Next Article in Journal
The Influence of Heat and Surface Treatment on the Functional Properties of Ti6Al4V Alloy Samples Obtained by Additive Technology for Applications in Personalized Implantology
Previous Article in Journal
System for the Acquisition and Analysis of Maintenance Data of Railway Traffic Control Devices
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Application of Machine Learning Algorithms for Predicting the Dynamic Stiffness of Rail Pads Based on Static Stiffness and Operating Conditions

LADICIM (Laboratory of Materials Science and Engineering), E.T.S. de Ingenieros de Caminos, Canales y Puertos, University of Cantabria, Av./Los Castros 44, 39005 Santander, Spain
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(15), 8310; https://doi.org/10.3390/app15158310
Submission received: 4 July 2025 / Revised: 23 July 2025 / Accepted: 24 July 2025 / Published: 25 July 2025

Abstract

The vertical stiffness of railway tracks is crucial for ensuring safe and efficient rail transport. Rail-pad dynamic stiffness is a key component influencing track performance. Determining the dynamic stiffness of rail pads poses a challenge because it depends not only on the material and geometry of the rail pad but also on the testing conditions, due to the non-linear material response. To address this issue, a methodology is proposed in this paper to estimate dynamic stiffness using static stiffness measurements. This approach enables the prediction of dynamic stiffness for different situations from a single laboratory test. This study further examines whether this correlation remains valid for different types of rail pads, even when their mechanical behavior has been degraded by temperature, wear, or chemical agents. Experiments were conducted under varying temperatures and on rail pads that underwent mechanical and chemical degradation. The analysis assesses the validity of the static-to-dynamic stiffness correlation under degraded conditions and investigates the influence of each testing condition on the ability to estimate dynamic stiffness from static stiffness and operational parameters. The findings provide insights into the reliability of this predictive model and highlight the impact of degradation mechanisms on the dynamic behavior of rail pads. This research enhances the understanding of rail pad performance and offers a practical approach for evaluating dynamic stiffness. By considering all of the variables used in the analysis, the approach achieves R2 values of up to 0.99, which carries significant implications for track design and maintenance.

1. Introduction

Railway systems are a key global player in the transition toward sustainable transportation, playing a crucial role in the transportation of goods and passengers from an environmental perspective. Despite carrying 9% of the world’s freight and passengers, rail accounts for only 3% of emissions from the transport sector [1]. This highlights its high energy efficiency and lower carbon footprint compared to other modes of transport, positioning it as a strategic tool to promote a modal shift toward low-carbon solutions. Furthermore, rail not only mitigates the impact of climate change but also strengthens economic and social sustainability by providing a reliable, safe, and scalable means of transportation to meet the present mobility needs.
Given that railways are considered a strategic mode of transportation, it is essential to design an optimized maintenance and operation strategy to ensure their long-term sustainability and efficiency. In this context, Pita et al. [2] highlight the impact of vertical track stiffness on the degradation of high-speed railway lines, emphasizing the importance of rail pads as key components for optimizing stiffness and reducing infrastructure wear. Rail pads, located between the rail and the sleeper, are key components in railway infrastructure designed to provide cushioning and support, as seen in Figure 1. Their stiffness plays a critical role in determining the vertical stiffness of the track, which is essential for ensuring stability and performance. In addition to this, rail pads are crucial for absorbing vibrations and protecting the infrastructure from the damaging effects of dynamic loads generated by rail traffic [3].
Rail pads, typically made of polymeric materials, exhibit highly non-linear behavior, with mechanical properties dependent not only on the type of material used but also on the applied load conditions. Additionally, the polymeric materials used in these components are not entirely stable over time. External factors, such as temperature, mechanical degradation as a consequence of fatigue, or chemical deterioration due to UV or hydrocarbons exposure, can significantly modify their mechanical properties, reducing their effectiveness in vibration damping and wear resistance [4,5,6]. In this context, some authors, such as Padhi et al. [7], have analyzed how the dynamic stiffness and damping capacity of rail pads vary with factors such as temperature, load frequency, and material wear. Similarly, Wei et al. [8] analyzed, in their study, how the stiffness of rail pads is affected by temperature variations, especially at certain frequencies, highlighting the importance of proper material property selection to optimize performance. Other authors have studied the effects of chemical degradation; for example, there is literature on the effect of UV light on polymeric materials. In this regard, Youn et al. [9] observed a significant reduction in the contact angle, an increase in cracking, and the formation of a calcareous layer on the surface. This implies a loss of electrical resistance and changes in the mechanical properties of the material. Likewise, Sainz-aja et al. [10] examined how exposure to hydrocarbons affects the mechanical properties of rail pads, demonstrating that hydrocarbon exposure has a significant effect on dynamic stiffness, especially in those made of EPDM. Rivas et al. [11] present a systematic and quantitative characterization of how mechanical aging, UV radiation, and hydrocarbon exposure affect the dynamic stiffness of EPDM and EVA rail pads and demonstrate how these variables significantly influence that stiffness.
This evidence shows that evaluating the dynamic behavior of these type of materials under different possible operating conditions and after exposure to degradation factors is a challenge that requires a large number of experimental tests, as well as subsequent analysis, representing a considerable consumption of resources. For instance, Carrascal et al. [12] used static tests at different temperatures and dynamic tests also at different temperatures, with 1000 sinusoidal load cycles ranging from 20 to 95 kN at a frequency of 5 Hz, to demonstrate that pads made of thermoplastic elastomers exhibit stiffness variations due to cyclic loading from train passage and wear factors. Additionally, [13] carried out a wide-band experimental characterization of Vossloh 300 rail pads and integrated it into a 3D explicit LS-DYNA model simulating a high-speed wheel running over a corrugated rail. They concluded that design models and maintenance plans must incorporate the frequency- and environment-dependent variation in pad stiffness and damping to predict in-service behavior reliably. Until now, it had not been possible to develop a model capable of estimating the dynamic stiffness of rail pads for different combinations of operating and degradation conditions. Recent advances in machine learning enable the proposal of a model capable of predicting dynamic stiffness based on static stiffness, operating conditions, and degradation effects. This type of approach has been successfully used by other authors, such as Ferreño et al. [14], who employed gradient boosting algorithms to predict the mechanical properties of rail pads under service conditions, considering factors such as temperature, frequency, and toe load. Other authors’ recent contributions extend this line of work. Guillén et al. [15] optimize sensorized rail pads for real-time monitoring and predictive maintenance. Mohammadzadeh et al. [16] show, using machine learning algorithms, a strong correlation between vertical ballast stiffness and track longitudinal. Malekjafarian et al. [17] propose an ANN-based drive-by approach that detects track-layer stiffness loss from in-service train accelerations, and Ramos et al. [18] develop random forest models that reliably predict long-term permanent deformation of ballasted tracks, offering a decision-support tool for life-cycle maintenance. Carrascal et al. [12] show that quantifying dynamic stiffness across all operating scenarios requires an extensive laboratory-testing campaign, whereas Ferreño et al. [14] demonstrate how the mechanical properties of rail pads can be accurately estimated through machine learning models. Other recent contributions, such as by Guillén et al. [15], Mohammadzadeh et al. [16], Malekjafarian et al. [17], and Ramos et al. [18], confirm the potential of ML to capture complex relationships within railway infrastructure—from real-time monitoring of sensorized rail pads and the correlation between ballast stiffness and track geometry to drive-by detection of stiffness loss and long-term prediction of permanent settlement. Taken together, these studies reveal a clear shift toward being data-driven, ML-based approaches that can predict dynamic stiffness more accurately and thereby optimize both design and maintenance strategies for railway superstructures.
This study aims to develop a machine learning (ML) model to estimate the dynamic stiffness of rail pads using minimal information, overcoming the limitations associated with the need to perform specific tests for each combination of frequency, toe load, and load range. In the case of static stiffness, it is possible to calculate its values for different conditions of toe load and load range with a single laboratory test. This work proposes using static stiffness under different test conditions as a basis for estimating dynamic stiffness values, resulting in a significant reduction in the number of laboratory tests required for dynamic stiffness characterization. To ensure a generalizable model, a wide range of case studies were evaluated, which involved performing a total of 1080 dynamic stiffness tests and only 30 static stiffness tests. Specifically, three types of rail pads were selected, covering a broad spectrum of stiffnesses, as detailed in Section 2 (Materials). The test conditions followed the values specified in the standards [19,20], and the pads were evaluated in the following different states: new, tested at high and low temperatures, and aged under mechanical and chemical conditions, as described in Section 3 (Methods). The results of the predictive models, along with their quality depending on the parameters used, are presented in Section 4 (Results). Finally, the conclusions derived from this study are summarized in Section 5 (Conclusions).

2. Material

Three different types of commercial railway pads, supplied by the company PANDROL, were used, which are shown in Figure 2. Henceforth, they are referred to as pads A, B, and C.
  • Pad A: Rail pad manufactured from EPDM (ethylene propylene diene monomer), with circular protrusions measuring 2.5 mm in height and a total thickness of 11 mm;
  • Pad B: Rail pad manufactured from EPDM (ethylene propylene diene monomer), with circular protrusions measuring 1.5 mm in height and a total thickness of 11 mm;
  • Pad C: Rail pad manufactured from EVA (ethylene vinyl acetate), with circular protrusions measuring 2 mm in height and a total thickness of 10 mm.

3. Methods

To ensure that the results of this study are generalizable and cover a wide range of conditions, a total of 10 experimental scenarios were defined, as detailed in Table 1. For each of these scenarios, the following two types of tests were conducted: static and dynamic. In total, 30 static stiffness tests and 1080 dynamic stiffness tests were performed, as detailed in Section 3.2.

3.1. Test Conditions

The three types of pads were subjected to different exposure conditions, as follows: non-degraded pads tested at various temperatures, mechanical degradation, and chemical degradation. All degradation tests were conducted at 20 °C, with tests on non-degraded pads at 20 °C serving as the reference condition. Table 1 summarizes the different conditions imposed.
  • Tests at various temperatures were conducted to simulate the following three possible scenarios: reference temperature (20 °C), low temperature (0 °C), and high temperature (50 °C);
  • Mechanical degradation occurs as a consequence of the continuous passage of trains and was experimentally reproduced by applying cyclic loads on the rail pad under typical high-speed loading conditions, namely, a load amplitude of 31.5 kN, pre-load of 18 kN, and frequency of 5 Hz. The level of damage was controlled through the number of cycles applied considering the following three levels: 250,000; 750,000; and 2,000,000 cycles;
  • Chemical conditions aimed to simulate the effect induced by different agents to which the rail pad are potentially exposed during their in-service life on the track. Two types of aging were imposed, exposure to ultraviolet light and to hydrocarbons. In both cases, the effect was analyzed at 100 and 500 h of exposure.

3.2. Mechanical Characterization: Static and Dynamic Stiffnesses

Stiffness tests were performed with a universal servo-hydraulic testing machine equipped with a load cell with a capacity of ±100 kN, as shown in Figure 3. The deformation of the rail pads was measured by 4 linear variable differential transformers (LVDTs) located on a metallic base that mimicked the geometry of the sleeper. The loads were applied to the rail pads by means of a UIC60 rail sample, which was connected to the test machine by means of a ball joint to ensure the verticality of the applied load. During the stiffness tests, the vertical load and rail pad deformation were recorded, the latter being the average of the four LVDTs.
As defined in standards EN 13481-2 [19] and EN 13146-9 [20], the rail pads’ static stiffness was determined by applying three loading and unloading ramps between 1 and 90 kN, with a load rate of 2 kN/s. The dynamic stiffness was obtained by applying 1000 sinusoidal load cycles to the rail pad. The force–displacement curves that had to be recorded to determine the static and dynamic pad stiffnesses are shown in Figure 4a,b, respectively. The static stiffness, kst, was obtained as the ratio between the load range and the displacement range during the last load ramp, as shown in Figure 4, and written as Equation (1), as follows:
k s t = F f i n a l F i n i t i a l D f i n a l D i n i t i a l
The dynamic stiffness of the rail pads was obtained as the ratio between the average load range and the average displacement range of the last 100 cycles of each test, written as Equation (2), as follows:
k d y n = F m a x ¯ F m i n ¯ D m a x ¯ D m i n ¯

3.3. Machine Learning Analysis

Three types of models were designed to predict the dynamic stiffness. The key difference lies in the inputs used. The first model considers the inputs’ static stiffness, frequency, toe load, and load amplitude. The second model includes the kind of rail pad as an additional input. The last model includes, additionally, information regarding the variables corresponding to the test conditions mentioned in Table 1. For each model type, various ML algorithms were evaluated to determine their predictive performance. The selection of algorithms in machine learning depends significantly on the characteristics of the dataset, as different types of tasks (classification, regression, and clustering) and dataset properties (size, balance, and type of variables) can favor the performance of certain algorithms over others, as can be seen in other studies [14,16,17,18,21]. Some of the most widely used models are as follows:
  • Multiple Linear Regression (LR): predicts a dependent variable using two or more independent variables under the assumption of a linear relationship between them;
  • Lasso (L1 Regression): A linear regression model that introduces an L1 penalty in the cost function. This penalty forces some coefficients to become exactly zero, effectively performing feature selection and helping to reduce overfitting;
  • Ridge (L2 Regression): A linear regression model that adds an L2 penalty to the cost function. This penalty shrinks the magnitude of coefficients, reducing variance and mitigating multicollinearity without setting any coefficients to zero. Unlike Lasso, it does not eliminate features entirely;
  • k-Nearest Neighbors (KNN): classifies a sample by assigning it the most common label or value among its k-nearest neighbors within the feature space;
  • Decision Tree (DT): a tree-based structure where each node represents a feature, each branch represents a decision, and each leaf represents an outcome;
  • Random Forest (RF): a technique that combines multiple decision trees to enhance prediction accuracy by reducing overfitting and increasing robustness;
  • Gradient Boosting (GB): a sequential method that builds predictive models, where each iteration focuses on correcting the errors of the previous model to improve prediction accuracy.
Validation involves dividing the dataset into training and testing subsets to accurately assess the model’s predictive performance. The selected machine learning algorithms were chosen due to their capability to adapt effectively to the specific characteristics of the dataset, thereby optimizing the results. The quality of each predictive model is evaluated by comparing its predicted outputs against the actual values, using performance metrics such as the following:
  • Root Mean Squared Error (RMSE): A metric that measures the average difference between the values predicted by a model and the actual values. It is calculated by taking the square root of the average of the squared errors;
  • Mean Absolute Error (MAE): a metric that measures the average of the absolute differences between predictions and actual values;
  • Coefficient of Determination (R2): measures how well a model explains the variability of the data.
Once the highest-performing algorithms were selected, we fine-tuned their hyper-parameters through a grid search implemented with GridSearchCV (part of the scikit-learn package). Grid search builds a cartesian product of user-defined candidate values, for example, number of trees {100, 300, 500} and maximum depth {None, 10, 20}, in a random forest and evaluates every combination via cross-validation. This procedure allows for finding the best combination of hyperparameters from a predefined set.

4. Results

Figure 5 shows the relationship between static stiffness and dynamic stiffness for the three types of rail pads studied. The graph reveals a positive and mostly linear correlation between the two variables, with some scatter and outliers that will be analyzed later.
Type A rail pads, marked with red circles, show a more concentrated distribution, with static stiffness values ranging from 15 to 70 kN/mm and dynamic stiffness between 15 and 130 kN/mm. The relationship between the two stiffnesses is primarily linear under all conditions. Type B rail pads (green triangles) show greater dispersion, with static stiffness ranging from 20 to 165 kN/mm and dynamic stiffness values reaching up to nearly 500 kN/mm. Although a linear trend is also observed, the variability in the relationship between static and dynamic stiffnesses is more pronounced, indicating less predictable behavior compared to Type A rail pads. Finally, the Type C rail pads (blue squares) exhibit the widest range in both static stiffness (75 to nearly 300 kN/mm) and dynamic stiffness (125 to 925 kN/mm). For this type of rail pad, an increasing trend in dynamic stiffness with static stiffness is observed. However, unlike Type A and B pads, the relationship appears to follow a non-linear pattern and shows significant dispersion under certain conditions.
The results suggest that Type C rail pads, which is the stiffest family, exhibit greater variability and non-linear behavior in the relationship between static and dynamic stiffnesses, in contrast to Type A pads, the least stiff, where dynamic stiffness depends almost exclusively on static stiffness, showing a linear relationship with low variability. Type B pads, with intermediate stiffness, display predominantly linear behavior and greater dispersion than Type A pads, although without reaching the non-linearity and variability observed in Type C pads. These results indicate that additional variables beyond static stiffness influence the dynamic stiffness of all rail pads, with their impact varying depending on the pad type. This suggests that the effects of mechanical and environmental conditions differ depending on the geometry and material of the rail pad. Based on the observations and previous discussions, different ML models have been trained and evaluated to predict dynamic stiffness.
In the initial analysis, only the operating conditions were used as input variables, which include frequency (Hz), toe load (kN), and amplitude (kN), along with static stiffness (kN/mm). Information regarding the test conditions listed in Table 1 and the type of rail pad was excluded from this initial analysis.
Table 2 collects the mean squared error, mean absolute error, coefficient of determination, and root mean squared error. These metrics were applied to the following models: linear regression, Lasso, Ridge, RF, KNN, and GB.
Subsequently, an independent model was created for each type of rail pad, using the same input variables as in the previous model. The objective was to evaluate the model’s ability to generalize. The results are presented in Table 3.
The results indicate that the decision-tree-based models (RF and GB) and KNN achieved the best performances, prompting the exclusion of linear models from further analysis. To enhance performance, the hyperparameters of RF, GB, and KNN were optimized using a grid search (GridSearchCV), revealing that GB outperformed the others. Consequently, GB was selected for subsequent analyses. Finally, Table 4 summarizes the obtained hyperparameters, and Table 5 presents the metrics.
These results indicate a strong ability to predict the dynamic stiffness from the static stiffness and the experimental conditions to which the rail pads were subjected, even without considering the specific degradation condition of the test. Based on the results obtained, it is possible to conclude that dynamic stiffness values can be estimated under any condition solely from the mechanical parameters and static stiffness, with reduced error if the model identifies the type of rail pad as a variable. The model achieves an RMSE of 4.43 kN/mm for pad A, 27.09 kN/mm for pad B, and 21.83 kN/mm for pad C, demonstrating the robustness of the prediction, even when test conditions are not directly considered in the model.
While it has been demonstrated that it is possible to estimate dynamic stiffness based on static stiffness and operating conditions, it is essential to deepen the study of rail pad degradation and its influence on prediction accuracy. Therefore, the type and degree of aging for each variable are now incorporated, independently analyzing the impact of each type of aging on each pad, with the objective of understanding the magnitude and nature of their influence on predictions.
Figure 6 shows a matrix of subplots that provides more detailed insight into the data, which are separated by aging type and show the relationship between static and dynamic stiffnesses for the three types of rail pads (A, B, and C) and the following four test conditions: temperature (Temp), mechanical load (Mec), ultraviolet radiation exposure (UV), and hydrocarbon exposure (HC). Each column corresponds to a type of rail pad, while each row represents a type of test condition.
The main patterns observed in Figure 6 are described below:
  • Temperature Tests (Row 1): Rail pad A shows a linear relationship with little dispersion. In contrast, rail pad B appears to exhibit non-linear behavior under condition T1, while remaining mostly linear under condition T2. For Type C rail pads, under condition T1, the relationship between static and dynamic stiffnesses appears inversely proportional, with dynamic stiffness decreasing as static stiffness increases. This pad shows the greatest variation in behavior among the conditions. The stiffening effect observed under condition T2, related to the behavior of rail pads at low temperatures, has been noted by other authors. For example, Wei et al. [22] observed that rail pads of types WJ-7, WJ-8, and Vossloh 300, used in high-speed railway systems in China, show a significant increase in dynamic stiffness at low temperatures. These rail pads exhibited glass transition temperatures of around −40 °C to −45 °C.
  • Mechanical Fatigue Tests (Row 2): Rail pads A and B show a more linear behavior with limited dispersion. In contrast, Type C rail pads, made from EVA, exhibit greater variability and display non-linear behavior.
  • Ultraviolet-Light Exposure Tests (Row 3):Type A and B rail pads show a predominantly increasing linear trend with low dispersion, while Type C rail pads, again, exhibit greater dispersion in dynamic stiffness values and non-linear behavior.
  • Hydrocarbon Exposure Tests (Row 4): Type A rail pads, as in previous cases, exhibit linear behavior with low dispersion. Type B pads show linear behavior under condition HC2 and non-linear behavior under condition HC1. Type C pads show low dispersion under condition HC1, with increased dispersion under condition HC2. This condition has been studied and explained in greater depth by other authors [10].
In this way, introducing a new dummy variable representing the test condition to which each rail pad was subjected should improve the model’s performance. To achieve this, an optimized GB model was re-evaluated for each rail pad, yielding the results shown in the Table 6 and Table 7.
It can be observed that the test conditions introduce valuable new information into the model, improving the results and reducing error in all cases.
To verify that the models were not overfitting due to the presence of only three frequency values (5, 10, and 20 Hz) while the rest of the variables remained constant, a model was trained using only the data corresponding to a frequency of 5 Hz.
As shown in Table 8, despite being trained on only one-third of the data, the results remain robust, indicating that overfitting is not occurring due to this factor. However, it can be observed that frequency is a particularly relevant variable for rail pad C, which is consistent with the findings from the analyses in Figure 7 and Figure 8. This is evident as the deterioration in the metrics for rail pad C is even greater than the deterioration observed when the test condition variable was removed from the analysis, as shown by comparing the results with those in Table 4.
To interpret the impact of the different variables on the predictions of the optimized GB model, the SHAP (SHapley Additive exPlanations) interpretability technique was used. This methodology quantifies the effect of each feature on the model’s output, providing a clear explanation of how the input variables influence the prediction of the target variable, in this case, dynamic stiffness. The SHAP bar chart presented in Figure 7 shows the average importance of each variable in the model. In this chart, the features are ordered from the highest to lowest impact on the prediction. This allows us to identify the most relevant factors in the estimation of dynamic stiffness.
Based on Figure 7, it can be observed that rail pads A and B, both made of EPDM but differing in the thickness of their protuberances, as explained in Section 2, rely primarily on just two variables to predict their dynamic stiffness. In pad B, the toe load stands out with an importance far exceeding that of the other variables, while in pad A, static stiffness is the dominant factor, closely followed by the toe load. In contrast, the C pad, fabricated from EVA, distributes its predictive weight across a broader set of parameters; aside from toe load and static stiffness, frequency and temperature play prominent roles, with mechanical degradation and exposure to hydrocarbons also showing a modest influence. This differs notably from pads A and B, where those factors contribute to a much lesser extent.
To analyze how each feature influences the prediction, Figure 8 presents SHAP dependence plots for the features used by the model, showing their direct contribution to the predictions of dynamic stiffness. In this way, we can observe the magnitude and direction of the influence of each feature on the prediction and make comparisons across the different rail pads (A, B, and C).
Static stiffness, with values ranging from 15 kN/mm to 275 kN/mm, shows a strong positive linear correlation with dynamic stiffness for rail pads A and C. In contrast, rail pad B exhibits a flatter relationship, consistent with Figure 7, where the impact of static stiffness on the model’s results was much lower than for pads A and C. This suggests a weaker relationship between static and dynamic stiffnesses for this rail pad. Additionally, a set of points between 50 and 100 kN/mm for pad C displays anomalous behavior, corresponding to the effect of a temperature of 0 °C, indicating a clear change in behavior under this condition.
Toe load, which ranges from 1 kN to 25 kN, exhibits a similar behavior across all rail pads, with consistent stiffening as its value increases. Notably, for pad B, in the range of 1 to 18 kN, the effect on the model is between −100 kN and 0 kN. However, for values of 25 kN, it has a strong positive effect on the model, with SHAP values ranging from 125 to 150 kN/mm.
Frequency, ranging from 5 Hz to 20 Hz, has an impact similar to that of toe load, with consistent stiffening of the rail pads as its value increases. However, its influence is smaller in magnitude than that of toe load for all pads, which aligns with Figure 7, where this variable showed less impact than toe load across all pads.
Mechanical degradation shows a stiffening effect on the rail pads as mechanical fatigue increases. Pads subjected to a low number of cycles or no degradation exhibit a negative impact on predicted dynamic stiffness, while as the number of cycles increases, the impact becomes positive. For pad B, this effect is more pronounced, with SHAP values reaching up to +40 kN/mm, suggesting that this pad is more sensitive to the accumulation of mechanical damage. There is also greater dispersion of SHAP values for a high number of cycles across all pads.
Amplitude also has a consistent stiffening effect on all rail pads. However, this impact is much more pronounced for pad B, both in magnitude and dispersion. For this pad, SHAP values range from −60 kN for an amplitude of 15.5 kN to +40 kN for 31.5 kN. In comparison, the impact on the other pads is significantly lower and shows less dispersion, indicating that pad B is more sensitive to changes in amplitude.
Hydrocarbon exposure has a mixed effect. For pads A and B, made of EPDM, an increase in stiffness is observed as the exposure time increases. Conversely, for pad C, made of EVA, the opposite effect is seen, with a softening of the pad as the exposure time increases. Furthermore, for pad C, there is significant data dispersion, which increases with longer exposure times.
Environmental variables, such as ultraviolet light exposure and temperature, have a relatively small impact on the model’s predictions. However, it is interesting to note that, for pad C, temperature has a significant effect at low values, with SHAP values ranging from 30 to 100 kN/mm for exposures at 0 °C. This suggests that, for pad C, low temperatures can have a stiffening effect. For pads A and B, this effect is almost negligible. Regarding ultraviolet light exposure, the SHAP values for all three pads are close to zero across the entire range, indicating a minimal influence of this variable on dynamic stiffness.

5. Conclusions

This study implemented machine learning models to predict the dynamic stiffness of elastomeric (EPDM and EVA) railway pads, which are key components for vibration absorption and for the proper functioning of railway superstructures. These pads are subjected to various mechanical and environmental conditions that influence their stiffness, making accurate evaluation essential for optimizing performance and service life. By leveraging a dataset of 1080 dynamic stiffness tests and 30 static stiffness tests conducted under different operating and degradation conditions, predictive models were developed and optimized, significantly reducing the need for extensive laboratory testing. The degradation conditions studied included different temperatures (0 °C, 20 °C, and 50 °C), mechanical fatigue (250,000; 750,000; and 2,000,000 cycles), and chemical exposure (UV radiation and exposure to hydrocarbons for 100 and 500 h). The results confirm that static stiffness is strongly correlated with dynamic stiffness, though the strength of this relationship varies by pad type and degradation condition. Type A rail pads exhibit a nearly linear correlation with low dispersion, whereas Type B and, especially, Type C pads show increased variability and non-linearity, highlighting the influence of additional factors beyond static stiffness.
Among the machine learning models evaluated, gradient boosting achieved the best predictive performance, with R2 values of nearly 0.99 on the test dataset when all test conditions were included as input variables. The final models demonstrated excellent predictive capabilities, achieving root mean squared errors of 2.69 kN/mm for pad A, 8.35 kN/mm for pad B, and 12.02 kN/mm for pad C. These results highlight the robustness of the method and the importance of considering different degradation mechanisms, such as temperature, mechanical fatigue, and chemical exposure. The predictive models showed a strong capacity to assess variations in dynamic stiffness across different mechanical scenarios and in the presence of various environmental and degradation factors. This accuracy enables the implementation of predictive maintenance strategies, optimizing resource utilization by reducing both inspection frequency and preventive replacements, thus contributing to a more efficient and sustainable management of railway infrastructure.
Furthermore, the SHAP interpretability analysis provided valuable information on the relative influence of various operational and environmental factors on the performance of railway pads. Toe load and static stiffness were identified as the most influential variables in predicting dynamic stiffness. However, temperature had a more significant impact on Type C pads, made of EVA, compared to Pads A and B, made of EPDM, highlighting the importance of proper material selection according to operating conditions. The SHAP interpretability analyses confirmed these findings, showing that low temperatures (0 °C) increase the SHAP value of Pad C’s stiffness by up to 100 kN/mm.
This research provides a scalable, data-driven methodology for evaluating rail pad performance, contributing to the development of more efficient and sustainable railway systems. Although studies such as that by Wei et al. [8] show that factors like temperature induce non-linear variations in the static stiffness of materials other than those used in the present research, such as chloroprene rubber (CR) and thermoplastic polyurethane elastomer (TPE), suggesting that the proposed approach could be extrapolated to a broader range of materials. Future work should extend its application to different materials, investigate the combined effects of multiple degradation agents, study the possibility of extrapolating results among pads, and validate the models under real service conditions.

Author Contributions

Conceptualization, I.R., J.A.S.-A. and V.C.; methodology, I.R., J.A.S.-A., D.F., V.C., I.C., J.C. and S.D.; software, I.R. and J.A.S.-A.; validation, I.R., J.A.S.-A., D.F., V.C., I.C., J.C. and S.D.; investigation, I.R.; data curation, I.R.; writing—original draft, I.R.; writing—review & editing, I.R., J.A.S.-A. and D.F.; funding acquisition, D.F., I.C., J.C. and S.D. All authors have read and agreed to the published version of the manuscript.

Funding

This publication is part of the R&D&I project PID2021-128031OB-I00, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data will be made available upon request.

Acknowledgments

This work was supported by PANDROL, who provided the rail pads for the experimental tests.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

  1. Robinson, M.; Schut, D. Rail as the sustainable backbone of the energy-efficient transport chain: A world view. OIDA Int. J. Sustain. Dev. 2014, 7, 19–30. [Google Scholar]
  2. Pita, A.L.; Teixeira, P.F.; Robuste, F. High speed and track deterioration: The role of vertical stiffness of the track. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2004, 218, 31–40. [Google Scholar] [CrossRef]
  3. Sol-Sánchez, M.; Moreno-Navarro, F.; Rubio-Gámez, M.C. The Use of Elastic Elements in Railway Tracks: A State of the Art Review; Elsevier Ltd.: Amsterdam, The Netherlands, 2015. [Google Scholar] [CrossRef]
  4. Carrascal, I.A.; Pérez, A.; Casado, J.A.; Diego, S.; Polanco, J.A.; Ferreño, D.; Martín, J.J. Development of Metal Rubber Pads for High Speed Railways. WIT Trans. Built Environ. 2018, 181, 487–498. [Google Scholar] [CrossRef]
  5. Li, Q.; Dai, B.; Zhu, Z.; Thompson, D.J. Improved indirect measurement of the dynamic stiffness of a rail fastener and its dependence on load and frequency. Constr. Build. Mater. 2021, 304, 124588. [Google Scholar] [CrossRef]
  6. Sadeghi, J.; Seyedkazemi, M.; Khajehdezfuly, A. Nonlinear simulation of vertical behavior of railway fastening system. Eng. Struct. 2020, 209, 110340. [Google Scholar] [CrossRef]
  7. Padhi, S.; Sharma, S.; Patel, Y. Rail Pad Dynamic Properties: A Review. Lecture Notes in Mechanical Engineering; Springer: Singapore, 2022; pp. 57–70. [Google Scholar] [CrossRef]
  8. Wei, K.; Liu, Z.-X.; Liang, Y.-C.; Wang, P. An investigation into the effect of temperature-dependent stiffness of rail pads on vehicle-track coupled vibrations. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2016, 231, 444–454. [Google Scholar] [CrossRef]
  9. Youn, B.-H.; Huh, C.-S. Surface degradation of HTV silicone rubber and EPDM used for outdoor insulators under accelerated ultraviolet weathering condition. IEEE Trans. Dielectr. Electr. Insul. 2005, 12, 1015–1024. [Google Scholar] [CrossRef]
  10. Sainz-Aja, J.A.; Carrascal, I.A.; Ferreño, D.; Casado, J.; Diego, S.; Pombo, J.; Rivas, I. Impact of hydrocarbon exposure on the mechanical properties of rail pads. Constr. Build. Mater. 2024, 419, 135561. [Google Scholar] [CrossRef]
  11. Rivas, I.; Sainz-Aja, J.A.; Ferreño, D.; Carrascal, I.; Casado, J.; Diego, S. Influence of Aging Conditions on the Dynamic Stiffness of EPDM and EVA Rail Pads. Appl. Sci. 2025, 15, 4394. [Google Scholar] [CrossRef]
  12. Carrascal, I.; Casado, J.; Polanco, J.; Gutiérrez-Solana, F. Dynamic behaviour of railway fastening setting pads. Eng. Fail. Anal. 2007, 14, 364–373. [Google Scholar] [CrossRef]
  13. Xu, J.; Wang, K.; Liang, X.; Gao, Y.; Liu, Z.; Chen, R.; Wang, P.; Xu, F.; Wei, K. Influence of viscoelastic mechanical properties of rail pads on wheel and corrugated rail rolling contact at high speeds. Tribol. Int. 2020, 151, 106523. [Google Scholar] [CrossRef]
  14. Ferreño, D.; Sainz-Aja, J.A.; Carrascal, I.A.; Cuartas, M.; Pombo, J.; Casado, J.A.; Diego, S. Prediction of mechanical properties of rail pads under in-service conditions through machine learning algorithms. Adv. Eng. Softw. 2021, 151, 102927. [Google Scholar] [CrossRef]
  15. Guillén, A.; Guerrero-Bustamante, O.; Iglesias, G.R.; Moreno-Navarro, F.; Sol-Sánchez, M. Design of Sensorized Rail Pads for Real-Time Monitoring and Predictive Maintenance of Railway Infrastructure. Infrastructures 2025, 10, 45. [Google Scholar] [CrossRef]
  16. Mohammadzadeh, S.; Heydari, H.; Karimi, M.; Mosleh, A. Correlation Analysis of Railway Track Alignment and Ballast Stiffness: Comparing Frequency-Based and Machine Learning Algorithms. Algorithms 2024, 17, 372. [Google Scholar] [CrossRef]
  17. Malekjafarian, A.; Sarrabezolles, C.-A.; Khan, M.A.; Golpayegani, F. A Machine-Learning-Based Approach for Railway Track Monitoring Using Acceleration Measured on an In-Service Train. Sensors 2023, 23, 7568. [Google Scholar] [CrossRef] [PubMed]
  18. Ramos, A.; Correia, A.G.; Nasrollahi, K.; Nielsen, J.C.; Calçada, R. Machine Learning Models for Predicting Permanent Deformation in Railway Tracks. Transp. Geotech. 2024, 47, 101289. [Google Scholar] [CrossRef]
  19. UNE. UNE-EN 13481-2:2023 Aplicaciones Ferroviarias. Vía. Requisitos de Funcionamiento para los Conjuntos de Sujeción. Parte 2: Conjuntos de Sujeción Para las Traviesas de Hormigón en vías con Balasto. 2023, AENOR, Madrid. Available online: https://tienda.aenor.com/norma-une-en-13481-2-2023-n0071035?gad_source=1&gclid=CjwKCAjwpbi4BhByEiwAMC8JnVwUiq-CJauwoQ9MpA_i8A3x8S7wYJ8hxRyAEbSqRHbG-ZmKPa_r3xoCR3kQAvD_BwE&gclsrc=aw.ds (accessed on 15 October 2024).
  20. UNE. Aplicaciones Ferroviarias Vía Métodos de Ensayo de los Sistemas de Fijación Parte 9: Determinación de la Rigidez. Available online: https://www.une.org/encuentra-tu-norma/busca-tu-norma/norma/?c=N0064825 (accessed on 27 December 2022).
  21. Kaewunruen, S.; Sresakoolchai, J.; Huang, J.; Zhu, Y.; Ngamkhanong, C.; Remennikov, A.M. Machine Learning Based Design of Railway Prestressed Concrete Sleepers. Appl. Sci. 2022, 12, 10311. [Google Scholar] [CrossRef]
  22. Wei, K.; Yang, Q.; Dou, Y.; Wang, F.; Wang, P. Experimental investigation into temperature- and frequency-dependent dynamic properties of high-speed rail pads. Constr. Build. Mater. 2017, 151, 848–858. [Google Scholar] [CrossRef]
Figure 1. Picture with the main elements of a railway fastening system.
Figure 1. Picture with the main elements of a railway fastening system.
Applsci 15 08310 g001
Figure 2. Pictures showing the different types of rail pads used in this study.
Figure 2. Pictures showing the different types of rail pads used in this study.
Applsci 15 08310 g002
Figure 3. Picture of the stiffness test on a universal servo-hydraulic testing machine.
Figure 3. Picture of the stiffness test on a universal servo-hydraulic testing machine.
Applsci 15 08310 g003
Figure 4. Graphs showing the force–displacement curves from static (a) and dynamic (b) stiffness tests.
Figure 4. Graphs showing the force–displacement curves from static (a) and dynamic (b) stiffness tests.
Applsci 15 08310 g004
Figure 5. Graph showing the relationship obtained between static stiffness and dynamic stiffness for all tests.
Figure 5. Graph showing the relationship obtained between static stiffness and dynamic stiffness for all tests.
Applsci 15 08310 g005
Figure 6. Matrix of subplots with the relationship between static and dynamic stiffnesses for each rail pad and condition.
Figure 6. Matrix of subplots with the relationship between static and dynamic stiffnesses for each rail pad and condition.
Applsci 15 08310 g006
Figure 7. SHAP chart with the impact of each variable on the model predictions: (a) Pad A; (b) Pad B; (c) Pad C.
Figure 7. SHAP chart with the impact of each variable on the model predictions: (a) Pad A; (b) Pad B; (c) Pad C.
Applsci 15 08310 g007aApplsci 15 08310 g007b
Figure 8. SHAP dependence plots of model features and their contribution to dynamic stiffness prediction: (a) static stiffness; (b) toe load; (c) amplitude; (d) frequency; (e) mechanical degradation (cycles); (f) temperature; (g) hydrocarbon exposure; (h) ultraviolet light exposure.
Figure 8. SHAP dependence plots of model features and their contribution to dynamic stiffness prediction: (a) static stiffness; (b) toe load; (c) amplitude; (d) frequency; (e) mechanical degradation (cycles); (f) temperature; (g) hydrocarbon exposure; (h) ultraviolet light exposure.
Applsci 15 08310 g008
Table 1. Summary of conditions imposed on the rail pads.
Table 1. Summary of conditions imposed on the rail pads.
Type of DegradationDegreeCode
Reference conditions20 °CR
Temperature conditions50 °CT1
0 °CT2
Mechanical degradation250,000 cyclesM1
750,000 cyclesM2
2,000,000 cyclesM3
Chemical
degradation
Exposure to UV light100 hUV1
500 hUV2
Exposure to hydrocarbons100 hHC1
500 hHC2
Table 2. Scores on the test set for the general model.
Table 2. Scores on the test set for the general model.
LRLassoRidgeRFKNNGB
MSE (kN2/mm2)4392.754388.434389.191848.434594.292388.02
MAE (kN/mm)45.7445.4345.7221.7337.3726.45
R20.87020.87030.87040.94540.86430.9294
RMSE (kN/mm)66.2766.2466.2542.9967.7848.86
Table 3. Scores on the test set for each type of rail pad.
Table 3. Scores on the test set for each type of rail pad.
LRLassoRidgeRFKNNGB
Pad AMSE (kN2/mm2)55.6458.3855.6130.1555.0331.98
MAE (kN/mm)4.884.664.863.364.603.70
R20.90340.89870.90350.94760.90450.9445
RMSE (kN/mm)7.457.647.455.497.415.65
Pad BMSE (kN2/mm2)1155.61131.61148.35852.63643.88782.77
MAE (kN/mm)28.6928.4128.5615.8014.2914.20
R20.84730.8500.84830.88730.91490.8966
RMSE (kN/mm)33.9933.6333.8829.1925.3727.97
Pad CMSE (kN2/mm2)2796.32736.42782.40992.292378.87725.10
MAE (kN/mm)38.0237.5837.9022.8934.2919.98
R20.83360.83710.83440.94090.85840.9568
RMSE (kN/mm)52.8852.3152.7431.5048.7726.92
Table 4. Parameters obtained after optimization for each of the GB models.
Table 4. Parameters obtained after optimization for each of the GB models.
Learning RateMax DepthMin Samples Splitn EstimatorsSubsample
Pad A0.056107000.6
Pad B0.255126000.7
Pad C0.15158000.7
Table 5. Metrics of the tests for each of the GB models.
Table 5. Metrics of the tests for each of the GB models.
Pad APad BPad C
MSE (kN2/mm2)19.64734.25476.68
MAE (kN/mm)2.7814.0315.28
RMSE (kN/mm)4.4327.0921.83
R20.96650.90300.9716
Table 6. Parameters obtained for each of the GB models.
Table 6. Parameters obtained for each of the GB models.
Learning RateMax DepthMin Samples Splitn EstimatorsSubsample
Pad A0.055108000.6
Pad B0.01528000.8
Pad C0.15108000.8
Table 7. Metrics of the tests for each of the GB models.
Table 7. Metrics of the tests for each of the GB models.
Pad APad BPad C
MSE: kN2/mm27.2569.81144.68
RMSE: kN/mm2.698.3512.02
MAE: kN/mm1.704.859.37
R20.98740.9910.991
Table 8. Metrics of the tests for each GB model across all frequencies and at 5 Hz.
Table 8. Metrics of the tests for each GB model across all frequencies and at 5 Hz.
All Frequencies5 Hz
Pad AMSE: kN2/mm27.2514.58
MAE: kN/mm1.702.67
RMSE: kN/mm2.63.81
R20.98740.9746
Pad BMSE: kN2/mm269.81211.85
MAE: kN/mm4.858.38
RMSE: kN/mm8.3514.55
R20.9910.9720
Pad CMSE: kN2/mm2144.681832.7
MAE: kN/mm9.3736.20
RMSE: kN/mm12.0242.81
R20.9910.8909
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.

Share and Cite

MDPI and ACS Style

Rivas, I.; Sainz-Aja, J.A.; Ferreño, D.; Calzada, V.; Carrascal, I.; Casado, J.; Diego, S. Application of Machine Learning Algorithms for Predicting the Dynamic Stiffness of Rail Pads Based on Static Stiffness and Operating Conditions. Appl. Sci. 2025, 15, 8310. https://doi.org/10.3390/app15158310

AMA Style

Rivas I, Sainz-Aja JA, Ferreño D, Calzada V, Carrascal I, Casado J, Diego S. Application of Machine Learning Algorithms for Predicting the Dynamic Stiffness of Rail Pads Based on Static Stiffness and Operating Conditions. Applied Sciences. 2025; 15(15):8310. https://doi.org/10.3390/app15158310

Chicago/Turabian Style

Rivas, Isaac, Jose A. Sainz-Aja, Diego Ferreño, Víctor Calzada, Isidro Carrascal, Jose Casado, and Soraya Diego. 2025. "Application of Machine Learning Algorithms for Predicting the Dynamic Stiffness of Rail Pads Based on Static Stiffness and Operating Conditions" Applied Sciences 15, no. 15: 8310. https://doi.org/10.3390/app15158310

APA Style

Rivas, I., Sainz-Aja, J. A., Ferreño, D., Calzada, V., Carrascal, I., Casado, J., & Diego, S. (2025). Application of Machine Learning Algorithms for Predicting the Dynamic Stiffness of Rail Pads Based on Static Stiffness and Operating Conditions. Applied Sciences, 15(15), 8310. https://doi.org/10.3390/app15158310

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop