Abstract
THE EXTENSION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS OF THE THEORY OF OSCILLATIONS OF BODIES WITH HETEROGENEITY
Grigoriy Zrazhevsky and Vera Zrazhevska*
ABSTRACT
The article considers the problem of determining the shifts of natural frequencies and changes of mode shape functions for low-frequency longitudinal vibration of an one-dimensional elastic rod containing local stepwise heterogeneities (defects). It is assumed that defects have small linear dimensions in comparison with the rod length and are characterized by a changes in Young's modulus. A new method to obtain the exact solution of the problem (method of solution extension), is proposed in the article. This method is like to the partial domain method and is based on the analytical extension of the solution from the uniform area to the non-uniform area by placing a point singularity on the non-uniform area. This method allows to obtain a solution of the problem for a rod with many defects with different parameters. The solution of the problem is constructed in the form of analytical series expansion according to the characteristic length of the heterogeneity which is considered small with respect to the wavelength. An infinite recursive system of boundary value problems with point discontinuities is obtained. The system allows to obtain a solution of the problem with a given accuracy.
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