#
Definition of Damage Indices for Railway Axle Bearings: Results of Long-Lasting Tests^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Tests on Railway Axle Bearings

#### 2.1. Experimental Data Measurements

#### 2.2. Test Cycles and Operating Conditions

## 3. Application of Improved Envelope-Based Algorithms for Bearing Diagnostics and for the Definition of Fault Indices

#### 3.1. Basics of Envelope Analysis and of Squared Envelope Spectrum

- 1.
- The residual signal $x(t)$ is filtered by means of a band-pass filter. Actually, this is the most critical part of the “envelope analysis” algorithm, because, generally, the signal is influenced by many modulations, caused by other components of the mechanical system and by environmental or mechanical noise, at low and medium frequencies. Thus, the signal has to filtered in the neighborhood of the frequency of the carrier ${f}_{c}$, which is nearby on of the resonance frequency of the system:$$x(t)\stackrel{\Im}{\to}X\left(f\right)\stackrel{Filtering}{\to}{X}^{F}\left(f\right)\stackrel{{\Im}^{-1}}{\to}{x}^{F}(t)$$
- 2.
- Then, the “analytic signal” ${x}_{a}^{F}(t)$ of the filtered signal is calculated. The analytic signal is defined as a complex-valued function with no negative frequency components. The real part of the analytic signal is given by the filtered signal ${x}^{F}(t)$, while the imaginary part is the Hilbert transform ${\tilde{x}}^{F}(t)$ of the filtered signal:$${x}_{a}^{F}(t)={x}^{F}(t)+\mathrm{i}\xb7{\tilde{x}}^{F}(t)$$
- 3.
- The “envelope” of the filtered signal $\mathrm{env}\left[{x}^{F}(t)\right]$ is defined as the absolute value of the analytic signal:$$\mathrm{env}\left[{x}^{F}(t)\right]=\left|{x}_{a}^{F}(t)\right|=\sqrt{{\left({x}^{F}(t)\right)}^{2}+{\left({\tilde{x}}^{F}(t)\right)}^{2}}$$

- 4.
- The “envelope spectrum” (ES) is obtained by applying the Fourier transform to the square of the envelope:$$\mathrm{ES}=\Im \left[{\mathrm{env}}_{2}\left[{x}^{F}(t)\right]\right]=\Im \left[{\left|{x}_{a}^{F}(t)\right|}^{2}\right]$$

#### 3.2. Definition of the Indices for the Selection of the Optimal Frequency Band for Signal Filtering and for the Monitoring of the Fault Severity

- 5.
- Root mean square:$$\mathrm{RMS}\left(x\right)=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}}$$
- 6.
- Kurtosis:$$\kappa \left(x\right)=\frac{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{4}}}{{\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}\right)}^{2}}$$
- 7.
- Band kurtosis:$${\kappa}_{l,h}=\frac{{\displaystyle \sum _{k=0}^{h-l}{\mathrm{SES}}_{l,h}[k]}}{{\mathrm{SES}}_{l,h}[0]}$$
- 8.
- Kurtosis of the SES:$${\kappa}_{p,q}\left({\mathrm{SES}}_{l,h}\right)$$
- 9.
- Ratio of Content Component (RCC), associated to the cyclic band $q,p$:$${\mathrm{RCC}}_{l,h}^{p,q}=\frac{{\displaystyle \sum _{k=p}^{q}{\mathrm{SES}}_{l,h}[k]}}{{\displaystyle \sum _{k=0}^{h-l}{\mathrm{SES}}_{l,h}[k]}}$$
- 10.
- The range l,h is assumed equal to 10% of the fault frequency. SES peak at the damaged frequency:$$\mathrm{SES}\left({f}_{fault}\right)$$
- 11.
- Normalized SES peak at the damaged frequency:$$\frac{\mathrm{SES}\left({f}_{fault}\right)}{\mathrm{SES}\left(f=0\right)}$$
- 12.
- STR index (SES to Threshold Ratio):$$\mathrm{STR}[{f}_{fault}]=\frac{\mathrm{SES}[{f}_{fault}]}{{\mathrm{TH}}_{SES}[{f}_{fault}]}$$

- 13.
- Continuous phase status (CPS):$$\mathrm{CPS}\left({\mathrm{env}}_{0}({x}^{F})\right)=\frac{dp}{2\pi \mathsf{\Omega}},dp=\frac{angle{(x}_{a}{(\mathrm{env}}_{0}\left)\right)}{dt},{\mathrm{env}}_{0}={x}_{a}\left(\left|{x}_{a}(x)\right|-\mathrm{mean}\left(\left|{x}_{a}(x)\right|\right)\right)$$
- 14.
- Spectral entropy (SE):$$\mathrm{SE}=\mathrm{Re}\left[\frac{-{\displaystyle \sum _{i=1}^{n}\left({p}_{i}\xb7{\mathrm{log}}_{2}\left({p}_{i}\right)\right)}}{{\mathrm{log}}_{2}n}\right],{p}_{i}=\frac{{X}^{F}(i)}{{\displaystyle \sum _{j=1}^{n}{X}^{F}(j)}},{X}^{F}=\Im \left[{x}^{F}\right]$$

## 4. Results of the Long-Lasting Tests

#### 4.1. Definition of the Optimal Frequency Band of the Filter for the Left Bearing

#### 4.2. Fault Detection and Localization for the Left Bearing

- the RMS of the raw signal shows a drop after about 1/8 of the total bearing cycles, then it is almost constant for the main bearing life and increases at the end of acquisitions;
- the increase of RMS at the end of acquisition is also highlighted by the decrease of the kurtosis;
- the spectral entropy SE continuously decreases with time, indicating that the energy is more concentrated at only a few frequencies (fault frequencies) as happens for a damaged bearing;
- the continuous phase status CPS continuously increases with time;
- all the indices that depend on the fault frequency (BPFO) increase in Figure 8.

#### 4.3. Definition of the Optimal Frequency Band of the Filter for the Right Bearing

#### 4.4. Fault Detection and Localization for the Right Bearing

## 5. Conclusions

- increasing the sampling frequency up to 25 kHz in order to include higher frequency resonances;
- increasing the time window of acquisition in order to improve the frequency resolution of the spectra;
- including the axial load information (amplitude and direction), in order to pre-process and classify the data, to compare records acquired in similar conditions. In this way, it is expected to produce more clear trends of the indices of the damage.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Outline of the main components of a test-rig compliant with EN 12082:2017 standard. A couple of axle bearings are simultaneously tested (TB, in yellow) and are installed on the extremities of a stub shaft, under vertical and axial loading.

**Figure 2.**Test cycles. From the top: rotational speed, radial load, and axial load (adapted from EN 12082:2017).

**Figure 3.**Definition of the optimal frequency band for filtering the residual signal according to the RCC index—left TB.

**Figure 4.**Definition of the optimal frequency band for filtering the residual signal according to the SES index—left TB.

**Figure 5.**Values of the normalized SES for the different data sets—left TB. The axis on the left of the base plane is relative to the vibration records sorted in chronological order.

**Figure 6.**Normalized SES map, i.e., top view of Figure 5—left TB.

**Figure 10.**Definition of the optimal frequency band for filtering the residual signal according to the RCC index—right TB.

**Figure 11.**Definition of the optimal frequency band for filtering the residual signal according to the SES index—right TB.

**Figure 12.**Values of the normalized SES for the different data sets, first frequency band—right TB. The axis on the left of the base plane is relative to the vibration records sorted in chronological order.

**Figure 13.**Normalized SES map, i.e., top view of Figure 12, first frequency band—right TB.

**Table 1.**Fault characteristic frequencies of test and support bearings at 1371 rpm (rated rotational speed).

Fault | Fault Type | Test Bearings—TB | Support Bearings—SB |
---|---|---|---|

BPFI | Fault on the inner race | 12.710 (Hz) | 5.0577 (Hz) |

BPFO | Fault on the outer race | 10.280 (Hz) | 7.9423 (Hz) |

BSF | Fault on the rolling element | 4.634 (Hz) | 2.1424 (Hz) |

FTF | Fault on the cage | 0.4475 (Hz) | 0.3888 (Hz) |

**Table 2.**List of symbols used in Figure 2 (adapted from EN 12082:2017).

Symbol | Unit | Description |
---|---|---|

F_{a} | (N) | axial test force |

F_{an} | (N) | nominal axial test force |

F_{r} | (N) | radial test force |

n | (rpm) | rotational test speed |

n_{test} | (rpm) | nominal rotational test speed |

t_{1} | (s) | time of one test cycle |

t_{2} | (s) | time of one elementary trip |

t_{3} | (s) | ramp up or ramp down time from n = 0 → n = n_{test} or n = n_{test} → n = 0 during one elementary trip |

t_{4} | (s) | time at rotational speed ntest during one elementary trip |

t_{5} | (s) | stop time (n = 0) |

t_{6} | (s) | time of one half load cycle of the alternating axial test force |

t_{7} | (s) | time during which axial test force is applied (including ramp up and ramp down) within the period t6 |

t_{8} | (s) | ramp up or ramp down time from F_{a} = 0 → F_{a} = F_{an} or F_{a} = F_{an} → F_{a} = 0 during one half load cycle of the alternating axial test force |

t_{9} | (s) | axial test force recovery time |

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**MDPI and ACS Style**

Pennacchi, P.; Chatterton, S.; Vania, A.; Massocchi, D.
Definition of Damage Indices for Railway Axle Bearings: Results of Long-Lasting Tests. *Machines* **2021**, *9*, 12.
https://doi.org/10.3390/machines9010012

**AMA Style**

Pennacchi P, Chatterton S, Vania A, Massocchi D.
Definition of Damage Indices for Railway Axle Bearings: Results of Long-Lasting Tests. *Machines*. 2021; 9(1):12.
https://doi.org/10.3390/machines9010012

**Chicago/Turabian Style**

Pennacchi, Paolo, Steven Chatterton, Andrea Vania, and Davide Massocchi.
2021. "Definition of Damage Indices for Railway Axle Bearings: Results of Long-Lasting Tests" *Machines* 9, no. 1: 12.
https://doi.org/10.3390/machines9010012