Reliability test method for rubber V belt
By establishing a degradation model and optimized test plan for rubber V-band, the problems of traditional reliability verification tests ignore individual performance differences and low evaluation accuracy are solved, and more efficient and accurate reliability evaluation is achieved.
Patent Information
- Application Number
- CN202510124838.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-26
- Publication Date
- 2025-05-27
AI Technical Summary
The reliability verification test of traditional rubber V-band ignores individual product performance differences, and is high in testing, time-consuming and low evaluation accuracy.
By establishing a degradation model of rubber V-band, selecting appropriate degradation process models, randomizing model parameters to characterize individual performance differences, constructing asymptotic variance of p-quantile life, and optimizing the degradation experimental scheme using genetic algorithms.
It effectively solves the problems of high cost, long time and low evaluation accuracy of rubber V-band reliability testing, and provides a more accurate and efficient reliability evaluation method.
Smart Images
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Abstract
Description
Technical Field
[0001] The invention relates to the field of reliability evaluation of rubber V-belts, and more particularly to a reliability test method for rubber V-belts. Background Art
[0002] The transmission performance of rubber V-belts plays an important role in ensuring the reliability and safety of belt transmission systems. Therefore, reliability verification tests are an indispensable part of the research and development and production of V-belt products. They are used to determine whether the products have reached the required reliability level during the development and production stages. However, under the traditional given sampling scheme, reliability verification tests based on life tests often ignore individual performance differences of products, and have problems such as high test costs, long test time, and low evaluation accuracy. Summary of the invention
[0003] In view of the shortcomings of the prior art, the present invention aims to provide a reliability test method for a rubber V-belt. By accurately establishing a degradation model of a rubber V-belt, and reasonably determining the optimization objective function and decision variables, a mathematical model for optimizing the degradation test scheme is established, and a genetic algorithm is used to solve the mathematical model to propose an optimal test scheme.
[0004] To achieve the above object, the present invention provides the following technical solution: a reliability test method for a rubber V-belt, comprising the following steps:
[0005] Step 1: Select the degradation index of the rubber V-belt, design the performance degradation test, and obtain the performance degradation data; Step 2: Based on the performance degradation data obtained in step 1, select a suitable degradation process model, and randomize the model parameters to characterize the performance differences between individuals, and establish a degradation model of the rubber V-belt that takes into account the individual performance differences;
[0006] Step 3, construct the asymptotic variance of the p-quantile life of the rubber V-belt, and minimize the asymptotic variance of the 0.1-quantile life as the optimization objective function;
[0007] Step 4: determine the decision variables and use the total cost of the experiment as a constraint;
[0008] Step five: Establish a mathematical model for the optimal design of the degradation test, use a genetic algorithm to solve the optimization model, and optimize the performance degradation test plan.
[0009] As a further improvement of the present invention, the specific method of selecting the degradation index of the rubber V-belt in step 1, designing the performance degradation test, and obtaining the performance degradation data is as follows:
[0010] Based on historical data and the main failure forms of rubber V-belts during operation, the failure index of rubber V-belts is selected, and a degradation test is designed to measure the performance degradation data of the test V-belts at the same time interval, as shown in the following formula:
[0011] n is the number of V-belts tested, m is the number of measurements;
[0012] Y ij is the performance degradation data of the rubber V-belt. From the degradation data, it can be obtained that within the time interval Δt, the degradation increment is: ΔY ij =Y i,j+1 -Y i,j
[0013] Wherein, i is the serial number of the tested V-belt, j+1, j are two adjacent test moments of the i-th tested V-belt. As a further improvement of the present invention, the specific steps of establishing the degradation model of the rubber V-belt considering individual performance differences in step 2 are as follows:
[0014] Step 21: The performance degradation process of the rubber V-belt is regarded as an inverse Gaussian process. The degradation amount at time t is Y(t), where the degradation amount Y(t) has the following properties:
[0015] For Y(0) = 0, it holds with probability 1;
[0016] For any t 4 >t 3 >t 2 >t 1 , there is Y(t 4 )-Y(t 3 )≥0,Y(t 2 )-Y(t 1 )≥0, and Y(t 4 )-Y(t 3 ) and Y(t 2 )-Y(t 1 ) are independent of each other, and the degradation increment obeys the inverse Gaussian distribution, which is recorded as: ΔY(t)~IG(ΔΛ(t),η[ΔΛ(t)] 2 );
[0017] Step 22: Assume that the degradation increment u in time Δt follows an inverse Gaussian distribution: ΔY(t)~IG(μΔΛ(t),λ[ΔΛ(t)] 2 ),μ,λ>0, where IG(μΔΛ(t),λ[ΔΛ(t)] 2 ) represents an inverse Gaussian distribution with mean μΔΛ(t) and variance ΔΛ(t) / λ; μ and λ are constants; Λ(t) represents a parameter function, Λ(0) = 0, ΔΛ(t) = Λ(t+Δt)-Λ(t);
[0018] Step 2 and 3, randomize the parameter μ related to the degradation rate, assuming that the parameter μ follows a truncated normal distribution, that is, μ~TN(ω,κ -2 );
[0019] The probability density function of μ is:
[0020] When μ is a random variable, the probability density function of Y(t) is;
[0021]
[0022] Step 24: Assume the failure threshold is Y D , when the degradation reaches the failure threshold for the first time, the product is judged to be failed, then the reliability function of the product at time t is:
[0023]
[0024] Where Φ(·) is the standard normal distribution function;
[0025] Step 25: Use the Markov chain Monte Carlo method based on Bayesian theory to estimate the model parameters θ = (ω, κ, λ, q) and obtain the joint posterior distribution of the model parameters:
[0026]
[0027] As a further improvement of the present invention, the specific method of estimating the model parameters θ=(ω, κ, λ, q) using the Markov chain Monte Carlo method based on Bayesian theory in step 25 is:
[0028] Assume that there are n samples for degradation test, for the i-th sample, at t i1 ,t i2 ,…,t im Time to carry out i The degradation data is tested, and the performance degradation data is recorded as Y ij , i = 1, 2, ..., n, j = 1, 2, ..., m, the degradation increment of the i-th sample is denoted as y ij , j=1,2,…,m i The following degradation increment data can be obtained from the above degradation data:
[0029]
[0030] According to the definition of inverse Gaussian process, y i j follows an inverse Gaussian distribution: in, Λ(·) is a parameter function, let Λ(t) = t q; Establish the following likelihood equation:
[0031]
[0032] The log-likelihood function is:
[0033]
[0034] Assume that the joint prior distribution of the inverse Gaussian model is:
[0035] π(θ k )=π(ω k )π(κ k )π(λ k )π(q k );
[0036] According to the Bayesian formula, the joint posterior distribution of the model parameters can be obtained as:
[0037]
[0038] As a further improvement of the present invention, in step three, the asymptotic variance of the p quantile life of the rubber V-belt is constructed, and the asymptotic variance of the 0.1 quantile life is minimized as the optimization objective function. Specifically, the V optimal criterion is derived from the relevant information matrix, and the asymptotic variance of the P quantile life is minimized as the optimization objective.
[0039] As a further improvement of the present invention, the P quantile life expression adopts an approximate method, specifically: taking the mean of the parameter μ as the true value of the model, the asymptotic variance Avar(ξ P ) is constructed as follows:
[0040]
[0041] Where H(θ) is For the one-dimensional derivative matrix of θ, H′(θ) is its transposed matrix, I(θ) is the Fisher information matrix, H(θ), The expression of the Fisher information matrix I(θ) is as follows:
[0042]
[0043] The log-likelihood function is:
[0044]
[0045] As a further improvement of the present invention, the decision variables in step 4 include: the total number of degradation test samples n, the measurement frequency m, and the time interval between two adjacent measurements f; the degradation test scheme can be expressed as P = {n, m, f}, and the value in P determines the asymptotic variance Avar(ξ P ), which also determines the total cost of the degradation test C total and total test time.
[0046] As a further improvement of the present invention, the total cost C of the degradation test total It can be expressed as:
[0047] C total (n,m)=C sa n+C me mn+fC t mn
[0048] In the formula, C sa is the cost of a single sample, C me is the cost of measuring degradation data once, C t It is the electricity consumption per unit test time.
[0049] As a further improvement of the present invention, the mathematical model for establishing the optimal design of the degradation test in step 5 is as follows:
[0050] Minimize
[0051] Avar(ξ p )
[0052] Subject to
[0053] C total (n,m)≤C D
[0054] N L ≤n≤N U
[0055] M L ≤m≤M U
[0056] F L ≤f≤F U ;
[0057] Among them, C D Estimate the total cost of the experiment.
[0058] As a further improvement of the present invention, the model is solved using the genetic algorithm toolkit in Matlab:
[0059] Under unconstrained conditions, the variation trend of the asymptotic variance of the 0.1 quantile life of the rubber V-belt is determined under different test scheme combinations of test sample number n and measurement frequency m.
[0060] Taking the total test cost as the constraint condition, find the test plan with the smallest asymptotic variance as the optimal test plan, and determine the number of test samples n of the optimal test plan * , measuring frequency m * and the time interval f between two adjacent measurements * .
[0061] The beneficial effects of the present invention are as follows: by accurately establishing a degradation model of a rubber V-belt, and reasonably determining the optimization objective function and decision variables, a mathematical model for optimizing the degradation test scheme is established, and a genetic algorithm is used to solve the mathematical model, and an optimal test scheme is proposed, thereby effectively solving the current problems of high cost, long time consumption and large errors in reliability evaluation results of reliability tests for rubber V-belt products. BRIEF DESCRIPTION OF THE DRAWINGS
[0062] Figure 1 It is a schematic diagram of the reliability evaluation process of the present invention;
[0063] Figure 2 Schematic diagram of the incremental change curve of the rubber V-belt slip rate degradation in the embodiment of the present invention;
[0064] Figure 3 is the asymptotic variance value of the 0.1 quantile life under different test samples and degradation data measurement times in the embodiment of the present invention. DETAILED DESCRIPTION
[0065] The present invention will be further described below in detail with reference to the embodiments shown in the accompanying drawings.
[0066] Reference Figure 1 As shown, a reliability test method of a rubber V-belt in this embodiment includes the following steps:
[0067] Step S1, selecting a degradation index of a rubber V-belt, designing a performance degradation test, and obtaining performance degradation data;
[0068] In step S1, the present invention takes the degradation data of slip rate collected by the dynamic fatigue test of the rubber V-belt under the same test standard as an example. The degradation test uses samples of the same model under the same test conditions, and collects slip rate data for all samples once every 10 hours;
[0069] The two performance degradation data of different samples obtained through degradation tests are shown in the following formula:
[0070]
[0071] Yij is the degradation data set, is the number of test V-belts, and m is the number of measurements;
[0072] like Figure 2 As shown. From the degradation data, the degradation increment within the time interval Δt can be obtained as:
[0073] ΔY ij =Y i,j+1 -Y i,j ;
[0074] Wherein, i is the number of tested V-belts, j+1,j are the two adjacent test moments of the i-th tested V-belt.
[0075] Step S2 includes:
[0076] The performance degradation process of the rubber V-belt is regarded as an inverse Gaussian process. The degradation amount at time t is Y(t), which has the following properties:
[0077] For Y(0) = 0, it holds with probability 1;
[0078] For any t 4 >t 3 >t 2 >t 1 , there is Y(t 4 )-Y(t 3 )≥0,Y(t 2 )-Y(t 1 )≥0, and Y(t 4 )-Y(t 3 ) and Y(t 2 )-Y(t 1 ) are independent of each other;
[0079] The degradation increment obeys the inverse Gaussian distribution, which is expressed as: ΔY(t)~IG(ΔΛ(t),η[ΔΛ(t)] 2 );
[0080] Assume that the degradation increment u within Δt time follows the inverse Gaussian distribution:
[0081] ΔY(t)~IG(μΔΛ(t),λ[ΔΛ(t)] 2 ),μ,λ>0, where IG(μΔΛ(t),λ[ΔΛ(t)] 2 ) represents an inverse Gaussian distribution with mean μΔΛ(t) and variance ΔΛ(t) / λ; μ and λ are constants; Λ(t) represents a parameter function, Λ(0)=0, ΔΛ(t)=Λ(t+Δt)-Λ(t).
[0082] Considering the individual differences of products, the parameter μ related to the degradation rate is randomized, assuming that the parameter μ obeys the truncated normal distribution, that is, μ~TN(ω,κ-2 );
[0083] The probability density function of μ is:
[0084] When μ is a random variable, the probability density function of Y(t) is;
[0085]
[0086] Assume that the failure threshold is Y D , when the degradation reaches the failure threshold for the first time, the product is judged to be failed, then the reliability function of the product at time t is:
[0087]
[0088] Where Φ(·) is the standard normal distribution function.
[0089] The model parameters θ = (ω, κ, λ, q) are estimated using the Markov chain Monte Carlo method based on Bayesian theory.
[0090] For the i-th sample at t i1 ,t i2 ,…,t im Time to carry out i The degradation data is tested, and the performance degradation data is recorded as Y ij , i = 1, 2, ..., n, j = 1, 2, ..., m, the degradation increment of the i-th sample is denoted as y ij , j=1,2,…,m i The following degradation increment data can be obtained from the above degradation data:
[0091]
[0092] According to the definition of inverse Gaussian process, y i j follows an inverse Gaussian distribution: in, Λ(·) is a parameter function, let Λ(t) = t q ; Establish the following likelihood equation:
[0093]
[0094] The log-likelihood function is:
[0095]
[0096] Assume that the joint prior distribution of the inverse Gaussian model is:
[0097] π(θ k )=π(ω k )π(κk )π(λ k )π(q k );
[0098] According to the Bayesian formula, the joint posterior distribution of the model parameters can be obtained as:
[0099]
[0100] Thus, the estimation results of the parameters θ = (ω, κ, λ, q) are obtained as shown in Table 1.
[0101] Table 1
[0102]
[0103] Step S3 includes:
[0104] Construct an optimization model. Adopting the V optimal criterion, the asymptotic variance of the P quantile life is minimized as the optimization target. The P quantile life expression can be expressed by an approximate method. Taking the mean of the parameter μ as the true value of the model, the asymptotic variance Avar(ξ P ) is constructed as follows:
[0105]
[0106] Where H(θ) is For the one-dimensional derivative matrix of θ, H′(θ) is its transposed matrix, and I(θ) is the Fisher information matrix. The expression of the Fisher information matrix I(θ) is as follows:
[0107]
[0108] The terms in H(θ) are:
[0109]
[0110] The log-likelihood function is:
[0111]
[0112] The specific expression of each element in the Fisher information matrix I(θ) is:
[0113]
[0114]
[0115] Step S4 includes:
[0116] Determine the decision variables and take the total test cost as a constraint. The decision variables include: the total number of degradation test samples n, the measurement frequency m, and the time interval between two adjacent measurements f, denoted by (F L ,F U ) is the boundary condition of the sampling time interval f; let (M L ,M U ) is the boundary condition for each sample to be tested m times; similarly, (N L ,N U ) is the boundary condition of the number of samples for the degradation test; the degradation test plan can be expressed as P = {n, m, f}, and the unit price of the test sample C sa =20, labor cost for a single measurement C me =10, the power of the test machine is 15KW, the average industrial electricity price is 0.62 yuan per kWh, and the hourly electricity consumption is C t =10.2.
[0117] The total cost of degradation testing can be expressed as:
[0118] C total (n,m)=C sa n+C me mn+fC t mn
[0119] In the formula, C sa is the cost of a single sample, C me is the cost of measuring degradation data once, C t It is the electricity consumption per unit test time.
[0120] The total cost of the experiment is C D , the mathematical model of degradation test optimization design is established as follows:
[0121] Minimize
[0122] Avar(ξ p )
[0123] Subject to
[0124] C total (n,m)≤C D
[0125] N L ≤n≤N U
[0126] M L ≤m≤M U
[0127] F L ≤f≤F U
[0128] Step S5 includes:
[0129] Under unconstrained conditions, the asymptotic variance of the 0.1 quantile life of the rubber V-belt and its changing trend are determined under different test sample numbers n and measurement frequencies m. The results are as follows: Figure 3 shown.
[0130] The asymptotic variance of the 0.1 quantile life gradually decreases with the increase of sample size and measurement frequency. The asymptotic variance of the 0.1 quantile life of the product is very sensitive to the changes in sample size and measurement frequency in the initial stage, and the asymptotic variance decreases very quickly. When the number of experimental samples is greater than 5 and the measurement frequency is greater than 20 times, as the number of samples and the number of measurements increase, the asymptotic variance of the product life has become flat.
[0131] Taking the total test cost as the constraint condition, the objective function is used as the fitness function of the genetic algorithm. Based on the global search for the optimal test variable in the solution space obtained by the genetic algorithm in the previous step, the test scheme with the minimum asymptotic variance is obtained as the optimal test scheme. Determine the number of test samples n for the optimal test scheme * , measuring frequency m * and the time interval f between two adjacent measurements * Substituting the total test costs of 4000 yuan, 5000 yuan, 6000 yuan, 7000 yuan, and 8000 yuan into the model as constraints, the optimal degradation test plan and the asymptotic variance of the 0.1 quantile life are shown in Table 2.
[0132] Table 2
[0133]
[0134] According to the results, the degradation test scheme with the minimum asymptotic variance of the 0.1 quantile reliability life of the rubber V-belt under different test cost conditions and the asymptotic variance of the 0.1 quantile reliability life of the product under this scheme can be obtained. More samples and measurements can reduce the asymptotic variance of the 0.1 quantile reliability life of the product and improve the accuracy of reliability assessment, but at the same time, as the number of samples and the number of measurements increase, the total test cost and test time will increase. By optimizing the design of the reliability test, the optimal test scheme can be obtained within the specified test cost.
[0135] 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 reliability test method for a rubber V-belt, characterized in that: The steps include: Step 1: Select the degradation index of the rubber V-belt, design the performance degradation test, and obtain the performance degradation data; Step 2: Based on the performance degradation data obtained in step 1, a suitable degradation process model is selected, and the model parameters are randomized to characterize the performance differences between individuals, so as to establish a degradation model of the rubber V-belt that takes into account the individual performance differences; Step 3, construct the asymptotic variance of the p-quantile life of the rubber V-belt, and minimize the asymptotic variance of the 0.1-quantile life as the optimization objective function; Step 4: determine the decision variables and use the total cost of the experiment as a constraint; Step five: Establish a mathematical model for the optimal design of the degradation test, use a genetic algorithm to solve the optimization model, and optimize the performance degradation test plan.
2. The reliability test method of the rubber V-belt according to claim 1, characterized in that: The specific method of selecting the degradation index of the rubber V-belt, designing the performance degradation test, and obtaining the performance degradation data in step 1 is as follows: Based on historical data and the main failure forms of rubber V-belts during operation, the failure index of rubber V-belts is selected, and a degradation test is designed to measure the performance degradation data of the test V-belts at the same time interval, as shown in the following formula: n is the number of V-belts tested, m is the number of measurements; Y ij is the performance degradation data of the rubber V-belt. From the degradation data, it can be obtained that within the time interval Δt, the degradation increment is: ΔY ij =Y i,j+1 -Y i,j Wherein, i is the serial number of the tested V-belt, j+1,j are the two adjacent test moments of the i-th tested V-belt.
3. The reliability test method of the rubber V-belt according to claim 1 or 2, characterized in that: The specific steps of establishing the degradation model of the rubber V-belt considering individual performance differences in step 2 are as follows: Step 21: The performance degradation process of the rubber V-belt is regarded as an inverse Gaussian process. The degradation amount at time t is Y(t), where the degradation amount Y(t) has the following properties: For Y(0) = 0, the probability is 1; For any t4>t3>t2>t1, Y(t4)-Y(t3)≥0,Y(t2)-Y(t1)≥0, and Y(t4)-Y(t3) and Y(t2)-Y(t1) are independent of each other. The degenerate increment follows the inverse Gaussian distribution, which is denoted as: ΔY(t)~IG(ΔΛ(t),η[ΔΛ(t)] 2 ); Step 22: Assume that the degradation increment u in time Δt follows an inverse Gaussian distribution: ΔY(t)~IG(μΔΛ(t),λ[ΔΛ(t)] 2 ),μ,λ>0, where IG(μΔΛ(t),λ[ΔΛ(t)] 2 ) represents an inverse Gaussian distribution with mean μΔΛ(t) and variance ΔΛ(t) / λ; μ and λ are constants; Λ(t) represents a parameter function, Λ(0) = 0, ΔΛ(t) = Λ(t+Δt)-Λ(t); Step 2 and 3, randomize the parameter μ related to the degradation rate, assuming that the parameter μ follows a truncated normal distribution, that is, μ~TN(ω,κ -2 ); The probability density function of μ is: When μ is a random variable, the probability density function of Y(t) is; Step 24: Assume the failure threshold is Y D , when the degradation reaches the failure threshold for the first time, the product is judged to be failed, then the reliability function of the product at time t is: Where Φ(·) is the standard normal distribution function; Step 25: Use the Markov chain Monte Carlo method based on Bayesian theory to estimate the model parameters θ = (ω, κ, λ, q) and obtain the joint posterior distribution of the model parameters:
4. The reliability test method of the rubber V-belt according to claim 3, characterized in that: The specific method of estimating the model parameters θ=(ω,κ,λ,q) using the Markov chain Monte Carlo method based on Bayesian theory in step 25 is: Assume that there are n samples for degradation test, for the i-th sample, i1 ,t i2 ,…,t im Time to carry out i The degradation data is tested, and the performance degradation data is recorded as Y ij , i = 1, 2, ..., n, j = 1, 2, ..., m, the degradation increment of the i-th sample is denoted as y ij , The following degradation increment data can be obtained from the above degradation data: According to the definition of inverse Gaussian process, y ij Obey the inverse Gaussian distribution: in, Λ(·) is a parameter function, let Λ(t) = t q ; Establish the following likelihood equation: The log-likelihood function is: Assume that the joint prior distribution of the inverse Gaussian model is: π(θ k )=π(ω k )π(κ k )π(λ k )π(q k ); According to the Bayesian formula, the joint posterior distribution of the model parameters can be obtained as:
5. The reliability test method of a rubber V-belt according to claim 1 or 2, characterized in that: In the step three, the asymptotic variance of the p quantile life of the rubber V-belt is constructed, and the asymptotic variance of the 0.1 quantile life is minimized as the optimization objective function. Specifically, the V optimal criterion is derived from the relevant information matrix, and the asymptotic variance of the P quantile life is minimized as the optimization objective.
6. The reliability test method of the rubber V-belt according to claim 5, characterized in that: The P quantile life expression adopts an approximate method, specifically: Taking the mean of the parameter μ as the true value of the model, the asymptotic variance Avar(ξ P ) is constructed as follows: Where H(θ) is For the one-dimensional derivative matrix of θ, H′(θ) is its transposed matrix, I(θ) is the Fisher information matrix, H(θ), The expression of the Fisher information matrix I(θ) is as follows: The log-likelihood function is:
7. The reliability test method of a rubber V-belt according to claim 1 or 2, characterized in that: The decision variables in step 4 include: the total number of degradation test samples n, the measurement frequency m, and the time interval between two adjacent measurements f; the degradation test scheme can be expressed as P = {n, m, f}, and the value in P determines the asymptotic variance Avar(ξ P ), which also determines the total cost of the degradation test C total and total test time.
8. The reliability test method of the rubber V-belt according to claim 7, characterized in that: The total cost of the degradation test is C total It can be expressed as: C total (n,m)=C sa n+C me mn+fC t mn In the formula, C sa is the cost of a single sample, C me is the cost of measuring degradation data once, C t It is the electricity consumption per unit test time.
9. The reliability test method of the rubber V-belt according to claim 8, characterized in that: The mathematical model for establishing the optimal design of the degradation test in step 5 is as follows: Minimize Avar(ξ p ) Subject to C total (n,m)≤C D N L ≤n≤N U M L ≤m≤M U F L ≤f≤F U ; Among them, C D Estimate the total cost of the experiment.
10. The reliability test method of the rubber V-belt according to claim 9, characterized in that: The model is solved using the genetic algorithm toolkit in Matlab: Under unconstrained conditions, the variation trend of the asymptotic variance of the 0.1 quantile life of the rubber V-belt is determined under different test scheme combinations of test sample number n and measurement frequency m. Taking the total test cost as the constraint condition, find the test plan with the smallest asymptotic variance as the optimal test plan, and determine the number of test samples n of the optimal test plan * , measuring frequency m * and the time interval f between two adjacent measurements * .