Saturday, August 28, 2021

TheGoofy's clock, and a digression on balance wheels

 One of the oldest clock designs on thingiverse is this model by user TheGoofy. It has many makes and several remixes. I made this once before several years ago, and never got it to run reliably. At the time, I had a printer which was much less accurate and well-tuned than my current one. TheGoofy notes in his description that it was designed for older, more inaccurate printers. I think that I couldn't make it work as a combination of what my printer was able to produce and my ability to figure out what was going wrong and fix it.

I recently had another go, with some success. Here are some pictures and video:

The video was taken while I was still tuning it, and it's quite obvious that the beat and timing are off. Here is a later version that is a bit better:

It worked pretty much straight off with only a few changes. Some of  the parts came out undersized. In most cases this does not matter, but in the pentagonal connector between the escapement spring and the balance wheel, it is important to get a good fit. The original spring is 1mm thick, and I found that the coils flopped around too much. I found a remix with a 1.3mm spring. The end of the spring has a triangular piece which fits into the frame, and this was also far too loose. At first I held it in place with some tape, and then adjusted the model to make a tighter fit. One other adaptation I needed was to drill out the holes for the arbors in all the gears and moving parts, as they were too tight. My preferred way of doing this is with a drill bit in a pin vise. It allows you to go slowly and carefully control the drill so that you don't end up skewing the hole.

I used the v1 ratchet and no planetary drum. This gives the shortest running time in the sense that the weight falls a greater distance for a given run time. As I was regarding this a more of a clock demonstrator rather than something I intend to use as a real timepiece, I didn't worry about this too much. I like the idea of the servo driven version, and may try it out later. I found that the clock ran quite reliably with 600g of weight. I think 500g is also OK, but less than that wasn't enough. Note that TheGoofy recommends 1.2kg, and that may be needed for a different ratchet/drum combination: for a higher gear ratio you need more weight, and also get greater run time. With the version I used the weight dropped about 5 cm in 10 minutes (measured very approximately).

One issue I had is that the clock would sometimes stop dead. It took only a slight touch on the balance wheel to get it going again. After a while I realized that this was because I had some screws for setting the time on the balance wheel, and they protruded just enough to occasionally catch on one of the gears. Countersinking the holes on the balance wheel so I could screw them in a tiny amount further was all that was needed to fix this.

I also tried a variant version of the anchor in response to getting an occasional stall. I'm fairly sure that something about this throws the beat out (that is, the ticks and tocks are uneven), and I switched back to the original one.

There are two features of this clock which make it different from the Peterson and A26 clocks. It has a seconds hand, with a little extra mechanical complexity as a result. More importantly, the timing element is a spring/balance wheel combination, with an anchor between this and the escapement. I think this is called a Swiss or lever escapement. It's a much more compact arrangement than using a pendulum. In a 3D printed version, it's less practical as the spring will wear out over time. It's also harder to tune the period. I haven't found any very good guide on this, so here is my understanding of the physics and some observations about the practical reality.

Some noodling about balance wheels

In theory, the balance wheel acts as a harmonic oscillator. Wikipedia gives an expression for the period. The important factors are:

  • it is inversely proportional to the square root of the spring stiffness. So a thicker spring makes the period shorter, resulting in less time between ticks. It speeds up the clock, making it run faster.
  • it is proportional to the square root of the moment of inertia of the balance wheel. If you imagine the balance wheel as being made up of lots of tiny masses, the moment of inertia is then the sum of each mass times the square of its distance from the axis (mr^2). So a heavier balance wheel or moving some of the mass outwards makes the period longer and the clock runs slower.
The balance wheel has its own intrinsic mass, and also has eight holes round it which you can insert screws into. The screws can be moved in or out to a small degree and you can choose how many to use provided they are kept in pairs. So by adding or removing screws or adjusting their position, you have some control over the timing.

Now this is all for an idealized system, and in a clock there are at least a couple of things that might make it different. I don't have the skill to analyze this in detail, but my thoughts are:
  • gravity is acting on the screws attached to the balance wheel. The direction of the gravitational force relative to the balance wheel changes as the balance wheel moves. So the resulting moment on the balance wheel is also different. This implies you get different effects depending on which of the balance screws you use.
  • when the nub on the spring hits the anchor, it loses some energy, and it then gains some energy back as the trailing edge of the anchor hits it. It also interrupts the smooth motion. I've no idea how this would affect the period, if at all.
Note that you will find some descriptions on the web of adjusting the position and setting of certain balance screws having a special effect. I think this information needs to be taken carefully as it is often referring to a bimetallic balance wheel, which behaves differently from the kind we are using.

I did a few experiments to see how changing the weights on the balance wheel affected the period, by adding and removing screws. There are eight holes for screws round the balance wheel. If you imagine it vertically, I'll call the top pair A, the next one down B, the ones just below the midpoint C, and the pair at the bottom D. Note that this depends on exactly the orientation you choose for the balance wheel relative to the spring. You can also position it so that there is a screw at the top and bottom, and there are other orientations which are unbalanced.

I initially used M2.5x4 screws. I timed how long one rotation of the seconds hand took. The spring for these experiment is 1.3mm thick, though I think as a result of the slicing parameters it is probably more like 1.25mm. I noticed that with weight higher up, the movement of the spring was less regular. It looked in some cases as if it was about to tangle (one loop catching on the next one). Also, I doubt that I was setting the screws to a consistent depth in the experiments, and as noted this affect r in the mr^2 computation of the moment of inertia.

Here are some results:
  • screws in A, B and C: 92 seconds per revolution of the seconds hand (and somewhat uneven).
  • screws in B and C: 77s.
  • screws in C only: 71s.
  • screws in D only: 56s. Note that this case has the same theoretical moment of inertia as the previous one, and so supports some of my speculation above.
  • no screws at all: 75s. On a second run I got 68s.
The last reading is a bit odd. I think what is happening here is that the mass of the balance wheel is now so low that it can't transfer enough momentum to the anchor and hence to the escapement wheel. Some of the beats were noticeably uneven. I think the other results are somewhat consistent, in that I reran the timing for a couple of the cases and got the same result to within a second.

I now changed the spring to a thicker one, 1.5mm deep, with these results:
  • D: 56s.
  • C: 61s.
Another little bit of physics here. For a spring with a circular cross section, the stiffness varies as the diameter of the wire to the 4th power. The spring in this clock has a rectangular cross-section and we are changing only one dimension of it, so it's a reasonable guess that the stiffness should change as the square of the depth. The period varies as the inverse square root of the spring thickness, so it should be linear (inversely, that is) in the thickness. We went from 1.25mm thick (nominally 1.3mm) to 1.5, so a factor of 1.25/1 = 0.833. And in the D test, the change in period is 56/66 = 0.848. So it looks like physics is working.

The answer

The main reason for the analysis above is that I haven't understood how to tune balance wheels in the past and I wanted to work through the logic. The short summary is:
  • stiffer spring means slower.
  • more weights means slower.
  • position of the weights matters.
The actual configuration I ended up with was a 1.5mm spring, balance wheel oriented with screw holes on the vertical axis (different to the A/B/C/D configuration I described above), and no weights at all. This gave me pretty close to 1 minute per rotation of the seconds hand. I saw some variation across measurements at different times. Also, the clock sometimes runs smoothly and sometime stutters a bit. I think this is probably when I hit spots on the gears which I hadn't finished well - I didn't clean everything up very carefully.

This is another nice design - thanks TheGoofy (aka Christoph Laimer). Looking over the last three clocks, you might be able to see there is a progression, from an easy hours+minutes pendulum design, to a slightly harder hours+minutes horizontal pendulum design, and now to hours+minutes+seconds balance wheel. I have a few ideas for what I would like to try next.

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