The statement of the problem of vibrations of a beam with a moving spring–loaded support carrying an attached mass is obtained. When the support is not absolutely rigid, energy exchange occurs through the moving boundary. In this regard, there is a difficulty in writing the boundary conditions. To formulate the problem, we used the variational principle of Hamilton. In this case, the viscoelastic properties of the beam material are taken into account. The problem posed includes the differential equation of vibrations, initial conditions for the bent axis of the beam and for the added mass, 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.
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