# MEASURING CLOCK ENERGY

I don't have any previous experience with time measurements, and clocks, but for some reason I did find the subject of clock noise fascinating enough to make me purchase a synchronome on ebay and attempt some measurements to satisfy my curiosity.

Let me describe what I did about the energetics of this clock, it helped me to develop some intuition about it and hopefully will encourage others to report their findings.

I have tested first the pendulum energy losses as a function of the swing amplitude for the pendulum alone, starting larger than usual oscillations and letting them decay while measuring the pendulum speed.

I have chosen to measure the oscillation speed at its max, when the pendulum is in the middle of its oscillation, with an optical sensor, a Sharp, GP1A57HRJ00F from Sparkfun https://www.sparkfun.com/products/9299 which has the property of outputing a digital signal compatible with Arduino.

The sensor has been carefully positioned in the middle of the oscillation by balancing the pulses symmetry between the right and left going swing on an a digital oscilloscope while tilting the sensor base which is mounted on 3 small springs (not visible in the picture).

The sensor circuitry is the one of the Sparkfun breakout board.

The light beam is interrupted by a pin of known diameter (D) coaxial with the pendulum rod, in my case 3.4 mm OD (see pictur above).

I found acceptable to construct it with a toothpick located inside a black heat-shrink tube shrunk to the pendulum rod threaded end as shown in the image. This simple design has the advantage that if the pendulum swings out of center thie flexible plastic tip bends away without causing mechanical damage.

The pendulum max speed vmax is very approximately equal to the ratio of the pin diameter, 3.4 mm in my case, to the measured interception time (ΔT) of the light beam , vmax=D/ΔT.

The kinematics of the harmonic oscillator, the same of the pendulum aside from the circular error, tells us that the maximum speed and the swing amplitude are related by the pendulum frequency F, vmax = ω xmax, ω= 2 π F, the frequency F in the case of the Synchronome is equal to 1/2, so xmax = vmax/π .

Measuring the pendulum speed yields also the measurement of the angle of the pendulum swing θ equal in degrees to 180/π xmax = 180/π vmax/2 π F = 9.12 vmax T where T is the pendulum period, T=2 s in the case of the Synchronome.

The data acquisition is a rudimentary one, it consists of an Arduino Uno running a super simple program:

where the key instruction pulseIn(pin,LOW,1500000) returns the length of the negative going sensor pulse in microseconds with a 1.5 10^6 μs of time out.

The noInterrupts() instruction is not essential but improves the accuracy and repeatability of the measurement by a factor of 10 in the author's experience. The calibration of my 3 arduinos when measuring 20 ms pulses was within 30 ppm.

The delay(10000) instruction line is there only to reduce the number of measurements displayed on the PC terminal with the PC time stamp (an arduino terminal option) which witnesses the actual delay between measurements. More measurements would produce a better statistics but measurent parsing allows for a quick spreadsheet analysis .

Arduino's clock accuracy, from which the accuracy of these measurements depends, doesn't look relevant to me for the present purpose but for the curious ones I have found a reference http://jorisvr.nl/article/arduino-frequency

A typical output is as follows

the time stamp of the measurement is followed on the same line by the measure in microseconds of the pendulum tip interrupting the photosensor light beam.

For a start I would suggest to copy and paste to a spreadsheet these numbers for analysis

The time span of this measurement for a good range of angles can be hours, typical Qs being in the thousands and the period is 2s.

After acquiring enough data check the regularity of the time stamps, get the average constant delay between measurements in seconds and evaluate Q following these steps:

1. check that the max delay (ΔT) in the set is << 2 x arcsin (D/2) / π , if not throw away the rest of the data, the oscillation amplitude has gotten smaller than the sensor pin and the pulse is meaningless
2. average the delays (ΔT) over N samples and select equally N-spaced values from the delT dataset to make the sample more manageable on a spreadsheet and multiply the micros ΔT by 1000 (Δ)
3. make a column with the pendulum max speed (vmax), D/Δ. If D is in mm, like in my case where it's 3.4 mm, vmax will be in m/s.
4. Assuming it's the field standard I adopted the Q definition from "My own right time" by P.Woodward, Q= 2πΕ/ΔE, where ΔE is the clock energy loss in one period, in our case 2s.
5. make a column with Q =( Tstamp*N/Cp)*2π*(Δi+1)^2/[(Δi)^2-(Δi+1)^2], Tstamp*N/Cp is the number of clock periods (Cp) between measurements, (Δi+1)^2/[(Δi)^2-(Δi+1)^2] is equal to E/ΔE since in our case Δ is the only time varying component of the kinetic energy 1/2 m v^2.
6. The knowledge of the max pendulum velocity yields also the oscillation angle. From harmonic oscillator kinematics maximum pendulum excursion and max half angle are equal to vmax/2πF. Since for a synchronome, a 1 s clock, the frequency F=.5 Hz, the oscillation half angle is = =vmax/π . A confusion can arise from the fact that a 1 s clock, 1 s for single swing, completes its full, two swings, cycle in 2 s.

## CLOCK ENERGY AND POWER

The typical oscillation amplitude for Synchronome 2177 while powered in steady state is 2.2° , its bob energy is around 50 mJ and the pendulum dissipated power, at a Q of 7000, is roughly 25 microW.

With the counting wheel connected Q drops to 4000 and the power rises in the region of 50 microW.

For regular mechanical pendulums these measurements are interesting because they yield the amplitude of the oscillations and the energy, from which one can calculate the driving mechanism power and the effect of any improvement in the clock gear chain.

All formulae should be checked, it's all very recent and I tend to work out things by myself and make errors. All suggestions, criticisms and questions are welcome, publicly or privately (pepicima@gmail.com).

Here is some progress in energy measurements, a couple of microcontrollers stuck on a wooden board next to the clock with SD cards (example of Arduino program) to store lots of data and Labview to analyze them, (example of Labview program) on top of EXCEL of course. I am not a Labview expert programmer at all but I must admit that I can effortlessly put together pretty large programs, it feels like a very convenient platform for this kind of work. As an example take this video https://www.youtube.com/watch?v=6KAltAiLE8U to analyze the frequency spectrum of recently recorded free pendulum amplitudes, it took about an hour to do the whole work. The first graph shows the photo-interrupt signal, red dots, proportional to the inverse oscillation amplitude. Superimposed is the white line, a 3rd degree polynomial best fit.

The second graph shows the difference between the two, the error if you want, with a slow component and a fast 1/4 Hz one. It's shown more clearly in the bottom left graph and in the time domain window to its right.

It's just an example to show how powerful the system is, in practice it has been useful to show that the 1/4Hz signal is an artifact which can be reduced by centering the photo detector to the pendulum vertical.

I used this setup to measure the pendulum amplitude decay in 4 different cases:

1) free pendulum, 2) free pendulum pushing the synchronome only toothed wheel 3) the wheel and the gravity arm latch, 4) this last setup with a jeweled gathering click, the other data all employed a home made steel one.

The useful working range is the one of the middle of the graph, between 2º and 3º, at 1.7º the stone gathering arm misses the tooth as one can see it in the abrupt change of slope of the yellow curve which returns to the one of free pendulum.

The jeweled gathering tool is unexpectedly worse than the steel one, in practice is very similar, the real losses seem to be in the wheel.

I tried some counterweighting, wrapping soldering wire to the ratchet lever, just enough to keep the ratchet working

and it makes a very noticeable difference, the steady state angle went from 1.9º to 2.7º

if I put the new data in the old free pendulum dissipation graph it shows that most of the counting wheel mess comes from the ratchet, Which is still there doing its job by the way