Energy and Momentum Change Laws for Two-Dimensional Elastic Systems with Moving Objects

Abstract— The article considers a self-consistent problem on the dynamic behavior of a deformable system consisting of a two-dimensional elastic guide (subsystem 1) and a one-dimensional elastic object moving continuously along it (subsystem 2). Local and global energy and wave momentum change laws are presented for the case when the Lagrangians of the contacting subsystems depend on generalized coordinates and their derivatives lower than the second order with respect to all the spatiotemporal variables. The conditions of radiation in the considered class of systems are discussed. A comparative analysis of both dispersion and energy characteristics of bending waves propagating in plates is carried out for two different models. The critical velocities of a constant load moving along these plates are found. The dependence of critical velocities on the rigidity coefficient of an elastic base and the physicomechanical properties of a plate is established. The principal possibility of converting the energy of two-dimensional elastic guide oscillations into the energy of the translational motion of a one-dimensional object is demonstrated. The wave pressure force expressed in a universal form through the two-dimensional system Lagrangian acts as a mediator of such conversion. The dependence of the coefficient of wave energy conversion into the energy of the translational motion of an absolutely rigid fastening on its motion velocity and two-dimensional system parameters is constructed.