshing stiffness of the face gear pair.

[0078] Step S400: Calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation.

[0079] First, establish the elastohydrodynamic lubrication control equations, which include the Reynolds equation, film thickness equation, deformation equation, viscosity-pressure equation, density-pressure equation, and load balance equation. Then, discretize the equations using the finite difference method and perform iterative solutions until the pressure convergence condition and load balance condition are satisfied. The pressure convergence condition is that the pressure difference obtained from two consecutive iterations is less than a preset threshold, and the load balance condition means that the pressure value and the external load satisfy the load smoothing equation. Finally, obtain the final pressure distribution and film thickness including elastic deformation. Calculate the oil film stiffness of the face gear and spur gear based on the final pressure distribution and film thickness.

[0080] Step S500: Use the oil film stiffness to correct the meshing stiffness of the face gear pair to obtain the target meshing stiffness of the face gear pair.

[0081] Replace the contact stiffness with the oil film stiffness and calculate the single tooth meshing stiffness considering the oil film stiffness. Then, superimpose all the single tooth meshing stiffnesses to obtain the time-varying meshing stiffness of the face gear pair considering the lubrication factor at the meshing interface.

[0082] In some embodiments, step S200 may include but is not limited to steps S210 to S220:

[0083] Step S210: Conduct a contact analysis on the geometric parameters of multiple face gear pairs to obtain target parameters, where the target parameters include the first contact area.

[0084] In this embodiment, to conduct a contact analysis on the geometric parameters of multiple face gear pairs, the tooth surface contact points and meshing trajectories can be obtained by first establishing the tooth surface equations of the face gear and spur gear, and then based on the Hertz contact theory, calculate the principal curvature radii and major and minor semi-axes of the face gear and spur gear to obtain the first contact area.

[0085] Specifically, the conducting a contact analysis on the geometric parameters of multiple face gear pairs to obtain target parameters, where the target parameters include the first contact area, includes:

[0086] Establish the face gear tooth surface equation and spur gear tooth surface equation based on the geometric parameters of multiple face gear pairs;

[0087] According to the face gear tooth surface equation and the spur gear tooth surface equation, and based on the meshing principle, obtain the tooth surface contact points and meshing trajectories;

[0088] Calculate the curvature radii of the face gear and the spur gear at the tooth surface contact points, and based on a preset first load distribution ratio and the Hertz contact theory, calculate the major and minor semi-axes of the meshing trajectory to obtain the first contact area.

[0089] In this embodiment, based on the geometric parameters of multiple said face gear pairs, a face gear tooth surface equation and a spur gear tooth surface equation are established. The face gear tooth surface equation can be derived from the tooth surface equation of the gear shaper cutter, while the spur gear tooth surface equation can be represented by the involute tooth surface equation. The tooth surface contact point refers to the position point where the two tooth surfaces actually contact when the face gear meshes with the spur gear. The meshing locus refers to the set of all contact points when the face gear meshes with the spur gear, forming a continuous locus line. Since under no-load conditions, the tooth surface contact points of the face gear and the spur gear will form a continuous approximately elliptical curve, the first contact area of the approximate ellipse can be obtained through the tooth surface contact points and the meshing locus.

[0090] Specifically, the face gear tooth surface is enveloped by the tooth surface of the gear shaper cutter according to the relative motion law of gear shaping. As Figure 2 shown, the gear shaper cutter and the face gear rotate about their own axes with angular velocities and respectively, and the angular velocities satisfy Equation (1):

[0091] (1);

[0092] In the formula, and are the number of teeth of the face gear and the gear shaper cutter respectively, is the tooth number ratio. To meet the performance requirements of contact, usually take the number of teeth of the gear shaper cutter minus the number of teeth of the spur gear equal to 1 - 3 to achieve point meshing between the spur gear and the face gear. The tooth surface equations of the gear shaper cutter and the spur gear are the same, both using the involute tooth surface equation. During the machining process, in addition to the two gears rotating about their own axes, the gear shaper cutter also needs to make a feed motion along the generatrix direction of the face gear cone.

[0093] Establish a gear shaper cutter envelope machining face gear coordinate system as shown in Figure 3 , where S 2( x 2, y 2, z 2) and S s ( x s , y s , z s ) are the moving coordinate systems fixed to the face gear and the gear shaper cutter respectively, S p ( x p , y p , z p) is the fixed coordinate system fixedly connected to the face gear bracket, S m ( x m , y m , z m ) is the fixed coordinate system fixedly connected to the shaper cutter machine base, and Sm and Sp are both auxiliary coordinate systems used to simplify the derivation of coordinate transformation. The shaper cutter rotation angle and the face gear rotation angle satisfy the relationship: , in the figure γ m is the shaft intersection angle of the face gear drive, and the face gear tooth surface can be expressed as Equation (2) in S 2:

[0094] (2);

[0095] where is the shaper cutter tooth surface equation, is the unit normal vector of the shaper cutter tooth surface, is the relative velocity between the shaper cutter and the face gear on the shaper cutter tooth surface, and are involute tooth surface parameters, is the meshing equation. , , , are transformation matrices, specifically shown in Equations (3) to (6) as follows:

[0096] is the transformation matrix from the coordinate system S s ( x s , y s , z s ) to S m ( x m , y m , z m ) :

[0097] (3);

[0098] is the transformation matrix from the coordinate system Sm ( xm, ym, zm ) to Sp (xp, yp, zp Transformation matrix of

[0099] (4);

[0100] is the coordinate system S p ( x p , y p , z p ) to S 2( x 2, y 2, z 2):

[0101] (5);

[0102] is the coordinate system of the gear shaper S s ( x s , y s , z s ) to the face gear coordinate system S 2( x 2, y 2, z 2):

[0103] (6);

[0104] To solve the meshing equation, the relative velocity of the gear shaper and the face gear on the tooth surface of the gear shaper needs to be solved first. The angular velocity magnitudes of the gear shaper and the face gear around their own rotation axes under S s are respectively , then the relative velocity of the two gears at the contact point is expressed as Equation (7):

[0105] (7);

[0106] Furthermore, according to the meshing equation and obtain , The relationship among the three parameters, eliminating , using and The tooth surface equation of the face gear is expressed as Equation (8) with two parameters:

[0107] (8);

[0108] Wherein:

[0109] ;

[0110] ;

[0111] ;

[0112] is the angular value from the symmetric line of the tooth groove to the starting point of the involute.

[0113] Thus, the tooth surface equation of the face gear is obtained. The algorithm for gear meshing simulation is based on the equation describing the continuous contact of two tooth surfaces, that is, the position vectors of the tangent contact points are equal and the unit normal vectors are equal in the same coordinate system, satisfying the meshing principle. The following analyzes the tooth surface contact trajectory under no load, and is based on the following assumptions: 1) Assume that the gear is a rigid body, no deformation occurs during the meshing process, and the influence of the deformation of the shaft and bearings is ignored; 2) Assume that the tooth surface is completely precise, and the temperature rise and the change of the tooth surface shape during the meshing process are not considered.

[0114] According to the formation of point contact of the face gear, the derivation of the tooth surface equation of the spur gear is the same as that of the tooth surface equation of the gear shaper cutter. Thus, the tooth surface equation of the spur gear can be obtained, expressed as , similarly, the solution of the unit normal vector of the tooth surface of the spur gear is the same as that of the unit normal vector of the tooth surface of the gear shaper cutter, denoted as . Now establish the installation coordinate system , and analyze the contact between the face gear and the spur gear in the coordinate system . Then its tooth surface equation and unit normal vector equation are Equation (9):

[0115] (9);

[0116] Among them, (1) and (2) in the above formula represent the spur gear and the face gear respectively, , are the rotation angles of the spur gear and the face gear respectively, , are the coordinate transformation matrices from the coordinate system S 1 to S f and S 2 to S f respectively, and are specifically expressed as shown in Equation (10) and Equation (11):

[0117] (10);

[0118] (11);

[0119] Among them, is the complementary angle of the shaft intersection angle. According to the meshing principle, Equation (12) can be obtained as follows:

[0120] (12);

[0121] Since the magnitudes of the unit normal vectors are all 1, the above equation contains only five independent equations. By taking the face gear rotation angle as the input parameter, the other five parameters can be solved. Substituting them into the tooth surface equation, the tooth surface contact points and the meshing locus can be obtained. In addition, the possible edge contact situation of the face gear pair must be considered, that is, the condition of continuous contact between the curve and the surface, which can be expressed as Equation (13):

[0122] (13);

[0123] Combining the gear contact area and the edge contact to obtain the potential contact trace of the face gear pair, and obtaining the face gear contact ratio according to the rotation angle of the gear during movement obtained from the contact analysis, as shown in Equation (14):

[0124] (14);

[0125] In the formula, and respectively represent the rotation angles corresponding to the initial position and the end position of the face gear.

[0126] Obtaining the transmission error of the face gear pair according to the rotation angle of the gear during movement obtained from the contact analysis, as shown in Equation (15):

[0127] (15);

[0128] In the formula, and respectively represent the rotation angles corresponding to the initial position of the spur gear.

[0129] The face gear and the spur gear are in point contact, and the instantaneous contact area can be approximated as an ellipse. The direction with the minimum induced normal curvature at the contact point is the major axis direction of the instantaneous contact ellipse, and the direction with the maximum induced normal curvature is the minor axis direction of the instantaneous contact ellipse. Before applying the Hertz contact analysis method to calculate the major and minor semi-axes of the elliptical contact of the face gear, the principal curvatures of the face gear at each meshing point need to be calculated first.

[0130] As Figure 4 (a) shows, for any point on the face gear, since the working tooth surface equation of the face gear can be expressed as , the tangent vectors of the two coordinate curves on the surface can be calculated as Equation (16):

[0131] (16);

[0132] The tangent vector in any direction within its tangent plane can be expressed as Equation (17):

[0133] (17);

[0134] where the included angle between the tangent vectors and at a certain point p has a definite value, while the value of the angle v can vary with the change in the direction of t, as shown in Equation (18): u (18);

[0135] (18);

[0136] According to the second fundamental form, the second fundamental quantities are respectively Equation (19):

[0137] (19);

[0138] where the vectors , and are the second-order derivative vectors of the tooth surface, and n is the normal vector of the tooth surface at the meshing point p . From the acceleration vector expression, the normal curvature expression of the surface can be deduced as shown in Equation (20):

[0139] (20);

[0140] where , , are as shown in Equation (21):

[0141] , (21);

[0142] Taking the derivative of Equation (20) and setting the derivative to 0, Equation (22) is obtained:

[0143] (22);

[0144] where u there are two solutions. Assuming one solution is u 1, then the other solution is u 2 = u 1 + 90°. Substituting the obtained solutions into Equation (17) and Equation (20), two corresponding principal directions and principal curvatures can be obtained. Thus, the principal curvatures and the corresponding principal directions at the meshing point of the face gear are obtained ( , and , ), similarly, the principal curvatures and the corresponding principal directions at the meshing point of the spur gear can be obtained ( , and , ).

[0145] As Figure 4 (b) shows, the angle between the major axis of the elliptical contact and the principal direction of the face gear tooth surface is , and the angle between the principal directions of the face gear and the spur gear is . The induced normal curvature on the major axis of the elliptical contact is Δ K , as shown in Equation (23):

[0146] (23);

[0147] According to differential geometry knowledge, the directions of the major and minor axes of the contact ellipse correspond to the directions where the induced normal curvature takes the minimum and maximum values, respectively. Therefore, let = 0, then the angle between the direction of the minor axis of the contact ellipse and can be obtained, and the definition of its principal directions x and y is Equation (24):

[0148] (24);

[0149] Then the radius of curvature of the face gear and the spur gear at the minor axis of the contact ellipse is Equation (25):

[0150] , (25);

[0151] The radius of curvature at the major axis of the contact ellipse is Equation (26):

[0152] , (26);

[0153] When the tooth surfaces of the spur gear and the face gear are non-coordinating surfaces and the radius of curvature is large enough, expanding the surface equation according to the Maclaurin series, the terms above the second order can be omitted, and the polynomial expression (27) of the two surface equations in any coordinate system can be obtained:

[0154] (27);

[0155] Among them, A 1, B 1, C 1 and A 2, B 2, C 2 are all constants, and the gap between the surfaces can be written as Equation (28):

[0156] (28);

[0157] Among them, A 、 B The constants are related to the geometric shapes of the two objects, and their values are calculated by Equation (29):

[0158] (29);

[0159] Among them, , and , represent the radii of curvature at the meshing point of the face gear and the spur gear. When using their principal radii of curvature to represent the initial clearance between the two surfaces, the radius of curvature of the convex surface is taken as positive, and the radius of curvature of the concave surface is taken as negative.

[0160] Calculate the major and minor semi-axes of the elliptical contact a and b The values of need to rely on the Hertz elastic contact theory Equation (30):

[0161] (30);

[0162] Among them, is the comprehensive elastic modulus, , and are the elastic modulus and Poisson's ratio of the spur gear and the face gear respectively.

[0163] P is the load, that is, the load of the target contact area calculated based on the target load distribution ratio, and are the parameters for calculating a and b as shown in Equation (31):

[0164] (31);

[0165] Among them, K ( e )、 E ( e ) are the first elliptic integral and the second elliptic integral respectively, e is the ellipticity. Usually, the following relationship is used to solve the ellipticity, as shown in Equation (32):

[0166] (32);

[0167] The first contact area can be obtained according to the above steps.

[0168] Step S220: Adjust the first contact area according to the gear boundary conditions to obtain the target contact area of the meshing interface, where the target contact area is an elliptical area.

[0169] In this embodiment, by adjusting the first contact area according to the gear boundary conditions, removing the parts exceeding the tooth width, tooth tip and tooth root, and retaining the contact area within the actual working range of the gear, the target contact area can be obtained. The target contact area is an elliptical area, which describes the contact range between the face gear and the spur gear during the actual meshing process.

[0170] Specifically, preset the initial value of the tooth contact load distribution ratio to 1, obtain the first contact area through step S210. As Figure 5 shown, judge the boundary conditions of the contact area, and cut the first contact area according to the gear boundary conditions (the parts exceeding the tooth width, tooth tip and tooth root) to obtain the target contact area.

[0171] In this embodiment, the elliptical contact area is obtained through contact analysis, which can calculate the meshing stiffness and tooth surface contact stress more accurately, thereby improving the accuracy of the dynamic analysis of the gear pair; adjusting the contact area according to the gear boundary conditions avoids the situation that the contact area exceeds the actual size of the gear, ensures that the calculation results are valid within the actual working range of the gear, and avoids errors.

[0172] In some embodiments, step S300 may include but is not limited to steps S310 to S390:

[0173] Step S310: Divide the tooth surface of the face gear into multiple sliced tooth surfaces along the tooth width direction within the target contact area;

[0174] Step S320: Calculate the bending stiffness, shear stiffness and axial compression stiffness of each sliced tooth surface respectively;

[0175] Step S330: Adjust the boundary of the target contact area through the deformation coordination condition to obtain the second contact area, and calculate the second load distribution ratio according to the second contact area, where the deformation coordination condition is the geometric condition satisfied by the displacements or strains of each part during the deformation process of the face gear pair;

[0176] Step S340: Determine whether the difference between the second load distribution ratio and the first load distribution ratio is greater than or equal to a first preset threshold. If the difference is greater than or equal to the first preset threshold, use the second load distribution ratio as the first load distribution ratio, and go to execute the step of calculating the major and minor axes of the meshing trajectory based on the preset first load distribution ratio and Hertz contact theory to obtain the first contact area, until the difference between the second load distribution ratio and the first load distribution ratio is less than the first preset threshold, and use the second load distribution ratio obtained in the last iteration as the target load distribution ratio;

[0177] Step S350: Integrate the bending stiffness of each slice tooth surface to obtain the total bending stiffness of all slice tooth surfaces;

[0178] Step S360: Integrate the shear stiffness of each slice tooth surface to obtain the total shear stiffness of all slice tooth surfaces;

[0179] Step S370: Integrate the axial compression stiffness of each slice tooth surface to obtain the total axial compression stiffness of all slice tooth surfaces;

[0180] Step S380: Calculate based on the target load distribution ratio to obtain the Hertz contact stiffness of the face gear pair;

[0181] Step S390: Calculate based on the Hertz contact stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness of the face gear pair to obtain the meshing stiffness of the face gear pair.

[0182] In this embodiment, the calculation steps of the meshing stiffness of the face gear pair are as follows: Obtain the second contact area (contact ellipse) according to the parameters of the face gear pair, and preset the first load distribution ratio (the initial value of the tooth contact load distribution ratio is preset to 1). Divide the face gear tooth surface into multiple slice tooth surfaces along the tooth width direction to calculate the stiffness of each slice separately. The number of slice divisions can be adjusted according to the accuracy requirements. Calculate the bending stiffness, shear stiffness, and axial compression stiffness of each slice tooth surface through finite element analysis or analytical methods respectively. Adjust the boundary of the target contact area through the deformation coordination condition to obtain the second contact area; the deformation coordination condition ensures that the displacements or strains of each part of the gear satisfy the geometric conditions during the deformation process; and calculate the second load distribution ratio according to the second contact area. Judge whether the difference between the second load distribution ratio and the first load distribution ratio is greater than or equal to the first preset threshold. If the difference is greater than or equal to the first preset threshold, take the second load distribution ratio as the first load distribution ratio, and repeat the steps of slice division and stiffness calculation until the difference is less than the first preset threshold; then take the second load distribution ratio as the target load distribution ratio. Integrate the bending stiffness, shear stiffness, and axial compression stiffness of each slice tooth surface respectively to obtain the total bending stiffness, total shear stiffness, and total axial compression stiffness of all slice tooth surfaces. Solve the load based on the target load distribution ratio, and obtain the Hertz contact stiffness according to the applied load. Finally, calculate the meshing stiffness of the face gear pair according to the Hertz contact stiffness, total bending stiffness, total shear stiffness, and total axial compression stiffness of the face gear pair.

[0183] Specifically, the calculation of obtaining the meshing stiffness of the face gear pair according to the Hertz contact stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness of the face gear pair includes:

[0184] Calculate the load of the target contact area based on the target load distribution ratio, and obtain the Hertz contact stiffness of the face gear pair according to the load;

[0185] Calculate the first meshing stiffness of single-tooth meshing based on the Hertz contact stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness. The face gear pair includes multiple single-tooth meshing;

[0186] Calculate the meshing stiffness of the face gear pair according to the deformation coordination relationship for the first meshing stiffness of single-tooth meshing.

[0187] In this embodiment, the load distribution in this area is calculated based on the target load distribution ratio, and the Hertz contact stiffness of the face gear pair is calculated according to the load distribution and the geometric parameters of the contact area by using the Hertz contact theory. The Hertz contact stiffness reflects the supporting ability of the oil film in the contact area. The meshing stiffness of the face gear pair consists of four parts: Hertz contact stiffness, total bending stiffness, total shear stiffness, and total axial compression stiffness. The equivalent meshing points of the face gear and the spur gear are sliced and divided to calculate the bending stiffness, shear stiffness, and axial compression stiffness of each slice respectively; then, according to the load distribution and geometric parameters on the slice, the bending stiffness, shear stiffness, and axial compression stiffness of each slice are calculated by using the formula; the boundary conditions of the contact area are judged, necessary trimming and adjustment are carried out, the load distribution ratio is recalculated, and iteration is performed until the load distribution ratio converges. Finally, the total bending stiffness, total shear stiffness, and total axial compression stiffness are obtained by summing the bending stiffness, shear stiffness, and axial compression stiffness of all slices respectively, and the Hertz contact stiffness, total bending stiffness, total shear stiffness, and total axial compression stiffness are substituted into the formula for calculation to obtain the meshing stiffness of single-tooth meshing (the first meshing stiffness); then, the single-tooth meshing stiffness is substituted into the deformation coordination relationship to obtain the meshing stiffness of the face gear pair.

[0188] Specifically, as Figure 6 shown, the equivalent meshing points on the face gear tooth surface are sliced and divided (the same applies to the spur gear).

[0189] The on each slice are solved respectively, as shown in Equations (33) to (34):

[0190] (33);

[0191] (34);

[0192] In the formula, , , are the bending potential energy, shear potential energy, and axial compression potential energy respectively, h x is the distance between the meshing point and the symmetry line of the tooth; t is the distance between the meshing point and the base circle, F is the mutual acting force at the meshing point, and the direction along the meshing line can be decomposed into the radial force F a and the tangential force F b , E is the elastic modulus, G is the shear modulus, b is the tooth width after slicing.

[0193] The bending stiffness of the gear is obtained by summing the stiffness of each slice. k b , shear stiffness k a , axial compression stiffness k s . As shown in Equation (35):

[0194] (35);

[0195] In the formula, n is the number of slices.

[0196] The load is solved through the load distribution ratio L sf , and the Hertz contact stiffness is obtained according to the applied load k h . As shown in Equations (36) to (37):

[0197] (36);

[0198] (37);

[0199] To sum up, the total meshing stiffness of a pair of gear pairs can be expressed as Equation (38):

[0200] (38);

[0201] In the formula, p and g are the driving gear and the driven gear respectively.

[0202] As Figure 7 shown, the contact range is reselected through the deformation coordination relationship to obtain a new load distribution ratio.

[0203] Through the deformation coordination relationship, the calculated contact ratio and the boundary conditions of the contact range are judged, which can be expressed as Equation (39):

[0204] (39);

[0205] In the formula, represents the transmission error between each tooth pair.

[0206] When the tooth pairs of the face gear pair are meshing, the load is shared by multiple tooth pairs, and the contact force can be expressed as Equations (40) to (41):

[0207] (40);

[0208] (41);

[0209] Substitute Equation (39) and Equation (40) into Equation (41) to obtain Equation (42):

[0210] (42);

[0211] where, ET 12 = - , and the same applies to others.

[0212] Furthermore, the load distribution ratio under this condition can be obtained, as shown in Equation (43):

[0213] (43);

[0214] When the face gear pair enters the double-tooth region, the meshing stiffness and load distribution ratio of the double-tooth region can also be obtained through the deformation coordination relationship, as shown in Equations (44) to (45):

[0215] (44);

[0216] (45);

[0217] The meshing stiffness results obtained from Equation (42) and Equation (44) may be less than 0 or smaller than the single-tooth stiffness, and need to be judged and corrected:

[0218] ① When k < 0, exclude all contact points.

[0219] ② When k < k e , if the gear is in the three-tooth region, it becomes the double-tooth region; if the gear is in the double-tooth region, it becomes the single-tooth region.

[0220] Finally, the contact range (target contact area) and load distribution ratio (target load distribution ratio) are obtained again.

[0221] If the difference in the load distribution ratio is less than 0.001 (the first preset threshold), the iteration is completed, and the meshing stiffness of the face gear pair is calculated according to the contact analysis and the deformation coordination relationship.

[0222] In this embodiment, the contact area is adjusted through the deformation coordination condition to ensure that the displacements or strains of each part satisfy the geometric conditions, and the deformation of each part during the gear meshing process is more accurately simulated; the load distribution ratio is adjusted iteratively to gradually approach the true value and improve the convergence of the calculation; the tooth surface is divided into multiple slices, and the stiffness of each slice is calculated separately, which simplifies the calculation process; at the same time, the bending stiffness, shear stiffness, axial compression stiffness, and Hertz contact stiffness are considered, more comprehensively reflecting the characteristics of the gear meshing stiffness.

[0223] In some embodiments, step S400 may further include but is not limited to steps S410 to S480:

[0224] Step S410, calculate according to the first oil film pressure value and the first minimum film thickness to obtain the first oil film thickness;

[0225] Step S420, obtain the first lubricating oil viscosity value and the first lubricating oil density value according to the first oil film thickness;

[0226] Step S430, obtain the second oil film pressure value according to the first lubricating oil viscosity value and the first lubricating oil density value;

[0227] Step S440, if the difference between the second oil film pressure value and the first oil film pressure value is greater than or equal to the second preset threshold, then obtain the second lubricating oil viscosity value and the second lubricating oil density value according to the second oil film pressure value, and use the second lubricating oil viscosity value as the first lubricating oil viscosity value, and use the second lubricating oil density value as the first lubricating oil density value, and transfer to the step of obtaining the second oil film pressure value according to the first lubricating oil viscosity value and the first lubricating oil density value for execution, and use the obtained second oil film pressure value as the target oil film pressure value until the difference between the second oil film pressure value and the first oil film pressure value is less than the second preset threshold;

[0228] Step S450, if the difference between the second oil film pressure value and the first oil film pressure value is less than the second preset threshold, then judge whether the target oil film pressure value meets the load balance condition according to the load balance equation;

[0229] Step S460, if the target oil film pressure value does not meet the load balance condition, then calculate the third oil film pressure value and the second minimum film thickness by adjusting the rigid normal displacement of the face gear pair, and use the third oil film pressure value as the first oil film pressure value, and use the second minimum film thickness as the first minimum film thickness, and transfer to the step of calculating the first oil film thickness according to the first oil film pressure value and the first minimum film thickness for execution until the load balance condition is met, and use the third oil film pressure value obtained in the last iteration as the target oil film pressure value;

[0230] Step S470, if the target oil film pressure value meets the load balance condition, then use the latest first oil film thickness as the target oil film thickness;

[0231] Step S480, calculate the oil film stiffness of the meshing interface according to the target oil film thickness and the target oil film pressure value.

[0232] In this embodiment, the first oil film pressure value is a preset initial oil film pressure distribution, and the Hertz contact pressure can be used as the initial value of the first oil film pressure; the first minimum film thickness H 00 can be calculated using the film thickness equation to obtain the initial film thickness distribution. The first lubricating oil viscosity value and the first lubricating oil density value can be calculated according to the first oil film pressure value using the viscosity-pressure equation and the density-pressure equation. Substitute the first lubricating oil viscosity value and the first lubricating oil density value into the Reynolds equation to solve for the new oil film pressure distribution and obtain the second oil film pressure value. Calculate the difference between the second oil film pressure value and the first oil film pressure value, and compare it with the second preset threshold. If the difference is less than the second preset threshold, the iteration terminates, and the oil film pressure distribution at this time is the target oil film pressure value, where the target oil film pressure value refers to the oil film pressure distribution at the end of the iteration. If the difference is greater than or equal to the second preset threshold, the second oil film pressure value is used as the new first oil film pressure value and the iterative calculation continues. After the iteration ends, determine whether the current situation satisfies the load balance condition according to the target oil film pressure value and the external load. The load balance condition requires that the integral value of the calculated oil film pressure distribution over the entire contact area reaches equilibrium with the external load. If the load balance condition is not satisfied, the rigid normal displacement of the face gear pair needs to be adjusted, and the oil film pressure and film thickness are recalculated. The adjusted rigid normal displacement is used as the new first minimum film thickness, and the process returns to the first step for iterative calculation until the load balance condition is satisfied. The oil film thickness obtained from the last iteration under the load balance condition is used as the target oil film thickness, and the oil film stiffness of the meshing interface is calculated using the oil film stiffness formula based on the target oil film thickness and the target pressure value.

[0233] As shown in FIGS. 8(a) and 8(b), by comparing the meshing stiffness of the face gear pair considering the influence of the oil film at the meshing interface and the meshing stiffness of the face gear pair without considering the influence of the oil film at the meshing interface, it can be seen that the oil film has a greater influence on the meshing stiffness of the face gear pair. Therefore, when calculating the gear meshing stiffness, the influence of the oil film thickness on gear meshing needs to be considered.

[0234] Specifically, in the installation coordinate system, the kinematic and dynamic relationships during the meshing of the face gear and the spur gear can be used to obtain the motion parameters required for elastohydrodynamic lubrication calculation.

[0235] In the installation coordinate system, the face gear and the spur gear rotate around the offset axis with angular velocities and respectively. The surface velocity vectors and can be expressed as Equation (46):

[0236] (46);

[0237] Among them, r represents the vector of the tooth surface meshing point. The slide-roll ratio is given by Equation (47):

[0238] (47);

[0239] Decompose the surface velocity vector into two parts, the component along the normal direction of the meshing point and the component in the tangential plane ( i = p , g ), and decompose it along the major and minor axes directions of the contact ellipse. Then there is Equation (48):

[0240] (48);

[0241] Use the velocity component in the tangential plane to calculate the relative velocity along the major and minor axes directions of the contact ellipse. It is expressed by the equation as Equation (49):

[0242] (49);

[0243] Similarly, the entrainment velocity is expressed as Equation (50):

[0244] (50);

[0245] Then, the angles between the relative sliding velocity and the entrainment velocity and the minor axis of the contact ellipse are given by Equation (51):

[0246] (51).

[0247] So far, the kinematic parameters of the face gear pair required for elastohydrodynamic lubrication calculation have been obtained, including: the relative velocity components of the face gear and the spur gear in the major and minor axis directions of the contact ellipse, the entrainment velocity components of the face gear and the spur gear in the major and minor axis directions of the contact ellipse, and the angles between the relative sliding velocity and the entrainment velocity and the minor axis of the contact ellipse.

[0248] In addition, for elastohydrodynamic lubrication calculation, it is also necessary to obtain the working conditions parameters and lubrication parameters of the face gear pair. The working conditions parameters include but are not limited to rotational speed and load, and the lubrication parameters include but are not limited to viscosity-pressure coefficient, lubricating oil ambient viscosity, and lubricating oil ambient density. Exemplarily, Table 2 and Table 3 list the working conditions parameters, lubrication parameters of the face gear pair, and a corresponding value-taking method.

[0249] Table 2

[0250]

[0251] Table 3

[0252]

[0253] Then, by constructing the elastohydrodynamic lubrication control equations, the oil film stiffness of the face gear and the spur gear is calculated. The elastohydrodynamic lubrication control equations include the Reynolds equation, the film thickness equation, the deformation equation, the viscosity-pressure equation, the density-pressure equation, and the load balance equation.

[0254] The Reynolds equation is the core equation in the theory of hydrodynamic lubrication and is used to describe the pressure distribution and film thickness variation in the lubricating film. Since the pressure and film thickness in the contact zone are controlled by the Reynolds equation, the key to the elastohydrodynamic lubrication problem is to solve the Reynolds equation. The Reynolds equations for elliptical contact elastohydrodynamic lubrication are all two-dimensional problems. The isothermal elliptical contact quasi-steady Reynolds equation considering the effect of the entrainment angle is Equation (52):

[0255] (52);

[0256] where, is the lubricating film thickness, is the pressure, is the lubricant density, is the lubricant viscosity.

[0257] The film thickness equation is used to describe the variation law of the lubricating film thickness. For the elliptical contact elastohydrodynamic lubrication model, the curvature radii of the equivalent contact principal surfaces of the two contacting bodies are selected as the z axis and y axis. At this time, the film thickness equation can be written as Equation (53):

[0258] (53);

[0259] where, represents the rigid normal displacement of the two tooth surfaces, and represent the geometry of the local elliptical contact zone, and represent the curvature radii in the directions of the major and minor axes of the contact ellipse. As shown in Equation (54):

[0260] (54);

[0261] where, the " " sign is because along the minor axis direction of the contact ellipse, the bending directions of the tooth surfaces of the face gear and the spur gear are opposite. The " " sign is because along the major axis direction of the contact ellipse, the bending directions of the tooth surfaces of the face gear and the spur gear are the same.

[0262] The deformation equation is used to describe the elastic deformation of the lubricating film caused by pressure and temperature changes. In elastohydrodynamic lubrication calculations, the elastic deformation of the surface needs to be considered. Therefore, an elastic deformation term must be included in the film thickness formula, as shown in Equation (55):

[0263] (55);

[0264] where, and represents the surface elastic deformation. For elliptical contacts, the Boussinesq integral can be used, as shown in Equation (56):

[0265] (56);

[0266] The viscosity-pressure equation is used to describe the relationship between the lubricant viscosity and pressure. For heavy-load hydrodynamic lubrication, especially in the elastohydrodynamic lubrication state, the viscosity-pressure characteristic is one of the crucial factors. Under isothermal conditions, the viscosity of the lubricating oil is only controlled by pressure. In this embodiment, the Roeland equation is used to represent the influence relationship of pressure on viscosity, as shown in Equation (57):

[0267] (57);

[0268] where, represents the ambient viscosity, is the dimensionless viscosity-pressure index, is the viscosity-pressure coefficient, generally taken as 2.2×10 -8 m 2 N -1 .

[0269] The density-pressure equation is used to describe the relationship between the lubricant density and pressure. The density of the lubricant increases with the increase of pressure. The relationship between density and pressure is described by Dowson-Higginson, as shown in Equation (58):

[0270] (58);

[0271] where the pressure shall be in Pa, is the density of the lubricant under standard atmospheric pressure.

[0272] The load balance equation is used to ensure the formation and stability of the oil film. The numerical solution problem of elastohydrodynamic lubrication needs to be carried out under the given load condition, and the calculated pressure must satisfy the load balance condition equation. During the iterative solution process of the Reynolds equation, multiple sets of oil film pressure distributions will be generated, and all of them should satisfy the load balance equation, so as to confirm that the correct elastohydrodynamic pressure solution has been obtained. The load balance equation requires that the integral value of the oil film pressure within the calculation domain reaches equilibrium with the external load. Therefore, only when the load balance condition is satisfied can the pressure distribution be correctly calculated. The load balance condition is shown in Equation (59):

[0273] (59);

[0274] where, is the external load, and the external load can be set as required according to the actual situation.

[0275] The control equation of elastohydrodynamic lubrication is highly nonlinear. After dimensionlessization, the finite difference method is used to discretize the equation for iterative solution. Through the analysis of the time-varying meshing characteristics of the face gear and the solution of the motion parameters, the relevant parameters (motion parameters, working conditions parameters, and lubrication parameters) of the transmission meshing period of the face gear pair are obtained. Then, by giving an initial pressure distribution (the first oil film pressure value) and an initial minimum film thickness H 00 (the first minimum film thickness), the Hertz contact pressure can be calculated based on the Hertz contact theory as the initial pressure distribution. Using the film thickness equation, combined with the initial pressure distribution and the initial minimum film thickness, the lubricating film thickness distribution within the contact area can be calculated; the viscosity and density of the lubricating oil will change with the change of pressure, so it is necessary to calculate the viscosity and density of the lubricating oil at different pressures according to the viscosity-pressure equation. Substitute the calculated film thickness, lubricating oil viscosity value (the first lubricating oil viscosity value), and lubricating oil density value (the first lubricating oil density value) into the Reynolds equation to solve the new pressure distribution. This process usually uses the finite difference method for discretization and iterative solution. Subsequently, continuously iterate and correct the previous H 0 (dimensionless rigid normal displacement h 0), recalculate the elastic deformation and film thickness in the contact area until the pressure difference obtained from two consecutive iterations is less than the preset threshold (the second preset threshold), and at this time the iteration ends. Finally, the final pressure distribution (the target oil film pressure value) including elastic deformation and the final film thickness (the target oil film thickness) are calculated.

[0276] According to the calculated final pressure distribution and final film thickness including elastic deformation, calculate the oil film stiffness at the meshing interface. The oil film stiffness refers to the ability of the oil film to resist deformation. Since the oil film can bear a large pressure, it has a large support stiffness. The calculation formula of the oil film stiffness is shown in Equation (60):

[0277] (60).

[0278] In this embodiment, by constructing an elastohydrodynamic lubrication model, the lubrication state during gear meshing can be more accurately described, including oil film thickness, pressure distribution, viscosity change, etc. After considering the lubrication factors, the calculated meshing stiffness is more in line with the actual situation and can more accurately reflect the dynamic characteristics of the gear pair. The elastohydrodynamic lubrication model has a high calculation efficiency and can quickly obtain the calculation results of the oil film stiffness, saving calculation time. Compared with the traditional finite element method, it can reduce the calculation cost.

[0279] In some embodiments, step S500 may include but is not limited to steps S510 to S520:

[0280] Step S510, calculate based on the oil film stiffness, the total bending stiffness, the total shear stiffness, and the total axial compression stiffness to obtain the second meshing stiffness of the single-tooth meshing;

[0281] Step S520, calculate the second meshing stiffness of the single-tooth meshing according to the deformation coordination relationship to obtain the target meshing stiffness of the face gear pair.

[0282] In this embodiment, using the oil film stiffness to correct the meshing stiffness can more accurately reflect the dynamic characteristics of the gear pair. Replace the Hertz contact stiffness with the oil film stiffness and calculate in combination with the total bending stiffness, the total shear stiffness, and the total axial compression stiffness to obtain the second meshing stiffness of the single-tooth meshing. The second meshing stiffness of the single-tooth meshing is the single-tooth meshing stiffness considering the lubrication factors at the meshing interface. Substitute the second meshing stiffness of the single-tooth meshing of the face gear pair into the deformation coordination relationship for calculation to obtain the target meshing stiffness of the face gear pair. The target meshing stiffness is the meshing stiffness of the face gear pair considering the lubrication factors at the meshing interface. It considers the influence of the oil film stiffness and is more in line with the actual working conditions.

[0283] Specifically, as Figure 9 shown, the oil film can be regarded as parallel "springs" at each node in the pressure distribution area, solve the stiffness of these "springs" for calculation, and then achieve parallel superposition to obtain the oil film stiffness in the entire area. The calculation formula of the oil film thickness includes the part of the contact deformation of the gear solid surface. The oil film stiffness includes the stiffness of the oil film itself and the stiffness of the solid contact. Therefore, when calculating the face gear meshing stiffness considering the oil film stiffness, it is necessary to replace the Hertz contact stiffness with the oil film stiffness. The single-tooth meshing stiffness considering the oil film stiffness can be expressed as Equation (61):

[0284] (61);

[0285] Substitute Equation (61) into Equations (39) to (45) for calculation to obtain the time-varying mesh stiffness of the face gear pair considering the lubrication factor at the meshing interface.

[0286] In this embodiment, by using the oil film stiffness to correct the mesh stiffness of the face gear pair, the calculation result can be made more in line with the actual situation and more accurately reflect the dynamic characteristics of the gear pair. The oil film stiffness takes into account the influence of factors such as the viscosity, pressure, and density of the lubricant on the mesh stiffness, and can calculate the mesh stiffness of the gear pair more precisely.

[0287] In the embodiment of the present application, by obtaining the geometric parameters of multiple face gear pairs, determining the target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the multiple face gear pairs; calculating the mesh stiffness of the face gear and the spur gear in the target contact area; calculating the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation; using the oil film stiffness to correct the mesh stiffness of the face gear pair to obtain the target mesh stiffness of the face gear pair; by simulating the lubrication state of the gear meshing process and combining the meshing effect of the gears to solve the mesh stiffness, considering the influence of the lubrication factor at the meshing interface on the mesh stiffness of the face gear, thereby improving the calculation accuracy of the mesh stiffness of the face gear. The embodiment of the present application can simulate the lubrication state of the gear meshing process and combine the meshing effect of the gears to solve the mesh stiffness, which is more in line with the actual situation and the calculation result is more accurate; compared with the traditional finite element method, the calculation efficiency is higher, saving time cost; and there is no need for complex finite element software, the operation is simple and convenient, and the use threshold is reduced.

[0288] Please refer to Figure 10 , the embodiment of the present application also provides a face gear mesh stiffness calculation device 600 considering the lubrication factor at the meshing interface, which can implement the above-mentioned face gear mesh stiffness calculation method considering the lubrication factor at the meshing interface. The device includes:

[0289] An acquisition module 10, configured to acquire geometric parameters of multiple face gear pairs, where the face gear pair includes a face gear and a spur gear;

[0290] A contact area calculation module 20, configured to determine a target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the multiple face gear pairs;

[0291] A mesh stiffness calculation module 30, configured to calculate the mesh stiffness of the face gear and the spur gear in the target contact area;

[0292] An oil film stiffness calculation module 40, configured to calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation;

[0293] A correction module 50, configured to correct the mesh stiffness of the face gear pair by using the oil film stiffness to obtain the target mesh stiffness of the face gear pair.

[0294] The specific implementation manner of the face gear meshing stiffness calculation device considering the lubrication factor of the meshing interface is basically the same as the specific embodiment of the face gear meshing stiffness calculation method considering the lubrication factor of the meshing interface, and will not be elaborated here.

[0295] The embodiment of the present application also provides an electronic device, which includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements the above-mentioned face gear meshing stiffness calculation method considering the lubrication factor of the meshing interface. The electronic device can be any intelligent terminal including a tablet computer, a vehicle-mounted computer, etc.

[0296] Please refer to Figure 11 , Figure 11 , which schematically shows the hardware structure of the electronic device in another embodiment. The electronic device includes:

[0297] A processor 701, which can be implemented in a general-purpose CPU (Central Processing Unit), a microprocessor, an application-specific integrated circuit (ASIC), or one or more integrated circuits, etc., and is used to execute relevant programs to implement the technical solutions provided by the embodiments of the present application;

[0298] A memory 702, which can be implemented in the form of a read-only memory (ROM), a static storage device, a dynamic storage device, or a random access memory (RAM), etc. The memory 702 can store an operating system and other application programs. When implementing the technical solutions provided by the embodiments of this specification through software or firmware, the relevant program codes are stored in the memory 702, and the processor 701 is used to call and execute the face gear meshing stiffness calculation method considering the lubrication factor of the meshing interface in the embodiments of the present application;

[0299] An input / output interface 703, which is used to implement information input and output;

[0300] A communication interface 704, which is used to implement communication interaction between this device and other devices, and can implement communication through a wired method (such as USB, network cable, etc.) or through a wireless method (such as mobile network, WIFI, Bluetooth, etc.);

[0301] A bus 705, which transmits information between various components of the device (such as the processor 701, the memory 702, the input / output interface 703, and the communication interface 704);

[0302] Among them, the processor 701, the memory 702, the input / output interface 703, and the communication interface 704 are communicatively connected to each other inside the device through the bus 705.

[0303] An embodiment of the present application also provides a computer-readable storage medium. The computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, it implements the above-mentioned face gear meshing stiffness calculation method considering the lubrication factor of the meshing interface.

[0304] As a non-transitory computer-readable storage medium, the memory can be used to store non-transitory software programs and non-transitory computer-executable programs. In addition, the memory may include high-speed random access memory, and may also include non-transitory memory, such as at least one magnetic disk storage device, a flash memory device, or other non-transitory solid-state storage devices. In some embodiments, the memory may optionally include a memory remotely disposed relative to the processor, and these remote memories may be connected to the processor through a network. Examples of the above-mentioned network include but are not limited to the Internet, an enterprise intranet, a local area network, a mobile communication network, and combinations thereof.

[0305] The face gear meshing stiffness calculation method, the face gear meshing stiffness calculation device, the electronic device, and the storage medium considering the lubrication factor of the meshing interface provided by the embodiments of the present application obtain geometric parameters of a plurality of face gear pairs, where the face gear pairs include face gears and spur gears; determine a target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the plurality of face gear pairs; calculate the meshing stiffness of the face gear and the spur gear in the target contact area; calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation; and correct the meshing stiffness of the face gear pair by using the oil film stiffness to obtain the target meshing stiffness of the face gear pair. The embodiments of the present application can improve the calculation accuracy of the face gear meshing stiffness in the time-varying meshing stiffness.

[0306] The embodiments described in the embodiments of the present application are for more clearly illustrating the technical solutions of the embodiments of the present application, and do not constitute a limitation to the technical solutions provided by the embodiments of the present application. Those skilled in the art will know that with the evolution of technology and the emergence of new application scenarios, the technical solutions provided by the embodiments of the present application are equally applicable to similar technical problems.

[0307] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separated, that is, they may be located in one place, or may be distributed to multiple network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0308] Those of ordinary skill in the art will understand that all or some of the steps in the methods disclosed above, and the functional modules / units in systems and devices, can be implemented as software, firmware, hardware, or a suitable combination thereof.

[0309] As used in the description of the present application and the above-mentioned drawings, the terms "first", "second", "third", "fourth", etc. (if any) are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that the data used in this way can be interchanged under appropriate circumstances so that the embodiments of the present application described herein can be implemented in an order different from those illustrated or described herein. In addition, the terms "comprising" and "having" and any variations thereof are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or device that comprises a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units not clearly listed or inherent to these processes, methods, products, or devices.

[0310] It should be understood that in the present application, "at least one (item)" means one or more, and "a plurality" means two or more. "And / or" is used to describe the association relationship of associated objects and indicates that three relationships may exist. For example, "A and / or B" may mean: only A exists, only B exists, and both A and B exist simultaneously. Here, A and B can be singular or plural. The character " / " generally indicates that the associated objects before and after are in an "or" relationship. "At least one (one) of the following" or a similar expression means any combination of these items, including any combination of single items (ones) or plural items (ones). For example, at least one (one) of a, b, or c can mean: a, b, c, "a and b", "a and c", "b and c", or "a and b and c", where a, b, and c can be single or multiple.

[0311] In the several embodiments provided in the present application, it should be understood that the disclosed devices and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the above-mentioned unit division is only a logical function division, and there may be other division methods in actual implementation. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed coupling, direct coupling, or communication connection to each other can be through some interfaces, and the indirect coupling or communication connection of devices or units can be in an electrical, mechanical, or other form.

[0312] The units described above as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed over multiple network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0313] In addition, each functional unit in various embodiments of the present application may be integrated into a processing unit, may exist separately as individual physical units, or two or more units may be integrated into one unit. The above-mentioned integrated units can be implemented in the form of hardware or in the form of software functional units.

[0314] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present application, in essence, or the part that contributes to the prior art, or all or part of this technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes multiple instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods in various embodiments of the present application. The foregoing storage medium includes: various media such as USB flash drives, mobile hard disks, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical discs that can store programs.

[0315] The preferred embodiments of the embodiments of the present application have been described above with reference to the drawings, but this does not limit the scope of the rights of the embodiments of the present application. Any modification, equivalent replacement, and improvement made by those skilled in the art without departing from the scope and essence of the embodiments of the present application shall be within the scope of the rights of the embodiments of the present application.

Claims

1. A calculation method for the meshing stiffness of face gears considering the lubrication factors of the meshing interface, characterized in that The method includes: Obtaining geometric parameters of a plurality of face gear pairs, where the face gear pairs include face gears and spur gears; Determining a target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the plurality of face gear pairs; Calculating the meshing stiffness of the face gear and the spur gear within the target contact area; Calculating the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation; Using the oil film stiffness to correct the meshing stiffness of the face gear pair to obtain the target meshing stiffness of the face gear pair; The determining the target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the plurality of face gear pairs includes: Performing contact analysis on the geometric parameters of the plurality of face gear pairs to obtain target parameters, where the target parameters include a first contact area; Adjusting the first contact area according to the gear boundary conditions to obtain the target contact area of the meshing interface, and the target contact area is an elliptical area.

2. The method according to claim 1, wherein The performing contact analysis on the geometric parameters of the plurality of face gear pairs to obtain target parameters includes: Establishing a face gear tooth surface equation and a spur gear tooth surface equation based on the geometric parameters of the plurality of face gear pairs; According to the face gear tooth surface equation and the spur gear tooth surface equation, and based on the meshing principle, obtaining tooth surface contact points and a meshing locus; Calculating the curvature radii of the face gear and the spur gear at the tooth surface contact points, and calculating the major and minor semi-axes of the meshing locus based on a preset first load distribution ratio and Hertz contact theory to obtain the first contact area.

3. The method according to claim 2, characterized in that, The calculating the meshing stiffness of the face gear and the spur gear within the target contact area includes: Dividing the tooth surface of the face gear into a plurality of sliced tooth surfaces along the tooth width direction within the target contact area; Calculating the bending stiffness of each sliced tooth surface, the shear stiffness of each sliced tooth surface, and the axial compression stiffness of each sliced tooth surface respectively; Adjusting the boundary of the target contact area through the deformation coordination condition to obtain a second contact area, and calculating a second load distribution ratio according to the second contact area, where the deformation coordination condition is a geometric condition satisfied by the displacements or strains of each part during the deformation process of the face gear pair; Judging whether the difference between the second load distribution ratio and the first load distribution ratio is greater than or equal to a first preset threshold. If the difference is greater than or equal to the first preset threshold, taking the second load distribution ratio as the first load distribution ratio, and turning to execute the step of calculating the major and minor semi-axes of the meshing locus based on the preset first load distribution ratio and Hertz contact theory to obtain the first contact area until the difference between the second load distribution ratio and the first load distribution ratio is less than the first preset threshold, and taking the second load distribution ratio obtained in the last iteration as the target load distribution ratio; Integrating the bending stiffness of each sliced tooth surface to obtain the total bending stiffness of all sliced tooth surfaces; Integrating the shear stiffness of each sliced tooth surface to obtain the total shear stiffness of all sliced tooth surfaces; Integrate the axial compression stiffness of each of the sliced tooth surfaces to obtain the total axial compression stiffness of all the sliced tooth surfaces; Calculate based on the target load distribution ratio to obtain the Hertz contact stiffness of the face gear pair; Calculate according to the Hertz contact stiffness, the total bending stiffness, the total shear stiffness and the total axial compression stiffness of the face gear pair to obtain the meshing stiffness of the face gear pair.

4. The method according to claim 3, characterized in that The calculating according to the Hertz contact stiffness, the total bending stiffness, the total shear stiffness and the total axial compression stiffness of the face gear pair to obtain the meshing stiffness of the face gear pair includes: Calculate the load of the target contact area based on the target load distribution ratio, and obtain the Hertz contact stiffness of the face gear pair according to the load of the target contact area; Calculate based on the Hertz contact stiffness, the total bending stiffness, the total shear stiffness and the total axial compression stiffness to obtain the first meshing stiffness of single-tooth meshing, and the face gear pair includes a plurality of the single-tooth meshing; Calculate the meshing stiffness of the face gear pair according to the deformation coordination relationship for the first meshing stiffness of the single-tooth meshing.

5. The method according to claim 4, characterized in that, The correcting the meshing stiffness of the face gear pair by using the oil film stiffness to obtain the target meshing stiffness of the face gear pair includes: Calculate based on the oil film stiffness, the total bending stiffness, the total shear stiffness and the total axial compression stiffness to obtain the second meshing stiffness of the single-tooth meshing; Calculate the target meshing stiffness of the face gear pair according to the deformation coordination relationship for the second meshing stiffness of the single-tooth meshing.

6. The method according to claim 1, characterized in that, The calculating the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation includes: Calculate according to the first oil film pressure value and the first minimum film thickness to obtain the first oil film thickness; Obtain the first lubricating oil viscosity value and the first lubricating oil density value according to the first oil film thickness; Obtain the second oil film pressure value according to the first lubricating oil viscosity value and the first lubricating oil density value; If the difference between the second oil film pressure value and the first oil film pressure value is greater than or equal to the second preset threshold, then obtain the second lubricating oil viscosity value and the second lubricating oil density value according to the second oil film pressure value, take the second lubricating oil viscosity value as the first lubricating oil viscosity value, take the second lubricating oil density value as the first lubricating oil density value, go to the step of obtaining the second oil film pressure value according to the first lubricating oil viscosity value and the first lubricating oil density value, and take the obtained second oil film pressure value as the target oil film pressure value until the difference between the second oil film pressure value and the first oil film pressure value is less than the second preset threshold; If the difference between the second oil film pressure value and the first oil film pressure value is less than the second preset threshold, then judge whether the target oil film pressure value meets the load balance condition according to the load balance equation; If the target oil film pressure value does not satisfy the load balance condition, calculate the third oil film pressure value and the second minimum film thickness by adjusting the rigid normal displacement of the face gear pair, take the third oil film pressure value as the first oil film pressure value, take the second minimum film thickness as the first minimum film thickness, and execute the step of calculating the first oil film thickness according to the first oil film pressure value and the first minimum film thickness until the load balance condition is satisfied, and take the third oil film pressure value obtained in the last iteration as the target oil film pressure value; If the target oil film pressure value satisfies the load balance condition, take the latest first oil film thickness as the target oil film thickness; Calculate the oil film stiffness of the meshing interface according to the target oil film thickness and the target oil film pressure value.

7. A face gear meshing stiffness calculation device considering lubrication factors at the meshing interface, characterized in that The device includes: An acquisition module, configured to acquire geometric parameters of a plurality of face gear pairs, where the face gear pair includes a face gear and a spur gear; A contact area calculation module, configured to determine a target contact area of the meshing interface between the face gear and the spur gear according to the geometric parameters of the plurality of face gear pairs; perform contact analysis on the geometric parameters of the plurality of face gear pairs to obtain target parameters, where the target parameters include a first contact area; adjust the first contact area according to the gear boundary conditions to obtain the target contact area of the meshing interface, and the target contact area is an elliptical area; A meshing stiffness calculation module, configured to calculate the meshing stiffness of the face gear and the spur gear within the target contact area; An oil film stiffness calculation module, configured to calculate the oil film stiffness of the face gear and the spur gear based on the elastohydrodynamic lubrication control equation; A correction module, configured to correct the meshing stiffness of the face gear pair by using the oil film stiffness to obtain the target meshing stiffness of the face gear pair.

8. An electronic device, characterized in that, The electronic device includes a memory and a processor, the memory stores a computer program, and when the processor executes the computer program, it implements the face gear meshing stiffness calculation method considering lubrication factors of the meshing interface according to any one of claims 1 to 6.

9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the face gear meshing stiffness calculation method considering lubrication factors of the meshing interface according to any one of claims 1 to 6.

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