Geodesic
Investigation of dynamic stability of pipeline
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 2, pp. 106-114

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The paper presents a mathematical model of an elastic pipeline, which is a hollow rod with the fluid (gas) running inside it. The article is devoted to the problem of the dynamic stability of the pipeline. Linear and non-linear models described by partial differential equations for the unknown function, i.e. the displacement of the pipeline points from the equilibrium state. By means of designed functional Lyapunov type, stability theorems were formulated and analytical stability conditions for the parameters of the mechanical system and different types of an initial conditions were founded. The obtained stability conditions are sufficient but not necessary. A mathematical software package was developed to solve this problem. This package allows to find an approximate numerical solution of differential equation for describing pipeline model and plot a stability region appropriate to both sufficient and necessary stability conditions. Full coverage to the design a numerical search algorithm for these regions was given. The obtained numerical results were compared with analytical stability conditions. The influence of the model parameters variation on the stability was studyied.
Keywords: elastic pipeline, dynamics, stability, partial differential equations, numerical methods, galerkin method. 
@article{SVMO_2016_18_2_a12,
     author = {P. A. Vel'misov and A. V. Korneev and S. V. Kireev},
     title = {Investigation of dynamic stability of pipeline},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {106--114},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a12/}
}
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P. A. Vel'misov; A. V. Korneev; S. V. Kireev. Investigation of dynamic stability of pipeline. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 2, pp. 106-114. http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a12/