Mechanical vibrations of regular systems with self-similar structure are considered. In such structures, each successive cell is geometrically similar to the previous one. Thus, the translational symmetry is accompanied by the similarity of operation or/and rotation of neighborhood cells (screw symmetry). As an example, longitudinal vibrations of a beam with stepwise section are considered. For this structure the dispersion equations is obtained. It was found that the natural frequencies for self similar structures coincide with corresponding ones for regular mechanical band filter. More over, the pass band depends on similarity coefficient: the higher this coefficient, the narrower the pass band is. In the limit, for a sufficiently high similarity coefficient, the system behaves as a one-frequency one. As for natural modes the amplitudes for each semi-wave linearly increase proportionally to the similarity coefficient, the phase shifts constantly increase
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