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

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SUMMARY

This discussion focuses on fitting damped harmonic motion data using Mathematica's FindFit function. The user initially encounters an error due to an extra layer of braces in the data structure, which prevents proper input into FindFit. After correcting the data format, the user explores using FindMinimum to specify initial values for parameters, ultimately leading to a successful fit by adjusting the model to include a phase parameter.

PREREQUISITES
  • Understanding of damped harmonic motion equations
  • Familiarity with Mathematica syntax and functions, particularly FindFit and FindMinimum
  • Knowledge of parameter estimation techniques in curve fitting
  • Basic grasp of mathematical modeling and error minimization
NEXT STEPS
  • Learn how to structure data for Mathematica functions, specifically for FindFit
  • Explore the use of FindMinimum for parameter optimization in Mathematica
  • Investigate the role of initial values in curve fitting and how to set them in Mathematica
  • Study the impact of phase parameters in oscillatory models and their implementation in Mathematica
USEFUL FOR

Researchers, physicists, and engineers involved in modeling oscillatory systems, as well as Mathematica users looking to enhance their data fitting skills.

globs123
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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!
 
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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.
[tex] \cos \left(\left(\sqrt{\frac{k}{m}-\left(\frac{R}{2 m}\right)^2}\right) t + \phi \right)[/tex]

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.