# Identification of the lateral track irregularities from the behavior of the train's axles on a surrogate railway dynamics model

Malek Chihaoui, Denis Duhamel, Christine Funfschilling, Guillaume Perrin

► To cite this version:

Malek Chihaoui, Denis Duhamel, Christine Funfschilling, Guillaume Perrin. Identification of the lateral track irregularities from the behavior of the train's axles on a surrogate railway dynamics model. UNCECOMP 2025, 6th International Conference on Uncertainty Quantification in Computational Science and Engineering, Jun 2025, Rhodes, Greece. pp.669 - 678, 10.7712/120225.12385.25031. hal-05459880

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# IDENTIFICATION OF THE LATERAL TRACK IRREGULARITIES FROM THE BEHAVIOR OF THE TRAIN'S AXLES ON A SURROGATE RAILWAY DYNAMICS MODEL

M.Chihaoui¹,², D. Duhamel¹, C.Funfschilling² and G. Perrin³

¹Laboratoire Navier, École nationale des ponts et chaussées
6 and 8 Avenue Blaise-Pascal, 77455 Champs-sur-Marne, France
e-mail: {malek.chihaoui,denis.duhamel}@enpc.fr

²SNCF, Direction Technologies, Innovation et Projets Groupe
1 to 9 Avenue François Mitterrand, 93210 Saint-Denis, France
e-mail: {malek.chihaoui,christine.funfschilling}@sncf.fr

³COSYS, Université Gustave Eiffel, 5 Boulevard Descartes, 77420 Champs-sur-Marne, France
e-mail: {guillaume.perrin}@univ-eiffel.fr

**Abstract.** This work aims at the reverse identification of the lateral offset irregularity from the kinematics of the axle on a simplified train surrogate model. The fundamental goal of this study is to have a better understanding of the dynamics of the train which would ultimately allow to identify the track irregularities from axle-boxes acceleration measurements on commercial trains. First, a known surrogate model of the dynamics of the axle from the literature is adapted to include a train carbody and the railway track irregularities and then the inversion of the system is achieved so that a relationship between the kinematics that would be measured on the axle and the lateral track irregularity is established.

**Keywords:** track irregularities, lateral offset, railway maintenance, inverse problem, linear estimation, train accelerations.
# 1 INTRODUCTION

Irregularities in railway tracks are critical factors that must be routinely monitored to maintain comfortable, safe, and punctual train operations. These geometrical irregularities and their classification is represented on figure 1. Additionally, understanding axle-rail contact forces and track irregularities is vital for analyzing the dynamic behavior and mechanical degradation of both trains and tracks. This data serves as a key input for various computational models, including those addressing rail and wheel damage, track settlement, and wheel wear.

Currently, specialized trains are used to measure and record track geometry, requiring dedi-

<figure><img src="image_2.png" /><figcaption>Figure 1: Classification of track geometric irregularities</figcaption></figure>

cated personnel and equipment. Installing accelerometers on commercial trains and using the recorded accelerations to assess track geometry could enable more frequent and cost-effective monitoring, improving the organization and optimization of maintenance operations.

Recent studies [1, 2, 3, 4], as well as a patent [5] have explored the reconstruction of vertical track irregularities using on-board accelerometers. This involves double integration of the measured accelerations, allowing the vertical track offset of each rail to be estimated from the vertical position of the corresponding wheel. However, this method cannot be directly applied to lateral track irregularities. This is because lateral movement of the train's axles is subject to a clearance (or play) in the wheel-rail contact, enabling the axles to shift laterally between the rails. Consequently, while lateral axle position is strongly influenced by lateral irregularities, it cannot be directly inferred from them as it can for vertical irregularities. The axle's lateral dynamics between the tracks are complex, depending heavily on the train's mechanical properties and speed. Additionally, the train-track system exhibits significant non-linearity and is influenced by various uncertainties, both internal (e.g., mass, suspension characteristics) and external (e.g., wind, rain), which further complicates the analysis.

In this work, the Klingel formulas [6], which describe the natural oscillations of axles on rails, are utilized and adapted to develop a surrogate model for railway dynamics. A linear surrogate model is constructed, incorporating the train carbody and track geometry, including certain irregularities. Then, an analytical inversion of the system is performed, allowing the deduction of the lateral offset irregularity from both the axle's kinematics and the known track layout parameters. The relationship is also identified using linear regression, and both estimators are then compared.
# 2 SURROGATE MODEL CREATION

## 2.1 Lone axle's dynamics on straight railway tracks

To begin, we consider a perfectly straight track with no geometrical defects. We focus on the kinematics of a single axle of mass $m$ and inertia $j$ along $\vec{z}$ with a conical wheel profile of constant conicity $c$, which is defined from the wheel conicity angle $\alpha$ as $c = \tan\alpha$ as shown on both figures 2 and 3. The conical profile of the wheels means that the rolling radius is not constant and changes according to the lateral position taken by the axle, as detailed in figure 2. When centered, the rolling radius is $r_0$. Moreover, the axle is subjected to a constant longitudinal speed

<figure><img src="image_3.png" /><figcaption>Figure 2: Rolling radius of the wheel</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Diagram of the motion of the axle</figcaption></figure>

$v\vec{x}$ and a constant wheel rotation speed $\dot{\rho} = \frac{v}{r_0}$. In order to take into account the interaction between the axle and the rail, it is necessary to calculate the friction forces and therefore identify the slip speed at each of the right and left wheels $\vec{v}_{slip,r/l}$. The full proof of how the creep force is obtained will not be detailed here, as it is well-documented in the literature [7, 8, 9]. However, a brief overview is provided. The relationship between the radius of each wheel and its lateral position enable to calculate the circumferential or rotation-induced speed of the contact point on the wheel at the contact interface $\vec{v}_{circ,r/l}$.

$$ \vec{v}_{circ,r/l} = v\left(1 \pm \frac{c}{r_0}y\right)\left(\vec{x} + \varphi\vec{y}\right) \quad (1) $$

Whereas the translational speed at each contact interface $\vec{v}_{transl,r/l}$ is calculated using the kinematics of the axle, which is assumed not to roll but to slide perfectly on the rail.

$$ \vec{v}_{transl,r/l} = (v \mp b_0 \dot{\varphi}) \vec{x} + \dot{y} \vec{y} \quad (2) $$

The slip speed is deduced from the difference between these two speeds at each wheel/rail interface points $O_r$ and $O_l$ which will be considered to be at a distance of $b_0$ from the axle's center of mass.

$$ \vec{v}_{slip,r/l} = \vec{v}_{transl,r/l} - \vec{v}_{circ,r/l} = \mp(b_0\dot{\varphi} + \frac{vc}{r_0}y)\vec{x} + (\dot{y} - \varphi v)\vec{y} \quad (3) $$
The creep forces are computed on each wheel directly from the slip speeds using the linear Kalker creep law. $\kappa_{11}$ and $\kappa_{22}$ are respectively the longitudinal and lateral creep coefficients

$$ \vec{F}_{creep,r/l} = -\frac{1}{v} \begin{bmatrix} \kappa_{11} & 0 \\ 0 & \kappa_{22} \end{bmatrix} \cdot \vec{v}_{slip,r/l} \quad (4) $$

These forces at each wheel create a lateral creep force as well as a momentum along the vertical axis.

$$ \begin{bmatrix} F_{creep} \\ M_{creep} \end{bmatrix} = - \begin{bmatrix} \frac{2\kappa_{22}}{v} & 0 \\ 0 & \frac{2b_0^2\kappa_{11}}{v} \end{bmatrix} \begin{bmatrix} \dot{y} \\ \dot{\varphi} \end{bmatrix} - \begin{bmatrix} 0 & -2\kappa_{22} \\ 2\kappa_{11}\frac{b_0c}{r_0} & 0 \end{bmatrix} \begin{bmatrix} y \\ \varphi \end{bmatrix} \quad (5) $$

Using Newton's second law of motion and this force, the dynamics equations governing a lone axle's movement can be deduced. This lateral dynamics surrogate model has been widely used in the literature, particularly for studying stability and calculating the critical speeds of trains. Typically, links to a virtual, motionless carbody centered between the rails are added, incorporating stiffness and damping elements. Indeed, with this model and at speeds below this critical speed, the axle tends to behave like a damped oscillator, swinging sideways before stabilizing in a centered lateral position.

## 2.2 Addition of the carbody, track parameters and irregularities

In our case, a free carbody with a lateral position of $y_c$ and a yaw angle of $\varphi_c$ of mass $m_c$ and inertia $j_c$ along $\vec{z}$ is added to the surrogate model. It is linked to the axle at each of the points $O_{r/l}$ with a longitudinal and a lateral stiffness $K_x$ and $K_y$ as well as a longitudinal and lateral damping $C_x$ and $C_y$. These will create a lateral force and a torque along the vertical axis acting on both the axle and the carbody that can be written as follows:

$$ \begin{bmatrix} F_{carbody/axle} \\ M_{carbody/axle} \end{bmatrix} = - \begin{bmatrix} F_{axle/carbody} \\ M_{axle/carbody} \end{bmatrix} = \begin{bmatrix} 2C_y & 0 \\ 0 & 2b_0^2 C_x \end{bmatrix} \begin{bmatrix} \dot{y} - \dot{y}_c \\ \dot{\varphi} - \dot{\varphi}_c \end{bmatrix} + \begin{bmatrix} 2K_y & 0 \\ 0 & 2b_0^2 K_x \end{bmatrix} \begin{bmatrix} y - y_c \\ \varphi - \varphi_c \end{bmatrix} \quad (6) $$

The next step involves incorporating the track layout parameters and irregularities into the model. These parameters will cause changes in the position of the rails along the curvilinear abscissa, affecting the creep forces. The relative lateral position of the axle in relation to the rail is denoted as $y_r$, which depends on the absolute lateral position of the axle $y$, the lateral offset irregularity $d_3$, and the effective roll of the rails $\theta_d$, as shown in Figure 4.

$$ y \approx y_r + d_3 + r_0\theta_d \quad (7) $$

The track's effective cross-level angle $\theta_d$, which incorporates both the track layout cross-level $c_{track}$ and the irregularity $d_2$, is defined by $\theta_d = \tan^{-1}(\frac{c_{track}+d_2}{2b_0})$. Since we are focusing on modeling only the lateral behavior of the train, the vertical offset and its irregularity $d_1$ will be ignored because their impact on the dynamics is purely vertical. Additionally, gauge irregularity $d_4$ will be disregarded for now, as it affects the system only through non-linearities [8]. The creep forces will then be computed using $y_r$ instead of $y$ in Equation 5.

$$ \begin{bmatrix} F_{creep/axle} \\ M_{creep/axle} \end{bmatrix} = - \begin{bmatrix} \frac{2\kappa_{22}}{v} & 0 \\ 0 & \frac{2b_0^2\kappa_{11}}{v} \end{bmatrix} \begin{bmatrix} \dot{y}_r \\ \dot{\varphi} \end{bmatrix} - \begin{bmatrix} 0 & -2\kappa_{22} \\ 2\kappa_{11}\frac{b_0c}{r_0} & 0 \end{bmatrix} \begin{bmatrix} y_r \\ \varphi \end{bmatrix} \quad (8) $$

The last railways parameter to take into consideration is the curvature of the track $\gamma$. Since, the frame we are computing the inertias in is non-Galilean, the centrifugal force needs to be taken
<figure><img src="image_5.png" /><figcaption>Figure 4: Diagram of the lateral motion of the axle</figcaption></figure>

into consideration for both the axle and the carbody.

$$ \begin{bmatrix} F_{centr/axle} \\ M_{centr/axle} \end{bmatrix} = \begin{bmatrix} -mv^2\gamma \\ -jv\dot{\gamma} \end{bmatrix}, \quad \begin{bmatrix} F_{centr/carbody} \\ M_{centr/carbody} \end{bmatrix} = \begin{bmatrix} -m_c v^2 \gamma \\ -j_c v \dot{\gamma} \end{bmatrix} \qquad (9) $$

Adding up all the mentioned forces in equations 6, 8, and 9, a 4 degrees of freedom model is obtained:

$$ \underline{\underline{M}}\ddot{\underline{U}} + \underline{\underline{C}}\dot{\underline{U}} + \underline{\underline{K}}\underline{U} = \underline{\underline{B}}_2\dot{\underline{E}} + \underline{\underline{B}}_1\underline{E}, \quad \underline{U} = \begin{bmatrix} y \\ y_c \\ \varphi \\ \varphi_c \end{bmatrix}, \quad \underline{\underline{E}} = \begin{bmatrix} d_3 \\ \theta_d \\ \gamma \end{bmatrix} \qquad (10) $$

The mass, damping, stiffness and forcing matrices are defined as follows:

$$ \underline{\underline{M}} = \frac{1}{2} \begin{bmatrix} m & 0 & 0 & 0 \\ 0 & m_c & 0 & 0 \\ 0 & 0 & j & 0 \\ 0 & 0 & 0 & j_c \end{bmatrix}, \quad \underline{\underline{C}} = \begin{bmatrix} C_y + \frac{\kappa_{22}}{v} & -C_y & 0 & 0 \\ -C_y & C_y & 0 & 0 \\ 0 & 0 & b_0^2(C_x + \frac{\kappa_{11}}{v}) & -b_0^2 C_x \\ 0 & 0 & -b_0^2 C_x & b_0^2 C_x \end{bmatrix} \qquad (11) $$

$$ \underline{\underline{K}} = \begin{bmatrix} K_y & -K_y & -\kappa_{22} & 0 \\ -K_y & K_y & 0 & 0 \\ \kappa_{11} \frac{b_0 c}{r_0} & 0 & b_0^2 K_x & -b_0^2 K_x \\ 0 & 0 & -b_0^2 K_x & b_0^2 K_x \end{bmatrix}, \qquad (12) $$

$$ \underline{\underline{B}}_1 = \begin{bmatrix} 0 & 0 & -\frac{1}{2}mv^2 \\ 0 & 0 & -\frac{1}{2}m_c v^2 \\ \kappa_{11} \frac{b_0 c}{r_0} & \kappa_{11} b_0 c & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad \underline{\underline{B}}_2 = \begin{bmatrix} \frac{\kappa_{22}}{v} & \frac{\kappa_{22} r_0}{v} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{2}jv \\ 0 & 0 & -\frac{1}{2}j_c v \end{bmatrix} \qquad (13) $$

# 3 KINEMATICS AND INVERSION OF THE SYSTEM

## 3.1 Measurable kinematical quantities

The resulting system has 4 degrees of freedom and 3 forcing parameters. The whole purpose of this study is to identify the lateral offset irregularity $d_3$ which is one of the forcing parameters
or inputs to the system from axlebox acceleration measurements. Furthermore, an analysis of the identification of the axle's kinematics shows that the lateral position $y$ and the roll of the axle $\theta$ can be deduced respectively from the lateral accelerations and the difference of vertical accelerations. However, identifying the axle's yaw angle $\phi$ using the difference in longitudinal accelerations is far more challenging, as its amplitude is very small and easily obscured by noise and sensor movements. In addition, the curvature $\gamma$ will be considered known along the entire track. The axle's roll $\theta$ although not directly appearing in equation 10 can be used since it can be expressed using other parameters if the dynamical deformation of the rail is neglected. It is composed by the roll of the rail $\theta_d$ that includes both the cross-level layout and the irregularity as well as the roll of the axle relative to the rail $\theta_r$.

$$ \theta = \theta_d + \theta_r \qquad (14) $$

The relative roll $\theta_r$ as demonstrated in figure 5 is linked to the relative lateral displacement $y_r$ through the following relation.

$$ \theta_r = -\frac{c}{b_0}y_r \qquad (15) $$

<figure><img src="image_6.png" /><figcaption>Figure 5: Diagram showing the isolated relative lateral displacement</figcaption></figure>

## 3.2 System inversion

Let $\omega$ be the angular frequency and $\hat{U}$ the Fourier transform of $U$. With the use of the Fourier transform on equation 10 and with a matrix inversion, it can be written as follows :

$$ \hat{U} = (\underline{\underline{K}} - \omega^2 \underline{\underline{M}} + i\omega \underline{\underline{C}})^{-1} (\underline{\underline{B}}_1 + i\omega \underline{\underline{B}}_2) \hat{E} \qquad (16) $$

Extracting the first line from this equation and noting $\underline{\underline{A}} = (\underline{\underline{K}} - \omega^2 \underline{\underline{M}} + i\omega \underline{\underline{C}})^{-1} (\underline{\underline{B}}_1 + i\omega \underline{\underline{B}}_2)$, The lateral position $\hat{y}$ can be written as such :

$$ \hat{y} = \underline{\underline{A}}(1, 1)\hat{d}_3 + \underline{\underline{A}}(1, 2)\hat{\theta}_d + \underline{\underline{A}}(1, 3)\hat{\gamma} \qquad (17) $$

Using equations 7, 14 and 15, $\theta_d$ is formulated as :

$$ \theta_d = \frac{1}{1 + \frac{r_0 c}{b_0}} \theta + \frac{\frac{c}{b_0}}{1 + \frac{r_0 c}{b_0}} y - \frac{\frac{c}{b_0}}{1 + \frac{r_0 c}{b_0}} d_3 \qquad (18) $$
Then, 17 and 18 are combined and $\hat{d}_3$ is isolated. The transfer functions obtained are shown on figur 6.

$$ \hat{d}_3 = \hat{H}_y \cdot \hat{y} + \hat{H}_\theta \cdot \hat{\theta} + \hat{H}_\gamma \cdot \hat{\gamma} \quad (19) $$

The first thing that stands out is the spikes that both $\hat{H}_y$ and $\hat{H}_\gamma$ reach around the wavelength $\lambda = 40m$. These spikes are linked to a known resonance phenomenon of trains which is the haunting oscillation. Then, one can notice that $\hat{H}_\theta$ has a constant norm and phase of $\pi$. This is due to the fact that $d_3$ and $\theta_d$ intervene in the same way in the equations and that $\underline{A}(1, 1)$ and $\underline{A}(1, 2)$ are the same up to a scaling factor.

<figure><img src="image_7.png" /><figcaption>Figure 6: Theoretical transfer functions obtained by inverting the system</figcaption></figure>

With the use of a reverse Fourier transform, the lateral offset irregularity $d_3$ at each point of the curvilinear abscissa $s$ can be deduced by convolution from the two measured kinematics quantities $y$ and $\theta$ as well as the known curvature $\gamma$ and their corresponding kernels $H_y$, $H_\theta$ and $H_\gamma$ which are the reverse Fourier transforms of the transfer functions $\hat{H}_y$, $\hat{H}_\theta$ and $\hat{H}_\gamma$. These convolution kernels are displayed on figur 7.

$$ d_3 = H_y * y + H_\theta * \theta + H_\gamma * \gamma \quad (20) $$

The obtained kernels are very different from each others. As expected $H_\theta$ is a Dirac delta function. $H_y$ and $H_\gamma$ are however very different. $H_y$ is discontinuous at the $s = 0$ and very limited in space as it gets close to null for $|s| > 50m$. This is the complete opposite for $H_\gamma$ whose value diminishes very slowly as $s$ decreases. This means that for these theoretical kernels to be used, they need to be very wide.

## 4 LATERAL TRACK IRREGULARITY IDENTIFICATION ON A NUMERICAL EXPERIMENT

A numerical experience is carried out where the reaction of the surrogate model to a real measured forcing is computed. Measured track irregularities $d_2$ and $d_3$ and track designs $c_{track}$ and $\gamma$ on the french railway network as well as realistic high speed train suspensions parameters, conicity and creep coefficient are employed. Then, the kernels derived from the system
<figure><img src="image_8.png" /><figcaption>Figure 7: Theoretical kernels obtained by inverting the system</figcaption></figure>

inversion are used to try to estimate the lateral offset irregularity $d_3$. These kernels were found to give a precise estimation only if their half width $l$ is large enough as the lateral offset identification requires the knowledge of the track curvature $\gamma$ for a long distance as it appears on 7. For half widths of 300m and 1000m, respective RMS errors of 3.4mm and 1.9mm are found. An identification error of $\epsilon_{d_3} = 0.25mm$ is only obtained for kernels with a half width of at least 3000m. These errors are summarized on table 1.

On the other hand, the kernels can also be identified using a linear regression algorithm. For this purpose, ridge regression with a regularization parameter $\alpha$ has been chosen as it gives the most accurate results. Let $X = (y, \theta, \gamma)$. The ridge kernels are obtained as follows :

$$H_{ridge} = \operatorname{argmin}_H \left\{ \sum_s |d_3(s) - H \cdot X [s-l, s+l]|^2 + \alpha |H|^2 \right\} \quad (21)$$

The kernels given by this method give good estimations of the lateral offset irregularity $d_3$ with much smaller half widths. with $l = 300m$ and $1000m$ we obtain respective RMS identification errors of 0.1mm and 0.03mm which should be compared to the RMS value of the lateral offset irregularity in the database used $RMS(d_3) = 2mm$. The ones with a half-width of $l = 1000m$ are shown on 8 and the identification of the lateral offset irregularity is on figure 9. The obtained kernels closely resemble the theoretical ones, but they appear as if noise has been added. In fact, since they are constructed based on statistical correlation, they depend not only on the dynamics equation but also on the characteristics of the given input signals. Unlike the theoretical kernels, they are also adapted to the chosen half-width and with $l = 1000m$ the identification of $d_3$ is practically perfect.

<table><caption>Table 1: Summary of the RMS identification errors of $d_3$</caption><thead><tr><th>Kernels used</th><th>Theoretical</th><th>Theoretical</th><th>Theoretical</th><th>Ridge regression</th><th>Ridge regression</th></tr></thead><tbody><tr><td>Half width l(m)</td><td>300</td><td>1000</td><td>3000</td><td>300</td><td>1000</td></tr><tr><td>RMSE ε<sub>d3</sub>(mm)</td><td>3.4</td><td>1.9</td><td>0.24</td><td>0.10</td><td>0.03</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>Figure 8: Empirical kernels obtained by Ridge regression</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9: Lateral offset identification using the ridge identifier kernel with a half-width of $l = 1000m$</figcaption></figure>

# 5 CONCLUSIONS

In conclusion, based on the well-known single-axle lateral dynamics model, we have developed a railway dynamics surrogate model that accounts for the presence of a carbody, the track layout, and geometrical irregularities. The resulting system was inverted, and the kinematics of the axle were analyzed, leading to the conclusion that the lateral track irregularity $d_3$ can be identified solely using the lateral position of the axle $y$ and its roll $\theta$, both of which can be reconstructed from axle-box accelerations, along with the known curvature of the track $\gamma$. Subsequently, the use of these theoretically identifiable kernels was compared to kernels derived through linear regression. The next step in this work is to study the stability and sensitivity of the system. Following this, realistic non-linearities will be introduced incrementally, and their impact on the identification of the lateral offset irregularity will be analyzed.

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