The solution of the problem of free circulation of circular arcs by numerical analytical boundary elements method

Abstract

The work is devoted to determining the natural frequencies and vibration modes of a circular arch using the numerical-analytical boundary elements method. A differential equation of the natural vibrations of the arch is obtained, a complete system of its fundamental solutions is determined, a transcendental frequency equation is constructed, the arch is calculated based on the obtained analytical dependencies, and the results are compared with the results of finite element analysis. For 10 possible combinations of the roots of the equation of the characteristic equation, 360 fundamental functions are calculated. An example of calculating a circular arch for free vibrations by the method of the authors is considered. The search for the roots of the transcendental frequency equation can be carried out by the method of successive approximations, using any programming environment. Here used MATLAB. As a result of the calculation, the first five frequencies of the natural vibrations of the arch were obtained. Their comparison with those calculated in the ANSYS program shows that the spectrum of natural frequencies calculated by the boundary element method is slightly lower (except for the first frequency) than the spectrum calculated by the finite element method, which indicates greater reliability of the results of the boundary element method.