ASYMPTOTIC METHOD IN SIMULATION OF NONLINEAR TRANSVERSE OSCILLATIONS OF AN ELASTIC MOVABLE 1D BODY

Abstract

Mathematical modeling of oscillations of systems moving along their geometric axis is an urgent issue both theoretically and from the point of view of practical applications. Therefore, studying the amplitude-frequency characteristics of nonlinear transverse oscillations of such systems is a difficult, but at the same time very important task. The paper presents and analyzes a certain mathematical model of nonlinear transverse vibrations of a 1D body moving along its axis. Models for both non-resonant and resonant oscillation modes are considered. Well-known approaches and methods of nonlinear mechanics were used in the research - the asymptotic method of integration of differential equations, the method of development by a small parameter, the Fourier method. The result of applying these approaches was obtaining a system of ratios that allow determining the amplitude-frequency characteristics of the dynamic process. These relations are formulated in the form of ordinary differential equations that can be integrated. The generally obtained mathematical models are formulated in such a way as to study both resonant and non-resonant modes of oscillations. As a result of the obtained amplitude-frequency ratios, it became possible to assess the influence of various parameters (physical-mechanical, kinematic, etc.) on the oscillatory process. In turn, this made it possible to solve the problem of selecting and optimizing parameters for specific technological systems in order to avoid various negative destructive phenomena during operation and undesirable modes of oscillations (resonance, beating oscillations, disruption of oscillations). In addition, the ratios obtained in the work make it possible to study the influence of the parameters of the moving medium on the nature of the change in the frequency and amplitude of oscillations, to consider the movement at the support points of the medium, to study the method of fastening the beam supports.

The practical application of the results obtained in the work is to obtain relatively simple mathematical models for determining the amplitude-frequency characteristics of the corresponding 1D body-type structures, which can be studied by design engineers. Such mathematical models are, on the one hand, key to the study of the dynamics of moving media, and on the other hand, they can be adequately calculated from the point of view of engineering practice. As a result, the dependencies obtained in this article allow designers with sufficient adequacy and high accuracy to take into account the influence of the characteristics given in the work and to predict dynamic phenomena in oscillating 1D body-type systems. In addition, during engineering calculations of various mechanical systems of this type, the resulting dependencies can be used to optimize parameters to avoid negative destructive phenomena during operation.