Dynamic instability of flexible plane into acoustic flow

Abstract

As a feedback system a flexible lifting structure may be dynamically unstable for certain relations between speed, bending stiffness, density and material of the structure. Dynamic aeroelasticity phenomena are caused by interaction of the inertia, aerodynamic and elastic forces. This work presents the mathematical model of the oscillatory motion in the subsonic flow as a mixed boundary value problem of aeroelasticity of the elastically deformable thin plate. The aerodynamic approach has been applied to solve the represented problem, namely the problem of bending the plate was considered as an auxiliary problem to determine the perturbed velocity of the plate points, forming a distribution of fluid pressure on the plate. The basic idea of the approach to the solving of the coupled problem is to ensure equivalence and keep the specific character for the methods developed in the basic sciences (aerodynamics, elasticity theory). In the case of hinge for any distribution of the transverse load the solution of the problem of cylindrical bending of plate was obtained by the method of Green's functions using Krylov’s beam functions. In the space of generalized functions of slow growth the aerodynamic part of the related problems is reduced to a singular integral equation by using of integral calculus and the theory of complex variable functions. The results of numerical experiments confirm the existence of the critical conditions in which there is a sharp change in the aerodynamic load on the plate. Values of critical numbers at which loss of stability occurs depend on the properties of the plate (hardness, weight), the properties of the liquid and the oscillation frequency. Each eigenvalue corresponds to the normal function defining the shape deformations related to certain type of natural fluctuations. Behavior of obtained own deformation forms is consistent with the character of the normal functions described in the Bisplingoff’s monograph "Aeroelasticity". The questions of the dynamic interaction quality of the system "elastic plane - compressible flow" considering in this work are part of a comprehensive study of aeroelastic behavior of the lifting structure in motion in continuous medium.