1. Introduction
The basic task of the braking system is to stop the vehicle, reduce its speed, and prevent the vehicle from rolling away when stationary [
1]. Friction disc brakes are a common solution today due to the effectiveness of the braking process, low cost, and high reliability [
2]. Despite the enormous progress in the development of the automotive industry, brake pads are commonly used actuators of braking systems [
3]. Currently, vehicles are becoming heavier and faster, so their operational efficiency must increase [
4]. The wear of pads used in real conditions is characterized by a number of non-linear, multi-dimensional factors [
5]. When braking, the vehicle generates friction force in the actuator systems, which results in a reduction of its kinetic energy. The change in kinetic energy leads to an increase in thermal energy occurring mainly on brake discs and pads [
6].
The mechanisms of wear and degradation of brake discs and pads have received considerable attention in the literature, highlighting the importance of the materials used in production. D.K. Kolluri et al. [
7] examined the effect of graphite particle size on the heating of brake discs. They observed that in composites, the use of small graphite particles compared to large particles improves the thermal properties of discs. The use of a copper-metal matrix for brake pads shows better tribological properties [
8,
9]. New materials are also used. The authors of Ref. [
10] used the newly developed DB-1 material and compared it to the materials currently used. In laboratory tests, they simulated the loads of high-speed trains. The new material has a very good coefficient of friction and is characterized by a lower wear rate due to the fine particles. There are publications in the literature regarding the use of natural material admixtures in effective braking systems. The authors [
11] replaced synthetic fibers with hemp fibers. The use of natural hemp fibers reduced the specific wear rate while obtaining a consistent coefficient of friction for brake pads.
The aspect of accumulating significant heat energy in the braking system can prove very dangerous and shorten the life of brake pads and discs. The overheating phenomenon can occur due to the design of brake discs or pads made of materials with low thermal conductivity.
The paper [
12] compared brake pads made from three asbestos-free composites. These composites contained different proportions of steel fibers 30, 35, and 41% and synthetic materials 11, 6, and 0%. The composite with 41% and 0% showed the best thermal stability and thermal conductivity. The effect of prolonged thermal loads on hardened steel causes a loss of mechanical properties [
13]. As the authors point out, this is due to the material’s susceptibility to tempering after heating. Improving the resistance to tempering is to increase the amount of such chemical elements as molybdenum and vanadium.
Articles [
14] describe the problem of overuse of the braking system, which is caused by an incorrect driving style. Prolonged use of the brake while driving downhill can cause this and contribute to complete pad wear [
15]. This problem is particularly dangerous for trucks with trailers or semi-trailers [
16]. As the authors point out [
17] (p. 1) “… when using the lowest-priced brake discs and brake pads, a substantial reduction in their efficiency can occur if braking intensively or over a long period”. Overheating of the brake system significantly reduces the friction coefficient of the brake pad against the brake disc. This forces an increase in braking force to achieve the same braking torque and results in accelerated wear [
18,
19]. Corrosion of the braking system also adversely affects the vehicle’s braking behavior. The primary corrosion factor is the composition of the brake pad and disc materials [
20]. The formation of corrosion intensifies during frequent changes in humidity and temperature difference between the brake disc and pad, which promotes a reduction in the friction coefficient [
21,
22].
Precision in the installation of new brake pads is also of great importance. When replacing new brake pads, a caliper that has not been cleaned of corrosion can result in faulty seating of the friction element and faster wear and vibration [
23]. The research topics presented above show the complexity of the braking phenomenon and the presence of many factors that affect brake pad wear. It should be noted that most of the research conducted was carried out in the laboratory and not on vehicles in real operating conditions. In the case of managing a fleet of multiple vehicles, the ability to estimate brake pad wear would make it possible to optimize the replacement schedule and identify the causes of rapid brake system wear. Currently, the most commonly used predictor is the vehicle distance traveled [
24].
In a very interesting study [
25], the authors measured the brake pad and disc wear on real objects. The study lasted 2 years, and 20 cars were analyzed. The influence of the type of traffic (urban–urban) and calendar month on the wear of the above-mentioned elements was determined. The study concluded that the wear and tear of the studied brake system components are influenced by the type of vehicle traffic and the season and are significantly statistical. However, the above work did not take into account many operational factors such as the driver’s driving style, kilometers traveled, and vehicle load.
Some researchers have based their brake pad wear estimation results on machine learning methods [
26,
27]. Good results were obtained for XGBoost + Logistic Regression and XGBoost + Deep Recurrent Neural Network—accuracy of 70% and 85%. The disadvantages of these methods are the need to collect a very large number of data and the selection of optimal configurations of processing methods. In the study [
28], frictional thermal energy and car braking analysis were used to determine wear. The results showed that the decisive factor in pad wear is the vehicle’s initial speed.
The Archard equation is a popular method for estimating brake pad wear. Kenneth Ma et al. tried to estimate brake pad wear based on the Archard equation. The estimation error on a real-world car proved to be very large, and as the authors stated “…without access to the associated usage data, accurate validation of the prediction cannot be carried out” [
24] (p. 12). Trucks carrying oversize loads experience frequent changes in the load carried. The weight of the load can vary up to 300% of the vehicle’s weight. Therefore, in this case, estimating brake pad wear is important and should be based on reliable data. A key factor for fleet managers is the use of data stored in the vehicle’s controllers. This allows verification of the driver’s driving style and monitoring of the truck’s operating data. OBD2 On-board vehicle diagnostics installed by manufacturers in each vehicle were used to download the data. Chunyu Yu et al. stated that “It is difficult to describe wear by using general formulas fully in engineering, because the wear characterization is related to several factors that are complex, nonlinear and multidimensional” [
5] (p. 2).
Therefore, the purpose of this publication is to create highly qualitative predictions of brake pad wear based on three stochastic quantitative models based on multiple regression. The matrix of explanatory variables based on real data takes into account the number of vehicle load ranges determined on a 5-point scale and the driver’s style of using the brake pedal determined on a 4-point scale. In this work, “the driver’s style of using the brake pedal” is understood as the number and intensity of pressing the brake pedal.
4. Results and Discussion
4.1. Results of the Initial Grouping of Source Data
The first part of sorting the data was to group them according to the assumptions shown in
Table 1. The results of grouping the data according to the adopted algorithm are shown in
Table 2.
A preliminary analysis of the grouped data is shown in
Table 3, which involves calculating the coefficient of variation value as a measure of dispersion [
29,
31].
The results obtained in the range of less than Var-co. < 10% were eliminated [
32]. Analysis of the results presented in
Table 3 postulates the elimination of the X
9 variable. Therefore, after the first stage of selection, the variables X
i = {X
1, X
2,…, X
8} were used for further model construction.
4.2. Linear Model Estimation and Verification Results
The constructed linear model, Formula (1) for X
i = {X
1, X
2,…, X
8}, was verified at the significance level α = 0.05. The model’s coefficient of determination is
= 0.926 (coefficient of convergence φ
2 = 5.3%). The model explains 92.6% of the variability of the studied trait, this indicates a good fit of the model to the empirical data (
Table 4).
The F statistic, given the truth of the null hypothesis, has an F Snedecor distribution with 8 degrees of freedom of the numerator and 20 degrees of freedom of the denominator. The empirical value of the statistic is F = 44.56, and the corresponding critical level of significance F = 4.413 × 10−11, which is less than the accepted significance level α = 0.05. We therefore reject hypothesis H0 in favor of H1. There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X1, X2,…, X8}. We test the significance of the individual regression coefficients.
The statistic at the truth of the null hypotheses has a Student’s t-distribution with 20 degrees of freedom. The empirical values of the t-Student’s statistic and the corresponding values of the critical level of significance (
p-value) are shown in
Table 4. There are no grounds for rejecting the hypothesis that the model constants ɑ
i = {α
0,…,α
8}\{α
2,α
5} are insignificant, i.e., equal to zero (the values of the critical level of significance for these coefficients are greater than the accepted level of significance α = 0.05). There is no basis for rejecting the hypothesis that the variables X
i = {X
1,…,X
8}\{X
2,X
5} are insignificant. The current model structure is flawed.
Re-Selection of the Linear Model Class
For the new linear model, Formula (1) for X
i = {X
2,X
5}, model fitting was carried out at a significance level of ɑ = 0.05. The coefficient of the model is
= 0.927 (coefficient of convergence φ
2 = 6.8%). The model explains 92.7% of the variation in the studied trait. This shows a good fit of the model to the empirical data (
Table 5).
The F statistic, with the null hypothesis being true, has a Snedecor F distribution with 2 degrees of freedom of the numerator and 26 degrees of freedom of the denominator. The empirical value of the statistic is F = 178.177. The critical level of significance F = 6.65 × 10
−16 is less than the accepted level of significance α = 0.05. We therefore reject hypothesis H
0 in favor of H
1. There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables X
i = {X
2,X
5}. For each coefficient of the regression model α
i, we test the hypothesis of its significance. We verify it with a statistic of Student’s t-distribution with 26 degrees of freedom. The empirical values of the t-Student’s statistic and the corresponding values of the critical level of significance (
p-value) are shown in
Table 5. All the coefficients of the model are significantly different from zero (the values of the critical level of significance are less than the accepted level of significance α = 0.05). There are no grounds for rejecting the hypothesis that all coefficients of the tested model are significantly different from zero. The current structure of the model is correct and is represented by the relation (21) and geometric interpretation in
Figure 2:
4.3. Estimation and Verification Results of Inverted Linear Model
For the inverted linear model, Formula (2) for X
i = {X
1,…,X
8}, the results of parameter estimation are presented in
Table 6.
The constructed linear inverse model was verified at the significance level α = 0.05. The model’s coefficient of determination is 0.961 (coefficient of convergence φ2 = 2.8%). The model explains 96.1% of the variation in the studied trait, this indicates a good fit of the model to the empirical data. The significance of the regression coefficients was tested, and we formulated hypotheses for a linear model. The F statistic, with the null hypothesis being true, has a Snedecor F distribution with 8 degrees of freedom of the numerator and 20 degrees of freedom of the denominator. The empirical value of the statistic is F = 86.598, and the corresponding critical level of significance F = 8.01 × 10−14. The level is less than the accepted significance level α = 0.05. We reject hypothesis H0 in favor of H1. There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X1, X2,…,X8}.
For each coefficient of the regression model, we hypothesize as for a linear model. The empirical values of the t-Student’s statistic and the corresponding values of the critical level of significance (
p-value) are shown in
Table 6. No basis for rejecting the hypothesis that the model constants α = {α
0,…,α
8}\{α
2,α
3} are insignificant, i.e., equal to zero; incorrect model structure.
New Inverse Linear Model
For the new inverted linear model, Formula (2) for X
i = {X
2,X
3}, model fitting was carried out at a significance level of α = 0.05. Based on the verified data, we determine the parameter values of the new inverted linear model (
Table 7).
The model fit was verified at a significance level of α = 0.05. The coefficient of the model is = 0.962 (coefficient of convergence φ2 = 3.5%). Conclusion: The model explains 96.2% of the variability of the studied trait. This shows a very good fit of the model to the empirical data. We hypothesize that the coefficients of the regression model are not significant. The F statistic, with the null hypothesis being true, has a Snedecor F distribution with 2 degrees of freedom of the numerator and 26 degrees of freedom of the denominator. The empirical value of the statistic is F = 355.962, and the corresponding critical significance level F = 1.92 × 10−19. The level is less than the accepted significance level α = 0.05. We therefore reject hypothesis H0 in favor of H1. There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X2,X3}.
For each coefficient of the regression model α
i, we test the hypothesis of its significance. We verify it with a statistic with a Student’s t-distribution with 26 degrees of freedom. The empirical values of the t-Student’s statistic and the corresponding values of the critical level of significance (
p-value) are shown in
Table 7. All the coefficients of the model are significantly different from zero (the values of the critical level of significance are less than the accepted level of significance α = 0.05). There is no basis for rejecting the hypothesis that all model coefficients are significantly different from zero. The current structure of the inverse linear model is correct and is represented by the relation (22) and geometric interpretation in
Figure 3:
4.4. Results of Estimation and Verification of the Power Model
For the power (quasi-linear) model specified by Formula (14), the parameter values are shown in
Table 8.
We will verify the constructed model at the significance level α = 0.05. The model’s coefficient of determination is = 0.941 (coefficient of convergence φ2 = 4.1% φ2 = 4.2%). The model explains 94.1% of the variation of the studied trait, this is a good fit of the model to the empirical data. The significance of the regression coefficients was checked. We put hypotheses as for previous models. The F statistic, with the null hypothesis being true, has an F Snedecor distribution with 8 degrees of freedom of the numerator and 20 degrees of freedom of the denominator. The empirical value of the statistic is F = 56.656, and the corresponding critical level of significance F = 4.62 × 10−12. The level is less than the accepted significance level α = 0.05. We reject hypothesis H0 in favor of H1. There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X1, X2,…,X8}.
For each coefficient of the regression model, we pose hypotheses as for previous models. The empirical values of the Student’s
t-statistic and the corresponding values of the critical level of significance (
p-value) are shown in
Table 8. There are no grounds for rejecting the hypothesis that the model constants ɑ = {α
0,…,α
8}\{α
2} are insignificant, i.e., equal to zero (the values of the critical level of significance for these coefficients are greater than the accepted level of significance α = 0.05). There are no grounds for rejecting the hypothesis that the variables X
i = {X
1,…,X
8}\{X
2} are insignificant. The current model structure is incorrect.
Determination of the New Power Model
The new quasi-linear power model is described by Equation (23):
Estimation of structural parameters for the presented model is provided in
Table 9.
We verify the fit of the model at a significance level of α = 0.05. The coefficient of the model is 0.901 (convergence rate φ2 = 9.6%). The model explains 90.1% of the variability of the studied trait. This indicates a good fit of the model to the empirical data. We hypothesize that the coefficients of the regression model are not significant. The F statistic, given the truth of the null hypothesis, has an F Snedecor distribution with 1 degree of freedom of the numerator and 27 degrees of freedom of the denominator. The empirical value of the statistic is F = 254.865, and the corresponding critical level of significance F = 2.821 × 10−15 is less than the accepted significance level α = 0.05. We reject hypothesis H0 in favor of H1. There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables X2.
For the regression model coefficient α
2, we test the hypothesis of its significance. We verify it with a statistic of
t-Student’s distribution with 27 degrees of freedom. The empirical values of the
t-Student’s statistic and the corresponding values of the critical level of significance (
p-value) are shown in
Table 9. The coefficients of the model are significantly different from zero (the values of the critical level of significance are less than the accepted level of significance α = 0.05). There are no grounds for rejecting the hypothesis that the coefficients of the tested model are significantly different from zero. The current structure of the quasi-linear power model is correct. After simple transformations, we obtain a power model in the form (24) and geometric interpretation in
Figure 4:
4.5. Results of Random Component Analysis of the Models
In the least squares method, as mentioned earlier, the Gauss–Markov assumptions must be met.
The random components of the residuals of the obtained models (21), (22), and (24) were analyzed, and the results are shown in
Table 10.
4.5.1. Results of the Hypothesis of Normality of the Random Components of the Residuals
Hypothesis H0 was set: The random components have a normal distribution—linear model N(0; 2583.179); inverse linear model N(0; 3.83 × 10−7); power model N(0; 0.048).
An analysis of
Table 11 shows that there is no basis for rejecting the hypothesis that the random components of the models have a normal distribution, for the N(0, 2583.179) linear model and the N(0, 0.048) power model. Note that for the inverse linear model N(0; 0.048) value W < W
α, and this means that there are grounds for rejecting hypothesis H
0. The large difference between the distribution of the residuals and the normal distribution may disturb the assessment of the significance of the coefficients of the individual variables of the model. Therefore, the inverted linear model was rejected.
4.5.2. Results of the Hypothesis of Autocorrelation of the Random Components of the Residuals
Autocorrelation is the interdependence of random components and is clearly undesirable. The results are presented in
Table 12.
There is no basis for rejecting hypothesis H0 about the lack of autocorrelation of random components of order one.
4.5.3. Results of the Hypothesis of Randomness of the Components of the Residuals
Hypothesis H
0 was set: The error of the model residuals is random. The data are shown in
Table 13.
The empirical value of the statistic does not fall into the critical area. There is no basis for rejecting the hypothesis H0 that the distribution of the components of the model residuals is random.
4.5.4. Results of the Hypothesis on the Symmetry of the Random Components of the Residuals
The critical area of the test is two-sided, and the results are shown in
Table 14.
The determined empirical value of the statistics is smaller in absolute value t than the critical value tα. There are no grounds to reject hypothesis H0 in favor of hypothesis H1.
4.5.5. Results of the Hypothesis of Homoskedasticity of the Random Components of the Residuals
The results of the Breusch–Pagan test are shown in
Table 15.
Analyzing
Table 15, it should be noted that only the linear model meets the criterion set. The computational value of the χ
2 statistic is less than the critical value of χ
2α. Therefore, there is no basis for rejecting hypothesis H
0 about the constancy of the variance of the model’s residuals. For the power model, the value of χ
2 is greater than the critical value of χ
2α. Therefore, there are grounds to reject the hypothesis of constancy of variance in favor of hypothesis H
1. Therefore, the power model was rejected.
The verification shows that only the linear model described by Equation (21) is characterized by high cognitive quality.
The linear model shows that the explanatory variable X5 has the greatest impact on brake pad wear. The coefficient for X5 is 5 times greater than the coefficient for X2. The correct interpretation of the regression model requires compliance with the ceteris paribus condition, i.e., changing only one explanatory variable. The presented model shows that each braking of the X2 translates into 44 km of brake pad durability. Each increase in kilometers traveled <20% GVW increases the durability of the brake pads by a further 234 km.
4.6. Results of the Evaluation of the Models Used
Based on the created models, brake pad wear was predicted for four example trucks, which are presented in
Table 16.
The forecast results are presented in
Table 16, and they show high compliance between the linear model and the actual wear of brake pads for K1, K2, K3, and K4 trucks. The model is characterized by very low MAPE prediction errors, less than <15%. For K1 and K2 vehicles operating in the same conditions as T
1 … T
29 trucks, the error is less than <5%, which means that MAPE is excellent. For K3 and K4 trucks, which are operated in different conditions and with different loads, the MAPE error was 5.21% and 7.03%, respectively. This result indicates a very good prediction of brake pad wear.