Hybrid system of differential equations describing systems of solids attached to a Timoshenko beam

The article proposes a generalized mathematical model described by a hybrid system of differential equations of a prescribed structure for one class of mechanical systems consisting of a system of connected solids, elastically attached to a Timoshenko beam. Theoretical background of the study of free vibrations has been developed for a generalized mathematical model, in particular, an analytical and numerical method for constructing a frequency equation based on the consideration of a boundary value problem for the corresponding hybrid system of differential equations. In this case, natural frequencies are in fact eigenvalues for which there exists a solution to a boundary value problem. A calculated example is given that shows the reliability and versatility of the proposed method for studying free vibrations of mechanical systems, which are systems of connected solids elastically attached to a Timoshenko beam.