Cutting Room Floor: Stiffness

This is part of my series of material that I edited out from my books prior to publishing. I usually cut whole sections like this when I realize they’re redundant or don’t fit well in the context. This is copy/pasted directly from my draft files with no additional editing, just for fun. 

This excerpt is from 2014, shortly after I first started writing my 3D printer book. This was back when I was playing around with different writing styles, so don’t judge my finished books by what you read here! I actually ended up throwing out almost everything I wrote in the first year (around 55,000 words of draft copy without illustrations).

This was also long before I pinned down a good outline. At the time, I thought I could fit everything important into about 70,000 words. Ha. Volume 1 alone — frames and motion hardware — is longer than that. As of Volume 1 going to press, I’m now expecting Volumes 1-5 to sum to about 300,000 words in total.

The matter of printer stiffness is a subject of great debate amongst the 3DP hobbyist community. Not so much whether it is good or bad — universal acclaim indicates stiffness is a virtue — but the debate rages on the why and the how and the how much. The truth of the situation requires an understanding of what happens to the entire printer when the motion mechanism moves.

Let’s examine a few limiting cases to consider the issue. 

First, imagine a 3d printer constructed of marshmallows and rubber bands. Such a structure would sag under its own weight, jiggle when the gantry moves, and so on. Clearly there cannot be sufficient positional accuracy for good results if the printer cannot even maintain its own shape. Any attempt by the drive system to rapidly accelerate the extruder carriage will cause the frame to flex enormously in the opposite direction. (Every action has an equal and opposite reaction.) Clearly some stiffness is required. 

Now imagine a printer frame carved out of solid unobtainium, securely encased in solid rock deep underground. The frame cannot move at all. It is almost perfectly rigid. (Nothing is perfectly rigid.) Aside from obvious issues of practicality, such a printer would probably work quite well. No jiggling, no sagging. When the carriage is accelerated, the reaction force is entirely supported by the frame and rock with negligible deflection. But this is not the optimal design! Surprisingly, there is another extreme case which has superior performance.

Now imagine a 3d printer with another almost-perfectly-rigid unobtanium frame, but orbiting the Earth in microgravity. No atmosphere, no other objects around to push on or react against. Now, when the carriage is accelerated, the frame accelerates in the opposite direction! Visualize this for a moment — as the carriage moves, the entire printer reacts. 

[diagram of printer floating in space]

Just for the sake of simplicity, let’s make a few assumptions:

  • The carriage and frame have the same mass (just to make the math easy)
  • The motion mechanism is entirely symmetrical around the center of mass (meaning we can neglect twisting/bending moments from off-center motion)

Ok, so maybe that doesn’t sound very simple, but because of the symmetry we can now approximate this system’s behavior as two equal point masses: the carriage, and the frame. And you’re in orbit alongside, watching.

[free-body diagram of space printer]

When the carriage accelerates in the +X direction, the frame accelerates an equal and opposite amount. (Because they have the same mass in our example) Pretty simple stuff — this is the same effect that propels rockets. Except unlike a classic Physics 101 rocket ship reaction mass problem, our carriage and frame are coupled together and the carriage must deccelerate to a stop before it hits the frame. When the carriage moves, the frame moves. And when the carriage stops, the frame stops. So the printer operates normally, as far as extruder positioning is concerned.

But what’s fascinating about this space-printer is that the carriage and frame each do half the moving. If you command the extruder to move to position +X relative to the frame, then from your perspective the carriage will move by +1/2*X and the frame will move by -1/2*X. The relative displacement is correct, but the absolute displacement (from your inertial reference frame) is only half. Which means the absolute velocity is only half as fast. Which means the absolute acceleration is only half. Which means the force required to accelerate the carriage is only half.

Only half what? Only half as much as the printer encased in rock. For the rock-encased model, a motion of X in time deltaT, beginning and ending at a standing stop, requires acceleration equal to:

[math here]

Specifically, the three-axis motion mechanism must be sufficiently rigid to minimize unwanted deflection. Unwanted deflection causes 

That’s where it ends; I never finished the thought. Probably would have continued into a discussion of print flaws.