Reliability evaluation method based on multiple performance degradation parameters of harmonic reducer

Abstract

The present invention discloses a reliability assessment method based on the multivariate performance degradation parameters of a harmonic reducer. The method collects multivariate performance degradation parameter data of the harmonic reducer through experiments, then performs Wiener process modeling based on the performance degradation parameter increments. The distribution of the Wiener process increments is only related to the time difference, so it is a homogeneous independent incremental process and a normal process. Therefore, a goodness-of-fit plot is used to verify that the parameters of the performance degradation parameter increments should satisfy the normal distribution and be distributed on both sides of the curve, thereby verifying the accuracy of the degradation process modeling results. Variance analysis is used to examine whether there are significant differences in the degradation performance parameters of different harmonic reducer levels under experimental testing. Furthermore, a Copula function is used to describe the correlation between the various multivariate performance degradation parameters of the harmonic reducer. Finally, the reliability of the multivariate performance parameters of the harmonic reducer is evaluated based on the reliability model.

Description

Technical Field

[0001] The present invention relates to the technical field of reliability assessment, and in particular to a reliability assessment method based on multiple performance degradation parameters of a harmonic reducer. Background Art

[0002] Harmonic reducers have been widely used in various mechanical systems, including engineering equipment and aerospace, becoming indispensable power transmission devices. Harmonic reducers are primarily composed of a rigid wheel, a flexspline, and a wave generator, along with a cross bearing and upper and lower housings. Power transmission is primarily achieved through the elastic deformation of the intermediate flexible member. Compared to other reducers, harmonic reducers offer advantages such as small size, high load capacity, compact structure, and high transmission accuracy, making them widely used in drive mechanisms. Therefore, reliability assessment of harmonic reducers is crucial.

[0003] Because harmonic reducers are reliable, long-lasting mechanical products with complex internal structures, conventional reliability assessment techniques for these high-reliability products require long test analysis cycles and high funding. Furthermore, harmonic reducer testing processes are characterized by a small number of test samples and slow performance degradation. Consequently, there are issues with small sample sizes, such as insufficient performance degradation data collected within a limited timeframe. Differences exist between the various harmonic reducers tested simultaneously, and their multivariate performance degradation processes also vary, introducing significant cognitive uncertainty and often resulting in less credible reliability assessment results.

[0004] On the other hand, existing evaluation technologies have a single means of evaluating the reliability of harmonic reducers, and usually process each parameter into a single performance parameter or consider the degradation process of each performance parameter separately. However, under the use of multiple working conditions, harmonic reducers usually have multiple performance parameters that degrade simultaneously. This single degradation performance parameter evaluation method often ignores the mutual coupling relationship between performance parameters. The correlation between various performance parameters will have a significant impact on the reliability evaluation of harmonic reducers. Therefore, in order to analyze the performance degradation data of small samples of harmonic reducer tests and the reliability problem of single degradation parameter evaluation, it is urgent to propose a method for predicting the reliability evaluation of multiple performance degradation parameters of harmonic reducers. The present invention makes full use of existing degradation data information to establish a multi-dimensional degradation performance parameter reliability evaluation model for harmonic reducers. Summary of the Invention

[0005] The purpose of the present invention is to overcome the above-mentioned defects and propose a reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer.

[0006] The present invention adopts the following technical solutions:

[0007] The invention specifically provides a method for collecting multivariate performance degradation parameter data of a harmonic reducer through experiments, and then performing Wiener process modeling using the performance degradation parameter increments. The distribution of the Wiener process increments is only related to the time difference, so it is a homogeneous independent increment process and a normal process. Therefore, a goodness of fit plot is used to test whether the parameters of the performance degradation parameter increments should satisfy the normal distribution and be distributed on both sides of the curve, thereby verifying the accuracy of the degradation process modeling results.

[0008] A variance analysis was performed to examine whether there were significant differences in the degradation performance parameters of the different harmonic reducer levels under test, verifying that the sample data of the different harmonic reducer levels under test had no impact on the overall data evaluation. While ensuring the accuracy of the expected and variance point estimates, the bootstrap method was used to resample the sample size to obtain more accurate estimates of the unknown parameters of the marginal distribution. The obtained unknown parameter estimates of the marginal distribution were then introduced into the Copula density function, and maximum likelihood estimation was performed to obtain the unknown parameter estimates of the Copula function.

[0009] In addition, a Copula function was used to describe the correlation between the various performance degradation parameters of the harmonic reducer. The correlation coefficients between the original degradation parameters were calculated and compared with the commonly used Kendall rank correlation coefficient τ to identify the most suitable Copula function. The closest Copula function was then selected using the minimum squared Euclidean distance. Finally, the reliability of the harmonic reducer's multivariate performance parameters was evaluated based on the reliability model.

[0010] A reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer comprises the following steps:

[0011] 1) Data distribution and statistical analysis of multivariate performance degradation parameters:

[0012] 1.1) Multivariate performance degradation parameter modeling and goodness-of-fit test:

[0013] When collecting multivariate performance degradation parameter analysis during the test, if each performance degradation parameter sample is randomly obtained from the overall sample and the obtained performance degradation parameters are fitted at the same time; after the Wiener process modeling is performed, the goodness of fit of the multivariate performance degradation parameter increments is tested to test the normal distribution of the degradation performance parameter increments and verify the accuracy of the degradation process modeling results;

[0014] 1.2) Multivariate performance degradation parameter variance analysis:

[0015] If the sample data of different harmonic reducers have no effect on the overall data evaluation under the test, then there is no significant difference between the between-group variance and the within-group variance; divide the between-group variance by the within-group variance to obtain the ratio F. The distribution of the ratio F obeys the F distribution, so the ratio F has a corresponding significant probability P value on the F distribution; under the significance level α, when the value Fα(s-1,ns) in the F distribution table is greater than the ratio F or when the P value is greater than the significance level of the hypothesis test, it means that there is no significant difference between the between-group variance and the within-group variance;

[0016] 2) Perform maximum likelihood estimation on multiple performance degradation parameters of the harmonic reducer to determine the unknown parameter estimates of the marginal distribution. Substitute the unknown parameter estimates of the obtained marginal distribution into the Copula density function and perform maximum likelihood estimation to obtain the unknown parameter estimates of the Copula function.

[0017] 3) Based on the data characteristics of the multivariate performance degradation parameters of the harmonic reducer, while ensuring the accuracy of the expectation and variance point estimates, in order to more accurately describe the characteristics of the product population characterized by the differences between individual samples, the bootstrap method is used for resampling to obtain more accurate estimates of the unknown parameters of the marginal distribution. The unknown parameter estimates of the obtained marginal distribution are then substituted into the Copula density function, and the maximum likelihood estimation is performed to obtain the unknown parameter estimates of the Copula function;

[0018] 4) Select a suitable Copula function to characterize the multivariate performance degradation parameters:

[0019] 4.1) Multivariate performance degradation parameter correlation coefficient test method:

[0020] First, calculate the correlation coefficient between the original degradation quantities, and then compare it with the correlation coefficient of commonly used Copula functions, including Gaussian, Frank, Clayton, and Gumbel. Select the Copula function with the correlation coefficient closest to the original data.

[0021] 4.2) Euclidean distance square calculation:

[0022] The maximum likelihood estimation method is used to obtain the parameter values of the Copula model, and the Euclidean distance between each commonly used Copula function and the empirical Copula distribution function is calculated. The Copula function with the smallest square of the Euclidean distance will have the best fitting effect on the original data.

[0023] 5) Establish a reliability evaluation model for the multivariate performance degradation parameter data of the harmonic reducer to predict the service life of the harmonic reducer under normal working conditions.

[0024] Furthermore, in step 1.1), the Wiener process is used to model the process of rigidity and backlash degradation of the harmonic reducer, respectively, which refers to X1=μ1t+σ1B(t) and X2=μ2t+σ2B(t), where B(t) is a standard Brownian motion form, μ1 and σ1 represent the backlash degradation rate and backlash diffusion coefficient of the harmonic reducer, respectively; μ2 and σ2 represent the rigidity degradation rate and rigidity diffusion coefficient of the harmonic reducer, respectively, and t=1, 2, ..., n represents the measurement time;

[0025] According to the performance of the Wiener process, the degradation increment ΔX ij Obey the normal distribution N(μΔt ij ,σ 2 Δt ij ), after modeling the Wiener process, the goodness of fit test of the incremental degradation parameters of the multivariate performance is conducted to test the normal distribution of the incremental degradation parameters. The parameters that satisfy the normal probability distribution of the incremental degradation amount should be distributed on both sides of the curve, verifying the accuracy of the results of the degradation process modeling.

[0026] Furthermore, in step 2), the maximum likelihood estimation method is used to solve the unknown parameter estimation value, and the process is as follows:

[0027] Two-dimensional Sklar theorem: Let H(·) be a two-dimensional distribution function, and F1(x1) and F2(x2) be univariate distribution functions. Then there must exist a Copula function C(·) that satisfies:

[0028] H(x1,x2)=C(F1(x1),F2(x2))

[0029] Let C(μ, v, α) be the Copula distribution function, μ = F1(x1; θ1), v = μ = F2(x2; θ2) are the probability distribution functions of the continuous random variables X and Y respectively, f1(x1; θ1), f2(x2; θ2) are the corresponding probability density functions respectively, where θ1 and θ2 are unknown; then according to Sklar theorem, the joint distribution function of (X, Y) is:

[0030] H(x1,x2,θ1,θ2)=C(F1(x1;θ1),F2(x2;θ2);α)

[0031] Then, the corresponding joint density function is obtained from the formula of Sklar's theorem:

[0032] h(x1,x2,θ1,θ2)=C[(F1(x1;θ1),F2(x2;θ2);α)]·f1(x1;θ1)·f2(x2;θ2)

[0033] Then we can further get (x 1i ,y 2j),j=1,2,...,n are the likelihood functions of the sample points:

[0034]

[0035] Then, by taking the logarithm of the above formula, we can get the likelihood function as follows:

[0036]

[0037] Then find the maximum points of the corresponding log-likelihood functions and obtain the maximum likelihood estimates of θ1 and θ2:

[0038]

[0039] pass and Then we get the maximum likelihood estimate of the unknown function parameter of Copula:

[0040]

[0041] Based on the theory of the above formula, this process is explained in detail. According to the performance of the Wiener process, the degradation increment ΔX ij Obey the normal distribution N(μΔt ij ,σ 2 Δt ij ), where Δt ij =t i,j -t i.j-1 ,i=1,2,...,n,j=1,2,...,m,Since the increment of degradation quantity conforms to the normal distribution, we can get t ij The probability density function of :

[0042]

[0043] The likelihood function is further obtained through the probability density function:

[0044] Then find the maximum point of the logarithmic likelihood function and obtain the following system of equations:

[0045]

[0046] Solving the system of equations, we get the maximum likelihood estimates of the parameters μ and σ: and

[0047]

[0048] Solve the unknown parameter estimates based on the density function of Copula:

[0049]

[0050] Furthermore, the specific process of step 4.1) is as follows:

[0051] Assume that F(x) and G(y) are X i and Y i (i=1, 2, ...), C(u, v) represents a binary copula function, and u=F(x) and v=G(y). The following is a detailed description of the above related row metrics.

[0052] 4.1.1) Kendall rank correlation coefficient τ:

[0053] The Kendall rank correlation coefficient τ is defined as:

[0054]

[0055] The value range of τ is between -1 and 1. When the value of τ is closer to 1 or -1, the correlation between the variables is stronger. When τ = 1, it means a perfect positive correlation; when τ = 0, it means no correlation; when τ = -1, it means a perfect negative correlation.

[0056] 4.1.2) Tail coefficient correlation coefficient λ:

[0057] The upper and lower tail correlation coefficients are defined as:

[0058]

[0059]

[0060] When λ up ∈(0,1], X and Y are called upper tail correlation. lo ∈(0,1], it is called the lower tail correlation λ up =0,λ lo = 0 means that the upper and lower tails of X and Y are independent;

[0061] Based on the Copula function, the tail correlation coefficient is expressed as:

[0062]

[0063]

[0064] in,

[0065] There is a certain conversion relationship between the correlation measurement index of the Copula function and each Copula function. By calculating the parameters of the Copula function, the value of the correlation measurement is derived.

[0066] Furthermore, in step 4.2):

[0067] The definition of Euclidean square distance is: Let (x i ,y i )(j=1,2,...,n) is a sample from the overall sample data (X,Y). When the empirical distribution functions of X and Y are H(x) and G(y) respectively, they are converted to uniform distributions. The empirical Copula function of the sample is defined as:

[0068]

[0069] Where: I(·) is the characteristic function; u,v∈[0,1], when H(x i )≤u, otherwise If the Copula joint distribution function value C(u i ,v i ) is expressed as, then the square of the Euclidean distance can be defined as:

[0070]

[0071] Furthermore, the specific process of step 5) is as follows:

[0072] According to the properties of the Wiener process, the degradation increment is: ΔX(t)~N(μΔt,σ 2 Δt), the corresponding cumulative distribution function is:

[0073]

[0074] Assume that the failure threshold of the performance degradation process is D, and the time when the performance parameter change reaches the failure threshold is defined as the lifespan T, then T = inf{t|X(t) = D, t ≥ 0}, where inf{·} represents the lower bound of a set. Through mathematical calculation, the distribution function F(t) of the lifespan T is obtained as:

[0075]

[0076] The calculation of the harmonic reducer backlash degradation failure distribution function is:

[0077]

[0078] in The estimated values of the backlash degradation rate and diffusion coefficient of the harmonic reducer are respectively, Φ(·) is the standard normal distribution function, D1 is the degradation failure threshold of the harmonic reducer, and the reliability function model of the backlash factor of the harmonic reducer is constructed as follows:

[0079]

[0080] The failure distribution function of the harmonic reducer's rigidity degradation is:

[0081]

[0082] in The estimated values of the harmonic reducer rigidity degradation rate and the harmonic reducer rigidity diffusion coefficient are respectively, Φ(·) is the standard normal distribution function, D2 is expressed as the degradation failure threshold of the harmonic reducer, and the reliability function model of the harmonic reducer rigidity factor is constructed as follows:

[0083]

[0084] The performance degradation trajectory of the two performance parameters of the harmonic reducer, backlash and stiffness, at time t can be expressed as X(t) = (X1(t), X2(t)). The corresponding failure thresholds are expressed as D = (D1, D2), and the lifespan is T = min(T1, T2). The reliability of the harmonic reducer is then expressed as:

[0085] R(t)=P(min(T1,T2>t))=P(T1>t,T2>t)

[0086] =P(X1(t)<D1,X2(t)<D2)

[0087] If the performance degradation parameters are independent of each other, the reliability of the harmonic reducer can be expressed as:

[0088] R(t)=R1(t)×R2(t)

[0089] The corresponding binary correlation reliability function can be expressed by the Copula function as follows:

[0090] R(t)=P(X1(t)≤D1,X2(t)≤D2)

[0091] =1-C(F1(t),F2(t))

[0092] =1-F1(t)-F2(t)+C α (F1(t),F2(t))

[0093] =R1(t)+R2(t)+C α (F1(t),F2(t))-1

[0094] Where C(·) is a binary Copula function with marginal distribution functions F1(t) and F2(t), F1(t) is the failure distribution function of the backlash degradation of the harmonic reducer, and F2(t) is the failure distribution function of the rigidity degradation of the harmonic reducer. The value is the estimated value of the correlation coefficient between the backlash degradation and rigidity degradation of the harmonic reducer.

[0095] From the above scheme, it can be seen that the advantages of the present invention are:

[0096] 1) The simultaneous degradation of multiple performance parameters of the harmonic reducer is studied, the correlation between the multiple performance parameters is considered, and the corresponding related theoretical Copula function is introduced to describe the correlation relationship between the parameters, so that the reliability evaluation results are more accurate.

[0097] 2) For harmonic reducers, which are products with high lifespan and high reliability, the degradation parameters of various samples are different under complex working conditions. Through variance analysis of performance degradation parameters, it is examined whether there are obvious differences in the various degradation performance parameters at different harmonic reducer levels.

[0098] 3) While ensuring the accuracy of the expected and variance point estimates, the bootstrap method is used to resample the sample size to obtain more accurate estimates of the unknown parameters of the marginal distribution. The unknown parameter estimates of the obtained marginal distribution are then introduced into the Copula density function part, and maximum likelihood estimation is performed to obtain the unknown parameter estimates of the Copula function.

[0099] 4) A reliability assessment model based on Copula function to describe the multivariate performance degradation parameter process of the harmonic reducer is given, and various selection methods of Copula function are analyzed. The feasibility and practicality of the present invention are verified through examples. At the same time, the proposed method and model can also be applied to other products, providing technical support and engineering application examples for the reliability assessment of other products. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figure 1 is a flow chart of the reliability assessment method of the present invention;

[0101] Figure 2 is a graph of degradation parameters of the backlash of the harmonic reducer according to an embodiment of the present invention;

[0102] Figure 3 is a degradation parameter diagram of the rigidity of the harmonic reducer according to an embodiment of the present invention;

[0103] Figure 4 This is a general diagram of the goodness of fit test of the normal distribution of the incremental parameter of the backlash degradation amount of the present invention;

[0104] Figure 5 This is a general diagram of the goodness of fit test of the normal distribution of the rigidity degradation increment parameter of the present invention;

[0105] Figure 6 It is a reliability curve diagram of the present invention.

[0106] Specific implementation steps

[0107] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0108] The present invention is a reliability evaluation method based on multiple performance parameters of a harmonic reducer, and its main features include the following steps:

[0109] Step 1: Data distribution and statistical analysis of multivariate performance degradation parameters

[0110] (1) Collection of multivariate performance degradation parameters and goodness-of-fit testing

[0111] Assuming that each performance degradation parameter sample is randomly selected from the overall sample, and the obtained performance degradation parameters are fitted at the same time or approximately at the same time, it is assumed that the degradation performance parameter data statistics of m harmonic reducer products measured at time point t are shown in the following table:

[0112]

[0113] Wherein, m = 1, 2, ..., n represents the number of harmonic reducers; t = 1, 2, ..., n represents the measurement time; performance 1 to performance n represent the performance degradation parameter data corresponding to each harmonic reducer at each measurement time;

[0114] The experiment specifically provides the collection of multivariate performance degradation parameter data of the harmonic reducer. This invention uses the Wiener process to model the process of rigidity and backlash degradation of the harmonic reducer, which means X1=μ1t+σ1B(t) and X2=μ2t+σ2B(t). In the formula, B(t) is the standard Brownian motion form, μ1 and σ1 represent the backlash degradation rate and backlash diffusion coefficient of the harmonic reducer respectively; μ2 and σ2 represent the rigidity degradation rate and rigidity diffusion coefficient of the harmonic reducer respectively.

[0115] According to the performance of the Wiener process, the degradation increment ΔX ij Obey the normal distribution N(μΔt ij ,σ 2 Δt ij ), after modeling the Wiener process, the goodness of fit test of the incremental degradation parameters of the multivariate performance is conducted to test the normal distribution of the incremental degradation parameters. The parameters that satisfy the normal probability distribution of the incremental degradation amount should be distributed on both sides of the curve, verifying the accuracy of the results of the degradation process modeling.

[0116] (2) Variance analysis of multivariate performance degradation parameters

[0117] If the performance degradation parameter factors of some samples at different harmonic reducer test levels have no effect on the overall data, then there is no significant difference between the between-group variance and the within-group variance. Divide the between-group variance by the within-group variance to obtain the ratio F. The distribution of the ratio F follows the F distribution, so the ratio F has a corresponding significant probability P value in the F distribution. The following is the variance analysis table:

[0118]

[0119] Among them S T Can reflect the differences between all performance parameters, so S T It is called the total deviation; the difference between the sample observation value and the sample mean caused by random error, S E It is called the sum of squares of errors; it is caused by the difference in the effect of the level and the random error, the difference between the sample mean under the level and the total mean of the data, S A It is called the sum of squares of the effect of a factor; S E =S E / (ns) are the mean squares of SA and SE respectively.

[0120] At the significance level α, when the value F in the F distribution table is checked, α When the variance analysis (s-1,ns) is greater than the ratio F in the above table, or the P value is greater than the significance level α of the hypothesis test, it means that there is no significant difference between the variance between groups and the variance within groups, that is, the sample data of different harmonic reducer levels under the test have no effect on the overall data evaluation.

[0121] Step 2: Perform maximum likelihood estimation on the multiple performance degradation parameters of the harmonic reducer to determine the unknown parameter estimates of the marginal distribution, bring the unknown parameter estimates of the obtained marginal distribution into the Copula density function, and perform maximum likelihood estimation to obtain the unknown parameter estimates of the Copula function; the present invention uses the maximum likelihood estimation method to solve the unknown parameter estimates, and the process is as follows:

[0122] Two-dimensional Sklar theorem: Let H(·) be a two-dimensional distribution function, and F1(x1) and F2(x2) be univariate distribution functions. Then there must exist a Copula function C(·) that satisfies:

[0123] H(x1,x2)=C(F1(x1),F2(x2))

[0124] Let C(μ, v, α) be the Copula distribution function, μ = F1(x1; θ1), v = μ = F2(x2; θ2) be the probability distribution functions of the continuous random variables X and Y respectively, and f1(x1; θ1) and f2(x2; θ2) be the corresponding probability density functions respectively, where θ1 and θ2 are unknown. Then, according to Sklar's theorem, the joint distribution function of (X, Y) is:

[0125] H(x1,x2,θ1,θ2)=C(F1(x1;θ1),F2(x2;θ2);α)

[0126] Then, according to the formula of Sklar's theorem, the corresponding joint density function can be obtained as:

[0127] h(x1,x2,θ1,θ2)=C[(F1(x1;θ1),F2(x2;θ2);α)]·f1(x1;θ1)·f2(x2;θ2)

[0128] Then we can further get (x 1i ,y 2j ),j=1,2,...,n are the likelihood functions of the sample points:

[0129]

[0130] Then, by taking the logarithm of the above formula, we can get the likelihood function as follows:

[0131]

[0132] Then find the maximum points of the corresponding log-likelihood functions and obtain the maximum likelihood estimates of θ1 and θ2:

[0133]

[0134] pass and Then we can get the maximum likelihood estimate of the unknown function parameter of Copula:

[0135]

[0136] Based on the theory of the above formula, this process is explained in detail. According to the performance of the Wiener process, the degradation increment ΔX ij Obey the normal distribution N(μΔt ij ,σ 2 Δt ij ), where Δt ij =t i,j -t i.j-1 ,i=1,2,...,n,j=1,2,...,m,Since the increment of degradation quantity conforms to the normal distribution, we can get tij The probability density function of :

[0137]

[0138] The likelihood function is further obtained through the probability density function:

[0139]

[0140] Then find the maximum point of the logarithmic likelihood function and obtain the following system of equations:

[0141]

[0142] Solve the system of equations. The maximum likelihood estimates of the parameters μ and σ are obtained as and

[0143]

[0144] Solve the unknown parameter estimates based on the density function of Copula:

[0145]

[0146] Step 3: Bootstrap method for multiple performance degradation parameters

[0147] This paper aims to improve the accuracy of reliability assessment by addressing the inaccurate unknown parameter estimation of small sample data using maximum likelihood. While ensuring the accuracy of the expected and variance point estimates, the Bootstrap method is used to further resample and estimate the unknown parameter and ensure that the unknown parameter estimate is within the confidence interval. The estimates of the Bootstrap method are compared with those of the classical statistical method, and then an example is used to illustrate that the Bootstrap method estimates are more accurate. The following table shows the comparison of parameter estimates:

[0148]

[0149] The following table estimates the unknown parameter estimates of the Copula function:

[0150]

[0151] The estimated values of the Copula function parameters and the estimated values of the backlash and stiffness parameters obtained using the Bootstrap method are introduced into the joint Copula distribution function.

[0152] Step 4: Select a suitable Copula function to characterize the multivariate performance degradation parameters (1) First calculate the correlation coefficient between the original degradation quantities, then compare it with the correlation coefficients of several commonly used Copula functions and select the Copula function that is closer.

[0153] Assume that F(x) and G(y) are X i and Y i (i=1, 2, ...), C(u, v) represents a binary copula function, and u=F(x) and v=G(y). The following is a detailed description of the above related row metrics.

[0154] (1) Kendall rank correlation coefficient τ

[0155] The Kendall rank correlation coefficient τ is defined as:

[0156]

[0157] The value range of τ is between -1 and 1. The closer the τ value is to 1 or -1, the stronger the correlation between the variables. When τ = 1, it means a complete positive correlation; when τ = 0, it means no correlation; when τ = -1, it means a complete negative correlation.

[0158] (2) Tail coefficient correlation coefficient λ

[0159] The upper and lower tail correlation coefficients are defined as:

[0160]

[0161]

[0162] When λ up ∈(0,1], X and Y are called upper tail correlation. lo ∈(0,1], it is called the lower tail correlation λ up =0,λ lo = 0 means that the upper and lower tails of X and Y are independent.

[0163] Based on the Copula function, the tail correlation coefficient can be expressed as:

[0164]

[0165]

[0166] in,

[0167] There is a certain conversion relationship between the correlation measurement index of the Copula function and each Copula function. By calculating the parameters of the Copula function, the value of the correlation measurement can be derived;

[0168] The conversion relationship is shown in the following table:

[0169]

[0170] Among them, ρ, α are the parameters of each Copula function model; D1 is expressed as

[0171] The Copula function has obvious advantages in correlation analysis. It can not only accurately express the dependence between multiple performance parameters, but also derive corresponding correlation metrics to quantitatively describe the degree of correlation between various parameters, mainly including the Kendall rank correlation coefficient τ, the tail correlation coefficient λ and the correlation coefficient ρ.

[0172] (2) Euclidean distance square calculation

[0173] Use the maximum likelihood estimation method to obtain the unknown parameter estimates of the Copula model. Calculate the Euclidean distance between each commonly used Copula function and the empirical Copula distribution function. Select the Copula function with the smallest squared Euclidean distance, as this Copula function will best fit the original performance degradation parameters.

[0174] The definition of Euclidean square distance is: Let (x i ,y i )(j=1,2,...,n) is a sample from the overall sample data (X,Y). When the empirical distribution functions of X and Y are H(x) and G(y) respectively, they are converted to uniform distributions. The empirical Copula function of the sample is defined as:

[0175]

[0176] Where: I(·) is the characteristic function; u,v∈[0,1], when H(x i )≤u, otherwise If the Copula joint distribution function value C(u i ,v i ) is expressed as, then the square of the Euclidean distance can be defined as:

[0177]

[0178] Step 5: Establish a reliability assessment model for the multiple performance degradation parameters of the harmonic reducer; predict the service life of the harmonic reducer under normal working conditions.

[0179] According to the properties of the Wiener process, the degradation increment is: ΔX(t)~N(μΔt,σ 2 Δt) ， The corresponding cumulative distribution function is:

[0180]

[0181] Assume that the failure threshold of the performance degradation process is D, and the time when the performance parameter change reaches the failure threshold is defined as the lifespan T, then T = inf{t|X(t) = D, t ≥ 0}, where inf{·} represents the lower bound of a set. Through mathematical calculation, the distribution function F(t) of the lifespan T can be obtained as:

[0182]

[0183] The calculation of the harmonic reducer backlash degradation failure distribution function is:

[0184]

[0185] in The estimated values of the backlash degradation rate and diffusion coefficient of the harmonic reducer are respectively, Φ(·) is the standard normal distribution function, D1 is the degradation failure threshold of the harmonic reducer, and the reliability function model of the backlash factor of the harmonic reducer is constructed as follows:

[0186]

[0187] The failure distribution function of the harmonic reducer's rigidity degradation is:

[0188]

[0189] in The estimated values of the harmonic reducer rigidity degradation rate and the harmonic reducer rigidity diffusion coefficient are respectively, Φ(·) is the standard normal distribution function, D2 is expressed as the degradation failure threshold of the harmonic reducer, and the reliability function model of the harmonic reducer rigidity factor is constructed as follows:

[0190]

[0191] The performance degradation trajectory of the two performance parameters of the harmonic reducer, backlash and stiffness, at time t can be expressed as X(t) = (X1(t), X2(t)). The corresponding failure thresholds can be expressed as D = (D1, D2), and the lifespan is T = min(T1, T2). The reliability of the harmonic reducer can be expressed as:

[0192] R(t)=P(min(T1,T2>t))=P(T1>t,T2>t)

[0193] =P(X1(t)<D1,X2(t)<D2)

[0194] If the performance degradation parameters are independent of each other, the reliability of the harmonic reducer can be expressed as:

[0195] R(t)=R1(t)×R2(t)

[0196] The corresponding binary correlation reliability function can be expressed by the Copula function as follows:

[0197] R(t)=P(X1(t)≤D1,X2(t)≤D2)

[0198] =1-C(F1(t),F2(t))

[0199] =1-F1(t)-F2(t)+C α (F1(t),F2(t))

[0200] =R1(t)+R2(t)+C α (F1(t),F2(t))-1

[0201] Where C(·) is a binary Copula function with marginal distribution functions F1(t) and F2(t), F1(t) is the failure distribution function of the backlash degradation of the harmonic reducer, and F2(t) is the failure distribution function of the rigidity degradation of the harmonic reducer. The value is the estimated value of the correlation coefficient between the backlash degradation and rigidity degradation of the harmonic reducer; Example

[0202] Step 1: Data distribution and statistical analysis of multivariate performance degradation parameters

[0203] (1) Collection of multivariate performance degradation parameters and goodness-of-fit testing

[0204] Assuming that each performance degradation parameter sample is randomly selected from the overall sample, the performance degradation parameters obtained from the test are processed at the same time or approximately at the same time, the two performance degradation parameters, including the backlash degradation parameter of the harmonic reducer, are as follows: Figure 2 As shown, the rigidity degradation parameters of the harmonic reducer are as follows Figure 3 As shown;

[0205] After the Wiener process modeling is performed and the goodness of fit graph is used for testing, the parameters that meet the normal distribution degradation increment should be distributed on both sides of the curve to verify the accuracy of the degradation process modeling results. The goodness of fit test of the harmonic reducer degradation amount is performed separately, and the normal goodness of fit test probability graphs of the backlash and rigidity degradation increment of the harmonic reducer are shown as follows: Figure 4 As shown and Figure 5 As shown;

[0206] (2) Variance analysis of multivariate performance degradation parameters

[0207] There are differences in the degradation parameters of each sample of the harmonic reducer under complex working conditions. Through variance analysis, it is examined whether there are obvious differences in the degradation performance parameters under different harmonic reducer levels.

[0208] The following table shows the variance analysis table of harmonic reducer backlash:

[0209] Sources of variance sum of squares degrees of freedom mean square Ratio F P-value Factor A 5.0440e-5 5 1.0089e-5 0.93 0.467 error 7.1550e-4 66 1.0841e-5 sum 7.6594e-4 71

[0210] At the significance level α = 0.05, according to the value F in the F distribution table, 0.05 (5,66)=4.33>F=0.93, or at the significance level of the hypothesis test of P=0.467>α=0.050, it shows that there is no significant difference between the variance between groups and the variance within groups, that is, under the experimental test, the sample data of different harmonic reducer levels have no effect on the overall data evaluation.

[0211] The following table shows the variance analysis table of the harmonic reducer rigidity:

[0212] Sources of variance sum of squares degrees of freedom mean square Ratio F P-value Factor B 5.9280e-3 5 1.1860e-3 2.23 0.061 error 3.5067e-2 66 5.3100e-4 sum 4.0995e-2 71

[0213] At the significance level α = 0.05, according to the value F in the F distribution table, 0.05 (5,66)=4.33>F=2.23, or the significance level of the hypothesis test of P=0.061>α=0.050, indicating that there is no significant difference between the variance between groups and the variance within groups, that is, the sample data of different harmonic reducer levels under the test have no effect on the overall data evaluation;

[0214] Step 2: Perform maximum likelihood estimation on multiple performance degradation parameters to determine the unknown parameter estimates of the marginal distribution;

[0215] The following table shows the parameter estimates of the performance degradation parameters:

[0216]

[0217] The unknown parameters of the obtained marginal distribution are brought into the Copula density function part, and the maximum likelihood estimation is performed to obtain the unknown parameter estimates of the Copula function.

[0218] Step 3: Bootstrap method of multiple performance degradation parameters.

[0219] The Bootstrap method is used to obtain the inaccurate improvement of unknown parameters for the small sample data of multivariate performance degradation parameters.

[0220] The following table shows a comparison of the parameter estimates:

[0221]

[0222]

[0223] While ensuring the accuracy of the expected and variance point estimates, the Bootstrap method is used to further resample and estimate the unknown parameter, ensuring that the unknown parameter estimate is within the confidence interval to improve the accuracy of the reliability assessment. The unknown parameter estimate of the Copula is then recalculated.

[0224] The following table shows the estimated values of the unknown parameters of the re-estimated Copula function:

[0225]

[0226] The unknown parameter estimates of the Copula function and the parameter estimates of the backlash and stiffness obtained using the Bootstrap method are introduced into the joint Copula distribution function.

[0227] Step 4: Select a suitable Copula function to characterize the multivariate performance degradation parameters

[0228] (1) First calculate the correlation coefficient between the original degradation quantities, then compare it with the correlation coefficients of several commonly used Copula functions and select the Copula function that is closer;

[0229] The calculated correlation coefficients are shown in the following table:

[0230] Copula Function Kendall τ <![CDATA[Upper tail correlation coefficient λ up > <![CDATA[Lower tail correlation coefficient λ lo > Gaussian 0.4201 0 0 Frank <![CDATA[ 0.4656 ]]> 0 0 Clayton 0.4208 0 0.6206 Gumbe] 0.4176 0.5577 0 Raw data 0.4749 0.5610 0

[0231] Through the analysis of the tabular data, it was found that the correlation coefficient between the Kendall rank of the Frank Copula function (0.4656) and the original data Kendall rank (0.4749) was closest.

[0232] (2) Euclidean distance square calculation τ

[0233] Calculate the Euclidean distance between each commonly used Copula function and the empirical Copula distribution function;

[0234] The following table shows the calculated Euclidean distance squared:

[0235] Function Type Gaussian Frank Clayton Gumbel <![CDATA[Euclidean distance d 2 > 0.0170 <![CDATA[ 0.0153 ]]> 0.0221 0.0261

[0236] Through the analysis of the tabular data, it was found that the Frank Copula function had the smallest squared Euclidean distance of 0.00153, and the Frank Copula function had the best fitting effect on the original performance degradation parameters.

[0237] Step 5: Establish a reliability assessment model for the multiple performance degradation parameters of the harmonic reducer; predict the service life of the harmonic reducer under normal working conditions.

[0238] The calculation of the harmonic reducer backlash degradation failure distribution function is:

[0239]

[0240] The reliability function model of the backlash factor of the harmonic reducer is constructed as follows:

[0241]

[0242] The failure distribution function of the harmonic reducer's rigidity degradation is:

[0243]

[0244] The reliability function model of the harmonic reducer stiffness factor is constructed as follows:

[0245]

[0246] If the performance degradation parameters are independent of each other, the reliability of the harmonic reducer can be expressed as:

[0247] R(t)=R1(t)×R2(t)

[0248] The joint distribution function of backlash and rigidity degradation failure of harmonic reducer based on FrankCopula function is:

[0249]

[0250] Reliability probability function of harmonic reducer The function curve is as follows Figure 6 As shown, according to the remaining life probability function of the harmonic reducer The reliability evaluation of the harmonic reducer can be completed.

[0251] According to the reliability curve Figure 6 The results show that using the reliability of a single performance parameter overestimates product reliability, while assuming the degradation of two performance parameters is independent of each other significantly underestimates product reliability. This leads to a certain deviation from the reliability calculated using the multivariate joint distribution function of the FrankCopula function. Therefore, it is necessary to establish a multi-degradation joint distribution function for harmonic reducers. This demonstrates that in actual operation, the multivariate performance degradation parameter model is more consistent with engineering practice than the single degradation parameter models.

[0252] For those skilled in the art, the examples provided here are intended to help readers better understand the principles of the present invention. Based on the guidance of the present invention, the scope of protection of the present invention is not limited to the statements and examples herein. Other persons skilled in the art may make changes, modifications, substitutions and deformations to the implementation methods without departing from the principles of the present invention, which still fall within the scope of protection of the present invention.

Claims

1. A reliability assessment method based on multiple performance degradation parameters of a harmonic reducer, characterized by comprising the following steps: 1) Data distribution and statistical analysis of multivariate performance degradation parameters: 1.1) Multivariate performance degradation parameter modeling and goodness-of-fit test: When collecting multivariate performance degradation parameter analysis during the test, if each performance degradation parameter sample is randomly obtained from the overall sample and the obtained performance degradation parameters are fitted at the same time; after the Wiener process modeling is performed, the goodness of fit of the multivariate performance degradation parameter increments is tested to test the normal distribution of the degradation performance parameter increments and verify the accuracy of the degradation process modeling results; 1.2) Multivariate performance degradation parameter variance analysis: If the sample data of different harmonic reducers have no effect on the overall data evaluation under the test, then there is no significant difference between the variance between groups and the variance within groups; divide the variance between groups by the variance within groups to get the ratio F, and the distribution of the ratio F obeys the F distribution, so the ratio F has a corresponding significant probability P value in the F distribution; under the significance level α, when the value F in the F distribution table is checked, α When (s-1,ns) is greater than the ratio F or when the P value is greater than the significance level of the hypothesis test, it means that there is no significant difference between the variance between groups and the variance within groups; 2) Perform maximum likelihood estimation on multiple performance degradation parameters of the harmonic reducer to determine the unknown parameter estimates of the marginal distribution. Substitute the unknown parameter estimates of the obtained marginal distribution into the Copula density function and perform maximum likelihood estimation to obtain the unknown parameter estimates of the Copula function. 3) Based on the data characteristics of the multivariate performance degradation parameters of the harmonic reducer, while ensuring the accuracy of the expectation and variance point estimates, in order to more accurately describe the characteristics of the product population characterized by the differences between individual samples, the bootstrap method is used for resampling to obtain more accurate estimates of the unknown parameters of the marginal distribution. The unknown parameter estimates of the obtained marginal distribution are then substituted into the Copula density function, and the maximum likelihood estimation is performed to obtain the unknown parameter estimates of the Copula function; 4) Select a suitable Copula function to characterize the multivariate performance degradation parameters: 4.1) Multivariate performance degradation parameter correlation coefficient test method: First, the correlation coefficient between the original degradation quantities is calculated, and then compared with the correlation coefficient of commonly used Copula functions, including Gaussian, Frank, Clayton, and Gumbel. After the correlation calculation, the Copula function with the closest correlation coefficient to the original data is selected. 4.2) Euclidean distance square calculation: The maximum likelihood estimation method is used to obtain the parameter values of the Copula model, and the Euclidean distance between each commonly used Copula function and the empirical Copula distribution function is calculated. The Copula function with the smallest square of the Euclidean distance will have the best fitting effect on the original data. 5) Establish a reliability evaluation model for the multivariate performance degradation parameter data of the harmonic reducer to predict the service life of the harmonic reducer under normal working conditions.

2. A reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer according to claim 1, characterized in that: In step 1.1), the Wiener process is used to model the process of rigidity and backlash degradation of the harmonic reducer, which refers to X1=μ1t+σ1B(t) and X2=μ2t+σ2B(t), where B(t) is a standard Brownian motion form, μ1 and σ1 represent the backlash degradation rate and backlash diffusion coefficient of the harmonic reducer, respectively; μ2 and σ2 represent the rigidity degradation rate and rigidity diffusion coefficient of the harmonic reducer, respectively, and t=1, 2, ..., n represents the measurement time; According to the performance of the Wiener process, the degradation increment ΔX ij Obey the normal distribution N(μΔt ij ,σ 2 Δt ij ), after modeling the Wiener process, the goodness of fit test of the incremental degradation parameters of the multivariate performance is conducted to test the normal distribution of the incremental degradation parameters. The parameters that satisfy the normal probability distribution of the incremental degradation amount should be distributed on both sides of the curve, verifying the accuracy of the results of the degradation process modeling.

3. The reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer according to claim 1 is characterized in that: In step 2), the maximum likelihood estimation method is used to solve the unknown parameter estimation value, and the process is as follows: Two-dimensional Sklar theorem: Let H(·) be a two-dimensional distribution function, and F1(x1) and F2(x2) be univariate distribution functions. Then there must exist a Copula function C(·) that satisfies: H(x1,x2)=C(F1(x1),F2(x2)) Let C(μ, v, α) be the Copula distribution function, μ = F1(x1; θ1), v = μ = F2(x2; θ2) are the probability distribution functions of the continuous random variables X and Y respectively, f1(x1; θ1), f2(x2; θ2) are the corresponding probability density functions respectively, where θ1 and θ2 are unknown; then according to Sklar theorem, the joint distribution function of (X, Y) is: H(x1,x2,θ1,θ2)=C(F1(x1;θ1),F2(x2;θ2);α) Then, the corresponding joint density function is obtained from the formula of Sklar's theorem: h(x1,x2,θ1,θ2)=C[(F1(x1;θ1),F2(x2;θ2);α)]·f1(x1;θ1)·f2(x2;θ2) Then we can further get (x 1i ,y 2j ),j=1,2,...,n are the likelihood functions of the sample points: Then, by taking the logarithm of the above formula, we can get the likelihood function as follows: Then find the maximum points of the corresponding log-likelihood functions and obtain the maximum likelihood estimates of θ1 and θ2: pass and Then we get the maximum likelihood estimate of the unknown function parameter of Copula: Based on the theory of the above formula, this process is explained in detail. According to the performance of the Wiener process, the degradation increment ΔX ij Obey the normal distribution N(μΔt ij ,σ 2 Δt ij ), where Δt ij =t i,j -t i.j-1 ,i=1,2,...,n,j=1,2,...,m,Since the increment of degradation quantity conforms to the normal distribution, we can get t ij The probability density function of : The likelihood function is further obtained through the probability density function: Then find the maximum point of the logarithmic likelihood function and obtain the following system of equations: Solving the system of equations, we get the maximum likelihood estimates of the parameters μ and σ: and Solve the unknown parameter estimates based on the density function of Copula:

4. The reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer according to claim 1 is characterized in that: The specific process of step 4.1) is as follows: Assume that F(x) and G(y) are X i and Y i (i = 1, 2, ...), C (u, v) represents a binary copula function, and u = F (x) and v = G (y); the above related row metrics are described in detail below; 4.1.1) Kendall rank correlation coefficient τ: The Kendall rank correlation coefficient τ is defined as: The value range of τ is between -1 and 1. When the value of τ is closer to 1 or -1, the correlation between the variables is stronger. When τ = 1, it means a perfect positive correlation; when τ = 0, it means no correlation; when τ = -1, it means a perfect negative correlation. 4.1.2) Tail coefficient correlation coefficient λ: The upper and lower tail correlation coefficients are defined as: When λ up ∈(0,1], X and Y are called upper tail correlation. lo ∈(0,1], it is called the lower tail correlation λ up =0,λ lo = 0 means that the upper and lower tails of X and Y are independent; Based on the Copula function, the tail correlation coefficient is expressed as: in, There is a certain conversion relationship between the correlation measurement index of the Copula function and each Copula function. By calculating the parameters of the Copula function, the value of the correlation measurement is derived.

5. The reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer according to claim 1 is characterized in that: In the step 4.2): The definition of Euclidean square distance is: Let (x i ,y i ), i = 1, 2, ..., n is a sample from the overall sample data (X, Y). When the empirical distribution functions of X and Y are H(x) and G(y) respectively, they are converted to uniform distributions, and the empirical Copula function of the sample is defined as: Where: I(·) is the characteristic function; u,v∈[0,1], when H(x i )≤u, otherwise If the Copula joint distribution function value C(u i ,v i ) is represented by , then the square of the Euclidean distance is defined as:

6. The reliability evaluation method based on multiple performance degradation parameters of a harmonic reducer according to claim 1 is characterized in that: The specific process of step 5) is as follows: According to the properties of the Wiener process, the degradation increment is: ΔX(t)~N(μΔt,σ 2 Δt), the corresponding cumulative distribution function is: Assume that the failure threshold of the performance degradation process is D, and the time when the performance parameter change reaches the failure threshold is defined as the lifespan T, then T = inf{t|X(t) = D, t ≥ 0}, where inf{·} represents the lower bound of a set. Through mathematical calculation, the distribution function F(t) of the lifespan T is obtained as: The calculation of the harmonic reducer backlash degradation failure distribution function is: in The estimated values of the backlash degradation rate and diffusion coefficient of the harmonic reducer are respectively, Φ(·) is the standard normal distribution function, D1 is the degradation failure threshold of the harmonic reducer, and the reliability function model of the backlash factor of the harmonic reducer is constructed as follows: The failure distribution function of the harmonic reducer's rigidity degradation is: in The estimated values of the harmonic reducer rigidity degradation rate and the harmonic reducer rigidity diffusion coefficient are respectively, Φ(·) is the standard normal distribution function, D2 is expressed as the degradation failure threshold of the harmonic reducer, and the reliability function model of the harmonic reducer rigidity factor is constructed as follows: The performance degradation trajectory of the two performance parameters of the harmonic reducer, backlash and stiffness, at time t is expressed as X(t) = (X1(t), X2(t)). The corresponding failure thresholds are expressed as D = (D1, D2), and the life is T = min(T1, T2). The reliability of the harmonic reducer is expressed as: R(t)=P(min(T1,T2>t))=P(T1>t,T2>t) =P(X1(t)<D1,X2(t)<D2) If the performance degradation parameters are independent of each other, the reliability of the harmonic reducer can be expressed as: R(t)=R1(t)×R2(t) The corresponding binary correlation reliability function is expressed by the Copula function as follows: R(t)=P(X1(t)≤D1,X2(t)≤D2) =1-C(F1(t),F2(t)) =1-F1(t)-F2(t)+C α (F1(t),F2(t)) =R1(t)+R2(t)+C α (F1(t),F2(t))-1 Where C(·) is a binary Copula function with marginal distribution functions F1(t) and F2(t), F1(t) is the failure distribution function of the backlash degradation of the harmonic reducer, and F2(t) is the failure distribution function of the rigidity degradation of the harmonic reducer. The value is the estimated value of the correlation coefficient between the backlash degradation and rigidity degradation of the harmonic reducer.

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