Nonlinear Oscillations of a Plate Resting on a Nonlinear Elastic Foundation and Forming the Bottom of a Plane Channel Filled with a Viscous Gas

This paper proposes a mathematical model for nonlinear oscillations of a Kirchhoff plate resting on an elastic foundation with hardening cubic nonlinearity and interacting with a pulsat- ing layer of viscous gas. The plate is the bottom of a narrow plane channel filled with the viscous gas; the upper channel wall is rigid. Within this model, the aeroelastic response and phase response of the plate to pressure pulsation at the channel ends are determined and investigated. The formulated model allows us to simultaneously study the effect on the plate vibrations of its dimensions and the material physical properties, the nonlinearity of the plate elastic foundation, the inertia of gas motion, as well as the gas compressibility and its dissipative properties. The model was developed based on the formulation and solution of the nonlinear boundary value problem of mathematical physics. The equation of plate dynamics together with the equations of viscous gas dynamics for the case of barotropic compressible medium, as well as boundary conditions at the channel ends and gas contact surfaces with the channel walls, constitute this coupled problem of aeroelasticity. The gas dynamics was considered similarly to the hydrody- namic lubrication theory, but with retention of inertial terms. Using the perturbation method, the asymptotic analysis of the aeroelasticity problem is carried out, which made it possible to linearize the equations of dynamics for the thin layer of viscous gas and solve them by the iter- ation method. As a result, the law of gas pressure distribution along the plate was determined and the original coupled problem was reduced to the study of a nonlinear integro-differential equation describing the aeroelastic oscillations of the plate. The use of the Bubnov – Galerkin method to study the obtained equation led us to reduce the original problem to the study of the generalized Duffing equation. The application of the harmonic balance method allowed us to determine the primary aeroelastic and phase responses of the plate in the form of implicit functions. A numerical study of these responses was carried out to evaluate the influence of the plate’s nonlinear-elastic foundation, gas motion inertia and its compressibility.