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The main aim of the lectures is to present an approach for identification of transverse cracks and crack-like defects in rods and beams by means of natural frequencies of longitudinal and transverse vibration. Plan of the lectures is as follows. First, we give the mathematical formulation of the problems under consideration. Then we recall some mathematical tools that are necessary for solving the problems. The mathematical tools, in particular, include elements of the theory of generalized functions, some problems of the theory of entire functions, inverse Sturm-Liouville problem and inverse spectral problem for an ordinary differential equation of fourth order on the finite interval. Finally, we will apply the mathematical apparatus to the problems of identification transverse cracks in rods by means of natural frequencies of longitudinal and transverse vibration.

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Generalized Functions and Their Applications to Direct and Inverse Vibration Problems

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  • 发表日期：

    2025年

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  • Efim Shifrin

摘要

The main aim of the lectures is to present an approach for identification of transverse cracks and crack-like defects in rods and beams by means of natural frequencies of longitudinal and transverse vibration. Plan of the lectures is as follows. First, we give the mathematical formulation of the problems under consideration. Then we recall some mathematical tools that are necessary for solving the problems. The mathematical tools, in particular, include elements of the theory of generalized functions, some problems of the theory of entire functions, inverse Sturm-Liouville problem and inverse spectral problem for an ordinary differential equation of fourth order on the finite interval. Finally, we will apply the mathematical apparatus to the problems of identification transverse cracks in rods by means of natural frequencies of longitudinal and transverse vibration.