Ball screw pair load distribution calculation method based on comprehensive process deviation

Abstract

The application discloses a kind of based on comprehensive process deviation's ball screw pair load distribution calculation method, mainly solve in actual comprehensive process deviation ball screw pair load distribution.It is realized step as follows: input ball screw pair parameters and process deviation value, according to ball and raceway contact deformation theory and ball center and raceway arc center coordinate geometric position relationship calculation ball and raceway axial deformation after actual contact angle, again according to ball screw pair axial force balance relationship and the contact deformation coordination relationship of adjacent ball and raceway, can list axial force balance equation and adjacent ball axial contact deformation relationship iterative equation, simultaneous solution can be based on process deviation's ball screw pair load distribution result is solved.The application perfects ball screw pair load distribution model, can directly solve considering the influence of ball diameter error, lead error, pitch diameter error, raceway arc radius error and raceway arc center offset error on load distribution result.

Description

Technical Field

[0001] The invention belongs to the technical field of ball screw pair transmission, and in particular to a method for calculating the load distribution of a ball screw pair based on comprehensive process deviation. Background Art

[0002] As a core functional component of CNC machine tools, the ball screw pair is one of the key components that ensure the precision and rigidity of industrial machine tools. In actual production, process variations can cause deviations between the actual and theoretical dimensions of the ball screw pair, resulting in a decrease in actual load-bearing performance. The load distribution model of the ball screw pair is fundamental for analyzing its load-bearing capacity. Therefore, establishing an accurate load distribution model for the ball screw pair to describe the inherent laws governing the impact of process variations on load-bearing performance is crucial for improving the load-bearing characteristics of the ball screw pair.

[0003] In recent years, theoretical research on the load distribution model of ball screw pairs has achieved certain results. Xue song Mei et al. (MEI Xuesong, TSUTSUMI M, TAO Tao, et al. Study on the load distribution of ball screws with errors [J]. Mechanism and Machine Theory, 2003, 38 (11): 1257-1269) studied and derived a load distribution model for ball screw pairs based on ball diameter errors, laying the foundation for related research. Izawa and Shimoda (M. Izawa, H. Shimoda. Study on the load distribution in the ball screw [J], Japan Jou Precision Machine 42 (11) (1976) 1021-1028) conducted a force analysis on the balls and raceways, derived the mechanical equations and the average load of the ball screw pair on half the lead, and further solved the load distribution of all the balls in the ball screw pair using the interpolation calculation method. Zhao Jiajia (Zhao Jiajia, Lin Mingxing, Song Xianchun, et al. Modeling and analysis of full-ball load distribution of ball screw pairs under composite loads [J]. Journal of Mechanical Engineering, 2020, 56(17): 126-136) considered the situation where the ball screw pair is subjected to composite radial and axial loads, derived the full-ball load distribution model of the ball screw pair, and analyzed the influence of ball diameter error on the load distribution on this basis. Zeng Shiqiang (Zeng Shiqiang, Zhou Changguang, Wang Kai, et al. Study on the influence of rolling element and raceway error on the load distribution of ball screw pairs [J]. Mechanical Design and Manufacturing Engineering, 2021, 50(5): 20-28) established a load distribution model of ball screw pairs based on ball diameter and raceway error.

[0004] The aforementioned studies on ball screw load distribution models have mostly focused on the impact of ball diameter error on load distribution. The models established only consider the impact of a single error on load distribution, without considering the coupling relationship between multiple errors. Currently, there is no research on ball screw load distribution based on comprehensive process deviations. Summary of the Invention

[0005] The present invention aims to address the shortcomings of existing technologies by providing a method for calculating the load distribution of a ball screw pair based on comprehensive process deviations. The load distribution of a ball screw pair is solved based on the contact force and deformation equations between the balls and raceways of the ball screw pair, the axial force balance equation, the iterative relationship between the contact deformation of adjacent balls, and Hertz contact theory. This method provides a reference for analyzing and improving the transmission precision design and load-bearing capacity assessment of ball screw pairs.

[0006] The technical solution to achieve the purpose of the present invention is: a method for calculating the load distribution of a ball screw pair based on comprehensive process deviation, the method comprising:

[0007] The screw and nut part between the centers of two adjacent balls is regarded as a force unit, and the screw and nut part between the center of the i-th ball and the center of the (i-1)-th ball is called the i-th force unit; the initial value of the axial deformation of the i-th force unit is set to δ a0 ;

[0008] Step 1: Calculate the actual contact angle α between the i-th ball and the raceway after axial deformation i ;

[0009] Step 2, based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i ;

[0010] Step 3: transform the normal contact deformation δ into i Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness;

[0011] Step 4: Calculate the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively. si and F ni ;

[0012] Step 5: Output the above solution results.

[0013] Furthermore, in step 1, the actual contact angle α between the i-th ball and the raceway after axial deformation is solved: i The calculation formula is:

[0014]

[0015] in,

[0016] D=r s +r n -2r b

[0017] Where Δr is the ball radius error; Δr is the difference between the actual value and the theoretical value; ΔL = ΔP / Z is the lead error, ΔP is the lead error, and Z is the number of balls loaded in one cycle; Δd ​​= (d0-d s ) / 2 is the mean diameter error, d0, d s are the theoretical and actual median diameters respectively; Δn=r0-r s The radius errors of the left and right arcs of the raceway, r0, r s are the theoretical and actual left and right arc radii, respectively; Δe is the offset error of the raceway curvature center; α0 is the design contact angle between the screw and the nut; λ is the helix angle; δ ai is the axial deformation value of the i-th load unit; r s 、r n 、r b They are the radii of the screw raceway, nut raceway and ball respectively.

[0018] Furthermore, the step 2 is based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i , specifically including:

[0019] Step 2-1: Generate the axial force balance equation based on the axial force balance relationship of the ball screw pair;

[0020] Step 2-2: Based on the normal cross-sectional diagram of the ball and raceway, analyze the geometric position relationship between the ball center, the screw and the nut raceway curvature center of the i-th force unit of the ball screw pair before and after loading, and generate the axial deformation equation of the i-th ball and raceway;

[0021] Step 2-3, based on the contact deformation coordination relationship between adjacent balls and raceways, generate the iterative equation of the axial contact deformation relationship between the i-th and (i-1)-th force units, and convert the actual contact angle α i Substituting into the equation, we can obtain the normal contact deformation δ of the i-th ball and raceway i .

[0022] Furthermore, step 2-1 specifically includes:

[0023] The axial force of the screw pair remains balanced, which can be expressed as:

[0024]

[0025] Where, F a is the axial force; Q i is the normal contact force of the i-th ball; α i is the contact angle between the i-th ball and the raceway; λ is the helix angle of the raceway; M is the number of load-bearing balls;

[0026] Then the equilibrium equation of the axial force of the i-th load unit is:

[0027]

[0028] in,

[0029]

[0030]

[0031]

[0032]

[0033]

[0034] Where k s (e), k n (e) are the first elliptic integrals of the contact side between the screw and the nut; are the minor semi-axis coefficients of the contact ellipse between the ball, the screw and the nut respectively; ∑ρ s ,∑ρ n are the principal curvatures of the contact points between the ball, the screw and the nut respectively; E′ is the equivalent elastic modulus; μ s , μ n are the Poisson's ratios of the contact sides of the ball, screw and nut respectively; E s , E n are the elastic modulus of the contact side between the ball and the screw and the nut; D w is the ball diameter; D pw f is the pitch circle diameter of the ball center; rs and f rn are the ratios of the screw raceway and nut raceway to the ball diameter respectively; Q m is the normal contact force of the mth ball, δ m is the normal contact deformation of the mth ball and raceway.

[0035] Furthermore, the axial deformation equation of the i-th ball and raceway in step 2-2 is:

[0036]

[0037] in,

[0038]

[0039] Where, δ s，i , δ n，i are the normal contact deformations of the screw raceway and the nut raceway of the i-th force unit respectively.

[0040] Furthermore, in step 2-3, based on the contact deformation coordination relationship between adjacent balls and raceways, the iterative equation for the axial contact deformation relationship between the i-th and (i-1)-th force units is:

[0041] Δs i +Δn i =(δ s,i-1 +δ n,i-1 )sinα i-1 cosλ-(δ s，i +δ n，i )sinα i cosλ

[0042]

[0043] Where, δ s，i-1 , δ n，i-1 are the normal contact deformations of the screw raceway and nut raceway of the (i-1)th force unit;

[0044] According to Hooke's law of material mechanics, the axial deformations of the screw and nut of the i-th force unit are Δs i , Δn i , expressed as:

[0045]

[0046]

[0047] in,

[0048]

[0049] Where ΔL si , ΔL ni are the axial distances between the screw and the nut between two adjacent balls, A s , A n are the cross-sectional areas of the screw and nut respectively; P h is the lead of the ball screw pair; Z is the number of balls loaded in one cycle.

[0050] Furthermore, the actual axial deformation value δ is verified in step 3. ai The correctness of the

[0051] δ ai and the initial value of axial deformation δa0 If the difference is less than the set minimum value ε, the actual axial deformation value δ is output. ai If the difference is greater than the set minimum value ε, the initial value of the axial deformation δ is reset. a0 , until the difference is less than the set minimum value ε.

[0052] Furthermore, the normal contact load F between the ball of the i-th force unit and the screw raceway and nut raceway in step 4 is si and F ni The calculation formula is:

[0053] F si =F ni =K*δ i 3 / 2 .

[0054] Compared with the prior art, the present invention has the following significant advantages:

[0055] (1) The present invention is a load distribution calculation method that takes into account the ball diameter error, lead error, mean diameter error, raceway arc diameter error, and raceway curvature center offset error. The load distribution under ideal conditions is more in line with the actual contact deformation and contact load of the ball and raceway, and provides a reference basis for analyzing and improving the transmission precision design and load-bearing capacity evaluation of the ball screw pair.

[0056] (2) The present invention establishes the relationship between ball diameter error, lead error, mean diameter error, raceway arc diameter error, raceway curvature center offset error and the load distribution of the ball screw pair, and can calculate the influence of each process deviation on the load distribution separately, and can compare the influence of these five process deviations on the load distribution.

[0057] (3) The load distribution calculation results of the present invention are substituted into the national standard ball screw pair axial static stiffness formula to make the theoretical axial static stiffness results more consistent with the experimental axial static stiffness results, and the experimental results have a consistency of 87.85% to 98.7%.

[0058] The present invention is further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 The present invention is a flowchart of a method for calculating the load distribution of a ball screw pair based on comprehensive process deviations in one embodiment.

[0060] Figure 2 This is an analysis diagram of the force unit of a ball screw pair in one embodiment.

[0061] Figure 3 This is a force analysis diagram of a ball screw pair in one embodiment.

[0062] Figure 4 Schematic diagram of the geometric positions of the ball centers, screw and nut raceway curvature centers of the ball screw pair before and after loading based on process deviation in one embodiment.

[0063] Figure 5 Schematic diagram of the coordinates of the ball centers, screw and nut raceway curvature centers of the ball screw pair before and after loading based on process deviation in one embodiment.

[0064] Figure 6 Schematic diagram of the deformation coordination relationship between adjacent balls in one embodiment.

[0065] Figure 7 Schematic diagram of the load distribution results of a ball screw pair based on process deviation in one embodiment.

[0066] Figure 8 Schematic diagram of the theoretical and experimental results of the axial static stiffness of a ball screw pair under different axial forces in one embodiment. DETAILED DESCRIPTION

[0067] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0068] It should be noted that if there are descriptions involving "first", "second", etc. in the embodiments of the present invention, the descriptions of "first", "second", etc. are only for descriptive purposes and cannot be understood as indicating or implying their relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined as "first" and "second" may explicitly or implicitly include at least one of such features. In addition, the technical solutions between the various embodiments can be combined with each other, but they must be based on the ability of ordinary technicians in this field to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be deemed that such a combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

[0069] In one embodiment, combined Figure 1 , provides a method for calculating the load distribution of a ball screw pair based on comprehensive process deviation, the method comprising:

[0070] like Figure 2 As shown in the figure, the screw and nut part between the two adjacent ball centers is regarded as a force unit, and the screw and nut part between the i-th ball center and the (i-1)-th ball center is called the i-th force unit; the initial value of the axial deformation of the i-th force unit is set to δ a0 ;

[0071] Step 1: Calculate the actual contact angle α between the i-th ball and the raceway after axial deformation i ;

[0072] Step 2, based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i ;

[0073] Step 3: transform the normal contact deformation δ into i Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness;

[0074] Step 4: Calculate the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively. si and F ni ;

[0075] Step 5: Output the above solution results.

[0076] Furthermore, in one embodiment, the actual contact angle α between the i-th ball and the raceway after axial deformation is solved in step 1: i The calculation formula is:

[0077]

[0078] in,

[0079] D=r s +r n -2r b

[0080] Where Δr is the ball radius error; Δr is the difference between the actual value and the theoretical value; ΔL = ΔP / Z is the lead error, ΔP is the lead error, and Z is the number of balls loaded in one cycle; Δd ​​= (d0-d s ) / 2 is the mean diameter error, d0, d s are the theoretical and actual median diameters respectively; Δn=r0-r s The radius errors of the left and right arcs of the raceway, r0, r s are the theoretical and actual left and right arc radii, respectively; Δe is the offset error of the raceway curvature center; α0 is the design contact angle between the screw and the nut; λ is the helix angle; δ ai is the axial deformation value of the i-th load unit; r s 、r n 、r b They are the radii of the screw raceway, nut raceway and ball respectively.

[0081] Furthermore, in one embodiment, the step 2 is based on the actual contact angle αi Solve the normal contact deformation δ of the i-th ball and raceway i , specifically including:

[0082] Step 2-1, such as Figure 3 As shown in the figure, according to the axial force balance relationship of the ball screw pair, the balance equation of the axial force is generated;

[0083] Step 2-2, such as Figure 4 、 Figure 5 As shown, based on the normal cross-section diagram of the ball and raceway and the schematic diagram of the center coordinates of the ball and raceway, the geometric position relationship between the ball center, the screw and the nut raceway curvature center of the i-th force unit of the ball screw pair before and after loading is analyzed, and the axial deformation equation of the i-th ball and raceway is generated;

[0084] Steps 2-3, such as Figure 6 As shown in the figure, according to the contact deformation coordination relationship between adjacent balls and raceways, the iterative equation of the axial contact deformation relationship of the i-th and (i-1)-th force units is generated, and the actual contact angle α is converted to i Substituting into the equation, we can obtain the normal contact deformation δ of the i-th ball and raceway i .

[0085] Here, step 2-1 specifically includes:

[0086] The axial force of the screw pair remains balanced, which can be expressed as:

[0087]

[0088] Where, F a is the axial force; Q i is the normal contact force of the i-th ball; α i is the contact angle between the i-th ball and the raceway; λ is the helix angle of the raceway; M is the number of load-bearing balls;

[0089] Then the equilibrium equation of the axial force of the i-th load unit is:

[0090]

[0091] in,

[0092]

[0093]

[0094]

[0095]

[0096]

[0097] Where k s (e), k n (e) are the first elliptic integrals of the contact side between the screw and the nut; are the minor semi-axis coefficients of the contact ellipse between the ball, the screw and the nut respectively; ∑ρ s ,∑ρ n are the principal curvatures of the contact points between the ball, the screw and the nut respectively; E′ is the equivalent elastic modulus; μ s , μ n are the Poisson's ratios of the contact sides of the ball, screw and nut respectively; E s , E n are the elastic modulus of the contact side between the ball and the screw and the nut; D w is the ball diameter; D pw f is the pitch circle diameter of the ball center; rs and f rn are the ratios of the screw raceway and nut raceway to the ball diameter respectively; Q m is the normal contact force of the mth ball, δ m is the normal contact deformation of the mth ball and raceway.

[0098] Here, the axial deformation equation of the i-th ball and raceway in step 2-2 is:

[0099]

[0100] in,

[0101]

[0102] Where, δ s，i , δ n，i are the normal contact deformations of the screw raceway and nut raceway of the i-th force unit respectively

[0103] Here, in step 2-3, based on the contact deformation coordination relationship between adjacent balls and raceways, the iterative equation for the axial contact deformation relationship between the i-th and (i-1)-th force units is:

[0104] Δs i +Δn i =(δ s，i-1 +δ n，i-1 )sinα i-1 cosλ-(δ s，i +δ n，i )sinα i cosλ

[0105]

[0106] Where, δ s，i-1 , δ n，i-1are the normal contact deformations of the screw raceway and nut raceway of the (i-1)th force unit;

[0107] According to Hooke's law of material mechanics, the axial deformations of the screw and nut of the i-th force unit are Δs i , Δn i Expressed as:

[0108]

[0109]

[0110] in,

[0111]

[0112] Where ΔL si , ΔL ni are the axial distances between the screw and the nut between two adjacent balls, A s , A n are the cross-sectional areas of the screw and nut respectively; P h is the lead of the ball screw pair; Z is the number of balls loaded in one cycle.

[0113] Furthermore, in one embodiment, the actual axial deformation value δ is verified in step 3. ai The correctness of the

[0114] δ ai and the initial value of axial deformation δ a0 If the difference is less than the set minimum value ε, the actual axial deformation value δ is output. ai If the difference is greater than the set minimum value ε, the initial value of the axial deformation δ is reset. a0 , until the difference is less than the set minimum value ε.

[0115] Furthermore, in one embodiment, the normal contact load F between the ball of the i-th force unit and the screw raceway and the nut raceway in step 4 is si and F ni The calculation formula is:

[0116] F si =F ni =K*δ i 3 / 2 .

[0117] In one embodiment, a ball screw pair load distribution calculation system based on comprehensive process deviation is provided, the system comprising:

[0118] The first module is used to solve the actual contact angle α between the i-th ball and the raceway after axial deformation.i ;

[0119] The second module is used to calculate the contact angle α based on the actual contact angle α. i Solve the normal contact deformation δ of the i-th ball and raceway i ;

[0120] The third module is used to transform the normal contact deformation δ according to the geometric relationship i Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness;

[0121] The fourth module is used to solve the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively si and F ni ;

[0122] The fifth module is used to output all the above solution results.

[0123] Regarding the specific limitations of the ball screw pair load distribution calculation system based on comprehensive process deviations, please refer to the limitations of the ball screw pair load distribution calculation method based on comprehensive process deviations above, which will not be repeated here. The various modules in the above-mentioned ball screw pair load distribution calculation system based on comprehensive process deviations can be implemented in whole or in part through software, hardware, and a combination thereof. The above-mentioned modules can be embedded in or independent of the processor in the computer device in hardware form, or can be stored in the memory of the computer device in software form, so that the processor can call and execute the operations corresponding to the above modules.

[0124] In one embodiment, a computer device is provided, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, the following is achieved:

[0125] The screw and nut part between the centers of two adjacent balls is regarded as a force unit, and the screw and nut part between the center of the i-th ball and the center of the (i-1)-th ball is called the i-th force unit; the initial value of the axial deformation of the i-th force unit is set to δ a0 ;

[0126] Step 1: Calculate the actual contact angle α between the i-th ball and the raceway after axial deformation i ;

[0127] Step 2, based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i ;

[0128] Step 3: transform the normal contact deformation δ intoi Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness;

[0129] Step 4: Calculate the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively. si and F ni ;

[0130] Step 5: Output the above solution results.

[0131] For the specific limitations of each step, please refer to the limitations of the ball screw pair load distribution calculation method based on comprehensive process deviations mentioned above, which will not be repeated here.

[0132] In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, the computer program implements:

[0133] The screw and nut part between the centers of two adjacent balls is regarded as a force unit, and the screw and nut part between the center of the i-th ball and the center of the (i-1)-th ball is called the i-th force unit; the initial value of the axial deformation of the i-th force unit is set to δ a0 ;

[0134] Step 1: Calculate the actual contact angle α between the i-th ball and the raceway after axial deformation i ;

[0135] Step 2, based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i ;

[0136] Step 3: transform the normal contact deformation δ into i Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness;

[0137] Step 4: Calculate the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively. si and F ni ;

[0138] Step 5: Output the above solution results.

[0139] For the specific limitations of each step, please refer to the limitations of the ball screw pair load distribution calculation method based on comprehensive process deviations mentioned above, which will not be repeated here.

[0140] As a specific example, the present invention is further verified and explained in one of the embodiments.

[0141] Taking the GZ3206T ball screw pair as an example, the process error value of the GZ3206T ball screw pair product is measured using a profilometer. Its parameters and error values ​​are shown in Table 1.

[0142] Table 1 Parameters and error values

[0143]

[0144]

[0145] like Figure 1 As shown in FIG, the ball screw pair load distribution calculation method based on comprehensive process deviation includes the following steps:

[0146] S1: Calculate the actual contact angle α between the i-th ball and the raceway after axial deformation i :

[0147] In order to facilitate the analysis of the contact load between the ball and the raceway, the screw and nut part between the centers of two adjacent balls is regarded as a force unit. The screw and nut part between the center of the i-th ball and the center of the (i-1)-th ball is called the i-th force unit. The initial value of the axial deformation of the i-th force unit is set to δ a0 , and the actual contact angle calculation formula can be substituted to obtain the actual contact angle α between the i-th ball and the raceway after axial deformation. i ;

[0148] The actual contact angle α between the i-th ball and the raceway after axial deformation in S1 i The calculation formula is:

[0149]

[0150] D=O s O n =r s +r n -2r b (2)

[0151] Where Δr is the difference between the actual and theoretical ball radius; ΔL = ΔP / Z; ΔP is the lead error, Z is the number of loads on the ball in one cycle; Δd ​​= (d0-d s ) / 2 is the mean diameter error, d0, d s are the theoretical and actual median diameters respectively; Δn=r0-r s The radius errors of the left and right arcs of the raceway, r0, r sare the theoretical and actual left and right arc radii, respectively; Δe is the offset error of the raceway curvature center; α0 is the design contact angle between the screw and the nut; λ is the helix angle; δ ai is the axial deformation value of the i-th load unit; r n 、r b They are the radii of the screw raceway, nut raceway and ball respectively.

[0152] According to the parameters and process error values ​​of the GZ3206T ball screw pair in Table 1, substitute them into equations (1) and (2), and set the initial value of the axial deformation of the i-th force unit to be δ a0 is 0.03, the actual contact angle α between the i-th ball and the raceway after axial deformation can be obtained. i ;

[0153] The actual contact angle α between the i-th ball and the raceway after axial deformation is preliminarily obtained in this example. i It is 45.497.

[0154] S2: Solve the normal contact deformation δ of the i-th ball and raceway i :

[0155] According to the axial force balance relationship of the ball screw pair, the axial force balance equation can be listed. According to the normal cross-section diagram of the ball and raceway, the geometric position relationship between the ball center, screw and nut raceway curvature center of the i-th force unit before and after the ball screw pair is loaded is analyzed, and the axial deformation equation is listed. Then, according to the contact deformation coordination relationship between adjacent balls and raceways, the axial contact deformation relationship iterative equation of the i-th and (i-1)-th force units can be listed, and the actual contact angle αi solved by S1 is substituted into the equation to obtain the normal contact deformation δ of the i-th ball and raceway. i :

[0156] The normal contact deformation δ of the i-th ball and raceway in S2 is i The calculation formula is:

[0157] The axial force of the screw pair remains balanced, which can be expressed as:

[0158]

[0159] Where, F a is the axial force; Q i is the normal contact force of the i-th ball; α i is the contact angle between the ith ball and the raceway; λ is the helix angle of the raceway; and M is the number of load-bearing balls.

[0160] In this example, F aSet it to 1 / 3 of the rated static load of the screw pair, which is 26036N; λ is 3.4155°, and substitute it into formula (3); the axial force balance equation can be listed.

[0161] The axial force of the i-th load unit is expressed as:

[0162]

[0163] Based on Hertz contact theory, the relationship between load and deformation is:

[0164]

[0165]

[0166]

[0167]

[0168]

[0169] Where k s (e), k n (e) is the first elliptic integral of the contact side between the screw and the nut; is the semi-minor axis coefficient of the contact ellipse between the ball, screw and nut; ∑ρ s ,∑ρ n is the sum of the principal curvatures of the contact points between the ball, the screw and the nut; E′ is the equivalent elastic modulus; μ s , μ n are the Poisson's ratios of the contact sides of the ball, screw and nut respectively; E s , E n are the elastic modulus of the contact side between the ball and the screw and the nut; D w is the ball diameter; D pw f is the pitch circle diameter of the ball center; rs and f rn are the ratios of the screw raceway and nut raceway to the ball diameter, respectively.

[0170] In this case, μ s , μ n is 0.29; E s , E n is 210000 MPa. Substituting it into equations (5)-(9) yields the load and deformation equations of the Hertz contact theory.

[0171] The contact deformation of the i-th ball and the raceway is:

[0172]

[0173]

[0174] According to the contact deformation coordination relationship of the ball screw pair, the iterative equation of the axial contact deformation relationship between the i-th and (i-1)-th force units is:

[0175] Δs i +Δn i =(δ s，i-1 +δ n，i-1 )sin α-i 1cosλ-(δ s，i +δ n，i )sinα i cosλ (12)

[0176]

[0177] According to Hooke's law of material mechanics, the axial deformation of the i-th force unit of the screw and nut is Δs i , Δn i , which can be expressed as:

[0178]

[0179]

[0180]

[0181] Where ΔL si , ΔL ni are the axial distances between the screw and the nut between two adjacent balls, A s , A n are the cross-sectional areas of the screw and nut respectively; P h is the lead of the ball screw pair; Z is the number of balls loaded in one cycle.

[0182] In this example, the actual contact angle α between the i-th ball and the raceway after axial deformation obtained by S1 is i Substituting 45.497° into equations (3), (4), (11), and (12) respectively, we can obtain the axial deformation equilibrium equation and the deformation iteration equation of adjacent force units.

[0183] The normal contact deformation δ of the ith ball and raceway can be obtained by combining equations (1) to (11): i .

[0184] The final calculated load distribution of the ball screw pair based on process deviation in this example is as follows: Figure 7 shown.

[0185] Substituting the load distribution calculation results into the national standard ball screw pair axial static stiffness formula, the theoretical calculation results and the experimental calculation results are compared. Figure 8As shown, it can be seen that the experimental results are consistent with each other by 87.85% to 98.7%.

[0186] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only illustrative of the principles of the present invention. Without departing from the spirit and scope of the present invention, any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included in the scope of protection of the present invention.

Claims

1. A method for calculating the load distribution of a ball screw pair based on comprehensive process deviation, characterized in that: The method comprises: The screw and nut part between the centers of two adjacent balls is regarded as a force unit, and the screw and nut part between the center of the i-th ball and the center of the (i-1)-th ball is called the i-th force unit; the initial value of the axial deformation of the i-th force unit is set to δ a0 ; Step 1: Calculate the actual contact angle α between the i-th ball and the raceway after axial deformation i The calculation formula is: in, D＝r s +r n -2r b Where Δr is the ball radius error; Δr is the difference between the actual value and the theoretical value; ΔL = ΔP / Z is the lead error, ΔP is the stroke error, and Z is the number of balls loaded in one cycle; Δd ​​= (d0-d s ) / 2 is the mean diameter error, d0, d s are the theoretical and actual median diameters respectively; Δn=r0-r s The radius errors of the left and right arcs of the raceway, r0, r s are the theoretical and actual left and right arc radii, respectively; Δe is the offset error of the raceway curvature center; α0 is the design contact angle between the screw and the nut; λ is the helix angle; δ ai is the axial deformation value of the i-th load unit; r s 、r n 、r b are the radii of the screw raceway, nut raceway and ball respectively; Step 2, based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i ; Step 3: transform the normal contact deformation δ into i Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness; Step 4: Calculate the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively. si and F ni ; Step 5: Output the above solution results.

2. The ball screw pair load distribution calculation method based on comprehensive process deviation according to claim 1 is characterized in that: Step 2 is based on the actual contact angle α i Solve the normal contact deformation δ of the i-th ball and raceway i , specifically including: Step 2-1: Generate the axial force balance equation based on the axial force balance relationship of the ball screw pair; Step 2-2: Based on the normal cross-sectional diagram of the ball and raceway, analyze the geometric position relationship between the ball center, the screw and the nut raceway curvature center of the i-th force unit of the ball screw pair before and after loading, and generate the axial deformation equation of the i-th ball and raceway; Step 2-3, based on the contact deformation coordination relationship between adjacent balls and raceways, generate the iterative equation of the axial contact deformation relationship between the i-th and (i-1)-th force units, and convert the actual contact angle α i Substituting into the equation, we can obtain the normal contact deformation δ of the i-th ball and raceway i .

3. The ball screw pair load distribution calculation method based on comprehensive process deviation according to claim 2 is characterized in that: Step 2-1 specifically includes: The axial force of the screw pair remains balanced, which can be expressed as: Where, F a is the axial force; Q i is the normal contact force of the i-th ball; α i is the contact angle between the i-th ball and the raceway; λ is the helix angle of the raceway; M is the number of load-bearing balls; Then the equilibrium equation of the axial force of the i-th load unit is: in, Where k s (e), k n (e) are the first elliptic integrals of the contact side between the screw and the nut; are the minor semi-axis coefficients of the contact ellipse between the ball, the screw and the nut respectively; ∑ρ s ,∑ρ n are the principal curvatures of the contact points between the ball, the screw and the nut respectively; E' is the equivalent elastic modulus; μ s ,μ n are the Poisson's ratios of the contact sides of the ball, screw and nut respectively; E s ,E n are the elastic modulus of the contact side between the ball and the screw and the nut respectively; D w is the ball diameter; D pw f is the pitch circle diameter of the ball center; rs and f rn are the ratios of the screw raceway and nut raceway to the ball diameter respectively; Q m is the normal contact force of the mth ball, δ m is the normal contact deformation of the mth ball and raceway.

4. The ball screw pair load distribution calculation method based on comprehensive process deviation according to claim 3 is characterized in that: The axial deformation equation of the i-th ball and raceway in step 2-2 is: in, Where, δ s,i , δ n,i are the normal contact deformations of the screw raceway and the nut raceway of the i-th force unit respectively.

5. The ball screw pair load distribution calculation method based on comprehensive process deviation according to claim 4 is characterized in that: In step 2-3, based on the contact deformation coordination relationship between adjacent balls and raceways, the iterative equations for the axial contact deformation relationship of the i-th and i-1-th force units are: Δs i +Δn i =(δ s,i-1 +d n,i-1 )sina i-1 cosλ-(δ s,i +d n,i )sina i cosλ Where, δ s,i-1 , δ n,i-1 are the normal contact deformations of the screw raceway and nut raceway of the (i-1)th force unit; According to Hooke's law of material mechanics, the axial deformations of the screw and nut of the i-th force unit are Δs i , Δn i , expressed as: in, Where ΔL si , ΔL ni are the axial distances between the screw and the nut between two adjacent balls, A s , A n are the cross-sectional areas of the screw and nut respectively; P h is the lead of the ball screw pair; Z is the number of balls loaded in one cycle.

6. The ball screw pair load distribution calculation method based on comprehensive process deviation according to claim 1 is characterized in that: Verify the actual axial deformation value δ as described in step 3 ai The correctness of the δ ai and the initial value of axial deformation δ a0 If the difference is less than the set minimum value ε, the actual axial deformation value δ is output. ai If the difference is greater than the set minimum value ε, the initial value of the axial deformation δ is reset. a0 , until the difference is less than the set minimum value ε.

7. The method for calculating the load distribution of a ball screw pair based on comprehensive process deviation according to claim 3, characterized in that: The normal contact load F between the ball of the i-th force unit and the screw raceway and nut raceway in step 4 si and F ni The calculation formula is: F si =F ni =K*δ i 3 / 2 。 8. A ball screw pair load distribution calculation system based on comprehensive process deviation according to the method of any one of claims 1 to 7, characterized in that: The system comprises: The first module is used to solve the actual contact angle α between the i-th ball and the raceway after axial deformation. i ; The second module is used to calculate the contact angle α based on the actual contact angle α. i Solve the normal contact deformation δ of the i-th ball and raceway i ; The third module is used to transform the normal contact deformation δ according to the geometric relationship i Converted to actual axial deformation value δ ai , and verify the actual axial deformation value δ ai Correctness; The fourth module is used to solve the normal contact load F between the ball and the screw raceway and the nut raceway of the i-th force unit respectively si and F ni ; The fifth module is used to output all the above solution results.

9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the computer program, the method according to any one of claims 1 to 7 is implemented.

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