3-1-Mma-Spring Mass Damping
Data Analysis -Student Version

PROF Emeritus Brian Winkel, Department of Mathematical Sciences,
United States Military Academy, West Point NY 10996, USA 
BrianWinkel@simiode.org
10 October 2014

We study a spring mass system as depicted below with a spring attached to a horizontal rod on a ring stand and a Vernier Motion Detector under the mass to capture its displacement from a static equilibrium value. The motion detector is turned on and the spring is stretched downward and released. We collect the data and ignore (edit out) data from hand placement until a reasonable point in the motion when the oscillation is clear and clean. That cleaned up data we move to this Mathematica notebook.

Collection of data and discussion.

We  attempt to model a spring mass damper using the differential equation,

                                 m y’’(t) + c y’(t) + m y(t) = 0,     y(0) = (Subscript[y, 0]),     y’(0) = (Subscript[v, 0])

where y(t) is displacement from static equilibrium of a bouncing spring. y(t) > 0 is downward from the static equilibrium and y’(t) > 0 is moving downward. Also,  y(0) = (Subscript[y, 0]) is the initial position and y’(0) = (Subscript[v, 0]) is the initial velocity.

We are using a known mass of  200 g  with four 3” by 5” index cards taped to the bottom of the mass to offer more resistance for total mass of m =  0.20626 kg and unknown values of a damping constant, c  N/(m/s), and spring constant,  k N/m.

Collection Period:  30 seconds @ 50 data points per second for total number of 1,500 observations in the set data.  9 May 2013 collection date at our home office. We used the Vernier Go!Motion collection device and a spring with a brass mass of 200 g.

EXECUTE these cells to get original data into this notebook. We have  1500 data points from sampling the position of the mass every 0.02 seconds for 30 seconds.   However, the same data and its final form as described below is available in an Excel spreadsheet 3-1-S-SpringMassDampingAnalysis-StudentVersion.

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We prepare the data for analysis.

We clean up and standardize the data so it has a displacement from its average (static equilibrium) value into a final data set, sdata, using the data set subdata, en route as we prepare the final data set for analysis. 

Basically, we find a symmetric difference in velocities about the second data point and use this as the initial velocity from our capture data occurring at the second data point as our new start. We do this as we are capturing data, particularly a velocity we can use as an initial velocity, while in motion, not at the instant of release.

We take the average value of the heights and subtract it off so as to center the displacement (about the static equilibrium) of the data.

Length[data]

Ave = Sum[data[[i,2]],{i,1, Length[data]}]/Length[data]

We subtract off the clock time to zero out the first observation. That is, we actually start our data for modeling from time t = 0.2 seconds (our tenth incremental observation with increments of 0.2 s) of the total 30 second data capture period, when the spring is moving smoothly and not irregularly from the starting release activity.

subdata = Table[{data[[i,1]]-data[[10,1]],data[[i,2]]- Ave},{i,10, Length[data]}];

dataTot = subdata; (*** we hold on to ALL the original data points, shifted, for possible future reference. ***)

Length[subdata]

We offer a plot of the displacement in m vs. time in s.  We do this first for the first 10 second period.

ListPlot[subdata,PlotRange->{{0,10},{-.1,.1}},PlotStyle->PointSize[.006],AxesLabel->{"Time (s)","Displacement (m)"}]

Of course, a plot of all the data, while showing the complete actions of our mass displacement is somewhat congested!

dataTotPlot = ListPlot[subdata,PlotStyle->PointSize[.006],AxesLabel->{"Time (s)","Displacement (m)"}]

We determine an initial velocity at a smooth point in the data, by taking the symmetric difference in velocity between points 2 and 3 and between points 1 and 2.

v0 = ((subdata[[3,2]] - subdata[[2,2]])/(subdata[[3,1]] - subdata[[2,1]])+ (subdata[[2,2]] - subdata[[1,2]])/(subdata[[2,1]] - subdata[[1,1]]))/2

We form our final variation of our data, called sdata, having shifted time to start with our second data point whose average velocity approximation is a good initial velocity for our differential equation modeling and whose initial time for the first observation is t = 0 s.

sdata = Table[subdata[[i]]-{subdata[[2,1]],0},{i,2, Length[subdata]}];

We finally determine the initial position of our displacement for this final data set, sdata.

y0 = sdata[[1,2]]

And we know our initial velocity from above. So we are ready to analyze the data and model it.

v0

y0 > 0 means our springs is BELOW  the static equilibrium at our starting time and v0 > 0 means  it is heading up.  We see that in our sdataPlot below.

We determine the time of our last data point of sdata we use in our modeling, then “end” of our time.

tend = sdata[[Length[sdata],1]]

sdataPlot = ListPlot[sdata,PlotRange->{{0,tend},{-.1,.1}},PlotStyle->PointSize[.006],AxesLabel->{"Time (s)","Displacement (m)"}]

SmallerPlotsdataPlot = ListPlot[sdata,PlotRange->{{0,0.4tend},{-.1,.1}},PlotStyle->PointSize[.006],AxesLabel->{"Time (s)","Displacement (m)"}]