The Modeling of Nonlinear Rotational Vibration in Periodic Medium with Infinite Number of Degrees of Freedom

The subject of the work is hypothetical 2D periodic medium with an infinite number of beams, each with a single degree of freedom, which allows on the rotation of the single plate around its center of gravity. Interaction by electrostatic forces of beams to each other, so that rotation of any of them induces rotation of the adjacent beams has been assumed. The motivation to undertake research in this field and adoption of such assumptions are potentially possible mechanism of optical phenomena in the atmosphere. Thus, in case the physical atmospheric phenomena, the hypothetical beams would be implemented by electrically charged plates of ice crystals. The aim of the work is to obtain the continuous nonlinear vibration model of such a medium. Large angle of rotation of plates was assumed, however each beam interacts with a 4-neighbors. The finite difference method and certain heuristic assumptions about the superposition of interactions have been used for modeling. As a result, the differential equation describing the behavior of such a medium in a continuous manner has been obtained. It is shown that under those conditions obtained final model equation is similar to the sine-Gordon equation. The next part of the work is possible examples of solutions coming from such a model for different sets of input data.