SPATIAL VIBRATIONS OF A PIPELINE HAVING A SLIDING SUPPORT UNDER THE ACTION OF VARIABLE INTERNAL PRESSURE

Abstract

This paper studies spatial vibrations of a pipeline having a sliding 'weightless' support under the action of variable internal pressure. The pipeline vibrations occur around an axis running through two supports, one of which is assumed to be immovable and the other slides freely on an ideally smooth horizontal plane. The pipe filled with an incompressible transport medium is surrounded by a viscous incompressible fluid. The internal pressure in the pipe-line is given by the harmonic law. Gravitational forces, Coriolis inertial forces, Archimedes buoyant force, viscous drag forces and those associated with the acceleration of the pipe transverse motion in the surrounding medium are taken into account. In this case the veloc-ity of the transport medium motion, flow friction and linear inertial forces are not taken into account. Deformations of the pipe due to its exit out of the flexure plane are assumed to be small. That is why, spatial vibrational movements of the pipeline are considered to consist of transferable rotational movements around the axis running through the supports and rela-tive flexural displacements in the flexure plane. As a result of these assumptions, the solution to the problem is reduced to integrating a set of two nonlinear partial differential equations describing rotational and flexural vibrations of the pipeline. The function that satisfies the boundary conditions is taken according to the first basic mode. Then using the Bubnov-Galerkin procedure, this set is reduced to two nonlinear ordinary differential equations regarding the bending angle and deflection at mid-span of the pipe in terms of time. The Runge-Kutt numerical method is applied to integrate the resultant set of equations under specific initial conditions. This numerical solution is then subjected to the discrete Fourier transform and Poincare mapping. Calculations were performed for a steel pipeline filled with fluid and gas. The pipeline executed vibrations in air and water media. The magnitude of the average, the amplitude and the initial phase of the internal pressure variable component took the only one value each, and the frequency of this pressure took three values. The computational results are given as time-dependence graphs of the mid-span deflection dynamic constituent, bending angle and sliding support displacement, their phase pattern, Fourier spectra and Poincare maps. The mid-span trajectories are also constructed to give a more visual representation of the pipeline in space over a given time interval.