About Vibrations of Multiple Solids Attached toan Elastic Rod Considering Initial Conditions

This work considers the initial-boundary problem of vibrations of a me-chanical system in the form of an elastic rod with an arbitrary number of elasticallylongitudinally fixed solids. The solution of a hybrid system of differential equationsdescribing the motion of a mechanical system, including both ordinary differentialequations and partial derivatives, is understood in a generalized sense. The applicationof the concept of a generalized solution is due to the presence in the equations of theDirac delta function, which must be taken into account at the points of connection tothe beam of bodies. A well-known substitution is made that reduces the hybrid systemof differential equations to a system of amplitude equations for solids and a rod. Bytransforming these equations, an orthogonality-type condition is obtained. Solutions aredecomposed into Fourier series by constant amplitudes of solid bodies and eigenformsof the rod with a variable coefficient depending on time. A methodology for determin-ing these variable coefficients, which depend on the natural frequencies of the me-chanical system and the vibration forms of the rod, the amplitudes of solids and theinitial displacements of solids and the rod, is presented. Hilbert space was introducedwith a given system of orthogonal unit vectors, which made it possible to express un-known amplitudes of solids through the initial displacements of solids and the rod. The result of the work was solving the problem in the form of decompositions into Fourierseries in a closed form, which makes it possible to make numerical calculations if theproblem is solved for the natural frequencies and forms of oscillations of the consid-ered mechanical system.