Reliability Analysis Method of Ball Screw Pair Based on Stress-Strength Model and Interference Theory

Abstract

The present invention discloses a reliability analysis method for a ball screw pair based on a stress-strength model and interference theory, comprising: calculating the mean and standard deviation of each key process parameter of the ball screw pair according to design requirements and processing technical conditions; solving the mean and standard deviation of the rated dynamic load and equivalent dynamic load respectively according to the Taylor series expansion; calculating the reliability index, setting the calculation step size and calculation accuracy, and calculating the probability of the random variable falling into the total failure area. When the accuracy requirements are met, the failure probability of the stress-strength interference model is solved. The reliability of the ball screw pair calculated by the present invention takes into account the uncertainty of the design and processing of the process parameters of the ball screw pair and the random uncertainty of the load, calculates the distribution range of the rated dynamic load and equivalent dynamic load of the ball screw pair, and then calculates the failure probability of the interference area. Based on the interference area of ​​the rated dynamic load and the equivalent dynamic load, a new reliability calculation method for the ball screw pair is provided, which is of great significance for improving the performance and increasing the reliability of the ball screw pair.

Description

Technical Field

[0001] The present invention belongs to the technical field of reliability optimization design and reliability growth of ball screw pairs, in particular to a reliability analysis method of ball screw pairs based on a stress-strength model and interference theory. Background Art

[0002] As a core functional component of CNC machine tools, the ball screw pair is one of the key components that ensure the precision and rigidity of industrial machine tools. In a narrow sense, the reliability of a ball screw pair refers to its ability to operate normally and maintain a certain level of accuracy and performance under specified conditions within a specified timeframe. This directly impacts the normal operation of mechanical transmission systems. Currently, little research has been conducted on the reliability of ball screw pairs in China, which is one of the main reasons hindering the rapid development of my country's rolling functional component and high-end CNC machine tool industries. Therefore, analyzing and optimizing the reliability of ball screw pairs is of great significance for improving the reliability of both ball screw pairs and the reliability of the entire CNC machine tool.

[0003] The stress-strength model and interference theory are methods for determining reliability based on the degree of interference between stress distribution and strength distribution. The analysis and design methods based on the stress-strength model and interference theory are commonly used methods in reliability analysis and design. In order to calculate the reliability of ball screw pairs, Wei Zongping et al. (Analysis of the sensitivity and reliability of the axial contact static stiffness of double-nut gasket preloaded ball screw pairs [J]. Mechanical Transmission, 2017, 41(11): 45-49, 147) used the interval theory method to analyze and calculate the reliability of the axial contact static stiffness of the screw. The reliability calculated by this method is different from the conventionally defined reliability. It is not based on the axioms of probability theory, so the correctness of the results is difficult to guarantee. Jia Dawei et al. (A structural reliability analysis method based on a convex set-probability hybrid model [J]. Journal of Solid Mechanics, 2020, 41(5): 470-484) proposed a structural reliability analysis method based on a convex set-probability hybrid model, which is essentially also an interval theory method. Zhang Yimin et al. (Analysis of Static Stiffness Reliability and Sensitivity of Single-nut Ball Screw Pair [J]. Journal of Harbin Institute of Technology, 2016, 48(7):140-144) used the improved first-order second moment method to study the reliability of the axial static stiffness of the single-nut ball screw pair. This method has high requirements on the amount of information of the parameters. Wang Dan et al. (Analysis of Kinematic Reliability of Ball Screw Pair Based on Monte Carlo Method [J]. Journal of Northeastern University (Natural Science Edition), 2012, 33(8):1179-1181, 1185) used the Monte Carlo simulation method to analyze the axial static stiffness of the ball screw pair, but it required a large sample size. Huang Kuan et al. (Analysis of Ball Screw Reliability Based on Quasi-Monte Carlo Method [J]. Combined Machine Tools and Automated Machining Technology, 2012(2):1-4) proposed a quasi-Monte Carlo method based on Halton sequence, which overcomes these shortcomings but only considers the uncertainty of the parameters.

[0004] Calculations of ball screw reliability have only considered the impact of inherent parameter uncertainty on axial static stiffness reliability, without examining the impact of ball screw parameter uncertainty and the dynamic behavior of the screw under actual operating conditions on life reliability. Currently, there is no research on ball screw reliability analysis based on stress-strength models and interference theory. Summary of the Invention

[0005] The purpose of the present invention is to address the problems existing in the above-mentioned prior art and provide a reliability analysis method for ball screw pairs based on stress-strength model and interference theory. The reliability analysis of the ball screw pairs is performed by taking into account the dynamic behavior of the ball screw pairs and combining the design formula of the ball screw pairs.

[0006] The technical solution to achieve the purpose of the present invention is: a ball screw pair reliability analysis method based on a stress-strength model and interference theory, the method comprising the following steps:

[0007] Step 1: Calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ;

[0008] Step 2: Calculate the mean value μ of the rated dynamic load using the Taylor series expansion s and standard deviation σ s ;

[0009] Step 3: Calculate the mean value μ of the equivalent dynamic load r and standard deviation σ r ;

[0010] Step 4: solve the reliability index and set the calculation step size and calculation accuracy requirements;

[0011] Step 5, calculate the probability that the random variable X falls within any circle;

[0012] Step 6: Calculate the probability that the random variable X falls in the total failure area, which is the failure probability of the stress-strength interference model;

[0013] Step 7: Determine whether the calculation accuracy meets the set calculation accuracy requirement. If so, execute step 8; otherwise, return to step 5.

[0014] Step 8: Output the failure probability of the stress-strength interference model.

[0015] Furthermore, in step 1, the key process parameter values ​​of the ball screw pair are processed as random processes, and the corresponding distribution is normal distribution, with a mean μ x and standard deviation σ x Specifically:

[0016] μ x =(μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α )

[0017] σ x =(σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α )

[0018] Among them, (μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β,μ α ), (σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α ) are the parameters of ball diameter D w , pitch circle diameter D pw , Screw raceway radius r s , nut raceway radius r n , the mean and standard deviation of the helix angle β and the contact angle α between the ball and the raceway.

[0019] Furthermore, in step 2, the mean value μ of the rated dynamic load is solved according to the Taylor series expansion: s and standard deviation σ s , specifically:

[0020] The model for the dynamic load rating is:

[0021] C a =f(D w ,D pw ,r s ,r n ,β,α…)=C i ·i 0.85

[0022] in,

[0023]

[0024] in,

[0025]

[0026] Where C a is the rated dynamic load; i is the number of balls in the bearing circle; f is the geometric coefficient; α is the contact angle between the ball and the raceway; z1 is the number of effective bearing balls in each ball circle; D w Ball diameter; D pw is the pitch diameter; β is the helix angle; f1, f2, f3 are geometric shape coefficients; γ is the structural coefficient; r s is the screw raceway radius; r n is the nut raceway radius; f rs 、f rn are the adaptability of the screw raceway and the nut raceway respectively; z u is the number of unloaded balls in each ball ring;

[0027] Substituting the mean value of each process parameter into the corresponding process parameter variable in the rated dynamic load model, the mean value μ of the rated dynamic load can be obtained. s for:

[0028] μ s =f(μ x )=f(μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α )

[0029] Standard deviation of rated dynamic load σ s According to the Taylor series expansion and ignoring the higher-order terms, the approximate value is obtained from the following formula:

[0030]

[0031] Where x i Take D w 、D pw 、r s 、r n , β and α, σ xi Represents x i The corresponding standard deviation, n represents x i The number of values ​​​​of .

[0032] Furthermore, the mean value μ of the equivalent dynamic load is solved in step 3 r and standard deviation σ r , specifically including:

[0033] The model of equivalent dynamic load is:

[0034]

[0035] in,

[0036]

[0037] Where, F m is the equivalent dynamic load; n m is the equivalent speed; F j is the load value within a working cycle, n j is the speed value of the jth working unit in a working cycle; q j is the proportion of load or speed in a working cycle; n' is the number of working units in a working cycle;

[0038] Then the mean value of the equivalent dynamic load μ r for:

[0039] μ r =F m

[0040] Standard deviation σ of equivalent dynamic load r for:

[0041]

[0042] Furthermore, step 4 solves the stress-strength interference model reliability index and sets the calculation step size and calculation accuracy requirements, specifically including:

[0043] If the rated dynamic load and equivalent dynamic load are normally distributed, then the variable distribution of the stress-strength interference model is also normally distributed, with mean and standard deviation μ respectively. z =(μ s -μ r )and Its reliability index β is:

[0044]

[0045] Set r0 = β, r i =r0+i×Δr, where Δr is the calculation step size, r0 is the initial radius of any circle, r i is the radius of the i-th iteration;

[0046] Set the calculation precision requirement P * .

[0047] Furthermore, the calculation of the probability of the random variable X falling within any circle in step 5 specifically includes:

[0048] The probability density function of the random variable X is:

[0049] f(x)=x·exp(-x 2 / 2),x>0

[0050] In the formula, x represents only the function variable;

[0051] The distribution function of X is:

[0052] F(x)=P(X<x)=1-exp(-x 2 / 2),x>0

[0053] Then the random variable X falls within the radius r i The probability P(X<r i )for:

[0054]

[0055] The random variable X falls on any ring (r i-1 , r i ) with a probability P i for:

[0056]

[0057] Furthermore, the probability of the random variable X falling into the total failure region calculated in step 6 is the failure probability of the stress-strength interference model, which specifically includes:

[0058] Any ring (r i-1 , r i ) area S i for:

[0059]

[0060] Where Δr = r i -r i-1 is the difference between the radii of two adjacent circles, i.e., the calculation step length;

[0061] The ring (r i-1 , r i ) The area of ​​the failure zone for:

[0062]

[0063] in,

[0064]

[0065] The random variable X is in the ith ring (r i-1 , r i ) falls within the failure zone in the interference zone i for:

[0066]

[0067] Then the probability F of the random variable X falling into the total failure area, that is, the failure probability of the stress-strength interference model is:

[0068]

[0069] Where n" represents the number of rings.

[0070] Furthermore, the step 7 determines whether the calculation accuracy meets the set calculation accuracy requirement. If so, step 8 is executed; otherwise, step 5 is returned to, which specifically includes:

[0071] The judgment is based on:

[0072] P(X>r i )<P *

[0073] Where, P(X>r i ) indicates that the random variable X falls within a radius r i The probability of being outside the circle, P * Indicates the calculation accuracy requirements set;

[0074] When the above judgment criteria are not met, let r0 = r i , i=i+1, repeat steps 5 to 7, otherwise, the calculation stops.

[0075] Compared with the prior art, the present invention has the following significant advantages:

[0076] (1) The present invention takes into account the influence of multiple process parameters of the ball screw pair on the reliability, and establishes a stress-strength interference model of the rated dynamic load and equivalent dynamic load of the ball screw pair, which is of great significance to the improvement of the reliability of the ball screw pair.

[0077] (2) The present invention applies a new solution of the stress-strength interference model, which can efficiently solve the failure probability with various accuracy requirements compared to the graphical method.

[0078] (3) The standard deviation of multiple process parameters of the ball screw pair is related to the standard deviation of the rated dynamic load using the Taylor series expansion method, which has certain guiding significance for solving the reliability of the ball screw pair.

[0079] The present invention is further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 The figure is a flow chart of the reliability analysis method of a ball screw pair based on the stress-strength model and interference theory of the present invention.

[0081] Figure 2 Graph showing the load spectrum of a ball screw pair in one embodiment.

[0082] Figure 3 1 is a diagram of a rated-equivalent dynamic load interference model, i.e., a stress-strength interference model, in one embodiment. DETAILED DESCRIPTION

[0083] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0084] It should be noted that if there are descriptions involving "first", "second", etc. in the embodiments of the present invention, the descriptions of "first", "second", etc. are only for descriptive purposes and cannot be understood as indicating or implying their relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined as "first" and "second" may explicitly or implicitly include at least one of such features. In addition, the technical solutions between the various embodiments can be combined with each other, but they must be based on the ability of ordinary technicians in this field to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be deemed that such a combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

[0085] In one embodiment, combined Figure 1 , provides a ball screw pair reliability analysis method based on stress-strength model and interference theory, characterized in that the method comprises the following steps:

[0086] Step 1: Calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ;

[0087] Step 2: Calculate the mean value μ of the rated dynamic load using the Taylor series expansion s and standard deviation σ s ;

[0088] Step 3: Calculate the mean value μ of the equivalent dynamic load r and standard deviation σ r ;

[0089] Step 4: solve the reliability index of the stress-strength interference model and set the calculation step size and calculation accuracy requirements;

[0090] Step 5, calculate the probability that the random variable X falls within any circle;

[0091] Step 6: Calculate the probability that the random variable X falls in the total failure area, which is the failure probability of the stress-strength interference model;

[0092] Step 7: Determine whether the calculation accuracy meets the set calculation accuracy requirement. If so, execute step 8; otherwise, return to step 5.

[0093] Step 8: Output the failure probability of the stress-strength interference model.

[0094] Furthermore, in one embodiment, in step 1, the key process parameter values ​​of the ball screw pair are processed as random processes, and the corresponding distribution is normal distribution, and its mean μ x and standard deviation σ x Specifically:

[0095] μ x =(μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α )

[0096] σ x =(σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α )

[0097] Among them, (μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α ), (σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α ) are the parameters of ball diameter D w , pitch circle diameter D pw , Screw raceway radius r s , nut raceway radius r n , the mean and standard deviation of the helix angle β and the contact angle α between the ball and the raceway.

[0098] Furthermore, in one embodiment, the step 2 is to solve the mean value μ of the rated dynamic load according to the Taylor series expansion. s and standard deviation σ s , specifically:

[0099] The model for the dynamic load rating is:

[0100] C a =f(D w ,D pw ,r s ,r n ,β,α…)=C i ·i 0.85

[0101] in,

[0102]

[0103] in,

[0104]

[0105]

[0106] Where C a is the rated dynamic load; i is the number of balls in the bearing circle; f is the geometric coefficient; α is the contact angle between the ball and the raceway; z1 is the number of effective bearing balls in each ball circle; D w Ball diameter; D pw is the pitch diameter; β is the helix angle; f1, f2, f3 are geometric shape coefficients; γ is the structural coefficient; r s is the screw raceway radius; r n is the nut raceway radius; f rs 、f rn are the adaptability of the screw raceway and the nut raceway respectively; z u is the number of unloaded balls in each ball ring;

[0107] Substituting the mean value of each process parameter into the corresponding process parameter variable in the rated dynamic load model, the mean value μ of the rated dynamic load can be obtained. s for:

[0108] μ s =f(μ x )=f(μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α )

[0109] Standard deviation of rated dynamic load σ s According to the Taylor series expansion and ignoring the higher-order terms, the approximate value is obtained from the following formula:

[0110]

[0111] Where x i Take D w 、D pw 、r s 、r n , β and α, σ xi Represents x i The corresponding standard deviation, n represents x i The number of values ​​​​of .

[0112] Furthermore, in one embodiment, the step 3 of solving the mean value μ of the equivalent dynamic load is r and standard deviation σ r , specifically including:

[0113] The model of equivalent dynamic load is:

[0114]

[0115] in,

[0116]

[0117] Where, F m is the equivalent dynamic load; n m is the equivalent speed; F j is the load value within a working cycle, n j is the speed value of the jth working unit in a working cycle; q j is the proportion of load or speed in a working cycle; n' is the number of working units in a working cycle;

[0118] Then the mean value of the equivalent dynamic load μ r for:

[0119] μ r =F m

[0120] Standard deviation σ of equivalent dynamic load r for:

[0121]

[0122] Furthermore, in one embodiment, the step 4 of solving the stress-strength interference model reliability index and setting the calculation step size and calculation accuracy requirements specifically includes:

[0123] Since the rated dynamic load and equivalent dynamic load are normally distributed, the variable distribution of the stress-strength interference model is also normally distributed, and its mean and standard deviation are μ z =(μ s -μ r )and Its reliability index β is:

[0124]

[0125] Set r0 = β, r i =r0+i×Δr, where Δr is the calculation step size, r0 is the initial radius of any circle, r i is the radius of the i-th iteration;

[0126] Set the calculation precision requirement P * .

[0127] Here, the calculation step length Δr will affect the calculation speed and calculation accuracy. The larger the Δr, the higher the calculation accuracy. The more complex the calculation, the slower the calculation. The calculation result accuracy P * It is the judgment basis of the calculation process. When the accuracy requirement is met, the calculation ends.

[0128] Furthermore, in one embodiment, the step 5 of calculating the probability of the random variable X falling within any circle specifically includes:

[0129] The probability density function of the random variable X is:

[0130] f(x)=x·exp(-x 2 / 2),x>0

[0131] In the formula, x represents only the function variable;

[0132] The distribution function of X is:

[0133] F(x)=P(X<x)=1-exp(-x 2 / 2),x>0

[0134] Then the random variable X falls within the radius r i The probability P(X<r i )for:

[0135]

[0136] The random variable X falls on any ring (r i-1 , r i ) with a probability P i for:

[0137]

[0138] Furthermore, in one embodiment, the calculation of the probability of the random variable X falling within the total failure region in step 6 is the failure probability of the stress-strength interference model, which specifically includes:

[0139] Any ring (r i-1 , r i ) area S i for:

[0140]

[0141] Where Δr = r i -r i-1 is the difference between the radii of two adjacent circles, i.e., the calculation step length;

[0142] The ring (r i-1 , r i ) The area of ​​the failure zone for:

[0143]

[0144] in,

[0145]

[0146] The random variable X is in the ith ring (r i-1 , r i ) falls within the failure zone in the interference zone i for:

[0147]

[0148] Then the probability F of the random variable X falling into the total failure area, that is, the failure probability of the stress-strength interference model is:

[0149]

[0150] Where n" represents the number of rings.

[0151] Furthermore, in one embodiment, the step 7 determines whether the calculation accuracy meets the set calculation accuracy requirement. If so, the step 8 is executed; otherwise, the step 5 is returned to, which specifically includes:

[0152] The judgment is based on:

[0153] P(X>r i )<P *

[0154] Where, P(X>r i ) indicates that the random variable X falls within a radius r i The probability of being outside the circle, P * Indicates the calculation accuracy requirements set;

[0155] When the above judgment criteria are not met, let r0 = r i , i=i+1, repeat steps 5 to 7, otherwise, the calculation stops.

[0156] In one embodiment, a ball screw pair reliability analysis system based on a stress-strength model and interference theory is provided, the system comprising the following steps executed in sequence:

[0157] The first module is used to calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ;

[0158] The second module is used to solve the mean value μ of the rated dynamic load based on the Taylor series expansion s and standard deviation σ s ;

[0159] The third module is used to solve the mean value μ of the equivalent dynamic load r and standard deviation σ r ;

[0160] The fourth module is used to solve the reliability index of the stress-strength interference model and set the calculation step size and calculation accuracy requirements;

[0161] The fifth module is used to calculate the probability that the random variable X falls within any circle;

[0162] The sixth module is used to calculate the probability that the random variable X falls into the total failure area, which is the failure probability of the stress-strength interference model;

[0163] The seventh module is used to determine whether the calculation accuracy meets the set calculation accuracy requirements. If so, the eighth module is executed, otherwise it returns to the fifth module;

[0164] The eighth module is used to output the failure probability of the stress-strength interference model.

[0165] Regarding the specific limitations of the ball screw pair reliability analysis system based on the stress-strength model and interference theory, please refer to the limitations of the ball screw pair reliability analysis method based on the stress-strength model and interference theory above, which will not be repeated here. Each module in the above-mentioned ball screw pair reliability analysis system based on the stress-strength model and interference theory can be implemented in whole or in part through software, hardware and their combination. The above-mentioned modules can be embedded in or independent of the processor in the computer device in the form of hardware, or can be stored in the memory of the computer device in the form of software, so that the processor can call and execute the operations corresponding to the above modules.

[0166] In one embodiment, a computer device is provided, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, the following is achieved:

[0167] Step 1: Calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ;

[0168] Step 2: Calculate the mean value μ of the rated dynamic load using the Taylor series expansion s and standard deviation σ s ;

[0169] Step 3: Calculate the mean value μ of the equivalent dynamic load r and standard deviation σ r ;

[0170] Step 4: solve the reliability index of the stress-strength interference model and set the calculation step size and calculation accuracy requirements;

[0171] Step 5, calculate the probability that the random variable X falls within any circle;

[0172] Step 6: Calculate the probability that the random variable X falls in the total failure area, which is the failure probability of the stress-strength interference model;

[0173] Step 7: Determine whether the calculation accuracy meets the set calculation accuracy requirement. If so, execute step 8; otherwise, return to step 5.

[0174] Step 8: Output the failure probability of the stress-strength interference model.

[0175] For the specific limitations of each step, please refer to the limitations of the ball screw pair reliability analysis method based on the stress-strength model and interference theory mentioned above, which will not be repeated here.

[0176] In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, the computer program implements:

[0177] Step 1: Calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ;

[0178] Step 2: Calculate the mean value μ of the rated dynamic load using the Taylor series expansion s and standard deviation σ s ;

[0179] Step 3: Calculate the mean value μ of the equivalent dynamic load r and standard deviation σ r ;

[0180] Step 4: solve the reliability index of the stress-strength interference model and set the calculation step size and calculation accuracy requirements;

[0181] Step 5, calculate the probability that the random variable X falls within any circle;

[0182] Step 6: Calculate the probability that the random variable X falls in the total failure area, which is the failure probability of the stress-strength interference model;

[0183] Step 7: Determine whether the calculation accuracy meets the set calculation accuracy requirement. If so, execute step 8; otherwise, return to step 5.

[0184] Step 8: Output the failure probability of the stress-strength interference model.

[0185] For the specific limitations of each step, please refer to the limitations of the ball screw pair reliability analysis method based on the stress-strength model and interference theory mentioned above, which will not be repeated here.

[0186] As a specific example, the present invention is further verified and explained in one of the embodiments.

[0187] Taking the GZ3206T ball screw pair as an example, a profilometer was used to measure the process parameters of a batch of ball screw pair products of the same model, and the mean and standard deviation of the ball screw pair process parameters were statistically calculated as shown in Table 1.

[0188] Table 1 Ball screw pair parameters

[0189]

[0190]

[0191] like Figure 1 As shown in FIG, the reliability analysis method of the ball screw pair based on the stress-strength model and interference theory includes the following steps:

[0192] S1: Calculate the mean μ of each key process parameter of the ball screw pair based on design requirements and processing technical conditions x and standard deviation σ x ;

[0193] μ x =(μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α ) (1)

[0194] σ x =(σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α ) (2)

[0195] Among them, μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α ,σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α The parameters are ball diameter D w , pitch circle diameter D pw , Screw raceway radius r s , nut raceway radius r n, the mean and standard deviation of the helix angle β and the contact angle α between the ball and the raceway.

[0196] Substitute the ball screw pair parameter table in Table 1 into equations (1) and (2); the mean value μ of each key process parameter can be obtained. x and standard deviation σ x ;

[0197] The mean values ​​μ of the key process parameters preliminarily obtained in this example are x The corresponding values ​​are (4.763, 33, 2.572, 2.572, 3.317, 45), and the standard deviation σ x The corresponding values ​​are (0.024, 0.165, 0.0129, 0.0129, 0.0166, 0.225) respectively.

[0198] S2: Solve the mean value μ of the rated dynamic load based on the Taylor series expansion s and standard deviation σ s ;

[0199] The model for the dynamic load rating is:

[0200] C a =f(D w ,D pw ,r s ,r n ,β,α…)=C i ·i 0.85 (3)

[0201] In the formula

[0202]

[0203] in

[0204]

[0205]

[0206] Where C a is the rated dynamic load; i is the number of balls in the bearing circle; f is the geometric coefficient; α is the contact angle between the ball and the raceway; z1 is the number of effective bearing balls in each ball circle; D w Ball diameter; D pw is the pitch diameter; β is the helix angle; f1, f2, f3 are geometric shape coefficients; γ is the structural coefficient; r s is the screw raceway radius; r n is the nut raceway radius; f rs 、f rn are the adaptability of the screw raceway and the nut raceway respectively; z uThe number of unloaded balls in each ball ring.

[0207] Substituting the mean value of each process parameter into the corresponding process parameter variable in the rated dynamic load model, the mean value μ of the rated dynamic load can be obtained. s for:

[0208] μ s =f(μ x )=f(μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α ) (15)

[0209] Since the rated dynamic load is related to multiple parameters, each process parameter is independent of each other, and the coefficient of variation of each parameter is very small, the standard deviation of the rated dynamic load σ s The approximate value can be obtained by Taylor series expansion and omitting higher-order terms:

[0210]

[0211] Where x i =(D w ,D pw ,r s ,r n ,β,α).

[0212] In this embodiment, the mean μ of each key process parameter obtained by S1 is x The corresponding values ​​are (4.763, 33, 2.572, 2.572, 3.317, 45). Substituting them into equations (3) to (15), we can calculate the mean value μ of the rated dynamic load. s =29471N; the standard deviation σ of each key process parameter obtained by S1 x The corresponding values ​​are (0.024, 0.165, 0.0129, 0.0129, 0.0166, 0.225) respectively; substituting them into equations (3) to (14), (16), we can calculate the standard deviation σ of the rated dynamic load. s :250.0920.

[0213] S3: Solve for the mean value μ of the equivalent dynamic load r and standard deviation σ r ;

[0214] The model of equivalent dynamic load is:

[0215]

[0216] in

[0217]

[0218] Where, F m is the equivalent dynamic load; n m is the equivalent speed; F j is the load value within a working cycle, n j is the speed value of the jth working unit in a working cycle; q j is the proportion of load or speed in a working cycle; n' is the number of working units in a working cycle.

[0219] The mean value of the equivalent dynamic load μ r for:

[0220] μ r =F m (19)

[0221] Standard deviation σ of equivalent dynamic load r for:

[0222]

[0223] like Figure 2 As shown in the figure, it is the load spectrum curve of the GZ3206T ball screw pair. Substituting its working load into formulas (17) to (19), the mean value of the equivalent dynamic load μ can be obtained. r =16918N; Substituting into formula (20), we can get the standard deviation σ of the equivalent dynamic load. r It is: 9673.6.

[0224] S4: Solve the reliability index and set the calculation step size and calculation accuracy requirements;

[0225] Since the rated dynamic load and equivalent dynamic load are normally distributed, the variable distribution of the interference model is also normally distributed, with mean and standard deviation μ z =(μ s -μ r )and Its reliability index β is:

[0226]

[0227] Set r0 = β, r i = r0 + i × Δr; the calculation step Δr will affect the calculation speed and calculation accuracy. The smaller Δr is, the higher the calculation accuracy is. The more complex the calculation is, the slower the calculation is. The calculation result accuracy P is * It is the judgment basis of the calculation process. When the accuracy requirement is met, the calculation ends.

[0228] like Figure 3As shown in the rated-equivalent dynamic load interference model, the mean value μ of the rated dynamic load and equivalent dynamic load obtained from S2 to S4 is s 、μ r , standard deviation σ s , σ r ; Substituting into formula (21), the reliability index β can be obtained as: 1.2974; setting Δr to: 0.0005; P * 1×10 -12 ;i=1.

[0229] S5: Calculate the probability that the random variable X falls within any circle;

[0230] The probability density function of the random variable X is:

[0231] f(x)=x·exp(-x 2 / 2),x>0 (22)

[0232] The distribution function of X is:

[0233] F(x)=P(X<x)=1-exp(-x 2 / 2),x>0 (23)

[0234] Then the random variable X falls within the radius r i The probability of being inside the circle is:

[0235]

[0236] The random variable X falls on any ring (r i-1 , r i ) with a probability P i for:

[0237]

[0238] Substituting r0 and r1 obtained by S4 into formula (25) we can get P i is: 2.7956×10 -4 .

[0239] S6. Calculate the probability that the random variable X falls within the total failure region, which is the failure probability of the stress-strength interference model;

[0240] Any ring (r i-1 , r i ) area S i for:

[0241]

[0242] Where Δr = r i -r i-1is the difference between the radii of two adjacent circles, which is the calculation step size.

[0243] The ring (r i-1 , r i) The area of ​​the inner failure zone for:

[0244]

[0245] in,

[0246]

[0247] The random variable X is in the ith ring (r i-1 , r i ) falls within the failure zone in the interference zone i for:

[0248]

[0249] Then the probability F of the random variable X falling into the total failure area, that is, the failure probability of the stress-strength interference model is:

[0250]

[0251] Solve S4 and S5 for r0, r1, P i Substituting into formulas (26) to (30), we can get F as: 1.647×10 -6 .

[0252] S7: Determine whether the calculation accuracy meets the set accuracy requirements;

[0253] The judgment is based on:

[0254] P(X>r i )<P *

[0255] When the judgment requirement is not met, r0=r i , i=i+1, repeat steps S5 to S7, otherwise, the calculation stops.

[0256] We obtain P(X>r1)=0.4307 <P * , the requirement is not met, so r0=r1; i=2 is substituted into steps S5~S7 to continue solving.

[0257] S8: Output failure probability of stress-strength interference model.

[0258] When the calculation accuracy requirements of S7 are met, the probability F of the output random variable X falling into the total failure area is the failure probability of the stress-strength interference model.

[0259] When the calculation accuracy requirement of S7 is met, the output of F solved by S6 is 0.0972526, then the failure probability of the stress interference-strength interference model is 9.725%, and the reliability of the ball screw pair is 90.275%.

[0260] The reliability of the ball screw pair calculated by the present invention takes into account the uncertainty of the design and processing of the process parameters of the ball screw pair and the random uncertainty of the load, calculates the distribution range of the rated dynamic load and the equivalent dynamic load of the ball screw pair, and then calculates the failure probability of the interference area. Based on the interference area of ​​the rated dynamic load and the equivalent dynamic load, a new method for calculating the reliability of the ball screw pair is provided, which is of great significance for improving the performance and increasing the reliability of the ball screw pair.

[0261] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only illustrative of the principles of the present invention. Without departing from the spirit and scope of the present invention, any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included in the scope of protection of the present invention.

Claims

1. A ball screw pair reliability analysis method based on stress-strength model and interference theory, characterized in that: The method comprises the following steps: Step 1: Calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ; Step 2: Calculate the mean value μ of the rated dynamic load using the Taylor series expansion s and standard deviation σ s ; Step 3: Calculate the mean value μ of the equivalent dynamic load r and standard deviation σ r ; Step 4: solve the reliability index of the stress-strength interference model and set the calculation step size and calculation accuracy requirements; Step 5, calculate the probability that the random variable X falls within any circle; Step 6: Calculate the probability that the random variable X falls in the total failure area, which is the failure probability of the stress-strength interference model; Step 7: Determine whether the calculation accuracy meets the set calculation accuracy requirement. If so, execute step 8; otherwise, return to step 5. Step 8: Output the failure probability of the stress-strength interference model; In step 1, the key process parameter values ​​of the ball screw pair are treated as random processes, and the corresponding distribution is normal distribution, with a mean μ x and standard deviation σ x Specifically: m x =(μ Dw ,m Dpw ,m rs ,m rn ,m β ,m α ) s x =(s Dw ,s Dpw ,s rs ,s rn ,s β ,s α ) Among them, (μ Dw ,μ Dpw ,μ rs ,μ rn ,μ β ,μ α ), (σ Dw ,σ Dpw ,σ rs ,σ rn ,σ β ,σ α ) are the parameters of ball diameter D w , pitch circle diameter D pw , Screw raceway radius r s , nut raceway radius r n , the mean and standard deviation of the helix angle β and the contact angle α between the ball and the raceway; Step 2: Solve the mean value μ of the rated dynamic load according to the Taylor series expansion s and standard deviation σ s , specifically: The model for the dynamic load rating is: C a =f(D w ,D pw ,r s ,r n ,β,α…)=C i ·i 0.85 in, in, Where C a is the rated dynamic load; i is the number of balls in the bearing circle; f is the geometric coefficient; α is the contact angle between the ball and the raceway; z1 is the number of effective bearing balls in each ball circle; D w Ball diameter; D pw is the pitch diameter; β is the helix angle; f1, f2, f3 are geometric shape coefficients; γ is the structural coefficient; r s is the screw raceway radius; r n is the nut raceway radius; f rs 、f rn are the adaptability of the screw raceway and the nut raceway respectively; z u is the number of unloaded balls in each ball ring; Substituting the mean value of each process parameter into the corresponding process parameter variable in the rated dynamic load model, the mean value μ of the rated dynamic load can be obtained. s for: m s =f(μ x )=f(μ Dw ,m Dpw ,m rs ,m rn ,m β ,m α ) Standard deviation of rated dynamic load σ s According to the Taylor series expansion and ignoring the higher-order terms, the approximate value is obtained from the following formula: Where x i Take D w 、D pw 、r s、 r n , β and α, σ xi Represents x i The corresponding standard deviation, n represents x i The number of values ​​of ; Step 3 solves the mean value μ of the equivalent dynamic load r and standard deviation σ r , specifically including: The model of equivalent dynamic load is: in, Where, F m is the equivalent dynamic load; n m is the equivalent speed; F j is the load value within a working cycle, n j is the speed value of the jth working unit in a working cycle; q j is the proportion of load or speed in a working cycle; n' is the number of working units in a working cycle; Then the mean value of the equivalent dynamic load μ r for: m r =F m Standard deviation σ of equivalent dynamic load r for:

2. The ball screw pair reliability analysis method based on stress-strength model and interference theory according to claim 1 is characterized in that: Step 4 solves the stress-strength interference model reliability index and sets the calculation step size and calculation accuracy requirements, specifically including: If the rated dynamic load and equivalent dynamic load are normally distributed, then the variable distribution of the stress-strength interference model is also normally distributed, with mean and standard deviation μ respectively. z =(μ s -μ r )and Its reliability index β is: Set r0 = β, r i =r0+i×Δr, where Δr is the calculation step size, r0 is the initial radius of any circle, r i is the radius of the i-th iteration; Set the calculation precision requirement P * .

3. The ball screw pair reliability analysis method based on stress-strength model and interference theory according to claim 2, characterized in that: Step 5 calculates the probability of the random variable X falling within any circle, specifically including: The probability density function of the random variable X is: f(x)=x·exp(-x 2 / 2),x>0 In the formula, x represents only the function variable; The distribution function of X is: F(x)=P(X<x)=1-exp(-x 2 / 2),x>0 Then the random variable X falls within the radius r i The probability P(X<r i )for: The random variable X falls on any ring (r i-1 , r i ) with a probability P i for:

4. The ball screw pair reliability analysis method based on stress-strength model and interference theory according to claim 3 is characterized in that: The probability of the random variable X falling into the total failure zone calculated in step 6 is the failure probability of the stress-strength interference model, which specifically includes: Any ring (r i-1 , r i ) area S i for: Where Δr = r i -r i-1 is the difference between the radii of two adjacent circles, i.e., the calculation step length; The ring (r i-1 , r i ) The area of ​​the failure zone for: in, The random variable X is in the ith ring (r i-1 , r i ) falls within the failure zone in the interference zone i for: Then the probability F of the random variable X falling into the total failure area, that is, the failure probability of the stress-strength interference model is: Where n" represents the number of rings.

5. The ball screw pair reliability analysis method based on stress-strength model and interference theory according to claim 4, characterized in that: In step 7, it is determined whether the calculation accuracy meets the set calculation accuracy requirement. If so, step 8 is executed; otherwise, the process returns to step 5, which specifically includes: The judgment is based on: P(X>r i )<P * Where, P(X>r i ) indicates that the random variable X falls within a radius of r i The probability of being outside the circle, P * Indicates the calculation accuracy requirements set; When the above judgment criteria are not met, let r0 = r i , i=i+1, repeat steps 5 to 7, otherwise, the calculation stops.

6. A ball screw pair reliability analysis system based on a stress-intensity model and interference theory according to the method of any one of claims 1 to 5, characterized in that: The system includes the following steps: The first module is used to calculate the mean μ of each key process parameter of the ball screw pair according to the design requirements and processing technical conditions. x and standard deviation σ x ; The second module is used to solve the mean value μ of the rated dynamic load based on the Taylor series expansion s and standard deviation σ s ; The third module is used to solve the mean value μ of the equivalent dynamic load r and standard deviation σ r ; The fourth module is used to solve the reliability index of the stress-strength interference model and set the calculation step size and calculation accuracy requirements; The fifth module is used to calculate the probability that the random variable X falls within any circle; The sixth module is used to calculate the probability that the random variable X falls into the total failure area, which is the failure probability of the stress-strength interference model; The seventh module is used to determine whether the calculation accuracy meets the set calculation accuracy requirements. If so, the eighth module is executed, otherwise it returns to the fifth module; The eighth module is used to output the failure probability of the stress-strength interference model.

7. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the computer program, the method according to any one of claims 1 to 5 is implemented.

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