Study on monitoring broken rails of heavy haul railway based on ultrasonic guided wave

Real-time monitoring of broken rails in heavy haul railways is crucial for ensuring the safe operation of railway lines. U78CrV steel is a common material used for heavy haul line rails in China. In this study, the semi-analytical finite element (SAFE) method is employed to calculate the dispersion curves and modal shapes of ultrasonic guided waves in U78CrV heavy steel rails. Guided wave modes that are suitable for detecting rail cracks across the entire cross-section are selected based on the total energy of each mode and the vibration energy in the rail head, web, and foot. The excitation method for the chosen mode is determined by analyzing the energy distribution of the mode shape on the rail surface. The ultrasonic guided wave (UGW) signal in the rail is analyzed using ANSYS three-dimensional finite element simulation. The group velocity of the primary mode in the guided wave signal is obtained through continuous wavelet transform to confirm the existence of the selected mode. It is validated that the selected mode can be excited by examining the similarity between the vibration shapes of a specific rail section and all modal vibration shapes obtained through SAFE. Through simulation and field verification, the guided wave mode selected and excited in this study demonstrates good sensitivity to cracks at the rail head, web, and foot, and it can propagate over distances exceeding 1 km in the rail. By detecting the reflected signal of the selected mode or the degree of attenuation of the transmitted wave, long-distance monitoring of broken rails in heavy-haul railway tracks can be achieved.


Dispersion curve of guided wave in U78CrV rail
According to the dimensional specifications of the 75 kg/m rail, the cross-section of the rail is drawn in SolidWorks.The rail section height is 192 mm, the rail head width is 75 mm, and the rail foot width is 150 mm.The coordinate data of the section is imported into MATLAB and plotted to represent the rail cross-section Ω.The cross-section is then discretized into triangular grid elements Ω e using the PDE discretization tool of MATLAB.The division effect is shown in Fig. 1.The section is divided into 550 units consisting of 340 nodes.
The direction of guided wave propagation is defined as the z-direction, and the rail section is defined as the Oxy plane, where the y-direction represents the vertical direction.According to the SAFE theory 38 , the harmonic displacement, stress, and strain field components of each point in the rail can be expressed as: The relationship between stress and strain is given by σ = C ε, and C is the constant elastic matrix of rail.
The displacement field can express the strain at any point in the rail as: where: (1) www.nature.com/scientificreports/ The amplitude variation of the section displacement field can be expressed by Eq. ( 3).
where ξ is the wave number in the z-direction, and ω is the angular frequency.According to Eq. ( 4), the displacement of any point within the discrete triangular element can be expressed using the shape function as: where N(x, y) is the shape function matrix, q (e) is the node displacement vector.The shape function of each node can be calculated from the coordinates of the node in the section 39 .Based on the Hamiltonian principle, the general homogeneous wave equation of UGW can be deduced 39 : In the equations, K 1 , K 2 , K 3 are three stiffness matrices of size 3n × 3n, where n is the total number of nodes.These matrices can be calculated directly according to the material parameters of U78CrV.M and U are the mass matrix and displacement matrix, respectively.The wave equation precisely describes the dispersion characteristics and multimodality of UGW.By introducing an auxiliary matrix of size 3n × 3n, the imaginary part in Eq. ( 5) can be eliminated.
After introducing the auxiliary matrix, Eq. ( 5) becomes: In the equations, K 2 = T T K 2 T −i , Û = TU .Equation ( 7) can be reformulated as a first-order eigenvalue system: In the equations: For each ω, we can obtain 6n eigenvalues of the wavenumber ξ from Eq. ( 7).Among these, the real eigenvalues correspond to propagating guided wave modes, while the complex eigenvalues correspond to evanescent waves.By iterating through the values of ω, we can obtain the ξ − ω relationship, from which we can solve for the phase velocity dispersion curve of ultrasonic guided waves.The phase velocity Cp is defined as follows: Based on Eq. ( 8), we can obtain the phase velocity dispersion curve of ultrasonic guided waves in the crosssection of U78CrV steel rails, as shown in Fig. 2. www.nature.com/scientificreports/From the dispersion curve, it can be observed that there are multiple guided wave modes in the steel rail.As the frequency of the guided waves changes, the modes of the guided waves also change.Moreover, with an increase in frequency, the number of guided wave modes increases.
Group velocity is defined as the derivative of angular frequency to wave number : And finally, the calculation formula of group velocity is 28 : where K2 is a symmetric matrix for undamped motion, Û is a new nodal displacement vector, ÛR represents the right eigenvector and Û L represents the left eigenvector ω, ξ , ÛL , ÛR can be obtained when solving the characteristic equation.The dispersion curve of guided wave group velocity in U78CrV rail is obtained by taking them into Eq.( 10), as shown in Fig. 3.
Based on the node displacement information in the eigenvectors solved in the previous step, the vibration mode shapes of all guided wave modes can be plotted.Taking the guided wave frequency of 35kHz as an example, there are 23 propagation modes in the rail at this frequency, and the vibration shape diagram of each mode is shown in Fig. 4.

Mode selection
According to the previous research 40 , when using UGW to detect the long-distance rail breakage of U78CrV rail, selecting 35kHz as the guided wave frequency has a good detection effect.To ensure that the exciting mode can cover the whole section of the rail, the mode requires a large vibration amplitude at the rail's head, web, and foot.At the frequency of 35kHz, there are 23 guided wave modes in the rail.The rail section height of 138-192 mm is defined as the rail head area, 33-138 mm as the rail web area, and 0-33 mm as the rail foot area.Define the vibration displacement of the No. i node of mode m as ( u m xi , u m yi , u m zi ), and the average vibration energy of mode m is defined as shown in Eq. ( 8).
According to the vibration data of all nodes in each area, the average vibration energy of 23 modes at the rail head, rail web, and rail foot, as well as the vibration energy of each mode in x, y, and z-directions are calculated respectively.The corresponding results are presented in the form of histograms, as shown in Fig. 5.
We selected the five modes with the highest total energy for in-depth analysis.Theses modes are mode 7, mode 9, mode 12, mode 16 and mode 18.The mode shapes are shown in Fig. 6, where Cp is the phase velocity of the mode in m/s.
Observing the vibration shapes of the five modes, it can be seen that mode 7, mode 12, and mode 16 exhibit many distorted forms, making them difficult to excite in the field environment.Additionally, the energy of mode 18 is very limited at the rail head and rail web, which may result in the omission of defects.Therefore, mode 9 is finally selected.The modes in the dispersion curve are separated, and the phase velocity and group velocity dispersion curves of mode 9 are drawn separately, as shown in Fig. 7.The blue line and red line in the figure represent the phase velocity and group velocity curves of mode 9, respectively.When the frequency is 35kHz, the phase velocity and group velocity of mode 9 solved by SAFE are 3571m/s and 2669m/s, respectively.www.nature.com/scientificreports/According to the field environment, the nodes that the UGW transducer can be installed on the outer surface of the rail section are selected, as shown in Fig. 8.There are 28 nodes symmetrically on the rail surface that meet the requirements.To choose the best excitation point, the vibration displacement of mode 9 in the x, y, and z-directions at 28 nodes is calculated, and the histogram is plotted, as shown in Fig. 9.The point with a large displacement is selected as the excitation point.It can be seen from the figure that the displacement of mode 9 in the z direction is the largest at nodes 118 and 84.Select node 84 (x = 10.4mm,y = 88mm) to excite in the z-direction (longitudinal direction of the rail).The following ANSYS simulation analysis is conducted to verify whether mode 9 can be excited here.

Mode verification
ANSYS is a commonly used three-dimensional simulation and analysis software.The rail vibration data are obtained through the transient dynamic simulation calculation of ANSYS, the group velocity is solved by wavelet transform, and the vibration shape of the rail section is analyzed to determine the guided wave mode in the vibration signal.www.nature.com/scientificreports/

Mode verification based on group velocity
A 5m long three-dimensional rail model is established by SolidWorks, as shown in Fig. 10.The rail model is meshed using Hypermesh software, and each section is divided into 480 nodes.The meshing effect is shown in Fig. 11.
The material properties of the U78CrV rail are assigned to all grid units, and the rail material parameters are shown in Table 1.
The established mesh model is imported into the ANSYS software for transient dynamic analysis.During the simulation, the element type used is the 3D solid element SOLID45 with 8 nodes.The material properties are as follows: material density is 7850 kg/m 3 , elastic modulus E is 208 GPa, Poisson's ratio ν is 0.33.The material is modeled as an isotropic material, MAT1, with properties that do not change with temperature.The excited ultrasonic guided wave modes are analyzed using ANSYS.According to the analysis results in section "Mode selection", at the middle of the rail web(x = 10.4mm,y = 88mm), a five-cycle sinusoidal signal modulated by the Hanning window is applied longitudinally along the rail, as shown in Fig. 12   The vibration data of 240 nodes at the same horizontal position as the excitation point and within the range of 1184-3104 mm from the excitation point are extracted, and continuous wavelet transform is performed on the data to obtain the wavelet time-frequency diagram, as shown in Fig. 14.
The "distance" in the figure represents the distance from the receiving node to the excitation point.The point with the highest frequency component is marked in the figure.It can be seen from the figure that the frequency of the signal is concentrated near 35kHz, and the calculated speed of the guided wave energy concentration point     reaching each node is shown in Table 2, which is very close to the group speed of 2669m/s of mode 9 calculated by the SAFE method.Therefore, the mode propagating in the rail is mainly mode 9.

Mode verification based on mode shapes
The Sect. 2080mm from the excitation point was selected as a specific cross-section for mode validation, named Ω.The theoretical group velocity of mode 9 solved by the SAFE method is 2669 m/s.According to the group velocity, it can be calculated that the time for mode 9 to reach the section Ω is about 0.000779s.
The vibration shapes of section Ω near the time of T = 0.000779 s are drawn, as shown in Fig. 15.The vibration shape at t = 0.000773 is selected for analysis after comparison.
The vibration shapes of 23 modes existing in the rail at 35kHz solved by the SAFE method are drawn and compared with the vibration shapes of section Ω at t = 0.000773s.The comparison diagram is shown in Fig. 16, with the vibration shape of section Ω at 0.000773 s in the right part of each subplot and the vibration shapes of each mode in the left part.
The 340 nodes of the section are numbered, and the vibration data of the section Ω at that time are extracted according to the three directions of x, y, and z and formed into the vector q Ω : t = 2080 mm 2669 m/s = 0.000779 s  Similarly, the vibration data of the 23 modes are extracted in the x, y, and z-directions to form onedimensional data and normalized to form the vector q m : The difference in vibration between the section Ω and the mode m is expressed in terms of the Manhattan distance, i.e.
A difference value is obtained for each mode, and the 23 difference values are sorted in ascending order to get the results shown in Table 3.
It can be seen from the table that when t = 0.000773, the difference between the vibration of section Ω and mode 9 is the smallest; that is, the vibration shape of section Ω is most similar to that of mode 9. Figure 17 shows the comparison between the vibration of mode 9 and the vibration of section Ω at the moment 0.000773s.
By comparing the group velocity and vibration shape diagrams, it can be demonstrated that the guided wave mode shown in the mode 9 vibration shape can be excited in the 75 kg/m rail by applying a five-cycle sinusoidal signal modulated by the Hanning window along the longitudinal rail direction at the node position (x = 10.4 mm, y = 88 mm).

Simulation analysis of rail defect detection
To further verify whether the mode excited at the node (x = 10.4mm,y = 88mm) has good sensitivity to the defects at the rail's head, web, and foot, finite element simulation experiments were conducted (Supplementary Information).The exciting, receiving, and defect positions for the finite element simulation are shown in Fig. 18.
The guided wave excitation point T1 is located at the rail waist of the rail end, and the receiving point R1 is set 1m away from the excitation point to receive the echo signal generated by the guided wave at the defect.At a distance of 2m from the T1 point, the rail head, web, and foot defects are set, respectively.A receiving point R2 is set at 4m to receive the transmitted wave signal after the guided wave passes through the defect.
The five-cycle sinusoidal signal modulated by the Hanning window is applied at the rail web with a center frequency of 35 kHz.The simulation steps are shown in Table 4; four simulations were conducted, the first simulation was performed on the intact rail, and the remaining three were conducted with defects at the rail head, rail web, and rail foot, respectively.
The crack settings are shown in Fig. 19.Four groups of signals are collected at the R1 point, the first group is the signal collected on the intact rail, and the second to fourth groups are echo signals of T1 excited guided wave signals after encountering defects at the rail head, rail web, and rail foot.By comparing the difference between the signals of the second to fourth groups and the signals of the first group, the sensitivity of the guided wave excited at the T1 position to the defects in different places was analyzed.
The left plot in Fig. 20 shows the crack reflection signals received by R1 when the defect is at the rail head, rail web and rail foot, respectively, indicated by the red line.It is compared with the reflected wave at the corresponding position when there is no defect.The right side shows the result of the difference made between the two sets of signals.
It can be seen that the guided wave signals excited at point T1 show clear echo signals at the locations of cracks on the rail head, web, and foot.
Four groups of signals are collected at the R2 point, the first group is the signal collected on the intact rail, and the 2nd-4th groups are the transmitted waves of the guided wave signal excited at T1 after passing through the defects at the rail head rail web and rail foot, respectively.Comparing the signal attenuation of groups 2-4 with group 1 of the T1-excited guided wave signal after passing through the defects at different locations is analyzed.
The left plot in Fig. 21 shows the crack transmission signals received by R2 when the defect is at the rail head, rail web and rail foot, respectively, indicated by the red line.It is also compared with the transmitted wave at the corresponding position when there is no defect.The right side shows the result of the difference made between the two sets of signals.Based on the signal analysis from Fig. 20, the selected modes exhibit reflection coefficients of 0.105, 0.21, and 0.15 for defects at the rail head, rail web, and rail foot, respectively.This indicates that the chosen modes are sensitive to defects at these locations.Additionally, it can be observed from the figures that the strength of the defect echoes follows the order of rail web, rail foot, and rail head, consistent with the energy analysis shown in Fig. 5, further confirming that the excited mode primarily corresponds to Mode 9.
From Fig. 21, it can be observed that the selected modes exhibit different degrees of attenuation when passing through defects at different positions.Considering both the reflection and attenuation signals, by applying the excitation signal at the node positions x = 10.4mm and y = 88mm ,it is possible to detect significant cracks in the rail by analyzing the echo of the defect or analyzing the attenuation of the transmitted wave..

Experimental results
Field experiments are conducted to further validate the effectiveness of the excitation mode applied to the actual rail line.The installation of the sandwich piezoelectric transducers in the field is illustrated in Fig. 22.  www.nature.com/scientificreports/Firstly, the defect detection experiment was carried out.It is impossible to make holes or cut out other defects on the actual operating line.Therefore, the preset earthing hole on the line was taken as the defect point.The line earthing holes are shown in Fig. 23.
The experimental setup is shown in Fig. 24.The exciting transducer is installed at point t1, as shown in Fig. 22.The receiving transducer r1 is 6m away from the exciting transducer, the first earthing hole h1 is 18.8m away from the exciting transducer, and the second grounding hole h2 is 119.13maway from the exciting transducer.
The field signal collected by the oscilloscope is shown in Fig. 25.The echo signals of defects h1 and h2 can be observed.
To verify the long-distance propagation performance of guided wave modes excited at the rail web, a 1km attenuation experiment was carried out on the Beijing Circular Railway (Supplementary Information).The transducer was installed at the rail web, and the excitation signal shown in Fig. 12 was applied along the longitudinal direction of the rail.
To verify the long-distance propagation performance of guided wave modes excited at the rail web, an attenuation experiment was conducted on the Beijing Circular Railway with a radius of 1.5 km.The transducer was installed at the rail web, and the excitation signal shown in Fig. 12 was applied along the longitudinal direction of the rail.
In the experiment, the amplitude of the excitation signal is ± 125V, and a receiving point is set at every 50m.We continued to use the excitation signal as shown in Fig. 12, exciting at a frequency of 35 kHz, and made efforts to ensure that the excitation point was the same as the selected excitation point in the simulation experiments.At each receiving point, three groups of data are measured, and the signal strength of the receiving point is determined by calculating the average peak energy value across the three groups of data.The data are shown in Table 5.Through experimental measurement, the amplitude of the guided wave signal can still reach more than 4.7mV after 1km propagation.
The data in Table 5 shows that the attenuation curve is fitted exponentially.The fitting effect is shown in Fig. 26, and the energy attenuation coefficient is calculated to be about 0.0078.The attenuation experiment shows that the mode selected in this study can receive an effective signal at 1km when the excitation signal amplitude is ± 125V, indicating that this mode can be used for long-distance rail flaw detection.
To explore the influence of environmental factors on the attenuation of guided wave signals, we conducted additional experiments on two different railway tracks.One experiment was conducted on a straight track, while the other was conducted on a track with smaller radius curves.In these experiments, we used higherprecision three-axis accelerometer sensors and increased the excitation voltage to ± 300V.After excitation, signals were collected at different distances.The experimental data on a straight track were shown in Table 6 and the attenuation curves of the second experiment were plotted as shown in Fig. 27.
From Table 6, it can be observed that on the straight track segment with good operating conditions, the guided wave modes excited by this method exhibit smaller attenuation.Guided wave signals can still be detected at a distance of 3 km.www.nature.com/scientificreports/In Fig. 27, we can see that on curved track segments, the attenuation of guided wave signals increases.This indicates that the method proposed in this study is still influenced by environmental factors such as curves.To address this issue, we can increase sensor density in curved sections according to the actual condition of the track.

Conclusion
In recent years, ultrasonic guided wave (UGW) has become an essential tool for non-destructive testing of railways.Aiming at the application of UGW in long-distance rail breakage monitoring of U78CrV rail, the main contributions of this paper are as follows.

Figure 2 .
Figure 2. Dispersion curve of phase velocity in U78CrV rail.

Figure 4 .
Figure 4. Guided wave modes in rails at 35 kHz.

Figure 5 .
Figure 5. Vibration energy distribution of 23 guided wave modes.

Figure 6 .
Figure 6.Vibration shapes diagram of the selected five modes.
. The left figure shows the position of the excitation point and the right figure shows the excitation signal.The deformation process of rail with time obtained by simulation is shown in Fig.13.

Figure 7 .
Figure 7. Phase velocity and group velocity dispersion curve of mode 9.

Figure 8 .
Figure 8.The node position where the transducer can be installed.

Figure 9 .
Figure 9. Vibration displacement of mode 9 at each node.

Figure 11 .
Figure 11.3D mesh generation effect of rail.

Figure 12 .
Figure 12.Excitation position (the left) and excitation signal (the right).

Figure 16 .
Figure 16.Comparison diagram of vibration shapes of section Ω and various modes at 0.000773 s.

Figure 17 .
Figure 17.Comparison diagram of vibration shape of section Ω and mode 9 at 0.000773 s.

Table 4 .Figure 19 .
Figure 19.Settings of rail cracks at different positions.

Figure 20 .
Figure 20.Comparison and differential analysis of reflected waves from defect and no defect at different positions.

Figure 21 .
Figure 21.Comparison and differential analysis of transmitted waves from defect and no defect at different positions.

Figure 26 .
Figure 26.Attenuation curve on the Beijing Circular Railway.

Table 1 .
Material parameters of U78CrV rail.

Table 2 .
Group speed of the guided wave.

Table 3 .
Difference value (D-value) between section Ω and the vibration shape of each mode at 0.000773 s.

Table 5 .
Data from the field experiments.

Table 6 .
Experimental data on a straight track.