Clock 4 pendulum pivot

In an effort to reduce the pendulum pivot friction for clock 4, I have decided to try setting the pivots on metal points rather than wood.  The points and matching cups are cut from brass.

While a little hard to see, the pins mount in holes drilled in the end of the pendulum's hook-shaped protrusion.  Once mounted, I marked where they sat on the hanger, and drilled holes to receive the cups.  It sounds more complicated than it is; here is a zoom-in of the installation.

In the process of installing the pins, I snapped off the pendulum's hook.  So that is now gluing.  Once it's dry, I'll be able to tell if this helps to reduce the pendulum's running friction.

Update: after drying, for the unloaded pendulum, I get a median of 71 full periods for the amplitude to halve.  That puts the unloaded Q = 320, which is a little bit better than before.

Wooden astrolabe

I have wanted an astrolabe for a long time and decided to make one as a small project.  After reading Chaucer's Treatise on the Astrolabe -- which is still a very clear manual for the instrument's use -- I had the plan fixed in my mind.

The finished product works nicely and looks smart.  I can usually measure the time from the stars or sun to within about 10 minutes, and can measure true north within about 5 degrees or so.

As many sources on the internet point out (correctly!) that the astrolabe is a stereographic projection of the sky onto a plane that is tangent to the earth at one of the poles.  For northern hemisphere astrolabes, such as mine, the plane is tangent to the north pole, and the projection point is the south pole.  That makes the north pole (and the north star) the center of the instrument.  Since stereographic projection turns circles on the earth into circles on the projection plane, nearly everything sketched on the astrolabe is also a circle.  For instance, both the equator and the ecliptic, which is the path that the sun appears to move through the sky, are both circles.  Since the ecliptic is almost concentric with the equator, but twisted off the equator by about 23.5 degrees (the tropics!), the ecliptic looks like an offset circle on the astrolabe.

I could do all these projections by geometric constructions, but decided that merely projecting points was easier.  This I did in python, and to keep organized, I chose to design all of the curves and scales in a Jupyter notebook.  The notebook produces SVG files as output that contain the various curves, stars, and scales, all at a fixed scale for printing.  I edited each of the files by hand to add some more difficult annotations or to make aesthetic adjustments.  For instance, the back of the instrument has an equation of time, to which I added some small glosses for "sun fast" and "sun slow" as well as the build date.

The front of the instrument consists of the rete (a simplified star chart), the tympan (a replaceable model of the sky's azimuth and elevation curves for local latitude), and a scale around the outer edge for time and compass directions.  I used this file as the source of my star chart, from which I produced the rete file.

With the rete file in hand, I manually selected the ten brightest stars, and shaped the pointers.  The idea is that the outer two rings go on the body of the instrument, while the rete, proper, starts at the inner two rings.  The picture below is an earlier revision, with somewhat different scales on the rete.  It also contains both front and back pointers.

This earlier revision uses mean solar time, from which the true position of the sun cannot be read directly.  You need to use the equation of time to make this adjustment.  I found that was too error prone.  I prefer to have the front of the astrolabe show the true position of everything, and then correct for mean solar time afterwards if desired. 

I printed two copies of this file, so that I would have clean copies of each for construction.

You need one tympan for each latitude.  This one is for my local latitude.

This file contains the same outer scales as the rete so that the pages can all be scaled the same.  These outer two scales are cut off and disposed, which is why I left some intersections.  The bright red mark is the location of true north, common to all files.

It wasn't too difficult to arrange the lines of constant azimuth and elevation, though I noticed that there is very little documentation about how the "unequal hours" lines are traced.  After playing with the models a bit, I realized that these lines are the horizon line rotated about the local north direction, not rotated about true north. 

The unequal hours aren't particularly in a modern instrument, but were used for reckoning time in Italy until the introduction of weight-driven clocks.  The idea is that day and night are divided into twelve hours of equal length, starting at sunset.  The hours are therefore of unequal length throughout the year.  During the day, the unequal hours can be read from the position of the sun.  At night, the astrolabe is more useful.  By turning the rete so that the stars are oriented correctly, the position of the sun in one of the unequal hours tells you the time.  At least on my instrument, the sketching the unequal hours seemed to occupy unused space in a pleasing way. 

The instrument was built using my usual paper-on-wood scroll saw technique.  I used 1/8" birch plywood for the flat pieces.  The tympan is merely a laminated sheet of paper, so that it is thin and sturdy.  The two pointers were cut from oak. 

Here is the astrolabe disassembled.

The instrument has a brass pin that holds all the parts on the common center (the north pole).  The back pointer has a cutout that sets the pin into place.

This is important because you simultaneously want one edge of the pointer to align with the center of the mounting hole -- so that you can sight across it and then read an elevation on the scale -- and you want the pin there too.  The pin has to fit back into the pointer to give clearance for the sight line.

The front pointer has a similar construction, but I made a small brass button to keep the pin end.  Once the pin is installed, you merely bend the tip of the pin to retain it.  The marks along the front pointer measure declination -- angular distance from the celestial equator.

Finally, I added a thumb ring that sets through a larger pin.  I turned this with a small flourish, and silver soldered the ring closed.

Two easy projects

Sometimes it's fun just to make simple projects.  Here are two I built this weekend.  A wooden yo-yo

and a sundial for my office.

The sundial is intended to be mounted on the wall, which does not lie in a cardinal direction.  It's therefore what is called "vertical declining" sundial.  My office wall is parallel to 130 degrees, so the gnomon lies off center and the spacing of the hour lines isn't uniform.  I built it according to the description given in

A. Waugh, Sundials: Their Theory and Construction, Dover, 1973.

Hopefully it'll work!

Clock 4 current power consumption

Continuing the thoughts from the previous post... How much power does the current Clock 4 pendulum and count wheel consume?  Especially, how much weight is really necessary to drive it?

I'll treat the pendulum rod and bob as two separate weights...

Rod = 28.86 oz = 0.818 kg, centered at 24" = 0.61 m
Bob = 26.75 oz = 0.758 kg, centered at 45" = 1.14 m

Potential energy for a swinging weight = m g L (1-cos(angle))

The amount of energy at the top of the test swing (4.8 degrees) is

( 0.818 kg * 0.61 m + 0.758 kg * 1.14 m ) * 9.8 N/kg * ( 1 - cos (4.8 degrees) ) = 0.046850 J

At the bottom of the test swing (2.4 degrees), the energy is

( 0.818 kg * 0.61 m + 0.758 kg * 1.14 m ) * 9.8 N/kg * ( 1 - cos (2.4 degrees) ) = 0.011718 J.

Assuming one period of the pendulum is 2 seconds (it's not, but will eventually be):

• The unloaded pendulum takes 65 periods to consume that energy = 0.270 mW
• The pendulum driving the pulling pallet consumes this energy in 54 periods = 0.325 mW
• The complete count wheel assembly consumes this energy in 50 periods = 0.351 mW

We can conclude that

• The count wheel assembly consumes 0.081 mW,
• of which 0.026 mW is due to the backstop.

These power figures are somewhat in line with my previous clocks.  Clock 1 runs on 0.5 mW and Clock 3 runs on 0.8 mW.  So thus far, Clock 3 is more efficient by a bit.

Assume that the escapement is triggered once per minute, is geared through a 10:1 gear mesh, and is driven by a 1" diameter barrel.  How much weight is required for all of these power requirements?

The weight falls at an average speed of pi * 0.0254 m / (36000 s) = 2.216e-6 m/s.

Thus, it takes

• 12.4 kg = 27.4 lb to drive the unloaded pendulum,
• 3.7 kg = 8.2 lb to drive the count wheel (without the pendulum), and
• 16.1 kg = 35.5 lb to drive the pendulum and count wheel assembly.

Way too high, I think!  I need to either improve the pendulum's Q or scrap the idea of the 10:1 gear mesh.

For testing purposes, if I were to drive the clock from the pin escape wheel directly, which has a 3/4" pinion, the weight falls at an average speed of pi * 0.75 in * 0.0254 m/in / (3600 s) = 1.6624e-05 m/s.  The amount of weight necessary to drive the pendulum and count wheel assembly becomes 2.16 kg = 4.8 lb.  (This may not be entirely safe since the pin escape wheel arbor isn't very strong.)

Clock 4 pendulum measurements

Here are some measurements of the clock 4 pendulum, trying to get a handle on its performance issues.  Woodward is adamant that limiting count wheel friction was major concern in his designs.  He employed a number of countermeasures, including anti-friction rollers, a polished acrylic count wheel, lightweight stainless steel pallets, and the merest hint of watch oil.  I don't know that my situation calls for such measures, but I figured I ought to investigate.

The pendulum is a solid square black walnut rod about 2" on a side, and is 48" from knife edge to bottom.  It weighs 28.86 oz, which is fairly uniformly distributed along its length.  The pendulum was fitted with a crude bob constructed of short copper-clad steel rods bound together with a rubber band located 45" (on center) from the knife edge, weighing 26.75 oz.

I measured pendulum amplitudes as deflections from equilibrium. 

Provided the amplitude is greater than 2.5" (3 degrees) and less than 5" (6 degrees), the count wheel advances reliably.  The count wheel does not advance at all when the amplitude is less than 2.25" (2.6 degrees). Double counting occurs when the amplitude is greater than 5.5" (6.6 degrees).

Here are counts of pendulum full periods starting at 4" (4.8 degrees) and ending at 2" (2.4 degrees), which is basically a half-time.  Pendulum Q can be estimated from this by Q = 4.532 * number of periods to halve the amplitude.

• Unloaded pendulum: 65, 72, 68.  Median Q = 308
• Pendulum driving pull pallet and count wheel, but no backstop: 58, 54, 54.  Median Q = 245
• Pendulum driving count wheel normally: 52, 45, 50.  Median Q = 226.

This indicates a count wheel-only reliable run time of about 100 seconds, which I've confirmed approximately on previous days.  If you push it a bit, you can sometimes do better on occasion.

There definitely is a noticeable change in loaded Q caused by driving the count wheel, as Woodward warns.  But, the unloaded Q figures are probably the source of my trouble, though.  The unloaded Q is around the same as a marine chronometer's balance (and not a good one at that), and that needs an impulse every period to keep running!  (I already know that the clock can run if it impulses every second.... it's not supposed to do that, though!)

Although this is probably excessive for my needs, Woodward has a table that lists a "heavy seconds pendulum" at Q = 15 000.  I think I need a better resonator!

Triggering the escapement certainly consumes energy, possibly a large amount of energy.  But it's unclear how exactly to measure that accurately...

Clock 4 escapement triggers

The next step of constructing Clock 4 is the intermittent triggering mechanism.  Once per minute (one rotation of the count wheel), it triggers the impulse hook to grab one pin of the pin escape wheel.

The impulse hook was cut from the pendulum rod (mostly for aesthetics).  This version has a brass hook rooted in the block and anchored with super glue.  I later replaced this with a stiffer steel one.

The hook is counterweighted by filling the wooden block with lead.  This was sufficient for the brass hook, but not for the steel hook.  I added a screw and nut outrigger counterweight for that.  The steel hook turned out to be a good idea because the brass one was really very pliant, and was getting distorted by each impulse. 

The hook assembly rides on a brass pin on the pendulum.

The impulse hook is triggered by an assembly that sits behind the count wheel.  The straight segment gets grabbed by the count wheel driving pallet (hook) once a minute, and pushes the flat segment against the impulse hook to engage it.

I also made a wood and brass key to wind the clock.

Here is a video the escapement being triggered successfully from the count wheel.  (Hemostats are useful to keep parts in place...)

This has taken the past two days to get it adjusted.  Here is a video of an amusing -- and vexing -- fail mechanism.  Watch to the end... it gets worse!

Next up: the pendulum is indeed not running long enough (as the previous post probably suggests...).  I suspect I do need to increase the weight of the pendulum bob, regardless of the power needs, and figure out how to reduce the friction.  That first requires finding where the friction is...

Should you add more weight to the pendulum bob when the clock doesn't run?

Short answer: no!

Medium answer: adding more weight to the bob increases the per-period energy requirement of the clock, but only up to a limit.  So once you have the clock running reliably, you can (and should) add more weight to the bob to improve its timekeeping stability.

When clock doesn't run because not enough energy is getting refreshed into the pendulum (or other resonator), it's tempting to look for easy fixes.  Adding more weight to the pendulum bob certainly delays the inevitable, since the clock will run longer before stopping.  But it has a certain futile feel to it... will you ever add enough so that it will never stop?  Sadly, you cannot.  Fixing the frictional losses (best) or increasing the drive power are the only solutions.

The reason is a straightforward derivation ending in a simple formula.

First of all, let the angular deflection of the pendulum be A = A(t), a function of time t.  For small angles, this is governed by

A'' + (c/m) A' + (g/L) A = 0,

where m is the pendulum bob mass, g is the acceleration due to gravity, L is the length of the pendulum, and c is the frictional loss constant.  By the usual process for solving such a differential equation, the envelope of the oscillations will naturally decay like

A(t) = exp( - ct/(2m) ) A(0) ( .. trig functions .. ).

If T is the time of one period, the max amplitude is given by

A(T) = exp( -cT/(2m) ) A(0).

Now switching to discuss energy, the height of the pendulum is found by a little geometry...

... to be given by

h = L(1-cos A(t)).

Therefore, the energy change from one period to the next is

dE = mgL(cos(A(T)) - cos(A(0))) = mgL(cos(exp( -cT/(2m) ) A(0)) - cos(A(0)))

again applying a small angle approximation,

This expression is the one we're after, and really we want to know how it changes as we change m.  Clearly at m = 0, the change in energy is zero. Taylor expanding in m, we have the behavior for large m is approximately

Here is a plot of the overall behavior

The takeaway is that as you increase the mass of the pendulum bob, you must supply more energy to sustain oscillations, but only up to a limit.  For stable, reliable operation, you should ensure that the drive supplies at least that limiting amount of energy first first, before increasing bob weight.  Minimizing frictional losses should be the first priority -- which decreases c -- before trying to increase drive weight.  Only after the clock runs reliably should you attempt to increase the pendulum bob weight.

Clock 4 detent works!

Remaking the detent a few times did the trick.  Each time it worked a little better than the one before, as I flushed out the bugs.  I had a scare where I damaged the gate, but a little super glue seems to be holding it together.  Here is the detent that finally does the job.

The detent properly releases one pin at a time when recoiled by hand with a weight of just about 2.2 lb on the great wheel.

I did not have to modify the pin wheel. It is helpful to have a banking so the detent is held in a convenient position if all the weight is removed.  This also gave a good opportunity for testing the winding mechanism, which does indeed work.