STABILITY INVESTIGATION OF PANEL AXIAL MOVEMENT TAKING INTO ACCOUNT HYDROTHERMOELASTIC INTERACTION

Abstract

An axial movement of a web under axial tension is considered. The web is modelled by a thermoelastic continuous panel (beam), which is supported by a system of fixed rollers realizing the simply supported conditions at the ends of panel spans. In the process of axial movement the panel provides transverse thermoelastic vibrations, which are described by applications of Eulerian coordinates. Arising transverse vibrations define the local accelerations, the Coriolis accelerations and the centripetal accelerations. It is supposed that the thermal loading on the panel leads to a stationary temperature distribution on the panel thickness and to corresponding temperature stresses and strains. It is treated that the axial movement and arising transverse vibrations are realized with interaction of panel and external medium modelled by an ideal fluid moving with a constant velocity. The exact integrodifferetial expression received from the theory of thin aerodynamic profiles is applied for the reaction of a fluid moving with a fixed velocity. The approximate middle value approximating the fluid reaction by a differential expression is defined. It makes possible to reduce the integrodifferential equation of panel dynamics to the simplified differential equation. The described model supposes that the hydrothermoelastic deformations are small and it corresponds to approximation of added mass of fluid used for hydrothermoelastic reaction. This leads to essential simplifications of the developing hydrothermoelastic model and it makes possible the application of analytical and semi-analytical approaches for the analysis of dynamics and stability of the considered system. For convenience the non-dimensional variables and defining parameters are introduced. The numerical realization of the hydrothermoelastic model describing the non-stationary behavior of a panel moving in the ideal fluid flow is based on the application of semi-discrete scheme of Galerkin's method. With the help of two shape functions the problem of non-stationary dynamics describing by the partial differential equation is reduced to integration of the system of two ordinary differential equations. As another application of the hydrothermoelastic model it is considered the problem of static instability of a moving panel in a flow of ideal fluid. The introduction of the auxiliary variable makes possible to formulate the boundary eigenvalue problem and to find the critical values of static instability parameters in the analytic form. Thus, the effectiveness of the described hydrothermoelastic model is illustrated by the examples of semi-discrete analysis of non-stationary vibrations and the analytic solution of the stability loss problem under the critical temperatures and velocities of system movement.