A numerical calculation method for stress on spline teeth considering parallel misalignment

Abstract

A method for numerically calculating stress on spline teeth, taking parallel misalignment into account, is disclosed. The method includes: determining the spline tooth side clearance based on the parallel misalignment of the spline, wherein the spline includes z pairs of teeth, each pair of teeth including external spline teeth and internal spline tooth spaces; constructing a load distribution model for the spline within the micro-segments by axially dividing the spline into n micro-segments within the contact length, wherein the contact length is the length of the contact portion between the external and internal splines of the spline; determining the number of meshing teeth in the spline based on the tooth side clearance and the load distribution model, wherein the number of meshing teeth is the number of meshing teeth in the z pairs of teeth, wherein the external spline teeth and the internal spline tooth spaces mesh; determining the first load of each pair of meshing teeth in each micro-segment based on the number of meshing teeth and the load distribution model; and determining the contact stress distribution of each pair of meshing teeth in each micro-segment based on the first load of each pair of meshing teeth in each micro-segment. This method can accurately calculate contact stress.

Description

Technical Field

[0001] The present disclosure generally relates to the field of mechanical design technology, and more particularly, to a method for numerically calculating stress on spline teeth considering parallel misalignment. Background Art

[0002] Mechanical systems are developing towards high precision, high reliability, and long life. Involute splines are a crucial component of mechanical transmission systems, playing a key role in torque transmission. They are widely used in transmissions such as aircraft engines, automobiles, and machine tools. Involute splines consist of internal and external splines. The external splines, which contain multiple teeth, mesh with a corresponding number of internal spline teeth to transmit torque at the power output.

[0003] However, in actual use, splines have a lot of wear and failure problems. Studies have shown that parallel misalignment is one of the main factors that lead to spline wear and failure. Spline parallel misalignment is mainly caused by factors such as assembly errors, structural deformation and shaft system working conditions. When parallel misalignment occurs between the inner and outer spline shafts, the rotation centers of the inner and outer splines no longer coincide, resulting in the load transmitted by the spline being borne by only a few teeth, increasing the stress on the tooth surface, and the misalignment causes the tooth surfaces of the mating splines to deflect, further affecting the distribution of contact stress on the teeth, making the wear position and wear amount of the tooth surface complex and variable, greatly accelerating the wear process of the spline. According to the classical Archard wear theory, contact stress is a key variable for accurately calculating wear. Therefore, in order to accurately evaluate the wear life of the spline, it is very important to obtain a method for calculating the contact stress of the spline teeth that takes into account the influence of parallel misalignment. Summary of the Invention

[0004] The present disclosure provides a numerical calculation method for stress on spline teeth considering parallel misalignment, which fully considers the influence of parallel misalignment on the uneven distribution of spline tooth side clearance. According to the deformation characteristics of the spline shaft, a load distribution model of the spline can be constructed to determine the number of meshing teeth and the distribution of load along the axial direction when the spline has a certain amount of parallel misalignment, so that the contact stress on the spline teeth can be obtained efficiently and accurately without the help of finite element simulation.

[0005] In a general aspect, a method for numerically calculating stress on spline teeth taking parallel misalignment into account is provided, comprising: determining the tooth side clearance of the spline based on the parallel misalignment of the spline, wherein the spline includes z pairs of teeth, each pair of teeth including external spline teeth and internal spline tooth grooves; constructing a load distribution model of the spline on the micro-segments by axially dividing the spline into n micro-segments within the contact length, wherein the contact length is the length of the contact portion between the external spline and the internal spline of the spline; determining the number of meshing teeth of the spline based on the tooth side clearance and the load distribution model, wherein the number of meshing teeth is the number of meshing teeth in the z pairs of teeth, and the external spline teeth and the internal spline tooth grooves of the meshing teeth are meshed; determining the first load of each pair of meshing teeth on each micro-segment based on the number of meshing teeth and the load distribution model; determining the contact stress distribution of each pair of meshing teeth on each micro-segment based on the first load of each pair of meshing teeth on each micro-segment.

[0006] Optionally, constructing the load distribution model of the spline on the micro-segment includes: constructing a first quantitative relationship between the second load of the spline on each micro-segment and the torsion angle of the internal spline and the external spline in each micro-segment; constructing a second quantitative relationship between the deformation generated by the meshing teeth of the spline at each micro-segment and the second load; constructing a third quantitative relationship between the first displacement of the internal spline and the external spline in each micro-segment and the deformation; constructing a fourth quantitative relationship between the first displacement, the deformation and the second displacement at the starting end of the external spline load application; based on the first quantitative relationship, the second quantitative relationship, the third quantitative relationship and the fourth quantitative relationship, the load distribution model is obtained.

[0007] Optionally, the first quantitative relationship is expressed by the following first equation:

[0008]

[0009] Among them, θ ext，j represents the torsion angle of the external spline at the jth differential segment, θ int，j represents the torsion angle of the internal spline at the jth differential segment, represents the second load of the spline on the pth differential segment, Δl represents the length of the differential segment, G ext represents the shear modulus of the external spline material, J ext Represents the polar moment of inertia of the external spline section, G int Indicates the shear modulus of the internal spline material, J int represents the polar moment of inertia of the internal spline section;

[0010] The second quantitative relationship is expressed by the following second equation:

[0011]

[0012] in, represents the second load of the spline on the jth differential segment, δ j Indicates The deformation of the meshing teeth at the jth differential segment under the action of E represents the meshing stiffness, r R Indicates the spline pitch circle radius;

[0013] The third quantitative relationship is expressed by the following third equation:

[0014]

[0015] Where Δx in，j represents the first displacement of the internal spline in the jth differential segment, Δx ext，j represents the first displacement of the external spline in the jth differential segment, r f，int Indicates the root circle radius of the internal spline tooth, r f，ext Indicates the root circle radius of the external spline;

[0016] The fourth quantitative relationship is expressed by the following fourth equation:

[0017]

[0018] Wherein, Δx represents the second displacement of the starting end of the external spline load application.

[0019] Optionally, determining the number of meshing teeth of the spline based on the tooth side clearance and the load distribution model includes: arranging the tooth side clearances in ascending order to obtain sorted tooth side clearances c1, c2, ..., c z , where cx is the tooth side clearance of the first pair of teeth, c2 is the tooth side clearance of the second pair of teeth, and c z is the tooth side clearance of the zth pair of teeth; in the numerical range [2, z], values ​​are taken for k in ascending order, so as to iteratively perform the following steps based on the sorted tooth side clearance until the preset condition is met: determine the second displacement Δx at the starting end of the external spline load application when the meshing deformation of the k-1th pair of teeth reaches the kth pair of teeth just in contact k-1 and meshing stiffness E k-1 , where Δx k-1 =(c k -c k-1 ), E k-1 =(k-1)E0, where E0 represents the single tooth meshing stiffness; Δx k-1 and E k-1 Substituting into the load distribution model, we get the k-1th pair of teeth meshing and the second displacement is Δx k-1 Load distribution on each micro-segment in, Based on load distribution Determine the total load M transmitted by the spline when the k-1th pair of teeth is engaged k-1 ,in, in, It represents the second load on the jth differential segment when the k-1th pair of teeth are engaged.

[0020] Optionally, the preset condition is M1+M2+…+M k-1 >M, wherein M represents an external load, wherein the determining the number of meshing teeth of the spline based on the tooth side clearance and the load distribution model further includes: when the preset condition is met, the value of k is k real Determined as the number of meshing teeth.

[0021] Optionally, the determining of the first load of each pair of meshing teeth on each micro-segment based on the number of meshing teeth and the load distribution model includes: determining the kth real Total load transmitted by the spline when the teeth are meshing and meshing stiffness in, Will and Substitute into the load distribution model and get the kth real Load distribution on each micro-segment when teeth are meshing in, According to the load distribution of each pair of teeth when meshing, determine the first load of the i-th pair of meshing teeth on the j-th micro-segment in, Where i≤k real , j≤n.

[0022] Optionally, the contact stress distribution of each pair of meshing teeth on each micro-segment is determined based on the first load of each pair of meshing teeth on each micro-segment, including: determining the nominal tangential force acting on each micro-segment of each pair of meshing teeth based on the first load; determining the relative offset angle of each pair of meshing teeth on each micro-segment, and equating the contact model of the spline to an indenter model; based on the nominal tangential force and the relative offset angle, combined with the indenter model, determining the contact stress distribution of each pair of meshing teeth on each micro-segment.

[0023] Optionally, determining the nominal tangential force acting on each micro-segment of each pair of meshing teeth based on the first load includes: determining the nominal tangential force by the following fifth equation:

[0024]

[0025] in, represents the nominal tangential force acting on the jth differential segment of the i-th pair of meshing teeth.

[0026] Optionally, the relative offset angle includes a first offset angle, a second offset angle and a third offset angle, wherein the first offset angle is caused by bending deformation, shear deformation and foundation tilt, the second offset angle is caused by contact deformation, and the third offset angle is caused by radial expansion and contraction deformation.

[0027] Optionally, determining the contact stress distribution of each pair of meshing teeth in each micro-segment based on the nominal tangential force and the relative offset angle in combination with the indenter model includes: determining the contact stress distribution by the following sixth equation:

[0028]

[0029] in, represents the contact stress distribution of the i-th pair of meshing teeth on the j-th differential segment, x represents the tooth height, T represents the elastic modulus, represents the relative offset angle of the i-th pair of meshing teeth on the j-th differential segment, a represents the radial coordinate value of the contact point between the internal spline tooth top and the external spline in the indenter model, b represents the radial coordinate value of the contact point between the external spline tooth top and the internal spline in the indenter model, w x represents the load action angle of the spline tooth surface, and v represents the Poisson's ratio.

[0030] According to the numerical calculation method of stress on spline teeth considering parallel misalignment in the embodiment of the present disclosure, it is possible to fully consider the influence of parallel misalignment on the uneven distribution of spline tooth side clearance. Based on the obtained uneven tooth side clearance at a certain parallel misalignment, a load distribution model of the spline teeth along the axial direction is established to determine the actual number of meshing teeth of a spline with a certain contact length and the distribution of load along the axial direction. Without the aid of finite element simulation, the contact stress of multiple teeth of the spline along the axial direction can be obtained by a numerical calculation method. On the one hand, it fully considers the difference in load on different teeth, and takes into account the uneven distribution of load along the axial direction caused by the torsional deformation of the spline shaft, so that the obtained results are closer to the actual load conditions. On the other hand, the load on the spline teeth is determined by the constructed load distribution model. On the basis of meeting the reasonable and feasible results, it is more efficient than the finite element and experimental methods, and can quickly determine the contact stress of the spline tooth surface with different parallel misalignment amounts, thereby effectively guiding the wear reliability analysis of the spline.

[0031] Additional aspects and / or advantages of the present general inventive concept will be set forth in part in the following description and in part will be apparent from the description, or may be learned through practice of the present general inventive concept. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] The above and other objects and features of the embodiments of the present disclosure will become more apparent through the following description in conjunction with the accompanying drawings showing the embodiments, in which:

[0033] Figure 1 is a flow chart illustrating a method for numerically calculating stress on spline teeth taking parallel misalignment into consideration according to an embodiment of the present disclosure;

[0034] Figure 2 is a schematic diagram illustrating a spline load distribution model according to an embodiment of the present disclosure;

[0035] Figure 3 is a schematic diagram illustrating a spline contact model according to an embodiment of the present disclosure;

[0036] Figure 4 is a schematic diagram illustrating an indenter model according to an embodiment of the present disclosure;

[0037] Figure 5 is a diagram illustrating load distribution on the first four pairs of meshing teeth of a spline according to an embodiment of the present disclosure;

[0038] Figure 6 is a diagram illustrating load distribution on the teeth when a spline has five pairs of teeth meshing according to an embodiment of the present disclosure;

[0039] Figure 7 is a diagram illustrating load distribution on each tooth of a spline according to an embodiment of the present disclosure;

[0040] Figure 8 is a diagram illustrating nominal tangential force distribution on each tooth of a spline according to an embodiment of the present disclosure;

[0041] Figure 9 is a diagram illustrating contact stress on a spline tooth surface obtained by finite element simulation according to an embodiment of the present disclosure;

[0042] Figure 10 FIG. 1 is a diagram showing a spline tooth micro segment S according to an embodiment of the present disclosure. 30 Comparison chart of contact stress results;

[0043] Figure 11 FIG. 1 is a diagram showing a spline tooth micro segment S according to an embodiment of the present disclosure. 29 Comparison chart of contact stress results;

[0044] Figure 12 FIG. 1 is a diagram showing a spline tooth micro segment S according to an embodiment of the present disclosure. 28 Comparison chart of contact stress results. DETAILED DESCRIPTION

[0045] The following detailed description is provided to help the reader gain a comprehensive understanding of the methods, devices and / or systems described herein. However, various changes, modifications and equivalents of the methods, devices and / or systems described herein will be clear after understanding the disclosure of the present application. For example, the order of operations described herein is merely an example and is not limited to those orders set forth herein, but can be changed as will be clear after understanding the disclosure of the present application, except for operations that must occur in a specific order. In addition, for greater clarity and conciseness, descriptions of features known in the art may be omitted.

[0046] The features described herein can be implemented in different forms and should not be construed as limited to the examples described herein. Rather, the examples described herein are provided to illustrate only some of the many possible ways to implement the methods, devices, and / or systems described herein, which will become clear after understanding the disclosure of this application.

[0047] Unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which the present disclosure pertains after understanding the present disclosure. Unless expressly defined otherwise herein, terms (such as those defined in general dictionaries) should be interpreted as having a meaning consistent with their meaning in the context of the relevant art and the present disclosure, and should not be interpreted in an idealized or overly formal manner.

[0048] Furthermore, in describing the examples, when it is deemed that a detailed description of well-known related structures or functions would cause ambiguous interpretation of the present disclosure, such detailed description will be omitted.

[0049] The following will refer to Figures 1 to 12 A method for numerically calculating stress on spline teeth considering parallel misalignment according to an embodiment of the present disclosure is described in detail.

[0050] Figure 1 is a flowchart illustrating a method for numerically calculating stress on spline teeth taking parallel misalignment into consideration according to an embodiment of the present disclosure.

[0051] Reference Figure 1 In step S101, the spline tooth side clearance can be determined based on the parallel misalignment of the spline. Here, the spline includes z pairs of teeth, each pair of teeth including external spline teeth and internal spline tooth spaces. Furthermore, those skilled in the art can obtain the spline tooth side clearance based on the journal "Engineering Failure Analysis", Volume 131, 2022, "Spline Wear Life Prediction Considering Multiple Errors", which will not be further described in this disclosure.

[0052] Next, in step S102, the spline is divided into n micro-segments along the axial direction within the contact length, and a load distribution model for the spline is constructed within each micro-segment. Here, the contact length is the length of the contact portion between the outer and inner splines of the spline. Furthermore, when constructing the load distribution model, it is assumed that only elastic deformation occurs during the spline load transfer process. Based on the theory of shaft torsional deformation in material mechanics and the displacement equilibrium condition, a load distribution model for the spline along the axial direction in each micro-segment is constructed.

[0053] Next, in step S103, the number of meshing teeth of the spline can be determined based on the tooth side clearance and the load distribution model. Here, the number of meshing teeth refers to the number of meshing teeth in the z-pair of teeth, where the external spline teeth and internal spline tooth spaces mesh. Furthermore, the number of meshing teeth of the spline can be determined using a numerical iteration method based on the tooth side clearance, combined with the load distribution model and load balance conditions.

[0054] Next, in step S104, the first load of each pair of meshing teeth on each micro-segment may be determined based on the number of meshing teeth and the load distribution model. Here, the loads distributed on different teeth may be determined according to the gap increment when different teeth are meshed.

[0055] Next, in step S105, the contact stress distribution of each pair of meshing teeth in each micro-segment can be determined based on the first load on each micro-segment for each pair of meshing teeth. Here, the relative deflection angle of the spline tooth mating surface under a certain load can be calculated according to the journal "Journal of Mechanical Engineering Science," Vol. 220, No. 12, 2006. The contact stress distribution of each axial segment of the spline teeth can be determined based on the stamping model and the first load on the teeth obtained in step S104.

[0056] According to an embodiment of the present disclosure, in the above-mentioned step S102, when constructing the load distribution model of the spline on the micro-segment, a first quantitative relationship between the second load of the spline on each micro-segment and the torsion angle of the inner spline and the outer spline in each micro-segment can be constructed; a second quantitative relationship between the deformation generated by the meshing teeth of the spline at each micro-segment and the second load can be constructed; a third quantitative relationship between the first displacement of the inner spline and the outer spline in each micro-segment and the deformation can be constructed; a fourth quantitative relationship between the first displacement, the deformation and the second displacement at the starting end of the load application of the outer spline can be constructed; and a load distribution model can be obtained based on the first quantitative relationship, the second quantitative relationship, the third quantitative relationship and the fourth quantitative relationship.

[0057] Figure 2 Schematic diagram showing the load distribution model of the spline according to an embodiment of the present disclosure. Here, in the process of the spline transmitting the load, the supporting structure of the external spline teeth and the internal spline tooth grooves (i.e., the circular shaft and the cylinder) will produce torsional deformation, and the teeth and tooth grooves themselves will also undergo elastic deformation. Figure 2 As shown, the spline with an axial contact length of L can be divided into n micro segments S1, S2, ..., S n , assuming that the torsional deformation of the support structure and the elastic deformation of the tooth / tooth groove in each micro-segment remain unchanged, define θ int，j and θ ext，j (j=1, 2, ..., n) are the torsion angles of the inner spline and outer spline at the jth micro-segment respectively, and the load distributed to the jth micro-segment of the inner and outer splines is According to material mechanics, the first quantitative relationship can be expressed by the following first equation (1):

[0058]

[0059] Here, θ ext，j represents the torsion angle of the external spline at the jth differential segment, θ int，j represents the torsion angle of the internal spline at the jth differential segment, represents the second load of the spline on the pth differential segment, Δl represents the length of the differential segment, G ext represents the shear modulus of the external spline material, J ext Represents the polar moment of inertia of the external spline section, G int Indicates the shear modulus of the internal spline material, J int Represents the polar moment of inertia of the internal spline section.

[0060] According to the definition of spline tooth meshing stiffness, it can be obtained that under load Under the action of j , that is, the second quantitative relationship can be expressed by the following second equation (2):

[0061]

[0062] here, represents the second load of the spline on the jth differential segment, δ j Indicates The deformation of the meshing teeth at the jth differential segment under the action of E represents the meshing stiffness, r R Indicates the spline pitch circle radius.

[0063] like Figure 2 As shown, in order to facilitate the description of the deformation relationship of each spline position, you can refer to Figure 2 In the lower left part of the figure, the spline meshing process is treated equivalently, where the spring unit represents the teeth and tooth grooves, and the structure connected to the spring represents the spline shaft and sleeve. Figure 2 In the lower right part, when the load is transmitted through the external spline, the load is transmitted to the internal spline through the teeth, and the spline teeth and the tooth support structure are deformed. Assume that under the action of external load M, S1, S2, ..., S on the internal and external splines are deformed. n The first displacements generated by the segments are Δx int，1 , Δx int，2 ,...,Δx int，n and Δx ext，1 , Δx ext，2 ,...,Δx ext，n , then the third quantitative relationship can be expressed by the following third equation (3):

[0064]

[0065] Here, Δx int,j represents the first displacement of the internal spline in the jth differential segment, Δx ext，j represents the first displacement of the external spline in the jth differential segment, r f，int Indicates the root circle radius of the internal spline tooth, r f，ext Indicates the root circle radius of the external spline tooth.

[0066] Assume that under the action of external load, the elastic deformation of the meshing teeth on each micro segment is δ1, δ2, ..., δ n , the second displacement of the starting end of the external spline load application is Δx, then according to the deformation coordination condition, the fourth quantitative relationship can be expressed by the following fourth equation (4):

[0067]

[0068] As an example, substituting equations (1) to (3) into equation (4), we can get the equation containing n+1 unknowns (respectively and Δx), further combined with the mechanical equilibrium condition The second load on each micro-segment along the axial direction of the spline can be obtained

[0069] According to an embodiment of the present disclosure, in the above step S103, the tooth side clearances a1, a2, ..., a z Arrange in ascending order to obtain the sorted tooth side clearance c1, c2, ..., c z Here, c1 is the tooth side clearance of the first pair of teeth, c2 is the tooth side clearance of the second pair of teeth, and so on. z is the tooth side clearance of the zth pair of teeth. Next, k can be selected from small to large values ​​in the numerical range [2, z] to iteratively perform the following steps 1) to 3) based on the sorted tooth side clearance until the preset conditions are met:

[0070] 1) Determine the second displacement Δx of the starting end of the external spline load application when the k-1th pair of teeth is deformed to the point where the kth pair of teeth just contacts k-1 and meshing stiffness E k-1 , where Δx k-1 =(c k -c k-1 ), E k-1 =(k-1)E0, further, E0 represents the single tooth meshing stiffness;

[0071] 2) Set Δx k-1 and E k-1 Substituting into the load distribution model, we get the k-1th pair of teeth meshing and the second displacement is Δx k-1 Load distribution on each micro-segment here,

[0072] 3) Based on load distribution Determine the total load M transmitted by the spline when the k-1th pair of teeth is engaged k-1 ,here, Furthermore, It represents the second load on the jth differential segment when the k-1th pair of teeth are engaged.

[0073] Here, when the preset conditions are met (i.e. the above iteration ends), the value of k can be real Determined as the number of meshing teeth. Further, the preset condition is M1+M2+…+M k-1 >M, where M represents external load.

[0074] In one possible implementation, under the action of external load M, assuming that there are k pairs of teeth meshing, in the process of spline load transmission, the first pair of teeth with the smallest gap meshes first, and when the first pair of teeth meshes and deforms to the point where the second pair of teeth just contacts, the displacement of the load input end is Δx1 = (c2-c1), and when the second pair of teeth meshes and deforms to the point where the third pair of teeth just contacts, the displacement of the load input end is Δx2 = (c3-c2), and so on. When the k-1th pair of teeth meshes and deforms to the point where the kth pair of teeth just contacts, the displacement of the load input end is Δx k-1 =(c k -c k-1 ); convert Δx1, Δx2,..., Δx k-1 As a known condition, combined with E k = kE0 meshing stiffness calculation method, substitute into the above equations (1 to (4), solve the n-variable linear equation containing n unknowns, and you can get the displacement x1, Δx2, ..., Δx when there are different pairs of teeth meshing. k-1 The load distribution of the spline along the axial direction is:

[0075]

[0076] Then the total load transmitted by the spline when all the above teeth are engaged at the same time is:

[0077]

[0078] Since the spline has a small elastic deformation, the load applied to the teeth can be linearly superimposed. The external load required when the spline deforms from the meshing of the k-1 pair of teeth to the initial contact of the k pair of teeth is no greater than the external load M, that is, M1+M2+…+M k -1≤M, therefore, k=2~z can be sequentially substituted into the above formulas for iterative calculation, first making M1+M2+…+M k -1≤M does not hold true, k is the number of meshing teeth of the spline real .

[0079] According to an embodiment of the present disclosure, after determining the actual number of meshing teeth k of the spline real After that, assuming the spline contains k real The displacement generated when the teeth mesh simultaneously is Δx k , obviously Δx k is not greater than the gap difference between the k+1th pair of teeth and the kth pair of teeth, then in the above step S104, the kth pair of teeth can be determined according to the load balance condition. real Total load transmitted by the spline when the teeth are meshing and meshing stiffness here, Then, you can and Substitute into the load distribution model and get the kth real Load distribution on each micro-segment when teeth are meshing here, Then, the first load of the i-th pair of meshing teeth on the j-th micro-segment can be determined based on the load distribution of each pair of teeth when meshing. here, Furthermore, i≤k real , j≤n.

[0080] Figure 3 is a schematic diagram illustrating a spline contact model according to an embodiment of the present disclosure; and, Figure 4 is a schematic diagram illustrating an indenter model according to an embodiment of the present disclosure.

[0081] According to the embodiment of the present disclosure, the equivalent assumption for the spline contact model in the journal "Journal of Mechanical Engineering Science", Vol. 220, No. 12, 2006, "Determining both radial pressure distribution and torsional stiffness of involute spline couplings" can be used to convert the spline contact model into a fixed contact model. Figure 3 The spline contact model shown is equivalent to Figure 4 The indenter model shown. Figure 4 , the contact point between the inner spline tooth top and the outer spline is a, the contact point between the outer spline tooth top and the inner spline is b, the nominal tangential force acting on the outer spline tooth is F, and it acts at point c; since the inner spline tooth top and the outer spline tooth top contain geometric mutations, the contact stress at a and b is unbounded, then the relative offset angle γ between the mutually meshing spline contact tooth pairs is γ=γ1+γ2+γ3, that is, the relative offset angle γ may include a first offset angle γ1, a second offset angle γ2 and a third offset angle γ3, the first offset angle γ1 is caused by bending deformation, shear deformation and basic inclination, the second offset angle γ2 is caused by contact deformation, and the third offset angle γ3 is caused by radial expansion and contraction deformation, here, γ1, γ2 and γ3 can be calculated based on the method in the above-mentioned journal, and the present disclosure will not repeat them here.

[0082] Therefore, in the above step S105, the nominal tangential force acting on each micro-segment of each pair of meshing teeth can be determined based on the first load; then, the relative offset angle of each pair of meshing teeth on each micro-segment is determined, and the contact model of the spline is equivalent to an indenter model; then, based on the nominal tangential force and the relative offset angle, combined with the indenter model, the contact stress distribution of each pair of meshing teeth on each micro-segment is determined. Here, the nominal tangential force can be determined by the following fifth equation (7):

[0083]

[0084] here, represents the nominal tangential force acting on the jth differential segment of the i-th pair of meshing teeth.

[0085] On this basis, according to the solution of the Riemann-Hilbert problem for the half-plane case in elasticity theory, the contact stress distribution can be determined by the following sixth equation (8):

[0086]

[0087] in, represents the contact stress distribution of the i-th pair of meshing teeth on the j-th differential segment, x represents the tooth height, T represents the elastic modulus, represents the relative offset angle of the i-th pair of meshing teeth on the j-th differential segment, a represents the radial coordinate value of the contact point between the internal spline tooth top and the external spline in the indenter model, b represents the radial coordinate value of the contact point between the external spline tooth top and the internal spline in the indenter model, w x represents the load action angle of the spline tooth surface, and v represents the Poisson's ratio.

[0088] In order to better understand the above embodiments, Figures 5 to 12 Here, a 24-tooth spline is used as an example to determine the distribution of contact stress on the spline tooth surface. The specific parameters and performance of the selected involute spline are shown in Tables 1 and 2.

[0089] Table 1 Spline model parameters

[0090]

[0091] Table 2 Spline material parameters

[0092]

[0093] When the parallel misalignment e of the spline is 0.04 mm, the tooth side clearance [a1, a2, ..., a 24 ]As shown in Table 3.

[0094] Table 3 Spline tooth side clearance

[0095]

[0096]

[0097] Next, assuming that only elastic deformation occurs during the spline load transfer process, based on the theory of shaft torsional deformation in material mechanics, the spline contact process along the axial direction is divided into 30 micro-segments S1, S2, ..., S 30, and based on the displacement equilibrium condition, a load distribution calculation model on each micro-segment along the spline axis is established.

[0098] Next, the tooth side clearance [a1, a2, ..., a 24 ] are sorted from small to large as [c1, c2, ..., c 24 ], as shown in Table 4.

[0099] Table 4 Spline tooth side clearance sorted from small to large

[0100]

[0101] With the help of the above load distribution model and according to the load balance condition, the actual number of meshing teeth k of the spline is determined through numerical iteration. real is 5.

[0102] Next, according to the number of meshing teeth k of the spline real With the help of the above load distribution model, the gap increments at different tooth meshing times are taken as the displacement of the input end, that is, Δx1 = 0.0016mm, Δx2 = 0mm, Δx3 = 0.0046mm and Δx4 = 0mm are substituted into the load distribution model respectively to obtain the load distribution on the teeth when there are 1 pair, 2 pairs, 3 pairs and 4 pairs of teeth meshing. and like Figure 5 As shown. Figure 5 , since Δx2 and Δx4 are zero, there is no torsional deformation, so the load on the teeth is zero when 2 pairs and 4 pairs of teeth are engaged at the same time.

[0103] The residual load on the fifth pair of teeth can be further obtained as M5 = M-(M1+M2+M3+M4) = 53.8 Nm. Assuming that under the action of load M5, the displacement generated when the spline has 5 pairs of teeth meshing is Δx5, Substituting this into the load distribution model according to the present disclosure, it can be solved that Δx5 = 0.0015 mm.

[0104] When there are 5 pairs of meshing teeth, the load distribution on the teeth is as follows Figure 6 As shown. Further, the loads distributed on different teeth can be obtained, as Figure 7 As shown. Figure 7 , since c2=c3,c4=c5,then the loads on the 2nd and 3rd pairs of teeth and the 4th and 5th pairs of teeth are the same. Furthermore, the nominal tangential force on the spline teeth can be determined as Figure 8 shown.

[0105] Next, the relative deflection angle of the spline tooth mating surface under a certain load can be determined based on the calculation method of the spline mating surface deflection angle in the above journal. Taking the first pair of teeth (i.e., the teeth that bear the largest load) as an example, its deflection angle is shown in Table 5.

[0106] Table 5 Deflection angle of the first pair of teeth

[0107]

[0108] On the one hand, with the help of finite element simulation, the finite element model of the spline shown in Table 1 and Table 2 is established, and the contact stress of the meshing teeth along the axial direction of 10mm from the load input end is intercepted, as shown in Figure 9 On the other hand, according to the stamping model, combined with the determined nominal tangential force and relative offset angle, the contact stress on the spline teeth is calculated based on the solution of the Riemann-Hilbert problem for the half-plane case in elastic theory. Taking the first pair of teeth with a larger load as an example, the S near the load input end is calculated. 30 、S 29 and S 28 The contact stress at , comparing the finite element simulation and the numerical calculation results disclosed in this disclosure, as shown Figures 10 to 12 shown.

[0109] Reference Figures 10 to 12 The predicted value of the contact stress on the first pair of teeth with the largest load is in good agreement with the finite element simulation value, and is relatively consistent at the edge and middle sections of the tooth contact. Therefore, the calculation results according to the embodiments of the present disclosure can provide data support for spline wear prediction.

[0110] According to the numerical calculation method of stress on spline teeth considering parallel misalignment in the embodiment of the present disclosure, it is possible to fully consider the influence of parallel misalignment on the uneven distribution of spline tooth side clearance. Based on the obtained uneven tooth side clearance at a certain parallel misalignment, a load distribution model of the spline teeth along the axial direction is established to determine the actual number of meshing teeth of a spline with a certain contact length and the distribution of load along the axial direction. Without the aid of finite element simulation, the contact stress of multiple teeth of the spline along the axial direction can be obtained by a numerical calculation method. On the one hand, it fully considers the difference in load on different teeth, and takes into account the uneven distribution of load along the axial direction caused by the torsional deformation of the spline shaft, so that the obtained results are closer to the actual load conditions. On the other hand, the load on the spline teeth is determined by the constructed load distribution model. On the basis of meeting the reasonable and feasible results, it is more efficient than the finite element and experimental methods, and can quickly determine the contact stress of the spline tooth surface with different parallel misalignment amounts, thereby effectively guiding the wear reliability analysis of the spline.

[0111] While some embodiments of the present disclosure have been shown and described, it will be appreciated by those skilled in the art that changes may be made to these embodiments without departing from the principles and spirit of the disclosure, the scope of which is defined by the claims and their equivalents.

Claims

1. A numerical calculation method for stress on spline teeth considering parallel misalignment, characterized in that: include: Determining a tooth side clearance of the spline according to a parallel misalignment of the spline, wherein the spline includes z pairs of teeth, each pair of teeth including an external spline tooth and an internal spline tooth groove; A load distribution model of the spline on the micro-segments is constructed by dividing the spline into n micro-segments along the axial direction within the contact length, wherein the contact length is the length of the contact portion between the outer spline and the inner spline of the spline; Determining the number of meshing teeth of the spline based on the tooth side clearance and the load distribution model, wherein the number of meshing teeth is the number of meshing teeth in the z pairs of teeth, wherein the external spline teeth and the internal spline tooth grooves of the meshing teeth mesh; determining a first load of each pair of meshing teeth on each micro-segment based on the number of meshing teeth and the load distribution model; Based on the first load of each pair of meshing teeth on each micro segment, a contact stress distribution of each pair of meshing teeth on each micro segment is determined.

2. The method according to claim 1, wherein The constructing of the load distribution model of the spline on the micro-segment comprises: Establishing a first quantitative relationship between a second load of the spline on each micro segment and a torsion angle of the inner spline and the outer spline in each micro segment; Establishing a second quantitative relationship between the deformation amount generated by the meshing teeth of the spline at each micro-segment and the second load; Constructing a third quantitative relationship between the first displacement amount and the deformation amount of the internal spline and the external spline in each micro-segment; Establishing a fourth quantitative relationship between the first displacement, the deformation, and a second displacement at the starting end of the external spline load application; The load distribution model is obtained based on the first quantitative relationship, the second quantitative relationship, the third quantitative relationship, and the fourth quantitative relationship.

3. The method according to claim 2, wherein The first quantitative relationship is expressed by the following first equation: Among them, θ ext,j represents the torsion angle of the external spline at the jth differential segment, θ int,j represents the torsion angle of the internal spline at the jth differential segment, represents the second load of the spline on the pth differential segment, Δl represents the length of the differential segment, G ext represents the shear modulus of the external spline material, J ext Represents the polar moment of inertia of the external spline section, G int Indicates the shear modulus of the internal spline material, J int represents the polar moment of inertia of the internal spline section; The second quantitative relationship is expressed by the following second equation: in, represents the second load of the spline on the jth differential segment, δ j Indicates The deformation of the meshing teeth at the jth differential segment under the action of E represents the meshing stiffness, r R Indicates the spline pitch circle radius; The third quantitative relationship is expressed by the following third equation: Where Δx int,j represents the first displacement of the internal spline in the jth differential segment, Δx ext,j represents the first displacement of the external spline in the jth differential segment, r f,int Indicates the root circle radius of the internal spline tooth, r f,ext Indicates the root circle radius of the external spline; The fourth quantitative relationship is expressed by the following fourth equation: Wherein, Δx represents the second displacement of the starting end of the external spline load application.

4. The method according to claim 3, wherein The determining the number of meshing teeth of the spline based on the tooth side clearance and the load distribution model includes: Arrange the tooth side clearances in ascending order to obtain sorted tooth side clearances c1, c2, ..., c z , where c1 is the tooth side clearance of the first pair of teeth, c2 is the tooth side clearance of the second pair of teeth, and C z is the tooth side clearance of the zth pair of teeth; In the numerical range [2, z], values ​​of k are selected from small to large, so as to iteratively perform the following steps based on the sorted tooth side clearance until the preset conditions are met: Determine the second displacement Δx of the external spline load application starting point when the k-1 pair of teeth are deformed from engagement to the k pair of teeth just contacting k-1 and meshing stiffness E k-1 , where Δx k-1 =(c k -c k-1 ), E k-1 =(k-1)E0, where E0 represents the single tooth meshing stiffness; Δx k-1 and E k-1 Substituting into the load distribution model, we get the k-1th pair of teeth meshing and the second displacement is Δx k-1 Load distribution on each micro-segment in, Based on load distribution Determine the total load M transmitted by the spline when the k-1th pair of teeth is engaged k-1 ,in, in, It represents the second load on the jth differential segment when the k-1th pair of teeth are engaged.

5. The method according to claim 4, wherein The preset condition is M1+M2+…+M k-1 >M, where M represents an external load, wherein determining the number of meshing teeth of the spline based on the tooth side clearance and the load distribution model further includes: When the preset conditions are met, the value of k is k real Determined as the number of meshing teeth.

6. The method according to claim 5, wherein The determining, based on the number of meshing teeth and the load distribution model, a first load of each pair of meshing teeth on each micro-segment comprises: Determine the kth real Total load transmitted by the spline when the teeth are meshing and meshing stiffness in, Will and Substitute into the load distribution model and get the kth real Load distribution on each micro-segment when teeth are meshing in, According to the load distribution of each pair of teeth when meshing, determine the first load of the i-th pair of meshing teeth on the j-th micro-segment in, Where i≤k real , j≤n.

7. The method according to claim 6, wherein The determining of the contact stress distribution of each pair of meshing teeth in each micro-segment based on the first load of each pair of meshing teeth in each micro-segment comprises: determining a nominal tangential force acting on each micro-segment of each pair of meshing teeth based on the first load; Determine the relative offset angle of each pair of meshing teeth in each micro-segment, and equate the contact model of the spline to an indenter model; Based on the nominal tangential force and the relative offset angle, combined with the indenter model, the contact stress distribution of each pair of meshing teeth in each micro-segment is determined.

8. The method according to claim 7, wherein Determining the nominal tangential force acting on each micro-segment of each pair of meshing teeth based on the first load includes: The nominal tangential force is determined by the following fifth equation: in, represents the nominal tangential force acting on the jth differential segment of the i-th pair of meshing teeth.

9. The method according to claim 7, wherein The relative offset angle includes a first offset angle, a second offset angle and a third offset angle. The first offset angle is caused by bending deformation, shear deformation and foundation tilt, the second offset angle is caused by contact deformation, and the third offset angle is caused by radial expansion and contraction deformation.

10. The method according to claim 8, wherein The determining of the contact stress distribution of each pair of meshing teeth in each micro-segment based on the nominal tangential force and the relative offset angle in combination with the indenter model includes: The contact stress distribution is determined by the following sixth equation: in, represents the contact stress distribution of the i-th pair of meshing teeth on the j-th differential segment, x represents the tooth height, T represents the elastic modulus, represents the relative offset angle of the i-th pair of meshing teeth on the j-th differential segment, a represents the radial coordinate value of the contact point between the internal spline tooth top and the external spline in the indenter model, b represents the radial coordinate value of the contact point between the external spline tooth top and the internal spline in the indenter model, w x represents the load action angle of the spline tooth surface, and v represents the Poisson's ratio.

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