Re: DATA on collapsing WTC

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

Glenn Carlson writes:

> ... Ludwik concludes that the tower was not in free-fall with
> an acceleration of magnitude of approximately 0.3g - 0.7g. I
> conclude that the tower was in free-fall with an acceleration
> of 0.95g - 1.05g.

I will offer my analysis (below) and conclusion that the data
strongly supports the following model:

For t < 2 seconds, ~constant acceleration of 0.2 - 0.3 g
For t > 2 seconds, ~constant acceleration of 0.4 - 0.5 g

> Ludwik's approach and findings--
>
> Taking the data during the collapse in small contiguous sets
> of 9-15 datapoints, an average time and vertical position is
> calculated. This reduced the amount of data from 120
> datapoints to 8-13 datapoints.

IMO this approach is to be avoided (as I mentioned in a previous
related thread called--"Air Resistance" in December 1997) as it
takes us one step further away from the raw data.  It is (almost?)
always preferable to fit unmassaged data if possible and allow an
appropriate chi-squared analysis to offer its own opinion on the
adequacy of the model.

> Taking the data during the collapse I calculated the time
> since the beginning of the collapse and the magnitude of the
> tower's displacement.  Using all of the approximately 120
> datapoints I calculated a best power law of the form y=at^n,
> where y is the magnitude of the displacement and t is the time
> since the beginning of the collapse.

I would not expect this to work very well as the results will
depend critically on one's choice of the moment at which the
collapse begins.  I played with this approach as well and,
depending upon that choice, I got exponents from around 1.3 to
over 3.  Furthermore, a chi-squared analysis reveals the
inadequacy of the model.

> After t=1s, I calculate a best fit curve of y=4.7t^1.9
> (R^2=0.99); I conclude the tower was in free-fall with an
> acceleration of a=2*4.7=9.4 m/s2

Isn't your data in feet?  Did you convert?  If not you should have
concluded that a is approximately 0.3 g, right?

Here's what I did:

I first fit all 149 pts without trying to determine at what point
the collapse began.  (From a purely visual inspection of the data,
it isn't obvious to me that the collapse wasn't already underway
at "t = 0."  Indeed, data I will offer in a moment suggests that
it might have been.)  I also calculated a chi-squared per point
value by assuming a best case scenario in which the y data had an
uncertainty of half a pixel (1.13 ft).  A good model will yield a
chi-squared per point of around 1.

The result was

t = 0 to 5 seconds

   y = -11.0 t^2 + 12.3 t - 2.6 with chi^2/pt = 6.5

These results suggest
 (1) an acceleration of 22 ft/s^2 (0.69 g)
 (2) an initial upward velocity of 12.3 ft/s (!!)
 (3) either
   (a) a significantly larger uncertainty in the y-values
       (this is certainly plausible) or
   (b) a bad choice of model (i.e., not constant acceleration)

I then looked at the residuals which, as brian w has pointed out,
look somewhat sinusoidal.  In fact, what the residuals indicate is
that the model gives y-values that are too low (i.e., more
negative than the data) near t = 0, 2, and 5 seconds and too large
(i.e., less negative than the data) near t = 1 and 4 seconds.
This suggested to me that we might be seeing two periods of
constant acceleration (0-2 s and 2-5 s) during which the
acceleration is constant with the first period having a lower
acceleration than the second period.  If this were the case,
trying to fit the *entire* period to a *single* acceleration would
give precisely the type of residual signature that appeared.

My fits yielded the following results:

t = 0 to 2 seconds

   y = -4.21 t^2 - 0.025 t + 0.380 with chi^2/pt = 2.0

t = 2 to 5 seconds

   y = -7.10 t^2 - 15.9 t + 45.3 with chi^2/pt = 1.3

These results suggest
 (1) The model is quite good. (Recall that I assigned a somewhat
     optimistic uncertainty to the y-values and still achieved a
     chi-squared per point that approaches its expected value.)
 (2) The building *may* already have been collapsing at t = 0.
     (Note the negative initial velocity at t = 0.)
 (3) The acceleration during the first two seconds of data was
     roughly constant at ~ 8 ft/s.  I'll be conservative here and
     say 0.2 to 0.3 g.
 (4) The acceleration during the last three seconds of data was
     roughly constant at ~ 14 ft/s.  I'll be conservative again and
     say 0.4 to 0.5 g.

I have put my spreadsheet analysis on the web for anyone who might
be interested at

  <http://www.csupomona.edu/~ajm/special/collapse.xls>

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm