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+# 2024-05-03
+
+## Deadbeat escapement
+
+I still haven't reread the section in the book, but I was thinking overnight.
+
+I think you draw a circle at the diameter of the tips of your escape wheel teeth (the
+tips are the only part that interact).
+
+Decide how many teeth you want the anchor to span (I think in the book they said this
+should be halfway between 2 teeth, for example 7.5).
+
+Divide that by the total number of teeth to work out what fraction of the wheel you are
+spanning, and therefore what angle. 7.5 / 30 = 0.25, or 90 degrees.
+
+Imagine a 90 degree sector of the escape wheel, centred on the Y axis. Where the sector arms
+meet the escape wheel diameter is where your pallets are going to be.
+
+Imagine tangent lines to these points, where the tangent lines intersect the Y axis is your anchor pivot.
+
+(Is this point where the start, middle, or end of the impulse occurs?)
+
+And then for definitions I think:
+
+ * lift: angle by which the anchor rotates during the impulse
+ * drop: angle by which the escape wheel rotates between the end of one impulse and the start of the next locking
+
+So let's say we want 4 degrees lift (I think the book said 4 degrees was sensible? can't remember),
+and let's say that the points at the end of our sector arms
+are the middle of our impulse.
+
+So now imagine 2 line segments passing through our anchor pivot, but 2 degrees either side of the tangent line (for
+4 degrees total). These line segments are the same length as the length between the anchor pivot and the point
+where the pallet interacts. That means that they will meet the escape wheel at the same place, but when the anchor
+has rotated +/- 2 degrees.
+
+You'll notice 1 line is clear of the escape wheel teeth and one line interferes with them, this is what we want.
+Now draw a line segment connecting the endpoints of these 2 segments. That line segment is your impulse pallet.
+
+So how do we look at this point?
+
+OK, no good: https://img.incoherency.co.uk/5430
+
+That puts our pallet face almost radial to the escape wheel, obviously, which is no good.
+
+OK, alternative plan. Given that we know how much "drop" we want (let's say 2 degrees), and we know how
+large is the angle between adjacent teeth (360/30 = 12 degrees), then we know the impulse needs to span
+10 degrees. So we can draw a 10 degree sector centered on the radial line on the escape wheel,
+and the intersection of the arms of that sector will intersect the lines at +/- 2 degrees of the tangent:
+
+https://img.incoherency.co.uk/5431
+
+Much more convincing!
+
+Now we need to draw an arc at the left-hand edge of this pallet face, centered on the anchor pivot,
+and that's our frictional rest pallet. It needs to be long enough that at maximum amplitude the
+escape wheel doesn't fall off it. Let's say maximum amplitude is 20 degrees, make this arc join
+up with a line 20 degrees away from the tangent line:
+
+https://img.incoherency.co.uk/5432
+
+Very good, that's the entry pallet working faces done.
+
+Now what if we do the same for the exit pallet?
+
+https://img.incoherency.co.uk/5433
+
+Looks believable I think. And this is the anchor in the dead centre position.
+
+Obviously there is some freedom in terms of which parts you make line up.
+For example I have made the impulse faces centered on the point on the escape wheel diameter that I care
+about, symmetrically, but you might alternatively want to make the locking faces equally far away from
+the anchor pivot, or you might want to go halfway between the two options, or something else. This
+definitely wants to be configurable in the simulator.
+
+I'm going to draw a quick escape wheel in FreeCAD and check that this escapement looks like it would work.
+
+No, it looks like something is wrong. I think the impulse faces are too long:
+
+https://img.incoherency.co.uk/5434
+
+You can see that a tooth is engaged with the entry pallet impulse face, but another tooth is already
+at about the right place for locking on the exit pallet.
+
+Was it a mistake to make the impulse faces span 7.5 teeth? Maybe it should be an integer number.
+
+7/30 = 84 degrees, let's try that.
+
+Obviously wrong, because it means when a tooth is at the centre of the entry pallet impulse face then
+another tooth will be at the centre of the exit pallet impulse face, so it can't possibly work.
+
+https://img.incoherency.co.uk/5435
+
+Oh! By making the impulse face span 10 degrees, with 2 degrees drop, I am basically saying that when a tooth
+escapes a pallet, I want there to be already another tooth dropping onto the same pallet. That is not right,
+I want the next tooth to drop onto the other pallet. So instead of saying I have 12 degrees between teeth,
+I think I actually say I have only 6 degrees between teeth, because they drop onto the pallets alternately.
+So the impulse face should span only 4 degrees.
+
+https://img.incoherency.co.uk/5436
+
+Looks perfect. Playing around with rotations in FreeCAD, it looks like it works almost exactly
+perfectly, except that at the point where a tooth escapes, the next tooth lands at the very corner of
+the other pallet, instead of landing safely on the locking face.
+
+So I think that says that we want the impulse pallet to be biased to be slightly *after* the tangent line.
+If I change +/- 2 degrees to 2.5 on one side and 1.5 on the other, does that look better?
+That now means that the impulse will be biased slightly after the dead centre position of the pendulum.
+
+Yes, perfect, this seems to work well now. So one of our parameters is "locking safety angle", there's
+probably a better word for it.
+
+So the obvious parameters are:
+
+ * how many teeth we span (usually halfway between 2 integers)
+ * lift angle (of anchor)
+ * drop angle (of escape wheel)
+ * locking safety angle (of anchor)
+
+And maybe to allow biasing the difference between "equal length from anchor pivot to impulse face" and
+"equal length from anchor pivot to locking face", we just change the sector of the teeth we span. Instead
+of making it symmetrical, you could rotate it slightly to the left to push the entry pallet further from
+the anchor pivot and slightly to the right to push the exit pallet further from the anchor pivot.
+Would you call that a "sector offset angle"?
+
+## Simulator
+
+So I can use the info above to calculate the geometry for the impulse faces and locking faces.
+But how do I turn that into a solid object?
+
+Firstly, the locking faces need to be broken up into a series of straight lines, because I can only specify
+polygons. Secondly, the exit pallet locking face will probably need to be multiple polygons because I can
+only specify *convex* polygons. And thirdly, I need to give the parts some thickness.
+
+I saw this [Graham's deadbeat escapement model](https://exhibitions.abdn.ac.uk/university-collections/file/5323):
+
+https://img.incoherency.co.uk/5438
+
+That looks quite smart (albeit that it doesn't look *right* - isn't the anchor pivot much too far from the escape wheel?). We can make it look like that by letting the entry pallet's locking face get
+longer, and then laying the rest of it out with arcs centered on the escape wheel and anchor:
+
+https://img.incoherency.co.uk/5437
+
+Why would you want the pivot to be further away than the intersection of the tangent lines? Would you want
+that, or is it just an error? And, if you do want that, how would you express it in the simulator?
+Add a parameter for "anchor pivot offset"?