matlabmatplotlibgnuplot
How to determine the damping coefficient from data in Matlab or other graphing software
I recently did an experiment where I measured the output signal of an RLC circuit which behaves exactly like a damped free harmonic oscillator. I want to to dermine the damping ratio from the data. My first idea was to use a decaying exponential fit on the enveloppe, as depicted in the following Figure.
After many hours of despair, I managed to use the Curve Fitting tool provided by Matlab to add an exponential fit, but I had to select manually the peaks to take into account. Is there any way to achieve this same result but with code ?
I mean, I know there has to be a way, but I've been struggling for more time than should be legally permitted, so If any kind soul could help me with a script, I'd warmheartedly welcome it. I can use either Gnuplot, python or Matlab.
Thanks in advance
Here is a link to the full data file, since SO doesn't let me dump the whole content here data file link
time[0] Y[0] time[1] Y[1]
747979 -4.694474E-1 -5.114054E-1
747979 -4.267468E-1 -4.689731E-1
747979 -4.694474E-1 -4.689731E-1
747979 -4.694474E-1 -5.114054E-1
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748075 5.553671E-1 1.228318E+0
748075 5.553671E-1 1.228318E+0
748075 5.553671E-1 1.228318E+0
748075 5.126665E-1 1.185885E+0
748075 5.553671E-1 1.228318E+0
748076 5.553671E-1 1.228318E+0
748076 5.553671E-1 1.185885E+0
748076 5.553671E-1 1.185885E+0
748076 5.553671E-1 1.228318E+0
748076 5.553671E-1 1.185885E+0
748077 5.553671E-1 1.185885E+0
748077 5.553671E-1 1.185885E+0
748077 5.553671E-1 1.185885E+0
748077 5.553671E-1 1.185885E+0
748077 5.553671E-1 1.185885E+0
748078 5.553671E-1 1.185885E+0
748078 5.126665E-1 1.185885E+0
748078 5.553671E-1 1.143453E+0
748078 5.553671E-1 1.185885E+0
748078 5.553671E-1 1.185885E+0
748079 5.553671E-1 1.143453E+0
748079 5.553671E-1 1.143453E+0
748079 5.553671E-1 1.143453E+0
748079 5.553671E-1 1.143453E+0
748079 5.553671E-1 1.143453E+0
748080 5.553671E-1 1.143453E+0
748080 5.553671E-1 1.101021E+0
748080 5.553671E-1 1.143453E+0
748080 5.553671E-1 1.101021E+0
748080 5.553671E-1 1.101021E+0
748081 5.553671E-1 1.101021E+0
748081 5.553671E-1 1.101021E+0
748081 5.553671E-1 1.101021E+0
748081 5.553671E-1 1.101021E+0
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748082 5.553671E-1 1.101021E+0
748082 5.553671E-1 1.101021E+0
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748082 5.553671E-1 1.058589E+0
748082 5.553671E-1 1.058589E+0
748083 5.553671E-1 1.058589E+0
748083 5.553671E-1 1.101021E+0
748083 5.553671E-1 1.058589E+0
748083 5.553671E-1 1.058589E+0
748083 5.553671E-1 1.058589E+0
748084 5.553671E-1 1.058589E+0
748084 5.553671E-1 1.016156E+0
748084 5.553671E-1 1.058589E+0
748084 5.553671E-1 1.016156E+0
748084 5.126665E-1 1.016156E+0
748085 5.553671E-1 1.016156E+0
748085 5.553671E-1 1.016156E+0
748085 5.553671E-1 1.016156E+0
748085 5.553671E-1 1.016156E+0
748085 5.553671E-1 1.016156E+0
748086 5.553671E-1 1.016156E+0
748086 5.553671E-1 9.737240E-1
748086 5.553671E-1 9.737240E-1
748086 5.553671E-1 9.737240E-1
748086 5.553671E-1 9.737240E-1
748087 5.553671E-1 9.737240E-1
748087 5.553671E-1 9.737240E-1
748087 5.553671E-1 9.737240E-1
748087 5.553671E-1 9.312917E-1
748087 5.553671E-1 9.312917E-1
748088 5.553671E-1 9.312917E-1
748088 5.553671E-1 9.312917E-1
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748088 5.553671E-1 9.312917E-1
748088 5.553671E-1 9.312917E-1
748089 5.553671E-1 9.312917E-1
748089 5.553671E-1 8.888594E-1
748089 5.553671E-1 9.312917E-1
748089 5.553671E-1 9.312917E-1
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748090 5.553671E-1 9.312917E-1
748090 5.553671E-1 8.888594E-1
748090 5.553671E-1 8.888594E-1
748090 5.553671E-1 8.888594E-1
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748091 5.553671E-1 8.888594E-1
748091 5.553671E-1 8.888594E-1
748091 5.553671E-1 8.464272E-1
748091 5.553671E-1 8.888594E-1
748091 5.553671E-1 8.888594E-1
748092 5.553671E-1 8.464272E-1
748092 5.553671E-1 8.464272E-1
748092 5.553671E-1 8.464272E-1
748092 5.126665E-1 8.464272E-1
748092 5.553671E-1 8.464272E-1
748093 5.553671E-1 8.039949E-1
748093 5.553671E-1 8.039949E-1
748093 5.126665E-1 7.615626E-1
748093 5.126665E-1 7.615626E-1
748093 5.126665E-1 8.039949E-1
748094 5.126665E-1 8.039949E-1
748094 5.126665E-1 7.615626E-1
748094 5.126665E-1 7.615626E-1
748094 5.553671E-1 7.615626E-1
748094 5.553671E-1 7.615626E-1
748095 5.126665E-1 7.615626E-1
748095 5.553671E-1 7.615626E-1
748095 5.553671E-1 7.615626E-1
748095 5.553671E-1 7.615626E-1
748095 5.126665E-1 7.615626E-1
748096 5.553671E-1 7.615626E-1
748096 5.553671E-1 7.191304E-1
748096 5.553671E-1 7.191304E-1
748096 5.553671E-1 7.191304E-1
748096 5.553671E-1 7.191304E-1
748097 5.553671E-1 7.191304E-1
748097 5.126665E-1 6.766981E-1
748097 5.553671E-1 6.766981E-1
748097 5.553671E-1 6.766981E-1
748097 5.553671E-1 6.766981E-1
748098 5.553671E-1 6.766981E-1
748098 5.553671E-1 6.766981E-1
748098 5.553671E-1 6.766981E-1
748098 5.553671E-1 6.766981E-1
748098 5.553671E-1 6.766981E-1
748099 5.553671E-1 6.342658E-1
748099 5.553671E-1 6.342658E-1
748099 5.553671E-1 6.342658E-1
748099 5.553671E-1 6.342658E-1
748099 5.553671E-1 6.342658E-1
748100 5.553671E-1 5.918336E-1
748100 5.553671E-1 5.918336E-1
748100 5.553671E-1 5.918336E-1
748100 5.553671E-1 5.918336E-1
748100 5.553671E-1 5.918336E-1
748101 5.553671E-1 5.918336E-1
748101 5.553671E-1 5.918336E-1
748101 5.553671E-1 5.918336E-1
748101 5.553671E-1 5.494013E-1
748101 5.553671E-1 5.918336E-1
748102 5.553671E-1 5.069690E-1
748102 5.553671E-1 5.494013E-1
748102 5.553671E-1 5.494013E-1
748102 5.553671E-1 5.494013E-1
748102 5.553671E-1 5.494013E-1
748103 5.553671E-1 5.494013E-1
748103 5.553671E-1 5.069690E-1
748103 5.553671E-1 5.069690E-1
748103 5.553671E-1 5.069690E-1
748103 5.553671E-1 5.069690E-1
748104 5.553671E-1 4.645368E-1
748104 5.553671E-1 4.645368E-1
748104 5.553671E-1 4.645368E-1
748104 5.553671E-1 4.645368E-1
748104 5.553671E-1 4.645368E-1
748105 5.553671E-1 4.221045E-1
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748105 5.553671E-1 4.221045E-1
748105 5.553671E-1 4.221045E-1
748105 5.553671E-1 4.221045E-1
748106 5.126665E-1 4.221045E-1
748106 5.553671E-1 4.221045E-1
748106 5.553671E-1 4.221045E-1
748106 5.553671E-1 4.221045E-1
748106 5.553671E-1 4.221045E-1
748107 5.553671E-1 4.221045E-1
748107 5.553671E-1 4.221045E-1
748107 5.553671E-1 3.796722E-1
748107 5.553671E-1 4.221045E-1
748107 5.553671E-1 3.796722E-1
748108 5.553671E-1 3.796722E-1
748108 5.553671E-1 3.796722E-1
748108 5.553671E-1 3.796722E-1
748108 5.553671E-1 3.796722E-1
748108 5.553671E-1 3.796722E-1
748109 5.553671E-1 3.796722E-1
748109 5.553671E-1 3.372400E-1
748109 5.553671E-1 3.796722E-1
748109 5.553671E-1 3.372400E-1
748109 5.553671E-1 3.372400E-1
748110 5.553671E-1 3.372400E-1
748110 5.553671E-1 3.372400E-1
748110 5.553671E-1 3.372400E-1
748110 5.553671E-1 3.372400E-1
748110 5.553671E-1 2.948077E-1
748111 5.553671E-1 2.523754E-1
748111 5.553671E-1 2.523754E-1
748111 5.553671E-1 2.523754E-1
748111 5.553671E-1 2.948077E-1
748111 5.553671E-1 2.523754E-1
748112 5.553671E-1 2.523754E-1
748112 5.553671E-1 2.523754E-1
748112 5.553671E-1 2.523754E-1
748112 5.553671E-1 2.523754E-1
748112 5.553671E-1 2.523754E-1
748113 5.553671E-1 2.099432E-1
748113 5.553671E-1 2.523754E-1
748113 5.553671E-1 2.099432E-1
748113 5.553671E-1 2.523754E-1
748113 5.553671E-1 2.099432E-1
748114 5.553671E-1 2.099432E-1
748114 5.553671E-1 2.099432E-1
748114 5.553671E-1 2.099432E-1
748114 5.553671E-1 2.099432E-1
748114 5.553671E-1 2.099432E-1
748115 5.553671E-1 2.099432E-1
748115 5.553671E-1 1.675109E-1
748115 5.553671E-1 2.099432E-1
748115 5.553671E-1 2.099432E-1
748115 5.553671E-1 1.675109E-1
748116 5.553671E-1 1.675109E-1
748116 5.553671E-1 1.675109E-1
748116 5.126665E-1 1.675109E-1
748116 5.553671E-1 1.675109E-1
748116 5.553671E-1 1.675109E-1
748117 5.553671E-1 1.250786E-1
748117 5.553671E-1 1.250786E-1
Solution
Fitting an oscillating function can work but can also get difficult.
First, you need a function of which you think your data follows. In your case this could be:
f(x) = A * exp(-b*(x-x0)) * sin(2*pi*(x-x0)/T) + y0
Now, it is important to get good starting values.
- y0: y-offset is easy, your data tends to y=0.5
- x0: x-offset is easy as well, read the first y=0.5 crossing from your data, approx. at x0=22040
- A: is approx. 0.8 (0.5 to the first maximum at ca. 1.3)
- T: between 22040 and approx. 23480 you count 10 periods, so approx.
T = (23480-22040)/10 = 144 - b: For the maxima you have:
A_n = A_0 * exp(-b *(xmn-xm0). And hence,b = -ln(A_n/A_0)/(xmn-xm0). So, take the amplitude of first maximumA_0and e.g. ofA_4(of course, subtracting offset y0). And you get approx.b = -ln(0.8/0.17)/(22650-22075) = 0.0027
If you have an interactive terminal, e.g. like wxt you can easily read out these values with the mouse cursor.
Generally, if you run a fit in the gnuplot console it will either a) converge or b) run "forever" or c) stop with some error. In case of b) you could interrupt (pressing Ctrl+C) at least with gnuplot 5.4.1 and let you show the current values. If they are reasonably good you could stop and use these values as starting values.
If you have good starting values, e.g. with the procedures above, the fit might converge relatively fast. Actually, in the example below the parameter x0 is not used for varying during the fit, but it could be varied as well.
Script:
### fitting damped oscillating data
reset session
FILE = "SO74694089.dat"
f(x) = A * exp(-b*(x-x0)) * sin(2*pi*(x-x0)/T) + y0
# starting values estimates
A = 0.8
b = 0.0027
x0 = 22040
T = 144
y0 = 0.5
set yrange[-0.6:1.4]
set xrange[22000:]
set grid
set samples 1000
set fit brief nolog
fit f(x) FILE u 1:3 via A,b,T,y0
plot FILE u 1:3 w l, \
f(x) lc "red"
### end of script
Result:
Final set of parameters Asymptotic Standard Error
======================= ==========================
A = 0.882658 +/- 0.002062 (0.2336%)
b = 0.00278188 +/- 1.023e-05 (0.3679%)
T = 144.17 +/- 0.02687 (0.01864%)
y0 = 0.508991 +/- 0.0003925 (0.07712%)
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