Machine Learning Methods as a Tool for Analysis and Prediction of Impact Resistance of Rubber–Textile Conveyor Belts

Abstract

Rubber–textile conveyor belts are an important element of large-scale transport systems, which in many cases are subjected to excessive dynamic loads. Assessing the impact resistance of them is essential for ensuring their reliability and longevity. The article focuses on the use of machine learning methods as one of the approaches to the analysis and prediction of the impact resistance of rubber–textile conveyor belts. Based on the data obtained from the design properties of conveyor belts and experimental testing conditions, four models were created (regression model, decision tree regression model, random forest model, ANN model), which are used to analyze and predict the impact force of the force acting on the conveyor belt during material impact. Each model was trained on training data and validated on test data. The performance of each model was evaluated using standard metrics and model indicators. The results of the model analysis show that the most powerful model, ANN, explains up to 99.6% of the data variability. The second-best model is the random forest model and then the regression model. The least suitable choice for predicting the impact force is the regression tree.

1. Introduction

In modern industry, efficient and reliable material handling is a fundamental pillar of productivity and economic efficiency, from raw material extraction to processing plants and logistics centers. Conveyor belts play an essential role in this context, forming large-scale transport systems that enable the smooth and continuous movement of various materials over varying distances, often under challenging conditions. Given their critical function in transporting large volumes, continuous monitoring and diagnosis of conveyor failures and their components are essential for uninterrupted operation [1].

The issue of impact resistance of conveyor belts and thereby the issue of damage to conveyor belts has been addressed by many authors both theoretically and experimentally. Ballhaus [2] observed the impact forces of a falling object in free-fall onto a moving conveyor belt. The aim of the research was to determine the effects of selected parameters (speed of belt movement, impact height, thickness of the damping layer) on the wear of the conveyor belt. Hardygóra [3] states that basic experimental methods for assessing the reliability and quality of rubber–textile conveyor belts include monitoring the impact resistance to belt puncture or splitting, testing belt fatigue, and examining the joins.

The development of damage to the conveyor belt carcass on a belt conveyor in an underground mine was studied by Kirjanów-Błażej et al. [4]. The analysis of the damage progression was conducted not only using a linear regression model but also with a quadratic regression model. The creation of regression models for determining impact forces at the point of impact of the material is used in the papers [5,6]. Taraba et al. [7] uses the finite element method in modeling the deformation conditions of the steel cords in a conveyor belt. Bocko et al. [8] tests the mechanical properties of a rubber conveyor belt using its impact load. Internal damage to the conveyor belts was evaluated by computed tomography. Fedorko et al. [9] use industrial tomography to examine damage to the carcass of a conveyor belt. The forces acting on the conveyor belt during its operation influence its susceptibility to damage, service life, and generally affect its condition. The influence of various factors on the reliable operation of belt conveyors was addressed by the authors in the article [10]. In the articles [11,12] the contact and tensile forces acting on the conveyor belt were analyzed and evaluated. Using the vibration frequencies of the conveyor, the increase in tensile force was determined in the paper [13]. The causes of dynamic stress in a loaded conveyor belt were examined by the authors in an article [14,15]. Dabek et al. [16] addressed the possibility of using non-contact methods, which among other things use the measurement of the compressive force on the tensioning device of the conveyor. A tension meter-based system helps to solve the issue of monitoring the tension and wear of the conveyor belt in the article [17].

Despite these valuable efforts, all previous studies on conveyor belt impact resistance, wear, and service life often rely on procedures with limited capabilities for accurate prediction or comprehensive damage assessment. This limitation highlights the critical need to augment classical approaches with advanced modeling and analytical methods that can provide a deeper understanding and more precise forecasting of conveyor belt damage. Recognizing this need, incorporating machine learning (ML) into conveyor belt systems represents a promising approach to significantly enhance system reliability and maintenance efficiency. ML plays a pivotal role in data analysis, particularly when the goal is to extract predictions, uncover patterns, or reveal hidden connections from vast and complex datasets that would be challenging for human perception alone. In this context, ML models are adept at learning intricate patterns within data to optimize process parameters, predict equipment wear, and inform proactive maintenance strategies [18].

This article builds on research into monitoring the severity of damage to rubber–textile conveyor belts, which is closely related to their impact resistance and the monitoring of their wear and damage. The article systematically compares four different ML approaches (regression analysis, decision tree regression, random forest and ANN) to predict the impact resistance of rubber–textile conveyor belts based on experimental data. It uses a complex set of input parameters and evaluates not only the accuracy of the models, but also their practical applicability. The primary aim of the article is to provide a complete and comprehensive comparison of selected ML approaches, which can significantly aid the development of a methodology for predicting the impact resistance of conveyor belts. Furthermore, our work aims to:

Identify and quantify the relationships between diverse input parameters and conveyor belt impact resistance, leveraging the strengths of various ML models (regression for linear relationships, tree models and neural networks for complex, nonlinear interactions).

Increase prediction reliability through systematic cross-model testing on identical, real-world experimental data, fostering confidence in the chosen model.

Precisely determine the significance of input variables (e.g., drop height, load weight, impactor type, belt strength), offering a deeper understanding of underlying damage mechanisms and enabling optimized belt design and operational parameters.

Demonstrate the flexibility and adaptability of various algorithms to different conditions and belt types, enhancing practical applicability across diverse industrial environments.

Contribute to increased efficiency and reduced costs by enabling more accurate maintenance planning, minimizing unplanned downtime, and extending belt life.

The paper is organized as follows: In Section 2 (Related Work) is presented an overview of the current state of knowledge in the field of machine learning and diagnostics in the area of conveyor belts. In Section 3 (Materials and Methods) is described the method of data acquisition and the evaluation methods used. Section 4 (Result) and Section 5 (Discussion) present the results of the analysis and comparison of the predictive capabilities of the selected models, including the evaluation of the significance of input variables. Section 6 (Conclusions) summarizes the main findings and contributions of the study, while Section 7 (Future study opportunities and limitations) is devoted to practical implications and directions for future research.

2. Related Work

2.1. Literature Review

The area of diagnosis, monitoring, and prediction of damage to conveyor belts has undergone significant development in recent years, primarily thanks to progress in machine learning (ML) methods, advanced sensor technologies, and innovative analytical approaches. Traditionally, it relies on conventional methods, such as visual inspection often supplemented by sensors [19], ultrasonic measurements, or 2D laser systems for measuring belt thickness [20]. For example, Leite et al. [21] successfully applied 2D lasers and statistical methods for the detection of longitudinal cracks. Although these techniques provided a detailed overview of the material’s condition, their main limitations lay in insufficient automation and low time efficiency, which made them less suitable for large-scale and continuous industrial applications. Similarly, metrotomography offers exceptional precision in the analysis of the internal structure of belts [22]; however, its use is limited by laboratory conditions and high demands on resources. These inherent limitations of traditional approaches underscored the urgent need for the development of more sophisticated and automated diagnostic solutions for industry.

Despite the shift towards advanced methods, classical statistical predictive models, such as regression and linear discriminant analysis (LDA) [23,24], still retain their place in certain applications, especially where model transparency and rapid application without extensive datasets are critical. These methods, including logistic regression and various classification models, have been successfully applied in the analysis of defects in rubber composites [25] and rubber–textile conveyor belts [26]. Complex evaluation of conveyor belt im-pact resistance was also realized using canonical correlation analysis [27]. However, despite their contribution to identifying correlations in data, their sensitivity to nonlinearities and strong assumptions about data structure often limit their ability to effectively solve complex and dynamic problems in modern industrial systems. This challenge led to the need for a transition to approaches that can better process nonlinear and extensive data, which corresponds with the development of information technologies and the advent of ma-chine learning-based solutions [28].

These persistent challenges led to the massive development of modern diagnostic approaches based on artificial neural networks (ANN) and deep learning (DL). These methodologies have demonstrated an excellent ability to process high-dimensional and complex data, significantly automating and increasing the precision of damage detection. For example, a study [29] effectively demonstrated the detection of conveyor belt damage using a two-layer neural network, pointing to the potential of ANN in automating defect identification and contributing to predictive maintenance. Similarly, Olchówka et al. [30] focused on detecting steel cord faults using statistical analysis and neural networks. The expansion of the spectrum of ML models, including decision trees, Random Forest, convolutional neural networks (CNN), and generative adversarial networks (GAN), subsequently enabled efficient processing of extensive operational data and offers the possibility of continuous, contactless monitoring [31,32,33,34,35,36]. An example of practical deployment is the faster and lighter detection method based on improved YOLOv5 [37] for foreign objects in coal mine belt conveyors, which excels in rapid and accurate localization of even small defects in real-time. In the broader context of ML application in industry, Elahi et al. [18] demonstrated the capabilities of ANN in predicting energy consumption, while Gunckel et al. [38] proposed a comprehensive framework for the creation and validation of predictive models in mining belt systems. These works collectively emphasize critical factors for successful industrial deployment, such as model robustness against data noise, adaptation to changing operating conditions, and the need for iterative model tuning.

The synergy of advanced sensor systems (e.g., industrial cameras, 2D lasers) and deep learning algorithms (e.g., CNN, GAN, hybrid approaches) [39,40,41,42] is particularly beneficial for more demanding applications, such as contactless detection of cracks and surface defects. Wang et al. [42], for example, successfully applied machine vision for crack detection in a demanding and contaminated industrial environment, thus confirming the high reliability of this technology. The effectiveness of combining UHF RFID (Ultra High Frequency Radio Frequency Identification) with neural networks for fast and accurate detection of belt faults was also demonstrated by [41]. The latest research continues to push the boundaries of defect detection, with deep learning proving to be a reliable tool for damage detection in mining conveyors [43,44], and Yuan et al. [45] also contributed to the progress. The integration of multiple visual models to increase safety was presented by the authors in [46]. Although these solutions significantly contributed to reducing inspection error rates, especially in demanding industrial operations, a key and persistent challenge remains the need for extensive, high-quality annotated training datasets and continuous model calibration for adaptation to changing operating conditions. To solve the problem of limited datasets for rare damage, methods like meta-learning and few-shot learning [47] are emerging, with the meta-learning model achieving significantly better accuracy with a small number of training examples compared to conventional methods. Most ML and artificial intelligence (AI) applications today indeed aim for increased detection accuracy even in environments with disruptive factors (contamination, vibrations); however, a limit remains the need for extensive, high-quality annotated datasets, validation across belt types, and adaptability to changing operating conditions [38,48].

In addition to damage detection, machine learning is increasingly used for operational optimization and risk assessment. Köken [49], for example, illustrated how genetic algorithms in combination with neural networks can estimate the necessary power for conveyors, which is crucial for energy efficiency. Burduk et al. [28] presented a comprehensive framework for risk assessment in mining conveyor systems, emphasizing the importance of a systematic approach to safety and reliability. In the area of load analysis and signal classification, deep learning has proven to be extremely effective. Zvirblis et al. [50] investigated deep learning models for precise classification of rubber belt loading, which is crucial for the development of intelligent monitoring systems with real-time response and optimization capabilities.

In summary, and in the context of persistent challenges, the transition from traditional to advanced ML/AI methodologies, comprehensive review frameworks, and robust sensory systems has fundamentally strengthened the possibilities of continuous monitoring, predictive maintenance, effective lifespan analysis, and optimization of conveyor belts across the full range of operating conditions [51].

2.2. Research Gap and Contribution/Novelty of This Study

Despite the advancements, a significant gap remains in the quantitative, systematic evaluation of the impact of key physical parameters on the impact force and resistance of belts under real (experimental) conditions. Despite advancements in monitoring and predictive maintenance of conveyor belts, a significant gap still exists in the quantitative and systematic evaluation of the impact of key parameters on the impact force and resistance of belts under controlled, yet real experimental conditions. This is an area where comprehensive and precise modeling of all relevant factors still lag. Most works declare the significance of drop height, load weight, or belt strength [24,52], but the precise determination of their influence within a complex model or experiment is rare. Truly detailed analytical and experimental evaluation is still the subject of newer works [53,54,55]. However, significant research results in the field of conveyor belt impact resistance have also been brought by systematically oriented studies [56,57,58,59]. Komander et al. [56] thoroughly analyze methods for evaluating the impact resistance of belts to the dynamic action of concentrated load and provide a detailed experimental description of the influence of individual material parameters on belt lifetime. Further studies [57,58] focus on quantifying the influence of structural properties and operating conditions on the fatigue strength of joints or the degradation and damage of belts, with emphasis placed on evaluating belt behavior under combined or repeated impact loads. The results of laboratory tests of joints and critical areas of belts [59] thoroughly analyze the resistance of textile belts to damage in harsh conditions of underground mines. These latest analytical and experimental publications have significantly advanced the understanding of the influence of individual parameters on impact resistance; however, the need for quantitative modeling of a wider range of factors—and their integration into complex predictive frameworks—still persists.

The study’s results significantly refine existing knowledge about the influence of individual input parameters on conveyor belt impact resistance. While many existing (and also our previous) studies focused primarily on the question “Which input parameter influences conveyor belt impact resistance?”, this study provides an answer to a fundamentally more detailed question: “How, or to what extent, do individual input parameters influence conveyor belt impact resistance?”—and that systematically and across multiple machine learning model approaches.

In comparison with available scientific works, this study contributes to the development of knowledge in the field of conveyor belt impact resistance prediction in the following ways:

Complex comparison of ML approaches: We perform a deep analysis and comparison of four different machine learning approaches (regression analysis, decision tree, random forest, and artificial neural network). This gives us a comprehensive view of their ability to capture different types of relationships—from linear to complex nonlinear and interactive relationships typical for material mechanics and impact dynamics. Systematic testing on identical data increases prediction reliability and allows for the selection of the most robust model for specific applications.

Use of original experimental data: The data used in the study is unique, obtained directly by experiment on a special device from testing real rubber–textile belts. This aspect significantly increases the relevance and credibility of the results.

In-depth analysis of model performance: We compare the performance of algorithms not only based on standard metrics (MSE, RMSE, R²), but also based on a thorough analysis of residuals and monitoring of systematic errors. This detailed approach provides a more comprehensive view of the robustness and reliability of predictive capabilities, which is crucial for practical applications.

Quantified evaluation of the significance of input variables: Unlike many works where the evaluation of variable significance is superficial, our study provides a comprehensive view of the input variables and their quantified influence on impact resistance. Model outputs allow for explicitly determining the importance weights of individual parameters (e.g., drop height, load weight, impactor type, belt strength). Thanks to these findings, it is possible to objectively recommend design modifications or maintenance strategies based on truly dominant damage factors, leading to more effective optimization.

The overall benefits of this study stem from the ability of various ML models to comprehensively and accurately analyze experimental and operational data, thereby supporting more efficient management and optimization of conveyor systems. The obtained knowledge can serve as a basis for the development of intelligent monitoring and predictive maintenance systems in industrial practice, which will significantly increase the reliability and efficiency of conveyor belt operation. At the same time, the results of this work open up space for further research focused on model optimization and extending their use to other types of technical equipment. In this way, the study aims to fill the existing research gap in the area of monitoring impact force and impact resistance of conveyor belts and to advance knowledge towards higher applicability of ML in engineering diagnostics of belt devices.

3. Materials and Methods

3.1. Data Acquisition

In order to meet different requirements, conveyor belts vary in material, structure, and specific properties. The basic structural elements of the conveyor belt include the carcass (textile, steel) and covering layers (upper and lower covering layers), the thickness of which is influenced by the properties of the transported material. An important element is the adhesive compound, which ensures a firm connection between the upper cover layer, the lower cover layer and the edge of the conveyor belt with the carcass, and at the same time ensures a firm connection of the individual plies of the conveyor belt carcass with each other. The most commonly used conveyor belts include rubber–textile conveyor belts (Figure 1). The carcass consists of several textile plies formed by combining polyester (E) and polyamide (P) fibers. The bottom layer (C, Figure 1) is thinner, while the top layer (B, Figure 1), which is in contact with the transported material, is thicker.

To determine the prediction of the impact force and its accuracy based on the input parameters and to determine the performance of individual models, data from the testing of three different types of conveyor belts (CB) were used, the basic parameters of which are listed in Table 1. Due to the complexity of the evaluation of the experiment, conveyor belts with three different strengths were selected, but with the same type of carcass and with the same category of cover layer.

Impact tests were performed using a test device (Figure 2), which enables the simulation of the impact of the material on the conveyor belt and is described in detail in the papers [60,61].

During the tests, the parameters were weight, height, impactor of the hammer, belt strength. The algorithm for the implementation of the experiment is shown in Figure 3. The weight of the falling material was simulated by changing the weight of the hammer across the range 50–100 kg at intervals of 10 kg. Additional weights were added to set the required weight. The impact height varied across the interval from 0.2 to 1.6 m with a step of 0.2 m.

The impactor of the hammer (Figure 4) simulated the type of material falling on the conveyor belt. The hemispherical impactor simulates the impact of non-cohesive crumbled material, and the pyramidal-shaped impactor simulates the impact of a cohesive sharp-edged material.

Significant damage (Figure 5) defines the degree of damage that, in the case of practical use of the conveyor belt, causes the conveyor belt to be disabled or necessitates its immediate repair. In this case, it is simultaneous damage to at least one covering layer (upper or lower) and the carcass of the conveyor belt.

3.2. Assessment Methods

The evaluation and analysis of the collected data were carried out using basic statistical methods, specifically descriptive statistics and statistical hypothesis testing (Shapiro–Wilk test for normality, paired t-test, and paired Wilcoxon test). In hypothesis testing, the decision to accept or reject the null hypothesis was based on the p-value. If the p-value was less than the chosen significance level α, the null hypothesis was rejected in favor of the alternative hypothesis. If the p-value was equal to or greater than the chosen significance level α, the null hypothesis was not rejected.

Machine learning (ML) is a key field for gaining valuable insights and predictions. It is a subfield of artificial intelligence that enables computer systems to learn from data without explicit programming and then identify patterns and make decisions. A wide range of methods can be used to solve regression tasks, such as predicting the continuous value of a target variable, from classical regression analysis, through decision trees to artificial neural networks [62].

3.2.1. Regression Analysis

The relationship between the output (dependent) $Y$ and k-input (independent) variables $X_1,X_2,\dots .,X_k$ can be expressed using the model

(1)$Y={\beta }_0+\sum _{i=1}^k{\beta }_i{X}_i+\epsilon ,$

where ${\beta }_0$ and ${\beta }_i$ for i = 1,…, k are the parameters (coefficients) of the regression model and ε is the random error of the model.

The F-test can be used to assess whether the model statistically significantly explains the variability of the dependent variable [63]. Statistical significance tests of the regression parameter are used to verify the statistical significance of individual parameters of the regression model. The strength of the dependence of variable Y on the effect of k independent variables is expressed by means of a multiple coefficient of determination $R^2$. The coefficient takes values from the interval ⟨0; 1⟩. The closer the value is to 1, the tighter the dependency.

3.2.2. Decision Tree Regression

The use of decision trees is among the basic methods of data mining. A decision tree is a graphical tool that is used to predict and classify and to facilitate decision-making for various decision-making problems [64]. According to the type of dependent variable, we divide them into classification trees and regression trees. Regression trees are used in cases where the target variable is contiguous. The model divides the data space into several ranges according to the values of the input variables, and assigns the resulting value in each region as the average of the outputs in that branch. The advantage is simplicity, interpretability and the ability to work with nonlinear relationships. However, the disadvantage is high variability and a tendency towards he input variables represent the height of the hammer’s impact, the weight of the falling hammer’s impact on the conveyor, especially with small samples or more complex data.

3.2.3. Random Forest

A random forest is a method that creates many decision trees during training and combines their outputs for classification or regression. This method reduces the risk of overlearning and increases the accuracy of prediction. It is part of the category of ensemble learning, where multiple models (in this case, decision trees) are combined to achieve greater accuracy and robustness [65]. Unlike a single decision tree, random forest is more stable and robust because randomly combining trees reduces errors caused by variability in training data. The disadvantages include lower interpretability of the resulting model and higher computational demands.

3.2.4. Artificial Neural Networks

Artificial neural networks (ANNs) model the relationships between input and output variables using interconnected “neurons” in layers. They are powerful computing tools inspired by the structure of the human brain. They consist of interconnected neurons arranged in layers, adjusting the weights of the connections during training to minimize prediction errors. As an essential element of artificial intelligence and machine learning, neural networks are designed to recognize patterns, make decisions, and predict outcomes based on input data [32,66].

The performance of the models and the accuracy of the prediction is verified using the mean absolute error (MAE), the mean squared error (MSE), the root mean squared error (RMSE) and the coefficient of determination (R²). It holds that:

(2)$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_i-{\widehat{y}}_i\right|,$

(3)$MSE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2,RMSE=\sqrt{MSE}$

(4)$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{\sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}}$

where $y_i$ is the real (measured) value, ${\widehat{y}}_i$ is the theoretical (predicted) value, $n$ is the number of observations, $\overline{y}$ is the average value of the measured values.

To determine the models, the R program (version 4.4.3) [67] was used with suitable libraries: regression model (stats, qqplot2, car), decision tree regression model (rpart, partykit), random forest model (qqplot2, randomForest, rpart) and ANN model (nnet, caret, neuralnet).

4. Result

4.1. Definition of Variables and Selection of ML Algorithms

The aim of the research is to quantify the influence of the selected parameters on the impact force that arises when the material hits the conveyor belt. The input variables represent the height of the hammer’s impact, the weight of the falling hammer’s impact on the conveyor belt, the type of impactor of the hammer and the strength of the conveyor belt. A basic overview of input and output variables is in Table 2.

The relationships between these variables were investigated using various machine learning methods: multiple regression model, decision tree regression, random forest, and artificial neural network. In all cases, the process of creating the model was performed according to the schema in Figure 6.

The measured data are divided into two groups: the training group (70% of the data) and the test group (30% of the data). The training group is used to create a model, while the test group is used to evaluate the quality of the model. Each model was created on a training group consisting of 70% of randomly selected experimental measurements (260 measurements). The accuracy and quality of the model was verified on the test group, which consisted of the remaining 30% of the experimental measurements (112 measurements). The performance of individual models was evaluated using standard metrics: MAE, MSE, RMSE and R².

Each of the considered algorithms, which result in 4 different models (regression model, decision tree regression model, random forest model and ANN model), represents a different approach to solving regression problems with different levels of complexity, interpretability, and ability to capture nonlinear relationships:

The regression model is the result of a basic method (regression analysis) that enables simple interpretation of the relationships between the input and output variables and is traditionally used in technical applications.

The decision tree regression model is a simple and interpretable model, often used for rapid data segmentation and basic prediction.

The random forest model is the result of a method that improves the robustness and accuracy of prediction by combining multiple trees, thereby overcoming the limitations of individual tree models.

The ANN model has been included as a result of an advanced machine learning algorithm that can efficiently model comprehensive and nonlinear relationships in data and has proven itself in many engineering applications in recent years.

The choice of these algorithms thereby enables a comprehensive comparison from simpler to more complex algorithms that cover a wide range of approaches used in current research on the impact resistance of conveyor belts. In addition, these models are well supported by the available libraries and enable efficient training and validation on an experimental database.

4.2. Result of ML Techniques

4.2.1. Regression Analysis (Regression Model)

In monitoring the influence of input variables (height of fall H, weight of falling hammer W, type of impactor I and strength of the conveyor belt S) on the output variable $F$. We will start from a general model for each type of conveyor belt in the form of

(5)$F=f\left(H,W,I,S\right),$

The best regression model for both output variables, regardless of the strength of the conveyor belt, is in the form

(6)$F={\beta }_0+{\beta }_{1}H+{\beta }_{2}W+{\beta }_{3}I+{\beta }_{4}S+\epsilon ,$

where ${\beta }_0$, ${\beta }_i$, $i=1, 2, 3, 4$ are the parameters of the models, W is the weight of the falling hammer, H is the impact height, I is the type of the impactor of the hammer, S is the strength of the conveyor belt and ε is the random error. The variable I (impactor) is a dichotomous variable that takes the values I = 0 in the case of a spherical impactor and I = 1 in the case of a pyramidal impactor. Point estimates of model parameters are determined by the least squares method.

The model was created using a training group of samples, which consisted of 70% of the samples (260 samples). Point estimates of the parameters of the regression model, standardized model parameters together with the statistical significance of the parameters (p-value) are in Table 3. The statistical significance of the overall model is tested through the F-test. The level of dependence of variable Y on the effect of k independent variables is expressed using the multiple coefficient of determination $R^2$. The coefficient takes values from the interval ⟨0; 1⟩. The closer the value is to 1, the tighter the dependency. The tables determine the values of the multiple coefficient of determination $R^2$ and the statistical significance of the model (p-value).

It turns out that input variables significantly affect the impact force in each type of conveyor belt. The strength of the dependence of the relationship is expressed through a multiple coefficient of determination R² and the coefficient of determination is equal to the value 0.936. This means that 93.6% of the variability of the variable Impact Force can be explained by the proposed regression model.

The Variance Inflation Factor (VIF) was used to check the multicollinearity of the variables. By multicollinearity we mean a strong correlation between independent variables. Strong multicollinearity reduces the accuracy of estimation of the regression coefficients, and the estimates obtained from the model are not reliable. If VIF = 1, then there is no correlation between one independent variable and the other independent variables. The generally accepted rule is that the problem of multicollinearity arises if any VIF coefficient is greater than 10. The values of the VIF coefficient are approximately equal to 1, which means that there is almost no multicollinearity (dependence) between a given input variable and the other input variables. We can assume that the model is stable and the parameters of the model are reliably estimated.

Because the variables in the regression analysis are expressed in different units, standardized regression coefficients were used to compare the relative significance of individual input variables ${\beta }_i^{*}$, $i=1,\dots ,k$. The standardized regression coefficient ${\beta }_i^{*}$ expresses the extent to which the output variable $Y$ changes with a unit change in the standard deviation of the independent variable $X_i$ provided that the other input variables are constant. The higher the absolute value of the coefficient ${\beta }_i^{*}$, the stronger the influence of the independent variable in question on the output variable. The values of the parameters of the obtained models show that the input variables have a positive effect on the value of the impact force. From the analysis of standardized Beta coefficients (column Beta, Table 2), we can obtain the relative importance of the variables. In all cases, the greatest influence is the impact height (it has the highest value of standardized coefficients), followed by the weight of the falling hammer. The type of impactor has by far the lowest influence.

The accuracy of the model for prediction was verified using a test group consisting of 30% of all samples (112 samples). Figure 7 shows a graph of the actual and predicted (theoretical) values that were obtained from the test group. On the x-axis, the actual values of the impact force are shown, and on the y-axis, the predicted values of the impact force. The line is a reference line and represents an ideal prediction where the actual values are equal to the predicted values. The distance of each point from the line represents the prediction error (residual) for a given value. The farther the point is from the line, the greater the prediction error for this particular case.

To evaluate the quality of the created regression model, the mean squared error (MSE) and the mean absolute error (MAE) were used. The results of MSE = 2.901 and MAE = 1.328 show that the created model is good and can explain a significant part of the variability. The value of R² is 0.936, which means that the resulting regression tree can explain 93.62% of the variability of the target variable F on the test data. This suggests that the regression model is able to capture the relationship between the input variables (H, W, I, S) and the output variable F very well.

4.2.2. Decision Tree Regression (Decision Tree Regression Model)

The decision tree regression model for the analysis of the impact force of conveyor belts is again created using the same training set of samples as in the regression model. The decision criterion is the continuous variable Impact Force (F). The input variables are the height of the impact (H), the weight of the falling hammer (W), the type of impactor (I) and the strength of the conveyor belt (S). The first two variables were applied to some level of the tree (Figure 8). The decision tree regression model was built up to the maximum possible depth or until the fulfillment of the default criteria of the minimum number of observations in the node (minsplit = 20) and the minimum number of observations in the leaf node (minbucket = 7). This approach leads to an extremely complex tree that can potentially be trained on training data. During the tree-building process, and then using the printcp() function, the rpart algorithm performs an internal k-fold cross-validation, by default 10 times (xval = 10). For each possible substructure of the tree (each possible pruned tree), the relative error of cross-validation (X-val Relative Error) is calculated. The primary objective was to minimize this relative error of cross-validation. Based on the results, the value of the complexity parameter (CP) was selected that corresponded to the lowest average error of cross-validation (in our case, CP = 0.01 for a 9-node tree). The maximum number of levels (branches) of the tree from the root node to the outermost leaf node was set at 30.

The importance of the variables (H = 76, W = 23) in the context of a decision tree regression indicates that the most important variable in the decision tree is the height of impact (H). In second place is the variable weight of the falling material (W). Two variables were not used in the creation of the model: the type of impactor and the strength of the conveyor belt. The variables weight of the falling hammer and strength of the conveyor belt, according to the model output, did not bring any improvement in node subdivision and model creation, which may indicate their negligible impact on the regression decision model and on prediction.

Despite the fact that the strength of the conveyor belt (S) and the impactor of the hammer (I) are also statistically significant variables in the regression model, these variables are not present in the decision tree regression. This may also be due to the fact that it is the height and weight variables that can explain most of the variability of the observed impact force, and the contribution from other variables such as impactor and strength are then not decisive in the formation of the tree.

The first branching is based on the variable height (H, Figure 8). If the height is less than 0.5 m (17% of all tested samples found in the given group), the impact force is an average of 6.6 kN. If the impact height is in the range from 0.5 m to 1.1 m and the weight is less than 75 kg (or more than 75 kg), then the average impact force is 10 kN and there are 33 samples (13%) in the group (or 16 kN, in the group with 25 samples, 10%). A detailed overview of the rules can be found in Table 4.

The accuracy of the model for prediction was verified using a test group. The results MSE = 9.22 and MAE = 2.46 show that the created model is good and can explain a significant part of the variability. The value for R² is 0.85, which means that the resulting regression tree can explain 85% of the variability in the target variable F on the test data. This suggests that the regression tree is able to capture the relationship between the input variables (H, W, I, S) and the output variable F very well.

Figure 9 shows a graph of the actual values and predicted values that were obtained from the test group. The actual impact force values are shown on the x axis and the predicted impact force values are shown on the y axis. The green line represents an ideal prediction.

4.2.3. Random Forest (Random Forest Model)

The random forest model predicts the variable F based on all other available variables (H, W, I, S). The random Forest was compiled from 500 individual decision trees (ntree = 500). For each node split within each tree, 2 variables (mtry = 2) were randomly selected from the four available predictors (H, W, I, S). The used value of mtry = 2 corresponds with common recommendations for random forest regression, which recommend setting mtry approximately to the square root of the number of input variables. Based on the performed sensitivity analysis of the hyperparameter mtry, we found that higher values, specifically 3 and 4, bring slightly better results in the form of lower RMSE and higher coefficient of determination R². However, the difference in model performance between these values and the mtry = 2 value is not very significant. At the same time, it is shown that increasing the number of trees above 500 no longer brings significant improvement. Graphical representation of the sensitivity of random forest model performance to changes in hyperparameters mtry and ntree is in Figure 10.

During the training, the ability to calculate the importance of variables was activated, which enables later analysis of the impact of individual inputs on the model. Other hyperparameters, such as the minimum number of data points in a leaf node (nodesize = 5), the maximum number of leaf nodes (maxnodes = NULL), and the maximum tree depth (maxdepth = NULL), were left at their default values, which means that trees grow to full depth without restrictions. The importance of variables for model creation and for the prediction of impact force is determined using the Percent Increase in Mean Squared Error (%IncMSE) indicator, which is used by the random forest to determine the importance of input variables for the prediction of the output variable (Figure 11).

A high value indicates that the variable is very important for the prediction, and a low value means that the variable has little effect on the prediction. In our case, we found:

Variable H is the most important variable, its permutation value is almost 126, which means that the variable is extremely critical for the ability of the Random Forest model to make accurate predictions of the impact force. At the same time, it means that the impact height is the dominant factor and other variables, although important, have a smaller impact on the overall variability of F compared to the impact height.

Variable W is the second most important variable, its permutation value is 90, the variable has a very significant impact on the prediction of the model.

Variable S is the third most important variable; its permutation value is 19. Although it is less important than h and m, it still has a measurable and positive effect on the model. This is a key difference from the original single regression tree, where variable S was of negligible importance. Random Forest, due to its structure (many trees, random selection of variables when dividing), was able to better identify and evaluate the contribution of variable S.

Variable I has minimal, almost zero importance: Its value is very close to zero and even slightly below zero (−2.6). This means that its contribution to forecasts is negligible. Removing variable K can even lead to a better model result.

Figure 12 shows a graph of the actual values and predicted values that were obtained from the test group. The line represents the ideal prediction, and the red symbols represent those predictions with the largest errors. The chart shows the Top 10 largest errors in the prediction, the values of which are also in Table 5.

The percentage of variability that the Random Forest model can explain is 96.28%, which has a similar function to the coefficient of determination in the regression model. A value of 96.28% means that the model can capture almost the entire variance (variability) in the impact force values. The results MSE = 1.76 and MAE = 1.01 show that the created model is better for prediction than previous models.

4.2.4. Artificial Neural Networks (ANN Model)

The ANN model was created in the R environment using the NeuralNetTools and caret packages. The network architecture consisted of a single input layer with 4 neurons that correspond to the input variables (H, W, I, S). This was followed by one hidden layer with 5 neurons, ensuring complete transformations of the input data. The output layer contained a single neuron that represented the predicted continuous output variable F (Figure 13). A linear activation function (linout = TRUE) was used for the output layer, suitable for regression tasks where the goal is to predict a continuous value. The key hyperparameters used during the training included the range of initial random weights (rang = 0.1), the maximum number of iterations (maxit = 500), and the regularization parameter (decay = 0.01), which helps prevent the model from becoming overtrained. The training was performed using 5-fold cross-validation, which enabled a more reliable evaluation of the generalization ability of the model. The results showed that this architecture provides a good ability to approximate the relationships between inputs and outputs on the given dataset.

Each line between the neurons represents a connection, with the values assigned to them indicating the weights of the connection. Where the line and the weights are black it indicates positive weights, i.e., if the value of the neuron at the beginning of the line increases, the value of the neuron at the end of the line also increases (with other inputs constant). The blue lines represent negative weights (if the value of the neuron at the beginning of the line increases, the value of the neuron at the end of the line decreases with other inputs constant). The number of iterations, or steps, that the training model performed is 24,463. The Error value represents the Sum of Squared Errors (SSE) that the network reached on the training data at the end of the optimization process. A lower value means better adaptation of the model to the training data.

The importance of individual input variables in the creation of the model is determined using weights of importance (Figure 14) The most important variable is the height of the impact of the hammer (0.367), which contributes 36.7% to the overall decision. With a very small difference in importance, the strength of the conveyor belt follows (0.346). A variable with a lower weight is weight of the falling hammer, with (0.229). The least relevant variable is considered to be the impactor of the hammer (0.057).

Figure 15 shows a graph of the actual values and predicted values that were obtained from the test group. The line represents the ideal prediction.

The accuracy of the model for prediction was verified using the test group. The results MSE = 0.248 and MAE = 0.383 show that the model created is very good and can explain a significant part of the variability. The value for R² is 0.996, which means that the resulting ANN model can explain up to 99.6% of the variability in the target variable F on the test data. The low values of the error metrics (MSE, MAE, RMSE) confirm that the ANN predictions are extremely close to the actual values.

4.3. Comparison of Result of ML Techniques

To compare the created models, we simulated 10 different measurements with different input parameters. Table 6 displays the predicted values of the impact force depending on the model used.

Summary results of the accuracy of the models for predicting the impact force of all models considered are presented in Table 7.

The ANN model is the most powerful model, it has the lowest MSE, MAE and RMSE values and at the same time the highest R² values, explaining almost the entire variability of the data.

The second-best model is the random forest model. Although its error metrics are higher than those of ANN, they are still significantly lower than those of linear regression and the regression tree. The value of R² = 0.963 confirms that the Random Forest model explains much of the variability of the data. Random Forest is a very strong model that has shown excellent predictive ability, much better than a single regression tree. Due to its structure (many trees, random selection of variables for division), it was able to better identify and evaluate the contribution of variable S. It can be a good substitute for the ANN model.

In the case of the regression model, the results are not bad. The value of R² is close to 0.95, indicating that there is a fairly strong linear relationship between the predictors and the target variable. Linear regression is a good baseline model, and its performance here shows that the data also has a linear component. It is suitable for a simpler and more illustrative interpretation of the model.

The regression tree has the worst results. It was found that the regression tree is the least suitable choice for prediction in this case, although it also serves as the basis for the Random Forest method.

Table 8 provides a final summary and comparison of individual models based on the selected desired characteristics.

In the case where we require the most accurate model regardless of complexity, the ANN model is the best. With a model that gives good results and is a little simpler, the random forest model is advantageous. The linear regression model gives slightly worse results, but due to its simplicity and practicality it is sufficient in many cases. With linear regression, the model equation is easy to interpret, and the regression tree also shows branching and decision-making. The random forest model and the ANN model are more difficult to interpret.

In the next step, residual analysis was performed for all the models obtained. Residuals were defined as the difference between the experimental (actual) F_E and theoretical (predicted) values of the impact force for each model case (F_R, F_T, F_RF, F_ANN). For each model, the basic statistical characteristics of the residuals (mean, median, variance, Max-Min range) were calculated and their normality was tested. The results of the residual analysis are shown in Table 9.

The average residual value is close to zero. Through testing (single-sample t-test, or Wilcoxon’s test depending on whether the sample has a normal distribution), it was tested whether the mean (or median) of the residuals was different from zero to a statistically significant level. The results of the tests are in Table 10.

It is shown that the random forest model, ANN model and equally the decision tree regression model do not have systematic errors (shifts in residuals around zero), which is helpful. Conversely, the regression model has a systematic shift (overestimation), which could be a problem in its application.

In addition, the variance of the residuals was compared between the two selected sets. Where both sets of residuals are from normal distributions, the F-test (comparison of the residuals of the regression model and the decision tree regression model) is used, in other cases the Levene test is used. The results are shown in Table 11.

In almost all cases (except for E_E,R vs. E_E,RF), the p-value is less than the significance level α, which means that we reject the null hypothesis of equal variances.

The ANN model is the best model in terms of residual variance, followed by the random forest model, and the tree model is the worst. The F-test (or Levene test) statistically confirms that these differences are significant and are not due to chance.

5. Discussion

5.1. Discussion and Interpretation of Results

Based on the results, we can draw the following conclusions:

The ANN model achieved the highest accuracy on the test data (R² = 0.996), but in practice its use may be limited by the need for more training data and lower interpretability of the results. Similar findings are present on the high accuracy of ANN; however, limitations in the area of interpretability and the need for large amounts of training data are reported by [48,68]. In conditions with less data or significantly different operating conditions, it may be more appropriate to use a more robust random forest model that is less prone to overlearning and better manages input variability. For higher robustness and the ability to handle input variability, it is also appropriate to consider the use of random forest models in industrial practice, as documented by articles [34,69].

The regression model and the regression tree model provide simpler and more transparent relationships between inputs and output, which is advantageous when there is a need for quick diagnostics or explanation of decisions in industrial practice. On the other hand, these models may have a limited ability to capture complex nonlinear relationships. The contribution and suitability of regression and decision trees for quick diagnostics and explainability are confirmed by [24,70].

When deployed in industry, it is also necessary to take into account the robustness of the models against noise, the ability to adapt to new types of damage or changes in operating conditions. The ANN model is sensitive to the quality of the input data and may require more frequent recalibration. The random forest model is more computationally intensive, which can be limiting when the fast processing of large volumes of data in real time is required. These aspects are pointed out by, for example, [31,34,69].

In terms of the importance of input variables in determining the conveyor belt impact resistance model, there are significant differences between the models. It is important to realize that each model determines the importance or significance of the variables to predict the output variable in a different way, and therefore we will not consider absolute values but the order of importance. In all the models, the impact height of the hammer has the most significant influence, and in all models, the negligible influence of the impactor of the hammer on the prediction is shown. A summary of the importance of the input variables in the creation of the model and in the prediction of the impact force is in Table 12.

The impact height is the most significant input in all models. This parameter most influences the resulting impact force and its leading position corresponds to basic physical expectations. The dominant influence of the impact height on belt impact resistance is in agreement with the results [57,68]. Weight of the hammer is the second most significant parameter in most models, but in ANN models it is only in third place. The impactor type is insignificant in all models, which means that its variability does not bring a fundamental change in prediction. This result is consistent with research findings [53,54] and also has practical significance—it allows for simplifying experiments as well as belt design in terms of robustness against the variability of falling material. Belt strength (CB Strength) manifests as a significant parameter in nonlinear models (random forest, ANN) (in 2nd/3rd place), while in the decision tree it has no significant influence.

The results of this study build upon and significantly extend systematic knowledge about the influence of key parameters (impact height, weight of impacting material, strength and shape of impactor) on the impact force and overall resistance of rubber–textile conveyor belts in real operating conditions. The results correspond with research findings [56,57].

The interpretation of the results shows that compared to previously published studies, where ML models were used more for damage detection or classification of belt conditions [32,38], the study systematically demonstrates that ANN and random forest are significantly more accurate also in quantitative prediction of impact resistance based on physical parameters. The results clearly illustrate not only the higher accuracy of ANN, but also its increased sensitivity to the quality of training data and lower interpretability compared to traditional models (regression, decision tree).

Compared to the work [13,27], where “drop height” was designated as a key input, the study not only confirms these conclusions, but also allows for precisely quantifying that the significance of this input is consistently dominant in all considered model approaches. According to [27], models with included “drop height” significantly improved the accuracy of conveyor belt lifespan prediction. We also demonstrated that belt strength is significant especially in nonlinear models (random forest, ANN), which was not explicitly highlighted in previous studies. In contrast, the impactor type (shape of the falling load) was identified as negligible in all models, which is consistent with the findings [53,54,55].

Through this approach, we simultaneously extend and refine current knowledge:

Not only quality, but also model selection significantly influences the significance of input variables (in some models, neglected inputs may turn out to be important and vice versa), which is also emphasized in [5,23,24,25,57].

Nonlinear ML models (random forest model, ANN model) are able to uncover significant relationships that linearized methods do not identify. Similarly, studies [68,69,70] state that these approaches are especially suitable when modeling complex systems with multiple inputs and unclear dependencies.

Knowing the precise order of significance of individual input variables allows for more effective experiment design or optimization of maintenance strategy in industrial practice. This approach is also supported by other experimental and model works in the field [27,33,57], which document that quality analysis of input importance leads to resource savings, better decisions, and more targeted interventions in planning monitoring or predictive maintenance.

5.2. Proposal for Integration of ML Models into Industrial DSS

Decision support systems (DSS) are computer systems designed to help managers and users make more informed decisions. They do this by collecting, analyzing, and visualizing complex data, and also enable the simulation of various scenarios, thereby improving the quality and speed of decision-making processes. The practical application of the proposed ML models offers wide possibilities for implementation in the area of monitoring and management of conveyor systems, while specific use depends on the needs of the given enterprise.

The created models for predicting the impact resistance of conveyor belts open up wide possibilities for practical application. They can be easily integrated into existing intelligent monitoring and decision support systems in an industrial DSS environment. These systems are crucial for effective predictive maintenance and full utilization of the potential of Industry 4.0 [71,72,73,74,75,76].

The integration of ML models into industrial control structures would enable continuous collection and processing of sensor data in real-time. This includes, for example, measurements of impact forces, belt strength, or various operating parameters [74,75,77]. Data would typically be collected using standard industrial communication protocols, such as Open-Platform Communications Unified Architecture (OPC UA) or Message-Queuing Telemetry Transport (MQTT), which are designed for robust and efficient information transfer in Industrial Internet of Things (IIoT) environments [78]. Subsequently, data would be pre-processed directly on-site, for example, by noise filtering, normalization, or aggregation, to be ready for trained models.

For fast processing and latency minimization, models with low computational demands could run directly on edge devices, such as industrial computers or programmable logic controllers with extended capacity, located near the belts. This is critical for dynamic industrial applications [74,79]. For more complex analyses or retraining of models, or for advanced functions such as a digital twin of a conveyor belt [28,77], aggregated data would be transferred to centralized cloud platforms or local data centers. Model outputs—for example, Remaining Useful Life (RUL), risk level of failure, or specific recommendations for maintenance—would be processed by the DSS module. Subsequently, they would serve for automated generation of recommendations for service interventions, early warnings before critical values are exceeded, or optimized maintenance planning [68,69,78,80,81].

Key to effective management is a clear and immediate presentation of these data. It would be realized through intuitive dashboards, which are part of existing HMI (Human–Machine Interface) or SCADA (Supervisory Control and Data Acquisition) systems, or as specialized web applications. These interfaces, supplemented by alarm systems (e.g., SMS notifications, email alerts, or visual/audible signals), would allow operators to quickly identify risky situations and make informed decisions [72,74]. Overall, integrating ML solutions into DSS provides operators and managers with objective real-time decision-making bases, leading to more efficient resource utilization and reduced operating costs. Early risk identification thus allows for more precise maintenance planning and a reduction in unplanned downtimes, which fundamentally increases the reliability and effectiveness of conveyor system operation [68,69,80]. However, it is important to emphasize that the practical implementation of these advanced systems requires a detailed analysis of specific operating conditions and close cooperation with experts in automation, IT (Information Technology) infrastructure, and maintenance. This also includes addressing challenges associated with integration into often older IT/OT (Operational Technology) infrastructure, ensuring robust cybersecurity of data and systems, as well as the potential use of explainable artificial intelligence (XAI) to increase user trust in model decisions in critical industrial applications [82,83,84]. Thorough consideration of the specifics of each industrial environment is essential for successful deployment.

6. Conclusions

Various machine learning methods were used to analyze and predict the impact resistance of rubber–textile conveyor belts. Based on experimental testing and the processing of the obtained data, four models were created. Comparing these selected machine learning methods and tools provides a deeper understanding of their ability to model the complex relationships between variables in practice. The study’s results thus not only confirm previous assumptions, but also provide precise, quantifiable recommendations for further research and industrial applications. Such a comparison and precise determination of the order of importance of individual inputs extends the current state of knowledge in the area of conveyor-belt impact resistance. The obtained results thus provide a more detailed insight into the significance of specific parameters, which ultimately increases the accuracy and effectiveness of prediction and optimization of maintenance and lifespan of conveyor belts in real operating conditions.

The knowledge gained can serve as the basis for the development of intelligent monitoring and predictive maintenance systems in industrial practice, increasing the reliability and efficiency of conveyor system operation.

7. Future Study Opportunities and Limitations

New and promising machine learning algorithms suitable for investigating the impact resistance of conveyor belts include, for example, deep learning (DL), which has proven itself in the processing of image and sensory data, and which can also be beneficial in damage analysis and evaluation of the impact resistance of belts. Furthermore, there are Reinforcement Learning (RL) algorithms that bring improvements in the adaptability and reliability of models in dynamic and complex conditions, which can be useful in the control and optimization of industrial processes. A combination of multiple algorithms (ensemble or hybrid approaches) is also significant, as it can increase the accuracy and robustness of predictions. Their effective application should include the expansion of input data with other operational and sensory variables that can increase the reliability of predictions in real industrial conditions. However, it is important to realize that the deployment of ML algorithms in real operations requires compatibility with existing infrastructure and processes, which can be technically and organizationally challenging.

Although the results of this study demonstrated a high level of accuracy and practical applicability of the proposed machine learning models in predicting the impact resistance of rubber–textile conveyor belts, it should be emphasized that the experimental part was performed on only three types of belts with identical carcass structure. This limitation may limit the general validity and applicability of the results to a wider range of belts used in different industrial conditions. Therefore, it is necessary to expand the research in the future to include other types of belts with different configurations and material properties in different operating and environmental conditions, in order to ensure the universality of the proposed models. This approach will contribute to a better understanding of the behavior of conveyor belts under impact loads and increase the practical value of the knowledge gained. For confirmation of the models’ robustness, further verification with larger datasets and diverse characteristics of conveyor belts and different operational parameters is recommended. This will allow for verifying the universality and reliability of the solution in real applications across various industrial environments.

Author Contributions

Conceptualization, M.A., D.M. and A.G.; Methodology, M.A. and A.G.; Software, M.A.; Validation, M.A., Z.K. and A.G.; Formal analysis, A.G. and D.M.; Investigation, M.A. and A.G.; Resources, A.G.; Data curation, M.A. and Z.K.; Writing—original draft preparation, M.A., A.G. and D.M.; Writing—review and editing, A.G. and D.M.; Visualization, M.A.; Supervision, D.M.; Project administration, D.M.; Funding acquisition, D.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

This work was financially supported by the Slovak Grant Agency under the grants APVV-23-0342, VEGA 1/0728/24 and VEGA 1/0191/24.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Rubber–textile conveyor belt structure (A—carcass; B—top cover layer; C—bottom cover layer; D—textile plies).

Figure 2 Schema of test equipment.

Figure 3 Algorithm of the tests performed.

Figure 4 Impactor (detail of the fixing of the spherical and pyramidal impactors).

Figure 5 Significant damage.

Figure 6 Principle of model creation.

Figure 7 Prediction vs. Reality (regression model).

Figure 8 Decision tree regression model.

Figure 9 Prediction vs. Reality (decission tree regression model).

Figure 10 Sensitivity of Random Forest Model Performance.

Figure 11 Importance of variables—random forest model.

Figure 12 Prediction vs. reality (random forest model).

Figure 13 ANN model.

Figure 14 Importance of variables—ANN model (output from the program R).

Figure 15 Prediction vs. reality (ANN model).

Table 1

The basic parameters of conveyor belts.

Belt Type	CB Carcass	Cover Layer Category	Number of Textile Plies	CB Carcass Weight(kg·m−2)	CB Carcass Thickness(mm)	Thickness of Cover LayersUpper + Lower(mm)	CB Tensile Strength(N·mm−1)
P630/3, 4 + 2	polyamide fibers in warp and weft	AA-abrasion resistant	3	5.0	4.7	4 + 2	630
P1000/4, 6 + 4			4	6.0	5.4	6 + 4	1000
P2000/4, 10 + 6			4	9.5	9.1	10 + 6	2000

Table 2

Overview of input and output variables.

Variables	Description
Dependent Variables
H	Impact height of the hammer (m)
W	Weight of the falling hammer (kg)
I	Type of impactor of the hammer (spherical, pyramidal)
S	Conveyor belt strength (N·mm−1)
Independent Variables
F	Impact force (kN)

Table 3

Regression Model—Impact Force (α = 0.05).

Parameters	Coefficients	Standardized Coefficients Beta	t Stat	VIF	p-Value
Model: Impact Force (p-value < 10−16), R2 = 0.936
Intercept	−16.21	1.5 × 10−17	−21.447		<10−16
H	9.533	0.910	56.586	1.03	<10−16
W	0.254	0.496	29.723	1.11	<10−16
I	0.908	0.057	3.464	1.09	0.0006
S	0.0018	0.131	8.282	1.00	<10−16

Table 4

Overview of rules.

Rule	ImpactForce (kN)	Number of Samples	Rule	ImpactForce (kN)	Number of Samples
H < 0.5 m	6.57	45 (17%)	1.1 ≤ H < 1.7, W < 85 kg	19.78	26 (10%)
0.5 ≤ H < 1.1 m, W < 75 kg	10.36	33 (13%)	1.1 ≤ H < 1.7, W ≥ 85 kg	24.11	12 (5%)
0.5 ≤ H < 1.1 m, W ≥ 75 kg	15.78	25 (10%)	1.7 ≤ H < 2.1 m	26.45	24 (10%)
1.1 ≤ H < 1.9 m, W < 65 kg	15.01	35 (13%)	H ≥ 2.1 m, W ≥ 65 kg	29.99	26 (10%)
H ≥ 1.9 m, W < 65 kg	19.92	34 (13%)	X	X	X

Table 5

Top 10 Prediction Errors Compared to Real Values.

No.	H (m)	W (kg)	S (N·mm−1)	I	F (kN)	Fp (kN)	Error
1.	1.6	100	2000	0	30.718	26.656	4.452
2.	1.6	100	1000	0	30.026	25.981	4.045
3.	2.0	80	1000	0	34.451	30.543	3.908
4.	2.0	90	2000	0	31.632	28.286	3.346
5.	2.6	60	2000	1	26.103	23.647	2.456
6.	2.6	60	2000	0	25.775	23.325	2.450
7.	2.4	80	630	1	28.422	25.986	2.436
8.	2.2	90	630	0	28.647	26.289	2.357
9.	2.4	80	1000	0	32.458	30.217	2.241
10.	1.2	50	630	0	9.229	11.463	2.234

Note: Fp—predicted values of impact force, F—empirical values of impact force.

Table 6

Prediction of impact force (n = 10 measurements).

Input Variable				Prediction—Impact Force (kN)
H (m)	W (kg)	I	S (N·mm−1)	Regression Model	Decision Tree Regression Model	Random Forest Model	ANN Model
0.5	95	Sphere	1000	14.462	15.780	10.429	12.706
1.1	70	Sphere	1000	13.828	19.575	15.645	16.020
1.5	65	Sphere	2000	18.133	19.575	15.844	18.219
1.8	85	Sphere	630	23.662	25.950	22.279	24.034
2.1	55	Sphere	630	18.897	19.918	14.721	16.405
0.5	95	Pyramid	1000	15.357	15.780	10.637	13.934
1.1	70	Pyramid	1000	14.736	19.575	15.796	16.826
1.5	65	Pyramid	2000	19.041	19.575	15.939	19.049
1.8	85	Pyramid	630	24.570	25.950	22.189	24.314
2.1	55	Pyramid	630	19.805	19.918	14.646	16.566

Table 7

Final summary of model accuracy.

Parameter	Regression Model	Decision Tree Regression Model	Random Forest Model	ANN Model
MSE	2.901	9.172	1.76	0.248
MAE	1.328	2.442	1.01	0.383
RMSE	1.703	3.029	1.22	0.498
R2	0.952	0.848	0.963	0.996

Table 8

Final summary of models.

Model	Prediction Accuracy	Interpretation	Robustness	Complexity
Regression model	Medium	Simple	Low	Low
Decision tree regression model	Lower	Simple	Low	Low
Random forest model	High	Medium complex	High	Medium
ANN model	Highest	Complex	High	High

Table 9

Characteristics of residuals.

Rezidual	Label of Residual	Mean	Median	Variance	Standard Deviation	Max–Min	Shapiro–Wilk Test (p-Value)
FE − FR	EE,R	0.389	0.392	2.679	1.637	8.652	0.531
FE − FT	EE,T	0.132	0.099	9.643	3.015	13.612	0.506
FE − FRF	EE,RF	0.099	0.015	1.993	1.412	9.427	0.029
FE − FANN	EE,ANN	0.056	0.037	0.452	0.672	6.144	2 × 10−11

Note: F_E—experimental value of impact force, F_R—theoretical value of impact force obtained by the regression model, F_T—theoretical value of impact force obtained by the regression tree, F_RF—theoretical value of impact force obtained by the random forest model, F_ANN—theoretical value of impact force obtained by the ANN model.

Table 10

Residual Test Results (α = 0.05).

Rezidual	Normal Distribution	Test	p-Value	Shift in Residuals from Zero	Interpretation
EE,R	Yes	t-test	0.013	Yes (kladný)	Systematic overestimation of results
EE,T	Yes	t-test	0.655	No	No systematic shift in mean of residuals
EE,RF	No	Wilcoxon	0.716	No	No systematic shift in median of residuals
EE,ANN	No	Wilcoxon	0.767	No	No systematic shift in median of residuals

Table 11

Residual variance Testing (α = 0.05).

Difference	Test	F Value	p-Value	Significance of Differences	Interpretation
EE,R vs. EE,T	F-test	0.28	6.9 × 10−11	Yes	The tree has much greater variance of errors than regression
EE,R vs. EE,RF	Levene	2.97	0.086	No	The difference in errors is not statistically significant
EE,R vs. EE,ANN	Levene	62.65	1.2 × 10−13	Yes	Regression has much greater variance of errors than ANN
EE,T vs. EE,RF	Levene	55.12	2.4 × 10−12	Yes	The tree has much greater variance of errors than random forest
EE,T vs. EE,ANN	Levene	132.45	<10−16	Yes	The tree has extremely greater variance of errors than ANN
EE,RF vs. EE,ANN	Levene	38.05	3 × 10−9	Yes	Random forest has much greater variance of errors than ANN

Table 12

Importance of input variables.

	Importance of Variables—Order
Model	Regression Model	Decision Tree Regression Model	Random Forest Model	ANN Model
Impact height	1.	1.	1.	1.
Weight of the hammer	2.	2.	2.	3.
Impactor of the hammer	4.	Insignificant	Insignificant	Insignificant
CB Strength	3.	Insignificant	3.	2.

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