Global study of the simple pendulum by the homotopy analysis method

Abstract

Techniques are developed to find all periodic solutions in the simple pendulum by means of the homotopy analysis method (HAM). This involves the solution of the equations of motion in two different coordinate representations. Expressions are obtained for the cycles and periods of oscillations with a high degree of accuracy in the whole range of amplitudes. Moreover, the convergence of the method is easily checked. The aim of this work is to show how the dynamics of a simple example of oscillatory systems may be studied globally with the HAM and to incentivize the interest of advanced undergraduate students in this type of techniques.

The physics of falling chimney stacks

“…For θ 0 ≡ 0 the exact solution of equation ( 5) is θ (t π ) = 4 arctan (e −ω 0 tπ ) [23,24] where t π is the time from θ (0) = π. If the toppling motion starts at t = (t π + t 0 ) = 0 with θ (0) = θ 0 , where θ 0 ≪ 1, (i.e.…”

Characterizing the movement of a falling rigid rod

“…Analytical and approximate solutions for the differential equation of the pendulum can be found in the literature (see for instance references [1,2,3,4,5,6,7] among many others, and references therein). These solutions are based on Jacobi elliptic functions [8].…”

“…This paper is focused on the experimental study of the fall of a rigid rod, which can freely rotate around an articulated joint at the lowest point, whose fundamental equations are the same as for a pendulum. Analytical and approximate solutions for the differential equation of the pendulum can be found in the literature (see for instance [1][2][3][4][5][6][7] among many others, and references therein). These solutions are based on Jacobi elliptic functions [8].…”