Forces of inertia of the cleaning piston when passing through an open section of the main gas pipeline

Abstract

The task of determining the forced oscillations of the gas pipeline open section during the passage of the cleaning piston through it belongs to such type of problems, which deal with determining of forced oscillations of one-dimensional elastic objects under the action of a moving inertial load. There are currently two approaches of solving such problems. The first one involves the integration of the differential equation into partial derivatives and the solution of the problem is a superposition of eigenvalues ​​and accompanying oscillations. The second approach does not involve the integration of the differential equation in partial derivatives. It includes methods of generalized coordinates, generalized displacements, as well as various numerical methods. Neither the first nor the second method is simple. Therefore, a combined method is proposed, which consists of two mathematical models. The first model includes the integration of the differential equation in partial derivatives, but without taking into account the force of inertia of the cleaning piston. The second mathematical model consists of two stages. At the first stage, when integrating the equation in partial derivatives, an integral equation is obtained, in which the unknown function is the inertia force of the cleaning piston. At the second stage, this equation is solved approximately by a numerical method and the deflection of the gas pipeline axis and bending moments along its open section are determined. The aim of this article is to obtain an integral equation in which the unknown function is the force of inertia of the cleaning piston. To obtain this equation, inhomogeneous differential equation is solved in partial derivatives for the deflection of the axis of the gas pipeline, in which in its right part, in addition to the gravity force of the piston, there is an unknown function of its inertia force. This problem, as in the case without taking into account the force of inertia, was solved by the Fourier method. Here, the right part of the equation was decomposed into an infinite series, which is the sum of the products of the eigenfunctions of free oscillations of the pipeline section and the unknown function of time. After finding this function, the time function in the Fourier method was determined, and hence the solution of the problem in the form of an infinite series, the terms of which decrease rapidly, was obtained. Using the solution of this problem, we receive an integral equation in which the unknown function is a function of the inertia force of the cleaning piston.