The main thing this exercise highlighted to me is that I don’t actually remember some pretty commonly used beam bending equations. This is probably because I have been over-reliant on the fact that I can easily look these equations up on the Internet. But as Prof. Slocum says, nothing beats having the common relationships at your fingertips, so I am going to pursue a two-pronged approach of committing the most common ones to memory and reacquainting myself with the fundamental Euler-Bernoulli relationships so I can derive the more unusual ones easily.
This week I had to take apart my Hario Slim coffee grinder for cleaning, as the taste of my morning coffees have started to suffer from build-up of stale grounds in the grinder. I took the opportunity to analyze the mechanism using some concepts we learned in class. Here is some casual analysis I did:
Coffee nerds (like me) want to produce grounds with consistent particle sizes that can be adjusted with high resolution. For a conical burr grinder, the sensitive directions are:
- translation along the “power” axis (controls grind size)
- rotation about the two orthogonal axes (determines burr wobble, and controls grind consistency)
Since this grinder is driven by a hand crank, there is a significant cyclical parasitic load that comes from the user bearing down on the handle while rotating it. This load contributes to shaft wobble by taking up radial clearances in the 2 bearings, as well as at the interface between shaft and burr. For what I imagine to be ease-of-manufacturing, the steel shaft is coupled to the ceramic burr via a snap-fit plastic part. There is significant clearance at this interface, and I think this may be the biggest contributor to wobble.
I wonder whether this coupling can be improved through the use of elastic averaging, which would provide a relatively precise connection while accommodating the inherently loose tolerances achievable in a cheap sintered ceramic part.
Speaking of elastic averaging, I have found that the grinder produces more consistent grounds when adjusted for fine grinds (e.g. for espresso) than when it is adjusted for coarser grinds (e.g. for a French press). I have a theory for why this is so. In operation, the conical burr is subjected to a collection of random (technically just chaotic) forces from the interactions between coffee beans and the burr. When the burrs are closer together for a fine grind, the tight space between the two burrs is filled by a large number of smaller particles. Conversely, when the burrs are further apart for a coarse grind, the space between them is typically occupied by a smaller number of larger particles. This means that the burr experiences a much more uniform force distribution when grinding finely, since it is essentially averaging over a large number.
The PUPS this week forced me to take a step back and look at the desk design problem from a high level. I concluded that I have prematurely focused in on a desk-mounted design previously. Although I still think that strategy would be the easiest way to achieve my functional requirements surrounding compactness, I will be revisiting alternative strategies and concepts in the next couple of weeks to make sure I am giving each idea the due consideration before selecting one.
I fabricated my linear motion axis (boxway) from scrap plywood and a plywood-oriented strand board laminate scrounged from around campus. As mentioned in my previous post, I wanted to keep the slider cross-sectional dimensions to a minimum of 1″ x 1″ in order to accommodate the flange nut when I incorporate the lead screw. Unfortunately, the only sufficiently thick material I could find was the plywood-oriented strand board laminate. This forced me to use the porous and irregular surface of cut oriented strand board as bearing surfaces instead of a smoother material. To compensate for the surface asperities and higher coefficient of friction associated with this material, I increased the radial clearance from my design value of 0.005″ to 0.01″.
After cutting the component pieces to size with the table saw, I glued up the assembly using copier paper (thickness = 0.0035″) as shims to achieve the necessary clearances. For example, a 3-layer stack of paper brings me within 0.0005″ of my design clearances. A mistake I made during this step was neglecting to account for the thickness of the glue layer, this ended up causing my boxway to have excessive radial clearance, increasing error motions.
I tested the geometric error in my linear motion axis using a laser pointer. I moved the slider between extreme positions on the axis of travel, while applying slight moments to take up the angular “backlash” caused by radial clearance. The position of the projected beam on a surface 4845 mm away was recorded between each adjustment.
The maximum lateral displacement of the laser beam was 109 mm, which corresponds to an angular error of 1.29°. This is the total side-to-side rotation, which we expect to be twice that predicted by our deterministic geometric error analysis utilizing radial clearances. For my boxway with 0.01″ of radial clearance, I predicted a sine error of 48.5 mm when measured 4845 mm away. The actual value is slightly higher than expected, which I attribute to the mistake I made in not accounting for thicknesses of adhesive layers as well as imperfect clamping during the glue up.
The fabrication process went smoothly and I was very satisfied with the product. The “snap” provided by the magnets used for preload is particularly satisfying. I often find myself toying with this coupling, and can see myself keeping this around as a conversation piece for a long time to come.
As before, I tested the angular repeatability of this coupling using my trusty laser pointer. One issue I ran into was the difficulty of rigidly fixturing a disc-shaped object in my apartment where I had minimal tools. I ended up attaching the bottom disc (with the grooves) to my desk using three strips of gaffer tape spaced 60° apart, taking advantage of the tape’s flexibility to place the disc under quasi-exact constraint. This is possible because the tape is virtually incapable of applying any lateral or compressive forces due to its flexibility.
Over 8 cycles, I obtained a group of points all lying within 2 mm of each other on a wall 2.9 m away. This suggests an angular error of only 0.04° — a 15-fold improvement over the melon-and-cardboard mock-up! I think at least part of this error can be attributed to the imperfect fixturing — impact loads when the balls engage the grooves may have caused the fixed disc to move slightly. The actual angular error is probably even lower!