Research on the influence of curved rail corrugation on the system dynamic characteristics of subway

Abstract

Using the multi-body dynamics simulation software Simpack, a dynamic model of the subway vehicle-track system with elastic wheelsets on curved lines is established. The influence of corrugation excitation with different characteristics on wheel-rail forces and wear, as well as the vertical vibration characteristics of the system, are studied. The results indicate that when the vehicle passes a straight creep point with a corrugation depth of 0.20 mm, the wheel-rail force reaches a value of 0 and exceeds the safety limit, resulting in decreased vehicle stability. When the vehicle passes the middle section of the curve and the corrugation depth is 0.20 mm, stress concentration occurs on the left wheel flange and right wheel tread, and the maximum contact stress is greater than the material yield limit, which leads to increased wheel-rail wear. The variation of corrugation wavelength causes a change in the peak range of vehicle structural vibration, and short-wavelength corrugation excitation leads to an increase in the peak energy, resulting in more obvious resonance. It is recommended to control the corrugation depth within 0.14 mm and conduct timely inspection and rail grinding of the line.

Keywords

Rail corrugation dynamic characteristics wheel-rail wear vibration characteristics

Introduction

As a critical component of urban public transportation, metro plays a vital role in alleviating traffic congestion, improving travel efficiency and fostering urban spatial optimization. With the rapid development of urban rail networks, the construction and operation of subway systems are facing more and more challenges. Rail corrugation has emerged as a significant issue, attracting considerable attention in recent years.^1–3

Rail corrugation refers to periodic longitudinal undulations on the rail surface after new rails are put into service. Such corrugation will lead to a series of hazards such as deterioration of wheel-rail contact relationship, increase of system vibration noise, and reduction of service life of rails, which will seriously affect the stable and comfortable operation of metros. As shown in Figure 1, investigations show that corrugation areas often have high rates of fastener failure, such as broken or detached clips and T-bolts.⁴ Additionally, rail fatigue cracks increase the risk of track fracture. Rail corrugation can damage critical track components and affect vehicle parts. Tests show that vertical vibrations of axle boxes and vehicles are much more intense in areas with corrugation, confirming severe wheel-rail interactions. The maintenance department also indicate failures such as broken speed sensor brackets at the axle and heavy flange wear, which seriously affects driving safety.⁵ Therefore, studying the impact of rail corrugation on vehicle dynamics in subway curves is crucial.

Recently, many scholars have studied the formation mechanism of rail corrugation and its impact on the dynamic performance of vehicle-rail systems through theoretical research and field tests. In terms of theoretical studies, Ling et al.⁶ developed a dynamic model of the vehicle and damping track system. The damping effects of metro vehicles passing through different kinds of damped tracks under corrugation are compared in detail. Yan et al.⁷ established a vehicle-rail dynamic model of the trapezoidal sleepers. The effects of sleepers with different design parameters on rail vibration and noise are investigated through time-frequency analysis Based on a genetic algorithm, the performance of the trapezoidal sleeper is optimized to reduce rail corrugation. Cheng et al.⁸ adopted a vehicle-rail coupling model based on a frequency domain simulation method. The wheel-rail vertical force and vibration were calculated. The formation of corrugation is affected by the peak of vertical force, and the growth trend is affected by the contact state and friction parameters. Shen et al.⁹ used numerical analysis to establish a coupled wheel-rail dynamic model. The USFD was imported to reproduce the evolution of rail corrugation. The RLBM of the wheelset is derived to be the main factor contributing to the formation of the long-wavelength corrugation. Based on the actual parameters of the Chinese high-speed railway, Niu¹⁰ established the track model with rail corrugation. The stability and dynamic response characteristics of the model were studied through validation with measured data. An Index is also proposed to quantify the extent of damage to the track. Liu et al.¹¹ established a wheel-rail spatial coupling vibration model and concluded that vertical vibration of the vehicle-rail system, torsional vibration of the wheelset and stick-slip vibration is the main reason leading to the rail corrugation. Li et al.¹² established a 3D finite element model of the short-track sleeper. The mode shape and frequency response characteristics of the track structure are studied. Based on the linear analysis theory in the frequency domain, Li¹³ proposed a numerical model of metro rail corrugation, derived from different rail wear characteristics.

In terms of field tests, Choi et al.¹⁴ studied corrugation data for slab track. Frequency analysis was applied to evaluate rail corrugation, concluding that the corrugation is caused by wheel excitation on the rail. Dong et al.¹⁵ conducted a 1.5-year study on two stations of Chengdu Metro and found that 30 mm wavelength corrugation occurred on the inner rail on a 350m radius curve track. With the increase in service time, the corrugation expands to the transition curve and the wavelength changes. Song et al.¹⁶ conducted a 2.5-year study on a subway and recorded the surface wear characteristics of curved tracks with a radius of 600m. It was found that the outer rail corrugation occurred between 12 and 17 months of the operation, while the inner rail did not. Wang et al.¹⁷ investigated and collected data from 14 subway lines in China, and redefined the classification of corrugation based on feature information by the t-SNE method and unsupervised clustering algorithm DBSCAN. Yao¹⁸ investigated the rail corrugation data and obtained the effectiveness of rail grinding measures based on the measured data. Chen et al.¹⁹ tested the frequency response function and vibration attenuation rate of the rail and monitored the roughness of the rail running belt. Tang et al.²⁰ investigated the rail corrugation, obtained the rail surface irregularity and analyzed the corrugation characteristics. The cause of rail corrugation on a 300 m spring-steel floating slab track has been derived mainly from the P2 resonance between the unsprung mass and the track structure at 62 Hz.

To study the cause or influence of the corrugation on the dynamic characteristics of the wheel-rail system, it is necessary to comprehensively consider the relationship and the dynamic interaction of the vehicle-rail system based on the vehicle-track coupling dynamics theory. Sun²¹ considered vehicle components as rigid bodies and ignored the elastic deformations. The dynamic response of the multi-rigid body vibration system under continuous harmonic excitation is analyzed. Liu²² established a rigid-flexible coupling model with flexible wheelset, adopted the two-layer discrete track model and added corrugation excitation to analyze the vibration response. Li²³ established a vehicle-rail dynamic model, studied the influence of such parameters as lateral and longitudinal rigidity of the bogie system, friction coefficient, curve radius, superelevation, gauge on the corrugation of curved track by orthogonal test. Tang et al.²⁴ established a finite element model including a wheel, track and fastener, simulated the e-type elastic rod fastener with spring and damping element. The response of the wheel and fastening system when passing through the small radius curve is analyzed. Yang et al.²⁵ modeled a single slab with finite element software and established a vehicle-track-slab coupling dynamic model based on D'A’s principle. The resonance relationship between the track slab and the rail corrugation is studied by modal analysis. Tang et al.²⁶ established a finite element model of wheelset and track. The relationship between coupled vibration of wheel-rail system and rail corrugation is studied.

By combining vehicle-track coupling dynamics theory with finite element and multi-body system dynamics, vehicle-track coupling model including rail corrugation and other factors can be established accurately for efficient real-time calculation and analysis. The dynamic model of vehicle-track coupling system with elastic wheelset is established based on actual parameters, focusing on the rail corrugation on curved line. Based on vehicle system dynamics and wheel-rail self-excited vibration theory, the rail corrugation with different characteristics is taken as the track irregularity excitation. The dynamic response of key structure under the corrugation is analyzed from the time domain and frequency domain.

Model establishment and track parameter selection

Establishment of vehicle dynamics model

The B-type metro vehicle is taken as the research object, with the speed of 80 km/h.^27–29 Based on the metro line, Simpack is used to establish vehicle-track system dynamic model, as shown in Figure 2.

One of the key components affecting the safety and stability of vehicle operation is the wheelset. The calculation of wear and wheel-rail creepage will be directly influenced by its elastic deformation. In addition, the rigid wheelset cannot effectively reflect their high-frequency vibration characteristics. Hypermesh is used to establish the elastic wheelset model, and Ansys is used to analyze the relationship between its vibration characteristics and frequency.

It is difficult to fully model the actual structure of the subway vehicle due to its complex structure. To simplify calculations and improve computational efficiency, structures that have little impact on output results are removed, while ignoring the interaction between vehicles.³⁰

Selection of track parameters

In Simpack, incentives are divided into two types: Track Related and Rail Related. Track Related includes excitation of lateral, vertical, torsional and gauge. Rail-Related include excitation of lateral left and right, vertical left and right, roll left and right. Different wave depths and wavelengths were obtained based on measured data and control variates. The dynamic performance of vehicle systems under excitations of different corrugations are analyzed.

Rail corrugation is a continuous harmonic disturbance, and periodic cosine function can be used to describe its characteristic³¹:

Z (t) = \frac{1}{2} a (1 - \cos ω t) (0 \leq t \leq \frac{n L}{v})

(1)

where Z is the vertical depth of the corrugation in m, a is the depth of irregularity in m, ω is the angular frequency in rad/s, t is the time variable in s, v is the speed of vehicle in km/h, n is the number of excitations, and L is the wavelength of irregularity in m.

The measured waveform is shown in Figure 3. Figure 3(a) shows a short-wave corrugation with a wavelength of 50 mm and depth of 0.02 mm; Figure 3(b) shows a long-wavelength corrugation with a wavelength of 160 mm and depth of 0.04 mm.

Influence of different corrugation wave depth on dynamic performance

In the example of 50 mm wavelength, four types of excitations with wave depths of 0.02, 0.08, 0.14, and 0.20 mm were obtained using control variates. The depth of corrugation is divided based on measured data, and the influence of different corrugation depths on the dynamic performance of vehicle systems is explored.

Impact on wheel-rail vertical force

The variation of wheel-rail vertical force is shown in Figure 4. With the increase of depth, the amplitude of vertical force increases accordingly and fluctuates around 80 kN. When the depth reaches 0.20 mm, the maximum value of wheel-rail vertical force is 230.83 kN, which is greater than the force limit of 170 kN and the safety limit of 35.8%. When the depth is 0.02, 0.08 and 0.14 mm, the maximum values of force are 90.97, 116.37 and 124.10 kN respectively, which are within the safety limits. In addition, force appears at 0 point within 5∼12.5 s when the depth is 0.20 mm, which indicates that the wheel-rail is not contacted. During this period, the vehicle passes through the transition curve and circular curve. Due to the change of curvature and superelevation, the stability of the vehicle is poor and the vehicle is unstable.

Impact on wheel-rail corrugation and contact stress

Wheel-rail contact and wear are important factors affecting vehicle operation. At present, the most widely used algorithm for solving creep force problems in contact areas is the Fastsim algorithm based on Kalker’s simplified theory.^32,33

This algorithm simplifies the contact area to an oval shape and divides it into cells along the rolling direction. The length variation of the cell center along the x and y directions is dx and dy. The tangential stress near the front cell n is denoted by pi (n), so the next cell n+1 is denoted by:

{\begin{array}{l} {p_{x}}^{(n + 1)} = {p_{x}}^{(n)} + (\frac{ξ_{x}}{L_{x}} - \frac{y φ}{L_{y}}) d x \\ {p_{y}}^{(n + 1)} = {p_{y}}^{(n)} + (\frac{ξ_{y}}{L_{y}} - \frac{x φ}{L_{y}}) d y \end{array}

(2)

where

ξ_{x}

and

ξ_{y}

are the longitudinal and lateral creepage,

φ

is the spin creepage, and L is the compliance coefficient.

The combined tangential stress at this point is:

{p_{τ}}^{(n + 1)} = \sqrt{{({p_{x}}^{(n + 1)})}^{2} + {({p_{y}}^{(n + 1)})}^{2}}

(3)

If ${p_{τ}}^{(n + 1)}$ satisfies the boundary condition of friction theory, $p_{τ} \leq μ p_{z}$ , where $p_{z}$ represents the vertical stress in Pa at that point. It signifies that the point is in the adhesion zone, with both longitudinal and lateral creepage rates being 0. Otherwise, the point is in the sliding zone and requires correction based on conditions.

The longitudinal and lateral creep forces and spin torque within the contact area are integrated after obtaining the stress distribution. Revised Elkins index is used as an evaluation indicator for wear:

W = T_{x} ξ_{x} + T_{y} ξ_{y} + M φ

(4)

where

T_{x}

and

T_{y}

are the longitudinal and the lateral creep force respectively in N, and M is the spin creep force in N.

The variation of wear number is shown in Figure 5. The number increases with the increase of corrugation depth and fluctuates around 60N. When passing through the circular curve section at 11.9 s, the number fluctuates most violently. When the depth is 0.20 mm, the number is 0, which can endanger the operating safety. The maximum under four depths are 63.0, 71.5, 80.0 and 88.5 N respectively, with an average growth rate of 10.6%. It indicates that with the increase of wave depth, the wear is more serious, and timely maintenance shall be carried out.

The wheel stress at 11.9s is further analyzed to obtain the contact stress under different wave depth. As shown in Figure 6, the maximum stress of the left wheel occurs in the flange and the stress of the right is in the tread, which indicates that the flange clings. With the increase of corrugation depth, the stress value of each wheel increases accordingly. When the depth is 0.02 mm, the maximum stress at the left wheel flange is 1600 MPa, which is far more than the yield limit of 551.6 MPa,³⁴ 3 times the allowable value. The stress at the left tread is 530 MPa, within the safety range. The stress at the right tread is 750 MPa, exceeding the allowable value of 29.9%.

The maximum stress of the left wheel is mainly concentrated near the flange at 11.9 s, while the stress of the right is mainly concentrated on the tread. The vehicle is passing the middle section of the circular curve. The left wheel clings to the track, and the stress concentration occurs due to the centrifugal action, which causes corrugation aggravation.

Impact on derailment coefficient and wheel load reduction rate

Derailment coefficient is the basic index to evaluate whether the flange climbs on the rail and derails under the action of force. Wheel unloading ratio is used to assist in determining the derailment phenomenon caused by excessive wheel load reduction, and a comprehensive evaluation of the two is necessary to ensure the safety of vehicle operation.³⁵ When derailment coefficient is less than 0.8 and wheel unloading ratio is less than 0.6, vehicle operation meets safety standards³⁶ referring to GB/T 5599 – 2019.

The response comparison of derailment coefficient and wheel unloading ratio under different is shown in Figure 7. With the increase in depth, both indicators increase. When the wave depth is 0.02 mm, 0.08 mm and 0.14 mm, the maximum derailment coefficients are 0.17, 0.26 and 0.35 respectively. Values of the wheel unloading ratio are 0.196, 0.231 and 0.455 respectively, which are lower than the safety limit and less likely to derail. When the depth is 0.20 mm, the maximum derailment coefficient is 0.40, which is lower than the safety limit. The maximum wheel unloading ratio is 1, which is doubled compared with the 0.08 mm wave depth. And the value exceeds the safety limit, with derailment risk.

The maximum derailment coefficient under the four wave depths occurs around 11.9 s. At this time, the vehicle is passing the circular curve section, indicating that the set track superelevation fails to balance the centrifugal force of the vehicle, which is easy to derail. The superelevation is suggested to be adjusted properly to ensure the safety. The maximum wheel unloading ratio occurs at 5.5 s, 11.9 s and 13.5∼16 s. At 5.5 s, the vehicle runs to the straight-spiral point, and at 13.5 s, the vehicle runs to the point of spiral to curve. Due to the change of curvature, the vertical and lateral force on the wheel track change, and the wheel unloading ratio appears a dangerous point.

The influence of corrugation wavelength on vehicle dynamic performance

Rail corrugation is divided into short-wavelength and long-wavelength corrugation according to wavelength, with the former being more common. It occurs on both straight and curved lines, with a wavelength range of 25-100 mm. Long-wavelength corrugation with a wavelength range of 100-1500 mm is usually found on the outer rail of slight radius curves in heavy-haul lines. Based on a corrugation depth of 0.08 mm, long-wavelength excitations with wavelengths of 50 mm and 160 mm are used to investigate the effect of different corrugation wavelengths on the dynamic performance of the vehicle system.

Analysis of vertical vibration on system

The vertical acceleration of the vehicle passing lines with different wavelength corrugation in Figure 8. Compared with long-wavelength, the vertical acceleration fluctuation of vehicle system structure under short-wavelength excitation is larger, and the frequency of maximum values increases. When the wavelength is 50 mm, the maximum vertical acceleration of the carbody is 1.12 m/s², exceeding the safety limit (1.0 m/s²) by 12%. The maximum acceleration corresponding to a wavelength of 160 mm is 0.61 m/s². When the wavelength is 50 and 160 mm, the maximum vertical acceleration of the bogie frame is 11.63 m/s² and 1.65 m/s², while the maximum acceleration on the wheelset is 27.83 m/s² and 4.42 m/s².

In the subway vehicle system, the amplitude of vertical acceleration from the wheelset to the carbody gradually decreases, and when transmitted to the carbody, the vibration amplitude has significantly decreased. Compared to long-wavelength excitation, short-wavelength causes severe vertical fluctuations and larger amplitudes in the system structure. The vibration of vehicle systems is significantly affected by rail corrugation with short-wavelength. In summary, more attention should be paid to the development of short-wavelength corrugation on the track, and timely detection and rail grinding should be carried out.

Analysis of the vibration characteristics on system

To investigate the vibration characteristics and transmission properties of vehicle structures, a sampling frequency of 1000 Hz was set to obtain the frequency-domain characteristics of the vehicle system from 0 to 500 Hz, as shown in Figure 9. In addition, Figure 10 shows the vibration of the vehicle structure from 0 to 40 Hz.

According to Figure 10, there are three similar vibration frequencies (3.35 Hz, 8.68 Hz and 35.05 Hz) in the wheelset, bogie frame and carbody. These are natural vibrations caused by the length of the track plate, wheel rotation and sleeper spacing³⁷:

(1) The vibration frequency caused by the length of the track slab is about 3.35 Hz, and the vibration frequency f₁ can be obtained by formula (5).

f_{1} = (v / 3.6) / l

(5)

where l is the length of track slab in m, 6.5 m; v = 80 km/h. It can be concluded that f₁ is 3.42 Hz.

(2) The vibration frequency caused by wheel rotation is about 8.68 Hz, and the rotation frequency f₂ can be obtained from formula (6).

f_{2} = (v / 3.6) / (π D)

(6)

where D is the wheel diameter in m, 0.84 m; v = 80 km/h. It can be obtained that f₂ is 8.43 Hz.

(3) The vibration frequency caused by sleeper spacing is about 35.05 Hz, and the vibration frequency f₃ can be obtained by formula (7).

f_{3} = (v / 3.6) / c

(7)

where c is the sleeper spacing in m, 0.625 m; v = 80 km/h. It can be concluded that f3 is 35.5 Hz.

The vertical vibration frequencies caused by the length of the track plate, wheel rotation and sleeper spacing are transmitted upwards to the carbody without being suppressed by the vehicle suspension system.

Compared with long-wavelength corrugation, vehicle system components exhibit higher vertical vibration energy under short-wavelength excitation, primarily due to induced high-frequency periodic vibrations in the vehicle-track system. When the excitation frequency approaches critical natural frequencies of the coupled system, structural resonance emerges, leading to significant amplification of structural vibration energy peaks. The vibration of elastic wheelsets is mainly concentrated in the range of 0-300 Hz, with multiple vibration peaks occurring within 100 Hz. In addition, there are frequencies that are similar to the first and second vertical vibration modes (69 Hz and 201 Hz) of the wheelset. The multiple vibration peaks of the bogie frame are in the range of 0-200 Hz. As the vibration frequency increases, the energy corresponding to the peak gradually decreases. The vibration frequency of the carbody is concentrated within 50 Hz, mainly low-frequency vibration. As shown in Figure 9, the vehicle structure exhibits peaks at vibration frequencies of 8.75 Hz, 17.4 Hz, 61.65 Hz and 112.8 Hz. Except the vibration frequency 8.75 Hz caused by wheel rotation, the vehicle system has obvious resonance at the vibration frequency of 17.40 and 61.65 Hz. To avoid resonance of adjacent components, optimizing the vehicle structure is recommended.

Comparison between simulation results and measured data

The simulation condition in this paper is consistent with the measured operation condition in Reference 38. Based on the simulation results and measured data obtained from the same wavelength excitation, the dynamic response is compared and analyzed to verify the correctness of the simulation results.

The acceleration of the vehicle structure is taken as the average of two test results, and the dynamic indicators are the simulation results, as shown in Table 1. Firstly, the simulation results of acceleration differ from the measured data by 0.52%–21.36%, which is relatively slight. The variation in vertical acceleration may be caused by differences in simulated corrugation depth and measured data. Secondly, in terms of dynamic response, the two are relatively close. The derailment coefficient is less than 0.8 and the wheel unloading rate is less than 0.6, indicating that the vehicle operation meets safety standards. Through data comparison, the correctness and feasibility of the simulation method have been verified, laying a theoretical foundation for subsequent research on the causes of rail corrugation and system transmission characteristics.

Table 1.

Comparison of relevant data.

Dynamic response	Simulation data	Measured data
Vehicle acceleration (m/s²)	1.03	0.81
Bogie acceleration (m/s²)	11.59	11.53
Wheel acceleration (m/s²)	79.15	83.27
Derailment coefficient	0.39	0.37
Wheel-load reduction	0.45	0.47
Wheel-rail vertical force (kN)	107	110

Conclusion

A dynamic model of the vehicle-track coupling system, including elastic wheelsets, has been established. Based on vehicle dynamics theory and the finite element method, rail corrugation with different characteristics is used as track excitation to analyze the dynamic response of the system structure under corrugation excitation. The key findings of the present work are as follows:

(1) Simulation results show that, when the vehicle passes a straight creep point (5.5 s) and the corrugation depth is 0.20 mm, the wheel-rail vertical force reaches a value of 0 (no contact between the wheel and rail) and exceeds the safety limit (170 kN). Due to changes in curvature and the influence of superelevation, the stability of the vehicle is poor at this location, and derailment is prone to occur. It should be given special attention.

(2) In terms of wheel-rail wear, as the corrugation depth increases, the wheel-rail wear becomes more severe. When the vehicle passes the middle section of the curve (11.9 s) and the corrugation depth is 0.20 mm, the maximum stress on the left wheel is concentrated near the wheel flange, while the maximum stress on the right wheel appears at the tread. Stress concentration occurs on both sides of the wheels, and the maximum stress value is greater than the yield strength of the material (551.6 MPa), which will lead to increased wheel-rail wear.

(3) The line, with short-wavelength corrugation, causes more severe vibrations in the vehicle structure, and the vibration acceleration transmitted to the carbody exceeds the safety limit. The vibration frequencies (3.42 Hz, 8.43 Hz, and 35.5 Hz) caused by the length of the track plate, wheel rotation, and sleeper spacing are not suppressed by the suspension system, but are transmitted upwards to the carbody. It is significant resonance for system at vibration frequencies of 17.4 Hz and 61.62 Hz, which can be avoided through structural optimization design. In addition, the dynamic simulation results are close to the measured data, which verifies the correctness and feasibility of the simulation method.

In summary, the increase in the corrugation depth leads to abnormal wheel-rail contact and intensified wear. Based on rail transit maintenance experience, relevant documents and simulation results, it is recommended to control the corrugation depth within 0.14 mm and conduct timely inspection and rail grinding of the line according to the curve radius, operation frequency and wheel-rail force test results. The variation of corrugation wavelength causes a change in the peak range of vehicle structural vibration, and short-wavelength corrugation excitation leads to an increase in the energy of the vibration peak, resulting in more obvious resonance. Therefore, it is suggested to conduct moderate speed restriction in the line section with obvious corrugation to avoid resonance between train operation frequency and track excitation frequency.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest concerning 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 authors thank the financial support from the Science and Technology Planning Project in Xuzhou (No. KC22293), Jiangsu Province College Student Innovation Training Program (202310320108Y), Innovative Training Project for College Students in China (202410320025Z).

ORCID iDs

Xinlei He

Xiang Li

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. Study on effect of rail corrugation on vehicle vibration and dynamic stress of frame. MD Thesis. Beijing, China: Beijing Jiaotong University, 2014.