Once the dynamic properties of the elastomeric material of the ERS were investigated experimentally, a modified Zener model for its frequency- and preload-dependent stiffness and damping was identified and validated with the laboratory test results. Similar to the determination of the dynamic properties, the model was also developed for unit length of the elastomeric element. The model of unit length of the ERS was subsequently set up with which time domain simulations were performed to illustrate the effect of the non-linear dynamic behaviour of the elastomeric elements in the train–track dynamic interaction.
3.1. Non-Linear Mechanical Model of Elastomeric Material
The macro-mechanical model of unit length of the elastomeric material of the ERS was a modified Zener model [
25,
26] and its structure is presented in
Figure 4.
The variable
z represents the deformation of the elastomeric material, and the
F represents the external force to realise the deformation. In the case of the described laboratory tests, the applied external force had two components: the static preload
P and the mono-harmonic force
F1 cosΩ
t. The reaction force generated by the elastomeric material had the same magnitude and an opposite direction of the external force. Parameters
k1*,
k2* and
c2* stand for the values of the springs and damper, respectively. The original Zener model was linear in its formulation; the non-linearity is herein included considering the effect of the static preload as a simple multiplication factor of the base parameter (stiffness or damping), which are described as:
where
k1,0,
k2,0, and
c2,0, are constant reference values of
k1*,
k2*, and
c2*;
P is the static preload applied on the ERS;
F1 is the amplitude of the dynamic load; Ω is the dynamic load frequency with the unit of rad/s;
P0 is a constant reference preload; and
x is a non-dimensional constant number. All model parameter values refer to unit length of the elastomeric material of the ERS.
The equivalent stiffness (
kequiv.) and damping (
cequiv.) of the non-linear macro-mechanical model, for a certain static preload and dynamic force frequency, are computed as:
The values of the model parameters,
k1,0,
k2,0,
c2,0, and
x, are determined through a minimisation procedure of the difference between the modelled and experimental equivalent stiffness and damping for all test frequencies and preloads. The constant reference preload
P0 is set as 64 kN. The objective function of the minimisation procedure is defined as:
where
i and
j are the indices of the preload case and test frequency, respectively;
np and
nΩ are the total number of the preload cases and test frequencies, respectively;
kequiv.,num. and
cequiv.,num. are the equivalent stiffness and damping computed by the numerical model; and
kequiv.,exp. and
cequiv.,exp. are the equivalent stiffness and damping computed by the experimental results. A weight of 10 was applied to the modelling error of the equivalent damping with respect to the equivalent stiffness for a better performance of the minimisation procedure due to the difference between the orders of magnitude of the two parameters. The identified model parameter values are presented in
Table 3.
The force–displacement cycles were reconstructed numerically using the identified model parameters and compared to the experimental ones in
Figure 5 for two test conditions. Note that when the numerical results are compared to the experimental ones, the model parameter values, used to obtain the numerical results, refer to a 750 mm segment of the ERS.
The similarity of the measured and reconstructed loops indicates the ability of the model to reproduce the equivalent stiffness and damping at different frequencies and preloads. For a better illustration of the entire ensemble of test results, in
Figure 6, the experimental and modelled equivalent stiffness (upper plot of each part) and viscous damping (lower plot of each part) of unit length of the elastomeric element of all test conditions are compared.
It can be observed that the modelled equivalent stiffness and viscous damping are quite similar to the experimental ones, in terms of both the absolute values and the variation trend as in function of preload and frequency. The best comparison is found for the damping, while for the stiffness, the maximum preload is better matched with respect to the lower preloads. It is worth noting that the modelled equivalent stiffness and damping are derived of different test conditions but are computed by a single set of model parameters. Therefore, it is acceptable that not all modelled data match the experimental ones.
In order to verify whether the model could predict the dynamic behaviour of the elastomeric element, the reaction force generated by the elastomeric element of the 750 mm ERS sample to a specific rail vertical displacement time history is predicted by performing a time domain simulation with the developed model and compared to the corresponding experimental result. Referring to the scheme of the model presented in
Figure 4, the input of the simulation is the deformation time history of the elastomeric material
z, while the output is the external force
F to realise such deformation (the reaction force has the same magnitude and an opposite direction of the external force). The rail displacement time history is obtained by converting the static rail deflection distribution along the track, given by a single-beam Winkler model, on which wheelset loads are modelled as a sequence of discrete loads, with a constant train velocity to simulate the passage of a train. The scheme of the Winkler model with the wheelset loads is shown in
Figure 7a together with the resultant static rail deflection distribution along the track, whose conversion to the deformation time history of the elastomeric material is illustrated in
Figure 7b. The values of the wheelset loads
Q (110 kN), wheelbase
pw (3 m), and bogie base
pb (19 m) refer to a passenger coach of ETR 500. The train velocity
v is set at 200 km/h.
The profile of the rail deformation time history is equivalent to that of the static rail deflection distribution along the track—the only difference is the conversion from the spatial coordinate to time through the constant velocity.
The external force can be numerically integrated with the following differential equation, which can be obtained from
Figure 4, that describes the relationship between the external force
F and the deformation of the elastomeric material
z:The calculations of k1*, k2*, and c2* are presented in Equation (1). Since the external force F(t) to be integrated is not a combination of a static preload and a mono-harmonic force, the static preload P in the calculation of k1*, k2*, and c2* are substituted by the instantaneous value of the external force F(t).
A corresponding laboratory test with the same input rail vertical displacement time history was performed with the test rig shown in
Figure 2, and the external force to realise the displacement was measured.
The experimental and numerically simulated external force time histories are compared in
Figure 8. The visualisation of the time history was limited to a time interval corresponding to the passage of a single bogie because it was verified that the portion of the time history related to the passage of a single bogie is independent from the other ones.
The simulated external force was highly similar to the experimental one. Regarding the trend of the time history, it correctly simulated that the second valley of the external force had a lower magnitude than the first one and the positive force after the two valleys corresponding to the passages of the two wheelsets. Furthermore, the simulated external force was very close to the experimental one in terms of the absolute value.
3.2. Models of Unit-Length of Reference ERS
After developing the model of the elastomeric element, the model of unit length of the ERS was developed, where a mass representing the rail was added on the top of the model of the elastomeric element. In order to investigate the effect of the preload- and frequency-dependent stiffness and viscous damping of the elastomeric element on train–track dynamic interaction, its effect on the dynamic behaviour of the reference ERS was firstly studied. Specifically, the response of the ERS to a harmonic force applied on the rail head was simulated (the scheme of the simulation is illustrated in
Figure 9a). For comparison, simulations were also performed with a linear spring–damper model (
Figure 9b), where the dependence of dynamic properties on preload and frequency were not considered, as it usually happens for classical track model. The stiffness and damping of the linear spring–damper model were 80 MN m
−2 and 10
−3 MN m
−2 s, respectively, referring to unit length of the ERS. The chosen stiffness value is regarded as representative of different frequencies and preloads according to the experimental data (see
Figure 3).
According to the composition of the two models of unit-length of the reference ERS, the dynamic behaviour of the models was apparently strongly influenced by the assigned values of the model parameters. Therefore, for an easier interpretation of the simulation results, in
Figure 10, the equivalent stiffness (upper plot) and viscous damping (lower plot) computed by the non-linear and linear models of unit length of the elastomeric element of ERS in the frequency range of 0 ÷ 250 Hz are compared. It can be observed that the dynamic stiffness of the non-linear model, with a constant preload, increased rapidly at a low-frequency range, i.e., approximately 0 ÷ 20 Hz, and then approached an asymptote. This asymptote increased with the preload according to the model’s formulation. On the contrary, the linear model had a fixed stiffness regardless of the frequency and preload. Concerning the equivalent damping, that of the non-linear model decreased from 1 MN m
−2 s to 10
−4 MN m
−2 s (note that the plot is in logarithmic scale) in the considered frequency range while the linear model had a constant value of 10
−3 MN m
−2 s. From this last figure, it is clear that the use of a viscous damping not depending on frequency can be set only corresponding to one frequency, and as a consequence, there is an overestimation of the damping beyond the setting frequency and an underestimation below the same value.
The large difference in the viscous damping of the two models can have a strong effect on the simulated rail response, both on the transient phase and the steady state phase.
Figure 11 represents the simulated free response of the rail subjected to a constant force of 55 kN with null initial conditions.
The response of the linear model had a period of approximately 0.0054 s (185 Hz) and decreased logarithmically with time. The 185 Hz corresponds to the natural frequency of the linear model of the ERS if it was regarded as a single degree of freedom system. Instead, the response of the non-linear model had a shorter period and a much larger amplitude, even though the simulation conditions are identical. Furthermore, the response seemed to not decrease with time in the presented time window. In fact, the response obtained with the non-linear model also decayed with time, but the decay rate was much lower than that of the linear model. It took 7.6 s for the non-linear model to reach a 90% decrease in the vibration amplitude while the time needed for the linear model was about 0.3 s. This can be attributed to the much lower equivalent viscous damping value of the non-linear model compared to the linear one in the frequency range around 185 Hz, as shown in the lower plot of
Figure 10.
In
Figure 12, results of the steady state response of the rail to the harmonic force (see
Figure 9) computed by the non-linear and linear models are presented.
For the non-linear model, time domain simulation was performed where the rail was subjected to a harmonic force with preload, as shown in
Figure 9a. The ratio of the amplitude between the steady-state response, synchronous with the forcing frequency, and that of the external force was calculated (upper plot of each part) as well as the phase delay between the two signals (lower plot of each part). Both elaborated results were compared to those of the linear model. In
Figure 12a, the results of the non-linear model are obtained with a preload of 6 kN and an amplitude of the harmonic force equal of 1 kN. The external harmonic frequency varies from 10 Hz to 500 Hz with a step of 2 Hz. The result of the linear model is substantially its frequency response function in terms of the rail displacement and is obtained with frequency domain calculation. Regarding the magnitude, the results can be roughly divided into three sections according to the frequency response function of the linear model: a quasi-static zone (approximately 0 ÷ 150 Hz), a resonance zone (approximately 150 ÷ 200 Hz), and a seismic zone (over 200 Hz). The curve of the non-linear model is different from the linear one, mainly in the quasi-static zone and the resonance zone. More specifically, the difference in the resonance zone is more obvious. According to the interpretation of the frequency response function of the linear model, the quasi-static zone is dominated by the stiffness of the model, the resonance zone is dominated by the viscous damping, and the seismic zone is dominated by the mass property. Consequently, it is reasonable that the two curves coincide in the seismic zone since the mass per unit length of the two models are identical. Similarly, the difference is mainly in the resonance zone since the most principal difference of the dynamic behaviour of the two models is the viscous damping (according to the lower plot of
Figure 10). Meanwhile, the difference regarding the frequency of the peak values, about 4 Hz, is limited, since the stiffness values, unlike the damping values, of the two models have the same order of magnitude. Regarding the phase delay, the curves obtained from the two models are similar. That of the linear model decreased from 0 rad to −
, crossing the resonance zone near 185 Hz.
In
Figure 12b, the results of the non-linear model obtained with different preloads are presented (6 kN, 18 kN, 37 kN, 55 kN). The magnitude of the harmonic force equals 1 kN for all preloads. The external harmonic frequency varies from 150 Hz to 210 Hz with a step of 2 Hz. The result of the linear model is identical to the one in
Figure 12a. The frequency range is the resonance zone according to the linear model. Regarding the magnitude ratio, the higher the preload, the higher the peak value and higher the corresponding frequency. The peak values of the curves obtained with the non-linear model are approximately 3 times that of the linear model (note that the plot is in logarithmic scale).
To study the effect of the dynamical properties of the elastomeric element of the ERS on the rail response in the case of the train–track dynamic interaction, a time domain simulation similar to the one presented in
Figure 9 was performed with both the linear and non-linear model. The model compositions remained invariant while the constant force component was substituted by a quasi-static force time history simulating the passage of a train (ETR 500 coach at 72 km/h). The dynamic force component had an amplitude of 20 N and a frequency of 170 Hz, simulating a dynamic force caused by a rail roughness with a wavelength of 120 mm, which is a typical wavelength of short pitch corrugation. The results are presented in
Figure 13 and limited to a 0.6 s time window centred at the time instant of the passage of a bogie in part (a). For a better visualisation of the high-frequency vibration, the time histories are limited to the time window of 1.7 ÷ 1.8 s in part (b).
Regarding the low-frequency response associated with the passage of the wheelsets (approximately 5 Hz), the non-linear model predicted a larger displacement due to the lower stiffness in the concerned frequency range. The difference between the maximum displacements was about 11%. For the high-frequency vibration associated with the dynamic force component, the amplitude obtained with the non-linear model was about 3 times that obtained with the linear model, which is quite close to the magnitude ratio presented in
Figure 12. The effect is more observable before and after the passage of the wheelsets and less obvious during the passage due to the rapid large-scale displacement of the rail.
To extend the effect of the modelled non-linear dynamic behaviour on the rail response to the study of railway issues, taking the rail noise emission and the transmitted force to the subgrade as example, the spectra of the rail velocity and the transmitted force are computed and presented in
Figure 14a,b, respectively. The upper plots illustrate the spectra in the frequency range of 0 to 50 Hz while the lower plots illustrate the spectra in the frequency range of 150 to 200 Hz, where the high-frequency dynamical force is superposed. All spectra are based on a time window of 3 s centred at the instant of the passage of the bogie.
The low-frequency range contribution relates to the passage of the wheelsets, while the main peaks located at 170 Hz in the high-frequency range are associated with the superposed dynamical force representing the effect of the short pitch corrugation. Only for the spectra obtained with the non-linear model can a peak located around 173 Hz be observed, corresponding to the transient response of the rail.
The patterns of
Figure 14a,b are almost identical with the two models in the low-frequency range, since the response of the rail is dominated by the equivalent stiffness, which is only slightly different. Meanwhile, the magnitude of the peaks at high frequency differs, since the equivalent viscous damping of the two models are remarkably different in the concerned frequency range.