CN105136097A

Movatterモバイル変換

Info

Publication number: CN105136097A
Application number: CN201510512754.XA
Authority: CN
Inventors: 姜洪奎; 李彦凤; 马建春; 许向荣; 宋现春; 宋一顺; 马凡营; 杜伟; 荣柏松
Original assignee: Shandong Jianzhu University
Current assignee: Shandong Jianzhu University
Priority date: 2015-08-19
Filing date: 2015-08-19
Publication date: 2015-12-09
Anticipated expiration: 2035-08-19
Also published as: CN105136097B

Abstract

本发明提供一种非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法。主要包括以下步骤：(1)测量滚珠半径r_b、滚珠丝杠副的径向间隙S_d，丝杠滚道半径r_s，螺母滚道半径r_n、丝杠侧滚道设计接触角α_s和螺母侧滚道设计接触角α_n，求取初始接触角α₀；(2)利用赫兹理论计算滚珠与滚道之间的法向变形量δ_ns和δ_nn；(3)建立初始接触角α₀和弹性变形接触角α的关系式，结合得到的初始接触角α₀，计算非等适应比滚道的滚珠丝杠副弹性变形接触角α。

The invention provides a method for determining the elastic deformation contact angle of a ball screw pair with non-equal adaptation ratio raceways. It mainly includes the following steps: (1) Measure the ball radius r_b , the radial clearance S_d of the ball screw pair, the screw raceway radius r_s , the nut raceway radius r_n , and the screw side raceway design contact angle α_s Calculate the initial contact angle α₀ with the design contact angle α_n of the raceway on the nut side; (2) Calculate the normal deformation between the ball and the raceway δ_ns and δ_nn by using Hertz theory; (3) Establish the initial contact angle The relationship between α₀ and elastic deformation contact angle α is combined with the obtained initial contact angle α₀ to calculate the elastic deformation contact angle α of the ball screw pair with non-equal adaptation ratio raceways.

Description

Translated fromChinese

非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法Determination Method of Elastic Deformation Contact Angle of Ball Screw Pair with Non-equal Adaptation Ratio Raceways

技术领域technical field

本发明公开了一种非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法。The invention discloses a method for determining the elastic deformation contact angle of a ball screw pair with non-equal adaptive ratio raceways.

背景技术Background technique

近年来，由于滚珠丝杠副传动效率高，且能够消除传动间隙、精度高以及运动平稳，因而在数控机床和机械制造中得到广泛应用。弹性变形接触角是影响滚珠丝杠副效率和柔顺性的一个重要因素。In recent years, due to the high transmission efficiency of the ball screw pair, the ability to eliminate the transmission gap, high precision and smooth movement, it has been widely used in CNC machine tools and machinery manufacturing. The elastic deformation contact angle is an important factor affecting the efficiency and compliance of the ball screw assembly.

目前所有关滚珠丝杠副弹性变形接触角的计算方法往往参考文献《Rollingbearinganalysis》的计算方法。而该文献《Rollingbearinganalysis》是针对滚珠轴承的弹性接触角的算法，对于滚珠丝杠副的弹性变形接触角的计算不能完全适用。参考文献《Rollingbearinganalysis》计算方法的基本思路如下：首先对弹性变形接触角等于45度的预紧载荷作用下的滚珠的变形协调公式，进行微积分，从而得出制造误差对接触角的影响。这些已有研究成果对滚珠丝杠副的设计和应用都有一定的指导意义，但文献《Rollingbearinganalysis》的理论分析都是基于滚珠两侧滚道的适应比相同的情况，并未涉及到非等适应比(即螺母侧滚道适应比、丝杠侧滚道适应比不同)或者两侧曲率半径差别较大的情况下。而事实上滚珠丝杠副在实际加工中，其法向截面往往是非等适应比的。因此建立非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法是亟待解决的问题。At present, all the calculation methods of the elastic deformation contact angle of the ball screw pair often refer to the calculation method of the literature "Rolling bearing analysis". However, the document "Rollingbearing analysis" is an algorithm for the elastic contact angle of the ball bearing, which cannot be fully applied to the calculation of the elastic deformation contact angle of the ball screw pair. The basic idea of the calculation method of the reference "Rollingbearing analysis" is as follows: First, the calculus is performed on the deformation coordination formula of the ball under the action of the preload of the elastic deformation contact angle equal to 45 degrees, so as to obtain the influence of manufacturing error on the contact angle. These existing research results have certain guiding significance for the design and application of the ball screw pair, but the theoretical analysis of the document "Rollingbearing analysis" is based on the same adaptation ratio of the raceways on both sides of the ball, and does not involve unequal When the adaptation ratio (that is, the adaptation ratio of the nut side raceway and the screw side raceway adaptation ratio is different) or the curvature radius of the two sides is greatly different. In fact, in the actual processing of the ball screw pair, its normal section is often non-equal adaptation ratio. Therefore, it is an urgent problem to establish a method to determine the elastic deformation contact angle of the ball screw pair with non-equal adaptation ratio raceways.

发明内容Contents of the invention

为了解决上述问题，本发明提供一种非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法，包括以下步骤：In order to solve the above problems, the present invention provides a method for determining the elastic deformation contact angle of a ball screw pair with non-equal adaptive ratio raceways, which includes the following steps:

(1)使用千分尺或工具显微镜测量滚珠半径r_b、使用千分尺或微分表测量滚珠丝杠副的径向间隙S_d，利用轮廓仪测量丝杠滚道半径r_s，螺母滚道半径r_n、丝杠侧滚道设计接触角α_s和螺母侧滚道设计接触角α_n，求取初始接触角α₀；(1) Use a micrometer or a tool microscope to measure the ball radius r_b , use a micrometer or a differential table to measure the radial clearance S_d of the ball screw pair, use a profiler to measure the screw raceway radius r_s , the nut raceway radius r_n , Calculate the initial contact angle α₀ from the design contact angle α_s of the screw side raceway and the design contact angle α_n of the nut side raceway;

(2)利用赫兹理论计算滚珠与滚道之间的法向变形量δ_ns和δ_nn；(2) Using Hertz theory to calculate the normal deformation between the ball and the raceway δ_ns and δ_nn ;

(3)建立初始接触角α₀和弹性变形接触角α的关系式，结合得到的初始接触角α₀，计算非等适应比滚道的滚珠丝杠副弹性变形接触角α。(3) Establish the relationship between the initial contact angle α₀ and the elastic deformation contact angle α, and combine the obtained initial contact angle α₀ to calculate the elastic deformation contact angle α of the ball screw pair with non-equal adaptation ratio raceways.

上述确定方法中，优选的是，所述步骤(1)的具体方法为：获取初始接触角cosα₀：In the above determination method, preferably, the specific method of the step (1) is: to obtain the initial contact angle cosα₀ :

使用轮廓仪测量点O_s的坐标(x_s,y_s)，发生平动后滚珠螺母侧右半外圆曲率中心点O_n的坐标(x_n',y_n')，滚珠半径r_b、丝杠滚道半径r_s和螺母滚道半径r_n，根据三角投影关系可以得出，初始接触角α₀为：Use a profiler to measure the coordinates (x_s , y_s ) of the point O_s , the coordinates (x_n ', y_n ') of the center point O_n of the curvature of the right half outer circle on the side of the ball nut after translation, the ball radius r_b , The radius r_s of the screw raceway and the radius r_n of the nut raceway can be obtained according to the triangular projection relationship, and the initial contact angle α₀ is:

cosα₀＝(y_s-y'_n)/(r_n+r_s-2r_b)(1)cosα₀ =(y_s -y'_n )/(r_n +r_s -2r_b )(1)

由于滚珠与滚道之间的几何约束关系不变，点O₀(x₀,y₀)位于两个圆的相切点处，则：Since the geometric constraints between the ball and the raceway remain unchanged, the point O₀ (x₀ ,y₀ ) is located at the tangent point of the two circles, then:

(x₀-x'_n)²+(y₀-y'_n)²＝(r_n-r_b)²(2)(x₀ -x'_n )² +(y₀ -y'_n )² ＝(r_n -r_b )² (2)

(x₀-x_s)²+(y₀-y_s)²＝(r_s-r_b)²(3)(x₀ -x_s )² +(y₀ -y_s )² ＝(r_s -r_b )² (3)

将公式(2)(3)代入公式(1)可以得出：Substituting formulas (2)(3) into formula (1) yields:

${cosα cosα}_{00} = = \frac{(({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{s the s} + + (({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{n no} - - \frac{55}{22} {S S}_{d d}}{{r r}_{n no} + + {r r}_{s the s} - - 22 {r r}_{b b}} . . - - - - - - ((44))$

上述确定方法中，优选的是，所述步骤(2)中，滚珠与滚道之间的法向变形量的具体计算方法为：In the above determination method, preferably, in the step (2), the specific calculation method of the normal deformation between the ball and the raceway is:

由赫兹理论可得出，滚珠与滚道之间的法向变形量δ_ns和δ_nn为：According to the Hertz theory, the normal deformations δ_ns and δ_nn between the ball and the raceway are:

${δ δ}_{n no i i} = = 2.79 2.79 \times \times 1010^{- - 44} {δ δ}_{i i}^{* *} {Q Q}_{i i}^{\frac{22}{33}} {(({Σρ Σρ}_{i i}))}^{\frac{11}{33}} - - - - - - ((55))$

其中，i＝n表示螺母侧，i＝s表示丝杠侧，Q_i为单个滚珠承受的法向载荷，Σρ_i为滚珠与滚道接触表面的曲率和。Among them, i=n represents the nut side, i=s represents the screw side, Q_i is the normal load borne by a single ball, and Σρ_i is the sum of the curvatures of the contact surface between the ball and the raceway.

上述确定方法中，优选的是，所述步骤(2)中，单个滚珠承受的法向载荷Q_i的具体计算方法为：In the above-mentioned determination method, preferably, in the step (2), the specific calculation method of the normal load Q_i borne by a single ball is:

${Q Q}_{i i} = = \frac{{F f}_{a a}}{z z {cosλsinα cosλsinα}_{00}} - - - - - - ((66))$

其中F_a为丝杠副所承受的轴向预紧载荷，z为单个螺母内的承载滚珠个数，λ为螺旋升角。Among them, F_a is the axial pretightening load borne by the screw pair, z is the number of loaded balls in a single nut, and λ is the helix angle.

上述的确定方法中，优选的是，所述步骤(3)的具体方法为：In the above-mentioned determining method, preferably, the specific method of the step (3) is:

初始接触角α₀弹性变形接触角α之间的关系式：The relationship between the initial contact angle α₀ elastic deformation contact angle α:

(r_n+r_s-2r_b)cosα＝(r_s+r_n-2r_b+δ_ns+δ_nn)cosα₀(7)(r_n +r_s -2r_b )cosα＝(r_s +r_n -2r_b +δ_ns +δ_nn )cosα₀ (7)

计算非等适应比滚道的滚珠丝杠副弹性变形接触角：Calculate the elastic deformation contact angle of the ball screw pair with non-equal adaptation ratio raceways:

将公式(4)(5)(6)代入接触角公式(7)得：Substituting the formula (4)(5)(6) into the contact angle formula (7):

$\frac{{F f}_{a a}}{z z c c o o s the s λ λ} = = s the s i i n no α α {((\frac{3.58422 3.58422 \times \times 1010^{33}}{{δ δ}_{s the s}^{* *} {(({Σρ Σρ}_{s the s}))}^{\frac{11}{33}} + + {δ δ}_{n no}^{* *} {(({Σρ Σρ}_{n no}))}^{\frac{11}{33}}} ((\frac{c c o o s the s α α}{{cosα cosα}_{00}} - - 11))))}^{1.5 1.5} - - - - - - ((88))$

其中：in:

${Σρ Σρ}_{s the s} = = \frac{11}{{r r}_{b b}} + + \frac{11}{{r r}_{s the s}} + + \frac{11}{{r r}_{b b}} + + \frac{{cosα cosα}_{s the s} c c o o s the s λ λ}{{r r}_{s the s} - - {r r}_{b b} {cosα cosα}_{s the s}}$

${Σρ Σρ}_{n no} = = \frac{11}{{r r}_{b b}} - - \frac{11}{{r r}_{n no}} + + \frac{11}{{r r}_{b b}} + + \frac{{cosα cosα}_{n no} c c o o s the s λ λ}{{r r}_{n no} - - {r r}_{b b} {cosα cosα}_{n no}}$

$\begin{matrix} {δ δ}_{s the s}^{* *} = = \frac{22 ((1.5277 1.5277 + + 0.6023 0.6023 l l n no \frac{{R R}_{y the y s the s}}{{R R}_{x x s the s}}))}{π π} \\ ((\frac{π π}{22 {((1.0339 1.0339 {((\frac{{R R}_{y the y s the s}}{{R R}_{x x s the s}}))}^{0.636 0.636}))}^{22} ((1.0003 1.0003 + + 0.5968 0.5968 \frac{{R R}_{x x s the s}}{{R R}_{y the y s the s}}))})) \end{matrix}$

$\begin{matrix} {δ δ}_{s the s}^{* *} = = \frac{22 ((1.5277 1.5277 + + 0.6023 0.6023 l l n no \frac{{R R}_{y the y n no}}{{R R}_{x x n no}}))}{π π} \\ ((\frac{π π}{22 {((1.0339 1.0339 {((\frac{{R R}_{y the y n no}}{{R R}_{x x n no}}))}^{0.636 0.636}))}^{22} ((1.0003 1.0003 + + 0.5968 0.5968 \frac{{R R}_{x x n no}}{{R R}_{y the y n no}}))})) \end{matrix}$

${R R}_{x x s the s}^{- - 11} = = \frac{11}{{r r}_{b b}} + + \frac{11}{{r r}_{s the s}}$

${R R}_{y the y s the s}^{- - 11} = = \frac{11}{{r r}_{b b}} + + \frac{c c o o s the s α α c c o o s the s λ λ}{{r r}_{s the s} - - {r r}_{b b} c c o o s the s α α}$

${R R}_{x x n no}^{- - 11} = = \frac{11}{{r r}_{b b}} - - \frac{11}{{r r}_{n no}}$

${R R}_{y the y n no}^{- - 11} = = \frac{11}{{r r}_{b b}} + + \frac{c c o o s the s α α c c o o s the s λ λ}{{r r}_{n no} - - {r r}_{b b} c c o o s the s α α}$

公式(8)为非线性方程，采用数值求解方法Newton-Raphson求解出弹性变形接触角α。Formula (8) is a nonlinear equation, and the elastic deformation contact angle α is obtained by using the numerical solution method Newton-Raphson.

本发明的有益效果：滚珠丝杠副在实际加工中，其法向截面往往是非等适应比的滚道，本发明首创性提出了非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法，为非等适比滚道的滚珠丝杠副设计与制造提供了详实的理论基础，进一步地保证并提高非等适应比滚道滚珠丝杠副的使用性能。Beneficial effects of the present invention: in the actual processing of the ball screw pair, its normal section is often a non-equal adaptation ratio raceway. The determination method provides a detailed theoretical basis for the design and manufacture of ball screw pairs with non-equal proportion raceways, and further ensures and improves the performance of ball screw pairs with non-equal proportion raceways.

附图说明Description of drawings

图1为垫片式预紧滚珠丝杠副结构示意图；Figure 1 is a schematic diagram of the structure of the gasket type preloaded ball screw pair;

图2为消除径向间隙前后滚珠与滚道接触位移示意图；Figure 2 is a schematic diagram of the contact displacement between the ball and the raceway before and after eliminating the radial clearance;

图3为预载后滚珠与滚道之间的弹性变形协调几何关系图；Figure 3 is a diagram of the coordinated geometric relationship between the elastic deformation of the ball and the raceway after preloading;

图4为本发明的非等适应比滚道的滚珠丝杠副弹性变形接触角α测量计算流程图。Fig. 4 is a flow chart of measuring and calculating the elastic deformation contact angle α of the ball screw pair with non-equal adaptation ratio raceways of the present invention.

具体实施方式Detailed ways

一、关于滚珠与滚道之间的几何分析：1. About the geometrical analysis between the ball and the raceway:

滚珠丝杠副的最常用的预紧方式是垫片式预紧，其结构形式如图1所示。即在螺母A和螺母B之间增加弹性垫片，以消除滚珠丝杠副的径向和轴向间隙。The most commonly used preloading method of the ball screw pair is the gasket type preloading, and its structure is shown in Figure 1. That is, an elastic washer is added between nut A and nut B to eliminate the radial and axial clearance of the ball screw pair.

本发明研究的对象是非等适应比滚道的垫片式双螺母滚珠丝杠副，因此假设在滚珠与两侧滚道发生弹性变形之前，螺母侧滚道沿轴向平动，丝杠侧滚道固定不动。参考图2和图3所示，设滚珠中心在径向方向上的最低点O_0s作为坐标原点，法截面的对称线为y轴，与y轴相垂直的轴为x轴。考虑径向间隙的存在，滚珠中心在径向方向上的最高点为O_0n，点O_0s与点O_0n在y轴的距离为径向间隙的一半，即S_d/2，Sd为径向间隙。设丝杠侧左半外圆曲率中心为O_s，螺母发生平动以前的螺母侧右半外圆曲率中心为O_n，发生平动后滚珠螺母侧右半外圆曲率中心为O′_n，平动距离为δ_a。滚珠在位置1处与螺母滚道之间的接触角为设计接触角α_n，在位置2处与螺母滚道之间的接触角为设计接触角α_s。螺母发生平动时，滚珠与螺母滚道始终相切，因此其中心轨迹为以点O′_n圆心、半径为r_n-r_b的圆；滚珠与丝杠滚道始终相切，滚珠中心的轨迹为以点O_s为圆心、半径为r_s-r_b的圆。丝杠发生变形前，滚珠与两侧滚道同时接触达到力平衡，因此滚珠中心的位置O₀位于两个圆的交点处，且与两圆的中心O_s和O′_n共线，即两圆的相切点。根据这一几何约束关系可以得出径向间隙、曲率半径、设计接触角和滚珠之间的关系。The research object of the present invention is the washer-type double-nut ball screw pair with non-equal adaptive ratio raceways. Therefore, it is assumed that before the elastic deformation of the balls and the raceways on both sides, the nut side raceway translates along the axial direction, and the screw rolls sideways. The way is fixed. Referring to Figures 2 and 3, the lowest point O_0s of the center of the ball in the radial direction is used as the coordinate origin, the symmetry line of the normal section is the y-axis, and the axis perpendicular to the y-axis is the x-axis. Considering the existence of the radial gap, the highest point of the ball center in the radial direction is O_0n , the distance between point O_0s and point O_0n on the y-axis is half of the radial gap, that is, S_d /2, and Sd is the radial gap. Let the center of curvature of the left half of the outer circle on the screw side be O_s , the center of curvature of the right half of the outer circle on the nut side before translation is O_n , and the center of curvature of the right half of the outer circle on the ball nut side after translation is O′_n . The translational distance is δ_a . The contact angle between the ball at position 1 and the nut raceway is the design contact angle α_n , and the contact angle between the ball at position 2 and the nut raceway is the design contact angle α_s . When the nut is in translation, the ball and the nut raceway are always tangent, so its center track is a circle with the center of point O′_n and the radius r_n -r_b ; the ball and the screw raceway are always tangent, and the center of the ball The trajectory is a circle with the point O_s as the center and the radius r_s -r_b . Before the deformation of the lead screw, the ball contacts with the raceways on both sides simultaneously to achieve force balance, so the position O₀ of the center of the ball is located at the intersection of the two circles, and is collinear with the centers O_s and O′_n of the two circles, that is, the two Tangent point of the circle. According to this geometric constraint relationship, the relationship among radial clearance, radius of curvature, design contact angle and ball can be obtained.

设滚珠中心的位置点O₀的坐标为(x_o，y_o)，点O_0s的坐标为(0，0)，点O_0n的坐标为(0，S_d/2)；Let the coordinates of the ball center point O₀ be (x_o , y_o ), the coordinates of point O_0s be (0, 0), and the coordinates of point O_0n be (0, S_d /2);

设螺母右侧面圆心右半外圆曲率中心为O_n，滚珠中心位于点O_0n处时，滚珠与螺母侧滚道右半外圆接触，直线O_0nO_n与y轴之间夹角为螺母侧设计接触角α_n，设点O_n的坐标为(x_n，y_n)，因为O_nO_0n的距离为滚道半径与滚珠半径之差r_n-r_b，所以Suppose the center of curvature of the right half of the outer circle on the right side of the nut is O_n , when the center of the ball is at point O_0n , the ball is in contact with the right half of the outer circle of the raceway on the nut side, and the angle between the straight line O_0n O_n and the y-axis is The contact angle α_n is designed on the nut side, and the coordinates of the point O_n are (x_n , y_n ), because the distance of O_n O_0n is the difference between the raceway radius and the ball radius r_n -r_b , so

$\{\begin{matrix} {x x}_{n no} = = - - (({r r}_{n no} - - {r r}_{b b})) {sinα sinα}_{n no} \\ {y the y}_{n no} = = {S S}_{d d} / / 22 - - (({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{n no} \end{matrix} - - - - - - ((11))$

螺母沿轴向平动距离为δ_a，发生平动后滚珠螺母侧右半外圆曲率中心为点O_n‘的坐标为(x′_n，y′_n)。The translation distance of the nut along the axial direction is δ_a , and the center of curvature of the right half of the outer circle on the side of the ball nut after translation occurs, and the coordinates of point O_n ' are (x′_n , y′_n ).

$\{\begin{matrix} {x x}_{n no}^{' '} = = - - (({r r}_{n no} - - {r r}_{b b})) {sinα sinα}_{n no} - - {δ δ}_{a a} \\ {y the y}_{n no}^{' '} = = {S S}_{d d} / / 22 - - (({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{n no} \end{matrix} - - - - - - ((22))$

滚珠中心位于点O_s处时，滚珠与丝杠侧滚道右半外圆接触，直线O_0sO_s与y轴之间夹角为螺母侧设计接触角α_s，设点O_s的坐标为(x_s，y_s)，因为O_0sO_s的距离为滚道半径与滚珠半径之差r_s-r_b，所以When the center of the ball is at the point O_s , the ball is in contact with the right half outer circle of the screw side raceway, the angle between the straight line O_0s O_s and the y-axis is the design contact angle α_s on the nut side, and the coordinates of the set point O_s are (x_s , y_s ), because the distance of O_0s O_s is the difference between the raceway radius and the ball radius r_s -r_b , so

$\{\begin{matrix} {x x}_{s the s} = = (({r r}_{s the s} - - {r r}_{b b})) {sinα sinα}_{s the s} \\ {y the y}_{s the s} = = (({r r}_{s the s} - - {r r}_{b b})) {cosα cosα}_{s the s} \end{matrix} - - - - - - ((33))$

滚珠与滚道发生变形前，设滚珠中心在点O₀处达到力平衡，滚珠与两侧滚道同时接触。考虑到滚珠在丝杠滚道内径向均布，忽略重力对滚珠平衡状态的影响。滚珠在两个接触点的作用下达到平衡，滚珠中心点O₀，丝杠侧左半圆曲率中心O_s和螺母侧右半圆曲率中心O′_n共线，也就是说滚珠与两侧滚道之间的接触角相等，即初始接触角α₀。同时滚珠与滚道之间的几何约束关系不变，所以点O₀位于两个圆的相切点处。设点O₀的坐标为(x₀，y₀)，则同时满足以下公式。Before the deformation of the ball and the raceway, assume that the center of the ball reaches a force balance at point O₀ , and the ball contacts the raceways on both sides at the same time. Considering that the balls are evenly distributed radially in the screw raceway, the influence of gravity on the balance state of the balls is ignored. The ball reaches balance under the action of two contact points, the ball center point O₀ , the center of curvature of the left semicircle on the screw side O_s and the center of curvature of the right semicircle O'_n on the nut side are collinear, that is to say, the distance between the ball and the raceways on both sides The contact angle between them is equal, that is, the initial contact angle α₀ . At the same time, the geometric constraint relationship between the ball and the raceway remains unchanged, so the point O₀ is located at the tangent point of the two circles. Assuming that the coordinates of the point O₀ are (x₀ , y₀ ), the following formulas are satisfied at the same time.

(x_o-x′_n)²+(y_o-y′_n)²＝(r_n-r_b)²(4)(x_o -x′_n )² +(y_o -y′_n )² ＝(r_n -r_b )² (4)

(x_o-x_s)²+(y_o-y_s)²＝(r_s-r_b)²(5)(x_o -x_s )² +(y_o -y_s )² ＝(r_s -r_b )² (5)

将公式(4)(5)代入公式(6)可以得出Substituting formula (4)(5) into formula (6) can get

$\frac{| | {y the y}_{o o} - - {y the y}_{s the s} | |}{| | {y the y}_{o o} - - {y the y}_{n no}^{' '} | |} = = \frac{{r r}_{s the s} - - {r r}_{b b}}{{r r}_{n no} - - {r r}_{b b}} - - - - - - ((77))$

${y the y}_{o o} = = \frac{(({r r}_{s the s} - - {r r}_{b b})) (({r r}_{n no} - - {r r}_{b b})) (({cosα cosα}_{s the s} - - {cosα cosα}_{n no})) + + (({r r}_{s the s} - - {r r}_{b b})) {S S}_{d d} / / 22}{(({r r}_{s the s} + + {r r}_{n no} - - 22 {r r}_{b b}))} - - - - - - ((88))$

根据三角投影关系可以得出，初始接触角α₀为According to the triangular projection relationship, it can be concluded that the initial contact angle α₀ is

cosα₀＝(y_s-y′_n)/(r_s+r_n-2r_b)(9)cosα₀ =(y_s -y′_n )/(r_s +r_n -2r_b )(9)

sinα₀＝(x_s-x′_n)/(r_s+r_n-2r_b)(10)sinα₀ =(x_s -x′_n )/(r_s +r_n -2r_b )(10)

将公式(4)(5)代入公式(9)可以得出Substituting formula (4)(5) into formula (9) can get

${cosα cosα}_{00} = = \frac{(({r r}_{s the s} - - {r r}_{b b})) {cosα cosα}_{s the s} + + (({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{n no} - - \frac{{s the s}_{d d}}{22}}{{r r}_{s the s} + + {r r}_{n no} - - 22 {r r}_{b b}} - - - - - - ((1111))$

将公式(2)(3)代入公式(10)可以得出轴向平动距离δ_a的计算公式Substituting formula (2) (3) into formula (10) can get the calculation formula of axial translation distance δ_a

δ_a＝(r_s+r_n-2r_b)sinα₀-(r_s-r_b)sinα_s-(r_n-r_b)sinα_n(12)δ_a ＝(r_s +r_n -2r_b )sinα₀ -(r_s -r_b )sinα_s -(r_n -r_b )sinα_n (12)

二、非等适应比滚珠弹性变形协调模型2. Non-equal adaptation ratio ball elastic deformation coordination model

如图3所示，在轴向预紧载荷的作用下，滚珠几何中心由原来的O₀移动到O₀‘，丝杠侧滚道左半外圆曲率中心为O_s移动到O_s‘，As shown in Figure 3, under the action of the axial preload, the geometric center of the ball moves from the original O₀ to O₀ ', and the center of curvature of the left half of the outer circle of the screw side raceway moves from O_s to O_s ',

$κ κ = = 1.0339 1.0339 {((\frac{{R R}_{y the y}}{{R R}_{x x}}))}^{0.636 0.636}$

$E E. = = 1.0003 1.0003 + + \frac{0.5968 0.5968}{\frac{{R R}_{y the y}}{{R R}_{x x}}}$

$F f = = 1.5277 1.5277 + + 0.6023 0.6023 l l n no ((\frac{{R R}_{y the y}}{{R R}_{x x}}))$

曲率函数 $F (ρ) = \frac{(ρ_{11} - ρ_{12}) + (ρ_{21} - ρ_{22})}{Σ ρ}$ curvature function $f (ρ) = \frac{(ρ_{11} - ρ_{12}) + (ρ_{twenty one} - ρ_{twenty two})}{Σ ρ}$

丝杠侧法向变形的计算Calculation of the lateral normal deformation of the lead screw

曲率半径radius of curvature

${ρ ρ}_{1212 s the s} = = \frac{22}{{d d}_{b b}}$

ρ_11s＝ρ_12sρ_11s = ρ_12s

${ρ ρ}_{21 twenty one s the s} = = - - \frac{11}{0.54 0.54 * * {d d}_{b b}}$

${ρ ρ}_{22 twenty two s the s} = = \frac{22 * * {cosα cosα}_{s the s} * * c c o o s the s λ λ}{d d - - {d d}_{b b} * * {cosα cosα}_{s the s}}$

式中α_s为丝杠侧设计接触角；λ为螺旋升角；d为丝杠公称直径；d_b为滚珠直径In the formula, α_s is the design contact angle of the screw side; λ is the helix angle; d is the nominal diameter of the screw; d_b is the diameter of the ball

其中 $λ = \arctan (\frac{L}{π d})$ in $λ = \arctan (\frac{L}{π d})$

L为丝杠副的导程。L is the lead of the screw pair.

各方向上的等效半径公式Equivalent Radius Formulas in All Directions

${R R}_{x x}^{- - 11} = = {ρ ρ}_{1111 s the s} + + {ρ ρ}_{21 twenty one s the s}$

${R R}_{y the y}^{- - 11} = = {ρ ρ}_{1212 s the s} + + {ρ ρ}_{22 twenty two s the s}$

${a a}_{s the s} = = 0.0236 0.0236 * * {a a}_{s the s}^{* *} {((\frac{Q Q}{Σ Σ ρ ρ}))}^{\frac{11}{33}}$

${b b}_{s the s} = = 0.0236 0.0236 * * {b b}_{s the s}^{* *} {((\frac{Q Q}{Σ Σ ρ ρ}))}^{\frac{11}{33}}$

${δ δ}_{n no s the s} = = 2.79 2.79 * * 1010^{- - 44} {δ δ}_{s the s}^{* *} {Q Q}^{\frac{22}{33}} {((Σ Σ ρ ρ))}^{\frac{11}{33}}$

其中in

${a a}_{s the s}^{* *} = = {((\frac{22 {k k}^{22} E E.}{π π}))}^{\frac{11}{33}}$

${b b}_{s the s}^{* *} = = {((\frac{22 E E.}{π π k k}))}^{\frac{11}{33}}$

${δ δ}_{s the s}^{* *} = = \frac{22 F f}{π π} {((\frac{π π}{22 {k k}^{22} E E.}))}^{\frac{11}{33}}$

对于椭圆接触区域，最大压应力出现在几何中心，其大小为For an elliptical contact region, the maximum compressive stress occurs at the geometric center with a magnitude of

${σ σ}_{s the s m m a a x x} = = \frac{33 Q Q}{22 {πa πa}_{s the s} {b b}_{s the s}}$

螺母侧法向变形的计算Calculation of Nut Side Normal Deformation

曲率半径radius of curvature

${ρ ρ}_{1212 n no} = = \frac{22}{{d d}_{b b}}$

ρ_11n＝ρ_12nρ_11n = ρ_12n

${ρ ρ}_{21 twenty one n no} = = - - \frac{11}{0.54 0.54 * * {d d}_{b b}}$

${ρ ρ}_{22 twenty two n no} = = - - \frac{22 * * {cosα cosα}_{n no} * * c c o o s the s λ λ}{d d + + {d d}_{b b} * * {cosα cosα}_{n no}}$

式中α_n为螺母侧设计接触角；λ为螺旋升角；d为丝杠公称直径；d_b为滚珠直径In the formula, α_n is the design contact angle of the nut side; λ is the helix angle; d is the nominal diameter of the screw; d_b is the diameter of the ball

其中in

$λ λ = = arctan arctan ((\frac{L L}{π π d d}))$

L为丝杠副的导程。L is the lead of the screw pair.

各方向上的等效半径公式Equivalent Radius Formulas in All Directions

${R R}_{x x}^{- - 11} = = {ρ ρ}_{1111 n no} + + {ρ ρ}_{21 twenty one n no}$

${R R}_{y the y}^{- - 11} = = {ρ ρ}_{1212 n no} + + {ρ ρ}_{22 twenty two n no}$

${a a}_{n no} = = 0.0236 0.0236 * * {a a}_{n no}^{* *} {((\frac{Q Q}{Σ Σ ρ ρ}))}^{\frac{11}{33}}$

${b b}_{n no} = = 0.0236 0.0236 * * {b b}_{n no}^{* *} {((\frac{Q Q}{Σ Σ ρ ρ}))}^{\frac{11}{33}}$

${δ δ}_{n no} = = 2.79 2.79 * * 1010^{- - 44} * * {δ δ}_{n no}^{* *} {Q Q}^{\frac{22}{33}} {((Σ Σ ρ ρ))}^{\frac{11}{33}}$

其中in

${a a}_{n no}^{* *} = = {((\frac{22 {k k}^{22} E E.}{π π}))}^{\frac{11}{33}}$

${b b}_{n no}^{* *} = = {((\frac{22 E E.}{π π k k}))}^{\frac{11}{33}}$

${δ δ}_{n no}^{* *} = = \frac{22 F f}{π π} {((\frac{π π}{22 {k k}^{22} E E.}))}^{\frac{11}{33}}$

${σ σ}_{n no max max} = = \frac{33 Q Q}{22 {πa πa}_{s the s} {b b}_{s the s}}$

1、当适应比相等时，与《Rollingbearinganalysis》的计算结果相同。1. When the adaptation ratio is equal, it is the same as the calculation result of "Rollingbearing analysis".

2、若滚道半径R和钢球半径rb的误差分布方向相同时，即R-rb＝定值，则误差相互抵消，对变形接触角没有影响。2. If the error distribution directions of the raceway radius R and the steel ball radius rb are the same, that is, R-rb=fixed value, the errors cancel each other out and have no effect on the deformation contact angle.

3、dR和drb对变形接触角的影响是导程、径向间隙的4倍以上，故滚道半径R和钢球半径的公差要比导程、余隙控制得更严格才好。3. The influence of dR and drb on the deformation contact angle is more than 4 times of the lead and radial clearance, so the tolerance of the raceway radius R and the ball radius should be more strictly controlled than the lead and clearance.

丝杠滚道采用较小适应比、螺母滚道采用较大适应比，两侧的应力值趋于相等，相对于等适应比滚道的滚珠丝杠副，非等适应比滚道的滚珠丝杠副更加耐磨损、寿命更长。The screw raceway adopts a smaller adaptation ratio, the nut raceway adopts a larger adaptation ratio, and the stress values on both sides tend to be equal. The bar pair is more wear-resistant and has a longer life.

参考图4所示，本实施例提供的非等适应比滚道的滚珠丝杠副弹性变形接触角的确定方法，步骤具体为：Referring to Figure 4, the method for determining the elastic deformation contact angle of a ball screw pair with non-equal adaptation ratio raceways provided by this embodiment, the specific steps are as follows:

步骤1：获取初始接触角cosα₀：Step 1: Get the initial contact angle cosα₀ :

使用千分尺或工具显微镜测量滚珠半径r_b、使用千分尺或微分表测量滚珠丝杠副的径向间隙Sd，利用轮廓仪测量丝杠滚道半径r_s，螺母滚道半径r_n、丝杠侧滚道设计接触角α_s和螺母侧滚道设计接触角α_n，则初始接触角α₀为Use a micrometer or a tool microscope to measure the ball radius r_b , use a micrometer or a differential table to measure the radial clearance Sd of the ball screw pair, use a profiler to measure the screw raceway radius r_s , the nut raceway radius r_n , and the screw side roll The design contact angle α_s of the raceway and the design contact angle α_n of the nut side raceway, then the initial contact angle α₀ is

${cosα cosα}_{00} = = \frac{(({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{s the s} + + (({r r}_{n no} - - {r r}_{b b})) {cosα cosα}_{n no} - - \frac{55}{22} {S S}_{d d}}{{r r}_{n no} + + {r r}_{s the s} - - 22 {r r}_{b b}} - - - - - - ((1313))$

步骤2：获得δ_ns和δ_nn：Step 2: Obtain δ_ns and δ_nn :

由赫兹理论可得出，滚珠与滚道之间的法向变形量δ_ns和δ_nn为According to the Hertz theory, the normal deformations δ_ns and δ_nn between the ball and the raceway are

${δ δ}_{n no i i} = = 2.79 2.79 \times \times 1010^{- - 44} {δ δ}_{i i}^{* *} {Q Q}_{i i}^{\frac{22}{33}} {(({Σρ Σρ}_{i i}))}^{\frac{11}{33}} - - - - - - ((1414))$

其中，i＝n表示螺母侧，i＝s表示丝杠侧，Q_i为单个滚珠承受的法向载荷，Σρ_i为滚珠与滚道接触表面的曲率和，Among them, i=n represents the nut side, i=s represents the screw side, Q_i is the normal load borne by a single ball, Σρ_i is the curvature sum of the contact surface between the ball and the raceway,

单个滚珠承受的法向载荷Q_i的计算公式为The calculation formula of the normal load Q_i borne by a single ball is

${Q Q}_{i i} = = \frac{{F f}_{a a}}{z z {cosλsinα cosλsinα}_{00}} - - - - - - ((1515))$

其中F_a为丝杠副所承受的轴向预紧载荷，z为单个螺母内的承载滚珠个数，λ为螺旋升角；Among them, F_a is the axial pretightening load borne by the screw pair, z is the number of loaded balls in a single nut, and λ is the helix angle;

步骤3：得到初始接触角α₀弹性变形接触角α之间的关系式：Step 3: Obtain the relationship between the initial contact angle α₀ and the elastic deformation contact angle α:

(r_n+r_s-2r_b)cosα＝(r_s+r_n-2r_b+δ_ns+δ_nn)cosα₀(16)(r_n +r_s -2r_b )cosα＝(r_s +r_n -2r_b +δ_ns +δ_nn )cosα₀ (16)

步骤4：计算非等适应比滚道的滚珠丝杠副弹性变形接触角：Step 4: Calculate the elastic deformation contact angle of the ball screw pair with non-equal adaptation ratio raceways:

将公式(13)(14)(15)代入接触角公式(16)得Substituting the formula (13)(14)(15) into the contact angle formula (16) to get

$\frac{{F f}_{a a}}{z z c c o o s the s λ λ} = = s the s i i n no α α {((\frac{3.58422 3.58422 \times \times 1010^{33}}{{δ δ}_{s the s}^{* *} {(({Σρ Σρ}_{s the s}))}^{\frac{11}{33}} + + {δ δ}_{n no}^{* *} {(({Σρ Σρ}_{n no}))}^{\frac{11}{33}}} ((\frac{c c o o s the s α α}{{cosα cosα}_{00}} - - 11))))}^{1.5 1.5} - - - - - - ((1717))$

其中in

${R R}_{y the y s the s}^{- - 11} = = \frac{11}{{r r}_{b b}} + + \frac{{cosα cosα}_{s the s} c c o o s the s λ λ}{{r r}_{s the s} - - {r r}_{b b} {cosα cosα}_{s the s}}$

${R R}_{y the y n no}^{- - 11} = = \frac{11}{{r r}_{b b}} + + \frac{{cosα cosα}_{n no} c c o o s the s λ λ}{{r r}_{n no} - - {r r}_{b b} {cosα cosα}_{n no}}$

公式(17)为非线性方程，采用数值求解方法Newton-Raphson求解出弹性变形接触角α。Formula (17) is a nonlinear equation, and the elastic deformation contact angle α is obtained by using the numerical solution method Newton-Raphson.

上述虽然结合附图对本发明的具体实施方式进行了描述，但并非对本发明保护范围的限制，所属领域技术人员应该明白，在本发明的技术方案的基础上，本领域技术人员不需要付出创造性劳动即可做出的各种修改或变形仍在本发明的保护范围以内。Although the specific implementation of the present invention has been described above in conjunction with the accompanying drawings, it does not limit the protection scope of the present invention. Those skilled in the art should understand that on the basis of the technical solution of the present invention, those skilled in the art do not need to pay creative work Various modifications or variations that can be made are still within the protection scope of the present invention.