Equations of Motion Formulation of a Pendulum Containing N-point Masses

TL;DR

This paper develops a compact, efficient formulation of the equations of motion for an n-point mass pendulum using Lagrange mechanics and vector methods, and explores the limit as the number of masses approaches infinity.

Contribution

It introduces a unified, concise equation formulation for n-point mass pendulums and analyzes their behavior as the number of masses becomes very large.

Findings

Derived a compact form of equations of motion for n-point mass pendulums.

Showed that as the number of masses approaches infinity, the system converges to a hanging rope.

Compared efficiency of the new formulation with traditional methods.

Abstract

This paper presents a general formulation of equations of motion of a pendulum with n point mass by use of two different methods. The first one is obtained by using Lagrange Mechanics and mathematical induction(inspection), and the second one is derived by defining a vector. Today, these equations can be obtained by employing numerous programs; however, this study gives a very compact form of these equations that is more efficient than solving Euler-Lagrange Equations for every pendulum with more complex structures than simple or double pendulum. Additionally, we investigate what will happen to our n-point mass system when we take limit as number of point masses goes infinity under well-defined assumptions. We find out that it converges to hanging rope system.