RE: [tuning] Digest Number 3427

Hi Carl,

Yes, you're right, they are decaying Lissajous figures.

Since -
1) a harmonograph simply combines the motions of two pendulums
at right angles to each other; and
2) the motion of a pendulum is given by
x = a sin (bt + c),
where x is the horizontal displacement from rest and a, b and c
are constants, provided x remains relatively small; then -

You can program these motions by any program that plots points
(x, y) in the Cartesian plane at time t with coordinates x and y
given by -
x = a sin (bt + c),
y = d sin (et + f),
where x is the horizontal displacement from rest,
y is the vertical displacement from rest, and
a, b, c, d, e and f are constants.

We had fun in high school physics making a simple harmonograph as
follows -

A. Take a length of string, cord or fishing line ("the cord").
B. Slide a pierced weight, for example a lead fishing weight, to the middle
of the cord.
C. Secure the weight in that position, say by tying a knot around it.
D. Tie another (firm but not immovable) knot partway towards the ends of
the cord.
E. Suspend the weight from the cord, by fixing the two ends several inches
apart.
F. Pull the weight diagonally away from the line joining the two fixed
points.
G. Watch it swing!

Here's a rough ASCII picture -
\ /
\ /
\ /
\ /
\ /
| |
| |
| |
0

H. Untie the not tied instep D, and retie it at a different point.
I. You see that you effectively have a double pendulum - one with two
different lengths simultaneously.
J. If you're handy, you can replace the weight by a small bag full of fine
sand, with a small hole
in it to trace patterns.

Regards,
Yahya

-----Original Message-----
________________________________________________________________________
Date: Thu, 03 Mar 2005 17:03:37 -0800
From: Carl Lumma <ekin@lumma.org>
Subject: Re: harmonograph

>> Anybody know anything about this?
>>
>> http://www.amazon.com/exec/obidos/tg/detail/-/0802714099/
>
>I have a copy on my bookshelf -- what kind of things would
>you like to know about it?

Everything! Mainly, how could I program a computer to make
these visualizations.

-Carl

________________________________________________________________________

[and later]

Oh, is there more than one kind in the book? I'm clueless about
it. Just looked interesting. They look something like lissajous
curves, but the amazon page claims they're "unique". Anyway, if
you know what they are, please spill the beans.

-Carl

________________________________________________________________________

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>Hi Carl,
>
>Yes, you're right, they are decaying Lissajous figures.
>
>Since -
>1) a harmonograph simply combines the motions of two pendulums
> at right angles to each other; and
>2) the motion of a pendulum is given by
> x = a sin (bt + c),
> where x is the horizontal displacement from rest and a, b and c
> are constants, provided x remains relatively small; then -
>
>You can program these motions by any program that plots points
>(x, y) in the Cartesian plane at time t with coordinates x and y
>given by -
> x = a sin (bt + c),
> y = d sin (et + f),
> where x is the horizontal displacement from rest,
> y is the vertical displacement from rest, and
> a, b, c, d, e and f are constants.

Aha! Thanks Yahya!

-Carl