Rail Corrugation Characteristics of Cologne Egg Fastener Section in Small Radius Curve

By analyzing the measured data of rail corrugation in a small radius curve track with Cologne fasteners, the typical passing frequencies of corrugation were obtained. /en, according to actual line conditions, the vehicle-track coupled model and rail material friction and wear model were established and simulated, and rail corrugation characteristics of the curved section were studied. /e results show that the variation amplitude of creep force on the inside of a guiding wheelset is large, and it coincides with that of the saturated creep force partially and the coincident part appears periodically. /e variation amplitude of creep force on the outside of the guiding wheelset is small, and it is approximately equal to that of the saturated creep force. In addition, the variation range of the wear depth curve on inner rail is mainly changing from 0 μm to 0.0014 μm, showing periodic wavy wear, and when the creep force is equal to the saturated creep force, the wear depth reaches the peak. /e variation range of the wear depth curve on outer rail is mainly changing from 0.0005 μm to 0.0008 μm, showing uniform wear. Combined with the predicted profile of rail wear, it can be seen that the inner rail mainly suffers from corrugation on the top of rail and the outer rail mainly suffers from uniform wear on the side, which results in serious corrugation on inner rail and slight corrugation on outer rail. /e frequency characteristics analysis of wear shows that the inner rail wear has characteristic frequencies similar to the passing frequencies. Modal analysis results show that the vibration of inner rail at the characteristic frequencies is greater than that of outer rail, which lead to the corrugation of the corresponding frequencies more easily. /e wear growth rates at the characteristic frequencies are relatively large, which indicates that the wear at the corresponding frequencies will continue to develop and eventually form corrugation.


Introduction
Corrugation is a kind of defect on rail heads, which is characterized by periodic irregularity of rail surface, and its shape is similar to that of wave. Corrugation occurs to varying degrees in metro systems all over the world, especially in small radius curve sections [1][2][3][4]. e wavelength of corrugation is generally between 20 mm and 300 mm, and the wave depth is not more than 1 mm [5]. Generally speaking, if the recurrence period of rail corrugation in a small radius curve section is not less than half a year, it can be regarded as normal corrugation. If rail corrugation occurs on a large radius curve section or straight line and the recurrence period is less than four months, it is considered as abnormal corrugation [6]. Rail corrugation has a serious negative impact on vehicles, tracks, and surrounding environment.
Since rail corrugation was first discovered in 1895, people have observed and investigated corrugation for more than 100 years. During this period, a lot of results on theoretical models, indoor tests, and field tests of corrugation had been presented. Table 1 lists a part of researchers and their corresponding research results. e existing results on corrugation are mainly based on time domain (nonlinear) and frequency domain (linear). Based on the nonlinear vehicle-track interaction, the nonlinear factors were introduced into the calculation model of rail corrugation for the first time by Igeland and Ilias [12].
Nielsen [23] used the nonlinear rail wear model to study the generation of corrugation on straight lines, and validated the model with the field test on Dutch railways. Sun et al. [24,25] proposed a simplified feedback model of corrugation including the delay of wheel transfer time. Stability analysis was used to determine the growth of corrugation when multiple wheelsets passed. Sheng et al. [26] used an unsteady two-dimensional method and a three-dimensional FAST-SIM model to calculate the wheel-rail tangential creep force and predicted the development trend of corrugation through repeated loading on wheelsets. Chen et al. [27] founded the finite element time domain model of the wheelset passing sharp radius curve. e simulation results showed that the wheel-rail self-induced vibration is an important cause of curve corrugation. Tassilly and Vincent [28] studied the transfer function among the wheel, rail initial roughness, and wear rate by using the linear model of corrugation. It was illustrated that the increase of rail corrugation in some frequency bands was mainly due to the combined effect of creepage between wheels and rails, wheel-rail parameters, and initial irregularity of rail surface. Based on the previous linear theoretical model, Müller [29] assumed that the wheel is in a static state, and the rail surface irregularity was loaded between wheels and rails to simulate the dynamic characteristics of wheels passing track. Li [30] used the finite element model to analyze the relationship between passing frequencies of corrugation and natural frequencies of track structure. However, subject to the complexity of interaction between wheels and rails and the influence of various factors, it is still an open problem on building convincing explanations for the formation and evolution mechanisms of corrugation.
Based on the measured data of corrugation obtained from our experiments, a serious corrugation phenomenon appears on the curve section of the Cologne Egg fastener track, as an unusually severe one in comparison with the ones on the straight line. More generally, the inner rail of the curve track with a radius less than 350 m produces corrugation of almost 100%, while outer rail seldom produces corrugation or has slight corrugation. In this paper, the corrugation characteristics of inner and outer rails in the small radius curve section of the Cologne fastener track are investigated by both computational simulations and analysis of the measured data. en, the vehicle-track coupled model and the rail material friction and wear model are established, and the wheel-rail creep characteristics are analyzed in time domain. Combined with the prediction profile of rail wear, the phenomena of serious corrugation on inner rail and slight corrugation on outer rail in the small radius curve section are explained. Finally, by using the frequency domain method, the development characteristics of wear are analyzed by using wear growth rate and the wear growth rates at the characteristic frequencies to develop and eventually form corrugation are discussed.

Characteristics of Measured Corrugation
e operating vehicle in the corrugation section is a metro type A vehicle. e curve radius is 350 m and foundation under rails is the integral track bed structure. e irregularity of measured corrugation (inner rail) and one-third octave wavelength spectrum of irregularity grades are shown in Figures 1(a) and 1(b). e characteristic wavelengths shown in Figure 1(b) are 20 mm, 31.5 mm, 63∼80 mm. e running speed of the vehicle is about 40 km/h, and the passing frequency f corresponding to the specific wavelength λ can be obtained according to the following equation: Table 1: History of corrugation investigations.

Refs
Results and conclusions Johnson and Gray [7] Rail corrugation induced by wheel-rail contact resonance Grassie et al. [8][9][10] Wavelength fixed mechanism and damage mechanism, frequency fixed mechanism Wang et al. [11] When the line was not smooth, the torsional vibration of wheelset would lead to the formation of corrugation Igeland and Ilias [12] For the first time, nonlinear factors were introduced into the model of corrugation calculation Valdivia et al. [13] Corrugation was caused by the interaction of wheel-rail dynamics, wheel-rail rolling contact, and material mechanics Dinna et al. [14] Corrugation was caused by actions of periodic track impedance and high creepage when vehicle passes curve Nielsen [15] Wheel-rail rolling contact could filter the rail surface roughness Jin et al. [16] Based on wheel-rail dynamics, the calculation model of wear type corrugation was established and verified Wu and ompson [17] Based on the two-dimensional unsteady contact model and wear model, the locomotive dynamic model was established in the frequency domain Wen [18] Combining the vertical coupled dynamics of vehicle-rail, unsteady rolling finite element model and twodimensional rolling contact model of cater, the calculation model of corrugation of plastic flow pattern was established Wu et al. [19] Fractal description of rail corrugation was established Zhang et al. [20] Causes and treatments for corrugation developed on Egg fastener track Li et al. [21] Using field observations, experiments and numerical study were conducted to investigate the occurrence mechanism of corrugation Cui et al. [22] Corrugation was caused by friction, self-excited vibration between wheels and rails Fang et al. [4] Formation mechanism and mitigation measures of short pitch corrugation 2 Shock and Vibration where v is the speed. e passing frequencies are � 139 ∼ 176 Hz. (2) Further, the field measurement is carried out as shown in Figure 2; the wavelength ranges of corrugation are about 20∼35 mm and 55∼85 mm, which are close to that listed in Figure 1(b), indicating that the numerical calculation result is correct.

Vehicle Model.
e main parameters of a metro type A vehicle are shown in Table 2. e software UM (Universal Mechanism) is used to build the vehicle model of a multibody system, referring to the literature [31] for details.

Track Model.
e track model is a flexible track model and parameter values of the track structure are shown in Table 3.
e detailed establishment process of the track model is shown in the literature [31].

Wheel-Rail Contact Model.
e wheel-rail contact model adopted the Kik-Piotrowski contact model, which has the characteristics of fast calculation speed and high reliability [32]. e vehicle-track space coupled model diagram is shown in Figure 3.

Wear Calculation Model
where P is the friction power, which is the sum of the product of creep forces and creepages in all directions, as shown in the following equation: In the UM software, the simulation can output the friction power in the contact area at any time, calculate the friction work in the corresponding time step, and then calculate the total wear amount at that time.
Considering Archard's wear model, the mathematic model of the rail material wear can be expressed as follows: where Δm is the total wear amount produced at position x and K stands for the wear ratio coefficient, which is 1 × 10 −9 kg/Nm [30]. After the total wear amount is obtained, the wear depth d(x) of rail surface corresponding to the position at that time can be calculated according to the following equation: where A represents the area of contact zone and ρ is the density of rail material, taking 7800 kg/m 3 .
In the abovementioned vehicle-track model, the four wheels are running on each rail at the same time, and the wear depth corresponding to each wheel is accumulated at the fixed position of the rail so that the wear depth Δ D(x) under single operation can be expressed as follows: where d i (x), (i � 1, 2, 3, 4) is the wear depth of each wheel (see Figure 3) at the fixed rail position. By adding the wear depth Δ D(x) calculated by the single operation as irregularity to rail surface, a new rail profile D 1 (x) can be obtained after the single operation: where D(x) is the initial irregularity of rail surface. In what follows, the next wear amount under the influence of previous wear can be obtained by calculating again. Repeating the process, the superimposed wear on rail surface can be obtained for many times. e flow chart of wear superposition calculation is shown in Figure 4. By using the Fourier transform of rail profile D n (x) after n times operation, the corresponding rail profile amplitude D nf (iω) in frequency domain can be obtained. Compared with the initial irregularity, the wear growth rate G f at frequency f is defined as follows [33,34]: where D 0 (iω) is the profile amplitude of initial irregularity of rail surface after the Fourier transform, α is the sensitivity of wear, K b is the sensitivity of transverse frictional power induced wear variations to vertical wheel-rail contact deflection variations, K ci is the modal sensitivity of the wheel-rail displacement to a change in longitudinal rail profile, n is the number of wheel passing, ξ i is the modal damping ratio, ω i is the modal natural frequency, ω is the angular frequency, for the frequency domain, λ is the ratio of the current speed to the mean speed, and p(λ) is the probability distribution of λ. According to equation (9), the wear growth rate G f is a function of frequency f. If wear growth rate is positive, it means that rail surface irregularity will continue to develop at the corresponding frequencies. e larger the numerical values, the faster the development, and finally the corrugation will be formed.

Creep
Characteristics. e radius of research curve line is 350 m and the vehicle speed is 40 km/h. In order to investigate the wheel-rail creep characteristics, the software UM is used to transform the data of measured corrugation into the irregularity spectrum, and then the generated irregularity spectrum is applied to left and right rail surfaces. e creep force and saturated creep force between guiding wheelsets and rails and corresponding rail wear are simulated and calculated. e data curves are shown in Figures 5  and 6. e results in Figures 5 and 6 can be summarized as follows: (1) e variation amplitude of creep force on the inside of the guiding wheelset is large, which is partially coincident with that of the saturated creep force and the coincident part appears periodically. e variation amplitude of creep force on the outside of the guiding wheelset is small, which is approximately equal to that of saturated creep force. (2) e variation period of the wear depth curve on inner rail is consistent with that of the creep force curve (there is a certain phase difference), and the variation range changes from 0 μm to 0.0014 μm and shows a periodic wavy wear. When the creep force is equal to the saturated creep force, the wear depth reaches the peak. e variation range of the wear depth curve on outer rail is small, which mainly changes from 0.0005 μm to 0.0008 μm, and implies a kind of uniform wear. erefore, the inner rail tends to have corrugation while the outer rail tends to have uniform corrugation, which is consistent with the fact that corrugation mostly occurs on inner rail of the small radius curve section.

Prediction Profile of Rail Wear.
In order to further explain the serious corrugation on the inner rail and slight corrugation on the outer rail, the wear position of the rail profile is predicted and analyzed. According to the literature   Shock and Vibration [35], the maximum rail wear depth 0.1 mm is selected as the basis for profile renewal, and the rail profile is smoothed.

Shock and Vibration
Taking the update times as 10, the profile contrast diagram and cumulative wear contrast diagram of the inner and outer rails are obtained as shown in Figures 7 and 8. In order to analyze the reasons for the difference of wears between inner and outer rails, the wheel-rail geometric contact relationship in the simulation process is studied. Since the geometric parameters involved in the  Shock and Vibration wheel-rail space geometric contact relationship are in the principle functions of the wheelset transverse displacement and contact angle (angle between wheel-rail contact surface and horizontal surface), several geometric parameters can be simplified to functions of wheelset transverse displacement and contact angle [36] when studying the difference of wear distribution between inner and outer rails. Figure 9 shows the change of wheelset transverse displacement during the operation of vehicle, and Figure 10 shows the change of the corresponding contact angle. When the vehicle is passing on the small radius curve section, the wheelset traverse displacement is large, and it is in the range of 9∼10 mm, and the contact angle changes slightly, approaching zero. Based on the matching relationship between the LMA wheel tread and CN60 rail profile, the transient contact solutions of the wheelset passing curve with different transverse displacements and contact angles can be obtained, as shown in Figure 11. e analysis shows that when the wheelset transverse displacement reaches 9∼10 mm and the contact angle approaches zero, the wheel flange attaches to the rail. Increasing the wheelset transverse displacement will cause the wheel flange to move toward outer rail, resulting in a large outer wheelrail contact angle, which is prone to rail side wear. e inner wheel-rail contact angle is smaller, and wear mainly occurs on the middle part of the rail head. is can also be derived from the wheel-rail contact diagram shown in Figure 12.
Furthermore, combined with the wheel-rail contact patch diagram shown in Figure 13, it can be obtained that the outer wheel-rail contact patch is mainly located in the shoulder area, resulting in large wear in the corresponding area, which is consistent with the wear distribution of outer rail shown in Figure 7. e inner wheel-rail contact patch mainly distributes in the middle part of the rail head, making the wear of the corresponding area large, which is also consistent with the wear distribution of inner rail shown in Figure 8. erefore, from the angle of wheel-rail geometric contact relationship, the difference of wear distribution between inner and outer rails in a small radius curve can be well explained.
According to the predicted profile of rail wear, the outer rail wear mainly occurs on the rail shoulder, while the inner rail wear mainly occurs on the middle of the rail head. Combining the conclusion in Section 5.1 (the inner rail tends to have corrugation while the outer rail tends to have uniform wear), the inner rail mainly has corrugation on the top of rail, and the outer rail mainly has uniform wear on the side, which results in serious corrugation on inner rail and slight corrugation on outer rail.

Frequency-Domain Characteristics of Wear Growth.
According to the data of measured corrugation in Section 2, the maximum passing frequency is about 556 Hz, so the frequency range in frequency domain analysis is set to 0∼600 Hz. e circular curve (R � 350 m) with a length of 200 m is selected for simulation, the measured rail surface irregularity is taken as the initial irregularity to calculate the wear amount (the additional wear generated by train running, considering 4 wheel runs on a rail) under a single operation, and the characteristic frequencies of rail corrugation (inner rail) can be obtained by the Fourier transform of wear results, as shown in Figure 14.

Shock and Vibration
As can be seen from Figure 14, the basic frequency of characteristic frequencies of rail wear is 35.1 Hz, which is caused by the wheel-rail low frequency impact force P2. When the wheel passes the low joint, weld, corrugation section, or wheel scratch, there will be two impact forces: high-frequency impact force P1 and low-frequency impact force P2. Force P1 is the impact force caused by the prominent surface defects such as joints and wheel flat. e P1 resonance can be regarded as the vibration caused by the wheel-rail contact deformation between unsprung mass and rail, and its frequency range is 500 ∼ 1000 Hz. It has a very short acting time and usually causes damage to the surfaces of wheels and rails. Force P2 is the impact force caused by the inevitable track irregularity and wheel circumferential irregularity. e P2 resonance can be regarded as the vibration of unsprung mass and track as the whole mass on the resilient foundation under rails, and its frequency range is 20∼100 Hz. It has a long acting time, which can be transmitted to the vehicle parts upward and to the slab downward [37,38]. Besides, it can be seen from Figure 14 that all other arising frequencies are harmonics of the frequency of 35 Hz, which is considered to be caused by the main frequency separation phenomenon of wheel-rail system responses under the wheel rotation effect [39,40]. Meanwhile, the inner rail wear contains characteristic frequencies (142 Hz, 177 Hz, 355 Hz, and 560 Hz) similar to the passing frequencies of measured corrugation (139∼176 Hz, 353 Hz, and 556 Hz).
Due to fixed frequency characteristics of corrugation, the vibration characteristics of the wheelset-track structure are generally considered as the main cause of corrugation occurrence at the specific frequencies. Using the finite element software ABAQUS, the track structure model with fastener spacing 0.6 m, track centerline length 36 m, curve radius 350 m, and wheelset model with a diameter 840 mm is established, as shown in Figure 15. e wheel tread is the LMA tread and cross-section size of the slab is 2700 mm × 300 mm. Shock and Vibration rough the modal analysis of the abovementioned model, the natural frequencies of the wheelset-track system and corresponding modes can be obtained, and the modes close to the passing frequencies are shown in Figure 16.
It can be seen from the analysis that the vibration of the inner rail is more violent than that of the outer rail at characteristic frequencies of the wheel-track system, which makes the inner rail more likely to produce the corrugation of corresponding frequencies.
is is consistent with the conclusion in Section 5.2 that the inner rail corrugation is serious and the outer rail corrugation is slight.
In order to measure the development level of wear at different frequencies, the wear growth rates at different frequencies of the wear amount (inner rail) relative to the initial irregularity are calculated according to equation (9), as shown in Figure 17.
It can be seen from Figure 17 that the wear growth rates at the characteristic frequencies of 142 Hz, 177 Hz, 355 Hz, and 560 Hz are relatively high, and these frequencies are similar to the passing frequencies. is indicates that the wear of corresponding frequencies will continue to develop and finally form corrugation. Outer rail Inner rail Figure 15: Wheelset-track coupled model.

Conclusions
Based on the measured data in the practical work conditions, the vehicle-track model and the rail material friction and the wear model are built. Further, corrugation characteristics of the Cologne Egg fastener track in the small radius curve are investigated. e following conclusions can be drawn: (1) e variation amplitude of creep force on the inside of the guiding wheelset is large, which is partially coincident with that of the saturated creep force and the coincident part appears periodically. e variation amplitude of creep force on the outside of the guiding wheelset is small, which is approximately equal to that of the saturated creep force. In addition, the variation period of the wear depth curve on inner rail is consistent with that of the creep force curve (there is a certain phase difference), the variation range is mainly changing from 0 μm to 0.0014 μm, showing periodic wavy wear, and when the creep force is equal to the saturated creep force, the wear depth reaches the peak. e variation range of the wear depth curve on outer rail is small, mainly changing from 0.0005 μm to 0.0008 μm, showing uniform wear. (2) Combining with the prediction profile of rail wear, it can be seen that side wear mainly occurs on the outer rail, while the corrugation tends to occur on inner rail, which results in serious corrugation on inner rail and slight corrugation on outer rail. (3) e inner rail wear in the small radius curve contains the characteristic frequencies (142 Hz, 177 Hz, 355 Hz, and 560 Hz) similar to the passing frequencies (139∼176 Hz, 353 Hz, and 556 Hz). e simulated results of modal analysis show that the vibration of inner rail is more violent than that of outer rail at the characteristic frequencies of the wheel-track system, which makes inner rail more likely to produce corrugation of corresponding frequencies. e wear growth rates at characteristic frequencies are relatively high, which indicates that the wear at corresponding frequencies will continue to develop and eventually form corrugation.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.