Forced vibrations of arch systems in its plane

Abstract

The work is devoted to solving the problem of forced oscillations of a circular arch using the numerical-analytical method of boundary elements. The algorithm of the method can conditionally be divided into two parts - analytical and numerical. The first of this two parts involves obtaining in an analytical form a complete system of fundamental solutions of the original differential equation, constructing the Green’s function and the components of the vector of external loads, which are the problems to be solved in this article. An ordinary sixth-order differential equation is obtained that describes the forced oscillations of an arch in its plane. It differs from the similar equation for free oscillations obtained earlier only in the presence of the right-hand side. This means that, as with free oscillations, 10 solutions are possible here, and the analytical expressions derived from 360 fundamental functions for these solutions remain unchanged. For one of the variants of the roots of the characteristic equation, an analytical expression of the Green’s function is constructed, a connection is established between the Green’s function and one of the fundamental functions, which is also valid for other values of the roots of the characteristic equation. Using impulse functions and splines, the arch load vector is constructed. The presented work implements the analytical component of the numerical-analytical method of boundary elements. The numerical implementation of the algorithm and the comparison of the results with the results of finite element analysis determine the direction of further research. It is noted that the cost of computer resources when implementing a program for calculating an arch system using the boundary element method is minimal, since it is necessary to solve a system of only twelve algebraic equations, which is significantly less than when using the finite element method. The results obtained allow us to perform dynamic calculations for forced vibrations any arched systems of arbitrary configuration.