The laws of variation of energy and momentum for two-dimensional elastic systems with moving objects

The self-consistent problem of dynamic behavior of a deformable system consisting of a two-dimensional elastic guide (subsystem 1) and a one-dimensional elastic object (subsystem 2) continuously moving along it is considered. Local and global laws of change in energy and wave momentum are presented in the case when the Lagrangians of the contacting subsystems depend on generalized coordinates and their derivatives of no more than the second order with respect to all space-time variables. Radiation conditions in the considered class of systems are discussed. A comparative analysis of both dispersion and energy characteristics of flexural waves propagating in plates described by two different models is carried out. Critical velocities of motion of a constant load along these plates are found. The dependence of critical velocities on the rigidity coefficient of the elastic foundation and the physical and mechanical properties of the plate is established. The fundamental possibility of converting the energy of oscillations of a two-dimensional elastic guide into the energy of translational motion of a one-dimensional object is demonstrated. The force caused by the wave pressure acts as a mediator of such transformation, the expression for which is obtained in a universal form through the Lagrangian of a two-dimensional system. The dependence of the coefficient of transformation of wave energy into the energy of translational motion of an absolutely rigid fastening on the speed of its motion and the parameters of a two-dimensional system is constructed.