Abstract
This research aims to develop and validate a method for tacho-less automatic detection of faults in rolling bearings for mechanical systems comprised by gears. The proposed method was based on the application of some mode decomposition technique in order to extract monocomponent signals from the vibration and to calculate an indicator of the modulation produced by the rolling element bearing fault. The computation of this indicator was performed by means of Lock-in Amplifiers, which are used in order to extract, through a synchronous approach, spectral components from non-stationary signals. A novel algorithm, previously applied on gear fault detection, was adapted and used in order to estimate the rotational speed. The effectiveness of the method was assessed through experiments with real signals. Besides, the capability of the indicator for serving as a relative measure of the fault severity was verified.
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Abbreviations
- \({a}_{k}\) :
-
Amplitude modulation function related to the shaft rotational speed
- \({A}_{a}\) :
-
Amplitude modulation of the vibration produced by a faulty rolling element bearing
- \({A}_{i}\) :
-
Amplitude of the modulation produced by the input gear
- \({A}_{o}\) :
-
Amplitude of the modulation produced by the output gear
- \({X}_{k}\) :
-
Amplitude of the \(k\)-th harmonic of the gear mesh frequency
- \(\mathcal{P}\left(t\right)\) :
-
Angle of a frequency modulation signal
- \({\theta }_{i}\left(n\right)\) :
-
Angle of the analytical signal of the monocomponent signal \(i\)
- \({\theta }_{r}\left(n\right)\) :
-
Angle of the rotational speed signal
- \({\overline{w} }_{j}\) :
-
Average angular speed between the \(j\)-th and the \((j-1)\)-th impulse produced by a faulty rolling element bearing
- \(\Theta \) :
-
Average occurrence period in angular domain
- \({\alpha }_{c}\) :
-
Characteristic coefficient for the cage fault
- \({\alpha }_{i}\) :
-
Characteristic coefficient for the inner race fault
- \({\alpha }_{o}\) :
-
Characteristic coefficient for the outer race fault
- \({\alpha }_{r}\) :
-
Characteristic coefficient for the rolling element fault
- \(\beta \) :
-
Contact angle
- \(x\left(n\right)\) :
-
Digital model of the gear vibration
- \({\varPsi }_{i}\) :
-
Equivalent jitter in the angular domain
- \(E\{\cdot \}\) :
-
Expected value operator
- \(\alpha \) :
-
Fault characteristic coefficient
- \({f}_{res}\) :
-
Frequency of a monocomponent signal
- \({f}_{c}\) :
-
Frequency of the carrier signal
- \({f}_{mod}\) :
-
Frequency of the modulating signal
- \({m}_{V}\) :
-
Gear ratio
- \({\widehat{\tilde{g }}}_{i}(n)\) :
-
Hilbert transform of the monocomponent signal \(i\)
- \(s\left(n\right)\) :
-
Impulse response stimulated by a local rolling element bearing defect
- \({\varphi }_{k}\) :
-
Initial phase of the \(k\)-th harmonic of the gear mesh frequency
- \(\mathcal{F}(t)\) :
-
Instantaneous frequency of a frequency modulation signal
- LIA:
-
Lock-in Amplifier
- \(Pd\) :
-
Mean diameter
- \({\tilde{g }}_{i}(n)\) :
-
Monocomponent signal \(i\)
- \({{D}_{N}({f}_{c},{f}_{mod})}_{3}\) :
-
Normalized modulation detection index
- \(K\) :
-
Number of harmonics of the digital model of the gear vibration
- \(Nb\) :
-
Number of rolling elements
- \({f}_{o}\) :
-
Rotational speed of the output gear
- \({f}_{r}\) :
-
Rotational speed
- \({T}_{s}\) :
-
Sampling period
- SNR:
-
Signal-to-Noise Rate
- SSD:
-
Singular Spectrum Decomposition
- \({{\widehat{D}}_{N}({f}_{c},{f}_{mod})}_{3}\) :
-
Smoothed version of the normalized modulation detection index
- TARSE:
-
Tacho-less Automatic Rotational Speed Estimation
- \(T\) :
-
Teeth number of the gear
- \({T}_{i}\) :
-
Time instant of impulse occurrence due to a vibration produced by a faulty rolling element bearing
- \(w(n)\) :
-
Vibration background noise
- \(x\left(n\right)\) :
-
Vibration produced by a faulty rolling element bearing
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The authors would like to thank data-acoustics.com for supplying the real vibration signals used in this work.
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RodrÃguez, A., Hernández, F., Ruiz, M. (2022). Automatic Detection of Rolling Element Bearing Faults to Be Applied on Mechanical Systems Comprised by Gears. In: Chaari, F., Leskow, J., Wylomanska, A., Zimroz, R., Napolitano, A. (eds) Nonstationary Systems: Theory and Applications. WNSTA 2021. Applied Condition Monitoring, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-82110-4_12
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