The nonlinear curves between the external static loads of Thermoplastic Polyurethane Elastomer (TPE) rail pads and their compressive deformations were measured. A finite element model (FEM) for a rail-fastener system was produced to determine the nonlinear compressive deformations of TPE rail pads and their nonlinear static stiffness under the static vehicle weight and the preload of rail fastener. Next, the vertical vehicle-track coupled model was employed to investigate the influence of the amplitude- and frequency-dependent stiffness of TPE rail pads on the vehicle-track random vibration. It is found that the static stiffness of TPE rail pads ranges from 19.1 to 37.9 kN/mm, apparently different from the classical secant stiffness of 26.7 kN/mm. Additionally, compared with the nonlinear amplitude- and frequency-dependent stiffness of rail pads, the classical secant stiffness would not only severely underestimate the random vibration acceleration levels of wheel-track coupled system at frequencies of 65–150 Hz but also alter the dominant frequency-distribution of vehicle wheel and steel rail. Considering that these frequencies of 65–150 Hz are the dominant frequencies of ground vibration accelerations caused by low-speed railway, the nonlinear amplitude- and frequency-dependent stiffness of rail pads should be taken into account in prediction of environment vibrations due to low-speed railway.
Urban railway traffic influences the environment by emissions of ground-borne vibration (1–80 Hz) and structure-borne noise (16–250 Hz). Vibrations and noises can sometimes reach such a high level that can hardly be tolerated by neighboring residents especially in heavily populated urban environments. Consequently, the issue of train-induced vibration has received increasing attention, particularly as people become more aware of environmental issues. Moreover, as new lines are proposed, noise and vibration are important aspects that require careful consideration in the planning stage and often form the basis of objections to new rail development.
Numerous efforts have been made to accurately predict environmental vibration and noise generated by urban railway. Nielsen et al. presented a hybrid model for the prediction of ground-borne vibration due to discrete wheel and rail irregularities [
As stated previously, the dynamic component of the vertical wheel-rail contact force due to out-of-round wheels and rail irregularities is an important source of ground vibration and structure-borne noise. This is especially true when the design speed of railway is below the wave velocities in the soil, such as a subway or tram, where the free-field response can be dominated by the dynamic loads [
In recent decades, a large proportion of models and algorithms have been introduced and developed to calculate the dynamic random loads of a vehicle-track coupled system. In general, a vehicle-track coupled model is composed of a vehicle model, a track model, and a wheel-track coupled relation. The vehicle model has evolved from a multibody model [
Although there have been a large number of vehicle-track coupled models and the corresponding algorithms, there remains a discrepancy between the predicted and measured vibration responses for a vehicle-track coupled system, especially in the frequency domain. The problem probably is related to the calculation parameters used in the vehicle-track coupled models. In a vehicle-track coupled system, there are inevitably polymer materials for vibration attenuation, such as rail pads, under sleeper pads [
In recent years, the influence of the frequency- and amplitude-dependent dynamic parameters of polymer materials in vehicle-track coupled system has been investigated. However, there are still some unsolved issues. Wei et al. used a frequency-domain algorithm of vehicle-track coupled system and the existing experimental results of frequency-dependent stiffness of rail pads to investigate the influence of frequency-dependent stiffness of rail pads on the frequency-domain distribution of vibrations created by subway in the frequency range of 0~200 Hz [
For the purpose of investigation into the effect of the nonlinear amplitude- and frequency-dependent dynamic stiffness of rail pads on the frequency-domain random vibration responses of vehicle-track coupled system, the rail pads of Thermoplastic Polyurethane Elastomer (TPE) typically used in Chinese subway fasteners were chosen as the focus of this study. Firstly, the nonlinear curves between the static loads of TPE rail pads and their corresponding compressive deformations were measured with the universal testing machine (Section
The nonlinear curves between the static loads of TPE rail pads and their corresponding compressive deformations were measured with a universal testing machine (Figure
Load testing of TPE rail pads: (a) universal testing machine and (b) test setup.
First, a piece of the prototype TPE rail pad was installed between the loading plate of universal testing machine and the bearing plate (Figure
The procedure outlined above represents the test procedure for a single piece of TPE rail pad. In this research, a total of three pieces of TPE rail pads were measured to minimize the experimental error from a single piece of rail pad.
A load-deformation curve was obtained from each of the three TPE specimens. In order to eliminate experimental error, the three load-deformation curves were averaged; the resulting load-deformation curve is shown in Figure
The average load-deformation curve of three TPE rail pads.
In previous research, the static stiffness of rail pads was simply regarded as the linear secant stiffness, which can be calculated with
In order to accurately obtain the variation of the static stiffness of the test pads with the external load amounts, the load-deformation curve in Figure
The stiffness-deformation curve of TPE rail pads.
According to the test results presented in Section
The nonlinear static FEM of the rail-fastener system was established by using commercial software (ANSYS). In this FEM model, a steel rail of 60 kg/m with fasteners installed at the interval of 0.6 m was simulated using Beam4 and Combin39 elements in ANSYS, respectively. Combin39 elements are capable of simulating the nonlinear relation between the loads and the deformations. The vertical stiffness of a rail-fastener system is composed of the spring stiffness and the pad stiffness. Since the spring stiffness is only 0.5~1.2 kN/mm, the stiffness of the rail pad can be approximately regarded as the stiffness of the entire fastener system. Thus, according to the nonlinear relationship between the external loads and the compressive deformations of rail pads, Combin39 elements were used to simulate the nonlinear mechanical behavior of the entire fastener system.
In the nonlinear static analysis, the loads imposed on a rail involved the rail weight (60 kg/m), the preload on each fastener (20 kN), and the half of vehicle weight (320 kN). The distance between the four wheels is listed in Table
The parameters of Chinese “Type A” subway vehicle.
Parameters | Magnitude |
---|---|
Mass of vehicle body |
38500 |
Mass of vehicle bogie |
2980 |
Mass of vehicle wheelset |
1350 |
The moment of inertia of vehicle body |
2.5 × 106 |
The moment of inertia of vehicle bogie |
3.6 × 103 |
The stiffness and of the primary suspension |
2.1 × 106 |
The damping of the primary suspension |
4.9 × 104 |
The stiffness of the secondary suspension |
2.5 × 106 |
The damping of the secondary suspension |
2.0 × 105 |
The length between two bogie centers in a vehicle/(m) | 18 |
The vehicle wheelbase/(m) | 2.4 |
Based on the nonlinear static results of the FEM for the rail-fastener system, the compressive deformations of TPE rail pads at the various positions of a rail under the preload of the fasteners and the half of vehicle weight are shown in Figure
The compressive deformations of TPE rail pads at the different positions of a rail under a rail-fastener preload of 20 kN and the half of vehicle weight (320 kN).
It can be observed from Figure
In light of the fact that the compressive deformations of TPE rail pads under any one bogie will essentially have the same distribution (see Figure
The static stiffness of TPE rail pads at different positions under a bogie.
The vertical vehicle-track coupled model applied for calculation of the random vibration of the vehicle-track system due to track irregularity has been reported by Wei et al. [
The parameters of the monolithic nonballasted track.
Components of track | Parameters | Magnitude |
---|---|---|
Steel rail | Young’s modulus |
2.06 × 1011 |
Area moment of inertia |
3.22 × 10−5 | |
The mass in one meter |
60.64 | |
|
||
Rail pad | The stiffness |
Variation with amounts and the frequencies of loads |
The damping |
7.5 × 104 |
Vehicle-track coupled dynamic model.
Additionally, in the vertical vehicle-track coupled model, the linear wheel/rail contact stiffness
In (
Thus, the linear wheel/rail contact stiffness
In the dynamic analysis, there is no relative movement between vehicle and track, only irregularity movement of the track. The positions between the vehicle wheels and the rail fasteners in the vertical vehicle-track coupled model are the same as those in the nonlinear static analysis of the rail-fastener FEM (Section
Based on the experimental results obtained in other studies [
In (
In order to compare the influence of the linear secant stiffness, the nonlinear frequency-dependent stiffness, and the nonlinear amplitude- and frequency-dependent stiffness of rail pads on random vibration of a vehicle-track coupled system, three calculation cases were designed on basis of the nonlinear static results of the rail-fastener system, as shown in Table
The calculation cases.
Case |
|
|
---|---|---|
1 | 1.4 × 26.7 kN = 37.4 kN | 0 |
2 | 1.4 × 26.7 kN = 37.4 kN | 0.15 |
3 | 1.4 × (19.1~37.9 kN) = 26.7~53.1 kN | 0.15 |
In Case 1, the dynamic stiffness of all rail pads in the vehicle-track coupled model is considered to be 1.4 times the linear secant static stiffness calculated with (
Considering that the study was mainly focused on the influence of the amplitude-dependent stiffness of rail pads on the random vibration of a vehicle-track coupled system,
According to the designed cases listed in Table
Power spectral density (PSD) of the vertical random vibration acceleration of a vehicle body due to the linear secant stiffness, the nonlinear frequency-dependent stiffness, and the nonlinear amplitude- and frequency-dependent stiffness of rail pads.
Power spectral density (PSD) of the vertical random vibration acceleration of a vehicle bogie due to the linear secant stiffness, the nonlinear frequency-dependent stiffness, and the nonlinear amplitude- and frequency-dependent stiffness of rail pads.
Power spectral density (PSD) of the vertical random vibration acceleration of a vehicle wheel due to the linear secant stiffness, the nonlinear frequency-dependent stiffness, and the nonlinear amplitude- and frequency-dependent stiffness of rail pads.
Power spectral density (PSD) of the vertical random vibration acceleration of a steel rail due to the linear secant stiffness, the nonlinear frequency-dependent stiffness, and the nonlinear amplitude- and frequency-dependent stiffness of rail pads.
It can be observed from Figure
It is also found from Figure
Similarly, the calculated dominant frequency domain of vehicle bogie in this paper also accords with the summary about the actual vibration generated by railway in [
Upon further observation, it is found that, within the frequency scope of 20~50 Hz, the vertical random vibration responses of the vehicle bogie in Case 1 are highest, followed by those in Case 2, with the responses in Case 3 being the lowest. For example, the PSD of the vertical random vibration acceleration of the vehicle bogie is 0.087 m2/s4/Hz at 37 Hz (point “A” in Figure
Due to the little effect of the three types of rail pad stiffness on the random vibrations at frequencies of 20~50 Hz, the random vibrations at frequencies higher than 50 Hz are the subsequent focus in this study.
It is clear from Figure
It is found that the dominant frequency of vertical random vibration acceleration of vehicle wheel is 57.6 Hz, which has a corresponding maximal PSD of 4.2 m2/s4/Hz in frequency domain in Case 1. In Case 2, the maximal PSD of vertical random vibration acceleration of vehicle wheel increases to 7.0 m2/s4/Hz which is 1.7 times the maximal PSD for Case 1; the corresponding dominant frequency also rises to 66.7 Hz which is 9.1 Hz higher than the dominant frequency in Case 1. Compared with Case 2, the maximal PSD of vertical random vibration acceleration of the vehicle wheel further increases by about 10% to become 7.8 m2/s4/Hz in Case 3, and the corresponding dominant frequency further rises by 8.4 Hz to become 75.1 Hz. It can be concluded that the nonlinear frequency-dependent stiffness or the nonlinear amplitude- and frequency-dependent stiffness of rail pads not only increases the frequency-domain amplitudes of random vibration responses of the vehicle wheel, in a certain frequency domain (50~150 Hz in the cases considered in this paper), but also moves the frequency-distribution of random vibration responses of the vehicle wheel to shift to the higher frequencies.
It can be seen from Figure
Similar to the variation of random vibration responses in a vehicle wheel with three types of the stiffness of rail pads, the influence of the nonlinear amplitude- and frequency-dependent stiffness of rail pads on the random vibration responses of steel rail is the most conspicuous within the frequency scope of 65~185 Hz, the influence of the nonlinear frequency-dependent stiffness of rail pads is secondary, and yet the influence of the linear secant stiffness of rail pads is relatively minimal.
In the three cases presented in this paper, the maximal PSD of vertical random vibration acceleration of the steel rail is 4.1 m2/s4/Hz with a dominant frequency of 76.1 Hz in Case 1. The maximal PSDs of vertical random vibration acceleration of a steel rail and their corresponding dominant frequencies increase to 6.2 m2/s4/Hz at 84.7 Hz in Case 2 and to 6.9 m2/s4/Hz at 85.0 Hz in Case 3, respectively. It is concluded that, contrasted with the linear secant stiffness, both the nonlinear frequency-dependent stiffness and the nonlinear amplitude- and frequency-dependent stiffness of rail pads significantly enlarge the random vibration of the vehicle wheel and steel rail in a certain frequency range (65~150 Hz in the calculation example presented in this paper). Moreover, the effect of the nonlinear amplitude- and frequency-dependent stiffness of rail pads is obviously more significant than the influence of the nonlinear frequency-dependent stiffness of rail pads.
Using rail pads of Thermoplastic Polyurethane Elastomer (TPE) usually used in Chinese subway fasteners as an example, the relationship between the static loads of TPE rail pads and their corresponding compressive deformations was measured with the universal testing machine. The nonlinear static analysis with application of a rail-fastener FEM was performed to quantify the compressive deformations of rail pads and their corresponding static stiffness under the static vehicle weight and the preload of rail fasteners. Next, based on the nonlinear static results of the rail-fastener system, the effect of the linear secant stiffness, the nonlinear frequency-dependent stiffness, and the nonlinear amplitude- and frequency-dependent stiffness of rail pads on the frequency-domain random vibration responses of vehicle body, bogie, wheel, and steel rail was investigated. According to the results presented in this study, some conclusions and suggestions are summarized as follows: The relationship between the static loads of TPE rail pads and their corresponding compressive deformations as measured with the universal testing machine is apparently nonlinear. With application of the nonlinear static FEM of a rail-fastener system, it can be found that the static stiffness of TPE rail pads is 19.1 kN/mm under a rail-fastener preload of 20 kN, and the maximum static stiffness of TPE rail pads is 37.9 kN/mm under both a vehicle weight of 640 kN and rail fasteners preload of 20 kN. However, in previous research, the linear secant static stiffness of TPE rail pads was found to be 26.7 kN/mm, which is apparently different from the actual nonlinear static stiffness. Therefore, since the rail pads have a strong nonlinear feature, it is unreasonable to use the linear secant stiffness without considering the variation of rail pad stiffness with the external load amounts. The random vibration responses in a vehicle-track coupled system demonstrate that the influence of the variation in stiffness of rail pads on the vertical random vibration of the vehicle body is negligible. Compared with the linear secant stiffness of rail pads, for the cases considered in this paper, the nonlinear frequency-dependent stiffness or the nonlinear amplitude- and frequency-dependent stiffness of rail pads has no influence whatsoever on the random vibration levels of the vehicle bogie, wheel, and steel rail in the frequencies below 20 Hz, and it slightly reduces these responses in the frequency range between 20 and 50 Hz. But it drastically enhances the random vibration levels at frequencies of 65~150 Hz, which is none other than the dominant frequency domain of the environment vibration acceleration caused by low-speed urban railway. The nonlinear frequency-dependent stiffness or the nonlinear amplitude- and frequency-dependent stiffness of rail pads not only significantly enlarges the frequency-domain amplitudes of the random vibration responses of vehicle wheel and steel rail but also shifts the dominant frequency-distribution of the vehicle wheel and steel rail to the higher frequencies. Moreover, the effect of the nonlinear amplitude- and frequency-dependent stiffness of rail pads is more significant than the influence of the nonlinear frequency-dependent stiffness of rail pads.
In summary, it can be concluded that if there are polymer materials with the strong nonlinear stiffness in a vehicle or track system, the nonlinear amplitude- and frequency-dependent stiffness of these materials must be taken into consideration in order to precisely predict the random vibration responses of vehicle-track coupled system.
The authors declare that they have no competing interests.
This research was supported by National Natural Science Foundation of China (Grant no. 51578468), Fundamental Research Funds for the Central Universities of China (Grant no. 2682015CX087), National Outstanding Youth Science Foundation of China (Grant no. 51425804), and Joint Funds from both Chinese High-Speed Railway Company and the National Natural Science Foundation of China (Grant nos. U1234201 and U1434201).
The authors would like to thank Chang-sheng Zhou, Ying-chun Liang, and Zi-xuan Liu for their assistance in laboratory experiment and data processing.