Mathematica How do I fit damped harmonic motion data (x vs. t) using Mathematica?

AI Thread Summary
To fit damped harmonic motion data in Mathematica, the initial error arises from an extra layer of braces in the data structure, which must be corrected to a proper format. The user also faces challenges with the FindFit function, as it does not allow for the specification of initial values, which can hinder convergence. An alternative approach using FindMinimum is suggested, which requires defining a squared error function to minimize. Additionally, the model equation may need adjustment to include a phase parameter and ensure proper placement of the time variable in the cosine function. Overall, with the right modifications, fitting the data can be successfully achieved in Mathematica.
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How do I fit the data given below into the standard form (ignore the part saying to use MathCad):

[PLAIN]http://screensnapr.com/u/apmqkd.png

Code: 
{{{0.002, 0.726}, {0.022, 0.739}, {0.042, 0.75}, {0.062, 
   0.759}, {0.082, 0.768}, {0.102, 0.776}, {0.121, 0.785}, {0.141, 
   0.794}, {0.161, 0.802}, {0.181, 0.81}, {0.201, 0.818}, {0.221, 
   0.826}, {0.24, 0.834}, {0.26, 0.84}, {0.28, 0.847}, {0.3, 
   0.854}, {0.32, 0.859}, {0.339, 0.864}, {0.359, 0.869}, {0.379, 
   0.873}, {0.399, 0.877}, {0.419, 0.879}, {0.439, 0.882}, {0.458, 
   0.883}, {0.478, 0.884}, {0.498, 0.885}, {0.518, 0.884}, {0.538, 
   0.883}, {0.557, 0.882}, {0.577, 0.88}, {0.597, 0.877}, {0.617, 
   0.874}, {0.636, 0.87}, {0.656, 0.865}, {0.676, 0.86}, {0.696, 
   0.855}, {0.716, 0.849}, {0.735, 0.843}, {0.755, 0.836}, {0.775, 
   0.829}, {0.795, 0.822}, {0.815, 0.814}, {0.834, 0.806}, {0.854, 
   0.799}, {0.874, 0.791}, {0.894, 0.783}, {0.913, 0.774}, {0.933, 
   0.766}, {0.953, 0.759}, {0.973, 0.751}, {0.993, 0.743}, {1.012, 
   0.736}, {1.032, 0.729}, {1.052, 0.722}, {1.072, 0.716}, {1.091, 
   0.71}, {1.111, 0.704}, {1.131, 0.699}, {1.151, 0.694}, {1.171, 
   0.689}, {1.19, 0.685}, {1.21, 0.683}, {1.23, 0.68}, {1.25, 
   0.675}, {1.27, 0.676}, {1.289, 0.675}, {1.309, 0.673}, {1.329, 
   0.676}, {1.349, 0.674}, {1.369, 0.675}, {1.388, 0.68}, {1.408, 
   0.68}, {1.428, 0.683}, {1.448, 0.689}, {1.468, 0.693}, {1.487, 
   0.698}, {1.507, 0.703}, {1.527, 0.708}, {1.547, 0.714}, {1.567, 
   0.719}, {1.586, 0.725}, {1.606, 0.732}, {1.626, 0.739}, {1.646, 
   0.746}, {1.666, 0.753}, {1.686, 0.76}, {1.705, 0.768}, {1.725, 
   0.775}, {1.745, 0.782}, {1.765, 0.789}, {1.785, 0.796}, {1.804, 
   0.803}, {1.824, 0.81}, {1.844, 0.816}, {1.864, 0.822}, {1.884, 
   0.828}, {1.904, 0.834}, {1.923, 0.839}, {1.943, 0.843}, {1.963, 
   0.848}, {1.983, 0.852}, {2.003, 0.855}, {2.023, 0.858}, {2.042, 
   0.861}, {2.062, 0.863}, {2.082, 0.864}, {2.102, 0.865}, {2.121, 
   0.865}, {2.141, 0.865}, {2.161, 0.865}, {2.181, 0.863}, {2.201, 
   0.861}, {2.221, 0.859}, {2.24, 0.857}, {2.26, 0.854}, {2.28, 
   0.85}, {2.3, 0.846}, {2.319, 0.841}, {2.339, 0.837}, {2.359, 
   0.832}, {2.379, 0.826}, {2.399, 0.82}, {2.418, 0.815}, {2.438, 
   0.808}, {2.458, 0.802}, {2.478, 0.795}, {2.498, 0.789}, {2.517, 
   0.782}, {2.537, 0.776}, {2.557, 0.769}, {2.577, 0.762}, {2.596, 
   0.756}, {2.616, 0.75}, {2.636, 0.744}, {2.656, 0.738}, {2.675, 
   0.731}, {2.695, 0.727}, {2.715, 0.722}, {2.735, 0.717}, {2.755, 
   0.712}, {2.774, 0.708}, {2.794, 0.705}, {2.814, 0.702}, {2.834, 
   0.699}, {2.854, 0.697}, {2.873, 0.695}, {2.893, 0.694}, {2.913, 
   0.693}, {2.933, 0.693}, {2.953, 0.693}, {2.972, 0.694}, {2.992, 
   0.695}, {3.012, 0.696}, {3.032, 0.698}, {3.052, 0.701}, {3.071, 
   0.704}, {3.091, 0.707}, {3.111, 0.71}, {3.131, 0.712}, {3.151, 
   0.718}, {3.17, 0.724}, {3.19, 0.728}, {3.21, 0.733}, {3.23, 
   0.739}, {3.25, 0.744}, {3.27, 0.75}, {3.289, 0.756}, {3.309, 
   0.762}, {3.329, 0.768}, {3.349, 0.774}, {3.369, 0.78}, {3.388, 
   0.786}, {3.408, 0.792}, {3.428, 0.797}, {3.448, 0.803}, {3.468, 
   0.808}, {3.488, 0.813}, {3.507, 0.818}, {3.527, 0.822}, {3.547, 
   0.827}, {3.567, 0.831}, {3.587, 0.835}, {3.606, 0.838}, {3.626, 
   0.841}, {3.646, 0.844}, {3.666, 0.846}, {3.686, 0.847}, {3.705, 
   0.848}, {3.725, 0.849}, {3.745, 0.85}, {3.765, 0.85}, {3.785, 
   0.849}, {3.804, 0.848}, {3.824, 0.847}, {3.844, 0.845}, {3.864, 
   0.843}, {3.884, 0.84}, {3.903, 0.837}, {3.923, 0.834}, {3.943, 
   0.831}, {3.963, 0.827}, {3.983, 0.823}, {4.002, 0.818}, {4.022, 
   0.814}, {4.042, 0.809}, {4.062, 0.804}, {4.081, 0.798}, {4.101, 
   0.793}, {4.121, 0.788}, {4.141, 0.782}, {4.161, 0.777}, {4.18, 
   0.771}, {4.2, 0.766}, {4.22, 0.761}, {4.24, 0.755}, {4.26, 
   0.75}, {4.279, 0.745}, {4.299, 0.741}, {4.319, 0.736}, {4.339, 
   0.732}, {4.358, 0.728}, {4.378, 0.724}, {4.398, 0.721}, {4.418, 
   0.718}, {4.438, 0.716}, {4.457, 0.713}, {4.477, 0.711}, {4.497, 
   0.71}, {4.517, 0.708}, {4.537, 0.707}, {4.556, 0.706}, {4.576, 
   0.705}, {4.596, 0.704}, {4.616, 0.705}, {4.636, 0.707}, {4.655, 
   0.708}, {4.675, 0.712}, {4.695, 0.712}, {4.715, 0.715}, {4.735, 
   0.72}, {4.754, 0.723}, {4.774, 0.727}, {4.794, 0.731}, {4.814, 
   0.735}, {4.834, 0.739}, {4.854, 0.744}, {4.873, 0.749}, {4.893, 
   0.753}, {4.913, 0.758}, {4.933, 0.763}, {4.953, 0.768}, {4.972, 
   0.773}, {4.992, 0.778}, {5.012, 0.783}}}

Graph display of above data:
[PLAIN]http://screensnapr.com/u/9k82rt.png

My attempt in Mathematica:
Code: 
FindFit[data, 
 A*E^(-(R/(2 m))*t)*Cos[Sqrt[k/m - (R/(2 m))^2*t]] + B + c*t, {A, R, 
  m, k, B, c}, t]

I get this error:
"FindFit::fitd: First argument {{{0.002,0.726},{0.022,0.739},{0.042,0.75},<<5>>,{0.161,0.802},{0.181,0.81},<<244>>}} in FindFit is not a list or a rectangular array. >>"



Can this be done using Mathematica?

Thank you!
 
Last edited by a moderator:
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First, your error message: Notice you have an extra layer of {} so you do not have {{x1,y1},{x2,y2}...} but {{{x1,y1},{x2,y2}...}. Get rid of that extra layer and you will make the error go away. One way of doing this is to substitute data[[1]] for data in your Mathematica code, that is what I did. Even with that fix I couldn't get your code to converge.

Now on to bigger things. I don't believe FindFit allows you to specify initial values. If you can find a way to do that then maybe that is your solution. You can include constraints along with your equation in FindFit and you could try to use that to force each of the parameters into a range, but I'm not convinced.

FindMinimum allows you to specify initial values, in fact it demands them. Perhaps that will help you. But that raises the question "what do you want to find the minimum of?"

Ponder this for a while:
y[r_, m_, a_, k_, b_, c_, t_] := a E^(-r/(2m))Cos[Sqrt[k/m - (r/(2m))^2t]] + b + c t;
squarederror = Total[Map[(y[r, m, a, k, b, c, #[[1]]] - #[[2]])^2 &, data[[1]]]];
FindMinimum[squarederror, {{r, .1}, {m, 1}, {a, 1}, {k, 2}, {b, .75}, {c, 0}}, MaxIterations -> 100]

But thus far I haven't been able to find initial values that will get it to converge.
 
Are you sure that equation is right? I htink there should be a phi_0 phase parameter as well if I am not mistaken. Also, the "t" in the cosine MUST be outside that square root, otherwise itll have units.
<br /> \cos \left(\left(\sqrt{\frac{k}{m}-\left(\frac{R}{2 m}\right)^2}\right) t + \phi \right)<br />

model = A*E^(-(R/(2 m))*t)*Cos[Sqrt[k/m - (R/(2 m))^2] t + \[Phi] ] +
B + c t
sln = FindFit[data,
model, {A, {R, 1/5}, {m, 1/2}, {k, 5}, B, {\[Phi], -1.5}, c}, t];
Show[ListPlot[data], Plot[model /. sln, {t, 0, 5}]]
sln

That fits quite nicely.
 

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