Modeling Aeroelastic Response of the Channel Wall Having the Suspension with the Softening Cubic Nonlinearity

This paper deals with the formulation and solution of an aeroelasticity problem for the study of nonlinear vibrations of a channel wall having a suspension with softening cubic nonlinearity. The narrow channel formed by two parallel rectangular rigid walls and filled with a pulsating viscous gas was investigated. The upper wall of the channel was assumed to be motionless, while the lower one has the nonlinear-elastic suspension that allows it to oscillate in the vertical direction. The wall oscillations are due to a given harmonic law of pressure variation at the ends of the channel. The plane nonlinear aeroelasticity problem for the channel bottom wall interacting with the viscous gas layer was formulated. Due to the narrowness of the channel, the transition to the equations of dynamics of the thin viscous gas layer was realized. Asymptotic analysis of the formulated problem of aeroelasticity by the perturbation method was carried out and linearized equations of dynamics for viscous gas were obtained. The pressure of the viscous gas in the channel was determined by the iteration method. As a result, the generalized Duffing equation for the aeroelastic vibrations of the channel wall was derived. This equation was solved by the harmonic balance method, and the aeroelastic response and phase response of the considered channel wall were found. The results of the analysis for the behavior of the obtained responses and possible transitions to special cases for incompressible gas and linear characteristic of the suspension stiffness of the channel wall were presented.