Clock 4 now runs with intermittent impulsing

"Intermittent" can be a problem, but not in this case!  Based on the numerous power budget calculations I've done, impulsing the pendulum in Clock 4 every minute is too much to ask.  After having gotten the escapement to impulse every period (2 seconds) reliably, with run times around 8 hours, it seemed like the right time to go back to trying to get the intermittent part working again.  Especially, the run times without intermittent escaping were limited by drive cord length -- I had a four fall pulley in place for the clock to run that long. 

Therefore, I added more deep cuts to the count wheel, now five in total.

This means that the escapement should be triggered every 30/5 = 6 pendulum periods, or every 12 seconds.  The pin wheel has 30 pins, so will then have a period of 12 seconds * 30 = 360 seconds = 6 minutes.  The pin wheel is driven through a 1:10 mesh for the drive wheel, so it should make a rotation every hour.  I can therefore drive the minute hand from the drive wheel, although it will run counter clockwise.

With some tuning, the Clock 4 runs with 8 lb of drive weight, directly driving a barrel of 1.2 inches.  The clock's run isn't perfect, as (1) the count wheel double counts immediately following an impulse and (2) sometimes this double-counting skips over an impulse.

But given these issues, Theodore measures the following periods in current configuration:

• 53 seconds for the count wheel

• 4 minutes 24 seconds for pin wheel

Given these measurements the drive barrel will make one rotation about every 44 minutes.  In that time, the weight will have dropped 3.7 inches. 

Thus the power consumed is:

3.7 inches / (12 in/ft)  * 8 lb / (44 min * (60 s/min)) = 9.4 * 10^(-4) ft lb / s = 1.28 mW

Laser cut equatorial sundials!

Just for fun, here are two laser cut sundials!

They are made from the 20190612_equatorial.svg file in my github repo.  You can't see the shadow of the gnomon on the clear one, but you can totally see it projected (correctly) on the ground!  The etching on the white background didn't show up initially, so I set some ink onto.  It's nothing fancy; I just smeared black whiteboard marker ink over the face of the dial, and then wiped off the excess ink with my hand.  It's not waterproof...

More power calculations with Woodward's intermittent grasshopper

By joining the count wheel pusher lever of the the Woodward escapement to the escapement trigger, you can make the escapement trigger once per period.  This is the most frequent that the intermittent grasshopper can be triggered.  Triggering every period already happened by accident, but I decided to force it to occur by linking the mechanisms together without using the count wheel.  This way, I could debug the escapement mechanism... and there were indeed problems there.  I think I've resolved them, and this modified mechanism reliably runs until the weight hits the floor.

Currently, the mechanism runs on 1 lb 14 oz, falling 3.9 inches every 10 minutes.  Converting to standard units, this means that the weight falls

3.9 inches * 25.4 mm/inch / (10 min * 60 s/min) = 0.17 mm/s
1 lb 14 oz = 0.85 kg = 8.3 N

Thus the power consumption is 0.17 mm/s * 8.3 N = 1.37 mW.

This is substantially more pessimistic than my previous figure of 0.325 mW averaged over one minute for the count wheel assembly.  This is even with an improvement resulting from a few changes I made.  The pendulum is now hung from two sharp brass points resting in brass cups.

This new hanger ensures a positive positional lock and a definite axis of rotation for the pendulum with substantially less friction than before.

I also made a number of small improvements including reshaping one of the pin wheel pinion teeth, aligning the impulse hook, and stopping the detent's fall a bit earlier.  Finally, I removed every other pin in the pin wheel, which means that the period of the pin wheel is one minute.

Update: 5/13/2019.
By clipping off the tail of the locking detent to make it somewhat more delicately balanced, I can reduce the drive weight by 6.5 oz.  Thus, the power consumption is

3.9 inches * 25.4 mm/inch / (10 min * 60 s/min) * (1.47 lb * 4.43 N/lb) = 1.08 mW.

"Hidden" power losses in Woodward's escapement

I have been thinking about how my implementation of Woodward's intermittent grasshopper is not successfully transferring enough power to run.  The pendulum power requirements I computed earlier are certainly sobering -- even though this clock's pendulum already appears to be substantially more efficient than my other clocks -- they are not the only power loss. 

Since the escapement runs intermittently, I can dramatically increase the power supplied by the escapement by repeatedly triggering the escapement.   In one experiment I tried, the triggering mechanism got jammed, which triggered an impulse every swing.  This was enough to run the mechanism until the pin escape wheel got stuck. 

But even when triggered every swing, the power supplied is only marginally sufficient, even setting aside the pendulum losses.  If I look at the amplitude immediately before and after an impulse, it's not noticeably different.  This indicates that there are substantial additional losses that occur during the triggering and impulse.

What can cause this? The most obvious (though probably not the only) energy losses are caused by the fact that it takes a definite amount of energy (force times distance) from the pendulum to move both the triggering lever and impulse hook.  Since both of these fall back to their original positions after the impulse without returning this energy to the pendulum, all of this energy is lost!  So, the return counterweights should be heavy enough to ensure a reliable positive action, but otherwise as light as possible. 

(I recall now a similar issue with Clock 3, where my initial attempt at a sprung detent resulted in too much energy loss.  Since I couldn't get the wooden spring weak enough without breaking it, I ended up opting for a light counterweight.  It's probably still too heavy, and may account for much of the need for a heavy drive weight.)

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.

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.

Clock 4 detent issues

I installed a new(er) 1/3 hp 1725 rpm motor on my lathe since the bearings on the old one were dead.  It runs much better than before!

The detent mechanism for Clock 4 uses a gate invented by Philip Woodward (I think).  The detent sits on a pivot near the pin escape wheel.

The detent is fairly long, but just press fit into the frame.

The detent is cut from a small piece of white oak.

Here is the detent after shaping.

There are many issues with the detent, and it doesn't run at the moment:

• The gate is very thin.  I broke two detents already
• Woodward didn't seem to bank his detent, but it looks like I need to since wood has more flexibility than metal
• The catch for holding the pin is very touchy as to how deep it is.  Woodward suggests that it might work as just a small depression, but this caused the pins to jump out.  Too deep, and they can't clear when the escape wheel recoiled... in which case the pins stick.
• The pins of the escape wheel are too inaccurate in their placement
• The pins of the escape wheel are too inaccurate in their vertical alignment
• The pins of the escape wheel are not all the same diameter (because some of them split in the process of being installed).
• The relative positioning of the catch and the gate slot is quite delicate, and there isn't much clearance.
• The counterweight portion of the detent governs how much weight is needed to run the escapement.  This needs to be very light.

A few times, I could feel the escapement "almost working" under my hand, but it wasn't consistent enough to run under a weight.

Clock 4 pin escape wheel

One of the defining features of Woodward's intermittent grasshopper escapement is the large pin escape wheel.  It is intended to be let off once every minute, so there are sixty pins.  After considering the possibilities for how to index, drill, and make the pins... here is how I proceeded.

Since I print the pattern and glue it to the wood, indexing is "sort of" not a problem.  I started by center punching each pin location.

Since I don't have a drill press, I set up the lathe to index each of the punch locations and drill as well.  The spacing between pins may be accurate enough, because when the drilled holes make their way around to the indexing pin, they're slightly off from the center punch dents.  But it at least this setup ensures that the pins are all the same distance from the center of the wheel.

After initially thinking of steel, then brass pins, I decided that metal pins might be rather loud.  So instead, the pins are made from toothpicks.  They are roughly cut to the right size.

Then they're staked into position...

... the end sticking out the bottom clipped off ...

... and then carefully planed flush with a chisel. 

After all this, I went through and cut off any pins that were longer than the rest, probably to within a 0.5 millimeter or so. 

Here is the escape wheel trial fit in the frame.

Clock 4 mounting and clicks

Unlike Clocks 1 and 3, but like Clock 2, I plan to make Clock 4 wall mounted.

 
My wall mounting plans are to use a French cleat, since this makes it easy to remove the clock, and it's sturdy.  I attached a cleat to the back of the clock...

... but it was unstable since the plate is wide and the pendulum is off center.  So to stabilize, I added a small dowel to the back of the plate that grips the bottom of the cleat on the wall.

This means that I'm confined to use a particular size cleat (on the wall), but other dowel locations can be added easily.

The other thing I wanted was for the clicks and click springs to be carved from a single piece of wood.  I tried this on Clock 3 with oak springs, but they were very stiff and eventually broke.  Now I'm trying a pair of walnut springs, each a little lighter than the single oak I used previously. 

They're glued to the drive wheel.  I thought about offsetting them, which would lead to smoother winding, but I had trouble keeping the mechanism stable while setting up the glue. Hopefully they'll stand up to use.

Clock 4 drive assembly

Previously, I haven't made key-wound mechanisms, so Clock 4 is to be driven by a roughly 1" barrel wound with a key.  The drive wheel rides loosely on the barrel arbor, which is (at least for now) friction-locked onto the barrel. 

The barrel arbor has a 1/8" plain pivot that fits a hole in the back plate.  To engage the key, there is a cross drilling for a steel pin.  The drive wheel is supported in a cock screwed onto the back plate.

The drive wheel meshes with a pinion that directly drives the escape wheel. 

The pinion rides on a small arbor that itself screws into the back plate.

I will cut away some of the cock to allow clearance for the escape wheel, which also will carry the minute hand (directly).  I also plan to try Aaron Dodd Crane's daisy wheel motion work to drive the hour hand coaxially.

Count wheels and pendulums

Philip Woodward invented many interesting clock mechanisms, which are based around intermittent escaping.  To time the intervals between impulses, his escapements use count wheels.  But he cautions that a poorly-designed count wheel can dramatically alter the Q of the pendulum, and cause reliability problems.  Since my next clock is planned to use Woodward's intermittent grasshopper, impulsing once per minute, I wanted to make sure that the count wheel worked well on its own.

Realizing that my previous clock #1 pendulum has a low unloaded Q because I chose to suspend it in a plain brass pivot, I tried a knife edge suspension.  To hang the pendulum, I took a piece of wood with a branch at a right angle and shaped it into a strong bracket.

The bracket has a slot cut into it to receive the pendulum's knife edge. The pendulum knife edge is crude at this point, and not very artistic.

Unloaded, the Q is around 300-500 with some steel weight I tied onto the bottom. 

For the count wheel, I cut a simple 30 tooth ratchet wheel from 1/8" plywood.  I tried to get the tooth spacing about what would correspond to a degree or two of pendulum amplitude about 6" from the suspension.

The count wheel is driven by two wire lever pallets.

The left pallet attaches to a hole in the pendulum and pulls the count wheel to advance it. The right pallet attaches to a separate anchor point, and serves as the backstop.

Here is the assembly, ready for testing.

Starting from a comfortable amplitude, which pushes the backstop about halfway back, the mechanism will run reliably for somewhat longer than 2 minutes.

Starting from just below an amplitude causing double counting, it will run for over 3 minutes.  This is heartening, because it indicates that impulsing every one minute is feasible, because there is plenty of extra energy available to let off the escapement (not built yet).

Metronome flasher

Here is a circuit Edwin built this morning.  It's not complicated, but it is clever...

He figured out that the metal mechanical parts of the metronome are all in contact with one another.  He attached a wire to the winding knob, which was therefore attached to the pendulum.  By aligning the pendulum carefully with a snap circuits wire, the metronome intermittently completes the circuit.

Here is a video of it in action!

Clock 3 cased and installed

Clock 3 is finally complete, and is now installed in an oak frame/case in my office.  The case has a matching French cleat so the movement is easy to remove for debugging.  I also installed a dial that is perched on pins on the shelf, which allows it to be easily removed by lifting it off the movement. 

It is now driven by a 10 pound bag of lead pellets (intended for scuba diving) with a two-fall pulley.  It runs for 16 hours on a wind.  This short run time is well enough so that it shouldn't bother my office neighbors.  It's a bit noisy, but now that it is no longer mounted on a hollow cavity, it's quieter than before.

Moving the clock from my cool, damp basement to the warm, dry office did require some adjustments...  The movement's frame has the grain going horizontally, which meant that it shrank vertically a bit.  This caused the impulse pin to bind on two things:
1. The detent tip, which required shifting the detent back slightly
2. The trailing edge of one escape wheel tooth, which required a small amount of filing.

The great wheel also fouled on the frame -- an indication that my "fancy" milled frame wasn't a good idea -- since it appears that the frame has shrunk vertically.  To compensate, I carved the milling back further.

The great wheel pivot, which I had previously needed to move (and I wondered why!) had to be set back in its original position as well.

Additionally, many friction-fit parts loosened and required the use of super glue to anchor:
1. The impulse pin
2. The hour finger
3. The dial pins 

Clock 3 drive updates

Now that Clock #3's escapement seems to be working, I replaced the center pinion with a v-pulley.  I made the v-pulley from two 1/8" plywood disks in which the edges were beveled in the lathe.  I used hot glue to adhere the disks together, and used a heat gun to ensure the pulley was evenly glued.  The v-pulley was secured to the center wheel by a pin.  It seemed that this would grip an 80 lb monofilament fishing line if a counterweight was used.  The line settles into the groove for an effective radius of 0.75 inches.  The clock ran for over an hour on 3.25 lb drive and 0.5 lb counterweight, which works out to just about 2 inch points of torque on the center wheel. 

I also noticed previously that escape wheel wheel drifted up on its arbor, so I added a small wooden washer cap that press fit to the arbor.  This seemed to work well enough.

Clock 3 torsion pendulum updates

I made several updates to Clock #3 over the weekend... in the end, it is running with about 1.75 inch-pounds of torque on the center wheel. 

Paradoxically perhaps, I found that it runs better with the right angle transmission meshing at the top rather than the bottom...

But then I found that the pivot below was unnecessary.  This reduced friction somewhat, and lengthening the pivot considerably was helpful. 

Initially this seemed to reduce the needed torque to around 1 inch pound.  With two pounds on the balance (one pound is shown above), this seemed very stable.  The center pivot was set in the wood frame.  Unfortunately, but since I had to make several drillings to get the depth correct, the hole walls were weak and eventually split.  So I inserted a brass bushing...

This bushing was not depthed correctly, so I had to drill out and shim the hole, so it looks less nice than it does above.  With the bushing, the running torque is back to 1.75 inch pounds...

Additionally, I found that occasionally the escape wheel would skip.  The reason is the when the wheel rides up on the locking detent (black arrow), it deflects the detent too far and the wheel slips past...

Clock 3 torsion pendulum Q

So the pivots on Clock #3 clearly are consuming energy... I tried the torsion pendulum outside the frame (basically just excluding the pivot).  I counted 70 periods before the amplitude halved... which yields a Q of roughly 315.  But there is considerable wobble since the balance itself is badly out of poise.  I tried to clean this up a bit on the lathe and redrill the center hole (which is misaligned).  Repositioning the suspension hanger also helped, so now the rod spins vertically without much wobble except when the impulse is given.

This seemed to help a bit, as the necessary drive torque has dropped to 1.75 inch-pounds, or about 0.34 mW.  This is about 2/3 as much power as Clock #1 uses, which gives basically one day of runtime with about the biggest weight I'm comfortable with.

Much better, but I suspect that there is still easily fixable power loss due to the wobble at impulse.  If the pendulum weight is placed lower (on a longer rod), this should be reduced, and might help matters further.