Modeling of Hydroelastic Vibrations of the Channel Wall on an Elastic Foundation with Softening Nonlinearity for Predicting the Nonlinear Response of the Channel Wall

Cyber-physical systems are widely used to monitor and support production processes, which requires the development of mathematical models for reliable prediction of the interaction of heterogeneous media and structural elements. The chapter deals with modeling the interaction of a viscous fluid with an elastic structure on a nonlinear-elastic foundation. The problem of hydroelasticity for two walls between which there is a viscous fluid is considered. One of the walls is a plate on an elastic foundation with softening cubic nonlinearity, while the opposite wall is a rigid vibrating die. The hydroelasticity problem is formulated including the system of equations of viscous fluid, Kirchhoff plate resting on softening nonlinear-elastic foundation, and boundary conditions at the channel ends and the contact surfaces of heterogeneous media. Asymptotic analysis of the formulated problem has been performed and distribution of fluid pressure in the channel has been found. As a result, the original problem is reduced to a single equation of hydroelastic vibrations of the plate. For its solution, the Bubnov-Galerkin method and the harmonic balance method are applied, which allowed obtaining and numerically investigating the nonlinear hydroelastic and phase response of the plate. We show that the dynamics of the considered oscillating system is described by the generalized Duffing equation. Calculations have revealed the possibility of the appearance of unstable vibrations in the region of resonance frequencies.