Abstract

The study of possibilities of geometric modeling of non-chaotic periodic paths of motion of a load of a swinging spring and its variants has been continued. In literature, a swinging spring is considered as a kind of mathematical pendulum which consists of a point load attached to a massless spring. The second end of the spring is fixed motionless. Pendular oscillations of the spring in a vertical plane are considered in conditions of maintaining straightness of its axis. The searched path of the spring load was modeled using Lagrange second-degree equations. Urgency of the topic is determined by the need to study conditions of dissociation from chaotic oscillations of elements of mechanical structures including springs, namely definition of rational parameter values to provide periodic paths of their oscillations. Swinging springs can be used as mechanical illustrations in the study of complex technological processes of dynamic systems when nonlinearly coupled oscillatory components of the system exchange energy with each other. The obtained results make it possible to add periodic curves as «parameters» in a graphic form to the list of numerical parameters of the swinging spring. That is, to determine numerical values of the parameters that would ensure existence of a predetermined form of the periodic path of motion of the spring load. An example of calculation of the load mass was considered based on the known stiffness of the spring, its length without load, initial conditions of initialization of oscillations as well as (attention!) the form of periodic path of this load. Periodic paths of the load motion for the swinging spring modifications (such as suspension to the movable carriage whose axis coincides with the mathematical pendulum) and two swinging springs with a common moving load and with different mounting points were obtained. The obtained results are illustrated by computer animation of oscillations of corresponding swinging springs and their varieties. The results can be used as a paradigm for studying nonlinear coupled systems as well as for calculation of variants of mechanical devices where springs influence oscillation of their elements and in cases when it is necessary to separate from chaotic motions of loads and provide periodic paths of their motion in technologies using mechanical devices

Highlights

  • In the present-day sense, a complex technological process can be interpreted as a dynamic system consisting of nonlinearly coupled oscillating components

  • It turned out that the most intense oscillation of a swing occurs when oscillations at the angle of attack occur with a frequency twice the frequency of lateral oscillations [4]. These examples explain when it is expedient to take into account energy exchange between its components within the framework of a dynamic system

  • It would be advisable to conduct studies aimed at geometric modeling of periodic paths of motion of the swinging spring load as well as varieties of swinging spring designs

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Summary

Introduction

In the present-day sense, a complex technological process can be interpreted as a dynamic system consisting of nonlinearly coupled oscillating components. The method consists in determining total system energy and correct estimation of energy values in time as well as their connection with each component To illustrate this approach, a two-dimensional spring pendulum is used as a mechanical model of studying several non-linearly connected systems. It turned out that the most intense oscillation of a swing occurs when oscillations at the angle of attack occur with a frequency twice the frequency of lateral oscillations [4] These examples explain when it is expedient to take into account energy exchange between its components (longitudinal and transverse oscillations) within the framework of a dynamic system (ship or aircraft). In a typical two-dimensional model, flexible thread can simultaneously perform transverse oscillations in its plane (analogous to angular oscillations of the loaded swinging spring) and pendulum oscillations which connect support attachments (analogous to vertical oscillations) [6], for example, wires of high-voltage lines whose state is in­ fluenced by wind gusts. It would be advisable to conduct studies aimed at geometric modeling of periodic paths of motion of the swinging spring load as well as varieties of swinging spring designs

Literature review and problem statement
The aim and objectives of the study
Conclusions
Energy distribution in intrinsically coupled systems
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