Study on the long-distance settlement threshold of high-speed railway subgrade based on vehicle responses

Abstract

The long-distance and large-scale subgrade settlement is a key issue in high-speed railway (HSR). This work focuses on the long-distance settlement threshold for the HSR subgrade. A Green’s function-based hybrid dynamic model with high efficiency is established to achieve the long-distance analysis of the vehicle-track-subgrade interaction system with a wide range of subgrade settlement. Based on the analysis results, the Wavelet decomposition technique is applied to extract the low-frequency components of vehicle responses excited by the additional deformation of rails induced by the long-distance subgrade settlement. The simulation results show that the low-frequency vibrations of vehicle are mainly excited by the additional deformation of rails induced by settlement. Based on the correlation between the vehicle responses and the second derivative of additional deformation of rails, the threshold of long-distance settlement can be determined. The vehicle response and subgrade settlement signal are linked in the time-frequency domain, and the relationship between system excitation and response is clearly revealed. This paper provides a referenceable analysis framework that is convenient for engineers in the control for the amplitude of large-scale subgrade settlement in HSR.

Keywords

railway subgrade settlement vehicle-track-subgrade interaction settlement threshold time-frequency analysis

1. Introduction

China HSR has entered the stage of long-term safe and stable operation from large-scale construction. However, the settlement of the subgrade of existing railway lines in service is a key scientific issue that affects the long-term safe and stable operation of HSR (Zhai et al., 2014). Subgrade settlement as a major defect of HSR has a non-negligible impact on the vehicle-track interaction (Paixao et al., 2015).

In general, the profile of subgrade settlement can be depicted as a long-wavelength signal, and it is an important external excitation that influences the dynamic behavior of high-speed vehicles and tracks (Ren et al., 2023; Shan et al., 2021; Guo and Zhai, 2018). Despite the past efforts to understand and investigate the effect of different wavelength-amplitude settlements on the vehicle-track dynamic interaction, most researchers concentrated on analyzing the single cosine type of settlement (Cai et al., 2022; Cui and Ling, 2021). The study of single cosine-type settlement is beneficial for obtaining regular conclusions. For example, the positive correlation between the wavelength-amplitude characteristics of settlement and the system responses has been reported (Chen and Fang, 2021). Moreover, the cumulative settlement has been predicted in the cosine-type settlement region by Guo and Zhai (2018). The subgrade settlement generates the additional deformation of track structures, and the additional deformation is regarded as the geometric excitation of wheel-rail interaction, which influences the system vibration characteristics. Karis et al. (2019) found a correlation between vehicle responses and track irregularities, but they thought that such a correlation exists only in the results of numerical simulation. They conducted a comparison of the simulation and measured results and found the measured data was irregular. The correlation between the vehicle acceleration and the long-wavelength settlement is also applied in the identification of subgrade settlement (Ren et al., 2023).

At present, most of the current research conducted by scholars can derive the controlled indicators of subgrade settlement for a specific problem, such as the additional stress of soil (Jiang et al., 2019), dynamic performance of the vehicle-track system (Zhang et al., 2022) and the damage of track slab (Cui and Ling, 2021). However, the above research focused on the small area of subgrade settlement. As for the HSR operation departments, they are more concerned about the settlement of the whole railway line, which includes long-distance and large-scale settlement areas. Zhai et al. (2014) reported a 4-km subgrade settlement in Beijing-Tianjin intercity HSR, in which the maximum settlement reaches 28 mm. Moreover, according to other literature (Zhao, 2016), the wavelength of subgrade settlement of China HSR ballastless track is more than 20m. Therefore, the threshold of long-wavelength subgrade settlement becomes a key scientific problem. Moreover, most research neglects the frequency information of the relationship between the settlement and the system response. Obviously, using the time-frequency method to depict the influence of subgrade settlement is more proper, such as the method adopted in the identification of track welded irregularity (Lou et al., 2023).

The large-scale subgrade settlement is a complex issue. On the one hand, the irregular shape of long-distance settlement has the upper arch type in small areas. On the other hand, the long-term analysis of the vehicle-track-subgrade (VTS) dynamic interaction under the large-scale settlement is difficult: in previous studies, the wavelength of settlement that can be considered in the model generally does not reach thousands of meters.

To achieve the long-distance analysis for large-scale subgrade settlement, this paper establishes a VTS dynamic model and based on this model, the correlation between the vehicle responses and settlement is investigated in detail. The novelty of this work lies on the analysis of VTS dynamic interaction subject to a long-distance subgrade settlement through a Green’s function-based hybrid dynamic model, and by introducing the time-frequency combined technique, the low-frequency components of vehicle vibrations excited by the settlement are extracted and associated to the settlement characteristics. The influence of subgrade settlement on vehicle vibrations is revealed from the perspective of the time-frequency correlation, and the amplitude-wavelength threshold of long-wave settlement is determined in a more convincing way than the machine learning method in some research (Ren et al., 2023; Chen et al., 2023).

2. Establishment of the VTS dynamic interaction

The establishment of a fully coupled VTS dynamic interaction model is necessary for analyzing and depicting the dynamic performance of the vehicle running through rail transportation. In this section, the VTS dynamic model with high efficiency is founded on the previous model basis (Zhai et al., 2009; Xu et al., 2020).

2.1. The basic configuration of the VTS dynamic interaction

Figure 1 shows the basic configuration of the VTS dynamic interaction system. The vehicle is modeled as multi-rigid-bodies, namely, the car body, front and rear bogies, and four wheelsets. Each body has 6 degrees of freedom (DOFs), namely, longitudinal-, lateral-, vertical-, yaw-, rolling-, and pitch-motion. The typical ballastless track system can be modeled by finite element (FE), in which the rail is modeled by Bernoulli–Euler beam elements, and the track slab and base plate are modeled as plate elements by the Mindlin plate theory. The railway subgrade is modeled by the 8-nodes solid elements. The interaction system between the track slab-base plate, and base plate-subgrade are modeled as spring-dashpot elements.

The dynamic matrices can be derived by the energy variation method improved by Zeng (2000). The total potential energy of the rigid-elastic system can be expressed as

Π_{d} = U_{ε} + V + V_{m} + V_{c}

(1)

where

Π_{d}

is the total potential energy of the rigid-elastic system;

U_{ε}

is the total strain energy of the system;

V

is the potential energy of external force;

V_{m}

is the negative value of work done by the inertia force;

V_{c}

is the negative value of work done by the damping force.

The dynamic matrices can be derived by the energy variation method, as given by

δ Π_{d} = δ (U_{ε} + V + V_{m} + V_{c}) = 0

(2)

where

δ

denotes the variation symbol.

The global matrices can be assembled by the “set-in-right-position” rule proposed by Zeng and Yang (1986). The basic modeling work has been elaborated in the previous work (Li et al., 2022a; Xu et al., 2020, 2021).

2.2. Solution method for VTS dynamic interaction

To achieve the high-efficient analysis of the VTS interaction with large DOFs, a Green’s function-based hybrid model is established. Different from other high-efficient techniques, for example, the 2.5D finite element method (Yang and Hung, 2001) and the periodic finite element model (Degrande et al., 2006), the hybrid model can describe the complex wheel-rail nonlinear contact behavior by the nonlinear Hertz contact and the creep theory. Depending on the frequency characteristics, the VTS interaction system can be classified into two subsystems, namely, the vehicle-rail interaction system and the slab track-soil system. The high-frequency vehicle-rail interaction system is solved by time-domain numerical integration, and the low-frequency slab track-subgrade system is solved by Green’s function method, as schematically shown in Figure 2. The hybrid model retains the ability to analyze the dynamic nonlinear contact between wheel and rail (Xu et al., 2021), and it can improve the efficiency without iterative procedures in implicit integration schemes.

The Green’s function matrices of the slab track-subgrade interaction system in the fastener points can be expressed as

G_{f} (x_{0} + c t, t) = [\begin{array}{l} G_{f, 11} (x_{0} + c t, t) & \dots & G_{f, 1 N} (x_{0} + c t, t) \\ ⋮ & ⋱ & ⋮ \\ G_{f, N 1} (x_{0} + c t, t) & \dots & G_{f, N N} (x_{0} + c t, t) \end{array}]

(3)

where

G

is the Green’s function,

x_{0}

is the position of the fastener force; the subscript “

f

” denotes the fastener, and “

N

” denotes the total number of fasteners.

G_{f, 1 N}

represents the unit impulse response of the first fastener when the unit impulse load is applied in the

N

th fastener;

c

is the vehicle speed and “

x_{0} + c t

” is a symbol of space coordinates that reflect the location of the fastener in the moving calculation boundary.

G_{f, 1 N}

can be derived by

G_{f, 1 N} (t) = F^{- 1} [H_{f, 1 N} (i Ω)]

(4)

where

H

is the receptance of track slab-subgrade system in the fastener points;

F

is the symbol of Fourier transform;

Ω

is the frequency samples with an interval of

Δ Ω

. The receptance of the fastener points of the track slab-subgrade interaction can be obtained by

H_{f, 1 N} (i Ω) = {(K_{T S} - M_{T S} + i C_{T S})}^{- 1} \tilde{F} (i Ω)

(5)

where

\tilde{F}

is the unit load vector applied in the fastener points;

K

M

, and

C

denote the stiffness, mass, and damping matrices of the track slab-subgrade interaction system; the subscript “

T S

” represents the “track slab-subgrade interaction system.”

The response of fastener points in the slab track-subgrade system can be calculated by the Duhamel integral. To ensure numerical accuracy, the solution method proposed by Mazilu (2010) is adopted, as follows:

X_{T S} (t_{n}) = d t \sum_{i = 1}^{n} | \frac{G (t_{n} - t_{i - 1}) P (t_{i}) + G (t_{n} - t_{i}) P (t_{i - 1})}{2} + \frac{[G (t_{n} - t_{i}) - G (t_{n} - t_{i - 1})] [P (t_{i}) - P (t_{i - 1})]}{3} |

(6)

where the

P

is the external load vector;

d t = t_{i} - t_{i - 1}

is the time interval;

X_{T S}

is the dynamic displacement of the track slab-subgrade interaction system. The velocity and acceleration of the system can be calculated by

{\begin{cases} {\dot{X}}_{T S} (t_{n}) = d t \sum_{i = 1}^{n} | \frac{\dot{G} (t_{n} - t_{i - 1}) P (t_{i}) + \dot{G} (t_{n} - t_{i}) P (t_{i - 1})}{2} + \frac{[\dot{G} (t_{n} - t_{i}) - \dot{G} (t_{n} - t_{i - 1})] [P (t_{i}) - P (t_{i - 1})]}{3} | \\ {\ddot{X}}_{T S} (t_{n}) = d t \sum_{i = 1}^{n} | \frac{\ddot{G} (t_{n} - t_{i - 1}) P (t_{i}) + \ddot{G} (t_{n} - t_{i}) P (t_{i - 1})}{2} + \frac{[\ddot{G} (t_{n} - t_{i}) - \ddot{G} (t_{n} - t_{i - 1})] [P (t_{i}) - P (t_{i - 1})]}{3} | \end{cases}

(7)

where

{\dot{X}}_{T S}

and

{\ddot{X}}_{T S}

are the velocity and acceleration of track slab-subgrade interaction system, respectively;

\dot{G}

and

\ddot{G}

denote the first derivative and second derivative of Green’s function; the subscript “

T S

” denotes the track slab-subgrade system.

The track slab-subgrade system described by using Green's function avoids the inverse calculation of the dynamic system with large DOFs, which can effectively improve computational efficiency. Especially for the long-distance analysis in this paper, the hybrid method is important for saving the computational source.

The compatibility between the two subsystems is maintained by the equilibrium of the fasteners’ interaction forces, as follows:

{\begin{cases} F_{f, s} (t) = k_{f} (t) (X_{f, V R} (t) - X_{f, T S} (t)) \\ F_{f, d} (t) = c_{f} (t) ({\dot{X}}_{f, V R} (t) - {\dot{X}}_{f, T S} (t)) \end{cases}

(8)

where

F

is the force vector;

X_{f, V R}

and

X_{f, V R}

are the fastener point’s displacement of vehicle-rail interaction system and track slab-subgrade interaction system, respective;

{\dot{X}}_{f, V R}

and

{\dot{X}}_{f, T S}

are the corresponding velocity of the two sub-systems;

k_{f}

and

c_{f}

are the stiffness and damping coefficients of spring-dashpot elements, respectively; the subscripts “

f

”, “

V R

”, “

s

”, and “

d

” denote the fastener points, vehicle-rail system, spring element, and dashpot element, respectively.

Based on the interaction forces calculated by equation (8), the dynamic equation of motion of the vehicle-rail interaction system can be expressed as

[\begin{array}{l} K_{V V} & K_{V R} \\ K_{R V} & K_{R R} \end{array}] {\begin{cases} X_{V} \\ X_{R} \end{cases}} + [\begin{array}{l} C_{V V} & C_{V R} \\ C_{R V} & C_{R R} \end{array}] {\begin{cases} {\dot{X}}_{V} \\ {\dot{X}}_{R} \end{cases}} + [\begin{array}{l} M_{VV} & 0 \\ 0 & M_{R R} \end{array}] {\begin{cases} {\ddot{X}}_{V} \\ {\ddot{X}}_{R} \end{cases}} = {\begin{cases} F_{V} \\ F_{R} \end{cases}} + {\begin{cases} 0 \\ F_{f, s} + F_{f, d} \end{cases}}

(9)

where

F_{V}

and

F_{R}

are the wheel-rail interaction forces, which had been elaborated in (Xu et al., 2021). The dynamic equation can be solved by implicit integration, such as the Park method (Park, 1975).

2.3. Cyclic calculation method

Generally, to guarantee computational efficiency, the scale of the model boundary is limited to less than a certain region. Thus, it is hard to achieve the long-length computation of the VTS interaction system. In this paper, the cyclic calculation method (Xu et al., 2018) is applied to achieve the long-length computation of the VTS dynamic model.

As shown in Figure 3, the cyclic calculation method is a cyclic boundary method, its main methodology is limiting the calculation boundary to a certain finite region near the moving vehicle. The mapping relationship between the boundary local coordinate system $X' Y' Z' - O$ and the global coordinate system $X Y Z - O$ can be depicted as

{\begin{cases} n_{ccb, O^{'}}^{Θ} = n_{global, O}^{Θ} - n_{global, O}^{Θ} (L_{4} - L_{0}) \\ n_{ccb, X^{'}}^{Θ} = n_{global, X}^{Θ} - n_{global, X}^{Θ} (L_{4} - L_{0}) \\ n_{ccb, Y^{'}}^{Θ} = n_{global, Y}^{Θ} \\ n_{ccb, Z^{'}}^{Θ} = n_{global, Z}^{Θ} \end{cases}

(10)

where

n_{ccb, Π}^{Θ} (Π = O', X', Y', Z')

is the element number of the calculation boundary in the coordinate system

X' Y' Z' - O

;

n_{global, Π}^{Θ} (Π = O, X, Y, Z)

denotes the DOF number of the calculation boundary in the global system

X Y Z - O

;

L_{4}

is the spatial coordinate of the 4th wheelset;

L_{0}

is the distance of the 4th wheelset from the calculation boundary; the subscripts “

ccb

” denotes the DOF location of cyclic calculation boundary.

2.4. Model validation

To validate the presented hybrid model, the FE model of VTS has been established. The FE model is solved by the Park implicit integration method. In the hybrid model, the vehicle-rail interaction system is solved by the Park method, and the track slab-subgrade interaction system is solved by Green’s function method. The time-step size is 0.0002 s and the vehicle speed is 300 km/h. Figures 4 and 5 display the comparison results between the two models. It can be seen that the calculation results of the two models are approximate, indicating that the hybrid model established in this paper is effective. In terms of computational efficiency, the hybrid model takes 8 min, while the FE model takes 3.5 h.

3. Analysis framework

To demonstrate the effect of long-wavelength subgrade settlement on the dynamic behavior of high-speed vehicles from the frequency domain perspective, this section presents a referable analysis framework that can better address this issue. It can be organized as the following steps:

Step 1

Selecting the measured data of subgrade settlement from HSR and calculating the additional deformation of rails caused by settlement using the iterative method proposed by Li et al. (2022b).

Step 2

The calculated additional deformation of the rails (low-frequency excitation) is used as the excitation of the wheel-rail interaction system. Simultaneously, the original track irregularity (high-frequency excitation) is considered to simulate a more realistic vehicle running condition.

Step 3

The implementation of the long-distance analysis of the VTS dynamic interaction system is conducted by using the hybrid model established in Section 2.

Step 4

Extracting the vehicle responses obtained in Step 3 and investigating the correlation between the vehicle responses and the additional deformation of the rails induced by subgrade settlement. The specific methodologies such as the time-frequency analysis and coherence analysis are introduced and described in Section 4.

Step 5

According to vehicle running safety and passenger’s comfort, the threshold of subgrade settlement varying from different long-wavelength can be determined.

Through the above procedure, the analysis of VTS dynamic interaction under the long-distance settlement can be achieved. The overall procedure is schematically illustrated by the flow chart in Figure 6.

4. Numerical study

In this section, an illustrative numerical study is conducted to demonstrate the effectiveness of the analysis framework proposed in Section 3. Moreover, the threshold of the amplitude of long-distance subgrade settlement is analyzed by the time-frequency analysis techniques.

4.1. The time history of track irregularity sample

As the vital excitation of wheel-rail interaction, the track irregularity is classified as two components, namely, (i) the original track irregularity and (ii) additional track irregularity caused by the settlement, denoted as I₁ and I₂, respectively.

The original track irregularity (I₁) with the wavelength of 0.5–200 m is produced by the ballastless track power spectral density (PSD) of China HSR (PSD of Ballastless Track Irregularities of High-Speed Railway, 2014). The expression of the PSD can be written as follows:

S (Ω) = \frac{A}{Ω^{n}}

(11)

where

S (Ω)

is the power spectral density,

Ω

is the spatial frequency (1/m), A and n are the fitting coefficients, which are listed in Table 1.

Table 1.

Fitting coefficients of ballastless track power spectral density.

Fitting coefficient	2 m ≤ λ ≤ 6.5 m	6.5 m ≤ λ ≤ 21.1 m	21.1 m ≤ λ ≤ 53.5 m	53.5 m ≤ λ ≤ 200 m
A	3.9488 × 10⁻⁴	1.9784 × 10⁻²	3.5588 × 10⁻³	1.0544 × 10⁻⁴
n	3.4516	1.3643	1.9271	3.3891

In this work, the spatial samples of rail irregularities are generated by the triangular series method, which can be formulated as

w (x) = \sum_{k = 1}^{N} \sqrt{2 S (w_{k}) Δ w} \cos (w_{k} + φ_{k})

(12)

where

w (x)

denotes the generated rail irregularity series;

S (w_{k})

denotes the power spectral density function for

w (x)

;

w_{k} (k = 1,2, . . . N)

is the frequency series with an bandwidth of

Δ w

;

φ_{k}

is the phase of the

k

th frequency.

In which, λ is the wavelength of the original track irregularity.

The time history of the original track irregularity is shown in Figure 7. For the simulation of long-distance settlement, the settlement curve of the Wuqing section of the Beijing-Tianjin intercity HSR reported in the literature (Zhai et al., 2014) is taken as an example. The iterative method proposed in the literature (Li et al., 2022b) is used to calculate the additional irregularity (I₂) caused by settlement, as shown in Figure 8(a). In this paper, the additional deformation of rails is regarded as the additional track irregularity excitation in wheel-rail interaction. The combined track irregularity (I₁ + I₂) curve is shown in Figure 8(b).

4.2. Effect of subgrade settlement on vehicle response

To investigate the effect of subgrade settlement on vehicle response, the calculation results with and without subgrade settlement are compared, as shown in Figures 9 and 10. From the comparative results, a conclusion similar to those in the literature (Li et al., 2022a, 2022) can be drawn: when the vehicle speed is constant, the subgrade settlement mainly affects the acceleration of the vehicle; for the wheel-rail force, the difference is relatively small. The difference in the car body acceleration shows that the long-distance settlement is a long-wavelength excitation for the vehicle system. Based on this conclusion, a correlation study between settlement and vehicle response can be carried out.

4.3. Positive correlation between the vehicle responses and settlement-induced track irregularity

Several literatures (Karis et al., 2019; Gou et al., 2020; Gou et al., 2019) have reported there is a positive correlation between the vehicle response and the second derivative (SD) of the track irregularity. By using this positive correlation, the relationship between car body acceleration, I₁ and I₂ was analyzed.

Taking the car body acceleration as an example, Figure 11 shows the vertical car body acceleration and the SD of I₁ + I₂. From the results, it can be seen that it is difficult to observe a clear correlation from the time domain due to the high-frequency components of the SD of I₁ + I₂. For high-speed railways, it is well-known that the track irregularity generally belongs to medium and short waves with wavelengths below 50 m. To eliminate the influence of high-frequency components, the frequency band of I₁ and I₂ exciting the system vibrations should be determined.

Figure 12 displays the Fourier spectrum of car body acceleration, SD of I₁ + I₂, SD of I₁ and SD of I₂. It can be observed from the comparison results that the additional track irregularity I₂ caused by the settlement is in the low-frequency band. The low-frequency components of car body acceleration are mainly contributed by I₂. For the high-frequency vibrations, it is mainly excited by the original track irregularity I₁. When the VTS dynamic system is only excited by the original track irregularity, the coherence function increases obviously at the frequency of 0.0057 (1/m). To further validate the low-frequency vibrations that are excited by the subgrade settlement “I₂”, the coherence functions between the car body accelerations and the SD of I₁, I₂ are calculated, as shown in Figure 13. It can be seen that the coherence between car body acceleration and SD of I₂ is strongest in the frequency band 0–0.0061 (1/m), the values of the coherence function are close to 1. On the contrary, the coherence between car body acceleration and SD of I₁ is weak in the frequency band 0–0.0061 (1/m). The results of the coherence show that I₂ excites the low-frequency vibration of the vehicle system. Therefore, the low-frequency components of vehicle accelerations that are most relevant to the subgrade settlement signal by the time-frequency method can be extracted.

In conclusion, it can be considered that the additional track irregularity caused by long-distance settlement has an obvious impact on the vehicle's low-frequency vibration. Therefore, the low-frequency components of the vehicle response can be extracted for analysis using a similar filtering method. To investigate the relationship between SD of I₂ and low-frequency components of car body acceleration separately, the signal components in the 0–0.0061 (1/m) frequency band are extracted using Wavelet decomposition (Daubechies 1992). In this case study, the Wavelet basis function used for the Wavelet decomposition is “db6” type, and the number of decompositions is 12. The approximate signal of the car body acceleration signal and SD of I₂ signal after the Wavelet decomposition are shown in Figure 14. It can be seen that the car body acceleration and the SD of I₂ have a similarity in the time domain. It shows that the low-frequency components of vehicle acceleration have a mapping relationship with the long-wavelength settlement. Generally, the mapping relationship between them can be depicted by a linear function because the SD of the settlement and the time history of vehicle acceleration are both signals that approximate zero means. Utilizing the least square method to fit the car body acceleration and the SD of I₂, as shown in Figure 15(a).

From the fitting results, it can be seen that the car body acceleration has a positive correlation with the SD of I₂. To further validate the positive correlation between the car body acceleration and SD of I₂, the cosine type of track irregularity is taken as an instance, where the wavelength of track irregularity is 200 m (spatial frequency < 0.0061 (1/m)), and the amplitude is 20 mm. Figure 15(b) shows the car body acceleration and the SD of track irregularity, it can be seen that the calculated samples fit well with the first-order function “y = 206.1x-9.625e⁶”, which illustrates the car body acceleration has a positive correlation with the SD of additional track irregularity caused by settlement.

4.4. The control for long-distance settlement

The threshold of settlement can be determined according to the running safety of high-speed vehicle. According to China specification “GB/T 5599-2019” (Specification for dynamic performance assessment and testing verification of rolling stock, 2019), the limit of vertical acceleration of high-speed train is 0.2551 g. By substituting “y < 0.2551” into “y = 206.1x-9.625e^-6”, the inequation can be obtained, as follows

0.2551 > 206.1 x - 9.625 e^{- 6}

(13)

Then, constructing a cosine wave function with zeros mean value to depict the general settlement wave shape, as follows:

ℏ = a \cos (\frac{2 π}{L} \bar{x})

(14)

where

a

and

L

denote the amplitude and wavelength of the cosine wave.

The SD of the cosine wave $ℏ$ can be calculated analytically

ℏ^{″} = - {(\frac{2 π}{L})}^{2} a \cos (\frac{2 π}{L} \bar{x})

(15)

By substituting equation (12) into equation (10), the inequation can be written as

\frac{(0.2551 + 9.625 e^{- 6}) L^{2}}{206.1 {(2 π)}^{2}} > - a \cos (\frac{2 π}{L} \bar{x})

(16)

For a conservative estimation, the maximum value of the right-hand term of the inequality is

a

, therefore, the inequation can be written as

\frac{(0.2551 + 9.625 e^{- 6}) L^{2}}{206.1 {(2 π)}^{2}} > a

(17)

The critical $a$ $L$ curve and its corresponding safety region are shown in Figure 15(a).

Moreover, the threshold of settlement can be determined according to the passenger’s comfort. The Sperling index presented in the China specification “GB/T 5599-2019” (Specification for dynamic performance assessment and testing verification of rolling stock, 2019) is selected as the basis. The Sperling index of car body acceleration from 5.9 to 20 Hz can be defined as

W = 7.08 \sqrt[10]{0.325 Ω V A^{3}}

(18)

where

Ω

is the spatial frequency (Unit: 1/m), which can be calculated by

Ω = \frac{1}{L}

;

A

denotes the car body acceleration (Unit: g);

V

is the vehicle speed (Unit: m/s).

According to the specification, the Sperling index needs to be less than 3. Using the same method of equations (13)–(16), the threshold corresponding to different settlement wavelengths can be obtained by the following inequation:

7.08 \sqrt[10]{\frac{0.325 V}{L} {({(\frac{2 π}{L})}^{2} a \times 206.1 - 9.625 e^{- 6})}^{3}} < 3.0

(19)

The critical $a$ $L$ curve and its corresponding safety region are shown in Figure 16(b). It can be seen that the control of the threshold of settlement amplitude is stricter when considering passenger comfort. When the passenger’s comfort is not considered, the threshold of settlement is 4 times larger than the result considering the passenger’s comfort (wavelength 20–160 m).

5. Conclusion

This paper establishes a VTS dynamic interaction system and applies it to investigate the correlation between additional deformation of rails induced by subgrade settlement and the vehicle responses. The low-frequency components of vehicle responses induced by the additional deformation of rails are extracted by Wavelet decomposition. Through the fitting function, the threshold of long-wavelength subgrade settlement is determined in terms of both vehicle running safety and passenger’s comfort, and the control for long-distance subgrade settlement is achieved. The following conclusions are drawn from the numerical study:

(1) The subgrade settlement is a low-frequency excitation for the VTS dynamic interaction system. The second derivative of the additional deformation of the rail induced by subgrade settlement has a strong coherence with the car body acceleration in the low-frequency domain.

(2) The settlement thresholds determined by vehicle running safety are greater than those determined by passenger’s comfort. This paper conservatively suggests that the threshold of long-wavelength settlement should be determined by passenger’s comfort. For example, if the wavelength is 50 m, the settlement should be controlled within 21.55 mm; if the wavelength is 100 m, the settlement should be controlled within 108.4 mm.

(3) The work in this paper has some reference value for the control of long-distance subgrade settlement. However, the control of settlement cannot only consider the vehicle running safety and passenger’s comfort, there are many other indexes that need to be studied in future work.

Footnotes

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: This study is supported by the Research Funds for the Central Universities of Central South University (No. 2021zzts0769); the National Natural Science Foundation of China (Grant Nos. 52378468; 52008404; U1934217; U1734208); the Fundamental Project for Key Laboratory of High Speed Railway Line Engineering of the Ministry of Education; Central South University Innovation-Driven Research programme (2023CXQD072); Science and Technology Research and Development Program Project of China railway group limited (Major Special Project, NO: 2020-Special-02; 2021-Major-02; 2021-Special-08); the National Natural Science Foundation of Hunan Province (Grant Nos. 2022JJ20071; 2021JJ30850); the Science and Technology Research and Development Program Project of China National Railway Group Co., LTD (Grant Nos. L2022G007; L2023G007).

ORCID iD

Zheng Li

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