A mathematical model of the layered plate throwing by detonation products

This study aims to investigate the forced vibrations caused by a time-harmonic force from a pre-stressed bi-layered plate resting on a rigid foundation under the action of a time-harmonic pointwise loading. Our investigation was conducted according to a piecewise homogeneous body model utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies. Throughout this study, we assumed that there is complete contact between the plate and the rigid foundation. The purpose of this study is threefold: the development of a mathematical model to investigate the dynamic response of the pre-stressed bi-layered plate, the analysis of the frequency response of the plate under consideration, and finally, demonstrating the relationship between the initial stress and the dimensionless frequency of the plate. We solved the mathematical model by employing the finite element method. We present our numerical results on the dynamic behavior of the plate. In particular, we have shown that an increase in the values of the aspect ratio of a plate under fixed thickness leads to a decrease in the normal stress resonance values.

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Mathematical Model of Oscillations of a Three-Layered Channel Wall Possessing a Compressible Core and Interacting with a Pulsating Viscous Liquid Layer

Herald of the Bauman Moscow State Technical University Series Instrument Engineering ◽

10.18698/0236-3933-2019-6-4-18 ◽

2019 ◽

pp. 4-18

Author(s):

E.D. Grushenkova ◽

L.I. Mogilevich ◽

V.S. Popov ◽

A.V. Khristoforova

Keyword(s):

Mathematical Model ◽

Viscous Fluid ◽

Channel Wall ◽

Stokes Equations ◽

Fluid Layer ◽

Fluid Pressure ◽

Pressure Change ◽

Narrow Channel ◽

Distribution Functions ◽

Layered Plate

The paper deals with the formulation of a mathematical model to study a dynamics interaction of a three-layered channel wall with a pulsating viscous fluid layer in a channel. The narrow channel formed by two parallel walls was considered. The lower channel wall was a three-layered plate with a compressible core, and the upper one was absolutely rigid. The face sheets of the three-layered plate satisfied Kirchhoff's hypotheses. The plate core was considered rigid taking into account its compression in the transverse direction. Plate deformations were assumed to be small. The continuity conditions of displacements are satisfied at the layers' boundaries of the three-layered plate. The oscillations of the three-layered channel wall occurred under the action of a given law of pressure pulsation at the channel edges. The dynamics of the viscous incompressible fluid layer within the framework of a creeping motion was considered. The formulated mathematical model consisted of the dynamics equations of the three-layered plate with compressible core, Navier --- Stokes equations, and the continuity equation. The boundary conditions of the model were the conditions at the plate edges, the no-slip conditions at the channel walls and the conditions for pressure at the channel edges. The steady-state harmonic oscillations were investigated and longitudinal displacements and deflections of the plate face sheets were determined. Frequency-dependent distribution functions of amplitudes of plate layers displacements were introduced. These functions allow us to investigate the dynamic response of the channel wall and the fluid pressure change in the channel. The elaborated model can be used for the evolution of non-destructive testing of elastic three-layered elements contacting with a viscous fluid layer and being part of the lubrication, damping or cooling systems of modern instruments and units.

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