On one solution of the problem of oscillations of mechanical systems with moving boundaries

The problem of oscillations of bodies with moving boundaries, formulated as a differential equation with boundary and initial conditions, is a non-classical generalization of a hyperbolic problem. To facilitate the construction of a solution to this problem and to justify the choice of the solution type, equivalent integro-differential equations with symmetric and non-stationary kernels and non-stationary integration limits are constructed. The advantages of the integro-differential equation method are revealed when moving to more complex dynamic systems carrying concentrated masses oscillating under the action of moving loads.
The method is extended to a wider class of model boundary value problems that take into account bending rigidity, resistance of the external environment and the rigidity of the base of the oscillating object. The solution is given in dimensionless variables with an accuracy of up to the values of the second order of smallness of relatively small parameters characterizing the velocity of the boundary. An approximate solution is found for the problem of transverse vibrations of a viscoelastic beam with bending rigidity, taking into account the action of damping forces.