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.
Abstract 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 estimation of the instantaneous spectral density of periodically random processes (PCRP) and its Fourier coefficients in continuous and discrete cases is considered. The formulae for calculation of the mean square error dependencies on parameters of processing and signals are obtained. The aliasing effects are analyzed. The example of estimation of PCRP spectral function for vibration signal of defect rotary unit is given.
Technical Diagnostics and Non-Destructive Testing, 2022, №04. TEKHNICHESKAYA DIAGNOSTIKA I NERAZRUSHAYUSHCHIY KONTROL Quarterly scientific-technical and production journal. ISSN: 0235-3474. Established in 1989. Subscription index: 74475. State Registration Certificate KV 4787 of 09.01.2001. All rights reserved Publisher: B.E.Paton Electric Welding Institute, National Academy of Sciences of Ukraine. Main topics of the magazine: theoretical and practical articles on technical diagnosing and non-destructive testing; flaw detection, residual life of structures. Each issue features: scientific articles; manufacturing experiences; reviews; exhibitions, conferences; advertisements.
Hilbert transform of periodically non-stationary random signal (PNRS) is analyzed. The relations for covariance and spectral components of PNRS and its Hilbert transform are obtained. The analytic signal and envelope properties for multi-and monocomponent signal are analyzed.
Use of a probabilistic model of vibration signals in the form of a periodically correlated random process allows new opportunities for diagnosing defects in rotating mechanisms at the early stages of their occurrence. Importance of multichannel simultaneous acquisition of vibration diagnostic signals for their join analysis grounded. General requirements to specialized portable devices for acquisition and preprocessing of the vibration diagnostic signals are considered.
The result of the application of the spectral methods for searching and analysis of the hidden periodicities in the properties of the vibration signals of the rotary machinery which allow to discover the spectral band of the vibration periodical non-stationarity, to determine the spectrum forms, to evaluate the power of their time variation. The typical peculiarities of the defects are established on this basis, the new diagnostic indicators are formed for their detection and typification.
The periodically non-stationary random signals (PNRSs), whose carrier harmonics are mo-dulated by jointly stationary high-frequency random processes are analyzed. 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 random process is periodically non-stationary of the second order only in the case when some of the cross-covariance functions of its modulation processes are not equal to zero. The correlations of the PNRP spectral harmonics and the correlations of the modulating processes in series representation are equivalent. Evaluating the specific features of the auto- and cross-covariances for modulating processes as well as contribution of each pair to the covariance component values allows us to detect defects at early stages.
The use of two different techniques for the analysis of vibration signals, whose carrier harmo¬nics are modulated by high-frequency narrow-band random processes is discussed. Periodically non-stationary random processes (PNRP) are suitable models for description of vibration signals of damaged mechanism. A proposed processing technique can be considered as an alternative to squared envelope analysis, kurtosis techniques, squared envelope spectrum (SES) and its use in the analysis of a vibration signal is discussed. It is shown that the spectral estimates obtained by the envelope square method are biased and inconsistent. The possibility of obtaining of the unbiased estimates by the PNVP method even for a signal/noise ratio equal to 0.07 has been demonstrated.
The strict characteristic and the comparative analysis of the developed by authors the coherent and the component methods for statistical analysis of periodically correlated random processes (PCRP) — the mathematical models of the stochastic oscillations — for unknown period of the non-stationarity is given. The comparison of efficiencies of the proposed methods for period estimation with the given in literature for individual cases of PCRP is provided.
