Modeling Nonlinear Oscillations for the Wall of a Narrow Channel Interacting with Viscous Liquid

Abstract

AbstractIn this paper, we considered nonlinear vibrations of a rigid plate induced by pulsating liquid pressure. The rigid plate is supported by a spring and is the top wall of a narrow parallel-walled channel filled with viscous liquid. We assumed the opposite channel wall is motionless, and the law of liquid pressure change at the channel ends is given as harmonic one, as well as the supporting spring possesses a hardening cubic nonlinearity. The plane coupled problem of hydroelasticity consisting of the dynamics equations for a viscous incompressible liquid and the channel wall motion equation as a spring-mass system, as well as the boundary conditions for the pressure at the channel ends and the liquid velocity at the channel walls, was formulated for the channel under consideration. Due to the narrowness of the channel, the liquid motion in it was studied as a creeping one. The hydrodynamic parameters of the liquid layer in the channel were determined, which made it possible to find the driving force acting on the channel wall, as well as the damping coefficient due to the liquid squeezing by the channel wall. As a result, the Duffing equation was obtained for the study of the channel wall nonlinear oscillations. Using the harmonic balance method, a solution to this equation was found and the study of the channel wall hydroelastic response for the case of anharmonic vibrations was carried out.KeywordsNonlinear oscillationsHydroelasticityParallel-walled channelViscous fluidMathematical modeling