Three-parameter bifurcation problem of elongated plate in a supersonic gas flow

Background. Aeroelasticity problems that are essentially the bifurcation ones, began to be studied in the late 30-ies of the last century, but for their investigation the methods of the bifurcation theory were not applied. The author proposes a technique that allows to explore divergence of an elongated plate in a supersonic gas flow with the corresponding nonlinear ODE of the fourth order, depending on the bifurcation parameters in the exact formulation. In this paper the boundary value problem for a fourth-order differential equation describing the static buckling of the flow around an elastic plate supersonic flow is investigated. The author also proposes an algorithm that allows to explore in the exact formulation the problem of the divergence of a thin flexible elongated elastic simply supported plate, compressed (stretched) by external boundary strains under a low normal load. The dependence of the differential equation on the bifurcation parameters is expressed in terms of the roots of the characteristic equation of the linearized problem, which can be calculated with any degree of accuracy. This representation allows to find the critical bifurcation surfaces and curves in the points' area of which there is developed the asymptotics of branching solutions in the form of converging in the small parameters of the bifurcation parameter deviations from their critical values. Thus, the corresponding small-norm solutions of functional spaces are determined in contrast to many studies that give a qualitative picture of either making or applying the element methods. Materials and methods. The Lyapunov-Schmidt method of the branching theory of nonlinear equations is for the first time applied to problems of divergence of an elongated plate in a supersonic gas flow. Fredholmness of linearization is proved by constructing the Green's function, which is performed for the first time for this type of problems. Results. The problem of divergence of an elongated plate in a supersonic gas flow, described by a nonlinear ODE of the fourth order, depending on the bifurcation parameters, is investigated. The critical manifolds are determined. In the points' area thereof the asymptotics of branching solutions in the form of converging in the small parameters of the bifurcation parameter deviations from their critical values is constructed. The Fredholmness of the linearized problem is proved by constructing the Green's function. Conclusions. The methods, developed in the article, allow to calculate the exact asymptotics of branching stationary or oscillatory solutions in models of aeroelasticity as convergent series in small deviations from the bifurcation parameters.