Accelerated life test method for small sample of mechanical and electrical product

Abstract

The invention discloses a mechanical and electrical product small sample accelerated life test method comprising the following specific steps: 1, carrying out an accelerated life test, and evaluating a reliable life single-side confidence limit of a mechanical and electrical product under an accelerated stress level; 2, preliminarily evaluating the reliable life single-side confidence lower limit of the mechanical and electrical product under the normal use stress level; and 3, accurately evaluating the reliable life single-side confidence lower limit of the mechanical and electrical product under the normal use stress level. According to the method, the accelerated life test of the mechanical and electrical product can be designed and developed under the small sample condition, the single-side confidence lower limit of the reliable life of the mechanical and electrical product under the normal use stress level can be evaluated without introducing additional assumptions, and the requirements of high confidence and high reliability in engineering can be met while a large number of tests are saved.

Description

[Technical field]

[0001] The invention belongs to the field of accelerated life test design and evaluation, and in particular relates to a small sample accelerated life test method for electromechanical products. [Background technology]

[0002] As an important part of aerospace, transportation and other fields, electromechanical products are developing in the direction of high reliability and long life. In order to carry out reliability assessment more quickly and efficiently, accelerated life tests are often used in engineering practice to shorten the test cycle. However, traditional accelerated life tests generally need to be carried out under multiple accelerated stress levels, and still face challenges such as large sample demand, long test cycle and high cost. In addition, the traditional accelerated life test evaluation method needs to introduce multiple assumptions and has large errors. It can usually only make a point estimate of the reliable life of electromechanical products under normal use stress levels, but it is difficult to make a high-confidence and high-reliability assessment of its unilateral confidence lower limit, which cannot fully meet engineering needs.

[0003] To this end, the present invention proposes a small sample accelerated life test method for electromechanical products whose life follows a two-parameter Weibull distribution and can undergo a constant truncated life test. This method only requires the accelerated life test to be carried out at two accelerated stress levels, and without introducing additional assumptions, the one-sided confidence lower limit of the reliable life of electromechanical products with high confidence and high reliability under normal use stress levels can be evaluated. [Summary of the invention]

[0004] (1) The purpose of the present invention is to provide a small sample accelerated life test method for electromechanical products. By selecting two accelerated stress levels and performing constant truncated accelerated life tests respectively, the one-sided confidence limit of the reliable life of the electromechanical product under the two accelerated stress levels is evaluated, and based on the established mathematical relationship between the one-sided confidence lower limit of the reliable life under the normal use stress level and the one-sided confidence limit of the reliable life under the two accelerated stress levels, a calculation formula for the one-sided confidence lower limit of the reliable life of the electromechanical product with high confidence and high reliability under the normal use stress level is given.

[0005] (2) Technical solution:

[0006] The present invention is a small sample accelerated life test method for electromechanical products, which comprises the following three steps:

[0007] Step 1: Conduct accelerated life tests to evaluate the one-sided confidence limit of the reliable life of electromechanical products under accelerated stress levels;

[0008] Step 2: Preliminary evaluation of the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress level;

[0009] Step 3: Accurately evaluate the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress levels.

[0010] Among them, in step 1, "conduct accelerated life test to evaluate the one-sided confidence limit of the reliable life of electromechanical products under accelerated stress level", the method is as follows:

[0011] Assume that the electromechanical product is under stress level S i The life t under the condition follows a two-parameter Weibull distribution, and its reliability function is:

[0012]

[0013] In the formula, α i and β i They are stress levels S i Shape and scale parameters of the lower lifetime distribution.

[0014] Under the premise of keeping the failure mode and failure mechanism consistent with the normal use stress level S0, two accelerated stress levels S2 and S1 (S0<S1<S2) are selected, and the electromechanical product is subjected to constant truncation test with n samples and r (r≥2) failure number, thus obtaining the accelerated stress level S i The test data under (i=1,2) is:

[0015] t i1 ≤t i2 ≤…≤t ir =t i(r+1) =…=t in (2)

[0016] The electromechanical product is under accelerated stress level S i The reliable life t with lower confidence level γ and reliability R Ri One-sided confidence lower limit t RLi,γ It can be given by the following formula:

[0017]

[0018]

[0019]

[0020] satisfy:

[0021] P(t RLi,γ ≤t Ri )=γ (6)

[0022] In the formula, is the χ with 2r degrees of freedom 2 Distribution ** The lower quantile, The confidence level is γ ** The shape parameter α i One-sided lower confidence limit, is α i is a strictly monotonically increasing function of χ with 2r-2 degrees of freedom. 2 distributed, is its inverse function, where is as of t ij Accelerated stress level S i The generalized total test time under γ ** It can be obtained by the following method: First calculate γ * :

[0023]

[0024]

[0025] γ k =1-(k-0.5) / M,k=1,2,…,M (9)

[0026] In the formula, F 2r (·) is the χ with 2r degrees of freedom 2 Distribution cumulative distribution function, M is set according to the numerical calculation accuracy requirements (such as 10 3 , 10 4 Then, adjust γ in equation (7) to a certain value γ ** , so that the obtained γ * is equal to the preset confidence level γ, at this time γ ** This is the desired value.

[0027] The electromechanical product is subjected to the accelerated stress level S i The reliable life t with lower confidence level γ and reliability R Ri One-sided upper confidence limit t RUi,γ It is given by the following formula:

[0028] t RUi,γ =t RLi,1-γ (10)

[0029] satisfy:

[0030] P(t RUi,γ ≥t Ri )=γ (11)

[0031] Among them, in step 2, the "preliminary assessment of the one-sided confidence lower limit of the reliable life of electromechanical products under normal use stress level" is done as follows:

[0032] Assume that the reliable life of electromechanical products is t Ri With stress level Si There is the following linear relationship between them:

[0033]

[0034] In the formula, Is the stress level S of the electromechanical product i The reliability life is R, a and b are unknown parameters. is the stress level S i A known function of , which can be determined based on the stress type.

[0035] According to the reliability life t of electromechanical products with a reliability of R under accelerated stress levels S2 and S1 R2 ,t R1 , formula (12) can also be transformed into:

[0036] lnt Ri =ω(S i )lnt R1 +[1-ω(S i )]lnt R2 (13)

[0037]

[0038] In the formula, ω(S i ) is the stress difference coefficient.

[0039] Therefore, the reliability life t of the electromechanical product with a reliability of R under the normal use stress level S0 (S2>S1>S0) is R0 It can be expressed as:

[0040] lnt R0 =ω(S0)lnt R1 +[1-ω(S0)]lnt R2 (15)

[0041] In the formula, ω(S0)>1. On this basis, the reliable life t of the electromechanical product with a confidence level of γ and a reliability of R under the normal use stress level S0 can be calculated. R0 One-sided confidence lower limit t RL0,γ Conduct an initial assessment:

[0042]

[0043] Among them, in step 3, the method of "accurately evaluating the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress level" is as follows:

[0044] Furthermore, the one-sided confidence lower limit of the reliable life given by equation (16) can be accurately calculated by the following formula: The confidence γ′ of

[0045]

[0046] Its numerical calculation formula is:

[0047]

[0048]

[0049]

[0050] γ l =1-(l-0.5) / N,l=1,2,…,N (21)

[0051] γ m =1-(m-0.5) / N,m=1,2,…,N (22)

[0052] In the formula, I() is an indicative function. If the inequality in the brackets holds, I() takes the value of 1, otherwise it takes the value of 0; Yes R1 The confidence level is γ l The one-sided lower confidence limit of ; Yes R2 The confidence level is γ m The one-sided confidence lower limit of can be given by step 1. In actual calculation, first select N according to the accuracy requirement, and substitute it into formula (21) and formula (22) to obtain γ l , γ m , and then use formula (19) and formula (20) to calculate Finally, substitute into equation (18) to obtain the confidence γ′.

[0053] Then, adjust equation (18) γ to γ″ in the formula, so that the obtained γ′ is equal to the preset confidence level γ. At this time, the one-sided confidence lower limit t of the reliable life of the electromechanical product with confidence level γ and reliability R under the normal use stress level S0 can be obtained. RL0,γ for:

[0054]

[0055] satisfy:

[0056] P(t RL0,γ ≤t R0 )=γ (24)

[0057] Calculated with formula (16) In comparison, the t given by formula (23) RL0,γ The confidence level of is equal to γ, while the confidence level of the former is at least γ, that is, t RL0,γWith higher accuracy, closer to the true value of reliable life t R0 .

[0058] Through the above three steps, based on the constant-truncated accelerated life test data under two accelerated stress levels, a high-confidence and high-reliability assessment of the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress levels is achieved.

[0059] The beneficial effects of the present invention are as follows: the present invention proposes a small sample accelerated life test design and reliability assessment method for electromechanical products whose life follows a two-parameter Weibull distribution and can undergo a constant-truncation life test. Without introducing additional assumptions, the one-sided confidence lower limit of the reliable life of the electromechanical products under normal use stress levels can be assessed, which saves a large number of tests while meeting engineering requirements for high confidence and high reliability.

Brief Description of the Drawings

[0060] Figure 1 It is a schematic diagram of the process of the present invention.

[0061] Figure 2 The constant truncation test results of a certain type of harmonic reducer at a torque level of 75N·m are shown.

[0062] Figure 3 These are the constant truncation test results of a certain type of harmonic reducer at a torque level of 50 N·m. [Specific implementation method]

[0063] The present invention is described in detail below with reference to the accompanying drawings.

[0064] The present invention proposes a small sample accelerated life test method for electromechanical products, the flow chart of which is as follows: Figure 1 shown.

[0065] In the present invention, an example of a small sample accelerated life test of a certain type of harmonic reducer is used to illustrate the implementation of the present invention, and the specific steps are as follows:

[0066] Among them, in step 1, "conduct accelerated life test to evaluate the one-sided confidence limit of the reliable life of electromechanical products under accelerated stress level", the method is as follows:

[0067] It is known that the life of a certain type of harmonic reducer follows a two-parameter Weibull distribution. An accelerated life test is now carried out on it using torque as the accelerating stress. Two accelerated stress levels S2 = 75N·m and S1 = 50N·m are selected. Under each accelerated stress level, a constant truncated life test is carried out on the harmonic reducer with a sample number of n = 6 and a failure number of r = 4. The test results are as follows: Figure 2 , Figure 3 shown.

[0068] According to the test results, the upper limit of the reliable life of the harmonic reducer with confidence level γ = 0.90 and reliability level R = 0.99 is obtained under S2 = 75 N·m and S1 = 50 N·m. RU2,γ , lower limit t RL1,γ for:

[0069] t RU2,γ =[-2×ln(0.99)×2.77×10 20 / 4.06] 1 / 6.607 =556h (25)

[0070] t RL1,γ =[-2×ln(0.99)×1.58×10 14 / 12.44] 1 / 3.952 =769h (26)

[0071] Among them, in step 2, the "preliminary assessment of the one-sided confidence lower limit of the reliable life of electromechanical products under normal use stress level" is done as follows:

[0072] Using the inverse power law model lnt Ri =a+blnS i As an acceleration model, it is preliminarily evaluated that the one-sided confidence lower limit of the reliable life of the harmonic reducer with a confidence level of 0.9 and a reliability of 0.99 under the normal use stress level S0 = 25N·m is:

[0073]

[0074] Among them, in step 3, the method of "accurately evaluating the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress level" is as follows:

[0075] Further precise calculation The confidence γ′ is:

[0076] γ′=0.9428>0.90 (28)

[0077] Then, adjust γ in formula (27) to γ″=0.8360, so that the corresponding γ′ is equal to 0.90, and the one-sided confidence lower limit of the reliable life of the harmonic reducer with a confidence level of 0.90 and a reliability of 0.99 under the normal use stress level S0=25N·m can be obtained as follows:

[0078] t RL0,γ =exp[2.710×ln(971)+(1-2.710)×ln(509)]=2925h(29).

Claims

1. A small sample accelerated life test method for electromechanical products, characterized in that: The steps include: Step 1: Conduct accelerated life tests to evaluate the one-sided confidence limit of the reliable life of electromechanical products under accelerated stress levels; Step 2: Preliminary evaluation of the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress level; Step 3: Accurately evaluate the one-sided lower confidence limit of the reliable life of electromechanical products under normal use stress levels.

2. The small sample accelerated life test method for electromechanical products according to claim 1, characterized in that: In step 1, specifically: Assume that the electromechanical product is at stress level S i The life t under this condition obeys the two-parameter Weibull distribution, and the reliability function is: In the formula, α i and β i They are stress levels S i shape parameter and scale parameter of the lower lifetime distribution; Under the premise of keeping the failure mode and failure mechanism consistent with the normal use stress level S0, two accelerated stress levels S2 and S1 are selected to conduct constant truncation tests on the electromechanical product with n samples and r failure numbers, respectively, thus obtaining the accelerated stress level S i The following test data are: t i1 ≤t i2 ≤…≤t ir =t i(r+1) =…=t in (2) Among them, S0<S1<S2, r≥2, i=1,2.

3. The small sample accelerated life test method for electromechanical products according to claim 2, characterized in that: Mechanical and electrical products under accelerated stress level S i The reliable life t with lower confidence level γ and reliability R Ri One-sided confidence lower limit t RLi,γ Given by: satisfy: P(t RLi,γ ≤t Ri )=γ (6) In the formula, is the χ with 2r degrees of freedom 2 Distribution ** The lower quantile, The confidence level is γ ** The shape parameter α i One-sided lower confidence limit, is α i is a strictly monotonically increasing function of χ with 2r-2 degrees of freedom. 2 distributed, is its inverse function, where is as of t ij Accelerated stress level S i The generalized total test time under .

4. The small sample accelerated life test method for electromechanical products according to claim 3, characterized in that: γ ** Obtained by the following method; Calculate γ * : γ k =1-(k-0.5) / M,k=1,2,…,M (9) In the formula, F 2r (·) is the χ with 2r degrees of freedom 2 The cumulative distribution function is calculated, and M is selected according to the numerical calculation accuracy requirements; γ in formula (7) is adjusted to a certain value γ ** , so that the obtained γ * is equal to the preset confidence level γ, at this time γ ** This is the desired value.

5. The method for accelerating the life test of a small sample of electromechanical products according to claim 4, characterized in that: The electromechanical product is subjected to the accelerated stress level S i The reliable life t with lower confidence level γ and reliability R Ri One-sided upper confidence limit t RUi,γ It is given by the following formula: t RUi,γ =t RLi,1-γ (10) satisfy: P(t RUi,γ ≥t Ri )=γ (11).

6. The small sample accelerated life test method for electromechanical products according to claim 1, characterized in that: In step 2, specifically: let the reliable life of electromechanical products t Ri With stress level S i There is the following linear relationship between them: In the formula, Is the stress level S of the electromechanical product i The reliability life is R, a and b are unknown parameters. is the stress level S i A known function of , determined according to the stress type.

7. The small sample accelerated life test method for electromechanical products according to claim 6, characterized in that: According to the reliability life t of electromechanical products with a reliability of R under accelerated stress levels S2 and S1 R2 ,t R1 , and transform equation (12) into: lnt Ri =ω(S i )ln t R1 +[1-ω(S i )]ln t R2 (13) In the formula, ω(S i ) is the stress difference coefficient.

8. The method for accelerating the life test of a small sample of electromechanical products according to claim 7, characterized in that: The reliability life t of the electromechanical product with a reliability of R under the normal use stress level S0 R0 It is expressed as: ln t R0 =ω(S0)ln t R1 +[1-ω(S0)]ln t R2 (15) Where S2>S1>S0, ω(S0)>1; the reliable life t of the electromechanical product with confidence γ and reliability R under the normal use stress level S0 is R0 One-sided confidence lower limit t RL0,γ Conduct an initial assessment:

9. The method for accelerating the life test of a small sample of electromechanical products according to claim 8, characterized in that: In step 3, specifically: the one-sided confidence lower limit of the reliable life given by equation (16) is accurately calculated by the following formula: The confidence γ′ of The numerical calculation formula is: c l =1-(l-0.5) / N,l=1,2,…,N (21) γ m =1-(m-0.5) / N,m=1,2,…,N (22) In the formula, I() is an indicative function. If the inequality in the brackets holds, I() takes the value of 1, otherwise it takes the value of 0; Yes R1 The confidence level is γ l The one-sided lower confidence limit of ; Yes R2 The confidence level is γ m The one-sided lower confidence limit for .

10. The method for accelerating the life test of a small sample of electromechanical products according to claim 9, characterized in that: Adjustment type (18) γ to γ″ in the figure, so that the obtained γ′ is equal to the preset confidence level γ, and the one-sided confidence lower limit t of the reliable life of the electromechanical product with confidence level γ and reliability R under the normal use stress level S0 is obtained. RL0,γ for: satisfy: P(t RL0,γ ≤t R0 )=γ (24) Calculated with formula (16) In comparison, the t given by formula (23) RL0,γ The confidence level of is equal to γ, while the confidence level of the former is at least γ, that is, t RL0,γ With higher accuracy, closer to the true value of reliable life t R0 .

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