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

Mykola Surianinov,Yurii Krutii,Vladimir Osadchiy,Anatolii Kovrov

doi:10.1051/e3sconf/202021102022

Mykola Surianinov, Yurii Krutii + Show 2 more

Open Access

https://doi.org/10.1051/e3sconf/202021102022

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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.

Highlights

Arch structures are used in various fields, such as shipbuilding, aviation, and mechanical engineering
The algorithm of application of numerical-analytical boundary elements method (NA BEM) can be divided into two components: "analytical" and "numerical"
The first of these is to obtain the analytical expressions of fundamental orthonormal functions, the Green function and the vector of external loads for all possible roots of the characteristic equation corresponding to the differential equation of the problem [11,12,13,14]

Summary

Introduction

Arch structures are used in various fields, such as shipbuilding, aviation, and mechanical engineering. The appearance of new materials has stimulated the use of arch structures in new fields, such as medicine, bioengineering, vibration diagnostics, or the production of nanocomposites. Arches have been especially widely used in construction, since their use makes it possible to cover long spans. The first began to erect arches of large size Romans (world-renowned aqueduct Pon-duGar). Well-known old wooden and stone arch bridges in Russia, France, China and many other countries. Modern construction tends to increase the use of steel arches ― Abu Dhabi Airport Terminal, Mike O'Callaghan-Pete Tillman Memorial Bridge in America, Charlie's Rail Tunnel in the UK, a new bridge over Ob in Russia and many other unique structures [1]

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