Variational formulation of the problem on vibrations of a beam with a moving spring-loaded support

We pose the problem of vibrations of a beam with a moving spring-loaded support carrying an attached mass. When the support is not absolutely rigid, energy exchange occurs through the moving boundary. As a result, there is a difficulty in writing the boundary conditions. To pose the problem, we use Hamilton's variational principle and take the viscoelastic properties of the beam material into account. The problem posed includes a differential equation for vibrations, initial conditions for a bent axis of the beam and for the added mass, and boundary conditions. The conditions on the moving boundary are written as ratios between the values of the function and its derivatives to the left and right of the boundary.