In operational diagnostics and recognition of states of complex technical systems, an important task is to identify small time-determined changes in complex measured diagnostic signals of the controlled object. For these purposes, the signal is transformed into a small-sized image in the diagnostic feature space, moving along trajectories of different shapes, depending on the nature and magnitude of the changes. It is important to identify stable and deterministic patterns of changes in these complex-shaped diagnostic signals. Identification of such patterns largely depends on the principles of constructing a small-sized feature space. In the article, the space of decomposition coefficients of the measured signal in the adaptive orthonormal basis of canonical transformations is considered as such a space. In this case, the basis is constructed based on a representative sample of realizations of the controlled signal for various states of the system using the proposed algorithm. The identified shapes of the trajectories of the images correspond to specific types of deterministic changes in the signal. Analytical functional dependencies were discovered linking a specific type of signal change with the shape of the trajectory of the image in the feature space. The proposed approach, when used, simplifies modeling, operational diagnostics and condition monitoring during the implementation of, for example, low-frequency diagnostics and defectoscopy of structures, vibration diagnostics, monitoring of the stress state of an object by analyzing the time characteristics of response functions to impact.

Keywords: modeling, functional dependencies, state recognition, diagnostic image, image movement trajectories, small changes in diagnostic signals, canonical decomposition basis, analytical description of image trajectory

The article considers a variant of constructing a digital twin model for diagnostics of operation of a mechanical gear transmission - a reducer consisting of a pair of cylindrical gears. The basis of the considered digital twin model is an adaptive learning vibration mathematical model of the static operating mode when gears rotate at a constant speed. The vibration signals of the reducer, recorded by accelerometers and effective for detecting and diagnosing faults, are used as the main information measured at the facility. The detected faults are cracking, wear, chipping and pitting of teeth. A special feature of the implemented digital twin is the ability to simultaneously detect faults of several teeth at once, both on the driving and driven gears, by time and frequency characteristics, as well as the ability to determine the main technical data of a specific reducer by vibration characteristics.

Keywords: diagnostic mathematical model, digital twin, vibration diagnostics, mechanical gearbox, tooth defect, diagnostic signal, vibration power spectrum, synchronously averaged time characteristic

A new mathematical apparatus is proposed for monitoring the adequacy of the choice of signal sampling interval from the point of view of taking into account the main high-frequency components and identifying the possibilities of increasing it. It is based on the construction of special aliasing grams based on measured signal samples. Aliasing grams are graphs of standard deviations of the amplitude spectra of a conventionally reference discrete signal, specified with the highest sampling frequency, and auxiliary discrete signals obtained over the same observation interval, but with lower sampling frequencies. By analyzing such graphs, it is easy to identify sampling frequencies that lead to the appearance of the aliasing effect in the case of sampling, and, consequently, to distortion of the signal spectrum. To speed up and simplify the construction of aliasinggrams, it is proposed to use as auxiliary signals obtained from the reference one by thinning. It has been shown that this device is also effective in the case of the spectrum spreading effect. It can be used in self-learning measuring systems.

Keywords: sampling interval, aliasing, amplitude spectrum, aliasing-gram, sample decimation, spectrum spreading