Flexural vibrations of the pipeline on elastic supports with moving fluid

In the work investigated the flexural vibrations of the pipeline. Parts of the pipeline on both sides of the sagging section have elastic supports. It is assumed that a constant longitudinal force acts along the neutral line. An incompressible fluid flows through the pipe at a constant average speed. The influence of internal pressure in the pipe on these oscillations is taken into account. The direct problem of determining the eigenfrequencies of flexural vibrations of the pipeline by the Kirchhoff model using Ferrari formulas is solved. The frequency spectrum is determined depending on the fluid pressure, the elasticity of the supports, the velocity of the fluid through the pipe. Particular and limiting cases are considered, for example, when the stiffness of the supports is very large and when they are very small. Graphs of the dependence of the first and second eigenfrequencies on the velocity of the transported liquid at different values of the liquid density parameter are constructed. It is shown that with the growth of the velocity parameter there is a decrease in the natural frequencies of flexural vibrations of the pipeline, and the faster the higher the density parameter of the liquid. It is determined that with an increase in the mass of the liquid per unit length of the pipeline there is a decrease in the natural frequencies of bending vibrations of the pipe. It is found that with the increase in the mass flow through the pipe, the natural frequencies of bending oscillations also decrease. It is confirmed that the frequencies of flexural vibrations of the pipeline are the same for the cases of pipe fastening "rigid fixing–rigid fixing“ and ”free end–free end". The results of the study will contribute to the development of methods of acoustic diagnostics and non-destructive testing and can find technical application for monitoring and diagnostics of pipeline systems.