SNG 9: Foldable Umbrella

I own a foofoo non-circular umbrella from Senz. My defense for the foofoo-ness is that I got it as a gift through my frequent flyer program. But it does work quite well on two counts:

  1. It behaves like an airfoil whose geometry you can adjust by tilting the umbrella differently. This makes it a lot easier to hold on to in high winds
  2. The asymmetrical design is ergonomically sound. My view is that unless it is part of an umbrella hat, there aren’t many good functional reasons why umbrellas should be circular.

This week, I geek out on the folding mechanism in this umbrella. Each arm of the umbrella is made up of a series of pin-ended aluminum members (C-channel sections in pictures) joined in series. Each alternate member is then connected with a slender spring tempered wire. Basically, each section is part of a local 4-bar linkage.


The interesting thing about this mechanism is that it exploits the buckling behavior of the spring tempered wires in its design. This marks a departure from the traditional “classroom” engineering approach where buckling is not considered until late in the design process when one verifies that the specified members are not going to buckle. This approach reminds me of a piece of MIT IP from Prof. Brian Wardle’s lab that I wrote about at work (as a technology licensing intern for the MIT TLO), where buckling is exploited to create 3D MEMS geometry using conventional 2D microfabrication workflows such as CMOS.

Bi-stable Mechanism

Back to the umbrella. In the folded configuration, the spring wires are in a relaxed state, free to rattle around in the pin-supports at their ends. In the fully open configuration, they are under tension — applying the preload to hold the mechanism in the open position and keep the fabric web taut. In between these two configurations, the wires experience compression loads large enough to cause them to buckle but not enough to cause them to yield. This allows them to act as compressive springs in this intermediate region. The result, then, is a bi-stable mechanism that stays fully open or fully closed, but is unstable in intermediate states. This is great design, and I believe not something every umbrella does! The stability of the 2 functionally relevant positions means that the latches and ball detents used to keep the umbrella open or closed actually experience very little load, allowing lighter and more easily actuated components to be used.

Variable Spring Rate

Another benefit of exploiting buckling is that you can get the spring wires to act as springs with different stiffnesses in tension vs. compression. When the umbrella is open, the wires are under axial tension with very high stiffness — great for standing up against wind loads and keeping the umbrella open. Conversely, when the mechanism is neither fully open nor fully closed, you want a spring that is strong enough to nudge it towards a stable conformation, but not one that would poke a user’s eye out. This is achieved by allowing the wires to enter the buckled state, which places them under pin-ended bending instead of axial compression — significantly reducing stiffness.  This is an excellent example of the use of self-help in mechanical design!

Buckling Direction

We have seen how buckling can be used to great effect, but it is fundamentally a phenomenon drive by instability. This makes the math a bit more complicated (i suspect this is why it isn’t classically popular in “school” engineering design) and means that it is not straightforward to predict the buckled shape of structural members. In an umbrella, this poses a number of risks, including the creation of unpredictable pinch points and inadvertent puncturing of the fabric web by a wire buckling in the wrong direction. The way the designers of this umbrella have dealt with this is by entrapping the spring wires within the adjacent channel at their midpoints using a tab. This biases the members and forces them to buckle in a predictable direction. Again, a nice touch. A minor drawback to this is that it sets up a sliding contact between the aluminum channel and the hardened steel wire in the middle of the umbrella-opening stroke, which is starting to create squeaking noises as repeated use starts to wear through the phosphate coating on the wires. On the bright side, that squeak was what prompted me to grease the contact points and, in the process, geek out about this mechanism!

SNG 6: Hand Soap Pump Dispenser

This week, I chose to geek out on hand soap pump bottles. First, some context: I ran out of hand soap this past week and had to get a new bottle. For the past couple of days, I found myself repeatedly getting annoyed by the behavior of the pump dispenser on the new bottle. Every time I tried to pump out some soap, the plunger seizes up at the top of the stroke and forces me to push harder than I otherwise would. At some threshold force, the plunger would release and I would end up with a much bigger glob of soap than I needed.

From a business standpoint, this is quite brilliant for the manufacturer since they are essentially exploiting stick-slip behavior to make people use more of their product. On the other hand, annoying your customers is probably not the smartest business decision, so I think this is likely an engineering mishap instead of a calculated move…

My old bottle of soap from the same company dispensed soap smoothly, so I took a closer look at the pump assemblies to look for any differences. Like all piston pumps, these things had a piston running in a bore with a check valve (in this case a plastic ball design) on one end. When the plunger is depressed, the check valve closes and soap in the cylinder is forced out through the nozzle. On the upstroke, driven by a compression spring, the plunger pulls a vacuum which unseats the ball to open the check valve, sucking more soap into the cylinder. Once the plunger returns to its neutral position, gravity reseats the ball and closes the check valve, keeping soap in the cylinder in anticipation of the next use.

The key difference I found between the two pump assemblies is that the unit that came with the new bottle had significantly more angular free play in the plunger stem. When a user applies an eccentric load via the cantilevered portion of the plunger, which happens almost every time you try to actuate the plunger using one hand, this free play results in pseudo-rigid body rotation about the center of the piston. Since the force applied by the user remains in a more-or-less downward direction, this rotation results in an increase in the length of the moment arm relative to the centre-of-stiffness of the piston. This moment arm grows in proportion to the sine of the rotation angle, and I think it was sufficient to push the bearing into operating under a stick-slip regime, which results in the annoying behavior.

This phenomenon reminds me of a behavior I encountered back when I was doing research on the mechanics of corn stalk lodging. Corn plants are essentially cantilever beams pseudo-rigidly supported at the ground, with a heavy ear of grain around three-quarters of the way up. Under neutral conditions, the heavy ear is roughly above the root of the plant. In this configuration, the ear primarily exerts an axial compressive force on the stalk. However, when the plant experiences a lateral wind load, the stalk flexes and increases the eccentricity of the heavy ear. The weight of the ear now exerts a bending moment on the stalk.

Just like the hand pump dispenser, the moment arm through which the weight of the ear acts grows in proportion to the sine of the angle. In the case of corn, the wind load experienced by the plant is also reduced by this bending since the flexed plant presents a smaller drag area. However, the height reduction is proportional to the cosine of the angle — that is, the drag area declines much slower than the ear weight eccentricity grows. This has a significant effect on the lodging resistance of corn plants.

I think these examples demonstrate how second-order effects can affect the functional outcomes of engineered systems. We usually focus only on first-order effects when doing preliminary design, and for good reason. But it is important to keep in mind that the classical small-deflection assumptions doesn’t always hold and that it is often useful to take a step back and just think qualitatively about how the system would behave if these assumptions are relaxed.

SNG 5: Park Tool Tire Levers

I had to change out a flat tire on my bike this weekend, and I took the opportunity to take a closer look at the tire levers I was using. These tire levers from Park Tool has nifty feature where they snap together. This is very useful in keeping them together in a messy tube change kit.

The way they attach to each other is through a Lego-esque snap-fit feature molded into the part. Unlike Lego, however, the boss on the bottom lever only makes contact with the mating lever at the two cylindrical faces. And therein lies the reason why these levers don’t align perfectly every time, unlike Lego. As we saw a few weeks ago, Lego bricks make use of elastically averaged interfaces to achieve amazing performance (at least for an injection-molded plastic brick). Like Lego bricks, these levers are held together by local elastic deformation; but unlike Lego bricks, these only make contact at a small number of points (2 in this case). Obviously, averaging over a small number of contact points results in significantly lower repeatability.

Another issue with this interface is that it has a large amount of wiggle in it. That is, not only do the levers not line up the same way every time you snap them together, but they also rotate appreciably (~0.5 mm) relative to each other while engaged. This is a result of using only 2 contact points. As we know, 3 points of contact are necessary to fully constrain a planar part (the local elastic deformation provides the preload in this case). Unfortunately, this wiggle does detract slightly from the sense of quality you otherwise get from using these otherwise excellent tools.

Well, if anyone from Park Tool is reading this: when you next get a new mold, include a third point of contact!

SNG 3: Hario Slim Coffee Grinder

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.

SNG 1: SOG PowerLock Multitool

SOG PowerLock Multitool (Photo by Shien Yang Lee, CC-BY-SA 4.0)

The star of this week’s Seek and Geek is a multitool I own. It uses a patented configuration of levers to achieve greater mechanical advantage. Compared to conventional pliers, which are made up of two class 1 levers pivoting about the same point, the handles of this tool pivot about a pair of pins (with formed heads) attached to the horizontal link in the image and attach to the jaws via the partially obscured button head socket cap screws.

Sprockets (Photo by Shien Yang Lee, CC-BY-SA 4.0)

As a 5-bar linkage, this mechanism should have 2 degrees of freedom according to Gruebler’s Equation. But the designers have incorporated a pair or sprockets that couples the motion of the handles, thereby eliminating one degree of freedom. I think this is to maintain the familiar usage pattern that people have come to associate with pliers and to facilitate the relatively fine manipulation of small parts that needlenose pliers often get used for.

I have noticed a couple of issues from using this tool. The greater mechanical advantage obviously comes with the side effect of requiring larger hand movements to move the jaws by a given amount. “Twice the cutting and gripping force” sounds great — it was partially why I bought it in the first place — but I have since come to realize that the limiting factor with using pliers is usually not how hard you can squeeze but how far you can open the jaws with one hand.

Another annoyance is the backlash introduced by the additional joints between handles and jaws. Normal pliers, even ones with worn pivots, primarily have out-of-plane free play. These have noticeable backlash around the actuation axis, which can be frustrating when trying to fine-tune gripping force on compliant or fragile parts.