# Multidimensional Data Interpretation of Vibration Signals Registered in Different Locations for System Condition Monitoring of a Three-Stage Gear Transmission Operating under Difficult Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Object of Study

- -
- 1st stage—bevel gear;
- -
- 2nd stage—cylindrical gear;
- -
- 3rd stage—planetary gear.

## 3. Research Design and Sensor Placements

## 4. Method of Analysis

_{x}) is the most useful, because it is directly related to the energy content of the vibration profile. It relates to the power of the wave. The root-mean-square value is one of the important factors for machinery condition monitoring. The standard deviation (std

_{x}) is a measure of the variability of a signal about its mean value ($\overline{x}$). The standard deviation is invariant under changes in location, and scales directly with the scale of the random variable. The mean and standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a “natural” measure of statistical dispersion if the center of the data is measured about the mean. For a vibration signal with a mean value of zero, the standard deviation is equal to the root-mean-square value of the signal. In probability theory and statistics, the coefficient of variation (CoV

_{x}) is a standardized measure of the dispersion of a probability distribution. The coefficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data [28,29]. It is widely used to express the precision and repeatability of an assay. Therefore, it can also be a diagnostic measure, assuming that the symptoms of successive failures increase the vibration signal dispersion. A sample time-varying vibration signal with calculated quantity estimators has been depicted in Figure 6.

^{2}complex operations for a signal containing N samples. This motivates the development of the fast Fourier transforms, a family of efficient implementations of the DFT for different composites of N. In particular, we examined the radix-2 Cooley–Tukey FFT algorithm [35], which reduces the number of complex operations required for Nlog

_{2}(N) [36]. The most commonly used FFT algorithm is named after J.W. Cooley, an employee of IBM, and J.W. Tukey, a statistician, who jointly developed an implementation of the FFT for high-speed computers in 1965 [37,38,39].

_{n}}, the DFT expresses them as a sequence {X

_{k}} of complex numbers, representing the amplitude and phase of different sinusoidal components of the input signal [36].

- (t − b)—window width;
- w(t)—Window function;
- R—Hop size between successive DFTs.

## 5. Results

## 6. Conclusions

- (1)
- To analyze and evaluate the information content in waveform vibration signals recorded at various measuring points, quantitative estimators were determined. The estimators were selected to represent the energy content of the signal, the distribution of the maximum values, and measures of dissipation and dispersion.
- (2)
- The largest range of extreme values was obtained for the coefficient of variation (CoV)—the percentage range was 418% (for the X-axis vibration); the smallest was obtained for RMS—74% (for the X-axis vibration). Each range of percentage values was much too large, which confirmed the large influence of the location of the measuring point on the obtained results of the vibration estimators.
- (3)
- The data shown in Figure 18 confirmed the significant dispersion of values. It is visible in the figure that, assuming certain SHM control thresholds, there was a high risk of exceeding their values and of an incorrect alarm condition that would result only from the incorrect selection of the sensor location (not damage).
- (4)
- FFT enabled the identification of components related to the dynamics of the gear transmission operation. This allowed for the identification of parameters of kinematic nodes and other dynamic phenomena, including changes in machine parameters resulting from lab damage. In the case of overlapping of other components in the spectra, the diagnostic result may have been difficult or erroneous.
- (5)
- A strong correlation of the characteristic frequencies was clearly visible (Figure 19); however, significant differences could be observed when comparing the amplitude values of the dominant FFT components. In the cases of points 4 and 5 in the FFT spectra, local extremes in the frequency bands were also visible, but were negligibly small for points 1–3.
- (6)
- For the multidimensional analysis of the signal, the time distribution of individual frequency components was clearly visible, which was not always constant. Such information about changes in the amplitudes of specific frequency components over time may also be essential in SHM systems, especially in the diagnostic aspect of complex machines operating in heterogeneous and difficult conditions, such as mining conveyors.
- (7)
- The STFT distributions of vibration signals at various measurement points clearly showed quantitative and qualitative differences in vibrations, as well as the presence of local extremes occurring only for a specific time and frequency, which were masked and invisible in waveforms and FFT spectrums.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**FFT spectra of vibration and main components correlated with the dynamics of the machine operation: (

**a**)—spectrum of Z axis vibration, (

**b**)—spectrum of Y axis vibration.

**Figure 9.**STFT distribution of the vibration signal of the tested transmission gear with view of frequency unstable components.

**Figure 19.**Comparison of the spectra of vibrations along the vertical axis (Z) determined for signals recorded at various measurement points.

Transmission Type: | Three-Stage Transmission of Bevel–Cylindrical–Planetary Configuration |
---|---|

Total ratio | 39.326 |

Transmission’s input shaft torque | 650 Nm |

Transmission’s input shaft rotational speed | 1470 rpm |

Transmission’s output shaft torque | 25,550 Nm |

Transmission’s output shaft rotational speed | 37.38 rpm |

Parameters: | Value |
---|---|

PCB 356A02 sensitivity (±10%) | 10 mV/g |

PCB 356A02 measurement range | ±500 g pk |

PCB 356A02 nonlinearity (400 g) | ≤1% |

PCB 356A02 temperature range | −54 to 121 °C |

Ni 9233 ADC resolution | 24 bits |

Ni 9233 number of channels | 4 analog inputs |

Ni 9233 type of ADC | Delta-Sigma |

Ni 9233 sampling mode | Simultaneous |

**Table 3.**Quantity estimators of waveforms of vibrations in three axes for five different measurement points.

max_{x} | RMS_{x} | std_{x} | CoV_{x} | max_{y} | RMS_{y} | std_{y} | CoV_{y} | max_{z} | RMS_{z} | std_{z} | CoV_{z} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Point 1 | 12.158 | 1.327 | 1.327 | 55.160 | 13.772 | 1.693 | 1.693 | 67.742 | 6.318 | 0.829 | 0.829 | 29.153 |

Point 2 | 7.080 | 0.834 | 0.834 | 44.979 | 11.861 | 1.202 | 1.202 | 52.506 | 4.603 | 0.625 | 0.618 | 7.029 |

Point 3 | 14.034 | 1.233 | 1.233 | 715.034 | 11.195 | 0.803 | 0.803 | 24.132 | 6.463 | 0.627 | 0.627 | 114.519 |

Point 4 | 8.644 | 0.584 | 0.584 | 4822.723 | 10.572 | 0.914 | 0.913 | 20.534 | 7.159 | 0.580 | 0.580 | 36.202 |

Point 5 | 19.070 | 1.049 | 1.049 | 79.211 | 19.795 | 0.988 | 0.988 | 140.693 | 23.566 | 1.557 | 1.557 | 53.122 |

mean | 12.197 | 1.005 | 1.005 | 1143.421 | 13.439 | 1.120 | 1.119 | 61.121 | 9.622 | 0.844 | 0.842 | 48.005 |

std | 4.229 | 0.270 | 0.270 | 1857.106 | 3.354 | 0.315 | 0.315 | 43.509 | 7.023 | 0.367 | 0.368 | 36.398 |

range | 11.991 | 0.743 | 0.742 | 4777.744 | 9.223 | 0.889 | 0.890 | 120.158 | 18.963 | 0.977 | 0.977 | 107.490 |

variance | 17.885 | 0.073 | 0.073 | 3,448,842.351 | 11.250 | 0.099 | 0.099 | 1893.039 | 49.317 | 0.135 | 0.135 | 1324.810 |

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**MDPI and ACS Style**

Wojnar, G.; Burdzik, R.; Wieczorek, A.N.; Konieczny, Ł.
Multidimensional Data Interpretation of Vibration Signals Registered in Different Locations for System Condition Monitoring of a Three-Stage Gear Transmission Operating under Difficult Conditions. *Sensors* **2021**, *21*, 7808.
https://doi.org/10.3390/s21237808

**AMA Style**

Wojnar G, Burdzik R, Wieczorek AN, Konieczny Ł.
Multidimensional Data Interpretation of Vibration Signals Registered in Different Locations for System Condition Monitoring of a Three-Stage Gear Transmission Operating under Difficult Conditions. *Sensors*. 2021; 21(23):7808.
https://doi.org/10.3390/s21237808

**Chicago/Turabian Style**

Wojnar, Grzegorz, Rafał Burdzik, Andrzej N. Wieczorek, and Łukasz Konieczny.
2021. "Multidimensional Data Interpretation of Vibration Signals Registered in Different Locations for System Condition Monitoring of a Three-Stage Gear Transmission Operating under Difficult Conditions" *Sensors* 21, no. 23: 7808.
https://doi.org/10.3390/s21237808