Dynamic Response of Train–Ballastless Track Caused by Failure in Cement–Asphalt Mortar Layer

Abstract

Cement–asphalt (CA) mortar voids in earth’s structure are prone to inducing abnormal vibrations in vehicle and track systems and are more difficult to recognize. In this paper, a vehicle–ballastless track coupling model considering cement–asphalt mortar voids is established and the accuracy of the model is verified. There are two main novel results: (1) The displacement of the track slab in the ballastless track structure is more sensitive to the void length. Voids can lead to blocked vibration transmission between the ballastless track slab and concrete base. (2) The wheel–rail vibration acceleration is particularly sensitive to voids in cement–asphalt mortar, making the bogie pendant acceleration a key indicator for detecting such voids through amplitude changes. Additionally, the train body pendant acceleration provides valuable feedback on the cyclic characteristics associated with single-point damage in the cement–asphalt mortar, thereby enhancing the accuracy of dynamic inspections for vehicles. In the sensitivity ordering of the identification indexes of voids, the bogie’s vertical acceleration in high-speed trains > the nodding acceleration of the bogie > the vehicle’s vertical acceleration. Adaptive suspension parameters can be designed to accommodate changes in track stiffness.

1. Introduction

The cement–asphalt (CA) mortar layer is a key structural element in the ballastless track systems of high-speed railways. This structural layer is used in railway tracks in China, Japan, Italy, and other countries. As the duration of the use of the CA mortar layer has increased and the base material itself has deteriorated [1,2], a state of void failure has been observed in ballastless tracks in various countries [3,4,5,6,7,8,9]. In China, the cement–asphalt (CA) mortar layer is widely used in the railway track system, specifically in the China Railway Track System II (CRTS-II) structure. The CRTS-II structure consists of three components: the track slab, the CA mortar layer, and the concrete base. However, over time, numerous voids have been observed in the CA mortar layer at project sites, as shown in Figure 1. This layer, located between the track slab and the concrete base, plays a crucial role in vibration damping and energy absorption. Consequently, the presence of voids in the CA mortar layer directly compromises the vibration-damping function of the ballastless track structure. Additionally, the lack of elasticity causes inconsistencies in the stiffness of the track structure along the railway’s direction. This inconsistency leads to a stronger impact from the operation of high-speed trains, posing safety risks to passengers. Furthermore, the voids exacerbate the dynamic effects of the high-speed trains, increasing the likelihood of damage to other components of the ballastless track.

The cement–asphalt mortar layer functions as an elastic adjustment layer, playing key roles such as buffering and vibration reduction [8,10]. However, with the prolonged operation of high-speed railway ballastless tracks, the CA mortar layer experiences various forms of damage, including gaps, mud pumping, spalling, and voids [11,12,13], due to long-term train vibrations, temperature fluctuations, humidity, and other external factors. These operational defects ultimately lead to the formation of voids in the CA mortar layer. The vehicle–ballastless track system operates as an integrated unit, coupled through wheel–rail interactions [14]. As a crucial elastic adjustment layer, any weakening of the CA mortar’s vibration-reducing function can cause abnormal dynamic interactions within the system, posing a long-term threat to both traffic safety and the service life of the structure [15,16,17,18,19].

Extensive research has been conducted on the damage and vibration behavior of ballastless track composite structures [20]. The High-Speed Railway Design Code [21] specifies that the design life of ballastless track structures is 60 years, but due to long-term cyclic loading from trains and the compounded effects of damage, such as from the CA mortar layer, the actual service life of ballastless tracks is significantly shortened. Firstly, for the failure problem of the CRTS-II ballastless track structure, research on the damage mechanism of the concrete [22,23] and CA mortar of CRTS-II ballastless tracks is focused on environmental temperature [15], trains’ dynamic loads [24,25], track irregularity [26], material fatigue and impact damage [27,28], foundation deformation [29], and other aspects.

Currently, the main research is focused on the damage mechanisms of ballastless tracks’ concrete and CA mortar layers, as well as the application of some non-destructive testing means to identify CA mortar void lesions [30]. But, the working efficiency is very low. However, in recent years, line infrastructure damage detection based on moving vehicles has gradually become a research hotspot. Kordestani et al. [31] used the acceleration response law to identify bridge damage. There are fewer studies on the comparison of vibration characteristics of vehicle–ballastless track systems caused by different failure modes in CA mortar [32].

In this paper, to address the abnormal vibration problem induced by varying CA mortar cavity lengths, a vehicle–ballastless track–earth structure model considering CA mortar failure is developed based on multi-body dynamics and the finite element method. The effects of four types of short-shaped void distributions on the dynamic response of the rail and ballastless track are investigated. Subsequently, the internal vibration transfer state of the ballastless track structure under CA mortar void conditions is analyzed using an innovative transfer function. Finally, a comparative analysis is performed to identify differences in the wheel–rail force, train body vibration, and bogie acceleration using high-speed trains, and a method is proposed to adapt the suspension parameters of high-speed trains to changes in track stiffness.

2. Numerical Modeling and Methods

The research approach and analysis process of this thesis is shown in Figure 2a. The model established in this paper contains the vehicle system, track system, and subgrade; the vehicle system is considered modeled as a pendant model with 10 degrees of freedom (DOF), the rails are modeled using a bending beam model, the track system is modeled using a solid model, and the subgrade is represented as springs and damping to achieve the support of the subgrade [33,34]. A combination of a vehicle and ballastless track is produced through wheel–rail contact. After forming the overall stiffness, damping, and mass matrices again, the solution is performed using the backward difference method. The accuracy of the model is verified. On this basis, parametric analysis of CA mortar void length is performed to compare the dynamic response of the vehicle and ballastless track and to propose a more sensitive index for CA mortar failure. The system dynamics model is shown in Figure 2b.

(1)

Train system

Since damage to the CA mortar layer leads to the deterioration of the stiffness and damping of this layer, which mainly affects the vertical dynamic behavior, the vehicle system was modeled using a 10 DOF multi-rigid-body dynamics model, taking into account the solution efficiency and the properties of the analytical problem. The solution variables of the vehicle system dynamics are the nodding and sinking of the vehicle body, the nodding and sinking of the front and rear bogies, and the nodding and sinking of the four wheel pairs. The vibration equation of the system is as follows (1):

$$\left[\mathbf{M}\right]\left[\ddot{\mathbf{z}}\right]+\left[\mathbf{C}\right]\left[\dot{\mathbf{z}}\right]+\left[\mathbf{K}\right]\left[\mathbf{z}\right]=\left[\mathbf{F}\right]$$

(1)

where $[\mathbf{M}]$ represents the mass matrix of the train body, bogie, and wheelset; $[\mathbf{K}]$ is the stiffness matrix of the vehicle assembly; $[\mathbf{C}]$ is the damping matrix of the vehicle; $[\mathbf{z}],[\dot{\mathbf{z}}],[\ddot{\mathbf{z}}]$ represent the displacement matrix, velocity matrix, and acceleration matrix of the vehicle system, respectively; $[\mathbf{F}]$ is the load vector of the vehicle; and the model parameters for the numerical calculation of the vehicle are shown in the table below. The computational parameters of the system dynamics model are shown in

Table 1. Calculation parameters of China railway high-speed train-3 (CRH3) (Lei [35]).

Parameters	Value
Half-car body mass, $M_c$ (kg)	20,000
Bogie mass, $M_t$ (kg)	3200
Wheelset mass, $M_w$ (kg)	2400
Primary suspension stiffness, $k_{s1}$ (MN/m)	2.08
Primary suspension damping, $c_{s1}$ ($\mathrm{k}\mathrm{N}·\mathrm{s}/\mathrm{m}$)	100
Secondary suspension stiffness, $k_{s2}$ (MN/m)	0.8
Secondary suspension damping, $c_{s2}$ ($\mathrm{k}\mathrm{N}·\mathrm{s}/\mathrm{m}$)	120
Train body pitch moment of inertia, $J_c$ ($\mathrm{k}\mathrm{g}·{\mathrm{m}}^2)$	$5.47\times {10}^5$
Bogie pitch moment of inertia, $J_t$ $(\mathrm{k}\mathrm{g}·{\mathrm{m}}^2)$	6800
Wheel base, (m)	2.5
Bogie center distance, (m)	17.375
Axle weight, (t)	14
Wheel–rail contact ratio, G ($\mathrm{m}/{\mathrm{N}}^{{\displaystyle \frac{2}{3}}}$)	$3.86R^{-0.115}\times {10}^{-8}$
Wheel rolling radius, R$\left(\mathrm{m}\right)$	0.457

. The matrix of the train system as follows:

$$\begin{gathered} \begin{array}{l}\mathbf{M}=diag\left\{{\mathbf{J}}_{\mathbf{c}}{\mathbf{M}}_{\mathbf{c}}{\mathbf{M}}_{\mathbf{t}}{\mathbf{J}}_{\mathbf{t}}{\mathbf{M}}_{\mathbf{w}\mathbf{1}}{\mathbf{M}}_{\mathbf{w}\mathbf{2}}{\mathbf{M}}_{\mathbf{w}\mathbf{3}}{\mathbf{M}}_{\mathbf{w}\mathbf{4}}\right\}\\ {\mathbf{K}}_{\mathbf{u}}=\left[\begin{array}{cccccccccc}\mathbf{2}{\mathbf{K}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& -{\mathbf{K}}_{\mathbf{s}\mathbf{2}}& -{\mathbf{K}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ & \mathbf{2}{\mathbf{K}}_{\mathbf{s}\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}^{\mathbf{2}}& {\mathbf{K}}_{\mathbf{s}\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}& -{\mathbf{K}}_{\mathbf{s}\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ & & \mathbf{2}{\mathbf{K}}_{\mathbf{s}\mathbf{1}}+{\mathbf{K}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& -{\mathbf{K}}_{\mathbf{s}\mathbf{1}}& -{\mathbf{K}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \mathbf{0}\\ & & & \mathbf{2}{\mathbf{K}}_{\mathbf{s}\mathbf{1}}+{\mathbf{K}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& -{\mathbf{K}}_{\mathbf{s}\mathbf{1}}& -{\mathbf{K}}_{\mathbf{s}\mathbf{1}}\\ & & & & \mathbf{2}{\mathbf{K}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}^{\mathbf{2}}& \mathbf{0}& {\mathbf{K}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}& -{\mathbf{K}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}& \mathbf{0}& \mathbf{0}\\ & & & & & \mathbf{2}{\mathbf{K}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}^{\mathbf{2}}& \mathbf{0}& \mathbf{0}& {\mathbf{K}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}& -{\mathbf{K}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}\\ & & & & & & {\mathbf{K}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ & & & & & & {\mathbf{K}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \mathbf{0}& \\ & & & & & & & {\mathbf{K}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \\ & & & & & & & & {\mathbf{K}}_{\mathbf{s}\mathbf{1}}& \end{array}\right]\\ \\ {\mathbf{C}}_{\mathbf{u}}=\left[\begin{array}{cccccccccc}\mathbf{2}{\mathbf{C}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& -{\mathbf{C}}_{\mathbf{s}\mathbf{2}}& -{\mathbf{C}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ & \mathbf{2}{\mathbf{C}}_{\mathbf{s}\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}^{\mathbf{2}}& {\mathbf{C}}_{\mathbf{s}\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}& -{\mathbf{C}}_{\mathbf{s}\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ & & \mathbf{2}{\mathbf{C}}_{\mathbf{s}\mathbf{1}}+{\mathbf{C}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& -{\mathbf{C}}_{\mathbf{s}\mathbf{1}}& -{\mathbf{C}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \mathbf{0}\\ & & & \mathbf{2}{\mathbf{C}}_{\mathbf{s}\mathbf{1}}+{\mathbf{C}}_{\mathbf{s}\mathbf{2}}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& -{\mathbf{C}}_{\mathbf{s}\mathbf{1}}& -{\mathbf{C}}_{\mathbf{s}\mathbf{1}}\\ & & & & \mathbf{2}{\mathbf{C}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}^{\mathbf{2}}& \mathbf{0}& {\mathbf{C}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}& -{\mathbf{C}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}& \mathbf{0}& \mathbf{0}\\ & & & & & \mathbf{2}{\mathbf{C}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}^{\mathbf{2}}& \mathbf{0}& \mathbf{0}& {\mathbf{C}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}& -{\mathbf{C}}_{\mathbf{s}\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}\\ & & & & & & {\mathbf{C}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ & & & & & & & {\mathbf{C}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}& \mathbf{0}\\ & & & & & & & & {\mathbf{C}}_{\mathbf{s}\mathbf{1}}& \mathbf{0}\\ & & & & & & & & {\mathbf{C}}_{\mathbf{s}\mathbf{1}}& \end{array}\right]\\ \mathbf{Z}={\left\{\begin{array}{llllllllll}{\mathbf{v}}_{\mathbf{c}}\hfill & {\mathbf{\beta }}_{\mathbf{c}}\hfill & {\mathbf{v}}_{\mathbf{t}\mathbf{1}}\hfill & {\mathbf{v}}_{\mathbf{t}\mathbf{2}}\hfill & {\mathbf{\beta }}_{\mathbf{t}\mathbf{1}}\hfill & {\mathbf{\beta }}_{\mathbf{t}\mathbf{2}}\hfill & {\mathbf{v}}_{\mathbf{w}\mathbf{1}}\hfill & {\mathbf{v}}_{\mathbf{w}\mathbf{2}}\hfill & {\mathbf{v}}_{\mathbf{w}\mathbf{3}}\hfill & {\mathbf{v}}_{\mathbf{w}\mathbf{4}}\hfill \end{array}\right\}}^{\mathbf{T}}\\ \mathbf{F}={\left\{\begin{array}{cccccccccc}{-\mathbf{M}}_{\mathbf{c}}\mathbf{g}& \mathbf{0}& {-\mathbf{M}}_{\mathbf{t}}\mathbf{g}& {-\mathbf{M}}_{\mathbf{t}}\mathbf{g}& \mathbf{0}& \mathbf{0}& {\mathbf{P}}_{\mathbf{1}}& {\mathbf{P}}_{\mathbf{2}}& {\mathbf{P}}_{\mathbf{3}}& {\mathbf{P}}_{\mathbf{4}}\end{array}\right\}}^{\mathbf{T}}\end{array} \end{gathered}$$

M_c and J_c represent the mass and rolling moment of the inertia of the rigid body of the car, respectively; M_t and J_t refer to the mass and rolling moment of the inertia of the bogies; and M_wi denotes the mass of the ith wheel. (v_c) and (${\beta }_{\mathrm{c}}$) represent the vertical displacement and pitch angle of the car body, respectively; (($v_{\mathrm{t}1}$, ${\beta }_{t1}$)) and (($v_{\mathrm{t}1}$, ${\beta }_{t2}$)) indicate the vertical displacement and pitch angle of the front and rear bogies, respectively; and ($v_{wi}$) denotes the vertical displacement of the ith wheel. Detailed equations for the damping matrix [C] and the stiffness matrix [K] can be found in Lei [35].

(2)

Track system

Computational simulations of rails are generally simplified using the Euler–Bernoulli beam model, with Euler beam governing equations [35]:

$$E_r{I}_r{\displaystyle \frac{{\partial }^4{w}_r}{\partial x^4}}+m_r{\displaystyle \frac{{\partial }^2{w}_r}{\partial t^2}}+c_r\left({\displaystyle \frac{\partial w_r}{\partial t}}-{\displaystyle \frac{\partial w_s}{\partial t}}\right)+k_r\left(w_r-w_s\right)=-F\left(t\right)\delta \left(x-vt\right)$$

(2)

where $w_r$ and $w_s$ represent the vertical deformation of the rail and track slab, respectively; $E_r{I}_r$ represents the bending stiffness of the rail; $m_r$ represents the mass per unit length of the rail; $k_r$ and $c_r$ represent the fastener stiffness coefficient and damping coefficient; $F\left(t\right)$ is the wheel–rail contact force; $V$ is the speed of the train; and $\delta$ is the Dirac function. The calculated parameters of the ballastless track model are shown in

Table 2. Calculation parameters of track structure (Lei [35]).

Parameters	Value
Mass of rail (kg/m)	60
Flexural stiffness of rail beam (kN·m2)	6756
Fastener stiffness (kN/m)	60
Fastener damping ($\mathrm{M}\mathrm{N}·\mathrm{s}/\mathrm{m}$)	47.7
Density of track slab (kg/m3)	2500
Elastic modulus of track slab (MPa)	36,000
Poisson’s ratio of track slab	0.2
CA mortar stiffness (MN/m)	900
CA mortar damping ($\mathrm{k}\mathrm{N}·\mathrm{s}/\mathrm{m}$)	83
Density of concrete base (kg/m3)	2500
Elastic modulus of concrete base (MPa)	34,000
Poisson’s ratio of concrete base	0.2

. A beam element model was used for discretization to obtain the mass, stiffness, and damping matrices.

The fastener system and the CA mortar layer mainly provide the vertical elasticity of the superstructure, which was modeled by spring-damping. The track slab and concrete base were modeled as if they were continuous elastic bodies, and the track slab and concrete base are described by Navier’s elastodynamic equations, both of which are expressed in a unified expression in tensor form [36]:

$${\sigma }_{ij,j}-\rho {\ddot{u}}_i=0$$

(3)

where $\sigma$ represents the stress tensor, $\rho$ is the density, and u is the displacement.

As shown in Figure 3, CA mortar voids were modeled by putting the stiffness of the CA mortar at 0 in this range (k = 0, c = 0), considering that the CA mortar in the void area could not provide support.

(3)

Wheel–rail contact behavior

In the vertical vibration analysis, the current mainstream wheel–rail Hertzian contact theory was used, and the Hertzian contact theory was chosen because it allows wheel–rail separation and the occurrence of rail hopping. Hertz contact is established under the assumption that under the condition of small deformation, the wheel and the rail are regarded as isotropic elastic cylinders. According to the Hertz contact theory to achieve the coupling of the vehicle system and the rail system, the wheel–rail contact force can be expressed as follows [35]:

$${\mathrm{F}}_{\mathrm{W}-\mathrm{R}}=\left\{\begin{array}{l}{G}^{{\displaystyle \frac{2}{3}}}{\left[\left|{\mathrm{z}}_{\mathrm{Wi}}{-\mathrm{z}}_{\mathrm{Ri}}-\eta \right|\right]}^{{\displaystyle \frac{3}{2}}}, {\mathrm{z}}_{\mathrm{Wi}}{-\mathrm{z}}_{\mathrm{Ri}}\le 0\\ 0 , {\mathrm{z}}_{\mathrm{Wi}}{-\mathrm{z}}_{\mathrm{Ri}}>0\end{array}\right.$$

(4)

where z represents the displacements of wheels and rails at the coordinate i, respectively, and G is the contact deflection coefficient. $\eta$ is the value of the irregularity of the rail surface at different locations on the track.

The track irregularity used the ballastless track spectrum from China’s high-speed rail network and applied the triangular gradient method to generate track height variations. The resulting track height deviations, generated by the program, are shown in Figure 4 [35].

(4)

Model solver

The total mass matrix, damping matrix, and stiffness matrix were constructed using the conventional finite element method. The time integration of the governing equations was performed using the implicit backward difference method. Specifically, the second-order backward difference method was employed for time integration. Further details of the backward difference method can be found in [37,38]. A time step of 0.0001 s was used for solving the model. For nonlinear analysis, the Newton–Raphson method was applied. For the ballastless track slabs and concrete bases, the mesh size was set to 0.1 m. The length of the numerical model was 150 m. The solid structure was discretized using first-order Lagrangian elements. The observation location of the dynamic response of the ballastless track structure was placed in the middle of the CA mortar void.

3. Model Validation

The simulated train speed was 72 km/h, and the rail surface was smooth. The initial conditions for displacement, velocity, and acceleration were all set to zero. A comparison of the dynamic response of high-speed trains under gravity was conducted. The model’s calculation parameters were consistent with those in the literature. The vehicle used was of the CRH3 type, and the ballastless track followed the CRTS-II type. All computational parameters in the validation model were taken from the literature [39]. The vehicle–track coupling dynamics calculation procedure employed the backward difference method to solve the problem, with a time step of 0.0001 s. The physical system used for model validation is shown in Figure 5.

The calculation procedure developed in this paper was compared with the results from the literature [39]. The comparison, shown in Figure 6, indicates that the results of the numerical method are in good agreement, demonstrating the accuracy of the proposed calculation approach.

In addition, in order to fully verify the accuracy of the model, different control parameters were selected and the calculated parameters were consistent with the original literature. From the comparison in

Table 3. Comparison of the results of some calculations.

Parameter	Ref. [40]	Ref. [41]	Ref. [39]	Present Work
Speed (km/h)	300	300	300	300
Rail displacement (mm)	1.3	1.34	\	1.23
Dynamic stress of subgrade surface (kPa)	14.6	14.03	\	13.81
Displacement of track slab (mm)	/	/	0.69	0.61

, we find that the calculated results are very close to each other, indicating the accuracy of the model in this paper.

4. Results and Discussion

In the following analysis, the vehicle, ballastless track, and subgrade calculation parameters are all from the literature [35]. A CA mortar void is a kind of imperceptible disease; therefore, analyzing the dynamic response of the vehicle–track system and its degree of change is of great significance in exploring the identification of this kind of disease. However, in the operation of high-speed railways, the problem of the small length of voids is generally hidden, and this potential damage has not been noticed so far; therefore, the extent of its impact on the dynamic performance of high-speed trains and ballastless track systems needs to be investigated in order to better identify its characteristics. Therefore, a total of four analytical conditions were set up using this model, namely the following: (1) CA mortar non-void + random irregularities; (2) CA mortar void 100 cm + random irregularities; (3) CA mortar void 200 cm + random irregularities; and (4) CA mortar void 300 cm + random irregularities.

4.1. Dynamic Response of Rail Track

(1)

Wheelset–rail system

Figure 7a shows the dynamic response behavior of the wheel–rail system of the vehicle under different line foundation conditions. From the analysis of the figure, it is clear that the value of the wheel–rail force was 73.56 kN when there were no voids in the CA mortar, which was small compared to the different lengths of voids, and the values of the wheel–rail force were 74.61 kN, 80.83 kN, and 88.92 kN for the change in the length of voids from 1 m to 3 m. The percentage of the increase in the value of the wheel–rail force compared to the absence of voids was 1.43%, 9.88%, and 20.88%.

Figure 7b shows the amplitude of the wheelset vibration acceleration as the development of the void was 1.62 m/s², 5.19 m/s², 9.81 m/s², and 17.54 m/s². With the development of the length of the void, the wheelset vibration acceleration increased by 220.37%, 505.55%, and 982.82%, respectively. The vertical vibration acceleration of the wheelset exhibited an obvious amplification law for the void in the CA mortar, and this feature was very significant for the response to void defects in the CA mortar.

Figure 8 shows the rail vibration acceleration and the time curve of rail dynamic displacement. From the perspective of rail vibration acceleration, the peak acceleration reached 60 m/s². When comparing the three working conditions (non-void and void), the acceleration of the rail did not exhibit a significant difference. The maximum values of rail vibration acceleration were 63.6 m/s², 55.1 m/s², 53.81 m/s², and 42.01 m/s², with corresponding percentage changes of 13.36%, 15.39%, and 33.95%.

Regarding the rail vibration displacement, this index clearly reflected the presence of voids. The dynamic displacement of the rail was 1.11 mm in the non-void condition. However, as the void length increased to 1 m, 2 m, and 3 m, the vertical displacement of the rail increased to 1.27 mm, 1.5 mm, and 1.83 mm, respectively. The percentage increases in displacement were 14.4%, 34.14%, and 64.86%, respectively.

(2)

Ballastless track system

Figure 9 illustrates the dynamic response of the track slab. First, the vibration acceleration of the track slab is analyzed. The maximum acceleration at the surface of the track slab was 11.77 m/s² when there was no void. As the void length increased, the vibration acceleration of the track slab reached 14.90 m/s², 27.43 m/s², and 17.80 m/s², respectively. This was due to the void length reaching up to 3 m. When the distance between the adjacent axles of the same bogie was 2.5 m, interference between the front and rear wheels occurred, resulting in an increase in dynamic displacement while causing acceleration to decrease.

Regarding the dynamic displacement of the track slab, Figure 9 clearly shows that the change in void length significantly impacted the deformation of the track slab. As the void length in the CA mortar increased, the bottom face-to-face support became localized. With an increasing void span, the displacement of the track slab increased substantially. The dynamic settlement displacement of the track slab was 0.52 mm, 0.60 mm, 0.84 mm, and 1.33 mm, corresponding to increases of 15.38%, 61.54%, and 155.77%, respectively. The dynamic displacement of the track slab strongly reflected the deformation response due to the CA mortar void. Therefore, the dynamic displacement of the track slab could be recorded by the dynamic acquisition device of the high-speed train, which could be used to better identify the CA mortar layer void and assess its damage range.

Figure 10 presents the dynamic response of the concrete base. The vibration acceleration on the surface of the concrete base was 9.77 m/s² when there was complete contact. As the void length increased from 1 m to 3 m, the acceleration decreased to 7.45 m/s², 7.08 m/s², and 6.46 m/s², respectively. The reduction percentages were 23.75%, 27.53%, and 33.88%. This decrease in peak acceleration was attributed to the reduced effective energy transfer downward due to the expansion of the void area. Regarding the displacement of the concrete base, the dynamic displacement was 0.48 mm, 0.46 mm, 0.38 mm, and 0.33 mm for the condition of no void and void lengths of 1 m, 2 m, and 3 m, respectively. The reduction percentages were 4.17%, 20.83%, and 31.25%.

4.2. Transfer Function of Ballastless Track

The effect of the CA mortar void damage on the system’s dynamic response was influenced by several factors. The vibration transfer function was based on the frequency response characteristics between the system’s substructures, which describe the intrinsic properties of vibration transfer between different components within the system in the frequency domain. This understanding is crucial for analyzing the role of various parameters within the system, which can be derived directly from the input–output frequency response relationships of the computational mode.

$$H(\omega )={\displaystyle \frac{\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{+\infty }Y}(t)e^{-j2\pi \omega t}dt}{\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{+\infty }X}(t)e^{-j2\pi \omega t}dt}}={\displaystyle \frac{Y(\omega )}{X(\omega )}}$$

(5)

where H(w) is the vibration transfer function; Y(t) and X(t) are the input and output time-domain signals of the system; and, correspondingly, Y(w) and X(w) are the input and output Fourier transform frequency-domain signals of the system. Therefore, in this paper, the transfer function calculation program was implemented using Python 3.7, and the transfer function was directly output through the results of the vehicle–rail coupling calculation model.

Figure 11 shows the amplitude–frequency response curves of the transfer function between the vibration displacements of the track slab and the concrete base. From the transfer function characterization model, the values of the transfer function were generally in the range of 0.87–1 under the condition of no void, indicating that the vibration distribution of the ballasted track system under optimal contact conditions was excellent. As the length of the void increased, the transfer function values varied within the following intervals: 0.43–0.98, 0.27–0.74, 0.08–0.32, and 0.08–0.32, respectively.

Figure 12 shows the transfer function of acceleration between the track slab and the concrete base. The transfer function ranged from 0.81 to 1 when there was no void. When the void was 1 m, the transfer function ranged from 0.66 to 0.98. For a void of 2 m, the transfer function ranged from 0.34 to 0.75, and for a void of 3 m, it ranged from 0.07 to 0.30. Overall, the degradation of the transfer function due to the removal of CA mortar was significant and clearly observed across all frequency-domain stages.

4.3. Dynamic Response of Vehicle

Figure 13a shows the vertical vibration acceleration of the bogie. The acceleration exhibited a significant increasing trend at the same position. When the deflection grew, the acceleration increased by 1.592 m/s², 1.597 m/s², 2.104 m/s², and 3.281 m/s², respectively. The vertical acceleration of the bogie increased by 106.09% when the deflection length reached 3 m.

Figure 13b illustrates the nodding acceleration of the bogie. The analysis indicates that the nodding acceleration exhibited more pronounced abnormal characteristics when the void length was 3 m. Conversely, the nodding acceleration showed some memory characteristics in the normal section. As the bogie passed through the void region, the nodding acceleration behaved similarly to that observed in the normal section without a void. However, in the void area, the nodding acceleration exhibited a different response amplitude compared to the normal section, which aided in identifying the CA mortar damage.

Figure 13c presents the body’s vertical acceleration. The time-domain characteristics of the body’s vertical acceleration reveal clear periodic behavior. When the first wheelset of the body passed through the de-embedded area, it triggered an increase in the body’s vibration acceleration. As different bogies of the same body passed through the same area, the body experienced vertical swaying at the same time. The periodic swaying occurred over a distance of 17.5 m, which corresponded exactly to the distance between the front and rear bogie centers. This characteristic is useful for high-speed dynamic inspection vehicles to identify CA mortar damage, although it may cause a bumpy experience for passenger vehicle operation.

Figure 14 presents an analysis of the damping parameters for the primary and secondary suspensions of the vehicle. Although identifying damage to the substructure beneath the rails through the dynamic response of inspection vehicles is desirable, locomotive vehicles can adaptively enhance stiffness compensation during operation to meet ride comfort requirements. From the perspective of general passenger vehicles, computational analysis showed that increasing the damping parameters of the primary suspension system did not significantly reduce the vertical acceleration of the vehicle body. However, a 10% increase in the damping of the secondary suspension significantly decreased vertical acceleration. When the secondary suspension was increased by 20%, the reduction in acceleration compensated for the stiffness irregularities of the substructure. Therefore, designing an adaptive suspension system that automatically senses changes in substructure stiffness and makes real-time compensations holds promise for improving the operational comfort of passenger vehicles.

By summarizing the vibration indicators of each subsystem in

Table 4. Statistical analysis of system dynamic response.

Index	Complete Contact (0 m)	Void—1 m	Void—2 m	Void—3 m	Statistically Significant $\mathbf{(}\mathbf{0} \mathbf{m}\to \mathbf{3} \mathbf{m}\mathbf{)}$
Wheel–rail force (kN)	73.56	74.61	80.83	88.92	20.88%
Acceleration of wheelset (m/s2)	1.62	5.19	9.81	17.54	982.71%
Rail displacement (mm)	1.11	1.27	1.50	1.83	64.86%
Rail acceleration (m/s2)	63.6	55.10	53.81	42.01	33.94%
Track slab displacement (mm)	0.52	0.60	0.84	1.33	155.76%
Track slab acceleration (m/s2)	11.77	14.90	27.43	17.80	51.23%
Concrete base displacement (mm)	0.48	0.46	0.38	0.33	31.25%
Concrete base acceleration (m/s2)	9.77	7.45	7.08	6.46	33.87%
Bogie rotation acceleration (m/s2)	2.75	2.54	2.31	1.12	59.27%
Bogie vertical acceleration (m/s2)	1.592	1.597	2.104	3.281	106.09%
Train body vertical acceleration (m/s2)	0.26	0.27	0.29	0.32	23.07%

, statistical analysis revealed that the wheelset acceleration was most sensitive to voids in the CA mortar layer, followed by the dynamic displacement of the track slab. Additionally, the vertical acceleration of the high-speed train bogie showed significant variations. These indicators were useful for detecting hidden voids in the CA mortar layer.

5. Conclusions

This paper establishes a vehicle–ballastless track dynamic coupling model and uses an FEM solution and compares the accuracy of the calculation results. Different void lengths in CA mortar are compared and the effects of the railway track’s dynamic response and transfer function between the ballastless track layers are analyzed. The main conclusions are as follows:

(1) The vibration acceleration of the high-speed train wheelsets was sensitive to the change in the CA mortar void length. It could increase from 1.62 m/s² under complete contact to 17.54 m/s². The displacement of the track slab in the ballastless track structure was more sensitive to the void length.

(2) In the sensitivity ordering of the identification indexes of the voids, the bogie’s vertical acceleration in the high-speed train > the nodding acceleration of the bogie > the vehicle’s vertical acceleration. When the length of the void was 3 m, the bogie’s vertical acceleration increased by 106.09%.

(3) Under the complete-contact condition, the value of the dynamic displacement transfer function between the track slab and concrete base was basically between 0.87 and 1. When the void length increased, the transfer function varies in the intervals of (0.43–0.98), (0.27–0.74), and (0.08–0.32), respectively. The values of the acceleration transfer function between the track slab and the concrete base varied between 0.81 and 1, and when the length of the void increased, it varied between 0.66 and 0.98, 0.34 and 0.75, and 0.07 and 0.30.

Although this paper numerically analyzes the CA mortar layer voids in ballastless track structures, there are still some limitations. At present, there are many ballastless track structure types in the world, among which CRTS-I, CRTS-II, and CRTS-III exist in China, and this paper only analyzes the CRTS-II type. In addition, because in real engineering high-speed trains and ballastless track structures are three-dimensional and the geometry of ballastless track structures is also very complicated, in future research, the three-dimensional characteristics of ballastless track structures can be further considered, and the damage model of concrete can be embedded in the numerical model so as to analyze it at a deeper level.