Reliability calculation method for gear racks considering block wear and material uncertainty

Abstract

The present invention discloses a gear rack reliability calculation method that considers the wear and material uncertainty of the pressing block. The method considers the wear degradation behavior of the pressing block caused by continuous operation, reflects the wear amount of the pressing block on the influence of the gear tooth meshing stress through the change of the spring elastic force, and then constructs simulation models of different wear states to calculate the maximum contact stress of the gear and rack, and establishes an engineering proxy model of the relationship between the maximum equivalent contact stress of the gear and rack and the maximum contact stress of the gear and rack and the wear depth. The method also considers the uncertainty characteristics of material parameters caused by lax raw material quality control or insufficient data, further establishes a gear rack reliability model, and combines the gear rack reliability model with the linear assumption of wear depth to finally construct a time-varying reliability model. The present invention can more accurately predict the reliable life level of the gear rack, thereby providing better technical support for the formulation of gear rack maintenance and guarantee plans.

Description

Technical Field

[0001] The invention belongs to the technical field of gear rack reliability calculation and analysis, and in particular relates to a gear rack reliability calculation method taking into account pressing block wear and material uncertainty. Background Art

[0002] As a key component of mechanical transmission systems, rack and pinion systems are widely used in precision machinery such as steering gears, CNC machine tools, and machining centers. Their reliability directly impacts the safe and stable operation of these machines. However, the reliability level of rack and pinion systems is not constant; it exhibits time-varying characteristics due to internal degradation mechanisms. The most critical factor is the wear of the clamp, which limits the meshing of the rack and pinion. This wear manifests itself as the clamp undergoes continuous wear during the rack's cyclical motion. This in turn causes changes in the clamp spring force, affecting the proper meshing of the gear teeth and ultimately putting the gear and rack teeth at risk of overstress fracture. Furthermore, due to lax raw material quality control or insufficient data, material parameters can exhibit uncertainties, leading to deviations from the design nominal values after actual manufacturing, further increasing the risk of overstress fracture. Therefore, comprehensively considering the continuous degradation of clamp wear and the uncertainties of the material during the design phase and developing a lifecycle-based time-varying reliability calculation method is crucial for ensuring safe and reliable rack and pinion operation.

[0003] However, current reliability research on rack-and-pinion mechanisms mainly focuses on selecting appropriate safety factors for rack-and-pinion models based on relevant mechanical design principles and product design specifications, or on providing reliability indicators under constant conditions based on stress-strength theory. A full-lifecycle time-varying reliability calculation method for degradation mechanisms has not yet been proposed, nor has the uncertainty characteristics of relevant material parameters been comprehensively considered. Summary of the Invention

[0004] The purpose of the present invention is to address the defects of the prior art and to propose a gear rack reliability calculation method that takes into account the wear of the pressing block and the uncertainty of the material.

[0005] The present invention provides a method for calculating the reliability of a gear rack taking into account the wear of a pressing block and material uncertainty, comprising the following steps:

[0006] S1: Use the Archard wear model to calculate the wear depth d of the pressing block under a single reciprocating motion of the rack;

[0007] S2: Based on the total number of cycles t of the gear rack in the entire life cycle total , determine the maximum wear depth D of the block max , and in 0~D max Evenly select n wear depth sample points D within the rangei ,i=0,1,…,n-1;

[0008] S3: Each wear depth sample point D i Combined with the spring compression deformation l0 when the pressure block is not worn, the spring force F exerted on the pressure block at different wear depths is determined. i ;

[0009] S4: Construct a gear-rack meshing stress simulation model considering different wear depths of the pressing block, analyze the gear-rack equivalent contact stress, and then extract the maximum equivalent contact stress value σ of the rack under different wear depths of the pressing block. 齿条,max,i and the maximum equivalent contact stress value of the gear σ 齿轮,max,i Among them, in the gear rack meshing stress simulation model under different wear depths of the pressure block, except for the spring force F i Except for the different normal contact pressures between the pressure block and the rack, the other constraints and load conditions are the same, and the mesh division is also the same;

[0010] S5: Combine the wear depth sample points D i and the corresponding σ 齿条,max,i and σ 齿轮,max,i , select σ 齿条,max,i and σ 齿轮,max,i For the larger value, the Lagrange interpolation polynomial is used to establish the engineering proxy model of the maximum equivalent contact stress of the gear rack;

[0011] S6: The yield strength σ of the gear and rack material S Assumed to be normally distributed The gear and rack are made of the same material, and the yield strength of the gear and rack materials is Standard deviation of yield strength of gear and rack materials and an engineering proxy model of the maximum equivalent contact stress of the gear rack to establish a reliability model;

[0012] S7: Based on the wear depth d of the pressing block under a single reciprocating motion of the rack obtained in step S1, a relationship between the number of cycles t and the wear depth D of the pressing block is established, thereby providing a time-varying reliability model of the gear rack.

[0013] Preferably, the Archard wear model selected in step S1 is:

[0014]

[0015] Where p is the positive pressure between the pressing block and the rack, L is the rack stroke, H is the hardness of the pressing block material, and K is the wear coefficient of the pressing block material.

[0016] Preferably, the maximum wear depth D in step S2 maxThe relationship with the single wear depth d is:

[0017] D max =d×t total

[0018] Preferably, the wear depth sample point D in step S2 i The selection method is:

[0019]

[0020] Where n≥4.

[0021] Preferably, the spring force F exerted on the pressing block under different wear conditions in step S3 i Expressed as:

[0022] F i =k(l0+D i )

[0023] Where k is the elastic constant of the spring.

[0024] Preferably, the maximum equivalent contact stress engineering proxy model of the gear rack established using the Lagrange interpolation polynomial in step S5 is:

[0025]

[0026] Where, σ max (D) is the maximum equivalent contact stress of the gear rack corresponding to the wear depth of the pressure block D, σ max,i is σ 齿条,max,i and σ 齿轮,max,i The larger value in .

[0027] Preferably, the standard deviation in step S6 Expressed as:

[0028]

[0029] Where δ is the dispersion coefficient.

[0030] Preferably, the reliability model in step S6 is:

[0031]

[0032] Where, R(D) is the reliability of the block when the wear depth is D, Φ N Normal distribution The probability distribution of

[0033] Preferably, the relationship between the number of cycles t and the wear depth D of the pressing block in step S7 is set to be linear: D=d×t, then the time-varying reliability model is:

[0034]

[0035] Compared with the prior art, the present invention has the following beneficial effects:

[0036] The present invention takes into account the wear degradation behavior of the pressure block caused by continuous operation, reflects the wear amount of the pressure block in the influence on the gear meshing stress through the change of the spring elastic force, and then constructs simulation models of different wear states to calculate the maximum contact stress of the gear and rack, and establishes an engineering proxy model of the relationship between the maximum equivalent contact stress of the gear and rack and the maximum contact stress of the gear and rack and the wear depth. Based on this, it also takes into account the uncertainty characteristics of material parameters caused by lax raw material quality control or insufficient data, and further establishes a gear rack reliability model. Combined with the gear rack reliability model and the linear assumption of wear depth, a time-varying reliability model is finally constructed. The present invention makes up for the shortcomings of other current methods that cannot comprehensively consider the continuous degradation process of wear depth and the uncertain characteristics of the material in the design stage. It can more accurately predict the reliable life level of the gear rack, and thus provide better technical support for the formulation of gear rack maintenance and guarantee plans. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is a flow chart of the present invention.

[0038] Figure 2 4 is a discrete relationship curve diagram of the reliability and cycle number of the gear rack mechanism of the steering system in the embodiment of the present invention. DETAILED DESCRIPTION

[0039] The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0040] The present invention considers the gear rack reliability calculation method of the pressure block wear and material uncertainty, such as Figure 1 As shown, the following steps are included:

[0041] S1: According to the wear mechanism of the press block, the Archard wear model is used to calculate the wear depth d of the press block under a single reciprocating motion of the rack;

[0042] S2: Based on the total number of cycles t of the gear rack in the entire life cycle total , determine the maximum wear depth D of the block max , and in 0~D max Evenly select n wear depth sample points D within the range i ,i=0,1,…,n-1;

[0043] S3: Each wear depth sample point D iCombined with the spring compression deformation l0 when the pressure block is not worn, the spring force F exerted on the pressure block at different wear depths is determined. i ;

[0044] S4: Construct (can be constructed in ANSYS software) a gear rack meshing stress simulation model considering different wear depths of the pressing block, analyze the gear rack equivalent contact stress, and then extract the maximum equivalent contact stress value σ of the rack under different wear depths of the pressing block 齿条,max,i and the maximum equivalent contact stress value of the gear σ 齿轮,max,i Among them, in the gear rack meshing stress simulation model under different wear depths of the pressure block, except for the spring force F i Except for the different normal contact pressures between the pressure block and the rack, the other constraints and load conditions are the same, and the mesh division is also the same;

[0045] S5: Combine the wear depth sample points D i and the corresponding σ 齿条,max,i and σ 齿轮,max,i , select σ 齿条,max,i and σ 齿轮,max,i For the larger value, the Lagrange interpolation polynomial is used to establish the engineering proxy model of the maximum equivalent contact stress of the gear rack;

[0046] S6: The yield strength σ of the gear and rack material S Assumed to be normally distributed The gear and rack are made of the same material, and the yield strength of the gear and rack materials is Standard deviation of yield strength of gear and rack materials and an engineering proxy model of the maximum equivalent contact stress of the gear rack to establish a reliability model;

[0047] S7: Based on the wear depth d of the pressing block under a single reciprocating motion of the rack obtained in step S1, a relationship between the number of cycles t and the wear depth D of the pressing block is established, thereby providing a time-varying reliability model of the gear rack.

[0048] Preferably, the Archard wear model selected in step S1 is:

[0049]

[0050] Where p is the positive pressure between the pressure block and the rack; L is the rack stroke; H is the hardness of the pressure block material (the hardness of the rack material is generally greater than that of the pressure block material or the same material is used), and K is the wear coefficient of the pressure block material.

[0051] Preferably, the maximum wear depth D in step S2 max The relationship with the single wear depth d is:

[0052] D max =d×t total

[0053] Preferably, the wear depth sample point D in step S2 i The selection method is:

[0054]

[0055] Where n is the number of selected sample points, which is usually not less than 4.

[0056] Preferably, the spring force F exerted on the pressing block under different wear conditions in step S3 i Expressed as:

[0057] F i =k(l0+D i )

[0058] Where k is the elastic constant of the spring.

[0059] Preferably, the maximum equivalent contact stress engineering proxy model of the gear rack established using the Lagrange interpolation polynomial in step S5 is:

[0060]

[0061] Where, σ max (D) is the maximum equivalent contact stress of the gear rack corresponding to the wear depth of the pressure block D, σ max,i is σ 齿条,max,i and σ 齿轮,max,i The larger value in .

[0062] Preferably, the standard deviation in step S6 Expressed as:

[0063]

[0064] Where δ is the dispersion coefficient, which reflects the dispersion degree of the material's yield strength.

[0065] Preferably, the reliability model in step S6 is:

[0066]

[0067] Where, R(D) is the reliability of the block when the wear depth is D, Φ N Normal distribution The probability distribution of

[0068] Preferably, the relationship between the number of cycles t and the wear depth D of the pressing block in step S7 is set to be linear: D=d×t, then the time-varying reliability model is:

[0069]

[0070] The following will further explain in detail the specific implementation steps of the present invention in conjunction with the reliability calculation process of a selected steering system rack and pinion mechanism. The specific implementation steps are as follows:

[0071] Step S1: Based on the material manual, the hardness H of the briquette material in this embodiment is designed to be 340 MPa and the wear coefficient K is 1.27×10 -8 The positive pressure p between the pressing block and the rack is designed to be 800 MPa, and the rack stroke L is 30 mm. According to the pressing block wear mechanism, the Archard wear model is used to calculate the wear depth d of the pressing block under a single reciprocating motion of the rack. The calculation results are as follows:

[0072]

[0073] Step S2: Based on the full life cycle load spectrum of the steering system, it is known that the total number of cycles is 600,000 times, and the maximum wear depth D is calculated. max Therefore, 5 wear depth sample points are evenly selected, namely 0, 0.1344mm, 0.2688mm, 0.4032mm, and 0.5376mm.

[0074] Step S3: Assuming the initial spring compression deformation to be 20 mm and the elastic coefficient to be 78,000 N / m, the spring compression deformations corresponding to the five wear depth sample points are 20, 20.1344 mm, 20.2688 mm, 20.4032 mm, and 20.5376 mm, respectively. Furthermore, the spring forces acting on the pressure blocks corresponding to the five wear depth sample points are calculated to be 1560 N, 1570 N, 1581 N, 1591 N, and 1602 N, respectively.

[0075] Step S4: The influence of the above wear is reflected in the gear rack meshing stress simulation model through the spring elastic force, and the gear rack meshing stress simulation models under the constructed 5 wear depth sample points are respectively established. In each gear rack meshing stress simulation model, except for the spring elastic force F i The other constraints, load conditions and mesh divisions are the same except that the contact normal pressure between the pressing block and the rack is different. The equivalent contact stress of the gear rack is analyzed, and then the maximum equivalent contact stress value σ of the rack under different wear depths of the pressing block is extracted. 齿条,max,i and the maximum equivalent contact stress value of the gear σ 齿轮,max,i , as shown in Table 1.

[0076] Table 1

[0077]

[0078] Step S5: Based on the simulation analysis results, the rack is selected as the reliability modeling object, and a 5th-order Lagrange interpolation polynomial is constructed as an engineering proxy model for the maximum equivalent contact stress of the gear rack as follows:

[0079] σ max (D)=924.43l0(D)+934.97l1(D)+949.41l2(D)+968.97l3(D)+995.58l4(D)

[0080]

[0081] Step S6: The gear rack material is 40Cr, and the yield strength mean μ is determined from the material manual. σS is 1020 MPa, and considering the dispersion characteristics of material parameters, the yield strength is set to obey the normal distribution and the dispersion coefficient δ is 0.035, that is, σ S ~N(1020,35.7 2 ), and combined with the engineering proxy model of the maximum equivalent contact stress of the gear rack, the reliability model is constructed as follows:

[0082]

[0083] Step S7: Based on the wear depth of the pressing block obtained in step S1 under a single reciprocating motion of the rack, d=8.96×10 - 7 mm, it can be seen that the wear depth D of the press block at the cycle number t is 8.96×10 -7 ×tmm, and thus the time-varying reliability model of the gear rack related to the actual cycle number t is established as follows:

[0084]

[0085] Through the above-mentioned time-varying reliability model of the gear rack, the reliability of the gear rack mechanism of the steering system under the corresponding cycle number t can be clarified; the discrete relationship curve between the reliability R(t) of the gear rack mechanism of the steering system and the cycle number t is drawn, as shown in Figure 2 As shown in the figure, with the increase of wear, the reliability of the gear rack mechanism of the steering system shows a downward trend.

[0086] Finally, it should be noted that the embodiments described above are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some or all of the technical features therein. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A gear rack reliability calculation method considering block wear and material uncertainty is characterized by: The following steps are involved: S1: Use the Archard wear model to calculate the wear depth d of the pressing block under a single reciprocating motion of the rack; S2: Based on the total number of cycles t of the gear rack in the entire life cycle total , determine the maximum wear depth D of the block max , and in 0~D max Evenly select n wear depth sample points D within the range i ,i=0,1,…,n-1; S3: Each wear depth sample point D i Combined with the spring compression deformation l0 when the pressure block is not worn, the spring force F exerted on the pressure block at different wear depths is determined. i ; S4: Construct a gear-rack meshing stress simulation model considering different wear depths of the pressing block, analyze the gear-rack equivalent contact stress, and then extract the maximum equivalent contact stress value σ of the rack under different wear depths of the pressing block. 齿条,max,i and the maximum equivalent contact stress value of the gear σ 齿轮,max,i Among them, in the gear rack meshing stress simulation model under different wear depths of the pressure block, except for the spring force F i Except for the different normal contact pressures between the pressure block and the rack, the other constraints and load conditions are the same, and the mesh division is also the same; S5: Combine the wear depth sample points D i and the corresponding σ 齿条,max,i and σ 齿轮,max,i , select σ 齿条,max,i and σ 齿轮,max,i For the larger value, the Lagrange interpolation polynomial is used to establish the engineering proxy model of the maximum equivalent contact stress of the gear rack; S6: The yield strength σ of the gear and rack material S Assumed to be normally distributed The gear and rack are made of the same material, and the yield strength of the gear and rack materials is Standard deviation of yield strength of gear and rack materials and an engineering proxy model of the maximum equivalent contact stress of the gear rack to establish a reliability model; S7: Based on the wear depth d of the pressing block under a single reciprocating motion of the rack obtained in step S1, a relationship between the number of cycles t and the wear depth D of the pressing block is established, thereby providing a time-varying reliability model of the gear rack.

2. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 1 is characterized by: The Archard wear model selected in step S1 is: Where p is the positive pressure between the pressing block and the rack, L is the rack stroke, H is the hardness of the pressing block material, and K is the wear coefficient of the pressing block material.

3. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 1 is characterized by: The maximum wear depth D in step S2 max The relationship with the single wear depth d is: D max =d×t total 。 4. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 1 is characterized in that: The wear depth sample point D in step S2 i The selection method is: Where n≥4.

5. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 1 is characterized in that: The spring force F exerted on the pressing block under different wear states in step S3 i Expressed as: F i =k(l0+D i ) Where k is the elastic constant of the spring.

6. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 1 is characterized in that: The engineering proxy model of the maximum equivalent contact stress of the gear rack established using the Lagrange interpolation polynomial in step S5 is: Where σ max (D) is the maximum equivalent contact stress of the gear rack corresponding to the wear depth of the pressure block D, σ max,i is σ 齿条,max,i and σ 齿轮,max,i The larger value in .

7. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 1 is characterized in that: Standard deviation in step S6 Expressed as: Where δ is the dispersion coefficient.

8. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 7 is characterized in that: The reliability model in step S6 is: Where, R(D) is the reliability of the block when the wear depth is D, Φ N Normal distribution The probability distribution of .

9. The gear rack reliability calculation method considering pressing block wear and material uncertainty according to claim 8 is characterized in that: Assuming that the relationship between the number of cycles t and the wear depth D of the pressing block in step S7 is linear: D = d × t, the time-varying reliability model is:

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