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.

The covariance structure for vibrations of the noising mechanism for a port crane and a wind turbine with gear tooth breakage are studied on the basis of their models in the form of periodically non-stationary random processes (PNRPs). The investigation is performed by the authors using PNRP statistical analysis methods. The estimations of the basic frequencies of the vibration’s deterministic components and the time changes of the power for the stochastic part (the PNRP variance) are the key issues of this analysis. In the present paper, the basic frequency estimators are obtained using the least squares technique which are characterized by the maximum of the signal-to-noise ratio as compared with techniques known in the literature. Using the basic frequency, values for the Fourier coefficients for the mean and covariance functions are calculated and the dependencies of the forenamed moment functions on time and lag are ascertained. The amplitude spectra of the deterministic oscillations and the time changes of the power of the stochastic part are presented. These spectra are considered as the symptomatic features of the mechanism’s state of health. It is revealed that the high-frequency modulation of PNRP carrier harmonics is narrow-band for both distributed and localized faults. Using bandpass filtering and Hilbert transform, the quadratures for each high-frequency component are separated and analyzed. It is shown that the auto- and cross-covariance structure of the high-frequency component quadratures characterizes the specific features of the different types of faults. Using the cross-covariance map for high-frequency components, the peculiarities of the bandpass filtering of vibrations for a damaged wind turbine’s gearbox as a PNRP’s is considered. It is shown that the filtering bandwidth cannot be narrower than 14 times the basic frequency. This bandwidth essentially exceeds that recommended in the literature (two to three times the basic frequency).

The coherent estimators of probabilistic characteristics of periodically correlated random processes with unknown period have been investigated. It is shown that these estimators are asymptotically unbiased and consistent. In a first approximation formulas were obtained for the bias and dispersion of estimators defining the impact of the preliminary determination of the period on the value of estimation error.