Investigation of rolling contact fatigue cracks using the transmitter-receiver eddy current testing under moving conditions

ABSTRACT Rolling contact fatigue (RCF) cracks online detection using eddy current testing (ECT) is in urgent demand. However, RCF cracks detection and evaluation in this way under moving conditions remains challenging due to the velocity effect in ECT. This paper aims to study the response of a transmitter-receiver eddy current probe to cracks under moving conditions and evaluate the depth and inclination angle of RCF cracks. In this paper, a high-speed eddy current testing system is developed to experimentally investigate the influence of coil gap, detection speed, and the lift-off on the eddy current probe’s response under moving conditions. In addition, the temporal and amplitude features of the eddy current signal are extracted to characterise the depth and inclination angle of the RCF cracks. The experimental results indicate the eddy current probe’s response can be improved by increasing the coil gap (coil centre distance) suitably, which can be done to compensate for the attenuation of the eddy current signal caused by detection speed and lift-off. The probe’s response hardly changes with an increase in detection speed when the driver and pick-up coils of the eddy current probe completely overlap. The crack depth and inclination angle can be evaluated under moving conditions. 


Introduction
The steel rail is probably the most important component of the railway system since its main functions are to transfer the wheel force to the track bed and to guide the train.Rails are subjected to complex and variable mechanical loads in harsh working environments, leading to the emergence of damage with time [1,2].Of such damages, rolling contact fatigue (RCF) cracks are currently considered to be the most serious.Because RCF cracks can be extended into the rail head, leading to rail fractures [3], it is necessary to study non-destructive testing (NDT) for RCF inspections [4].However, with the detection time between running trains lessening due to the increases in train speed, manual NDT technology cannot meet the need for effective rail inspection at high speed [5,6].Thus, far more research is required to be done to improve the reliability of NDT technology for high-speed rail inspection.
New NDT technologies are now being investigated for rail inspection.Conventional ultrasonic testing (UT) methods have been used for online rails detection [7], but it is almost impossible for them to distinguish surface defects and subsurface defects within a 4 mm depth, and they are limited to low-speed detection.Although guided-wave UT technology can detect surface and subsurface defects, it is not suitable for high-speed inspection and can only detect large transverse rail head defects [8,9].Magnetic flux leakage (MFL) technology also is broadly used as a complementary method to ultrasonic testing for rail inspection; nevertheless, MFL is also negatively affected due to a decrease in magnetic flux density caused by increasing detection speed [10][11][12].In recent years, visual camera systems for rail inspection have been implemented.Although the inspection speed of these systems can vary from 1 to 320 km/h, the vision systems cannot gather information about the depth of the defect [13,14].In addition, some researchers have attempted to apply new NDT technologies to rail inspection.Peng J et al [15] and Liu [16] researched eddy current pulsed thermography for RCF inspection, although the cost of this is higher than that of traditional technologies for rail detection.Because eddy current testing has a better ability to detect RCF cracks, surface defects [17], and grinding marks, it can now be applied to compensate for the drawbacks of the UT method for rail inspection.Therefore, eddy current testing is one of the promising NDT techniques for high-speed rail inspection systems.
Thus far, many scholars have carried out some useful research on high-speed eddy current testing.Yuan et al [18] proposed a novel DC electromagnetic probe based on the drag effect at high speed that was used to quantify cracks.Their investigation indicated that this probe can quantify cracks at 20 m/s when the crack depth is deeper than 1 mm.Piao et al [19] designed an eddy current probe that employs horizontal U-shaped magnets and a magnetic sensor, utilising the wake effect caused by the relative motion between the magnet and the rail track to detect the rail head.This research showed that the higher the detection speed, the greater the response of the magnetic sensor.However, these probes, based on the motion-induced eddy current principle, have low sensitivity at low speeds.As a result, research is needed on the speed effect in eddy current testing.Xu et al [20] illustrated that the amplitude of the eddy current probe decreases approximately linearly as the speed increases and achieves a relationship between crack signal feature, inspection speed, and crack depth.Xu et al [21] proposed a multi-frequency composited electromagnetic detection method that can also suppress the attenuation of the eddy current signal as speed increases.Yuan et al [22] investigated the velocity effect of the PEC technique, their results indicated that the velocity effect can change the baseline of the detection signal in the PEC inspection system.In addition, Piao et al [23] proposed a novel method to discriminate inner/outer diameter defects by taking advantage of the conductivity-dependent and permeability-dependent distribution of induced eddy current on the inner diameter surface of steel pipes in high-speed eddy current testing.Besides this, many studies [24,25] show that the higher the working frequency is, the smaller the speed effect is.However, in the above studies, improving the eddy current probe's response to crack under moving conditions and suppressing the attenuation of the eddy current signal caused by detection speed and lift-off have not been further studied.Elsewhere, many researchers have also studied the characterisation of RCF crack depth and inclination angle.Zhu J et al [26] evaluated eight spatial and temporal eddy current pulsed thermography features of idealised and real RCF cracks, with the results showing that longer time slots and pulse durations make the relations clearer.Shen J. et al [27] used an alternating current field measurement sensor to predict the size of RCF cluster crack dimensions and the influence of crack length and vertical angle.Tong Z. et al [28] proposed a hybrid processing method combining principal component analysis and 2D wavelet transformation to acquire the spatial features of the thermographic image sequences and realised quantitative mapping of the depth profile of fatigue cracks.Although a large number of studies of the depth and inclination angle of RCF cracks have been carried out, further research is needed to quantify the depth and inclination angle of RCF cracks under moving conditions.To sum up, suppressing the attenuation of an eddy current signal caused by speed and lift-off to improve the response of an eddy current probe and quantifying the depth and inclination angle of RCF cracks under moving conditions remain challenges for online rail detection.
To solve the problem of eddy current signal attenuation caused by speed and lift-off, as mentioned above, in this paper, the relationship between coil gap, detection speed, lift-off, and eddy current probe's response is studied experimentally and theoretically under moving conditions.The experimental results indicate that increasing the coil gap appropriately can reduce the proportion of direct coupling to improve the response of the eddy current probe to cracks under moving conditions.This can be used to compensate for the attenuation of the probe's response caused by speed and lift-off.On the other hand, the change of indirect coupling is least affected by detection speed when the driver and pickup coils of the eddy current probe completely overlap, which causes the response of the eddy current probe to cracks to hardly decrease with an increase in detection speed.Based on the analysis above, the driver and the pick-up coils mentioned in the previous work [29] are connected in series as the new driver coil, and the new pick-up coil is identical to the new driver coil.With the above optimisation, the coil gap can be adjusted from 0 mm to meet the requirement of investigating the influence of detection speed on the probe's response with different coil gaps.In addition, the response of the optimised eddy current probe provides multiple features, including temporal and amplitude ones, that can be used to evaluate the depth, and inclination angle of RCF cracks and measure detection speed.
The rest of the paper is organised as follows.The theoretical analysis of the transmitter-receiver eddy current testing under motion conditions and the optimised method of the previously proposed probe [29] are described in section 2. Section 3 illustrates the experimental system and setup.Section 4 conducts experimental validation and analysis of the experimental result.Section 5 establishes a model for quantifying crack depth and inclination angle using features extracted from the response of the optimised probe.Finally, section 6 draws a conclusion and future work.

Theoretical analysis of eddy current testing under motion conditions
The eddy current probe based on the transmitter-receiver (Tx-Rx) model consists of a driver coil and a pick-up coil, which can be equivalent to the transform circuit model, as shown in Figure 1.The driver coil will direct couple with the pick-up coil through the primary magnetic field (B 1 ) in the absence of a specimen.The B 1 can produce electromotive force in the pick-up coil.The direct coupling can be described by the direct mutual inductance M. When the eddy current probe is close to the electrical conductor, according to Lenz's law, the eddy current induced by B 1 in the electrical conductor will generate secondary magnetic fields (B 2 ) that also can produce electromotive force in the pick-up coil.That is, the driver coil also indirectly couples with the pick-up coil through the electrical conductor, which can be described as the indirect mutual inductance M 0 .The output (V o Þ of the pick-up coil in the presence of the specimen, can be expressed as [30]: where L 1 , L 2 are the self-inductance of Tx and Rx, r 1 ,r 2 are the internal resistance of Tx and Rx, l 1 ; l 2 are self-coupling of Tx and Rx through the specimen, i 1 ,i 2 are the current of Tx and Rx and M 0 is the indirect mutual inductance, and M is the direct mutual inductance that is negatively correlated with the distance (coil gap) between the driver coil and the pick-up coil.
According to references [30,31] and Equation 9), the V O is induced by the primary magnetic field and secondary magnetic fields, which are affected by the excitation current (i 1 ), direct mutual inductance (M), and indirect mutual inductance (M 0 Þ.The secondary magnetic fields will change under moving conditions. Under motion conditions, it is necessary to take time to generate an eddy current field that induces the secondary magnetic field.The faster the detection speed is, the shorter the eddy current generation time is, which leads to a decrease in the secondary magnetic field intensity.In addition, the motion-induced eddy current (MIEC) will be produced in the specimen because of the relative motion, and MIEC distributed in the rear part of the driver coil along the moving direction will increase the eddy current caused by induced by B 1 [22], as shown in Figure 2. The above-mentioned decrease in the secondary magnetic field intensity and the change in MIEC are both caused by a change in speed.If the speed is constant, the secondary magnetic field intensity will not change, due to relative motion.Hence, the influence of the coil gap on the output of the pick-up coil can be similar to that of the coil gap when stationary, when the detection speed is constant.In  When the detection speed increases, the secondary magnetic field intensity decreases, and the voltage induced by the secondary magnetic field in the pick-up coil is different due to the different coil gaps.A specific analysis of this is as follows: In eddy current testing, the specimen is equivalent to the RL circuit in the circuit model of the transformer, as shown in Figure 1 (b).It is necessary to take a certain length of time for the eddy current generated in the specimen to reach the maximum value [32,33].The eddy current on the specimen decreases with the increase in the detection speed (V 1 -V 4 ), since the faster the detection speed is, the shorter the time for eddy current generation is, which will lead to a weaker eddy current intensity, as shown in Figure 2.
On the other hand, a certain amount of time is needed for the eddy currents to dissipate [32,33] when the driver coil moves on the specimen.If the coil gap is 0 mm, the pick-up coil can immediately sense the eddy current that has hardly dissipated, which makes the indirect coupling hardly decreases, as shown in Figure 3 (a).Compared with the coil gap at 0 mm, the pick-up coil needs to take more time to sense the eddy current when the coil gap is x mm (x > 0), as shown in Figure 3 (b).Based on the above analysis, the attenuation of the V 0 is at a minimum with the increase in detection speed when the coil gap is 0 mm.The larger the coil gap, the greater the attenuation of the output (V o ) of the pick-up  coil.However, the MIEC distributed in the rear part of the driver coil along the moving direction can compensate for the attenuation of the eddy current.Finally, the attenuation of the V o caused by an increase in detection speed first increases before it decreases.

Optimized method of eddy current probe for RCF detection
The eddy current probe was designed in the previous paper [29] to detect RCF cracks.As shown in Figure 4 (a), the driver coil contains four square coils arranged at the four corners of the probe.Moreover, the pick-up coil comprises two 8-shaped coils directly connected from end to end, in an orthogonal arrangement.The number of turns of the driver coil and the pick-up coil is 30 and 34, respectively.
According to the theoretical analysis in section 2.1, the output of the pick-up will be affected by detection speed, since the detection speed can affect the magnetic field induced by the eddy current under moving conditions.To verify the above analysis, experiments were designed to study the influence of the coil gap on the probe's response under moving conditions and of speed on the probe's response at different coil gaps.Therefore, the coil gap should change from 0 mm in the experiment.In this section, the probe in the previous paper is optimised to meet the requirements for the coil gap changes.The optimised method is as follows.
The series connection of the driver and the 8-shaped coils in the X direction from the previous probe will generate a unidirectional eddy current since the direction of the eddy current on the specimen generated by the driver coil and the 8-shaped coil in the X direction are the same.Moreover, the above structure is then connected in series with the 8-shaped coil in the Y direction, and finally the unidirectional eddy current on the specimen will be generated, as shown in Figure 4 (b).The optimised probe is composed of a new driver and new pick-up coils, and they are overlapped up and down.The structure of the new driver coil and new pick-up coil is identical.As described in reference [29], the size of the optimised probe is 65 � 65 mm.When the unidirectional eddy current interacts with the crack, the new pick-up coil senses the eddy current disturbances in the X and Y directions.This working principle is the same as that described in the previous paper [29].Hence, the response of the optimised probe is similar to that of the probe described in the previous paper [29], which has multiple features (temporal and amplitude features) that can be used to characterise the depth and inclination angle of an RCF crack.

Experimental system
The high-speed eddy current testing system included an eddy current testing system and a high-speed turntable, as shown in Figure 5(a).The eddy current testing system consisted of an eddy current probe, a conditioning circuit based on AD698, a data acquisition card, and a computer, as described in reference [29].Different from the reference [29], the eddy current probe was the optimised probe in section 2.2.The sampling frequency of the data acquisition card was 800 kHz.The high-speed turntable, with a diameter of 1.5 m, simulated the actual rail inspection at different detection speeds.Its speed can be adjusted to between 5 km/h and 300 km/h.Because the speed of a track detection car is generally not more than 80 km/h, the detection speed in this experiment ranged from 10 km/h to 80 km/h.The actual experimental platform and surface crack are shown in Figure 5(b).
To study the response of the optimised probe to RCF cracks.There are 29 surface cracks of different sizes that were cut into the turntable surface.The length of all the cracks ran through the surface with widths ranging from 0.2 mm to 0.8 mm and depths from 2 mm to 8 mm.In addition, the angles with speed direction and the surface of the turntable ranged from 30 � to 90 � .These cracks were numbered from 1 to 29, and information about them is shown in Table 1.The distance between the adjacent cracks of 1-20 was 100 mm, and that between the adjacent cracks 20-29 was less than 20 mm.

Experimental setup
The experiment aimed to identify the influence of coil gap, lift-off, and detection speed on the response of the transmitter-receiver eddy current probe, which was based on the high-speed eddy current testing system described in section 3.1.The optimised probe was placed on the side of the turntable as shown in Figure 5.The working frequency of the optimised probe was 17 kHz.The AD698 can generate a driver current of 0.02 A.
Optimized probes with different coil gaps had been designed for this experiment.When the coil gap was 0 mm, the driver coil and pick-up coil completely overlapped, as shown in Figure 7 (a).Coil gaps of is 15 mm, 30 mm, and 45 mm are shown in Figure 7  (b), (c), and (d), respectively.
To study the relationship between coil gap, lift-off, and the response of the optimised probe in high-speed eddy current testing, three experiments were conducted using the optimised probe with different coil gaps, as follows: (1) When the detection speed was constant and lift-off was 0 mm, the optimised probe with different coil gaps was used to detect the cracks on the turntable surface.This experiment aimed to study the influence of the coil gap on the response of the optimised probe when the detection speed was constant.(2) When the lift-off was 0 mm, the optimised probe with different coil gaps was used to detect the cracks on the turntable surface when the detection speed increased.This experiment aimed to study the influence of the detection speed on the response of the optimised probe with different coil gaps.(3) When the detection speed was constant and the coil gap was 15 mm, an optimised probe with different lift-offs was used to detect the cracks on the turntable surface.This experiment aimed to study the influence of lift-off on the response of the optimised probe when the detection speed was constant.

Experimental results and analysis
This section provides the experimental results of section 3.2 and analyses the influence of coil gap, detection speed, and lift-off on the response of the optimised probe under moving conditions.In addition, the probe response for the crack with different depths and inclination angles will be used to characterise the depth and inclination angle of the surface crack under moving conditions in section 5 When the coil gap of the optimised probe was 0 mm and the detection speed was 10 km/h, the detection signal could be acquired.However, the detection signal included a periodic baseline drift due to the uneven rotation of the turntable.The empirical mode decomposition method [34] was used to eliminate the baseline drift.The detection signal is shown in Figure 8.For these surface cracks on the turntable surface, the high-speed eddy current testing system could detect all cracks.The signal number in Figure 8 corresponds to the crack number in Table 1.The distance between the adjacent cracks of No. 1-20 was 100 mm, and the distance between the adjacent cracks of No. 20-24 and No. 25-29 was less than 20 mm.However, the minimum length of the optimised probes with different coil gaps in the detection direction was 65 mm.Therefore, the optimised probes could not distinguish the probe's response to cracks No. 20-24 and 25-29.To identify the influence of the coil gap, detection speed, and lift-off on the response of the optimised probe under moving conditions, four surface cracks with different depths on the turntable were selected, and their number is 4-7, as shown in Table 1 and Figure 6.Since the detection speed is a variable in sections 4 and 5, the abscissa of the eddy current signal was set as the time.The response of the optimised probe for cracks with different depths is shown in Figure 9, the crack number is 5, 6, 7. The green dotted line indicates the probe's response to a crack whose depth is 6 mm.The response of the optimised probe has five peaks, with the coordinates of these peaks being (t 1 , p 1 ), (t 2 , p 2 ), (t 3 , p 3 ), (t 4 , p 4 ), and (t 5 , p 5 ) respectively.The abscissa and ordinate represent the time and amplitude respectively.
The response of the optimised probe will reach a peak when the crack passes through a different position of the optimised probe, as described in [29].The time (t) and amplitude (p) of the eddy current signal at high detection speed contain the information of the RCF crack.ΔP represents the amplitude difference between the peak and the peak of the eddy current signal as an amplitude feature.∆t represents the time difference between the peak and the peak of the eddy current signal as a temporal feature.In this section, the ∆P max is selected to represent the response of the optimised probe.The ΔP and Δt are used to characterise RCF crack parameters in section 5.The ΔP and Δt, ∆P max can be expressed as:

Influence of the coil gap on the response of the optimised probe at constant detection speed
In the experiment described in section 3.2, the detection speed was 10 km/h, 30 km/h, 60 km/h, and 80 km/h respectively.An optimised probe with different coil gaps was used to detect the surface cracks on the turntable.The coil gaps were set at 0 mm, 15 mm, 30 mm, and 45 mm respectively, as shown in Figure 7. Take the surface crack whose depth is 2 mm and width is 0.4 mm as an example in this section.The influence of the coil gap on the response of the optimised probe at constant detection speed was identified, as shown in Figure 10.
The experimental results indicated that the ∆P max increased when the coil gap increased from 0 mm to 30 mm, and the ∆P max decreased when the coil gap continued to increase.The minimum of ∆P max can be obtained when the coil gap = 0 mm.
According to the theoretical analysis in section 2.1, when the detection speed is constant, the influence of the coil gap on the response of the optimised probe can be similar to that of the coil gap when stationary.Increasing the coil gap results in reduced direct coupling, which can improve the optimised probe's response to cracks.The indirect coupling decreases when the coil gap continues to increase, which reduces the probe's response to cracks.Therefore the response of the optimised probe to cracks initially increases before it decreases when the coil gap increases.
The experimental results show that the eddy current probe's response to cracks can be improved by properly increasing the coil gap when the detection speed is constant.

Influence of detection speed on the response of the optimised probe with different coil gaps
Based on the experiment described in section 3.2, this section studies the influence of detection speed on the response of the optimised probe with different coil gaps to cracks.To analyse the above influence, surface cracks with a width of 0.4 mm and depths of 2 mm, 4 mm, 6 mm, and 8 mm are taken as examples.The changing trend of ∆P max concerning the detection speed at different coil gaps is plotted, as shown in Figure 11.From Figure 11, the optimised probe's response to the cracks is lowest when the coil gap is 0 mm for all detection speeds during this experiment.In addition, it appears that ΔP max decreases as speed increases for cracks of all depths.
The attenuation rate (AR speed ) is defined to analyse the influence of detection speed on the response of the optimised probe with different coil gaps.The AR speed can be expressed as: where ΔP max 10km=h is the response of the probe when the detection speed is 10 km/h and ΔP max 80km=h is that when the detection speed is 80 km/h.
Table 2 shows the attenuation rate of the optimised probe's response with different coil gaps when detection speed increase.It indicated that attenuation of the optimised probe response is minimum when the coil gap is 0 mm.The attenuation rate increases sharply as the coil gap increases.The attenuation of the optimised probe response is maximum when the coil gap is 15 mm.When the coil gap continues to increase, the attenuation rate decreases, since the MIEC distributed in the rear part of the driver coil supplements the eddy current field induced by the B 1 .Despite the existence of MIEC, when the coil gap is 45 mm, the attenuation rate of the probe's response is still 5-10 times greater than when the coil gap is 0 mm.When the driver and pick coils completely overlap, the pick-up coil can immediately sense the secondary magnetic field (B 2 ) induced by the eddy current.Hence, although the response of the optimised probe with a coil gap of 0 mm is also at a minimum, the probe's response to crack is hardly affected by the detection speed.This is useful for further improving the detection speed.

Influence of lift-off on the response of the optimised probe at constant detection speed
Based on the experiment described in section 3.2, this section will study the influence of lift-off on the response of the optimised probe to cracks at constant detection speed.When the speed is constant, the lift-offs are 0 mm, 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm respectively.Moreover, in this experiment, the coil gap of the probe is 15 mm and the detection speeds are 10 km/h, 20 km/h, 40 km/h, 60 km/h, and 80 km/h respectively.Take the surface crack whose depth is 2 mm and width is 0.4 mm as an example for analysis.The influence of lift-off on the response of the optimised probe at constant detection speed is shown in Figure 12. Figure 12 indicates that the optimised probe response decrease as lift-off increase at a certain detection speed, since the eddy current intensity decrease with an increase in lift-off.According to the experimental results in this section, although both the detection speed and the lift-off will cause the attenuation of the eddy current probe's response to cracks, an appropriate increase in the coil gap will improve the probe's response.
In actual rail detection, there is an urgent need to suppress the attenuation of a probe's response caused by detection speed and lift-off.Therefore, the optimal combination of parameters (lift-off and coil gap) for the probe is coil gap = 0 mm and lift-off = 0 mm.

Quantitative evaluation of rail cracks
According to the analysis in section 4.2, the attenuation of the current signal is minimum with an increase in detection speed, when the coil gap is 0 mm.Therefore, an optimised probe with a 0 mm coil gap was adopted to quantify the depth and inclination angle of the cracks in this section.
For the cracks (cracks number is 4-7) with different depths, an increase in the crack depth will lead to an increase in eddy current disturbance, which in turn leads to the increase of probe response.On the other hand, the time interval of the probe's response peak will decrease with an increase in detection speed.According to the above analysis, the model for evaluating crack depth is derived by combining the temporal and amplitude feature under moving conditions, which can quantify the crack depth without knowing the detection speed.
Furthermore, for the cracks (cracks number is 8-11) with different inclination angles, the greater the angle is, the shorter the time it will take for the probe to detect the cracks when the detection speed is constant.According to the above analysis, the temporal feature can be extracted to quantify the inclination angle of the crack when the detection speed is constant and known.

Evaluation of surface cracks with different depths
Although the temporal and amplitude features extracted in section 4 contain information about the crack, the appropriate feature selected is useful for quantifying the crack.From these features in section 4, (t 2 , p 2 ), (t 3 , p 3 ), and (t 4 , p 4 ) are selected since their peaks come from the detection region that can completely cover the crack.As described in Equation 9), the ΔP k can be expressed as: Figure 14(a) presents the ΔP k for cracks with different depths when the detection speed is 10 km/h.As can be seen from Figure 14(a), ΔP 1 and ΔP 2 are in approximately linear growth as depth increases.The fitting coefficient (R 2 ) of the first-order function fitted by ΔP 1 and ΔP 2 at different detection speeds are shown in Table 3, which indicates that the fitting coefficients of ΔP 2 are generally larger than those of ΔP 1 .Thus, the ΔP 2 is selected to establish an evaluation model Figure 13.As shown in Figure 14(b), the ΔP 2 and crack depth is fitted by a first-order function at different speeds.The first-order function can be expressed as: where d is the crack depth, k is the slope of the ΔP 2 fitting curve and b is a variable.As can be seen from Figure 14(b), k and b change with the increase of speed (v); that is, k and b are functions with speed (v) as independent variables and can be expressed as k=k(v), b=b (v).The relationship between ΔP 2 and d can be derived as: On the other side, the temporal feature is related to the size of the eddy current probe and detection speed and represents the time it takes for the probe to scan for cracks.Δt 0 ¼ t 4 À t 2 is the time difference when the probe's response reaches a peak.Because the size of the probe is constant, Δt 0 is negatively correlated with detection speed.The Δt 0 and detection speed (v) can be fitted as a monotonically decreasing function (v ¼v(Δt)) using MATLAB.
The v ¼v(Δt) is substituted into (9).Thus, the functional relationship between ΔP 2 and crack depth d is obtained, as follows:  It can be seen from Equation 9) that the crack depth can be derived from the Δt and ΔP 2 obtained from the probe's response without being affected by speed.
To validate the appeal evaluation model, cracks with depths of 2 mm, 4 mm, 6 mm and 8 mm were evaluated at different detection speeds.The evaluation results at different detection speeds are shown in Figures 15(a-d) .The maximum evaluation error for cracks with depths of 2 mm, 4 mm, 6 mm, and 8 mm at different speeds are −0.36 mm, 0.67 mm, 0.77 mm, and −0.64 mm respectively Figure 15(e) shows the error bar of the evaluation results for crack depth at a different detection speed.As can be seen from Figure 15(e) , the root means square errors (RMSE) of evaluation for crack depth at different speeds are 0.04 mm, 0.27 mm, 0.36 mm, 0.22 mm The evaluation results verify that the effect of the detection speed on the quantification results for crack depth is not more than 0.36 mm.The evaluation model can be applied for the optimisation of online rail defect detection under moving conditions.

Evaluation of surface cracks with different inclination angles
To evaluate the crack inclination angle, four surface cracks were selected (cracks 8, 9, 10, and 11) with different inclination angles on the turntable, as described in Figure 6 and Table 1.
According to Figure 6(a) and Table 1, the depth of the cracks was constant (4 mm), and the inclination angle (θ) can be expressed as: where θ is negatively correlated with slanting length (L) and dis the depth of the crack.The time required for the probe to scan each crack was different due to different slanting lengths.Hence, a temporal feature was selected to evaluate the inclination angles.
Of temporal features in section 4, Δt 1 ¼ t 5 À t 2 , Δt 2 ¼ t 5 À t 3 , and Δt 3 ¼ t 5 À t 4 were first selected, as shown in Figure 13.Because the relationship between Δt 1 ,Δt 2 , Δt 3 and the inclination angle is similar, Δt 2 was taken as an example for analysis.As shown in Figure 16, the Δt 2 decreases as the inclination angle increases.The inclination angle increase causes the slanting length to decrease, which leads to a decrease in the temporal feature.Because the time required for the probe to scan cracks is reduced.The inclination angle evaluation model from the curve fitting tool apps' in the MATLAB software was used to compute the model's fitting parameters.
Here, the above evaluation model, whereby the detection speed is 60 km/h, is taken as an example for verification.The raw data of Δt 2 are substituted in the evaluation model and the evaluation results of the inclination angle are calculated.The evaluation results and errors of the evaluation results of the crack inclination angle at 60 km/h are shown in Table 4.
Table 4 shows the errors of the evaluation results of the crack inclination angle, which reveal that the maximum error is 4.2 � , and the minimum relative error is 2.5 � .The evaluation results verify the correctness of the evaluation model for the crack inclination angle.The evaluation model has the potential to evaluate the crack inclination angle when the detection speed is constant.

Conclusion and future work
In this paper, the influence of coil gap, detection speed, and lift-off on the response of the transmitter-receiver eddy current probe under moving conditions has been investigated experimentally.Multiple features of the eddy current signal have been extracted to characterise the depth and inclination angle of the cracks in high-speed eddy current testing.The following conclusions can be drawn as follows:  (1) Under the condition of constant detection speed, increasing the coil gap suitably can reduce the direct coupling, which can improve the response of the eddy current probe to cracks.This method is beneficial for optimising and improving the eddy current probe's response at high detection speed and high lift-off; (2) Under the condition of increasing detection speed, the attenuation of the eddy current probe's response is minimal when the coil gap is 0 mm.Because the change of indirect coupling with detection speed is minimal when the coil gap is 0 mm, it has great application prospects to further improve the detection speed in the actual rail detection.(3) The response of the optimised probe provides multiple features (the temporal and amplitude features) for the quantifying depth and inclination angle of cracks under moving conditions.The amplitude feature has a linear relationship with the crack depth, which can be combined with the temporal feature to characterise the crack depth without knowing the detection speed.In addition, the temporal feature decreases monotonically with the inclination angle of the crack when the detection speed is constant.
In addition to detection speed, lift-off also is an important factor leading to the attenuation of eddy current signal in online rail defect detection.Hence, lift-off will be measured to compensate for the attenuation of the eddy current signal in future work.Moreover, the identification of complex-shaped defects such as pit erosion and RCF clustered crack will be studied in future work.

Figure 1 .
Figure 1.(a) ECT probe with the specimen and (b) Equivalent circuits of the ECT probe with the specimen.

section 4 . 1 .
the influence of the coil gap on the output of the pick-up coil is studied at constant detection speed.

Figure 2 .
Figure 2. Electromagnetic field in high-speed eddy current testing.

Figure 3 .
Figure 3. Indirect coupling between the specimen and the probe with different coil gap.

Figure 4 .
Figure 4. (a) the eddy current probe in the previous paper, (b) the optimized ECT probe.

Figure 5 .
Figure 5.The high-speed eddy current testing experimental system (a)the block diagram of the experimental system, (b) the actual experimental platform.

Figure 6 .
Figure 6.(a) Schematic and (b) physical diagram of the cracks with different inclination angles and depths.

Figure 7 .
Figure 7.The optimised probe with different coil gaps in high-speed eddy current testing.

Figure 8 .
Figure 8.The detection results of the surface cracks on the turntable at 10 km/h.

Figure 9 .
Figure 9.The response of the optimized probe to surface cracks with different depths.

Figure 10 .
Figure 10.The influence of the coil gap on the response of the optimized probe at high speed.

Figure 11 .
Figure 11.The attenuation of the response of the optimized probe for cracks with different depths with an increase of detection speed (a) depth=2 mm, (b) depth=4 mm, (c) depth=6 mm, (d) depth=8 mm.

Figure 12 .
Figure 12.The decrease of the response of the optimized probe with an increase in lift-off.

13 .
(a)the amplitude feature and (b) the temporal feature of the optimized probe's response.

Figure
Figure ΔP k for cracks different depths the relationship between ΔP k different depths, (b) the relationship between ΔP and different depths at different speed.

15 .
The evaluation of cracks with a depth of (a) 2 mm, (b) 4 mm, (c) 6 mm, (d) 8 mm at different speeds, (e) the error bar of the evaluation results.

Figure 16 .
Figure 16.The t 2 fitting curves with the inclination angle of cracks under different speeds.

Table 1 .
The parameters of cracks on the turntable.

Table 2 .
The attenuation rate of probe response with an increase of speed for different coil gap.

Table 4 .
The evaluation results and errors of the evaluation