HOW TO PERFORM A STATISTICAL ANALYSIS OF NON-DESTRUCTIVE DEGRADATION DATA TO STUDY CRACK GROWTH IN WIND BLADES AS A FUNCTION OF THE NUMBER OF CYCLES

Objective: The aim of this paper is to show the application of the Exponential Regression Model to study the failure time of a system over time. The data was taken from the paper (Wang et al. , 2021). Theoretical framework: The linear regression model, where the response is associated with the explanatory variables by means of a linear model, is the best known. To formulate the model, it is necessary to specify a deterministic component and a random (stochastic) component. The exponential model should be used when it is assumed that risk is constant over time. Once the model has been specified, its parameters are estimated. In the absence of normal errors and especially in the presence of censoring, a more appropriate option is the maximum likelihood method. As the equations are non-linear in their parameters and do not have an analytical solution, it is necessary to use the Newton-Raphson numerical method. Due to the simplicity of the exponential regression model, few situations in practice are adequately adjusted by this model. The Weibull regression model is widely used in survival analysis (Bastos Lyra et al. , 2008) Method


RESUMEN
Objetivo: El objetivo de este trabajo es mostrar la aplicación del Modelo de Regresión Exponencial para estudiar el tiempo de fallo de un sistema a lo largo del tiempo.Los datos se tomaron del artículo (Wang et al., 2021).

INTRODUCTION
During the last century, statistics revolutionized science by presenting useful models that modernized the research process in the direction of better research parameters, making it possible toguide decision making in a wide variety of areas.Statistical methods were developed as a mixture of science and logic for the solution and investigation of problems in various areas of human knowledge (Abraão et al., 2024;da Motta Reis et al., 2023;N. A. S. Sampaio, Mazza, Sérgio, et al., 2024;N. A. S. Sampaio, Mazza, Siqueira, et al., 2024;Silva et al., 2023).
Launching a new product and/or process usually involves working with a large number of variables.Proper planning of the experiments that must be used to manipulate these variables and arrive at the desired answers is indispensable for obtaining reliable results and carrying out robust statistical analyses.It is therefore no longer possible to develop products and processes empirically, as was done in the past.Fierce competition, the spread of high-tech processes and the responsibility of the scientific community make such procedures impossible.Optimizing processes requires more than ever a robust and efficient statistical study (de Souza Sampaio et al., 2022;Espuny et al., 2023;Leoni et al., 2017;Mazza et al., 2022Mazza et al., , 2024;;Saboia et al., 2024;Sales et al., 2021;Siqueira et al., 2024).
Most reliability models which include aspects of uncertainty use probability theory to quantify information about these uncertain aspects.For example, to take a future lifetime T of 4 a component into account in a model for system reliability, a probability distribution for T can be used, which e.g. can be assumed to belong to a certain class of parametric distributions such as Weibull, Gamma, or Lognormal.Such a choice of a single distribution is often highly subjective, even if there is a fair amount of relevant data available there are still many possible distributions which are equally well supported by the data (Coolen, 2004).The aim of this article is to present the results obtained by applying the Exponential Regression Model to a case study with data from the aforementioned paper (N. A. de S. Sampaio et al., 2022;Wang et al., 2021).

THEORETICAL BACKGROUND
Launching a new product and/or process usually involves working with a large number of variables.Proper planning of the experiments that must be used to manipulate these variables and arrive at the desired answers is indispensable for obtaining reliable results and carrying out consistent statistical analyses.In this context, it is no longer possible to develop products and processes empirically, as was done in the past.Fierce competition, the spread of technological processes and the responsibility of the scientific community now make such procedures impossible.Optimizing processes requires more than ever a robust and efficient statistical study (de Souza Sampaio et al., 2022;Espuny et al., 2023;Sales et al., 2022;Siqueira et al., 2024) The increase in demand for better performing products, services and processes at competitive costs has led to a need to increase accessibility to the various elements of the production chain, combined with an increase in the reliability of these elements.The reliability of a system is associated with the successful operation of a product or system in the absence of failures.Therefore, methodologies that are consistent in defining the right maintenance policies, with a consequent reduction in operating costs, have become necessary and imperative for the success of the business.Traditional maintenance methods that use non-repairable models can lead to wasted resources and high maintenance costs.Reliability analysis of complex repairable systems is an important but computationally intensive task, as it is characterized by stochastic processes, imperfect maintenance and susceptible to degradation over time (Deus et al., 2024;Sampaio, Mazza, Sérgio, et al., 2024;Sampaio, Mazza, Siqueira, et al., 2024).The Weibull regression model is widely used in survival analysis (Bastos Lyra et al., 2008).

MATERIALS AND METHODS
This work can be classified as applied research, as it aims to provide improvements in the current literature, with normative empirical objectives, aiming at the development of policies and strategies that improve the current situation (Bertrand & Fransoo, 2002).The approach to the problem is quantitative, as is the modeling and simulation research method.
The research stages were carried out following the sequence shown in Figure 1.

Research method steps
Step 1: Selection of experimental data for the case study.
Step 4: Discussion and interpretation of the results and their importance for similar statistical applications.
Step 5: Critical analysis of the data Discussion.

CASE 1
The case shows a case study carried out in the paper (Wang et al., 2021)All the data was obtained from that article but the discussion is based on the use of exponential regression, as it seems to be the most appropriate for the data.In the specific case of this article, five turbine blades are tested for crack propagation.The units are tensioned and checked every 100,000 6 cycles to observe the crack length and failure is defined as a crack greater than or equal to 30mm.Table 1 shows the test results for the five units for each cycle.

CASE 1
The data obtained in Table 1 was processed using Minitab-19 software and the initial objective was to mathematically model the exponential regression of crack size as a function of time.The five turbine blades are tested for crack propagation.The test units are cyclically stressed and inspected every 100,000 cycles for crack length.Failure is defined as a crack length of 30 mm or more.Reliability and lifetime analysis of components and systems cannot always be done with failure time data.Estimates can be obtained by measuring the degradation or deterioration suffered by components and systems before they suffer total failure.In this case, you need a characteristic that degrades or deteriorates over time and that can be measured and quantified.It is important to conceptualize what level of degradation is undesirable (i.e.makes the component or system a failure).In these cases, the response variable is not the final failure (failed/not failed) but the level of the undesired characteristic that can be measured quantitatively.Now the 5 equations will be solved for the value of y being 30mm and thus find the value of x in all of them.Using Minitab software, the equations were solved and the results are shown in Table 2.The software solves equation (1) and isolates the variable x to obtain equation ( 2 cycles and so on for all units.Now that the values for a crack size of 30 mm have been predicted, an analysis can be carried out to see which Probability Distribution is best suited to studying this Reliability problem.These calculated Cycles must be considered as the Failure Time.Once again, the software is used to test the quality of the fit using the Anderson-Darling statistic.This criterion is used to choose the smallest possible parameter, but this should not be the only criterion, taking into account the easiest method to represent, as it often needs to be handled professionally by someone less experienced with statistical processing.Table 3 shows the values for each type of Distribution that is available in the Software.The ones with the lowest values are 3-Parameter Loglogistic (2,41) and Logistic (2,41).
However, the parameters of the Weibull (2.45) and Normal (2.443) distributions are a little higher, but they are much more widely used and have a lot of material available in the scientific literature.It is clear that either of the two distributions mentioned above can perfectly well be used to study the trica problem in this context.

FINAL CONSIDERATIONS
In conclusion, the Weibull and Normal Distribution Models are suitable for studying the crack problem and Unit 4 takes the most Cycles to fail (with 594.4281Cycles) and produces a crack size of 30 mm.
How to Perform a Statistical Analysis of Non-Destructive Degradation Data to Study Crack Growth in Wind Blades as a Function of the Number of Cycles ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-11 | e08192 | 2024.
he linear regression model, where the response is associated with the explanatory variables by means of a linear model, is the best known.To formulate the model, it is necessary to specify a deterministic component and a random (stochastic) component.The exponential model should be used when it is assumed that risk is constant over time.Once the model has How to Perform a Statistical Analysis of Non-Destructive Degradation Data to Study Crack Growth in Wind Blades as a Function of the Number of Cycles ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-11 | e08192 | 2024.5 been specified, its parameters are estimated.In the absence of normal errors and especially in the presence of censoring, a more appropriate option is the maximum likelihood method.As the equations are non-linear in their parameters and do not have an analytical solution, it is necessary to use the Newton-Raphson numerical method.Due to the simplicity of the exponential regression model, few situations in practice are adequately adjusted by this model.
How to Perform a Statistical Analysis of Non-Destructive Degradation Data to Study Crack Growth in Wind Blades as a Function of the Number of Cycles ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-11 | e08192 | 2024.
Figure-2 shows the Fitted Line Graph of Unit 1 as a function of the number of Cycles with the parameters a and b seen in equation 1.

Figure 2
Figure 2Adjusted Line Graph of Unit A as a function of the number of Cycles

Figure 3
Figure 3Adjusted Line Graph of Unit ,C, D and E as a function of the number of Cycles a Statistical Analysis of Non-Destructive Degradation Data to Study Crack Growth in Wind Blades as a Function of the Number of Cycles ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-11 | e08192 | 2024.9 So, for example, for Unit 1, a crack size of 30 mm gives a number of cracks of 509.8247

Table 1
Test results for the five units for each cycle

Table 2
Test results for the five units for each cycle

Table 3
Fit Quality Test Table