Boundary-Value Problems of the Dynamic Behavior of Two-Dimensional Elastic Systems with Moving Objects

The interdependent dynamic behavior of a two-dimensional elastic system in the form of a one-dimensional mechanical object moving on a band is considered. The Lagrangian density of the two-dimensional system depends on the generalized coordinates and their derivatives up to and including the second order, and the Lagrangian of the moving object as one of the generalized coordinates contains the motion law, which is an unknown function of this problem. The physically and mathematically correct conditions at the moving boundary have been found as a result of formulating the self-consistent boundary-value problem based on the Hamilton variational principle. The problem of the unseparated motion of a rod, which performs bending and torsional vibrations, along a plate with consideration for the rotational inertia of its components is formulated as an example. The differential and integral laws of the change in energy and wave momentum are derived for both the entire complex system and its isolated parts. The relationships true at the moving boundary are established between the components of the energy flux density vector and the wave momentum flux density tensor.