Fifth-order AGM-formula for the period of a large-angle pendulum

Abstract

In this paper, an approximate algebraic formula for calculating the period of a large-angle pendulum was developed based on fifth-order iteration of the arithmetic-geometric mean (AGM) formula for the complete elliptic integral of the first kind. The present formula is capable of estimating the period of the nonlinear pendulum for the entire range of possible amplitudes i.e. 0 ∘ < A < 180 ∘, but it is particularly useful for large-angle ( 90 ∘ < A ≤ 170 ∘) and extremely large-angle ( 170 ∘ < A ≤ 179.9 ∘) oscillations. The accuracy of the present formula was tested using exact solution, numerical solution and other published large-angle formulas. It was observed that the present formula is several orders more accurate than the numerical solution and the other published formulas. The maximum error of the present formula for amplitudes up to 179.9 ∘ was found to be 2.93 × 10 - 6 %. The present formula can be used for pedagogical purpose because of its simplicity.

Keywords

• large-angle pendulum
• arithmetic-geometric mean
• elliptic integral
• nonlinear oscillations