Qualitative Effects of Vibrations of Reinforcing Ring Elements with Attached Mass, as a Special Case of an Infinitely Long Thin Circular Cylindrical Shell

This paper investigates the effect of a small attached mass on the frequency and form of flexural vibrations of an infinitely long circular cylindrical shell-ring, under plane deformation conditions. The equations of motion for transverse vibrations are used as a mathematical model. These equations are obtained from analogous equations of the Donnell–Mushtari–Vlasov theory of shallow shells. A new approach to the construction of a mathematical model is proposed by the author. This approach suggests that the attached mass in the linear formulation leads to the interaction between the conjugate and the radial flexural forms of vibrations. The new system of dynamic equations obtained by the Bubnov-Galerkin method suggests that such additional inclusions lead to the connectedness and interaction of low-frequency flexural vibrations of the shell with high-frequency radial vibrations. In this case the radial vibrations act as an additional inertial connection between the conjugate flexural forms. It is shown that the radial vibrations are inconspicuous. However, by taking them into account, it is possible to arrive at a qualitative conclusion about the impact of the wave formation parameter on the smaller of the split eigenfrequencies. This parameter depends on the relative thickness of the ring. This means that the effect of reducing the smaller of the split eigenfrequencies of predominantly flexural vibrations depends not only on the attached mass, as it is assumed at the moment, but also on the geometric and wave parameters of the shell. It is established that at certain values of the wave formation parameter, the frequencies and amplitudes of predominantly radial vibrations can be commensurate with the frequencies and amplitudes of predominantly flexural vibrations. This means that when the shell is subjected to dynamic impact, the resonance effect may occur not only at frequencies of the flexural vibrations, as follows from the known analytical solutions, but also at frequencies corresponding to the radial vibrations. The results and conclusions obtained are in qualitative agreement with the available experimental data and numerical calculations. These results can be generalized to the case of vibrations of thin circular cylindrical shells of finite length that carry attached mass.

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