Emulation Of A Harmonograph by Andrew Purdam Brought To You By Wabmo
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- To stop and start the Harmonograph, click the mouse button in the applet, then or press the space-bar.
- To tweak the parameters (and to load some presets), click the right mouse button, or press Shift-Space, and shortly a dialog will come up. This will let you twiddle away at any of the 22 parameters used.
- If you want to print or save what you see, you'll need a screen capture program. Windows has this built in, simply by pressing the [Print Screen] button. The screen will then be saved on the clipboard for pasting to your favourite graphics program. To actually print onto paper.
History:
The harmonograph was pioneered by the French physicist, Jules Antoine Lissajous in 1857. The first harmonograph
actually used a light beam on a screen (funny how history repeats itself!) Following the invention of the harmonograph it
became a very popular device and was found in many homes. After the early 1900s it decreased in popularity and is rarely
seen today, except in hands-on science centres.
The Physics of the Harmonograph
General Description:
The Harmonograph is basically a large, heavy, rectangular platform suspended at all four corners by wires. It can
basically wobble forwards and backwards, side to side, and rotate a bit too. A pen is held stationary whilst a piece
of paper attached to the surface of the Harmonograph oscillates according to how it was started. The oscillations decay,
with the resulting curves getting smaller and smaller, spiralling in a highly wobbly way.
Arm-wavy Physical Model:
Roughly speaking, the movement of the Harmonograph can be approximated into
three not-quite independent parts
- A pendulum along an X axis, centred on its centre of mass
- A pendulum along a Y axis, centred on its centre of mass
- A rotational oscillation, also centred about its centre of mass
There are some more assumptions made at this point:
- The X and Y oscillations are nearly the same frequency. This is due to the
fact that the weight of each "pendulum" is the same, and more importantly,
that each pendulum is of the same "length of string".
- The rotational oscillation S is double the frequency of the X and Ys. In
fact, the rotational oscillation in real life depends on the rotational
inertia of the table. This alters depending on where ballast (heavy weights)
are placed on the platform. If they are placed in the middle, the ratio is 1:2
(double), if they are placed at the corners, the ratio is 2:3.
- Because the pen is situated a relatively long distance from the centre of
mass, the S oscillation can be simplified to a second oscillation in the X or
Y. Otherwise, feel free to add this oscillation in both the X and Y
directions.
- I think it could be argued that the rest is just adjustments in relative
phases of each component.
The Mathematics
So then the Harmonograph can be plotted over time as follows:
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x1(t) =
Ax(t)sin(wxt+px)
y(t) =
Ay(t)sin(wyt+py)
x2(t) =
As(t)sin(wst+ps)
x(t) = x1(t)+x2(t), |
where....
A is the amplitude of the oscillation
w is frequency
p is the phase
s is the rotational factor |
All amplitudes and phases are independent. To make things a little more
interesting, wy is allowed to be slightly different in
relation to wx and ws is also allowed to be
slightly different to double wx. Lastly, these amplitudes all
decay over time.
Ax(t) = Ax(0)e-icdt where
dx is x's decay constant and c is a common factor for
all decay equations. This of course comes down to an iterative thing given by
Ax(t+1) = Ax(t) *
(1-dx)
ie. the amplitude this time is the amplitude last
time multiplied by a number slightly less than 1.
Can be downloaded as an application on download page