A mathematical model for a rotary harmonograph

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.