CN114626248B - Spiral spring reliability evaluation method based on multi-stress accelerated degradation data - Google Patents
Spiral spring reliability evaluation method based on multi-stress accelerated degradation data Download PDFInfo
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Abstract
The invention provides a method for evaluating reliability of a spiral spring based on multi-stress accelerated degradation data, which comprises the following implementation steps of: firstly, the following steps: collecting data of a multi-stress accelerated degradation test; II, secondly, the method comprises the following steps: constructing a multi-stress accelerated degradation model based on the organic fusion of the acceleration models; thirdly, the method comprises the following steps: estimating parameters of a multi-stress accelerated degradation model; fourthly, the method comprises the following steps: evaluating the reliability of the spiral spring under normal stress; according to the method, a multi-stress acceleration model is constructed by utilizing an acceleration model organic fusion method, a spiral spring multi-stress acceleration degradation model is formed by combining a degradation model, and parameters of the multi-stress acceleration degradation model are estimated by utilizing a two-stage least square method, so that the reliability level of the spiral spring under normal stress is given, and the use efficiency evaluation and examination of the spiral spring are improved; the method provided by the invention has the advantages of simple and understandable principle and calculation, easy realization, convenience for engineering technicians to master and use, and convenience for application and popularization.
Description
Technical Field
The invention provides a spiral spring reliability evaluation method based on multi-stress accelerated degradation data, which is a spiral spring reliability evaluation method based on organic fusion of an acceleration model; aiming at the compression permanent deformation rate data of the spiral spring under the acceleration of temperature, humidity and mechanical stress, a multi-stress accelerated degradation model is constructed through the organic fusion of acceleration models, parameters of the multi-stress accelerated degradation model are estimated by using a two-stage least square estimation method, and finally, the reliability evaluation of the spiral spring is completed based on the normal stress level; the method is suitable for the related technical fields of product degradation reliability evaluation and the like.
Background
The spiral spring is a spiral spring formed by coiling a spring wire, and is stretched and compressed to generate elastic deformation, so that the aim of storing energy is fulfilled; the spiral spring is an elastic element widely used in the mechanical and electronic industry, provides a power source for sealing, resetting, driving, buffering and the like of a device, and is affected by stress relaxation, and the elastic performance of the spiral spring is gradually reduced along with the increase of the service time, so that the normal operation of the mechanical and electronic device is affected; in order to identify the change of the elastic property of the spiral spring in the using process, a stress relaxation test is required to be carried out on the spiral spring and test data are analyzed;
because the spiral spring generally has the characteristics of high reliability and long service life, effective elastic performance degradation data are difficult to obtain in a short time in a conventional test and even a single stress acceleration test, and the reliability evaluation of the spiral spring is seriously influenced; in consideration of the fact that the spiral spring is always in a stressed stretching or compression state in the actual use process and the elastic property degradation process of the spiral spring is easily influenced by various types of stress such as environmental temperature, humidity and the like, the multi-stress accelerated degradation test is carried out on the spiral spring, so that the elastic property degradation process of the spiral spring can be accelerated, the actual use environment of the spiral spring can be closer to the stress, and the effectiveness of test data is improved; however, in engineering application, only an acceleration model considering the influence of temperature, humidity or temperature and mechanical stress exists, and an acceleration model capable of simultaneously describing the influence of temperature, humidity and mechanical stress on the elastic performance degradation process of the spiral spring is lacked, so that the reliability evaluation of the spiral spring cannot be carried out by fully utilizing multiple stress degradation test data, and the use efficiency evaluation and examination of the spiral spring are influenced;
based on the method, the invention provides a spiral spring reliability evaluation method based on multi-stress accelerated degradation data, namely a spiral spring multi-stress accelerated degradation model construction and reliability evaluation method based on acceleration model fusion.
Disclosure of Invention
(1) The purpose of the invention is as follows:
the invention provides a reliability evaluation method of a spiral spring based on multi-stress accelerated degradation data, aiming at the problem that the reliability evaluation of the spiral spring cannot be effectively carried out by utilizing the multi-stress accelerated degradation data due to the lack of an acceleration model which simultaneously considers the influence of temperature, humidity and mechanical stress on the elastic performance degradation process of the spiral spring, and the reliability evaluation method of the spiral spring comprises the steps of collecting the multi-stress accelerated degradation test data, constructing a multi-stress accelerated degradation model, estimating model parameters and evaluating the reliability; and representing the elastic performance of the spiral spring through the compression permanent deformation rate, further constructing a multi-stress accelerated degradation model based on the organic fusion of the accelerated model, performing two-stage least square estimation on model parameters, and finally evaluating the reliability of the spiral spring under the normal stress level.
(2) The technical scheme is as follows:
based on the theory and thought, the invention provides a method for evaluating the reliability of a spiral spring based on multi-stress accelerated degradation data, which comprises the following specific implementation steps:
the method comprises the following steps: multi-stress accelerated degradation test data collection
Distributing the spiral springs to different acceleration stress level combinations for testing, and collecting free length data of all samples at different moments under different acceleration stress levels; assume a total of R stress level combinations, denoted as S r (ii) a R =1,2, …, R }, where S r ＝(T r ，RH r ，MS r ) Denotes the r-th stress level combination, T r Representing the value of the absolute temperature in the r-th stress level combination, RH r Represents the value of the relative humidity in the r-th stress level combination, MS r Representing the value of the mechanical stress in the r-th stress level combination; therefore, the j-th test data of the i-th sample at the r-th stress level combination is recorded as:
{(t r，j ，H r，i，j )；r＝1，2，…，R；i＝0，1，…，M r ；j＝1，2，…，N r } (1)
in the formula, t r，j Denotes the time of the j-th test under the r-th stress level combination, H r，i，j Represents the free length measurement of the ith sample in the jth test under the jth stress level combination, R represents the number of all stress level combinations, M r Represents the sample size at the r-th stress level combination，N r Representing the total number of measurements of the sample at the r-th stress level combination; in particular, when i =0, the corresponding H r，0，j Representing the initial free length of the ith sample at the combination of the r-th stress levels;
furthermore, according to practical engineering experience, the stress relaxation of a coil spring is often characterized by a compression set ratio, which is defined as:
in the formula, y r，i，j Denotes the compression set, H, of the ith sample in the jth test at the combination of the r-th stress levels r，0，j Denotes the initial free length, H, of the ith sample at the r-th stress level combination r，i，j Representing the free length of the jth sample measured at the jth time under the ith stress level combination; determining the failure threshold value of the compression set rate of the spiral spring as D according to the corresponding national standard and the requirement of a user, namely considering that the spiral spring fails when the compression set rate is higher than D;
step two: multi-stress accelerated degradation model structure based on accelerated model organic fusion
In engineering, when the combination of stress levels is S r The degradation model of the compression set rate of the coil spring is:
y(t|S r )＝α 0 +α 1 (S r )×ln(t)+ε (3)
wherein t represents an accelerated degradation test time of the coil spring, S r Representing a combination of stress levels, y (t | S) r ) Representing a combination of stress levels S r Compression set, alpha, of the lower coil spring at time t 1 (S r ) For the degenerate model (3) to combine S at the stress level r Acceleration coefficient of 0 Is a parameter in the degeneration model (3), and epsilon represents a measurement error and follows a normal distribution
The main stress types influencing the degradation process of the compression set rate of the spiral spring are temperature, humidity and mechanical stress, wherein the influence mode of the temperature and the humidity on the degradation process of the compression set rate of the spiral spring is characterized by utilizing an Ailin model shown as a formula (4):
in the formula, T r And RH r Respectively, the absolute temperature and the relative humidity, alpha, at the r-th stress level 1 (T r ，RH r ) Denotes absolute temperature and relative humidity as T r And RH r Acceleration factor of time, k =8.617 × 10 -5 eV/K, representing Boltzmann constant, β 0 、β 1 、β 2 Three parameters in the model (4); the mode of influence of temperature and mechanical stress on the degradation process of the compression set rate of a coil spring is characterized by using a Boltzmann-Arrhenius-Zhurkov model as shown in formula (5):
in the formula, T r And MS r Respectively, the absolute temperature and the mechanical stress at the r-th stress level, alpha 1 (T r ，MS r ) Denotes absolute temperature and mechanical stress respectively as T r And MS r Acceleration coefficient of time, p 0 And ρ 1 Two parameters in the model (5);
therefore, in order to integrate the influence of three stresses of temperature, humidity and mechanical stress on the compression set of the spiral spring and keep the interaction among the stresses unchanged, the three-stress acceleration model structure is realized by utilizing logarithmic transformation and exponential inverse transformation based on an acceleration model organic fusion method, and the specific process is as follows:
(1) Accelerated model log transformation and merging
First, logarithmic transformation is performed on the model (4) and the model (5), and the two models are combined to obtain a logarithmic form acceleration model as shown below:
in the formula, alpha 1 (T r ，RH r ，MS r ) Denotes absolute temperature, relative humidity and mechanical stress respectively as T r 、RH r And MS r Acceleration coefficient of time, beta 0 、β 1 、β 2 For three parameters in the model (4), p 0 、ρ 1 Two parameters in the model (5);
(2) Accelerating model exponential inverse transformation and merging same-class items
Then, an exponential inverse transformation is performed on model (6), resulting in an accelerated model as follows:
let μ = β 0 ×ρ 0 The obtained acceleration model comprehensively considering the influence of temperature, humidity and mechanical stress is as follows:
in the formula, alpha 1 (T r ，RH r ，MS r ) Denotes absolute temperature, relative humidity and mechanical stress respectively as T r 、RH r And MS r Acceleration factor of time; beta is a 0 、β 1 、β 2 、ρ 0 Four parameters in the model;
(3) Multi-stress accelerated degradation model structure
According to the degeneration model (3) and the acceleration model (8), the compression set rate of the spiral spring at the temperature T is obtained r Humidity RH r And mechanical stress MS r The following multi-stress accelerated degradation model is:
in the formula, alpha 0 、μ、β 1 、β 2 、ρ 1 For the four parameters in the model, epsilon represents the measurement error;
step three: multi-stress accelerated degradation model parameter estimation
Based on the multi-stress accelerated degradation model (9) and the compression permanent deformation rate data of the spiral spring, a least square estimation method is used for carrying out two-stage parameter estimation on the multi-stress accelerated degradation model (9), and the carrying out process is as follows:
(1) Degradation model parameter estimation
In the first stage, let θ r，1 ＝[α r，0 ，α 1 (S r )] T Representing the combination S of the degradation model (3) at the stress level r A vector of unknown parameters, where r，0 Representing a parameter alpha 0 At stress level combination S r The value of S is then combined with the stress level based on the least squares estimation principle r The following compression set data of each spiral spring are obtained, and the parameter estimation result is as follows:
in the formula (I), the compound is shown in the specification,
denotes alpha r，0 、α 1 (S r ) Is evaluated based on the evaluation of->
Wherein the content of the first and second substances,
representing combinations S of stress levels r All samples are at t r，j Average compression set at time, R =1,2, …, R, j =1,2, …, N r (ii) a Thus, the parameter α 0 Is greater than or equal to>
Comprises the following steps:
in addition, let
Representing a combination S of stress levels based on a degradation model (3) and a parameter estimation result (10) r The next ith sample is at t r，j The parameter is then obtained at the time of the predicted compression set>
Is greater than or equal to>
Comprises the following steps:
in the formula (I), the compound is shown in the specification,
denotes the standard deviation sigma of the measurement error epsilon ε Estimate of (a), y r，i，j Representing combinations S of stress levels r The next ith sample is at t r，j The actual compression set at a time is greater or less than>
Representing combinations S of stress levels r The next ith sample is at t r，j Predicting the compression set rate at a moment;
(2) Accelerating model parameter estimation
In the second stage, let θ 2 ＝[μ，ρ 1 ，β 1 ，β 2 ] T Representing the combination S of the acceleration model (8) at the stress level r The vector formed by the unknown parameters is based on the least square estimation principle and each stress level combination S r Alpha of 1 (S r ) The estimation result of the obtained parameters is as follows:
in the formula (I), the compound is shown in the specification,
representing model parameters mu, beta 1 、β 2 、ρ 1 Is estimated by the estimation of (a) a,
in the formula, T r 、RH r 、MS r Respectively representing temperature, humidity and mechanical stress at the r-th stress level combination, S r ＝(T r ，RH r ，MS r )，
Denotes alpha 1 (S r ) R =1,2, …, R;
so far, the estimation of unknown parameters in the multi-stress accelerated degradation model (9) is completed;
step four: coil spring reliability evaluation under normal stress
After the construction and parameter estimation of the multi-stress accelerated degradation model of the compression permanent deformation rate of the coil spring are completed, the normal stress level S of the coil spring is based 0 ＝(T 0 ，RH 0 ，MS 0 ) And obtaining a reliability function of the coil spring under normal stress with a compression permanent deformation rate failure threshold value D as follows:
wherein R (t | S) 0 ) Indicates the normal stress level S 0 The reliability of the lower coil spring at time t,
as a model parameter alpha 0 、μ、β 1 、β 2 、ρ 1 Estimate of (A), T 0 、RH 0 、MS 0 Respectively, temperature, humidity and mechanical stress at normal stress levels, S 0 ＝(T r ，RH r ，MS r )，/>
Denotes the standard deviation sigma of the measurement error epsilon ε ψ (-) is the cumulative distribution function of a standard normal distribution;
thus, the reliability evaluation of the coil spring under normal stress is completed.
(3) The advantages and the effects are as follows:
the invention relates to a method for evaluating the reliability of a spiral spring based on multi-stress accelerated degradation data, which has the advantages that:
(1) aiming at accelerated degradation test data of the spiral spring under temperature, humidity and mechanical stress, a multi-stress accelerated degradation model is constructed by utilizing an accelerated model organic fusion method, a spiral spring multi-stress accelerated degradation model is formed by combining the accelerated degradation model, and parameters of the multi-stress accelerated degradation model are estimated by utilizing a two-stage least square method, so that the reliability level of the spiral spring under normal stress is given, and the use efficiency identification and examination of the spiral spring are improved;
(2) the method provided by the invention has the advantages of simple and understandable principle and calculation, easy realization, convenience for engineering technicians to master and use, and convenience for application and popularization.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a graph of reliability corresponding to the present invention (a diagram illustrating the reliability evaluation results of a certain type of coil spring).
Detailed Description
The present invention will be described in further detail below with reference to fig. 1 by taking a certain type of coil spring used in a certain type of elastic safety valve product as an example.
In a stress relaxation test of a coil spring, 24 coil springs were equally divided into 8 groups, and accelerated degradation tests were performed under 8 combinations of temperature, humidity, and mechanical stress levels, respectively, and the stress level conditions in each stress level combination are shown in table 1. During the test, the free length of the 24 coil springs was measured to obtain the compression set rate data of the coil springs. The test was carried out for a total of 120 days, during which free length data of all springs were measured on days 1, 3, 7, 15, 30, 50, 70, 95, 120, respectively. In addition, the normal stress level of the spiral spring is S in engineering 0 (T 0 ＝333.15K,RH0＝50％,MS0＝8N。
TABLE 1 stress level combinations for stress relaxation tests of certain types of coil springs
| Stress level combination serial number | Absolute temperature (K) | Relative humidity (%) | Mechanical stress (N) |
| 1 | 343.15 | 70 | 10 |
| 2 | 343.15 | 70 | 8 |
| 3 | 343.15 | 90 | 10 |
| 4 | 343.15 | 90 | 8 |
| 5 | 353.15 | 70 | 10 |
| 6 | 353.15 | 70 | 8 |
| 7 | 353.15 | 90 | 10 |
| 8 | 353.15 | 90 | 8 |
Therefore, the invention provides a reliability evaluation method of a spiral spring based on multi-stress accelerated degradation data, the operation flow is shown in figure 1, and the specific steps are as follows:
the method comprises the following steps: multi-stress accelerated degradation test data collection
After the free length data of the coil spring was obtained through the experiment, the free length data was converted into compression set data by the formula (2), and the results are shown in table 2.
TABLE 2 compression set test data for certain type of coil spring
Step two: multi-stress accelerated degradation model structure based on accelerated model organic fusion
The main acceleration stresses of this type of coil spring are known from table 1 as temperature, humidity and mechanical stress. Therefore, based on an acceleration model organic fusion method, a spiral spring multi-stress accelerated degradation model under the acceleration of temperature, humidity and mechanical stress is constructed by combining degradation models of the compression permanent deformation rate of the spiral spring under each stress, and is as follows:
wherein y (t | S) r ) Representing a combination of stress levels S r Compression set, alpha, of the lower coil spring at time t 0 、μ、β 1 、β 2 、ρ 1 For four parameters in the model, T r 、RH r 、MS r Respectively representing temperature, humidity and mechanical stress at the r-th stress level combination, S r ＝(T r ，RH r ，MS r ) ε represents the measurement error;
step three: multi-stress accelerated degradation model parameter estimation
Based on the test data of the compression permanent deformation rate of the coil spring shown in the table (2), the parameter alpha in the multi-stress accelerated degradation model (15) is carried out by using a two-stage least square estimation method 0 、μ、ρ 1 、β 1 、β 2 And variance of ε
(ii) is estimated; the least squares estimation results of the multi-stress accelerated degradation model parameters obtained by solving equations (10) to (13) are shown in table 3.
TABLE 3 estimation results of multiple stress accelerated degradation model parameters
Therefore, the multi-stress accelerated degradation model of the compression set of the coil spring is obtained as follows:
in the formula, T r 、RH r 、MS r Respectively representing temperature, humidity and mechanical stress at the r-th stress level combination;
step four: coil spring reliability evaluation under normal stress
After a multi-stress accelerated degradation model of the compression permanent deformation rate of the spiral spring is obtained, reliability evaluation under normal stress can be carried out according to the failure threshold value of the spiral spring and the formula (16); according to engineering experience, determining that the failure threshold value of the compression set rate of the spiral spring is 13%, namely when the compression set rate exceeds 13%, determining that the spiral spring cannot provide enough elastic force to ensure the normal operation of the elastic safety valve, and according to the multi-stress accelerated degradation model (16) obtained in the step three, obtaining the reliability function of the spiral spring under normal stress as follows:
in the formula, T 0 、RH 0 、MS 0 Respectively, temperature, humidity and mechanical stress under normal stress. The corresponding reliability curve is shown in fig. 2; therefore, the reliability evaluation of the spiral spring under normal stress is completed;
in summary, the invention provides a coil spring reliability assessment method based on multi-stress accelerated degradation data, and relates to a coil spring reliability assessment method based on an accelerated model organic fusion and a two-stage least square estimation method; aiming at the stress relaxation test data of the spiral spring under temperature, humidity and mechanical stress, the method utilizes the change process of the compression permanent deformation rate to represent the elastic performance degradation process of the spiral spring, further constructs a multi-stress accelerated degradation model through the organic fusion of accelerated models, carries out model parameter estimation by using a two-stage least square estimation method, and finally evaluates the reliability of the spiral spring under the normal stress level, thereby effectively improving the practical use efficiency identification and examination of the spiral spring.
Claims (1)
1. A reliability evaluation method of a spiral spring based on multi-stress accelerated degradation data is characterized by comprising the following steps: the method comprises the following specific steps:
the method comprises the following steps: multi-stress accelerated degradation test data collection
Distributing the spiral springs to different acceleration stress level combinations for testing, and collecting free length data of all samples at different moments under different acceleration stress levels; let a total of R stress level combinations be denoted as { S r (ii) a R =1,2, …, R }, where S r ＝(T r ，RH r ，MS r ) Denotes the r-th stress level combination, T r Representing the value of the absolute temperature in the r-th stress level combination, RH r Representing the value of the relative humidity in the r-th combination of stress levels, MS r Representing the value of the mechanical stress in the r-th stress level combination; therefore, the j-th test data of the i-th sample at the r-th stress level combination is recorded as:
{(t r，j ，H r，i，j )；r＝1，2，…，R；i＝0，1，…，M r ；j＝1，2，…，N r } (1)
in the formula, t r，j Denotes the time of the j-th test under the r-th stress level combination, H r，i，j Represents the free length of the ith sample in the jth test under the jth stress level combinationMeasured values, R representing the number of combinations of all stress levels, M r Denotes the sample size, N, at the r-th combination of stress levels r Representing the total number of measurements of the sample at the r-th stress level combination; when i =0, the corresponding H r，0，j Representing the initial free length of the ith sample at the r-th stress level combination;
furthermore, according to practical engineering experience, the stress relaxation of a coil spring is often characterized by a compression set ratio, which is defined as:
in the formula, y r，i，j Denotes the compression set, H, of the ith sample in the jth test at the combination of the r-th stress levels r，0，j Denotes the initial free length, H, of the ith sample at the r-th stress level combination r，i，j Representing the free length of the jth sample measured at the jth time under the ith stress level combination; determining that the failure threshold value of the compression set rate of the spiral spring is D, namely considering that the spiral spring fails when the compression set rate is higher than D;
step two: multi-stress accelerated degradation model structure based on organic fusion of acceleration models
When the combination of stress levels is S r The degradation model of the compression set of the coil spring is:
y(t|S r )＝α 0 +α 1 (S r )×ln(t)+ε (3)
wherein t represents an accelerated degradation test time of the coil spring, S r Representing a combination of stress levels, y (t | S) r ) Representing a combination of stress levels S r Compression set, alpha, of the lower coil spring at time t 1 (S r ) For the degenerate model (3) to combine S at the stress level r Acceleration coefficient of 0 Is a parameter in the degeneration model (3), and epsilon represents a measurement error and follows a normal distribution
The stress types influencing the degradation process of the compression set rate of the spiral spring are temperature, humidity and mechanical stress, wherein the influence mode of the temperature and the humidity on the degradation process of the compression set rate of the spiral spring is characterized by utilizing an Ailin model shown as a formula (4):
in the formula, T r And RH r Respectively, the absolute temperature and the relative humidity, alpha, at the r-th stress level 1 (T r ，RH r ) Denotes absolute temperature and relative humidity as T r And RH r Acceleration factor of time, k =8.617 × 10 -5 eV/K, representing Boltzmann constant, β 0 、β 1 、β 2 Three parameters in the model (4); the mode of influence of temperature and mechanical stress on the degradation process of the compression set rate of a coil spring is characterized by using a Boltzmann-Arrhenius-Zhurkov model as shown in formula (5):
in the formula, T r And MS r Respectively, the absolute temperature and the mechanical stress at the r-th stress level, alpha 1 (T r ，MS r ) Denotes absolute temperature and mechanical stress respectively as T r And MS r Acceleration coefficient of time, p 0 And ρ 1 Two parameters in the model (5);
therefore, in order to integrate the influence of three stresses of temperature, humidity and mechanical stress on the compression set of the spiral spring and keep the interaction among the stresses unchanged, the three-stress acceleration model structure is realized by utilizing logarithmic transformation and exponential inverse transformation based on an acceleration model organic fusion method, and the specific process is as follows:
(1) Accelerated model log transformation and merging
First, logarithmic transformation is performed on the model (4) and the model (5), and the two models are combined to obtain a logarithmic form acceleration model as shown below:
in the formula, alpha 1 (T r ，RH r ，MS r ) Denotes absolute temperature, relative humidity and mechanical stress respectively as T r 、RH r And MS r Acceleration coefficient of time, beta 0 、β 1 、β 2 For three parameters in the model (4), p 0 、ρ 1 Two parameters in the model (5);
(2) Accelerating model exponential inverse transformation and merging same-class items
Then, an exponential inverse transformation is performed on model (6), resulting in an accelerated model as follows:
let μ = β 0 ×ρ 0 Obtaining an acceleration model comprehensively considering the influence of temperature, humidity and mechanical stress as follows:
in the formula, alpha 1 (T r ，RH r ，MS r ) Denotes absolute temperature, relative humidity and mechanical stress respectively as T r 、RH r And MS r Acceleration factor of time; beta is a 0 、β 1 、β 2 、ρ 0 Four parameters in the model;
(3) Multi-stress accelerated degradation model structure
According to the degeneration model (3) and the acceleration model (8), the compression set rate of the spiral spring at the temperature T is obtained r Humidity RH r And mechanical stress MS r The following multi-stress accelerated degradation model is:
in the formula, alpha 0 、μ、β 1 、β 2 、ρ 1 For the four parameters in the model, epsilon represents the measurement error;
step three: multi-stress accelerated degradation model parameter estimation
Based on the multi-stress accelerated degradation model (9) and the compression permanent deformation rate data of the spiral spring, a least square estimation method is used for carrying out two-stage parameter estimation on the multi-stress accelerated degradation model (9), and the carrying out process is as follows:
(1) Degradation model parameter estimation
In the first stage, let θ r，1 ＝[α r，0 ，α 1 (S r )] T Representing the combination S of the degradation model (3) at the stress level r A vector of unknown parameters, wherein r，0 Representing a parameter alpha 0 At stress level combination S r The value of S is then combined with the stress level based on the least squares estimation principle r The compression set rate data of each spiral spring is as follows, and the estimation result of the parameters is as follows:
in the formula (I), the compound is shown in the specification,
denotes alpha r，0 、α 1 (S r ) Is evaluated based on the evaluation of->
Wherein the content of the first and second substances,
representing combinations S of stress levels r All samples are at t r，j Average compression set at time, R =1,2, …, R, j =1,2, …, N r (ii) a Thus, the parameter α 0 Is greater than or equal to>
Comprises the following steps:
in addition, let
Representing a combination S of stress levels based on a degradation model (3) and a parameter estimation result (10) r The next ith sample is at t r，j A compression set rate predicted at a time is determined, a parameter->
Evaluation value of>
Comprises the following steps:
in the formula (I), the compound is shown in the specification,
denotes the standard deviation sigma of the measurement error epsilon ε Estimate of (a), y r，i，j Representing combinations S of stress levels r The next ith sample is at t r，j The actual compression set at a time is greater or less than>
Representing combinations S of stress levels r The next ith sample is at t r，j Predicting the compression set rate at a moment;
(2) Accelerating model parameter estimation
In the second stage, let θ 2 ＝[μ，ρ 1 ，β 1 ，β 2 ] T Representing the combination S of the acceleration model (8) at the stress level r The vector formed by the unknown parameters is based on the least square estimation principle and each stress level combination S r Alpha of 1 (S r ) The estimation result of the obtained parameters is as follows:
in the formula (I), the compound is shown in the specification,
representing model parameters mu, beta 1 、β 2 、ρ 1 (ii) an estimate of (d);
in the formula, T r 、RH r 、MS r Respectively representing temperature, humidity and mechanical stress at the r-th stress level combination, S r ＝(T r ，RH r ，MS r )，
Denotes alpha 1 (S r ) R =1,2, …, R;
so far, the estimation of unknown parameters in the multi-stress accelerated degradation model (9) is completed;
step four: coil spring reliability evaluation under normal stress
Multi-stress acceleration to achieve compression set of coil springsAfter the degradation model is constructed and the parameters are estimated, the normal stress level S is based on the spiral spring 0 ＝(T 0 ，RH 0 ，MS 0 ) And obtaining a reliability function of the coil spring under normal stress with a compression permanent deformation rate failure threshold value D as follows:
wherein R (t | S) 0 ) Indicates the normal stress level S 0 The reliability of the lower coil spring at time t,
as a model parameter alpha 0 、μ、β 1 、β 2 、ρ 1 Estimate of (A), T 0 、RH 0 、MS 0 Respectively, temperature, humidity and mechanical stress at normal stress levels, S 0 ＝(T r ，RH r ，MS r )，/>
Denotes the standard deviation sigma of the measurement error epsilon ε ψ (-) is a cumulative distribution function of a standard normal distribution;
thus, the reliability evaluation under the normal stress of the coil spring is completed.
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