1. Introduction
High-speed rail tracks are mainly divided into two types: ballast track and ballastless track. Ballastless track, compared with ballast track, reduces the maintenance workload and cost and increases driving safety and stability of driving; therefore, the ballastless track is becoming increasingly common in high-speed rail [
1]. At present, most of the high-speed rail track construction in China, Japan, Germany, and other countries is for ballastless tracks [
2]. In recent years, the increase in the speed of high-speed rail has posed new challenges to the safety, comfort, and durability of high-speed rail [
3]. Therefore, it is important to establish a reliable ballastless track model for high-speed rail, develop accurate and efficient calculation methods, and study the dynamic performance of high-speed rail ballastless track systems under complex operating environments for the development of ballastless track for high-speed railways in China.
Most of the existing studies on ballastless tracks are based on the assumption of the continuous support of track beams, ignoring the additional dynamic response generated by discrete track pads on the track system. To explain the effect of discrete support, Sheng et al. [
4] investigated the dynamic response of an infinitely long Timoshenko beam supported by discrete sleepers and subject to wheel movement. Dai et al. [
5] proposed a cyclic calculation method based on the moving element method (MEM) and used the Euler beam theory to investigate the dynamic response of a high-speed rail system under the discrete support of trackpads. Based on this cyclic calculation method, Lei et al. [
6] investigated the excitation of the train-track system by sinusoidal wave discontinuities, substituting a three-layer Timoshenko beam model for the Euler beam model and accounting for the impact of discontinuous supports. Moreover, the issue of discontinuities between neighboring track slabs has been oversimplified in most research. To analyze the dynamic response of a ballastless track system, for instance, Yang et al. [
7] suggested a vehicle-track finite element model, where the discontinuity between consecutive track plates was omitted to reduce the thermal expansion effect. According to the study, precast track slabs exhibit stiffness discontinuities at the slab end joints under typical vehicle operation, which poses a serious risk to the track’s structural integrity. In the subject of steel-spring floating plate tracks for urban rail transport, where shear hinges are preferred between adjacent floating plates, the issue of discontinuity between track plates has received much attention in previous studies. To examine the restraint impact of shear hinges between track slabs, Yang et al. [
8] simulated shear hinges using vertical and transverse shear springs. Wei et al.’s [
9] simulation of shear hinges using bending shear spring damping units allowed them to determine the ideal set of parameters for shear hinges between track slabs. In the field of ballastless tracks, there are fewer studies on inter-slab discontinuity; although, some researchers have considered this issue, such as Hussein et al. [
10] and Sadheghi et al. [
11], who modeled the effect of inter-slab discontinuity, but they did not disclose model specifics. In order to address this issue, Xu et al. [
12] established an infinitely long beam model for vehicle-track interaction and proposed a cyclic calculation method based on the Euler beam theory for the track plate discontinuity problem, both of which have significant implications for the advancement of ballastless track models.
The finite element unit method (FEM) is still the primary technique for analyzing vehicle-track vibration at this time. However, analysis using FEM faces the issue of high computational effort and poor analysis efficiency due to the appreciable rise in the train operating speed [
13]. To solve these difficulties in FEM, Andersen et al. [
14] provided a finite element equation for an infinite Euler beam on a linear viscoelastic Kelvin foundation under simple harmonic moving loads; using this method, they obtained the equations for the immobile point loads of orbital beams under kinematic loads. Inspired by these ideas, Koh et al. [
15] used moving coordinates to solve moving vehicle-track vibration problems and named the method the moving element method (MEM). Since then, many researchers have applied MEM to solving vehicle-track problems. Dai et al. [
16] modeled the ballasted track model in three layers: a continuous rail layer, a discrete sleeper layer, and a track bed; the dynamic response of the train-track system was calculated using the moving unit method. Lei et al. [
17] established a ballastless track unit with a three-layer Euler beam model as the basis, derived the corresponding stiffness, mass, and damping matrices, and investigated the dynamic characteristics of the train and ballastless track system, but they neglected the effect of discrete supports and the non-linear relationship between wheel and rail contact. It has been shown through existing domestic and international research that the use of MEM simulations to calculate the dynamic response of the track can significantly reduce the size of the track calculation model, providing more efficient, accurate, and effective calculations.
Track unevenness has a significant effect on the coupled vibration of the vehicle-track system and is very harmful to the high-speed rail track system. In this regard, a large amount of research has been produced by academics at home and abroad. In the time domain dynamic analysis, the most commonly used methods to describe track unevenness are the deterministic function method and the stochastic process method. The deterministic function method assumes that the track irregularity has a sine or cosine curve shape; for instance, Khajenuezrury et al. [
18] proposed a two-dimensional Timoshenko beam vehicle-track coupling numerical model to investigate the influence of simple harmonic function track unevenness on the track mat stiffness. In the stochastic process method, a time-domain sample of the stochastic unevenness is extracted from the power spectral density (PSD) of the track; for instance, Lei et al. [
6] treated the track vertical unevenness as a Gauss smooth stochastic process and investigated the dynamic response of the vehicle and track system. Yang et al. [
7] used the Fourier inversion method based on the German track unevenness spectrum to obtain the track spectrum at wavelengths greater than 1 m for the effect of random process irregularities on the dynamic response of the ballastless track. Lai et al. [
19] selected a high-speed rail track and bridge system for nonlinear dynamic analysis, proposed the distribution law of the seismic residual shift of high-speed rail track and a bridge system under different seismic intensities, and introduced the evaluation method of the seismic power spectrum density (PSD) of track unevenness. Zhu et al. [
20] proposed an effective method to predict the ground vibrations caused by metro trains, taking into account random track unevenness. In order to study the impact of ballastless track unevenness on China’s high-speed railways, the China Academy of Railway Sciences proposed a ballastless track unevenness spectrum for China’s high-speed railways in 2014, which better reflects the current overall design and construction level of ballastless tracks in China [
21].
In this article, on the basis of the moving unit method, a vibration analysis model of the coupled high-speed rail vehicle-ballastless track system is established and a MATLAB calculation and analysis program is developed, taking into account the discontinuity of the prefabricated track plate and the discrete support of the trackpad in CRTS II ballast-less track. The effects of different travel speeds, the consideration of track slab discontinuities, and the use of the random track unevenness conditions generated by the high and low unevenness spectrum of Chinese high-speed rail ballastless track on the dynamic response of high-speed rail ballastless track systems are investigated.
3. Establishment of the MEM Model
The MEM principle is as follows. First, intercept a section of track and discrete it into a finite number of beam units, then place the point of the wheel-rail contact force in the middle of the whole track so that it meets a certain distance upstream and downstream, thus ignoring the boundary effect; at this time, the point on the coordinate axis
x-axis and the corresponding coordinates are fixed. Then, take point a on the fixed coordinate axis
x-axis, whose coordinates are
xa, and define the moving coordinate axis
r-axis. The origin of the
r-axis follows the train load movement, then point a corresponds to the moving coordinate
ra on the
r-axis as shown in Equation (5):
Substituting Equation (5) into Equations (2)–(4), applying the chain rule, and using a moving coordinate system instead of a fixed coordinate system, the discrete dynamic equations for the new three-layer Euler beam ballast track are derived as shown in Equations (6)–(8):
where
Ls denotes the spacing between two adjacent track pads,
Rj denotes the distance the
jth wheel pair runs,
Fcj denotes the contact force between the
jth wheel pair and the track, and
δ(
x) denotes the Dirac function.
The C-ELE model diagram is shown in
Figure 3, and a typical orbital unit of length l is taken for analysis.
The derivation process of Lei [
6] can be referred to be based on the moving coordinates and by introducing the form function
N. From the form function
N, the matrices
Nr,
Nt, and
Nb can be obtained. The Euler beam model for the first layer of the rail is first calculated by multiplying Equation (6) by a weighting function and integrating over one unit length cell. Galerkin’s method, the unit mass matrix
Mr, the unit damping matrix
Cr, and the unit stiffness matrix
Kr are obtained for the first layer of the C-ELE rail based on the track dynamics equations. Using the same method, the corresponding matrices for the second layer of the rail as well as the third layer of the rail can be found. Combining the unit matrices of the three-layer Euler beam model, the unit mass matrix
Me, unit damping matrix
Ce, and unit stiffness matrix
Ke of the C-ELE can be obtained.
The P-ELE is distributed at the joint of the two adjacent track slabs of the track slab Euler beam with the expansion joint located inside the cell; the P-ELE model is shown in
Figure 4.
In the study of the ballastless track model, the ballastless track is cut into one track slab ignoring the expansion joints between the slabs, while assuming that the cross sections of the track slabs are identical in terms of the mechanical properties and structural shape. This shows that the physical properties of the P-ELE and C-ELE can be generalized, except for the difference in the continuity and degrees of freedom, so the transformation method for degenerating the P-ELE to the C-ELE, reversing the transformation, can be used to obtain the corresponding matrix of the P-ELE by deriving the cell mass, damping, and stiffness matrices of the C-ELE. The unit mass, damping, and stiffness matrices for the second layer of the rail after truncation of the
Iw-long track plate, adding the effects of the shear spring
Kss and the damper
Css, provides the cell damping and stiffness matrices of the P-ELE as shown in Equations (9)–(11).
Additionally, the shear spring
Kss and the damper
Css can be obtained by means of Equations (12) and (13). Referring to the model of Xu [
12],
Nss = [1–1].
Additionally, because the discontinuity of the track plate affects only the second layer of the track plate in the dynamic equations, the unit mass, damping, and stiffness matrices of the first and third layers remain unchanged. Through the addition of the cell mass matrix, the cell damping matrix, and the cell stiffness matrix of each layer of the P-ELE, we obtain the P-ELE cell mass matrix , cell damping matrix , and cell stiffness matrix .
The ballastless track is laid by numerous prefabricated track slabs. The unit mass matrix, unit damping matrix, and unit stiffness matrix of each part are combined to obtain the total mass matrix
MD, total damping matrix
CD, and total stiffness matrix
KD of the ballastless track model to establish the dynamic equations of the ballastless track model as shown in Equation (14):
where
,
and
denote the acceleration vector, velocity vector, and displacement vector of the ballast track model, respectively.
FD denotes the force vector of the ballastless track model.
In addition, if a combined unit consists entirely of C-ELE, then the ballastless track model constructed using this combined unit does not take into account the discontinuity of the ballastless track precast track slab, i.e., the track slab is continuous, and this part of the track slab structure is equivalent to a long beam with a constant full distance stiffness. According to this principle, a ballastless track model with continuous track plates can be built, and thus a coupled vehicle-track model with continuous track plates can be constructed, and the specific modeling process is not repeated.
Combining Equations (1) and (14), i.e., the vehicle model dynamic equations and the ballastless track model dynamic equations, the coupled vehicle-ballastless track system equations are obtained as shown in Equation (15):
4. Analysis of the Effect of Travel Speed on the Dynamic Response of the System
The constructed model was used, and the specific parameters were obtained by referring to the literature [
6]. To account for the dynamic response of the ballastless track, the track length was set to 140 m, and speeds of 160 km/h, 200 km/h, 250 km/h, 300 km/h, and 350 km/h were chosen as the conditions for the different speeds. To account for the impact of track unevenness and to minimize computational effort, a simpler harmonic function model is used, i.e., a sine wave with a wavelength of 1.0 m and an amplitude of 0.3 mm, the model of which is shown in Equation (16).
where
yt denotes the degree of track irregularity, and
A and
B denote the amplitude and wavelength of track irregularity, respectively.
In order to fully demonstrate the variation between the calculated results for the track plate discontinuity and the track plate continuity cases, the coupled vehicle-ballastless track model with the track plate discontinuity mentioned in the previous chapter of this paper was also used for comparison with the model without the track plate discontinuity in the numerical simulations. After the results stabilized, the steady-state output 3 s data at the first wheel-rail contact point are plotted separately.
4.1. Comparative Analysis of Vertical Acceleration of Ballastless Track
4.1.1. Vertical Acceleration of Rail
The peak vertical acceleration of the rail under different speed conditions with discontinuous track plates is extracted and compared with the peak vertical acceleration of the rail under the same speed with continuous track plates. The comparison results are shown in
Figure 5.
It is understandable that the vertical acceleration of the steel rails increases with the increase in the travel speed, but not proportionally. The reason for this is that the short-wave discontinuity has a significant excitation effect on the steel rails’ acceleration, which is also evidenced in the literature [
6]. In addition, the vertical acceleration of the steel rails in the track plate discontinuity condition is greater and significantly different than in the track plate continuity condition, with an average difference of about 132 m/s
2.
4.1.2. Vertical Acceleration of Rail
The peak vertical acceleration of the track plate under different speed conditions with discontinuous track plates is compared with the peak vertical acceleration of the track plate under continuous track plate conditions with the same speed. The results of the comparison are shown in
Figure 6.
It is obvious that the vertical acceleration of the track plate increases with the increase in the travel speed, but this occurs disproportionately, and the reason for the declining section is the same as that of the rail section. The vertical acceleration of the rail slab under discontinuous conditions is greater than under continuous conditions and has a significant difference, with an average difference of approximately 43 m/s2. The former has a more dramatic dynamic response and is more significantly stimulated by the speed change.
4.1.3. Vertical Acceleration of the Foundation
The peak vertical acceleration of the concrete foundation for different speed conditions with discontinuous track slabs is extracted and compared with the peak vertical acceleration of the concrete foundation for the same speed with continuous track slabs as shown in
Figure 7.
It is obvious that the effect of travel speed on the vertical acceleration of the concrete foundation is small, and the vertical acceleration of the concrete foundation increases with the travel speed disproportionately, yet there is still a decreasing section, but this is basically flat, meaning that the fluctuation is reduced. The vertical acceleration of the concrete foundation under discontinuous track slab conditions is greater than under continuous track slab conditions but similar, with an average difference of approximately 3 m/s2, indicating that the discontinuity of the track slab does not lead to significant changes in the concrete acceleration with speed.
4.2. Analysis of the Vertical Acceleration of the Car Body Compared to the Wheel-Rail Contact Force
4.2.1. Vertical Acceleration of the Vehicle Body
The peak vertical acceleration of the car body under different speed conditions with discontinuous track plates is extracted and compared with the peak vertical acceleration of the rail under continuous track plates at the same speed as shown in
Figure 8.
Obviously, the general trend is that the vertical acceleration of the vehicle body increases with increasing travel speed. In addition, whether or not the discontinuity of the track plate is considered has no significant effect on the vertical acceleration of the rail because the difference between the two is very small; the reason for this is that the primary and secondary suspension system of the CRH3 vehicle has an effective dampening effect on the transmission of vibrations from the track plate. From the above analysis, it can be seen that the discontinuity of the track plate basically does not affect the comfort of passengers riding on high-speed railway vehicles.
4.2.2. Wheel-Rail Contact Forces
The peak wheel-rail contact forces are extracted for different speed conditions with discontinuous track plates and compared with the peak wheel-rail contact forces for the same speed with continuous track plates, as shown in
Figure 9.
It is obvious that the wheel-rail contact force increases with increasing speed, but not proportionally. The wheel-rail contact force for the discontinuous track plate condition is greater than for the continuous track plate condition, with an average difference of approximately 12 kN. The reason for this is that the discontinuous track plate corresponds to multiple plates connected by shear springs and dampers, whereas the continuous track plate corresponds to a single beam, the overall stiffness of which is weaker than the latter, making the dynamic response more significant in the discontinuous track plate condition.