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Abstract

One of the problems with high-speed tracks in service is the damage of their cement asphalt mortar layer. The development of cement asphalt mortar damage can lead to gradually failure of the slab supporting. In fact, the cement asphalt mortar damage is uniform. The effect of cement asphalt mortar damage location seriously affects the dynamic behavior of the track and even operational safety. It is important to clearly understand the effect of cement asphalt mortar damage location on dynamics, which is helpful to direct track maintenance. In this study, a three-dimensional coupling dynamic model of a vehicle and a slab track is developed. Using the proposed model, the wheel–rail contact forces and the displacements and accelerations of the track are calculated to study the influences of the size and location of the damaged cement asphalt mortar zone on the dynamic characteristics of the track system.

Keywords

Vehicle/track interaction wheel/rail force track dynamics cement asphalt mortar damage

Introduction

A new type of slab track, the so-called China Railway Track System I (CRTS I), is developed and has been applied in the high-speed lines of Chongqing–Suining, Shijiazhuang–Taiyuan, Shanghai–Nanjing, and Harbin–Dalian railways in China.¹ During their operation, different degrees of damages occurred on the cement asphalt mortar (CAM) filling layers of the tracks. The damages include cracks, shelling, and deterioration due to the effect of vadose water in the tracks, as shown in Figure 1.^2,3 The research on the vehicle–track coupling system considering CAM damage is scarce. Xiang et al.⁴ studied the effect of the voided slab induced by the deterioration of the CAM layer on the vibration responses of the slab track with speed variation. Based on the perspective of system energy, he used the Wilson-θ numerical integral method to solve the track vibration equations. Wang et al.⁵ studied the effect of CAM debonding on the dynamic properties of the CRTS II slab track. He used the LS-DYNA to solve the dynamic equations. The authors of both these studies treated the rail as a continuous Euler beam, and their models considered the vertical vibration only. Liu et al.⁶ developed a vehicle–track vertical coupling dynamic finite element method (FEM) model to analyze the influence of the contact loss length on the dynamic characteristics of vehicle and track subsystems at different train speeds. The location of contact loss length was not discussed, and an extreme situation of wheel/rail separation and track displacement beyond limit caused by disabled CAM was also discussed. Zhu and Cai⁷ investigated interface damage and its effect on the vibrations of a slab track under different temperature and vehicle dynamic loads. The loads were obtained using the developed vertical vehicle–track coupling dynamic model and the track model was developed using ABAQUS software. The model assumed that the influence of temperature is important to CAM damage after a period of time. However, when CAM damage had already occurred, its effect on train running safety was not discussed. Han et al.⁸ studied the effect of softening of CAM on vehicle operation safety with considering track dynamics in curve and lateral dynamics. However, the influence of sizes and locations of the damaged CAM zone on the dynamic characteristics of wheel/rail contact force and the track displacement was not discussed.

Figure 1.

Damages of the CAM.^2,3

This article develops a three-dimensional (3D) coupling dynamic model of a high-speed vehicle and a CRTS I slab track. Using the proposed model, the wheel–rail contact forces, the displacements, and the accelerations of the track are calculated in the case of CAM damage at high speed. The influences of sizes and locations of the damaged CAM zone on the dynamic characteristics of the track system are analyzed.

Vehicle–track coupling dynamic model considering disabled CAM

Figure 2 illustrates the coupling dynamic model of a high-speed vehicle and the CRTS I slab track to study the effect of disabled CAM layer on the dynamic behavior of the track. In the numerical simulation, the supporting stiffness under the slab is considered as being zero to approximately simulate the CAM damage. A moving rail support is adopted as a new vehicle–track coupling interface excitation model (called as “tracking window”).^9–12 This excitation model is more close to the real moving vehicle under the excitation of the discrete sleepers, and it can save about 30% of computational time compared to the moving vehicle model. The vehicle–track coupling system equations are solved by means of a new explicit integration method.¹³

Figure 2.

Vehicle–track coupling dynamic model considering disabled CAM.

Dynamic model of vehicle subsystem

According to the vehicle dynamic theory,¹⁴ a high-speed railway vehicle is considered as a rigid multi-body model, in which the car body is supported on two double-axle bogies with the primary and secondary suspension systems. For the connecting parts (primary vertical damper, secondary lateral damper, secondary yaw damper, and lateral stopping block) with non-linear characteristics, a piecewise linear simulation is used. As shown in Figure 2, each component of the vehicle has five degrees of freedom (DOFs): the lateral displacement, the vertical displacement, the roll angle, the yaw angle, and the pitch angle. The total DOFs of the vehicle are 35. Based on the coordinate system moving along the track at a constant speed of the vehicle, the equations of the vehicle subsystem can be described using second-order differential equations in the time domain

$M_{v} {\overset{\cdot\cdot}{u}}_{v} + C_{v} {\overset{\cdot}{u}}_{v} + K_{v} u_{v} = F_{v}$ (1)

where M_v is the mass matrix of the vehicle; C_v and K_v are the damping and stiffness matrices, respectively; ${\overset{\cdot\cdot}{u}}_{v}$ , ${\overset{\cdot}{u}}_{v}$ , and $u_{v}$ are the vectors of displacements, velocities, and accelerations of the vehicle subsystem, respectively; and F_v is the vector of generalized loads acting on the vehicle subsystem.

Dynamic model of slab track subsystem

The dynamic model of the slab track subsystem includes rails, fastener systems, slabs, CAM layers, and concrete base, as shown in Figure 2. The rail is treated as a continuous Timoshenko beam resting on rail pads, and the lateral, vertical, and torsional motions of rails are simultaneously taken into account.¹⁵ The slabs and the concrete base are modeled using 3D FEM. The rail fastener systems and the CAM layer are modeled using periodic discrete viscoelastic units.

The vibration of the slab can be described using second-order differential equations in terms of generalized coordinates, as expressed by equation (2)

${[M]}_{i} {\overset{\cdot\cdot}{u}}_{i} + {[C]}_{i} {\overset{\cdot}{u}}_{i} + {[K]}_{i} {u}_{i} = {F_{rs}}_{i} + {F_{sc}}_{i}$ (2)

where [M]_i, [C]_i, and [K]_i are, respectively, the mass, damping, and stiffness matrices of the ith slab; ${\overset{\cdot\cdot}{u}}_{i}$ , ${\overset{\cdot}{u}}_{i}$ , and ${u}_{i}$ are, respectively, the acceleration vector, velocity vector, and displacement vector; ${F_{rs}}_{i}$ is the load vector between the rail and the ith slab; and ${F_{sc}}_{i}$ is the load vector between the slab and the concrete base.

Equation (2) is decoupled to equation (3) according to the modal superposition principle

$\begin{matrix} {[M_{n}]}_{i} {{\overset{\cdot\cdot}{u}}_{n}}_{i} + {[C_{n}]}_{i} {{\overset{\cdot}{u}}_{n}}_{i} + {[K_{n}]}_{i} {u_{n}}_{i} = {F_{rsn}}_{i} + {F_{scn}}_{i} \\ (n = 1 ~ N_{mode}) \end{matrix}$ (3)

where ${[M_{n}]}_{i}$ , ${[C_{n}]}_{i}$ , and ${[K_{n}]}_{i}$ are, respectively, the generalized regular mass, damping, and stiffness matrices of the ith slab. ${{\overset{\cdot\cdot}{u}}_{n}}_{i}$ , ${{\overset{\cdot}{u}}_{n}}_{i}$ , and ${u_{n}}_{i}$ are, respectively, the generalized regular acceleration, velocity, and displacement vectors. ${F_{rsn}}_{i}$ is the generalized regular load vector between the rail and the ith slab. ${F_{scn}}_{i}$ is the generalized regular load vector between the slab and the concrete base. Number n is the regular modal order. N_mode is the considered total number of modal order.

The generalized regular mass, damping, and stiffness matrices and generalized regular load vectors are solved by equation (4)

${\begin{matrix} \begin{matrix} {[M_{n}]}_{i} = {Φ}_{n}^{T} {[M]}_{i} {Φ}_{n}, & {[C_{n}]}_{i} = {Φ}_{n}^{T} {[C]}_{i} {Φ}_{n}, & {[K_{n}]}_{i} = {Φ}_{n}^{T} {[K]}_{i} {Φ}_{n} \end{matrix} \\ [P_{n}]_{i} = {Φ}_{n}^{T} {F^{rs}}_{i} + {Φ}_{n}^{T} {F^{g}}_{i} \end{matrix}$ (4)

where ${Φ}_{n}$ is the regular modal vector, which can be calculated by means of ANSYS to obtain the first 20 order modes, which include 6 rigid and 14 elastic modes. The length of the slab is 4.962 m. The geometric dimensioning of its cross section is 0.19 m × 2.4 m.

The model of the concrete base is similar to that of the slab. The length of the concrete base is 60 m. The geometric dimensioning of its cross section is 0.3 m × 3 m.

Model of wheel–rail in rolling contact

Wheel/rail dynamic interaction modeling is the key of vehicle–track coupling dynamic model. In this article, when calculating the dynamic response of the vehicle–track system, the tracing curve method¹⁶ is adopted to locate the wheel–rail spatial contact geometry. Therefore, the method of two-dimensional (2D) scanning can be replaced by one-dimensional (1D) scanning through a tracing curve. It can greatly reduce the computational time. The non-linear Hertzian elastic contact theory is used to calculate the wheel–rail normal contact forces. The tangential wheel–rail creep forces are calculated using the Shen–Hedrick–Elkins non-linear theory.¹⁷

Figure 3 shows the rail and slab displacements calculated by the proposed 3D coupling dynamic model to discuss the reasonability about the model before analysis. From Figure 3, it can be seen that the displacements have four peaks corresponding to four wheelsets of one vehicle, the distances between wheels and bogies consistent with the set of vehicle parameters and the basic physical concept. The amplitude of rail displacement is close to the results reported previously.^7,14 Also the amplitude of slab displacement is close to the results reported previously.⁸ Through discussion and analysis of the basic results of the developed model, it can be seen that the model developed and used in this article is reasonable.

Figure 3.

Rail and slab displacements based on the proposed 3D coupling dynamic model.

Results and analysis

Using the vehicle–track coupling dynamic model, the dynamic responses of the track system influenced by the sizes and locations of the damaged CAM zone are analyzed, as shown in Figure 4. Figure 4(a) illustrates the size change of the damaged CAM zone. A1–A3 represent different sizes, the length of which are 2 sleeper pitches (1.258 m), 4 sleeper pitches (2.516 m), and 6 sleeper pitches (3.774 m), respectively. And A0 represents the case without the CAM damage. Figure 4(b) illustrates the location change of the damaged CAM zone without size change. P1–P4 represent different locations, the length of which are 2 sleeper pitches (1.258 m). And P0 represents the case without the CAM damage. Figure 4(c) illustrates the position of the damaged CAM zone occurred at the two ends (each end 0.629 m) of the slab. Figure 4(d) illustrates the position of the damaged CAM zone occurred at the two sides (each side 0.45 m) of the slab.

Figure 4.

Changes of size and location of the damaged CAM zone: (a) area change, (b) location change I, (c) location change II, and (d) location change III.

Effect of the damaged CAM area

In this analysis, a high-speed train runs on a tangent slab track at 250 km/h. The track is symmetrical with respect to its centerline, so here the dynamic responses of the left side of the track are presented. Taking the leading wheelset as the example, Figure 5 shows the wheel/rail normal forces with the traveling distance in the cases without the CAM damage (A0) and with the different damaged areas (A1–A3). As shown in Figure 5, when the wheelset is about to pass over the damaged area, the wheel/rail normal force starts deloading. When the wheelset is passing through the area, the wheel/rail normal force increases significantly and its maximum value occurs in the middle of the damaged area. With the increase of the damaged CAM area, the wheel/rail normal force increases.

Figure 5.

Wheel/rail normal forces of the leading wheelset in the case of the damaged CAM area.

Figure 6 shows the maximum wheel/rail normal forces of the four wheelsets in the cases of the different damaged CAM areas. W1 (W3) and W2 (W4) indicate the leading wheelset and trailing wheelset of Bogie 1 (2), respectively. When the damaged area reaches A3 (longitudinal length 3.72 m > wheelbase 2.5 m), the leading wheelset has not yet left the damaged area; however, the trailing wheelset has entered the damaged area. In such a situation, the maximum wheel/rail normal force of the trailing wheelset is greater than that of the leading wheel.

Figure 6.

Maximum wheel/rail normal force with the increase of damaged CAM area.

Figure 7(a) shows the vertical displacement of the rail at the first wheelset with the traveling distance in the cases of CAM damage (A0, A1, A2, A3, and A3). As shown in Figure 7(a), with the increase of the damaged CAM area, the vertical displacement of the rail increases. The vertical displacement of the rail in the case of A3 increases most obviously.

Figure 7.

Vertical displacement changes of the rail influenced by the damaged CAM area: (a) under the leading wheelset and (b) in the damaged CAM zone.

Figure 7(b) shows the vertical displacement of the rail in the middle of the damaged CAM zone. In Figure 7(b), there are four evident displacement peaks, which denote the largest displacements of the rail at the four wheels W1, W2, W3, and W4. The displacement of the rail in the damaged CAM zone is much greater than that in the case without the CAM damage. With the increase of the damaged CAM area, the vertical displacement of the rail increases. In the case of A3, the vertical displacement of the rail increases obviously. The rail displacement under W2 is greater than that under W1, and the displacement under W4 is greater than that under W3.

The vertical displacement of rail is usually less than the benchmark 1.5 mm and shall not be greater than the maximum limit of 2 mm.¹⁸ In the case of A3, the vertical displacement of the rail exceeds the maximum limit.

Figure 8(a) shows the vertical displacement of the slab at the fasteners in the cases of A0–A3. As shown in Figure 8(a), when the vehicle passes over the damaged CAM zone, the vertical displacement of the slab increases obviously. With the increase of the damaged CAM area, the vertical displacement of the slab increases. The vertical displacement of the slab in the case of A3 increases most obviously.

Figure 8.

Vertical displacement of the slab influenced by the damaged CAM area: (a) at the fasteners and (b) in the damaged CAM zone.

Figure 8(b) shows the vertical displacement of the slab in the middle of the damaged CAM zone. As shown in Figure 8(b), there are four obvious displacement peaks when the vehicle passes over the monitoring points. The displacement of the slab in the damaged CAM zone is significantly greater than that in the case without the CAM damage. With the increase of the damaged CAM area, the vertical displacement of the slab increases. In the case of A3, the vertical displacement of the slab increases obviously. At the same time, the displacement peaks contain the combined contributions by the passing wheels W1 (W3) and W2 (W4).

The vertical displacement in the middle of the slab is usually less than the benchmark 0.2 mm and shall not be greater than the maximum limit of 0.3 mm.¹⁸ In the case of A2 and A3, the vertical displacement in the middle of the slab exceeds the maximum limit.

Effect of location of the damaged CAM zone

Figure 9 shows the wheel/rail normal forces with the traveling distance in the case that considers the normal CAM (A0) and the different positions (P1–P4) of the damaged CAM zones. For simplicity, the case is indicated by (I). The damaged CAM areas occur from the end to the middle of a slab along the longitudinal direction. As shown in Figure 9, when the wheelset is about to pass over the damaged zone, the wheel/rail normal force appears deloading in the first place. Then when the wheelset passes over the damaged CAM zone, the wheel/rail normal force increases significantly. The damage to the CAM occurring at the end of the slab (P1) causes the most significant wheel/rail normal force.

Figure 9.

Wheel/rail normal force change of the leading wheelset in (I).

Figure 10 shows the vertical displacements of the rail with the traveling distance in (I). When the damaged CAM zone occurs at the end of the slab (P1), the vertical displacement of the rail is the most significant, and when the damage occurs in the middle of the slab (P4), the vertical displacement of the rail is the second significant, which does not exceed the benchmark.

Figure 10.

Vertical displacement of the rail under the leading wheelset in (I).

Figure 11 shows the vertical displacements of the slab at the fasteners in the longitudinal direction positions in (I). The damaged CAM zone (P1) causes the most significant vertical displacement of the slab, which exceeds the benchmark 0.4 and the maximum limit 0.5 (standard in the end of the slab).¹⁸

Figure 11.

Vertical displacement of the slab at the fasteners in (I).

Figure 12 shows the wheel/rail normal force with the traveling distance in the case which considers the normal CAM, the damaged CAM area occurring at the rear of the slab, and at both the front and rear of the slab. For simplicity, the case is denoted using (II). In Figure 12, the black line indicates the result of the normal CAM, the dashed line represents the result of the damaged CAM zone at the rear, and the dash dot line represents the results at the front and the rear. When the vehicle passes over the damaged CAM zone at the rear, it is found that the wheel/rail normal forces of the two kinds of damaged CAM zones are basically identical. However, in the damaged CAM zone in the front of the slab with the vehicle passing over it, the fluctuating amplitudes of the wheel/rail normal force are enlarged again.

Figure 12.

Wheel/rail normal force of the leading wheelset in (II).

Figures 13 and 14 show the vertical displacements of the rail and slab with the traveling distance in the same case as in Figure 12. When the vehicle passes over the damaged CAM zone in the rear or front, the vertical displacements of rail/slab in the two damaged CAM zones are close and have a large peak (Peak 1). When the damage occurs in both the rear and front ends of the slab, the vertical displacement of the rail/slab has two high peaks (Peak 1 and Peak 2).

Figure 13.

Vertical displacement of the rail under leading wheelset in (II).

Figure 14.

Vertical displacement of the slab at the fasteners in (II).

Figure 15 shows the wheel/rail normal force acting on the left rail with the traveling distance in the case that includes the normal CAM and the damaged CAM on the left/right side of the slab. For simplicity, the case is denoted using (III). As shown in Figure 15, the dashed line represents the damaged CAM zone on the left side and the dash dot line represents the damaged CAM zone on the left and right sides. The influence of case (III) on the wheel/rail normal force is not obvious. Enlarging the damaged zone, when the vehicle passes over it, the fluctuating amplitude of the wheel/rail normal force increases and that of the wheel/rail normal force in the case of the damaged CAM areas on the left and right sides is larger than that on the left side.

Figure 15.

Wheel/rail normal force of leading wheelset in (III).

Figure 16 shows the vertical displacement of the left rail with the traveling distance in (III). As shown in Figure 16, when the vehicle passes the damaged CAM zone, the rail displacement increases, and after a stable period, it begins to decrease. The rail displacement in the case of the damaged CAM zone on the left side is slightly larger than that on the left and right sides at the same time. The main reason is that the damage only on the left side causes the unbalance supporting between the left and right sides of the track.

Figure 16.

Vertical displacement of the rail under leading wheelset in (III).

Figure 17 shows the slab vertical displacement under the left rail in the fasteners in (III). As shown in Figure 17, when the vehicle passes over the damaged CAM zone, the slab vertical displacements at its two ends are larger than those in its middle. The slab displacement in the damaged CAM zone on the left side of the slab is slightly larger than that on the left and right sides of the slab at the same time. The main reason is that the damage only on the left side causes the unbalance supporting between the left and right sides of the slab.

Figure 17.

Vertical displacement of the slab at the fasteners in (III).

Figure 18 shows the maximum vertical displacement of the slab surface in the lateral direction in (III). As shown in Figure 18, when the vehicle passes over the cross section of the slab, a part of the slab with the damaged CAM areas bends down significantly and the rest of the slab bends upward compared to the slab without the CAM damage.

Figure 18.

Vertical displacement of the slab in the lateral direction in (III).

It can be seen from Figures 16 –18 that the displacements of rail and slab do not exceed the benchmark and the maximum limit. However, this kind of damage (case III) is common in practice, and it can easily develop into a comprehensive damage. So this kind of damage is still worthy of our attention.

Conclusion

A 3D coupling dynamic model of a vehicle and a CRTS I slab track is developed. The developed model is to study the effect of the location of the damaged CAM zone on the track dynamics. The following conclusions can be drawn:

When the vehicle passes over the damaged CAM zone, the dynamic responses of the track increase significantly and their maximum values appear in the middle of the damaged zone. With the increase of the damaged CAM area, the rail displacement increases from 0.5 to 2.25 mm and the slab displacement increases from 0.1 to 1.75 mm. And it is recommended that the longitudinal length of the damaged CAM zone should not be larger than the wheelbase (2.5 m) of the vehicle because vertical rail displacement (2.25 mm) and slab displacement (1.75 mm) with the CAM damage in this case are much larger than the corresponding maximum limits.

The CAM damage that occurred at one end (1.258 m × 2.4 m) of the slab causes the most significant dynamic responses of the track, compared to the damaged CAM zones occurring in the other places. The vertical displacement of the slab with the CAM damage occurred at one end is 0.8 mm, which is beyond the maximum limit. The difference of slab displacement between the CAM damage occurred at one end and that in the other places is about 0.7–0.8 mm.

When the vehicle passes over the CAM damage (one end: 0.629 m × 2.4 m) occurs at the rear or front of slab, wheel/rail normal force, rail and slab displacement occurs significant peaks. When the CAM damage occurs at both ends, the second significant peak occurs in the wheel/rail normal force and track dynamic, and the fluctuating amplitudes of track dynamic are enlarged again at the second damage zone. It should be avoided that CAM appears in more than two damages in the longitudinal direction when the railway train is operated at a high speed.

When the CAM damage occurs on the left or right side (one side: 0.45 m × 4.962 m), a part of the slab with CAM damage bends down significantly and the rest of the slab bends upward compared to the slab without CAM damage. The dynamic responses of the track with the CAM damage on the one side of the slab are slightly larger than those on the left and right sides of the slab at the same time. The difference is smaller than 0.01 mm. However, both rail and slab displacements are not beyond the limit.

Footnotes

Handling editor: Aki Mikkola

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: The financial support was provided by the National Natural Science Foundation of China (Nos U1434201 and 51475390) and the National Key R&D Program of China (Nos 2016YFB1200503-02,2016YFB1200506-08,and 2016YFE0205200).

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