Improved calculation method of contact deformation of ball screw pair based on fractal theory

Abstract

The application discloses a kind of based on fractal theory's ball screw pair contact deformation improved calculation method, applied to ball screw pair bearing performance technical field, it includes: based on fractal theory, micro contact mechanics theory and the cosine function to micro asperity elastic-plastic stage research three common constructions micro asperity contact model;Based on Hertz contact theory, hornton elastic-plastic contact theory and yield characteristic Mises criterion, construct macro level ball screw pair deformation model;Optimize macro level ball screw pair deformation model in combination with effective ball number;Establish ball screw pair contact deformation overall calculation model.The application improves the accuracy of deformation amount calculation in the deformation process of ball screw pair, better guide the process and design application of relevant functional components.

Description

Technical Field

[0001] This invention relates to the field of ball screw pair load-bearing performance technology, and more specifically to an improved calculation method for contact deformation of ball screw pairs based on fractal theory. Background Technology

[0002] Current research on the load-bearing characteristics of ball screw pairs mainly focuses on 30% of static loads. However, in practical applications, static stiffness is rarely used; dynamic loads are primarily employed. In studying the deformation mechanism of ball screw pairs, there is a significant discrepancy between the theoretically calculated load and deformation in traditional contact models and the actual load and deformation during operation. To more accurately calculate the contact deformation mechanism of ball screw pairs, an improved calculation method based on fractal theory is proposed, which plays a crucial role in studying the load-bearing characteristics of ball screw pairs.

[0003] Currently, ball screw pairs only exhibit micro-protrusion deformation mechanisms and macroscopic deformation mechanisms during load-bearing deformation processes. There is limited research on the combination of these two mechanisms, with related directions mainly focusing on the calculation of individual deformation amounts. There is very little research on the overall deformation mechanism at both the microscopic and macroscopic levels.

[0004] Therefore, an improved calculation method for contact deformation of ball screw pairs based on fractal theory is proposed to solve the difficulties existing in the prior art, which is an urgent problem to be solved by those skilled in the art. Summary of the Invention

[0005] In view of this, the present invention provides an improved calculation method for contact deformation of ball screw pairs based on fractal theory, which improves the accuracy of deformation calculation during the deformation process of ball screw pairs and better guides the process and design application of related functional components.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] An improved calculation method for contact deformation of ball screw pairs based on fractal theory includes the following steps:

[0008] S1. Based on fractal theory, micro-contact mechanics theory, and the study of the elastic-plastic stage of micro-convex bodies using cosine functions, a micro-convex body contact model was jointly constructed.

[0009] S2. Based on Hertz contact theory, Hornton elastoplastic contact theory and Mises criterion for yield characteristics, a macroscopic deformation model of the ball screw pair is constructed.

[0010] S3. Optimize the deformation model of the ball screw pair at the macro level by combining the effective number of balls;

[0011] S4. Establish an overall calculation model for the contact deformation of the ball screw pair.

[0012] Optionally, in S1, the surface morphology parameters of the micro-convex body are characterized according to fractal theory, specifically including:

[0013] The WM function is used to characterize the microstructure of rough surfaces, as shown below:

[0014]

[0015] Where z(x) is the height of the two-dimensional curved profile; G is the characteristic height scale of the rough surface; D is the fractal dimension of the two-dimensional surface profile; n is the frequency exponent of the micro-protrusions on the contact surface; and γ n For the roughness spectrum of a rough surface, n m The lowest frequency exponent of all collected values, the lowest cutoff frequency of the surface profile is expressed as: L represents the length of the collected sample.

[0016] Optionally, S1 also includes fractal parameter extraction using structural parameters, as shown below:

[0017]

[0018] lg S(σ)=(4-2D)lgσ+lgB+2(D-1)lg G

[0019]

[0020] Where σ=n0△L, B=Γ(2D-3)sin[(D-1.5)π] / (4-2D)lnγ, n0 represents the count of sampling points, and △L represents the sampling interval.

[0021] Optionally, in S2, a macroscopic deformation model of the ball screw pair is constructed based on Hertz contact theory, Hornton elastoplastic contact theory, and the Mises criterion for yield characteristics, as follows:

[0022] Relationship between normal load and deformation in the elastic stage:

[0023]

[0024] Where K(e) is the first-type elliptic integral; m a m b ∑ρ is the coefficient related to the eccentricity of the ellipse; ∑ρ is the curvature of the contact surface.

[0025] Optionally, S2 also includes the relationship between yield stress and maximum contact stress as shown in the Mises criterion:

[0026]

[0027] Where: k is the reliability coefficient of the material properties;

[0028] The relationship between deformation during the loading process is as follows:

[0029]

[0030] Optionally, S2 also includes consideration of the impact of manufacturing errors in the ball screw pair on the number of loads it can bear:

[0031]

[0032] Where: f p Where F0 is the accuracy tolerance factor, and R is the axial force. n / s C represents the axial static stiffness of the ball nut body and the ball screw. E The typical value is 0.4643. Y s Y n These are the auxiliary values ​​for the first and second types of integrals for the nut and the lead screw, respectively.

[0033] Optionally, in S3, the deformation model of the ball screw pair at the macroscopic level is optimized by combining the effective number of balls, as shown below:

[0034] The axial static load of the ball screw pair can be obtained according to ISO3408-5-2006:

[0035]

[0036] Where: i is the number of ball bearing revolutions, C s C is the rated dynamic load per turn of the lead screw. n This is the rated dynamic load per turn of the nut.

[0037] Optionally, the overall calculation model for the contact deformation of the ball screw pair in S4 is as follows:

[0038] When δ1≤R:

[0039] δ1=a l / 2πR

[0040]

[0041] When the micro-protrusion is subjected to a load Q, the macroscopic surface also experiences the same axial force:

[0042]

[0043] δ=δ1+δ2

[0044] When δ1+δ2<δ≤δ q hour:

[0045]

[0046] When δ > δ q hour:

[0047]

[0048] As can be seen from the above technical solution, compared with the prior art, the present invention discloses an improved calculation method for contact deformation of ball screw pairs based on fractal theory, which has the following beneficial effects:

[0049] 1) This paper uses fractal theory and micro-contact mechanics theory to study the elastic-plastic deformation stage based on cosine function according to the shape of micro-convex body, in addition to the traditional elastic and plastic deformation stages, and constructs a complete micro-level ball screw pair contact deformation model.

[0050] 2) Based on Hertz contact theory, Hornton elastoplastic contact theory and the Mises criterion for yield characteristics, this paper constructs a macroscopic deformation model of the ball screw pair;

[0051] 3) The contact deformation of the ball screw pair is optimized based on the effective number of balls and the deformation stage of the micro-protrusion, so as to more accurately predict the load-bearing capacity of the ball screw pair. Attached Figure Description

[0052] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0053] Figure 1 A flowchart illustrating an improved calculation method for contact deformation of ball screw pairs based on fractal theory, provided by this invention;

[0054] Figure 2 This is a microscopic morphology diagram of the ball screw assembly of the present invention;

[0055] Figure 3 This is a diagram showing the solution of the fractal parameters of this invention;

[0056] Figure 4 This is a diagram showing the overall deformation of the ball screw assembly of the present invention. Detailed Implementation

[0057] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0058] Reference Figure 1 As shown, this invention discloses an improved calculation method for contact deformation of ball screw pairs based on fractal theory, including the following steps:

[0059] S1. Based on fractal theory, micro-contact mechanics theory, and the study of the elastic-plastic stage of micro-convex bodies using cosine functions, a micro-convex body contact model was jointly constructed.

[0060] S2. Based on Hertz contact theory, Hornton elastoplastic contact theory and Mises criterion for yield characteristics, a macroscopic deformation model of the ball screw pair is constructed.

[0061] S3. Optimize the deformation model of the ball screw pair at the macro level by combining the effective number of balls;

[0062] S4. Establish an overall calculation model for the contact deformation of the ball screw pair.

[0063] Furthermore, in S1, the surface morphology parameters of the micro-convex body are characterized according to fractal theory, specifically including:

[0064] like Figure 2 As shown, the WM function is used to characterize the microstructure of a rough surface, expressed as follows:

[0065]

[0066] Where z(x) is the height of the two-dimensional curved profile; G is the characteristic height scale of the rough surface, and the larger G is, the larger the surface scale; D is the fractal dimension of the two-dimensional surface profile, which is affected by the roughness of the contact surface, and the larger D is, the rougher the surface; n is the frequency exponent of the micro-protrusions on the contact surface, γ n For the roughness spectrum of a rough surface, γ = 1.5 when the rough profile follows a normal distribution. m Given the lowest frequency exponent of all collected values, the lowest cutoff frequency of the surface profile can be expressed as: L represents the sample length, and all collected values ​​are affected by the resolution of the measuring device.

[0067] Furthermore, S1 also includes fractal parameter extraction using structural parameters, such as... Figure 3 As shown, it represents the following:

[0068]

[0069] lg S(σ)=(4-2D)lgσ+lgB+2(D-1)lg G

[0070]

[0071] Where σ=n0△L, B=Γ(2D-3)sin[(D-1.5)π] / (4-2D)lnγ, n0 represents the count of sampling points, and △L represents the sampling interval.

[0072] Specifically, step 1, the three-stage model of micro-convexity contact deformation, includes:

[0073] when a<a ec At this time, the micro-protrusion is in an elastic deformation state, and the contact load of a single micro-protrusion is:

[0074]

[0075] Where: K is the maximum contact factor, a Poisson's ratio function of the two contact surfaces, K = 0.545 + 0.41v; H is the contact hardness of the friction interface; θ is a material property, related to the contact materials themselves, θ = H / E; and E is the comprehensive elastic modulus. E1 and E2 are the elastic moduli, and v1 and v2 are the Poisson's ratios of the two contact materials, respectively.

[0076] when a ec <a<a pc At this time, the micro-protrusion is in an elastoplastic deformation state, and the contact load of a single micro-protrusion is:

[0077]

[0078] in: C2 = F e (a ec )-F p (a pc ).

[0079] When a>a pc At this time, the micro-protrusion is in a state of plastic deformation, and the contact load of a single micro-protrusion is:

[0080]

[0081] Furthermore, in step 1, based on fractal theory, microscopic contact mechanics theory, and the study of the elastoplastic stage of micro-convex bodies using cosine functions, a contact model for micro-convex bodies was constructed, which is expressed as follows:

[0082]

[0083] In the formula, F rTo withstand axial loads, F e F ep F p These represent elasticity, elastoplasticity, and plasticity, respectively; D is the fractal dimension; G is the fractal roughness; H is the surface hardness; and E is the elastic modulus. Indicates the number of micro-convexities, a e a ep a p These represent the contact areas for elasticity, elastoplasticity, and plasticity, respectively.

[0084] Furthermore, based on Hertz contact theory, Hornton elastoplastic contact theory, and the Mises criterion for yield characteristics, a macroscopic deformation model of the ball screw pair is constructed in S2, as follows:

[0085] Relationship between normal load and deformation in the elastic stage:

[0086]

[0087] Where K(e) is the first-type elliptic integral; m a m b ∑ρ is the coefficient related to the eccentricity of the ellipse; ∑ρ is the curvature of the contact surface.

[0088] Furthermore, S2 also includes the relationship between yield stress and maximum contact stress as shown in the Mises criterion:

[0089]

[0090] Where: k is the reliability coefficient of the material properties;

[0091] The relationship between deformation during the loading process is as follows:

[0092]

[0093] Furthermore, S2 also includes consideration of the impact of manufacturing errors in the ball screw pair on the number of loads it can bear:

[0094]

[0095] Where: f p Where F0 is the accuracy tolerance factor, and R is the axial force. n / s C represents the axial static stiffness of the ball nut body and the ball screw. E The typical value is 0.4643. Y s Y n These are the auxiliary values ​​for the first and second types of integrals for the nut and the lead screw, respectively.

[0096] Specifically, the axial force borne by each ball:

[0097]

[0098] Where: a is the contact angle, and γ is the lead angle.

[0099] Furthermore, in S3, the deformation model of the ball screw pair at the macroscopic level is optimized by combining the effective number of balls, as shown below:

[0100] The axial static load of the ball screw pair can be obtained according to ISO3408-5-2006:

[0101]

[0102] Where: i is the number of ball bearing revolutions, C s C is the rated dynamic load per turn of the lead screw. n This is the rated dynamic load per turn of the nut.

[0103] Specifically, consider the impact of manufacturing errors in the ball screw assembly on the number of loads it can withstand:

[0104]

[0105] Where: f p F is the accuracy tolerance factor. a To withstand axial force, F a According to the specifications in JBT13815-2020 for ball screw pairs, the dynamic load C is... a 30%. R n / s C represents the axial static stiffness of the ball nut body and the ball screw. E The typical value is 0.4643. Y s Y n These are the auxiliary values ​​for the first and second types of integrals for the nut and the lead screw, respectively.

[0106] The axial force borne by each ball:

[0107]

[0108] Where: a is the contact angle, and γ is the lead angle.

[0109] Furthermore, the overall calculation model for the contact deformation of the ball screw pair in S4 is as follows:

[0110] When δ1≤R:

[0111]

[0112] When the micro-protrusion is subjected to a load Q, the macroscopic surface also experiences the same axial force:

[0113]

[0114] δ=δ1+δ2

[0115] When δ1+δ2<δ≤δ q hour:

[0116]

[0117] When δ > δ q hour:

[0118]

[0119] Specifically, the plastic deformation of a ball screw pair from the initial contact stage is due to the combined effects of micro-protrusion deformation and macroscopic deformation. When the load begins to be applied, the deformation generated by micro-protrusion contact and the deformation generated by Hertz contact together contribute to the screw's load-bearing deformation. As the load increases, the micro-protrusion reaches its maximum deformation and ceases to deform; the deformation then becomes solely that generated by Hertz contact theory. After reaching the elastic limit, the model transitions to elastoplastic deformation.

[0120] like Figure 4 As shown in the figure, as a specific example, this invention conducts rigidity tests on a 2010 model ball screw from a domestic manufacturer to accurately determine the relationship between load-bearing capacity and deformation, verifying the overall deformation model of the ball screw pair at both the micro and macro levels. The traditional contact model is represented by the blue line. Compared to experimental data, the traditional contact model differs significantly, with an error of 71.87%. This is due to manufacturing errors in the ball screw pair, such as inconsistent ball size, lead error, and pitch diameter error, leading to a reduction in the effective load-bearing balls. Therefore, effective ball optimization calculations are performed, as shown by the green line in the figure. Compared to the actual model, the error is 11.16%, a 60% reduction compared to the traditional contact model. Furthermore, considering the presence of micro-protrusions on the raceway surface, the concept of incorporating micro-protrusions is represented by the red line segment as the final model. This model has an error of only 4.08% compared to the actual model, improving the accuracy of the contact deformation mechanism of the ball screw pair.

[0121] This invention applies fractal theory and cosine function to ball screws for microscopic research, and Hertz contact theory, Hornton elastoplastic contact theory and Mises criterion for yield characteristics for macroscopic research. It also considers the effective number of balls and the deformation mechanism of micro-protrusions to study the screw, establishes a complete deformation model, and accurately predicts the deformation mechanism of the ball screw pair under load.

[0122] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0123] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those 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 invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An improved calculation method for contact deformation of ball screw pairs based on fractal theory, characterized in that, Includes the following steps: S1. Based on fractal theory, micro-contact mechanics theory, and the study of the elastic-plastic stage of micro-convex bodies using cosine functions, a micro-convex body contact model was jointly constructed. S2. Based on Hertz contact theory, Hornton elastoplastic contact theory and Mises criterion for yield characteristics, a macroscopic deformation model of the ball screw pair is constructed. S3. Optimize the deformation model of the ball screw pair at the macro level by combining the effective number of balls; S4. Establish an overall calculation model for the contact deformation of the ball screw pair; The overall calculation model for contact deformation of the ball screw pair in S4 is as follows: when hour: When the micro-protrusion is subjected to a load Q, the macroscopic surface also experiences the same axial force: when hour: when hour: In the formula, For a first-kind complete elliptic integral, E is the elastic modulus; The coefficient is related to the eccentricity of the ellipse; Let the curvature of the contact surface be , D For fractal dimension, G For fractal roughness, H Surface hardness, Indicates the number of micro-protrusions. These represent the contact areas of elasticity, elastoplasticity, and plasticity, respectively. It is an elastic-plastic bearing capacity.

2. The improved calculation method for contact deformation of ball screw pairs based on fractal theory according to claim 1, characterized in that, In S1, the surface morphology parameters of the micro-convex bodies are characterized according to fractal theory, specifically including: The WM function is used to characterize the microstructure of rough surfaces, as shown below: in, The height of the two-dimensional curved section; The characteristic height scale of the rough surface. The fractal dimension of the two-dimensional surface profile. n The frequency index of the micro-protrusions on the contact surface. This represents the surface roughness spectrum. n m The lowest frequency exponent of all collected values, the lowest cutoff frequency of the surface profile is expressed as: , L The length of the sample collected.

3. The improved calculation method for contact deformation of ball screw pairs based on fractal theory according to claim 2, characterized in that, S1 also includes fractal parameter extraction using structural parameters, as shown below: in, , , This represents the count of sampling points. Indicates the sampling interval.

4. The improved calculation method for contact deformation of ball screw pairs based on fractal theory according to claim 1, characterized in that, Based on Hertz contact theory, Hornton elastoplastic contact theory, and the Mises criterion for yield characteristics, a macroscopic deformation model of the ball screw pair is constructed in S2, as follows: Relationship between normal load and deformation in the elastic stage: in, For a first-kind complete elliptic integral, E is the elastic modulus; The coefficient is related to the eccentricity of the ellipse; Let be the curvature of the contact surface.

5. The improved calculation method for contact deformation of ball screw pairs based on fractal theory according to claim 4, characterized in that, S2 also includes the relationship between yield stress and maximum contact stress as shown in the Mises criterion: in: The reliability coefficient of the material properties; The relationship between deformation during the loading process is as follows: 。 6. The improved calculation method for contact deformation of ball screw pairs based on fractal theory according to claim 4, characterized in that, S2 also includes consideration of the impact of manufacturing errors in the ball screw pair on the number of loads it can bear: in: For accuracy tolerance coefficient, To withstand axial force, For the axial static stiffness of the ball nut body and the ball screw, The value is 0.4643. , , These are the auxiliary values ​​for the first and second types of integrals, respectively, for the nut and the lead screw. Contact angle, The lead angle.

7. The improved calculation method for contact deformation of ball screw pairs based on fractal theory according to claim 1, characterized in that, In S3, the deformation model of the ball screw pair at the macro level is optimized by combining the effective number of balls, as shown below: The axial static load of the ball screw pair can be obtained according to ISO3408-5-2006: in: To accommodate the number of ball bearing revolutions, This is the rated dynamic load for a single turn of the leadscrew. This is the rated dynamic load per turn of the nut.

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