Method for calculating static stiffness of four-point contact type ball screw pair

Abstract

The invention discloses a method for calculating static stiffness of a four-point contact type ball screw pair, and is applied to the technical field of bearing performance of ball screw pairs. Comprising the following steps: performing contact mechanical analysis on the four-point contact type ball screw pair to obtain contact stress and contact load at a contact part; based on a Hertz contact theory, establishing a ball-raceway contact deformation model of the four-point contact type ball screw pair; a four-point contact type ball screw pair static rigidity model is established in combination with a deformation coordination equation, a static equilibrium equation and the like; and establishing a static stiffness calculation model of the four-point contact type ball screw pair, and performing calculation. According to the method, the static rigidity calculation model of the four-point contact type ball screw pair is obtained through contact characteristic analysis and calculation, the blank in the aspect of rigidity research of the four-point contact type ball screw pair is filled up, and the method has great significance in bearing characteristic analysis of the ball screw pair; and the actual requirement of current high-end numerical control machine tool equipment on the high rigidity of the ball screw pair under the low-speed working condition is met.

Description

Technical Field

[0001] The invention relates to the technical field of bearing performance of a ball screw pair, and more particularly to a method for calculating the static stiffness of a four-point contact ball screw pair. Background Art

[0002] The ball screw pair uses balls as the transmission medium, and has high transmission efficiency, large output torque, and relatively high rigidity. Therefore, it can stably output high-precision power under high-load and high-speed transmission conditions. With the emergence of different working conditions and load requirements, the requirements for the stiffness of the ball screw pair are also constantly increasing. At present, the research on the stiffness of the ball screw pair is mainly aimed at the two-point contact ball screw pair, that is, the research on the single arc ball screw pair, and there is less research on the static stiffness of the four-point contact ball screw pair. At the same time, due to the current situation of technical information blockade and confidentiality, there is no static stiffness calculation method for the four-point contact ball screw pair. Therefore, how to provide a static stiffness calculation method for the four-point contact ball screw pair is a problem that technicians in this field urgently need to solve. Summary of the invention

[0003] In view of this, the present invention provides a method for calculating the static stiffness of a four-point contact ball screw pair. A static stiffness calculation model of a four-point contact ball screw pair is obtained by contact characteristic analysis and calculation, thereby solving the problem that there is no calculation method for the static stiffness of a four-point contact ball screw pair in the prior art.

[0004] In order to achieve the above object, the present invention provides the following technical solutions:

[0005] A method for calculating the static stiffness of a four-point contact ball screw pair comprises the following steps:

[0006] S1. Conduct contact mechanics analysis on the four-point contact ball screw pair to obtain the contact stress and contact load at the contact point;

[0007] S2. Based on Hertz contact theory, a four-point contact ball screw pair ball-raceway contact deformation model is established;

[0008] S3. Combine the deformation coordination equation and the static equilibrium equation to establish a static stiffness model of a four-point contact ball screw pair;

[0009] S4. Establish a static stiffness calculation model for a four-point contact ball screw pair.

[0010] Optionally, S1 is specifically as follows: a single four-point contact ball, a ball screw pair and a nut are selected as research objects, the contact point between the four-point contact ball and the raceway is regarded as a contact ellipse contact of two contact bodies, the contact stress at the contact point is calculated according to the contact ellipse distribution, the contact load of the AC contact pair is Q, and the contact load of the BD contact pair is Q′.

[0011] Optionally, the contact load is specified as:

[0012] Q=K a ·δ 3 / 2

[0013] Q′=K a δ′ 3 / 2

[0014] In the formula, K a is the axial contact deformation coefficient of the single nut ball screw pair, δ is the contact deformation of the AC contact pair, and δ′ is the contact deformation of the BD contact pair.

[0015] Optionally, the contact deformation model of the ball-raceway of the four-point contact ball screw pair is as follows:

[0016]

[0017]

[0018] Where ce is the material coefficient, ρ s and ρ n are the inverse of the curvature radius at the contact point of the ball-screw raceway and the ball-nut raceway, respectively, s and Y n are the auxiliary values ​​of the first and second elliptic integrals of the screw and nut in Hertz contact theory, d b is the ball diameter, d0 is the pitch circle diameter of the ball center, t b is the raceway curvature ratio, f rs is the ratio of the screw raceway to the ball diameter, f rn is the ratio of the nut raceway to the ball diameter, α0 is the initial contact angle, α1 is the contact angle of the AC contact pair after the screw is deformed by force, α2 is the contact angle of the BD contact pair after the screw is deformed by force, λ is the helix angle, sin τs 、sin τn It can be obtained by the ratio of the long and short radii of the ball-raceway contact ellipse.

[0019] Optionally, the deformation of the combined deformation coordination equation in S3 is specifically:

[0020] δ=O sl O nr′ -O sl O nr =[(O sr O nr ·sinα0+δ α ) 2 +(O sr O nr ·cosα0+δi ) 2 ] 1 / 2 -O s 1O nl

[0021] δ′=O sr O nl′ -O sr O nl =[(O sr O nl ·sinα0-δ α ) 2 +(O sr O nl ·cosα0+δ i ) 2 ] 1 / 2 -O sr O nl

[0022] O sl O nr =O sr O nl =r s +r n -d b

[0023]

[0024] In the formula, O sl and O sr The center of curvature of the left and right raceways of the screw before bearing the axial load, O sl′ and O sr′ The center of curvature of the left and right raceways of the screw after bearing the axial load, O nr and O nl The center of curvature of the left and right raceways of the nut before the axial load is applied, O nr′ and O nl′ is the center of curvature of the left and right raceways of the nut after bearing the axial load, δ α is the axial displacement of the screw after the axial force is applied, δ i is the radial displacement of the screw after the axial force is applied, r s is the curvature radius of the screw channel, r n is the radius of curvature of the nut groove; when the ball screw is subjected to axial force, the contact deformation of the AC contact pair increases, the contact load Q also increases, the contact deformation of the BD contact pair decreases, the contact load Q′ also decreases, the contact deformation of the AC contact pair is δ, and the contact deformation of the BD contact pair is δ′.

[0025] Optionally, S3 is specifically as follows: combining the physical equation and the static equilibrium equation, a static stiffness model of a four-point contact ball screw pair is established. The ball screw pair is subjected to an external axial force F a The static balance of the applied axial load is considered to be uniformly distributed. The force on each ball is F. The axial component of the contact load applied by the steel ball to the screw raceway and the nut raceway is balanced with the axial force F. Specifically, the axial static balance equation can be established as follows:

[0026]

[0027] F=(Q·sinα1-Q′·sinα2)

[0028] Where z is the number of balls in the ball screw pair, F a is the external axial force of the screw, and F is the force on a single ball.

[0029] Optionally, S4 is specifically: obtaining the axial displacement of the screw relative to the ball under the action of the axial force F, and then obtaining the axial static stiffness K of the four-point contact ball screw pair:

[0030] K=dF / dδ a .

[0031] It can be seen from the above technical solution that, compared with the prior art, the present invention provides a method for calculating the static stiffness of a four-point contact ball screw pair, which has the following beneficial effects:

[0032] 1. The present invention combines Hertz contact theory and analyzes the contact of a four-point contact ball screw pair according to the principle of force deformation, thus filling the gap in related theories;

[0033] 2. The present invention constructs a macroscopic deformation model of a four-point contact ball screw pair based on the Hertz contact theory, which facilitates the subsequent study of the load-bearing characteristics and dynamic characteristics of the four-point contact ball screw pair;

[0034] 3. The present invention proposes a static stiffness calculation model for a four-point contact ball screw pair, which is aimed at the current working condition requirements of a low-speed and high-rigidity ball screw pair. The ball contacts the raceway at four points, which greatly improves the stiffness of the ball screw pair. At the same time, it can more accurately predict the degree of improvement in the bearing capacity of the ball screw pair.

[0035] 4. The present invention proposes a static stiffness model of a four-point contact ball screw pair, which is different from the current common theoretical research on two-point contact ball screw pairs, greatly improves the load-bearing characteristics of the ball screw pair, and fills the theoretical and experimental research on four-point contact ball screw pairs. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying creative work.

[0037] Figure 1 The present invention provides a method flow for calculating and testing the static stiffness of a ball screw pair;

[0038] Figure 2 is a relationship diagram between the axial deformation of the ball of the ball screw pair and the normal deformation of the contact point in an embodiment of the present invention;

[0039] Figure 3 It is a comparison diagram of the static stiffness theoretical model and the test curve of the ball screw pair in the embodiment of the present invention. DETAILED DESCRIPTION

[0040] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0041] The embodiment of the present invention discloses a method for calculating the static stiffness of a four-point contact ball screw pair. Figure 1 As shown, the following steps are included:

[0042] S1. Conduct contact mechanics analysis on the four-point contact ball screw pair to obtain the contact stress and contact load at the contact point;

[0043] S2. Based on Hertz contact theory, a four-point contact ball screw pair ball-raceway contact deformation model is established;

[0044] S3. Combine the deformation coordination equation and the static equilibrium equation to establish a static stiffness model of a four-point contact ball screw pair;

[0045] S4. Establish a static stiffness calculation model for a four-point contact ball screw pair.

[0046] Furthermore, S1 is specifically as follows: a single four-point contact ball, a ball screw pair and a nut are selected as the research objects, the contact point between the four-point contact ball and the raceway is regarded as the contact ellipse contact of two contact bodies, and the contact stress at the contact point is calculated according to the contact ellipse distribution, such as Figure 2 As shown, the contact load of the AC contact pair is Q, and the contact load of the BD contact pair is Q′.

[0047] Furthermore, the contact load is specifically:

[0048] Q=Ka · δ 3 / 2

[0049] Q′=K a ·δ′ 3 / 2

[0050] In the formula, K a is the axial contact deformation coefficient of the single-nut ball screw pair, which is only related to the structural dimensions and material parameters of the ball screw pair and has nothing to do with the axial load. δ is the contact deformation of the AC contact pair, and δ′ is the contact deformation of the BD contact pair.

[0051] Furthermore, the contact deformation model of the ball-raceway of the four-point contact ball screw pair is as follows:

[0052]

[0053] Where ce is the material coefficient, ρ s and ρ n are the inverse of the curvature radius at the contact point of the ball-screw raceway and the ball-nut raceway, respectively, s and Y n are the auxiliary values ​​of the first and second elliptic integrals of the screw and nut in Hertz contact theory, d b is the ball diameter, d0 is the pitch circle diameter of the ball center, t b is the raceway curvature ratio, f rs is the ratio of the screw raceway to the ball diameter, f rn is the ratio of the nut raceway to the ball diameter, α0 is the initial contact angle, α1 is the contact angle of the AC contact pair after the screw is deformed by force, α2 is the contact angle of the BD contact pair after the screw is deformed by force, λ is the helix angle, sin τs 、sin τn It can be obtained by the ratio of the long and short radii of the ball-raceway contact ellipse.

[0054] Furthermore, the deformation of the combined deformation coordination equation in S3 is specifically:

[0055] δ=O sl O nr′ -O sl O nr =[(O sl O nr ·sinα0+δ α ) 2 +(O sl O nr ·cosα0+δ i ) 2] 1 / 2 -O sl O nr

[0056] δ′=O sr O nr′ -O sr O nl =[(O sr O nl ·csinα0-δ α ) 2 +(O sr O nl ·cosα0+δ i ) 2 ] 1 / 2 -O sr O nl

[0057] O sl O nr =O sr O nl =r s +r n -d b

[0058]

[0059] In the formula, O sl and O sr The center of curvature of the left and right raceways of the screw before bearing the axial load, O sl′ and O sr′ The center of curvature of the left and right raceways of the screw after bearing the axial load, O nr and O nl The center of curvature of the left and right raceways of the nut before the axial load is applied, O nr′ and O nl′ is the center of curvature of the left and right raceways of the nut after bearing the axial load, δ α is the axial displacement of the screw after the axial force is applied, δ i is the radial displacement of the screw after the axial force is applied, r s is the curvature radius of the screw channel, r n is the radius of curvature of the nut groove; when the ball screw is subjected to axial force, the contact deformation of the AC contact pair increases, the contact load Q also increases, the contact deformation of the BD contact pair decreases, the contact load Q′ also decreases, the contact deformation of the AC contact pair is δ, and the contact deformation of the BD contact pair is δ′.

[0060] Furthermore, S3 is specifically as follows: combining the physical equation and the static equilibrium equation, a four-point contact ball screw pair static stiffness model is established, and the static equilibrium of the ball screw pair under the action of an external axial force is considered. Considering that the external axial load is a uniform load distribution, the force on each ball is F, and the axial component of the contact load applied by the steel ball to the screw raceway and the nut raceway is balanced with the axial force F. Specifically, the axial static equilibrium equation can be established as follows:

[0061]

[0062] F=(Q·sinα1-Q′sinα2)

[0063] Where z is the number of balls in the ball screw pair, F a is the external axial force of the screw, and F is the force on a single ball.

[0064] Further, S4 is specifically as follows: the axial displacement of the screw relative to the ball under the action of the axial force F is obtained, and then the axial static stiffness K of the four-point contact ball screw pair can be obtained:

[0065] K=dF / dδ a .

[0066] Furthermore, in an embodiment of the present invention, when conducting a static stiffness test of a ball screw pair, all components are correctly installed on the table, and a suitable load is selected as the maximum loading load of the static stiffness test process. In this test, 30% of the dynamic load of the ball screw pair is selected, and a suitable load is selected as the preload for the static stiffness test. In this test, 1KN is selected as the preload, and the sample is preloaded at least 3 times to eliminate the gaps between the components; after preloading, it is unloaded, and the sensor indication is cleared to zero; a continuous loading method is used to conduct a static stiffness test of the ball screw pair; during the test, loading is performed uniformly from zero at a certain speed until the preset maximum load value, and the measurement values ​​of the loading force sensor and the four displacement sensors are automatically and synchronously collected by the measurement and control system during the loading process; when loaded to the preset maximum load, unloading begins, and the above method is also used to collect data during unloading, and the theoretical model is compared with the experimental curve. Figure 3 shown.

[0067] In the embodiment of the present invention, the preload is 1 KN, the preset maximum load is 11 KN, the rated dynamic load is 36.7 KN, the rated static load is 91 KN, the loaded static stiffness is 332.7992 N / μm, and the unloaded static stiffness is 353.7065 N / μm.

[0068] The parameters of the lead screw are shown in Table 1:

[0069] Table 1

[0070] Screw Model hj-4010-yu Experiment time 2024-8-2 Lead(mm) 10 Manufacturer Hanjiang Machine Tool Nominal diameter(mm) 40 Nut diameter(mm) 82 Accuracy level 3 Flange thickness(mm) 18

[0071] The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the various embodiments can be referenced to each other.

[0072] The above description of the disclosed embodiments enables one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for calculating the static stiffness of a four-point contact ball screw pair, characterized in that: The following steps are involved: S1. Conduct contact mechanics analysis on the four-point contact ball screw pair to obtain the contact stress and contact load at the contact point; S2. Based on Hertz contact theory, a four-point contact ball screw pair ball-raceway contact deformation model is established; S3. Combine the deformation coordination equation and the static equilibrium equation to establish a static stiffness model of a four-point contact ball screw pair; S4. Establish a static stiffness calculation model for a four-point contact ball screw pair.

2. A method for calculating static stiffness of a four-point contact ball screw pair according to claim 1, characterized in that: S1 is specifically as follows: a single four-point contact ball, a ball screw pair and a nut are selected as the research objects, and the contact point between the four-point contact ball and the raceway is regarded as a contact ellipse contact of two contact bodies. The contact stress at the contact point is calculated according to the contact ellipse distribution. The contact load of the AC contact pair is Q, and the contact load of the BD contact pair is Q′.

3. A method for calculating static stiffness of a four-point contact ball screw pair according to claim 2, characterized in that: The contact load is: Q=K a ·δ 3 / 2 Q′=K a ·δ′ 3 / 2 In the formula, K a is the axial contact deformation coefficient of the single nut ball screw pair, δ is the contact deformation of the AC contact pair, and δ′ is the contact deformation of the BD contact pair.

4. A method for calculating static stiffness of a four-point contact ball screw pair according to claim 3, characterized in that: The contact deformation model of the ball-raceway of the four-point contact ball screw pair is as follows: Where ce is the material coefficient, ρ s and ρ n are the inverse of the curvature radius at the contact point of the ball-screw raceway and the ball-nut raceway, respectively, s and Y n are the auxiliary values ​​of the first and second elliptic integrals of the screw and nut in Hertz contact theory, d b is the ball diameter, d0 is the pitch circle diameter of the ball center, t b is the raceway curvature ratio, f rs is the ratio of the screw raceway to the ball diameter, f rn is the ratio of the nut raceway to the ball diameter, α0 is the initial contact angle, α1 is the contact angle of the AC contact pair after the screw is deformed by force, α2 is the contact angle of the BD contact pair after the screw is deformed by force, λ is the helix angle, sin τs 、sin τn Calculated by the ratio of the long and short radii of the ball-raceway contact ellipse.

5. A method for calculating static stiffness of a four-point contact ball screw pair according to claim 4, characterized in that: The deformation of the combined deformation coordination equation in S3 is specifically: δ=O sl THE nr′ -THE sl THE nr =[(O sr THE nl ·sinα0+δ α ) 2 +(O sl THE nr ·cosα0+δ i ) 2 ] 1 / 2 -THE sl THE nr δ′=O sr THE nl′ -THE sr THE nl =[(O sr THE nl ·sinα0-δ α ) 2 +(O sr THE nl ·cosα0+δ i ) 2 ] 1 / 2 -THE sr THE nl O sl O nr =O sr O nl =r s +r n -d b In the formula, O sl and O sr The center of curvature of the left and right raceways of the screw before bearing the axial load, O sl′ and O sr′ The center of curvature of the left and right raceways of the screw after bearing the axial load, O nr and O nl The center of curvature of the left and right raceways of the nut before the axial load is applied, O nr′ and O nl′ is the center of curvature of the left and right raceways of the nut after bearing the axial load, δ α is the axial displacement of the screw after the axial force is applied, δ i is the radial displacement of the screw after the axial force is applied, r s is the curvature radius of the screw channel, r n is the radius of curvature of the nut groove; when the ball screw is subjected to axial force, the contact deformation of the AC contact pair increases, the contact load Q also increases, the contact deformation of the BD contact pair decreases, the contact load Q′ also decreases, the contact deformation of the AC contact pair is δ, and the contact deformation of the BD contact pair is δ′.

6. A method for calculating static stiffness of a four-point contact ball screw pair according to claim 5, characterized in that: S3 is as follows: Combine the physical equation and the static equilibrium equation to establish a static stiffness model of the four-point contact ball screw pair. The ball screw pair is subjected to an external axial force F a The static balance under the action of the external axial load is considered to be a uniform load distribution. The force on each ball is F. The axial component of the contact load applied by the steel ball to the screw raceway and the nut raceway is balanced with the axial force F. Specifically, the axial static balance equation can be established as follows: F=(Q·sinα1-Q′sinα2) Where z is the number of balls in the ball screw pair, F a is the external axial force of the screw, and F is the force on a single ball.

7. A method for calculating static stiffness of a four-point contact ball screw pair according to claim 6, characterized in that: S4 is specifically: obtaining the axial displacement of the screw relative to the ball under the action of the axial force F and then obtaining the axial static stiffness K of the four-point contact ball screw pair: K=dF / dδ a 。

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