The system is coupled and joined by three rigid rods of equal lengths, hinged to each of the masses. Despite the complexity of the system we will obtain, with the help of the Euler Lagrange (E-L) equations, the equation of motion of each of the individual masses (assumed to be equal). We consider the problem of analyzing the dynamics of a triple pendulum as shown in Figure 1. 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. Maliki and Okereke investigated the stability analysis of certain third order linear and nonlinear ordinary differential equations. The stability analysis of the system was investigated by the direct method and it was observed that the coupled system is asymptotically stable for the strictly negative roots and strongly unstable for the positive roots.
![polyroots mathcad polyroots mathcad](https://konspekta.net/infopediasu/baza14/274739350256.files/image275.jpg)
Laplace transform was also used to get the analytical solution of the system.
![polyroots mathcad polyroots mathcad](https://www.mathcadhelp.com/wp-content/uploads/2015/11/A-solve-block-with-both-equations-and-inequalities.-300x2231-150x150.jpg)
Maliki and Nwoba studied a mathematical model of a coupled system of harmonic oscillators using the generalized coordinates and Euler-Lagrange equation. They derived the stability conditions and found stable and unstable region on the frequency response curve. The approximate methods were used to find the periodic solutions. The damping coefficient and excitation amplitude are assumed to be small. Seyrania and Wang studied the stability of periodic solutions of the harmonically excited Duffing’s equation with the direct application of the Lyapunov theorem. For the second method of Lyapunov (indirect method), the idea of system linearization around a given point is used and local stability within small stability regions is possibly achieved. It has two approaches: indirect and direct methods. Indeed stability plays a central role in system engineering, especially in the field of control system and automation, with regards to both dynamics and control.Ĭhutiphon suggested Lyapunov stability as a general and useful approach to analyze the stability of nonlinear systems. Of importance is the notion of stability of a given dynamical system, where we would be concerned with the stability of some critical point of the system. The fundamental idea of the Lagrangean approach to mechanics is to reformulate the equations of motion in terms of the dynamical variables that describe the degree of freedom, and thereby incorporate constraint forces into the definition of the degrees of freedom rather than explicitly including them as forces in Newton’s second law. The equations of motion are often derived by the Euler-Lagrange equations. One of the most important stages in the analysis of any mechanical model is to establish and find the solution of the dynamical equations which are referred to as equations of motion. They are mostly represented as nonlinear dynamical systems. The dynamics of coupled bodies and oscillators is significant in mechanics, engineering, electronics as well as biological systems. Sources for each of the oscillating masses showing that the system under investigation Furthermore, the resulting phase portrait analysis depicted spiral It is discovered that theĬoupled rigid pendulum gives rise to irregular oscillations with ever increasingĪmplitude. The graphical profiles for each generalized coordinates representing theĪngles measured with respect to the vertical axis. Finally, we performed MathCAD simulation of the resulting ordinaryĭifferential equations, describing the dynamics of the system and obtained Linearization method and subsequently confirmed our result by phase portraitĪnalysis. Proceeded to study the stability of the dynamical systems using the Jacobian Using the generalized coordinates and the Euler-Lagrange equations.
POLYROOTS MATHCAD FREE
Is free to oscillate in the vertical plane. System involving coupled rigid bodies consisting of three equal masses joinedīy three rigid rods of equal lengths, hinged at each of their bases.
![polyroots mathcad polyroots mathcad](https://file.elecfans.com/web1/M00/5F/09/pIYBAFt8PcyAG-Q0AAA5BJ-Brek333.jpg)
In this research article, we investigate the stability of a complex dynamical