Four-Mode Model of Dynamics of Distributed Systems

摘要

Distributed systems are widely used in practice. These are cosmic ligaments in the near-Earth space with the length of tens of kilometers. They approximate reinforced concrete piles in a soil when calculating the stress-strain state and assessing the technical conditions; pipelines both in an 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 are known. To solve the problems of the dynamics of such systems in various environments the known mathematical models are not quite correct in terms of taking into account the variety of wave processes. This 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 fields 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 limitation and hyperbolicity are fulfilled. Together with the boundary and initial conditions the model can be used to describe the 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 flows, etc. For two-mode spatial reduction of the model the theorem about correctness of Cauchy problem has been considered. The model has been tested based on numerical solution of the spatial problem of the propagation of four waves of three types: longitudinal, configurational in the direction of the normal and binormal and the torsioned ones. The necessary quantitative estimates of the twist angle and the torque for the specific distributed system in the liquid flow were determined using the numerical algorithm and the program based on the finite-difference method.