Method for analyzing reliability of power tool rest structure under hybrid uncertainty

Abstract

The invention discloses a method for analyzing the reliability of a power tool rest structure under hybrid uncertainty, and the method comprises the steps: proposing an improved KKT condition to remarkably improve the search efficiency of sample points through deep analysis of a key region where the precision of a model is influenced by the mixing of random and interval variables; and based on a learning function dynamic weight learning function, deeply searching an optimal sample point, and adaptively balancing global exploration and local development in a Kriging agent model construction process to avoid falling into local optimum. A power knife rest gear transmission structure case verifies that the performance function calling times are averagely reduced by 21.4 times compared with an existing method while the failure probability interval evaluation precision is guaranteed, the calculation efficiency is improved by about 34%, and the method can be widely applied to reliability optimization design of complex structures such as aerospace equipment and heavy machinery.

Description

Technical Field

[0001] The present invention belongs to the technical field of structural analysis of functional components of numerically controlled machine tools, and in particular relates to an adaptive structural reliability analysis method based on a Kriging model and a dynamic weighted projection profile. Background Art

[0002] The power tool holder is a key functional component of high-end CNC machine tools such as turning centers and turning-milling composite machining centers. Uncertain factors such as bearing stiffness, material properties, and complex load conditions in the power tool holder structure affect the overall reliability of the machine tool. Analyzing and determining the impact of these uncertain factors on the structure has important research and application value.

[0003] In the structural reliability analysis under mixed random and cognitive variables, many studies have referred to the reliability analysis methods under random variables and focused on finding sample points near the limit state surface to update the model. Although this can ensure the accuracy of the evaluation results, it does not fully consider the characteristics of reliability analysis under mixed variable problems. Currently, the key areas that affect the accuracy of the model under mixed variables have not been deeply studied. Therefore, how to lock the more critical approximate area based on the properties of cognitive variables (such as interval variables) and focus the sampling process on this area to maintain accuracy while reducing computational costs still requires in-depth research. Summary of the Invention

[0004] The purpose of the present invention is to solve the above problems and provide a method for analyzing the reliability of a power tool holder structure under mixed uncertainty.

[0005] A reliability analysis method for a power tool holder structure under mixed uncertainty includes:

[0006] S1. Generate candidate sample points and obtain a sample library of power tool holder structural parameters:

[0007] The structural parameters of the power tool holder include n random variables and m interval vectors, which are generated by MCS method. candidate sample points, and obtain the candidate sample set Ω;

[0008] For n random variables, generate according to the probability distribution function of each variable samples; for m interval variables, generate uniformly within the value interval of each interval variable samples;

[0009] S2. Generate the initial experimental design DoE sample and obtain the initial training sample library of the power tool holder structural parameters:

[0010] For n random variables, according to the 5-Sigma criterion, use the LHS method to calculate the For m interval variables, LHS method is used to generate them respectively in Endogenous;

[0011] The number of initial training sample points is set to max{12,n+m+2}, and the new sample points are added to the training sample set S;

[0012] S3. Construct a Kriging model based on the training sample set of the power tool holder structure and calculate the predicted mean and variance of each point:

[0013] Calculate the corresponding performance function response value based on the initial DoE , where 0.003 is the failure threshold of the power tool holder structure, The displacement of the output shaft end of the power tool holder is obtained through finite element analysis, and the Kriging model is constructed;

[0014] According to the current Kriging model, the predicted mean and predicted variance of each point in the candidate sample point set S are calculated, and the failure probability of the power tool holder structure is estimated to obtain the failure probability of the structure;

[0015] S4. Determine whether the model meets the stopping criteria:

[0016] By using the constructed Kriging model to calculate the predicted mean and predicted variance of each point in the candidate sample point set Ω, the stopping criterion is used to determine whether the model meets the accuracy requirements.

[0017] If the model meets the stopping criteria, stop updating the model and execute step S6; otherwise, execute step S5;

[0018] The stopping criterion is to ensure that the model is stable before stopping. When the stopping criterion of the U learning function and the stopping criterion PU are met at the same time, the model update stops;

[0019] S5. Get new training sample points:

[0020] 1) For a power tool holder structure containing random and interval variables, its function The limit state surface of can be described in Cartesian coordinate system as:

[0021] ;

[0022] 2) For a function with n-dimensional random variables and m-dimensional interval variables, divide the uncertainty space into x and y subspaces and define a projection plane parallel to the n random variables , project the limit state surface along the y subspace direction to On the limit state surface, two projection boundaries are obtained. The projection contour on the limit state surface can be expressed as:

[0023] ;

[0024] ;

[0025] 3) The value of the interval variable y at the point on the projected contour satisfies the KKT condition:

[0026] a. The function takes the maximum value, satisfy:

[0027] ;

[0028] b. The function takes the minimum value, satisfy:

[0029] ;

[0030] Where, represents the true response value of the functional function, and Represent interval variables The lower and upper bounds of the j-th element of and Respectively and The jth element in ;

[0031] c. Relax the original KKT condition to search for points within a certain range near the projected contour:

[0032] ;

[0033] ;

[0034] Where, represents the Kriging model prediction value of the performance function, represents the boundary search range of the jth interval variable, represents the gradient search range of the jth interval variable; where, and The expression is as follows:

[0035] ;

[0036] ;

[0037] ;

[0038] Where m is the number of interval variables, is the function for taking the median;

[0039] d. Construct a learning function, namely:

[0040] ;

[0041] ;

[0042] Where, represents the absolute value of the predicted value of the performance function, represents the predicted standard deviation of the performance function, k is the number of model updates, m is the number of interval variables, and n is the number of random variables. It is a function that represents the degree of proximity between the sample point and the potential projection contour. The smaller its value is, the closer it is to the potential projection contour curve. The expression is as follows:

[0043] ;

[0044] Where, The function that represents the proximity of the jth interval variable to the interval boundary or interval extreme point is expressed as follows:

[0045] ;

[0046] e. Find the best sample point After that, the sample points Conduct a local re-search for the center;

[0047] For the random variable part, the search range is , for The value of the i-th random variable in , is the standard deviation of the i-th random variable;

[0048] For interval variables, the search range is as follows:

[0049] ;

[0050] Where, for The value of the jth interval variable in ;

[0051] f. Perform local sampling on the search range, extract 10,000 sample points by the LHS method, and calculate the learning function value respectively, select the smallest one as the new training sample point to add to the training sample point set to update the Kriging model; execute step S3;

[0052] S6. Evaluate the interval of failure probability:

[0053] By using the MCS method, the maximum and minimum values ​​of the failure probability can be estimated respectively to obtain the upper and lower limits of the failure probability interval. When the sample size is large enough, the minimum value of the failure probability is It can be estimated by the following formula:

[0054] ;

[0055] Likewise, the maximum failure probability It can be estimated by the following formula:

[0056] ;

[0057] Where, is an indicator function, which indicates whether the minimum value of the function is in the failure domain when the random variable is x. It can be expressed as:

[0058] ;

[0059] When random and interval variables are mixed, the function The value of is an interval, which is divided into three cases according to whether the interval contains 0 value:

[0060] (1) The maximum value of the function is less than 0, that is , at this time the value of the function must be less than 0;

[0061] (2) The minimum value of the functional function is greater than 0, that is , at this time the value of the function must be greater than 0;

[0062] (3) The maximum value of the function is greater than 0, and the minimum value is less than 0, that is, , , at this time the value of the function must be greater than 0.

[0063] According to the number of sample points in case (1), the minimum value of the failure probability is solved ; The number of sample points in cases (2) and (3) to find the maximum failure probability ,

[0064] S7. Verify the results and calculate whether the coefficient of variation meets the requirements:

[0065] To ensure that the number of sample points in the sample point set Ω is sufficient for the evaluation of the failure probability, it is necessary to calculate the coefficient of variation of the upper and lower bounds of the failure probability:

[0066] ;

[0067] ;

[0068] When the coefficient of variation is less than 0.05, it is considered that the number of sample points is sufficient; execute step S8;

[0069] Otherwise, the candidate sample point set Ω is expanded and step S3 is executed;

[0070] S8. After the Kriging model is constructed, the failure probability interval value that meets the accuracy requirements is obtained, and the reliability analysis of the power tool holder structure under mixed uncertainty is completed;

[0071] The stopping criterion of the U learning function described in step S4 is:

[0072] , ;

[0073] The impact of the optimal sample point on the model can be determined by the degree of change in the estimated failure probability before and after the optimal sample point is added:

[0074] ;

[0075] Where, represents the estimated failure probability after the mth adaptive addition;

[0076] if Less than a certain threshold twice in a row , it is believed that most of the points that have a greater impact on the model are added to the training sample point set;

[0077] The stopping criterion PU can be expressed as:

[0078] ;

[0079] The certain threshold value described in step S4 Take 0.01;

[0080] Step S5 Take 0.01; when the overall gradient is small, take ;

[0081] The results of MCS will be passed The maximum and minimum values ​​of the function are obtained by taking the random variable sample points on the interval variable. uniform samples to calculate, where m is the number of sample points in the interval;

[0082] The random variables in step S1 include: the stiffness of the driving gear, transition gear, transition gear and output gear of the power tool holder structure; the interval variables include: the feed force in the x and y directions of the power tool holder structure;

[0083] Step S1 generates by MCS method Random sample points, the maximum and minimum values ​​of the function are obtained by taking uniform samples to calculate, where m is the number of sample points in the interval;

[0084] The Kriging model was established using the DACE toolbox in MATLAB.

[0085] The present invention provides a method for analyzing the reliability of a power tool holder structure under mixed uncertainty. The method is an adaptive structural reliability analysis method based on the Kriging model and the dynamic weight projection profile under a mixture of random and interval variables. The method deeply analyzes the key areas that affect the accuracy of the model when random and interval variables are mixed, and converts the approximation of the area into an approximation of the projection profile, thereby improving the search efficiency of the optimal sample point. Then, a balance is achieved between the global and local structures of the model by introducing a dynamic weight learning function. Finally, the dynamic weight learning function is used to further search in the local area to obtain the optimal sample point.

[0086] Beneficial effects of the present invention:

[0087] 1. The stopping criterion based on prediction uncertainty (PU criterion) enables the model to stop in time when the accuracy is met, further improving the analysis efficiency.

[0088] 2. By introducing a dynamic weight learning function, a balance is achieved between the global and local construction of the model, which improves the search efficiency of the optimal sample point.

[0089] 3. Use the dynamic weight learning function to further search in the local area to obtain the best sample point, thereby reducing the number of function calls while improving accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] Figure 1 The flowchart of the adaptive structural reliability analysis method based on Kriging model and dynamic weighted projection contour considering the mixture of random and interval variables;

[0091] Figure 2 It is a three-dimensional diagram of the machine tool power tool holder;

[0092] Figure 3 This is the result of the dynamic simulation of the gear transmission structure of the machine tool power tool holder;

[0093] Figure 4 This is the total displacement result diagram of the power tool holder output position. DETAILED DESCRIPTION

[0094] Example 1 A method for analyzing the reliability of a power tool holder structure under mixed uncertainty

[0095] like Figure 1, a reliability analysis method for power tool holder structure under mixed uncertainty, the steps are as follows:

[0096] S1. Generate candidate sample points and obtain a sample library of power tool holder structural parameters:

[0097] The structural parameters of the power tool holder include n random variables and m interval vectors, which are generated by MCS method. candidate sample points, and obtain the candidate sample point set Ω. For n random variables, generate according to the probability distribution function of each variable samples; for m interval variables, generate uniformly within the value interval of each interval variable samples.

[0098] S2. Generate the initial experimental design DoE sample and obtain the initial training sample library of the power tool holder structural parameters:

[0099] In order to establish the initial Kriging model, initial training sample points are required. Through the initial experimental design, the initial training sample point set is uniformly generated in the entire uncertainty space. The method for generating training sample points is as follows: for the n random variable parts, according to the 5-Sigma criterion, the LHS method is used to generate the initial training sample points in the first and second random variables. For m interval variables, LHS method is used to generate them respectively in The number of initial training sample points is usually set to max{12,n+m+2}, and the new sample points are added to the training sample point set S.

[0100] S3. Construct a Kriging model based on the training sample set of the power tool holder structure and calculate the predicted mean and variance of each point:

[0101] Calculate the corresponding performance function response value based on the initial DoE , where 0.003 is the failure threshold of the power tool holder structure, The displacement of the output shaft end of the power tool holder is obtained through finite element analysis, and the Kriging model is constructed;

[0102] In the present invention, the DACE toolbox in MATLAB is used to establish the Kriging model.

[0103] According to the current Kriging model, the predicted mean and predicted variance of each point in the candidate sample point set S are calculated, and the failure probability of the structure is estimated to obtain the failure probability of the structure.

[0104] S4. Determine whether the model meets the stopping criteria:

[0105] The constructed Kriging model is used to calculate the predicted mean and predicted variance of each point in the candidate sample point set Ω, and the stopping criterion is used to determine whether the model meets the accuracy requirements.

[0106] In the present invention, in order to avoid the problem of incorrect stopping due to low model accuracy in the early stage of model training, it is ensured that the model is stable before stopping, and the model update is stopped when the stopping criteria of the U learning function and the stopping criteria PU are met at the same time.

[0107] The stopping criterion of the U learning function is:

[0108] ,

[0109] The impact of the optimal sample point on the model can be determined by the degree of change in the estimated failure probability before and after the optimal sample point is added:

[0110]

[0111] Where, It represents the estimated failure probability after the mth adaptive addition. Less than a certain threshold twice in a row , it can be considered that most of the points that have a greater impact on the model are added to the training sample point set, and there are fewer points with higher uncertainty. The value can well reflect the accuracy of the model.

[0112] To ensure that the model stops when the accuracy is high, At the same time, it ensures that the point with the highest uncertainty in the model will not have a significant impact on the model. Therefore, the new stopping criterion PU can be expressed as:

[0113]

[0114] In the present invention, Take 0.01.

[0115] If the model meets the stopping criteria, the updating of the model is stopped and step S6 is executed; otherwise, step S5 is executed.

[0116] S5. Get new training sample points:

[0117] For the reliability analysis of the power tool holder structure under a mixture of random and interval variables, in addition to constructing the limit state surface, it is also necessary to construct a high-precision projection contour. Therefore, when adaptively adding sample points, you can focus on the vicinity of the projection contour.

[0118] For a structure containing random and interval variables, its function The limit state surface of can be described in Cartesian coordinate system as:

[0119]

[0120] For a function with n-dimensional random variables and m-dimensional interval variables, the uncertainty space is divided into x and y subspaces, and a projection plane parallel to the n random variables is defined. , project the limit state surface along the y subspace direction to On the limit state surface, two projection boundaries are obtained. The projection contour on the limit state surface can be expressed as:

[0121]

[0122]

[0123] The points on the projected contour are not only located on the limit state surface, but also at the extreme points of the performance function, so the values ​​of the interval variable y at these points satisfy the KKT conditions (Karush–Kuhn–Tucker conditions):

[0124] The function takes the maximum value, satisfy:

[0125]

[0126] The function takes the minimum value, satisfy:

[0127]

[0128] Where, represents the true response value of the functional function, and Represent interval variables The lower and upper bounds of the j-th element of and Respectively and The jth element in .

[0129] Based on this, a sample point search strategy based on the projection contour can be obtained to obtain points near the projection contour. On this basis, in order to avoid the problem of too few points near the projection contour when the variable dimensions are large, the sample point search strategy is improved and the search range is appropriately relaxed according to the variable dimensions.

[0130] Since the extracted candidate sample point set is discrete, it is very difficult to find a point exactly on the projection contour directly through the original KKT. In order to reduce the search difficulty, the original KKT condition is relaxed to search for points within a certain range near the projection contour:

[0131]

[0132]

[0133] Where, represents the Kriging model prediction value of the performance function, represents the boundary search range of the jth interval variable, Represents the gradient search range of the jth interval variable. and The expression is as follows:

[0134]

[0135]

[0136]

[0137] Where m is the number of interval variables, is the function for finding the median. The search range for the boundaries of the interval variable The gradient search range of interval variables increases with the number of interval variables to reduce the difficulty of searching and avoid too few sample points when there are many interval variables. It can be automatically adjusted based on the gradient information of each point in the current model. Usually, Take 0.01 to filter out sample points near the extreme point. However, when the overall gradient is small, many sample points will be searched out. Therefore, when the overall gradient is small, take Through the above improved sample point search strategy, we can Search for the candidate sample points closest to the projection contour points, forming a sample point set ,When selecting the best sample points, selecting them from this set of points can help ,construct high-precision projection contour curves and improve ,modeling efficiency.

[0138] The sample points in the candidate sample point set Ω are preliminarily screened by the improved sample point search strategy to obtain the sample point set Then, based on the projection contour method, a point adding method for the mixed case of random and interval variables is proposed. The point adding strategy of the functional function in this method is mainly divided into two stages: the early stage focuses on the global exploration of the limit state surface, and this stage is mainly to construct the approximate limit state surface; the later stage focuses on the local exploration of the projection contour. In this stage, points that are more helpful in constructing the projection contour will be given higher weights. It can be seen that this method can balance global accuracy and local accuracy at the same time.

[0139] Based on the above analysis, a learning function is constructed, namely:

[0140]

[0141]

[0142] Where, represents the absolute value of the predicted value of the performance function, represents the predicted standard deviation of the performance function, k is the number of model updates, m is the number of interval variables, and n is the number of random variables. It is a function that represents the degree of proximity between the sample point and the potential projection contour. The smaller its value is, the closer it is to the potential projection contour curve. The expression is as follows:

[0143]

[0144] Where, The function that represents the proximity of the jth interval variable to the interval boundary or interval extreme point is expressed as follows:

[0145]

[0146] The smaller the value, the closer the j-th interval variable value of the sample point is to the interval boundary or the smaller the gradient is, that is, the closer it is to the potential projection contour.

[0147] The best sample points selected by the learning function are all from the candidate sample point set. Due to the number of sample points in the candidate sample point set, the sample points near the projection contour are relatively rare or have a low density compared to the entire value range. Therefore, there may be some points near the best sample point that are not included in the candidate sample point set. They are not only close to the best sample point, but also close to the best sample point. , and with Compared with the previous example, the probability of the symbol being predicted incorrectly is higher or closer to the potential projection contour, so it is obviously more efficient to add these sample points to the training sample point set to update the Kriging model. Based on this idea, when finding the best sample point through the learning function Afterwards, it will be Further search is performed nearby to find new training sample points to improve the convergence speed of the method in this section. The following describes the implementation method of the search:

[0148] Find the best sample point After that, the sample points For the random variable part, the search range is , for The value of the i-th random variable in , is the standard deviation of the i-th random variable. For the interval variable, the search range is as follows:

[0149]

[0150] Where, for Next, perform local sampling on the search range, extract 10,000 sample points using the LHS method, calculate their learning function values, and select the smallest one as a new training sample point to add to the training sample point set and update the model. Then execute step S3.

[0151] S6. Evaluate the interval of failure probability:

[0152] By using the MCS method, the maximum and minimum values ​​of the failure probability can be estimated respectively to obtain the upper and lower limits of the failure probability interval. When the sample size is large enough, the minimum value of the failure probability is It can be estimated by the following formula:

[0153] ;

[0154] Likewise, the maximum failure probability It can be estimated by the following formula:

[0155] ;

[0156] Where, is an indicator function, which indicates whether the minimum value of the function is in the failure domain when the random variable is x. It can be expressed as:

[0157] ;

[0158] When random and interval variables are mixed, the function The value of is an interval, which is divided into three cases according to whether the interval contains 0 value:

[0159] (1) The maximum value of the function is less than 0, that is , at this time the value of the function must be less than 0;

[0160] (2) The minimum value of the functional function is greater than 0, that is , at this time the value of the function must be greater than 0;

[0161] (3) The maximum value of the function is greater than 0, and the minimum value is less than 0, that is, , at this time the value of the function must be greater than 0.

[0162] Find the minimum value of failure probability When , we need to pay attention to the number of sample points in the first case and solve the maximum failure probability When , we need to pay attention to the number of sample points in the first and third cases. In this invention, the result of MCS will be The maximum and minimum values ​​of the function are obtained by taking the random variable sample points on the interval variable. The calculation is done using uniform samples, where m is the number of sample points in the interval.

[0163] Conventional MCS needs to traverse all interval sample points when solving the upper and lower limits of failure probability, so it is necessary to call This will consume a lot of computing resources. and When calculating the maximum value of the function, we only need to solve the sign of the maximum value point of the function without knowing the specific value. Therefore, when calculating the maximum value of the function, if both positive and negative values ​​appear at the same time, we can determine it as the third case mentioned above without traversing This can reduce the computational burden and improve computational efficiency to a certain extent.

[0164] S7. Verify the results and calculate whether the coefficient of variation meets the requirements:

[0165] To ensure that the number of sample points in the sample point set Ω is sufficient for the evaluation of the failure probability, it is necessary to calculate the coefficient of variation of the upper and lower bounds of the failure probability:

[0166] ;

[0167] ;

[0168] When the coefficient of variation is less than 0.05, it is considered that the number of sample points is sufficient and the evaluation of the failure probability has high robustness. Otherwise, the candidate sample point set Ω is expanded and step S3 is executed.

[0169] S8. The Kriging model is constructed and the failure probability interval value that meets the accuracy is obtained, completing the structural reliability analysis under mixed uncertainty.

[0170] Example 2: Power tool holder structure implementation case

[0171] The gear transmission structure of the machine tool tool holder power head is as follows Figure 2 As shown in the figure, when the structure is running, the uncertain factors in the structure will cause the output shaft to vibrate, which will reduce the transmission accuracy of the system. According to the machining accuracy requirements in actual engineering, when the maximum displacement of the power head simulation result is When it is greater than 0.003 mm, the structure is considered to be failed. Therefore, the functional function response value corresponding to the gear transmission structure of the tool holder power head of the machine tool can be expressed as:

[0172]

[0173] The value of this function can be obtained through transient dynamic finite element analysis, and its displacement analysis results are as follows: Figure 3 As shown, the radial displacement of the power head output position is as follows Figure 4 shown.

[0174] In this case, there are 11 uncertain variables, including 7 random variables and 2 interval variables. The detailed distribution information of the random variables is as follows:

[0175]

[0176] where K b1 -K b4 The bearing stiffness of the four gear shafts, namely the driving gear, small transition gear, large transition gear, and output gear, k g is the gear pair stiffness, Motor speed, M is the working torque of the power head.

[0177] The distribution information of interval variables is:

[0178]

[0179] Among them, F x and F y is the feed force in the x and y directions.

[0180] In this case, the MCS method is used to generate For each random sample point, 400 uniform samples are taken on the interval variable to calculate the maximum and minimum values ​​of the performance function, and the upper and lower bounds of the failure probability are calculated. For both the POAL-Kriging and AK-DWP methods, the LHS method is used to generate 12 initial training sample points.

[0181] The detailed results of several analysis methods are:

[0182]

[0183] It can be seen that the number of function calls of the POAL-Kriging method is 12+51.5 times, while the number of function calls of the AK-DWP method proposed in this paper is only 12+29.1 times. Obviously, the AK-DWP method greatly reduces the number of function calls and improves the analysis efficiency.

[0184] In terms of accuracy, the minimum failure probability calculated by the MCS method is 0.458×10 -2 The maximum failure probability is 0.561×10 -2 Taking the calculation results of the MCS method as the standard, the relative error of the calculation results of the POAL-Kriging method is and The relative errors of the calculation results of the AK-DWP method are 0.94% and 2.26% respectively. and are 0.43% and 1.79% respectively. It can be seen that the AK-DWP method solves and The accuracy of the AK-DWP method is better than that of the POAL-Kriging method. Obviously, in this case, the accuracy of the AK-DWP method is higher than that of the POAL-Kriging method.

Claims

1. A reliability analysis method for a power tool holder structure under mixed uncertainty, characterized by: S1. Generate candidate sample points and obtain a sample library of power tool holder structural parameters: The structural parameters of the power tool holder include n random variables and m interval vectors, which are generated by MCS method. candidate sample points, and obtain the candidate sample set Ω; For n random variables, generate according to the probability distribution function of each variable samples; for m interval variables, generate uniformly within the value interval of each interval variable samples; S2. Generate initial DoE samples and obtain the initial training sample library of the power tool holder structural parameters: For n random variables, according to the 5-Sigma criterion, use the LHS method to calculate the For m interval variables, LHS method is used to generate them respectively in Endogenous; The number of initial training sample points is set to max{12,n+m+2}, and the new sample points are added to the training sample set S; S3. Construct a Kriging model based on the training sample set of the power tool holder structure and calculate the predicted mean and variance of each point: Calculate the corresponding performance function response value based on the initial DoE , where 0.003 is the failure threshold of the power tool holder structure, The displacement of the output shaft end of the power tool holder is obtained through finite element analysis and the Kriging model is constructed; According to the current Kriging model, the predicted mean and predicted variance of each point in the candidate sample point set S are calculated, and the failure probability of the power tool holder structure is estimated to obtain the failure probability of the structure; S4. Determine whether the model meets the stopping criteria: By using the constructed Kriging model to calculate the predicted mean and predicted variance of each point in the candidate sample point set Ω, the stopping criterion is used to determine whether the model meets the accuracy requirements; If the model meets the stopping criteria, stop updating the model and execute step S6; otherwise, execute step S5; The stopping criterion is to ensure that the model is stable before stopping. When the stopping criterion of the U learning function and the stopping criterion PU are met at the same time, the model update stops; S5. Get new training sample points: 1) For a power tool holder structure containing random and interval variables, its function The limit state surface of can be described in Cartesian coordinate system as: ； 2) For a function with n-dimensional random variables and m-dimensional interval variables, divide the uncertainty space into x and y subspaces and define a projection plane parallel to the n random variables , project the limit state surface along the y subspace direction to On the limit state surface, two projection boundaries are obtained. The projection contour on the limit state surface can be expressed as: ； ； 3) The value of the interval variable y at the point on the projected contour satisfies the KKT condition: a. The function takes the maximum value, satisfy: ； b. The function takes the minimum value, satisfy: ； Where, represents the true response value of the functional function, and Represent interval variables The lower and upper bounds of the j-th element of and Respectively and The jth element in ; c. Relax the original KKT condition to search for points within a certain range near the projected contour: ； ； Where, represents the Kriging model prediction value of the performance function, represents the boundary search range of the jth interval variable, represents the gradient search range of the jth interval variable; where, and The expression is as follows: ； ； ； Where m is the number of interval variables, is the function for taking the median; d. Construct a learning function, namely: ； ； Where, represents the absolute value of the predicted value of the performance function, represents the predicted standard deviation of the performance function, k is the number of model updates, m is the number of interval variables, and n is the number of random variables. It is a function that represents the degree of proximity between the sample point and the potential projection contour. The smaller its value is, the closer it is to the potential projection contour curve. The expression is as follows: ； Where, The function that represents the proximity of the jth interval variable to the interval boundary or interval extreme point is expressed as follows: ； e. Find the best sample point After that, the sample points Conduct a local re-search for the center; For the random variable part, the search range is , for The value of the i-th random variable in , is the standard deviation of the i-th random variable; For interval variables, the search range is as follows: ； Where, for The value of the jth interval variable in ; f. Perform local sampling on the search range, extract 10,000 sample points by the LHS method, and calculate the learning function value respectively, select the smallest one as the new training sample point to add to the training sample point set to update the Kriging model; execute step S3; S6. Evaluate the interval of failure probability: By using the MCS method, the maximum and minimum values ​​of the failure probability can be estimated respectively to obtain the upper and lower limits of the failure probability interval. When the sample size is large enough, the minimum value of the failure probability is It can be estimated by the following formula: ； Likewise, the maximum failure probability It can be estimated by the following formula: ； Where, is an indicator function, which indicates whether the minimum value of the function is in the failure domain when the random variable is x. It can be expressed as: ； When random and interval variables are mixed, the function The value of is an interval, which is divided into three cases according to whether the interval contains 0 value: (1) The maximum value of the function is less than 0, that is , at this time the value of the function must be less than 0; (2) The minimum value of the functional function is greater than 0, that is , at this time the value of the function must be greater than 0; (3) The maximum value of the function is greater than 0, and the minimum value is less than 0, that is, , , at this time the value of the function must be greater than 0; According to the number of sample points in case (1), the minimum value of the failure probability is solved ; The number of sample points in cases (2) and (3) to find the maximum failure probability , S7. Verify the results and calculate whether the coefficient of variation meets the requirements: To ensure that the number of sample points in the sample point set Ω is sufficient for the evaluation of the failure probability, it is necessary to calculate the coefficient of variation of the upper and lower bounds of the failure probability: ； ； When the coefficient of variation is less than 0.05, it is considered that the number of sample points is sufficient; execute step S8; Otherwise, the candidate sample point set Ω is expanded and step S3 is executed; S8. The Kriging model is constructed and the failure probability interval value that meets the accuracy is obtained, completing the reliability analysis of the power tool holder structure under mixed uncertainty.

2. The method for analyzing the reliability of a power tool holder structure under mixed uncertainty according to claim 1, characterized in that: The stopping criterion of the U learning function described in step S4 is: ， ； The impact of the optimal sample point on the model can be determined by the degree of change in the estimated failure probability before and after the optimal sample point is added: ； Where, represents the estimated failure probability after the mth adaptive addition; if Less than a certain threshold twice in a row , it is believed that most of the points that have a greater impact on the model are added to the training sample point set; The stopping criterion PU can be expressed as: 。 3. The method for analyzing the reliability of a power tool holder structure under mixed uncertainty according to claim 2, characterized in that: The certain threshold value described in step S4 Take 0.

01.

4. The method for analyzing the reliability of a power tool holder structure under mixed uncertainty according to claim 3 is characterized by: Step S5 Take 0.01; when the overall gradient is small, take .

5. The method for analyzing the reliability of a power tool holder structure under mixed uncertainty according to claim 1, 2, 3 or 4, characterized in that: The random variables described in step S1 include: the driving gear, transition gear, transition gear and output gear stiffness of the power tool holder structure; and the interval variables include: the feed force in the x and y directions of the power tool holder structure.

6. The method for analyzing the reliability of a power tool holder structure under mixed uncertainty according to claim 5, characterized in that: Step S1 generates by MCS method Random sample points, the maximum and minimum values ​​of the function are obtained by taking uniform samples to calculate, where m is the number of sample points in the interval.

7. The method for analyzing the reliability of a power tool holder structure under mixed uncertainty according to claim 6, characterized in that: The Kriging model was established using the DACE toolbox in MATLAB.

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