The Inverse Spectral Problems in the Detection of the Defects and Heterogeneities of the Civil Structures

Abstract

In this paper we study how to detect physical and geometrical heterogeneities of the civil structures by the resonance frequencies of the torsional vibrations of its separate elements. Defects in a material of the element produce the changes of the mass density and alter the values of the resonance frequencies of the free vibrations. The mathematical model is based on the linear partial differential equation for motion of elastic elements in the assumption of heterogeneity of the density distribution along its length. Using the separation of variables we reduce the problem to the calculation of zeros of the transcendental resonance equation and find the resonance frequencies of free torsional vibrations. Then we solve the inverse spectral problem: restoration of the density distribution along the length of element if the several resonance frequencies are known. This solution can be used for the non-destructive control. The given element of the structure is subjected to the vibrations. The location of the defect and its quantitative characteristics are determined by the deviations of the values for the resonance frequencies obtained in the certain moment of time from the corresponding perfect values. The simple engineering formulas for the location of the cracks and its parameters are derived.