Deterministic and stochastic methods combining while solving the problem of defectoscopy of an elastic rod


  • G. M. Zrazhevsky Taras Shevchenko National University of Kyiv
  • V. F. Zrazhevska National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37, Prosp. Peremohy, Kyiv, Ukraine, 03056



harmonic rod oscillations, natural frequencies, flaw detection, Bootstrap-aggregated Regression Trees


The paper considers the problem of natural harmonic oscillations of an elastic rod with stress-free ends in the presence of one or a set of defects. Defects are modeled by the inhomogeneity of the Young's modulus. The location of the defects, their geometric size, which is considered small, and the change in elastic properties are the parameters of the defects. The analysis of natural frequency shifts caused by the defect of the rod is the subject of the study. The aim of the work is a mathematical substantiation for the construction of fast and stable algorithms for determining the defect parameters of elastic bodies by analyzing free oscillations. The paper uses and compares fundamentally different research methods. The first methods are classical mathematical methods of mechanics, applied to the analysis of deterministic systems and based on analytical studies combined with numerical implementation. In contrast, a composite machine learning meta-algorithm used in standard statistical classification and regression - Bootstrap-aggregated Regression Trees (BART) - is used to solve the inverse problem. When comparing the constructed algorithms, the statistical method Sampling was used, which allowed to quantify the accuracy and stability of the algorithms.

Pages of the article in the issue: 35 - 38

Language of the article: Ukrainian


KAPLUNOV, J., PRIKAZCHIKOV., D., SERGUSHOVA, O. (2016). Journal of Sound and Vibration, 366, p. 264–276. Available from:

RUBIO, L., FERNÁNDEZ-SÁEZ, J., MORASSI, A. (2015). The full nonlinear crack detection problem in uniform vibrating rods. Journal of Sound and Vibration, 339, p. 99–111. Available from:

SOLOVIEV A.N., PARINOV I.A., CHERPAKOV A.V., CHAIKA YU.A., ROZHKOV E.V. Analysis of oscillation forms at defect identification in node of truss based on finite element modeling (2018). Materials Physics and Mechanics, 37, p. 192-197.

ZRAZHEVSKY, G., ZRAZHEVSKA, V. Obtaining and investigation of the integral representation of solution and boundary integral equation for the non-stationary problem of thermal conductivity (2016). Eureka: Physics and Engineering, 6, p. 53–58. doi: 10.21303/2461-4262.2016.00216

ZRAZHEVSKY, G., GOLODNIKOV, A., URYASEV, S. (2019). Mathematical Methods to Find Optimal Control of Oscillations of a Hinged Beam (Deterministic Case). Cybernetics and Systems Analysis, 55 (6), p. 1009-1026. doi: 10.1007/s10559-019-00211-x

ZRAZHEVSKY, G., ZRAZHEVSKA, V. (2020). The extension method for solving boundary value problems of the theory of oscillations of bodies with heterogeneity World Journal of Engineering Research and Technology 6 (2), p. 503-514.

ZRAZHEVSKY G.M., ZRAZHEVSKA V.F. (2021) Modeling of Finite Inhomogeneities by Discret Singularities. Journal of Computational and Applied Mathematics. 1 (135) р. 138-144.




How to Cite

Zrazhevsky, G. M., & Zrazhevska, V. F. (2021). Deterministic and stochastic methods combining while solving the problem of defectoscopy of an elastic rod. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (4), 35–38.



Differential equations, mathematical physics and mechanics