Energy distribution in intrinsically coupled systems: The spring pendulum paradigm

Abstract

Intrinsically nonlinear coupled systems present different oscillating components that exchange energy among themselves. We present a new approach to deal with such energy exchanges and to investigate how it depends on the system control parameters. The method consists in writing the total energy of the system, and properly identifying the energy terms for each component and, especially, their coupling. To illustrate the proposed approach, we work with the bi-dimensional spring pendulum, which is a paradigm to study nonlinear coupled systems, and is used as a model for several systems. For the spring pendulum, we identify three energy components, resembling the spring and pendulum like motions, and the coupling between them. With these analytical expressions, we analyze the energy exchange for individual trajectories, and we also obtain global characteristics of the spring pendulum energy distribution by calculating spatial and time average energy components for a great number of trajectories (periodic, quasi-periodic and chaotic) throughout the phase space. Considering an energy term due to the nonlinear coupling, we identify regions in the parameter space that correspond to strong and weak coupling. The presented procedure can be applied to nonlinear coupled systems to reveal how the coupling mediates internal energy exchanges, and how the energy distribution varies according to the system parameters.

Development of a method for computer simulation of a swinging spring load movement path

“…What is essential in this process is the fact that the system components perform energy exchange with each other. Analysis of such energy exchange processes is presented in [1] in order to find out how all this depends on the system control parameters. To illustrate this, authors use a swinging spring as a paradigm for studying nonlinear coupled systems.…”

“…Nonlinear coupled systems with interacting subsystems are present in many fields: from physics and engineering to biology and social sciences. Examples of coupled systems include wave unification in plasma physics, laser pumping, biological oscillatory nets, neural nets, and genetic chains (corresponding references are given in [1]).…”

Synthesis and classification of periodic motion trajectories of the swinging spring load

“…As usual, the methods of dynamic system solution based on ideas of a swinging spring use coordinates that determine spring and pendulum motion [1][2][3]. In this case, possibility of representing a Hamiltonian in the form of sum of three members corresponding to the energies associated with motions of the spring, the pendulum and the component of their connection are foreseen.…”

“…Besides, within the framework of a dynamic system, its components can exchange energy with each other. An approach to solving the class of problems associated with the phenomenon of energy exchange between components is considered in [1][2][3]. The issues of dependence of this action on the system control parameters are studied.…”

“…A large number of possible implementations of the idea of oscillation of a swinging spring are given in [1][2][3]. For example, oscillations of swinging springs are directly related to many mechanical dynamic systems.…”