PREDICTION AND MEASUREMENTS OF
VIBRATIONS FROM A RAILWAY
TRACK LYING ON A SEMI-INFINITE
PEATY GROUND

Benoit Picoux, René Rotinat, Jean-Pierre Regoin, Donatien Le Houédec

Ecole Centrale de Nantes

Laboratoire Mécanique et Matériaux

1 rue de la Noë BP 92 101

44321 Nantes Cedex 3 FRANCE

Phone: (33) 2 40 37 25 54

Fax: (33) 2 40 37 25 73

E-mail: benoit.picoux@ec-nantes.fr

ABSTRACT

In this paper we introduce a semi-analytical wave propagation model subjected to a set of
harmonical moving loads and based on a calculation method of the dynamic stiffness matrix of
the ground. In order to model a complete railway system, we also take into account the effect of
a simple railroad model including rails, sleepers and ballast especially conceived for the study of
low vibration frequencies. The priority has been given to a simple formulation based on the
principle of Fourier transform complying with a good numerical efficiency and giving quick
solutions. Besides, *in situ* measurements for a soft soil near a railway track were carried out and
will be used to validate the numerical implementation. Then numerical and experimental results
constitute a significant basis of useful data to, on the one hand, characterize the response of the
environment of tracks and, on the other hand, appreciate the importance of the speed and
weight on the behaviour of structure.

KEYWORDS

Wave propagation, railway track, moving load, Fourier transform, soft soil, *in situ* measurements, optical methods.
# 1. INTRODUCTION

Along a railroad in the North West of France, observations revealed the existence of high displacements of the ground surface under a railway traffic. These vibrations are all the more harmful, for the environment of the track and for the track itself, that the speed and the weight of the train are important, particularly when the speed of the train exceeds the low speed of surface waves in soft ground. These vibrations entail high displacements in the track (mainly for the rail) not acceptable for the safety and the speed of the railway traffic. Possible long term effects are possible for the track foundation, the rail or for the neighbouring structures. Aware of these problems, the French Ministry of Environment has financed a research program for the topics "Noises and vibrations". These researches are destined to better apprehend the phenomenon of vibration propagation from railway vehicles moving at high and constant speed. The project partnership implies the French National Railway Society and an European geophysical office. In a first time the main objective consists in conceive simulation and estimation methods for vibrations near the railroad. Then the study of a simple two-dimensional model of track - soil interaction allows us to determine resonance frequencies of the system and the critical speed of the train. Also a parametric analysis of the track components and the ground parameters is performed. Besides, effects of intrinsic parameters modification of the track can be studied (strengthening of the platform by adding rails, reduction of the distance between sleepers). In a second time, in *situ* measurements are undertaken in order to validate the modelling and to obtain useful and suitable data to apprehend the phenomena of wave propagation from vehicles (especially trains) moving at constant speed. This last study is still in progress and only the first main results are presented here.

# 2. PREDICTION MODEL

Phenomena of wave propagation in ground have been in the past the subject of several researches [1] associating numerical simulations [2] as well as *in situ* measurements [3]. In these analysis, mechanical characteristics and behaviour laws of the different soils play effectively an important role for the relevance and the validity of the proposed model [4]. So hypotheses have to be chosen with carefulness. Since some decades, an other essential factor is constituted by the speed of the load moving often at the surface of the ground [5] or possibly in the basement (tunnels for example). The value of this speed is compared to that of the Rayleigh wave. This kind of waves plays a preponderant role in the observable vibration amplitude [6] during the passage of a train.

A railway model [7] lying on a layered soil medium and subjected to a railway traffic can be developed using the formalism of *Fourier transform* [2]. Fig. 1 describes the complete model with two distinct parts : the rail lying on sleepers through railpads, the ground constituted by the ballast and n layers. Fixed reference linked to the ground and mobile reference linked to the single and rectangular harmonical load are noted respectively (O,x₁,x₃) and (O,x,z). A parametric analysis of the different track elements and a vertical displacement evaluation according to the speed and the composition of trains can be given by this model.

The rail is represented as an infinite *Euler beam* with the following characteristics : mass $m_R$, Young's modulus E and moment of inertia I. Railpads and sleepers are represented as a continuous mass – spring system [2], (mass $m_S$, stiffness $k_P$ and damping $\eta_P$). For this kind of
modelling the continuity is justified taking into account the studied wavelengths (frequency range under 100 Hz).

<figure><img src="image_1.png" /><figcaption>Figure 1: Description of prediction model</figcaption></figure>

For the rail described by an Euler beam an [2], we obtain the following equation :

$$EI \frac{\partial^4 w_R(x_1, t)}{\partial x_1^4} + m_R \frac{\partial^2 w_R(x_1, t)}{\partial t^2} + k_P [w_R(x_1, t) - w_S(x_1, t)] = \begin{cases} \sigma_{TR AIN} (x_1, t) & \text{if } |x_1 - ct| < b \\ 0 & \text{else} \end{cases} \quad \{1\}$$

where $\sigma_{TR AIN}(x_1, t)$ is the train Cauchy stress equal to $P_0 e^{i\Omega_0 t} / 2b$ for harmonical a rectangular load (fig. 1).

The equation for sleepers is written :

$$m_S \frac{\partial^2 w_S(x_1, t)}{\partial t^2} + k_P [w_S(x_1, t) - w_R(x_1, t)] = -F_S(x_1, t) \quad \{2\}$$

This structure is lying on a multi-layered ground of thickness $h_n$ that first represents the ballast (mechanical parameters : $E_b, v_b, \rho_b, \eta_b$). Below the ballast, each soil layer is described by its mechanical parameters $E_n, v_n, \rho_n, \eta_n$. Under the last layer spreads a halfspace.

For an elastic, homogeneous and isotropic layer, we write for the equation of the movement of movement :

$$(\lambda + \mu) \nabla(\nabla \cdot U) + \mu \Delta^2 U = \rho \frac{\partial^2 U}{\partial t^2} \quad \{3\}$$

and for the behaviour law :

$$\sigma_{ij} = \lambda \delta_{ij} (\nabla \cdot U) + \mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \quad \{4\}$$
To define elastic volume waves (P-waves and S-waves), we usually use *Helmoltz's potentials* $\phi$ et $\psi$ such that :

$$ U = \nabla\phi + \nabla \wedge \Psi. \qquad \{5\} $$

The typical train excitation is composed of a set of vertical components that symbolise the effect of each engine and carriage wheel on the rail. Each of these forces is taken as harmonic. Finally, $w_R$ and $w_S$ represent respectively the vertical displacements of rail and sleepers. $F_R$ is the force exerted by the rail on sleepers and $F_S$ the force exerted on the ballast allowing the contact between the structure and the layered ground.

The resolution method [6] is described in fig. 2 where the fitted phase angle is defined by *Helmoltz's potentials*, such as:

$$ \begin{cases} \phi = A e^{\alpha_p (z-h)} + C e^{-\alpha_p z} \\ \psi = B e^{\alpha_S (z-h)} + D e^{-\alpha_S z} \end{cases} \quad \text{where } \alpha_i = \sqrt{\beta^2 - \frac{\Omega_0 - \beta c}{c_i}}, \quad i = P, S. \qquad \{6\} $$

$c_P(E, \rho, v)$ is the speed of compression waves (P-waves) and $c_S(E, \rho, v)$ the speed of shear waves (S-waves) (e.g. Table I). In fact, the application of this phase translation gives a good conditioning of the rigidity matrix of the ground. A change of variables (translation in the mobile reference linked to the load) and functions (evaluation of the steady state solution) as well as the use of spatial *Fourier transform* lead to the calculation of vertical displacement in the wave number domain (fig. 2).

<figure><img src="image_2.png" /><figcaption>Figure 2 : Resolution method</figcaption></figure>

After writing equation concerning with the wave motion, the behaviour law, the boundary conditions for each layer [3] and track equations [2] a relationship between displacements and stresses at each layer interface (including the ballast) can be written.
This relationship is given in a matrix form :

$$
\begin{bmatrix}
T_{11}^{\text{Bal}} & T_{21}^{\text{Bal}} & T_{31}^{\text{Bal}} & T_{41}^{\text{Bal}} & 0 & 0 \\
T_{12}^{\text{Bal}} & T_{22}^{\text{Bal}} & T_{32}^{\text{Bal}} & T_{42}^{\text{Bal}} & 0 & 0 \\
T_{13}^{\text{Bal}} & T_{23}^{\text{Bal}} & T_{33}^{\text{Bal}} + T_{11}^{\text{n}} & T_{43}^{\text{Bal}} + T_{21}^{\text{n}} & T_{31}^{\text{n}} & T_{41}^{\text{n}} \\
T_{14}^{\text{Bal}} & T_{24}^{\text{Bal}} & T_{34}^{\text{Bal}} + T_{12}^{\text{n}} & T_{44}^{\text{Bal}} + T_{22}^{\text{n}} & T_{32}^{\text{n}} & T_{42}^{\text{n}} \\
0 & 0 & T_{13}^{\text{n}} & T_{23}^{\text{Bal}} & T_{33}^{\text{n}} + R_{11} & T_{43}^{\text{n}} + R_{21} \\
0 & 0 & T_{14}^{\text{n}} & T_{24}^{\text{Bal}} & T_{34}^{\text{n}} + R_{12} & T_{44}^{\text{n}} + R_{22}
\end{bmatrix}
=
\begin{bmatrix}
i\bar{w}^*(0) \\
\bar{u}^*(0) \\
i\bar{w}^*(h) \\
\bar{u}^*(h) \\
i\bar{w}^*(h^n) \\
\bar{u}^*(h^n)
\end{bmatrix}
=
\begin{bmatrix}
-i\bar{\sigma}_{zz}^*(0) \\
-i\bar{\sigma}_{xz}^*(0) \\
0 \\
0 \\
0 \\
0
\end{bmatrix}
=
\begin{bmatrix}
i\bar{F}_S^*(\beta) \\
0 \\
0 \\
0 \\
0 \\
0
\end{bmatrix}
\quad \{7\}
$$

where [T<sup>BAL</sup>] is the rigidity matrix of ballast, [T<sup>n</sup>] is the rigidity matrix of n ground layers and
[R] is the rigidity matrix of halfspace. The rigidity matrix [T] is obtained by the product of the
inverse of the displacement matrix [Q] by the stress matrix [S] whose coefficients depend on
Lamé coefficients, layer thickness h, excitation frequency Ω<sub>0</sub>, load speed c, and wave number β.
All these coefficients are complex because of hysteretic damping considered in our model.

Thus, vertical displacements at the surface of ballast and ground can be written, respectively :

$$
\bar{w}_{z=0}^*(\beta) = [T]_{11}^{-1} \bar{F}_S^*(\beta) \quad \{8\}
$$

$$
\bar{w}_{z=h}^{*}(\beta) = [T]_{31}^{-1} \bar{F}_{S}^{*}(\beta) \quad \{8\}
$$

By assuming that the system of sleepers is continuous in front of the wavelength involved, the
system can be written in the following equations:

$$
\left\{
\begin{array}{l}
(\mathrm{EI}\beta^4 - m_R \Omega_0^2 + k_P) \bar{w}_R^*(\beta) - k_P \bar{w}_S^*(\beta)] = \bar{\sigma}_{\mathrm{TR AIN}}(\beta) \\
\quad - k_P \bar{w}_R^*(\beta) + (k_P - m_S \Omega_0^2) \bar{w}_S^*(\beta) = -\bar{F}_S^*(\beta) \\[1em]
\bar{w}_S^*(\beta) = [T]_{11}^{-1} \bar{F}_S^*(\beta)
\end{array}
\right.
\qquad \{9\}
$$

of which are deduced displacement at the soil surface:

$$
\bar{w}_{z=h}^*(\beta) = [T]_{31}^{-1} \frac{k_P \bar{\sigma}_{\text{TR AIN}}(\beta)}{[(EI\beta^4 - m_R\Omega_0^2 + k_P)(k_P - m_S\Omega_0^2) - k_P^2][T]_{11}^{-1} + (EI\beta^4 - m_R\Omega_0^2 + k_P)} \quad \{10\}
$$

then $\bar{w}_{z=h}^*(\beta)$ represents the vertical displacement in the wave number domain under the ballast (ground surface). An analytical solution exists if the soil is considered as infinite in the positive z-direction. In this case, the *Rayleigh's* function involving an infinite vertical displacement is obtained by the cancellation of the denominator, deducing exactly the speed of *Rayleigh* waves in the medium [1]. In this case, a residual method by integration in the complex plane gives the actual vertical displacements. In the case of a layered soil, actual displacements are calculated numerically by an inverse *Fourier* transform.
### 3. NUMERICAL RESULTS

For the numerical simulation, we use the following data for the structure. First, for the steel rail, the Young's modulus is equal to $2,11.10^{11}$ N/m², the moment of inertia to $3055 \text{ cm}^4$ and the mass per unit of track length to 120 kg/m. The stiffness of the pads between the rail and sleepers is equal to $3,5.10^8$ N/m with a damping coefficient equal to 0,15. The mass of sleepers per unit of track length is equal to 490 kg/m.

Let us consider for the numerical investigation the case of a relatively multi-layered soft soil (before considering a peaty soil). In the following example, a 7 m soft layer is lying on a halfspace. Mechanical parameters for this example are summarized in Table 1.

<table><caption>Table 1: Parameters for numerical investigation</caption><thead><tr><th></th><th>h (m)</th><th>E (N/m²)</th><th>ν</th><th>ρ (kg/m³)</th><th>η</th><th>c<sub>P</sub> (m/s)</th><th>c<sub>S</sub> (m/s)</th></tr></thead><tbody><tr><td>Ballast</td><td>0,8</td><td>400.10<sup>6</sup></td><td>0,3</td><td>1800</td><td>0,1</td><td>547</td><td>292</td></tr><tr><td>Layer</td><td>7</td><td>269.10<sup>6</sup></td><td>0,257</td><td>1550</td><td>0,1</td><td>459</td><td>263</td></tr><tr><td>Substratum</td><td>∞</td><td>2040.10<sup>6</sup></td><td>0,179</td><td>2450</td><td>0,1</td><td>950</td><td>594</td></tr></tbody></table>

A harmonic unit load is acting on the rail surface. In fig. 3, maximum displacements at the soil surface are plotted for the conceivable speed regimes ("static" $c = 0$, sub-Rayleigh $c = 0,5 \times C_R$ and super-Rayleigh $c = 1,5 \times C_R$). These displacements are symmetrical as compared to the load when the speed is equal to zero. More important amplitude are obtained when the load is moving. For the super-Rayleigh regime, displacements behind the load are most important and oscillating. In this last regime *Mach* cones are observed in presence of front waves moving with the load. These cones are due to the P- and S-waves and exist in the ground (fig. 4) but also at the soil surface due to the preponderant R-waves.

<figure><img src="image_3.png" /><figcaption>Figure 3: Maximum displacements for the three speed regimes (static: solid line, sub-Rayleigh: dotted line, super-Rayleigh: dash-dotted line)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 4: Displacements against depth in super-Rayleigh regime.</figcaption></figure>
The analytical expression of stress for a train can be written easily in the wave number domain thanks to symmetries noticed previously. Fig. 5 describes the vertical stress function (Cauchy stress) for a typical “Corail” train composed of an engine and two carriages. This function, involving a same low and identical excitation frequency for all loads (wheels) is introduced in the model of track – soil interaction. Displacement generated at the ground surface by the train are plotted in fig. 8 according to the different regimes of speed.

When the train doesn't move on the track, the ground undergoes a displacement due to the train weight on the track (static regime). The look of the curve obtained for this regime is preserved until a speed equal to about 0.5 time the speed of the surface Rayleigh wave. The dynamic regime leads to more large vibrations ahead and behind the wheel load (sub-Rayleigh regime) and only behind the wheel load (super-Rayleigh regime).

Finally, the developed model allows us to evaluate numerically vertical, longitudinal and lateral displacements for all point of the ground surface and at the interface of each layer as well as the vertical displacement of each track element.

<figure><img src="image_5.png" /><figcaption>Figure 5 : Actual train Cauchy stress $\sigma_{\text{TRAIN}}$ for a current French train</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 6 : Maximum vertical displacement at the soil surface near the track for current train</figcaption></figure>

# 4. EXPERIMENTAL INVESTIGATIONS

In complement of simulation studies, *in situ* measurements have been undertaken in partnership with the French National Railway Society (S.N.C.F.) and a geophysical office (Soletanche - Bachy). The three sites are situated in the valley of river *Somme* along the railway line Amiens - Boulogne (France) and are concerning with different states (current track with vibration problems, modified track with vibration problems, current track without vibration problem). These sites correspond to a ground composed of a peaty and clayey soil (a few meters deep) lying on a limestone substratum (fig. 7).

A sounding test indicates a saturated soil already at a weak depth (amplified state in consequence of floods during the spring 2001). A parameter study of the ground undertaken by seismic
measurements shows a critical speed near 100 m/s while the studied trains are moving with sub-Rayleigh speeds (20 to 50 m/s).

For the two problematical sites, during of the passage of a heavy train, the railway platform and the ground undergo displacements in the order of some millimeters (with a maximum able to reach ten millimeters for the rail). The railway structure is composed of a rail UIC60, bi-block sleepers U41, pads and a ballast. Measurements have been performed in the end of July 2001 for the three sites, the measures linked to vibrations being coupled with other tests in the aim to determine the mechanical characteristics of ground.

<figure><img src="image_7.png" /><figcaption>Figure 7: Mechanical properties of ground</figcaption></figure>

Measurements give us a lot of informations about lateral and vertical accelerations of track elements and on the soil surface. With the help of a rapid numerical camera, a laser velocimeter, and several accelerometers, all displacements can be evaluated for each loading and for different speeds (fig. 8).

<figure><img src="image_8.png" /><figcaption>Figure 8: Description of experimental investigation</figcaption></figure>
To evaluate in-situ rails vibrations, an experimental device has been developed. In association with accelerometer and velocimeter, an optical technique is used for the local plane displacement measurement. This method is based on the tracking of markers which are simply deposited with a felt pen or painted on a surface (see fig. 9). A high speed CCD video camera (MotionVision CA-D6) with a 532×512 pixel spatial resolution and a data acquisition card associated with a software (SYSMAT Industries VNR250) are used to monitor the studied surface, to transfer and to record the corresponding images to a Personal Computer. With this arrangement, the maximum acquisition frequency is equal to 250 frames per second. In this case, the acquirement time is close to 7 seconds, either approximately 1,800 images.

Then the device allows to measure longitudinal and vertical displacements of the rail as well as deformations. Besides, it gives us an accurate value of the train speed thanks to an infrared detector cutting the traffic way. As the camera is fixed in the vicinity of the track (2 meters), an accelerometer is laid on the ground near the latter in order to correct displacements transmitted from the ground to the optical system. The Laser velocimeter provides indications on lateral rail displacements (due to a light track curve). This information will be necessary to estimate the slipping problem. An accelerometer is magnetized to the rail and can be used in a vertical or lateral position to confirm previous results. An other accelerometer is glued on a sleeper and allows to determine some information on pads behaviour. Finally five accelerometers are laid on the ground and the ballast. Distances between each of them can vary from a site to an other (function of estimated wavelengths and possibility of arrangement).

For the whole of the three sites, measurements were obtained for a thirty of trains, offering a parametric diversity taking into account the speed of the train and the traffic type (freight, “autorail” or “corail”).

## 5. EXPERIMENTAL RESULTS

Displacements of each track element and displacements on the soil surface were calculated with a reasonable accuracy from recorded accelerations. A particular attention is borne upon the test procedure, the transducers used to record the accelerations and the test parameters such as the sample rate.

Thanks to an asynchronous recording between the numerical and analogical channels, a sample rate of 1000 samples per second is used. So, the acceleration data from accelerometer measurements can be integrated to obtain successfully velocity and displacement data using two numerical integrations.

A small DC offset can be noted in the data due to the bias in the signal conditioning and causes a linear trend in the velocity results as well as in the displacement results. Consequently, digital filter are carefully for removing extraneous errors to the signals. For all measurement data, the trend is removed using a high pass *Butterworth* filter before each integration.

The analysis of all images in fig. 9 is realised in differed time thanks to a program which calculates automatically for each image the mark centre and their positions. The localisation of the centre is done by the calculation of the mark barycentre weighted by the light intensity. The lower limit of the light intensity must be chosen in order to extract correctly the mark from the background. The accuracy is a function of the mark diameter. Local plane rail displacements are finally and easily deduced from co-ordinates of the marks.
<figure><img src="image_9.png" /><figcaption>Figure 9: Numerical analysis process for the evaluation of vertical rail displacements</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 10: Vertical displacements measured by numerical video camera ("Corail" train composed of 2 engines and 6 carriages)</figcaption></figure>

Results of this analysis for a Corail train passage at two different speeds are plotted in figure 11. At 70 km/h, vertical rail displacements (maximum 3 mm) are smaller than for 135 km/h (maximum 9 mm) and are only quasi-static (deformation under each wheel). Fig. 11 shows displacements (absolute mean value) of each track element against the kind of traffic. Note that, in each element, displacements are greater for heavy traffic like a Corail train. The same figure gives an evaluation of mitigation between track elements. Finally fig. 12 shows the shape and amplitude of displacements at the soil surface in two perpendicular points from track near the embankment. Maximum displacements (peak to peak) are then about 1 mm (already important for this low train speed).

<figure><img src="image_11.png" /><figcaption>Figure 11: Vertical displacements of track elements for different traffic passage.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 12: Vertical displacements of ground surface in 2 perpendicular points near the track for a "Corail" train passage.</figcaption></figure>
## 6. CONCLUSION

A two dimensional model of track - layered ground interaction has been developed for the study of low frequency vibration in a railway system. This representation can easily be extended to a three dimensional model. Besides, in situ measurements has allowed to make an evaluation of vertical and lateral displacements in the track and on the ground for different traffics and various speeds according to the speed range of traffic on this railway line.

Today, other researches are necessary to insure the global comparison between results of the numerical simulation with these experimental results. Already these *in situ* measurements constitute an indispensable database not only for this comparison but also to provide tools of estimation according to the traffic configuration and characteristics of the ground surrounding the railway track.

In the framework of this comparison, for the modified site where a palliative solution is focused on a duplication of rails combined with a reduction of sleepers spacing, the measured attenuation of vibration (about twenty percent) has been able to be already justified with the help of the numerical simulation.

## ACKNOWLEDGMENTS

Authors wishes to acknowledge the French Minister of environment government for its financial support under research grant n°DGAD/SRAE/99107 and concerning with the propagation of waves in ground due to a railway traffic moving at high speed.

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