It was tantalizing to see the responses to Hugh Hunt's pointer for data
on a Trinity clock restart for October 2009, in connection with seeking
a value for Q.
Though the various determinations of a restart Q were scattered (from
~1000 to 2000) they all were remarkably lower than the Q values found
earlier for free decay of the pendulum. (I calculated 6094 for the decay
on that same morning! )
Why? If a small but constant fraction of the available energy is lost at
each decay cycle, then during restart, not only that loss needs to be
made good, but the remainder of the escapement's contribution is all
that remains to increase the amplitude. This seems like a plausible
reason for the discrepancy between the decay Q and the recovery Q. I
was particularly taken with Bernard's sharing of a method using the
transit time of a narrow vane at peak pendulum velocity on successive
cycles as tokens for kinetic energy and energy loss per cycle (in the
form of the square and the difference of successive squares) though it
was not difficult to see that the mechanization of this fine idea
introduced two discrepancies which led him to a somewhat high dynamic
value for Q.
I was curious to understand how Drumheller found a way to compensate
the peak velocity value in the face of decreasing amplitude, where more
of the slower swing was included. I was interested to hear of Rawling's
Q estimator - using the cycle count to half amplitude - which naturally
led to a review of the multiplier between time to half life and
exponential time constant tau. This was pertinent to the very snappy
response to Hugh's note which Bernard offered (within the hour?) where
from a decent exponential model and a reasonable value of time constant
of ~26 minutes for the October 2009 recovery data to which Hugh alluded,
he "Qestimated" (cute!) a value Q~1000 whereas I show it closer to
Q=1633 using his time constant, and Q=1566 using my similar model and
Rawling's estimator. Even Donald Polvani's carefully computed value
Q=2100 is far below Hugh's value for Q during decay (Q=5441). I notice
that Doug Bateman keeper of the Westminster clock (aka Big Ben) alluded
to a high Q value of the unloaded undriven pendulum there. Though he
objected to Hugh's calculation of Q including all losses (not least the
fly clutch) I saw that simply as a comparable calculation to Bernard's
original citation of a text problem which also did the very same
inclusion of all system components (though one supposes there was no
lossy vane involved in the text example).
Finally, I should again say thanks to the contributors to this thread
for providing a week of harmless computation for me: - Matlab, NLReg
(Non linear Regression Analysis by Sherrod) and the old TI Scientific
just didn't know what hit them! Sincerely Brian Whatcott