An optimization design method for helical spring degradation test based on equally spaced degradation increments

Abstract

The present invention proposes a method for optimizing the degradation test of coil springs based on equally spaced degradation increments. First, based on the historical degradation test data of the coil springs, a Wiener process model is established and the degradation model parameters are estimated; second, the expected degradation amount of the next measurement point is determined based on the current sampling data; then, based on the Bayesian theory, the degradation model parameters are updated, the measurement time of the next sampling point is inferred, and the actual degradation amount of the sampling point is measured. By comparing the deviation between the expected degradation amount and the actual degradation amount of the sampling point, it is determined whether to terminate the test; finally, the pseudo-life of the coil spring is calculated based on the parameter estimate obtained when the test is stopped, and the optimization design method for the degradation test of the coil spring with equally spaced degradation increments is completed, thereby improving the accuracy of life assessment and test efficiency. The present invention effectively solves the problem that the pseudo-life of the product predicted by the reliability degradation test analysis of the degraded product is actually different from the actual life, which brings additional risks to the product reliability assessment.

Description

Technical Field

[0001] The present invention proposes a method for optimizing the degradation test of a coil spring based on equally spaced degradation increments. First, based on the historical degradation test data of the coil spring, a Wiener process model is established and the degradation model parameters are estimated; second, the expected degradation amount of the next measurement point is determined based on the current sampling data; then, based on the Bayesian theory, the degradation model parameters are updated, the measurement time of the next sampling point is calculated, and the actual degradation amount of the sampling point is measured. By comparing the deviation between the expected degradation amount and the actual degradation amount of the sampling point, it is determined whether to terminate the test; finally, the pseudo-life of the coil spring is calculated based on the parameter estimate obtained when the test is stopped, and the optimization design method for the degradation test of the coil spring with equally spaced degradation increments is completed, thereby improving the accuracy of life assessment and test efficiency. The present invention effectively solves the problem that there is a real deviation between the pseudo-life of the coil spring product predicted by the reliability degradation test and the actual life, which brings additional risks to the product reliability assessment. The present invention is applicable to related technical fields such as reliability verification and life assessment. Background Art

[0002] A spring is a mechanical component that utilizes elasticity to function. Made of elastic material, it deforms under external force and returns to its original shape when the force is removed. Springs perform measurement, compression, resetting, cushioning, and energy storage functions. Coil springs, on the other hand, withstand torsional deformation. Due to their strong compressive resistance, coil springs are crucial to the aerospace industry. Fatigue failure of coil springs is directly related to the reliability of aircraft engines and automatic weapons. Failure of a coil spring in any critical location requires significant time and cost for repair, and can even result in serious accidents. Therefore, condition reliability assessment and life prediction of coil springs are crucial for improving the operational reliability of corresponding aviation equipment and reducing maintenance costs.

[0003] With the improvement of product reliability, a large number of high-reliability, long-life products, such as coil springs, are widely used. However, for these high-reliability, long-life products, obtaining sufficient failure data through life testing or accelerated life testing is difficult within limited time and cost. Therefore, traditional modeling methods based on failure data are not suitable for life assessment of these products. In this case, product performance degradation data can provide important information for life assessment. Because the degradation of product performance characteristics in real-world environments is affected by multiple factors and inevitably exhibits randomness, stochastic process models are often used to describe the product performance degradation process. However, on the one hand, current performance degradation data for coil spring products is often obtained through sampling at equal intervals, which is simple to operate but requires a long test time. On the other hand, existing methods fail to consider other useful information besides field degradation data when predicting the life of coil springs, resulting in significant deviations between the predicted and actual product life, posing significant risks to product reliability assessment and life prediction.

[0004] To mitigate this risk and accurately predict the true product lifespan, this paper proposes a method for optimizing the design of degradation tests for coil springs based on equally spaced degradation increments. Based on degradation test data from coil springs, this method employs a Wiener process model, maximum likelihood estimation, and Bayesian methodologies to design a specific test flow for predicting the pseudo-lifespan of coil springs using degradation tests with equally spaced degradation increments. This method effectively improves lifespan prediction accuracy and test efficiency, providing technical support for reliability testing. Summary of the Invention

[0005] (1) The purpose of the present invention: In order to solve the problem that the pseudo-life of the product obtained based on degradation test analysis deviates seriously from the actual life, which brings additional risks to product reliability assessment, the present invention provides a method for optimizing the degradation test of a coil spring based on equally spaced degradation increments. First, based on the historical degradation test data of the coil spring, a Wiener process model is established and the degradation model parameters are estimated; second, the expected degradation amount of the next measurement point is determined based on the current sampling data; then, based on the Bayesian theory, the degradation model parameters are updated, the measurement time of the next sampling point is calculated, and the actual degradation amount of the point is measured. By comparing the deviation between the expected degradation amount and the actual degradation amount of the sampling point, it is determined whether to terminate the test; finally, the pseudo-life of the coil spring is calculated based on the parameter estimation value obtained when the test is stopped, and the degradation test of the coil spring with equally spaced degradation increments is completed.

[0006] (2) Technical Solution: Based on the above problems, the present invention provides a method for optimizing the degradation test of coil springs based on the degradation increments of historical degradation test data of coil springs at equal intervals. The theoretical basis required is as follows:

[0007] As a typical random process, the Wiener process model has become one of the most widely used models for reliability analysis and life assessment based on performance degradation. Assuming that X(t) is the performance degradation of the coil spring at time t, {X(t), t≥0} obeys the Wiener process, its degradation mathematical model can be expressed as

[0008] X(t)=→t+σB(t), (1) where B(t) represents the standard Brownian motion and follows a normal distribution with mean 0 and variance t, i.e., B(t)~N(0, t). Then X(t) follows a normal distribution with mean μt and variance σ 2 Normal distribution of t, that is, X(t)~N(μt,σ 2 t), where μ is the drift coefficient, σ 2 is the diffusion coefficient. The Wiener process satisfies the following properties:

[0009] 1) X(0) = 0;

[0010] 2) In any two non-intersecting time ranges, the increments have stationary independence (independence: the increments are independent of each other; stationarity: the distribution of the increment process does not change with the starting point);

[0011] 3) For any t i >t i-1 ≥0, ΔX i =X(t i )-X(t i-1 ) obeys the mean μΔt i , the variance is σ 2 αt i The normal distribution of ΔX i ~N(μΔt i ,σ 2 Δt i ), where μ represents the drift coefficient, σ 2 represents the diffusion coefficient, t i and t i-1 They represent the time of the system’s i-th and i-1-th measurements, respectively. X(t i ) and X(t i-1 ) represent time t i and time t i-1 The degradation level of the system when ΔX i Indicates [t i-1 , t i ] the increment of degradation level during the time interval Δt i =t i -t i-1 .

[0012] According to the definition of degradation model (1), the degradation amount X(t) obeys the normal distribution, that is, X(t)~N(μt,σ 2 t), the probability density function is

[0013]

[0014] Formula (2) does not consider the randomness of parameters, but in practice, the degradation process of coil springs has individual differences. Therefore, according to the Bayesian viewpoint, the model parameters μ and σ can be 2 As a random variable. Since σ 2 >0,1 / σ 2 >0, then we can assume 1 / σ 2 The prior distribution of follows the Gamma distribution, then σ 2 Obey the inverse Gamma distribution with shape parameter a0 and scale parameter b0, that is, σ 2 ~IG(a0, b0). The general form of the probability density function of the inverse Gamma distribution IG(α, λ) is

[0015]

[0016] Among them, the gamma function α is the shape parameter, λ is the scale parameter, and α>0, λ>0.

[0017] Therefore, according to σ 2 Obeying the inverse Gamma distribution IG(a0, b0), we can get σ 2 The probability density function of

[0018]

[0019] Where, shape parameter a0>0, scale parameter b0>0, gamma function Furthermore, in σ 2 Under given conditions, μ obeys the mean μ0 and variance η0σ 2 Normal distribution, that is, μ|σ 2 ~N(μ0,η0σ 2 ), where η0 is the variance coefficient.

[0020] Since μ, σ 2 The prior distribution of is the normal-inverse Gamma distribution, that is,

[0021]

[0022] Among them, a0, b0, μ0, and η0 are four hyperparameters, a0 represents the shape parameter, b0 represents the scale parameter, μ0 represents the drift coefficient, and η0 represents the variance coefficient.

[0023] Note j is the jth measurement moment of the sample, X j is the performance degradation of the sample obtained at the jth measurement, j = 1, 2, ... n, n is the total number of measurements of the degradation test. If the test is carried out at the time [t1, t2, ..., t j ]A set of degradation increment data [ΔX1, ΔX2, …, ΔX j ], where ΔX j =X j -X j-1 , the measurement time intervals are [Δt1, Δt2, …, Δt j ], Δt j =t j -t j-1 , then μ, σ 2 The posterior distribution of is also normal-inverse Gamma distribution, that is,

[0024]

[0025] Among them, a j , b j , μ j , η j are the four hyperparameters of the posterior distribution. After substituting the degradation increment data [ΔX1, ΔX2, ..., ΔX j ], the variance coefficient Drift coefficient Shape parameters scale parameter η j-1 、μ j-1 、a j-1 、b j-1 Respectively represent the data calculated by the previous j-1 measurements [ΔX1, ΔX2, ..., ΔX j-1 ]Substitute the updated variance coefficient, drift coefficient, shape parameter and scale parameter of the posterior distribution into formula (6); They represent the degradation increment data [ΔX1, ΔX2, ..., ΔX j ], the mean and variance of the measurement time interval Δt j =t j -t j-1 .

[0026] Based on the above theoretical basis, the specific implementation steps of the present invention are as follows:

[0027] Step 1: Wiener process model parameter estimation.

[0028] First, determine the performance degradation model of the coil spring. Assume that X(t) is the performance degradation of the coil spring at time t, {X(t), t≥0} follows a Wiener process, and its degradation X(t) follows a process with mean μt and variance σ 2 Normal distribution of t, that is, X(t)~N(μt,σ 2 t), where μ is the drift coefficient and σ is the diffusion coefficient.

[0029] Secondly, obtain the historical performance degradation data of the same type of products. Assume that there are N coil spring samples in the historical data, record k as the serial number of the sample in the historical data, t k，p and X(t k，p ) represent the sample k in the historical data at the pth measurement time and the corresponding performance degradation, k = 1, 2, ..., N, p = 1, 2, ..., m k , where m k Represents the total number of measurements of sample k. Note that the initial measurement time and initial degradation amount of each sample are both 0, that is, t k，1 =0, X(t k，1 )=0. Thus we can get the sample k at time t k，p To time t k，p+1 Time interval Δt k，p =t k，p+1 -t k，p , and sample k at time interval Δt k，p The degradation increment ΔX k，p =X(t k，p+1 )-X(t k，p ).

[0030] Finally, estimate the drift coefficient of sample k and diffusion coefficient remember and represent the drift coefficient and diffusion coefficient of historical sample k respectively, and Respectively and According to the performance degradation model (1), the maximum likelihood estimation method is used to obtain the estimated values of the drift coefficient and diffusion coefficient of sample k in the historical data. The formula is as follows:

[0031]

[0032]

[0033] Among them, m k represents the total number of measurements of sample k, and N represents the total number of coil spring samples in the historical data.

[0034] Step 2: Determine the expected degradation amount at the next measurement point based on the sampled data.

[0035] Assume that t0, X0, t1, and X1 represent the test start time, initial value of performance degradation, first measurement time, and actual degradation amount corresponding to the first measurement time of the coil spring, respectively; Δt1 and ΔX1 represent the first time interval and actual degradation increment of the coil spring in the degradation test, respectively, where Δt1 = t1-t0 and ΔX1 = X1-X0. j′ , Δt j′ , X j′ , ΔX j′ They represent the j′th measurement time, measurement time interval, actual degradation amount and actual degradation increment of the coil spring in the degradation test, They represent the expected degradation amount and expected degradation increment of the coil spring in the j′th degradation test, respectively, where n is the total number of measurements when the product fails.

[0036] According to the product history information (historical test measurement interval, total number of measurements when historical samples fail), the first measurement time t1 of the product degradation test, the total number of measurements n when the product fails, and the failure threshold C are determined, where the failure threshold C means that the coil spring product is recorded as failed when the degradation amount reaches C for the first time. At the beginning of the test, the initial value of the product performance degradation t0=0 and the degradation amount X0=0 are recorded. Based on the historical measurement interval Δτ, the first measurement time of the test is determined and the sampling (t1, X1) is completed, where t1=t0+Δτ. After completing the first sampling, the degradation increments can be divided into equal intervals according to the first degradation amount data and the failure threshold C, and the expected degradation amount of the second measurement point can be calculated. That is, starting from the second sampling point, if the actual degradation amount X of the j′-1th measurement is known, j′-1 , then the expected degradation of the j′th sampling can be calculated according to formula (9):

[0037]

[0038] Among them, the expected degradation increment It represents the difference between the expected degradation amount at the j′th time and the actual degradation amount at the j′-1th time.

[0039] Step 3: Update the degradation model parameters based on Bayesian theory and complete the degradation test with equally spaced degradation increments.

[0040] Through steps 1 and 2, the estimated values of the drift coefficient and diffusion coefficient of the historical sample k can be obtained and and the expected degradation of the j′th sampling In order to reduce the deviation between the expected degradation amount and the actual degradation amount, the Bayesian theory is used to update the degradation model parameters during the test. First, based on the estimated value and Parameter estimation is performed on the prior distribution parameter values a0, b0, μ0, η0, where a0, b0, μ0, η0 represent the shape parameter, scale parameter, drift coefficient, and variance coefficient of the prior distribution, respectively. Secondly, the posterior distribution hyperparameters are updated based on the estimated values of the prior distribution hyperparameters. Finally, the next measurement moment is calculated based on the estimated values of the posterior distribution hyperparameters and sampling is completed to obtain the actual degradation amount. The size of the predicted deviation is compared to see if it is lower than the threshold. If the deviation is lower than the threshold, the test can be terminated. Otherwise, the test continues until the product fails.

[0041] First, based on the estimated and The four hyperparameters a0, b0, μ0, and η0 of the prior distribution are estimated. a0, b0, μ0, and η0 represent the shape parameter, scale parameter, drift coefficient, and variance coefficient of the prior distribution, respectively. According to formulas (4) and (5), the model parameters μ and σ can be obtained: 2 The joint prior distribution of

[0042]

[0043] Among them, the unknown parameters a0, b0, μ0, η0 are the shape parameter, scale parameter, drift coefficient and variance coefficient respectively. According to formula (10), the likelihood function of the unknown parameters can be obtained as

[0044]

[0045] Taking the logarithm of formula (11), we can get the log-likelihood function as

[0046]

[0047] Furthermore, the first-order partial derivative equation of formula (12) for unknown parameters a0, b0, μ0, η0 can be expressed as

[0048]

[0049]

[0050]

[0051]

[0052] Among them, the gamma function ψ(a0)=dln(Γ(a0)) / dx, N is the total number of samples in the historical data, a0, b0, μ0, η0 are the shape parameter, scale parameter, drift coefficient and variance coefficient of the prior distribution respectively.

[0053] Therefore, based on the maximum likelihood estimation method, solving formulas (13)-(16) can obtain the estimated values of the four hyperparameters a0, b0, μ0, and η0 of the prior distribution.

[0054] Secondly, the estimated values of the four hyperparameters based on the prior distribution Estimate the posterior distribution hyperparameter a j′ , b j′ , μ j′ , η j′ , j′=2, 3, ... n. Among them, a j′ , b j′ , μ j′ , μ j′ They represent the degradation increment data [ΔX1, ΔX2, …, ΔX j′ Substitute into formula (6) to update the shape parameter, scale parameter, drift coefficient and variance coefficient of the posterior distribution. According to the estimated values of the four hyperparameters of the prior distribution Substituting into formula (5) we can get the prior distribution as

[0055]

[0056] From step 2, we can get the first measurement moment and the first actual degradation amount in the test as (t1, X1). The first actual degradation increment ΔX1 = X1-X0 can be calculated, and the measurement time interval Δt1 = t1-t0. Substituting the actual degradation increment data ΔX1 into formula (6) can obtain the posterior distribution:

[0057]

[0058] Among them, a1, b1, μ1, η1 represent the four hyperparameters of the posterior distribution obtained by substituting the first actual degradation increment ΔX1 into formula (6), a1 is the shape parameter, b1 is the scale parameter, μ1 is the drift coefficient, and η1 is the variance coefficient. Therefore, according to formula (6), the estimated values of a1, b1, μ1, and η1 can be calculated as They are

[0059]

[0060] Furthermore, starting from the second sampling point, if the actual degradation amount measured at the j′th sampling time is (t j′ , X j′), j′=1, 2, ... n, calculate the actual degradation increment ΔX at that moment j′ =X j′ -X j′-1 , measuring time interval Δt j′ =t j′ -t j′-1 , then the actual degradation increment data calculated by the first j′ measurements is [ΔX1, ΔX2, …, ΔX j′ ], substituting into formula (6) we can get the posterior distribution as

[0061]

[0062] Among them, a j′ , b j′ , μ j′ , η j′ Indicates that the data [ΔX1, ΔX2, …, ΔX j′ ]Substitute the four hyperparameters of the posterior distribution, a j′ is the shape parameter, b j′ is the scale parameter, μ j′ is the drift coefficient, η j′ is the coefficient of variance.

[0063] Therefore, according to formula (6), we can calculate a j′ , b j′ , μ j′ , η j′ Estimated value of They are

[0064]

[0065] Where, The data calculated for the first j′-1 measurements [ΔX1, ΔX2, …, ΔX j′-1 ] Substitute the updated shape parameter, scale parameter, drift coefficient and variance coefficient estimates obtained from the posterior distribution, j′=2,3,...n, and They are the degradation increment data [ΔX1, ΔX2, ..., ΔX j′ ]’s mean and variance.

[0066] Finally, the degradation test of the equally spaced degradation increments is completed. The degradation model parameter value after each update is used as the prior distribution when the next sampling point is updated, so as to infer the test time of the next measurement point according to the performance degradation model (1). After completing the first sampling calculation, the parameter estimation value is obtained. After that, the measurement time of the next measurement point can be calculated Therefore, if the estimated value of the model drift coefficient is obtained after completing the j′-1th measurement calculation Then the measurement time t of the j′th measurement point is j′ The specific calculation formula is

[0067]

[0068] Where, is the expected degradation amount of the jth measurement point, Drift coefficient estimate calculated for the j′-1th measurement.

[0069] Further, starting from the second sampling point, according to the test time t of the j′th measurement point j′ Complete sampling and get data (t j′ , X j′ ), compare the actual degradation amount X measured at the j′th time j′ The calculated expected degradation Deviation Δω j′ , the specific formula is as follows:

[0070]

[0071] According to engineering experience, given the deviation threshold D, if the deviation calculated by the j′th measurement satisfies Δω j′ ≤D, the test is terminated, otherwise continue measuring until the product fails and terminate the test. The actual total number of sampling times at the end of the test is recorded as n o , the first n o Degradation increment data calculated by the measurement Substituting into formula (21) we can obtain the parameter estimates of shape parameter, scale parameter, drift coefficient and variance coefficient

[0072] Step 4: Calculate the pseudo life of the coil spring based on the equally spaced degradation increment data.

[0073] The estimated model parameters obtained by updating when the experiment stops in step 3 Get product pseudo lifespan The drift coefficient estimate obtained from the final update Using the performance degradation model (1), the pseudo life of the product can be calculated as

[0074]

[0075] Where C represents the product failure threshold, n o Indicates the actual total number of samples taken when the experiment is terminated.

[0076] (3) Advantages:

[0077] The present invention proposes a method for optimizing the degradation test of a coil spring based on equally spaced degradation increments, which has the following advantages:

[0078] ① The present invention proposes a degradation test optimization design method with equally spaced degradation increments, starting from coil springs and based on product degradation data, based on the Wiener process and Bayesian theory. This method solves the problems of large deviation between the pseudo-life and the true life obtained from reliability degradation test analysis of degraded products, as well as excessively long test time.

[0079] ② The method proposed in the present invention is simple to calculate, easy to implement, and more in line with engineering practice, convenient for engineering technicians to master and use, and the method is scientific and easy to apply and promote. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 is a flow chart of the method of the present invention.

[0081] Figure 2 This is a graph showing the change in the permanent compression set of a coil spring over time at 80°C.

[0082] Figure 3 It is the predicted deviation diagram of degradation amount of sample 1 in the optimized experimental design.

[0083] Figure 4 It is the predicted deviation diagram of degradation amount of sample 2 in the optimized experimental design.

[0084] Figure 5 It is the predicted deviation diagram of degradation amount of optimized experimental design sample 3. DETAILED DESCRIPTION

[0085] The present invention proposes a method for optimizing the degradation test of a coil spring based on equally spaced degradation increments. The flow chart is as follows: Figure 1 shown.

[0086] The present invention will be further described below in detail using a 60Si2Mn coil spring product as an example.

[0087] Based on engineering experience, under working conditions, the spring is in a compressed state for a long time. Its deformation recovery performance will gradually deteriorate with working time, which is reflected in the shortening of the spring length when no external force is applied. The compression permanent deformation rate describes the degradation performance of the coil spring. The specific formula is:

[0088]

[0089] Where C(t) is the compression set at time t, H0 is the initial free length, H(t) is the free length after mechanical stress is removed at time t, and h is the fixed height to which the coil spring is compressed. A coil spring fails when the compression set exceeds 10%.

[0090] The fluctuation of the initial free length will cause the randomness of the initial mechanical pressure applied to the coil spring, thereby affecting the degradation performance of the coil spring. Therefore, the 15 measured initial free length values are shown in Table 1.

[0091] Table 1 Initial free length of 15 coil springs (mm)

[0092] 28.05 27.57 27.25 27.88 27.86 28.18 27.89 28.08 27.85 27.60 28.44 27.81 27.92 27.99 27.45

[0093] Figure 2 This figure shows the time-dependent compression set trends for 15 coil spring samples, measured at 80°C every 250 hours until 4000 hours. The following historical information is available: the total number of coil spring measurements, n, is 16; the measurement interval, Δτ, is 250; and the coil spring failure threshold, C, is 10.

[0094] The present invention proposes a method for optimizing the degradation test of a coil spring based on equally spaced degradation increments. The specific implementation steps are as follows:

[0095] Step 1: Wiener process model parameter estimation

[0096] Assume that X(t) is the performance degradation of the coil spring at time t, {X(t), t≥0} obeys the Wiener process, and its degradation X(t) obeys the normal distribution, that is, X(t)~N(μt,σ 2 t). Where μ is the drift coefficient, σ 2 is the diffusion coefficient. and Respectively represent the drift coefficient and diffusion coefficient of each sample in the historical data, and Respectively and The estimated value of t k，p and X k，p It represents the performance degradation of sample k at the time of the pth measurement and the total number of historical measurements of the coil spring n, so that the performance degradation of sample k at time t can be obtained. k，p-1 To time t k，p Time interval Δt k，p =t k，p -t k，p-1 , and sample k at time interval Δt k，p The degradation increment ΔX k，p =X(t k，p )-X(t k，p-1 ), p=1, 2, ..., n.

[0097] Therefore, Figure 2Substituting the 15 sets of historical data shown in the figure into formulas (7) and (8) yields the estimated drift coefficient and diffusion coefficient for each sample. Right now

[0098]

[0099]

[0100] Where Δt k，p =t k，p -t k，p-1 , k=1,2,…,15, represents sample i at time t k，p-1 To time t k，p time interval.

[0101] The specific calculation results are shown in Table 2.

[0102] Table 2 Estimated values of drift coefficient and diffusion coefficient of coil spring samples

[0103]

[0104]

[0105] Step 2: Determine the expected degradation at the next measurement point based on the sampled data

[0106] There are three coil spring products in this test. Assume that t i，0 , X i，0 , t i，1 , X i，1 They represent the test start time, initial value of performance degradation, first measurement time and actual degradation corresponding to the first measurement time of product i, Δt i，1 , ΔX i，1 They represent the first time interval and actual degradation increment of product i in the degradation test, respectively, where Δt i，1 =t i，1 -t i，0 , ΔX i，1 =X i，1 X i，0 , i=1,2,3. Use t i，j′ , Δt i，j′ , X i，j′ , ΔX i，j′ They represent the j′th measurement time, measurement time interval, actual degradation amount and actual degradation increment of product i during the degradation test, respectively. They represent the expected degradation amount and expected degradation increment of product i during the j′th degradation test, respectively, where ΔX i，j′ =X i，j′ -X i，j′ -1, i = 1, 2, 3; j′ = 2, 3, ..., n, where n is the total number of test measurements when the product fails.

[0107] At the beginning of the degradation test, the initial degradation amount of each product is recorded as 0, that is, t i，0 =0,X i，0 = 0. According to the initial test information n, Δτ, C, we can get the first measurement point time t of product i in the test i，1 =t i，0 +Δτ=250h, the spring length is measured and the degradation (permanent deformation rate) is calculated as (X 1，1 , X 2，1 , X 3，1 )=(0.4824, 0.5549, 0.3647).

[0108] After completing the first measurement, the degradation increments can be divided into equal intervals according to the first degradation data to calculate the expected degradation of the second measurement point. Further, starting from the second measurement point, if the current sampling is the j′-1 sampling, the degradation obtained by measurement is X i，j′-1 , then according to formula (9), the expected degradation of the j′th measurement point of sample i can be obtained The calculation formula is

[0109]

[0110] Among them, the expected degradation increment represents the difference between the expected degradation of sample i at the j′th time and the actual degradation at the j′-1th time. The first degradation amount calculated based on the three sample measurements is (X 1，1 , X 2，1 , X 3，1 )=(01.4824, 0.5549, 0.3647), the first expected degradation increments can be calculated as Substituting into formula (9), the expected degradation of the second measurement point can be obtained as follows: Therefore, starting from the second measurement point, the j′th expected degradation value of sample i can be calculated based on the j′-1th measurement data Then the expected degradation data set of sample i can be obtained Among them, i=1, 2, 3.

[0111] Step 3: Update the degradation model parameters based on Bayesian theory and complete the degradation test with equally spaced degradation increments.

[0112] First, based on the estimated and Parameter estimation is performed on the hyperparameters a0, b0, μ0, η0 of the prior distribution, where a0, b0, μ0, η0 represent the shape parameter, scale parameter, drift coefficient, and variance coefficient of the prior distribution respectively. Based on the estimated values of the degradation model parameters of each sample in the historical data According to formulas (13)-(16), the estimated values of the shape parameter, scale parameter, drift coefficient and variance coefficient in the prior distribution are calculated.

[0113]

[0114] Second, based on the estimated The posterior distribution hyperparameter a for each sample i，j′ , b i，j′ , μ i，j′ , η i，j′ Perform parameter estimation, where i = 1, 2, 3; j′ = 2, 3, ..., n. From step 2, we can get the first expected degradation increment of sample i in the test is According to formula (19), the updated shape parameter a can be obtained by substituting it into the posterior distribution: i，1 , scale parameter b i，1 , drift coefficient μ i，1 and the coefficient of variance η i，1 Estimated value of They are

[0115]

[0116] Furthermore, starting from the second measurement point, when the degradation increment data [ΔX1, ΔX2, ..., ΔX j′ ], product data and historical data are integrated and the Wiener process model parameters are gradually updated using the Bayesian method. According to formula (21), the shape parameter a of the posterior distribution of the j′th update of sample i can be obtained i，j′ , scale parameter b i，j′ , drift coefficient μ i，j′ and the coefficient of variance η i，j′ Estimated value of

[0117] Then, the degradation test of equally spaced degradation increments is completed. Taking the second measurement point as an example, the degradation test process of equally spaced degradation increments is described in detail. When product i obtains the posterior distribution hyperparameter estimate after the first measurement is completed According to formula (22), the time value of the second measurement point of sample i (t 1，2 , t 2，2 , t 3，2 ) are (557h, 501h, 437h); then, sample i is at ti，2 The second sampling is performed at the moment to measure the actual degradation amount (X 1，2 , X 2，2 , X 3，2 )=(1.2116, 1.4000, 0.8324), respectively, and the expected degradation Repeat the above steps and compare the actual degradation measurement data X obtained by the j′th measurement of sample i according to formula (23) i，j′ and expected degradation Deviation Δω ij′ for

[0118]

[0119] Furthermore, based on engineering experience, a deviation threshold D = 0.03 is given. If the deviation calculated by the j′th measurement of sample i satisfies Δω ij′ ≤D, end the test, otherwise continue to measure the product failure and end the test. oi is the actual total number of sampling times when the sample i terminates the test, i = 1, 2, 3, that is (n o1 , n o2 , n o3 )=(15,11,15), then the first n samples of sample i oi Degradation increment data calculated from the measurement After substituting into formula (21), we can obtain the estimated values of the shape parameter, scale parameter, drift coefficient and variance coefficient of the posterior distribution: They are

[0120]

[0121] Specific test data and calculation results are shown in Table 3-8 and Figure 3-5 shown.

[0122] Step 4: Calculate the pseudo life of the coil spring based on the equally spaced degradation increment data.

[0123] According to step 3, the model parameters are updated when the test stops. Get the pseudo lifespan of product i According to the failure threshold C of the coil spring product and the estimated value of the drift coefficient Based on the performance degradation model (1), the sample pseudo life formula can be calculated as follows:

[0124]

[0125] Where C represents the product failure threshold, n oi is the actual total number of sampling times when the test of sample i is terminated, i = 1, 2, 3. According to formula (33), the pseudo lifespans of the three products are

[0126] To demonstrate the optimization effect of the present invention in improving the accuracy of life prediction and shortening the test time, a traditional test with equal intervals was added as a control test. Based on historical product information, the control test was set as follows: equal interval Δτ = 250h, total number of test sampling times n = 16, and total test time T0 = 4000h. i，j , ΔX′ i，j They represent the actual degradation amount and actual degradation increment of coil spring product i during the control test, where i = 1, 2, 3; j = 1, 2, ..., n. After completing n sampling times, according to the performance degradation model (1), the product pseudo life formula can be obtained as

[0127]

[0128] in, It represents the estimated drift coefficient value calculated after the product i completes the test with equal time intervals. n represents the total number of test measurements, and n = 16, i = 1, 2, 3.

[0129] Therefore, the pseudo life of the control test of product i is calculated according to formula (34): Specific test data and calculation results are shown in Tables 9-11 and 12.

[0130] Table 3 Measurement data of degradation test optimization design sample 1

[0131]

[0132] Table 4 Measurement data of degradation test optimization design sample 2

[0133]

[0134] Table 5 Measurement data of degradation test optimization design sample 3

[0135]

[0136]

[0137] Table 6 Parameter estimates for degradation test optimization design sample 1

[0138]

[0139] Table 7 Parameter estimates for degradation test optimization design sample 2

[0140]

[0141]

[0142] Table 8 Parameter estimates for degradation test optimization design sample 3

[0143]

[0144] Table 9 Degradation test data of sample 1 measured at equal time intervals

[0145] Number of measurements j 0 1 2 3 4 5 6 7 8 Time / h 0 250 500 750 1000 1250 1500 1750 2000 <![CDATA[X j ]]> 0 0.4823 1.0155 1.7453 2.4340 3.0418 3.7712 4.4085 4.9507 Number of measurements j 9 10 11 12 13 14 15 16 Time / h 2250 2500 2750 3000 3250 3500 3750 4000 <![CDATA[X j ]]> 5.8353 6.4169 7.0328 7.5122 8.0324 8.6060 9.3474 9.8285

[0146] Table 10 Degradation test data of sample 2 measured at equal time intervals

[0147] <![CDATA[Number of measurements j > 0 1 2 3 4 5 6 7 8 Time / h 0 250 500 750 1000 1250 1500 1750 2000 <![CDATA[X j ]]> 0 0.5549 1.4000 1.9421 2.4308 3.1354 3.6085 3.9363 4.5863 <![CDATA[Number of measurements j > 9 10 11 12 13 14 15 16 Time / h 2250 2500 2750 3000 3250 3500 3750 4000 <![CDATA[X j ]]> 5.1073 5.8002 6.3278 6.8798 7.3570 8.0444 8.5612 9.2033

[0148] Table 11 Degradation test data of sample 3 measured at equal time intervals

[0149] <![CDATA[Number of measurements j > 0 1 2 3 4 5 6 7 8 Time / h 0 250 500 750 1000 1250 1500 1750 2000 <![CDATA[X j ]]> 0 0.3647 0.9934 1.6898 2.3049 2.9311 3.5428 4.0982 4.4998 <![CDATA[Number of measurements j > 9 10 11 12 13 14 15 16 Time / h 2250 2500 2750 3000 3250 3500 3750 4000 <![CDATA[X j ]]> 5.1130 5.7706 6.3811 6.9111 7.3442 8.1137 8.7030 9.3638

[0150] According to the two approaches of equal interval degradation increment degradation test design and equal time interval degradation test design, degradation data of three coil spring samples were obtained. Among them, Table 3-8 shows the process and calculation process of the entire degradation test optimization design; Figure 3-5 The deviation between the actual degradation amount and the expected degradation amount during the degradation test is shown. Figure 3-5 It can be concluded that the prediction results obtained from the optimization test are more accurate; and Tables 9-11 are the control test data, that is, the permanent deformation degradation data of the coil spring samples measured at equal intervals.

[0151] Table 12 Comparison of sample life prediction effects under degradation test optimization design methods

[0152]

[0153] The “experiment time reduction rate” in Table 12 indicates the reduction rate of the total experimental time of the optimized design experiment compared with the control experiment. The calculation formula is:

[0154]

[0155] As shown in Table 12, based on a comparison between the optimized design test and the control test, the optimized design proposed in this invention reduced the total test time by 1.8%, 36.1%, and 8.3%, respectively, and the total number of sampling times was reduced by 2, 6, and 2, respectively. Furthermore, the deviation in the product pseudo-life prediction was effectively controlled, decreasing by 78 hours, 6 hours, and 6 hours, respectively. These results demonstrate that this optimized design method can effectively shorten degradation test time and reduce the number of sampling times, while significantly improving the accuracy of life prediction.

[0156] In summary, this invention effectively addresses the discrepancy between the pseudo-lifespan predicted by reliability degradation tests for coil springs and their actual lifespan, which poses additional risks to product reliability assessments. This invention effectively improves lifespan prediction accuracy and test efficiency, providing technical support for reliability testing.

Claims

1. A method for optimizing the degradation test of a coil spring based on equally spaced degradation increments, characterized in that: The method includes the following steps: Step 1: Wiener process model parameter estimation; first, determine the performance degradation model of the coil spring; second, obtain the historical performance degradation data of the same type of products; finally, estimate the drift coefficient of sample k and diffusion coefficient Step 2: Determine the expected degradation amount at the next measurement point based on the sampled data; Step 3: Update the degradation model parameters based on Bayesian theory and complete the degradation test with equally spaced degradation increments; Step 4: Calculate the pseudo life of the coil spring based on the equally spaced degradation increment data; Among them, in step 2, use t j ,Δt j ,X j ,ΔX j They represent the measurement time, measurement time interval, actual degradation amount and actual degradation increment of the coil spring during the degradation test, denote the expected degradation amount and expected degradation increment of the coil spring during the degradation test, respectively, where Δt j =t j -t j-1 ,ΔX j =X j -X j-1 , n is the total number of test measurements when the product fails; the initial measurement time t0 of the product working environment degradation test, the total number of measurements n when the product fails, and the failure threshold C are determined based on the product history information. C indicates that the coil spring product is recorded as failed when the degradation reaches this value; at the beginning of the test, the initial value of the performance degradation of the product is measured to be t0 = 0, and the degradation is recorded as X0 = 0; based on the historical measurement interval time Δτ, the first measurement time of the test is determined and the sampling is completed (t1, X1), where t1 = t0 + Δτ; starting from the first sampling point, the degradation data of the current sampling point is divided into equally spaced degradation increments to obtain the expected degradation amount of the next measurement point; if the current sampling is the j-1th sampling, the actual degradation amount obtained by measurement is X j-1 , then the expected degradation amount of the jth sampling point is The calculation formula is: Among them, the expected degradation increment It represents the increment of the expected degradation amount at the jth time and the actual degradation amount at the j-1th time; Among them, the estimated values of the four hyperparameters based on the prior distribution Estimate the posterior distribution hyperparameter a j ,b j ,μ j ,η j ,j=1,2,…n; where a j ,b j ,μ j ,η j They represent the degradation increment data [ΔX1, ΔX2,…, ΔX j ]After substitution, update the shape parameter, scale parameter, drift coefficient and variance coefficient of the posterior distribution; according to the estimated values of the four hyperparameters of the prior distribution The prior distribution is obtained as: When the degradation test of the coil spring begins, the actual degradation amount measured at the jth sampling moment is (t j ,X j ), j=1,2,…n, calculate the degradation increment ΔX at that moment j =X j -X j-1 , measurement interval time period Δt j =t j -t j-1 , then the degradation increment data [ΔX1, ΔX2,…, ΔX j ], the posterior distribution is: Among them, a j ,b j ,μ j ,η j Indicates that the data [ΔX1, ΔX2,…, ΔX j ]Substitute the four hyperparameters of the posterior distribution, a j is the shape parameter, b j is the scale parameter, μ j is the drift coefficient, η j is the variance coefficient; μ is the drift coefficient, σ is the diffusion coefficient; a j ,b j ,μ j ,η j Estimated value of They are: Where, The data calculated for the first j-1 measurements [ΔX1, ΔX2,…, ΔX j-1 ] is substituted into the updated estimates of the shape parameter, scale parameter, drift coefficient and variance coefficient obtained from the posterior distribution, and They are the degradation increment data [ΔX1, ΔX2,…, ΔX j ]’s mean and variance.

2. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 1, characterized in that: In step 1, let X(t) be the performance degradation of the coil spring at time t, {X(t), t≥0} obeys the Wiener process, and its degradation X(t) obeys the normal distribution, that is, X(t)~N(μt,σ 2 t); where μ is the drift coefficient and σ is the diffusion coefficient.

3. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 1 or 2, characterized in that: In step 1, suppose there are N coil spring samples in the historical data, k is the serial number of the sample in the historical data, t k,p and X(t k,p ) represent the sample k in the historical data at the pth measurement time and the corresponding performance degradation, k = 1, 2, ..., N, p = 1, 2, ..., m k , where m k Represents the total number of measurements of sample k; the initial measurement time and initial degradation amount of each sample are both 0, that is, t k,1 =0, X(t k,1 )=0; thus we get sample k at time t k,p To time t k,p+1 Time period length Δt k,p =t k,p+1 -t k,p , and sample k at time interval Δt k,p The degradation increment ΔX k,p =X(t k,p+1 )-X(t k,p ).

4. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 3, characterized in that: In step one, remember and represent the drift coefficient and diffusion coefficient of historical sample k respectively, and Respectively and Parameter estimates; Based on the performance degradation model, the maximum likelihood estimation method is used to obtain the estimated values of the drift coefficient and diffusion coefficient of sample k in the historical data The formula is as follows: Among them, m k Indicates the total number of measurements of sample k.

5. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 1, characterized in that: In step 3, the estimated values of the drift coefficient and diffusion coefficient of the historical sample k are obtained through steps 1 and 2. and And the expected degradation of the jth sampling In order to reduce the prediction deviation between the expected degradation amount and the actual degradation amount, the Bayesian theory is used to update the degradation model parameters during the test. First, based on the estimated value and The parameters of the prior distribution parameters a0, b0, μ0, and η0 are estimated. Secondly, the posterior distribution hyperparameters are updated based on the estimated values of the prior distribution hyperparameters. Finally, the next measurement moment is calculated based on the estimated values of the posterior distribution hyperparameters and the actual degradation amount is obtained by sampling. The size of the predicted deviation is compared to see if it is lower than the threshold. If the deviation is lower than the threshold, the test is terminated. Otherwise, the test continues until the product fails.

6. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 5, characterized in that: Based on estimates and Parameter estimation is performed on the four hyperparameters a0, b0, μ0, η0 of the prior distribution; a0, b0, μ0, η0 represent the shape parameter, scale parameter, drift coefficient and variance coefficient respectively; the model parameters μ and σ are obtained 2 The joint prior distribution of is: Among them, the unknown parameters a0, b0, μ0, η0 are shape parameter, scale parameter, drift coefficient and variance coefficient respectively; the unknown parameter likelihood function is obtained as: Taking the logarithm, we get the log-likelihood function: Furthermore, the first-order partial derivative equation for unknown parameters a0, b0, μ0, η0 is expressed as: Among them, the gamma function ψ(x)=dln(Γ(x)) / dx, where N is the total number of samples in the historical data; Therefore, based on the maximum likelihood estimation method, the estimated values of the four hyperparameters a0, b0, μ0, and η0 of the prior distribution are obtained.

7. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 1, characterized in that: The degradation model parameter value after each update is used as the prior distribution when the next sampling point is updated, thereby inferring the test time of the next measurement point; If the estimated drift coefficient of the model is obtained after completing the j-1th measurement calculation Then the measurement time t of the jth measurement point is j The specific calculation formula is: Where, is the expected degradation amount of the jth measurement point, The drift coefficient estimate calculated for the j-1th measurement; Furthermore, according to the test time t of the jth measurement point j Complete sampling and get data (t j ,X j ), compare the actual degradation amount X measured at the jth time j The calculated expected degradation Deviation Δω j , the formula is as follows: Given a deviation threshold D, if the deviation calculated by the jth measurement satisfies Δω j ≤D, the test is terminated, otherwise the measurement is continued until the product fails and the test is terminated; the actual total number of sampling times at the end of the test is recorded as n o , the first n o Degradation increment data calculated by the measurement Substitute the parameter estimates of shape parameter, scale parameter, drift coefficient and variance coefficient into 8. The method for optimizing the degradation test of a coil spring based on equally spaced degradation increments according to claim 7, characterized in that: In step 4, the estimated model parameters are updated according to the test stop in step 3. Get product pseudo lifespan The drift coefficient estimate obtained from the final update The pseudo life of the product is calculated using the performance degradation model as follows: Where C represents the product failure threshold, n o Indicates the actual total number of samples taken when the experiment is terminated.

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