Identification of Cross-Section Defects of the Rod by Using Eigenfrequencies and Features of the Shape of Longitudinal O
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entification of Cross-Section Defects of the Rod by Using Eigenfrequencies and Features of the Shape of Longitudinal Oscillations L. D. Akulenko1, 2 , A. A. Gavrikov1* , and S. V. Nesterov1 1
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia 2 Bauman Moscow State Technical University, ul. 2-ya Baumanskaya 5, Moscow, 105005 Russia Received April 28, 2019; revised May 15, 2019; accepted June 3, 2019
Abstract—A method is proposed for solving inverse flaw detection problems for rods performing longitudinal oscillations. Based on the modeling of the cross-sectional defect as a known function, the main parameters characterizing it, such as the location and volume of the two lowest oscillation frequencies of the free and cantilevered rods, will be approximately determined. Using numerical simulations, it is shown that to satisfactorily determine the properties of a defect, it is sufficient to use several lower frequencies. A method for identifying a defect by one lowest frequency of a free rod is also proposed, provided the known location of the defect is applied, for which it is necessary to determine the characteristics of the oscillation modes. The results of an experimental study are presented. DOI:
Keywords: Sturm-Liouville problem, inverse problem, defects, natural oscillations, longitudinal oscillations.
1. INTRODUCTION The problem of identifying defects in the cross section of a round rod by measuring the natural frequencies (NF) of longitudinal vibrations and comparing the measured frequencies with the NF of the reference rod is solved. In the classical theory of oscillations [1] it is usually assumed that there is sufficiently complete information about the oscillatory system and the main difficulty lies in constructing an adequate model and solving the equation corresponding to it. When solving inverse problems, however, some a priori assumptions about the system under study are also usually introduced, in this case, about the type of defect. For example, in numerous publications, cross-sectional defects are modeled by extension-compression springs. Sufficiently comprehensive reviews of this approach are given in [2, 3]. To solve the inverse problem, here we can use both explicit expressions [4] and analytic [4] and differential properties of natural frequencies [5]. Note that in the last study, to determine the location of the defect, a derivative of the eigenvalue used in the present work to solve the direct problem was introduced [6–8]. In this article, we use the analytical approximation [9], which models defects in the cross section of the rod, which allows us to obtain simple formulas for solving the inverse problem, that is, determination of defects and their sizes from measurements of longitudinal oscillations of the same rod with defects under the boundary conditions of free-free and cantilever fixing. Defects are assumed to be small enough, which causes a small change in NF, but leading to a change in both the
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