Petr Alexandrovich Velmisov, Ulyanovsk State Technical University, Doctor of Physics and Mathematics, Professor; graduated from the Faculty of Mechanics and Mathematics of Saratov State University; Head of the Department of Higher Mathematics at Ulyanovsk State Technical University; an author of articles and monographs in the field of aerohydromechanics, aerohydroelasticity, and mathematical modeling. [email: velmisov@ulstu.ru]P. Velmisov, Andrei Viktorovich Korneev, Ulyanovsk State Technical University, graduated from the Faculty of Information Systems and Technologies at Ulyanovsk State Technical University; PostGraduate Student at the Department of Higher Mathematics of Ulyanovsk State Technical University; an author of articles in the field of aerohydroelasticity, optimal control, and algorithms development. [email: a.korneev1@gmail.com]A. Korneev


Mathematical Modeling in the Problem of Dynamic Stability of a Pipeline
The paper presents mathematical models for a viscoelastic pipeline that is a hollow rod containing flowing the fluid (gas). The article is devoted to the problem of the dynamic stability of a pipeline. Linear and nonlinear models describe partial differential equations for an unknown function (the displacement of the pipeline points from the equilibrium state). By means of Lyapunov functionals designed stability theorems are formulated and analytical stability conditions for the parameters of the mechanical system and different types of initial conditions are found. The obtained stability conditions are sufficient but not necessary. A mathematical software package is developed to solve this problem. It allows to find an approximate numerical solution of differential equation for describing pipeline vibration and to plot a stability area appropriate to both sufficient and necessary stability conditions. A numerical experiment of stability areas designing is conducted on the basis of the software package. The obtained numerical results are interpreted and compared with analytical stability conditions. The influence of the model parameters variation on the stability is researched. Mathematical modeling, viscoelastic pipeline, aerohydroelasticity, stability, functional, partial differential equations, numerical methods, galerkin method.

