Mechanical structure reliability analysis method based on time-varying mixed Copula function

Abstract

The invention relates to the field of mechanical engineering, in particular to a mechanical structure reliability analysis method based on a time-varying mixed Copula function, which comprises the following steps of: firstly, obtaining structural stress and structural strength of a mechanical structure in a service period through finite element analysis or test, and calculating edge distribution characteristics of the mechanical structure and degradation amount of the structural strength along with time; the method comprises the following steps of: determining the structure reliability in a single failure mode and a multi-failure mode, further obtaining joint distribution and determining the type of a Copula function, judging the characteristics of the failure mode, respectively calculating the structure reliability in the single failure mode and the multi-failure mode, and finally, considering the correlation among multiple components, establishing a multi-component multi-failure mode T-Mc-Copula function, and calculating the structure reliability of the multi-component multi-failure mode T-Mc-Copula function. According to the method, the time-varying mixed Copula function can be used for representing the change rule of the correlation between the stress and the structural strength of the mechanical structure, the correlation of multiple failure modes and the correlation among multiple components along with time, a more accurate structural reliability analysis result is obtained, and the reasonability of reliability calculation of the mechanical structure is ensured.

Description

Technical Field

[0001] The present invention relates to the field of mechanical structure reliability analysis. Background Art

[0002] In the field of mechanical engineering, high reliability requirements for mechanical structures and systems are a crucial task throughout the entire equipment design process. However, with the increasing complexity of structures (the complexity of structural functional states, the interaction of multiple failure modes, the multi-source nature of influencing factors, etc.) and the extreme nature of service environments (multiple physical fields with time-varying loads such as thermal stress, vibration, and mechanical stress), mechanical equipment performance degradation involves multiple dynamic failure modes, and these failure modes are correlated, bringing new difficulties and challenges to structural reliability analysis. Mechanical structural failures typically involve multiple potential failure modes. Due to the homology of structural / system random variables (such as external loads, geometric parameters, and material properties), failure modes will have varying degrees of correlation, meaning that one failure may accelerate (or slow down) the occurrence of another. For major technical equipment, the reliability of multiple components under multiple failure modes is fundamental to ensuring their safe service.

[0003] Currently, among the structural reliability analysis methods that consider correlation, the methods based on deterministic objects have poor universality, the methods based on linear correlation description have large analysis errors, and the methods based on Copula correlation description lack consideration of performance degradation, multiple failures, and the relationship between multiple components. Summary of the Invention

[0004] In order to overcome the problems of poor universality, large errors, lack of consideration of performance degradation, multiple failures, and relationships between multiple components in traditional mechanical structure reliability analysis, the present invention provides a mechanical structure reliability analysis method based on time-varying hybrid Copula function.

[0005] The technical solution adopted by the present invention to achieve the above-mentioned object is: a mechanical structure reliability analysis method based on time-varying hybrid Copula function, comprising the following steps:

[0006] S1. Obtain the structural stress dataset and structural strength dataset of the mechanical structure to be analyzed during service through finite element analysis or experiments;

[0007] S2. Calculate the structural stress edge distribution characteristics, structural strength edge distribution characteristics, and structural strength degradation amount based on the structural stress data set and the structural strength data set, and then calculate the joint distribution of structural stress and structural strength;

[0008] S3. Determine the failure mode based on the structural stress data set and the structural strength data set. If it is a single component single failure mode, establish the T-Copula function under the failure mode and check the fitting accuracy to calculate the structural reliability. If it is a single component multiple failure modes, establish the T-Copula function under each failure mode respectively. f - Copula function and test the fitting accuracy to obtain structural reliability;

[0009] S4. Considering the correlation and strength degradation of multiple components, establish the multi-component multi-failure mode T-M c - Copula function and test the fitting accuracy to calculate the structural reliability.

[0010] Preferably, step S2 includes:

[0011] S2-1. Calculate the structural stress edge distribution characteristics based on the structural stress dataset and the structural strength dataset. and structural strength edge distribution characteristics , and Probability density function via Johnson SB distribution and the cumulative distribution function Mathematically expressed, the expression is: ; ;

[0012] in, , is a continuous shape parameter that controls the skewness of the distribution, is a continuous shape parameter that controls the kurtosis of the distribution, is the cumulative distribution function of the standard normal distribution, is the scale parameter, are continuous position parameters;

[0013] S2-2. According to the temporal variation characteristics of the structural strength data set, the random degradation of the structural strength is calculated by the linear gamma random process. The expression is: ; ;

[0014] in, is the expected value of random degradation of structural strength, is the variance of random degradation of structural strength, is a gamma random process, For time, is the shape function coefficient, is the size parameter coefficient;

[0015] S2-3, let the random variable in the data set be , and are two sets of sample values, if ,but and is positively correlated, if ,but and The rank correlation coefficient is used to measure the consistency of the structural stress data set and the structural strength data set to determine their correlation. The expression is: ;

[0016] in, is the length of the sample, ;

[0017] S2-4. Calculate the joint distribution of structural stress and structural strength : .

[0018] Preferably, step S3 includes:

[0019] S3-1. Based on the characteristics of structural stress edge distribution , structural strength edge distribution characteristics and the joint distribution of structural stress and structural strength , determine the Copula function type : ; ;

[0020] in, is the correlation parameter, are the marginal distribution function values of two random variables, ;

[0021] Based on the structural stress data set and the structural strength data set, the failure mode of the structure to be analyzed is determined. The structural stress data set and the structural strength data set are compared point by point to identify the area where the stress exceeds the strength. If the exceeded area is concentrated in a single location, it is a single failure mode. If the exceeded area is dispersed in multiple locations, it is a multiple failure mode.

[0022] S3-2. If it is a single failure mode, establish the limit state equation of the failure mode : ;

[0023] in, is the structural stress that varies with time, For structural strength subject to gamma degradation;

[0024] According to S3-1 and The correlation parameter and , establish a time-varying Clopula function, and consider the time-varying characteristics when the correlation degree parameter is expressed as , that is, T-Copula function ;

[0025] The T-Copula function goodness of fit is tested by the statistical square difference formula: ;

[0026] in, is the empirical distribution function;

[0027] Calculate structural reliability under single component single failure mode : ;

[0028] S3-3. If there are multiple failure modes, establish the limit state equation for each failure mode : ;

[0029] According to S3-1 and The correlation parameter and , establish the time-varying Clopula function of each failure mode, namely T-Copula function , and calculate the limit state function value under each failure mode according to the limit state equation ;

[0030] Consider the correlation between failure modes and select the best one The T-Copula function establishes a time-varying mixed Copula function with multiple failure modes, namely TM f -Copula function: ;

[0031] in, is the T-Copula function of the correlation of multiple failure modes of a single component, is the weight coefficient of each Copula function, is the time-varying related parameter in each Copula function;

[0032] By Akaike Information Criteria InspectionTMf -Copula function goodness of fit: ;

[0033] in, is the residual sum of squares, is the sample size, For TM f -The number of parameters to be estimated in the Copula function;

[0034] Calculate structural reliability under multiple failure modes of a single component : ;

[0035] in, for The distribution function of is the failure probability, For difference calculation, .

[0036] Preferably, step S4 includes: evaluating the correlation between components based on the time-varying hybrid reliability values of each component under multiple failure modes, and selecting the best TM f -Copula function, constructing TM between multiple components c -Copula function: ;

[0037] By Akaike Information Criteria InspectionTM c -Copula function goodness of fit: ;

[0038] Assume the number of multiple components is The mechanical structure is calculated to be Reliability of each component : .

[0039] Compared with the prior art, the present invention has the following beneficial effects:

[0040] The present invention uses a time-varying mixed Copula function to characterize the stress-strength correlation of mechanical structures, the correlation of multiple failure modes, and the temporal variation of the correlation between multiple components, thereby obtaining more accurate structural reliability analysis results with good universality and small error, thus ensuring the rationality of the mechanical structure reliability calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1This is a flow chart of a mechanical structure reliability analysis method based on a time-varying mixed Copula function according to an embodiment of the present invention;

[0042] Figure 2 1 is a schematic diagram of the distribution of tooth surface contact strength data of two failure modes of the driving wheel in an embodiment of the present invention;

[0043] Figure 3 1 is a schematic diagram of the distribution of contact stress data on the tooth surface of the driving wheel in two failure modes according to an embodiment of the present invention;

[0044] Figure 4 1 is a schematic diagram of the distribution of tooth surface bending strength data of two failure modes of the driving wheel in an embodiment of the present invention;

[0045] Figure 5 1 is a schematic diagram of the distribution of bending stress data on the tooth surface of the driving wheel in two failure modes according to an embodiment of the present invention;

[0046] Figure 6 Schematic diagram of a Copula function representing correlation in an embodiment of the present invention;

[0047] Figure 7 1 is a schematic diagram of the reliability calculation results of the driving wheel under the influence of stress-strength correlation in an embodiment of the present invention;

[0048] Figure 8 1 is a schematic diagram of the reliability calculation results of the driving wheel under the influence of failure mode correlation in an embodiment of the present invention;

[0049] Figure 9 2 is a schematic diagram of the reliability calculation results of the gear transmission system under the influence of the correlation between multiple parts in an embodiment of the present invention. DETAILED DESCRIPTION

[0050] like Figure 1 As shown, an embodiment of the present invention provides a mechanical structure reliability analysis method based on a time-varying mixed Copula function, comprising the following steps:

[0051] S1. Use finite element analysis or experiments to obtain the structural stress data set of the mechanical structure to be analyzed during its service life. and structural strength datasets ;

[0052] S2. Calculate the structural stress edge distribution characteristics, structural strength edge distribution characteristics, and structural strength degradation amount based on the structural stress data set and the structural strength data set, and then calculate the joint distribution of structural stress and structural strength;

[0053] S2-1. Calculate the structural stress edge distribution characteristics based on the structural stress dataset and the structural strength dataset. and structural strength edge distribution characteristics , and Probability density function via Johnson SB distribution and the cumulative distribution function Mathematically expressed, the expression is: ; ;

[0054] in, , is a continuous shape parameter that controls the skewness of the distribution, is a continuous shape parameter that controls the kurtosis of the distribution, is the cumulative distribution function of the standard normal distribution, is the scale parameter, are continuous positional parameters.

[0055] S2-2. According to the temporal variation characteristics of the structural strength data set, the random degradation of the structural strength is calculated by the linear gamma random process. The expression is: ; ;

[0056] in, is the expected value of random degradation of structural strength, is the variance of random degradation of structural strength, is a gamma random process, For time, is the shape function coefficient, is the size parameter coefficient.

[0057] S2-3, let the random variable in the data set be , and are two sets of sample values, if ,but and is positively correlated, if ,but and The rank correlation coefficient is used to measure the consistency of the structural stress data set and the structural strength data set to determine their correlation. The expression is: ;

[0058] in, is the length of the sample, .

[0059] S2-4. Calculate the joint distribution of structural stress and structural strength : .

[0060] S3. Determine the failure mode based on the structural stress data set and the structural strength data set. If it is a single component single failure mode, establish the T-Copula function under the failure mode and check the fitting accuracy to calculate the structural reliability. If it is a single component multiple failure modes, establish the T-Copula function under each failure mode respectively. f - Copula function and test the fitting accuracy to obtain structural reliability;

[0061] S3-1. In order to more accurately characterize the correlation between structural stress and structural strength, according to the structural stress edge distribution characteristics , structural strength edge distribution characteristics and the joint distribution of structural stress and structural strength , determine the Copula function type : ; ;

[0062] in, is the correlation parameter, are the marginal distribution function values of two random variables, ;

[0063] Based on the structural stress data set and the structural strength data set, the failure mode of the structure to be analyzed is determined. The structural stress data set and the structural strength data set are compared point by point to identify the area where the stress exceeds the strength. If the exceeded area is concentrated in a single location, it is a single failure mode. If the exceeded area is scattered in multiple locations, it is a multiple failure mode.

[0064] S3-2. If it is a single failure mode, establish the limit state equation of the failure mode : ;

[0065] in, is the structural stress that varies with time, For structural strength subject to gamma degradation;

[0066] According to S3-1 and The correlation parameter and , establish a time-varying Clopula function, and consider the time-varying characteristics when the correlation degree parameter is expressed as , that is, T-Copula function ;

[0067] The T-Copula function goodness of fit is tested by the statistical square difference formula: ;

[0068] in, is the empirical distribution function, The smaller the value, the better the T-Copula function represents the structural stress and strength-related characteristics, and the more accurate the reliability calculation under the single failure mode.

[0069] Calculate structural reliability under single component single failure mode : .

[0070] S3-3. If there are multiple failure modes, establish the limit state equation for each failure mode: ;

[0071] According to S3-1 and The correlation parameter and , establish the time-varying Clopula function of each failure mode, namely T-Copula function , and calculate the limit state function value under each failure mode according to the limit state equation ;

[0072] Consider the correlation between failure modes and select the best one The T-Copula function establishes a time-varying mixed Copula function with multiple failure modes, namely TM f -Copula function: ;

[0073] in, TM for correlation of multiple failure modes for a single component f -Copula function, is the weight coefficient of each Copula function, is the time-varying related parameter in each Copula function;

[0074] By Akaike Information Criteria InspectionTM f -Copula function goodness of fit: ;

[0075] in, is the residual sum of squares, is the sample size, For TM f -The number of parameters to be estimated in the Copula function, The larger the value, the higher the TM f The better the Copula function characterizes the time-varying correlation characteristics of multiple failure modes, the more accurate the reliability calculation results under multiple failure modes;

[0076] Calculate structural reliability under multiple failure modes of a single component : ;

[0077] in, for The distribution function of is the failure probability, For difference calculation, .

[0078] S4. Considering the correlation and strength degradation of multiple components, establish the multi-component multi-failure mode T-M c - Copula function and test the fitting accuracy to obtain structural reliability;

[0079] According to the time-varying mixed reliability value of each component under multiple failure modes, the correlation between the components is evaluated and the best one is selected. T-Copula function to build TM between multiple components c -Copula function: ;

[0080] By Akaike Information Criteria InspectionTM c -Copula function goodness of fit: ;

[0081] Assume the number of multiple components is The mechanical structure is calculated to be Reliability of each component : .

[0082] This embodiment provides the following calculation example:

[0083] Assuming that the main failure modes of the driving wheel are tooth surface contact fatigue and tooth root bending fatigue, and the failure mode of the driven wheel is tooth surface contact fatigue, the relevant parameter values are shown in the following table: ;

[0084] When constructing the limit state function for the three failure modes of the gear transmission system mentioned above, it is assumed that the structural strength and structural stress obey the normal distribution, the coefficient of variation is 0.001, and the gamma random process is used to represent the strength degradation. The strength and stress data are obtained. Taking the driving wheel as an example, Figure 2 As shown in the figure, the coordinate frequency represents the number of data points falling in the corresponding interval. Assuming that the strength stress value is 340-380 MPa and the time is 300-400 h, the frequency is 65, which means that within this stress range and time range, there are 65 tooth surface contact strength data observations in the data set that fall into this interval.

[0085] According to the above step S2-3, the rank correlation coefficient is used to measure the consistency, and the stress and strength rank correlation coefficient of the driving gear tooth surface contact fatigue failure mode is 0.3175, the stress and strength rank correlation coefficient of the driving gear tooth root bending fatigue failure mode is 0.9621, and the stress and strength rank correlation coefficient of the driven gear tooth surface contact fatigue failure mode is 0.8543, indicating that the correlations are all positive.

[0086] According to the above steps S2-S3, the Copula function type is determined and the corresponding reliability is calculated. The Frank Copula function is used to characterize the correlation between tooth surface contact fatigue and tooth root bending fatigue, and the statistical square tolerance method is used to test the function fitting goodness of fit. RSS=0.12, as shown in Figure 3 As shown in the figure, it can be seen that the characterization of the tail correlation characteristics of the limit state functions of the two failure behaviors is relatively realistic. Although the correlation representation in the middle part is slightly weaker, the overall representation of the correlation characteristics is relatively accurate.

[0087] Use Gumbel Copula function, Clayton Copula function and Frank Copula function to construct time-varying mixed Copula function, and calculate the corresponding weights to obtain TM f -Copula: ;

[0088] Finally, the reliability results of the driving wheel with and without considering the stress-strength correlation are obtained as follows: Figure 4 As shown in the figure, after considering the stress-strength correlation characteristics, the reliability is reduced regardless of tooth surface contact or tooth root bending, and the reliability decrease trend is greater after considering the stress-strength correlation, that is, the surface stress-strength correlation promotes the reliability calculation results under any failure mode of the gear.

[0089] According to the above step S3, the AIC criterion is used to test the goodness of fit of the time-varying mixed Copula function, which is 0.87. The reliability comparison results of considering the failure mode correlation and not considering the failure mode correlation are calculated as follows: Figure 5As shown in the figure, the two failure modes of tooth surface contact and tooth root bending affect each other, aggravating the gear failure process. The reliability under a single failure mode is too optimistic and cannot truly reflect the actual reliability of the gear during operation.

[0090] According to the above step S4, the reliability of the transmission system considering the component correlation is calculated as follows: Figure 6 As shown in the figure, the influence of multi-component correlation characteristics on reliability is similar to the influence of multi-failure mode correlation characteristics. The reliability calculated for a single component is higher than the reliability after considering the component correlation.

[0091] The present invention is described by way of example, and those skilled in the art will appreciate that various changes or equivalent substitutions may be made to these features and embodiments without departing from the spirit and scope of the present invention. Furthermore, under the teachings of the present invention, these features and embodiments may be modified to suit specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are intended to be within the scope of the present invention.

Claims

1. A mechanical structure reliability analysis method based on time-varying mixed Copula function, characterized in that: The following steps are involved: S1. Obtain the structural stress dataset and structural strength dataset of the mechanical structure to be analyzed during service through finite element analysis or experiments; S2. Calculate the structural stress edge distribution characteristics, structural strength edge distribution characteristics, and structural strength degradation amount based on the structural stress data set and the structural strength data set, and then calculate the joint distribution of structural stress and structural strength; S3. Determine the failure mode based on the structural stress data set and the structural strength data set. If it is a single component single failure mode, establish the T-Copula function under the failure mode and check the fitting accuracy to calculate the structural reliability. If it is a single component multiple failure modes, establish the T-Copula function under each failure mode respectively. f - Copula function and test the fitting accuracy to obtain structural reliability; S4. Considering the correlation and strength degradation of multiple components, establish the multi-component multi-failure mode T-M c - Copula function and test the fitting accuracy to calculate the structural reliability.

2. The mechanical structure reliability analysis method based on time-varying mixed Copula function according to claim 1 is characterized in that: The step S2 comprises: S2-1. Calculate the structural stress edge distribution characteristics based on the structural stress dataset and the structural strength dataset. and structural strength edge distribution characteristics , and Through the probability density function and the cumulative distribution function Mathematically expressed, the expression is: ； ； in, , is a continuous shape parameter that controls the skewness of the distribution, is a continuous shape parameter that controls the kurtosis of the distribution, is the cumulative distribution function of the standard normal distribution, is the scale parameter, are continuous position parameters; S2-2. Calculate the random degradation of structural strength based on the temporal variation characteristics of the structural strength data set: ； ； in, is the expected value of random degradation of structural strength, is the variance of random degradation of structural strength, is a gamma random process, For time, is the shape function coefficient, is the size parameter coefficient; S2-3, let the random variable in the data set be , calculate the joint distribution of structural stress and structural strength : 。 3. The mechanical structure reliability analysis method based on time-varying mixed Copula function according to claim 2 is characterized in that: The step S2-3 includes measuring the consistency of the structural stress data set and the structural strength data set by using the rank correlation coefficient to determine their correlation. The expression is: ； in, is the length of the sample, .

4. The mechanical structure reliability analysis method based on time-varying mixed Copula function according to claim 1 is characterized in that: The step S3 includes: according to the structural stress edge distribution characteristics , structural strength edge distribution characteristics and the joint distribution of structural stress and structural strength , determine the Copula function type : ； ； in, is the correlation parameter, are the marginal distribution function values of two random variables, ; Based on the structural stress data set and the structural strength data set, the failure mode of the structure to be analyzed is determined. The structural stress data set and the structural strength data set are compared point by point to identify the area where the stress exceeds the strength. If the exceeded area is concentrated in a single location, it is a single failure mode. If the exceeded area is scattered in multiple locations, it is a multiple failure mode.

5. The mechanical structure reliability analysis method based on time-varying mixed Copula function according to claim 4 is characterized in that: If it is a single failure mode, the limit state equation of this failure mode is for: ； in, is the structural stress that varies with time, For structural strength subject to gamma degradation; Establish a time-varying Clopula function, and consider the time-varying characteristics when the correlation parameter is expressed as , that is, T-Copula function ; The T-Copula function goodness of fit is tested by the statistical square difference formula: ； in, is the empirical distribution function; Calculate structural reliability under single component single failure mode : 。 6. The mechanical structure reliability analysis method based on time-varying mixed Copula function according to claim 4 is characterized in that: If there are multiple failure modes, the limit state equations for each failure mode are for: ； Establish the time-varying Clopula function of each failure mode, namely T-Copula function , and calculate the limit state function value under each failure mode according to the limit state equation ; Consider the correlation between failure modes and select the best one The T-Copula function establishes a time-varying mixed Copula function with multiple failure modes, namely TM f -Copula function: ； in, TM for correlation of multiple failure modes for a single component f -Copula function, is the weight coefficient of each Copula function, is the time-varying related parameter in each Copula function; By Akaike Information Criteria InspectionTM f -Copula function goodness of fit: ； in, is the residual sum of squares, is the sample size, For TM f -The number of parameters to be estimated in the Copula function; Calculate structural reliability under multiple failure modes of a single component : ； in, for The distribution function of is the failure probability, For difference calculation, .

7. The mechanical structure reliability analysis method based on time-varying mixed Copula function according to claim 1 is characterized in that: The step S4 includes: selecting T-Copula function to build TM between multiple components c -Copula function: ； By Akaike Information Criteria InspectionTM c -Copula function goodness of fit: ； Assume the number of multiple components is The mechanical structure is calculated to be Reliability of each component : 。

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