# Derivation of the dynamic equation for a geometrically nonlinear plate interacting with a thin layer of a viscous incompressible fluid

DOI: 10.34759/trd-2021-121-06

### Аuthors

Gyagyaeva A. G.1*, Kondratov D. V.2**, Mogilevich L. I.3***

1. Russian Presidential Academy of National Economy and Public Administration, 23/25, Cathedral str. Saratov, 410031, Russia
2. Saratov State University named after N. G. Chernyshevsky, 83, Astrakhanskaya str., Saratov, 410012, Russia
3. Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

*e-mail: gyagaevaa@mail.ru
**e-mail: kondratovdv@yandex.ru
***e-mail: mogilevichli@gmail.com

### Abstract

At present, different methods for solving the problem of hydroelasticity are applied. They allow solving a set of problems in question be means of assumptions and limitations with a required precision and reliability. To make engineer decisions while examining hydroelasticity problems, specialized computer software is used. It allows simplifying the calculating process of hydroelasticity problems and demonstrating a graphical solution of the problems. Thin-wall structures interact with viscous incompressible fluid, particularly plates and shell structures. They are used in machine industry, in instrument making industry, in aviation and space industry. Thus, one needs to build mathematical models and apply program systems to make decisions in designing machines and devices.The article consider the model of a mechanical system consisting of an absolutely rigid body (vibrator); an elastic rib, rectangular plate (stator) and viscous incompressible fluid. Using the Hamilton’s variation principle, the equation for the elastic geometrically nonlinear plate dynamics is derived. The mathematical model of the presented mechanical system consists of the incompressible fluid dynamics equations, the elastic geometrically nonlinear ribbed plate dynamic equations, the equation of motion for the absolutely rigid vibrator and the corresponding boundary conditions. The program for the analytical derivation of the dynamic equations for an elastic irregular nonlinear stator was developed. It takes into account the boundary conditions of a free support and the conditions of free fluid flow on the ends. The constructed mathematical model can be applied for the design of hydrodynamic supports. The mathematical model can be used in designing of hydrodynamic bearings that are operated in machine industry, in instrument making industry, in aviation and space industry.

### Keywords:

viscous incompressible fluid, elastic geometrically irregular plate, absolutely rigid vibrator

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