# Feasibility of Applying Mel-Frequency Cepstral Coefficients in a Drive-by Damage Detection Methodology for High-Speed Railway Bridges

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## Abstract

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## 1. Introduction

^{®}[64]. Different damage scenarios are simulated, as well as external excitations such as measurement noises and different levels of track irregularities. Finally, the key results are summarized in the last section, and some prospects are highlighted. The results presented in this study should be seen as a first attempt to link cepstrum-based features in an HSR drive-by damage detection approach.

## 2. Methodology

#### 2.1. Vehicle-Track-Bridge Dynamic Interaction System

#### 2.1.1. Vehicle Model

#### 2.1.2. Track Model

#### 2.1.3. Bridge Model

#### 2.1.4. Equations of Motion for the VTBI System

#### 2.2. Irregularity of Track Vertical Profile

#### 2.3. Drive-by Damage Detection Using MFCC

#### 2.3.1. Vehicle’s Acceleration Segmentation Short-Term Spectra

#### 2.3.2. Adapted Mel-Frequency Scale and Frequency Warping

#### 2.3.3. Mel-Frequency Cepstral Coefficients (MFCC)

#### 2.3.4. Damage Index

## 3. Numerical Simulations: Results and Discussion

^{®}[64], and the feasibility of using damage features extracted from MFCC in indirect SHM of HSR infrastructure is numerically evaluated in this section for two different levels of track conditions: low disturbance (LD) and high disturbance (HD) (as described in Section 2.2), in addition to the case where no irregularity is considered (NC), for train speeds of $144\mathrm{k}\mathrm{m}/\mathrm{h}$ and $288\mathrm{k}\mathrm{m}/\mathrm{h}$. To analyze the sensitivity of the method regarding the intensity of the damage, five damage scenarios were simulated on the bridge, assuming bending stiffness reductions of ${f}_{sr}=5\%,10\%,15\%,20\%$ and 25% in a region of the Euler-Bernoulli beam with $1.2\mathrm{m}$ length (The bridge span is $L=45\mathrm{m}$). On the other hand, to verify the sensitivity of MFFC-based DI as to the damage location, these damage scenarios were considered for different cases, in which these regions were located around the cross-section situated at the midspan and at 1/4 of the span from the left support.

#### 3.1. Dynamic Analysis

#### 3.1.1. Bridge Response

#### 3.1.2. Vehicle Response

- As shown in Figure 9, different damage conditions are not visibly distinguished by the bogie responses. However, due to the reduction provided by the secondary suspension system in both the frequencies and the amplitude of the carbody’s responses relative to those of the bogie, different damage conditions were picked up by the car body’s vertical accelerations, albeit subtly. For this reason, the methodology proposed in this paper was applied, concerning the carbody accelerations.
- Figure 8a,b shows that the differences between the responses to damage severity become less apparent with increasing speed. This aspect represents the main challenge to be overcome in developing vibration-based drive-by damage detection methodologies for indirect SHM of HSR bridges.

#### 3.2. Performance of the Proposed Drive-by Damage Detection Using MFCC

- The output dataset corresponding to the time-histories of the carbody’s vertical acceleration recorded at each case were corrupted by 5% white noise, to simulate the effect of measurement disturbances.
- No bandpass filters were used in the signal processing, since this procedure is already covered in the frequency warping stage, with the filter bank application.
- The filter bank was configured with 50 triangular filters, whose corresponding 50 frequency values in Hertz, equally spaced on a Mel-frequency scale, had a cut-off frequency of 60 Hz.
- For the segmentation/windowing of the carbody’s signal recorded, the STFT was set up with a window function length of ${2}^{11}$ samples (22 frames) to guarantee enough segments for damage identification on the bridge. Moreover, as the window functions taper off at the edges to avoid spectral ringing, the corresponding overlapping used was 34/64 of the windowing length.

#### Damage Detection Performance

## 4. Conclusions and Prospects

- The DI values acquired from MFCC offer robust information regarding the location of the damage. In all circumstances, the DI values reliably reflect the location and severity of the damage.
- The amplitude of the feature extracted from MFCC correlates with the vehicle’s speed over the bridge and the degree of track irregularities. The DI values are more sensitive to excitation sources caused by track irregularities than those induced by damage.
- The results indicate that the distribution patterns of DI differ concerning different damage locations, with a tendency to exhibit peaks near these damaged zones.
- In this paper, no attempt has been made to apply any dimensionality reduction techniques to the extracted features, such as principal component analysis, in which a linear mapping of the MFCC can be performed to extract the components more likely to be influenced by structural properties than by other excitation sources. Neither have more sophisticated techniques such as deep learning or time series analysis been applied, to compare the statistical distributions of the MFCC and extract damage location from them. These prospects will be investigated in the future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**2-D vehicle-track-bridge dynamic interaction system: (

**a**) vehicle model, (

**b**) track model, and (

**c**) bridge model.

**Figure 4.**Frequency warping: (

**a**) Mel-frequency mapping using Equation (5), and (

**b**) correspondent triangular filter bank.

**Figure 5.**Idealized cross-section of the bridge numerical model (Adapted from [77]).

**Figure 6.**Time-histories of the bridge vertical displacements at the midspan for different track and damage conditions; and for different operating speeds: (

**a**) $v=144\mathrm{km}/\mathrm{h}$ and (

**b**) $v=288\mathrm{km}/\mathrm{h}$. (The y-axis was reversed, according to the positive direction adopted in the VBI modeling).

**Figure 7.**Time-histories of the bridge vertical accelerations at the midspan for different track and damage conditions, and for different operating speeds: (

**a**) $v=144\mathrm{km}/\mathrm{h}$ and (

**b**) $v=288\mathrm{km}/\mathrm{h}$. (The y-axis was reversed, according to the positive direction adopted in the VBI modeling).

**Figure 8.**Time-histories of the carbody vertical accelerations for different track and damage conditions, and for different operating speeds: (

**a**) $v=144\mathrm{km}/\mathrm{h}$ and (

**b**) $v=288\mathrm{km}/\mathrm{h}$. (The y-axis was reversed, according to the positive direction adopted in the VBI modeling).

**Figure 9.**Time-histories of the front bogie vertical accelerations for different track and damage conditions, and for different operating speeds: (

**a**) $v=144\mathrm{km}/\mathrm{h}$ and (

**b**) $v=288\mathrm{km}/\mathrm{h}$. (The y-axis was reversed, according to the positive direction adopted in the VBI modeling).

**Figure 10.**DI values under different damage conditions at 1/4 of the span, for the vehicle running at $v=144\mathrm{km}/\mathrm{h}$, and under different levels of track irregularities: (

**a**) NC, (

**b**) LD, and (

**c**) HD.

**Figure 11.**DI values under different damage conditions at 1/4 of the span, for the vehicle running at $v=288\mathrm{km}/\mathrm{h}$, and under different levels of track irregularities: (

**a**) NC, (

**b**) LD, and (

**c**) HD.

**Figure 12.**DI values under different damage conditions at the midspan, for the vehicle running at $v=144\mathrm{km}/\mathrm{h}$, and under different levels of track irregularities: (

**a**) NC, (

**b**) LD, and (

**c**) HD.

**Figure 13.**DI values under different damage conditions at the midspan, for the vehicle running at $v=288\mathrm{km}/\mathrm{h}$, and under different levels of track irregularities: (

**a**) NC, (

**b**) LD, and (

**c**) HD.

**Figure 14.**DI values under different damage conditions at 1/4 of the span and at the midspan, for the vehicle running at $v=144\mathrm{km}/\mathrm{h}$, and under different levels of track irregularities: (

**a**) NC, (

**b**) LD, and (

**c**) HD.

**Table 1.**Roughness coefficients and cut-off frequencies for German track irregularity PSDs [72].

Track Class | ${\mathbf{\Omega}}_{\mathit{c}}(\mathbf{rad}/\mathbf{m})$ | ${\mathbf{\Omega}}_{\mathit{r}}(\mathbf{rad}/\mathbf{m})$ | ${\mathit{A}}_{\mathit{V}}$$({\mathbf{m}}^{2}\times \mathbf{rad}/\mathbf{m})$ |
---|---|---|---|

Low disturbance | 0.8246 | 0.0206 | 4.032 × 10^{−7} |

High disturbance | 0.8246 | 0.0206 | 1.080 × 10^{−6} |

Properties | Notation | Value | Unit | |
---|---|---|---|---|

Deck | Reinforced concrete density | ${\rho}_{c}$ | 2500 | kg/m^{3} |

Reinforced concrete area | ${A}_{c}$ | 6.0191 | m^{2} | |

Ballast [72] | Ballast density | ${\rho}_{b}$ | 1750 | kg/m^{3} |

Ballast area | ${A}_{b}$ | 3.6315 | m^{2} | |

Sleeper [72] | Sleeper mass (half) | ${m}_{s}$ | 170 | kg |

Sleeper spacing (half) | ${l}_{s}$ | 0.60 | m | |

Rail [72] | Rail mass per unit length (per rail seat) | ${\overline{m}}_{r}$ | 60.64 | kg/m |

Euler-Bernoulli beam: | ||||

Mass per unit length | $\overline{m}$ | 13,350 | kg/m | |

Modulus of elasticity | $E$ | 39 | GPa | |

Cross-section moment of inertia | $I$ | 2.60 | m^{4} | |

Span length | $L$ | 45 | m |

**Table 3.**Mechanical, geometrical, and suspension properties of the 2D vehicle model (ICE3 Velaro) [69].

Properties | Notation | Value | Unit | |
---|---|---|---|---|

Concentrated masses | Carbody | ${m}_{c}$ | 47,800 | kg |

Bogies | ${m}_{b}$ | 3500 | kg | |

Wheelsets | ${m}_{w}$ | 1800 | kg | |

Rotary inertia | Carbody | ${J}_{c}$ | 1.96 × 10^{6} | kg·m^{2} |

Bogies | ${J}_{b}$ | 1715 | kg·m^{2} | |

Stiffness coefficient ^{(}*^{)} | Primary suspension | ${k}_{1y}$ | 2.40 × 10^{6} | N/m |

Secondary suspension | ${c}_{1y}$ | 2.00 × 10^{4} | N·s/m | |

Damping coefficient ^{(}*^{)} | Primary suspension | ${k}_{2y}$ | 7.00 × 10^{5} | N/m |

Secondary suspension | ${c}_{2y}$ | 4.00 × 10^{4} | N·s/m | |

Distances | ||||

Distance between bogie centers in one car | ${L}_{c}$ | 17.375 | m | |

Distance between wheelset axles in one bogie | ${L}_{b}$ | 2.50 | m | |

Distance between bogie centers from adjacent cars | ${L}_{ac}$ | 7.40 | m |

^{(}*

^{)}For the 2-D model, the stiffness and damping coefficients of each spring-damper set in the suspension systems correspond to the equivalent constants of a parallel arrangement of the axle boxes at both sides of each axle. Thus, the assumed values were taken from the reference values (given per axle box) multiplied by two for these properties.

Vertical bending mode | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th |

Frequency (Hz) | 2.14 | 8.55 | 19.24 | 34.21 | 53.45 | 76.96 | 104.75 | 136.82 |

Mode | Frequency (Hz) | ||
---|---|---|---|

Carbody | Front Bogie | Rear Bogie | |

Vertical (bouncing) | 0.804 | 6.317 | 6.323 |

Pitching | 1.089 | 10.525 | 10.525 |

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## Share and Cite

**MDPI and ACS Style**

de Souza, E.F.; Bittencourt, T.N.; Ribeiro, D.; Carvalho, H.
Feasibility of Applying Mel-Frequency Cepstral Coefficients in a Drive-by Damage Detection Methodology for High-Speed Railway Bridges. *Sustainability* **2022**, *14*, 13290.
https://doi.org/10.3390/su142013290

**AMA Style**

de Souza EF, Bittencourt TN, Ribeiro D, Carvalho H.
Feasibility of Applying Mel-Frequency Cepstral Coefficients in a Drive-by Damage Detection Methodology for High-Speed Railway Bridges. *Sustainability*. 2022; 14(20):13290.
https://doi.org/10.3390/su142013290

**Chicago/Turabian Style**

de Souza, Edson Florentino, Túlio Nogueira Bittencourt, Diogo Ribeiro, and Hermes Carvalho.
2022. "Feasibility of Applying Mel-Frequency Cepstral Coefficients in a Drive-by Damage Detection Methodology for High-Speed Railway Bridges" *Sustainability* 14, no. 20: 13290.
https://doi.org/10.3390/su142013290