Method for detecting elongation at break of spinning product

Abstract

The invention provides a spinning product elongation at break detection method. The spinning product elongation at break detection method comprises the following steps: a sample preparation and pretreatment step, a sample form measurement step and a mathematical modeling step; a step of calibrating tensile test equipment and setting parameters; a stretching test execution and data acquisition step; calculating and preliminarily analyzing the elongation at break; a stress and strain relation analysis step; a step of data stability analysis; an abnormal value detection and processing step; a step of result comprehensive analysis and report generation; according to the method, accurate analysis of the relationship between the yarn form and the stress and strain is realized, and the accuracy of a test result is improved; stability and abnormal value detection analysis is carried out on the data, so that the reliability and stability of the test data are ensured; and powerful support is provided for performance evaluation and improvement of the yarn. The method has feasibility and practicability, and the efficiency and accuracy of detecting the elongation at break of the spinning product can be remarkably improved.

Description

Technical Field

[0001] The invention specifically relates to a method for detecting elongation at break of a spinning product. Background Art

[0002] Elongation at break refers to the ratio of the elongation of a material when it is stretched to break to its initial length, usually expressed as a percentage. It is an important indicator of the softness and elasticity of a material. The greater the elongation at break, the better the softness and elasticity of the material. In textile processing, fibers with high elongation at break feel softer, can buffer stress, and reduce lint and breakage. The elongation at break of ordinary textile fibers is generally between 10% and 30%. For industrial strong yarns, high breaking strength and low elongation at break are required to prevent product deformation.

[0003] In the field of spinning, there is currently no method for detecting the elongation at break of spinning products that is accurate, reliable and stable. Therefore, the present application proposes a method for detecting the elongation at break of spinning products to solve this problem. Summary of the invention

[0004] The purpose of the present invention is to provide a method for detecting the elongation at break of a spinning product in view of the deficiencies in the prior art, and the method for detecting the elongation at break of a spinning product can well solve the above-mentioned problems.

[0005] In order to achieve the above requirements, the technical solution adopted by the present invention is: to provide a method for detecting the elongation at break of a spinning product, the method for detecting the elongation at break of a spinning product comprises the following steps:

[0006] S1: Sample preparation and pretreatment steps;

[0007] Randomly select at least 30 samples from the spinning products to be tested, ensuring that the samples cover different batches and different locations to ensure the representativeness of the results. Place the samples at a temperature of 20±2℃ and a relative humidity of 65±3% for at least 48 hours to eliminate the impact of the environment on the test results;

[0008] S2: Steps for sample morphology measurement and mathematical modeling;

[0009] A high-precision optical microscope was used to measure the diameter and cross-sectional shape parameters of each sample. Based on the measured data, a mathematical model with an elliptical cross-sectional shape was established using the curve fitting technique in calculus. The equation is expressed as follows:

[0010]

[0011] Where a and b are the major and minor axes of the fractional superellipse, respectively;

[0012] α(x), β(y), γ(y), and δ(x) are fractional functions of x and y;

[0013] x and y are the coordinates of any point on the cross section;

[0014] By combining the calculus of variations with fractional calculus, the actual measured data are fitted to obtain the optimal values ​​of a, b, α(x), β(y), γ(y), and δ(x);

[0015] S3: Steps for calibrating and setting parameters of the tensile test equipment;

[0016] The tensile test equipment was calibrated using standard samples to ensure the accuracy and stability of the equipment. The tensile speed was set to a constant value of 50 mm / min, and the sampling frequency of the data acquisition system was set to 100 Hz to ensure that the subtle changes of the yarn during the stretching process could be captured.

[0017] S4: Steps for executing the tensile test and collecting data;

[0018] The pretreated samples were installed on the tensile test equipment and the tensile test was carried out according to the set parameters. During the test, the data acquisition system recorded the tensile force, elongation and time data in real time. Each sample was tested at least five times to reduce the influence of random errors.

[0019] S5: Steps for calculating elongation at break and conducting preliminary analysis;

[0020] According to the data obtained from the tensile test, the elongation at break of each sample is calculated. The elongation at break calculation formula is:

[0021]

[0022] Conduct a preliminary analysis on the calculated elongation at break to check the rationality and consistency of the data;

[0023] S6: Steps for stress-strain relationship analysis;

[0024] Using the differential method in calculus, the stress-strain relationship of the yarn during the stretching process is analyzed. The stress-strain relationship of the yarn conforms to the fractional-order nonlinear elastic model, and its mathematical expression is:

[0025]

[0026] Where σ is stress, λ is the elongation ratio (λ = 1 + ∈, ∈ is strain), C1 and C2 are material constants, α(λ), β(λ -1 ),γ(λ),δ(λ -1) is a fractional function of λ. The data obtained from the tensile test are combined with the fractional-order nonlinear least squares method and the variational principle to fit C1, C2, α(λ), β(λ -1 ),γ(λ),δ(λ -1 )’s optimal value;

[0027] S7: Steps for data stability analysis;

[0028] The ANOVA analysis method was used to analyze the stability of the test results between different batches and different locations, and the total variation was decomposed into inter-group variation and intra-group variation. By comparing the ratio of inter-group variation to intra-group variation, it was determined whether the variation from different sources was significant. The Bayesian mixed effects variance analysis was used, and its mathematical model was:

[0029] Y ijk ~Normal(μ+A i +B j +(AB) ij +S ijk , σ 2 );

[0030] Among them, Y ijk is the observed value of the kth repeated measurement in the i-th batch, j-th position;

[0031] μ is the population mean;

[0032] A i It is a batch effect;

[0033] B j It is the position effect;

[0034] (AB) ij is the interaction effect of batch and position;

[0035] S ijk It is the within-subject effect, that is, the repeated measurement effect within the same batch and the same location;

[0036] σ 2 is the error variance;

[0037] S8: Step of performing outlier detection and processing;

[0038] The box plot method in statistics is used to detect outliers in the test data. By calculating the quartiles Q1, Q2, Q3 and the interquartile range IQR = Q3-Q1 of the data, a box plot is drawn and outliers are identified. For the detected outliers, necessary processing is performed including deletion and replacement.

[0039] S9: Steps for comprehensive analysis of results and report generation;

[0040] Conduct a comprehensive analysis of the test data, including the calculation of the average, standard deviation, maximum and minimum values ​​of the elongation at break, evaluate the performance of the yarn, and generate a test report based on the analysis results.

[0041] The advantages of the spinning product elongation at break detection method are as follows:

[0042] This method solves several key problems in the existing elongation at break test of spinning products: first, through the application of calculus formulas, the accurate analysis of yarn morphology and stress-strain relationship is achieved, and the accuracy of the test results is improved; second, the stability and outlier detection analysis of the data are performed using statistical formulas to ensure the reliability and stability of the test data; finally, through detailed comprehensive analysis and report generation, it provides strong support for the performance evaluation and improvement of yarns. This method is feasible and practical, and can significantly improve the efficiency and accuracy of the elongation at break test of spinning products. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The same reference numerals are used in these drawings to represent the same or similar parts. The exemplary embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0044] Figure 1 The figure schematically shows a flow chart of a method for detecting elongation at break of a spinning product according to an embodiment of the present application. DETAILED DESCRIPTION

[0045] In order to make the objectives, technical solutions and advantages of the present application more clear, the present application is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0046] In the following description, references to "one embodiment", "an embodiment", "an example", "an example", etc. indicate that the embodiment or example described in this way may include specific features, structures, characteristics, properties, elements or limitations, but not every embodiment or example necessarily includes the specific features, structures, characteristics, properties, elements or limitations. In addition, repeated use of the phrase "according to one embodiment of the present application" may refer to the same embodiment, but does not necessarily refer to the same embodiment.

[0047] For the sake of simplicity, certain technical features well known to those skilled in the art are omitted in the following description.

[0048] According to one embodiment of the present application, a method for detecting elongation at break of a spinning product is provided. Figure 1 As shown, the following steps are included:

[0049] S1: Sample preparation and pretreatment steps;

[0050] At least 30 samples are randomly selected from the spinning products to be tested, ensuring that the samples cover different batches and different locations to ensure the representativeness of the results. The samples are placed at a temperature of 20±2℃ and a relative humidity of 65±3% for at least 48 hours to eliminate the impact of the environment on the test results. Sample pretreatment is a key step to ensure the accuracy of the test. By placing it under standard temperature and humidity conditions, fluctuations in yarn performance caused by environmental changes can be eliminated. At the same time, the cleaning step removes impurities that may affect the test results.

[0051] S2: Steps for sample morphology measurement and mathematical modeling;

[0052] A high-precision optical microscope was used to measure the diameter and cross-sectional shape parameters of each sample. Based on the measured data, a mathematical model with an elliptical cross-sectional shape was established using the curve fitting technique in calculus. The equation is expressed as follows:

[0053]

[0054] Where a and b are the major and minor axes of the fractional superellipse, respectively;

[0055] α(x), β(y), γ(y), and δ(x) are fractional functions of x and y;

[0056] x and y are the coordinates of any point on the cross section;

[0057] By combining the calculus of variations with fractional calculus, the actual measured data are fitted to obtain the optimal values ​​of a, b, α(x), β(y), γ(y), and δ(x);

[0058] S3: Steps for calibrating and setting parameters of the tensile test equipment;

[0059] Use standard samples to calibrate the tensile test equipment to ensure the accuracy and stability of the equipment. Set the tensile speed to a constant value of 50mm / min and the sampling frequency of the data acquisition system to 100Hz to ensure that the subtle changes of the yarn during the stretching process can be captured. Equipment calibration is the basis for ensuring the accuracy of the test. The accuracy and stability of the equipment can be verified by testing standard samples. The setting of the tensile speed and sampling frequency is determined according to the characteristics of the yarn material and the test requirements.

[0060] S4: Steps for executing the tensile test and collecting data;

[0061] The pretreated samples are installed on the tensile test equipment and the tensile test is carried out according to the set parameters. During the test, the data acquisition system records the tensile force, elongation and time data in real time. Each sample is tested at least five times to reduce the influence of random errors. The tensile test is a key step in evaluating the elongation at break of the yarn. By recording the key data during the stretching process in real time, the elongation at break of the yarn can be calculated. Repeated testing can improve the reliability of the test results.

[0062] S5: Steps for calculating elongation at break and conducting preliminary analysis;

[0063] According to the data obtained from the tensile test, the elongation at break of each sample is calculated. The elongation at break calculation formula is:

[0064]

[0065] Perform a preliminary analysis on the calculated elongation at break to check the rationality and consistency of the data; elongation at break is an important indicator for evaluating yarn performance. By calculating the ratio of elongation at break to initial length, the elongation at break of the yarn can be obtained. Preliminary analysis helps to discover abnormal data or potential problems in the test process.

[0066] S6: Steps for stress-strain relationship analysis;

[0067] Using the differential method in calculus, the stress-strain relationship of the yarn during the stretching process is analyzed. The stress-strain relationship of the yarn conforms to the fractional-order nonlinear elastic model, and its mathematical expression is:

[0068]

[0069] Where σ is stress, λ is the elongation ratio (λ = 1 + ∈, ∈ is strain), C1 and C2 are material constants, α(λ), β(λ -1 ),γ(λ),δ(λ -1 ) is a fractional function of λ. The data obtained from the tensile test are combined with the fractional-order nonlinear least squares method and the variational principle to fit C1, C2, α(λ), β(λ -1 ),γ(λ),δ(λ -1 )’s optimal value;

[0070] S7: Steps for data stability analysis;

[0071] The ANOVA analysis method was used to analyze the stability of the test results between different batches and different locations, and the total variation was decomposed into inter-group variation and intra-group variation. By comparing the ratio of inter-group variation to intra-group variation, it was determined whether the variation from different sources was significant. The Bayesian mixed effects variance analysis was used, and its mathematical model was:

[0072] Y ijk ~Normal(μ+A i +B j +(AB) ij +S ijk ,σ 2 );

[0073] Among them, Y ijk is the observed value of the kth repeated measurement in the i-th batch, j-th position;

[0074] μ is the population mean;

[0075] A i It is a batch effect;

[0076] B j It is the position effect;

[0077] (AB) ij is the interaction effect of batch and position;

[0078] S ijk It is the within-subject effect, that is, the repeated measurement effect within the same batch and the same location;

[0079] σ 2 is the error variance;

[0080] S8: Step of performing outlier detection and processing;

[0081] The box plot method in statistics is used to detect outliers in the test data. By calculating the quartiles Q1, Q2, Q3 and the interquartile range IQR=Q3-Q1 of the data, a box plot is drawn and outliers are identified. For the detected outliers, necessary processing is performed including deletion and replacement. Outlier detection is an important step to ensure the accuracy of test data. The box plot method can intuitively identify outliers in the data and perform necessary processing.

[0082] S9: Steps for comprehensive analysis of results and report generation;

[0083] Comprehensive analysis of the test data, including calculation of the average, standard deviation, maximum and minimum values ​​of the elongation at break, evaluates the performance of the yarn, and generates a test report based on the analysis results. Comprehensive analysis of results and report generation are the summary stage of the test work. By comprehensively analyzing the test data, we can fully understand the performance characteristics of the yarn and provide a basis for subsequent improvement and optimization.

[0084] According to an embodiment of the present application, the step S1 of the method for detecting the elongation at break of spinning products is a step of preparing and pretreating samples, and further includes: cleaning the samples to remove surface impurities to ensure the accuracy of the test.

[0085] According to one embodiment of the present application, the mathematical model established in step S2 of the spinning product elongation at break detection method is expressed as follows:

[0086]

[0087] Where a and b are the major and minor axes of the fractional superellipse, respectively;

[0088] α(x), β(y), γ(y), and δ(x) are fractional functions of x and y;

[0089] x and y are the coordinates of any point on the cross section;

[0090] The actual measured data are fitted by combining the calculus of variations with fractional calculus to obtain the optimal values ​​of a, b, α(x), β(y), γ(y), and δ(x).

[0091] According to one embodiment of the present application, the mathematical model established in step S2 of the spinning product elongation at break detection method is expressed as follows:

[0092]

[0093] in:

[0094] a and b: the reference values ​​of the major and minor semi-axes of the fractional generalized superellipse, respectively;

[0095] α(x,y), β(x,y), γ(x,y), δ(x,y): are fractional functions of x and y, indicating the complexity of the cross-sectional shape as it changes with position;

[0096] x and y: coordinates of any point on the cross section;

[0097] The actual measured data are fitted by the variational method combined with fractional calculus to obtain the optimal values ​​of a, b, α(x, y), β(x, y), γ(x, y), and δ(x, y). The periodic characteristics of the cross-sectional shape are analyzed by Fourier transform, and the nonlinear characteristics of the cross-sectional shape are extracted by combining Hilbert-Huang transform.

[0098] According to one embodiment of the present application, the stress-strain relationship of the yarn in step S6 of the spinning product elongation at break detection method conforms to the mathematical expression of the fractional-order nonlinear elastic model:

[0099]

[0100] Where σ is stress, λ is the elongation ratio (λ = 1 + ∈, ∈ is strain), C1 and C2 are material constants, α(λ), β(λ -1 ),γ(λ),δ(λ -1 ) is a fractional function of λ. The data obtained from the tensile test are combined with the fractional-order nonlinear least squares method and the variational principle to fit C1, C2, α(λ), β(λ -1 ),γ(λ),δ(λ -1 ) is the optimal value.

[0101] According to one embodiment of the present application, the stress-strain relationship of the yarn in step S6 of the spinning product elongation at break detection method conforms to the mathematical expression of the fractional-order nonlinear elastic model:

[0102]

[0103] in:

[0104] σ(t): stress at time t;

[0105] K1(t-τ) and K2(t-τ): kernel functions related to time and strain history, respectively, representing the viscoelastic properties of the material;

[0106] λ(τ): elongation ratio at time τ;

[0107] α(λ(τ)) and β(λ(τ) -1 ): It is a fractional function of the elongation ratio, indicating the nonlinear characteristics of the stress-strain relationship;

[0108] γ(τ): is a fractional function with respect to time, representing the non-integer derivative of the strain rate;

[0109] d γ(τ) / dτ γ(τ) : represents the fractional derivative, which is used to describe the non-integer order changes of strain rate.

[0110] According to one embodiment of the present application, in the spinning product elongation at break detection method, in S7, whether the variation from different sources is significant is judged, and Bayesian mixed effects variance analysis is adopted, and its mathematical model is:

[0111] Y ijk ~Normal(μ+A i +B j +(AB) ij +S ijk ,σ 2 );

[0112] Among them, Y ijkis the observed value of the kth repeated measurement in the i-th batch, j-th position;

[0113] μ is the population mean;

[0114] A i It is a batch effect;

[0115] B j It is the position effect;

[0116] (AB) ij is the interaction effect of batch and position;

[0117] S ijk It is the within-subject effect, that is, the repeated measurement effect within the same batch and the same location;

[0118] σ 2 is the error variance.

[0119] According to an embodiment of the present application, in the method for detecting the elongation at break of a spinning product, in step S7, whether the variation from different sources is significant is determined, and a Bayesian mixed effects variance analysis is used, and its mathematical model is:

[0120] g(μ ijk )=η+A i +B j +(AB) ij +S ijk +Z l β l ;

[0121] in:

[0122] g(μ ijk ): Link function, used to convert the expected mean μ ijk Linked to linear predictor;

[0123] η: intercept term;

[0124] A i : Batch effect, indicating the impact of the i-th batch on the results;

[0125] B j : Position effect, indicating the influence of the jth position on the result;

[0126] (AB) ij : The interactive effect of batch and position;

[0127] S ijk : Within-subject effect, which means repeated measurement effect within the same batch and the same location;

[0128] Z l : Design matrix, representing other covariates that may affect the results;

[0129] β l : is the coefficient of the covariate.

[0130] The above-mentioned embodiments only represent several implementation methods of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. It should be pointed out that, for a person skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the claims.

Claims

1. A method for detecting elongation at break of a spinning product, characterized in that: The steps include: S1: Sample preparation and pretreatment steps; Randomly select at least 30 samples from the spinning products to be tested, ensuring that the samples cover different batches and different locations to ensure the representativeness of the results. Place the samples at a temperature of 20±2℃ and a relative humidity of 65±3% for at least 48 hours to eliminate the impact of the environment on the test results; S2: Steps for sample morphology measurement and mathematical modeling; The diameter and cross-sectional shape parameters of each sample were measured using a high-precision optical microscope. Based on the measured data, a mathematical model of the cross-sectional shape of the elliptical yarn was established using the curve fitting technology in calculus. S3: Steps for calibrating and setting parameters of the tensile test equipment; The tensile test equipment was calibrated using standard samples to ensure the accuracy and stability of the equipment. The tensile speed was set to a constant value of 50 mm / min, and the sampling frequency of the data acquisition system was set to 100 Hz to ensure that the subtle changes of the yarn during the stretching process could be captured. S4: Steps for executing the tensile test and collecting data; The pretreated samples were installed on the tensile test equipment and the tensile test was carried out according to the set parameters. During the test, the data acquisition system recorded the tensile force, elongation and time data in real time. Each sample was tested at least five times to reduce the influence of random errors. S5: Steps for calculating elongation at break and conducting preliminary analysis; According to the data obtained from the tensile test, the elongation at break of each sample is calculated. The elongation at break calculation formula is: Conduct a preliminary analysis on the calculated elongation at break to check the rationality and consistency of the data; S6: Steps for stress-strain relationship analysis; Using the differential method in calculus, the stress-strain relationship of the yarn during the stretching process is analyzed. The stress-strain relationship of the yarn conforms to the fractional-order nonlinear elastic model. S7: Steps for data stability analysis; The ANOVA analysis method was used to analyze the stability of the test results between different batches and different locations, and the total variation was decomposed into inter-group variation and intra-group variation. By comparing the ratio of inter-group variation to intra-group variation, it was determined whether the variation from different sources was significant. S8: Step of performing outlier detection and processing; The box plot method in statistics is used to detect outliers in the test data. By calculating the quartiles Q1, Q2, Q3 and the interquartile range IQR = Q3-Q1 of the data, a box plot is drawn and outliers are identified. For the detected outliers, necessary processing is performed including deletion and replacement. S9: Steps for comprehensive analysis of results and report generation; Conduct a comprehensive analysis of the test data, including the calculation of the average, standard deviation, maximum and minimum values ​​of the elongation at break, evaluate the performance of the yarn, and generate a test report based on the analysis results.

2. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that: The step S1 is a step of preparing and pre-treating the sample, and also includes: cleaning the sample to remove surface impurities to ensure the accuracy of the test.

3. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that: The mathematical model established in step S2 is expressed as follows: Where a and b are the major and minor axes of the fractional superellipse, respectively; α(x), β(y), γ(y), and δ(x) are fractional functions of x and y; x and y are the coordinates of any point on the cross section; The actual measured data are fitted by combining the calculus of variations with fractional calculus to obtain the optimal values ​​of a, b, α(x), β(y), γ(y), and δ(x).

4. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that: The mathematical model established in step S2 is expressed as follows: in: a and b: the reference values ​​of the major and minor semi-axes of the fractional generalized superellipse, respectively; α(x,y), β(x,y), γ(x,y), δ(x,y): are fractional functions of x and y, indicating the complexity of the cross-sectional shape as it changes with position; x and y: coordinates of any point on the cross section; The actual measured data are fitted by the variational method combined with fractional calculus to obtain the optimal values ​​of a, b, α(x, y), β(x, y), γ(x, y), and δ(x, y). The periodic characteristics of the cross-sectional shape are analyzed by Fourier transform, and the nonlinear characteristics of the cross-sectional shape are extracted by combining Hilbert-Huang transform.

5. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that: The stress-strain relationship of the yarn in step S6 conforms to the mathematical expression of the fractional-order nonlinear elastic model: Where σ is stress, λ is the elongation ratio (λ = 1 + ∈, ∈ is strain), C1 and C2 are material constants, α(λ), β(λ -1 ),γ(λ),δ(λ -1 ) is a fractional function of λ. The data obtained from the tensile test are combined with the fractional-order nonlinear least squares method and the variational principle to fit C1, C2, α(λ), β(λ -1 ),γ(λ),δ(λ -1 ) is the optimal value.

6. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that: The stress-strain relationship of the yarn in step S6 conforms to the mathematical expression of the fractional-order nonlinear elastic model: in: σ(t): stress at time t; K1(t-τ) and K2(t-τ): kernel functions related to time and strain history, respectively, representing the viscoelastic properties of the material; λ(τ): elongation ratio at time τ; α(λ(τ)) and β(λ(τ) -1 : It is a fractional function of the elongation ratio, indicating the nonlinear characteristics of the stress-strain relationship; γ(τ): is a fractional function with respect to time, representing the non-integer derivative of the strain rate; d γ(τ) / dτ γ(τ) : represents the fractional derivative, which is used to describe the non-integer order changes of strain rate.

7. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that In step S7, whether the variation from different sources is significant is determined by using Bayesian mixed effects variance analysis, and its mathematical model is: Y ijk ~Normal(μ+A i +B j +(AB) ij +S ijk ,s 2 ); Among them, Y ijk is the observed value of the kth repeated measurement in the i-th batch, j-th position; μ is the population mean; A i It is a batch effect; B j It is the position effect; (AB) ij is the interaction effect of batch and position; S ijk It is the within-subject effect, that is, the repeated measurement effect within the same batch and the same location; σ 2 is the error variance.

8. The method for detecting elongation at break of a spinning product according to claim 1, characterized in that: In step S7, whether the variation from different sources is significant is determined by using Bayesian mixed effects variance analysis, and its mathematical model is: g(μ ijk )=η+A i +B j +(AB) ij +S ijk +Z l b l ; in: g(μ ijk ): Link function, used to convert the expected mean μ ijk Linked to linear predictor; η: intercept term; A i : Batch effect, indicating the impact of the i-th batch on the results; B j : Position effect, indicating the influence of the jth position on the result; (AB ij ): the interaction effect between batch and position; S ijk : Within-subject effect, which means repeated measurement effect within the same batch and the same location; Z l : Design matrix, representing other covariates that may affect the results; β l : is the coefficient of the covariate.

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