Mathematical Simulation of Nonlinear Vibrations of a Channel Wall Interacting with a Vibrating die Via Viscous Liquid Layer

Abstract

The article considers the developed mathematical model and investigates the dynamics of the interaction of a channel wall supported by a nonlinear spring with a vibrating opposite wall through a viscous fluid layer filling the channel. A flat slotted channel formed by two absolutely rigid rectangular walls, parallel to each other was investigated. One of the channel dimensions in the plan was much larger than the other, which leads to the transition to a plane problem. The bottom channel wall rested on a spring with a cubic nonlinear characteristic, and the upper wall was a die oscillating according to a given law. The gap between the walls was assumed to be much smaller than the channel longitudinal dimension, and the amplitudes of wall vibrations were much less than the channel gap. The movement of the viscous fluid in the channel was considered to be creeping. The mathematical model of the channel under consideration consisted of an equation of the dynamics of a single-mass system with a spring having a cubic nonlinearity, as well as the Navier --- Stokes and continuity equations, supple-mented by the boundary conditions for fluid nonslip on the channel walls and its free outflow at the ends. The steady-state nonlinear vibrations of the bottom channel wall at the fundamental frequency were studied, and its hydroelastic response was determined. The proposed model can be used to study nonlinear vibrations of elastically fixed elements that are in contact with liquid and are parts of modern devices and assemblies