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Ермолицкая А. Д. Ermolitskaya A. D. 2025 5 Keywords: transverse vibrations, rods, concentrated loads, internal friction, forced vibrations, Laplace operational calculus. Transverse vibrations are encountered in the calculation of various kinds of beams subjected to the action of variable transverse forces, in the study of vibrations of bridges and railway rails under the action of moving loads, in the study of vibrations arising in the hull of ships under the action of the forces of inertia of unbalanced parts of machines, etc. Vibrations can be conditionally divided into two types: free and forced. Free oscillations occur without external excitation forces, while forced oscillations occur under the action of external loads that change with time. In the thesis work forced oscillations are considered. We will assume that the oscillations occur in one of the main planes of the rod. In this case we will deal with plane bending. The bending plane will be taken as the XY coordinate plane. To determine the equation of transverse vibrations in the XY plane of a rod of constant cross-section without taking into account damping, inertia of section rotation and shear forces, it is necessary to solve the differential equation, provided that the rod has axes of symmetry and is not subjected to concentrated loads: 4 4 + 2 2 = 0 , (1) where — bending stiffness in the vibration plane, ( , ) — displacement of the neutral axis point of the rod with abscissa x, – unit volume mass, – cross-sectional area, — time, If we introduce a dimensionless coordinate = , equation (1) takes the form: 4 4 + 4 2 2 = 0 (2) The solution (1) is obtained in the form: ( , ) = ( cos + sin ) ( ) , (3) where ( ) = ch + sh + cos + sin . The arbitrary constants , , , are determined from the boundary conditions. We briefly review the derivation of the equations of transverse vibrations of rods with consideration of damping for various boundary conditions and find their analytical solutions. It is assumed that the internal friction forces will be proportional to the strain rate [5]. Therefore, introducing internal friction, let us replace Hooke's law = by the dependence