Non-Linear Oscillations of the Channel Wall Filled with a Viscous Liquid Induced by the Vibrating Foundation

Abstract

The paper considers the issues of mathematical modelling nonlinear vibrations for the wall of a narrow parallel walled channel filled with a viscous incompressible liquid. The upper channel wall is a rigid rectangular plate supported by nonlinear spring with a cubic restoring force, while the bottom one is a fixed rigid rectangular plate. The case of nonlinear oscillations of the upper channel wall due to channel’s foundation vibration has been investigated in the frame of hydroelasticity problem. The main attention is paid to the consideration of steady-state nonlinear oscillations for the upper channel wall and the creeping motion of the liquid in the channel. The liquid layer reaction acting on the upper channel wall is found and the channel’s wall nonlinear oscillation equation is obtained taking into account the energy dissipation due to the liquid viscosity. It was shown that this equation coincided with the Duffing one. The solution of the nonlinear oscillation equation was carried out by the harmonic balance method. Based on the found solution, the hydroelastic response of the channel wall on a nonlinear elastic suspension to the channel foundation vibration was determined.