Solution of the initial-boundary value problem for oscillations of the cascade rigid body systems on an Euler-Bernoulli beam

The paper investigates the natural vibrations of a cascade system of solids mounted on an Euler-Bernoulli beam. A hybrid system of differential equations describing the vibrations of a given mechanical system is derived using Hamilton's variational principle. The solution of this system is understood in a generalized sense. The problem is set for the natural frequencies of the mechanical system, the method for obtaining the equation for the frequencies and forms of natural vibrations is indicated. The orthogonality condition is derived and the initial-boundary value problem is solved with the derivation of formulas for the displacements of the points of the beam axis depending on their coordinates and time, as well as the displacements of an arbitrary number of solid bodies that form a cascade system depending on time in the form of finite series. In this case, the solution of the initial-boundary value problem for fixed physical parameters of the mechanical system is determined by the type of boundary conditions at the ends of the beam, as well as by the choice of initial conditions.