Dynamical stability of an elastic element of a flow channel

Background. The purpose of this work is to research the dynamic stability of an elastic element of a wall of a flow channel with a subsonic stream of gas or liquid in it. Materials and methods. Gas or liquid influence (in the model of ideal compressed environment) on the constructions is defined from the asymptotic linear equations of aerohydromechanics. For the description of dynamics of the elastic element representing an elastic plate, the linear theory of a solid deformable body was used. According to the specified assumptions the mathematical model of a channel containing an elastic element on one of walls, with a subsonic stream of gas or liquid, was constructed. The model was described by the related system of differential equations with partial derivatives, containing both the equation of movement of the gas-liquid environment and the equation of dynamics of a deformable element, for two unknown functions - the potential of gas velocity and deformations of an elastic element. Determination of stability of an elastic body corresponds to the concept of stability of dynamic systems across Lyapunov. Research of stability was conducted on the basis of creation of the “mixed” functionals of Lyapunov type for the received related system of equations. Results. The mathematical model of a channel containing an elastic element on one of walls, with a flow of a subsonic stream of gas or liquid, was constructed. On the basis of the constructed functionals the dynamic stability of an elastic element was investigated. The sufficient stability conditions were received. The conditions impose restrictions on velocity of a uniform stream of gas, squeezing (stretching) effort of an element, flexural rigidity of an element and other parameters of the mechanical system. For the concrete example of mechanical systems the stability area on the plane of two parameters “squeezing effort-stream velocity” was constructed. Conclusions. The received sufficient stability conditions, imposing restrictions on parameters of the mechanical system, provide stability of fluctuations of an elastic element, namely: small deformations of an elastic element in the initial timepoint (i.e. small initial deviations from the position of balance) 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 an elastic element.