Stability of solutions of initial boundary value problems of aerohydroelasticity

P. A. Velmisov ; A. V. Ankilov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, Tome 59 (2016), pp. 35-52

Résumé

At designing structures and devices interacting with the flow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The definition of stability of an elastic body, taken in the article, corresponds to the Lyapunov's concept of stability of dynamical system. On the base of a proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic flow of gas or liquid (in an ideal model of a incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensification of technological processes, for example, the process of mixing is considered. The action of these devices is based on the oscillations of elastic elements at the flowing around a of gas or liquid flow. The dynamic stability of the elastic element, located on one wall of the flow channel with the subsonic flow of gas or liquid (in an ideal model of a compressible environment) is investigated. The both models is described by coupled nonlinear system of differential equations for the unknown functions – the potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functionals, the sufficient conditions of the stability, impose restrictions on the free-stream velocity of the gas, the flexural stiffness of the elastic element, and other parameters of the mechanical system is obtained. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.