The Use of Rocking Springs as a Mechanical Model of Modern Technological Processes as Dynamic Systems

Abstract

The paper considers the approach to solving a class of problems, when within a certain dynamic system its nonlinearly connected oscillatory components can exchange energy with each other. Many examples of such problems are given in [1,2]. At the same time, the dependence of the energy exchange action on the system control parameters is investigated. The problem is to determine the total energy of the system and correctly estimate the energy values over time, as well as their relationship for each of the components.To illustrate this approach, a two-dimensional spring pendulum is used as a mechanical model for the study of several nonlinearly coupled systems. The twodimensional spring pendulum in idealized form consists of a "point" load of mass m attached to the end of a weightless spring with a stiffness k and a length h in the unloaded state. The other end of the spring is fixed. The oscillating system formed in this way should move only in the vertical plane, while keeping the spring axis rectilinear. Point load simultaneously participates in two types of oscillations: springlike -when moving along the rectilinear axis of the spring, and pendulum-like -when it oscillates in conjunction with its axis. This type of oscillating system in the literature is called a swinging spring. With the help of a rocking spring, the exchange of energies between transverse (pendulum) and longitudinal (spring) oscillations is clearly illustrated. The influence of the initial conditions of oscillation initiation must also be taken into account. Of particular importance is the study of the condition of the resonance state of the oscillating spring. That is, when the frequency of longitudinal oscillations will differ a multiple of the frequency of transverse oscillations. In addition to the common "classical" case (2: 1 resonance), it is advisable to solve problems with other values of the frequency ratio. For example, there is a need [3] to build trajectories of cargo for cases of such resonances: 2: 1, 7: 3, 9: 4, 11: 2 and others. The found geometrical forms of a trajectory of movement of cargo of a rocking spring [4] with the set parameters will help to define characteristics of a decision of the chosen problem.In [1,5] a large number of possible implementations based on the application of the idea of oscillating spring oscillations are given. A significant part of this list is