Determining frequencies of transverse vibrations for crossovers and dead ends of gas pipelines

The paper presents a stationary equation for bending deformations of a hollow rod derived by means of variational calculus. Further, the authors introduce into consideration an inertial term as consistent with a standard procedure and obtain the wave equation for pipe bending vibrations. Applying the method of separation of variables, the resulting hyperbolic equation of vibrations is reduced to an ordinary fourth-order differential equation for a standing wave on the axial line of the pipe. Fundamental solutions to the latter equation are referred to as the Krylov functions, while the standing wave is represented as a linear combination of two independent Krylov functions. The solution to the obtained homogeneous equation is only found at certain values of characteristic parameters which are amounted to a countable set for each case of fixed ends of the pipeline segment. Thus, the whole frequency spectrum of the pipe bending vibrations is determined, and the main vibration mode is revealed for each case of fixed pipeline ends.