GEOMETRIC MODELING OF THE TRAJECTORY CARGO SWINGING SPRING WITH MOVABLE SUSPENSION

Abstract

The geometric modeling of pendulum oscillation in the vertical plane of the cargo suspended by a suspended spring, which, when moving, retains the rectinity of its axis. In the literature, this kind of pendulum is called the swinging spring (Swinging Spring). The specified spring model is widely used as a mechanical model of more complex processes in nature and technology. Just as the famous pendulum of Kapitsi allows you to explain some fundamental concepts. In our case, we will talk about processes with internal nonlinearly related systems for the provision of different oscillatory components. At the same time, which is essential, the components of the system exchange the energy among themselves. Often the authors use the swinging spring as a paradigm to study nonlinear related systems. For swinging springs, three energy components are identified, similar to the movement of the spring, the movement of the pendulum, as well as the relationship between these movements. The presented procedure can be applied, in principle, to arbitrary nonlinear associated systems to show how the connection is mediated by internal energy exchanges and how the energy distribution varies according to the system parameters.

In the work using a computer, the cargo trajectory of the swinging spring is modeled using the weights of the cargo, the rigidity of the spring and its length in the unloaded state. In addition, the initial values ​​of the initiation parameters of the oscillations of the swinging spring are used: the angle of deviation of the spring axis from the vertical, the rate of changes in the magnitude of this angle, as well as the springs elongation parameter and the rate of elongation change. Calculations are made using the Lagrange equation of the second kind. The options for finding a nonchaotic trajectory of point load of a swinging spring with movable (along the coordinate axes) of the mounting point are considered.

Keywords: pendulum oscillations with moving point of suspension, periodic trajectory of movement, swinging spring, second-kind Lagrange equation.