The Wheel–Rail Contact Force for a Heavy-Load Train Can Be Measured Using a Collaborative Calibration Algorithm

Abstract

The wheel–rail contact force is a crucial indicator for ensuring the secure operation of a heavy-load train. However, obtaining the real-time wheel–rail contact force of a heavy-load train is a challenging task as, due to safety considerations, it is not possible to install instrumented wheelsets on heavy-load trains. This work presents a novel approach to quantify the wheel–rail contact force of a heavy-load train, utilizing a cooperative calibration methodology. First, a ground measurement platform for the wheel–rail contact force of a heavy-load train is constructed on a selected rail section. The railway inspection car’s wheel–rail contact force measurement system is fine-tuned using a multilayer perceptron calibration approach, and the ground platform then uses the results to fine-tune the railway inspection car’s examined wheelset. Second, based on actual measured data on the wheel–rail contact force of a heavy-load train, and using the golden jackal optimization algorithm, the multilayer perceptron correction approach is employed to create a data relationship mapping model. This model correlates the corrected data on the wheel–rail contact force obtained from the railway inspection car with the wheel–rail contact force of a heavy-haul train with an axle load of 25 tons, and the precision of the mapping is guaranteed. Finally, by combining the wheel–rail contact force correction method for the railway inspection car and the contact force mapping model between the railway inspection car and the heavy-load train, collaborative calibration of the wheel–rail contact force of the heavy-load train is realized. The experimental results under two working conditions show that this method can realize high-precision, real-time measurement of the wheel–rail contact force of a heavy-load train. For the working condition of a straight-line section, the calibration error was within 1.593 kN, and the MAPE was 0.105%; for the working condition of a curved-line section, the calibration error was 2.344 kN, and the RMSE was 184.72 N.

1. Introduction

For modern rail vehicles, the interaction between the wheels and the rails affects the vehicles’ safety and stability, as well as the life of the railway line. A major manifestation of wheel–rail interaction is force. The term “contact force” refers to the force that occurs between the wheel tread and the rail surface when a train passes. It is mainly composed of vertical force, lateral force, and longitudinal force [1,2,3,4]. The vertical wheel–rail contact force immediately impacts the train’s dynamic reaction, while the lateral force directly affects its derailment coefficient on a curved line. Therefore, to maintain safe and smooth train operation, the wheel–rail contact force must be strictly monitored [5,6,7,8,9]. However, measuring the wheel–rail contact force directly is challenging; thus, it is crucial to develop methods for accurate and constant measurement during train operation [10,11,12,13,14,15,16,17].

The persistent efforts of many scholars have led to numerous studies on wheel–rail contact force measurement, which is done in two different ways: directly or indirectly. The direct method refers to measurements made using an instrumented wheelset or by attaching strain gauges to the rails. However, the complex calibration process and subsequent maintenance work make such methods difficult to apply. The indirect approach involves utilizing the dynamic load recognition method to accurately determine the value of the wheel–rail contact force by measuring the response of the vehicle [18].

Several studies have been carried out in recent years with the aim of achieving a continuous measurement of wheel–rail contact forces. Di et al. [19] designed and studied a calibration test bench of an instrumented wheelset, while Jin et al. [20] explored a calibration method for the high-speed slabbing web of an instrumented wheelset. Li et al. [21] constructed a load spectrum and a force spectrum for a high-speed axle at 400 km/h by applying an instrumented wheelset. In early research, although continuous correction was carried out for wheel–rail contact force data from instrumented wheelsets, there was always a margin of error due to technical limitations. Zhang et al. [22] studied calibration technologies and developed a calibration test bench for instrumented wheelsets.

With the advancement of artificial intelligence technology in recent years, there has been an increasing application of intelligent algorithms in the sphere of railway transportation [23]. Ujjan et al. [24] investigated the application of improved deep neural networks in wheel–rail contact force measurement. Zhang et al. [25] used the NARX model to enhance the prediction accuracy of the nonlinear wheel–rail contact force exerted by a train on a bridge with an uneven track. Liu et al. [26] devised a technique called the wavenumber domain method (WDM) to detect the waveform of the vertical force between the wheel and rail by analyzing the monitoring data of the track’s dynamic reaction. This method of identifying the contact force between wheels and rails was subjected to rigorous field testing on a track, and the outcomes demonstrated its accuracy and practicality. However, there are a scarcity of studies that specifically address real-time adjustment methods of the wheel–rail contact force for heavy-load trains in practical situations.

For safety reasons, instrumented wheelsets cannot be mounted on heavy-load trains, and the existing methods of measuring wheel–rail contact force cannot easily achieve fast, precise, and continuous wheel–rail contact force measurements for heavy-load trains. For this reason, inspired by the powerful multilayer nonlinear map-ping capability of deep learning, a golden-jackal-optimized multilayer perception machine (GJO-MLP) collaborative calibration model for the wheel–rail contact force of heavy-load trains is proposed in this paper. A railway inspection vehicle’s wheel–rail contact force data are used to determine a heavy-load train’s wheel–rail contact force. By mapping the wheel–rail contact force connection with the proposed model, the actual continuous wheel–rail contact force of a heavy-load train may be calculated, aiding studies on the operation and maintenance of such trains. The overall flow of the algorithm is shown in Figure 1.

In summary, the main contributions of this study are as follows:

(1)

A wheel–rail contact force measurement platform is built for heavy-load trains. A railway inspection vehicle’s instrumented wheelset and a steel rail ground sensor device are combined. By using data from two different sets of signal acquisition equipment, the error caused by a single type of equipment is effectively eliminated.

(2)

Using the experimental site’s characteristics, a data relationship mapping model is developed for the ground-measured wheel–rail contact force and railway inspection vehicle measurements of heavy-load trains. Based on a multilayer perceptron (MLP), this model explores the independent characteristics of intelligent agents in the golden jackal algorithm to provide the MLP with greater training efficiency and maps the wheel–rail contact force data for two working circumstances.

(3)

A collaborative calibration algorithm for the wheel–rail contact force of heavy-load trains is suggested using the wheel–rail contact force data calibration method from the instrumented wheelset on the railway inspection vehicle and the data relationship mapping model. This collaborative calibration can be used to measure a heavy-load train’s wheel–rail contact force by detecting the railway inspection vehicle’s force as it passes by. This overcomes the problem of directly measuring the force.

2. Hardware Design and Data Acquisition

2.1. Equipment

The testing took place on China’s Baotou–Shenmu Railway. The railway inspection vehicle’s detection device was used to collect train wheel–rail contact force signals. The detection system consisted of hardware and system software. The hardware included the instrumented wheelset, signal adaptor, data acquisition, processing computer, etc. Figure 2 shows the instrumented wheelset installation positions.

The instrumented wheelset is a bridge composed of 8 strain gauges, used to collect the wheel–rail contact force signal. To enable transmission of the test signal from the outside of the wheel to the inside of the wheel, there are 2 through-holes on the wheel spoke. The signals on the rotating wheel are transmitted out using telemetry equipment. Finally, approximate sine (cosine) signals with simple harmonic components can be obtained.

Strain gauges mounted on the instrumented wheelset rotate at high speed with the wheels, so collector rings are used to measure the stress of the rotating parts. Since the instrumented wheelset collection ring is frequently put on the bearing’s inner ring and the wire penetrates the shaft, the wheel structure is harmed. Therefore, the measuring wheelset cannot be directly mounted on a heavy-load train.

Furthermore, 6 ground sampling places were chosen to capture ground data concerning the wheel–rail contact force of the genuine heavy-load train and the railway inspection vehicle, based on normal working conditions. These sampling points cover one or more typical working conditions; thus, the data collected here can effectively describe the operation of the train under these working conditions.

2.2. Data Acquisition and Preprocessing

The railway inspection vehicle’s central processing control computer receives wheel–rail contact force signals from the sensor on the instrumented wheelset, analyzes them with specific software, and outputs them as a force measurement.

The collection of wheel–rail contact force signals is mainly carried out in two scenarios; i.e., on a straight line and on a curved line with a small radius. On a straight-line section, the wheel–rail vertical force most strongly affects train operation; on a curved line with a small radius, the derailment coefficient—an important indicator of train operation safety—is greatly influenced by the wheel–rail contact force’s vertical and lateral components. Simultaneously, when a vehicle goes over a small-radius curve, centrifugal force, outer rail elevation, and other curve section parameters cause eccentric loading and significant wheel–rail vertical forces on both the inner and outer rails.

Thus, experiments were conducted to measure the wheel–rail vertical force on a straight rail segment and the vertical and lateral forces on a short-radius curved-line section. The small-radius curved segment had a 600 m left-turn radius, a 70 m superelevation, and a 140 m transition line.

The preprocessing of the collected data mainly included zero drift removal, deburring, filtering, and a short-time Fourier transform. Meanwhile, to ensure data stability, the initial 100 data points from each group were removed, and the collected signals from the stable stage were used as the input signals of the model. Outliers were detected by applying the 3σ principle and dealt with using the capping method.

3. Collaborative Calibration Algorithm to Determine the Wheel–Rail Contact Force of Heavy-Load Trains

3.1. Calibration Model of Wheel–Rail Contact Force from the Railway Inspection Vehicle

The railway inspection vehicle’s instrumented wheelset is prone to inaccuracies that occur due to vibration and weathering over time; therefore, the ground sensors must first be calibrated. The ground sensor data are then utilized as the standard values to rectify the railway inspection vehicle’s instrumented wheelset and reduce long-term usage errors. The MLP neural network model can be used to correct the data measured by the railway inspection vehicle’s instrumented wheelset and labelled by the ground sensor.

In this study, the applied MLP model has two hidden layers, and its specific hierarchical structure is 2–10–6–2; that is, it starts from the input layer, proceeds through two hidden layers, and finally reaches the output layer [27]. The full connection mode is applied between the layers. In addition, the sigmoid function serves as the activation function between layers in this investigation. The MLP model structure is shown in Figure 3.

Using the instrumented wheelset’s data on the wheel–rail contact force as the input vector, the calculation process of the MLP model from input to output is as follows:

$$S_b={\displaystyle \sum _{a=1}^u(W_{ab}{X}_a-{\theta }_b)},$$

(1)

$$T_c={\displaystyle \sum _{b=1}^h\left(W_{bc}{S}_b-{\theta }_c\right)},$$

(2)

$$O_k={\displaystyle \sum _{c=1}^i(W_{ck}{T}_c)-{{\theta }^{\prime }}_k},$$

(3)

In the output layer [28], $O_k$ represents the corrected wheel–rail contact force of the k^th node (k = 1, 2, ···, m, where m is the number of neurons); $W$ is the weight between two layers; and $\theta$ is the bias of each layer node. S, T, and O are the output values of each node in the first hidden layer, the second hidden layer, and the output layer, respectively; u, h, and i are the number of nodes in the first hidden layer, the second hidden layer, and the output layer, respectively.

3.2. Wheel–Rail Contact Force Relationship Mapping between the Railway Inspection Vehicle and Heavy-Load Train

The model input data are the measured wheel–rail contact forces of the railway inspection vehicle with adjustments as described in Section 2.1, and the labels are the ground sensor data of a heavy-load train with a 25 t axle load. GJO-MLP determines the relationship between these two sets of wheel–rail contact force data and creates a relationship mapping model for the railway inspection vehicle’s and heavy-load train’s wheel–rail contact forces.

Because of the significant disparity and variation in the time–frequency domain of the wheel–rail contact force data acquired by the ground sensor and the instrumented wheelset of the railway inspection vehicle, training the basic neural network model takes a long time and has low accuracy for heavy-load trains. Three MLP hyper-parameters—the learning rate, number of hidden layer neurons, and weight values—significantly affect model calibration [28]. Thus, the GJO optimization algorithm optimizes these three hyperparameters to enhance the MLP neural network and improve the accuracy of the wheel–rail contact force mapping model between the railway inspection vehicle and heavy-load train.

Nitish Chopra proposed the golden jackal optimization method in 2022. As a new intelligent optimization system, it mimics the golden jackal’s cooperative hunting style [29]. Searching for, rounding up, and assaulting prey comprise the GJO optimization method.

Figure 4 presents a flow diagram describing the wheel–rail contact force relationship mapping model for the railway inspection vehicle and heavy-load train based on GJO-MLP.

The optimization algorithm estimates the target fitness using the fitness function [29], which produces the target fitness value matrix:

$$F_{OA}=\left[\begin{array}{c}f(Y_{1,1},Y_{1,2},\cdots ,Y_{1,d})\\ f(Y_{2,1},Y_{2,2},\cdots ,Y_{2,d})\\ ⋮\\ f(Y_{n,1},Y_{n,2},\cdots ,Y_{n,d})\end{array}\right],$$

(4)

where $f\left(\cdot \right)$ is the fitness function, $Y_{i,j}$ is the position of the $j\mathrm{t}\mathrm{h}$(j = 1, 2, ···, d, where d represents the number of parameters that need to be optimized) variable of the object to be optimized corresponding to the $i\mathrm{t}\mathrm{h}$(i = 1, 2, ···, n, where n represents the number of targets) target.

The equation for determining the exact location [29] in the golden jackal algorithm is

$$Y_1(t)=Y_M(t)-E\left|r_1{Y}_M(t)-P(t)\right|,$$

(5)

$$Y_2(t)=Y_{FM}(t)-E\left|r_1{Y}_{FM}(t)-P(t)\right|,$$

(6)

where $P\left(t\right)$ represents the position of the target at the $t\mathrm{t}\mathrm{h}$ iteration; $r_1$ is a random number generated from a Levy distribution; E represents the target’s escape energy; $Y_M\left(t\right)$ and $Y_{FM}\left(t\right)$ represent the positions of male and female golden jackals, respectively, at the $t\mathrm{t}\mathrm{h}$ iteration; and $Y_1\left(t\right)$ and $Y_2\left(t\right)$ represent the updated positions of male and female golden jackals, respectively, in the tth iteration.

In summary, the formula used to update the position [29] of the golden jackal is

$$Y(t+1)={\displaystyle \frac{Y_1(t)+Y_2(t)}{2}}$$

(7)

3.3. Collaborative Calibration Algorithm of Wheel–Rail Contact Force for Heavy-Load Trains

In this study, the steps taken to collaboratively calibrate the wheel–rail contact force of a heavy-load train running on the Baotou–Shenmu Railway were as follows:

Step 1: Data preprocessing. Due to the different sampling frequencies of the two sets of wheel–rail contact force acquisition systems, the two sets of data cannot be aligned. To obtain a better analysis of wheel–rail contact force change, downsampling was carried out to preprocess the data collected at the ground monitoring points.

Finally, the data were normalized using the Z-score, which is a statistical measure that indicates how many standard deviations a data point is from the mean, i. e. quantifying how many standard deviations an element is to the mean. The outliers and medians were filtered out to remove the environmental interference noise signals, and the processed dataset was obtained.

Step 2: Dividing the dataset into separate parts. The preprocessed dataset was partitioned into a training set and a test set in a ratio of 4:1.

Step 3: Railway inspection vehicle wheel–rail contact force calibration. The MLP neural network was used to calibrate data from the railway inspection vehicle’s instrumented wheelset and ground sensors.

Step 4: Railway inspection vehicle/heavy-load train wheel–rail contact force data relationship mapping model. The relationship between the calibrated wheel–rail contact force data from the railway inspection vehicle and the data from the actual heavy-load train was determined using GJO-MLP. This was based on the calibrated wheel–rail contact force data obtained from the instrumented wheelset on the railway inspection vehicle and the actual heavy-load train data collected from ground sensors in the previous step.

Step 5: Comparative analysis of different collaborative calibration algorithms. By integrating Steps 3 and 4, a collaborative calibration algorithm was built to determine the wheel–rail contact force of a heavy-load train. Additionally, three different models were used to determine the link between the contact force data from the railway inspection vehicle and the heavy-load train. These models were the GA-BP, SSA-LSTM, and NGO-MLP algorithms. The information was derived solely from the data on the forces exerted between the wheel and rail, acquired from the Baotou–Shenmu Railway line; the calibration results of the four algorithms were compared with each other and with the actual monitored values of the wheel–rail contact force. The comparison proves that the proposed wheel–rail contact force data relationship mapping model based on GJO–MLP is reliable and that this collaborative calibration method is effective and advantageous.

4. Experiment Results and Analysis

In this section, the experimental setup is first introduced. Subsequently, the experimental findings are presented and scrutinized. For the experiments, the data collected via the ground monitoring system installed on China’s Baotou–Shenmu Railway line and via China Energy Baoshen Railway Group’s track inspection vehicle running on the same line section were selected for calibration and analysis under two typical working conditions. Additionally, the proposed GJO-MLP is compared with Genetic Algorithm-Backpropagation (GA-BP), Sparrow Search Algorithm-Long Short-Term Memory (SSA-LSTM), and Natural Gradient Optimization-Multilayer Perceptron (NGO-MLP). The performance of the algorithm is assessed using assessment metrics.

4.1. Test Details

4.1.1. Experimental Setup

For the experiment, 64-bit Windows 10, a 12th Gen Intel Core i5-12400F CPU (Santa Clara, CA, USA), and an NVIDIA RTX 3060 GPU (Santa Clara, CA, USA) were used, along with a starting learning rate of 0.005, a population size of 30, and a maximum iteration number of 200. Data were subjected to downsampling, a time–frequency domain transform, noise cancellation, and discretization.

The initial parameters of each algorithm were defined within a certain range to ensure fairness in the comparative experiments.

Table 1. Algorithm parameter settings.

Algorithm	Parameter	Value
GA-BP	Number of evolutionary iterations	200
	Population size	30
	Crossover probability	0.6
	Mutation probability	0.05
SSA-LSTM	Population size	30
	Maximum number of iterations	200
NGO-MLP	Population size	30
	Maximum number of iterations	200
GJO-MLP	Population size	30
	Maximum number of iterations	200

displays the fundamental parameters of each algorithm.

4.1.2. Dataset Introduction

A uniform standard was offered by Zhuzhou Times Electronic Technology Co., Ltd. (Zhuzhou, China) for the instrumented wheelset of the railway inspection vehicle and the ground sensor system utilized in this research. Using a 25 t axle load heavy-load train and the railway inspection vehicle, data were collected along the whole Baotou–Shenmu Railway line. The 192 km Baotou–Shenmu Railway in China connects Shenmu County, Shaanxi Province, to Baotou City, Inner Mongolia Autonomous Region. This is the northern corridor for coal exports from the Shen-fu-Dongsheng mines. On the heavy-haul Baotou–Shenmu Railway line, there are many typical application scenarios: curved-line sections with small radii, heavy-haul uphill sections, mixed passenger and freight line sections, etc. Therefore, this line is suitable as an experimental site. In this study, 6000 datapoints from Baotou–Shenmu Railway ground monitoring points and the instrumented wheelset of the railway inspection vehicle were collected from 15 July 2023 to 23 August 2023 and used in the algorithm correction analysis.

4.1.3. Evaluation Indicators

The evaluation indicators employed in this experiment to assess the wheel–rail contact force relationship mapping model for the railway inspection vehicle and heavy-load train were the root-mean-square error (RMSE), the mean absolute error (MAE), and the mean percentage error (MAPE). The RMSE was employed to quantify the discrepancy between the corrected values and the true values. The MAE was utilized to depict the actual extent of error in the corrected values, while the MAPE was employed to assess the precision of the model. Smaller values of the RMSE and MAE suggest greater model accuracy and increased trust in the model. A lower MAPE suggests a lesser percentage error of the model and greater confidence.

4.2. Experimental Analysis of Collaborative Calibration of the Wheel–Rail Contact Force for a Heavy-Load Train

4.2.1. Wheel–Rail Contact Force Calibration of a Railway Inspection Vehicle on the Baotou–Shenmu Railway Line

An MLP neural network was utilized to calibrate the wheel–rail contact force signals obtained from the instrumented wheelset of the railway inspection vehicle operating on the Baotou–Shenmu Railway line. This calibration was based on the data collected by the ground monitoring platform that corresponded to the railway inspection vehicle’s measurements. The results of the calibration are displayed in Figure 5.

The MLP neural network model better calibrated the railway inspection vehicle’s wheel–rail contact force after training and validation. The model’s prediction accuracy was 99.5% when compared to the measured data. This proves that the model can accurately capture wheel–rail contact force changes during train operation and produce precise calibration results.

The proposed model utilizes the MLP neural network’s exceptional attributes to autonomously acquire knowledge and detect nonlinear connections within the input data. Additionally, the MLP enhances the model’s calibration capability by computing the gradient of weights and biases and adjusting their values in accordance with the gradient’s direction.

4.2.2. Calibration of Wheel–Rail Contact Force on a Straight Section for a Heavy-Load Train

Table 2. Calibration results of wheel–rail contact force, obtained using 4 algorithms, for a heavy-load train running on the Baotou–Shenmu Railway.

Model	RMSE/N	MAE/kN	MAPE/%
GA-BP	4656.17	4.147	4.16
SSA-LSTM	3441.59	3.418	2.75
NGO-MLP	2206.46	2.384	1.83
GJO-MLP	1625.64	1.593	1.35

displays the absolute inaccuracy between the corrected values and real values of wheel–rail contact force for a heavy-load train on the Baotou–Shenmu Railway. The small absolute error values indicate the good performance of the algorithms. The error values of the four collaborative calibration algorithms used to obtain the wheel–rail contact force for a heavy-load train running on the Baotou–Shenmu Railway are shown in Figure 6. A lower error indicates improved algorithm performance.

By comprehensively considering the results in Table 2 and Figure 6, as well as the fairness of each algorithm, the conclusion can be drawn that compared with the other three algorithms, the proposed model based on GJO-MLP has a smaller absolute error in the calibration of the wheel–rail contact force for a heavy-load train running on the Baotou–Shenmu Railway. This result clearly demonstrates the advantage of GJO-MLP in building relational mapping models.

Because the optimization algorithm was used to optimize the weights, the training time of the MLP was greatly reduced to only 26 s. Compared with the original MLP network, the training time was cut down by about 50%, which is a very significant improvement. Compared with those of GA-BP and NGO-MLP, the training time of GJO-MLP was also improved by about 32% and 25%, respectively. Although the training time of the SSA-LSTM model was only 22 s, which is less than that of GJO-MLP, its other three indicators were all inferior to those of GJO-MLP. Hence, the GJO-MLP method presented in this research demonstrated superior performance when compared to the other models. It has a higher operating efficiency and is more suitable for use in establishing the wheel–rail contact force data relationship mapping model between a railway inspection vehicle and a heavy-load train.

To summarize, GJO–MLP has good reliability and feasibility when used for wheel–rail contact force collaborative calibration for a heavy-load train.

4.2.3. Calibration of Wheel–Rail Contact Force for a Heavy-Load Train Running on a Curved Line with a Small Radius

From the complex and versatile scenarios of the Baotou–Shenmu Railway, a curved-line section with a small radius of 600 m was selected as the source of measured vertical and lateral wheel–rail contact forces, which were then used as experimental data. In this experiment, the compared algorithms and corresponding parameters were consistent with those in Section 3.1.

Table 3. Lateral wheel–rail contact force calibration results from 4 algorithms for a heavy-load train on an r = 600 m curved line section.

Model	RMSE/N	MAE/kN	MAPE/%
GA-BP	7015.36	7.473	17.69
SSA-LSTM	4331.21	4.026	10.33
NGO-MLP	2789.88	2.728	6.71
GJO-MLP	1584.73	1.644	3.84

illustrates the absolute errors between the calibrated wheel–rail contact force obtained using the four algorithms and the real monitoring values from a heavy-load train running on the Baotou–Shenmu Railway. Figure 7 presents a comparison between the four algorithms in terms of the calibration results and actual monitoring values. Figure 8 provides the total calibration absolute errors for these four algorithms at position K126 on the Baotou–Shenmu Railway.

It can be seen in Figure 7 that GJO-MLP demonstrated a better calibration ability on data from the 600 m radius curved line section on the Baotou–Shenmu Railway. In this experiment, NGO-MLP was slightly inferior to GJO-MLP, while SSA-LSTM and GA-BP were relatively poor. The values in Table 3 indicate that GJO-MLP had the lowest RMSE and MAPE, outperforming the other three algorithms. From this and the information in Figure 8, it can be concluded that GJO-MLP had the smallest total absolute error. In summary, the collaborative calibration results from GJO-MLP for the data from a small-radius curved-line segment on the Baotou–Shenmu Railway are optimal and exhibit superior robustness.

Judging from the values of the three evaluation indicators, the wheel–rail contact force data relationship mapping model based on GJO-MLP maintained an average error of about 1.5 kN and an average absolute percentage error of within 4%. Although these findings are not as good as those for the straight-line section, the reliability of the proposed relationship mapping model is still verified.

Working conditions are split into left and right curve circumstances because high-load trains run on distinct curves and have varying vertical wheel–rail contact forces. Unlike on a straight-line section, on a curved section, the curve radius and superelevation affect the vertical wheel–rail contact force. Thus, curved-line working conditions need stricter wheel–rail contact force calibration results. Figure 9 shows the monitored vertical wheel–rail contact forces for left and right curve conditions and their calibrated results after GJO-MLP neural network training. The training results converged better and matched the projected wheel–rail vertical force in time, thus better reflecting the vertical wheel–rail force change for a heavy-load train moving through a bend.

5. Conclusions

To address the challenge of real-time wheel–rail contact force measurements for heavy-load trains, a collaborative calibration technique was proposed herein. This method is based on GJO-MLP and involves establishing a calibration model specifically for the wheel–rail contact force of heavy-load trains. The conclusions are as follows:

(1)

A data collection method to obtain the wheel–rail contact force of heavy-load trains was proposed. The method combines a ground monitoring system and a railway inspection vehicle. Through methods such as downsampling, the two sets of system data are effectively aligned.

(2)

A collaborative calibration algorithm for the wheel–rail contact force of a heavy-load train was established. When compared with three other algorithms, the calibration accuracy of the proposed GJO-MLP model was higher, with improved MAE and MAPE values. For the working condition of a straight-line section, the MAPE of the calibration results was 0.105%; for the working condition of a curved-line section, the RMSE of the calibration results was 184.72 N.

(3)

The heavy-load train wheel–rail contact force calibration model based on GJO–MLP maintained good robustness and generalization ability for different operating conditions and varied parameters. This means that this model has high reliability and adaptability in practical applications.

(4)

The empirical evidence garnered from field data elucidates that within railway sections characterized by reduced curvature radii, wheel sets exhibit a pronounced lateral displacement toward the inside of the curve. This phenomenon engenders an augmentation in the vibrational intensity of the inner rail, thereby imposing substantial ramifications on the operational safety of trains and the requisite maintenance of the track infrastructure. Subsequent to implementing corrective measures, a comprehensive assessment of designated performance indicators becomes feasible; these indicators are pivotal in ascertaining the capacity of trains to operate in a secure and stable manner.

This investigation has been confined to experimentation on the wheel–rail force characteristics. Prospective research endeavors should endeavor to integrate a broader spectrum of feature signals derived from heavy-haul trains, thereby augmenting the precision of the algorithm. Furthermore, the present study has exclusively scrutinized the wheel–rail contact forces encountered by heavy-haul trains in linear segments and sections with small-radius curves. Acknowledging that real-world train operations encompass a multitude of intricate operational scenarios, forthcoming studies are envisaged to explore such complex conditions comprehensively. This will facilitate the attainment of an accurate calibration of wheel–rail contact forces for heavy-haul trains operating along their entire route, thus contributing to a more holistic understanding of the subject matter.