Jonathan Lansey

Abstract:
An apparatus to draw to graph some lovely pictures.  (pictures are called Lissajous Figures, with damping).  Since I understood what was going on, I could change certain things to get different looking graphs.  In the end, I put the equations on paper then graphed them on a computer.
Page Map:
Basic Analysis,  Analog Output, Digital Graphs, the Apparatus.

Description: 2003 (click here to see a short video clip)
After a short push, it would swing around and around for 10 minutes then the picture was done.
Complete List of Materials:

• pen and paper
• strings
• 45 kg of weights
• a ladder
• tape
• paper clips
• thermometer.

Disclaimer: you will need to do some thinking.

The weights hung from 2 blue strings, see first figure at right.  (Ignore red and green for now).  When this contraption swings from left to right the effective radius is where the string is tied.  In the far right picture (when it swings in a perpendicular direction) the effective radius is the red point (center of mass).

Since these two radii are different, the period of the swing is different depending on the direction it is swung in.

Notice: the radius can be changed by tying the string to different weights.  Using the green string will result in a longer period than using the blue string.

Call the red arrows the y axis and the green arrows the x axis.  Since the pendulum is oscillating approximately harmonically it can be described by the following parametric equation:
b=(close to zero); multiply by (k*t) for damping; a=amplitude;
x(t)=a*cos(t*b)(k*t)
y(t)=a*cos(t)(k*t)
Some people say the damping should be logarithmic.  I think it should be linear because the main source of damping (kinetic friction of the pen sliding on the paper) is constant regardless of amplitude and velocity.  I cannot calculate the damping by the ladder, this is a problem.
Note: It's a good idea to have 't' go from - to 0.  This way the thing slows to a stop.

That is the type of pictures drawn in the museum.  My contraption was more complicated because it also oscillated along the blue arrows.  Unlike the others, the period of this oscillation depends mostly on the moment of inertia of the weights along a vertical axis.  I could change 'I' easily and very precisely by moving some smaller weight further or closer to the center (see the gray weight in the first figure).  It was convenient to set this oscillation to be around twice the other oscillations.

Assuming that the Blue arrow oscillation is harmonic (like a normal pendulum it is not, but very close at small angles) then the angle 's' that the pen will move is:
f=(constant, depends on 'I', f is almost 2);  j=damping (Note: this motion was somewhat less damped by the ladder than the green and red, so it gets its own constant);
s(t)=cos(t*f)*(j*t)
When the green and red arrows are zero:
r=(distance from center to pen); m=(amplitude);
y(t)=r*sin(cos(t*f)(j*t))m
x(t)=r*cos(cos(t*f)(j*t))m

To get the complicated motion simply add these to the original parametric formula.
x(t)=a*cos(t*b)(k*t)+r*cos(cos(t*f)(j*t))m
y(t)=a*cos(t)(k*t)+r*sin(cos(t*f)(j*t))m

So to get different output, I simply tied the string to a different weight (thus changing 'b') or moved the small weight around (changing 'f').

I could also push and twist harder or softer (changing a, and m respectively).  Another way to change the output without changing the string arrangement was to change the direction of the initial push (ranging between tangent to the center and directed at the center).  This simply adds two constants (h,g) on the inside of the parenthesis, one for initial twist position/velocity and another for initial amplitude/velocity.
x(t)=a*cos(t*b+h)(k*t)+r*cos(cos(t*f+g)(j*t))
y(t)=a*cos(t+h)(k*t)+r*sin(cos(t*f+g)(j*t))

Calculating all these constants would have been rather difficult considering that the only measuring device I had was a thermometer.  Instead I just guessed from the way the graphs looked (with practice, a quick look at the graph will shout out what is happening).

Heres an example.
With 2 Axis's of motion (red and green): x(t)=cos(t*1.03)*t
y(t)=2*cos(t)*t

or

x(t)=cos(t*1.005)*2*t
y(t)=cos(t)*2*t

With 3 Axis's of motion (red green & blue)
x(t)=cos(t*1.01)*2*t-25*sin(cos(t*2.06)*0.001*t)*5
y(t)=cos(t)*2*t+25*cos(cos(t*2.06)*0.001*t)*5