# Rational Resampling Ratio as Enhancement to Shaft Imbalance Detection

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Resampling

#### 2.2. Resampling Ratio

#### 2.3. Narrowband Component Extraction and Calculation Techniques

## 3. Experiment Description

#### 3.1. Mechanical Rig

#### 3.2. State of Shaft Imbalance

#### 3.3. Data Acquisition

^{®}MTN general-purpose vibration sensor was mounted at the top of the housing of the rolling-element bearing closest to the disc, where imbalance was introduced. Vibration signals were recorded using a 24-bit AVM4000

^{®}data collection unit. Analyzed data consisted of 235 signals, each of 10 s length, with a sampling rate equal to 25 kHz (AVM4000 supports 25, 50, and 100 kHz). The total number of points analyzed was equal to 235 [signals] × 10 [seconds] × 25,000 [samples/second] = 58,750,000 data points with double precision. Individual signals were collected periodically, ca. every minute.

#### 3.4. Characteristic Imbalance Component

## 4. Experiment Results

#### 4.1. Trend Analysis

- PP: peak-to-peak value, calculated in time domain;
- RMS: root mean square value, calculated in time domain;
- VRMS: velocity RMS, calculated in time domain, ISO 20816 (10–1000 Hz);
- Output_Shaft_x1: output shaft component in frequency spectrum;
- Output_Shaft_x1_IN: output shaft component in order spectrum calculated with respect to reference input shaft;
- Output_Shaft_x1_OUT: output shaft component in order spectrum calculated with respect to output shaft.

- ▪
- Output Shaft x1 [Hz]—classical approach without resampling;
- ▪
- Output Shaft x1 IN [ord]—classical approach with classical resampling;
- ▪
- Output Shaft x1 OUT [ord]—classical approach with new proposed resampling.

#### 4.2. Quantitative Assessment of Indicators

#### 4.3. Analytical Considerations

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Top: signal sampled with constant sampling frequency, bottom: signal sampled with a constant number of samples per rotation.

**Figure 9.**Broadband and narrowband HIs in detection of imbalance state. Signal index represents subsequent instances of vibration signals.

**Figure 10.**Narrowband output_shaft_x1 HIs in detection of imbalance state. Abscissa is the same as in Figure 9.

Parameter | Frequency Spectrum | Resampling Input Shaft | Resampling Output Shaft |
---|---|---|---|

Nominal length | 10 s | 10 s | 10 s |

Sampling frequency | 25,000 Hz | - | - |

Number of samples | 250,000 | N = 250,000 | N = 250,000 |

Input shaft nominal speed | 50 Hz | - | - |

Output shaft nominal speed | 50 × 23/67=17.16 Hz | - | - |

No. input shaft rotations | - | Nrot = 50 × 10 = 500 | - |

No. output shaft rotations | - | - | Nrot = 17.16 × 10 = 171 |

No. of samples per shaft rotation | - | Nsamp = N/Nrot = 500 | Nsamp = N/Nrot = 1461 |

Spectral range | 12,500 Hz | Nsamp/2 = 250 [ord] | Nsamp/2 = 730.5 [ord] |

Spectral resolution | 0.1 Hz | 1/Nrot = 1/500 = 0.002 | 1/Nrot = 1/171 = 0.00585 |

Spectral order–input shaft | - | 1 | 2.91 |

Spectral order–output shaft1 | 0.343 | 1 | |

Input shaft spectral index | 500 | 1/0.002 = 500 | 2.91/0.00585 = 500 |

Output shaft spectral index | 171 | 0.343/0.002 = 171 | 1/0.00585 = 171 |

Indicator | Amplitude RD | Variance |
---|---|---|

output_Shaft_x1 [Hz] | 625.5% | 0.00109 |

output_Shaft_x1_IN [ord] | 625.3% | 0.00366 |

output_Shaft_x1_OUT [ord] | 629.7% | 0.00091 |

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**MDPI and ACS Style**

Jablonski, A.
Rational Resampling Ratio as Enhancement to Shaft Imbalance Detection. *Sensors* **2023**, *23*, 1719.
https://doi.org/10.3390/s23031719

**AMA Style**

Jablonski A.
Rational Resampling Ratio as Enhancement to Shaft Imbalance Detection. *Sensors*. 2023; 23(3):1719.
https://doi.org/10.3390/s23031719

**Chicago/Turabian Style**

Jablonski, Adam.
2023. "Rational Resampling Ratio as Enhancement to Shaft Imbalance Detection" *Sensors* 23, no. 3: 1719.
https://doi.org/10.3390/s23031719