Oscillations of a Channel Wall on a Nonlinear Elastic Suspension Under the Action of a Pulsating Layer of Viscous Gas in the Channel

We consider aeroelastic oscillations of a rigid channel wall having an elastic suspension with hardening cubic nonlinearity and interacting with a pulsating layer of a viscous gas in the channel. The study is based on the reduction of the coupled boundary value problem of mathematical physics with the equations of viscous gas dynamics and the equation of rigid wall dynamics included as well as the corresponding boundary conditions to the Duffing oscillator equation. Initially, the aeroelastic oscillation problem is formulated for the channel wall under consideration and analyzed asymptotically by the perturbation method. As a result, the equations for the dynamics of a viscous compressible gas are linearized and then solved iteratively. The gas reaction force on the rigid wall is determined, and an equation is obtained for the aeroelastic oscillations of the channel wall; this equation is a generalized Duffing oscillator equation. Based on solving this equation by the harmonic balance method, we determine and study the nonlinear aeroelastic response of the channel wall as well as its phase response. It is shown that the allowance for compressibility of the viscous gas leads to an increase in the values of resonance frequencies and the wall oscillation amplitudes, as well as to an additional phase shift of the disturbing force, which is determined by the given profile of the pressure pulsation at the channel ends.