A mathematical model for a rotary harmonograph

Peter S. Cumber

doi:10.1177/03064190251380665

A mathematical model for a rotary harmonograph

Peter S. Cumber^*

^*Corresponding author for this work

School of Engineering & Physical Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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Abstract

In this paper a mathematical model for a rotary harmonograph with two or three pendulums is presented. The rotary harmonograph model is derived using first principles and is a complete model in the sense that it includes the equations of motion, the kinematic expressions relating the pen trajectory to the pendulum conditions, and a prescription of harmonograph initial conditions that mimics the way a physical harmonograph is initiated. The rotary harmonograph model is used to generate a number of geometric designs for the 2-pendulum and 3-pendulum harmonograph. The sensitivity of the geometric design produced to the natural frequencies, friction and pendulum initial conditions is investigated. The generated patterns are qualitatively similar to the output of physical harmonographs.

Original language	English
Journal	International Journal of Mechanical Engineering Education
Early online date	29 Sept 2025
DOIs	https://doi.org/10.1177/03064190251380665
Publication status	E-pub ahead of print - 29 Sept 2025

Keywords

Dynamic system
Lissajous curve
oscillating system
rotary harmonograph

ASJC Scopus subject areas

Education
Mechanical Engineering

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abstract = "In this paper a mathematical model for a rotary harmonograph with two or three pendulums is presented. The rotary harmonograph model is derived using first principles and is a complete model in the sense that it includes the equations of motion, the kinematic expressions relating the pen trajectory to the pendulum conditions, and a prescription of harmonograph initial conditions that mimics the way a physical harmonograph is initiated. The rotary harmonograph model is used to generate a number of geometric designs for the 2-pendulum and 3-pendulum harmonograph. The sensitivity of the geometric design produced to the natural frequencies, friction and pendulum initial conditions is investigated. The generated patterns are qualitatively similar to the output of physical harmonographs.",

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A mathematical model for a rotary harmonograph

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Keywords

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