ABSTRACT
We discuss the use of the Hilbert transform for the analysis of periodically non-stationary random signals (PNRSs), whose carrier harmonics are modulated by jointly stationary high-frequency narrow-band random processes. PNRS of this type are suitable models for numerous natural and man-made phenomena, including the vibration of a damaged mechanism. We show that the auto-covariance function of the signal and its Hilbert transform are the same, and that their cross-covariance functions differ only in their sign, meaning that the sum of squares of the signal and its Hilbert transform cannot be considered a 'squared envelope' and no new information is contained compared with the variance of the raw signal. A representation of the signal in the form of a superposition of high-frequency components is obtained and it is shown that these components are jointly periodically non-stationary random processes. The properties of the band-pass filtered signals are examined, and it is shown that band-pass filtering can reduce both the number of signal variance cyclic harmonics and their amplitudes. We show that it is possible to extract the quadratures of narrow-band high-frequency modulation processes using the Hilbert transform. The results obtained here theoretically substantiate the use of the Hilbert transform for the analysis of high-frequency modulation which occurs when a fault appears. They offer a new way to consider the traditional approach to vibration diagnosis. A processing technique that can be considered an alternative to envelope analysis is described, and its use in the analysis of a vibration signal is discussed.
ABSTRACT
It is shown that the models of gear pair vibration, proposed in literature, are particular cases of the bi-periodically correlated random processes (BPCRPs), which describe its stochastic recurrence with two periods. The possibility of vibration and analysis within the framework of BPCRP approximation, in the form of periodically correlated random processes (PCRPs), is grounded and the implementation of vibration processing procedures using PCRP techniques, which are worked out by the authors, is given. Searching for hidden periodicities of the first and the second orders was considered as the main issue of this approach. The estimation of the non-stationary period (basic frequency) allowed us to carry out a detailed analysis of the deterministic part, the covariance structure of the stochastic part, and to form, using their parameters, the sensitive indicators for fault detection. The results of the processing of the wind turbine gearbox vibration signals are presented. The amplitude spectra of the deterministic oscillations and the time changes of the stochastic part power for different fault stages are analyzed. The most efficient indicators, which are formed using the amplitude spectra for practical applications, are proposed. The presented approach was compared with known in literature cyclostationary analysis and envelope techniques, and its advantages are shown.
