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.