Determination of the speed and density of the fluid inside the pipeline on elastic supports

Abstract

The natural frequencies of the bending vibrations of the pipeline are investigated. The pipe sags over the obstacle and is under the action of tensile force. Outside the sagging area, the pipe rests on elastic supports. The fluid transported through the pipeline is under pressure. The direct problem was solved earlier, in this article, the inverse problem of identifying the speed and density of the transported fluid by the known natural frequencies of bending vibrations is solved. The equation of bending vibrations of a pipeline is described by the Kirchhoff model. The characteristic equation is solved using Ferrari formulas. The general decision is determined. We substitute the general solution into the boundary conditions and obtain a system of equations. This system gives a frequency equation, which is solved numerically on a developed program in the Maple package. The method of successive approximations is applied, after the third iteration, the accuracy of calculating the parameters of the velocity and density of the liquid is approximately 10−3. Thus, it was found that with an increase in the oscillation frequency, the density of the liquid inside the pipe decreases. It is determined that with increasing natural frequencies of pipe bending vibrations, the fluid velocity parameter increases. It is shown that the two lower frequencies of bending vibrations of the pipeline can be used to determine the parameters of the velocity and density of the liquid. The dependence of the mass flow rate of the liquid on the first natural frequency of the pipe oscillations is given. It is shown that with increasing frequency, the mass flow rate decreases. The research results will help the development of acoustic diagnostic methods and non-destructive testing methods and will find technical application for monitoring and diagnosing the state of pipeline systems.