The finite element simulation software ABAQUS/CAE was used for analysis, and the rail model CHN60AT was selected. The rail simulation model and cross section were shown in Fig. 1, and the three-dimensional rail model with a length of 1m was selected.

The material properties of the model are: density P = 7840kg/m3, Poisson ratio σ = 0.29, elastic modulus E = 210GPa.

The blue mark simulates the probe installed at the bottom of the rail, the excitation probe in the middle, the receiving probe on both sides, and the cut artificial rail bottom crack in the red frame,as shown in Fig. 2 and Fig. 3. A simulated crack is made at a distance of L1 = 423mm from the rail end face. The crack length is 150mm, the depth is 10mm, and the width is 1.5mm. The R-E spacing is changed by changing the length of LR − E while keeping L2 = 5mm unchanged.

According to the current inspection requirements of the railway department, the inspection depth required for the bottom crack of the rail is 10mm. Theoretically, the detected crack size should be ensured to be greater than half of the detection frequency wavelength [25], so the guided wave frequency of 200kHz is selected.

The simulation parameters of HWS excitation waveform are shown in Table 3,the Hanning window modulated signal (HWS) is used as the guided wave excitation signal. The initial interval of the R-E pitch is 40mm, and the step size is 20mm each time, until the R-E pitch is 140mm.

Table 3

Simulation parameters of HWS excitation waveform

No. | R − E interval (mm) | Excitation waveform | No. | R − E interval (mm) | Excitation waveform |

1 | 40 | HWS | 4 | 100 | HWS |

2 | 60 | HWS | 5 | 120 | HWS |

3 | 80 | HWS | 6 | 140 | HWS |

As shown in Fig. 4, Vpnd represents the non-damaged waveform, and Vpd represents the waveform after the crack. Cracks cause significant amplitude attenuation for the propagation of guided wave signals. With the gradual increase of the R-E distance, the propagation distance increases, and the amplitude of the received signal also decreases.

In order to describe the attenuation degree of guided wave signal in the propagation process, the attenuation coefficient α is used to represent it, and its formula is shown in Eq. (3) [25].

α = 20log10 (v0 /v1) (3)

Where, v0 represents the amplitude of the signal without attenuation, v1 represents the amplitude of the signal after attenuation. Here, Vpd is used to represent the peak value of guided wave signal passing through the rail bottom crack, and Vpnd is used to represent the peak value of guided wave signal not passing through the crack at the same distance.

αpd = 20log10(Vpnd/Vpd) (4)

Table 4

Vpnd and Vpd of HWS excitation waveform varying with distance

No. (S0 ) | Interval(mm) | Vpnd | Vpd | αpd (dB) |

1 | 40 | 2031.49 | 711.054 | 9.118 |

2 | 60 | 2030.34 | 719.075 | 9.016 |

3 | 80 | 2083.17 | 732.387 | 9.079 |

4 | 100 | 1509.10 | 656.726 | 7.227 |

5 | 120 | 1284.37 | 367.039 | 10.879 |

6 | 140 | 1107.31 | 440.769 | 8.001 |

As can be seen from Table 4, the average attenuation coefficient αpd of signal S0 without spread spectrum processing is about 8.89dB.

In addition, the increase of the propagation distance will also cause the attenuation of the guided wave signal. In order to describe the attenuation degree of guided wave propagation distance, the attenuation coefficient αL is expressed as:

αL = 20log10 \(\:\frac{\begin{array}{c}Vpnd(R-Emin)\\\:\end{array}}{Vpd(R-Emax)}\) (5)

Vpnd (R-Emax) indicates the Vpnd amplitude at a distance of 140mm, and Vpnd (R-Emin) indicates the Vpnd amplitude at a distance of 40mm.

The simulation results in Table 4 were analyzed, and the attenuation coefficient αL was calculated to be 5.271dB.

The greater the attenuation coefficient αL, the more serious the attenuation, that is, the more serious the influence of distance attenuation on the guided wave signal.

Barker code is used as the spread spectrum code to process the HWS excitation mode, and the processed Signal is used as the Direct Sequence spread spectrum signal (DSS). The simulation parameters are the same as Table 3 except for the excitation signal.

The received signal is de-amplified and related processing is carried out, and the processed signal is denoted as S1. The processed signal is shown in Fig. 5. After the de-amplification process, an obvious correlation peak is obtained, which is the information code of "1" in the excitation signal. After the signal passes through the crack, the crack reflects and scatters the guided wave signal to a certain extent, resulting in obvious distortion and energy attenuation of the guided wave signal. Therefore, the correlation between the signal after deamplification and the original excitation signal is weakened, and the information code "1" in the excitation signal cannot be fully recovered, and there is no high correlation peak in the signal. It is in sharp contrast to the direct wave processing signal without crack.

As can be seen from Fig. 6, when the distance is 100mm, Vpnd attenuates significantly, but there is still a large difference between Vpd and Vpnd, which reduces the influence caused by the increase of the propagation distance and further improves the detection range of cracks in the rail.

Calculated from Table 5, the mean attenuation coefficient αpd of signal S1 is 17.198dB, while the mean attenuation coefficient αpd of signal S0 without spread spectrum processing is 8.887dB, which increases the mean value of αpd by 8.311dB. The larger the attenuation coefficient αpd is, the more serious the amplitude attenuation of the signal after the crack, indicating that the crack in the bottom of the rail is more easily detected after the DSS signal is processed by spread spectrum.

As can be seen from Table 5, the calculated amplitudes of Vpnd at 40mm spacing and 140mm spacing show that the guided wave attenuation coefficient αL of this frequency is 4.977dB, which is 0.574dB lower than the signal attenuation coefficient αL = 5.271dB without spread spectrum processing.

Table 5

Vpnd and Vpd of DSS excitation waveform varying with distance

No. (S1 ) | Interval(mm) | Vpnd | Vpd | αpd (dB) |

1 | 40 | 54.985 | 8.811 | 15.904 |

2 | 60 | 54.635 | 5.791 | 19.494 |

3 | 80 | 53.273 | 5.978 | 18.999 |

4 | 100 | 32.003 | 5.781 | 14.864 |

5 | 120 | 31.891 | 3.769 | 18.549 |

6 | 140 | 31.003 | 5.278 | 15.379 |

The simulation results show that the spread spectrum technique can reduce the influence of distance attenuation on crack identification. At the same time, the amplitude of the signal decreases more after the crack, which increases the sensitivity of crack detection.