Applied Mathematics, 2018, 9, 210-222 http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385 DOI: 10.4236/am.2018.93016 Mar. 22, 2018 210 Applied Mathematics On the Stability Analysis of a Coupled Rigid Body Olaniyi S. Maliki, Victor O. Anozie Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria 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 ordi- nary differential equations, describing the dynamics of the system and ob- tained 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 in- vestigation is unstable. Keywords Coupled Rigid Body, Differential Equations, Stability, Phase Portrait, MathCAD Simulation 1. Introduction The dynamics of coupled bodies and oscillators is significant in mechanics, en- gineering, electronics as well as biological systems. They are mostly represented as nonlinear dynamical systems [1]. One of the most important stages in the analysis of any mechanical model is to establish and find the solution of the dy- namical equations which are referred to as equations of motion [2]. The equa- tions of motion are often derived by the Euler-Lagrange equations. The funda- How to cite this paper: Maliki, O.S. and Anozie, V.O. (2018) On the Stability Anal- ysis of a Coupled Rigid Body. Applied Mathematics, 9, 210-222. https://doi.org/10.4236/am.2018.93016 Received: December 7, 2017 Accepted: March 19, 2018 Published: March 22, 2018 Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access