The oscillations of a circular cylindrical shell of finite length with additional inertia mass are explored. The shell contacts with an elastic medium, which reaction is represented by Winkler model. Such systems can model various elements of buildings, hydro technical constructions, bridges, oil rigs, various pipes etc. The problem of shell vibrations is investigated using the theory of Kirchhoff-Love shells. The shell is loaded with concentrated mass in the form of zero width “mass belt”. The axisymmetric free harmonic oscillations of the shell are explored. In addition, a computer model using ANSYS software has been developed and the shell oscillations have been analyzed by the finite element method. At some values of the system’s parameters, its anomalous behavior occurs. The nature of the system in the vicinity of dispersion curves special points has been investigated. The influence of system parameters on these processes is analyzed using both approaches.