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Interesting article http://johncarlosbaez.wordpress.com/2014/07/18/the-harmonograph/

Modeling the harmonograph is more tricky than it might seem. Naively the curves originates from imperfect tuning of frequencies. Look at the pictures produced by 2D harmonograph from the old book. The nodes stays in the same place, which is impossible if the frequencies are not tuned. The amplitude obviously decreases, so we need friction. But if it affects the pendulums the same way the lines on the third figure always cross under 90 degrees. The angle in reality is drifting, while vertical axis of symmetry stays at the same place. We must have unequal friction in the pendulums!

Here are the pictures you can obtain using simple Mathematica code below.


Aren't they the same as on the picture? Can you reproduce those with unequal freqiencies?

First plot:

delta = 0.015;
lam = Exp[-delta t];
w1 = 2 Pi;
w2 = 3/2 2 Pi;
phi1 = Pi/2;
phi2 = Pi/2;
A1 = 1 lam;
A2 = 1 lam^2;
X[t_] := A1 Sin[w1 t + phi1];
Y[t_] := A2 Sin[w2 t + phi2];

ParametricPlot[{X[t], Y[t]  }, {t, 0, 40}]

Parameters for the other two:

delta = 0.015;
lam = Exp[-delta t];
w1 = 2 Pi;
w2 = 3/2 2 Pi;
phi1 = 0;
phi2 = 0;

delta = 0.015;
lam = Exp[-delta t];
w1 = 2 Pi;
w2 = 3/2 2 Pi;
phi1 = Pi/2;
phi2 = Pi/2 + Pi/8;