Mathematical models of nonlinear longitudinal-cross oscillations of object with moving borders

The nonlinear formulation of problems for describing longitudinal-cross oscillations of objects with moving borders is noted. These mathematical models consist of a system of two nonlinear partial differential equations with the higher time derivative of the second order and the fourth-order by the spatial variable. The nonlinear boundary conditions on moving boundary have a higher time derivative of the second order and the third-order by the spatial variable. The geometric nonlinearity, visco-elasticity, the flexural stiffness of the oscillating object and the elasticity of the substrate of object are taken into account. Boundary conditions in the case of energy exchange between the parts of the object on the left and right of the moving boundary are given. The moving boundary has got a joined mass. The elastic nature of borders joining is considered. The longitudinal-cross oscillations of objects with moving borders of high intensity can be described by the resulting differential model. The Hamilton's variational principle is used in the formulation of the problem.