Kurapov P.R. Modeling and methods of analysis of narrow-band periodically non-stationary random processes - a qualifying scientific work in the form of a manuscript. Dissertation for the degree of Doctor of Philosophy in specialty 113 "Applied Mathematics" (11 - Mathematics and Statistics). Lviv Polytechnic National University, Lviv, 2022.
The use of a probabilistic approach to the analysis of signals, in particular vibration signals of rotating nodes of mechanisms as well as fluctuations of the Earth's magnetic field, allows us to describe the behavior of a physical system in more detail, not limited to small physical details. The random nature of noise and interference observed during the operation of mechanical systems requires the use of probabilistic methods in the tasks of flaw detection of objects of long-term operation. Recently, a random process has become the main mathematical model for describing vibration signals - the carriers that accompany it. Statistical processing methods have found wide and effective application in radar, radio communications, radio physics, seismology, telemetry, vibration diagnostics, biomedical engineering, climatology. The properties of repeatability and stochasticity are characteristic of many random signals found in nature and human technical activity. Therefore, the use of a probabilistic approach based on models in the form of periodically correlated random processes (PCRP) for the analysis of such stochastic signals makes it possible to describe the properties of physical systems that cause them, taking into account both their deterministic part and random interference and disturbances that can carry both useful and insignificant information about the physical phenomenon.
The properties of the signal source within the framework of the correlation and spectral theory of the PCRP are described based on the analysis of the moment functions of the first and second order - the mathematical expectation, the correlation function, the spectral density and the coefficients of their expansion into a Fourier series. The parameters of signal processing using nonparametric methods are the length of the signal realization, the sampling interval, the truncation point of the correlogram, weighted correlation and spectral windows. The unaccounted part of the values of the autocorrelation function outside the window is considered equal to zero or is a periodic continuation of the taken into account part, which is usually a rough assumption and leads to distortion of estimates. When performing the dissertation work, a software product (PP) was created for the statistical analysis of the stationary coupled components of the PCRP. The developed software (SW) is used to calculate the statistical characteristics of the PCVP and calculate diagnostic indicators based on the study of their vibration signal. Software has been developed for simulating PCRP according to predefined probabilistic characteristics, which is a computer simulation model of a phenomenon with stochastic repeatability for verifying methods for isolating stationary coupled components of the PCRP signal. The theoretical and practical results obtained in the dissertation work have been used in the research work of the Department of "Methods and Means of Selection and Processing of Diagnostic Signals" of the G.V. Karpenko Institute of Physics and Mechanics of the NAS of Ukraine. The software and methodology developed by the author were used in the performance of scientific works “Development of information technologies and means of vibration diagnostics based on periodically non-stationary nonlinear models using the Hilbert transform” (state registration number 0119U101061), “Development of methods for correlation analysis of the polyrhythmic structure of vibration signals to increase the efficiency of diagnostics of elements of mechanisms with different rotation speeds” (0122U002138), as well as in the performance of the economic contract subject “Study of the technical condition of the gear reducer of the boom conveyor drive of the ship loading machine 18SP1 inventory number №000100”.
Key words: periodically correlated random processes, computer modeling, harmonic representation, parametric models, evaluation methods, vibration signals, Earth's magnetic field.
