MATHEMATICAL MODEL OF OSCILLATIONS IN THE CHAIN OF PHYSICAL OSCILLATORS

Abstract

The analysis of the oscillatory process in a linear system of spring oscillators from several spherical bodies is carried out. The case is considered when one end of the system is fixed, and force acts on the opposite end. The cause of oscillations is the initial displacement of one or more bodies from the equilibrium state or an external load. It is shown that at a sufficiently large amplitude of the oscillations, neighboring bodies can come into contact with each other, as a result of which their velocities will change stepwise in accordance with the laws of conservation of kinetic energy and conservation of momenta. The mathematical model of the process is presented in the form of a system of second-order differential equations. The solution to the problem was obtained using the numerical Rung-Kutta method. For calculations, a program was developed in the
C ++ algorithmic language, which allows you to build phase portraits taking into account the collision of neighboring elements, as well as determine the time dependences of the deviation and the speed of this deviation of each body. Variant calculations showed that the possibility of using a mathematical or physical model of the oscillatory process is determined by the ratio of the maximum amplitude to the diameter of the balls. If this ratio is much less than unity, you can use the model of mathematical oscillators. Otherwise, it is necessary to apply the model of physical oscillators. The adequacy of the physical model is maintained under any initial conditions and for any external load. The magnitude of the jumps usually increases in the direction from the first to the last oscillator, to which the force is applied. Fracture on the graph of the time dependence of the deviations of the balls from the equilibrium position at low load is usually manifested to a small extent. In this case, you should refer to the analysis of the phase portrait.

Key words: oscillatory process, mathematical model, oscillator, phase portrait, physical oscillator.