Oscillations properties of the dynamic fractal structures

A wide class of systems in nature and engineering consists of self-similar fractal structures, where each cell repeats the structure of the previous one on a certain scale. However, the well-known B. Mandelbrot's fractals that use the scaling of geometric parameters do not reflect the dynamic properties of the system well enough. Therefore, when studying the system dynamics, it is suggested that structures which have a scaling of dynamic, parameters, should be considered as a special class of vibratory system. Such class of structures where stiffness and inertial parameters change with the same scale factor from cell to cell can be called dynamic fractals. Research has shown that such structures have a number of specific wave and vibrational properties. Thus, the dynamic fractal is equivalent in terms of frequency to a certain periodic structure, which is a band-pass filter. It has been also shown that the dynamic fractal having a scale factor greater than one with increasing magnitudes of stiffness and inertial parameters along its length has good vibroisolation properties. The main advantage of such a lattice is intensive vibration attenuation in all frequency ranges including in resonant modes of the lattice, which favorably compares with periodic lattices. On the contrary, the dynamic fractal having a scale factor less than one with decreasing magnitudes of stiffness and inertial parameters along its length has the properties of amplifying the incoming signal along the structure, which can be used in control systems. We have also studied the oscillation properties of a branched self-similar structure, a dichotomous lattice. It was found that its calculation model in the low-frequency range can be represented as a dynamically fractal structure. The natural frequencies and vibration modes at bending oscillations are determined. The conditions of dynamical self-similarity for such a lattice, its dispersion equation, and a favorable frequency range depending on the scaling factor are obtained.