# Continuous Evaluation of Track Modulus from a Moving Railcar Using ANN-Based Techniques

^{*}

## Abstract

**:**

## 1. Introduction

_{rel}) between the rail/wheel contact plane, and the rail surface at a distance of 1.22 m from the nearest wheel to the sensor system. The MRail system has been tested over different railway lines in the USA and Canada for evaluating track conditions [18,19,20,21]. Results from the MRail field tests show that the system not only has the potential to identify the local track problems, i.e., muddy ballast, degraded joints, crushed rail head, broken ties, but also provides an opportunity to map the subgrade condition and assess the track performance along the railway line [22,23,24,25].

_{rel}data, and numerical approaches have been proposed to estimate the track modulus from Y

_{rel}[21,26]. The current study aims to propose a new and advanced approach for estimating track modulus statistical properties from Y

_{rel}data more accurately compared to previous studies. First, the details of the MRail system are briefly presented, and the numerical models developed by others and their shortcomings are discussed. Then, artificial neural networks (ANNs) are explained as the main tool to investigate the relationship between track modulus and Y

_{rel}data in this paper. Different methods for training the ANNs are used, and the effectiveness of the trained ANNs are investigated using error measurement parameters such as the coefficient of determination (R

^{2}), the root mean square error (RMSE), and mean absolute percentage error (MAPE). Suitable signatures of Y

_{rel}data are identified by conducting both statistical and frequency analysis. Feedforward neural networks are proposed as a function approximation technique to estimate the track modulus average (U

_{Ave}) and standard deviation (U

_{SD}) from Y

_{rel}data. To further investigate the effectiveness of the ANNs for estimating the track modulus, noisy finite element models (FEM) datasets are employed for training the ANNs. The accuracy of the track modulus estimations using these ANNs is also investigated using R

^{2}, RMSE and MAPE.

## 2. The Stiffness Measurement System and Numerical Simulations

#### 2.1. MRail Measurement System

_{rel}) between the rail surface and the rail/wheel contact plane, at a distance of 1.22 m from the nearest wheel to the acquisition system (Figure 1a). The sensors consist of two laser lines and a digital camera mounted on the side frame of the rail car (Figure 1b). The laser system projects two curves on the rail surface, whose minimum distance (d) is captured by the camera (Figure 1c). Subsequently, the distance between the camera and the rail surface (h) is computed by converting d. Finally, the relative deflection Y

_{rel}is calculated by subtracting h from (Y

_{rel}+ h), the fixed distance between the rail/wheel contact plane and the camera.

_{rel}[24].

#### 2.2. Winkler Model

^{0.25}, U is the track modulus, E is the modulus of the elasticity of the rail, and I is the second moment of area of the rail.

_{rel}can be calculated as the relative vertical deflection between the rail surface and the rail/wheel contact plane at a distance of 1.22 m from the nearest wheel (Figure 1a) [11,30]. The main shortcoming in this method is that the Winkler model assumes a track modulus is constant along the track while the field data shows that a track modulus stochastically varies along the track [31,32]. Therefore, the estimation of the track modulus from the Y

_{rel}measurements needs more advanced numerical models.

#### 2.3. Finite Element Model

_{rel}measurements. Fallah Nafari et al., developed 90 FEMs with a stochastically varying track modulus to facilitate a more detailed investigation of the relationship between the Y

_{rel}and the track modulus [21]. Datasets from the 90 FEMs were used for the study in this paper. Hence, the details of the models are discussed briefly. The models are developed using CSiBridge software, where each model includes a 180.8 m track structure with two rails, crossties, and spring supports [33]. To develop each model, a normal track modulus distribution is assumed and randomly selected numbers from this distribution are assigned to the spring supports along the track. Statistical properties of the assumed normal distributions are summarized in Table 1, and the applied loads are depicted in Figure 2. RE136 rail size and 0.508 m tie spacings are used in the models.

_{rel}values are calculated from the vertical deflection profile at every 0.3048 m ($\approx $1 ft) interval while the moving loads pass the track model. The dynamic effects of track–train interactions are not considered during the simulations due to the software’s limitation. This is acceptable within the scope of this study which mostly focuses on the Canadian freight lines where speeds are most likely lower than 65 km/h.

_{rel}output. Fallah Nafari et al., used basic statistical analysis and curve fitting approaches to study the relationship between the statistical properties of track modulus (U) and Y

_{rel}[21]. The results showed that the average and standard deviation of the track modulus over a track section length can be estimated from the average and standard deviation of Y

_{rel}over the same track section length. However, the estimation accuracy becomes lower by decreasing the track section length [21]. To overcome this shortcoming and increase the estimation accuracy of the track modulus, ANNs are proposed for the track modulus estimations in this study.

#### 2.4. Estimation of Track Modulus Average

#### 2.4.1. Multilayer Perceptron Artificial Neural Networks

_{ij}(n) (Figure 4b) as the level of connectivity.

#### 2.4.2. Estimation Procedure and Results

_{rel}data from the 180 m track models are divided into equivalent groups based on a track section length (e.g., 5 m, 10 m, etc.). Once the subgroups are defined, the average and standard deviation of Y

_{rel}in each subgroup are used as the networks’ inputs whereas the track modulus averages in the corresponding track segments are defined as the network’s outputs.

_{rel}data extracted from eighty-one FEMs (out of ninety FEMs) are used to train the neural network. The accuracy of the trained network is then tested using the remaining nine (unseen) FEMs. These nine FEMs are called “unseen models” hereafter as they are not used in training the network. To test the trained network, track modulus average is estimated from Y

_{rel}average and the standard deviation for the nine unseen models. The estimated track modulus average is then compared with the track modulus inputted initially into the FEMs to generate Y

_{rel}data. The effectiveness of the proposed network is measured based on three parameters: the coefficient of determination (R

^{2}), the root mean square error (RMSE), and the mean absolute percentage error (MAPE) [39]. These measures are described as follows:

^{2}is 0.95 for the case of the 10 m section length, which means that the estimated and inputted track modulus averages are well correlated. Moreover, the RMSE and MAPE are quite small, i.e., 2.81 MPa, and 6.99% respectively, considering that range of inputted track modulus average is 12.8 to 41.4 MPa. In addition to confirming the applicability of the Y

_{rel}data in indicating the track modulus information, the current methodology provides more accurate results than the other method in the literature [21]. As shown in Table 2, the R

^{2}value computed in the related study decreases as the length of the track segment reduces, whereas the R

^{2}in the current study is almost constant for cases with a 10 m track section and more.

_{rel}data extracted from the FEMs. This simulates the real-life condition in which the Y

_{rel}measurements are affected by parameters such as the resolution of the MRail measurement system, track irregularities, etc. The artificial noise was added based on Equation (5) [40]. An example of pure vs. noise-added Y

_{rel}is shown in Figure 6:

_{rel}is used to train new networks, and then the trained networks are used to estimate the track modulus average. The estimated track modulus is then compared with the inputted track modulus for each model and the error is reported in Table 3. From the table, the estimation of the track modulus average (U

_{ave}) from the noisy Y

_{rel}is still successful even for the short track section length of 10 m as R

^{2}is 0.95, and RMSE is 2.77 MPa. This demonstrates that the framework performs effectively even when the Y

_{rel}data contain noise, and thus is expected to work with real-life data.

#### 2.5. Estimation of Track Modulus Standard Deviation (U_{SD})

_{rel}data using statistical methods and curve-fitting approaches has not been successful for track section lengths shorter than 80 m [21]. Therefore, frequency characteristics of the deflection data are investigated in this study to increase the estimation accuracy of track modulus standard deviation. The coefficients associated with the Y

_{rel}frequency components are employed as one of the inputs to the ANNs, whose outputs are the track modulus standard deviation over different track section lengths. As demonstrated in Figure 7, Y

_{rel}and the track modulus data are divided into different subgroups based on various track section lengths (similar to the procedure used for estimating the track modulus average). Then, statistical analysis, fast Fourier transform, and a liftering technique are applied on the Y

_{rel}data in each subgroup to extract the average and standard deviation of the Y

_{rel}and average and the standard deviation of liftering the fast Fourier transform (FFT) coefficients. These parameters are used as the inputs of ANNs.

_{rel}data over a track section of 30 m for 81 models. As can be seen, the coefficients at higher orders are relatively small. This is undesirable for training the ANN due to possible bias. Therefore, the coefficients are processed using a liftering technique (Equation (6) to roughly normalize their variances) [41]:

_{rel}, and the average and standard deviation of the lifted FFT; ANN-2 with two inputs, i.e., mean and standard deviation of Y

_{rel}). In each case, the two networks are trained and tested multiple times and the mean and standard deviation of the performance parameters are computed and reported in Table 4. For the case of 5 m section length, for instance, the networks’ input, and output are first extracted based on the chosen section (5 m), then ANN-1 and ANN-2 networks are trained using the training data and tested against the data extracted from nine unseen FEMs.

_{SD}) can be estimated satisfactorily by both network configurations (ANN-1 and ANN-2). Even for the 10-m section length case, for instance, the coefficient of correlations between the actual U

_{SD}and the one estimated by the two networks are very high, e.g., 0.83 and 0.82 respectively. However, the networks with four inputs (ANN-1) slightly outperform the one with two inputs (ANN-2) regardless of the section lengths. Specifically, the RMSE and MAPE are always smaller than those arising from the trained networks whose inputs are the statistical properties of Y

_{rel}only (ANN-2). Values estimated using the networks with four inputs have relatively high R

^{2}in all cases showing that the methodology is successful. In particular, the R

^{2}is as high as 0.94 for the case of the 25 m section length and the RMSE is 1.83 MPa, which is a relatively small error considering that the maximum standard deviation of the inputted track modulus in the FEMs is 31.05 MPa. Moreover, the first network (ANN-1) provides more reliable results as the standard deviation of RMSE remains stable (varying from 0.11 to 0.17 MPa) and lower than those of ANN-2. Therefore, combining FFT and statistical analysis to configure the input for the networks noticeably improves the estimation accuracy, and increases the stability of the ANNs, the mapping function between the Y

_{rel}characteristics and the track modulus standard deviation (U

_{SD}). Most importantly, there is a big step forward in this paper compared to the previous study, where the R

^{2}coefficient is 0.748 even though the 40 m section length is used [21]. The performance of this estimation can be considered ineffective as the R

^{2}coefficient reduced significantly in shorter track segment cases (Table 4). Hence, considering the current results, it can be claimed that neural networks are more powerful for mapping the relationship between Y

_{rel}and track modulus, especially over the short track section lengths.

_{rel}). Similar to the procedure mentioned in the previous section, noise is added to the Y

_{rel}data from 90 models using Equation ((5). The dataset from 81 models is then used to train the networks using two approaches: networks with two inputs (average and standard deviation of Y

_{rel}) and networks with four inputs (average and standard deviation of Y

_{rel}and average and standard deviation of the lifted FFT). The developed networks are used to estimate the track modulus standard deviations over the different section lengths from the unseen Y

_{rel}data. The estimated values are compared with the standard deviation of track modulus inputted to FEMs and results are reported in Table 5. The results show that the proposed approaches work very well even when Y

_{rel}datasets are affected by noises. The R

^{2}is again higher than 0.90 when the 25 m or higher section lengths are utilized.

## 3. Conclusions

_{rel}data (a relative rail vertical deflection measured using the MRail system) for the track modulus estimations. The relationship between the statistical properties of the track modulus and the Y

_{rel}data were investigated using artificial neural networks (ANNs). Datasets from FEMs are used to train the ANNs in which their outputs are either the track modulus average or standard deviations. Both statistical and frequency analyses were conducted to identify the optimized inputs for the ANNs from the Y

_{rel}data. From the results, the track modulus average over a track section length of 10 m or longer is accurately estimated from the average and standard deviation of the Y

_{rel}data within the corresponding section length. Additionally, the standard deviation of the track modulus over a section length of 25 m or longer is estimated with an acceptable level of accuracy. It is also shown that the trained ANNs work very well for the track modulus estimations even when the Y

_{rel}values as the ANN inputs are affected by noise. The proposed ANNs are only applicable to a specific rail type and loading condition. Hence, a similar procedure should be followed to train the ANNs for different ranges of rail sections and loading types.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Demonstration of the MRail system (real-time vertical track deflection measurement system) for Y

_{rel}measurements: (

**a**) the measurement system on a rigid frame; (

**b**) the sensor system; and (

**c**) the projections of the laser lines on the railhead.

**Figure 3.**(

**a**) Track modulus inputted to the FEM (Mean = 41.4 MPa, COV = 0.25); and (

**b**) the extracted Y

_{rel.}

**Figure 5.**Moving average of the actual track modulus inputted to the FEMs vs. estimated values over: (

**a**) the 10 m section length; and (

**b**) the 20 m section length.

**Figure 9.**The actual track modulus standard deviation over the 25 m section length vs. the estimated values.

Track Modulus Average (MPa) | Coefficient of Variation (COV) | No. of Simulations |
---|---|---|

41.4 | 0.25; 0.5; 0.75 | 30 (10 simulations for each COV) |

27.6 | 0.25; 0.5; 0.75 | 30 (10 simulations for each COV) |

12.8 | 0.25; 0.5; 0.75 | 30 (10 simulations for each COV) |

Section Length (m) | MAPE * (%) | RMSE ** (MPa) | R^{2} | R^{2} in [21] |
---|---|---|---|---|

5 | 12.42 | 4.58 | 0.86 | 0.79 |

10 | 6.99 | 2.81 | 0.95 | 0.93 |

15 | 5.90 | 2.56 | 0.95 | N/A |

20 | 4.32 | 1.60 | 0.98 | 0.96 |

25 | 3.87 | 1.63 | 0.98 | N/A |

Section Length (m) | MAPE (%) | R^{2} | RMSE (MPa) | R^{2} in [21] |
---|---|---|---|---|

5 | 14.09 | 0.83 | 5.07 | 0.79 |

10 | 7.01 | 0.95 | 2.77 | 0.93 |

15 | 5.93 | 0.96 | 2.36 | - * |

20 | 6.07 | 0.97 | 1.98 | 0.96 |

25 | 3.98 | 0.98 | 1.53 | - * |

**Table 4.**Estimation accuracy of the U

_{SD}(no noise added, the standard deviation in the parenthesis).

Section Length (m) | Network Configuration | RMSE (MPa) | MAPE (%) | R^{2} | R^{2} in [21] |
---|---|---|---|---|---|

10 | ANN-1 | 3.00 (0.16 *) | 18.41 | 0.83 | 0.53 |

ANN-2 | 3.05 (0.22) | 19.12 | 0.82 | - | |

15 | ANN-1 | 2.36 (0.08) | 15.01 | 0.89 | - |

ANN-2 | 2.61 (0.39) | 15.79 | 0.87 | - | |

20 | ANN-1 | 2.23 (0.11) | 14.49 | 0.91 | 0.66 |

ANN-2 | 2.59 (0.89) | 14.47 | 0.88 | - | |

25 | ANN-1 | 1.83 (0.13) | 11.96 | 0.94 | - |

ANN-2 | 1.99 (0.30) | 11.72 | 0.92 | - | |

30 | ANN-1 | 2.08 (0.17) | 11.61 | 0.92 | - |

ANN-2 | 2.14 (0.44) | 11.79 | 0.91 | - |

Section Length (m) | Network Configuration | R^{2} | RMSE (MPa) | MAPE (%) |
---|---|---|---|---|

10 | ANN-1 | 0.82 | 3.06 | 20.12 |

ANN-2 | 0.81 | 3.14 | 19.64 | |

15 | ANN-1 | 0.87 | 2.59 | 16.30 |

ANN-2 | 0.87 | 2.64 | 16.23 | |

20 | ANN-1 | 0.89 | 2.42 | 16.13 |

ANN-2 | 0.89 | 2.45 | 14.71 | |

25 | ANN-1 | 0.94 | 1.86 | 11.96 |

ANN-2 | 0.93 | 1.88 | 11.73 | |

30 | ANN-1 | 0.94 | 1.84 | 10.43 |

ANN-2 | 0.93 | 1.89 | 10.95 |

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## Share and Cite

**MDPI and ACS Style**

Do, N.T.; Gül, M.; Nafari, S.F.
Continuous Evaluation of Track Modulus from a Moving Railcar Using ANN-Based Techniques. *Vibration* **2020**, *3*, 149-161.
https://doi.org/10.3390/vibration3020012

**AMA Style**

Do NT, Gül M, Nafari SF.
Continuous Evaluation of Track Modulus from a Moving Railcar Using ANN-Based Techniques. *Vibration*. 2020; 3(2):149-161.
https://doi.org/10.3390/vibration3020012

**Chicago/Turabian Style**

Do, Ngoan T., Mustafa Gül, and Saeideh Fallah Nafari.
2020. "Continuous Evaluation of Track Modulus from a Moving Railcar Using ANN-Based Techniques" *Vibration* 3, no. 2: 149-161.
https://doi.org/10.3390/vibration3020012