Dynamics and stability of the elastic aileron of an aircraft wing in a subsonic streamline

Background. At designing of constructions and devices interacting with a stream of gas, it is necessary to solve the problems connected with research of stability, demanded for their functioning and reliability of operation. On the one hand the influence of a stream can lead to the negative effects which causes infringement of necessary functional properties of elastic constructional elements, up to their destruction (for example, lead to a condition of instability owing to increase in amplitude or frequency of fluctuations to critically admissible values). On the other hand the phenomenon of initiation of fluctuations of elastic elements at the aerohydrodynamic influence, stated above as negative, is necessary for functioning of some technical devices. The purpose of this work is to research dynamics and dynamic stability of the elastic flap (aileron) of a wing taking into account a flow a subsonic stream of ideal gas (liquid). Materials and methods. Influence of gas or liquid (in a model of ideal incompressible environment) on constructions is defined from the asymptotic linear equations of aerohydromechanics. For the description of dynamics of the elastic aileron the linear theory of a solid deformable body is used. According to the specified assumptions the mathematical model of a wing with the elastic aileron taking into account a flow of a subsonic stream of gas or liquid is constructed. In the model the elastic communication of the aileron with a wing and the variable thickness of the aileron are considered. The model is described by the related system of differential equations in partial derivatives containing both the equations of movement of the gas-liquid environment and the equation of dynamics of the deformable aileron for two unknown functions - the potential of gas velocity and deformation of the aileron. On the basis of methods of the theory of complex variable functions it was succeeded to expel the potential of velocity from the system of equations and to consolidate the solution of the aerohydroelasticity problem to research the integro-differential equation containing only the unknown function of deformation of the aileron. Research of stability was conducted on the basis of creation of the positively certain functional corresponding to the received integro-differential equation with partial derivatives. Determination of stability of an elastic body corresponds to the concept of stability of dynamic systems by Lyapunov. The solution of the equation for the function of deformation of the aileron was constructed on the basis of Galerkin's method with carrying out a numerical experiment. Results. The mathematical model of a wing with the elastic aileron of variable thickness taking into account a flow of a subsonic stream of gas or liquid was constructed. The case of elastic fixing of one end and free fixing of another end of the aileron is considered. Dynamic stability of the aileron was investigated. On the basis of the constructed functional the sufficient conditions of dynamic stability imposing restrictions on velocity of a stream of gas, flexural rigidity of the aileron and other parameters of mechanical system were received. Research of dynamics of the aileron of variable thickness by Galerkin's method was conducted. For the concrete examples of mechanical systems the schedules of deformations of the elastic flap at various laws of plate thickness change and various velocities of a running stream were constructed. Conclusions. The received sufficient stability conditions, imposing restrictions on parameters of the mechanical system, provide stability of fluctuations of the elastic aileron, namely: small deformations of the aileron in the initial timepoint (i.e. small initial deviations from the position of balance) will correspond to small deformations at any timepoint. For the parameters which aren't meeting these conditions, it is impossible to make certain conclusions about stability of fluctuations of the aileron that is shown by concrete examples of mechanical systems.