Very basically a harmonograph works by giving the platfrom (or pendulum) a gentle push.

The pens on the harmograph are usually the main thing that slows down the movement. This is mostly due to friction coming from the sticky ink.

The pens act as a kind of brake to slow the platform down due to the friction of the ink or pigment. When using a ball point pen for example the ball rolls with ink, the ink is quite sticky and so drags on the paper and slows down the platform.

Also the hinges or joints of the pendulum have a small amount of friction, in addition to a small amount of air friction.

**For a simple lateral harmonograph...**

By changing the position of weights distributed around the platform changes the way the platform down. This is also called changing the overall platform's moment of inertia.

An example of the mathematical equations that can be used to described a harmonograph, and can be entered onto a graphing program could be:

x(t) = Ax(t) sin(wx t + px) + As(t) sin(ws t + ps)

y(t) = Ay(t) sin(wy t + py)

This type of equation is called a parametric equation. In this case it produces a 2D image. It is possible to produce a 3D version by adding z(t). This is a kind of solution to another kind of equation called a differential equation.