A method for the quantitative evaluation of rail corrugation from the vertical acceleration of the axle-box

Abstract

Corrugation has attracted increasing attention in railways due to its destructive nature, and axle-box acceleration can provide an effective method of detecting corrugation. However, there is a lack of direct relationship between axle-box acceleration and unacceptable levels of corrugation. Based on the excitation characteristics of corrugation, this study proposes a method to quantitatively evaluate corrugation via axle-box vertical acceleration (ABVA). First, a 3D wheel-rail dynamics model was established by using parameters from China’s high-speed railway, the transfer characteristics of the wheel-rail system were investigated by steady-state response analysis, and the influence of corrugation wavelength in the range of 40 to 300 mm on ABVA was revealed; Second, ABVA excited by corrugation with grinding limit was numerically simulated at a speed of 300 km/h, the map relationship between corrugation and ABVA was obtained, and then influences of natural frequency and contact behavior of wheel-rail system on the map relationship were discussed, a method based on the intervention value was introduced to quantify the degree of corrugation to be ground. Finally, field measurements in high-speed railways were carried out to verify the feasibility of the method, the results showed that the hit rate between the intervention value and the actual scenarios of corrugations was 90.0%, indicating that the proposed method has a good performance in quantitatively evaluating corrugations. This work can provide a scientific reference for decision-makings on whether or not the corrugation is severe enough to be ground.

Keywords

Corrugation numerical simulation axle-box vertical acceleration track management

1. Introduction

Corrugation refers to the uneven wear of the rail surface in the longitudinal direction, which can generate large dynamic wheel-rail forces and high-frequency vibrations (Klaus and Sebastian, 2017). Corrugation causes degradation of transport performance, resulting in poor ride quality and high maintenance costs. Severe corrugation can significantly reduce the service life of vehicles and tracks, and can even lead to unwanted derailments. However, the corrugation formation mechanisms are complex (Torstensson and Nielsen, 2015), there are no effective measures to prevent corrugation formation, and rail grinding operations is widely used to mitigate noise and vibration emissions. It is therefore important for track maintenance to find an effective method of evaluating whether corrugation is severe enough to be ground.

Railways require regular inspection to maintain acceptable levels of rail roughness in corrugated sections, and the parameters such as moving peak PPR or RMS (BS EN 13231-3, 2006), roughness level (BS EN 15610, 2009; ISO3095, 2013), and depth (TG/GW 115-2012, 2013) are used as acceptable level of corrugation, which can be measured directly by workers using corrugation analysis trolleys or other instruments. However, the detection results depend on the expertise of the operators and the work environment during the midnight maintenance gap. In general, on-board inspection equipment can detect corrugation quickly and objectively, so it is widely used in the railway industry. For example, track vertical irregularities can be obtained from the vertical acceleration of car-body (Haigermoser et al., 2015) or axle-box (Kaewunruen, 2018) by means of double integration in the time domain, rail roughness is then calculated by filtering, which is compared with the recommended limits in the Standard and the final decision is made. The laser camera technology developed in the last decade (Gazafrudi et al., 2020; Guerrieri et al., 2018) is a visual inspection for corrugation. A high-resolution camera is mounted on the bogie to capture images of the rail surface at high-frequency, and a 3D shape of the rail surface is reconstructed by laser peak extraction technology. Track vertical irregularity is extracted and fitted by a sinusoidal function, and then the depth and wavelength of the corrugation are estimated, and a decision is made on whether to intervene in the corrugation or not. Based on Empirical Wavelet Transform and on-board axle-box acceleration, a deep learning fault diagnosis method for the graded diagnosis of rail corrugation was introduced (Wang et al., 2024), which was developed to extract and classify features of corrugation, enhancing detection intelligence and automation. In addition to the above, acoustic sensors (Javad and Hasheminezhad, 2016), instrumented wheelsets (Nielsen, 2008), or other technologies can also identify corrugation, sound pressure level (ISO3095, 2013), 3σ value (Gullers et al., 2011; Jin, 2019), or root mean square (RMS) of wheel-rail vertical force (Li et al., 2019) with a cut-off frequency of 2 kHz.

Due to the ease of installation and maintenance of an acceleration sensor mounted on axle-box, the use of ABVA on in-service vehicles to identify corrugation is potentially more cost-effective than the above techniques (Weston et al., 2015). However, ABVA measures the vertical vibration excited by the rail roughness rather than the roughness itself, and there is an indirect correspondence between roughness and ABVA (Salvador et al., 2016), which is a key issue in actual track management. To evaluate corrugation, ABVA have previously been analyzed by many techniques, ranging from simple root-mean-square in the time domain (Bocciolone et al., 2007) to frequency derivative analysis (Abdulrazagh et al., 2021). For example, continuous wavelet coefficients were used to characterize the energy distributions of ABVA, and a recommended threshold was set when corrugation had to be ground (Caprioli et al., 2007). To suppress the influence of running speed, a modified index was defined as the RMS of the ABVA divided by the running speed, and the application of the modified index to the inner rail on a curved track was on-site verified in situ (Bocciolone et al., 2007). A corrugation index was introduced to ensure consistency of evaluation from different vehicles and sensor mounts (Niu et al., 2020), the relationship between corrugation depth and corrugation index was fitted, and the recommended limits were proposed according to the Chinese Rail Maintenance Regulations (TG/GW 115-2012, 2013).

In the available literature, the mapping relationship between corrugation and ABVA is fully studied, but the reverse mapping relationship is less involved, which is particularly important for railway maintenance to extract information from ABVA. Establishing relationship between them requires knowledge of the excitation characteristics from corrugation to ABVA. Wavelength and depth are two critical characteristics of corrugations. The wavelength is a sensitive factor in wheel–rail interaction (Wu et al., 2022) and can be determined numerically by the dominant excitation frequency and running speed (Klaus and Sebastian, 2017), both of which can be accurately measured during actual detection. Although it is theoretically feasible to establish the relationship via a large number of reference corrugations and the consequent ABVA, this process is quite a time consuming and laborious. This paper employs a 3D numerical wheel-rail mode, and investigates the response characteristics of ABVA by parameter variation, and proposes a corrugation evaluation method based on theoretical study and Chinese Rail Maintenance Regulations (TG/GW 115-2012, 2013), numerical simulations and field measurements were carried out to verify the feasibility. The paper is structured as follows: the 3D wheel-rail dynamics model with corrugated rail is presented in Section 2; the transfer characteristics and verification of the model are introduced in Section 3; the mapping relationship between corrugation and ABVA is discussed in Section 4; a corrugation evaluation method is introduced and verified in Section 5; and finally, some conclusions are given in Section 6.

2. Wheel-rail dynamics model and corrugated rail surface

2.1. Model of wheel-rail system

Corrugation appears on two tracks in high-speed railway, accompanied by periodic changes on rail surface, as shown in Figure 1(a), λ, 2 $w$ represents the wavelength of corrugation and the width of wheel-rail contact trajectory. Due to the complexity of the formation mechanism of corrugation, the distribution range of wavelength is wide. According to the field observation of corrugation section (Tian, 2014), the wavelength is mainly in the range of 40 to 300 mm, and the depth is mainly in the range of 0.03 to 0.24 mm. The bar chart of corrugation wavelength samples is shown in Figure 1(b).

It can be seen from Figure 1(b) that the proportion of corrugations with wavelength in the range of 110 to 130 mm is the highest, followed by corrugations with wavelength in the range of 60 to 80 mm. Therefore, the wheel-rail dynamics model is established to consider the corrugation with wavelength in the range of 40 to 300 mm, as shown in Figure 2, which assumes that corrugation is located on straight track. To improve the computational efficiency, the model is symmetrical along the centerline of the track, which includes half of wheelset, half of the car-body and bogie, single track and its associated fasteners, and slab. This study focuses on the ABVA in high frequency, so the car-body and bogie are lumped into a mass block M, which is supported in the vertical direction by the primary suspension. The primary suspension is modeled as spring elements with stiffness k₁ and damping c₁, applied to the end of the wheelset as shown in Figure 2(b). The lateral displacement of M is restricted as it has little influence on ABVA. The wheelset, rail and slab are modeled with actual geometry and material properties. The rolling circle radius of the wheelset is 430 mm and the tread type is LM_A, and the rail type is CN 60 with a 1:40 rail cant. The span between the fasteners is 648 mm and their lateral and longitudinal displacements are constrained as shown in Figure 2(a). A Cartesian coordinate system with its origin is located at the initial contact position between the rail and the wheelset, which runs along the x-axis direction, as shown in Figure 2.

To seek a trade-off between rail length and computational effort, rail length is set to 16.4 m, including 25 sets of sleepers, and both rail ends are restrained by symmetric constraints, which are satisfied with numerical solution accuracy (Beshbichi et al., 2019). A bilinear isotropic constitutive model simulates elastoplastic properties of wheel-rail materials (Resapu and Perumahanthi, 2020). The corrugation is placed on the ab-section of the rail with a length of 3 m, as shown in Figure 2(a). To ensure that the model is in a state of dynamic equilibrium when the wheelset arrives at point a, the length of the oa-section of the rail is set to 3 m after the trial calculation. The coefficient of friction is set to 0.3 (Hua et al., 2011; Wu et al., 2022). The wheel–rail interaction is treated as surface-to-surface contact with a penalty algorithm. To obtain a solution with sufficient accuracy within an acceptable time, eight-node hexahedral solid elements are used to partition the model, a multi-scale partitioning technique is adopted to refine the corrugated section s as shown in Figure 3(a). The fine grids are close to the wheel-rail contact area such as the wheel tread and the rail surface in the corrugated section. The mesh density is high and the minimum size near the contact area is 1 × 1 × 1 mm; the coarse grids are far from the wheel-rail contact area such as the non-corrugated rail, axle, and mortar. The mesh density is low and the mesh size is large. Fasteners are simulated by 5 × 11 spring-damping elements with total stiffness k₂ and total damping c₂, which are applied to corresponding nodes between rail and sleepers, as shown in Figure 3(a). The dynamic stiffness k₂ is related to the static stiffness of the rail pad and the wheel–rail interaction frequency (Thompson and Verheij, 1997), here k₂ is set to 1.5 times the static stiffness of the rail pad (Zhi-ping et al., 2019). The model parameters are listed in Table 1.

Table 1.

Parameters of the wheel-rail dynamics model.

Parameter		Value	Unit	Parameters		Value	Unit
Wheel and rail	Poisson’s ratio	0.3	–	Primary suspension	Vertical stiffness	1.04	kN/mm
	Density	7.8	mg/mm³		Vertical damping	20.0	Ns/mm
	Elastic Modulus	210.0	GPa		Lateral stiffness	0.25	kN/mm
	Tangential modulus	21.0	GPa	Fastener	Stiffness	22.0	kN/mm
	Yield strength	542.0	MPa	Fastener	Damping	47.7	Ns/mm
Track slab	Poisson’s ratio	0.25	–	Mass block (M)		5.9	10³ kg
	Density	2.4	mg/mm³	Rail mass per meter (m_r)		60.64	kg/m
	Elastic Modulus	34.5	GPa	Sleeper spacing (L_r)		650.0	mm
Mortar	Poisson’s ratio	0.2		Concrete bed	Poisson’s ratio	0.16	–
	Density	1.6	mg/mm³		Density	2.4	mg/mm³
	Elastic Modulus	8.0	GPa		Elastic Modulus	32.5	GPa
Static wheel weight (P₀)		68.0	kN	Rolling radius of wheel		430	mm
Wheel-rail friction coefficient		0.30	–	Hollow radius of Wheelset		30	mm

2.2. Corrugated rail model

As can be seen in Figure 1(a), there is a quasi-periodic trajectory on the rail surface in corrugated section (Grassie, 2009). The corrugation depth varies regularly with position in the longitudinal and horizontal directions, which can be described by a sinusoidal and parabolic function, respectively. In addition to the periodic corrugation irregularity, there are also random track short-wave irregularities on the rail surface in actual railways. Thus, the track irregularity on a corrugated rail is as follows

R = {\begin{cases} [R_{s} + R_{z}] * R_{y} & x_{b} \geq x \geq x_{a} \\ 0 & x < x_{a}, x > x_{b} \end{cases}

(1)

where x_a and x_b are abscissa of points a and b in Figure 2; R_y is a function to modify the corrugation depth in the horizontal direction; and R_z and R_s represent periodic corrugated irregularity and random track short-wave irregularity, respectively. The expressions for R_z and R_y are

R_{x} (x, y, z) = {\begin{cases} \frac{d}{2} (1 - \cos (\frac{2 π (x - x_{a})}{λ})) & x_{b} \geq x \geq x_{a} \\ 0 & x < x_{a}, x > x_{b} \end{cases}

(2)

where d and λ are the depth and the wavelength of corrugation, respectively.

R_{y} (x, y, z) = 1 - {(\frac{y}{w})}^{2} \begin{array}{l} | y | \leq w \end{array}

(3)

where w is half of the maximum width of the contact trajectory, as shown in Figure 1(a).

R_s is a sample generated by the Sato spectrum (Sato, 1977), whose expression is as follows:

s (Ω) = \frac{A}{Ω^{3}}

(4)

where Ω is the spatial frequency and A is the roughness coefficient and here A is set to 1 × 10⁻⁷.

Using equations (1)–(4), the expression for the irregularity of the corrugated rail surface is

R = [R_{s} + \frac{d}{2} (1 - \cos (\frac{2 π (x - x_{a})}{λ}))] ({1 - (\frac{y}{w})}^{2}) x_{b} \geq x \geq x_{a}, | y | \leq w

(5)

In the non-corrugated section, the rail surface is assumed to be smooth. R defined by equation (5) is built into the model by modifying the node coordinates, the specific process is as follows: First, all grids in the ab-section of the rail are extracted. Then the coordinates of the extracted nodes are modified by the corrugation depth calculated by equation (5). Finally, the corrugated rail is established. For example, a seven-cycle corrugation with a wavelength of 120 mm is shown in Figure 3(b). The maximum and minimum of corrugation are the trough and crest, respectively, and to enhance the visual effect, the depth d, the difference between trough and crest, is set to 4 mm. The color map changes from blue to red, and indicates that the corrugation depth varies from 0 to 4 mm. The red curves on the xoz and yoz planes are projections of R_x and R_y defined in equations (4) and (5), respectively. The red dotted thick lines on the xoy plane are projections of the corrugation boundaries on the rail surface, and the girdle shape of the corrugated rail surface is shown in the upper right of Figure 3(b).

The vibration behavior at the wheel–rail interface is also related to the wheelset and the model also considers the influence of wheelset out-of-roundness on ABVA. The out-of-roundness is measured from a wheelset on an inspection vehicle. The measured out-of-roundness is built into the tread of the wheelset, the establishment process is similar to the corrugated irregularity aforementioned.

Steady-state response analysis and transient dynamic analysis are employed to investigate the vertical vibration of axle-box excited by corrugation in the following section. The former is used to study the transfer characteristics of the wheel-rail system from a frequency domain perspective, while the latter is used to study the instantaneous vibration behavior of the axle-box from a time domain perspective.

3. Transfer characteristics and verification of numerical model

3.1. Steady-state transfer analyses

The steady-state response analysis is applied to obtain the transfer characteristics of the wheel-rail dynamics model. The force due to corrugation is applied to the nodes at the wheel-rail contact area, which is equivalent to a simple harmonic force with different frequencies. The receptance $H_{A} (ω)$ between corrugation and ABVA can be written (Carrigan and Talbot, 2023) as follows:

H_{A} (ω) = - ω^{2} (\frac{H_{w} (ω)}{H_{w} (ω) + H_{r} (ω) + k_{c}^{- 1}})

(6)

where

H_{w} (ω)

and

H_{r} (ω)

are the receptance of the wheel and the rail, respectively;

k_{c}^{- 1}

is the contact spring stiffness;

ω

is the radian frequency, can be calculated as follows (Grassie, 2009)

ω = 2 π f

(7)

f = \frac{v}{λ}

(8)

where f is the corrugation excitation frequency; λ is the wavelength; and v is the running speed.

The vertical acceleration of a node on the wheelset axis is extracted as a numerical simulation of ABVA in hereinafter. The $H_{A}$ of the numerical model is shown in Figure 4, which reflects the flexible transfer characteristics in vertical direction at 0∼2500 Hz, taking the excitation frequency f as the abscissa axis and $f = \frac{ω}{2 π}$ . The corrugation wavelength in high-speed railway is 40∼300 mm in Figure 1(b), the excitation frequency f is 277∼2084 Hz at a speed of 300 km/h according to equation (8), as shown in the light gray zone in Figure 4.

From Figure 4, it can be seen that the response characteristics of ABVA vary with the excitation frequency, this is to say, the sensitivity of ABVA varies with the corrugation wavelength at a fixed speed according to equation (8). The $H_{A}$ value in the high frequency range of 800∼2500 Hz is larger than that in the low frequency range of 0∼800 Hz. There are clearly five resonance peaks in the $H_{A}$ curve and the corresponding modal shapes are also shown in Figure 4, the vertical deformation is magnified by 210 times for a good visual effect. The frequencies in peak ①, ③, and ⑤ are 52.5 Hz, 421.9 Hz, and 1572.2 Hz, which are related to the wheel-rail coupling vibration. Meanwhile, the frequencies in peaks ② and ④ are related to the natural vibration of the rail, corresponding to its resonance frequency and vertical pinned–pinned resonance frequency, respectively. The former is caused by the stiffness of the fasteners, and the latter is caused by discontinuous support of the sleepers. The frequencies are estimated theoretically (Man, 2002) as follows:

f_{r} = \frac{1}{2 π} \sqrt{\frac{k_{r}}{m_{r}}}

(9)

f_{p p} = 10.2 L_{r}^{- 1.61} {(\frac{E I}{m_{r}})}^{0.33}

(10)

where

k_{r}

is the support stiffness per meter at rail base, L_r is the span of the adjacent sleeper, m_r is the rail mass per meter, and EI is the flexural stiffness of the rail.

The frequencies in peak ② and ④ are 117.2 Hz and 949.8 Hz, which are close to the theoretical values 119.7 Hz and 943.0 Hz, which are obtained from equations (9)–(10) and parameters in Table 1.

On the other hand, the maximum time increment is about 5 × 10⁻⁸ s, which is determined by the numerical model. Thus, the wheel-rail dynamic model can satisfy the numerical solution of ABVA excited by corrugation with wavelength in the range of 40 to 300 mm at a speed of 300 km/h.

3.2. Model verification

Transient dynamic analysis is used for model verification and corrugation-induced ABVA response investigation. Model verification is achieved by comparing measured ABVA and simulated results in the time–frequency domain. The measured ABVA is obtained from the axle-box acceleration sensor installed on the inspection vehicle, as shown in Figure 5(a), and the sampling frequency is 5000 Hz. Before applying periodic corrugated irregularity, out-of-roundness and random track short-wave irregularity are imposed on wheelset tread and rail, respectively. Out-of-roundness is actually measured from wheelset near axle-box acceleration sensor. It should be pointed out that the wheelsets of the inspection vehicle are strictly regulated to avoid serious out-of-roundness, which is an unfavorable excitation for the ABVA.

The simulated ABVA are shown in Figure 5(b) under 4 sets of load cases. In load case 1, there is smooth on rail surface and wheelset tread, ABVA shows a periodic low-frequency fluctuation in the range of −3 to 3 g, which is caused by the discontinuous support effect of sleepers; in load case 2, there is a random track short-wave irregularity on the rail surface, which is obtained from equation (4), and the amplitude is about 0.06 mm. The ABVA fluctuates in the range of −5 to 5 g, and it is obvious that there are more high-frequency components in the ABVA than that in load case 1; in load case 3, there is out-of-roundness on the wheel tread, the amplitude of which is less than 0.02 mm. The ABVA fluctuates in the range of −5 to 5 g and the fluctuation range is quite similar to that of load case 2; there is random track short-wave irregularity and out-of-roundness in the model at the same time, the high-frequency ABVA fluctuates in the range of −10 to 10 g, which is basically the same as the measured ABVA of inspection vehicle in non-corrugation railway. Therefore, the load case 4 is an initial boundary condition for further investigation by the wheel-rail dynamics model.

The corrugated section used for model validation is selected on a high-speed railway with a length of 25 m, as shown in Figure 6(a). The rail roughness is measured by CAT and the roughness level spectra are shown in Figure 6(b).

It is observed that the spectra have a local maximum with level in the order of 15 dB re 1 μm, which indicating that there is a dominant wavelength of 86 mm in the corrugated section. According to equation (8), the excitation frequency at the wheel–rail interface at a speed of 300 km/h is 969 Hz. The maximum depth (peak-to-peak value) is 0.21 mm located in the range of 14.2 to 17.7 m, as shown in the gray zone in Figure 6(b), where the roughness is used to validate the model according to equation (5). The simulated ABVA is shown by the dotted line in Figure 7(a), whose maximum and standard deviation are 124.4 g and 67.6 g, respectively. The corresponding measured ABVA from the inspection vehicle is also shown by the solid line in Figure 7(a), whose maximum and standard deviation are 124.2 g and 67.5 g, respectively. For comparison, the maximum error between the maximum and standard deviation of both is less than 2%. After time–frequency domain transformation (Kerschen et al., 2008), the time–frequency plots of both are shown in Figure 7(b) and (c). It can be seen from Figure 7 that there is good agreement in the time-varying properties and amplitudes of the two sets of ABVA. There are also similarities in the energy distribution characteristics in the frequency domain. There is an obvious energy concentration at 970 Hz, which is consistent with the frequency excited by the corrugation in Figure 6 at a running speed of 300 km/h. The difference between them is negligible in both time and frequency domain, indicating that the numerical model has sufficient accuracy to simulate the vibration behavior of the axle-box excited by corrugation.

The RMS of the ABVA can evaluate the excitation characteristics of the corrugation from an energy point of view (Caprioli et al., 2007). In the following, the RMS of ABVA is first calculated with a 1 m sliding Hanning window, then the 99.73rd percentile value of which is extracted and denoted as S_A based on the 3σ rule. Since S_A is the comprehensive response excited by periodic corrugation irregularity, random track short-wave irregularity and out-of-roundness, the S_A in load case 4 in Figure 5(b) is 2.5 g and is denoted as S₀. The S value, which is used as an indicator to quantitatively evaluate the severity of corrugation with wavelength λ and depth d, can be written as follows.

S_{λ, d} = S_{A} - S_{0}

(11)

4. Mapped relationships between corrugation and S value

According to Chinese Rail Maintenance Regulations (TG/GW 115-2012, 2013), corrugated rail with depth not less than 0.08 mm should be maintained by grinding, and corrugated rail with depth greater than or equal to 0.20 mm is considered as severely damaged. Therefore, in the following numerical analysis, the depth is set from 0.01 to 0.24 mm in following numerical analysis, the S value is shown in Figure 8 when λ is 60, 120, 180, and 240 mm, respectively. The abscissas of the vertical dashed and solid lines in Figure 8 are 0.08 mm and 0.24 mm, which are the thresholds for maintenance and severely damaged evaluation, respectively, as defined in (TG/GW 115-2012, 2013).

As shown in Figure 8, there is a significant positive correlation between S and d at a fixed wavelength, and the slope is related to λ. For example, the slope is largest when λ is 60 mm and the slope is smallest when λ is 240 mm. Consequently, for a specific d in the range of 0.01 to 0.24 mm, and the S_{60, d}> S_{120, d} > S_{180, d} > S_{240, d.} However, due to the influences of λ on ABVA, the relationship between S and d becomes unclear under different wavelengths, for example, S_120,0.20> S_60,0.08> S_180,0.16> S_240,0.20. To accurately evaluate the corrugation by ABVA, it is necessary to study the excitation characteristics of the wavelength at a fixed depth.

The S excited by corrugation with wavelengths in the range of 40 to 300 mm are shown in Figure 9(a), the depths are set to 0.08 mm and 0.20 mm, the wavelength step is 4 mm. The magnifications of S at the same wavelength as d increases from 0.08 mm to 0.20 mm are shown in Figure 9(b).

As shown in Figure 4, the ABVA receptance curve is related to the excitation frequency (wavelength) and the natural frequencies of the wheel-rail system. Accordingly, if the excitation frequency is close to the natural frequencies of the wheel-rail system, the excited vibration will be amplified due to the resonance effect at the wheel–rail interface and transferred to the axle-box, resulting in an increase in S at some wavelength. Therefore, the curves in Figure 8 are not smooth whether d is 0.08 mm or 0.20 mm. For example, there is an obvious inflection point on the curves in Figure 9(a) at a wavelength of 88 mm. That is, S increases with λ when λ < 88 mm, but decreases with λ when λ > 88 mm. According to equation (8), the excitation frequency f is approximately 946.9 Hz at a speed of 300 km/h when λ is 88 mm, which is close to the vertical pinned–pinned resonance frequency, shown as frequency peak ④ in Figure 4.

The frequency peak ⑤ in Figure 4 is 1572.2 Hz, corresponding to a wavelength of 53 mm at a speed of 300 km/h. However, there is no obvious corresponding feature on the S value in Figure 9(a). On the other hand, it can be seen in Figure 9(b) that the magnifications are in the range from 2.5 to 2.7 with a narrow distribution range when λ ≥ 120 mm, while the magnifications are in the range from 1.4 to 2. 4 with a dispersed distribution when λ < 120 mm. The above two phenomena are mainly related to the wheel-rail contact behavior excited by the corrugation. The vertical wheel-rail forces at wavelengths of 60, 120, and 180 mm, respectively, are shown in Figure 10. The corrugation-induced force fluctuates periodically around the static wheel weight, P₀, and the fluctuation increases with d. The minimum force becomes 0 kN in some areas in Figure 10(a) at d = 0.12 mm, which means that the wheel and rail instantaneously are out of contact. The number of wheel-rail uncontacted areas increases with d, and the minimum force is 0 kN almost within each periodic cycle at d ≥ 0.20 mm. Compared with the curve when λ is 60 mm in Figure 8, it can be seen that the change in wheel-rail contact state causes a decrease in the slope, and the slope is inversely proportional to the number of wheel-rail uncontacted areas. In Figure 10(b), the minimum force is 0 kN in a few areas at d = 0.24 mm, and in Figure 10(c), the minimum force is greater than 0 kN at d ≥ 0.20 mm at d < 0.24 mm. The slopes of the curves remain almost unchanged when λ is 120 and 180 mm in Figure 8. It can be inferred from Figures 8 and 10 that there are few wheel-rail uncontacted areas at λ ≥ 120 mm and d < 0.24 mm, under this premise, the slopes of curves remain unchanged in Figure 8 and the magnifications are in the range of 2.5 to 2.7 in Figure 9. It can also be inferred that the depth of the first occurrence of the wheel-rail uncontacted area is related to the wavelength when λ is less than 120 mm, that is, the shorter the wavelength is, the smaller the depth is, and the magnifications decrease gradually as λ decreases in Figure 9.

Due to the excitation characteristics of corrugation, the S value cannot be used to evaluate whether corrugation should be ground or not. For example, the values of S_60,0.06, S_120,0.10, S_180,0.15, and S_240,0.21 are all equal to 51.5 g, but the management for the above corrugations is different according to Chinese Rail Maintenance Regulations (TG/GW 115-2012, 2013): the corrugation with d = 0.06 mm doesn’t need to be ground, while other three corrugations need to be ground, the corrugation with d = 0.20 mm has the most urgent demand among them. In short, although the S value can identify whether corrugation exists on the rail surface, it can’t quantitatively evaluate the severity of corrugation and give a decision on whether the corrugation needs to be ground or not.

5. Evaluation method and its application

The growth of corrugation should be intervened when d reaches the limit for rail grinding. Based on the excitation characteristics from corrugation to ABVA, an intervention method is proposed by combining the use of S and λ to estimate d quantitatively, the intervention value is defined as follows:

I = {\begin{cases} \frac{{\bar{S}}_{λ}}{S_{λ, 0.08}} {\bar{S}}_{λ} < S_{λ, 0.08} \\ 1 + \frac{{\bar{S}}_{λ} - S_{λ, 0.08}}{S_{λ, 0.20} - S_{λ, 0.08}} S_{λ, 0.08} \leq {\bar{S}}_{λ} < S_{λ, 0.20} \\ 2 + \frac{{\bar{S}}_{λ} - S_{λ, 0.20}}{S_{λ, 0.20}} {\bar{S}}_{λ} \geq S_{λ, 0.20} \end{cases}

(12)

where

{\bar{S}}_{λ}

is the S value from measured ABVA according to equation (11).

λ

is the corrugation wavelength estimated from measured ABVA, the process is as follows: first the spectrum is extracted by time–frequency conversion of ABVA, and then the dominant excitation frequency is selected, finally λ is calculated from equation (8) and the running speed.

S_{λ, 0.08}

and

S_{λ, 0.20}

are the S value excited by corrugations with wavelength λ and depths of 0.08 mm and 0.20 mm, respectively, as shown in Figure 9(a) and listed in Appendix 1. The S value at wavelengths not listed in Appendix 1 can be obtained by interpolation method.

It can be seen from equation (12) that the I value quantifies the intervention degree of the corrugation:

➢ If $I < 1.0$ , there is corrugation on the rail surface and its depth has not reached the management limit for rail grinding, regular observations of the rail surface become increasingly necessary as I value approaches 1.0;

➢ If $1.0 \leq I < 2.0$ , the corrugation depth has exceeded the maintenance limit, and the greater I, the greater the need for rail grinding;

➢ If $I \geq 2.0$ , it is mandatory to grind the corrugated rail which has been severely damaged and may lead to undesirable consequences.

In order to verify the feasibility of the proposed method, experimental tests are carried out on high-speed railway by axle-box acceleration shown in Figure 5(a). The actual data processing flowchart is shown in Figure 11.

Unlike the numerical study above, actual railways have a long distance and are divided into several evaluation segments with a length of 20 m. The I value is calculated in each segment and adjacent segments are merged together as a corrugated section if $I \geq 0.2$ . $\bar{S_{0}}$ reflects the normal fluctuation level of the measured ABVA in the non-corrugated section. In actual detection, $\bar{S_{0}}$ is related not only to inspection vehicle and track, but also to test noise, running speed, etc. $\bar{S_{0}}$ is about 7.5 g at a speed of 300 km/h, which is slightly higher than the $\bar{S}$ of load case 4 in Figure 5(b).

A criterion for ABVA to identify whether corrugation exists or not is set as $\bar{S} \geq 3 \bar{S_{0}}$ based on empirical application, and the S value in each evaluation segment is shown in Figure 12(a). There are 155 segments with $\bar{S} \geq 3 \bar{S_{0}}$ where corrugation is likely to exist. The I value is calculated sequentially according to equation (12) and there are 30 sets of segments with $I \geq 0.5$ , as shown in Figure 12(b).

The rail roughness was measured in 30 segments in Figure 12(b) to validate the performance of the I value; the field measurements are shown in Table 2. The hit rate between the I value and the actual scenario of corrugation is 90.0% according to Table 2.

Table 2.

Comparison between intervention value and field scenarios.

I value	Number of segments	Field scenarios	Number of misjudgments	hit rate
I ⩾ 2.0	2	There are corrugations in 2 segments, and the depths are all greater than 0.20 mm	0	100%
0.1 ⩽ I < 2.0	15	There are corrugations in 15 segments, and the depths are greater than 0.08 mm in 14 segments	1	93.3%
0.9 ⩽ I < 1.0	3	There are corrugations in 3 segments, but the depth is greater 0.08 mm in 1 segment	1	66.7%
0.8 ⩽ I < 0.9	3	There are corrugations in 3 segments, and the depths are all less than 0.08 mm	0	100%
I < 0.8	7	There are corrugations in 7 segments, but the depth is greater than 0.08 mm in 1 segment	1	85.7%
Total	30	The I value matches field scenarios well in 27 segments and poorly in 3 segments	3	90.0%

The I value in segment No. 178 is the maximum, 2.13, and the corrugation depth is 0.22 mm, which is greater than the threshold of severely damaged rail, the field scenario is seen in Figure 6; the I value in segment No. 59 is 1.06, where the rail surface has obvious periodic corrugation wear, as shown in Figure 13(a), the length which is about 100 m and the maximum depth is 0.10 mm, which has exceeded the limit for rail grinding. The measured ABVA at a speed of 305 km/h is shown in Figure 13(b), there is an obvious energy concentration at 592 Hz in the time–frequency spectrum in Figure 13(c), which indicates that the dominant corrugation wavelength in this segment is about 143 mm.

It can be seen from Table 2 and the field measurements that the proposed method has a good performance in the quantitatively evaluating of corrugation on high-speed railway.

6. Conclusions

This paper aims to evaluate corrugation quantitatively based on ABVA, a 3D wheel-rail model with corrugated rail is established. Steady-state response analysis and transient dynamic analysis are introduced to investigate the vibration behavior of axle-box excited by the corrugation of interest. On this basis, a quantitative corrugation evaluation method is proposed, and its applicability is verified by field measurements. The following conclusions are drawn:

(1) The excitation characteristics and the contact behavior of the wheel-rail system influence the ABVA. There is a positive correlation between depth and ABVA at a fixed wavelength, but ABVA alone cannot directly evaluate corrugation, for example, S_60,0.06, S_120,0.10, S_180,0.15, and S_240,0.21 are all equal to 51.5 g, resulting in different management measures when treating the above corrugations. Wavelength and depth should be considered together to quantify the state of corrugation by ABVA.

(2) According to the excitation characteristics of corrugation on ABVA and the limit for rail grinding, ABVA excited by corrugation with wavelength range of 40 to 300 mm at 300 km/h are numerically simulated, A method of quantifying the intervention degree of corrugation is introduced. Filed measurements on high-speed railway show that the hit rate of the method is 90.0% in terms of evaluating corrugation quantitatively, and it is suitable as a reference for track maintenance to make decisions on corrugation.

(3) In practical application of the method, it is difficult to accurately estimate $\bar{S_{0}}$ due to factors, such as vehicle and track parameters, the installation of acceleration sensor, test noise and so on. In order to reduce omissions and judgments, it is recommended that I ⩾ 0.95 be used as the threshold for rail grinding. In this case, the hit rate in Table 2 can reach 96.7%.

(4) The method is introduced based on the numerical model and the measured ABVA from the axle-box acceleration sensor as shown in Figure 5(a). Meanwhile, there is a significant diversity in vehicle types, track substructures in China’s high-speed railways, which cause differences in excitation characteristics, and mapping relationship between corrugation and ‾S value. In the next step, numerical simulation and measurement experiments will be conducted to study the mapping relationship with the above parameters, and then improve the applicability of the present method in quantifying corrugation.

Footnotes

Acknowledgments

The authors thank all the anonymous reviewers and the editor for their constructive comments and suggestions on the original version of this paper; Thanks to Gaofeng Chu and Pei’ang Miao for their efforts in data collation and writing.

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 project has been supported by Fund of China Academy of Railway Sciences Group Corporation (2023YJ038).

ORCID iD

Liubin Niu

Appendix

λ/mm	d = 0.08 mm	d = 0.20 mm	λ/mm	d = 0.08 mm	d = 0.20 mm
40	45.21	69.11	172	23.38	62.44
44	52.74	83.19	176	27.30	70.92
48	61.78	94.77	180	26.12	70.00
52	64.66	107.40	184	22.50	61.43
56	63.26	107.27	188	26.52	69.37
60	65.12	113.40	192	23.10	59.60
64	67.35	111.27	196	22.74	55.94
68	65.59	119.08	200	25.57	66.56
72	67.92	121.14	204	25.51	68.35
76	61.54	117.13	208	20.44	52.64
80	60.91	119.50	212	22.66	59.66
84	62.72	118.64	216	22.08	58.88
88	64.58	130.16	220	19.55	52.64
92	62.92	116.81	224	19.32	50.07
96	57.31	125.98	228	21.69	56.27
100	53.16	114.74	232	24.60	64.67
104	49.99	113.31	236	21.43	57.42
108	46.87	109.29	240	17.59	46.88
112	43.94	108.90	244	18.45	50.85
116	40.48	100.59	248	22.45	60.10
120	38.59	98.76	252	24.26	63.56
124	36.29	93.82	256	21.49	56.09
128	35.14	89.00	260	17.26	45.42
132	37.21	98.09	264	18.00	46.89
136	33.75	87.27	268	21.09	53.93
140	32.67	84.68	272	23.30	59.65
144	34.04	86.89	276	24.27	63.16
148	30.05	78.54	280	18.62	50.12
152	30.86	80.83	284	16.10	44.04
156	29.21	75.59	288	15.98	43.23
160	31.20	81.95	292	17.62	48.68
164	25.83	67.39	296	19.30	53.20
168	29.47	76.32	300	20.41	55.67

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