# Train Wheel Condition Monitoring via Cepstral Analysis of Axle Box Accelerations

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## Abstract

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## Featured Application

**Online wheel condition monitoring for condition based and predictive maintenance.**

## Abstract

## 1. Introduction

## 2. Cepstral Analysis and Navewumber Domain

- The power spectrum ${\left|Y\left(k\right)\right|}^{2}$ of the function$$y\left(x\right)=s\left(x\right)\ast r\left(x\right),$$$${\left|Y\left(k\right)\right|}^{2}={\left|S\left(k\right)\right|}^{2}\xb7{\left|R\left(k\right)\right|}^{2},$$
- The logarithm of the power spectrum is taken:$$\mathrm{log}\left({\left|Y\left(k\right)\right|}^{2}\right)=\mathrm{log}\left({\left|S\left(k\right)\right|}^{2}\right)+\mathrm{log}\left({\left|R\left(k\right)\right|}^{2}\right).$$Thus, the components are coupled by addition.
- Applying the inverse Fourier transform to the logarithmic power spectrum $\mathrm{log}\left({\left|Y\left(k\right)\right|}^{2}\right)$ and finally squaring the results yield the cepstrum of $y\left(x\right)$:$${C}_{y}\left(\tilde{x}\right)={C}_{s}\left(\tilde{x}\right)+{C}_{r}\left(\tilde{x}\right)+\text{cross-product}\mathrm{term}.$$

## 3. Synthetic Data Analysis

#### 3.1. Synthetic Models

#### 3.2. Cepstrum Analysis of Synthetic Data

## 4. Experimental Data

#### 4.1. Data Acquisition

#### 4.2. Speed Estimation and Data Pre-Processing

#### 4.3. Cepstrum Analysis of Experimental Data

## 5. Discussion

#### 5.1. Wheel Condition Monitoring with ABA Sensors

#### 5.2. Navewumber Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Krummenacher, G.; Ong, C.S.; Koller, S.; Kobayashi, S.; Buhmann, J.M. Wheel Defect Detection with Machine Learning. IEEE Trans. Intell. Transport. Syst.
**2018**, 19, 1176–1187. [Google Scholar] [CrossRef] - Nielsen, J.C.O.; Johansson, A. Out-of-round railway wheels—A literature survey. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2000**, 214, 79–91. [Google Scholar] [CrossRef] - Bian, J.; Gu, Y.; Murray, M.H. A dynamic wheel-rail impact analysis of railway track under wheel flat by finite element analysis. Veh. Syst. Dyn.
**2013**, 51, 784–797. [Google Scholar] [CrossRef] [Green Version] - Bogacz, R.; Frischmuth, K. On dynamic effects of wheel-rail interaction in the case of Polygonalisation. Mech. Syst. Signal Process.
**2016**, 79, 166–173. [Google Scholar] [CrossRef] - Telliskivi, T.; Olofsson, U. Wheel-rail wear simulation. Wear
**2004**, 257, 1145–1153. [Google Scholar] [CrossRef] - Casanueva, C.; Enblom, R.; Stichel, S.; Berg, M. On integrated wheel and track damage prediction using vehicle–track dynamic simulations. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2017**, 231, 775–785. [Google Scholar] [CrossRef] - Mosleh, A.; Montenegro, P.; Alves Costa, P.; Calçada, R. An approach for wheel flat detection of railway train wheels using envelope spectrum analysis. Struct. Infrastruct. Eng.
**2020**, 202, 1–20. [Google Scholar] [CrossRef] - Molodova, M.; Li, Z.; Dollevoet, R. Axle box acceleration: Measurement and simulation for detection of short track defects. Wear
**2011**, 271, 349–356. [Google Scholar] [CrossRef] - Mori, H.; Sato, Y.; Ohno, H.; Tsunashima, H.; Saito, Y. Development of Compact Size Onboard Device for Condition Monitoring of Railway Tracks. J. Mech. Syst. Transp. Logist.
**2013**, 6, 142–149. [Google Scholar] [CrossRef] [Green Version] - Lederman, G.; Chen, S.; Garrett, J.; Kovačević, J.; Noh, H.Y.; Bielak, J. Track-monitoring from the dynamic response of an operational train. Mech. Syst. Signal Process.
**2017**, 87, 1–16. [Google Scholar] [CrossRef] [Green Version] - Baasch, B.; Roth, M.; Groos, J. In-service condition monitoring of rail tracks: On an on-board low-cost multi-sensor system for condition based maintenance of railway tracks. Int. Verk.
**2018**, 70, 76–79. [Google Scholar] - Niebling, J.; Baasch, B.; Kruspe, A. Analysis of Railway Track Irregularities with Convolutional Autoencoders and Clustering Algorithms. In Dependable Computing—EDCC 2020 Workshops; Bernardi, S., Vittorini, V., Flammini, F., Nardone, R., Marrone, S., Adler, R., Schneider, D., Schleiß, P., Nostro, N., Løvenstein Olsen, R., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 78–89. [Google Scholar]
- Li, C.; Luo, S.; Cole, C.; Spiryagin, M. An overview: Modern techniques for railway vehicle on-board health monitoring systems. Veh. Syst. Dyn.
**2017**, 55, 1045–1070. [Google Scholar] [CrossRef] - Bosso, N.; Gugliotta, A.; Zampieri, N. Wheel flat detection algorithm for onboard diagnostic. Measurement
**2018**, 123, 193–202. [Google Scholar] [CrossRef] - Heirich, O.; Steingass, A.; Lehner, A.; Strang, T. Velocity and location information from onboard vibration measurements of rail vehicles. In Proceedings of the 16th International Conference on Information Fusion, Istanbul, Turkey, 9–12 July 2013; pp. 1835–1840. [Google Scholar]
- Jia, S.; Dhanasekar, M. Detection of Rail Wheel Flats using Wavelet Approaches. Struct. Health Monit.
**2016**, 6, 121–131. [Google Scholar] [CrossRef] - Ye, Y.; Shi, D.; Krause, P.; Hecht, M. A data-driven method for estimating wheel flat length. Veh. Syst. Dyn.
**2019**, 213, 1–19. [Google Scholar] [CrossRef] - Bai, Y.; Yang, J.; Wang, J.; Li, Q. Intelligent Diagnosis for Railway Wheel Flat Using Frequency-Domain Gramian Angular Field and Transfer Learning Network. IEEE Access
**2020**, 8, 105118–105126. [Google Scholar] [CrossRef] - Childers, D.G.; Skinner, D.P.; Kemerait, R.C. The cepstrum: A guide to processing. Proc. IEEE
**1977**, 65, 1428–1443. [Google Scholar] [CrossRef] - Oppenheim, A.V.; Schafer, R.W. From frequency to quefrency: A history of the cepstrum. IEEE Signal Process. Mag.
**2004**, 21, 95–106. [Google Scholar] [CrossRef] - Bogert, B.P.; Healy, M.J.; Tukey, J.W. The quefrency alanysis of time series for echoes: Cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking. In Proceedings of the Symposium on Time Series Analysis, Brown-University, Providence, RI, USA, 11–14 June 1962; John Wiley & Sons: New York, NY, USA; London, UK, 1963; pp. 209–243. [Google Scholar]
- Bracciali, A.; Cascini, G. Detection of corrugation and wheelflats of railway wheels using energy and cepstrum analysis of rail acceleration. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**1997**, 211, 109–116. [Google Scholar] [CrossRef] - Teunissen, P.J.; Montenbruck, O. (Eds.) Springer Handbook of Global Navigation Satellite Systems; Springer International Publishing: Cham, Switzerland, 2017. [Google Scholar]
- Groves, P.D. Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd ed.; Artech House: Boston, MA, USA, 2013. [Google Scholar]
- Gustafsson, F. Statistical Sensor Fusion; Studentlitteratur: Lund, Sweden, 2010. [Google Scholar]
- Buttkus, B. Homomorphic Filtering—Theory and Practice. Geophys. Prospect.
**1975**, 23, 712–748. [Google Scholar] [CrossRef] - Ulrych, T.J. Application of homomorphic deconvolution to seismology. Geophysics
**1971**, 36, 650–660. [Google Scholar] [CrossRef] - Wu, T.; Thompson, D. Theoretical Investigation of Wheel/Rail Non-Linear Interaction due to Roughness Excitation. Veh. Syst. Dyn.
**2000**, 34, 261–282. [Google Scholar] [CrossRef]

**Figure 1.**Different synthetic data models according to Table 1.

**Figure 3.**(

**a**) Measurement train; (

**b**) Multi-sensor box inside the wagon; (

**c**) Accelerometer mounted at the axle box; (

**d**) Global navigation satellite system (GNSS) antenna (yellow box).

**Figure 5.**Axle-box accelerations (ABA) data (solid blue line) and speed data (dashed red line) in the time domain.

**Figure 7.**Cepstrum analysis of 40-m-long track segments; (

**a**) position of cepstrum peaks at navewumbers between two and four meters and (

**b**) corresponding peak amplitude.

Model Number | Wheel Flat Length in mm | Wheel Flat Depth in mm | Wheel Roughness Std in mm | Track Roughness Std in mm |
---|---|---|---|---|

1 | 50 | 0.3 | - | 0.01 |

2 | 50 | 0.6 | - | 0.01 |

3 | 100 | 0.3 | - | 0.01 |

4 | 50 | 0.3 | 0.01 | 0.01 |

5 | 50 | 0.3 | 0.05 | 0.01 |

6 | 50 | 0.3 | 0.01 | 0.002 |

Window Length in m | Number of Windows | Median Peak Position in m | Percentage of “Correctly” Depicted Peaks | Mean Amplitude of “Correctly” Depicted Peaks |
---|---|---|---|---|

10 | 2222 | 2.99 | 56.57 | 0.28 |

20 | 1112 | 2.99 | 79.77 | 0.52 |

40 | 557 | 2.99 | 86.71 | 0.53 |

80 | 279 | 2.99 | 86.02 | 0.52 |

160 | 140 | 2.99 | 89.29 | 0.51 |

320 | 71 | 2.99 | 84.51 | 0.56 |

640 | 36 | 2.99 | 88.89 | 0.52 |

1280 | 19 | 2.99 | 84.21 | 0.54 |

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**MDPI and ACS Style**

Baasch, B.; Heusel, J.; Roth, M.; Neumann, T.
Train Wheel Condition Monitoring via Cepstral Analysis of Axle Box Accelerations. *Appl. Sci.* **2021**, *11*, 1432.
https://doi.org/10.3390/app11041432

**AMA Style**

Baasch B, Heusel J, Roth M, Neumann T.
Train Wheel Condition Monitoring via Cepstral Analysis of Axle Box Accelerations. *Applied Sciences*. 2021; 11(4):1432.
https://doi.org/10.3390/app11041432

**Chicago/Turabian Style**

Baasch, Benjamin, Judith Heusel, Michael Roth, and Thorsten Neumann.
2021. "Train Wheel Condition Monitoring via Cepstral Analysis of Axle Box Accelerations" *Applied Sciences* 11, no. 4: 1432.
https://doi.org/10.3390/app11041432