I was thinking of SunShine’s flexure indicator 3-D printed from PLA, just using a couple of “blade flexures” that converge on an indicator needle, nearly at the same point, perhaps 0.1 mm apart; the needle is about 20 mm long. Although he doesn’t have the thing calibrated, he was able to use it to detect the thickness of a 50-micron-thick sheet of paper, which produced about a millimeter of movement.
(I’ve put quotes around “blade flexures” because each “blade” is made up of a set of parallel wiggle bars in order to allow them to connect to the same bar at almost the same point without interfering; “we stagger the layers”, as he says.)
In SunShine’s mechanism, most of the actual flexing takes place far away from the indicator itself, which unfortunately greatly reduces the amplification factor to only about 20:1; this could be remedied with a more rigid flexure design, at the cost of increasing plunger force, but a better flexure design is also possible. He also has sinned against flexures by making the plunger shaft a sliding contact with the 8mm indicator stem rather than using parallel blades or some similar prismatic flexure joint.
The total range of motion of his indicator needle is about 10 mm, which reduces the precision of readings available, even if you calibrate the device. It occurred to me that using the vernier principle it should be possible to make much smaller rotations easily visible. By printing a disc that rotates relative to a fixed disc with graduations at a slightly different frequency, you can visually see quite small rotations. Better still, I think, would be to print each disc with a series of holes or slits in it, rather than merely graduations on the surface, and place the two in sliding contact with one another, or a flexural approximation thereof, so that the place where the holes coincide moves around the dial much more rapidly than the holes themselves.
It ought to be possible to get 200-micron holes with conventional 3-D printing and laser-cutting processes, which ought to afford about 20-micron visible precision on the outer edge of the dial. If the mechanical advantage can be set to 50:1, this would provide 400-nanometer resolution.
As with a ruler or caliper, thermal expansion or contraction will introduce error in the measurement. However, uniform expansion either of the dials or of the plunger and stem poses no such risk, because such expansion doesn’t change the angles; it’s specifically the part of the plunger outside the stem, and to a much greater extent, the lever arm over which the plunger’s translation is transformed into rotation, which determines the calibration. Thus it should be possible to incorporate a small piece of wood, invar, glass, sapphire, carbon fiber, or fused quartz into that part of the movement, or build it like a gridiron pendulum, to get a measurement tool that is immune to such problems, differential though it is.
A simpler way to cancel thermal expansion in one dimension than a gridiron pendulum is with the following structure:
0 1 2 3
####### #######
A ######### ######### A
# #
## ##
B ##@@@@@## B
## @@@ ##
## ##
## ##
## ##
## ### ##
C ######### C
0 1 2 3
Here the # represents a material with a large thermal coefficient of expansion, and the @ represents a (normally more of a pain in the ass) material with a smaller but still positive TCE. There are six flexural joints in this setup: A1, A2, B1, B2, C1, and C2; let’s suppose that essentially all the flexion happens there, while the rest of the structure remains rigid. Consider the ratio of distances AB:AC. If this is the same as the ratio of TCEs between the two materials, then uniform heating will not change the distance A1A2. By putting B a little bit further down, we can get a negative coefficient of expansion for A1A2, which could be chosen, for example, to cancel the coefficient of expansion for A0A1 and A2A3, so that the distance A0A3 is invariant with uniform heating.
In this literal form the structure would not be very stable; in practice you would want to stiffen it. Making just B1 and/or C1 perfectly rigid would probably answer for many purposes, and even if C2 were rigid the structure might work adequately with the right corrections. If it could be arranged to be under constant compression, the low-expansion B1B2 member could possibly be a ball bearing (steel: 12 ppm/K), or a glass marble (8.5 ppm/K), perhaps held in two lengthwise V-grooves in the A1C1 and A2C2 members, so that any necessary rotation can happen by rolling, without stick-slip movement. Constant stress, whether compression or not, would probably rule out the use of low-melting and therefore high-creep materials like PLA.
Many materials might serve. Radial expansion for Douglas fir is given as 27 ppm/K, while parallel to the grain it is 3.5, but it is also sensitive to humidity. Brass is 19, aluminum 23, fused quartz 0.59.
Getting back to the indicator, a simple expedient might be to laser-cut the whole thing from one to three sheets of steel. Steel has significant thermal expansion and contraction, but it’s much smaller than that of many alternative materials; polypropylene’s TCE is given as 150, 12 times higher, and even PLA is about 40 ppm/K. And, unlike them, steel isn’t hygroscopic and doesn’t creep significantly at ambient temperature.