Rubber V belt service life prediction method based on Wiener process

Abstract

The invention discloses a method for predicting the service life of a rubber V belt based on a Wiener process, and the method comprises the following steps: designing a dynamic and static pre-experiment, analyzing a significant factor influence rule representing the service life of the V belt, and constructing a fatigue life unified representation model of multi-physics field coupling; the evolution rule of the health state along with time in the dynamic operation process of the V belt is explored, and a function relation expression of the dynamic failure index slip ratio, the tension force and the service life of the V belt is extracted; selecting a proper Copula function by using an AIC information criterion to describe related characteristics between the two performance indexes so as to obtain a joint probability density function of the service life of the rubber V belt; and establishing a final service life prediction model of the rubber V belt by combining a V belt physical property unified representation equation, a relational expression of the dynamic failure indexes and the service life and a Wiener process degradation model, updating unknown parameters by using a maximum likelihood estimation method, and predicting the service life of the rubber V belt. The method needs few training samples, is short in time consumption, is high in prediction precision, and is very high in practical value.

Description

Technical Field

[0001] The invention relates to the technical field of belt transmission life prediction, and more particularly to a rubber V-belt life prediction method based on Wiener process. Background Art

[0002] As the most widely used type of belt transmission, rubber V-belt transmission has the characteristics of simple structure, smooth transmission, low cost, no need for lubrication, buffering and vibration absorption, and easy maintenance. It is widely used in the field of mechanical power transmission in automobile transmission, agricultural machinery power transmission, wind turbine power generation and large-scale engineering machinery industries, and occupies a very important position. However, due to the influence of internal and external temperature and other environmental factors, the V-belt will degrade or fail with the cyclic effect of alternating stress during transmission, such as severe fatigue wear and even breakage. Failures caused by V-belts can reduce the working performance of the equipment at the least, or even cause the entire equipment or system to shut down and fail or even cause major safety accidents. Therefore, it is necessary to monitor the health status and predict the life of rubber V-belts to ensure that the equipment can operate reliably and stably, give early warnings, reduce the huge economic losses caused by sudden failures, and avoid major accidents such as casualties.

[0003] In recent years, with the development of life prediction and reliability research technology, data-driven life prediction methods have been widely used. Among them, the Wiener process has excellent statistical characteristics and the time distribution of the equipment performance index reaching failure for the first time has the characteristics of obeying the inverse Gaussian distribution. It is currently a widely used data-driven life prediction method. Due to the degradation process of rubber V-belts and the influence of complex working conditions, there are very few studies on the life prediction of V-belts, and the accuracy of life prediction using simple physical models is poor. Summary of the invention

[0004] In view of the shortcomings of the prior art, the object of the present invention is to provide a method for predicting the life of a rubber V-belt based on a Wiener process with high prediction accuracy.

[0005] To achieve the above object, the present invention provides the following technical solution: a method for predicting the life of a rubber V-belt based on a Wiener process, comprising the following steps:

[0006] Step 1: Collect and organize the static physical performance data of the rubber V-belt, analyze the correlation and find out the factors with significant influencing characteristics are the tensile strength, reference force elongation, elastic modulus and hardness of the V-belt;

[0007] Step 2: Design a V-belt dynamic fatigue pre-experiment based on existing standards, use static characteristic factors and experimental degradation data to analyze the performance degradation law of the V-belt, and construct a unified fatigue life characterization model coupled with multiple physical fields;

[0008] Step 3: Analyze the evolution of the health status of the V-belt over time during dynamic operation through the characterization model constructed in step 2, and extract the functional relationship between the dynamic failure index slip rate, tension force and V-belt life;

[0009] Step 4: Select the failure index slip rate and tension force as two performance indicators to characterize the health status of the V-belt, use the AIC information criterion to select a suitable Copula function to describe the correlation characteristics between the two performance indicators, and obtain the joint probability density function of the rubber V-belt life based on the Wiener process degradation modeling method;

[0010] Step 5: Combine the unified characterization equation of V-belt physical properties, the relationship between dynamic failure index and life, and the Wiener process degradation model to build a life prediction model for rubber V-belts, and use the maximum likelihood estimation method to update the unknown parameters of the model to predict the life of the V-belts.

[0011] As a further improvement of the present invention, the unified characterization model of fatigue life of multi-physical field coupling constructed in step 2 is as follows:

[0012] T=961.8-2.9X1-1393.6X2-0.8X3-7.8X4 (1)

[0013] Wherein, T is the life of the V-belt, H; X1 is the tensile strength of the V-belt, KN; X2 is the reference force elongation of the V-belt, %; X3 is the elastic modulus of the V-belt, MPa; X4 is the hardness of the V-belt.

[0014] As a further improvement of the present invention, the function relationship extracted in step 3 is as follows:

[0015]

[0016] In the above formula, η represents the slip rate of the V-belt; T represents the life of the V-belt, the unit is H; F represents the tension of the V-belt during the transmission process, the unit is N.

[0017] As a further improvement of the present invention, the unified characterization equation of the physical properties of the V-belt, the relationship between the dynamic failure index and the life, and the Wiener process degradation model in step 5 are as follows:

[0018] When the value of the rubber V-belt performance index exceeds the failure threshold ω (ω>0) for the first time, the V-belt transmission performance is considered to have failed. Since the two performance indexes of the V-belt are used at the same time to predict the life, it is defined that when any one of the performance indexes {X1(t), t≥0} and {X2(t), t≥0} exceeds its corresponding failure threshold ω1, ω2, the V-belt performance is considered to have failed. Taking the B-type V-belt as an example, the life of the rubber V-belt is defined as:

[0019] T=inf{t:X1(t)>0.02orX2(t)≤359} (3)

[0020] Wherein, T represents the life of the V-belt, and its unit is H; X1(t) represents the dynamic slip rate η of the V-belt; X2(t) represents the dynamic tension F of the V-belt.

[0021] As a further improvement of the present invention, the specific steps of obtaining the joint probability density function of the rubber V-belt life in step 4 are as follows:

[0022] Step 41: Model the V-belt performance indicators based on the Wiener process, and the time distribution of the first failure obeys the inverse Gaussian distribution, that is, the marginal probability density function of the life of the two performance indicators of the V-belt is:

[0023] In the above formula, k=1,2; ω1 is the failure threshold of the V-belt slip rate; ω2 is the failure threshold of the V-belt tension force;

[0024] Step 42, then use the Copula function:

[0025] F(x1,x2)=C(F(x1),F(x2);θ) (5)

[0026] In the above formula, F(x1) and F(x2) are the marginal distribution functions of the two performance indicators; F(x1, x2) is the joint distribution function; θ is the unknown parameter in the Copula function that describes the correlation between the two indicators;

[0027] Step 4.3: Finally, construct the joint probability density function of the marginal probability density functions of the two performance indicators:

[0028] In the above formula, F1(t) and F2(t) are the cumulative distribution functions of the marginal probability density functions of life corresponding to the two performance indicators, and c(F1(t), F2(t); θ) is the probability density function of C(F(x1), F(x2); θ).

[0029] As a further improvement of the present invention, in the Copula function used in step 4, the most suitable Copula function is selected using the AIC information criterion:

[0030] AIC=-2lnA+2m (7)

[0031] In the above formula, A is the maximum likelihood function corresponding to the model; m is the number of parameters in the model; the smaller the AIC value, the better the fitting effect. The binary Copula function with the smallest final AIC calculation value is selected to describe the correlation between the V-belt slip rate and tension.

[0032] As a further improvement of the present invention, the steps of constructing the life prediction model of the rubber V-belt in step 5 are as follows:

[0033] Step 5. Establish the life probability density function of the rubber V-belt:

[0034]

[0035] In the above formula, k = 1, 2, which respectively represent the slip rate and tension of the V-belt;

[0036] Step 52: Construct a life prediction model for the rubber V-belt:

[0037]

[0038] As a further improvement of the present invention, the specific steps of updating the unknown parameters of the model using the maximum likelihood estimation method in step 5 are as follows:

[0039] Step 53: First, according to the properties of the Wiener process, the degradation increment of the kth failure index of the V-belt in the i-th time interval is The corresponding probability density function is:

[0040] In the formula, ΔX ki is the increment of V-belt performance degradation; μ k is the drift coefficient of the model; σ k is the diffusion coefficient of the model; Step 54, the likelihood function expression of the Wiener process model parameters is obtained by equation (10):

[0041]

[0042] Step 55: Take the logarithm of both sides of equation (11) and get:

[0043]

[0044] Step 5 and 6: Use formula (12) to calculate μ k and σ k Find the partial derivative, set it to 0, and solve the equation. Get the drift coefficient μ k and the diffusion coefficient σ k The estimated values ​​are:

[0045]

[0046] In formula (13) and (14), it is the data sample size corresponding to the degradation amount of failure index;

[0047] Step 57: Get parameter μ k , σ kAfter estimating the value, the likelihood function L(θ) about the parameter θ is solved by formula (5), and the derivative of the parameter θ is taken and the derivative is set to 0 to obtain the parameter estimate

[0048]

[0049] Substitute the unknown parameter values ​​into formula (8) to obtain the life prediction probability density function of the rubber V-belt, and use the time corresponding to the maximum value of the probability density function as the predicted value T of the V-belt life:

[0050] T={t:f RUL(max) (t|w k ,μ k ,σ k ,θ)} (16)

[0051] Step 58, complete the prediction of V-belt life.

[0052] Beneficial effects of the present invention:

[0053] A comprehensive analysis of the physical performance indicators of rubber V-belts and the multi-environmental influencing factors of dynamic operation is conducted, a unified characterization model of V-belts under multi-physical field coupling conditions is established, and the life degradation mechanism under the physical performance conditions of the V-belts themselves is refined. Finally, a rubber V-belt life prediction method based on the Wiener process is proposed.

[0054] The present invention can provide a reference for life prediction research of V-belts and even the entire belt transmission industry, and lay a theoretical foundation for life optimization and performance improvement in the research and development of new V-belt products. It has important theoretical significance and engineering application value for improving the efficiency and service life of V-belt transmission, achieving high reliability of mechanical transmission systems, and reducing equipment maintenance and replacement costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 It is a flow chart of the present invention;

[0056] Figure 2 The mathematical relationship curves between the significant influencing factors and the life of rubber V-belts, where (a) is the relationship between tensile strength and life, (b) is the relationship between reference force elongation and life, (c) is the relationship between elastic modulus and life, and (d) is the relationship between hardness and life;

[0057] Figure 3 The graph shows the change of performance indexes during the degradation process of rubber V-belts, where (a) represents the slip rate and (b) represents the tension force;

[0058] Figure 4 The probability density function diagram for the prediction of rubber V-belt life;

[0059] Figure 5 This is the result diagram of rubber V-belt life prediction. DETAILED DESCRIPTION

[0060] The present invention will be further described below in conjunction with the embodiments given in the accompanying drawings.

[0061] Reference Figures 1 to 5 As shown, a method for predicting the life of a rubber V-belt based on a Wiener process in this embodiment includes the following steps:

[0062] S1. Collect and sort out the static physical performance data of rubber V-belts, analyze the correlation and find out the significant influencing characteristics are the tensile strength, reference force elongation, elastic modulus, hardness and other factors of the V-belts;

[0063] S2. Design a V-belt dynamic fatigue pre-experiment under the conditions of Zhejiang Manufacturing Standards, and use static characteristic factors and experimental degradation data to analyze the performance degradation law of the V-belt (see Appendix Figure 2 As shown), a unified characterization model of fatigue life coupled with multi-physics fields is constructed as follows, and subsequent steps are carried out based on this;

[0064] T=961.8-2.9X1-1393.6X2-0.8X3-7.8X4 (1)

[0065] Wherein, T is the life of the V-belt, H; X1 is the tensile strength of the V-belt, KN; X2 is the reference force elongation of the V-belt, %; X3 is the elastic modulus of the V-belt, MPa; X4 is the hardness of the V-belt.

[0066] S3. Analyze the evolution of the health status of the V-belt during dynamic operation over time (see Appendix Figure 3 shown),

[0067] The degradation laws of the two failure indicators, slip rate and tension force, in the dynamic fatigue pre-test were analyzed, and the mathematical model of the two failure indicators and the life of the V-belt was constructed by using the MATLAB curve fitting method:

[0068]

[0069] In the above formula, η represents the slip rate of the V-belt; T represents the life of the V-belt, H; F represents the tension of the V-belt during the transmission process, N.

[0070] S4. Select the failure index slip rate and tension as two performance indicators to characterize the health status of the V-belt. Use the AIC information criterion to select a suitable Copula function to describe the correlation characteristics between the two performance indicators. Based on the Wiener process degradation modeling method, the joint probability density function of the rubber V-belt life is obtained. The process is as follows:

[0071] In step S4, the V-belt performance index is modeled based on the Wiener process, and the time distribution of the first failure obeys the inverse Gaussian distribution, that is, the marginal probability density function of the two performance indexes of the V-belt corresponding to the life is:

[0072]

[0073] In the above formula, k=1,2; ω1 is the failure threshold of the V-belt slip rate; ω2 is the failure threshold of the V-belt tension force;

[0074] Then use the Copula function to construct the joint probability density function of the marginal probability density functions of the two performance indicators:

[0075]

[0076] In the above formula, F1(t) and F2(t) are the cumulative distribution functions of the marginal probability density functions of the life corresponding to the two performance indicators, and c(F1(t), F2(t); θ) is the probability density function of C(F(x1), F(x2); θ);

[0077] Among the commonly used binary Copula function forms, the most appropriate Copula function is selected using the AIC information criterion:

[0078] AIC=-2lnA+2m (5)

[0079] In the above formula, A is the maximum likelihood function corresponding to the model; m is the number of parameters in the model;

[0080] As shown in Table 1, the results of three commonly used Copula functions calculated using the AIC information criterion are shown. The smaller the AIC value, the more suitable the corresponding Copula function is.

[0081] Table 1

[0082]

[0083] From the results in the table, it can be seen that the AIC value corresponding to the Frank function is the smallest, so the Frank Copula function is selected to analyze the correlation characteristics between the V-belt slip rate and tension.

[0084] S5. The unified characterization equation of V-belt physical properties, the relationship between dynamic failure index and life, and the Wiener process degradation model are combined to construct a life prediction model for rubber V-belts. The maximum likelihood estimation method is used to update the unknown parameters of the model and predict the life of the V-belts.

[0085] In step S5, the life prediction process is as follows:

[0086] When the value of the rubber V-belt performance index exceeds the failure threshold ω (ω>0) for the first time, the V-belt transmission performance is considered to have failed. Since the two performance indexes of the V-belt are used at the same time to predict the life, it is defined that when any one of the performance indexes {X1(t), t≥0} and {X2(t), t≥0} exceeds its corresponding failure threshold ω1, ω2, the V-belt performance is considered to have failed. Taking the B-type V-belt as an example, the life of the rubber V-belt is defined as:

[0087] T=inf{t:X1(t)>0.02orX2(t)≤359} (6)

[0088] Where, T represents the life of the V-belt, H; X1(t) represents the dynamic slip rate η of the V-belt; X2(t) represents the dynamic tension F of the V-belt.

[0089] Then the life probability density function of the rubber V-belt is established as:

[0090]

[0091] In the above formula, k=1,2, which respectively represent the two performance indicators of the V-belt slip rate and tension.

[0092] The final life prediction model of the rubber V-belt is:

[0093]

[0094] Finally, the maximum likelihood function estimation method is used to update the model parameters. The process is as follows:

[0095] First, according to the properties of the Wiener process, the degradation increment of the kth failure index of the V-belt in the i-th time interval is The corresponding probability density function is:

[0096]

[0097] In the formula, ΔX ki is the increment of V-belt performance degradation; μ k is the drift coefficient of the model; σ k is the diffusion coefficient of the model.

[0098] The parameter likelihood function expression based on the Wiener process model is obtained from formula (9):

[0099]

[0100] Taking the logarithm of both sides of equation (10), we can get:

[0101]

[0102] According to formula (11), μ k and σ k Find the partial derivative, set it to 0, and solve the equation. Get the drift coefficient μ k and the diffusion coefficient σ k The estimated values ​​are:

[0103]

[0104] Formulas (12) and (13) are the data sample sizes corresponding to the failure index degradation.

[0105] Get the parameter μ k , σ k After estimating the value, the likelihood function L(θ) about the parameter θ is solved by formula (5), and the derivative of the parameter θ is taken and the derivative is set to 0 to obtain the parameter estimate

[0106]

[0107] Substitute the unknown parameter values ​​into formula (8) to obtain the life prediction probability density function of the rubber V-belt, and use the time corresponding to the maximum value of the probability density function as the predicted value Tk of the V-belt life:

[0108] T k ={t max :f RUL(max) (t|w k ,μ k ,σ k ,θ)} (15)

[0109] The present invention conducts a final rubber V-belt static test by design, collects test data and verifies the rubber V-belt life prediction method based on the Wiener process. The specific process is as follows:

[0110] (1) The tensile strength and reference force elongation of the V-belt are tested by a tensile testing machine, the elastic modulus of the V-belt is tested by an elastic modulus measuring instrument, and the hardness of the V-belt is measured by a Shore hardness tester; each static performance value is the average of three test values.

[0111] (2) Under the dynamic test conditions of Zhejiang manufacturing standard "T / ZZB 0060-2016 Ordinary V-belts for Transmission", a dynamic test of the V-belt was carried out using a dynamic fatigue testing machine. The corresponding V-belt slip rate and tension force values ​​were recorded every 15h, and the actual life value of the V-belt when it finally failed was collected.

[0112] (3) Using the AIC information criterion, select the appropriate Copula function to describe the correlation characteristics between performance indicators and obtain the joint probability density function of V-belt life:

[0113]

[0114] Then, the maximum likelihood estimation method is used to calculate the unknown parameters of the model and obtain the life probability density function of the rubber V-belt at different times (see Appendix Figure 4 As shown in the figure), the time corresponding to the maximum value of the probability density function is used as the remaining life prediction value at each monitoring time point, and the result of the rubber V-belt life prediction can be obtained (as shown in the figure). Figure 5 As shown), the feasibility and accuracy of the rubber V-belt life prediction method of the present invention are effectively verified.

[0115] In summary, a method for predicting the life of a rubber V-belt based on a Wiener process in this embodiment includes the following steps: designing dynamic and static pre-experiments, analyzing the influence of significant factors characterizing the life of the V-belt, and constructing a unified fatigue life characterization model coupled with multiple physical fields; exploring the evolution of the health state of the V-belt over time during dynamic operation, and refining the functional relationship between the dynamic failure index slip rate, tension and the life of the V-belt; selecting the failure index slip rate and tension as two performance degradation indicators of the life of the V-belt, and using the AIC information criterion to select a suitable Copula function to describe the correlation characteristics between the two performance indicators, thereby obtaining the joint probability density function of the life of the rubber V-belt; then jointly establishing the unified characterization equation of the physical properties of the V-belt, the relationship between the dynamic failure index and the life, and the Wiener process degradation model, to construct the final life prediction model of the rubber V-belt, and using the maximum likelihood estimation method to update the unknown parameters and predict the life of the rubber V-belt. The present invention requires fewer training samples, consumes less time, has higher prediction accuracy, and has extremely high practical value.

[0116] The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A method for predicting the life of a rubber V-belt based on a Wiener process, characterized in that: The steps include: Step 1: Collect and organize the static physical performance data of the rubber V-belt, analyze the correlation and find out the factors with significant influencing characteristics are the tensile strength, reference force elongation, elastic modulus and hardness of the V-belt; Step 2: Design a V-belt dynamic fatigue pre-experiment based on existing standards, use static characteristic factors and experimental degradation data to analyze the performance degradation law of the V-belt, and construct a unified fatigue life characterization model coupled with multiple physical fields; Step 3: Analyze the evolution of the health status of the V-belt over time during dynamic operation through the characterization model constructed in step 2, and extract the functional relationship between the dynamic failure index slip rate, tension force and V-belt life; Step 4: Select the failure index slip rate and tension force as two performance indicators to characterize the health status of the V-belt, use the AIC information criterion to select a suitable Copula function to describe the correlation characteristics between the two performance indicators, and obtain the joint probability density function of the rubber V-belt life based on the Wiener process degradation modeling method; Step 5: Combine the unified characterization equation of V-belt physical properties, the relationship between dynamic failure index and life, and the Wiener process degradation model to build a life prediction model for rubber V-belts, and use the maximum likelihood estimation method to update the unknown parameters of the model to predict the life of the V-belts.

2. The method for predicting the life of a rubber V-belt based on Wiener process according to claim 1 is characterized in that: The unified characterization model of fatigue life of multi-physics field coupling constructed in step 2 is as follows: T=961.8-2.9X1-1393.6X2-0.8X3-7.8X4 (1) Wherein, T is the life of the V-belt, H; X1 is the tensile strength of the V-belt, KN; X2 is the reference force elongation of the V-belt, %; X3 is the elastic modulus of the V-belt, MPa; X4 is the hardness of the V-belt.

3. The method for predicting the life of a rubber V-belt based on a Wiener process according to claim 1 or 2, characterized in that: The extracted functional relationship of step 3 is as follows: In the above formula, η represents the slip rate of the V-belt; T represents the life of the V-belt, the unit is H; F represents the tension of the V-belt during the transmission process, the unit is N.

4. The method for predicting the life of a rubber V-belt based on a Wiener process according to claim 1 or 2, characterized in that: In step 5, the unified characterization equation of V-belt physical properties, the relationship between dynamic failure index and life, and the Wiener process degradation model are as follows: When the value of the rubber V-belt performance index exceeds the failure threshold ω (ω>0) for the first time, the V-belt transmission performance is considered to have failed. Since the two performance indexes of the V-belt are used at the same time to predict the life, it is defined that when any one of the performance indexes {X1(t), t≥0} and {X2(t), t≥0} exceeds its corresponding failure threshold ω1, ω2, the V-belt performance is considered to have failed. Taking the B-type V-belt as an example, the life of the rubber V-belt is defined as: T=inf{t:X1(t)>0.02orX2(t)≤359} (3) Wherein, T represents the life of the V-belt, and its unit is H; X1(t) represents the dynamic slip rate η of the V-belt; X2(t) represents the dynamic tension F of the V-belt.

5. The method for predicting the life of a rubber V-belt based on Wiener process according to claim 1 or 2, characterized in that: The specific steps of obtaining the joint probability density function of the rubber V-belt life in step 4 are as follows: Step 41: Model the V-belt performance indicators based on the Wiener process, and the time distribution of the first failure obeys the inverse Gaussian distribution, that is, the marginal probability density function of the life of the two performance indicators of the V-belt is: In the above formula, k = 1, 2; ω1 is the failure threshold of the V-belt slip rate; ω2 is the failure threshold of the V-belt tension force; Step 42, then use the Copula function: F(x1,x2)=C(F(x1),F(x2);θ) (5) In the above formula, F(x1) and F(x2) are the marginal distribution functions of the two performance indicators; F(x1, x2) is the joint distribution function; θ is the unknown parameter in the Copula function that describes the correlation between the two indicators; Step 4.3: Finally, construct the joint probability density function of the marginal probability density functions of the two performance indicators: In the above formula, F1(t) and F2(t) are the cumulative distribution functions of the marginal probability density functions of life corresponding to the two performance indicators, and c(F1(t), F2(t); θ) is the probability density function of C(F(x1), F(x2); θ).

6. The method for predicting the life of a rubber V-belt based on Wiener process according to claim 5 is characterized in that: In the Copula function in step 4, the most appropriate Copula function is selected using the AIC information criterion: AIC=-2lnA+2m (7) In the above formula, A is the maximum likelihood function corresponding to the model; m is the number of parameters in the model; the smaller the AIC value, the better the fitting effect. The binary Copula function with the smallest final AIC calculation value is selected to describe the correlation between the V-belt slip rate and tension.

7. The method for predicting the life of a rubber V-belt based on Wiener process according to claim 5 is characterized in that: The steps of constructing the life prediction model of the rubber V-belt in step 5 are as follows: Step 5. Establish the life probability density function of the rubber V-belt as follows: In the above formula, k = 1, 2, which respectively represent the slip rate and tension of the V-belt; Step 52: Construct a life prediction model for the rubber V-belt:

8. The method for predicting the life of a rubber V-belt based on Wiener process according to claim 7 is characterized in that: The specific steps of updating the unknown parameters of the model using the maximum likelihood estimation method in step 5 are as follows: Step 53: First, according to the properties of the Wiener process, the degradation increment of the kth failure index of the V-belt in the i-th time interval is The corresponding probability density function is: Where ΔX ki is the increment of V-belt performance degradation; μ k is the drift coefficient of the model; σ k is the diffusion coefficient of the model; Step 54: The parameter likelihood function expression based on the Wiener process model is obtained from formula (10): Step 55: Take the logarithms of both sides of equation (11) and get: Step 5 and 6: Use formula (12) to calculate μ k and σ k Find the partial derivative, set it to 0, and solve the equation. Get the drift coefficient μ k and the diffusion coefficient σ k The estimated values ​​are: In formula (13) and (14), it is the data sample size corresponding to the degradation amount of failure index; Step 57: Get parameter μ k , σ k After estimating the value, the likelihood function L(θ) about the parameter θ is solved by formula (5), and the derivative of the parameter θ is taken and the derivative is set to 0 to obtain the parameter estimate Substitute the unknown parameter values ​​into formula (8) to obtain the life prediction probability density function of the rubber V-belt, and use the time corresponding to the maximum value of the probability density function as the predicted value T of the V-belt life: T={t:f RUL(max) (t|w k ,m k ,s k ,i)} (16) Step 58, complete the prediction of V-belt life.

Images