INTERACTION OF BENDING WAVES PROPAGATING IN AN INHOMOGENEOUS PLATE WITH AN OBSTACLE, WHICH IS A ROD RESTING ON A VISCOELASTIC FOUNDATION

Abstract

A single interaction of a wave propagating in an inhomogeneous plate in the form of an infinite strip with a one-dimensional distributed mechanical object is considered. A distributed object is understood as a rod lying on a viscoelastic foundation that performs bending and torsional vibrations. It is considered that the plate has different parameters to the left and to the right of the rod. Physically and mathematically correct conditions at the interface between a plate and a rod are obtained as a consequence of the formulation of contact problems of the dynamics of two-dimensional elastic systems with one-dimensional loads, based on the variational principle of Hamilton – Ostrogradsky. The frequencies and wave numbers of the secondary (reflected and transmitted) waves, as well as the critical frequency, below which the wave does not propagate in the plate, are determined. Based on the solution of the kinematics problem from a system of linear algebraic equations obtained from the boundary conditions, the coefficients of reflection and transmission of bending waves are found. These coefficients resonantly depend on the frequency of the incident wave. Calculated graphs of the transmission coefficient versus the frequency of the incident wave are given for various parameters of the rod. The conditions for self-isolation and reflectionless passage of a wave through an obstacle are determined. It has been established that the frequency of the maximum vibration isolation is located above the frequencies at which the waves completely pass through the obstacle. An expression is obtained for the force due to the pressure of bending waves on a one-dimensional object. Its constant component is calculated, which vanishes (for a homogeneous plate) in the absence of waves reflected from the obstacle. It is shown that in degenerate cases the results obtained coincide with previous studies by other authors.