There is more than one way to force a pendulum

Peter Cumber*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
64 Downloads (Pure)


The dynamics of a simple pendulum are often presented to undergraduate engineering students in introductory courses in dynamics. It is usually the first dynamic system considered by students that is modelled by a differential equation. This paper presents the standard material given to students. It is fair to say that students are accepting this material, but many do not fully appreciate its importance as a basis for the analysis of other dynamic models. Students also believe that the behaviour of a forced simple pendulum is obvious and is not worthy of a deep analysis. This paper presents many interesting results that are available in the open literature, but is not usually presented to undergraduate students. Some of these are counter-intuitive results such as unstable points becoming stable when forced and deterministic models giving chaotic pendulum trajectories. Many results demonstrate that simple pendulums are very interesting and worthy of analysis. The extension of a forced simple pendulum to an application that all students have experience of a child playing on a swing is presented. Only a limited analysis of the swing models is presented. This leaves some open questions for students and lecturers alike to explore.

Original languageEnglish
Pages (from-to)579-613
Number of pages35
JournalInternational Journal of Mathematical Education in Science and Technology
Issue number4
Early online date11 Mar 2022
Publication statusPublished - 21 Apr 2023


  • bifurcation
  • chaos
  • inverted pendulums
  • Mechanics

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Education
  • Applied Mathematics


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