Building of a Mathematical Model for Solving the Elastic–Gas-Dynamic Task of the Gas Lubrication Theory for Petal Bearings

Sergey Surovtcev, Elena Zadorozhnaya (), Nadezhda Khozenyuk and Oleg Mashkov
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Sergey Surovtcev: Department of Motor Transport, Institute of Engineering and Technology, South Ural State University, 76 Prospekt Lenina, 454080 Chelyabinsk, Russia
Elena Zadorozhnaya: Department of Motor Transport, Institute of Engineering and Technology, South Ural State University, 76 Prospekt Lenina, 454080 Chelyabinsk, Russia
Nadezhda Khozenyuk: Department of Motor Transport, Institute of Engineering and Technology, South Ural State University, 76 Prospekt Lenina, 454080 Chelyabinsk, Russia
Oleg Mashkov: Joint Stock Company “Special Design Bureau “Turbina”, 2-b, Prospekt Lenina, 454007 Chelyabinsk, Russia

Mathematics, 2023, vol. 11, issue 13, 1-19

Abstract: Petal gas bearings are widely used in industrial turbine installations. However, there are practically no mathematical models that take into account the nonlinear dependence of the elastic element deformations of the gas-dynamic bearing on the pressure. It is very difficult to obtain a converged solution with a given accuracy in an acceptable time. On this basis, a mathematical model of the motion of a gas-dynamic bearing with a petals package was proposed. The model takes into account the pressure dependence of the elastic element deflections of the bearing, as well as the bearing design features and its elastic elements. A feature of solving the presented task was the joint solution of two subtasks: gas lubrication and deformations of the elastic elements of the bearing. The problem was that the contact areas of the elastic elements are not known in advance. The search for deflections or the formulation of the elasticity problem was based on the Lagrange variational principle. Petals are in the shape of thin cylindrical shells. The solution was achieved by minimizing the potential energy of the system of deformed petal shells using the first-order gradient method. The solution to the gas-dynamics problem was achieved by applying an explicit finite-difference approximation in time. To implement the proposed model, a numerical algorithm was developed to solve the related tasks of dynamics and elasticity by combining the methods of variational calculus, optimization, and an explicit finite-difference time scheme. The order of accuracy of convergence of the solution corresponded to 10 −5 . The study demonstrated that the load-carrying capability of the bearing increased by 2–4 times with an increase in the number of petals. The results of experimental studies allowed us to estimate the interval for the ascent speed of the rotor, which was used as the initial conditions for numerical modeling.

Keywords: elastic–gas-dynamic task; first-order gradient method; variational principle of lagrange; finite difference method; petal bearing (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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