Track Model with Nonlinear Elastic Characteristic of the Rubber Rail Pad

This paper presents a new basic nonlinear track model consisting of an infinite EulerBernoulli beam (rail) resting on continuous foundation with two elastic layers (rail pad and ballast bed) and intermediate inertial layer (sleepers). The two elastic layers have bilinear elastic characteristic obtained from the load-displacement characteristic of the rail pad and ballast. A time-varying load with two components time-constant one and harmonic other, representing the wheel/rail contact force is considered as the track model input. Rail deflection due to the time-constant component of the load is obtained solving the nonlinear equations of the balance position. Subsequently, the structure of the nonhomogeneous foundation is determined. Dynamic response of the rail in terms of receptance due to the harmonic component of the load is calculated using the linearised track model with nonhomogeneous elastic characteristic. Influence of the time-constant component and the reflected waves due to the nonhomogeneous foundation are presented.


Introduction
Track modelling is crucial for the study of the interaction between the trains and track. This research topic has many technical applications because the railway traffic is developing unwanted vibration and noise which affect the track superstructure, the rolling stock, the passenger comfort, and the life conditions in the track area.   [8] The rail pads play an important role in the wheel/rail dynamics because they are the closest elastic elements to the wheel/rail contact patch. The rail pads are assembled between the steel rails and sleepers to distribute the wheel/rail load over a large surface avoiding the resultant fatigue and shock stresses. Usually, the rail pads are made of rubber, ethylene propylene diene monomer (EPDM), thermoplastic polyester elastomer (TPE), high-density polyethylene (HDPE), and ethylene-vinyl acetate (EVA), as presented in Figure 2 [7,8]. Such materials are well known in industry and these are extensively studied from many view points to solve various practical issues [9][10][11][12][13][14].
From mechanical viewpoint, the rail pad has nonlinear characteristics in both quasistatic behavior (load-deflection curve) and frequency response function as revealed in some research [8,[15][16][17].
The modelling of fastening systems, including the rail pad modelling has been extensively treated in papers like the references [18][19][20][21]. The main representation of the rail pad is the Kelvin-Voigt system which consists in a linear elastic element working in parallel with a viscous damper element [2]. This model can be applied in the analytical representation of the track. When the FEM is used to model the rails and sleepers, the rail pad should be modelled as a line of many discrete Kelvin-Voight systems or even as an area covered by such systems [18,21]. Other approach regarding the rail pad modelling is represented by the fractional derivative model [20]. The nonlinearity of the elastic characteristic of the rail pad is introduced via polynomial functions [22].
In this paper, a new basic analytical model of the track based on the approximation of the nonlinear elastic characteristic of the rail pad using the bilinear functions is presented. Also, the nonlinear elastic characteristic of the ballast bed [8] is introduced via a bilinear function. Rail pad and ballast damping is considered as hysteretic damping. In short, track model consists of a beam (rail) resting on a continuous foundation with two elastic layers, both with bilinear elastic characteristic. The rail is under a time-  Figure 3 shows the nonlinear elastic characteristic of a rail rubber pad and ballast. Rail rubber pad is part of the K-type rail fastening system [23] and its elastic characteristic consists of the low (elastic) and upper (rigid) limit curves (Figure 3a). Force-deformation diagram of a particular rail rubber pad must be situated between the two limit curves to be proper for using. The mean elastic characteristic is also shown in Figure 3a, and this is taken as reference in the followings. When mounted on the track, the rail rubber pad is prestressed by 10 kN, and the deformation of 0 mm in the figure corresponds to the prestressed condition. of the elastic characteristic of the rail rubber pad for the K-type rail fastening system (after [23]); (b) ballast (after [15])  (1) where Q is the force, dthe deformation and CHthe Hertzian constant; CH = 800 MN/m 3/2 .
These elastic characteristics could be included into the track model in a simplified form as two piecewise linear functions (bilinear approximation (Figure 4)).
where ke and kr are the 'elastic' and 'rigid' elastic constants of the bilinear characteristic, (wer, Qer = kewer) are the coordinates of the transition point from elastic to rigid portion of the bilinear characteristic, and (wm, Qm = Qer + kr(wmwer)) are the coordinates of the limit point of the bilinear characteristic. Construction of the bilinear characteristic may be obtained using the least square method, which means that the sum of the squared residuals must be minimised. The residual is the difference between the value of the exact characteristic, and the value of the bilinear approximation.
Results of the applying of the least square method is displayed in Figure 5, where the exact characteristics of the rail rubber pad and ballast and their bilinear approximations can be traced for the static load of 40 kN. Starting from the discrete points of the rail rubber pad elastic characteristic, many other points have been obtained by spline interpolation and then, the bilinear approximation has been calculated ( Figure 5a). The elastic characteristic of the ballast has analytical shape (Eq. 1) and this form has been used to obtain the bilinear approximation ( Figure 5b Errors introduced by the bilinear approximation does not exceed 6.9% for the rail rubber pad and 3.75% for the ballast. The RMS value of the errors is less than 3.0% for the rail rubber pad and 1.7 % for the ballast.

Track model
In this section, the track model depicted in Figure 6 is considered. The model is based on the symmetry property of the track with respect to the longitudinal axis, and due to that, only half of its structure is modelled. Influence of the spacing of the sleepers is neglected and, therefore, the model of the track is reduced to a beam on a continuous foundation with two elastic layers. The first elastic layer under the beam shapes the elasticity of the rail pads, while the second elastic layer shapes the ballast. The rail is assimilated with an infinite Euler-Bernoulli beam having constant section and whose linear mass is equal to the mass of the rail per unit length. The beam has the bending stiffness given by the moment of inertia of the cross section, I, and by the longitudinal modulus of elasticity of the material from which the rail is made, E. The internal friction in rail is neglected. The model of the rail pads is represented by Winkler foundation with bilinear characteristic and hysteretic damping. The elastic constants are k11 and k10, and the loss factors are 11 and 10. The stiffness jump occurs when the relative displacement between the rail and the sleepers is uo = wer1 and the elastic constant of the rail pad becomes k12 = k11 + k10. Corresponding, the loss factor takes the value of 12 = (k11 11 + k10 10)/k12.
The elastic and damping characteristics of the ballast are shaped similarly to the case of the rail pads. The elastic constants are k21 and k20, and the loss factors are h21 and h20 for ballast. The jump of stiffness occurs when the sleeper displacement is zo = wer2 and the elastic constant of the ballast is k22 = k21 + k20 and, the corresponding loss factor is 22 = (k21 21 + k20 20)/k22.
The values of the elastic constants kij with i, j = 1, 2 result from the above calculated stiffnesses of the rail pad and ballast divided to the sleeper bay, d, The elastic constants k10 and k20 are introduced due to the coherence reason of the track model. Knowing the k11,12 and k21,22 elastic constants given by Eq 3, the values of the k10 and k20 elastic constants result.
The bilinear characteristic of each elastic layer can be designed in terms of the uniform load [N/m] versus displacement (Figure 7). Transition points of coordinates (uo, q10) for the rail pad and (zo, q20) are related by 10  Next, corresponding to the results above, it considers that q10 > q20. According to this type of the track model, the sleepers are represented by an inertial layer that takes neither shear force nor bending moment. The mass per unit length of the inertial layer is calculated with the relation where Ms is the sleeper mass and d is the sleeper bay.
As the speed of trains is much slower than the speed of the propagation of the elastic waves through the track, the effect induced by the displacement of the force is neglected and, therefore, it assumes that the force has fixed support.
The origin of the reference system, Oxz, against which the movement of the beam and the inertial layer (of the sleepers) is studied, is in the neutral axis of the beam. The reference position is the position of the beam and the inertial layer under its own weight on the elastic foundation.
In a section x, the displacement of the beam is w(x, t), and that of the inertial layer, z(x, t). For simplicity, it is considered that the force acting on the beam has the support in the section of the origin of the reference system.
Given that the force Q(t) has two components, one static and the other dynamic, and the response of the beam and the inertial layer will have two components accordingly . The static components w(x) and z(x) can be calculated considering that only the static component of the force, Q, acts on the beam, while the dynamic components, w(x, t) and z(x, t), result as an effect of the motion of the system under the action of the dynamic component of the force, Q(t).
To find the balance position of the track under the static load Q, the nonlinear model of the track is used. Depending on the magnitude of the static load, three cases can be exposed: for small static load, the rail pad and ballast work on the elastic portion of their bilinear characteristics.
For medium values of the static load, the ballast works around the static load section on the rigid portion of the bilinear characteristic and on the elastic portion in rest; the rail pad works only on the elastic portion.
For high values of the static load (Figure 8), there is a region around the static load section of 2l1 length, where both rail pad and ballast work on the rigid portion of their bilinear characteristics. Beyond this region, there is other region of l2 length, where the ballast works on the rigid portion of its bilinear characteristic, meanwhile the rail pad works on the elastic portion. Further on, there is other region, the third, which expands to infinite, where both rail pad and ballast work on the elastic portion of the bilinear characteristics. Obviously, the track structure has the loading section as the centre of symmetry.
Next, the solution of the third case is shown because the fact that this case is usual. To this aim, the boundary condition method is applied.  (Figure 9). This characteristic shows the evenly distributed load acting on the rail depending on its displacement. There are three branches with the stiffnesses k1, k2 and k3 and two transition points of coordinates (w12, q12) and (w23, q23). First point corresponds to the point of transition in the ballast characteristic, and the second one corresponds to point of transition in the rail pad characteristic. It holds 11 21  11 22  12 22  1  2  3  11  21  11  22  12  22   12  20  23  10 , , , ,.
The rail and sleeper displacement for the two transition points can be calculated using the equations 21 11  21  12  23  11  22  22   11  21  12  23  22  22   1  ,  1 1 , In the following, only half of the track model is considered due to the symmetry ( Figure 10).  (10) where: The boundary conditions must be fulfilled -at x3 = 0, the slope is zero and the shear force is -Q/2 -at x3 = l1 and x2 = 0, the continuity conditions -at x2 = l2 and x1=0, the continuity conditions Inserting Eqs. (17) in the boundary conditions, a set of 11 nonlinear equations results with the unknowns Ai, where i = 1÷ 9 and l1 and l2. Solution can be obtained using the Newton-Raphson method.   (20) are the eigenvalues of the three differential equations (18). The complex amplitude of the rail displacement must verify the boundary conditions -at x = 0, the symmetry condition and the amplitude of the shear force is Q -at x = l1 and x = l1+l2, the continuity conditions Inserting Eqs. (19) in the boundary conditions (21,22), a set of 10 linear equations results with the unknowns Bi, where i = 1÷10. Solution to this system can be obtained numerically for any angular frequency  and then, the complex amplitude of the rail emerges from Eqs. (19).
Finally, the rail receptance results where w0 and z0 are the rail and sleeper deflection in acting section obtained from the nonlinear model, and ke is the equivalent stiffness 12 Similar, taking ke1 = k12 and ke2 = k22, the linear model with homogeneous foundation with 'rigidrigid' stiffness of the two elastic layers ('rigid-rigid' LMHF) can be used as comparison.

Results and discussions
In this section, the track model with bilinear characteristic for the rail pad and ballast is applied to bring in light the effect of the bilinear characteristics upon the dynamic response of the rail. The parameters for the track model are presented in Table 1.  Figure 13 shows the rail and sleepers deflection along the track due to the static load of 100 kN calculated with the nonlinear model and equivalent linear model with homogeneous foundation. First, the displacement of the rail is greater than the displacement of the sleepers, aspect which is justified by the effect of the elasticity of the rail pad. The curve showing the rail displacement calculated according to the nonlinear model shows the transition points first, in the rail pad characteristic and then, in the ballast characteristic. The results obtained with the two types of models are very close in the vicinity of the section in which the static force acts and then some differences appear. Thus, it is found that the displacement the rail and the sleepers along the track are smaller in the case of the equivalent linear model with homogeneous elastic characteristic. Consequently, it shows that this model overestimates the stiffness of the track. Both models with homogeneous foundations exhibit two resonance frequencies and one antiresonance frequency due to the dynamic absorber effect induced by the sleepers (Figure 14 a). The two resonance frequencies are at 60.7 and 145.5 Hz for the equivalent LMHF and at 76.6 and 436.8 Hz for the 'rigidrigid' LMHF. The antiresonance frequency is at 145.5 Hz for the equivalent LMHF and at 215.2 Hz for the 'rigid-rigid' LMHF.
The rail receptance calculated using LMNHF exhibits several peaks and dips. There is no match between the rail receptance derived from LMNHF and equivalent LMHF, excepting the high frequencies range. The explanation of former aspect lies in the fact that at high frequencies, inertial forces are prevalent and therefore the influence of elastic forces induced by the foundation (rail pad and ballast) is insignificant, and the rail receptance calculated with the two models is the same, depending on only the linear mass of the rail and frequency.
It should be noticed that the most important peaks calculated with LMNHF are at the resonance frequencies predicted by 'rigid-rigid' LMHF. Also, the main dip of the rail receptance computed using LMNHF is located at the same frequency as the antiresonance frequency of the 'rigid-rigid' LMHF. There are several similitudes between the rail receptance calculated with LMNHF and the ones calculated with 'rigid-rigid' LMHF: at low frequencies up to the first resonance frequency, between the antiresonance frequency and the second resonance frequency, and at the high frequencies. The main differences appear around the two resonance frequencies and within the range between the first resonance and the antiresonance.
The explanation for this dynamic behaviour of the rail lies in two aspects. First, it should be noted that in the vicinity of the section where the harmonic excitation force acts, both the rail pad and the ballast work on the rigid branch of the bilinear characteristic. As a result, the dynamic behaviour of the rail in https://doi.org /10. the excitation force section is strongly influenced by the rigid characteristics of the rail pad and the ballast, thus explaining the similarities above signalled. On the other hand, the change in the stiffness of the foundation due to the transition from the rigid to the elastic branch of the rail pad and ballast characteristics determines the reflected waves that change the dynamic behaviour of the rail as described.
Damping reduces the differences between the results delivered with the LMNHF and the "rigid-rigid" LMHF ( Figure 14b). However, the differences signalized in the rail receptance diagram when using the equivalent LMHF remain practically unchanged.  Figure 15 presents the rail receptance for the static load of 50 kN calculated applying the identic methods as above and using the same loss factors for the hysteretic damping. In this case, the distances l1 and l2 are shorter than those calculated when Q = 100 kN, respectively, l1 = 0.426 m and l2 = 0.126 m. Also, the elastic layers stiffness of the equivalent LMHF differ: k1e = 94.95 MN/m 2 and k2e = 32.78 MN/m 2 . This time, the rail receptance calculated with LMNHF is higher at low frequencies because the length of the stiff foundation is smaller, and the track becomes more flexible. There are distinct differences between the results obtained with the three models for a wideband frequency.

Conclusions
In this paper, a new track model incorporating the nonlinear elastic characteristics of the rubber rail pad and ballast is presented. The key features of the track model are the bilinear approximation of the nonlinear elastic characteristics and the manner in which the rail response is treated in the sense that it has two components corresponding to the static load due to the weight of the vehicle on the wheel, and the dynamic load caused by the wheel-rail interaction in the presence of unevenness.
The elastic characteristics of the rubber rail pad and ballast are described as nonlinear functions of static load depending on deflection. These nonlinear functions can be implemented in the track model using the bilinear approximation depending on the magnitude of the static load. Subsequently, the equivalent elastic characteristic of the track model is built in terms of uniform load versus rail displacement and three branches of constant stiffness each result. In this way, the track model under a static load is obtained. The nonlinear equations of equilibrium are solved using an iterative algorithm based on the Newton-Raphson method. Calculation of the rail and sleeper deflection allow to identify the position of the transition points between the branches of the equivalent elastic characteristic of the track. Further on, the track model is linearised around the equilibrium position under static load and the linearised model with nonhomogeneous foundation emerges.
Numerical application shows that the dynamic behaviour of the rail in terms of receptance depends on the static load and this dependence cannot be captured using other linear models with homogeneous foundation.
Further research is focused on the implementation of the nonlinear track model into the model of the wheel-rail interaction to outline the influence of the elastic characteristic of the rail rubber pad on the wheel-rail vibration.