Abstract
In this research article, we investigate the stability of a complex dynamical system involving coupled rigid bodies consisting of three equal masses joined by three rigid rods of equal lengths, hinged at each of their bases. The system is free to oscillate in the vertical plane. We obtained the equation of motion using the generalized coordinates and the Euler-Lagrange equations. We then proceeded to study the stability of the dynamical systems using the Jacobian linearization method and subsequently confirmed our result by phase portrait analysis. Finally, we performed MathCAD simulation of the resulting ordinary differential equations, describing the dynamics of the system and obtained the graphical profiles for each generalized coordinates representing the angles measured with respect to the vertical axis. It is discovered that the coupled rigid pendulum gives rise to irregular oscillations with ever increasing amplitude. Furthermore, the resulting phase portrait analysis depicted spiral sources for each of the oscillating masses showing that the system under investigation is unstable.
Highlights
The dynamics of coupled bodies and oscillators is significant in mechanics, engineering, electronics as well as biological systems
We investigate the stability of a complex dynamical system involving coupled rigid bodies consisting of three equal masses joined by three rigid rods of equal lengths, hinged at each of their bases
The study of coupled systems is useful in mechanics, electronics as well as biological systems
Summary
The dynamics of coupled bodies and oscillators is significant in mechanics, engineering, electronics as well as biological systems. Chutiphon [3] suggested Lyapunov stability as a general and useful approach to analyze the stability of nonlinear systems It has two approaches: indirect and direct methods. Maliki and Okereke [6] investigated the stability analysis of certain third order linear and nonlinear ordinary differential equations. They employed the method of phase portrait analysis and showed, using simulation that the Hartman-Groβman theorem is verified, for a second order linearized system, which approximates the nonlinear system, preserving the topological features
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