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
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bstract 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.
1 Introduction The issues of nonlinear rotational vibrations are not completely new. Rotary discrete systems, such as: Josephson Junction Ladder (JJL) were modeled by several authors, for example: [1, 2]. It has been widely shown, such that systems can be
A. Wirowski (✉) ⋅ P. Szczerba Department of Structural Mechanics, Technical University of Lodz, Lodz, Poland e-mail: [email protected] P. Szczerba e-mail: [email protected] © Springer International Publishing Switzerland 2016 J. Awrejcewicz (ed.), Dynamical Systems: Modelling, Springer Proceedings in Mathematics & Statistics 181, DOI 10.1007/978-3-319-42402-6_33
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described by the continuous equations of type sine-Gordon [3]. This leads to a number of mechanical phenomena, such as solitons and breathers [4], which may be present in these types of systems.
1.1
An Overview of Issues
In this paper, we model the medium mathematically similar to the JJL, but with completely different physical principles. Let us consider two-dimensional discrete system consisting of an infinite number of beams, with only one degree of freedom each. We assume that all the beams are equally charged electrically. Interactions between adjacent beams are induced by electrostatic forces. Therefore, the rotation of one beam causes a rotation of adjacent beams. As will be shown in this work, the model equation is similar to sine-Gordon equation. As in the case JJL, some solutions of obtained equations may have a form of solitons. However, despite the similarities with existing nonlinear mechanical equations, the resulting equation is completely new. Theoretical attempts to description this medium, appears in the papers on atmospheric
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