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Re: [Phys-L] Trinity Pendulum Q (was Re: checking the work for minus signs ... et cetera)

Thank you for responding, Hugh. Your working clarified that the cause of Bernard's rather high value for the Q of the Trinity Pendulum was his use of the escapement energy contribution at each cycle as about 0.0029 j in lieu of the 0.0038 j which would be needed to maintain a Q value as low (!!) as your free decay Q value. This escapement energy contribution value was evidently derived from a Trinity Clock site mention of the Denison gravity arms weighing 50 grams each and descending 3 mm.
Your note was also a useful reminder of Denison's use of an air-brake ('fly') with a lossy friction clutch to cope with the varying clock drive needed during the year while maintaining a sensibly constant impulse to the pendulum, and as important, eliminating the escapement skip not uncommon in tower clocks prior to 1860.
Thanks again
Brian Whatcott

On 5/17/2015 12:21 PM, Hugh Hunt wrote:
Dear Bernard, Doug, Brian

I'm very interested in your calculations. I am Keeper of the Clock at Trinity and Rick Lupton, who wrote the report, was my student.

The pendulum free decay (without drive) can be see here t clock change
<http://trin-hosts.trin.cam.ac.uk/clock/?menu_option=data&from=26/10/2008&to=26/10/2008&width=700&channel=amp&&xmax=1&skip=0&ymin=32&ymax=53.6>

Note that the decay is nicely exponential. The width of the plot is one hour, so the decay is from 53.6mrad to 29.8mrad in about 3050 seconds
Using the usual definition of Q from where it derived (Quality Factor for resonant circuits) then A2/A1 = exp(- zeta w T )
where A1 and A2 are two amplitudes separated by time T for a damped oscillator with resonance frequency w (rad/s)
The Damping Factor zeta = 1/(2Q)

So A1=53.6 A2=29.8 T=3050 s w = 2*pi/3 for our 3s pendulum (half period = 1.5s)
gives zeta = 0.0000919 or

Q = 5,441

Now, this Q is not the same as the running Q because of the additional energy losses in the drive train that are not present during a free-pendulum slow-down.

If you take the fact that the going weight (80kg) descends about 7 metres per week then the energy loss per cycle is about 0.028 J
The pendulum bob mass is about 104kg and the length is about 2.23m so at amplitude of 53.6mrad the peak kinetic energy is
KE = 0.5 * m * v^2 and v=L*theta * w
KE = 0.5 * 104 * (2.23 * 0.0536 * 2*pi/3)^2 = 3.26 J

So energy loss per cycle is 0.028 / 3.26 = 0.0086 = 2*pi / Q

so Q = 731

This shows that the escapement and drive mechanism is consuming 87% of the delivered energy.

Most of this is lost in the acceleration and impact of the going fly.

Does this accord with your own sums?

Best wishes

Hugh





--On 17 May 2015 00:10 -0500 brian whatcott <betwys1@sbcglobal.net> wrote:

On 5/16/2015 4:57 PM, Bernard Cleyet wrote:



On 2015, May 15, , at 12:06, brian whatcott <betwys1@sbcglobal.net> wrote:


Bernard,

I replicated your value for Q given the following assumptions which I presume you made:
1) the effective pendulum length l is obtainable from Period = 2.Pi. root(l/g)





yes


2) The value for g at Trinity is 9.8 N/kg.




yes, however in my method it "cancels". [except for the length, which changes little (0.2%)
as the change is ~ delta (g)/2 ]

I am supposing that by length, you mean the effective pendulum length for constant periodic time,
would be increased by 0.2% where g increases from 9.80 to 9.81 N/kg.
In contrast, I imagine delta L is proportional to delta g for constant period. Or is it?



The pendulum amplitude is 55mr.
3) The maximal value for pendulum energy occurs at the potential energy max.





? is ~ constant, and I understand your reasoning, which I used. BTW, there's a web site
that gives the E for M, g, A,and T; and the speed at BDC!

For a lossless pendulum, the kinetic energy max would equal the potential energy max.
In a practical pendulum, the total energy decays after the escapement's "push".
It is because the pendulum's energy decays during each beat that computing the potential
energy stored at the moment when the pendulum is still, does not completely define the kinetic
energy available at maximal speed.






4) Infinitely rigid (support), lossless pivot





Not necessary as the drive "cancels" all the dissipations.

While it is certainly the case that a pendulum maintained at constant amplitude has its losses
made good by the escapement, the potential energy calculated using Woodward's method
necessitates an assumption about how the dissipation occurs - if I postulate a pivot point with
great horizontal compliance, then the potential energy calculation which depends upon the bob's
(reduced) ascension would be reduced, which would have an obvious effect in reducing the value of
Q calculated for a given escapement energy contribution.





If I choose instead the moment where angular displacement = 0 mr
and an escapement increment has occurred, the maximal KINETIC energy is calculated by adding
half the escapement energy to the peak potential energy (the other half of the escapement's
energy contribution being dissipated as the bob returns to a 0 angular displacement.) This
changes the value for Q but only slightly.
Similarly, the amplitude of this clock varies: today (friday) it is 47 mr which has a much more
distinct effect on the value of Q obtained.






Yes, only v. ~ 17% high now.


Not quite sure about this comment: a reduced amplitude goes with a reduced
ascension, a reduced potential energy, hence a reduced kinetic energy, and for a constant
escapement energy input, a reduced Q figure. If your 'v' refers to the maximal bob speed, a 17%
reduction in amplitude would involve ~ 8% reduction in v, would it not?






I assume the Q was measured as a decay over "some" time with the pendulum moving the
gravity arms only, which is likely a problem. This is because I only know of two persons(1)
who've used the "bc" continuous Q measurement method, which finds the Q at the running
amplitude instead of an average.




Doug has shared his reservations about 'free-wheeling' the Trinity clock gravity arms. It would
be hard to disagree. As to continuous measurement of Q: this could be achieved by
continuously monitoring the decay of energy between escapement contributions - but this is not
something that you have done, as far as you have described it, so it is not quite a 'continuous'
method that you have outlined. You mentioned Doug's "Measuring Amplitude Velocity and Q. I
wonder if this is available on line at all?

Regards

Brian W




Moreover, in the Cambridge vicinity, g runs 9.81
See http://www.bgs.ac.uk/products/geophysics/landGravity.html for N52.33 W0.0

None of this explains the large difference between your value and that provided for Dr Hunt.
I notice that no estimation of the effect of suspension pivot rigidity in space is given,
other that a speculation about wind-driven deflection of the tower.

Brian W
[I see that Doug S Drumheller - Sandia emeritus and Hugh Hunt at Cambridge are both on your
copy list. It would be interesting if they care to contribute.]





(1) Drumheller, Douglas Measuring Amplitude, Velocity, and Q HSN 2012-4 The author
describes an improved version of the "bc" method w/ an example.
And B. Mumford who has incorporated it in one of his software versions accompanying the
MicroSet.

/snip/