THE INFLUENCE OF THE COMPRESSION FORCE (TENSION) OF THE SEED PIPE OF THE STRIP PLANT AND SPEEDS OF SEED MOVEMENT ON TRANSVERSE OSCILLATIONS

Abstract

The mathematical model of the nozzle-grain system corresponds to the physical process of bending vibrations of the nozzle along which the grain moves. It takes into account the nonlinear elastic properties of the pipe during its bending vibrations and the movement of grain along it. The flow of grain into the nozzle is modeled by an incompressible solid medium that moves relative to the nozzle. Such a nonlinear mathematical model is equivalent to a one-dimensional system with distributed parameters, besides, it takes into account the longitudinal component of the speed of movement of the distributed mass (grain) along the elastic body (pipe). The main difficulties of studying the dynamics of the mechanical system of nozzles – the flow of moving grain – are connected with the above factors. After all, within the limits of the linear theory of oscillations, it is not possible to explain a whole series of phenomena that occur in a similar type of mechanical systems. At the same time, the mathematical apparatus for studying nonlinear models of systems with distributed parameters is relatively fully developed only for their so-called quasi-linear counterparts of limited length, partially for systems with power-law nonlinearity and those close to them. For such systems, based on the general ideas of perturbation methods or their modifications, it is possible to construct asymptotic approximations that are sufficiently adequate to the dynamic process. As for continuous media characterized by longitudinal movement (in addition to a pipe with a flow of moving grain, such systems include rope hoists, flexible working elements of drive and transportation systems, etc.), they remain poorly researched primarily due to the lack of a perfect apparatus for analyzing them even linear mathematical models. However, their wide application in various branches of the national economy and technology has led to the fact that in recent decades, different approaches (numerical and analytical to the study of linear and nonlinear models of these systems have become widespread. In most cases, models of longitudinally moving systems (one-dimensional or twodimensional) were considered under the condition that their bending stiffness can be neglected. Such a reasonable, for a certain class of dynamic systems, simplification made it possible to extend the basic ideas of the wave theory of motion to the case of their longitudinal or transverse oscillations. However, for longitudinally moving systems with significant bending stiffness, the simplification of the physical, and hence the corresponding mathematical models, leads to certain inaccuracies in the determination of the main parameters that characterize the process as a whole. Taking into account the bending stiffness in longitudinally moving media leads to the construction of qualitatively new mathematical models of their dynamics. Finding analytical solutions corresponding to them, which would provide an opportunity to comprehensively analyze the process, is a complex mathematical problem. This means that the problem of analytical research of dynamic processes of systems characterized by longitudinal movement, taking into account their bending stiffness, remains open.