Five-mode quasilinear model of nonlinear dynamics of extended system
DOI:
https://doi.org/10.32347/2411-4049.2021.2.104-120Keywords:
mathematical modeling, extended systems, waves, ite difference method, loopingAbstract
Distributed systems are widely used in practice. These are cosmic ligaments in the near-Earth space with a length of tens of kilometers. They approximate reinforced concrete piles in the soil when calculating the stress-strain state and assessing the technical condition; pipelines both in air and in liquid, underwater towed systems. Known underwater airlift systems of great length for the extraction of minerals (nodules) from the ocean floor with a length of 5-10 km. To solve the problems of the dynamics of such systems in various environments, the well-known mathematical models are not quite correct from the point of view of taking into account the variety of wave processes. It determines the need to build refined wave models. A new quasilinear mathematical model, which describes the nonlinear four-mode dynamics of the distributed system in the spatially inhomogeneous field of mass and surface forces, has been obtained. It is described by a nonlinear system of twelve first-order partial differential equations. For it, the principles of ultimate and hyperbolicity are fulfilled. Together with the boundary and initial conditions, it can be used to describe dynamics and statics of geometrically and physically nonlinear rod elements, piles in the ground, crane equipment ropes, mine lifts, aerial cableways, towed systems in liquid and gas flow, etc. For two-mode spatial reduction of the model, the theorem about correctness of Cauchy problem has been considered. As a result of the calculations, the earlier assumptions about the movement of the cable along its initial configuration were changed as the length of the cable changed. It has been found out that this assumption is only true for the initial transition participant. It is also established that at a given tachogram in the configuration of the towed line, there is a point of inflection, which shifts from top to bottom when lifting it. It can be a factor in the looping, contributing to the breakage of the cable system during towing.
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